Novel Logarithmic Distance and Similarity Measures for Hesitant Fuzzy Sets and Their Application in Groundwater Depletion Analysis

Here, we propose logarithmic-based distance and similarity measures for Hesitant Fuzzy Sets (HFSs) as an extension of fuzzy set theory to better capture uncertainty in decision-making problems. Traditional distance and similarity measures often exhibit limitations, such as sensitivity to scale, complexity, and the curse of dimensionality, which reduce their effectiveness in real-world applications. The proposed logarithmic measures address these shortcomings by enhancing uncertainty handling. The theoretical properties of these measures are validated while demonstrating their boundedness, symmetry, and strong reflexivity. A comparative analysis of the proposed measures has been done with some existing measures. Furthermore, we apply these measures to analyze groundwater depletion in India utilizing real-world data to cluster regions based on depletion severity. The results demonstrate that our logarithmic measure offers a reliable and effective alternative to existing measures, significantly enhancing decision-making accuracy in uncertain environments.