Pricing Coordination in Supply Chains through Revenue Sharing Contracts

Transcription

1 Information and Management Sciences Volume 19, Number 1, pp , 2008 Pricing Coordination in Supply Chains through Revenue Sharing Contracts Yong-Wu Zhou Shanlin Yang Hefei University of Technology Hefei University of Technology P. R. China P. R. China Abstract In a decentralized supply chain, local decision-makers at different echelons of the supply chain usually pursue their own maximizing-profits. However, a decision that is locally optimal could be globally inefficient. Through the revenue sharing mechanism, this paper proposes a supply-chain contract model to coordinate the pricing decision in a multi-echelon supply chain with a deterministic price-sensitive demand. We identify the conditions under which the contract can achieve channel coordination as well as improve the profits of all the supply chain members, and also present how to modify the contract parameters so as to more evenly share the profit along the chain. Keywords: Decentralized Supply Chain, Revenue Sharing Contract, Pricing Coordination. 1. Introduction With increasing emphasis on the significance of effective supply chain management, many researchers have paid much attention to coordination issues between manufacturers (suppliers) and retailers (buyers) in a multi-echelon supply chain in recent years. A supply chain is coordinated if the decisions made by all partners maximize the total profit of the supply chain. This would occur if the supply chain is vertically integrated or centralized. In reality, however, the players (manufacturers, distributors, retailers, etc.) involved in the supply chain usually belong to different entities. Thus, they would strive to maximize their own profits rather than that of the whole channel when they make a decision. This independent decision-making would directly result in the supply chain lacking coordination because a decision that maximizes the retailer s profit might Received February 2006; Revised November 2006; Accepted September Supported in part by National Natural Science Foundation of China under grant numbers and , Program for New Century Excellent Talents in University under grant number NCET and a Foundation for the Author of National Excellent Doctoral Dissertation of PR China under grant number

2 32 Information and Management Sciences, Vol. 19, No. 1, March, 2008 not maximize the total profit of all members in the supply chain. Therefore, it is necessary to design coordination mechanisms that enable local decision-makers to pursue channel coordination. Much previous literature considered coordination issues in decentralized supply chains, and proposed various coordination mechanisms such as quantity discount, contracts, etc. While most of these papers employed quantity discounts to coordinate supply chain members, Giannoccaro and Pontrandolfo (2004) used the revenue sharing (abbreviated by RS) contracts proposed by Cachon and Lariviere (2000) as a tool to coordinate order decisions in a three-echelon supply chain. They assumed that the market demand is independent of the retailing price. Unlike Giannoccaro and Pontrandolfo, we consider in this paper a deterministic price-sensitive demand rate, and use the RS contract to coordinate the pricing decision behavior of supply chain partners, ensuring the supply chain to function as a centralized one. The RS contract in a two-echelon supply chain is characterized by the two parameters (w,φ): the supplier charges the retailer a wholesale price w less than the marginal cost C in change for a percentage (1 Φ) of the retailer s revenue. The condition w < C is to pledge channel coordination while Φ determines the distribution of total profits between the supplier and the retailer. The objectives of our study are twofold. First, we want to make sure, under a deterministic price-sensitive market demand, whether one can design the RS contracts to achieve channel coordination in both a two-echelon and a three-echelon supply chain. And if so, how can one do that. Second, we hope to identify the conditions under which such RS contracts would be beneficial to all supply chain partners Literature review Due to the increasing emphasis on improving supply chain management, a number of papers considered coordination issues in decentralized supply chains. Earlier researches on channel coordination include two main categories. The first category, from a marketing perspective, focuses on the coordination of pricing decisions, without inventory-related cost considerations. The most representative work on this topic was that done by Jeuland and Shugan (1983). They showed that a manufacturer, by designing a quantity discount schedule, could entice a retailer to choose the channel-optimal selling price. Moorthy (1987) presented that the manufacturer may also maximize both individual and channel profit through an appropriately

3 Pricing Coordination in Supply Chains through Revenue Sharing Contracts 33 designed two-part tariff. Ingene and Parry (1995) further discussed the coordination issues of pricing decisions in a decentralized system where a manufacturer sells through heterogeneous competing retailers. They showed that a manufacturer s quantity discounts could maximize channel profit with heterogeneous competing retailers, yet failed to show whether the manufacturer can simultaneously maximize individual profit as well. More recently, Balachannder and Srinivasan (1998) reported that, when a manufacturer supplies a product to heterogeneous retailers, quantity discounts could neither maximize the manufacturer s profit nor the channel s. Different from the previous works mentioned above, this paper will use the revenue sharing contracts to complete channel coordination. Another category, however, mainly focuses on the coordination of order decisions. Monahan (1984) presented a distribution system consisted of a single supplier and a single buyer, and showed that the supplier, offering the buyer a quantity discount schedule, could earn higher profits as a result of improved channel coordination. Lee and Rosenblatt (1986) generalized Monahan s model by relaxing Monahan s order-for-order assumption for the supplier. Dolan (1987) provided a survey on quantity discount literature. Some researchers further extended the above works to cases with homogeneous or heterogeneous buyers (Parlar and Wang, 1994; Weng, 1995; Kim and Hwang, 1988; Wang and Wu, 2000; Chen et al., 2001; Raju and Zhang, 2005), multiple products (Chen and Chen, 2005), and multi-period & probabilistic customer demand (Li and Liu, 2006) etc. Munson and Rosenblatt (2001) generalized the previous works to consider a three-level supply chain (supplier-manufacturer-retailer). They pointed out incorporating quantity discounts into both ends of the supply chain can significantly decrease costs compared to concentrating only on two levels (p373). Klastorin et al. (2002) considered the issues of order coordination in a decentralized multi-echelon supply chain where a manufacturer, who outsources the production of a product to an Original Equipment Manufacturer, supplies the product to multiple retailers. They proposed a price discount policy that can make retailers coordinate the timing of their orders with the manufacturer s order cycle. Unlike most of the existing quantity discount models concerning the coordination of order decisions, Viswanathan and Piplani (2001) presented a model in which a supplier offers a discount that induces all buyers to place orders only at times specified by the supplier (e.g., the first day of every month). Mishra (2004) generalized their model to allow for a selective discount policy that, if beneficial, excludes some buyers to minimize the supplier s total cost. Unlike the models mentioned in this paragraph, the present paper considers the price-sensitive demand rate and a different coordination mechanism.

