Inventory Management &

Risk Pooling

Introduction

General Motors in 1984:

Logistic network consisted of 20,000 supplier plants, 133

parts plants, 31 assembly plants, and 11,000 dealers.

Freight transportation costs were about $4.1 billion, of

which 60 percent for material shipments.

GM inventory was valued at $7.4 billion, of which 70

percent was WIP and the rest was finished vehicles.

Response:-Inventory Management in Supply Chain

Goals of Inventory Management

By effectively managing inventory: GM has reduced parts inventory and transportation costs

by 26% annually Xerox eliminated $700 million inventory from its supply

chain Wal-Mart became the largest retail company utilizing

efficient inventory management

Reduce Cost, Improve Service

Inventory Levels FinancialInvestment

Operational Need

Inventory

Where do we hold inventory?

Suppliers and manufacturers

warehouses and distribution centers

retailers

Types of Inventory: General classification

WIP

raw materials

finished goods

Functions of Inventory

To meet anticipated demand

To smooth production requirements

To decouple operations

To protect against stock-outs

To take advantage of order cycles

To help hedge against price increases

To take advantage of quantity discounts

Factors Affecting Inventory Policy

Demand Characteristics: known in advance or random

Lead Time

Number of Different Products Stored in the Warehouse

Economies of scale offered by suppliers & transport

companies

Length of Planning Horizon

Service level desired

1000 2000 3000 4000 5000 6000

0

50

100

150

200

250

300

350

Ordering (Acquisition)Costs

Holding or Carry

ing CostsTotal CostsEconomic Order Quantity

Economic Order Quantity Model

Assuming demand certainty

Trade-offs between setup costs and inventory holding costs, but ignores issues such as demand uncertainty and forecasting.

Single Period Model Without Initial Inventory

Case: Swimsuit Production

A company designs, produces, and sells summer fashion

items such as swinsuits.

The company has to commit itself six months before summer

to specific production quantities for all its products

– predicting demand for each product.

The trade-offs are clear: overestimating customer demand

will result in unsold inventory while underestimating

customer demand will lead to inventory stockouts and

loss of potential customers.

Demand forecast

forecast averages about 13,000

The marketing department uses historical data from the last five years, current economic conditions, and other factors to construct a probabilistic forecast of the demand.

11% 11%

28%

22%

18%

10%

0%

5%

10%

15%

20%

25%

30%

8000 10000 12000 14000 16000 18000

Unit sales

Swimsuit Costs

Production cost per unit (C): $80

Selling price per unit (S): $125

Salvage value per unit (V): $20

Fixed production cost (F): $100,000

Q is production quantity, D: demand

Profit = Revenue - Variable Cost - Fixed Cost + Salvage

Swimsuit Two Scenarios

Scenario One: Suppose you make 12,000 jackets and demand ends up

being 13,000 jackets. Profit = 125(12,000) - 80(12,000) - 100,000 = $440,000

Scenario Two: Suppose you make 12,000 jackets and demand ends up

being 11,000 jackets. Profit = 125(11,000) - 80(12,000) - 100,000 + 20(1000) =

$ 335,000

Swimsuit Best Questions ?

Find order quantity that maximizes weighted average profit?

Will this quantity be less than, equal to, or greater than average demand?

How much to Make?

Marginal cost Vs. marginal profit if extra jacket sold, profit is 125-80 = 45 if not sold, cost is 80-20 = 60

So we will make less than average

Swimsuit Expected Profit

Expected Profit

$0

$100,000

$200,000

$300,000

$400,000

8000 12000 16000 20000

Order Quantity

Pro

fit

Swimsuit : Important Observations

Tradeoff between ordering enough to meet demand and ordering too much

Several quantities have the same average profit Average profit does not tell the whole story 9000 and 16000 units lead to about the same average

profit, so which do we prefer?

Swimsuit Expected Profit

Expected Profit

$0

$100,000

$200,000

$300,000

$400,000

8000 12000 16000 20000

Order Quantity

Pro

fit

Case: Swimsuit Production

But Need to understand risk associated with certain

decisions.

