Competitive and Cooperative Inventory Policies in a Two-Stage ...

760 views

Published on

Published in: Technology, Business
  • Be the first to comment

Competitive and Cooperative Inventory Policies in a Two-Stage ...

  1. 1. Competitive and Cooperative Inventory Policies in a Two-Stage Supply Chain Gerard P. Cachon • Paul H. Zipkin ´ The Fuqua School of Business, Duke University, Durham, North Carolina 27708 gpc@mail.duke.edu zipkin@mail.duke.edu W e investigate a two-stage serial supply chain with stationary stochastic demand and fixed transportation times. Inventory holding costs are charged at each stage, and each stage may incur a consumer backorder penalty cost, e.g. the upper stage (the supplier) may dislike backorders at the lower stage (the retailer). We consider two games. In both, the stages independently choose base stock policies to minimize their costs. The games differ in how the firms track their inventory levels (in one, the firms are committed to tracking echelon inventory; in the other they track local inventory). We compare the policies chosen under this competitive regime to those selected to minimize total supply chain costs, i.e., the optimal solution. We show that the games (nearly always) have a unique Nash equilibrium, and it differs from the optimal solution. Hence, competition reduces efficiency. Furthermore, the two games’ equilibria are different, so the tracking method influences strategic behavior. We show that the system optimal solution can be achieved as a Nash equilibrium using simple linear transfer payments. The value of cooperation is context specific: In some settings competition increases total cost by only a fraction of a percent, whereas in other settings the cost increase is enormous. We also discuss Stackelberg equilibria. (Supply Chain; Game Theory; Multiechelon Inventory; Incentive Contracts) 1. Introduction Furthermore, to what extent will these temptations How should a supply chain manage inventory? If the lead to supply chain inefficiency? members care only about overall system performance, This paper studies the difference between global/ they should choose policies to minimize total costs, cooperative and independent/competitive optimiza- i.e., the optimal solution. While this approach is ap- tion in a serial supply chain with one supplier and one pealing, it harbors an important weakness. Each mem- retailer. (We assume there are two independent firms, ber may incur only a portion of the supply chain’s but the model also applies to independent agents costs, so the optimal solution may not minimize each within the same firm.) Consumer demand is stochas- member’s own costs. For example, a supplier may care tic, but independent and stationary across periods. more than a retailer about consumer backorders for There are inventory holding costs and consumer back- the supplier’s product, or the retailer’s cost to hold order penalty costs, but no ordering costs. There is a inventory may be higher than the supplier’s. While the constant transportation time between stages, and the firms may agree in principal to cooperate, each may supplier’s source has infinite capacity. Inventory is face a temptation to deviate from any agreement, to tracked using either echelon inventory or local inventory. reduce its own costs. Supposing each firm can antici- (A firm’s local inventory is its on-hand inventory, and pate these temptations, how will the firms behave? its echelon inventory is its local inventory plus all 0025-1909/99/4507/0936$05.00 Copyright © 1999, Institute for Operations Research 936 Management Science/Vol. 45, No. 7, July 1999 and the Management Sciences
  2. 2. CACHON AND ZIPKIN Competitive and Cooperative Inventory Policies inventory held lower in the supply chain.) In the of linear contracts that meet this objective, and briefly optimal solution, the firms choose base stock policies, discuss other techniques for aligning incentives. described in §3. These policies can be implemented by In these games neither player dominates the other, tracking either echelon inventory or local inventory. and the firms simultaneously choose their strategies. To model independent decision making we con- We also study Stackelberg versions of the games, in sider two games, the Echelon Inventory (EI) game and which one dominant player chooses its strategy before the Local Inventory (LI) game. In both games the firms the other. simultaneously choose their base stock levels. This is The next section reviews the related literature, and their only strategic decision, and it cannot be modified §3 formulates the model. Section 4 describes the once it is announced. The supplier pays holding costs system optimal solution, and §5 analyzes the two for inventory in its possession or in-transit to the games. Section 6 compares the games’ equilibria with the optimal solution. Section 7 describes contracts that retailer, and the retailer pays holding costs on units it make the system optimal solution a Nash equilibrium. possesses. Both firms are concerned about consumer Section 8 discusses the numerical study. Section 9 backorders; the supplier pays a consumer backorder analyzes the Stackelberg games, and §10 concludes. penalty as does the retailer. This is an important assumption, because it allows us to study how the firms’ relative preferences influence their strategic 2. Literature Review behavior and, in turn, the performance of the system. The literature on supply chain inventory management (Section 3 discusses this modeling issue.) mostly assumes policies are set by a central decision The EI and LI games differ in only one way: In the