## 6 Numerical Example
### The Decision-Making Steps  
Step 1: The risk of each company is evaluated by four experts based on their comprehensive situation, and the results are presented in Tables 4-7.  
Figure 2: Algorithm of Three-Way Decision.  
\begin{table}
\begin{tabular}{|c|c|c|} \hline \multicolumn{2}{|c|}{\(A_{1}\)} & \multicolumn{2}{|c|}{\(A_{2}\)} \\ \hline  & \(\Omega\) & \(-\Omega\) & \(\Omega\) & \(-\Omega\) \\ \hline \(a_{\mathcal{P}}\) & \((\{0.2,0.4\},\{0.18,0.27\})\) & \((\{0.12,0.4\},\{0.56,0.27\})\) & \((\{0.1,0.2\},\{0.27,0.18\})\) & \((\{0.05,0.2\},\{0.09,0.18\})\) \\ \hline \(a_{\mathcal{B}}\) & \((\{0.7,0.14\},\{0.21,0.08\})\) & \((\{0.64,0.14\},\{0.62,0.08\})\) & \((\{0.12,0.25\},\{0.14,0.21\})\) & \((\{0.12,0.3\},\{0.45,0.21\})\) \\ \hline \(a_{\mathcal{N}}\) & \((\{0.3,0.5\},\{0.37,0.12\})\) & \((\{0.3,0.24\},\{0.37,0.21\})\) & \((\{0.5,0.1\},\{0.22,0.19\})\) & \((\{0.15,0.1\},\{0.12,0.19\})\) \\ \hline \(a_{\mathcal{3}}\) & \((\{0.5,0.1\},\{0.15,0.19\})\) & \((\{0.13,0.18\},\{0.23,0.19\})\) & \((\{0.14,0.29\},\{0.12,0.22\})\) & \((\{0.24,0.29\},\{0.21,0.22\})\) \\ \hline \(a_{\mathcal{B}}\) & \((\{0.25,0.35\},\{0.12,0.62\})\) & \((\{0.25,0.35\},\{0.12,0.43\})\) & \((\{0.74,0.2\},\{0.15,0.25\})\) & \((\{0.45,0.2\},\{0.43,0.25\})\) \\ \hline \(a_{\mathcal{N}}\) & \((\{0.36,0.54\},\{0.17,0.27\})\) & \((\{0.36,0.67\},\{0.17,0.21\})\) & \((\{0.122,0.61\},\{0.27,0.13\},\{0.12,0.45\},\{0.31,0.13\})\) \\ \hline \end{tabular}
\end{table}
Table 6: The risk given by \(\mathrm{E}_{3}\).  
\begin{table}
\begin{tabular}{|c|c|c|c|} \hline \multicolumn{2}{|c|}{\(A_{1}\)} & \multicolumn{2}{|c|}{\(A_{2}\)} \\ \hline  & \(\Omega\) & \(-\Omega\) & \(\Omega\) \\ \hline \(a_{\mathcal{P}}\) & \((\{0.1,0.3\},\{0.1,0.4\})\) & \((\{0.12,0.31\},\{0.23,0.14\})\) & \((\{0.03,\{0.3,0.4\}\})\) & \((\{0.08,0.34\},\{0.23,0.4\})\) \\ \hline \(a_{\mathcal{B}}\) & \((\{0.2,0.1\},\{0.1,0.2\})\) & \((\{0.012,0.3\},\{0.08,0.2\})\) & \((\{0.10,1.02\},\{0.01\})\) & \((\{0.52,0.11\},\{0.5,0.1\})\) \\ \hline \(a_{\mathcal{N}}\) & \((\{0.2,0.3\},\{0.2,0.2\})\) & \((\{0.5,0.1\},\{0.2,0.34\})\) & \((\{0.0,0.1\},\{0.1,0.2\})\) & \((\{0.43,0.16\},\{0.1,0.14\})\) \\ \hline \multicolumn{2}{|c|}{\(A_{3}\)} & \multicolumn{2}{|c|}{\(A_{4}\)} \\ \hline \(a_{\mathcal{P}}\) & \((\{0.03,0.3\},\{0.01,0.1\})\) & \((\{0.34,0.2\},\{0.15,0.13\})\) & \((\{0.02,0.4\},\{0.10,2\})\) & \((\{0.10,0.2\},\{0.23\},\{0.40,0.08\})\) \\ \hline \(a_{\mathcal{B}}\) & \((\{0.2,0.5\},\{0.1,0.2\})\) & \((\{0.32,0.5\},\{0.09,0.2\})\) & \((\{0.06,\{0.1,0.2\}\},\{0.1,0.2\})\) & \((\{0.3,0.54\},\{0.1,0.2\})\) \\ \hline \(a_{\mathcal{N}}\) & \((\{0.01,0.1\},\{0.1,0.3\})\) & \((\{0.01,0.13\},\{0.1,0.23\})\) & \((\{0.01,0.02\},\{0.01,0.3\})\) & \((\{0.01,0.02\},\{0.34,0.23\})\) \\ \hline \end{tabular}
\end{table}
Table 4: The risk given by \(\mathrm{E}_{1}\).  
\begin{table}
\begin{tabular}{|c|c|c|c|} \hline \multicolumn{2}{|c|}{\(A_{1}\)} & \multicolumn{2}{|c|}{\(A_{2}\)} \\ \hline  & \(\Omega\) & \(-\Omega\) & \(\Omega\) \\ \hline \(a_{\mathcal{P}}\) & \((\{0.1,0.3\},\{0.1,0.4\})\) & \((\{0.12,0.31\},\{0.23,0.14\})\) & \((\{0.03,\{0.3,0.4\}\})\) & \((\{0.08,0.34\},\{0.23,0.4\})\) \\ \hline \Step 5. Based on the 3 WD rules (4)-(6), the corresponding decision rule for each company can be determined. Currently, the investment judgment of each company largely depends on the expected value of its expected losses. The final decision result of each company is presented in Table 10.  
Step 6. \(A_{1}\), \(A_{2}\), and \(A_{3}\) are chosen as the suitable companies for investment for [PERSON] X.