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 \(\beta \)-covering and PF \(\beta \)-neighborhood, we construct a Pythagorean fuzzy \(\beta \)-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.

具有决策理论粗糙集（DTRS）的三向决策（3WD）是解决风险决策问题的一种新方法。在DTRS中，这对于确定损失函数至关重要。毕达哥拉斯模糊（PF）集比直觉模糊集更有效地处理不确定性和不准确性。尽管研究人员已经将毕达哥拉斯模糊数（PFN）引入损失函数中，但对3WD和PF覆盖率的组合进行的研究仍是空白。有鉴于此，我们基于PF覆盖技术开发了带DTRS的3WD。首先，利用PF \（\ beta \）- covering和PF \（\ beta \）- Neighborhood的概念，我们构造了勾股模糊\（\ beta \）贝叶斯决策程序覆盖决策理论粗糙集（PFCDTRS）模型。然后，研究了与模型相关的预期损失的一些有趣特性。其次，基于PFN的隶属度和非隶属度，建立了解决PFN形式表示的预期损失的四种方法，并推导了相应的3WD。最后，我们开发了相应的算法来推导带有PFCDTRS的3WD，然后提供了一个实例来验证带有PFCDTRS的3WD的可行性和可靠性。通过比较所提出的方法和现有方法，我们得出结论，所提出的方法3和4优于现有方法。