maker to optimize total supply chain performance. EI game both firms are committed to tracking echelon Three exceptions are Lee and Whang (1996), Chen inventory, whereas in the LI game both firms track (1997), and Porteus (1997). local inventory. In Lee and Whang (1996), the firms use echelon The firms in each game play a Nash equilibrium. (A stock policies and all backorder penalties are charged pair of strategies is a Nash equilibrium, if each firm to the lowest stage. The upper stage incurs holding minimizes its own cost assuming the other player costs only. Therefore, with competitive selection of chooses its equilibrium strategy.) Thus, each firm policies, the upper stage carries no inventory, thereby makes an optimal decision given the behavior of the minimizing its own cost. They develop a nonlinear transfer payment contract that induces each firm to other firm, and therefore neither firm has an incentive choose the system optimal base stock policies. Our to deviate unilaterally from the equilibrium. model differs from theirs on several dimensions. We We find that in each game there is (usually) a assume the upper stage (the supplier) may care about unique Nash equilibrium. We compare the games’ consumer backorders, so it may carry inventory even equilibria to each other and to the optimal solution. when inventory policies are chosen competitively. The optimal solution is typically not a Nash equilib- Hence, the competitive decisions are nontrivial. We rium, so competitive decision making degrades sup- distinguish between echelon inventory and local in- ply chain efficiency. We evaluate the magnitude of ventory and investigate how these different methods this effect with an extensive numerical study. for tracking inventory influence strategic behavior. Implementation of the cooperative solution requires Finally, we develop linear transfer payment contracts. that the firms eliminate the incentives to deviate, i.e., They consider a setup cost at the upper echelon, while they should modify their costs so that the optimal we do not. Porteus (1997) studies a model similar to solution becomes a Nash equilibrium. This goal can be Lee and Whang’s model, but he proposes a different achieved by a contract that specifies linear transfer coordination scheme, called responsibility tokens. payments based on easily verifiable performance mea- Chen (1997) studies a game similar to the popular sures like inventory and backorders. We develop a set Beer Game (Sterman 1989), except that the demands in Management Science/Vol. 45, No. 7, July 1999 937
  3. 3. CACHON AND ZIPKIN Competitive and Cooperative Inventory Policies different periods are independent random variables There is a lead time for shipments from the source to with a common distribution that is known to all the supplier, L 2 , and from the supplier to the retailer, players. Unlike our players, his share the objective of L 1 . Each firm may order any nonnegative amount in minimizing total system costs; they have no compet- each period. There is no fixed cost for placing or ing interests. He outlines an accounting scheme that processing an order. Each firm pays a constant price allows each player to optimize its own costs and yet per unit ordered, so there are no quantity discounts. choose the system optimal solution. This scheme is The supplier is charged holding cost h 2 per period more complex than ours in some ways, though sim- for each unit in its stock or enroute to the retailer. The pler in others, as explained in §7. He also studies the retailer’s holding cost is h 2 h 1 per period for each behavior of boundedly rational players, whereas we unit in its stock. Assume h 2 0 and h 1 0. only assume rational players. Unmet demands are backlogged, and all backorders Several other papers address related issues, yet their are ultimately filled. Both the retailer and the supplier models are significantly different. Lippman and Mc- may incur costs when demand is backordered. The Cardle (1997) study competition between two or more retailer is charged p for each backorder, and the firms in a one-period setting, where a consumer may supplier (1 ) p, 0 1. The parameter p is the switch among firms to find available inventory. Parlar total system backorder cost, and specifies how this (1988) and Li (1992) also study the role of inventory in cost is divided among the firms. The parameter is the competition among retailers. In a multiechelon exogenous. model with multiple retailers, Muckstadt and Thomas These backorder costs have several possible interpre- (1980), Hausman and Erkip (1994), and Axsater (1996) ¨ tations, all standard. They may represent the costs of investigate a centralized control system that allows financing receivables, if customers pay only upon the each firm to optimize its own costs and still choose an fulfillment of demands. (This requires a discounted-cost outcome desirable to the central planner. The behavior model to represent exactly, but the approximation here of a central planner has also been investigated in is standard in the average-cost context, analogous to the settings with moral hazard (e.g., Porteus and Whang treatment of inventory financing costs.) Alternatively, 1991, Kouvelis and Lariviere 1996). Many papers in- they may be proxies for losses in customer good-will, vestigate how a supplier can induce a retailer to which in turn lead to long-run declines in demand. Such behave in a manner that is more favorable to the costs need not affect the firms equally, which is why we supplier (e.g., Donohue 1996, Tsay 1996, Ha 1996, Lal allow the flexibility to choose [0, 1]. Finally, they and