## 4 Construct AHP-EW Combined Weight Matter Element Extension Model
### AHP-EW Combined Weight Element Extension Model  
According to the matter element extension theory [22], the establishment steps of the matter element extension model are as follows:  
(1) Determine the classical domain:  
The classical domain matrix of the evaluation index can be expressed as a formula. \[R_{0t}=\left[N,C_{j},V_{0t}\right]= \begin{bmatrix}N&C_{1}&\langle a_{0t1},b_{0 t1}\rangle\\ &C_{2}&\langle a_{02},b_{0t2}\rangle\\ &\vdots&\vdots\\ &C_{n}&\langle a_{0 tn},b_{0 tn}\rangle\end{bmatrix}  \tag{15}\]  
In the formula, \(R_{0t}\) is the classical domain matter element; \(N\) is the whole matter element to be evaluated; \(C_{j}\) (j = 1, 2, \(\ldots\), n) is the evaluation index of the thing \(N\); \(V_{0 tj}=\langle a_{0 tj},\,b_{0 tj}\rangle\) is the evaluation index \(C_{j}\) of the classical domain matter element in t (t = 1, 2, \(\ldots\), g); the value range of the evaluation grade is the classical domain; \(a_{0 tj}\) is the lower limit of the classical domain; \(b_{0 tj}\) is the upper limit of the classical domain. (2) Determine the section domain:  
The section domain matrix of the evaluation index is expressed as the following formula:  
\[R_{q}=\left[N,C_{j},V_{qj}\right]= \begin{bmatrix}N&C_{1}&\langle a_{q1},b_{q1 }\rangle\\ &C_{2}&\langle a_{q2},b_{q2}\rangle\\ &\vdots&\vdots\\ &C_{n}&\langle a_{qn},b_{qn}\rangle\end{bmatrix}  \tag{16}\]  
In the formula, \(R_{q}\) is a nodal matter element; \(V_{qj}=\langle a_{qj},\,b_{qj}\rangle\) is the value range of \(N\) for evaluation indicators \(C_{j}\) (j = 1, 2, \(\ldots\), n), which is the section field. The value range of (n) is the nodal domain; \(\mathrm{a_{qj}}\) is the lower limit value of the nodal domain; \(\mathrm{b_{qj}}\) is the upper limit of the node domain. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{1}{c}{**Evaluating Indicator**} & \multicolumn{5}{c}{**Order of Evaluation**} \\ \cline{2-7}  & **1** & **II** & **III** & **IV** & **V** \\ \cline{2-7}  & **Very Low Risk** & **Low Risk** & **Intermediate Risk** & **High Risk** & **Very High Risk** \\ \hline Stable rock thickness \(\mathrm{U_{11}/m}\) & 0.8–1.0 & 0.6–0.8 & 0.4–0.6 & 0.2–0.4 & 0.0–0.2 \\ \hline Geological condition \(\mathrm{U_{12}}\) & 0.0–0.2 & 0.2–0.4 & 0.4–0.6 & 0.6–0.8 & 0.8–1.0 \\ \hline Cross-section shape of roadway \(\mathrm{U_{13}}\) & 0.8–1.0 & 0.6–0.8 & 0.4–0.6 & 0.2–0.4 & 0.0–0.2 \\ \hline Angle between maximum horizontal principal stress and roadway \(\mathrm{U_{14}/}^{\circ}\) & 0.0–0.2 & 0.2–0.4 & 0.4–0.6 & 0.6–0.8 & 0.8–1.0 \\ \hline The amount of displacement of roof strata \(\mathrm{U_{11}/m}\) & 0.0–0.14 & 0.14–0.28 & 0.28–0.42 & 0.42–0.56 & 0.56–1.00 \\ \hline Visible fracture development situation \(\mathrm{U_{22}/pcs}\) & 0.0–0.2 & 0.2–0.4 & 0.4–0.6 & 0.6–0.8 & 0.8–1.0 \\ \hline Compressive strength of roof & 0.8–1.0 & 0.6–0.8 & 0.4–0.6 & 0.2–0.4 & 0.0–0.2 \\ \hline Surrounding disturbance \(\mathrm{U_{31}}\) & 0.0–0.2 & 0.2–0.4 & 0.4–0.6 & 0.6–0.8 & 0.8–1.0 \\ \hline Maximum exposure time \(\mathrm{U_{32}/d}\) & 0.0–0.05 & 0.05–0.15 & 0.15–0.30 & 0.30–0.60 & 0.60–1.00 \\ \hline Supporting quality \(\mathrm{U_{33}}\) & 0.8–1.0 & 0.6–0.8 & 0.4–0.6 & 0.2–0.4 & 0.0–0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Evaluation index level of roadway roof fall risks without dimensional treatment.