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We consider a decentralized supply chain, whereby a supplier sells a product to a group of independent buyers, and develop a strategy for the supplier to offer an all-units price discount or cash rebate for orders that are synchronized with its replenishments. As synchronized orders can be met with inventory directly from receiving to shipping without warehousing, the proposed strategy streamlines system inventory flows to minimize inventory and, hence, the related costs. On the other hand, by increasing the replenishment interval of the supplier, the proposed strategy is able to induce buyers to order in large quantities and hence achieve the objectives of quantity discounts. We show that the proposed strategy can achieve nearly optimal (minimum) system cost, and is much more effective than the existing coordination strategies for decentralized supply chains in the literature.

Supply chain coordination has attracted considerable interest from both academics and practitioners. Many firms have realized that collaborating with their supply chain members can substantially improve their profits and enhance their strategic positioning. However, supply chain members are often separate economic entities that act independently and opportunistically to optimize their individual profits. The challenge, then, is to create useful and effective coordination mechanisms that are able to improve performance not only for the party that takes the initiative, but also for the entire supply chain (Li and Wang, 2007 and Arshinder et al., 2008). Previous research into this problem has focused on using quantity discount policies for suppliers to induce independent buyers to increase their order quantities so as to reduce ordering and order processing costs. Crowther (1964) first demonstrated that quantity discounts could substantially improve channel efficiency. Subsequently, various quantity discount policies were developed under different channel conditions (see, e.g. Dolan, 1978, Monahan, 1984, Lal and Staelin, 1984, Joglekar and Tharthare, 1990, Wang and Wu, 2000, Munson and Rosenblatt, 2001, Parlar and Wang, 1994, Weng, 1995, Wang, 2002, Altintas et al., 2008 and Bellantuono et al., 2009). These studies showed that quantity discounts are able to produce significant benefits for independent suppliers and buyers. However, the benefits from quantity discounts usually represent only a small portion of the maximum potential benefits of supply chain coordination (Wang, 2002). Consider a two-echelon supply chain whereby a supplier sells a product to a group of independent retailers. In addition to the benefits of quantity discounts, there are at least two other important benefits that the supplier can exploit. First, the supplier can induce buyers to synchronize their orders with its replenishments so that such orders can be met directly with inventory from receiving to shipping without warehousing, hence reducing its inventory and inventory holding cost. Secondly, the supplier can consolidate orders at common points in time so that such orders can be delivered and/or processed more efficiently, hence reducing its delivery and/or order processing costs. Recent studies have addressed this issue by combining quantity discounts and a power-of-two or integer-ratio policy (Wang, 2001 and Wang, 2004). A power-of-two or integer-ratio policy requires buyers to order in intervals that are power-of-two or integer-ratio multiples of the supplier's constant replenishment interval, and is able to achieve at least 98% of optimality for a centralized supply chain (Roundy, 1985 and Roundy, 1986). These studies advance the understanding of supply chain coordination by showing that the combination of quantity discounts and time coordination that synchronizes supply chain inventory decisions and activities can produce significant benefits over traditional quantity discounts. However, there is still a significant benefit that cannot be obtained by these mechanisms. This study develops the following new strategy: the supplier takes the initiative to set up a constant replenishment interval and provides an all-units price discount or cash rebate for orders that can be delivered at the arrival points of its replenishments. As an example, let the supplier order once a week and receive deliveries on Monday. The supplier then offers a price discount or cash rebate for every unit that can be delivered on Monday. Buyers remain independent economic agents and act rationally to minimize their relevant costs. The supplier offers a sufficiently large benefit to induce buyers to comply with the coordination mechanism and imposes no other conditions on their ordering decisions. Following previous quantity discount studies, we develop the supplier's optimal strategy and the buyers' optimal ordering decisions as (equilibrium) solutions to a Stackelberg game, in which the supplier acts as the leader and the buyers act as followers (Tirole, 1988). The proposed strategy differs from a coordination strategy that combines quantity discounts and time coordination in several aspects. First, the supplier offers a quantity discount for any order at any point in time that meets a minimum quantity requirement under the latter strategy (Chen et al., 2001 and Wang, 2004). In contrast, the supplier offers a price discount for every unit that is ordered at (and only at) an arrival point of a replenishment from the outside source under the former strategy. We refer to this discount scheme as time discount. Second, while the two incentive schemes share a common purpose of inducing buyers to comply with the coordination mechanism, the primary purpose of a quantity discount is to induce buyers to increase their order