## 2 Literature Review
### Train Timetabling Problem  
TTP includes periodic timetabling, aperiodic timetabling, and mixed timetabling. Mixed timetabling is a combination of periodic timetabling and aperiodic timetabling. Many researchers use the periodic event scheduling problem (PESP) model to study the periodic TTP. In 1996, [PERSON] et al. [22] proposed that different types of trains can have different operation cycles. According to different types of trains, the operation line of each type of train shall schedule the timetable according to their respective cycles. The model stipulates that each type of train has the same starting and ending points, train routes, stop plans. [PERSON] et al. [23] found that the train running time is fixed in some models but in the actual situation they are unfixed. Thus, they estimated that the train running time is unfixed and then designed a model based on the PESP model. [PERSON] et al. [24] proposed a hybrid multivariable hydraulic algorithm to solve the PESP model. In the model, multiple kinds of trains are operated on a single-track railway. [PERSON] et al. [25] studied the periodic timetabling problem models actually applied in the Netherlands railways and the Berlin subway. They additionally considered passenger paths and line frequencies and then verified the model on the railway in Germany. A case study showed that the travel time of passengers is shorter in the extended model. Based on their previous studies, [PERSON] et al. [26] further studied the periodic train timetable's intersection and transfer connection. [PERSON] et al. [27] describe the method of applying the PESP to the existing periodic timetable scheduling problem, the shortcomings of the existing models are analyzed, and an extended model allowing partial deviation of the running time is proposed. [PERSON] et al. [28] divide the railway line into a series of independent units. They established a decision-making model for the time and priority of the train. [PERSON] et al. [29] established a model for scheduling timetables for different types of trains. In this model, the operation line of each train is scheduled according to its own period. Each type of train has the same starting and ending points, train path, stop plan and speed. [PERSON] et al. [30] established a cycle timetable optimization model for pursuing the minimum cycle time. They believe that a short cycle length can make the timetable more stable. [PERSON] et al [31] proposed a method of scheduling a robust timetable. They first generate a line plan, then schedule a timetable based on the line plan by PESP model, and feedback the quality of the line plan with the buffer time of the timetable. The variable time length of the critical line makes the method more flexible. They verified the algorithm with the railway network structure of Copenhagen, and the results show that robustness can be improved. [PERSON] et al. [32] transformed the mathematical model of PESP into a multi commodity network flow model with two coupled schedule networks and sidetrack capacity constraints. Through this model, they synchronize limited operational resources toward a master periodic schedule of transport services. In terms of aperiodic timetable, [PERSON] et al. [33] proposed a multi-objective model, in which fuel consumption cost is adapted to evaluate the operation department's satisfaction. [PERSON] et al. [34] proposed the discrete event model of trains to dispatch trains in the railway network. In this model, they adopted the local feedback-based travel advance strategy to deal with the disturbances in the timetable. Considering that both the section capacity and the station capacity are limited resources, [PERSON] et al. [35] add a set of operational constraints. The branch and bound algorithm and the Lagrangian relaxation algorithm are adapted to calculate the train timetable. [PERSON] et al. [36] put forward a model in which all variables are related to the train running time. They solved the model using heuristic and exact algorithms.