A multiple criteria hesitant fuzzy decision making with Shapley value-based VIKOR method

Hesitancy is the most common problem in decision making, for which hesitant fuzzy set can be considered as a suitable means allowing several possible degrees for an element to a set. And, the VIKOR method is an effective tool to determine a compromise solution by providing a maximum group utility for the majority and a minimum individual regret for the opponent in decision making, particularly in a situation where the decision maker is not able or does not know to express his/her preference. In many practical situations, the inter-dependent or interactive characteristics among criteria and preference of decision makers should be taken into account, we generally utilize the Choquet integral to depict the correlations and interactions in the decision process. When the Choquet integral is used to solve the correlative decision making problems, we ignore the importance of the ordered position of the element. In fact, each element has the same probability to be drawn and all permutations have the same probability. In this paper, we introduce the Shapley value to solve correlative problem, which is used to be interpreted as a kind of average value of the contribution of an element alone in all coalitions with the same position probability. Firstly, we present some concepts of hesitant fuzzy set and define the Shapley value-based L-p - metric (SLp,mu - metric). With the SLp,mu - metric, an extended VIKOR method is developed to deal with the correlative multiple criteria decision making (MCDM) problem under hesitant fuzzy environment. To comparative analysis, we also apply the TOPSIS method to solve the problem based on the Shapley value. Finally, a comparative analysis of the two methods is illustrated with a numerical example.