An innovative Fermatean fuzzy distance metric with its application in classification and bidirectional approximate reasoning

The concept of Fermatean fuzzy set offers a more comprehensive perspective on Pythagorean fuzzy sets, with greater potential for practical applications due to its broader scope compared to Pythagorean fuzzy sets. In contrast, distance measures serve as effective information measures in decision-making using deep learning methods. This research paper introduces a novel distance measure within the Fermatean fuzzy framework. This new distance measure exhibit enhanced performance indices compared to existing distance measures discussed in previous literature. The newly proposed distance operators based on Fermatean fuzzy sets offer a fresh approach by considering all three membership functions of Fermatean fuzzy sets, which differs from existing practices. This distance operator also incorporate all membership functions to prevent errors caused by exclusions, which have been observed in other distance operators. Several theorems are provided to validate the alignment of this new Fermatean fuzzy distance technique with properties expected of distance operators. The comparative analysis of the newly introduced Fermatean fuzzy distance measure with similar existing distance techniques is conducted through numerical comparison and also in relation to their applications, aiming to highlight the superiority of the proposed Pythagorean fuzzy distance techniques over the existing ones. The application of the proposed Fermatean fuzzy distance measure is demonstrated in scenarios involving pattern classification and bidirectional approximate reasoning.