Integrated Optimization of Stop Planning and Timetabling for Demand-Responsive Transport in High-Speed Railways

: The high-speed railways have made rapid developments in recent years. Fulﬁlling passenger demand and providing precise train services are the core problems to be solved in railway operation. This paper proposes an optimization strategy for demand-responsive transport to integrate train-stop planning and timetabling in high-speed railways. Passenger travel information, including their origins, destinations and expected departure times is taken as input. A mixed integer linear programming model is established to obtain an effective service plan, which consists of train stop pattern, passenger ride plan and train arrival/departure times at all stations. The optimization objective is to minimize the remaining passenger demand and train travel time. Finally, the proposed method is applied to a real-world case, and a series of several experiments are conducted to prove the efﬁciency and validity of the proposed model. The results suggest that the proposed approach could generate efﬁcient service plans which are responsive to passenger demand.


Introduction
There has been rapid development of the high-speed railways over the past few decades.With the expansion of the railway network and passenger demand, railway operators are facing more challenges in providing higher quality services to passengers.
Fulfilling passenger demand and providing transportation services accurately for the demand is the core problem in high-speed railways.Currently, railway operators have been using a 'planning-based' mode to provide transportation services, which forecasts the passenger demand for a certain time period and prepares train plans to match the infrastructure resources with the forecasted demand as much as possible.However, passenger demand typically has an uneven distribution, with highly heterogeneous spatial flow patterns and flow sizes, especially during the COVID-19 pandemic.Dynamic transport demand has not been accurately and opportunely grasped by operators.As a result, there are two phenomena occurring on the network simultaneously: "hard to get tickets" and "excess capacity".
To better respond to the passenger demand, the railway operators have been making unremitting efforts, aiming to adapt to the time-varying passenger demand by constantly shortening the adjustment period of train schedules.However, due to the low accuracy of passenger travel demand forecast, 'planning-based' train operation planning is always flawed during time horizons, when the passenger demand fluctuates greatly.A new 'demand-responsive' train planning concept provides an important idea to match passenger demand and transport resources dynamically to solve this problem.Its core mechanism is to improve the information exchange between the supply and demand sides.Once the travel demand information of passengers is obtained, the railway operators will respond to provide a service plan with a more effective allocation of resources.As two important parts of train planning, both stop planning have significant effects on train operation and service quality.determines the stop pattern of all trains, is an important factor departure and arrival times of each train at each station are determ timetabling.Thus, system efficiency, capacity utilization and regarding travel times are highly dependent on stop plans a demand-responsive transport, train planning also faces new cha above two parts are arranged in a sequential and progressive mann 1.However, with a predetermined stop plan, the solution space for As two important parts of train planning, both stop planning and train timetabling have significant effects on train operation and service quality.Stop planning, which determines the stop pattern of all trains, is an important factor in line planning.The departure and arrival times of each train at each station are determined in the process of timetabling.Thus, system efficiency, capacity utilization and sensitivity to delays regarding travel times are highly dependent on stop plans and timetables.Under demand-responsive transport, train planning also faces new challenges.Normally, the above two parts are arranged in a sequential and progressive manner, as shown in Figure 1.However, with a predetermined stop plan, the solution space for solving the timetabling problem is reduced.The phased design of stop planning and train timetabling usually cannot generate a best solution with time-dependent passenger flow demand.Therefore, it is necessary to optimize the integration of both stop planning and train timetabling for demand-responsive transport in high-speed railways.

Literature Review
Over the past few decades, scholars have done lots of research on the problem of line planning and train timetabling considering passenger demand.In this part, we review the recent studies in demand-responsive transportation, integrated optimization of stop planning and train timetabling and train timetabling with time-dependent passenger demand.

