A Unified Parametric Divergence Operator for Fermatean Fuzzy Environment and Its Applications in Machine Learning and Intelligent Decision-Making

Uncertainty and ambiguity are pervasive in real-world intelligent systems, necessitating advanced mathematical frameworks for effective modeling and analysis. Fermatean fuzzy sets (FFSs), as a recent extension of classical fuzzy theory, provide enhanced flexibility for representing complex uncertainty. In this paper, we propose a unified parametric divergence operator for FFSs, which comprehensively captures the interplay among membership, non-membership, and hesitation degrees. The proposed operator is rigorously analyzed with respect to key mathematical properties, including non-negativity, non-degeneracy, and symmetry. Notably, several well-known divergence operators, such as Jensen-Shannon divergence, Hellinger distance, and -divergence, are shown to be special cases within our unified framework. Extensive experiments on pattern classification, hierarchical clustering, and multiattribute decision-making tasks demonstrate the competitive performance and stability of the proposed operator. These results confirm both the theoretical significance and practical value of our method for advanced fuzzy information processing in machine learning and intelligent decision-making.