The notion of fuzziness as defined in this paper relates to situations in which the source of imprecision is not a random variable or a stochastic process, but rather a class or classes which do not possess sharply defined boundaries, e.g., the “class of bald men,” or the “class of numbers which are much greater than 10,” or the “class of adaptive systems,” etc. A basic concept which makes it possible to treat fuzziness in a quantitative manner is that of a fuzzy set, that is, a class in which there may be grades of membership intermediate between full membership and non-membership. Thus, a fuzzy set is characterized by a membership function which assigns to each object its grade of membership (a number lying between 0 and 1) in the fuzzy set. After a review of some of the relevant properties of fuzzy sets, the notions of a fuzzy system and a fuzzy class of systems are introduced and briefly analyzed. The paper closes with a section dealing with optimization under fuzzy constraints in which an approach to...

Fuzzy sets and systems

The q-rung orthopair fuzzy sets (q-ROFs) are an important way to express uncertain information, and they are superior to the intuitionistic fuzzy sets and the Pythagorean fuzzy sets. Their eminent characteristic is that the sum of the qth power of the membership degree and the qth power of the degrees of non-membership is equal to or less than 1, so the space of uncertain information they can describe is broader. Under these environments, we propose the q-rung orthopair fuzzy weighted averaging operator and the q-rung orthopair fuzzy weighted geometric operator to deal with the decision information, and their some properties are well proved. Further, based on these operators, we presented two new methods to deal with the multi-attribute decision making problems under the fuzzy environment. Finally, we used some practical examples to illustrate the validity and superiority of the proposed method by comparing with other existing methods.

Some q‐Rung Orthopair Fuzzy Aggregation Operators and their Applications to Multiple‐Attribute Decision Making

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.

A New Generalized Pythagorean Fuzzy Information Aggregation Using Einstein Operations and Its Application to Decision Making

A hybrid PSO-GA algorithm for constrained optimization problems

Gradient-based optimizer: A new metaheuristic optimization algorithm

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.

Generalized Pythagorean Fuzzy Geometric Aggregation Operators Using Einstein t-Norm and t-Conorm for Multicriteria Decision-Making Process

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.

A Novel Correlation Coefficients between Pythagorean Fuzzy Sets and Its Applications to Decision-Making Processes

The objective of the present work is divided into two folds. Firstly, an interval-valued Pythagorean fuzzy set (IVPFS) has been introduced along with their two aggregation operators, namely, interval-valued Pythagorean fuzzy weighted average and weighted geometric operators for different IVPFS. Secondly, an improved accuracy function under IVPFS environment has been developed by taking the account of the unknown hesitation degree. The proposed function has been applied to decision making problems to show the validity, practicality and effectiveness of the new approach. A systematic comparison between the existing work and the proposed work has also been given.

Harish Garg

Papers

A New Generalized Pythagorean Fuzzy Information Aggregation Using Einstein Operations and Its Application to Decision Making

A hybrid PSO-GA algorithm for constrained optimization problems

Generalized Pythagorean Fuzzy Geometric Aggregation Operators Using Einstein t-Norm and t-Conorm for Multicriteria Decision-Making Process

A Novel Correlation Coefficients between Pythagorean Fuzzy Sets and Its Applications to Decision-Making Processes

A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem