Modeling and Heuristically Solving Group Train Operation Scheduling for Heavy-Haul Railway Transportation

Abstract

In light of the improvements to the capacity and timeliness of heavy-haul railway transportation that can be organized through group trains originating at a technical station, we address a group train operation scheduling problem with freight demand importance via a newly proposed mixed integer programming model and a simulated annealing algorithm. The optimization objective of the mixed integer programming model is to minimize the weighted sum of the transportation cost and the total cargo travel time under the condition of matching freight supply and demand within the optimization period. The main constraints are extracted from the supply and demand relations, the cargo delivery time commitment, the maintenance time, and the number of locomotives. A simulated annealing algorithm was constructed to generate the grouping scheme, the stopping scheme and the running schedule of group trains. A numerical experiment based on a real heavy-haul railway configuration was employed to verify the efficacy of the proposed model and heuristics algorithm. The results show that the proposed methodology can achieve high-quality solutions. The case results reveal that the freight volume increased by 2.03%, the departure cost decreased by CNY 337,000, the transportation cost which results from the difference in the supply and demand matching increased by CNY 27,764, and the total cargo travel time decreased by 40.9%, indicating that group train operation can create benefits for both railway enterprises and customers.

1. Introduction

For the past two decades, with the continuous development of heavy-haul railway transportation in China and significant rises in the planned annual transportation demand, it has been vital to improve heavy-haul railway transportation capacity. At present, two main solutions are used to improve the capacity of a heavy-haul railway. The first method is to increase the axle load or train length and, for example, run trains with a 25-ton axle load or 30-ton axle load, or use 20,000-ton or 30,000-ton combined heavy-haul trains [1]. The second way is to shorten the train running interval and optimize the running strategy [2]. The transportation capacity can be improved to some extent using the first scheme, but further increasing the axle load and train length will bring new problems to the existing heavy-haul railway system. If the axle load is increased, high investment is needed to strengthen and transform bridges and culverts, roadbeds and other infrastructure. If the train weight is increased, it is difficult to solve the problem of the longitudinal force of the train operation, which easily causes off-line accidents, accelerates rail fatigue and increases the probability of rail fracture. Furthermore, as the train length increases, there is a significant increase in the distance from the station to the departure track and an increase in the train assembling time.

Therefore, under the existing infrastructure conditions, the study and application of virtual coupling technology is necessary as it can reduce the train operation interval [3]. Virtual coupling is a signal control technology that enables several trains to operate coordinately and reduces tracking intervals, as well as achieving dynamic in-transit coupling and uncoupling. From the perspective of transportation organization, this kind of train operation organization with virtual coupling technology is called group train operation. Group trains refer to multiple trains that do not rely on physical connection, but directly carry out control information interaction between trains, which enables the safe following of multiple trains with dynamic speed limit, dynamic limit interval and collaborative control mode, simulating the train group composed of multiple trains into one train safe operation [4]. In the process of two or more trains running in the normal block system, the normal tracking distance between the trains is gradually shortened to the maximum distance range of two adjacent trains during running, and trains are then integrated into one group to realize dynamic marshalling. During the running of a group, the distance between trains is gradually expanded to the normal tracking distance to realize dynamic unmarshalling.

In the 14th Five-Year Plan for the Development of Modern Integrated Transportation System in China [5], the Chinese government proposed to adopt new technology and equipment for heavy-haul railway transportation to promote coal transportation capacity. The National Railway Administration in China has listed virtual coupling technology in its major scientific and technological innovation development plan. Shenhua Energy Company has invested a huge amount of money in research and the potential application of group train operation in Shuohuang’s heavy-haul railway special line.

Virtual coupling has been mentioned to potentially improve transportation capacity from the perspective of signal control [6]. However, few studies have focused on the issues of how much transportation capacity can be improved or whether transportation time and cost can be reduced from the perspective of the transportation organization of group trains. To address this gap in the current literature, this research aims to establish a mathematical model and an algorithm to optimize the scheduling of group trains operating on heavy-haul railways. The contribution of this research is at least three-fold:

• Define and describe the grouping scheme (group order, number of unit trains within the group) and running process of group trains in detail.
• Propose a mathematical model and an algorithm for optimizing group train scheduling for heavy-haul special lines.
• Investigate the influence of the freight demand importance on scheduling optimization.

