# Dynamic Risk Assessment of Ultra-Shallow-Buried and Large-Span Double-Arch Tunnel Construction
## 4 Construction Dynamic Risk Evaluation Index Weight  
### Benchmark Weight Determination  
According to the existing risk assessment hierarchical system, the commonly used and effective AHP method is used to determine the weight of the indexes. The AHP is a systematic analysis method that combines qualitative and quantitative analyses. The AHP hierarchies the problem, decomposes it into different components, and combines them in hierarchical clusters according to their mutual influence and affiliation. A judgment matrix is formed by comparing the relative importance of the indexes at each level. The relative importance of each index is quantified using a certain digital scale, which is called the judgment scale, between two indexes. The most commonly used judgment standard would use the numbers 1-9 to represent the relative importance of the indexes, that is, one represents equal importance, and nine represents extreme importance [50]. The degree of importance increases with the digital value, and the criteria are listed in Table 1.  
Figure 2: Risk evaluation index system of urban shallow-buried large-span double-arch tunnel construction.  
The importance of each evaluation index at the same level is compared in pairs in order to construct a judgment matrix. Assuming \(n\) factors U1, U2,..., Un in a level, scale aij with a value of 1-9 is used to reflect the relative importance of the comparison between Ui and Uj, and an \(n\)-th order judgment matrix is obtained, as shown in Equation (1).  
\[A=\left[\begin{array}{cc}a_{11}\cdots&a_{1n}\\ \vdots&\vdots\\ a_{n1}\cdots&a_{nn}\end{array}\right] \tag{1}\]  
The maximum eigenvalue (\(\lambda_{\text{max}}\)) and eigenvector (\(W\)) of the judgment matrix are obtained, and the eigenvector is normalized to obtain the relative weight of the risk indexes at the same level. Then, the consistency test is carried out to determine the logical rationality of the judgment matrix. The random consistency ratio (\(C_{R}\)) is used to check the consistency of the judgment matrix. When \(C_{R}\) is less than 0.1, the requirement is met.  
\[C_{R}=\frac{C_{I}}{R_{I}}, \tag{2}\]  
where \(R_{I}\) is the average random consistency index value of the judgment matrix, as shown in Table 2. \(C_{I}\) is the consistency index, which can be calculated by Equation (3).  
\[C_{I}=\frac{\lambda_{\text{max}}-n}{n-1}. \tag{3}\]  
Ten experienced tunnel design, geological exploration, and tunnel construction experts were selected in order to form an evaluation team. Through the table comparison method [51], the experts scored the importance of each level of risk factors. According to the experts' replies, a judgment matrix was constructed. According to the AHP method, the relative weights of the first- and second-level risk evaluation indexes were calculated, and the results are shown in Tables 3 and 4.  
\begin{table}
\begin{tabular}{c c} \hline
**Digital Scale** & **Definition** \\ \hline
1 & Ui is as important as Uj \\
3 & Ui is slightly more important than Uj \\
5 & Ui is obviously more important than Uj \\
7 & Ui is more important than Uj \\
9 & Ui is extremely important than Uj \\
2, 4, 6, 8 & Take the median value between the two adjacent degrees of the above comparison \\ \hline \end{tabular}
\end{table}
Table 1: Relative comparison scale.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline
**Matrix Order** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline \(R_{I}\) & 0 & 0.58 & 0.90 & 1.12 & 1.24 & 1.32 & 1.41 & 1.46 & 1.49 \\ \hline \end{tabular}
\end{table}
Table 2: Average random consistency index \(R_{I}\).  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Stage**} & \multicolumn{6}{c}{**First-Level Indexes**} & \multicolumn{1}{c}{\multirow{2}{*}{\(C_{I}\)}} & \multicolumn{1}{c}{\multirow{2}{*}{\(C_{R}\)}} & \multicolumn{1}{c}{\multirow{2}{*}{\(Weight\)}} \\ \cline{3-10}  & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{