Muhammad Sajjad Ali Khan

Interval-valued Pythagorean fuzzy GRA method for multiple-attribute decision making with incomplete weight information

In this paper, the concept of multiple‐attribute group decision‐making (MAGDM) problems with interval‐valued Pythagorean fuzzy information is developed, in which the attribute values are interval‐valued Pythagorean fuzzy numbers and the information about the attribute weight is incomplete. Since the concept of interval‐valued Pythagorean fuzzy sets is the generalization of interval‐valued intuitionistic fuzzy set. Thus, due the this motivation in this paper, the concept of interval‐valued Pythagorean fuzzy Choquet integral average (IVPFCIA) operator is introduced by generalizing the concept of interval‐valued intuitionistic fuzzy Choquet integral average operator. To illustrate the developed operator, a numerical example is also investigated. Extended the concept of traditional GRA method, a new extension of GRA method based on interval‐valued Pythagorean fuzzy information is introduced. First, utilize IVPFCIA operator to aggregate all the interval‐valued Pythagorean fuzzy decision matrices. Then, an optimization model based on the basic ideal of traditional grey relational analysis (GRA) method is established, to get the weight vector of the attributes. Based on the traditional GRA method, calculation steps for solving interval‐valued Pythagorean fuzzy MAGDM problems with incompletely known weight information are given. The degree of grey relation between every alternative and positive‐ideal solution and negative‐ideal solution is calculated. To determine the ranking order of all alternatives, a relative relational degree is defined by calculating the degree of grey relation to both the positive‐ideal solution and negative ideal solution simultaneously. Finally, to illustrate the developed approach a numerical example is to demonstrate its practicality and effectiveness.

Some Interval-Valued Pythagorean Fuzzy Einstein Weighted Averaging Aggregation Operators and Their Application to Group Decision Making

Abstract In this paper, we introduce the notion of Einstein aggregation operators, such as the interval-valued Pythagorean fuzzy Einstein weighted averaging aggregation operator and the interval-valued Pythagorean fuzzy Einstein ordered weighted averaging aggregation operator. We also discuss some desirable properties, such as idempotency, boundedness, commutativity, and monotonicity. The main advantage of using the proposed operators is that these operators give a more complete view of the problem to the decision makers. These operators provide more accurate and precise results as compared the existing method. Finally, we apply these operators to deal with multiple-attribute group decision making under interval-valued Pythagorean fuzzy information. For this, we construct an algorithm for multiple-attribute group decision making. Lastly, we also construct a numerical example for multiple-attribute group decision making.

Gray Method for Multiple Attribute Decision Making with Incomplete Weight Information under the Pythagorean Fuzzy Setting

Abstract In this study, we developed an approach to investigate multiple attribute group decision-making (MAGDM) problems, in which the attribute values take the form of Pythagorean fuzzy numbers whose information about attribute weights is incompletely known. First, the Pythagorean fuzzy Choquet integral geometric operator is utilized to aggregate the given decision information to obtain the overall preference value of each alternative by experts. In order to obtain the weight vector of the criteria, an optimization model based on the basic ideal of the traditional gray relational analysis method is established, and the calculation steps for solving Pythagorean fuzzy MAGDM problems with incompletely known weight information are given. The degree of gray relation between every alternative and positive-ideal solution and negative-ideal solution is calculated. Then, a relative relational degree is defined to determine the ranking order of all alternatives by calculating the degree of gray relation to both the positive-ideal solution and negative-ideal solution simultaneously. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.