## 2 Problem Statement and Assumptions  
Taking the passengers' expected departure times into account, this paper proposes an integrated optimization model to solve the problem of train timetabling and train stop planning. In our approach, passengers could pre-submit their travel information, including their expected departure times, origin and destination stations, while station, track and train operation information are also taken into account in the modeling. Finally, the efficient train service plan is obtained. In this process, there are several decisions that need to be made, including which stations to stop at for each train, what time to schedule departure and arrival for each train at each station, and which trains to get on for passenger assignment. We consider two optimization objectives. First, passenger transport demand may not be fully satisfied, so minimizing the remaining passenger demand is taken as one objective. Train travel time has a significant influence on passenger travel efficiency, so the second objective is to minimize train travel time. We present a simple case with four stations, three trains and 12 passengers to better illustrate the problem. Table 2 shows the passenger OD demand and their expected departure times. Figure 2 shows two service plans for the small case, of which the train service times are presented in Table 3. The horizontal and vertical perimeters of the gray coordinate grid indicate the time and station, respectively. The blue straight lines indicate the trains, and the arcs of different colors indicate the travel arcs of different passengers. We suppose that the train capacity is four persons. For passengers, the maximum deviation between their actual and expected departure times is set to 2 min. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Study** & **Problem Type** & **Model Type** & **Objective** & \begin{tabular}{c} **Passenger** \\ **Demand Type** \\ \end{tabular} & **Solution Method** \\ \hline [PERSON] et al. [24] & TT & MINLP & Passenger waiting time & Time-dependent & GAMS \\ [PERSON] et al. [17] & SP \& TT & MILP & Total dwelling time & Static & CPLEX \\ [PERSON] et al. [18] & SP \& TT & MILP & Profit of train & Static & Column-generation-based \\  & & & Empty-seat-hours, & & heuristic \\ [PERSON] and [PERSON] [22] & LP \& TT & MILP & train & Static & Iterative framework with \\  & & & travel time and & & GUROBI \\  & & & timetable robustness & & & \\ [PERSON] and [PERSON] [30] & SP \& TT & MILP & Transport profit & \begin{tabular}{c} Service interval \\ dependent \\ \end{tabular} & \begin{tabular}{c} Lagrangian \\ relaxation \\ \end{tabular} \\ [PERSON] et al. [31] & SP \& TT & MINLP & \begin{tabular}{c} Passenger travel \\ efficiency and train \\ running time \\ \end{tabular} & Time-dependent & ALNS \\ [PERSON] et al. [32] & SP \& TT & MILP & Train travel time & Time-dependent & CPLEX \\ [PERSON] et al. [34] & RP \& TT & MINLP & Operation cost and & Time-dependent & NSGA-II \\  & & & travel cost & & \\ This study & SP \& TT & MILP & \
\begin{tabular}{c} Remaining travel \\ demand and train \\ travel time \\ \end{tabular} &
\begin{tabular}{c} With expected \\ departure time \\ \end{tabular} & GUROBI \\ \hline \hline \end{tabular}  
* **Problem type:** LP—line planning; SP—stop planning; TT—train timetabling; RT—route planning; **Model type:** MILP—mixed integer linear programming; MINLP—mixed integer nonlinear programming; **Solution method**: NSGA-II—non-dominated sorted genetic algorithm-II; ALNS—adaptive large-scale neighborhood search.  
\end{table} 
Table 1: Summary of relevant studies. In Figure 2, there are three trains in both timetables. Train 2 stops at Stations \(B\) and \(C\) in Service plan I, but there is no stop in Service plan II. Passengers marked in red are assigned to Train 1 in both plans. In Figure 1(a), passengers marked in green, purple and orange are assigned to Train 2. In Figure 1(b), As Train 2 does not stop at both Stations \(B\) and \(C\), passengers marked in purple and orange cannot take this train.