1. Managing Uncertainty with Inventory I
John H. Vande Vate
Spring, 2007 1

2. Topics
Integrate Obermeyer (wholesaler) with the Retail Game (retail pricing)
Continuous Review Inventory Management
Periodic Review Inventory Management
Safety Lead Time 2

3. 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? 3

4. Estimating Demand How fast was it selling?
Average 209/week
So an estimate of season demand for this item is
2473 = 2000 – *209 4

5. New Estimate Should we order 1664? What are the issues?
If salvage value exceeds our cost?
If salvage value is less than our cost? 5

6. 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) 6

7. 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 7

8. Forecasts 8

9. Inferred Cum. Probability9

10. Net Profit as a function of Quantity
Gross Profits from sales
Net Profits
Costs of liquidations

11. (Selling Price– Salvage)
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 messier Some protection from vendor some times 11

12. The Value of P as a function of Average Selling Price
If Cost is $30 12

13. The Quantity as a function of Average Selling Price
If Cost is $30 13

14. Not Overly Sensitive
Differences are small 14

15. 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? 15

16. 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) 16

17. 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 17

18. Inventory Inventory On-hand Inventory Position: On-hand and on-order18

19. Continuous Review Basics
Order Up to Level
On Hand
Position
If lead time is long, …
EOQ
Lead Time
Inventory
Reorder Point
Actual Lead Time Demand
Order placed
Avg LT Demand
Safety Stock
Time 19

20. Assumptions Fixed Order Cost Constant average demand
Typically assume Normally distributed lead time demand 20

21. Safety Stock Basics Lead time demand N(m, s) Typically Normal with
Average lead time demand m
Standard Deviation in lead time demand s
Setting Safety Stock
Choose z from N(0,1) to get correct probability that lead time demand exceeds z,
Safety stock is zs 21

22. Only Variability in Demand
Sq. Root because we are adding up L independent (daily) demands.
If Lead Times are reliable
Average Lead Time Demand
L * D
Standard Deviation in lead time demand
sL = LsD
Sqrt of Lead time * Standard Deviation in demand
Units (Example)
L is the Lead Time in days,
sD is the standard deviation in daily demand 22

23. 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) 23

24. Example 3-7 page 61 24

25. Lead Time Variability If Lead Times are variable
D = Average (daily) demand
sD = 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
sL = Ls2D + D2 s2L
Remember: Std. Dev. in lead time demand drives safety stock 25

26. Levers to Pull Std. Dev. in lead time demand sL = Ls2D + D2 s2L
Reduce Lead Time
Reduce Variability in Lead Time
Reduce Variability in Demand 26

27. 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? 27

28. 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? 28

29. 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? 29

30. (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. 30

31. Order Up To Policy Stock on hand Time
Target Inventory Position
Reorder Point
Reorder Point
Order Quantity
Actual Lead Time Demand
Actual Lead Time Demand
Stock on hand
Lead Time
Actual Lead Time Demand
Actual Lead Time Demand
Order placed
Time
How much stock is available to cover demand in this period? 31

32. Order Up To Policy: Inventory
Reorder Point
Reorder Point
On Average this is the Expected demand between orders
Order Quantity
On Average this is the safety stock
Stock on hand
Time
So average on-hand inventory is DT/2+ss 32

33. Order Up To Policy: Inventory
Reorder Point
Reorder Point
After an order is placed, it is the Order up to level
Order Quantity
Before an order is placed it is smaller by the demand in the period
Stock on hand
Time
So average Pipeline inventory is S – DT/2 33

34. 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 sL
Demand per time unit has mean D, std dev sD
Assume demands in different periods are independent
Let sD denote the standard deviation in demand per unit time
Let sL denote the standard deviation in the lead time. 34

35. 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]) sD2 +D2 sL2
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? 35

36. 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]) sD2 +D2 sL2
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 TsD2/2 36

37. Example Time period is a day Frequency is once per week 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]) sD2 +D2 sL2 = (7+2)*672 + (1052)*22
= 40, ,100 = 84,501
Std Deviation = 291 37

38. Example Cont’d Expected Demand in T+L Variance in Demand in T+L
D (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]) sD2 +D2 sL2 = (7+28)*672 + (1052)*22
= 40, ,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 38

39. Example Cont’d
With weekly shipments: To have a 98% chance of no stockouts in a year, we need chance of no stockouts in a week
~ .98
With twice a week shipments, we need chance of no stockouts between two shipments
~ .98
With thrice a week shipments, we need chance of no stockouts between two shipments
~ .98 39

40. Example Cont’d Assume Demand in L+T is Normal
Hold risk constant 98% chance of no shortages all year 40

41. Lead time = 28 When lead time is long relative to T
Safety stock is less clear (Intervals of L+T overlap)
Very Conservative Estimate
Assume independence 41

42. 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 42

43. 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) 43

44. Safety Lead Time as a quantity
Safety Lead Time: The next X days of forecasted demand 44

45. 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 45

46. Safety Lead Time: The next X days of forecasted demand
If all goes as planned
Ship to this level
Planned Inventory
Safety Lead Time: The next X days of forecasted demand 46

47. Safety Stock Basics n customers Each with lead time demand N(m, s)
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 zs
Total safety stock is nzs 47

48. Safety Stock Basics Collective Lead time demand N(nm, ns)
This is true if their demands and lead times are independent!
Collective safety stock is nzs
Typically demands are negatively or positively correlated
What happens to the collective safety stock if demands are
positively correlated?
Negatively correlated? 48

49. Risk Pooling Case 3.3 p 64 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
Pooling Inventory can reduce safety stock 49

50. 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 50

51. Next Time Read Mass Customization Article
Read To Pull or Not To Pull by Spearman 51