# Rockburst Hazard Evaluation Using an Extended COPRAS Method with Interval-Valued Fuzzy Information
## 4 Case Study
### Evaluation of Rockburst Hazards  
#### 4.4.1 Express the Indicator Values with IVFNs  
In order to distinguish the hazard levels of seven sites, the samples with known hazard levels were determined according to the grading standards in Table 3. The standard samples with levels I, II, III and IV were denoted as \(C_{1}^{*}\), \(C_{II}^{*}\), \(C_{III}^{*}\) and \(C_{IV}^{*}\), respectively, as shown in Table 5.  
Then, according to Equations (16) and (17), the normalized decision-making matrix was obtained, as shown in Table 6.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(B_{1}\) & \(B_{2}\) & \(B_{3}\) & \(B_{4}\) & \(B_{5}\) & \(B_{6}\) \\ \hline \(C_{1}^{*}\) & [0, 0] & [0, 0] & [0, 0] & [0, 0] & [0, 0] & [1, 1] \\ \(C_{II}^{*}\) & [0.0821, 0.0821] & [0.0696, 0.0696] & [0.0534, 0.0534] & [0.2950, 0.2950] & [0.2149, 0.2149] & [0.3793, 0.3793] \\ \(C_{III}^{*}\) & [0.2053, 0.2053] & [0.1219, 0.1219] & [0.1601, 0.1601] & [0.4129, 0.4129] & [0.3223, 0.3223] & [0.1724, 0.1724] \\ \(C_{IV}^{*}\) & [0.4106, 0.4106] & [0.1741, 0.1741] & [0.2402, 0.2402] & [0.5309, 0.5309] & [0.5908, 0.5908] & [0, 0] \\ \(C_{1}\) & [0.2458, 0.2954] & [0.1048, 0.3367] & [0.1446, 0.3702] & [0.2332, 0.3313] & [0.2652, 0.5938] & [0.2939, 0.3104] \\ \(C_{2}\) & [0.2552, 0.4915] & [0.2998, 0.6605] & [0.2194, 0.5655] & [0.2993, 0.5886] & [0.4464, 0.5852] & [0.2962, 0.3882] \\ \(C_{3}\) & [0.2608, 0.3869] & [0.2065, 1] & [0.2175, 1] & [0.2314, 0.2699] & [0.5855, 0.5889] & [0.2924, 0.3083] \\ \(C_{4}\) & [0.1939, 0.2070] & [0.0958, 0.1953] & [0.2631, 0.4500] & [0.3671, 0.4473] & [0.7323, 0.7458] & [0.2340, 0.2383] \\ \(C_{5}\) & [0.1159, 0.5247] & [0.1574, 0.8109] & [0.2266, 0.8786] & [0.1457, 0.6963] & [0.4796, 1] & [0.1722, 0.3589] \\ \(C_{6}\) & [0.1068, 0.4699] & [0.0808, 0.0905] & [0.1329, 0.1596] & [0.2560, 0.6171] & [0.5123, 0.9871] & [0.1768, 0.3407] \\ \(C_{7}\) & [0.2269, 1] & [0.2458, 0.3468] & [0.2541, 0.6186] & [0.4654, 1] & [0.3524, 0.7197] & [0.2363, 0.4826] \\ \hline \hline \end{tabular}
\end{table}
Table 6: Normalized decision-making matrix.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Mining** & **Depth (m)** & **Lithology** & \(B_{1}\) & \(B_{2}\) & \(B_{3}\) & \(B_{4}\) & \(B_{5}\) & \(B_{6}\) \\ \hline \multirow{4}{*}{Sizhuang} & \(-660\) & Serieticized & [119.74, 143.88] & [3.01, 9.67] & [5.42, 13.87] & [7.91, 11.23] & [0.52, 0.55] & [4.26, 4.50] \\  & \(-700\) & Serieticized & [124.30, 239.42] & [8.61, 18.97] & [8.22, 21.19] & [10.15, 19.96] & [0.42, 0.54] & [4.29, 5.63] \\  & \(-740\) & Serietized & [127.03, 188.42] & [5.93, 28.72] & [8.15, 37.47] & [7.84, 9.15] & [0.52, 0.55] & [4.24, 4.47] \\ \cline{2-8}  & \(-660\) & Wedays & [94.46, 100.81] & [2.75, 5.61] & [9.86, 16.86] & [12.44, 15.17] & [0.68, 0.69] & [3.39, 3.46] \\  & \(-740\) & Wedays & [94.45, 255.54] & [4.52, 23.29] & [8.49, 32.92] & [4.94, 23.61] & [0.45, 0.93] & [2.50, 5.20] \\ \cline{2-8}  & \(-660\) & Wedays & [94.46, 100.81] & [2.75, 5.61] & [9.86, 16.86] & [12.44, 15.17] & [0.68, 0.69] & [3.39, 3.46] \\ \cline{2-8}  & \(-740\) & Wedays & [95.65, 45, 255.54] & [4.52, 23.29] & [8.49, 32.92] & [4.94, 23.61] & [0.45, 0.93] & [2.50, 5.20] \\ \cline{2-8}  & \(-660\) & Wedays & [94.46, 100.81] & [2.75, 5.61] & [9.86, 16.86] & [12.44, 15.17] & [0.68, 0.69] & [3.39, 3.46] \\ \cline{2-8}  & \(-740\) & Wedays & [95.65, 45, 255.54] & [4.52, 23.29] & [8.49, 32.92] & [4.94, 23.61] & [0.45, 0.93] & [2.50, 5.20] \\ \cline{2-8}  & \(-660\) & Wedays & [95.65, 45, 255.54] & [4.52, 23.29] & [8.49, 32.92] & [4.94, 23.61] & [0.48, 0.92] & [2.56, 4.94] \\ \cline{2-8}  & \(-820\) & Wedays & [95.65, 45, 255.54] & [4.52, 23.29] & [8.49, 32.92] & [4.98, 5.98] & [8.68, 20.92] & [0.48, 0.92] & [2.56, 4.94] \\ \hline \hline \end{tabular}
\end{table}
Table 4: Fuzzy indicator values.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(B_{1}\) & \(B_{2}\) & \(B_{3}\) & \(B_{4}\) & \(B_{5}\) & \(B_{6}\) \\ \hline \(C_{1}^{*}\) & [0, 0] & [0, 0] & [0, 0] & [0, 0] & [0, 0] & [14.5, 14.5] \\ \(C_{II}^{*}\) & [40, 40] & [2, 2] & [