Production_and_Operations_Management

PRODUCTION AND OPERATIONS MANAGEMENT Vol. 13, No. 1, Spring 2004, pp. 23–33 issn 1059-1478 04 1301 023$1.25 POMS © 2004 Production and Operations Management Society Revenue-Sharing vs. Wholesale-Price Contracts in Assembly Systems with Random Demand Yigal Gerchak • Yunzeng Wang Department of Industrial Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel Department of Operations, Weatherhead School of Management, Case Western Reserve University, Cleveland, Ohio 44106-7235 ssembly and kitting operations, as well as jointly sold products, are rather basic yet intriguing decentralized supply chains, where achieving coordination through appropriate incentives is very important, especially when demand is uncertain. We investigate two very distinct types of arrangements between an assembler/retailer and its suppliers. One scheme is a vendor-managed inventory with revenue sharing, and the other a wholesale-price driven contract. In the vmi case, each supplier faces strategic uncertainty as to the amounts of components, which need to be mated with its own, that other suppliers will deliver. We explore the resulting components’ delivery quantities equilibrium in this decentralized supply chain and its implications for participants’ and system’s expected profits. We derive the revenue shares the assembler should select in order to maximize its own profits. We then explore a revenue-plus-surplus-subsidy incentive scheme, where, in addition to a share of revenue, the assembler also provides a subsidy to component suppliers for their unsold components. We show that, by using this two-parameter contract, the assembler can achieve channel coordination and increase the profits of all parties involved. We then explore a wholesale-price-driven scheme, both as a single lever and in combination with buybacks. The channel performance of a wholesale-price-only scheme is shown to degrade with the number of suppliers, which is not the case with a revenue-share-only contract. Key words: supply chain management; assembly systems; revenue-sharing contract; wholesale-price contract Submissions and Acceptance: Received July 2002; revision received May 2003; accepted August 2003. A 1. Introduction The sales and profit of one supplier often depend on the delivery quantities and timing of complementary components or products delivered by other suppliers, as well as on realized demand. In assembly systems (e.g., personal computers), complete sets of components, supplied by various manufacturers, are needed to put units together. In distribution centers, packaging material is needed in proportion to the amount of hardware packaged and is typically purchased from a different supplier. At the retail level, some products that are almost always purchased jointly (e.g., solder and flux in plumbing stores; marscapone cream and savoiardi biscuit, used to make Tiramisu, in food ´ stores) could be produced and delivered to stores by different producers. Depending on the financial ar23 rangements or incentive system, one supplier operating in such an environment may then base its own production/delivery decisions on its anticipation of how much other suppliers of complementary components/items will deliver, as well as its forecast of demand. Such a decentralized supply chain gives rise to interesting strategic inter-supplier considerations in choosing their individual production and delivery quantities. Moreover, anticipating the suppliers’ behavior, the assembler or retailer will choose an incentive scheme to maximize its own expected profits. While in recent years operations management researchers have been stressing the importance of coordination mechanisms in decentralized supply chains (e.g., Cachon and Lariviere 2001; Lariviere 1999; Cor- 24 Gerchak and Wang: Revenue Sharing vs. Wholesale-Price Contracts Production and Operations Management 13(1), pp. 23–33, © 2004 Production and Operations Management Society bett and Tang 1999; Tsay, Nahmias, and Agrawal 1999; Cachon 2001 for comprehensive review), a decentralized assembly system of the type described above has not yet been explored. As stated, for example, by Cachon and Lariviere (2001): “In the GM example above, we are concerned only with ashtrays and not body panels or drive trains.” One exception is a recent study by Gurnani and Gerchak (1998), who addressed a scenario with random component production yields but known demand. The purpose of this paper is to study decentralized decisions and channel performance in assembly/joint-purchase systems with random demand. We explore and compare two types of settings. One is a vmi (vendor-managed-inventory) system with revenue sharing, and the other a wholesale-pricebased system. The vmi system is one where suppliers choose how much to deliver, and are paid only for units (of assembled/combined product) sold. Thus, here it is the retailer who sets the parameters (revenue shares), and the suppliers then decide delivery quantities. Cachon and Lariviere (2001) analyze contracts of this type, in the case of a single supplier. vmi systems of this type are common in retail settings (e.g., with dairy products), and seem to be used even in cases where complementary products are delivered by different suppliers. The above-mentioned ingredients of Tiramisu are a case in point. We note that there also ´ exist revenue-sharing contracts where the shares are selected by the supplier(s), while the retailer then selects the quantity (Pasternack 1999; Cachon and Lariviere 2000). Ours is a revenue-sharing scheme led by the downstream player (the retailer), accompanied by a vmi quantity