4 34 Information and Management Sciences, Vol. 19, No. 1, March, 2008 Various models of supply chain contracts can be found in the literature, including the backup agreements (Eppen and Iyer, 1997), the return policies (Emmons and Gilbert, 1998), the quantity flexibility contracts (Tsay, 1999), the incentive mechanisms (Lee and Whang, 1999), the revenue sharing contracts (Cachon and Lariviere, 2000;), etc. Interested readers can read a review by Cachon (2001) for a detailed survey of issues in general supply chain contracts. Most of contract models considered a demand independent of the retail price but this paper considers a price-dependent demand. 2. Assumptions and Notations 2.1. Assumptions 1) Section 3 considers a two-echelon supply chain in which a distributor supplies a single product to a retailer, while Section 4 considers a three-echelon system where a manufacturer provides an item to a distributor who then supplies it to a retailer. 2) The retailer in each of the two systems faces the deterministic price-sensitive market demand that is assumed to be of the following form: f(p) = a bp, where a (> 0) is a potential market demand, b (> 0) a slope parameter and p ( a/b) the retail price charged from customers. This demand pattern was used by many researchers like Ingene and Parry (1995), Balachannder and Srinivasan (1998), etc. 3) Before accepting the revenue sharing contract, the two members in the two-echelon channel follow the distributor-stackelberg market setting, in which the distributor announces a wholesale price and then the retailer in turn determines a retail price or an order volume. 4) All parties in the three-echelon channel follow the manufacturer-distributor-stackelberg market setting before accepting the revenue sharing contract. That is, the manufacturer first claims a wholesale price to the distributor after knowing the distributor s optimal reaction to any wholesale price set by him. The distributor then sets a wholesale price to the retailer after observing the retailer s optimal reaction to any wholesale price set by her Notations X = {M,D,R} is the set of members in the considered supply chain, where M represents a manufacturer, D a distributor and R a retailer. C x denotes the marginal cost of member x, where x X and (C R +C D +C M ) < a/b.

5 Pricing Coordination in Supply Chains through Revenue Sharing Contracts 35 Y = {c,m} is the set of the considered settings, where c represents an RS contract setting and m the *-Stackelberg market setting. Π I stands for the total profit of the virtually integrated firm, where all members in the channel are assumed to be willing to make pricing decision jointly; p I and q I are respectively the sales price and sales volume (hence order volume) of the virtually integrated firm. Π xy is the profit of member x in the supply chain in the setting y, x X, y Y. w xy is the wholesale price charged by member x in the setting y, x X \ {R}, y Y. p Ry is the retail price charged by the retailer in the setting y, y Y. Φ Dc is the quota that retailer R keeps his revenue under the RS contract. Φ Mc the quota that distributor D keeps the revenue of her in the RS contract. We will then present in Section 3 the RS mechanism concerning the two-echelon supply chain and Section 4 a coordinated pricing model for the three-echelon supply chain through the RS contract. 3. The Revenue Sharing Model in a Two-echelon Supply Chain In this section, we consider how the distributor in the above-mentioned two-echelon channel sets the RS contract to coordinate the two parties pricing so as to maximize the profit of the whole channel. Before presenting the RS contract model, we need to briefly review the joint and separate pricing decision of the two members, 3.1. Joint decision Consider a virtual situation that distributor D and retailer R belong to an integrated firm. In such a case, the two partners will make the pricing decision jointly. Thus, following the way similar to that used by Ingene and Parry (1995), Balachannder and Srinivasan (1998) etc, we have, when the retailing price is set as p I, the profit of the integrated firm in the form of Π I = (p I C R C D )(a bp I ) (1) It is clear that the integrated firm will set the optimal retailing price as p I = [a + b(c R + C D )]/(2b) (2)

6 36 Information and Management Sciences, Vol. 19, No. 1, March, 2008 The corresponding order quantity and the maximal profit of the integrated firm will be given respectively by q I = [a b(c R + C D )]/2; Π I = [a b(c R + C D )] 2 /(4b) (3) 3.2. Separate decision Consider now the practical situation that distributor D and retailer R make their pricing decisions independently. Based on assumptions in Section 2, before implementing the RS contract, distributor D as a price-leader declares a wholesale price w Dm and retailer R sets independently his optimal sales price p Rm to maximize his own profit after being informed of the wholesale price w Dm. Thus, for any given w Dm, profit-maximizing retailer R will set the sales price as p Rm = [a + b(w Dm + C R )]/(2b) (4) With this retail price, the quantity retailer R sells to customers (hence orders from distributor D) is given by q Rm = [a b(w Dm + C R )]/2 (5) This makes distributor D earn the following profit Π Dm = (w Dm C D )q Rm (6) Profit-maximizing distributor D will set her optimal wholesale price as w Dm = [a + b(c R + C D )]/(2b) (7) Hence, under the distributor-stackelberg market setting, the retail price and two partners profits are respectively p Rm = [3a + b(c R + C D )]/(4b); Π Rm = [a b(c R + C D )] 2 /(16b); Π Dm = [a b(c R + C D )] 2 /(8b) (8) Obviously, the lack of coordination between two partners pricing decisions makes the channel profit reduce by [a b(c R + C D )] 2 /(16b).

