Novel distance and similarity measures for hesitant fuzzy sets and their applications to multiple attribute decision making

Distance and similarity measures have recently been investigated in-depth within the context of hesitant fuzzy sets. By analyzing the existing studies concerning distance measures for hesitant fuzzy sets, we find that they have some limitations. To address the flaws, this study develops some novel distance measures for hesitant fuzzy sets, including the normalized Euclidean distance measure, the Hausdorff metric distance measure, the normalized generalized distance measure, and their corresponding weighted distance measures. The proposals of this study not only hold many ideal characteristics but also do not consider the lengths of hesitant fuzzy elements as well as the arrangement of their possible values. To deal with the situations where both of the universe of discourse and the weight of element are continuous, some continuous hesitant fuzzy distance measures are also investigated. Based on the relationship between distance measure and similarity measure, some novel similarity measures for hesitant fuzzy sets can be further deduced from the proposed distance measures. Finally, two numerical examples are given to demonstrate the applicability and validity of the proposed hesitant fuzzy distance measures.