New distance measure for Fermatean fuzzy sets and its application

As a new extended form of intuitionistic fuzzy sets, Fermatean fuzzy sets are powerful tools for describing vagueness and uncertainty in complex problems. In the method of handling Fermatean fuzzy information, the distance measure is an essential tool to depict the difference between two Fermatean fuzzy sets. However, how to accurately measure the distance between two Fermatean fuzzy sets is still a problem to be solved. In this paper, we devise two novel distance measure methods for Fermatean fuzzy sets. One is the distance measure of Fermatean fuzzy sets based on the Hellinger distance, which is called the FFSH distance. The other is the distance measure of Fermatean fuzzy sets based on the triangular divergence, which is called the FFSTD distance. Then, we prove that the proposed distance measure methods satisfy the axiomatic requirements of the distance function. Afterward, numerical examples are given to reveal that the proposed distance measures are more effective and reasonable than the normalized Euclidean distance measure, which can overcome the counter-intuitive situation. Besides, we utilize the proposed distance measure methods to address the problems of pattern recognition and medical diagnosis under Fermatean fuzzy environment and achieved excellent results. The experimental results illustrate that the proposed distance measure methods can efficiently handle the practical application under Fermatean fuzzy environment, and are more reliable than the normalized Euclidean distance measure.