## 5 Example Analysis
### Subjective Weight Calculation  
Experts on the working conditions and service environment of the bridge were invited to grade the Dongmen bridge according to the scale presented in Table 2. The class I indicators of the bridge risk in its period of operation are represented by \(C\), among which the natural disasters \(C_{1}\) include wind-induced disasters (\(C_{11}\)), earthquakes (\(C_{12}\)), geological hazards (\(C_{13}\)), floods (\(C_{14}\)), and blizzards (\(C_{15}\)). The relative importance of each risk indicator is summarized in Table 4. \begin{table}
\begin{tabular}{c c c c c} \hline \(\mathbf{C_{1}}\) & \(\mathbf{C_{11}}\) & \(\mathbf{C_{12}}\) & \(\mathbf{C_{13}}\) & \(\mathbf{C_{14}}\) & \(\mathbf{C_{15}}\) \\ \hline \(C_{11}\) & 1 & 1/3 & 1/3 & 1/4 & 1/4 \\ \(C_{12}\) & 3 & 1 & 4 & 3 & 3 \\ \(C_{13}\) & 3 & 1/4 & 1 & 1/2 & 1/2 \\ \(C_{14}\) & 4 & 1/3 & 2 & 1 & 3 \\ \(C_{15}\) & 4 & 1/3 & 2 & 1/3 & 1 \\ \hline \end{tabular}
\end{table} 
Table 4: Relative importance of natural disasters. The corresponding judgment matrix \(C_{1}\) is shown in Equation (33):  
\[C_{1}= \begin{bmatrix}1&\frac{1}{3}&\frac{1}{3}&\frac{1}{4}&\frac{1}{4}\\ 3&1&4&3&3\\ 3&\frac{1}{4}&1&\frac{1}{2}&\frac{1}{2}\\ 4&\frac{1}{3}&2&1&3\\ 4&\frac{1}{3}&2&\frac{1}{3}&1\end{bmatrix}  \tag{33}\]  
Normalizing each column element of the matrix yields matrix \(P_{1}\):  
\[P_{1}= \begin{bmatrix}0.0667&0.1481&0.0357&0.0492&0.0323\\ 0.2000&0.4444&0.4286&0.5902&0.3871\\ 0.2000&0.1111&0.1071&0.0984&0.0645\\ 0.2667&0.1481&0.2143&0.1967&0.3871\\ 0.2667&0.1481&0.2143&0.0656&0.1290\end{bmatrix}  \tag{34}\]  
The elements in the matrix \(P_{1}\) are added in rows to obtain the vector \(P_{1}\)', which is then normalized to obtain the eigenvector \(u_{1}\) of the judgment matrix \(C_{1}\):  
\[u_{1}= \begin{bmatrix}0.0664&0.4101&0.1162&0.2426&0.1647\end{bmatrix}  \tag{35}\]  
The maximum eigenvalue of matrix \(C_{1}\) is \(\lambda_{\max}=5.4192\), \(CI=\frac{5.4192-5}{4}=0.1048\), \(CR=\frac{0.1048}{1.12}=0.0936<0.1\); therefore, this matrix passes the consistency test. The accidents affecting the bridge in its period of operation in the category \(C_{2}\) include ship collisions with the bridge (\(C_{21}\)), fires (\(C_{22}\)), electrical shocks (\(C_{23}\)), falling objects (\(C_{24}\)), overload accidents (\(C_{25}\)), man-made damages (\(C_{26}\)), and vehicle collisions (\(C_{27}\)). The relative importance of each risk indicator is summarized in Table 5.  
The corresponding judgment matrix is expressed as shown in Equation (36):  
\[C_{2}=\begin{bmatrix}1&4&5&4&3&4&3\\ \frac{1}{4}&1&2&3&4&3&3\\ \frac{1}{5}&\frac{1}{2}&1&\frac{1}{2}&\frac{1}{3}&\frac{1}{4}\\ \frac{1}{4  
Normalizing each column element of the matrix gives  
\[P_{2}= \begin{bmatrix}0.3822&0.5926&0.2381&0.2424&0.3186&0.2697&0.3025\\ 0.0955&0.1481&0.0952&0.1818&0.4248&0.2022&0.3025\\ 0.0764&0.0741&0.0476&0.0303&0.0265&0.0225&0.0252\\ 0.0955&0.0494&0.0952&0.0606&0.0354&0.0337&0.0336\\ 0.1274&0.0370&0.1905&0.1818&0.1062&0.2022&0.2017\\ 0.0955&0.0494&0.1429&0.1212&0.0354&0.0674&0.0336\\ 0.1274&0.0494&0.1905&0.1818&0.0531&0.2022&0.1008\end{bmatrix}  \tag{37}\]  
Adding the elements in matrix \(P_{2}\) by rows gives the vector \({P_{2}}^{\prime}\), which is then normalized to obtain the eigenvector \(u_{2}\) of the judgment matrix \(C_{2}\):  
\[u_{2}= \begin{bmatrix}0.3351&0.2072&0.0432&0.0576&0.1495&0.0779&0.1293\end{bmatrix}  \tag{38}\]  
The maximum eigenvalue of matrix \(C_{2}\) is \(\lambda_{\text{max}}=7.7915\), \(CI=\frac{7.7915-7}{6}=0.1319\), \(CR=\frac{0.1319}{1.12}=0.0998<0.1\); therefore, this matrix passes the consistency test. The secondary indicators include natural disasters \(C_{1}\) and accidents \(C_{2}\), and the relative importance of these risk factors is presented in Table 6.  
The corresponding judgment matrix is shown in Equation (39):  
\[C= \begin{bmatrix}1&3\\ \frac{1}{3}&1\end{bmatrix}  \tag{39}\]  
Normalizing each column element of the matrix yields matrix \(P\):  
\[P= \begin{bmatrix}0.7500&0.7500\\ 0.2500&0.2500\end{bmatrix}