A frequency histogram provides information about

potential profit for the two given production

quantities, 9,000 units and 16,000 units. The

possible risk and possible reward increases as we

increase the production size.

Probability of Outcomes

0 0 0 0 0

0.11

0.89

0 00

0.11 0.11

0 0

0.28

0

0.22

0

0.28

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

-3E

+0

5

-2E

+0

5

-1E

+0

5

0

10

00

0

20

00

0

30

00

0

40

00

0

50

00

0

60

00

0

Cost

Pro

ba

bil

ity

Q =9000

Q =16000

Key Points from this Case

The optimal order quantity is not necessarily equal to

average forecast demand

The optimal quantity depends on the relationship between

marginal profit and marginal cost

As order quantity increases, average profit first increases

and then decreases

As production quantity increases, risk increases. In other

words, the probability of large gains and of large losses

increases

Single Period Model With Initial Inventory

Initial Inventory

Suppose that one of the jacket designs is a model produced last year.

Some inventory is left from last year Assume the same demand pattern as before If only old inventory is sold, no setup cost

Question: If there are 7000 units remaining, what should the company do? What should they do if there are 10,000 remaining?

Initial Inventory and Profit

0

100000

200000

300000

400000

500000

5000

6000

7000

8000

9000

1000

0

1100

0

1200

0

1300

0

1400

0

1500

0

1600

0

Production Quantity

Prof

it

The case motivates a powerful (s,S) inventory policy (or a min max policy): s is the reorder point and S is the order-up-to-level

Multi-Order Opportunities under Uncertainties

Inventory Policies

Continuous review policy in which inventory is reviewed every day and a decision

is made about whether and how much to order.

Periodic review policy in which the inventory level is reviewed at regular

intervals and an appropriate quantity is ordered after

each review.

Variable Demand with a Fixed ROP

Reorderpoint, R

Q

LT

Time

LT

Inve

nto

ry le

vel

0

Result of uncertainty

Reorder Point with a Safety Stock

Reorderpoint, R

Q

LT

Time

LT

Inve

nto

ry le

vel

0

Safety Stock

The amount of safety stock needed is based on the degree of uncertainty in the lead time demand and desired customer service level

Determinants of the Reorder Point

The rate of demand

The lead time

Demand and/or lead time variability

Stockout risk (safety stock)

Continuous Review Policy

AVG = Average daily demand faced STD = Standard deviation of daily demand faced L = Replenishment lead time h = Cost of holding one unit of the product per unit timeα = service level (the probability of stocking out is 1 – α)

hp

p

p =shortage cost

Continuous Review Policy

The inventory position at any point in time is the actual inventory at the warehouse plus items ordered by the distributor that have not yet arrived minus items that are backordered.

The reorder level, R consists of two components: the average inventory during lead time, which is the product of average daily demand and the lead time; and the safety stock, which is the amount of inventory that the distributor needs to keep at the warehouse and in the pipeline to protect against deviations from average demand during lead time.

Continuous Review Policy –Variable demand & fixed lead time

Average demand during lead time is exactly

Safety stock is

where z is a constant, referred to as the safety factor.

This constant is associated with the service level.

The reorder level is

Economic lot size is

LSTDz

AVGL

LSTDzAVGL

h

AVGKQ

2

Continuous Review Policy –Variable demand & fixed lead time

The expected level of inventory before receiving the order

is (lowest level i.e. Safety

Stock)

The expected level of inventory immediately after

receiving the order is (highest

level)

The average inventory level is the average of these two

values

LSTDzQ

LSTDzQ

2

LSTDz

In many situation, the lead time to the warehouse must be assumed to be normally distributed with average lead time denoted by AVGL and standard deviation denoted by STDL. In this case, the reorder point is calculated as

where AVG x AVGL represents average demand during lead time, &

is the standard deviation of demand during lead time. The amount of safety stock that has to be kept is equal to

222 STDLAVGSTDAVGLz

222 STDLAVGSTDAVGL

Continuous Review Policy –Variable demand & lead time

Periodic Review Policy

Inventory level is reviewed periodically at regular

intervals and an appropriate quantity so as to arrive at

base stock level is ordered after each review . Since inventory levels are reviewed at a periodic interval, the fixed

cost of placing an order is a sunk cost and hence can be ignored.