quantities, whereas the primary purpose of a time discount is to induce buyers to synchronize their orders with replenishments at the supplier. Finally, in response to the different incentive schemes that are adopted by the supplier, a buyer will always order in the same replenishment interval or order quantity under the latter strategy. However, a buyer may place a large order at a point with time discount and smaller orders subsequently under the former strategy. The former strategy provides a more effective mechanism to streamline inventory flow in the supply chain. We demonstrate in a numerical study that the proposed strategy is able to reduce system inventory by more than 6.6% and system inventory related cost by more than 4.6% as compared to the coordination strategy that combines quantity discounts and an integer-ratio policy. Furthermore, the proposed strategy is able to achieve a system cost that is not more than 2% higher than the minimum system cost in the numerical study. In addition, we also show that these observations are robust with respect to changes in the supply chain demand and cost structure. Therefore, the proposed strategy provides a highly effective coordination strategy for a decentralized supply chain. These findings provide useful insights into the coordination of decentralized supply chains. First, time coordination provides a significant dimension for supply chain coordination. The proposed strategy focuses on time coordination and is able to achieve almost minimum system cost. In contrast, a substantial portion of the potential supply chain coordination benefit cannot be achieved by quantity discounts. Second, the findings provide a new interpretation, as well as a new application strategy, for quantity discounts. The proposed strategy offers only a simple price discount and is able to achieve better results than a strategy that combines a full quantity discount schedule and an integer-ratio coordination scheme. This suggests that time coordination is able to achieve the benefits of quantity discounts. The remainder of the paper is organized as follows. Section 2 provides a brief review of the related literature. Section 3 develops a two-echelon supply chain model. Section 4 develops the optimal coordination strategy. The benefits of the strategy are demonstrated numerically in Section 5. Finally, concluding remarks are provided in Section 6.

In this paper, we develop a coordination strategy for a decentralized two-echelon supply chain that is able to induce maximum time coordination or synchronization of supply chain inventory activities. Under this strategy, the supplier sets up a constant replenishment interval and provides a sufficiently large all-units price discount or cash rebate to entice buyers to synchronize their orders with its replenishments. We show that not only can this strategy obtain substantial benefits as compared to existing coordination strategies in this literature. It can also attain nearly optimal (minimum) system cost and, therefore, provides a highly effective coordination strategy for a decentralized supply chain. Our findings offer useful managerial insights into the coordination of a decentralized supply chain. The proposed strategy is highly effective because of two reasons. First, it provides an effective way to streamline system inventory flows. By providing an incentive only for orders that are synchronized with the replenishments of the supplier, the proposed strategy entails a lower inventory level than that under simple integer-ratio time coordination. Second, the strategy induces buyers to order in large replenishment intervals and achieves the benefits of quantity discounts. This challenges the conventional wisdom of, and provides a new explanation and application to, traditional quantity discounts. The proposed strategy provides a feasible framework for the implementation of a cross-docking operation. Cross docking is the direct flow of merchandise or materials from the receiving process to the shipping process. Orders from different buyers and/or for different products are consolidated such that they are handled and delivered more efficiently. Richardson (1999) has reported substantial cost savings from cross docking. By adopting the supplier's replenishment cum coordination points as cross-docking points, the current model provides a plausible mechanism for a cross-docking operation to take place when the supplier and buyers are separate economic entities. In addition to a price discount or cash rebate, we note that suppliers in reality may also share a buyer's ordering and/or delivery cost if the timing of an order is desirable and/or impose a special order processing and/or delivery charge if the timing of an order is undesirable. In view of such practices, we considered two alternative incentive schemes in our research: (i) the supplier pays for or shares a percentage of the ordering cost for a coordinated order and (ii) the supplier imposes a charge as a percentage of its order processing cost for a non-coordinated order and provides an all-units price discount for a coordinated order. The percentage is common to all buyers in both cases. The price discount in the second alternative incentive scheme is necessary to ensure that buyers will not be worse off as compared to their EOQ policies and, thus, will comply with the coordination mechanism. The detailed derivations are omitted for simplicity of presentation but available upon request. Essentially, a simple all-units price discount or cash rebate is most flexible and generally most effective. As a result, these incentive schemes are not adopted. Future research may extend the coordination strategy to decentralized supply chains with price-elastic and/or uncertain demand.