Analysis of a Simple Cost Allocation Rule for Joint Replenishment

Abstract

We consider the infinite-horizon multiple retailer joint replenishment problem with first order interaction. In this model, the joint setup cost incurred by a group of retailers placing an order simultaneously consists of a group-independent major setup cost and retailer-specific minor setup costs. The goal is to determine an inventory replenishment policy that minimizes the long-run average system-wide cost. In this paper, we adopt a non-cooperative approach to study the joint replenishment game. We consider the allocation rule in which the major setup cost is split equally among the retailers who place an order together, and each retailer pays his own holding and minor setup costs. Given the preannounced allocation rule, each retailer determines his replenishment policy to minimize his own cost anticipating the other retailers' strategy. We show that a payoff dominant Nash equilibrium (N.E.) exists, and quantify the efficiency loss of the non-cooperative outcome relative to the social optimum. Although the worst-case ratio between the best decentralized outcome and the social optimum is O (sqrt{ln n}), where n is the number of retailers, numerical results suggest that the best equilibrium is near-optimal.