## 5 Example Analysis
### Comprehensive Evaluation  
Five experts with significant experience in bridge engineering were invited to grade each risk indicator, leading to the matrix in Equation (45):  
\[U=\begin{bmatrix}6&9&7&8&7&7&6&7&5&8&7&6\\ 7&8&7&8&6&8&7&7&6&7&6&7&8&7\\ 7&9&8&8&7&8&7&8&7&8&7&7\\ 6&8&7&7&6&7&6&7&6&7&8&8\\ 6&9&8&7&7&6&8&8&7&8&7&6\\ \end{bmatrix} \tag{45}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Criterion Layer** & **Indicators Layer** & **Probability** & **Loss** & **Weight** \\ \hline \multirow{8}{*}{Natural disaster} & Wind induced disaster & 0.6 & 40 & 0.08306 \\  & Earthquake & 0.5 & 50 & 0.08374 \\  & Geologic hazard & 0.5 & 30 & 0.08350 \\  & Flood & 0.8 & 40 & 0.08306 \\  & Bizzard & 0.6 & 40 & 0.08306 \\  & Ship collision bridge & 0.4 & 50 & 0.08374 \\  & Fire & 0.5 & 30 & 0.08350 \\  & Get an electric shock & 0.7 & 50 & 0.08306 \\ Accident & High falling objects & 0.6 & 40 & 0.08306 \\  & Overload accident & 0.5 & 60 & 0.08350 \\  & Man-made destruction & 0.5 & 50 & 0.08366 \\  & Vehicle impact Bridge & 0.6 & 40 & 0.08306 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Objective weights.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Criterion Layer** & **Indicator Layer** & **Subjective Weight** & **Objective Weight** & **Combination Weight** \\ \hline \multirow{8}{*}{Natural disaster} & Wind-induced disaster & 0.0498 & 0.08306 & 0.0747 \\  & Earthquake & 0.3076 & 0.08374 & 0.1399 \\  & Geologic hazard & 0.0872 & 0.08350 & 0.0844 \\  & Flood & 0.1820 & 0.08306 & 0.1079 \\  & Blizzard & 0.1235 & 0.08306 & 0.0932 \\  & Ship collision with bridge & 0.0838 & 0.08374 & 0.0838 \\  & Fire & 0.0518 & 0.08350 & 0.0756 \\  & Get an electric shock & 0.0108 & 0.08306 & 0.0649 \\ Accident & High falling objects & 0.0144 & 0.08306 & 0.0658 \\  & Overload accident & 0.0374 & 0.08350 & 0.0719 \\  & Man-made destruction & 0.0195 & 0.08366 & 0.0676 \\  & Vehicle impact on bridge & 0.0322 & 0.08306 & 0.0703 \\ \hline \hline \end{tabular}
\end{table}
Table 11: Combination weights.  
The grades given by the experts were substituted into the risk evaluation model, and the elements \(6,7,7,6\), and \(6\) in the first column were substituted into the five corresponding types of defined weighted functions in the risk evaluation model.  
When \(u_{11}=6\), \(f_{1}(u_{11})=\frac{6}{9}=0.667\), \(f_{2}(u_{11})=\frac{6}{7}=0.857\), \(f_{3}(u_{11})=-\frac{6}{5}+2=0.8\), \(f_{4}(u_{11})=-\frac{6}{3}+2=0\), and \(f_{5}(u_{11})=0\).  
When \(u_{21}=7\), \(f_{1}(u_{21})=\frac{7}{9}=0.778\), \(f_{2}(u_{21})=-\frac{7}{7}+2=1\), \(f_{3}(u_{21})=-\frac{7}{5}+2=0.6\), \(f_{4}(u_{21})=0\), and \(f_{5}(u_{21})=0\).  
The statistical values of class I are as follows: \(n_{11}=f_{1}(u_{11})+f_{1}(u_{21})+f_{1}(u_{31})+f_{1}(u_{41})+f_{1}(u_{51})\).  
The grey statistical values of the other four classes were solved by other methods, and the results are presented in Table 12.  
The evaluation weights belonging to the first evaluation standard are: \(v_{11}=\frac{n_{11}}{c_{11}}=\frac{3.556}{11.727}=0.303\), \(v_{12}=\frac{n_{12}}{c_{11}}=\frac{4.571}{11.727}=0.390\), \(v_{13}=\frac{n_{13}}{nc_{11}}=\frac{3.600}{11.727}=0.307\), \(v_{14}=\frac{n_{14}}{nc_{11}}=\frac{0.000}{11.727}=0.000\), and \(v_{15}=\frac{n_{15}}{n_{11}}=\frac{0.000}{11.727}=0.000\).  
Similarly, five evaluation weights belonging to other indicators were obtained.  
\[V=\begin{bmatrix}0.303&0.390&0.307&0.000&0.000\\ 0.476&0.384&0.140&0.000&0.000\\ 0.360&0.413&0.227&0.000&0.000\\ 0.377&0.408&0.214&0.000&0.000\\ 0.311&0.400&0.289&0.000&0.000\\ 0.340&0.388&0.272&0.000&0.000\\ 0.327&0.396&0.277&0.000&0.000\\ 0.360&0.413&0.227&0.000&0.000\\ 0.287&0.369&0.317&0.000&0.000\\ 0.377&0.408&0.214&0.000&0.000\\ 0.360&0.413&0.227&0.000&0.000\\ 0.360&0.413&0.227&0.000&0.000\\ 0.313&0.403&0.283&0.000&0.000\\ \end{bmatrix} \tag{46}\]