choice. At first thought, such an arrangement does not seem effective, and we initially explored it primarily since it has been observed in practice. The conclusions of how it will perform with rational suppliers are, however, quite surprising. The basic system we consider has component suppliers who, when choosing how much to produce, trade-off their production costs against the revenue, which is uncertain since: (i) the number of kits assembled, and hence everyone’s revenue, is constrained by the supplier who delivered the least; and (ii) demand is random. Each supplier knows the production costs and revenue share of others, and they all have the same probabilistic beliefs about demand. (The assumption that suppliers know each other’s cost and revenue share is not needed when the assembler optimally sets contract terms.) In such a setting, the decentralized Nash equilibrium will equal the smallest component lot size, determined by the solution of an appropriate newsvendor problem. We then consider the selection of the best revenue shares from the assembler’s point of view. We show that they are chosen such that all suppliers’ indepen- dent lot sizes are equal, and that the suppliers do not need to know each other’s and the assembler’s costs in choosing their individual production quantities. We further show that such an optimal quantity exists uniquely under a mild restriction on the demand distribution. In general, the above incentive scheme based on revenue share alone cannot coordinate the decentralized assembly system. That is, the final quantity delivered and assembled, which is determined by the revenue shares set by the assembler to maximize its own profit, is in general not equal to the quantity which would optimize the entire system. To achieve coordination, we then propose a revenue-plus-surplus-subsidy scheme where, in addition to a share of revenue, a supplier is partially paid by the assembler for its delivered components that are not sold. Surplus subsidies transfer some of the risk of demand uncertainty from the suppliers to the assembler. In that sense, they bear economic similarity to buybacks (to be discussed shortly) which transfer risk from retailer to manufacturer in wholesale-price-based contracts (Pasternack 1985; Lariviere 1999). We show that, in such an environment, there exists a continuum of supplier-specific, two-parameter contracts for the assembler to choose from to coordinate the decentralized system. We further demonstrate that each such coordinating contract simply corresponds to a different amount of profit (out of the maximum total channel profit) allocated to the supplier. Thus, in addition to achieving channel coordination, the assembler can also easily make sure that some or all parties involved improve their benefits, and no one is worse off (compared with any situation where coordination was not achieved). We then analyze a system with a more “conventional” wholesale-price-based contract. That system, a generalization of Lariviere and Porteus (2001) “selling to a newsvendor” model, works as follows. The n suppliers first simultaneously choose their individual component wholesale prices. The assembler then chooses the quantity to order from all suppliers. Under a mild restriction on demand distribution, we prove the existence, and in the case of identical suppliers also the uniqueness, of a Nash equilibrium. For a given total production cost, we show that the production quantity and channel profit are decreasing in the number of suppliers, which is not the case for the revenue-sharing contract. Using the exponential demand distribution, we provide an example for which the vmi with revenuesharing system dominates the wholesale-price-based system even for a single supplier, and increasingly so for more than one supplier. As is well known from the single-supplier literature, a wholesale price alone cannot coordinate the channel Gerchak and Wang: Revenue Sharing vs. Wholesale-Price Contracts Production and Operations Management 13(1), pp. 23–33, © 2004 Production and Operations Management Society 25 (e.g., Lariviere 1999). The natural additional financial lever to supplement it for achieving coordination is buybacks (returns). Unsold units are returned to the supplier for a pre-arranged price (Pasternack 1985; Emmons and Gilbert 1998; Donohue 2000). We thus endow our n-supplier wholesale-price-driven model with these additional supplier-specific levers, and show that the channel can be coordinated. The rest of the paper is organized as follows. We introduce the basic model and its centralized solution in Section 2. Section 3 discusses a decentralized system with vmi revenue-sharing contract. At first, only revenue shares are used, and exemplified. We then introduce the revenue-plus-surplus-subsidy incentive scheme and discuss its channel coordination property. Section 4 analyzes a wholesale-price-based contract. Initially, only wholesale prices are used. Then buybacks are added as a second lever. Some concluding remarks comparing the effectiveness of the two schemes and their informational requirements are provided in Section 5. costs do not exceed revenue), then since the expected profit function of (1) is concave, the optimal produc* tion quantity Q c for the centralized system satisfies the first-order condition n F Q* c i 0 ci . (2) * Substituting Qc Qc into (1), we can show that the optimal system-wide expected profit is given by Q* c Q* c 0 xf x dx. (3) * * Both the production quantity Qc and profit (Qc) are decreasing in the total unit production costs ¥n 0 ci , as i expected. We now consider decentralized systems. 