Based on decision theory rough sets (DTRSs), three-way decisions (TWDs) provide a risk decision method for solving multi-attribute decision making (MADM) problems. The loss function matrix of DTRS is the basis of this method. In order to better solve the uncertainty and ambiguity of the decision problem, we introduce the q-rung orthopair fuzzy numbers (q-ROFNs) into the loss function. Firstly, we introduce concepts of q-rung orthopair fuzzy β-covering (q-ROF β-covering) and q-rung orthopair fuzzy β-neighborhood (q-ROF β-neighborhood). We combine covering-based q-rung orthopair fuzzy rough set (Cq-ROFRS) with the loss function matrix of DTRS in the q-rung orthopair fuzzy environment. Secondly, we propose a new model of q-ROF β-covering DTRSs (q-ROFCDTRSs) and elaborate its relevant properties. Then, by using membership and non-membership degrees of q-ROFNs, five methods for solving expected losses based on q-ROFNs are given and corresponding TWDs are also derived. On this basis, we present an algorithm based on q-ROFCDTRSs for MADM. Then, the feasibility of these five methods in solving the MADM problems is verified by an example. Finally, the sensitivity of each parameter and the stability and effectiveness of these five methods are compared and analyzed.

中文翻译：

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基于覆盖的q阶正交对模糊粗糙集模型的决策理论粗糙集三向决策

基于决策理论粗糙集（DTRS），三向决策（TWD）提供了一种解决多属性决策（MADM）问题的风险决策方法。DTRS的损失函数矩阵是该方法的基础。为了更好地解决决策问题的不确定性和模糊性，我们将q阶正交对模糊数（q-ROFNs）引入到损失函数中。首先，我们介绍q-阶邻对模糊β-覆盖（q-ROFβ-覆盖）和q-阶邻对模糊β-覆盖（q-ROFβ-邻域）的概念。我们将基于覆盖的q阶正交对模糊粗糙集（Cq-ROFRS）与DTRS的损失函数矩阵组合在q阶正交对模糊环境中。其次，我们提出了一种新的q-ROFβ-覆盖DTRSs（q-ROFCDTRSs）模型，并阐述了其相关性质。然后，通过利用q-ROFN的隶属度和非隶属度，给出了基于q-ROFN的5种解决预期损失的方法，并推导了相应的TWDs。在此基础上，我们提出了一种基于q-ROFCDTRSs的MADM算法。然后，通过实例验证了这五种方法在解决MADM问题中的可行性。最后，对每个参数的敏感性以及这五种方法的稳定性和有效性进行了比较和分析。