2. Related Work

Few studies have been conducted on group trains, so we mainly analyzed the literature in the areas where relevant methods could be used. Pan et al. [7] optimized a train schedule and rolling stock cycle plan, proposed an integer linear programming model based on a column generation algorithm and verified this using real data from the Shanghai Metro. The results showed that the model could optimize passenger demand and cost-effectiveness. Wang et al. [8] established a joint optimization model based on large demand scenarios or complex operation schemes and used Lagrange relaxation and a branch constraint algorithm to generate train schedules. Then, they used real data for example analysis. Jia et al. [9] proposed a mixed integer linear programming (MILP) model for the last train timetabling problem. All Pareto optimal solutions of the MILP model were generated using the augmented ε-constraint method. The results of numerical testing of the network in the Chengdu urban rail transit showed that the model was effective. Meng et al. [10] constructed an integration model considering requirements, services and resources, then designed a Lagrange relaxation algorithm to solve this model, and finally obtained a well-designed train timetable.

Samà et al. [11] studied the problem of real-time train scheduling and route planning in the railway network, proposed mixed integer linear programming, and verified it using a practical case. Allafeepour et al. [12] planned train marshalling by allocating rail cars and locomotives simultaneously according to the schedule and proposed a mathematical model of mixed integer programming based on the profit maximization of railway enterprises. They finally verified the effectiveness of the model by using an example. Bijan et al. [13] proposed a mixed integer quadratic constrained programming (MIQCP) model to solve the allocation problem of freight train locomotives. Because of the non-deterministic polynomial hard (NP-hard) nature of the problem, two metaheuristic solutions were proposed, including an adaptive genetic algorithm and simulated annealing. Experimental analysis based on small- and large-scale cases was carried out to study the performance of the solution method. Xiao et al. [14] proposed the formulation and solution of the network problem of freight train marshalling planning for China Railway, established a comprehensive optimization model of a single-segment train and double-segment train marshalling scheme with the goal of minimizing the total train time consumption of each yard, and proposed a hybrid algorithm of a genetic algorithm and tabu search to solve the single-block train formation plan (TFP) model. Finally, the model and the solution method were verified for the actual railway network in China. Abuobidalla et al. [15] introduced the idea of a multi-commodity network flow and solved the optimal route and marshalling sequence of trains from the starting point to the destination according to the determination of cargo demand, considering train running schedule and yard capacity.

Bożejko et al. [16] considered the constraints of actual railway transport, such as passenger train schedules and so on. They designed an algorithm based on Dijkstra to determine the fastest route for freight trains in the railway network. Lidén et al. [17] comprehensively considered various constraints such as the continuous working time of railway staff and comprehensive maintenance windows and established a mixed integer programming model to solve the train operation scheme and the comprehensive optimization problem of the comprehensive maintenance window. Zhao et al. [18] optimized a train marshalling plan and rolling stock schedules. Their proposed model reduced fixed and operating costs and solved the problem of rolling stock shortage when the number of rolling stocks was limited. Zhang et al. [19] proposed a method to optimize route planning and train schedules simultaneously. The effectiveness of their method was then verified by numerical experiments. Li et al. [20] proposed a multi-objective train scheduling model that minimized energy, carbon emission cost and total passenger time. A fuzzy multi-objective optimization algorithm was applied to solve the model and a non-dominated schedule was obtained. Wei et al. [21] took the departure frequency as the decision variable, the passenger travel time and the minimum enterprise operating cost as the objective function and considered the allocation of passengers between different trains. Then, they built an optimization model of the train running scheme under the condition of a general tree line. They finally verified the effectiveness of the scheme by comparing it with the split-line scheme.