7 Pricing Coordination in Supply Chains through Revenue Sharing Contracts Revenue sharing contract model Next, we consider how distributor D designs the RS contract to modify retailer R s profit so that retailer R is willing to accept the RS contract to coordinate their pricing decision coherent with the channel-optimal one. As described in Cachon and Lariviere s paper (2000), the RS contract is such a coordination mechanism that distributor D offers retailer R a lower wholesale price w Dc in change for that retailer R returns distributor D a percentage (1 Φ Dc ) of his revenue. Under this RS contract, retailer R s profit is Π Rc = [Φ Dc p Rc (w Dc + C R )](a bp Rc ) (9) Hence, profit-maximizing retailer R will set the retail price as p Rc = [Φ Dc a + b(w Dc + C R )]/(2bΦ Dc ) (10) To achieve channel coordination, the optimal sales price set by retailer R must be consistent with the sales price of the integrated firm, i.e., p Rc = p I. It follows that under the RS contract distributor D should select the wholesale price as w Dc = Φ Dc (C R + C D ) C R (11) which is obviously less than distributor D s marginal cost C D. In other words, if the adopted RS contract satisfies (11), then channel coordination will be assured for any Φ Dc (0,1). In that case, distributor D s and retailer R s profits are respectively given by Π Dc = (1 Φ Dc )[a b(c R + C D )] 2 /(4b); Π Rc = Φ Dc [a b(c R + C D )] 2 /(4b) (12) Now the problem is how to choose Φ Dc so that both partners are willing to adopt the RS contract. From retailer R s perspective, he will accept the RS contract only if he can obtain higher profit under the RS contract than under the distributor-stackelberg market setting. That is, the following condition should be met. Π Rc > Π Rm, i.e., Φ Dc > 1/4 (13) Similarly, distributor D is willing to offer the RS contract to retailer R only if Π Dc > Π Dm, which gives Φ Dc < 1/2 (14)

8 38 Information and Management Sciences, Vol. 19, No. 1, March, 2008 Additionally, noting w Dc > 0 will give Φ Dc > C R /(C R + C D ) (15) From (14) and (15), one can easily infer that distributor D may not find an acceptable Φ Dc if C R /(C R + C D ) 1/2, i.e., C D C R. This indicates that the RS contract is able to achieve channel coordination as long as distributor D s marginal cost is greater than retailer R s. Under that case, the Φ Dc chosen by distributor D for the RS contract can only drop in the interval (Φ 0,1/2), where Φ 0 = max{1/4,c R /(C R + C D )}. Through some simple operations, one will obtain that the range of acceptable Φ Dc is (C R /(C R + C D ),1/2) if C R < C D 3C R and (1/4,1/2) if C D > 3C R. Figure 1(a). The set of acceptable para- Figure 1(b). The set of acceptable parmeters (w Dc,Φ Dc ) for C R < C D 3C R ameters(w Dc,Φ Dc ) for C D > 3C R The solid portions of Figures 1(a) and 1(b) intuitively describe the set of all RS contract parameters (w Dc,Φ Dc ) that achieve channel coordination and benefit two partners as well. In practice, the value of Φ Dc in the adopted RS contract depends on the bargaining power between two partners. 4. The Revenue Sharing Model in a Three-echelon Supply Chain Now, consider how to design the RS contract to achieve the perfect coordination of the considered three-echelon supply chain. Before doing so, we also need to briefly review the joint and separate pricing decision of all the partners in the chain Joint decision As analyzed in Section 3, if the three partners had belonged to a vertically integrated firm, then the optimal sales price of this integrated firm charged from customers would be set as p I = [a + b(c R + C D + C M )]/(2b) (16)

9 Pricing Coordination in Supply Chains through Revenue Sharing Contracts 39 This price of channel optimality results in the integrated firm selling the product of q I = [a b(c R +C D +C M )]/2; and obtaining the profit of Π I = [a b(c R +C D +C M )] 2 /(4b) Separate decision Consider now the situation that all the three parties make their pricing decisions independently. Under such a situation, each participator in the channel will determine his pricing policy to maximize his own profit. As described in Section 2, before implementing the RS contract, they follow the manufacturer-distributor-stackelberg market setting. That is, manufacturer M as the price-leader sets a wholesale price wmm to distributor D after knowing distributor D s optimal reaction to any given wmm. Distributor D then determines a wholesale price w Dm to retailer R after observing retailer R s optimal reaction to any given w Dm. Following the way similar to that in Section 3, one has no difficulty to derive that, under this manufacturer-distributor-stackelberg market setting, the optimal wholesale prices designed by manufacturer M and distributor D and the optimal retail price charged by retailer R from customers are respectively w Mm = [a + b(c R + C D + C M )]/(2b); w Dm = [3a + b(c R + C D + C M )]/(4b); p Rm = [7a + b(c R + C D + C M )]/(8b) (17a) (17b) (17c) With these pricing decisions, three partners (manafacturer M, distributor D and retailer R) will obtain respectively the following profits: Π Mm = [a b(c R + C D + C M )] 2 /(16b); Π Dm = [a b(c R + C D + C M )] 2 /(32b); Π Rm = [a b(c R + C D + C M )] 2 /(64b) (18a) (18b) (18c) Thus, under the manufacturer-distributor-stackelberg market setting, the whole channel achieves only the profit of 7[a b(c R + C D + C M )] 2 /(64b). Therefore, channel coordination can make the channel get more profits back in the threeechelon supply chain than in the two-echelon supply chain Revenue sharing contract model Consider now the issues of how to design an RS contract in the three-echelon supply chain to achieve channel coordination. Such an RS contract can be characterized as

