introduction to stochasticstochastic inventory mmodels ... · introduction to stochasticstochastic...

Introduction to StochasticIntroduction to Stochastic Inventory Models and SupplyInventory Models and Supply

Contracts

David Simchi-Levi

Professor of Engineering SystemsMassachusetts Institute of TechnologyMassachusetts Institute of Technology

�

�

�

Outline of the Presentation � Introduction

� The Effect of Demand Uncertainty( S) li �(s,S) Policy�Supply Contracts�Risk PoolingRisk Pooling

� Practical Issues in Inventory Practical Issues in Inventory Management

©Copyright 2002 D. Simchi-Levi

Sources:Plants vendors

portsRegional warehouses:

Stocking points

Field warehouses:Stocking points

Customers, demand

centers sinks

Transportation costsTransportation costs

Inventory & warehousingcosts

Inventory & warehousingcosts

Production/purchasecosts

Supply

Image by MIT OpenCourseWare.

Goals:Goals: Reduce Cost, Improve Service

• By effectively managing inventory: – Xerox eliminated $700 million inventory from its supply chain – Wal-Mart became the largest retail company utilizing

efficient inventory managementefficient inventory management – GM has reduced parts inventory and transportation costs by

26% annually


Goals:Goals: Reduce Cost, Improve Service

• By not managing inventory successfully – In 1994, “IBM continues to struggle with shortages in their

ThinkPad line” (WSJ, Oct 7, 1994) – In 1993, “Liz Claiborne said its unexpected earning decline is Liz Claiborne said its unexpected earning decline isIn 1993,

the consequence of higher than anticipated excess inventory” (WSJ, July 15, 1993)

– I 1993 “D ll C t di t l St k l DIn 1993, “Dell Computers predicts a loss; Stock plunges. Dellll acknowledged that the company was sharply off in its forecast of demand, resulting in inventory write downs” (WSJ, August 1993)


j

Understanding Inventory • The inventory policy is affected by:

– Demand Characteristics

– Lead Time

– Number of Products – Objectives

• Service level • Minimize costs

– Cost Structure


The Effect ofThe Effect of Demand Uncertainty

• Most companies treat the world as if it were predictable: – Production and inventory planning are based on

forecasts of demand made far in advance of the selling season

– CCompaniies are aware of demandd uncertaintyf d i when they create a forecast, but they design their planning process as if the forecast truly representsplanning process as if the forecast truly represents reality


Demand Forecast • The three principles of all forecasting

t h itechniques: – Forecasting is always wrong – The longer the forecast horizon the worst is the

forecast – Aggregate forecasts are more accurate


The Effect ofThe Effect of Demand Uncertainty

• Most companies treat the world as if it were predidicttable:bl – Production and inventory planning are based on forecasts of

demand made far in advance of the selling seasondemand made far in advance of the selling season – Companies are aware of demand uncertainty when they

create a forecast, but they design their planning process as if th f t t l t litifthe forecast truly represents reality

• Recent technological advances have increased the level of demand uncertainty:level of demand uncertainty: – Short product life cycles – Increasing product varietyyg p


SnowTime Sporting Goods • Fashion items have short life cycles, high

vari tiety off competitorstit • SnowTime Sporting Goods

– New designs are completed – One production opportunity – BBasedd on pastt salles, kknowlleddge of th f the iinddusttry,

and economic conditions, the marketing department has a probabilistic forecastdepartment has a probabilistic forecast

– The forecast averages about 13,000, but there is a chance that demand will be greater or less than thithis.


Supply Chain Time Lines

Jan 00 Jan 01 Jan 02 Design Production Retailing

Feb 00 Sep 00 Feb 01 Sep 01p pProduction


SnowTime Sporting Goods • Fashion items have short life cycles, high

vari tiety off competitorstit • SnowTime Sporting Goods

– New designs are completed – One production opportunity – BBasedd on pastt salles, kknowlleddge of th f the iinddusttry,

and economic conditions, the marketing department has a probabilistic forecastdepartment has a probabilistic forecast

– The forecast averages about 13,000, but there is a chance that demand will be greater or less than thithis.


