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Article Excerpt
1. Introduction
It is common in many firms that different divisions, plants, or stores of a firm are given local autonomy on decision making. However, it is in the interest of the firm to set incentive structures so that units maximizing their own local profits end up making decisions that benefit the whole firm. We consider a special instance of this problem, where a firm has multiple (two) locations producing the same product, with each location responsible for its own demand from a specific area. Should one of the locations be unable to meet its own demand in a given period, either because capacity that period was lower than expected or demand was higher, it can ask the other location for some goods to be transshipped, given they are available. Such situations are very common in practice with or without production. Despite using some form of transshipment, many firms make ad hoc determinations of transshipment decisions and do not consider what transfer pricing schemes would induce globally optimal behavior. For example, we recently worked with a diesel engine manufacturer that had multiple plants producing castings. The capacity of the plants making the castings in any week was random due to quality problems and, therefore, the desired quantities could not be produced and castings would be transshipped from one plant to another. Even though the situation occurred repeatedly, the decisions were made in an ad hoc fashion.
Research on how to achieve coordination between suppliers and buyers or to improve the performance of the whole supply chain has received considerable attention in several disciplines including economics, marketing, accounting, and operations management. Much work focuses on maximizing joint profit by using various pricing policies such as return policy (Pasternack 1985), franchising policy (Lal 1990), and quantity discount policy (Weng 1995), etc. The accounting literature has addressed the design of the internal accounting rules and transfer pricing schemes to mitigate incentive misalignment between divisions (e.g., Dorestani 2004, Wei 2004, Sahay 2003). In the operations management literature, different types of contracts are studied, such as price-scheme contracts like buy back (Krishnan et al. 2004) and two-part tariff (Corbett et al. 2004) and quantity-scheme contracts like quantity flexibility (see Tsay and Lovejoy 1999). All of these typically involve vertical coordination (see Lariviere 1999 for a comprehensive review). In this paper, however, we are interested in the lateral coordination of the retailers through transshipping the surplus inventories between them. Hence, the relevant literature includes also the transshipment literature, especially papers analyzing distribution with transshipment.
Most transshipment literature considers centralized systems; see Rudi et al. (2001) and Hu et al. (2004) for an overview of the literature. A few papers have focused on allocation mechanisms that induce first-best solutions in decentralized distribution systems with transshipment after demand is realized. These papers consider single-period models and focus on what mechanisms can be created to reallocate, after demand is realized, the units where they are being demanded. Anupindi et al. (2001) consider a distribution system with n independent retailers where the inventory is transshipped between retailers after demand realization, and the profits generated by such transshipment are allocated among the retailers. They suggest an allocation rule based on the dual price of the transshipment problem and show that such an allocation rule is always in the core of the cooperative transshipment game. Based on the dual price allocation, they establish an allocation rule that specifies the transshipment prices which are different for various possible combinations of residual demand/supply of the n retailers. Granot and Sosic (2003) extend the Anupindi et al. (2001) model by inserting another decision stage between production and transshipment: retailers decide how much of their residual supply/demand should be shared with others.
