The cooperative game theory consists of a set of players and utility function that has positive values for a subset of players, called coalition, in the game. The purpose of cost allocation method is to allocate the relevant cost among game players in a fair and rational way. Therefore, cost allocation method based on cooperative game theory has been applied in many areas for fair and reasonable cost allocation. On the other hand, the desirable characteristics of the cost allocation method are Pareto optimality, rationality, and marginality. Pareto optimality means that costs are entirely paid by participating players. Rationality means that by joining the grand coalition, players do not pay more than they would if they chose to be part of any smaller coalition of players. Marginality means that players are charged at least enough to cover their marginal costs. If these characteristics are all met, the solution of cost allocation method exists in the core. In this study, proportional method is applied to EOQ inventory game and EPQ inventory game with shortage. Proportional method is a method that allocates costs proportionally to a certain allocator. This method has been applied to a variety of problems because of its convenience and simple calculations. However, depending on what the allocator is used for, the proportional method has a weakness that its solution may not exist in the core. Three allocators such as demand, marginal cost, and cost are considered. We prove that the solution of the proportional method to demand and the proportional method to marginal cost for EOQ game and EPQ game with shortage is in the core. The counterexample also shows that the solution of the proportional method to cost does not exist in the core.

Cost allocation studies on the rational allocation method for the common cost of the joint products or services that provide different benefits to each economic entity under the constraints of the efficiency and fairness. Cooperative game theory is often used for cost allocation and studies on a fair and efficient allocation of the utility if some feasible utility for a whole or subset of the players in a game is given. This study shows a variety of cooperative game theory approaches and discusses the pros and cons of each approach.

A number of stores can save costs by forming a centralized inventory system. The number and location of centralized warehouses can be determined in minimizing total costs and then these cost reductions should be allocated fairly to each store. If no group of stores feel that they are paying to subsidize to the complementary set of stores, then that cost allocation is in core. That is, all stores become better off by forming a group. This study shows that there does not exist the core in some examples. Hence, we suggest a framework for the cost allocation both of the case in which the core exists and the core does not exist.