In most developed countries, modern industrialization systems are established based on reducing the involvement of new input factors and recycling waste. Since not all waste is recycled in waste recycling plants, there must be cooperation among waste producers to reduce the overall costs of processing non-recyclable waste. Many researchers have published several papers on this issue. The following is some of the previous research in the context of waste management: Cheng et al.[1] integrated multi-criteria decision analysis and linear programming to optimize the selection of a landfill site and the waste flows costs considered in the waste management problem; Karmperis et al.[2] reviewed multi-criteria decision analysis model and introduced the waste management bargaining game to support decision-makers; He et al.[3] proposed an agent-based waste treatment model as a complex adaptive system, and private operators set gate fees to vie for waste in low-information competition; Nguyen-Trong et al.[4] modeled the optimization of collection services on transportation of municipal solid waste; Feyzi et al.[5] provided a new MCDM framework and evaluated sustainability-based criteria to select the most appropriate site for solid waste incineration power plants.

Game theory is the analysis of a situation involving conflicting interests (as in business or military strategy) in terms of gains and losses among opposing players. The intention of game theory is to produce optimal decision-making of independent and competing actors in a strategic setting. The cooperative game has been applied to all situations in which product actions maximize utility and has shown to be a powerful tool for creating practical collaboration actions in a wide range of applications. According to the definition of cooperative game theory, this theory concentrates on mathematical models of interaction indices among rational participants (the players) of the modeled dispute (the game)[6]. In the garbage producers' conflict, the coalitional game structure was introduced [7-8]. In addition, types of directed cooperative gaming applications have been built [9-10] for waste management challenges. There are many important and interaction indices (see, eg [11-17]), one of them is the Shapley important index which is used to demonstrate the possible cost to waste producers in the event of cooperation, as well as to minimize the overall costs of processing non-recyclable trash during the collaboration between the among producers. The interaction phenomena among a set of players, which can be seen as an extension of the notion of value, has been applied to multi-criteria decision-making in the framework of aggregation by the Choquet integral [18-21]. The