Cost sharing in distribution problems for franchise operations

This paper studies the cost sharing problem in an inventory transportation system with multiple agents, where transportation costs are different for each agent. Orderings of a single item are placed jointly using an economic order quantity (EOQ) policy. Part of the ordering cost is shared, and part is specific to each agent and depends on the distance from the supplier (transportation cost). For this inventory situation, cooperation is not always profitable. We therefore examine when cooperation is profitable and how to divide the total cost in a way that ensures stability (no group of agents can improve by deviating from the total group) and computability. We use cooperative game theory to provide adequate answers to all those questions. We prove that if cooperation is profitable (the corresponding inventory game is subadditive), then we can always find coalitional stable allocations of the total cost (the core of the game is not empty). We further define two kinds of context-specific cost sharing rules and study their properties. The first one, which turns out to be coalitional stable (it always belongs to the core), is a cost sharing rule à la Shapley. The second one, simpler but not always coalitional stable, belongs to the family of proportional cost sharing rules.