I N V E N T O R Y C O N T R O L IN T H E R E T A I L S E C T O R A C A S E STUDY O F C A N A D I A N TIRE PACIFIC ASSOCIATES

by

BRIAN A N T H O N Y K A P A L K A

BSc(CE) The University of Manitoba 1992 BSc The University of Manitoba 1992

A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF

T H E REQUIREMENTS FOR T H E D E G R E E OF

MASTER OF SCIENCE (BUSINESS ADMINISTRATION)

in

T H E F A C U L T Y OF G R A D U A T E STUDIES

(Department of Commerce and Business Administration)

We accept this thesis as conforming to the required standard

T H E UNIVERSITY OF BRITISH COLUMBIA

April 1995

reg Brian Anthony Kapalka 1995

In presenting t h i s thesis i n p a r t i a l f u l f i l l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e for reference and study I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by h i s or her representatives I t i s understood that copying or pub l i c a t i o n of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission

Department of Cotv^errg- ftni)gt ^osmecs AAnmna-WoAor

The University of B r i t i s h Columbia Vancouver Canada

Date ZS A y r A 9 9 5

Abstract

Canadian Tire Pacific Associates owns and operates 21 retail stores in the lower

mainland of British Columbia and a central warehouse in Burnaby In this thesis we formulate

a single-product single-location model of its inventory system as a first step in developing an

integrated interactive inventory control system Specifically we formulate a Markov chain

model for a periodic review system with a deterministic lead time and lost sales The model

utilizes empirical demand data to calculate the long-run average cost of inventory for a given

(sS) policy We then develop a heuristic that locates a near optimal policy quickly The

heuristic incorporates a constraint on the customer service level makes use of an updating

technique for the transition probability matrix and is based on assumptions regarding the

properties of the solution space Next we create a prototype of the interface that enables

managers to use the model interactively Finally we compare the existing inventory policy to

the optimal policy for each of 420 products sold at one of the stores This thesis finds that

Canadian Tire Pacific Associates is currently holding excessively large in-store inventory and

that it could reduce its cost of inventory by approximately 40 to 50 We estimate that

implementing optimal inventory control in the stores would result in annual savings of between

$55 and $7 million

ii

Table of Contents

Abstract ii

Table of Contents iii

List of Tables v

List of Figures vi

Acknowledgement vii

I INTRODUCTION 1

II T H E INVENTORY SYSTEM A T CANADIAN TIRE PACIFIC ASSOCIATES 8 A Background 8 B The Problem 11 C The Project 15 D The Demand Data 17 E The Cost Data 18

III M O D E L FORMULATION 20 A Terminology 20 B Assumptions 21 C A Markov Chain Model 22 D The Transition Probability Matrix 25 E The Steady State Probabilities 27 F The Cost of Inventory 29 G The Customer Service Level 31 H A Methodology for Evaluating an (sS) Policy 34

IV A N ALGORITHM FOR OBTAINING T H E OPTIMAL (sS) POLICY 35 A Introduction 35 B A Grid Search 36 C A Technique for Updating the Transition Probability Matrix 41

A new policy (s+mS) 41 A new policy (sS+m) 42 The modified algorithm 43

D A Lower Bound on S 45 E A Heuristic Search 48

V T H E INTERFACE 52

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VI RESULTS A N D SENSITIVITY ANALYSIS 55 A Comparison of Current Policies to Optimal Policies 55 B Sensitivity Analysis on the Ordering Cost and Holding Rate 62 C Sensitivity Analysis on the Demand Probability Mass Function 65

VII CONCLUSION 70

Afterword 73

Bibliography 74

Appendix A A Portion of the Sales File for Product 200001 a 30-amp

Inline Fuse 76

Appendix B Sample Distribution Files 77

Appendix C A Proof that the Limiting Distribution is Independent of the

Initial State 79

Appendix D A Measure for the Steady State Customer Service Level 83

Appendix E The Calculation of the Conditional Expected Demand Not Satisfied

During a Period of T Consecutive Days 86

Appendix F Justification of the Updating Technique for the New Policy (s+mS) 87

Appendix G Justification of the Updating Technique for the New Policy (sS+m) 92

Appendix H A Hypothetical Consultation 97

Appendix I The Source Code for the Interface Module 104

Appendix J The Source Code for the Main Module 119

Appendix K Current and Optimal Policies for Product Category 20 at Store 6 131

iv

List o f Tables

Table 1 Location and particulars of the 21 stores 10

Table 2 Execution times of the algorithms for product 200001 a 30-amp inline fuse 39

Table 3 Execution times of the algorithms for product 206917 a 6

solder connector 40

Table 4 Products with insufficient existing policies 56

Table 5 Products with the largest potential absolute savings 60

Table 6 A comparison of optimal policies to true optimal policies 63

Table 7 The cost of existing policies and the relative savings of the optimal policies under various scenarios 64

Table 8 True optimal and optimal policies for each demand scenario 66

v

List of Figures

Figure 1 A typical sample path of the process with a 4-day review period

and a 2-day lead time 24

Figure 2 The flow chart of the grid search algorithm 38

Figure 3 The flow chart of the grid search algorithm utilizing the updating

technique for the transition probability matrix 44

Figure 4 An evaluation of the lower bound Sm i n 47

Figure 5 The flow chart of the heuristic algorithm 50

Figure 6 The distribution of savings of optimal policies in (a) dollars and

(b) percentage of current cost 59

Figure A-1 The title screen 97

Figure A-2 The main menu 98

Figure A-3 Calculating the optimal policy and evaluating the current policy 99

Figure A-4 Displaying the results 100

Figure A-5 Entering an alternate policy 101

Figure A-6 The particulars of the alternate policy 102

Figure A-7 The main menu revisited 103

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Acknowledgement

The completion of this thesis was made possible by the encouragement and assistance

of a number of people

I would like to express my sincere appreciation to my thesis supervisor Professor

Martin Puterman for all of his many efforts on my behalf His help advice patience charity

and tolerance were very much appreciated

I would like to acknowledge Professor Hong Chen and Professor Garland Chow for their

time and input as members of my thesis committee In addition I must acknowledge the

assistance of PhD student Kaan Katircioglu for his insight and help on this project

I offer many thanks to Professor Tom Ross for his kindness and friendship during the

past few years especially during the writing of this work I must also thank his family for

adopting me on many holidays I still owe thanks to Professor Slobodan Simonovic at the

University of Manitoba for influencing me to attend graduate school in the first place and for

helping me to obtain funding

I cannot thank my parents enough for their never-ending support both emotional and

financial Also to my friends especially Cathy Dave Lisa Steve and the Philbuds thanks

for giving me a life off campus and for picking up many a tab - the next one is on me

I would like to give special thanks to my good friend and fellow MSc student Paul

Crookshanks for allowing me to bounce ideas off of him and for being such a procrastinator

that despite my finishing a year late I only lost by two days

Financial assistance received from the Natural Science and Engineering Research

Council (NSERC) and from Canadian Tire Pacific Associates was greatly appreciated

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I INTRODUCTION

The importance of inventory management has grown significantly over the years

especially since the turn of this century In colonial times large inventories were viewed as

signs of wealth and therefore merchants and policy makers were not overly concerned with

controlling inventory However during the economic collapse of the 1930s managers began

to perceive the risks associated with holding large inventories As a result managers

emphasized rapid rates of inventory turnover (Silver and Peterson 1985) Following the

Second World War Arrow Harris and Marschak (1951) and Dvoretzky Kiefer and

Wolfowitz (1952ab) laid the basis for future developments in mathematical inventory theory

Shortly thereafter new inventory control methodologies were widely applied in the private

manufacturing sector More recently when inflation and interest rates soared during the 1970s

many organizations were forced to rethink their inventory strategies yet again Today the

control of inventory is a problem common to all organizations in any sector of the economy

Sir Graham Day chairman of Britains Cadbury-Schweppes P C L expressed this sentiment

when he stated I believe the easiest money any business having any inventory can save lies

with the minimization of that inventory (Day 1992) Lee and Nahmias (1993) and Silver and

Peterson (1985) provide a more detailed history of inventory management and control

Most inventory problems in the retail sector involve the control and management of a

large number of different products Ideally the inventory policies should be determined for all

products on a system-wide basis however because the number of products is often in the tens

of thousands formulating a model to do so is impractical Therefore single-product models

are frequently used in practice to determine policies for products separately (Lee and Nahmias

1993) Lee and Nahmias (1993) and Silver and Peterson (1985) provide a good overview of

1

single-product single-location models for a more complete review of the literature we refer

to them

The degree of complexity of a single-product model depends largely on the assumptions

made regarding the demand cost structure and physical characteristics of the inventory system

For example demand can be either deterministic meaning it is known with absolute certainty

or stochastic meaning it is unknown but follows a known probability distribution

Deterministic models tend to be relatively simple yet useful such as the basic EOQ model

However future demand is seldom known with certainty in practice and therefore the use of

these simple models is limited As a result the majority of research in inventory has focused

on stochastic demand models (Lee and Nahmias 1993)

Because the objective of most inventory models is to minimize the cost of inventory

assumptions regarding the structure of the cost are very important also Generally speaking

the cost of inventory has four components the purchaseproduction cost the orderingsetup

cost the holding cost and the shortage cost1 Each of these costs can be fixed proportional

nonlinear or some complex combination of these Further because nearly all inventory

problems require decisions be made sequentially over a period of time the costs can be either

discounted or averaged In the former case a discount factor is included in the cost function

in order to account for the time value of money However since most inventory problems are

on-going the long-run average cost is usually minimized (Lee and Nahmias 1993) In

practice this is achieved by minimizing the expected cost per time period under steady state

conditions

Finally the physical characteristics of the system are important in determining the

1 The cost of a stockout is often very difficult to estimate in practice As a result service levels are frequently used in its place When this is done a constraint is included in the model

2

complexity of the model For example the review process can be either continuous in which

case the inventory is reviewed continuously and orders are placed as soon as the stock reaches

a certain level or periodic in which case the inventory is reviewed at regular intervals and

orders can be placed only at those fixed times Also excess demand can be fully backordered

as is common in many manufacturing and wholesale systems completely lost as is common

in most retail environments or some combination of the two Lead time assumptions are also

important models with lead times are more complex than those in which the supply of

inventory is assumed to be instantaneous

Although the literature is dominated by single-product models (Lee and Nahmias 1993)

much recent work has focused on extending those models to multiple-product problems

However multiple-product models are inherently more complex than the simple-product

models In fact no efficient algorithms are currently available for determining fully optimal

strategies for most multiple-product problems (Federgruen 1993) As a result research has

focused on the development of easily computable approximation techniques Muckstadt and

Roundy (1993) provide a good overview of multiple-product andor multiple-location models

with deterministic demand although emphasis is placed on models for production and

distribution Federgruen (1993) extends the discussion to consider these models under uncertain

demand and also provides a complete list of references on the subject Silver and Peterson

(1985) discuss models for coordinated replenishments for both deterministic and stochastic

demand

In this thesis we examine the inventory system at Canadian Tire Pacific Associates

owner of 21 retail stores and a central warehouse Although this is a multiple-product

multiple-location single-warehouse problem and strictly speaking should be modelled as such

we analyze the inventory problem on a product-by-product and store-by-store basis We believe

3

that this simpler single-product single-location model captures the essentials of the problem for

two reasons First the coordination of replenishments for products andor locations is most

advantageous when significant savings are possible because of high first order costs and low

marginal costs for placing subsequent orders However because of the large number of

products and the physical distribution system in this problem trucks are dispatched to stores

on a regular basis Therefore the benefits of coordinating replenishments for this problem are

negligible Second we believe that controlling in-store inventory alone can result in substantial

savings to Canadian Tire Pacific Associates Further given the excessive amount of inventory

in the warehouse a single-location model assuming an infinite warehouse capacity is reasonable

Of course should inventory control in the warehouse become a priority a more elaborate model

would be required

More specifically we consider a periodic review single-product inventory model with

stationary independent stochastic demand a deterministic lead time and an infinite planning

horizon The probability mass function for the demand is based on empirical sales data

collected over a period of approximately ten months The cost of inventory includes a

proportional holding cost and a fixed ordering cost neither of which is discounted All excess

demand is completely lost The objective of the model is to minimize the long-run average cost

per unit time subject to a service level constraint on the fraction of demand satisfied directly

from in-store inventory In order to obtain meaningful cost comparisons the model calculates

the expected annual cost of inventory

For inventory problems like the one described above optimal (sS) policies can be

computed by Markovian decision process methods such as successive approximations or policy

iteration or by stationary Markovian methods However each of these methods generate

calculations that are complex and time consuming Therefore much research interest lies in

4

developing techniques to find approximate (sS) policies with little computational effort The

majority of the approximation techniques begin with an exact formulation of the problem and

then manipulate the problem by relaxations restrictions projections or cost approximations

(Federgruen 1993) Porteus (1985) presents three approximation techniques and compares

them to fourteen others for a periodic review single-product inventory model in which

shortages are fully backordered and a shortage cost is incurred Federgruen Groenevelt and

Tijms (1984) present an algorithm for finding approximate (sS) policies for a continuous review

model with backordering and a service level constraint Also Tijms and Groenevelt (1984)

present simple approximations for periodic review systems with lost sales and service level

constraints All three works find that the approximation techniques perform well However

the techniques are evaluated for a variety of demand parameters that all result in policies with

a value of S in the range of 50 to 200 Given the slower moving products at Canadian Tire 2

it is unclear how well such approximation techniques would perform Therefore we do not

attempt to apply or modify any of these approximation methods

Exact optimal policies for this problem can be obtained by formulating a constrained

Markov decision process and solving it However the data generating requirements and the

lack of a good upper bound on S make this formulation impractical Therefore we formulate

a Markov chain model for fixed values of s and S and then employ a search technique to find

the optimal values for s and S In order to improve the performance of the model we develop

a heuristic search that locates near optimal policies relatively quickly

The remaining chapters of this thesis are organized as follows In Chapter II an

overview of the existing inventory system at Canadian Tire is presented along with a description

2 For example very few of the products analyzed have optimal values of S exceeding 20 with the largest value of S equalling 69

5

of the problem Next a multi-phase project designed to develop a fully integrated interactive

inventory control system is described with the scope of the research reported in this thesis

completing the first phase of the project Finally a description of the demand and cost data is

given

In Chapter III we formulate a Markov chain model that can be used to calculate the cost

and service level associated with a given (sS) policy Based on several simplifying

assumptions the Markov chain is formulated and the equations for the corresponding transition

probability matrix are developed Then given the limiting probability distribution of the chain

the equations for the steady state annual cost of inventory and the corresponding service level

are obtained The chapter concludes with a brief summary of the model

In Chapter IV an algorithm that searches for the optimal (sS) policy is developed As

a starting point a simple grid search algorithm is developed and evaluated on the basis of

execution time In order to improve the algorithms speed a technique for updating the

transition probability matrix is presented and the improvement in the algorithm is noted Next

a theoretical lower bound on S is obtained although the corresponding reduction in the

execution time is minimal Finally based on two assumptions regarding the shape of the

feasible space a heuristic search is developed which improves the speed of the algorithm

significantly

In Chapter V a prototype of the inventory control systems interface is created In its

present form the interface allows managers to evaluate inventory policies interactively The

manager can set review parameters compare optimal policies and current policies and evaluate

alternate policies The chapter concludes by recommending specific options be including in

future versions to make the system more automated and user-friendly

In Chapter VI optimal policies for all products in one product category sold at one store

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are compared to existing policies on the basis of expected annual costs and customer service

levels The analysis suggests that existing policies result in a large excess inventory and that

implementing the optimal policies would reduce the annual cost of inventory by approximately

50 A sensitivity analysis on the cost estimates is performed and suggests that the model is

not sensitive to relative errors of approximately 20 in the cost estimates Finally a sensitivity

analysis on the demand probability mass function is performed The analysis indicates that the

model is sensitive at least in terms of the customer service level to slight changes in the tail

of the demand distributions of low-demand products

Finally in Chapter VII the results of the analysis are summarized and future work is

outlined In order to incorporate recent changes in the review procedures at several stores the

model must be modified to allow for the duration of the review period to exceed the lead time

We recommend that the second phase of the project proceed specifically that a data collection

and storage scheme be designed and that a forecasting model be developed Because of the

substantial savings that can be realized by reducing inventory only at the stores inventory

control at the warehouse should also be investigated

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II T H E I N V E N T O R Y S Y S T E M A T C A N A D I A N TIRE

A Background

Canadian Tire Pacific Associates owns and operates 21 retail stores in the lower

mainland of British Columbia The stores carry approximately 30000 active products3 that

are divided into 80 product categories The merchandise comprises primarily automotive parts

and accessories home hardware housewares and sporting goods Canadian Tire Pacific

Associates also owns a central warehouse in Burnaby at which inventory is received from

Toronto Inventory is then shipped periodically from the warehouse via truck to the local

stores In 1992 total sales at the stores exceeded $99 million and the value of inventory was

estimated to be $28 million of which approximately half was stored in the warehouse

On selected days of each week a stores inventory records are reviewed automatically

and orders for products are placed to the warehouse as required Management uses the term

flushing to describe the process of reviewing the inventory and producing the tickets for stock

owing and the term picking to describe the physical act of filling the order at the warehouse4

Whether or not an order is placed for a product depends upon its current inventory level its

target level and its minimum fill 5 If the target level less the current level exceeds the

minimum fill an order is placed to bring the stock up to the target level From the moment

3 Active products are those that are regularly restocked Canadian Tire Pacific Associates can choose from up to 50000 products

4 Note that the number of picks per week is equal to the number of times per week that the inventory records are reviewed At the beginning of this study eighteen stores had two picks per week while the remaining three stores had only one

5 The target level is the maximum number of items that should be on the shelf at one time and the minimum fill is the minimum number of items that should be ordered

8

the order is placed to the moment the product is reshelved takes between two to four days

depending on the store On average 11000 items are shipped from the warehouse at each

flush

Although a products minimum fill is the same for all stores its target level depends

upon the size of the store Based upon its sales volume and retail space a store is classified

as being one of five sizes small (S) medium (M) large (L) jumbo (J) and extra jumbo

(EJ) 6 Each product has associated with it a target level for each size of store

Consequently all stores of the same size have identical inventory policies Table 1 lists the

location size number of picks per week sales and floor space of each store as of October

1993

Periodically the inventory manager reviews the policies and when deemed necessary

adjusts them In this way the manager can account for seasonality in demand changes in mean

demand and any shortcomings of previous policies At the present time the review process

is not automated nor performed on-line The inventory manager uses Hxl5 computer

printouts with five products to a sheet to study the relevant information and on which to note

any changes Afterward the changes are recorded in the system and the new policies take

effect

Currently the inventory policies are based on rules of thumb developed through the

years and not on a statistical analysis of demand nor on a calculation of the cost of inventory

These rules of thumb incorporate a marketing philosophy which at times dominates the

inventory strategy The prevailing philosophy holds that a large in-store inventory instills

6 During the course of this study stores were resized several times At one point the entire class of medium stores was eliminated leaving only four sizes of stores On another occasion the store in Chilliwack was changed from being a small store to a large store

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Table 1 Location and particulars of the 21 stores

Store Location Store Size

Picks Per Week

1992 Sales ($)

Floor Space (ft2)

1 Chilliwack S 2 2901969 6678

2 Coquitlam M 2 4015303 6510

3 Langley EJ 2 8138673 15422

4 Scott Road EJ 2 7307820 15064

5 Maple Ridge L 2 4102110 6678

6 Kingsway L 2 4153736 6574

7 Richmond EJ 2 8175105 14961

8 North Vancouver M 2 3856571 6534

9 Guildford M 2 3714372 6567

10 Burnaby M 2 3532752 6983

11 New Westminster S 1 2417992 6470

12 Abbotsford EJ 2 7899398 15303

13 Hastings J 2 4647794 9849

14 King Edward EJ 2 3825372 25492

15 White Rock L 2 3960262 6870

16 Main amp Mountain S 1 2705602 6431

17 Heather J 2 4161227 9427

18 King George J 2 4606817 9640

19 Cambie J 2 5855022 9091

20 Lougheed M 1 3114180 6615

21 Burrard J 2 6553310 10608

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consumer confidence which in turn leads to increased sales In other words if a customer

perceives that Canadian Tire is in the hardware business he or she will choose to shop there

when meeting future needs However management has adopted this philosophy without any

analysis of the cost of holding excess inventory Only recently has management begun to

question the cost-effectiveness of allocating inventory based on these rules of thumb

Speculating that existing inventory practices might be deficient Mr Don Graham the

vice-president of Canadian Tire Pacific Associates commissioned a pilot study to examine the

inventory system and to propose alternate strategies7 The study using a steady state model

to represent a simplified inventory system suggested that existing policies lead to overly high

inventory levels More importantly the study outlined a methodology that would enable

management to determine inventory levels based on observed demand The pilot study

recommended that a more detailed model of the inventory system be formulated that the

methodology be expanded to incorporate the cost of inventory and that an algorithm be

developed that would quickly identify the optimal inventory policies

B The Problem

At the time of the pilot study management was concerned primarily with its inability

to distinguish between stores of the same size As mentioned in the previous section a

stores size is based solely on its sales volume and floor space and under existing practices

all stores of the same size have identical inventory policies for every product Management

realized that demand for some products might be a function of variables other than size such

7 The pilot study was performed by Martin Puterman a professor of Management Science and Kaan Katircioglu a doctoral student in Management Science both at the University of British Columbia The results of the study were presented at a U B C Transportation and Logistics Workshop on March 25 1993

11

as location For example the demand for sump pumps is likely to be very different in suburban

areas than in urban ones However management was unable to allocate more sump pumps to

large suburban stores than to large urban stores This inability led management to seek a

tool that would allow for a more effective allocation of inventory between stores of the same

size

By the end of the pilot study however management was less concerned with

reallocating inventory within a size of store than with reallocating inventory between sizes

of stores and with reducing overall inventory According to management assigning individual

inventory policies to each store appeared to be operationally infeasible Firstly management

believed that having individual policies would lead to confusion in the warehouse and to

increased shipping errors Exactly why this would occur is unclear the warehouse would

simply continue to fill orders from tickets that make no reference to the policy in place

Secondly and perhaps more tenably the computer technology at Canadian Tire Pacific

Associates did not allow for policies to vary across all stores Both the database used to store

the inventory data and the printouts used by the inventory manager allowed for only five

different policies per product all with the same minimum fill Moving to individual policies

would have required an overhaul of the existing information system Although management

saw this overhaul as inevitable it did not see the overhaul as imminent Therefore

management switched the focus of the study to developing a tool that would help reduce overall

inventory levels andor reallocate inventory more effectively between groups of stores

The subtle change in the studys objective prompted an important question how should

the stores be grouped The current practice of grouping stores based on size is problematic

for two reasons First as discussed above additional factors of demand such as location

cannot be taken into account Second grouping inherently results in a misallocation of

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inventory when some stores within a group have differing numbers of picks per week For

example the medium-sized Lougheed store which has one pick per week requires higher

inventory levels to achieve a given customer service level than the other medium stores

which all have two picks per week8 In fact in terms of customer service levels it might

be more appropriate to group the Lougheed store with the large stores having two picks per

week Ideally this artificial grouping of stores will be abandoned when the computer system

is upgraded and policies can then be set on a store by store basis

As the project proceeded management grew somewhat uneasy with the concept of using

an analytical tool to evaluate and set inventory policies According to both the pilot study and

the preliminary results of this study significantly reducing inventory would greatly reduce costs

while still providing a high level of customer service However management held that because

the model cannot take into account intangibles such as marketing considerations such a drastic

reduction in inventory would be very inappropriate In this light management viewed the

optimal policies as lower bounds for inventory levels these policies would have to be

modified upwards in order to be practical Therefore management contended that any

methodology or software produced during this study should provide the means to evaluate and

compare a number of alternate policies

Regardless of exactly how the tool is to be used there is currently a lack of historical

sales data from which to analyze the inventory system and determine optimal policies No sales

data had ever been collected prior to the pilot study and only a relatively small amount has

been collected since In order to proceed with any analysis of the inventory system a

8 If demand is a function of size then the Lougheed store must meet the same demand with one fewer pick per week This would lead to either increased probabilities of stockouts at the Lougheed store or higher inventory levels at all the other medium stores

13

substantial amount of additional data would be needed The existing information system at

Canadian Tire Pacific Associates does allow for sales data to be captured however the system

is limited by the amount of data it can capture at one time and by its storage capacity In order

to collect the amount of data required management would have to upgrade the system and

develop some kind of data collection scheme coupled with a data compression technique The

collection scheme must allow for characteristics of demand such as trends and seasonality to

be identified and the compression technique must allow for sales data for over 30000 products

to be stored for each store

Because many of the products sold by Canadian Tire have nonstationary demand as a

result of seasonality andor trend the inventory tool must be able to forecast demand for the

coming period Products such as sporting goods and outdoor equipment exhibit patterns of

demand that are very seasonal For example sales of snow shovels increase through the fall

peak in the winter and drop to zero by late spring At the same time some products might

become either more or less fashionable over time resulting in changes in mean demand

Setting inventory policies for any of these products based on observed demand would result in

insufficient stock if demand increased during the period or in excess stock if demand decreased

Therefore the tool must be able to forecast the demand for the period for which the policy is

intended

Toward the end of the project management began discussing the integration of the

interactive inventory tool with the existing system to form a new inventory control system The

new system would utilize a modified point of sales system to collect sales data continually

Based on the latest available data the system would forecast demand for the coming period and

calculate the optimal policies for every product at every store The system would then provide

all relevant information on-line for the inventory manager to review The manager could weigh

14

other information such as marketing considerations and examine alternate policies to determine

the effect on cost Finally the preferred policy would be set in place with the touch of a key

and information on the next product or store would be presented for review

C The Project

Development of the inventory control system envisioned by management requires the

completion of five tasks 1) the design of a data collection system that generates and stores the

necessary sales data 2) the development of a forecasting method that provides accurate

estimates of demand distributions 3) the formulation of a model that calculates the cost of

inventory and the customer service level for an inventory policy 4) the development of an

algorithm that finds the optimal policy for a product at a store and 5) the creation of an

interface that allows interactive and on-line consultation Ultimately the above components

would have to be integrated and incorporated in a manner that would ensure the compatibility

and reliability of the system

The scope of the research reported in this thesis completes the first phase of the project

specifically we formulate an inventory model develop an algorithm and create a prototype of

the interface Because of the changing environment at Canadian Tire Pacific Associates minor

modifications to both the model and the algorithm developed herein might be necessary to

reflect future changes in operational procedures Upon completion of phase one the interface

will be interactive allowing managers to select the store product and review scenario and then

to compare optimal policies with existing policies or with alternate policies Although the

prototype will be delivered to Canadian Tire Pacific Associates for ih-house testing at the end

of this phase the interface will require modification in subsequent phases as feedback from

management is received and as the system is brought on-line

15

In order to assess more fully the potential savings of implementing the inventory control

system this thesis includes a comparison of optimal policies9 and existing policies The

expected annual cost of inventory for an entire product category is calculated for both the

optimal policies and the existing policies Based on an analysis of these costs the potential

savings of implementing the optimal policies are extrapolated Further the thesis examines the

sensitivity of the model to variations in the estimates of demand and costs

Originally results from all 21 stores were to be included in the above analysis

However just prior to the analysis management changed the review procedure at several

stores the result of which being that three picks per week are now performed at those stores

Because one of the main assumptions of the model specifically that the length of the lead time

does not exceed the duration of the review period is violated under this new review procedure

those stores cannot be included in the analysis Therefore the analysis is based solely on the

results from store six the large store located on Kingsway Avenue10

For the purpose of the analysis sales data for product category 20 automotive and

electrical accessories were collected from all 21 stores This product category was selected

because the distribution of demand for products within it is non-seasonal and because few of

the products are ever placed on sale11 The lack of trend or seasonality eliminates the need

for forecasting demand the optimal policies can be determined based directly on observed

demand

9 That is those policies obtained from the model

1 0 Store six is one of the only stores whose review procedure is not expected to change from two picks per week

1 1 Products that were on sale at any time during the period were not included in the data set

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D The Demand Data

In total sales data for 520 products of which 458 are active products were collected

over a 307-day period from October 10 1992 to August 18 199312 The data was supplied

by Canadian Tire Pacific Associates in the form of a 10 MB text file with one line per day per

product recording sales from the stores13 Each line had the following format

Columns 1-6 part number Columns 7-9 Julian date14

Columns 10-18 cost of the item (to three decimal places) Columns 19-124 sales for stores 1 through 21 (5 columns per store)

In order to facilitate the analysis the file was sorted into 458 sales files with each of

these files containing the history of daily sales for one of the products Two changes were

made to the format of these files to improve readability 1) a column was added between the

product number and the Julian date and 2) all leading zeros were dropped Appendix A

contains a partial printout of the sales file for a typical product Finally a distribution file

summarizing the daily sales was created from each of the sales files Each distribution file

contains the products number the products cost and the number of days during which various

quantities of the product were sold at each store

Prior to the analysis the distribution files for a large number of products were

examined and based on the shapes of the histograms the products were divided into three

groups The first group low-demand comprises products that have demand on only a few

days with the daily demand seldom exceeding three items The second group medium-

1 2 Of the active products only 420 are sold at store 6

1 3 Strictly speaking a line was written to the file only on days on which at least one store registered a sale of the product

1 4 According to the Julian calendar days are numbered sequentially from 1 to 365 (366 during leap-years) beginning with the first of January

17

demand comprises products that have demand on approximately half of the days with the daily

demand seldom exceeding more than six items Finally the last group high-demand

comprises products that have demand on most days occasionally upwards of twenty items and

whose demand histograms resemble Poisson distributions The majority of the products

examined fell into the low and medium-demand groups only a few high-demand products

were observed Appendix B contains a printout of the distribution file for a typical product

in each group

For most products no theoretical distribution appeared to model the demand well

Demand for low and medium-demand products followed a non-standard distribution with

a mass at zero and a relatively small tail Although some theoretical distributions such as the

mixed-Poisson can be made to resemble this it was unclear how well they would perform

Further the effects of sampling errors from such non-standard distributions were unknown

On the other hand the standard Poisson distribution did appear to model high-demand

products well however only a few products fell into that group Therefore in the absence of

a good theoretical candidate the empirical demand function was chosen for the analysis The

empirical function was readily available from the distribution files and offered the best

estimate of the true underlying function

E The Cost Data

Along with the sales data management supplied estimates relating to the costs of holding

and ordering inventory Management estimates that the cost of holding an item in inventory

for one year is 30 of the items cost Incorporated in this holding rate are a number of costs

such as the opportunity cost of money invested and the costs incurred in running a store

Because it is the annual cost of inventory that is of interest either the compounded daily

18

holding rate or the simple daily holding rate can be used to determine the costs incurred during

a review period however the method of calculating and extrapolating the holding costs must

be consistent For the sake of simplicity a simple daily holding rate is used

In order to estimate the cost of ordering inventory management analyzed the various

tasks that result from an order being placed Using the average time required to fill and then

reshelve an order management obtained an estimate of the direct labour costs involved To

this management added an estimate of the cost of transporting an order from the warehouse to

the store Finally the estimate was adjusted slightly upward to account for the cost of breakage

and errors Based on the analysis management estimates the cost of placing an order for a

product to be $0085 regardless of the size of the order

19

III M O D E L F O R M U L A T I O N

A Terminology

In the previous chapter several terms were introduced to describe the inventory system

at Canadian Tire Pacific Associates These terms though perfectly adequate are not widely

used in the academic literature to describe inventory management problems Therefore in

order to avoid confusion and be consistent with the literature the following terms are used

throughout this thesis

The length of time between successive moments at which the

inventory position is reviewed15

The time at which a reorder decision is made

The length of time between the moment at which a decision to

place an order is made and the moment at which the order is

physically on the shelf

An operating policy in a periodic review inventory model in

which (i) the inventory position is reviewed every T days (ii) a

replenishment order is placed at a review only when the inventory

level x is at or below the reorder level s (gt 0) and (iii) a

replenishment order is placed for S-x units (S gt s)1 6

Lost Sales - The situation in which any demand during a stockout is lost

Review Period -

Decision Epoch

Lead Time -

(sS) Policy

1 5 With respect to the terminology used at Canadian Tire Pacific Associates a review period equals the number of days between flushes

1 6 With respect to the terminology used at Canadian Tire Pacific Associates S represents the target level and s represents the difference between the target level and the minimum fill

20

B Assumptions

In to order make the model of the inventory system mathematically manageable several

simplifying assumptions are made However we believe that the following assumptions do not

significantly reduce the accuracy of the model

1) Demand can be approximated very closely by sales during the period in which the data

is collected Generally inventory levels at the stores are very high and the number of

lost sales is assumed to be negligible In fact the inventory levels for a number of

products were tracked over time during the pilot study and not a single incidence of

stockout was observed Because all products appear to have similar excess inventory

it seems reasonable to assume that virtually all demand resulted in sales

2) The demands in successive periods are independent identically distributed random

variables Preliminary tests (Fishers exact tests) suggested no dependence between

sales on consecutive days and plots of sales over time indicated no seasonal pattern

3) Demand is independent across products That is we assume that sales of one product

do not affect nor can they be predicted from sales of another product

4) Unmet demand results in lost sales with no penalty other than the loss of the sale

5) The duration of the review period for a store is an integer value and is fixed over a

planning period For stores with two picks per week we assume that the inventory

is reviewed every four days rather than on a four day - three day cycle

6) The decision epoch is the start of a review period Review periods begin at midnight

at which time the inventory is reviewed and orders are placed instantaneously

7) The duration of the lead time for a store is deterministic and integer-valued In reality

the lead time for some stores was stochastic but management felt the variation could be

controlled and virtually eliminated

21

8) The lead time does not exceed the review period In other words items ordered at a

decision epoch arrive and are shelved before the next decision epoch In practice this

did not always occur

9) Orders arrive at the end of the lead time and are on the shelf to start the next day

10) Holding costs are incurred daily for all inventory held at the start of the day

11) The warehouse has an infinite capacity that is all orders to the warehouse are filled

Because of the high inventory levels in the warehouse stockouts in the warehouse are

very unlikely

C A Markov Chain Model

Consider the situation in which an (sS) policy is applied to an inventory system based

on the above assumptions Inventory is reviewed at the decision epoch for each of the periods

labelled t = 0 12 and replenishment occurs at the end of the lead time provided an order

was placed Demand during day d in period t is a random variable whose probability mass

function is independent of the day and period The function is given by

f^ik) = Pr ltd = k = ak for k = 0 1 2

where ak gt 0 and E ak = 1 Let T denote the duration of a review period and L denote the

length of the lead time Next define D T ( as the total demand during period t D L as the total

demand during the lead time in period t and D T L as the difference between D T and DLt

Assuming that successive demands pound f pound 2 are independent the probability mass function of

1 7 The aggregated demand variables are given by

^ = XXd and DT_Ut=J d= d= d=L+l

22

each aggregated demand variable can be obtained by convolving the probability mass function

of t d l s Finally let X denote the inventory on hand at the beginning of period t just prior to

the review

The process X is a discrete time stochastic process with the finite discrete state space

0 1 S A typical sample path of the process with a review period of four days and a

lead time of two days is depicted in Figure 1 As period t begins X units are on hand The

inventory is reviewed and because X is less than s an order is placed for (S-X) units Over

the next two days a demand of D L r occurs and the inventory drops accordingly With the

arrival of the order the inventory level increases by (S-X) at the start of the third day of the

period Over the remaining two days a demand of D T L is observed bring the inventory level

down to X + 1 Note that in this example an order would not be placed in period t+l because

X r + i exceeds s

The stock levels at the beginning of two consecutive review periods are related by the

balance equation

_K- DLy+ (s - v - DT-LX ixtlts ( 1 )

t + l ~ (Xt - DTtY ifXtgts

where the positive operator is interpreted as

= max a 0

1 8 For example the probability mass function of D T bdquo denoted fD T ( is the T-th fold convolution of f t Mathematically the function can be obtained using the recursive relation

and the identity f1^ = f5(

23

-2 I 8

gt o

r

CO

o a

CM

X

00

24

Because X + 1 depends only on Xt the process X satisfies the Markov property

PrXt+l=jX0 = i0Xt_l=it_vXt = i) = PrXt+1=jXt = i

and therefore is a discrete time Markov chain Further because the probability mass function

of the random variable pound is independent of t and thus so too are the probability mass functions

of the aggregated demand the Markov chain is stationary19

D The Transition Probability Matrix

The stationary transition probability p(gt is defined as the probability of X + 1 being in state

j given that X is in state i independent of the period t In the inventory model p-- represents

the probability of starting the next review period with j items on the shelf given there are i units

on hand at the start of the current period By convention the transition probabilities are

arranged in an (S + l)x(S + 1) matrix P such that (P)y = p- Thus the (i + l)st row of P contains

the probability distribution for the stock on hand at the beginning of the next period given that

there are i units on hand at the start of the current period The matrix P is known as the

transition probability matrix Note that since we are concerned with a steady state analysis we

need not consider the possibility of initial stock levels greater than S once the inventory level

is below S it can never again exceed S

The transition probability ptj is calculated by expanding (1) to include all possible

instances of demand that allow for a change in inventory from i units at the beginning of a

period to j units at the beginning of the next period For example suppose the inventory policy

(510) is in place 2 items are on hand at the start of the current review period and 7 items are

1 9 Henceforth the subscripts t and d are omitted from the daily and aggregate demand variables for the sake of brevity

25

on hand at the start of the next period Then only three scenarios are possible

1) No items are sold during the lead time (meaning that the stock level increases to 10

after replenishment) and 3 items are sold during the remainder of the period

2) One item is sold during the lead time and 2 items are sold during the remainder of

the period

3) Two items are sold during the lead time (though demand could exceed 2) and 1 item

is sold during the remainder of the period

The transition probability p 2 7 is the sum of the probabilities of the various scenarios and is

computed by

p2J = PrDL=0PrDT_L = 3] + PrDL = l] Pr[DT_L=2 + PrDL2] PrDT_L = l]

In general the transition probabilities for the inventory model are calculated as follows

Case 1 The lead time is less than the review period

For 0 lt lt s

Pij

i-l pound [PrDL=kPrDT_LgtS-k] + PrDLgtiPrDT_LgtS-i if j = 0

i - l

^[PriD^^PrlD^S-j-kn+PHD^PriD^S-j-i if 0 lt j ltL S-i (2) k=0 s-j

E [PrDL=kPrDT_L = S-j-k] if S-i ltj^S k=0

26

For s lt i lt S

PrDTzi]

0 p = PrDT=i-j)

ifj = 0 ifOltjzi ifiltjltS

(3)

Case 2 The lead time equals the review period

For 0 lt i lt s

Pa

0 PrDTzi) PrDT = S-j

ifOltLJlt S-i ifj = S-i if S-i ltj plusmn S

(4)

For s lt i lt S use (3)

E The Steady State Probabilities

Provided 1) there is a positive probability of non-zero demand during a review period

and 2) there is a positive probability of zero demand during a review period20 then the

Markov chain described by (1) has a limiting probability distribution ir = (ir0 TTu irs)

where 0 lt IT lt 1 for j = 0 1 S and E 7ry = 1 Further this limiting distribution is

independent of the initial state In other words the probability of finding the Markov chain in

a particular state in the long-run converges to a nonnegative number no matter in which state

the chain began at time 0 The limiting distribution is the unique nonnegative solution of the

equations

2 0 That is 0 lt PrD x = 0 lt 1 Given the nature of the problem this assumption is not very restrictive

27

^ = E ^ - far j = 0 l S (5)

s and pound nk = 1 (6)

For the proof that the limiting distribution is independent of the initial state and is obtained from

(5) and (6) the reader is directed to Appendix C

In terms of the inventory problem 7ry is the steady state probability of having j units of

inventory on hand at the decision epoch However some subsequent calculations are made

easier if the steady state probability of having j units of inventory on the shelf just after

replenishment is known Instead of reformulating the Markov chain these probabilities can be

obtained by shifting the limiting distribution TT Much in the same way that the transition

probabilities are calculated the shifted limiting distribution t] is obtained by accounting for

all possible instances of demand that allow for the shifted inventory position For example

suppose that the policy (510) has the limiting distribution TT In order to have say 8 units on

hand immediately after replenishment one of the following must occur

1) Ten items are on hand at the decision epoch and 2 items are sold during the lead time

2) Nine items are on hand at the decision epoch and 1 item is sold during the lead time

3) Eight items are on hand at the decision epoch and nothing is sold during the lead time

4) Either 2 3 4 or 5 items are on hand at the decision epoch and 2 items are sold during

the lead time

The steady state probability of having 8 units on hand just after replenishment is calculated as

n 8 = n10PrDL = 2] +iz9PrDL = l] +n8PrDL = 0 +[iz5 + iz4 + iz3]PrDL = 2 +n2PrDLgt2

28

In general the shifted steady state probabilities are given by

s pound nPrDLgti ifj = 0

i=s+l S

pound PrDL = i-j] ifl plusmnjltS-s i=max(s+l)

S = pound 7tjirZ)L = i-+7i 5 FrD i ^5 ifj = 5-5

i=max(s+l)

^ 7 1 ^ ^ = 1-7+ pound TtiFrZgti=5-y+iis_PrD1^S-7 ifS-sltjltS i=max(gt+l) i=S-j+l

(7)

F The Cost of Inventory

For this problem the cost of inventory has two components (i) a fixed ordering cost

cf incurred whenever an order is placed and (ii) a holding cost cv incurred daily for each item

held in inventory The expected cost of inventory for a review period can be calculated by

summing the costs associated with each inventory position under steady state conditions

Because a simple holding rate is used this cost is simply adjusted by number of review periods

in a year to obtain the expected annual cost of inventory

The ordering cost for a review period is relatively straight forward to calculate either

an order is placed in which case the order cost is incurred or no order is placed in which case

no cost is incurred According to the rules of the inventory policy orders are placed only if

the stock on hand at the beginning of a review period is less than or equal to s Further the

probability that the review period begins with the inventory in one of these positions is equal

to the sum of the first (s + 1) steady state probabilities Therefore the expected ordering cost

29

for a review period is given by

E[OC] = cfY n y=o

Unlike the simple structure of the ordering cost the holding cost is incurred daily for

each item held in inventory Therefore the daily holding costs associated with all possible

inventory positions must be weighted and summed to obtain the holding cost for the review

period In order to make the calculations less complicated a shifted review period beginning

immediately after replenishment and ending just prior to the next replenishment is considered

Define e(0 to be the expected number of items on hand to start day i of the shifted

review period and let g(i f) denote the expected number of items on hand to start day i given

that there were j items immediately following replenishment21 The function g(z j) is the sum

of the probabilities of possible demand weighted by the resulting stock levels that is

g(ij) = j^kPriD^j-k)

To obtain the expected number of items on hand to start day i independent of j the above

function is summed over all possible values of j and weighted by the corresponding steady state

probabilities yielding

s j ed) = pound r 5 gt P r Z ) = - pound

7=0 k=

2 1 In other words g(if) is the expected number of items on hand to start day i given e(0)=v

30

The expected holding cost for a review period is then given by

T

E[HC] = $gt v e(0 i=i

i=l =0 k=l

where c v equals the simple daily holding rate multiplied by the cost of the item

Although differently defined review periods are used to calculate the above expected

costs the costs can still be summed to obtain the expected cost of inventory for a review

period The expected costs are calculated based on steady state conditions and therefore

represent the costs expected to be incurred during any T consecutive days Summing the above

two expected costs and multiplying the sum by the number of review periods in a year yields

the expected annual cost of inventory

E[AC] = mdash[E[OC]+E[HC]]

T S j 1 ( g ) 365

j=0 i=l j=0 fc=l

G The Customer Service Level

There are many ways to define customer service and the most appropriate definition

depends upon the nature of the business and the perspective of management Two commonly

used definitions are 1) the proportion of periods in which all demand is met and 2) the

proportion of demand satisfied immediately from inventory Although the former definition is

often used because of its simplicity the latter definition is closer to what one commonly means

by service (Lee and Nahmias 1993) A third definition also relevant in retail environments

31

is the probability that lost sales do not exceed some critical number Other definitions such

as probability that backorders are filled within a certain amount of time are more applicable

to industrial and manufacturing settings where backlogging demand is common

Given the retail environment with the possibility of lost sales that exists at Canadian Tire

Pacific Associates management is concerned primarily with the fraction of sales that are lost

For example management views the situation in which demand exceeds stock by one unit every

period more favourably than that in which the same demand exceeds stock by three units every

other period For this reason the proportion of demand satisfied immediately from inventory

is the most appropriate measure of customer service

The customer service level for a single review period CSLRP is given by

r bdquo T _ Demand satisfied during RP Total demand during RP

Because we are interested in the long-run behaviour of the system a measure of the steady state

customer service C S L must be obtained As shown in Appendix D the correct measure is

poundpoundpound _ E[Demand satisfied during RP] E[Total demand during RP]

which can be rewritten as

pound pound pound _ j _ E[Demand not satisfied during RP] ^ E[Total demand during RP]

Because of the steady state conditions the above customer service level can be estimated

from any period of T consecutive days As was done for one of the cost calculations the T-day

period beginning just after replenishment is considered Suppose that there are j units of

inventory on hand immediately following replenishment Then the conditional customer service

32

level is

CSL j = 1 E[(DT-j)l

E[DT] (10)

The denominator in the second term is merely the expect demand during the period

E[DT] = 7gt (11)

where p is the mean of the random daily demand variable pound The numerator which is the

conditional expected demand that is not satisfied during the T-day period expands to

7-1

E[(DT-j)+] = T u - j + YU-k)PrDT = k] k=0

(12)

The reader is directed to Appendix E for a mathematical justification of (12) In order to obtain

the unconditional expected demand not satisfied (12) is applied to all possible values of j and

a weighted sum of the terms is calculated The result is given by

E[Demand not satisfied] = Y j =0

y-i Ti - j + 1pound(j-k)PrDT = k

S j-1

= - E[I] + Y0-k)Pr[DT=k) =0 it=0

(13)

where E[I] denotes the steady state inventory level Finally the customer service level is

obtained by the substituting (11) and (13) into (9) yielding

CSL = 1

s j-l

Ti - E[I] + Jgty Y(J-k)PrDT=k]

m n J2VjTU-k)PrDT=k Eil J=Q fc=o

(14)

33

H A Methodology for Evaluating an (sS) Policy

In this section a brief methodology summarizing the main results of the model is

presented The methodology can be applied to evaluate a given (sS) policy in terms of its cost

and its resulting customer service level Given the probability mass function of the daily

demand f the duration of the review period T the length of the lead time L the fixed cost

of placing an order cf and the daily unit cost of holding inventory cv

1) Calculate the aggregated demand probability mass functions fDT fDL and f D x L as the T-th

fold L-th fold and (T-L)th fold convolutions of f respectively

2) Generate the transition probability matrix using either equation (2) or (4) and equation (3)

3) Calculate the limiting distribution TT from (5) and (6)

4) Obtain the L-day shifted limiting distribution rj using (7)

5) Calculate the expected annual cost of inventory using (8)

6) Determine the customer service level using (14)

34

IV A N A L G O R I T H M F O R OBTAINING T H E O P T I M A L (sS) P O L I C Y

A Introduction

The methodology developed in the previous chapter can be used to calculate the cost and

service level associated with a given (sS) policy In this chapter an algorithm that finds the

feasible policy yielding the lowest cost is developed A policy is feasible if it satisfies both the

basic condition 0 lt s lt S and the imposed constraint on the customer service level2 2 In

order to find the optimal policy the algorithm must combine the methodology for evaluating

a single policy with a search technique

Search techniques are evaluated in terms of two somewhat antithetic criteria On the one

hand the search should locate the optimal solution or at least a near optimal solution on

the other hand the search should take as little time as possible Many search techniques

guarantee optimality provided the objective function and constraints exhibit certain properties

such as convexity However even if such a search technique also guarantees its termination

in polynomial time the search might take too long to be practical Alternatively some search

techniques provide reasonable though sub-optimal solutions very quickly These techniques

often called heuristics must be evaluated in terms of both their speed and the accuracy of their

solutions The choice of a heuristic over a search that guarantees optimality depends upon the

nature of the feasible space the need for accuracy and the cost of time

Given the complexity of the cost and customer service level functions it is very difficult

to show theoretically for what if any combinations of demand and costs the feasible space is

convex However a plot of the objective function for a single product may be obtained by

2 2 Management specified that each policy must provide a customer service level of at least 975 Therefore the algorithm must incorporate the constraint CSL gt 0975

35

evaluating the costs for a number of policies within a grid When the service level constraint

is included in these plots maps of the feasible space are obtained Examination of the

maps for several products suggests that the space is convex over most regions however

maps for some products reveal some regions of non-convexity Unfortunately there is no

way of knowing the degree of non-convexity for other products other stores or different costs

estimates without a more rigorous survey of the feasible space Therefore if one of the

techniques that guarantee optimality under the condition of convexity is employed the search

would have to be evaluated as a heuristic Instead as a first step to finding a practical search

technique a crude search that guarantees optimality under virtually all conditions is

investigated

B A Grid Search

The simplest search technique that guarantees optimality merely evaluates all policies in

the grid defined by 0 lt s lt S and 0 lt Sm i n lt S lt S m a x where Sm i n and S m a x are

some predefined values That is the grid search is a brute force method that examines every

policy within the grid 2 3 Although the number of policies that must be examined is large the

grid search is guaranteed to find the optimal policy provided that S o p t is contained in the grid

that is provided Sm i n lt S o p t lt Sm a x The grid search algorithm proceeds by generating the

transition probability matrix for the policy (0Smin) and then evaluating the service level

associated with that policy If the policy is feasible24 the cost associated with the policy is

2 3 The total number of policies evaluated is the sum of the first Sm a x integers less the sum of the first Smin-1 integers Numerically [(l2)(Sm a x)(Sm a x+l) - (l2)(Smin-l)(Smin)] policies are examined

2 4 That is if the service level is adequate

36

calculated and compared to the current lowest cost The procedure continues until all points

in the grid have been examined and the optimal policy has been found A flow chart of the grid

search algorithm is shown in Figure 2

In order to assess the grid search the algorithm was coded and compiled into an

executable program The program was run for two products (a medium-demand product and

a high-demand product) and with four values of Sm a x In each run Sm i n was taken to be zero

The following times were recorded for each trial

T P M - the amount of time spent generating the transition probability matrix

GAUSS - the amount of time spent solving for the steady state probabilities

COSTS - the amount of time spent calculating the expected annual costs

SERVICE - the amount of time spent determining the customer service levels

T O T A L - the total execution time of the program

The columns entitled Grid Search in Tables 2 and 3 summarize the results of the trials All

times are recorded in seconds as required by a 486DX33 MHz personal computer with an

empty buffer memory

Three results are evident from the tables First the vast majority of the execution time

is spent generating transition probability matrices and then solving for the steady state

probabilities with the time being split almost equally between the two tasks Second the total

execution time of the algorithm increases exponentially as Sm a x increases For product 206917

doubling S^ from 30 to 60 led to an increase in the execution time of more than 2100 (from

172 seconds to 3661 seconds) Third the grid search algorithm as it stands does not appear

to be adequate for the problem at hand Given the large number of products that must be

evaluated the algorithm is much too slow to be implementable

37

Calculate the demand probability functions

Initialize s = 0 S = Smin] cost = 10A5

s = s+1

Calculate the probability transition matrix for (sS)

Evaluate policy (sS) Is service level ^ 0975

No

No Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Is s = S-l

Yes

Is S = Smax No

Yes

Print results (sS) cost

s = 0 S = S+1

Figure 2 The flow chart of the grid search algorithm

38

Table 2 Execution times of the algorithms for product 200001 a 30-amp inline fuse

(in seconds)

-max Subroutine Grid Search Grid Search

Updating T P M Heuristic

10 T P M 006 010 001

GAUSS 010 006 001

COST 005 005 001

SERVICE 001 001 001

T O T A L 050 035 027

20 T P M 126 021 010

GAUSS 119 160 023

COST 027 026 001

SERVICE 017 016 001

T O T A L 318 236 055

30 T P M 703 096 010

GAUSS 757 801 093

COST 079 085 001

SERVICE 047 042 001

T O T A L 1664 1137 126

40 T P M 2257 163 015

GAUSS 2958 3003 206

COST 313 329 001

SERVICE 092 083 011

T O T A L 5871 3851 285

39

Table 3 Execution times of the algorithms for product 206917 a 6 solder connector

(in seconds)

bull-max Subroutine Grid Search Grid Search

Updating T P M Heuristic

30 T P M 717 067 006

GAUSS 762 880 093

COST 098 087 010

SERVICE 016 021 006

T O T A L 1720 1176 220

40 T P M 2390 157 022

GAUSS 2861 2911 209

COST 219 227 006

SERVICE 080 088 011

T O T A L 5827 3790 362

50 T P M 6691 413 010

GAUSS 7994 8201 472

COST 568 577 016

SERVICE 212 197 005

T O T A L 15840 10041 654

60 T P M 15509 894 040

GAUSS 18607 18951 876

COST 1367 1353 038

SERVICE 350 359 022

T O T A L 36614 22794 1120

40

C A Technique for Updating the Transition Probability Matrix

As was noted in the previous section approximately half of the execution time of the grid

search algorithm is spent generating transition probability matrices Furthermore the amount

of time increases exponentially as Sm a x increases This increase is exponential for two reasons

first the number of transition matrices that must be calculated is a function of S^ squared and

second the average size of the matrices and hence the average number of calculations per

matrix increases with Sm a x For example when S m a x equals 30 the transition probability matrix

must be generated 465 times with the largest matrix having dimensions (31x31) When Sm a x

equals 40 the matrix must be generated 820 times with 355 of the matrices having dimensions

larger than (31x31) The performance of the algorithm could be improved considerably if

either the number of matrices that must be generated or the number of calculations per matrix

is reduced This section introduces a technique that accomplishes the latter by updating the

transition probability matrix for a new policy instead of generating it anew for each policy

A new policy (s+mS)

Suppose the policy (sS) has associated with it the (S + l)x(S + l) transition probability

matrix P and the policy (s+mS) where m is any integer satisfying 0 lt s+m lt S has

associated with it the (S + l)x(S + 1) transition probability matrix J Then T can be obtained by

recalculating only m rows of P Recall that the equations for generating a transition

probability matrix are divided into two sets based upon the row being calculated equations (2)

or (4) for rows 0 through s and (3) for rows (s + 1) through S The equations themselves are

independent of s only the limits defining the cases contain s Therefore only the m rows

that are affected by the change in the limits need to be recalculated The following updating

technique can be used to obtain 7

41

1) If m gt 0 rows i = (s + 1) (s+2) (s+m) are recalculated using (2) or (4)

2) If m lt 0 rows i = (s-1) (s-2) (s-m) are recalculated using (3)

3) All other rows of T are identical to those of P

The reader is directed to Appendix F for a mathematical justification of the technique

A new policy (sS+m)

Suppose the policy (sS) has associated with it the (S + l)x(S + l) transition probability

matrix P and the policy (sS+m) where m is any positive integer has associated with it the

(S+m+l)x(S+m+l) transition probability matrix P Then T can be obtained by modifying

each row of P and adding m additional rows and columns The updating technique is not as

simple as that for a new policy (s+mS) for two reasons the dimensions of P and T differ and

the equations for generating a transition probability matrix contain S Nevertheless an updating

technique that requires much less computing time than generating the matrix anew can be

developed The following technique can be used to obtain P

1) For rows i from 0 through s

For columns j from 0 through m recalculate using (2) or (4)

For columns j from (m+1) through (S+m) enter the (j-m)th column entry of P

2) For rows i from (s + 1) through S

For columns j from 0 through S enter the corresponding value from P

For columns j from (S + 1) through (S+m) enter 0

3) For rows i from (S + 1) through (S+m) recalculate using (3)

The above technique is justified mathematically in Appendix G

42

The modified algorithm

The updating technique can be easily incorporated into the grid search algorithm with

only a slight change in the manner in which s is incremented Figure 3 shows the flow chart

of the new algorithm As before the algorithm was coded and compiled into an executable

program and the various execution times were recorded The results are listed in Tables 2 and

3 under the columns Grid Search Updating TPM

Updating the transition probability matrix does improve the performance of the

algorithm The time required to generate the transition probability matrices is reduced by

approximately 90 This translates to a reduction in total execution time of approximately

30 Not surprisingly the reductions are greater for large values of Sm a x However despite

this seemly significant improvement a decrease of only approximately 08 seconds is obtained

when S^ equals 20 Further when S ^ equals 30 the algorithm requires over 11 seconds

which is still much too high

The execution time is now dominated by the time required to solve for the steady state

probabilities The steady state probabilities could be solved by using approximation techniques

which are faster than solving the systems exactly however because the probabilities are often

very small numbers the approximation techniques would be unable to provide the required

accuracy The only remaining way to decrease the execution time is to reduce the number of

policies that are evaluated during the search This can be done by either reducing the size of

the grid being searched or reducing the number of policies being evaluated within the grid The

following section introduces a method of reducing the size of the grid by obtaining a lower

bound for S namely Sm i n

43

s = s+k

Calculate the demand probability functions

Initialize s = 0 S = Smin| cost = 105

k= 1

Update the probability transition matrix for (sS)

Evaluate policy (sS) Is service level ^ 0975

No

No

iNo

Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Ifk= 1 iss = S-l If k =-1 is s = 0

Yes

Is S = Smax No

Yes

Print results (sS) cost

If k=l s=s+l S = S+1

Figure 3 The flow chart of the grid search algorithm utilizing the updating technique for the transition probability matrix

44

D A Lower Bound on S

As mentioned earlier the size of the grid is specified by the number of policies within

the domain defined by 0 lt s lt S and 0 lt Smin lt S lt Smax Thus the size of the grid

and hence the total amount of computation time can be reduced by obtaining tight bounds on

S Unfortunately because of the extreme non-linearity of the objection function and the service

level constraint obtaining a mathematical upper bound on S is very difficult and beyond the

scope of this work This section introduces a method for obtaining a lower bound on S25

For a given value of S an upper bound on the customer service level can be found as

follows Suppose that there are S units of inventory on the shelf at the beginning of every

review period regardless of demand No (sS) policy can provide a higher level of customer

service because there exists a positive probability of beginning a review period with less than

S units on the shelf Thus the customer service level associated with starting each review

period with S units on the shelf is an upper bound on the customer service level for any (sS)

policy From (10) this level is

E[(DT-SY] CSL 5 = 1 - T mdash mdash - ( 1 5 gt

E[DT]

Replacing j with S in (12) and then substituting (12) and (11) into (15) yields

CSL 15 = mdash- - J2(S-k)PrDT = k] (16)

To obtain Smin use (16) to find the largest value of S with a customer service level less than

0975

There are two criteria for a good lower bound on S the bound must be no larger than

This method was proposed by Kaan Katircioglu

45

S o p t and it must be very close to So p t These criteria ensure that the optimal policy will be

contained in the search grid and that the size of the grid is reduced by as much as possible

Based on these criteria the above method of obtaining a lower bound is assessed A value for

S m i n is obtained for each of the products in product category 20 and compared with the

corresponding value of S o p t 2 6 The results are illustrated in Figure 4 which shows the number

of occurrences of various Sm i n and S o p t combinations Note that a point lying on the 45deg line

represents a perfect bound while a point lying close to the S o p t axis represents a very loose

bound The absence of any points above the 45deg line indicates that the first criterion is satisfied

in each instance the method did not generate a value of Sm i n exceeding the corresponding value

of So p t Moreover the bounds are reasonably tight for 85 of the products analyzed that

is for those products with S o p t equal to 10 or less27 However as S o p t increases the bounds

become steadily worse Because values of S o p t are likely to be larger for these products at the

ten Jumbo and Extra Jumbo stores and for other products with higher demand the lower bounds

are not expected to be very tight in general

Although obtaining the bound Sm i n does reduce the size of the grid the corresponding

reduction in the execution time of the algorithm is miriimal For example if S^ equals 60 and

a value of 30 is obtained for Sm i n the program would still require approximately 215 seconds

to search the grid (compared to the 228 seconds required without a lower bound on S) Because

the execution time increases exponentially with So p t a tight upper bound on S would likely have

a more significant impact on reducing the execution time of the algorithm Further research

on this issue is required

2 6 The optimal policy for each product was obtained using a grid search with an extremely large upper bound

2 7 On average Sm i n is approximately half of S o p t for these products

46

47

The only remaining approach to improving the algorithm is to modify the search so that

fewer policies within the grid are evaluated Toward this goal assumptions can be made with

respect to the feasible space and a heuristic search can be developed based on these

assumptions

E A Heuristic Search

As mentioned previously in this chapter a map of the feasible space for each of

several products was obtained Upon closer examination the maps illustrate two patterns

without exception for a given value of S both the cost and the customer service level increase

as s increases The following intuitive argument is offered as a possible explanation for the

observed behaviour When S is held constant increasing s results in more instances in which

an order is placed upon review Hence the average stock level would increase which in turn

would lead to an increase in the cost and the customer service level28 With this in mind a

heuristic search based on the following assumptions is developed for a given value of S both

the cost and the customer service level are monotonically increasing functions of s

In general the heuristic proceeds as follows Having just incremented S by one the

customer service level of the policy (sS) is evaluated If the policy is feasible s is decreased

until the lowest value of s producing a feasible policy is found If the policy is not feasible

s is increased until either the cost exceeds the current lowest cost for a feasible policy a

feasible policy is obtained or s equals S-l In the event that a feasible policy is found the cost

2 8 In order to prove that the customer service level increases as s increases it would be sufficient to show that the probability of placing an order increases as s increases Such a result might be possible if assumptions regarding the demand function are made Further if both the probability of placing an order and the average stock level increase then both the ordering cost and the holding cost would increase No proofs are offered here

48

of the policy is calculated if the cost of the policy is less than the current lowest cost the new

policy becomes the current best Finally S is increased by one and the process repeats

provided S does not exceed some predefined value Sm a x The flow chart of the algorithm

incorporating the heuristic search is shown in Figure 5 Note that the algorithm begins by

attempting to improve on the lower bound Sm i n the policy (Sm i n-lSm i n) is evaluated and if it

is not feasible S m i n is increased by one The improvement continues until a feasible policy is

found

As previously mentioned a heuristic search must be evaluated in terms of both its speed

and the accuracy of its solutions To evaluate the heuristic on the basis of speed the same

approach used to evaluate the previous algorithms is employed The columns entitled

Heuristic in Tables 2 and 3 contain the various execution times of the heuristic The heuristic

is a vast improvement over the previous search techniques the reduction in execution time is

enormous and becomes more so as S m a x increases For a S m a x of 20 the reduction is

approximately 75 of the time required by the grid search algorithm29 and for a S m a x of 60

that reduction soars to over 95 More importantly the heuristic can search a grid with a

of 40 in under 4 seconds or a grid with a S m a x of 60 in under 12 seconds These low execution

times should be sufficient to allow for a large number of products to be analyzed in a

reasonable amount of time

In order to further assess the practicability of the heuristic the total time required by the

heuristic to find solutions for the entire product category was determined and compared to the

corresponding time for the grid search algorithm With a priori knowledge of the optimal

solutions conservative yet reasonable values of S m a x were selected and used by both of the

2 9 Hereafter grid search algorithm refers to the algorithm utilizing the grid search the updating technique and the lower bound Sm i n

49

Calculate the demand probability functions

Calculate Smin S = Smin s = S-l

S = S+l s = S-l t

Update the probability transition matrix for (sS)

No Is service level pound 0975 Yes

No Is service level ^ 0975

Yes

Is cost lt cost No

Yes

Is s lt S-l No

Yes

s = s+l

Update the probability transition matrix for (sS)

Is service level pound 0975

Yes

Is cost lt cost No

Yes

s = s S = S cost = cost

s = s-l

No

Update the probability transition matrix for (sS)

S = S+l

Is s 2 0 Yes

Update the probability transition matrix for (sS)

Is service level pound 0975

No

s = s+l

No

No

Is S = Smax

^Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Print results (sS) cost

Figure 5 The flow chart of the heuristic algorithm

50

algorithms30 The grid search algorithm required approximately 9 hours and 41 minutes to

find the solutions while the heuristic required only slightly more than 23 minutes Based on

speed the heuristic is preferable to the grid search algorithm

Finally to evaluate the heuristic based on the accuracy of its solutions its solutions were

compared to the previously obtained optimal policies Without exception the heuristic

algorithm found the optimal policy Although there is no guarantee that the heuristic will find

the optimal solution for different products or different demand functions the heuristic appears

to work extremely well Based on the fast execution time and the apparent accuracy of the

solutions the heuristic will be used in the development of the inventory tool

3 0 For the test the following values of Sm a x were used 20 (273 products) 30 (19) 50 (23) 100 (3) and 125 (2)

51

V T H E I N T E R F A C E

With the model formulated and the algorithm developed the only remaining task in the

development of the prototype is the creation the interface In fact two interfaces are created

one that displays results in terms of minimum fills and target levels and one that displays

results in terms of reorder points and target levels To be consistent with the previous chapters

the latter interface is described here31

In its present form the interface allows the manager to set review parameters to

compare optimal and current policies and to evaluate alternate policies From the main menu

the manager selects the product store duration of the review period length of the lead time

and value of Sm a x Once the parameters are set the system calculates the optimal policy and

then evaluates the current policy displaying results as they are obtained In particular the

reorder point target level expected annual cost and expected customer service level are

displayed for each policy If a current policy meets the desired customer service level the

expected annual savings of the optimal policy are displayed in terms of both dollars and

percentage of current cost otherwise the value of the current customer service level is

displayed in red to indicate that it is deficient Finally the manager can choose to evaluate

alternate policies or to return to the main menu

Although the prototype is far from the ultimate integrated inventory control system it

is still valuable to management As mentioned in Chapter II inventory policies at Canadian

Tire Pacific Associates are currently based on rules of thumb that incorporate marketing

3 1 Because significant changes to the interface are expected as feedback from management is received and as the system is brought on-line only a brief description of the present interface is given

52

strategies The prototype provides management with a means of determining the proportion of

the cost of inventory that is not attributable to meeting customer demand For example

consider a hypothetical consultation in which the manager chooses to analyze product 200001

a 30-amp inline fuse at store 6 The prototype displays to the manager the current policy of

(1214) which provides a customer service level of 1000 at an annual cost of $748 and the

optimal policy of (39) which provides a customer service level of 977 at an annual cost of

only $319 If for some reason the manager feels the optimal policy is undesirable he or she

can investigate the additional cost of holding excess inventory Suppose the manager chooses

to evaluate the policy (610) The prototype evaluates the policy and displays the results the

customer service level is 998 and the annual cost is $441 With the alternate policy

virtually all demand is satisfied and the cost of inventory is still reduced by over 40 Thus

the manager ascertains that approximately 40 of the current cost of inventory for this product

is directly attributable to marketing considerations Using the prototype in this way

management can begin to understand better the nature of its inventory costs A description and

graphical depiction of the above hypothetical consultation are presented in Appendix H

As for the technical details of the system the prototype can run on any computer using

the DOS operating system having approximately 420K of available conventional memory and

attached to a V G A or SVGA monitor Further the prototype can be used on a machine without

a math coprocessor however performance is slower The prototype consists of two modules

linked together into one executable program The interface module contains the various

interactive subroutines and the main module contains the methodology of the model and the

logic of the heuristic search Both modules are coded in Fortran and compiled using Microsoft

Fortran 50 which provides many extensions to standard Fortran The source codes for the

interface module and the main module are listed in Appendix I and Appendix J

53

respectively

In addition to the above technical specifications the program requires the presence of

certain files in certain directories The program uses the distribution files mentioned in

Chapter II to read the unit costs and the empirical demand functions Each product has a

unique distribution file the name of which is the six digit product number followed by the

extension dst The text file containing the existing policies at each store must be located in

the same directory as the distribution files Finally the file tmsrbfon which contains the

font used by the interface must be located in the directory from which the program is executed

In its future versions the system should be more automated and more user-friendly

Rather than having the manager choose an upper bound for the search the algorithm should

incorporate a method of determining Sm a x For this to happen either a theoretical upper bound

must be obtained or a heuristic for terminating the search must be developed Also the amount

of time spent waiting for calculations to be performed can be reduced by reading the optimal

and current policies and their associated costs and customer service levels from a file or

database This solution file could be created any time prior to the consultation by running

the system in a batch mode The interface should allow the manager to change cost estimates

and service level values view results graphically and print results and summary documents

Finally the system should provide an option for automatically incrementing the product andor

store during a consultation This option would enable a manager to review quickly the

inventory policy of a product at each store

54

VI R E S U L T S AND SENSITIVITY ANALYSIS

A Comparison of Current Policies to Optimal Policies

In order to estimate the potential savings of implementing optimal policies the solutions

obtained in the Chapter IV are analyzed in more detail For each product the current policy

is evaluated and then compared to the optimal policy with respect to cost and customer service

level Finally the potential savings of the optimal policies are extrapolated to all products and

all stores

Of the 420 products examined each of 37 products has an existing policy that fails to

provide the desired customer service level of 975 However most of these policies just

fail to do so only 4 policies provide a customer service level of less than 90 Table 4

provides a list of these 37 products each with its current customer service level its current

annual cost and the additional cost of its optimal policy3 2 Two comments regarding Table

4 are in order First the current customer service level for product 203312 an amber

driveway marker is listed at a misleading 303 Examination of the demand distribution

reveals that this product was sold on only 1 of the 307 days during which sales data were

collected and on that day 13 of the markers were sold Clearly this data point constitutes

either an error or an unusual event the current policy of (34) does not allow for more than 4

items to be sold per review period Second the additional cost of the optimal policy for each

of 5 products is negative In other words for each of these products the current policy which

fails to provide the desired customer service level costs more than the optimal policy which

3 2 This is the difference between the expected annual cost of the optimal policy and the annual cost of the current policy It is the minimum additional cost required to increase the customer service level to at least 975

55

Table 4 Products with insufficient existing policies

Product Current CSL ()

Current Annual Cost ($)

Additional Cost of the Optimal Policy ($)

206932 974 204 -051

204570 973 222 -025

203053 973 248 009

206550 972 118 042

206547 972 105 035

206565 972 146 056

203702 972 340 157

206876 972 155 061

205752 972 330 153

205922 972 108 036

206857 972 180 073

205393 972 190 -003

201513 971 345 015

203405 970 255 037

203069 968 295 -048

203071 967 321 -001

205635 967 390 029

203914 966 460 160

204601 962 252 061

203326 960 144 014

202003 958 876 437

206516 958 112 029

206517 958 283 120

206350 958 333 147

205932 958 180 066

203936 952 598 294

56

Product Current CSL ()

Current Annual Cost ($)

Additional Cost of the Optimal Policy ($)

206932 974 204 -051

203939 948 413 193

203977 945 1216 1242

206551 945 132 077

205389 934 598 047

203590 927 1223 607

206567 919 122 070

203591 905 1228 834

205390 887 291 074

203318 833 136 084

206393 776 319 289

203312 303 237 511

provides at least the desired customer service level For the 36 products the total additional

cost of the optimal policies is $542033

Of the remaining 383 products most have an existing policy that provides a customer

service level well above 975 When the customer service levels are rounded to one decimal

each of 253 products has a customer service level of 1000 and each of 347 products has a

customer service level of at least 990 In contrast each of only 58 products has an optimal

policy that yields a customer service level of at least 99034 For each of 37 products the

3 3 The cost associated with product 203312 is not included in the total

3 4 Optimal policies do not have customer service levels of exactly 975 because the function is discrete For low-demand products a unit step in s or S results in a relatively large change in the customer service level

57

current policy is identical to the optimal policy

The extremely high level of service provided by most policies allows for substantial

savings in the cost of inventory With current policies the estimated annual cost of inventory

for all 420 products is $420743 whereas the expected annual cost of the optimal policies is

only $186884 This amounts to a savings of approximately $2340 or over 55 of the current

cost of inventory

In order to further assess the savings the difference between the costs of the current

policies and the costs of the optimal policies are examined in more detail For each of the 383

products whose existing policy provides a customer service level above 975 the difference

between the estimated annual cost of the existing policy and the expected annual cost of the

optimal policy is calculated The resulting histograms of the savings in (a) dollars and (b)

percentage of current cost are shown in Figure 6 The particulars for the current and optimal

policies for all 420 products are provided in Appendix K

As evident from Figure 6(a) the majority of the expected savings in absolute terms

comes from only six products In order to examine these products more closely the current

and optimal policies along with the associated costs and savings for each of the six products

are listed in Table 5 Each current policy is very different than its corresponding optimal

policy resulting in excessive inventory for each product In fact over half of the total expected

savings could be realized by implementing the optimal policy just for each of these six products

Interestingly five of the six products are sealed beam headlights

58

90

10 - 20 30 - 40 50 - 60 70 - 80 90- 100

Savings ()

(b)

Figure 6 The distribution of savings of optimal policies in (a) dollars and (b) percentage of current cost

59

Table 5 Products with the largest potential absolute savings

PRODUCT

CURRENT OPTIMAL SAVINGS

PRODUCT s S Cost ($)

C S L

()

s S Cost ($)

C S L () ($) ()

2036801 194 200 64014 1000 37 48 bull 9245 975 54769 856

2036772 94 100 23652 1000 18 22 3850 978 19802 837

2036723 78 90 23043 1000 19 23 4454 975 18589 807

2030944 145 175 19522 1000 34 45 3727 975 15795 809

2036425 48 60 10491 1000 13 17 2682 977 7809 744

2036556 68 80 7265 1000 14 22 1699 975 5566 766

1 H6054 rectangular sealed beam headlight 2 H4656 rectangular sealed beam headlight 3 6053 rectangular sealed beam headlight 4 9004 halogen bulb 5 4652 rectangular sealed beam headlight 6 6104 7 sealed beam headlight

Although Figure 6(a) seems to indicate that the majority of the current policies are close

in cost to the optimal policies this is very misleading For example suppose the optimal policy

for a product costs $1 and that the current policy costs $2 In absolute terms the optimal

policy saves only $1 but in relative terms the optimal policy saves 50 of the current cost

When savings are viewed in this manner as in Figure 6(b) it is clear that substantial savings

can be realized for the majority of the products In fact both the mode and the mean of the

savings are between 40 and 50 of the current costs

In order to extrapolate the above savings to obtain the potential savings for product

category 20 at all 21 stores the differences between stores must be considered During the

60

pilot study the current policy for each of several products at each size of store was evaluated

The study suggested that medium and large stores have a similar amount of excess

inventory and that jumbo and extra jumbo stores have much more excess inventory One

could reasonably expect that the additional excess inventory allows for higher savings to be

realized at the jumbo and extra jumbo stores Thus because 18 of the 21 stores are

medium or larger the expected 40-50 savings at store 6 can be viewed as a lower bound

on the potential savings to Canadian Tire Pacific Associates from implementing optimal policies

for product category 20 in all its stores

Finally in order to obtain the overall potential savings the expected savings from

product category 20 must be extrapolated to all product categories Ideally the proportion of

inventory comprising each product category and the relative savings that can be realized for

each category would be used to obtain a good estimate of the overall potential savings

However as this information is not known only a crude extrapolation is possible Thus the

analysis proceeds under the assumption that a 40-50 savings can be realized in each product

category35

Before an estimate of the total potential savings in absolute terms can be obtained the

proportion of the current inventory to which the savings apply must be determined Up to this

point the analysis has been concerned solely with reducing inventory at the stores However

it is inappropriate to perform further analysis without considering the entire inventory system

including the central warehouse Because the lead time for orders from Toronto is highly

variable and considerably greater than those at individual stores Canadian Tire Pacific

3 5 During the pilot study the current policies for a paint brush and an ironing board were evaluated and found to result in excess inventory similar to that of product category 20 Although this in no way justifies the assumption it does lend some support

61

Associates must keep a fairly high safety stock at the warehouse Without knowing the

proportion of inventory serving as the safety stock it is inappropriate to assume any reduction

in inventory at the warehouse is possible Therefore the relative savings of implementing

optimal policies is applied only to the inventory held in the stores36 Of the estimated $28

million in inventory approximately half is held in the stores therefore the 40-50 savings

translates to a savings of approximately $55 to $7 million per year

B Sensitivity Analysis on the Ordering Cost and Holding Rate

The above analysis was based on estimates of the ordering cost and the annual unit

holding rate37 provided by management As described in Chapter II these estimates

incorporate a number of costs that are difficult to quantify or that are speculative Therefore

there is some question as to the accuracy of the estimates In this section errors of +$0015

in the ordering cost and +5 in the holding cost are assumed in order to determine the

sensitivity of the model to errors in these variables38

The analysis proceeds as follows Based on an ordering cost of $0080 and a holding

rate of 30 the optimal policies from the previous section were obtained39 Then assuming

3 6 Of course this assumes that the excess inventory currently held in the stores is not required as part of the safety stock Based on the large amount of inventory at the warehouse this assumption seems reasonable

3 7 Hereafter simply referred to as the holding rate

3 8 An error of +$0015 in the ordering cost implies an actual ordering cost of $0070 a relative error of approximately 21 and an error of -$0015 implies an actual ordering cost of $0100 a relative error of 15 An error of +5 in the holding rate implies an actual rate of 25 a relative error of 20 and an error of -5 implies an actual holding rate of 35 a relative error of approximately 17

3 9 These policies are referred to as optimal policies for the duration of this chapter

62

the actual ordering cost andor holding rate are different from managements estimates the

heuristic is used to obtain the true optimal policies Finally the optimal policies are

evaluated based on the actual ordering cost and holding rate and then compared to the true

optimal policies For each possible combination of ordering cost and holding rate three

numbers are recorded the number of products with an optimal policy that differs from its

true optimal policy the total cost of implementing the optimal policies given the actual

ordering cost and holding rate and the difference between the cost of implementing the

optimal policies and the cost of implementing the true optimal policies The results of the

analysis are summarized in Table 6

Table 6 A comparison of optimal policies to true optimal policies

Actual Ordering Cost

$0070 $0085 $0100

25 5 products $155439

$002

51 products $160590

$092

88 products $165464

$449

Actual Holding Rate

30 54 products

$181470 $182

48 products $192026

$088

35 79 products

$207270 $574

48 products $212940

$136

2 products $218312

$000

4 0 The table should be read as follows If the actual ordering cost is $0070 and the actual holding rate is 25 then each of 5 products has an optimal policy that is different from its true optimal policy The total cost of implementing the optimal policies is $155439 which is $002 more than implementing the true optimal policies

63

The model appears to be quite insensitive to relative errors in the order of 20 Under

the worst case scenario when one of the variables is over-estimated and the other is undershy

estimated the optimal policies result in a total cost of inventory which is only approximately

3 higher than the cost of the true optimal policies Further because demand is not changed

by errors in these estimates each of the optimal policies still provides a customer service

level of at least 975 under all of the scenarios Finally although the cost of the optimal

policies varies significantly for the different scenarios the relative savings remain virtually

constant Table 7 lists the cost of existing policies and in parenthesis the relative savings of

the optimal policies for each of the different combinations of ordering cost and holding rate

The cost of the existing policies varies in the same manner as that of the optimal policies the

relative savings for each of the scenarios remains around 55 Therefore even in the presence

of such errors in the ordering cost and holding rate the projected savings of implementing the

optimal policies would be unchanged

Table 7 The cost of existing policies and the relative savings of the optimal policies under various scenarios

Actual Ordering Cost

$0070 $0085 $0100

25 $350153 (56)

$358464 (55)

$366764 (55)

Actual Holding Rate

30 $412438 (56)

$429048 (55)

35 $474725 (56)

$483026 (56)

$491320 (56)

64

C Sensitivity Analysis on the Demand Probability Mass Function

In this section the sensitivity of the model to slight changes in the demand probability

mass function is investigated As described in Chapter II the demand function used in the

model is based solely on observed sales however this empirical function is only one possible

instance of the underlying true demand function For example had the data been collected

over a period both starting and finishing one month earlier many of the empirical functions

would be slightly different Consequently depending upon the degree to which the empirical

functions vary and the sensitivity of the model to these variations the optimal policies from

the model might be inaccurate For example recall the case of the amber driveway marker

only thirteen markers were sold and all were sold on the same day Because the empirical

demand function is used the model calculates an optimal policy of (1213) clearly this

policy is impractical Although this example is extreme it does indicate the danger of using

unrealistic empirical demand functions41

As discussed in Chapter II three classes of products were identified namely low

medium and high-demand products For this analysis the same method of classification

is used Two products are selected from each of these groups and two new distribution files

are created for each of the six products For each product one of the files contains two

additional days of data with each day having a demand equal to the maximum observed demand

for that product In the second file only one additional day of data is added but the demand

on that day is two more than the maximum observed demand The situations that could give

4 1 Certainly a rigorous analysis to determine both the extent of variation in the empirical demand functions and the effect of using the empirical functions needs to be performed However such an analysis is beyond the scope of this thesis only a brief analysis that might suggest the degree to which the model is sensitive is performed here

65

rise to such changes in the demand data are termed Scenario 1 and Scenario 2 respectively42

The analysis proceeds in a manner similar to the one in the previous section Given the

demand that was originally observed the optimal policies were obtained Now assuming the

true demand function corresponds to the demand under Scenario 1 the true optimal policy

for each product is calculated Then each optimal policy is re-evaluated using the true

demand function and compared to the true optimal policy The analysis is then repeated

assuming the true demand function corresponds to the demand under Scenario 2 Table 8

lists the results for each product and scenario

Table 8 True optimal and optimal policies for each demand scenario

Product 202101 (low demand)

Scenario s s Cost ($) C S L ()

1 True optimal 1 2 470 993

Optimal 1 2 470 993

2 True optimal 2 3 663 976

Optimal 1 2 461 875

Product 203507 (low demand)

Scenario s S Cost ($) C S L ()

1 True optimal 0 3 334 981

Optimal 0 2 258 972

2 True optimal 2 3 508 980

Optimal 0 2 273 702

4 2 To be clear the scenarios are defined as follows Scenario 1 The addition of two days each with a demand of max Scenario 2 The addition of one day with a demand of max+2

where max is the maximum observed daily demand for each product in the sample period

66

Product 201032 (medium demand)

Scenario s S Cost ($) C S L ()

1 True optimal 6 12 1377 976

Optimal 6 11 1338 973

2 True optimal 6 13 1426 977

Optimal 6 11 1338 970

Product 203039 (medium demand)

Scenario s S Cost ($) CSL ()

1 True optimal 5 19 256 975

Optimal 5 16 251 969

2 True optimal 5 20 258 976

Optimal 5 16 249 968

Product 203677 (high demand)

Scenario s S Cost ($) CSL ()

1 True optimal 18 23 3981 977

Optimal 18 22 3817 972

2 True optimal 18 23 3995 978

Optimal 18 22 3831 973

Product 203680 (high demand)

Scenario s S Cost ($) CSL ()

1 True optimal 38 49 9498 976

Optimal 37 48 9175 971

2 True optimal 37 49 9484 975

Optimal 37 48 9206 972

67

The model is most sensitive to changes or errors in the demand functions of low-

demand products When only one additional day of demand for max+2 units is added to the

data the optimal policy for each of the low-demand products results in a very low customer

service level In fact for one of the products the customer service level drops to almost 70

The increase in demand does lower the customer service level for each of the other the

products but that decrease is slight

The above analysis indicates the need for some kind of smoothing or modelling of the

demand data especially for low-demand products The inherent variation between two

instances of a true demand function could very well result in more pronounced changes than

those resulting from either of the above scenarios Therefore it is likely that many optimal

policies especially those for low-demand products would actually provide a much lower or

much higher customer service level than expected However by smoothing the data the

variation between instances of a true demand function can be reduced and the desired

customer service levels can be achieved

For example suppose that the demand for a product is stationary that the average

demand during the previous period was unusually low and that the optimal policy for this

period is based on the demand during the previous period Because the demand is stationary

the probability that the demand during this period is higher than that during the past period is

more than 50 In other words the probability that the demand during this period is higher

than that for which the optimal policy provides the desired expected customer service level

is greater than 05 Thus expected customer service level for this period is actually less than

that specified Now suppose instead that the optimal policy is based on smoothed data which

reflects very closely the expected demand The probability that the demand during this period

is higher than that for which the optimal policy provides the desired expected customer service

68

level is virtually 05 Thus as a result of smoothing the data the specified customer service

level should be expected

69

VII C O N C L U S I O N

In this thesis we developed a single-product single-location model of the inventory

system at Canadian Tire Pacific Associates Specifically we formulated a Markov chain model

to determine the long-run average cost of inventory for fixed values of s and S and then

developed a heuristic search to find near optimal policies quickly Also as a first step toward

developing an integrated interactive inventory control system we created a prototype of the

interface

According to the analysis performed here Canadian Tire Pacific Associates is currently

holding an excessively large inventory in its stores This study finds that Canadian Tire Pacific

Associates could reduce the cost of inventory for product category 20 at store 6 by

approximately 50 and still maintain customer service levels of at least 975 Further the

mean and modal values of savings for these products are between 40 and 50 Given the

similar excess inventory at the other stores and in all product categories we believe that savings

in this range are possible for all products and stores We estimate that the application of

inventory control as described in this thesis would result in annual savings of $55 to $7

million

Given the large potential savings to Canadian Tire Pacific Associates we recommend

that management proceed with the next phase of the project Specifically we recommend that

the model algorithm and interface be refined further that a forecasting model be developed

and that a data collection and storage scheme be designed

In order to reflect operational changes at Canadian Tire Pacific Associates subsequent

to the formulation of the model modifications must be made to allow for the lead time to

exceed the review period Several of the stores now have three picks per week and as a

70

result orders placed for these stores at one decision epoch do not arrive until after the next

decision epoch Because this violates one of the models assumptions the model must be

modified accordingly Note however that with three picks per week and a review period

of four days orders from only the previous decision epoch are ever outstanding at the current

decision epoch

Also the model should be refined to reflect the fact that the review period is not fixed

over the planning period In its present form the model assumes a fixed review period

however inventory at stores with two picks per week is actually reviewed on a four day -

three day cycle and inventory at stores with three picks per week is reviewed on a three day

- two day - two day cycle Rather than obtaining the steady state costs for fixed review period

the model should calculate the steady state cost for a period of one week during which multiple

replenishments occur Although the solutions are not expected to differ significantly as a result

of the change the model would reflect the actual process more accurately

Further work is also required on the search technique for locating near optimal

policies Although the heuristic search described in this thesis appears to work well the

decision maker must provide an upper bound on S prior to the search In practice this upper

bound is generally not known in advance Thus if the heuristic is to be used either a tight

theoretical bound must be found or a termination heuristic must be developed Alternatively

a new heuristic search could be developed based on additional or different assumptions

regarding the properties of the feasible space For example although some regions of non-

convexity were found for some products much of the feasible space appeared to be convex

A heuristic that assumes a convex feasible space would not require an upper bound on S and

might perform very well However before such a heuristic is implemented further work is

necessary to determine the degree of non-convexity for a variety of demand functions and

71

ordering and holding costs

In addition to making the above modifications we recommend that Canadian Tire Pacific

Associates begin work on the remaining two components of the inventory control system First

in order to allow for products with seasonal demand the system requires a forecasting model

to provide accurate estimates of demand for the coming period Second the system needs a

data collection and storage module that systematically collects sales data for each active

product and for each store and then updates the corresponding demand distributions Because

approximately 25 of demand is expected to result in lost sales under the optimal policies the

sales data will have to be adjusted accordingly in the future Also because the model appears

to be sensitive to slight changes in the demand distribution for low-demand products the

module should incorporate a technique for smoothing the empirical distributions for these

products Further work should be undertaken to determine the extent of variation in the

empirical distributions and the most appropriate smoothing technique However because of

the large amount of data required to forecast seasonal demand and because of the current lack

of historical data we recommend that management proceed immediately with the design of a

data collection scheme

Finally two additional issues that are beyond the scope of the next phase should be

addressed at some point First a monitoring system will be required once inventory control

is implemented in the stores The monitoring system will ensure that optimal policies perform

as expected and will indicated whether or not demand is adversely affected by the lower levels

of in-store inventory Second given the degree to which in-store inventory can be reduced

inventory control in the warehouse should be investigated Because approximately half of the

existing inventory is held in the warehouse the additional savings from reducing system-wide

inventory could be great

72

Afterword

Just prior to the completion of the work described in this thesis Mr Don Graham

principle owner of Canadian Tire Pacific Associates sold his shares in the company Shortly

thereafter the results of this thesis were presented to the new management group

Unfortunately the new management did not consider development of an inventory control

system to be a priority Nonetheless the prototype was delivered to Canadian Tire Pacific

Associates for possible in-house testing however at the time of this writing - almost one year

after the meeting - Canadian Tire Pacific Associates has not decided to continue with the

project

73

Bibliography

Arrow K A T E Harris and J Marschak (1951) Optimal inventory policy Econometrica 19 250-272

Day G (1992) In a speech to the Empire Club of Canada Toronto quoted in When the JIT fits The Globe and Mail October 20 1992 B24 and in Puterman (1994)

Dvoretzky A J Kiefer and J Wolfowitz (1952a) The inventory problem I Case of known distributions of demand Econometrica 20 187-222

Dvoretzky A J Kiefer and J Wolfowitz (1952b) The inventory problem II Case of unknown distributions of demand Econometrica 20 450-466

Federgruen A (1993) Centralized Planning Models in SC Graves A H G Rinnooy Kan and P H Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Federgruen A H Groenevelt and H C Tijms (1984) Coordinated replenishments in a multi-item inventory system with compound Poisson demands and constant lead times Management Sci 30 344-357

Lee H L and S Nahmias (1993) Single-Product Single-Location Models in SC Graves A H G Rinnooy Kan and PH Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Muckstadt J A and R O Roundy (1993) Analysis of Multistage Production Systems in SC Graves A H G Rinnooy Kan and PH Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Porteus E L (1985) Numerical Comparisons of Inventory Policies for Periodic Review Systems Oper Res 33 134-152

Puterman M L (1994) Markov Decision Processes Discrete Stochastic Dynamic Programming Wiley New York

Silver E A and R Peterson (1985) Decision Systems for Inventory Management and Production Planning 2nd edition Wiley New York

Taylor H M and S Karlin (1994) An Introduction to Stochastic Modeling revised edition Academic Press San Diego

74

Tijms H C and H Groenevelt (1984) Simple approximations for the reorder point in periodic and continuous review (sS) inventory systems with service level constraints European J Oper Res 17 175-190

75

Appendix A

A Portion of the Sales File for Product 200001 a 30-amp Inline Fuse

200001 283 1190 0 0 2 0 0 1 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 200001 284 1190 2 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 200001 285 1190 0 0 4 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 286 1190 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 200001 287 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 200001 288 1190 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1 0 2 200001 289 1190 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 290 1190 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 200001 291 1190 0 0 2 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 200001 292 1190 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 200001 294 1190 0 0 0 1 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 1 200001 295 1190 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 200001 296 1190 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 200001 297 1190 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 200001 298 1190 0 0 1 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 200001 299 1190 0 0 1 0 0 0 1 0 1 0 0 2 0 0 0 0 0 2 1 1 0 200001 300 1190 0 0 1 0 0 0 0 1 0 0 0 0 0 2 2 0 0 0 0 3 0 200001 301 1190 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 200001 303 1190 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 200001 304 1190 1 0 0 0 0 0 0 0 1 0 0 2 0 0 0 0 0 0 0 0 0 200001 305 1190 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 200001 306 1190 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 2 0 0 1 0 200001 308 1190 0 0 0 1 0 1 0 0 0 2 0 1 0 0 0 0 1 1 1 0 0 200001 309 1190 1 3 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 200001 310 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 200001 311 1190 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 200001 312 1190 0 1 1 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 200001 313 1190 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 2 0 200001 314 1190 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 200001 315 1190 0 0 2 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 200001 316 1190 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 200001 317 1190 2 0 0 1 0 0 0 0 0 0 5 1 0 0 0 1 0 0 0 0 0 200001 318 1190 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 1 200001 319 1190 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 200001 320 1190 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 200001 321 1190 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 1 200001 322 1190 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 323 1190 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 200001 324 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 0 1 0 0 2 1 1 0 200001 325 1190 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3 200001 326 1190 0 0 1 0 1 0 0 0 2 0 0 1 0 0 0 1 0 1 0 0 1 200001 328 1190 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 200001 329 1190 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 200001 330 1190 0 0 1 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 200001 331 1190 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 200001 332 1190 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 200001 333 1190 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 200001 334 1190 0 0 2 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 200001 335 1190 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 200001 336 1190 0 0 0 1 0 0 3 0 0 0 0 0 2 0 0 0 0 0 0 0 0

(253 subsequent lines omitted)

76

Appendix B

Sample Distribution Files

a) A low-demand product - an automobile back-up alarm

PRODUCT NUMBER 202101 PRICE $ 684

DMD

3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

N 2 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 2 2 0 1

1 5 3 8 12 5 7 11 6 11 5 4 11 10 19 4 2 5 14 5 6 4

0 302 304 299 295 302 300 296 301 296 302 303 295 296 288 303 305 301 290 300 301 302

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

b) A medium-demand product - a 168 bulb

PRODUCT NUMBER 203039 PRICE $ 51

DMD

16 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 1 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0

9 0 0 2 5 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0

8 0 0 2 5 0 0 2 0 1 1 3 0 1 3 0 0 1 1 0 0 1

7 2 1 3 1 0 0 1 0 2 3 1 1 4 1 1 0 2 1 1 1 1

6 4 7 1 10 3 1 3 2 1 3 5 2 7 4 1 0 1 1 3 3 4

5 1 7 7 17 0 2 11 1 5 4 1 2 6 9 6 2 6 9 4 4 1

4 5 10 10 16 5 5 19 10 13 8 11 9 13 10 12 7 5 15 5 8 3

3 10 12 13 27 4 6 24 11 10 13 14 9 23 14 18 8 14 18 14 12 14

2 14 30 36 43 16 16 43 32 38 39 39 34 47 36 41 21 19 39 20 34 24

1 30 56 68 56 37 30 70 48 67 61 44 75 62 63 48 28 52 61 63 53 47

0 240 184 165 126 240 247 133 202 169 175 189 173 143 165 180 240 206 161 197 192 212

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

77

c) A high-demand product - a H4651 rectangular sealed beam headlight

PRODUCT NUMBER 203677 PRICE $ 857

DMD

14 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

13 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 2 3 0 0 1 0 0 0 0 1 1 1 0 0 0 0 2 0 0

11 0 1 3 1 0 0 4 1 0 0 0 0 0 0 0 0 1 0 0 0 1

10 0 0 4 4 1 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 1

9 0 3 14 11 0 1 7 1 0 2 0 8 0 1 0 0 0 2 1 0 0

8 0 4 12 16 1 0 18 4 3 5 0 2 1 4 2 1 1 0 0 1 3

7 1 2 12 13 1 6 19 3 5 9 0 11 2 5 4 0 2 3 3 1 4

6 2 9 22 21 6 10 29 12 10 11 4 24 3 7 10 4 7 11 6 3 14

5 7 13 35 33 8 16 29 19 18 8 3 26 15 26 15 12 9 14 13 11 19

4 19 35 34 43 32 30 41 31 32 47 11 42 20 33 30 31 20 31 22 21 31

3 32 63 49 54 42 60 51 40 51 35 33 55 39 51 39 38 39 57 58 40 54

2 46 59 56 55 73 63 61 65 63 61 63 54 68 85 65 53 72 69 70 77 63

1 109 79 42 40 80 64 32 80 74 77 99 55 97 57 83 93 80 75 81 97 73

0 91 39 21 13 63 57 11 51 51 52 94 29 61 35 59 75 76 45 51 56 44

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

78

Appendix C

A Proof that the Limiting Probability Distribution is Independent of the Initial State

In this appendix we show that provided 0 lt PrDT = 0 lt 1 the Markov chain

corresponding to an inventory model with a (sS) policy has a limiting probability distribution

7T where 0 lt TTJ lt 1 for y = 0 1 S that is independent of the initial state Further the

limiting distribution is the unique nonnegative solution of the equations

Tt P = TI (AD

s and J gt = 1 (A 2)

i=0

where P is the (S + l) x (S + l) transition probability matrix whose first n rows (and columns)

correspond to the n recurrent states in the chain and whose remaining rows (and columns) if

any correspond to transient states43 Prior to the proof several preliminary results are

obtained

RESULT 1 The Markov chain has exactly one positive recurrent class and perhaps some

transient states

First note that in the absence of any absorbing states a Markov chain with a discrete

finite state space must have at least one positive recurrent class containing at least two states

Now consider the Markov chain for an inventory model with a (sS) policy

a) State i gt s cannot be absorbing because there is a positive probability of reaching

j lt i (Since PrDT gt 0 gt 0)

4 3 Note that since the rows and columns of P are reordered the elements of it must be reordered accordingly

79

b) State lt s cannot be absorbing because by the rules of the ordering policy and

the assumption that PrDT = 0 gt 0 there is a positive probability reaching S

Therefore the chain has no absorbing states and must have at least one positive recurrent class

Next consider the subset of states A = i | i lt s Because the state space is finite

and PrDT gt 0 gt 0 the process must return to at least one state in A in other words at least

one state in A must be recurrent Let k G A be a member of a recurrent class dt Because

of the ordering policy and the assumption that PrDT = 0 gt 0 k -raquo S Also by definition

there must be a positive probability of returning to k from S S -gt k Therefore S G 3w

Further because S is reachable from all states in A i G A is either transient with i -raquo dt or

recurrent with pound iR

Now consider the subset of states B = | j gt s Because PrDT gt 0 gt 0 the

chain will eventually reach a state i G A If i is transient then j must be transient also with

y -gt 3 if i G 9 then j is either transient with j -gt dt or recurrent with j E dt Thus the

Markov chain has exactly one recurrent class and perhaps some transient states

RESULT 2 The limiting probability distribution ir for the Markov chain corresponding to the

nxn submatrix Pm = [Ptj] i j G dt is independent of the initial state and is the

unique nonnegative solution of the equations

TC-Prr = re (A3)

and J2 K = 1 ( A 4 )

jt=i

where the rows and columns of are numbered from 1 to n rather than

Since if i j i and j must be in the same equivalence class

80

according to their associated states

First note that both the original Markov chain and the one corresponding to the

submatrix P R R are aperiodic because PrDT = 0 gt 0 Therefore given that 9 is a positive

recurrent aperiodic class the limiting distribution ir is uniquely determined by (A3) and (A4)

(Taylor and Karlin 1994 Theorem 44 p213) Further because there are no absorbing states

or other recurrent classes in the original chain the process must eventually enter the recurrent

class 9 regardless of the initial state Therefore in the limit the probability of being in state

i G 9 is independent of the initial state

RESULT 3 The limiting probability distribution for the transient states is the null vector

independent of the initial state

This result follows directly from the definition of a transient state

PROOF Suppose the states are reordered such that the first n rows (and columns) of P

correspond to the n recurrent states of the Markov chain Then P can be partitioned into four

submatrices

PRK - the n x n submatrix [Py] i j E 9

P T T - the (S + 1-I) x (S + l-n) submatrix [Py] i j g 9

P T R - the (S + l-n) x n submatrix [Py] i pound 9 j G 9

P R T - the n x (S + l-n) submatrix [Py] i G 9 j pound 9t By definition this is a matrix

of zeros

(Al) can then be rewritten as

81

(ltR TtT) P P L TR TT

I-P RR 0

p I-P TR 1 TT

(TtR TCT)

(0- 0T)

where irR denotes the limiting probability distribution for the n recurrent states and irT denotes

the distribution for the remaining S + l-n transient states Solving for TT t gives

TCr-0 + itT-(I-PTI) = 0T

itT-(I-PJJ) = 0 r

i c r = (I-P^-Oj

Since (I - PTT)1 always exists (Puterman 1994 Proposition A3 p 593) irT = 0X which is

in agreement with Result 3 Now solving for 7rR yields

(A2) can be rewritten as

bullRPRR 1 1R (A5)

(A6) S + l

k + L k =1

k=l k=n+

k=i

where ir contains the appropriately reordered steady state probabilities Since (A5) and (A6)

are equivalent to (A3) and (A4) respectively Result 2 holds Therefore the Markov chain

corresponding to an inventory model with a (sS) policy has a limiting distribution that is

independent of the initial state and is the unique nonnegative solution to (Al) and (A2)

82

Appendix D

A Measure for the Steady State Customer Service Level

In this appendix we show that the appropriate measure for the steady state customer

service level is given by

_ E[Demand satisfied during RP] E[Total demand during RP]

Consider the following two measures of the steady state customer service level

Measure 1 Suppose the customer service level is calculated only after n review periods have

passed Denoting DS as the demand satisfied in review period i and DT as the total demand

in period i the customer service level after n periods CSL(n) is

DS + DS0 + + DSbdquo CSLn) = - -

DT + DT + + DT

Dividing both the numerator and denominator by n taking the limit as n -raquo oo and applying

the Law of Large Numbers yields

CSL = lim CSL(n) n-degdeg

- [DS + DS + + DS] = l i m ^ J 1 1

raquo~ - [DT + DT + + DT]

E[DS]

E[DT]

83

Measure 2 Suppose the customer service level is calculated for each of n review periods and

then averaged Thus the customer service level after n periods CSL(n) is given by

CSL(n) = -DS DS

DT DT DT_

Taking the limit as n -gt oo yields

CSL = lim CSL(n)

lim mdash DSl DS2

DT DT DT_

DS

DT

Now consider the scenario in which there is one item on the shelf to begin each review

period and in which demand is stochastic with PrDT = 2 = Vi and PrD x = 100 = Vi

According to Measure 1 the customer service level is

CSL - = ^ W + ^ W = plusmn bdquo 002 E[DT) 1 (2) + 1(100) 51

while according to Measure 2 the customer service level is

CSL = E DS DT

1(1) + i(-Ls

22) + 2 100

51 200

026

Clearly from managements point of view the first measure is more indicative of the

actual level of service being provided The problem with the second measure is that equal

84

weight is given to each ratio regardless of the magnitude of the numbers For example with

respect to Measure 2 the events LSLR = 12 and LSLR = 50100 are identical however

with respect to lost sales the second event is much worse than the first Therefore the proper

measure for the steady state customer service level is Measure 1

85

Appendix E

The Calculation of the Conditional Expected Demand Not Satisfied During a Period of T Consecutive Days

Denoting D x as the demand during a period of T consecutive days and j as the inventory

on hand at the beginning of the period the conditional expected demand not satisfied during the

period E[(DT-y)+] is calculated as follows

E[(DT-j)+] = pound ltk-j)Pr[DT = k k=j+l

= pound (k-j)PrDT = k] + J2(k-j)PrDT = k] - pound (fc-)VJgtr = fc k=j+l k=0 k=0

= J2(k-j)Pr[DT=k -J2(k-j)PrDT = k] k=0 k=0

= YkPrDT=k) - J2jPrDT=k - Yk-j)PrDT=k) Jfc=0 k=Q k=0

= Ti - j - Y(k-j)PrDT=k 7-1

= Tx - j + YltJ-k)PrDT = k jt=0

where fx is the mean daily demand

86

Appendix F

Justification of the Updating Technique for the New Policy (s+mS)

Let the probability transition matrix corresponding to the policy (sS) be denoted by the

(S + l)x(S + l) matrix P with elements py Define s = s+m where m is an integer satisfying

the condition 0 lt s+m lt S Let the probability transition matrix of the new policy (sS)

be denoted by the (S + 1)X(S + 1) matrix 7gt with elements py Using the equations developed

in Chapter III the elements of P are found as follows

CASE 1 The lead time is less than the review period

A m gt 0

For 0 lt i lt s

( i-i pound [PrDL=kPrDT_LgtS-k]] + PrDLziPrDT_LplusmnS-i k=0

i-l 53 [PrDL=kPrDT_L = S-j-k]] + PrDLziPr[DT_L = S-j-i) k=0 s-j

52 [Pr[DL = kPrDT_L = S-j-kn k=0

ifj = 6

ifOltjz S-i

if S-i ltj ltS

a) 0 lt lt s

Pij

= lt Pij

Pij

= pij

b) s lt i lt s This does not simplify but it is identical to (2)

87

For s lt i lt S

PrDTgt i

p = lt PrDT = i-j

0

Pfj ifj = 0

= lt Pij ifOltjlti

Pij if i ltj plusmn S

= pij

B m lt 0

ifj = 0

if 0 ltj lt i

if i ltj plusmn S

For 0 lt i lt s

i-l 53 [PrDL=kPrDT_LplusmnS-k]] + PrDLgtiPrDT_LplusmnS-i

i-l E [PrZgti = fciVZ)7W=S--fc] +PrD tgtiP-D 7 t = S-7-

p y laquoy = o

P i j ifOltjS-i

vPij ifS-iltjS

ifj = 0

ifOltjltS-i

if S-i ltjltS

For s lt i lt S

fPrJgtrgt- 17=0 = PrDT = i-j] ifOltjlti

0 ifiltjplusmnS

88

a) s lt lt s This does not simplify but it is identical to (3)

b) s lt i lt S

PrDTgti

= bull PrDT = i-j]

0

17=0

if0ltjplusmni

ifiltjS

= Pt

ifj = 0

if0ltjlti

ifiltjplusmnS

= Pi

CASE 2 The lead time is equals the review period

A m gt 0

For 0 lt i lt s

0 if0ltjlt S-i

PrDTgti ifj = S-i

PrDT=S-j] ifS-iltjltS

a) 0 lt i lt s

= bull y Pij

Pij

ifOplusmnjltS-i

ifj = S-i

ifS-iltjS

= Pij

b) s lt i lt s This does not simplify but it is identical to (4)

89

For s lt i lt S

PrDTgti)

Pr[DT=i-j

0

laquof 7=0

ifOltjplusmni

if i ltj lt S

Pa

Pij

if 7 = 0

if 0 lty lt i

ifiltjltS

B w lt 0

For 0 lt t lt s

ro PrJrgt PrDT = S-j

ifOzjltS-i

ifj = S-i

if S-i ltj lt S

Pij

ifOzjltS-i

ifj = S-i

if S-i ltj ltS

= Pi

For s lt i lt S

PrDTgti

= PrDT = i-j

0

i 7 = o ifOltjplusmni

ifiltjltS

a) s lt i lt s This does not simplify but it is identical to (3)

90

PrDTzi ifj = 0

PrDT = i-j] ifOltjplusmni

0 ifiltjltS

Py ifj = 0

ptj ifOltjplusmni

Pij if iltj plusmnS

Appendix G

Justification of the Updating Technique for the New Policy (sS+m)

Let the probability transition matrix corresponding to the policy (sS) be denoted by the

(S + l)x(S + l) matrix P with elements p(gt Define S = S+m where m is any non-negative

integer Let the probability transition matrix of the new policy (sS) be denoted by the

(S+m+l)x(S+m + l) matrix IP with elements ptj Then using the equations developed in

Chapter III the elements of 7 are found as follows

CASE 1 The lead time is less than the review period

For 0 lt i lt s

pound [PrDL=kPrDT_LplusmnS-k] + PrDLi)PrDT^Sgt- k=0

i-l pound [Pr[DL = kPrDT_L = S-j-k- +Pr[DLiPrDT_L = S-j-i Jt=0

s-i Y [Pr[DL=kPrDT_L = S-j-k] k=0

i - l

ifj=0

ifOltjltS-i

ifS-iltjltS

pound [PrDL = kPrDT_LS-k + Pr[DLi)PrDT_LplusmnS-i) if j = 0 fc=0

i - l = pound [PrDL=kPrDT_L = (S+m)-j-k] + PrDLgtiPr[DT_L = (S+m)-j-i ifOltj lt (J5+m)-i

if (S+m)-iltj lt(S+m)

k=0

i - l

pound (Sm)-j

pound [PrDL = kPrDT_L = (S+m)-j-k]] k=0

92

I-l

53 [PrDL = kPrDT_LgtS-k] + PrDLziPrDT_LS-i ifj = 0 k=0

i-l

Y[PrDL = kPrDT_L = S-(j-m)-k] + Pr[DLziPrDT_L = S-(j-m)-i] ifOltjzS+m-i k=0 S-(j-m)

53 [PrDL = kPrDT_L = S-(j-m)-k]] if S+m-i ltj lt S+m

i - l

53 [PrDL = kPrDT_LzS-k] + PrDLziPrDT_LzS-i] ifj = 0 =0

i - l

13 [Pr[DL = kPrDT_L = S-(j-m)-k] + Pr[DLziPr[DT_L = S-(j-m)-i ifOltjz S+m-i k=o and 0 z j-m

Pi(j-m)

Pi(j-m)

if 0 ltj lt S+m-i and 0ltj-mltS-i

if S+m-i ltj lt S+m and S-i lt j-m zS

i - l

Y [PrDL = k)PrDT_LzS-k] + PrDLi)PrDT_LplusmnS-i ifj = 0

k=0 i - l

53 [PrDL = kPrDT_L = S-(j-m)-k] + PrDLziPrDT_L = S-(j-m)-i ifOltjzm k=0

Piij-m)

Piij-m)

if mltj lt S+m-i

if S+m-i ltj lt S+m

i - l

YlPrDL = kPr[DT_L^S-k + PrDLi)PrDT_LS-i) ifj = 0

i - l

53 [PrDL = kPrDT_L = S-j-k] + PrDLgtiPrDT_L = S-j-i ifOltj i m k=0

Pi if mltj ltS

which for 0 lt j lt m is identical to (2)

93

For s lt i lt S

Pu =

PrDTgti

PrDT = i-j]

0

if0ltjlti

ifiltjltS

a) s lt i lt S

PrDTgti 7=0

PrDT = i -j) if0ltjlti

0 if i ltj lt S

0 ifSltjltSgt

ifj=0

if 0 ltj lt i

if i ltj plusmn S

0 ifSltjltS

ifOltjltS

0 ifSltjltSgt

b) S lt i lt S This does not simplify but it is identical to (3)

CASE 2 The lead time equals the review period

For 0 lt i lt s

0 ifO lt jltS-i

PrDTgti ifj=S-i

PrDT=S-j ifS-iltjltS

94

0 ifOplusmnjlt(S+m)-i

PrDT ifj = (S+m)-i

PrDT = (S+m) -j) if (S+m)-i ltj ltL (S+m)

0 if 0 lt jltS+m-i

PrDTzi] if j = S+m-i

PrDT = S-(j-m) if S+m-i ltj lt S+m

0 if 0 ltj lt S+m-i and 0 ^ j-m

Pi(j-m) if 0ltjlt S+m-i and 0ltj-mltS-i bull

Pi(j-m) if j = S+m-i and j-m = S-i

Pi(j-m) if S+m-i ltj z S+m and S-i lt j-m ^S

0 if 0 zj ltm

Pi(j-m) if mlt jlt S+m-i lt

Piij-m) if j = S+m-i

if S+m-i ltj lt S+m

1deg if 0 ltj ltm

[PiV-m) ifmltjltS

which for 0 lt i lt s is identical to (3)

For s lt lt S

PrDTzi

= PrDT = i-j)

0

ifj = 0

if 0 ltj ltL i

ifKjzS1

95

a) s lt i lt S

PrDTzi

PrDT = i-j]

0

0

17=0

if0ltjplusmni

ifiltjS

ifSltjltSgt

Pij 7=0

Pij if0ltjii

Pij if i ltj ltL S

0 ifSltjltS

Pij ifOzjiS

0 ifSltjltS

b) S lt i lt S This does not simplify but it is identical to (3)

96

Appendix H

A Hypothetical Consultation

In this appendix the interface during a hypothetical consultation is depicted The figures

accurately represent the various screens that are presented to the manager with the exception

that all lines and text are really white on a blue background During the consultation described

below only the choices made by the manager are hypothetical all the results are real

The title screen as depicted in Figure A - l appears initially upon execution of the

program In order to proceed to the main menu the manager presses the enter key

O I C S

Optimat Inventory Control System

For the exclusive use of

Canadian Tire Pacific Associates

Developed at The Faculty of Commerce

and Business Administration The University of British Columbia

Brian Kapalka Kaan Katircioghi Marty Puterman

Press ltEntergt to continue

Figure A - l The title screen

97

Main Menu 1 Select data directory

ctircdatadst

2 Select product and store combination Product 200001

3 Select scenario Review Period 4 Lead Time 3

4 Select search parameter Smax 25

5 Execute program

6 Exit to DOS

Please enter the desired option 5

Figure A-2 The main menu

Figure A-2 depicts the main menu of the program it is from here that the manager

enters all the review parameters If the manager wishes to change one of the parameters from

its default value he or she selects the appropriate number and enters the correct information

in the window presented The new settings will then be shown in place of the default values

In this example the relevant data files are located in the default directory ctiredatadst and

the manager has selected the productstore combination of product 200001 and store 6 Further

the manager has selected a review scenario of a 4-day review period and a 3-day lead time and

an upper search limit of 25 Finally in order to proceed with the analysis the manager

executes the main module of the program by selecting option 5

98

In presenting t h i s thesis i n p a r t i a l f u l f i l l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e for reference and study I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by h i s or her representatives I t i s understood that copying or pub l i c a t i o n of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission

Department of Cotv^errg- ftni)gt ^osmecs AAnmna-WoAor

The University of B r i t i s h Columbia Vancouver Canada

Date ZS A y r A 9 9 5

Abstract

Canadian Tire Pacific Associates owns and operates 21 retail stores in the lower

mainland of British Columbia and a central warehouse in Burnaby In this thesis we formulate

a single-product single-location model of its inventory system as a first step in developing an

integrated interactive inventory control system Specifically we formulate a Markov chain

model for a periodic review system with a deterministic lead time and lost sales The model

utilizes empirical demand data to calculate the long-run average cost of inventory for a given

(sS) policy We then develop a heuristic that locates a near optimal policy quickly The

heuristic incorporates a constraint on the customer service level makes use of an updating

technique for the transition probability matrix and is based on assumptions regarding the

properties of the solution space Next we create a prototype of the interface that enables

managers to use the model interactively Finally we compare the existing inventory policy to

the optimal policy for each of 420 products sold at one of the stores This thesis finds that

Canadian Tire Pacific Associates is currently holding excessively large in-store inventory and

that it could reduce its cost of inventory by approximately 40 to 50 We estimate that

implementing optimal inventory control in the stores would result in annual savings of between

$55 and $7 million

ii

Table of Contents

Abstract ii

Table of Contents iii

List of Tables v

List of Figures vi

Acknowledgement vii

I INTRODUCTION 1

II T H E INVENTORY SYSTEM A T CANADIAN TIRE PACIFIC ASSOCIATES 8 A Background 8 B The Problem 11 C The Project 15 D The Demand Data 17 E The Cost Data 18

III M O D E L FORMULATION 20 A Terminology 20 B Assumptions 21 C A Markov Chain Model 22 D The Transition Probability Matrix 25 E The Steady State Probabilities 27 F The Cost of Inventory 29 G The Customer Service Level 31 H A Methodology for Evaluating an (sS) Policy 34

IV A N ALGORITHM FOR OBTAINING T H E OPTIMAL (sS) POLICY 35 A Introduction 35 B A Grid Search 36 C A Technique for Updating the Transition Probability Matrix 41

A new policy (s+mS) 41 A new policy (sS+m) 42 The modified algorithm 43

D A Lower Bound on S 45 E A Heuristic Search 48

V T H E INTERFACE 52

iii

VI RESULTS A N D SENSITIVITY ANALYSIS 55 A Comparison of Current Policies to Optimal Policies 55 B Sensitivity Analysis on the Ordering Cost and Holding Rate 62 C Sensitivity Analysis on the Demand Probability Mass Function 65

VII CONCLUSION 70

Afterword 73

Bibliography 74

Appendix A A Portion of the Sales File for Product 200001 a 30-amp

Inline Fuse 76

Appendix B Sample Distribution Files 77

Appendix C A Proof that the Limiting Distribution is Independent of the

Initial State 79

Appendix D A Measure for the Steady State Customer Service Level 83

Appendix E The Calculation of the Conditional Expected Demand Not Satisfied

During a Period of T Consecutive Days 86

Appendix F Justification of the Updating Technique for the New Policy (s+mS) 87

Appendix G Justification of the Updating Technique for the New Policy (sS+m) 92

Appendix H A Hypothetical Consultation 97

Appendix I The Source Code for the Interface Module 104

Appendix J The Source Code for the Main Module 119

Appendix K Current and Optimal Policies for Product Category 20 at Store 6 131

iv

List o f Tables

Table 1 Location and particulars of the 21 stores 10

Table 2 Execution times of the algorithms for product 200001 a 30-amp inline fuse 39

Table 3 Execution times of the algorithms for product 206917 a 6

solder connector 40

Table 4 Products with insufficient existing policies 56

Table 5 Products with the largest potential absolute savings 60

Table 6 A comparison of optimal policies to true optimal policies 63

Table 7 The cost of existing policies and the relative savings of the optimal policies under various scenarios 64

Table 8 True optimal and optimal policies for each demand scenario 66

v

List of Figures

Figure 1 A typical sample path of the process with a 4-day review period

and a 2-day lead time 24

Figure 2 The flow chart of the grid search algorithm 38

Figure 3 The flow chart of the grid search algorithm utilizing the updating

technique for the transition probability matrix 44

Figure 4 An evaluation of the lower bound Sm i n 47

Figure 5 The flow chart of the heuristic algorithm 50

Figure 6 The distribution of savings of optimal policies in (a) dollars and

(b) percentage of current cost 59

Figure A-1 The title screen 97

Figure A-2 The main menu 98

Figure A-3 Calculating the optimal policy and evaluating the current policy 99

Figure A-4 Displaying the results 100

Figure A-5 Entering an alternate policy 101

Figure A-6 The particulars of the alternate policy 102

Figure A-7 The main menu revisited 103

vi

Acknowledgement

The completion of this thesis was made possible by the encouragement and assistance

of a number of people

I would like to express my sincere appreciation to my thesis supervisor Professor

Martin Puterman for all of his many efforts on my behalf His help advice patience charity

and tolerance were very much appreciated

I would like to acknowledge Professor Hong Chen and Professor Garland Chow for their

time and input as members of my thesis committee In addition I must acknowledge the

assistance of PhD student Kaan Katircioglu for his insight and help on this project

I offer many thanks to Professor Tom Ross for his kindness and friendship during the

past few years especially during the writing of this work I must also thank his family for

adopting me on many holidays I still owe thanks to Professor Slobodan Simonovic at the

University of Manitoba for influencing me to attend graduate school in the first place and for

helping me to obtain funding

I cannot thank my parents enough for their never-ending support both emotional and

financial Also to my friends especially Cathy Dave Lisa Steve and the Philbuds thanks

for giving me a life off campus and for picking up many a tab - the next one is on me

I would like to give special thanks to my good friend and fellow MSc student Paul

Crookshanks for allowing me to bounce ideas off of him and for being such a procrastinator

that despite my finishing a year late I only lost by two days

Financial assistance received from the Natural Science and Engineering Research

Council (NSERC) and from Canadian Tire Pacific Associates was greatly appreciated

vii

I INTRODUCTION

The importance of inventory management has grown significantly over the years

especially since the turn of this century In colonial times large inventories were viewed as

signs of wealth and therefore merchants and policy makers were not overly concerned with

controlling inventory However during the economic collapse of the 1930s managers began

to perceive the risks associated with holding large inventories As a result managers

emphasized rapid rates of inventory turnover (Silver and Peterson 1985) Following the

Second World War Arrow Harris and Marschak (1951) and Dvoretzky Kiefer and

Wolfowitz (1952ab) laid the basis for future developments in mathematical inventory theory

Shortly thereafter new inventory control methodologies were widely applied in the private

manufacturing sector More recently when inflation and interest rates soared during the 1970s

many organizations were forced to rethink their inventory strategies yet again Today the

control of inventory is a problem common to all organizations in any sector of the economy

Sir Graham Day chairman of Britains Cadbury-Schweppes P C L expressed this sentiment

when he stated I believe the easiest money any business having any inventory can save lies

with the minimization of that inventory (Day 1992) Lee and Nahmias (1993) and Silver and

Peterson (1985) provide a more detailed history of inventory management and control

Most inventory problems in the retail sector involve the control and management of a

large number of different products Ideally the inventory policies should be determined for all

products on a system-wide basis however because the number of products is often in the tens

of thousands formulating a model to do so is impractical Therefore single-product models

are frequently used in practice to determine policies for products separately (Lee and Nahmias

1993) Lee and Nahmias (1993) and Silver and Peterson (1985) provide a good overview of

1

single-product single-location models for a more complete review of the literature we refer

to them

The degree of complexity of a single-product model depends largely on the assumptions

made regarding the demand cost structure and physical characteristics of the inventory system

For example demand can be either deterministic meaning it is known with absolute certainty

or stochastic meaning it is unknown but follows a known probability distribution

Deterministic models tend to be relatively simple yet useful such as the basic EOQ model

However future demand is seldom known with certainty in practice and therefore the use of

these simple models is limited As a result the majority of research in inventory has focused

on stochastic demand models (Lee and Nahmias 1993)

Because the objective of most inventory models is to minimize the cost of inventory

assumptions regarding the structure of the cost are very important also Generally speaking

the cost of inventory has four components the purchaseproduction cost the orderingsetup

cost the holding cost and the shortage cost1 Each of these costs can be fixed proportional

nonlinear or some complex combination of these Further because nearly all inventory

problems require decisions be made sequentially over a period of time the costs can be either

discounted or averaged In the former case a discount factor is included in the cost function

in order to account for the time value of money However since most inventory problems are

on-going the long-run average cost is usually minimized (Lee and Nahmias 1993) In

practice this is achieved by minimizing the expected cost per time period under steady state

conditions

Finally the physical characteristics of the system are important in determining the

1 The cost of a stockout is often very difficult to estimate in practice As a result service levels are frequently used in its place When this is done a constraint is included in the model

2

complexity of the model For example the review process can be either continuous in which

case the inventory is reviewed continuously and orders are placed as soon as the stock reaches

a certain level or periodic in which case the inventory is reviewed at regular intervals and

orders can be placed only at those fixed times Also excess demand can be fully backordered

as is common in many manufacturing and wholesale systems completely lost as is common

in most retail environments or some combination of the two Lead time assumptions are also

important models with lead times are more complex than those in which the supply of

inventory is assumed to be instantaneous

Although the literature is dominated by single-product models (Lee and Nahmias 1993)

much recent work has focused on extending those models to multiple-product problems

However multiple-product models are inherently more complex than the simple-product

models In fact no efficient algorithms are currently available for determining fully optimal

strategies for most multiple-product problems (Federgruen 1993) As a result research has

focused on the development of easily computable approximation techniques Muckstadt and

Roundy (1993) provide a good overview of multiple-product andor multiple-location models

with deterministic demand although emphasis is placed on models for production and

distribution Federgruen (1993) extends the discussion to consider these models under uncertain

demand and also provides a complete list of references on the subject Silver and Peterson

(1985) discuss models for coordinated replenishments for both deterministic and stochastic

demand

In this thesis we examine the inventory system at Canadian Tire Pacific Associates

owner of 21 retail stores and a central warehouse Although this is a multiple-product

multiple-location single-warehouse problem and strictly speaking should be modelled as such

we analyze the inventory problem on a product-by-product and store-by-store basis We believe

3

that this simpler single-product single-location model captures the essentials of the problem for

two reasons First the coordination of replenishments for products andor locations is most

advantageous when significant savings are possible because of high first order costs and low

marginal costs for placing subsequent orders However because of the large number of

products and the physical distribution system in this problem trucks are dispatched to stores

on a regular basis Therefore the benefits of coordinating replenishments for this problem are

negligible Second we believe that controlling in-store inventory alone can result in substantial

savings to Canadian Tire Pacific Associates Further given the excessive amount of inventory

in the warehouse a single-location model assuming an infinite warehouse capacity is reasonable

Of course should inventory control in the warehouse become a priority a more elaborate model

would be required

More specifically we consider a periodic review single-product inventory model with

stationary independent stochastic demand a deterministic lead time and an infinite planning

horizon The probability mass function for the demand is based on empirical sales data

collected over a period of approximately ten months The cost of inventory includes a

proportional holding cost and a fixed ordering cost neither of which is discounted All excess

demand is completely lost The objective of the model is to minimize the long-run average cost

per unit time subject to a service level constraint on the fraction of demand satisfied directly

from in-store inventory In order to obtain meaningful cost comparisons the model calculates

the expected annual cost of inventory

For inventory problems like the one described above optimal (sS) policies can be

computed by Markovian decision process methods such as successive approximations or policy

iteration or by stationary Markovian methods However each of these methods generate

calculations that are complex and time consuming Therefore much research interest lies in

4

developing techniques to find approximate (sS) policies with little computational effort The

majority of the approximation techniques begin with an exact formulation of the problem and

then manipulate the problem by relaxations restrictions projections or cost approximations

(Federgruen 1993) Porteus (1985) presents three approximation techniques and compares

them to fourteen others for a periodic review single-product inventory model in which

shortages are fully backordered and a shortage cost is incurred Federgruen Groenevelt and

Tijms (1984) present an algorithm for finding approximate (sS) policies for a continuous review

model with backordering and a service level constraint Also Tijms and Groenevelt (1984)

present simple approximations for periodic review systems with lost sales and service level

constraints All three works find that the approximation techniques perform well However

the techniques are evaluated for a variety of demand parameters that all result in policies with

a value of S in the range of 50 to 200 Given the slower moving products at Canadian Tire 2

it is unclear how well such approximation techniques would perform Therefore we do not

attempt to apply or modify any of these approximation methods

Exact optimal policies for this problem can be obtained by formulating a constrained

Markov decision process and solving it However the data generating requirements and the

lack of a good upper bound on S make this formulation impractical Therefore we formulate

a Markov chain model for fixed values of s and S and then employ a search technique to find

the optimal values for s and S In order to improve the performance of the model we develop

a heuristic search that locates near optimal policies relatively quickly

The remaining chapters of this thesis are organized as follows In Chapter II an

overview of the existing inventory system at Canadian Tire is presented along with a description

2 For example very few of the products analyzed have optimal values of S exceeding 20 with the largest value of S equalling 69

5

of the problem Next a multi-phase project designed to develop a fully integrated interactive

inventory control system is described with the scope of the research reported in this thesis

completing the first phase of the project Finally a description of the demand and cost data is

given

In Chapter III we formulate a Markov chain model that can be used to calculate the cost

and service level associated with a given (sS) policy Based on several simplifying

assumptions the Markov chain is formulated and the equations for the corresponding transition

probability matrix are developed Then given the limiting probability distribution of the chain

the equations for the steady state annual cost of inventory and the corresponding service level

are obtained The chapter concludes with a brief summary of the model

In Chapter IV an algorithm that searches for the optimal (sS) policy is developed As

a starting point a simple grid search algorithm is developed and evaluated on the basis of

execution time In order to improve the algorithms speed a technique for updating the

transition probability matrix is presented and the improvement in the algorithm is noted Next

a theoretical lower bound on S is obtained although the corresponding reduction in the

execution time is minimal Finally based on two assumptions regarding the shape of the

feasible space a heuristic search is developed which improves the speed of the algorithm

significantly

In Chapter V a prototype of the inventory control systems interface is created In its

present form the interface allows managers to evaluate inventory policies interactively The

manager can set review parameters compare optimal policies and current policies and evaluate

alternate policies The chapter concludes by recommending specific options be including in

future versions to make the system more automated and user-friendly

In Chapter VI optimal policies for all products in one product category sold at one store

6

are compared to existing policies on the basis of expected annual costs and customer service

levels The analysis suggests that existing policies result in a large excess inventory and that

implementing the optimal policies would reduce the annual cost of inventory by approximately

50 A sensitivity analysis on the cost estimates is performed and suggests that the model is

not sensitive to relative errors of approximately 20 in the cost estimates Finally a sensitivity

analysis on the demand probability mass function is performed The analysis indicates that the

model is sensitive at least in terms of the customer service level to slight changes in the tail

of the demand distributions of low-demand products

Finally in Chapter VII the results of the analysis are summarized and future work is

outlined In order to incorporate recent changes in the review procedures at several stores the

model must be modified to allow for the duration of the review period to exceed the lead time

We recommend that the second phase of the project proceed specifically that a data collection

and storage scheme be designed and that a forecasting model be developed Because of the

substantial savings that can be realized by reducing inventory only at the stores inventory

control at the warehouse should also be investigated

7

II T H E I N V E N T O R Y S Y S T E M A T C A N A D I A N TIRE

A Background

Canadian Tire Pacific Associates owns and operates 21 retail stores in the lower

mainland of British Columbia The stores carry approximately 30000 active products3 that

are divided into 80 product categories The merchandise comprises primarily automotive parts

and accessories home hardware housewares and sporting goods Canadian Tire Pacific

Associates also owns a central warehouse in Burnaby at which inventory is received from

Toronto Inventory is then shipped periodically from the warehouse via truck to the local

stores In 1992 total sales at the stores exceeded $99 million and the value of inventory was

estimated to be $28 million of which approximately half was stored in the warehouse

On selected days of each week a stores inventory records are reviewed automatically

and orders for products are placed to the warehouse as required Management uses the term

flushing to describe the process of reviewing the inventory and producing the tickets for stock

owing and the term picking to describe the physical act of filling the order at the warehouse4

Whether or not an order is placed for a product depends upon its current inventory level its

target level and its minimum fill 5 If the target level less the current level exceeds the

minimum fill an order is placed to bring the stock up to the target level From the moment

3 Active products are those that are regularly restocked Canadian Tire Pacific Associates can choose from up to 50000 products

4 Note that the number of picks per week is equal to the number of times per week that the inventory records are reviewed At the beginning of this study eighteen stores had two picks per week while the remaining three stores had only one

5 The target level is the maximum number of items that should be on the shelf at one time and the minimum fill is the minimum number of items that should be ordered

8

the order is placed to the moment the product is reshelved takes between two to four days

depending on the store On average 11000 items are shipped from the warehouse at each

flush

Although a products minimum fill is the same for all stores its target level depends

upon the size of the store Based upon its sales volume and retail space a store is classified

as being one of five sizes small (S) medium (M) large (L) jumbo (J) and extra jumbo

(EJ) 6 Each product has associated with it a target level for each size of store

Consequently all stores of the same size have identical inventory policies Table 1 lists the

location size number of picks per week sales and floor space of each store as of October

1993

Periodically the inventory manager reviews the policies and when deemed necessary

adjusts them In this way the manager can account for seasonality in demand changes in mean

demand and any shortcomings of previous policies At the present time the review process

is not automated nor performed on-line The inventory manager uses Hxl5 computer

printouts with five products to a sheet to study the relevant information and on which to note

any changes Afterward the changes are recorded in the system and the new policies take

effect

Currently the inventory policies are based on rules of thumb developed through the

years and not on a statistical analysis of demand nor on a calculation of the cost of inventory

These rules of thumb incorporate a marketing philosophy which at times dominates the

inventory strategy The prevailing philosophy holds that a large in-store inventory instills

6 During the course of this study stores were resized several times At one point the entire class of medium stores was eliminated leaving only four sizes of stores On another occasion the store in Chilliwack was changed from being a small store to a large store

9

Table 1 Location and particulars of the 21 stores

Store Location Store Size

Picks Per Week

1992 Sales ($)

Floor Space (ft2)

1 Chilliwack S 2 2901969 6678

2 Coquitlam M 2 4015303 6510

3 Langley EJ 2 8138673 15422

4 Scott Road EJ 2 7307820 15064

5 Maple Ridge L 2 4102110 6678

6 Kingsway L 2 4153736 6574

7 Richmond EJ 2 8175105 14961

8 North Vancouver M 2 3856571 6534

9 Guildford M 2 3714372 6567

10 Burnaby M 2 3532752 6983

11 New Westminster S 1 2417992 6470

12 Abbotsford EJ 2 7899398 15303

13 Hastings J 2 4647794 9849

14 King Edward EJ 2 3825372 25492

15 White Rock L 2 3960262 6870

16 Main amp Mountain S 1 2705602 6431

17 Heather J 2 4161227 9427

18 King George J 2 4606817 9640

19 Cambie J 2 5855022 9091

20 Lougheed M 1 3114180 6615

21 Burrard J 2 6553310 10608

10

consumer confidence which in turn leads to increased sales In other words if a customer

perceives that Canadian Tire is in the hardware business he or she will choose to shop there

when meeting future needs However management has adopted this philosophy without any

analysis of the cost of holding excess inventory Only recently has management begun to

question the cost-effectiveness of allocating inventory based on these rules of thumb

Speculating that existing inventory practices might be deficient Mr Don Graham the

vice-president of Canadian Tire Pacific Associates commissioned a pilot study to examine the

inventory system and to propose alternate strategies7 The study using a steady state model

to represent a simplified inventory system suggested that existing policies lead to overly high

inventory levels More importantly the study outlined a methodology that would enable

management to determine inventory levels based on observed demand The pilot study

recommended that a more detailed model of the inventory system be formulated that the

methodology be expanded to incorporate the cost of inventory and that an algorithm be

developed that would quickly identify the optimal inventory policies

B The Problem

At the time of the pilot study management was concerned primarily with its inability

to distinguish between stores of the same size As mentioned in the previous section a

stores size is based solely on its sales volume and floor space and under existing practices

all stores of the same size have identical inventory policies for every product Management

realized that demand for some products might be a function of variables other than size such

7 The pilot study was performed by Martin Puterman a professor of Management Science and Kaan Katircioglu a doctoral student in Management Science both at the University of British Columbia The results of the study were presented at a U B C Transportation and Logistics Workshop on March 25 1993

11

as location For example the demand for sump pumps is likely to be very different in suburban

areas than in urban ones However management was unable to allocate more sump pumps to

large suburban stores than to large urban stores This inability led management to seek a

tool that would allow for a more effective allocation of inventory between stores of the same

size

By the end of the pilot study however management was less concerned with

reallocating inventory within a size of store than with reallocating inventory between sizes

of stores and with reducing overall inventory According to management assigning individual

inventory policies to each store appeared to be operationally infeasible Firstly management

believed that having individual policies would lead to confusion in the warehouse and to

increased shipping errors Exactly why this would occur is unclear the warehouse would

simply continue to fill orders from tickets that make no reference to the policy in place

Secondly and perhaps more tenably the computer technology at Canadian Tire Pacific

Associates did not allow for policies to vary across all stores Both the database used to store

the inventory data and the printouts used by the inventory manager allowed for only five

different policies per product all with the same minimum fill Moving to individual policies

would have required an overhaul of the existing information system Although management

saw this overhaul as inevitable it did not see the overhaul as imminent Therefore

management switched the focus of the study to developing a tool that would help reduce overall

inventory levels andor reallocate inventory more effectively between groups of stores

The subtle change in the studys objective prompted an important question how should

the stores be grouped The current practice of grouping stores based on size is problematic

for two reasons First as discussed above additional factors of demand such as location

cannot be taken into account Second grouping inherently results in a misallocation of

12

inventory when some stores within a group have differing numbers of picks per week For

example the medium-sized Lougheed store which has one pick per week requires higher

inventory levels to achieve a given customer service level than the other medium stores

which all have two picks per week8 In fact in terms of customer service levels it might

be more appropriate to group the Lougheed store with the large stores having two picks per

week Ideally this artificial grouping of stores will be abandoned when the computer system

is upgraded and policies can then be set on a store by store basis

As the project proceeded management grew somewhat uneasy with the concept of using

an analytical tool to evaluate and set inventory policies According to both the pilot study and

the preliminary results of this study significantly reducing inventory would greatly reduce costs

while still providing a high level of customer service However management held that because

the model cannot take into account intangibles such as marketing considerations such a drastic

reduction in inventory would be very inappropriate In this light management viewed the

optimal policies as lower bounds for inventory levels these policies would have to be

modified upwards in order to be practical Therefore management contended that any

methodology or software produced during this study should provide the means to evaluate and

compare a number of alternate policies

Regardless of exactly how the tool is to be used there is currently a lack of historical

sales data from which to analyze the inventory system and determine optimal policies No sales

data had ever been collected prior to the pilot study and only a relatively small amount has

been collected since In order to proceed with any analysis of the inventory system a