Staelin 1984, Moses and Seshadri 1996, Narayanan provide a crude approximation to lost sales. (It would be and Raman 1996, Pasternack 1985). Chen et al. (1997) better, of course, to model lost sales directly, but that study competitive selection of inventory policies in a introduces considerable analytical difficulties. Even the multiechelon model with deterministic demand. optimal policy is unknown.) In period t before demand define the following for 3. Model Description stage i: in-transit inventory, IT it ; echelon inventory level, Consider a one-product inventory system with one IL it , is all inventory at stage i or lower in the system supplier and one retailer. The supplier is Stage 2 and minus consumer backorders; local inventory level, IL it , the retailer is Stage 1. Time is divided into an infinite is inventory at stage i minus backorders at stage i (the number of discrete periods. Consumer demand at the supplier’s backorders are unfilled retailer orders); retailer is stochastic, independent across periods and echelon inventory position, IP it , IP it IL it IT it ; and stationary. The following is the sequence of events local inventory position, IP it , IP it IL it IT it . during a period: (1) shipments arrive at each stage; (2) Each firm uses a base stock policy. Using an echelon orders are submitted and shipments are released; (3) base stock level, each period the firm orders a suffi- consumer demand occurs; (4) holding and backorder cient amount to raise its echelon inventory position penalty costs are charged. plus outstanding orders to that level. A firm’s local 938 Management Science/Vol. 45, No. 7, July 1999
  4. 4. CACHON AND ZIPKIN Competitive and Cooperative Inventory Policies o base stock level is similar, except the local inventory In period t charge the supplier G 2 (IP 2t ), where position replaces the echelon inventory position. De- fine s i as stage i’s echelon base stock level and s i as its o G2 y ˆo E G2 y D L2 . local base stock level. o Let D denote random total demand over periods, The supplier’s optimal echelon base stock level, s 2 , o and denote mean total demand over periods. Let minimizes G 2 . and be the density and distribution functions of demand over periods, respectively. We assume 1 5. Echelon and Local Inventory ( x) is continuous, increasing, and differentiable for x 0, so the same is true of , 0. Furthermore, Games 1 (0) 0, so positive demand occurs in each period. In the Echelon Inventory (EI) game, the two stages are Math notation follows: [ x] max{0, x}; [ x] independent firms or players. In the game’s only max{0, x}; [a, b] is the closed interval from a to b; move, the players simultaneously choose their strate- and E[ x] is the expected value of x. A prime denotes gies, s i [0, S], where s i equals player i’s the derivative of a function of one variable. echelon base stock level, is player i’s strategy space, and S is a very large constant. (S is sufficiently large that it never constrains the players.) A joint strategy s 4. System Optimal Solution is a pair (s 1 , s 2 ). After their choices, the players The system optimal solution minimizes the total aver- implement their policies over an infinite horizon. In age cost per period. Clark and Scarf (1960), Feder- addition, all model parameters are common knowl- gruen and Zipkin (1984), and Chen and Zheng (1994) edge. demonstrate that an echelon base stock policy is In the Local Inventory (LI) game the supplier and optimal in this setting. The optimal solution is found the retailer choose local base stock levels, s 2 , s 1 . by allocating costs to the firms in a particular way. Again, strategies are chosen simultaneously, the play- Then, each firm chooses a policy that minimizes its ers are committed to their strategies over an infinite cost function. This section briefly outlines this method. horizon, and all parameters are common knowledge. ˆo Let G 1 (IL 1t D 1 ) equal the retailer’s charge in The players know which game they are playing; the period t, where choice between the EI and LI games is not one of their decisions. ˆo G1 x h1 x h2 p x . Define H i (s 1 , s 2 ) as player i’s expected per-period o Also in period t, define G 1 (IP 1t ) as the retailer’s cost when players use echelon base stock levels (s 1 , expected charge in period t L 1 , where s 2 ). When s 2 s 2 s 1 and s 1 s 1 , the local base stock pair (s 1 , s 2 ) is equivalent to (s 1 , s 2 ) in the sense that o G1 y ˆo E G1 y D L1 1 . H i (s 1 , s 2 ) H i (s 1 , s 2 s 1 ). Since any echelon base o o stock pair can be converted into an equivalent local Define s 1 as the value of y that minimizes G 1 ( y): pair, there is no need to define distinct cost functions h2 p with local arguments. We will frequently switch a pair L1 1 o s1 . (1) of base stock levels from one tracking method to h1 h2 p another to facilitate comparisons. Although there is This is the retailer’s optimal base stock level. Define little operational distinction between echelon and local the induced penalty function, base stock policies, we later show that they differ o o o o o strategically. (However, the operational equivalence G1 y G 1 min s 1 , y G1 s1 , of echelon and local base stocks does depend on the and define assumption of stationary demand. In a nonstationary demand environment, it may not be possible to run ˆo G2 y h2 y 1 o G1 y . the system optimally with local base stock policies.) Management Science/Vol. 45, No. 7, July 1999 939
