Method for three-way decisions using ideal TOPSIS solutions at Pythagorean fuzzy information

Pythagorean fuzzy sets (PFSs) as a new generalization of intuitionistic fuzzy sets (IFSs) can effectively handle uncertain information more flexibly in the process of decision making. As a natural extension of three-way decisions with decision-theoretic rough sets (DTRSs), this paper proposes a new model of three-way decisions and develops the corresponding decision-making procedure based on Pythagorean fuzzy information systems. With respect to the results reported in most of the existing papers, we consider a general situation that the information system does not have the class label. In this case, we can encounter two challenges and need to reinterpret the loss function and the conditional probability. Considering the properties of PFSs, we firstly introduce the Pythagorean fuzzy number (PFN) into DTRSs, which can provide a new interpretation for the loss function. Then, we construct a new model of Pythagorean fuzzy decision-theoretic rough sets (PFDTRSs) based on the Bayesian decision procedure. With respect to the conditional probability, we effectively utilize the technique for order preference by similarity to ideal solution (TOPSIS) method to estimate it. Furthermore, we design a decision-making procedure of three-way decisions-based ideal solutions in the Pythagorean fuzzy information system. Our proposed method not only takes the decision risk into consideration, but also tells us how to choose the action for each project and gives its corresponding semantic explanation, which can replenish the decision results of TOPSIS. Finally, we expound the application of three-way decisions by an example of the research and development (R&D) project selection and validate our method via the comparison analysis. (C) 2018 Elsevier Inc. All rights reserved.