3. Decentralized System with Revenue-Sharing Contracts 2. The Centralized System A final product faces a random demand D, with common-belief cdf F and pdf f. Assume that F has support on (0, ) with unique inverse, and is differentiable everywhere. Without loss of generality, the unit revenue from selling the product is normalized to be $1. The product consists of n components, or sets thereof (without loss of generality one unit of each) produced by independent suppliers. The components’ unit production costs are ci , i 1, . . . , n, and the assembler incurs a unit cost of c0 mating the components together (in a retail setting it is likely that c0 0). The decision variables are the suppliers’ components production/delivery quantities Qi , i 1, . . . , n, and the assembler’s assembly quantity Q0. All decisions have to be made before the demand is realized. For simplicity, assume that there are no holding costs or salvage values for unsold products or components. If the system were centralized, the decision maker would want to maximize the expected system-wide ... Qn profit. Clearly, one should set Q1 Q0 Qc, since any unmated or unassembled components will be wasted. The choice of optimal Qc would then be a simple newsvendor problem. The expected profit is: n Assume now that the component lot sizes Qi , i 1, . . . , n, and the assembly quantity Q0 are chosen by individual suppliers and the assembler, respectively. All parties (the suppliers and the assembler) are assumed to have the same belief concerning the demand distribution (i.e., “common knowledge” is assumed). We initially assume that all suppliers and the assembler know each other’s production costs. However, it becomes clear later from our model analyses (Proposition 3) that when the assembler sets the contract terms { i} to maximize its own profit, the suppliers do not need to know each other’s or the assembler’s costs in order to choose their individual Nash strategies (production quantities). All that is needed is that the assembler knows every supplier’s cost and that all suppliers know that the assembler knows their costs. 3.1. Basic Revenue-Sharing Contract A basic revenue-share contract specifies that, for each unit of final product sold, the assembler pays supplier i 1, . . . , n, 0 1, out of the $1 total revenue. i, i i Thus, the assembler keeps 0 1 ¥n 1 i for itself. i That revenue sharing scheme is known to the suppliers. Clearly, a necessary condition for each party to stay in business is i ci , i 0, 1, . . . , n. (4) Qc E i 0 ci Qc n min Qc , D Qc For such a revenue-sharing scheme, if one firm’s final revenue did not depend on other firms’ deliveries or assembly decision, it would solve its own newsvendor problem, i.e., it would produce QcF Qc , ci (1) F Q* i ci / i ci Qc i 0 0 xf x dx , i 1 0, 1, . . . , n. (5) where F 1 F. Assuming that ¥n i 0 $1 (i.e., that Suppose wlog that c1/ maxi {ci/ i}. Then, 26 Gerchak and Wang: Revenue Sharing vs. Wholesale-Price Contracts Production and Operations Management 13(1), pp. 23–33, © 2004 Production and Operations Management Society * Q1 mini {Qi* We then refer to supplier 1 as the }. “critical supplier.” Returning to the situation with inter-dependent revenues (and decisions), we now argue that, at equilibrium, all suppliers will deliver, and the assembler will * * assemble, no more than Q1. The reason: Q1 is the optimal amount for supplier 1, who thus clearly does not want to deliver more. The other suppliers (the assembler) would have liked to deliver (assemble) * more than Q1 if they had a chance to be paid for these extra units; but since profit is a function of the number of complete and assembled kits, there is no benefit for * the other suppliers to deliver more than Q1, and it is thus infeasible for the assembler to assemble more * * than Q1. Any amount in [0, Q1] is here a Nash equi* librium and Q1 is the one among them which maximizes all parties’ profits (i.e., a Pareto-optimal point). To summarize: Proposition 1. If c1/ 1 maxi {ci/ i}, all points in * * * [0, Q1] are Nash equilibria, and Qd Q1 is the Paretooptimal point among them. Note that this result does not really depend on whether decisions among firms are simultaneous or sequential. It also does not depend on the type of incentive system; it is a direct consequence of * Q1 mini {Qi* }. * We observe that since F(Qd) c1/ 1, then while n * *) F(Qc ¥i 0 ci , the decentralized solution Qd is, in * general, not equal to the system-optimal quantity Qc. In fact, since by assumption c1/ 1 maxi {c1/ i}, then n c1 i 0, 1, . . . , n, so c1 ¥n 0 i 1ci , i i 1 ¥i 0 ci. n n Thus, c1/ 1 ¥i 0 ci , since ¥i 0 i 1, and, hence, * * Qd Qc. We can show that the two quantities are equal if and only if i ci/¥n 0 ci for i 0, 1, . . . , n. So, i we further have the following proposition: Proposition 2. (1) The decentralized production quantity cannot be larger than the centralized quantity. That is, * * Qd Qc. (2) The decentralized production quantity will be the same as the centralized quantity if and only if the revenue allocation is such that the revenue share of each * * party is equal to its cost share. That is, Qd Qc iff ci/¥n 0 ci for i 0, 1, . . . , n. i i 3.1.1. Revenue Shares Maximizing Assembler’s Expected Profit. How is the assembler going to set the incentive scheme (i.e., 1, . . . , n , and so 0 1 ¥n 1 i i), if it has the power to do so' It is plausible to assume that it tries to maximize its own expected profit. So, we have a Stackelberg type game: First, the assembler sets the revenue shares, and then the sup* pliers decide the number of units Qd to deliver. Denote the assembler’s expected profit