  5. 5. CACHON AND ZIPKIN Competitive and Cooperative Inventory Policies For the EI game the best reply mapping for firm i is er’s costs, assuming retailer orders are shipped imme- a set-valued relationship associating each strategy s j , j diately, i, with a subset of according to the following a rules: s1 arg min G 1 y . y r1 s2 s1 |H 1 s 1 , s 2 min H 1 x, s 2 a Differentiation verifies that G 1 is strictly convex, so s 1 x a is determined by G 1 (s 1 ) 0, r2 s1 s2 |H 2 s 1 , s 2 min H 2 s 1 , x . x L1 1 a p s1 . h1 h2 p Likewise, for the LI game, the best reply mappings are The retailer’s true expected cost depends on both its r1 s2 s1 |H 1 s 1 , s 2 s1 min H 1 x, s 2 x own base stock as well as the supplier’s base stock. We x use a standard derivation. After the firms place their r2 s1 s2 |H 2 s 1 , s 2 s1 min H 2 s 1 , x s1 . orders in period t L 2 , the supplier’s echelon inven- x tory position equals s 2 . After inventory arrives in period t, but before period t demand, the supplier’s A pure strategy Nash equilibrium is a pair of e e echelon inventory level equals s 2 D L 2 . Hence, E[s 2 echelon base stock levels, (s 1, s 2), in the EI game, or L2 l l D ] is the supply chain’s expected inventory level local base stock levels, (s 1, s 2), in the LI game, such (average supply chain inventory minus average back- that each player chooses a best reply to the other orders). When s 2 D L2 s 1 , the supplier can player’s equilibrium base stock level: completely fill the retailer’s period t order, so IP 1t e s2 e r2 s 1 e s1 e r1 s 2 s 1 . When s 2 D L2 s 1 , the supplier cannot fill all of the retailer’s order, and IP 1t s2 D L 2 . Hence, l l l l s2 r2 s 1 s1 r1 s 2 . H 1 s 1, s 2 E G 1 min s 2 D L 2, s 1 (We do not consider mixed strategies. We generally L2 find a unique pure strategy equilibrium.) s2 s1 G1 s1 5.1. Actual Cost Functions L2 In each period, the retailer is charged h 1 h 2 per unit x G1 s2 x dx. s2 s 1 held in inventory and p per unit backordered. Define ˆ G 1 (IL 1t D 1 ) as the sum of these costs in period t, ˆ Define G 2 (IL 1t D 1 ) as the supplier’s actual period t backorder cost, ˆ G1 y h1 h2 y p y . ˆ G2 y 1 p y , Define G 1 (IP 1t ) as the retailer’s expected cost in period t L 1, and G 2 (IP 1t ) as the supplier’s expected period t L1 backorder cost, G1 y ˆ E G1 y D L1 1 L1 1 G2 y ˆ E G2 y D L1 1 . h1 h2 y h1 h2 p Define L1 1 x y x dx. ˆ H 2 s 1, x h2 L1 h2 x G2 s1 min x, 0 , y a so Define s 1 as the value that minimizes this function, that is, the base stock level that minimizes the retail- H 2 s 1, s 2 ˆ E H 2 s 1, s 2 s1 D L2 940 Management Science/Vol. 45, No. 7, July 1999
  6. 6. CACHON AND ZIPKIN Competitive and Cooperative Inventory Policies s2 s 1 for s 2 s 1 . The second term, E[G 2 (min{s 2 D L2, L1 L2 h2 h2 s2 s1 x x dx s 1 })], is convex, and strictly convex when s 1 0 and 0 s2 0. Hence, H 2 (s 1 , s 2 ) is strictly convex in s 2 0. L2 Consider H 1 . When s 1 s 2 , H 1 is constant with s2 s1 G2 s1 respect to s 1 . Assume s 1 s 2 and differentiate H 1 , L2 H1 L2 x G2 s2 x dx. s2 s1 G 1 s1 . s1 s2 s 1 a When s 2 s 1 , H 1 is decreasing for s 1 s 2 and The first term above is the expected holding cost for constant for s 1 s 2 . When s 2 a s 1 , H 1 is decreasing the units in-transit to the retailer (from Little’s Law), for s 1 a s 1 , increasing for s 1 a s1 s 2 , and constant the second term is the expected cost for inventory held for s 1 s 2 . Hence, H 1 is quasiconvex in s 1 . at the supplier and the final two terms are the ex- The following lemma characterizes the supplier’s pected backorder cost charged to the supplier. best reply mapping. We mentioned above the operational equivalence of local and echelon base stock policies when s 1 s 1 and Lemma 2. Assuming 1, r 2 (s 1 ) is a function, r 2 (s 1 ) s2 s1 s 2 . However, the change in player i’s cost s 1 , and 0 r 2 (s 1 ) 1. due to a shift in player j’s strategy depends on the Proof. From Lemma 1, H 2 is strictly convex in s 2 , inventory tracking method. For example, holding s 2 so r 2 (s 1 ) is a function (i.e., H 2 has a unique minimum) constant, the supplier’s expected on-hand inventory is and is determined by the first-order condition independent of s 1 , but when s 2 stays constant, the supplier’s inventory declines as s 1 increases. Further- H2 L2 L2 h2 s2 s1 x G 2 s2 x dx 0. more, the total system inventory depends on s 2 only. s2 s2 s 1 So holding s 2 fixed, the retailer’s s 1 only influences the allocation of inventory between the supplier and the This condition cannot hold at s 2 s 1 because then L2 retailer. However, holding s 2 fixed, the retailer can (s 2 s 1 ) 0 and G 2 ( y) 0. Therefore, s 2 r 2 (s 1 ) increase total system inventory by raising s 1 . s 1 . Given s 2 s 1 , from the implicit function theorem, 5.2. Echelon Inventory Game Equilibria with 2 2 Shared Backorder Costs H2 H2 r 2 s1 2 In this section, we assume that each firm incurs some s2 s1 s2 backorder cost, i.e., 0 1. (We subsequently 2 consider the extreme cases 0 and 1.) We begin H2 with some preliminary results on the players’ cost s2 s1 2 , (2) functions and best reply mappings. H2 L2 s2 s 1 x G 2 s2 x dx Lemma 1. Assuming 1, H 2 (s 1 , s 2 ) is strictly s2 s1 convex in s 2 , s 2 0, and H 1 (s 1 , s 2 ) is quasiconvex in s 1 . where L2 Proof. Fix D and s 1 . Consider the following 2 H2 L2 function of s 2 : 2 h2 G 2 s1 s2 s1 s2 h2 s2 s1 D L2 G 2 min s 2 D L 2, s 1 . L2 x G 2 s2 x dx, Both terms are convex, while the second term is s2 s 1 strictly convex in the interval s 2 [D L 2 , D L 2 s 1 ]. 2 L2 Now take the expectation over D . The first term, H2 L2 s2 s1 h2 G 2 s1 . h 2 E[(s 2 s1 D L 2 ) ], is convex, and strictly convex s2 s1 Management Science/Vol. 45, No. 7, July 1999 941