8000

100

120

0 1

0

SnowTime Demand Scenarios

Demand ScenariosDemand Scenarios

25% 30%

ity

10% 15% 20% 25%

Pro

babi

li

0% 5%

10%

0 0 0 0 0 0

00 0014

000

1600 80

0

SalesSales


SnowTime 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

• ProfitProfit = Revenue - Variable Cost - Fixed Cost + SalvageSalvage


SnowTime Best Solution

• Find order qquantityy that maximizes weighted average profit.

•• Question: Will this quantity be less Question: Will this quantity be less than, equal to, or greater than average d d? demand?


What to Make? • Question: Will this quantity be less

than equal to or greater than average than, equal to, or greater than average demand?

• Average demand is 13,100 • Look at marginal cost Vs marginalLook at marginal cost Vs. marginal

profit – ifif exttra jjackkett soldld, profitfit i is 125125-8080 = 4545 – if not sold, cost is 80-20 = 60

• So we will make less than average ©Copyright 2002 D. Simchi-Levi

– = - -

SnowTime Scenarios • Scenario One:

– Suppose you make 12 000 jackets andSuppose 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 andSuppose you make 12,000 jackets and

demand ends up being 11,000 jackets.– Profit = 125(11 000) - 80(12 000) - 100 000 + Profit 125(11,000) 80(12,000) 100,000 +

20(1000) = $ 335,000 ©Copyright 2002 D. Simchi-Levi

SnowTime Expected Profit

Expected ProfitExpected Profit

$400,000

$200 000

$300,000

Proof

it

$100,000

$200,000

$0 8000 12000 16000 20000

Order Quantity Order Quantity




$400,000

$200 000

$300,000

ofit

$100,000

$200,000

Pro

$0 8000 12000 16000 20000



SnowTime:SnowTime: Important Observations

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

• Several qquantities have the same averagge profit

• Average profit does not tell the whole storyAverage profit does not tell the whole story

• Q ti 9000 d 16000 Question: 9000 and 16000 unitits lead to about the same average profitfit, so whi hich do we preffer?h d ?


0 0 00

0

000

0000

0000

0000

-300

0 -1

00 0

100 0

0

300 0

0

500 0

0

Probability of Outcomes

100%

60%

80%

100%

Q=9000

20%

40%

60% Q=9000 Q=16000

0%

20%

Prob

abibilit

y

Cost


Key Insights from this Model • The optimal order quantity is not necessarily

equal to average forecast demandequal to average forecast demand • The optimal quantity depends on the

l ti hi b t i l fit drelationship 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 pprobabilityy of large gains and of large losses increases


Supply Contracts


Manufacturer Manufacturer DC Retail DC

Stores

Fixed Production Cost = $35

Wholesale Price = $80

Selling Price = $125Salvage Price = $20

Variable Production Cost = $100,000

20:00

20:00


010

012

0

0 1

0

Demand Scenarios

Demand ScenariosDemand Scenarios

25% 30%

ity

10% 15% 20% 25%

Pro

babi

li

0% 5%

10%

0 0 0 0 0 0

800 00 00

1400

016

00 800

Sales Sales


Distributor Expected Profit Expected Profit

400000

500000

100000

200000

300000

0

100000

6000 8000 10000 12000 14000 16000 18000 20000

Order Quantity


Supply Contracts (cont.)

• Distributor opptimal order qquantityy is 12,000 units

• Distributor expected profit is $470 000• Distributor expected profit is $470,000

• Manufacturer profit is $440,000 • Supply Chain Profit is $910,000

–IS there anything that the distributor andIS there anything that the distributor and manufacturer can do to increase the profit of both?


