Objective function. Within the CMLST network framework, the objective function for maximizing the total number of trains in the passenger flow section is stated in Eq (1). (1) Objective function (1) aims to maximize the number of trains for which stop plan k is not 0 (i.e., the trains are scheduled in the timetable).

Flow balance constraint. Since the model proposed in this paper is different from the typical space-time network model, the flow balance constraints of the model should be modified accordingly. To depict a feasible train path in the CMLST network, a set of flow balance constraints is constructed as follows: (2) (3) (4) Constraint (2) means that for each individual train l, if train l selects speed level p from set P and stop plan k from set , then and there should be one and only one arc selected from all the starting arcs of train l from to (ol,t) Otherwise, if for all p ∈ P and , then there should be no arc selected from all the starting arcs of train l.

Constraint (3) means that for each individual train l, if train l selects speed level p from set P and stop plan k from set , then and there should be one and only one arc selected from all the ending arcs of train l from (dl, t) to . Otherwise, if for all p ∈ P and , then no arc should be selected from the ending arcs of train l.

Constraint (4) means that for all vertices (s, t) ∈ V, the number of arcs from (s, t) to (s’, t’) should equal the number of arcs from (s’, t’) to (s, t).

These constraints ensure that when a train is scheduled, it will start from the dummy origin and end at the dummy sink after a series of network vertices, and the selected arcs will finally form a complete path.

Train selection constraint. The selection of trains in set L is not arbitrary. It must not only meet the service frequency demand of the stations but also consider the limitation of the number of consecutive trains and the resources of the EMU. A set of train selection constraints is constructed as follows: (5) (6) (7) (8) Constraint (5) means that for all trains in L, a combination of a speed level p and stop plan k must be selected, and if k = 0 in the selected combination, it means that the train will not be scheduled in the timetable.

Constraint (6) means that for intermediate station m, the number of trains stopping at station m () should be greater than or equal to the demand of station m.

Constraint (7) means that the number of trains with speed level p should be less than the maximum number of EMU resources.

Consecutive trains with the same speed level can improve the utilization rate of the capacity, but it will lead to a strong homogeneity of trains, which will cause concentrated train stops and long waiting times of overtaken low-speed trains at the stations. Constraint (8) limits the number of consecutive trains with the same speed level p in section e.

Train dwell constraint. The dwell of train l at station m should be consistent with the selected stop plan k, and the stop time should be between the maximum stop time and the minimum stop time of station m. At the same time, the additional starting and stopping time of train l at station m should be considered. A set of train dwell constraints is constructed as follows: (9) (10) (11) (12) Constraint (9) means that if train l with stop plan k does not stop at intermediate station m (), then train l cannot select the dwell arcs of intermediate station m.

Constraints (10) and (11) mean that if train l with stop plan k stops at intermediate station m (), then the number of dwell arcs of intermediate station m selected by train l should be greater than or equal to the shortest stop time gm and less than or equal to the longest stop time hm.

The dwell of train l at station m needs to include the process of starting and braking, which will affect the travel time in segment e. As shown in Fig 5, Train 1 does not stop at either Station 1 or Station 2, while Train 2 stops at both Station 1 and Station 2. Therefore, the travel time of Train 2 in section e is equal to the travel time of Train 1 plus the additional starting time αs at Station 1 and the additional stopping time βs at Station 2.

https://doi.org/10.1371/journal.pone.0264835.g005

The purpose of constraint (12) is to ensure that the difference between t and t’ of the selected arc (s,s’,t,t’) is equal to the standard travel time plus the additional starting and stopping time.

Train safety (time headway) constraint. To guarantee the safety of high-speed railway operation, there is a minimum time headway between the departure, arrival and passing through of two consecutive trains at the same station. In this paper, seven kinds of train safety constraints are considered in the CMLST network model, as follows: (13) (14) (15) (16) (17) (18) (19) Constraint (13) means that for segment e = (s,s’), time t and trains that depart from station s in the time range and arrive at station s’ at time t”, at most one segment arc a = (s,s’,t,t”) can be selected in order to ensure the minimum headway time between two trains departing from the same station s in succession.

Constraint (14) means that for segment e = (s, s’), time t and trains that arrive at station s’ in the time range and depart from station s at time t’, at most one segment arc a = (s,s’,t’,t”) can be selected in order to ensure the minimum headway time between two trains arriving at the same station s’ in succession.

Constraint (15) means that for segment e = (s, s’), time t and trains that pass through station s in the time range and arrive at station s’ at time t”, at most one segment arc a = (s,s’,t’,t”) can be selected in order to ensure the minimum headway time between two trains passing through the same station s in succession.

Constraint (16) means that for segment e = (s, s’), time t and two trains, if the first train departs from station s at time t, then the second train, which passes through station s at time t”, cannot select arcs in the time range . Otherwise, at most one arc can be selected by the latter train to ensure the minimum headway time between the two trains.

Constraint (17) means that for segment e = (s, s’), time t and two trains, if the first train passes through station s at time t, then the second train, which departs from station s at time t”, cannot select arcs in the time range . Otherwise, at most one arc can be selected by the latter train.

Constraint (18) means that for segment e = (s, s’), time t’ and two trains, if the first train arrives at station s’ at time t’, then the second train, which passes through station s’ at time t”‘, cannot select arcs in the time range . Otherwise, at most one arc can be selected by the latter train.

Constraint (19) means that for segment e = (s, s’), time t’ and two trains, if the former train passes through station s’ at time t’, then the latter train, which arrives at station s’ at time t”‘, cannot select arcs in time range . Otherwise, at most one arc can be selected by the latter train.

Train overtaking constraint. Train overtaking can only occur in a station, not in a segment; otherwise, there will be conflict.

Constraint (20) means that if segment arc a = (s,s’,t,t’) is selected by train l, then the arcs a’ = (s,s’,t’,t”) crossing arc a in the segment cannot be selected by other trains.

Cross-line train constraint. For high-speed railway passenger flow sections with long mileage, there are usually some trains from other lines called cross-line trains, which usually have a higher priority than the trains in the research section. In this model, cross-line train l with speed level and stop plan can be scheduled in advance by setting the relevant variable to 1, as shown in Eq (21). The arcs occupied by the cross-line trains in the network cannot be selected by other trains again. (21) Binary variable constraint. There are also binary definitional constraints for variables and .