10 40 Information and Management Sciences, Vol. 19, No. 1, March, 2008 follows. Retailer R would keep a quota Φ Dc of his revenue, giving the rest (1 Φ Dc ) to distributor D, and this would be balanced by a lower price w Dc charged by distributor D. Similarly, D would keep a quota Φ Mc of her revenue, giving the rest (1 Φ Mc ) to manufacturer M, and this would be also balanced by a lower price w Mc charged by M. Under this RS contract, the profit of retailer R is given by Π Rc = [Φ Dc p Rc (w Dc + C R )](a bp Rc ) (19) Profit-maximizing retailer R will determine the sales price as p Rc = [Φ Dc a + b(w Dc + C R )]/(2bΦ Dc ) (20) Hence, retailer R s sales quantity (or order quantity) is q = [Φ Dc a b(w Dc + C R )]/(2Φ Dc ) (21) Such an order quantity will lead to distributor D and manufacturer M respectively obtaining the following profits. Π Dc = {Φ Mc [(1 Φ Dc )p Rc + w Dc ] (w Mc + C D )}q ; Π Mc = {(1 Φ Mc )[(1 Φ Dc )p Rc + w Dc ] + w Mc C M }q (22a) (22b) Thus, profit-maximizing distributor D will charge the following wholesale price from retailer R. w Dc = {Φ Dc [Φ Mc Φ Dc a + b(w Mc + C D + Φ Mc C R )]}/[bφ Mc (1 + Φ Dc )] C R (23) Therefore, for any given (Φ Dc,Φ Mc ), to achieve channel coordination, the wholesale price w Dc set by distributor D has to enable retailer R set the optimal retailing price p Rc coherent with the optimal sales price of the integrated firm. That is, w Dc should satisfy p Rc = p I, which gives w Dc = Φ Dc (C R + C D + C M ) C R (24) Thus, manufacturer M who is located at the upstream market has to adjust the wholesale price w Mc so that profit-maximizing distributor D just sets the same wholesale price as one given by (24). This indicates that the wholesale price charged by manufacturer M from distributor D should satisfy the equation below. {Φ Dc [Φ Mc Φ Dc a + b(w Mc + C D + Φ Mc C R )]}/[bφ Mc (1 + Φ Dc )] = Φ Dc (C R + C D + C M )

11 Pricing Coordination in Supply Chains through Revenue Sharing Contracts 41 which gives w Mc = Φ Mc {(C D + C M ) Φ Dc [a b(c R + C D + C M )]/b} C D (25) After noting that both w Dc and w Mc must be positive, from (24) and (25) one has Φ Dc > γ, where γ = C R /(C R + C D + C M ) (26a) Φ Mc > f(φ Dc ), where f(φ Dc )=C D /{(C D +C M ) Φ Dc [a b(c R +C D +C M )]/b} (26b) Due to 0 < Φ Mc < 1, from (26b) it follows Φ Dc < β, where β = bc M /[a b(c R + C D + C M )] (26c) Therefore, to achieve channel coordination, an RS contract scenario must simultaneously satisfy (24), (25) and (26). We refer to such an RS contract that can achieve channel coordination as the feasible one. Combining (19) (22), (24) and (25), one can easily obtain that a given feasible RS contract {(w Dc,Φ Dc ), (w Mc,Φ Mc ) will bring the three partners (M,D,R) respectively the following profits. Π Rc = Φ Dc [a b(c R + C D + C M )] 2 /(4b); Π Dc = Φ Mc (1 + Φ Dc )[a b(c R + C D + C M )] 2 /(4b) Π Mc = [1 Φ Dc Φ Mc (1 + Φ Dc )][a b(c R + C D + C M )] 2 /(4b) (27a) (27b) (27c) For this feasible RS contract to be implemented or adopted, the chosen values of Φ Dc and Φ Mc must render all partners more profits under the adopted contract than in the manufacturer-distributor-stackelberg market setting. That is, the values of Φ Dc and Φ Mc must satisfy Π xc > Π xm for each x X. From (18) and (27), it follows Φ Dc > 1/16 Φ Mc > g(φ Dc ), where g(φ Dc ) = 1/[8(1 + Φ Dc )] Φ Mc < h(φ Dc ), where h(φ Dc ) = 1 + 7/[4(1 + Φ Dc )] (28a) (28b) (28c) Thus, an RS contract that achieves the channel coordination of three-echelon supply chain not only needs to satisfy (24) (26) but also (28). For making sure further how to design a feasible RS contract that can be implemented, we need to present several results below. Proposition 1 shows the condition under which there is no feasible RS contract. Proposition 2 shows the upper and lower bounds of the

