## 4 Example Application
### Reliability Analysis  
According to the sensitivity matrix, the upper-level model of the bi-level model for reliability analysis can be transformed into a simple linear programming problem.  
Taking the capacity state of the first road section (25, 25, 15, 15, 15, 15, 15), the reserve capacity coefficient under capacity constraint is \(\mu_{\gamma c}=1.5067\), the reserve capacity coefficient under travel time constraint is \(\mu_{\gamma t}=1.6866\), and the minimum value is taken as the initial reserve capacity coefficient \(\mu_{\gamma}{}^{(n)}=1.5067\).  
According to step 3 (see the algorithm design part for details), we can get \(\mu_{\gamma}{}^{(n+1)}=1.0062\), the convergence condition is \(\left|\mu_{\gamma}{}^{(n+1)}\mu_{\gamma}{}^{(n)}-\mu_{\gamma}{}^{(n)}\right|=0.0094\leq 0.01\), then getting the reserve capacity coefficient \(\mu_{\gamma}=1.5067\ast 1.0062=1.5160\).  
Similarly, the reserve capacity under other conditions can be obtained, as shown in Table 5. Then, through the approximation form of upper bound and lower bound function, the reliability of the example road network is 0.6224, as shown in Table 6 and Figure 3.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**States**} & \multicolumn{2}{c}{\(P_{\mathcal{S}}\)} & \multirow{2}{*}{**States**} & \multicolumn{2}{c}{\(P_{\mathcal{S}}\)} \\ \cline{2-2} \cline{4-5}  & **Upper Bound** & **Lower Bound** & & **Upper Bound** & **Lower Bound** \\ \hline (25,25,15,15,15,15) & 1 & 0.3024 & (25,25,7,5,15,15,7,5,15) & 0.6616 & 0.596 \\ (25,25,15,15,15,7,5,15) & 0.7984 & 0.3024 & (25,125,7,5,15,7,5,15) & 0.64 & 0.596 \\ (25,25,15,7,5,15,15) & 0.7984 & 0.432 & (25,25,7,5,7,5,15,15,15) & 0.64 & 0.6104 \\ (25,25,15,7,5,15,7,5,15) & 0.712 & 0.432 & (25,25,7,5,15,7,5,15,7,5,15) & 0.6304 & 0.6104 \\ (25,12,5,15,15,15,15,15) & 0.712 & 0.5076 & (25,12,5,7,5,15,15,15,15) & 0.6304 & 0.6188 \\ (25,12,5,15,15,15,15,7,5,15) & 0.6616 & 0.5076 & (25,12,5,7,5,15,15,7,5,15) & 0.6248 & 0.6188 \\ (25,25,7,5,15,15,15,15) & 0.6616 & 0.5412 & (25,12,5,7,5,7,5,15,15,15) & 0.6248 & 0.6224 \\ (25,12,5,15,7,5,15,15,15) & 0.6616 & 0.5736 & (25,12,5,7,5,7,5,15,7,5,15) & 0.6224 & 0.6224 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Reliability upper and lower bounds analysis based on road network capacity.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Road Section** & **Flow(pcu/h)** & **Road Section** & **Flow(pcu/h)** \\ \hline
1 & 16.2867 & 5 & 5.2772 \\
2 & 8.7133 & 6 & 6.4905 \\
3 & 7.5000 & 7 & 9.7228 \\
4 & 3.5095 & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Equilibrium solution of state 7.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Road Section** & **Flow(pcu/h)** & **Road Section** & **Flow(pcu/h)** \\ \hline
1 & 16.2867 & 5 & 5.2772 \\
2 & 8.7133 & 6 & 6.4905 \\
3 & 7.5000 & 7 & 9.7228 \\
4 & 3.5095 & & \\ \hline \hline \end{tabular}
\end{table}
Table 5: List of reserve capacity in capacity status of each road section.  
In order to improve the reliability, the capacity of all road sections is increased by 10%. It can be found that the range of reliability improvement is different, as shown in Table 7. The contribution of road is the largest. The improvement of some road sections has no contribution to the improvement of road network reliability. Therefore, to improve the reliability of the road network, it is necessary to upgrade the key sections.