11

Managing Uncertainty with Inventory I

John H. Vande Vate

Spring, 2007

22

Topics

• Integrate Obermeyer (wholesaler) with the Retail Game (retail pricing)

• Continuous Review Inventory Management

• Periodic Review Inventory Management

• Safety Lead Time

33

Item Season Sales1 10342 19423 10974 10685 15786 20007 14298 11459 1571

10 124811 200012 170813 177014 153715 161116 2000

Average 1546

The Retail Game Revisited

• How much inventory to bring to the market? 2000?

• What will demand be?

• How to estimate it?

That’s not demand! It’s supply

• How to estimate demand for this item?

44

Estimating Demand

• How fast was it selling?

Week Inventory Price Weeks Sales1 2000 60 942 1906 60 853 1821 60 1704 1651 60 1555 1496 60 1266 1370 60 647 1306 60 1058 1201 48 2299 972 48 253

10 719 48 17911 540 48 16312 377 48 22313 154 48 15414 0 48 015 0 48 016 0

Average 209/week

• So an estimate of season demand for this item is

• 2473 = 2000 – 154 + 3*209

55

New Estimate

• Should we order 1664?

• What are the issues?

• If salvage value exceeds our cost?

• If salvage value is less than our cost?

Item Season Sales1 10342 19423 10974 10685 15786 24737 14298 11459 1571

10 124811 249012 170813 177014 153715 161116 2927

Average 1664

66

Risk & Return

• Will Demand be 1664?• How to measure our uncertainty about

demand?– Method 1: Standard deviation of diverse

forecasts– Method 2: Historical A/F ratios + Point

forecast

• Trade off Risks (out of stock and overstock) vs Return (sales)

77

Measuring Risk and Return • Profit from the last item

$profit if demand is greater, $0 otherwise

• Expected Profit$profit*Probability demand is greater than our choice

• Risk posed by last item$risk if demand is smaller, $0 otherwise

• Expected Risk$risk*Probability demand is smaller than our choice

Example: risk = Salvage Value - CostWhat if Salvage Value > Cost?

88

Balancing Risk and Return• Expected Profit

$profit*Probability demand is greater than our choice

• Expected Risk$risk*Probability demand is smaller than

our choice

• How are probabilities related?

99

Risk & RewardDistribution

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 2 4 6 8 10 12

Prob. Outcome is smaller

Prob. Outcome is larger

Our choice

How are they related?

1010

Balance

• Expected Revenue$profit*(1- Probability demand is smaller than our

choice)

• Expected Risk$risk*Probability demand is smaller than our choice

• Set these equalprofit*(1-P) = risk*Pprofit = (profit+risk)*Pprofit/(profit + risk) = P = Probability demand is smaller

than our choice

1111

Distribution

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 2 4 6 8 10 12

Making the Choice

Prob. Demand is smaller

Our choice

profit/(profit - risk)

Cumulative Probability

1212

Swimsuit Case p 49

• Fixed Production Cost $100K• Variable Production Cost $80• Selling Price $125• Salvage Value $20• Profit is $125 - $80 = $45• Risk is $80 - $20 = $60• Profit + Risk is $125 - $20 = $105• Order to an expected stock out probability

57% = 1-$45/$105 = 1-43% • Several Sales Forecasts

1313

Forecasts

0%

5%

10%

15%

20%

25%

30%

8,000 10,000 12,000 14,000 16,000 18,000

1414

Inferred Cum. Probability

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

8,000 10,000 12,000 14,000 16,000 18,000

57% stockout:11,490 units

11490=10000+2000*

[43%-Pr(10000)]/[Pr(12000)-Pr(10000)]

43%

Net Profit as a function of Quantity

$(400)

$(300)

$(200)

$(100)

$-

$100

$200

$300

$400

$500

$600

8,000 9,000 10,000 11,000 12,000 13,000 14,000 15,000 16,000 17,000 18,000

Th

ou

san

ds

Gross Profits from sales

Costs of liquidations

Net Profits= Gr. Profits from sales – Cost of liquidation-fixed

cost

1616

What to order?