12 42 Information and Management Sciences, Vol. 19, No. 1, March, 2008 quota that retailer R will keep his own revenue under an adopted feasible RS contract. Proposition 3 further shows the condition under which a feasible RS contract can not be implemented. Proposition 1. If a/b (1+C M /C R )(C R +C D +C M ), there is no RS contract that can achieve channel coordination. Proof. If a/b (1 + C M /C R )(C R + C D + C M ), then γ β. Therefore, combining (26a) and (26c) will show the validity of Proposition 1. Proposition 2. Under any feasible RS contract to be implemented, retailer R can only keep greater than 6.25 and less than 62.5 percent of his revenue. Proof. Since (27a) shows clearly that only if Φ Dc > 1/16 a feasible RS contract can be accepted by retailer R, it is obvious that retailer R keeps greater than 6.25 percent of his revenue under the implemented RS contract. Additionally, (27b) and (27c) indicate that the other two partners are willing to adopt the feasible RS contract only if the following condition is met g(φ Dc ) < Φ Mc < h(φ Dc ) (29) From the definitions of g(φ Dc ) and h(φ Dc ), one easily knows that both functions are decreasing convex ones and have the same value at Φ Dc = 5/8. Their digraphs in the (Φ Dc Φ Mc ) plane are shown in Figure 2. From Figure 2 one can see that (29) will not be held if Φ Dc 5/8. So M or D will not accept the RS contract when Φ Dc 5/8. It implies that, under a feasible RS contract to be successfully implemented, retailer R can only keep less than 5/8 of his revenue. Figure 2. The diagrams of functions g(φ Dc ) and h(φ Dc ).

13 Pricing Coordination in Supply Chains through Revenue Sharing Contracts 43 Proposition 3. If the sum of manufacturer M s and distributor D s marginal costs does not exceed 3/5 of retailer R s marginal cost (i.e., C D + C M 3C R /5), any feasible RS contract can not be implemented successfully. Proof. Since C D + C M 3C R /5, γ 5/8. It, together with (26a), means that for every feasible RS contract, parameter Φ Dc might be greater than 5/8. Therefore, all of the feasible RS contracts will not be accepted by M or D, which shows Proposition 3. Proposition 4. For a three-echelon supply chain, if to more evenly share the profit along the chain, the contract parameter Φ Dc should be close to 1/3 and Φ Mc close to 1/4. Proof. Evenly sharing the profit along the chain means that each of the three partners will gain a profit close to 1/3 of the profit obtained by the virtually integrated firm. That is, the RS contract parameter (Φ Dc,Φ Dc ) should meet Π Rc Π I /3; Π Dc Π I /3 Substituting (27) and Π I into the above formula gives Φ Dc 1/3; Φ Mc 1/4 The proof of Proposition 4 is completed. Propositions 1 and 3 have shown that when a/b (1 + C M /C R )(C R + C D + C M ) or C D + C M 3C R /5, it is impossible for the considered three-echelon supply chain to implement RS contract to improve channel coordination. Therefore, without loss of generality, assume a/b < (1 + C M /C R )(C R + C D + C M ) (i.e., γ < β) and C D + C M > 3C R /5 in the analysis below. For convenience, divide the problem by two cases as follows. Case I. β 1 > γ Combining (26a) and (26c) will show that the Φ Dc designed by distributor D for the feasible RS contract can be chosen only in the interval (γ, 1). This, together with (24), indicates that the set of parameters (w Dc,Φ Dc ) designed by distributor D for the feasible RS contract can be represented intuitively by the thick line segment in Figure 3(a). For any pair of parameters (w Dc,Φ Dc ) chosen by distributor D, to assure channel coordination, the corresponding parameters (w Mc,Φ Mc ) designed by manufacturer M have to satisfy (25). Due to w Mc > 0 and 0 < Φ Mc < 1, the thick line segment L characterized by equation (25) in Figure 3(b) provides the correspondent set of parameters (w Mc,Φ Mc )

14 44 Information and Management Sciences, Vol. 19, No. 1, March, 2008 that manufacturer M can select so as to achieve channel coordination. If Φ Dc changes from γ to 1, line segment L will move continuously from line segment A to B. Therefore, the dashed area between line segment A and B in Figure 3(b) graphically depicts the set of all parameters (w Mc,Φ Mc ) which manufacturer M can choose for achieving channel coordination. Figure 3(a). The feasible area of para- Figure 3(b). The feasible area of parameters (w Dc,Φ Dc ) charged by D to achieve coordination. meters (w Mc,Φ Mc ) offered by M to achieve coordination. Although the feasible RS contracts mentioned above could achieve channel coordination, they might not be adopted because they probably do not benefit all partners. Next, we will further identify conditions under which the above RS contracts can be implemented. Set Ψ = max{1/16,γ}. Since C D + C M > 3C R /5, γ < 5/8. It means Ψ < 5/8. One has known from the earlier analysis that a feasible RS contract requires Φ Dc > γ and Φ Mc > f(φ Dc ), and that this RS contract also requires Φ Dc > 1/16 if retailer R is willing to accept it. Therefore, from retailer R s side, an implemented feasible RS contract requires Φ Dc > Ψ and Φ Mc > f(φ Dc ). If taking manufacturer M s and distributor D s benefits into account, the implemented feasible RS contract requires (Φ Dc,Φ Mc ) to be chosen only in the area characterized by Φ Dc > Ψ, Φ Mc > f(φ Dc ), Φ Mc > g(φ Dc ) and Φ Mc < h(φ Dc ) in the (Φ Dc Φ Mc ) plane. We refer to this area as the acceptable area. Identifying this acceptable area needs to draw three curves: Φ Mc = g(φ Dc ), Φ Mc = h(φ Dc ) and Φ Mc = f(φ Dc ) in the (Φ Dc Φ Mc ) plane. As shown in Figure 2, the first two are fixed in the plane. The third changes its location as the values of model parameters vary. However, after noting that the curve Φ Mc = f(φ Dc ) is an increasing convex one with its value changing from C D /(C D + C M ) to f(1) < 1, one can infer that the location relationship of the three curves can be depicted respectively by Figure 4(a) if f(ψ) h(ψ), Figure 4(b) if g(ψ) f(ψ) < h(ψ), Figure 4(c) if f(ψ) < g(ψ) and f(5/8) < h(5/8) = 1/13 and Figure 4(d) if f(ψ) < g(ψ) and f(5/8) > h(5/8).

