## 3 Methodology
### Notation  
The notations used in this study are given as follows:  
**Sets and indices**  
\(A\): Set of all links in the highway network, indexed by \(a\in A\). \(I\): Set of bridges to be maintained or set of maintenance activities, indexed by \(i\in I\). \(S\): Set of all crews, indexed by \(s\in S\). \(W\): Set of origin-destination (OD) pairs, indexed by \(w\in W\). \(K_{w}\): Set of paths that connect OD pair \(w\), indexed by \(k\in K_{w}\). **Parameters**  
\(d_{i}\): Duration of the maintenance activity \(i\). \(c_{i}\): Maintenance cost of bridge \(i\). \(B\): Availability of budget. \(T\): Length of a discrete time period. \(t\): Time period, \(t\ =\ 1,\ 2,\ \cdots,\ T\). \(q_{w}\): Traffic demands of OD pair \(w\). \(l_{a}\): Length of link \(a\). \(V_{0a}\): Pre-maintenance speed limit of link \(a\). Parameters of BRP function. BRP function proposed by U.S. Bureau of Public Roads is used to formulate the relationship between travel time and traffic flows. \(\alpha\) is the correction factor and \(\beta\) indicates the growth rate of travel time \(C_{0a}\) Pre-maintenance traffic capacity of link \(a\). Pre-maintenance traffic flow on link \(a\) under user equilibrium (UE) state. UE state is reached  
\(q_{a}^{*}\) when, for each OD pair, the actual route travel time experienced by travelers within a traffic  
network is equal and minimal. **Variables**  
\(Z\) Cumulative traffic delays in the network during the makespan. \(u_{0a}\) Pre-maintenance free-flow travel time of link \(a\). \(M\) Makespan of the maintenance scheduling scheme. \(U_{t}\) Travel time in the network at time \(t\). \(N\) Number of bridges maintained. \(U^{*}\) Pre-maintenance travel time in the network. \(c\) Total maintenance costs. \(q_{a}^{t}\) Traffic flow on link \(a\) at time \(t\). \(C_{a}^{t}\) Traffic capacity of link \(a\) at time \(t\). If the bridge \(i\) maintained at time \(t\) lies on link \(a\), \(C_{a}^  
where Equation (1) is the objective function, which guarantees the minimum cumulative traffic delays during the makespan \(M\). \(\sum_{t=1}^{M}U_{t}\) is the total travel time during the makespan \(M\). The other objective function in Equation (2) aims to cover as many bridges to be maintained as possible with the budget limit. Equation (3) defines the maintenance makespan as the maximum end time for all maintenance activities \(\max\limits_{i\in I,s\in S}\{\sum_{t=1}^{T}t\cdot x_{ist}+d_{i}\}\). Equation (4) is the budget constraint. \(\sum\limits_{i\in I}\sum_{t=\max\{1,t-d_{i}+\ 1\}}^{t}x_{ist}\) in Equation (5) is denoted as the number of ongoing maintenance activities carried out by crew \(s\) at time \(t\), i.e., each crew can maintain at most one bridge at time \(t\). Equation (6) represents that the amount of simultaneous maintenance activities cannot exceed the number of crews at time \(t\). Equation (7) defines the types of decision variables. **The lower-level model**  
\[\min\ U_{t}=\sum\limits_{a\in A}\int_{0}^{d_{a}^{t}}u_{a}^{t}(q)\mathrm{d}q,\ t =\ 1,\ 2,\ \cdots,\ T \tag{8}\]  
Subject to  
\[\sum\limits_{k\in K_{w}}h_{k}^{w,t}=\ q_{w},\ w\in W \tag{9}\]  
\[\sum\limits_{w\in W}\sum\limits_{k\in K_{w}}h_{k}^{w,t}\delta_{a,k}^{w,t}=\ q_{a}^{t},\ a\in A \tag{10}\]  
\[h_{k}^{w,t}\ \geq\ 0,\ w\in W,\ k\in K_{w} \tag{11}\]  
\[u_{0a}=\ l_{a}/V_{0a},\ \forall a\in A \tag{12}\]  
\[u_{a}^{t}(q)\ =\ u_{0a}\Big{[}1\ +\ \alpha\Big{(}q_{a}^{t}/C_{a}^{t}\Big{)}^{ \beta}\Big{]},\ \forall a\in A \tag{13}\]  
\[\delta_{a,k}^{w,t}\in\{0,1\},\ a\in A,\ w\in W,\ k\in K_{w} \tag{14}\]  
where Equations (8)-(11) define the Beckmann's transformation [32] to simulate the traffic assignment under UE state at time \(t\). Herein, the objective function in Equation (8) minimizes the total travel time in the network. Equation (9) denotes that the traffic demands of any OD pair are the sum of traffic flows on all paths linking the OD pair. Equation (10) indicates that the traffic volume on every single link is the sum of traffic volume on all paths on which the link lies.