• So, we want P to be (Selling Price – Cost)

(Selling Price– Salvage)

• Assume Cost = $30,

• But what’s the selling price?

• In a wholesale environment this is easier. In a retail environment, it is messierSome protection from vendor some times

Retail Game

1717

The Value of P as a function of Average Selling Price

• If Cost is $30

74%

76%

78%

80%

82%

84%

86%

88%

$48 $50 $52 $54 $56 $58 $60

Selling Price

P

(Selling Price – Cost)(Selling Price– Salvage)

1818

The Quantity as a function of Average Selling Price

• If Cost is $30

2,000

2,050

2,100

2,150

2,200

2,250

2,300

$48 $50 $52 $54 $56 $58 $60

P=Pr(D<=Q)N-1(P)=QMean:1664, stdev=555

1919

Not Overly Sensitive

$-

$5,000

$10,000

$15,000

$20,000

$25,000

$30,000

$35,000

$40,000

$45,000

$50,000

$48 $50 $52 $54 $56 $58 $60

2100 2150

2200 2250

Differences are small

2020

Extend Idea

• Ship too little, you have to EXPEDITE the rest

• Ship Q

• If demand < Q– We sell demand and salvage (Q – demand)

• If demand > Q– We sell demand and expedite (demand – Q)

• What’s the strategy?

2121

Same idea

• Ignore profit from sales – that’s independent of Q

• Focus on salvage and expedite costs• Look at last item

– Chance we salvage it is P– Chance we expedite it is (1-P)

• Balance these costs– Unit salvage cost * P = Unit expedite cost (1-P)– P = expedite/(expedite + salvage)

2222

Safety Stock

• Protection against variability– Variability in demand and

– Variability in lead time

– Typically described as days of supply

– Should be described as standard deviations in lead time demand

– Example: BMW safety stock • For axles only protects against lead time variability

• For option parts protects against usage variability too

2323

Inventory

• Inventory On-hand

• Inventory Position: On-hand and on-order

2424

Continuous Review Basics

Time

Inve

ntor

y

Safety Stock

Reorder Point

Order placed

Lead Time

Actual Lead Time Demand

Avg LT Demand

On Hand

Position

Order Up to Level

EOQIf lead time is long, …

2525

Assumptions

• Fixed Order Cost

• Constant average demand

• Typically assume Normally distributed lead time demand

2626

Safety Stock Basics

• Lead time demand N(, )

• Typically Normal with – Average lead time demand – Standard Deviation in lead time demand

• Setting Safety Stock– Choose z from N(0,1) to get correct

probability that lead time demand exceeds z,– Safety stock is z

2727

Only Variability in Demand

• If Lead Times are reliable– Average Lead Time Demand

L * D

– Standard Deviation in lead time demand

L = LD

– Sqrt of Lead time * Standard Deviation in demand

– Units (Example)• L is the Lead Time in days, • D is the standard deviation in daily demand

Sq. Root because we are adding up L independent (daily)

demands.

2828

Implementation

• Inventory On-hand• Inventory Position: On-hand and on-order• When Inventory Position reaches a re-order point

(ROP), order the EOQ• This takes the Inventory Position to the Order-

Up-To Level: EOQ + ROP• That’s because review is continuous – we always

re-order at the ROP• Often called a (Q,r) policy (when inventory

reaches r, order Q)

2929

Example 3-7 page 61

Order cost $4,500 (e.g., transport cost)Cost of TV $250Holding cost 18%Lead time 2 weeks

Month September October November December January February March April May June July AugustSales 200 152 100 221 287 176 151 198 246 309 98 156

Avg Monthly Demand 191.17 UnitsStd Deviation in Monthly Demand 66.53 Units

Avg Weekly Demand 44.12 UnitsStd Deviation in weekly Demand 32.09 Units

Service Level 97% This is the fraction of time we expect to run out of stock before the next order arrives

z value 1.88 Standard DeviationsSafety Stock 85.34 UnitsEconomic Order Quantity 677 UnitsReorder Point 174 UnitsOrder-Up-To Level 851 UnitsAverage Inventory Position 512.2 The inventory position rises and falls between the Reorder Point and the Order-Up-To LevelAverage Inventory On Hand 424.0 The inventory on hand rises and falls between the Safety Stock and the Economic Order Quantity plus the Safety StockAverage Pipeline Inventory 88.23 The difference is the average pipeline inventory

