## 4 Model Description  
In this study, the concept of RCPSP is used to find a proper plan for bridge maintenance. A one-time maintenance strategy of the bridge and its effectiveness are shown in Figure 1.  
The plan includes two periods. These are named the maintenance period and the effect period in this study. Additionally, the total period of a plan is named as the functional period in this study. Table 2 shows the parameters of the two periods corresponding to RCPSP. The cost incurred during the maintenance period is the construction cost, and the cost incurred during the effect period refers to the daily expenditure for maintenance operations. The overall duration (the aforementioned functional period) includes the maintenance period and the effect period. Considering the implementation time of maintenance work, this study concludes the following points based on [32]: if a bridge is maintained earlier, the operating period can be extended more, suggesting a longer effect period; meanwhile, if the maintenance is carried out late, the effect period becomes shorter. Figure 2 elaborates on this concept, with the horizontal and vertical axes referring to the time and the condition of the bridge, respectively. The larger the vertical axis value, the better \begin{table}
\begin{tabular}{c c} \hline \hline
**RCPSP** & **Bridge Maintenance in This Study** \\ \hline Activity & Plan \\ \hline Duration & Functional period \\ \hline Cost & Maintenance cost \\ \hline Facility Resource & Bridge component \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Relationship between bridge maintenance and resource-constrained project scheduling problem (RCPSP) framework. Figure 1: Illustration of bridge maintenance effect. \begin{table}
\begin{tabular}{c c c} \hline \hline  & \multicolumn{3}{c}{**Parameters in RCPSP**} \\ \hline Plan period & Cost & Duration \\ \hline Maintenance period & Maintenance cost & Maintenance duration \\ \hline Effect period & Daily maintenance cost & Operation duration after maintenance \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of two periods corresponding to RCPSP. the bridge's condition. However, the model retains flexibility in the setting of parameters, where the parameters can be adjusted according to the actual conditions. Moreover, as shown in Figure 2, this study classifies the implementation of the plan into three scenarios (i.e., early, middle, and late) and refers them as early, middle, and late plans, respectively. These plans are implemented on the bridge in its current state (i.e., early and middle plans) at three different time frames. Therefore, the selection of plans will have an impact on the remaining operation duration of the current plan of the bridge. Naturally, the earlier the plan is executed, the longer the remaining duration of the leading plan will be sacrificed. As shown in Figure 3, subsequent plans change the duration of the current plan. In general, the early plan significantly reduces the functional period of the leading plan, while the late plan does not affect it. Greater variation can be observed while reducing the functional period of the leading plan. When there are different leading programs and an early plan, a middle or late plan is selected, shown in Figure 4.  
Figure 3: Impact of different plans on the leading plan. Figure 2: Diagram of different plans. This difference can be represented by an influence matrix (Table 3). Each value in the matrix represents the amount of service life lost for each plan combination. The influence matrix can be used to find the best plan combination. Overall, this study proposes two models according to the actual requirements. The recursive yearly implementation process is shown in Figure 5. Subordinate authorities use Model-I to evaluate the annual maintenance cost for all bridges in the planning cycle, and use the results of the model as a reference basis for utilizing the following year's budget, though the result of Model-I is calculated in a budget-free setup.