by 0( 1, . . . , n). It faces the following optimization problem max 1, 0 n 1 ,..., n . . ., n E * c0 Qd 1 i 1 i * min Qd , D . (6) The following property partially characterizes the assembler’s optimal policy: Proposition 3. The assembler will always set n such that c1 / 1 1, ..., (7) ··· cn / n c0 / 0 . Proof. We first show that the revenue shares allocated to the n suppliers ought to be such that c1/ 1 ... cn/ n. Otherwise, assume wlog that c1/ 1 max {ci/ i : i 1, . . . , n} and c1/ 1 ci/ i for some i 1. Then, by reducing i to a value such that c1/ 1 ci/ i , the assembler will increase its own share c0 of the unit revenue without reducing the delivery quantity of complete sets of components, and, hence, it will increase its expected profit 0( 1, . . . , n) in (6). Now, for the second part in (7), assume otherwise, i.e., c1/ 1 . . . cn/ n c0/ 0. Then, we know from (5) that each component supplier is willing to deliver more than that the assembler is willing to assemble. Thus, by reducing the revenue share allocated to each ... supplier at least to a value such that c1/ 1 cn/ n c0/ 0, the assembler can again only improve its own expected profit. Proposition 3 implies that, when the assembler behaves rationally in setting the revenue shares, all suppliers will be “critical.” More importantly, it implies that, in determining their Nash strategy (Proposition 1), suppliers do not need to know each other’s and the assembler’s cost information, since they can simply infer from (7) each other’s cost/share ratio. Obviously, the assembler will need to know every supplier’s cost in order to be able to set the revenue share according to (7). Now, since all suppliers will be “critical,” the de* centralized production quantity Qd is determined by * Q1. Then, the n-dimensional problem in (6) reduces to a one-dimensional one, since it follows from (7) that n ¥n 1 i c1/¥n 0 ci. That is, i 1 ¥i 1 ci/c1 and also 1 i max c1 1 n c1 /¥i 0 ci 0 1 n E * c0 Q1 1 i 1 ci /c1 1 * min Q1 , D . (8) Since at suppliers’ optimum 1 c 1 /F (Q *), a mono1 tone increasing function, one can perform the optimization over Q * rather than over 1 (see Lariviere 1 and Porteus 2001 for a similar approach). We also Gerchak and Wang: Revenue Sharing vs. Wholesale-Price Contracts Production and Operations Management 13(1), pp. 23–33, © 2004 Production and Operations Management Society 27 know that Q * Q * when 1 c 1 /¥ n 0 c i . Thus, 1 c i suppressing the super/subscripts on Q, problem (8) becomes max 0 Q Q* c 0 Proposition 4. If f Q F Q Q 2 0 Q E c0 Q 1 ¥in 1 ci min Q, D FQ ¥in 1 ci FQ Q F x dx (12) c0 Q 1 F x dx. 0 (9) is increasing, then 0(Q) is concave and has a unique interior maximum which can be found by solving d 0(Q)/dQ 0. The assumption above is very weak. It is implied by the ifr (increasing failure rate) property (increasing f/F), and will be satisfied by essentially any practical unimodal demand distribution. Now, the assumption is equivalent to the first derivative of (12) being positive. That is, 2 f Q F Q f Q f Q F Q F x dx 0, Proposition 3 states that the revenue shares allocated among the n suppliers by the assembler will always be proportional to their production costs. As a result, if the total production cost, namely ¥n 1 ci , is kept as a coni stant, the number of suppliers n (and their relative production costs) will not affect the assembler’s order-size decision. This is also evident in (9), where the total cost ¥n 1 ci appears as a single parameter in the assembler’s i profit function. So, we have, Corollary 1. For a given total components production * cost ¥n 1 ci , the decentralized production quantity Qd and, i * hence, total channel profit (Qd) are not affected by the number of suppliers and the allocation of the total cost among them. Since the number of suppliers will not affect the decentralized decision, problem (9) is equivalent to the problem studied by Cachon and Lariviere (2001), where the downstream firm provides incentives to induce a single supplier to build up production capacity. It should be noted, however, that the observation in our decentralized assembly system that if the assembler acts optimally the problem becomes equivalent to one with a single supplier is a result rather than something that is obvious from the outset. Second, in the following, we derive a concavity condition, which is weaker than that obtained by Cachon and Lariviere. Now, the first order condition of optimality for problem (9) is d Q dQ 0 n Q 0 which is weaker than the following condition used by Cachon and Lariviere (2001; Theorem 2): 2 f Q F Q f Q f Q 0. We also note that the condition in Proposition 4 here differs from Lariviere and Porteus’s (2001) “increasing generalized failure rate” (increasing Qf/F); neither one of the two conditions implies the other. When { f(Q)/[F(Q)]2} Q F(x)dx is increasing, the 0 following properties can be established through (10): Corollary 2. The decentralized production quantity * * Qd and, hence, total channel profit (Qd) are (1) decreasing in ¥n 1 ci and c0; and (2) increasing in the ratio of c0/(c0 i ¥n 1 ci) for any given total costs c0 ¥n 1 ci. i i The managerial implication of part (2) above is that, when the party capturing most of the revenue also bears most of the channel cost, “double marginalization” will not cause significant deterioration in channel performance for decentralized supply chains. With i , i 1, . . . , n, being set by the assembler such that c1/ 1 . . . cn/ n c0/ 0, the decentral* ized production quantity Qd, obtained by solving (10), is the Newsvendor-optimal delivery quantity for each ... * of the suppliers, i.e., F(Qd) cn/ n. c1/ 