## 3 Calculate Index Weights  
### Analytic Hierarchy Process (AHP) to Calculate the Weight  
The analytic hierarchy process (AHP) [18] first divides the nature of the analysis object into related factors. Then, the related factors are classified to form a multi-level structure model. The relative importance of each factor is judged by experience or experts. According to the importance degree, the weight is obtained, and the characterization analysis is transformed into a quantitative analysis. The specific calculation steps are as follows:  
(1) Construct judgment matrix:  
The 1-9 scale method [19] was used to determine the importance of the evaluation index elements at each level. The judgment matrix expression (1) is established. \[A=\big{(}a_{ij}\big{)}_{n\times n} \tag{1}\]  
\(a_{ij}\) is the relative importance of risk factor \(i\) to risk factor \(j\) at this level. \[\mathrm{a_{ij}}=\frac{1}{\mathrm{a_{ji}}},\mathrm{a_{ii}}=1,(\mathrm{i}=1,2, \ldots,\mathrm{n}),(\mathrm{j}=1,2,\ldots,\mathrm{n})\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Evaluating Indicator** & **Overview of Indicators** & **Index Property** \\ \hline Angle between maximum & With the increase in the angle between the maximum & \\  & horizontal principal stress and the roadway, the & \\  & damage degree of the roof and floor of the roadway & \\  & and roadway U\({}_{14}\)/\({}^{\circ}\) & also increases. When the angle is 90\({}^{\circ}\), the damage to & \\  & the surrounding rock in the roadway is the most & \\  & & serious. \\ \hline The amount of & The subsidence of roof strata is an observation index & \\  & clearly stipulated in the \"Coal Mine Safety & \\  & displacement of roof strata & Regulations\". After it exceeds the critical deformation & Quantification \\  & U\({}_{21}\)/m & value, physical phenomena such as roof falls occur in the roadway. It is obtained by field measurement. & \\ \hline Visible fracture & The development of fissures reflects the stability of the & \\  & development situation & roof, which is accurately monitored by the roof peeper. & Quantification \\  & & \\  & The tensile strength of the roof rock characterizes the & \\  & bearing capacity of the roof. Due to the action of & \\  & gravity, the surrounding rock in the shallow area of the & \\  & roof rock U\({}_{23}\)/MPa & roof is subjected to tensile stress. If the tensile stress exceeds the tensile strength of the rock, a large number & Quantification \\  & of cracks will appear in the roof rock. & \\ \hline Water content of roof & Groundwater will weaken the mechanical properties of & \\  & rock mass. This leads to physical and chemical changes & \\  & rock U\({}_{24}\) & in the mechanical properties of the roof, reducing the & \\  & roof’s load bearing capacity. & \\ \hline Roof stability U\({}_{25}\) & The stability of coal mine roadway roofs directly affects the risk of roof falls. The better the stability of the roof, & Characterization \\  & the lower the risk of roof falls. & \\ \hline Surrounding & The greater the disturbance stress around the roadway, & \\  & disturbance U\({}_{31}\) & the more easily the roof of the roadway falls. & Characterization \\ \hline Maximum exposure time & The longer the roadway roof is exposed to air, the & \\  & lower the strength of the exposed rock mass, and the & \\  & worse the stability of the roof. & \\ \hline Supporting quality U\({}_{33}\) & The quality of the support directly reflects the stability & \\  & of the roadway roof. The better the support quality, the & \\  & better the stability of the roof. & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._(2) Calculate weight:  
After establishing the judgment matrix, the weight of each index is calculated by Expression (2). Common calculation methods include the feature method and root method. The root method is used in this paper. \[\mathrm{G_{i}^{\prime}=\frac{\overline{G_{i}^{\prime}}}{\sum_{i=1}^{n}\overline{ G_{i}^{\prime}}}} \tag{2}\]  
\(G^{\prime}{}_{i}\) is the weight of the ith evaluation index after unitization.