Schwerdfeger et al. [22] put forward the rail platooning problem, adopted a three-layered time-space network approach to solve the resulting optimization problem and finally compared the approach with a traditional solution and showed that the new method was effective. Li et al. [23] built an optimization model of the cross-border freight train operation scheme of the China–Laos railway passage based on the data of the current situation of China–Laos bilateral trade. They took the maximization of comprehensive transport income as the goal, considering the transport conditions of the passage. Then, they solved and verified the validity of the model. Sahli et al. [24] designed an effective and robust method to solve a freight rail transport scheduling problem and proposed a genetic algorithm to do this. Experiments showed that the method was feasible and effective. Xiao et al. [25] defined an integer programming optimization model to solve a block-to-training assignment problem, and proposed a heuristic method based on a genetic algorithm and tabu search to solve the model. The research solved the train assignment problem. Habiballahi et al. [26] proposed a hybrid integer quadratic constrained programming model and introduced two metaheuristic algorithms to solve small- and large-scale cases, thus solving a new locomotive assignment problem. Milenković et al. [27] considered the current reality of truck fleet management and proposed a decomposition optimization method. Their experimental results showed that their proposed method was practical compared with existing empirical methods. Abramov et al. [28] studied the practical impact of freight rates on traction fuel and energy expenditure and the optimization of freight train speed based on relevant constraints. Huang et al. [29] proposed a two-stage entropy method based on the historical data of an index system to select railway express service stations.

Based on the above research, this paper investigates a newly proposed train operation scheduling problem, where group control technology is taken into account for heavy-haul railway transportation with a focus on through-group trains originating at a technical station. The aim is to minimize the weighted sum of the transportation cost and the total cargo travel time of the heavy-haul railway under the condition that the cargo supply and demand match within the optimization period. A mixed integer programming model is established by fully considering the matching relationship between cargo supply and demand, the cargo delivery time commitment, the maintenance time, and the number of locomotives. A simulated annealing algorithm is constructed to obtain the grouping scheme, stopping scheme and running schedule of group trains.

3. Problem Statement

In our research, we assume a heavy-haul railway line with 1 + K stations. Through trains should be organized from the initial technical station within the optimization period to transport cargo, in an amount not less than the planned supply and meeting different freight demands, to each terminal station. The line and freight demand are shown in Figure 1.

Group control technology is adopted to organize the running of group trains on this line. However, considering the technical difficulty of using a mixed group at present, the plan uses single 5000-ton unit trains that run in groups at certain intervals, and the unit trains within the group automatically decouple as planned during the interval running. During the decoupling period, the leading train maintains its normal speed, and the trailing train leaves the group at the decoupling speed. The operation mode of the group train is shown in Figure 2. The illustration on the right in Figure 2 is a detailed description of the process in the circles surrounded by a black dotted line on the left.

The aim of this paper is to optimize and obtain the group train operation scheduling of this line by comprehensively considering the freight demand importance of reference indicators such as cargo demand at each terminal station, the latest arrival time at the station, customer level and transportation distance, and under the condition of satisfying the matching relationship between cargo supply and demand, the cargo delivery time commitment, the maintenance time, and the number of locomotives. The group train operation scheduling includes the grouping scheme (group order, number of unit trains within the group), the stopping scheme and the running schedule so as to minimize the weighted sum of the transportation cost and the total cargo travel time.

4. Optimization Model

4.1. Model Assumptions

Considering the actual production situation of special coal transport lines such as the Shuohuang heavy-haul railway and the generality and rigor of the research question, the following assumptions are made:

(1)

The cargo is of a single type along the line, and the quantity of cargo at the initial technical station is abundant;

(2)

The unit trains are sent in groups of the same weight with the same number in each group;

(3)

The average speed of a unit train within a group in the interval is constant.

(4)

The railway maintenance time is for daily maintenance, and the time is fixed.

4.2. Symbols and Variables

The symbols and decision variables are shown in

Table 1. Symbols and meanings.

Symbol	Meaning
$I$	Set of network stations
$i$	Number of stations, $i\in I$
$W$	Initial technical station
$Y_i$	Station $i$
$A$	Set of network section numbers
$a$	Number of railway sections, $a\in A$
$M$	Number of operating group trains in a single day
$m$	Number of groups, $m\in M$
$N$	Number of heavy-haul trains within the group
$n$	Number of trains, $n\in N$
$D$	Set of freight demand numbers
$d$	Number of freight demand, $d\in D$
$w_d$	Freight demand importance
$f_d$	Freight demand
$T_m$	Cargo travel time of the group $m$ which has the same destination
$T_{m,n}^{\prime }$	Cargo travel time of dispersed arriving train n in group m
$T_{m,n}^s$	Departure time of train $n$ in group $m$
$T_{m,n}^e$	Arrival time of train $n$ in group $m$
${TC}^s$	Start time of maintenance time
${TC}^e$	End time of maintenance time
$T_i^{fix}$	Latest arrival time at terminal station $i$
$l_a$	Length of the railway section $a$
${\phi }_i$	Start and end points of the $i-1$ and $i$ segments
$U_{m,n}$	Load of a unit train
$H$	Number of locomotives to be pulled by a unit train
$H_{max}$	Maximum number of locomotives on the network
$t_1$	Departure interval of adjacent groups
$t_2$	Departure interval of adjacent trains within the group
$V_{normal}$	Normal operating speed of the train section
$V_{away}$	Decoupling running speed
$S_d$	Actual supply of $f_d$ for freight demand
$P_d$	Planned supply of $f_d$ for freight demand
$E$	Transportation cost
$R$	Total cargo travel time
$C$	Departure cost of a train at the initial technical station
$\mathrm{c}$	Unit transportation cost
$g$	Weighting coefficient of transportation cost
$j$	Weighting coefficient of total cargo travel time

and

Table 2. Variables and meanings.

Variable	Meaning
${\epsilon }_i^d$	A 0–1 decision variable which is 1 if station $i$ is the terminal station of freight demand $f_d.$ Otherwise, it is 0.
${\alpha }_m^d$	A 0–1 decision variable which is 1 if the freight demand of the whole group $m$ is $f_d$. Otherwise, it is 0.
${\delta }_a^i$	A 0–1 decision variable which is 1 if the railway section $a$ belongs to the path of terminal station $i$. Otherwise, it is 0.
${\theta }_{m,n}^d$	A 0–1 decision variable which is 1 if train $n$ in group $m$ belongs to freight demand $d$. Otherwise, it is 0.

.

4.3. Mathematical Formulation

Group train operation can improve the transport capacity of a line [4]. In this paper, from the perspective of optimizing the transportation supply scheme based on the established freight demand importance, the weighted sum of the transportation cost and the total cargo travel time of a heavy-haul railway in the optimized period are taken as the goal, which is:

$$\min Z=g·E+j·R$$

(1)

The transportation cost in this paper is considered to be the departure cost of the train from the initial technical station and the transportation cost of the difference between the cargo transportation supply and demand, which is expressed as Equation (2). $E_{out}$ refers to the mechanical equipment cost and information equipment cost of the initial technical station generated before the train starts and does not change with the number of single departures. Assuming that the cost of a single departure is the constant $C$ [30], this is expressed in Equation (3). $E_{gap}$ represents the transportation cost when there is a difference between the actual cargo transportation supply and the planned supply [31], and this is related to the freight demand importance, the cargo tonnage kilometers, the unit transportation cost, and so on. This value is shown in Formula (4).

$$E=E_{out}+E_{gap}$$

(2)

$$E_{out}={\displaystyle \sum _1^M}C$$

(3)

$$E_{gap}={\displaystyle \sum _d^D}\left(S_d-P_d\right)·w_d·c·{\epsilon }_i^d·L_d$$

(4)

$$L_d={\displaystyle \sum _a}(l_a·{\delta }_a^i·{\epsilon }_i^d)$$

(5)

In Equation (4), $w_d$ is determined by a comprehensive consideration of the grey comprehensive evaluation method used in [31], and $L_d$ in Equation (5) represents the transportation distance of the cargo.

The total cargo travel time mainly consists of four parts: the cargo travel time of the whole group, which has the same destination ($T_z$), the interval of group departure ($T_q$), the interval of trains within the group ($T_l$), and the cargo travel time of the dispersed arriving trains ($T_f$). The formula is shown in Equation (6). In the formula, the cargo travel time of the whole group of arriving trains is calculated for the group, and the cargo travel time of the dispersed arriving trains is calculated for each train within the group.

$$R=T_z+T_q+T_l+T_f$$

(6)

The calculation formulas for each part of the time are described as follows:

The cargo travel time of the whole group of arriving trains $T_z$ refers to the sum of the cargo travel time of groups with the same departure and terminal stations. The calculation formula is shown in Equation (7).

$$T_z={\displaystyle \sum _{m=1}^M}{\displaystyle \sum _d^D}{\alpha }_m^d·T_m$$

(7)

$$T_m={\displaystyle \sum _a^A}{\delta }_a^i·\frac{l_a}{V_{normal}}$$

(8)

In Equation (8), $T_m$ represents the cargo travel time of the whole arriving unit group.