Demand-Responsive Transport
The core of the demand-responsive transportation system is the reservation processing model, which is essentially a joint optimization of demand and flexible planning.In recent years, demand-responsive transportation has become a hot issue internationally.Some relevant research has been carried out in many fields, such as urban public transportation, the most typical of which is defined as the Dial-A-Ride Problem (DARP).The system for solving this problem based on cars or small buses has been introduced in some European countries in recent years, initially aiming to serve people who travel in remote areas or at night time with low demand.
For the static DARP, Psaraftis [2] was the first to propose a dynamic programming strategy to address the single-vehicle problem.Cordeau and Laporte [3] solved a multivehicle problem with consideration of route duration, vehicle capacity, and the maximum passenger ride times, which were addressed using a tabu search algorithm.Parragh and Schmid [4] not only considered the heterogeneity of users and vehicles in the modeling, but also studied the algorithm design.A hybrid algorithm was developed to solve the DARP, which combined a large neighborhood search and column-generation method.With the objective of maximizing corporate profits, Parragh, et al. [5] formulated a mixed integer programming model to solve the DARP.Garaix, et al. [6] considered the DARP, aiming to maximize the passenger occupancy rate.On the basis of the static problem, many scholars have considered dynamic path planning in problems where a new user request is used as the event that triggers the update schedule.The vehicle paths as well as the service recipients are continuously updated to ensure that users are served as much as possible [7,8].Many scholars have also considered the stochastic in DARP.Ho and Haugland [9] considered user requires service only as a given probability to design the vehicle routes.Hyytiä, et al. [10] proposed a stochastic model, taking the blocking probability, the acceptance rate, and the average sojourn time into account to solve a singlevehicle DAR in a transport system.Schilde, et al. [11] extended two pairs of metaheuristic methods for dealing with stochastic, time-varying drive speeds for the situation of the dynamic DARP.Given the stochastic service and travel times, Shi, et al. [12] established a stochastic model applied to the path planning for the Home Health Aides service.Then, five approaches were adopted, respectively, to find solutions to the problem.Lu, et al. [13] built a multiple graph to depict the stochastic DARP, which was formulated by a two-stage recourse model.Authors used an iterative sampling-based method to address the proposed stochastic model, embedded with a heuristic search algorithm.These studies provide a good theoretical basis for developing demand-responsive transportation systems, but most studied a small-scale vehicle routing problem.Compared with the routing problem on the road networks, there is an obvious difference in the theory of the high-speed railway networks and the demand-responsive service, and there are some obstacles to its application to the railway system.

Integrated Optimization of Train Stop Planning and Timetabling
In the railway transportation system, train stop planning and timetabling are two fundamental problems.As train timetabling is directly influenced by the stop plan, the integrated optimization of the above two problems could provide a high-quality service for both passengers and operators; this is also confirmed in some studies.To minimize both user and operator costs, Kaspi and Raviv [14] established an optimization model integrating line plan and timetable, which was handled by a cross-entropy metaheuristic method.It was found that the average journey time of passengers decreased by 20%.Aiming to reduce passenger waiting times, in-vehicle travel times and train travel times in an urban rail transport system, Cao,et al. [15] established a 0-1 integer programming model considering the skip-stop operation strategy.They used the tabu search algorithm to address the problem.Borndörfer, et al. [16] studied periodic timetabling models integrated with passenger routing, which showed that different modeling methods of route planning had a notable effect on the service quality of the entire transport system.Yang, et al. [17] provided an integrated optimization method to decide the train stop pattern and timetable in high-speed railways.The objective function considered is to minimize the total dwell time at intermediate stations and total train delay time at their starting stations; they used the CPLEX optimizer to solve the proposed model.Yue, et al. [18] built a coordinationoptimization model considering the train stop pattern and train timetable together for the high-speed railway, and designed an algorithm based on the column generation technology to solve a large-scale train timetabling problem.Burggraeve, et al. [19] considered passenger robustness in the integration optimization of line plan and train timetable.They built an iterative and interactive framework for the integrated optimization model to generate a robust timetable with a lower cost.Schöbel [20] proposed an eigenmodel, which could determine line plan, train timetable, and vehicle schedule integrally.An iterative algorithm was designed as heuristics for this integrated problem.To schedule additional trains on the basis of the original timetable, Gao, et al. [21] presented a novel double-objective mixed integral programming model with consideration of acceleration/deceleration times and platform assignment at a macroscopic level.A three-phase optimization approach was applied to deal with this problem.Taking the robustness and regularity of the timetable and the passenger travel time into account, Yan and Goverde [22] proposed a multi-period model and a multi-frequency model.These two models worked iteratively to achieve more competitive planning in the rail transport system.

