Distance and similarity measures of hesitant fuzzy sets based on Hausdorff metric with applications to multi-criteria decision making and clustering

Abstract

Distance and similarity measures play a vital role to differentiate between two sets or objects. Different distance and similarity measures had been proposed for hesitant fuzzy sets (HFSs) in the literature, but either they are insufficient or not reflect desirable results. In this paper, the construction of new distance and similarity measures between HFSs based on Hausdorff metric is proposed. We first present a novel and simple method for calculating a distance between HFSs based on Hausdorff metric in a suitable and intuitive way. Two main features of the proposed approach are: (1) not necessary to add a minimum value, a maximum value or any value to the shorter one of hesitant fuzzy elements (HFEs) for extending it to the larger one of HFEs; and (2) no need to arrange HFEs either in ascending or descending order. This is because adding such values and arrangements of elements will not put any impact on final results. We then extend distance to similarity measure between HFSs. Furthermore, we claim some properties and also compare the proposed distance and similarity measures with existing methods. The comparison results demonstrate that the proposed distance and similarity measure are simpler, intuitive and better than most existing methods. Finally, we apply the proposed distance of HFSs to multi-criteria decision making and the similarity measure of HFSs to clustering.