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

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

3. 3 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?

4. 4 4 Estimating Demand How fast was it selling? Average 209/week So an estimate of season demand for this item is 2473 = 2000 – 154 + 3*209

5. 5 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?

6. 6 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)

7. 7 7 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 - Cost What if Salvage Value > Cost?

8. 8 8 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?

9. 9 9 Risk & Reward Prob. Outcome is smaller Prob. Outcome is larger Our choice How are they related?

10. 10 Balance Expected Revenue  $profit*(1- Probability demand is smaller than our choice) Expected Risk  $risk*Probability demand is smaller than our choice Set these equal  profit*(1-P) = risk*P  profit = (profit+risk)*P  profit/(profit + risk) = P = Probability demand is smaller than our choice

11. 11 Making the Choice Prob. Demand is smaller Our choice profit/(profit - risk) Cumulative Probability

12. 12 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

13. 13 Forecasts

14. 14 Inferred Cum. Probability 57% stockout: 11,490 units 11490=10000+ 2000* [43%-Pr(10000)]/ [Pr(12000)-Pr(10000)] 43%

15. Net Profit as a function of Quantity Gross Profits from sales Costs of liquidations Net Profits = Gr. Profits from sales – Cost of liquidation-fixed cost

16. 16 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 Retail Game

17. 17 The Value of P as a function of Average Selling Price If Cost is $30 Selling Price P (Selling Price – Cost) (Selling Price– Salvage)

18. 18 The Quantity as a function of Average Selling Price If Cost is $30 P=Pr(D<=Q) N -1 (P)=Q Mean:1664, stdev=555

19. 19 Not Overly Sensitive Differences are small

20. 20 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?

21. 21 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)

22. 22 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

23. 23 Inventory Inventory On-hand Inventory Position: On-hand and on-order

24. 24 Continuous Review Basics Time Inventory Safety Stock Reorder Point Order placed Lead Time Actual Lead Time Demand Avg LT Demand On Hand Position Order Up to Level EOQ If lead time is long, …

25. 25 Assumptions Fixed Order Cost Constant average demand Typically assume Normally distributed lead time demand

26. 26 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 

27. 27 Only Variability in Demand If Lead Times are reliable –Average Lead Time Demand L * D –Standard Deviation in lead time demand  L =  L  D –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.

28. 28 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)

29. 29 Example 3-7 page 61 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* AvgD ROP+EOQ [ROP+(EOQ+ROP)]/2 ss+EOQ/2

30. 30 Lead Time Variability If Lead Times are variable D = Average (daily) demand  D = Std. Dev. in (daily) demand L = Average lead time (days) s L = Std. Dev. in lead time (days) Average lead time demand –DL Std. Dev. in lead time demand –  L =  L  2 D + D 2 s 2 L Remember: Std. Dev. in lead time demand drives safety stock

31. 31 Levers to Pull Std. Dev. in lead time demand –  L =  L  2 D + D 2 s 2 L Reduce Lead Time Reduce Variability in Lead Time Reduce Variability in Demand

32. 32 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?

33. 33 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?

34. 34 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?

35. 35 (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.

36. 36 Order Up To Policy Time Stock on hand Reorder Point Order placed Lead Time Reorder Point Target Inventory Position Actual Lead Time Demand Order Quantity Actual Lead Time Demand How much stock is available to cover demand in this period?

37. 37 Order Up To Policy: Inventory Time Stock on hand 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

38. 38 Order Up To Policy: Inventory Time Stock on hand 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

39. 39 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  D  denote the standard deviation in demand per unit time Let  L  denote the standard deviation in the lead time.

40. 40 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 +D 2  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?

41. 41 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 +D 2  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 T  D 2 /2

42. 42 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 +D 2  L 2 = (7+2)*67 2 + (105 2 )*2 2  = 40,401 + 44,100 = 84,501  Std Deviation = 291

43. 43 Example Cont’d Expected 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])  D 2 +D 2  L 2 = (7+28)*67 2 + (105 2 )*2 2  = 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

44. 44 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 .9996 52 ~.98 With twice a week shipments, we need.9998 chance of no stockouts between two shipments .9998 104 ~.98 With thrice a week shipments, we need.9999 chance of no stockouts between two shipments .9999 156 ~.98

45. 45 Example Cont’d Assume Demand in L+T is Normal Hold risk constant 98% chance of no shortages all year NormInv(0.9996) S-D(T+E[L]) DT/2+ss OHI+DE[L]

46. 46 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

47. 47 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

48. 48 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)

49. 49 Safety Lead Time as a quantity Safety Lead Time: The next X days of forecasted demand

50. 50 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

51. 51 If all goes as planned Safety Lead Time: The next X days of forecasted demand Planned Inventory Ship to this level

52. 52 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 

53. 53 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?

54. 54 Risk Pooling Case 3.3 p 64 97% ss+ EOQ/2

55. 55 Risk Pooling Case 3.3 p 64 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

56. 56 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

57. 57 Next Time Read Mass Customization Article Read To Pull or Not To Pull by Spearman