  7. 7. CACHON AND ZIPKIN Competitive and Cooperative Inventory Policies The cross partial of H 2 is negative because G 2 0 Figure 1 Reaction Functions, 0.30, p 5, h 1 h2 0.5, L1 L2 1 h 2 and L 2 (s 2 s 1) 0 for s 2 s 1 . Since G 2 0, 0 r 2 (s 1 ) 1. Although in the EI game the supplier does not always fill the retailer’s orders immediately, the sup- plier’s echelon base stock level has little influence over the retailer’s strategy. Lemma 3. For the EI game, the retailer’s best reply mapping is a a s1 s2 s1 r1 s2 s 2, S s2 s1 . a Proof. Recall that G 1 ( y) is strictly convex and a minimized by y s 1 . Let x D L 2 . When s 2 x a s 1, s1 s 2 minimizes G 1 (min{s 2 x, s 1 }). When s 2 x Figure 2 Reaction Functions, 0.90, p 5, h 1 h2 0.5, a a s 1 , only s 1 s 1 minimizes G 1 (min{s 2 x, s 1 }). L1 L2 1 When s 2 s 1 , s 2 D L 2 s 1 , so r 1 (s 2 ) [s 2 , S]. When a a a a s 2 s 1 , only s 1 s 1 minimizes G 1 (min{s 2 x, s 1 }) for a all x, so r 1 (s 2 ) s 1. The retailer’s best reply is not necessarily unique, but there is only one Nash equilibrium. e Theorem 4. Assuming 0 1, in the EI game (s 1 a e a s 1, s 2 r 2 (s 1 )) is the unique Nash equilibrium. Proof. From Theorem 1.2 in Fudenberg and Tirole (1991), a pure strategy Nash equilibrium exists if (1) each player’s strategy space is a nonempty, compact convex subset of a Euclidean space, and (2) player i’s cost function is continuous in s and quasiconvex in s i . By the assumptions and Lemma 1, these conditions Lemma 5. H 2 (s 1 , s 1 s 2 ) is strictly convex in s 2 and are met, so there is at least one equilibrium. From H 1 (s 1 , s 1 s 2 ) is strictly convex in s 1 . e e e e e Lemma 2 in any equilibrium, (s 1, s 2), s 2 r 2 (s 1) s 1. e If s 2 a e s 1 , Lemma 3 implies s 1 e s 2, a contradiction. Proof. Set s 2 s 1 s 2 and s 1 s 1 . Differentiation e a e a of H 2 (s 1 , s 1 s 2 ) reveals that Hence s 2 s 1 , but from Lemma 3, this implies s 1 s 1 . e a Since r 2 is a function, there is only one s 2 r 2 (s 1 ). H 2 s 1, s 1 s2 H 2 s 1, s 2 Therefore, the equilibrium is unique. ; s2 s2 Figures 1 and 2 plot the firms’ reaction functions and the resulting equilibrium for two examples. 2 H 2 s 1, s 1 s2 2 H 2 s 1, s 2 2 2 . (3) s2 s2 5.3. Local Inventory Game Equilibria with Shared Backorder Costs From Lemma 1, H 2 (s 1 , s 2 ) is strictly convex in s 2 , so The analysis of the LI game also begins by character- H 2 (s 1 , s 1 s 2 ) is strictly convex in s 2 . Differentiate izing the cost functions and the best reply mappings. H 1 (s 1 , s 1 s 2 ), 942 Management Science/Vol. 45, No. 7, July 1999
  8. 8. CACHON AND ZIPKIN Competitive and Cooperative Inventory Policies H 1 s 1, s 1 s2 When s 2 0, L 2 (s 2 )G 1 (s 1 ) 0 and therefore r 1 (s 2 ) L2 s2 G 1 s1 1. s1 When 0 1, there is a unique Nash equilibrium L2 in the LI game. x G 1 s1 s2 x dx; l l s2 Theorem 8. Assuming 0 1, (s 1, s 2) is the unique Nash equilibrium. 2 H 1 s 1, s 1 s2 L2 2 s2 G 1 s1 Proof. Lemma 5 confirms the required conditions s1 for the existence of an equilibrium (in the proof of Theorem 4). First, from Lemma 6, r 2 (s 1 ) r 2 (s 1 ) s1 L2 x G 1 s1 s2 x dx. l 0, so s 2 0. Now, suppose there are two equilibria, l l s2 (s 1, s 2) and (s *, s *). Without loss of generality, assume 1 2 (4) l s 2 s *. From Lemma 7, this implies that s * s 1. From l 2 1 l Since G 1 is strictly convex, 2 H 1 (s 1 , s 1 2 s 2 )/ s 1 the same lemma, r 1 (s 2 ) 1, so s * 2 s* 1 s*2 s1 l l 0, which means that H 1 is strictly convex in s 1 . s2 s 2. But from Lemma 2, r 2 is increasing, so s * 2 l l The next two lemmas characterize the best reply s 2 implies that s * s 1, a contradiction. Hence, there 1 mappings. is a unique equilibrium. Figures 1 and 2 also display the reaction functions in Lemma 6. Assuming 1, r 2 (s 1 ) s1 r 2 (s 1 ); the LI game as well as the Nash equilibrium. r 2 (s 1 ) 0; and 1 r 2 (s 1 ) 0. 5.4. Equilibria Under Extreme Backorder Cost Proof. For the supplier r 2 (s 1 ) s1 r 2 (s 1 ), Allocations because H 2 (s 1 , s 1 s 2) H 2 (s 1 , s 2 ) whenever s 2 s1 Suppose the retailer is charged all of the backorder s 2 and s 1 s 1 . From Lemma 2, r 2 (s 1 ) s 1 , which costs, i.e., 1. In this situation, the Nash equilibrium implies that r 2 (s 1 ) 0. From the same lemma, 0 in the EI game is no longer unique. r 2 (s 1 ) 1. Also r 2 (s 1 ) r 2 (s 1 ) 1, so 1 r 2 (s 1 ) 0. Theorem 9. For 1, in the EI game the Nash e e e a a equilibria are (s 1 [s 2, S], s 2 [0, s 1 ]). Lemma 7. Assuming 0, r 1 (s 2 ) s . When s 2 1 0, 1 r 1 (s 2 ) 0; and when s 2 0, r 1 (s 2 ) 1. Proof. The existence proof in Theorem 4 applies even when 1, so a pure strategy equilibrium Proof. The retailer’s best reply is determined by exists. When 1, the supplier incurs no backorder the first order condition H 1 (s 1 , s 1 s 2 )/ s 1 0, see a costs, only holding costs. Hence, the supplier picks s 2 (4). When s 1 s 1 , G 1 (s 1 ) 0, and therefore H 1 (s 1 , s 1 a s 1 , i.e., r 2 (s 1 ) [0, s 1 ]. Suppose (s *, s *) is an 1 2 s 2 )/ s 1 0. Hence r 1 (s 2 ) s 1 . From the implicit a equilibrium, where s * 2 s 1 . From Lemma 3, r 1 (s *) 2 function theorem a s 1 , but an equilibrium only occurs when s * s *, so 2 1 a a s* 2 s 1 cannot be an equilibrium. Suppose s * 2 s 1. r 1 s2 a From Lemma 3, r 1 (s *) [s *, S], so for any s * s 1 , (s * 2 2 2 1 2 H 1 s 1, s 1 s2 2 H 1 s 1, s 1 s2 [s *, S], s *) is an equilibrium. 2 2 s1 s2 2 s1 In the LI game there is a unique equilibrium even when the retailer incurs all of the backorder cost. L2 s2 x G 1 s1 s2 x dx l . Theorem 10. Assuming 1, in the LI game (s 1 L2 L2 s2 G 1 s1 s2 x G 1 s1 s2 x dx l r 1 (0), s 2 0) is the unique Nash equilibrium. Assume s 2 0. Since L 2 (s 2 )G 1 (s 1 ) 0, the numer- Proof. When 1 the supplier chooses s 2 0. ator above is positive and the numerator equals the Since r 1 (s 2 ) is a function, r 1 (0) is unique. second term in the denominator, 1 r 1 (s 2 ) 0. When the supplier incurs all backorder costs, there Management Science/Vol. 45, No. 7, July 1999 943
