Bullwhip Effects

1. Bullwhip Effects JinfengYue

2. Lack of SC Coordination • Supply chain coordination – all stages in the supply chain take actions together (usually results in greater total supply chain profits) • SC coordination requires that each stage take into account the effects of its actions on the other stages • Lack of coordination results when: • Objectives of different stages conflict or • Information moving between stages is distorted

3. Bullwhip Effects, Hau Lee, etc. • Bullwhip Effect: “the phenomenon where orders to the supplier tend to have larger variance than sales to the buyer, and the distortion propagates upstream in an amplified form (variance amplification)”

4. Bullwhip Effects, Hau Lee, etc. • Fluctuations in orders increase as they move up the supply chain from retailers to wholesalers to manufacturers to suppliers • Distorts demand information within the supply chain, where different stages have very different estimates of what demand looks like • Results in a loss of supply chain coordination

5. Bullwhip Effect Figure

6. The Effect of Lack ofCoordination on Performance • Manufacturing cost (increases) • Inventory cost (increases) • Replenishment lead time (increases) • Transportation cost (increases) • Labor cost for shipping and receiving (increases) • Level of product availability (decreases) • Relationships across the supply chain (worsens) • Profitability (decreases) • The bullwhip effect reduces supply chain profitability by making it more expensive to provide a given level of product availability • How expansive it could be? 1993, $300 billion grocery industry, $75 to $100 inventory, 12.5% to 25% excess costs by it.

7. Beer Game Miller Lite

8. Beer Game Budweiser

9. Explanation • Forrester (1961): A series of case studies, the basic for and policies used by an organization can give rise to characteristic and undesirable behaviors in the supply chain. (solution: behavioral practice) • Sterman (1989): Bullwhip effect observed in Beer Game, and interprets the phenomenon as a consequence of players’ systematic irrational behavior, or “misperceptions of feedback” (players tends to disregard the inventory in pipeline they ordered earlier and keep on ordering more). (solution: individual education)

10. Economists Contribution • Holt et al. 1960; Blinder 1982; Blanchard 1983: The use of (s,S) type inventory policies by retailers results in the variance of replenishment orders exceeding the variance of demand. • Kahn (1987) shows that the presence of positive serial correlation in demand and backlogging also results in the bullwhip effect

11. Hau Lee et al. Explanation • Demand signal processing: demand is non-stationary and one uses past demand information to update forecasts • The rationing game: strategic ordering behavior of buyers when supply shortage is anticipated • Order batching (non-fixed order cost) • Price variations

12. Demand Signal Processing • Retailer-Supplier (applicable to wholesaler-distributor, distributor-manufacturer, manufacturer-parts supplier) relationship • Multi-period inventory model • Non-stationary demand • Order-up-to point for each period for the inventory system is also non-stationary

13. Demand Signal Processing --- Model • At the beginning of period t, order • Stbe the order-up-to point for period t. since • There is a delay of ν period between ordering and receiving the goods. • The goods ordered ν period ago arrive • Demand is realized, the availability inventory is used to meet the demand • Excess demand is backlogged

14. Demand Signal Processing --- Model • h --- unit holding cost • π--- unit shortage cost • c --- unit ordering cost (price) • Decision variable , the amount in stock plus on order (including those in transit) after the decision has been made in period t. • β--- cost discount factor per period

15. Demand Signal Processing --- Model • Demand follows correlated process where and

16. Demand Signal Processing --- Model • Cost minimization problem where

17. Demand Signal Processing --- Model • Theorem 1. In the above setting, we have (a) If 0 < ρ < 1, the variance of retail orders is strictly larger than that of retail sales, i.e. (b) If 0 < ρ < 1, the larger the replenishment lead time, the larger the variance of orders; i,.e., strictly increases in ν .

18. Demand Signal Processing --- Model • Proof. Heyman and Sobel (1984) show that the program can be solved by solving where

19. Demand Signal Processing --- Model • Proof (cont.) Optimal solution is where denotes the distribution function of

20. Demand Signal Processing --- Model • Proof (cont.) since , for Thus, at the decision point in period 1 is a random variable with

21. Demand Signal Processing --- Model • Proof (cont.) where and

22. Demand Signal Processing --- Model • Proof (cont.) Thus decision is where and Φ is standard normal

23. Demand Signal Processing --- Model • Proof (cont.) The optimal order quantity is and the variance is

24. Demand Signal Processing --- Model • Proof (cont.) since we have whenν increases, variance increases also. EOP

25. Rationing Game • Shortage, demand potentially exceeds supply (production capacity limitation or uncertainty of production yield) • The manufacturer would ration the supply of the product to satisfy the retailers’ orders. • To try to secure more units, each retailer will issue an order which exceeds in quantity what the retailer would order if the supply of the product is not unlimited.

26. Rationing Game --- Model • One manufacturer and N identical retailers ( n = 1, 2, …, N). • Retailer n first observes demand distribution Φ(.) and places an order zn at time 1, • Manufacturer delivers the product at time 1. • Manufacturer’s output μ is a random variable, distributed to F(.)

