A new order relation for Pythagorean fuzzy numbers and application to multi-attribute group decision making

This paper proposes a new order relation for Pythagorean fuzzy numbers (PFNs) and applies to multi-attribute group decision making (MAGDM). The main contributions are outlined as five aspects: (1) the concepts of relative distance and information reliability of PFN are proposed. Then, a new order relation is developed to compare PFNs. Moreover, the new order relation of PFNs is demonstrated to be an admissible order. (2) Knowledge measure of PFN is defined to describe the amount of information. The desirable properties of knowledge measure of PFN are studied concretely. (3) For MAGDM with PFNs, the comprehensive distance between individual Pythagorean fuzzy matrices and a mean one are defined. Then, the decision makers' weights are obtained by the comprehensive distances. Thus, a collective Pythagorean fuzzy matrix is derived by using the Pythagorean fuzzy weighted average operator. (4) To determine attribute weights, a multi-objective programming model is constructed by maximizing the overall knowledge measure of each alternative. This model is further transformed into a single-objective mathematical program to resolve. (5) According to the defined new order relation of PFNs, the ranking order of alternatives is generated by the comprehensive values of alternatives. Therefore, a new method is proposed to solve MAGDM with PFNs. Finally, an example of venture capital investment selection is provided to illustrate the effectiveness of the proposed method.