1 Their corresponding expected profits will be Q* d i i 0 c0 FQ i 1 ci fQ FQ 0(Q)/dQ Q 2 0 1 Note that at Q * at Q Qc, d Q /dQ i 1 F x dx 1 ¥n i ci 0. (10) 0, and 0, d n 0 xf x dx ci F Q* d Q* d xf x dx, 0 Q* c 0 ci f Qc* / F Qc* 2 0 F x dx 0. (11) i 1, . . . , n. (13) Thus, the following proposition follows immediately: Note that each supplier’s expected profit is also proportional to its marginal production cost, i.e., c1/ . . . cn/ n. 1 28 Gerchak and Wang: Revenue Sharing vs. Wholesale-Price Contracts Production and Operations Management 13(1), pp. 23–33, © 2004 Production and Operations Management Society Example 1. Assume that demand for the final product is exponentially distributed with a mean . This ifr distribution clearly satisfies (12). We compare the production quantity and profits of a centralized system with those of a decentralized one. Assuming that the total production cost per unit equals half of the unit revenue (i.e., c0 ¥n 1 ci $0.5), Figure 1 illusi trates how the deviations (as percentage) of the decentralized production quantity and system profit from their centralized counterparts change with the allocation of cost between the assembler and its suppliers (i.e., the ratio of c0/(c0 ¥n 1 ci)). (For the exponential i * * * distribution, it turns out that neither (Qc Qd)/Qc nor * * * [ (Qc) (Qd)]/ (Qc) depend on the mean demand level .) First, we see that, when the assembler bears a relatively low portion of the total channel costs, decentralized decisions can result in a channel profit which is over 20% lower than that of a centralized decision. Consistent with Corollary 2, Part 2, the decentralized system performance improves as the assembler bears a greater fraction of the total cost. * In the decentralized system, the channel profit (Qd) is shared by the assembler and its suppliers. Setting the scalar 1, Figure 2 illustrates that, as the assembler’s portion of cost increases, the total channel profit as well as that of the assembler increase while the (total) profit of suppliers decreases. This is again rather intuitive. 3.2. Surplus Subsidy and Channel Coordination Suppose now that, in addition to a revenue share i from sales of final product, the assembler will pay supplier i si per unit for its delivered components that are not sold—a surplus subsidy. To avoid trivial cases, we assume that i ci si , for i 1, . . . , n, and ¥n 1 i 1 c0. Note that a delivered component may end i up unsold either due to low demand or due to a shortage of mating components, or both. The surplus subsidy does not distinguish between causes. How- Figure 2 Revenue Sharing: Profits in the Decentralized System with c0 ¥in 1 ci 0.5 and 1 ever, as we shall see, rational suppliers will actually deliver equal amounts, so unmated components will actually not occur. Such a subsidy, in effect, transfers some of the risk due to uncertain demand from the suppliers to the assembler. Economically, this is similar to manufacturers’ reducing retailers’ risk by committing to buy-backs (returns) (Pasternack 1985; Lariviere 1999) within a different type of contract discussed in later sections. If supplier i’s revenue did not depend on other suppliers’ deliveries, it would now face the following newsvendor profit function: i Qi E ci Qi ci Qi i min Qi , D Qi si Qi QiF Qi D i 0 xf x dx Qi si 0 Qi x f x dx. (14) This yields its most desired delivery quantity Qi*, which satisfies F Q* i ci i Figure 1 Revenue Sharing: Deviation of the Decentralized System from the Centralized System, with c0 ¥in 1 ci 0.5 si , si i 1, . . . , n. (15) As expected, Qi* is increasing in si. But, since the over-delivery subsidy alone cannot help to fully recover the production costs of components (i.e., ci si), none of the suppliers will be willing to deliver more than anyone else. Furthermore, it is optimal for the assembler to set the incentives scheme ( i , si) for each i such that all suppliers willingly deliver the same quantity: by reducing either the revenue share i and/or subsidy si for those who are willing to deliver more than that the “critical” supplier’s newsvendor quantity, the assembler can only benefit itself. That is, Gerchak and Wang: Revenue Sharing vs. Wholesale-Price Contracts Production and Operations Management 13(1), pp. 23–33, © 2004 Production and Operations Management Society 29 i Proposition 5. The assembler will set ( i , si), 1, . . . , n, such that c1 1 when the channel is not coordinated and still leave itself with more. s1 s1 c2 2 s2 s2 ··· cn n sn . sn (16) 4. Wholesale Price Contract When the delivery quantity of each supplier in the decentralized system equals the centralized decision * Qc, we say that the supply chain is coordinated. Comparing (15) with (2), it is then obvious that: Proposition 6. To coordinate the system, the assembler only needs to set ( i , si) for each i, i 1, . . . , n, such that ci i si si n cj , j 0 or si ci i 1 ¥jn 0 cj . ¥jn 0 cj i (17) With a wholesale price contract, the n suppliers first simultaneously choose their individual component wholesale prices wi , i 1, 2, . . . , n, charged to the assembler; then, the assembler chooses a quantity Q ordered from all suppliers. Thus, mismatching of components will never happen in this environment. This setting is a multi-supplier generalization of Lariviere and Porteus (2001). When the wholesale prices wi , i 1, 2, . . . , n, are offered by the suppliers, the assembler faces the simple Newsvendor problem, n Equation (17) determines si as a function of the other variable i. That is, for any given revenue share i , as long as the corresponding surplus subsidy si is determined by (17), the resulting