# Rockburst Hazard Evaluation Using an Extended COPRAS Method with Interval-Valued Fuzzy Information
## 5 Discussion
### Comparison Analysis  
In this study, the COPRAS method was extended with IVFNs for rockburst hazard evaluation. To further verify the effectiveness of the proposed method, different methods were used for comparison. The extended TOPSIS method with IVFNs and extended VIKOR method with IVFNs were used to compare with the proposed method. The evaluation results are shown in Table 15, and the detailed calculation processes are indicated as follows. \begin{table}
\begin{tabular}{c c} \hline \(\theta\) & \multicolumn{2}{c}{**Ranking Results**} \\ \hline
0 & \(C_{5}>C_{3}>C_{7}>C_{2}>C_{\rm IV}^{*}>C_{4}>C_{1}>C_{6}>C_{\rm III}^{*}>C_{ \rm II}^{*}>C_{\rm I}^{*}\) \\
0.1 & \(C_{5}>C_{3}>C_{7}>C_{2}>C_{\rm IV}^{*}>C_{4}>C_{1}>C_{6}>C_{\rm III}^{*}>C_{ \rm II}^{*}>C_{\rm I}^{*}\) \\
0.2 & \(C_{5}>C_{3}>C_{7}>C_{2}>C_{\rm IV}^{*}>C_{4}>C_{1}>C_{6}>C_{\rm III}^{*}>C_{ \rm II}^{*}>C_{\rm I}^{*}\) \\
0.3 & \(C_{5}>C_{3}>C_{7}>C_{2}>C_{\rm IV}^{*}>C_{4}>C_{6}>C_{1}>C_{\rm III}^{*}>C_{ \rm II}^{*}>C_{\rm I}^{*}\) \\
0.4 & \(C_{5}>C_{3}>C_{7}>C_{3}>C_{2}>C_{4}>C_{6}>C_{1}>C_{\rm III}^{*}>C_{\rm II}^{*}>C _{\rm I}^{*}\) \\
0.5 & \(C_{5}>C_{7}>C_{3}>C_{\rm IV}^{*}>C_{2}>C_{4}>C_{6}>C_{1}>C_{\rm III}^{*}>C_{ \rm II}^{*}>C_{\rm I}^{*}\) \\
0.6 & \(C_{5}>C_{7}>C_{3}>C_{\rm IV}^{*}>C_{2}>C_{4}>C_{6}>C_{1}>C_{\rm III}^{*}>C_{ \rm II}^{*}>C_{\rm I}^{*}\) \\
0.7 & \(C_{5}>C_{7}>C_{3}>C_{\rm IV}^{*}>C_{2}>C_{4}>C_{6}>C_{1}>C_{\rm III}^{*}>C_{ \rm II}^{*}>C_{\rm I}^{*}\) \\
0.8 & \(C_{5}>C_{7}>C_{3}>C_{\rm IV}^{*}>C_{2}>C_{4}>C_{6}>C_{1}>C_{\rm III}^{*}>C_{ \rm II}^{*}>C_{\rm II}^{*}>C_{\rm I}^{*}\) \\
0.9 & \(C_{7}>C_{5}>C_{3}>C_{\rm IV}^{*}>C_{2}>C_{4}>C_{6}>C_{1}>C_{\rm III}^{*}>C_{ \rm II}^{*}>C_{\rm I}^{*}\) \\
1 & \(C_{7}>C_{5}>C_{3}>C_{\rm IV}^{*}>C_{2}>C_{4}>C_{6}>C_{1}>C_{\rm III}^{*}>C_{ \rm II}^{*}>C_{\rm I}^{*}\) \\ \hline \end{tabular}
\end{table} 
Table 14: Scores of samples. Figure 5: Half-widths of significance scores under different values of \(\theta\). When the extended TOPSIS method with IVFNs was applied, the fuzzy decision-making matrix was first obtained. Then, the ideal solution was determined as \(\widehat{a}_{j}^{+}=\{[0.4106,\) 1\(]\), [0.2998, 1\(]\), [0.2631, 1\(]\), [0.5309, 1\(]\), [0.7323, 1\(]\), [0, 0]\(]\), and the anti-ideal solution was determined as \(\widehat{a}_{j}^{-}=\{[0,\) 0\(]\), [0\(,\) 0\(]\), [0\(,\) 0\(]\), [0\(,\) 0\(]\), [1\(,\) 1]\(]\). Afterwards, the weighted distances of each sample from the ideal solution were calculated as \(\widehat{d}_{i}^{+}=\{[1.0517,\) 1.2560\(]\), [0.7878, 0.9226\(]\), [0.6493, 0.7547], [0.4840, 0.5625], [0.5642, 0.6626], [0.4208, 0.4956], [0.3643, 0.4144], [0.5315, 0.6239], [0.3172, 0.3689], [0.5458, 0.6541], [0.3518, 0.4351]\(]\), and the distances from the anti-ideal solution were calculated as \(\widehat{d}_{i}^{-}=\{[0,\) 0\(]\), [0.2741, 0.3450\(]\), [0.4258, 0.5281], [0.6157, 0.7475], [0.4961, 0.6024], [0.6377, 0.7689], [0.7118, 0.8710], [0.5420, 0.6563], [0.7710, 0.9286], [0.5308, 0.6307], [0.7472, 0.8746]\(]\). Finally, the relative closeness of the samples was determined as \(\widehat{R}_{i}=\{[0,\) 0\(]\), [0.2581, 0.2722\(]\), [0.3961, 0.4117\(]\), [0.5599, 0.5706], [0.4679, 0.4762], [0.6025, 0.6081], [0.6614, 0.6776], [0.5049, 0.5126], [0.7085, 0.7157], [0.4909, 0.4930], [0.6678, 0.6799]\(]\). Therefore, the corresponding ranking result was \(C_{5}>C_{7}>C_{3}>C_{2}>C_{7}^{*}>C_{4}>C_{6}\)\(>C_{1}^{*}>C_{1}^{*}\). When the extended VIKOR method with IVFNs was applied, the fuzzy decision-making matrix was first obtained. The ideal and anti-ideal solutions were the same as those determined with the extended TOPSIS method.