## 4 Model Description
### Model-I: Life Cycle Lifespan Evaluation Model
\[\forall_{i,j}\Big{(}PMF_{ij}=BS_{ij,1}\Big{)}\ i\ =\ 1\ \sim f,\ j\ =\ 1\ \sim r,\ k=1 \tag{9}\]  
Formula (9) indicates the end time of the current effect period, which is equal to the start time of the first plan. \[\forall_{i,j}\Big{(}PBD_{ij}=pbd_{ij}\Big{)} \tag{10}\]  
Formula (10) indicates that the current maintenance period parameter \(pbd_{ij}\) in the model is entered as the input into the current maintenance period variable \(PBD_{ij}\). Its value is more than 0 during maintenance, and 0 after maintenance. \[\forall_{i,j}\Big{(}PMD_{ij}\leq pmd_{ij}\Big{)} \tag{11}\]  
The remaining effect period of the current operation must also be entered as the input parameter into the model. Formula (11) indicates that the remaining time parameter \(pmd_{ij}\) in the model is entered as the input into the current remaining effect period variable \(PMD_{ij}\). However, since the remaining effect period may vary due to the selection of subsequent plans, \(PMD_{ij}\) is less than or equal to  
Formula (15) sets the maintenance period of the operation and its starting time as 0. \[ \begin{array}{l}\textit{if }\forall_{i,j,p}\left(DS_{ipp}=1\right)\\ \textit{then}\forall_{i,j,p}\left(BDS_{ijp}=bd_{ipp}\right)\\ \&\&\left(BUC_{ipp}=buc_{ipp}\right)\\ \&\left(BUR_{ipp}=1\right)\end{array}  \tag{16}\]  
Formula (16) indicates that when a facility is maintained, the maintenance time and maintenance budget will be brought into the model calculation. \(BUR_{ipp}=1\) indicates that the facility resources are being used. \[ \begin{array}{l}\textit{if }\forall_{i,j,p}\left(DS_{ipp}=1\right)\\ \textit{then}\forall_{i,j,p}\left(MDS_{ip}=md_{ipp}\right)\\ \&\left(MUC_{ip}=muc_{ipp}\right)\\ \&\left(MUR_{ipp}=1\right)\end{array}  \tag{17}\]  
Formula (17) also indicates that if a certain part of a bridge has been determined to be maintained according to a certain maintenance plan, then the corresponding maintenance period, cost, and resources required for the maintenance plan to keep the facility in effect are also determined. \[ \begin{array}{l}\textit{if }\forall_{i,j,p}\left(DS_{ip}=0\right)\\ \textit{then}\forall_{i,j,p}\left(BDS_{ijp}=0\right)\\ \&\left(BUC_{ipp}=0\right)\\ \&\left(BUR_{ip}=0\right)\end{array}  \tag{18}\]  
\[ \begin{array}{l}\textit{if }\forall_{i,j,p}\left(DS_{ip}=0\right)\\ \textit{then}\forall_{i,j,p}\left(MDS_{ip}=0\right)\\ \&\left(MUC_{ip}=0\right)\\ \&\left(MUR_{ip}=0\right)\end{array}  \tag{19}\]  
Formulas (18) and (19) indicate that if the maintenance plan has not occurred, the cost, maintenance period, and resource usage during the maintenance and effect periods will not occur. The function of these constraint equations is that when the budget is insufficient or if the bridge requiring maintenance cannot be maintained, the corresponding parameter values will not be included into the variables. \[\sum_{p=1}^{t}\sum_{j=1}^{r}\sum_{i=1}^{f}\left(burr_{ipp}\right)\geq\sum_{p=1 }^{t}\sum_{j=1}^{r}\sum_{i=1}^{f}\left(BUR_{ipp}\right) \tag{20}\]  
According to the resource constraints, the total amount of facility resources for the maintenance operation in formula (20) cannot exceed the upper limit of the original facility resources. \[\sum_{p=1}^{t}\sum_{j=1}^{r}\sum_{i=1}^{f}\left(mur_{ipp}\right)\geq\sum_{p=1 }^{t}\sum_{j=1}^{r}\sum_{i=1}^{f}\left(MUR_{ip}\right) \tag{21}\]  
Similarly, Formula (21) indicates that the total amount of facility resources for all operations, to keep the facility in the effect period, cannot exceed the upper limit of the original facility resources. \[\textit{cul}_{u}\geq\sum_{p=1}^{t}\sum_{j=1}^{r}\sum_{i=1}^{f}\left(BUC_{ipu }+MUC_{ipu}\right) \tag{22}\]  
Formula (22) indicates that the sum of the costs spent in all the maintenance and effect periods cannot exceed the budget ceiling for that year \(u\). Under the condition that only a single year is considered, the objective function is shown in (23). In response to possible budget shortfall, the objective formula adds the variable \(MUR_{ijp}\), which reflects whether facility resources are being used.