Supply Contracts


Stores





20:00

20:00



Retailer ProfiRetailer Profit (Buy Back=$55)

600,000

300 000

400,000

500,000

r Pro

fit

100,000

200,000

300,000

Reta

ile

0

6000

7000

800

000 00 00 00 00

160

018

00

090

00 0

100

1 10

120

130

140

150 00

1700

0

Order QuantityOrder Quantity


600 0

7000

8000

9000

10

000

1100

012

000

1300

014

000

1500

016

000

1700

018

000

Retailer ProfiRetailer Profit (Buy Back=$55)

600,000 $513 800

300 000

400,000

500,000

r Pro

fit

$513,800

100,000

200,000

300,000

Reta

ile

0

Order QuantityOrder Quantity


Manufacturer ProfitManufacturer Profit (Buy Back=$55)

600 , 000

400, 000

500, 000

,

rer

Prof

it

100 000

200, 000

300, 000

Man

ufac

tur

0

100 , 000

6000

70

00

8000

90

00

1000

0 11

000

1200

0 13

000

1400

0 15

000

1600

0 17

000

1800

0

M

6 7 8 9 10

11

12

13

14

15

16

17

18

Production Quantity


Manufacturer ProfitManufacturer Profit (Buy Back=$55)

600 , 000

400, 000

500, 000

,

rer

Prof

it $471,900

100 000

200, 000

300, 000

Man

ufac

tur

0

100 , 000

6000

70

00

8000

90

00

1000

0 11

000

1200

0 13

000

1400

0 15

000

1600

0 17

000

1800

0

M

6 7 8 9 10

11

12

13

14

15

16

17

18

Production Quantity


Supply Contracts



Stores





20:00

20:00


Retailer ProfiRetailer Profit (Wholesale Price $70, RS 15%)

600,000

400,000

500,000

,

Pro

fit

100 000

200,000

300,000

Ret

aile

r P

0

100,000

6000

7000

8000

9000

0000 00

020

0030

0040

0050

0060

0070

0080

00

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

Order Quantity


Retailer ProfiRetailer Profit (Wholesale Price $70, RS 15%)

600,000 $504 325

400,000

500,000

,

Pro

fit

$504,325

100 000

200,000

300,000

Ret

aile

r P

0

100,000

6000

7000

8000

9000

0000 00

020

0030

0040

0050

0060

0070

0080

00

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

Order Quantity


6 7 8 9 10

11

12

13

14

15

16

17

18

Manufacturer ProfitManufacturer Profit (Wholesale Price $70, RS 15%)

700,000

400 0 00

500, 000

600, 000

,

rer P

rofit

100 0 00

200, 000

300, 000

400 , 000

Man

ufac

tur

0

100 , 000

6000

70

00

8000

90

00

1000

0 11

000

1200

0 13

000

1400

0 15

000

1600

0 17

000

1800

0

M

Production Quantity


6 7 8 9 10

11

12

13

14

15

16

17

18

Manufacturer ProfitManufacturer Profit (Wholesale Price $70, RS 15%)

700,000

400 0 00

500, 000

600, 000

,

rer P

rofit

$481,375

100 0 00

200, 000

300, 000

400 , 000

Man

ufac

tur

0

100 , 000

6000

70

00

8000

90

00

1000

0 11

000

1200

0 13

000

1400

0 15

000

1600

0 17

000

1800

0

M

Production Quantity


Supply Contracts

Strategy Retailer Manufacturer Total Sequential Optimization 470,700 440,000 910,700 Buyback 513,800 471,900 985,700 Revenue Sharing 504,325 481,375 985,700


Supply Contracts



Stores





20:00

20:00


60

70

80

9 3 6 80

Supply Chain Profit

1,200,000

800,000

1,000,000

, ,

in P

rofit

200 000

400,000

600,000

uppl

y C

ha

0

200,000

6000

7000

8000

9000

0000

1000

2000

3000

4000

5000

6000

7000

8000

Su

010

011

0 12

01

014

015

01

0 17

01

Production Quantity


60

70

80

9 3 6 80

Supply Chain Profit

1,200,000 $1 014 500

800,000

1,000,000

, ,

in P

rofit

$1,014,500

200 000

400,000

600,000

uppl

y C

ha

0

200,000

6000

7000

8000

9000

0000

1000

2000

3000

4000

5000

6000

7000

8000

Su

010

011

0 12

01

014

015

01

0 17

01

Production Quantity


Supply Contracts

Strategy Retailer Manufacturer Total Sequential Optimization 470,700 440,000 910,700 Buyback Buyback 513 800 513,800 471 900 471,900 985 700 985,700 Revenue Sharing 504,325 481,375 985,700 Global Optimization 1,014,500


