A Three-Stage Model for a Decentralized Distribution System of Retailers

Abstract

We present and study a three-stage model of a decentralized distribution system consisting of n retailers, each of whom faces a stochastic demand for an identical product. In the first stage, before the demand is realized, each retailer independently orders her initial inventory. In the second stage, after the demand is realized, each retailer decides how much of her residual supply/demand she wants to share with the other retailers. In the third stage, residual inventories are transshipped to meet residual demands, and an additional profit is allocated. Our model is an extension of the two-stage model of Anupindi et al. (ABZ) (2001), which implicitly assumes that all residuals enter the transshipment stage. We show, however, that allocation rules in the third stage based on dual solutions, which were used in the ABZ model, may induce the retailers to hold back some of their residual supply/demand. In general, we study the effect of implementing various allocations rules in the third stage on the values of the residual supply/demand the retailers are willing to share with others in the second stage, and the trade-off involved in achieving an optimal solution for the corresponding centralized system.

Cost Allocation for Joint Replenishment Models

“…Applications of game theory in analyzing cooperations among supply chain agents can be found in Anupindi et al [2], Granot and Sošić, [8], and Nagarajan and Sošić [15]. For an excellent comprehensive review, see Nagarajan and Sošić [16].…”

Game Theory in Supply Chain Analysis

“…Because the Shapley value is unique, it has found numerous applications in economics and political sciences. So far, however, SCM applications are scarce: Except for discussion in Granot and Sosic [41] and analysis in Bartholdi and Kemahlioglu-Ziya [5], we are not aware of any other papers employing the concept of the Shapley value. Although uniqueness of the Shapely value is a convenient feature, caution should surely be taken with Shapley value: The Shapley value need not be in the core; hence, although the Shapely is appealing from the perspective of fairness, it may not be a reasonable prediction of the outcome of a game (i.e., because it is not in the core, there exists some subset of players that can deviate and improve their lots).…”

Game Theory in Supply Chain Analysis

“…Because the Shapley value is unique, it has found numerous applications in economics and political sciences. So far, however, SCM applications are scarce: except for discussion in Granot and Sosic (2001) we are not aware of any other papers employing the concept of the Shapley value. Although uniqueness of the Shapely value is a convenient feature, caution should surely be taken with Shapley value: the Shapley value need not be in the core, hence, although the Shapely is appealing from the perspective of fairness, it may not be a reasonable prediction of the outcome of a game (i.e., because it is not in the core, there exists some subset of players that can deviate and improve their lots).…”

“…Moreover, they find an allocation mechanism that is in the core and that allows them to achieve coordination, i.e., the first-best solution. Granot and Sosic (2001) analyze a similar problem but allow retailers to hold back the residual inventory. In their model there are actually three stages: inventory procurement, decision about how much inventory to share with others and finally the transshipment stage.…”