When self-disposing of radioactive waste, it is important to follow the acceptable concentration standards for each nuclide set by the Nuclear Safety and Security Commission (NSSC). Gamma-emitting nuclides can be easily analyzed with a simple pretreatment process, but beta-emitting nuclides require a chemical separation procedure to be analyzed for radiochemistry analysis. When analyzing betaemitting nuclides for the purpose of self-disposal, there may be difficulties in radiation detection after the chemical separation process. This is because the concentration of beta nuclides in the sample may be low and some of them may be lost during the chemical separation. Therefore, measurement method of gross-beta activity can be used instead of that of each nuclide to access the compliance of selfdisposal criteria. While a proportional counter is commonly used to measure gross-beta activity, liquid scintillation counting can also be used to measure gross-beta, and we plan to compare the results of both methods.

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.

If the coalition of players orders the product together, then they can reduce the inventory costs such as ordering cost and holding cost. Inventory game can be defined as the allocation of the inventory costs to the players in a fair and rational manner. The characteristics recommended for the solutions provided by the method for the inventory game are completeness, rationality, and marginality. The solutions that satisfy these characteristics are in the core, where the proportional method may depend on the allocator. This study has found out that the solutions of the proportional method with some allocators for economic order quantity model exist in the core.