The Cost Allocation Problem for the First Order Interaction Joint Replenishment Model

Abstract

We consider an infinite-horizon deterministic joint replenishment problem with first order interaction. Under this model, the setup transportation/reorder cost associated with a group of retailers placing an order at the same time equals some group-independent major setup cost plus retailer-dependent minor setup costs. In addition, each retailer is associated with a retailer-dependent holding-cost rate. The structure of optimal replenishment policies is not known, thus research has focused on optimal power-of-two (POT) policies. Following this convention, we consider the cost allocation problem of an optimal POT policy among the various retailers. For this sake, we define a characteristic function that assigns to any subset of retailers the average-time total cost of an optimal POT policy for replenishing the retailers in the subset, under the assumption that these are the only existing retailers. We show that the resulting transferable utility cooperative game with this characteristic function is concave. In particular, it is a totally balanced game, namely, this game and any of its subgames have nonempty core sets. Finally, we give an example for a core allocation and prove that there are infinitely many core allocations.

Cost Allocation for Joint Replenishment Models

“…We also show that a special case of this game is concave, which generalizes one of the main results of Anily and Haviv [1].…”

“…In particular, this paper is closely related to Anily and Haviv [1]. Their model is essentially the same as ours with two key differences.…”

“…Yet, the optimal policy for the joint replenishment problem with first order interaction is unknown. Anily and Haviv [1] focus, therefore, on a class of easyto-implement policies, called power-of-two policies, because it is well-known that optimal power-of-two policies are 98% effective; see Roundy [18] and Jackson et al [11]. Anily and Haviv show that under the optimal power-of-two policy, the cooperative game associated with the joint replenishment model with first-order interaction is concave and thus has a non-empty core.…”

“…Meca et al [13] study a special case of Anily and Haviv's model [1]. In the special case, all the minor setup costs are assumed to be zero.…”

“…Therefore it is always optimal for all the retailers to order together. In a recent paper, Dror and Hartman [5] study the same model as Anily and Haviv [1], but their focus is mainly on characterizing a necessary and sufficient condition, under which it is optimal for all the retailers to order together and the game to have a non-empty core.…”