## 2 Problem Description  
A crowded railway line linking the city center to the suburbs is studied in this paper, as shown in Figures 2 and 3. \(S\) is used to represent the stations where the trains will perform the task of loading passengers, i.e., \(S=\{1,2,3,\ldots,n,n+1,n+2,\ldots,2n\}\).  
In Figure 2, it is obvious that the number of trains running in both directions is the same, following the strategy called the T-TPOS. The T-TPOS embodies the advantages of being simple and easy to realize, but it also has many shortcomings when dealing with the problems caused by tidal passenger flow. If T-TPOS is implemented on tidal passenger flow lines, the total train capacities in the upstream direction (i.e., the main passenger flow direction) will not be able to meet the passenger demand and will lead to the waste of train utilization in the downstream direction (i.e., the non-main passenger flow direction).  
Figure 2: Illustration of train operation under T-TPOS.  
In order to solve the problem of uneven distribution of passenger numbers in opposing directions, timetable optimization based on the TUOS-ELS is studied to enhance the matching of train capacity and passenger demand. In Figure 2, the total number of trains running in both directions is eight. With the same total number of trains running, the TUOS-ELS in Figure 3 is adopted, where five trains operate closely in the upstream direction and three trains operate sparsely in the downstream direction. The flexible number of trains and the headway between two consecutive trains can accommodate uneven passenger demand. In addition, the determination of the number of trains plays an important role. The ELS is executed in the downstream direction, and the express trains can shorten the travel time from the first station to the terminal station. However, there are local trains running in both directions, maintaining all-stop behavior to ensure service quality. All-stop behavior means that the train stops at every station it passes through.  
This paper considers both passenger service quality and operating costs. When passengers wait on the platform, they only care about whether they can board the upcoming train and their travel time. Therefore, the number of stranded passengers on the platform and travel time of trains were chosen as indicators of service quality. The stopping costs were selected as operating costs. To conclude, this paper adopts a multi-strategy operation method and constructs a multi-objective MILP model to optimize the timetable, aiming to minimize the total number of stops, the number of stranded passengers, and the total travel time of trains.  
This paper focuses on the study of timetable optimization. The number of trains, the running time between stations, the dwell time, the departure time, and the passenger's traveling behavior are considered. To simplify the model, the proposed assumptions are illustrated as follows:  
**Assumption 1**.: _Urban rail lines do not provide sidings, do not support temporary train additions and subtractions, and do not support overtaking operations at any location on the line._  
**Assumption 2**.: _Passengers can be informed of stations where the upcoming train will not stop through their mobile devices as well as platform announcements. Passengers may only travel on trains that are valid for them, i.e., trains that stop at both the passenger's origin and destination, and that do not take into account the passenger's transfer behavior._  
**Assumption 3**.: _To simplify the passenger waiting model, it is assumed that the waiting passengers of different OD (Origin-Destination) pairs are mixed at the platform. When the train reaches the platform, the number of stranded passengers is random._