8 If demand is a function of size then the Lougheed store must meet the same demand with one fewer pick per week This would lead to either increased probabilities of stockouts at the Lougheed store or higher inventory levels at all the other medium stores

13

substantial amount of additional data would be needed The existing information system at

Canadian Tire Pacific Associates does allow for sales data to be captured however the system

is limited by the amount of data it can capture at one time and by its storage capacity In order

to collect the amount of data required management would have to upgrade the system and

develop some kind of data collection scheme coupled with a data compression technique The

collection scheme must allow for characteristics of demand such as trends and seasonality to

be identified and the compression technique must allow for sales data for over 30000 products

to be stored for each store

Because many of the products sold by Canadian Tire have nonstationary demand as a

result of seasonality andor trend the inventory tool must be able to forecast demand for the

coming period Products such as sporting goods and outdoor equipment exhibit patterns of

demand that are very seasonal For example sales of snow shovels increase through the fall

peak in the winter and drop to zero by late spring At the same time some products might

become either more or less fashionable over time resulting in changes in mean demand

Setting inventory policies for any of these products based on observed demand would result in

insufficient stock if demand increased during the period or in excess stock if demand decreased

Therefore the tool must be able to forecast the demand for the period for which the policy is

intended

Toward the end of the project management began discussing the integration of the

interactive inventory tool with the existing system to form a new inventory control system The

new system would utilize a modified point of sales system to collect sales data continually

Based on the latest available data the system would forecast demand for the coming period and

calculate the optimal policies for every product at every store The system would then provide

all relevant information on-line for the inventory manager to review The manager could weigh

14

other information such as marketing considerations and examine alternate policies to determine

the effect on cost Finally the preferred policy would be set in place with the touch of a key

and information on the next product or store would be presented for review

C The Project

Development of the inventory control system envisioned by management requires the

completion of five tasks 1) the design of a data collection system that generates and stores the

necessary sales data 2) the development of a forecasting method that provides accurate

estimates of demand distributions 3) the formulation of a model that calculates the cost of

inventory and the customer service level for an inventory policy 4) the development of an

algorithm that finds the optimal policy for a product at a store and 5) the creation of an

interface that allows interactive and on-line consultation Ultimately the above components

would have to be integrated and incorporated in a manner that would ensure the compatibility

and reliability of the system

The scope of the research reported in this thesis completes the first phase of the project

specifically we formulate an inventory model develop an algorithm and create a prototype of

the interface Because of the changing environment at Canadian Tire Pacific Associates minor

modifications to both the model and the algorithm developed herein might be necessary to

reflect future changes in operational procedures Upon completion of phase one the interface

will be interactive allowing managers to select the store product and review scenario and then

to compare optimal policies with existing policies or with alternate policies Although the

prototype will be delivered to Canadian Tire Pacific Associates for ih-house testing at the end

of this phase the interface will require modification in subsequent phases as feedback from

management is received and as the system is brought on-line

15

In order to assess more fully the potential savings of implementing the inventory control

system this thesis includes a comparison of optimal policies9 and existing policies The

expected annual cost of inventory for an entire product category is calculated for both the

optimal policies and the existing policies Based on an analysis of these costs the potential

savings of implementing the optimal policies are extrapolated Further the thesis examines the

sensitivity of the model to variations in the estimates of demand and costs

Originally results from all 21 stores were to be included in the above analysis

However just prior to the analysis management changed the review procedure at several

stores the result of which being that three picks per week are now performed at those stores

Because one of the main assumptions of the model specifically that the length of the lead time

does not exceed the duration of the review period is violated under this new review procedure

those stores cannot be included in the analysis Therefore the analysis is based solely on the

results from store six the large store located on Kingsway Avenue10

For the purpose of the analysis sales data for product category 20 automotive and

electrical accessories were collected from all 21 stores This product category was selected

because the distribution of demand for products within it is non-seasonal and because few of

the products are ever placed on sale11 The lack of trend or seasonality eliminates the need

for forecasting demand the optimal policies can be determined based directly on observed

demand

9 That is those policies obtained from the model

1 0 Store six is one of the only stores whose review procedure is not expected to change from two picks per week

1 1 Products that were on sale at any time during the period were not included in the data set

16

D The Demand Data

In total sales data for 520 products of which 458 are active products were collected

over a 307-day period from October 10 1992 to August 18 199312 The data was supplied

by Canadian Tire Pacific Associates in the form of a 10 MB text file with one line per day per

product recording sales from the stores13 Each line had the following format

Columns 1-6 part number Columns 7-9 Julian date14

Columns 10-18 cost of the item (to three decimal places) Columns 19-124 sales for stores 1 through 21 (5 columns per store)

In order to facilitate the analysis the file was sorted into 458 sales files with each of

these files containing the history of daily sales for one of the products Two changes were

made to the format of these files to improve readability 1) a column was added between the

product number and the Julian date and 2) all leading zeros were dropped Appendix A

contains a partial printout of the sales file for a typical product Finally a distribution file

summarizing the daily sales was created from each of the sales files Each distribution file

contains the products number the products cost and the number of days during which various

quantities of the product were sold at each store

Prior to the analysis the distribution files for a large number of products were

examined and based on the shapes of the histograms the products were divided into three

groups The first group low-demand comprises products that have demand on only a few

days with the daily demand seldom exceeding three items The second group medium-

1 2 Of the active products only 420 are sold at store 6

1 3 Strictly speaking a line was written to the file only on days on which at least one store registered a sale of the product

1 4 According to the Julian calendar days are numbered sequentially from 1 to 365 (366 during leap-years) beginning with the first of January

17

demand comprises products that have demand on approximately half of the days with the daily

demand seldom exceeding more than six items Finally the last group high-demand

comprises products that have demand on most days occasionally upwards of twenty items and

whose demand histograms resemble Poisson distributions The majority of the products

examined fell into the low and medium-demand groups only a few high-demand products

were observed Appendix B contains a printout of the distribution file for a typical product

in each group

For most products no theoretical distribution appeared to model the demand well

Demand for low and medium-demand products followed a non-standard distribution with

a mass at zero and a relatively small tail Although some theoretical distributions such as the

mixed-Poisson can be made to resemble this it was unclear how well they would perform

Further the effects of sampling errors from such non-standard distributions were unknown

On the other hand the standard Poisson distribution did appear to model high-demand

products well however only a few products fell into that group Therefore in the absence of

a good theoretical candidate the empirical demand function was chosen for the analysis The

empirical function was readily available from the distribution files and offered the best

estimate of the true underlying function

E The Cost Data

Along with the sales data management supplied estimates relating to the costs of holding

and ordering inventory Management estimates that the cost of holding an item in inventory

for one year is 30 of the items cost Incorporated in this holding rate are a number of costs

such as the opportunity cost of money invested and the costs incurred in running a store

Because it is the annual cost of inventory that is of interest either the compounded daily

18

holding rate or the simple daily holding rate can be used to determine the costs incurred during

a review period however the method of calculating and extrapolating the holding costs must

be consistent For the sake of simplicity a simple daily holding rate is used

In order to estimate the cost of ordering inventory management analyzed the various

tasks that result from an order being placed Using the average time required to fill and then

reshelve an order management obtained an estimate of the direct labour costs involved To

this management added an estimate of the cost of transporting an order from the warehouse to

the store Finally the estimate was adjusted slightly upward to account for the cost of breakage

and errors Based on the analysis management estimates the cost of placing an order for a

product to be $0085 regardless of the size of the order

19

III M O D E L F O R M U L A T I O N

A Terminology

In the previous chapter several terms were introduced to describe the inventory system

at Canadian Tire Pacific Associates These terms though perfectly adequate are not widely

used in the academic literature to describe inventory management problems Therefore in

order to avoid confusion and be consistent with the literature the following terms are used

throughout this thesis

The length of time between successive moments at which the

inventory position is reviewed15

The time at which a reorder decision is made

The length of time between the moment at which a decision to

place an order is made and the moment at which the order is

physically on the shelf

An operating policy in a periodic review inventory model in

which (i) the inventory position is reviewed every T days (ii) a

replenishment order is placed at a review only when the inventory

level x is at or below the reorder level s (gt 0) and (iii) a

replenishment order is placed for S-x units (S gt s)1 6

Lost Sales - The situation in which any demand during a stockout is lost

Review Period -

Decision Epoch

Lead Time -

(sS) Policy

1 5 With respect to the terminology used at Canadian Tire Pacific Associates a review period equals the number of days between flushes

1 6 With respect to the terminology used at Canadian Tire Pacific Associates S represents the target level and s represents the difference between the target level and the minimum fill

20

B Assumptions

In to order make the model of the inventory system mathematically manageable several

simplifying assumptions are made However we believe that the following assumptions do not

significantly reduce the accuracy of the model

1) Demand can be approximated very closely by sales during the period in which the data

is collected Generally inventory levels at the stores are very high and the number of

lost sales is assumed to be negligible In fact the inventory levels for a number of

products were tracked over time during the pilot study and not a single incidence of

stockout was observed Because all products appear to have similar excess inventory

it seems reasonable to assume that virtually all demand resulted in sales

2) The demands in successive periods are independent identically distributed random

variables Preliminary tests (Fishers exact tests) suggested no dependence between

sales on consecutive days and plots of sales over time indicated no seasonal pattern

3) Demand is independent across products That is we assume that sales of one product

do not affect nor can they be predicted from sales of another product

4) Unmet demand results in lost sales with no penalty other than the loss of the sale

5) The duration of the review period for a store is an integer value and is fixed over a

planning period For stores with two picks per week we assume that the inventory

is reviewed every four days rather than on a four day - three day cycle

6) The decision epoch is the start of a review period Review periods begin at midnight

at which time the inventory is reviewed and orders are placed instantaneously

7) The duration of the lead time for a store is deterministic and integer-valued In reality

the lead time for some stores was stochastic but management felt the variation could be

controlled and virtually eliminated

21

8) The lead time does not exceed the review period In other words items ordered at a

decision epoch arrive and are shelved before the next decision epoch In practice this

did not always occur

9) Orders arrive at the end of the lead time and are on the shelf to start the next day

10) Holding costs are incurred daily for all inventory held at the start of the day

11) The warehouse has an infinite capacity that is all orders to the warehouse are filled

Because of the high inventory levels in the warehouse stockouts in the warehouse are

very unlikely

C A Markov Chain Model

Consider the situation in which an (sS) policy is applied to an inventory system based

on the above assumptions Inventory is reviewed at the decision epoch for each of the periods

labelled t = 0 12 and replenishment occurs at the end of the lead time provided an order

was placed Demand during day d in period t is a random variable whose probability mass

function is independent of the day and period The function is given by

f^ik) = Pr ltd = k = ak for k = 0 1 2

where ak gt 0 and E ak = 1 Let T denote the duration of a review period and L denote the

length of the lead time Next define D T ( as the total demand during period t D L as the total

demand during the lead time in period t and D T L as the difference between D T and DLt

Assuming that successive demands pound f pound 2 are independent the probability mass function of

1 7 The aggregated demand variables are given by

^ = XXd and DT_Ut=J d= d= d=L+l

22

each aggregated demand variable can be obtained by convolving the probability mass function

of t d l s Finally let X denote the inventory on hand at the beginning of period t just prior to

the review

The process X is a discrete time stochastic process with the finite discrete state space

0 1 S A typical sample path of the process with a review period of four days and a

lead time of two days is depicted in Figure 1 As period t begins X units are on hand The

inventory is reviewed and because X is less than s an order is placed for (S-X) units Over

the next two days a demand of D L r occurs and the inventory drops accordingly With the

arrival of the order the inventory level increases by (S-X) at the start of the third day of the

period Over the remaining two days a demand of D T L is observed bring the inventory level

down to X + 1 Note that in this example an order would not be placed in period t+l because

X r + i exceeds s

The stock levels at the beginning of two consecutive review periods are related by the

balance equation

_K- DLy+ (s - v - DT-LX ixtlts ( 1 )

t + l ~ (Xt - DTtY ifXtgts

where the positive operator is interpreted as

= max a 0

1 8 For example the probability mass function of D T bdquo denoted fD T ( is the T-th fold convolution of f t Mathematically the function can be obtained using the recursive relation

and the identity f1^ = f5(

23

-2 I 8

gt o

r

CO

o a

CM

X

00

24

Because X + 1 depends only on Xt the process X satisfies the Markov property

PrXt+l=jX0 = i0Xt_l=it_vXt = i) = PrXt+1=jXt = i

and therefore is a discrete time Markov chain Further because the probability mass function

of the random variable pound is independent of t and thus so too are the probability mass functions

of the aggregated demand the Markov chain is stationary19

D The Transition Probability Matrix

The stationary transition probability p(gt is defined as the probability of X + 1 being in state

j given that X is in state i independent of the period t In the inventory model p-- represents

the probability of starting the next review period with j items on the shelf given there are i units

on hand at the start of the current period By convention the transition probabilities are

arranged in an (S + l)x(S + 1) matrix P such that (P)y = p- Thus the (i + l)st row of P contains

the probability distribution for the stock on hand at the beginning of the next period given that

there are i units on hand at the start of the current period The matrix P is known as the

transition probability matrix Note that since we are concerned with a steady state analysis we

need not consider the possibility of initial stock levels greater than S once the inventory level

is below S it can never again exceed S

The transition probability ptj is calculated by expanding (1) to include all possible

instances of demand that allow for a change in inventory from i units at the beginning of a

period to j units at the beginning of the next period For example suppose the inventory policy

(510) is in place 2 items are on hand at the start of the current review period and 7 items are

1 9 Henceforth the subscripts t and d are omitted from the daily and aggregate demand variables for the sake of brevity

25

on hand at the start of the next period Then only three scenarios are possible

1) No items are sold during the lead time (meaning that the stock level increases to 10

after replenishment) and 3 items are sold during the remainder of the period

2) One item is sold during the lead time and 2 items are sold during the remainder of

the period

3) Two items are sold during the lead time (though demand could exceed 2) and 1 item

is sold during the remainder of the period

The transition probability p 2 7 is the sum of the probabilities of the various scenarios and is

computed by

p2J = PrDL=0PrDT_L = 3] + PrDL = l] Pr[DT_L=2 + PrDL2] PrDT_L = l]

In general the transition probabilities for the inventory model are calculated as follows

Case 1 The lead time is less than the review period

For 0 lt lt s

Pij

i-l pound [PrDL=kPrDT_LgtS-k] + PrDLgtiPrDT_LgtS-i if j = 0

i - l

^[PriD^^PrlD^S-j-kn+PHD^PriD^S-j-i if 0 lt j ltL S-i (2) k=0 s-j

E [PrDL=kPrDT_L = S-j-k] if S-i ltj^S k=0

26

For s lt i lt S

PrDTzi]

0 p = PrDT=i-j)

ifj = 0 ifOltjzi ifiltjltS

(3)

Case 2 The lead time equals the review period

For 0 lt i lt s

Pa

0 PrDTzi) PrDT = S-j

ifOltLJlt S-i ifj = S-i if S-i ltj plusmn S

(4)

For s lt i lt S use (3)

E The Steady State Probabilities

Provided 1) there is a positive probability of non-zero demand during a review period

and 2) there is a positive probability of zero demand during a review period20 then the

Markov chain described by (1) has a limiting probability distribution ir = (ir0 TTu irs)

where 0 lt IT lt 1 for j = 0 1 S and E 7ry = 1 Further this limiting distribution is

independent of the initial state In other words the probability of finding the Markov chain in

a particular state in the long-run converges to a nonnegative number no matter in which state

the chain began at time 0 The limiting distribution is the unique nonnegative solution of the

equations

2 0 That is 0 lt PrD x = 0 lt 1 Given the nature of the problem this assumption is not very restrictive

27

^ = E ^ - far j = 0 l S (5)

s and pound nk = 1 (6)

For the proof that the limiting distribution is independent of the initial state and is obtained from

(5) and (6) the reader is directed to Appendix C

In terms of the inventory problem 7ry is the steady state probability of having j units of

inventory on hand at the decision epoch However some subsequent calculations are made

easier if the steady state probability of having j units of inventory on the shelf just after

replenishment is known Instead of reformulating the Markov chain these probabilities can be

obtained by shifting the limiting distribution TT Much in the same way that the transition

probabilities are calculated the shifted limiting distribution t] is obtained by accounting for

all possible instances of demand that allow for the shifted inventory position For example

suppose that the policy (510) has the limiting distribution TT In order to have say 8 units on

hand immediately after replenishment one of the following must occur

1) Ten items are on hand at the decision epoch and 2 items are sold during the lead time

2) Nine items are on hand at the decision epoch and 1 item is sold during the lead time

3) Eight items are on hand at the decision epoch and nothing is sold during the lead time

4) Either 2 3 4 or 5 items are on hand at the decision epoch and 2 items are sold during

the lead time

The steady state probability of having 8 units on hand just after replenishment is calculated as

n 8 = n10PrDL = 2] +iz9PrDL = l] +n8PrDL = 0 +[iz5 + iz4 + iz3]PrDL = 2 +n2PrDLgt2

28

In general the shifted steady state probabilities are given by

s pound nPrDLgti ifj = 0

i=s+l S

pound PrDL = i-j] ifl plusmnjltS-s i=max(s+l)

S = pound 7tjirZ)L = i-+7i 5 FrD i ^5 ifj = 5-5

i=max(s+l)

^ 7 1 ^ ^ = 1-7+ pound TtiFrZgti=5-y+iis_PrD1^S-7 ifS-sltjltS i=max(gt+l) i=S-j+l

(7)

F The Cost of Inventory

For this problem the cost of inventory has two components (i) a fixed ordering cost

cf incurred whenever an order is placed and (ii) a holding cost cv incurred daily for each item

held in inventory The expected cost of inventory for a review period can be calculated by

summing the costs associated with each inventory position under steady state conditions

Because a simple holding rate is used this cost is simply adjusted by number of review periods

in a year to obtain the expected annual cost of inventory

The ordering cost for a review period is relatively straight forward to calculate either

an order is placed in which case the order cost is incurred or no order is placed in which case

no cost is incurred According to the rules of the inventory policy orders are placed only if

the stock on hand at the beginning of a review period is less than or equal to s Further the

probability that the review period begins with the inventory in one of these positions is equal

to the sum of the first (s + 1) steady state probabilities Therefore the expected ordering cost

29

for a review period is given by

E[OC] = cfY n y=o

Unlike the simple structure of the ordering cost the holding cost is incurred daily for

each item held in inventory Therefore the daily holding costs associated with all possible

inventory positions must be weighted and summed to obtain the holding cost for the review

period In order to make the calculations less complicated a shifted review period beginning

immediately after replenishment and ending just prior to the next replenishment is considered

Define e(0 to be the expected number of items on hand to start day i of the shifted

review period and let g(i f) denote the expected number of items on hand to start day i given

that there were j items immediately following replenishment21 The function g(z j) is the sum

of the probabilities of possible demand weighted by the resulting stock levels that is

g(ij) = j^kPriD^j-k)

To obtain the expected number of items on hand to start day i independent of j the above

function is summed over all possible values of j and weighted by the corresponding steady state

probabilities yielding

s j ed) = pound r 5 gt P r Z ) = - pound

7=0 k=

2 1 In other words g(if) is the expected number of items on hand to start day i given e(0)=v

30

The expected holding cost for a review period is then given by

T

E[HC] = $gt v e(0 i=i

i=l =0 k=l

where c v equals the simple daily holding rate multiplied by the cost of the item

Although differently defined review periods are used to calculate the above expected

costs the costs can still be summed to obtain the expected cost of inventory for a review

period The expected costs are calculated based on steady state conditions and therefore

represent the costs expected to be incurred during any T consecutive days Summing the above

two expected costs and multiplying the sum by the number of review periods in a year yields

the expected annual cost of inventory

E[AC] = mdash[E[OC]+E[HC]]

T S j 1 ( g ) 365

j=0 i=l j=0 fc=l

G The Customer Service Level

There are many ways to define customer service and the most appropriate definition

depends upon the nature of the business and the perspective of management Two commonly

used definitions are 1) the proportion of periods in which all demand is met and 2) the

proportion of demand satisfied immediately from inventory Although the former definition is

often used because of its simplicity the latter definition is closer to what one commonly means

by service (Lee and Nahmias 1993) A third definition also relevant in retail environments

31

is the probability that lost sales do not exceed some critical number Other definitions such

as probability that backorders are filled within a certain amount of time are more applicable

to industrial and manufacturing settings where backlogging demand is common

Given the retail environment with the possibility of lost sales that exists at Canadian Tire

Pacific Associates management is concerned primarily with the fraction of sales that are lost

For example management views the situation in which demand exceeds stock by one unit every

period more favourably than that in which the same demand exceeds stock by three units every

other period For this reason the proportion of demand satisfied immediately from inventory

is the most appropriate measure of customer service

The customer service level for a single review period CSLRP is given by

r bdquo T _ Demand satisfied during RP Total demand during RP

Because we are interested in the long-run behaviour of the system a measure of the steady state

customer service C S L must be obtained As shown in Appendix D the correct measure is

poundpoundpound _ E[Demand satisfied during RP] E[Total demand during RP]

which can be rewritten as

pound pound pound _ j _ E[Demand not satisfied during RP] ^ E[Total demand during RP]

Because of the steady state conditions the above customer service level can be estimated

from any period of T consecutive days As was done for one of the cost calculations the T-day

period beginning just after replenishment is considered Suppose that there are j units of

inventory on hand immediately following replenishment Then the conditional customer service

32

level is

CSL j = 1 E[(DT-j)l

E[DT] (10)

The denominator in the second term is merely the expect demand during the period

E[DT] = 7gt (11)

where p is the mean of the random daily demand variable pound The numerator which is the

conditional expected demand that is not satisfied during the T-day period expands to

7-1

E[(DT-j)+] = T u - j + YU-k)PrDT = k] k=0

(12)

The reader is directed to Appendix E for a mathematical justification of (12) In order to obtain

the unconditional expected demand not satisfied (12) is applied to all possible values of j and

a weighted sum of the terms is calculated The result is given by

E[Demand not satisfied] = Y j =0

y-i Ti - j + 1pound(j-k)PrDT = k

S j-1

= - E[I] + Y0-k)Pr[DT=k) =0 it=0

(13)

where E[I] denotes the steady state inventory level Finally the customer service level is

obtained by the substituting (11) and (13) into (9) yielding

CSL = 1

s j-l

Ti - E[I] + Jgty Y(J-k)PrDT=k]

m n J2VjTU-k)PrDT=k Eil J=Q fc=o

(14)

33

H A Methodology for Evaluating an (sS) Policy

In this section a brief methodology summarizing the main results of the model is

presented The methodology can be applied to evaluate a given (sS) policy in terms of its cost

and its resulting customer service level Given the probability mass function of the daily

demand f the duration of the review period T the length of the lead time L the fixed cost

of placing an order cf and the daily unit cost of holding inventory cv

1) Calculate the aggregated demand probability mass functions fDT fDL and f D x L as the T-th

fold L-th fold and (T-L)th fold convolutions of f respectively

2) Generate the transition probability matrix using either equation (2) or (4) and equation (3)

3) Calculate the limiting distribution TT from (5) and (6)

4) Obtain the L-day shifted limiting distribution rj using (7)

5) Calculate the expected annual cost of inventory using (8)

6) Determine the customer service level using (14)

34

IV A N A L G O R I T H M F O R OBTAINING T H E O P T I M A L (sS) P O L I C Y

A Introduction

The methodology developed in the previous chapter can be used to calculate the cost and

service level associated with a given (sS) policy In this chapter an algorithm that finds the

feasible policy yielding the lowest cost is developed A policy is feasible if it satisfies both the

basic condition 0 lt s lt S and the imposed constraint on the customer service level2 2 In

order to find the optimal policy the algorithm must combine the methodology for evaluating

a single policy with a search technique

Search techniques are evaluated in terms of two somewhat antithetic criteria On the one

hand the search should locate the optimal solution or at least a near optimal solution on

the other hand the search should take as little time as possible Many search techniques

guarantee optimality provided the objective function and constraints exhibit certain properties

such as convexity However even if such a search technique also guarantees its termination

in polynomial time the search might take too long to be practical Alternatively some search

techniques provide reasonable though sub-optimal solutions very quickly These techniques

often called heuristics must be evaluated in terms of both their speed and the accuracy of their

solutions The choice of a heuristic over a search that guarantees optimality depends upon the

nature of the feasible space the need for accuracy and the cost of time

Given the complexity of the cost and customer service level functions it is very difficult

to show theoretically for what if any combinations of demand and costs the feasible space is

convex However a plot of the objective function for a single product may be obtained by

2 2 Management specified that each policy must provide a customer service level of at least 975 Therefore the algorithm must incorporate the constraint CSL gt 0975

35

evaluating the costs for a number of policies within a grid When the service level constraint

is included in these plots maps of the feasible space are obtained Examination of the

maps for several products suggests that the space is convex over most regions however

maps for some products reveal some regions of non-convexity Unfortunately there is no

way of knowing the degree of non-convexity for other products other stores or different costs

estimates without a more rigorous survey of the feasible space Therefore if one of the

techniques that guarantee optimality under the condition of convexity is employed the search

would have to be evaluated as a heuristic Instead as a first step to finding a practical search

technique a crude search that guarantees optimality under virtually all conditions is

investigated

B A Grid Search

The simplest search technique that guarantees optimality merely evaluates all policies in

the grid defined by 0 lt s lt S and 0 lt Sm i n lt S lt S m a x where Sm i n and S m a x are

some predefined values That is the grid search is a brute force method that examines every

policy within the grid 2 3 Although the number of policies that must be examined is large the

grid search is guaranteed to find the optimal policy provided that S o p t is contained in the grid

that is provided Sm i n lt S o p t lt Sm a x The grid search algorithm proceeds by generating the

transition probability matrix for the policy (0Smin) and then evaluating the service level

associated with that policy If the policy is feasible24 the cost associated with the policy is

2 3 The total number of policies evaluated is the sum of the first Sm a x integers less the sum of the first Smin-1 integers Numerically [(l2)(Sm a x)(Sm a x+l) - (l2)(Smin-l)(Smin)] policies are examined

2 4 That is if the service level is adequate

36

calculated and compared to the current lowest cost The procedure continues until all points

in the grid have been examined and the optimal policy has been found A flow chart of the grid

search algorithm is shown in Figure 2

In order to assess the grid search the algorithm was coded and compiled into an

executable program The program was run for two products (a medium-demand product and

a high-demand product) and with four values of Sm a x In each run Sm i n was taken to be zero

The following times were recorded for each trial

T P M - the amount of time spent generating the transition probability matrix

GAUSS - the amount of time spent solving for the steady state probabilities

COSTS - the amount of time spent calculating the expected annual costs

SERVICE - the amount of time spent determining the customer service levels

T O T A L - the total execution time of the program

The columns entitled Grid Search in Tables 2 and 3 summarize the results of the trials All

times are recorded in seconds as required by a 486DX33 MHz personal computer with an

empty buffer memory

Three results are evident from the tables First the vast majority of the execution time

is spent generating transition probability matrices and then solving for the steady state

probabilities with the time being split almost equally between the two tasks Second the total

execution time of the algorithm increases exponentially as Sm a x increases For product 206917

doubling S^ from 30 to 60 led to an increase in the execution time of more than 2100 (from

172 seconds to 3661 seconds) Third the grid search algorithm as it stands does not appear

to be adequate for the problem at hand Given the large number of products that must be

evaluated the algorithm is much too slow to be implementable

37

Calculate the demand probability functions

Initialize s = 0 S = Smin] cost = 10A5

s = s+1

Calculate the probability transition matrix for (sS)

Evaluate policy (sS) Is service level ^ 0975

No

No Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Is s = S-l

Yes

Is S = Smax No

Yes

Print results (sS) cost

s = 0 S = S+1

Figure 2 The flow chart of the grid search algorithm

38

Table 2 Execution times of the algorithms for product 200001 a 30-amp inline fuse

(in seconds)

-max Subroutine Grid Search Grid Search

Updating T P M Heuristic

10 T P M 006 010 001

GAUSS 010 006 001

COST 005 005 001

SERVICE 001 001 001

T O T A L 050 035 027

20 T P M 126 021 010

GAUSS 119 160 023

COST 027 026 001

SERVICE 017 016 001

T O T A L 318 236 055

30 T P M 703 096 010

GAUSS 757 801 093

COST 079 085 001

SERVICE 047 042 001

T O T A L 1664 1137 126

40 T P M 2257 163 015

GAUSS 2958 3003 206

COST 313 329 001

SERVICE 092 083 011

T O T A L 5871 3851 285

39

Table 3 Execution times of the algorithms for product 206917 a 6 solder connector

(in seconds)

bull-max Subroutine Grid Search Grid Search

Updating T P M Heuristic

30 T P M 717 067 006

GAUSS 762 880 093

COST 098 087 010

SERVICE 016 021 006

T O T A L 1720 1176 220

40 T P M 2390 157 022

GAUSS 2861 2911 209

COST 219 227 006

SERVICE 080 088 011

T O T A L 5827 3790 362

50 T P M 6691 413 010

GAUSS 7994 8201 472

COST 568 577 016

SERVICE 212 197 005

T O T A L 15840 10041 654

60 T P M 15509 894 040

GAUSS 18607 18951 876

COST 1367 1353 038

SERVICE 350 359 022

T O T A L 36614 22794 1120

40

C A Technique for Updating the Transition Probability Matrix

As was noted in the previous section approximately half of the execution time of the grid

search algorithm is spent generating transition probability matrices Furthermore the amount

of time increases exponentially as Sm a x increases This increase is exponential for two reasons

first the number of transition matrices that must be calculated is a function of S^ squared and

second the average size of the matrices and hence the average number of calculations per

matrix increases with Sm a x For example when S m a x equals 30 the transition probability matrix

must be generated 465 times with the largest matrix having dimensions (31x31) When Sm a x

equals 40 the matrix must be generated 820 times with 355 of the matrices having dimensions

larger than (31x31) The performance of the algorithm could be improved considerably if

either the number of matrices that must be generated or the number of calculations per matrix

is reduced This section introduces a technique that accomplishes the latter by updating the

transition probability matrix for a new policy instead of generating it anew for each policy

A new policy (s+mS)

Suppose the policy (sS) has associated with it the (S + l)x(S + l) transition probability

matrix P and the policy (s+mS) where m is any integer satisfying 0 lt s+m lt S has

associated with it the (S + l)x(S + 1) transition probability matrix J Then T can be obtained by

recalculating only m rows of P Recall that the equations for generating a transition

probability matrix are divided into two sets based upon the row being calculated equations (2)

or (4) for rows 0 through s and (3) for rows (s + 1) through S The equations themselves are

independent of s only the limits defining the cases contain s Therefore only the m rows

that are affected by the change in the limits need to be recalculated The following updating

technique can be used to obtain 7

41

1) If m gt 0 rows i = (s + 1) (s+2) (s+m) are recalculated using (2) or (4)

2) If m lt 0 rows i = (s-1) (s-2) (s-m) are recalculated using (3)

3) All other rows of T are identical to those of P

The reader is directed to Appendix F for a mathematical justification of the technique

A new policy (sS+m)

Suppose the policy (sS) has associated with it the (S + l)x(S + l) transition probability

matrix P and the policy (sS+m) where m is any positive integer has associated with it the

(S+m+l)x(S+m+l) transition probability matrix P Then T can be obtained by modifying

each row of P and adding m additional rows and columns The updating technique is not as

simple as that for a new policy (s+mS) for two reasons the dimensions of P and T differ and

the equations for generating a transition probability matrix contain S Nevertheless an updating

technique that requires much less computing time than generating the matrix anew can be

developed The following technique can be used to obtain P

1) For rows i from 0 through s

For columns j from 0 through m recalculate using (2) or (4)

For columns j from (m+1) through (S+m) enter the (j-m)th column entry of P

2) For rows i from (s + 1) through S

For columns j from 0 through S enter the corresponding value from P

For columns j from (S + 1) through (S+m) enter 0

3) For rows i from (S + 1) through (S+m) recalculate using (3)

The above technique is justified mathematically in Appendix G

42

The modified algorithm

The updating technique can be easily incorporated into the grid search algorithm with

only a slight change in the manner in which s is incremented Figure 3 shows the flow chart

of the new algorithm As before the algorithm was coded and compiled into an executable

program and the various execution times were recorded The results are listed in Tables 2 and

3 under the columns Grid Search Updating TPM

Updating the transition probability matrix does improve the performance of the

algorithm The time required to generate the transition probability matrices is reduced by

approximately 90 This translates to a reduction in total execution time of approximately

30 Not surprisingly the reductions are greater for large values of Sm a x However despite

this seemly significant improvement a decrease of only approximately 08 seconds is obtained

when S^ equals 20 Further when S ^ equals 30 the algorithm requires over 11 seconds

which is still much too high

The execution time is now dominated by the time required to solve for the steady state

probabilities The steady state probabilities could be solved by using approximation techniques

which are faster than solving the systems exactly however because the probabilities are often

very small numbers the approximation techniques would be unable to provide the required

accuracy The only remaining way to decrease the execution time is to reduce the number of

policies that are evaluated during the search This can be done by either reducing the size of

the grid being searched or reducing the number of policies being evaluated within the grid The

following section introduces a method of reducing the size of the grid by obtaining a lower

bound for S namely Sm i n

43

s = s+k

Calculate the demand probability functions

Initialize s = 0 S = Smin| cost = 105

k= 1

Update the probability transition matrix for (sS)

Evaluate policy (sS) Is service level ^ 0975

No

No

iNo

Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Ifk= 1 iss = S-l If k =-1 is s = 0

Yes

Is S = Smax No

Yes

Print results (sS) cost

If k=l s=s+l S = S+1

Figure 3 The flow chart of the grid search algorithm utilizing the updating technique for the transition probability matrix

44

D A Lower Bound on S

As mentioned earlier the size of the grid is specified by the number of policies within

the domain defined by 0 lt s lt S and 0 lt Smin lt S lt Smax Thus the size of the grid

and hence the total amount of computation time can be reduced by obtaining tight bounds on

S Unfortunately because of the extreme non-linearity of the objection function and the service

level constraint obtaining a mathematical upper bound on S is very difficult and beyond the

scope of this work This section introduces a method for obtaining a lower bound on S25

For a given value of S an upper bound on the customer service level can be found as

follows Suppose that there are S units of inventory on the shelf at the beginning of every

review period regardless of demand No (sS) policy can provide a higher level of customer

service because there exists a positive probability of beginning a review period with less than

S units on the shelf Thus the customer service level associated with starting each review

period with S units on the shelf is an upper bound on the customer service level for any (sS)

policy From (10) this level is

E[(DT-SY] CSL 5 = 1 - T mdash mdash - ( 1 5 gt

E[DT]

Replacing j with S in (12) and then substituting (12) and (11) into (15) yields

CSL 15 = mdash- - J2(S-k)PrDT = k] (16)

To obtain Smin use (16) to find the largest value of S with a customer service level less than

0975

There are two criteria for a good lower bound on S the bound must be no larger than

This method was proposed by Kaan Katircioglu

45

S o p t and it must be very close to So p t These criteria ensure that the optimal policy will be

contained in the search grid and that the size of the grid is reduced by as much as possible

Based on these criteria the above method of obtaining a lower bound is assessed A value for

S m i n is obtained for each of the products in product category 20 and compared with the

corresponding value of S o p t 2 6 The results are illustrated in Figure 4 which shows the number

of occurrences of various Sm i n and S o p t combinations Note that a point lying on the 45deg line

represents a perfect bound while a point lying close to the S o p t axis represents a very loose

bound The absence of any points above the 45deg line indicates that the first criterion is satisfied

in each instance the method did not generate a value of Sm i n exceeding the corresponding value

of So p t Moreover the bounds are reasonably tight for 85 of the products analyzed that

is for those products with S o p t equal to 10 or less27 However as S o p t increases the bounds

become steadily worse Because values of S o p t are likely to be larger for these products at the

ten Jumbo and Extra Jumbo stores and for other products with higher demand the lower bounds

are not expected to be very tight in general

Although obtaining the bound Sm i n does reduce the size of the grid the corresponding

reduction in the execution time of the algorithm is miriimal For example if S^ equals 60 and

a value of 30 is obtained for Sm i n the program would still require approximately 215 seconds

to search the grid (compared to the 228 seconds required without a lower bound on S) Because

the execution time increases exponentially with So p t a tight upper bound on S would likely have

a more significant impact on reducing the execution time of the algorithm Further research

on this issue is required

2 6 The optimal policy for each product was obtained using a grid search with an extremely large upper bound

2 7 On average Sm i n is approximately half of S o p t for these products

46

47

The only remaining approach to improving the algorithm is to modify the search so that

fewer policies within the grid are evaluated Toward this goal assumptions can be made with

respect to the feasible space and a heuristic search can be developed based on these

assumptions

E A Heuristic Search

As mentioned previously in this chapter a map of the feasible space for each of

several products was obtained Upon closer examination the maps illustrate two patterns

without exception for a given value of S both the cost and the customer service level increase

as s increases The following intuitive argument is offered as a possible explanation for the

observed behaviour When S is held constant increasing s results in more instances in which

an order is placed upon review Hence the average stock level would increase which in turn

would lead to an increase in the cost and the customer service level28 With this in mind a

heuristic search based on the following assumptions is developed for a given value of S both

the cost and the customer service level are monotonically increasing functions of s

In general the heuristic proceeds as follows Having just incremented S by one the

customer service level of the policy (sS) is evaluated If the policy is feasible s is decreased

until the lowest value of s producing a feasible policy is found If the policy is not feasible

s is increased until either the cost exceeds the current lowest cost for a feasible policy a

feasible policy is obtained or s equals S-l In the event that a feasible policy is found the cost

2 8 In order to prove that the customer service level increases as s increases it would be sufficient to show that the probability of placing an order increases as s increases Such a result might be possible if assumptions regarding the demand function are made Further if both the probability of placing an order and the average stock level increase then both the ordering cost and the holding cost would increase No proofs are offered here

48

of the policy is calculated if the cost of the policy is less than the current lowest cost the new

policy becomes the current best Finally S is increased by one and the process repeats

provided S does not exceed some predefined value Sm a x The flow chart of the algorithm

incorporating the heuristic search is shown in Figure 5 Note that the algorithm begins by

attempting to improve on the lower bound Sm i n the policy (Sm i n-lSm i n) is evaluated and if it

is not feasible S m i n is increased by one The improvement continues until a feasible policy is

found

As previously mentioned a heuristic search must be evaluated in terms of both its speed

and the accuracy of its solutions To evaluate the heuristic on the basis of speed the same

approach used to evaluate the previous algorithms is employed The columns entitled

Heuristic in Tables 2 and 3 contain the various execution times of the heuristic The heuristic

is a vast improvement over the previous search techniques the reduction in execution time is

enormous and becomes more so as S m a x increases For a S m a x of 20 the reduction is

approximately 75 of the time required by the grid search algorithm29 and for a S m a x of 60

that reduction soars to over 95 More importantly the heuristic can search a grid with a

of 40 in under 4 seconds or a grid with a S m a x of 60 in under 12 seconds These low execution

times should be sufficient to allow for a large number of products to be analyzed in a

reasonable amount of time

In order to further assess the practicability of the heuristic the total time required by the

heuristic to find solutions for the entire product category was determined and compared to the

corresponding time for the grid search algorithm With a priori knowledge of the optimal

solutions conservative yet reasonable values of S m a x were selected and used by both of the

2 9 Hereafter grid search algorithm refers to the algorithm utilizing the grid search the updating technique and the lower bound Sm i n

49

Calculate the demand probability functions

Calculate Smin S = Smin s = S-l

S = S+l s = S-l t

Update the probability transition matrix for (sS)

No Is service level pound 0975 Yes

No Is service level ^ 0975

Yes

Is cost lt cost No

Yes

Is s lt S-l No

Yes

s = s+l

Update the probability transition matrix for (sS)

Is service level pound 0975

Yes

Is cost lt cost No

Yes

s = s S = S cost = cost

s = s-l

No

Update the probability transition matrix for (sS)

S = S+l

Is s 2 0 Yes

Update the probability transition matrix for (sS)

Is service level pound 0975

No

s = s+l

No

No

Is S = Smax

^Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Print results (sS) cost

Figure 5 The flow chart of the heuristic algorithm

50

algorithms30 The grid search algorithm required approximately 9 hours and 41 minutes to

find the solutions while the heuristic required only slightly more than 23 minutes Based on

speed the heuristic is preferable to the grid search algorithm

Finally to evaluate the heuristic based on the accuracy of its solutions its solutions were

compared to the previously obtained optimal policies Without exception the heuristic

algorithm found the optimal policy Although there is no guarantee that the heuristic will find

the optimal solution for different products or different demand functions the heuristic appears

to work extremely well Based on the fast execution time and the apparent accuracy of the

solutions the heuristic will be used in the development of the inventory tool

3 0 For the test the following values of Sm a x were used 20 (273 products) 30 (19) 50 (23) 100 (3) and 125 (2)

51

V T H E I N T E R F A C E

With the model formulated and the algorithm developed the only remaining task in the

development of the prototype is the creation the interface In fact two interfaces are created

one that displays results in terms of minimum fills and target levels and one that displays

results in terms of reorder points and target levels To be consistent with the previous chapters

the latter interface is described here31

In its present form the interface allows the manager to set review parameters to

compare optimal and current policies and to evaluate alternate policies From the main menu

the manager selects the product store duration of the review period length of the lead time

and value of Sm a x Once the parameters are set the system calculates the optimal policy and

then evaluates the current policy displaying results as they are obtained In particular the

reorder point target level expected annual cost and expected customer service level are

displayed for each policy If a current policy meets the desired customer service level the

expected annual savings of the optimal policy are displayed in terms of both dollars and

percentage of current cost otherwise the value of the current customer service level is

displayed in red to indicate that it is deficient Finally the manager can choose to evaluate

alternate policies or to return to the main menu

Although the prototype is far from the ultimate integrated inventory control system it

is still valuable to management As mentioned in Chapter II inventory policies at Canadian

Tire Pacific Associates are currently based on rules of thumb that incorporate marketing

3 1 Because significant changes to the interface are expected as feedback from management is received and as the system is brought on-line only a brief description of the present interface is given

52

strategies The prototype provides management with a means of determining the proportion of

the cost of inventory that is not attributable to meeting customer demand For example

consider a hypothetical consultation in which the manager chooses to analyze product 200001

a 30-amp inline fuse at store 6 The prototype displays to the manager the current policy of

(1214) which provides a customer service level of 1000 at an annual cost of $748 and the

optimal policy of (39) which provides a customer service level of 977 at an annual cost of

only $319 If for some reason the manager feels the optimal policy is undesirable he or she

can investigate the additional cost of holding excess inventory Suppose the manager chooses

to evaluate the policy (610) The prototype evaluates the policy and displays the results the

customer service level is 998 and the annual cost is $441 With the alternate policy

virtually all demand is satisfied and the cost of inventory is still reduced by over 40 Thus

the manager ascertains that approximately 40 of the current cost of inventory for this product

is directly attributable to marketing considerations Using the prototype in this way

management can begin to understand better the nature of its inventory costs A description and

graphical depiction of the above hypothetical consultation are presented in Appendix H

As for the technical details of the system the prototype can run on any computer using

the DOS operating system having approximately 420K of available conventional memory and

attached to a V G A or SVGA monitor Further the prototype can be used on a machine without

a math coprocessor however performance is slower The prototype consists of two modules

linked together into one executable program The interface module contains the various

interactive subroutines and the main module contains the methodology of the model and the

logic of the heuristic search Both modules are coded in Fortran and compiled using Microsoft

Fortran 50 which provides many extensions to standard Fortran The source codes for the

interface module and the main module are listed in Appendix I and Appendix J

53

respectively

In addition to the above technical specifications the program requires the presence of

certain files in certain directories The program uses the distribution files mentioned in

Chapter II to read the unit costs and the empirical demand functions Each product has a

unique distribution file the name of which is the six digit product number followed by the

extension dst The text file containing the existing policies at each store must be located in

the same directory as the distribution files Finally the file tmsrbfon which contains the

font used by the interface must be located in the directory from which the program is executed

In its future versions the system should be more automated and more user-friendly

Rather than having the manager choose an upper bound for the search the algorithm should

incorporate a method of determining Sm a x For this to happen either a theoretical upper bound

must be obtained or a heuristic for terminating the search must be developed Also the amount

of time spent waiting for calculations to be performed can be reduced by reading the optimal

and current policies and their associated costs and customer service levels from a file or

database This solution file could be created any time prior to the consultation by running

the system in a batch mode The interface should allow the manager to change cost estimates

and service level values view results graphically and print results and summary documents

Finally the system should provide an option for automatically incrementing the product andor

store during a consultation This option would enable a manager to review quickly the

inventory policy of a product at each store

54

VI R E S U L T S AND SENSITIVITY ANALYSIS

A Comparison of Current Policies to Optimal Policies

In order to estimate the potential savings of implementing optimal policies the solutions

obtained in the Chapter IV are analyzed in more detail For each product the current policy

is evaluated and then compared to the optimal policy with respect to cost and customer service

level Finally the potential savings of the optimal policies are extrapolated to all products and

all stores

Of the 420 products examined each of 37 products has an existing policy that fails to

provide the desired customer service level of 975 However most of these policies just

fail to do so only 4 policies provide a customer service level of less than 90 Table 4

provides a list of these 37 products each with its current customer service level its current

annual cost and the additional cost of its optimal policy3 2 Two comments regarding Table

4 are in order First the current customer service level for product 203312 an amber

driveway marker is listed at a misleading 303 Examination of the demand distribution

reveals that this product was sold on only 1 of the 307 days during which sales data were

collected and on that day 13 of the markers were sold Clearly this data point constitutes

either an error or an unusual event the current policy of (34) does not allow for more than 4

items to be sold per review period Second the additional cost of the optimal policy for each

of 5 products is negative In other words for each of these products the current policy which

fails to provide the desired customer service level costs more than the optimal policy which

3 2 This is the difference between the expected annual cost of the optimal policy and the annual cost of the current policy It is the minimum additional cost required to increase the customer service level to at least 975

55

Table 4 Products with insufficient existing policies

Product Current CSL ()

Current Annual Cost ($)

Additional Cost of the Optimal Policy ($)

206932 974 204 -051

204570 973 222 -025

203053 973 248 009

206550 972 118 042

206547 972 105 035

206565 972 146 056

203702 972 340 157

206876 972 155 061

205752 972 330 153

205922 972 108 036

206857 972 180 073

205393 972 190 -003

201513 971 345 015

203405 970 255 037

203069 968 295 -048

203071 967 321 -001

205635 967 390 029

203914 966 460 160

204601 962 252 061

203326 960 144 014

202003 958 876 437

206516 958 112 029

206517 958 283 120

206350 958 333 147

205932 958 180 066

203936 952 598 294

56

Product Current CSL ()

Current Annual Cost ($)

Additional Cost of the Optimal Policy ($)

206932 974 204 -051

203939 948 413 193

203977 945 1216 1242

206551 945 132 077

205389 934 598 047

203590 927 1223 607

206567 919 122 070

203591 905 1228 834

205390 887 291 074

203318 833 136 084

206393 776 319 289

203312 303 237 511

provides at least the desired customer service level For the 36 products the total additional

cost of the optimal policies is $542033

Of the remaining 383 products most have an existing policy that provides a customer

service level well above 975 When the customer service levels are rounded to one decimal

each of 253 products has a customer service level of 1000 and each of 347 products has a

customer service level of at least 990 In contrast each of only 58 products has an optimal

policy that yields a customer service level of at least 99034 For each of 37 products the

3 3 The cost associated with product 203312 is not included in the total

3 4 Optimal policies do not have customer service levels of exactly 975 because the function is discrete For low-demand products a unit step in s or S results in a relatively large change in the customer service level

57

current policy is identical to the optimal policy

The extremely high level of service provided by most policies allows for substantial

savings in the cost of inventory With current policies the estimated annual cost of inventory

for all 420 products is $420743 whereas the expected annual cost of the optimal policies is

only $186884 This amounts to a savings of approximately $2340 or over 55 of the current

cost of inventory

In order to further assess the savings the difference between the costs of the current

policies and the costs of the optimal policies are examined in more detail For each of the 383

products whose existing policy provides a customer service level above 975 the difference

between the estimated annual cost of the existing policy and the expected annual cost of the

optimal policy is calculated The resulting histograms of the savings in (a) dollars and (b)

percentage of current cost are shown in Figure 6 The particulars for the current and optimal

policies for all 420 products are provided in Appendix K

As evident from Figure 6(a) the majority of the expected savings in absolute terms

comes from only six products In order to examine these products more closely the current

and optimal policies along with the associated costs and savings for each of the six products

are listed in Table 5 Each current policy is very different than its corresponding optimal

policy resulting in excessive inventory for each product In fact over half of the total expected

savings could be realized by implementing the optimal policy just for each of these six products

Interestingly five of the six products are sealed beam headlights

58

90

10 - 20 30 - 40 50 - 60 70 - 80 90- 100

Savings ()

(b)

Figure 6 The distribution of savings of optimal policies in (a) dollars and (b) percentage of current cost

59

Table 5 Products with the largest potential absolute savings

PRODUCT

CURRENT OPTIMAL SAVINGS

PRODUCT s S Cost ($)

C S L

()

s S Cost ($)

C S L () ($) ()

2036801 194 200 64014 1000 37 48 bull 9245 975 54769 856

2036772 94 100 23652 1000 18 22 3850 978 19802 837

2036723 78 90 23043 1000 19 23 4454 975 18589 807

2030944 145 175 19522 1000 34 45 3727 975 15795 809

2036425 48 60 10491 1000 13 17 2682 977 7809 744

2036556 68 80 7265 1000 14 22 1699 975 5566 766

1 H6054 rectangular sealed beam headlight 2 H4656 rectangular sealed beam headlight 3 6053 rectangular sealed beam headlight 4 9004 halogen bulb 5 4652 rectangular sealed beam headlight 6 6104 7 sealed beam headlight

Although Figure 6(a) seems to indicate that the majority of the current policies are close

in cost to the optimal policies this is very misleading For example suppose the optimal policy

for a product costs $1 and that the current policy costs $2 In absolute terms the optimal

policy saves only $1 but in relative terms the optimal policy saves 50 of the current cost

When savings are viewed in this manner as in Figure 6(b) it is clear that substantial savings

can be realized for the majority of the products In fact both the mode and the mean of the

savings are between 40 and 50 of the current costs

In order to extrapolate the above savings to obtain the potential savings for product

category 20 at all 21 stores the differences between stores must be considered During the

60

pilot study the current policy for each of several products at each size of store was evaluated

The study suggested that medium and large stores have a similar amount of excess

inventory and that jumbo and extra jumbo stores have much more excess inventory One

could reasonably expect that the additional excess inventory allows for higher savings to be

realized at the jumbo and extra jumbo stores Thus because 18 of the 21 stores are

medium or larger the expected 40-50 savings at store 6 can be viewed as a lower bound

on the potential savings to Canadian Tire Pacific Associates from implementing optimal policies

for product category 20 in all its stores

Finally in order to obtain the overall potential savings the expected savings from

product category 20 must be extrapolated to all product categories Ideally the proportion of

inventory comprising each product category and the relative savings that can be realized for

each category would be used to obtain a good estimate of the overall potential savings

However as this information is not known only a crude extrapolation is possible Thus the

analysis proceeds under the assumption that a 40-50 savings can be realized in each product

category35

Before an estimate of the total potential savings in absolute terms can be obtained the

proportion of the current inventory to which the savings apply must be determined Up to this

point the analysis has been concerned solely with reducing inventory at the stores However

it is inappropriate to perform further analysis without considering the entire inventory system

including the central warehouse Because the lead time for orders from Toronto is highly

variable and considerably greater than those at individual stores Canadian Tire Pacific

3 5 During the pilot study the current policies for a paint brush and an ironing board were evaluated and found to result in excess inventory similar to that of product category 20 Although this in no way justifies the assumption it does lend some support

61

Associates must keep a fairly high safety stock at the warehouse Without knowing the

proportion of inventory serving as the safety stock it is inappropriate to assume any reduction

in inventory at the warehouse is possible Therefore the relative savings of implementing

optimal policies is applied only to the inventory held in the stores36 Of the estimated $28

million in inventory approximately half is held in the stores therefore the 40-50 savings

translates to a savings of approximately $55 to $7 million per year

B Sensitivity Analysis on the Ordering Cost and Holding Rate

The above analysis was based on estimates of the ordering cost and the annual unit

holding rate37 provided by management As described in Chapter II these estimates

incorporate a number of costs that are difficult to quantify or that are speculative Therefore

there is some question as to the accuracy of the estimates In this section errors of +$0015

in the ordering cost and +5 in the holding cost are assumed in order to determine the

sensitivity of the model to errors in these variables38

The analysis proceeds as follows Based on an ordering cost of $0080 and a holding

rate of 30 the optimal policies from the previous section were obtained39 Then assuming

3 6 Of course this assumes that the excess inventory currently held in the stores is not required as part of the safety stock Based on the large amount of inventory at the warehouse this assumption seems reasonable

3 7 Hereafter simply referred to as the holding rate

3 8 An error of +$0015 in the ordering cost implies an actual ordering cost of $0070 a relative error of approximately 21 and an error of -$0015 implies an actual ordering cost of $0100 a relative error of 15 An error of +5 in the holding rate implies an actual rate of 25 a relative error of 20 and an error of -5 implies an actual holding rate of 35 a relative error of approximately 17

3 9 These policies are referred to as optimal policies for the duration of this chapter

62

the actual ordering cost andor holding rate are different from managements estimates the

heuristic is used to obtain the true optimal policies Finally the optimal policies are

evaluated based on the actual ordering cost and holding rate and then compared to the true

optimal policies For each possible combination of ordering cost and holding rate three

numbers are recorded the number of products with an optimal policy that differs from its

true optimal policy the total cost of implementing the optimal policies given the actual

ordering cost and holding rate and the difference between the cost of implementing the

optimal policies and the cost of implementing the true optimal policies The results of the

analysis are summarized in Table 6

Table 6 A comparison of optimal policies to true optimal policies

Actual Ordering Cost

$0070 $0085 $0100

25 5 products $155439

$002

51 products $160590

$092

88 products $165464

$449

Actual Holding Rate

30 54 products

$181470 $182

48 products $192026

$088

35 79 products

$207270 $574

48 products $212940

$136

2 products $218312

$000

4 0 The table should be read as follows If the actual ordering cost is $0070 and the actual holding rate is 25 then each of 5 products has an optimal policy that is different from its true optimal policy The total cost of implementing the optimal policies is $155439 which is $002 more than implementing the true optimal policies

63

The model appears to be quite insensitive to relative errors in the order of 20 Under

the worst case scenario when one of the variables is over-estimated and the other is undershy

estimated the optimal policies result in a total cost of inventory which is only approximately

3 higher than the cost of the true optimal policies Further because demand is not changed

by errors in these estimates each of the optimal policies still provides a customer service

level of at least 975 under all of the scenarios Finally although the cost of the optimal

policies varies significantly for the different scenarios the relative savings remain virtually

constant Table 7 lists the cost of existing policies and in parenthesis the relative savings of

the optimal policies for each of the different combinations of ordering cost and holding rate

The cost of the existing policies varies in the same manner as that of the optimal policies the

relative savings for each of the scenarios remains around 55 Therefore even in the presence

of such errors in the ordering cost and holding rate the projected savings of implementing the

optimal policies would be unchanged

Table 7 The cost of existing policies and the relative savings of the optimal policies under various scenarios

Actual Ordering Cost

$0070 $0085 $0100

25 $350153 (56)

$358464 (55)

$366764 (55)

Actual Holding Rate

30 $412438 (56)

$429048 (55)

35 $474725 (56)

$483026 (56)

$491320 (56)

64

C Sensitivity Analysis on the Demand Probability Mass Function

In this section the sensitivity of the model to slight changes in the demand probability

mass function is investigated As described in Chapter II the demand function used in the

model is based solely on observed sales however this empirical function is only one possible

instance of the underlying true demand function For example had the data been collected

over a period both starting and finishing one month earlier many of the empirical functions

would be slightly different Consequently depending upon the degree to which the empirical

functions vary and the sensitivity of the model to these variations the optimal policies from

the model might be inaccurate For example recall the case of the amber driveway marker

only thirteen markers were sold and all were sold on the same day Because the empirical

demand function is used the model calculates an optimal policy of (1213) clearly this

policy is impractical Although this example is extreme it does indicate the danger of using

unrealistic empirical demand functions41

As discussed in Chapter II three classes of products were identified namely low

medium and high-demand products For this analysis the same method of classification

is used Two products are selected from each of these groups and two new distribution files

are created for each of the six products For each product one of the files contains two

additional days of data with each day having a demand equal to the maximum observed demand

for that product In the second file only one additional day of data is added but the demand

on that day is two more than the maximum observed demand The situations that could give

4 1 Certainly a rigorous analysis to determine both the extent of variation in the empirical demand functions and the effect of using the empirical functions needs to be performed However such an analysis is beyond the scope of this thesis only a brief analysis that might suggest the degree to which the model is sensitive is performed here

65

rise to such changes in the demand data are termed Scenario 1 and Scenario 2 respectively42

The analysis proceeds in a manner similar to the one in the previous section Given the

demand that was originally observed the optimal policies were obtained Now assuming the

true demand function corresponds to the demand under Scenario 1 the true optimal policy

for each product is calculated Then each optimal policy is re-evaluated using the true

demand function and compared to the true optimal policy The analysis is then repeated

assuming the true demand function corresponds to the demand under Scenario 2 Table 8

lists the results for each product and scenario

Table 8 True optimal and optimal policies for each demand scenario

Product 202101 (low demand)

Scenario s s Cost ($) C S L ()

1 True optimal 1 2 470 993

Optimal 1 2 470 993

2 True optimal 2 3 663 976

Optimal 1 2 461 875

Product 203507 (low demand)

Scenario s S Cost ($) C S L ()

1 True optimal 0 3 334 981

Optimal 0 2 258 972

2 True optimal 2 3 508 980

Optimal 0 2 273 702

4 2 To be clear the scenarios are defined as follows Scenario 1 The addition of two days each with a demand of max Scenario 2 The addition of one day with a demand of max+2

where max is the maximum observed daily demand for each product in the sample period

66

Product 201032 (medium demand)

Scenario s S Cost ($) C S L ()

1 True optimal 6 12 1377 976

Optimal 6 11 1338 973

2 True optimal 6 13 1426 977

Optimal 6 11 1338 970

Product 203039 (medium demand)

Scenario s S Cost ($) CSL ()

1 True optimal 5 19 256 975

Optimal 5 16 251 969

2 True optimal 5 20 258 976

Optimal 5 16 249 968

Product 203677 (high demand)

Scenario s S Cost ($) CSL ()

1 True optimal 18 23 3981 977

Optimal 18 22 3817 972

2 True optimal 18 23 3995 978

Optimal 18 22 3831 973

Product 203680 (high demand)

Scenario s S Cost ($) CSL ()

1 True optimal 38 49 9498 976

Optimal 37 48 9175 971

2 True optimal 37 49 9484 975

Optimal 37 48 9206 972

67

The model is most sensitive to changes or errors in the demand functions of low-

demand products When only one additional day of demand for max+2 units is added to the

data the optimal policy for each of the low-demand products results in a very low customer

service level In fact for one of the products the customer service level drops to almost 70

The increase in demand does lower the customer service level for each of the other the

products but that decrease is slight

The above analysis indicates the need for some kind of smoothing or modelling of the

demand data especially for low-demand products The inherent variation between two

instances of a true demand function could very well result in more pronounced changes than

those resulting from either of the above scenarios Therefore it is likely that many optimal

policies especially those for low-demand products would actually provide a much lower or

much higher customer service level than expected However by smoothing the data the

variation between instances of a true demand function can be reduced and the desired

customer service levels can be achieved

For example suppose that the demand for a product is stationary that the average

demand during the previous period was unusually low and that the optimal policy for this

period is based on the demand during the previous period Because the demand is stationary

the probability that the demand during this period is higher than that during the past period is

more than 50 In other words the probability that the demand during this period is higher

than that for which the optimal policy provides the desired expected customer service level

is greater than 05 Thus expected customer service level for this period is actually less than

that specified Now suppose instead that the optimal policy is based on smoothed data which

reflects very closely the expected demand The probability that the demand during this period

is higher than that for which the optimal policy provides the desired expected customer service

68

level is virtually 05 Thus as a result of smoothing the data the specified customer service

level should be expected

69

VII C O N C L U S I O N

In this thesis we developed a single-product single-location model of the inventory

system at Canadian Tire Pacific Associates Specifically we formulated a Markov chain model

to determine the long-run average cost of inventory for fixed values of s and S and then

developed a heuristic search to find near optimal policies quickly Also as a first step toward

developing an integrated interactive inventory control system we created a prototype of the

interface

According to the analysis performed here Canadian Tire Pacific Associates is currently

holding an excessively large inventory in its stores This study finds that Canadian Tire Pacific

Associates could reduce the cost of inventory for product category 20 at store 6 by

approximately 50 and still maintain customer service levels of at least 975 Further the

mean and modal values of savings for these products are between 40 and 50 Given the

similar excess inventory at the other stores and in all product categories we believe that savings

in this range are possible for all products and stores We estimate that the application of

inventory control as described in this thesis would result in annual savings of $55 to $7

million

Given the large potential savings to Canadian Tire Pacific Associates we recommend

that management proceed with the next phase of the project Specifically we recommend that

the model algorithm and interface be refined further that a forecasting model be developed

and that a data collection and storage scheme be designed

In order to reflect operational changes at Canadian Tire Pacific Associates subsequent

to the formulation of the model modifications must be made to allow for the lead time to

exceed the review period Several of the stores now have three picks per week and as a

70

result orders placed for these stores at one decision epoch do not arrive until after the next

decision epoch Because this violates one of the models assumptions the model must be

modified accordingly Note however that with three picks per week and a review period

of four days orders from only the previous decision epoch are ever outstanding at the current

decision epoch

Also the model should be refined to reflect the fact that the review period is not fixed

over the planning period In its present form the model assumes a fixed review period

however inventory at stores with two picks per week is actually reviewed on a four day -

three day cycle and inventory at stores with three picks per week is reviewed on a three day

- two day - two day cycle Rather than obtaining the steady state costs for fixed review period

the model should calculate the steady state cost for a period of one week during which multiple

replenishments occur Although the solutions are not expected to differ significantly as a result

of the change the model would reflect the actual process more accurately

Further work is also required on the search technique for locating near optimal

policies Although the heuristic search described in this thesis appears to work well the

decision maker must provide an upper bound on S prior to the search In practice this upper

bound is generally not known in advance Thus if the heuristic is to be used either a tight

theoretical bound must be found or a termination heuristic must be developed Alternatively

a new heuristic search could be developed based on additional or different assumptions

regarding the properties of the feasible space For example although some regions of non-

convexity were found for some products much of the feasible space appeared to be convex

A heuristic that assumes a convex feasible space would not require an upper bound on S and

might perform very well However before such a heuristic is implemented further work is

necessary to determine the degree of non-convexity for a variety of demand functions and

71

ordering and holding costs

In addition to making the above modifications we recommend that Canadian Tire Pacific

Associates begin work on the remaining two components of the inventory control system First

in order to allow for products with seasonal demand the system requires a forecasting model

to provide accurate estimates of demand for the coming period Second the system needs a

data collection and storage module that systematically collects sales data for each active

product and for each store and then updates the corresponding demand distributions Because

approximately 25 of demand is expected to result in lost sales under the optimal policies the

sales data will have to be adjusted accordingly in the future Also because the model appears

to be sensitive to slight changes in the demand distribution for low-demand products the

module should incorporate a technique for smoothing the empirical distributions for these

products Further work should be undertaken to determine the extent of variation in the

empirical distributions and the most appropriate smoothing technique However because of

the large amount of data required to forecast seasonal demand and because of the current lack

of historical data we recommend that management proceed immediately with the design of a

data collection scheme

Finally two additional issues that are beyond the scope of the next phase should be

addressed at some point First a monitoring system will be required once inventory control

is implemented in the stores The monitoring system will ensure that optimal policies perform

as expected and will indicated whether or not demand is adversely affected by the lower levels

of in-store inventory Second given the degree to which in-store inventory can be reduced

inventory control in the warehouse should be investigated Because approximately half of the

existing inventory is held in the warehouse the additional savings from reducing system-wide

inventory could be great

72

Afterword

Just prior to the completion of the work described in this thesis Mr Don Graham

principle owner of Canadian Tire Pacific Associates sold his shares in the company Shortly

thereafter the results of this thesis were presented to the new management group

Unfortunately the new management did not consider development of an inventory control

system to be a priority Nonetheless the prototype was delivered to Canadian Tire Pacific

Associates for possible in-house testing however at the time of this writing - almost one year

after the meeting - Canadian Tire Pacific Associates has not decided to continue with the

project

73

Bibliography

Arrow K A T E Harris and J Marschak (1951) Optimal inventory policy Econometrica 19 250-272

Day G (1992) In a speech to the Empire Club of Canada Toronto quoted in When the JIT fits The Globe and Mail October 20 1992 B24 and in Puterman (1994)

Dvoretzky A J Kiefer and J Wolfowitz (1952a) The inventory problem I Case of known distributions of demand Econometrica 20 187-222

Dvoretzky A J Kiefer and J Wolfowitz (1952b) The inventory problem II Case of unknown distributions of demand Econometrica 20 450-466

Federgruen A (1993) Centralized Planning Models in SC Graves A H G Rinnooy Kan and P H Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Federgruen A H Groenevelt and H C Tijms (1984) Coordinated replenishments in a multi-item inventory system with compound Poisson demands and constant lead times Management Sci 30 344-357

Lee H L and S Nahmias (1993) Single-Product Single-Location Models in SC Graves A H G Rinnooy Kan and PH Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Muckstadt J A and R O Roundy (1993) Analysis of Multistage Production Systems in SC Graves A H G Rinnooy Kan and PH Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Porteus E L (1985) Numerical Comparisons of Inventory Policies for Periodic Review Systems Oper Res 33 134-152

Puterman M L (1994) Markov Decision Processes Discrete Stochastic Dynamic Programming Wiley New York

Silver E A and R Peterson (1985) Decision Systems for Inventory Management and Production Planning 2nd edition Wiley New York

Taylor H M and S Karlin (1994) An Introduction to Stochastic Modeling revised edition Academic Press San Diego

74

Tijms H C and H Groenevelt (1984) Simple approximations for the reorder point in periodic and continuous review (sS) inventory systems with service level constraints European J Oper Res 17 175-190

75

Appendix A

A Portion of the Sales File for Product 200001 a 30-amp Inline Fuse

200001 283 1190 0 0 2 0 0 1 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 200001 284 1190 2 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 200001 285 1190 0 0 4 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 286 1190 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 200001 287 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 200001 288 1190 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1 0 2 200001 289 1190 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 290 1190 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 200001 291 1190 0 0 2 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 200001 292 1190 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 200001 294 1190 0 0 0 1 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 1 200001 295 1190 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 200001 296 1190 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 200001 297 1190 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 200001 298 1190 0 0 1 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 200001 299 1190 0 0 1 0 0 0 1 0 1 0 0 2 0 0 0 0 0 2 1 1 0 200001 300 1190 0 0 1 0 0 0 0 1 0 0 0 0 0 2 2 0 0 0 0 3 0 200001 301 1190 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 200001 303 1190 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 200001 304 1190 1 0 0 0 0 0 0 0 1 0 0 2 0 0 0 0 0 0 0 0 0 200001 305 1190 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 200001 306 1190 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 2 0 0 1 0 200001 308 1190 0 0 0 1 0 1 0 0 0 2 0 1 0 0 0 0 1 1 1 0 0 200001 309 1190 1 3 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 200001 310 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 200001 311 1190 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 200001 312 1190 0 1 1 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 200001 313 1190 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 2 0 200001 314 1190 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 200001 315 1190 0 0 2 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 200001 316 1190 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 200001 317 1190 2 0 0 1 0 0 0 0 0 0 5 1 0 0 0 1 0 0 0 0 0 200001 318 1190 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 1 200001 319 1190 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 200001 320 1190 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 200001 321 1190 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 1 200001 322 1190 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 323 1190 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 200001 324 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 0 1 0 0 2 1 1 0 200001 325 1190 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3 200001 326 1190 0 0 1 0 1 0 0 0 2 0 0 1 0 0 0 1 0 1 0 0 1 200001 328 1190 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 200001 329 1190 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 200001 330 1190 0 0 1 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 200001 331 1190 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 200001 332 1190 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 200001 333 1190 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 200001 334 1190 0 0 2 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 200001 335 1190 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 200001 336 1190 0 0 0 1 0 0 3 0 0 0 0 0 2 0 0 0 0 0 0 0 0

(253 subsequent lines omitted)

76

Appendix B

Sample Distribution Files

a) A low-demand product - an automobile back-up alarm

PRODUCT NUMBER 202101 PRICE $ 684

DMD

3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

N 2 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 2 2 0 1

1 5 3 8 12 5 7 11 6 11 5 4 11 10 19 4 2 5 14 5 6 4

0 302 304 299 295 302 300 296 301 296 302 303 295 296 288 303 305 301 290 300 301 302

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

b) A medium-demand product - a 168 bulb

PRODUCT NUMBER 203039 PRICE $ 51

DMD

16 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 1 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0

9 0 0 2 5 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0

8 0 0 2 5 0 0 2 0 1 1 3 0 1 3 0 0 1 1 0 0 1

7 2 1 3 1 0 0 1 0 2 3 1 1 4 1 1 0 2 1 1 1 1

6 4 7 1 10 3 1 3 2 1 3 5 2 7 4 1 0 1 1 3 3 4

5 1 7 7 17 0 2 11 1 5 4 1 2 6 9 6 2 6 9 4 4 1

4 5 10 10 16 5 5 19 10 13 8 11 9 13 10 12 7 5 15 5 8 3

3 10 12 13 27 4 6 24 11 10 13 14 9 23 14 18 8 14 18 14 12 14

2 14 30 36 43 16 16 43 32 38 39 39 34 47 36 41 21 19 39 20 34 24

1 30 56 68 56 37 30 70 48 67 61 44 75 62 63 48 28 52 61 63 53 47

0 240 184 165 126 240 247 133 202 169 175 189 173 143 165 180 240 206 161 197 192 212

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

77

c) A high-demand product - a H4651 rectangular sealed beam headlight

PRODUCT NUMBER 203677 PRICE $ 857

DMD

14 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

13 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 2 3 0 0 1 0 0 0 0 1 1 1 0 0 0 0 2 0 0

11 0 1 3 1 0 0 4 1 0 0 0 0 0 0 0 0 1 0 0 0 1

10 0 0 4 4 1 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 1

9 0 3 14 11 0 1 7 1 0 2 0 8 0 1 0 0 0 2 1 0 0

8 0 4 12 16 1 0 18 4 3 5 0 2 1 4 2 1 1 0 0 1 3

7 1 2 12 13 1 6 19 3 5 9 0 11 2 5 4 0 2 3 3 1 4

6 2 9 22 21 6 10 29 12 10 11 4 24 3 7 10 4 7 11 6 3 14

5 7 13 35 33 8 16 29 19 18 8 3 26 15 26 15 12 9 14 13 11 19

4 19 35 34 43 32 30 41 31 32 47 11 42 20 33 30 31 20 31 22 21 31

3 32 63 49 54 42 60 51 40 51 35 33 55 39 51 39 38 39 57 58 40 54

2 46 59 56 55 73 63 61 65 63 61 63 54 68 85 65 53 72 69 70 77 63

1 109 79 42 40 80 64 32 80 74 77 99 55 97 57 83 93 80 75 81 97 73

0 91 39 21 13 63 57 11 51 51 52 94 29 61 35 59 75 76 45 51 56 44

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

78

Appendix C

A Proof that the Limiting Probability Distribution is Independent of the Initial State

In this appendix we show that provided 0 lt PrDT = 0 lt 1 the Markov chain

corresponding to an inventory model with a (sS) policy has a limiting probability distribution

7T where 0 lt TTJ lt 1 for y = 0 1 S that is independent of the initial state Further the

limiting distribution is the unique nonnegative solution of the equations

Tt P = TI (AD

s and J gt = 1 (A 2)

i=0

where P is the (S + l) x (S + l) transition probability matrix whose first n rows (and columns)

correspond to the n recurrent states in the chain and whose remaining rows (and columns) if

any correspond to transient states43 Prior to the proof several preliminary results are

obtained

RESULT 1 The Markov chain has exactly one positive recurrent class and perhaps some

transient states

First note that in the absence of any absorbing states a Markov chain with a discrete

finite state space must have at least one positive recurrent class containing at least two states

Now consider the Markov chain for an inventory model with a (sS) policy

a) State i gt s cannot be absorbing because there is a positive probability of reaching

j lt i (Since PrDT gt 0 gt 0)

4 3 Note that since the rows and columns of P are reordered the elements of it must be reordered accordingly

79

b) State lt s cannot be absorbing because by the rules of the ordering policy and

the assumption that PrDT = 0 gt 0 there is a positive probability reaching S

Therefore the chain has no absorbing states and must have at least one positive recurrent class

Next consider the subset of states A = i | i lt s Because the state space is finite

and PrDT gt 0 gt 0 the process must return to at least one state in A in other words at least

one state in A must be recurrent Let k G A be a member of a recurrent class dt Because

of the ordering policy and the assumption that PrDT = 0 gt 0 k -raquo S Also by definition

there must be a positive probability of returning to k from S S -gt k Therefore S G 3w

Further because S is reachable from all states in A i G A is either transient with i -raquo dt or

recurrent with pound iR

Now consider the subset of states B = | j gt s Because PrDT gt 0 gt 0 the

chain will eventually reach a state i G A If i is transient then j must be transient also with

y -gt 3 if i G 9 then j is either transient with j -gt dt or recurrent with j E dt Thus the

Markov chain has exactly one recurrent class and perhaps some transient states

RESULT 2 The limiting probability distribution ir for the Markov chain corresponding to the

nxn submatrix Pm = [Ptj] i j G dt is independent of the initial state and is the

unique nonnegative solution of the equations

TC-Prr = re (A3)

and J2 K = 1 ( A 4 )

jt=i

where the rows and columns of are numbered from 1 to n rather than

Since if i j i and j must be in the same equivalence class

80

according to their associated states

First note that both the original Markov chain and the one corresponding to the

submatrix P R R are aperiodic because PrDT = 0 gt 0 Therefore given that 9 is a positive

recurrent aperiodic class the limiting distribution ir is uniquely determined by (A3) and (A4)

(Taylor and Karlin 1994 Theorem 44 p213) Further because there are no absorbing states

or other recurrent classes in the original chain the process must eventually enter the recurrent

class 9 regardless of the initial state Therefore in the limit the probability of being in state

i G 9 is independent of the initial state

RESULT 3 The limiting probability distribution for the transient states is the null vector

independent of the initial state

This result follows directly from the definition of a transient state

PROOF Suppose the states are reordered such that the first n rows (and columns) of P

correspond to the n recurrent states of the Markov chain Then P can be partitioned into four

submatrices

PRK - the n x n submatrix [Py] i j E 9

P T T - the (S + 1-I) x (S + l-n) submatrix [Py] i j g 9

P T R - the (S + l-n) x n submatrix [Py] i pound 9 j G 9

P R T - the n x (S + l-n) submatrix [Py] i G 9 j pound 9t By definition this is a matrix

of zeros

(Al) can then be rewritten as

81

(ltR TtT) P P L TR TT

I-P RR 0

p I-P TR 1 TT

(TtR TCT)

(0- 0T)

where irR denotes the limiting probability distribution for the n recurrent states and irT denotes

the distribution for the remaining S + l-n transient states Solving for TT t gives

TCr-0 + itT-(I-PTI) = 0T

itT-(I-PJJ) = 0 r

i c r = (I-P^-Oj

Since (I - PTT)1 always exists (Puterman 1994 Proposition A3 p 593) irT = 0X which is

in agreement with Result 3 Now solving for 7rR yields

(A2) can be rewritten as

bullRPRR 1 1R (A5)

(A6) S + l

k + L k =1

k=l k=n+

k=i

where ir contains the appropriately reordered steady state probabilities Since (A5) and (A6)

are equivalent to (A3) and (A4) respectively Result 2 holds Therefore the Markov chain

corresponding to an inventory model with a (sS) policy has a limiting distribution that is

independent of the initial state and is the unique nonnegative solution to (Al) and (A2)

82

Appendix D

A Measure for the Steady State Customer Service Level

In this appendix we show that the appropriate measure for the steady state customer

service level is given by

_ E[Demand satisfied during RP] E[Total demand during RP]

Consider the following two measures of the steady state customer service level

Measure 1 Suppose the customer service level is calculated only after n review periods have

passed Denoting DS as the demand satisfied in review period i and DT as the total demand

in period i the customer service level after n periods CSL(n) is

DS + DS0 + + DSbdquo CSLn) = - -

DT + DT + + DT

Dividing both the numerator and denominator by n taking the limit as n -raquo oo and applying

the Law of Large Numbers yields

CSL = lim CSL(n) n-degdeg

- [DS + DS + + DS] = l i m ^ J 1 1

raquo~ - [DT + DT + + DT]

E[DS]

E[DT]

83

Measure 2 Suppose the customer service level is calculated for each of n review periods and

then averaged Thus the customer service level after n periods CSL(n) is given by

CSL(n) = -DS DS

DT DT DT_

Taking the limit as n -gt oo yields

CSL = lim CSL(n)

lim mdash DSl DS2

DT DT DT_

DS

DT

Now consider the scenario in which there is one item on the shelf to begin each review

period and in which demand is stochastic with PrDT = 2 = Vi and PrD x = 100 = Vi

According to Measure 1 the customer service level is

CSL - = ^ W + ^ W = plusmn bdquo 002 E[DT) 1 (2) + 1(100) 51

while according to Measure 2 the customer service level is

CSL = E DS DT

1(1) + i(-Ls

22) + 2 100

51 200

026

Clearly from managements point of view the first measure is more indicative of the

actual level of service being provided The problem with the second measure is that equal

84

weight is given to each ratio regardless of the magnitude of the numbers For example with

respect to Measure 2 the events LSLR = 12 and LSLR = 50100 are identical however

with respect to lost sales the second event is much worse than the first Therefore the proper

measure for the steady state customer service level is Measure 1

85

Appendix E

The Calculation of the Conditional Expected Demand Not Satisfied During a Period of T Consecutive Days

Denoting D x as the demand during a period of T consecutive days and j as the inventory

on hand at the beginning of the period the conditional expected demand not satisfied during the

period E[(DT-y)+] is calculated as follows

E[(DT-j)+] = pound ltk-j)Pr[DT = k k=j+l

= pound (k-j)PrDT = k] + J2(k-j)PrDT = k] - pound (fc-)VJgtr = fc k=j+l k=0 k=0

= J2(k-j)Pr[DT=k -J2(k-j)PrDT = k] k=0 k=0

= YkPrDT=k) - J2jPrDT=k - Yk-j)PrDT=k) Jfc=0 k=Q k=0

= Ti - j - Y(k-j)PrDT=k 7-1

= Tx - j + YltJ-k)PrDT = k jt=0

where fx is the mean daily demand

86

Appendix F

Justification of the Updating Technique for the New Policy (s+mS)

Let the probability transition matrix corresponding to the policy (sS) be denoted by the

(S + l)x(S + l) matrix P with elements py Define s = s+m where m is an integer satisfying

the condition 0 lt s+m lt S Let the probability transition matrix of the new policy (sS)

be denoted by the (S + 1)X(S + 1) matrix 7gt with elements py Using the equations developed

in Chapter III the elements of P are found as follows

CASE 1 The lead time is less than the review period

A m gt 0

For 0 lt i lt s

( i-i pound [PrDL=kPrDT_LgtS-k]] + PrDLziPrDT_LplusmnS-i k=0

i-l 53 [PrDL=kPrDT_L = S-j-k]] + PrDLziPr[DT_L = S-j-i) k=0 s-j

52 [Pr[DL = kPrDT_L = S-j-kn k=0

ifj = 6

ifOltjz S-i

if S-i ltj ltS

a) 0 lt lt s

Pij

= lt Pij

Pij

= pij

b) s lt i lt s This does not simplify but it is identical to (2)

87

For s lt i lt S

PrDTgt i

p = lt PrDT = i-j

0

Pfj ifj = 0

= lt Pij ifOltjlti

Pij if i ltj plusmn S

= pij

B m lt 0

ifj = 0

if 0 ltj lt i

if i ltj plusmn S

For 0 lt i lt s

i-l 53 [PrDL=kPrDT_LplusmnS-k]] + PrDLgtiPrDT_LplusmnS-i

i-l E [PrZgti = fciVZ)7W=S--fc] +PrD tgtiP-D 7 t = S-7-

p y laquoy = o

P i j ifOltjS-i

vPij ifS-iltjS

ifj = 0

ifOltjltS-i

if S-i ltjltS

For s lt i lt S

fPrJgtrgt- 17=0 = PrDT = i-j] ifOltjlti

0 ifiltjplusmnS

88

a) s lt lt s This does not simplify but it is identical to (3)

b) s lt i lt S

PrDTgti

= bull PrDT = i-j]

0

17=0

if0ltjplusmni

ifiltjS

= Pt

ifj = 0

if0ltjlti

ifiltjplusmnS

= Pi

CASE 2 The lead time is equals the review period

A m gt 0

For 0 lt i lt s

0 if0ltjlt S-i

PrDTgti ifj = S-i

PrDT=S-j] ifS-iltjltS

a) 0 lt i lt s

= bull y Pij

Pij

ifOplusmnjltS-i

ifj = S-i

ifS-iltjS

= Pij

b) s lt i lt s This does not simplify but it is identical to (4)

89

For s lt i lt S

PrDTgti)

Pr[DT=i-j

0

laquof 7=0

ifOltjplusmni

if i ltj lt S

Pa

Pij

if 7 = 0

if 0 lty lt i

ifiltjltS

B w lt 0

For 0 lt t lt s

ro PrJrgt PrDT = S-j

ifOzjltS-i

ifj = S-i

if S-i ltj lt S

Pij

ifOzjltS-i

ifj = S-i

if S-i ltj ltS

= Pi

For s lt i lt S

PrDTgti

= PrDT = i-j

0

i 7 = o ifOltjplusmni

ifiltjltS

a) s lt i lt s This does not simplify but it is identical to (3)

90

PrDTzi ifj = 0

PrDT = i-j] ifOltjplusmni

0 ifiltjltS

Py ifj = 0

ptj ifOltjplusmni

Pij if iltj plusmnS

Appendix G

Justification of the Updating Technique for the New Policy (sS+m)

Let the probability transition matrix corresponding to the policy (sS) be denoted by the

(S + l)x(S + l) matrix P with elements p(gt Define S = S+m where m is any non-negative

integer Let the probability transition matrix of the new policy (sS) be denoted by the

(S+m+l)x(S+m + l) matrix IP with elements ptj Then using the equations developed in

Chapter III the elements of 7 are found as follows

CASE 1 The lead time is less than the review period

For 0 lt i lt s

pound [PrDL=kPrDT_LplusmnS-k] + PrDLi)PrDT^Sgt- k=0

i-l pound [Pr[DL = kPrDT_L = S-j-k- +Pr[DLiPrDT_L = S-j-i Jt=0

s-i Y [Pr[DL=kPrDT_L = S-j-k] k=0

i - l

ifj=0

ifOltjltS-i

ifS-iltjltS

pound [PrDL = kPrDT_LS-k + Pr[DLi)PrDT_LplusmnS-i) if j = 0 fc=0

i - l = pound [PrDL=kPrDT_L = (S+m)-j-k] + PrDLgtiPr[DT_L = (S+m)-j-i ifOltj lt (J5+m)-i

if (S+m)-iltj lt(S+m)

k=0

i - l

pound (Sm)-j

pound [PrDL = kPrDT_L = (S+m)-j-k]] k=0

92

I-l

53 [PrDL = kPrDT_LgtS-k] + PrDLziPrDT_LS-i ifj = 0 k=0

i-l

Y[PrDL = kPrDT_L = S-(j-m)-k] + Pr[DLziPrDT_L = S-(j-m)-i] ifOltjzS+m-i k=0 S-(j-m)

53 [PrDL = kPrDT_L = S-(j-m)-k]] if S+m-i ltj lt S+m

i - l

53 [PrDL = kPrDT_LzS-k] + PrDLziPrDT_LzS-i] ifj = 0 =0

i - l

13 [Pr[DL = kPrDT_L = S-(j-m)-k] + Pr[DLziPr[DT_L = S-(j-m)-i ifOltjz S+m-i k=o and 0 z j-m

Pi(j-m)

Pi(j-m)

if 0 ltj lt S+m-i and 0ltj-mltS-i

if S+m-i ltj lt S+m and S-i lt j-m zS

i - l

Y [PrDL = k)PrDT_LzS-k] + PrDLi)PrDT_LplusmnS-i ifj = 0

k=0 i - l

53 [PrDL = kPrDT_L = S-(j-m)-k] + PrDLziPrDT_L = S-(j-m)-i ifOltjzm k=0

Piij-m)

Piij-m)

if mltj lt S+m-i

if S+m-i ltj lt S+m

i - l

YlPrDL = kPr[DT_L^S-k + PrDLi)PrDT_LS-i) ifj = 0

i - l

53 [PrDL = kPrDT_L = S-j-k] + PrDLgtiPrDT_L = S-j-i ifOltj i m k=0

Pi if mltj ltS

which for 0 lt j lt m is identical to (2)

93

For s lt i lt S

Pu =

PrDTgti

PrDT = i-j]

0

if0ltjlti

ifiltjltS

a) s lt i lt S

PrDTgti 7=0

PrDT = i -j) if0ltjlti

0 if i ltj lt S

0 ifSltjltSgt

ifj=0

if 0 ltj lt i

if i ltj plusmn S

0 ifSltjltS

ifOltjltS

0 ifSltjltSgt

b) S lt i lt S This does not simplify but it is identical to (3)

CASE 2 The lead time equals the review period

For 0 lt i lt s

0 ifO lt jltS-i

PrDTgti ifj=S-i

PrDT=S-j ifS-iltjltS

94

0 ifOplusmnjlt(S+m)-i

PrDT ifj = (S+m)-i

PrDT = (S+m) -j) if (S+m)-i ltj ltL (S+m)

0 if 0 lt jltS+m-i

PrDTzi] if j = S+m-i

PrDT = S-(j-m) if S+m-i ltj lt S+m

0 if 0 ltj lt S+m-i and 0 ^ j-m

Pi(j-m) if 0ltjlt S+m-i and 0ltj-mltS-i bull

Pi(j-m) if j = S+m-i and j-m = S-i

Pi(j-m) if S+m-i ltj z S+m and S-i lt j-m ^S

0 if 0 zj ltm

Pi(j-m) if mlt jlt S+m-i lt

Piij-m) if j = S+m-i

if S+m-i ltj lt S+m

1deg if 0 ltj ltm

[PiV-m) ifmltjltS

which for 0 lt i lt s is identical to (3)

For s lt lt S

PrDTzi

= PrDT = i-j)

0

ifj = 0

if 0 ltj ltL i

ifKjzS1

95

a) s lt i lt S

PrDTzi

PrDT = i-j]

0

0

17=0

if0ltjplusmni

ifiltjS

ifSltjltSgt

Pij 7=0

Pij if0ltjii

Pij if i ltj ltL S

0 ifSltjltS

Pij ifOzjiS

0 ifSltjltS

b) S lt i lt S This does not simplify but it is identical to (3)

96

Appendix H

A Hypothetical Consultation

In this appendix the interface during a hypothetical consultation is depicted The figures

accurately represent the various screens that are presented to the manager with the exception

that all lines and text are really white on a blue background During the consultation described

below only the choices made by the manager are hypothetical all the results are real

The title screen as depicted in Figure A - l appears initially upon execution of the

program In order to proceed to the main menu the manager presses the enter key

O I C S

Optimat Inventory Control System

For the exclusive use of

Canadian Tire Pacific Associates

Developed at The Faculty of Commerce

and Business Administration The University of British Columbia

Brian Kapalka Kaan Katircioghi Marty Puterman

Press ltEntergt to continue

Figure A - l The title screen

97

Main Menu 1 Select data directory

ctircdatadst

2 Select product and store combination Product 200001

3 Select scenario Review Period 4 Lead Time 3

4 Select search parameter Smax 25

5 Execute program

6 Exit to DOS

Please enter the desired option 5

Figure A-2 The main menu

Figure A-2 depicts the main menu of the program it is from here that the manager

enters all the review parameters If the manager wishes to change one of the parameters from

its default value he or she selects the appropriate number and enters the correct information

in the window presented The new settings will then be shown in place of the default values

In this example the relevant data files are located in the default directory ctiredatadst and

the manager has selected the productstore combination of product 200001 and store 6 Further

the manager has selected a review scenario of a 4-day review period and a 3-day lead time and

an upper search limit of 25 Finally in order to proceed with the analysis the manager

executes the main module of the program by selecting option 5

98

Abstract

Canadian Tire Pacific Associates owns and operates 21 retail stores in the lower

mainland of British Columbia and a central warehouse in Burnaby In this thesis we formulate

a single-product single-location model of its inventory system as a first step in developing an

integrated interactive inventory control system Specifically we formulate a Markov chain

model for a periodic review system with a deterministic lead time and lost sales The model

utilizes empirical demand data to calculate the long-run average cost of inventory for a given

(sS) policy We then develop a heuristic that locates a near optimal policy quickly The

heuristic incorporates a constraint on the customer service level makes use of an updating

technique for the transition probability matrix and is based on assumptions regarding the

properties of the solution space Next we create a prototype of the interface that enables

managers to use the model interactively Finally we compare the existing inventory policy to

the optimal policy for each of 420 products sold at one of the stores This thesis finds that

Canadian Tire Pacific Associates is currently holding excessively large in-store inventory and

that it could reduce its cost of inventory by approximately 40 to 50 We estimate that

implementing optimal inventory control in the stores would result in annual savings of between

$55 and $7 million

ii

Table of Contents

Abstract ii

Table of Contents iii

List of Tables v

List of Figures vi

Acknowledgement vii

I INTRODUCTION 1

II T H E INVENTORY SYSTEM A T CANADIAN TIRE PACIFIC ASSOCIATES 8 A Background 8 B The Problem 11 C The Project 15 D The Demand Data 17 E The Cost Data 18

III M O D E L FORMULATION 20 A Terminology 20 B Assumptions 21 C A Markov Chain Model 22 D The Transition Probability Matrix 25 E The Steady State Probabilities 27 F The Cost of Inventory 29 G The Customer Service Level 31 H A Methodology for Evaluating an (sS) Policy 34

IV A N ALGORITHM FOR OBTAINING T H E OPTIMAL (sS) POLICY 35 A Introduction 35 B A Grid Search 36 C A Technique for Updating the Transition Probability Matrix 41

A new policy (s+mS) 41 A new policy (sS+m) 42 The modified algorithm 43

D A Lower Bound on S 45 E A Heuristic Search 48

V T H E INTERFACE 52

iii

VI RESULTS A N D SENSITIVITY ANALYSIS 55 A Comparison of Current Policies to Optimal Policies 55 B Sensitivity Analysis on the Ordering Cost and Holding Rate 62 C Sensitivity Analysis on the Demand Probability Mass Function 65

VII CONCLUSION 70

Afterword 73

Bibliography 74

Appendix A A Portion of the Sales File for Product 200001 a 30-amp

Inline Fuse 76

Appendix B Sample Distribution Files 77

Appendix C A Proof that the Limiting Distribution is Independent of the

Initial State 79

Appendix D A Measure for the Steady State Customer Service Level 83

Appendix E The Calculation of the Conditional Expected Demand Not Satisfied

During a Period of T Consecutive Days 86

Appendix F Justification of the Updating Technique for the New Policy (s+mS) 87

Appendix G Justification of the Updating Technique for the New Policy (sS+m) 92

Appendix H A Hypothetical Consultation 97

Appendix I The Source Code for the Interface Module 104

Appendix J The Source Code for the Main Module 119

Appendix K Current and Optimal Policies for Product Category 20 at Store 6 131

iv

List o f Tables

Table 1 Location and particulars of the 21 stores 10

Table 2 Execution times of the algorithms for product 200001 a 30-amp inline fuse 39

Table 3 Execution times of the algorithms for product 206917 a 6

solder connector 40

Table 4 Products with insufficient existing policies 56

Table 5 Products with the largest potential absolute savings 60

Table 6 A comparison of optimal policies to true optimal policies 63

Table 7 The cost of existing policies and the relative savings of the optimal policies under various scenarios 64

Table 8 True optimal and optimal policies for each demand scenario 66

v

List of Figures

Figure 1 A typical sample path of the process with a 4-day review period

and a 2-day lead time 24

Figure 2 The flow chart of the grid search algorithm 38

Figure 3 The flow chart of the grid search algorithm utilizing the updating

technique for the transition probability matrix 44

Figure 4 An evaluation of the lower bound Sm i n 47

Figure 5 The flow chart of the heuristic algorithm 50

Figure 6 The distribution of savings of optimal policies in (a) dollars and

(b) percentage of current cost 59

Figure A-1 The title screen 97

Figure A-2 The main menu 98

Figure A-3 Calculating the optimal policy and evaluating the current policy 99

Figure A-4 Displaying the results 100

Figure A-5 Entering an alternate policy 101

Figure A-6 The particulars of the alternate policy 102

Figure A-7 The main menu revisited 103

vi

Acknowledgement

The completion of this thesis was made possible by the encouragement and assistance

of a number of people

I would like to express my sincere appreciation to my thesis supervisor Professor

Martin Puterman for all of his many efforts on my behalf His help advice patience charity

and tolerance were very much appreciated

I would like to acknowledge Professor Hong Chen and Professor Garland Chow for their

time and input as members of my thesis committee In addition I must acknowledge the

assistance of PhD student Kaan Katircioglu for his insight and help on this project

I offer many thanks to Professor Tom Ross for his kindness and friendship during the

past few years especially during the writing of this work I must also thank his family for

adopting me on many holidays I still owe thanks to Professor Slobodan Simonovic at the

University of Manitoba for influencing me to attend graduate school in the first place and for

helping me to obtain funding

I cannot thank my parents enough for their never-ending support both emotional and

financial Also to my friends especially Cathy Dave Lisa Steve and the Philbuds thanks

for giving me a life off campus and for picking up many a tab - the next one is on me

I would like to give special thanks to my good friend and fellow MSc student Paul

Crookshanks for allowing me to bounce ideas off of him and for being such a procrastinator

that despite my finishing a year late I only lost by two days

Financial assistance received from the Natural Science and Engineering Research

Council (NSERC) and from Canadian Tire Pacific Associates was greatly appreciated

vii

I INTRODUCTION

The importance of inventory management has grown significantly over the years

especially since the turn of this century In colonial times large inventories were viewed as

signs of wealth and therefore merchants and policy makers were not overly concerned with

controlling inventory However during the economic collapse of the 1930s managers began

to perceive the risks associated with holding large inventories As a result managers

emphasized rapid rates of inventory turnover (Silver and Peterson 1985) Following the

Second World War Arrow Harris and Marschak (1951) and Dvoretzky Kiefer and

Wolfowitz (1952ab) laid the basis for future developments in mathematical inventory theory

Shortly thereafter new inventory control methodologies were widely applied in the private

manufacturing sector More recently when inflation and interest rates soared during the 1970s

many organizations were forced to rethink their inventory strategies yet again Today the

control of inventory is a problem common to all organizations in any sector of the economy

Sir Graham Day chairman of Britains Cadbury-Schweppes P C L expressed this sentiment

when he stated I believe the easiest money any business having any inventory can save lies

with the minimization of that inventory (Day 1992) Lee and Nahmias (1993) and Silver and

Peterson (1985) provide a more detailed history of inventory management and control

Most inventory problems in the retail sector involve the control and management of a

large number of different products Ideally the inventory policies should be determined for all

products on a system-wide basis however because the number of products is often in the tens

of thousands formulating a model to do so is impractical Therefore single-product models

are frequently used in practice to determine policies for products separately (Lee and Nahmias

1993) Lee and Nahmias (1993) and Silver and Peterson (1985) provide a good overview of

1

single-product single-location models for a more complete review of the literature we refer

to them

The degree of complexity of a single-product model depends largely on the assumptions

made regarding the demand cost structure and physical characteristics of the inventory system

For example demand can be either deterministic meaning it is known with absolute certainty

or stochastic meaning it is unknown but follows a known probability distribution

Deterministic models tend to be relatively simple yet useful such as the basic EOQ model

However future demand is seldom known with certainty in practice and therefore the use of

these simple models is limited As a result the majority of research in inventory has focused

on stochastic demand models (Lee and Nahmias 1993)

Because the objective of most inventory models is to minimize the cost of inventory

assumptions regarding the structure of the cost are very important also Generally speaking

the cost of inventory has four components the purchaseproduction cost the orderingsetup

cost the holding cost and the shortage cost1 Each of these costs can be fixed proportional

nonlinear or some complex combination of these Further because nearly all inventory

problems require decisions be made sequentially over a period of time the costs can be either

discounted or averaged In the former case a discount factor is included in the cost function

in order to account for the time value of money However since most inventory problems are

on-going the long-run average cost is usually minimized (Lee and Nahmias 1993) In

practice this is achieved by minimizing the expected cost per time period under steady state

conditions

Finally the physical characteristics of the system are important in determining the

1 The cost of a stockout is often very difficult to estimate in practice As a result service levels are frequently used in its place When this is done a constraint is included in the model

2

complexity of the model For example the review process can be either continuous in which

case the inventory is reviewed continuously and orders are placed as soon as the stock reaches

a certain level or periodic in which case the inventory is reviewed at regular intervals and

orders can be placed only at those fixed times Also excess demand can be fully backordered

as is common in many manufacturing and wholesale systems completely lost as is common

in most retail environments or some combination of the two Lead time assumptions are also

important models with lead times are more complex than those in which the supply of

inventory is assumed to be instantaneous

Although the literature is dominated by single-product models (Lee and Nahmias 1993)

much recent work has focused on extending those models to multiple-product problems

However multiple-product models are inherently more complex than the simple-product

models In fact no efficient algorithms are currently available for determining fully optimal

strategies for most multiple-product problems (Federgruen 1993) As a result research has

focused on the development of easily computable approximation techniques Muckstadt and

Roundy (1993) provide a good overview of multiple-product andor multiple-location models

with deterministic demand although emphasis is placed on models for production and

distribution Federgruen (1993) extends the discussion to consider these models under uncertain

demand and also provides a complete list of references on the subject Silver and Peterson

(1985) discuss models for coordinated replenishments for both deterministic and stochastic

demand

In this thesis we examine the inventory system at Canadian Tire Pacific Associates

owner of 21 retail stores and a central warehouse Although this is a multiple-product

multiple-location single-warehouse problem and strictly speaking should be modelled as such

we analyze the inventory problem on a product-by-product and store-by-store basis We believe

3

that this simpler single-product single-location model captures the essentials of the problem for

two reasons First the coordination of replenishments for products andor locations is most

advantageous when significant savings are possible because of high first order costs and low

marginal costs for placing subsequent orders However because of the large number of

products and the physical distribution system in this problem trucks are dispatched to stores

on a regular basis Therefore the benefits of coordinating replenishments for this problem are

negligible Second we believe that controlling in-store inventory alone can result in substantial

savings to Canadian Tire Pacific Associates Further given the excessive amount of inventory

in the warehouse a single-location model assuming an infinite warehouse capacity is reasonable

Of course should inventory control in the warehouse become a priority a more elaborate model

would be required

More specifically we consider a periodic review single-product inventory model with

stationary independent stochastic demand a deterministic lead time and an infinite planning

horizon The probability mass function for the demand is based on empirical sales data

collected over a period of approximately ten months The cost of inventory includes a

proportional holding cost and a fixed ordering cost neither of which is discounted All excess

demand is completely lost The objective of the model is to minimize the long-run average cost

per unit time subject to a service level constraint on the fraction of demand satisfied directly

from in-store inventory In order to obtain meaningful cost comparisons the model calculates

the expected annual cost of inventory

For inventory problems like the one described above optimal (sS) policies can be

computed by Markovian decision process methods such as successive approximations or policy

iteration or by stationary Markovian methods However each of these methods generate

calculations that are complex and time consuming Therefore much research interest lies in

4

developing techniques to find approximate (sS) policies with little computational effort The

majority of the approximation techniques begin with an exact formulation of the problem and

then manipulate the problem by relaxations restrictions projections or cost approximations

(Federgruen 1993) Porteus (1985) presents three approximation techniques and compares

them to fourteen others for a periodic review single-product inventory model in which

shortages are fully backordered and a shortage cost is incurred Federgruen Groenevelt and

Tijms (1984) present an algorithm for finding approximate (sS) policies for a continuous review

model with backordering and a service level constraint Also Tijms and Groenevelt (1984)

present simple approximations for periodic review systems with lost sales and service level

constraints All three works find that the approximation techniques perform well However

the techniques are evaluated for a variety of demand parameters that all result in policies with

a value of S in the range of 50 to 200 Given the slower moving products at Canadian Tire 2

it is unclear how well such approximation techniques would perform Therefore we do not

attempt to apply or modify any of these approximation methods

Exact optimal policies for this problem can be obtained by formulating a constrained

Markov decision process and solving it However the data generating requirements and the

lack of a good upper bound on S make this formulation impractical Therefore we formulate

a Markov chain model for fixed values of s and S and then employ a search technique to find

the optimal values for s and S In order to improve the performance of the model we develop

a heuristic search that locates near optimal policies relatively quickly

The remaining chapters of this thesis are organized as follows In Chapter II an

overview of the existing inventory system at Canadian Tire is presented along with a description

2 For example very few of the products analyzed have optimal values of S exceeding 20 with the largest value of S equalling 69

5

of the problem Next a multi-phase project designed to develop a fully integrated interactive

inventory control system is described with the scope of the research reported in this thesis

completing the first phase of the project Finally a description of the demand and cost data is

given

In Chapter III we formulate a Markov chain model that can be used to calculate the cost

and service level associated with a given (sS) policy Based on several simplifying

assumptions the Markov chain is formulated and the equations for the corresponding transition

probability matrix are developed Then given the limiting probability distribution of the chain

the equations for the steady state annual cost of inventory and the corresponding service level

are obtained The chapter concludes with a brief summary of the model

In Chapter IV an algorithm that searches for the optimal (sS) policy is developed As

a starting point a simple grid search algorithm is developed and evaluated on the basis of

execution time In order to improve the algorithms speed a technique for updating the

transition probability matrix is presented and the improvement in the algorithm is noted Next

a theoretical lower bound on S is obtained although the corresponding reduction in the

execution time is minimal Finally based on two assumptions regarding the shape of the

feasible space a heuristic search is developed which improves the speed of the algorithm

significantly

In Chapter V a prototype of the inventory control systems interface is created In its

present form the interface allows managers to evaluate inventory policies interactively The

manager can set review parameters compare optimal policies and current policies and evaluate

alternate policies The chapter concludes by recommending specific options be including in

future versions to make the system more automated and user-friendly

In Chapter VI optimal policies for all products in one product category sold at one store

6

are compared to existing policies on the basis of expected annual costs and customer service

levels The analysis suggests that existing policies result in a large excess inventory and that

implementing the optimal policies would reduce the annual cost of inventory by approximately

50 A sensitivity analysis on the cost estimates is performed and suggests that the model is

not sensitive to relative errors of approximately 20 in the cost estimates Finally a sensitivity

analysis on the demand probability mass function is performed The analysis indicates that the

model is sensitive at least in terms of the customer service level to slight changes in the tail

of the demand distributions of low-demand products

Finally in Chapter VII the results of the analysis are summarized and future work is

outlined In order to incorporate recent changes in the review procedures at several stores the

model must be modified to allow for the duration of the review period to exceed the lead time

We recommend that the second phase of the project proceed specifically that a data collection

and storage scheme be designed and that a forecasting model be developed Because of the

substantial savings that can be realized by reducing inventory only at the stores inventory

control at the warehouse should also be investigated

7

II T H E I N V E N T O R Y S Y S T E M A T C A N A D I A N TIRE

A Background

Canadian Tire Pacific Associates owns and operates 21 retail stores in the lower

mainland of British Columbia The stores carry approximately 30000 active products3 that

are divided into 80 product categories The merchandise comprises primarily automotive parts

and accessories home hardware housewares and sporting goods Canadian Tire Pacific

Associates also owns a central warehouse in Burnaby at which inventory is received from

Toronto Inventory is then shipped periodically from the warehouse via truck to the local

stores In 1992 total sales at the stores exceeded $99 million and the value of inventory was

estimated to be $28 million of which approximately half was stored in the warehouse

On selected days of each week a stores inventory records are reviewed automatically

and orders for products are placed to the warehouse as required Management uses the term

flushing to describe the process of reviewing the inventory and producing the tickets for stock

owing and the term picking to describe the physical act of filling the order at the warehouse4

Whether or not an order is placed for a product depends upon its current inventory level its

target level and its minimum fill 5 If the target level less the current level exceeds the

minimum fill an order is placed to bring the stock up to the target level From the moment

3 Active products are those that are regularly restocked Canadian Tire Pacific Associates can choose from up to 50000 products

4 Note that the number of picks per week is equal to the number of times per week that the inventory records are reviewed At the beginning of this study eighteen stores had two picks per week while the remaining three stores had only one

5 The target level is the maximum number of items that should be on the shelf at one time and the minimum fill is the minimum number of items that should be ordered

8

the order is placed to the moment the product is reshelved takes between two to four days

depending on the store On average 11000 items are shipped from the warehouse at each

flush

Although a products minimum fill is the same for all stores its target level depends

upon the size of the store Based upon its sales volume and retail space a store is classified

as being one of five sizes small (S) medium (M) large (L) jumbo (J) and extra jumbo

(EJ) 6 Each product has associated with it a target level for each size of store

Consequently all stores of the same size have identical inventory policies Table 1 lists the

location size number of picks per week sales and floor space of each store as of October

1993

Periodically the inventory manager reviews the policies and when deemed necessary

adjusts them In this way the manager can account for seasonality in demand changes in mean

demand and any shortcomings of previous policies At the present time the review process

is not automated nor performed on-line The inventory manager uses Hxl5 computer

printouts with five products to a sheet to study the relevant information and on which to note

any changes Afterward the changes are recorded in the system and the new policies take

effect

Currently the inventory policies are based on rules of thumb developed through the

years and not on a statistical analysis of demand nor on a calculation of the cost of inventory

These rules of thumb incorporate a marketing philosophy which at times dominates the

inventory strategy The prevailing philosophy holds that a large in-store inventory instills

6 During the course of this study stores were resized several times At one point the entire class of medium stores was eliminated leaving only four sizes of stores On another occasion the store in Chilliwack was changed from being a small store to a large store

9

Table 1 Location and particulars of the 21 stores

Store Location Store Size

Picks Per Week

1992 Sales ($)

Floor Space (ft2)

1 Chilliwack S 2 2901969 6678

2 Coquitlam M 2 4015303 6510

3 Langley EJ 2 8138673 15422

4 Scott Road EJ 2 7307820 15064

5 Maple Ridge L 2 4102110 6678

6 Kingsway L 2 4153736 6574

7 Richmond EJ 2 8175105 14961

8 North Vancouver M 2 3856571 6534

9 Guildford M 2 3714372 6567

10 Burnaby M 2 3532752 6983

11 New Westminster S 1 2417992 6470

12 Abbotsford EJ 2 7899398 15303

13 Hastings J 2 4647794 9849

14 King Edward EJ 2 3825372 25492

15 White Rock L 2 3960262 6870

16 Main amp Mountain S 1 2705602 6431

17 Heather J 2 4161227 9427

18 King George J 2 4606817 9640

19 Cambie J 2 5855022 9091

20 Lougheed M 1 3114180 6615

21 Burrard J 2 6553310 10608

10

consumer confidence which in turn leads to increased sales In other words if a customer

perceives that Canadian Tire is in the hardware business he or she will choose to shop there

when meeting future needs However management has adopted this philosophy without any

analysis of the cost of holding excess inventory Only recently has management begun to

question the cost-effectiveness of allocating inventory based on these rules of thumb

Speculating that existing inventory practices might be deficient Mr Don Graham the

vice-president of Canadian Tire Pacific Associates commissioned a pilot study to examine the

inventory system and to propose alternate strategies7 The study using a steady state model

to represent a simplified inventory system suggested that existing policies lead to overly high

inventory levels More importantly the study outlined a methodology that would enable

management to determine inventory levels based on observed demand The pilot study

recommended that a more detailed model of the inventory system be formulated that the

methodology be expanded to incorporate the cost of inventory and that an algorithm be

developed that would quickly identify the optimal inventory policies

B The Problem

At the time of the pilot study management was concerned primarily with its inability

to distinguish between stores of the same size As mentioned in the previous section a

stores size is based solely on its sales volume and floor space and under existing practices

all stores of the same size have identical inventory policies for every product Management

realized that demand for some products might be a function of variables other than size such

7 The pilot study was performed by Martin Puterman a professor of Management Science and Kaan Katircioglu a doctoral student in Management Science both at the University of British Columbia The results of the study were presented at a U B C Transportation and Logistics Workshop on March 25 1993

11

as location For example the demand for sump pumps is likely to be very different in suburban

areas than in urban ones However management was unable to allocate more sump pumps to

large suburban stores than to large urban stores This inability led management to seek a

tool that would allow for a more effective allocation of inventory between stores of the same

size

By the end of the pilot study however management was less concerned with

reallocating inventory within a size of store than with reallocating inventory between sizes

of stores and with reducing overall inventory According to management assigning individual

inventory policies to each store appeared to be operationally infeasible Firstly management

believed that having individual policies would lead to confusion in the warehouse and to

increased shipping errors Exactly why this would occur is unclear the warehouse would

simply continue to fill orders from tickets that make no reference to the policy in place

Secondly and perhaps more tenably the computer technology at Canadian Tire Pacific

Associates did not allow for policies to vary across all stores Both the database used to store

the inventory data and the printouts used by the inventory manager allowed for only five

different policies per product all with the same minimum fill Moving to individual policies

would have required an overhaul of the existing information system Although management

saw this overhaul as inevitable it did not see the overhaul as imminent Therefore

management switched the focus of the study to developing a tool that would help reduce overall

inventory levels andor reallocate inventory more effectively between groups of stores

The subtle change in the studys objective prompted an important question how should

the stores be grouped The current practice of grouping stores based on size is problematic

for two reasons First as discussed above additional factors of demand such as location

cannot be taken into account Second grouping inherently results in a misallocation of

12

inventory when some stores within a group have differing numbers of picks per week For

example the medium-sized Lougheed store which has one pick per week requires higher

inventory levels to achieve a given customer service level than the other medium stores

which all have two picks per week8 In fact in terms of customer service levels it might

be more appropriate to group the Lougheed store with the large stores having two picks per

week Ideally this artificial grouping of stores will be abandoned when the computer system

is upgraded and policies can then be set on a store by store basis

As the project proceeded management grew somewhat uneasy with the concept of using

an analytical tool to evaluate and set inventory policies According to both the pilot study and

the preliminary results of this study significantly reducing inventory would greatly reduce costs

while still providing a high level of customer service However management held that because

the model cannot take into account intangibles such as marketing considerations such a drastic

reduction in inventory would be very inappropriate In this light management viewed the

optimal policies as lower bounds for inventory levels these policies would have to be

modified upwards in order to be practical Therefore management contended that any

methodology or software produced during this study should provide the means to evaluate and

compare a number of alternate policies

Regardless of exactly how the tool is to be used there is currently a lack of historical

sales data from which to analyze the inventory system and determine optimal policies No sales

data had ever been collected prior to the pilot study and only a relatively small amount has

been collected since In order to proceed with any analysis of the inventory system a

8 If demand is a function of size then the Lougheed store must meet the same demand with one fewer pick per week This would lead to either increased probabilities of stockouts at the Lougheed store or higher inventory levels at all the other medium stores

13

substantial amount of additional data would be needed The existing information system at

Canadian Tire Pacific Associates does allow for sales data to be captured however the system

is limited by the amount of data it can capture at one time and by its storage capacity In order

to collect the amount of data required management would have to upgrade the system and

develop some kind of data collection scheme coupled with a data compression technique The

collection scheme must allow for characteristics of demand such as trends and seasonality to

be identified and the compression technique must allow for sales data for over 30000 products

to be stored for each store

Because many of the products sold by Canadian Tire have nonstationary demand as a

result of seasonality andor trend the inventory tool must be able to forecast demand for the

coming period Products such as sporting goods and outdoor equipment exhibit patterns of

demand that are very seasonal For example sales of snow shovels increase through the fall

peak in the winter and drop to zero by late spring At the same time some products might

become either more or less fashionable over time resulting in changes in mean demand

Setting inventory policies for any of these products based on observed demand would result in

insufficient stock if demand increased during the period or in excess stock if demand decreased

Therefore the tool must be able to forecast the demand for the period for which the policy is

intended

Toward the end of the project management began discussing the integration of the

interactive inventory tool with the existing system to form a new inventory control system The

new system would utilize a modified point of sales system to collect sales data continually

Based on the latest available data the system would forecast demand for the coming period and

calculate the optimal policies for every product at every store The system would then provide

all relevant information on-line for the inventory manager to review The manager could weigh

14

other information such as marketing considerations and examine alternate policies to determine

the effect on cost Finally the preferred policy would be set in place with the touch of a key

and information on the next product or store would be presented for review

C The Project

Development of the inventory control system envisioned by management requires the

completion of five tasks 1) the design of a data collection system that generates and stores the

necessary sales data 2) the development of a forecasting method that provides accurate

estimates of demand distributions 3) the formulation of a model that calculates the cost of

inventory and the customer service level for an inventory policy 4) the development of an

algorithm that finds the optimal policy for a product at a store and 5) the creation of an

interface that allows interactive and on-line consultation Ultimately the above components

would have to be integrated and incorporated in a manner that would ensure the compatibility

and reliability of the system

The scope of the research reported in this thesis completes the first phase of the project

specifically we formulate an inventory model develop an algorithm and create a prototype of

the interface Because of the changing environment at Canadian Tire Pacific Associates minor

modifications to both the model and the algorithm developed herein might be necessary to

reflect future changes in operational procedures Upon completion of phase one the interface

will be interactive allowing managers to select the store product and review scenario and then

to compare optimal policies with existing policies or with alternate policies Although the

prototype will be delivered to Canadian Tire Pacific Associates for ih-house testing at the end

of this phase the interface will require modification in subsequent phases as feedback from

management is received and as the system is brought on-line

15

In order to assess more fully the potential savings of implementing the inventory control

system this thesis includes a comparison of optimal policies9 and existing policies The

expected annual cost of inventory for an entire product category is calculated for both the

optimal policies and the existing policies Based on an analysis of these costs the potential

savings of implementing the optimal policies are extrapolated Further the thesis examines the

sensitivity of the model to variations in the estimates of demand and costs

Originally results from all 21 stores were to be included in the above analysis

However just prior to the analysis management changed the review procedure at several

stores the result of which being that three picks per week are now performed at those stores

Because one of the main assumptions of the model specifically that the length of the lead time

does not exceed the duration of the review period is violated under this new review procedure

those stores cannot be included in the analysis Therefore the analysis is based solely on the

results from store six the large store located on Kingsway Avenue10

For the purpose of the analysis sales data for product category 20 automotive and

electrical accessories were collected from all 21 stores This product category was selected

because the distribution of demand for products within it is non-seasonal and because few of

the products are ever placed on sale11 The lack of trend or seasonality eliminates the need

for forecasting demand the optimal policies can be determined based directly on observed

demand

9 That is those policies obtained from the model

1 0 Store six is one of the only stores whose review procedure is not expected to change from two picks per week

1 1 Products that were on sale at any time during the period were not included in the data set

16

D The Demand Data

In total sales data for 520 products of which 458 are active products were collected

over a 307-day period from October 10 1992 to August 18 199312 The data was supplied

by Canadian Tire Pacific Associates in the form of a 10 MB text file with one line per day per

product recording sales from the stores13 Each line had the following format

Columns 1-6 part number Columns 7-9 Julian date14

Columns 10-18 cost of the item (to three decimal places) Columns 19-124 sales for stores 1 through 21 (5 columns per store)

In order to facilitate the analysis the file was sorted into 458 sales files with each of

these files containing the history of daily sales for one of the products Two changes were

made to the format of these files to improve readability 1) a column was added between the

product number and the Julian date and 2) all leading zeros were dropped Appendix A

contains a partial printout of the sales file for a typical product Finally a distribution file

summarizing the daily sales was created from each of the sales files Each distribution file

contains the products number the products cost and the number of days during which various

quantities of the product were sold at each store

Prior to the analysis the distribution files for a large number of products were

examined and based on the shapes of the histograms the products were divided into three

groups The first group low-demand comprises products that have demand on only a few

days with the daily demand seldom exceeding three items The second group medium-

1 2 Of the active products only 420 are sold at store 6

1 3 Strictly speaking a line was written to the file only on days on which at least one store registered a sale of the product

1 4 According to the Julian calendar days are numbered sequentially from 1 to 365 (366 during leap-years) beginning with the first of January

17

demand comprises products that have demand on approximately half of the days with the daily

demand seldom exceeding more than six items Finally the last group high-demand

comprises products that have demand on most days occasionally upwards of twenty items and

whose demand histograms resemble Poisson distributions The majority of the products

examined fell into the low and medium-demand groups only a few high-demand products

were observed Appendix B contains a printout of the distribution file for a typical product

in each group

For most products no theoretical distribution appeared to model the demand well

Demand for low and medium-demand products followed a non-standard distribution with

a mass at zero and a relatively small tail Although some theoretical distributions such as the

mixed-Poisson can be made to resemble this it was unclear how well they would perform

Further the effects of sampling errors from such non-standard distributions were unknown

On the other hand the standard Poisson distribution did appear to model high-demand

products well however only a few products fell into that group Therefore in the absence of

a good theoretical candidate the empirical demand function was chosen for the analysis The

empirical function was readily available from the distribution files and offered the best

estimate of the true underlying function

E The Cost Data

Along with the sales data management supplied estimates relating to the costs of holding

and ordering inventory Management estimates that the cost of holding an item in inventory

for one year is 30 of the items cost Incorporated in this holding rate are a number of costs

such as the opportunity cost of money invested and the costs incurred in running a store

Because it is the annual cost of inventory that is of interest either the compounded daily

18

holding rate or the simple daily holding rate can be used to determine the costs incurred during

a review period however the method of calculating and extrapolating the holding costs must

be consistent For the sake of simplicity a simple daily holding rate is used

In order to estimate the cost of ordering inventory management analyzed the various

tasks that result from an order being placed Using the average time required to fill and then

reshelve an order management obtained an estimate of the direct labour costs involved To

this management added an estimate of the cost of transporting an order from the warehouse to

the store Finally the estimate was adjusted slightly upward to account for the cost of breakage

and errors Based on the analysis management estimates the cost of placing an order for a

product to be $0085 regardless of the size of the order

19

III M O D E L F O R M U L A T I O N

A Terminology

In the previous chapter several terms were introduced to describe the inventory system

at Canadian Tire Pacific Associates These terms though perfectly adequate are not widely

used in the academic literature to describe inventory management problems Therefore in

order to avoid confusion and be consistent with the literature the following terms are used

throughout this thesis

The length of time between successive moments at which the

inventory position is reviewed15

The time at which a reorder decision is made

The length of time between the moment at which a decision to

place an order is made and the moment at which the order is

physically on the shelf

An operating policy in a periodic review inventory model in

which (i) the inventory position is reviewed every T days (ii) a

replenishment order is placed at a review only when the inventory

level x is at or below the reorder level s (gt 0) and (iii) a

replenishment order is placed for S-x units (S gt s)1 6

Lost Sales - The situation in which any demand during a stockout is lost

Review Period -

Decision Epoch

Lead Time -

(sS) Policy

1 5 With respect to the terminology used at Canadian Tire Pacific Associates a review period equals the number of days between flushes

1 6 With respect to the terminology used at Canadian Tire Pacific Associates S represents the target level and s represents the difference between the target level and the minimum fill

20

B Assumptions

In to order make the model of the inventory system mathematically manageable several

simplifying assumptions are made However we believe that the following assumptions do not

significantly reduce the accuracy of the model

1) Demand can be approximated very closely by sales during the period in which the data

is collected Generally inventory levels at the stores are very high and the number of

lost sales is assumed to be negligible In fact the inventory levels for a number of

products were tracked over time during the pilot study and not a single incidence of

stockout was observed Because all products appear to have similar excess inventory

it seems reasonable to assume that virtually all demand resulted in sales

2) The demands in successive periods are independent identically distributed random

variables Preliminary tests (Fishers exact tests) suggested no dependence between

sales on consecutive days and plots of sales over time indicated no seasonal pattern

3) Demand is independent across products That is we assume that sales of one product

do not affect nor can they be predicted from sales of another product

4) Unmet demand results in lost sales with no penalty other than the loss of the sale

5) The duration of the review period for a store is an integer value and is fixed over a

planning period For stores with two picks per week we assume that the inventory

is reviewed every four days rather than on a four day - three day cycle

6) The decision epoch is the start of a review period Review periods begin at midnight

at which time the inventory is reviewed and orders are placed instantaneously

7) The duration of the lead time for a store is deterministic and integer-valued In reality

the lead time for some stores was stochastic but management felt the variation could be

controlled and virtually eliminated

21

8) The lead time does not exceed the review period In other words items ordered at a

decision epoch arrive and are shelved before the next decision epoch In practice this

did not always occur

9) Orders arrive at the end of the lead time and are on the shelf to start the next day

10) Holding costs are incurred daily for all inventory held at the start of the day

11) The warehouse has an infinite capacity that is all orders to the warehouse are filled

Because of the high inventory levels in the warehouse stockouts in the warehouse are

very unlikely

C A Markov Chain Model

Consider the situation in which an (sS) policy is applied to an inventory system based

on the above assumptions Inventory is reviewed at the decision epoch for each of the periods

labelled t = 0 12 and replenishment occurs at the end of the lead time provided an order

was placed Demand during day d in period t is a random variable whose probability mass

function is independent of the day and period The function is given by

f^ik) = Pr ltd = k = ak for k = 0 1 2

where ak gt 0 and E ak = 1 Let T denote the duration of a review period and L denote the

length of the lead time Next define D T ( as the total demand during period t D L as the total

demand during the lead time in period t and D T L as the difference between D T and DLt

Assuming that successive demands pound f pound 2 are independent the probability mass function of

1 7 The aggregated demand variables are given by

^ = XXd and DT_Ut=J d= d= d=L+l

22

each aggregated demand variable can be obtained by convolving the probability mass function

of t d l s Finally let X denote the inventory on hand at the beginning of period t just prior to

the review

The process X is a discrete time stochastic process with the finite discrete state space

0 1 S A typical sample path of the process with a review period of four days and a

lead time of two days is depicted in Figure 1 As period t begins X units are on hand The

inventory is reviewed and because X is less than s an order is placed for (S-X) units Over

the next two days a demand of D L r occurs and the inventory drops accordingly With the

arrival of the order the inventory level increases by (S-X) at the start of the third day of the

period Over the remaining two days a demand of D T L is observed bring the inventory level

down to X + 1 Note that in this example an order would not be placed in period t+l because

X r + i exceeds s

The stock levels at the beginning of two consecutive review periods are related by the

balance equation

_K- DLy+ (s - v - DT-LX ixtlts ( 1 )

t + l ~ (Xt - DTtY ifXtgts

where the positive operator is interpreted as

= max a 0

1 8 For example the probability mass function of D T bdquo denoted fD T ( is the T-th fold convolution of f t Mathematically the function can be obtained using the recursive relation

and the identity f1^ = f5(

23

-2 I 8

gt o

r

CO

o a

CM

X

00

24

Because X + 1 depends only on Xt the process X satisfies the Markov property

PrXt+l=jX0 = i0Xt_l=it_vXt = i) = PrXt+1=jXt = i

and therefore is a discrete time Markov chain Further because the probability mass function

of the random variable pound is independent of t and thus so too are the probability mass functions

of the aggregated demand the Markov chain is stationary19

D The Transition Probability Matrix

The stationary transition probability p(gt is defined as the probability of X + 1 being in state

j given that X is in state i independent of the period t In the inventory model p-- represents

the probability of starting the next review period with j items on the shelf given there are i units

on hand at the start of the current period By convention the transition probabilities are

arranged in an (S + l)x(S + 1) matrix P such that (P)y = p- Thus the (i + l)st row of P contains

the probability distribution for the stock on hand at the beginning of the next period given that

there are i units on hand at the start of the current period The matrix P is known as the

transition probability matrix Note that since we are concerned with a steady state analysis we

need not consider the possibility of initial stock levels greater than S once the inventory level

is below S it can never again exceed S

The transition probability ptj is calculated by expanding (1) to include all possible

instances of demand that allow for a change in inventory from i units at the beginning of a

period to j units at the beginning of the next period For example suppose the inventory policy

(510) is in place 2 items are on hand at the start of the current review period and 7 items are

1 9 Henceforth the subscripts t and d are omitted from the daily and aggregate demand variables for the sake of brevity

25

on hand at the start of the next period Then only three scenarios are possible

1) No items are sold during the lead time (meaning that the stock level increases to 10

after replenishment) and 3 items are sold during the remainder of the period

2) One item is sold during the lead time and 2 items are sold during the remainder of

the period

3) Two items are sold during the lead time (though demand could exceed 2) and 1 item

is sold during the remainder of the period

The transition probability p 2 7 is the sum of the probabilities of the various scenarios and is

computed by

p2J = PrDL=0PrDT_L = 3] + PrDL = l] Pr[DT_L=2 + PrDL2] PrDT_L = l]

In general the transition probabilities for the inventory model are calculated as follows

Case 1 The lead time is less than the review period

For 0 lt lt s

Pij

i-l pound [PrDL=kPrDT_LgtS-k] + PrDLgtiPrDT_LgtS-i if j = 0

i - l

^[PriD^^PrlD^S-j-kn+PHD^PriD^S-j-i if 0 lt j ltL S-i (2) k=0 s-j

E [PrDL=kPrDT_L = S-j-k] if S-i ltj^S k=0

26

For s lt i lt S

PrDTzi]

0 p = PrDT=i-j)

ifj = 0 ifOltjzi ifiltjltS

(3)

Case 2 The lead time equals the review period

For 0 lt i lt s

Pa

0 PrDTzi) PrDT = S-j

ifOltLJlt S-i ifj = S-i if S-i ltj plusmn S

(4)

For s lt i lt S use (3)

E The Steady State Probabilities

Provided 1) there is a positive probability of non-zero demand during a review period

and 2) there is a positive probability of zero demand during a review period20 then the

Markov chain described by (1) has a limiting probability distribution ir = (ir0 TTu irs)

where 0 lt IT lt 1 for j = 0 1 S and E 7ry = 1 Further this limiting distribution is

independent of the initial state In other words the probability of finding the Markov chain in

a particular state in the long-run converges to a nonnegative number no matter in which state

the chain began at time 0 The limiting distribution is the unique nonnegative solution of the

equations

2 0 That is 0 lt PrD x = 0 lt 1 Given the nature of the problem this assumption is not very restrictive

27

^ = E ^ - far j = 0 l S (5)

s and pound nk = 1 (6)

For the proof that the limiting distribution is independent of the initial state and is obtained from

(5) and (6) the reader is directed to Appendix C

In terms of the inventory problem 7ry is the steady state probability of having j units of

inventory on hand at the decision epoch However some subsequent calculations are made

easier if the steady state probability of having j units of inventory on the shelf just after

replenishment is known Instead of reformulating the Markov chain these probabilities can be

obtained by shifting the limiting distribution TT Much in the same way that the transition

probabilities are calculated the shifted limiting distribution t] is obtained by accounting for

all possible instances of demand that allow for the shifted inventory position For example

suppose that the policy (510) has the limiting distribution TT In order to have say 8 units on

hand immediately after replenishment one of the following must occur

1) Ten items are on hand at the decision epoch and 2 items are sold during the lead time

2) Nine items are on hand at the decision epoch and 1 item is sold during the lead time

3) Eight items are on hand at the decision epoch and nothing is sold during the lead time

4) Either 2 3 4 or 5 items are on hand at the decision epoch and 2 items are sold during

the lead time

The steady state probability of having 8 units on hand just after replenishment is calculated as

n 8 = n10PrDL = 2] +iz9PrDL = l] +n8PrDL = 0 +[iz5 + iz4 + iz3]PrDL = 2 +n2PrDLgt2

28

In general the shifted steady state probabilities are given by

s pound nPrDLgti ifj = 0

i=s+l S

pound PrDL = i-j] ifl plusmnjltS-s i=max(s+l)

S = pound 7tjirZ)L = i-+7i 5 FrD i ^5 ifj = 5-5

i=max(s+l)

^ 7 1 ^ ^ = 1-7+ pound TtiFrZgti=5-y+iis_PrD1^S-7 ifS-sltjltS i=max(gt+l) i=S-j+l

(7)

F The Cost of Inventory

For this problem the cost of inventory has two components (i) a fixed ordering cost

cf incurred whenever an order is placed and (ii) a holding cost cv incurred daily for each item

held in inventory The expected cost of inventory for a review period can be calculated by

summing the costs associated with each inventory position under steady state conditions

Because a simple holding rate is used this cost is simply adjusted by number of review periods

in a year to obtain the expected annual cost of inventory

The ordering cost for a review period is relatively straight forward to calculate either

an order is placed in which case the order cost is incurred or no order is placed in which case

no cost is incurred According to the rules of the inventory policy orders are placed only if

the stock on hand at the beginning of a review period is less than or equal to s Further the

probability that the review period begins with the inventory in one of these positions is equal

to the sum of the first (s + 1) steady state probabilities Therefore the expected ordering cost

29

for a review period is given by

E[OC] = cfY n y=o

Unlike the simple structure of the ordering cost the holding cost is incurred daily for

each item held in inventory Therefore the daily holding costs associated with all possible

inventory positions must be weighted and summed to obtain the holding cost for the review

period In order to make the calculations less complicated a shifted review period beginning

immediately after replenishment and ending just prior to the next replenishment is considered

Define e(0 to be the expected number of items on hand to start day i of the shifted

review period and let g(i f) denote the expected number of items on hand to start day i given

that there were j items immediately following replenishment21 The function g(z j) is the sum

of the probabilities of possible demand weighted by the resulting stock levels that is

g(ij) = j^kPriD^j-k)

To obtain the expected number of items on hand to start day i independent of j the above

function is summed over all possible values of j and weighted by the corresponding steady state

probabilities yielding

s j ed) = pound r 5 gt P r Z ) = - pound

7=0 k=

2 1 In other words g(if) is the expected number of items on hand to start day i given e(0)=v

30

The expected holding cost for a review period is then given by

T

E[HC] = $gt v e(0 i=i

i=l =0 k=l

where c v equals the simple daily holding rate multiplied by the cost of the item

Although differently defined review periods are used to calculate the above expected

costs the costs can still be summed to obtain the expected cost of inventory for a review

period The expected costs are calculated based on steady state conditions and therefore

represent the costs expected to be incurred during any T consecutive days Summing the above

two expected costs and multiplying the sum by the number of review periods in a year yields

the expected annual cost of inventory

E[AC] = mdash[E[OC]+E[HC]]

T S j 1 ( g ) 365

j=0 i=l j=0 fc=l

G The Customer Service Level

There are many ways to define customer service and the most appropriate definition

depends upon the nature of the business and the perspective of management Two commonly

used definitions are 1) the proportion of periods in which all demand is met and 2) the

proportion of demand satisfied immediately from inventory Although the former definition is

often used because of its simplicity the latter definition is closer to what one commonly means

by service (Lee and Nahmias 1993) A third definition also relevant in retail environments

31

is the probability that lost sales do not exceed some critical number Other definitions such

as probability that backorders are filled within a certain amount of time are more applicable

to industrial and manufacturing settings where backlogging demand is common

Given the retail environment with the possibility of lost sales that exists at Canadian Tire

Pacific Associates management is concerned primarily with the fraction of sales that are lost

For example management views the situation in which demand exceeds stock by one unit every

period more favourably than that in which the same demand exceeds stock by three units every

other period For this reason the proportion of demand satisfied immediately from inventory

is the most appropriate measure of customer service

The customer service level for a single review period CSLRP is given by

r bdquo T _ Demand satisfied during RP Total demand during RP

Because we are interested in the long-run behaviour of the system a measure of the steady state

customer service C S L must be obtained As shown in Appendix D the correct measure is

poundpoundpound _ E[Demand satisfied during RP] E[Total demand during RP]

which can be rewritten as

pound pound pound _ j _ E[Demand not satisfied during RP] ^ E[Total demand during RP]

Because of the steady state conditions the above customer service level can be estimated

from any period of T consecutive days As was done for one of the cost calculations the T-day

period beginning just after replenishment is considered Suppose that there are j units of

inventory on hand immediately following replenishment Then the conditional customer service

32

level is

CSL j = 1 E[(DT-j)l

E[DT] (10)

The denominator in the second term is merely the expect demand during the period

E[DT] = 7gt (11)

where p is the mean of the random daily demand variable pound The numerator which is the

conditional expected demand that is not satisfied during the T-day period expands to

7-1

E[(DT-j)+] = T u - j + YU-k)PrDT = k] k=0

(12)

The reader is directed to Appendix E for a mathematical justification of (12) In order to obtain

the unconditional expected demand not satisfied (12) is applied to all possible values of j and

a weighted sum of the terms is calculated The result is given by

E[Demand not satisfied] = Y j =0

y-i Ti - j + 1pound(j-k)PrDT = k

S j-1

= - E[I] + Y0-k)Pr[DT=k) =0 it=0

(13)

where E[I] denotes the steady state inventory level Finally the customer service level is

obtained by the substituting (11) and (13) into (9) yielding

CSL = 1

s j-l

Ti - E[I] + Jgty Y(J-k)PrDT=k]

m n J2VjTU-k)PrDT=k Eil J=Q fc=o

(14)

33

H A Methodology for Evaluating an (sS) Policy

In this section a brief methodology summarizing the main results of the model is

presented The methodology can be applied to evaluate a given (sS) policy in terms of its cost

and its resulting customer service level Given the probability mass function of the daily

demand f the duration of the review period T the length of the lead time L the fixed cost

of placing an order cf and the daily unit cost of holding inventory cv

1) Calculate the aggregated demand probability mass functions fDT fDL and f D x L as the T-th

fold L-th fold and (T-L)th fold convolutions of f respectively

2) Generate the transition probability matrix using either equation (2) or (4) and equation (3)

3) Calculate the limiting distribution TT from (5) and (6)

4) Obtain the L-day shifted limiting distribution rj using (7)

5) Calculate the expected annual cost of inventory using (8)

6) Determine the customer service level using (14)

34

IV A N A L G O R I T H M F O R OBTAINING T H E O P T I M A L (sS) P O L I C Y

A Introduction

The methodology developed in the previous chapter can be used to calculate the cost and

service level associated with a given (sS) policy In this chapter an algorithm that finds the

feasible policy yielding the lowest cost is developed A policy is feasible if it satisfies both the

basic condition 0 lt s lt S and the imposed constraint on the customer service level2 2 In

order to find the optimal policy the algorithm must combine the methodology for evaluating

a single policy with a search technique

Search techniques are evaluated in terms of two somewhat antithetic criteria On the one

hand the search should locate the optimal solution or at least a near optimal solution on

the other hand the search should take as little time as possible Many search techniques

guarantee optimality provided the objective function and constraints exhibit certain properties

such as convexity However even if such a search technique also guarantees its termination

in polynomial time the search might take too long to be practical Alternatively some search

techniques provide reasonable though sub-optimal solutions very quickly These techniques

often called heuristics must be evaluated in terms of both their speed and the accuracy of their

solutions The choice of a heuristic over a search that guarantees optimality depends upon the

nature of the feasible space the need for accuracy and the cost of time

Given the complexity of the cost and customer service level functions it is very difficult

to show theoretically for what if any combinations of demand and costs the feasible space is

convex However a plot of the objective function for a single product may be obtained by

