Three-way decisions with decision-theoretic rough sets based on Pythagorean fuzzy covering

Three-way decisions (3WDs) with decision-theoretic rough sets (DTRSs) are a new method for solving problems of risky decision. In DTRSs, it is crucial to the determination of the loss function. Pythagorean fuzzy (PF) sets are a more powerful mathematical tool than intuitionistic fuzzy sets for dealing with uncertainty and inaccuracy. Although the researchers have introduced Pythagorean fuzzy numbers (PFNs) into the loss function, study on the combination of 3WDs and PF covering is still blank. In view of this, we develop 3WDs with DTRS based on PF covering. Firstly, by using the concepts of PF β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-covering and PF β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-neighborhood, we construct a Pythagorean fuzzy β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-covering decision-theoretic rough set (PFCDTRS) model as per Bayesian decision procedure. Then, some of interesting properties of the expected loss related to the model are investigated. Secondly, based on the membership degree and non-membership degree of PFNs, four methods to address the expected loss expressed in the form of PFNs are established and the corresponding 3WDs are also derived. Finally, we develop a corresponding algorithm for deriving 3WDs with PFCDTRS, and then, an example is provided to validate the feasibility and reliability of 3WDs with PFCDTRS. Compared the proposed methods with the existing methods, we conclude that the proposed Methods 3 and 4 are superior to the existing methods.