Bullwhip Effects Jinfeng Yue

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

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)”

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

Bullwhip Effect Figure

The Effect of Lack of Coordination 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.

Beer Game Miller Lite

Beer Game Budweiser

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)

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

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

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

Demand Signal Processing --- Model At the beginning of period t, order S t be 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

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

Demand Signal Processing --- Model Demand follows correlated process where and

Demand Signal Processing --- Model Cost minimization problem where

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.

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

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

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

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

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

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

Demand Signal Processing --- Model Proof (cont.) since we have when ν increases, variance increases also. EOP

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.

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

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 z i.

Rationing Game --- Model Identity retailers using symmetric Nash equilibrium:

Rationing Game --- Model Where Decision z i must be taken before the capacity μ is realized. Its first order condition is given by

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

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.

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?

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 ξ jk, which is i.i.d. with mean m and variance σ 2 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

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.

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.,

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

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)

Order Batching --- Positively Correlated Ordering In above formula, we obtain E(n) = N/R, Var(n) = N 2 /R(1-1/R) Let

Order Batching --- Positively Correlated Ordering We have and

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

Order Batching --- Balanced Order We have Then

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

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.

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.

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

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

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 c L, get as close as possible to the stock level S L, and at price c H, bring the stock level to S H, where S H < S L.

Price Variations Theorem 5. In above setting, Var[z t ] > Var[ ].

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.

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.

Managerial Explanation Rationing Game (cont.) 3. A contract that restricts the buyer’s flexibility (order quantities, free return and generous order cancellation policies all contribute to gaming).

Managerial Explanation Order Batching Batching of orders is a consequence of two factors: the periodic review process and the processing cost of a purchase transaction. 1. Providing the manufacturer with access to sell- through data and / or inventory data at the retailer level (create a production schedule determined by sales as opposed to orders)

Managerial Explanation Order Batching (cont.) 2. Lower the transaction cost by using ESI- based order transmission systems, computer assisted ordering (CAO). Resulting more frequent replenishment in small batches, which in turn leads to less distortion of demand information and more efficient delivery/production schedules.

Managerial Explanation Order Batching (cont.) 3. Manufacturers can influence buyers’ batching decisions in other ways: a. allow retailers to order an assortment of products to fill a truckload and offer the same volume discount. b. coordination of delivery schedules --- moves the channel away from random or correlated ordering to balanced ordering. 4. Third party logistics providers.

Managerial Explanation Price Variations Reduce the frequency as well as depth of manufacturer’s trade promotions (i.e., wholesale price discounts). Shift to the Every Day Low Price (EDLP) strategy.

Reference Lee, Padmanabhan, Whang (1997) Information Distortion in a Supply Chain: The Bullwhip Effect, Management Science, 43, 4,