Inventory Management &

Risk Pooling

Introduction

General Motors in 1984:z Logistic network consisted of 20,000 supplier plants, 133

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

z Freight transportation costs were about $4.1 billion, of which 60 percent for material shipments.

z 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

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

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

chainz Wal-Mart became the largest retail company utilizing

efficient inventory management

Reduce Cost, Improve Service

Inventory Levels FinancialInvestment

OperationalNeed

Inventory

z Where do we hold inventory?

z Suppliers and manufacturers

z warehouses and distribution centers

z retailers

z Types of Inventory: General classification

z WIP

z raw materials

z finished goods

Functions of Inventory

zTo meet anticipated demand

zTo smooth production requirements

zTo decouple operations

zTo protect against stock-outs

zTo take advantage of order cycles

zTo help hedge against price increases

zTo take advantage of quantity discounts

Factors Affecting Inventory Policy

z Demand Characteristics: known in advance or random

z Lead Time

z Number of Different Products Stored in the Warehouse

z Economies of scale offered by suppliers & transport

companies

z Length of Planning Horizon

z Service level desired

1000 2000 3000 4000 5000 6000

0

50

100

150

200

250

300350

Ordering (Acquisition)Costs

Holding

or Car

rying C

ostsTota

l 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 swimsuits.

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 underestimatingcustomer demand will lead to inventory stockouts and loss of potential customers.

Demand forecast

forecast averages about 13,100

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 18000Unit sales

Swimsuit Costs

z Production cost per unit (C): $80

z Selling price per unit (S): $125

z Salvage value per unit (V): $20

z Fixed production cost (F): $100,000

z Q is production quantity, D: demand

z Profit = Revenue - Variable Cost - Fixed Cost + Salvage

Swimsuit Two Scenarios

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

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

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

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

$ 335,000

Swimsuit Best Questions ?

z Find order quantity that maximizes weighted average profit?

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

How much to Make?

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

z 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

P

r

o

f

i

t

If Quantity ordered is 12000Profit = (0.78)*12000*125+ 8000*125+ 10000*125-80*12000-100000+4000*0.11*20+2000*0.11*20= 1170000+247500 960000 -100000 + 13200= 3070700

Swimsuit : Important Observations

z Tradeoff between ordering enough to meet demand and ordering too much

z Several quantities have the same average profitz Average profit does not tell the whole storyz 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

P

r

o

f

i

t

Case: Swimsuit Production

z But Need to understand risk associated with certain decisions.

z 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 . 8 9

0 00

0 . 11 0 . 11

0 0

0 . 2 8

0

0 . 2 2

0

0 . 2 8

0 . 11

0%10%20%30%40%50%60%70%80%90%

100%

-

3

0

0

0

0

0

-

2

0

0

0

0

0

-

1

0

0

0

0

0 0

1

0

0

0

0

0

2

0

0

0

0

0

3

0

0

0

0

0

4

0

0

0

0

0

5

0

0

0

0

0

6

0

0

0

0

0

Profit

P

r

o

b

a

b

i

l

i

t

y

Q =9000Q =16000

Key Points from this Case

z The optimal order quantity is not necessarily equal to average forecast demand

z The optimal quantity depends on the relationship between marginal profit and marginal cost

z As order quantity increases, average profit first increases and then decreases

z 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

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

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

z 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

0100000200000300000400000500000

5

0

0

0

6

0

0

0

7

0

0

0

8

0

0

0

9

0

0

0

1

0

0

0

0

1

1

0

0

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1

2

0

0

0

1

3

0

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1

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1

5

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1

6

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0

0

Production Quantity

P

r

o

f

i

t

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

Multi-Order Opportunities under Uncertainties

Inventory Policies

z Continuous review policyz in which inventory is reviewed every day and a

decision is made about whether and how much to order.

z Periodic review policyz 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

LTTime

LT

I

n

v

e

n

t

o

r

y

l

e

v

e

l

0

Result of uncertainty

Reorder Point with a Safety Stock

Reorderpoint, R

Q

LTTime

LT

I

n

v

e

n

t

o

r

y

l

e

v

e

l

0Safety 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

z The rate of demand

z The lead time

z Demand and/or lead time variability

z Stockout risk (safety stock)