2 2 Management specified that each policy must provide a customer service level of at least 975 Therefore the algorithm must incorporate the constraint CSL gt 0975

35

evaluating the costs for a number of policies within a grid When the service level constraint

is included in these plots maps of the feasible space are obtained Examination of the

maps for several products suggests that the space is convex over most regions however

maps for some products reveal some regions of non-convexity Unfortunately there is no

way of knowing the degree of non-convexity for other products other stores or different costs

estimates without a more rigorous survey of the feasible space Therefore if one of the

techniques that guarantee optimality under the condition of convexity is employed the search

would have to be evaluated as a heuristic Instead as a first step to finding a practical search

technique a crude search that guarantees optimality under virtually all conditions is

investigated

B A Grid Search

The simplest search technique that guarantees optimality merely evaluates all policies in

the grid defined by 0 lt s lt S and 0 lt Sm i n lt S lt S m a x where Sm i n and S m a x are

some predefined values That is the grid search is a brute force method that examines every

policy within the grid 2 3 Although the number of policies that must be examined is large the

grid search is guaranteed to find the optimal policy provided that S o p t is contained in the grid

that is provided Sm i n lt S o p t lt Sm a x The grid search algorithm proceeds by generating the

transition probability matrix for the policy (0Smin) and then evaluating the service level

associated with that policy If the policy is feasible24 the cost associated with the policy is

2 3 The total number of policies evaluated is the sum of the first Sm a x integers less the sum of the first Smin-1 integers Numerically [(l2)(Sm a x)(Sm a x+l) - (l2)(Smin-l)(Smin)] policies are examined

2 4 That is if the service level is adequate

36

calculated and compared to the current lowest cost The procedure continues until all points

in the grid have been examined and the optimal policy has been found A flow chart of the grid

search algorithm is shown in Figure 2

In order to assess the grid search the algorithm was coded and compiled into an

executable program The program was run for two products (a medium-demand product and

a high-demand product) and with four values of Sm a x In each run Sm i n was taken to be zero

The following times were recorded for each trial

T P M - the amount of time spent generating the transition probability matrix

GAUSS - the amount of time spent solving for the steady state probabilities

COSTS - the amount of time spent calculating the expected annual costs

SERVICE - the amount of time spent determining the customer service levels

T O T A L - the total execution time of the program

The columns entitled Grid Search in Tables 2 and 3 summarize the results of the trials All

times are recorded in seconds as required by a 486DX33 MHz personal computer with an

empty buffer memory

Three results are evident from the tables First the vast majority of the execution time

is spent generating transition probability matrices and then solving for the steady state

probabilities with the time being split almost equally between the two tasks Second the total

execution time of the algorithm increases exponentially as Sm a x increases For product 206917

doubling S^ from 30 to 60 led to an increase in the execution time of more than 2100 (from

172 seconds to 3661 seconds) Third the grid search algorithm as it stands does not appear

to be adequate for the problem at hand Given the large number of products that must be

evaluated the algorithm is much too slow to be implementable

37

Calculate the demand probability functions

Initialize s = 0 S = Smin] cost = 10A5

s = s+1

Calculate the probability transition matrix for (sS)

Evaluate policy (sS) Is service level ^ 0975

No

No Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Is s = S-l

Yes

Is S = Smax No

Yes

Print results (sS) cost

s = 0 S = S+1

Figure 2 The flow chart of the grid search algorithm

38

Table 2 Execution times of the algorithms for product 200001 a 30-amp inline fuse

(in seconds)

-max Subroutine Grid Search Grid Search

Updating T P M Heuristic

10 T P M 006 010 001

GAUSS 010 006 001

COST 005 005 001

SERVICE 001 001 001

T O T A L 050 035 027

20 T P M 126 021 010

GAUSS 119 160 023

COST 027 026 001

SERVICE 017 016 001

T O T A L 318 236 055

30 T P M 703 096 010

GAUSS 757 801 093

COST 079 085 001

SERVICE 047 042 001

T O T A L 1664 1137 126

40 T P M 2257 163 015

GAUSS 2958 3003 206

COST 313 329 001

SERVICE 092 083 011

T O T A L 5871 3851 285

39

Table 3 Execution times of the algorithms for product 206917 a 6 solder connector

(in seconds)

bull-max Subroutine Grid Search Grid Search

Updating T P M Heuristic

30 T P M 717 067 006

GAUSS 762 880 093

COST 098 087 010

SERVICE 016 021 006

T O T A L 1720 1176 220

40 T P M 2390 157 022

GAUSS 2861 2911 209

COST 219 227 006

SERVICE 080 088 011

T O T A L 5827 3790 362

50 T P M 6691 413 010

GAUSS 7994 8201 472

COST 568 577 016

SERVICE 212 197 005

T O T A L 15840 10041 654

60 T P M 15509 894 040

GAUSS 18607 18951 876

COST 1367 1353 038

SERVICE 350 359 022

T O T A L 36614 22794 1120

40

C A Technique for Updating the Transition Probability Matrix

As was noted in the previous section approximately half of the execution time of the grid

search algorithm is spent generating transition probability matrices Furthermore the amount

of time increases exponentially as Sm a x increases This increase is exponential for two reasons

first the number of transition matrices that must be calculated is a function of S^ squared and

second the average size of the matrices and hence the average number of calculations per

matrix increases with Sm a x For example when S m a x equals 30 the transition probability matrix

must be generated 465 times with the largest matrix having dimensions (31x31) When Sm a x

equals 40 the matrix must be generated 820 times with 355 of the matrices having dimensions

larger than (31x31) The performance of the algorithm could be improved considerably if

either the number of matrices that must be generated or the number of calculations per matrix

is reduced This section introduces a technique that accomplishes the latter by updating the

transition probability matrix for a new policy instead of generating it anew for each policy

A new policy (s+mS)

Suppose the policy (sS) has associated with it the (S + l)x(S + l) transition probability

matrix P and the policy (s+mS) where m is any integer satisfying 0 lt s+m lt S has

associated with it the (S + l)x(S + 1) transition probability matrix J Then T can be obtained by

recalculating only m rows of P Recall that the equations for generating a transition

probability matrix are divided into two sets based upon the row being calculated equations (2)

or (4) for rows 0 through s and (3) for rows (s + 1) through S The equations themselves are

independent of s only the limits defining the cases contain s Therefore only the m rows

that are affected by the change in the limits need to be recalculated The following updating

technique can be used to obtain 7

41

1) If m gt 0 rows i = (s + 1) (s+2) (s+m) are recalculated using (2) or (4)

2) If m lt 0 rows i = (s-1) (s-2) (s-m) are recalculated using (3)

3) All other rows of T are identical to those of P

The reader is directed to Appendix F for a mathematical justification of the technique

A new policy (sS+m)

Suppose the policy (sS) has associated with it the (S + l)x(S + l) transition probability

matrix P and the policy (sS+m) where m is any positive integer has associated with it the

(S+m+l)x(S+m+l) transition probability matrix P Then T can be obtained by modifying

each row of P and adding m additional rows and columns The updating technique is not as

simple as that for a new policy (s+mS) for two reasons the dimensions of P and T differ and

the equations for generating a transition probability matrix contain S Nevertheless an updating

technique that requires much less computing time than generating the matrix anew can be

developed The following technique can be used to obtain P

1) For rows i from 0 through s

For columns j from 0 through m recalculate using (2) or (4)

For columns j from (m+1) through (S+m) enter the (j-m)th column entry of P

2) For rows i from (s + 1) through S

For columns j from 0 through S enter the corresponding value from P

For columns j from (S + 1) through (S+m) enter 0

3) For rows i from (S + 1) through (S+m) recalculate using (3)

The above technique is justified mathematically in Appendix G

42

The modified algorithm

The updating technique can be easily incorporated into the grid search algorithm with

only a slight change in the manner in which s is incremented Figure 3 shows the flow chart

of the new algorithm As before the algorithm was coded and compiled into an executable

program and the various execution times were recorded The results are listed in Tables 2 and

3 under the columns Grid Search Updating TPM

Updating the transition probability matrix does improve the performance of the

algorithm The time required to generate the transition probability matrices is reduced by

approximately 90 This translates to a reduction in total execution time of approximately

30 Not surprisingly the reductions are greater for large values of Sm a x However despite

this seemly significant improvement a decrease of only approximately 08 seconds is obtained

when S^ equals 20 Further when S ^ equals 30 the algorithm requires over 11 seconds

which is still much too high

The execution time is now dominated by the time required to solve for the steady state

probabilities The steady state probabilities could be solved by using approximation techniques

which are faster than solving the systems exactly however because the probabilities are often

very small numbers the approximation techniques would be unable to provide the required

accuracy The only remaining way to decrease the execution time is to reduce the number of

policies that are evaluated during the search This can be done by either reducing the size of

the grid being searched or reducing the number of policies being evaluated within the grid The

following section introduces a method of reducing the size of the grid by obtaining a lower

bound for S namely Sm i n

43

s = s+k

Calculate the demand probability functions

Initialize s = 0 S = Smin| cost = 105

k= 1

Update the probability transition matrix for (sS)

Evaluate policy (sS) Is service level ^ 0975

No

No

iNo

Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Ifk= 1 iss = S-l If k =-1 is s = 0

Yes

Is S = Smax No

Yes

Print results (sS) cost

If k=l s=s+l S = S+1

Figure 3 The flow chart of the grid search algorithm utilizing the updating technique for the transition probability matrix

44

D A Lower Bound on S

As mentioned earlier the size of the grid is specified by the number of policies within

the domain defined by 0 lt s lt S and 0 lt Smin lt S lt Smax Thus the size of the grid

and hence the total amount of computation time can be reduced by obtaining tight bounds on

S Unfortunately because of the extreme non-linearity of the objection function and the service

level constraint obtaining a mathematical upper bound on S is very difficult and beyond the

scope of this work This section introduces a method for obtaining a lower bound on S25

For a given value of S an upper bound on the customer service level can be found as

follows Suppose that there are S units of inventory on the shelf at the beginning of every

review period regardless of demand No (sS) policy can provide a higher level of customer

service because there exists a positive probability of beginning a review period with less than

S units on the shelf Thus the customer service level associated with starting each review

period with S units on the shelf is an upper bound on the customer service level for any (sS)

policy From (10) this level is

E[(DT-SY] CSL 5 = 1 - T mdash mdash - ( 1 5 gt

E[DT]

Replacing j with S in (12) and then substituting (12) and (11) into (15) yields

CSL 15 = mdash- - J2(S-k)PrDT = k] (16)

To obtain Smin use (16) to find the largest value of S with a customer service level less than

0975

There are two criteria for a good lower bound on S the bound must be no larger than

This method was proposed by Kaan Katircioglu

45

S o p t and it must be very close to So p t These criteria ensure that the optimal policy will be

contained in the search grid and that the size of the grid is reduced by as much as possible

Based on these criteria the above method of obtaining a lower bound is assessed A value for

S m i n is obtained for each of the products in product category 20 and compared with the

corresponding value of S o p t 2 6 The results are illustrated in Figure 4 which shows the number

of occurrences of various Sm i n and S o p t combinations Note that a point lying on the 45deg line

represents a perfect bound while a point lying close to the S o p t axis represents a very loose

bound The absence of any points above the 45deg line indicates that the first criterion is satisfied

in each instance the method did not generate a value of Sm i n exceeding the corresponding value

of So p t Moreover the bounds are reasonably tight for 85 of the products analyzed that

is for those products with S o p t equal to 10 or less27 However as S o p t increases the bounds

become steadily worse Because values of S o p t are likely to be larger for these products at the

ten Jumbo and Extra Jumbo stores and for other products with higher demand the lower bounds

are not expected to be very tight in general

Although obtaining the bound Sm i n does reduce the size of the grid the corresponding

reduction in the execution time of the algorithm is miriimal For example if S^ equals 60 and

a value of 30 is obtained for Sm i n the program would still require approximately 215 seconds

to search the grid (compared to the 228 seconds required without a lower bound on S) Because

the execution time increases exponentially with So p t a tight upper bound on S would likely have

a more significant impact on reducing the execution time of the algorithm Further research

on this issue is required

2 6 The optimal policy for each product was obtained using a grid search with an extremely large upper bound

2 7 On average Sm i n is approximately half of S o p t for these products

46

47

The only remaining approach to improving the algorithm is to modify the search so that

fewer policies within the grid are evaluated Toward this goal assumptions can be made with

respect to the feasible space and a heuristic search can be developed based on these

assumptions

E A Heuristic Search

As mentioned previously in this chapter a map of the feasible space for each of

several products was obtained Upon closer examination the maps illustrate two patterns

without exception for a given value of S both the cost and the customer service level increase

as s increases The following intuitive argument is offered as a possible explanation for the

observed behaviour When S is held constant increasing s results in more instances in which

an order is placed upon review Hence the average stock level would increase which in turn

would lead to an increase in the cost and the customer service level28 With this in mind a

heuristic search based on the following assumptions is developed for a given value of S both

the cost and the customer service level are monotonically increasing functions of s

In general the heuristic proceeds as follows Having just incremented S by one the

customer service level of the policy (sS) is evaluated If the policy is feasible s is decreased

until the lowest value of s producing a feasible policy is found If the policy is not feasible

s is increased until either the cost exceeds the current lowest cost for a feasible policy a

feasible policy is obtained or s equals S-l In the event that a feasible policy is found the cost

2 8 In order to prove that the customer service level increases as s increases it would be sufficient to show that the probability of placing an order increases as s increases Such a result might be possible if assumptions regarding the demand function are made Further if both the probability of placing an order and the average stock level increase then both the ordering cost and the holding cost would increase No proofs are offered here

48

of the policy is calculated if the cost of the policy is less than the current lowest cost the new

policy becomes the current best Finally S is increased by one and the process repeats

provided S does not exceed some predefined value Sm a x The flow chart of the algorithm

incorporating the heuristic search is shown in Figure 5 Note that the algorithm begins by

attempting to improve on the lower bound Sm i n the policy (Sm i n-lSm i n) is evaluated and if it

is not feasible S m i n is increased by one The improvement continues until a feasible policy is

found

As previously mentioned a heuristic search must be evaluated in terms of both its speed

and the accuracy of its solutions To evaluate the heuristic on the basis of speed the same

approach used to evaluate the previous algorithms is employed The columns entitled

Heuristic in Tables 2 and 3 contain the various execution times of the heuristic The heuristic

is a vast improvement over the previous search techniques the reduction in execution time is

enormous and becomes more so as S m a x increases For a S m a x of 20 the reduction is

approximately 75 of the time required by the grid search algorithm29 and for a S m a x of 60

that reduction soars to over 95 More importantly the heuristic can search a grid with a

of 40 in under 4 seconds or a grid with a S m a x of 60 in under 12 seconds These low execution

times should be sufficient to allow for a large number of products to be analyzed in a

reasonable amount of time

In order to further assess the practicability of the heuristic the total time required by the

heuristic to find solutions for the entire product category was determined and compared to the

corresponding time for the grid search algorithm With a priori knowledge of the optimal

solutions conservative yet reasonable values of S m a x were selected and used by both of the

2 9 Hereafter grid search algorithm refers to the algorithm utilizing the grid search the updating technique and the lower bound Sm i n

49

Calculate the demand probability functions

Calculate Smin S = Smin s = S-l

S = S+l s = S-l t

Update the probability transition matrix for (sS)

No Is service level pound 0975 Yes

No Is service level ^ 0975

Yes

Is cost lt cost No

Yes

Is s lt S-l No

Yes

s = s+l

Update the probability transition matrix for (sS)

Is service level pound 0975

Yes

Is cost lt cost No

Yes

s = s S = S cost = cost

s = s-l

No

Update the probability transition matrix for (sS)

S = S+l

Is s 2 0 Yes

Update the probability transition matrix for (sS)

Is service level pound 0975

No

s = s+l

No

No

Is S = Smax

^Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Print results (sS) cost

Figure 5 The flow chart of the heuristic algorithm

50

algorithms30 The grid search algorithm required approximately 9 hours and 41 minutes to

find the solutions while the heuristic required only slightly more than 23 minutes Based on

speed the heuristic is preferable to the grid search algorithm

Finally to evaluate the heuristic based on the accuracy of its solutions its solutions were

compared to the previously obtained optimal policies Without exception the heuristic

algorithm found the optimal policy Although there is no guarantee that the heuristic will find

the optimal solution for different products or different demand functions the heuristic appears

to work extremely well Based on the fast execution time and the apparent accuracy of the

solutions the heuristic will be used in the development of the inventory tool

3 0 For the test the following values of Sm a x were used 20 (273 products) 30 (19) 50 (23) 100 (3) and 125 (2)

51

V T H E I N T E R F A C E

With the model formulated and the algorithm developed the only remaining task in the

development of the prototype is the creation the interface In fact two interfaces are created

one that displays results in terms of minimum fills and target levels and one that displays

results in terms of reorder points and target levels To be consistent with the previous chapters

the latter interface is described here31

In its present form the interface allows the manager to set review parameters to

compare optimal and current policies and to evaluate alternate policies From the main menu

the manager selects the product store duration of the review period length of the lead time

and value of Sm a x Once the parameters are set the system calculates the optimal policy and

then evaluates the current policy displaying results as they are obtained In particular the

reorder point target level expected annual cost and expected customer service level are

displayed for each policy If a current policy meets the desired customer service level the

expected annual savings of the optimal policy are displayed in terms of both dollars and

percentage of current cost otherwise the value of the current customer service level is

displayed in red to indicate that it is deficient Finally the manager can choose to evaluate

alternate policies or to return to the main menu

Although the prototype is far from the ultimate integrated inventory control system it

is still valuable to management As mentioned in Chapter II inventory policies at Canadian

Tire Pacific Associates are currently based on rules of thumb that incorporate marketing

3 1 Because significant changes to the interface are expected as feedback from management is received and as the system is brought on-line only a brief description of the present interface is given

52

strategies The prototype provides management with a means of determining the proportion of

the cost of inventory that is not attributable to meeting customer demand For example

consider a hypothetical consultation in which the manager chooses to analyze product 200001

a 30-amp inline fuse at store 6 The prototype displays to the manager the current policy of

(1214) which provides a customer service level of 1000 at an annual cost of $748 and the

optimal policy of (39) which provides a customer service level of 977 at an annual cost of

only $319 If for some reason the manager feels the optimal policy is undesirable he or she

can investigate the additional cost of holding excess inventory Suppose the manager chooses

to evaluate the policy (610) The prototype evaluates the policy and displays the results the

customer service level is 998 and the annual cost is $441 With the alternate policy

virtually all demand is satisfied and the cost of inventory is still reduced by over 40 Thus

the manager ascertains that approximately 40 of the current cost of inventory for this product

is directly attributable to marketing considerations Using the prototype in this way

management can begin to understand better the nature of its inventory costs A description and

graphical depiction of the above hypothetical consultation are presented in Appendix H

As for the technical details of the system the prototype can run on any computer using

the DOS operating system having approximately 420K of available conventional memory and

attached to a V G A or SVGA monitor Further the prototype can be used on a machine without

a math coprocessor however performance is slower The prototype consists of two modules

linked together into one executable program The interface module contains the various

interactive subroutines and the main module contains the methodology of the model and the

logic of the heuristic search Both modules are coded in Fortran and compiled using Microsoft

Fortran 50 which provides many extensions to standard Fortran The source codes for the

interface module and the main module are listed in Appendix I and Appendix J

53

respectively

In addition to the above technical specifications the program requires the presence of

certain files in certain directories The program uses the distribution files mentioned in

Chapter II to read the unit costs and the empirical demand functions Each product has a

unique distribution file the name of which is the six digit product number followed by the

extension dst The text file containing the existing policies at each store must be located in

the same directory as the distribution files Finally the file tmsrbfon which contains the

font used by the interface must be located in the directory from which the program is executed

In its future versions the system should be more automated and more user-friendly

Rather than having the manager choose an upper bound for the search the algorithm should

incorporate a method of determining Sm a x For this to happen either a theoretical upper bound

must be obtained or a heuristic for terminating the search must be developed Also the amount

of time spent waiting for calculations to be performed can be reduced by reading the optimal

and current policies and their associated costs and customer service levels from a file or

database This solution file could be created any time prior to the consultation by running

the system in a batch mode The interface should allow the manager to change cost estimates

and service level values view results graphically and print results and summary documents

Finally the system should provide an option for automatically incrementing the product andor

store during a consultation This option would enable a manager to review quickly the

inventory policy of a product at each store

54

VI R E S U L T S AND SENSITIVITY ANALYSIS

A Comparison of Current Policies to Optimal Policies

In order to estimate the potential savings of implementing optimal policies the solutions

obtained in the Chapter IV are analyzed in more detail For each product the current policy

is evaluated and then compared to the optimal policy with respect to cost and customer service

level Finally the potential savings of the optimal policies are extrapolated to all products and

all stores

Of the 420 products examined each of 37 products has an existing policy that fails to

provide the desired customer service level of 975 However most of these policies just

fail to do so only 4 policies provide a customer service level of less than 90 Table 4

provides a list of these 37 products each with its current customer service level its current

annual cost and the additional cost of its optimal policy3 2 Two comments regarding Table

4 are in order First the current customer service level for product 203312 an amber

driveway marker is listed at a misleading 303 Examination of the demand distribution

reveals that this product was sold on only 1 of the 307 days during which sales data were

collected and on that day 13 of the markers were sold Clearly this data point constitutes

either an error or an unusual event the current policy of (34) does not allow for more than 4

items to be sold per review period Second the additional cost of the optimal policy for each

of 5 products is negative In other words for each of these products the current policy which

fails to provide the desired customer service level costs more than the optimal policy which

3 2 This is the difference between the expected annual cost of the optimal policy and the annual cost of the current policy It is the minimum additional cost required to increase the customer service level to at least 975

55

Table 4 Products with insufficient existing policies

Product Current CSL ()

Current Annual Cost ($)

Additional Cost of the Optimal Policy ($)

206932 974 204 -051

204570 973 222 -025

203053 973 248 009

206550 972 118 042

206547 972 105 035

206565 972 146 056

203702 972 340 157

206876 972 155 061

205752 972 330 153

205922 972 108 036

206857 972 180 073

205393 972 190 -003

201513 971 345 015

203405 970 255 037

203069 968 295 -048

203071 967 321 -001

205635 967 390 029

203914 966 460 160

204601 962 252 061

203326 960 144 014

202003 958 876 437

206516 958 112 029

206517 958 283 120

206350 958 333 147

205932 958 180 066

203936 952 598 294

56

Product Current CSL ()

Current Annual Cost ($)

Additional Cost of the Optimal Policy ($)

206932 974 204 -051

203939 948 413 193

203977 945 1216 1242

206551 945 132 077

205389 934 598 047

203590 927 1223 607

206567 919 122 070

203591 905 1228 834

205390 887 291 074

203318 833 136 084

206393 776 319 289

203312 303 237 511

provides at least the desired customer service level For the 36 products the total additional

cost of the optimal policies is $542033

Of the remaining 383 products most have an existing policy that provides a customer

service level well above 975 When the customer service levels are rounded to one decimal

each of 253 products has a customer service level of 1000 and each of 347 products has a

customer service level of at least 990 In contrast each of only 58 products has an optimal

policy that yields a customer service level of at least 99034 For each of 37 products the

3 3 The cost associated with product 203312 is not included in the total

3 4 Optimal policies do not have customer service levels of exactly 975 because the function is discrete For low-demand products a unit step in s or S results in a relatively large change in the customer service level

57

current policy is identical to the optimal policy

The extremely high level of service provided by most policies allows for substantial

savings in the cost of inventory With current policies the estimated annual cost of inventory

for all 420 products is $420743 whereas the expected annual cost of the optimal policies is

only $186884 This amounts to a savings of approximately $2340 or over 55 of the current

cost of inventory

In order to further assess the savings the difference between the costs of the current

policies and the costs of the optimal policies are examined in more detail For each of the 383

products whose existing policy provides a customer service level above 975 the difference

between the estimated annual cost of the existing policy and the expected annual cost of the

optimal policy is calculated The resulting histograms of the savings in (a) dollars and (b)

percentage of current cost are shown in Figure 6 The particulars for the current and optimal

policies for all 420 products are provided in Appendix K

As evident from Figure 6(a) the majority of the expected savings in absolute terms

comes from only six products In order to examine these products more closely the current

and optimal policies along with the associated costs and savings for each of the six products

are listed in Table 5 Each current policy is very different than its corresponding optimal

policy resulting in excessive inventory for each product In fact over half of the total expected

savings could be realized by implementing the optimal policy just for each of these six products

Interestingly five of the six products are sealed beam headlights

58

90

10 - 20 30 - 40 50 - 60 70 - 80 90- 100

Savings ()

(b)

Figure 6 The distribution of savings of optimal policies in (a) dollars and (b) percentage of current cost

59

Table 5 Products with the largest potential absolute savings

PRODUCT

CURRENT OPTIMAL SAVINGS

PRODUCT s S Cost ($)

C S L

()

s S Cost ($)

C S L () ($) ()

2036801 194 200 64014 1000 37 48 bull 9245 975 54769 856

2036772 94 100 23652 1000 18 22 3850 978 19802 837

2036723 78 90 23043 1000 19 23 4454 975 18589 807

2030944 145 175 19522 1000 34 45 3727 975 15795 809

2036425 48 60 10491 1000 13 17 2682 977 7809 744

2036556 68 80 7265 1000 14 22 1699 975 5566 766

1 H6054 rectangular sealed beam headlight 2 H4656 rectangular sealed beam headlight 3 6053 rectangular sealed beam headlight 4 9004 halogen bulb 5 4652 rectangular sealed beam headlight 6 6104 7 sealed beam headlight

Although Figure 6(a) seems to indicate that the majority of the current policies are close

in cost to the optimal policies this is very misleading For example suppose the optimal policy

for a product costs $1 and that the current policy costs $2 In absolute terms the optimal

policy saves only $1 but in relative terms the optimal policy saves 50 of the current cost

When savings are viewed in this manner as in Figure 6(b) it is clear that substantial savings

can be realized for the majority of the products In fact both the mode and the mean of the

savings are between 40 and 50 of the current costs

In order to extrapolate the above savings to obtain the potential savings for product

category 20 at all 21 stores the differences between stores must be considered During the

60

pilot study the current policy for each of several products at each size of store was evaluated

The study suggested that medium and large stores have a similar amount of excess

inventory and that jumbo and extra jumbo stores have much more excess inventory One

could reasonably expect that the additional excess inventory allows for higher savings to be

realized at the jumbo and extra jumbo stores Thus because 18 of the 21 stores are

medium or larger the expected 40-50 savings at store 6 can be viewed as a lower bound

on the potential savings to Canadian Tire Pacific Associates from implementing optimal policies

for product category 20 in all its stores

Finally in order to obtain the overall potential savings the expected savings from

product category 20 must be extrapolated to all product categories Ideally the proportion of

inventory comprising each product category and the relative savings that can be realized for

each category would be used to obtain a good estimate of the overall potential savings

However as this information is not known only a crude extrapolation is possible Thus the

analysis proceeds under the assumption that a 40-50 savings can be realized in each product

category35

Before an estimate of the total potential savings in absolute terms can be obtained the

proportion of the current inventory to which the savings apply must be determined Up to this

point the analysis has been concerned solely with reducing inventory at the stores However

it is inappropriate to perform further analysis without considering the entire inventory system

including the central warehouse Because the lead time for orders from Toronto is highly

variable and considerably greater than those at individual stores Canadian Tire Pacific

3 5 During the pilot study the current policies for a paint brush and an ironing board were evaluated and found to result in excess inventory similar to that of product category 20 Although this in no way justifies the assumption it does lend some support

61

Associates must keep a fairly high safety stock at the warehouse Without knowing the

proportion of inventory serving as the safety stock it is inappropriate to assume any reduction

in inventory at the warehouse is possible Therefore the relative savings of implementing

optimal policies is applied only to the inventory held in the stores36 Of the estimated $28

million in inventory approximately half is held in the stores therefore the 40-50 savings

translates to a savings of approximately $55 to $7 million per year

B Sensitivity Analysis on the Ordering Cost and Holding Rate

The above analysis was based on estimates of the ordering cost and the annual unit

holding rate37 provided by management As described in Chapter II these estimates

incorporate a number of costs that are difficult to quantify or that are speculative Therefore

there is some question as to the accuracy of the estimates In this section errors of +$0015

in the ordering cost and +5 in the holding cost are assumed in order to determine the

sensitivity of the model to errors in these variables38

The analysis proceeds as follows Based on an ordering cost of $0080 and a holding

rate of 30 the optimal policies from the previous section were obtained39 Then assuming

3 6 Of course this assumes that the excess inventory currently held in the stores is not required as part of the safety stock Based on the large amount of inventory at the warehouse this assumption seems reasonable

3 7 Hereafter simply referred to as the holding rate

3 8 An error of +$0015 in the ordering cost implies an actual ordering cost of $0070 a relative error of approximately 21 and an error of -$0015 implies an actual ordering cost of $0100 a relative error of 15 An error of +5 in the holding rate implies an actual rate of 25 a relative error of 20 and an error of -5 implies an actual holding rate of 35 a relative error of approximately 17

3 9 These policies are referred to as optimal policies for the duration of this chapter

62

the actual ordering cost andor holding rate are different from managements estimates the

heuristic is used to obtain the true optimal policies Finally the optimal policies are

evaluated based on the actual ordering cost and holding rate and then compared to the true

optimal policies For each possible combination of ordering cost and holding rate three

numbers are recorded the number of products with an optimal policy that differs from its

true optimal policy the total cost of implementing the optimal policies given the actual

ordering cost and holding rate and the difference between the cost of implementing the

optimal policies and the cost of implementing the true optimal policies The results of the

analysis are summarized in Table 6

Table 6 A comparison of optimal policies to true optimal policies

Actual Ordering Cost

$0070 $0085 $0100

25 5 products $155439

$002

51 products $160590

$092

88 products $165464

$449

Actual Holding Rate

30 54 products

$181470 $182

48 products $192026

$088

35 79 products

$207270 $574

48 products $212940

$136

2 products $218312

$000

4 0 The table should be read as follows If the actual ordering cost is $0070 and the actual holding rate is 25 then each of 5 products has an optimal policy that is different from its true optimal policy The total cost of implementing the optimal policies is $155439 which is $002 more than implementing the true optimal policies

63

The model appears to be quite insensitive to relative errors in the order of 20 Under

the worst case scenario when one of the variables is over-estimated and the other is undershy

estimated the optimal policies result in a total cost of inventory which is only approximately

3 higher than the cost of the true optimal policies Further because demand is not changed

by errors in these estimates each of the optimal policies still provides a customer service

level of at least 975 under all of the scenarios Finally although the cost of the optimal

policies varies significantly for the different scenarios the relative savings remain virtually

constant Table 7 lists the cost of existing policies and in parenthesis the relative savings of

the optimal policies for each of the different combinations of ordering cost and holding rate

The cost of the existing policies varies in the same manner as that of the optimal policies the

relative savings for each of the scenarios remains around 55 Therefore even in the presence

of such errors in the ordering cost and holding rate the projected savings of implementing the

optimal policies would be unchanged

Table 7 The cost of existing policies and the relative savings of the optimal policies under various scenarios

Actual Ordering Cost

$0070 $0085 $0100

25 $350153 (56)

$358464 (55)

$366764 (55)

Actual Holding Rate

30 $412438 (56)

$429048 (55)

35 $474725 (56)

$483026 (56)

$491320 (56)

64

C Sensitivity Analysis on the Demand Probability Mass Function

In this section the sensitivity of the model to slight changes in the demand probability

mass function is investigated As described in Chapter II the demand function used in the

model is based solely on observed sales however this empirical function is only one possible

instance of the underlying true demand function For example had the data been collected

over a period both starting and finishing one month earlier many of the empirical functions

would be slightly different Consequently depending upon the degree to which the empirical

functions vary and the sensitivity of the model to these variations the optimal policies from

the model might be inaccurate For example recall the case of the amber driveway marker

only thirteen markers were sold and all were sold on the same day Because the empirical

demand function is used the model calculates an optimal policy of (1213) clearly this

policy is impractical Although this example is extreme it does indicate the danger of using

unrealistic empirical demand functions41

As discussed in Chapter II three classes of products were identified namely low

medium and high-demand products For this analysis the same method of classification

is used Two products are selected from each of these groups and two new distribution files

are created for each of the six products For each product one of the files contains two

additional days of data with each day having a demand equal to the maximum observed demand

for that product In the second file only one additional day of data is added but the demand

on that day is two more than the maximum observed demand The situations that could give

4 1 Certainly a rigorous analysis to determine both the extent of variation in the empirical demand functions and the effect of using the empirical functions needs to be performed However such an analysis is beyond the scope of this thesis only a brief analysis that might suggest the degree to which the model is sensitive is performed here

65

rise to such changes in the demand data are termed Scenario 1 and Scenario 2 respectively42

The analysis proceeds in a manner similar to the one in the previous section Given the

demand that was originally observed the optimal policies were obtained Now assuming the

true demand function corresponds to the demand under Scenario 1 the true optimal policy

for each product is calculated Then each optimal policy is re-evaluated using the true

demand function and compared to the true optimal policy The analysis is then repeated

assuming the true demand function corresponds to the demand under Scenario 2 Table 8

lists the results for each product and scenario

Table 8 True optimal and optimal policies for each demand scenario

Product 202101 (low demand)

Scenario s s Cost ($) C S L ()

1 True optimal 1 2 470 993

Optimal 1 2 470 993

2 True optimal 2 3 663 976

Optimal 1 2 461 875

Product 203507 (low demand)

Scenario s S Cost ($) C S L ()

1 True optimal 0 3 334 981

Optimal 0 2 258 972

2 True optimal 2 3 508 980

Optimal 0 2 273 702

4 2 To be clear the scenarios are defined as follows Scenario 1 The addition of two days each with a demand of max Scenario 2 The addition of one day with a demand of max+2

where max is the maximum observed daily demand for each product in the sample period

66

Product 201032 (medium demand)

Scenario s S Cost ($) C S L ()

1 True optimal 6 12 1377 976

Optimal 6 11 1338 973

2 True optimal 6 13 1426 977

Optimal 6 11 1338 970

Product 203039 (medium demand)

Scenario s S Cost ($) CSL ()

1 True optimal 5 19 256 975

Optimal 5 16 251 969

2 True optimal 5 20 258 976

Optimal 5 16 249 968

Product 203677 (high demand)

Scenario s S Cost ($) CSL ()

1 True optimal 18 23 3981 977

Optimal 18 22 3817 972

2 True optimal 18 23 3995 978

Optimal 18 22 3831 973

Product 203680 (high demand)

Scenario s S Cost ($) CSL ()

1 True optimal 38 49 9498 976

Optimal 37 48 9175 971

2 True optimal 37 49 9484 975

Optimal 37 48 9206 972

67

The model is most sensitive to changes or errors in the demand functions of low-

demand products When only one additional day of demand for max+2 units is added to the

data the optimal policy for each of the low-demand products results in a very low customer

service level In fact for one of the products the customer service level drops to almost 70

The increase in demand does lower the customer service level for each of the other the

products but that decrease is slight

The above analysis indicates the need for some kind of smoothing or modelling of the

demand data especially for low-demand products The inherent variation between two

instances of a true demand function could very well result in more pronounced changes than

those resulting from either of the above scenarios Therefore it is likely that many optimal

policies especially those for low-demand products would actually provide a much lower or

much higher customer service level than expected However by smoothing the data the

variation between instances of a true demand function can be reduced and the desired

customer service levels can be achieved

For example suppose that the demand for a product is stationary that the average

demand during the previous period was unusually low and that the optimal policy for this

period is based on the demand during the previous period Because the demand is stationary

the probability that the demand during this period is higher than that during the past period is

more than 50 In other words the probability that the demand during this period is higher

than that for which the optimal policy provides the desired expected customer service level

is greater than 05 Thus expected customer service level for this period is actually less than

that specified Now suppose instead that the optimal policy is based on smoothed data which

reflects very closely the expected demand The probability that the demand during this period

is higher than that for which the optimal policy provides the desired expected customer service

68

level is virtually 05 Thus as a result of smoothing the data the specified customer service

level should be expected

69

VII C O N C L U S I O N

In this thesis we developed a single-product single-location model of the inventory

system at Canadian Tire Pacific Associates Specifically we formulated a Markov chain model

to determine the long-run average cost of inventory for fixed values of s and S and then

developed a heuristic search to find near optimal policies quickly Also as a first step toward

developing an integrated interactive inventory control system we created a prototype of the

interface

According to the analysis performed here Canadian Tire Pacific Associates is currently

holding an excessively large inventory in its stores This study finds that Canadian Tire Pacific

Associates could reduce the cost of inventory for product category 20 at store 6 by

approximately 50 and still maintain customer service levels of at least 975 Further the

mean and modal values of savings for these products are between 40 and 50 Given the

similar excess inventory at the other stores and in all product categories we believe that savings

in this range are possible for all products and stores We estimate that the application of

inventory control as described in this thesis would result in annual savings of $55 to $7

million

Given the large potential savings to Canadian Tire Pacific Associates we recommend

that management proceed with the next phase of the project Specifically we recommend that

the model algorithm and interface be refined further that a forecasting model be developed

and that a data collection and storage scheme be designed

In order to reflect operational changes at Canadian Tire Pacific Associates subsequent

to the formulation of the model modifications must be made to allow for the lead time to

exceed the review period Several of the stores now have three picks per week and as a

70

result orders placed for these stores at one decision epoch do not arrive until after the next

decision epoch Because this violates one of the models assumptions the model must be

modified accordingly Note however that with three picks per week and a review period

of four days orders from only the previous decision epoch are ever outstanding at the current

decision epoch

Also the model should be refined to reflect the fact that the review period is not fixed

over the planning period In its present form the model assumes a fixed review period

however inventory at stores with two picks per week is actually reviewed on a four day -

three day cycle and inventory at stores with three picks per week is reviewed on a three day

- two day - two day cycle Rather than obtaining the steady state costs for fixed review period

the model should calculate the steady state cost for a period of one week during which multiple

replenishments occur Although the solutions are not expected to differ significantly as a result

of the change the model would reflect the actual process more accurately

Further work is also required on the search technique for locating near optimal

policies Although the heuristic search described in this thesis appears to work well the

decision maker must provide an upper bound on S prior to the search In practice this upper

bound is generally not known in advance Thus if the heuristic is to be used either a tight

theoretical bound must be found or a termination heuristic must be developed Alternatively

a new heuristic search could be developed based on additional or different assumptions

regarding the properties of the feasible space For example although some regions of non-

convexity were found for some products much of the feasible space appeared to be convex

A heuristic that assumes a convex feasible space would not require an upper bound on S and

might perform very well However before such a heuristic is implemented further work is

necessary to determine the degree of non-convexity for a variety of demand functions and

71

ordering and holding costs

In addition to making the above modifications we recommend that Canadian Tire Pacific

Associates begin work on the remaining two components of the inventory control system First

in order to allow for products with seasonal demand the system requires a forecasting model

to provide accurate estimates of demand for the coming period Second the system needs a

data collection and storage module that systematically collects sales data for each active

product and for each store and then updates the corresponding demand distributions Because

approximately 25 of demand is expected to result in lost sales under the optimal policies the

sales data will have to be adjusted accordingly in the future Also because the model appears

to be sensitive to slight changes in the demand distribution for low-demand products the

module should incorporate a technique for smoothing the empirical distributions for these

products Further work should be undertaken to determine the extent of variation in the

empirical distributions and the most appropriate smoothing technique However because of

the large amount of data required to forecast seasonal demand and because of the current lack

of historical data we recommend that management proceed immediately with the design of a

data collection scheme

Finally two additional issues that are beyond the scope of the next phase should be

addressed at some point First a monitoring system will be required once inventory control

is implemented in the stores The monitoring system will ensure that optimal policies perform

as expected and will indicated whether or not demand is adversely affected by the lower levels

of in-store inventory Second given the degree to which in-store inventory can be reduced

inventory control in the warehouse should be investigated Because approximately half of the

existing inventory is held in the warehouse the additional savings from reducing system-wide

inventory could be great

72

Afterword

Just prior to the completion of the work described in this thesis Mr Don Graham

principle owner of Canadian Tire Pacific Associates sold his shares in the company Shortly

thereafter the results of this thesis were presented to the new management group

Unfortunately the new management did not consider development of an inventory control

system to be a priority Nonetheless the prototype was delivered to Canadian Tire Pacific

Associates for possible in-house testing however at the time of this writing - almost one year

after the meeting - Canadian Tire Pacific Associates has not decided to continue with the

project

73

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74

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75

Appendix A

A Portion of the Sales File for Product 200001 a 30-amp Inline Fuse

200001 283 1190 0 0 2 0 0 1 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 200001 284 1190 2 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 200001 285 1190 0 0 4 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 286 1190 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 200001 287 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 200001 288 1190 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1 0 2 200001 289 1190 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 290 1190 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 200001 291 1190 0 0 2 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 200001 292 1190 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 200001 294 1190 0 0 0 1 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 1 200001 295 1190 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 200001 296 1190 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 200001 297 1190 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 200001 298 1190 0 0 1 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 200001 299 1190 0 0 1 0 0 0 1 0 1 0 0 2 0 0 0 0 0 2 1 1 0 200001 300 1190 0 0 1 0 0 0 0 1 0 0 0 0 0 2 2 0 0 0 0 3 0 200001 301 1190 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 200001 303 1190 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 200001 304 1190 1 0 0 0 0 0 0 0 1 0 0 2 0 0 0 0 0 0 0 0 0 200001 305 1190 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 200001 306 1190 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 2 0 0 1 0 200001 308 1190 0 0 0 1 0 1 0 0 0 2 0 1 0 0 0 0 1 1 1 0 0 200001 309 1190 1 3 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 200001 310 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 200001 311 1190 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 200001 312 1190 0 1 1 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 200001 313 1190 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 2 0 200001 314 1190 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 200001 315 1190 0 0 2 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 200001 316 1190 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 200001 317 1190 2 0 0 1 0 0 0 0 0 0 5 1 0 0 0 1 0 0 0 0 0 200001 318 1190 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 1 200001 319 1190 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 200001 320 1190 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 200001 321 1190 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 1 200001 322 1190 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 323 1190 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 200001 324 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 0 1 0 0 2 1 1 0 200001 325 1190 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3 200001 326 1190 0 0 1 0 1 0 0 0 2 0 0 1 0 0 0 1 0 1 0 0 1 200001 328 1190 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 200001 329 1190 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 200001 330 1190 0 0 1 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 200001 331 1190 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 200001 332 1190 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 200001 333 1190 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 200001 334 1190 0 0 2 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 200001 335 1190 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 200001 336 1190 0 0 0 1 0 0 3 0 0 0 0 0 2 0 0 0 0 0 0 0 0

(253 subsequent lines omitted)

76

Appendix B

Sample Distribution Files

a) A low-demand product - an automobile back-up alarm

PRODUCT NUMBER 202101 PRICE $ 684

DMD

3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

N 2 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 2 2 0 1

1 5 3 8 12 5 7 11 6 11 5 4 11 10 19 4 2 5 14 5 6 4

0 302 304 299 295 302 300 296 301 296 302 303 295 296 288 303 305 301 290 300 301 302

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

b) A medium-demand product - a 168 bulb

PRODUCT NUMBER 203039 PRICE $ 51

DMD

16 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 1 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0

9 0 0 2 5 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0

8 0 0 2 5 0 0 2 0 1 1 3 0 1 3 0 0 1 1 0 0 1

7 2 1 3 1 0 0 1 0 2 3 1 1 4 1 1 0 2 1 1 1 1

6 4 7 1 10 3 1 3 2 1 3 5 2 7 4 1 0 1 1 3 3 4

5 1 7 7 17 0 2 11 1 5 4 1 2 6 9 6 2 6 9 4 4 1

4 5 10 10 16 5 5 19 10 13 8 11 9 13 10 12 7 5 15 5 8 3

3 10 12 13 27 4 6 24 11 10 13 14 9 23 14 18 8 14 18 14 12 14

2 14 30 36 43 16 16 43 32 38 39 39 34 47 36 41 21 19 39 20 34 24

1 30 56 68 56 37 30 70 48 67 61 44 75 62 63 48 28 52 61 63 53 47

0 240 184 165 126 240 247 133 202 169 175 189 173 143 165 180 240 206 161 197 192 212

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

77

c) A high-demand product - a H4651 rectangular sealed beam headlight

PRODUCT NUMBER 203677 PRICE $ 857

DMD

14 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

13 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 2 3 0 0 1 0 0 0 0 1 1 1 0 0 0 0 2 0 0

11 0 1 3 1 0 0 4 1 0 0 0 0 0 0 0 0 1 0 0 0 1

10 0 0 4 4 1 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 1

9 0 3 14 11 0 1 7 1 0 2 0 8 0 1 0 0 0 2 1 0 0

8 0 4 12 16 1 0 18 4 3 5 0 2 1 4 2 1 1 0 0 1 3

7 1 2 12 13 1 6 19 3 5 9 0 11 2 5 4 0 2 3 3 1 4

6 2 9 22 21 6 10 29 12 10 11 4 24 3 7 10 4 7 11 6 3 14

5 7 13 35 33 8 16 29 19 18 8 3 26 15 26 15 12 9 14 13 11 19

4 19 35 34 43 32 30 41 31 32 47 11 42 20 33 30 31 20 31 22 21 31

3 32 63 49 54 42 60 51 40 51 35 33 55 39 51 39 38 39 57 58 40 54

2 46 59 56 55 73 63 61 65 63 61 63 54 68 85 65 53 72 69 70 77 63

1 109 79 42 40 80 64 32 80 74 77 99 55 97 57 83 93 80 75 81 97 73

0 91 39 21 13 63 57 11 51 51 52 94 29 61 35 59 75 76 45 51 56 44

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

78

Appendix C

A Proof that the Limiting Probability Distribution is Independent of the Initial State

In this appendix we show that provided 0 lt PrDT = 0 lt 1 the Markov chain

corresponding to an inventory model with a (sS) policy has a limiting probability distribution

7T where 0 lt TTJ lt 1 for y = 0 1 S that is independent of the initial state Further the

limiting distribution is the unique nonnegative solution of the equations

Tt P = TI (AD

s and J gt = 1 (A 2)

i=0

where P is the (S + l) x (S + l) transition probability matrix whose first n rows (and columns)

correspond to the n recurrent states in the chain and whose remaining rows (and columns) if

any correspond to transient states43 Prior to the proof several preliminary results are

obtained

RESULT 1 The Markov chain has exactly one positive recurrent class and perhaps some

transient states

First note that in the absence of any absorbing states a Markov chain with a discrete

finite state space must have at least one positive recurrent class containing at least two states

Now consider the Markov chain for an inventory model with a (sS) policy

a) State i gt s cannot be absorbing because there is a positive probability of reaching

j lt i (Since PrDT gt 0 gt 0)

4 3 Note that since the rows and columns of P are reordered the elements of it must be reordered accordingly

79

b) State lt s cannot be absorbing because by the rules of the ordering policy and

the assumption that PrDT = 0 gt 0 there is a positive probability reaching S

Therefore the chain has no absorbing states and must have at least one positive recurrent class

Next consider the subset of states A = i | i lt s Because the state space is finite

and PrDT gt 0 gt 0 the process must return to at least one state in A in other words at least

one state in A must be recurrent Let k G A be a member of a recurrent class dt Because

of the ordering policy and the assumption that PrDT = 0 gt 0 k -raquo S Also by definition

there must be a positive probability of returning to k from S S -gt k Therefore S G 3w

Further because S is reachable from all states in A i G A is either transient with i -raquo dt or

recurrent with pound iR

Now consider the subset of states B = | j gt s Because PrDT gt 0 gt 0 the

chain will eventually reach a state i G A If i is transient then j must be transient also with

y -gt 3 if i G 9 then j is either transient with j -gt dt or recurrent with j E dt Thus the

Markov chain has exactly one recurrent class and perhaps some transient states

RESULT 2 The limiting probability distribution ir for the Markov chain corresponding to the

nxn submatrix Pm = [Ptj] i j G dt is independent of the initial state and is the

unique nonnegative solution of the equations

TC-Prr = re (A3)

and J2 K = 1 ( A 4 )

jt=i

where the rows and columns of are numbered from 1 to n rather than

Since if i j i and j must be in the same equivalence class

80

according to their associated states

First note that both the original Markov chain and the one corresponding to the

submatrix P R R are aperiodic because PrDT = 0 gt 0 Therefore given that 9 is a positive

recurrent aperiodic class the limiting distribution ir is uniquely determined by (A3) and (A4)

(Taylor and Karlin 1994 Theorem 44 p213) Further because there are no absorbing states

or other recurrent classes in the original chain the process must eventually enter the recurrent

class 9 regardless of the initial state Therefore in the limit the probability of being in state

i G 9 is independent of the initial state

RESULT 3 The limiting probability distribution for the transient states is the null vector

independent of the initial state

This result follows directly from the definition of a transient state

PROOF Suppose the states are reordered such that the first n rows (and columns) of P

correspond to the n recurrent states of the Markov chain Then P can be partitioned into four

submatrices

PRK - the n x n submatrix [Py] i j E 9

P T T - the (S + 1-I) x (S + l-n) submatrix [Py] i j g 9

P T R - the (S + l-n) x n submatrix [Py] i pound 9 j G 9

P R T - the n x (S + l-n) submatrix [Py] i G 9 j pound 9t By definition this is a matrix

of zeros

(Al) can then be rewritten as

81

(ltR TtT) P P L TR TT

I-P RR 0

p I-P TR 1 TT

(TtR TCT)

(0- 0T)

where irR denotes the limiting probability distribution for the n recurrent states and irT denotes

the distribution for the remaining S + l-n transient states Solving for TT t gives

TCr-0 + itT-(I-PTI) = 0T

itT-(I-PJJ) = 0 r

i c r = (I-P^-Oj

Since (I - PTT)1 always exists (Puterman 1994 Proposition A3 p 593) irT = 0X which is

in agreement with Result 3 Now solving for 7rR yields

(A2) can be rewritten as

bullRPRR 1 1R (A5)

(A6) S + l

k + L k =1

k=l k=n+

k=i

where ir contains the appropriately reordered steady state probabilities Since (A5) and (A6)

are equivalent to (A3) and (A4) respectively Result 2 holds Therefore the Markov chain

corresponding to an inventory model with a (sS) policy has a limiting distribution that is

independent of the initial state and is the unique nonnegative solution to (Al) and (A2)

82

Appendix D

A Measure for the Steady State Customer Service Level

In this appendix we show that the appropriate measure for the steady state customer

service level is given by

_ E[Demand satisfied during RP] E[Total demand during RP]

Consider the following two measures of the steady state customer service level

Measure 1 Suppose the customer service level is calculated only after n review periods have

passed Denoting DS as the demand satisfied in review period i and DT as the total demand

in period i the customer service level after n periods CSL(n) is

DS + DS0 + + DSbdquo CSLn) = - -

DT + DT + + DT

Dividing both the numerator and denominator by n taking the limit as n -raquo oo and applying

the Law of Large Numbers yields

CSL = lim CSL(n) n-degdeg

- [DS + DS + + DS] = l i m ^ J 1 1

raquo~ - [DT + DT + + DT]

E[DS]

E[DT]

83

Measure 2 Suppose the customer service level is calculated for each of n review periods and

then averaged Thus the customer service level after n periods CSL(n) is given by

CSL(n) = -DS DS

DT DT DT_

Taking the limit as n -gt oo yields

CSL = lim CSL(n)

lim mdash DSl DS2

DT DT DT_

DS

DT

Now consider the scenario in which there is one item on the shelf to begin each review

period and in which demand is stochastic with PrDT = 2 = Vi and PrD x = 100 = Vi

According to Measure 1 the customer service level is

CSL - = ^ W + ^ W = plusmn bdquo 002 E[DT) 1 (2) + 1(100) 51

while according to Measure 2 the customer service level is

CSL = E DS DT

1(1) + i(-Ls

22) + 2 100

51 200

026

Clearly from managements point of view the first measure is more indicative of the

actual level of service being provided The problem with the second measure is that equal

84

weight is given to each ratio regardless of the magnitude of the numbers For example with

respect to Measure 2 the events LSLR = 12 and LSLR = 50100 are identical however

with respect to lost sales the second event is much worse than the first Therefore the proper

measure for the steady state customer service level is Measure 1

85

Appendix E

The Calculation of the Conditional Expected Demand Not Satisfied During a Period of T Consecutive Days

Denoting D x as the demand during a period of T consecutive days and j as the inventory

on hand at the beginning of the period the conditional expected demand not satisfied during the

period E[(DT-y)+] is calculated as follows

E[(DT-j)+] = pound ltk-j)Pr[DT = k k=j+l

= pound (k-j)PrDT = k] + J2(k-j)PrDT = k] - pound (fc-)VJgtr = fc k=j+l k=0 k=0

= J2(k-j)Pr[DT=k -J2(k-j)PrDT = k] k=0 k=0

= YkPrDT=k) - J2jPrDT=k - Yk-j)PrDT=k) Jfc=0 k=Q k=0

= Ti - j - Y(k-j)PrDT=k 7-1

= Tx - j + YltJ-k)PrDT = k jt=0

where fx is the mean daily demand

86

Appendix F

Justification of the Updating Technique for the New Policy (s+mS)

Let the probability transition matrix corresponding to the policy (sS) be denoted by the

(S + l)x(S + l) matrix P with elements py Define s = s+m where m is an integer satisfying

the condition 0 lt s+m lt S Let the probability transition matrix of the new policy (sS)

be denoted by the (S + 1)X(S + 1) matrix 7gt with elements py Using the equations developed

in Chapter III the elements of P are found as follows

CASE 1 The lead time is less than the review period

A m gt 0

For 0 lt i lt s

( i-i pound [PrDL=kPrDT_LgtS-k]] + PrDLziPrDT_LplusmnS-i k=0

i-l 53 [PrDL=kPrDT_L = S-j-k]] + PrDLziPr[DT_L = S-j-i) k=0 s-j

52 [Pr[DL = kPrDT_L = S-j-kn k=0

ifj = 6

ifOltjz S-i

if S-i ltj ltS

a) 0 lt lt s

Pij

= lt Pij

Pij

= pij

b) s lt i lt s This does not simplify but it is identical to (2)

87

For s lt i lt S

PrDTgt i

p = lt PrDT = i-j

0

Pfj ifj = 0

= lt Pij ifOltjlti

Pij if i ltj plusmn S

= pij

B m lt 0

ifj = 0

if 0 ltj lt i

if i ltj plusmn S

For 0 lt i lt s

i-l 53 [PrDL=kPrDT_LplusmnS-k]] + PrDLgtiPrDT_LplusmnS-i

i-l E [PrZgti = fciVZ)7W=S--fc] +PrD tgtiP-D 7 t = S-7-

p y laquoy = o

P i j ifOltjS-i

vPij ifS-iltjS

ifj = 0

ifOltjltS-i

if S-i ltjltS

For s lt i lt S

fPrJgtrgt- 17=0 = PrDT = i-j] ifOltjlti

0 ifiltjplusmnS

88

a) s lt lt s This does not simplify but it is identical to (3)

b) s lt i lt S

PrDTgti

= bull PrDT = i-j]

0

17=0

if0ltjplusmni

ifiltjS

= Pt

ifj = 0

if0ltjlti

ifiltjplusmnS

= Pi

CASE 2 The lead time is equals the review period

A m gt 0

For 0 lt i lt s

0 if0ltjlt S-i

PrDTgti ifj = S-i

PrDT=S-j] ifS-iltjltS

a) 0 lt i lt s

= bull y Pij

Pij

ifOplusmnjltS-i

ifj = S-i

ifS-iltjS

= Pij

b) s lt i lt s This does not simplify but it is identical to (4)

89

For s lt i lt S

PrDTgti)

Pr[DT=i-j

0

laquof 7=0

ifOltjplusmni

if i ltj lt S

Pa

Pij

if 7 = 0

if 0 lty lt i

ifiltjltS

B w lt 0

For 0 lt t lt s

ro PrJrgt PrDT = S-j

ifOzjltS-i

ifj = S-i

if S-i ltj lt S

Pij

ifOzjltS-i

ifj = S-i

if S-i ltj ltS

= Pi

For s lt i lt S

PrDTgti

= PrDT = i-j

0

i 7 = o ifOltjplusmni

ifiltjltS

a) s lt i lt s This does not simplify but it is identical to (3)

90

PrDTzi ifj = 0

PrDT = i-j] ifOltjplusmni

0 ifiltjltS

Py ifj = 0

ptj ifOltjplusmni

Pij if iltj plusmnS

Appendix G

Justification of the Updating Technique for the New Policy (sS+m)

Let the probability transition matrix corresponding to the policy (sS) be denoted by the

(S + l)x(S + l) matrix P with elements p(gt Define S = S+m where m is any non-negative

integer Let the probability transition matrix of the new policy (sS) be denoted by the

(S+m+l)x(S+m + l) matrix IP with elements ptj Then using the equations developed in

Chapter III the elements of 7 are found as follows

CASE 1 The lead time is less than the review period

For 0 lt i lt s

pound [PrDL=kPrDT_LplusmnS-k] + PrDLi)PrDT^Sgt- k=0

i-l pound [Pr[DL = kPrDT_L = S-j-k- +Pr[DLiPrDT_L = S-j-i Jt=0

s-i Y [Pr[DL=kPrDT_L = S-j-k] k=0

i - l

ifj=0

ifOltjltS-i

ifS-iltjltS

pound [PrDL = kPrDT_LS-k + Pr[DLi)PrDT_LplusmnS-i) if j = 0 fc=0

i - l = pound [PrDL=kPrDT_L = (S+m)-j-k] + PrDLgtiPr[DT_L = (S+m)-j-i ifOltj lt (J5+m)-i

if (S+m)-iltj lt(S+m)

k=0

i - l

pound (Sm)-j

pound [PrDL = kPrDT_L = (S+m)-j-k]] k=0

92

I-l

53 [PrDL = kPrDT_LgtS-k] + PrDLziPrDT_LS-i ifj = 0 k=0

i-l

Y[PrDL = kPrDT_L = S-(j-m)-k] + Pr[DLziPrDT_L = S-(j-m)-i] ifOltjzS+m-i k=0 S-(j-m)

53 [PrDL = kPrDT_L = S-(j-m)-k]] if S+m-i ltj lt S+m

i - l

53 [PrDL = kPrDT_LzS-k] + PrDLziPrDT_LzS-i] ifj = 0 =0

i - l

13 [Pr[DL = kPrDT_L = S-(j-m)-k] + Pr[DLziPr[DT_L = S-(j-m)-i ifOltjz S+m-i k=o and 0 z j-m

Pi(j-m)

Pi(j-m)

if 0 ltj lt S+m-i and 0ltj-mltS-i

if S+m-i ltj lt S+m and S-i lt j-m zS

i - l

Y [PrDL = k)PrDT_LzS-k] + PrDLi)PrDT_LplusmnS-i ifj = 0

k=0 i - l

53 [PrDL = kPrDT_L = S-(j-m)-k] + PrDLziPrDT_L = S-(j-m)-i ifOltjzm k=0

Piij-m)

Piij-m)

if mltj lt S+m-i

if S+m-i ltj lt S+m

i - l

YlPrDL = kPr[DT_L^S-k + PrDLi)PrDT_LS-i) ifj = 0

i - l

53 [PrDL = kPrDT_L = S-j-k] + PrDLgtiPrDT_L = S-j-i ifOltj i m k=0

Pi if mltj ltS

which for 0 lt j lt m is identical to (2)

93

For s lt i lt S

Pu =

PrDTgti

PrDT = i-j]

0

if0ltjlti

ifiltjltS

a) s lt i lt S

PrDTgti 7=0

PrDT = i -j) if0ltjlti

0 if i ltj lt S

0 ifSltjltSgt

ifj=0

if 0 ltj lt i

if i ltj plusmn S

0 ifSltjltS

ifOltjltS

0 ifSltjltSgt

b) S lt i lt S This does not simplify but it is identical to (3)

CASE 2 The lead time equals the review period

For 0 lt i lt s

0 ifO lt jltS-i

PrDTgti ifj=S-i

PrDT=S-j ifS-iltjltS

94

0 ifOplusmnjlt(S+m)-i

PrDT ifj = (S+m)-i

PrDT = (S+m) -j) if (S+m)-i ltj ltL (S+m)

0 if 0 lt jltS+m-i

PrDTzi] if j = S+m-i

PrDT = S-(j-m) if S+m-i ltj lt S+m

0 if 0 ltj lt S+m-i and 0 ^ j-m

Pi(j-m) if 0ltjlt S+m-i and 0ltj-mltS-i bull

Pi(j-m) if j = S+m-i and j-m = S-i

Pi(j-m) if S+m-i ltj z S+m and S-i lt j-m ^S

0 if 0 zj ltm

Pi(j-m) if mlt jlt S+m-i lt

Piij-m) if j = S+m-i

if S+m-i ltj lt S+m

1deg if 0 ltj ltm

[PiV-m) ifmltjltS

which for 0 lt i lt s is identical to (3)

For s lt lt S

PrDTzi

= PrDT = i-j)

0

ifj = 0

if 0 ltj ltL i

ifKjzS1

95

a) s lt i lt S

PrDTzi

PrDT = i-j]

0

0

17=0

if0ltjplusmni

ifiltjS

ifSltjltSgt

Pij 7=0

Pij if0ltjii

Pij if i ltj ltL S

0 ifSltjltS

Pij ifOzjiS

0 ifSltjltS

b) S lt i lt S This does not simplify but it is identical to (3)

96

Appendix H

A Hypothetical Consultation

In this appendix the interface during a hypothetical consultation is depicted The figures

accurately represent the various screens that are presented to the manager with the exception

that all lines and text are really white on a blue background During the consultation described

below only the choices made by the manager are hypothetical all the results are real

The title screen as depicted in Figure A - l appears initially upon execution of the

program In order to proceed to the main menu the manager presses the enter key

O I C S

Optimat Inventory Control System

For the exclusive use of

Canadian Tire Pacific Associates

Developed at The Faculty of Commerce

and Business Administration The University of British Columbia

Brian Kapalka Kaan Katircioghi Marty Puterman

Press ltEntergt to continue

Figure A - l The title screen

97

Main Menu 1 Select data directory

ctircdatadst

2 Select product and store combination Product 200001

3 Select scenario Review Period 4 Lead Time 3

4 Select search parameter Smax 25

5 Execute program

6 Exit to DOS

Please enter the desired option 5

Figure A-2 The main menu

Figure A-2 depicts the main menu of the program it is from here that the manager

enters all the review parameters If the manager wishes to change one of the parameters from

its default value he or she selects the appropriate number and enters the correct information

in the window presented The new settings will then be shown in place of the default values

In this example the relevant data files are located in the default directory ctiredatadst and

the manager has selected the productstore combination of product 200001 and store 6 Further

the manager has selected a review scenario of a 4-day review period and a 3-day lead time and

an upper search limit of 25 Finally in order to proceed with the analysis the manager

executes the main module of the program by selecting option 5

98

Table of Contents

Abstract ii

Table of Contents iii

List of Tables v

List of Figures vi

Acknowledgement vii

I INTRODUCTION 1

II T H E INVENTORY SYSTEM A T CANADIAN TIRE PACIFIC ASSOCIATES 8 A Background 8 B The Problem 11 C The Project 15 D The Demand Data 17 E The Cost Data 18

III M O D E L FORMULATION 20 A Terminology 20 B Assumptions 21 C A Markov Chain Model 22 D The Transition Probability Matrix 25 E The Steady State Probabilities 27 F The Cost of Inventory 29 G The Customer Service Level 31 H A Methodology for Evaluating an (sS) Policy 34

IV A N ALGORITHM FOR OBTAINING T H E OPTIMAL (sS) POLICY 35 A Introduction 35 B A Grid Search 36 C A Technique for Updating the Transition Probability Matrix 41

A new policy (s+mS) 41 A new policy (sS+m) 42 The modified algorithm 43

D A Lower Bound on S 45 E A Heuristic Search 48

V T H E INTERFACE 52

iii

VI RESULTS A N D SENSITIVITY ANALYSIS 55 A Comparison of Current Policies to Optimal Policies 55 B Sensitivity Analysis on the Ordering Cost and Holding Rate 62 C Sensitivity Analysis on the Demand Probability Mass Function 65

VII CONCLUSION 70

Afterword 73

Bibliography 74

Appendix A A Portion of the Sales File for Product 200001 a 30-amp

Inline Fuse 76

Appendix B Sample Distribution Files 77

Appendix C A Proof that the Limiting Distribution is Independent of the

Initial State 79

Appendix D A Measure for the Steady State Customer Service Level 83

Appendix E The Calculation of the Conditional Expected Demand Not Satisfied

During a Period of T Consecutive Days 86

Appendix F Justification of the Updating Technique for the New Policy (s+mS) 87

Appendix G Justification of the Updating Technique for the New Policy (sS+m) 92

Appendix H A Hypothetical Consultation 97

Appendix I The Source Code for the Interface Module 104

Appendix J The Source Code for the Main Module 119

Appendix K Current and Optimal Policies for Product Category 20 at Store 6 131

iv

List o f Tables

Table 1 Location and particulars of the 21 stores 10

Table 2 Execution times of the algorithms for product 200001 a 30-amp inline fuse 39

Table 3 Execution times of the algorithms for product 206917 a 6

solder connector 40

Table 4 Products with insufficient existing policies 56

Table 5 Products with the largest potential absolute savings 60

Table 6 A comparison of optimal policies to true optimal policies 63

Table 7 The cost of existing policies and the relative savings of the optimal policies under various scenarios 64

Table 8 True optimal and optimal policies for each demand scenario 66

v

List of Figures

Figure 1 A typical sample path of the process with a 4-day review period

and a 2-day lead time 24

Figure 2 The flow chart of the grid search algorithm 38

Figure 3 The flow chart of the grid search algorithm utilizing the updating

technique for the transition probability matrix 44

Figure 4 An evaluation of the lower bound Sm i n 47

Figure 5 The flow chart of the heuristic algorithm 50

Figure 6 The distribution of savings of optimal policies in (a) dollars and

(b) percentage of current cost 59

Figure A-1 The title screen 97

Figure A-2 The main menu 98

Figure A-3 Calculating the optimal policy and evaluating the current policy 99

Figure A-4 Displaying the results 100

Figure A-5 Entering an alternate policy 101

Figure A-6 The particulars of the alternate policy 102

Figure A-7 The main menu revisited 103

vi

Acknowledgement

The completion of this thesis was made possible by the encouragement and assistance

of a number of people

I would like to express my sincere appreciation to my thesis supervisor Professor

Martin Puterman for all of his many efforts on my behalf His help advice patience charity

and tolerance were very much appreciated

I would like to acknowledge Professor Hong Chen and Professor Garland Chow for their

time and input as members of my thesis committee In addition I must acknowledge the

assistance of PhD student Kaan Katircioglu for his insight and help on this project

I offer many thanks to Professor Tom Ross for his kindness and friendship during the

past few years especially during the writing of this work I must also thank his family for

adopting me on many holidays I still owe thanks to Professor Slobodan Simonovic at the

University of Manitoba for influencing me to attend graduate school in the first place and for

helping me to obtain funding

I cannot thank my parents enough for their never-ending support both emotional and

financial Also to my friends especially Cathy Dave Lisa Steve and the Philbuds thanks

for giving me a life off campus and for picking up many a tab - the next one is on me

I would like to give special thanks to my good friend and fellow MSc student Paul

Crookshanks for allowing me to bounce ideas off of him and for being such a procrastinator

that despite my finishing a year late I only lost by two days

Financial assistance received from the Natural Science and Engineering Research

Council (NSERC) and from Canadian Tire Pacific Associates was greatly appreciated

vii

I INTRODUCTION

The importance of inventory management has grown significantly over the years

especially since the turn of this century In colonial times large inventories were viewed as

signs of wealth and therefore merchants and policy makers were not overly concerned with

controlling inventory However during the economic collapse of the 1930s managers began

to perceive the risks associated with holding large inventories As a result managers

emphasized rapid rates of inventory turnover (Silver and Peterson 1985) Following the

Second World War Arrow Harris and Marschak (1951) and Dvoretzky Kiefer and

Wolfowitz (1952ab) laid the basis for future developments in mathematical inventory theory

Shortly thereafter new inventory control methodologies were widely applied in the private

manufacturing sector More recently when inflation and interest rates soared during the 1970s

many organizations were forced to rethink their inventory strategies yet again Today the

control of inventory is a problem common to all organizations in any sector of the economy

Sir Graham Day chairman of Britains Cadbury-Schweppes P C L expressed this sentiment

when he stated I believe the easiest money any business having any inventory can save lies

with the minimization of that inventory (Day 1992) Lee and Nahmias (1993) and Silver and

Peterson (1985) provide a more detailed history of inventory management and control

Most inventory problems in the retail sector involve the control and management of a

large number of different products Ideally the inventory policies should be determined for all

products on a system-wide basis however because the number of products is often in the tens

of thousands formulating a model to do so is impractical Therefore single-product models

are frequently used in practice to determine policies for products separately (Lee and Nahmias

1993) Lee and Nahmias (1993) and Silver and Peterson (1985) provide a good overview of

1

single-product single-location models for a more complete review of the literature we refer

to them

The degree of complexity of a single-product model depends largely on the assumptions

made regarding the demand cost structure and physical characteristics of the inventory system

For example demand can be either deterministic meaning it is known with absolute certainty

or stochastic meaning it is unknown but follows a known probability distribution

Deterministic models tend to be relatively simple yet useful such as the basic EOQ model

However future demand is seldom known with certainty in practice and therefore the use of

these simple models is limited As a result the majority of research in inventory has focused

on stochastic demand models (Lee and Nahmias 1993)

Because the objective of most inventory models is to minimize the cost of inventory

assumptions regarding the structure of the cost are very important also Generally speaking

the cost of inventory has four components the purchaseproduction cost the orderingsetup

cost the holding cost and the shortage cost1 Each of these costs can be fixed proportional

nonlinear or some complex combination of these Further because nearly all inventory

problems require decisions be made sequentially over a period of time the costs can be either

discounted or averaged In the former case a discount factor is included in the cost function

in order to account for the time value of money However since most inventory problems are

on-going the long-run average cost is usually minimized (Lee and Nahmias 1993) In

practice this is achieved by minimizing the expected cost per time period under steady state

conditions

Finally the physical characteristics of the system are important in determining the

1 The cost of a stockout is often very difficult to estimate in practice As a result service levels are frequently used in its place When this is done a constraint is included in the model

2

complexity of the model For example the review process can be either continuous in which

case the inventory is reviewed continuously and orders are placed as soon as the stock reaches

a certain level or periodic in which case the inventory is reviewed at regular intervals and

orders can be placed only at those fixed times Also excess demand can be fully backordered

as is common in many manufacturing and wholesale systems completely lost as is common

in most retail environments or some combination of the two Lead time assumptions are also

important models with lead times are more complex than those in which the supply of

inventory is assumed to be instantaneous

Although the literature is dominated by single-product models (Lee and Nahmias 1993)

much recent work has focused on extending those models to multiple-product problems

However multiple-product models are inherently more complex than the simple-product

models In fact no efficient algorithms are currently available for determining fully optimal

strategies for most multiple-product problems (Federgruen 1993) As a result research has

focused on the development of easily computable approximation techniques Muckstadt and

Roundy (1993) provide a good overview of multiple-product andor multiple-location models

with deterministic demand although emphasis is placed on models for production and

distribution Federgruen (1993) extends the discussion to consider these models under uncertain

demand and also provides a complete list of references on the subject Silver and Peterson

(1985) discuss models for coordinated replenishments for both deterministic and stochastic

demand

In this thesis we examine the inventory system at Canadian Tire Pacific Associates

owner of 21 retail stores and a central warehouse Although this is a multiple-product

multiple-location single-warehouse problem and strictly speaking should be modelled as such

we analyze the inventory problem on a product-by-product and store-by-store basis We believe

3

that this simpler single-product single-location model captures the essentials of the problem for

two reasons First the coordination of replenishments for products andor locations is most

advantageous when significant savings are possible because of high first order costs and low

marginal costs for placing subsequent orders However because of the large number of

products and the physical distribution system in this problem trucks are dispatched to stores

on a regular basis Therefore the benefits of coordinating replenishments for this problem are

negligible Second we believe that controlling in-store inventory alone can result in substantial

savings to Canadian Tire Pacific Associates Further given the excessive amount of inventory

in the warehouse a single-location model assuming an infinite warehouse capacity is reasonable

Of course should inventory control in the warehouse become a priority a more elaborate model

would be required

More specifically we consider a periodic review single-product inventory model with

stationary independent stochastic demand a deterministic lead time and an infinite planning

horizon The probability mass function for the demand is based on empirical sales data

collected over a period of approximately ten months The cost of inventory includes a

proportional holding cost and a fixed ordering cost neither of which is discounted All excess

demand is completely lost The objective of the model is to minimize the long-run average cost

per unit time subject to a service level constraint on the fraction of demand satisfied directly

from in-store inventory In order to obtain meaningful cost comparisons the model calculates

the expected annual cost of inventory

For inventory problems like the one described above optimal (sS) policies can be

computed by Markovian decision process methods such as successive approximations or policy

iteration or by stationary Markovian methods However each of these methods generate

calculations that are complex and time consuming Therefore much research interest lies in

4

developing techniques to find approximate (sS) policies with little computational effort The

majority of the approximation techniques begin with an exact formulation of the problem and

then manipulate the problem by relaxations restrictions projections or cost approximations

(Federgruen 1993) Porteus (1985) presents three approximation techniques and compares

them to fourteen others for a periodic review single-product inventory model in which

shortages are fully backordered and a shortage cost is incurred Federgruen Groenevelt and

Tijms (1984) present an algorithm for finding approximate (sS) policies for a continuous review

model with backordering and a service level constraint Also Tijms and Groenevelt (1984)

present simple approximations for periodic review systems with lost sales and service level

constraints All three works find that the approximation techniques perform well However

the techniques are evaluated for a variety of demand parameters that all result in policies with

a value of S in the range of 50 to 200 Given the slower moving products at Canadian Tire 2

it is unclear how well such approximation techniques would perform Therefore we do not

attempt to apply or modify any of these approximation methods

Exact optimal policies for this problem can be obtained by formulating a constrained

Markov decision process and solving it However the data generating requirements and the

lack of a good upper bound on S make this formulation impractical Therefore we formulate

a Markov chain model for fixed values of s and S and then employ a search technique to find

the optimal values for s and S In order to improve the performance of the model we develop

a heuristic search that locates near optimal policies relatively quickly

The remaining chapters of this thesis are organized as follows In Chapter II an

overview of the existing inventory system at Canadian Tire is presented along with a description

2 For example very few of the products analyzed have optimal values of S exceeding 20 with the largest value of S equalling 69

5

of the problem Next a multi-phase project designed to develop a fully integrated interactive

inventory control system is described with the scope of the research reported in this thesis

completing the first phase of the project Finally a description of the demand and cost data is

given

In Chapter III we formulate a Markov chain model that can be used to calculate the cost

and service level associated with a given (sS) policy Based on several simplifying

assumptions the Markov chain is formulated and the equations for the corresponding transition

probability matrix are developed Then given the limiting probability distribution of the chain

the equations for the steady state annual cost of inventory and the corresponding service level

are obtained The chapter concludes with a brief summary of the model

In Chapter IV an algorithm that searches for the optimal (sS) policy is developed As

a starting point a simple grid search algorithm is developed and evaluated on the basis of

execution time In order to improve the algorithms speed a technique for updating the

transition probability matrix is presented and the improvement in the algorithm is noted Next

a theoretical lower bound on S is obtained although the corresponding reduction in the

execution time is minimal Finally based on two assumptions regarding the shape of the

feasible space a heuristic search is developed which improves the speed of the algorithm

significantly

In Chapter V a prototype of the inventory control systems interface is created In its

present form the interface allows managers to evaluate inventory policies interactively The

manager can set review parameters compare optimal policies and current policies and evaluate

alternate policies The chapter concludes by recommending specific options be including in

future versions to make the system more automated and user-friendly

In Chapter VI optimal policies for all products in one product category sold at one store

6

are compared to existing policies on the basis of expected annual costs and customer service

levels The analysis suggests that existing policies result in a large excess inventory and that

implementing the optimal policies would reduce the annual cost of inventory by approximately

50 A sensitivity analysis on the cost estimates is performed and suggests that the model is

not sensitive to relative errors of approximately 20 in the cost estimates Finally a sensitivity

analysis on the demand probability mass function is performed The analysis indicates that the

model is sensitive at least in terms of the customer service level to slight changes in the tail

of the demand distributions of low-demand products

Finally in Chapter VII the results of the analysis are summarized and future work is

outlined In order to incorporate recent changes in the review procedures at several stores the

model must be modified to allow for the duration of the review period to exceed the lead time

We recommend that the second phase of the project proceed specifically that a data collection

and storage scheme be designed and that a forecasting model be developed Because of the

substantial savings that can be realized by reducing inventory only at the stores inventory

control at the warehouse should also be investigated

7

II T H E I N V E N T O R Y S Y S T E M A T C A N A D I A N TIRE

A Background

Canadian Tire Pacific Associates owns and operates 21 retail stores in the lower

mainland of British Columbia The stores carry approximately 30000 active products3 that

are divided into 80 product categories The merchandise comprises primarily automotive parts

and accessories home hardware housewares and sporting goods Canadian Tire Pacific

Associates also owns a central warehouse in Burnaby at which inventory is received from

Toronto Inventory is then shipped periodically from the warehouse via truck to the local

stores In 1992 total sales at the stores exceeded $99 million and the value of inventory was

estimated to be $28 million of which approximately half was stored in the warehouse

On selected days of each week a stores inventory records are reviewed automatically

and orders for products are placed to the warehouse as required Management uses the term

flushing to describe the process of reviewing the inventory and producing the tickets for stock

owing and the term picking to describe the physical act of filling the order at the warehouse4

Whether or not an order is placed for a product depends upon its current inventory level its

target level and its minimum fill 5 If the target level less the current level exceeds the

minimum fill an order is placed to bring the stock up to the target level From the moment

3 Active products are those that are regularly restocked Canadian Tire Pacific Associates can choose from up to 50000 products

4 Note that the number of picks per week is equal to the number of times per week that the inventory records are reviewed At the beginning of this study eighteen stores had two picks per week while the remaining three stores had only one

5 The target level is the maximum number of items that should be on the shelf at one time and the minimum fill is the minimum number of items that should be ordered

8

the order is placed to the moment the product is reshelved takes between two to four days

depending on the store On average 11000 items are shipped from the warehouse at each

flush

Although a products minimum fill is the same for all stores its target level depends

upon the size of the store Based upon its sales volume and retail space a store is classified

as being one of five sizes small (S) medium (M) large (L) jumbo (J) and extra jumbo

(EJ) 6 Each product has associated with it a target level for each size of store

Consequently all stores of the same size have identical inventory policies Table 1 lists the

location size number of picks per week sales and floor space of each store as of October

1993

Periodically the inventory manager reviews the policies and when deemed necessary

adjusts them In this way the manager can account for seasonality in demand changes in mean

demand and any shortcomings of previous policies At the present time the review process

is not automated nor performed on-line The inventory manager uses Hxl5 computer

printouts with five products to a sheet to study the relevant information and on which to note

any changes Afterward the changes are recorded in the system and the new policies take

effect

Currently the inventory policies are based on rules of thumb developed through the

years and not on a statistical analysis of demand nor on a calculation of the cost of inventory

These rules of thumb incorporate a marketing philosophy which at times dominates the

inventory strategy The prevailing philosophy holds that a large in-store inventory instills

6 During the course of this study stores were resized several times At one point the entire class of medium stores was eliminated leaving only four sizes of stores On another occasion the store in Chilliwack was changed from being a small store to a large store

9

Table 1 Location and particulars of the 21 stores

Store Location Store Size

Picks Per Week

1992 Sales ($)

Floor Space (ft2)

1 Chilliwack S 2 2901969 6678

2 Coquitlam M 2 4015303 6510

3 Langley EJ 2 8138673 15422

4 Scott Road EJ 2 7307820 15064

5 Maple Ridge L 2 4102110 6678

6 Kingsway L 2 4153736 6574

7 Richmond EJ 2 8175105 14961

8 North Vancouver M 2 3856571 6534

9 Guildford M 2 3714372 6567

10 Burnaby M 2 3532752 6983

11 New Westminster S 1 2417992 6470

12 Abbotsford EJ 2 7899398 15303

13 Hastings J 2 4647794 9849

14 King Edward EJ 2 3825372 25492

15 White Rock L 2 3960262 6870

16 Main amp Mountain S 1 2705602 6431

17 Heather J 2 4161227 9427

18 King George J 2 4606817 9640

19 Cambie J 2 5855022 9091

20 Lougheed M 1 3114180 6615

21 Burrard J 2 6553310 10608

10

consumer confidence which in turn leads to increased sales In other words if a customer

perceives that Canadian Tire is in the hardware business he or she will choose to shop there

when meeting future needs However management has adopted this philosophy without any

analysis of the cost of holding excess inventory Only recently has management begun to

question the cost-effectiveness of allocating inventory based on these rules of thumb

Speculating that existing inventory practices might be deficient Mr Don Graham the

vice-president of Canadian Tire Pacific Associates commissioned a pilot study to examine the

inventory system and to propose alternate strategies7 The study using a steady state model

to represent a simplified inventory system suggested that existing policies lead to overly high

inventory levels More importantly the study outlined a methodology that would enable

management to determine inventory levels based on observed demand The pilot study

recommended that a more detailed model of the inventory system be formulated that the

methodology be expanded to incorporate the cost of inventory and that an algorithm be

developed that would quickly identify the optimal inventory policies

B The Problem

At the time of the pilot study management was concerned primarily with its inability

to distinguish between stores of the same size As mentioned in the previous section a

stores size is based solely on its sales volume and floor space and under existing practices

all stores of the same size have identical inventory policies for every product Management

realized that demand for some products might be a function of variables other than size such

7 The pilot study was performed by Martin Puterman a professor of Management Science and Kaan Katircioglu a doctoral student in Management Science both at the University of British Columbia The results of the study were presented at a U B C Transportation and Logistics Workshop on March 25 1993

11

as location For example the demand for sump pumps is likely to be very different in suburban

areas than in urban ones However management was unable to allocate more sump pumps to

large suburban stores than to large urban stores This inability led management to seek a

tool that would allow for a more effective allocation of inventory between stores of the same

size

By the end of the pilot study however management was less concerned with

reallocating inventory within a size of store than with reallocating inventory between sizes

of stores and with reducing overall inventory According to management assigning individual

inventory policies to each store appeared to be operationally infeasible Firstly management

believed that having individual policies would lead to confusion in the warehouse and to

increased shipping errors Exactly why this would occur is unclear the warehouse would

simply continue to fill orders from tickets that make no reference to the policy in place

Secondly and perhaps more tenably the computer technology at Canadian Tire Pacific

Associates did not allow for policies to vary across all stores Both the database used to store

the inventory data and the printouts used by the inventory manager allowed for only five

different policies per product all with the same minimum fill Moving to individual policies

would have required an overhaul of the existing information system Although management

saw this overhaul as inevitable it did not see the overhaul as imminent Therefore

management switched the focus of the study to developing a tool that would help reduce overall

inventory levels andor reallocate inventory more effectively between groups of stores

The subtle change in the studys objective prompted an important question how should

the stores be grouped The current practice of grouping stores based on size is problematic

for two reasons First as discussed above additional factors of demand such as location

cannot be taken into account Second grouping inherently results in a misallocation of

12

inventory when some stores within a group have differing numbers of picks per week For

example the medium-sized Lougheed store which has one pick per week requires higher

inventory levels to achieve a given customer service level than the other medium stores

which all have two picks per week8 In fact in terms of customer service levels it might

be more appropriate to group the Lougheed store with the large stores having two picks per

week Ideally this artificial grouping of stores will be abandoned when the computer system

is upgraded and policies can then be set on a store by store basis

As the project proceeded management grew somewhat uneasy with the concept of using

an analytical tool to evaluate and set inventory policies According to both the pilot study and

the preliminary results of this study significantly reducing inventory would greatly reduce costs

while still providing a high level of customer service However management held that because

the model cannot take into account intangibles such as marketing considerations such a drastic

reduction in inventory would be very inappropriate In this light management viewed the

optimal policies as lower bounds for inventory levels these policies would have to be

modified upwards in order to be practical Therefore management contended that any

methodology or software produced during this study should provide the means to evaluate and

compare a number of alternate policies

Regardless of exactly how the tool is to be used there is currently a lack of historical

sales data from which to analyze the inventory system and determine optimal policies No sales

data had ever been collected prior to the pilot study and only a relatively small amount has

been collected since In order to proceed with any analysis of the inventory system a

8 If demand is a function of size then the Lougheed store must meet the same demand with one fewer pick per week This would lead to either increased probabilities of stockouts at the Lougheed store or higher inventory levels at all the other medium stores

13

substantial amount of additional data would be needed The existing information system at

Canadian Tire Pacific Associates does allow for sales data to be captured however the system

is limited by the amount of data it can capture at one time and by its storage capacity In order

to collect the amount of data required management would have to upgrade the system and

develop some kind of data collection scheme coupled with a data compression technique The

collection scheme must allow for characteristics of demand such as trends and seasonality to

be identified and the compression technique must allow for sales data for over 30000 products

to be stored for each store

Because many of the products sold by Canadian Tire have nonstationary demand as a

result of seasonality andor trend the inventory tool must be able to forecast demand for the

coming period Products such as sporting goods and outdoor equipment exhibit patterns of

demand that are very seasonal For example sales of snow shovels increase through the fall

peak in the winter and drop to zero by late spring At the same time some products might

become either more or less fashionable over time resulting in changes in mean demand

Setting inventory policies for any of these products based on observed demand would result in

insufficient stock if demand increased during the period or in excess stock if demand decreased

Therefore the tool must be able to forecast the demand for the period for which the policy is

intended

Toward the end of the project management began discussing the integration of the

interactive inventory tool with the existing system to form a new inventory control system The

new system would utilize a modified point of sales system to collect sales data continually

Based on the latest available data the system would forecast demand for the coming period and

calculate the optimal policies for every product at every store The system would then provide

all relevant information on-line for the inventory manager to review The manager could weigh

14

other information such as marketing considerations and examine alternate policies to determine

the effect on cost Finally the preferred policy would be set in place with the touch of a key

and information on the next product or store would be presented for review

C The Project

Development of the inventory control system envisioned by management requires the

completion of five tasks 1) the design of a data collection system that generates and stores the

necessary sales data 2) the development of a forecasting method that provides accurate

estimates of demand distributions 3) the formulation of a model that calculates the cost of

inventory and the customer service level for an inventory policy 4) the development of an

algorithm that finds the optimal policy for a product at a store and 5) the creation of an

interface that allows interactive and on-line consultation Ultimately the above components

would have to be integrated and incorporated in a manner that would ensure the compatibility

and reliability of the system

The scope of the research reported in this thesis completes the first phase of the project

specifically we formulate an inventory model develop an algorithm and create a prototype of

the interface Because of the changing environment at Canadian Tire Pacific Associates minor

modifications to both the model and the algorithm developed herein might be necessary to

reflect future changes in operational procedures Upon completion of phase one the interface

will be interactive allowing managers to select the store product and review scenario and then

to compare optimal policies with existing policies or with alternate policies Although the

prototype will be delivered to Canadian Tire Pacific Associates for ih-house testing at the end

of this phase the interface will require modification in subsequent phases as feedback from

management is received and as the system is brought on-line

15

In order to assess more fully the potential savings of implementing the inventory control

system this thesis includes a comparison of optimal policies9 and existing policies The

expected annual cost of inventory for an entire product category is calculated for both the

optimal policies and the existing policies Based on an analysis of these costs the potential

savings of implementing the optimal policies are extrapolated Further the thesis examines the

sensitivity of the model to variations in the estimates of demand and costs

Originally results from all 21 stores were to be included in the above analysis

However just prior to the analysis management changed the review procedure at several

stores the result of which being that three picks per week are now performed at those stores

Because one of the main assumptions of the model specifically that the length of the lead time

does not exceed the duration of the review period is violated under this new review procedure

those stores cannot be included in the analysis Therefore the analysis is based solely on the

results from store six the large store located on Kingsway Avenue10

For the purpose of the analysis sales data for product category 20 automotive and

electrical accessories were collected from all 21 stores This product category was selected

because the distribution of demand for products within it is non-seasonal and because few of

the products are ever placed on sale11 The lack of trend or seasonality eliminates the need

for forecasting demand the optimal policies can be determined based directly on observed

demand

9 That is those policies obtained from the model

1 0 Store six is one of the only stores whose review procedure is not expected to change from two picks per week

1 1 Products that were on sale at any time during the period were not included in the data set

16

D The Demand Data

In total sales data for 520 products of which 458 are active products were collected

over a 307-day period from October 10 1992 to August 18 199312 The data was supplied

by Canadian Tire Pacific Associates in the form of a 10 MB text file with one line per day per

product recording sales from the stores13 Each line had the following format

Columns 1-6 part number Columns 7-9 Julian date14

Columns 10-18 cost of the item (to three decimal places) Columns 19-124 sales for stores 1 through 21 (5 columns per store)

In order to facilitate the analysis the file was sorted into 458 sales files with each of

these files containing the history of daily sales for one of the products Two changes were

made to the format of these files to improve readability 1) a column was added between the

product number and the Julian date and 2) all leading zeros were dropped Appendix A

contains a partial printout of the sales file for a typical product Finally a distribution file

summarizing the daily sales was created from each of the sales files Each distribution file

contains the products number the products cost and the number of days during which various

quantities of the product were sold at each store

Prior to the analysis the distribution files for a large number of products were

examined and based on the shapes of the histograms the products were divided into three

groups The first group low-demand comprises products that have demand on only a few

days with the daily demand seldom exceeding three items The second group medium-

1 2 Of the active products only 420 are sold at store 6

1 3 Strictly speaking a line was written to the file only on days on which at least one store registered a sale of the product

1 4 According to the Julian calendar days are numbered sequentially from 1 to 365 (366 during leap-years) beginning with the first of January

17

demand comprises products that have demand on approximately half of the days with the daily

demand seldom exceeding more than six items Finally the last group high-demand

comprises products that have demand on most days occasionally upwards of twenty items and

whose demand histograms resemble Poisson distributions The majority of the products

examined fell into the low and medium-demand groups only a few high-demand products

were observed Appendix B contains a printout of the distribution file for a typical product

in each group

For most products no theoretical distribution appeared to model the demand well

Demand for low and medium-demand products followed a non-standard distribution with

a mass at zero and a relatively small tail Although some theoretical distributions such as the

mixed-Poisson can be made to resemble this it was unclear how well they would perform

Further the effects of sampling errors from such non-standard distributions were unknown

On the other hand the standard Poisson distribution did appear to model high-demand

products well however only a few products fell into that group Therefore in the absence of

a good theoretical candidate the empirical demand function was chosen for the analysis The

empirical function was readily available from the distribution files and offered the best

estimate of the true underlying function

E The Cost Data

Along with the sales data management supplied estimates relating to the costs of holding

and ordering inventory Management estimates that the cost of holding an item in inventory

for one year is 30 of the items cost Incorporated in this holding rate are a number of costs

such as the opportunity cost of money invested and the costs incurred in running a store

Because it is the annual cost of inventory that is of interest either the compounded daily

18

holding rate or the simple daily holding rate can be used to determine the costs incurred during

a review period however the method of calculating and extrapolating the holding costs must

be consistent For the sake of simplicity a simple daily holding rate is used

In order to estimate the cost of ordering inventory management analyzed the various

tasks that result from an order being placed Using the average time required to fill and then

reshelve an order management obtained an estimate of the direct labour costs involved To

this management added an estimate of the cost of transporting an order from the warehouse to

the store Finally the estimate was adjusted slightly upward to account for the cost of breakage

and errors Based on the analysis management estimates the cost of placing an order for a

product to be $0085 regardless of the size of the order

19

III M O D E L F O R M U L A T I O N

A Terminology

In the previous chapter several terms were introduced to describe the inventory system

at Canadian Tire Pacific Associates These terms though perfectly adequate are not widely

used in the academic literature to describe inventory management problems Therefore in

order to avoid confusion and be consistent with the literature the following terms are used

throughout this thesis

The length of time between successive moments at which the

inventory position is reviewed15

The time at which a reorder decision is made

The length of time between the moment at which a decision to

place an order is made and the moment at which the order is

physically on the shelf

An operating policy in a periodic review inventory model in

which (i) the inventory position is reviewed every T days (ii) a

replenishment order is placed at a review only when the inventory

level x is at or below the reorder level s (gt 0) and (iii) a

replenishment order is placed for S-x units (S gt s)1 6

Lost Sales - The situation in which any demand during a stockout is lost

Review Period -

Decision Epoch

Lead Time -

(sS) Policy

1 5 With respect to the terminology used at Canadian Tire Pacific Associates a review period equals the number of days between flushes

1 6 With respect to the terminology used at Canadian Tire Pacific Associates S represents the target level and s represents the difference between the target level and the minimum fill

20

B Assumptions

In to order make the model of the inventory system mathematically manageable several

simplifying assumptions are made However we believe that the following assumptions do not

significantly reduce the accuracy of the model

1) Demand can be approximated very closely by sales during the period in which the data

is collected Generally inventory levels at the stores are very high and the number of

lost sales is assumed to be negligible In fact the inventory levels for a number of

products were tracked over time during the pilot study and not a single incidence of

stockout was observed Because all products appear to have similar excess inventory

it seems reasonable to assume that virtually all demand resulted in sales

2) The demands in successive periods are independent identically distributed random

variables Preliminary tests (Fishers exact tests) suggested no dependence between

sales on consecutive days and plots of sales over time indicated no seasonal pattern

3) Demand is independent across products That is we assume that sales of one product

do not affect nor can they be predicted from sales of another product

4) Unmet demand results in lost sales with no penalty other than the loss of the sale

5) The duration of the review period for a store is an integer value and is fixed over a

planning period For stores with two picks per week we assume that the inventory

is reviewed every four days rather than on a four day - three day cycle

6) The decision epoch is the start of a review period Review periods begin at midnight

at which time the inventory is reviewed and orders are placed instantaneously

7) The duration of the lead time for a store is deterministic and integer-valued In reality

the lead time for some stores was stochastic but management felt the variation could be

controlled and virtually eliminated

21

8) The lead time does not exceed the review period In other words items ordered at a

decision epoch arrive and are shelved before the next decision epoch In practice this

did not always occur

9) Orders arrive at the end of the lead time and are on the shelf to start the next day

10) Holding costs are incurred daily for all inventory held at the start of the day

11) The warehouse has an infinite capacity that is all orders to the warehouse are filled

Because of the high inventory levels in the warehouse stockouts in the warehouse are

very unlikely

C A Markov Chain Model

Consider the situation in which an (sS) policy is applied to an inventory system based

on the above assumptions Inventory is reviewed at the decision epoch for each of the periods

labelled t = 0 12 and replenishment occurs at the end of the lead time provided an order

was placed Demand during day d in period t is a random variable whose probability mass

function is independent of the day and period The function is given by

f^ik) = Pr ltd = k = ak for k = 0 1 2

where ak gt 0 and E ak = 1 Let T denote the duration of a review period and L denote the

length of the lead time Next define D T ( as the total demand during period t D L as the total

demand during the lead time in period t and D T L as the difference between D T and DLt

Assuming that successive demands pound f pound 2 are independent the probability mass function of

1 7 The aggregated demand variables are given by

^ = XXd and DT_Ut=J d= d= d=L+l

22

each aggregated demand variable can be obtained by convolving the probability mass function

of t d l s Finally let X denote the inventory on hand at the beginning of period t just prior to

the review

The process X is a discrete time stochastic process with the finite discrete state space

0 1 S A typical sample path of the process with a review period of four days and a

lead time of two days is depicted in Figure 1 As period t begins X units are on hand The

inventory is reviewed and because X is less than s an order is placed for (S-X) units Over

the next two days a demand of D L r occurs and the inventory drops accordingly With the

arrival of the order the inventory level increases by (S-X) at the start of the third day of the

period Over the remaining two days a demand of D T L is observed bring the inventory level

down to X + 1 Note that in this example an order would not be placed in period t+l because

X r + i exceeds s

The stock levels at the beginning of two consecutive review periods are related by the

balance equation

_K- DLy+ (s - v - DT-LX ixtlts ( 1 )

t + l ~ (Xt - DTtY ifXtgts

where the positive operator is interpreted as

= max a 0

1 8 For example the probability mass function of D T bdquo denoted fD T ( is the T-th fold convolution of f t Mathematically the function can be obtained using the recursive relation

and the identity f1^ = f5(

23

-2 I 8

gt o

r

CO

o a

CM

X

00

24

Because X + 1 depends only on Xt the process X satisfies the Markov property

PrXt+l=jX0 = i0Xt_l=it_vXt = i) = PrXt+1=jXt = i

and therefore is a discrete time Markov chain Further because the probability mass function

of the random variable pound is independent of t and thus so too are the probability mass functions

of the aggregated demand the Markov chain is stationary19

D The Transition Probability Matrix

The stationary transition probability p(gt is defined as the probability of X + 1 being in state

j given that X is in state i independent of the period t In the inventory model p-- represents

the probability of starting the next review period with j items on the shelf given there are i units

on hand at the start of the current period By convention the transition probabilities are

arranged in an (S + l)x(S + 1) matrix P such that (P)y = p- Thus the (i + l)st row of P contains

the probability distribution for the stock on hand at the beginning of the next period given that

there are i units on hand at the start of the current period The matrix P is known as the

transition probability matrix Note that since we are concerned with a steady state analysis we

need not consider the possibility of initial stock levels greater than S once the inventory level

is below S it can never again exceed S

The transition probability ptj is calculated by expanding (1) to include all possible

instances of demand that allow for a change in inventory from i units at the beginning of a

period to j units at the beginning of the next period For example suppose the inventory policy

(510) is in place 2 items are on hand at the start of the current review period and 7 items are

1 9 Henceforth the subscripts t and d are omitted from the daily and aggregate demand variables for the sake of brevity

25

on hand at the start of the next period Then only three scenarios are possible

1) No items are sold during the lead time (meaning that the stock level increases to 10

after replenishment) and 3 items are sold during the remainder of the period

2) One item is sold during the lead time and 2 items are sold during the remainder of

the period

3) Two items are sold during the lead time (though demand could exceed 2) and 1 item

is sold during the remainder of the period

The transition probability p 2 7 is the sum of the probabilities of the various scenarios and is

computed by

p2J = PrDL=0PrDT_L = 3] + PrDL = l] Pr[DT_L=2 + PrDL2] PrDT_L = l]

In general the transition probabilities for the inventory model are calculated as follows

Case 1 The lead time is less than the review period

For 0 lt lt s

Pij

i-l pound [PrDL=kPrDT_LgtS-k] + PrDLgtiPrDT_LgtS-i if j = 0

i - l

^[PriD^^PrlD^S-j-kn+PHD^PriD^S-j-i if 0 lt j ltL S-i (2) k=0 s-j

E [PrDL=kPrDT_L = S-j-k] if S-i ltj^S k=0

26

For s lt i lt S

PrDTzi]

0 p = PrDT=i-j)

ifj = 0 ifOltjzi ifiltjltS

(3)

Case 2 The lead time equals the review period

For 0 lt i lt s

Pa

0 PrDTzi) PrDT = S-j

ifOltLJlt S-i ifj = S-i if S-i ltj plusmn S

(4)

For s lt i lt S use (3)

E The Steady State Probabilities

Provided 1) there is a positive probability of non-zero demand during a review period

and 2) there is a positive probability of zero demand during a review period20 then the

Markov chain described by (1) has a limiting probability distribution ir = (ir0 TTu irs)

where 0 lt IT lt 1 for j = 0 1 S and E 7ry = 1 Further this limiting distribution is

independent of the initial state In other words the probability of finding the Markov chain in

a particular state in the long-run converges to a nonnegative number no matter in which state

the chain began at time 0 The limiting distribution is the unique nonnegative solution of the

equations

2 0 That is 0 lt PrD x = 0 lt 1 Given the nature of the problem this assumption is not very restrictive

27

^ = E ^ - far j = 0 l S (5)

s and pound nk = 1 (6)

For the proof that the limiting distribution is independent of the initial state and is obtained from

(5) and (6) the reader is directed to Appendix C

In terms of the inventory problem 7ry is the steady state probability of having j units of

inventory on hand at the decision epoch However some subsequent calculations are made

easier if the steady state probability of having j units of inventory on the shelf just after

replenishment is known Instead of reformulating the Markov chain these probabilities can be

obtained by shifting the limiting distribution TT Much in the same way that the transition

probabilities are calculated the shifted limiting distribution t] is obtained by accounting for

all possible instances of demand that allow for the shifted inventory position For example

suppose that the policy (510) has the limiting distribution TT In order to have say 8 units on

hand immediately after replenishment one of the following must occur

1) Ten items are on hand at the decision epoch and 2 items are sold during the lead time

2) Nine items are on hand at the decision epoch and 1 item is sold during the lead time

3) Eight items are on hand at the decision epoch and nothing is sold during the lead time

4) Either 2 3 4 or 5 items are on hand at the decision epoch and 2 items are sold during

the lead time

The steady state probability of having 8 units on hand just after replenishment is calculated as

n 8 = n10PrDL = 2] +iz9PrDL = l] +n8PrDL = 0 +[iz5 + iz4 + iz3]PrDL = 2 +n2PrDLgt2

28

In general the shifted steady state probabilities are given by

s pound nPrDLgti ifj = 0

i=s+l S

pound PrDL = i-j] ifl plusmnjltS-s i=max(s+l)

S = pound 7tjirZ)L = i-+7i 5 FrD i ^5 ifj = 5-5

i=max(s+l)

^ 7 1 ^ ^ = 1-7+ pound TtiFrZgti=5-y+iis_PrD1^S-7 ifS-sltjltS i=max(gt+l) i=S-j+l

(7)

F The Cost of Inventory

For this problem the cost of inventory has two components (i) a fixed ordering cost

cf incurred whenever an order is placed and (ii) a holding cost cv incurred daily for each item

held in inventory The expected cost of inventory for a review period can be calculated by

summing the costs associated with each inventory position under steady state conditions

Because a simple holding rate is used this cost is simply adjusted by number of review periods

in a year to obtain the expected annual cost of inventory

The ordering cost for a review period is relatively straight forward to calculate either

an order is placed in which case the order cost is incurred or no order is placed in which case

no cost is incurred According to the rules of the inventory policy orders are placed only if

the stock on hand at the beginning of a review period is less than or equal to s Further the

probability that the review period begins with the inventory in one of these positions is equal

to the sum of the first (s + 1) steady state probabilities Therefore the expected ordering cost

29

for a review period is given by

E[OC] = cfY n y=o

Unlike the simple structure of the ordering cost the holding cost is incurred daily for

each item held in inventory Therefore the daily holding costs associated with all possible

inventory positions must be weighted and summed to obtain the holding cost for the review

period In order to make the calculations less complicated a shifted review period beginning

immediately after replenishment and ending just prior to the next replenishment is considered

Define e(0 to be the expected number of items on hand to start day i of the shifted

review period and let g(i f) denote the expected number of items on hand to start day i given

that there were j items immediately following replenishment21 The function g(z j) is the sum

of the probabilities of possible demand weighted by the resulting stock levels that is

g(ij) = j^kPriD^j-k)

To obtain the expected number of items on hand to start day i independent of j the above

function is summed over all possible values of j and weighted by the corresponding steady state

probabilities yielding

s j ed) = pound r 5 gt P r Z ) = - pound

7=0 k=

2 1 In other words g(if) is the expected number of items on hand to start day i given e(0)=v

30

The expected holding cost for a review period is then given by

T

E[HC] = $gt v e(0 i=i

i=l =0 k=l

where c v equals the simple daily holding rate multiplied by the cost of the item

Although differently defined review periods are used to calculate the above expected

costs the costs can still be summed to obtain the expected cost of inventory for a review

period The expected costs are calculated based on steady state conditions and therefore

represent the costs expected to be incurred during any T consecutive days Summing the above

two expected costs and multiplying the sum by the number of review periods in a year yields

the expected annual cost of inventory

E[AC] = mdash[E[OC]+E[HC]]

T S j 1 ( g ) 365

j=0 i=l j=0 fc=l

G The Customer Service Level

There are many ways to define customer service and the most appropriate definition

depends upon the nature of the business and the perspective of management Two commonly

used definitions are 1) the proportion of periods in which all demand is met and 2) the

proportion of demand satisfied immediately from inventory Although the former definition is

often used because of its simplicity the latter definition is closer to what one commonly means

by service (Lee and Nahmias 1993) A third definition also relevant in retail environments

31

is the probability that lost sales do not exceed some critical number Other definitions such

as probability that backorders are filled within a certain amount of time are more applicable

to industrial and manufacturing settings where backlogging demand is common

Given the retail environment with the possibility of lost sales that exists at Canadian Tire

Pacific Associates management is concerned primarily with the fraction of sales that are lost

For example management views the situation in which demand exceeds stock by one unit every

period more favourably than that in which the same demand exceeds stock by three units every

other period For this reason the proportion of demand satisfied immediately from inventory

is the most appropriate measure of customer service

The customer service level for a single review period CSLRP is given by

r bdquo T _ Demand satisfied during RP Total demand during RP

Because we are interested in the long-run behaviour of the system a measure of the steady state

customer service C S L must be obtained As shown in Appendix D the correct measure is

poundpoundpound _ E[Demand satisfied during RP] E[Total demand during RP]

which can be rewritten as

pound pound pound _ j _ E[Demand not satisfied during RP] ^ E[Total demand during RP]

Because of the steady state conditions the above customer service level can be estimated

from any period of T consecutive days As was done for one of the cost calculations the T-day

period beginning just after replenishment is considered Suppose that there are j units of

inventory on hand immediately following replenishment Then the conditional customer service

32

level is

CSL j = 1 E[(DT-j)l

E[DT] (10)

The denominator in the second term is merely the expect demand during the period

E[DT] = 7gt (11)

where p is the mean of the random daily demand variable pound The numerator which is the

conditional expected demand that is not satisfied during the T-day period expands to

7-1

E[(DT-j)+] = T u - j + YU-k)PrDT = k] k=0

(12)

The reader is directed to Appendix E for a mathematical justification of (12) In order to obtain

the unconditional expected demand not satisfied (12) is applied to all possible values of j and

a weighted sum of the terms is calculated The result is given by

E[Demand not satisfied] = Y j =0

y-i Ti - j + 1pound(j-k)PrDT = k

S j-1

= - E[I] + Y0-k)Pr[DT=k) =0 it=0

(13)

where E[I] denotes the steady state inventory level Finally the customer service level is

obtained by the substituting (11) and (13) into (9) yielding

CSL = 1

s j-l

Ti - E[I] + Jgty Y(J-k)PrDT=k]

m n J2VjTU-k)PrDT=k Eil J=Q fc=o

(14)

33

H A Methodology for Evaluating an (sS) Policy

In this section a brief methodology summarizing the main results of the model is

presented The methodology can be applied to evaluate a given (sS) policy in terms of its cost

and its resulting customer service level Given the probability mass function of the daily

demand f the duration of the review period T the length of the lead time L the fixed cost

of placing an order cf and the daily unit cost of holding inventory cv

1) Calculate the aggregated demand probability mass functions fDT fDL and f D x L as the T-th

fold L-th fold and (T-L)th fold convolutions of f respectively

2) Generate the transition probability matrix using either equation (2) or (4) and equation (3)

3) Calculate the limiting distribution TT from (5) and (6)

4) Obtain the L-day shifted limiting distribution rj using (7)

5) Calculate the expected annual cost of inventory using (8)

6) Determine the customer service level using (14)

34

IV A N A L G O R I T H M F O R OBTAINING T H E O P T I M A L (sS) P O L I C Y

A Introduction

The methodology developed in the previous chapter can be used to calculate the cost and

service level associated with a given (sS) policy In this chapter an algorithm that finds the

feasible policy yielding the lowest cost is developed A policy is feasible if it satisfies both the

basic condition 0 lt s lt S and the imposed constraint on the customer service level2 2 In

order to find the optimal policy the algorithm must combine the methodology for evaluating

a single policy with a search technique

Search techniques are evaluated in terms of two somewhat antithetic criteria On the one

hand the search should locate the optimal solution or at least a near optimal solution on

the other hand the search should take as little time as possible Many search techniques

guarantee optimality provided the objective function and constraints exhibit certain properties

such as convexity However even if such a search technique also guarantees its termination

in polynomial time the search might take too long to be practical Alternatively some search

techniques provide reasonable though sub-optimal solutions very quickly These techniques

often called heuristics must be evaluated in terms of both their speed and the accuracy of their

solutions The choice of a heuristic over a search that guarantees optimality depends upon the

nature of the feasible space the need for accuracy and the cost of time

Given the complexity of the cost and customer service level functions it is very difficult

to show theoretically for what if any combinations of demand and costs the feasible space is

convex However a plot of the objective function for a single product may be obtained by

2 2 Management specified that each policy must provide a customer service level of at least 975 Therefore the algorithm must incorporate the constraint CSL gt 0975

35

evaluating the costs for a number of policies within a grid When the service level constraint

is included in these plots maps of the feasible space are obtained Examination of the

maps for several products suggests that the space is convex over most regions however

maps for some products reveal some regions of non-convexity Unfortunately there is no

way of knowing the degree of non-convexity for other products other stores or different costs

estimates without a more rigorous survey of the feasible space Therefore if one of the

techniques that guarantee optimality under the condition of convexity is employed the search

would have to be evaluated as a heuristic Instead as a first step to finding a practical search

technique a crude search that guarantees optimality under virtually all conditions is

investigated

B A Grid Search

The simplest search technique that guarantees optimality merely evaluates all policies in

the grid defined by 0 lt s lt S and 0 lt Sm i n lt S lt S m a x where Sm i n and S m a x are

some predefined values That is the grid search is a brute force method that examines every

policy within the grid 2 3 Although the number of policies that must be examined is large the

grid search is guaranteed to find the optimal policy provided that S o p t is contained in the grid

that is provided Sm i n lt S o p t lt Sm a x The grid search algorithm proceeds by generating the

transition probability matrix for the policy (0Smin) and then evaluating the service level

associated with that policy If the policy is feasible24 the cost associated with the policy is

2 3 The total number of policies evaluated is the sum of the first Sm a x integers less the sum of the first Smin-1 integers Numerically [(l2)(Sm a x)(Sm a x+l) - (l2)(Smin-l)(Smin)] policies are examined

2 4 That is if the service level is adequate

36

calculated and compared to the current lowest cost The procedure continues until all points

in the grid have been examined and the optimal policy has been found A flow chart of the grid

search algorithm is shown in Figure 2

In order to assess the grid search the algorithm was coded and compiled into an

executable program The program was run for two products (a medium-demand product and

a high-demand product) and with four values of Sm a x In each run Sm i n was taken to be zero

The following times were recorded for each trial

T P M - the amount of time spent generating the transition probability matrix

GAUSS - the amount of time spent solving for the steady state probabilities

COSTS - the amount of time spent calculating the expected annual costs

SERVICE - the amount of time spent determining the customer service levels

T O T A L - the total execution time of the program

The columns entitled Grid Search in Tables 2 and 3 summarize the results of the trials All

times are recorded in seconds as required by a 486DX33 MHz personal computer with an

empty buffer memory

Three results are evident from the tables First the vast majority of the execution time

is spent generating transition probability matrices and then solving for the steady state

probabilities with the time being split almost equally between the two tasks Second the total

execution time of the algorithm increases exponentially as Sm a x increases For product 206917

doubling S^ from 30 to 60 led to an increase in the execution time of more than 2100 (from

172 seconds to 3661 seconds) Third the grid search algorithm as it stands does not appear

to be adequate for the problem at hand Given the large number of products that must be

evaluated the algorithm is much too slow to be implementable

37

Calculate the demand probability functions

Initialize s = 0 S = Smin] cost = 10A5

s = s+1

Calculate the probability transition matrix for (sS)

Evaluate policy (sS) Is service level ^ 0975

No

No Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Is s = S-l

Yes

Is S = Smax No

Yes

Print results (sS) cost

s = 0 S = S+1

Figure 2 The flow chart of the grid search algorithm

38

Table 2 Execution times of the algorithms for product 200001 a 30-amp inline fuse

(in seconds)

-max Subroutine Grid Search Grid Search

Updating T P M Heuristic

10 T P M 006 010 001

GAUSS 010 006 001

COST 005 005 001

SERVICE 001 001 001

T O T A L 050 035 027

20 T P M 126 021 010

GAUSS 119 160 023

COST 027 026 001

SERVICE 017 016 001

T O T A L 318 236 055

30 T P M 703 096 010

GAUSS 757 801 093

COST 079 085 001

SERVICE 047 042 001

T O T A L 1664 1137 126

40 T P M 2257 163 015

GAUSS 2958 3003 206

COST 313 329 001

SERVICE 092 083 011

T O T A L 5871 3851 285

39

Table 3 Execution times of the algorithms for product 206917 a 6 solder connector

(in seconds)

bull-max Subroutine Grid Search Grid Search

Updating T P M Heuristic

30 T P M 717 067 006

GAUSS 762 880 093

COST 098 087 010

SERVICE 016 021 006

T O T A L 1720 1176 220

40 T P M 2390 157 022

GAUSS 2861 2911 209

COST 219 227 006

SERVICE 080 088 011

T O T A L 5827 3790 362

50 T P M 6691 413 010

GAUSS 7994 8201 472

COST 568 577 016

SERVICE 212 197 005

T O T A L 15840 10041 654

60 T P M 15509 894 040

GAUSS 18607 18951 876

COST 1367 1353 038

SERVICE 350 359 022

T O T A L 36614 22794 1120

40

C A Technique for Updating the Transition Probability Matrix

As was noted in the previous section approximately half of the execution time of the grid

search algorithm is spent generating transition probability matrices Furthermore the amount

of time increases exponentially as Sm a x increases This increase is exponential for two reasons

first the number of transition matrices that must be calculated is a function of S^ squared and

second the average size of the matrices and hence the average number of calculations per

matrix increases with Sm a x For example when S m a x equals 30 the transition probability matrix

must be generated 465 times with the largest matrix having dimensions (31x31) When Sm a x

equals 40 the matrix must be generated 820 times with 355 of the matrices having dimensions

larger than (31x31) The performance of the algorithm could be improved considerably if

either the number of matrices that must be generated or the number of calculations per matrix

is reduced This section introduces a technique that accomplishes the latter by updating the

transition probability matrix for a new policy instead of generating it anew for each policy

A new policy (s+mS)

Suppose the policy (sS) has associated with it the (S + l)x(S + l) transition probability

matrix P and the policy (s+mS) where m is any integer satisfying 0 lt s+m lt S has

associated with it the (S + l)x(S + 1) transition probability matrix J Then T can be obtained by

recalculating only m rows of P Recall that the equations for generating a transition

probability matrix are divided into two sets based upon the row being calculated equations (2)

or (4) for rows 0 through s and (3) for rows (s + 1) through S The equations themselves are

independent of s only the limits defining the cases contain s Therefore only the m rows

that are affected by the change in the limits need to be recalculated The following updating

technique can be used to obtain 7

41

1) If m gt 0 rows i = (s + 1) (s+2) (s+m) are recalculated using (2) or (4)

2) If m lt 0 rows i = (s-1) (s-2) (s-m) are recalculated using (3)

3) All other rows of T are identical to those of P

The reader is directed to Appendix F for a mathematical justification of the technique

A new policy (sS+m)

Suppose the policy (sS) has associated with it the (S + l)x(S + l) transition probability

matrix P and the policy (sS+m) where m is any positive integer has associated with it the

(S+m+l)x(S+m+l) transition probability matrix P Then T can be obtained by modifying

each row of P and adding m additional rows and columns The updating technique is not as

simple as that for a new policy (s+mS) for two reasons the dimensions of P and T differ and

the equations for generating a transition probability matrix contain S Nevertheless an updating

technique that requires much less computing time than generating the matrix anew can be

developed The following technique can be used to obtain P

1) For rows i from 0 through s

For columns j from 0 through m recalculate using (2) or (4)

For columns j from (m+1) through (S+m) enter the (j-m)th column entry of P

2) For rows i from (s + 1) through S

For columns j from 0 through S enter the corresponding value from P

For columns j from (S + 1) through (S+m) enter 0

3) For rows i from (S + 1) through (S+m) recalculate using (3)

The above technique is justified mathematically in Appendix G

42

The modified algorithm

The updating technique can be easily incorporated into the grid search algorithm with

only a slight change in the manner in which s is incremented Figure 3 shows the flow chart

of the new algorithm As before the algorithm was coded and compiled into an executable

program and the various execution times were recorded The results are listed in Tables 2 and

3 under the columns Grid Search Updating TPM

Updating the transition probability matrix does improve the performance of the

algorithm The time required to generate the transition probability matrices is reduced by

approximately 90 This translates to a reduction in total execution time of approximately

30 Not surprisingly the reductions are greater for large values of Sm a x However despite

this seemly significant improvement a decrease of only approximately 08 seconds is obtained

when S^ equals 20 Further when S ^ equals 30 the algorithm requires over 11 seconds

which is still much too high

The execution time is now dominated by the time required to solve for the steady state

probabilities The steady state probabilities could be solved by using approximation techniques

which are faster than solving the systems exactly however because the probabilities are often

very small numbers the approximation techniques would be unable to provide the required

accuracy The only remaining way to decrease the execution time is to reduce the number of

policies that are evaluated during the search This can be done by either reducing the size of

the grid being searched or reducing the number of policies being evaluated within the grid The

following section introduces a method of reducing the size of the grid by obtaining a lower

bound for S namely Sm i n

43

s = s+k

Calculate the demand probability functions

Initialize s = 0 S = Smin| cost = 105

k= 1

Update the probability transition matrix for (sS)

Evaluate policy (sS) Is service level ^ 0975

No

No

iNo

Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Ifk= 1 iss = S-l If k =-1 is s = 0

Yes

Is S = Smax No

Yes

Print results (sS) cost

If k=l s=s+l S = S+1

Figure 3 The flow chart of the grid search algorithm utilizing the updating technique for the transition probability matrix

44

D A Lower Bound on S

As mentioned earlier the size of the grid is specified by the number of policies within

the domain defined by 0 lt s lt S and 0 lt Smin lt S lt Smax Thus the size of the grid

and hence the total amount of computation time can be reduced by obtaining tight bounds on

S Unfortunately because of the extreme non-linearity of the objection function and the service

level constraint obtaining a mathematical upper bound on S is very difficult and beyond the

scope of this work This section introduces a method for obtaining a lower bound on S25

For a given value of S an upper bound on the customer service level can be found as

follows Suppose that there are S units of inventory on the shelf at the beginning of every

review period regardless of demand No (sS) policy can provide a higher level of customer

service because there exists a positive probability of beginning a review period with less than

S units on the shelf Thus the customer service level associated with starting each review

period with S units on the shelf is an upper bound on the customer service level for any (sS)

policy From (10) this level is

E[(DT-SY] CSL 5 = 1 - T mdash mdash - ( 1 5 gt

E[DT]

Replacing j with S in (12) and then substituting (12) and (11) into (15) yields

CSL 15 = mdash- - J2(S-k)PrDT = k] (16)

To obtain Smin use (16) to find the largest value of S with a customer service level less than

0975

There are two criteria for a good lower bound on S the bound must be no larger than

This method was proposed by Kaan Katircioglu

45

S o p t and it must be very close to So p t These criteria ensure that the optimal policy will be

contained in the search grid and that the size of the grid is reduced by as much as possible

Based on these criteria the above method of obtaining a lower bound is assessed A value for

S m i n is obtained for each of the products in product category 20 and compared with the

corresponding value of S o p t 2 6 The results are illustrated in Figure 4 which shows the number

of occurrences of various Sm i n and S o p t combinations Note that a point lying on the 45deg line

represents a perfect bound while a point lying close to the S o p t axis represents a very loose

bound The absence of any points above the 45deg line indicates that the first criterion is satisfied

in each instance the method did not generate a value of Sm i n exceeding the corresponding value

of So p t Moreover the bounds are reasonably tight for 85 of the products analyzed that

is for those products with S o p t equal to 10 or less27 However as S o p t increases the bounds

become steadily worse Because values of S o p t are likely to be larger for these products at the

ten Jumbo and Extra Jumbo stores and for other products with higher demand the lower bounds

are not expected to be very tight in general

Although obtaining the bound Sm i n does reduce the size of the grid the corresponding

reduction in the execution time of the algorithm is miriimal For example if S^ equals 60 and

a value of 30 is obtained for Sm i n the program would still require approximately 215 seconds

to search the grid (compared to the 228 seconds required without a lower bound on S) Because

the execution time increases exponentially with So p t a tight upper bound on S would likely have

a more significant impact on reducing the execution time of the algorithm Further research

on this issue is required

2 6 The optimal policy for each product was obtained using a grid search with an extremely large upper bound

2 7 On average Sm i n is approximately half of S o p t for these products

46

47

The only remaining approach to improving the algorithm is to modify the search so that

fewer policies within the grid are evaluated Toward this goal assumptions can be made with

respect to the feasible space and a heuristic search can be developed based on these

assumptions

E A Heuristic Search

As mentioned previously in this chapter a map of the feasible space for each of

several products was obtained Upon closer examination the maps illustrate two patterns

without exception for a given value of S both the cost and the customer service level increase

as s increases The following intuitive argument is offered as a possible explanation for the

observed behaviour When S is held constant increasing s results in more instances in which

an order is placed upon review Hence the average stock level would increase which in turn

would lead to an increase in the cost and the customer service level28 With this in mind a

heuristic search based on the following assumptions is developed for a given value of S both

the cost and the customer service level are monotonically increasing functions of s

In general the heuristic proceeds as follows Having just incremented S by one the

customer service level of the policy (sS) is evaluated If the policy is feasible s is decreased

until the lowest value of s producing a feasible policy is found If the policy is not feasible

s is increased until either the cost exceeds the current lowest cost for a feasible policy a

feasible policy is obtained or s equals S-l In the event that a feasible policy is found the cost

2 8 In order to prove that the customer service level increases as s increases it would be sufficient to show that the probability of placing an order increases as s increases Such a result might be possible if assumptions regarding the demand function are made Further if both the probability of placing an order and the average stock level increase then both the ordering cost and the holding cost would increase No proofs are offered here

48

of the policy is calculated if the cost of the policy is less than the current lowest cost the new

policy becomes the current best Finally S is increased by one and the process repeats

provided S does not exceed some predefined value Sm a x The flow chart of the algorithm

incorporating the heuristic search is shown in Figure 5 Note that the algorithm begins by

attempting to improve on the lower bound Sm i n the policy (Sm i n-lSm i n) is evaluated and if it

is not feasible S m i n is increased by one The improvement continues until a feasible policy is

found

As previously mentioned a heuristic search must be evaluated in terms of both its speed

and the accuracy of its solutions To evaluate the heuristic on the basis of speed the same

approach used to evaluate the previous algorithms is employed The columns entitled

Heuristic in Tables 2 and 3 contain the various execution times of the heuristic The heuristic

is a vast improvement over the previous search techniques the reduction in execution time is

enormous and becomes more so as S m a x increases For a S m a x of 20 the reduction is

approximately 75 of the time required by the grid search algorithm29 and for a S m a x of 60

that reduction soars to over 95 More importantly the heuristic can search a grid with a

of 40 in under 4 seconds or a grid with a S m a x of 60 in under 12 seconds These low execution

times should be sufficient to allow for a large number of products to be analyzed in a

reasonable amount of time

In order to further assess the practicability of the heuristic the total time required by the

heuristic to find solutions for the entire product category was determined and compared to the

corresponding time for the grid search algorithm With a priori knowledge of the optimal

solutions conservative yet reasonable values of S m a x were selected and used by both of the

2 9 Hereafter grid search algorithm refers to the algorithm utilizing the grid search the updating technique and the lower bound Sm i n

49

Calculate the demand probability functions

Calculate Smin S = Smin s = S-l

S = S+l s = S-l t

Update the probability transition matrix for (sS)

No Is service level pound 0975 Yes

No Is service level ^ 0975

Yes

Is cost lt cost No

Yes

Is s lt S-l No

Yes

s = s+l

Update the probability transition matrix for (sS)

Is service level pound 0975

Yes

Is cost lt cost No

Yes

s = s S = S cost = cost

s = s-l

No

Update the probability transition matrix for (sS)

S = S+l

Is s 2 0 Yes

Update the probability transition matrix for (sS)

Is service level pound 0975

No

s = s+l

No

No

Is S = Smax

^Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Print results (sS) cost

Figure 5 The flow chart of the heuristic algorithm

50

algorithms30 The grid search algorithm required approximately 9 hours and 41 minutes to

find the solutions while the heuristic required only slightly more than 23 minutes Based on

speed the heuristic is preferable to the grid search algorithm

Finally to evaluate the heuristic based on the accuracy of its solutions its solutions were

compared to the previously obtained optimal policies Without exception the heuristic

algorithm found the optimal policy Although there is no guarantee that the heuristic will find

the optimal solution for different products or different demand functions the heuristic appears

to work extremely well Based on the fast execution time and the apparent accuracy of the

solutions the heuristic will be used in the development of the inventory tool

3 0 For the test the following values of Sm a x were used 20 (273 products) 30 (19) 50 (23) 100 (3) and 125 (2)

51

V T H E I N T E R F A C E

With the model formulated and the algorithm developed the only remaining task in the

development of the prototype is the creation the interface In fact two interfaces are created

one that displays results in terms of minimum fills and target levels and one that displays

results in terms of reorder points and target levels To be consistent with the previous chapters

the latter interface is described here31

In its present form the interface allows the manager to set review parameters to

compare optimal and current policies and to evaluate alternate policies From the main menu

the manager selects the product store duration of the review period length of the lead time

and value of Sm a x Once the parameters are set the system calculates the optimal policy and

then evaluates the current policy displaying results as they are obtained In particular the

reorder point target level expected annual cost and expected customer service level are

displayed for each policy If a current policy meets the desired customer service level the

expected annual savings of the optimal policy are displayed in terms of both dollars and

percentage of current cost otherwise the value of the current customer service level is

displayed in red to indicate that it is deficient Finally the manager can choose to evaluate

alternate policies or to return to the main menu

Although the prototype is far from the ultimate integrated inventory control system it

is still valuable to management As mentioned in Chapter II inventory policies at Canadian

Tire Pacific Associates are currently based on rules of thumb that incorporate marketing

3 1 Because significant changes to the interface are expected as feedback from management is received and as the system is brought on-line only a brief description of the present interface is given

52

strategies The prototype provides management with a means of determining the proportion of

the cost of inventory that is not attributable to meeting customer demand For example

consider a hypothetical consultation in which the manager chooses to analyze product 200001

a 30-amp inline fuse at store 6 The prototype displays to the manager the current policy of

(1214) which provides a customer service level of 1000 at an annual cost of $748 and the

optimal policy of (39) which provides a customer service level of 977 at an annual cost of

only $319 If for some reason the manager feels the optimal policy is undesirable he or she

can investigate the additional cost of holding excess inventory Suppose the manager chooses

to evaluate the policy (610) The prototype evaluates the policy and displays the results the

customer service level is 998 and the annual cost is $441 With the alternate policy

virtually all demand is satisfied and the cost of inventory is still reduced by over 40 Thus

the manager ascertains that approximately 40 of the current cost of inventory for this product

is directly attributable to marketing considerations Using the prototype in this way

management can begin to understand better the nature of its inventory costs A description and

graphical depiction of the above hypothetical consultation are presented in Appendix H

As for the technical details of the system the prototype can run on any computer using

the DOS operating system having approximately 420K of available conventional memory and

attached to a V G A or SVGA monitor Further the prototype can be used on a machine without

a math coprocessor however performance is slower The prototype consists of two modules

linked together into one executable program The interface module contains the various

interactive subroutines and the main module contains the methodology of the model and the

logic of the heuristic search Both modules are coded in Fortran and compiled using Microsoft

Fortran 50 which provides many extensions to standard Fortran The source codes for the

interface module and the main module are listed in Appendix I and Appendix J

53

respectively

In addition to the above technical specifications the program requires the presence of

certain files in certain directories The program uses the distribution files mentioned in

Chapter II to read the unit costs and the empirical demand functions Each product has a

unique distribution file the name of which is the six digit product number followed by the

extension dst The text file containing the existing policies at each store must be located in

the same directory as the distribution files Finally the file tmsrbfon which contains the

font used by the interface must be located in the directory from which the program is executed

In its future versions the system should be more automated and more user-friendly

Rather than having the manager choose an upper bound for the search the algorithm should

incorporate a method of determining Sm a x For this to happen either a theoretical upper bound

must be obtained or a heuristic for terminating the search must be developed Also the amount

of time spent waiting for calculations to be performed can be reduced by reading the optimal

and current policies and their associated costs and customer service levels from a file or

database This solution file could be created any time prior to the consultation by running

the system in a batch mode The interface should allow the manager to change cost estimates

and service level values view results graphically and print results and summary documents

Finally the system should provide an option for automatically incrementing the product andor

store during a consultation This option would enable a manager to review quickly the

inventory policy of a product at each store

54

VI R E S U L T S AND SENSITIVITY ANALYSIS

A Comparison of Current Policies to Optimal Policies

In order to estimate the potential savings of implementing optimal policies the solutions

obtained in the Chapter IV are analyzed in more detail For each product the current policy

is evaluated and then compared to the optimal policy with respect to cost and customer service

level Finally the potential savings of the optimal policies are extrapolated to all products and

all stores

Of the 420 products examined each of 37 products has an existing policy that fails to

provide the desired customer service level of 975 However most of these policies just

fail to do so only 4 policies provide a customer service level of less than 90 Table 4

provides a list of these 37 products each with its current customer service level its current

annual cost and the additional cost of its optimal policy3 2 Two comments regarding Table

4 are in order First the current customer service level for product 203312 an amber

driveway marker is listed at a misleading 303 Examination of the demand distribution

reveals that this product was sold on only 1 of the 307 days during which sales data were

collected and on that day 13 of the markers were sold Clearly this data point constitutes

either an error or an unusual event the current policy of (34) does not allow for more than 4

items to be sold per review period Second the additional cost of the optimal policy for each

of 5 products is negative In other words for each of these products the current policy which

fails to provide the desired customer service level costs more than the optimal policy which

3 2 This is the difference between the expected annual cost of the optimal policy and the annual cost of the current policy It is the minimum additional cost required to increase the customer service level to at least 975

55

Table 4 Products with insufficient existing policies

Product Current CSL ()

Current Annual Cost ($)

Additional Cost of the Optimal Policy ($)

206932 974 204 -051

204570 973 222 -025

203053 973 248 009

206550 972 118 042

206547 972 105 035

206565 972 146 056

203702 972 340 157

206876 972 155 061

205752 972 330 153

205922 972 108 036

206857 972 180 073

205393 972 190 -003

201513 971 345 015

203405 970 255 037

203069 968 295 -048

203071 967 321 -001

205635 967 390 029

203914 966 460 160

204601 962 252 061

203326 960 144 014

202003 958 876 437

206516 958 112 029

206517 958 283 120

206350 958 333 147

205932 958 180 066

203936 952 598 294

56

Product Current CSL ()

Current Annual Cost ($)

Additional Cost of the Optimal Policy ($)

206932 974 204 -051

203939 948 413 193

203977 945 1216 1242

206551 945 132 077

205389 934 598 047

203590 927 1223 607

206567 919 122 070

203591 905 1228 834

205390 887 291 074

203318 833 136 084

206393 776 319 289

203312 303 237 511

provides at least the desired customer service level For the 36 products the total additional

cost of the optimal policies is $542033

Of the remaining 383 products most have an existing policy that provides a customer

service level well above 975 When the customer service levels are rounded to one decimal

each of 253 products has a customer service level of 1000 and each of 347 products has a

customer service level of at least 990 In contrast each of only 58 products has an optimal

policy that yields a customer service level of at least 99034 For each of 37 products the

3 3 The cost associated with product 203312 is not included in the total

3 4 Optimal policies do not have customer service levels of exactly 975 because the function is discrete For low-demand products a unit step in s or S results in a relatively large change in the customer service level

57

current policy is identical to the optimal policy

The extremely high level of service provided by most policies allows for substantial

savings in the cost of inventory With current policies the estimated annual cost of inventory

for all 420 products is $420743 whereas the expected annual cost of the optimal policies is

only $186884 This amounts to a savings of approximately $2340 or over 55 of the current

cost of inventory

In order to further assess the savings the difference between the costs of the current

policies and the costs of the optimal policies are examined in more detail For each of the 383

products whose existing policy provides a customer service level above 975 the difference

between the estimated annual cost of the existing policy and the expected annual cost of the

optimal policy is calculated The resulting histograms of the savings in (a) dollars and (b)

percentage of current cost are shown in Figure 6 The particulars for the current and optimal

policies for all 420 products are provided in Appendix K

As evident from Figure 6(a) the majority of the expected savings in absolute terms

comes from only six products In order to examine these products more closely the current

and optimal policies along with the associated costs and savings for each of the six products

are listed in Table 5 Each current policy is very different than its corresponding optimal

policy resulting in excessive inventory for each product In fact over half of the total expected

savings could be realized by implementing the optimal policy just for each of these six products

Interestingly five of the six products are sealed beam headlights

58

90

10 - 20 30 - 40 50 - 60 70 - 80 90- 100

Savings ()

(b)

Figure 6 The distribution of savings of optimal policies in (a) dollars and (b) percentage of current cost

59

Table 5 Products with the largest potential absolute savings

PRODUCT

CURRENT OPTIMAL SAVINGS

PRODUCT s S Cost ($)

C S L

()

s S Cost ($)

C S L () ($) ()

2036801 194 200 64014 1000 37 48 bull 9245 975 54769 856

2036772 94 100 23652 1000 18 22 3850 978 19802 837

2036723 78 90 23043 1000 19 23 4454 975 18589 807

2030944 145 175 19522 1000 34 45 3727 975 15795 809

2036425 48 60 10491 1000 13 17 2682 977 7809 744

2036556 68 80 7265 1000 14 22 1699 975 5566 766

1 H6054 rectangular sealed beam headlight 2 H4656 rectangular sealed beam headlight 3 6053 rectangular sealed beam headlight 4 9004 halogen bulb 5 4652 rectangular sealed beam headlight 6 6104 7 sealed beam headlight

Although Figure 6(a) seems to indicate that the majority of the current policies are close

in cost to the optimal policies this is very misleading For example suppose the optimal policy

for a product costs $1 and that the current policy costs $2 In absolute terms the optimal

policy saves only $1 but in relative terms the optimal policy saves 50 of the current cost

When savings are viewed in this manner as in Figure 6(b) it is clear that substantial savings

can be realized for the majority of the products In fact both the mode and the mean of the

savings are between 40 and 50 of the current costs

In order to extrapolate the above savings to obtain the potential savings for product

category 20 at all 21 stores the differences between stores must be considered During the

60

pilot study the current policy for each of several products at each size of store was evaluated

The study suggested that medium and large stores have a similar amount of excess

inventory and that jumbo and extra jumbo stores have much more excess inventory One

could reasonably expect that the additional excess inventory allows for higher savings to be

realized at the jumbo and extra jumbo stores Thus because 18 of the 21 stores are

medium or larger the expected 40-50 savings at store 6 can be viewed as a lower bound

on the potential savings to Canadian Tire Pacific Associates from implementing optimal policies

for product category 20 in all its stores

Finally in order to obtain the overall potential savings the expected savings from

product category 20 must be extrapolated to all product categories Ideally the proportion of

inventory comprising each product category and the relative savings that can be realized for

each category would be used to obtain a good estimate of the overall potential savings

