## 2 Literature Review  
The line planning problem (LPP) is a critical issue in railway planning which can balance demand and operation facilities. Line planning is usually based on the time-space distribution of demand, line capacity, and vehicle resources, which determines the train routes, frequency, stop patterns, and composition mode. The effectiveness of a line plan is often judged by the level of service and the associated travel costs from the passenger point of view and by the operation costs and facility utilization factors from the operators' point of view [8]. In general, the development of LPP has experienced a process from simple to complex, from static demand to dynamic demand, from single line planning optimization to integrate optimization of another operational planning process. There have been numerous LPP studies that focus on sub-processes, such as those that are mainly concerning route design [9], frequency setting [10; 11], stop planning [12; 13], and a combination of the above [14; 15]. Below, we briefly introduce the critical literature that is relevant to this paper. LPP in transportation can go back to 1965 [16], which introduced the basic theoretical framework. Since then, numerous studies and reviews [6; 8; 17] have been conducted on LPP for public transportation. Initially, LPP was always designed with a static demand because of its complexity, e.g., a day, a certain part of the day, an hour [6; 18; 19]. The line planning literature has immensely progressed by considering the demand for transport as static, and problem formulations suppose that the demand rate stays the same for a given origin-destination during the planning horizon. The intercity railway systems are subject to a high level of variation and fluctuation in demand over the day. When this variation and fluctuation are observed in both time and space, it is crucial to develop line plans that are responsive to demand. Then, some scholars questioned the appropriateness of using the same line plan over the day [8; 20]. When formulating the train line planning, the railway department always considers the impact of the fluctuation characteristics in both time and space of the passenger demand. For example, the \"time-varying\" characteristics of passenger flow demand urge the railway department to operate multi-period line planning under the flexible composition mode. The \"space non-equilibrium\" characteristics may cause the imbalance distribution of the number of carriage resources at different stations. In this case, the railway department also needs to consider the dynamic resources when formulating line planning. So, when the demand is extremely unsteady and asymmetric, the traditional line planning methods are not convincing and an approach that can adapt to the demand fluctuation is needed. Some scholars have taken the average daily passenger flow as the basis and considered the elastic characteristics of demand to a certain extent [7; 15; 21]. Recently, some studies have gradually focused on the time variability of demand [6; 22; 23; 24; 25] and they always apply the multi-period planning method to remedy any defects. While a multi-period planning approach is a remedy for the demand fluctuation, the problem becomes more complicated due to the coupling constraints [26]. Although constraints for a period allocate essential resources to activities, coupling constraints transfer the resources among activities from one period to the next. In this respect, a multi-period LPP comes inherently with resource constraints and their allocation throughout periods because a multi-period line plan needs to be feasible during each period concerning the associated resources as well. This necessity leads to the integration of line planning decisions with decisions on allocating resources. Integrations between the different levels of decision-making in transportation systems planning have been a topic of general interest in the literature.