The interval of group departure $T_q$ refers to the sum of the departure intervals of adjacent groups. The formula is:

$$T_q={\displaystyle \sum _{m=1}^{M-1}}{t}_1$$

(9)

The interval of trains within the group $T_l$ refers to the sum of the departure intervals of adjacent trains within a group. The formula is:

$$T_l={\displaystyle \sum _{m=1}^M}\sum _{n=1}^{N-1}{t}_2$$

(10)

The cargo travel time of dispersed arriving trains $T_f$ refers to the sum of the cargo travel time of group trains with different arrivals. The formula is shown in Equation (11).

$$T_f={\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{n=1}^N}{\displaystyle \sum _d^D}\left({1-\alpha }_m^d\right)·{\theta }_{m,n}^d·T_{m,n}^{\prime }$$

(11)

$$T_{m,n}^{\prime }={\displaystyle \sum _{a\in \left(A-{\phi }_i\right)}}{\delta }_a^i·\frac{l_a}{V_{normal}}+\frac{l_{{\phi }_i}}{V_{away}}$$

(12)

4.4. Constraints

In order to meet the situation where the cargo supply is not less than the demand, the actual supply for each freight demand should not be less than the planned supply, so the constraint is:

$$S_d\ge P_d$$

(13)

$$S_d={\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{n=1}^N}{U_{m,n}·\theta }_{m,n}^d$$

(14)

The cargo delivery time commitment should meet the time constraint at each terminal station and should not exceed the latest arrival time [32], so the constraint is:

$${\epsilon }_i^d·T_{m,n}^e·{\theta }_{m,n}^d\le T_i^{fix}$$

(15)

$$T_{m,1}^s\ge {TC}^e$$

(16)

$$T_{m,N}^e\le {TC}^s$$

(17)

The number of locomotives used in the optimization period should not be greater than the number of available locomotives [33], so the constraint is:

$${\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{n=1}^N}H\le H_{max}$$

(18)

The planned and the actual supply for each freight demand should be an integer greater than 0, so the constraints are:

$$S_d\ge 0$$

(19)

$$P_d\ge 0$$

(20)

$S_d$ and $P_d$ are both integers.

5. Simulated Annealing Algorithm

5.1. Algorithm Framework

The model constructed in this paper is a mixed integer nonlinear multi-objective programming model. It is an NP-hard problem. In view of its strong local search ability and short running time, a simulated annealing algorithm was selected to solve the problem.

5.1.1. Generate Initial Solution

The initial solution of the algorithm is the group train operation scheduling of the special line of the heavy-haul railway $\left\{\alpha ,\beta ,\gamma ,{T_{start},T}_{end}\right\}$. The generation rules are as follows.

(1) The binary combination of ${(M,N)}^0$ is generated randomly in the range of [$M^{min}$, $M^{max}$] and [$N^{min}$, $N^{max}$], and the number of the initial group $M$ and the number of trains in the group $N$ are obtained by decoding.

(2) Based on the number of the initial group $M^0$ and the number of unit trains in the group $N^0$, the group scheme is randomly generated (the two-dimensional matrix ${\alpha }^0$ and ${\beta }^0$ representing the whole group or the dispersed arrival information of the group).

(3) Based on the relevant parameters of the road network $\left\{l_a,L_d,V_{normal},V_{away},d,U_{m,n},{TC}^e,{TC}^s\right\}$, the decision variables $\left\{{\epsilon }_i^d,{\delta }_a^i,{\theta }_{m,n}^d,{\alpha }_m^d\right\}$ and the group schemes, ${\alpha }^0$ and ${\beta }^0$ are found using step (2). Then, the train stopping scheme of the group (a three-dimensional matrix ${\gamma }^0$ representing the specific arrival information of each train in the group) and the starting schedule $T_{start}^0$ and $T_{end}^0$ are generated.