Train Timetabling with Time-Dependent Passenger Demand
The above integrated studies are often based on historical demand data and ignore the time-dependent nature of passenger travel demand.Recently, more and more researchers have focused on the optimization of demand-oriented train timetables to systematically provide satisfactory services to meet time-dependent passenger flow demand.Niu and Zhou [23] introduced the latest arrival time of the boarding passengers to obtain a timevarying waiting time by calculating the effective loading time period.A nonlinear programming model was established to solve the timetabling problem, intending to reduce the passenger waiting time in an oversaturated urban rail corridor.Then, Niu, et al. [24] took train skip-stop patterns into account in the timetabling problem under time-varying passenger demand.Oriented to the dynamic travel demand, Barrena,et al. [25] formulated three train timetabling models and took the minimum average passenger waiting time as the objective.A fast adaptive large neighborhood search was proposed for real-scale case application with faster solution efficiency in the study [26].Niu, et al. [27] focused on the timetabling problem of two interconnected high-speed railway lines.A nonlinear programming model was developed based on time-dependent origin-destination passenger demand.Considering the train speed profiles of different train driving strategies, Yin, et al. [28] established an integrated model for the train timetabling problem to reduce energy consumption and the waiting time of passengers.A Lagrangian relaxation-based heuristic algorithm was implemented to find a good plan within a short computational time.Zhu,et al. [29] formulated a bi-level programming model to deal with the train timetabling problem for an urban rail line.Authors introduced a two-phase genetic algorithm combining the successive average approach to solve the model.For the high-speed railway system, Meng and Zhou [30] described the relationship between train service times and passenger behavioral responses, and proposed an integrated optimization model, taking passenger demand, train service and resource utilization into account.Dong, et al. [31] built a combination model to optimize both train stop plan and timetable with dynamic travel demand.Then, an extended adaptive large-scale neighborhood search algorithm was designed to find the optimal solution.For the combined optimization of train stop plan and timetable, Qi, et al. [32] and Zhang et al. [33] proposed an integrated optimization strategy, considering the passenger expected departure intervals between their origins to destinations.Yang, et al. [34] integrated the train route selection based on the optimized train timetable under dynamic passenger demand, and adopted a modified non-dominated sorted genetic algorithm-II to address the bi-objective optimization problem.
Recent studies on stop planning and timetabling are shown in Table 1.According to the above literature review, little research has considered demand-responsive transportation in high-speed railways and completed the integrated optimization study of the train stop plan and train timetable with more realistic conditions.Few studies have considered minimum remaining passenger demand as one of the objectives.Previous studies have placed some limitations on this problem, with methods such as predefined timetables.Some actual operation factors are also ignored in the previous studies, such as the additional acceleration time and deceleration time, different headway types and overtaking modes between trains.To offer a high-efficiency service for passengers and operators, this paper proposes an integrated optimization approach of train stop planning and timetabling, considering passenger demand with expected departure time.The application of demandresponse transport on high-speed railways with more practical constraints is also trialed in this study.

The Proposed Methods
The main contributions of this study are as follows: (1) This study proposes a demand-responsive high-speed railway service plan optimization approach.This method takes the travel demand information of passengers as input, including their expected departure times, origin and destination stations.With the operation information of the railway line, an efficient train service plan is generated, including passenger ride plan, train stop plan and train timetable.We also considered the tradeoff between minimizing the remaining passenger demand and train travel time in the optimization objective.
(2) A mixed integer linear programming (MILP) model is developed in this study.In the process of developing the model, we consider passenger assignment in the integrated construction.A decision variable is used to represent the time-dependent passenger flow specific to each train.We consider more practical factors in terms of timetable constraints, including additional acceleration/deceleration time and headways, under different operating conditions.
(3) The proposed approach is applied to real case studies on a Chinese railway corridor.Several experiments are set up to prove the effectiveness of the proposed integration method.We adopt the GUROBI optimizer to solve the proposed model.The computational results indicate that efficient solutions are generated by the proposed method with an acceptable computation time.
The remainder of this study is structured as follows.Section 2 presents the problem statement and makes some assumptions.Section 3 describes the details of the integrated optimization model.In Section 4, the proposed approach is applied to a real case, and the results are presented to prove the effectiveness of the model.Finally, several conclusions and future research works are given in Section 5.

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.