Supply Contracts: Key Insights

• Effective supppplyy contracts allow supplypp y chain partners to replace sequential optimization by global optimization optimization by global optimization

• Buy Back and Revenue Sharing contracts achihieve thithis objecti tive thth roughh riskbj i k sharing

• No one has an incentive to deviate from the contract terms from the contract terms


Supply Contracts: Case Study • Example: Demand for a movie newly released video

cassette typically starts high and decreases rapidlycassette typically starts high and decreases rapidly– Peak demand last about 10 weeks

• Blockbuster purchases a copy from a studio for $65Blockbuster purchases a copy from a studio for $65 and rent for $3 – Hence, retailer must rent the tape at least 22 times before

earning profitearning profit • Retailers cannot justify purchasing enough to cover

the ppeak demand – In 1998, 20% of surveyed customers reported that they

could not rent the movie they wanted


– -

Supply Contracts: Case Study • Starting in 1998 Blockbuster entered a revenue sharing

agreementt with th ith the major sttudiosj di– Studio charges $8 per copy – Blockbuster pays 30-45% of its rental income Blockbuster pays 30 45% of its rental income

• Even if Blockbuster keeps only half of the rental income, the breakeven point is 6 rental per copythe breakeven point is 6 rental per copy

• The impact of revenue sharing on Blockbuster was dramatic – Rentals increased by 75% in test markets – Market share increased from 25% to 31% (The 2nd largest

ret il tailer, HHollywood E d Entterttaiinment C t Corp has 5% 5% market shhare)ll h k t )


t t

What are the drawbacks of RS? • Administrative Cost

L it b ht b th i d d t id t il h– Lawsuit brought by three independent video retailers who complained that they had been excluded from receiving the benefits of revenue sharing was dismissed (June 2002)

– Th W l Di C h d Bl kb iThe Walt Disney Company has sued Blockbuster accusing them of cheating its video unit of approximately $120 million under a four year revenue sharing agreement (January 2003)

• Impact on sales effort – Retailers have incentive to push products with higher profit

marginsmargins – Automotive industry: automobile sales depends on retail effort


What are the drawbacks of RS? • Retailer may carry substitute or

complementary products from other suppliers – One supplier offers revenue sharing while

the other does not • Substitute products: retail will push the product

with high margin • CCompllementtary prodductts: rettail iler may di discountt

the product offered under revenue sharing to motivate sales of the other product motivate sales of the other product


SnowTime Costs: Initial Inventory • 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

• ProfitProfit = Revenue - Variable Cost - Fixed Cost + SalvageSalvage




$400,000

$200 000

$300,000

Proof

it

$100,000

$200,000

$0 8000 12000 16000 20000



Initial Inventory • Suppose that one of the jacket designs is a

model produced last year. • Some inventoryy is left from last yyear • Assume the same demand pattern as before

• If only old inventory is sold If only old inventory is sold, no setup cost• no setup cost

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


Initial Inventory and Profit

500000

300000 400000 500000

it

100000 200000 300000

Prof

i

0

000 500 000 500

00 00 00 00

50006500800095001100012500 1400015500

Production Quantityy


Initial Inventory and Profit

500000

300000 400000 500000

t

100000 200000 300000

Prof

i

0 100000

00

00

00

00

00

00

00

00

00

00

00

00

500

600

700

800

900

1000

11

00

1200

13

00

1400

15

00

1600

Production Quantity Production Quantity


eo n e o e p

(s, S) Policies • For some starting inventory levels, it is better to not

st ttart prodducti tion • If we start, we always produce to the same level • h ( Ss, ) li f h i lThus, we use an ( S) policy. If the inventory level i l is

below s, we produce up to S. • i the de point d S i th d to le els is the reorder point, and S is the order-up-to level• The difference between the two levels is driven by

the fixed costs associated with orderingthe fixed costs associated with ordering, transportation, or manufacturing


o pe o p o e o ep en

A Multi-Period Inventory Model

• Often, there are multiple reorder opportunities

• Consider a central distribution facility which orders from a manufacturer and delivers to retailers. The di dist ib tributtor periiodidi ll cally pll eaces ordders tto replenil ishh it its inventory


•

Case Study: Electronic ComponentCase Study: Electronic Component Distributor

• Electronic Compponent Distributor • Parent company HQ in Japan with

worldworld-wide manufacturingwide manufacturing • All products manufactured by parent

company One central warehouse in U SOne central warehouse in U.S.