This level of the inventory position should be enough to

protect the warehouse against shortages until the next order

arrives, that is to cover demand during a period of r + L

days, with r being the length of review period and L being

the lead time.

Periodic Review Policy

Thus, the base-stock level should include two components: average demand during an interval of r + L days, which is equal to

and the safety stock, which is calculated as

where z is a safety factor.

AVGLr )(

LrSTDz

Periodic Review Policy

Maximum inventory level is achieved immediately after receiving an order, while the minimum level of inventory is achieved just before receiving an order.

It is easy to see that the expected level of inventory after receiving an order is

while the expected level of inventory before an order arrives is just the safety stock

Hence, the average inventory level is the average of these two values

LrSTDzAVGr

LrSTDz

LrSTDzAVGr

2

RISK POOLING

Risk Pooling Consider these two systems:

Market Two

Supplier

Warehouse One

Warehouse Two

Market One

Market Two

Supplier Warehouse

Market One

Questions: Q1: For the same service level, which system will require more inventory?Q2: For the same total inventory level, which system will have better service?

What is Risk Pooling?

The idea behind risk pooling is to redesign the supply chain,

the production process, or the product to either reduce the

uncertainty the firm faces or to hedge uncertainty so that

the firm is in a better position to mitigate the consequence

of uncertainty.

• Location pooling

• Product pooling

• Lead Time pooling

• Capacity pooling

Lead Time Pooling

Store 1

Sup

plie

r

Store 100

8-week lead time

Lead Time Pooling

Store 1

Sup

plie

r

Store 100

8-week lead time

Retail DC

1-week lead time

Capacity Pooling

3 Links – no flexibility

Capacity Pooling

9 Links – Total Flexibility

Advantages / Disadvantages

Advantages Disadvantages

Location Pooling reduce demand variabilitycreates distance between inventory and

customers

 

reduce expected inventory investment needed to achieve a target service level  

Product Pooling reduction in demand variability potentially degrades product functionality

 better performance in terms of

matching supply and demand  

Lead Time Pooling decrease lead time extra costs of operating distribution center

  keep inventory closer to customer additional transportation costs

  reduce inventory investment  

Capacity Pooling accommodate demand uncertainty large costs to have flexibility

Summary Risk Pooling

Risk-pooling strategies are most effective when demands

are negatively correlated because then the uncertainty with

total demand is much less than the uncertainty with any

individual item/location

Risk-pooling strategies do not help reduce pipeline

inventory

Risk-pooling strategies can be used to reduce inventory

while maintaining the same service or they can be used to

increase service while holding the same inventory

Example

Decentralized system: total SS = 47.88

total avg. invent. = 179

Safety Stock SS = z ·STD · L

Reorder Point R = AVG·L + SSOrder Quantity Q = sqrt(2*C0*AVG/h)Order-up-to-level R + QAverage Inventory SS + Q/2

AVG STD SS R QOrder-

up-to LevelAverage

Inventory

Warehouse 1 39.3 13.2 25.08 65 132 197 91

Warehouse 2 38.6 12.0 22.8 62 131 193 88

CentralizedWarehouse

77.9 20.7 39.35 118 186 304 132

Service Level:97% k=1.88Lead Time= 1 week

Q/2+SS

Risk Pooling – Effect of Correlation

The benefits of risk pooling depend on the behavior of demand from one market relative to the demand from another market.

WarehouseMarket 1

Market 2

D1+D2: (, 2)

Calculating demand variability of centralized system

Warehouse 1

Warehouse 2

Market 1

Market 2

D1: (1, 12)

D2: (2, 22)

2 = 1

2 + 22 + 212,

where -1 12

= 12 + 2

2 + 212,

where -1 1

: correlation coefficient of D1, D2

1+ 2 1+ 2

Conclusions: 1. Stdev of aggregated demand is less than the sum of stdev of individual demands2. If demands are independent or negatively correlated, the std of aggregated demand is much less

Conclusions: 1. Stdev of aggregated demand is less than the sum of stdev of individual demands2. If demands are independent or negatively correlated, the std of aggregated demand is much less

1. If D1, D2 positively correlated, > 02. If D1, D2 are independent, = 03. If D1, D2 negatively correlated, < 0

= 1 + 2

= ??

