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...much optimal inventory should be kept at different stages of an integrated supply chain. A more recent stream of research recognizes that most real supply chain operations may not be integrated and that decentralized decision making takes place in practice. In this perspective, inventory-replenishment decisions of different players (members of the chain), made in a noncooperative and decentralized manner, are modeled and analyzed. In this paper, we pursue this analysis for a supply chain consisting of a capacitated supplier and a downstream retailer.
The retailer, in our model, faces a stationary random customer demand and replenishes from its supplier using a base stock policy. The supplier, who operates a capacitated manufacturing facility also uses a base stock policy for its internal replenishment. In the decentralized setting, both the supplier and the retailer choose their base stock levels independently in order to minimize their respective inventory-related costs. This decentralized and noncooperative operation is inefficient in terms of the total supply chain costs with respect to a fully integrated operation. In the first part of this paper, we focus on understanding the causes of this inefficiency and assessing its magnitude. In the second part, we study a number of simple contracts that can be used to overcome the inefficiency of the decentralized system. Although our main focus is on Nash games and equilibria, we also briefly investigate the Stackelberg equilbria where one of the parties is the dominant member of the supply chain.
The paper is structured as follows: Section 2 presents the literature review. The main model is presented in Section 3. Section 4 focuses on the analysis of the decentralized supply chain in the framework of a Nash game. In Section 5 we present and analyze contracts and investigate related coordination issues. Section 6 summarizes our results on Stackelberg games and Section 7 presents the conclusions.
2. Literature review
The effects of decentralized decision making in supply chains have been investigated in several papers in recent years. In particular, a number of papers have studied supply chains with random demand in a single-period setting based on generalizations of the newsvendor framework. Review papers by Lariviere (1999), Tsay et al. (1999), Cachon (2002) and Cachon and Netessine (2004) provide comprehensive pointers to this literature.
Papers that investigate decentralized supply chains by using stochastic models in an infinite-horizon setting are relatively fewer. A number of these papers study the uncapacitated multiechelon system (the Clark and Scarf model). Chen (1999) and Lee and Whang (1999) focus on coordination mechanisms that use nonlinear pricing schemes. Cachon and Zipkin (1999) study the two-stage decentralized supply chain in detail and look into coordination issues through linear transfer payments.
There are also a few recent papers that study capacitated supply systems in a decentralized setting. The underlying models in this framework are built on the make-to-stock queue where the supplier's capacity is modeled by the server of a queueing system (see Buzacott and Shanthikumar (1993)). Cachon (1999) studies a supplier-retailer system with lost sales where each party controls its own inventories. Caldentey and Wein (2003) study a similar system with backorders where the customer backorder cost is shared by both parties. In this paper, the supplier sets the capacity level and the retailer controls the inventories in the system. Gupta and Weerawat (2003) study a supplier-manufacturer system in a manufacture-to-order environment and focus on coordination issues through revenue sharing contracts. Elahi et al. (2003) investigate a model in which multiple capacitated suppliers compete for the demand coming from a single buyer (manufacturer).
Our paper is closely related to Cachon and Zipkin (1999), Cachon (1999), and Caldentey and Wein (2003). All of these papers study two-stage systems, analyze the decentralized chain and its performance and investigate coordination by linear transfer payments and we follow the same general path. As in Cachon and Zipkin (1999), in our two-stage system, both parties are responsible for their own inventory costs and a portion of the total customer backorder cost. Our cost structure is identical, but the supplier in our case is capacitated. As in Cachon (1999), in our model the transportation times between the supplier's inventories and the retailer are assumed to be negligible. This makes the analysis of the centralized system tractable and enables us to obtain analytical results even on the decentralized system. In contrast with Cachon (1999) our system experiences back-orders and the capacity/queueing effects are manifested in a much sharper manner. Finally, as in Caldentey and Wein (2003) our supplier is capacitated and both parties share the backorder cost but both parties may keep inventory in our setting. We also employ a discrete state-space model (i.e., with integer inventory levels) as opposed to working with a continuous approximation as in Caldentey and Wein (2003).
The positioning of our paper with respect to the above three papers can be summarized as follows: for the two-stage system with a capacitated supplier we obtain explicit analytical results on the decentralized system. The corresponding analysis in Cachon (1999) for the identical system with lost sales mostly relies on numerical calculations essentially due to the difficulty of the centralized system therein (i.e., there is no explicit analytical expression for the optimal base stock level, or the optimal supply chain profit in the system with lost sales). This enables us to obtain simple and explicit characterizations of equilibrium behavior in the decentralized system even in the case of unequal holding costs (not treated in Cachon (1999). In this sense, the analytical simplicity and transparency of our results are comparable to those of Caldentey and Wein (2003) whose focus is different. Apart from reaching simple and exact analytical characterizations, there is another important reason for looking at the backorder version of the model in Cachon (1999)). The effect of limited capacity and its consequences in terms of replenishment delays are dampened for the lost sales system because some arrivals are lost. These effects usually appear in a more distinguishing manner in the backorder system where the average cost per unit time goes to infinity as the utilization rate approaches unity.
Finally, it should be noted that our assumption of negligible transportation times (as in Cachon (1999)) between the supplier's inventories and the retailer is crucial for the tractability of the analysis. The corresponding system with two capacitated suppliers in tandem cannot be analyzed exactly even in the centralized case (see Buzacott et al. (1992) for approximations). The other problem for this system is the lack of a complete characterization of the optimal inventory policy. It is known that base stock policies are not optimal and only a partial characterization of the optimal inventory control policy is available (see Veatch and Wein (1994) or Karaesmen and Dallery (2000)). Jemai (2003) presents a numerical investigation of the Nash game when both parties use base stock policies.
3. Modeling assumptions and notations
We consider a two-stage supply chain consisting of a manufacturing stage (stage 1) and a retail stage (stage 2) which satisfies end-customer demand. Both stages have their own separate inventories. In addition, the manufacturing stage has a limited production capacity. The retail stage is replenished from the manufacturer's inventory. End-customer demand arrives in single units according to a Poisson process with rate [lambda]. The manufacturer processes items one by one using a single resource. Item processing times have an exponential distribution with rate [mu]. Let [rho] = [lambda]/[mu] be the utilization rate of the manufacturer.
All customer demand that cannot be satisfied from inventory can be backordered. Both the retailer and the manufacturer manage their own inventories according to base stock policies...
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