Pythagorean Membership Grades, Complex Numbers, and Decision Making

TLDR

The authors aim to introduce Pythagorean membership grades, examine their negation and set operations, and apply them to multicriteria decision making using geometric mean and ordered weighted geometric aggregation. They develop the theoretical framework by defining Pythagorean fuzzy subsets, establishing closure under Π‑i numbers, and employing these operators for criteria aggregation. The study demonstrates that Pythagorean membership grades correspond to complex numbers, specifically forming a subclass known as Π‑i numbers.

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.