However as this information is not known only a crude extrapolation is possible Thus the

analysis proceeds under the assumption that a 40-50 savings can be realized in each product

category35

Before an estimate of the total potential savings in absolute terms can be obtained the

proportion of the current inventory to which the savings apply must be determined Up to this

point the analysis has been concerned solely with reducing inventory at the stores However

it is inappropriate to perform further analysis without considering the entire inventory system

including the central warehouse Because the lead time for orders from Toronto is highly

variable and considerably greater than those at individual stores Canadian Tire Pacific

3 5 During the pilot study the current policies for a paint brush and an ironing board were evaluated and found to result in excess inventory similar to that of product category 20 Although this in no way justifies the assumption it does lend some support

61

Associates must keep a fairly high safety stock at the warehouse Without knowing the

proportion of inventory serving as the safety stock it is inappropriate to assume any reduction

in inventory at the warehouse is possible Therefore the relative savings of implementing

optimal policies is applied only to the inventory held in the stores36 Of the estimated $28

million in inventory approximately half is held in the stores therefore the 40-50 savings

translates to a savings of approximately $55 to $7 million per year

B Sensitivity Analysis on the Ordering Cost and Holding Rate

The above analysis was based on estimates of the ordering cost and the annual unit

holding rate37 provided by management As described in Chapter II these estimates

incorporate a number of costs that are difficult to quantify or that are speculative Therefore

there is some question as to the accuracy of the estimates In this section errors of +$0015

in the ordering cost and +5 in the holding cost are assumed in order to determine the

sensitivity of the model to errors in these variables38

The analysis proceeds as follows Based on an ordering cost of $0080 and a holding

rate of 30 the optimal policies from the previous section were obtained39 Then assuming

3 6 Of course this assumes that the excess inventory currently held in the stores is not required as part of the safety stock Based on the large amount of inventory at the warehouse this assumption seems reasonable

3 7 Hereafter simply referred to as the holding rate

3 8 An error of +$0015 in the ordering cost implies an actual ordering cost of $0070 a relative error of approximately 21 and an error of -$0015 implies an actual ordering cost of $0100 a relative error of 15 An error of +5 in the holding rate implies an actual rate of 25 a relative error of 20 and an error of -5 implies an actual holding rate of 35 a relative error of approximately 17

3 9 These policies are referred to as optimal policies for the duration of this chapter

62

the actual ordering cost andor holding rate are different from managements estimates the

heuristic is used to obtain the true optimal policies Finally the optimal policies are

evaluated based on the actual ordering cost and holding rate and then compared to the true

optimal policies For each possible combination of ordering cost and holding rate three

numbers are recorded the number of products with an optimal policy that differs from its

true optimal policy the total cost of implementing the optimal policies given the actual

ordering cost and holding rate and the difference between the cost of implementing the

optimal policies and the cost of implementing the true optimal policies The results of the

analysis are summarized in Table 6

Table 6 A comparison of optimal policies to true optimal policies

Actual Ordering Cost

$0070 $0085 $0100

25 5 products $155439

$002

51 products $160590

$092

88 products $165464

$449

Actual Holding Rate

30 54 products

$181470 $182

48 products $192026

$088

35 79 products

$207270 $574

48 products $212940

$136

2 products $218312

$000

4 0 The table should be read as follows If the actual ordering cost is $0070 and the actual holding rate is 25 then each of 5 products has an optimal policy that is different from its true optimal policy The total cost of implementing the optimal policies is $155439 which is $002 more than implementing the true optimal policies

63

The model appears to be quite insensitive to relative errors in the order of 20 Under

the worst case scenario when one of the variables is over-estimated and the other is undershy

estimated the optimal policies result in a total cost of inventory which is only approximately

3 higher than the cost of the true optimal policies Further because demand is not changed

by errors in these estimates each of the optimal policies still provides a customer service

level of at least 975 under all of the scenarios Finally although the cost of the optimal

policies varies significantly for the different scenarios the relative savings remain virtually

constant Table 7 lists the cost of existing policies and in parenthesis the relative savings of

the optimal policies for each of the different combinations of ordering cost and holding rate

The cost of the existing policies varies in the same manner as that of the optimal policies the

relative savings for each of the scenarios remains around 55 Therefore even in the presence

of such errors in the ordering cost and holding rate the projected savings of implementing the

optimal policies would be unchanged

Table 7 The cost of existing policies and the relative savings of the optimal policies under various scenarios

Actual Ordering Cost

$0070 $0085 $0100

25 $350153 (56)

$358464 (55)

$366764 (55)

Actual Holding Rate

30 $412438 (56)

$429048 (55)

35 $474725 (56)

$483026 (56)

$491320 (56)

64

C Sensitivity Analysis on the Demand Probability Mass Function

In this section the sensitivity of the model to slight changes in the demand probability

mass function is investigated As described in Chapter II the demand function used in the

model is based solely on observed sales however this empirical function is only one possible

instance of the underlying true demand function For example had the data been collected

over a period both starting and finishing one month earlier many of the empirical functions

would be slightly different Consequently depending upon the degree to which the empirical

functions vary and the sensitivity of the model to these variations the optimal policies from

the model might be inaccurate For example recall the case of the amber driveway marker

only thirteen markers were sold and all were sold on the same day Because the empirical

demand function is used the model calculates an optimal policy of (1213) clearly this

policy is impractical Although this example is extreme it does indicate the danger of using

unrealistic empirical demand functions41

As discussed in Chapter II three classes of products were identified namely low

medium and high-demand products For this analysis the same method of classification

is used Two products are selected from each of these groups and two new distribution files

are created for each of the six products For each product one of the files contains two

additional days of data with each day having a demand equal to the maximum observed demand

for that product In the second file only one additional day of data is added but the demand

on that day is two more than the maximum observed demand The situations that could give

4 1 Certainly a rigorous analysis to determine both the extent of variation in the empirical demand functions and the effect of using the empirical functions needs to be performed However such an analysis is beyond the scope of this thesis only a brief analysis that might suggest the degree to which the model is sensitive is performed here

65

rise to such changes in the demand data are termed Scenario 1 and Scenario 2 respectively42

The analysis proceeds in a manner similar to the one in the previous section Given the

demand that was originally observed the optimal policies were obtained Now assuming the

true demand function corresponds to the demand under Scenario 1 the true optimal policy

for each product is calculated Then each optimal policy is re-evaluated using the true

demand function and compared to the true optimal policy The analysis is then repeated

assuming the true demand function corresponds to the demand under Scenario 2 Table 8

lists the results for each product and scenario

Table 8 True optimal and optimal policies for each demand scenario

Product 202101 (low demand)

Scenario s s Cost ($) C S L ()

1 True optimal 1 2 470 993

Optimal 1 2 470 993

2 True optimal 2 3 663 976

Optimal 1 2 461 875

Product 203507 (low demand)

Scenario s S Cost ($) C S L ()

1 True optimal 0 3 334 981

Optimal 0 2 258 972

2 True optimal 2 3 508 980

Optimal 0 2 273 702

4 2 To be clear the scenarios are defined as follows Scenario 1 The addition of two days each with a demand of max Scenario 2 The addition of one day with a demand of max+2

where max is the maximum observed daily demand for each product in the sample period

66

Product 201032 (medium demand)

Scenario s S Cost ($) C S L ()

1 True optimal 6 12 1377 976

Optimal 6 11 1338 973

2 True optimal 6 13 1426 977

Optimal 6 11 1338 970

Product 203039 (medium demand)

Scenario s S Cost ($) CSL ()

1 True optimal 5 19 256 975

Optimal 5 16 251 969

2 True optimal 5 20 258 976

Optimal 5 16 249 968

Product 203677 (high demand)

Scenario s S Cost ($) CSL ()

1 True optimal 18 23 3981 977

Optimal 18 22 3817 972

2 True optimal 18 23 3995 978

Optimal 18 22 3831 973

Product 203680 (high demand)

Scenario s S Cost ($) CSL ()

1 True optimal 38 49 9498 976

Optimal 37 48 9175 971

2 True optimal 37 49 9484 975

Optimal 37 48 9206 972

67

The model is most sensitive to changes or errors in the demand functions of low-

demand products When only one additional day of demand for max+2 units is added to the

data the optimal policy for each of the low-demand products results in a very low customer

service level In fact for one of the products the customer service level drops to almost 70

The increase in demand does lower the customer service level for each of the other the

products but that decrease is slight

The above analysis indicates the need for some kind of smoothing or modelling of the

demand data especially for low-demand products The inherent variation between two

instances of a true demand function could very well result in more pronounced changes than

those resulting from either of the above scenarios Therefore it is likely that many optimal

policies especially those for low-demand products would actually provide a much lower or

much higher customer service level than expected However by smoothing the data the

variation between instances of a true demand function can be reduced and the desired

customer service levels can be achieved

For example suppose that the demand for a product is stationary that the average

demand during the previous period was unusually low and that the optimal policy for this

period is based on the demand during the previous period Because the demand is stationary

the probability that the demand during this period is higher than that during the past period is

more than 50 In other words the probability that the demand during this period is higher

than that for which the optimal policy provides the desired expected customer service level

is greater than 05 Thus expected customer service level for this period is actually less than

that specified Now suppose instead that the optimal policy is based on smoothed data which

reflects very closely the expected demand The probability that the demand during this period

is higher than that for which the optimal policy provides the desired expected customer service

68

level is virtually 05 Thus as a result of smoothing the data the specified customer service

level should be expected

69

VII C O N C L U S I O N

In this thesis we developed a single-product single-location model of the inventory

system at Canadian Tire Pacific Associates Specifically we formulated a Markov chain model

to determine the long-run average cost of inventory for fixed values of s and S and then

developed a heuristic search to find near optimal policies quickly Also as a first step toward

developing an integrated interactive inventory control system we created a prototype of the

interface

According to the analysis performed here Canadian Tire Pacific Associates is currently

holding an excessively large inventory in its stores This study finds that Canadian Tire Pacific

Associates could reduce the cost of inventory for product category 20 at store 6 by

approximately 50 and still maintain customer service levels of at least 975 Further the

mean and modal values of savings for these products are between 40 and 50 Given the

similar excess inventory at the other stores and in all product categories we believe that savings

in this range are possible for all products and stores We estimate that the application of

inventory control as described in this thesis would result in annual savings of $55 to $7

million

Given the large potential savings to Canadian Tire Pacific Associates we recommend

that management proceed with the next phase of the project Specifically we recommend that

the model algorithm and interface be refined further that a forecasting model be developed

and that a data collection and storage scheme be designed

In order to reflect operational changes at Canadian Tire Pacific Associates subsequent

to the formulation of the model modifications must be made to allow for the lead time to

exceed the review period Several of the stores now have three picks per week and as a

70

result orders placed for these stores at one decision epoch do not arrive until after the next

decision epoch Because this violates one of the models assumptions the model must be

modified accordingly Note however that with three picks per week and a review period

of four days orders from only the previous decision epoch are ever outstanding at the current

decision epoch

Also the model should be refined to reflect the fact that the review period is not fixed

over the planning period In its present form the model assumes a fixed review period

however inventory at stores with two picks per week is actually reviewed on a four day -

three day cycle and inventory at stores with three picks per week is reviewed on a three day

- two day - two day cycle Rather than obtaining the steady state costs for fixed review period

the model should calculate the steady state cost for a period of one week during which multiple

replenishments occur Although the solutions are not expected to differ significantly as a result

of the change the model would reflect the actual process more accurately

Further work is also required on the search technique for locating near optimal

policies Although the heuristic search described in this thesis appears to work well the

decision maker must provide an upper bound on S prior to the search In practice this upper

bound is generally not known in advance Thus if the heuristic is to be used either a tight

theoretical bound must be found or a termination heuristic must be developed Alternatively

a new heuristic search could be developed based on additional or different assumptions

regarding the properties of the feasible space For example although some regions of non-

convexity were found for some products much of the feasible space appeared to be convex

A heuristic that assumes a convex feasible space would not require an upper bound on S and

might perform very well However before such a heuristic is implemented further work is

necessary to determine the degree of non-convexity for a variety of demand functions and

71

ordering and holding costs

In addition to making the above modifications we recommend that Canadian Tire Pacific

Associates begin work on the remaining two components of the inventory control system First

in order to allow for products with seasonal demand the system requires a forecasting model

to provide accurate estimates of demand for the coming period Second the system needs a

data collection and storage module that systematically collects sales data for each active

product and for each store and then updates the corresponding demand distributions Because

approximately 25 of demand is expected to result in lost sales under the optimal policies the

sales data will have to be adjusted accordingly in the future Also because the model appears

to be sensitive to slight changes in the demand distribution for low-demand products the

module should incorporate a technique for smoothing the empirical distributions for these

products Further work should be undertaken to determine the extent of variation in the

empirical distributions and the most appropriate smoothing technique However because of

the large amount of data required to forecast seasonal demand and because of the current lack

of historical data we recommend that management proceed immediately with the design of a

data collection scheme

Finally two additional issues that are beyond the scope of the next phase should be

addressed at some point First a monitoring system will be required once inventory control

is implemented in the stores The monitoring system will ensure that optimal policies perform

as expected and will indicated whether or not demand is adversely affected by the lower levels

of in-store inventory Second given the degree to which in-store inventory can be reduced

inventory control in the warehouse should be investigated Because approximately half of the

existing inventory is held in the warehouse the additional savings from reducing system-wide

inventory could be great

72

Afterword

Just prior to the completion of the work described in this thesis Mr Don Graham

principle owner of Canadian Tire Pacific Associates sold his shares in the company Shortly

thereafter the results of this thesis were presented to the new management group

Unfortunately the new management did not consider development of an inventory control

system to be a priority Nonetheless the prototype was delivered to Canadian Tire Pacific

Associates for possible in-house testing however at the time of this writing - almost one year

after the meeting - Canadian Tire Pacific Associates has not decided to continue with the

project

73

Bibliography

Arrow K A T E Harris and J Marschak (1951) Optimal inventory policy Econometrica 19 250-272

Day G (1992) In a speech to the Empire Club of Canada Toronto quoted in When the JIT fits The Globe and Mail October 20 1992 B24 and in Puterman (1994)

Dvoretzky A J Kiefer and J Wolfowitz (1952a) The inventory problem I Case of known distributions of demand Econometrica 20 187-222

Dvoretzky A J Kiefer and J Wolfowitz (1952b) The inventory problem II Case of unknown distributions of demand Econometrica 20 450-466

Federgruen A (1993) Centralized Planning Models in SC Graves A H G Rinnooy Kan and P H Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Federgruen A H Groenevelt and H C Tijms (1984) Coordinated replenishments in a multi-item inventory system with compound Poisson demands and constant lead times Management Sci 30 344-357

Lee H L and S Nahmias (1993) Single-Product Single-Location Models in SC Graves A H G Rinnooy Kan and PH Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Muckstadt J A and R O Roundy (1993) Analysis of Multistage Production Systems in SC Graves A H G Rinnooy Kan and PH Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Porteus E L (1985) Numerical Comparisons of Inventory Policies for Periodic Review Systems Oper Res 33 134-152

Puterman M L (1994) Markov Decision Processes Discrete Stochastic Dynamic Programming Wiley New York

Silver E A and R Peterson (1985) Decision Systems for Inventory Management and Production Planning 2nd edition Wiley New York

Taylor H M and S Karlin (1994) An Introduction to Stochastic Modeling revised edition Academic Press San Diego

74

Tijms H C and H Groenevelt (1984) Simple approximations for the reorder point in periodic and continuous review (sS) inventory systems with service level constraints European J Oper Res 17 175-190

75

Appendix A

A Portion of the Sales File for Product 200001 a 30-amp Inline Fuse

200001 283 1190 0 0 2 0 0 1 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 200001 284 1190 2 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 200001 285 1190 0 0 4 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 286 1190 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 200001 287 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 200001 288 1190 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1 0 2 200001 289 1190 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 290 1190 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 200001 291 1190 0 0 2 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 200001 292 1190 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 200001 294 1190 0 0 0 1 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 1 200001 295 1190 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 200001 296 1190 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 200001 297 1190 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 200001 298 1190 0 0 1 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 200001 299 1190 0 0 1 0 0 0 1 0 1 0 0 2 0 0 0 0 0 2 1 1 0 200001 300 1190 0 0 1 0 0 0 0 1 0 0 0 0 0 2 2 0 0 0 0 3 0 200001 301 1190 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 200001 303 1190 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 200001 304 1190 1 0 0 0 0 0 0 0 1 0 0 2 0 0 0 0 0 0 0 0 0 200001 305 1190 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 200001 306 1190 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 2 0 0 1 0 200001 308 1190 0 0 0 1 0 1 0 0 0 2 0 1 0 0 0 0 1 1 1 0 0 200001 309 1190 1 3 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 200001 310 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 200001 311 1190 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 200001 312 1190 0 1 1 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 200001 313 1190 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 2 0 200001 314 1190 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 200001 315 1190 0 0 2 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 200001 316 1190 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 200001 317 1190 2 0 0 1 0 0 0 0 0 0 5 1 0 0 0 1 0 0 0 0 0 200001 318 1190 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 1 200001 319 1190 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 200001 320 1190 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 200001 321 1190 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 1 200001 322 1190 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 323 1190 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 200001 324 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 0 1 0 0 2 1 1 0 200001 325 1190 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3 200001 326 1190 0 0 1 0 1 0 0 0 2 0 0 1 0 0 0 1 0 1 0 0 1 200001 328 1190 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 200001 329 1190 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 200001 330 1190 0 0 1 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 200001 331 1190 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 200001 332 1190 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 200001 333 1190 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 200001 334 1190 0 0 2 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 200001 335 1190 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 200001 336 1190 0 0 0 1 0 0 3 0 0 0 0 0 2 0 0 0 0 0 0 0 0

(253 subsequent lines omitted)

76

Appendix B

Sample Distribution Files

a) A low-demand product - an automobile back-up alarm

PRODUCT NUMBER 202101 PRICE $ 684

DMD

3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

N 2 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 2 2 0 1

1 5 3 8 12 5 7 11 6 11 5 4 11 10 19 4 2 5 14 5 6 4

0 302 304 299 295 302 300 296 301 296 302 303 295 296 288 303 305 301 290 300 301 302

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

b) A medium-demand product - a 168 bulb

PRODUCT NUMBER 203039 PRICE $ 51

DMD

16 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 1 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0

9 0 0 2 5 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0

8 0 0 2 5 0 0 2 0 1 1 3 0 1 3 0 0 1 1 0 0 1

7 2 1 3 1 0 0 1 0 2 3 1 1 4 1 1 0 2 1 1 1 1

6 4 7 1 10 3 1 3 2 1 3 5 2 7 4 1 0 1 1 3 3 4

5 1 7 7 17 0 2 11 1 5 4 1 2 6 9 6 2 6 9 4 4 1

4 5 10 10 16 5 5 19 10 13 8 11 9 13 10 12 7 5 15 5 8 3

3 10 12 13 27 4 6 24 11 10 13 14 9 23 14 18 8 14 18 14 12 14

2 14 30 36 43 16 16 43 32 38 39 39 34 47 36 41 21 19 39 20 34 24

1 30 56 68 56 37 30 70 48 67 61 44 75 62 63 48 28 52 61 63 53 47

0 240 184 165 126 240 247 133 202 169 175 189 173 143 165 180 240 206 161 197 192 212

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

77

c) A high-demand product - a H4651 rectangular sealed beam headlight

PRODUCT NUMBER 203677 PRICE $ 857

DMD

14 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

13 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 2 3 0 0 1 0 0 0 0 1 1 1 0 0 0 0 2 0 0

11 0 1 3 1 0 0 4 1 0 0 0 0 0 0 0 0 1 0 0 0 1

10 0 0 4 4 1 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 1

9 0 3 14 11 0 1 7 1 0 2 0 8 0 1 0 0 0 2 1 0 0

8 0 4 12 16 1 0 18 4 3 5 0 2 1 4 2 1 1 0 0 1 3

7 1 2 12 13 1 6 19 3 5 9 0 11 2 5 4 0 2 3 3 1 4

6 2 9 22 21 6 10 29 12 10 11 4 24 3 7 10 4 7 11 6 3 14

5 7 13 35 33 8 16 29 19 18 8 3 26 15 26 15 12 9 14 13 11 19

4 19 35 34 43 32 30 41 31 32 47 11 42 20 33 30 31 20 31 22 21 31

3 32 63 49 54 42 60 51 40 51 35 33 55 39 51 39 38 39 57 58 40 54

2 46 59 56 55 73 63 61 65 63 61 63 54 68 85 65 53 72 69 70 77 63

1 109 79 42 40 80 64 32 80 74 77 99 55 97 57 83 93 80 75 81 97 73

0 91 39 21 13 63 57 11 51 51 52 94 29 61 35 59 75 76 45 51 56 44

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

78

Appendix C

A Proof that the Limiting Probability Distribution is Independent of the Initial State

In this appendix we show that provided 0 lt PrDT = 0 lt 1 the Markov chain

corresponding to an inventory model with a (sS) policy has a limiting probability distribution

7T where 0 lt TTJ lt 1 for y = 0 1 S that is independent of the initial state Further the

limiting distribution is the unique nonnegative solution of the equations

Tt P = TI (AD

s and J gt = 1 (A 2)

i=0

where P is the (S + l) x (S + l) transition probability matrix whose first n rows (and columns)

correspond to the n recurrent states in the chain and whose remaining rows (and columns) if

any correspond to transient states43 Prior to the proof several preliminary results are

obtained

RESULT 1 The Markov chain has exactly one positive recurrent class and perhaps some

transient states

First note that in the absence of any absorbing states a Markov chain with a discrete

finite state space must have at least one positive recurrent class containing at least two states

Now consider the Markov chain for an inventory model with a (sS) policy

a) State i gt s cannot be absorbing because there is a positive probability of reaching

j lt i (Since PrDT gt 0 gt 0)

4 3 Note that since the rows and columns of P are reordered the elements of it must be reordered accordingly

79

b) State lt s cannot be absorbing because by the rules of the ordering policy and

the assumption that PrDT = 0 gt 0 there is a positive probability reaching S

Therefore the chain has no absorbing states and must have at least one positive recurrent class

Next consider the subset of states A = i | i lt s Because the state space is finite

and PrDT gt 0 gt 0 the process must return to at least one state in A in other words at least

one state in A must be recurrent Let k G A be a member of a recurrent class dt Because

of the ordering policy and the assumption that PrDT = 0 gt 0 k -raquo S Also by definition

there must be a positive probability of returning to k from S S -gt k Therefore S G 3w

Further because S is reachable from all states in A i G A is either transient with i -raquo dt or

recurrent with pound iR

Now consider the subset of states B = | j gt s Because PrDT gt 0 gt 0 the

chain will eventually reach a state i G A If i is transient then j must be transient also with

y -gt 3 if i G 9 then j is either transient with j -gt dt or recurrent with j E dt Thus the

Markov chain has exactly one recurrent class and perhaps some transient states

RESULT 2 The limiting probability distribution ir for the Markov chain corresponding to the

nxn submatrix Pm = [Ptj] i j G dt is independent of the initial state and is the

unique nonnegative solution of the equations

TC-Prr = re (A3)

and J2 K = 1 ( A 4 )

jt=i

where the rows and columns of are numbered from 1 to n rather than

Since if i j i and j must be in the same equivalence class

80

according to their associated states

First note that both the original Markov chain and the one corresponding to the

submatrix P R R are aperiodic because PrDT = 0 gt 0 Therefore given that 9 is a positive

recurrent aperiodic class the limiting distribution ir is uniquely determined by (A3) and (A4)

(Taylor and Karlin 1994 Theorem 44 p213) Further because there are no absorbing states

or other recurrent classes in the original chain the process must eventually enter the recurrent

class 9 regardless of the initial state Therefore in the limit the probability of being in state

i G 9 is independent of the initial state

RESULT 3 The limiting probability distribution for the transient states is the null vector

independent of the initial state

This result follows directly from the definition of a transient state

PROOF Suppose the states are reordered such that the first n rows (and columns) of P

correspond to the n recurrent states of the Markov chain Then P can be partitioned into four

submatrices

PRK - the n x n submatrix [Py] i j E 9

P T T - the (S + 1-I) x (S + l-n) submatrix [Py] i j g 9

P T R - the (S + l-n) x n submatrix [Py] i pound 9 j G 9

P R T - the n x (S + l-n) submatrix [Py] i G 9 j pound 9t By definition this is a matrix

of zeros

(Al) can then be rewritten as

81

(ltR TtT) P P L TR TT

I-P RR 0

p I-P TR 1 TT

(TtR TCT)

(0- 0T)

where irR denotes the limiting probability distribution for the n recurrent states and irT denotes

the distribution for the remaining S + l-n transient states Solving for TT t gives

TCr-0 + itT-(I-PTI) = 0T

itT-(I-PJJ) = 0 r

i c r = (I-P^-Oj

Since (I - PTT)1 always exists (Puterman 1994 Proposition A3 p 593) irT = 0X which is

in agreement with Result 3 Now solving for 7rR yields

(A2) can be rewritten as

bullRPRR 1 1R (A5)

(A6) S + l

k + L k =1

k=l k=n+

k=i

where ir contains the appropriately reordered steady state probabilities Since (A5) and (A6)

are equivalent to (A3) and (A4) respectively Result 2 holds Therefore the Markov chain

corresponding to an inventory model with a (sS) policy has a limiting distribution that is

independent of the initial state and is the unique nonnegative solution to (Al) and (A2)

82

Appendix D

A Measure for the Steady State Customer Service Level

In this appendix we show that the appropriate measure for the steady state customer

service level is given by

_ E[Demand satisfied during RP] E[Total demand during RP]

Consider the following two measures of the steady state customer service level

Measure 1 Suppose the customer service level is calculated only after n review periods have

passed Denoting DS as the demand satisfied in review period i and DT as the total demand

in period i the customer service level after n periods CSL(n) is

DS + DS0 + + DSbdquo CSLn) = - -

DT + DT + + DT

Dividing both the numerator and denominator by n taking the limit as n -raquo oo and applying

the Law of Large Numbers yields

CSL = lim CSL(n) n-degdeg

- [DS + DS + + DS] = l i m ^ J 1 1

raquo~ - [DT + DT + + DT]

E[DS]

E[DT]

83

Measure 2 Suppose the customer service level is calculated for each of n review periods and

then averaged Thus the customer service level after n periods CSL(n) is given by

CSL(n) = -DS DS

DT DT DT_

Taking the limit as n -gt oo yields

CSL = lim CSL(n)

lim mdash DSl DS2

DT DT DT_

DS

DT

Now consider the scenario in which there is one item on the shelf to begin each review

period and in which demand is stochastic with PrDT = 2 = Vi and PrD x = 100 = Vi

According to Measure 1 the customer service level is

CSL - = ^ W + ^ W = plusmn bdquo 002 E[DT) 1 (2) + 1(100) 51

while according to Measure 2 the customer service level is

CSL = E DS DT

1(1) + i(-Ls

22) + 2 100

51 200

026

Clearly from managements point of view the first measure is more indicative of the

actual level of service being provided The problem with the second measure is that equal

84

weight is given to each ratio regardless of the magnitude of the numbers For example with

respect to Measure 2 the events LSLR = 12 and LSLR = 50100 are identical however

with respect to lost sales the second event is much worse than the first Therefore the proper

measure for the steady state customer service level is Measure 1

85

Appendix E

The Calculation of the Conditional Expected Demand Not Satisfied During a Period of T Consecutive Days

Denoting D x as the demand during a period of T consecutive days and j as the inventory

on hand at the beginning of the period the conditional expected demand not satisfied during the

period E[(DT-y)+] is calculated as follows

E[(DT-j)+] = pound ltk-j)Pr[DT = k k=j+l

= pound (k-j)PrDT = k] + J2(k-j)PrDT = k] - pound (fc-)VJgtr = fc k=j+l k=0 k=0

= J2(k-j)Pr[DT=k -J2(k-j)PrDT = k] k=0 k=0

= YkPrDT=k) - J2jPrDT=k - Yk-j)PrDT=k) Jfc=0 k=Q k=0

= Ti - j - Y(k-j)PrDT=k 7-1

= Tx - j + YltJ-k)PrDT = k jt=0

where fx is the mean daily demand

86

Appendix F

Justification of the Updating Technique for the New Policy (s+mS)

Let the probability transition matrix corresponding to the policy (sS) be denoted by the

(S + l)x(S + l) matrix P with elements py Define s = s+m where m is an integer satisfying

the condition 0 lt s+m lt S Let the probability transition matrix of the new policy (sS)

be denoted by the (S + 1)X(S + 1) matrix 7gt with elements py Using the equations developed

in Chapter III the elements of P are found as follows

CASE 1 The lead time is less than the review period

A m gt 0

For 0 lt i lt s

( i-i pound [PrDL=kPrDT_LgtS-k]] + PrDLziPrDT_LplusmnS-i k=0

i-l 53 [PrDL=kPrDT_L = S-j-k]] + PrDLziPr[DT_L = S-j-i) k=0 s-j

52 [Pr[DL = kPrDT_L = S-j-kn k=0

ifj = 6

ifOltjz S-i

if S-i ltj ltS

a) 0 lt lt s

Pij

= lt Pij

Pij

= pij

b) s lt i lt s This does not simplify but it is identical to (2)

87

For s lt i lt S

PrDTgt i

p = lt PrDT = i-j

0

Pfj ifj = 0

= lt Pij ifOltjlti

Pij if i ltj plusmn S

= pij

B m lt 0

ifj = 0

if 0 ltj lt i

if i ltj plusmn S

For 0 lt i lt s

i-l 53 [PrDL=kPrDT_LplusmnS-k]] + PrDLgtiPrDT_LplusmnS-i

i-l E [PrZgti = fciVZ)7W=S--fc] +PrD tgtiP-D 7 t = S-7-

p y laquoy = o

P i j ifOltjS-i

vPij ifS-iltjS

ifj = 0

ifOltjltS-i

if S-i ltjltS

For s lt i lt S

fPrJgtrgt- 17=0 = PrDT = i-j] ifOltjlti

0 ifiltjplusmnS

88

a) s lt lt s This does not simplify but it is identical to (3)

b) s lt i lt S

PrDTgti

= bull PrDT = i-j]

0

17=0

if0ltjplusmni

ifiltjS

= Pt

ifj = 0

if0ltjlti

ifiltjplusmnS

= Pi

CASE 2 The lead time is equals the review period

A m gt 0

For 0 lt i lt s

0 if0ltjlt S-i

PrDTgti ifj = S-i

PrDT=S-j] ifS-iltjltS

a) 0 lt i lt s

= bull y Pij

Pij

ifOplusmnjltS-i

ifj = S-i

ifS-iltjS

= Pij

b) s lt i lt s This does not simplify but it is identical to (4)

89

For s lt i lt S

PrDTgti)

Pr[DT=i-j

0

laquof 7=0

ifOltjplusmni

if i ltj lt S

Pa

Pij

if 7 = 0

if 0 lty lt i

ifiltjltS

B w lt 0

For 0 lt t lt s

ro PrJrgt PrDT = S-j

ifOzjltS-i

ifj = S-i

if S-i ltj lt S

Pij

ifOzjltS-i

ifj = S-i

if S-i ltj ltS

= Pi

For s lt i lt S

PrDTgti

= PrDT = i-j

0

i 7 = o ifOltjplusmni

ifiltjltS

a) s lt i lt s This does not simplify but it is identical to (3)

90

PrDTzi ifj = 0

PrDT = i-j] ifOltjplusmni

0 ifiltjltS

Py ifj = 0

ptj ifOltjplusmni

Pij if iltj plusmnS

Appendix G

Justification of the Updating Technique for the New Policy (sS+m)

Let the probability transition matrix corresponding to the policy (sS) be denoted by the

(S + l)x(S + l) matrix P with elements p(gt Define S = S+m where m is any non-negative

integer Let the probability transition matrix of the new policy (sS) be denoted by the

(S+m+l)x(S+m + l) matrix IP with elements ptj Then using the equations developed in

Chapter III the elements of 7 are found as follows

CASE 1 The lead time is less than the review period

For 0 lt i lt s

pound [PrDL=kPrDT_LplusmnS-k] + PrDLi)PrDT^Sgt- k=0

i-l pound [Pr[DL = kPrDT_L = S-j-k- +Pr[DLiPrDT_L = S-j-i Jt=0

s-i Y [Pr[DL=kPrDT_L = S-j-k] k=0

i - l

ifj=0

ifOltjltS-i

ifS-iltjltS

pound [PrDL = kPrDT_LS-k + Pr[DLi)PrDT_LplusmnS-i) if j = 0 fc=0

i - l = pound [PrDL=kPrDT_L = (S+m)-j-k] + PrDLgtiPr[DT_L = (S+m)-j-i ifOltj lt (J5+m)-i

if (S+m)-iltj lt(S+m)

k=0

i - l

pound (Sm)-j

pound [PrDL = kPrDT_L = (S+m)-j-k]] k=0

92

I-l

53 [PrDL = kPrDT_LgtS-k] + PrDLziPrDT_LS-i ifj = 0 k=0

i-l

Y[PrDL = kPrDT_L = S-(j-m)-k] + Pr[DLziPrDT_L = S-(j-m)-i] ifOltjzS+m-i k=0 S-(j-m)

53 [PrDL = kPrDT_L = S-(j-m)-k]] if S+m-i ltj lt S+m

i - l

53 [PrDL = kPrDT_LzS-k] + PrDLziPrDT_LzS-i] ifj = 0 =0

i - l

13 [Pr[DL = kPrDT_L = S-(j-m)-k] + Pr[DLziPr[DT_L = S-(j-m)-i ifOltjz S+m-i k=o and 0 z j-m

Pi(j-m)

Pi(j-m)

if 0 ltj lt S+m-i and 0ltj-mltS-i

if S+m-i ltj lt S+m and S-i lt j-m zS

i - l

Y [PrDL = k)PrDT_LzS-k] + PrDLi)PrDT_LplusmnS-i ifj = 0

k=0 i - l

53 [PrDL = kPrDT_L = S-(j-m)-k] + PrDLziPrDT_L = S-(j-m)-i ifOltjzm k=0

Piij-m)

Piij-m)

if mltj lt S+m-i

if S+m-i ltj lt S+m

i - l

YlPrDL = kPr[DT_L^S-k + PrDLi)PrDT_LS-i) ifj = 0

i - l

53 [PrDL = kPrDT_L = S-j-k] + PrDLgtiPrDT_L = S-j-i ifOltj i m k=0

Pi if mltj ltS

which for 0 lt j lt m is identical to (2)

93

For s lt i lt S

Pu =

PrDTgti

PrDT = i-j]

0

if0ltjlti

ifiltjltS

a) s lt i lt S

PrDTgti 7=0

PrDT = i -j) if0ltjlti

0 if i ltj lt S

0 ifSltjltSgt

ifj=0

if 0 ltj lt i

if i ltj plusmn S

0 ifSltjltS

ifOltjltS

0 ifSltjltSgt

b) S lt i lt S This does not simplify but it is identical to (3)

CASE 2 The lead time equals the review period

For 0 lt i lt s

0 ifO lt jltS-i

PrDTgti ifj=S-i

PrDT=S-j ifS-iltjltS

94

0 ifOplusmnjlt(S+m)-i

PrDT ifj = (S+m)-i

PrDT = (S+m) -j) if (S+m)-i ltj ltL (S+m)

0 if 0 lt jltS+m-i

PrDTzi] if j = S+m-i

PrDT = S-(j-m) if S+m-i ltj lt S+m

0 if 0 ltj lt S+m-i and 0 ^ j-m

Pi(j-m) if 0ltjlt S+m-i and 0ltj-mltS-i bull

Pi(j-m) if j = S+m-i and j-m = S-i

Pi(j-m) if S+m-i ltj z S+m and S-i lt j-m ^S

0 if 0 zj ltm

Pi(j-m) if mlt jlt S+m-i lt

Piij-m) if j = S+m-i

if S+m-i ltj lt S+m

1deg if 0 ltj ltm

[PiV-m) ifmltjltS

which for 0 lt i lt s is identical to (3)

For s lt lt S

PrDTzi

= PrDT = i-j)

0

ifj = 0

if 0 ltj ltL i

ifKjzS1

95

a) s lt i lt S

PrDTzi

PrDT = i-j]

0

0

17=0

if0ltjplusmni

ifiltjS

ifSltjltSgt

Pij 7=0

Pij if0ltjii

Pij if i ltj ltL S

0 ifSltjltS

Pij ifOzjiS

0 ifSltjltS

b) S lt i lt S This does not simplify but it is identical to (3)

96

Appendix H

A Hypothetical Consultation

In this appendix the interface during a hypothetical consultation is depicted The figures

accurately represent the various screens that are presented to the manager with the exception

that all lines and text are really white on a blue background During the consultation described

below only the choices made by the manager are hypothetical all the results are real

The title screen as depicted in Figure A - l appears initially upon execution of the

program In order to proceed to the main menu the manager presses the enter key

O I C S

Optimat Inventory Control System

For the exclusive use of

Canadian Tire Pacific Associates

Developed at The Faculty of Commerce

and Business Administration The University of British Columbia

Brian Kapalka Kaan Katircioghi Marty Puterman

Press ltEntergt to continue

Figure A - l The title screen

97

Main Menu 1 Select data directory

ctircdatadst

2 Select product and store combination Product 200001

3 Select scenario Review Period 4 Lead Time 3

4 Select search parameter Smax 25

5 Execute program

6 Exit to DOS

Please enter the desired option 5

Figure A-2 The main menu

Figure A-2 depicts the main menu of the program it is from here that the manager

enters all the review parameters If the manager wishes to change one of the parameters from

its default value he or she selects the appropriate number and enters the correct information

in the window presented The new settings will then be shown in place of the default values

In this example the relevant data files are located in the default directory ctiredatadst and

the manager has selected the productstore combination of product 200001 and store 6 Further

the manager has selected a review scenario of a 4-day review period and a 3-day lead time and

an upper search limit of 25 Finally in order to proceed with the analysis the manager

executes the main module of the program by selecting option 5

98

VI RESULTS A N D SENSITIVITY ANALYSIS 55 A Comparison of Current Policies to Optimal Policies 55 B Sensitivity Analysis on the Ordering Cost and Holding Rate 62 C Sensitivity Analysis on the Demand Probability Mass Function 65

VII CONCLUSION 70

Afterword 73

Bibliography 74

Appendix A A Portion of the Sales File for Product 200001 a 30-amp

Inline Fuse 76

Appendix B Sample Distribution Files 77

Appendix C A Proof that the Limiting Distribution is Independent of the

Initial State 79

Appendix D A Measure for the Steady State Customer Service Level 83

Appendix E The Calculation of the Conditional Expected Demand Not Satisfied

During a Period of T Consecutive Days 86

Appendix F Justification of the Updating Technique for the New Policy (s+mS) 87

Appendix G Justification of the Updating Technique for the New Policy (sS+m) 92

Appendix H A Hypothetical Consultation 97

Appendix I The Source Code for the Interface Module 104

Appendix J The Source Code for the Main Module 119

Appendix K Current and Optimal Policies for Product Category 20 at Store 6 131

iv

List o f Tables

Table 1 Location and particulars of the 21 stores 10

Table 2 Execution times of the algorithms for product 200001 a 30-amp inline fuse 39

Table 3 Execution times of the algorithms for product 206917 a 6

solder connector 40

Table 4 Products with insufficient existing policies 56

Table 5 Products with the largest potential absolute savings 60

Table 6 A comparison of optimal policies to true optimal policies 63

Table 7 The cost of existing policies and the relative savings of the optimal policies under various scenarios 64

Table 8 True optimal and optimal policies for each demand scenario 66

v

List of Figures

Figure 1 A typical sample path of the process with a 4-day review period

and a 2-day lead time 24

Figure 2 The flow chart of the grid search algorithm 38

Figure 3 The flow chart of the grid search algorithm utilizing the updating

technique for the transition probability matrix 44

Figure 4 An evaluation of the lower bound Sm i n 47

Figure 5 The flow chart of the heuristic algorithm 50

Figure 6 The distribution of savings of optimal policies in (a) dollars and

(b) percentage of current cost 59

Figure A-1 The title screen 97

Figure A-2 The main menu 98

Figure A-3 Calculating the optimal policy and evaluating the current policy 99

Figure A-4 Displaying the results 100

Figure A-5 Entering an alternate policy 101

Figure A-6 The particulars of the alternate policy 102

Figure A-7 The main menu revisited 103

vi

Acknowledgement

The completion of this thesis was made possible by the encouragement and assistance

of a number of people

I would like to express my sincere appreciation to my thesis supervisor Professor

Martin Puterman for all of his many efforts on my behalf His help advice patience charity

and tolerance were very much appreciated

I would like to acknowledge Professor Hong Chen and Professor Garland Chow for their

time and input as members of my thesis committee In addition I must acknowledge the

assistance of PhD student Kaan Katircioglu for his insight and help on this project

I offer many thanks to Professor Tom Ross for his kindness and friendship during the

past few years especially during the writing of this work I must also thank his family for

adopting me on many holidays I still owe thanks to Professor Slobodan Simonovic at the

University of Manitoba for influencing me to attend graduate school in the first place and for

helping me to obtain funding

I cannot thank my parents enough for their never-ending support both emotional and

financial Also to my friends especially Cathy Dave Lisa Steve and the Philbuds thanks

for giving me a life off campus and for picking up many a tab - the next one is on me

I would like to give special thanks to my good friend and fellow MSc student Paul

Crookshanks for allowing me to bounce ideas off of him and for being such a procrastinator

that despite my finishing a year late I only lost by two days

Financial assistance received from the Natural Science and Engineering Research

Council (NSERC) and from Canadian Tire Pacific Associates was greatly appreciated

vii

I INTRODUCTION

The importance of inventory management has grown significantly over the years

especially since the turn of this century In colonial times large inventories were viewed as

signs of wealth and therefore merchants and policy makers were not overly concerned with

controlling inventory However during the economic collapse of the 1930s managers began

to perceive the risks associated with holding large inventories As a result managers

emphasized rapid rates of inventory turnover (Silver and Peterson 1985) Following the

Second World War Arrow Harris and Marschak (1951) and Dvoretzky Kiefer and

Wolfowitz (1952ab) laid the basis for future developments in mathematical inventory theory

Shortly thereafter new inventory control methodologies were widely applied in the private

manufacturing sector More recently when inflation and interest rates soared during the 1970s

many organizations were forced to rethink their inventory strategies yet again Today the

control of inventory is a problem common to all organizations in any sector of the economy

Sir Graham Day chairman of Britains Cadbury-Schweppes P C L expressed this sentiment

when he stated I believe the easiest money any business having any inventory can save lies

with the minimization of that inventory (Day 1992) Lee and Nahmias (1993) and Silver and

Peterson (1985) provide a more detailed history of inventory management and control

Most inventory problems in the retail sector involve the control and management of a

large number of different products Ideally the inventory policies should be determined for all

products on a system-wide basis however because the number of products is often in the tens

of thousands formulating a model to do so is impractical Therefore single-product models

are frequently used in practice to determine policies for products separately (Lee and Nahmias

1993) Lee and Nahmias (1993) and Silver and Peterson (1985) provide a good overview of

1

single-product single-location models for a more complete review of the literature we refer

to them

The degree of complexity of a single-product model depends largely on the assumptions

made regarding the demand cost structure and physical characteristics of the inventory system

For example demand can be either deterministic meaning it is known with absolute certainty

or stochastic meaning it is unknown but follows a known probability distribution

Deterministic models tend to be relatively simple yet useful such as the basic EOQ model

However future demand is seldom known with certainty in practice and therefore the use of

these simple models is limited As a result the majority of research in inventory has focused

on stochastic demand models (Lee and Nahmias 1993)

Because the objective of most inventory models is to minimize the cost of inventory

assumptions regarding the structure of the cost are very important also Generally speaking

the cost of inventory has four components the purchaseproduction cost the orderingsetup

cost the holding cost and the shortage cost1 Each of these costs can be fixed proportional

nonlinear or some complex combination of these Further because nearly all inventory

problems require decisions be made sequentially over a period of time the costs can be either

discounted or averaged In the former case a discount factor is included in the cost function

in order to account for the time value of money However since most inventory problems are

on-going the long-run average cost is usually minimized (Lee and Nahmias 1993) In

practice this is achieved by minimizing the expected cost per time period under steady state

conditions

Finally the physical characteristics of the system are important in determining the

1 The cost of a stockout is often very difficult to estimate in practice As a result service levels are frequently used in its place When this is done a constraint is included in the model

2

complexity of the model For example the review process can be either continuous in which

case the inventory is reviewed continuously and orders are placed as soon as the stock reaches

a certain level or periodic in which case the inventory is reviewed at regular intervals and

orders can be placed only at those fixed times Also excess demand can be fully backordered

as is common in many manufacturing and wholesale systems completely lost as is common

in most retail environments or some combination of the two Lead time assumptions are also

important models with lead times are more complex than those in which the supply of

inventory is assumed to be instantaneous

Although the literature is dominated by single-product models (Lee and Nahmias 1993)

much recent work has focused on extending those models to multiple-product problems

However multiple-product models are inherently more complex than the simple-product

models In fact no efficient algorithms are currently available for determining fully optimal

strategies for most multiple-product problems (Federgruen 1993) As a result research has

focused on the development of easily computable approximation techniques Muckstadt and

Roundy (1993) provide a good overview of multiple-product andor multiple-location models

with deterministic demand although emphasis is placed on models for production and

distribution Federgruen (1993) extends the discussion to consider these models under uncertain

demand and also provides a complete list of references on the subject Silver and Peterson

(1985) discuss models for coordinated replenishments for both deterministic and stochastic

demand

In this thesis we examine the inventory system at Canadian Tire Pacific Associates

owner of 21 retail stores and a central warehouse Although this is a multiple-product

multiple-location single-warehouse problem and strictly speaking should be modelled as such

we analyze the inventory problem on a product-by-product and store-by-store basis We believe

3

that this simpler single-product single-location model captures the essentials of the problem for

two reasons First the coordination of replenishments for products andor locations is most

advantageous when significant savings are possible because of high first order costs and low

marginal costs for placing subsequent orders However because of the large number of

products and the physical distribution system in this problem trucks are dispatched to stores

on a regular basis Therefore the benefits of coordinating replenishments for this problem are

negligible Second we believe that controlling in-store inventory alone can result in substantial

savings to Canadian Tire Pacific Associates Further given the excessive amount of inventory

in the warehouse a single-location model assuming an infinite warehouse capacity is reasonable

Of course should inventory control in the warehouse become a priority a more elaborate model

would be required

More specifically we consider a periodic review single-product inventory model with

stationary independent stochastic demand a deterministic lead time and an infinite planning

horizon The probability mass function for the demand is based on empirical sales data

collected over a period of approximately ten months The cost of inventory includes a

proportional holding cost and a fixed ordering cost neither of which is discounted All excess

demand is completely lost The objective of the model is to minimize the long-run average cost

per unit time subject to a service level constraint on the fraction of demand satisfied directly

from in-store inventory In order to obtain meaningful cost comparisons the model calculates

the expected annual cost of inventory

For inventory problems like the one described above optimal (sS) policies can be

computed by Markovian decision process methods such as successive approximations or policy

iteration or by stationary Markovian methods However each of these methods generate

calculations that are complex and time consuming Therefore much research interest lies in

4

developing techniques to find approximate (sS) policies with little computational effort The

majority of the approximation techniques begin with an exact formulation of the problem and

then manipulate the problem by relaxations restrictions projections or cost approximations

(Federgruen 1993) Porteus (1985) presents three approximation techniques and compares

them to fourteen others for a periodic review single-product inventory model in which

shortages are fully backordered and a shortage cost is incurred Federgruen Groenevelt and

Tijms (1984) present an algorithm for finding approximate (sS) policies for a continuous review

model with backordering and a service level constraint Also Tijms and Groenevelt (1984)

present simple approximations for periodic review systems with lost sales and service level

constraints All three works find that the approximation techniques perform well However

the techniques are evaluated for a variety of demand parameters that all result in policies with

a value of S in the range of 50 to 200 Given the slower moving products at Canadian Tire 2

it is unclear how well such approximation techniques would perform Therefore we do not

attempt to apply or modify any of these approximation methods

Exact optimal policies for this problem can be obtained by formulating a constrained

Markov decision process and solving it However the data generating requirements and the

lack of a good upper bound on S make this formulation impractical Therefore we formulate

a Markov chain model for fixed values of s and S and then employ a search technique to find

the optimal values for s and S In order to improve the performance of the model we develop

a heuristic search that locates near optimal policies relatively quickly

The remaining chapters of this thesis are organized as follows In Chapter II an

overview of the existing inventory system at Canadian Tire is presented along with a description

2 For example very few of the products analyzed have optimal values of S exceeding 20 with the largest value of S equalling 69

5

of the problem Next a multi-phase project designed to develop a fully integrated interactive

inventory control system is described with the scope of the research reported in this thesis

completing the first phase of the project Finally a description of the demand and cost data is

given

In Chapter III we formulate a Markov chain model that can be used to calculate the cost

and service level associated with a given (sS) policy Based on several simplifying

assumptions the Markov chain is formulated and the equations for the corresponding transition

probability matrix are developed Then given the limiting probability distribution of the chain

the equations for the steady state annual cost of inventory and the corresponding service level

are obtained The chapter concludes with a brief summary of the model

In Chapter IV an algorithm that searches for the optimal (sS) policy is developed As

a starting point a simple grid search algorithm is developed and evaluated on the basis of

execution time In order to improve the algorithms speed a technique for updating the

transition probability matrix is presented and the improvement in the algorithm is noted Next

a theoretical lower bound on S is obtained although the corresponding reduction in the

execution time is minimal Finally based on two assumptions regarding the shape of the

feasible space a heuristic search is developed which improves the speed of the algorithm

significantly

In Chapter V a prototype of the inventory control systems interface is created In its

present form the interface allows managers to evaluate inventory policies interactively The

manager can set review parameters compare optimal policies and current policies and evaluate

alternate policies The chapter concludes by recommending specific options be including in

future versions to make the system more automated and user-friendly

In Chapter VI optimal policies for all products in one product category sold at one store

6

are compared to existing policies on the basis of expected annual costs and customer service

levels The analysis suggests that existing policies result in a large excess inventory and that

implementing the optimal policies would reduce the annual cost of inventory by approximately

50 A sensitivity analysis on the cost estimates is performed and suggests that the model is

not sensitive to relative errors of approximately 20 in the cost estimates Finally a sensitivity

analysis on the demand probability mass function is performed The analysis indicates that the

model is sensitive at least in terms of the customer service level to slight changes in the tail

of the demand distributions of low-demand products

Finally in Chapter VII the results of the analysis are summarized and future work is

outlined In order to incorporate recent changes in the review procedures at several stores the

model must be modified to allow for the duration of the review period to exceed the lead time

We recommend that the second phase of the project proceed specifically that a data collection

and storage scheme be designed and that a forecasting model be developed Because of the

substantial savings that can be realized by reducing inventory only at the stores inventory

control at the warehouse should also be investigated

7

II T H E I N V E N T O R Y S Y S T E M A T C A N A D I A N TIRE

A Background

Canadian Tire Pacific Associates owns and operates 21 retail stores in the lower

mainland of British Columbia The stores carry approximately 30000 active products3 that

are divided into 80 product categories The merchandise comprises primarily automotive parts

and accessories home hardware housewares and sporting goods Canadian Tire Pacific

Associates also owns a central warehouse in Burnaby at which inventory is received from

Toronto Inventory is then shipped periodically from the warehouse via truck to the local

stores In 1992 total sales at the stores exceeded $99 million and the value of inventory was

estimated to be $28 million of which approximately half was stored in the warehouse

On selected days of each week a stores inventory records are reviewed automatically

and orders for products are placed to the warehouse as required Management uses the term

flushing to describe the process of reviewing the inventory and producing the tickets for stock

owing and the term picking to describe the physical act of filling the order at the warehouse4

Whether or not an order is placed for a product depends upon its current inventory level its

target level and its minimum fill 5 If the target level less the current level exceeds the

minimum fill an order is placed to bring the stock up to the target level From the moment

3 Active products are those that are regularly restocked Canadian Tire Pacific Associates can choose from up to 50000 products

4 Note that the number of picks per week is equal to the number of times per week that the inventory records are reviewed At the beginning of this study eighteen stores had two picks per week while the remaining three stores had only one

5 The target level is the maximum number of items that should be on the shelf at one time and the minimum fill is the minimum number of items that should be ordered

8

the order is placed to the moment the product is reshelved takes between two to four days

depending on the store On average 11000 items are shipped from the warehouse at each

flush

Although a products minimum fill is the same for all stores its target level depends

upon the size of the store Based upon its sales volume and retail space a store is classified

as being one of five sizes small (S) medium (M) large (L) jumbo (J) and extra jumbo

(EJ) 6 Each product has associated with it a target level for each size of store

Consequently all stores of the same size have identical inventory policies Table 1 lists the

location size number of picks per week sales and floor space of each store as of October

1993

Periodically the inventory manager reviews the policies and when deemed necessary

adjusts them In this way the manager can account for seasonality in demand changes in mean

demand and any shortcomings of previous policies At the present time the review process

is not automated nor performed on-line The inventory manager uses Hxl5 computer

printouts with five products to a sheet to study the relevant information and on which to note

any changes Afterward the changes are recorded in the system and the new policies take

effect

Currently the inventory policies are based on rules of thumb developed through the

years and not on a statistical analysis of demand nor on a calculation of the cost of inventory

These rules of thumb incorporate a marketing philosophy which at times dominates the

inventory strategy The prevailing philosophy holds that a large in-store inventory instills

6 During the course of this study stores were resized several times At one point the entire class of medium stores was eliminated leaving only four sizes of stores On another occasion the store in Chilliwack was changed from being a small store to a large store

9

Table 1 Location and particulars of the 21 stores

Store Location Store Size

Picks Per Week

1992 Sales ($)

Floor Space (ft2)

1 Chilliwack S 2 2901969 6678

2 Coquitlam M 2 4015303 6510

3 Langley EJ 2 8138673 15422

4 Scott Road EJ 2 7307820 15064

5 Maple Ridge L 2 4102110 6678

6 Kingsway L 2 4153736 6574

7 Richmond EJ 2 8175105 14961

8 North Vancouver M 2 3856571 6534

9 Guildford M 2 3714372 6567

10 Burnaby M 2 3532752 6983

11 New Westminster S 1 2417992 6470

12 Abbotsford EJ 2 7899398 15303

13 Hastings J 2 4647794 9849

14 King Edward EJ 2 3825372 25492

15 White Rock L 2 3960262 6870

16 Main amp Mountain S 1 2705602 6431

17 Heather J 2 4161227 9427

18 King George J 2 4606817 9640

19 Cambie J 2 5855022 9091

20 Lougheed M 1 3114180 6615

21 Burrard J 2 6553310 10608

10

consumer confidence which in turn leads to increased sales In other words if a customer

perceives that Canadian Tire is in the hardware business he or she will choose to shop there

when meeting future needs However management has adopted this philosophy without any

analysis of the cost of holding excess inventory Only recently has management begun to

question the cost-effectiveness of allocating inventory based on these rules of thumb

Speculating that existing inventory practices might be deficient Mr Don Graham the

vice-president of Canadian Tire Pacific Associates commissioned a pilot study to examine the

inventory system and to propose alternate strategies7 The study using a steady state model

to represent a simplified inventory system suggested that existing policies lead to overly high

inventory levels More importantly the study outlined a methodology that would enable

management to determine inventory levels based on observed demand The pilot study

recommended that a more detailed model of the inventory system be formulated that the

methodology be expanded to incorporate the cost of inventory and that an algorithm be

developed that would quickly identify the optimal inventory policies

B The Problem

At the time of the pilot study management was concerned primarily with its inability

to distinguish between stores of the same size As mentioned in the previous section a

stores size is based solely on its sales volume and floor space and under existing practices

all stores of the same size have identical inventory policies for every product Management

realized that demand for some products might be a function of variables other than size such

7 The pilot study was performed by Martin Puterman a professor of Management Science and Kaan Katircioglu a doctoral student in Management Science both at the University of British Columbia The results of the study were presented at a U B C Transportation and Logistics Workshop on March 25 1993

11

as location For example the demand for sump pumps is likely to be very different in suburban

areas than in urban ones However management was unable to allocate more sump pumps to

large suburban stores than to large urban stores This inability led management to seek a

tool that would allow for a more effective allocation of inventory between stores of the same

size

By the end of the pilot study however management was less concerned with

reallocating inventory within a size of store than with reallocating inventory between sizes

of stores and with reducing overall inventory According to management assigning individual

inventory policies to each store appeared to be operationally infeasible Firstly management

believed that having individual policies would lead to confusion in the warehouse and to

increased shipping errors Exactly why this would occur is unclear the warehouse would

simply continue to fill orders from tickets that make no reference to the policy in place

Secondly and perhaps more tenably the computer technology at Canadian Tire Pacific

Associates did not allow for policies to vary across all stores Both the database used to store

the inventory data and the printouts used by the inventory manager allowed for only five

different policies per product all with the same minimum fill Moving to individual policies

would have required an overhaul of the existing information system Although management

saw this overhaul as inevitable it did not see the overhaul as imminent Therefore

management switched the focus of the study to developing a tool that would help reduce overall

inventory levels andor reallocate inventory more effectively between groups of stores

The subtle change in the studys objective prompted an important question how should

the stores be grouped The current practice of grouping stores based on size is problematic

for two reasons First as discussed above additional factors of demand such as location

cannot be taken into account Second grouping inherently results in a misallocation of

12

inventory when some stores within a group have differing numbers of picks per week For

example the medium-sized Lougheed store which has one pick per week requires higher

inventory levels to achieve a given customer service level than the other medium stores

which all have two picks per week8 In fact in terms of customer service levels it might

be more appropriate to group the Lougheed store with the large stores having two picks per

week Ideally this artificial grouping of stores will be abandoned when the computer system

is upgraded and policies can then be set on a store by store basis

As the project proceeded management grew somewhat uneasy with the concept of using

an analytical tool to evaluate and set inventory policies According to both the pilot study and

the preliminary results of this study significantly reducing inventory would greatly reduce costs

while still providing a high level of customer service However management held that because

the model cannot take into account intangibles such as marketing considerations such a drastic

reduction in inventory would be very inappropriate In this light management viewed the

optimal policies as lower bounds for inventory levels these policies would have to be

modified upwards in order to be practical Therefore management contended that any

methodology or software produced during this study should provide the means to evaluate and

compare a number of alternate policies

Regardless of exactly how the tool is to be used there is currently a lack of historical

sales data from which to analyze the inventory system and determine optimal policies No sales

data had ever been collected prior to the pilot study and only a relatively small amount has

been collected since In order to proceed with any analysis of the inventory system a

8 If demand is a function of size then the Lougheed store must meet the same demand with one fewer pick per week This would lead to either increased probabilities of stockouts at the Lougheed store or higher inventory levels at all the other medium stores

13

substantial amount of additional data would be needed The existing information system at

Canadian Tire Pacific Associates does allow for sales data to be captured however the system

is limited by the amount of data it can capture at one time and by its storage capacity In order

to collect the amount of data required management would have to upgrade the system and

develop some kind of data collection scheme coupled with a data compression technique The

collection scheme must allow for characteristics of demand such as trends and seasonality to

be identified and the compression technique must allow for sales data for over 30000 products

to be stored for each store

Because many of the products sold by Canadian Tire have nonstationary demand as a

result of seasonality andor trend the inventory tool must be able to forecast demand for the

coming period Products such as sporting goods and outdoor equipment exhibit patterns of

demand that are very seasonal For example sales of snow shovels increase through the fall

peak in the winter and drop to zero by late spring At the same time some products might

become either more or less fashionable over time resulting in changes in mean demand

Setting inventory policies for any of these products based on observed demand would result in

insufficient stock if demand increased during the period or in excess stock if demand decreased

Therefore the tool must be able to forecast the demand for the period for which the policy is

intended

Toward the end of the project management began discussing the integration of the

interactive inventory tool with the existing system to form a new inventory control system The

new system would utilize a modified point of sales system to collect sales data continually

Based on the latest available data the system would forecast demand for the coming period and

calculate the optimal policies for every product at every store The system would then provide

all relevant information on-line for the inventory manager to review The manager could weigh

14

other information such as marketing considerations and examine alternate policies to determine

the effect on cost Finally the preferred policy would be set in place with the touch of a key

and information on the next product or store would be presented for review

C The Project

Development of the inventory control system envisioned by management requires the

completion of five tasks 1) the design of a data collection system that generates and stores the

necessary sales data 2) the development of a forecasting method that provides accurate

estimates of demand distributions 3) the formulation of a model that calculates the cost of

inventory and the customer service level for an inventory policy 4) the development of an

algorithm that finds the optimal policy for a product at a store and 5) the creation of an

interface that allows interactive and on-line consultation Ultimately the above components

would have to be integrated and incorporated in a manner that would ensure the compatibility

and reliability of the system

The scope of the research reported in this thesis completes the first phase of the project

specifically we formulate an inventory model develop an algorithm and create a prototype of

the interface Because of the changing environment at Canadian Tire Pacific Associates minor

modifications to both the model and the algorithm developed herein might be necessary to

reflect future changes in operational procedures Upon completion of phase one the interface

will be interactive allowing managers to select the store product and review scenario and then

to compare optimal policies with existing policies or with alternate policies Although the

prototype will be delivered to Canadian Tire Pacific Associates for ih-house testing at the end

of this phase the interface will require modification in subsequent phases as feedback from

management is received and as the system is brought on-line

15

In order to assess more fully the potential savings of implementing the inventory control

system this thesis includes a comparison of optimal policies9 and existing policies The

expected annual cost of inventory for an entire product category is calculated for both the

optimal policies and the existing policies Based on an analysis of these costs the potential

savings of implementing the optimal policies are extrapolated Further the thesis examines the

sensitivity of the model to variations in the estimates of demand and costs

Originally results from all 21 stores were to be included in the above analysis

However just prior to the analysis management changed the review procedure at several

stores the result of which being that three picks per week are now performed at those stores

Because one of the main assumptions of the model specifically that the length of the lead time

does not exceed the duration of the review period is violated under this new review procedure

those stores cannot be included in the analysis Therefore the analysis is based solely on the

results from store six the large store located on Kingsway Avenue10

For the purpose of the analysis sales data for product category 20 automotive and

electrical accessories were collected from all 21 stores This product category was selected

because the distribution of demand for products within it is non-seasonal and because few of

the products are ever placed on sale11 The lack of trend or seasonality eliminates the need

for forecasting demand the optimal policies can be determined based directly on observed

demand

9 That is those policies obtained from the model

1 0 Store six is one of the only stores whose review procedure is not expected to change from two picks per week

1 1 Products that were on sale at any time during the period were not included in the data set

16

D The Demand Data

In total sales data for 520 products of which 458 are active products were collected

over a 307-day period from October 10 1992 to August 18 199312 The data was supplied

by Canadian Tire Pacific Associates in the form of a 10 MB text file with one line per day per

product recording sales from the stores13 Each line had the following format

Columns 1-6 part number Columns 7-9 Julian date14

Columns 10-18 cost of the item (to three decimal places) Columns 19-124 sales for stores 1 through 21 (5 columns per store)

In order to facilitate the analysis the file was sorted into 458 sales files with each of

these files containing the history of daily sales for one of the products Two changes were

made to the format of these files to improve readability 1) a column was added between the

product number and the Julian date and 2) all leading zeros were dropped Appendix A

contains a partial printout of the sales file for a typical product Finally a distribution file

summarizing the daily sales was created from each of the sales files Each distribution file

contains the products number the products cost and the number of days during which various

quantities of the product were sold at each store

Prior to the analysis the distribution files for a large number of products were

examined and based on the shapes of the histograms the products were divided into three

groups The first group low-demand comprises products that have demand on only a few

days with the daily demand seldom exceeding three items The second group medium-

1 2 Of the active products only 420 are sold at store 6

1 3 Strictly speaking a line was written to the file only on days on which at least one store registered a sale of the product

1 4 According to the Julian calendar days are numbered sequentially from 1 to 365 (366 during leap-years) beginning with the first of January

17

demand comprises products that have demand on approximately half of the days with the daily

demand seldom exceeding more than six items Finally the last group high-demand

comprises products that have demand on most days occasionally upwards of twenty items and

whose demand histograms resemble Poisson distributions The majority of the products

examined fell into the low and medium-demand groups only a few high-demand products

were observed Appendix B contains a printout of the distribution file for a typical product

in each group

For most products no theoretical distribution appeared to model the demand well

Demand for low and medium-demand products followed a non-standard distribution with

a mass at zero and a relatively small tail Although some theoretical distributions such as the

mixed-Poisson can be made to resemble this it was unclear how well they would perform

Further the effects of sampling errors from such non-standard distributions were unknown

On the other hand the standard Poisson distribution did appear to model high-demand

products well however only a few products fell into that group Therefore in the absence of

a good theoretical candidate the empirical demand function was chosen for the analysis The

empirical function was readily available from the distribution files and offered the best

estimate of the true underlying function

E The Cost Data

Along with the sales data management supplied estimates relating to the costs of holding

and ordering inventory Management estimates that the cost of holding an item in inventory

for one year is 30 of the items cost Incorporated in this holding rate are a number of costs

such as the opportunity cost of money invested and the costs incurred in running a store

Because it is the annual cost of inventory that is of interest either the compounded daily

18

holding rate or the simple daily holding rate can be used to determine the costs incurred during

a review period however the method of calculating and extrapolating the holding costs must

be consistent For the sake of simplicity a simple daily holding rate is used

In order to estimate the cost of ordering inventory management analyzed the various

tasks that result from an order being placed Using the average time required to fill and then

reshelve an order management obtained an estimate of the direct labour costs involved To

this management added an estimate of the cost of transporting an order from the warehouse to

the store Finally the estimate was adjusted slightly upward to account for the cost of breakage

and errors Based on the analysis management estimates the cost of placing an order for a

product to be $0085 regardless of the size of the order

19

III M O D E L F O R M U L A T I O N

A Terminology

In the previous chapter several terms were introduced to describe the inventory system

at Canadian Tire Pacific Associates These terms though perfectly adequate are not widely

used in the academic literature to describe inventory management problems Therefore in

order to avoid confusion and be consistent with the literature the following terms are used

throughout this thesis

The length of time between successive moments at which the

inventory position is reviewed15

The time at which a reorder decision is made

The length of time between the moment at which a decision to

place an order is made and the moment at which the order is

physically on the shelf

An operating policy in a periodic review inventory model in

which (i) the inventory position is reviewed every T days (ii) a

replenishment order is placed at a review only when the inventory

level x is at or below the reorder level s (gt 0) and (iii) a

replenishment order is placed for S-x units (S gt s)1 6

Lost Sales - The situation in which any demand during a stockout is lost

Review Period -

Decision Epoch

Lead Time -

(sS) Policy

1 5 With respect to the terminology used at Canadian Tire Pacific Associates a review period equals the number of days between flushes

1 6 With respect to the terminology used at Canadian Tire Pacific Associates S represents the target level and s represents the difference between the target level and the minimum fill

20

B Assumptions

In to order make the model of the inventory system mathematically manageable several

simplifying assumptions are made However we believe that the following assumptions do not

significantly reduce the accuracy of the model

1) Demand can be approximated very closely by sales during the period in which the data

is collected Generally inventory levels at the stores are very high and the number of

lost sales is assumed to be negligible In fact the inventory levels for a number of

products were tracked over time during the pilot study and not a single incidence of

stockout was observed Because all products appear to have similar excess inventory

it seems reasonable to assume that virtually all demand resulted in sales

2) The demands in successive periods are independent identically distributed random

variables Preliminary tests (Fishers exact tests) suggested no dependence between

sales on consecutive days and plots of sales over time indicated no seasonal pattern

3) Demand is independent across products That is we assume that sales of one product

do not affect nor can they be predicted from sales of another product

4) Unmet demand results in lost sales with no penalty other than the loss of the sale

5) The duration of the review period for a store is an integer value and is fixed over a

planning period For stores with two picks per week we assume that the inventory

is reviewed every four days rather than on a four day - three day cycle

6) The decision epoch is the start of a review period Review periods begin at midnight

at which time the inventory is reviewed and orders are placed instantaneously

7) The duration of the lead time for a store is deterministic and integer-valued In reality

the lead time for some stores was stochastic but management felt the variation could be

controlled and virtually eliminated

21

8) The lead time does not exceed the review period In other words items ordered at a

decision epoch arrive and are shelved before the next decision epoch In practice this

did not always occur

9) Orders arrive at the end of the lead time and are on the shelf to start the next day

10) Holding costs are incurred daily for all inventory held at the start of the day

11) The warehouse has an infinite capacity that is all orders to the warehouse are filled

Because of the high inventory levels in the warehouse stockouts in the warehouse are

very unlikely

C A Markov Chain Model

Consider the situation in which an (sS) policy is applied to an inventory system based

on the above assumptions Inventory is reviewed at the decision epoch for each of the periods

labelled t = 0 12 and replenishment occurs at the end of the lead time provided an order

was placed Demand during day d in period t is a random variable whose probability mass

function is independent of the day and period The function is given by

f^ik) = Pr ltd = k = ak for k = 0 1 2

where ak gt 0 and E ak = 1 Let T denote the duration of a review period and L denote the

length of the lead time Next define D T ( as the total demand during period t D L as the total

demand during the lead time in period t and D T L as the difference between D T and DLt

Assuming that successive demands pound f pound 2 are independent the probability mass function of

1 7 The aggregated demand variables are given by

^ = XXd and DT_Ut=J d= d= d=L+l

22

each aggregated demand variable can be obtained by convolving the probability mass function

of t d l s Finally let X denote the inventory on hand at the beginning of period t just prior to

the review

The process X is a discrete time stochastic process with the finite discrete state space

0 1 S A typical sample path of the process with a review period of four days and a

lead time of two days is depicted in Figure 1 As period t begins X units are on hand The

inventory is reviewed and because X is less than s an order is placed for (S-X) units Over

the next two days a demand of D L r occurs and the inventory drops accordingly With the

arrival of the order the inventory level increases by (S-X) at the start of the third day of the

period Over the remaining two days a demand of D T L is observed bring the inventory level

down to X + 1 Note that in this example an order would not be placed in period t+l because

X r + i exceeds s

The stock levels at the beginning of two consecutive review periods are related by the

balance equation

_K- DLy+ (s - v - DT-LX ixtlts ( 1 )

t + l ~ (Xt - DTtY ifXtgts

where the positive operator is interpreted as

= max a 0

1 8 For example the probability mass function of D T bdquo denoted fD T ( is the T-th fold convolution of f t Mathematically the function can be obtained using the recursive relation

and the identity f1^ = f5(

23

-2 I 8

gt o

r

CO

o a

CM

X

00

24

Because X + 1 depends only on Xt the process X satisfies the Markov property

PrXt+l=jX0 = i0Xt_l=it_vXt = i) = PrXt+1=jXt = i

and therefore is a discrete time Markov chain Further because the probability mass function

of the random variable pound is independent of t and thus so too are the probability mass functions

of the aggregated demand the Markov chain is stationary19

D The Transition Probability Matrix

The stationary transition probability p(gt is defined as the probability of X + 1 being in state

j given that X is in state i independent of the period t In the inventory model p-- represents

the probability of starting the next review period with j items on the shelf given there are i units

on hand at the start of the current period By convention the transition probabilities are

arranged in an (S + l)x(S + 1) matrix P such that (P)y = p- Thus the (i + l)st row of P contains

the probability distribution for the stock on hand at the beginning of the next period given that

there are i units on hand at the start of the current period The matrix P is known as the

transition probability matrix Note that since we are concerned with a steady state analysis we

need not consider the possibility of initial stock levels greater than S once the inventory level

is below S it can never again exceed S

The transition probability ptj is calculated by expanding (1) to include all possible

instances of demand that allow for a change in inventory from i units at the beginning of a

period to j units at the beginning of the next period For example suppose the inventory policy

(510) is in place 2 items are on hand at the start of the current review period and 7 items are

1 9 Henceforth the subscripts t and d are omitted from the daily and aggregate demand variables for the sake of brevity

25

on hand at the start of the next period Then only three scenarios are possible

1) No items are sold during the lead time (meaning that the stock level increases to 10

after replenishment) and 3 items are sold during the remainder of the period

2) One item is sold during the lead time and 2 items are sold during the remainder of

the period

3) Two items are sold during the lead time (though demand could exceed 2) and 1 item

is sold during the remainder of the period

The transition probability p 2 7 is the sum of the probabilities of the various scenarios and is

computed by

p2J = PrDL=0PrDT_L = 3] + PrDL = l] Pr[DT_L=2 + PrDL2] PrDT_L = l]

In general the transition probabilities for the inventory model are calculated as follows

Case 1 The lead time is less than the review period

For 0 lt lt s

Pij

i-l pound [PrDL=kPrDT_LgtS-k] + PrDLgtiPrDT_LgtS-i if j = 0

i - l

^[PriD^^PrlD^S-j-kn+PHD^PriD^S-j-i if 0 lt j ltL S-i (2) k=0 s-j

E [PrDL=kPrDT_L = S-j-k] if S-i ltj^S k=0

26

For s lt i lt S

PrDTzi]

0 p = PrDT=i-j)

ifj = 0 ifOltjzi ifiltjltS

(3)

Case 2 The lead time equals the review period

For 0 lt i lt s

Pa

0 PrDTzi) PrDT = S-j

ifOltLJlt S-i ifj = S-i if S-i ltj plusmn S

(4)

For s lt i lt S use (3)

E The Steady State Probabilities

Provided 1) there is a positive probability of non-zero demand during a review period

and 2) there is a positive probability of zero demand during a review period20 then the

Markov chain described by (1) has a limiting probability distribution ir = (ir0 TTu irs)

where 0 lt IT lt 1 for j = 0 1 S and E 7ry = 1 Further this limiting distribution is

independent of the initial state In other words the probability of finding the Markov chain in

a particular state in the long-run converges to a nonnegative number no matter in which state

the chain began at time 0 The limiting distribution is the unique nonnegative solution of the

equations

2 0 That is 0 lt PrD x = 0 lt 1 Given the nature of the problem this assumption is not very restrictive

27

^ = E ^ - far j = 0 l S (5)

s and pound nk = 1 (6)

For the proof that the limiting distribution is independent of the initial state and is obtained from

(5) and (6) the reader is directed to Appendix C

In terms of the inventory problem 7ry is the steady state probability of having j units of

inventory on hand at the decision epoch However some subsequent calculations are made

easier if the steady state probability of having j units of inventory on the shelf just after

replenishment is known Instead of reformulating the Markov chain these probabilities can be

obtained by shifting the limiting distribution TT Much in the same way that the transition

probabilities are calculated the shifted limiting distribution t] is obtained by accounting for

all possible instances of demand that allow for the shifted inventory position For example

suppose that the policy (510) has the limiting distribution TT In order to have say 8 units on

hand immediately after replenishment one of the following must occur

1) Ten items are on hand at the decision epoch and 2 items are sold during the lead time

2) Nine items are on hand at the decision epoch and 1 item is sold during the lead time

3) Eight items are on hand at the decision epoch and nothing is sold during the lead time

4) Either 2 3 4 or 5 items are on hand at the decision epoch and 2 items are sold during

the lead time

The steady state probability of having 8 units on hand just after replenishment is calculated as

n 8 = n10PrDL = 2] +iz9PrDL = l] +n8PrDL = 0 +[iz5 + iz4 + iz3]PrDL = 2 +n2PrDLgt2

28

In general the shifted steady state probabilities are given by

s pound nPrDLgti ifj = 0

i=s+l S

pound PrDL = i-j] ifl plusmnjltS-s i=max(s+l)

S = pound 7tjirZ)L = i-+7i 5 FrD i ^5 ifj = 5-5

i=max(s+l)

^ 7 1 ^ ^ = 1-7+ pound TtiFrZgti=5-y+iis_PrD1^S-7 ifS-sltjltS i=max(gt+l) i=S-j+l

(7)

F The Cost of Inventory

For this problem the cost of inventory has two components (i) a fixed ordering cost

cf incurred whenever an order is placed and (ii) a holding cost cv incurred daily for each item

held in inventory The expected cost of inventory for a review period can be calculated by

summing the costs associated with each inventory position under steady state conditions

Because a simple holding rate is used this cost is simply adjusted by number of review periods

in a year to obtain the expected annual cost of inventory

The ordering cost for a review period is relatively straight forward to calculate either

an order is placed in which case the order cost is incurred or no order is placed in which case

no cost is incurred According to the rules of the inventory policy orders are placed only if

the stock on hand at the beginning of a review period is less than or equal to s Further the

probability that the review period begins with the inventory in one of these positions is equal

to the sum of the first (s + 1) steady state probabilities Therefore the expected ordering cost

29

for a review period is given by

E[OC] = cfY n y=o

Unlike the simple structure of the ordering cost the holding cost is incurred daily for

each item held in inventory Therefore the daily holding costs associated with all possible

inventory positions must be weighted and summed to obtain the holding cost for the review

period In order to make the calculations less complicated a shifted review period beginning

immediately after replenishment and ending just prior to the next replenishment is considered

Define e(0 to be the expected number of items on hand to start day i of the shifted

review period and let g(i f) denote the expected number of items on hand to start day i given

that there were j items immediately following replenishment21 The function g(z j) is the sum

of the probabilities of possible demand weighted by the resulting stock levels that is

g(ij) = j^kPriD^j-k)

To obtain the expected number of items on hand to start day i independent of j the above

function is summed over all possible values of j and weighted by the corresponding steady state

probabilities yielding

s j ed) = pound r 5 gt P r Z ) = - pound

7=0 k=

2 1 In other words g(if) is the expected number of items on hand to start day i given e(0)=v

30

The expected holding cost for a review period is then given by

T

E[HC] = $gt v e(0 i=i

i=l =0 k=l

where c v equals the simple daily holding rate multiplied by the cost of the item

Although differently defined review periods are used to calculate the above expected

costs the costs can still be summed to obtain the expected cost of inventory for a review

period The expected costs are calculated based on steady state conditions and therefore

represent the costs expected to be incurred during any T consecutive days Summing the above

two expected costs and multiplying the sum by the number of review periods in a year yields

the expected annual cost of inventory

E[AC] = mdash[E[OC]+E[HC]]

T S j 1 ( g ) 365

j=0 i=l j=0 fc=l

G The Customer Service Level

There are many ways to define customer service and the most appropriate definition

depends upon the nature of the business and the perspective of management Two commonly

used definitions are 1) the proportion of periods in which all demand is met and 2) the

proportion of demand satisfied immediately from inventory Although the former definition is

often used because of its simplicity the latter definition is closer to what one commonly means

by service (Lee and Nahmias 1993) A third definition also relevant in retail environments

31

is the probability that lost sales do not exceed some critical number Other definitions such

as probability that backorders are filled within a certain amount of time are more applicable

to industrial and manufacturing settings where backlogging demand is common

Given the retail environment with the possibility of lost sales that exists at Canadian Tire

Pacific Associates management is concerned primarily with the fraction of sales that are lost

For example management views the situation in which demand exceeds stock by one unit every

period more favourably than that in which the same demand exceeds stock by three units every

other period For this reason the proportion of demand satisfied immediately from inventory

is the most appropriate measure of customer service

The customer service level for a single review period CSLRP is given by

r bdquo T _ Demand satisfied during RP Total demand during RP

Because we are interested in the long-run behaviour of the system a measure of the steady state

customer service C S L must be obtained As shown in Appendix D the correct measure is

poundpoundpound _ E[Demand satisfied during RP] E[Total demand during RP]

which can be rewritten as

pound pound pound _ j _ E[Demand not satisfied during RP] ^ E[Total demand during RP]

Because of the steady state conditions the above customer service level can be estimated

from any period of T consecutive days As was done for one of the cost calculations the T-day

period beginning just after replenishment is considered Suppose that there are j units of

inventory on hand immediately following replenishment Then the conditional customer service

32

level is

CSL j = 1 E[(DT-j)l

E[DT] (10)

The denominator in the second term is merely the expect demand during the period

E[DT] = 7gt (11)

where p is the mean of the random daily demand variable pound The numerator which is the

conditional expected demand that is not satisfied during the T-day period expands to

7-1

E[(DT-j)+] = T u - j + YU-k)PrDT = k] k=0

(12)

The reader is directed to Appendix E for a mathematical justification of (12) In order to obtain

the unconditional expected demand not satisfied (12) is applied to all possible values of j and

a weighted sum of the terms is calculated The result is given by

E[Demand not satisfied] = Y j =0

y-i Ti - j + 1pound(j-k)PrDT = k

S j-1

= - E[I] + Y0-k)Pr[DT=k) =0 it=0

(13)

where E[I] denotes the steady state inventory level Finally the customer service level is

obtained by the substituting (11) and (13) into (9) yielding

CSL = 1

s j-l

Ti - E[I] + Jgty Y(J-k)PrDT=k]

m n J2VjTU-k)PrDT=k Eil J=Q fc=o

(14)

33

H A Methodology for Evaluating an (sS) Policy

In this section a brief methodology summarizing the main results of the model is

presented The methodology can be applied to evaluate a given (sS) policy in terms of its cost

and its resulting customer service level Given the probability mass function of the daily

demand f the duration of the review period T the length of the lead time L the fixed cost

of placing an order cf and the daily unit cost of holding inventory cv

1) Calculate the aggregated demand probability mass functions fDT fDL and f D x L as the T-th

fold L-th fold and (T-L)th fold convolutions of f respectively

2) Generate the transition probability matrix using either equation (2) or (4) and equation (3)

3) Calculate the limiting distribution TT from (5) and (6)

4) Obtain the L-day shifted limiting distribution rj using (7)

5) Calculate the expected annual cost of inventory using (8)

6) Determine the customer service level using (14)

34

IV A N A L G O R I T H M F O R OBTAINING T H E O P T I M A L (sS) P O L I C Y

A Introduction

The methodology developed in the previous chapter can be used to calculate the cost and

service level associated with a given (sS) policy In this chapter an algorithm that finds the

feasible policy yielding the lowest cost is developed A policy is feasible if it satisfies both the

basic condition 0 lt s lt S and the imposed constraint on the customer service level2 2 In

order to find the optimal policy the algorithm must combine the methodology for evaluating

a single policy with a search technique

Search techniques are evaluated in terms of two somewhat antithetic criteria On the one

hand the search should locate the optimal solution or at least a near optimal solution on

the other hand the search should take as little time as possible Many search techniques

guarantee optimality provided the objective function and constraints exhibit certain properties

such as convexity However even if such a search technique also guarantees its termination

in polynomial time the search might take too long to be practical Alternatively some search

techniques provide reasonable though sub-optimal solutions very quickly These techniques

often called heuristics must be evaluated in terms of both their speed and the accuracy of their

solutions The choice of a heuristic over a search that guarantees optimality depends upon the

nature of the feasible space the need for accuracy and the cost of time

Given the complexity of the cost and customer service level functions it is very difficult

to show theoretically for what if any combinations of demand and costs the feasible space is

convex However a plot of the objective function for a single product may be obtained by

2 2 Management specified that each policy must provide a customer service level of at least 975 Therefore the algorithm must incorporate the constraint CSL gt 0975

35

evaluating the costs for a number of policies within a grid When the service level constraint

is included in these plots maps of the feasible space are obtained Examination of the

maps for several products suggests that the space is convex over most regions however

maps for some products reveal some regions of non-convexity Unfortunately there is no

way of knowing the degree of non-convexity for other products other stores or different costs

estimates without a more rigorous survey of the feasible space Therefore if one of the

techniques that guarantee optimality under the condition of convexity is employed the search

would have to be evaluated as a heuristic Instead as a first step to finding a practical search

technique a crude search that guarantees optimality under virtually all conditions is

investigated

B A Grid Search

The simplest search technique that guarantees optimality merely evaluates all policies in

the grid defined by 0 lt s lt S and 0 lt Sm i n lt S lt S m a x where Sm i n and S m a x are

some predefined values That is the grid search is a brute force method that examines every

policy within the grid 2 3 Although the number of policies that must be examined is large the

grid search is guaranteed to find the optimal policy provided that S o p t is contained in the grid

that is provided Sm i n lt S o p t lt Sm a x The grid search algorithm proceeds by generating the

transition probability matrix for the policy (0Smin) and then evaluating the service level

associated with that policy If the policy is feasible24 the cost associated with the policy is

2 3 The total number of policies evaluated is the sum of the first Sm a x integers less the sum of the first Smin-1 integers Numerically [(l2)(Sm a x)(Sm a x+l) - (l2)(Smin-l)(Smin)] policies are examined

2 4 That is if the service level is adequate

36

calculated and compared to the current lowest cost The procedure continues until all points

in the grid have been examined and the optimal policy has been found A flow chart of the grid

search algorithm is shown in Figure 2

In order to assess the grid search the algorithm was coded and compiled into an

executable program The program was run for two products (a medium-demand product and

a high-demand product) and with four values of Sm a x In each run Sm i n was taken to be zero

The following times were recorded for each trial

T P M - the amount of time spent generating the transition probability matrix

GAUSS - the amount of time spent solving for the steady state probabilities

COSTS - the amount of time spent calculating the expected annual costs

SERVICE - the amount of time spent determining the customer service levels

T O T A L - the total execution time of the program

The columns entitled Grid Search in Tables 2 and 3 summarize the results of the trials All

times are recorded in seconds as required by a 486DX33 MHz personal computer with an

empty buffer memory

Three results are evident from the tables First the vast majority of the execution time

is spent generating transition probability matrices and then solving for the steady state

probabilities with the time being split almost equally between the two tasks Second the total

execution time of the algorithm increases exponentially as Sm a x increases For product 206917

doubling S^ from 30 to 60 led to an increase in the execution time of more than 2100 (from

172 seconds to 3661 seconds) Third the grid search algorithm as it stands does not appear

to be adequate for the problem at hand Given the large number of products that must be

evaluated the algorithm is much too slow to be implementable

37

Calculate the demand probability functions

Initialize s = 0 S = Smin] cost = 10A5

s = s+1

Calculate the probability transition matrix for (sS)

Evaluate policy (sS) Is service level ^ 0975

No

No Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Is s = S-l

Yes

Is S = Smax No

Yes

Print results (sS) cost

s = 0 S = S+1

Figure 2 The flow chart of the grid search algorithm

38

Table 2 Execution times of the algorithms for product 200001 a 30-amp inline fuse

(in seconds)

-max Subroutine Grid Search Grid Search

Updating T P M Heuristic

10 T P M 006 010 001

GAUSS 010 006 001

COST 005 005 001

SERVICE 001 001 001

T O T A L 050 035 027

20 T P M 126 021 010

GAUSS 119 160 023

COST 027 026 001

SERVICE 017 016 001

T O T A L 318 236 055

30 T P M 703 096 010

GAUSS 757 801 093

COST 079 085 001

SERVICE 047 042 001

T O T A L 1664 1137 126

40 T P M 2257 163 015

GAUSS 2958 3003 206

COST 313 329 001

SERVICE 092 083 011

T O T A L 5871 3851 285

39

Table 3 Execution times of the algorithms for product 206917 a 6 solder connector

(in seconds)

bull-max Subroutine Grid Search Grid Search

Updating T P M Heuristic

30 T P M 717 067 006

GAUSS 762 880 093

COST 098 087 010

SERVICE 016 021 006

T O T A L 1720 1176 220

40 T P M 2390 157 022

GAUSS 2861 2911 209

COST 219 227 006

SERVICE 080 088 011

T O T A L 5827 3790 362

50 T P M 6691 413 010

GAUSS 7994 8201 472

COST 568 577 016

SERVICE 212 197 005

T O T A L 15840 10041 654

60 T P M 15509 894 040

GAUSS 18607 18951 876

COST 1367 1353 038

SERVICE 350 359 022

T O T A L 36614 22794 1120

40

C A Technique for Updating the Transition Probability Matrix

As was noted in the previous section approximately half of the execution time of the grid

search algorithm is spent generating transition probability matrices Furthermore the amount

of time increases exponentially as Sm a x increases This increase is exponential for two reasons

first the number of transition matrices that must be calculated is a function of S^ squared and

second the average size of the matrices and hence the average number of calculations per

matrix increases with Sm a x For example when S m a x equals 30 the transition probability matrix

must be generated 465 times with the largest matrix having dimensions (31x31) When Sm a x

equals 40 the matrix must be generated 820 times with 355 of the matrices having dimensions

larger than (31x31) The performance of the algorithm could be improved considerably if

either the number of matrices that must be generated or the number of calculations per matrix

is reduced This section introduces a technique that accomplishes the latter by updating the

transition probability matrix for a new policy instead of generating it anew for each policy

A new policy (s+mS)

Suppose the policy (sS) has associated with it the (S + l)x(S + l) transition probability

matrix P and the policy (s+mS) where m is any integer satisfying 0 lt s+m lt S has

associated with it the (S + l)x(S + 1) transition probability matrix J Then T can be obtained by

recalculating only m rows of P Recall that the equations for generating a transition

probability matrix are divided into two sets based upon the row being calculated equations (2)

or (4) for rows 0 through s and (3) for rows (s + 1) through S The equations themselves are

independent of s only the limits defining the cases contain s Therefore only the m rows

that are affected by the change in the limits need to be recalculated The following updating

technique can be used to obtain 7

41

1) If m gt 0 rows i = (s + 1) (s+2) (s+m) are recalculated using (2) or (4)

2) If m lt 0 rows i = (s-1) (s-2) (s-m) are recalculated using (3)

3) All other rows of T are identical to those of P

The reader is directed to Appendix F for a mathematical justification of the technique

A new policy (sS+m)

Suppose the policy (sS) has associated with it the (S + l)x(S + l) transition probability

matrix P and the policy (sS+m) where m is any positive integer has associated with it the

(S+m+l)x(S+m+l) transition probability matrix P Then T can be obtained by modifying

each row of P and adding m additional rows and columns The updating technique is not as

simple as that for a new policy (s+mS) for two reasons the dimensions of P and T differ and

the equations for generating a transition probability matrix contain S Nevertheless an updating

technique that requires much less computing time than generating the matrix anew can be

developed The following technique can be used to obtain P

1) For rows i from 0 through s

For columns j from 0 through m recalculate using (2) or (4)

For columns j from (m+1) through (S+m) enter the (j-m)th column entry of P

2) For rows i from (s + 1) through S

For columns j from 0 through S enter the corresponding value from P

For columns j from (S + 1) through (S+m) enter 0

3) For rows i from (S + 1) through (S+m) recalculate using (3)

The above technique is justified mathematically in Appendix G

42

The modified algorithm

The updating technique can be easily incorporated into the grid search algorithm with

only a slight change in the manner in which s is incremented Figure 3 shows the flow chart

of the new algorithm As before the algorithm was coded and compiled into an executable

program and the various execution times were recorded The results are listed in Tables 2 and

3 under the columns Grid Search Updating TPM

Updating the transition probability matrix does improve the performance of the

algorithm The time required to generate the transition probability matrices is reduced by

approximately 90 This translates to a reduction in total execution time of approximately

30 Not surprisingly the reductions are greater for large values of Sm a x However despite

this seemly significant improvement a decrease of only approximately 08 seconds is obtained

when S^ equals 20 Further when S ^ equals 30 the algorithm requires over 11 seconds

which is still much too high

The execution time is now dominated by the time required to solve for the steady state

probabilities The steady state probabilities could be solved by using approximation techniques

which are faster than solving the systems exactly however because the probabilities are often

very small numbers the approximation techniques would be unable to provide the required

accuracy The only remaining way to decrease the execution time is to reduce the number of

policies that are evaluated during the search This can be done by either reducing the size of

the grid being searched or reducing the number of policies being evaluated within the grid The

following section introduces a method of reducing the size of the grid by obtaining a lower

bound for S namely Sm i n

43

s = s+k

Calculate the demand probability functions

Initialize s = 0 S = Smin| cost = 105

k= 1

Update the probability transition matrix for (sS)

Evaluate policy (sS) Is service level ^ 0975

No

No

iNo

Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Ifk= 1 iss = S-l If k =-1 is s = 0

Yes

Is S = Smax No

Yes

Print results (sS) cost

If k=l s=s+l S = S+1

Figure 3 The flow chart of the grid search algorithm utilizing the updating technique for the transition probability matrix

44

D A Lower Bound on S

As mentioned earlier the size of the grid is specified by the number of policies within

the domain defined by 0 lt s lt S and 0 lt Smin lt S lt Smax Thus the size of the grid

and hence the total amount of computation time can be reduced by obtaining tight bounds on

S Unfortunately because of the extreme non-linearity of the objection function and the service

level constraint obtaining a mathematical upper bound on S is very difficult and beyond the

scope of this work This section introduces a method for obtaining a lower bound on S25

For a given value of S an upper bound on the customer service level can be found as

follows Suppose that there are S units of inventory on the shelf at the beginning of every

review period regardless of demand No (sS) policy can provide a higher level of customer

service because there exists a positive probability of beginning a review period with less than

S units on the shelf Thus the customer service level associated with starting each review

period with S units on the shelf is an upper bound on the customer service level for any (sS)

policy From (10) this level is

E[(DT-SY] CSL 5 = 1 - T mdash mdash - ( 1 5 gt

E[DT]

Replacing j with S in (12) and then substituting (12) and (11) into (15) yields

CSL 15 = mdash- - J2(S-k)PrDT = k] (16)

To obtain Smin use (16) to find the largest value of S with a customer service level less than

0975

There are two criteria for a good lower bound on S the bound must be no larger than

This method was proposed by Kaan Katircioglu

45

S o p t and it must be very close to So p t These criteria ensure that the optimal policy will be

contained in the search grid and that the size of the grid is reduced by as much as possible

Based on these criteria the above method of obtaining a lower bound is assessed A value for

S m i n is obtained for each of the products in product category 20 and compared with the

corresponding value of S o p t 2 6 The results are illustrated in Figure 4 which shows the number

of occurrences of various Sm i n and S o p t combinations Note that a point lying on the 45deg line

represents a perfect bound while a point lying close to the S o p t axis represents a very loose

bound The absence of any points above the 45deg line indicates that the first criterion is satisfied

in each instance the method did not generate a value of Sm i n exceeding the corresponding value

of So p t Moreover the bounds are reasonably tight for 85 of the products analyzed that

is for those products with S o p t equal to 10 or less27 However as S o p t increases the bounds

become steadily worse Because values of S o p t are likely to be larger for these products at the

ten Jumbo and Extra Jumbo stores and for other products with higher demand the lower bounds

are not expected to be very tight in general

Although obtaining the bound Sm i n does reduce the size of the grid the corresponding

reduction in the execution time of the algorithm is miriimal For example if S^ equals 60 and

a value of 30 is obtained for Sm i n the program would still require approximately 215 seconds

to search the grid (compared to the 228 seconds required without a lower bound on S) Because

the execution time increases exponentially with So p t a tight upper bound on S would likely have

a more significant impact on reducing the execution time of the algorithm Further research

on this issue is required

2 6 The optimal policy for each product was obtained using a grid search with an extremely large upper bound

2 7 On average Sm i n is approximately half of S o p t for these products

46

47

The only remaining approach to improving the algorithm is to modify the search so that

fewer policies within the grid are evaluated Toward this goal assumptions can be made with

respect to the feasible space and a heuristic search can be developed based on these

assumptions

E A Heuristic Search

As mentioned previously in this chapter a map of the feasible space for each of

several products was obtained Upon closer examination the maps illustrate two patterns

without exception for a given value of S both the cost and the customer service level increase

as s increases The following intuitive argument is offered as a possible explanation for the

observed behaviour When S is held constant increasing s results in more instances in which

an order is placed upon review Hence the average stock level would increase which in turn

would lead to an increase in the cost and the customer service level28 With this in mind a

heuristic search based on the following assumptions is developed for a given value of S both

the cost and the customer service level are monotonically increasing functions of s

In general the heuristic proceeds as follows Having just incremented S by one the

customer service level of the policy (sS) is evaluated If the policy is feasible s is decreased

until the lowest value of s producing a feasible policy is found If the policy is not feasible

s is increased until either the cost exceeds the current lowest cost for a feasible policy a

feasible policy is obtained or s equals S-l In the event that a feasible policy is found the cost

2 8 In order to prove that the customer service level increases as s increases it would be sufficient to show that the probability of placing an order increases as s increases Such a result might be possible if assumptions regarding the demand function are made Further if both the probability of placing an order and the average stock level increase then both the ordering cost and the holding cost would increase No proofs are offered here

48

of the policy is calculated if the cost of the policy is less than the current lowest cost the new

policy becomes the current best Finally S is increased by one and the process repeats

provided S does not exceed some predefined value Sm a x The flow chart of the algorithm

incorporating the heuristic search is shown in Figure 5 Note that the algorithm begins by

attempting to improve on the lower bound Sm i n the policy (Sm i n-lSm i n) is evaluated and if it

is not feasible S m i n is increased by one The improvement continues until a feasible policy is

found

As previously mentioned a heuristic search must be evaluated in terms of both its speed

and the accuracy of its solutions To evaluate the heuristic on the basis of speed the same

approach used to evaluate the previous algorithms is employed The columns entitled

Heuristic in Tables 2 and 3 contain the various execution times of the heuristic The heuristic

is a vast improvement over the previous search techniques the reduction in execution time is

enormous and becomes more so as S m a x increases For a S m a x of 20 the reduction is

approximately 75 of the time required by the grid search algorithm29 and for a S m a x of 60

that reduction soars to over 95 More importantly the heuristic can search a grid with a

of 40 in under 4 seconds or a grid with a S m a x of 60 in under 12 seconds These low execution

times should be sufficient to allow for a large number of products to be analyzed in a

reasonable amount of time

In order to further assess the practicability of the heuristic the total time required by the

heuristic to find solutions for the entire product category was determined and compared to the

corresponding time for the grid search algorithm With a priori knowledge of the optimal

solutions conservative yet reasonable values of S m a x were selected and used by both of the

2 9 Hereafter grid search algorithm refers to the algorithm utilizing the grid search the updating technique and the lower bound Sm i n

49

Calculate the demand probability functions

Calculate Smin S = Smin s = S-l

S = S+l s = S-l t

Update the probability transition matrix for (sS)

No Is service level pound 0975 Yes

No Is service level ^ 0975

Yes

Is cost lt cost No

Yes

Is s lt S-l No

Yes

s = s+l

Update the probability transition matrix for (sS)

Is service level pound 0975

Yes

Is cost lt cost No

Yes

s = s S = S cost = cost

s = s-l

No

Update the probability transition matrix for (sS)

S = S+l

Is s 2 0 Yes

Update the probability transition matrix for (sS)

Is service level pound 0975

No

s = s+l

No

No

Is S = Smax

^Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Print results (sS) cost

Figure 5 The flow chart of the heuristic algorithm

50

algorithms30 The grid search algorithm required approximately 9 hours and 41 minutes to

find the solutions while the heuristic required only slightly more than 23 minutes Based on

speed the heuristic is preferable to the grid search algorithm

Finally to evaluate the heuristic based on the accuracy of its solutions its solutions were

compared to the previously obtained optimal policies Without exception the heuristic

algorithm found the optimal policy Although there is no guarantee that the heuristic will find

the optimal solution for different products or different demand functions the heuristic appears

to work extremely well Based on the fast execution time and the apparent accuracy of the

solutions the heuristic will be used in the development of the inventory tool

3 0 For the test the following values of Sm a x were used 20 (273 products) 30 (19) 50 (23) 100 (3) and 125 (2)

51

V T H E I N T E R F A C E

With the model formulated and the algorithm developed the only remaining task in the

development of the prototype is the creation the interface In fact two interfaces are created

one that displays results in terms of minimum fills and target levels and one that displays

results in terms of reorder points and target levels To be consistent with the previous chapters

the latter interface is described here31

In its present form the interface allows the manager to set review parameters to

compare optimal and current policies and to evaluate alternate policies From the main menu

the manager selects the product store duration of the review period length of the lead time

and value of Sm a x Once the parameters are set the system calculates the optimal policy and

then evaluates the current policy displaying results as they are obtained In particular the

reorder point target level expected annual cost and expected customer service level are

displayed for each policy If a current policy meets the desired customer service level the

expected annual savings of the optimal policy are displayed in terms of both dollars and

percentage of current cost otherwise the value of the current customer service level is

displayed in red to indicate that it is deficient Finally the manager can choose to evaluate

alternate policies or to return to the main menu

Although the prototype is far from the ultimate integrated inventory control system it

is still valuable to management As mentioned in Chapter II inventory policies at Canadian

Tire Pacific Associates are currently based on rules of thumb that incorporate marketing

3 1 Because significant changes to the interface are expected as feedback from management is received and as the system is brought on-line only a brief description of the present interface is given

52

strategies The prototype provides management with a means of determining the proportion of

the cost of inventory that is not attributable to meeting customer demand For example

consider a hypothetical consultation in which the manager chooses to analyze product 200001

a 30-amp inline fuse at store 6 The prototype displays to the manager the current policy of

(1214) which provides a customer service level of 1000 at an annual cost of $748 and the

optimal policy of (39) which provides a customer service level of 977 at an annual cost of

only $319 If for some reason the manager feels the optimal policy is undesirable he or she

can investigate the additional cost of holding excess inventory Suppose the manager chooses

to evaluate the policy (610) The prototype evaluates the policy and displays the results the

customer service level is 998 and the annual cost is $441 With the alternate policy

virtually all demand is satisfied and the cost of inventory is still reduced by over 40 Thus

the manager ascertains that approximately 40 of the current cost of inventory for this product

is directly attributable to marketing considerations Using the prototype in this way

management can begin to understand better the nature of its inventory costs A description and

graphical depiction of the above hypothetical consultation are presented in Appendix H

As for the technical details of the system the prototype can run on any computer using

the DOS operating system having approximately 420K of available conventional memory and

attached to a V G A or SVGA monitor Further the prototype can be used on a machine without

a math coprocessor however performance is slower The prototype consists of two modules

linked together into one executable program The interface module contains the various

interactive subroutines and the main module contains the methodology of the model and the

logic of the heuristic search Both modules are coded in Fortran and compiled using Microsoft

Fortran 50 which provides many extensions to standard Fortran The source codes for the

interface module and the main module are listed in Appendix I and Appendix J

53

respectively

In addition to the above technical specifications the program requires the presence of

certain files in certain directories The program uses the distribution files mentioned in

Chapter II to read the unit costs and the empirical demand functions Each product has a

unique distribution file the name of which is the six digit product number followed by the

extension dst The text file containing the existing policies at each store must be located in

the same directory as the distribution files Finally the file tmsrbfon which contains the

font used by the interface must be located in the directory from which the program is executed

In its future versions the system should be more automated and more user-friendly

Rather than having the manager choose an upper bound for the search the algorithm should

incorporate a method of determining Sm a x For this to happen either a theoretical upper bound

must be obtained or a heuristic for terminating the search must be developed Also the amount

of time spent waiting for calculations to be performed can be reduced by reading the optimal

and current policies and their associated costs and customer service levels from a file or

database This solution file could be created any time prior to the consultation by running

the system in a batch mode The interface should allow the manager to change cost estimates

and service level values view results graphically and print results and summary documents

Finally the system should provide an option for automatically incrementing the product andor

store during a consultation This option would enable a manager to review quickly the

inventory policy of a product at each store

54

VI R E S U L T S AND SENSITIVITY ANALYSIS

A Comparison of Current Policies to Optimal Policies

In order to estimate the potential savings of implementing optimal policies the solutions

obtained in the Chapter IV are analyzed in more detail For each product the current policy

is evaluated and then compared to the optimal policy with respect to cost and customer service

level Finally the potential savings of the optimal policies are extrapolated to all products and

all stores

Of the 420 products examined each of 37 products has an existing policy that fails to

provide the desired customer service level of 975 However most of these policies just

fail to do so only 4 policies provide a customer service level of less than 90 Table 4

provides a list of these 37 products each with its current customer service level its current

annual cost and the additional cost of its optimal policy3 2 Two comments regarding Table

4 are in order First the current customer service level for product 203312 an amber

driveway marker is listed at a misleading 303 Examination of the demand distribution

reveals that this product was sold on only 1 of the 307 days during which sales data were

collected and on that day 13 of the markers were sold Clearly this data point constitutes

either an error or an unusual event the current policy of (34) does not allow for more than 4

items to be sold per review period Second the additional cost of the optimal policy for each

of 5 products is negative In other words for each of these products the current policy which

fails to provide the desired customer service level costs more than the optimal policy which

3 2 This is the difference between the expected annual cost of the optimal policy and the annual cost of the current policy It is the minimum additional cost required to increase the customer service level to at least 975

55

Table 4 Products with insufficient existing policies

Product Current CSL ()

Current Annual Cost ($)

Additional Cost of the Optimal Policy ($)

206932 974 204 -051

204570 973 222 -025

203053 973 248 009

206550 972 118 042

206547 972 105 035

206565 972 146 056

203702 972 340 157

206876 972 155 061

205752 972 330 153

205922 972 108 036

206857 972 180 073

205393 972 190 -003

201513 971 345 015

203405 970 255 037

203069 968 295 -048

203071 967 321 -001

205635 967 390 029

203914 966 460 160

204601 962 252 061

203326 960 144 014

202003 958 876 437

206516 958 112 029

206517 958 283 120

206350 958 333 147

205932 958 180 066

203936 952 598 294

56

Product Current CSL ()

Current Annual Cost ($)

Additional Cost of the Optimal Policy ($)

206932 974 204 -051

203939 948 413 193

203977 945 1216 1242

206551 945 132 077

205389 934 598 047

203590 927 1223 607

206567 919 122 070

203591 905 1228 834

205390 887 291 074

203318 833 136 084

206393 776 319 289

203312 303 237 511

provides at least the desired customer service level For the 36 products the total additional

cost of the optimal policies is $542033

Of the remaining 383 products most have an existing policy that provides a customer

service level well above 975 When the customer service levels are rounded to one decimal

each of 253 products has a customer service level of 1000 and each of 347 products has a

customer service level of at least 990 In contrast each of only 58 products has an optimal

policy that yields a customer service level of at least 99034 For each of 37 products the

3 3 The cost associated with product 203312 is not included in the total

3 4 Optimal policies do not have customer service levels of exactly 975 because the function is discrete For low-demand products a unit step in s or S results in a relatively large change in the customer service level

57

current policy is identical to the optimal policy

The extremely high level of service provided by most policies allows for substantial

savings in the cost of inventory With current policies the estimated annual cost of inventory

for all 420 products is $420743 whereas the expected annual cost of the optimal policies is

only $186884 This amounts to a savings of approximately $2340 or over 55 of the current

cost of inventory

In order to further assess the savings the difference between the costs of the current

policies and the costs of the optimal policies are examined in more detail For each of the 383

products whose existing policy provides a customer service level above 975 the difference

between the estimated annual cost of the existing policy and the expected annual cost of the

optimal policy is calculated The resulting histograms of the savings in (a) dollars and (b)

percentage of current cost are shown in Figure 6 The particulars for the current and optimal

policies for all 420 products are provided in Appendix K

As evident from Figure 6(a) the majority of the expected savings in absolute terms

comes from only six products In order to examine these products more closely the current

and optimal policies along with the associated costs and savings for each of the six products

are listed in Table 5 Each current policy is very different than its corresponding optimal

policy resulting in excessive inventory for each product In fact over half of the total expected

savings could be realized by implementing the optimal policy just for each of these six products

Interestingly five of the six products are sealed beam headlights

58

90

10 - 20 30 - 40 50 - 60 70 - 80 90- 100

Savings ()

(b)

Figure 6 The distribution of savings of optimal policies in (a) dollars and (b) percentage of current cost

59

Table 5 Products with the largest potential absolute savings

PRODUCT

CURRENT OPTIMAL SAVINGS

PRODUCT s S Cost ($)

C S L

()

s S Cost ($)

C S L () ($) ()

2036801 194 200 64014 1000 37 48 bull 9245 975 54769 856

2036772 94 100 23652 1000 18 22 3850 978 19802 837

2036723 78 90 23043 1000 19 23 4454 975 18589 807

2030944 145 175 19522 1000 34 45 3727 975 15795 809

2036425 48 60 10491 1000 13 17 2682 977 7809 744

2036556 68 80 7265 1000 14 22 1699 975 5566 766

1 H6054 rectangular sealed beam headlight 2 H4656 rectangular sealed beam headlight 3 6053 rectangular sealed beam headlight 4 9004 halogen bulb 5 4652 rectangular sealed beam headlight 6 6104 7 sealed beam headlight

Although Figure 6(a) seems to indicate that the majority of the current policies are close

in cost to the optimal policies this is very misleading For example suppose the optimal policy

for a product costs $1 and that the current policy costs $2 In absolute terms the optimal

policy saves only $1 but in relative terms the optimal policy saves 50 of the current cost

When savings are viewed in this manner as in Figure 6(b) it is clear that substantial savings

can be realized for the majority of the products In fact both the mode and the mean of the

savings are between 40 and 50 of the current costs

In order to extrapolate the above savings to obtain the potential savings for product

category 20 at all 21 stores the differences between stores must be considered During the

60

pilot study the current policy for each of several products at each size of store was evaluated

The study suggested that medium and large stores have a similar amount of excess

inventory and that jumbo and extra jumbo stores have much more excess inventory One

could reasonably expect that the additional excess inventory allows for higher savings to be

realized at the jumbo and extra jumbo stores Thus because 18 of the 21 stores are

medium or larger the expected 40-50 savings at store 6 can be viewed as a lower bound

on the potential savings to Canadian Tire Pacific Associates from implementing optimal policies

for product category 20 in all its stores

Finally in order to obtain the overall potential savings the expected savings from

product category 20 must be extrapolated to all product categories Ideally the proportion of

inventory comprising each product category and the relative savings that can be realized for

each category would be used to obtain a good estimate of the overall potential savings

However as this information is not known only a crude extrapolation is possible Thus the

analysis proceeds under the assumption that a 40-50 savings can be realized in each product

category35

Before an estimate of the total potential savings in absolute terms can be obtained the

proportion of the current inventory to which the savings apply must be determined Up to this

point the analysis has been concerned solely with reducing inventory at the stores However

it is inappropriate to perform further analysis without considering the entire inventory system

including the central warehouse Because the lead time for orders from Toronto is highly

variable and considerably greater than those at individual stores Canadian Tire Pacific

3 5 During the pilot study the current policies for a paint brush and an ironing board were evaluated and found to result in excess inventory similar to that of product category 20 Although this in no way justifies the assumption it does lend some support

61

Associates must keep a fairly high safety stock at the warehouse Without knowing the

proportion of inventory serving as the safety stock it is inappropriate to assume any reduction

in inventory at the warehouse is possible Therefore the relative savings of implementing

optimal policies is applied only to the inventory held in the stores36 Of the estimated $28

million in inventory approximately half is held in the stores therefore the 40-50 savings

translates to a savings of approximately $55 to $7 million per year

B Sensitivity Analysis on the Ordering Cost and Holding Rate

The above analysis was based on estimates of the ordering cost and the annual unit

holding rate37 provided by management As described in Chapter II these estimates

incorporate a number of costs that are difficult to quantify or that are speculative Therefore

there is some question as to the accuracy of the estimates In this section errors of +$0015

in the ordering cost and +5 in the holding cost are assumed in order to determine the

sensitivity of the model to errors in these variables38

The analysis proceeds as follows Based on an ordering cost of $0080 and a holding

rate of 30 the optimal policies from the previous section were obtained39 Then assuming

3 6 Of course this assumes that the excess inventory currently held in the stores is not required as part of the safety stock Based on the large amount of inventory at the warehouse this assumption seems reasonable

3 7 Hereafter simply referred to as the holding rate

3 8 An error of +$0015 in the ordering cost implies an actual ordering cost of $0070 a relative error of approximately 21 and an error of -$0015 implies an actual ordering cost of $0100 a relative error of 15 An error of +5 in the holding rate implies an actual rate of 25 a relative error of 20 and an error of -5 implies an actual holding rate of 35 a relative error of approximately 17

3 9 These policies are referred to as optimal policies for the duration of this chapter

62

the actual ordering cost andor holding rate are different from managements estimates the

heuristic is used to obtain the true optimal policies Finally the optimal policies are

evaluated based on the actual ordering cost and holding rate and then compared to the true

optimal policies For each possible combination of ordering cost and holding rate three

numbers are recorded the number of products with an optimal policy that differs from its

true optimal policy the total cost of implementing the optimal policies given the actual

ordering cost and holding rate and the difference between the cost of implementing the

optimal policies and the cost of implementing the true optimal policies The results of the

analysis are summarized in Table 6

Table 6 A comparison of optimal policies to true optimal policies

Actual Ordering Cost

$0070 $0085 $0100

25 5 products $155439

$002

51 products $160590

$092

88 products $165464

$449

Actual Holding Rate

30 54 products

$181470 $182

48 products $192026

$088

35 79 products

$207270 $574

48 products $212940

$136

2 products $218312

$000

4 0 The table should be read as follows If the actual ordering cost is $0070 and the actual holding rate is 25 then each of 5 products has an optimal policy that is different from its true optimal policy The total cost of implementing the optimal policies is $155439 which is $002 more than implementing the true optimal policies

63

The model appears to be quite insensitive to relative errors in the order of 20 Under

the worst case scenario when one of the variables is over-estimated and the other is undershy

estimated the optimal policies result in a total cost of inventory which is only approximately

3 higher than the cost of the true optimal policies Further because demand is not changed

by errors in these estimates each of the optimal policies still provides a customer service

level of at least 975 under all of the scenarios Finally although the cost of the optimal

policies varies significantly for the different scenarios the relative savings remain virtually

constant Table 7 lists the cost of existing policies and in parenthesis the relative savings of

the optimal policies for each of the different combinations of ordering cost and holding rate

The cost of the existing policies varies in the same manner as that of the optimal policies the

relative savings for each of the scenarios remains around 55 Therefore even in the presence

of such errors in the ordering cost and holding rate the projected savings of implementing the

optimal policies would be unchanged

Table 7 The cost of existing policies and the relative savings of the optimal policies under various scenarios

Actual Ordering Cost

$0070 $0085 $0100

25 $350153 (56)

$358464 (55)

$366764 (55)

Actual Holding Rate

30 $412438 (56)

$429048 (55)

35 $474725 (56)

$483026 (56)

$491320 (56)

64

C Sensitivity Analysis on the Demand Probability Mass Function

In this section the sensitivity of the model to slight changes in the demand probability

mass function is investigated As described in Chapter II the demand function used in the

model is based solely on observed sales however this empirical function is only one possible

instance of the underlying true demand function For example had the data been collected

over a period both starting and finishing one month earlier many of the empirical functions

would be slightly different Consequently depending upon the degree to which the empirical

functions vary and the sensitivity of the model to these variations the optimal policies from

the model might be inaccurate For example recall the case of the amber driveway marker

only thirteen markers were sold and all were sold on the same day Because the empirical

demand function is used the model calculates an optimal policy of (1213) clearly this

policy is impractical Although this example is extreme it does indicate the danger of using

unrealistic empirical demand functions41

As discussed in Chapter II three classes of products were identified namely low

medium and high-demand products For this analysis the same method of classification

is used Two products are selected from each of these groups and two new distribution files

are created for each of the six products For each product one of the files contains two

additional days of data with each day having a demand equal to the maximum observed demand

for that product In the second file only one additional day of data is added but the demand

on that day is two more than the maximum observed demand The situations that could give

4 1 Certainly a rigorous analysis to determine both the extent of variation in the empirical demand functions and the effect of using the empirical functions needs to be performed However such an analysis is beyond the scope of this thesis only a brief analysis that might suggest the degree to which the model is sensitive is performed here

65

rise to such changes in the demand data are termed Scenario 1 and Scenario 2 respectively42

The analysis proceeds in a manner similar to the one in the previous section Given the

demand that was originally observed the optimal policies were obtained Now assuming the

true demand function corresponds to the demand under Scenario 1 the true optimal policy

for each product is calculated Then each optimal policy is re-evaluated using the true

demand function and compared to the true optimal policy The analysis is then repeated

assuming the true demand function corresponds to the demand under Scenario 2 Table 8

lists the results for each product and scenario

Table 8 True optimal and optimal policies for each demand scenario

Product 202101 (low demand)

Scenario s s Cost ($) C S L ()

1 True optimal 1 2 470 993

Optimal 1 2 470 993

2 True optimal 2 3 663 976

Optimal 1 2 461 875

Product 203507 (low demand)

Scenario s S Cost ($) C S L ()

1 True optimal 0 3 334 981

Optimal 0 2 258 972

2 True optimal 2 3 508 980

Optimal 0 2 273 702

4 2 To be clear the scenarios are defined as follows Scenario 1 The addition of two days each with a demand of max Scenario 2 The addition of one day with a demand of max+2

where max is the maximum observed daily demand for each product in the sample period

66

Product 201032 (medium demand)

Scenario s S Cost ($) C S L ()

1 True optimal 6 12 1377 976

Optimal 6 11 1338 973

2 True optimal 6 13 1426 977

Optimal 6 11 1338 970

Product 203039 (medium demand)

Scenario s S Cost ($) CSL ()

1 True optimal 5 19 256 975

Optimal 5 16 251 969

2 True optimal 5 20 258 976

Optimal 5 16 249 968

Product 203677 (high demand)

Scenario s S Cost ($) CSL ()

1 True optimal 18 23 3981 977

Optimal 18 22 3817 972

2 True optimal 18 23 3995 978

Optimal 18 22 3831 973

Product 203680 (high demand)

Scenario s S Cost ($) CSL ()

1 True optimal 38 49 9498 976

Optimal 37 48 9175 971

2 True optimal 37 49 9484 975

Optimal 37 48 9206 972

67

The model is most sensitive to changes or errors in the demand functions of low-

demand products When only one additional day of demand for max+2 units is added to the

data the optimal policy for each of the low-demand products results in a very low customer

service level In fact for one of the products the customer service level drops to almost 70

The increase in demand does lower the customer service level for each of the other the

products but that decrease is slight

The above analysis indicates the need for some kind of smoothing or modelling of the

demand data especially for low-demand products The inherent variation between two

instances of a true demand function could very well result in more pronounced changes than

those resulting from either of the above scenarios Therefore it is likely that many optimal

policies especially those for low-demand products would actually provide a much lower or

much higher customer service level than expected However by smoothing the data the

variation between instances of a true demand function can be reduced and the desired

customer service levels can be achieved

For example suppose that the demand for a product is stationary that the average

demand during the previous period was unusually low and that the optimal policy for this

period is based on the demand during the previous period Because the demand is stationary

the probability that the demand during this period is higher than that during the past period is

more than 50 In other words the probability that the demand during this period is higher

than that for which the optimal policy provides the desired expected customer service level

is greater than 05 Thus expected customer service level for this period is actually less than

that specified Now suppose instead that the optimal policy is based on smoothed data which

reflects very closely the expected demand The probability that the demand during this period

is higher than that for which the optimal policy provides the desired expected customer service

68

level is virtually 05 Thus as a result of smoothing the data the specified customer service

level should be expected

69

VII C O N C L U S I O N

In this thesis we developed a single-product single-location model of the inventory

system at Canadian Tire Pacific Associates Specifically we formulated a Markov chain model

to determine the long-run average cost of inventory for fixed values of s and S and then

developed a heuristic search to find near optimal policies quickly Also as a first step toward

developing an integrated interactive inventory control system we created a prototype of the

interface

According to the analysis performed here Canadian Tire Pacific Associates is currently

holding an excessively large inventory in its stores This study finds that Canadian Tire Pacific

Associates could reduce the cost of inventory for product category 20 at store 6 by

approximately 50 and still maintain customer service levels of at least 975 Further the

mean and modal values of savings for these products are between 40 and 50 Given the

similar excess inventory at the other stores and in all product categories we believe that savings

in this range are possible for all products and stores We estimate that the application of

inventory control as described in this thesis would result in annual savings of $55 to $7

million

Given the large potential savings to Canadian Tire Pacific Associates we recommend

that management proceed with the next phase of the project Specifically we recommend that

the model algorithm and interface be refined further that a forecasting model be developed

and that a data collection and storage scheme be designed

In order to reflect operational changes at Canadian Tire Pacific Associates subsequent

to the formulation of the model modifications must be made to allow for the lead time to

exceed the review period Several of the stores now have three picks per week and as a

70

result orders placed for these stores at one decision epoch do not arrive until after the next

decision epoch Because this violates one of the models assumptions the model must be

modified accordingly Note however that with three picks per week and a review period

of four days orders from only the previous decision epoch are ever outstanding at the current

decision epoch

Also the model should be refined to reflect the fact that the review period is not fixed

over the planning period In its present form the model assumes a fixed review period

however inventory at stores with two picks per week is actually reviewed on a four day -

three day cycle and inventory at stores with three picks per week is reviewed on a three day

- two day - two day cycle Rather than obtaining the steady state costs for fixed review period

the model should calculate the steady state cost for a period of one week during which multiple

replenishments occur Although the solutions are not expected to differ significantly as a result

of the change the model would reflect the actual process more accurately

Further work is also required on the search technique for locating near optimal

policies Although the heuristic search described in this thesis appears to work well the

decision maker must provide an upper bound on S prior to the search In practice this upper

bound is generally not known in advance Thus if the heuristic is to be used either a tight

theoretical bound must be found or a termination heuristic must be developed Alternatively

a new heuristic search could be developed based on additional or different assumptions

regarding the properties of the feasible space For example although some regions of non-

convexity were found for some products much of the feasible space appeared to be convex

A heuristic that assumes a convex feasible space would not require an upper bound on S and

might perform very well However before such a heuristic is implemented further work is

necessary to determine the degree of non-convexity for a variety of demand functions and

71

ordering and holding costs

In addition to making the above modifications we recommend that Canadian Tire Pacific

Associates begin work on the remaining two components of the inventory control system First

in order to allow for products with seasonal demand the system requires a forecasting model

to provide accurate estimates of demand for the coming period Second the system needs a

data collection and storage module that systematically collects sales data for each active

product and for each store and then updates the corresponding demand distributions Because

approximately 25 of demand is expected to result in lost sales under the optimal policies the

sales data will have to be adjusted accordingly in the future Also because the model appears

to be sensitive to slight changes in the demand distribution for low-demand products the

module should incorporate a technique for smoothing the empirical distributions for these

products Further work should be undertaken to determine the extent of variation in the

empirical distributions and the most appropriate smoothing technique However because of

the large amount of data required to forecast seasonal demand and because of the current lack

of historical data we recommend that management proceed immediately with the design of a

data collection scheme

Finally two additional issues that are beyond the scope of the next phase should be

addressed at some point First a monitoring system will be required once inventory control

is implemented in the stores The monitoring system will ensure that optimal policies perform

as expected and will indicated whether or not demand is adversely affected by the lower levels

of in-store inventory Second given the degree to which in-store inventory can be reduced

inventory control in the warehouse should be investigated Because approximately half of the

existing inventory is held in the warehouse the additional savings from reducing system-wide

inventory could be great

72

Afterword

Just prior to the completion of the work described in this thesis Mr Don Graham

principle owner of Canadian Tire Pacific Associates sold his shares in the company Shortly

thereafter the results of this thesis were presented to the new management group

Unfortunately the new management did not consider development of an inventory control

system to be a priority Nonetheless the prototype was delivered to Canadian Tire Pacific

Associates for possible in-house testing however at the time of this writing - almost one year

after the meeting - Canadian Tire Pacific Associates has not decided to continue with the

project

73

Bibliography

Arrow K A T E Harris and J Marschak (1951) Optimal inventory policy Econometrica 19 250-272

Day G (1992) In a speech to the Empire Club of Canada Toronto quoted in When the JIT fits The Globe and Mail October 20 1992 B24 and in Puterman (1994)

Dvoretzky A J Kiefer and J Wolfowitz (1952a) The inventory problem I Case of known distributions of demand Econometrica 20 187-222

Dvoretzky A J Kiefer and J Wolfowitz (1952b) The inventory problem II Case of unknown distributions of demand Econometrica 20 450-466

Federgruen A (1993) Centralized Planning Models in SC Graves A H G Rinnooy Kan and P H Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Federgruen A H Groenevelt and H C Tijms (1984) Coordinated replenishments in a multi-item inventory system with compound Poisson demands and constant lead times Management Sci 30 344-357

Lee H L and S Nahmias (1993) Single-Product Single-Location Models in SC Graves A H G Rinnooy Kan and PH Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Muckstadt J A and R O Roundy (1993) Analysis of Multistage Production Systems in SC Graves A H G Rinnooy Kan and PH Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Porteus E L (1985) Numerical Comparisons of Inventory Policies for Periodic Review Systems Oper Res 33 134-152

Puterman M L (1994) Markov Decision Processes Discrete Stochastic Dynamic Programming Wiley New York

Silver E A and R Peterson (1985) Decision Systems for Inventory Management and Production Planning 2nd edition Wiley New York

Taylor H M and S Karlin (1994) An Introduction to Stochastic Modeling revised edition Academic Press San Diego

74

Tijms H C and H Groenevelt (1984) Simple approximations for the reorder point in periodic and continuous review (sS) inventory systems with service level constraints European J Oper Res 17 175-190

75

Appendix A

A Portion of the Sales File for Product 200001 a 30-amp Inline Fuse

200001 283 1190 0 0 2 0 0 1 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 200001 284 1190 2 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 200001 285 1190 0 0 4 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 286 1190 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 200001 287 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 200001 288 1190 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1 0 2 200001 289 1190 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 290 1190 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 200001 291 1190 0 0 2 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 200001 292 1190 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 200001 294 1190 0 0 0 1 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 1 200001 295 1190 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 200001 296 1190 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 200001 297 1190 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 200001 298 1190 0 0 1 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 200001 299 1190 0 0 1 0 0 0 1 0 1 0 0 2 0 0 0 0 0 2 1 1 0 200001 300 1190 0 0 1 0 0 0 0 1 0 0 0 0 0 2 2 0 0 0 0 3 0 200001 301 1190 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 200001 303 1190 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 200001 304 1190 1 0 0 0 0 0 0 0 1 0 0 2 0 0 0 0 0 0 0 0 0 200001 305 1190 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 200001 306 1190 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 2 0 0 1 0 200001 308 1190 0 0 0 1 0 1 0 0 0 2 0 1 0 0 0 0 1 1 1 0 0 200001 309 1190 1 3 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 200001 310 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 200001 311 1190 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 200001 312 1190 0 1 1 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 200001 313 1190 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 2 0 200001 314 1190 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 200001 315 1190 0 0 2 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 200001 316 1190 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 200001 317 1190 2 0 0 1 0 0 0 0 0 0 5 1 0 0 0 1 0 0 0 0 0 200001 318 1190 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 1 200001 319 1190 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 200001 320 1190 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 200001 321 1190 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 1 200001 322 1190 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 323 1190 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 200001 324 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 0 1 0 0 2 1 1 0 200001 325 1190 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3 200001 326 1190 0 0 1 0 1 0 0 0 2 0 0 1 0 0 0 1 0 1 0 0 1 200001 328 1190 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 200001 329 1190 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 200001 330 1190 0 0 1 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 200001 331 1190 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 200001 332 1190 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 200001 333 1190 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 200001 334 1190 0 0 2 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 200001 335 1190 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 200001 336 1190 0 0 0 1 0 0 3 0 0 0 0 0 2 0 0 0 0 0 0 0 0

(253 subsequent lines omitted)

76

Appendix B

Sample Distribution Files

a) A low-demand product - an automobile back-up alarm

PRODUCT NUMBER 202101 PRICE $ 684

DMD

3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

N 2 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 2 2 0 1

1 5 3 8 12 5 7 11 6 11 5 4 11 10 19 4 2 5 14 5 6 4

0 302 304 299 295 302 300 296 301 296 302 303 295 296 288 303 305 301 290 300 301 302

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

b) A medium-demand product - a 168 bulb

PRODUCT NUMBER 203039 PRICE $ 51

DMD

16 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 1 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0

9 0 0 2 5 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0

8 0 0 2 5 0 0 2 0 1 1 3 0 1 3 0 0 1 1 0 0 1

7 2 1 3 1 0 0 1 0 2 3 1 1 4 1 1 0 2 1 1 1 1

6 4 7 1 10 3 1 3 2 1 3 5 2 7 4 1 0 1 1 3 3 4

5 1 7 7 17 0 2 11 1 5 4 1 2 6 9 6 2 6 9 4 4 1

4 5 10 10 16 5 5 19 10 13 8 11 9 13 10 12 7 5 15 5 8 3

3 10 12 13 27 4 6 24 11 10 13 14 9 23 14 18 8 14 18 14 12 14

2 14 30 36 43 16 16 43 32 38 39 39 34 47 36 41 21 19 39 20 34 24

1 30 56 68 56 37 30 70 48 67 61 44 75 62 63 48 28 52 61 63 53 47

0 240 184 165 126 240 247 133 202 169 175 189 173 143 165 180 240 206 161 197 192 212

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

77

c) A high-demand product - a H4651 rectangular sealed beam headlight

PRODUCT NUMBER 203677 PRICE $ 857

DMD

14 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

13 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 2 3 0 0 1 0 0 0 0 1 1 1 0 0 0 0 2 0 0

11 0 1 3 1 0 0 4 1 0 0 0 0 0 0 0 0 1 0 0 0 1

10 0 0 4 4 1 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 1

9 0 3 14 11 0 1 7 1 0 2 0 8 0 1 0 0 0 2 1 0 0

8 0 4 12 16 1 0 18 4 3 5 0 2 1 4 2 1 1 0 0 1 3

7 1 2 12 13 1 6 19 3 5 9 0 11 2 5 4 0 2 3 3 1 4

6 2 9 22 21 6 10 29 12 10 11 4 24 3 7 10 4 7 11 6 3 14

5 7 13 35 33 8 16 29 19 18 8 3 26 15 26 15 12 9 14 13 11 19

4 19 35 34 43 32 30 41 31 32 47 11 42 20 33 30 31 20 31 22 21 31

3 32 63 49 54 42 60 51 40 51 35 33 55 39 51 39 38 39 57 58 40 54

2 46 59 56 55 73 63 61 65 63 61 63 54 68 85 65 53 72 69 70 77 63

1 109 79 42 40 80 64 32 80 74 77 99 55 97 57 83 93 80 75 81 97 73

0 91 39 21 13 63 57 11 51 51 52 94 29 61 35 59 75 76 45 51 56 44

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

78

Appendix C

A Proof that the Limiting Probability Distribution is Independent of the Initial State

In this appendix we show that provided 0 lt PrDT = 0 lt 1 the Markov chain

corresponding to an inventory model with a (sS) policy has a limiting probability distribution

7T where 0 lt TTJ lt 1 for y = 0 1 S that is independent of the initial state Further the

limiting distribution is the unique nonnegative solution of the equations

Tt P = TI (AD

s and J gt = 1 (A 2)

i=0

where P is the (S + l) x (S + l) transition probability matrix whose first n rows (and columns)

correspond to the n recurrent states in the chain and whose remaining rows (and columns) if

any correspond to transient states43 Prior to the proof several preliminary results are

obtained

RESULT 1 The Markov chain has exactly one positive recurrent class and perhaps some

transient states

First note that in the absence of any absorbing states a Markov chain with a discrete

finite state space must have at least one positive recurrent class containing at least two states

Now consider the Markov chain for an inventory model with a (sS) policy

a) State i gt s cannot be absorbing because there is a positive probability of reaching

j lt i (Since PrDT gt 0 gt 0)

4 3 Note that since the rows and columns of P are reordered the elements of it must be reordered accordingly

79

b) State lt s cannot be absorbing because by the rules of the ordering policy and

the assumption that PrDT = 0 gt 0 there is a positive probability reaching S

Therefore the chain has no absorbing states and must have at least one positive recurrent class

Next consider the subset of states A = i | i lt s Because the state space is finite

and PrDT gt 0 gt 0 the process must return to at least one state in A in other words at least

one state in A must be recurrent Let k G A be a member of a recurrent class dt Because

of the ordering policy and the assumption that PrDT = 0 gt 0 k -raquo S Also by definition

there must be a positive probability of returning to k from S S -gt k Therefore S G 3w

Further because S is reachable from all states in A i G A is either transient with i -raquo dt or

recurrent with pound iR

Now consider the subset of states B = | j gt s Because PrDT gt 0 gt 0 the

chain will eventually reach a state i G A If i is transient then j must be transient also with

y -gt 3 if i G 9 then j is either transient with j -gt dt or recurrent with j E dt Thus the

Markov chain has exactly one recurrent class and perhaps some transient states

RESULT 2 The limiting probability distribution ir for the Markov chain corresponding to the

nxn submatrix Pm = [Ptj] i j G dt is independent of the initial state and is the

unique nonnegative solution of the equations

TC-Prr = re (A3)

and J2 K = 1 ( A 4 )

jt=i

where the rows and columns of are numbered from 1 to n rather than

Since if i j i and j must be in the same equivalence class

80

according to their associated states

First note that both the original Markov chain and the one corresponding to the

submatrix P R R are aperiodic because PrDT = 0 gt 0 Therefore given that 9 is a positive

recurrent aperiodic class the limiting distribution ir is uniquely determined by (A3) and (A4)

(Taylor and Karlin 1994 Theorem 44 p213) Further because there are no absorbing states

or other recurrent classes in the original chain the process must eventually enter the recurrent

class 9 regardless of the initial state Therefore in the limit the probability of being in state

i G 9 is independent of the initial state

RESULT 3 The limiting probability distribution for the transient states is the null vector

independent of the initial state

This result follows directly from the definition of a transient state

PROOF Suppose the states are reordered such that the first n rows (and columns) of P

correspond to the n recurrent states of the Markov chain Then P can be partitioned into four

submatrices

PRK - the n x n submatrix [Py] i j E 9

P T T - the (S + 1-I) x (S + l-n) submatrix [Py] i j g 9

P T R - the (S + l-n) x n submatrix [Py] i pound 9 j G 9

P R T - the n x (S + l-n) submatrix [Py] i G 9 j pound 9t By definition this is a matrix

of zeros

(Al) can then be rewritten as

81

(ltR TtT) P P L TR TT

I-P RR 0

p I-P TR 1 TT

(TtR TCT)

(0- 0T)

where irR denotes the limiting probability distribution for the n recurrent states and irT denotes

the distribution for the remaining S + l-n transient states Solving for TT t gives

TCr-0 + itT-(I-PTI) = 0T

itT-(I-PJJ) = 0 r

i c r = (I-P^-Oj

Since (I - PTT)1 always exists (Puterman 1994 Proposition A3 p 593) irT = 0X which is

in agreement with Result 3 Now solving for 7rR yields

(A2) can be rewritten as

bullRPRR 1 1R (A5)

(A6) S + l

k + L k =1

k=l k=n+

k=i

where ir contains the appropriately reordered steady state probabilities Since (A5) and (A6)

are equivalent to (A3) and (A4) respectively Result 2 holds Therefore the Markov chain

corresponding to an inventory model with a (sS) policy has a limiting distribution that is

independent of the initial state and is the unique nonnegative solution to (Al) and (A2)

82

Appendix D

A Measure for the Steady State Customer Service Level

In this appendix we show that the appropriate measure for the steady state customer

service level is given by

_ E[Demand satisfied during RP] E[Total demand during RP]

Consider the following two measures of the steady state customer service level

Measure 1 Suppose the customer service level is calculated only after n review periods have

passed Denoting DS as the demand satisfied in review period i and DT as the total demand

in period i the customer service level after n periods CSL(n) is

DS + DS0 + + DSbdquo CSLn) = - -

DT + DT + + DT

Dividing both the numerator and denominator by n taking the limit as n -raquo oo and applying

the Law of Large Numbers yields

CSL = lim CSL(n) n-degdeg

- [DS + DS + + DS] = l i m ^ J 1 1

raquo~ - [DT + DT + + DT]

E[DS]

E[DT]

