A Pythagorean fuzzy set (PFS) is a powerful tool for depicting fuzziness and uncertainty. This model is more flexible and practical as compared to an intuitionistic fuzzy model. This paper proposes a new graph, called Pythagorean fuzzy graph (PFG). We investigate some properties of our proposed graphs. We determine the degree and total degree of a vertex of PFGs. Furthermore, we present the concept of Pythagorean fuzzy preference relations (PFPRs). In particular, we solve decision-making problems, including evaluation of hospitals, partner selection in supply chain management, and electronic learning main factors evaluation by using PFGs.

This paper presents a new method to estimate the unknown values in incomplete interval-valued fuzzy preference relations (IVFPRs). The method is based on the min-consistency and is used to develop the algorithm for group decision making (GDM) dealing with incomplete IVFPRs.

The capability of complex fuzzy sets plays a valuable role in resolving many real-life problems. In this paper, we present the compositions of complex fuzzy relations by using the idea of implication operators and max-product compositions of complex fuzzy relations and illustrate these compositions with concrete examples. The converse of these newly invented triangular compositions in terms of compositions of the converse relations is also defined. We also study the interactions with the union and intersection. The main goal of this article is to present a new technique to enhance medical diagnostic models that can assist in improving the features of healthcare systems. We utilize these compositions to diagnose diseases in patients on the basis of the intensity of symptoms.

The interrelationship of notions is presented in fuzzy transitivity and monotonicity of fuzzy similarity measures. It is observed that the axiom of fuzzy transitivity may replace that of monotonicity in the definition of fuzzy similarity measures.