## 2 Literature Review
### Integrated Optimization of Line Planning and Timetabling  
In recent years, the integrated optimization of line planning and timetabling has become popular ([PERSON] et al., 2018; [PERSON] et al., 2019; [PERSON] et al., 2020). Most research on the integrated optimization of frequency and timetabling assumes that the demand is evenly distributed in each period during the study period ([PERSON] et al., 2018; [PERSON] et al., 2018). This strategy is easy to apply and performs well when the train capacity is sufficient under uncongested conditions ([PERSON] et al., 2018). To adapt to time-dependent passenger demands under congested conditions, [PERSON], [PERSON] et al. (2018) presented a nonlinear non-convex programming model to design timetables with the objective of minimizing the total passenger travel time (TTT) by considering variable numbers of trains, train running times, and train dwell times.  
In research on the integrated optimization of operation zone and timetabling, [PERSON], [PERSON] et al. (2018) developed a mixed-integer nonlinear programming (MINLP) model for train scheduling with full-length and short-turn train services to minimize the headway deviation and number of trains. They solved the main decision variables, i.e., the types of train services, departure times, and arrival times, for each train at the origin and destination station based on the predefined headway obtained by passenger demand analysis using the CPLEX solver. The performance of the model, including its reduction in the utilization of trains, was evaluated through a case study on Beijing Subway Line 4.  
Research on the integrated optimization of stopping patterns and timetabling is mainly conducted in the field of railways ([PERSON] et al., 2018; [PERSON] et al., 2018; [PERSON] et al., 2019; [PERSON] et al., 2020). [PERSON] et al. (2020) developed a mixed-integer programming model to describe the problem of high-speed railways with the aim of minimizing the train dwell time at intermediate stations and delays in expected departure time.  
They used continuous variables to represent the train arrival and departure times and 0-1 variables to represent the train stop plan and indicator of train departure order. Meanwhile, in the field of urban rail transit, [PERSON] et al. [41] described the integrated problem under a stop-skipping strategy as a bi-level MINLP model to reduce passenger travel time and energy consumption with penalties that ensured the service level of passengers on the last train. The high-level decision variable represented train stopping patterns, and the low-level variable represented the speed of the train speed holding phase and departure times. In the model, trains could only skip the stations that belonged to the predefined skipping set with no overtaking of other trains, and passengers could only take one train to arrive at their destination, i.e., transfer between different trains along the line was not allowed.  
Most research on the integrated optimization of train formation and timetabling focuses on the field of public buses and railways and includes frequency optimization, because train formation is usually combined with frequency to determine the capacity distribution [42]. [PERSON] et al. [10] proposed three flexible timetable optimization methods based on hybrid vehicle size, large vehicle size, and small vehicle size to minimize the vehicle operational costs and TTT by optimizing the vehicle size and scheduling interval (similar to headway in urban rail transit), which determine the frequency. The results of a case study indicated that the hybrid vehicle size model outperformed the other two models in both total time and total cost.