Expected Departure Time
Passenger OD Demand 2 q AD = 4 6 q AD = 2, q AB = 2 10 q CD = 2 12 q BD = 2  In Figure 2, there are three trains in both timetables.Train 2 stops at Stations B and in Service plan I, but there is no stop in Service plan II.Passengers marked in red assigned to Train 1 in both plans.In Figure 2a, passengers marked in green, purple a orange are assigned to Train 2. In Figure 2b, As Train 2 does not stop at both Station and C, passengers marked in purple and orange cannot take this train.Within allowable departure time deviation, Train 3 could transport passengers in purple to th destination.However, passengers marked in orange will not be satisfied, because ther no train serving both their origin and destination stations.
The stop numbers and objective values of two service plans are presented in Tabl In the Service plan I, there are no remaining passengers, the train travel time is 21, and number of train stops is 3.In the Service plan II, there are 2 remaining passengers, train travel time is 20, and the number of train stops is 1.Although there are no remain passengers in Service plan I, the train travel time is longer due to more stops.Service p  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 2a, passengers marked in green, purple and orange are assigned to Train 2. In Figure 2b, As Train 2 does not stop at both Stations B and C, passengers marked in purple and orange cannot take this train.Within the allowable departure time deviation, Train 3 could transport passengers in purple to their destination.However, passengers marked in orange will not be satisfied, because there is no train serving both their origin and destination stations.
The stop numbers and objective values of two service plans are presented in Table 4.In the Service plan I, there are no remaining passengers, the train travel time is 21, and the number of train stops is 3.In the Service plan II, there are 2 remaining passengers, the train travel time is 20, and the number of train stops is 1.Although there are no remaining passengers in Service plan I, the train travel time is longer due to more stops.Service plan II has a higher operation efficiency, but it fails to meet all passenger travel demand.The results of the above small case shows that the service plan has a significant influence on passenger travel and train operation efficiency.Therefore, it is urgent to develop an optimization method to generate the efficient train service plan to reduce both train travel time and remaining passenger demand.
To simplify the modeling process, we made the following assumptions.
Assumption 1.The railway line studied in this paper is a double-track railway, and its upstream and downstream systems are completely independent.We only consider one train running direction.
Assumption 2. The capacity of section and station is not considered in this study.The capacity constraint is assumed to be always satisfied.

Assumption 3.
We assume that all passengers only get on trains that run directly to their destination stations, and they will not transfer between trains.

Mathematical Model
This section will introduce the proposed mixed integer linear programming model, including decision variables, sets and parameters, objective functions and systematic constraints.

Notations
The notations involved in the model are summarized in Table 5.

Model Formulation
In this part, we describe the specific construction process of the model, including the objective function and the associated constraints.In the objective function, we consider the remaining passenger demand and train travel time.There are eight types of constraints taken into account in the modeling process, including passenger flow constraints, seat capacity constraints, passenger departure time range constraints, number of train stops constraints, section running time constraints, dwelling time constraints and headway constraints.

Objective Function
In train service planning, passenger travel demand should be satisfied as much as possible, and the efficiency of train operation should also be considered.Minimizing the weighted sum of the remaining passenger demand (RPD) and train travel time (TTT) is taken as the objective function, as shown in Equation (1).

Parameters Definition r l,m
The pure running time of trains from station m to station m + 1, m ∈ {1, 2, .The total RRD is formulated as Equation (2).Fewer remaining passengers means that more travel demand is satisfied.When a passenger submits a reservation request, there is a possibility that it will be rejected due to the limited capacity of train seats.The passenger cannot get the corresponding train service and must remain.Therefore, minimizing the number of remaining passengers is considered one of the objectives in this paper.Equation (3) describes the total TTT.Shorter train travel time indicates that the train could complete its journey within a shorter period, which has a significant impact on the efficiency of train operations.More efficient train operations will not only reduce travel time for passengers but also reduce costs for operators.Therefore, minimizing the train travel time is considered another objective in this paper.θ 1 and θ 2 represent the weights of two parts of the objective, respectively.They are expected to be set with different values, since the weights in the objective often have an obvious influence on the final decision.The value of θ 1 needs to be increased if a demand-oriented service plan is required, and the value of θ 2 needs to be increased if an efficiency -oriented service plan is required.