  9. 9. CACHON AND ZIPKIN Competitive and Cooperative Inventory Policies is a unique equilibrium in both games and they are dH 2 H2 H2 r2 s1 H2 identical. ds 1 s1 s2 s1 s1 e e Theorem 11. Assuming 0, (s 1 0, s 2 r 2 (0)) L2 s2 s1 h2 G 2 s1 , l is the unique Nash equilibrium in the EI game, and (s 1 e l s 1, s 2 e s2 l s 1) is the unique Nash equilibrium of the since H 2 (s 1 , r 2 (s 1 ))/ s 2 0. From Lemma 2, r 2 (s 1 ) LI game. s 1 , so L 2 (s 2 s 1) 0, and G 2 0, so dH 2 /ds 1 0. Thus, the supplier’s cost declines as s 1 increases. Proof. Since the retailer incurs no backorder cost l e l e l Since s 1 s 1, H 2 is lower at s 1. s1 s1 0. The supplier’s best reply mapping is a Why does the supplier prefer the LI game equilib- e l function in either game, so s 2 r 2 (0). Furthermore, s 2 rium? The supplier always prefers the retailer to e e s2 s 1. increase its base stock, thereby increasing the retailer’s inventory and decreasing the supplier’s backorder 6. Comparing Equilibria costs. The retailer always chooses a lower base stock in This section compares the equilibria in the LI and EI the EI game than it does in the LI game, hence the games to each other as well as to the optimal solution. supplier is always better off in the LI game. To facilitate these comparisons, convert the LI game 6.2. Competitive Equilibria and the Optimal l l equilibrium, (s 1, s 2), into the equivalent pair of eche- Solution l l l l l l lon base stock levels, (s 1, s 2), where s 1 s 1 and s 2 s 1 In the EI game the retailer’s base stock level is lower l s 2. than in the optimal solution. 6.1. Competitive Equilibria Theorem 14. In an EI game equilibrium, the retailer’s The firms choose higher base stock levels in the LI base stock level is lower than in the optimal solution. game than in the EI game. ˆo ˆ Proof. Note that G 1 ( x) G 1 ( x), for all x. Hence, o o Theorem 12. Assuming 0 1, the base stock G ( y) G 1 ( y), for all y. Since both G 1 ( y) and G 1 ( y) 1 o a e levels for both firms are higher in the LI game equilibrium are increasing in y, s 1 s1 s 1. l o l o l than in the EI game equilibrium, i.e., s 2 e l s 2 and s 1 e s 1. In the LI game either s 1 s 1 or s 1 s 1 is possible. l o e a (In Figure 1 the retailer chooses s 1 s 1 , but in Figure Proof. The equilibrium in the EI game is (s 1 s 1, 2 s1 l o s 1 .) However, when backorder costs are e e a s2 r 2 (s 1)). From Lemma 7, r 1 (s 2 ) s 1 , which implies charged to the supplier, the supplier’s base stock level l e a that s 1 s1 s 1 . From Lemma 2, r 2 (s 1 ) is increasing is lower than in the system optimal solution in both l l a e in s 1 , so s 2 r 2 (s 1) r 2 (s 1 ) s 2. games. The retailer’s cost in the LI game equilibrium can be more or less than in the EI game equilibrium. (The Theorem 15. Assuming 1, the supplier’s base numerical study confirms this.) However, the supplier stock level in both the LI and the EI equilibria is lower than has a definite preference for the LI game. in the system optimal solution. l e Theorem 13. Assuming 0 1, the supplier’s Proof. In any equilibrium s 2 s 2, so it is sufficient o l cost in the LI game equilibrium is lower than its cost in the to show that s 2 s 2. For x s 1, EI game equilibrium. ˆ H2 G 2 s1 x 1 p Proof. In the EI game the supplier chooses r 2 (s 1 ). x In the LI game the supplier chooses r 2 (s 1 ) s 1 as its and local base stock level and the equivalent echelon base stock level is r 2 (s 1 ). Differentiate the supplier’s cost ˆo o o G2 x h2 G1 x h2 G1 x function with respect to the retailer’s base stock level, assuming the supplier chooses s 2 r 2 (s 1 ): p. 944 Management Science/Vol. 45, No. 7, July 1999
  10. 10. CACHON AND ZIPKIN Competitive and Cooperative Inventory Policies For s1 x 0 p L2 L 1 1 l s1 . h1 h2 p ˆ H2 G 2 s1 x 1 p, 0 l It is possible that s 1 s 1 because L 2 L 1 1 stochasti- o x L1 1 cally dominates and p/(h 1 h2 p) (h 2 ˆo and again G 2 ( x) p. For x ˆ 0, H 2 / x h 2 and o o p)/(h 1 h 2 p). For the supplier s 2 s 1 when G 2 o o o (s 2 s 1) 0. This occurs precisely when ˆo G2 x h2 o G1 x h 2, L2 L 1 1 o p o ˆo s1 . with strict inequality for x s 1 . So in all cases G 2 ( x) h1 h2 p ˆ H 2 / x, with strict inequality for s 1 x o s 1. o o l l o In that case, s 2 s1 s1 s 2. Therefore, G 2 ( x) H 2 / x, with strict inequality for o l s2 s 1 . So, s 2 s 2. Recall that the supplier’s echelon base stock deter- 7. Cooperative Inventory Policies mines the supply chain’s average inventory level. According to Theorem 17, the optimal solution is From Theorem 15 it follows that in either game’s virtually never a Nash equilibrium. Hence, the firms equilibrium the supply chain’s average inventory can lower total costs by acting cooperatively. level will be lower than in the optimal solution, There are several methods that enable the firms to suggesting that competition will also tend to lower the minimize total costs and still remain confident that the supply chain’s average inventory. The numerical other firm will not deviate from this agreement. For o o study confirms this observation. instance, the firms could contract to choose (s 1 , s 2 ) as When the supplier incurs no backorder costs, the their base stock levels. But, since each firm has an supplier’s base stock level is no greater than in the incentive to deviate from this contract (because it is system optimal solution. not a Nash equilibrium), the contract must also specify