15 Pricing Coordination in Supply Chains through Revenue Sharing Contracts 45 Observing Figures 4(a) 4(d), one can easily find out: (1) The acceptable area is vacuous if f(ψ) h(ψ). Therefore, any feasible RS contract can not be accepted by all partners if f(ψ) h(ψ); (2) The acceptable area is enclosed by line Φ Dc = Ψ and two curves: Φ Mc = f(φ Dc ) and Φ Mc = h(φ Dc ) in the (Φ Dc Φ Mc ) plane if g(ψ) f(ψ) < h(ψ); If f(ψ) < g(ψ), then the acceptable area is enclosed by line Φ Dc = Ψ and two curves: Φ Mc = g(φ Dc ) and Φ Mc = h(φ Dc ) in the (Φ Dc Φ Mc ) plane if f(5/8) < 1/13; otherwise enclosed by line Φ Dc = Ψ and three curves: Φ Mc = f(φ Dc ), Φ Mc = g(φ Dc ) and Φ Mc = h(φ Dc ) in the (Φ Dc Φ Mc ) plane. 4(a) 4(b) 4(c) 4(d) Figure 4. The location relationship of the three curves: Φ Mc = g(φ Dc ), Φ Mc = h(φ Dc ) and Φ Mc = f(φ Dc ). Summarizing the above analysis, one can obtain that under Case I the feasible RS contract can be implemented only if Φ Dc belongs to the interval (Ψ,θ) if f(ψ) < h(ψ); otherwise any feasible RS contract can not be implemented, where θ = min{u,5/8} and u denotes the root to equation f(φ Dc ) h(φ Dc ) = 0 in the interval (Ψ,1). Therefore, if

16 46 Information and Management Sciences, Vol. 19, No. 1, March, 2008 there is any feasible RS contract adopted by all partners, the thick line segment between lines Φ Dc = Ψ and Φ Dc = θ in Figure 3(a) provides all acceptable options (w Dc,Φ Dc ) for the distributor-retailer interface, whereas the dashed area between two thin line segments (characterized respectively by equation (25) with Φ Dc = Ψ and Φ Dc = θ) in Figure 3(b) provides all acceptable options (w Mc,Φ Mc ) for the manufacturer-distributor interface. Case II. γ < β < 1 Following the way similar to that used in Case I, one easily knows that under such a situation the thick line segment in Figure 5(a) provides distributor D with the feasible values (w Dc,Φ Dc ) she can choose. For any Φ Dc chosen by distributor D, the correspondent line segment L (characterized by (25)) in Figure 5(b) shows the feasible values (w Mc,Φ Mc ) available to be selected by M to achieve channel coordination. With Φ Dc varying in (γ,β), the dotted area between line segments A and B in Figure 5(d) graphically depicts the set of all feasible values (w Mc,Φ Mc ) which manufacturer M offers distributor D to achieve channel coordination. Figure 5(a). The feasible area of para- Figure 5(b). The feasible area of parameters (w Dc,Φ Dc ) charged by D to achieve coordination. meters (w Mc,Φ Mc ) offered by M to achieve coordination. Similarly, if f(ψ) h(ψ), all of the feasible RS contracts shown above can not be implemented because they can not benefit all partners. If f(ψ) < h(ψ), the feasible RS contract can be implemented only if Φ Dc belongs to the interval (Ψ,θ), where θ = min{u,5/8} and u denotes the root to equation f(φ Dc ) h(φ Dc ) = 0 in the interval (Ψ,β). Therefore, if there is any feasible RS contract adopted by all partners, the thick line segment between lines Φ Dc = Ψ and Φ Dc = θ in Figure 5(a) graphically describes all possible options (w Dc,Φ Dc ) for the distributor-retailer interface, whereas the dotted area between two thin line segments in Figure 5(b) provides all acceptable options (w Mc,Φ Mc )

