## 4 Fuzzy Hierarchical Comprehensive Evaluation of Impact Risk in Yanbei Coal Mine 4.1. Mine Overview
### AHP Calculation of Index Weight in Index Layer  
The scaling method of 1-9 and its reciprocal was adopted for the value of the elements of the judgment matrix, and the ratio of the two elements was used [47] as follows:  
\[y_{\rm ij}=\frac{x_{i}}{x_{j}} \tag{13}\]  
\(x_{i}\): the weight or score value of the \(i\)-th factor (indicator) affecting the energy and frequency of microseismic events; \(x_{j}\): the weight or score value of the \(j\)-th factor (index) affecting the energy and frequency of microseismic events; \(y_{ij}\): the element value of the judgment matrix. The element value calculated by Equation (13) is rounded. When rounded, the value is only rounded, but the maximum value cannot exceed 9. If it is expressed by a fraction and its denominator is the reciprocal of \(y_{ij}<1\), take the whole, and only enter when taking the whole. According to the above method, the scale weight (scoring value) of three indexes is evaluated, which is used to determine the importance of the evaluation index, and then build the judgment matrix, as shown in Table 3.  
The \"sum-product method\" introduced already was used to calculate the weight of factors, and the weight of factors was obtained using Equations (6)-(10), as follows. (1) Normalize the evaluation index judgment matrix of the criterion layer according to Equation (6):  
\[\sum_{i=1}^{n}b_{i1}=1+\frac{1}{2}+\frac{1}{4}=1.750\ \ \overline{b}_{11}=\frac{b_{11}}{ \sum_{i=1}^{n}b_{i1}}=\frac{1}{1.750}=0.571\]  
\[\overline{b}_{21}=\frac{b_{21}}{\sum_{i=1}^{n}b_{i1}}=\frac{0.500}{1.750}=0.286 \ \ \overline{b}_{31}=\frac{b_{31}}{\sum_{i=1}^{n}b_{i1}}=\frac{0.250}{1.750}=0.143\]  
\[\sum_{i=1}^{n}b_{i2}=2+1+\frac{1}{2}=3.500\ \ \overline{b}_{12}=\frac{b_{12}}{ \sum_{i=1}^{n}b_{i2}}=\frac{2}{3.500}=0.571\]  
\[\overline{b}_{22}=\frac{b_{22}}{\sum_{i=1}^{n}b_{i2}}=\frac{1}{3.500}=0.286 \ \ \overline{b}_{32}=\frac{b_{32}}{\sum_{i=1}^{n}b_{i2}}=\frac{0.5}{3.500}=0.143\]  
\[\sum_{i=1}^{n}b_{i3}=4+2+1=7\ \ \ \ \overline{b}_{13}=\frac{b_{13}}{\sum_{i=1}^{ n}b_{i3}}=\frac{4}{7}=0.571\]  
\[\overline{b}_{23}=\frac{b_{23}}{\sum_{i=1}^{n}b_{i2}}=\frac{2}{7}=0.286\ \ \ \ \ \overline{b}_{33}=\frac{b_{33}}{\sum_{i=1}^{n}b_{i3}}=\frac{1}{7}=0.143\]  
(2) Construct the judgment matrix after normalization:  
\[ \begin{pmatrix}0.571&0.571&0.571\\ 0.286&0.286&0.286\\ 0.143&0.143&0.143\end{pmatrix} \]  
(3) Add the elements of raws of the normalized judgment matrix:  
\[\overline{W}_{1}=\sum_{j=1}^{n}b_{1j}=0.571+0.571+0.571=1.713\]  
\[\overline{W}_{2}=\sum_{j=1}^{n}b_{1j}=0.286+0.286+0.286=0.858\]  
\[\overline{W}_{3}=\sum_{j=1}^{n}b_{3j}=0.143+0.143+0.143=0.429\]  
Then normalize the vector = [1.713, 0.858, 0.429], as follows:  
\[\sum_{j=1}^{n}\overline{W}_{j}=1.713+0.858+0.429=3.000;\ \ \ W_{1}=\frac{ \overline{W}_{1}}{\sum_{j=1}^{n}\overline{W}_{j}}\ \ =\frac{1.713}{3}=0.571;\]  
\[W_{2}=\frac{\overline{W}_{2}}{\sum_{j=1}^{n}\overline{W}_{j}}=\frac{0.858}{3} =0.286\ \ ;\ \ \ \ W_{3}=\frac{\overline{W}_{3}}{\sum_{j=1}^{n}\overline{W}_{j}}=\frac{0.429 }{3}=0.143\ \ ;\]  
Thus, the eigenvector = [0.571,0.286,0.143]. \begin{table}
\begin{tabular}{c c c c} \hline
**Judgment Matrix** & **B\({}_{1}\)** & **B\({}_{2}\)** & **B\({}_{3}\)** \\ \hline B\({}_{1}\) & 1 & 2 & 4 \\ B\({}_{2}\) & 1/2 & 1 & 2 \\ B\({}_{3}\) & 1/4 & 1/2 & 1 \\ \hline \end{tabular}
\end{table} 
Table 3: Judgment matrix of evaluation index of the criterion layer. (4) Calculate the maximum eigen root \(\lambda_{\max}\) of the judgment matrix:  
\[AW= \begin{pmatrix}1&2&4\\ \frac{1}{2}&1&2\\ \frac{1}{4}&\frac{1}{2}&1\end{pmatrix}\begin{pmatrix}0.571\\ 0.286\\ 0.143\end{pmatrix}