## 4 Model Description
### Model-I: Life Cycle Lifespan Evaluation Model  
All model parameters and variables are shown in Table 4. To solve the long-term maintenance budget allocation problem, Model-I maximizes the functional periods of all bridges from the perspective of life cycle under a given planning time. Therefore, through the concept of limited resource scheduling, the related constraints are described as follows:  
\[\forall_{i,j,k}\big{(}MS_{ijk}=BS_{ijk}+BD_{ijk}\big{)} \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Statement** \\ \hline \(f\) & Total number of considered bridges \\ \(r\) & Total number of considered bridge components \\ \(t\) & Total number of plans \\ \(c\) & Number of plan cycles in a life cycle \\ \(q\) & Number of years in a life cycle \\ \(i\) & Bridge number, where i = 1, 2, 3,..., f \\ \(j\) & Bridge component number, where, j = 1, 2, 3,..., r \\ \(k\) & Plan cycle number, where k = 1, 2, 3,..., c \\ \(p\) & Plan selection, where p = 1, 2, 3,..., t, for example, early/middle/late \\ \(u\) & Year number, where u = 1, 2, 3,..., q \\ \(m\) & Current plan selection, where m = 1,2,3,..., f, for example, early/middle/late \\ \(w_{i}\) & Weight of bridge \\ \(bd_{ijp}\) & A plan’s maintenance period for a facility (i.e., bridge component) \\ \(md_{ip}\) & A plan’s effect period for a facility \\ \(pbd_{ij}\) & Facility maintenance period in the current plan \\ \(pmd_{ij}\) & Facility effect period in the current plan \\ \(rul_{j,k}\) & Resource limit in a year \\ \(buc_{ip}\) & Facility maintenance cost of a plan \\ \(muc_{ip}\) & Facility effect period cost per year \\ \(burip\) & Possible maintenance period of the plan for each facility \\ \(mur_{ip}\) & Possible effect period of the plan for each facility \\ \(cul_{u}\) & Budget limit in year \(u\) \\ \hline \hline \multicolumn{2}{c}{Variable} & Statement \\ \hline \(BD_{ijk}\) & Maintenance period for a facility (component \(j\) of bridge \(i\) in plan cycle \(k\)) \\ \(MD_{ijk}\) & Effect period for a facility (component \(j\) of bridge \(i\) in plan cycle \(k\)) \\ \(BS_{ijk}\) & Start time of a plan’s maintenance period (component \(j\) of bridge \(i\) in plan cycle \(k\)) \\ \(MS_{ijk}\) & Start time of a plan’s effect period (component \(j\) of bridge \(i\) in plan cycle \(k\)) \\ \(BF_{ijk}\) & End time of a plan’s maintenance period (component \(j\) of bridge \(i\) in plan cycle \(k\)) \\ \(MT_{ijk}\) & End time of a plan’s effect period (component \(j\) of bridge \(i\) in plan cycle \(k\)) \\ \(PBD_{ij}\) & Maintenance period for a facility in the current state (component \(j\) of bridge \(i\)) \\ \(PMD_{ij}\) & Effect period for a facility in the current state (component \(j\) of bridge \(i\)) \\ \(PBS_{ij}\) & Start time of maintenance period for a facility in the current state (component \(j\) of bridge \(i\)) \\ \(PMS_{ij}\) & Start time of effect period for a facility in the current state (component \(j\) of bridge \(i\)) \\ \(PBF_{ij}\) & End time of maintenance period for a facility in the current state (component \(j\) of bridge \(i\)) \\ \(PMF_{ij}\) & End time of effect period for a facility in the current state (component \(j\) of bridge \(i\)) \\ \(BSS_{ipp}\) & Start time of a plan’s maintenance period for Model-II (component \(j\) of bridge \(i\) with plan selection \(p\)) \\ \(MSS_{ipp}\) & Start time of a plan’s effect period for Model-II (component \(j\) of bridge \(i\) with plan selection \(p\)) \\ \(RUB_{ijku}\) & Use of resources when a bridge component is in the maintenance period (component \(j\) of bridge \(i\) in plan cycle \(k\) in year \(u\)) \\ \(RUM_{ijku}\) & Use of resources when a bridge component is in the effect period (component \(j\) of bridge \(i\) in plan cycle \(k\) in year \(u\)) \\ \(TU_{ijk}\) & Plan choice (component \(j\) of bridge \(i\) in plan cycle \(k\)) \\ \(PTU_{ijk}\) & Current plan (component \(j\) of bridge \(i\) in plan cycle \(k\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Model parameters and variables.