Pythagorean hesitant fuzzy sets and their application to group decision making with incomplete weight information

Pythagorean fuzzy sets (PFSs), hesitant fuzzy sets (HFSs) and intuitionistic hesitant fuzzy sets (IHFSs) have attracted more and more scholars' attention due to their powerfulness in expressing vagueness and uncertainty. Intuitionistic hesitant fuzzy set satisfies the condition that the sum of its membership's degrees is less than or equal to one. However, there may be a situation where the decision maker may provide the degree of membership and nonmembership of a particular attribute in such a way that their sum is greater than 1. To overcome this shortcoming, in this paper we introduce the concept of Pythagorean hesitant fuzzy set (PHFS) which is the generalization of intuitionistic hesitant fuzzy set under the restriction that the square sum of its membership degrees is less than or equal to 1. We discuss some properties of PHFS. We define score and accuracy degree of the Pythagorean hesitant fuzzy numbers (PHFNs) for comparison in Pythagorean hesitant fuzzy numbers. Also in decision making with PHFSs, aggregation operators play a very important role since they can be used to synthesize multidimensional evaluation values represented as Pythagorean hesitant fuzzy valued into collective values. We develop distance measure between PHFNs. Under PHFS environments, we develop aggregation operators namely, Pythagorean hesitant fuzzy weighted averaging (PHFWA), Pythagorean hesitant fuzzy weighted geometric (PHFWG). We develop the maximizing deviation method for solving MADM problems, in which the evaluation information provided by the decision maker is expressed in Pythagorean hesitant fuzzy numbers and the information about attribute weights is incomplete. The main advantage of these operators is that it is to provide more accurate and precious results. Furthermore, we developed these operators are applied to decision-making problems in which experts provide their preferences in the Pythagorean hesitant fuzzy environment to show the validity, practicality and effectiveness of the new approach.