A critical analysis and proposed method for fermatean fuzzy sets

Since K. T. Atanassov introduced the intuitionistic fuzzy set (IFS), numerous extensions and and methodologies have been developed and applied in practical contexts. Among these, the Fermatean fuzzy set (FFS) has been recognized as particularly adaptable for handling the uncertainty inherent in multi-criteria decision-making (MCDM) problems due to its expanded domain space. Although various aggregation operators (AOs) have been proposed for FFSs, some of them yield unreasonable or counter-intuitive results in certain situations. In this paper, a detailed review and critical analysis are conducted to reveal the shortcomings of existing approaches. To address these issues, new operational rules (ORs) for FFSs are proposed, and their desirable mathematical properties are formally established. The advantages of the proposed method are demonstrated through several numerical examples that highlight its improved behavior compared with existing approaches. Furthermore, to validate the reliability and practical relevance of the proposed method, it is applied to selected MCDM and pattern recognition problems.