## 2 Literature Review  
The studies of network-level bridge maintenance optimization can be classified into two groups, in terms of the length of time horizon: single-stage maintenance problem and multi-stage maintenance problem [6; 7; 8]. This research falls in the first group. For the single-stage maintenance problem, it aims to identify the prioritization of a set of bridges to be maintained in a network under budget limits. Combinatorial optimization theory is the most used tool in dealing with this problem, which formulates the problem as linear or nonlinear integer programming or mixed-integer programming models. For instance, [PERSON] and [PERSON] solve the bridge annual maintenance prioritization using multi-objective combinatorial optimization, which minimizes the largest condition index and the present value of maintenance cost as well as maximizes the smallest safety index [6]. Additionally, the maximization of bridge quality [9], maximization of tolerable condition index [10], maximization of service life [7; 11], minimum user cost in the bridge network [12], minimum cumulative expected failure cost [13], maximum structural reliability [14], etc. are also considered to be objective functions in the literature. Recently, the concept of resilience was highlighted in the studies of infrastructure systems, and the maximization of the bridge network resilience was also taken into consideration. For example, [PERSON] and [PERSON] define bridge network resilience as the integral in time of the network functionality and present a linear programming model with two conflicting objectives, i.e., maximum resilience and minimum maintenance cost to generate an optimal bridge maintenance schedule [15]. In general, two approaches exist to solve the single-stage maintenance problem: the top-down approach and the bottom-up approach [16; 17]. The top-down approach assumes that all the facilities in the transportation network have homogeneous features, e.g., traffic patterns, properties of performance deterioration [18], while the bottom-up approach assumes that the facilities have different characteristics, which can capture facility-specific features [19; 20]. [PERSON] and [PERSON] apply a top-down approach to solve selecting and scheduling maintenance activities on networks of transportation facilities under uncertainty [21]. [PERSON] and [PERSON] deal with the problem of optimizing bridge maintenance and replacement decisions over a finite planning horizon using a bottom-up solution method [22]. The multi-stage maintenance problem is usually formulated as dynamic programming models. Numerous studies employ Markov decision processes (MDP) and its variants to generate a long-term maintenance plan [23; 24]. MDP simulates the discrete performance conditions of transportation facilities with the state transition equation. [PERSON] et al. propose a Markov chain model to simulate the propagation phase of performance deterioration over time, which is incorporated into modeling the optimal maintenance strategy [25]. [PERSON] and [PERSON] put forward a partially observable Markov decision processes model combined with a non-linear action model for the optimal decision making of transportation infrastructure [26]. Additionally, Monte Carlo simulation also has been applied to solve the multi-stage maintenance problem [27; 28; 29]. However, the maintenance plans obtained by the above methods are randomized policies that are probabilistic in nature so that they cannot determine the specific maintenance strategy of every single facility in practice. To fill this gap, some other researchers adopt the approximate dynamic programming (ADP) models. For example, [PERSON] introduces an ADP-based model to overcome the network-level infrastructure maintenance problem under budgetary restrictions and resource constraints [30].