Continuous Review Policy

zAVG = Average daily demand faced zSTD = Standard deviation of daily demand faced zL = Replenishment lead time zh = Cost of holding one unit of the product per unit timez = service level (the probability of stocking out is 1 )

hpp+=p =shortage cost

Continuous Review Policy

z 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.

z 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

z Average demand during lead time is exactly

z Safety stock is

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

This constant is associated with the service level.

z The reorder level is

z Economic lot size is

LSTDz AVGL

LSTDzAVGL +

hAVGKQ = 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

z 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 * 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 STDLAVGSTDAVGLzAVGLAVG ++=

222 STDLAVGSTDAVGLz +

222 STDLAVGSTDAVGL +

Continuous Review Policy Variable demand & lead time

Periodic Review Policy

z 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 . z 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.

z This level of the inventory position should be enough to protectthe warehouse against shortages until the next order arrives.

z The next order should cover demand during a period of r + Ldays, with r being the length of review period and L being the lead time.

Periodic Review Policy

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

z and the safety stock, which is calculated aswhere z is a safety factor.

AVGLr + )(

LrSTDz +

Periodic Review Policy (with SS)

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 Poolingz Consider these two systems:

Market Two

SupplierWarehouse One

Warehouse Two

Market One

Market Two

Supplier WarehouseMarket 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

S

u

p

p

l

i

e

r

Store 100

8-week lead time

Lead Time Pooling

Store 1

S

u

p

p

l

i

e

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 functionalitybetter 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 Poolingz 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

z Risk-pooling strategies do not help reduce pipeline inventory

z 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 = 47.88total avg. invent. = 179

Safety Stock SS = z STD LReorder Point R = AVGL + SSOrder Quantity Q = sqrt(2*C0*AVG/h)Order-up-to-level R + QAverage Inventory SS + Q/2

AVG STD SS R Q Order-up-to Level AverageInventory

Warehouse 1 39.3 13.2 25.08 65 132 197 91

Warehouse 2 38.6 12.0 22.8 62 131 193 88CentralizedWarehouse 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 Correlationz 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 = 12 + 22 + 212, where -1 12 = 12 + 22 + 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 demands

2. If demands are independent ornegatively correlated, the std of aggregated demand is much less

Conclusions: 1. Stdev of aggregated demand is

less than the sum of stdev of individual demands

2. If demands are independent ornegatively 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 Variationz The higher the C.V. of demand observed in one

market, the greater the benefit from risk pooling

z COV= Standard deviation/Avg. demand

DecentralizedCentralized

Facility/Labor cost

Outbound transportation cost(from warehouses to retailers)

Responsiveness to customers(lead time)

Inbound transportation cost(from factories to warehouses)

Safety Stock

Centralized vs. Decentralized

Overhead Costs

Service Level

Echelon Inventory System

Supplier

Warehouse

Retailers

Warehouse echelon

inventoryWarehouse echelon lead

time

Managing Inventory in the Supply Chainz 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 +=

THANKYOU

Inventory Management & Risk PoolingIntroductionGoals of Inventory Management InventoryFunctions of InventoryFactors Affecting Inventory PolicySingle Period Model Without Initial InventoryCase: Swimsuit ProductionDemand forecastSwimsuit CostsSwimsuit Two ScenariosSwimsuit Best Questions ?How much to Make?Swimsuit Expected ProfitSwimsuit : Important ObservationsSwimsuit Expected ProfitCase: Swimsuit ProductionProbability of OutcomesKey Points from this CaseSingle Period Model With Initial InventoryInitial InventoryInitial Inventory and ProfitMulti-Order Opportunities under UncertaintiesInventory PoliciesVariable Demand with a Fixed ROPReorder Point with a Safety StockDeterminants of the Reorder PointContinuous Review PolicyContinuous Review PolicyContinuous Review Policy Variable demand & fixed lead timeContinuous Review Policy Variable demand & fixed lead timeContinuous Review Policy Variable demand & lead timePeriodic Review PolicyPeriodic Review PolicyPeriodic Review Policy (with SS)Periodic Review PolicyRISK POOLINGRisk PoolingWhat is Risk Pooling?Lead Time PoolingLead Time PoolingCapacity Pooling Capacity Pooling Advantages / Disadvantages Summary Risk PoolingRisk Pooling Effect of Correlation Risk Pooling Effect of Coefficient of VariationEchelon Inventory SystemManaging Inventory in the Supply ChainTHANKYOU