The Choquet integral-based Shapley function for n-person cooperative games with probabilistic hesitant fuzzy coalitions

Abstract

The use of probabilistic hesitant fuzzy elements (PHFEs) to represent the alliance status in cooperative games can describe the player's participation levels and associated probability. This paper proposes an effective Shapley function for probabilistic hesitant fuzzy cooperative games based on generalized Choquet integral. We first define the probabilistic hesitant fuzzy coalition of cooperative games. After that, a supplementary model of incomplete probability information is established. Without losing any probability information, an adjustment algorithm is developed to derive the consistent probability of coalitions. Then the characteristic function of probabilistic hesitant fuzzy cooperative games is constructed based on generalized Choquet integral. The Choquet integral-based Shapley function is presented according to the proposed characteristic function. Some desirable properties of Shapley function are discussed in detail. The CIG algorithm is established for generating consistent imputation of cooperative games. Finally, the proposed algorithm is applied to the cooperative production of alloy materials.