Optimizing Mixed Group Train Operation for Heavy-Haul Railway Transportation: A Case Study in China

Abstract

Group train operation (GTO) applications have reduced the tracking intervals for overloaded trains, and can affect the efficiency of rail transport. In this paper, we first analyze the differences between GTO and traditional operation (TO). A new mathematical model and simulated annealing algorithm are then used to study the problem of mixed group train operation. The optimization objective of this model is to maximize the transportation volume of special heavy-haul railway lines within the optimization period. The main constraint conditions are extracted from the maintenance time, the minimum ratio of freight volume, and the committed arrival time at each station. A simulated annealing algorithm is constructed to generate the mixed GTO plan. Through numerical experiments conducted on actual heavy-haul railway structures, we validate the effectiveness of the proposed model and meta-heuristic algorithm. The results of the first contrastive experiment show that the freight volume for group trains is 37.5% higher than that of traditional trains, and the second experiment shows a 30.6% reduction in the time during which the line is occupied by trains in GTO. These findings provide compelling evidence that GTO can effectively enhance the capacity and reduce the transportation time cost of special heavy-haul railway lines.

1. Introduction

In recent years, the direction of development of China’s railway freight has been towards the logistics of high-value goods and heavy load transportation of bulk goods. An economic analysis by Burlington Northern and Santa Fe Railway (BNSF) has shown that the best economic benefits can be achieved when using a fleet of 120 heavy-haul trains with a capacity of 112.5 tons per unit, representing a saving of 5.2% in transport costs per year.

Researchers have devoted extensive efforts to improving railway freight transport. There are two main approaches to increasing the capacity of heavy-haul railways. The first is to increase the axle load or the train length, which involves the operation of a 25-ton axle load train, a 30-ton axle load train, or a 20,000-ton or 30,000-ton combined heavy-haul train [1]. The second approach is to shorten the interval between train running times and to optimize the operational strategy [2]. Although the first approach can improve the transportation capacity to some extent, further increasing the axle load and train length can impose new problems on the existing facilities of heavy-haul railways. The second approach can be achieved by adding a small number of devices to the existing signal system, without the need to build new basic signal equipment. It is therefore more practical to consider the second approach of reducing the time interval. However, the limitations of signaling technology have always posed an obstacle to the development of heavy haul railways.

Chinese researchers have begun to study and apply group control technology to heavy-haul railways. This is a kind of signal control technology that allows multiple heavy-haul trains to run in coordination, and can be used to shorten their tracking intervals and realize dynamic marshaling and unmarshaling in transit. The National Railway Administration has referred to group control signal technology in its major scientific and technological innovation development plan. In addition, Shenhua Energy Company has invested a huge amount of money into research and development in regard to group control signal technology. On the northern line of BaoShen Railway, the verification of operational experiments in heavy-haul trains has been carried out using group control technology. From the perspective of the transportation organization, this kind of train operation organization with group control technology is called GTO [3]. It can be seen from Figure 1 that the tracking interval for GTO is shorter than in TO ($L_{unit}$ means the interval distance between the adjacent trains in GTO and $L_{traditional}$ represents the interval distance between the adjacent trains in TO).

A reduction in tracking intervals and unmarshaling in transit can have positive effects in terms of optimizing the transportation organization of heavy-haul railways. However, current transportation organization has not yet been optimized based on this technology. The introduction of group control signal technology will cause some difficulties for the current operational plan for heavy-haul railways. The existing plan needs to be optimized, and the existing model cannot be applied to the planning of group control operations. The aim of this study is therefore to optimize these operations from the perspective of the transportation organization. We also introduce some policy parameters to demonstrate that in GTO, the transportation capacity of a heavy-haul special line can be improved, and the benefits can be increased. Some suggestions are also put forward for railway enterprises.

2. Literature Review

Due to the importance of virtual coupling and the optimization of railway transportation, a huge number of studies related to these topics have been carried out over the past few decades. Although virtual coupling and group control technology are based on similar principles in terms of signal control, group control technology is a Chinese innovation specifically designed for heavy-haul railway applications.

GTO has been significantly influenced by the advanced principles of group control technology, leading to a fundamental shift away from the traditional signal block system. This transformation has effectively resulted in a reduction in the intervals between trains, and has catalyzed the emergence of a group mode of operation. The following studies concentrate on the model of a railway line and its associated algorithm.

Aoun et al. [4] analyzed the attractiveness of virtual coupling using several mathematical methods, and their results show that virtual coupling has the potential to improve current train operations. Most research in the field of railway virtual coupling is related to signal control [5,6,7,8,9].