83

Measure 2 Suppose the customer service level is calculated for each of n review periods and

then averaged Thus the customer service level after n periods CSL(n) is given by

CSL(n) = -DS DS

DT DT DT_

Taking the limit as n -gt oo yields

CSL = lim CSL(n)

lim mdash DSl DS2

DT DT DT_

DS

DT

Now consider the scenario in which there is one item on the shelf to begin each review

period and in which demand is stochastic with PrDT = 2 = Vi and PrD x = 100 = Vi

According to Measure 1 the customer service level is

CSL - = ^ W + ^ W = plusmn bdquo 002 E[DT) 1 (2) + 1(100) 51

while according to Measure 2 the customer service level is

CSL = E DS DT

1(1) + i(-Ls

22) + 2 100

51 200

026

Clearly from managements point of view the first measure is more indicative of the

actual level of service being provided The problem with the second measure is that equal

84

weight is given to each ratio regardless of the magnitude of the numbers For example with

respect to Measure 2 the events LSLR = 12 and LSLR = 50100 are identical however

with respect to lost sales the second event is much worse than the first Therefore the proper

measure for the steady state customer service level is Measure 1

85

Appendix E

The Calculation of the Conditional Expected Demand Not Satisfied During a Period of T Consecutive Days

Denoting D x as the demand during a period of T consecutive days and j as the inventory

on hand at the beginning of the period the conditional expected demand not satisfied during the

period E[(DT-y)+] is calculated as follows

E[(DT-j)+] = pound ltk-j)Pr[DT = k k=j+l

= pound (k-j)PrDT = k] + J2(k-j)PrDT = k] - pound (fc-)VJgtr = fc k=j+l k=0 k=0

= J2(k-j)Pr[DT=k -J2(k-j)PrDT = k] k=0 k=0

= YkPrDT=k) - J2jPrDT=k - Yk-j)PrDT=k) Jfc=0 k=Q k=0

= Ti - j - Y(k-j)PrDT=k 7-1

= Tx - j + YltJ-k)PrDT = k jt=0

where fx is the mean daily demand

86

Appendix F

Justification of the Updating Technique for the New Policy (s+mS)

Let the probability transition matrix corresponding to the policy (sS) be denoted by the

(S + l)x(S + l) matrix P with elements py Define s = s+m where m is an integer satisfying

the condition 0 lt s+m lt S Let the probability transition matrix of the new policy (sS)

be denoted by the (S + 1)X(S + 1) matrix 7gt with elements py Using the equations developed

in Chapter III the elements of P are found as follows

CASE 1 The lead time is less than the review period

A m gt 0

For 0 lt i lt s

( i-i pound [PrDL=kPrDT_LgtS-k]] + PrDLziPrDT_LplusmnS-i k=0

i-l 53 [PrDL=kPrDT_L = S-j-k]] + PrDLziPr[DT_L = S-j-i) k=0 s-j

52 [Pr[DL = kPrDT_L = S-j-kn k=0

ifj = 6

ifOltjz S-i

if S-i ltj ltS

a) 0 lt lt s

Pij

= lt Pij

Pij

= pij

b) s lt i lt s This does not simplify but it is identical to (2)

87

For s lt i lt S

PrDTgt i

p = lt PrDT = i-j

0

Pfj ifj = 0

= lt Pij ifOltjlti

Pij if i ltj plusmn S

= pij

B m lt 0

ifj = 0

if 0 ltj lt i

if i ltj plusmn S

For 0 lt i lt s

i-l 53 [PrDL=kPrDT_LplusmnS-k]] + PrDLgtiPrDT_LplusmnS-i

i-l E [PrZgti = fciVZ)7W=S--fc] +PrD tgtiP-D 7 t = S-7-

p y laquoy = o

P i j ifOltjS-i

vPij ifS-iltjS

ifj = 0

ifOltjltS-i

if S-i ltjltS

For s lt i lt S

fPrJgtrgt- 17=0 = PrDT = i-j] ifOltjlti

0 ifiltjplusmnS

88

a) s lt lt s This does not simplify but it is identical to (3)

b) s lt i lt S

PrDTgti

= bull PrDT = i-j]

0

17=0

if0ltjplusmni

ifiltjS

= Pt

ifj = 0

if0ltjlti

ifiltjplusmnS

= Pi

CASE 2 The lead time is equals the review period

A m gt 0

For 0 lt i lt s

0 if0ltjlt S-i

PrDTgti ifj = S-i

PrDT=S-j] ifS-iltjltS

a) 0 lt i lt s

= bull y Pij

Pij

ifOplusmnjltS-i

ifj = S-i

ifS-iltjS

= Pij

b) s lt i lt s This does not simplify but it is identical to (4)

89

For s lt i lt S

PrDTgti)

Pr[DT=i-j

0

laquof 7=0

ifOltjplusmni

if i ltj lt S

Pa

Pij

if 7 = 0

if 0 lty lt i

ifiltjltS

B w lt 0

For 0 lt t lt s

ro PrJrgt PrDT = S-j

ifOzjltS-i

ifj = S-i

if S-i ltj lt S

Pij

ifOzjltS-i

ifj = S-i

if S-i ltj ltS

= Pi

For s lt i lt S

PrDTgti

= PrDT = i-j

0

i 7 = o ifOltjplusmni

ifiltjltS

a) s lt i lt s This does not simplify but it is identical to (3)

90

PrDTzi ifj = 0

PrDT = i-j] ifOltjplusmni

0 ifiltjltS

Py ifj = 0

ptj ifOltjplusmni

Pij if iltj plusmnS

Appendix G

Justification of the Updating Technique for the New Policy (sS+m)

Let the probability transition matrix corresponding to the policy (sS) be denoted by the

(S + l)x(S + l) matrix P with elements p(gt Define S = S+m where m is any non-negative

integer Let the probability transition matrix of the new policy (sS) be denoted by the

(S+m+l)x(S+m + l) matrix IP with elements ptj Then using the equations developed in

Chapter III the elements of 7 are found as follows

CASE 1 The lead time is less than the review period

For 0 lt i lt s

pound [PrDL=kPrDT_LplusmnS-k] + PrDLi)PrDT^Sgt- k=0

i-l pound [Pr[DL = kPrDT_L = S-j-k- +Pr[DLiPrDT_L = S-j-i Jt=0

s-i Y [Pr[DL=kPrDT_L = S-j-k] k=0

i - l

ifj=0

ifOltjltS-i

ifS-iltjltS

pound [PrDL = kPrDT_LS-k + Pr[DLi)PrDT_LplusmnS-i) if j = 0 fc=0

i - l = pound [PrDL=kPrDT_L = (S+m)-j-k] + PrDLgtiPr[DT_L = (S+m)-j-i ifOltj lt (J5+m)-i

if (S+m)-iltj lt(S+m)

k=0

i - l

pound (Sm)-j

pound [PrDL = kPrDT_L = (S+m)-j-k]] k=0

92

I-l

53 [PrDL = kPrDT_LgtS-k] + PrDLziPrDT_LS-i ifj = 0 k=0

i-l

Y[PrDL = kPrDT_L = S-(j-m)-k] + Pr[DLziPrDT_L = S-(j-m)-i] ifOltjzS+m-i k=0 S-(j-m)

53 [PrDL = kPrDT_L = S-(j-m)-k]] if S+m-i ltj lt S+m

i - l

53 [PrDL = kPrDT_LzS-k] + PrDLziPrDT_LzS-i] ifj = 0 =0

i - l

13 [Pr[DL = kPrDT_L = S-(j-m)-k] + Pr[DLziPr[DT_L = S-(j-m)-i ifOltjz S+m-i k=o and 0 z j-m

Pi(j-m)

Pi(j-m)

if 0 ltj lt S+m-i and 0ltj-mltS-i

if S+m-i ltj lt S+m and S-i lt j-m zS

i - l

Y [PrDL = k)PrDT_LzS-k] + PrDLi)PrDT_LplusmnS-i ifj = 0

k=0 i - l

53 [PrDL = kPrDT_L = S-(j-m)-k] + PrDLziPrDT_L = S-(j-m)-i ifOltjzm k=0

Piij-m)

Piij-m)

if mltj lt S+m-i

if S+m-i ltj lt S+m

i - l

YlPrDL = kPr[DT_L^S-k + PrDLi)PrDT_LS-i) ifj = 0

i - l

53 [PrDL = kPrDT_L = S-j-k] + PrDLgtiPrDT_L = S-j-i ifOltj i m k=0

Pi if mltj ltS

which for 0 lt j lt m is identical to (2)

93

For s lt i lt S

Pu =

PrDTgti

PrDT = i-j]

0

if0ltjlti

ifiltjltS

a) s lt i lt S

PrDTgti 7=0

PrDT = i -j) if0ltjlti

0 if i ltj lt S

0 ifSltjltSgt

ifj=0

if 0 ltj lt i

if i ltj plusmn S

0 ifSltjltS

ifOltjltS

0 ifSltjltSgt

b) S lt i lt S This does not simplify but it is identical to (3)

CASE 2 The lead time equals the review period

For 0 lt i lt s

0 ifO lt jltS-i

PrDTgti ifj=S-i

PrDT=S-j ifS-iltjltS

94

0 ifOplusmnjlt(S+m)-i

PrDT ifj = (S+m)-i

PrDT = (S+m) -j) if (S+m)-i ltj ltL (S+m)

0 if 0 lt jltS+m-i

PrDTzi] if j = S+m-i

PrDT = S-(j-m) if S+m-i ltj lt S+m

0 if 0 ltj lt S+m-i and 0 ^ j-m

Pi(j-m) if 0ltjlt S+m-i and 0ltj-mltS-i bull

Pi(j-m) if j = S+m-i and j-m = S-i

Pi(j-m) if S+m-i ltj z S+m and S-i lt j-m ^S

0 if 0 zj ltm

Pi(j-m) if mlt jlt S+m-i lt

Piij-m) if j = S+m-i

if S+m-i ltj lt S+m

1deg if 0 ltj ltm

[PiV-m) ifmltjltS

which for 0 lt i lt s is identical to (3)

For s lt lt S

PrDTzi

= PrDT = i-j)

0

ifj = 0

if 0 ltj ltL i

ifKjzS1

95

a) s lt i lt S

PrDTzi

PrDT = i-j]

0

0

17=0

if0ltjplusmni

ifiltjS

ifSltjltSgt

Pij 7=0

Pij if0ltjii

Pij if i ltj ltL S

0 ifSltjltS

Pij ifOzjiS

0 ifSltjltS

b) S lt i lt S This does not simplify but it is identical to (3)

96

Appendix H

A Hypothetical Consultation

In this appendix the interface during a hypothetical consultation is depicted The figures

accurately represent the various screens that are presented to the manager with the exception

that all lines and text are really white on a blue background During the consultation described

below only the choices made by the manager are hypothetical all the results are real

The title screen as depicted in Figure A - l appears initially upon execution of the

program In order to proceed to the main menu the manager presses the enter key

O I C S

Optimat Inventory Control System

For the exclusive use of

Canadian Tire Pacific Associates

Developed at The Faculty of Commerce

and Business Administration The University of British Columbia

Brian Kapalka Kaan Katircioghi Marty Puterman

Press ltEntergt to continue

Figure A - l The title screen

97

Main Menu 1 Select data directory

ctircdatadst

2 Select product and store combination Product 200001

3 Select scenario Review Period 4 Lead Time 3

4 Select search parameter Smax 25

5 Execute program

6 Exit to DOS

Please enter the desired option 5

Figure A-2 The main menu

Figure A-2 depicts the main menu of the program it is from here that the manager

enters all the review parameters If the manager wishes to change one of the parameters from

its default value he or she selects the appropriate number and enters the correct information

in the window presented The new settings will then be shown in place of the default values

In this example the relevant data files are located in the default directory ctiredatadst and

the manager has selected the productstore combination of product 200001 and store 6 Further

the manager has selected a review scenario of a 4-day review period and a 3-day lead time and

an upper search limit of 25 Finally in order to proceed with the analysis the manager

executes the main module of the program by selecting option 5

98

List o f Tables

Table 1 Location and particulars of the 21 stores 10

Table 2 Execution times of the algorithms for product 200001 a 30-amp inline fuse 39

Table 3 Execution times of the algorithms for product 206917 a 6

solder connector 40

Table 4 Products with insufficient existing policies 56

Table 5 Products with the largest potential absolute savings 60

Table 6 A comparison of optimal policies to true optimal policies 63

Table 7 The cost of existing policies and the relative savings of the optimal policies under various scenarios 64

Table 8 True optimal and optimal policies for each demand scenario 66

v

List of Figures

Figure 1 A typical sample path of the process with a 4-day review period

and a 2-day lead time 24

Figure 2 The flow chart of the grid search algorithm 38

Figure 3 The flow chart of the grid search algorithm utilizing the updating

technique for the transition probability matrix 44

Figure 4 An evaluation of the lower bound Sm i n 47

Figure 5 The flow chart of the heuristic algorithm 50

Figure 6 The distribution of savings of optimal policies in (a) dollars and

(b) percentage of current cost 59

Figure A-1 The title screen 97

Figure A-2 The main menu 98

Figure A-3 Calculating the optimal policy and evaluating the current policy 99

Figure A-4 Displaying the results 100

Figure A-5 Entering an alternate policy 101

Figure A-6 The particulars of the alternate policy 102

Figure A-7 The main menu revisited 103

vi

Acknowledgement

The completion of this thesis was made possible by the encouragement and assistance

of a number of people

I would like to express my sincere appreciation to my thesis supervisor Professor

Martin Puterman for all of his many efforts on my behalf His help advice patience charity

and tolerance were very much appreciated

I would like to acknowledge Professor Hong Chen and Professor Garland Chow for their

time and input as members of my thesis committee In addition I must acknowledge the

assistance of PhD student Kaan Katircioglu for his insight and help on this project

I offer many thanks to Professor Tom Ross for his kindness and friendship during the

past few years especially during the writing of this work I must also thank his family for

adopting me on many holidays I still owe thanks to Professor Slobodan Simonovic at the

University of Manitoba for influencing me to attend graduate school in the first place and for

helping me to obtain funding

I cannot thank my parents enough for their never-ending support both emotional and

financial Also to my friends especially Cathy Dave Lisa Steve and the Philbuds thanks

for giving me a life off campus and for picking up many a tab - the next one is on me

I would like to give special thanks to my good friend and fellow MSc student Paul

Crookshanks for allowing me to bounce ideas off of him and for being such a procrastinator

that despite my finishing a year late I only lost by two days

Financial assistance received from the Natural Science and Engineering Research

Council (NSERC) and from Canadian Tire Pacific Associates was greatly appreciated

vii

I INTRODUCTION

The importance of inventory management has grown significantly over the years

especially since the turn of this century In colonial times large inventories were viewed as

signs of wealth and therefore merchants and policy makers were not overly concerned with

controlling inventory However during the economic collapse of the 1930s managers began

to perceive the risks associated with holding large inventories As a result managers

emphasized rapid rates of inventory turnover (Silver and Peterson 1985) Following the

Second World War Arrow Harris and Marschak (1951) and Dvoretzky Kiefer and

Wolfowitz (1952ab) laid the basis for future developments in mathematical inventory theory

Shortly thereafter new inventory control methodologies were widely applied in the private

manufacturing sector More recently when inflation and interest rates soared during the 1970s

many organizations were forced to rethink their inventory strategies yet again Today the

control of inventory is a problem common to all organizations in any sector of the economy

Sir Graham Day chairman of Britains Cadbury-Schweppes P C L expressed this sentiment

when he stated I believe the easiest money any business having any inventory can save lies

with the minimization of that inventory (Day 1992) Lee and Nahmias (1993) and Silver and

Peterson (1985) provide a more detailed history of inventory management and control

Most inventory problems in the retail sector involve the control and management of a

large number of different products Ideally the inventory policies should be determined for all

products on a system-wide basis however because the number of products is often in the tens

of thousands formulating a model to do so is impractical Therefore single-product models

are frequently used in practice to determine policies for products separately (Lee and Nahmias

1993) Lee and Nahmias (1993) and Silver and Peterson (1985) provide a good overview of

1

single-product single-location models for a more complete review of the literature we refer

to them

The degree of complexity of a single-product model depends largely on the assumptions

made regarding the demand cost structure and physical characteristics of the inventory system

For example demand can be either deterministic meaning it is known with absolute certainty

or stochastic meaning it is unknown but follows a known probability distribution

Deterministic models tend to be relatively simple yet useful such as the basic EOQ model

However future demand is seldom known with certainty in practice and therefore the use of

these simple models is limited As a result the majority of research in inventory has focused

on stochastic demand models (Lee and Nahmias 1993)

Because the objective of most inventory models is to minimize the cost of inventory

assumptions regarding the structure of the cost are very important also Generally speaking

the cost of inventory has four components the purchaseproduction cost the orderingsetup

cost the holding cost and the shortage cost1 Each of these costs can be fixed proportional

nonlinear or some complex combination of these Further because nearly all inventory

problems require decisions be made sequentially over a period of time the costs can be either

discounted or averaged In the former case a discount factor is included in the cost function

in order to account for the time value of money However since most inventory problems are

on-going the long-run average cost is usually minimized (Lee and Nahmias 1993) In

practice this is achieved by minimizing the expected cost per time period under steady state

conditions

Finally the physical characteristics of the system are important in determining the

1 The cost of a stockout is often very difficult to estimate in practice As a result service levels are frequently used in its place When this is done a constraint is included in the model

2

complexity of the model For example the review process can be either continuous in which

case the inventory is reviewed continuously and orders are placed as soon as the stock reaches

a certain level or periodic in which case the inventory is reviewed at regular intervals and

orders can be placed only at those fixed times Also excess demand can be fully backordered

as is common in many manufacturing and wholesale systems completely lost as is common

in most retail environments or some combination of the two Lead time assumptions are also

important models with lead times are more complex than those in which the supply of

inventory is assumed to be instantaneous

Although the literature is dominated by single-product models (Lee and Nahmias 1993)

much recent work has focused on extending those models to multiple-product problems

However multiple-product models are inherently more complex than the simple-product

models In fact no efficient algorithms are currently available for determining fully optimal

strategies for most multiple-product problems (Federgruen 1993) As a result research has

focused on the development of easily computable approximation techniques Muckstadt and

Roundy (1993) provide a good overview of multiple-product andor multiple-location models

with deterministic demand although emphasis is placed on models for production and

distribution Federgruen (1993) extends the discussion to consider these models under uncertain

demand and also provides a complete list of references on the subject Silver and Peterson

(1985) discuss models for coordinated replenishments for both deterministic and stochastic

demand

In this thesis we examine the inventory system at Canadian Tire Pacific Associates

owner of 21 retail stores and a central warehouse Although this is a multiple-product

multiple-location single-warehouse problem and strictly speaking should be modelled as such

we analyze the inventory problem on a product-by-product and store-by-store basis We believe

3

that this simpler single-product single-location model captures the essentials of the problem for

two reasons First the coordination of replenishments for products andor locations is most

advantageous when significant savings are possible because of high first order costs and low

marginal costs for placing subsequent orders However because of the large number of

products and the physical distribution system in this problem trucks are dispatched to stores

on a regular basis Therefore the benefits of coordinating replenishments for this problem are

negligible Second we believe that controlling in-store inventory alone can result in substantial

savings to Canadian Tire Pacific Associates Further given the excessive amount of inventory

in the warehouse a single-location model assuming an infinite warehouse capacity is reasonable

Of course should inventory control in the warehouse become a priority a more elaborate model

would be required

More specifically we consider a periodic review single-product inventory model with

stationary independent stochastic demand a deterministic lead time and an infinite planning

horizon The probability mass function for the demand is based on empirical sales data

collected over a period of approximately ten months The cost of inventory includes a

proportional holding cost and a fixed ordering cost neither of which is discounted All excess

demand is completely lost The objective of the model is to minimize the long-run average cost

per unit time subject to a service level constraint on the fraction of demand satisfied directly

from in-store inventory In order to obtain meaningful cost comparisons the model calculates

the expected annual cost of inventory

For inventory problems like the one described above optimal (sS) policies can be

computed by Markovian decision process methods such as successive approximations or policy

iteration or by stationary Markovian methods However each of these methods generate

calculations that are complex and time consuming Therefore much research interest lies in

4

developing techniques to find approximate (sS) policies with little computational effort The

majority of the approximation techniques begin with an exact formulation of the problem and

then manipulate the problem by relaxations restrictions projections or cost approximations

(Federgruen 1993) Porteus (1985) presents three approximation techniques and compares

them to fourteen others for a periodic review single-product inventory model in which

shortages are fully backordered and a shortage cost is incurred Federgruen Groenevelt and

Tijms (1984) present an algorithm for finding approximate (sS) policies for a continuous review

model with backordering and a service level constraint Also Tijms and Groenevelt (1984)

present simple approximations for periodic review systems with lost sales and service level

constraints All three works find that the approximation techniques perform well However

the techniques are evaluated for a variety of demand parameters that all result in policies with

a value of S in the range of 50 to 200 Given the slower moving products at Canadian Tire 2

it is unclear how well such approximation techniques would perform Therefore we do not

attempt to apply or modify any of these approximation methods

Exact optimal policies for this problem can be obtained by formulating a constrained

Markov decision process and solving it However the data generating requirements and the

lack of a good upper bound on S make this formulation impractical Therefore we formulate

a Markov chain model for fixed values of s and S and then employ a search technique to find

the optimal values for s and S In order to improve the performance of the model we develop

a heuristic search that locates near optimal policies relatively quickly

The remaining chapters of this thesis are organized as follows In Chapter II an

overview of the existing inventory system at Canadian Tire is presented along with a description

2 For example very few of the products analyzed have optimal values of S exceeding 20 with the largest value of S equalling 69

5

of the problem Next a multi-phase project designed to develop a fully integrated interactive

inventory control system is described with the scope of the research reported in this thesis

completing the first phase of the project Finally a description of the demand and cost data is

given

In Chapter III we formulate a Markov chain model that can be used to calculate the cost

and service level associated with a given (sS) policy Based on several simplifying

assumptions the Markov chain is formulated and the equations for the corresponding transition

probability matrix are developed Then given the limiting probability distribution of the chain

the equations for the steady state annual cost of inventory and the corresponding service level

are obtained The chapter concludes with a brief summary of the model

In Chapter IV an algorithm that searches for the optimal (sS) policy is developed As

a starting point a simple grid search algorithm is developed and evaluated on the basis of

execution time In order to improve the algorithms speed a technique for updating the

transition probability matrix is presented and the improvement in the algorithm is noted Next

a theoretical lower bound on S is obtained although the corresponding reduction in the

execution time is minimal Finally based on two assumptions regarding the shape of the

feasible space a heuristic search is developed which improves the speed of the algorithm

significantly

In Chapter V a prototype of the inventory control systems interface is created In its

present form the interface allows managers to evaluate inventory policies interactively The

manager can set review parameters compare optimal policies and current policies and evaluate

alternate policies The chapter concludes by recommending specific options be including in

future versions to make the system more automated and user-friendly

In Chapter VI optimal policies for all products in one product category sold at one store

6

are compared to existing policies on the basis of expected annual costs and customer service

levels The analysis suggests that existing policies result in a large excess inventory and that

implementing the optimal policies would reduce the annual cost of inventory by approximately

50 A sensitivity analysis on the cost estimates is performed and suggests that the model is

not sensitive to relative errors of approximately 20 in the cost estimates Finally a sensitivity

analysis on the demand probability mass function is performed The analysis indicates that the

model is sensitive at least in terms of the customer service level to slight changes in the tail

of the demand distributions of low-demand products

Finally in Chapter VII the results of the analysis are summarized and future work is

outlined In order to incorporate recent changes in the review procedures at several stores the

model must be modified to allow for the duration of the review period to exceed the lead time

We recommend that the second phase of the project proceed specifically that a data collection

and storage scheme be designed and that a forecasting model be developed Because of the

substantial savings that can be realized by reducing inventory only at the stores inventory

control at the warehouse should also be investigated

7

II T H E I N V E N T O R Y S Y S T E M A T C A N A D I A N TIRE

A Background

Canadian Tire Pacific Associates owns and operates 21 retail stores in the lower

mainland of British Columbia The stores carry approximately 30000 active products3 that

are divided into 80 product categories The merchandise comprises primarily automotive parts

and accessories home hardware housewares and sporting goods Canadian Tire Pacific

Associates also owns a central warehouse in Burnaby at which inventory is received from

Toronto Inventory is then shipped periodically from the warehouse via truck to the local

stores In 1992 total sales at the stores exceeded $99 million and the value of inventory was

estimated to be $28 million of which approximately half was stored in the warehouse

On selected days of each week a stores inventory records are reviewed automatically

and orders for products are placed to the warehouse as required Management uses the term

flushing to describe the process of reviewing the inventory and producing the tickets for stock

owing and the term picking to describe the physical act of filling the order at the warehouse4

Whether or not an order is placed for a product depends upon its current inventory level its

target level and its minimum fill 5 If the target level less the current level exceeds the

minimum fill an order is placed to bring the stock up to the target level From the moment

3 Active products are those that are regularly restocked Canadian Tire Pacific Associates can choose from up to 50000 products

4 Note that the number of picks per week is equal to the number of times per week that the inventory records are reviewed At the beginning of this study eighteen stores had two picks per week while the remaining three stores had only one

5 The target level is the maximum number of items that should be on the shelf at one time and the minimum fill is the minimum number of items that should be ordered

8

the order is placed to the moment the product is reshelved takes between two to four days

depending on the store On average 11000 items are shipped from the warehouse at each

flush

Although a products minimum fill is the same for all stores its target level depends

upon the size of the store Based upon its sales volume and retail space a store is classified

as being one of five sizes small (S) medium (M) large (L) jumbo (J) and extra jumbo

(EJ) 6 Each product has associated with it a target level for each size of store

Consequently all stores of the same size have identical inventory policies Table 1 lists the

location size number of picks per week sales and floor space of each store as of October

1993

Periodically the inventory manager reviews the policies and when deemed necessary

adjusts them In this way the manager can account for seasonality in demand changes in mean

demand and any shortcomings of previous policies At the present time the review process

is not automated nor performed on-line The inventory manager uses Hxl5 computer

printouts with five products to a sheet to study the relevant information and on which to note

any changes Afterward the changes are recorded in the system and the new policies take

effect

Currently the inventory policies are based on rules of thumb developed through the

years and not on a statistical analysis of demand nor on a calculation of the cost of inventory

These rules of thumb incorporate a marketing philosophy which at times dominates the

inventory strategy The prevailing philosophy holds that a large in-store inventory instills

6 During the course of this study stores were resized several times At one point the entire class of medium stores was eliminated leaving only four sizes of stores On another occasion the store in Chilliwack was changed from being a small store to a large store

9

Table 1 Location and particulars of the 21 stores

Store Location Store Size

Picks Per Week

1992 Sales ($)

Floor Space (ft2)

1 Chilliwack S 2 2901969 6678

2 Coquitlam M 2 4015303 6510

3 Langley EJ 2 8138673 15422

4 Scott Road EJ 2 7307820 15064

5 Maple Ridge L 2 4102110 6678

6 Kingsway L 2 4153736 6574

7 Richmond EJ 2 8175105 14961

8 North Vancouver M 2 3856571 6534

9 Guildford M 2 3714372 6567

10 Burnaby M 2 3532752 6983

11 New Westminster S 1 2417992 6470

12 Abbotsford EJ 2 7899398 15303

13 Hastings J 2 4647794 9849

14 King Edward EJ 2 3825372 25492

15 White Rock L 2 3960262 6870

16 Main amp Mountain S 1 2705602 6431

17 Heather J 2 4161227 9427

18 King George J 2 4606817 9640

19 Cambie J 2 5855022 9091

20 Lougheed M 1 3114180 6615

21 Burrard J 2 6553310 10608

10

consumer confidence which in turn leads to increased sales In other words if a customer

perceives that Canadian Tire is in the hardware business he or she will choose to shop there

when meeting future needs However management has adopted this philosophy without any

analysis of the cost of holding excess inventory Only recently has management begun to

question the cost-effectiveness of allocating inventory based on these rules of thumb

Speculating that existing inventory practices might be deficient Mr Don Graham the

vice-president of Canadian Tire Pacific Associates commissioned a pilot study to examine the

inventory system and to propose alternate strategies7 The study using a steady state model

to represent a simplified inventory system suggested that existing policies lead to overly high

inventory levels More importantly the study outlined a methodology that would enable

management to determine inventory levels based on observed demand The pilot study

recommended that a more detailed model of the inventory system be formulated that the

methodology be expanded to incorporate the cost of inventory and that an algorithm be

developed that would quickly identify the optimal inventory policies

B The Problem

At the time of the pilot study management was concerned primarily with its inability

to distinguish between stores of the same size As mentioned in the previous section a

stores size is based solely on its sales volume and floor space and under existing practices

all stores of the same size have identical inventory policies for every product Management

realized that demand for some products might be a function of variables other than size such

7 The pilot study was performed by Martin Puterman a professor of Management Science and Kaan Katircioglu a doctoral student in Management Science both at the University of British Columbia The results of the study were presented at a U B C Transportation and Logistics Workshop on March 25 1993

11

as location For example the demand for sump pumps is likely to be very different in suburban

areas than in urban ones However management was unable to allocate more sump pumps to

large suburban stores than to large urban stores This inability led management to seek a

tool that would allow for a more effective allocation of inventory between stores of the same

size

By the end of the pilot study however management was less concerned with

reallocating inventory within a size of store than with reallocating inventory between sizes

of stores and with reducing overall inventory According to management assigning individual

inventory policies to each store appeared to be operationally infeasible Firstly management

believed that having individual policies would lead to confusion in the warehouse and to

increased shipping errors Exactly why this would occur is unclear the warehouse would

simply continue to fill orders from tickets that make no reference to the policy in place

Secondly and perhaps more tenably the computer technology at Canadian Tire Pacific

Associates did not allow for policies to vary across all stores Both the database used to store

the inventory data and the printouts used by the inventory manager allowed for only five

different policies per product all with the same minimum fill Moving to individual policies

would have required an overhaul of the existing information system Although management

saw this overhaul as inevitable it did not see the overhaul as imminent Therefore

management switched the focus of the study to developing a tool that would help reduce overall

inventory levels andor reallocate inventory more effectively between groups of stores

The subtle change in the studys objective prompted an important question how should

the stores be grouped The current practice of grouping stores based on size is problematic

for two reasons First as discussed above additional factors of demand such as location

cannot be taken into account Second grouping inherently results in a misallocation of

12

inventory when some stores within a group have differing numbers of picks per week For

example the medium-sized Lougheed store which has one pick per week requires higher

inventory levels to achieve a given customer service level than the other medium stores

which all have two picks per week8 In fact in terms of customer service levels it might

be more appropriate to group the Lougheed store with the large stores having two picks per

week Ideally this artificial grouping of stores will be abandoned when the computer system

is upgraded and policies can then be set on a store by store basis

As the project proceeded management grew somewhat uneasy with the concept of using

an analytical tool to evaluate and set inventory policies According to both the pilot study and

the preliminary results of this study significantly reducing inventory would greatly reduce costs

while still providing a high level of customer service However management held that because

the model cannot take into account intangibles such as marketing considerations such a drastic

reduction in inventory would be very inappropriate In this light management viewed the

optimal policies as lower bounds for inventory levels these policies would have to be

modified upwards in order to be practical Therefore management contended that any

methodology or software produced during this study should provide the means to evaluate and

compare a number of alternate policies

Regardless of exactly how the tool is to be used there is currently a lack of historical

sales data from which to analyze the inventory system and determine optimal policies No sales

data had ever been collected prior to the pilot study and only a relatively small amount has

been collected since In order to proceed with any analysis of the inventory system a

8 If demand is a function of size then the Lougheed store must meet the same demand with one fewer pick per week This would lead to either increased probabilities of stockouts at the Lougheed store or higher inventory levels at all the other medium stores

13

substantial amount of additional data would be needed The existing information system at

Canadian Tire Pacific Associates does allow for sales data to be captured however the system

is limited by the amount of data it can capture at one time and by its storage capacity In order

to collect the amount of data required management would have to upgrade the system and

develop some kind of data collection scheme coupled with a data compression technique The

collection scheme must allow for characteristics of demand such as trends and seasonality to

be identified and the compression technique must allow for sales data for over 30000 products

to be stored for each store

Because many of the products sold by Canadian Tire have nonstationary demand as a

result of seasonality andor trend the inventory tool must be able to forecast demand for the

coming period Products such as sporting goods and outdoor equipment exhibit patterns of

demand that are very seasonal For example sales of snow shovels increase through the fall

peak in the winter and drop to zero by late spring At the same time some products might

become either more or less fashionable over time resulting in changes in mean demand

Setting inventory policies for any of these products based on observed demand would result in

insufficient stock if demand increased during the period or in excess stock if demand decreased

Therefore the tool must be able to forecast the demand for the period for which the policy is

intended

Toward the end of the project management began discussing the integration of the

interactive inventory tool with the existing system to form a new inventory control system The

new system would utilize a modified point of sales system to collect sales data continually

Based on the latest available data the system would forecast demand for the coming period and

calculate the optimal policies for every product at every store The system would then provide

all relevant information on-line for the inventory manager to review The manager could weigh

14

other information such as marketing considerations and examine alternate policies to determine

the effect on cost Finally the preferred policy would be set in place with the touch of a key

and information on the next product or store would be presented for review

C The Project

Development of the inventory control system envisioned by management requires the

completion of five tasks 1) the design of a data collection system that generates and stores the

necessary sales data 2) the development of a forecasting method that provides accurate

estimates of demand distributions 3) the formulation of a model that calculates the cost of

inventory and the customer service level for an inventory policy 4) the development of an

algorithm that finds the optimal policy for a product at a store and 5) the creation of an

interface that allows interactive and on-line consultation Ultimately the above components

would have to be integrated and incorporated in a manner that would ensure the compatibility

and reliability of the system

The scope of the research reported in this thesis completes the first phase of the project

specifically we formulate an inventory model develop an algorithm and create a prototype of

the interface Because of the changing environment at Canadian Tire Pacific Associates minor

modifications to both the model and the algorithm developed herein might be necessary to

reflect future changes in operational procedures Upon completion of phase one the interface

will be interactive allowing managers to select the store product and review scenario and then

to compare optimal policies with existing policies or with alternate policies Although the

prototype will be delivered to Canadian Tire Pacific Associates for ih-house testing at the end

of this phase the interface will require modification in subsequent phases as feedback from

management is received and as the system is brought on-line

15

In order to assess more fully the potential savings of implementing the inventory control

system this thesis includes a comparison of optimal policies9 and existing policies The

expected annual cost of inventory for an entire product category is calculated for both the

optimal policies and the existing policies Based on an analysis of these costs the potential

savings of implementing the optimal policies are extrapolated Further the thesis examines the

sensitivity of the model to variations in the estimates of demand and costs

Originally results from all 21 stores were to be included in the above analysis

However just prior to the analysis management changed the review procedure at several

stores the result of which being that three picks per week are now performed at those stores

Because one of the main assumptions of the model specifically that the length of the lead time

does not exceed the duration of the review period is violated under this new review procedure

those stores cannot be included in the analysis Therefore the analysis is based solely on the

results from store six the large store located on Kingsway Avenue10

For the purpose of the analysis sales data for product category 20 automotive and

electrical accessories were collected from all 21 stores This product category was selected

because the distribution of demand for products within it is non-seasonal and because few of

the products are ever placed on sale11 The lack of trend or seasonality eliminates the need

for forecasting demand the optimal policies can be determined based directly on observed

demand

9 That is those policies obtained from the model

1 0 Store six is one of the only stores whose review procedure is not expected to change from two picks per week

1 1 Products that were on sale at any time during the period were not included in the data set

16

D The Demand Data

In total sales data for 520 products of which 458 are active products were collected

over a 307-day period from October 10 1992 to August 18 199312 The data was supplied

by Canadian Tire Pacific Associates in the form of a 10 MB text file with one line per day per

product recording sales from the stores13 Each line had the following format

Columns 1-6 part number Columns 7-9 Julian date14

Columns 10-18 cost of the item (to three decimal places) Columns 19-124 sales for stores 1 through 21 (5 columns per store)

In order to facilitate the analysis the file was sorted into 458 sales files with each of

these files containing the history of daily sales for one of the products Two changes were

made to the format of these files to improve readability 1) a column was added between the

product number and the Julian date and 2) all leading zeros were dropped Appendix A

contains a partial printout of the sales file for a typical product Finally a distribution file

summarizing the daily sales was created from each of the sales files Each distribution file

contains the products number the products cost and the number of days during which various

quantities of the product were sold at each store

Prior to the analysis the distribution files for a large number of products were

examined and based on the shapes of the histograms the products were divided into three

groups The first group low-demand comprises products that have demand on only a few

days with the daily demand seldom exceeding three items The second group medium-

1 2 Of the active products only 420 are sold at store 6

1 3 Strictly speaking a line was written to the file only on days on which at least one store registered a sale of the product

1 4 According to the Julian calendar days are numbered sequentially from 1 to 365 (366 during leap-years) beginning with the first of January

17

demand comprises products that have demand on approximately half of the days with the daily

demand seldom exceeding more than six items Finally the last group high-demand

comprises products that have demand on most days occasionally upwards of twenty items and

whose demand histograms resemble Poisson distributions The majority of the products

examined fell into the low and medium-demand groups only a few high-demand products

were observed Appendix B contains a printout of the distribution file for a typical product

in each group

For most products no theoretical distribution appeared to model the demand well

Demand for low and medium-demand products followed a non-standard distribution with

a mass at zero and a relatively small tail Although some theoretical distributions such as the

mixed-Poisson can be made to resemble this it was unclear how well they would perform

Further the effects of sampling errors from such non-standard distributions were unknown

On the other hand the standard Poisson distribution did appear to model high-demand

products well however only a few products fell into that group Therefore in the absence of

a good theoretical candidate the empirical demand function was chosen for the analysis The

empirical function was readily available from the distribution files and offered the best

estimate of the true underlying function

E The Cost Data

Along with the sales data management supplied estimates relating to the costs of holding

and ordering inventory Management estimates that the cost of holding an item in inventory

for one year is 30 of the items cost Incorporated in this holding rate are a number of costs

such as the opportunity cost of money invested and the costs incurred in running a store

Because it is the annual cost of inventory that is of interest either the compounded daily

18

holding rate or the simple daily holding rate can be used to determine the costs incurred during

a review period however the method of calculating and extrapolating the holding costs must

be consistent For the sake of simplicity a simple daily holding rate is used

In order to estimate the cost of ordering inventory management analyzed the various

tasks that result from an order being placed Using the average time required to fill and then

reshelve an order management obtained an estimate of the direct labour costs involved To

this management added an estimate of the cost of transporting an order from the warehouse to

the store Finally the estimate was adjusted slightly upward to account for the cost of breakage

and errors Based on the analysis management estimates the cost of placing an order for a

product to be $0085 regardless of the size of the order

19

III M O D E L F O R M U L A T I O N

A Terminology

In the previous chapter several terms were introduced to describe the inventory system

at Canadian Tire Pacific Associates These terms though perfectly adequate are not widely

used in the academic literature to describe inventory management problems Therefore in

order to avoid confusion and be consistent with the literature the following terms are used

throughout this thesis

The length of time between successive moments at which the

inventory position is reviewed15

The time at which a reorder decision is made

The length of time between the moment at which a decision to

place an order is made and the moment at which the order is

physically on the shelf

An operating policy in a periodic review inventory model in

which (i) the inventory position is reviewed every T days (ii) a

replenishment order is placed at a review only when the inventory

level x is at or below the reorder level s (gt 0) and (iii) a

replenishment order is placed for S-x units (S gt s)1 6

Lost Sales - The situation in which any demand during a stockout is lost

Review Period -

Decision Epoch

Lead Time -

(sS) Policy

1 5 With respect to the terminology used at Canadian Tire Pacific Associates a review period equals the number of days between flushes

1 6 With respect to the terminology used at Canadian Tire Pacific Associates S represents the target level and s represents the difference between the target level and the minimum fill

20

B Assumptions

In to order make the model of the inventory system mathematically manageable several

simplifying assumptions are made However we believe that the following assumptions do not

significantly reduce the accuracy of the model

1) Demand can be approximated very closely by sales during the period in which the data

is collected Generally inventory levels at the stores are very high and the number of

lost sales is assumed to be negligible In fact the inventory levels for a number of

products were tracked over time during the pilot study and not a single incidence of

stockout was observed Because all products appear to have similar excess inventory

it seems reasonable to assume that virtually all demand resulted in sales

2) The demands in successive periods are independent identically distributed random

variables Preliminary tests (Fishers exact tests) suggested no dependence between

sales on consecutive days and plots of sales over time indicated no seasonal pattern

3) Demand is independent across products That is we assume that sales of one product

do not affect nor can they be predicted from sales of another product

4) Unmet demand results in lost sales with no penalty other than the loss of the sale

5) The duration of the review period for a store is an integer value and is fixed over a

planning period For stores with two picks per week we assume that the inventory

is reviewed every four days rather than on a four day - three day cycle

6) The decision epoch is the start of a review period Review periods begin at midnight

at which time the inventory is reviewed and orders are placed instantaneously

7) The duration of the lead time for a store is deterministic and integer-valued In reality

the lead time for some stores was stochastic but management felt the variation could be

controlled and virtually eliminated

21

8) The lead time does not exceed the review period In other words items ordered at a

decision epoch arrive and are shelved before the next decision epoch In practice this

did not always occur

9) Orders arrive at the end of the lead time and are on the shelf to start the next day

10) Holding costs are incurred daily for all inventory held at the start of the day

11) The warehouse has an infinite capacity that is all orders to the warehouse are filled

Because of the high inventory levels in the warehouse stockouts in the warehouse are

very unlikely

C A Markov Chain Model

Consider the situation in which an (sS) policy is applied to an inventory system based

on the above assumptions Inventory is reviewed at the decision epoch for each of the periods

labelled t = 0 12 and replenishment occurs at the end of the lead time provided an order

was placed Demand during day d in period t is a random variable whose probability mass

function is independent of the day and period The function is given by

f^ik) = Pr ltd = k = ak for k = 0 1 2

where ak gt 0 and E ak = 1 Let T denote the duration of a review period and L denote the

length of the lead time Next define D T ( as the total demand during period t D L as the total

demand during the lead time in period t and D T L as the difference between D T and DLt

Assuming that successive demands pound f pound 2 are independent the probability mass function of

1 7 The aggregated demand variables are given by

^ = XXd and DT_Ut=J d= d= d=L+l

22

each aggregated demand variable can be obtained by convolving the probability mass function

of t d l s Finally let X denote the inventory on hand at the beginning of period t just prior to

the review

The process X is a discrete time stochastic process with the finite discrete state space

0 1 S A typical sample path of the process with a review period of four days and a

lead time of two days is depicted in Figure 1 As period t begins X units are on hand The

inventory is reviewed and because X is less than s an order is placed for (S-X) units Over

the next two days a demand of D L r occurs and the inventory drops accordingly With the

arrival of the order the inventory level increases by (S-X) at the start of the third day of the

period Over the remaining two days a demand of D T L is observed bring the inventory level

down to X + 1 Note that in this example an order would not be placed in period t+l because

X r + i exceeds s

The stock levels at the beginning of two consecutive review periods are related by the

balance equation

_K- DLy+ (s - v - DT-LX ixtlts ( 1 )

t + l ~ (Xt - DTtY ifXtgts

where the positive operator is interpreted as

= max a 0

1 8 For example the probability mass function of D T bdquo denoted fD T ( is the T-th fold convolution of f t Mathematically the function can be obtained using the recursive relation

and the identity f1^ = f5(

23

-2 I 8

gt o

r

CO

o a

CM

X

00

24

Because X + 1 depends only on Xt the process X satisfies the Markov property

PrXt+l=jX0 = i0Xt_l=it_vXt = i) = PrXt+1=jXt = i

and therefore is a discrete time Markov chain Further because the probability mass function

of the random variable pound is independent of t and thus so too are the probability mass functions

of the aggregated demand the Markov chain is stationary19

D The Transition Probability Matrix

The stationary transition probability p(gt is defined as the probability of X + 1 being in state

j given that X is in state i independent of the period t In the inventory model p-- represents

the probability of starting the next review period with j items on the shelf given there are i units

on hand at the start of the current period By convention the transition probabilities are

arranged in an (S + l)x(S + 1) matrix P such that (P)y = p- Thus the (i + l)st row of P contains

the probability distribution for the stock on hand at the beginning of the next period given that

there are i units on hand at the start of the current period The matrix P is known as the

transition probability matrix Note that since we are concerned with a steady state analysis we

need not consider the possibility of initial stock levels greater than S once the inventory level

is below S it can never again exceed S

The transition probability ptj is calculated by expanding (1) to include all possible

instances of demand that allow for a change in inventory from i units at the beginning of a

period to j units at the beginning of the next period For example suppose the inventory policy

(510) is in place 2 items are on hand at the start of the current review period and 7 items are

1 9 Henceforth the subscripts t and d are omitted from the daily and aggregate demand variables for the sake of brevity

25

on hand at the start of the next period Then only three scenarios are possible

1) No items are sold during the lead time (meaning that the stock level increases to 10

after replenishment) and 3 items are sold during the remainder of the period

2) One item is sold during the lead time and 2 items are sold during the remainder of

the period

3) Two items are sold during the lead time (though demand could exceed 2) and 1 item

is sold during the remainder of the period

The transition probability p 2 7 is the sum of the probabilities of the various scenarios and is

computed by

p2J = PrDL=0PrDT_L = 3] + PrDL = l] Pr[DT_L=2 + PrDL2] PrDT_L = l]

In general the transition probabilities for the inventory model are calculated as follows

Case 1 The lead time is less than the review period

For 0 lt lt s

Pij

i-l pound [PrDL=kPrDT_LgtS-k] + PrDLgtiPrDT_LgtS-i if j = 0

i - l

^[PriD^^PrlD^S-j-kn+PHD^PriD^S-j-i if 0 lt j ltL S-i (2) k=0 s-j

E [PrDL=kPrDT_L = S-j-k] if S-i ltj^S k=0

26

For s lt i lt S

PrDTzi]

0 p = PrDT=i-j)

ifj = 0 ifOltjzi ifiltjltS

(3)

Case 2 The lead time equals the review period

For 0 lt i lt s

Pa

0 PrDTzi) PrDT = S-j

ifOltLJlt S-i ifj = S-i if S-i ltj plusmn S

(4)

For s lt i lt S use (3)

E The Steady State Probabilities

Provided 1) there is a positive probability of non-zero demand during a review period

and 2) there is a positive probability of zero demand during a review period20 then the

Markov chain described by (1) has a limiting probability distribution ir = (ir0 TTu irs)

where 0 lt IT lt 1 for j = 0 1 S and E 7ry = 1 Further this limiting distribution is

independent of the initial state In other words the probability of finding the Markov chain in

a particular state in the long-run converges to a nonnegative number no matter in which state

the chain began at time 0 The limiting distribution is the unique nonnegative solution of the

equations

2 0 That is 0 lt PrD x = 0 lt 1 Given the nature of the problem this assumption is not very restrictive

27

^ = E ^ - far j = 0 l S (5)

s and pound nk = 1 (6)

For the proof that the limiting distribution is independent of the initial state and is obtained from

(5) and (6) the reader is directed to Appendix C

In terms of the inventory problem 7ry is the steady state probability of having j units of

inventory on hand at the decision epoch However some subsequent calculations are made

easier if the steady state probability of having j units of inventory on the shelf just after

replenishment is known Instead of reformulating the Markov chain these probabilities can be

obtained by shifting the limiting distribution TT Much in the same way that the transition

probabilities are calculated the shifted limiting distribution t] is obtained by accounting for

all possible instances of demand that allow for the shifted inventory position For example

suppose that the policy (510) has the limiting distribution TT In order to have say 8 units on

hand immediately after replenishment one of the following must occur

1) Ten items are on hand at the decision epoch and 2 items are sold during the lead time

2) Nine items are on hand at the decision epoch and 1 item is sold during the lead time

3) Eight items are on hand at the decision epoch and nothing is sold during the lead time

4) Either 2 3 4 or 5 items are on hand at the decision epoch and 2 items are sold during

the lead time

The steady state probability of having 8 units on hand just after replenishment is calculated as

n 8 = n10PrDL = 2] +iz9PrDL = l] +n8PrDL = 0 +[iz5 + iz4 + iz3]PrDL = 2 +n2PrDLgt2

28

In general the shifted steady state probabilities are given by

s pound nPrDLgti ifj = 0

i=s+l S

pound PrDL = i-j] ifl plusmnjltS-s i=max(s+l)

S = pound 7tjirZ)L = i-+7i 5 FrD i ^5 ifj = 5-5

i=max(s+l)

^ 7 1 ^ ^ = 1-7+ pound TtiFrZgti=5-y+iis_PrD1^S-7 ifS-sltjltS i=max(gt+l) i=S-j+l

(7)

F The Cost of Inventory

For this problem the cost of inventory has two components (i) a fixed ordering cost

cf incurred whenever an order is placed and (ii) a holding cost cv incurred daily for each item

held in inventory The expected cost of inventory for a review period can be calculated by

summing the costs associated with each inventory position under steady state conditions

Because a simple holding rate is used this cost is simply adjusted by number of review periods

in a year to obtain the expected annual cost of inventory

The ordering cost for a review period is relatively straight forward to calculate either

an order is placed in which case the order cost is incurred or no order is placed in which case

no cost is incurred According to the rules of the inventory policy orders are placed only if

the stock on hand at the beginning of a review period is less than or equal to s Further the

probability that the review period begins with the inventory in one of these positions is equal

to the sum of the first (s + 1) steady state probabilities Therefore the expected ordering cost

29

for a review period is given by

E[OC] = cfY n y=o

Unlike the simple structure of the ordering cost the holding cost is incurred daily for

each item held in inventory Therefore the daily holding costs associated with all possible

inventory positions must be weighted and summed to obtain the holding cost for the review

period In order to make the calculations less complicated a shifted review period beginning

immediately after replenishment and ending just prior to the next replenishment is considered

Define e(0 to be the expected number of items on hand to start day i of the shifted

review period and let g(i f) denote the expected number of items on hand to start day i given

that there were j items immediately following replenishment21 The function g(z j) is the sum

of the probabilities of possible demand weighted by the resulting stock levels that is

g(ij) = j^kPriD^j-k)

To obtain the expected number of items on hand to start day i independent of j the above

function is summed over all possible values of j and weighted by the corresponding steady state

probabilities yielding

s j ed) = pound r 5 gt P r Z ) = - pound

7=0 k=

2 1 In other words g(if) is the expected number of items on hand to start day i given e(0)=v

30

The expected holding cost for a review period is then given by

T

E[HC] = $gt v e(0 i=i

i=l =0 k=l

where c v equals the simple daily holding rate multiplied by the cost of the item

Although differently defined review periods are used to calculate the above expected

costs the costs can still be summed to obtain the expected cost of inventory for a review

period The expected costs are calculated based on steady state conditions and therefore

represent the costs expected to be incurred during any T consecutive days Summing the above

two expected costs and multiplying the sum by the number of review periods in a year yields

the expected annual cost of inventory

E[AC] = mdash[E[OC]+E[HC]]

T S j 1 ( g ) 365

j=0 i=l j=0 fc=l

G The Customer Service Level

There are many ways to define customer service and the most appropriate definition

depends upon the nature of the business and the perspective of management Two commonly

used definitions are 1) the proportion of periods in which all demand is met and 2) the

proportion of demand satisfied immediately from inventory Although the former definition is

often used because of its simplicity the latter definition is closer to what one commonly means

by service (Lee and Nahmias 1993) A third definition also relevant in retail environments

31

is the probability that lost sales do not exceed some critical number Other definitions such

as probability that backorders are filled within a certain amount of time are more applicable

to industrial and manufacturing settings where backlogging demand is common

Given the retail environment with the possibility of lost sales that exists at Canadian Tire

Pacific Associates management is concerned primarily with the fraction of sales that are lost

For example management views the situation in which demand exceeds stock by one unit every

period more favourably than that in which the same demand exceeds stock by three units every

other period For this reason the proportion of demand satisfied immediately from inventory

is the most appropriate measure of customer service

The customer service level for a single review period CSLRP is given by

r bdquo T _ Demand satisfied during RP Total demand during RP

Because we are interested in the long-run behaviour of the system a measure of the steady state

customer service C S L must be obtained As shown in Appendix D the correct measure is

poundpoundpound _ E[Demand satisfied during RP] E[Total demand during RP]

which can be rewritten as

pound pound pound _ j _ E[Demand not satisfied during RP] ^ E[Total demand during RP]

Because of the steady state conditions the above customer service level can be estimated

from any period of T consecutive days As was done for one of the cost calculations the T-day

period beginning just after replenishment is considered Suppose that there are j units of

inventory on hand immediately following replenishment Then the conditional customer service

32

level is

CSL j = 1 E[(DT-j)l

E[DT] (10)

The denominator in the second term is merely the expect demand during the period

E[DT] = 7gt (11)

where p is the mean of the random daily demand variable pound The numerator which is the

conditional expected demand that is not satisfied during the T-day period expands to

7-1

E[(DT-j)+] = T u - j + YU-k)PrDT = k] k=0

(12)

The reader is directed to Appendix E for a mathematical justification of (12) In order to obtain

the unconditional expected demand not satisfied (12) is applied to all possible values of j and

a weighted sum of the terms is calculated The result is given by

E[Demand not satisfied] = Y j =0

y-i Ti - j + 1pound(j-k)PrDT = k

S j-1

= - E[I] + Y0-k)Pr[DT=k) =0 it=0

(13)

where E[I] denotes the steady state inventory level Finally the customer service level is

obtained by the substituting (11) and (13) into (9) yielding

CSL = 1

s j-l

Ti - E[I] + Jgty Y(J-k)PrDT=k]

m n J2VjTU-k)PrDT=k Eil J=Q fc=o

(14)

33

H A Methodology for Evaluating an (sS) Policy

In this section a brief methodology summarizing the main results of the model is

presented The methodology can be applied to evaluate a given (sS) policy in terms of its cost

and its resulting customer service level Given the probability mass function of the daily

demand f the duration of the review period T the length of the lead time L the fixed cost

of placing an order cf and the daily unit cost of holding inventory cv

1) Calculate the aggregated demand probability mass functions fDT fDL and f D x L as the T-th

fold L-th fold and (T-L)th fold convolutions of f respectively

2) Generate the transition probability matrix using either equation (2) or (4) and equation (3)

3) Calculate the limiting distribution TT from (5) and (6)

4) Obtain the L-day shifted limiting distribution rj using (7)

5) Calculate the expected annual cost of inventory using (8)

6) Determine the customer service level using (14)

34

IV A N A L G O R I T H M F O R OBTAINING T H E O P T I M A L (sS) P O L I C Y

A Introduction

The methodology developed in the previous chapter can be used to calculate the cost and

service level associated with a given (sS) policy In this chapter an algorithm that finds the

feasible policy yielding the lowest cost is developed A policy is feasible if it satisfies both the

basic condition 0 lt s lt S and the imposed constraint on the customer service level2 2 In

order to find the optimal policy the algorithm must combine the methodology for evaluating

a single policy with a search technique

Search techniques are evaluated in terms of two somewhat antithetic criteria On the one

hand the search should locate the optimal solution or at least a near optimal solution on

the other hand the search should take as little time as possible Many search techniques

guarantee optimality provided the objective function and constraints exhibit certain properties

such as convexity However even if such a search technique also guarantees its termination

in polynomial time the search might take too long to be practical Alternatively some search

techniques provide reasonable though sub-optimal solutions very quickly These techniques

often called heuristics must be evaluated in terms of both their speed and the accuracy of their

solutions The choice of a heuristic over a search that guarantees optimality depends upon the

nature of the feasible space the need for accuracy and the cost of time

Given the complexity of the cost and customer service level functions it is very difficult

to show theoretically for what if any combinations of demand and costs the feasible space is

convex However a plot of the objective function for a single product may be obtained by

2 2 Management specified that each policy must provide a customer service level of at least 975 Therefore the algorithm must incorporate the constraint CSL gt 0975

35

evaluating the costs for a number of policies within a grid When the service level constraint

is included in these plots maps of the feasible space are obtained Examination of the

maps for several products suggests that the space is convex over most regions however

maps for some products reveal some regions of non-convexity Unfortunately there is no

way of knowing the degree of non-convexity for other products other stores or different costs

estimates without a more rigorous survey of the feasible space Therefore if one of the

techniques that guarantee optimality under the condition of convexity is employed the search

would have to be evaluated as a heuristic Instead as a first step to finding a practical search

technique a crude search that guarantees optimality under virtually all conditions is

investigated

B A Grid Search

The simplest search technique that guarantees optimality merely evaluates all policies in

the grid defined by 0 lt s lt S and 0 lt Sm i n lt S lt S m a x where Sm i n and S m a x are

some predefined values That is the grid search is a brute force method that examines every

policy within the grid 2 3 Although the number of policies that must be examined is large the

grid search is guaranteed to find the optimal policy provided that S o p t is contained in the grid

that is provided Sm i n lt S o p t lt Sm a x The grid search algorithm proceeds by generating the

transition probability matrix for the policy (0Smin) and then evaluating the service level

associated with that policy If the policy is feasible24 the cost associated with the policy is

2 3 The total number of policies evaluated is the sum of the first Sm a x integers less the sum of the first Smin-1 integers Numerically [(l2)(Sm a x)(Sm a x+l) - (l2)(Smin-l)(Smin)] policies are examined

2 4 That is if the service level is adequate

36

calculated and compared to the current lowest cost The procedure continues until all points

in the grid have been examined and the optimal policy has been found A flow chart of the grid

search algorithm is shown in Figure 2

In order to assess the grid search the algorithm was coded and compiled into an

executable program The program was run for two products (a medium-demand product and

a high-demand product) and with four values of Sm a x In each run Sm i n was taken to be zero

The following times were recorded for each trial

T P M - the amount of time spent generating the transition probability matrix

GAUSS - the amount of time spent solving for the steady state probabilities

COSTS - the amount of time spent calculating the expected annual costs

SERVICE - the amount of time spent determining the customer service levels

T O T A L - the total execution time of the program

The columns entitled Grid Search in Tables 2 and 3 summarize the results of the trials All

times are recorded in seconds as required by a 486DX33 MHz personal computer with an

empty buffer memory

Three results are evident from the tables First the vast majority of the execution time

is spent generating transition probability matrices and then solving for the steady state

probabilities with the time being split almost equally between the two tasks Second the total

execution time of the algorithm increases exponentially as Sm a x increases For product 206917

doubling S^ from 30 to 60 led to an increase in the execution time of more than 2100 (from

172 seconds to 3661 seconds) Third the grid search algorithm as it stands does not appear

to be adequate for the problem at hand Given the large number of products that must be

evaluated the algorithm is much too slow to be implementable

37

Calculate the demand probability functions

Initialize s = 0 S = Smin] cost = 10A5

s = s+1

Calculate the probability transition matrix for (sS)

Evaluate policy (sS) Is service level ^ 0975

No

No Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Is s = S-l

Yes

Is S = Smax No

Yes

Print results (sS) cost

s = 0 S = S+1

Figure 2 The flow chart of the grid search algorithm

38

Table 2 Execution times of the algorithms for product 200001 a 30-amp inline fuse

(in seconds)

-max Subroutine Grid Search Grid Search

Updating T P M Heuristic

10 T P M 006 010 001

GAUSS 010 006 001

COST 005 005 001

SERVICE 001 001 001

T O T A L 050 035 027

20 T P M 126 021 010

GAUSS 119 160 023

COST 027 026 001

SERVICE 017 016 001

T O T A L 318 236 055

30 T P M 703 096 010

GAUSS 757 801 093

COST 079 085 001

SERVICE 047 042 001

T O T A L 1664 1137 126

40 T P M 2257 163 015

GAUSS 2958 3003 206

COST 313 329 001

SERVICE 092 083 011

T O T A L 5871 3851 285

39

Table 3 Execution times of the algorithms for product 206917 a 6 solder connector

(in seconds)

bull-max Subroutine Grid Search Grid Search

Updating T P M Heuristic

30 T P M 717 067 006

GAUSS 762 880 093

COST 098 087 010

SERVICE 016 021 006

T O T A L 1720 1176 220

40 T P M 2390 157 022

GAUSS 2861 2911 209

COST 219 227 006

SERVICE 080 088 011

T O T A L 5827 3790 362

50 T P M 6691 413 010

GAUSS 7994 8201 472

COST 568 577 016

SERVICE 212 197 005

T O T A L 15840 10041 654

60 T P M 15509 894 040

GAUSS 18607 18951 876

COST 1367 1353 038

SERVICE 350 359 022

T O T A L 36614 22794 1120

40

C A Technique for Updating the Transition Probability Matrix

As was noted in the previous section approximately half of the execution time of the grid

search algorithm is spent generating transition probability matrices Furthermore the amount

of time increases exponentially as Sm a x increases This increase is exponential for two reasons

first the number of transition matrices that must be calculated is a function of S^ squared and

second the average size of the matrices and hence the average number of calculations per

matrix increases with Sm a x For example when S m a x equals 30 the transition probability matrix

must be generated 465 times with the largest matrix having dimensions (31x31) When Sm a x

equals 40 the matrix must be generated 820 times with 355 of the matrices having dimensions

larger than (31x31) The performance of the algorithm could be improved considerably if

either the number of matrices that must be generated or the number of calculations per matrix

is reduced This section introduces a technique that accomplishes the latter by updating the

transition probability matrix for a new policy instead of generating it anew for each policy

A new policy (s+mS)

Suppose the policy (sS) has associated with it the (S + l)x(S + l) transition probability

matrix P and the policy (s+mS) where m is any integer satisfying 0 lt s+m lt S has

associated with it the (S + l)x(S + 1) transition probability matrix J Then T can be obtained by

recalculating only m rows of P Recall that the equations for generating a transition

probability matrix are divided into two sets based upon the row being calculated equations (2)

or (4) for rows 0 through s and (3) for rows (s + 1) through S The equations themselves are

independent of s only the limits defining the cases contain s Therefore only the m rows

that are affected by the change in the limits need to be recalculated The following updating

technique can be used to obtain 7

41

1) If m gt 0 rows i = (s + 1) (s+2) (s+m) are recalculated using (2) or (4)

2) If m lt 0 rows i = (s-1) (s-2) (s-m) are recalculated using (3)

3) All other rows of T are identical to those of P

The reader is directed to Appendix F for a mathematical justification of the technique

A new policy (sS+m)

Suppose the policy (sS) has associated with it the (S + l)x(S + l) transition probability

matrix P and the policy (sS+m) where m is any positive integer has associated with it the

(S+m+l)x(S+m+l) transition probability matrix P Then T can be obtained by modifying

each row of P and adding m additional rows and columns The updating technique is not as

simple as that for a new policy (s+mS) for two reasons the dimensions of P and T differ and

the equations for generating a transition probability matrix contain S Nevertheless an updating

technique that requires much less computing time than generating the matrix anew can be

developed The following technique can be used to obtain P

1) For rows i from 0 through s

For columns j from 0 through m recalculate using (2) or (4)

For columns j from (m+1) through (S+m) enter the (j-m)th column entry of P

2) For rows i from (s + 1) through S

For columns j from 0 through S enter the corresponding value from P

For columns j from (S + 1) through (S+m) enter 0

3) For rows i from (S + 1) through (S+m) recalculate using (3)

The above technique is justified mathematically in Appendix G

42

The modified algorithm

The updating technique can be easily incorporated into the grid search algorithm with

only a slight change in the manner in which s is incremented Figure 3 shows the flow chart

of the new algorithm As before the algorithm was coded and compiled into an executable

program and the various execution times were recorded The results are listed in Tables 2 and

3 under the columns Grid Search Updating TPM

Updating the transition probability matrix does improve the performance of the

algorithm The time required to generate the transition probability matrices is reduced by

approximately 90 This translates to a reduction in total execution time of approximately

30 Not surprisingly the reductions are greater for large values of Sm a x However despite

this seemly significant improvement a decrease of only approximately 08 seconds is obtained

when S^ equals 20 Further when S ^ equals 30 the algorithm requires over 11 seconds

which is still much too high

The execution time is now dominated by the time required to solve for the steady state

probabilities The steady state probabilities could be solved by using approximation techniques

which are faster than solving the systems exactly however because the probabilities are often

very small numbers the approximation techniques would be unable to provide the required

accuracy The only remaining way to decrease the execution time is to reduce the number of

policies that are evaluated during the search This can be done by either reducing the size of

the grid being searched or reducing the number of policies being evaluated within the grid The

following section introduces a method of reducing the size of the grid by obtaining a lower

bound for S namely Sm i n

43

s = s+k

Calculate the demand probability functions

Initialize s = 0 S = Smin| cost = 105

k= 1

Update the probability transition matrix for (sS)

Evaluate policy (sS) Is service level ^ 0975

No

No

iNo

Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Ifk= 1 iss = S-l If k =-1 is s = 0

Yes

Is S = Smax No

Yes

Print results (sS) cost

If k=l s=s+l S = S+1

Figure 3 The flow chart of the grid search algorithm utilizing the updating technique for the transition probability matrix

44

D A Lower Bound on S

As mentioned earlier the size of the grid is specified by the number of policies within

the domain defined by 0 lt s lt S and 0 lt Smin lt S lt Smax Thus the size of the grid

and hence the total amount of computation time can be reduced by obtaining tight bounds on

S Unfortunately because of the extreme non-linearity of the objection function and the service

level constraint obtaining a mathematical upper bound on S is very difficult and beyond the

scope of this work This section introduces a method for obtaining a lower bound on S25

For a given value of S an upper bound on the customer service level can be found as

follows Suppose that there are S units of inventory on the shelf at the beginning of every

review period regardless of demand No (sS) policy can provide a higher level of customer

service because there exists a positive probability of beginning a review period with less than

S units on the shelf Thus the customer service level associated with starting each review

period with S units on the shelf is an upper bound on the customer service level for any (sS)

policy From (10) this level is

E[(DT-SY] CSL 5 = 1 - T mdash mdash - ( 1 5 gt

E[DT]

Replacing j with S in (12) and then substituting (12) and (11) into (15) yields

CSL 15 = mdash- - J2(S-k)PrDT = k] (16)

To obtain Smin use (16) to find the largest value of S with a customer service level less than

0975

There are two criteria for a good lower bound on S the bound must be no larger than

This method was proposed by Kaan Katircioglu

45

S o p t and it must be very close to So p t These criteria ensure that the optimal policy will be

contained in the search grid and that the size of the grid is reduced by as much as possible

Based on these criteria the above method of obtaining a lower bound is assessed A value for

S m i n is obtained for each of the products in product category 20 and compared with the

corresponding value of S o p t 2 6 The results are illustrated in Figure 4 which shows the number

of occurrences of various Sm i n and S o p t combinations Note that a point lying on the 45deg line

represents a perfect bound while a point lying close to the S o p t axis represents a very loose

bound The absence of any points above the 45deg line indicates that the first criterion is satisfied

in each instance the method did not generate a value of Sm i n exceeding the corresponding value

of So p t Moreover the bounds are reasonably tight for 85 of the products analyzed that

is for those products with S o p t equal to 10 or less27 However as S o p t increases the bounds

become steadily worse Because values of S o p t are likely to be larger for these products at the

ten Jumbo and Extra Jumbo stores and for other products with higher demand the lower bounds

are not expected to be very tight in general

Although obtaining the bound Sm i n does reduce the size of the grid the corresponding

reduction in the execution time of the algorithm is miriimal For example if S^ equals 60 and

a value of 30 is obtained for Sm i n the program would still require approximately 215 seconds

to search the grid (compared to the 228 seconds required without a lower bound on S) Because

the execution time increases exponentially with So p t a tight upper bound on S would likely have

a more significant impact on reducing the execution time of the algorithm Further research

on this issue is required

2 6 The optimal policy for each product was obtained using a grid search with an extremely large upper bound

2 7 On average Sm i n is approximately half of S o p t for these products

46

47

The only remaining approach to improving the algorithm is to modify the search so that

fewer policies within the grid are evaluated Toward this goal assumptions can be made with

respect to the feasible space and a heuristic search can be developed based on these

assumptions

E A Heuristic Search

As mentioned previously in this chapter a map of the feasible space for each of

several products was obtained Upon closer examination the maps illustrate two patterns

without exception for a given value of S both the cost and the customer service level increase

as s increases The following intuitive argument is offered as a possible explanation for the

observed behaviour When S is held constant increasing s results in more instances in which

an order is placed upon review Hence the average stock level would increase which in turn

would lead to an increase in the cost and the customer service level28 With this in mind a

heuristic search based on the following assumptions is developed for a given value of S both

the cost and the customer service level are monotonically increasing functions of s

In general the heuristic proceeds as follows Having just incremented S by one the

customer service level of the policy (sS) is evaluated If the policy is feasible s is decreased

until the lowest value of s producing a feasible policy is found If the policy is not feasible

s is increased until either the cost exceeds the current lowest cost for a feasible policy a

feasible policy is obtained or s equals S-l In the event that a feasible policy is found the cost

2 8 In order to prove that the customer service level increases as s increases it would be sufficient to show that the probability of placing an order increases as s increases Such a result might be possible if assumptions regarding the demand function are made Further if both the probability of placing an order and the average stock level increase then both the ordering cost and the holding cost would increase No proofs are offered here

48

of the policy is calculated if the cost of the policy is less than the current lowest cost the new

policy becomes the current best Finally S is increased by one and the process repeats

provided S does not exceed some predefined value Sm a x The flow chart of the algorithm

incorporating the heuristic search is shown in Figure 5 Note that the algorithm begins by

attempting to improve on the lower bound Sm i n the policy (Sm i n-lSm i n) is evaluated and if it

is not feasible S m i n is increased by one The improvement continues until a feasible policy is

found

As previously mentioned a heuristic search must be evaluated in terms of both its speed

and the accuracy of its solutions To evaluate the heuristic on the basis of speed the same

approach used to evaluate the previous algorithms is employed The columns entitled

Heuristic in Tables 2 and 3 contain the various execution times of the heuristic The heuristic

is a vast improvement over the previous search techniques the reduction in execution time is

enormous and becomes more so as S m a x increases For a S m a x of 20 the reduction is

approximately 75 of the time required by the grid search algorithm29 and for a S m a x of 60

that reduction soars to over 95 More importantly the heuristic can search a grid with a

of 40 in under 4 seconds or a grid with a S m a x of 60 in under 12 seconds These low execution

times should be sufficient to allow for a large number of products to be analyzed in a

reasonable amount of time

In order to further assess the practicability of the heuristic the total time required by the

heuristic to find solutions for the entire product category was determined and compared to the

corresponding time for the grid search algorithm With a priori knowledge of the optimal

solutions conservative yet reasonable values of S m a x were selected and used by both of the

2 9 Hereafter grid search algorithm refers to the algorithm utilizing the grid search the updating technique and the lower bound Sm i n

49

Calculate the demand probability functions

Calculate Smin S = Smin s = S-l

S = S+l s = S-l t

Update the probability transition matrix for (sS)

No Is service level pound 0975 Yes

No Is service level ^ 0975

Yes

Is cost lt cost No

Yes

Is s lt S-l No

Yes

s = s+l

Update the probability transition matrix for (sS)

Is service level pound 0975

Yes

Is cost lt cost No

Yes

s = s S = S cost = cost

s = s-l

No

Update the probability transition matrix for (sS)

S = S+l

Is s 2 0 Yes

Update the probability transition matrix for (sS)

Is service level pound 0975

No

s = s+l

No

No

Is S = Smax

^Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Print results (sS) cost

Figure 5 The flow chart of the heuristic algorithm

50

algorithms30 The grid search algorithm required approximately 9 hours and 41 minutes to

find the solutions while the heuristic required only slightly more than 23 minutes Based on

speed the heuristic is preferable to the grid search algorithm

Finally to evaluate the heuristic based on the accuracy of its solutions its solutions were

compared to the previously obtained optimal policies Without exception the heuristic

algorithm found the optimal policy Although there is no guarantee that the heuristic will find

the optimal solution for different products or different demand functions the heuristic appears

to work extremely well Based on the fast execution time and the apparent accuracy of the

solutions the heuristic will be used in the development of the inventory tool

3 0 For the test the following values of Sm a x were used 20 (273 products) 30 (19) 50 (23) 100 (3) and 125 (2)

51

V T H E I N T E R F A C E

With the model formulated and the algorithm developed the only remaining task in the

development of the prototype is the creation the interface In fact two interfaces are created

one that displays results in terms of minimum fills and target levels and one that displays

results in terms of reorder points and target levels To be consistent with the previous chapters

the latter interface is described here31

In its present form the interface allows the manager to set review parameters to

compare optimal and current policies and to evaluate alternate policies From the main menu

the manager selects the product store duration of the review period length of the lead time

and value of Sm a x Once the parameters are set the system calculates the optimal policy and

then evaluates the current policy displaying results as they are obtained In particular the

reorder point target level expected annual cost and expected customer service level are

displayed for each policy If a current policy meets the desired customer service level the

expected annual savings of the optimal policy are displayed in terms of both dollars and

percentage of current cost otherwise the value of the current customer service level is

displayed in red to indicate that it is deficient Finally the manager can choose to evaluate

alternate policies or to return to the main menu

Although the prototype is far from the ultimate integrated inventory control system it

is still valuable to management As mentioned in Chapter II inventory policies at Canadian

Tire Pacific Associates are currently based on rules of thumb that incorporate marketing

3 1 Because significant changes to the interface are expected as feedback from management is received and as the system is brought on-line only a brief description of the present interface is given

52

strategies The prototype provides management with a means of determining the proportion of

the cost of inventory that is not attributable to meeting customer demand For example

consider a hypothetical consultation in which the manager chooses to analyze product 200001

a 30-amp inline fuse at store 6 The prototype displays to the manager the current policy of

(1214) which provides a customer service level of 1000 at an annual cost of $748 and the

optimal policy of (39) which provides a customer service level of 977 at an annual cost of

only $319 If for some reason the manager feels the optimal policy is undesirable he or she

can investigate the additional cost of holding excess inventory Suppose the manager chooses

to evaluate the policy (610) The prototype evaluates the policy and displays the results the

customer service level is 998 and the annual cost is $441 With the alternate policy

virtually all demand is satisfied and the cost of inventory is still reduced by over 40 Thus

the manager ascertains that approximately 40 of the current cost of inventory for this product

is directly attributable to marketing considerations Using the prototype in this way

management can begin to understand better the nature of its inventory costs A description and

graphical depiction of the above hypothetical consultation are presented in Appendix H

As for the technical details of the system the prototype can run on any computer using

the DOS operating system having approximately 420K of available conventional memory and

attached to a V G A or SVGA monitor Further the prototype can be used on a machine without

a math coprocessor however performance is slower The prototype consists of two modules

linked together into one executable program The interface module contains the various

interactive subroutines and the main module contains the methodology of the model and the

logic of the heuristic search Both modules are coded in Fortran and compiled using Microsoft

Fortran 50 which provides many extensions to standard Fortran The source codes for the

interface module and the main module are listed in Appendix I and Appendix J

53

respectively

In addition to the above technical specifications the program requires the presence of

certain files in certain directories The program uses the distribution files mentioned in

Chapter II to read the unit costs and the empirical demand functions Each product has a

unique distribution file the name of which is the six digit product number followed by the

extension dst The text file containing the existing policies at each store must be located in

the same directory as the distribution files Finally the file tmsrbfon which contains the

font used by the interface must be located in the directory from which the program is executed

In its future versions the system should be more automated and more user-friendly

Rather than having the manager choose an upper bound for the search the algorithm should

incorporate a method of determining Sm a x For this to happen either a theoretical upper bound

must be obtained or a heuristic for terminating the search must be developed Also the amount

of time spent waiting for calculations to be performed can be reduced by reading the optimal

and current policies and their associated costs and customer service levels from a file or

database This solution file could be created any time prior to the consultation by running

the system in a batch mode The interface should allow the manager to change cost estimates

and service level values view results graphically and print results and summary documents

Finally the system should provide an option for automatically incrementing the product andor

store during a consultation This option would enable a manager to review quickly the

inventory policy of a product at each store

54

VI R E S U L T S AND SENSITIVITY ANALYSIS

A Comparison of Current Policies to Optimal Policies

In order to estimate the potential savings of implementing optimal policies the solutions

obtained in the Chapter IV are analyzed in more detail For each product the current policy

is evaluated and then compared to the optimal policy with respect to cost and customer service

level Finally the potential savings of the optimal policies are extrapolated to all products and

all stores

Of the 420 products examined each of 37 products has an existing policy that fails to

provide the desired customer service level of 975 However most of these policies just

fail to do so only 4 policies provide a customer service level of less than 90 Table 4

provides a list of these 37 products each with its current customer service level its current

annual cost and the additional cost of its optimal policy3 2 Two comments regarding Table

4 are in order First the current customer service level for product 203312 an amber

driveway marker is listed at a misleading 303 Examination of the demand distribution

reveals that this product was sold on only 1 of the 307 days during which sales data were

collected and on that day 13 of the markers were sold Clearly this data point constitutes

either an error or an unusual event the current policy of (34) does not allow for more than 4

items to be sold per review period Second the additional cost of the optimal policy for each

of 5 products is negative In other words for each of these products the current policy which

fails to provide the desired customer service level costs more than the optimal policy which

3 2 This is the difference between the expected annual cost of the optimal policy and the annual cost of the current policy It is the minimum additional cost required to increase the customer service level to at least 975

55

Table 4 Products with insufficient existing policies

Product Current CSL ()

Current Annual Cost ($)

Additional Cost of the Optimal Policy ($)

206932 974 204 -051

204570 973 222 -025

203053 973 248 009

206550 972 118 042

206547 972 105 035

206565 972 146 056

203702 972 340 157

206876 972 155 061

205752 972 330 153

205922 972 108 036

206857 972 180 073

205393 972 190 -003

201513 971 345 015

203405 970 255 037

203069 968 295 -048

203071 967 321 -001

205635 967 390 029

203914 966 460 160

204601 962 252 061

203326 960 144 014

202003 958 876 437

206516 958 112 029

206517 958 283 120

206350 958 333 147

205932 958 180 066

203936 952 598 294

56

Product Current CSL ()

Current Annual Cost ($)

Additional Cost of the Optimal Policy ($)

206932 974 204 -051

203939 948 413 193

203977 945 1216 1242

206551 945 132 077

205389 934 598 047

203590 927 1223 607

206567 919 122 070

203591 905 1228 834

205390 887 291 074

203318 833 136 084

206393 776 319 289

203312 303 237 511

provides at least the desired customer service level For the 36 products the total additional

cost of the optimal policies is $542033

Of the remaining 383 products most have an existing policy that provides a customer

service level well above 975 When the customer service levels are rounded to one decimal

each of 253 products has a customer service level of 1000 and each of 347 products has a

customer service level of at least 990 In contrast each of only 58 products has an optimal

policy that yields a customer service level of at least 99034 For each of 37 products the

3 3 The cost associated with product 203312 is not included in the total

3 4 Optimal policies do not have customer service levels of exactly 975 because the function is discrete For low-demand products a unit step in s or S results in a relatively large change in the customer service level

57

current policy is identical to the optimal policy

The extremely high level of service provided by most policies allows for substantial

savings in the cost of inventory With current policies the estimated annual cost of inventory

for all 420 products is $420743 whereas the expected annual cost of the optimal policies is

only $186884 This amounts to a savings of approximately $2340 or over 55 of the current

cost of inventory

In order to further assess the savings the difference between the costs of the current

policies and the costs of the optimal policies are examined in more detail For each of the 383

products whose existing policy provides a customer service level above 975 the difference

between the estimated annual cost of the existing policy and the expected annual cost of the

optimal policy is calculated The resulting histograms of the savings in (a) dollars and (b)

percentage of current cost are shown in Figure 6 The particulars for the current and optimal

policies for all 420 products are provided in Appendix K

As evident from Figure 6(a) the majority of the expected savings in absolute terms

comes from only six products In order to examine these products more closely the current

and optimal policies along with the associated costs and savings for each of the six products

are listed in Table 5 Each current policy is very different than its corresponding optimal

policy resulting in excessive inventory for each product In fact over half of the total expected

savings could be realized by implementing the optimal policy just for each of these six products

Interestingly five of the six products are sealed beam headlights

58

90

10 - 20 30 - 40 50 - 60 70 - 80 90- 100

Savings ()

(b)

Figure 6 The distribution of savings of optimal policies in (a) dollars and (b) percentage of current cost

59

Table 5 Products with the largest potential absolute savings

PRODUCT

CURRENT OPTIMAL SAVINGS

PRODUCT s S Cost ($)

C S L

()

s S Cost ($)

C S L () ($) ()

2036801 194 200 64014 1000 37 48 bull 9245 975 54769 856

2036772 94 100 23652 1000 18 22 3850 978 19802 837

2036723 78 90 23043 1000 19 23 4454 975 18589 807

2030944 145 175 19522 1000 34 45 3727 975 15795 809

2036425 48 60 10491 1000 13 17 2682 977 7809 744

2036556 68 80 7265 1000 14 22 1699 975 5566 766

1 H6054 rectangular sealed beam headlight 2 H4656 rectangular sealed beam headlight 3 6053 rectangular sealed beam headlight 4 9004 halogen bulb 5 4652 rectangular sealed beam headlight 6 6104 7 sealed beam headlight

Although Figure 6(a) seems to indicate that the majority of the current policies are close

in cost to the optimal policies this is very misleading For example suppose the optimal policy

for a product costs $1 and that the current policy costs $2 In absolute terms the optimal

policy saves only $1 but in relative terms the optimal policy saves 50 of the current cost

When savings are viewed in this manner as in Figure 6(b) it is clear that substantial savings

can be realized for the majority of the products In fact both the mode and the mean of the

savings are between 40 and 50 of the current costs

In order to extrapolate the above savings to obtain the potential savings for product

category 20 at all 21 stores the differences between stores must be considered During the

60

pilot study the current policy for each of several products at each size of store was evaluated

The study suggested that medium and large stores have a similar amount of excess

inventory and that jumbo and extra jumbo stores have much more excess inventory One

could reasonably expect that the additional excess inventory allows for higher savings to be

realized at the jumbo and extra jumbo stores Thus because 18 of the 21 stores are

medium or larger the expected 40-50 savings at store 6 can be viewed as a lower bound

on the potential savings to Canadian Tire Pacific Associates from implementing optimal policies

for product category 20 in all its stores

Finally in order to obtain the overall potential savings the expected savings from

product category 20 must be extrapolated to all product categories Ideally the proportion of

inventory comprising each product category and the relative savings that can be realized for

each category would be used to obtain a good estimate of the overall potential savings

However as this information is not known only a crude extrapolation is possible Thus the

analysis proceeds under the assumption that a 40-50 savings can be realized in each product

category35

Before an estimate of the total potential savings in absolute terms can be obtained the

proportion of the current inventory to which the savings apply must be determined Up to this

point the analysis has been concerned solely with reducing inventory at the stores However

it is inappropriate to perform further analysis without considering the entire inventory system

including the central warehouse Because the lead time for orders from Toronto is highly

variable and considerably greater than those at individual stores Canadian Tire Pacific

3 5 During the pilot study the current policies for a paint brush and an ironing board were evaluated and found to result in excess inventory similar to that of product category 20 Although this in no way justifies the assumption it does lend some support

61

Associates must keep a fairly high safety stock at the warehouse Without knowing the

proportion of inventory serving as the safety stock it is inappropriate to assume any reduction

in inventory at the warehouse is possible Therefore the relative savings of implementing

optimal policies is applied only to the inventory held in the stores36 Of the estimated $28

million in inventory approximately half is held in the stores therefore the 40-50 savings

translates to a savings of approximately $55 to $7 million per year

B Sensitivity Analysis on the Ordering Cost and Holding Rate

The above analysis was based on estimates of the ordering cost and the annual unit

holding rate37 provided by management As described in Chapter II these estimates

incorporate a number of costs that are difficult to quantify or that are speculative Therefore

there is some question as to the accuracy of the estimates In this section errors of +$0015

in the ordering cost and +5 in the holding cost are assumed in order to determine the

sensitivity of the model to errors in these variables38

The analysis proceeds as follows Based on an ordering cost of $0080 and a holding

rate of 30 the optimal policies from the previous section were obtained39 Then assuming

3 6 Of course this assumes that the excess inventory currently held in the stores is not required as part of the safety stock Based on the large amount of inventory at the warehouse this assumption seems reasonable

3 7 Hereafter simply referred to as the holding rate

3 8 An error of +$0015 in the ordering cost implies an actual ordering cost of $0070 a relative error of approximately 21 and an error of -$0015 implies an actual ordering cost of $0100 a relative error of 15 An error of +5 in the holding rate implies an actual rate of 25 a relative error of 20 and an error of -5 implies an actual holding rate of 35 a relative error of approximately 17

3 9 These policies are referred to as optimal policies for the duration of this chapter

62

the actual ordering cost andor holding rate are different from managements estimates the

heuristic is used to obtain the true optimal policies Finally the optimal policies are

evaluated based on the actual ordering cost and holding rate and then compared to the true

optimal policies For each possible combination of ordering cost and holding rate three

numbers are recorded the number of products with an optimal policy that differs from its

true optimal policy the total cost of implementing the optimal policies given the actual

ordering cost and holding rate and the difference between the cost of implementing the

optimal policies and the cost of implementing the true optimal policies The results of the

analysis are summarized in Table 6

Table 6 A comparison of optimal policies to true optimal policies

Actual Ordering Cost

$0070 $0085 $0100

25 5 products $155439

$002

51 products $160590

$092

88 products $165464

$449

Actual Holding Rate

30 54 products

$181470 $182

48 products $192026

$088

35 79 products

$207270 $574

48 products $212940

$136

2 products $218312

$000

4 0 The table should be read as follows If the actual ordering cost is $0070 and the actual holding rate is 25 then each of 5 products has an optimal policy that is different from its true optimal policy The total cost of implementing the optimal policies is $155439 which is $002 more than implementing the true optimal policies

63

The model appears to be quite insensitive to relative errors in the order of 20 Under

the worst case scenario when one of the variables is over-estimated and the other is undershy

estimated the optimal policies result in a total cost of inventory which is only approximately

3 higher than the cost of the true optimal policies Further because demand is not changed

by errors in these estimates each of the optimal policies still provides a customer service

level of at least 975 under all of the scenarios Finally although the cost of the optimal

policies varies significantly for the different scenarios the relative savings remain virtually

constant Table 7 lists the cost of existing policies and in parenthesis the relative savings of

the optimal policies for each of the different combinations of ordering cost and holding rate

The cost of the existing policies varies in the same manner as that of the optimal policies the

relative savings for each of the scenarios remains around 55 Therefore even in the presence

of such errors in the ordering cost and holding rate the projected savings of implementing the

optimal policies would be unchanged

Table 7 The cost of existing policies and the relative savings of the optimal policies under various scenarios

Actual Ordering Cost

$0070 $0085 $0100

25 $350153 (56)

$358464 (55)

$366764 (55)

Actual Holding Rate

30 $412438 (56)

$429048 (55)

35 $474725 (56)

$483026 (56)

$491320 (56)

64

C Sensitivity Analysis on the Demand Probability Mass Function

In this section the sensitivity of the model to slight changes in the demand probability

mass function is investigated As described in Chapter II the demand function used in the

model is based solely on observed sales however this empirical function is only one possible

instance of the underlying true demand function For example had the data been collected

over a period both starting and finishing one month earlier many of the empirical functions

would be slightly different Consequently depending upon the degree to which the empirical

functions vary and the sensitivity of the model to these variations the optimal policies from

the model might be inaccurate For example recall the case of the amber driveway marker

only thirteen markers were sold and all were sold on the same day Because the empirical

demand function is used the model calculates an optimal policy of (1213) clearly this

policy is impractical Although this example is extreme it does indicate the danger of using

unrealistic empirical demand functions41

As discussed in Chapter II three classes of products were identified namely low

medium and high-demand products For this analysis the same method of classification

is used Two products are selected from each of these groups and two new distribution files

are created for each of the six products For each product one of the files contains two

additional days of data with each day having a demand equal to the maximum observed demand

for that product In the second file only one additional day of data is added but the demand

on that day is two more than the maximum observed demand The situations that could give

4 1 Certainly a rigorous analysis to determine both the extent of variation in the empirical demand functions and the effect of using the empirical functions needs to be performed However such an analysis is beyond the scope of this thesis only a brief analysis that might suggest the degree to which the model is sensitive is performed here

65

rise to such changes in the demand data are termed Scenario 1 and Scenario 2 respectively42

The analysis proceeds in a manner similar to the one in the previous section Given the

demand that was originally observed the optimal policies were obtained Now assuming the

true demand function corresponds to the demand under Scenario 1 the true optimal policy

for each product is calculated Then each optimal policy is re-evaluated using the true

demand function and compared to the true optimal policy The analysis is then repeated

assuming the true demand function corresponds to the demand under Scenario 2 Table 8

lists the results for each product and scenario

Table 8 True optimal and optimal policies for each demand scenario

Product 202101 (low demand)

Scenario s s Cost ($) C S L ()

1 True optimal 1 2 470 993

Optimal 1 2 470 993

2 True optimal 2 3 663 976

Optimal 1 2 461 875

Product 203507 (low demand)

Scenario s S Cost ($) C S L ()

1 True optimal 0 3 334 981

Optimal 0 2 258 972

2 True optimal 2 3 508 980

Optimal 0 2 273 702

4 2 To be clear the scenarios are defined as follows Scenario 1 The addition of two days each with a demand of max Scenario 2 The addition of one day with a demand of max+2

where max is the maximum observed daily demand for each product in the sample period

66

Product 201032 (medium demand)

Scenario s S Cost ($) C S L ()

1 True optimal 6 12 1377 976

Optimal 6 11 1338 973

2 True optimal 6 13 1426 977

Optimal 6 11 1338 970

Product 203039 (medium demand)

Scenario s S Cost ($) CSL ()

1 True optimal 5 19 256 975

Optimal 5 16 251 969

2 True optimal 5 20 258 976

Optimal 5 16 249 968

Product 203677 (high demand)

Scenario s S Cost ($) CSL ()

1 True optimal 18 23 3981 977

Optimal 18 22 3817 972

2 True optimal 18 23 3995 978

Optimal 18 22 3831 973

Product 203680 (high demand)

Scenario s S Cost ($) CSL ()

1 True optimal 38 49 9498 976

Optimal 37 48 9175 971

2 True optimal 37 49 9484 975

Optimal 37 48 9206 972

67

The model is most sensitive to changes or errors in the demand functions of low-

demand products When only one additional day of demand for max+2 units is added to the

data the optimal policy for each of the low-demand products results in a very low customer

service level In fact for one of the products the customer service level drops to almost 70

The increase in demand does lower the customer service level for each of the other the

products but that decrease is slight

The above analysis indicates the need for some kind of smoothing or modelling of the

demand data especially for low-demand products The inherent variation between two

instances of a true demand function could very well result in more pronounced changes than

those resulting from either of the above scenarios Therefore it is likely that many optimal

policies especially those for low-demand products would actually provide a much lower or

much higher customer service level than expected However by smoothing the data the

variation between instances of a true demand function can be reduced and the desired

customer service levels can be achieved

For example suppose that the demand for a product is stationary that the average

demand during the previous period was unusually low and that the optimal policy for this

period is based on the demand during the previous period Because the demand is stationary

the probability that the demand during this period is higher than that during the past period is

more than 50 In other words the probability that the demand during this period is higher

than that for which the optimal policy provides the desired expected customer service level

is greater than 05 Thus expected customer service level for this period is actually less than

that specified Now suppose instead that the optimal policy is based on smoothed data which

reflects very closely the expected demand The probability that the demand during this period

is higher than that for which the optimal policy provides the desired expected customer service

68

level is virtually 05 Thus as a result of smoothing the data the specified customer service

level should be expected

69

VII C O N C L U S I O N

In this thesis we developed a single-product single-location model of the inventory

system at Canadian Tire Pacific Associates Specifically we formulated a Markov chain model

to determine the long-run average cost of inventory for fixed values of s and S and then

developed a heuristic search to find near optimal policies quickly Also as a first step toward

developing an integrated interactive inventory control system we created a prototype of the

interface

According to the analysis performed here Canadian Tire Pacific Associates is currently

holding an excessively large inventory in its stores This study finds that Canadian Tire Pacific

Associates could reduce the cost of inventory for product category 20 at store 6 by

approximately 50 and still maintain customer service levels of at least 975 Further the

mean and modal values of savings for these products are between 40 and 50 Given the

similar excess inventory at the other stores and in all product categories we believe that savings

in this range are possible for all products and stores We estimate that the application of

inventory control as described in this thesis would result in annual savings of $55 to $7

million

Given the large potential savings to Canadian Tire Pacific Associates we recommend

that management proceed with the next phase of the project Specifically we recommend that

the model algorithm and interface be refined further that a forecasting model be developed

and that a data collection and storage scheme be designed

In order to reflect operational changes at Canadian Tire Pacific Associates subsequent

to the formulation of the model modifications must be made to allow for the lead time to

exceed the review period Several of the stores now have three picks per week and as a

70

result orders placed for these stores at one decision epoch do not arrive until after the next

decision epoch Because this violates one of the models assumptions the model must be

modified accordingly Note however that with three picks per week and a review period

of four days orders from only the previous decision epoch are ever outstanding at the current

decision epoch

Also the model should be refined to reflect the fact that the review period is not fixed

over the planning period In its present form the model assumes a fixed review period

however inventory at stores with two picks per week is actually reviewed on a four day -

three day cycle and inventory at stores with three picks per week is reviewed on a three day

- two day - two day cycle Rather than obtaining the steady state costs for fixed review period

the model should calculate the steady state cost for a period of one week during which multiple

replenishments occur Although the solutions are not expected to differ significantly as a result

of the change the model would reflect the actual process more accurately

Further work is also required on the search technique for locating near optimal

policies Although the heuristic search described in this thesis appears to work well the

decision maker must provide an upper bound on S prior to the search In practice this upper

bound is generally not known in advance Thus if the heuristic is to be used either a tight

theoretical bound must be found or a termination heuristic must be developed Alternatively

a new heuristic search could be developed based on additional or different assumptions

regarding the properties of the feasible space For example although some regions of non-

convexity were found for some products much of the feasible space appeared to be convex

A heuristic that assumes a convex feasible space would not require an upper bound on S and

might perform very well However before such a heuristic is implemented further work is

necessary to determine the degree of non-convexity for a variety of demand functions and

71

ordering and holding costs

In addition to making the above modifications we recommend that Canadian Tire Pacific

Associates begin work on the remaining two components of the inventory control system First

in order to allow for products with seasonal demand the system requires a forecasting model

to provide accurate estimates of demand for the coming period Second the system needs a

data collection and storage module that systematically collects sales data for each active

product and for each store and then updates the corresponding demand distributions Because

approximately 25 of demand is expected to result in lost sales under the optimal policies the

sales data will have to be adjusted accordingly in the future Also because the model appears

to be sensitive to slight changes in the demand distribution for low-demand products the

module should incorporate a technique for smoothing the empirical distributions for these

products Further work should be undertaken to determine the extent of variation in the

empirical distributions and the most appropriate smoothing technique However because of

the large amount of data required to forecast seasonal demand and because of the current lack

of historical data we recommend that management proceed immediately with the design of a

data collection scheme

Finally two additional issues that are beyond the scope of the next phase should be

addressed at some point First a monitoring system will be required once inventory control

is implemented in the stores The monitoring system will ensure that optimal policies perform

as expected and will indicated whether or not demand is adversely affected by the lower levels

of in-store inventory Second given the degree to which in-store inventory can be reduced

inventory control in the warehouse should be investigated Because approximately half of the

existing inventory is held in the warehouse the additional savings from reducing system-wide

inventory could be great

72

Afterword

Just prior to the completion of the work described in this thesis Mr Don Graham

principle owner of Canadian Tire Pacific Associates sold his shares in the company Shortly

thereafter the results of this thesis were presented to the new management group

Unfortunately the new management did not consider development of an inventory control

system to be a priority Nonetheless the prototype was delivered to Canadian Tire Pacific

Associates for possible in-house testing however at the time of this writing - almost one year

after the meeting - Canadian Tire Pacific Associates has not decided to continue with the

project

73

Bibliography

Arrow K A T E Harris and J Marschak (1951) Optimal inventory policy Econometrica 19 250-272

Day G (1992) In a speech to the Empire Club of Canada Toronto quoted in When the JIT fits The Globe and Mail October 20 1992 B24 and in Puterman (1994)

Dvoretzky A J Kiefer and J Wolfowitz (1952a) The inventory problem I Case of known distributions of demand Econometrica 20 187-222

Dvoretzky A J Kiefer and J Wolfowitz (1952b) The inventory problem II Case of unknown distributions of demand Econometrica 20 450-466

Federgruen A (1993) Centralized Planning Models in SC Graves A H G Rinnooy Kan and P H Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Federgruen A H Groenevelt and H C Tijms (1984) Coordinated replenishments in a multi-item inventory system with compound Poisson demands and constant lead times Management Sci 30 344-357

Lee H L and S Nahmias (1993) Single-Product Single-Location Models in SC Graves A H G Rinnooy Kan and PH Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Muckstadt J A and R O Roundy (1993) Analysis of Multistage Production Systems in SC Graves A H G Rinnooy Kan and PH Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Porteus E L (1985) Numerical Comparisons of Inventory Policies for Periodic Review Systems Oper Res 33 134-152

Puterman M L (1994) Markov Decision Processes Discrete Stochastic Dynamic Programming Wiley New York

Silver E A and R Peterson (1985) Decision Systems for Inventory Management and Production Planning 2nd edition Wiley New York

Taylor H M and S Karlin (1994) An Introduction to Stochastic Modeling revised edition Academic Press San Diego

74

Tijms H C and H Groenevelt (1984) Simple approximations for the reorder point in periodic and continuous review (sS) inventory systems with service level constraints European J Oper Res 17 175-190

75

Appendix A

A Portion of the Sales File for Product 200001 a 30-amp Inline Fuse

200001 283 1190 0 0 2 0 0 1 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 200001 284 1190 2 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 200001 285 1190 0 0 4 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 286 1190 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 200001 287 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 200001 288 1190 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1 0 2 200001 289 1190 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 290 1190 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 200001 291 1190 0 0 2 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 200001 292 1190 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 200001 294 1190 0 0 0 1 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 1 200001 295 1190 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 200001 296 1190 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 200001 297 1190 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 200001 298 1190 0 0 1 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 200001 299 1190 0 0 1 0 0 0 1 0 1 0 0 2 0 0 0 0 0 2 1 1 0 200001 300 1190 0 0 1 0 0 0 0 1 0 0 0 0 0 2 2 0 0 0 0 3 0 200001 301 1190 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 200001 303 1190 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 200001 304 1190 1 0 0 0 0 0 0 0 1 0 0 2 0 0 0 0 0 0 0 0 0 200001 305 1190 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 200001 306 1190 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 2 0 0 1 0 200001 308 1190 0 0 0 1 0 1 0 0 0 2 0 1 0 0 0 0 1 1 1 0 0 200001 309 1190 1 3 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 200001 310 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 200001 311 1190 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 200001 312 1190 0 1 1 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 200001 313 1190 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 2 0 200001 314 1190 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 200001 315 1190 0 0 2 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 200001 316 1190 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 200001 317 1190 2 0 0 1 0 0 0 0 0 0 5 1 0 0 0 1 0 0 0 0 0 200001 318 1190 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 1 200001 319 1190 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 200001 320 1190 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 200001 321 1190 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 1 200001 322 1190 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 323 1190 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 200001 324 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 0 1 0 0 2 1 1 0 200001 325 1190 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3 200001 326 1190 0 0 1 0 1 0 0 0 2 0 0 1 0 0 0 1 0 1 0 0 1 200001 328 1190 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 200001 329 1190 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 200001 330 1190 0 0 1 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 200001 331 1190 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 200001 332 1190 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 200001 333 1190 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 200001 334 1190 0 0 2 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 200001 335 1190 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 200001 336 1190 0 0 0 1 0 0 3 0 0 0 0 0 2 0 0 0 0 0 0 0 0

(253 subsequent lines omitted)

76

Appendix B

Sample Distribution Files

a) A low-demand product - an automobile back-up alarm

PRODUCT NUMBER 202101 PRICE $ 684

DMD

3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

N 2 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 2 2 0 1

1 5 3 8 12 5 7 11 6 11 5 4 11 10 19 4 2 5 14 5 6 4

0 302 304 299 295 302 300 296 301 296 302 303 295 296 288 303 305 301 290 300 301 302

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

b) A medium-demand product - a 168 bulb

PRODUCT NUMBER 203039 PRICE $ 51

DMD

16 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 1 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0

9 0 0 2 5 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0

8 0 0 2 5 0 0 2 0 1 1 3 0 1 3 0 0 1 1 0 0 1

7 2 1 3 1 0 0 1 0 2 3 1 1 4 1 1 0 2 1 1 1 1

6 4 7 1 10 3 1 3 2 1 3 5 2 7 4 1 0 1 1 3 3 4

5 1 7 7 17 0 2 11 1 5 4 1 2 6 9 6 2 6 9 4 4 1

4 5 10 10 16 5 5 19 10 13 8 11 9 13 10 12 7 5 15 5 8 3

3 10 12 13 27 4 6 24 11 10 13 14 9 23 14 18 8 14 18 14 12 14

2 14 30 36 43 16 16 43 32 38 39 39 34 47 36 41 21 19 39 20 34 24

1 30 56 68 56 37 30 70 48 67 61 44 75 62 63 48 28 52 61 63 53 47

0 240 184 165 126 240 247 133 202 169 175 189 173 143 165 180 240 206 161 197 192 212

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

77

c) A high-demand product - a H4651 rectangular sealed beam headlight

PRODUCT NUMBER 203677 PRICE $ 857

DMD

14 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

13 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 2 3 0 0 1 0 0 0 0 1 1 1 0 0 0 0 2 0 0

11 0 1 3 1 0 0 4 1 0 0 0 0 0 0 0 0 1 0 0 0 1

10 0 0 4 4 1 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 1

9 0 3 14 11 0 1 7 1 0 2 0 8 0 1 0 0 0 2 1 0 0

8 0 4 12 16 1 0 18 4 3 5 0 2 1 4 2 1 1 0 0 1 3

7 1 2 12 13 1 6 19 3 5 9 0 11 2 5 4 0 2 3 3 1 4

6 2 9 22 21 6 10 29 12 10 11 4 24 3 7 10 4 7 11 6 3 14

5 7 13 35 33 8 16 29 19 18 8 3 26 15 26 15 12 9 14 13 11 19

4 19 35 34 43 32 30 41 31 32 47 11 42 20 33 30 31 20 31 22 21 31

3 32 63 49 54 42 60 51 40 51 35 33 55 39 51 39 38 39 57 58 40 54

2 46 59 56 55 73 63 61 65 63 61 63 54 68 85 65 53 72 69 70 77 63

1 109 79 42 40 80 64 32 80 74 77 99 55 97 57 83 93 80 75 81 97 73

0 91 39 21 13 63 57 11 51 51 52 94 29 61 35 59 75 76 45 51 56 44

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

78

Appendix C

A Proof that the Limiting Probability Distribution is Independent of the Initial State

In this appendix we show that provided 0 lt PrDT = 0 lt 1 the Markov chain

corresponding to an inventory model with a (sS) policy has a limiting probability distribution

7T where 0 lt TTJ lt 1 for y = 0 1 S that is independent of the initial state Further the

limiting distribution is the unique nonnegative solution of the equations

Tt P = TI (AD

s and J gt = 1 (A 2)

i=0

where P is the (S + l) x (S + l) transition probability matrix whose first n rows (and columns)

correspond to the n recurrent states in the chain and whose remaining rows (and columns) if

any correspond to transient states43 Prior to the proof several preliminary results are

obtained

RESULT 1 The Markov chain has exactly one positive recurrent class and perhaps some

transient states

First note that in the absence of any absorbing states a Markov chain with a discrete

finite state space must have at least one positive recurrent class containing at least two states

Now consider the Markov chain for an inventory model with a (sS) policy

a) State i gt s cannot be absorbing because there is a positive probability of reaching

j lt i (Since PrDT gt 0 gt 0)

4 3 Note that since the rows and columns of P are reordered the elements of it must be reordered accordingly

79

b) State lt s cannot be absorbing because by the rules of the ordering policy and

the assumption that PrDT = 0 gt 0 there is a positive probability reaching S

Therefore the chain has no absorbing states and must have at least one positive recurrent class

Next consider the subset of states A = i | i lt s Because the state space is finite

and PrDT gt 0 gt 0 the process must return to at least one state in A in other words at least

one state in A must be recurrent Let k G A be a member of a recurrent class dt Because

of the ordering policy and the assumption that PrDT = 0 gt 0 k -raquo S Also by definition

there must be a positive probability of returning to k from S S -gt k Therefore S G 3w

Further because S is reachable from all states in A i G A is either transient with i -raquo dt or

recurrent with pound iR

Now consider the subset of states B = | j gt s Because PrDT gt 0 gt 0 the

chain will eventually reach a state i G A If i is transient then j must be transient also with

y -gt 3 if i G 9 then j is either transient with j -gt dt or recurrent with j E dt Thus the

Markov chain has exactly one recurrent class and perhaps some transient states

RESULT 2 The limiting probability distribution ir for the Markov chain corresponding to the

nxn submatrix Pm = [Ptj] i j G dt is independent of the initial state and is the

unique nonnegative solution of the equations

TC-Prr = re (A3)

and J2 K = 1 ( A 4 )

jt=i

where the rows and columns of are numbered from 1 to n rather than

Since if i j i and j must be in the same equivalence class

80

according to their associated states

First note that both the original Markov chain and the one corresponding to the

submatrix P R R are aperiodic because PrDT = 0 gt 0 Therefore given that 9 is a positive

recurrent aperiodic class the limiting distribution ir is uniquely determined by (A3) and (A4)

(Taylor and Karlin 1994 Theorem 44 p213) Further because there are no absorbing states

or other recurrent classes in the original chain the process must eventually enter the recurrent

class 9 regardless of the initial state Therefore in the limit the probability of being in state

i G 9 is independent of the initial state

RESULT 3 The limiting probability distribution for the transient states is the null vector

independent of the initial state

This result follows directly from the definition of a transient state

PROOF Suppose the states are reordered such that the first n rows (and columns) of P

correspond to the n recurrent states of the Markov chain Then P can be partitioned into four

submatrices

PRK - the n x n submatrix [Py] i j E 9

P T T - the (S + 1-I) x (S + l-n) submatrix [Py] i j g 9

P T R - the (S + l-n) x n submatrix [Py] i pound 9 j G 9

P R T - the n x (S + l-n) submatrix [Py] i G 9 j pound 9t By definition this is a matrix

of zeros

(Al) can then be rewritten as

81

(ltR TtT) P P L TR TT

I-P RR 0

p I-P TR 1 TT

(TtR TCT)

(0- 0T)

where irR denotes the limiting probability distribution for the n recurrent states and irT denotes

the distribution for the remaining S + l-n transient states Solving for TT t gives

TCr-0 + itT-(I-PTI) = 0T

itT-(I-PJJ) = 0 r

i c r = (I-P^-Oj

Since (I - PTT)1 always exists (Puterman 1994 Proposition A3 p 593) irT = 0X which is

in agreement with Result 3 Now solving for 7rR yields

(A2) can be rewritten as

bullRPRR 1 1R (A5)

(A6) S + l

k + L k =1

k=l k=n+

k=i

where ir contains the appropriately reordered steady state probabilities Since (A5) and (A6)

are equivalent to (A3) and (A4) respectively Result 2 holds Therefore the Markov chain

corresponding to an inventory model with a (sS) policy has a limiting distribution that is

independent of the initial state and is the unique nonnegative solution to (Al) and (A2)

82

Appendix D

A Measure for the Steady State Customer Service Level

In this appendix we show that the appropriate measure for the steady state customer

service level is given by

_ E[Demand satisfied during RP] E[Total demand during RP]

Consider the following two measures of the steady state customer service level

Measure 1 Suppose the customer service level is calculated only after n review periods have

passed Denoting DS as the demand satisfied in review period i and DT as the total demand

in period i the customer service level after n periods CSL(n) is

DS + DS0 + + DSbdquo CSLn) = - -

DT + DT + + DT

Dividing both the numerator and denominator by n taking the limit as n -raquo oo and applying

the Law of Large Numbers yields

CSL = lim CSL(n) n-degdeg

- [DS + DS + + DS] = l i m ^ J 1 1

raquo~ - [DT + DT + + DT]

E[DS]

E[DT]

83

Measure 2 Suppose the customer service level is calculated for each of n review periods and

then averaged Thus the customer service level after n periods CSL(n) is given by

CSL(n) = -DS DS

DT DT DT_

Taking the limit as n -gt oo yields

CSL = lim CSL(n)

lim mdash DSl DS2

DT DT DT_

DS

DT

Now consider the scenario in which there is one item on the shelf to begin each review

period and in which demand is stochastic with PrDT = 2 = Vi and PrD x = 100 = Vi

According to Measure 1 the customer service level is

CSL - = ^ W + ^ W = plusmn bdquo 002 E[DT) 1 (2) + 1(100) 51

while according to Measure 2 the customer service level is

CSL = E DS DT

1(1) + i(-Ls

22) + 2 100

51 200

026

Clearly from managements point of view the first measure is more indicative of the

actual level of service being provided The problem with the second measure is that equal

84

weight is given to each ratio regardless of the magnitude of the numbers For example with

respect to Measure 2 the events LSLR = 12 and LSLR = 50100 are identical however

with respect to lost sales the second event is much worse than the first Therefore the proper

measure for the steady state customer service level is Measure 1

85

Appendix E

The Calculation of the Conditional Expected Demand Not Satisfied During a Period of T Consecutive Days

Denoting D x as the demand during a period of T consecutive days and j as the inventory

on hand at the beginning of the period the conditional expected demand not satisfied during the

period E[(DT-y)+] is calculated as follows

E[(DT-j)+] = pound ltk-j)Pr[DT = k k=j+l

= pound (k-j)PrDT = k] + J2(k-j)PrDT = k] - pound (fc-)VJgtr = fc k=j+l k=0 k=0

= J2(k-j)Pr[DT=k -J2(k-j)PrDT = k] k=0 k=0

= YkPrDT=k) - J2jPrDT=k - Yk-j)PrDT=k) Jfc=0 k=Q k=0

= Ti - j - Y(k-j)PrDT=k 7-1

= Tx - j + YltJ-k)PrDT = k jt=0

where fx is the mean daily demand

86

Appendix F

Justification of the Updating Technique for the New Policy (s+mS)

Let the probability transition matrix corresponding to the policy (sS) be denoted by the

(S + l)x(S + l) matrix P with elements py Define s = s+m where m is an integer satisfying

the condition 0 lt s+m lt S Let the probability transition matrix of the new policy (sS)

be denoted by the (S + 1)X(S + 1) matrix 7gt with elements py Using the equations developed

in Chapter III the elements of P are found as follows

CASE 1 The lead time is less than the review period

A m gt 0

For 0 lt i lt s

( i-i pound [PrDL=kPrDT_LgtS-k]] + PrDLziPrDT_LplusmnS-i k=0

i-l 53 [PrDL=kPrDT_L = S-j-k]] + PrDLziPr[DT_L = S-j-i) k=0 s-j

52 [Pr[DL = kPrDT_L = S-j-kn k=0

ifj = 6

ifOltjz S-i

if S-i ltj ltS

a) 0 lt lt s

Pij

= lt Pij

Pij

= pij

b) s lt i lt s This does not simplify but it is identical to (2)

87

For s lt i lt S

PrDTgt i

p = lt PrDT = i-j

0

Pfj ifj = 0

= lt Pij ifOltjlti

Pij if i ltj plusmn S

= pij

B m lt 0

ifj = 0

if 0 ltj lt i

if i ltj plusmn S

For 0 lt i lt s

i-l 53 [PrDL=kPrDT_LplusmnS-k]] + PrDLgtiPrDT_LplusmnS-i

i-l E [PrZgti = fciVZ)7W=S--fc] +PrD tgtiP-D 7 t = S-7-

p y laquoy = o

P i j ifOltjS-i

vPij ifS-iltjS

ifj = 0

ifOltjltS-i

if S-i ltjltS

For s lt i lt S

fPrJgtrgt- 17=0 = PrDT = i-j] ifOltjlti

0 ifiltjplusmnS

88

a) s lt lt s This does not simplify but it is identical to (3)

b) s lt i lt S

PrDTgti

= bull PrDT = i-j]

0

17=0

if0ltjplusmni

ifiltjS

= Pt

ifj = 0

if0ltjlti

ifiltjplusmnS

= Pi

CASE 2 The lead time is equals the review period

A m gt 0

For 0 lt i lt s

0 if0ltjlt S-i

PrDTgti ifj = S-i

PrDT=S-j] ifS-iltjltS

a) 0 lt i lt s

= bull y Pij

Pij

ifOplusmnjltS-i

ifj = S-i

ifS-iltjS

= Pij

b) s lt i lt s This does not simplify but it is identical to (4)

89

For s lt i lt S

PrDTgti)

Pr[DT=i-j

0

laquof 7=0

ifOltjplusmni

if i ltj lt S

Pa

Pij

if 7 = 0

if 0 lty lt i

ifiltjltS

B w lt 0

For 0 lt t lt s

ro PrJrgt PrDT = S-j

ifOzjltS-i

ifj = S-i

if S-i ltj lt S

Pij

ifOzjltS-i

ifj = S-i

if S-i ltj ltS

= Pi

For s lt i lt S

PrDTgti

= PrDT = i-j

0

i 7 = o ifOltjplusmni

ifiltjltS

a) s lt i lt s This does not simplify but it is identical to (3)

90

PrDTzi ifj = 0

PrDT = i-j] ifOltjplusmni

0 ifiltjltS

Py ifj = 0

ptj ifOltjplusmni

Pij if iltj plusmnS

Appendix G

Justification of the Updating Technique for the New Policy (sS+m)

Let the probability transition matrix corresponding to the policy (sS) be denoted by the

(S + l)x(S + l) matrix P with elements p(gt Define S = S+m where m is any non-negative

integer Let the probability transition matrix of the new policy (sS) be denoted by the

(S+m+l)x(S+m + l) matrix IP with elements ptj Then using the equations developed in

Chapter III the elements of 7 are found as follows

CASE 1 The lead time is less than the review period

For 0 lt i lt s

pound [PrDL=kPrDT_LplusmnS-k] + PrDLi)PrDT^Sgt- k=0

i-l pound [Pr[DL = kPrDT_L = S-j-k- +Pr[DLiPrDT_L = S-j-i Jt=0

s-i Y [Pr[DL=kPrDT_L = S-j-k] k=0

i - l

ifj=0

ifOltjltS-i

ifS-iltjltS

pound [PrDL = kPrDT_LS-k + Pr[DLi)PrDT_LplusmnS-i) if j = 0 fc=0

i - l = pound [PrDL=kPrDT_L = (S+m)-j-k] + PrDLgtiPr[DT_L = (S+m)-j-i ifOltj lt (J5+m)-i

if (S+m)-iltj lt(S+m)

k=0

i - l

pound (Sm)-j

pound [PrDL = kPrDT_L = (S+m)-j-k]] k=0

92

I-l

53 [PrDL = kPrDT_LgtS-k] + PrDLziPrDT_LS-i ifj = 0 k=0

i-l

Y[PrDL = kPrDT_L = S-(j-m)-k] + Pr[DLziPrDT_L = S-(j-m)-i] ifOltjzS+m-i k=0 S-(j-m)

53 [PrDL = kPrDT_L = S-(j-m)-k]] if S+m-i ltj lt S+m

i - l

53 [PrDL = kPrDT_LzS-k] + PrDLziPrDT_LzS-i] ifj = 0 =0

i - l

13 [Pr[DL = kPrDT_L = S-(j-m)-k] + Pr[DLziPr[DT_L = S-(j-m)-i ifOltjz S+m-i k=o and 0 z j-m

Pi(j-m)

Pi(j-m)

if 0 ltj lt S+m-i and 0ltj-mltS-i

if S+m-i ltj lt S+m and S-i lt j-m zS

i - l

Y [PrDL = k)PrDT_LzS-k] + PrDLi)PrDT_LplusmnS-i ifj = 0

k=0 i - l

53 [PrDL = kPrDT_L = S-(j-m)-k] + PrDLziPrDT_L = S-(j-m)-i ifOltjzm k=0

Piij-m)

Piij-m)

if mltj lt S+m-i

if S+m-i ltj lt S+m

i - l

YlPrDL = kPr[DT_L^S-k + PrDLi)PrDT_LS-i) ifj = 0

i - l

53 [PrDL = kPrDT_L = S-j-k] + PrDLgtiPrDT_L = S-j-i ifOltj i m k=0

Pi if mltj ltS

which for 0 lt j lt m is identical to (2)

93

For s lt i lt S

Pu =

PrDTgti

PrDT = i-j]

0

if0ltjlti

ifiltjltS

a) s lt i lt S

PrDTgti 7=0

PrDT = i -j) if0ltjlti

0 if i ltj lt S

0 ifSltjltSgt

ifj=0

if 0 ltj lt i

if i ltj plusmn S

0 ifSltjltS

ifOltjltS

0 ifSltjltSgt

b) S lt i lt S This does not simplify but it is identical to (3)

CASE 2 The lead time equals the review period

For 0 lt i lt s

0 ifO lt jltS-i

PrDTgti ifj=S-i

PrDT=S-j ifS-iltjltS

94

0 ifOplusmnjlt(S+m)-i

PrDT ifj = (S+m)-i

PrDT = (S+m) -j) if (S+m)-i ltj ltL (S+m)

0 if 0 lt jltS+m-i

PrDTzi] if j = S+m-i

PrDT = S-(j-m) if S+m-i ltj lt S+m

0 if 0 ltj lt S+m-i and 0 ^ j-m

Pi(j-m) if 0ltjlt S+m-i and 0ltj-mltS-i bull

Pi(j-m) if j = S+m-i and j-m = S-i

Pi(j-m) if S+m-i ltj z S+m and S-i lt j-m ^S

0 if 0 zj ltm

Pi(j-m) if mlt jlt S+m-i lt

Piij-m) if j = S+m-i

if S+m-i ltj lt S+m

1deg if 0 ltj ltm

[PiV-m) ifmltjltS

which for 0 lt i lt s is identical to (3)

For s lt lt S

PrDTzi

= PrDT = i-j)

0

ifj = 0

if 0 ltj ltL i

ifKjzS1

95

a) s lt i lt S

PrDTzi

PrDT = i-j]

0

0

17=0

if0ltjplusmni

ifiltjS

ifSltjltSgt

Pij 7=0

Pij if0ltjii

Pij if i ltj ltL S

0 ifSltjltS

Pij ifOzjiS

0 ifSltjltS

b) S lt i lt S This does not simplify but it is identical to (3)

96

Appendix H

A Hypothetical Consultation

In this appendix the interface during a hypothetical consultation is depicted The figures

accurately represent the various screens that are presented to the manager with the exception

that all lines and text are really white on a blue background During the consultation described

below only the choices made by the manager are hypothetical all the results are real

The title screen as depicted in Figure A - l appears initially upon execution of the

program In order to proceed to the main menu the manager presses the enter key

O I C S

Optimat Inventory Control System

For the exclusive use of

Canadian Tire Pacific Associates

Developed at The Faculty of Commerce

and Business Administration The University of British Columbia

Brian Kapalka Kaan Katircioghi Marty Puterman

Press ltEntergt to continue

Figure A - l The title screen

97

Main Menu 1 Select data directory

ctircdatadst

2 Select product and store combination Product 200001

3 Select scenario Review Period 4 Lead Time 3

4 Select search parameter Smax 25

5 Execute program

6 Exit to DOS

Please enter the desired option 5

Figure A-2 The main menu

Figure A-2 depicts the main menu of the program it is from here that the manager

enters all the review parameters If the manager wishes to change one of the parameters from

its default value he or she selects the appropriate number and enters the correct information

in the window presented The new settings will then be shown in place of the default values

In this example the relevant data files are located in the default directory ctiredatadst and

the manager has selected the productstore combination of product 200001 and store 6 Further

the manager has selected a review scenario of a 4-day review period and a 3-day lead time and

an upper search limit of 25 Finally in order to proceed with the analysis the manager

executes the main module of the program by selecting option 5

98

List of Figures

Figure 1 A typical sample path of the process with a 4-day review period

and a 2-day lead time 24

Figure 2 The flow chart of the grid search algorithm 38

Figure 3 The flow chart of the grid search algorithm utilizing the updating

technique for the transition probability matrix 44

Figure 4 An evaluation of the lower bound Sm i n 47

Figure 5 The flow chart of the heuristic algorithm 50

Figure 6 The distribution of savings of optimal policies in (a) dollars and

(b) percentage of current cost 59

Figure A-1 The title screen 97

Figure A-2 The main menu 98

Figure A-3 Calculating the optimal policy and evaluating the current policy 99

Figure A-4 Displaying the results 100

Figure A-5 Entering an alternate policy 101

Figure A-6 The particulars of the alternate policy 102

Figure A-7 The main menu revisited 103

vi

Acknowledgement

The completion of this thesis was made possible by the encouragement and assistance

of a number of people

I would like to express my sincere appreciation to my thesis supervisor Professor

Martin Puterman for all of his many efforts on my behalf His help advice patience charity

and tolerance were very much appreciated

I would like to acknowledge Professor Hong Chen and Professor Garland Chow for their

time and input as members of my thesis committee In addition I must acknowledge the

assistance of PhD student Kaan Katircioglu for his insight and help on this project

I offer many thanks to Professor Tom Ross for his kindness and friendship during the

past few years especially during the writing of this work I must also thank his family for

adopting me on many holidays I still owe thanks to Professor Slobodan Simonovic at the

University of Manitoba for influencing me to attend graduate school in the first place and for

helping me to obtain funding

I cannot thank my parents enough for their never-ending support both emotional and

financial Also to my friends especially Cathy Dave Lisa Steve and the Philbuds thanks

for giving me a life off campus and for picking up many a tab - the next one is on me

I would like to give special thanks to my good friend and fellow MSc student Paul

Crookshanks for allowing me to bounce ideas off of him and for being such a procrastinator

that despite my finishing a year late I only lost by two days

Financial assistance received from the Natural Science and Engineering Research

Council (NSERC) and from Canadian Tire Pacific Associates was greatly appreciated

vii

I INTRODUCTION

The importance of inventory management has grown significantly over the years

especially since the turn of this century In colonial times large inventories were viewed as

signs of wealth and therefore merchants and policy makers were not overly concerned with

controlling inventory However during the economic collapse of the 1930s managers began

to perceive the risks associated with holding large inventories As a result managers

emphasized rapid rates of inventory turnover (Silver and Peterson 1985) Following the

Second World War Arrow Harris and Marschak (1951) and Dvoretzky Kiefer and

Wolfowitz (1952ab) laid the basis for future developments in mathematical inventory theory

Shortly thereafter new inventory control methodologies were widely applied in the private

manufacturing sector More recently when inflation and interest rates soared during the 1970s

many organizations were forced to rethink their inventory strategies yet again Today the

control of inventory is a problem common to all organizations in any sector of the economy

Sir Graham Day chairman of Britains Cadbury-Schweppes P C L expressed this sentiment

when he stated I believe the easiest money any business having any inventory can save lies

with the minimization of that inventory (Day 1992) Lee and Nahmias (1993) and Silver and

Peterson (1985) provide a more detailed history of inventory management and control

Most inventory problems in the retail sector involve the control and management of a

large number of different products Ideally the inventory policies should be determined for all

products on a system-wide basis however because the number of products is often in the tens

of thousands formulating a model to do so is impractical Therefore single-product models

are frequently used in practice to determine policies for products separately (Lee and Nahmias

1993) Lee and Nahmias (1993) and Silver and Peterson (1985) provide a good overview of

1

single-product single-location models for a more complete review of the literature we refer

to them

The degree of complexity of a single-product model depends largely on the assumptions

made regarding the demand cost structure and physical characteristics of the inventory system

For example demand can be either deterministic meaning it is known with absolute certainty

or stochastic meaning it is unknown but follows a known probability distribution

Deterministic models tend to be relatively simple yet useful such as the basic EOQ model

However future demand is seldom known with certainty in practice and therefore the use of

these simple models is limited As a result the majority of research in inventory has focused

on stochastic demand models (Lee and Nahmias 1993)

Because the objective of most inventory models is to minimize the cost of inventory

assumptions regarding the structure of the cost are very important also Generally speaking

the cost of inventory has four components the purchaseproduction cost the orderingsetup

cost the holding cost and the shortage cost1 Each of these costs can be fixed proportional

nonlinear or some complex combination of these Further because nearly all inventory

problems require decisions be made sequentially over a period of time the costs can be either

discounted or averaged In the former case a discount factor is included in the cost function

in order to account for the time value of money However since most inventory problems are

on-going the long-run average cost is usually minimized (Lee and Nahmias 1993) In

practice this is achieved by minimizing the expected cost per time period under steady state

conditions

Finally the physical characteristics of the system are important in determining the

1 The cost of a stockout is often very difficult to estimate in practice As a result service levels are frequently used in its place When this is done a constraint is included in the model

2

complexity of the model For example the review process can be either continuous in which

case the inventory is reviewed continuously and orders are placed as soon as the stock reaches

a certain level or periodic in which case the inventory is reviewed at regular intervals and

orders can be placed only at those fixed times Also excess demand can be fully backordered

as is common in many manufacturing and wholesale systems completely lost as is common

in most retail environments or some combination of the two Lead time assumptions are also

important models with lead times are more complex than those in which the supply of

inventory is assumed to be instantaneous

Although the literature is dominated by single-product models (Lee and Nahmias 1993)

much recent work has focused on extending those models to multiple-product problems

However multiple-product models are inherently more complex than the simple-product

models In fact no efficient algorithms are currently available for determining fully optimal

strategies for most multiple-product problems (Federgruen 1993) As a result research has

focused on the development of easily computable approximation techniques Muckstadt and

Roundy (1993) provide a good overview of multiple-product andor multiple-location models

with deterministic demand although emphasis is placed on models for production and

distribution Federgruen (1993) extends the discussion to consider these models under uncertain

demand and also provides a complete list of references on the subject Silver and Peterson

(1985) discuss models for coordinated replenishments for both deterministic and stochastic

demand

In this thesis we examine the inventory system at Canadian Tire Pacific Associates

owner of 21 retail stores and a central warehouse Although this is a multiple-product

multiple-location single-warehouse problem and strictly speaking should be modelled as such

we analyze the inventory problem on a product-by-product and store-by-store basis We believe

3

that this simpler single-product single-location model captures the essentials of the problem for

two reasons First the coordination of replenishments for products andor locations is most

advantageous when significant savings are possible because of high first order costs and low

marginal costs for placing subsequent orders However because of the large number of

products and the physical distribution system in this problem trucks are dispatched to stores

on a regular basis Therefore the benefits of coordinating replenishments for this problem are

negligible Second we believe that controlling in-store inventory alone can result in substantial

savings to Canadian Tire Pacific Associates Further given the excessive amount of inventory

in the warehouse a single-location model assuming an infinite warehouse capacity is reasonable

Of course should inventory control in the warehouse become a priority a more elaborate model

would be required

More specifically we consider a periodic review single-product inventory model with

stationary independent stochastic demand a deterministic lead time and an infinite planning

horizon The probability mass function for the demand is based on empirical sales data

collected over a period of approximately ten months The cost of inventory includes a

proportional holding cost and a fixed ordering cost neither of which is discounted All excess

demand is completely lost The objective of the model is to minimize the long-run average cost

per unit time subject to a service level constraint on the fraction of demand satisfied directly

from in-store inventory In order to obtain meaningful cost comparisons the model calculates

the expected annual cost of inventory

For inventory problems like the one described above optimal (sS) policies can be

computed by Markovian decision process methods such as successive approximations or policy

iteration or by stationary Markovian methods However each of these methods generate

calculations that are complex and time consuming Therefore much research interest lies in

4

developing techniques to find approximate (sS) policies with little computational effort The

majority of the approximation techniques begin with an exact formulation of the problem and

then manipulate the problem by relaxations restrictions projections or cost approximations

(Federgruen 1993) Porteus (1985) presents three approximation techniques and compares

them to fourteen others for a periodic review single-product inventory model in which

shortages are fully backordered and a shortage cost is incurred Federgruen Groenevelt and

Tijms (1984) present an algorithm for finding approximate (sS) policies for a continuous review

model with backordering and a service level constraint Also Tijms and Groenevelt (1984)

present simple approximations for periodic review systems with lost sales and service level

constraints All three works find that the approximation techniques perform well However

the techniques are evaluated for a variety of demand parameters that all result in policies with

a value of S in the range of 50 to 200 Given the slower moving products at Canadian Tire 2

it is unclear how well such approximation techniques would perform Therefore we do not

attempt to apply or modify any of these approximation methods

Exact optimal policies for this problem can be obtained by formulating a constrained

Markov decision process and solving it However the data generating requirements and the

lack of a good upper bound on S make this formulation impractical Therefore we formulate

a Markov chain model for fixed values of s and S and then employ a search technique to find

the optimal values for s and S In order to improve the performance of the model we develop

a heuristic search that locates near optimal policies relatively quickly

The remaining chapters of this thesis are organized as follows In Chapter II an

overview of the existing inventory system at Canadian Tire is presented along with a description

2 For example very few of the products analyzed have optimal values of S exceeding 20 with the largest value of S equalling 69

5

of the problem Next a multi-phase project designed to develop a fully integrated interactive

inventory control system is described with the scope of the research reported in this thesis

completing the first phase of the project Finally a description of the demand and cost data is

given

In Chapter III we formulate a Markov chain model that can be used to calculate the cost

and service level associated with a given (sS) policy Based on several simplifying

assumptions the Markov chain is formulated and the equations for the corresponding transition

probability matrix are developed Then given the limiting probability distribution of the chain

the equations for the steady state annual cost of inventory and the corresponding service level

are obtained The chapter concludes with a brief summary of the model

In Chapter IV an algorithm that searches for the optimal (sS) policy is developed As

a starting point a simple grid search algorithm is developed and evaluated on the basis of

execution time In order to improve the algorithms speed a technique for updating the

transition probability matrix is presented and the improvement in the algorithm is noted Next

a theoretical lower bound on S is obtained although the corresponding reduction in the

execution time is minimal Finally based on two assumptions regarding the shape of the

feasible space a heuristic search is developed which improves the speed of the algorithm

significantly

In Chapter V a prototype of the inventory control systems interface is created In its

present form the interface allows managers to evaluate inventory policies interactively The

manager can set review parameters compare optimal policies and current policies and evaluate

alternate policies The chapter concludes by recommending specific options be including in

future versions to make the system more automated and user-friendly

In Chapter VI optimal policies for all products in one product category sold at one store

6

are compared to existing policies on the basis of expected annual costs and customer service

levels The analysis suggests that existing policies result in a large excess inventory and that

implementing the optimal policies would reduce the annual cost of inventory by approximately

50 A sensitivity analysis on the cost estimates is performed and suggests that the model is

not sensitive to relative errors of approximately 20 in the cost estimates Finally a sensitivity

analysis on the demand probability mass function is performed The analysis indicates that the

model is sensitive at least in terms of the customer service level to slight changes in the tail

of the demand distributions of low-demand products

Finally in Chapter VII the results of the analysis are summarized and future work is

outlined In order to incorporate recent changes in the review procedures at several stores the

model must be modified to allow for the duration of the review period to exceed the lead time

We recommend that the second phase of the project proceed specifically that a data collection

and storage scheme be designed and that a forecasting model be developed Because of the

substantial savings that can be realized by reducing inventory only at the stores inventory

control at the warehouse should also be investigated

7

II T H E I N V E N T O R Y S Y S T E M A T C A N A D I A N TIRE

A Background

Canadian Tire Pacific Associates owns and operates 21 retail stores in the lower

mainland of British Columbia The stores carry approximately 30000 active products3 that

are divided into 80 product categories The merchandise comprises primarily automotive parts

and accessories home hardware housewares and sporting goods Canadian Tire Pacific

Associates also owns a central warehouse in Burnaby at which inventory is received from

Toronto Inventory is then shipped periodically from the warehouse via truck to the local

stores In 1992 total sales at the stores exceeded $99 million and the value of inventory was

estimated to be $28 million of which approximately half was stored in the warehouse

On selected days of each week a stores inventory records are reviewed automatically

and orders for products are placed to the warehouse as required Management uses the term

flushing to describe the process of reviewing the inventory and producing the tickets for stock

owing and the term picking to describe the physical act of filling the order at the warehouse4

Whether or not an order is placed for a product depends upon its current inventory level its

target level and its minimum fill 5 If the target level less the current level exceeds the

minimum fill an order is placed to bring the stock up to the target level From the moment

3 Active products are those that are regularly restocked Canadian Tire Pacific Associates can choose from up to 50000 products

4 Note that the number of picks per week is equal to the number of times per week that the inventory records are reviewed At the beginning of this study eighteen stores had two picks per week while the remaining three stores had only one

5 The target level is the maximum number of items that should be on the shelf at one time and the minimum fill is the minimum number of items that should be ordered

8

the order is placed to the moment the product is reshelved takes between two to four days

depending on the store On average 11000 items are shipped from the warehouse at each

flush

Although a products minimum fill is the same for all stores its target level depends

upon the size of the store Based upon its sales volume and retail space a store is classified

as being one of five sizes small (S) medium (M) large (L) jumbo (J) and extra jumbo

(EJ) 6 Each product has associated with it a target level for each size of store

Consequently all stores of the same size have identical inventory policies Table 1 lists the

location size number of picks per week sales and floor space of each store as of October

1993

Periodically the inventory manager reviews the policies and when deemed necessary

adjusts them In this way the manager can account for seasonality in demand changes in mean

demand and any shortcomings of previous policies At the present time the review process

is not automated nor performed on-line The inventory manager uses Hxl5 computer

printouts with five products to a sheet to study the relevant information and on which to note

any changes Afterward the changes are recorded in the system and the new policies take

effect

Currently the inventory policies are based on rules of thumb developed through the

years and not on a statistical analysis of demand nor on a calculation of the cost of inventory

These rules of thumb incorporate a marketing philosophy which at times dominates the

inventory strategy The prevailing philosophy holds that a large in-store inventory instills

6 During the course of this study stores were resized several times At one point the entire class of medium stores was eliminated leaving only four sizes of stores On another occasion the store in Chilliwack was changed from being a small store to a large store

9

Table 1 Location and particulars of the 21 stores

Store Location Store Size

Picks Per Week

1992 Sales ($)

Floor Space (ft2)

1 Chilliwack S 2 2901969 6678

2 Coquitlam M 2 4015303 6510

3 Langley EJ 2 8138673 15422

4 Scott Road EJ 2 7307820 15064

5 Maple Ridge L 2 4102110 6678

6 Kingsway L 2 4153736 6574

7 Richmond EJ 2 8175105 14961

8 North Vancouver M 2 3856571 6534

9 Guildford M 2 3714372 6567

10 Burnaby M 2 3532752 6983

11 New Westminster S 1 2417992 6470

12 Abbotsford EJ 2 7899398 15303

13 Hastings J 2 4647794 9849

14 King Edward EJ 2 3825372 25492

15 White Rock L 2 3960262 6870

16 Main amp Mountain S 1 2705602 6431

17 Heather J 2 4161227 9427

18 King George J 2 4606817 9640

19 Cambie J 2 5855022 9091

20 Lougheed M 1 3114180 6615

21 Burrard J 2 6553310 10608

10

consumer confidence which in turn leads to increased sales In other words if a customer

perceives that Canadian Tire is in the hardware business he or she will choose to shop there

when meeting future needs However management has adopted this philosophy without any

analysis of the cost of holding excess inventory Only recently has management begun to

question the cost-effectiveness of allocating inventory based on these rules of thumb

