Journal of Prime Research in Mathematics

# Similarity measures for picture hesitant fuzzy sets and their applications in pattern recognition

Zeeshan Ahmad
Department of Mathematics & Statistics, International Islamic University Islamabad, Pakistan.
Tahir Mahmood
Department of Mathematics & Statistics, International Islamic University Islamabad, Pakistan.
Muhammad Saad
Department of Mathematics & Statistics, International Islamic University Islamabad, Pakistan.
Naeem Jan
Department of Mathematics & Statistics, International Islamic University Islamabad, Pakistan.
Kifayat Ullah
Department of Mathematics & Statistics, International Islamic University Islamabad, Pakistan.

$$^{5}$$Corresponding Author: : kifayat.phdma72@iiu.edu.pk

### Abstract

The concept of picture hesitant fuzzy set (PHFS) is a generalization of picture fuzzy set (PFS) and hesitant fuzzy set (HFS). Such kind of idea can be very helpful in problems where opinions are of more than two types i.e. yes, no, abstinence and refusal. The goal of this manuscript is to introduce the concept of similarity measures for PHFSs as a generalization of the similarity measures for PFS. We studied the basic concepts of  PFSs, HFSs and PHFSs. We proposed some similarity measures for PHFSs such as cosine similarity measure, set-theoretic similarity measure and grey similarity measure for PHFSs. Some weighted similarity measures are also proposed where weight of the attributes is considered. Then these similarity measures for PHFSs are applied to a building material recognition problem. Finally, a comparative study of similarity measures of PHFSs is established with similarity measures of PFSs, HFSs, IFSs and IHFSs and the advantages of new work are studied.