# A Three-Way Acceleration Approach for Interval-Valued Multi-Attribute Decision-Making Problems
## 5 Experiment and Analysis
### Sensitivity Analysis of \(\psi\) and \(Pr(\tilde{\mathcal{C}}|[\chi_{i}])\)  
Adjustment coefficient \(\psi\) is set to a fixed number in the above experiment. We assume that \(\psi\) is changeable and then discuss the impact of different values of \(\psi\) on the decision-making results. Different results with changeable \(\psi\) are listed as follows, in which the \begin{table}
\begin{tabular}{l l l} \hline \hline
**Different Weight Methods** & **Ranking Results** & **Optimal Object** \\ \hline w1 & \(\chi_{3}\succ\chi_{8}\succ\chi_{1}\succ\chi_{10}\succ\chi_{9}\succ\chi_{5} \succ\chi_{6}\succ\chi\succ\chi_{2}\succ\chi_{4}\) & \(\chi_{3}\) \\ w2 & \(\chi_{3}\succ\chi_{8}\succ\chi_{1}\succ\chi_{10}\succ\chi_{9}\succ\chi_{6} \succ\chi\succ\chi_{5}\succ\chi_{2}\succ\chi_{4}\) & \(\chi_{3}\) \\ w3 & \(\chi_{3}\succ\chi_{8}\succ\chi_{1}\succ\chi_{10}\succ\chi_{9}\succ\chi_{6} \succ\chi\succ\chi_{5}\succ\chi_{2}\succ\chi_{4}\) & \(\chi_{3}\) \\ w4 & \(\chi_{3}\succ\chi_{8}\succ\chi_{1}\succ\chi_{10}\succ\chi_{5}\succ\chi_{9} \succ\chi_{6}\succ\chi\succ\chi_{7}\succ\chi_{2}\succ\chi_{4}\) & \(\chi_{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The ranking results of different weight. Figure 4: Comparisons of attribute weight. Figure 5: Dominance relations of objects. conditional probability \(Pr(\tilde{C}|[\chi_{i}])\) is set to be 0.36 and \(\psi\) increases from 0.40 to 0.50 with step size 0.01. When \(\psi\in[0.40,0.43]\), all objects are included in \(BND(\tilde{C})\); when \(\psi\) is 0.44, \(\chi_{1}\), \(\chi_{2}\), and \(\chi_{3}\in NEG(\tilde{C})\), the rest of the objects are in \(BND(\tilde{C})\); when \(\psi\) is 0.45, \(\chi_{1}\), \(\chi_{2}\), \(\chi_{3}\), and \(\chi_{7}\in NEG(\tilde{C})\), the rest of the objects are in \(BND(\tilde{C})\); when \(\psi\) is 0.46, \(\chi_{4}\), \(\chi_{5}\), \(\chi_{6}\), and \(\chi_{9}\in BND(\tilde{C})\), the rest of the objects are in \(NEG(\tilde{C})\); when \(\psi\) is 0.47, \(\chi_{4}\), \(\chi_{5}\) and \(\chi_{6}\in BND(\tilde{C})\), the rest of the objects are in \(NEG(\tilde{C})\); when \(\psi\) is 0.48, \(\chi_{5}\) and \(\chi_{6}\in BND(\tilde{C})\), the rest of the objects are in \(NEG(\tilde{C})\); when \(\psi\) is 0.49, \(\chi_{6}\in POS(\tilde{C})\), \(\chi_{5}\in BND(\tilde{C})\), the rest of the objects are in \(NEG(\tilde{C})\); when \(\psi\) is 0.50, \(\chi_{6}\in POS(\tilde{C})\), the rest of the objects are in \(NEG(\tilde{C})\). The above analysis shows that the distribution of objects is different for different adjustment coefficients \(\psi\). To show the above description more vividly, we use Figure 6 to describe the number of objects in different regions. According to Figure 6, the number of objects in NEG is always not less than that in POS under different \(\psi\) values. When the \(\psi\) value increases, the size of BND decreases while the sizes of POS and NEG increase. Figure 7 shows the effect of \(\psi\) on the maximum number of objects. Since the number of objects in our experimental data is 10, the maximum number of objects is 10 under the normal rank. In a specific area, the maximum number of objects is less than or equal to the number of all objects (10) in \(POS\), \(BND\), or \(NEG\). According to Figure 8, computational complexity can be reduced significantly by ranking only a part of all the candidates, especially when the number of candidates is large. The meanings of the dotted lines in Figure 8 are the same as those in Figure 7. Assuming that the number of global objects is 10, then a set \(\{R_{1},R_{2},\ldots,R_{20}\}\) containing 20 random numbers that are generated from the range of 0 to 10 to represent the number of candidates needed. In the decision-making process, the number of candidates needed is less than or equal to that of all objects, further illustrating the importance of division for decision-making efficiency. Figure 6: The number of objects in three regions based on different \(\psi\).