## 4 Case Study
### Results  
The OMSS generated by the proposed model is demonstrated in Figure 2. Figure 2a shows the start and end time of the maintenance activity for each bridge, where the number on each bar is the duration of each maintenance activity. Figure 2b indicates the assignment of all the maintenance activities, where the letter and number on each bar are the bridge ID and maintenance duration, respectively, and the job sequence of each crew is ordered from left to right. Specifically, crew 1 undertakes the maintenance activities of bridge B and H, crew 2 is responsible for the maintenance activities of bridge L, E, and G, and bridge N and J are assigned to crew 3. The entire makespan of the maintenance schedule is 25 days. From the simulation of traffic evolution during the makespan, we know that the travel time in the network has a significant day-to-day fluctuation, which is shown in Figure 3. Apparently, the travel time changes at the start and end time of every single maintenance activity. Compared to the travel time under UE state in the pre-maintenance period, the maintenance activities add up to 22,610 h of traffic delays. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(i\) & \(d_{i}\) (day) & \(c_{i}\) (Fund \(-\) Unit) & \(i\) & \(d_{i}\) (day) & \(c_{i}\) (Fund \(-\) Unit) \\ \hline A & 15 & 260 & H & 13 & 230 \\ B & 12 & 240 & I & 15 & 120 \\ C & 11 & 170 & J & 7 & 190 \\ D & 17 & 280 & K & 12 & 230 \\ E & 9 & 260 & L & 5 & 210 \\ F & 6 & 190 & N & 16 & 140 \\ G & 11 & 180 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Maintenance duration and maintenance cost of each bridge. Figure 2: Results of the OMSS. (**a**) Time sequence of maintenance activities; (**b**) Job sequence of crews. In the bridge maintenance practice, two empirical strategies, i.e., flow-first scheduling strategy (FFSS) [38] and worst-first scheduling strategy (WFSS) [39] are two commonly adopted strategies. FFSS determines the time sequence of bridge maintenance activities by traffic flow from highest to lowest. WFSS determines the time sequence of maintenance activities by bridge performance from lowest to highest. Additionally, both the two strategies try to cover as many bridges as possible under the budget limit. Table 4 presents the traffic flow and performance ranking of each bridge in the pre-maintenance period. According to Table 4, FFSS and WFSS generate the maintenance schedules as in Figures 4 and 5. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(i\) & \(a\) & \(q_{a}^{*}\) **(veh/day)** & \(R\) & \(i\) & \(a\) & \(q_{a}^{*}\) **(veh/day)** & \(R\) \\ \hline \multirow{2}{*}{A} & 1–2 & 3647 & \multirow{2}{*}{7} & \multirow{2}{*}{H} & 8–9 & 1377 & \multirow{2}{*}{9} \\  & 2–1 & 2752 & & & 9–8 & 1491 & \\ \hline \multirow{2}{*}{B} & 2–3 & 1231 & \multirow{2}{*}{10} & \multirow{2}{*}{I} & 9–10 & 1973 & \multirow{2}{*}{1681} \\  & 3–2 & 1127 & & & 10–9 & 1681 & \\ \hline \multirow{2}{*}{C} & 3–4 & 1486 & \multirow{2}{*}{3} & \multirow{2}{*}{J} & 11–13 & 3007 & \multirow{2}{*}{6} \\  & 4–3 & 1265 & & & 13–11 & 2461 & \\ \hline \multirow{2}{*}{D} & 3–4 & 2065 & \multirow{2}{*}{13} & \multirow{2}{*}{K} & 12–14 & 2091 & \multirow{2}{*}{2} \\  & 4–3 & 2329 & & & 14–12 & 2177 & \\ \hline \multirow{2}{*}{E} & 4–5 & 1863 & \multirow{2}{*}{12} & \multirow{2}{*}{L} & 1–14 & 4210 & \multirow{2}{*}{11} \\  & 5–4 & 1463 & & & 14–1 & 2806 & \\ \hline \multirow{2}{*}{F} & 5–7 & 2149 & \multirow{2}{*}{4} & \multirow{2}{*}{N} & 1–14 & 3157 & \multirow{2}{*}{8} \\  & 7–5 & 1432 & & & 14–1 & 3859 & \\ \hline \multirow{2}{*}{G} & 6–10 & 2852 & \multirow{2}{*}{5} & \multirow{2}{*}{} & \multirow{2}{*}{} & \multirow{2}{*}{} \\  & 10–6 & 2530 & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Traffic flow and performance ranking (\(R\)) of each bridge. Figure 3: Day-to-day fluctuation of travel time in the network. Table 5 lists the results of the OMSS, FFSS and WFSS respectively.