(4) If the above generated opening strategy $\left\{{\alpha }^0,{\beta }^0,{\gamma }^0,T_{start}^0,T_{end}^0\right\}$ satisfies the constraint Formulas (13)–(20), it can be used as the initial solution $\left\{X_0|X_0={\alpha }^0,{\beta }^0,{\gamma }^0,T_{start}^0,T_{end}^0\right\}$ of the algorithm. Otherwise, the algorithm returns to step (1) to regenerate.

5.1.2. Energy Function

In order to reduce the complexity of the model solving, the output solutions $\left\{X|X=\alpha ,\beta ,\gamma ,{T_{start},T}_{end}\right\}$ are made to meet the set constraint conditions (13)–(20). Then, they are directly substituted into Equations (1)–(12) to solve the problem, and the target value is the current system energy value $Z\left(X\right)$.

5.1.3. Generate Neighborhood Solutions

For the initial ${(M,N)}^0$ binary combination obtained in Section 5.1.1, probabilistic perturbation is performed to obtain ${(M,N)}^i$. After decoding, the updated opening group $M$ and the number of trains in group N are obtained. Then, according to steps (2)–(4) in Section 5.1.1, group schemes ${\alpha }^i$ and ${\beta }^i$, stopping schemes ${\gamma }^i$ and running schedule $T_{start}^i$ and $T_{end}^i$ are generated. If the result of this strategy does not satisfy the constraint conditions (13)–(21), the probabilistic random value perturbation strategy is re-applied to the initial ${(M,N)}^0$ binary combination until the opening strategy $\left\{{\alpha }^i,{\beta }^i,{\gamma }^i,T_{start}^i,T_{end}^i\right\}$ satisfies the constraint conditions completely. Then, the neighborhood solution $\left\{X_i|X_i={\alpha }^i,{\beta }^i,{\gamma }^i,T_{start}^i,T_{end}^i\right\}$ is the output.

5.2. Steps of the Proposed Algorithm

Step 1 Determine the initial temperature $T_0$. Record the current number for the temperature drop k = 0 and then go to Step 2.

Step 2 Record the number of iterations as n = 0. At the initial temperature, generate the initial solution $X_0$ and calculate the energy function value according to the method in Section 5.1.1, then go to Step 3.

Step 3 After k times of cooling, record the solution $X_{n-1}$ after the (n−1)th iteration at the temperature $T_k$. In order to obtain the neighborhood solution $X_n$, use the random disturbance update $(M,N)$ combination to calculate the corresponding energy function value $Z\left(X_n\right)$. Then go to Step 4.

Step 4 Use the Metropolis criterion to test the current solution $X_n$. If $Z\left(X_n\right)-Z\left(X_{n-1}\right)$ is less than 0, accept $X_n$ unconditionally in place of $X_{n-1}$. If $Z\left(X_n\right)-Z\left(X_{n-1}\right)$ is greater than 0, accept the neighborhood solution with probability ${\mathsf{\gamma }}_n\left(T_k\right)$. The formula for solving ${\mathsf{\gamma }}_n\left(T_k\right)$ is:

$${\mathsf{\gamma }}_n\left(T_k\right)=\min \left\{1,exp\frac{\mathrm{Z}\left(X_{n-1}\right)-\mathrm{Z}\left(X_n\right)}{T_k}\right\}$$

Then, go to Step 5.

Step 5 Under the corresponding temperature, set the upper limit of iterative steps to $\stackrel{-}{\lambda }$. When the temperature iterations reach $\stackrel{-}{\lambda }$, go to Step 6. Otherwise, return to Step 3.

Step 6 Algorithm convergence termination decision. The termination criterion set in this paper is that the external temperature is lower than the specified threshold, and the algorithm ends when the criterion is met. Otherwise, go to Step 7.

Step 7 Lower the temperature according to the following rules. The formula is:

$$T_{k+1}=\rho ·T_k$$

(21)

where $\rho$ is equal to 0.9.

