Multi-attribute group decision-making problem based on some induced Einstein aggregation operators under complex fuzzy environment

Abstract

The aim of the paper is to introduce some complex Einstein aggregation operators for aggregating the different complex Pythagorean fuzzy sets (CPFSs) by considering the dependency between the pairs of its membership degrees. In the existing studies of fuzzy and its extensions, the uncertainties present in the data are handled with the help of degrees of membership that are the subset of real numbers, which may also loss some valuable data and hence consequently affect the decision results. A modification to these, complex Pythagorean fuzzy set handles the uncertainties with the degree whose ranges are extended from real subset to the complex subset with unit disc and hence handle the two dimensional information in a single set. Thus motivated by this and this paper we present some novel Einstein aggregation operators, namely complex Pythagorean fuzzy Einstein weighted averaging (CPFEWA) operator, complex Pythagorean fuzzy Einstein ordered weighted averaging (CPFEOWA) operator, complex Pythagorean fuzzy Einstein hybrid averaging (CPFEHA) operator, induced complex Pythagorean fuzzy Einstein ordered weighted averaging (I-CPFEOWA) operator, and induced complex Pythagorean fuzzy Einstein hybrid averaging (I-CPFEHA) operator. Also develop some of their desirable properties. Furthermore, based on these operators a multi-attribute group decision making problems developed. An illustrative example related to the selection of the best alternative is considered to show the effectiveness, of the novel developed methods.