# A Three-Way Acceleration Approach for Interval-Valued Multi-Attribute Decision-Making Problems
## 5 Experiment and Analysis
### A Numerical Example
When \(Pr(\tilde{C}\mid[\chi_{i}])\in[0.3.0.33]\), all objects are included in \(NEG(\tilde{C})\); when \(Pr(\tilde{C}\mid[\chi_{i}])\in[0.34,\,0.35]\), all objects are included in \(NEG(\tilde{C})\) except \(\chi_{5}\); when \(Pr(\tilde{C}\mid[\chi_{i}])\) is 0.38, \(\chi_{5}\) and \(\chi_{6}\in POS(\tilde{C})\), \(\chi_{2}\) and \(\chi_{9}\in BND(\tilde{C})\), the rest of the objects are in \(NEG(\tilde{C})\); when \(Pr(\tilde{C}\mid[\chi_{i}])\) is 0.39, \(\chi_{4}\), \(\chi_{5}\) and \(\chi_{6}\in POS(\tilde{C})\), \(\chi_{8}\) and \(\chi_{9}\in BND(\tilde{C})\), the rest of the objects are in \(NEG(\tilde{C})\); when \(Pr(\tilde{C}\mid[\chi_{i}])\) is 0.40, \(\chi_{4}\), \(\chi_{5}\), \(\chi_{6}\) and \(\chi_{9}\in POS(\tilde{C})\), \(\chi_{7}\), \(\chi_{8}\) and \(\chi_{9}\in BND(\tilde{C})\), the rest of the objects are in \(NEG(\tilde{C})\); when \(Pr(\tilde{C}\mid[\chi_{i}])\) is 0.41, \(\chi_{1}\), \(\chi_{2}\) and \(\chi_{3}\in NEG(\tilde{C})\), \(\chi_{7}\) and \(\chi_{10}\in BND(\tilde{C})\), the rest of the objects are in \(POS(\tilde{C})\); when \(Pr(\tilde{C}\mid[\chi_{i}])\in[0.42,0.43]\), \(\chi_{1}\), \(\chi_{2}\) and \(\chi_{3}\in BND(\tilde{C})\), the rest of the objects are in \(POS(\tilde{C})\); when \(Pr(\tilde{C}\mid[\chi_{i}])\in[0.44,\,0.45]\), all objects are included in \(POS(\tilde{C})\). \begin{table}
\begin{tabular}{c c c c} \hline  & & \(\tilde{C}\) & \(\
eg\tilde{C}\) \\ \hline \(\chi_{1}\) & \(a_{p}\) & 0 & 0.0826 \\  & \(a_{B}\) & 0.558 & 0.0405 \\  & \(a_{N}\) & 0.1139 & 0 \\ \hline \(\chi_{2}\) & \(a_{p}\) & 0 & 0.0776 \\  & \(a_{B}\) & 0.0525 & 0.0380 \\  & \(a_{N}\) & 0.1071 & 0 \\ \hline \(\chi_{3}\) & \(a_{p}\) & 0 & 0.0854 \\  & \(a_{B}\) & 0.0574 & 0.0418 \\  & \(a_{N}\) & 0.1171 & 0 \\ \hline \(\chi_{4}\) & \(a_{p}\) & 0 & 0.0769 \\  & \(a_{B}\) & 0.0616 & 0.0377 \\  & \(a_{N}\) & 0.1257 & 0 \\ \hline \(\chi_{5}\) & \(a_{p}\) & 0 & 0.07941 \\  & \(a_{B}\) & 0.0664 & 0.0389 \\  & \(a_{N}\) & 0.1356 & 0 \\ \hline \(\chi_{6}\) & \(a_{p}\) & 0 & 0.0569 \\  & \(a_{B}\) & 0.0526 & 0.0279 \\  & \(a_{N}\) & 0.1074 & 0 \\ \hline \(\chi_{7}\) & \(a_{p}\) & 0 & 0.0866 \\  & \(a_{B}\) & 0.0616 & 0.0424 \\  & \(a_{N}\) & 0.1257 & 0 \\ \hline \(\chi_{8}\) & \(a_{p}\) & 0 & 0.0686 \\  & \(a_{B}\) & 0.0507 & 0.0336 \\  & \(a_{N}\) & 0.1035 & 0 \\ \hline \(\chi_{9}\) & \(a_{p}\) & 0 & 0.0768 \\  & \(a_{B}\) & 0.0592 & 0.0376 \\  & \(a_{N}\) & 0.1207 & 0 \\ \hline \(\chi_{10}\) & \(a_{p}\) & 0 & 0.0781 \\  & \(a_{B}\) & 0.0564 & 0.0383 \\  & \(a_{N}\) & 0.1152 & 0 \\ \hline \end{tabular}
\end{table} 
Table 5: The table of loss functions. The distribution of objects under different conditional probabilities is depicted in Figure 3. In Figure 3, we use three colors to represent the regions of objects under different conditional probabilities. When the conditional probabilities are different, the region of an object changes. For example, when the conditional probability is between 0.3 and 0.41, \(\chi_{1}\) is in the positive region; when the conditional probability is between 0.41 and 0.43, \(\chi_{1}\) is in the boundary region; when the conditional probability is between 0.43 and 0.45, \(\chi_{1}\) is in the negative region. We can also find from Figure 3 that all objects are in the positive region when the conditional probability is between 0.43 and 0.45. All objects are in the negative region when the conditional probability is between 0.3 and 0.34. According to the above division of all candidates, we can get the dominance relation \(POS(\hat{C})\succ BND(\hat{C})\succ NEG(\hat{C})\). Instead of the method in [22; 32], to solve the MADM problems efficiently, our approach is to get the final ranked result utilizing ranking the divided alternatives.