Statistical Inventory control models

Using Excel

Learning objective

After this class the students should be able to:

• calculate the appropriate order quantity in the face of uncertain demand using Excel and Cumulative Probability for Newsboy Model simplified.

Time management

The expected time to deliver this module is 50 minutes. 30 minutes are reserved for team practices and exercises and 20 minutes for lecture.

Introduction

We will study situations in which inventory cannot be

carried from period to period similar to Newsboys

Model.

• perishable products are fruits and vegetables in

supermarkets.

• products that rapidly become obsolete, such as fashion

items, and

• those that are bought for specific time periods, such as a

promotional sale for a holiday.

The Strawberry Ordering Model Cora, buyer for the Fresh Foods supermarket, is

considering the computer specifications for the ordering of strawberries.

Baskets of strawberries are delivered daily:

• If she orders too few, there will be many stockouts, sales will be lost, and profit will be low.

• If she orders too many, there will be a surplus of strawberries in the evening that will have to be unloaded to canneries at a large discount.

What quantity should Cora order?.

Data

Each basket of strawberries sells for $6.00,

the cost is $4.00, and the salvage value of any surplus sold to

a cannery is $3.00. So, each unit sold brings a profit of

$2.00, and each unit salvaged leads to a loss of $1.00.

Data

basket of strawberries price: $6.00, basket of strawberries cost: $4.00, and the salvage value: $3.00. Each unit sold brings a profit of $2.00,

and each unit salvaged leads to a loss of $1.00.

Decision tree

Cora knows from past computer records that most daily sales are between 11 and 20 baskets, so she has 10 alternatives for the order quantity: 11, 12, . . . , 20. This decision tree visually represents her choices and possible outcomes.

Dealing with uncertainty

Cora do not have enough information to derive a sophisticated probability distribution, then…

She assumes a uniform distribution and sets the probability of each of the ten values equal to 0.1.

The model

2

3

4

5

6

7

A B C D

$4.00 $4.00$6.00 $6.00$3.00 $3.00$2.00 =D4-D3

-$1.00 =D5-D3

Model InputsCostPriceScrap ValueProfitScrap Loss

Go to worksheet

General Profit function

shortageof cost hoverage of costkunit per costc

unit per pricepprofitP

shortageDQifQDhQcpoverageDQifDQkQcp

P

)()()()()()(

Individual Profits

111213141516171819202122

A B C D E F G H I J K L

11 12 13 14 15 16 17 18 19 2011 22 21 20 19 18 17 16 15 14 1312 22 24 23 22 21 20 19 18 17 1613 22 24 26 25 24 23 22 21 20 1914 22 24 26 28 27 26 25 24 23 2215 22 24 26 28 30 29 28 27 26 2516 22 24 26 28 30 32 31 30 29 2817 22 24 26 28 30 32 34 33 32 3118 22 24 26 28 30 32 34 36 35 3419 22 24 26 28 30 32 34 36 38 3720 22 24 26 28 30 32 34 36 38 40

Individual Profits

Dem

and

- D

Order quantity - Q

)QDfP ,(C13 =IF($B13<=C$12,$C$6*$B13,$C$6*C$12)+IF($B13< C$12,(C$12- $B13)*$C$7,0)

Probability

262728293031323334353637

A B C D E F G H I J K L

11 12 13 14 15 16 17 18 19 2011 10% 10% 10% 10% 10% 10% 10% 10% 10% 10%12 10% 10% 10% 10% 10% 10% 10% 10% 10% 10%13 10% 10% 10% 10% 10% 10% 10% 10% 10% 10%14 10% 10% 10% 10% 10% 10% 10% 10% 10% 10%15 10% 10% 10% 10% 10% 10% 10% 10% 10% 10%16 10% 10% 10% 10% 10% 10% 10% 10% 10% 10%17 10% 10% 10% 10% 10% 10% 10% 10% 10% 10%18 10% 10% 10% 10% 10% 10% 10% 10% 10% 10%19 10% 10% 10% 10% 10% 10% 10% 10% 10% 10%20 10% 10% 10% 10% 10% 10% 10% 10% 10% 10%

ProbabilityOrder quantity - Q

Dem

and

- D

Expected Profits

41424344454647484950515253

A B C D E F G H I J K L

11 12 13 14 15 16 17 18 19 2011 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.312 2.2 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.613 2.2 2.4 2.6 2.5 2.4 2.3 2.2 2.1 2.0 1.914 2.2 2.4 2.6 2.8 2.7 2.6 2.5 2.4 2.3 2.215 2.2 2.4 2.6 2.8 3.0 2.9 2.8 2.7 2.6 2.516 2.2 2.4 2.6 2.8 3.0 3.2 3.1 3.0 2.9 2.817 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.3 3.2 3.118 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.5 3.419 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 3.720 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0

Total 22.0 23.7 25.1 26.2 27.0 27.5 27.7 27.6 27.2 26.5

Profit expected

Dem

and

- D

Order quantity - Q

Optimum727ProfitExpected17Q .*

0.0

5.0

10.015.0

20.0

25.0

30.0

11 12 13 14 15 16 17 18 19 20

Order quantity

Exp

ecte

d P

rofi

ts

A mathematical shortcut

The solution method used to help Beth just described enumerates all alternatives and selects the best one.

• This "brute force" approach is not practical when there are too many alternatives.

• Fortunately, there is a mathematical procedure for finding the optimal order quantity.

Notation

P=Unit sales price C=Unit cost S=Unit salvage value CF=Critical factor

The critical factor is calculated as CF= (P- C)/(P- S)

Procedure to find Q*

Plot the cumulative probability distribution of demand.

Mark point A on the y-axis at the value of CF.

Move horizontally to point B on the curve. Drop vertically to point C on the x-axis.

The point immediately to the right is Q*.

The strawberry problem

P= $6.00

C = $4.00

S = $3.00

CF= (6 ‑ 4)/(6 ‑ 3)=2/3

Q* = 17 baskets

Exercise

Robin Lowe, a buyer at the Newstorm Department Store, must decide how many high‑fashion hats to order.

The unit sales price P = $125; The cost C = $60, and there is no salvage value because Robin

does not want any of the high‑fashion item sold by some discount house.

Exercise

Probability equally

distributed

<100 0.0%Between 100 and 109 0.5%Between 110 and 125 5.0%Between 126 and 150 0.6%

>150 0.0%

Demand

How many hats should she order?

Use the method used in this class to solve this problem (20 minutes)

Reflections

Each team is invited to analyze the following insights, based on the statistical model (10) minutes):

1. “Cycle stock increase as replenishment frequency decrease”

2. “Safety stock provide a buffer against stockout”

Reference

Operations Management Using Excel .Weida; Richardson and Vazsony, Duxbury, 2001, Chapter 6, p.136-143