## 4 Model Description
### Model-I: Life Cycle Lifespan Evaluation Model
Formula (1) indicates the starting time at which any plan enters the effect period upon maintenance completion, thus ensuring the above definition that a plan must consist of two periods, i.e., the maintenance and effect periods. \[\forall_{i,j}\big{(}BS_{ij,k+1}=MS_{ijk}+MD_{ijk}\big{)} \tag{2}\]  
Formula (2) indicates that the starting time of the next plan is equal to the end of the defined service life of the previous plan, thus confirming that the plans need to be continuous and uninterrupted in the time series. \[\forall_{i,j}\big{(}MF_{ijc}\geq q\big{)}\;for\;k\;=\;c \tag{3}\]  
Formula (3) ensures the solution meets the defined considered years. \[ \begin{array}{l}if\;\forall_{i,j,k}\big{(}TUI_{ijk,k-1}=p\big{)}\;\&\;\Big{(} TUI_{ijk}=m\big{)}\\ then\;\forall_{i,j,k,p,m}\left(MD_{ij,k-1}\leq MD_{ijk,k-1}-DC_{ijpm}\right)\\ i\;=\;1\;\sim f\;,\;j\;=\;1\sim r,\;k\;=\;2\;\sim c,\;p\;=\;1\;\sim t,\;m\;=\; 1\;\sim t\end{array}  \tag{4}\]  
Formula (4) indicates that the effect period for a bridge is defined after the selection of the current and subsequent operation plans. \(DC_{ijpm}\) is the influence parameter and its values are shown in Table 3.  
\[ \begin{array}{l}if\;\forall_{i,j,k}\big{(}TUI_{ijk}=p\big{)}\\ then\;\forall_{i,j,k}\left(BD_{ijk}=\mathrm{bd}_{ijP}\right)\;and\;\left(MD_{ ijk}\leq md_{ijP}\right)\end{array}  \tag{5}\]  
Formula (5) indicates that when a plan \(p\) is selected, its maintenance period and service life are also defined. \[\sum_{u=1}^{q}\sum_{k=1}^{c}\sum_{j=1}^{r}\sum_{i=1}^{f}\big{(}TUI_{ijku}\big{)} \geq\sum_{u=1}^{q}\sum_{k=1}^{c}\sum_{j=1}^{r}\sum_{i=1}^{f}\big{(}RUB_{ijku} +RUM_{ijku}\big{)} \tag{6}\]  
Formula (6) sets the upper limit of resource usage. The resources here, refer to bridges, so that each facility in use (i.e., a facility undergoing a maintenance plan) cannot be used by another plan. This ensures that the plans will not be repeated at the same time. \[\forall_{i,j}\big{(}PBS_{ij}\big{)}=0 \tag{7}\]  
Formula (7) indicates that the start time for the current maintenance activity must be 0, regardless of whether the plan is implemented in the maintenance period or in the effect period, the current operation must be controlled at the onset of the planning period. If the bridge is being constructed using a maintenance plan, the model will enter the maintenance period of the plan. If the plan has \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Statement** \\ \hline \(DC_{ijpm}\) & Duration in the Influence Matrix (component \(j\) of bridge \(i\) to plan selection \(p\) from current plan selection \(m\)) \\ \(DS_{ijp}\) & Binary variable representing whether a facility is under maintenance, \(DS_{ijp}\in[0,\;1]\) \\ \(BD_{ijp}\) & (component \(j\) of bridge \(i\) with plan selection \(p\)) \\ \(BD_{ijp}\) & Maintenance duration for Model-II (component \(j\) of bridge \(i\) with plan selection \(p\)) \\ \(MD_{ijp}\) & Effect duration for Model-II (component \(i\) of bridge \(i\) with plan selection \(p\)) \\ \(BUC_{ijpm}\) & Maintenance cost for Model-II (component \(j\) of bridge \(i\) with plan selection \(p\) in year \(u\)) \\ \(MUC_{ijpm}\) & Maintenance and effect period cost per year for a facility (component \(j\) of bridge \(i\) with plan selection \(p\) in year \(u\)) \\ \(BUR_{ijp}\) & Binary variable representing whether facility resources are being used for facility maintenance period, \(BUR_{ijp}\in[0,\;1]\) (component \(j\) of bridge \(i\) with plan selection \(p\)) \\ \(MUR_{ijp}\) & Binary variable representing whether facility resources are being used for facility maintenance and effect period, \(MUR_{ijp}\in[0,\;1]\) (component \(j\) of bridge \(i\) with plan selection \(p\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._been implemented and is in the effect period, the maintenance duration is 0 and the remaining effect period is entered. \[\forall_{i,j}\Big{(}PBF_{ij}=PMS_{ij}\Big{)} \tag{8}\]  
Formula (8) ensures that the end of the maintenance periods of all bridges equals to the beginning of their effect periods.