Enhancing Decision-Making Clarity: Layered Set-Based Similarity Measures for Probabilistic Hesitant Fuzzy Sets

Many existing similarity measures for probabilistic hesitant fuzzy sets (PHFSs) fail to distinguish between the hesitant fuzzy set and probability components, potentially leading to misinterpretations. Since probability and hesitant fuzzy concepts represent different semantic meanings, disregarding these distinctions can result in unrealistic outcomes. To address this issue, we first evaluate the similarity of the probabilistic component of PHFSs and incorporate this value as a coefficient in the similarity measure formula for the hesitant fuzzy component. This approach yields a comprehensive similarity measure that effectively captures both probabilistic and hesitant fuzzy aspects of PHFSs. Building on this framework, we refine traditional set-based similarity measures—Jaccard, Dice, and Cosine—for PHFSs and introduce layered set-based similarity measures (LSSMs), which externally integrate distinct semantic components. The effectiveness of LSSM is demonstrated by extending the CRITIC technique for decision-making, allowing for the determination of criteria weights even when information is incomplete or uncertain. We further validate the proposed measures using Spearman's correlation coefficient to analyze ranking stability and context sensitivity, facilitating their selection for specific decision scenarios. Finally, we apply the developed measures to optimize electric vehicle charging infrastructure in a large-scale problem, showcasing their practical applicability.