•

a

Case Study: The Supply Chain OutboundInbound

Distributor Factoryy

End UserDC DC

4) Production 1) OrderPlanningg 3) Order

ProcessingProcessingProcessing

2) Forecasting ReplenishmentReplenishment Order Flow Order Flow

Product Flow ©Copyright 2002 D. Simchi-Levi

0

AprMay Ju

n Ju

lAugSep OctNovDec Ja

nFeb Mar

Demand Variability: Example 1

P d t D Product Demandd

225250

150

75 100

150 125 104

100 150 200

Demand (000's) 75

50 61 48 53 45

0 50

100(000's)

M hMonth


Demand Variability: Example 1 Histogram for Value of Orders Placed in a Week

20

25

ncy

5

10

15

Freq

uen

0

$25,0

00

$50,0

00

$75,0

00

00,00

0

25,00

0

50,00

0

75,00

0

00,00

0

$25

$50

$75

$100

$125

$150

$175

$200

Value of Orders Placed in a Week


Reminder:Reminder: The Normal Distribution

St d d D i ti 5Standard Deviation = 5

Standard Deviation = 10Standard Deviation 10

0 10 20 30 40 50 60

Average = 30

0 10 20 30 40 50 60


The DC holds inventory to:

•• Satisfy demand during lead time Satisfy demand during lead time

• Protect against demand uncertainty

• Balance fixed costs and holding costs


e o

•

The Multi-Period Inventory Model • Normally distributed random demand • Fixed order cost plus a cost proportional to amount

ordered. • Inventory cost iis chhargedd per iitem per uniit tiime • If an order arrives and there is no inventory, the

o d i torder is lol st • The distributor has a required service level. This is

expressed as the the likelihood that the distributorexpressed as the the likelihood that the distributor will not stock out during lead time.

• IntuitivelyIntuitively, how will this effect our policy?how will this effect our policy?


A View of (s, S) Policy In

vent

oory

Levve

l

SS

s

Inventory Position

Lead Time LeadTime

0 Time


The (s,S) Policy • (s, S) Policy: Whenever the inventory position

drops below a certain level, s, we order to raise the inventory position to level S.

• The reorder point is a function of: – The Lead Time – Average demand – Demand variabilityDemand variability – Service level


Notation • AVG = average daily demand • STD = standard deviation of daily demand • LT = replenishment lead time in days • h = holding cost of one unit for one day • SL = service level (for example, 95%). This implies

that the probability of stocking out is 100%-SL (for example, 5%)

• AlAlso, ththe IInventtory PPositi ition att any ti time iis th the actual inventory plus items already ordered, but not yet delivered yet delivered.


Analysis • The reorder point has two components:

– To account for average demand during lead time: LT×AVG

– To account for deviations from average (we call this safetyTo account for deviations from average (we call this safety stock)

z × STD × √LT where z is chosen from statistical tables to ensure that thewhere z is chosen from statistical tables to ensure that the probability of stockouts during leadtime is 100%-SL.


Example • The distributor has historically observed weekly

ddemandd off: AVG = 44.6 STD = 32.1

Replenishment lead time is 2 weeks and desired Replenishment lead time is 2 weeks, and desired service level SL = 97%

• Average demand during lead time is:Average demand during lead time is: 44.6 × 2 = 89.2

• Safetyy Stock is: 1.88 × 32.1 × √2 = 85.3

• Reorder point is thus 175, or about 3.9 weeks of supply at warehouse and in the pipeline


Fixed Order Schedule

• Suppppose the distributor pplaces orders everyy month

• What policy should the distributor use?

• What about the fixed cost?