Model assumes constant average monthly sales with

variability around that average: no seasonality or

growth in our sales

NormInv(0.97)√(L) D

ss+ L* AvgDROP+EOQ

[ROP+(EOQ+ROP)]/2

ss+EOQ/2

3030

Lead Time Variability

If Lead Times are variable• D = Average (daily) demand• D = Std. Dev. in (daily) demand• L = Average lead time (days)• sL = Std. Dev. in lead time (days)• Average lead time demand

– DL

• Std. Dev. in lead time demand– L = L2

D + D2 s2L

• Remember: Std. Dev. in lead time demand drives safety stock

3131

Levers to Pull

• Std. Dev. in lead time demand– L = L2

D + D2 s2L

Reduce Lead Time

Reduce Variability

in Lead Time

Reduce Variability in Demand

3232

Periodic Review

• Orders can only be triggered at certain times

• Examples– Batched transmissions (e.g., every night,

week, …)– Imposed by transportation (e.g., weekly

vessel)

• Examples of Continuous Review?

3333

No Ordering Cost

• Example?

• Cost typically viewed as – Inventory cost

• Service Level seen as a constraint– Probability of stock out in an order cycle

• Key Assumption: NO COST TO CHANGE ORDER SIZE

• Is this typically the case?

3434

Order-Up-To Policy

• Order-up-to Policy: At each period place an order to bring inventory position up to a level S

• What problem might we encounter?

3535

(S,s) Policy

• To avoid small orders

• In each period, if the inventory position is below s, place an order to bring it up to S.

3636

Order Up To Policy

Time

Sto

ck o

n ha

ndReorder Point

Order placed

Lead Time

Reorder Point

Target Inventory Position

Actual Lead Time Demand

Actual Lead Time Demand

Order Quantity

Actual Lead Time Demand

Actual Lead Time Demand

How much stock is available to cover demand in this period?

3737

Order Up To Policy: Inventory

Time

Sto

ck o

n ha

ndReorder Point Reorder Point

On Average this is the Expected

demand between orders

Order Quantity

So average on-hand inventory is DT/2+ss

On Average this is the safety stock

3838

Order Up To Policy: Inventory

Time

Sto

ck o

n ha

ndReorder Point Reorder Point

After an order is placed, it is the

Order up to level

Order Quantity

So average Pipeline inventory is S – DT/2

Before an order is placed it is smaller by the demand in

the period

3939

Safety Stock in Periodic Review

• Probability of stock out is the probability demand in T+L exceeds the order up to level, S

• Set a time unit, e.g., days• T = Time between orders (fixed)• L = Lead time, mean E[L], std dev L

• Demand per time unit has mean D, std dev D

• Assume demands in different periods are independent• Let Ddenote the standard deviation in demand per unit

time• Let Ldenote the standard deviation in the lead time.

4040

Safety Stock in Periodic Review

• Probability of stock out is the probability demand in T+L exceeds the order up to level, S

• Expected Demand in T + L D(T+E[L])

• Variance in Demand in T+L (T+E[L]) D

2 +D2 L

2

• Order Up to Level: S= D(T+E[L]) + safety stock• Question: What happens to service level if we

hold safety stock constant, but increase frequency?