At the temperature $T_k$, the temperature is renewed after cooling. At this time, record the number of coolings k = k + 1, set the number of iteration steps n = 0, return to Step 3, and carry out cyclic iteration under the new temperature.

6. Numerical Experiment

6.1. Case Description

To facilitate the analysis, a simplified calculation example is made based on a coal heavy-haul railway in China. Figure 3 shows a diagram of the simplified line, which has one initial technical station and six terminal stations.

Table 3. Parameters of the stations.

Station	Station Center	One-Day Freight Demand (in Ten Thousand Tons)	Latest Arrival Time	Station Level
$W$	DK0 + 00	−42	-	First-class
$Y_1$	DK16 + 00	+6	18:00	Third-class
$Y_2$	DK137 + 00	+7	19:00	Second-class
$Y_3$	DK267 + 00	+7	20:00	Second-class
$Y_4$	DK434 + 00	+8	21:00	First-class
$Y_5$	DK630 + 00	+6	22:00	Third-class
$Y_6$	DK811 + 00	+8	23:00	Second-class

shows the parameters of the site, and

Table 4. Variables and values.

Variable	Value
Normal operating speed of the train section (km∙h−1)	100
Decoupling running speed (km∙h−1)	80
Departure cost of the train at the initial technical station (CNY)	5000
Unit transportation cost (CNY)	533
Start time of maintenance	24:00
End time of maintenance	02:00 (the next day)
Departure intervals of adjacent groups (minutes)	18
Departure intervals of adjacent in-group trains (minutes)	6
Departure intervals of adjacent traditional trains (minutes)	10
Weighting coefficient $g,j$	0.5, 0.5

shows other parameters.

For the first part of station center (such as DK A + B) in Table 3, A is in kilometers. For the last part of station center, B is in meters. According to the Measures for Grading of National Railway Stations, station levels can be divided into special class, first class, second class, third class, fourth class and fifth class, and the smaller the number is, the higher the level is. This is generally stipulated according to the cargo volume and technical operation size.

Using reference [34], $w_d$ is determined by the analytic hierarchy process and the grey comprehensive evaluation method, which is shown in

Table 5. The freight demand importance.

Freight Demand ${\mathit{f}}_{\mathit{d}}$	${\mathit{f}}_1$	${\mathit{f}}_2$	${\mathit{f}}_3$	${\mathit{f}}_4$	${\mathit{f}}_5$	${\mathit{f}}_6$
Importance degree $w_d$	0.5387	0.5695	0.5229	0.8135	0.3996	0.6927

.

6.2. Solution Results

6.2.1. Group Train Operation Scheduling

The scheduling diagram of group trains is shown in Figure 4. The algorithm convergence process is shown in Figure 5a, and the results are shown in

Table 6. The cost of the group train operation scheduling.

Result	Value
Number of group trains	11
Number of trains within a group	8
Number of locomotives per unit	88
Total cargo travel time (hours)	196.945
Supply and demand matching difference in transportation cost (CNY)	27,764
Departure cost (CNY)	55,000

. In Figure 4, different colors represent trains with different freight demand importance. For trains that arrive at a different station from the first train in the group, the lines displayed in their last section are not parallel to the lines in the previous sections. That is because they need to leave the group at a lower decoupling running speed before the first train in the group reaches the terminal station. As can be seen from Figure 5a, the proposed algorithm converges to the result about 150 times, which is 65.5% better than the initial solution.

6.2.2. Traditional Train Operation Scheduling

The traditional train operation plan is to start with a 5000-ton unit train by default, that is, n = 1, and the first train of the group is the last train. The results of the group train operation scheduling are compared with those of the traditional operation scheduling, and the difference is reflected in the second objective of the model, which is:

$$R^{\prime }=T_z^{\prime }+T_q^{\prime }$$

(22)

According to the actual situation of the current operation, the departure interval of the trains is set at 10 min. The algorithm convergence process is shown in Figure 5b, and the results are shown in

Table 7. The cost of group train operation scheduling.

