Modified Zhang and Xu's distance measure for Pythagorean fuzzy sets and its application to pattern recognition problems.

The concept of distance between Pythagorean fuzzy sets (PFSs) has been proven to be relevant in the applications of PFSs as seen in the literature. The main purpose of this paper is to show that Zhang and Xu's distance measure between PFSs fails the conditions of distance measure; hence, it is not an appropriate distance measure for PFSs. Some numerical examples are used to validate this stance. In order to remedy this shortcoming, Zhang and Xu's distance measure for PFSs is normalised/modified to cater for the limitation by employing the technique used to normalise both Hamming and Euclidean distances between intuitionistic fuzzy sets by Szmidt and Kacprzyk. The modified Zhang and Xu's distance measure for PFSs satisfies the conditions of the axiomatic definition of distance measure for PFSs; hence, it is an appropriate/reliable distance measure for PFSs. Finally, the modified Zhang and Xu's distance measure for PFSs is applied to pattern recognition problems of classification of building materials and mineral fields. [ABSTRACT FROM AUTHOR]