Speculating that existing inventory practices might be deficient Mr Don Graham the

vice-president of Canadian Tire Pacific Associates commissioned a pilot study to examine the

inventory system and to propose alternate strategies7 The study using a steady state model

to represent a simplified inventory system suggested that existing policies lead to overly high

inventory levels More importantly the study outlined a methodology that would enable

management to determine inventory levels based on observed demand The pilot study

recommended that a more detailed model of the inventory system be formulated that the

methodology be expanded to incorporate the cost of inventory and that an algorithm be

developed that would quickly identify the optimal inventory policies

B The Problem

At the time of the pilot study management was concerned primarily with its inability

to distinguish between stores of the same size As mentioned in the previous section a

stores size is based solely on its sales volume and floor space and under existing practices

all stores of the same size have identical inventory policies for every product Management

realized that demand for some products might be a function of variables other than size such

7 The pilot study was performed by Martin Puterman a professor of Management Science and Kaan Katircioglu a doctoral student in Management Science both at the University of British Columbia The results of the study were presented at a U B C Transportation and Logistics Workshop on March 25 1993

11

as location For example the demand for sump pumps is likely to be very different in suburban

areas than in urban ones However management was unable to allocate more sump pumps to

large suburban stores than to large urban stores This inability led management to seek a

tool that would allow for a more effective allocation of inventory between stores of the same

size

By the end of the pilot study however management was less concerned with

reallocating inventory within a size of store than with reallocating inventory between sizes

of stores and with reducing overall inventory According to management assigning individual

inventory policies to each store appeared to be operationally infeasible Firstly management

believed that having individual policies would lead to confusion in the warehouse and to

increased shipping errors Exactly why this would occur is unclear the warehouse would

simply continue to fill orders from tickets that make no reference to the policy in place

Secondly and perhaps more tenably the computer technology at Canadian Tire Pacific

Associates did not allow for policies to vary across all stores Both the database used to store

the inventory data and the printouts used by the inventory manager allowed for only five

different policies per product all with the same minimum fill Moving to individual policies

would have required an overhaul of the existing information system Although management

saw this overhaul as inevitable it did not see the overhaul as imminent Therefore

management switched the focus of the study to developing a tool that would help reduce overall

inventory levels andor reallocate inventory more effectively between groups of stores

The subtle change in the studys objective prompted an important question how should

the stores be grouped The current practice of grouping stores based on size is problematic

for two reasons First as discussed above additional factors of demand such as location

cannot be taken into account Second grouping inherently results in a misallocation of

12

inventory when some stores within a group have differing numbers of picks per week For

example the medium-sized Lougheed store which has one pick per week requires higher

inventory levels to achieve a given customer service level than the other medium stores

which all have two picks per week8 In fact in terms of customer service levels it might

be more appropriate to group the Lougheed store with the large stores having two picks per

week Ideally this artificial grouping of stores will be abandoned when the computer system

is upgraded and policies can then be set on a store by store basis

As the project proceeded management grew somewhat uneasy with the concept of using

an analytical tool to evaluate and set inventory policies According to both the pilot study and

the preliminary results of this study significantly reducing inventory would greatly reduce costs

while still providing a high level of customer service However management held that because

the model cannot take into account intangibles such as marketing considerations such a drastic

reduction in inventory would be very inappropriate In this light management viewed the

optimal policies as lower bounds for inventory levels these policies would have to be

modified upwards in order to be practical Therefore management contended that any

methodology or software produced during this study should provide the means to evaluate and

compare a number of alternate policies

Regardless of exactly how the tool is to be used there is currently a lack of historical

sales data from which to analyze the inventory system and determine optimal policies No sales

data had ever been collected prior to the pilot study and only a relatively small amount has

been collected since In order to proceed with any analysis of the inventory system a

8 If demand is a function of size then the Lougheed store must meet the same demand with one fewer pick per week This would lead to either increased probabilities of stockouts at the Lougheed store or higher inventory levels at all the other medium stores

13

substantial amount of additional data would be needed The existing information system at

Canadian Tire Pacific Associates does allow for sales data to be captured however the system

is limited by the amount of data it can capture at one time and by its storage capacity In order

to collect the amount of data required management would have to upgrade the system and

develop some kind of data collection scheme coupled with a data compression technique The

collection scheme must allow for characteristics of demand such as trends and seasonality to

be identified and the compression technique must allow for sales data for over 30000 products

to be stored for each store

Because many of the products sold by Canadian Tire have nonstationary demand as a

result of seasonality andor trend the inventory tool must be able to forecast demand for the

coming period Products such as sporting goods and outdoor equipment exhibit patterns of

demand that are very seasonal For example sales of snow shovels increase through the fall

peak in the winter and drop to zero by late spring At the same time some products might

become either more or less fashionable over time resulting in changes in mean demand

Setting inventory policies for any of these products based on observed demand would result in

insufficient stock if demand increased during the period or in excess stock if demand decreased

Therefore the tool must be able to forecast the demand for the period for which the policy is

intended

Toward the end of the project management began discussing the integration of the

interactive inventory tool with the existing system to form a new inventory control system The

new system would utilize a modified point of sales system to collect sales data continually

Based on the latest available data the system would forecast demand for the coming period and

calculate the optimal policies for every product at every store The system would then provide

all relevant information on-line for the inventory manager to review The manager could weigh

14

other information such as marketing considerations and examine alternate policies to determine

the effect on cost Finally the preferred policy would be set in place with the touch of a key

and information on the next product or store would be presented for review

C The Project

Development of the inventory control system envisioned by management requires the

completion of five tasks 1) the design of a data collection system that generates and stores the

necessary sales data 2) the development of a forecasting method that provides accurate

estimates of demand distributions 3) the formulation of a model that calculates the cost of

inventory and the customer service level for an inventory policy 4) the development of an

algorithm that finds the optimal policy for a product at a store and 5) the creation of an

interface that allows interactive and on-line consultation Ultimately the above components

would have to be integrated and incorporated in a manner that would ensure the compatibility

and reliability of the system

The scope of the research reported in this thesis completes the first phase of the project

specifically we formulate an inventory model develop an algorithm and create a prototype of

the interface Because of the changing environment at Canadian Tire Pacific Associates minor

modifications to both the model and the algorithm developed herein might be necessary to

reflect future changes in operational procedures Upon completion of phase one the interface

will be interactive allowing managers to select the store product and review scenario and then

to compare optimal policies with existing policies or with alternate policies Although the

prototype will be delivered to Canadian Tire Pacific Associates for ih-house testing at the end

of this phase the interface will require modification in subsequent phases as feedback from

management is received and as the system is brought on-line

15

In order to assess more fully the potential savings of implementing the inventory control

system this thesis includes a comparison of optimal policies9 and existing policies The

expected annual cost of inventory for an entire product category is calculated for both the

optimal policies and the existing policies Based on an analysis of these costs the potential

savings of implementing the optimal policies are extrapolated Further the thesis examines the

sensitivity of the model to variations in the estimates of demand and costs

Originally results from all 21 stores were to be included in the above analysis

However just prior to the analysis management changed the review procedure at several

stores the result of which being that three picks per week are now performed at those stores

Because one of the main assumptions of the model specifically that the length of the lead time

does not exceed the duration of the review period is violated under this new review procedure

those stores cannot be included in the analysis Therefore the analysis is based solely on the

results from store six the large store located on Kingsway Avenue10

For the purpose of the analysis sales data for product category 20 automotive and

electrical accessories were collected from all 21 stores This product category was selected

because the distribution of demand for products within it is non-seasonal and because few of

the products are ever placed on sale11 The lack of trend or seasonality eliminates the need

for forecasting demand the optimal policies can be determined based directly on observed

demand

9 That is those policies obtained from the model

1 0 Store six is one of the only stores whose review procedure is not expected to change from two picks per week

1 1 Products that were on sale at any time during the period were not included in the data set

16

D The Demand Data

In total sales data for 520 products of which 458 are active products were collected

over a 307-day period from October 10 1992 to August 18 199312 The data was supplied

by Canadian Tire Pacific Associates in the form of a 10 MB text file with one line per day per

product recording sales from the stores13 Each line had the following format

Columns 1-6 part number Columns 7-9 Julian date14

Columns 10-18 cost of the item (to three decimal places) Columns 19-124 sales for stores 1 through 21 (5 columns per store)

In order to facilitate the analysis the file was sorted into 458 sales files with each of

these files containing the history of daily sales for one of the products Two changes were

made to the format of these files to improve readability 1) a column was added between the

product number and the Julian date and 2) all leading zeros were dropped Appendix A

contains a partial printout of the sales file for a typical product Finally a distribution file

summarizing the daily sales was created from each of the sales files Each distribution file

contains the products number the products cost and the number of days during which various

quantities of the product were sold at each store

Prior to the analysis the distribution files for a large number of products were

examined and based on the shapes of the histograms the products were divided into three

groups The first group low-demand comprises products that have demand on only a few

days with the daily demand seldom exceeding three items The second group medium-

1 2 Of the active products only 420 are sold at store 6

1 3 Strictly speaking a line was written to the file only on days on which at least one store registered a sale of the product

1 4 According to the Julian calendar days are numbered sequentially from 1 to 365 (366 during leap-years) beginning with the first of January

17

demand comprises products that have demand on approximately half of the days with the daily

demand seldom exceeding more than six items Finally the last group high-demand

comprises products that have demand on most days occasionally upwards of twenty items and

whose demand histograms resemble Poisson distributions The majority of the products

examined fell into the low and medium-demand groups only a few high-demand products

were observed Appendix B contains a printout of the distribution file for a typical product

in each group

For most products no theoretical distribution appeared to model the demand well

Demand for low and medium-demand products followed a non-standard distribution with

a mass at zero and a relatively small tail Although some theoretical distributions such as the

mixed-Poisson can be made to resemble this it was unclear how well they would perform

Further the effects of sampling errors from such non-standard distributions were unknown

On the other hand the standard Poisson distribution did appear to model high-demand

products well however only a few products fell into that group Therefore in the absence of

a good theoretical candidate the empirical demand function was chosen for the analysis The

empirical function was readily available from the distribution files and offered the best

estimate of the true underlying function

E The Cost Data

Along with the sales data management supplied estimates relating to the costs of holding

and ordering inventory Management estimates that the cost of holding an item in inventory

for one year is 30 of the items cost Incorporated in this holding rate are a number of costs

such as the opportunity cost of money invested and the costs incurred in running a store

Because it is the annual cost of inventory that is of interest either the compounded daily

18

holding rate or the simple daily holding rate can be used to determine the costs incurred during

a review period however the method of calculating and extrapolating the holding costs must

be consistent For the sake of simplicity a simple daily holding rate is used

In order to estimate the cost of ordering inventory management analyzed the various

tasks that result from an order being placed Using the average time required to fill and then

reshelve an order management obtained an estimate of the direct labour costs involved To

this management added an estimate of the cost of transporting an order from the warehouse to

the store Finally the estimate was adjusted slightly upward to account for the cost of breakage

and errors Based on the analysis management estimates the cost of placing an order for a

product to be $0085 regardless of the size of the order

19

III M O D E L F O R M U L A T I O N

A Terminology

In the previous chapter several terms were introduced to describe the inventory system

at Canadian Tire Pacific Associates These terms though perfectly adequate are not widely

used in the academic literature to describe inventory management problems Therefore in

order to avoid confusion and be consistent with the literature the following terms are used

throughout this thesis

The length of time between successive moments at which the

inventory position is reviewed15

The time at which a reorder decision is made

The length of time between the moment at which a decision to

place an order is made and the moment at which the order is

physically on the shelf

An operating policy in a periodic review inventory model in

which (i) the inventory position is reviewed every T days (ii) a

replenishment order is placed at a review only when the inventory

level x is at or below the reorder level s (gt 0) and (iii) a

replenishment order is placed for S-x units (S gt s)1 6

Lost Sales - The situation in which any demand during a stockout is lost

Review Period -

Decision Epoch

Lead Time -

(sS) Policy

1 5 With respect to the terminology used at Canadian Tire Pacific Associates a review period equals the number of days between flushes

1 6 With respect to the terminology used at Canadian Tire Pacific Associates S represents the target level and s represents the difference between the target level and the minimum fill

20

B Assumptions

In to order make the model of the inventory system mathematically manageable several

simplifying assumptions are made However we believe that the following assumptions do not

significantly reduce the accuracy of the model

1) Demand can be approximated very closely by sales during the period in which the data

is collected Generally inventory levels at the stores are very high and the number of

lost sales is assumed to be negligible In fact the inventory levels for a number of

products were tracked over time during the pilot study and not a single incidence of

stockout was observed Because all products appear to have similar excess inventory

it seems reasonable to assume that virtually all demand resulted in sales

2) The demands in successive periods are independent identically distributed random

variables Preliminary tests (Fishers exact tests) suggested no dependence between

sales on consecutive days and plots of sales over time indicated no seasonal pattern

3) Demand is independent across products That is we assume that sales of one product

do not affect nor can they be predicted from sales of another product

4) Unmet demand results in lost sales with no penalty other than the loss of the sale

5) The duration of the review period for a store is an integer value and is fixed over a

planning period For stores with two picks per week we assume that the inventory

is reviewed every four days rather than on a four day - three day cycle

6) The decision epoch is the start of a review period Review periods begin at midnight

at which time the inventory is reviewed and orders are placed instantaneously

7) The duration of the lead time for a store is deterministic and integer-valued In reality

the lead time for some stores was stochastic but management felt the variation could be

controlled and virtually eliminated

21

8) The lead time does not exceed the review period In other words items ordered at a

decision epoch arrive and are shelved before the next decision epoch In practice this

did not always occur

9) Orders arrive at the end of the lead time and are on the shelf to start the next day

10) Holding costs are incurred daily for all inventory held at the start of the day

11) The warehouse has an infinite capacity that is all orders to the warehouse are filled

Because of the high inventory levels in the warehouse stockouts in the warehouse are

very unlikely

C A Markov Chain Model

Consider the situation in which an (sS) policy is applied to an inventory system based

on the above assumptions Inventory is reviewed at the decision epoch for each of the periods

labelled t = 0 12 and replenishment occurs at the end of the lead time provided an order

was placed Demand during day d in period t is a random variable whose probability mass

function is independent of the day and period The function is given by

f^ik) = Pr ltd = k = ak for k = 0 1 2

where ak gt 0 and E ak = 1 Let T denote the duration of a review period and L denote the

length of the lead time Next define D T ( as the total demand during period t D L as the total

demand during the lead time in period t and D T L as the difference between D T and DLt

Assuming that successive demands pound f pound 2 are independent the probability mass function of

1 7 The aggregated demand variables are given by

^ = XXd and DT_Ut=J d= d= d=L+l

22

each aggregated demand variable can be obtained by convolving the probability mass function

of t d l s Finally let X denote the inventory on hand at the beginning of period t just prior to

the review

The process X is a discrete time stochastic process with the finite discrete state space

0 1 S A typical sample path of the process with a review period of four days and a

lead time of two days is depicted in Figure 1 As period t begins X units are on hand The

inventory is reviewed and because X is less than s an order is placed for (S-X) units Over

the next two days a demand of D L r occurs and the inventory drops accordingly With the

arrival of the order the inventory level increases by (S-X) at the start of the third day of the

period Over the remaining two days a demand of D T L is observed bring the inventory level

down to X + 1 Note that in this example an order would not be placed in period t+l because

X r + i exceeds s

The stock levels at the beginning of two consecutive review periods are related by the

balance equation

_K- DLy+ (s - v - DT-LX ixtlts ( 1 )

t + l ~ (Xt - DTtY ifXtgts

where the positive operator is interpreted as

= max a 0

1 8 For example the probability mass function of D T bdquo denoted fD T ( is the T-th fold convolution of f t Mathematically the function can be obtained using the recursive relation

and the identity f1^ = f5(

23

-2 I 8

gt o

r

CO

o a

CM

X

00

24

Because X + 1 depends only on Xt the process X satisfies the Markov property

PrXt+l=jX0 = i0Xt_l=it_vXt = i) = PrXt+1=jXt = i

and therefore is a discrete time Markov chain Further because the probability mass function

of the random variable pound is independent of t and thus so too are the probability mass functions

of the aggregated demand the Markov chain is stationary19

D The Transition Probability Matrix

The stationary transition probability p(gt is defined as the probability of X + 1 being in state

j given that X is in state i independent of the period t In the inventory model p-- represents

the probability of starting the next review period with j items on the shelf given there are i units

on hand at the start of the current period By convention the transition probabilities are

arranged in an (S + l)x(S + 1) matrix P such that (P)y = p- Thus the (i + l)st row of P contains

the probability distribution for the stock on hand at the beginning of the next period given that

there are i units on hand at the start of the current period The matrix P is known as the

transition probability matrix Note that since we are concerned with a steady state analysis we

need not consider the possibility of initial stock levels greater than S once the inventory level

is below S it can never again exceed S

The transition probability ptj is calculated by expanding (1) to include all possible

instances of demand that allow for a change in inventory from i units at the beginning of a

period to j units at the beginning of the next period For example suppose the inventory policy

(510) is in place 2 items are on hand at the start of the current review period and 7 items are

1 9 Henceforth the subscripts t and d are omitted from the daily and aggregate demand variables for the sake of brevity

25

on hand at the start of the next period Then only three scenarios are possible

1) No items are sold during the lead time (meaning that the stock level increases to 10

after replenishment) and 3 items are sold during the remainder of the period

2) One item is sold during the lead time and 2 items are sold during the remainder of

the period

3) Two items are sold during the lead time (though demand could exceed 2) and 1 item

is sold during the remainder of the period

The transition probability p 2 7 is the sum of the probabilities of the various scenarios and is

computed by

p2J = PrDL=0PrDT_L = 3] + PrDL = l] Pr[DT_L=2 + PrDL2] PrDT_L = l]

In general the transition probabilities for the inventory model are calculated as follows

Case 1 The lead time is less than the review period

For 0 lt lt s

Pij

i-l pound [PrDL=kPrDT_LgtS-k] + PrDLgtiPrDT_LgtS-i if j = 0

i - l

^[PriD^^PrlD^S-j-kn+PHD^PriD^S-j-i if 0 lt j ltL S-i (2) k=0 s-j

E [PrDL=kPrDT_L = S-j-k] if S-i ltj^S k=0

26

For s lt i lt S

PrDTzi]

0 p = PrDT=i-j)

ifj = 0 ifOltjzi ifiltjltS

(3)

Case 2 The lead time equals the review period

For 0 lt i lt s

Pa

0 PrDTzi) PrDT = S-j

ifOltLJlt S-i ifj = S-i if S-i ltj plusmn S

(4)

For s lt i lt S use (3)

E The Steady State Probabilities

Provided 1) there is a positive probability of non-zero demand during a review period

and 2) there is a positive probability of zero demand during a review period20 then the

Markov chain described by (1) has a limiting probability distribution ir = (ir0 TTu irs)

where 0 lt IT lt 1 for j = 0 1 S and E 7ry = 1 Further this limiting distribution is

independent of the initial state In other words the probability of finding the Markov chain in

a particular state in the long-run converges to a nonnegative number no matter in which state

the chain began at time 0 The limiting distribution is the unique nonnegative solution of the

equations

2 0 That is 0 lt PrD x = 0 lt 1 Given the nature of the problem this assumption is not very restrictive

27

^ = E ^ - far j = 0 l S (5)

s and pound nk = 1 (6)

For the proof that the limiting distribution is independent of the initial state and is obtained from

(5) and (6) the reader is directed to Appendix C

In terms of the inventory problem 7ry is the steady state probability of having j units of

inventory on hand at the decision epoch However some subsequent calculations are made

easier if the steady state probability of having j units of inventory on the shelf just after

replenishment is known Instead of reformulating the Markov chain these probabilities can be

obtained by shifting the limiting distribution TT Much in the same way that the transition

probabilities are calculated the shifted limiting distribution t] is obtained by accounting for

all possible instances of demand that allow for the shifted inventory position For example

suppose that the policy (510) has the limiting distribution TT In order to have say 8 units on

hand immediately after replenishment one of the following must occur

1) Ten items are on hand at the decision epoch and 2 items are sold during the lead time

2) Nine items are on hand at the decision epoch and 1 item is sold during the lead time

3) Eight items are on hand at the decision epoch and nothing is sold during the lead time

4) Either 2 3 4 or 5 items are on hand at the decision epoch and 2 items are sold during

the lead time

The steady state probability of having 8 units on hand just after replenishment is calculated as

n 8 = n10PrDL = 2] +iz9PrDL = l] +n8PrDL = 0 +[iz5 + iz4 + iz3]PrDL = 2 +n2PrDLgt2

28

In general the shifted steady state probabilities are given by

s pound nPrDLgti ifj = 0

i=s+l S

pound PrDL = i-j] ifl plusmnjltS-s i=max(s+l)

S = pound 7tjirZ)L = i-+7i 5 FrD i ^5 ifj = 5-5

i=max(s+l)

^ 7 1 ^ ^ = 1-7+ pound TtiFrZgti=5-y+iis_PrD1^S-7 ifS-sltjltS i=max(gt+l) i=S-j+l

(7)

F The Cost of Inventory

For this problem the cost of inventory has two components (i) a fixed ordering cost

cf incurred whenever an order is placed and (ii) a holding cost cv incurred daily for each item

held in inventory The expected cost of inventory for a review period can be calculated by

summing the costs associated with each inventory position under steady state conditions

Because a simple holding rate is used this cost is simply adjusted by number of review periods

in a year to obtain the expected annual cost of inventory

The ordering cost for a review period is relatively straight forward to calculate either

an order is placed in which case the order cost is incurred or no order is placed in which case

no cost is incurred According to the rules of the inventory policy orders are placed only if

the stock on hand at the beginning of a review period is less than or equal to s Further the

probability that the review period begins with the inventory in one of these positions is equal

to the sum of the first (s + 1) steady state probabilities Therefore the expected ordering cost

29

for a review period is given by

E[OC] = cfY n y=o

Unlike the simple structure of the ordering cost the holding cost is incurred daily for

each item held in inventory Therefore the daily holding costs associated with all possible

inventory positions must be weighted and summed to obtain the holding cost for the review

period In order to make the calculations less complicated a shifted review period beginning

immediately after replenishment and ending just prior to the next replenishment is considered

Define e(0 to be the expected number of items on hand to start day i of the shifted

review period and let g(i f) denote the expected number of items on hand to start day i given

that there were j items immediately following replenishment21 The function g(z j) is the sum

of the probabilities of possible demand weighted by the resulting stock levels that is

g(ij) = j^kPriD^j-k)

To obtain the expected number of items on hand to start day i independent of j the above

function is summed over all possible values of j and weighted by the corresponding steady state

probabilities yielding

s j ed) = pound r 5 gt P r Z ) = - pound

7=0 k=

2 1 In other words g(if) is the expected number of items on hand to start day i given e(0)=v

30

The expected holding cost for a review period is then given by

T

E[HC] = $gt v e(0 i=i

i=l =0 k=l

where c v equals the simple daily holding rate multiplied by the cost of the item

Although differently defined review periods are used to calculate the above expected

costs the costs can still be summed to obtain the expected cost of inventory for a review

period The expected costs are calculated based on steady state conditions and therefore

represent the costs expected to be incurred during any T consecutive days Summing the above

two expected costs and multiplying the sum by the number of review periods in a year yields

the expected annual cost of inventory

E[AC] = mdash[E[OC]+E[HC]]

T S j 1 ( g ) 365

j=0 i=l j=0 fc=l

G The Customer Service Level

There are many ways to define customer service and the most appropriate definition

depends upon the nature of the business and the perspective of management Two commonly

used definitions are 1) the proportion of periods in which all demand is met and 2) the

proportion of demand satisfied immediately from inventory Although the former definition is

often used because of its simplicity the latter definition is closer to what one commonly means

by service (Lee and Nahmias 1993) A third definition also relevant in retail environments

31

is the probability that lost sales do not exceed some critical number Other definitions such

as probability that backorders are filled within a certain amount of time are more applicable

to industrial and manufacturing settings where backlogging demand is common

Given the retail environment with the possibility of lost sales that exists at Canadian Tire

Pacific Associates management is concerned primarily with the fraction of sales that are lost

For example management views the situation in which demand exceeds stock by one unit every

period more favourably than that in which the same demand exceeds stock by three units every

other period For this reason the proportion of demand satisfied immediately from inventory

is the most appropriate measure of customer service

The customer service level for a single review period CSLRP is given by

r bdquo T _ Demand satisfied during RP Total demand during RP

Because we are interested in the long-run behaviour of the system a measure of the steady state

customer service C S L must be obtained As shown in Appendix D the correct measure is

poundpoundpound _ E[Demand satisfied during RP] E[Total demand during RP]

which can be rewritten as

pound pound pound _ j _ E[Demand not satisfied during RP] ^ E[Total demand during RP]

Because of the steady state conditions the above customer service level can be estimated

from any period of T consecutive days As was done for one of the cost calculations the T-day

period beginning just after replenishment is considered Suppose that there are j units of

inventory on hand immediately following replenishment Then the conditional customer service

32

level is

CSL j = 1 E[(DT-j)l

E[DT] (10)

The denominator in the second term is merely the expect demand during the period

E[DT] = 7gt (11)

where p is the mean of the random daily demand variable pound The numerator which is the

conditional expected demand that is not satisfied during the T-day period expands to

7-1

E[(DT-j)+] = T u - j + YU-k)PrDT = k] k=0

(12)

The reader is directed to Appendix E for a mathematical justification of (12) In order to obtain

the unconditional expected demand not satisfied (12) is applied to all possible values of j and

a weighted sum of the terms is calculated The result is given by

E[Demand not satisfied] = Y j =0

y-i Ti - j + 1pound(j-k)PrDT = k

S j-1

= - E[I] + Y0-k)Pr[DT=k) =0 it=0

(13)

where E[I] denotes the steady state inventory level Finally the customer service level is

obtained by the substituting (11) and (13) into (9) yielding

CSL = 1

s j-l

Ti - E[I] + Jgty Y(J-k)PrDT=k]

m n J2VjTU-k)PrDT=k Eil J=Q fc=o

(14)

33

H A Methodology for Evaluating an (sS) Policy

In this section a brief methodology summarizing the main results of the model is

presented The methodology can be applied to evaluate a given (sS) policy in terms of its cost

and its resulting customer service level Given the probability mass function of the daily

demand f the duration of the review period T the length of the lead time L the fixed cost

of placing an order cf and the daily unit cost of holding inventory cv

1) Calculate the aggregated demand probability mass functions fDT fDL and f D x L as the T-th

fold L-th fold and (T-L)th fold convolutions of f respectively

2) Generate the transition probability matrix using either equation (2) or (4) and equation (3)

3) Calculate the limiting distribution TT from (5) and (6)

4) Obtain the L-day shifted limiting distribution rj using (7)

5) Calculate the expected annual cost of inventory using (8)

6) Determine the customer service level using (14)

34

IV A N A L G O R I T H M F O R OBTAINING T H E O P T I M A L (sS) P O L I C Y

A Introduction

The methodology developed in the previous chapter can be used to calculate the cost and

service level associated with a given (sS) policy In this chapter an algorithm that finds the

feasible policy yielding the lowest cost is developed A policy is feasible if it satisfies both the

basic condition 0 lt s lt S and the imposed constraint on the customer service level2 2 In

order to find the optimal policy the algorithm must combine the methodology for evaluating

a single policy with a search technique

Search techniques are evaluated in terms of two somewhat antithetic criteria On the one

hand the search should locate the optimal solution or at least a near optimal solution on

the other hand the search should take as little time as possible Many search techniques

guarantee optimality provided the objective function and constraints exhibit certain properties

such as convexity However even if such a search technique also guarantees its termination

in polynomial time the search might take too long to be practical Alternatively some search

techniques provide reasonable though sub-optimal solutions very quickly These techniques

often called heuristics must be evaluated in terms of both their speed and the accuracy of their

solutions The choice of a heuristic over a search that guarantees optimality depends upon the

nature of the feasible space the need for accuracy and the cost of time

Given the complexity of the cost and customer service level functions it is very difficult

to show theoretically for what if any combinations of demand and costs the feasible space is

convex However a plot of the objective function for a single product may be obtained by

2 2 Management specified that each policy must provide a customer service level of at least 975 Therefore the algorithm must incorporate the constraint CSL gt 0975

35

evaluating the costs for a number of policies within a grid When the service level constraint

is included in these plots maps of the feasible space are obtained Examination of the

maps for several products suggests that the space is convex over most regions however

maps for some products reveal some regions of non-convexity Unfortunately there is no

way of knowing the degree of non-convexity for other products other stores or different costs

estimates without a more rigorous survey of the feasible space Therefore if one of the

techniques that guarantee optimality under the condition of convexity is employed the search

would have to be evaluated as a heuristic Instead as a first step to finding a practical search

technique a crude search that guarantees optimality under virtually all conditions is

investigated

B A Grid Search

The simplest search technique that guarantees optimality merely evaluates all policies in

the grid defined by 0 lt s lt S and 0 lt Sm i n lt S lt S m a x where Sm i n and S m a x are

some predefined values That is the grid search is a brute force method that examines every

policy within the grid 2 3 Although the number of policies that must be examined is large the

grid search is guaranteed to find the optimal policy provided that S o p t is contained in the grid

that is provided Sm i n lt S o p t lt Sm a x The grid search algorithm proceeds by generating the

transition probability matrix for the policy (0Smin) and then evaluating the service level

associated with that policy If the policy is feasible24 the cost associated with the policy is

2 3 The total number of policies evaluated is the sum of the first Sm a x integers less the sum of the first Smin-1 integers Numerically [(l2)(Sm a x)(Sm a x+l) - (l2)(Smin-l)(Smin)] policies are examined

2 4 That is if the service level is adequate

36

calculated and compared to the current lowest cost The procedure continues until all points

in the grid have been examined and the optimal policy has been found A flow chart of the grid

search algorithm is shown in Figure 2

In order to assess the grid search the algorithm was coded and compiled into an

executable program The program was run for two products (a medium-demand product and

a high-demand product) and with four values of Sm a x In each run Sm i n was taken to be zero

The following times were recorded for each trial

T P M - the amount of time spent generating the transition probability matrix

GAUSS - the amount of time spent solving for the steady state probabilities

COSTS - the amount of time spent calculating the expected annual costs

SERVICE - the amount of time spent determining the customer service levels

T O T A L - the total execution time of the program

The columns entitled Grid Search in Tables 2 and 3 summarize the results of the trials All

times are recorded in seconds as required by a 486DX33 MHz personal computer with an

empty buffer memory

Three results are evident from the tables First the vast majority of the execution time

is spent generating transition probability matrices and then solving for the steady state

probabilities with the time being split almost equally between the two tasks Second the total

execution time of the algorithm increases exponentially as Sm a x increases For product 206917

doubling S^ from 30 to 60 led to an increase in the execution time of more than 2100 (from

172 seconds to 3661 seconds) Third the grid search algorithm as it stands does not appear

to be adequate for the problem at hand Given the large number of products that must be

evaluated the algorithm is much too slow to be implementable

37

Calculate the demand probability functions

Initialize s = 0 S = Smin] cost = 10A5

s = s+1

Calculate the probability transition matrix for (sS)

Evaluate policy (sS) Is service level ^ 0975

No

No Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Is s = S-l

Yes

Is S = Smax No

Yes

Print results (sS) cost

s = 0 S = S+1

Figure 2 The flow chart of the grid search algorithm

38

Table 2 Execution times of the algorithms for product 200001 a 30-amp inline fuse

(in seconds)

-max Subroutine Grid Search Grid Search

Updating T P M Heuristic

10 T P M 006 010 001

GAUSS 010 006 001

COST 005 005 001

SERVICE 001 001 001

T O T A L 050 035 027

20 T P M 126 021 010

GAUSS 119 160 023

COST 027 026 001

SERVICE 017 016 001

T O T A L 318 236 055

30 T P M 703 096 010

GAUSS 757 801 093

COST 079 085 001

SERVICE 047 042 001

T O T A L 1664 1137 126

40 T P M 2257 163 015

GAUSS 2958 3003 206

COST 313 329 001

SERVICE 092 083 011

T O T A L 5871 3851 285

39

Table 3 Execution times of the algorithms for product 206917 a 6 solder connector

(in seconds)

bull-max Subroutine Grid Search Grid Search

Updating T P M Heuristic

30 T P M 717 067 006

GAUSS 762 880 093

COST 098 087 010

SERVICE 016 021 006

T O T A L 1720 1176 220

40 T P M 2390 157 022

GAUSS 2861 2911 209

COST 219 227 006

SERVICE 080 088 011

T O T A L 5827 3790 362

50 T P M 6691 413 010

GAUSS 7994 8201 472

COST 568 577 016

SERVICE 212 197 005

T O T A L 15840 10041 654

60 T P M 15509 894 040

GAUSS 18607 18951 876

COST 1367 1353 038

SERVICE 350 359 022

T O T A L 36614 22794 1120

40

C A Technique for Updating the Transition Probability Matrix

As was noted in the previous section approximately half of the execution time of the grid

search algorithm is spent generating transition probability matrices Furthermore the amount

of time increases exponentially as Sm a x increases This increase is exponential for two reasons

first the number of transition matrices that must be calculated is a function of S^ squared and

second the average size of the matrices and hence the average number of calculations per

matrix increases with Sm a x For example when S m a x equals 30 the transition probability matrix

must be generated 465 times with the largest matrix having dimensions (31x31) When Sm a x

equals 40 the matrix must be generated 820 times with 355 of the matrices having dimensions

larger than (31x31) The performance of the algorithm could be improved considerably if

either the number of matrices that must be generated or the number of calculations per matrix

is reduced This section introduces a technique that accomplishes the latter by updating the

transition probability matrix for a new policy instead of generating it anew for each policy

A new policy (s+mS)

Suppose the policy (sS) has associated with it the (S + l)x(S + l) transition probability

matrix P and the policy (s+mS) where m is any integer satisfying 0 lt s+m lt S has

associated with it the (S + l)x(S + 1) transition probability matrix J Then T can be obtained by

recalculating only m rows of P Recall that the equations for generating a transition

probability matrix are divided into two sets based upon the row being calculated equations (2)

or (4) for rows 0 through s and (3) for rows (s + 1) through S The equations themselves are

independent of s only the limits defining the cases contain s Therefore only the m rows

that are affected by the change in the limits need to be recalculated The following updating

technique can be used to obtain 7

41

1) If m gt 0 rows i = (s + 1) (s+2) (s+m) are recalculated using (2) or (4)

2) If m lt 0 rows i = (s-1) (s-2) (s-m) are recalculated using (3)

3) All other rows of T are identical to those of P

The reader is directed to Appendix F for a mathematical justification of the technique

A new policy (sS+m)

Suppose the policy (sS) has associated with it the (S + l)x(S + l) transition probability

matrix P and the policy (sS+m) where m is any positive integer has associated with it the

(S+m+l)x(S+m+l) transition probability matrix P Then T can be obtained by modifying

each row of P and adding m additional rows and columns The updating technique is not as

simple as that for a new policy (s+mS) for two reasons the dimensions of P and T differ and

the equations for generating a transition probability matrix contain S Nevertheless an updating

technique that requires much less computing time than generating the matrix anew can be

developed The following technique can be used to obtain P

1) For rows i from 0 through s

For columns j from 0 through m recalculate using (2) or (4)

For columns j from (m+1) through (S+m) enter the (j-m)th column entry of P

2) For rows i from (s + 1) through S

For columns j from 0 through S enter the corresponding value from P

For columns j from (S + 1) through (S+m) enter 0

3) For rows i from (S + 1) through (S+m) recalculate using (3)

The above technique is justified mathematically in Appendix G

42

The modified algorithm

The updating technique can be easily incorporated into the grid search algorithm with

only a slight change in the manner in which s is incremented Figure 3 shows the flow chart

of the new algorithm As before the algorithm was coded and compiled into an executable

program and the various execution times were recorded The results are listed in Tables 2 and

3 under the columns Grid Search Updating TPM

Updating the transition probability matrix does improve the performance of the

algorithm The time required to generate the transition probability matrices is reduced by

approximately 90 This translates to a reduction in total execution time of approximately

30 Not surprisingly the reductions are greater for large values of Sm a x However despite

this seemly significant improvement a decrease of only approximately 08 seconds is obtained

when S^ equals 20 Further when S ^ equals 30 the algorithm requires over 11 seconds

which is still much too high

The execution time is now dominated by the time required to solve for the steady state

probabilities The steady state probabilities could be solved by using approximation techniques

which are faster than solving the systems exactly however because the probabilities are often

very small numbers the approximation techniques would be unable to provide the required

accuracy The only remaining way to decrease the execution time is to reduce the number of

policies that are evaluated during the search This can be done by either reducing the size of

the grid being searched or reducing the number of policies being evaluated within the grid The

following section introduces a method of reducing the size of the grid by obtaining a lower

bound for S namely Sm i n

43

s = s+k

Calculate the demand probability functions

Initialize s = 0 S = Smin| cost = 105

k= 1

Update the probability transition matrix for (sS)

Evaluate policy (sS) Is service level ^ 0975

No

No

iNo

Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Ifk= 1 iss = S-l If k =-1 is s = 0

Yes

Is S = Smax No

Yes

Print results (sS) cost

If k=l s=s+l S = S+1

Figure 3 The flow chart of the grid search algorithm utilizing the updating technique for the transition probability matrix

44

D A Lower Bound on S

As mentioned earlier the size of the grid is specified by the number of policies within

the domain defined by 0 lt s lt S and 0 lt Smin lt S lt Smax Thus the size of the grid

and hence the total amount of computation time can be reduced by obtaining tight bounds on

S Unfortunately because of the extreme non-linearity of the objection function and the service

level constraint obtaining a mathematical upper bound on S is very difficult and beyond the

scope of this work This section introduces a method for obtaining a lower bound on S25

For a given value of S an upper bound on the customer service level can be found as

follows Suppose that there are S units of inventory on the shelf at the beginning of every

review period regardless of demand No (sS) policy can provide a higher level of customer

service because there exists a positive probability of beginning a review period with less than

S units on the shelf Thus the customer service level associated with starting each review

period with S units on the shelf is an upper bound on the customer service level for any (sS)

policy From (10) this level is

E[(DT-SY] CSL 5 = 1 - T mdash mdash - ( 1 5 gt

E[DT]

Replacing j with S in (12) and then substituting (12) and (11) into (15) yields

CSL 15 = mdash- - J2(S-k)PrDT = k] (16)

To obtain Smin use (16) to find the largest value of S with a customer service level less than

0975

There are two criteria for a good lower bound on S the bound must be no larger than

This method was proposed by Kaan Katircioglu

45

S o p t and it must be very close to So p t These criteria ensure that the optimal policy will be

contained in the search grid and that the size of the grid is reduced by as much as possible

Based on these criteria the above method of obtaining a lower bound is assessed A value for

S m i n is obtained for each of the products in product category 20 and compared with the

corresponding value of S o p t 2 6 The results are illustrated in Figure 4 which shows the number

of occurrences of various Sm i n and S o p t combinations Note that a point lying on the 45deg line

represents a perfect bound while a point lying close to the S o p t axis represents a very loose

bound The absence of any points above the 45deg line indicates that the first criterion is satisfied

in each instance the method did not generate a value of Sm i n exceeding the corresponding value

of So p t Moreover the bounds are reasonably tight for 85 of the products analyzed that

is for those products with S o p t equal to 10 or less27 However as S o p t increases the bounds

become steadily worse Because values of S o p t are likely to be larger for these products at the

ten Jumbo and Extra Jumbo stores and for other products with higher demand the lower bounds

are not expected to be very tight in general

Although obtaining the bound Sm i n does reduce the size of the grid the corresponding

reduction in the execution time of the algorithm is miriimal For example if S^ equals 60 and

a value of 30 is obtained for Sm i n the program would still require approximately 215 seconds

to search the grid (compared to the 228 seconds required without a lower bound on S) Because

the execution time increases exponentially with So p t a tight upper bound on S would likely have

a more significant impact on reducing the execution time of the algorithm Further research

on this issue is required

2 6 The optimal policy for each product was obtained using a grid search with an extremely large upper bound

2 7 On average Sm i n is approximately half of S o p t for these products

46

47

The only remaining approach to improving the algorithm is to modify the search so that

fewer policies within the grid are evaluated Toward this goal assumptions can be made with

respect to the feasible space and a heuristic search can be developed based on these

assumptions

E A Heuristic Search

As mentioned previously in this chapter a map of the feasible space for each of

several products was obtained Upon closer examination the maps illustrate two patterns

without exception for a given value of S both the cost and the customer service level increase

as s increases The following intuitive argument is offered as a possible explanation for the

observed behaviour When S is held constant increasing s results in more instances in which

an order is placed upon review Hence the average stock level would increase which in turn

would lead to an increase in the cost and the customer service level28 With this in mind a

heuristic search based on the following assumptions is developed for a given value of S both

the cost and the customer service level are monotonically increasing functions of s

In general the heuristic proceeds as follows Having just incremented S by one the

customer service level of the policy (sS) is evaluated If the policy is feasible s is decreased

until the lowest value of s producing a feasible policy is found If the policy is not feasible

s is increased until either the cost exceeds the current lowest cost for a feasible policy a

feasible policy is obtained or s equals S-l In the event that a feasible policy is found the cost

2 8 In order to prove that the customer service level increases as s increases it would be sufficient to show that the probability of placing an order increases as s increases Such a result might be possible if assumptions regarding the demand function are made Further if both the probability of placing an order and the average stock level increase then both the ordering cost and the holding cost would increase No proofs are offered here

48

of the policy is calculated if the cost of the policy is less than the current lowest cost the new

policy becomes the current best Finally S is increased by one and the process repeats

provided S does not exceed some predefined value Sm a x The flow chart of the algorithm

incorporating the heuristic search is shown in Figure 5 Note that the algorithm begins by

attempting to improve on the lower bound Sm i n the policy (Sm i n-lSm i n) is evaluated and if it

is not feasible S m i n is increased by one The improvement continues until a feasible policy is

found

As previously mentioned a heuristic search must be evaluated in terms of both its speed

and the accuracy of its solutions To evaluate the heuristic on the basis of speed the same

approach used to evaluate the previous algorithms is employed The columns entitled

Heuristic in Tables 2 and 3 contain the various execution times of the heuristic The heuristic

is a vast improvement over the previous search techniques the reduction in execution time is

enormous and becomes more so as S m a x increases For a S m a x of 20 the reduction is

approximately 75 of the time required by the grid search algorithm29 and for a S m a x of 60

that reduction soars to over 95 More importantly the heuristic can search a grid with a

of 40 in under 4 seconds or a grid with a S m a x of 60 in under 12 seconds These low execution

times should be sufficient to allow for a large number of products to be analyzed in a

reasonable amount of time

In order to further assess the practicability of the heuristic the total time required by the

heuristic to find solutions for the entire product category was determined and compared to the

corresponding time for the grid search algorithm With a priori knowledge of the optimal

solutions conservative yet reasonable values of S m a x were selected and used by both of the

2 9 Hereafter grid search algorithm refers to the algorithm utilizing the grid search the updating technique and the lower bound Sm i n

49

Calculate the demand probability functions

Calculate Smin S = Smin s = S-l

S = S+l s = S-l t

Update the probability transition matrix for (sS)

No Is service level pound 0975 Yes

No Is service level ^ 0975

Yes

Is cost lt cost No

Yes

Is s lt S-l No

Yes

s = s+l

Update the probability transition matrix for (sS)

Is service level pound 0975

Yes

Is cost lt cost No

Yes

s = s S = S cost = cost

s = s-l

No

Update the probability transition matrix for (sS)

S = S+l

Is s 2 0 Yes

Update the probability transition matrix for (sS)

Is service level pound 0975

No

s = s+l

No

No

Is S = Smax

^Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Print results (sS) cost

Figure 5 The flow chart of the heuristic algorithm

50

algorithms30 The grid search algorithm required approximately 9 hours and 41 minutes to

find the solutions while the heuristic required only slightly more than 23 minutes Based on

speed the heuristic is preferable to the grid search algorithm

Finally to evaluate the heuristic based on the accuracy of its solutions its solutions were

compared to the previously obtained optimal policies Without exception the heuristic

algorithm found the optimal policy Although there is no guarantee that the heuristic will find

the optimal solution for different products or different demand functions the heuristic appears

to work extremely well Based on the fast execution time and the apparent accuracy of the

solutions the heuristic will be used in the development of the inventory tool

3 0 For the test the following values of Sm a x were used 20 (273 products) 30 (19) 50 (23) 100 (3) and 125 (2)

51

V T H E I N T E R F A C E

With the model formulated and the algorithm developed the only remaining task in the

development of the prototype is the creation the interface In fact two interfaces are created

one that displays results in terms of minimum fills and target levels and one that displays

results in terms of reorder points and target levels To be consistent with the previous chapters

the latter interface is described here31

In its present form the interface allows the manager to set review parameters to

compare optimal and current policies and to evaluate alternate policies From the main menu

the manager selects the product store duration of the review period length of the lead time

and value of Sm a x Once the parameters are set the system calculates the optimal policy and

then evaluates the current policy displaying results as they are obtained In particular the

reorder point target level expected annual cost and expected customer service level are

displayed for each policy If a current policy meets the desired customer service level the

expected annual savings of the optimal policy are displayed in terms of both dollars and

percentage of current cost otherwise the value of the current customer service level is

displayed in red to indicate that it is deficient Finally the manager can choose to evaluate

alternate policies or to return to the main menu

Although the prototype is far from the ultimate integrated inventory control system it

is still valuable to management As mentioned in Chapter II inventory policies at Canadian

Tire Pacific Associates are currently based on rules of thumb that incorporate marketing

3 1 Because significant changes to the interface are expected as feedback from management is received and as the system is brought on-line only a brief description of the present interface is given

52

strategies The prototype provides management with a means of determining the proportion of

the cost of inventory that is not attributable to meeting customer demand For example

consider a hypothetical consultation in which the manager chooses to analyze product 200001

a 30-amp inline fuse at store 6 The prototype displays to the manager the current policy of

(1214) which provides a customer service level of 1000 at an annual cost of $748 and the

optimal policy of (39) which provides a customer service level of 977 at an annual cost of

only $319 If for some reason the manager feels the optimal policy is undesirable he or she

can investigate the additional cost of holding excess inventory Suppose the manager chooses

to evaluate the policy (610) The prototype evaluates the policy and displays the results the

customer service level is 998 and the annual cost is $441 With the alternate policy

virtually all demand is satisfied and the cost of inventory is still reduced by over 40 Thus

the manager ascertains that approximately 40 of the current cost of inventory for this product

is directly attributable to marketing considerations Using the prototype in this way

management can begin to understand better the nature of its inventory costs A description and

graphical depiction of the above hypothetical consultation are presented in Appendix H

As for the technical details of the system the prototype can run on any computer using

the DOS operating system having approximately 420K of available conventional memory and

attached to a V G A or SVGA monitor Further the prototype can be used on a machine without

a math coprocessor however performance is slower The prototype consists of two modules

linked together into one executable program The interface module contains the various

interactive subroutines and the main module contains the methodology of the model and the

logic of the heuristic search Both modules are coded in Fortran and compiled using Microsoft

Fortran 50 which provides many extensions to standard Fortran The source codes for the

interface module and the main module are listed in Appendix I and Appendix J

53

respectively

In addition to the above technical specifications the program requires the presence of

certain files in certain directories The program uses the distribution files mentioned in

Chapter II to read the unit costs and the empirical demand functions Each product has a

unique distribution file the name of which is the six digit product number followed by the

extension dst The text file containing the existing policies at each store must be located in

the same directory as the distribution files Finally the file tmsrbfon which contains the

font used by the interface must be located in the directory from which the program is executed

In its future versions the system should be more automated and more user-friendly

Rather than having the manager choose an upper bound for the search the algorithm should

incorporate a method of determining Sm a x For this to happen either a theoretical upper bound

must be obtained or a heuristic for terminating the search must be developed Also the amount

of time spent waiting for calculations to be performed can be reduced by reading the optimal

and current policies and their associated costs and customer service levels from a file or

database This solution file could be created any time prior to the consultation by running

the system in a batch mode The interface should allow the manager to change cost estimates

and service level values view results graphically and print results and summary documents

Finally the system should provide an option for automatically incrementing the product andor

store during a consultation This option would enable a manager to review quickly the

inventory policy of a product at each store

54

VI R E S U L T S AND SENSITIVITY ANALYSIS

A Comparison of Current Policies to Optimal Policies

In order to estimate the potential savings of implementing optimal policies the solutions

obtained in the Chapter IV are analyzed in more detail For each product the current policy

is evaluated and then compared to the optimal policy with respect to cost and customer service

level Finally the potential savings of the optimal policies are extrapolated to all products and

all stores

Of the 420 products examined each of 37 products has an existing policy that fails to

provide the desired customer service level of 975 However most of these policies just

fail to do so only 4 policies provide a customer service level of less than 90 Table 4

provides a list of these 37 products each with its current customer service level its current

annual cost and the additional cost of its optimal policy3 2 Two comments regarding Table

4 are in order First the current customer service level for product 203312 an amber

driveway marker is listed at a misleading 303 Examination of the demand distribution

reveals that this product was sold on only 1 of the 307 days during which sales data were

collected and on that day 13 of the markers were sold Clearly this data point constitutes

either an error or an unusual event the current policy of (34) does not allow for more than 4

items to be sold per review period Second the additional cost of the optimal policy for each

of 5 products is negative In other words for each of these products the current policy which

fails to provide the desired customer service level costs more than the optimal policy which

3 2 This is the difference between the expected annual cost of the optimal policy and the annual cost of the current policy It is the minimum additional cost required to increase the customer service level to at least 975

55

Table 4 Products with insufficient existing policies

Product Current CSL ()

Current Annual Cost ($)

Additional Cost of the Optimal Policy ($)

206932 974 204 -051

204570 973 222 -025

203053 973 248 009

206550 972 118 042

206547 972 105 035

206565 972 146 056

203702 972 340 157

206876 972 155 061

205752 972 330 153

205922 972 108 036

206857 972 180 073

205393 972 190 -003

201513 971 345 015

203405 970 255 037

203069 968 295 -048

203071 967 321 -001

205635 967 390 029

203914 966 460 160

204601 962 252 061

203326 960 144 014

202003 958 876 437

206516 958 112 029

206517 958 283 120

206350 958 333 147

205932 958 180 066

203936 952 598 294

56

Product Current CSL ()

Current Annual Cost ($)

Additional Cost of the Optimal Policy ($)

206932 974 204 -051

203939 948 413 193

203977 945 1216 1242

206551 945 132 077

205389 934 598 047

203590 927 1223 607

206567 919 122 070

203591 905 1228 834

205390 887 291 074

203318 833 136 084

206393 776 319 289

203312 303 237 511

provides at least the desired customer service level For the 36 products the total additional

cost of the optimal policies is $542033

Of the remaining 383 products most have an existing policy that provides a customer

service level well above 975 When the customer service levels are rounded to one decimal

each of 253 products has a customer service level of 1000 and each of 347 products has a

customer service level of at least 990 In contrast each of only 58 products has an optimal

policy that yields a customer service level of at least 99034 For each of 37 products the

3 3 The cost associated with product 203312 is not included in the total

3 4 Optimal policies do not have customer service levels of exactly 975 because the function is discrete For low-demand products a unit step in s or S results in a relatively large change in the customer service level

57

current policy is identical to the optimal policy

The extremely high level of service provided by most policies allows for substantial

savings in the cost of inventory With current policies the estimated annual cost of inventory

for all 420 products is $420743 whereas the expected annual cost of the optimal policies is

only $186884 This amounts to a savings of approximately $2340 or over 55 of the current

cost of inventory

In order to further assess the savings the difference between the costs of the current

policies and the costs of the optimal policies are examined in more detail For each of the 383

products whose existing policy provides a customer service level above 975 the difference

between the estimated annual cost of the existing policy and the expected annual cost of the

optimal policy is calculated The resulting histograms of the savings in (a) dollars and (b)

percentage of current cost are shown in Figure 6 The particulars for the current and optimal

policies for all 420 products are provided in Appendix K

As evident from Figure 6(a) the majority of the expected savings in absolute terms

comes from only six products In order to examine these products more closely the current

and optimal policies along with the associated costs and savings for each of the six products

are listed in Table 5 Each current policy is very different than its corresponding optimal

policy resulting in excessive inventory for each product In fact over half of the total expected

savings could be realized by implementing the optimal policy just for each of these six products

Interestingly five of the six products are sealed beam headlights

58

90

10 - 20 30 - 40 50 - 60 70 - 80 90- 100

Savings ()

(b)

Figure 6 The distribution of savings of optimal policies in (a) dollars and (b) percentage of current cost

59

Table 5 Products with the largest potential absolute savings

PRODUCT

CURRENT OPTIMAL SAVINGS

PRODUCT s S Cost ($)

C S L

()

s S Cost ($)

C S L () ($) ()

2036801 194 200 64014 1000 37 48 bull 9245 975 54769 856

2036772 94 100 23652 1000 18 22 3850 978 19802 837

2036723 78 90 23043 1000 19 23 4454 975 18589 807

2030944 145 175 19522 1000 34 45 3727 975 15795 809

2036425 48 60 10491 1000 13 17 2682 977 7809 744

2036556 68 80 7265 1000 14 22 1699 975 5566 766

1 H6054 rectangular sealed beam headlight 2 H4656 rectangular sealed beam headlight 3 6053 rectangular sealed beam headlight 4 9004 halogen bulb 5 4652 rectangular sealed beam headlight 6 6104 7 sealed beam headlight

Although Figure 6(a) seems to indicate that the majority of the current policies are close

in cost to the optimal policies this is very misleading For example suppose the optimal policy

for a product costs $1 and that the current policy costs $2 In absolute terms the optimal

policy saves only $1 but in relative terms the optimal policy saves 50 of the current cost

When savings are viewed in this manner as in Figure 6(b) it is clear that substantial savings

can be realized for the majority of the products In fact both the mode and the mean of the

savings are between 40 and 50 of the current costs

In order to extrapolate the above savings to obtain the potential savings for product

category 20 at all 21 stores the differences between stores must be considered During the

60

pilot study the current policy for each of several products at each size of store was evaluated

The study suggested that medium and large stores have a similar amount of excess

inventory and that jumbo and extra jumbo stores have much more excess inventory One

could reasonably expect that the additional excess inventory allows for higher savings to be

realized at the jumbo and extra jumbo stores Thus because 18 of the 21 stores are

medium or larger the expected 40-50 savings at store 6 can be viewed as a lower bound

on the potential savings to Canadian Tire Pacific Associates from implementing optimal policies

for product category 20 in all its stores

Finally in order to obtain the overall potential savings the expected savings from

product category 20 must be extrapolated to all product categories Ideally the proportion of

inventory comprising each product category and the relative savings that can be realized for

each category would be used to obtain a good estimate of the overall potential savings

However as this information is not known only a crude extrapolation is possible Thus the

analysis proceeds under the assumption that a 40-50 savings can be realized in each product

category35

Before an estimate of the total potential savings in absolute terms can be obtained the

proportion of the current inventory to which the savings apply must be determined Up to this

point the analysis has been concerned solely with reducing inventory at the stores However

it is inappropriate to perform further analysis without considering the entire inventory system

including the central warehouse Because the lead time for orders from Toronto is highly

variable and considerably greater than those at individual stores Canadian Tire Pacific

3 5 During the pilot study the current policies for a paint brush and an ironing board were evaluated and found to result in excess inventory similar to that of product category 20 Although this in no way justifies the assumption it does lend some support

61

Associates must keep a fairly high safety stock at the warehouse Without knowing the

proportion of inventory serving as the safety stock it is inappropriate to assume any reduction

in inventory at the warehouse is possible Therefore the relative savings of implementing

optimal policies is applied only to the inventory held in the stores36 Of the estimated $28

million in inventory approximately half is held in the stores therefore the 40-50 savings

translates to a savings of approximately $55 to $7 million per year

B Sensitivity Analysis on the Ordering Cost and Holding Rate

The above analysis was based on estimates of the ordering cost and the annual unit

holding rate37 provided by management As described in Chapter II these estimates

incorporate a number of costs that are difficult to quantify or that are speculative Therefore

there is some question as to the accuracy of the estimates In this section errors of +$0015

in the ordering cost and +5 in the holding cost are assumed in order to determine the

sensitivity of the model to errors in these variables38

The analysis proceeds as follows Based on an ordering cost of $0080 and a holding

rate of 30 the optimal policies from the previous section were obtained39 Then assuming

3 6 Of course this assumes that the excess inventory currently held in the stores is not required as part of the safety stock Based on the large amount of inventory at the warehouse this assumption seems reasonable

3 7 Hereafter simply referred to as the holding rate

3 8 An error of +$0015 in the ordering cost implies an actual ordering cost of $0070 a relative error of approximately 21 and an error of -$0015 implies an actual ordering cost of $0100 a relative error of 15 An error of +5 in the holding rate implies an actual rate of 25 a relative error of 20 and an error of -5 implies an actual holding rate of 35 a relative error of approximately 17

3 9 These policies are referred to as optimal policies for the duration of this chapter

62

the actual ordering cost andor holding rate are different from managements estimates the

heuristic is used to obtain the true optimal policies Finally the optimal policies are

evaluated based on the actual ordering cost and holding rate and then compared to the true

optimal policies For each possible combination of ordering cost and holding rate three

numbers are recorded the number of products with an optimal policy that differs from its

true optimal policy the total cost of implementing the optimal policies given the actual

ordering cost and holding rate and the difference between the cost of implementing the

optimal policies and the cost of implementing the true optimal policies The results of the

analysis are summarized in Table 6

Table 6 A comparison of optimal policies to true optimal policies

Actual Ordering Cost

$0070 $0085 $0100

25 5 products $155439

$002

51 products $160590

$092

88 products $165464

$449

Actual Holding Rate

30 54 products

$181470 $182

48 products $192026

$088

35 79 products

$207270 $574

48 products $212940

$136

2 products $218312

$000

4 0 The table should be read as follows If the actual ordering cost is $0070 and the actual holding rate is 25 then each of 5 products has an optimal policy that is different from its true optimal policy The total cost of implementing the optimal policies is $155439 which is $002 more than implementing the true optimal policies

63

The model appears to be quite insensitive to relative errors in the order of 20 Under

the worst case scenario when one of the variables is over-estimated and the other is undershy

estimated the optimal policies result in a total cost of inventory which is only approximately

3 higher than the cost of the true optimal policies Further because demand is not changed

by errors in these estimates each of the optimal policies still provides a customer service

level of at least 975 under all of the scenarios Finally although the cost of the optimal

policies varies significantly for the different scenarios the relative savings remain virtually

constant Table 7 lists the cost of existing policies and in parenthesis the relative savings of

the optimal policies for each of the different combinations of ordering cost and holding rate

The cost of the existing policies varies in the same manner as that of the optimal policies the

relative savings for each of the scenarios remains around 55 Therefore even in the presence

of such errors in the ordering cost and holding rate the projected savings of implementing the

optimal policies would be unchanged

Table 7 The cost of existing policies and the relative savings of the optimal policies under various scenarios

Actual Ordering Cost

$0070 $0085 $0100

25 $350153 (56)

$358464 (55)

$366764 (55)

Actual Holding Rate

30 $412438 (56)

$429048 (55)

35 $474725 (56)

$483026 (56)

$491320 (56)

64

C Sensitivity Analysis on the Demand Probability Mass Function

In this section the sensitivity of the model to slight changes in the demand probability

mass function is investigated As described in Chapter II the demand function used in the

model is based solely on observed sales however this empirical function is only one possible

instance of the underlying true demand function For example had the data been collected

over a period both starting and finishing one month earlier many of the empirical functions

would be slightly different Consequently depending upon the degree to which the empirical

functions vary and the sensitivity of the model to these variations the optimal policies from

the model might be inaccurate For example recall the case of the amber driveway marker

only thirteen markers were sold and all were sold on the same day Because the empirical

demand function is used the model calculates an optimal policy of (1213) clearly this

policy is impractical Although this example is extreme it does indicate the danger of using

unrealistic empirical demand functions41

As discussed in Chapter II three classes of products were identified namely low

medium and high-demand products For this analysis the same method of classification

is used Two products are selected from each of these groups and two new distribution files

are created for each of the six products For each product one of the files contains two

additional days of data with each day having a demand equal to the maximum observed demand

for that product In the second file only one additional day of data is added but the demand

on that day is two more than the maximum observed demand The situations that could give

4 1 Certainly a rigorous analysis to determine both the extent of variation in the empirical demand functions and the effect of using the empirical functions needs to be performed However such an analysis is beyond the scope of this thesis only a brief analysis that might suggest the degree to which the model is sensitive is performed here

65

rise to such changes in the demand data are termed Scenario 1 and Scenario 2 respectively42

The analysis proceeds in a manner similar to the one in the previous section Given the

demand that was originally observed the optimal policies were obtained Now assuming the

true demand function corresponds to the demand under Scenario 1 the true optimal policy

for each product is calculated Then each optimal policy is re-evaluated using the true

demand function and compared to the true optimal policy The analysis is then repeated

assuming the true demand function corresponds to the demand under Scenario 2 Table 8

lists the results for each product and scenario

Table 8 True optimal and optimal policies for each demand scenario

Product 202101 (low demand)

Scenario s s Cost ($) C S L ()

1 True optimal 1 2 470 993

Optimal 1 2 470 993

2 True optimal 2 3 663 976

Optimal 1 2 461 875

Product 203507 (low demand)

Scenario s S Cost ($) C S L ()

1 True optimal 0 3 334 981

Optimal 0 2 258 972

2 True optimal 2 3 508 980

Optimal 0 2 273 702

4 2 To be clear the scenarios are defined as follows Scenario 1 The addition of two days each with a demand of max Scenario 2 The addition of one day with a demand of max+2

where max is the maximum observed daily demand for each product in the sample period

66

Product 201032 (medium demand)

Scenario s S Cost ($) C S L ()

1 True optimal 6 12 1377 976

Optimal 6 11 1338 973

2 True optimal 6 13 1426 977

Optimal 6 11 1338 970

Product 203039 (medium demand)

Scenario s S Cost ($) CSL ()

1 True optimal 5 19 256 975

Optimal 5 16 251 969

2 True optimal 5 20 258 976

Optimal 5 16 249 968

Product 203677 (high demand)

Scenario s S Cost ($) CSL ()

1 True optimal 18 23 3981 977

Optimal 18 22 3817 972

2 True optimal 18 23 3995 978

Optimal 18 22 3831 973

Product 203680 (high demand)

Scenario s S Cost ($) CSL ()

1 True optimal 38 49 9498 976

Optimal 37 48 9175 971

2 True optimal 37 49 9484 975

Optimal 37 48 9206 972

67

The model is most sensitive to changes or errors in the demand functions of low-

demand products When only one additional day of demand for max+2 units is added to the

data the optimal policy for each of the low-demand products results in a very low customer

service level In fact for one of the products the customer service level drops to almost 70

The increase in demand does lower the customer service level for each of the other the

products but that decrease is slight

The above analysis indicates the need for some kind of smoothing or modelling of the

demand data especially for low-demand products The inherent variation between two

instances of a true demand function could very well result in more pronounced changes than

those resulting from either of the above scenarios Therefore it is likely that many optimal

policies especially those for low-demand products would actually provide a much lower or

much higher customer service level than expected However by smoothing the data the

variation between instances of a true demand function can be reduced and the desired

customer service levels can be achieved

For example suppose that the demand for a product is stationary that the average

demand during the previous period was unusually low and that the optimal policy for this

period is based on the demand during the previous period Because the demand is stationary

the probability that the demand during this period is higher than that during the past period is

more than 50 In other words the probability that the demand during this period is higher

than that for which the optimal policy provides the desired expected customer service level

is greater than 05 Thus expected customer service level for this period is actually less than

that specified Now suppose instead that the optimal policy is based on smoothed data which

reflects very closely the expected demand The probability that the demand during this period

is higher than that for which the optimal policy provides the desired expected customer service

68

level is virtually 05 Thus as a result of smoothing the data the specified customer service

level should be expected

69

VII C O N C L U S I O N

In this thesis we developed a single-product single-location model of the inventory

system at Canadian Tire Pacific Associates Specifically we formulated a Markov chain model

to determine the long-run average cost of inventory for fixed values of s and S and then

developed a heuristic search to find near optimal policies quickly Also as a first step toward

developing an integrated interactive inventory control system we created a prototype of the

interface

According to the analysis performed here Canadian Tire Pacific Associates is currently

holding an excessively large inventory in its stores This study finds that Canadian Tire Pacific

Associates could reduce the cost of inventory for product category 20 at store 6 by

approximately 50 and still maintain customer service levels of at least 975 Further the

mean and modal values of savings for these products are between 40 and 50 Given the

similar excess inventory at the other stores and in all product categories we believe that savings

in this range are possible for all products and stores We estimate that the application of

inventory control as described in this thesis would result in annual savings of $55 to $7

million

Given the large potential savings to Canadian Tire Pacific Associates we recommend

that management proceed with the next phase of the project Specifically we recommend that

the model algorithm and interface be refined further that a forecasting model be developed

and that a data collection and storage scheme be designed

In order to reflect operational changes at Canadian Tire Pacific Associates subsequent

to the formulation of the model modifications must be made to allow for the lead time to

exceed the review period Several of the stores now have three picks per week and as a

70

result orders placed for these stores at one decision epoch do not arrive until after the next

decision epoch Because this violates one of the models assumptions the model must be

modified accordingly Note however that with three picks per week and a review period

of four days orders from only the previous decision epoch are ever outstanding at the current

decision epoch

Also the model should be refined to reflect the fact that the review period is not fixed

over the planning period In its present form the model assumes a fixed review period

however inventory at stores with two picks per week is actually reviewed on a four day -

three day cycle and inventory at stores with three picks per week is reviewed on a three day

- two day - two day cycle Rather than obtaining the steady state costs for fixed review period

the model should calculate the steady state cost for a period of one week during which multiple

replenishments occur Although the solutions are not expected to differ significantly as a result

of the change the model would reflect the actual process more accurately

Further work is also required on the search technique for locating near optimal

policies Although the heuristic search described in this thesis appears to work well the

decision maker must provide an upper bound on S prior to the search In practice this upper

bound is generally not known in advance Thus if the heuristic is to be used either a tight

theoretical bound must be found or a termination heuristic must be developed Alternatively

a new heuristic search could be developed based on additional or different assumptions

regarding the properties of the feasible space For example although some regions of non-

convexity were found for some products much of the feasible space appeared to be convex

A heuristic that assumes a convex feasible space would not require an upper bound on S and

might perform very well However before such a heuristic is implemented further work is

necessary to determine the degree of non-convexity for a variety of demand functions and

71

ordering and holding costs

In addition to making the above modifications we recommend that Canadian Tire Pacific

Associates begin work on the remaining two components of the inventory control system First

in order to allow for products with seasonal demand the system requires a forecasting model

to provide accurate estimates of demand for the coming period Second the system needs a

data collection and storage module that systematically collects sales data for each active

product and for each store and then updates the corresponding demand distributions Because

approximately 25 of demand is expected to result in lost sales under the optimal policies the

sales data will have to be adjusted accordingly in the future Also because the model appears

to be sensitive to slight changes in the demand distribution for low-demand products the

module should incorporate a technique for smoothing the empirical distributions for these

products Further work should be undertaken to determine the extent of variation in the

empirical distributions and the most appropriate smoothing technique However because of

the large amount of data required to forecast seasonal demand and because of the current lack

of historical data we recommend that management proceed immediately with the design of a

data collection scheme

Finally two additional issues that are beyond the scope of the next phase should be

addressed at some point First a monitoring system will be required once inventory control

is implemented in the stores The monitoring system will ensure that optimal policies perform

as expected and will indicated whether or not demand is adversely affected by the lower levels

of in-store inventory Second given the degree to which in-store inventory can be reduced

inventory control in the warehouse should be investigated Because approximately half of the

existing inventory is held in the warehouse the additional savings from reducing system-wide

inventory could be great

72

Afterword

Just prior to the completion of the work described in this thesis Mr Don Graham

principle owner of Canadian Tire Pacific Associates sold his shares in the company Shortly

thereafter the results of this thesis were presented to the new management group

Unfortunately the new management did not consider development of an inventory control

system to be a priority Nonetheless the prototype was delivered to Canadian Tire Pacific

Associates for possible in-house testing however at the time of this writing - almost one year

after the meeting - Canadian Tire Pacific Associates has not decided to continue with the

project

73

Bibliography

Arrow K A T E Harris and J Marschak (1951) Optimal inventory policy Econometrica 19 250-272

Day G (1992) In a speech to the Empire Club of Canada Toronto quoted in When the JIT fits The Globe and Mail October 20 1992 B24 and in Puterman (1994)

Dvoretzky A J Kiefer and J Wolfowitz (1952a) The inventory problem I Case of known distributions of demand Econometrica 20 187-222

Dvoretzky A J Kiefer and J Wolfowitz (1952b) The inventory problem II Case of unknown distributions of demand Econometrica 20 450-466

Federgruen A (1993) Centralized Planning Models in SC Graves A H G Rinnooy Kan and P H Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Federgruen A H Groenevelt and H C Tijms (1984) Coordinated replenishments in a multi-item inventory system with compound Poisson demands and constant lead times Management Sci 30 344-357

Lee H L and S Nahmias (1993) Single-Product Single-Location Models in SC Graves A H G Rinnooy Kan and PH Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Muckstadt J A and R O Roundy (1993) Analysis of Multistage Production Systems in SC Graves A H G Rinnooy Kan and PH Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Porteus E L (1985) Numerical Comparisons of Inventory Policies for Periodic Review Systems Oper Res 33 134-152

Puterman M L (1994) Markov Decision Processes Discrete Stochastic Dynamic Programming Wiley New York

Silver E A and R Peterson (1985) Decision Systems for Inventory Management and Production Planning 2nd edition Wiley New York

Taylor H M and S Karlin (1994) An Introduction to Stochastic Modeling revised edition Academic Press San Diego

74

Tijms H C and H Groenevelt (1984) Simple approximations for the reorder point in periodic and continuous review (sS) inventory systems with service level constraints European J Oper Res 17 175-190

75

Appendix A

A Portion of the Sales File for Product 200001 a 30-amp Inline Fuse

200001 283 1190 0 0 2 0 0 1 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 200001 284 1190 2 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 200001 285 1190 0 0 4 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 286 1190 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 200001 287 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 200001 288 1190 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1 0 2 200001 289 1190 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 290 1190 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 200001 291 1190 0 0 2 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 200001 292 1190 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 200001 294 1190 0 0 0 1 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 1 200001 295 1190 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 200001 296 1190 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 200001 297 1190 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 200001 298 1190 0 0 1 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 200001 299 1190 0 0 1 0 0 0 1 0 1 0 0 2 0 0 0 0 0 2 1 1 0 200001 300 1190 0 0 1 0 0 0 0 1 0 0 0 0 0 2 2 0 0 0 0 3 0 200001 301 1190 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 200001 303 1190 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 200001 304 1190 1 0 0 0 0 0 0 0 1 0 0 2 0 0 0 0 0 0 0 0 0 200001 305 1190 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 200001 306 1190 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 2 0 0 1 0 200001 308 1190 0 0 0 1 0 1 0 0 0 2 0 1 0 0 0 0 1 1 1 0 0 200001 309 1190 1 3 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 200001 310 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 200001 311 1190 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 200001 312 1190 0 1 1 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 200001 313 1190 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 2 0 200001 314 1190 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 200001 315 1190 0 0 2 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 200001 316 1190 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 200001 317 1190 2 0 0 1 0 0 0 0 0 0 5 1 0 0 0 1 0 0 0 0 0 200001 318 1190 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 1 200001 319 1190 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 200001 320 1190 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 200001 321 1190 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 1 200001 322 1190 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 323 1190 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 200001 324 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 0 1 0 0 2 1 1 0 200001 325 1190 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3 200001 326 1190 0 0 1 0 1 0 0 0 2 0 0 1 0 0 0 1 0 1 0 0 1 200001 328 1190 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 200001 329 1190 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 200001 330 1190 0 0 1 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 200001 331 1190 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 200001 332 1190 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 200001 333 1190 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 200001 334 1190 0 0 2 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 200001 335 1190 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 200001 336 1190 0 0 0 1 0 0 3 0 0 0 0 0 2 0 0 0 0 0 0 0 0

(253 subsequent lines omitted)

76

Appendix B

Sample Distribution Files

a) A low-demand product - an automobile back-up alarm

PRODUCT NUMBER 202101 PRICE $ 684

DMD

3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

N 2 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 2 2 0 1

1 5 3 8 12 5 7 11 6 11 5 4 11 10 19 4 2 5 14 5 6 4

0 302 304 299 295 302 300 296 301 296 302 303 295 296 288 303 305 301 290 300 301 302

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

b) A medium-demand product - a 168 bulb

PRODUCT NUMBER 203039 PRICE $ 51

DMD

16 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 1 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0

9 0 0 2 5 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0

8 0 0 2 5 0 0 2 0 1 1 3 0 1 3 0 0 1 1 0 0 1

7 2 1 3 1 0 0 1 0 2 3 1 1 4 1 1 0 2 1 1 1 1

6 4 7 1 10 3 1 3 2 1 3 5 2 7 4 1 0 1 1 3 3 4

5 1 7 7 17 0 2 11 1 5 4 1 2 6 9 6 2 6 9 4 4 1

4 5 10 10 16 5 5 19 10 13 8 11 9 13 10 12 7 5 15 5 8 3

3 10 12 13 27 4 6 24 11 10 13 14 9 23 14 18 8 14 18 14 12 14

2 14 30 36 43 16 16 43 32 38 39 39 34 47 36 41 21 19 39 20 34 24

1 30 56 68 56 37 30 70 48 67 61 44 75 62 63 48 28 52 61 63 53 47

0 240 184 165 126 240 247 133 202 169 175 189 173 143 165 180 240 206 161 197 192 212

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

77

c) A high-demand product - a H4651 rectangular sealed beam headlight

PRODUCT NUMBER 203677 PRICE $ 857

DMD

14 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

13 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 2 3 0 0 1 0 0 0 0 1 1 1 0 0 0 0 2 0 0

11 0 1 3 1 0 0 4 1 0 0 0 0 0 0 0 0 1 0 0 0 1

10 0 0 4 4 1 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 1

9 0 3 14 11 0 1 7 1 0 2 0 8 0 1 0 0 0 2 1 0 0

8 0 4 12 16 1 0 18 4 3 5 0 2 1 4 2 1 1 0 0 1 3

7 1 2 12 13 1 6 19 3 5 9 0 11 2 5 4 0 2 3 3 1 4

6 2 9 22 21 6 10 29 12 10 11 4 24 3 7 10 4 7 11 6 3 14

5 7 13 35 33 8 16 29 19 18 8 3 26 15 26 15 12 9 14 13 11 19

4 19 35 34 43 32 30 41 31 32 47 11 42 20 33 30 31 20 31 22 21 31

3 32 63 49 54 42 60 51 40 51 35 33 55 39 51 39 38 39 57 58 40 54

2 46 59 56 55 73 63 61 65 63 61 63 54 68 85 65 53 72 69 70 77 63

1 109 79 42 40 80 64 32 80 74 77 99 55 97 57 83 93 80 75 81 97 73

0 91 39 21 13 63 57 11 51 51 52 94 29 61 35 59 75 76 45 51 56 44

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

78

Appendix C

A Proof that the Limiting Probability Distribution is Independent of the Initial State

In this appendix we show that provided 0 lt PrDT = 0 lt 1 the Markov chain

corresponding to an inventory model with a (sS) policy has a limiting probability distribution

7T where 0 lt TTJ lt 1 for y = 0 1 S that is independent of the initial state Further the

limiting distribution is the unique nonnegative solution of the equations

Tt P = TI (AD

s and J gt = 1 (A 2)

i=0

where P is the (S + l) x (S + l) transition probability matrix whose first n rows (and columns)

correspond to the n recurrent states in the chain and whose remaining rows (and columns) if

any correspond to transient states43 Prior to the proof several preliminary results are

obtained

RESULT 1 The Markov chain has exactly one positive recurrent class and perhaps some

transient states

First note that in the absence of any absorbing states a Markov chain with a discrete

finite state space must have at least one positive recurrent class containing at least two states

Now consider the Markov chain for an inventory model with a (sS) policy

a) State i gt s cannot be absorbing because there is a positive probability of reaching

j lt i (Since PrDT gt 0 gt 0)

4 3 Note that since the rows and columns of P are reordered the elements of it must be reordered accordingly

79

b) State lt s cannot be absorbing because by the rules of the ordering policy and

the assumption that PrDT = 0 gt 0 there is a positive probability reaching S

Therefore the chain has no absorbing states and must have at least one positive recurrent class

Next consider the subset of states A = i | i lt s Because the state space is finite

and PrDT gt 0 gt 0 the process must return to at least one state in A in other words at least

one state in A must be recurrent Let k G A be a member of a recurrent class dt Because

of the ordering policy and the assumption that PrDT = 0 gt 0 k -raquo S Also by definition

there must be a positive probability of returning to k from S S -gt k Therefore S G 3w

Further because S is reachable from all states in A i G A is either transient with i -raquo dt or

recurrent with pound iR

Now consider the subset of states B = | j gt s Because PrDT gt 0 gt 0 the

chain will eventually reach a state i G A If i is transient then j must be transient also with

y -gt 3 if i G 9 then j is either transient with j -gt dt or recurrent with j E dt Thus the

Markov chain has exactly one recurrent class and perhaps some transient states

RESULT 2 The limiting probability distribution ir for the Markov chain corresponding to the

nxn submatrix Pm = [Ptj] i j G dt is independent of the initial state and is the

unique nonnegative solution of the equations

TC-Prr = re (A3)

and J2 K = 1 ( A 4 )

jt=i

where the rows and columns of are numbered from 1 to n rather than

Since if i j i and j must be in the same equivalence class

80

according to their associated states

First note that both the original Markov chain and the one corresponding to the

submatrix P R R are aperiodic because PrDT = 0 gt 0 Therefore given that 9 is a positive

recurrent aperiodic class the limiting distribution ir is uniquely determined by (A3) and (A4)

(Taylor and Karlin 1994 Theorem 44 p213) Further because there are no absorbing states

or other recurrent classes in the original chain the process must eventually enter the recurrent

class 9 regardless of the initial state Therefore in the limit the probability of being in state

i G 9 is independent of the initial state

RESULT 3 The limiting probability distribution for the transient states is the null vector

independent of the initial state

This result follows directly from the definition of a transient state

PROOF Suppose the states are reordered such that the first n rows (and columns) of P

correspond to the n recurrent states of the Markov chain Then P can be partitioned into four

submatrices

PRK - the n x n submatrix [Py] i j E 9

P T T - the (S + 1-I) x (S + l-n) submatrix [Py] i j g 9

P T R - the (S + l-n) x n submatrix [Py] i pound 9 j G 9

P R T - the n x (S + l-n) submatrix [Py] i G 9 j pound 9t By definition this is a matrix

of zeros

(Al) can then be rewritten as

81

(ltR TtT) P P L TR TT

I-P RR 0

p I-P TR 1 TT

(TtR TCT)

(0- 0T)

where irR denotes the limiting probability distribution for the n recurrent states and irT denotes

the distribution for the remaining S + l-n transient states Solving for TT t gives

TCr-0 + itT-(I-PTI) = 0T

itT-(I-PJJ) = 0 r

i c r = (I-P^-Oj

Since (I - PTT)1 always exists (Puterman 1994 Proposition A3 p 593) irT = 0X which is

in agreement with Result 3 Now solving for 7rR yields

(A2) can be rewritten as

bullRPRR 1 1R (A5)

(A6) S + l

k + L k =1

k=l k=n+

k=i

where ir contains the appropriately reordered steady state probabilities Since (A5) and (A6)

are equivalent to (A3) and (A4) respectively Result 2 holds Therefore the Markov chain

corresponding to an inventory model with a (sS) policy has a limiting distribution that is

independent of the initial state and is the unique nonnegative solution to (Al) and (A2)

82

Appendix D

A Measure for the Steady State Customer Service Level

In this appendix we show that the appropriate measure for the steady state customer

service level is given by

_ E[Demand satisfied during RP] E[Total demand during RP]

Consider the following two measures of the steady state customer service level

Measure 1 Suppose the customer service level is calculated only after n review periods have

passed Denoting DS as the demand satisfied in review period i and DT as the total demand

in period i the customer service level after n periods CSL(n) is

DS + DS0 + + DSbdquo CSLn) = - -

DT + DT + + DT

Dividing both the numerator and denominator by n taking the limit as n -raquo oo and applying

the Law of Large Numbers yields

CSL = lim CSL(n) n-degdeg

- [DS + DS + + DS] = l i m ^ J 1 1

raquo~ - [DT + DT + + DT]

E[DS]

E[DT]

83

Measure 2 Suppose the customer service level is calculated for each of n review periods and

then averaged Thus the customer service level after n periods CSL(n) is given by

CSL(n) = -DS DS

DT DT DT_

Taking the limit as n -gt oo yields

CSL = lim CSL(n)

lim mdash DSl DS2

DT DT DT_

DS

DT

Now consider the scenario in which there is one item on the shelf to begin each review

period and in which demand is stochastic with PrDT = 2 = Vi and PrD x = 100 = Vi

According to Measure 1 the customer service level is

CSL - = ^ W + ^ W = plusmn bdquo 002 E[DT) 1 (2) + 1(100) 51

while according to Measure 2 the customer service level is

CSL = E DS DT

1(1) + i(-Ls

22) + 2 100

51 200

026

Clearly from managements point of view the first measure is more indicative of the

actual level of service being provided The problem with the second measure is that equal

84

weight is given to each ratio regardless of the magnitude of the numbers For example with

respect to Measure 2 the events LSLR = 12 and LSLR = 50100 are identical however

with respect to lost sales the second event is much worse than the first Therefore the proper

measure for the steady state customer service level is Measure 1

85

Appendix E

The Calculation of the Conditional Expected Demand Not Satisfied During a Period of T Consecutive Days

Denoting D x as the demand during a period of T consecutive days and j as the inventory

on hand at the beginning of the period the conditional expected demand not satisfied during the

period E[(DT-y)+] is calculated as follows

E[(DT-j)+] = pound ltk-j)Pr[DT = k k=j+l

= pound (k-j)PrDT = k] + J2(k-j)PrDT = k] - pound (fc-)VJgtr = fc k=j+l k=0 k=0

= J2(k-j)Pr[DT=k -J2(k-j)PrDT = k] k=0 k=0

= YkPrDT=k) - J2jPrDT=k - Yk-j)PrDT=k) Jfc=0 k=Q k=0

= Ti - j - Y(k-j)PrDT=k 7-1

= Tx - j + YltJ-k)PrDT = k jt=0

where fx is the mean daily demand

86

Appendix F

Justification of the Updating Technique for the New Policy (s+mS)

Let the probability transition matrix corresponding to the policy (sS) be denoted by the

(S + l)x(S + l) matrix P with elements py Define s = s+m where m is an integer satisfying

the condition 0 lt s+m lt S Let the probability transition matrix of the new policy (sS)

be denoted by the (S + 1)X(S + 1) matrix 7gt with elements py Using the equations developed

in Chapter III the elements of P are found as follows

CASE 1 The lead time is less than the review period

A m gt 0

For 0 lt i lt s

( i-i pound [PrDL=kPrDT_LgtS-k]] + PrDLziPrDT_LplusmnS-i k=0

i-l 53 [PrDL=kPrDT_L = S-j-k]] + PrDLziPr[DT_L = S-j-i) k=0 s-j

52 [Pr[DL = kPrDT_L = S-j-kn k=0

ifj = 6

ifOltjz S-i

if S-i ltj ltS

a) 0 lt lt s

Pij

= lt Pij

Pij

= pij

b) s lt i lt s This does not simplify but it is identical to (2)

87

For s lt i lt S

PrDTgt i

p = lt PrDT = i-j

0

Pfj ifj = 0

= lt Pij ifOltjlti

Pij if i ltj plusmn S

= pij

B m lt 0

ifj = 0

if 0 ltj lt i

if i ltj plusmn S

For 0 lt i lt s

i-l 53 [PrDL=kPrDT_LplusmnS-k]] + PrDLgtiPrDT_LplusmnS-i

i-l E [PrZgti = fciVZ)7W=S--fc] +PrD tgtiP-D 7 t = S-7-

p y laquoy = o

P i j ifOltjS-i

vPij ifS-iltjS

ifj = 0

ifOltjltS-i

if S-i ltjltS

For s lt i lt S

fPrJgtrgt- 17=0 = PrDT = i-j] ifOltjlti

0 ifiltjplusmnS

88

a) s lt lt s This does not simplify but it is identical to (3)

b) s lt i lt S

PrDTgti

= bull PrDT = i-j]

0

17=0

if0ltjplusmni

ifiltjS

= Pt

ifj = 0

if0ltjlti

ifiltjplusmnS

= Pi

CASE 2 The lead time is equals the review period

A m gt 0

For 0 lt i lt s

0 if0ltjlt S-i

PrDTgti ifj = S-i

PrDT=S-j] ifS-iltjltS

a) 0 lt i lt s

= bull y Pij

Pij

ifOplusmnjltS-i

ifj = S-i

ifS-iltjS

= Pij

b) s lt i lt s This does not simplify but it is identical to (4)

89

For s lt i lt S

PrDTgti)

Pr[DT=i-j

0

laquof 7=0

ifOltjplusmni

if i ltj lt S

Pa

Pij

if 7 = 0

if 0 lty lt i

ifiltjltS

B w lt 0

For 0 lt t lt s

ro PrJrgt PrDT = S-j

ifOzjltS-i

ifj = S-i

if S-i ltj lt S

Pij

ifOzjltS-i

ifj = S-i

if S-i ltj ltS

= Pi

For s lt i lt S

PrDTgti

= PrDT = i-j

0

i 7 = o ifOltjplusmni

ifiltjltS

a) s lt i lt s This does not simplify but it is identical to (3)

90

PrDTzi ifj = 0

PrDT = i-j] ifOltjplusmni

0 ifiltjltS

Py ifj = 0

ptj ifOltjplusmni

Pij if iltj plusmnS

Appendix G

Justification of the Updating Technique for the New Policy (sS+m)

Let the probability transition matrix corresponding to the policy (sS) be denoted by the

(S + l)x(S + l) matrix P with elements p(gt Define S = S+m where m is any non-negative

integer Let the probability transition matrix of the new policy (sS) be denoted by the

(S+m+l)x(S+m + l) matrix IP with elements ptj Then using the equations developed in

Chapter III the elements of 7 are found as follows

CASE 1 The lead time is less than the review period

For 0 lt i lt s

pound [PrDL=kPrDT_LplusmnS-k] + PrDLi)PrDT^Sgt- k=0

i-l pound [Pr[DL = kPrDT_L = S-j-k- +Pr[DLiPrDT_L = S-j-i Jt=0

s-i Y [Pr[DL=kPrDT_L = S-j-k] k=0

i - l

ifj=0

ifOltjltS-i

ifS-iltjltS

pound [PrDL = kPrDT_LS-k + Pr[DLi)PrDT_LplusmnS-i) if j = 0 fc=0

i - l = pound [PrDL=kPrDT_L = (S+m)-j-k] + PrDLgtiPr[DT_L = (S+m)-j-i ifOltj lt (J5+m)-i

if (S+m)-iltj lt(S+m)

k=0

i - l

pound (Sm)-j

pound [PrDL = kPrDT_L = (S+m)-j-k]] k=0

92

I-l

53 [PrDL = kPrDT_LgtS-k] + PrDLziPrDT_LS-i ifj = 0 k=0

i-l

Y[PrDL = kPrDT_L = S-(j-m)-k] + Pr[DLziPrDT_L = S-(j-m)-i] ifOltjzS+m-i k=0 S-(j-m)

53 [PrDL = kPrDT_L = S-(j-m)-k]] if S+m-i ltj lt S+m

i - l

53 [PrDL = kPrDT_LzS-k] + PrDLziPrDT_LzS-i] ifj = 0 =0

i - l

13 [Pr[DL = kPrDT_L = S-(j-m)-k] + Pr[DLziPr[DT_L = S-(j-m)-i ifOltjz S+m-i k=o and 0 z j-m

Pi(j-m)

Pi(j-m)

if 0 ltj lt S+m-i and 0ltj-mltS-i

if S+m-i ltj lt S+m and S-i lt j-m zS

i - l

Y [PrDL = k)PrDT_LzS-k] + PrDLi)PrDT_LplusmnS-i ifj = 0

k=0 i - l

53 [PrDL = kPrDT_L = S-(j-m)-k] + PrDLziPrDT_L = S-(j-m)-i ifOltjzm k=0

Piij-m)

Piij-m)

if mltj lt S+m-i

if S+m-i ltj lt S+m

i - l

YlPrDL = kPr[DT_L^S-k + PrDLi)PrDT_LS-i) ifj = 0

i - l

53 [PrDL = kPrDT_L = S-j-k] + PrDLgtiPrDT_L = S-j-i ifOltj i m k=0

Pi if mltj ltS

which for 0 lt j lt m is identical to (2)

93

For s lt i lt S

Pu =

PrDTgti

PrDT = i-j]

0

if0ltjlti

ifiltjltS

a) s lt i lt S

PrDTgti 7=0

PrDT = i -j) if0ltjlti

0 if i ltj lt S

0 ifSltjltSgt

ifj=0

if 0 ltj lt i

if i ltj plusmn S

0 ifSltjltS

ifOltjltS

0 ifSltjltSgt

b) S lt i lt S This does not simplify but it is identical to (3)

CASE 2 The lead time equals the review period

For 0 lt i lt s

0 ifO lt jltS-i

PrDTgti ifj=S-i

PrDT=S-j ifS-iltjltS

94

0 ifOplusmnjlt(S+m)-i

PrDT ifj = (S+m)-i

PrDT = (S+m) -j) if (S+m)-i ltj ltL (S+m)

0 if 0 lt jltS+m-i

PrDTzi] if j = S+m-i

PrDT = S-(j-m) if S+m-i ltj lt S+m

0 if 0 ltj lt S+m-i and 0 ^ j-m

Pi(j-m) if 0ltjlt S+m-i and 0ltj-mltS-i bull

Pi(j-m) if j = S+m-i and j-m = S-i

Pi(j-m) if S+m-i ltj z S+m and S-i lt j-m ^S

0 if 0 zj ltm

Pi(j-m) if mlt jlt S+m-i lt

Piij-m) if j = S+m-i

if S+m-i ltj lt S+m

1deg if 0 ltj ltm

[PiV-m) ifmltjltS

which for 0 lt i lt s is identical to (3)

For s lt lt S

PrDTzi

= PrDT = i-j)

0

ifj = 0

if 0 ltj ltL i

ifKjzS1

95

a) s lt i lt S

PrDTzi

PrDT = i-j]

0

0

17=0

if0ltjplusmni

ifiltjS

ifSltjltSgt

Pij 7=0

Pij if0ltjii

Pij if i ltj ltL S

0 ifSltjltS

Pij ifOzjiS

0 ifSltjltS

b) S lt i lt S This does not simplify but it is identical to (3)

96

Appendix H

A Hypothetical Consultation

In this appendix the interface during a hypothetical consultation is depicted The figures

accurately represent the various screens that are presented to the manager with the exception

that all lines and text are really white on a blue background During the consultation described

below only the choices made by the manager are hypothetical all the results are real

The title screen as depicted in Figure A - l appears initially upon execution of the

program In order to proceed to the main menu the manager presses the enter key

O I C S

Optimat Inventory Control System

For the exclusive use of

Canadian Tire Pacific Associates

Developed at The Faculty of Commerce

and Business Administration The University of British Columbia

Brian Kapalka Kaan Katircioghi Marty Puterman

Press ltEntergt to continue

Figure A - l The title screen

97

Main Menu 1 Select data directory

ctircdatadst

2 Select product and store combination Product 200001

3 Select scenario Review Period 4 Lead Time 3

4 Select search parameter Smax 25

5 Execute program

6 Exit to DOS

Please enter the desired option 5

Figure A-2 The main menu

Figure A-2 depicts the main menu of the program it is from here that the manager

enters all the review parameters If the manager wishes to change one of the parameters from

its default value he or she selects the appropriate number and enters the correct information

in the window presented The new settings will then be shown in place of the default values

In this example the relevant data files are located in the default directory ctiredatadst and

the manager has selected the productstore combination of product 200001 and store 6 Further

the manager has selected a review scenario of a 4-day review period and a 3-day lead time and

an upper search limit of 25 Finally in order to proceed with the analysis the manager

executes the main module of the program by selecting option 5

98

Acknowledgement

The completion of this thesis was made possible by the encouragement and assistance

of a number of people

I would like to express my sincere appreciation to my thesis supervisor Professor

Martin Puterman for all of his many efforts on my behalf His help advice patience charity

and tolerance were very much appreciated

I would like to acknowledge Professor Hong Chen and Professor Garland Chow for their

time and input as members of my thesis committee In addition I must acknowledge the

assistance of PhD student Kaan Katircioglu for his insight and help on this project

I offer many thanks to Professor Tom Ross for his kindness and friendship during the

past few years especially during the writing of this work I must also thank his family for

adopting me on many holidays I still owe thanks to Professor Slobodan Simonovic at the

University of Manitoba for influencing me to attend graduate school in the first place and for

helping me to obtain funding

I cannot thank my parents enough for their never-ending support both emotional and

financial Also to my friends especially Cathy Dave Lisa Steve and the Philbuds thanks

for giving me a life off campus and for picking up many a tab - the next one is on me

I would like to give special thanks to my good friend and fellow MSc student Paul

Crookshanks for allowing me to bounce ideas off of him and for being such a procrastinator

that despite my finishing a year late I only lost by two days

Financial assistance received from the Natural Science and Engineering Research

Council (NSERC) and from Canadian Tire Pacific Associates was greatly appreciated

vii

I INTRODUCTION

The importance of inventory management has grown significantly over the years

especially since the turn of this century In colonial times large inventories were viewed as

signs of wealth and therefore merchants and policy makers were not overly concerned with

controlling inventory However during the economic collapse of the 1930s managers began

to perceive the risks associated with holding large inventories As a result managers

emphasized rapid rates of inventory turnover (Silver and Peterson 1985) Following the

Second World War Arrow Harris and Marschak (1951) and Dvoretzky Kiefer and

Wolfowitz (1952ab) laid the basis for future developments in mathematical inventory theory

Shortly thereafter new inventory control methodologies were widely applied in the private

manufacturing sector More recently when inflation and interest rates soared during the 1970s

many organizations were forced to rethink their inventory strategies yet again Today the

control of inventory is a problem common to all organizations in any sector of the economy

Sir Graham Day chairman of Britains Cadbury-Schweppes P C L expressed this sentiment

when he stated I believe the easiest money any business having any inventory can save lies

with the minimization of that inventory (Day 1992) Lee and Nahmias (1993) and Silver and

Peterson (1985) provide a more detailed history of inventory management and control

Most inventory problems in the retail sector involve the control and management of a

large number of different products Ideally the inventory policies should be determined for all

products on a system-wide basis however because the number of products is often in the tens

of thousands formulating a model to do so is impractical Therefore single-product models

are frequently used in practice to determine policies for products separately (Lee and Nahmias

1993) Lee and Nahmias (1993) and Silver and Peterson (1985) provide a good overview of

1

single-product single-location models for a more complete review of the literature we refer

to them

The degree of complexity of a single-product model depends largely on the assumptions

made regarding the demand cost structure and physical characteristics of the inventory system

For example demand can be either deterministic meaning it is known with absolute certainty

or stochastic meaning it is unknown but follows a known probability distribution

Deterministic models tend to be relatively simple yet useful such as the basic EOQ model

However future demand is seldom known with certainty in practice and therefore the use of

these simple models is limited As a result the majority of research in inventory has focused

on stochastic demand models (Lee and Nahmias 1993)

Because the objective of most inventory models is to minimize the cost of inventory

assumptions regarding the structure of the cost are very important also Generally speaking

the cost of inventory has four components the purchaseproduction cost the orderingsetup

cost the holding cost and the shortage cost1 Each of these costs can be fixed proportional

nonlinear or some complex combination of these Further because nearly all inventory

problems require decisions be made sequentially over a period of time the costs can be either

discounted or averaged In the former case a discount factor is included in the cost function

in order to account for the time value of money However since most inventory problems are

on-going the long-run average cost is usually minimized (Lee and Nahmias 1993) In

practice this is achieved by minimizing the expected cost per time period under steady state

conditions

Finally the physical characteristics of the system are important in determining the

1 The cost of a stockout is often very difficult to estimate in practice As a result service levels are frequently used in its place When this is done a constraint is included in the model

2

complexity of the model For example the review process can be either continuous in which

case the inventory is reviewed continuously and orders are placed as soon as the stock reaches

a certain level or periodic in which case the inventory is reviewed at regular intervals and

orders can be placed only at those fixed times Also excess demand can be fully backordered

as is common in many manufacturing and wholesale systems completely lost as is common

in most retail environments or some combination of the two Lead time assumptions are also

important models with lead times are more complex than those in which the supply of

inventory is assumed to be instantaneous

Although the literature is dominated by single-product models (Lee and Nahmias 1993)

much recent work has focused on extending those models to multiple-product problems

However multiple-product models are inherently more complex than the simple-product

models In fact no efficient algorithms are currently available for determining fully optimal

strategies for most multiple-product problems (Federgruen 1993) As a result research has

focused on the development of easily computable approximation techniques Muckstadt and

Roundy (1993) provide a good overview of multiple-product andor multiple-location models

with deterministic demand although emphasis is placed on models for production and

distribution Federgruen (1993) extends the discussion to consider these models under uncertain

demand and also provides a complete list of references on the subject Silver and Peterson

(1985) discuss models for coordinated replenishments for both deterministic and stochastic

demand

In this thesis we examine the inventory system at Canadian Tire Pacific Associates

owner of 21 retail stores and a central warehouse Although this is a multiple-product

multiple-location single-warehouse problem and strictly speaking should be modelled as such

we analyze the inventory problem on a product-by-product and store-by-store basis We believe

3

that this simpler single-product single-location model captures the essentials of the problem for

two reasons First the coordination of replenishments for products andor locations is most

advantageous when significant savings are possible because of high first order costs and low

marginal costs for placing subsequent orders However because of the large number of

products and the physical distribution system in this problem trucks are dispatched to stores

on a regular basis Therefore the benefits of coordinating replenishments for this problem are

negligible Second we believe that controlling in-store inventory alone can result in substantial

savings to Canadian Tire Pacific Associates Further given the excessive amount of inventory

in the warehouse a single-location model assuming an infinite warehouse capacity is reasonable

Of course should inventory control in the warehouse become a priority a more elaborate model

would be required

More specifically we consider a periodic review single-product inventory model with

stationary independent stochastic demand a deterministic lead time and an infinite planning

horizon The probability mass function for the demand is based on empirical sales data

collected over a period of approximately ten months The cost of inventory includes a

proportional holding cost and a fixed ordering cost neither of which is discounted All excess

demand is completely lost The objective of the model is to minimize the long-run average cost

per unit time subject to a service level constraint on the fraction of demand satisfied directly

from in-store inventory In order to obtain meaningful cost comparisons the model calculates

the expected annual cost of inventory

For inventory problems like the one described above optimal (sS) policies can be

computed by Markovian decision process methods such as successive approximations or policy

iteration or by stationary Markovian methods However each of these methods generate

calculations that are complex and time consuming Therefore much research interest lies in

4

developing techniques to find approximate (sS) policies with little computational effort The

majority of the approximation techniques begin with an exact formulation of the problem and

then manipulate the problem by relaxations restrictions projections or cost approximations

(Federgruen 1993) Porteus (1985) presents three approximation techniques and compares

them to fourteen others for a periodic review single-product inventory model in which

shortages are fully backordered and a shortage cost is incurred Federgruen Groenevelt and

Tijms (1984) present an algorithm for finding approximate (sS) policies for a continuous review

model with backordering and a service level constraint Also Tijms and Groenevelt (1984)

present simple approximations for periodic review systems with lost sales and service level

constraints All three works find that the approximation techniques perform well However

the techniques are evaluated for a variety of demand parameters that all result in policies with

a value of S in the range of 50 to 200 Given the slower moving products at Canadian Tire 2

it is unclear how well such approximation techniques would perform Therefore we do not

attempt to apply or modify any of these approximation methods

Exact optimal policies for this problem can be obtained by formulating a constrained

Markov decision process and solving it However the data generating requirements and the

lack of a good upper bound on S make this formulation impractical Therefore we formulate

a Markov chain model for fixed values of s and S and then employ a search technique to find

the optimal values for s and S In order to improve the performance of the model we develop

a heuristic search that locates near optimal policies relatively quickly

The remaining chapters of this thesis are organized as follows In Chapter II an

overview of the existing inventory system at Canadian Tire is presented along with a description

2 For example very few of the products analyzed have optimal values of S exceeding 20 with the largest value of S equalling 69

5

of the problem Next a multi-phase project designed to develop a fully integrated interactive

inventory control system is described with the scope of the research reported in this thesis

completing the first phase of the project Finally a description of the demand and cost data is

given

In Chapter III we formulate a Markov chain model that can be used to calculate the cost

and service level associated with a given (sS) policy Based on several simplifying

assumptions the Markov chain is formulated and the equations for the corresponding transition

probability matrix are developed Then given the limiting probability distribution of the chain

the equations for the steady state annual cost of inventory and the corresponding service level

are obtained The chapter concludes with a brief summary of the model

In Chapter IV an algorithm that searches for the optimal (sS) policy is developed As

a starting point a simple grid search algorithm is developed and evaluated on the basis of

execution time In order to improve the algorithms speed a technique for updating the

transition probability matrix is presented and the improvement in the algorithm is noted Next

a theoretical lower bound on S is obtained although the corresponding reduction in the

execution time is minimal Finally based on two assumptions regarding the shape of the

feasible space a heuristic search is developed which improves the speed of the algorithm

significantly

In Chapter V a prototype of the inventory control systems interface is created In its

present form the interface allows managers to evaluate inventory policies interactively The

manager can set review parameters compare optimal policies and current policies and evaluate

alternate policies The chapter concludes by recommending specific options be including in

future versions to make the system more automated and user-friendly

In Chapter VI optimal policies for all products in one product category sold at one store

6

are compared to existing policies on the basis of expected annual costs and customer service

levels The analysis suggests that existing policies result in a large excess inventory and that

implementing the optimal policies would reduce the annual cost of inventory by approximately

50 A sensitivity analysis on the cost estimates is performed and suggests that the model is

not sensitive to relative errors of approximately 20 in the cost estimates Finally a sensitivity

analysis on the demand probability mass function is performed The analysis indicates that the

model is sensitive at least in terms of the customer service level to slight changes in the tail

of the demand distributions of low-demand products

Finally in Chapter VII the results of the analysis are summarized and future work is

outlined In order to incorporate recent changes in the review procedures at several stores the

model must be modified to allow for the duration of the review period to exceed the lead time

We recommend that the second phase of the project proceed specifically that a data collection

and storage scheme be designed and that a forecasting model be developed Because of the

substantial savings that can be realized by reducing inventory only at the stores inventory

control at the warehouse should also be investigated

7

II T H E I N V E N T O R Y S Y S T E M A T C A N A D I A N TIRE

A Background

Canadian Tire Pacific Associates owns and operates 21 retail stores in the lower

mainland of British Columbia The stores carry approximately 30000 active products3 that

are divided into 80 product categories The merchandise comprises primarily automotive parts

and accessories home hardware housewares and sporting goods Canadian Tire Pacific

Associates also owns a central warehouse in Burnaby at which inventory is received from

Toronto Inventory is then shipped periodically from the warehouse via truck to the local

stores In 1992 total sales at the stores exceeded $99 million and the value of inventory was

estimated to be $28 million of which approximately half was stored in the warehouse

On selected days of each week a stores inventory records are reviewed automatically

and orders for products are placed to the warehouse as required Management uses the term

flushing to describe the process of reviewing the inventory and producing the tickets for stock

owing and the term picking to describe the physical act of filling the order at the warehouse4

Whether or not an order is placed for a product depends upon its current inventory level its

target level and its minimum fill 5 If the target level less the current level exceeds the

minimum fill an order is placed to bring the stock up to the target level From the moment

3 Active products are those that are regularly restocked Canadian Tire Pacific Associates can choose from up to 50000 products

4 Note that the number of picks per week is equal to the number of times per week that the inventory records are reviewed At the beginning of this study eighteen stores had two picks per week while the remaining three stores had only one

5 The target level is the maximum number of items that should be on the shelf at one time and the minimum fill is the minimum number of items that should be ordered

8

the order is placed to the moment the product is reshelved takes between two to four days

depending on the store On average 11000 items are shipped from the warehouse at each

flush

Although a products minimum fill is the same for all stores its target level depends

upon the size of the store Based upon its sales volume and retail space a store is classified

as being one of five sizes small (S) medium (M) large (L) jumbo (J) and extra jumbo

(EJ) 6 Each product has associated with it a target level for each size of store

Consequently all stores of the same size have identical inventory policies Table 1 lists the

location size number of picks per week sales and floor space of each store as of October

1993

Periodically the inventory manager reviews the policies and when deemed necessary

adjusts them In this way the manager can account for seasonality in demand changes in mean

demand and any shortcomings of previous policies At the present time the review process

is not automated nor performed on-line The inventory manager uses Hxl5 computer

printouts with five products to a sheet to study the relevant information and on which to note

any changes Afterward the changes are recorded in the system and the new policies take

effect

Currently the inventory policies are based on rules of thumb developed through the

years and not on a statistical analysis of demand nor on a calculation of the cost of inventory

These rules of thumb incorporate a marketing philosophy which at times dominates the

inventory strategy The prevailing philosophy holds that a large in-store inventory instills

6 During the course of this study stores were resized several times At one point the entire class of medium stores was eliminated leaving only four sizes of stores On another occasion the store in Chilliwack was changed from being a small store to a large store

9

Table 1 Location and particulars of the 21 stores

Store Location Store Size

Picks Per Week

1992 Sales ($)

Floor Space (ft2)

1 Chilliwack S 2 2901969 6678

2 Coquitlam M 2 4015303 6510

3 Langley EJ 2 8138673 15422

4 Scott Road EJ 2 7307820 15064

5 Maple Ridge L 2 4102110 6678

6 Kingsway L 2 4153736 6574

7 Richmond EJ 2 8175105 14961

8 North Vancouver M 2 3856571 6534

9 Guildford M 2 3714372 6567

10 Burnaby M 2 3532752 6983

11 New Westminster S 1 2417992 6470

12 Abbotsford EJ 2 7899398 15303

13 Hastings J 2 4647794 9849

14 King Edward EJ 2 3825372 25492

15 White Rock L 2 3960262 6870

16 Main amp Mountain S 1 2705602 6431

17 Heather J 2 4161227 9427

18 King George J 2 4606817 9640

19 Cambie J 2 5855022 9091

20 Lougheed M 1 3114180 6615

21 Burrard J 2 6553310 10608

10

consumer confidence which in turn leads to increased sales In other words if a customer

perceives that Canadian Tire is in the hardware business he or she will choose to shop there

when meeting future needs However management has adopted this philosophy without any

analysis of the cost of holding excess inventory Only recently has management begun to

question the cost-effectiveness of allocating inventory based on these rules of thumb

Speculating that existing inventory practices might be deficient Mr Don Graham the

vice-president of Canadian Tire Pacific Associates commissioned a pilot study to examine the

inventory system and to propose alternate strategies7 The study using a steady state model

to represent a simplified inventory system suggested that existing policies lead to overly high

inventory levels More importantly the study outlined a methodology that would enable

management to determine inventory levels based on observed demand The pilot study

recommended that a more detailed model of the inventory system be formulated that the

methodology be expanded to incorporate the cost of inventory and that an algorithm be

developed that would quickly identify the optimal inventory policies

B The Problem

At the time of the pilot study management was concerned primarily with its inability

to distinguish between stores of the same size As mentioned in the previous section a

stores size is based solely on its sales volume and floor space and under existing practices

all stores of the same size have identical inventory policies for every product Management

realized that demand for some products might be a function of variables other than size such

7 The pilot study was performed by Martin Puterman a professor of Management Science and Kaan Katircioglu a doctoral student in Management Science both at the University of British Columbia The results of the study were presented at a U B C Transportation and Logistics Workshop on March 25 1993

11

as location For example the demand for sump pumps is likely to be very different in suburban

areas than in urban ones However management was unable to allocate more sump pumps to

large suburban stores than to large urban stores This inability led management to seek a

tool that would allow for a more effective allocation of inventory between stores of the same

size

By the end of the pilot study however management was less concerned with

reallocating inventory within a size of store than with reallocating inventory between sizes

of stores and with reducing overall inventory According to management assigning individual

inventory policies to each store appeared to be operationally infeasible Firstly management

believed that having individual policies would lead to confusion in the warehouse and to

increased shipping errors Exactly why this would occur is unclear the warehouse would

simply continue to fill orders from tickets that make no reference to the policy in place

Secondly and perhaps more tenably the computer technology at Canadian Tire Pacific

Associates did not allow for policies to vary across all stores Both the database used to store

the inventory data and the printouts used by the inventory manager allowed for only five

different policies per product all with the same minimum fill Moving to individual policies

would have required an overhaul of the existing information system Although management

saw this overhaul as inevitable it did not see the overhaul as imminent Therefore

management switched the focus of the study to developing a tool that would help reduce overall

inventory levels andor reallocate inventory more effectively between groups of stores

The subtle change in the studys objective prompted an important question how should

the stores be grouped The current practice of grouping stores based on size is problematic

for two reasons First as discussed above additional factors of demand such as location

cannot be taken into account Second grouping inherently results in a misallocation of

12

inventory when some stores within a group have differing numbers of picks per week For

example the medium-sized Lougheed store which has one pick per week requires higher

inventory levels to achieve a given customer service level than the other medium stores

which all have two picks per week8 In fact in terms of customer service levels it might

be more appropriate to group the Lougheed store with the large stores having two picks per

week Ideally this artificial grouping of stores will be abandoned when the computer system

is upgraded and policies can then be set on a store by store basis

As the project proceeded management grew somewhat uneasy with the concept of using

an analytical tool to evaluate and set inventory policies According to both the pilot study and

the preliminary results of this study significantly reducing inventory would greatly reduce costs

while still providing a high level of customer service However management held that because

the model cannot take into account intangibles such as marketing considerations such a drastic

reduction in inventory would be very inappropriate In this light management viewed the

optimal policies as lower bounds for inventory levels these policies would have to be

modified upwards in order to be practical Therefore management contended that any

methodology or software produced during this study should provide the means to evaluate and

compare a number of alternate policies

Regardless of exactly how the tool is to be used there is currently a lack of historical

sales data from which to analyze the inventory system and determine optimal policies No sales

data had ever been collected prior to the pilot study and only a relatively small amount has

been collected since In order to proceed with any analysis of the inventory system a

8 If demand is a function of size then the Lougheed store must meet the same demand with one fewer pick per week This would lead to either increased probabilities of stockouts at the Lougheed store or higher inventory levels at all the other medium stores

13

substantial amount of additional data would be needed The existing information system at

Canadian Tire Pacific Associates does allow for sales data to be captured however the system

is limited by the amount of data it can capture at one time and by its storage capacity In order

to collect the amount of data required management would have to upgrade the system and

develop some kind of data collection scheme coupled with a data compression technique The

collection scheme must allow for characteristics of demand such as trends and seasonality to

be identified and the compression technique must allow for sales data for over 30000 products

to be stored for each store

Because many of the products sold by Canadian Tire have nonstationary demand as a

result of seasonality andor trend the inventory tool must be able to forecast demand for the

coming period Products such as sporting goods and outdoor equipment exhibit patterns of

demand that are very seasonal For example sales of snow shovels increase through the fall

peak in the winter and drop to zero by late spring At the same time some products might

become either more or less fashionable over time resulting in changes in mean demand

Setting inventory policies for any of these products based on observed demand would result in

insufficient stock if demand increased during the period or in excess stock if demand decreased

Therefore the tool must be able to forecast the demand for the period for which the policy is

intended

Toward the end of the project management began discussing the integration of the

interactive inventory tool with the existing system to form a new inventory control system The

new system would utilize a modified point of sales system to collect sales data continually

Based on the latest available data the system would forecast demand for the coming period and

calculate the optimal policies for every product at every store The system would then provide

all relevant information on-line for the inventory manager to review The manager could weigh

14

other information such as marketing considerations and examine alternate policies to determine

the effect on cost Finally the preferred policy would be set in place with the touch of a key

and information on the next product or store would be presented for review

C The Project

Development of the inventory control system envisioned by management requires the

completion of five tasks 1) the design of a data collection system that generates and stores the

necessary sales data 2) the development of a forecasting method that provides accurate

estimates of demand distributions 3) the formulation of a model that calculates the cost of

inventory and the customer service level for an inventory policy 4) the development of an

algorithm that finds the optimal policy for a product at a store and 5) the creation of an

interface that allows interactive and on-line consultation Ultimately the above components

would have to be integrated and incorporated in a manner that would ensure the compatibility

and reliability of the system

The scope of the research reported in this thesis completes the first phase of the project

specifically we formulate an inventory model develop an algorithm and create a prototype of

the interface Because of the changing environment at Canadian Tire Pacific Associates minor

modifications to both the model and the algorithm developed herein might be necessary to

reflect future changes in operational procedures Upon completion of phase one the interface

will be interactive allowing managers to select the store product and review scenario and then

to compare optimal policies with existing policies or with alternate policies Although the

prototype will be delivered to Canadian Tire Pacific Associates for ih-house testing at the end

of this phase the interface will require modification in subsequent phases as feedback from

management is received and as the system is brought on-line

15

In order to assess more fully the potential savings of implementing the inventory control

system this thesis includes a comparison of optimal policies9 and existing policies The

expected annual cost of inventory for an entire product category is calculated for both the

optimal policies and the existing policies Based on an analysis of these costs the potential

savings of implementing the optimal policies are extrapolated Further the thesis examines the

sensitivity of the model to variations in the estimates of demand and costs

Originally results from all 21 stores were to be included in the above analysis

However just prior to the analysis management changed the review procedure at several

stores the result of which being that three picks per week are now performed at those stores

Because one of the main assumptions of the model specifically that the length of the lead time

does not exceed the duration of the review period is violated under this new review procedure

those stores cannot be included in the analysis Therefore the analysis is based solely on the

results from store six the large store located on Kingsway Avenue10

For the purpose of the analysis sales data for product category 20 automotive and

electrical accessories were collected from all 21 stores This product category was selected

because the distribution of demand for products within it is non-seasonal and because few of

the products are ever placed on sale11 The lack of trend or seasonality eliminates the need

for forecasting demand the optimal policies can be determined based directly on observed

demand

9 That is those policies obtained from the model

1 0 Store six is one of the only stores whose review procedure is not expected to change from two picks per week

1 1 Products that were on sale at any time during the period were not included in the data set

16

D The Demand Data

In total sales data for 520 products of which 458 are active products were collected

over a 307-day period from October 10 1992 to August 18 199312 The data was supplied

by Canadian Tire Pacific Associates in the form of a 10 MB text file with one line per day per

product recording sales from the stores13 Each line had the following format

Columns 1-6 part number Columns 7-9 Julian date14

Columns 10-18 cost of the item (to three decimal places) Columns 19-124 sales for stores 1 through 21 (5 columns per store)

In order to facilitate the analysis the file was sorted into 458 sales files with each of

these files containing the history of daily sales for one of the products Two changes were

made to the format of these files to improve readability 1) a column was added between the

product number and the Julian date and 2) all leading zeros were dropped Appendix A

contains a partial printout of the sales file for a typical product Finally a distribution file

summarizing the daily sales was created from each of the sales files Each distribution file

contains the products number the products cost and the number of days during which various

quantities of the product were sold at each store

Prior to the analysis the distribution files for a large number of products were

examined and based on the shapes of the histograms the products were divided into three

groups The first group low-demand comprises products that have demand on only a few

days with the daily demand seldom exceeding three items The second group medium-

1 2 Of the active products only 420 are sold at store 6

1 3 Strictly speaking a line was written to the file only on days on which at least one store registered a sale of the product

1 4 According to the Julian calendar days are numbered sequentially from 1 to 365 (366 during leap-years) beginning with the first of January

17

demand comprises products that have demand on approximately half of the days with the daily

demand seldom exceeding more than six items Finally the last group high-demand

comprises products that have demand on most days occasionally upwards of twenty items and

whose demand histograms resemble Poisson distributions The majority of the products

examined fell into the low and medium-demand groups only a few high-demand products

were observed Appendix B contains a printout of the distribution file for a typical product

in each group

For most products no theoretical distribution appeared to model the demand well

Demand for low and medium-demand products followed a non-standard distribution with

a mass at zero and a relatively small tail Although some theoretical distributions such as the

mixed-Poisson can be made to resemble this it was unclear how well they would perform

Further the effects of sampling errors from such non-standard distributions were unknown

On the other hand the standard Poisson distribution did appear to model high-demand

products well however only a few products fell into that group Therefore in the absence of

a good theoretical candidate the empirical demand function was chosen for the analysis The

empirical function was readily available from the distribution files and offered the best

estimate of the true underlying function

E The Cost Data

Along with the sales data management supplied estimates relating to the costs of holding

and ordering inventory Management estimates that the cost of holding an item in inventory

for one year is 30 of the items cost Incorporated in this holding rate are a number of costs

such as the opportunity cost of money invested and the costs incurred in running a store

Because it is the annual cost of inventory that is of interest either the compounded daily

18

holding rate or the simple daily holding rate can be used to determine the costs incurred during

a review period however the method of calculating and extrapolating the holding costs must

be consistent For the sake of simplicity a simple daily holding rate is used

In order to estimate the cost of ordering inventory management analyzed the various

tasks that result from an order being placed Using the average time required to fill and then

reshelve an order management obtained an estimate of the direct labour costs involved To

this management added an estimate of the cost of transporting an order from the warehouse to

the store Finally the estimate was adjusted slightly upward to account for the cost of breakage

and errors Based on the analysis management estimates the cost of placing an order for a

product to be $0085 regardless of the size of the order

19

III M O D E L F O R M U L A T I O N

A Terminology

In the previous chapter several terms were introduced to describe the inventory system

at Canadian Tire Pacific Associates These terms though perfectly adequate are not widely

used in the academic literature to describe inventory management problems Therefore in

order to avoid confusion and be consistent with the literature the following terms are used

throughout this thesis

The length of time between successive moments at which the

inventory position is reviewed15

The time at which a reorder decision is made

The length of time between the moment at which a decision to

place an order is made and the moment at which the order is

physically on the shelf

An operating policy in a periodic review inventory model in

which (i) the inventory position is reviewed every T days (ii) a

replenishment order is placed at a review only when the inventory

level x is at or below the reorder level s (gt 0) and (iii) a

replenishment order is placed for S-x units (S gt s)1 6

Lost Sales - The situation in which any demand during a stockout is lost

Review Period -

Decision Epoch

Lead Time -

(sS) Policy

1 5 With respect to the terminology used at Canadian Tire Pacific Associates a review period equals the number of days between flushes

1 6 With respect to the terminology used at Canadian Tire Pacific Associates S represents the target level and s represents the difference between the target level and the minimum fill

20

B Assumptions

In to order make the model of the inventory system mathematically manageable several

simplifying assumptions are made However we believe that the following assumptions do not

significantly reduce the accuracy of the model

1) Demand can be approximated very closely by sales during the period in which the data

is collected Generally inventory levels at the stores are very high and the number of

lost sales is assumed to be negligible In fact the inventory levels for a number of

products were tracked over time during the pilot study and not a single incidence of

stockout was observed Because all products appear to have similar excess inventory

it seems reasonable to assume that virtually all demand resulted in sales

2) The demands in successive periods are independent identically distributed random

variables Preliminary tests (Fishers exact tests) suggested no dependence between

sales on consecutive days and plots of sales over time indicated no seasonal pattern

3) Demand is independent across products That is we assume that sales of one product

do not affect nor can they be predicted from sales of another product

4) Unmet demand results in lost sales with no penalty other than the loss of the sale

5) The duration of the review period for a store is an integer value and is fixed over a

planning period For stores with two picks per week we assume that the inventory

is reviewed every four days rather than on a four day - three day cycle

6) The decision epoch is the start of a review period Review periods begin at midnight

at which time the inventory is reviewed and orders are placed instantaneously

7) The duration of the lead time for a store is deterministic and integer-valued In reality

the lead time for some stores was stochastic but management felt the variation could be

controlled and virtually eliminated

21

8) The lead time does not exceed the review period In other words items ordered at a

decision epoch arrive and are shelved before the next decision epoch In practice this

did not always occur

9) Orders arrive at the end of the lead time and are on the shelf to start the next day

10) Holding costs are incurred daily for all inventory held at the start of the day

11) The warehouse has an infinite capacity that is all orders to the warehouse are filled

Because of the high inventory levels in the warehouse stockouts in the warehouse are

very unlikely

C A Markov Chain Model

Consider the situation in which an (sS) policy is applied to an inventory system based

on the above assumptions Inventory is reviewed at the decision epoch for each of the periods

labelled t = 0 12 and replenishment occurs at the end of the lead time provided an order

was placed Demand during day d in period t is a random variable whose probability mass

function is independent of the day and period The function is given by

f^ik) = Pr ltd = k = ak for k = 0 1 2

where ak gt 0 and E ak = 1 Let T denote the duration of a review period and L denote the

length of the lead time Next define D T ( as the total demand during period t D L as the total

demand during the lead time in period t and D T L as the difference between D T and DLt

Assuming that successive demands pound f pound 2 are independent the probability mass function of

1 7 The aggregated demand variables are given by

^ = XXd and DT_Ut=J d= d= d=L+l

22

each aggregated demand variable can be obtained by convolving the probability mass function

of t d l s Finally let X denote the inventory on hand at the beginning of period t just prior to

the review

The process X is a discrete time stochastic process with the finite discrete state space

0 1 S A typical sample path of the process with a review period of four days and a

lead time of two days is depicted in Figure 1 As period t begins X units are on hand The

inventory is reviewed and because X is less than s an order is placed for (S-X) units Over

the next two days a demand of D L r occurs and the inventory drops accordingly With the

arrival of the order the inventory level increases by (S-X) at the start of the third day of the

period Over the remaining two days a demand of D T L is observed bring the inventory level

down to X + 1 Note that in this example an order would not be placed in period t+l because

X r + i exceeds s

The stock levels at the beginning of two consecutive review periods are related by the

balance equation

_K- DLy+ (s - v - DT-LX ixtlts ( 1 )

t + l ~ (Xt - DTtY ifXtgts

where the positive operator is interpreted as

= max a 0

1 8 For example the probability mass function of D T bdquo denoted fD T ( is the T-th fold convolution of f t Mathematically the function can be obtained using the recursive relation

and the identity f1^ = f5(

23

-2 I 8

gt o

r

CO

o a

CM

X

00

24

Because X + 1 depends only on Xt the process X satisfies the Markov property

PrXt+l=jX0 = i0Xt_l=it_vXt = i) = PrXt+1=jXt = i

and therefore is a discrete time Markov chain Further because the probability mass function

of the random variable pound is independent of t and thus so too are the probability mass functions

of the aggregated demand the Markov chain is stationary19

D The Transition Probability Matrix

The stationary transition probability p(gt is defined as the probability of X + 1 being in state

j given that X is in state i independent of the period t In the inventory model p-- represents

the probability of starting the next review period with j items on the shelf given there are i units

on hand at the start of the current period By convention the transition probabilities are

arranged in an (S + l)x(S + 1) matrix P such that (P)y = p- Thus the (i + l)st row of P contains

the probability distribution for the stock on hand at the beginning of the next period given that

there are i units on hand at the start of the current period The matrix P is known as the

transition probability matrix Note that since we are concerned with a steady state analysis we

need not consider the possibility of initial stock levels greater than S once the inventory level

is below S it can never again exceed S

The transition probability ptj is calculated by expanding (1) to include all possible

instances of demand that allow for a change in inventory from i units at the beginning of a

period to j units at the beginning of the next period For example suppose the inventory policy

(510) is in place 2 items are on hand at the start of the current review period and 7 items are

1 9 Henceforth the subscripts t and d are omitted from the daily and aggregate demand variables for the sake of brevity

25

on hand at the start of the next period Then only three scenarios are possible

1) No items are sold during the lead time (meaning that the stock level increases to 10

after replenishment) and 3 items are sold during the remainder of the period

2) One item is sold during the lead time and 2 items are sold during the remainder of

the period

3) Two items are sold during the lead time (though demand could exceed 2) and 1 item

is sold during the remainder of the period

The transition probability p 2 7 is the sum of the probabilities of the various scenarios and is

computed by

p2J = PrDL=0PrDT_L = 3] + PrDL = l] Pr[DT_L=2 + PrDL2] PrDT_L = l]

In general the transition probabilities for the inventory model are calculated as follows

Case 1 The lead time is less than the review period

For 0 lt lt s

Pij

i-l pound [PrDL=kPrDT_LgtS-k] + PrDLgtiPrDT_LgtS-i if j = 0

i - l

^[PriD^^PrlD^S-j-kn+PHD^PriD^S-j-i if 0 lt j ltL S-i (2) k=0 s-j

E [PrDL=kPrDT_L = S-j-k] if S-i ltj^S k=0

26

For s lt i lt S

PrDTzi]

0 p = PrDT=i-j)

ifj = 0 ifOltjzi ifiltjltS

(3)

Case 2 The lead time equals the review period

For 0 lt i lt s

Pa

0 PrDTzi) PrDT = S-j

ifOltLJlt S-i ifj = S-i if S-i ltj plusmn S

(4)

For s lt i lt S use (3)

E The Steady State Probabilities

Provided 1) there is a positive probability of non-zero demand during a review period

and 2) there is a positive probability of zero demand during a review period20 then the

Markov chain described by (1) has a limiting probability distribution ir = (ir0 TTu irs)

where 0 lt IT lt 1 for j = 0 1 S and E 7ry = 1 Further this limiting distribution is

independent of the initial state In other words the probability of finding the Markov chain in

a particular state in the long-run converges to a nonnegative number no matter in which state

the chain began at time 0 The limiting distribution is the unique nonnegative solution of the

equations

2 0 That is 0 lt PrD x = 0 lt 1 Given the nature of the problem this assumption is not very restrictive

27

^ = E ^ - far j = 0 l S (5)

s and pound nk = 1 (6)

For the proof that the limiting distribution is independent of the initial state and is obtained from

(5) and (6) the reader is directed to Appendix C

In terms of the inventory problem 7ry is the steady state probability of having j units of

inventory on hand at the decision epoch However some subsequent calculations are made

easier if the steady state probability of having j units of inventory on the shelf just after

replenishment is known Instead of reformulating the Markov chain these probabilities can be

obtained by shifting the limiting distribution TT Much in the same way that the transition

probabilities are calculated the shifted limiting distribution t] is obtained by accounting for

all possible instances of demand that allow for the shifted inventory position For example

suppose that the policy (510) has the limiting distribution TT In order to have say 8 units on

hand immediately after replenishment one of the following must occur

1) Ten items are on hand at the decision epoch and 2 items are sold during the lead time

2) Nine items are on hand at the decision epoch and 1 item is sold during the lead time

3) Eight items are on hand at the decision epoch and nothing is sold during the lead time

4) Either 2 3 4 or 5 items are on hand at the decision epoch and 2 items are sold during

the lead time

The steady state probability of having 8 units on hand just after replenishment is calculated as

n 8 = n10PrDL = 2] +iz9PrDL = l] +n8PrDL = 0 +[iz5 + iz4 + iz3]PrDL = 2 +n2PrDLgt2

28

In general the shifted steady state probabilities are given by

s pound nPrDLgti ifj = 0

i=s+l S

pound PrDL = i-j] ifl plusmnjltS-s i=max(s+l)

S = pound 7tjirZ)L = i-+7i 5 FrD i ^5 ifj = 5-5

i=max(s+l)

^ 7 1 ^ ^ = 1-7+ pound TtiFrZgti=5-y+iis_PrD1^S-7 ifS-sltjltS i=max(gt+l) i=S-j+l

(7)

F The Cost of Inventory

For this problem the cost of inventory has two components (i) a fixed ordering cost

cf incurred whenever an order is placed and (ii) a holding cost cv incurred daily for each item

held in inventory The expected cost of inventory for a review period can be calculated by

summing the costs associated with each inventory position under steady state conditions

Because a simple holding rate is used this cost is simply adjusted by number of review periods

in a year to obtain the expected annual cost of inventory

The ordering cost for a review period is relatively straight forward to calculate either

an order is placed in which case the order cost is incurred or no order is placed in which case

no cost is incurred According to the rules of the inventory policy orders are placed only if

the stock on hand at the beginning of a review period is less than or equal to s Further the

probability that the review period begins with the inventory in one of these positions is equal

to the sum of the first (s + 1) steady state probabilities Therefore the expected ordering cost

29

for a review period is given by

E[OC] = cfY n y=o

Unlike the simple structure of the ordering cost the holding cost is incurred daily for

each item held in inventory Therefore the daily holding costs associated with all possible

inventory positions must be weighted and summed to obtain the holding cost for the review

period In order to make the calculations less complicated a shifted review period beginning

immediately after replenishment and ending just prior to the next replenishment is considered

Define e(0 to be the expected number of items on hand to start day i of the shifted

review period and let g(i f) denote the expected number of items on hand to start day i given

that there were j items immediately following replenishment21 The function g(z j) is the sum

of the probabilities of possible demand weighted by the resulting stock levels that is

g(ij) = j^kPriD^j-k)

To obtain the expected number of items on hand to start day i independent of j the above

function is summed over all possible values of j and weighted by the corresponding steady state

probabilities yielding

s j ed) = pound r 5 gt P r Z ) = - pound

7=0 k=

2 1 In other words g(if) is the expected number of items on hand to start day i given e(0)=v

30

The expected holding cost for a review period is then given by

T

E[HC] = $gt v e(0 i=i

i=l =0 k=l

where c v equals the simple daily holding rate multiplied by the cost of the item

Although differently defined review periods are used to calculate the above expected

costs the costs can still be summed to obtain the expected cost of inventory for a review

period The expected costs are calculated based on steady state conditions and therefore

represent the costs expected to be incurred during any T consecutive days Summing the above

two expected costs and multiplying the sum by the number of review periods in a year yields

the expected annual cost of inventory

E[AC] = mdash[E[OC]+E[HC]]

T S j 1 ( g ) 365

j=0 i=l j=0 fc=l

G The Customer Service Level

There are many ways to define customer service and the most appropriate definition

depends upon the nature of the business and the perspective of management Two commonly

used definitions are 1) the proportion of periods in which all demand is met and 2) the

proportion of demand satisfied immediately from inventory Although the former definition is

often used because of its simplicity the latter definition is closer to what one commonly means

by service (Lee and Nahmias 1993) A third definition also relevant in retail environments

31

is the probability that lost sales do not exceed some critical number Other definitions such

as probability that backorders are filled within a certain amount of time are more applicable

to industrial and manufacturing settings where backlogging demand is common

Given the retail environment with the possibility of lost sales that exists at Canadian Tire

Pacific Associates management is concerned primarily with the fraction of sales that are lost

For example management views the situation in which demand exceeds stock by one unit every

period more favourably than that in which the same demand exceeds stock by three units every

other period For this reason the proportion of demand satisfied immediately from inventory

is the most appropriate measure of customer service

The customer service level for a single review period CSLRP is given by

r bdquo T _ Demand satisfied during RP Total demand during RP

Because we are interested in the long-run behaviour of the system a measure of the steady state

customer service C S L must be obtained As shown in Appendix D the correct measure is

poundpoundpound _ E[Demand satisfied during RP] E[Total demand during RP]

which can be rewritten as

pound pound pound _ j _ E[Demand not satisfied during RP] ^ E[Total demand during RP]

Because of the steady state conditions the above customer service level can be estimated

from any period of T consecutive days As was done for one of the cost calculations the T-day

period beginning just after replenishment is considered Suppose that there are j units of

inventory on hand immediately following replenishment Then the conditional customer service

32

level is

CSL j = 1 E[(DT-j)l

E[DT] (10)

The denominator in the second term is merely the expect demand during the period

E[DT] = 7gt (11)

where p is the mean of the random daily demand variable pound The numerator which is the

conditional expected demand that is not satisfied during the T-day period expands to

7-1

E[(DT-j)+] = T u - j + YU-k)PrDT = k] k=0

(12)

The reader is directed to Appendix E for a mathematical justification of (12) In order to obtain

the unconditional expected demand not satisfied (12) is applied to all possible values of j and

a weighted sum of the terms is calculated The result is given by

E[Demand not satisfied] = Y j =0

y-i Ti - j + 1pound(j-k)PrDT = k

S j-1

= - E[I] + Y0-k)Pr[DT=k) =0 it=0

(13)

where E[I] denotes the steady state inventory level Finally the customer service level is

obtained by the substituting (11) and (13) into (9) yielding

CSL = 1

s j-l

Ti - E[I] + Jgty Y(J-k)PrDT=k]

m n J2VjTU-k)PrDT=k Eil J=Q fc=o

(14)

33

H A Methodology for Evaluating an (sS) Policy

In this section a brief methodology summarizing the main results of the model is

presented The methodology can be applied to evaluate a given (sS) policy in terms of its cost

and its resulting customer service level Given the probability mass function of the daily

demand f the duration of the review period T the length of the lead time L the fixed cost

of placing an order cf and the daily unit cost of holding inventory cv

1) Calculate the aggregated demand probability mass functions fDT fDL and f D x L as the T-th

fold L-th fold and (T-L)th fold convolutions of f respectively

2) Generate the transition probability matrix using either equation (2) or (4) and equation (3)

3) Calculate the limiting distribution TT from (5) and (6)

4) Obtain the L-day shifted limiting distribution rj using (7)

5) Calculate the expected annual cost of inventory using (8)

6) Determine the customer service level using (14)

34

IV A N A L G O R I T H M F O R OBTAINING T H E O P T I M A L (sS) P O L I C Y

A Introduction

The methodology developed in the previous chapter can be used to calculate the cost and

service level associated with a given (sS) policy In this chapter an algorithm that finds the

feasible policy yielding the lowest cost is developed A policy is feasible if it satisfies both the

basic condition 0 lt s lt S and the imposed constraint on the customer service level2 2 In

order to find the optimal policy the algorithm must combine the methodology for evaluating

a single policy with a search technique

Search techniques are evaluated in terms of two somewhat antithetic criteria On the one

hand the search should locate the optimal solution or at least a near optimal solution on

the other hand the search should take as little time as possible Many search techniques

guarantee optimality provided the objective function and constraints exhibit certain properties

such as convexity However even if such a search technique also guarantees its termination

in polynomial time the search might take too long to be practical Alternatively some search

techniques provide reasonable though sub-optimal solutions very quickly These techniques

often called heuristics must be evaluated in terms of both their speed and the accuracy of their

solutions The choice of a heuristic over a search that guarantees optimality depends upon the

nature of the feasible space the need for accuracy and the cost of time

Given the complexity of the cost and customer service level functions it is very difficult

to show theoretically for what if any combinations of demand and costs the feasible space is

convex However a plot of the objective function for a single product may be obtained by

2 2 Management specified that each policy must provide a customer service level of at least 975 Therefore the algorithm must incorporate the constraint CSL gt 0975

35

evaluating the costs for a number of policies within a grid When the service level constraint

is included in these plots maps of the feasible space are obtained Examination of the

maps for several products suggests that the space is convex over most regions however

maps for some products reveal some regions of non-convexity Unfortunately there is no

way of knowing the degree of non-convexity for other products other stores or different costs

estimates without a more rigorous survey of the feasible space Therefore if one of the

techniques that guarantee optimality under the condition of convexity is employed the search

would have to be evaluated as a heuristic Instead as a first step to finding a practical search

technique a crude search that guarantees optimality under virtually all conditions is

investigated

B A Grid Search

The simplest search technique that guarantees optimality merely evaluates all policies in

the grid defined by 0 lt s lt S and 0 lt Sm i n lt S lt S m a x where Sm i n and S m a x are

some predefined values That is the grid search is a brute force method that examines every

policy within the grid 2 3 Although the number of policies that must be examined is large the

grid search is guaranteed to find the optimal policy provided that S o p t is contained in the grid

that is provided Sm i n lt S o p t lt Sm a x The grid search algorithm proceeds by generating the

transition probability matrix for the policy (0Smin) and then evaluating the service level

associated with that policy If the policy is feasible24 the cost associated with the policy is