Passenger Flow Constraints
Passengers expecting to depart from a station at a certain time may be assigned to different trains, and passenger demand may not be met in full.For each expected departure time, constraint (4) specifies that all passengers assigned to the train are equal to the total passenger demand, minus the remaining passengers.Constraint (5) indicates that all passengers will only be assigned to the trains that could direct them to their destination stations.

Seat Capacity Constraint
Train seat capacity refers to the maximum amount of passengers that each train could carry.Constraint (6) ensures that the number of passengers in each successive section of each train does not extend beyond the maximum train seat capacity.

Passenger Departure Time Range Constraints
For passengers, the deviation of their actual and expected departure times should not be too large; this has a great impact on the satisfaction of passengers.Constraint ( 7) limits the deviation of the passenger's actual and expected departure times within a given range.Constraints (8) and (9) represent that if there are passengers assigned on train i who expect to depart from station m to station n at time t,x m,n i,t equals 1; otherwise, x m,n i,t equals 0.

Number of Train Stops Constraints
There needs to be some restrictions on the number of train stops.Too many stops will reduce operational efficiency, and fewer stops may fail to meet the passenger travel demand.Constraint (10) means that the number of stops per train is limited to a reasonable range.

Section Running Time Constraint
The section running time refers to the time length of the train running between two adjacent stations, which generally consists of the section's pure running time and additional times for acceleration and deceleration.The section's pure running time is determined according to the train running speed and section mileage.The additional acceleration and deceleration times are caused by the train speed increasing from zero to running speed when the train departs, and the train speed decreasing to zero when the train stops.Constraint (11) shows the calculation of the train running time between two consecutive stations, which consists of the pure running time and possible additional time for acceleration and deceleration.

Dwelling Time Constraints
The dwelling time is expected to be within a certain range.If the dwelling time is too short, it may not be enough for passengers to board and alight; a dwelling time that is too long will increase the travel time.Constraint (12) indicates that the train dwelling time cannot be less than the minimum value.Similarly, Constraint (13) indicates that the train dwelling time cannot exceed the maximum value.

Headway Constraints
Headway constraints could ensure safe operation at stations and maintain a safe distance between adjacent trains in sections.In actual operation of high-speed railways, the train headway is not a fixed value; it is different due to the stop status of adjacent trains at the station.Therefore, we handle the train headway more precisely in this model.Figures 3  and 4 illustrate the different types of headway with four types each of departure and arrival headway.The corresponding relationship between different types of headway and the stop status of trains is shown in Tables 6 and 7.

Arrival Headway Type
As shown in Figures 3 and 4, the above headway only takes effect when train i departs before train j; however, the sequence of trains in the model is not fixed.To ensure the reasonableness of the constraint, we introduce a binary variable y m ij , which equals 1 when train i departs from station m earlier than train j; otherwise, it is 0.
Therefore, the headway constraints are expressed as Constraints ( 14) and (15).The variable y m ij is expected to satisfy Constraint (16).Theoretically, the right side of the Constraints ( 14) and ( 15) is one minimum headway only when y m ij is 1.Moreover, this set of constraints can also avoid conflicts between trains in sections.

Computational Experiments
To demonstrate the performance of the model proposed in Section 3, a real-scale instance is tested in this part.We select one of the lines in China's high-speed railway network, denoted as Line L. Line L includes 10 stations, which are denoted as Stations A, B, . . ., J. We carry out a series of experiments on the selected lines.All tests in the experiment were implemented by Python 3.8 and GUROBI 9.1.2,on a personal computer which is equipped with Intel Core I7-6700K CPUs and 16 GB RAM.

Input Data and Parameters
Before optimization, some basic data need to be input, including train operation data, passenger OD demand and other parameters involved in the model.
In this experiment, we designed a 4-h train timetable from 7:00-11:00.Passenger OD demand during this time horizon on a weekday in 2018 is shown in Table 8.It can be found that the largest passenger demand is 1735 between station A and station J. Except for the terminal stations, Stations D, E and G have a relatively large passenger demand.Considering that the passenger demand in Table 8 is not correlated with the expected departure time, we assume that the expected departure times during the whole study period obey the normal distribution [31] in order to obtain the passenger expected departure times.Then, the passenger demand with expected departure time is simulated by Equations ( 17) and (18).
At each station, p(t) indicates the proportion of passengers expecting to depart at time t; ϕ(t) presents the number of passengers expecting to depart at time t; and q indicates the total passenger demand for an OD pair.
In this way, the number of passengers with the expected departure times from one station to another are obtained.Distances and pure train running times of all sections are shown in Table 9, and other parameters involved in the model are listed in Table 10.

