Hesitant fuzzy multi-attribute decision-making based on the minimum deviation method

As a powerful and efficient tool in expressing the indeterminate and fuzzy information, the hesitant fuzzy set (HFS) has shown its increasing importance. It was first proposed by Torra and Narukawa, permitting the membership degree of an element to a set of several possible values. In this paper, based on the idea of minimum deviation between the subjective and the objective preferences, we first develop two methods to determine the attribute weights under the hesitant fuzzy environment. To do so, we present the concept of hesitant fuzzy expected value and then establish several optimization models to gain the attribute weights. After that, we use the information aggregation techniques to integrate the hesitant fuzzy attribute values or their expected values, and then sort the alternatives by the overall values. Moreover, we generalize these two methods to interval-valued HFSs, and a numerical example is utilized to show the detailed implementation procedure and effectiveness of our methods in solving multi-attribute decision-making problems with hesitant fuzzy or interval-valued hesitant fuzzy information.