2 3 The total number of policies evaluated is the sum of the first Sm a x integers less the sum of the first Smin-1 integers Numerically [(l2)(Sm a x)(Sm a x+l) - (l2)(Smin-l)(Smin)] policies are examined

2 4 That is if the service level is adequate

36

calculated and compared to the current lowest cost The procedure continues until all points

in the grid have been examined and the optimal policy has been found A flow chart of the grid

search algorithm is shown in Figure 2

In order to assess the grid search the algorithm was coded and compiled into an

executable program The program was run for two products (a medium-demand product and

a high-demand product) and with four values of Sm a x In each run Sm i n was taken to be zero

The following times were recorded for each trial

T P M - the amount of time spent generating the transition probability matrix

GAUSS - the amount of time spent solving for the steady state probabilities

COSTS - the amount of time spent calculating the expected annual costs

SERVICE - the amount of time spent determining the customer service levels

T O T A L - the total execution time of the program

The columns entitled Grid Search in Tables 2 and 3 summarize the results of the trials All

times are recorded in seconds as required by a 486DX33 MHz personal computer with an

empty buffer memory

Three results are evident from the tables First the vast majority of the execution time

is spent generating transition probability matrices and then solving for the steady state

probabilities with the time being split almost equally between the two tasks Second the total

execution time of the algorithm increases exponentially as Sm a x increases For product 206917

doubling S^ from 30 to 60 led to an increase in the execution time of more than 2100 (from

172 seconds to 3661 seconds) Third the grid search algorithm as it stands does not appear

to be adequate for the problem at hand Given the large number of products that must be

evaluated the algorithm is much too slow to be implementable

37

Calculate the demand probability functions

Initialize s = 0 S = Smin] cost = 10A5

s = s+1

Calculate the probability transition matrix for (sS)

Evaluate policy (sS) Is service level ^ 0975

No

No Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Is s = S-l

Yes

Is S = Smax No

Yes

Print results (sS) cost

s = 0 S = S+1

Figure 2 The flow chart of the grid search algorithm

38

Table 2 Execution times of the algorithms for product 200001 a 30-amp inline fuse

(in seconds)

-max Subroutine Grid Search Grid Search

Updating T P M Heuristic

10 T P M 006 010 001

GAUSS 010 006 001

COST 005 005 001

SERVICE 001 001 001

T O T A L 050 035 027

20 T P M 126 021 010

GAUSS 119 160 023

COST 027 026 001

SERVICE 017 016 001

T O T A L 318 236 055

30 T P M 703 096 010

GAUSS 757 801 093

COST 079 085 001

SERVICE 047 042 001

T O T A L 1664 1137 126

40 T P M 2257 163 015

GAUSS 2958 3003 206

COST 313 329 001

SERVICE 092 083 011

T O T A L 5871 3851 285

39

Table 3 Execution times of the algorithms for product 206917 a 6 solder connector

(in seconds)

bull-max Subroutine Grid Search Grid Search

Updating T P M Heuristic

30 T P M 717 067 006

GAUSS 762 880 093

COST 098 087 010

SERVICE 016 021 006

T O T A L 1720 1176 220

40 T P M 2390 157 022

GAUSS 2861 2911 209

COST 219 227 006

SERVICE 080 088 011

T O T A L 5827 3790 362

50 T P M 6691 413 010

GAUSS 7994 8201 472

COST 568 577 016

SERVICE 212 197 005

T O T A L 15840 10041 654

60 T P M 15509 894 040

GAUSS 18607 18951 876

COST 1367 1353 038

SERVICE 350 359 022

T O T A L 36614 22794 1120

40

C A Technique for Updating the Transition Probability Matrix

As was noted in the previous section approximately half of the execution time of the grid

search algorithm is spent generating transition probability matrices Furthermore the amount

of time increases exponentially as Sm a x increases This increase is exponential for two reasons

first the number of transition matrices that must be calculated is a function of S^ squared and

second the average size of the matrices and hence the average number of calculations per

matrix increases with Sm a x For example when S m a x equals 30 the transition probability matrix

must be generated 465 times with the largest matrix having dimensions (31x31) When Sm a x

equals 40 the matrix must be generated 820 times with 355 of the matrices having dimensions

larger than (31x31) The performance of the algorithm could be improved considerably if

either the number of matrices that must be generated or the number of calculations per matrix

is reduced This section introduces a technique that accomplishes the latter by updating the

transition probability matrix for a new policy instead of generating it anew for each policy

A new policy (s+mS)

Suppose the policy (sS) has associated with it the (S + l)x(S + l) transition probability

matrix P and the policy (s+mS) where m is any integer satisfying 0 lt s+m lt S has

associated with it the (S + l)x(S + 1) transition probability matrix J Then T can be obtained by

recalculating only m rows of P Recall that the equations for generating a transition

probability matrix are divided into two sets based upon the row being calculated equations (2)

or (4) for rows 0 through s and (3) for rows (s + 1) through S The equations themselves are

independent of s only the limits defining the cases contain s Therefore only the m rows

that are affected by the change in the limits need to be recalculated The following updating

technique can be used to obtain 7

41

1) If m gt 0 rows i = (s + 1) (s+2) (s+m) are recalculated using (2) or (4)

2) If m lt 0 rows i = (s-1) (s-2) (s-m) are recalculated using (3)

3) All other rows of T are identical to those of P

The reader is directed to Appendix F for a mathematical justification of the technique

A new policy (sS+m)

Suppose the policy (sS) has associated with it the (S + l)x(S + l) transition probability

matrix P and the policy (sS+m) where m is any positive integer has associated with it the

(S+m+l)x(S+m+l) transition probability matrix P Then T can be obtained by modifying

each row of P and adding m additional rows and columns The updating technique is not as

simple as that for a new policy (s+mS) for two reasons the dimensions of P and T differ and

the equations for generating a transition probability matrix contain S Nevertheless an updating

technique that requires much less computing time than generating the matrix anew can be

developed The following technique can be used to obtain P

1) For rows i from 0 through s

For columns j from 0 through m recalculate using (2) or (4)

For columns j from (m+1) through (S+m) enter the (j-m)th column entry of P

2) For rows i from (s + 1) through S

For columns j from 0 through S enter the corresponding value from P

For columns j from (S + 1) through (S+m) enter 0

3) For rows i from (S + 1) through (S+m) recalculate using (3)

The above technique is justified mathematically in Appendix G

42

The modified algorithm

The updating technique can be easily incorporated into the grid search algorithm with

only a slight change in the manner in which s is incremented Figure 3 shows the flow chart

of the new algorithm As before the algorithm was coded and compiled into an executable

program and the various execution times were recorded The results are listed in Tables 2 and

3 under the columns Grid Search Updating TPM

Updating the transition probability matrix does improve the performance of the

algorithm The time required to generate the transition probability matrices is reduced by

approximately 90 This translates to a reduction in total execution time of approximately

30 Not surprisingly the reductions are greater for large values of Sm a x However despite

this seemly significant improvement a decrease of only approximately 08 seconds is obtained

when S^ equals 20 Further when S ^ equals 30 the algorithm requires over 11 seconds

which is still much too high

The execution time is now dominated by the time required to solve for the steady state

probabilities The steady state probabilities could be solved by using approximation techniques

which are faster than solving the systems exactly however because the probabilities are often

very small numbers the approximation techniques would be unable to provide the required

accuracy The only remaining way to decrease the execution time is to reduce the number of

policies that are evaluated during the search This can be done by either reducing the size of

the grid being searched or reducing the number of policies being evaluated within the grid The

following section introduces a method of reducing the size of the grid by obtaining a lower

bound for S namely Sm i n

43

s = s+k

Calculate the demand probability functions

Initialize s = 0 S = Smin| cost = 105

k= 1

Update the probability transition matrix for (sS)

Evaluate policy (sS) Is service level ^ 0975

No

No

iNo

Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Ifk= 1 iss = S-l If k =-1 is s = 0

Yes

Is S = Smax No

Yes

Print results (sS) cost

If k=l s=s+l S = S+1

Figure 3 The flow chart of the grid search algorithm utilizing the updating technique for the transition probability matrix

44

D A Lower Bound on S

As mentioned earlier the size of the grid is specified by the number of policies within

the domain defined by 0 lt s lt S and 0 lt Smin lt S lt Smax Thus the size of the grid

and hence the total amount of computation time can be reduced by obtaining tight bounds on

S Unfortunately because of the extreme non-linearity of the objection function and the service

level constraint obtaining a mathematical upper bound on S is very difficult and beyond the

scope of this work This section introduces a method for obtaining a lower bound on S25

For a given value of S an upper bound on the customer service level can be found as

follows Suppose that there are S units of inventory on the shelf at the beginning of every

review period regardless of demand No (sS) policy can provide a higher level of customer

service because there exists a positive probability of beginning a review period with less than

S units on the shelf Thus the customer service level associated with starting each review

period with S units on the shelf is an upper bound on the customer service level for any (sS)

policy From (10) this level is

E[(DT-SY] CSL 5 = 1 - T mdash mdash - ( 1 5 gt

E[DT]

Replacing j with S in (12) and then substituting (12) and (11) into (15) yields

CSL 15 = mdash- - J2(S-k)PrDT = k] (16)

To obtain Smin use (16) to find the largest value of S with a customer service level less than

0975

There are two criteria for a good lower bound on S the bound must be no larger than

This method was proposed by Kaan Katircioglu

45

S o p t and it must be very close to So p t These criteria ensure that the optimal policy will be

contained in the search grid and that the size of the grid is reduced by as much as possible

Based on these criteria the above method of obtaining a lower bound is assessed A value for

S m i n is obtained for each of the products in product category 20 and compared with the

corresponding value of S o p t 2 6 The results are illustrated in Figure 4 which shows the number

of occurrences of various Sm i n and S o p t combinations Note that a point lying on the 45deg line

represents a perfect bound while a point lying close to the S o p t axis represents a very loose

bound The absence of any points above the 45deg line indicates that the first criterion is satisfied

in each instance the method did not generate a value of Sm i n exceeding the corresponding value

of So p t Moreover the bounds are reasonably tight for 85 of the products analyzed that

is for those products with S o p t equal to 10 or less27 However as S o p t increases the bounds

become steadily worse Because values of S o p t are likely to be larger for these products at the

ten Jumbo and Extra Jumbo stores and for other products with higher demand the lower bounds

are not expected to be very tight in general

Although obtaining the bound Sm i n does reduce the size of the grid the corresponding

reduction in the execution time of the algorithm is miriimal For example if S^ equals 60 and

a value of 30 is obtained for Sm i n the program would still require approximately 215 seconds

to search the grid (compared to the 228 seconds required without a lower bound on S) Because

the execution time increases exponentially with So p t a tight upper bound on S would likely have

a more significant impact on reducing the execution time of the algorithm Further research

on this issue is required

2 6 The optimal policy for each product was obtained using a grid search with an extremely large upper bound

2 7 On average Sm i n is approximately half of S o p t for these products

46

47

The only remaining approach to improving the algorithm is to modify the search so that

fewer policies within the grid are evaluated Toward this goal assumptions can be made with

respect to the feasible space and a heuristic search can be developed based on these

assumptions

E A Heuristic Search

As mentioned previously in this chapter a map of the feasible space for each of

several products was obtained Upon closer examination the maps illustrate two patterns

without exception for a given value of S both the cost and the customer service level increase

as s increases The following intuitive argument is offered as a possible explanation for the

observed behaviour When S is held constant increasing s results in more instances in which

an order is placed upon review Hence the average stock level would increase which in turn

would lead to an increase in the cost and the customer service level28 With this in mind a

heuristic search based on the following assumptions is developed for a given value of S both

the cost and the customer service level are monotonically increasing functions of s

In general the heuristic proceeds as follows Having just incremented S by one the

customer service level of the policy (sS) is evaluated If the policy is feasible s is decreased

until the lowest value of s producing a feasible policy is found If the policy is not feasible

s is increased until either the cost exceeds the current lowest cost for a feasible policy a

feasible policy is obtained or s equals S-l In the event that a feasible policy is found the cost

2 8 In order to prove that the customer service level increases as s increases it would be sufficient to show that the probability of placing an order increases as s increases Such a result might be possible if assumptions regarding the demand function are made Further if both the probability of placing an order and the average stock level increase then both the ordering cost and the holding cost would increase No proofs are offered here

48

of the policy is calculated if the cost of the policy is less than the current lowest cost the new

policy becomes the current best Finally S is increased by one and the process repeats

provided S does not exceed some predefined value Sm a x The flow chart of the algorithm

incorporating the heuristic search is shown in Figure 5 Note that the algorithm begins by

attempting to improve on the lower bound Sm i n the policy (Sm i n-lSm i n) is evaluated and if it

is not feasible S m i n is increased by one The improvement continues until a feasible policy is

found

As previously mentioned a heuristic search must be evaluated in terms of both its speed

and the accuracy of its solutions To evaluate the heuristic on the basis of speed the same

approach used to evaluate the previous algorithms is employed The columns entitled

Heuristic in Tables 2 and 3 contain the various execution times of the heuristic The heuristic

is a vast improvement over the previous search techniques the reduction in execution time is

enormous and becomes more so as S m a x increases For a S m a x of 20 the reduction is

approximately 75 of the time required by the grid search algorithm29 and for a S m a x of 60

that reduction soars to over 95 More importantly the heuristic can search a grid with a

of 40 in under 4 seconds or a grid with a S m a x of 60 in under 12 seconds These low execution

times should be sufficient to allow for a large number of products to be analyzed in a

reasonable amount of time

In order to further assess the practicability of the heuristic the total time required by the

heuristic to find solutions for the entire product category was determined and compared to the

corresponding time for the grid search algorithm With a priori knowledge of the optimal

solutions conservative yet reasonable values of S m a x were selected and used by both of the

2 9 Hereafter grid search algorithm refers to the algorithm utilizing the grid search the updating technique and the lower bound Sm i n

49

Calculate the demand probability functions

Calculate Smin S = Smin s = S-l

S = S+l s = S-l t

Update the probability transition matrix for (sS)

No Is service level pound 0975 Yes

No Is service level ^ 0975

Yes

Is cost lt cost No

Yes

Is s lt S-l No

Yes

s = s+l

Update the probability transition matrix for (sS)

Is service level pound 0975

Yes

Is cost lt cost No

Yes

s = s S = S cost = cost

s = s-l

No

Update the probability transition matrix for (sS)

S = S+l

Is s 2 0 Yes

Update the probability transition matrix for (sS)

Is service level pound 0975

No

s = s+l

No

No

Is S = Smax

^Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Print results (sS) cost

Figure 5 The flow chart of the heuristic algorithm

50

algorithms30 The grid search algorithm required approximately 9 hours and 41 minutes to

find the solutions while the heuristic required only slightly more than 23 minutes Based on

speed the heuristic is preferable to the grid search algorithm

Finally to evaluate the heuristic based on the accuracy of its solutions its solutions were

compared to the previously obtained optimal policies Without exception the heuristic

algorithm found the optimal policy Although there is no guarantee that the heuristic will find

the optimal solution for different products or different demand functions the heuristic appears

to work extremely well Based on the fast execution time and the apparent accuracy of the

solutions the heuristic will be used in the development of the inventory tool

3 0 For the test the following values of Sm a x were used 20 (273 products) 30 (19) 50 (23) 100 (3) and 125 (2)

51

V T H E I N T E R F A C E

With the model formulated and the algorithm developed the only remaining task in the

development of the prototype is the creation the interface In fact two interfaces are created

one that displays results in terms of minimum fills and target levels and one that displays

results in terms of reorder points and target levels To be consistent with the previous chapters

the latter interface is described here31

In its present form the interface allows the manager to set review parameters to

compare optimal and current policies and to evaluate alternate policies From the main menu

the manager selects the product store duration of the review period length of the lead time

and value of Sm a x Once the parameters are set the system calculates the optimal policy and

then evaluates the current policy displaying results as they are obtained In particular the

reorder point target level expected annual cost and expected customer service level are

displayed for each policy If a current policy meets the desired customer service level the

expected annual savings of the optimal policy are displayed in terms of both dollars and

percentage of current cost otherwise the value of the current customer service level is

displayed in red to indicate that it is deficient Finally the manager can choose to evaluate

alternate policies or to return to the main menu

Although the prototype is far from the ultimate integrated inventory control system it

is still valuable to management As mentioned in Chapter II inventory policies at Canadian

Tire Pacific Associates are currently based on rules of thumb that incorporate marketing

3 1 Because significant changes to the interface are expected as feedback from management is received and as the system is brought on-line only a brief description of the present interface is given

52

strategies The prototype provides management with a means of determining the proportion of

the cost of inventory that is not attributable to meeting customer demand For example

consider a hypothetical consultation in which the manager chooses to analyze product 200001

a 30-amp inline fuse at store 6 The prototype displays to the manager the current policy of

(1214) which provides a customer service level of 1000 at an annual cost of $748 and the

optimal policy of (39) which provides a customer service level of 977 at an annual cost of

only $319 If for some reason the manager feels the optimal policy is undesirable he or she

can investigate the additional cost of holding excess inventory Suppose the manager chooses

to evaluate the policy (610) The prototype evaluates the policy and displays the results the

customer service level is 998 and the annual cost is $441 With the alternate policy

virtually all demand is satisfied and the cost of inventory is still reduced by over 40 Thus

the manager ascertains that approximately 40 of the current cost of inventory for this product

is directly attributable to marketing considerations Using the prototype in this way

management can begin to understand better the nature of its inventory costs A description and

graphical depiction of the above hypothetical consultation are presented in Appendix H

As for the technical details of the system the prototype can run on any computer using

the DOS operating system having approximately 420K of available conventional memory and

attached to a V G A or SVGA monitor Further the prototype can be used on a machine without

a math coprocessor however performance is slower The prototype consists of two modules

linked together into one executable program The interface module contains the various

interactive subroutines and the main module contains the methodology of the model and the

logic of the heuristic search Both modules are coded in Fortran and compiled using Microsoft

Fortran 50 which provides many extensions to standard Fortran The source codes for the

interface module and the main module are listed in Appendix I and Appendix J

53

respectively

In addition to the above technical specifications the program requires the presence of

certain files in certain directories The program uses the distribution files mentioned in

Chapter II to read the unit costs and the empirical demand functions Each product has a

unique distribution file the name of which is the six digit product number followed by the

extension dst The text file containing the existing policies at each store must be located in

the same directory as the distribution files Finally the file tmsrbfon which contains the

font used by the interface must be located in the directory from which the program is executed

In its future versions the system should be more automated and more user-friendly

Rather than having the manager choose an upper bound for the search the algorithm should

incorporate a method of determining Sm a x For this to happen either a theoretical upper bound

must be obtained or a heuristic for terminating the search must be developed Also the amount

of time spent waiting for calculations to be performed can be reduced by reading the optimal

and current policies and their associated costs and customer service levels from a file or

database This solution file could be created any time prior to the consultation by running

the system in a batch mode The interface should allow the manager to change cost estimates

and service level values view results graphically and print results and summary documents

Finally the system should provide an option for automatically incrementing the product andor

store during a consultation This option would enable a manager to review quickly the

inventory policy of a product at each store

54

VI R E S U L T S AND SENSITIVITY ANALYSIS

A Comparison of Current Policies to Optimal Policies

In order to estimate the potential savings of implementing optimal policies the solutions

obtained in the Chapter IV are analyzed in more detail For each product the current policy

is evaluated and then compared to the optimal policy with respect to cost and customer service

level Finally the potential savings of the optimal policies are extrapolated to all products and

all stores

Of the 420 products examined each of 37 products has an existing policy that fails to

provide the desired customer service level of 975 However most of these policies just

fail to do so only 4 policies provide a customer service level of less than 90 Table 4

provides a list of these 37 products each with its current customer service level its current

annual cost and the additional cost of its optimal policy3 2 Two comments regarding Table

4 are in order First the current customer service level for product 203312 an amber

driveway marker is listed at a misleading 303 Examination of the demand distribution

reveals that this product was sold on only 1 of the 307 days during which sales data were

collected and on that day 13 of the markers were sold Clearly this data point constitutes

either an error or an unusual event the current policy of (34) does not allow for more than 4

items to be sold per review period Second the additional cost of the optimal policy for each

of 5 products is negative In other words for each of these products the current policy which

fails to provide the desired customer service level costs more than the optimal policy which

3 2 This is the difference between the expected annual cost of the optimal policy and the annual cost of the current policy It is the minimum additional cost required to increase the customer service level to at least 975

55

Table 4 Products with insufficient existing policies

Product Current CSL ()

Current Annual Cost ($)

Additional Cost of the Optimal Policy ($)

206932 974 204 -051

204570 973 222 -025

203053 973 248 009

206550 972 118 042

206547 972 105 035

206565 972 146 056

203702 972 340 157

206876 972 155 061

205752 972 330 153

205922 972 108 036

206857 972 180 073

205393 972 190 -003

201513 971 345 015

203405 970 255 037

203069 968 295 -048

203071 967 321 -001

205635 967 390 029

203914 966 460 160

204601 962 252 061

203326 960 144 014

202003 958 876 437

206516 958 112 029

206517 958 283 120

206350 958 333 147

205932 958 180 066

203936 952 598 294

56

Product Current CSL ()

Current Annual Cost ($)

Additional Cost of the Optimal Policy ($)

206932 974 204 -051

203939 948 413 193

203977 945 1216 1242

206551 945 132 077

205389 934 598 047

203590 927 1223 607

206567 919 122 070

203591 905 1228 834

205390 887 291 074

203318 833 136 084

206393 776 319 289

203312 303 237 511

provides at least the desired customer service level For the 36 products the total additional

cost of the optimal policies is $542033

Of the remaining 383 products most have an existing policy that provides a customer

service level well above 975 When the customer service levels are rounded to one decimal

each of 253 products has a customer service level of 1000 and each of 347 products has a

customer service level of at least 990 In contrast each of only 58 products has an optimal

policy that yields a customer service level of at least 99034 For each of 37 products the

3 3 The cost associated with product 203312 is not included in the total

3 4 Optimal policies do not have customer service levels of exactly 975 because the function is discrete For low-demand products a unit step in s or S results in a relatively large change in the customer service level

57

current policy is identical to the optimal policy

The extremely high level of service provided by most policies allows for substantial

savings in the cost of inventory With current policies the estimated annual cost of inventory

for all 420 products is $420743 whereas the expected annual cost of the optimal policies is

only $186884 This amounts to a savings of approximately $2340 or over 55 of the current

cost of inventory

In order to further assess the savings the difference between the costs of the current

policies and the costs of the optimal policies are examined in more detail For each of the 383

products whose existing policy provides a customer service level above 975 the difference

between the estimated annual cost of the existing policy and the expected annual cost of the

optimal policy is calculated The resulting histograms of the savings in (a) dollars and (b)

percentage of current cost are shown in Figure 6 The particulars for the current and optimal

policies for all 420 products are provided in Appendix K

As evident from Figure 6(a) the majority of the expected savings in absolute terms

comes from only six products In order to examine these products more closely the current

and optimal policies along with the associated costs and savings for each of the six products

are listed in Table 5 Each current policy is very different than its corresponding optimal

policy resulting in excessive inventory for each product In fact over half of the total expected

savings could be realized by implementing the optimal policy just for each of these six products

Interestingly five of the six products are sealed beam headlights

58

90

10 - 20 30 - 40 50 - 60 70 - 80 90- 100

Savings ()

(b)

Figure 6 The distribution of savings of optimal policies in (a) dollars and (b) percentage of current cost

59

Table 5 Products with the largest potential absolute savings

PRODUCT

CURRENT OPTIMAL SAVINGS

PRODUCT s S Cost ($)

C S L

()

s S Cost ($)

C S L () ($) ()

2036801 194 200 64014 1000 37 48 bull 9245 975 54769 856

2036772 94 100 23652 1000 18 22 3850 978 19802 837

2036723 78 90 23043 1000 19 23 4454 975 18589 807

2030944 145 175 19522 1000 34 45 3727 975 15795 809

2036425 48 60 10491 1000 13 17 2682 977 7809 744

2036556 68 80 7265 1000 14 22 1699 975 5566 766

1 H6054 rectangular sealed beam headlight 2 H4656 rectangular sealed beam headlight 3 6053 rectangular sealed beam headlight 4 9004 halogen bulb 5 4652 rectangular sealed beam headlight 6 6104 7 sealed beam headlight

Although Figure 6(a) seems to indicate that the majority of the current policies are close

in cost to the optimal policies this is very misleading For example suppose the optimal policy

for a product costs $1 and that the current policy costs $2 In absolute terms the optimal

policy saves only $1 but in relative terms the optimal policy saves 50 of the current cost

When savings are viewed in this manner as in Figure 6(b) it is clear that substantial savings

can be realized for the majority of the products In fact both the mode and the mean of the

savings are between 40 and 50 of the current costs

In order to extrapolate the above savings to obtain the potential savings for product

category 20 at all 21 stores the differences between stores must be considered During the

60

pilot study the current policy for each of several products at each size of store was evaluated

The study suggested that medium and large stores have a similar amount of excess

inventory and that jumbo and extra jumbo stores have much more excess inventory One

could reasonably expect that the additional excess inventory allows for higher savings to be

realized at the jumbo and extra jumbo stores Thus because 18 of the 21 stores are

medium or larger the expected 40-50 savings at store 6 can be viewed as a lower bound

on the potential savings to Canadian Tire Pacific Associates from implementing optimal policies

for product category 20 in all its stores

Finally in order to obtain the overall potential savings the expected savings from

product category 20 must be extrapolated to all product categories Ideally the proportion of

inventory comprising each product category and the relative savings that can be realized for

each category would be used to obtain a good estimate of the overall potential savings

However as this information is not known only a crude extrapolation is possible Thus the

analysis proceeds under the assumption that a 40-50 savings can be realized in each product

category35

Before an estimate of the total potential savings in absolute terms can be obtained the

proportion of the current inventory to which the savings apply must be determined Up to this

point the analysis has been concerned solely with reducing inventory at the stores However

it is inappropriate to perform further analysis without considering the entire inventory system

including the central warehouse Because the lead time for orders from Toronto is highly

variable and considerably greater than those at individual stores Canadian Tire Pacific

3 5 During the pilot study the current policies for a paint brush and an ironing board were evaluated and found to result in excess inventory similar to that of product category 20 Although this in no way justifies the assumption it does lend some support

61

Associates must keep a fairly high safety stock at the warehouse Without knowing the

proportion of inventory serving as the safety stock it is inappropriate to assume any reduction

in inventory at the warehouse is possible Therefore the relative savings of implementing

optimal policies is applied only to the inventory held in the stores36 Of the estimated $28

million in inventory approximately half is held in the stores therefore the 40-50 savings

translates to a savings of approximately $55 to $7 million per year

B Sensitivity Analysis on the Ordering Cost and Holding Rate

The above analysis was based on estimates of the ordering cost and the annual unit

holding rate37 provided by management As described in Chapter II these estimates

incorporate a number of costs that are difficult to quantify or that are speculative Therefore

there is some question as to the accuracy of the estimates In this section errors of +$0015

in the ordering cost and +5 in the holding cost are assumed in order to determine the

sensitivity of the model to errors in these variables38

The analysis proceeds as follows Based on an ordering cost of $0080 and a holding

rate of 30 the optimal policies from the previous section were obtained39 Then assuming

3 6 Of course this assumes that the excess inventory currently held in the stores is not required as part of the safety stock Based on the large amount of inventory at the warehouse this assumption seems reasonable

3 7 Hereafter simply referred to as the holding rate

3 8 An error of +$0015 in the ordering cost implies an actual ordering cost of $0070 a relative error of approximately 21 and an error of -$0015 implies an actual ordering cost of $0100 a relative error of 15 An error of +5 in the holding rate implies an actual rate of 25 a relative error of 20 and an error of -5 implies an actual holding rate of 35 a relative error of approximately 17

3 9 These policies are referred to as optimal policies for the duration of this chapter

62

the actual ordering cost andor holding rate are different from managements estimates the

heuristic is used to obtain the true optimal policies Finally the optimal policies are

evaluated based on the actual ordering cost and holding rate and then compared to the true

optimal policies For each possible combination of ordering cost and holding rate three

numbers are recorded the number of products with an optimal policy that differs from its

true optimal policy the total cost of implementing the optimal policies given the actual

ordering cost and holding rate and the difference between the cost of implementing the

optimal policies and the cost of implementing the true optimal policies The results of the

analysis are summarized in Table 6

Table 6 A comparison of optimal policies to true optimal policies

Actual Ordering Cost

$0070 $0085 $0100

25 5 products $155439

$002

51 products $160590

$092

88 products $165464

$449

Actual Holding Rate

30 54 products

$181470 $182

48 products $192026

$088

35 79 products

$207270 $574

48 products $212940

$136

2 products $218312

$000

4 0 The table should be read as follows If the actual ordering cost is $0070 and the actual holding rate is 25 then each of 5 products has an optimal policy that is different from its true optimal policy The total cost of implementing the optimal policies is $155439 which is $002 more than implementing the true optimal policies

63

The model appears to be quite insensitive to relative errors in the order of 20 Under

the worst case scenario when one of the variables is over-estimated and the other is undershy

estimated the optimal policies result in a total cost of inventory which is only approximately

3 higher than the cost of the true optimal policies Further because demand is not changed

by errors in these estimates each of the optimal policies still provides a customer service

level of at least 975 under all of the scenarios Finally although the cost of the optimal

policies varies significantly for the different scenarios the relative savings remain virtually

constant Table 7 lists the cost of existing policies and in parenthesis the relative savings of

the optimal policies for each of the different combinations of ordering cost and holding rate

The cost of the existing policies varies in the same manner as that of the optimal policies the

relative savings for each of the scenarios remains around 55 Therefore even in the presence

of such errors in the ordering cost and holding rate the projected savings of implementing the

optimal policies would be unchanged

Table 7 The cost of existing policies and the relative savings of the optimal policies under various scenarios

Actual Ordering Cost

$0070 $0085 $0100

25 $350153 (56)

$358464 (55)

$366764 (55)

Actual Holding Rate

30 $412438 (56)

$429048 (55)

35 $474725 (56)

$483026 (56)

$491320 (56)

64

C Sensitivity Analysis on the Demand Probability Mass Function

In this section the sensitivity of the model to slight changes in the demand probability

mass function is investigated As described in Chapter II the demand function used in the

model is based solely on observed sales however this empirical function is only one possible

instance of the underlying true demand function For example had the data been collected

over a period both starting and finishing one month earlier many of the empirical functions

would be slightly different Consequently depending upon the degree to which the empirical

functions vary and the sensitivity of the model to these variations the optimal policies from

the model might be inaccurate For example recall the case of the amber driveway marker

only thirteen markers were sold and all were sold on the same day Because the empirical

demand function is used the model calculates an optimal policy of (1213) clearly this

policy is impractical Although this example is extreme it does indicate the danger of using

unrealistic empirical demand functions41

As discussed in Chapter II three classes of products were identified namely low

medium and high-demand products For this analysis the same method of classification

is used Two products are selected from each of these groups and two new distribution files

are created for each of the six products For each product one of the files contains two

additional days of data with each day having a demand equal to the maximum observed demand

for that product In the second file only one additional day of data is added but the demand

on that day is two more than the maximum observed demand The situations that could give

4 1 Certainly a rigorous analysis to determine both the extent of variation in the empirical demand functions and the effect of using the empirical functions needs to be performed However such an analysis is beyond the scope of this thesis only a brief analysis that might suggest the degree to which the model is sensitive is performed here

65

rise to such changes in the demand data are termed Scenario 1 and Scenario 2 respectively42

The analysis proceeds in a manner similar to the one in the previous section Given the

demand that was originally observed the optimal policies were obtained Now assuming the

true demand function corresponds to the demand under Scenario 1 the true optimal policy

for each product is calculated Then each optimal policy is re-evaluated using the true

demand function and compared to the true optimal policy The analysis is then repeated

assuming the true demand function corresponds to the demand under Scenario 2 Table 8

lists the results for each product and scenario

Table 8 True optimal and optimal policies for each demand scenario

Product 202101 (low demand)

Scenario s s Cost ($) C S L ()

1 True optimal 1 2 470 993

Optimal 1 2 470 993

2 True optimal 2 3 663 976

Optimal 1 2 461 875

Product 203507 (low demand)

Scenario s S Cost ($) C S L ()

1 True optimal 0 3 334 981

Optimal 0 2 258 972

2 True optimal 2 3 508 980

Optimal 0 2 273 702

4 2 To be clear the scenarios are defined as follows Scenario 1 The addition of two days each with a demand of max Scenario 2 The addition of one day with a demand of max+2

where max is the maximum observed daily demand for each product in the sample period

66

Product 201032 (medium demand)

Scenario s S Cost ($) C S L ()

1 True optimal 6 12 1377 976

Optimal 6 11 1338 973

2 True optimal 6 13 1426 977

Optimal 6 11 1338 970

Product 203039 (medium demand)

Scenario s S Cost ($) CSL ()

1 True optimal 5 19 256 975

Optimal 5 16 251 969

2 True optimal 5 20 258 976

Optimal 5 16 249 968

Product 203677 (high demand)

Scenario s S Cost ($) CSL ()

1 True optimal 18 23 3981 977

Optimal 18 22 3817 972

2 True optimal 18 23 3995 978

Optimal 18 22 3831 973

Product 203680 (high demand)

Scenario s S Cost ($) CSL ()

1 True optimal 38 49 9498 976

Optimal 37 48 9175 971

2 True optimal 37 49 9484 975

Optimal 37 48 9206 972

67

The model is most sensitive to changes or errors in the demand functions of low-

demand products When only one additional day of demand for max+2 units is added to the

data the optimal policy for each of the low-demand products results in a very low customer

service level In fact for one of the products the customer service level drops to almost 70

The increase in demand does lower the customer service level for each of the other the

products but that decrease is slight

The above analysis indicates the need for some kind of smoothing or modelling of the

demand data especially for low-demand products The inherent variation between two

instances of a true demand function could very well result in more pronounced changes than

those resulting from either of the above scenarios Therefore it is likely that many optimal

policies especially those for low-demand products would actually provide a much lower or

much higher customer service level than expected However by smoothing the data the

variation between instances of a true demand function can be reduced and the desired

customer service levels can be achieved

For example suppose that the demand for a product is stationary that the average

demand during the previous period was unusually low and that the optimal policy for this

period is based on the demand during the previous period Because the demand is stationary

the probability that the demand during this period is higher than that during the past period is

more than 50 In other words the probability that the demand during this period is higher

than that for which the optimal policy provides the desired expected customer service level

is greater than 05 Thus expected customer service level for this period is actually less than

that specified Now suppose instead that the optimal policy is based on smoothed data which

reflects very closely the expected demand The probability that the demand during this period

is higher than that for which the optimal policy provides the desired expected customer service

68

level is virtually 05 Thus as a result of smoothing the data the specified customer service

level should be expected

69

VII C O N C L U S I O N

In this thesis we developed a single-product single-location model of the inventory

system at Canadian Tire Pacific Associates Specifically we formulated a Markov chain model

to determine the long-run average cost of inventory for fixed values of s and S and then

developed a heuristic search to find near optimal policies quickly Also as a first step toward

developing an integrated interactive inventory control system we created a prototype of the

interface

According to the analysis performed here Canadian Tire Pacific Associates is currently

holding an excessively large inventory in its stores This study finds that Canadian Tire Pacific

Associates could reduce the cost of inventory for product category 20 at store 6 by

approximately 50 and still maintain customer service levels of at least 975 Further the

mean and modal values of savings for these products are between 40 and 50 Given the

similar excess inventory at the other stores and in all product categories we believe that savings

in this range are possible for all products and stores We estimate that the application of

inventory control as described in this thesis would result in annual savings of $55 to $7

million

Given the large potential savings to Canadian Tire Pacific Associates we recommend

that management proceed with the next phase of the project Specifically we recommend that

the model algorithm and interface be refined further that a forecasting model be developed

and that a data collection and storage scheme be designed

In order to reflect operational changes at Canadian Tire Pacific Associates subsequent

to the formulation of the model modifications must be made to allow for the lead time to

exceed the review period Several of the stores now have three picks per week and as a

70

result orders placed for these stores at one decision epoch do not arrive until after the next

decision epoch Because this violates one of the models assumptions the model must be

modified accordingly Note however that with three picks per week and a review period

of four days orders from only the previous decision epoch are ever outstanding at the current

decision epoch

Also the model should be refined to reflect the fact that the review period is not fixed

over the planning period In its present form the model assumes a fixed review period

however inventory at stores with two picks per week is actually reviewed on a four day -

three day cycle and inventory at stores with three picks per week is reviewed on a three day

- two day - two day cycle Rather than obtaining the steady state costs for fixed review period

the model should calculate the steady state cost for a period of one week during which multiple

replenishments occur Although the solutions are not expected to differ significantly as a result

of the change the model would reflect the actual process more accurately

Further work is also required on the search technique for locating near optimal

policies Although the heuristic search described in this thesis appears to work well the

decision maker must provide an upper bound on S prior to the search In practice this upper

bound is generally not known in advance Thus if the heuristic is to be used either a tight

theoretical bound must be found or a termination heuristic must be developed Alternatively

a new heuristic search could be developed based on additional or different assumptions

regarding the properties of the feasible space For example although some regions of non-

convexity were found for some products much of the feasible space appeared to be convex

A heuristic that assumes a convex feasible space would not require an upper bound on S and

might perform very well However before such a heuristic is implemented further work is

necessary to determine the degree of non-convexity for a variety of demand functions and

71

ordering and holding costs

In addition to making the above modifications we recommend that Canadian Tire Pacific

Associates begin work on the remaining two components of the inventory control system First

in order to allow for products with seasonal demand the system requires a forecasting model

to provide accurate estimates of demand for the coming period Second the system needs a

data collection and storage module that systematically collects sales data for each active

product and for each store and then updates the corresponding demand distributions Because

approximately 25 of demand is expected to result in lost sales under the optimal policies the

sales data will have to be adjusted accordingly in the future Also because the model appears

to be sensitive to slight changes in the demand distribution for low-demand products the

module should incorporate a technique for smoothing the empirical distributions for these

products Further work should be undertaken to determine the extent of variation in the

empirical distributions and the most appropriate smoothing technique However because of

the large amount of data required to forecast seasonal demand and because of the current lack

of historical data we recommend that management proceed immediately with the design of a

data collection scheme

Finally two additional issues that are beyond the scope of the next phase should be

addressed at some point First a monitoring system will be required once inventory control

is implemented in the stores The monitoring system will ensure that optimal policies perform

as expected and will indicated whether or not demand is adversely affected by the lower levels

of in-store inventory Second given the degree to which in-store inventory can be reduced

inventory control in the warehouse should be investigated Because approximately half of the

existing inventory is held in the warehouse the additional savings from reducing system-wide

inventory could be great

72

Afterword

Just prior to the completion of the work described in this thesis Mr Don Graham

principle owner of Canadian Tire Pacific Associates sold his shares in the company Shortly

thereafter the results of this thesis were presented to the new management group

Unfortunately the new management did not consider development of an inventory control

system to be a priority Nonetheless the prototype was delivered to Canadian Tire Pacific

Associates for possible in-house testing however at the time of this writing - almost one year

after the meeting - Canadian Tire Pacific Associates has not decided to continue with the

project

73

Bibliography

Arrow K A T E Harris and J Marschak (1951) Optimal inventory policy Econometrica 19 250-272

Day G (1992) In a speech to the Empire Club of Canada Toronto quoted in When the JIT fits The Globe and Mail October 20 1992 B24 and in Puterman (1994)

Dvoretzky A J Kiefer and J Wolfowitz (1952a) The inventory problem I Case of known distributions of demand Econometrica 20 187-222

Dvoretzky A J Kiefer and J Wolfowitz (1952b) The inventory problem II Case of unknown distributions of demand Econometrica 20 450-466

Federgruen A (1993) Centralized Planning Models in SC Graves A H G Rinnooy Kan and P H Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Federgruen A H Groenevelt and H C Tijms (1984) Coordinated replenishments in a multi-item inventory system with compound Poisson demands and constant lead times Management Sci 30 344-357

Lee H L and S Nahmias (1993) Single-Product Single-Location Models in SC Graves A H G Rinnooy Kan and PH Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Muckstadt J A and R O Roundy (1993) Analysis of Multistage Production Systems in SC Graves A H G Rinnooy Kan and PH Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Porteus E L (1985) Numerical Comparisons of Inventory Policies for Periodic Review Systems Oper Res 33 134-152

Puterman M L (1994) Markov Decision Processes Discrete Stochastic Dynamic Programming Wiley New York

Silver E A and R Peterson (1985) Decision Systems for Inventory Management and Production Planning 2nd edition Wiley New York

Taylor H M and S Karlin (1994) An Introduction to Stochastic Modeling revised edition Academic Press San Diego

74

Tijms H C and H Groenevelt (1984) Simple approximations for the reorder point in periodic and continuous review (sS) inventory systems with service level constraints European J Oper Res 17 175-190

75

Appendix A

A Portion of the Sales File for Product 200001 a 30-amp Inline Fuse

200001 283 1190 0 0 2 0 0 1 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 200001 284 1190 2 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 200001 285 1190 0 0 4 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 286 1190 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 200001 287 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 200001 288 1190 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1 0 2 200001 289 1190 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 290 1190 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 200001 291 1190 0 0 2 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 200001 292 1190 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 200001 294 1190 0 0 0 1 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 1 200001 295 1190 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 200001 296 1190 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 200001 297 1190 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 200001 298 1190 0 0 1 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 200001 299 1190 0 0 1 0 0 0 1 0 1 0 0 2 0 0 0 0 0 2 1 1 0 200001 300 1190 0 0 1 0 0 0 0 1 0 0 0 0 0 2 2 0 0 0 0 3 0 200001 301 1190 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 200001 303 1190 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 200001 304 1190 1 0 0 0 0 0 0 0 1 0 0 2 0 0 0 0 0 0 0 0 0 200001 305 1190 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 200001 306 1190 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 2 0 0 1 0 200001 308 1190 0 0 0 1 0 1 0 0 0 2 0 1 0 0 0 0 1 1 1 0 0 200001 309 1190 1 3 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 200001 310 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 200001 311 1190 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 200001 312 1190 0 1 1 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 200001 313 1190 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 2 0 200001 314 1190 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 200001 315 1190 0 0 2 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 200001 316 1190 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 200001 317 1190 2 0 0 1 0 0 0 0 0 0 5 1 0 0 0 1 0 0 0 0 0 200001 318 1190 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 1 200001 319 1190 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 200001 320 1190 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 200001 321 1190 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 1 200001 322 1190 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 323 1190 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 200001 324 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 0 1 0 0 2 1 1 0 200001 325 1190 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3 200001 326 1190 0 0 1 0 1 0 0 0 2 0 0 1 0 0 0 1 0 1 0 0 1 200001 328 1190 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 200001 329 1190 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 200001 330 1190 0 0 1 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 200001 331 1190 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 200001 332 1190 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 200001 333 1190 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 200001 334 1190 0 0 2 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 200001 335 1190 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 200001 336 1190 0 0 0 1 0 0 3 0 0 0 0 0 2 0 0 0 0 0 0 0 0

(253 subsequent lines omitted)

76

Appendix B

Sample Distribution Files

a) A low-demand product - an automobile back-up alarm

PRODUCT NUMBER 202101 PRICE $ 684

DMD

3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

N 2 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 2 2 0 1

1 5 3 8 12 5 7 11 6 11 5 4 11 10 19 4 2 5 14 5 6 4

0 302 304 299 295 302 300 296 301 296 302 303 295 296 288 303 305 301 290 300 301 302

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

b) A medium-demand product - a 168 bulb

PRODUCT NUMBER 203039 PRICE $ 51

DMD

16 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 1 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0

9 0 0 2 5 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0

8 0 0 2 5 0 0 2 0 1 1 3 0 1 3 0 0 1 1 0 0 1

7 2 1 3 1 0 0 1 0 2 3 1 1 4 1 1 0 2 1 1 1 1

6 4 7 1 10 3 1 3 2 1 3 5 2 7 4 1 0 1 1 3 3 4

5 1 7 7 17 0 2 11 1 5 4 1 2 6 9 6 2 6 9 4 4 1

4 5 10 10 16 5 5 19 10 13 8 11 9 13 10 12 7 5 15 5 8 3

3 10 12 13 27 4 6 24 11 10 13 14 9 23 14 18 8 14 18 14 12 14

2 14 30 36 43 16 16 43 32 38 39 39 34 47 36 41 21 19 39 20 34 24

1 30 56 68 56 37 30 70 48 67 61 44 75 62 63 48 28 52 61 63 53 47

0 240 184 165 126 240 247 133 202 169 175 189 173 143 165 180 240 206 161 197 192 212

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

77

c) A high-demand product - a H4651 rectangular sealed beam headlight

PRODUCT NUMBER 203677 PRICE $ 857

DMD

14 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

13 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 2 3 0 0 1 0 0 0 0 1 1 1 0 0 0 0 2 0 0

11 0 1 3 1 0 0 4 1 0 0 0 0 0 0 0 0 1 0 0 0 1

10 0 0 4 4 1 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 1

9 0 3 14 11 0 1 7 1 0 2 0 8 0 1 0 0 0 2 1 0 0

8 0 4 12 16 1 0 18 4 3 5 0 2 1 4 2 1 1 0 0 1 3

7 1 2 12 13 1 6 19 3 5 9 0 11 2 5 4 0 2 3 3 1 4

6 2 9 22 21 6 10 29 12 10 11 4 24 3 7 10 4 7 11 6 3 14

5 7 13 35 33 8 16 29 19 18 8 3 26 15 26 15 12 9 14 13 11 19

4 19 35 34 43 32 30 41 31 32 47 11 42 20 33 30 31 20 31 22 21 31

3 32 63 49 54 42 60 51 40 51 35 33 55 39 51 39 38 39 57 58 40 54

2 46 59 56 55 73 63 61 65 63 61 63 54 68 85 65 53 72 69 70 77 63

1 109 79 42 40 80 64 32 80 74 77 99 55 97 57 83 93 80 75 81 97 73

0 91 39 21 13 63 57 11 51 51 52 94 29 61 35 59 75 76 45 51 56 44

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

78

Appendix C

A Proof that the Limiting Probability Distribution is Independent of the Initial State

In this appendix we show that provided 0 lt PrDT = 0 lt 1 the Markov chain

corresponding to an inventory model with a (sS) policy has a limiting probability distribution

7T where 0 lt TTJ lt 1 for y = 0 1 S that is independent of the initial state Further the

limiting distribution is the unique nonnegative solution of the equations

Tt P = TI (AD

s and J gt = 1 (A 2)

i=0

where P is the (S + l) x (S + l) transition probability matrix whose first n rows (and columns)

correspond to the n recurrent states in the chain and whose remaining rows (and columns) if

any correspond to transient states43 Prior to the proof several preliminary results are

obtained

RESULT 1 The Markov chain has exactly one positive recurrent class and perhaps some

transient states

First note that in the absence of any absorbing states a Markov chain with a discrete

finite state space must have at least one positive recurrent class containing at least two states

Now consider the Markov chain for an inventory model with a (sS) policy

a) State i gt s cannot be absorbing because there is a positive probability of reaching

j lt i (Since PrDT gt 0 gt 0)

4 3 Note that since the rows and columns of P are reordered the elements of it must be reordered accordingly

79

b) State lt s cannot be absorbing because by the rules of the ordering policy and

the assumption that PrDT = 0 gt 0 there is a positive probability reaching S

Therefore the chain has no absorbing states and must have at least one positive recurrent class

Next consider the subset of states A = i | i lt s Because the state space is finite

and PrDT gt 0 gt 0 the process must return to at least one state in A in other words at least

one state in A must be recurrent Let k G A be a member of a recurrent class dt Because

of the ordering policy and the assumption that PrDT = 0 gt 0 k -raquo S Also by definition

there must be a positive probability of returning to k from S S -gt k Therefore S G 3w

Further because S is reachable from all states in A i G A is either transient with i -raquo dt or

recurrent with pound iR

Now consider the subset of states B = | j gt s Because PrDT gt 0 gt 0 the

chain will eventually reach a state i G A If i is transient then j must be transient also with

y -gt 3 if i G 9 then j is either transient with j -gt dt or recurrent with j E dt Thus the

Markov chain has exactly one recurrent class and perhaps some transient states

RESULT 2 The limiting probability distribution ir for the Markov chain corresponding to the

nxn submatrix Pm = [Ptj] i j G dt is independent of the initial state and is the

unique nonnegative solution of the equations

TC-Prr = re (A3)

and J2 K = 1 ( A 4 )

jt=i

where the rows and columns of are numbered from 1 to n rather than

Since if i j i and j must be in the same equivalence class

80

according to their associated states

First note that both the original Markov chain and the one corresponding to the

submatrix P R R are aperiodic because PrDT = 0 gt 0 Therefore given that 9 is a positive

recurrent aperiodic class the limiting distribution ir is uniquely determined by (A3) and (A4)

(Taylor and Karlin 1994 Theorem 44 p213) Further because there are no absorbing states

or other recurrent classes in the original chain the process must eventually enter the recurrent

class 9 regardless of the initial state Therefore in the limit the probability of being in state

i G 9 is independent of the initial state

RESULT 3 The limiting probability distribution for the transient states is the null vector

independent of the initial state

This result follows directly from the definition of a transient state

PROOF Suppose the states are reordered such that the first n rows (and columns) of P

correspond to the n recurrent states of the Markov chain Then P can be partitioned into four

submatrices

PRK - the n x n submatrix [Py] i j E 9

P T T - the (S + 1-I) x (S + l-n) submatrix [Py] i j g 9

P T R - the (S + l-n) x n submatrix [Py] i pound 9 j G 9

P R T - the n x (S + l-n) submatrix [Py] i G 9 j pound 9t By definition this is a matrix

of zeros

(Al) can then be rewritten as

81

(ltR TtT) P P L TR TT

I-P RR 0

p I-P TR 1 TT

(TtR TCT)

(0- 0T)

where irR denotes the limiting probability distribution for the n recurrent states and irT denotes

the distribution for the remaining S + l-n transient states Solving for TT t gives

TCr-0 + itT-(I-PTI) = 0T

itT-(I-PJJ) = 0 r

i c r = (I-P^-Oj

Since (I - PTT)1 always exists (Puterman 1994 Proposition A3 p 593) irT = 0X which is

in agreement with Result 3 Now solving for 7rR yields

(A2) can be rewritten as

bullRPRR 1 1R (A5)

(A6) S + l

k + L k =1

k=l k=n+

k=i

where ir contains the appropriately reordered steady state probabilities Since (A5) and (A6)

are equivalent to (A3) and (A4) respectively Result 2 holds Therefore the Markov chain

corresponding to an inventory model with a (sS) policy has a limiting distribution that is

independent of the initial state and is the unique nonnegative solution to (Al) and (A2)

82

Appendix D

A Measure for the Steady State Customer Service Level

In this appendix we show that the appropriate measure for the steady state customer

service level is given by

_ E[Demand satisfied during RP] E[Total demand during RP]

Consider the following two measures of the steady state customer service level

Measure 1 Suppose the customer service level is calculated only after n review periods have

passed Denoting DS as the demand satisfied in review period i and DT as the total demand

in period i the customer service level after n periods CSL(n) is

DS + DS0 + + DSbdquo CSLn) = - -

DT + DT + + DT

Dividing both the numerator and denominator by n taking the limit as n -raquo oo and applying

the Law of Large Numbers yields

CSL = lim CSL(n) n-degdeg

- [DS + DS + + DS] = l i m ^ J 1 1

raquo~ - [DT + DT + + DT]

E[DS]

E[DT]

83

Measure 2 Suppose the customer service level is calculated for each of n review periods and

then averaged Thus the customer service level after n periods CSL(n) is given by

CSL(n) = -DS DS

DT DT DT_

Taking the limit as n -gt oo yields

CSL = lim CSL(n)

lim mdash DSl DS2

DT DT DT_

DS

DT

Now consider the scenario in which there is one item on the shelf to begin each review

period and in which demand is stochastic with PrDT = 2 = Vi and PrD x = 100 = Vi

According to Measure 1 the customer service level is

CSL - = ^ W + ^ W = plusmn bdquo 002 E[DT) 1 (2) + 1(100) 51

while according to Measure 2 the customer service level is

CSL = E DS DT

1(1) + i(-Ls

22) + 2 100

51 200

026

Clearly from managements point of view the first measure is more indicative of the

actual level of service being provided The problem with the second measure is that equal

84

weight is given to each ratio regardless of the magnitude of the numbers For example with

respect to Measure 2 the events LSLR = 12 and LSLR = 50100 are identical however

with respect to lost sales the second event is much worse than the first Therefore the proper

measure for the steady state customer service level is Measure 1

85

Appendix E

The Calculation of the Conditional Expected Demand Not Satisfied During a Period of T Consecutive Days

Denoting D x as the demand during a period of T consecutive days and j as the inventory

on hand at the beginning of the period the conditional expected demand not satisfied during the

period E[(DT-y)+] is calculated as follows

E[(DT-j)+] = pound ltk-j)Pr[DT = k k=j+l

= pound (k-j)PrDT = k] + J2(k-j)PrDT = k] - pound (fc-)VJgtr = fc k=j+l k=0 k=0

= J2(k-j)Pr[DT=k -J2(k-j)PrDT = k] k=0 k=0

= YkPrDT=k) - J2jPrDT=k - Yk-j)PrDT=k) Jfc=0 k=Q k=0

= Ti - j - Y(k-j)PrDT=k 7-1

= Tx - j + YltJ-k)PrDT = k jt=0

where fx is the mean daily demand

86

Appendix F

Justification of the Updating Technique for the New Policy (s+mS)

Let the probability transition matrix corresponding to the policy (sS) be denoted by the

(S + l)x(S + l) matrix P with elements py Define s = s+m where m is an integer satisfying

the condition 0 lt s+m lt S Let the probability transition matrix of the new policy (sS)

be denoted by the (S + 1)X(S + 1) matrix 7gt with elements py Using the equations developed

in Chapter III the elements of P are found as follows

CASE 1 The lead time is less than the review period

A m gt 0

For 0 lt i lt s

( i-i pound [PrDL=kPrDT_LgtS-k]] + PrDLziPrDT_LplusmnS-i k=0

i-l 53 [PrDL=kPrDT_L = S-j-k]] + PrDLziPr[DT_L = S-j-i) k=0 s-j

52 [Pr[DL = kPrDT_L = S-j-kn k=0

ifj = 6

ifOltjz S-i

if S-i ltj ltS

a) 0 lt lt s

Pij

= lt Pij

Pij

= pij

b) s lt i lt s This does not simplify but it is identical to (2)

87

For s lt i lt S

PrDTgt i

p = lt PrDT = i-j

0

Pfj ifj = 0

= lt Pij ifOltjlti

Pij if i ltj plusmn S

= pij

B m lt 0

ifj = 0

if 0 ltj lt i

if i ltj plusmn S

For 0 lt i lt s

i-l 53 [PrDL=kPrDT_LplusmnS-k]] + PrDLgtiPrDT_LplusmnS-i

i-l E [PrZgti = fciVZ)7W=S--fc] +PrD tgtiP-D 7 t = S-7-

p y laquoy = o

P i j ifOltjS-i

vPij ifS-iltjS

ifj = 0

ifOltjltS-i

if S-i ltjltS

For s lt i lt S

fPrJgtrgt- 17=0 = PrDT = i-j] ifOltjlti

0 ifiltjplusmnS

88

a) s lt lt s This does not simplify but it is identical to (3)

b) s lt i lt S

PrDTgti

= bull PrDT = i-j]

0

17=0

if0ltjplusmni

ifiltjS

= Pt

ifj = 0

if0ltjlti

ifiltjplusmnS

= Pi

CASE 2 The lead time is equals the review period

A m gt 0

For 0 lt i lt s

0 if0ltjlt S-i

PrDTgti ifj = S-i

PrDT=S-j] ifS-iltjltS

a) 0 lt i lt s

= bull y Pij

Pij

ifOplusmnjltS-i

ifj = S-i

ifS-iltjS

= Pij

b) s lt i lt s This does not simplify but it is identical to (4)

89

For s lt i lt S

PrDTgti)

Pr[DT=i-j

0

laquof 7=0

ifOltjplusmni

if i ltj lt S

Pa

Pij

if 7 = 0

if 0 lty lt i

ifiltjltS

B w lt 0

For 0 lt t lt s

ro PrJrgt PrDT = S-j

ifOzjltS-i

ifj = S-i

if S-i ltj lt S

Pij

ifOzjltS-i

ifj = S-i

if S-i ltj ltS

= Pi

For s lt i lt S

PrDTgti

= PrDT = i-j

0

i 7 = o ifOltjplusmni

ifiltjltS

a) s lt i lt s This does not simplify but it is identical to (3)

90

PrDTzi ifj = 0

PrDT = i-j] ifOltjplusmni

0 ifiltjltS

Py ifj = 0

ptj ifOltjplusmni

Pij if iltj plusmnS

Appendix G

Justification of the Updating Technique for the New Policy (sS+m)

Let the probability transition matrix corresponding to the policy (sS) be denoted by the

(S + l)x(S + l) matrix P with elements p(gt Define S = S+m where m is any non-negative

integer Let the probability transition matrix of the new policy (sS) be denoted by the

(S+m+l)x(S+m + l) matrix IP with elements ptj Then using the equations developed in

Chapter III the elements of 7 are found as follows

CASE 1 The lead time is less than the review period

For 0 lt i lt s

pound [PrDL=kPrDT_LplusmnS-k] + PrDLi)PrDT^Sgt- k=0

i-l pound [Pr[DL = kPrDT_L = S-j-k- +Pr[DLiPrDT_L = S-j-i Jt=0

s-i Y [Pr[DL=kPrDT_L = S-j-k] k=0

i - l

ifj=0

ifOltjltS-i

ifS-iltjltS

pound [PrDL = kPrDT_LS-k + Pr[DLi)PrDT_LplusmnS-i) if j = 0 fc=0

i - l = pound [PrDL=kPrDT_L = (S+m)-j-k] + PrDLgtiPr[DT_L = (S+m)-j-i ifOltj lt (J5+m)-i

if (S+m)-iltj lt(S+m)

k=0

i - l

pound (Sm)-j

pound [PrDL = kPrDT_L = (S+m)-j-k]] k=0

92

I-l

53 [PrDL = kPrDT_LgtS-k] + PrDLziPrDT_LS-i ifj = 0 k=0

i-l

Y[PrDL = kPrDT_L = S-(j-m)-k] + Pr[DLziPrDT_L = S-(j-m)-i] ifOltjzS+m-i k=0 S-(j-m)

53 [PrDL = kPrDT_L = S-(j-m)-k]] if S+m-i ltj lt S+m

i - l

53 [PrDL = kPrDT_LzS-k] + PrDLziPrDT_LzS-i] ifj = 0 =0

i - l

13 [Pr[DL = kPrDT_L = S-(j-m)-k] + Pr[DLziPr[DT_L = S-(j-m)-i ifOltjz S+m-i k=o and 0 z j-m

Pi(j-m)

Pi(j-m)

if 0 ltj lt S+m-i and 0ltj-mltS-i

if S+m-i ltj lt S+m and S-i lt j-m zS

i - l

Y [PrDL = k)PrDT_LzS-k] + PrDLi)PrDT_LplusmnS-i ifj = 0

k=0 i - l

53 [PrDL = kPrDT_L = S-(j-m)-k] + PrDLziPrDT_L = S-(j-m)-i ifOltjzm k=0

Piij-m)

Piij-m)

if mltj lt S+m-i

if S+m-i ltj lt S+m

i - l

YlPrDL = kPr[DT_L^S-k + PrDLi)PrDT_LS-i) ifj = 0

i - l

53 [PrDL = kPrDT_L = S-j-k] + PrDLgtiPrDT_L = S-j-i ifOltj i m k=0

Pi if mltj ltS

which for 0 lt j lt m is identical to (2)

93

For s lt i lt S

Pu =

PrDTgti

PrDT = i-j]

0

if0ltjlti

ifiltjltS

a) s lt i lt S

PrDTgti 7=0

PrDT = i -j) if0ltjlti

0 if i ltj lt S

0 ifSltjltSgt

ifj=0

if 0 ltj lt i

if i ltj plusmn S

0 ifSltjltS

ifOltjltS

0 ifSltjltSgt

b) S lt i lt S This does not simplify but it is identical to (3)

CASE 2 The lead time equals the review period

For 0 lt i lt s

0 ifO lt jltS-i

PrDTgti ifj=S-i

PrDT=S-j ifS-iltjltS

94

0 ifOplusmnjlt(S+m)-i

PrDT ifj = (S+m)-i

PrDT = (S+m) -j) if (S+m)-i ltj ltL (S+m)

0 if 0 lt jltS+m-i

PrDTzi] if j = S+m-i

PrDT = S-(j-m) if S+m-i ltj lt S+m

0 if 0 ltj lt S+m-i and 0 ^ j-m

Pi(j-m) if 0ltjlt S+m-i and 0ltj-mltS-i bull

Pi(j-m) if j = S+m-i and j-m = S-i

Pi(j-m) if S+m-i ltj z S+m and S-i lt j-m ^S

0 if 0 zj ltm

Pi(j-m) if mlt jlt S+m-i lt

Piij-m) if j = S+m-i

if S+m-i ltj lt S+m

1deg if 0 ltj ltm

[PiV-m) ifmltjltS

which for 0 lt i lt s is identical to (3)

For s lt lt S

PrDTzi

= PrDT = i-j)

0

ifj = 0

if 0 ltj ltL i

ifKjzS1

95

a) s lt i lt S

PrDTzi

PrDT = i-j]

0

0

17=0

if0ltjplusmni

ifiltjS

ifSltjltSgt

Pij 7=0

Pij if0ltjii

Pij if i ltj ltL S

0 ifSltjltS

Pij ifOzjiS

0 ifSltjltS

b) S lt i lt S This does not simplify but it is identical to (3)

96

Appendix H

A Hypothetical Consultation

In this appendix the interface during a hypothetical consultation is depicted The figures

accurately represent the various screens that are presented to the manager with the exception

that all lines and text are really white on a blue background During the consultation described

below only the choices made by the manager are hypothetical all the results are real

The title screen as depicted in Figure A - l appears initially upon execution of the

program In order to proceed to the main menu the manager presses the enter key

O I C S

Optimat Inventory Control System

For the exclusive use of

Canadian Tire Pacific Associates

Developed at The Faculty of Commerce

and Business Administration The University of British Columbia

Brian Kapalka Kaan Katircioghi Marty Puterman

Press ltEntergt to continue

Figure A - l The title screen

97

Main Menu 1 Select data directory

ctircdatadst

2 Select product and store combination Product 200001

3 Select scenario Review Period 4 Lead Time 3

4 Select search parameter Smax 25

5 Execute program

6 Exit to DOS

Please enter the desired option 5

Figure A-2 The main menu

Figure A-2 depicts the main menu of the program it is from here that the manager

enters all the review parameters If the manager wishes to change one of the parameters from

its default value he or she selects the appropriate number and enters the correct information

in the window presented The new settings will then be shown in place of the default values

In this example the relevant data files are located in the default directory ctiredatadst and

the manager has selected the productstore combination of product 200001 and store 6 Further

the manager has selected a review scenario of a 4-day review period and a 3-day lead time and

an upper search limit of 25 Finally in order to proceed with the analysis the manager

executes the main module of the program by selecting option 5

98

I INTRODUCTION

The importance of inventory management has grown significantly over the years

especially since the turn of this century In colonial times large inventories were viewed as

signs of wealth and therefore merchants and policy makers were not overly concerned with

controlling inventory However during the economic collapse of the 1930s managers began

to perceive the risks associated with holding large inventories As a result managers

emphasized rapid rates of inventory turnover (Silver and Peterson 1985) Following the

Second World War Arrow Harris and Marschak (1951) and Dvoretzky Kiefer and

Wolfowitz (1952ab) laid the basis for future developments in mathematical inventory theory

Shortly thereafter new inventory control methodologies were widely applied in the private

manufacturing sector More recently when inflation and interest rates soared during the 1970s

many organizations were forced to rethink their inventory strategies yet again Today the

control of inventory is a problem common to all organizations in any sector of the economy

Sir Graham Day chairman of Britains Cadbury-Schweppes P C L expressed this sentiment

when he stated I believe the easiest money any business having any inventory can save lies

with the minimization of that inventory (Day 1992) Lee and Nahmias (1993) and Silver and

Peterson (1985) provide a more detailed history of inventory management and control

Most inventory problems in the retail sector involve the control and management of a

large number of different products Ideally the inventory policies should be determined for all

products on a system-wide basis however because the number of products is often in the tens

of thousands formulating a model to do so is impractical Therefore single-product models

are frequently used in practice to determine policies for products separately (Lee and Nahmias

1993) Lee and Nahmias (1993) and Silver and Peterson (1985) provide a good overview of

1

single-product single-location models for a more complete review of the literature we refer

to them

The degree of complexity of a single-product model depends largely on the assumptions

made regarding the demand cost structure and physical characteristics of the inventory system

For example demand can be either deterministic meaning it is known with absolute certainty

or stochastic meaning it is unknown but follows a known probability distribution

Deterministic models tend to be relatively simple yet useful such as the basic EOQ model

However future demand is seldom known with certainty in practice and therefore the use of

these simple models is limited As a result the majority of research in inventory has focused

on stochastic demand models (Lee and Nahmias 1993)

Because the objective of most inventory models is to minimize the cost of inventory

assumptions regarding the structure of the cost are very important also Generally speaking

the cost of inventory has four components the purchaseproduction cost the orderingsetup

cost the holding cost and the shortage cost1 Each of these costs can be fixed proportional

nonlinear or some complex combination of these Further because nearly all inventory

problems require decisions be made sequentially over a period of time the costs can be either

discounted or averaged In the former case a discount factor is included in the cost function

in order to account for the time value of money However since most inventory problems are

on-going the long-run average cost is usually minimized (Lee and Nahmias 1993) In

practice this is achieved by minimizing the expected cost per time period under steady state

conditions

Finally the physical characteristics of the system are important in determining the

1 The cost of a stockout is often very difficult to estimate in practice As a result service levels are frequently used in its place When this is done a constraint is included in the model

2

complexity of the model For example the review process can be either continuous in which

case the inventory is reviewed continuously and orders are placed as soon as the stock reaches

a certain level or periodic in which case the inventory is reviewed at regular intervals and

orders can be placed only at those fixed times Also excess demand can be fully backordered

as is common in many manufacturing and wholesale systems completely lost as is common

in most retail environments or some combination of the two Lead time assumptions are also

important models with lead times are more complex than those in which the supply of

inventory is assumed to be instantaneous

Although the literature is dominated by single-product models (Lee and Nahmias 1993)

much recent work has focused on extending those models to multiple-product problems

However multiple-product models are inherently more complex than the simple-product

models In fact no efficient algorithms are currently available for determining fully optimal

strategies for most multiple-product problems (Federgruen 1993) As a result research has

focused on the development of easily computable approximation techniques Muckstadt and

Roundy (1993) provide a good overview of multiple-product andor multiple-location models

with deterministic demand although emphasis is placed on models for production and

distribution Federgruen (1993) extends the discussion to consider these models under uncertain

demand and also provides a complete list of references on the subject Silver and Peterson

(1985) discuss models for coordinated replenishments for both deterministic and stochastic

demand

In this thesis we examine the inventory system at Canadian Tire Pacific Associates

owner of 21 retail stores and a central warehouse Although this is a multiple-product

multiple-location single-warehouse problem and strictly speaking should be modelled as such

we analyze the inventory problem on a product-by-product and store-by-store basis We believe

3

that this simpler single-product single-location model captures the essentials of the problem for

two reasons First the coordination of replenishments for products andor locations is most

advantageous when significant savings are possible because of high first order costs and low

marginal costs for placing subsequent orders However because of the large number of

products and the physical distribution system in this problem trucks are dispatched to stores

on a regular basis Therefore the benefits of coordinating replenishments for this problem are

negligible Second we believe that controlling in-store inventory alone can result in substantial

savings to Canadian Tire Pacific Associates Further given the excessive amount of inventory

in the warehouse a single-location model assuming an infinite warehouse capacity is reasonable

Of course should inventory control in the warehouse become a priority a more elaborate model

would be required

More specifically we consider a periodic review single-product inventory model with

stationary independent stochastic demand a deterministic lead time and an infinite planning

horizon The probability mass function for the demand is based on empirical sales data

collected over a period of approximately ten months The cost of inventory includes a

proportional holding cost and a fixed ordering cost neither of which is discounted All excess

demand is completely lost The objective of the model is to minimize the long-run average cost

per unit time subject to a service level constraint on the fraction of demand satisfied directly

from in-store inventory In order to obtain meaningful cost comparisons the model calculates

the expected annual cost of inventory

For inventory problems like the one described above optimal (sS) policies can be

computed by Markovian decision process methods such as successive approximations or policy

iteration or by stationary Markovian methods However each of these methods generate

calculations that are complex and time consuming Therefore much research interest lies in

4

developing techniques to find approximate (sS) policies with little computational effort The

majority of the approximation techniques begin with an exact formulation of the problem and

then manipulate the problem by relaxations restrictions projections or cost approximations

(Federgruen 1993) Porteus (1985) presents three approximation techniques and compares

them to fourteen others for a periodic review single-product inventory model in which

shortages are fully backordered and a shortage cost is incurred Federgruen Groenevelt and

Tijms (1984) present an algorithm for finding approximate (sS) policies for a continuous review

model with backordering and a service level constraint Also Tijms and Groenevelt (1984)

present simple approximations for periodic review systems with lost sales and service level

constraints All three works find that the approximation techniques perform well However

the techniques are evaluated for a variety of demand parameters that all result in policies with

a value of S in the range of 50 to 200 Given the slower moving products at Canadian Tire 2

it is unclear how well such approximation techniques would perform Therefore we do not

attempt to apply or modify any of these approximation methods

Exact optimal policies for this problem can be obtained by formulating a constrained

Markov decision process and solving it However the data generating requirements and the

lack of a good upper bound on S make this formulation impractical Therefore we formulate

a Markov chain model for fixed values of s and S and then employ a search technique to find

the optimal values for s and S In order to improve the performance of the model we develop

a heuristic search that locates near optimal policies relatively quickly

The remaining chapters of this thesis are organized as follows In Chapter II an

overview of the existing inventory system at Canadian Tire is presented along with a description

2 For example very few of the products analyzed have optimal values of S exceeding 20 with the largest value of S equalling 69

5

of the problem Next a multi-phase project designed to develop a fully integrated interactive

inventory control system is described with the scope of the research reported in this thesis

completing the first phase of the project Finally a description of the demand and cost data is

given

In Chapter III we formulate a Markov chain model that can be used to calculate the cost

and service level associated with a given (sS) policy Based on several simplifying

assumptions the Markov chain is formulated and the equations for the corresponding transition

probability matrix are developed Then given the limiting probability distribution of the chain

the equations for the steady state annual cost of inventory and the corresponding service level

are obtained The chapter concludes with a brief summary of the model

In Chapter IV an algorithm that searches for the optimal (sS) policy is developed As

a starting point a simple grid search algorithm is developed and evaluated on the basis of

execution time In order to improve the algorithms speed a technique for updating the

transition probability matrix is presented and the improvement in the algorithm is noted Next

a theoretical lower bound on S is obtained although the corresponding reduction in the

execution time is minimal Finally based on two assumptions regarding the shape of the

feasible space a heuristic search is developed which improves the speed of the algorithm

significantly

In Chapter V a prototype of the inventory control systems interface is created In its

present form the interface allows managers to evaluate inventory policies interactively The

manager can set review parameters compare optimal policies and current policies and evaluate

alternate policies The chapter concludes by recommending specific options be including in

future versions to make the system more automated and user-friendly

In Chapter VI optimal policies for all products in one product category sold at one store

6

are compared to existing policies on the basis of expected annual costs and customer service

levels The analysis suggests that existing policies result in a large excess inventory and that

implementing the optimal policies would reduce the annual cost of inventory by approximately

50 A sensitivity analysis on the cost estimates is performed and suggests that the model is

not sensitive to relative errors of approximately 20 in the cost estimates Finally a sensitivity

analysis on the demand probability mass function is performed The analysis indicates that the

model is sensitive at least in terms of the customer service level to slight changes in the tail

of the demand distributions of low-demand products

Finally in Chapter VII the results of the analysis are summarized and future work is

outlined In order to incorporate recent changes in the review procedures at several stores the

model must be modified to allow for the duration of the review period to exceed the lead time

We recommend that the second phase of the project proceed specifically that a data collection

and storage scheme be designed and that a forecasting model be developed Because of the

substantial savings that can be realized by reducing inventory only at the stores inventory

control at the warehouse should also be investigated

7

II T H E I N V E N T O R Y S Y S T E M A T C A N A D I A N TIRE

A Background

Canadian Tire Pacific Associates owns and operates 21 retail stores in the lower

mainland of British Columbia The stores carry approximately 30000 active products3 that

are divided into 80 product categories The merchandise comprises primarily automotive parts

and accessories home hardware housewares and sporting goods Canadian Tire Pacific

Associates also owns a central warehouse in Burnaby at which inventory is received from

Toronto Inventory is then shipped periodically from the warehouse via truck to the local

stores In 1992 total sales at the stores exceeded $99 million and the value of inventory was

estimated to be $28 million of which approximately half was stored in the warehouse

On selected days of each week a stores inventory records are reviewed automatically

and orders for products are placed to the warehouse as required Management uses the term

flushing to describe the process of reviewing the inventory and producing the tickets for stock

owing and the term picking to describe the physical act of filling the order at the warehouse4

Whether or not an order is placed for a product depends upon its current inventory level its

target level and its minimum fill 5 If the target level less the current level exceeds the

minimum fill an order is placed to bring the stock up to the target level From the moment

3 Active products are those that are regularly restocked Canadian Tire Pacific Associates can choose from up to 50000 products

4 Note that the number of picks per week is equal to the number of times per week that the inventory records are reviewed At the beginning of this study eighteen stores had two picks per week while the remaining three stores had only one

5 The target level is the maximum number of items that should be on the shelf at one time and the minimum fill is the minimum number of items that should be ordered

8

the order is placed to the moment the product is reshelved takes between two to four days

depending on the store On average 11000 items are shipped from the warehouse at each

flush

Although a products minimum fill is the same for all stores its target level depends

upon the size of the store Based upon its sales volume and retail space a store is classified

as being one of five sizes small (S) medium (M) large (L) jumbo (J) and extra jumbo

(EJ) 6 Each product has associated with it a target level for each size of store

Consequently all stores of the same size have identical inventory policies Table 1 lists the

location size number of picks per week sales and floor space of each store as of October

1993

Periodically the inventory manager reviews the policies and when deemed necessary

adjusts them In this way the manager can account for seasonality in demand changes in mean

demand and any shortcomings of previous policies At the present time the review process

is not automated nor performed on-line The inventory manager uses Hxl5 computer

printouts with five products to a sheet to study the relevant information and on which to note

any changes Afterward the changes are recorded in the system and the new policies take

effect

Currently the inventory policies are based on rules of thumb developed through the

years and not on a statistical analysis of demand nor on a calculation of the cost of inventory

These rules of thumb incorporate a marketing philosophy which at times dominates the

inventory strategy The prevailing philosophy holds that a large in-store inventory instills

6 During the course of this study stores were resized several times At one point the entire class of medium stores was eliminated leaving only four sizes of stores On another occasion the store in Chilliwack was changed from being a small store to a large store

9

Table 1 Location and particulars of the 21 stores

Store Location Store Size

Picks Per Week

1992 Sales ($)

Floor Space (ft2)

1 Chilliwack S 2 2901969 6678

2 Coquitlam M 2 4015303 6510

3 Langley EJ 2 8138673 15422

4 Scott Road EJ 2 7307820 15064

5 Maple Ridge L 2 4102110 6678

6 Kingsway L 2 4153736 6574

7 Richmond EJ 2 8175105 14961

8 North Vancouver M 2 3856571 6534

9 Guildford M 2 3714372 6567

10 Burnaby M 2 3532752 6983

11 New Westminster S 1 2417992 6470

12 Abbotsford EJ 2 7899398 15303

13 Hastings J 2 4647794 9849

14 King Edward EJ 2 3825372 25492

15 White Rock L 2 3960262 6870

16 Main amp Mountain S 1 2705602 6431

17 Heather J 2 4161227 9427

18 King George J 2 4606817 9640

19 Cambie J 2 5855022 9091

20 Lougheed M 1 3114180 6615

21 Burrard J 2 6553310 10608

10

consumer confidence which in turn leads to increased sales In other words if a customer

perceives that Canadian Tire is in the hardware business he or she will choose to shop there

when meeting future needs However management has adopted this philosophy without any

analysis of the cost of holding excess inventory Only recently has management begun to

question the cost-effectiveness of allocating inventory based on these rules of thumb

Speculating that existing inventory practices might be deficient Mr Don Graham the

vice-president of Canadian Tire Pacific Associates commissioned a pilot study to examine the

inventory system and to propose alternate strategies7 The study using a steady state model

to represent a simplified inventory system suggested that existing policies lead to overly high

inventory levels More importantly the study outlined a methodology that would enable

management to determine inventory levels based on observed demand The pilot study

recommended that a more detailed model of the inventory system be formulated that the

methodology be expanded to incorporate the cost of inventory and that an algorithm be

developed that would quickly identify the optimal inventory policies

B The Problem

At the time of the pilot study management was concerned primarily with its inability

to distinguish between stores of the same size As mentioned in the previous section a

stores size is based solely on its sales volume and floor space and under existing practices

all stores of the same size have identical inventory policies for every product Management

realized that demand for some products might be a function of variables other than size such

7 The pilot study was performed by Martin Puterman a professor of Management Science and Kaan Katircioglu a doctoral student in Management Science both at the University of British Columbia The results of the study were presented at a U B C Transportation and Logistics Workshop on March 25 1993

11

as location For example the demand for sump pumps is likely to be very different in suburban

areas than in urban ones However management was unable to allocate more sump pumps to

large suburban stores than to large urban stores This inability led management to seek a

tool that would allow for a more effective allocation of inventory between stores of the same

size

By the end of the pilot study however management was less concerned with

reallocating inventory within a size of store than with reallocating inventory between sizes

of stores and with reducing overall inventory According to management assigning individual

inventory policies to each store appeared to be operationally infeasible Firstly management

believed that having individual policies would lead to confusion in the warehouse and to

increased shipping errors Exactly why this would occur is unclear the warehouse would

simply continue to fill orders from tickets that make no reference to the policy in place

Secondly and perhaps more tenably the computer technology at Canadian Tire Pacific

Associates did not allow for policies to vary across all stores Both the database used to store

the inventory data and the printouts used by the inventory manager allowed for only five

different policies per product all with the same minimum fill Moving to individual policies

would have required an overhaul of the existing information system Although management

saw this overhaul as inevitable it did not see the overhaul as imminent Therefore

management switched the focus of the study to developing a tool that would help reduce overall

inventory levels andor reallocate inventory more effectively between groups of stores

The subtle change in the studys objective prompted an important question how should

the stores be grouped The current practice of grouping stores based on size is problematic

for two reasons First as discussed above additional factors of demand such as location

cannot be taken into account Second grouping inherently results in a misallocation of

12

inventory when some stores within a group have differing numbers of picks per week For

example the medium-sized Lougheed store which has one pick per week requires higher

inventory levels to achieve a given customer service level than the other medium stores

which all have two picks per week8 In fact in terms of customer service levels it might

be more appropriate to group the Lougheed store with the large stores having two picks per

week Ideally this artificial grouping of stores will be abandoned when the computer system

is upgraded and policies can then be set on a store by store basis

As the project proceeded management grew somewhat uneasy with the concept of using

an analytical tool to evaluate and set inventory policies According to both the pilot study and

the preliminary results of this study significantly reducing inventory would greatly reduce costs

while still providing a high level of customer service However management held that because

the model cannot take into account intangibles such as marketing considerations such a drastic

reduction in inventory would be very inappropriate In this light management viewed the

optimal policies as lower bounds for inventory levels these policies would have to be

modified upwards in order to be practical Therefore management contended that any

methodology or software produced during this study should provide the means to evaluate and

compare a number of alternate policies

Regardless of exactly how the tool is to be used there is currently a lack of historical

sales data from which to analyze the inventory system and determine optimal policies No sales

data had ever been collected prior to the pilot study and only a relatively small amount has

been collected since In order to proceed with any analysis of the inventory system a

8 If demand is a function of size then the Lougheed store must meet the same demand with one fewer pick per week This would lead to either increased probabilities of stockouts at the Lougheed store or higher inventory levels at all the other medium stores

13

substantial amount of additional data would be needed The existing information system at

Canadian Tire Pacific Associates does allow for sales data to be captured however the system

is limited by the amount of data it can capture at one time and by its storage capacity In order

to collect the amount of data required management would have to upgrade the system and

develop some kind of data collection scheme coupled with a data compression technique The

collection scheme must allow for characteristics of demand such as trends and seasonality to

be identified and the compression technique must allow for sales data for over 30000 products

to be stored for each store

Because many of the products sold by Canadian Tire have nonstationary demand as a

result of seasonality andor trend the inventory tool must be able to forecast demand for the

coming period Products such as sporting goods and outdoor equipment exhibit patterns of

demand that are very seasonal For example sales of snow shovels increase through the fall

peak in the winter and drop to zero by late spring At the same time some products might

become either more or less fashionable over time resulting in changes in mean demand

Setting inventory policies for any of these products based on observed demand would result in

insufficient stock if demand increased during the period or in excess stock if demand decreased

Therefore the tool must be able to forecast the demand for the period for which the policy is

intended

Toward the end of the project management began discussing the integration of the

interactive inventory tool with the existing system to form a new inventory control system The

new system would utilize a modified point of sales system to collect sales data continually

Based on the latest available data the system would forecast demand for the coming period and

calculate the optimal policies for every product at every store The system would then provide

all relevant information on-line for the inventory manager to review The manager could weigh

14

other information such as marketing considerations and examine alternate policies to determine

the effect on cost Finally the preferred policy would be set in place with the touch of a key

and information on the next product or store would be presented for review

C The Project

Development of the inventory control system envisioned by management requires the

completion of five tasks 1) the design of a data collection system that generates and stores the

necessary sales data 2) the development of a forecasting method that provides accurate

estimates of demand distributions 3) the formulation of a model that calculates the cost of

inventory and the customer service level for an inventory policy 4) the development of an

algorithm that finds the optimal policy for a product at a store and 5) the creation of an

interface that allows interactive and on-line consultation Ultimately the above components

would have to be integrated and incorporated in a manner that would ensure the compatibility

and reliability of the system

The scope of the research reported in this thesis completes the first phase of the project

specifically we formulate an inventory model develop an algorithm and create a prototype of

the interface Because of the changing environment at Canadian Tire Pacific Associates minor

modifications to both the model and the algorithm developed herein might be necessary to

reflect future changes in operational procedures Upon completion of phase one the interface

will be interactive allowing managers to select the store product and review scenario and then

to compare optimal policies with existing policies or with alternate policies Although the

prototype will be delivered to Canadian Tire Pacific Associates for ih-house testing at the end

of this phase the interface will require modification in subsequent phases as feedback from

management is received and as the system is brought on-line

15

In order to assess more fully the potential savings of implementing the inventory control

system this thesis includes a comparison of optimal policies9 and existing policies The

expected annual cost of inventory for an entire product category is calculated for both the

optimal policies and the existing policies Based on an analysis of these costs the potential

savings of implementing the optimal policies are extrapolated Further the thesis examines the

sensitivity of the model to variations in the estimates of demand and costs

Originally results from all 21 stores were to be included in the above analysis

However just prior to the analysis management changed the review procedure at several

stores the result of which being that three picks per week are now performed at those stores

Because one of the main assumptions of the model specifically that the length of the lead time

does not exceed the duration of the review period is violated under this new review procedure

those stores cannot be included in the analysis Therefore the analysis is based solely on the

results from store six the large store located on Kingsway Avenue10

For the purpose of the analysis sales data for product category 20 automotive and

electrical accessories were collected from all 21 stores This product category was selected

because the distribution of demand for products within it is non-seasonal and because few of

the products are ever placed on sale11 The lack of trend or seasonality eliminates the need

for forecasting demand the optimal policies can be determined based directly on observed

demand

9 That is those policies obtained from the model

1 0 Store six is one of the only stores whose review procedure is not expected to change from two picks per week

1 1 Products that were on sale at any time during the period were not included in the data set

16

D The Demand Data

In total sales data for 520 products of which 458 are active products were collected

over a 307-day period from October 10 1992 to August 18 199312 The data was supplied

by Canadian Tire Pacific Associates in the form of a 10 MB text file with one line per day per

product recording sales from the stores13 Each line had the following format

Columns 1-6 part number Columns 7-9 Julian date14

Columns 10-18 cost of the item (to three decimal places) Columns 19-124 sales for stores 1 through 21 (5 columns per store)

In order to facilitate the analysis the file was sorted into 458 sales files with each of

these files containing the history of daily sales for one of the products Two changes were

made to the format of these files to improve readability 1) a column was added between the

product number and the Julian date and 2) all leading zeros were dropped Appendix A

contains a partial printout of the sales file for a typical product Finally a distribution file

summarizing the daily sales was created from each of the sales files Each distribution file

contains the products number the products cost and the number of days during which various

quantities of the product were sold at each store

Prior to the analysis the distribution files for a large number of products were

examined and based on the shapes of the histograms the products were divided into three

groups The first group low-demand comprises products that have demand on only a few

days with the daily demand seldom exceeding three items The second group medium-

1 2 Of the active products only 420 are sold at store 6

1 3 Strictly speaking a line was written to the file only on days on which at least one store registered a sale of the product

1 4 According to the Julian calendar days are numbered sequentially from 1 to 365 (366 during leap-years) beginning with the first of January

17

demand comprises products that have demand on approximately half of the days with the daily

demand seldom exceeding more than six items Finally the last group high-demand

comprises products that have demand on most days occasionally upwards of twenty items and

whose demand histograms resemble Poisson distributions The majority of the products

examined fell into the low and medium-demand groups only a few high-demand products

were observed Appendix B contains a printout of the distribution file for a typical product

in each group

For most products no theoretical distribution appeared to model the demand well

Demand for low and medium-demand products followed a non-standard distribution with

a mass at zero and a relatively small tail Although some theoretical distributions such as the

mixed-Poisson can be made to resemble this it was unclear how well they would perform

Further the effects of sampling errors from such non-standard distributions were unknown

On the other hand the standard Poisson distribution did appear to model high-demand

products well however only a few products fell into that group Therefore in the absence of

a good theoretical candidate the empirical demand function was chosen for the analysis The

empirical function was readily available from the distribution files and offered the best

estimate of the true underlying function

E The Cost Data

Along with the sales data management supplied estimates relating to the costs of holding

and ordering inventory Management estimates that the cost of holding an item in inventory

for one year is 30 of the items cost Incorporated in this holding rate are a number of costs

such as the opportunity cost of money invested and the costs incurred in running a store

Because it is the annual cost of inventory that is of interest either the compounded daily

18

holding rate or the simple daily holding rate can be used to determine the costs incurred during

a review period however the method of calculating and extrapolating the holding costs must

be consistent For the sake of simplicity a simple daily holding rate is used

In order to estimate the cost of ordering inventory management analyzed the various

tasks that result from an order being placed Using the average time required to fill and then

reshelve an order management obtained an estimate of the direct labour costs involved To

this management added an estimate of the cost of transporting an order from the warehouse to

the store Finally the estimate was adjusted slightly upward to account for the cost of breakage

and errors Based on the analysis management estimates the cost of placing an order for a

product to be $0085 regardless of the size of the order

19

III M O D E L F O R M U L A T I O N

A Terminology

In the previous chapter several terms were introduced to describe the inventory system

at Canadian Tire Pacific Associates These terms though perfectly adequate are not widely

used in the academic literature to describe inventory management problems Therefore in

order to avoid confusion and be consistent with the literature the following terms are used

throughout this thesis

The length of time between successive moments at which the

inventory position is reviewed15

The time at which a reorder decision is made

The length of time between the moment at which a decision to

place an order is made and the moment at which the order is

physically on the shelf

An operating policy in a periodic review inventory model in

which (i) the inventory position is reviewed every T days (ii) a

replenishment order is placed at a review only when the inventory

level x is at or below the reorder level s (gt 0) and (iii) a

replenishment order is placed for S-x units (S gt s)1 6

Lost Sales - The situation in which any demand during a stockout is lost

Review Period -

Decision Epoch

Lead Time -

(sS) Policy

1 5 With respect to the terminology used at Canadian Tire Pacific Associates a review period equals the number of days between flushes

1 6 With respect to the terminology used at Canadian Tire Pacific Associates S represents the target level and s represents the difference between the target level and the minimum fill

20

B Assumptions

In to order make the model of the inventory system mathematically manageable several

simplifying assumptions are made However we believe that the following assumptions do not

significantly reduce the accuracy of the model

1) Demand can be approximated very closely by sales during the period in which the data

is collected Generally inventory levels at the stores are very high and the number of

lost sales is assumed to be negligible In fact the inventory levels for a number of

products were tracked over time during the pilot study and not a single incidence of

stockout was observed Because all products appear to have similar excess inventory

it seems reasonable to assume that virtually all demand resulted in sales

2) The demands in successive periods are independent identically distributed random

variables Preliminary tests (Fishers exact tests) suggested no dependence between

sales on consecutive days and plots of sales over time indicated no seasonal pattern

3) Demand is independent across products That is we assume that sales of one product

do not affect nor can they be predicted from sales of another product

4) Unmet demand results in lost sales with no penalty other than the loss of the sale

5) The duration of the review period for a store is an integer value and is fixed over a

planning period For stores with two picks per week we assume that the inventory

is reviewed every four days rather than on a four day - three day cycle

6) The decision epoch is the start of a review period Review periods begin at midnight

at which time the inventory is reviewed and orders are placed instantaneously

7) The duration of the lead time for a store is deterministic and integer-valued In reality

the lead time for some stores was stochastic but management felt the variation could be

controlled and virtually eliminated

21

8) The lead time does not exceed the review period In other words items ordered at a

decision epoch arrive and are shelved before the next decision epoch In practice this

did not always occur

9) Orders arrive at the end of the lead time and are on the shelf to start the next day

10) Holding costs are incurred daily for all inventory held at the start of the day

11) The warehouse has an infinite capacity that is all orders to the warehouse are filled

Because of the high inventory levels in the warehouse stockouts in the warehouse are

very unlikely

C A Markov Chain Model

Consider the situation in which an (sS) policy is applied to an inventory system based

on the above assumptions Inventory is reviewed at the decision epoch for each of the periods

labelled t = 0 12 and replenishment occurs at the end of the lead time provided an order

was placed Demand during day d in period t is a random variable whose probability mass

function is independent of the day and period The function is given by

f^ik) = Pr ltd = k = ak for k = 0 1 2

where ak gt 0 and E ak = 1 Let T denote the duration of a review period and L denote the

length of the lead time Next define D T ( as the total demand during period t D L as the total

demand during the lead time in period t and D T L as the difference between D T and DLt

Assuming that successive demands pound f pound 2 are independent the probability mass function of

1 7 The aggregated demand variables are given by

^ = XXd and DT_Ut=J d= d= d=L+l

22

each aggregated demand variable can be obtained by convolving the probability mass function

of t d l s Finally let X denote the inventory on hand at the beginning of period t just prior to

the review

The process X is a discrete time stochastic process with the finite discrete state space

0 1 S A typical sample path of the process with a review period of four days and a

lead time of two days is depicted in Figure 1 As period t begins X units are on hand The

inventory is reviewed and because X is less than s an order is placed for (S-X) units Over

the next two days a demand of D L r occurs and the inventory drops accordingly With the

arrival of the order the inventory level increases by (S-X) at the start of the third day of the

period Over the remaining two days a demand of D T L is observed bring the inventory level

down to X + 1 Note that in this example an order would not be placed in period t+l because

X r + i exceeds s

The stock levels at the beginning of two consecutive review periods are related by the

balance equation

_K- DLy+ (s - v - DT-LX ixtlts ( 1 )

t + l ~ (Xt - DTtY ifXtgts

where the positive operator is interpreted as

= max a 0

1 8 For example the probability mass function of D T bdquo denoted fD T ( is the T-th fold convolution of f t Mathematically the function can be obtained using the recursive relation

and the identity f1^ = f5(

23

-2 I 8

gt o

r

CO

o a

CM

X

00

24

Because X + 1 depends only on Xt the process X satisfies the Markov property

PrXt+l=jX0 = i0Xt_l=it_vXt = i) = PrXt+1=jXt = i

and therefore is a discrete time Markov chain Further because the probability mass function

of the random variable pound is independent of t and thus so too are the probability mass functions

of the aggregated demand the Markov chain is stationary19

D The Transition Probability Matrix

The stationary transition probability p(gt is defined as the probability of X + 1 being in state

j given that X is in state i independent of the period t In the inventory model p-- represents

the probability of starting the next review period with j items on the shelf given there are i units

on hand at the start of the current period By convention the transition probabilities are

arranged in an (S + l)x(S + 1) matrix P such that (P)y = p- Thus the (i + l)st row of P contains

the probability distribution for the stock on hand at the beginning of the next period given that

there are i units on hand at the start of the current period The matrix P is known as the

transition probability matrix Note that since we are concerned with a steady state analysis we

need not consider the possibility of initial stock levels greater than S once the inventory level

is below S it can never again exceed S

The transition probability ptj is calculated by expanding (1) to include all possible

instances of demand that allow for a change in inventory from i units at the beginning of a

period to j units at the beginning of the next period For example suppose the inventory policy

(510) is in place 2 items are on hand at the start of the current review period and 7 items are

1 9 Henceforth the subscripts t and d are omitted from the daily and aggregate demand variables for the sake of brevity

25

on hand at the start of the next period Then only three scenarios are possible

1) No items are sold during the lead time (meaning that the stock level increases to 10

after replenishment) and 3 items are sold during the remainder of the period

2) One item is sold during the lead time and 2 items are sold during the remainder of

the period

3) Two items are sold during the lead time (though demand could exceed 2) and 1 item

is sold during the remainder of the period

The transition probability p 2 7 is the sum of the probabilities of the various scenarios and is

computed by

p2J = PrDL=0PrDT_L = 3] + PrDL = l] Pr[DT_L=2 + PrDL2] PrDT_L = l]

In general the transition probabilities for the inventory model are calculated as follows

Case 1 The lead time is less than the review period

For 0 lt lt s

Pij

i-l pound [PrDL=kPrDT_LgtS-k] + PrDLgtiPrDT_LgtS-i if j = 0

i - l

^[PriD^^PrlD^S-j-kn+PHD^PriD^S-j-i if 0 lt j ltL S-i (2) k=0 s-j

E [PrDL=kPrDT_L = S-j-k] if S-i ltj^S k=0

26

For s lt i lt S

PrDTzi]

0 p = PrDT=i-j)

ifj = 0 ifOltjzi ifiltjltS

(3)

Case 2 The lead time equals the review period

For 0 lt i lt s

Pa

0 PrDTzi) PrDT = S-j

ifOltLJlt S-i ifj = S-i if S-i ltj plusmn S

(4)

For s lt i lt S use (3)

E The Steady State Probabilities

Provided 1) there is a positive probability of non-zero demand during a review period

and 2) there is a positive probability of zero demand during a review period20 then the

Markov chain described by (1) has a limiting probability distribution ir = (ir0 TTu irs)

where 0 lt IT lt 1 for j = 0 1 S and E 7ry = 1 Further this limiting distribution is

independent of the initial state In other words the probability of finding the Markov chain in

a particular state in the long-run converges to a nonnegative number no matter in which state

the chain began at time 0 The limiting distribution is the unique nonnegative solution of the

equations

2 0 That is 0 lt PrD x = 0 lt 1 Given the nature of the problem this assumption is not very restrictive

27

^ = E ^ - far j = 0 l S (5)

s and pound nk = 1 (6)

For the proof that the limiting distribution is independent of the initial state and is obtained from

(5) and (6) the reader is directed to Appendix C

In terms of the inventory problem 7ry is the steady state probability of having j units of

inventory on hand at the decision epoch However some subsequent calculations are made

easier if the steady state probability of having j units of inventory on the shelf just after

replenishment is known Instead of reformulating the Markov chain these probabilities can be

obtained by shifting the limiting distribution TT Much in the same way that the transition

probabilities are calculated the shifted limiting distribution t] is obtained by accounting for

all possible instances of demand that allow for the shifted inventory position For example

suppose that the policy (510) has the limiting distribution TT In order to have say 8 units on

hand immediately after replenishment one of the following must occur

1) Ten items are on hand at the decision epoch and 2 items are sold during the lead time

2) Nine items are on hand at the decision epoch and 1 item is sold during the lead time

3) Eight items are on hand at the decision epoch and nothing is sold during the lead time

4) Either 2 3 4 or 5 items are on hand at the decision epoch and 2 items are sold during

the lead time

The steady state probability of having 8 units on hand just after replenishment is calculated as

n 8 = n10PrDL = 2] +iz9PrDL = l] +n8PrDL = 0 +[iz5 + iz4 + iz3]PrDL = 2 +n2PrDLgt2

28

In general the shifted steady state probabilities are given by

s pound nPrDLgti ifj = 0

i=s+l S

pound PrDL = i-j] ifl plusmnjltS-s i=max(s+l)

S = pound 7tjirZ)L = i-+7i 5 FrD i ^5 ifj = 5-5

i=max(s+l)

^ 7 1 ^ ^ = 1-7+ pound TtiFrZgti=5-y+iis_PrD1^S-7 ifS-sltjltS i=max(gt+l) i=S-j+l

(7)

F The Cost of Inventory

For this problem the cost of inventory has two components (i) a fixed ordering cost

cf incurred whenever an order is placed and (ii) a holding cost cv incurred daily for each item

held in inventory The expected cost of inventory for a review period can be calculated by

summing the costs associated with each inventory position under steady state conditions

Because a simple holding rate is used this cost is simply adjusted by number of review periods

in a year to obtain the expected annual cost of inventory

The ordering cost for a review period is relatively straight forward to calculate either

an order is placed in which case the order cost is incurred or no order is placed in which case

no cost is incurred According to the rules of the inventory policy orders are placed only if

the stock on hand at the beginning of a review period is less than or equal to s Further the

probability that the review period begins with the inventory in one of these positions is equal

to the sum of the first (s + 1) steady state probabilities Therefore the expected ordering cost

29

for a review period is given by

E[OC] = cfY n y=o

Unlike the simple structure of the ordering cost the holding cost is incurred daily for

each item held in inventory Therefore the daily holding costs associated with all possible

inventory positions must be weighted and summed to obtain the holding cost for the review

period In order to make the calculations less complicated a shifted review period beginning

immediately after replenishment and ending just prior to the next replenishment is considered

Define e(0 to be the expected number of items on hand to start day i of the shifted

review period and let g(i f) denote the expected number of items on hand to start day i given

that there were j items immediately following replenishment21 The function g(z j) is the sum

of the probabilities of possible demand weighted by the resulting stock levels that is

g(ij) = j^kPriD^j-k)

To obtain the expected number of items on hand to start day i independent of j the above

function is summed over all possible values of j and weighted by the corresponding steady state

probabilities yielding

s j ed) = pound r 5 gt P r Z ) = - pound

7=0 k=

2 1 In other words g(if) is the expected number of items on hand to start day i given e(0)=v

30

The expected holding cost for a review period is then given by

T

E[HC] = $gt v e(0 i=i

i=l =0 k=l

where c v equals the simple daily holding rate multiplied by the cost of the item

Although differently defined review periods are used to calculate the above expected

costs the costs can still be summed to obtain the expected cost of inventory for a review

period The expected costs are calculated based on steady state conditions and therefore

represent the costs expected to be incurred during any T consecutive days Summing the above

two expected costs and multiplying the sum by the number of review periods in a year yields

the expected annual cost of inventory

E[AC] = mdash[E[OC]+E[HC]]

T S j 1 ( g ) 365

j=0 i=l j=0 fc=l

G The Customer Service Level

There are many ways to define customer service and the most appropriate definition

depends upon the nature of the business and the perspective of management Two commonly

used definitions are 1) the proportion of periods in which all demand is met and 2) the

proportion of demand satisfied immediately from inventory Although the former definition is

often used because of its simplicity the latter definition is closer to what one commonly means

by service (Lee and Nahmias 1993) A third definition also relevant in retail environments

31

is the probability that lost sales do not exceed some critical number Other definitions such

as probability that backorders are filled within a certain amount of time are more applicable

to industrial and manufacturing settings where backlogging demand is common

Given the retail environment with the possibility of lost sales that exists at Canadian Tire

Pacific Associates management is concerned primarily with the fraction of sales that are lost

For example management views the situation in which demand exceeds stock by one unit every

period more favourably than that in which the same demand exceeds stock by three units every

other period For this reason the proportion of demand satisfied immediately from inventory

is the most appropriate measure of customer service

The customer service level for a single review period CSLRP is given by

r bdquo T _ Demand satisfied during RP Total demand during RP

Because we are interested in the long-run behaviour of the system a measure of the steady state

customer service C S L must be obtained As shown in Appendix D the correct measure is

poundpoundpound _ E[Demand satisfied during RP] E[Total demand during RP]

which can be rewritten as

pound pound pound _ j _ E[Demand not satisfied during RP] ^ E[Total demand during RP]

Because of the steady state conditions the above customer service level can be estimated

from any period of T consecutive days As was done for one of the cost calculations the T-day

period beginning just after replenishment is considered Suppose that there are j units of

inventory on hand immediately following replenishment Then the conditional customer service

32

level is

CSL j = 1 E[(DT-j)l

E[DT] (10)

The denominator in the second term is merely the expect demand during the period

E[DT] = 7gt (11)

where p is the mean of the random daily demand variable pound The numerator which is the

conditional expected demand that is not satisfied during the T-day period expands to

7-1

E[(DT-j)+] = T u - j + YU-k)PrDT = k] k=0

(12)

The reader is directed to Appendix E for a mathematical justification of (12) In order to obtain

the unconditional expected demand not satisfied (12) is applied to all possible values of j and

a weighted sum of the terms is calculated The result is given by

E[Demand not satisfied] = Y j =0

y-i Ti - j + 1pound(j-k)PrDT = k

S j-1

= - E[I] + Y0-k)Pr[DT=k) =0 it=0

(13)

where E[I] denotes the steady state inventory level Finally the customer service level is

obtained by the substituting (11) and (13) into (9) yielding

CSL = 1

s j-l

Ti - E[I] + Jgty Y(J-k)PrDT=k]

m n J2VjTU-k)PrDT=k Eil J=Q fc=o

(14)

33

H A Methodology for Evaluating an (sS) Policy

In this section a brief methodology summarizing the main results of the model is

presented The methodology can be applied to evaluate a given (sS) policy in terms of its cost

and its resulting customer service level Given the probability mass function of the daily

demand f the duration of the review period T the length of the lead time L the fixed cost

of placing an order cf and the daily unit cost of holding inventory cv

1) Calculate the aggregated demand probability mass functions fDT fDL and f D x L as the T-th

fold L-th fold and (T-L)th fold convolutions of f respectively

2) Generate the transition probability matrix using either equation (2) or (4) and equation (3)

3) Calculate the limiting distribution TT from (5) and (6)

4) Obtain the L-day shifted limiting distribution rj using (7)

5) Calculate the expected annual cost of inventory using (8)

6) Determine the customer service level using (14)

34

IV A N A L G O R I T H M F O R OBTAINING T H E O P T I M A L (sS) P O L I C Y

A Introduction

The methodology developed in the previous chapter can be used to calculate the cost and

service level associated with a given (sS) policy In this chapter an algorithm that finds the

feasible policy yielding the lowest cost is developed A policy is feasible if it satisfies both the

basic condition 0 lt s lt S and the imposed constraint on the customer service level2 2 In

order to find the optimal policy the algorithm must combine the methodology for evaluating

a single policy with a search technique

Search techniques are evaluated in terms of two somewhat antithetic criteria On the one

hand the search should locate the optimal solution or at least a near optimal solution on

the other hand the search should take as little time as possible Many search techniques

guarantee optimality provided the objective function and constraints exhibit certain properties

such as convexity However even if such a search technique also guarantees its termination

in polynomial time the search might take too long to be practical Alternatively some search

techniques provide reasonable though sub-optimal solutions very quickly These techniques

often called heuristics must be evaluated in terms of both their speed and the accuracy of their

solutions The choice of a heuristic over a search that guarantees optimality depends upon the

nature of the feasible space the need for accuracy and the cost of time

Given the complexity of the cost and customer service level functions it is very difficult

to show theoretically for what if any combinations of demand and costs the feasible space is

convex However a plot of the objective function for a single product may be obtained by

2 2 Management specified that each policy must provide a customer service level of at least 975 Therefore the algorithm must incorporate the constraint CSL gt 0975

35

evaluating the costs for a number of policies within a grid When the service level constraint

is included in these plots maps of the feasible space are obtained Examination of the

maps for several products suggests that the space is convex over most regions however

maps for some products reveal some regions of non-convexity Unfortunately there is no

way of knowing the degree of non-convexity for other products other stores or different costs

estimates without a more rigorous survey of the feasible space Therefore if one of the

techniques that guarantee optimality under the condition of convexity is employed the search

would have to be evaluated as a heuristic Instead as a first step to finding a practical search

technique a crude search that guarantees optimality under virtually all conditions is

investigated

B A Grid Search

The simplest search technique that guarantees optimality merely evaluates all policies in

the grid defined by 0 lt s lt S and 0 lt Sm i n lt S lt S m a x where Sm i n and S m a x are

some predefined values That is the grid search is a brute force method that examines every

policy within the grid 2 3 Although the number of policies that must be examined is large the

grid search is guaranteed to find the optimal policy provided that S o p t is contained in the grid

that is provided Sm i n lt S o p t lt Sm a x The grid search algorithm proceeds by generating the

transition probability matrix for the policy (0Smin) and then evaluating the service level

associated with that policy If the policy is feasible24 the cost associated with the policy is

2 3 The total number of policies evaluated is the sum of the first Sm a x integers less the sum of the first Smin-1 integers Numerically [(l2)(Sm a x)(Sm a x+l) - (l2)(Smin-l)(Smin)] policies are examined

2 4 That is if the service level is adequate

36

calculated and compared to the current lowest cost The procedure continues until all points

in the grid have been examined and the optimal policy has been found A flow chart of the grid

search algorithm is shown in Figure 2

In order to assess the grid search the algorithm was coded and compiled into an

executable program The program was run for two products (a medium-demand product and

a high-demand product) and with four values of Sm a x In each run Sm i n was taken to be zero

The following times were recorded for each trial

T P M - the amount of time spent generating the transition probability matrix

GAUSS - the amount of time spent solving for the steady state probabilities

COSTS - the amount of time spent calculating the expected annual costs

SERVICE - the amount of time spent determining the customer service levels

T O T A L - the total execution time of the program

The columns entitled Grid Search in Tables 2 and 3 summarize the results of the trials All

times are recorded in seconds as required by a 486DX33 MHz personal computer with an

empty buffer memory

Three results are evident from the tables First the vast majority of the execution time

is spent generating transition probability matrices and then solving for the steady state

probabilities with the time being split almost equally between the two tasks Second the total

execution time of the algorithm increases exponentially as Sm a x increases For product 206917

doubling S^ from 30 to 60 led to an increase in the execution time of more than 2100 (from

172 seconds to 3661 seconds) Third the grid search algorithm as it stands does not appear

to be adequate for the problem at hand Given the large number of products that must be

evaluated the algorithm is much too slow to be implementable

37

Calculate the demand probability functions

Initialize s = 0 S = Smin] cost = 10A5

s = s+1

Calculate the probability transition matrix for (sS)

Evaluate policy (sS) Is service level ^ 0975

No

No Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Is s = S-l

Yes

Is S = Smax No

Yes

Print results (sS) cost

s = 0 S = S+1

Figure 2 The flow chart of the grid search algorithm

38

Table 2 Execution times of the algorithms for product 200001 a 30-amp inline fuse

(in seconds)

-max Subroutine Grid Search Grid Search

Updating T P M Heuristic

10 T P M 006 010 001

GAUSS 010 006 001

COST 005 005 001

SERVICE 001 001 001

T O T A L 050 035 027

20 T P M 126 021 010

GAUSS 119 160 023

COST 027 026 001

SERVICE 017 016 001

T O T A L 318 236 055

30 T P M 703 096 010

GAUSS 757 801 093

COST 079 085 001

SERVICE 047 042 001

T O T A L 1664 1137 126

40 T P M 2257 163 015

GAUSS 2958 3003 206

COST 313 329 001

SERVICE 092 083 011

T O T A L 5871 3851 285

39

Table 3 Execution times of the algorithms for product 206917 a 6 solder connector

(in seconds)

bull-max Subroutine Grid Search Grid Search

Updating T P M Heuristic

30 T P M 717 067 006

GAUSS 762 880 093

COST 098 087 010

SERVICE 016 021 006

T O T A L 1720 1176 220

40 T P M 2390 157 022

GAUSS 2861 2911 209

COST 219 227 006

SERVICE 080 088 011

T O T A L 5827 3790 362

50 T P M 6691 413 010

GAUSS 7994 8201 472

COST 568 577 016

SERVICE 212 197 005

T O T A L 15840 10041 654

60 T P M 15509 894 040

GAUSS 18607 18951 876

COST 1367 1353 038

SERVICE 350 359 022

T O T A L 36614 22794 1120

40

C A Technique for Updating the Transition Probability Matrix

As was noted in the previous section approximately half of the execution time of the grid

search algorithm is spent generating transition probability matrices Furthermore the amount

of time increases exponentially as Sm a x increases This increase is exponential for two reasons

first the number of transition matrices that must be calculated is a function of S^ squared and

second the average size of the matrices and hence the average number of calculations per

matrix increases with Sm a x For example when S m a x equals 30 the transition probability matrix

must be generated 465 times with the largest matrix having dimensions (31x31) When Sm a x

equals 40 the matrix must be generated 820 times with 355 of the matrices having dimensions

larger than (31x31) The performance of the algorithm could be improved considerably if

either the number of matrices that must be generated or the number of calculations per matrix

is reduced This section introduces a technique that accomplishes the latter by updating the

transition probability matrix for a new policy instead of generating it anew for each policy

A new policy (s+mS)

Suppose the policy (sS) has associated with it the (S + l)x(S + l) transition probability

matrix P and the policy (s+mS) where m is any integer satisfying 0 lt s+m lt S has

associated with it the (S + l)x(S + 1) transition probability matrix J Then T can be obtained by

recalculating only m rows of P Recall that the equations for generating a transition

probability matrix are divided into two sets based upon the row being calculated equations (2)

or (4) for rows 0 through s and (3) for rows (s + 1) through S The equations themselves are

independent of s only the limits defining the cases contain s Therefore only the m rows

that are affected by the change in the limits need to be recalculated The following updating

technique can be used to obtain 7

41

1) If m gt 0 rows i = (s + 1) (s+2) (s+m) are recalculated using (2) or (4)

2) If m lt 0 rows i = (s-1) (s-2) (s-m) are recalculated using (3)

3) All other rows of T are identical to those of P

The reader is directed to Appendix F for a mathematical justification of the technique

A new policy (sS+m)

Suppose the policy (sS) has associated with it the (S + l)x(S + l) transition probability

matrix P and the policy (sS+m) where m is any positive integer has associated with it the

(S+m+l)x(S+m+l) transition probability matrix P Then T can be obtained by modifying

each row of P and adding m additional rows and columns The updating technique is not as

simple as that for a new policy (s+mS) for two reasons the dimensions of P and T differ and

the equations for generating a transition probability matrix contain S Nevertheless an updating

technique that requires much less computing time than generating the matrix anew can be

developed The following technique can be used to obtain P

1) For rows i from 0 through s

For columns j from 0 through m recalculate using (2) or (4)

For columns j from (m+1) through (S+m) enter the (j-m)th column entry of P

2) For rows i from (s + 1) through S

For columns j from 0 through S enter the corresponding value from P

For columns j from (S + 1) through (S+m) enter 0

3) For rows i from (S + 1) through (S+m) recalculate using (3)

The above technique is justified mathematically in Appendix G

42

The modified algorithm

The updating technique can be easily incorporated into the grid search algorithm with

only a slight change in the manner in which s is incremented Figure 3 shows the flow chart

of the new algorithm As before the algorithm was coded and compiled into an executable

program and the various execution times were recorded The results are listed in Tables 2 and

3 under the columns Grid Search Updating TPM

Updating the transition probability matrix does improve the performance of the

algorithm The time required to generate the transition probability matrices is reduced by

approximately 90 This translates to a reduction in total execution time of approximately

30 Not surprisingly the reductions are greater for large values of Sm a x However despite

this seemly significant improvement a decrease of only approximately 08 seconds is obtained

when S^ equals 20 Further when S ^ equals 30 the algorithm requires over 11 seconds

which is still much too high

The execution time is now dominated by the time required to solve for the steady state

probabilities The steady state probabilities could be solved by using approximation techniques

which are faster than solving the systems exactly however because the probabilities are often

very small numbers the approximation techniques would be unable to provide the required

accuracy The only remaining way to decrease the execution time is to reduce the number of

policies that are evaluated during the search This can be done by either reducing the size of

the grid being searched or reducing the number of policies being evaluated within the grid The

following section introduces a method of reducing the size of the grid by obtaining a lower

bound for S namely Sm i n

43

s = s+k

Calculate the demand probability functions

Initialize s = 0 S = Smin| cost = 105

k= 1

Update the probability transition matrix for (sS)

Evaluate policy (sS) Is service level ^ 0975

No

No

iNo

Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Ifk= 1 iss = S-l If k =-1 is s = 0

Yes

Is S = Smax No

Yes

Print results (sS) cost

If k=l s=s+l S = S+1

Figure 3 The flow chart of the grid search algorithm utilizing the updating technique for the transition probability matrix

44

D A Lower Bound on S

As mentioned earlier the size of the grid is specified by the number of policies within

the domain defined by 0 lt s lt S and 0 lt Smin lt S lt Smax Thus the size of the grid

and hence the total amount of computation time can be reduced by obtaining tight bounds on

S Unfortunately because of the extreme non-linearity of the objection function and the service

level constraint obtaining a mathematical upper bound on S is very difficult and beyond the

scope of this work This section introduces a method for obtaining a lower bound on S25

For a given value of S an upper bound on the customer service level can be found as

follows Suppose that there are S units of inventory on the shelf at the beginning of every

review period regardless of demand No (sS) policy can provide a higher level of customer

service because there exists a positive probability of beginning a review period with less than

S units on the shelf Thus the customer service level associated with starting each review

period with S units on the shelf is an upper bound on the customer service level for any (sS)

policy From (10) this level is

E[(DT-SY] CSL 5 = 1 - T mdash mdash - ( 1 5 gt

E[DT]

Replacing j with S in (12) and then substituting (12) and (11) into (15) yields

CSL 15 = mdash- - J2(S-k)PrDT = k] (16)

To obtain Smin use (16) to find the largest value of S with a customer service level less than

0975

There are two criteria for a good lower bound on S the bound must be no larger than

This method was proposed by Kaan Katircioglu

45

S o p t and it must be very close to So p t These criteria ensure that the optimal policy will be

contained in the search grid and that the size of the grid is reduced by as much as possible

Based on these criteria the above method of obtaining a lower bound is assessed A value for

S m i n is obtained for each of the products in product category 20 and compared with the

corresponding value of S o p t 2 6 The results are illustrated in Figure 4 which shows the number

of occurrences of various Sm i n and S o p t combinations Note that a point lying on the 45deg line

represents a perfect bound while a point lying close to the S o p t axis represents a very loose

bound The absence of any points above the 45deg line indicates that the first criterion is satisfied

in each instance the method did not generate a value of Sm i n exceeding the corresponding value

of So p t Moreover the bounds are reasonably tight for 85 of the products analyzed that

is for those products with S o p t equal to 10 or less27 However as S o p t increases the bounds

become steadily worse Because values of S o p t are likely to be larger for these products at the

ten Jumbo and Extra Jumbo stores and for other products with higher demand the lower bounds

are not expected to be very tight in general

Although obtaining the bound Sm i n does reduce the size of the grid the corresponding

reduction in the execution time of the algorithm is miriimal For example if S^ equals 60 and

a value of 30 is obtained for Sm i n the program would still require approximately 215 seconds

to search the grid (compared to the 228 seconds required without a lower bound on S) Because

the execution time increases exponentially with So p t a tight upper bound on S would likely have

a more significant impact on reducing the execution time of the algorithm Further research

on this issue is required

2 6 The optimal policy for each product was obtained using a grid search with an extremely large upper bound

2 7 On average Sm i n is approximately half of S o p t for these products

46

47

The only remaining approach to improving the algorithm is to modify the search so that

fewer policies within the grid are evaluated Toward this goal assumptions can be made with

respect to the feasible space and a heuristic search can be developed based on these

assumptions

E A Heuristic Search

As mentioned previously in this chapter a map of the feasible space for each of

several products was obtained Upon closer examination the maps illustrate two patterns

without exception for a given value of S both the cost and the customer service level increase

as s increases The following intuitive argument is offered as a possible explanation for the

observed behaviour When S is held constant increasing s results in more instances in which

an order is placed upon review Hence the average stock level would increase which in turn

would lead to an increase in the cost and the customer service level28 With this in mind a

heuristic search based on the following assumptions is developed for a given value of S both

the cost and the customer service level are monotonically increasing functions of s

In general the heuristic proceeds as follows Having just incremented S by one the

customer service level of the policy (sS) is evaluated If the policy is feasible s is decreased

until the lowest value of s producing a feasible policy is found If the policy is not feasible

s is increased until either the cost exceeds the current lowest cost for a feasible policy a

feasible policy is obtained or s equals S-l In the event that a feasible policy is found the cost

2 8 In order to prove that the customer service level increases as s increases it would be sufficient to show that the probability of placing an order increases as s increases Such a result might be possible if assumptions regarding the demand function are made Further if both the probability of placing an order and the average stock level increase then both the ordering cost and the holding cost would increase No proofs are offered here

48

of the policy is calculated if the cost of the policy is less than the current lowest cost the new

policy becomes the current best Finally S is increased by one and the process repeats

provided S does not exceed some predefined value Sm a x The flow chart of the algorithm

incorporating the heuristic search is shown in Figure 5 Note that the algorithm begins by

attempting to improve on the lower bound Sm i n the policy (Sm i n-lSm i n) is evaluated and if it

is not feasible S m i n is increased by one The improvement continues until a feasible policy is

found

As previously mentioned a heuristic search must be evaluated in terms of both its speed

and the accuracy of its solutions To evaluate the heuristic on the basis of speed the same

approach used to evaluate the previous algorithms is employed The columns entitled

Heuristic in Tables 2 and 3 contain the various execution times of the heuristic The heuristic

is a vast improvement over the previous search techniques the reduction in execution time is

enormous and becomes more so as S m a x increases For a S m a x of 20 the reduction is

approximately 75 of the time required by the grid search algorithm29 and for a S m a x of 60

that reduction soars to over 95 More importantly the heuristic can search a grid with a

of 40 in under 4 seconds or a grid with a S m a x of 60 in under 12 seconds These low execution

times should be sufficient to allow for a large number of products to be analyzed in a

reasonable amount of time

In order to further assess the practicability of the heuristic the total time required by the

heuristic to find solutions for the entire product category was determined and compared to the

corresponding time for the grid search algorithm With a priori knowledge of the optimal

solutions conservative yet reasonable values of S m a x were selected and used by both of the

2 9 Hereafter grid search algorithm refers to the algorithm utilizing the grid search the updating technique and the lower bound Sm i n

49

Calculate the demand probability functions

Calculate Smin S = Smin s = S-l

S = S+l s = S-l t

Update the probability transition matrix for (sS)

No Is service level pound 0975 Yes

No Is service level ^ 0975

Yes

Is cost lt cost No

Yes

Is s lt S-l No

Yes

s = s+l

Update the probability transition matrix for (sS)

Is service level pound 0975

Yes

Is cost lt cost No

Yes

s = s S = S cost = cost

s = s-l

No

Update the probability transition matrix for (sS)

S = S+l

Is s 2 0 Yes

Update the probability transition matrix for (sS)

Is service level pound 0975

No

s = s+l

No

No

Is S = Smax

^Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Print results (sS) cost

Figure 5 The flow chart of the heuristic algorithm

50

algorithms30 The grid search algorithm required approximately 9 hours and 41 minutes to

find the solutions while the heuristic required only slightly more than 23 minutes Based on

speed the heuristic is preferable to the grid search algorithm

Finally to evaluate the heuristic based on the accuracy of its solutions its solutions were

compared to the previously obtained optimal policies Without exception the heuristic

algorithm found the optimal policy Although there is no guarantee that the heuristic will find

the optimal solution for different products or different demand functions the heuristic appears

to work extremely well Based on the fast execution time and the apparent accuracy of the

solutions the heuristic will be used in the development of the inventory tool

3 0 For the test the following values of Sm a x were used 20 (273 products) 30 (19) 50 (23) 100 (3) and 125 (2)

51

V T H E I N T E R F A C E

With the model formulated and the algorithm developed the only remaining task in the

development of the prototype is the creation the interface In fact two interfaces are created

one that displays results in terms of minimum fills and target levels and one that displays

results in terms of reorder points and target levels To be consistent with the previous chapters

the latter interface is described here31

In its present form the interface allows the manager to set review parameters to

compare optimal and current policies and to evaluate alternate policies From the main menu

the manager selects the product store duration of the review period length of the lead time

and value of Sm a x Once the parameters are set the system calculates the optimal policy and

then evaluates the current policy displaying results as they are obtained In particular the

reorder point target level expected annual cost and expected customer service level are

displayed for each policy If a current policy meets the desired customer service level the

expected annual savings of the optimal policy are displayed in terms of both dollars and

percentage of current cost otherwise the value of the current customer service level is

displayed in red to indicate that it is deficient Finally the manager can choose to evaluate

alternate policies or to return to the main menu

Although the prototype is far from the ultimate integrated inventory control system it

is still valuable to management As mentioned in Chapter II inventory policies at Canadian

Tire Pacific Associates are currently based on rules of thumb that incorporate marketing

3 1 Because significant changes to the interface are expected as feedback from management is received and as the system is brought on-line only a brief description of the present interface is given

52

strategies The prototype provides management with a means of determining the proportion of

the cost of inventory that is not attributable to meeting customer demand For example

consider a hypothetical consultation in which the manager chooses to analyze product 200001

a 30-amp inline fuse at store 6 The prototype displays to the manager the current policy of

(1214) which provides a customer service level of 1000 at an annual cost of $748 and the

optimal policy of (39) which provides a customer service level of 977 at an annual cost of

only $319 If for some reason the manager feels the optimal policy is undesirable he or she

can investigate the additional cost of holding excess inventory Suppose the manager chooses

to evaluate the policy (610) The prototype evaluates the policy and displays the results the

customer service level is 998 and the annual cost is $441 With the alternate policy

virtually all demand is satisfied and the cost of inventory is still reduced by over 40 Thus

the manager ascertains that approximately 40 of the current cost of inventory for this product

is directly attributable to marketing considerations Using the prototype in this way

management can begin to understand better the nature of its inventory costs A description and

graphical depiction of the above hypothetical consultation are presented in Appendix H

As for the technical details of the system the prototype can run on any computer using

the DOS operating system having approximately 420K of available conventional memory and

attached to a V G A or SVGA monitor Further the prototype can be used on a machine without

a math coprocessor however performance is slower The prototype consists of two modules

linked together into one executable program The interface module contains the various

interactive subroutines and the main module contains the methodology of the model and the

logic of the heuristic search Both modules are coded in Fortran and compiled using Microsoft

Fortran 50 which provides many extensions to standard Fortran The source codes for the

interface module and the main module are listed in Appendix I and Appendix J

53

respectively

In addition to the above technical specifications the program requires the presence of

certain files in certain directories The program uses the distribution files mentioned in

Chapter II to read the unit costs and the empirical demand functions Each product has a

unique distribution file the name of which is the six digit product number followed by the

extension dst The text file containing the existing policies at each store must be located in

the same directory as the distribution files Finally the file tmsrbfon which contains the

font used by the interface must be located in the directory from which the program is executed

In its future versions the system should be more automated and more user-friendly

Rather than having the manager choose an upper bound for the search the algorithm should

incorporate a method of determining Sm a x For this to happen either a theoretical upper bound

must be obtained or a heuristic for terminating the search must be developed Also the amount

of time spent waiting for calculations to be performed can be reduced by reading the optimal

and current policies and their associated costs and customer service levels from a file or

database This solution file could be created any time prior to the consultation by running

the system in a batch mode The interface should allow the manager to change cost estimates

and service level values view results graphically and print results and summary documents

Finally the system should provide an option for automatically incrementing the product andor

store during a consultation This option would enable a manager to review quickly the

inventory policy of a product at each store

54

VI R E S U L T S AND SENSITIVITY ANALYSIS

A Comparison of Current Policies to Optimal Policies

In order to estimate the potential savings of implementing optimal policies the solutions

obtained in the Chapter IV are analyzed in more detail For each product the current policy

is evaluated and then compared to the optimal policy with respect to cost and customer service

level Finally the potential savings of the optimal policies are extrapolated to all products and

all stores

Of the 420 products examined each of 37 products has an existing policy that fails to

provide the desired customer service level of 975 However most of these policies just

fail to do so only 4 policies provide a customer service level of less than 90 Table 4

provides a list of these 37 products each with its current customer service level its current

annual cost and the additional cost of its optimal policy3 2 Two comments regarding Table

4 are in order First the current customer service level for product 203312 an amber

driveway marker is listed at a misleading 303 Examination of the demand distribution

reveals that this product was sold on only 1 of the 307 days during which sales data were

collected and on that day 13 of the markers were sold Clearly this data point constitutes

either an error or an unusual event the current policy of (34) does not allow for more than 4

items to be sold per review period Second the additional cost of the optimal policy for each

of 5 products is negative In other words for each of these products the current policy which

fails to provide the desired customer service level costs more than the optimal policy which

3 2 This is the difference between the expected annual cost of the optimal policy and the annual cost of the current policy It is the minimum additional cost required to increase the customer service level to at least 975

55

Table 4 Products with insufficient existing policies

Product Current CSL ()

Current Annual Cost ($)

Additional Cost of the Optimal Policy ($)

206932 974 204 -051

204570 973 222 -025

203053 973 248 009

206550 972 118 042

206547 972 105 035

206565 972 146 056

203702 972 340 157

206876 972 155 061

205752 972 330 153

205922 972 108 036

206857 972 180 073

205393 972 190 -003

201513 971 345 015

203405 970 255 037

203069 968 295 -048

203071 967 321 -001

205635 967 390 029

203914 966 460 160

204601 962 252 061

203326 960 144 014

202003 958 876 437

206516 958 112 029

206517 958 283 120

206350 958 333 147

205932 958 180 066

203936 952 598 294

56

Product Current CSL ()

Current Annual Cost ($)

Additional Cost of the Optimal Policy ($)

206932 974 204 -051

203939 948 413 193

203977 945 1216 1242

206551 945 132 077

205389 934 598 047

203590 927 1223 607

206567 919 122 070

203591 905 1228 834

205390 887 291 074

203318 833 136 084

206393 776 319 289

203312 303 237 511

provides at least the desired customer service level For the 36 products the total additional

cost of the optimal policies is $542033

Of the remaining 383 products most have an existing policy that provides a customer

service level well above 975 When the customer service levels are rounded to one decimal

each of 253 products has a customer service level of 1000 and each of 347 products has a

customer service level of at least 990 In contrast each of only 58 products has an optimal

policy that yields a customer service level of at least 99034 For each of 37 products the

3 3 The cost associated with product 203312 is not included in the total

3 4 Optimal policies do not have customer service levels of exactly 975 because the function is discrete For low-demand products a unit step in s or S results in a relatively large change in the customer service level

57

current policy is identical to the optimal policy

The extremely high level of service provided by most policies allows for substantial

savings in the cost of inventory With current policies the estimated annual cost of inventory

for all 420 products is $420743 whereas the expected annual cost of the optimal policies is

only $186884 This amounts to a savings of approximately $2340 or over 55 of the current

cost of inventory

In order to further assess the savings the difference between the costs of the current

policies and the costs of the optimal policies are examined in more detail For each of the 383

products whose existing policy provides a customer service level above 975 the difference

between the estimated annual cost of the existing policy and the expected annual cost of the

optimal policy is calculated The resulting histograms of the savings in (a) dollars and (b)

percentage of current cost are shown in Figure 6 The particulars for the current and optimal

policies for all 420 products are provided in Appendix K

As evident from Figure 6(a) the majority of the expected savings in absolute terms

comes from only six products In order to examine these products more closely the current

and optimal policies along with the associated costs and savings for each of the six products

are listed in Table 5 Each current policy is very different than its corresponding optimal

policy resulting in excessive inventory for each product In fact over half of the total expected

savings could be realized by implementing the optimal policy just for each of these six products

Interestingly five of the six products are sealed beam headlights

58

90

10 - 20 30 - 40 50 - 60 70 - 80 90- 100

Savings ()

(b)

Figure 6 The distribution of savings of optimal policies in (a) dollars and (b) percentage of current cost

59

Table 5 Products with the largest potential absolute savings

PRODUCT

CURRENT OPTIMAL SAVINGS

PRODUCT s S Cost ($)

C S L

()

s S Cost ($)

C S L () ($) ()

2036801 194 200 64014 1000 37 48 bull 9245 975 54769 856

2036772 94 100 23652 1000 18 22 3850 978 19802 837

2036723 78 90 23043 1000 19 23 4454 975 18589 807

2030944 145 175 19522 1000 34 45 3727 975 15795 809

2036425 48 60 10491 1000 13 17 2682 977 7809 744

2036556 68 80 7265 1000 14 22 1699 975 5566 766

1 H6054 rectangular sealed beam headlight 2 H4656 rectangular sealed beam headlight 3 6053 rectangular sealed beam headlight 4 9004 halogen bulb 5 4652 rectangular sealed beam headlight 6 6104 7 sealed beam headlight

Although Figure 6(a) seems to indicate that the majority of the current policies are close

in cost to the optimal policies this is very misleading For example suppose the optimal policy

for a product costs $1 and that the current policy costs $2 In absolute terms the optimal

policy saves only $1 but in relative terms the optimal policy saves 50 of the current cost

When savings are viewed in this manner as in Figure 6(b) it is clear that substantial savings

can be realized for the majority of the products In fact both the mode and the mean of the

savings are between 40 and 50 of the current costs

In order to extrapolate the above savings to obtain the potential savings for product

category 20 at all 21 stores the differences between stores must be considered During the

60

pilot study the current policy for each of several products at each size of store was evaluated

The study suggested that medium and large stores have a similar amount of excess

inventory and that jumbo and extra jumbo stores have much more excess inventory One

could reasonably expect that the additional excess inventory allows for higher savings to be

realized at the jumbo and extra jumbo stores Thus because 18 of the 21 stores are

medium or larger the expected 40-50 savings at store 6 can be viewed as a lower bound

on the potential savings to Canadian Tire Pacific Associates from implementing optimal policies

for product category 20 in all its stores

Finally in order to obtain the overall potential savings the expected savings from

product category 20 must be extrapolated to all product categories Ideally the proportion of

inventory comprising each product category and the relative savings that can be realized for

each category would be used to obtain a good estimate of the overall potential savings

However as this information is not known only a crude extrapolation is possible Thus the

analysis proceeds under the assumption that a 40-50 savings can be realized in each product

category35

Before an estimate of the total potential savings in absolute terms can be obtained the

proportion of the current inventory to which the savings apply must be determined Up to this

point the analysis has been concerned solely with reducing inventory at the stores However

it is inappropriate to perform further analysis without considering the entire inventory system

including the central warehouse Because the lead time for orders from Toronto is highly

variable and considerably greater than those at individual stores Canadian Tire Pacific

3 5 During the pilot study the current policies for a paint brush and an ironing board were evaluated and found to result in excess inventory similar to that of product category 20 Although this in no way justifies the assumption it does lend some support

61

Associates must keep a fairly high safety stock at the warehouse Without knowing the

proportion of inventory serving as the safety stock it is inappropriate to assume any reduction

in inventory at the warehouse is possible Therefore the relative savings of implementing

optimal policies is applied only to the inventory held in the stores36 Of the estimated $28

million in inventory approximately half is held in the stores therefore the 40-50 savings

translates to a savings of approximately $55 to $7 million per year

B Sensitivity Analysis on the Ordering Cost and Holding Rate

The above analysis was based on estimates of the ordering cost and the annual unit

holding rate37 provided by management As described in Chapter II these estimates

incorporate a number of costs that are difficult to quantify or that are speculative Therefore

there is some question as to the accuracy of the estimates In this section errors of +$0015

in the ordering cost and +5 in the holding cost are assumed in order to determine the

sensitivity of the model to errors in these variables38

The analysis proceeds as follows Based on an ordering cost of $0080 and a holding

rate of 30 the optimal policies from the previous section were obtained39 Then assuming

3 6 Of course this assumes that the excess inventory currently held in the stores is not required as part of the safety stock Based on the large amount of inventory at the warehouse this assumption seems reasonable

3 7 Hereafter simply referred to as the holding rate

3 8 An error of +$0015 in the ordering cost implies an actual ordering cost of $0070 a relative error of approximately 21 and an error of -$0015 implies an actual ordering cost of $0100 a relative error of 15 An error of +5 in the holding rate implies an actual rate of 25 a relative error of 20 and an error of -5 implies an actual holding rate of 35 a relative error of approximately 17

3 9 These policies are referred to as optimal policies for the duration of this chapter

62

the actual ordering cost andor holding rate are different from managements estimates the

heuristic is used to obtain the true optimal policies Finally the optimal policies are

evaluated based on the actual ordering cost and holding rate and then compared to the true

optimal policies For each possible combination of ordering cost and holding rate three

numbers are recorded the number of products with an optimal policy that differs from its

true optimal policy the total cost of implementing the optimal policies given the actual

ordering cost and holding rate and the difference between the cost of implementing the

optimal policies and the cost of implementing the true optimal policies The results of the

analysis are summarized in Table 6

Table 6 A comparison of optimal policies to true optimal policies

Actual Ordering Cost

$0070 $0085 $0100

25 5 products $155439

$002

51 products $160590

$092

88 products $165464

$449

Actual Holding Rate

30 54 products

$181470 $182

48 products $192026

$088

35 79 products

$207270 $574

48 products $212940

$136

2 products $218312

$000

4 0 The table should be read as follows If the actual ordering cost is $0070 and the actual holding rate is 25 then each of 5 products has an optimal policy that is different from its true optimal policy The total cost of implementing the optimal policies is $155439 which is $002 more than implementing the true optimal policies

63

The model appears to be quite insensitive to relative errors in the order of 20 Under

the worst case scenario when one of the variables is over-estimated and the other is undershy

estimated the optimal policies result in a total cost of inventory which is only approximately

3 higher than the cost of the true optimal policies Further because demand is not changed

by errors in these estimates each of the optimal policies still provides a customer service

level of at least 975 under all of the scenarios Finally although the cost of the optimal

policies varies significantly for the different scenarios the relative savings remain virtually

constant Table 7 lists the cost of existing policies and in parenthesis the relative savings of

the optimal policies for each of the different combinations of ordering cost and holding rate

The cost of the existing policies varies in the same manner as that of the optimal policies the

relative savings for each of the scenarios remains around 55 Therefore even in the presence

of such errors in the ordering cost and holding rate the projected savings of implementing the

optimal policies would be unchanged

Table 7 The cost of existing policies and the relative savings of the optimal policies under various scenarios