27. Rationing Game --- Model • If total order Q exceeds manufacturer’s total output μ, it is • Retailer i will receive due to allocation. • is the expected cost of retailer i with order quantity zi .

28. Rationing Game --- Model • Identity retailers using symmetric Nash equilibrium:

29. Rationing Game --- Model • Where • Decision zimust be taken before the capacity μ is realized. • Its first order condition is given by

30. Rationing Game --- Model • Optimal solution satisfies: • Theorem 2: In above setting, z’ <= z*, where z’ is the solution of the newsvendor problem.

31. Rationing Game • Bullwhip Effect by rationing: Retailers’ equilibrium order quantity may identical or close to the newsvendor solution for low-demand periods, while it will be larger than the newsvendor solution for high-demand periods. Hence, the variance is amplified at the retailer.

32. Combining Rationing Game and Demand Signal • If the retailer observe a demand increasing signal, what will happen? • If it is extended to three layers, four layers, five layers, …, what will happen?

33. Order Batching • N retailers each used periodical review system with the review cycle equal to R periods. • The demands for retailer j in period k is ξjk , which is i.i.d. with mean m and variance σ2 for each retailer. • Depending on whether and how retailers’ order cycles are dependent or correlated, consider three cases (a) random ordering, (b) (positively) correlated ordering, and (c) balanced ordering

34. Order Batching --- Random Ordering • (a) Random Ordering: Demands from retailers are independent Let n be a random variable denoting the number of retailers who order in a randomly chosen period. n is a binomial variable with parameters N and 1/R. for i = 1, 2, …, N.

35. Order Batching --- Random Ordering • Hence, E(n) = N/R and Var(n) = N(1/R)(1-1/R), Let denote the total orders from n retailers in period t, i.e.,

36. Order Batching --- Random Ordering • Then by “law of total expectation”, we have And

37. Order Batching --- Positively Correlated Ordering • (Positively Correlated Ordering), considering extreme case in which all retailers order in the same period (e.g., when R is a week, all retailers order on Monday with probability 1/R and not on other days of the week)

38. Order Batching --- Positively Correlated Ordering • In above formula, we obtain E(n) = N/R, Var(n) = N2/R(1-1/R) • Let

39. Order Batching --- Positively Correlated Ordering • We have and

40. Order Batching --- Balanced Order • Suppose N = MR + k, where M and k are integers and 0 <= k <= R. All N retailers are divided into R groups: k groups of size (M + 1), and (R – k) groups of size M. e.g. R = 5 days (one week), N = 23, k = 3, M = 4. Monday, Tuesday, Wednesday, (M+1) = 5 retailers per day; Thursday and Friday, M = 4 retailers per day. Total = 5*3 + 2*4= 23 retailers

41. Order Batching --- Balanced Order • We have Then

42. Order Batching --- Balanced Order • Let denote the total orders from n retailers in period t, i.e • We have and

43. Order Batching --- Conclusion • Theorem 3. (1) (2) where , , and are the random variables denoting the orders from N retailers, respectively, under correlated ordering, random ordering, and balanced ordering. In all cases, the variability of demand experienced by the supplier is higher than that experienced by the retailers.

44. Order Batching --- Conclusion • When N = MR + k, and k = 0, “perfect balanced” or “completely synchronized” retailer ordering can be achieved. The variability experienced by the supplier and the retailers are identical, and bullwhip effect disappears.

45. Price Variations • A retailer faces independent and identically distributed demand with density function φ(.) each period. Sole manufacturing source alternates between two prices cL and cH over time, where cL < cH. With probability q (or 1-q, respectively) the price in a period will be cL (or cH , respectively).

46. Price Variations • The retailer’s inventory problem where Vi (i = H, L) denotes the minimal expected discounted cost incurred throughout an infinite horizon when the current price is ci.

47. Price Variations • L(.) is the sum of one-period inventory and shortage costs at a given level of inventory y, and is given by • Theorem 4. The following policy is optimal to the problem in last slide: At price cL, get as close as possible to the stock level SL, and at price cH, bring the stock level to SH, where SH < SL.

48. Price Variations • Theorem 5. In above setting, Var[zt] > Var[ ].

49. Managerial Explanation • For Demand Signal Processing 1. Demand Information Sharing (the manufacturer access to the demand data at the retail outlet, EDI--- Electronic Data Interchange); 2. Implement centralized multi-echelon inventory control system, Vendor-Managed-Inventory, Continuous Replenishment Program; 3. Direct Marketing Channel (Dell); 4. Shortening the lead time.

50. Managerial Explanation • Rationing Game 1. In shortage, allocate the supply in proportion to the retailer’s market share in the previous period; 2. To avoid retailers’ self-protection against imaginary shortage, as opposed to shortage, the manufacturer shares production and inventory information with downstream members of the supply chain.