Base-Stock Policy Target Inventory

Units Expected UBUnits Expected UB

,

r1 r2 r3 Time

Expected LB

L L L


Risk Pooling • Consider these two systems:

Supplier

Warehouse One Market One

Market Two Warehouse Two

Market One

M k t T

Supplier Warehouse

Market Two


Risk Pooling • For the same service level, which system will

require more inventory? Why? • For the same total inventoryy level ,, which

system will have better service? Why?

• What are the factors that affect theseWhat are the factors that affect theseanswers?


g

Risk Pooling Example • Compare the two systems:

– two products – maintain 97% service level – $60 order cost – $.27 weekly holding cost – $1.05 transportation cost per unit in decentralized

system, $1.10 in centralized system – 1 week lead time


Risk Pooling Example

Week 1 2 3 4 5 6 7 8

Prod A, Market 1

33 45 37 38 55 30 18 58

Prod A, Market 2

46 35 41 40 26 48 18 55

Prod B, Market 1

0 2 3 0 0 1 3 0

Product B Product B, Market 2

22 44 00 00 33 11 00 00


Risk Pooling Example Warehouse Product AVG STD CV

Market 1 A 39.3 13.2 .34

Market 2 A 38.6 12.0 .31

Market 1 B 1.125 1.36 1.21

Market 2 B 1.25 1.58 1.26


Risk Pooling Example Warehouse Product AVG STD CV s S Avg.

InvenInven. % Dec Dec.

Market 1 A 39.3 13.2 .34 65 158 91

Market 2 A 38.6 12.0 .31 62 154 88

Market 1 B 1.125 1.36 1.21 4 26 15 Market 2 B 1.25 1.58 1.26 5 27 15 Cent. A 77.9 20.7 .27 118 226 132 26% Cent B 2.375 1.9 .81 6 37 20 33%


Risk Pooling:Risk Pooling: Important Observations

• Centralizing inventory control reduces both safety stock and average inventory level for the same service level.

• This works best for – Higgh coefficient of variation,, which reduces

required safety stock. – Negatively correlated demand. Why?

• What other kinds of risk pooling will we see?


–

To Centralize or not to Centralize

• What is the effect on:– Safety stock? – Service level?Service level? – Overhead? – Lead time? – Transpportation Costs?


Inventory Management: Best Practice

• Periodic inventoryy review policy (59%)) p y ( • Tight management of usage rates, lead

times and safety stock (46%)times and safety stock (46%) • ABC approach (37%) • Reduced safety stock levels (34%) • Shift more inventory Shift more inventory, or inventory• or inventory

ownership, to suppliers (31%)• Quantitative approaches (33%)


to o e 3 o e a e

Changes In Inventory Turnover

• Inventoryy turns increased byy 30% from 1995 to 1998

• Inventory turns increased by 27% from• Inventory turns increased by 27% from 1998 to 2000

• Overall the increase is from 8.0 turns per year to over 13 per year over a fivepe yea pe yeayear period ending in year 2000.


Industry

Inventory Turnover Ratio Upper

Quartile Median Lower

Quartile Dairy Products 34.4 19.3 9.2

Electronic Component 9.8 5.7 3.7 Electronic Computers 9 4 5 3 3 5 Electronic Computers 9.4 5.3 3.5

Books: publishing 9.8 2.4 1.3

Household audio & video equipment

6.2 3.4 2.3

Household electrical 8.0 5.0 3.8 appliances

8.0 5.0 3.8

Industrial chemical 10.3 6.6 4.4


Factors that Drive Reduction inFactors that Drive Reduction in Inventory

• Top management emphasis on inventory reduction (19%)(19%)

• Number of SKUs in the warehouse (10%) d f i %) • Improved forecasting ((7%)

• Use of sophisticated inventory management software (6%)(6%)

• Coordination among supply chain members (6%) • OthOthers


Factors that will Drive InventoryFactors that will Drive Inventory Turns Change by 2000

• Better software for inventory management (16.2%)• Reduced lead time (15%) • Improved forecasting (10.7%) • Application of SCM principals (9.6%) • More attention to inventory management (6.6%) • Reduction in SKU (5.1%) • Others


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ESD.273J / 1.270J Logistics and Supply Chain Management Fall 2009

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