e o l o a penalty for deviations. Such stipulations are hard to Theorem 16. Assuming 1, s 2 s 2 , and s 2 s 2. enforce. e e a o o Proof. In the EI game s 2 s1 s1 s1 s 2, Alternatively, the firms could write a contract that e o l l hence s 2 s 2 . When 1, in the LI game s 2 s 1, so specifies transfer payments which eliminate incentives l o the proof of Theorem 15 demonstrates s 2 s 2. to deviate from the optimal solution. There are several schemes to achieve this goal: a per period fee for the Theorem 17. Assuming 1, the system optimal supplier’s backorder; a per unit fee for each unit the solution is not a Nash equilibrium. supplier does not ship immediately; a per unit fee per e l o Proof. From Theorem 15, s 2 s2 s 2 , so the consumer backorder; or a subsidy for each unit of optimal solution is not a Nash equilibrium in either inventory in the system. (Nonlinear payment sched- game. ules, as in Lee and Whang (1996), could also be When the supplier incurs no backorder costs, the considered, but these are necessarily more cumber- o system optimal solution can be a Nash equilibrium some. For instance, the induced penalty function G 1 is under a very special condition. nonlinear.) Transfer payments can also be imposed on the retailer, e.g. a subsidy on retailer inventories, or a Theorem 18. Assuming 1, the system optimal subsidy on backorders. solution is a Nash equilibrium in the LI game only when Payments could also be based on the inventory and p backorder levels that would have occurred had the L2 L 1 1 o s1 . supplier performed certain actions. Chen (1997) uses h1 h2 p this approach. Define accounting inventory and ac- Proof. When 1, the LI game Nash equilibrium counting backorders as the inventory and backorder l l l l is (s 1, s 2 s 1). Solving for s 1, levels, assuming the supplier fills retailer orders im- Management Science/Vol. 45, No. 7, July 1999 945
  11. 11. CACHON AND ZIPKIN Competitive and Cooperative Inventory Policies mediately. Suppose the supplier pays all of the retail- T s 1, s 2 E 2 s2 D L2 er’s actual costs, and the supplier charges the retailer h 1 per unit of accounting inventory and h 2 p per T1 s1 min 0, s 2 D L2 accounting backorder. Then, the retailer will choose o s 1 . Since the supplier incurs all actual costs, and the 2 L2 x x s 2 dx L2 s2 T1 s1 o o retailer chooses s 1 , the supplier chooses s 2 . With this s2 scheme, the retailer’s decision is independent of the supplier’s; there is no strategic interaction between the L2 firms. However, this approach creates a challenging x T1 s1 s2 x dx. s2 accounting problem. We study linear transfer payments based on actual Note that s 1 influences the retailer inventory and inventory and backorder levels. While our approach backorders, but not the supplier’s backorders. Let avoids the problem of tracking accounting inventory H ic (s 1 , s 1 s 2 ) be player i’s costs after accounting for and backorders, Chen’s method uses only cost param- the transfer payment, eters. Ours also requires a demand parameter. Thus, c H 1 s 1, s 1 s2 H 1 s 1, s 1 s2 T s 1, s 2 , his technique may be easier to implement in some situations, ours in others. c H 2 s 1, s 1 s2 H 2 s 1, s 1 s2 T s 1, s 2 . 7.1. Linear Contracts We wish to determine the set of contracts, ( 1, 2, 1), Suppose the firms track local inventory and they c c such that (s 1 , s 2 ) is a Nash equilibrium for the cost adopt a transfer payment contract with constant pa- functions H ic (s 1 , s 1 c s 2 ), where s 1 o c s 1 , and s 1 s2c rameters ( 1, 2, 1). This contract specifies that the o c c s 2 . With these contracts the firms can choose (s 1 , s 2 ), period t transfer payment from the supplier to the thereby minimizing total costs, and also be assured retailer is that no player has an incentive to deviate. I 1 1t 2 B 2t 1 B 1t , To find the desired set of contracts, first assume that H ic is strictly convex in s i , given that player j chooses where I 1t is the retailer’s on-hand inventory, and B it is s jc , j i. Then determine the contracts in which s ic stage i’s backorders, all measured at the end of the satisfies player i’s first order condition, thereby mini- period. There are no a priori sign restrictions on these mizing player i’s cost. Finally, determine the subset of parameters, e.g. 1 0 represents a holding cost these contracts that also satisfy the original strict subsidy to the retailer and 1 0 represents a holding convexity assumption. fee. (We later impose some restrictions on the param- The following are the first order conditions: eters.) We also assume the optimal solution is com- c H1 mon knowledge. 0 L2 s2 G 1 s1 T 1 s1 Define T 1 (IP 1t ) as the expected transfer payment in s1 period t L 1 due to retailer inventory and back- orders, where L2 x G 1 s1 s2 x L1 1 L1 1 s2 T1 y E 1 y D 1 y D y L1 1 T 1 s1 s2 x dx; (5) 1 c H2 L2 0 2 h2 2 s2 1 1 x y L1 1 x dx. s2 y L2 Define T(s 1 , s 2 ) as the expected per period transfer x G 2 s1 s2 x payment from the supplier to the retailer, s2 946 Management Science/Vol. 45, No. 7, July 1999