17 Pricing Coordination in Supply Chains through Revenue Sharing Contracts 47 for the manufacturer-distributor interface. Likewise, the values of Φ Dc and Φ Mc in the adopted three-echelon RS contract still depend on the bargaining power of all partners. To sum up, the procedure for designing the proposed three-echelon RS contract can be simply summarized as the following algorithm. Algorithm: Step 1 Input the values of parameters of the model; Step 2 If a/b (1 + C M /C R )(C R + C D + C M ), there is no feasible RS contract, Stop; Otherwise, go to Step 3. Step 3 Calculate γ and β from (26a) and (26b), respectively. Let Ψ = max{1/16, γ}. If f(ψ) h(ψ), there is also no feasible RS contract, Stop. Otherwise, go to Step 4. Step 4 If γ < 1 β, find the root, u, to equation f(x) h(x) = 0 in the interval (Ψ,1) and go to Step 5; If γ < β < 1, determine the root, u, to equation f(x) h(x) = 0 in the interval (Ψ,β) and go to Step 5. Step 5 Let θ = min{u,5/8}, then in the interval (Ψ,θ) choose a Φ Dc (its value dependent of all members negotiation ability). Go to Step 6. Step 6 Once Φ Dc is determined, the corresponding w Dc and w Mc can be got respectively from (24) and (25) while Φ Mc is selected from the interval (max(f(φ Dc ), g(φ Dc )),h(φ Dc )) in terms of three partners bargaining power. Then, (w Dc, Φ Dc,w Mc,Φ Mc ) will be an RS contract that can be implemented. Stop. 5. Numerical Examples In this section, we only illustrate the proposed three-echelon RS contract model through numerical examples. For this purpose, three sets of the assumed data are shown in Table 1. The first set denotes a three-echelon supply chain with the retailer s marginal cost of C R = 1, the distributor s of C D = 1, the manufacturer s of C M = 5, a potential market demand of a = 150 and a price-sensitive parameter of b = 3. The second one presents a similar supply chain just slightly lower in potential market demand, while the third one, compared with the other two, is substantially lower in retailer s and distributor s marginal costs and potential market demand but higher in the value of parameter b. Following the algorithm depicted in Section 4, one easily know that the RS contracts can not be used to coordinate the three-echelon channel with the first set of

18 48 Information and Management Sciences, Vol. 19, No. 1, March, 2008 data due to a/b = 50 > (1 + C M /C R )(C R + C D + C M ) = 42. For the system with Data Set 2, although we have a/b = 40 < (1 + C M /C R )(C R + C D + C M ) = 42, there is still not any feasible RS contract acceptable to all members because of Ψ = 1/7 and f(ψ) = > h(ψ) = In contrast, the channel with Data Set 3 may be perfectly coordinated through the RS contract because one has a/b = 10 < (1 + C M /C R )(C R + C D + C M ) = 270.3, Ψ = 1/16 and f(ψ) = < h(ψ) = According to the above algorithm, the parameter Φ Dc can be only chosen in the interval (Ψ,u) = (0.0625,0.6115) while the three partners set the RS contract. Given Φ Dc = 50%, they can only select the parameter Φ Mc in the interval (max(f(φ Dc ),g(φ Dc )),h(φ Dc )) = (0.0833,0.1667). If Φ Mc = 15% is negotiated, then the ultimate RS contract will be (w Dc,Φ Dc,w Mc,Φ Mc ) = (2.45,50%,0.228,15%). Table 2 shows all partners profits in four different RS contract settings and the manufacturer-distributor-stackelberg setting. From Table 2, one has no difficulty to notice that in contract setting (a), which is characterized by (w Dc,Φ Dc,w Mc,Φ Mc ) = (2.45,50%,0.228,15%), retailer R has the strongest bargaining power. In contract setting (b) given by (w Dc,Φ Dc,w Mc,Φ Mc ) = (0.43, 10%, 0.983, 25%), manufacturer M has the highest power. In contract setting (c) characterized by (w Dc,Φ Dc,w Mc,Φ Mc ) = (0.43,10%,2.165,50%), distributor D has the greatest bargaining power. Finally, contract setting (d), which is given by (w Dc,Φ Dc, w Mc,Φ Mc ) = (1.569,32%,0.724, 25%), shows the situation in which each partner has an equivalent power. Among all four RS contracts mentioned above, each one achieves channel coordination and makes every partner gain a profit higher than in the manufacturerdistributor-stackelberg setting as well. Table 3 shows some sensitivity analysis of parameters in the model. The analysis is implemented by changing the value of one parameter at every time and letting the remaining parameters keep the same values as in Data Set 3. It can be observed from Table 3 that, when Φ Dc and Φ Mc in the implemented RS contract are kept the same as in contract setting (b), all the three members profits will increase as a increases but decrease as other parameters increase, and that the three partners profits are slightly sensitive to C R and C D as compared to other parameters. Table 1. Problem data. Parameters C R C D C M a b Data set Data set Data set

19 Pricing Coordination in Supply Chains through Revenue Sharing Contracts 49 Table 2. All members profits under various settings. Integrated M-D-S Contract (a) Contract (b) Contract (c) Contract (d) firm setting (2.45, 50%, 0.228, 15%) (0.43, 10%, 0.983, 25%) (0.43, 10%, 2.165, 50%) (1.569, 32%, 0.724, 25%) Π M Π D Π R Table 3. Table 3 Impact of change of parameters value on RS contract and three members profit. Φ Dc = 10%; Φ Mc = 25% (Ψ, θ) Π R Π D Π M 0.1 (0.0625, ) (0.0625, ) C R 0.12 (0.0625, ) (0.0625, ) (0.0625, ) (0.0625, ) (0.0625, ) C D 0.24 (0.0625, ) (0.0625, ) (0.0625, ) (0.0625, ) (0.0625, ) C M 5.0 (0.0625, ) (0.0625, 0.625) (0.0625, 0.625) (0.0625, ) (0.0625, ) a 120 (0.0625, ) (0.0625, ) (0.0625, ) (0.0625, ) (0.0625, 0.625) b 12 (0.0625, 0.625) (0.0625, 0.625) (0.0625, 0.625) Conclusion This paper has considered the pricing coordination issues of a decentralized supply chain from the marketing perspective. Based on a revenue sharing mechanism, a contract model is proposed to coordinate the pricing decisions of partners in both a two-echelon and a three-echelon supply chain facing deterministic price-sensitive demand. The paper

