## 1 Introduction  
In recent years, the network scale of HSR has developed rapidly, and the length of the HSR line also increased. More and more passengers are choosing to travel by HSR. As the number of HSR passengers has increased, some new operational requirements have emerged. In normal times, the demand for passengers to travel by HSR does not fluctuate much. However, the demand fluctuates because of holidays or special activities. Train timetables are designed to meet the travel demands of passengers, which state the departure time and the arrival time of all train operation lines. HSR operation departments need to change the HSR train timetable in time to respond to the travel demands of passengers. Timely adjustment of the timetable according to the number of passengers can make the arrangement of the operation department more aligned with the demands of passengers, and save the cost of operating a railway line for the operation department. The train timetable contains relevant operational plans, such as train-set circulation plans. The train-set circulation plan is a part of the train timetable, which is a detailed plan of how to use the train-sets when implementing the timetable. The schematic diagram of train-set circulation plan is shown in Figure 1.  
As shown in Figure 1, there are four train operation lines in the timetable, representing four operation tasks. A train-set can only perform one operation at a time. Train-set 1 performs operation line 1 first, and train-set 2 performs operation line 2. After the completion of one operation line, the train needs a period to replenish materials and clean the carriage, to perform the next operation line. This period is called the turn-back time. We can see that one train-set can perform multiple tasks in one day. Train-set 1 will perform operation line 4 after completing operation line 1. Similarly, train-set 2 will perform operation line 3 after completing operation line 2. As a result, a timetable requires fewer train-sets than the operation lines it contains. Such a plan containing the corresponding relationship between train-set and operation line is called a train-set circulation plan. Passengers need to get on or off the train, so, train-set needs to stop at a station to serve passengers. If a train-set does not stop at the station, we call the train-set pass through the station. When the old timetable is switched to the new timetable, the operation plan related to the timetable also needs to be switched. For the operation department, the number of train-sets required by each station to implement the new timetable may be different from the old timetable. This means that in terms of the location of the train-set, each station has two states. The state after the implementation of the old timetable is the old state, and the state required by the implementation of the new timetable is called the new state. The schematic diagram of the old state and the new state is shown in Figure 2.  
In Figure 2 we can see that in the old state, station A has two train-sets and station G has three train-sets. However, in the new state, station A needs three train-sets and station G only needs two train-sets. The time between the end time of the old timetable and the start time of the new timetable is called transition time. To meet the number requirement of train-sets in each station in the new timetable, the operation department has two methods. The first is to use spare train-sets to perform operation lines (method 1). Second, at the end of the day, trains will be transferred from nearby stations that have more train-sets to those that lack them (method 2). The schematic diagram of the timetable switching method is shown in Figure 3.  
Figure 1: The schematic diagram of train-set circulation plan. As shown in Figure 3, there is a spare train-set in station G and station A is short of a train-set. To make up for this train-set, the backup train-set at station A can be used, or the spared train-set can be transferred from station G to station A during the transition time.