# A Status Evaluation of Rock Instability in Metal Mines Based on the SPA-IAHP-PCN Model
## 4 Results and Discussion
### Calculation of the Weight  
To ensure the scientific rigor and accuracy of the index weights, 10 experienced experts in the field were invited to evaluate the weights of the indices using the 1-9 scaling rules. Each judgment value based on the 1/9-9 scale required unanimous consensus among all participating experts; otherwise, it was subject to re-evaluation. The details of the 10 experts are provided in Table 4.  
Based on the understanding and analysis of the selected parameters, and considering the specific conditions of the mine, experts evaluated the four parameters in the established evaluation index system: cumulative number of events (\(A_{1}\)), cumulative microseismic event energy (\(A_{2}\)), cumulative apparent stress (\(A_{3}\)), and cumulative logarithmic seismic moment (\(A_{4}\)). The interval pairwise comparison matrix for these parameters is shown below. \[A= \begin{bmatrix}\begin{bmatrix}1,1\\ \frac{4}{3},2\\ \frac{9}{5},5\\ \end{bmatrix}&\begin{bmatrix}1,1\\ \frac{4}{3},2\\ \frac{9}{5},4\\ \frac{9}{2},2\\ \end{bmatrix}&\begin{bmatrix}1,1\\ \frac{1}{1},1\\ \frac{3}{2},2\\ \end{bmatrix}&\begin{bmatrix}1,2\\ \frac{1}{1},1\\ \end{bmatrix}\end{bmatrix} \]  
And the \(A_{L}\) and the \(A_{U}\) can be obtained according to Equation (3). \[A^{L}= \begin{bmatrix}1&\frac{2}{3}&\frac{5}{6}&\frac{1}{2}\\ \frac{4}{3}&1&\frac{3}{4}&\frac{1}{3}\\ \frac{7}{5}&\frac{6}{1}&\frac{1}{2}\\ \frac{3}{2}&2&\frac{3}{2}&1\end{bmatrix} ,A^{U}= \begin{bmatrix}1&\frac{3}{4}& \frac{7}{9}&\frac{2}{3}\\ \frac{3}{2}&1&\frac{5}{6}&\frac{1}{2}\\ \frac{6}{4}&\frac{5}{3}&1&\frac{5}{2}\\ 2&3&2&1\end{bmatrix} \] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Expert** & \begin{tabular}{c} **Professional** \\ **Position** \\ \end{tabular} & \
\begin{tabular}{c} **Education** \\ **Background** \\ \end{tabular} &
\begin{tabular}{c} **Work Experience** \\ **(Year)** \\ \end{tabular} \\ \hline Expert 1 & Student & Master & 7 \\ \hline Expert 2 & Worker & Bachelor & 16 \\ \hline Expert 3 & Worker & Master & 14 \\ \hline Expert 4 & Engineer & Master & 26 \\ \hline Expert 5 & Engineer & PhD & 31 \\ \hline Expert 6 & Engineer & Master & 37 \\ \hline Expert 7 & Professor & PhD & 15 \\ \hline Expert 8 & Professor & PhD & 20 \\ \hline Expert 9 & Professor & PhD & 32 \\ \hline Expert 10 & Professor & PhD & 44 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The introduction of experts. \begin{table}
\begin{tabular}{c c} \hline \hline \multirow{2}{*}{Status evaluation of rock instability of metal mine} & \multicolumn{2}{c}{Cumulative Number of Events \(A_{1}\)} \\ \cline{2-3}  & \multicolumn{1}{c}{\multirow{2}{*}{\begin{tabular}{c} Cumulative Microseismic Event Energy \(A_{2}\) \\ \end{tabular} }} \\ \cline{2-3}  & \multicolumn{1}{c}{\multirow{2}{*}{\
\begin{tabular}{c} Cumulative Apparent Stress \(A_{3}\) \\ \end{tabular} }} \\ \cline{2-3}  & \multicolumn{1}{c}{\multirow{2}{*}{
\begin{tabular}{c} Cumulative Logarithmic Seismic Moment \(A_{4}\) \\ \end{tabular} }} \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The status evaluation index system of rock instability of metal mine. Next, the standardized eigenvectors \(\beta^{L}\) and \(\beta^{U}\) corresponding to \(A^{L}\) and \(A^{U}\), respectively, are obtained by the calculation of Matlab software. \[\beta^{L}=(0.1870,0.1993,0.2398,0.3739)\]  
\[\beta^{U}=(0.1757,0.1941,0.2209,0.4093)\]  
Then, the coefficients \(k\) = 0.9537 and \(h\) = 1.0365 are calculated according to Equations (4) and (5). Based on the above calculation, the \(W_{i}\) is obtained. \[W_{i}= \begin{bmatrix}w_{i}^{L},w_{i}^{U}\end{bmatrix} = \begin{bmatrix}kg^{L},k \beta^{U}\end{bmatrix} = \begin{bmatrix}0.1784&0.1821\\ 0.1901&0.2012\\ 0.2287&0.2290\\ 0.3566&0.4242\end{bmatrix} \]  
Subsequently, the coefficients are checked by Equation (6). Meanwhile, consistency is obtained by Equation (7). Comparing the consistency value (\(z^{*}\) = 0.0827) with Table 1, it is clear that the consistency check result meets the requirement of the algorithm. After that, the \(\mu_{i}\) is obtained based on Equation (8).