AN ENHANCED DIVERGENCE-BASED DISTANCE MEASURE FOR INTUITIONISTIC FUZZY SETS WITH HESITATION INFORMATIONAND ITS EXTENSIONS TO INTERVAL-VALUEDAND PICTURE FUZZYENVIRONMENTS

Abstract

Distance measures for intuitionistic fuzzy sets (IFSs) are central tools for pattern recognition, clustering, and multi-attribute decision making (MADM) under uncertainty. Although numerous divergence-based and geometric distance mea sures have been proposed in the literature, most of them either neglect the hesita tion degree, which carries genuine epistemic information, or fail to discriminate between sets that exhibit equal membership-non-membership differences but distinct hesitation profiles. To overcome these limitations, this paper proposes a novel enhanced divergence-based distance measure D+ L for IFSs that incor porates an explicit hesitation term derived from a modified Kullback–Leibler divergence. The measure is constructed from a single, symmetric core function and is shown to satisfy all four axiomatic requirements of an IFS distance metric, namely boundedness, separability, symmetry, and monotonicity. The proposed measure is further extended to two important generalizations of intuitionistic fuzzy theory: a six-term version DIV+ L for interval-valued intuitionistic fuzzy sets (IVIFSs) that fully exploits both the lower and upper bounds of member ship, non-membership and hesitation intervals, and a three-component version DP L for picture fuzzy sets (PFSs) that handles positive, neutral, and negative memberships. Six classical benchmark cases and two innovation-management decision problems are recomputed entirely from scratch using the proposed measure. Comparative analysis with twelve existing measures shows that D+ L resolves the counter-intuitive ties that plague competing measures, distinguishes hesitation-sensitive cases that earlier divergence-based measures could not, and yields stable rankings in TOPSIS-based MADM. The results confirm that the enhanced measure is mathematically rigorous, computationally concise, and practically effective.