## 4 Numerical Example
### Bridge System Reliability Estimation Obtained by Considering Multiple Failure Criteria  
After calculating the system failure probability corresponding to each failure criterion, the AHP-EW method was used to evaluate the bridge system reliability by considering multiple failure criteria. The hierarchical model shown in tabl5 was established. The target layer was the bridge system reliability obtained by considering multiple failure criteria, and the index layer contained each failure criterion of the bridge system. Thirteen experts in the field of civil engineering were selected to score each index based on their experience and knowledge. The possible scores ranged from 1 to 5. The higher the score was, the higher the probability that the bridge system would fail according to this failure criterion. The subjective weight and objective weight of each index were calculated using the AHP and EW method, respectively. Then, the comprehensive weight was obtained according to Equation (23). The selected indexes and their weights are shown in Table 14. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{\(Z_{i}Z_{j}Z_{k}\)} & \multicolumn{4}{c}{**Copula Parameters \(\theta\)**} \\ \cline{2-7}  & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** \\ \hline \(Z_{1}Z_{2}Z_{3}\) & 0.9787 & 7.356 & 6.462 & 26.21 \\ \(Z_{2}Z_{3}Z_{4}\) & 0.9783 & 7.121 & 6.434 & 25.74 \\ \(Z_{3}Z_{4}Z_{5}\) & 0.9778 & 7.233 & 6.367 & 25.33 \\ \(Z_{4}Z_{5}Z_{6}\) & 0.9780 & 7.266 & 6.437 & 25.63 \\ \(Z_{5}Z_{6}Z_{7}\) & 0.9780 & 7.131 & 6.431 & 25.32 \\ \(Z_{6}Z_{7}Z_{8}\) & 0.9775 & 7.003 & 6.408 & 25.38 \\ \(Z_{7}Z_{8}Z_{9}\) & 0.9783 & 7.207 & 6.488 & 25.88 \\ \hline \end{tabular}
\end{table} 
Table 11: The copula parameters for the candidate copulas. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{\(Z_{i}Z_{j}Z_{k}\)} & \multicolumn{2}{c}{**[PERSON]**} & \multicolumn{2}{c}{**[PERSON]**} & \multicolumn{2}{c}{**[PERSON]**} & \multicolumn{2}{c}{**[PERSON]**} \\ \cline{2-7}  & **AIC** & **BIC** & **AIC** & **BIC** & **AIC** & **BIC** & **AIC** & **BIC** \\ \hline \(Z_{1}Z_{2}Z_{3}\) & \(-6595.721\) & \(-6590.813\) & \(-5367.287\) & \(-5362.379\) & \(-6181.589\) & \(-6176.681\) & \(-6011.271\) & \(-6006.363\) \\ \(Z_{2}Z_{3}Z_{4}\) & \(-6551.639\) & \(-6546.731\) & \(-5266.967\) & \(-5262.06\) & \(-6184.301\) & \(-6179.393\) & \(-5952.425\) & \(-5947.517\) \\ \(Z_{3}Z_{4}Z_{5}\) & \(-6510.834\) & \(-6505.926\) & \(-5310.706\) & \(-5305.798\) & \(-6142.439\) & \(-6137.531\) & \(-5903.524\) & \(-5898.616\) \\ \(Z_{4}Z_{5}Z_{6}\) & \(-6528.413\) & \(-6523.505\) & \(-5328.672\) & \(-5323.764\) & \(-6172.587\) & \(-6167.679\) & \(-5936.418\) & \(-5931.51\) \\ \(Z_{5}Z_{6}Z_{7}\) & \(-6527.538\) & \(-6522.631\) & \(-5265.767\) & \(-5260.859\) & \(-6179.003\) & \(-6174.095\) & \(-5903.251\) & \(-5898.343\) \\ \(Z_{6}Z_{7}Z_{8}\) & \(-6481.984\) & \(-6477.076\) & \(-5216.417\) & \(-5211.509\) & \(-6153.824\) & \(-6148.916\) & \(-5892.996\) & \(-5888.088\) \\ \(Z_{7}Z_{8}Z_{9}\) & \(-6551.996\) & \(-6547.088\) & \(-5298.768\) & \(-5293.860\) & \(-6218.126\) & \(-6213.218\) & \(-5955.734\) & \(-5950.826\) \\ \hline \end{tabular}
\end{table} 
Table 12: The AIC and BIC values for the candidate copulas. \begin{table}
\begin{tabular}{c c c c} \hline \(Z_{i}Z_{j}Z_{k}\) & **Joint Failure Probability \(P_{f}\)** & \(Z_{i}Z_{j}Z_{k}\) & **Joint Failure Probability \(P_{f}\)** \\ \hline \(Z_{1}Z_{2}Z_{3}\) & \(3.961\times 10^{-8}\) & \(Z_{5}Z_{6}Z_{7}\) & \(6.062\times 10^{-7}\) \\ \(Z_{2}Z_{3}Z_{4}\) & \(1.414\times 10^{-7}\) & \(Z_{6}Z_{7}Z_{8}\) & \(1.404\times 10^{-7}\) \\ \(Z_{3}Z_{4}Z_{5}\) & \(6.058\times 10^{-7}\) & \(Z_{7}Z_{8}Z_{9}\) & \(3.943\times 10^{-8}\) \\ \(Z_{4}Z_{5}Z_{6}\) & \(2.783\times 10^{-6}\) & & \\ \hline \end{tabular}
\end{table} 
Table 13: The joint failure probability of each group of failure modes based on a Gaussian copula.