Discrete similarity measures on Pythagorean fuzzy sets and its applications to medical diagnosis and clustering problems

Abstract

Pythagorean fuzzy sets are an extension of intuitionistic fuzzy sets and are more efficient from an application perspective. Though the Pythagorean fuzzy sets are more informative, not much work on similarity measures is available in the literature. Furthermore, existing similarity measures are not efficient. Also, the containment property in Pythagorean fuzzy units is not correctly defined or ineffective. As a result, the existing similarity measures do not reflect appropriate information between the Pythagorean fuzzy sets. The scalar function of the Yager is mainly used for verifying the validity of similarity measures. Most of the existing similarity measures do not conform to the Yager scalar function. Hence, the existing similarity measures exhibit some discrepancies. Furthermore, the existing similarity measures are inconsistent in determining the similarity in intuitionsitic and Pythagorean fuzzy sets. In some real-world modeling issues, past, present, and cross-time information are essential. However, such information is missing in the existing similarity measures. Therefore, in this paper, two new measures of similarity are being developed based on the deviation of the parameters: membership degree, nonmembership degree, strength of commitment, direction of commitment, and cross-time evaluation factors. Under this construction, the proposed similarity measures effectively measure the similarity between the Pythagorean fuzzy sets. Furthermore, the newly defined containment property is also reflected in the proposed similarity measures, which were a limitation in most cases. Moreover, Yager's scalar function is also reflected by the proposed similarity measures. The complement of given information is also essential in some real-world problems. However, such information is incomplete in the theory of Pythagorean fuzzy sets. Hence, the complement of the Pythagorean fuzzy set is being redefined, and a few related results on similarity measures are proposed. Finally, the proposed similarity measures are tested for applicability to medical diagnosis and clustering problems through some hypothetical case studies.