## 2 Problem Description  
In this section, a detailed description of the problem and assumptions is given. A railway corridor consists of a set of stations and segments that connect two consecutive stations. Figure 1 shows a double-track railway corridor from station 1 to station \(S\). The segments between two consecutive stations are indexed as \(\{1\,,\,2\,,\ldots,N\}\). Since the study environment is a high-speed railway corridor,which is always double-track, we only consider our problem on the outbound-track railway corridor. For the outbound railway corridor, station 1 is the starting station of the railway corridor and station \(S\) is the ending station. The starting station of the railway corridor can only be selected as the origin station of an outbound trip, and the ending station can only be selected as the terminal station of a trip. There are two sets of trains (\(K_{1}\cup K_{2}\)) in this problem, wherein \(K_{1}\) is the set of existing trains and \(K_{2}\) is the set of additional trains. In peak periods, existing trains cannot meet the passenger demand. As shown in Figure 1, the blue trains represent existing trains in the off-peak period. To provide more transport capacity, we will insert some additional trains--the red trains in Figure 1.  
Our goal is to design an operational plan for additional trains. The operational plan includes the number of trains, operational zones, stop plans, timetables, and passenger distribution for trains. The reason why all plans for the number of trains, operational zones, stop plans, timetables, and passenger distribution are jointly optimized is that these plans are closely connected. In the following, we will explain the connection between them. A small example is shown in Figure 2 to illustrate the impact of the operational zone and stop plans on the distribution of passenger demand. As shown in Figure 2, a railway corridor with five stations and four segments is considered. In Figure 2, there are five lines with arrows to represent the operational plans for trains. The arrows indicate the running direction of the train. The endpoint of a line represents the origin station or terminal station. A solid dot in the line denotes the train stopping and a hollow dot denotes a non-stopping train. The black line represents the operational plan for the existing train. The red lines represent four potential operational plans for the additional train. We assume only one existing train runs on this corridor, and the operational zone is from station 1 to station 5 in this example. The capacity of each train is \(N\). The passenger demand for each OD pair is listed in Table 1. In Table 1, \(N\) denotes that \(N\) passengers want to travel from their origin station to the terminal station. Obviously, the best stopping plan for the existing train is to stop at station 3. \(N\) passengers take the existing train to travel from station 1 to station 3, then another \(N\) passengers take it to travel from station 3 to station 5. However, the existing train still cannot meet the passenger demand, so adding an additional train is necessary. Four potential plans for the additional train are considered, as shown in Figure 2. We compare these different plans for the additional train. Several indexes of these plans are summarized in Table 2. We set the operating cost of the existing trains to \(E\), because the operational zone and stop plan for the existing train are fixed. In this example, we assume that the operating cost of additional trains is related to the number of segments of their operational zone and the number of stops. The operational cost of each segment is \(c\), and the operational cost of each stop is \(s\). First, we compare plan 1 and plan 2, which demonstrate the impact of the stop plan on the distribution of passenger demand and operating cost. As shown in Figure 2, plan 1 and plan 2 have the same operational zone from station 1 to station 4 but different stop plans.