## 4 Demonstration
### Demonstration of Advantages  
\(\sigma\) is called the risk coefficient, which indicates the extent to which a user obtains information. The larger the \(\sigma\) value, the more sufficient the information obtained. The second demonstration experiment will demonstrate that the proposed approach is superior to the approaches based on multi-attribute two-way decisions when the \(\sigma\) value is small. According to Table 3 and (3), (4), and (5), and assuming \(Pr(C|[x])=0.5\), then the expected loss values of decision \(POS(C)\) (i.e., \(a_{P}\)), \(BND(C)\) (i.e., \(a_{B}\)), and \(NEG(C)\) (i.e., \(a_{N}\)) are, respectively:  
\[\text{A1}:\ R(a_{N}|[x])=0.5x_{ij}\] \[\text{A2}:\ R(a_{B}|[x])=0.5\sigma\] \[\text{A2}\text{-1}:\ (\text{A2 curve at }\sigma=0);\text{A2} \text{-2}:\ (\text{A2 curve at }\sigma=0.1);\] \[\text{A2}\text{-3}:\ (\text{A2 curve at }\sigma=0.2);\text{A2} \text{-4}:\ (\text{A2 curve at }\sigma=0.3);\] \[\text{A2}\text{-5}:\ (\text{A2 curve at }\sigma=0.4);\text{A2} \text{-6}:\ (\text{A2 curve at }\sigma=0.5);\] \[\text{A3}:\ R(a_{P}|[x])=0.5(1-x_{ij})\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Example** & **AM Machine Selection Approach** & **Generated Rank of Alternative AM Machines** \\ \hline \multirow{3}{*}{Example 1} & [PERSON] and [PERSON] [12] & \(A_{3}>A_{5}>A_{4}>A_{1}>A_{6}>A_{2}\) \\  & [PERSON] et al. [47] & \(A_{3}>A_{4}>A_{1}>A_{5}>A_{6}>A_{2}\) \\  & The proposed approach & \(NEG(C)=\{A_{2},\ A_{6}\}\), \(BND(C)=\{A_{1},\ A_{3}\}\), \(POS(C)=\{\ A_{4},\ A_{5}\}\) \\ \hline \multirow{3}{*}{Example 2} & [PERSON] and [PERSON] [14] & \(A_{5}>A_{1}>A_{2}>A_{4}>A_{3}>A_{6}\) \\  & [PERSON] et al. [22] & \(A_{5}>A_{1}>A_{2}>A_{4}>A_{3}>A_{6}\) \\  & [PERSON] and [PERSON] [10] & \(A_{6}>A_{4}>A_{5}>A_{1}>A_{2}>A_{3}\) \\  & [PERSON] and [PERSON] [28] & \(A_{6}>A_{5}>A_{1}>A_{4}>A_{2}>A_{3}\) \\  & [PERSON] et al. [29] & \(A_{6}>A_{5}>A_{4}>A_{1}>A_{2}>A_{3}\) \\  & [PERSON] et al. [47] & \(A_{5}>A_{1}>A_{2}>A_{4}>A_{3}>A_{6}\) \\  & The proposed approach & \(NEG(C)=\{\varnothing\}\), \(BND(C)=\{A_{3}\}\), \(POS(C)=\{A_{1},\ A_{2},\ A_{4},\ A_{5},\ A_{6}\}\) \\ \hline \multirow{3}{*}{Example 3} & [PERSON] and [PERSON] [28] & \(A_{1}>A_{2}>A_{3}>A_{4}>A_{5}>A_{6}\) \\  & [PERSON] et al. [47] & \(A_{2}>A_{1}>A_{4}>A_{5}>A_{6}\) \\  & The proposed approach & \(NEG(C)=\{A_{6}\}\), \(BND(C)=\{A_{3}\}\), \(POS(C)=\{A_{1},\ A_{2},\ A_{4},\ A_{5}\}\) \\ \hline \multirow{3}{*}{Example 4} & [PERSON] et al. [42] & \(A_{3}>A_{5}>A_{2}>A_{4}>A_{1}>A_{6}\) \\  & [PERSON] et al. [47] & \(A_{3}>A_{5}>A_{2}>A_{4}>A_{1}>A_{6}\) \\ \cline{1-1}  & The proposed approach & \(NEG(C)=\{A_{6}\}\), \(BND(C)=\{A_{1}\}\), \(POS(C)=\{A_{2},\ A_{3},\ A_{4},\ A_{5}\}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Results of the comparison experiment. The dominant values of decision \(a_{B}\) under different \(\sigma\) are, respectively:  
\[ \begin{array}{ll}\text{B1}:&R(a_{N}|[x])-R(a_{B}|[x])\text{ when }x_{ij}\in[0,\;0.5]\\ &R(a_{P}|[x])-R(a_{B}|[x])\text{ when }x_{ij}\in[0.5,\;1];\;(\sigma=0)\\ \text{B2}:&R(a_{N}|[x])-R(a_{B}|[x])\text{ when }x_{ij}\in[0,\;0.5]\\ &R(a_{P}|[x])-R(a_{B}|[x])\text{ when }x_{ij}\in[0.5,\;1];\;(\sigma=0.1)\\ \text{B3}:&R(a_{N}|[x])-R(a_{B}|[x])\text{ when }x_{ij}\in[0,\;0.5]\\ &R(a_{P}|[x])-R(a_{B}|[x])\text{ when }x_{ij}\in[0.5,\;1];\;(\sigma=0.2)\\ \text{B4}:&R(a_{N}|[x])-R(a_{B}|[x])\text{ when }x_{ij}\in[0,\;0.5]\\ &R(a_{P}|[x])-R(a_{B}|[x])\text{ when }x_{ij}\in[0.5,\;1];\;(\sigma=0.3)\\ \text{B5}:&R(a_{N}|[x])-R(a_{B}|[x])\text{ when }x_{ij}\in[0,\;0.5]\\ &R(a_{P}|[x])-R(a_{B}|[x])\text{ when }x_{ij}\in[0.5,\;1];\;(\sigma=0.4)\\ \text{B6}:&R(a_{N}|[x])-R(a_{B}|[x])\text{ when }x_{ij}\in[0,\;0.5]\\ &R(a_{P}|[x])-R(a_{B}|[x])\text{ when }x_{ij}\in[0.5,\;1];\;(\sigma=0.5)\\ \end{array} \]  
The experimental results are shown in Figures 8 and 9.