Hesitant Fuzzy Variable and Distribution

Abstract

In recent decades, the hesitant fuzzy set theory has been used as a main tool to describe the hesitant fuzzy phenomenon, which usually exists in multiple attributes of decision making. However, in the general case concerning numerous decision-making problems, values of attributes are real numbers, and some decision makers are hesitant about these values. Consequently, the possibility of taking a number contains several possible values in the real number interval [0, 1]. As a result, the hesitant possibility of hesitant fuzzy events cannot be inferred from the given hesitant fuzzy set which only presents the hesitant membership degree with respect to an element belonging to this one. To address this problem, this paper explores the axiomatic system of the hesitant possibility measure from which the hesitant fuzzy theory is constructed. Firstly, a hesitant possibility measure from the pattern space to the power set of [0, 1] is defined, and some properties of this measure are discussed. Secondly, a hesitant fuzzy variable, which is a symmetric set-valued function on the hesitant possibility measure space, is proposed, and the distribution of this variable and one of its functions are studied. Finally, two examples are shown in order to explain the practical applications of the hesitant fuzzy variable in the hesitant fuzzy graph model and decision-making considering hesitant fuzzy attributes. The relevant research results of this paper provide an important mathematical tool for hesitant fuzzy information processing from another new angle different from the theory of hesitant fuzzy sets, and can be utilized to solve decision problems in light of the hesitant fuzzy value of multiple attributes.