contract ( i , si) will coordinate the decentralized system. Thus, for each supplier i, there actually exists a continuum of contracts that can coordinate the supply chain. Note that if i is close to ci then si will also be close to ci; that will correspond to small profit margin but also low downside risk. Second, we note that for coordination purposes, the contract ( i , si) of one supplier does not have to depend on those of other suppliers. Thus, the assembler can negotiate the contracts independently with different suppliers. These two properties make this two-parameter contract structure especially attractive from a practical point of view. When the supply chain is coordinated through the incentive scheme in (17), supplier i’s expected profit * can be calculated by substituting Qi Qc and si( i) n n (ci ¥j 0 cj) into (14). After some i ¥j 0 cj)/(1 algebra, we have ci 1 ¥ jn Q* c 0 max Q 0 Q E i 1 wi c0 Q min D, Q , (19) and its optimal order quantity will be n Q F 1 i 1 wi c0 . (20) In the simultaneous sub-game of choosing component wholesale prices, all suppliers know the production quantity decision made by the assembler. Obviously, we require wi ci , i 1, 2, . . . , n, and ¥n 1 i wi c0 1 to ensure that every one remains in business. Then, comparing (20) with (2), we have that the decentralized production quantity in this setting will again never be more than the centralized quantity * Qc. Now, for given wholesale prices of all other suppliers wi , i j, supplier j would choose its price wj to maximize its own profit. That is, by (20) n i i 1 cj xf x dx. 0 (18) max wj j wj wj cj Q wj cj F j 1 i 1 wi c0 , (21) Thus, supplier i’s profit i is determined solely by its revenue share i. With coordination achieved, the supply chain reaches its highest possible total profit. Now, the best strategy for the assembler is simply to try to allocate as low revenue shares i and, hence, as little profits, as possible to each of the suppliers; the rest of the maximum channel profit goes to itself. Obviously, this revenue-plus-surplus-subsidy contract dominates the revenue-only contract and any other contract types that cannot achieve channel coordination. For example, due to the “profit surplus” generated through coordination, the assembler can allocate to each supplier at least the same profit as 1, 2, . . . , n. Since there is a one-to-one correspondence between wj and Q for given wi , i j, i.e., wj F(Q) (¥i j wi c0), choosing a value for wj is equivalent to choosing a corresponding value for Q. Thus, the optimization over wj in (21) can equivalently be written as the following optimization over Q: max Q j Q FQ i j wi c0 j cj Q, 1, 2, . . . , n. (22) This transformation helps us to characterize the prop- 30 Gerchak and Wang: Revenue Sharing vs. Wholesale-Price Contracts Production and Operations Management 13(1), pp. 23–33, © 2004 Production and Operations Management Society erty of j , since we can show from (22) that the firstorder condition of optimality is given by d j dQ F Q 1 f Q Q F Q wi i j wholesale prices as in (20), Proposition 7 implies that there exists an equilibrium production quantity for the decentralized system. 4.1. Identical Suppliers Assume that the n suppliers are identical in terms of their production costs, i.e., ci c for all i. From (21), we can check that 2 c0 cj 0. (23) Assume that [ f(Q)/F(Q)]Q is increasing in Q. Define Q as the value of Q such that [ f(Q)/F(Q)]Q 1, if there exists such a Q on (0, ) for the given distribution function; otherwise, define Q to be . Then, the first term in (23) is positive and decreasing in Q for Q Q, and is negative for Q Q. Thus, we have the following Lemma, where the second part follows from the one-to-one and monotone correspondence between wj and Q: Lemma 1. If [f(Q)/F(Q)]Q is increasing, then j is pseudo-concave in Q as defined in (22) and, hence, pseudoconcave in wj as defined in (21). Note that problem (22) is equivalent to a singlesupplier-manufacturer system studied by Lariviere and Porteus (2001). Lariviere and Porteus prove that if [ f(Q)/F(Q)]Q is increasing, the support of the demand distribution F is of the form [a, b) with 0 a b . They also show that the optimal solution for the supplier may actually be given by Q a, instead of the solution given by the first order condition of (23) (in our case here). This may happen when the demand is guaranteed to exceed some large value (i.e., a large value of a) and has a very small coefficient of variation. An obvious and extreme example of it is when one faces a constant demand D a 0 with probability 1. Readers are referred to Lariviere and Porteus (2001) for detailed discussions on related technical issues. On the other hand, since our main purpose in this paper is to generate managerial insights by comparing the two types of contractual arrangements (i.e., revenue-sharing vs. wholesale-price) in decentralized assembly systems, we will assume, for the rest of the paper, that the first order condition (23) always provides the optimal solution. Now, wj , j 1, 2, . . . , n, is constrained to be in [cj , 1 c0], which is a nonempty, compact and convex set. The payoff function j in (21) is continuous in wj , assuming that the demand distribution function F is continuous. Then, Lemma 1 guarantees the existence of a Nash equilibrium (Theorem 1.2 of Fudenberg and Tirole 1991). That is, Proposition 7. If [f(Q)/F(Q)]Q is increasing, there exists a pure-strategy Nash equilibrium for the game of choosing the wholesale prices by the component suppliers. Since the production quantity Q chosen by the assembler is uniquely determined by the sum of the w j 2 j 2 j wj wk 1 f ¥in 1 wi c0 0, j k. This eliminates the existence of non-symmetric equilibrium (Theorem 4.1 of Anupindi, Bassok, and Zemel 1999). Thus, in equilibrium, we will always have wi w for all i.1 Now, from (20) we have w [F(Q) c0]/n. Substituting wi w [F(Q) c0]/n for all i together with cj c into (23) and letting it equal zero, the equilibrium production quantity Q can be found by solving F Q 1 n f Q Q F Q nc c0 . (24) Now, if [ f(Q)/F(Q)]Q is increasing, the left-hand side ˆ ˆ of (24) is decreasing in Q for Q Q, where Q is such ˆ ˆ ˆ that n[ f(Q)/F(Q)]Q 1 if it exists and, otherwise, ˆ define Q to be . Furthermore, when Q 0, the left-hand side equals 1 (the unit product revenue), which is bigger than nc c0 (the total production ˆ cost), and as Q 3 Q, the left-hand becomes zero. Thus, the solution to (24) will be unique and finite. That is, Proposition 8. With identical suppliers, if [f(Q)/ F(Q)]Q is increasing, there exists a unique decentralized production quantity. The following properties regarding the equilibrium quantity can be obtained from (24): Corollary 3. The decentralized production quantity * * Qd and, hence, the total channel profit (Qd) (1) are decreasing in the component cost c and assembly cost c0 for any given number of suppliers; (2) are decreasing in the number of suppliers n for any given total production cost nc c0; (3) do not change with the allocation of production costs between the assembler and the suppliers, keeping the total costs nc c0 fixed. We note that while part (1) here is rather intuitive, parts (2) and (3) may not be so. Also, parts (2) and (3) contrast sharply with properties of the revenue-sharing systems, where the number of suppliers does not affect the production quantity (Corollary 1), while the 1 We again assume that the demand distribution F is such that the first order conditions always provide the optimal solution for the suppliers. An extreme counterexample is the case of constant demand, where one can easily see that any point (w1, w2, . . . , wn) such n that ¥i 1 wi 1 c0 and wi ci for all i is a Nash equilibrium. Gerchak and Wang: Revenue Sharing vs. Wholesale-Price Contracts Production and Operations Management 13(1), pp. 23–33, © 2004 Production and Operations Management Society 31 Figure 3 Performance Comparison of Systems with nc 1 c0 0.5 and a wholesale price wi for all units ordered, supplier i, i 1, 2, . . . , n, also agrees to pay back the assembler si per unit for any quantity (ordered by the firm) above the realized demand. To avoid trivial cases, the following relationships must hold: wi ci and wi si for i 1, 2, . . . , n, and ¥n 1 wi c0 1. i Now the assembler faces the following Newsvendor profit function: 0 Q n n E i 1 wi n c0 Q min Q, D i 1 Q si Q D allocation of costs between the assembler and suppliers does (Corollary 2). These structural differences suggest that the choice of incentive schemes (revenueshare vs. wholesale price) can be critical to system performance. Example 2. As in Example 1, we assume the expo1 nential demand distribution. Setting the scalar and the total production cost nc c0 0.5, Figure 3 shows that the production quantity (hence, systemwide performance) decreases dramatically with the number of suppliers in the decentralized system with wholesale price contract. Also shown in Figure 3 are the production quantities for the same system with the revenue-share contract (solving (10) for Q with c0/(nc c0) 0 and c0/(nc c0) 0.7, respectively) and centralized control. First of all, for all these latter cases, system performance does not change with the number of suppliers in the system, while for the wholesale price contract case, it does not change with the cost allocation between the assembler and suppliers. More surprisingly, the system with a revenue-share contract always performs better than one with wholesale price contract. That is, the worst case (i.e., with c0/(nc c0) 0) of the revenue-share systems dominates the best case (i.e., with n 1) of the wholesale price systems. Our extensive numerical testing seems to confirm that this conclusion is always true with the exponential demand distribution. Other distribution functions are yet to be explored. The managerial implications of this conclusion can be significant. 4.2. Inventory Buy-Back Policy and Channel Coordination Pasternack (1985) showed that, when a single supplier wholesales to a retailer/manufacturer, a properly designed inventory buy-back/return policy can coordinate the supply chain. In the following, we show how such a policy can be adopted to coordinate our multi-supplier, decentralized assembly system. Assume that, while initially charging the assembler wi i 1 c0 Q 0 n xf x dx QF Q Q si i 1 0 Q x f x dx. (25) Its optimal order quantity satisfies F Q ¥ in 1 wi 1 c0 ¥ in 1 ¥ in si 1 si . (26) Comparing (26) with (2), we have Proposition 9. The assembler will choose the system* wide optimal production quantity Qc, if and only if ¥ in 1 wi 1 c0 ¥ in 1 ¥ in si 1 si n ci . i 0 (27) How can the channel be coordinated' Assume that, for any value of wholesale price wj charged to the assembler, each supplier j, j 1, 2, . . . , n, further agrees to pay the assembler a corresponding buy-back price of sj wj 1 1 ¥ in ci wj 1/n ¥ in 1 c i . 1 ¥ in 0 c i (28) 0 Then, one can easily verify that condition (27) is always satisfied and, thus, the channel is coordinated! Now, with the channel coordinated (i.e., the produc* tion quantity being Qc and, hence, the total channel profit pie being at its maximum size), supplier j’s profit is given by Q* c j wj wj cj Q* c sj 0 Q* c x f x dx, j 1, 2, . . . , n. In conjunction with (28), we can show that j(wj) is just a linear function of its wholesale price wj. Thus, in 32 Gerchak and Wang: Revenue Sharing vs. Wholesale-Price Contracts Production and Operations Management 13(1), pp. 23–33, © 2004 Production and Operations Management Society reality, each supplier simply tries to force as high a wholesale price as possible on the assembler. In equilibrium, every supplier will indeed choose to offer a channel coordinating contract. This can be seen as follows. Suppose that all other suppliers, except supplier 1 (wlog), have offered their coordinating contracts according to (28). In choosing its offer, supplier 1 faces nothing but a downstream Newsvendor who takes into account all other suppliers’ contracts as its parameters (in addition to its own production cost c0 and selling price of the final product). It is well known that, in such a single-supplier, single-Newsvendor-retailer type of supply chain, supplier 1 will maximize its own profit by offering a channel coordination contract (i.e., according to (28)). That is, the offering of coordinating contracts by every supplier forms an equilibrium. In equilibrium, the set of coordinating contracts {wj: j 1, 2, . . . , n} represents an allocation of the total channel profit among the n suppliers, with the assembler always left with (almost) zero profit (or its minimum reservation profit). Obviously, there exists a continuum of such contracts and, hence, a continuum of equilibrium allocation of channel profit. 5. Concluding Remarks Ours is the first study of coordination in decentralized assembly/joint-purchase systems with random demand. The basic observation that with either singlelever contract the decentralized inventory levels are less than the centralized ones (Proposition 2 and Corollary 3) should be viewed in the context of products’ complementarity vs. substitution. Ours is a perfect complementarity setting. For a similar observation in a wholesale-price-driven assemble-to-order system, see Dong and Lee (1999, Lemma 2). In a substitution environment, on the other hand (e.g., Mahajan and van Ryzin 1999), the typical conclusion is that the decentralized inventory levels will be higher than systemoptimal. Viewed in a retail context, our complementary products were assumed to be jointly purchased in every case. There are products, however, like coffee and filters, which, although perfectly complementary, are not always purchased together or at fixed ratios (when you have guests you consume coffee faster than filters relative to when you make coffee just for yourself). The implications of this weaker form of purchase-coupling need to be explored. Delegating the inventory management of complementary products to individual, non-communicating suppliers, while used in practice, at first does not sound natural. Yet the observation that at equilibrium all suppliers will deliver the same quantity “removes the danger” of mismatches. Furthermore, the perfor- mance of such a system is independent of the number of suppliers, and an additional lever, surplus subsidy, will coordinate the channel. The performance of a wholesale-price-based system, on the other hand, does degrade with the number of suppliers, and appears (in our example) to be inferior to the vmi one even for a small number of suppliers. That said, the practicality and relative attractiveness of the schemes would also heavily depend on informational assumptions/requirements and monitoring and enforcement issues. We shall only discuss the informational requirements. Again, at first, the vmi system “sounds” like the one that will require more information about costs; after all, in deciding how much to deliver, a supplier in a vmi system needs to contemplate the other suppliers’ choices, and these are naturally based on their costs. But, as we pointed out after Proposition 3, a supplier can infer other suppliers’ cost/share ratios and determine his Nash strategy (Proposition 1) without explicit knowledge of others’, or assembler’s, costs. On the other hand, within the wholesale-price-based contract, although a supplier’s optimization problem does not explicitly involve other suppliers’ costs, in this game, all suppliers’ optimality conditions (reaction curves) will have to be solved simultaneously; thus, each supplier will need to use information about other suppliers’, as well as assembler’s, costs. However, once the wholesale prices were chosen and announced, the assembler will not need to know the suppliers’ costs to select its order quantity; on the other hand, the revenue-share selecting firm will need to know the suppliers’ costs. Thus, the vmi system requires the assembler to have more information about the suppliers’ costs than does the wholesale-price-system, but the latter requires the suppliers to be better informed. As a future research direction, it would be interesting to extend the models studied here to situations where the firms involved have asymmetric information about the demand forecast or each other’s costs. Acknowledgments The authors wish to thank Martin Lariviere and Yehuda Bassok for useful comments. They are grateful to Joseph Geunes and Anant Balakrishnan—the Guest Editors, and two anonymous referees for their helpful suggestions. References Anupindi, R., Y. Bassok, E. Zemel. 1999. Study of Decentralized Distribution Systems: Part II—Application, working paper, Kellogg Graduate School of Management, Northwestern University, Evanston, IL. Cachon, G. P. 2001. Supply Chain Coordination with Contracts, working paper, The Wharton School, University of Pennsylvania. Cachon, G. P., M. A. 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