Decision-theoretic rough sets under Pythagorean fuzzy information

Decision-theoretic rough sets (DTRSs), which provide a classical model of three-way decisions (3WDs), play an important role in risk decision-making problems. The risk is associated with the loss function of DTRSs, which is evaluated by the decision makers. As a new extension of fuzzy sets, Pythagorean fuzzy sets can handle uncertain information more flexibly than intuitionistic fuzzy sets in the process of decision making and it gives a new measure for the determination of loss functions of DTRSs. More specifically, we take into account the loss functions of DTRSs with Pythagorean fuzzy numbers and propose a Pythagorean fuzzy decision-theoretic rough set (PFDTRS) model. Some properties of the expected losses are carefully investigated. Then we further design three approaches for deriving 3WDs with the PFDTRS model. The group decision making (GDM) based on the PFDTRS model is also discussed. It provides a novel interpretation for the determination of loss functions. With the aid of the Pythagorean fuzz weighted averaging operator, we aggregate the loss functions, as suggested by the all experts, which support a coherent way of designing information granules in the presence of numerics. An algorithm for 3WDs in GDM based on the PFDTRS model is designed. Then, an example is presented to elaborate on 3WDs with the PFDTRS model.