## 1 Introduction  
With the aging of a bridge, the deterioration of materials and downgrading service level, as well as natural disasters such as typhoons and earthquakes, all increase the potential for bridge damage and require frequent and routine maintenance. For preventing such disasters caused by unsafe implementation of bridge maintenance, governments invest in bridge maintenance projects every year to ensure the safety of people and property. For example, in the United States, the cost of repairing and replacing deteriorating bridges was approximated as USS 100 billion in 2010 [1]. Therefore, the cost optimization of bridge structure maintenance becomes an important challenge in bridge management problems [2]. In recent years, it has been theoretically assumed that the budget plan should be prepared from the life cycle perspective [3; 4; 5; 6; 7; 8], where the ideal budget plan is somewhat unrealistic in terms of being applicable to the real world. However, solutions to resolve the mismatch between the planned budget and the awarded budget have been rarely discussed in previous studies related to bridge maintenance problems. Thus, with practical considerations, a crucial question arises, the question of exploring how to allocate the budget where the life cycle maintenance plan violates the current budgeting policy. Such policy only complies with the needs of short-term budget expenditure and not the assessment of concerns that apply to the long-term life cycle. Thus, there exits difficulties in budget implementation, owing to the different perspectives of budget planners and budget reviewers. In Taiwan, budgeting is prepared based on a bottom-up flow. Competent authorities (e.g., budget planners) submit proposed annual budgets to higher-level authorities (e.g., budget reviewers), according to their own current situation. Subsequently, the higher-level authorities review the submitted plans, contemplate annual expenditures holistically, and allocate limited budgets. As a result, the budget plan for the following year may not necessarily be approved, as has been the case for previous proposals regarding each facility. This may be due to insufficient funds in the current year, policy changes, and other reasons, resulting in some facilities failing to obtain funds or a sufficient budget. Budget allocation does not meet the requirements of lower-level authorities (budget recipients). Thus, they must readjust maintenance plans accordingly to achieve maximum benefits, where some plans may be evaluated through a life cycle point of view. Despite a reasonable bridge maintenance plan based on a life cycle based analysis, the required budget can often not be fully obtained owing to the aforementioned practical circumstances, which fit into the domain of resource-constrained project scheduling problems (RCPSP) [9; 10; 11; 12]. The RCPSP mainly focuses on a contractor's constraints in terms of resource possession and the available quantity of usable resources. Thus, an upper limit is set regarding the usage of each type of resource, and the project is then scheduled. Therefore, if each bridge requiring maintenance is regarded as a resource according to the RCPSP definitions, as long as the resource is being used continuously, the serviceable condition of all the bridges can be ensured. Thus, the concept of bridge maintenance scheduling is consistent with the goal of resolving the RCPSP. In addition, project budget limitation in RCPSP is similar to the condition of limited budget utilization caused by the present reduced maintenance budget. Therefore, this study proposes the application of the RCPSP in solving maintenance scheduling problems for bridges under budget constraints. To aid problem solving, two-stage optimization models are presented in this paper. IBM ILOG CPLEX Optimizer software and the constraint programming (CP) approach is adopted for model development [13].