  12. 12. CACHON AND ZIPKIN Competitive and Cooperative Inventory Policies T 1 s1 s2 x dx. (6) T 1 s1 s2 x dx 0; L2 o o Define 2 (s 2 s 1 ). (This is the supplier’s 2 H2 c L2 in-stock probability, essentially its fill rate.) Further- 2 h2 2 s2 s2 more, the supplier’s first order condition in the opti- mal solution is L2 L2 o o x G 2 s1 s2 x 0 p p h2 s2 s1 h1 h2 p s2 L2 L1 1 o T 1 s1 s2 x dx 0. x s2 x dx, so so 2 1 The first inequality reduces to or, h1 h2 p 1 1 0. L2 L1 1 o p p h2 2 Substituting (8) yields 1 h1 h 2 and p. For x s2 x dx . (7) 1 h1 h2 p the supplier, sufficient conditions are so 2 so 1 Using (7), (5), and (6) yield the following two equa- 1 p 1 1 0; tions in three unknowns, h2 2 1 p 1 p 1 p 1 1 , (8) 1 p L1 1 o s1 0. h1 h2 1 1 h2 1 Combining the first inequality with (8) yields 1 0 2 h2 1 2 . (9) and 1 (1 ) p. The second inequality, along h1 h2 2 with (8) and (9), yields 2 0. It remains to ensure that the costs functions are These are quite reasonable conditions: The first indeed strictly convex. requires that the retailer’s inventory subsidy not elim- Theorem 19. When the firms choose ( 1 , 2 , 1 ) to inate retailer holding costs; the second stipulates that satisfy (8) and (9), and the following additional restrictions the supplier be penalized for its backorders; and the apply third states that the supplier should not fully reim- burse the retailer’s backorder costs, and the retailer (i) h 1 h2 1 0 should not overcompensate the supplier’s backorder costs. (ii) 2 0 To help interpret these results, consider the three extreme contracts where one of the parameters is set to (iii) p 1 1 p, zero: c c then the optimal policy (s , s ) is a Nash equilibrium. 1 2 2 Proof. When the following second order condi- i) 1 0 2 h2 1 1 p 1 2 tions are satisfied, H ic is strictly convex in s i , assuming sj s jc , j i: ii) 1 h1 h2 2 0 1 p 2 c H1 L2 2 2 s2 G 1 s1 T 1 s1 iii) 1 1 h1 h2 2 h2 1 0. s1 1 2 (Of these three contracts, the second does not meet the L2 x G 1 s1 s2 x conditions in Theorem 19, because the supplier fully s2 compensates the retailer for all of its costs. The retail- Management Science/Vol. 45, No. 7, July 1999 947
  13. 13. CACHON AND ZIPKIN Competitive and Cooperative Inventory Policies o er’s incentive to choose the optimal policy is weak: s 1 ing to the optimal solution Pareto dominates any is a Nash equilibrium strategy, but any s 1 is too.) other. Hence, the players can coordinate on this equi- With the first contract the retailer fully reimburses librium. (There is experimental evidence that players the supplier for the supplier’s consumer backorder coordinate on a Pareto dominant equilibrium when penalty. However, the supplier still carries inventory they are able to converse before playing the game, e.g. because it pays a penalty for its local backorders. With Cooper et al. 1989, Cachon and Camerer 1996.) the third contract the supplier subsidizes the retailer’s Although total costs decline when the firms coordi- holding costs, but not fully (provided 0). In nate, one firm’s cost may increase. This firm will be addition, the supplier is penalized for its backorders, unwilling to participate in the contract unless it re- but less than in the first contract. When the retailer ceives an additional transfer payment. To maintain the incurs all backorder costs (i.e., 1), only a supplier strategic balance of the contract, this payment should backorder penalty is required, 2 2 h 2 /(1 2 ). be independent of all other costs and actions. For Incidentally, (8) and (9) can be written example, the firms could transfer a fixed fee each period. Alternatively, one could seek a contract ( 1, 2, p 1 p 1 ) such that each firm’s cost is no greater than in the 1 h1 h2 p h1 h2 1 p 1 original Nash equilibrium. h1 h2 Finally, this analysis assumes the firms use local base stock levels. In this context local policies have 2 2 . several advantages over echelon policies. Recall that s 1 1 h2 1 2 has no influence on 2 B 2t , the supplier’s backorder h1 h2 penalty, when firms use local base stock levels. How- Consider the first identity. The quantity p ever, with echelon stock base stock levels, s 1 does 1 is the retailer’s backorder cost, and its holding cost, includ- influence 2 B 2t , holding s 2 constant. This can create a ing transfer payments, is perverse incentive. Suppose 2 is large. By increasing s1 s 2 , the retailer can make B 2t arbitrarily large 1 (assuming S, the limit on s 1 , is large too). The 2 B 2t h1 h2 1 , h1 h2 transfer payment could easily dominate the additional retailer inventory cost. (Furthermore, once s 1 s 2 , the which is written more simply as h 1 h2 1 . So the retailer can increase s 1 without increasing its inven- left-hand side is the retailer’s critical ratio. The right- tory.) There is a solution to this problem. The transfer hand side is the critical ratio for the total system costs o payment could assume that the retailer chooses s 1 s 1 . controlled by the retailer. These ratios must be iden- Hence, the retailer would receive no additional benefit tical to induce the retailer to minimize total system o by raising s 1 above s 1 . Clearly this increases the costs. The second identity specifies a critical ratio for complexity of the contract. Local measurements avoid the supplier. 2 is its (local) backorder cost. The this problem altogether. holding cost h 2 is multiplied by the factor (1 1 /(h 1 h 2 )), which is the fraction of actual holding costs paid by the retailer, taking the transfer payment into account. Thus, this rule effectively reduces both 8. Numerical Study The system optimal solution is virtually never a Nash stages’ holding costs by the same fraction. equilibrium, but how large is the difference between 7.2. Additional Contracting Issues their costs? To answer this question, we conducted a Theorem 19 details the contracts that make the opti- numerical study. mal solution a Nash equilibrium, but this does not One period demand is normally distributed with imply a unique equilibrium. Nevertheless, even if there mean 1 and standard deviation 1/4. (There is only a were additional Nash equilibria, the one correspond- tiny probability of negative demand.) The remaining 948 Management Science/Vol. 45, No. 7, July 1999

×