4141

Impact of Frequency• What if we double frequency, but hold safety stock

constant?• Expected Demand in T/2 + L

D(T/2+E[L])

• Variance in Demand in T/2+L (T/2+E[L]) D

2 +D2 L

2

• Order Up to Level: S = D(T/2+E[L]) + safety stock

But now we face the risk of failure twice as often

This is reduced by

TD2/2

4242

Example• Time period is a day• Frequency is once per week

T = 7

• Daily demand Average 105 Std Dev 67

• Lead time Average 2 days Std Dev 2 days

• Expected Demand in T+L D (T + E[L]) = 105 (7 + 2) = 945

• Variance in Demand in T+L (T+E[L]) D

2 +D2 L

2 = (7+2)*672 + (1052)*22

= 40,401 + 44,100 = 84,501 Std Deviation = 291

4343

Example Cont’d

Expected Demand in T+LD (T + E[L]) = 105 (7 + 2) = 945 If we ship twice a week this drops to 578 If we ship thrice a week this drops to 456

• Variance in Demand in T+L (T+E[L]) D

2 +D2 L

2 = (7+28)*672 + (1052)*22

= 40,401 + 44,100 = 84,501

Std Deviation = 291 If we ship twice a week this drops to 262 If we ship thrice a week this drops to 252

4444

Example Cont’d

• With weekly shipments: To have a 98% chance of no stockouts in a year, we need .9996 chance of no stockouts in a week .999652 ~ .98

• With twice a week shipments, we need .9998 chance of no stockouts between two shipments .9998104 ~ .98

• With thrice a week shipments, we need .9999 chance of no stockouts between two shipments .9999156 ~ .98

4545

Example Cont’d

• Assume Demand in L+T is Normal

• Hold risk constant 98% chance of no shortages all year

Once a week Twice a week Thrice a weekD(T+E[L]) 945 578 455 Std dev in Demand 291 262 252 Order up to Level 1,920 1,506 1,392 Safety Stock 975 928 937 On Hand Inventory 1,342 1,112 1,060 % Reduction 0% 17.1% 21.0%Average Inventory 1,552 1,322 1,270 % Reduction - 14.8% 18.2%

NormInv(0.9996) S-D(T+E[L])

DT/2+ssOHI+DE[L]

4646

Lead time = 28

• When lead time is long relative to T

• Safety stock is less clear (Intervals of L+T overlap)

• Very Conservative Estimate Once a week Twice a week Thrice a week

D(T+E[L]) 3,675 3,308 3,185 Std dev in Demand 449 431 425 Order up to Level 5,179 4,832 4,764 Safety Stock 1,504 1,525 1,579 On Hand Inventory 1,871 1,708 1,701 % Reduction 0% 8.7% 9.1%Average Inventory 4,811 4,648 4,641 % Reduction 0.0% 3.4% 3.5%

Assume independence

4747

Lead time = 28

• When lead time is long relative to T

• Safety stock is less clear (Intervals of L+T overlap)

• Aggressive Estimate: Hold safety stock constant

Once a week Twice a week Thrice a weekD(T+E[L]) 3,675 3,308 3,185 Std dev in Demand 449 431 425 Order up to Level 5,179 4,811 4,689 Safety Stock 1,504 1,504 1,504 On Hand Inventory 1,871 1,688 1,626 % Reduction 0.0% 9.8% 13.1%Average Inventory 4,811 4,628 4,566 % Reduction 0.0% 3.8% 5.1%

4848

Periodic Review against a Forecast

• A forecast of day-to-day or week-to-week requirements

• Two sources of error– Forecast error (from demand variability)– Lead time variability

• Safety Lead Time replaces/augments Safety Stock• Example 6 days Safety Lead Time• Safety Lead Time translates into a quantity through

the forecast, e.g., the next 6 days of forecasted requirements (remember the forecast changes)

4949

Safety Lead Time as a quantity

0

100

200

300

400

500

600

700

Safety Lead Time: The next X days of forecasted demand

5050

The Ship-to-Forecast Policy• Periodic shipments every T days

• Safety lead time of S days

• Each shipment is planned so that after it arrives we should have S + T days of coverage.

