The objective of this article is to extend and present an idea related to weighted aggregated operators from fuzzy to Pythagorean fuzzy sets (PFSs). The main feature of the PFS is to relax the condition that the sum of the degree of membership functions is less than one with the square sum of the degree of membership functions is less than one. Under these environments, aggregator operators, namely, Pythagorean fuzzy Einstein weighted averaging (PFEWA), Pythagorean fuzzy Einstein ordered weighted averaging (PFEOWA), generalized Pythagorean fuzzy Einstein weighted averaging (GPFEWA), and generalized Pythagorean fuzzy Einstein ordered weighted averaging (GPFEOWA), are proposed in this article. Some desirable properties corresponding to it have also been investigated. Furthermore, these operators are applied to decision-making problems in which experts provide their preferences in the Pythagorean fuzzy environment to show the validity, practicality, and effectiveness of the new approach. Finally, a systematic comparison between the existing work and the proposed work has been given.

The objective of this paper is to present some series of geometric-aggregated operators under Pythagorean fuzzy environment by relaxing the condition that the sum of the degree of membership functions is less than one with the square sum of the degree of membership functions is less than one. Under these environments, aggregator operators, namely, Pythagorean fuzzy Einstein weighted geometric, Pythagorean fuzzy Einstein ordered weighted geometric, generalized Pythagorean fuzzy Einstein weighted geometric, and generalized Pythagorean fuzzy Einstein ordered weighted geometric operators, are proposed in this paper. Some of its properties have also been investigated in details. Finally, an illustrative example for multicriteria decision-making problems of alternatives is taken to demonstrate the effectiveness of the approach.

In this article, a new linguistic Pythagorean fuzzy set (LPFS) is presented by combining the concepts of a Pythagorean fuzzy set and linguistic fuzzy set. LPFS is a better way to deal with the uncertain and imprecise information in decision making, which is characterized by linguistic membership and nonmembership degrees. Some of the basic operational laws, score, and accuracy functions are defined to compare the two or more linguistic Pythagorean fuzzy numbers and their properties are investigated in detail. Based on the norm operations, some series of the linguistic Pythagorean weighted averaging and geometric aggregation operators, named as linguistic Pythagorean fuzzy weighted average and geometric, ordered weighted average and geometric with linguistic Pythagorean fuzzy information are proposed. Furthermore, a multiattribute decision-making method is established based on these operators. Finally, an illustrative example is used to illustrate the applicability and validity of the proposed approach and compare the results with the existing methods to show the effectiveness of it.

Pythagorean fuzzy set (PFS) is one of the most successful in terms of representing comprehensively uncertain and vague information. Considering that the correlation coefficient plays an important role in statistics and engineering sciences, in this paper, after pointing out the weakness of the existing correlation coefficients between intuitionistic fuzzy sets (IFSs), we propose a novel correlation coefficient and weighted correlation coefficient formulation to measure the relationship between two PFSs. Pairs of membership, nonmembership, and hesitation degree as a vector representation with the two elements have been considered during formulation. Numerical examples of pattern recognition and medical diagnosis have been taken to demonstrate the efficiency of the proposed approach. Results computed by the proposed approach are compared with the existing indices.