Correlation Coefficients of Interval-Valued Hesitant Fuzzy Sets and Their Application Based on the Shapley Function

Interval-valued hesitant fuzzy sets permit the membership degree of an element to have several possible interval values in [0, 1] rather than real numbers, which can well deal with inherent hesitancy and uncertainty in the human decision-making process. In this paper, we first point out the issue of the existing correlation coefficients of interval-valued hesitant fuzzy sets and then define several new ones, which do not have to consider the lengths of interval-valued hesitant fuzzy elements and the arrangement of their possible interval values. Since the assumption that the elements in a set are independent is usually violated, we further define several Shapley weighted correlation coefficients, which consider their interactions. To deal with the situations where the elements are correlative and the weight formation is incompletely known, models for the optimal fuzzy measures on a feature set and on an attribute set are established, respectively. Finally, a procedure to pattern recognition and multiattribute decision making with incomplete weight information and interactive conditions is developed. Meanwhile, the corresponding examples are provided to show the practicality and feasibility of the proposed procedures. (C) 2015 Wiley Periodicals, Inc.