Pythagorean Membership Grades, Complex Numbers, and Decision Making

Abstract

We describe the idea of Pythagorean membership grades and the related idea of Pythagorean fuzzy subsets. We focus on the negation and its relationship to the Pythagorean theorem. We look at the basic set operations for the case of Pythagorean fuzzy subsets. A relationship is shown between Pythagorean membership grades and complex numbers. We specifically show that Pythagorean membership grades are a subclass of complex numbers called Π‐i numbers. We investigate operations that are closed under Π‐i numbers. We consider the problem of multicriteria decision making with satisfactions expressed as Pythagorean membership grades, Π‐i numbers. We look at the use of the geometric mean and ordered weighted geometric operator for aggregating criteria satisfaction.

New Operations for Interval-valued Pythagorean Fuzzy Set

“…Consequently, the evaluation and decision of the given emergency plans with simulations is considered essential for the disaster management of mine accidents [42]. 4 , A 5 } to be considered for an explosion accident in the coal mine. The expert chooses the decision parameters set C = {C 1 ,C 2 ,C 3 ,C 4 } to be the noxious gas concentration C 1 (denoted as gas), reducing casualty of current events C 2 (denoted as casualty), the smoke and the dust level C 3 (denoted as smoke), the feasibility of rescue operations C 4 (denoted as feasibility) and repairing facility damages caused by the emergency C 5 (denoted as facility).…”

“…Yager [4] gave an example to illustrate this situation: an expert gives his support for membership of an alternative is √ 3 2 and his against membership is 1 2 . Due to the sum of two values is bigger than 1, they are not available for IFS, but they are feasible for PFS since (…”

A Novel Approach to Decision-Making with Pythagorean Fuzzy Information

“…An IFS is able to describe the things from three aspects of superiority, inferiority and hesitation, which are usually represented by the intuitionistic fuzzy numbers (IFNs) [3]. To capture more useful information under imprecise and uncertain circumstances, Yager [4][5][6] recently proposed the concept of Pythagorean fuzzy set (PFS) as a new evaluation format, which is characterized by the membership and the non-membership degree satisfying the condition that their square sum is not greater than 1. Zhang and Xu [7] provided the detailed mathematical expression for PFS and put forward the concept of Pythagorean fuzzy number (PFN).…”

Interval-valued Pythagorean fuzzy Einstein hybrid weighted averaging aggregation operator and their application to group decision making

“…Like intuitionistic fuzzy aggregation operators, Pythagorean fuzzy aggregation operators are also become an interesting and important area for research, after the advent of Pythagorean fuzzy set theory. Several researchers in [19][20][21][22][23][24][25][26][27][28] introduced many aggregation operators for decision using Pythagorean fuzzy information.…”