Whereas the above two papers focus on transshipment prices that are determined after demand is realized, we focus on a model that specifies linear coordinating transshipment prices in advance of demand/capacity realization. The advantage of this approach is that it is described by only a pair of prices, which do not depend on the realization of the random variables. (The possibilities of various demand scenarios still need to be considered in setting the transhipment prices, but this is likely to be simpler in practice than plant managers jointly considering all possible demands and prices. Additionally, the possibility of multiple coordinating transfer prices for a single demand scenario is usually present with demand/capacity-dependent pricing and this may lead to additional negotiations, which is another complicating factor.) Our paper is most closely related to that of Rudi et al. (2001), who address the question of the existence of such coordinating predetermined linear transshipment prices in a model with infinite capacity. In their paper, they claim to show that such coordinating prices always exist. In this paper, we present a counterexample to their result and show that such coordinating prices may easily fail to exist in a range of cases. We also derive a sufficient and necessary condition for the existence of coordinating prices. (A special case of our condition in Theorem 1, for the case without capacity uncertainty, has previously been derived in an unpublished manuscript by Anupindi et al. 1999. (1))
In our model, we allow for uncertain capacity at each of the locations, which is not considered in any of the above papers. Capacity uncertainty has been modeled in the literature in two main ways. One approach, e.g., Henig and Gerchak (1990), is to use stochastically proportional random yield. This approach assumes that a certain number of units are released for production and a random fraction at the end turns out to be good units. Another approach is to consider capacity in a given time interval as a random variable; see Ciarallo et al. (1994), Duenyas et al. (1997), and Hu et al. (2004). This approach assumes that the firm is uncertain about the maximum number of good units it can produce during a fixed interval of time, e.g., a shift. The amount the firm will be able to produce in a period is the minimum of its production plan and the realization of its capacity. Hu et al. (2004) use such an approach to consider a multiperiod centralized production and lateral transshipment model. We use a similar approach here, but focus on a decentralized system.
The rest of this paper is organized as follows. In [section]2, we present our model and notation. In [section]3, we provide sufficient and necessary conditions under which coordinating transshipment prices will exist and also present a counterexample to a result provided in Rudi et al. (2001). Section 4 is devoted to analyzing how sensitive the existence and magnitude of coordinating transshipment prices are to problem parameters. We show that coordinating prices may sometimes exist for only a narrow range of problem parameters. In fact, we are able to show that in some cases, even though coordinating prices exist for specific actual production costs and profit margins, any deviation in one of these values results in nonexistence of coordinating prices. We also explore the impact of demand and capacity variability on the magnitude of coordinating transshipment prices. Section 5 contains concluding remarks.
2. Model
We consider a two-location production/inventory model with transshipment, where the production decisions are made locally. Our model is a generalization of that in Rudi et al. (2001), in that we allow production capacity to be uncertain, whereas in their model they implicitly assume capacity to be infinite. The two locations, indexed by i = 1, 2, operate independently and each optimizes its own individual profits by deciding its target production quantity [Q.sub.i]. First, each location makes a target production decision to try to produce [Q.sub.i] units. When making the target production decision, the locations do not know the capacity or demand realization, but capacity distributions at each location and the joint demand distribution at the two locations are common knowledge. We denote [C.sub.i] and [D.sub.i] as the random capacity for production and random demand in location i. We assume that demand in location i has probability distribution function (p.d.f.) of [g.sub.i], cumulative distribution function (c.d.f.) of [G.sub.i], and support [[L.sub.i], [H.sub.i]]. The capacity distribution for location i has p.d.f. [w.sub.i], c.d.f. [W.sub.i], and [bar.W.sub.i] = 1 - [W.sub.i]. We assume that the two capacities are independent of each other and of the demands, while the two locations' demands are allowed to be correlated with correlation p. After target production decisions, each location realizes its capacity, thus achieving its actual production level min {[Q.sub.i], [C.sub.i]} with unit production cost [c.sub.i]. With the demand realization, each location satisfies its own demand first. Location i obtains revenue [r.sub.i] > [c.sub.i] for every unit sold, is penalized [b.sub.i] [greater than or equal to] for every unit of unmet demand, and can salvage unsold units at unit price [s.sub.i] < [c.sub.i]. If one location has extra inventory after its demand is satisfied and the other location faces a shortage, then inventory is transshipped from the location with the surplus to the location with the shortage at a predetermined transshipment price. For each unit transshipped from location i to j, location i charges transshipment price [p.sub.ij] and pays the associated transshipment cost [[tau].sub.ij]. Throughout this paper, we assume that i, j = 1, 2 and j = 3 - i. For analytical convenience, the cumulative distribution functions of demand and of capacity are assumed to be twice differentiable and strictly increasing on the convex envelope of their support.
The following assumptions are the same as those made in Rudi et al. (2001) (the notation...
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