Actual Ordering Cost

$0070 $0085 $0100

25 $350153 (56)

$358464 (55)

$366764 (55)

Actual Holding Rate

30 $412438 (56)

$429048 (55)

35 $474725 (56)

$483026 (56)

$491320 (56)

64

C Sensitivity Analysis on the Demand Probability Mass Function

In this section the sensitivity of the model to slight changes in the demand probability

mass function is investigated As described in Chapter II the demand function used in the

model is based solely on observed sales however this empirical function is only one possible

instance of the underlying true demand function For example had the data been collected

over a period both starting and finishing one month earlier many of the empirical functions

would be slightly different Consequently depending upon the degree to which the empirical

functions vary and the sensitivity of the model to these variations the optimal policies from

the model might be inaccurate For example recall the case of the amber driveway marker

only thirteen markers were sold and all were sold on the same day Because the empirical

demand function is used the model calculates an optimal policy of (1213) clearly this

policy is impractical Although this example is extreme it does indicate the danger of using

unrealistic empirical demand functions41

As discussed in Chapter II three classes of products were identified namely low

medium and high-demand products For this analysis the same method of classification

is used Two products are selected from each of these groups and two new distribution files

are created for each of the six products For each product one of the files contains two

additional days of data with each day having a demand equal to the maximum observed demand

for that product In the second file only one additional day of data is added but the demand

on that day is two more than the maximum observed demand The situations that could give

4 1 Certainly a rigorous analysis to determine both the extent of variation in the empirical demand functions and the effect of using the empirical functions needs to be performed However such an analysis is beyond the scope of this thesis only a brief analysis that might suggest the degree to which the model is sensitive is performed here

65

rise to such changes in the demand data are termed Scenario 1 and Scenario 2 respectively42

The analysis proceeds in a manner similar to the one in the previous section Given the

demand that was originally observed the optimal policies were obtained Now assuming the

true demand function corresponds to the demand under Scenario 1 the true optimal policy

for each product is calculated Then each optimal policy is re-evaluated using the true

demand function and compared to the true optimal policy The analysis is then repeated

assuming the true demand function corresponds to the demand under Scenario 2 Table 8

lists the results for each product and scenario

Table 8 True optimal and optimal policies for each demand scenario

Product 202101 (low demand)

Scenario s s Cost ($) C S L ()

1 True optimal 1 2 470 993

Optimal 1 2 470 993

2 True optimal 2 3 663 976

Optimal 1 2 461 875

Product 203507 (low demand)

Scenario s S Cost ($) C S L ()

1 True optimal 0 3 334 981

Optimal 0 2 258 972

2 True optimal 2 3 508 980

Optimal 0 2 273 702

4 2 To be clear the scenarios are defined as follows Scenario 1 The addition of two days each with a demand of max Scenario 2 The addition of one day with a demand of max+2

where max is the maximum observed daily demand for each product in the sample period

66

Product 201032 (medium demand)

Scenario s S Cost ($) C S L ()

1 True optimal 6 12 1377 976

Optimal 6 11 1338 973

2 True optimal 6 13 1426 977

Optimal 6 11 1338 970

Product 203039 (medium demand)

Scenario s S Cost ($) CSL ()

1 True optimal 5 19 256 975

Optimal 5 16 251 969

2 True optimal 5 20 258 976

Optimal 5 16 249 968

Product 203677 (high demand)

Scenario s S Cost ($) CSL ()

1 True optimal 18 23 3981 977

Optimal 18 22 3817 972

2 True optimal 18 23 3995 978

Optimal 18 22 3831 973

Product 203680 (high demand)

Scenario s S Cost ($) CSL ()

1 True optimal 38 49 9498 976

Optimal 37 48 9175 971

2 True optimal 37 49 9484 975

Optimal 37 48 9206 972

67

The model is most sensitive to changes or errors in the demand functions of low-

demand products When only one additional day of demand for max+2 units is added to the

data the optimal policy for each of the low-demand products results in a very low customer

service level In fact for one of the products the customer service level drops to almost 70

The increase in demand does lower the customer service level for each of the other the

products but that decrease is slight

The above analysis indicates the need for some kind of smoothing or modelling of the

demand data especially for low-demand products The inherent variation between two

instances of a true demand function could very well result in more pronounced changes than

those resulting from either of the above scenarios Therefore it is likely that many optimal

policies especially those for low-demand products would actually provide a much lower or

much higher customer service level than expected However by smoothing the data the

variation between instances of a true demand function can be reduced and the desired

customer service levels can be achieved

For example suppose that the demand for a product is stationary that the average

demand during the previous period was unusually low and that the optimal policy for this

period is based on the demand during the previous period Because the demand is stationary

the probability that the demand during this period is higher than that during the past period is

more than 50 In other words the probability that the demand during this period is higher

than that for which the optimal policy provides the desired expected customer service level

is greater than 05 Thus expected customer service level for this period is actually less than

that specified Now suppose instead that the optimal policy is based on smoothed data which

reflects very closely the expected demand The probability that the demand during this period

is higher than that for which the optimal policy provides the desired expected customer service

68

level is virtually 05 Thus as a result of smoothing the data the specified customer service

level should be expected

69

VII C O N C L U S I O N

In this thesis we developed a single-product single-location model of the inventory

system at Canadian Tire Pacific Associates Specifically we formulated a Markov chain model

to determine the long-run average cost of inventory for fixed values of s and S and then

developed a heuristic search to find near optimal policies quickly Also as a first step toward

developing an integrated interactive inventory control system we created a prototype of the

interface

According to the analysis performed here Canadian Tire Pacific Associates is currently

holding an excessively large inventory in its stores This study finds that Canadian Tire Pacific

Associates could reduce the cost of inventory for product category 20 at store 6 by

approximately 50 and still maintain customer service levels of at least 975 Further the

mean and modal values of savings for these products are between 40 and 50 Given the

similar excess inventory at the other stores and in all product categories we believe that savings

in this range are possible for all products and stores We estimate that the application of

inventory control as described in this thesis would result in annual savings of $55 to $7

million

Given the large potential savings to Canadian Tire Pacific Associates we recommend

that management proceed with the next phase of the project Specifically we recommend that

the model algorithm and interface be refined further that a forecasting model be developed

and that a data collection and storage scheme be designed

In order to reflect operational changes at Canadian Tire Pacific Associates subsequent

to the formulation of the model modifications must be made to allow for the lead time to

exceed the review period Several of the stores now have three picks per week and as a

70

result orders placed for these stores at one decision epoch do not arrive until after the next

decision epoch Because this violates one of the models assumptions the model must be

modified accordingly Note however that with three picks per week and a review period

of four days orders from only the previous decision epoch are ever outstanding at the current

decision epoch

Also the model should be refined to reflect the fact that the review period is not fixed

over the planning period In its present form the model assumes a fixed review period

however inventory at stores with two picks per week is actually reviewed on a four day -

three day cycle and inventory at stores with three picks per week is reviewed on a three day

- two day - two day cycle Rather than obtaining the steady state costs for fixed review period

the model should calculate the steady state cost for a period of one week during which multiple

replenishments occur Although the solutions are not expected to differ significantly as a result

of the change the model would reflect the actual process more accurately

Further work is also required on the search technique for locating near optimal

policies Although the heuristic search described in this thesis appears to work well the

decision maker must provide an upper bound on S prior to the search In practice this upper

bound is generally not known in advance Thus if the heuristic is to be used either a tight

theoretical bound must be found or a termination heuristic must be developed Alternatively

a new heuristic search could be developed based on additional or different assumptions

regarding the properties of the feasible space For example although some regions of non-

convexity were found for some products much of the feasible space appeared to be convex

A heuristic that assumes a convex feasible space would not require an upper bound on S and

might perform very well However before such a heuristic is implemented further work is

necessary to determine the degree of non-convexity for a variety of demand functions and

71

ordering and holding costs

In addition to making the above modifications we recommend that Canadian Tire Pacific

Associates begin work on the remaining two components of the inventory control system First

in order to allow for products with seasonal demand the system requires a forecasting model

to provide accurate estimates of demand for the coming period Second the system needs a

data collection and storage module that systematically collects sales data for each active

product and for each store and then updates the corresponding demand distributions Because

approximately 25 of demand is expected to result in lost sales under the optimal policies the

sales data will have to be adjusted accordingly in the future Also because the model appears

to be sensitive to slight changes in the demand distribution for low-demand products the

module should incorporate a technique for smoothing the empirical distributions for these

products Further work should be undertaken to determine the extent of variation in the

empirical distributions and the most appropriate smoothing technique However because of

the large amount of data required to forecast seasonal demand and because of the current lack

of historical data we recommend that management proceed immediately with the design of a

data collection scheme

Finally two additional issues that are beyond the scope of the next phase should be

addressed at some point First a monitoring system will be required once inventory control

is implemented in the stores The monitoring system will ensure that optimal policies perform

as expected and will indicated whether or not demand is adversely affected by the lower levels

of in-store inventory Second given the degree to which in-store inventory can be reduced

inventory control in the warehouse should be investigated Because approximately half of the

existing inventory is held in the warehouse the additional savings from reducing system-wide

inventory could be great

72

Afterword

Just prior to the completion of the work described in this thesis Mr Don Graham

principle owner of Canadian Tire Pacific Associates sold his shares in the company Shortly

thereafter the results of this thesis were presented to the new management group

Unfortunately the new management did not consider development of an inventory control

system to be a priority Nonetheless the prototype was delivered to Canadian Tire Pacific

Associates for possible in-house testing however at the time of this writing - almost one year

after the meeting - Canadian Tire Pacific Associates has not decided to continue with the

project

73

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74

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75

Appendix A

A Portion of the Sales File for Product 200001 a 30-amp Inline Fuse

200001 283 1190 0 0 2 0 0 1 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 200001 284 1190 2 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 200001 285 1190 0 0 4 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 286 1190 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 200001 287 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 200001 288 1190 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1 0 2 200001 289 1190 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 290 1190 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 200001 291 1190 0 0 2 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 200001 292 1190 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 200001 294 1190 0 0 0 1 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 1 200001 295 1190 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 200001 296 1190 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 200001 297 1190 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 200001 298 1190 0 0 1 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 200001 299 1190 0 0 1 0 0 0 1 0 1 0 0 2 0 0 0 0 0 2 1 1 0 200001 300 1190 0 0 1 0 0 0 0 1 0 0 0 0 0 2 2 0 0 0 0 3 0 200001 301 1190 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 200001 303 1190 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 200001 304 1190 1 0 0 0 0 0 0 0 1 0 0 2 0 0 0 0 0 0 0 0 0 200001 305 1190 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 200001 306 1190 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 2 0 0 1 0 200001 308 1190 0 0 0 1 0 1 0 0 0 2 0 1 0 0 0 0 1 1 1 0 0 200001 309 1190 1 3 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 200001 310 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 200001 311 1190 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 200001 312 1190 0 1 1 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 200001 313 1190 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 2 0 200001 314 1190 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 200001 315 1190 0 0 2 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 200001 316 1190 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 200001 317 1190 2 0 0 1 0 0 0 0 0 0 5 1 0 0 0 1 0 0 0 0 0 200001 318 1190 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 1 200001 319 1190 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 200001 320 1190 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 200001 321 1190 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 1 200001 322 1190 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 323 1190 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 200001 324 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 0 1 0 0 2 1 1 0 200001 325 1190 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3 200001 326 1190 0 0 1 0 1 0 0 0 2 0 0 1 0 0 0 1 0 1 0 0 1 200001 328 1190 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 200001 329 1190 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 200001 330 1190 0 0 1 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 200001 331 1190 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 200001 332 1190 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 200001 333 1190 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 200001 334 1190 0 0 2 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 200001 335 1190 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 200001 336 1190 0 0 0 1 0 0 3 0 0 0 0 0 2 0 0 0 0 0 0 0 0

(253 subsequent lines omitted)

76

Appendix B

Sample Distribution Files

a) A low-demand product - an automobile back-up alarm

PRODUCT NUMBER 202101 PRICE $ 684

DMD

3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

N 2 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 2 2 0 1

1 5 3 8 12 5 7 11 6 11 5 4 11 10 19 4 2 5 14 5 6 4

0 302 304 299 295 302 300 296 301 296 302 303 295 296 288 303 305 301 290 300 301 302

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

b) A medium-demand product - a 168 bulb

PRODUCT NUMBER 203039 PRICE $ 51

DMD

16 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 1 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0

9 0 0 2 5 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0

8 0 0 2 5 0 0 2 0 1 1 3 0 1 3 0 0 1 1 0 0 1

7 2 1 3 1 0 0 1 0 2 3 1 1 4 1 1 0 2 1 1 1 1

6 4 7 1 10 3 1 3 2 1 3 5 2 7 4 1 0 1 1 3 3 4

5 1 7 7 17 0 2 11 1 5 4 1 2 6 9 6 2 6 9 4 4 1

4 5 10 10 16 5 5 19 10 13 8 11 9 13 10 12 7 5 15 5 8 3

3 10 12 13 27 4 6 24 11 10 13 14 9 23 14 18 8 14 18 14 12 14

2 14 30 36 43 16 16 43 32 38 39 39 34 47 36 41 21 19 39 20 34 24

1 30 56 68 56 37 30 70 48 67 61 44 75 62 63 48 28 52 61 63 53 47

0 240 184 165 126 240 247 133 202 169 175 189 173 143 165 180 240 206 161 197 192 212

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

77

c) A high-demand product - a H4651 rectangular sealed beam headlight

PRODUCT NUMBER 203677 PRICE $ 857

DMD

14 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

13 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 2 3 0 0 1 0 0 0 0 1 1 1 0 0 0 0 2 0 0

11 0 1 3 1 0 0 4 1 0 0 0 0 0 0 0 0 1 0 0 0 1

10 0 0 4 4 1 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 1

9 0 3 14 11 0 1 7 1 0 2 0 8 0 1 0 0 0 2 1 0 0

8 0 4 12 16 1 0 18 4 3 5 0 2 1 4 2 1 1 0 0 1 3

7 1 2 12 13 1 6 19 3 5 9 0 11 2 5 4 0 2 3 3 1 4

6 2 9 22 21 6 10 29 12 10 11 4 24 3 7 10 4 7 11 6 3 14

5 7 13 35 33 8 16 29 19 18 8 3 26 15 26 15 12 9 14 13 11 19

4 19 35 34 43 32 30 41 31 32 47 11 42 20 33 30 31 20 31 22 21 31

3 32 63 49 54 42 60 51 40 51 35 33 55 39 51 39 38 39 57 58 40 54

2 46 59 56 55 73 63 61 65 63 61 63 54 68 85 65 53 72 69 70 77 63

1 109 79 42 40 80 64 32 80 74 77 99 55 97 57 83 93 80 75 81 97 73

0 91 39 21 13 63 57 11 51 51 52 94 29 61 35 59 75 76 45 51 56 44

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

78

Appendix C

A Proof that the Limiting Probability Distribution is Independent of the Initial State

In this appendix we show that provided 0 lt PrDT = 0 lt 1 the Markov chain

corresponding to an inventory model with a (sS) policy has a limiting probability distribution

7T where 0 lt TTJ lt 1 for y = 0 1 S that is independent of the initial state Further the

limiting distribution is the unique nonnegative solution of the equations

Tt P = TI (AD

s and J gt = 1 (A 2)

i=0

where P is the (S + l) x (S + l) transition probability matrix whose first n rows (and columns)

correspond to the n recurrent states in the chain and whose remaining rows (and columns) if

any correspond to transient states43 Prior to the proof several preliminary results are

obtained

RESULT 1 The Markov chain has exactly one positive recurrent class and perhaps some

transient states

First note that in the absence of any absorbing states a Markov chain with a discrete

finite state space must have at least one positive recurrent class containing at least two states

Now consider the Markov chain for an inventory model with a (sS) policy

a) State i gt s cannot be absorbing because there is a positive probability of reaching

j lt i (Since PrDT gt 0 gt 0)

4 3 Note that since the rows and columns of P are reordered the elements of it must be reordered accordingly

79

b) State lt s cannot be absorbing because by the rules of the ordering policy and

the assumption that PrDT = 0 gt 0 there is a positive probability reaching S

Therefore the chain has no absorbing states and must have at least one positive recurrent class

Next consider the subset of states A = i | i lt s Because the state space is finite

and PrDT gt 0 gt 0 the process must return to at least one state in A in other words at least

one state in A must be recurrent Let k G A be a member of a recurrent class dt Because

of the ordering policy and the assumption that PrDT = 0 gt 0 k -raquo S Also by definition

there must be a positive probability of returning to k from S S -gt k Therefore S G 3w

Further because S is reachable from all states in A i G A is either transient with i -raquo dt or

recurrent with pound iR

Now consider the subset of states B = | j gt s Because PrDT gt 0 gt 0 the

chain will eventually reach a state i G A If i is transient then j must be transient also with

y -gt 3 if i G 9 then j is either transient with j -gt dt or recurrent with j E dt Thus the

Markov chain has exactly one recurrent class and perhaps some transient states

RESULT 2 The limiting probability distribution ir for the Markov chain corresponding to the

nxn submatrix Pm = [Ptj] i j G dt is independent of the initial state and is the

unique nonnegative solution of the equations

TC-Prr = re (A3)

and J2 K = 1 ( A 4 )

jt=i

where the rows and columns of are numbered from 1 to n rather than

Since if i j i and j must be in the same equivalence class

80

according to their associated states

First note that both the original Markov chain and the one corresponding to the

submatrix P R R are aperiodic because PrDT = 0 gt 0 Therefore given that 9 is a positive

recurrent aperiodic class the limiting distribution ir is uniquely determined by (A3) and (A4)

(Taylor and Karlin 1994 Theorem 44 p213) Further because there are no absorbing states

or other recurrent classes in the original chain the process must eventually enter the recurrent

class 9 regardless of the initial state Therefore in the limit the probability of being in state

i G 9 is independent of the initial state

RESULT 3 The limiting probability distribution for the transient states is the null vector

independent of the initial state

This result follows directly from the definition of a transient state

PROOF Suppose the states are reordered such that the first n rows (and columns) of P

correspond to the n recurrent states of the Markov chain Then P can be partitioned into four

submatrices

PRK - the n x n submatrix [Py] i j E 9

P T T - the (S + 1-I) x (S + l-n) submatrix [Py] i j g 9

P T R - the (S + l-n) x n submatrix [Py] i pound 9 j G 9

P R T - the n x (S + l-n) submatrix [Py] i G 9 j pound 9t By definition this is a matrix

of zeros

(Al) can then be rewritten as

81

(ltR TtT) P P L TR TT

I-P RR 0

p I-P TR 1 TT

(TtR TCT)

(0- 0T)

where irR denotes the limiting probability distribution for the n recurrent states and irT denotes

the distribution for the remaining S + l-n transient states Solving for TT t gives

TCr-0 + itT-(I-PTI) = 0T

itT-(I-PJJ) = 0 r

i c r = (I-P^-Oj

Since (I - PTT)1 always exists (Puterman 1994 Proposition A3 p 593) irT = 0X which is

in agreement with Result 3 Now solving for 7rR yields

(A2) can be rewritten as

bullRPRR 1 1R (A5)

(A6) S + l

k + L k =1

k=l k=n+

k=i

where ir contains the appropriately reordered steady state probabilities Since (A5) and (A6)

are equivalent to (A3) and (A4) respectively Result 2 holds Therefore the Markov chain

corresponding to an inventory model with a (sS) policy has a limiting distribution that is

independent of the initial state and is the unique nonnegative solution to (Al) and (A2)

82

Appendix D

A Measure for the Steady State Customer Service Level

In this appendix we show that the appropriate measure for the steady state customer

service level is given by

_ E[Demand satisfied during RP] E[Total demand during RP]

Consider the following two measures of the steady state customer service level

Measure 1 Suppose the customer service level is calculated only after n review periods have

passed Denoting DS as the demand satisfied in review period i and DT as the total demand

in period i the customer service level after n periods CSL(n) is

DS + DS0 + + DSbdquo CSLn) = - -

DT + DT + + DT

Dividing both the numerator and denominator by n taking the limit as n -raquo oo and applying

the Law of Large Numbers yields

CSL = lim CSL(n) n-degdeg

- [DS + DS + + DS] = l i m ^ J 1 1

raquo~ - [DT + DT + + DT]

E[DS]

E[DT]

83

Measure 2 Suppose the customer service level is calculated for each of n review periods and

then averaged Thus the customer service level after n periods CSL(n) is given by

CSL(n) = -DS DS

DT DT DT_

Taking the limit as n -gt oo yields

CSL = lim CSL(n)

lim mdash DSl DS2

DT DT DT_

DS

DT

Now consider the scenario in which there is one item on the shelf to begin each review

period and in which demand is stochastic with PrDT = 2 = Vi and PrD x = 100 = Vi

According to Measure 1 the customer service level is

CSL - = ^ W + ^ W = plusmn bdquo 002 E[DT) 1 (2) + 1(100) 51

while according to Measure 2 the customer service level is

CSL = E DS DT

1(1) + i(-Ls

22) + 2 100

51 200

026

Clearly from managements point of view the first measure is more indicative of the

actual level of service being provided The problem with the second measure is that equal

84

weight is given to each ratio regardless of the magnitude of the numbers For example with

respect to Measure 2 the events LSLR = 12 and LSLR = 50100 are identical however

with respect to lost sales the second event is much worse than the first Therefore the proper

measure for the steady state customer service level is Measure 1

85

Appendix E

The Calculation of the Conditional Expected Demand Not Satisfied During a Period of T Consecutive Days

Denoting D x as the demand during a period of T consecutive days and j as the inventory

on hand at the beginning of the period the conditional expected demand not satisfied during the

period E[(DT-y)+] is calculated as follows

E[(DT-j)+] = pound ltk-j)Pr[DT = k k=j+l

= pound (k-j)PrDT = k] + J2(k-j)PrDT = k] - pound (fc-)VJgtr = fc k=j+l k=0 k=0

= J2(k-j)Pr[DT=k -J2(k-j)PrDT = k] k=0 k=0

= YkPrDT=k) - J2jPrDT=k - Yk-j)PrDT=k) Jfc=0 k=Q k=0

= Ti - j - Y(k-j)PrDT=k 7-1

= Tx - j + YltJ-k)PrDT = k jt=0

where fx is the mean daily demand

86

Appendix F

Justification of the Updating Technique for the New Policy (s+mS)

Let the probability transition matrix corresponding to the policy (sS) be denoted by the

(S + l)x(S + l) matrix P with elements py Define s = s+m where m is an integer satisfying

the condition 0 lt s+m lt S Let the probability transition matrix of the new policy (sS)

be denoted by the (S + 1)X(S + 1) matrix 7gt with elements py Using the equations developed

in Chapter III the elements of P are found as follows

CASE 1 The lead time is less than the review period

A m gt 0

For 0 lt i lt s

( i-i pound [PrDL=kPrDT_LgtS-k]] + PrDLziPrDT_LplusmnS-i k=0

i-l 53 [PrDL=kPrDT_L = S-j-k]] + PrDLziPr[DT_L = S-j-i) k=0 s-j

52 [Pr[DL = kPrDT_L = S-j-kn k=0

ifj = 6

ifOltjz S-i

if S-i ltj ltS

a) 0 lt lt s

Pij

= lt Pij

Pij

= pij

b) s lt i lt s This does not simplify but it is identical to (2)

87

For s lt i lt S

PrDTgt i

p = lt PrDT = i-j

0

Pfj ifj = 0

= lt Pij ifOltjlti

Pij if i ltj plusmn S

= pij

B m lt 0

ifj = 0

if 0 ltj lt i

if i ltj plusmn S

For 0 lt i lt s

i-l 53 [PrDL=kPrDT_LplusmnS-k]] + PrDLgtiPrDT_LplusmnS-i

i-l E [PrZgti = fciVZ)7W=S--fc] +PrD tgtiP-D 7 t = S-7-

p y laquoy = o

P i j ifOltjS-i

vPij ifS-iltjS

ifj = 0

ifOltjltS-i

if S-i ltjltS

For s lt i lt S

fPrJgtrgt- 17=0 = PrDT = i-j] ifOltjlti

0 ifiltjplusmnS

88

a) s lt lt s This does not simplify but it is identical to (3)

b) s lt i lt S

PrDTgti

= bull PrDT = i-j]

0

17=0

if0ltjplusmni

ifiltjS

= Pt

ifj = 0

if0ltjlti

ifiltjplusmnS

= Pi

CASE 2 The lead time is equals the review period

A m gt 0

For 0 lt i lt s

0 if0ltjlt S-i

PrDTgti ifj = S-i

PrDT=S-j] ifS-iltjltS

a) 0 lt i lt s

= bull y Pij

Pij

ifOplusmnjltS-i

ifj = S-i

ifS-iltjS

= Pij

b) s lt i lt s This does not simplify but it is identical to (4)

89

For s lt i lt S

PrDTgti)

Pr[DT=i-j

0

laquof 7=0

ifOltjplusmni

if i ltj lt S

Pa

Pij

if 7 = 0

if 0 lty lt i

ifiltjltS

B w lt 0

For 0 lt t lt s

ro PrJrgt PrDT = S-j

ifOzjltS-i

ifj = S-i

if S-i ltj lt S

Pij

ifOzjltS-i

ifj = S-i

if S-i ltj ltS

= Pi

For s lt i lt S

PrDTgti

= PrDT = i-j

0

i 7 = o ifOltjplusmni

ifiltjltS

a) s lt i lt s This does not simplify but it is identical to (3)

90

PrDTzi ifj = 0

PrDT = i-j] ifOltjplusmni

0 ifiltjltS

Py ifj = 0

ptj ifOltjplusmni

Pij if iltj plusmnS

Appendix G

Justification of the Updating Technique for the New Policy (sS+m)

Let the probability transition matrix corresponding to the policy (sS) be denoted by the

(S + l)x(S + l) matrix P with elements p(gt Define S = S+m where m is any non-negative

integer Let the probability transition matrix of the new policy (sS) be denoted by the

(S+m+l)x(S+m + l) matrix IP with elements ptj Then using the equations developed in

Chapter III the elements of 7 are found as follows

CASE 1 The lead time is less than the review period

For 0 lt i lt s

pound [PrDL=kPrDT_LplusmnS-k] + PrDLi)PrDT^Sgt- k=0

i-l pound [Pr[DL = kPrDT_L = S-j-k- +Pr[DLiPrDT_L = S-j-i Jt=0

s-i Y [Pr[DL=kPrDT_L = S-j-k] k=0

i - l

ifj=0

ifOltjltS-i

ifS-iltjltS

pound [PrDL = kPrDT_LS-k + Pr[DLi)PrDT_LplusmnS-i) if j = 0 fc=0

i - l = pound [PrDL=kPrDT_L = (S+m)-j-k] + PrDLgtiPr[DT_L = (S+m)-j-i ifOltj lt (J5+m)-i

if (S+m)-iltj lt(S+m)

k=0

i - l

pound (Sm)-j

pound [PrDL = kPrDT_L = (S+m)-j-k]] k=0

92

I-l

53 [PrDL = kPrDT_LgtS-k] + PrDLziPrDT_LS-i ifj = 0 k=0

i-l

Y[PrDL = kPrDT_L = S-(j-m)-k] + Pr[DLziPrDT_L = S-(j-m)-i] ifOltjzS+m-i k=0 S-(j-m)

53 [PrDL = kPrDT_L = S-(j-m)-k]] if S+m-i ltj lt S+m

i - l

53 [PrDL = kPrDT_LzS-k] + PrDLziPrDT_LzS-i] ifj = 0 =0

i - l

13 [Pr[DL = kPrDT_L = S-(j-m)-k] + Pr[DLziPr[DT_L = S-(j-m)-i ifOltjz S+m-i k=o and 0 z j-m

Pi(j-m)

Pi(j-m)

if 0 ltj lt S+m-i and 0ltj-mltS-i

if S+m-i ltj lt S+m and S-i lt j-m zS

i - l

Y [PrDL = k)PrDT_LzS-k] + PrDLi)PrDT_LplusmnS-i ifj = 0

k=0 i - l

53 [PrDL = kPrDT_L = S-(j-m)-k] + PrDLziPrDT_L = S-(j-m)-i ifOltjzm k=0

Piij-m)

Piij-m)

if mltj lt S+m-i

if S+m-i ltj lt S+m

i - l

YlPrDL = kPr[DT_L^S-k + PrDLi)PrDT_LS-i) ifj = 0

i - l

53 [PrDL = kPrDT_L = S-j-k] + PrDLgtiPrDT_L = S-j-i ifOltj i m k=0

Pi if mltj ltS

which for 0 lt j lt m is identical to (2)

93

For s lt i lt S

Pu =

PrDTgti

PrDT = i-j]

0

if0ltjlti

ifiltjltS

a) s lt i lt S

PrDTgti 7=0

PrDT = i -j) if0ltjlti

0 if i ltj lt S

0 ifSltjltSgt

ifj=0

if 0 ltj lt i

if i ltj plusmn S

0 ifSltjltS

ifOltjltS

0 ifSltjltSgt

b) S lt i lt S This does not simplify but it is identical to (3)

CASE 2 The lead time equals the review period

For 0 lt i lt s

0 ifO lt jltS-i

PrDTgti ifj=S-i

PrDT=S-j ifS-iltjltS

94

0 ifOplusmnjlt(S+m)-i

PrDT ifj = (S+m)-i

PrDT = (S+m) -j) if (S+m)-i ltj ltL (S+m)

0 if 0 lt jltS+m-i

PrDTzi] if j = S+m-i

PrDT = S-(j-m) if S+m-i ltj lt S+m

0 if 0 ltj lt S+m-i and 0 ^ j-m

Pi(j-m) if 0ltjlt S+m-i and 0ltj-mltS-i bull

Pi(j-m) if j = S+m-i and j-m = S-i

Pi(j-m) if S+m-i ltj z S+m and S-i lt j-m ^S

0 if 0 zj ltm

Pi(j-m) if mlt jlt S+m-i lt

Piij-m) if j = S+m-i

if S+m-i ltj lt S+m

1deg if 0 ltj ltm

[PiV-m) ifmltjltS

which for 0 lt i lt s is identical to (3)

For s lt lt S

PrDTzi

= PrDT = i-j)

0

ifj = 0

if 0 ltj ltL i

ifKjzS1

95

a) s lt i lt S

PrDTzi

PrDT = i-j]

0

0

17=0

if0ltjplusmni

ifiltjS

ifSltjltSgt

Pij 7=0

Pij if0ltjii

Pij if i ltj ltL S

0 ifSltjltS

Pij ifOzjiS

0 ifSltjltS

b) S lt i lt S This does not simplify but it is identical to (3)

96

Appendix H

A Hypothetical Consultation

In this appendix the interface during a hypothetical consultation is depicted The figures

accurately represent the various screens that are presented to the manager with the exception

that all lines and text are really white on a blue background During the consultation described

below only the choices made by the manager are hypothetical all the results are real

The title screen as depicted in Figure A - l appears initially upon execution of the

program In order to proceed to the main menu the manager presses the enter key

O I C S

Optimat Inventory Control System

For the exclusive use of

Canadian Tire Pacific Associates

Developed at The Faculty of Commerce

and Business Administration The University of British Columbia

Brian Kapalka Kaan Katircioghi Marty Puterman

Press ltEntergt to continue

Figure A - l The title screen

97

Main Menu 1 Select data directory

ctircdatadst

2 Select product and store combination Product 200001

3 Select scenario Review Period 4 Lead Time 3

4 Select search parameter Smax 25

5 Execute program

6 Exit to DOS

Please enter the desired option 5

Figure A-2 The main menu

Figure A-2 depicts the main menu of the program it is from here that the manager

enters all the review parameters If the manager wishes to change one of the parameters from

its default value he or she selects the appropriate number and enters the correct information

in the window presented The new settings will then be shown in place of the default values

In this example the relevant data files are located in the default directory ctiredatadst and

the manager has selected the productstore combination of product 200001 and store 6 Further

the manager has selected a review scenario of a 4-day review period and a 3-day lead time and

an upper search limit of 25 Finally in order to proceed with the analysis the manager

executes the main module of the program by selecting option 5

98

single-product single-location models for a more complete review of the literature we refer

to them

The degree of complexity of a single-product model depends largely on the assumptions

made regarding the demand cost structure and physical characteristics of the inventory system

For example demand can be either deterministic meaning it is known with absolute certainty

or stochastic meaning it is unknown but follows a known probability distribution

Deterministic models tend to be relatively simple yet useful such as the basic EOQ model

However future demand is seldom known with certainty in practice and therefore the use of

these simple models is limited As a result the majority of research in inventory has focused

on stochastic demand models (Lee and Nahmias 1993)

Because the objective of most inventory models is to minimize the cost of inventory

assumptions regarding the structure of the cost are very important also Generally speaking

the cost of inventory has four components the purchaseproduction cost the orderingsetup

cost the holding cost and the shortage cost1 Each of these costs can be fixed proportional

nonlinear or some complex combination of these Further because nearly all inventory

problems require decisions be made sequentially over a period of time the costs can be either

discounted or averaged In the former case a discount factor is included in the cost function

in order to account for the time value of money However since most inventory problems are

on-going the long-run average cost is usually minimized (Lee and Nahmias 1993) In

practice this is achieved by minimizing the expected cost per time period under steady state

conditions

Finally the physical characteristics of the system are important in determining the

1 The cost of a stockout is often very difficult to estimate in practice As a result service levels are frequently used in its place When this is done a constraint is included in the model

2

complexity of the model For example the review process can be either continuous in which

case the inventory is reviewed continuously and orders are placed as soon as the stock reaches

a certain level or periodic in which case the inventory is reviewed at regular intervals and

orders can be placed only at those fixed times Also excess demand can be fully backordered

as is common in many manufacturing and wholesale systems completely lost as is common

in most retail environments or some combination of the two Lead time assumptions are also

important models with lead times are more complex than those in which the supply of

inventory is assumed to be instantaneous

Although the literature is dominated by single-product models (Lee and Nahmias 1993)

much recent work has focused on extending those models to multiple-product problems

However multiple-product models are inherently more complex than the simple-product

models In fact no efficient algorithms are currently available for determining fully optimal

strategies for most multiple-product problems (Federgruen 1993) As a result research has

focused on the development of easily computable approximation techniques Muckstadt and

Roundy (1993) provide a good overview of multiple-product andor multiple-location models

with deterministic demand although emphasis is placed on models for production and

distribution Federgruen (1993) extends the discussion to consider these models under uncertain

demand and also provides a complete list of references on the subject Silver and Peterson

(1985) discuss models for coordinated replenishments for both deterministic and stochastic

demand

In this thesis we examine the inventory system at Canadian Tire Pacific Associates

owner of 21 retail stores and a central warehouse Although this is a multiple-product

multiple-location single-warehouse problem and strictly speaking should be modelled as such

we analyze the inventory problem on a product-by-product and store-by-store basis We believe

3

that this simpler single-product single-location model captures the essentials of the problem for

two reasons First the coordination of replenishments for products andor locations is most

advantageous when significant savings are possible because of high first order costs and low

marginal costs for placing subsequent orders However because of the large number of

products and the physical distribution system in this problem trucks are dispatched to stores

on a regular basis Therefore the benefits of coordinating replenishments for this problem are

negligible Second we believe that controlling in-store inventory alone can result in substantial

savings to Canadian Tire Pacific Associates Further given the excessive amount of inventory

in the warehouse a single-location model assuming an infinite warehouse capacity is reasonable

Of course should inventory control in the warehouse become a priority a more elaborate model

would be required

More specifically we consider a periodic review single-product inventory model with

stationary independent stochastic demand a deterministic lead time and an infinite planning

horizon The probability mass function for the demand is based on empirical sales data

collected over a period of approximately ten months The cost of inventory includes a

proportional holding cost and a fixed ordering cost neither of which is discounted All excess

demand is completely lost The objective of the model is to minimize the long-run average cost

per unit time subject to a service level constraint on the fraction of demand satisfied directly

from in-store inventory In order to obtain meaningful cost comparisons the model calculates

the expected annual cost of inventory

For inventory problems like the one described above optimal (sS) policies can be

computed by Markovian decision process methods such as successive approximations or policy

iteration or by stationary Markovian methods However each of these methods generate

calculations that are complex and time consuming Therefore much research interest lies in

4

developing techniques to find approximate (sS) policies with little computational effort The

majority of the approximation techniques begin with an exact formulation of the problem and

then manipulate the problem by relaxations restrictions projections or cost approximations

(Federgruen 1993) Porteus (1985) presents three approximation techniques and compares

them to fourteen others for a periodic review single-product inventory model in which

shortages are fully backordered and a shortage cost is incurred Federgruen Groenevelt and

Tijms (1984) present an algorithm for finding approximate (sS) policies for a continuous review

model with backordering and a service level constraint Also Tijms and Groenevelt (1984)

present simple approximations for periodic review systems with lost sales and service level

constraints All three works find that the approximation techniques perform well However

the techniques are evaluated for a variety of demand parameters that all result in policies with

a value of S in the range of 50 to 200 Given the slower moving products at Canadian Tire 2

it is unclear how well such approximation techniques would perform Therefore we do not

attempt to apply or modify any of these approximation methods

Exact optimal policies for this problem can be obtained by formulating a constrained

Markov decision process and solving it However the data generating requirements and the

lack of a good upper bound on S make this formulation impractical Therefore we formulate

a Markov chain model for fixed values of s and S and then employ a search technique to find

the optimal values for s and S In order to improve the performance of the model we develop

a heuristic search that locates near optimal policies relatively quickly

The remaining chapters of this thesis are organized as follows In Chapter II an

overview of the existing inventory system at Canadian Tire is presented along with a description

2 For example very few of the products analyzed have optimal values of S exceeding 20 with the largest value of S equalling 69

5

of the problem Next a multi-phase project designed to develop a fully integrated interactive

inventory control system is described with the scope of the research reported in this thesis

completing the first phase of the project Finally a description of the demand and cost data is

given

In Chapter III we formulate a Markov chain model that can be used to calculate the cost

and service level associated with a given (sS) policy Based on several simplifying

assumptions the Markov chain is formulated and the equations for the corresponding transition

probability matrix are developed Then given the limiting probability distribution of the chain

the equations for the steady state annual cost of inventory and the corresponding service level

are obtained The chapter concludes with a brief summary of the model

In Chapter IV an algorithm that searches for the optimal (sS) policy is developed As

a starting point a simple grid search algorithm is developed and evaluated on the basis of

execution time In order to improve the algorithms speed a technique for updating the

transition probability matrix is presented and the improvement in the algorithm is noted Next

a theoretical lower bound on S is obtained although the corresponding reduction in the

execution time is minimal Finally based on two assumptions regarding the shape of the

feasible space a heuristic search is developed which improves the speed of the algorithm

significantly

In Chapter V a prototype of the inventory control systems interface is created In its

present form the interface allows managers to evaluate inventory policies interactively The

manager can set review parameters compare optimal policies and current policies and evaluate

alternate policies The chapter concludes by recommending specific options be including in

future versions to make the system more automated and user-friendly

In Chapter VI optimal policies for all products in one product category sold at one store

6

are compared to existing policies on the basis of expected annual costs and customer service

levels The analysis suggests that existing policies result in a large excess inventory and that

implementing the optimal policies would reduce the annual cost of inventory by approximately

50 A sensitivity analysis on the cost estimates is performed and suggests that the model is

not sensitive to relative errors of approximately 20 in the cost estimates Finally a sensitivity

analysis on the demand probability mass function is performed The analysis indicates that the

model is sensitive at least in terms of the customer service level to slight changes in the tail

of the demand distributions of low-demand products

Finally in Chapter VII the results of the analysis are summarized and future work is

outlined In order to incorporate recent changes in the review procedures at several stores the

model must be modified to allow for the duration of the review period to exceed the lead time

We recommend that the second phase of the project proceed specifically that a data collection

and storage scheme be designed and that a forecasting model be developed Because of the

substantial savings that can be realized by reducing inventory only at the stores inventory

control at the warehouse should also be investigated

7

II T H E I N V E N T O R Y S Y S T E M A T C A N A D I A N TIRE

A Background

Canadian Tire Pacific Associates owns and operates 21 retail stores in the lower

mainland of British Columbia The stores carry approximately 30000 active products3 that

are divided into 80 product categories The merchandise comprises primarily automotive parts

and accessories home hardware housewares and sporting goods Canadian Tire Pacific

Associates also owns a central warehouse in Burnaby at which inventory is received from

Toronto Inventory is then shipped periodically from the warehouse via truck to the local

stores In 1992 total sales at the stores exceeded $99 million and the value of inventory was

estimated to be $28 million of which approximately half was stored in the warehouse

On selected days of each week a stores inventory records are reviewed automatically

and orders for products are placed to the warehouse as required Management uses the term

flushing to describe the process of reviewing the inventory and producing the tickets for stock

owing and the term picking to describe the physical act of filling the order at the warehouse4

Whether or not an order is placed for a product depends upon its current inventory level its

target level and its minimum fill 5 If the target level less the current level exceeds the

minimum fill an order is placed to bring the stock up to the target level From the moment

3 Active products are those that are regularly restocked Canadian Tire Pacific Associates can choose from up to 50000 products

4 Note that the number of picks per week is equal to the number of times per week that the inventory records are reviewed At the beginning of this study eighteen stores had two picks per week while the remaining three stores had only one

5 The target level is the maximum number of items that should be on the shelf at one time and the minimum fill is the minimum number of items that should be ordered

8

the order is placed to the moment the product is reshelved takes between two to four days

depending on the store On average 11000 items are shipped from the warehouse at each

flush

Although a products minimum fill is the same for all stores its target level depends

upon the size of the store Based upon its sales volume and retail space a store is classified

as being one of five sizes small (S) medium (M) large (L) jumbo (J) and extra jumbo

(EJ) 6 Each product has associated with it a target level for each size of store

Consequently all stores of the same size have identical inventory policies Table 1 lists the

location size number of picks per week sales and floor space of each store as of October

1993

Periodically the inventory manager reviews the policies and when deemed necessary

adjusts them In this way the manager can account for seasonality in demand changes in mean

demand and any shortcomings of previous policies At the present time the review process

is not automated nor performed on-line The inventory manager uses Hxl5 computer

printouts with five products to a sheet to study the relevant information and on which to note

any changes Afterward the changes are recorded in the system and the new policies take

effect

Currently the inventory policies are based on rules of thumb developed through the

years and not on a statistical analysis of demand nor on a calculation of the cost of inventory

These rules of thumb incorporate a marketing philosophy which at times dominates the

inventory strategy The prevailing philosophy holds that a large in-store inventory instills

6 During the course of this study stores were resized several times At one point the entire class of medium stores was eliminated leaving only four sizes of stores On another occasion the store in Chilliwack was changed from being a small store to a large store

9

Table 1 Location and particulars of the 21 stores

Store Location Store Size

Picks Per Week

1992 Sales ($)

Floor Space (ft2)

1 Chilliwack S 2 2901969 6678

2 Coquitlam M 2 4015303 6510

3 Langley EJ 2 8138673 15422

4 Scott Road EJ 2 7307820 15064

5 Maple Ridge L 2 4102110 6678

6 Kingsway L 2 4153736 6574

7 Richmond EJ 2 8175105 14961

8 North Vancouver M 2 3856571 6534

9 Guildford M 2 3714372 6567

10 Burnaby M 2 3532752 6983

11 New Westminster S 1 2417992 6470

12 Abbotsford EJ 2 7899398 15303

13 Hastings J 2 4647794 9849

14 King Edward EJ 2 3825372 25492

15 White Rock L 2 3960262 6870

16 Main amp Mountain S 1 2705602 6431

17 Heather J 2 4161227 9427

18 King George J 2 4606817 9640

19 Cambie J 2 5855022 9091

20 Lougheed M 1 3114180 6615

21 Burrard J 2 6553310 10608

10

consumer confidence which in turn leads to increased sales In other words if a customer

perceives that Canadian Tire is in the hardware business he or she will choose to shop there

when meeting future needs However management has adopted this philosophy without any

analysis of the cost of holding excess inventory Only recently has management begun to

question the cost-effectiveness of allocating inventory based on these rules of thumb

Speculating that existing inventory practices might be deficient Mr Don Graham the

vice-president of Canadian Tire Pacific Associates commissioned a pilot study to examine the

inventory system and to propose alternate strategies7 The study using a steady state model

to represent a simplified inventory system suggested that existing policies lead to overly high

inventory levels More importantly the study outlined a methodology that would enable

management to determine inventory levels based on observed demand The pilot study

recommended that a more detailed model of the inventory system be formulated that the

methodology be expanded to incorporate the cost of inventory and that an algorithm be

developed that would quickly identify the optimal inventory policies

B The Problem

At the time of the pilot study management was concerned primarily with its inability

to distinguish between stores of the same size As mentioned in the previous section a

stores size is based solely on its sales volume and floor space and under existing practices

all stores of the same size have identical inventory policies for every product Management

realized that demand for some products might be a function of variables other than size such

7 The pilot study was performed by Martin Puterman a professor of Management Science and Kaan Katircioglu a doctoral student in Management Science both at the University of British Columbia The results of the study were presented at a U B C Transportation and Logistics Workshop on March 25 1993

11

as location For example the demand for sump pumps is likely to be very different in suburban

areas than in urban ones However management was unable to allocate more sump pumps to

large suburban stores than to large urban stores This inability led management to seek a

tool that would allow for a more effective allocation of inventory between stores of the same

size

By the end of the pilot study however management was less concerned with

reallocating inventory within a size of store than with reallocating inventory between sizes

of stores and with reducing overall inventory According to management assigning individual

inventory policies to each store appeared to be operationally infeasible Firstly management

believed that having individual policies would lead to confusion in the warehouse and to

increased shipping errors Exactly why this would occur is unclear the warehouse would

simply continue to fill orders from tickets that make no reference to the policy in place

Secondly and perhaps more tenably the computer technology at Canadian Tire Pacific

Associates did not allow for policies to vary across all stores Both the database used to store

the inventory data and the printouts used by the inventory manager allowed for only five

different policies per product all with the same minimum fill Moving to individual policies

would have required an overhaul of the existing information system Although management

saw this overhaul as inevitable it did not see the overhaul as imminent Therefore

management switched the focus of the study to developing a tool that would help reduce overall

inventory levels andor reallocate inventory more effectively between groups of stores

The subtle change in the studys objective prompted an important question how should

the stores be grouped The current practice of grouping stores based on size is problematic

for two reasons First as discussed above additional factors of demand such as location

cannot be taken into account Second grouping inherently results in a misallocation of

12

inventory when some stores within a group have differing numbers of picks per week For

example the medium-sized Lougheed store which has one pick per week requires higher

inventory levels to achieve a given customer service level than the other medium stores

which all have two picks per week8 In fact in terms of customer service levels it might

be more appropriate to group the Lougheed store with the large stores having two picks per

week Ideally this artificial grouping of stores will be abandoned when the computer system

is upgraded and policies can then be set on a store by store basis

As the project proceeded management grew somewhat uneasy with the concept of using

an analytical tool to evaluate and set inventory policies According to both the pilot study and

the preliminary results of this study significantly reducing inventory would greatly reduce costs

while still providing a high level of customer service However management held that because

the model cannot take into account intangibles such as marketing considerations such a drastic

reduction in inventory would be very inappropriate In this light management viewed the

optimal policies as lower bounds for inventory levels these policies would have to be

modified upwards in order to be practical Therefore management contended that any

methodology or software produced during this study should provide the means to evaluate and

compare a number of alternate policies

Regardless of exactly how the tool is to be used there is currently a lack of historical

sales data from which to analyze the inventory system and determine optimal policies No sales

data had ever been collected prior to the pilot study and only a relatively small amount has

been collected since In order to proceed with any analysis of the inventory system a

8 If demand is a function of size then the Lougheed store must meet the same demand with one fewer pick per week This would lead to either increased probabilities of stockouts at the Lougheed store or higher inventory levels at all the other medium stores

13

substantial amount of additional data would be needed The existing information system at

Canadian Tire Pacific Associates does allow for sales data to be captured however the system

is limited by the amount of data it can capture at one time and by its storage capacity In order

to collect the amount of data required management would have to upgrade the system and

develop some kind of data collection scheme coupled with a data compression technique The

collection scheme must allow for characteristics of demand such as trends and seasonality to

be identified and the compression technique must allow for sales data for over 30000 products

to be stored for each store

Because many of the products sold by Canadian Tire have nonstationary demand as a

result of seasonality andor trend the inventory tool must be able to forecast demand for the

coming period Products such as sporting goods and outdoor equipment exhibit patterns of

demand that are very seasonal For example sales of snow shovels increase through the fall

peak in the winter and drop to zero by late spring At the same time some products might

become either more or less fashionable over time resulting in changes in mean demand

Setting inventory policies for any of these products based on observed demand would result in

insufficient stock if demand increased during the period or in excess stock if demand decreased

Therefore the tool must be able to forecast the demand for the period for which the policy is

intended

Toward the end of the project management began discussing the integration of the

interactive inventory tool with the existing system to form a new inventory control system The

new system would utilize a modified point of sales system to collect sales data continually

Based on the latest available data the system would forecast demand for the coming period and

calculate the optimal policies for every product at every store The system would then provide

all relevant information on-line for the inventory manager to review The manager could weigh

14

other information such as marketing considerations and examine alternate policies to determine

the effect on cost Finally the preferred policy would be set in place with the touch of a key

and information on the next product or store would be presented for review

C The Project

Development of the inventory control system envisioned by management requires the

completion of five tasks 1) the design of a data collection system that generates and stores the

necessary sales data 2) the development of a forecasting method that provides accurate

estimates of demand distributions 3) the formulation of a model that calculates the cost of

inventory and the customer service level for an inventory policy 4) the development of an

algorithm that finds the optimal policy for a product at a store and 5) the creation of an

interface that allows interactive and on-line consultation Ultimately the above components

would have to be integrated and incorporated in a manner that would ensure the compatibility

and reliability of the system

The scope of the research reported in this thesis completes the first phase of the project

specifically we formulate an inventory model develop an algorithm and create a prototype of

the interface Because of the changing environment at Canadian Tire Pacific Associates minor

modifications to both the model and the algorithm developed herein might be necessary to

reflect future changes in operational procedures Upon completion of phase one the interface

will be interactive allowing managers to select the store product and review scenario and then

to compare optimal policies with existing policies or with alternate policies Although the

prototype will be delivered to Canadian Tire Pacific Associates for ih-house testing at the end

of this phase the interface will require modification in subsequent phases as feedback from

management is received and as the system is brought on-line

15

In order to assess more fully the potential savings of implementing the inventory control

system this thesis includes a comparison of optimal policies9 and existing policies The

expected annual cost of inventory for an entire product category is calculated for both the

optimal policies and the existing policies Based on an analysis of these costs the potential

savings of implementing the optimal policies are extrapolated Further the thesis examines the

sensitivity of the model to variations in the estimates of demand and costs

Originally results from all 21 stores were to be included in the above analysis

However just prior to the analysis management changed the review procedure at several

stores the result of which being that three picks per week are now performed at those stores

Because one of the main assumptions of the model specifically that the length of the lead time

does not exceed the duration of the review period is violated under this new review procedure

those stores cannot be included in the analysis Therefore the analysis is based solely on the

results from store six the large store located on Kingsway Avenue10

For the purpose of the analysis sales data for product category 20 automotive and

electrical accessories were collected from all 21 stores This product category was selected

because the distribution of demand for products within it is non-seasonal and because few of

the products are ever placed on sale11 The lack of trend or seasonality eliminates the need

for forecasting demand the optimal policies can be determined based directly on observed

demand

9 That is those policies obtained from the model

1 0 Store six is one of the only stores whose review procedure is not expected to change from two picks per week

1 1 Products that were on sale at any time during the period were not included in the data set

16

D The Demand Data

In total sales data for 520 products of which 458 are active products were collected

over a 307-day period from October 10 1992 to August 18 199312 The data was supplied

by Canadian Tire Pacific Associates in the form of a 10 MB text file with one line per day per

product recording sales from the stores13 Each line had the following format

Columns 1-6 part number Columns 7-9 Julian date14

Columns 10-18 cost of the item (to three decimal places) Columns 19-124 sales for stores 1 through 21 (5 columns per store)

In order to facilitate the analysis the file was sorted into 458 sales files with each of

these files containing the history of daily sales for one of the products Two changes were

made to the format of these files to improve readability 1) a column was added between the

product number and the Julian date and 2) all leading zeros were dropped Appendix A

contains a partial printout of the sales file for a typical product Finally a distribution file

summarizing the daily sales was created from each of the sales files Each distribution file

contains the products number the products cost and the number of days during which various

quantities of the product were sold at each store

Prior to the analysis the distribution files for a large number of products were

examined and based on the shapes of the histograms the products were divided into three

groups The first group low-demand comprises products that have demand on only a few

days with the daily demand seldom exceeding three items The second group medium-

1 2 Of the active products only 420 are sold at store 6

1 3 Strictly speaking a line was written to the file only on days on which at least one store registered a sale of the product

1 4 According to the Julian calendar days are numbered sequentially from 1 to 365 (366 during leap-years) beginning with the first of January

17

demand comprises products that have demand on approximately half of the days with the daily

demand seldom exceeding more than six items Finally the last group high-demand

comprises products that have demand on most days occasionally upwards of twenty items and

whose demand histograms resemble Poisson distributions The majority of the products

examined fell into the low and medium-demand groups only a few high-demand products

were observed Appendix B contains a printout of the distribution file for a typical product

in each group

For most products no theoretical distribution appeared to model the demand well

Demand for low and medium-demand products followed a non-standard distribution with

a mass at zero and a relatively small tail Although some theoretical distributions such as the

mixed-Poisson can be made to resemble this it was unclear how well they would perform

Further the effects of sampling errors from such non-standard distributions were unknown

On the other hand the standard Poisson distribution did appear to model high-demand

products well however only a few products fell into that group Therefore in the absence of

a good theoretical candidate the empirical demand function was chosen for the analysis The

empirical function was readily available from the distribution files and offered the best

estimate of the true underlying function

E The Cost Data

Along with the sales data management supplied estimates relating to the costs of holding

and ordering inventory Management estimates that the cost of holding an item in inventory

for one year is 30 of the items cost Incorporated in this holding rate are a number of costs

such as the opportunity cost of money invested and the costs incurred in running a store

Because it is the annual cost of inventory that is of interest either the compounded daily

18

holding rate or the simple daily holding rate can be used to determine the costs incurred during

a review period however the method of calculating and extrapolating the holding costs must

be consistent For the sake of simplicity a simple daily holding rate is used

In order to estimate the cost of ordering inventory management analyzed the various

tasks that result from an order being placed Using the average time required to fill and then

reshelve an order management obtained an estimate of the direct labour costs involved To

this management added an estimate of the cost of transporting an order from the warehouse to

the store Finally the estimate was adjusted slightly upward to account for the cost of breakage

and errors Based on the analysis management estimates the cost of placing an order for a

product to be $0085 regardless of the size of the order

19

III M O D E L F O R M U L A T I O N

A Terminology

In the previous chapter several terms were introduced to describe the inventory system

at Canadian Tire Pacific Associates These terms though perfectly adequate are not widely

used in the academic literature to describe inventory management problems Therefore in

order to avoid confusion and be consistent with the literature the following terms are used

throughout this thesis

The length of time between successive moments at which the

inventory position is reviewed15

The time at which a reorder decision is made

The length of time between the moment at which a decision to

place an order is made and the moment at which the order is

physically on the shelf

An operating policy in a periodic review inventory model in

which (i) the inventory position is reviewed every T days (ii) a

replenishment order is placed at a review only when the inventory

level x is at or below the reorder level s (gt 0) and (iii) a

replenishment order is placed for S-x units (S gt s)1 6

Lost Sales - The situation in which any demand during a stockout is lost

Review Period -

Decision Epoch

Lead Time -

(sS) Policy

1 5 With respect to the terminology used at Canadian Tire Pacific Associates a review period equals the number of days between flushes

1 6 With respect to the terminology used at Canadian Tire Pacific Associates S represents the target level and s represents the difference between the target level and the minimum fill

20

B Assumptions

In to order make the model of the inventory system mathematically manageable several

simplifying assumptions are made However we believe that the following assumptions do not

significantly reduce the accuracy of the model

1) Demand can be approximated very closely by sales during the period in which the data

is collected Generally inventory levels at the stores are very high and the number of

lost sales is assumed to be negligible In fact the inventory levels for a number of

products were tracked over time during the pilot study and not a single incidence of

stockout was observed Because all products appear to have similar excess inventory

it seems reasonable to assume that virtually all demand resulted in sales

2) The demands in successive periods are independent identically distributed random

variables Preliminary tests (Fishers exact tests) suggested no dependence between

sales on consecutive days and plots of sales over time indicated no seasonal pattern

3) Demand is independent across products That is we assume that sales of one product

do not affect nor can they be predicted from sales of another product

4) Unmet demand results in lost sales with no penalty other than the loss of the sale

5) The duration of the review period for a store is an integer value and is fixed over a

planning period For stores with two picks per week we assume that the inventory

is reviewed every four days rather than on a four day - three day cycle

6) The decision epoch is the start of a review period Review periods begin at midnight

at which time the inventory is reviewed and orders are placed instantaneously

7) The duration of the lead time for a store is deterministic and integer-valued In reality

the lead time for some stores was stochastic but management felt the variation could be

controlled and virtually eliminated

21

8) The lead time does not exceed the review period In other words items ordered at a

decision epoch arrive and are shelved before the next decision epoch In practice this

did not always occur

9) Orders arrive at the end of the lead time and are on the shelf to start the next day

10) Holding costs are incurred daily for all inventory held at the start of the day

11) The warehouse has an infinite capacity that is all orders to the warehouse are filled

Because of the high inventory levels in the warehouse stockouts in the warehouse are

very unlikely

C A Markov Chain Model

Consider the situation in which an (sS) policy is applied to an inventory system based

on the above assumptions Inventory is reviewed at the decision epoch for each of the periods

labelled t = 0 12 and replenishment occurs at the end of the lead time provided an order

was placed Demand during day d in period t is a random variable whose probability mass

function is independent of the day and period The function is given by

f^ik) = Pr ltd = k = ak for k = 0 1 2

where ak gt 0 and E ak = 1 Let T denote the duration of a review period and L denote the

length of the lead time Next define D T ( as the total demand during period t D L as the total

demand during the lead time in period t and D T L as the difference between D T and DLt

Assuming that successive demands pound f pound 2 are independent the probability mass function of

1 7 The aggregated demand variables are given by

^ = XXd and DT_Ut=J d= d= d=L+l

22

each aggregated demand variable can be obtained by convolving the probability mass function

of t d l s Finally let X denote the inventory on hand at the beginning of period t just prior to

the review

The process X is a discrete time stochastic process with the finite discrete state space

0 1 S A typical sample path of the process with a review period of four days and a

lead time of two days is depicted in Figure 1 As period t begins X units are on hand The

inventory is reviewed and because X is less than s an order is placed for (S-X) units Over

the next two days a demand of D L r occurs and the inventory drops accordingly With the

arrival of the order the inventory level increases by (S-X) at the start of the third day of the

period Over the remaining two days a demand of D T L is observed bring the inventory level

down to X + 1 Note that in this example an order would not be placed in period t+l because

X r + i exceeds s

The stock levels at the beginning of two consecutive review periods are related by the

balance equation

_K- DLy+ (s - v - DT-LX ixtlts ( 1 )

t + l ~ (Xt - DTtY ifXtgts

where the positive operator is interpreted as

= max a 0

1 8 For example the probability mass function of D T bdquo denoted fD T ( is the T-th fold convolution of f t Mathematically the function can be obtained using the recursive relation

and the identity f1^ = f5(

23

-2 I 8

gt o

r

CO

o a

CM

X

00

24

Because X + 1 depends only on Xt the process X satisfies the Markov property

PrXt+l=jX0 = i0Xt_l=it_vXt = i) = PrXt+1=jXt = i

and therefore is a discrete time Markov chain Further because the probability mass function

of the random variable pound is independent of t and thus so too are the probability mass functions

of the aggregated demand the Markov chain is stationary19

D The Transition Probability Matrix

The stationary transition probability p(gt is defined as the probability of X + 1 being in state

j given that X is in state i independent of the period t In the inventory model p-- represents

the probability of starting the next review period with j items on the shelf given there are i units

on hand at the start of the current period By convention the transition probabilities are

arranged in an (S + l)x(S + 1) matrix P such that (P)y = p- Thus the (i + l)st row of P contains

the probability distribution for the stock on hand at the beginning of the next period given that

there are i units on hand at the start of the current period The matrix P is known as the

transition probability matrix Note that since we are concerned with a steady state analysis we

need not consider the possibility of initial stock levels greater than S once the inventory level

is below S it can never again exceed S

The transition probability ptj is calculated by expanding (1) to include all possible

instances of demand that allow for a change in inventory from i units at the beginning of a

period to j units at the beginning of the next period For example suppose the inventory policy

(510) is in place 2 items are on hand at the start of the current review period and 7 items are

1 9 Henceforth the subscripts t and d are omitted from the daily and aggregate demand variables for the sake of brevity

25

on hand at the start of the next period Then only three scenarios are possible

1) No items are sold during the lead time (meaning that the stock level increases to 10

after replenishment) and 3 items are sold during the remainder of the period

2) One item is sold during the lead time and 2 items are sold during the remainder of

the period

3) Two items are sold during the lead time (though demand could exceed 2) and 1 item

is sold during the remainder of the period

The transition probability p 2 7 is the sum of the probabilities of the various scenarios and is

computed by

p2J = PrDL=0PrDT_L = 3] + PrDL = l] Pr[DT_L=2 + PrDL2] PrDT_L = l]

In general the transition probabilities for the inventory model are calculated as follows

Case 1 The lead time is less than the review period

For 0 lt lt s

Pij

i-l pound [PrDL=kPrDT_LgtS-k] + PrDLgtiPrDT_LgtS-i if j = 0

i - l

^[PriD^^PrlD^S-j-kn+PHD^PriD^S-j-i if 0 lt j ltL S-i (2) k=0 s-j

E [PrDL=kPrDT_L = S-j-k] if S-i ltj^S k=0

26

For s lt i lt S

PrDTzi]

0 p = PrDT=i-j)

ifj = 0 ifOltjzi ifiltjltS

(3)

Case 2 The lead time equals the review period

For 0 lt i lt s

Pa

0 PrDTzi) PrDT = S-j

ifOltLJlt S-i ifj = S-i if S-i ltj plusmn S

(4)

For s lt i lt S use (3)

E The Steady State Probabilities

Provided 1) there is a positive probability of non-zero demand during a review period

and 2) there is a positive probability of zero demand during a review period20 then the

Markov chain described by (1) has a limiting probability distribution ir = (ir0 TTu irs)

where 0 lt IT lt 1 for j = 0 1 S and E 7ry = 1 Further this limiting distribution is

independent of the initial state In other words the probability of finding the Markov chain in

a particular state in the long-run converges to a nonnegative number no matter in which state

the chain began at time 0 The limiting distribution is the unique nonnegative solution of the

equations

2 0 That is 0 lt PrD x = 0 lt 1 Given the nature of the problem this assumption is not very restrictive

27

^ = E ^ - far j = 0 l S (5)

s and pound nk = 1 (6)

For the proof that the limiting distribution is independent of the initial state and is obtained from

(5) and (6) the reader is directed to Appendix C

In terms of the inventory problem 7ry is the steady state probability of having j units of

inventory on hand at the decision epoch However some subsequent calculations are made

easier if the steady state probability of having j units of inventory on the shelf just after

replenishment is known Instead of reformulating the Markov chain these probabilities can be

obtained by shifting the limiting distribution TT Much in the same way that the transition

probabilities are calculated the shifted limiting distribution t] is obtained by accounting for

all possible instances of demand that allow for the shifted inventory position For example

suppose that the policy (510) has the limiting distribution TT In order to have say 8 units on

hand immediately after replenishment one of the following must occur

1) Ten items are on hand at the decision epoch and 2 items are sold during the lead time

2) Nine items are on hand at the decision epoch and 1 item is sold during the lead time

3) Eight items are on hand at the decision epoch and nothing is sold during the lead time

4) Either 2 3 4 or 5 items are on hand at the decision epoch and 2 items are sold during

the lead time

The steady state probability of having 8 units on hand just after replenishment is calculated as

n 8 = n10PrDL = 2] +iz9PrDL = l] +n8PrDL = 0 +[iz5 + iz4 + iz3]PrDL = 2 +n2PrDLgt2

28

In general the shifted steady state probabilities are given by

s pound nPrDLgti ifj = 0

i=s+l S

pound PrDL = i-j] ifl plusmnjltS-s i=max(s+l)

S = pound 7tjirZ)L = i-+7i 5 FrD i ^5 ifj = 5-5

i=max(s+l)

^ 7 1 ^ ^ = 1-7+ pound TtiFrZgti=5-y+iis_PrD1^S-7 ifS-sltjltS i=max(gt+l) i=S-j+l

(7)

F The Cost of Inventory

For this problem the cost of inventory has two components (i) a fixed ordering cost

cf incurred whenever an order is placed and (ii) a holding cost cv incurred daily for each item

held in inventory The expected cost of inventory for a review period can be calculated by

summing the costs associated with each inventory position under steady state conditions

Because a simple holding rate is used this cost is simply adjusted by number of review periods

in a year to obtain the expected annual cost of inventory

The ordering cost for a review period is relatively straight forward to calculate either

an order is placed in which case the order cost is incurred or no order is placed in which case

no cost is incurred According to the rules of the inventory policy orders are placed only if

the stock on hand at the beginning of a review period is less than or equal to s Further the

probability that the review period begins with the inventory in one of these positions is equal

to the sum of the first (s + 1) steady state probabilities Therefore the expected ordering cost

29

for a review period is given by

E[OC] = cfY n y=o

Unlike the simple structure of the ordering cost the holding cost is incurred daily for

each item held in inventory Therefore the daily holding costs associated with all possible

inventory positions must be weighted and summed to obtain the holding cost for the review

period In order to make the calculations less complicated a shifted review period beginning

immediately after replenishment and ending just prior to the next replenishment is considered

Define e(0 to be the expected number of items on hand to start day i of the shifted

review period and let g(i f) denote the expected number of items on hand to start day i given

that there were j items immediately following replenishment21 The function g(z j) is the sum

of the probabilities of possible demand weighted by the resulting stock levels that is

g(ij) = j^kPriD^j-k)

To obtain the expected number of items on hand to start day i independent of j the above

function is summed over all possible values of j and weighted by the corresponding steady state

probabilities yielding

s j ed) = pound r 5 gt P r Z ) = - pound

7=0 k=

2 1 In other words g(if) is the expected number of items on hand to start day i given e(0)=v

30

The expected holding cost for a review period is then given by

T

E[HC] = $gt v e(0 i=i

i=l =0 k=l

where c v equals the simple daily holding rate multiplied by the cost of the item

Although differently defined review periods are used to calculate the above expected

costs the costs can still be summed to obtain the expected cost of inventory for a review

period The expected costs are calculated based on steady state conditions and therefore

represent the costs expected to be incurred during any T consecutive days Summing the above

two expected costs and multiplying the sum by the number of review periods in a year yields

the expected annual cost of inventory

E[AC] = mdash[E[OC]+E[HC]]

T S j 1 ( g ) 365

j=0 i=l j=0 fc=l

G The Customer Service Level

There are many ways to define customer service and the most appropriate definition

depends upon the nature of the business and the perspective of management Two commonly

used definitions are 1) the proportion of periods in which all demand is met and 2) the

proportion of demand satisfied immediately from inventory Although the former definition is

often used because of its simplicity the latter definition is closer to what one commonly means

by service (Lee and Nahmias 1993) A third definition also relevant in retail environments

31

is the probability that lost sales do not exceed some critical number Other definitions such

as probability that backorders are filled within a certain amount of time are more applicable

to industrial and manufacturing settings where backlogging demand is common

Given the retail environment with the possibility of lost sales that exists at Canadian Tire

Pacific Associates management is concerned primarily with the fraction of sales that are lost

For example management views the situation in which demand exceeds stock by one unit every

period more favourably than that in which the same demand exceeds stock by three units every

other period For this reason the proportion of demand satisfied immediately from inventory

is the most appropriate measure of customer service

The customer service level for a single review period CSLRP is given by

r bdquo T _ Demand satisfied during RP Total demand during RP

Because we are interested in the long-run behaviour of the system a measure of the steady state

customer service C S L must be obtained As shown in Appendix D the correct measure is

poundpoundpound _ E[Demand satisfied during RP] E[Total demand during RP]

which can be rewritten as

pound pound pound _ j _ E[Demand not satisfied during RP] ^ E[Total demand during RP]

Because of the steady state conditions the above customer service level can be estimated

from any period of T consecutive days As was done for one of the cost calculations the T-day

period beginning just after replenishment is considered Suppose that there are j units of

inventory on hand immediately following replenishment Then the conditional customer service

32

level is

CSL j = 1 E[(DT-j)l

E[DT] (10)

The denominator in the second term is merely the expect demand during the period

E[DT] = 7gt (11)

where p is the mean of the random daily demand variable pound The numerator which is the

conditional expected demand that is not satisfied during the T-day period expands to

7-1

E[(DT-j)+] = T u - j + YU-k)PrDT = k] k=0

(12)

The reader is directed to Appendix E for a mathematical justification of (12) In order to obtain

the unconditional expected demand not satisfied (12) is applied to all possible values of j and

a weighted sum of the terms is calculated The result is given by

E[Demand not satisfied] = Y j =0

y-i Ti - j + 1pound(j-k)PrDT = k

S j-1

= - E[I] + Y0-k)Pr[DT=k) =0 it=0

(13)

where E[I] denotes the steady state inventory level Finally the customer service level is

obtained by the substituting (11) and (13) into (9) yielding

CSL = 1

s j-l

Ti - E[I] + Jgty Y(J-k)PrDT=k]

m n J2VjTU-k)PrDT=k Eil J=Q fc=o

(14)

33

H A Methodology for Evaluating an (sS) Policy

In this section a brief methodology summarizing the main results of the model is

presented The methodology can be applied to evaluate a given (sS) policy in terms of its cost

and its resulting customer service level Given the probability mass function of the daily

demand f the duration of the review period T the length of the lead time L the fixed cost

of placing an order cf and the daily unit cost of holding inventory cv

1) Calculate the aggregated demand probability mass functions fDT fDL and f D x L as the T-th

fold L-th fold and (T-L)th fold convolutions of f respectively

2) Generate the transition probability matrix using either equation (2) or (4) and equation (3)

3) Calculate the limiting distribution TT from (5) and (6)

4) Obtain the L-day shifted limiting distribution rj using (7)

5) Calculate the expected annual cost of inventory using (8)

6) Determine the customer service level using (14)

34

IV A N A L G O R I T H M F O R OBTAINING T H E O P T I M A L (sS) P O L I C Y

A Introduction

The methodology developed in the previous chapter can be used to calculate the cost and

service level associated with a given (sS) policy In this chapter an algorithm that finds the

feasible policy yielding the lowest cost is developed A policy is feasible if it satisfies both the

basic condition 0 lt s lt S and the imposed constraint on the customer service level2 2 In

order to find the optimal policy the algorithm must combine the methodology for evaluating

a single policy with a search technique

Search techniques are evaluated in terms of two somewhat antithetic criteria On the one

hand the search should locate the optimal solution or at least a near optimal solution on

the other hand the search should take as little time as possible Many search techniques

guarantee optimality provided the objective function and constraints exhibit certain properties

such as convexity However even if such a search technique also guarantees its termination

in polynomial time the search might take too long to be practical Alternatively some search

techniques provide reasonable though sub-optimal solutions very quickly These techniques

often called heuristics must be evaluated in terms of both their speed and the accuracy of their

solutions The choice of a heuristic over a search that guarantees optimality depends upon the

nature of the feasible space the need for accuracy and the cost of time

Given the complexity of the cost and customer service level functions it is very difficult

to show theoretically for what if any combinations of demand and costs the feasible space is

convex However a plot of the objective function for a single product may be obtained by

2 2 Management specified that each policy must provide a customer service level of at least 975 Therefore the algorithm must incorporate the constraint CSL gt 0975

35

evaluating the costs for a number of policies within a grid When the service level constraint

is included in these plots maps of the feasible space are obtained Examination of the

maps for several products suggests that the space is convex over most regions however

maps for some products reveal some regions of non-convexity Unfortunately there is no

way of knowing the degree of non-convexity for other products other stores or different costs

estimates without a more rigorous survey of the feasible space Therefore if one of the

techniques that guarantee optimality under the condition of convexity is employed the search

would have to be evaluated as a heuristic Instead as a first step to finding a practical search

technique a crude search that guarantees optimality under virtually all conditions is

investigated

B A Grid Search

The simplest search technique that guarantees optimality merely evaluates all policies in

the grid defined by 0 lt s lt S and 0 lt Sm i n lt S lt S m a x where Sm i n and S m a x are

some predefined values That is the grid search is a brute force method that examines every

policy within the grid 2 3 Although the number of policies that must be examined is large the

grid search is guaranteed to find the optimal policy provided that S o p t is contained in the grid

that is provided Sm i n lt S o p t lt Sm a x The grid search algorithm proceeds by generating the

transition probability matrix for the policy (0Smin) and then evaluating the service level

associated with that policy If the policy is feasible24 the cost associated with the policy is

2 3 The total number of policies evaluated is the sum of the first Sm a x integers less the sum of the first Smin-1 integers Numerically [(l2)(Sm a x)(Sm a x+l) - (l2)(Smin-l)(Smin)] policies are examined

2 4 That is if the service level is adequate

36

calculated and compared to the current lowest cost The procedure continues until all points

in the grid have been examined and the optimal policy has been found A flow chart of the grid

search algorithm is shown in Figure 2

In order to assess the grid search the algorithm was coded and compiled into an

executable program The program was run for two products (a medium-demand product and

a high-demand product) and with four values of Sm a x In each run Sm i n was taken to be zero

The following times were recorded for each trial

T P M - the amount of time spent generating the transition probability matrix

GAUSS - the amount of time spent solving for the steady state probabilities

COSTS - the amount of time spent calculating the expected annual costs

SERVICE - the amount of time spent determining the customer service levels

T O T A L - the total execution time of the program

The columns entitled Grid Search in Tables 2 and 3 summarize the results of the trials All

times are recorded in seconds as required by a 486DX33 MHz personal computer with an

empty buffer memory

Three results are evident from the tables First the vast majority of the execution time

is spent generating transition probability matrices and then solving for the steady state

probabilities with the time being split almost equally between the two tasks Second the total

execution time of the algorithm increases exponentially as Sm a x increases For product 206917

doubling S^ from 30 to 60 led to an increase in the execution time of more than 2100 (from

172 seconds to 3661 seconds) Third the grid search algorithm as it stands does not appear

to be adequate for the problem at hand Given the large number of products that must be

evaluated the algorithm is much too slow to be implementable

37

Calculate the demand probability functions

Initialize s = 0 S = Smin] cost = 10A5

s = s+1

Calculate the probability transition matrix for (sS)

Evaluate policy (sS) Is service level ^ 0975

No

No Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Is s = S-l

Yes

Is S = Smax No

Yes

Print results (sS) cost

s = 0 S = S+1

Figure 2 The flow chart of the grid search algorithm

38

Table 2 Execution times of the algorithms for product 200001 a 30-amp inline fuse

(in seconds)

-max Subroutine Grid Search Grid Search

Updating T P M Heuristic

10 T P M 006 010 001

GAUSS 010 006 001

COST 005 005 001

SERVICE 001 001 001

T O T A L 050 035 027

20 T P M 126 021 010

GAUSS 119 160 023

COST 027 026 001

SERVICE 017 016 001

T O T A L 318 236 055

30 T P M 703 096 010

GAUSS 757 801 093

COST 079 085 001

SERVICE 047 042 001

T O T A L 1664 1137 126

40 T P M 2257 163 015

GAUSS 2958 3003 206

COST 313 329 001

SERVICE 092 083 011

T O T A L 5871 3851 285

39

Table 3 Execution times of the algorithms for product 206917 a 6 solder connector

(in seconds)

bull-max Subroutine Grid Search Grid Search

Updating T P M Heuristic

30 T P M 717 067 006

GAUSS 762 880 093

COST 098 087 010

SERVICE 016 021 006

T O T A L 1720 1176 220

40 T P M 2390 157 022

GAUSS 2861 2911 209

COST 219 227 006

SERVICE 080 088 011

T O T A L 5827 3790 362

50 T P M 6691 413 010

GAUSS 7994 8201 472

COST 568 577 016

SERVICE 212 197 005

T O T A L 15840 10041 654

60 T P M 15509 894 040

GAUSS 18607 18951 876

COST 1367 1353 038

SERVICE 350 359 022

T O T A L 36614 22794 1120

40

C A Technique for Updating the Transition Probability Matrix

As was noted in the previous section approximately half of the execution time of the grid

search algorithm is spent generating transition probability matrices Furthermore the amount

of time increases exponentially as Sm a x increases This increase is exponential for two reasons

first the number of transition matrices that must be calculated is a function of S^ squared and

second the average size of the matrices and hence the average number of calculations per

matrix increases with Sm a x For example when S m a x equals 30 the transition probability matrix

must be generated 465 times with the largest matrix having dimensions (31x31) When Sm a x

equals 40 the matrix must be generated 820 times with 355 of the matrices having dimensions

larger than (31x31) The performance of the algorithm could be improved considerably if

either the number of matrices that must be generated or the number of calculations per matrix

is reduced This section introduces a technique that accomplishes the latter by updating the

transition probability matrix for a new policy instead of generating it anew for each policy

A new policy (s+mS)

Suppose the policy (sS) has associated with it the (S + l)x(S + l) transition probability

matrix P and the policy (s+mS) where m is any integer satisfying 0 lt s+m lt S has

associated with it the (S + l)x(S + 1) transition probability matrix J Then T can be obtained by

recalculating only m rows of P Recall that the equations for generating a transition

probability matrix are divided into two sets based upon the row being calculated equations (2)

or (4) for rows 0 through s and (3) for rows (s + 1) through S The equations themselves are

independent of s only the limits defining the cases contain s Therefore only the m rows

that are affected by the change in the limits need to be recalculated The following updating

technique can be used to obtain 7

41

1) If m gt 0 rows i = (s + 1) (s+2) (s+m) are recalculated using (2) or (4)

2) If m lt 0 rows i = (s-1) (s-2) (s-m) are recalculated using (3)

3) All other rows of T are identical to those of P

The reader is directed to Appendix F for a mathematical justification of the technique

A new policy (sS+m)

Suppose the policy (sS) has associated with it the (S + l)x(S + l) transition probability

matrix P and the policy (sS+m) where m is any positive integer has associated with it the

(S+m+l)x(S+m+l) transition probability matrix P Then T can be obtained by modifying

each row of P and adding m additional rows and columns The updating technique is not as

simple as that for a new policy (s+mS) for two reasons the dimensions of P and T differ and

the equations for generating a transition probability matrix contain S Nevertheless an updating

technique that requires much less computing time than generating the matrix anew can be

developed The following technique can be used to obtain P

1) For rows i from 0 through s

For columns j from 0 through m recalculate using (2) or (4)

For columns j from (m+1) through (S+m) enter the (j-m)th column entry of P

2) For rows i from (s + 1) through S

For columns j from 0 through S enter the corresponding value from P

For columns j from (S + 1) through (S+m) enter 0

3) For rows i from (S + 1) through (S+m) recalculate using (3)

The above technique is justified mathematically in Appendix G

42

The modified algorithm

The updating technique can be easily incorporated into the grid search algorithm with

only a slight change in the manner in which s is incremented Figure 3 shows the flow chart

of the new algorithm As before the algorithm was coded and compiled into an executable

program and the various execution times were recorded The results are listed in Tables 2 and

3 under the columns Grid Search Updating TPM

Updating the transition probability matrix does improve the performance of the

algorithm The time required to generate the transition probability matrices is reduced by

approximately 90 This translates to a reduction in total execution time of approximately

30 Not surprisingly the reductions are greater for large values of Sm a x However despite

this seemly significant improvement a decrease of only approximately 08 seconds is obtained

when S^ equals 20 Further when S ^ equals 30 the algorithm requires over 11 seconds

which is still much too high

The execution time is now dominated by the time required to solve for the steady state

probabilities The steady state probabilities could be solved by using approximation techniques

which are faster than solving the systems exactly however because the probabilities are often

very small numbers the approximation techniques would be unable to provide the required

accuracy The only remaining way to decrease the execution time is to reduce the number of

policies that are evaluated during the search This can be done by either reducing the size of

the grid being searched or reducing the number of policies being evaluated within the grid The

following section introduces a method of reducing the size of the grid by obtaining a lower

bound for S namely Sm i n

43

s = s+k

Calculate the demand probability functions

Initialize s = 0 S = Smin| cost = 105

k= 1

Update the probability transition matrix for (sS)

Evaluate policy (sS) Is service level ^ 0975

No

No

iNo

Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Ifk= 1 iss = S-l If k =-1 is s = 0

Yes

Is S = Smax No

Yes

Print results (sS) cost

If k=l s=s+l S = S+1

Figure 3 The flow chart of the grid search algorithm utilizing the updating technique for the transition probability matrix

44

D A Lower Bound on S

As mentioned earlier the size of the grid is specified by the number of policies within

the domain defined by 0 lt s lt S and 0 lt Smin lt S lt Smax Thus the size of the grid

and hence the total amount of computation time can be reduced by obtaining tight bounds on

S Unfortunately because of the extreme non-linearity of the objection function and the service

level constraint obtaining a mathematical upper bound on S is very difficult and beyond the

scope of this work This section introduces a method for obtaining a lower bound on S25

For a given value of S an upper bound on the customer service level can be found as

follows Suppose that there are S units of inventory on the shelf at the beginning of every

review period regardless of demand No (sS) policy can provide a higher level of customer

service because there exists a positive probability of beginning a review period with less than

S units on the shelf Thus the customer service level associated with starting each review

period with S units on the shelf is an upper bound on the customer service level for any (sS)

policy From (10) this level is

E[(DT-SY] CSL 5 = 1 - T mdash mdash - ( 1 5 gt

E[DT]

Replacing j with S in (12) and then substituting (12) and (11) into (15) yields

CSL 15 = mdash- - J2(S-k)PrDT = k] (16)

To obtain Smin use (16) to find the largest value of S with a customer service level less than

0975

There are two criteria for a good lower bound on S the bound must be no larger than

This method was proposed by Kaan Katircioglu

45

S o p t and it must be very close to So p t These criteria ensure that the optimal policy will be

contained in the search grid and that the size of the grid is reduced by as much as possible

Based on these criteria the above method of obtaining a lower bound is assessed A value for

S m i n is obtained for each of the products in product category 20 and compared with the

corresponding value of S o p t 2 6 The results are illustrated in Figure 4 which shows the number

of occurrences of various Sm i n and S o p t combinations Note that a point lying on the 45deg line

represents a perfect bound while a point lying close to the S o p t axis represents a very loose

bound The absence of any points above the 45deg line indicates that the first criterion is satisfied

in each instance the method did not generate a value of Sm i n exceeding the corresponding value

of So p t Moreover the bounds are reasonably tight for 85 of the products analyzed that

is for those products with S o p t equal to 10 or less27 However as S o p t increases the bounds

become steadily worse Because values of S o p t are likely to be larger for these products at the

ten Jumbo and Extra Jumbo stores and for other products with higher demand the lower bounds

are not expected to be very tight in general

Although obtaining the bound Sm i n does reduce the size of the grid the corresponding

reduction in the execution time of the algorithm is miriimal For example if S^ equals 60 and

a value of 30 is obtained for Sm i n the program would still require approximately 215 seconds

to search the grid (compared to the 228 seconds required without a lower bound on S) Because

the execution time increases exponentially with So p t a tight upper bound on S would likely have

a more significant impact on reducing the execution time of the algorithm Further research

on this issue is required

2 6 The optimal policy for each product was obtained using a grid search with an extremely large upper bound

2 7 On average Sm i n is approximately half of S o p t for these products

46

47

The only remaining approach to improving the algorithm is to modify the search so that

fewer policies within the grid are evaluated Toward this goal assumptions can be made with

respect to the feasible space and a heuristic search can be developed based on these

assumptions

E A Heuristic Search

As mentioned previously in this chapter a map of the feasible space for each of

several products was obtained Upon closer examination the maps illustrate two patterns

without exception for a given value of S both the cost and the customer service level increase

as s increases The following intuitive argument is offered as a possible explanation for the

observed behaviour When S is held constant increasing s results in more instances in which

an order is placed upon review Hence the average stock level would increase which in turn

would lead to an increase in the cost and the customer service level28 With this in mind a

heuristic search based on the following assumptions is developed for a given value of S both

the cost and the customer service level are monotonically increasing functions of s

In general the heuristic proceeds as follows Having just incremented S by one the

customer service level of the policy (sS) is evaluated If the policy is feasible s is decreased

until the lowest value of s producing a feasible policy is found If the policy is not feasible

s is increased until either the cost exceeds the current lowest cost for a feasible policy a

feasible policy is obtained or s equals S-l In the event that a feasible policy is found the cost

2 8 In order to prove that the customer service level increases as s increases it would be sufficient to show that the probability of placing an order increases as s increases Such a result might be possible if assumptions regarding the demand function are made Further if both the probability of placing an order and the average stock level increase then both the ordering cost and the holding cost would increase No proofs are offered here

48

of the policy is calculated if the cost of the policy is less than the current lowest cost the new

policy becomes the current best Finally S is increased by one and the process repeats

provided S does not exceed some predefined value Sm a x The flow chart of the algorithm

incorporating the heuristic search is shown in Figure 5 Note that the algorithm begins by

attempting to improve on the lower bound Sm i n the policy (Sm i n-lSm i n) is evaluated and if it

is not feasible S m i n is increased by one The improvement continues until a feasible policy is

found

As previously mentioned a heuristic search must be evaluated in terms of both its speed

and the accuracy of its solutions To evaluate the heuristic on the basis of speed the same

approach used to evaluate the previous algorithms is employed The columns entitled

Heuristic in Tables 2 and 3 contain the various execution times of the heuristic The heuristic

is a vast improvement over the previous search techniques the reduction in execution time is

enormous and becomes more so as S m a x increases For a S m a x of 20 the reduction is

approximately 75 of the time required by the grid search algorithm29 and for a S m a x of 60

that reduction soars to over 95 More importantly the heuristic can search a grid with a

of 40 in under 4 seconds or a grid with a S m a x of 60 in under 12 seconds These low execution

times should be sufficient to allow for a large number of products to be analyzed in a

reasonable amount of time

In order to further assess the practicability of the heuristic the total time required by the

heuristic to find solutions for the entire product category was determined and compared to the

corresponding time for the grid search algorithm With a priori knowledge of the optimal

solutions conservative yet reasonable values of S m a x were selected and used by both of the

2 9 Hereafter grid search algorithm refers to the algorithm utilizing the grid search the updating technique and the lower bound Sm i n

49

Calculate the demand probability functions

Calculate Smin S = Smin s = S-l

S = S+l s = S-l t

Update the probability transition matrix for (sS)

No Is service level pound 0975 Yes

No Is service level ^ 0975

Yes

Is cost lt cost No

Yes

Is s lt S-l No

Yes

s = s+l

Update the probability transition matrix for (sS)

Is service level pound 0975

Yes

Is cost lt cost No

Yes

s = s S = S cost = cost

s = s-l

No

Update the probability transition matrix for (sS)

S = S+l

Is s 2 0 Yes

Update the probability transition matrix for (sS)

Is service level pound 0975

No

s = s+l

No

No

Is S = Smax

^Yes

Is cost lt cost

Yes

s = s S = S cost = cost

Print results (sS) cost

Figure 5 The flow chart of the heuristic algorithm

50

algorithms30 The grid search algorithm required approximately 9 hours and 41 minutes to

find the solutions while the heuristic required only slightly more than 23 minutes Based on

speed the heuristic is preferable to the grid search algorithm

Finally to evaluate the heuristic based on the accuracy of its solutions its solutions were

compared to the previously obtained optimal policies Without exception the heuristic

algorithm found the optimal policy Although there is no guarantee that the heuristic will find

the optimal solution for different products or different demand functions the heuristic appears

to work extremely well Based on the fast execution time and the apparent accuracy of the

solutions the heuristic will be used in the development of the inventory tool

3 0 For the test the following values of Sm a x were used 20 (273 products) 30 (19) 50 (23) 100 (3) and 125 (2)

51

V T H E I N T E R F A C E

With the model formulated and the algorithm developed the only remaining task in the

development of the prototype is the creation the interface In fact two interfaces are created

one that displays results in terms of minimum fills and target levels and one that displays

results in terms of reorder points and target levels To be consistent with the previous chapters

the latter interface is described here31

In its present form the interface allows the manager to set review parameters to

compare optimal and current policies and to evaluate alternate policies From the main menu

the manager selects the product store duration of the review period length of the lead time

and value of Sm a x Once the parameters are set the system calculates the optimal policy and

then evaluates the current policy displaying results as they are obtained In particular the

reorder point target level expected annual cost and expected customer service level are

displayed for each policy If a current policy meets the desired customer service level the

expected annual savings of the optimal policy are displayed in terms of both dollars and

percentage of current cost otherwise the value of the current customer service level is

displayed in red to indicate that it is deficient Finally the manager can choose to evaluate

alternate policies or to return to the main menu

Although the prototype is far from the ultimate integrated inventory control system it

is still valuable to management As mentioned in Chapter II inventory policies at Canadian

Tire Pacific Associates are currently based on rules of thumb that incorporate marketing

3 1 Because significant changes to the interface are expected as feedback from management is received and as the system is brought on-line only a brief description of the present interface is given

52

strategies The prototype provides management with a means of determining the proportion of

the cost of inventory that is not attributable to meeting customer demand For example

consider a hypothetical consultation in which the manager chooses to analyze product 200001

a 30-amp inline fuse at store 6 The prototype displays to the manager the current policy of

(1214) which provides a customer service level of 1000 at an annual cost of $748 and the

optimal policy of (39) which provides a customer service level of 977 at an annual cost of

only $319 If for some reason the manager feels the optimal policy is undesirable he or she

can investigate the additional cost of holding excess inventory Suppose the manager chooses

to evaluate the policy (610) The prototype evaluates the policy and displays the results the

customer service level is 998 and the annual cost is $441 With the alternate policy

virtually all demand is satisfied and the cost of inventory is still reduced by over 40 Thus

the manager ascertains that approximately 40 of the current cost of inventory for this product

is directly attributable to marketing considerations Using the prototype in this way

management can begin to understand better the nature of its inventory costs A description and

graphical depiction of the above hypothetical consultation are presented in Appendix H

As for the technical details of the system the prototype can run on any computer using

the DOS operating system having approximately 420K of available conventional memory and

attached to a V G A or SVGA monitor Further the prototype can be used on a machine without

a math coprocessor however performance is slower The prototype consists of two modules

linked together into one executable program The interface module contains the various

interactive subroutines and the main module contains the methodology of the model and the

logic of the heuristic search Both modules are coded in Fortran and compiled using Microsoft

Fortran 50 which provides many extensions to standard Fortran The source codes for the

interface module and the main module are listed in Appendix I and Appendix J

53

respectively

In addition to the above technical specifications the program requires the presence of

certain files in certain directories The program uses the distribution files mentioned in

Chapter II to read the unit costs and the empirical demand functions Each product has a

unique distribution file the name of which is the six digit product number followed by the

extension dst The text file containing the existing policies at each store must be located in

the same directory as the distribution files Finally the file tmsrbfon which contains the

font used by the interface must be located in the directory from which the program is executed

In its future versions the system should be more automated and more user-friendly

Rather than having the manager choose an upper bound for the search the algorithm should

incorporate a method of determining Sm a x For this to happen either a theoretical upper bound

must be obtained or a heuristic for terminating the search must be developed Also the amount

of time spent waiting for calculations to be performed can be reduced by reading the optimal

and current policies and their associated costs and customer service levels from a file or

database This solution file could be created any time prior to the consultation by running

the system in a batch mode The interface should allow the manager to change cost estimates

and service level values view results graphically and print results and summary documents

Finally the system should provide an option for automatically incrementing the product andor

store during a consultation This option would enable a manager to review quickly the

inventory policy of a product at each store

54

VI R E S U L T S AND SENSITIVITY ANALYSIS

A Comparison of Current Policies to Optimal Policies

In order to estimate the potential savings of implementing optimal policies the solutions

obtained in the Chapter IV are analyzed in more detail For each product the current policy

is evaluated and then compared to the optimal policy with respect to cost and customer service

level Finally the potential savings of the optimal policies are extrapolated to all products and

all stores

Of the 420 products examined each of 37 products has an existing policy that fails to

provide the desired customer service level of 975 However most of these policies just

fail to do so only 4 policies provide a customer service level of less than 90 Table 4

provides a list of these 37 products each with its current customer service level its current

annual cost and the additional cost of its optimal policy3 2 Two comments regarding Table

4 are in order First the current customer service level for product 203312 an amber

driveway marker is listed at a misleading 303 Examination of the demand distribution

reveals that this product was sold on only 1 of the 307 days during which sales data were

collected and on that day 13 of the markers were sold Clearly this data point constitutes

either an error or an unusual event the current policy of (34) does not allow for more than 4

items to be sold per review period Second the additional cost of the optimal policy for each

of 5 products is negative In other words for each of these products the current policy which

fails to provide the desired customer service level costs more than the optimal policy which

3 2 This is the difference between the expected annual cost of the optimal policy and the annual cost of the current policy It is the minimum additional cost required to increase the customer service level to at least 975

55

Table 4 Products with insufficient existing policies

Product Current CSL ()

Current Annual Cost ($)

Additional Cost of the Optimal Policy ($)

206932 974 204 -051

204570 973 222 -025

203053 973 248 009

206550 972 118 042

206547 972 105 035

206565 972 146 056

203702 972 340 157

206876 972 155 061

205752 972 330 153

205922 972 108 036

206857 972 180 073

205393 972 190 -003

201513 971 345 015

203405 970 255 037

203069 968 295 -048

203071 967 321 -001

205635 967 390 029

203914 966 460 160

204601 962 252 061

203326 960 144 014

202003 958 876 437

206516 958 112 029

206517 958 283 120

206350 958 333 147

205932 958 180 066

203936 952 598 294

56

Product Current CSL ()

Current Annual Cost ($)

Additional Cost of the Optimal Policy ($)

206932 974 204 -051

203939 948 413 193

203977 945 1216 1242

206551 945 132 077

205389 934 598 047

203590 927 1223 607

206567 919 122 070

203591 905 1228 834

205390 887 291 074

203318 833 136 084

206393 776 319 289

203312 303 237 511

provides at least the desired customer service level For the 36 products the total additional

cost of the optimal policies is $542033

Of the remaining 383 products most have an existing policy that provides a customer

service level well above 975 When the customer service levels are rounded to one decimal

each of 253 products has a customer service level of 1000 and each of 347 products has a

customer service level of at least 990 In contrast each of only 58 products has an optimal

policy that yields a customer service level of at least 99034 For each of 37 products the

3 3 The cost associated with product 203312 is not included in the total

3 4 Optimal policies do not have customer service levels of exactly 975 because the function is discrete For low-demand products a unit step in s or S results in a relatively large change in the customer service level

57

current policy is identical to the optimal policy

The extremely high level of service provided by most policies allows for substantial

savings in the cost of inventory With current policies the estimated annual cost of inventory

for all 420 products is $420743 whereas the expected annual cost of the optimal policies is

only $186884 This amounts to a savings of approximately $2340 or over 55 of the current

cost of inventory

In order to further assess the savings the difference between the costs of the current

policies and the costs of the optimal policies are examined in more detail For each of the 383

products whose existing policy provides a customer service level above 975 the difference

between the estimated annual cost of the existing policy and the expected annual cost of the

optimal policy is calculated The resulting histograms of the savings in (a) dollars and (b)

percentage of current cost are shown in Figure 6 The particulars for the current and optimal

policies for all 420 products are provided in Appendix K

As evident from Figure 6(a) the majority of the expected savings in absolute terms

comes from only six products In order to examine these products more closely the current

and optimal policies along with the associated costs and savings for each of the six products

are listed in Table 5 Each current policy is very different than its corresponding optimal

policy resulting in excessive inventory for each product In fact over half of the total expected

savings could be realized by implementing the optimal policy just for each of these six products

Interestingly five of the six products are sealed beam headlights

58

90

10 - 20 30 - 40 50 - 60 70 - 80 90- 100

Savings ()

(b)

Figure 6 The distribution of savings of optimal policies in (a) dollars and (b) percentage of current cost

59

Table 5 Products with the largest potential absolute savings

PRODUCT

CURRENT OPTIMAL SAVINGS

PRODUCT s S Cost ($)

C S L

()

s S Cost ($)

C S L () ($) ()

2036801 194 200 64014 1000 37 48 bull 9245 975 54769 856

2036772 94 100 23652 1000 18 22 3850 978 19802 837

2036723 78 90 23043 1000 19 23 4454 975 18589 807

2030944 145 175 19522 1000 34 45 3727 975 15795 809

2036425 48 60 10491 1000 13 17 2682 977 7809 744

2036556 68 80 7265 1000 14 22 1699 975 5566 766

1 H6054 rectangular sealed beam headlight 2 H4656 rectangular sealed beam headlight 3 6053 rectangular sealed beam headlight 4 9004 halogen bulb 5 4652 rectangular sealed beam headlight 6 6104 7 sealed beam headlight

Although Figure 6(a) seems to indicate that the majority of the current policies are close

in cost to the optimal policies this is very misleading For example suppose the optimal policy

for a product costs $1 and that the current policy costs $2 In absolute terms the optimal

policy saves only $1 but in relative terms the optimal policy saves 50 of the current cost

When savings are viewed in this manner as in Figure 6(b) it is clear that substantial savings

can be realized for the majority of the products In fact both the mode and the mean of the

savings are between 40 and 50 of the current costs

In order to extrapolate the above savings to obtain the potential savings for product

category 20 at all 21 stores the differences between stores must be considered During the

60

pilot study the current policy for each of several products at each size of store was evaluated

The study suggested that medium and large stores have a similar amount of excess

inventory and that jumbo and extra jumbo stores have much more excess inventory One

could reasonably expect that the additional excess inventory allows for higher savings to be

realized at the jumbo and extra jumbo stores Thus because 18 of the 21 stores are

medium or larger the expected 40-50 savings at store 6 can be viewed as a lower bound

on the potential savings to Canadian Tire Pacific Associates from implementing optimal policies

for product category 20 in all its stores

Finally in order to obtain the overall potential savings the expected savings from

product category 20 must be extrapolated to all product categories Ideally the proportion of

inventory comprising each product category and the relative savings that can be realized for

each category would be used to obtain a good estimate of the overall potential savings

However as this information is not known only a crude extrapolation is possible Thus the

analysis proceeds under the assumption that a 40-50 savings can be realized in each product

category35

Before an estimate of the total potential savings in absolute terms can be obtained the

proportion of the current inventory to which the savings apply must be determined Up to this

point the analysis has been concerned solely with reducing inventory at the stores However

it is inappropriate to perform further analysis without considering the entire inventory system

including the central warehouse Because the lead time for orders from Toronto is highly

variable and considerably greater than those at individual stores Canadian Tire Pacific

3 5 During the pilot study the current policies for a paint brush and an ironing board were evaluated and found to result in excess inventory similar to that of product category 20 Although this in no way justifies the assumption it does lend some support

61

Associates must keep a fairly high safety stock at the warehouse Without knowing the

proportion of inventory serving as the safety stock it is inappropriate to assume any reduction

in inventory at the warehouse is possible Therefore the relative savings of implementing

optimal policies is applied only to the inventory held in the stores36 Of the estimated $28

million in inventory approximately half is held in the stores therefore the 40-50 savings

translates to a savings of approximately $55 to $7 million per year

B Sensitivity Analysis on the Ordering Cost and Holding Rate

The above analysis was based on estimates of the ordering cost and the annual unit

holding rate37 provided by management As described in Chapter II these estimates

incorporate a number of costs that are difficult to quantify or that are speculative Therefore

there is some question as to the accuracy of the estimates In this section errors of +$0015

in the ordering cost and +5 in the holding cost are assumed in order to determine the

sensitivity of the model to errors in these variables38

The analysis proceeds as follows Based on an ordering cost of $0080 and a holding

rate of 30 the optimal policies from the previous section were obtained39 Then assuming

3 6 Of course this assumes that the excess inventory currently held in the stores is not required as part of the safety stock Based on the large amount of inventory at the warehouse this assumption seems reasonable

3 7 Hereafter simply referred to as the holding rate

3 8 An error of +$0015 in the ordering cost implies an actual ordering cost of $0070 a relative error of approximately 21 and an error of -$0015 implies an actual ordering cost of $0100 a relative error of 15 An error of +5 in the holding rate implies an actual rate of 25 a relative error of 20 and an error of -5 implies an actual holding rate of 35 a relative error of approximately 17

3 9 These policies are referred to as optimal policies for the duration of this chapter

62

the actual ordering cost andor holding rate are different from managements estimates the

heuristic is used to obtain the true optimal policies Finally the optimal policies are

evaluated based on the actual ordering cost and holding rate and then compared to the true

optimal policies For each possible combination of ordering cost and holding rate three

numbers are recorded the number of products with an optimal policy that differs from its

true optimal policy the total cost of implementing the optimal policies given the actual

ordering cost and holding rate and the difference between the cost of implementing the

optimal policies and the cost of implementing the true optimal policies The results of the

analysis are summarized in Table 6

Table 6 A comparison of optimal policies to true optimal policies

Actual Ordering Cost

$0070 $0085 $0100

25 5 products $155439

$002

51 products $160590

$092

88 products $165464

$449

Actual Holding Rate

30 54 products

$181470 $182

48 products $192026

$088

35 79 products

$207270 $574

48 products $212940

$136

2 products $218312

$000

4 0 The table should be read as follows If the actual ordering cost is $0070 and the actual holding rate is 25 then each of 5 products has an optimal policy that is different from its true optimal policy The total cost of implementing the optimal policies is $155439 which is $002 more than implementing the true optimal policies

63

The model appears to be quite insensitive to relative errors in the order of 20 Under

the worst case scenario when one of the variables is over-estimated and the other is undershy

estimated the optimal policies result in a total cost of inventory which is only approximately

3 higher than the cost of the true optimal policies Further because demand is not changed

by errors in these estimates each of the optimal policies still provides a customer service

level of at least 975 under all of the scenarios Finally although the cost of the optimal

policies varies significantly for the different scenarios the relative savings remain virtually

constant Table 7 lists the cost of existing policies and in parenthesis the relative savings of

the optimal policies for each of the different combinations of ordering cost and holding rate

The cost of the existing policies varies in the same manner as that of the optimal policies the

relative savings for each of the scenarios remains around 55 Therefore even in the presence

of such errors in the ordering cost and holding rate the projected savings of implementing the

optimal policies would be unchanged

Table 7 The cost of existing policies and the relative savings of the optimal policies under various scenarios

Actual Ordering Cost

$0070 $0085 $0100

25 $350153 (56)

$358464 (55)

$366764 (55)

Actual Holding Rate

30 $412438 (56)

$429048 (55)

35 $474725 (56)

$483026 (56)

$491320 (56)

64

C Sensitivity Analysis on the Demand Probability Mass Function

In this section the sensitivity of the model to slight changes in the demand probability

mass function is investigated As described in Chapter II the demand function used in the

model is based solely on observed sales however this empirical function is only one possible

instance of the underlying true demand function For example had the data been collected

over a period both starting and finishing one month earlier many of the empirical functions

would be slightly different Consequently depending upon the degree to which the empirical

functions vary and the sensitivity of the model to these variations the optimal policies from

the model might be inaccurate For example recall the case of the amber driveway marker

only thirteen markers were sold and all were sold on the same day Because the empirical

demand function is used the model calculates an optimal policy of (1213) clearly this

policy is impractical Although this example is extreme it does indicate the danger of using

unrealistic empirical demand functions41

As discussed in Chapter II three classes of products were identified namely low

medium and high-demand products For this analysis the same method of classification

is used Two products are selected from each of these groups and two new distribution files

are created for each of the six products For each product one of the files contains two

additional days of data with each day having a demand equal to the maximum observed demand

for that product In the second file only one additional day of data is added but the demand

on that day is two more than the maximum observed demand The situations that could give

4 1 Certainly a rigorous analysis to determine both the extent of variation in the empirical demand functions and the effect of using the empirical functions needs to be performed However such an analysis is beyond the scope of this thesis only a brief analysis that might suggest the degree to which the model is sensitive is performed here

65

rise to such changes in the demand data are termed Scenario 1 and Scenario 2 respectively42

The analysis proceeds in a manner similar to the one in the previous section Given the

demand that was originally observed the optimal policies were obtained Now assuming the

true demand function corresponds to the demand under Scenario 1 the true optimal policy

for each product is calculated Then each optimal policy is re-evaluated using the true

demand function and compared to the true optimal policy The analysis is then repeated

assuming the true demand function corresponds to the demand under Scenario 2 Table 8

lists the results for each product and scenario

Table 8 True optimal and optimal policies for each demand scenario

Product 202101 (low demand)

Scenario s s Cost ($) C S L ()

1 True optimal 1 2 470 993

Optimal 1 2 470 993

2 True optimal 2 3 663 976

Optimal 1 2 461 875

Product 203507 (low demand)

Scenario s S Cost ($) C S L ()

1 True optimal 0 3 334 981

Optimal 0 2 258 972

2 True optimal 2 3 508 980

Optimal 0 2 273 702

4 2 To be clear the scenarios are defined as follows Scenario 1 The addition of two days each with a demand of max Scenario 2 The addition of one day with a demand of max+2

where max is the maximum observed daily demand for each product in the sample period

66

Product 201032 (medium demand)

Scenario s S Cost ($) C S L ()

1 True optimal 6 12 1377 976

Optimal 6 11 1338 973

2 True optimal 6 13 1426 977

Optimal 6 11 1338 970

Product 203039 (medium demand)

Scenario s S Cost ($) CSL ()

1 True optimal 5 19 256 975

Optimal 5 16 251 969

2 True optimal 5 20 258 976

Optimal 5 16 249 968

Product 203677 (high demand)

Scenario s S Cost ($) CSL ()

1 True optimal 18 23 3981 977

Optimal 18 22 3817 972

2 True optimal 18 23 3995 978

Optimal 18 22 3831 973

Product 203680 (high demand)

Scenario s S Cost ($) CSL ()

1 True optimal 38 49 9498 976

Optimal 37 48 9175 971

2 True optimal 37 49 9484 975

Optimal 37 48 9206 972

67

The model is most sensitive to changes or errors in the demand functions of low-

demand products When only one additional day of demand for max+2 units is added to the

data the optimal policy for each of the low-demand products results in a very low customer

service level In fact for one of the products the customer service level drops to almost 70

The increase in demand does lower the customer service level for each of the other the

products but that decrease is slight

The above analysis indicates the need for some kind of smoothing or modelling of the

demand data especially for low-demand products The inherent variation between two

instances of a true demand function could very well result in more pronounced changes than

those resulting from either of the above scenarios Therefore it is likely that many optimal

policies especially those for low-demand products would actually provide a much lower or

much higher customer service level than expected However by smoothing the data the

variation between instances of a true demand function can be reduced and the desired

customer service levels can be achieved

For example suppose that the demand for a product is stationary that the average

demand during the previous period was unusually low and that the optimal policy for this

period is based on the demand during the previous period Because the demand is stationary

the probability that the demand during this period is higher than that during the past period is

more than 50 In other words the probability that the demand during this period is higher

than that for which the optimal policy provides the desired expected customer service level

is greater than 05 Thus expected customer service level for this period is actually less than

that specified Now suppose instead that the optimal policy is based on smoothed data which

reflects very closely the expected demand The probability that the demand during this period

is higher than that for which the optimal policy provides the desired expected customer service

68

level is virtually 05 Thus as a result of smoothing the data the specified customer service

level should be expected

69

VII C O N C L U S I O N

In this thesis we developed a single-product single-location model of the inventory

system at Canadian Tire Pacific Associates Specifically we formulated a Markov chain model

to determine the long-run average cost of inventory for fixed values of s and S and then

developed a heuristic search to find near optimal policies quickly Also as a first step toward

developing an integrated interactive inventory control system we created a prototype of the

interface

According to the analysis performed here Canadian Tire Pacific Associates is currently

holding an excessively large inventory in its stores This study finds that Canadian Tire Pacific

Associates could reduce the cost of inventory for product category 20 at store 6 by

approximately 50 and still maintain customer service levels of at least 975 Further the

mean and modal values of savings for these products are between 40 and 50 Given the

similar excess inventory at the other stores and in all product categories we believe that savings

in this range are possible for all products and stores We estimate that the application of

inventory control as described in this thesis would result in annual savings of $55 to $7

million

Given the large potential savings to Canadian Tire Pacific Associates we recommend

that management proceed with the next phase of the project Specifically we recommend that

the model algorithm and interface be refined further that a forecasting model be developed

and that a data collection and storage scheme be designed

In order to reflect operational changes at Canadian Tire Pacific Associates subsequent

to the formulation of the model modifications must be made to allow for the lead time to

exceed the review period Several of the stores now have three picks per week and as a

70

result orders placed for these stores at one decision epoch do not arrive until after the next

decision epoch Because this violates one of the models assumptions the model must be

modified accordingly Note however that with three picks per week and a review period

of four days orders from only the previous decision epoch are ever outstanding at the current

decision epoch

Also the model should be refined to reflect the fact that the review period is not fixed

over the planning period In its present form the model assumes a fixed review period

however inventory at stores with two picks per week is actually reviewed on a four day -

three day cycle and inventory at stores with three picks per week is reviewed on a three day

- two day - two day cycle Rather than obtaining the steady state costs for fixed review period

the model should calculate the steady state cost for a period of one week during which multiple

replenishments occur Although the solutions are not expected to differ significantly as a result

of the change the model would reflect the actual process more accurately

Further work is also required on the search technique for locating near optimal

policies Although the heuristic search described in this thesis appears to work well the

decision maker must provide an upper bound on S prior to the search In practice this upper

bound is generally not known in advance Thus if the heuristic is to be used either a tight

theoretical bound must be found or a termination heuristic must be developed Alternatively

a new heuristic search could be developed based on additional or different assumptions

regarding the properties of the feasible space For example although some regions of non-

convexity were found for some products much of the feasible space appeared to be convex

A heuristic that assumes a convex feasible space would not require an upper bound on S and

might perform very well However before such a heuristic is implemented further work is

necessary to determine the degree of non-convexity for a variety of demand functions and

71

ordering and holding costs

In addition to making the above modifications we recommend that Canadian Tire Pacific

Associates begin work on the remaining two components of the inventory control system First

in order to allow for products with seasonal demand the system requires a forecasting model

to provide accurate estimates of demand for the coming period Second the system needs a

data collection and storage module that systematically collects sales data for each active

product and for each store and then updates the corresponding demand distributions Because

approximately 25 of demand is expected to result in lost sales under the optimal policies the

sales data will have to be adjusted accordingly in the future Also because the model appears

to be sensitive to slight changes in the demand distribution for low-demand products the

module should incorporate a technique for smoothing the empirical distributions for these

products Further work should be undertaken to determine the extent of variation in the

empirical distributions and the most appropriate smoothing technique However because of

the large amount of data required to forecast seasonal demand and because of the current lack

of historical data we recommend that management proceed immediately with the design of a

data collection scheme

Finally two additional issues that are beyond the scope of the next phase should be

addressed at some point First a monitoring system will be required once inventory control

is implemented in the stores The monitoring system will ensure that optimal policies perform

as expected and will indicated whether or not demand is adversely affected by the lower levels

of in-store inventory Second given the degree to which in-store inventory can be reduced

inventory control in the warehouse should be investigated Because approximately half of the

existing inventory is held in the warehouse the additional savings from reducing system-wide

inventory could be great

72

Afterword

Just prior to the completion of the work described in this thesis Mr Don Graham

principle owner of Canadian Tire Pacific Associates sold his shares in the company Shortly

thereafter the results of this thesis were presented to the new management group

Unfortunately the new management did not consider development of an inventory control

system to be a priority Nonetheless the prototype was delivered to Canadian Tire Pacific

Associates for possible in-house testing however at the time of this writing - almost one year

after the meeting - Canadian Tire Pacific Associates has not decided to continue with the

project

73

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Lee H L and S Nahmias (1993) Single-Product Single-Location Models in SC Graves A H G Rinnooy Kan and PH Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Muckstadt J A and R O Roundy (1993) Analysis of Multistage Production Systems in SC Graves A H G Rinnooy Kan and PH Zipkin (eds) Logistics of Production and Inventory Handbooks in Operations Research and Management Science vol 4 North-Holland Amsterdam

Porteus E L (1985) Numerical Comparisons of Inventory Policies for Periodic Review Systems Oper Res 33 134-152

Puterman M L (1994) Markov Decision Processes Discrete Stochastic Dynamic Programming Wiley New York

Silver E A and R Peterson (1985) Decision Systems for Inventory Management and Production Planning 2nd edition Wiley New York

Taylor H M and S Karlin (1994) An Introduction to Stochastic Modeling revised edition Academic Press San Diego

74

Tijms H C and H Groenevelt (1984) Simple approximations for the reorder point in periodic and continuous review (sS) inventory systems with service level constraints European J Oper Res 17 175-190

75

Appendix A

A Portion of the Sales File for Product 200001 a 30-amp Inline Fuse

200001 283 1190 0 0 2 0 0 1 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 200001 284 1190 2 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 200001 285 1190 0 0 4 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 286 1190 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 200001 287 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 200001 288 1190 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1 0 2 200001 289 1190 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 290 1190 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 200001 291 1190 0 0 2 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 200001 292 1190 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 200001 294 1190 0 0 0 1 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 1 200001 295 1190 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 200001 296 1190 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 200001 297 1190 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 200001 298 1190 0 0 1 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 200001 299 1190 0 0 1 0 0 0 1 0 1 0 0 2 0 0 0 0 0 2 1 1 0 200001 300 1190 0 0 1 0 0 0 0 1 0 0 0 0 0 2 2 0 0 0 0 3 0 200001 301 1190 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 200001 303 1190 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 200001 304 1190 1 0 0 0 0 0 0 0 1 0 0 2 0 0 0 0 0 0 0 0 0 200001 305 1190 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 200001 306 1190 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 2 0 0 1 0 200001 308 1190 0 0 0 1 0 1 0 0 0 2 0 1 0 0 0 0 1 1 1 0 0 200001 309 1190 1 3 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 200001 310 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 200001 311 1190 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 200001 312 1190 0 1 1 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 200001 313 1190 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 2 0 200001 314 1190 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 200001 315 1190 0 0 2 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 200001 316 1190 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 200001 317 1190 2 0 0 1 0 0 0 0 0 0 5 1 0 0 0 1 0 0 0 0 0 200001 318 1190 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 1 200001 319 1190 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 200001 320 1190 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 200001 321 1190 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 1 200001 322 1190 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 200001 323 1190 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 200001 324 1190 0 0 2 0 0 0 0 0 0 1 0 0 0 0 1 0 0 2 1 1 0 200001 325 1190 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3 200001 326 1190 0 0 1 0 1 0 0 0 2 0 0 1 0 0 0 1 0 1 0 0 1 200001 328 1190 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 200001 329 1190 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 200001 330 1190 0 0 1 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 200001 331 1190 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 200001 332 1190 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 200001 333 1190 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 200001 334 1190 0 0 2 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 200001 335 1190 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 200001 336 1190 0 0 0 1 0 0 3 0 0 0 0 0 2 0 0 0 0 0 0 0 0

(253 subsequent lines omitted)

76

Appendix B

Sample Distribution Files

a) A low-demand product - an automobile back-up alarm

PRODUCT NUMBER 202101 PRICE $ 684

DMD

3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

N 2 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 2 2 0 1

1 5 3 8 12 5 7 11 6 11 5 4 11 10 19 4 2 5 14 5 6 4

0 302 304 299 295 302 300 296 301 296 302 303 295 296 288 303 305 301 290 300 301 302

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

b) A medium-demand product - a 168 bulb

PRODUCT NUMBER 203039 PRICE $ 51

DMD

16 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 1 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0

9 0 0 2 5 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0

8 0 0 2 5 0 0 2 0 1 1 3 0 1 3 0 0 1 1 0 0 1

7 2 1 3 1 0 0 1 0 2 3 1 1 4 1 1 0 2 1 1 1 1

6 4 7 1 10 3 1 3 2 1 3 5 2 7 4 1 0 1 1 3 3 4

5 1 7 7 17 0 2 11 1 5 4 1 2 6 9 6 2 6 9 4 4 1

4 5 10 10 16 5 5 19 10 13 8 11 9 13 10 12 7 5 15 5 8 3

3 10 12 13 27 4 6 24 11 10 13 14 9 23 14 18 8 14 18 14 12 14

2 14 30 36 43 16 16 43 32 38 39 39 34 47 36 41 21 19 39 20 34 24

1 30 56 68 56 37 30 70 48 67 61 44 75 62 63 48 28 52 61 63 53 47

0 240 184 165 126 240 247 133 202 169 175 189 173 143 165 180 240 206 161 197 192 212

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

77

c) A high-demand product - a H4651 rectangular sealed beam headlight

PRODUCT NUMBER 203677 PRICE $ 857

DMD

14 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

13 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 2 3 0 0 1 0 0 0 0 1 1 1 0 0 0 0 2 0 0

11 0 1 3 1 0 0 4 1 0 0 0 0 0 0 0 0 1 0 0 0 1

10 0 0 4 4 1 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 1

9 0 3 14 11 0 1 7 1 0 2 0 8 0 1 0 0 0 2 1 0 0

8 0 4 12 16 1 0 18 4 3 5 0 2 1 4 2 1 1 0 0 1 3

7 1 2 12 13 1 6 19 3 5 9 0 11 2 5 4 0 2 3 3 1 4

6 2 9 22 21 6 10 29 12 10 11 4 24 3 7 10 4 7 11 6 3 14

5 7 13 35 33 8 16 29 19 18 8 3 26 15 26 15 12 9 14 13 11 19

4 19 35 34 43 32 30 41 31 32 47 11 42 20 33 30 31 20 31 22 21 31

3 32 63 49 54 42 60 51 40 51 35 33 55 39 51 39 38 39 57 58 40 54

2 46 59 56 55 73 63 61 65 63 61 63 54 68 85 65 53 72 69 70 77 63

1 109 79 42 40 80 64 32 80 74 77 99 55 97 57 83 93 80 75 81 97 73

0 91 39 21 13 63 57 11 51 51 52 94 29 61 35 59 75 76 45 51 56 44

STORE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

78

Appendix C

A Proof that the Limiting Probability Distribution is Independent of the Initial State

In this appendix we show that provided 0 lt PrDT = 0 lt 1 the Markov chain

corresponding to an inventory model with a (sS) policy has a limiting probability distribution

7T where 0 lt TTJ lt 1 for y = 0 1 S that is independent of the initial state Further the

limiting distribution is the unique nonnegative solution of the equations

Tt P = TI (AD

s and J gt = 1 (A 2)

i=0

where P is the (S + l) x (S + l) transition probability matrix whose first n rows (and columns)

correspond to the n recurrent states in the chain and whose remaining rows (and columns) if

any correspond to transient states43 Prior to the proof several preliminary results are

obtained

RESULT 1 The Markov chain has exactly one positive recurrent class and perhaps some

transient states

First note that in the absence of any absorbing states a Markov chain with a discrete

finite state space must have at least one positive recurrent class containing at least two states

Now consider the Markov chain for an inventory model with a (sS) policy

a) State i gt s cannot be absorbing because there is a positive probability of reaching

j lt i (Since PrDT gt 0 gt 0)

4 3 Note that since the rows and columns of P are reordered the elements of it must be reordered accordingly

79

b) State lt s cannot be absorbing because by the rules of the ordering policy and

the assumption that PrDT = 0 gt 0 there is a positive probability reaching S

Therefore the chain has no absorbing states and must have at least one positive recurrent class

Next consider the subset of states A = i | i lt s Because the state space is finite

and PrDT gt 0 gt 0 the process must return to at least one state in A in other words at least

one state in A must be recurrent Let k G A be a member of a recurrent class dt Because

of the ordering policy and the assumption that PrDT = 0 gt 0 k -raquo S Also by definition

there must be a positive probability of returning to k from S S -gt k Therefore S G 3w

Further because S is reachable from all states in A i G A is either transient with i -raquo dt or

recurrent with pound iR

Now consider the subset of states B = | j gt s Because PrDT gt 0 gt 0 the

chain will eventually reach a state i G A If i is transient then j must be transient also with

y -gt 3 if i G 9 then j is either transient with j -gt dt or recurrent with j E dt Thus the

Markov chain has exactly one recurrent class and perhaps some transient states

RESULT 2 The limiting probability distribution ir for the Markov chain corresponding to the

nxn submatrix Pm = [Ptj] i j G dt is independent of the initial state and is the

unique nonnegative solution of the equations

TC-Prr = re (A3)

and J2 K = 1 ( A 4 )

jt=i

where the rows and columns of are numbered from 1 to n rather than

Since if i j i and j must be in the same equivalence class

80

according to their associated states

First note that both the original Markov chain and the one corresponding to the

submatrix P R R are aperiodic because PrDT = 0 gt 0 Therefore given that 9 is a positive

recurrent aperiodic class the limiting distribution ir is uniquely determined by (A3) and (A4)

(Taylor and Karlin 1994 Theorem 44 p213) Further because there are no absorbing states

or other recurrent classes in the original chain the process must eventually enter the recurrent

class 9 regardless of the initial state Therefore in the limit the probability of being in state

i G 9 is independent of the initial state

RESULT 3 The limiting probability distribution for the transient states is the null vector

independent of the initial state

This result follows directly from the definition of a transient state

PROOF Suppose the states are reordered such that the first n rows (and columns) of P

correspond to the n recurrent states of the Markov chain Then P can be partitioned into four

submatrices

PRK - the n x n submatrix [Py] i j E 9

P T T - the (S + 1-I) x (S + l-n) submatrix [Py] i j g 9

P T R - the (S + l-n) x n submatrix [Py] i pound 9 j G 9

P R T - the n x (S + l-n) submatrix [Py] i G 9 j pound 9t By definition this is a matrix

of zeros

(Al) can then be rewritten as

81

(ltR TtT) P P L TR TT

I-P RR 0

p I-P TR 1 TT

(TtR TCT)

(0- 0T)

where irR denotes the limiting probability distribution for the n recurrent states and irT denotes

the distribution for the remaining S + l-n transient states Solving for TT t gives

TCr-0 + itT-(I-PTI) = 0T

itT-(I-PJJ) = 0 r

i c r = (I-P^-Oj

Since (I - PTT)1 always exists (Puterman 1994 Proposition A3 p 593) irT = 0X which is

in agreement with Result 3 Now solving for 7rR yields

(A2) can be rewritten as

bullRPRR 1 1R (A5)

(A6) S + l

k + L k =1

k=l k=n+

k=i

where ir contains the appropriately reordered steady state probabilities Since (A5) and (A6)

are equivalent to (A3) and (A4) respectively Result 2 holds Therefore the Markov chain

corresponding to an inventory model with a (sS) policy has a limiting distribution that is

independent of the initial state and is the unique nonnegative solution to (Al) and (A2)

82

Appendix D

A Measure for the Steady State Customer Service Level

In this appendix we show that the appropriate measure for the steady state customer

service level is given by

_ E[Demand satisfied during RP] E[Total demand during RP]

Consider the following two measures of the steady state customer service level

Measure 1 Suppose the customer service level is calculated only after n review periods have

passed Denoting DS as the demand satisfied in review period i and DT as the total demand

in period i the customer service level after n periods CSL(n) is

DS + DS0 + + DSbdquo CSLn) = - -

DT + DT + + DT

Dividing both the numerator and denominator by n taking the limit as n -raquo oo and applying

the Law of Large Numbers yields

CSL = lim CSL(n) n-degdeg

- [DS + DS + + DS] = l i m ^ J 1 1

raquo~ - [DT + DT + + DT]

E[DS]

E[DT]

83

Measure 2 Suppose the customer service level is calculated for each of n review periods and

then averaged Thus the customer service level after n periods CSL(n) is given by

CSL(n) = -DS DS

DT DT DT_

Taking the limit as n -gt oo yields

CSL = lim CSL(n)

lim mdash DSl DS2

DT DT DT_

DS

DT

Now consider the scenario in which there is one item on the shelf to begin each review

period and in which demand is stochastic with PrDT = 2 = Vi and PrD x = 100 = Vi

According to Measure 1 the customer service level is

CSL - = ^ W + ^ W = plusmn bdquo 002 E[DT) 1 (2) + 1(100) 51

while according to Measure 2 the customer service level is

CSL = E DS DT

1(1) + i(-Ls

22) + 2 100

51 200

026

Clearly from managements point of view the first measure is more indicative of the

actual level of service being provided The problem with the second measure is that equal

84

weight is given to each ratio regardless of the magnitude of the numbers For example with

respect to Measure 2 the events LSLR = 12 and LSLR = 50100 are identical however

with respect to lost sales the second event is much worse than the first Therefore the proper

measure for the steady state customer service level is Measure 1

85

Appendix E

The Calculation of the Conditional Expected Demand Not Satisfied During a Period of T Consecutive Days

Denoting D x as the demand during a period of T consecutive days and j as the inventory

on hand at the beginning of the period the conditional expected demand not satisfied during the

period E[(DT-y)+] is calculated as follows

E[(DT-j)+] = pound ltk-j)Pr[DT = k k=j+l

= pound (k-j)PrDT = k] + J2(k-j)PrDT = k] - pound (fc-)VJgtr = fc k=j+l k=0 k=0

= J2(k-j)Pr[DT=k -J2(k-j)PrDT = k] k=0 k=0

= YkPrDT=k) - J2jPrDT=k - Yk-j)PrDT=k) Jfc=0 k=Q k=0

= Ti - j - Y(k-j)PrDT=k 7-1

= Tx - j + YltJ-k)PrDT = k jt=0

where fx is the mean daily demand

86

Appendix F

Justification of the Updating Technique for the New Policy (s+mS)

Let the probability transition matrix corresponding to the policy (sS) be denoted by the

(S + l)x(S + l) matrix P with elements py Define s = s+m where m is an integer satisfying

the condition 0 lt s+m lt S Let the probability transition matrix of the new policy (sS)

be denoted by the (S + 1)X(S + 1) matrix 7gt with elements py Using the equations developed

in Chapter III the elements of P are found as follows

CASE 1 The lead time is less than the review period

A m gt 0

For 0 lt i lt s

( i-i pound [PrDL=kPrDT_LgtS-k]] + PrDLziPrDT_LplusmnS-i k=0

i-l 53 [PrDL=kPrDT_L = S-j-k]] + PrDLziPr[DT_L = S-j-i) k=0 s-j

52 [Pr[DL = kPrDT_L = S-j-kn k=0

ifj = 6

ifOltjz S-i

if S-i ltj ltS

a) 0 lt lt s

Pij

= lt Pij

Pij

= pij

b) s lt i lt s This does not simplify but it is identical to (2)

87

For s lt i lt S

PrDTgt i

p = lt PrDT = i-j

0

Pfj ifj = 0

= lt Pij ifOltjlti

Pij if i ltj plusmn S

= pij

B m lt 0

ifj = 0

if 0 ltj lt i

if i ltj plusmn S

For 0 lt i lt s

i-l 53 [PrDL=kPrDT_LplusmnS-k]] + PrDLgtiPrDT_LplusmnS-i

i-l E [PrZgti = fciVZ)7W=S--fc] +PrD tgtiP-D 7 t = S-7-

p y laquoy = o

P i j ifOltjS-i

vPij ifS-iltjS

ifj = 0

ifOltjltS-i

if S-i ltjltS

For s lt i lt S

fPrJgtrgt- 17=0 = PrDT = i-j] ifOltjlti

0 ifiltjplusmnS

88

a) s lt lt s This does not simplify but it is identical to (3)

b) s lt i lt S

PrDTgti

= bull PrDT = i-j]

0

17=0

if0ltjplusmni

ifiltjS

= Pt

ifj = 0

if0ltjlti

ifiltjplusmnS

= Pi

CASE 2 The lead time is equals the review period

A m gt 0

For 0 lt i lt s

0 if0ltjlt S-i

PrDTgti ifj = S-i

PrDT=S-j] ifS-iltjltS

a) 0 lt i lt s

= bull y Pij

Pij

ifOplusmnjltS-i

ifj = S-i

ifS-iltjS

= Pij

b) s lt i lt s This does not simplify but it is identical to (4)

89

For s lt i lt S

PrDTgti)

Pr[DT=i-j

0

laquof 7=0

ifOltjplusmni

if i ltj lt S

Pa

Pij

if 7 = 0

if 0 lty lt i

ifiltjltS

B w lt 0

For 0 lt t lt s

ro PrJrgt PrDT = S-j

ifOzjltS-i

ifj = S-i

if S-i ltj lt S

Pij

ifOzjltS-i

ifj = S-i

if S-i ltj ltS

= Pi

For s lt i lt S

PrDTgti

= PrDT = i-j

0

i 7 = o ifOltjplusmni

ifiltjltS

a) s lt i lt s This does not simplify but it is identical to (3)

90

PrDTzi ifj = 0

PrDT = i-j] ifOltjplusmni

0 ifiltjltS

Py ifj = 0

ptj ifOltjplusmni

Pij if iltj plusmnS

Appendix G

Justification of the Updating Technique for the New Policy (sS+m)

Let the probability transition matrix corresponding to the policy (sS) be denoted by the

(S + l)x(S + l) matrix P with elements p(gt Define S = S+m where m is any non-negative

integer Let the probability transition matrix of the new policy (sS) be denoted by the

(S+m+l)x(S+m + l) matrix IP with elements ptj Then using the equations developed in

Chapter III the elements of 7 are found as follows

CASE 1 The lead time is less than the review period

For 0 lt i lt s

pound [PrDL=kPrDT_LplusmnS-k] + PrDLi)PrDT^Sgt- k=0

i-l pound [Pr[DL = kPrDT_L = S-j-k- +Pr[DLiPrDT_L = S-j-i Jt=0

s-i Y [Pr[DL=kPrDT_L = S-j-k] k=0

i - l

ifj=0

ifOltjltS-i

ifS-iltjltS

pound [PrDL = kPrDT_LS-k + Pr[DLi)PrDT_LplusmnS-i) if j = 0 fc=0

i - l = pound [PrDL=kPrDT_L = (S+m)-j-k] + PrDLgtiPr[DT_L = (S+m)-j-i ifOltj lt (J5+m)-i

if (S+m)-iltj lt(S+m)

k=0

i - l

pound (Sm)-j

pound [PrDL = kPrDT_L = (S+m)-j-k]] k=0

92

I-l

53 [PrDL = kPrDT_LgtS-k] + PrDLziPrDT_LS-i ifj = 0 k=0

i-l

Y[PrDL = kPrDT_L = S-(j-m)-k] + Pr[DLziPrDT_L = S-(j-m)-i] ifOltjzS+m-i k=0 S-(j-m)

53 [PrDL = kPrDT_L = S-(j-m)-k]] if S+m-i ltj lt S+m

i - l

53 [PrDL = kPrDT_LzS-k] + PrDLziPrDT_LzS-i] ifj = 0 =0

i - l

13 [Pr[DL = kPrDT_L = S-(j-m)-k] + Pr[DLziPr[DT_L = S-(j-m)-i ifOltjz S+m-i k=o and 0 z j-m

Pi(j-m)

Pi(j-m)

if 0 ltj lt S+m-i and 0ltj-mltS-i

if S+m-i ltj lt S+m and S-i lt j-m zS

i - l

Y [PrDL = k)PrDT_LzS-k] + PrDLi)PrDT_LplusmnS-i ifj = 0

k=0 i - l

53 [PrDL = kPrDT_L = S-(j-m)-k] + PrDLziPrDT_L = S-(j-m)-i ifOltjzm k=0

Piij-m)

Piij-m)

if mltj lt S+m-i

if S+m-i ltj lt S+m

i - l

YlPrDL = kPr[DT_L^S-k + PrDLi)PrDT_LS-i) ifj = 0

i - l

53 [PrDL = kPrDT_L = S-j-k] + PrDLgtiPrDT_L = S-j-i ifOltj i m k=0

Pi if mltj ltS

which for 0 lt j lt m is identical to (2)

93

For s lt i lt S

Pu =

PrDTgti

PrDT = i-j]

0

if0ltjlti

ifiltjltS

a) s lt i lt S

PrDTgti 7=0

PrDT = i -j) if0ltjlti

0 if i ltj lt S

0 ifSltjltSgt

ifj=0

if 0 ltj lt i

if i ltj plusmn S

0 ifSltjltS

ifOltjltS

0 ifSltjltSgt

b) S lt i lt S This does not simplify but it is identical to (3)

CASE 2 The lead time equals the review period

For 0 lt i lt s

0 ifO lt jltS-i

PrDTgti ifj=S-i

PrDT=S-j ifS-iltjltS

94

0 ifOplusmnjlt(S+m)-i

PrDT ifj = (S+m)-i

PrDT = (S+m) -j) if (S+m)-i ltj ltL (S+m)

0 if 0 lt jltS+m-i

PrDTzi] if j = S+m-i

PrDT = S-(j-m) if S+m-i ltj lt S+m

0 if 0 ltj lt S+m-i and 0 ^ j-m

Pi(j-m) if 0ltjlt S+m-i and 0ltj-mltS-i bull

Pi(j-m) if j = S+m-i and j-m = S-i

Pi(j-m) if S+m-i ltj z S+m and S-i lt j-m ^S

0 if 0 zj ltm

Pi(j-m) if mlt jlt S+m-i lt

Piij-m) if j = S+m-i

if S+m-i ltj lt S+m

1deg if 0 ltj ltm

[PiV-m) ifmltjltS

which for 0 lt i lt s is identical to (3)

For s lt lt S

PrDTzi

= PrDT = i-j)

0

ifj = 0

if 0 ltj ltL i

ifKjzS1

95

a) s lt i lt S

PrDTzi

PrDT = i-j]

0

0

17=0

if0ltjplusmni

ifiltjS

ifSltjltSgt

Pij 7=0

Pij if0ltjii

Pij if i ltj ltL S

0 ifSltjltS

Pij ifOzjiS

0 ifSltjltS

b) S lt i lt S This does not simplify but it is identical to (3)

96

Appendix H

A Hypothetical Consultation

In this appendix the interface during a hypothetical consultation is depicted The figures

accurately represent the various screens that are presented to the manager with the exception

that all lines and text are really white on a blue background During the consultation described

below only the choices made by the manager are hypothetical all the results are real

The title screen as depicted in Figure A - l appears initially upon execution of the

program In order to proceed to the main menu the manager presses the enter key

O I C S

Optimat Inventory Control System

For the exclusive use of

Canadian Tire Pacific Associates

Developed at The Faculty of Commerce

and Business Administration The University of British Columbia

Brian Kapalka Kaan Katircioghi Marty Puterman

Press ltEntergt to continue

Figure A - l The title screen

97

Main Menu 1 Select data directory

ctircdatadst

2 Select product and store combination Product 200001

3 Select scenario Review Period 4 Lead Time 3

4 Select search parameter Smax 25

5 Execute program

6 Exit to DOS

Please enter the desired option 5

Figure A-2 The main menu

Figure A-2 depicts the main menu of the program it is from here that the manager

enters all the review parameters If the manager wishes to change one of the parameters from

its default value he or she selects the appropriate number and enters the correct information

in the window presented The new settings will then be shown in place of the default values

In this example the relevant data files are located in the default directory ctiredatadst and

the manager has selected the productstore combination of product 200001 and store 6 Further

the manager has selected a review scenario of a 4-day review period and a 3-day lead time and

an upper search limit of 25 Finally in order to proceed with the analysis the manager

executes the main module of the program by selecting option 5

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