20 50 Information and Management Sciences, Vol. 19, No. 1, March, 2008 reveals under what conditions the three-echelon channel can implement the RS contract to achieve channel coordination. Simultaneously, the present paper also shows how to design such an RS contract to achieve channel coordination. It has been verified from numerical examples that a suitable contract designed by the proposed method can give every partner more profit than that in a manufacturer/distributor-stackelberg setting. References [1] Balachander, S. and Srinivasan, K., Quantity discounts, manufacturer and channel profit maximization: impact of retailer heterogeneity, Marketing Letters, Vol. 9, No. 2, pp , [2] Cachon, G. and Lariviere, M. A., Supply chain coordination with revenue sharing contracts: strengths and limitations, Working paper, The Wharton School of Business, University of Pennsylvania, Philadelphia, [3] Cachon, G., Supply chain coordination with contracts, Working paper, The Wharton School of Business, University of Pennsylvania, Philadelphia, [4] Chen, F., Federgruen, A. and Zheng, Y., Coordination mechanisms for a distribution system with one supplier and multiple retailers, Management Science, Vol. 47, No. 5, pp , [5] Chen, T.-H. and Chen, J.-M., Optimizing supply chain collaboration based on joint replenishment and channel coordination, Transportation Research Part E, Vol. 41, pp , [6] Dolan, R.J., Quantity discounts: managerial issues and research opportunities, Marketing Science, Vol. 6, No. 1, pp.1-27, [7] Emmons, H. and Gilbert, S.M., The role of returns policies in pricing and inventory decisions for catalogue goods, Management Science, Vol. 44, No. 2, pp , [8] Eppen, G.D. and Iyer, A.V., Backup agreements in fashion buying-the value of upstream flexibility, Management Science, Vol. 43, No. 11, pp , [9] Giannoccaro, I. and Pontrandolfo, P., Supply chain coordination by revenue sharing contracts, International Journal of Production Economics, Vol. 89, pp , [10] Ingene, A.C. and Parry, M.E., Channel coordination when retailers compete, Marketing Science, Vol. 14, pp , [11] Jeuland, A.P. and Shugan, S.M., Managing channel profits, Marketing Science, Vol. 2, pp , [12] Klastorin, T.D., Moinzadeh, K. and Joong Son, Coordinating orders in supply chains through price discounts, IIE Transactions Vol. 34, pp , [13] Kim, K.H. and Hwang, H., An incremental discount pricing schedule with multiple customers and single price break, European Journal of Operational Research, Vol. 35, pp.71-79, [14] Lee, H. and Whang, S., Decentralized multi-echelon supply chains: Incentives and information, Management Science, Vol. 45, No. 5, (1999). [15] Li, J. and Liu, L., Supply chain coordination with quantity discount policy, International Journal of Production Economics, 101, pp.89-98, [16] Mishra, A.K., Selective discount for supplier-buyer coordination using common replenishment epochs, European Journal of Operational Research, Vol. 153, pp , [17] Moorthy, K., Managing channel profits: Comments, Marketing Science, Vol. 6, pp , [18] Munson, C. and Rosenblatt, M.J., Coordinating a three-level supply chain with quantity discounts, IIE Transactions, Vol. 33, pp , [19] Parlar, M. and Wang, Q., Discounting decisions in a supplier-buyer relationship with a linear buyer s demand, IIE Transactions, Vol. 26, pp.34-41, 1994.

21 Pricing Coordination in Supply Chains through Revenue Sharing Contracts 51 [20] Raju, J. and Zhang, Z.J., Channel coordination in the presence of a dominant retailer, Marketing Science, Vol. 24, No. 2, pp , [21] Tsay, A., The quantity flexibility contract and supplier-customer incentives, Management Science, Vol. 45, No. 10, [22] Tsay, A.A. and Lovejoy, W.S., Quantity flexibility contracts and supply chain performance, Manufacturing and Service Operation Management, Vol. 1, No. 2, pp , [23] Viswanathan, S. and Piplani, R., Coordinating supply chain inventories through common replenishment epochs, European Journal of Operational Research, Vol. 129, pp , [24] Wang, Q. and Wu, Z., Improving a supplier s quantity discount gain from many different buyers, IIE Transactions, Vol. 32, pp , [25] Weng, Z.K., Channel coordination and quantity discounts, Management Science, Vol. 41, pp , Authors Information Yong-Wu Zhou is currently a professor in Institute of Logistics and Supply Chain Management, Hefei University of Technology. He received his Ph.D. in Management Science from Hefei University of Technology. His research interests are inventory control model, logistics and supply chain management, operations research, etc. He has published more than 80 papers in such journals as IIE Transactions, European Journal of Operational Research, Journal of the Operational Research Society, International Journal of Production Economics, Computers & Operations Research, etc. Institute of Logistics and Supply Chain Management, Hefei University of Technology, Hefei, Anhui, P.R.China. ywzhou@mail.hf.ah.cn TEL : Shanlin Yang is currently a professor in the Department of Information Management and Information System, Hefei University of Technology, Hefei, Anhui, P.R.C.. He received his M.S. degree in Computer Science and Technology from Hefei University of Technology. His research interests includes information management, decision support system, etc. Department of Information Management and Information System, Hefei University of Technology, Hefei, Anhui, P.R.C. slyang@mail.hf.ah.cn TEL :