Rational divergence measures on intuitionistic fuzzy sets

This paper investigates the theoretical foundations and practical applications of rational divergence measures within the framework of fuzzy set theory. Unlike traditional distance metrics, rational divergence measures are characterized by their functional form, typically expressed as a ratio of membership functions or set cardinalities, which allows for a more nuanced quantification of the "informational gap" between fuzzy sets. The study explores the axiomatic properties of these measures, focusing on their ability to handle non-linearity and uncertainty in complex data structures. By examining rational forms of divergence, such as those derived from the Jaccard-like ratios or specialized f-divergences, this work demonstrates how these measures overcome the limitations of standard Euclidean distances in high-dimensional fuzzy spaces. Key emphasis is placed on their role in multi-criteria decision-making (MCDM) and pattern recognition, where the rational expression of divergence provides a more stable and intuitive measure of dissimilarity. Furthermore, the paper provides a comparative analysis of different rational divergence formulations, evaluating their sensitivity to membership fluctuations and their performance in clustering algorithms. The results suggest that rational divergence measures offer superior discriminative power, making them a tool for modeling expert knowledge and imprecise information in modern intelligent systems. Many authors investigated the possibilities how two fuzzy sets can be compared. The basic study of fuzzy sets theory was introduced by Lotfi Zadeh in 1965. We discuss the divergences defined on more general objects, namely intuitionistic fuzzy sets (IFSs). We have focused on special class of divergences, since some restriction conditions are necessary. This approach to the divergence measure is motivated by class of the rational similarity measures between fuzzy subsets expressed using some set operations (namely intersection, complement, difference, and symmetric difference) and their scalar cardinalities. In this study, we have considered the value of divergence between IFSs as a Σ-count of two scalar cardinalities, i.e. as a pair of real numbers.