Result	Value
Number of group trains	84
Number of trains within a group	1
Number of locomotives per unit	84
Total cargo travel time (hours)	333.28
Supply and demand matching difference in transportation cost (CNY)	0
Departure cost (CNY)	420,000

. As can be seen from Figure 5b, the algorithm designed by us converges to the result about 180 times, which is 24.4% better than the initial solution. It can be seen from the convergence results that the algorithm designed in this study has an effective solving ability. The scheduling diagram of traditional trains is shown in Figure 6. Trains with the same freight demand importance have the same color. There is no decoupling circumstance, so all the lines are parallel. By comparing Figure 4 and Figure 6, we can find that the time range required by traditional trains is larger than that required by group trains.

The results comparison of the two strategies is shown in

Table 8. Cost comparison of the two strategies.

Target	Group Train Operation Scheduling	Traditional Train Operation Scheduling
Target value	100,470	309,984
Transportation cost (CNY)	82,764	420,000
Supply and demand matching difference in transportation cost (CNY)	27,764	0
Departure cost (CNY)	55,000	420,000
The total cargo travel time (hours)	196.945	333.280

and

Table 9. Comparison of supply and demand relationship between the two strategies.

Freight Demand ${\mathit{f}}_{\mathit{d}}$	Transport Path $\mathit{W}-{\mathit{Y}}_{\mathit{i}}$	Group Train Operation Scheduling		Traditional Train Operation Scheduling		Planned Supply ${\mathit{P}}_{\mathit{d}}$
		${\mathit{S}}_{\mathit{d}}$	$∆$	${\mathit{S}}_{\mathit{d}}$	$∆$
$f_1$	$W-Y_1$	7.5	+1.5	6	0	6
$f_2$	$W-Y_2$	7	0	7	0	7
$f_3$	$W-Y_3$	7	0	7	0	7
$f_4$	$W-Y_4$	8	0	8	0	8
$f_5$	$W-Y_5$	6.5	+0.5	6	0	6
$f_6$	$W-Y_6$	8	0	8	0	8

. By calculating the data in Table 9, under the condition of lower transportation cost, the tonnage kilometers of cargo transported with the group train operation scheduling increases by 3.39 million-ton kilometers, which is an increase of 2.03%.

Compared with traditional trains, group trains are built in a unit of multiple trains. Therefore, the number of departures is lower. Because the cost of departures is only affected by the number of departures, the results show that the cost is reduced. Because the group train adopts vehicle–vehicle communication technology, it does not need to stop in the process. Only the train that is about to arrive at the terminal station will slow down and leave the group, which does not affect the operation of the trains ahead. Furthermore, the interval between group trains is smaller than that of the traditional running mode. Therefore, compared with the traditional strategy, the transit time of cargo will be reduced.

7. Conclusions

In this study, a group train operation scheduling problem considering freight demand importance in a heavy-haul railway transportation system was formulated and addressed. Specifically, the aim of the proposed approach was to improve the heavy-haul railway transportation volume and reduce the total cargo travel time by using group control technology and optimizing the operation scheduling of group trains. The study illustrated the characteristics of group trains with direct departure from a technical station, and freight demand importance was introduced, which considers freight demand, the cargo delivery time commitment, customer level and transportation distance as reference indicators. An optimization model and a simulated annealing algorithm were developed to generate the grouping scheme and the stopping scheme, as well as the running schedule.

The numerical results show that adopting the optimized group train operation scheduling potentially improves the freight volume and reduces the total cargo travel time. From the real case study, compared with traditional train operation, the tonnage kilometers of cargo transported with the optimized group train operation scheduling increases by 3.39 million-ton kilometers in one day, which is an increase of 2.03%; the total cargo travel time decreases by 40.9%. The departure cost decreases by CNY 337,000, the transportation cost, which results from the difference in the supply and demand matching, increases by CNY 27,764. These results indicate that the group train operation can create benefits for both railway enterprises and customers.

This work has opened several directions for future research on group train operation scheduling. First, the model and algorithm proposed in this paper mainly consider a single point radial freight demand, and a single group and fixed inter-group departure interval on a heavy-haul railway special line. In the future, further research can be carried out using radial freight demand at each point, a mixing situation and the optimization of the inter-group departure interval. Second, the proposed approach only takes into account the operation scheduling for a grouping scheme, stopping scheme and running schedule based on the origin and destination demand, but another research direction could be to develop methods to coordinate this with track utilization optimization at technical stations.