Results and Analysis
With the given operation information and the passenger OD demand, we expect to obtain a more efficient timetable using the proposed method.In this experiment, we set weights θ 1 and θ 2 as 10 and 1, respectively.A result was obtained with the gap 14.9% after 30,000 s.The RPD and TTT included in the objective are 1 and 1606, respectively.Figure 5 shows the optimized train stop plans, where the solid dot "•" indicates the stations that the train services.It can be seen that Train 6 and Train 9 have the largest number of stops; they both have 8 stops on the railway line.Train 11 has the least number of stops.
The relationship between the stop frequency and passenger demand is shown in Figure 6, which indicates that the frequency of train stops and the size of passenger demand at stations are closely related.Except for the terminal stations, Stations D, E and G have a relatively high stop frequency, which is in line with the given larger passenger travel demand of these three stations.The lower passenger demand also results in fewer stops at Stations C, F and H. Experimental results show that the scale of passenger demand determines the stopping pattern of trains to a certain extent.
Figure 7 shows the obtained train timetable in the experiment.The horizontal axis indicates the time, the vertical axis indicates the station, and the lines of different colors indicate different train services in this figure.It can be seen that Trains 4-11 in the dashed box are in denser departure compared with other trains.The peak hour for the given passenger demand is 8:30-9:30.The timetable structure obtained is relatively consistent with the passenger demand distribution.The expected departure times of passengers also affect the structure of the timetable.Besides the number of stops, overtaking will also increase the train travel time.In this test result, four instances of overtaking occurred.
and G have a relatively high stop frequency, which is in line with the g passenger travel demand of these three stations.The lower passenger demand in fewer stops at Stations C , F and H . Experimental results show that passenger demand determines the stopping pattern of trains to a certain exte  To further track the passenger distribution per train, we give the seat occupancy during each serviced section for each train, as shown in Figure 8.The seat occupancy can be calculated by Equation (19): where ω s i represents the seat occupancy of train i between station s and s + 1. Passenger distribution in each train is closely related to the passenger flow scale of different OD and the distribution in the time dimension.It can be seen that passengers on Trains 1-8 are mainly concentrated in the second half of the train journey.Trains 9 and 10 are more crowded in the middle of their journey.The passenger distribution of the remaining trains is relatively balanced, among which Train 11 has the largest seat occupancy rate.The average occupancy rate of all trains is more than 70%, which means that train resources are well utilized.To further track the passenger distribution per train, we give the seat occupancy during each serviced section for each train, as shown in Figure 8.The seat occupancy can be calculated by Equation ( 19):

Sensitivity Analysis
The weight of the objectives in the presented model has an effect on the op results.Finally, we performed a sensitivity analysis to compare this impact.experiments using the data in the above case are proposed, in which 1 θ is di 2 θ is fixed to 1. Table 11 shows the experimental results.With increasing θ travel time becomes longer, and remaining passenger demand becomes less.W larger, it is necessary to increase the stop frequency to transport more passeng train travel time will become longer and the remaining passenger demand w Hence, train travel time and remaining passenger demand can be balanced by the weights in the objective.

Sensitivity Analysis
The weight of the objectives in the presented model has an effect on the optimization results.Finally, we performed a sensitivity analysis to compare this impact.A series of experiments using the data in the above case are proposed, in which θ 1 is different and θ 2 is fixed to 1. Table 11 shows the experimental results.With increasing θ 1 , the train travel time becomes longer, and remaining passenger demand becomes less.When θ 1 is larger, it is necessary to increase the stop frequency to transport more passengers, so the train travel time will become longer and the remaining passenger demand will be less.Hence, train travel time and remaining passenger demand can be balanced by adjusting the weights in the objective.

