Abstract
The framework of the T-spherical fuzzy set is a recent development in fuzzy set theory that can describe imprecise events using four types of membership grades with no restrictions. The purpose of this manuscript is to point out the limitations of the existing intuitionistic fuzzy Einstein averaging and geometric operators and to develop some improved Einstein aggregation operators. To do so, first some new operational laws were developed for T-spherical fuzzy sets and their properties were investigated. Based on these new operations, two types of Einstein aggregation operators are proposed namely the Einstein interactive averaging aggregation operators and the Einstein interactive geometric aggregation operators. The properties of the newly developed aggregation operators were then investigated and verified. The T-spherical fuzzy aggregation operators were then applied to a multi-attribute decision making (MADM) problem related to the degree of pollution of five major cities in China. Actual datasets sourced from the UCI Machine Learning Repository were used for this purpose. A detailed study was done to determine the most and least polluted city for different perceptions for different situations. Several compliance tests were then outlined to test and verify the accuracy of the results obtained via our proposed decision-making algorithm. It was proved that the results obtained via our proposed decision-making algorithm was fully compliant with all the tests that were outlined, thereby confirming the accuracy of the results obtained via our proposed method.
Highlights
Zadeh [1] first introduced a formal tool to deal with the uncertainties and imprecision that occurs in real-life situations and called this as a fuzzy set (FS)
Yager [4,5] introduced a concept of a Pythagorean fuzzy set (PyFS) in which he relaxed this limitation by defining the sum of the squares of the membership function and non-membership function must lie within the interval [0, 1]
The T-spherical fuzzy aggregation operators were applied to a multi-a ribute decision making (MADM) problem related to the degree of pollution of five major cities in China
Summary
Zadeh [1] first introduced a formal tool to deal with the uncertainties and imprecision that occurs in real-life situations and called this as a fuzzy set (FS). Yager [4,5] introduced a concept of a Pythagorean fuzzy set (PyFS) in which he relaxed this limitation by defining the sum of the squares of the membership function and non-membership function must lie within the interval [0, 1] This presents the decision makers with wider options when modelling a situation using PyFS, yet there are still restrictions as the decision makers are only free to assign values that fit a certain condition. The PFS model has a similar restriction to the IFS model, in which the sum of the membership, abstinence, and non-membership functions must lie within the interval of [0, 1] To further overcome this issue, Mahmood et al [9] introduced the concept of spherical fuzzy sets (SFSs) in which they relaxed this condition so that the sum of the squares of these three membership values must lie within the interval of [0, 1].
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