1+2

10-1

22

21

P.C.N.C. Ind.As (safety) stock is based on standard deviation

Square Root Law:Square Root Law: stock for combined demands usually less than the combined stocks

Risk Pooling – Effect of Coefficient of Variation

The higher the C.V. of demand observed in one market, the greater the benefit from risk pooling

COV= Standard deviation/Avg. demand

DecentralizedCentralized

Inbound transportation cost (from factories to warehouses)

Facility/Labor cost

Outbound transportation cost (from warehouses to retailers)

Safety Stock

Responsiveness to customers (lead time)

Centralized vs. Decentralized

Overhead Costs

Service Level

Case Study

# below stage = processing time # in white box = CST In this solution, inventory is held of finished

product and its raw materials

PART 1DALLAS ($260)

157

8

PART 2CHARLESTON ($7)

14

PART 4BALTIMORE ($220)

5

PART 3AUSTIN ($2)

14

6

8

5

PART 5CHICAGO ($155)

45

PART 7CHARLESTON ($30)

14

PART 6CHARLESTON ($2)

32

8

0

14

55

1445

14

32

(Adapted from Simchi-Levi, Chen, and Bramel, The Logic of Logistics, Springer, 2004)

A Pure Pull System

Produce to orderLong CST to customerNo inventory held in system

PART 1DALLAS ($260)

157

8

PART 2CHARLESTON ($7)

14

PART 4BALTIMORE ($220)

5

PART 3AUSTIN ($2)

14

6

8

5

PART 5CHICAGO ($155)

45

PART 7CHARLESTON ($30)

14

PART 6CHARLESTON ($2)

32

8

77

14

55

1445

14

32

A Pure Push System

Produce to forecastZero CST to customerHold lots of finished goods inventory

PART 1DALLAS ($260)

157

8

PART 2CHARLESTON ($7)

14

PART 4BALTIMORE ($220)

5

PART 3AUSTIN ($2)

14

6

8

5

PART 5CHICAGO ($155)

45

PART 7CHARLESTON ($30)

14

PART 6CHARLESTON ($2)

32

8

0

14

55

1445

14

32

A Hybrid Push-Pull System

Part of system operated produce-to-stock, part produce-to-order

Moderate lead time to customer

PART 1DALLAS ($260)

157

8

PART 2CHARLESTON ($7)

14

PART 4BALTIMORE ($220)

5

PART 3AUSTIN ($2)

14

6

8

5

PART 5CHICAGO ($155)

45

PART 7CHARLESTON ($30)

14

PART 6CHARLESTON ($2)

32

8

30

7

8

945

14

32

push/pull boundary

CST vs. Inventory Cost

$0

$2,000

$4,000

$6,000

$8,000

$10,000

$12,000

$14,000

0 10 20 30 40 50 60 70 80

Committed Lead Time to Customer (days)

Inve

nto

ry C

ost

($/

year

)

Push System

Pull System

Push-Pull System

Echelon Inventory System

Supplier

Warehouse

Retailers

Warehouse echelon

inventoryWarehouse

echelon lead time

Managing Inventory in the Supply Chain

How should the reorder point associated with the warehouse

echelon inventory position be calculated? The reorder point

is

where Le = echelon lead time, defined as the lead time between the

retailers and the warehouse plus the lead time between the

warehouse and its supplier

AVG = average demand across all retailers (i.e., the

average of the aggregate demand)

STD = standard deviation of (aggregate) demand across

all retailers

ee LSTDzAVGLs

Forecasting

Recall the three rulesNevertheless, forecast is criticalGeneral Overview:

Judgment methodsMarket research methodsTime Series methodsCausal methods

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