• Coverage: Inventory on hand should meet S+T days of forecasted demand

5151

If all goes as planned

0

100

200

300

400

500

600

700

Safety Lead Time: The next X days of forecasted demand

Planned Inventory

Ship to this level

5252

Safety Stock Basics

• n customers

• Each with lead time demand N(, )

• Individual safety stock levels– Choose z from N(0,1) to get correct

probability that lead time demand exceeds z,– Safety stock for each customer is z– Total safety stock is nz

5353

Safety Stock Basics

• Collective Lead time demand N(n, n)• This is true if their demands and lead times are

independent!• Collective safety stock is nz• Typically demands are negatively or positively

correlated• What happens to the collective safety stock if

demands are – positively correlated?– Negatively correlated?

5454

Risk Pooling Case 3.3 p 64Historical Data for Product A

1 2 3 4 5 6 7 8Massachusetts 33 45 37 38 55 30 18 58New Jersey 46 35 41 40 26 48 18 55Pooled 79 80 78 78 81 78 36 113

Average Std DevCoeff of

Var

Avg. Lead time

DemandSafety Stock

Reorder Point EOQ

Order Up To Level

Avg. Inventory

Massachusetts 39.25 13.18 0.34 39.25 24.78 64 132 196 91New Jersey 38.63 12.05 0.31 38.63 22.66 61 131 192 88Pooled 77.88 20.71 0.27 77.88 38.95 117 186 303 132

Historical Data for Product B1 2 3 4 5 6 7 8

Massachusetts 0 2 3 0 0 1 3 0New Jersey 2 4 0 0 3 1 0 0Pooled 2 6 3 0 3 2 3 0

Average Std DevCoeff of

Var

Avg. Lead time

DemandSafety Stock

Reorder Point EOQ

Order Up To Level

Avg. Inventory

Massachusetts 1.13 1.36 1.21 1.13 2.55 4 22 26 14New Jersey 1.25 1.58 1.26 1.25 2.97 4 24 28 15Pooled 2.38 1.92 0.81 2.38 3.62 6 32 38 20

Inventory ComparisonMassachusetts New Jersey Total Pooled Reduction

Product A 91 88 179 132 26%Product B 14 15 28 20 30%Total 105 103 207 152 27%

Week

Week

97%

ss+EOQ/2

5555

Risk Pooling Case 3.3 p 64Historical Data for Product A

1 2 3 4 5 6 7 8Massachusetts 33 45 37 38 55 30 18 58New Jersey 46 35 41 40 26 48 18 55Pooled 79 80 78 78 81 78 36 113

Average Std DevCoeff of

Var

Avg. Lead time

DemandSafety Stock

Reorder Point EOQ

Order Up To Level

Avg. Inventory

Massachusetts 39.25 13.18 0.34 39.25 24.78 64 132 196 91New Jersey 38.63 12.05 0.31 38.63 22.66 61 131 192 88Pooled 77.88 20.71 0.27 77.88 38.95 117 186 303 132

Historical Data for Product B1 2 3 4 5 6 7 8

Massachusetts 0 2 3 0 0 1 3 0New Jersey 2 4 0 0 3 1 0 0Pooled 2 6 3 0 3 2 3 0

Average Std DevCoeff of

Var

Avg. Lead time

DemandSafety Stock

Reorder Point EOQ

Order Up To Level

Avg. Inventory

Massachusetts 1.13 1.36 1.21 1.13 2.55 4 22 26 14New Jersey 1.25 1.58 1.26 1.25 2.97 4 24 28 15Pooled 2.38 1.92 0.81 2.38 3.62 6 32 38 20

Inventory ComparisonMassachusetts New Jersey Total Pooled Reduction

Product A 91 88 179 132 26%Product B 14 15 28 20 30%Total 105 103 207 152 27%

Week

Week

Pooling Inventory can reduce safety stock

The impact is less than the sqrt of 2 law

It predicts that if 2 DCs need 47 units then a single DC will need

33

The impact is greater than the sqrt of 2 law

It predicts that if 2 DCs need 5.5 units

then a single DC will need 4

5656

Inventory (Risk) Pooling

• Centralizing inventory can reduce safety stock

• Best results with high variability and uncorrelated or negatively correlated demands

• Postponement ~ risk pooling across products

5757

Next Time

• Read Mass Customization Article

• Read To Pull or Not To Pull by Spearman