Coalition formation and cost allocation for joint replenishment systems

This paper focuses on the issues of coalition formation and cost allocation in a joint replenishment system involving a set of independent and freely interacting retailers purchasing an item from one supplier to meet a deterministic demand. The papers dealing with this problem are mainly focused on supperadditive games, where the cost savings associated with a coalition increase with the number of players in the coalition. The most relevant question addressed then is how to allocate the savings to the players. In this paper, we propose to go further by dealing with a non-supperadditive game, where a set of independent retailers have the common understanding to share the cost savings according to the cost-based proportional rule. In this setting, the global cost optimization is no longer a relevant approach to identify appealing coalitions for any retailer. Here, we provide an iterative procedure to form the so-called efficient coalition structure and we show that this coalition structure has the nice properties of being (i) weakly stable in the sense of the coalition structure core and (ii) strongly stable under a given assumption. An exact fractional programming based solution is also given to generate such efficient coalitions.