Conclusions
This paper developed an integrated approach to optimize train stop planning and train timetabling with a demand-responsive mode in high-speed railways.Taking passenger demand with expected departure times and infrastructure conditions into account, we developed a mixed integer linear programming model to determine a high-quality service plan including stop patterns, arrival/departure times at stations of all trains and passenger ride plans.In this modeling process, minimizing the remaining passenger demand and total train travel time were considered to be the optimization objectives.Finally, the proposed method was applied to a real-scale instance study.A series of several experiments were set up to prove the effectiveness of the constructed model.The results show that a service plan suitable for passenger travel demand could be obtained by the proposed approach, while reducing the amount of remaining passengers and train travel time.Moreover, we separately analyzed the impact of variable weight coefficients θ of the objectives on the results.The remaining passenger demand and train travel time are mutually interrelated, and operators need to find a balance between them.
Future studies could pay attention to the following three areas: (1) In the method of this paper, the number and operation range of trains are fixed.In order to obtain a more flexible service plan, a variable number of trains and an unfixed scope of train operation need to be considered in future research.(2) More efficient solving algorithms could be developed to deal with the model.Although the proposed model could be solved using the exact solver, it performs poorly in terms of computational efficiency for large-scale application in the real world.(3) Only one train operation direction is considered in the process of modeling.The problem of scheduling in two directions with rolling stock needs to be studied in the future.

Figure 2 .
Figure 2. Two types of service plans with the same passenger demand.

Figure 2 .
Figure 2. Two types of service plans with the same passenger demand.

Figure 5 .
Figure 5. Optimized train stop plan in the experiment.

Figure 5 .
Figure 5. Optimized train stop plan in the experiment.

Figure 6 .
Figure 6.Relationship between stop frequency and passenger demand.

Figure 6 .
Figure 6.Relationship between stop frequency and passenger demand.

Figure 7
Figure 7 shows the obtained train timetable in the experiment.The horizontal axis indicates the time, the vertical axis indicates the station, and the lines of different colors indicate different train services in this figure.It can be seen that Trains 4-11 in the dashed box are in denser departure compared with other trains.The peak hour for the given passenger demand is 8:30-9:30.The timetable structure obtained is relatively consistent with the passenger demand distribution.The expected departure times of passengers also affect the structure of the timetable.Besides the number of stops, overtaking will also increase the train travel time.In this test result, four instances of overtaking occurred.

Figure 7 .
Figure 7. Time-distance diagram of trains in the experiment.

Figure 7 .
Figure 7. Time-distance diagram of trains in the experiment.

Figure 8 .
Figure 8. Seat occupancy of each train in the experiment.

Figure 8 .
Figure 8. Seat occupancy of each train in the experiment.

Table 1 .
Summary of relevant studies.

Table 2 .
Passenger OD demand with expected departure time.

Table 3 .
Timetables of the two service plans.

Table 3 .
Timetables of the two service plans.

Table 4 .
Relevant indexes of timetable.

Table 5 .
Sets, parameters and variables used in the model.
Minimum headway between two adjacent trains, of which the former one departs from the station and the latter one passes through the same station.hpdMinimumheadwaybetweentwoadjacent trains, of which the former one passes through the station and the latter one departs from the same station.hppMinimumheadway between two adjacent trains, which both pass through the same station.hddMinimumheadway between two adjacent trains, which both depart from the same station.hapMinimumheadway between two adjacent trains, of which the former one arrives at the station and the latter one passes through the same station.Number of passengers assigned on train i who expect to travel from station m to station n at time t.Binary variable : if there are passengers assigned on train i who expect to travel from station m to station n at time t h paMinimum headway between two adjacent trains, of which the former one passes through the station and the latter one arrives at the same station.h aa Minimum headway between two adjacent trains, which both arrive at the same station.i Arrival time of train i at station m, m ∈ {2, 3, . . . ,|S|}.ij Binary variable : if train i departs from station m earlier than train j, equals 1; otherwise, 0.

Table 6 .
Four types of departure headway according to the value of p m i and p m j .

Table 7 .
Four types of arrival headway according to the value of p m i and p m j .

Table 9 .
Distance and pure running time in each section.

Table 10 .
Parameters involved in the model.

Table 11 .
The results of the series of experiments.
ining Passenger Train Travel Time Number of Stops Gap Com

Table 11 .
The results of the series of experiments.