## 2 Preliminaries
### Three-Way Decision Based on Dtrs  
DTRS is a well-known model for three-way decision-making based on Bayesian decision theory that minimizes the risk of multiple decisions [38]. The resulting approach is similar to hypothesis testing in statistics. In hypothesis testing, a hypothesis is accepted if there is sufficient evidence supporting it, rejected if there is sufficient evidence refuting it, and needs further evaluation if there is insufficient evidence supporting or refuting it. This interpretation justifies three-way decision-making based on the risk or cost of various decisions and requires an understanding of the cost of acquiring and applying evidence. The 3 WDM with DTRS theory [38] is succinctly explained here. It begins with a set of states \((\Omega,\ -\Omega)\) designating, respectively, that components are in \(\Omega\) and not in \(\Omega\). For both of these states, a series of actions is taken as \(\mathcal{A}c=\{a_{\text{P}},\ a_{\text{B}},\ a_{\text{N}}\}\), where \(a_{\text{P}},\ a_{\text{B}}\) and \(\ a_{\text{N}}\), respectively, represent the classification of an object \(e\)'s acceptance \((\mathrm{e}\in\mathcal{P}os(\Omega))\), deferment \((\mathrm{e}\in\mathcal{B}nd(\Omega))\), and rejection \((\mathrm{e}\in\mathcal{N}eg(\Omega))\) decision. The positive region \(\mathcal{P}os(\Omega)\), boundary region \(\mathcal{B}nd(\Omega)\), and negative region \(\mathcal{N}eg(\Omega)\) are three disjoint regions. Moreover, as indicated in Table 2, a matrix \(\mathcal{M}=\{\xi_{\sigma\uparrow}\}_{3\times 2}\ (\sigma=\text{P},\ \text{B},\ \text{N},\ and\ \tau=\text{P},\ \text{N})\) provides the cost parameters. The costs associated with the actions \(a_{\text{P}},\ a_{\text{B}}\), and \(a_{\text{N}}\) when an element goes to \(\Omega\) are \(\xi_{\mathcal{P}\mathcal{P}}\), \(\xi_{\mathcal{B}\mathcal{P}}\), and \(\xi_{\mathcal{N}\mathcal{P}}\). However, the expenses for the corresponding three actions are denoted by \(\xi_{\mathcal{P}\mathcal{N}}\), \(\xi_{\mathcal{B}\mathcal{N}}\), and \(\xi_{\mathcal{N}\mathcal{N}}\) when an item does not belong to \(\Omega\). Since \(\mathcal{P}r(\Omega|[e])+\mathcal{P}r(\
eg\Omega|[e])=1\), by using the Bayesian risk decision theory [38], for the element \(e\), the classification losses \(\mathbb{R}(a_{\sigma}|[e])\) associated with the three actions are expressed as follows:  
\[\mathbb{R}(a_{\text{P}}|[e]) =\xi_{\mathcal{P}\mathcal{P}}\mathcal{P}r(\Omega|[e])+\xi_{ \mathcal{P}\mathcal{N}}\mathcal{P}r(\
eg\Omega|[e])\] \[\mathbb{R}(a_{\text{B}}|[e]) =\xi_{\mathcal{B}\mathcal{P}}\mathcal{P}r(\Omega|[e])+\xi_{ \mathcal{B}\mathcal{N}}\mathcal{P}r(\
eg\Omega|[e])\]  
For the minimum-loss decisions, the DTRS theory presents the following decision rules:  
* [(1)] _If_ \(\mathbb{R}(a_{\text{P}}|[e])\leq\mathbb{R}(a_{\text{B}}|[e])\) _and_ \(\mathbb{R}(a_{\text{P}}|[e])\leq\mathbb{R}(a_{\text{N}}|[e])\)_, then_ \(\mathrm{e}\ \in\ \mathcal{P}os(\Omega)\)_._ \begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{Actions}\backslash\mathbf{States}\) & \(\mathbf{\Omega}\) & \(\
eg\mathbf{\Omega}\) \\ \hline \(\boldsymbol{a_{\text{P}}}\) & \(\xi_{\mathcal{P}\mathcal{P}}\) & \(\xi_{\mathcal{P}\mathcal{N}}\) \\ \hline \(\boldsymbol{a_{\text{B}}}\) & \(\xi_{\mathcal{P}\mathcal{P}}\) & \(\xi_{\mathcal{B}\mathcal{N}}\) \\ \hline \(\boldsymbol{a_{\text{N}}}\) & \(\xi_{\mathcal{N}\mathcal{P}}\) & \(\xi_{\mathcal{N}\mathcal{N}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Cost parameter matrix. 2. If \(\mathbb{R}(a_{\rm B}|[{\rm e}])\leq\mathbb{R}(a_{\rm P}|[{\rm e}])\) and \(\mathbb{R}(a_{\rm B}|[{\rm e}])\leq\mathbb{R}(a_{\rm N}|[{\rm e}])\), then \({\rm e}\ \in\ \mathcal{B}nd(\Omega)\). 3. If \(\mathbb{R}(a_{\rm N}|[{\rm e}])\leq\mathbb{R}(a_{\rm P}|[{\rm e}])\) and \(\mathbb{R}(a_{\rm N}|[{\rm e}])\leq\mathbb{R}(a_{\rm B}|[{\rm e}])\), then \({\rm e}\ \in\ \mathcal{N}eg(\Omega)\). According to the Bayesian decision process [38], always choose the action plan with the lowest decision-making risk as the first choice. **3.