## 6 Conclusions  
Because of climate change and increasing truck-loads, bridges are more inclined to suffer from rapid performance deterioration. Hence, periodic maintenance activities are highly needed to improve the bearing capacity and extend the service life of bridges. However, the available budget cannot cover all the bridges in a network. In this situation, it is necessary for transportation agencies to develop an optimal maintenance scheduling strategy with limited available resources. Most of the existing studies define this problem as optimal allocation of the limited funds, which can identify which bridges to maintain but cannot determine the time sequence of maintenance activities for the bridges accurately. This research focuses on the optimal maintenance scheduling strategy for bridge networks. We formulate this problem as a bi-level model. The upper-level model is a multi-objective nonlinear programming model, which minimizes the total traffic delays during the maintenance period and maximizes the number of bridges to be maintained subject to the budget limit and the number of crews. In the lower-level, the users' route choice following the upper-level decision is simulated using a modified user equilibrium model. Then, a SAA is employed to solve the model. Finally, the proposed method is illustrated on a highway bridge network to generate an optimal maintenance scheduling strategy. The key findings are summarized as follows: \begin{table}
\begin{tabular}{c c c c} \hline \hline \(B\) & **1500** & **2000** & **2500** \\ \hline \(c\) & 1450 & 1970 & 1970 \\ \hline \(Z\) & 21,978 & 39,158 & 39,158 \\ \hline \(M\) & 25 & 37 & 37 \\ \hline \(N\) & 7 & 10 & 10 \\ \hline \multirow{3}{*}{OMSS} & Crew 1: B, H & Crew 1: B, E, N & Crew 1: B, E, N \\  & Crew 2: L, E, G & Crew 2: C, G, I & Crew 2: C, G, I \\ \cline{1-1}  & Crew 3: N, J & Crew 3: H, J, K, L & Crew 3: H, J, K, L \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Results of the OMSS for different budget limits. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\rho\). & **0.3** & **0.5** & **0.7** \\ \hline \(c\) & 1450 & 1450 & 1260 \\ \hline \(Z\) & 29,851 & 21,978 & 19,657 \\ \hline \(M\) & 29 & 25 & 22 \\ \hline \(N\) & 7 & 7 & 6 \\ \hline \multirow{3}{*}{OMSS} & Crew 1: B, N & Crew 1: B, H & Crew 1: C, G \\  & Crew 2: C, D & Crew 2: L, E, G & Crew 2: E, K \\ \cline{1-1}  & Crew 3: G, H, L & Crew 3: N, J & Crew 3: H, J \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Results of the OMSS for different \(\rho\). 1. Compared with the two empirical strategies, i.e., FFSS and WFSS, the optimal maintenance scheduling strategy generated by the proposed method has an advantage in saving maintenance cost, reducing traffic delays, minimizing makespan and can provide a potential aid for transportation agencies in making efficient maintenance decisions. 2. Traffic demand has a significant impact on the optimal maintenance scheduling strategy including time sequence and job sequence. For the bridge network with heavy traffic, avoiding simultaneous maintenance activities of bridges on two parallel links can reduce traffic delays during the maintenance period. 3. More crews can shorten the entire maintenance period apparently, but cannot always guarantee minimal traffic delays. Because more crews mean more bridges being maintained simultaneously, which aggravate traffic congestion. Only when the manpower and funds match can the optimal maintenance scheduling strategy be improved. 4. Decision maker's preference also affects both the time sequence and job sequence. Hence, decision makers should consider a reasonable tradeoff between the traffic delays and the maximum number of bridges to be maintained. The results of the case study show that the proposed model can generate the optimal maintenance scheduling strategy for a bridge network taking both the traffic delays and the maximum number of bridges to be maintained into account, which provides a reference for transportation agencies in bridge management.