Novel distance and similarity measures on hesitant fuzzy sets with applications to clustering analysis

Abstract

Distance and similarity measures are fundamentally important in a variety of scientific fields such as clustering analysis, pattern recognition and decision making, etc. In this paper, by analyzing the existing distance and similarity measures between hesitant fuzzy sets, we show that they are not reasonable in some situations. To this end, we propose a novel concept of hesitancy index of hesitant fuzzy set to measure the hesitancy degree among the possible values in each hesitant fuzzy element of the hesitant fuzzy set. By taking their hesitancy indices into account, new methods for measuring the distances between hesitant fuzzy sets are proposed and their properties are discussed. According to the relationship between the distance measure and the similarity measure, two novel similarity measures for hesitant fuzzy sets are further developed. Afterwards, we propound a novel hesitant fuzzy clustering algorithm on the basis of the novel similarity measures for classifying objects with hesitant fuzzy sets. At length, a real-life example is given to illustrate the detailed implementation process of the proposed clustering approach, and a comparative study on the same example is conducted.