A New Similarity Measure of Hesitant Fuzzy Sets and Its Application

Similarity measure is an tool to evaluate the degree of similarity between two objects. In hesitant fuzzy sets, several similarity measures have been proposed, but all of them need that two hesitant fuzzy elements h1(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{1}(x)$$\end{document} and h2(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{2}(x)$$\end{document} have the same cardinality. In practice, however, the cardinality of hesitant fuzzy element h1(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{1}(x)$$\end{document} may not equal to the cardinality of hesitant fuzzy element h2(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{2}(x)$$\end{document} for some x. In this case, to use existing similarity measures, one must extend the shorter hesitant fuzzy element by adding values subjectively until both of them have the same cardinality. Thus, different results may obtained by different adding values, this is not allowed in some problems, like in pattern recognition problem. In this paper, a new similarity measure between hesitant fuzzy sets which can overcome the defect of existing similarity measures is proposed. Additionally, some properties of the proposed similarity measure are discussed. Furthermore, numerical experiments are conducted to show the rationality and validity of our proposed similarity measure. Finally, the proposed similarity measure is applied to a real classification problem, and the performance of the proposed similarity measure is compared with the existing similarity measure. The application results show that the proposed similarity measure has a well performance.