Various distance and similarity measures have been proposed for hesitant fuzzy sets (HFSs) in the literature. However, some of these approaches are either inadequate or fail to produce reliable results for different scenarios. In this paper, we introduce a novel methodology for computing distance and similarity measures between two HFSs based on an axiomatic framework. A key challenge arises when the lengths of two hesitant fuzzy elements (HFEs) differ. Traditionally, it is assumed that a pessimist would repeatedly add the minimum value, while an optimist would add the maximum value until the HFEs are of equal length. However, this approach may introduce bias and is not intuitively acceptable. To overcome this limitation, we propose an innovative and intuitive technique that ensures fairness by repeatedly adding zero to equalize HFE lengths. This method aligns with intuition and satisfies all axioms of distance and similarity measures. Several numerical examples demonstrate its effectiveness compared to existing methods. Additionally, we apply our approach to develop a hesitant fuzzy TODIM (HF-TODIM) model for interactive and multi-criteria decision-making. To validate its applicability, we use it to evaluate different livestock species and identify the most profitable option. The results confirm that our method is well-suited for handling complex and uncertain hesitant fuzzy information in a balanced and intuitive manner.
Hussain et al. (Sun,) studied this question.