Flexible forms of virtual coupling have posed great challenges in terms of the optimization of railway transportation. Today, research on railway operation plans based on virtual coupling is still in its initial stages, and existing studies of train operation plans using this approach are mainly related to the field of urban rail transportation. Lee et al. [10] proposed an algorithm for generating a virtual coupling train operation scheme, in which trains ran according to a fixed periodic stopping scheme. Gallo et al. [11] proposed a model based on a mathematical planning formula to optimize the number of trains in a one-way loop and to avoid wasted train capacity. Zhou [12] constructed a bi-level optimization model based on the size of virtual trains, which took into consideration the constraints on both the train departure frequency and the number of trains, and designed a solution method based on multi-population genetic algorithm. Xu [13] proposed an optimization model for a Y-type routing scheme that could ensure a relative balance between the load factors on various sections of the line while reducing the travel costs to passengers and the operating costs to enterprises. Chen et al. [3] proposed a mixed integer programming model aiming at transportation cost and travel time, and solved the fixed group train operation plan.

In order to study the operation plan of heavy-haul railway trains based on coupling technology, it is necessary to refer to research related to the optimization of transportation organization using existing technology. Wolfinger et al. [14] proposed a route planning method to minimize the cost and optimize the solution for the long-distance transportation of goods, which considered multimodal forms of transport. Qi et al. [15] optimized and solved a stopping plan and schedule for a railway transportation problem by establishing a mathematical model that considered the changing demand from passengers. Zhen et al. [16] established a two-stage stochastic integer linear programming model to maximize the expected net operating profit of a high-speed railway in order to solve the problem of allocating freight flow and transportation resources. Shao et al. [17] comprehensively optimized a train schedule and stopping plan with the aim of minimizing the weighted values of travel time and equity performance. Zhao et al. [18] proposed a multi-objective mixed-integer nonlinear programming model based on a space-time network, with the aim of minimizing costs while simultaneously optimizing a train schedule and train coupling plan. Zhang et al. [19] set up a mixed-integer linear programming model with the aim of minimizing both the train running cost and the total travel time, and solved for the optimal passenger allocation plan, train running schedule and stopping plan. Wang et al. [20] set up a mixed-integer linear programming model with the goal of minimizing the weighted sum of the total residence time and transfer time for the total number of rail vehicles, and proposed an iterative search algorithm to solve for a train combination scheme for a heavy-duty marshaling station.

Table 1. Characteristics of some recent studies.

Problem Type	Model Type	Objective	Solution	Reference
TSP&TT	MIP	Transportation cost and total travel time	MH	Chen et al. [3]
LP	LP	Transportation cost	ILS	Wolfinger et al. [14]
TSP&TT	ILP	Total travel time	CPLEX	Qi et al. [15]
TRA&FFA	TSILP	Expected net operational profit	MH	Zhen et al. [16]
TT&TSP	MILP	Total travel time, train running time, and equity performance	ALNS	Shao et al. [17]
TT&ITSP	MINLP	Total passenger waiting time, rolling stock, etc.	CPLEX&BARON	Zhao et al. [18]
LP&TT	MILP	Operation cost and total travel time	C++ with GUROBI	Zhang et al. [19]
TC	MILP	Total railcar dwell time and total railcar extra transfer time	ILS	Wang et al. [20]

Problem type: TSP—train stop planning; TT—train timetabling; LP—line planning; TRA—transportation resource arrangement; FFA—freight flow allocation; ITSP—integrated train service planning; TC—train combination. Model type: MIP—mixed integer programming; LP—linear programming; ILP—integer linear programming; TSILP—two-stage stochastic integer linear programming; MILP—mixed-integer linear programming; MINLP—mixed integer nonlinear programming. Solution: ILS—iterated local search; MH—meta-heuristic algorithm; ALNS—adaptive large-scale neighbourhood search.

summarizes the studies described above.

In the realm of transportation organization optimization, numerous scholars have significantly advanced the resolution of intricate problems through the utilization of mathematical modeling and solving techniques. However, when confronted with challenges characterized by either extensive scale or heightened complexity, the focus of research has progressively shifted towards the application of metaheuristic algorithms, notably the simulated annealing algorithm. Sirdey et al. [21] were dedicated to addressing the NP-hard resource-constrained scheduling problem and proposed a metaheuristic algorithm based on simulated annealing, demonstrating its practical capability to generate acceptable solutions in a rigorously defined sense. Burduk et al. [22] proposed a simulated annealing algorithm to solve the transport-production optimization task, and proved the ability and efficiency of the algorithm in solving the problem. Ceschia et al. [23] proposed a meta-heuristic algorithm based on simulated annealing to solve the static problems defined by INRC-II, and proved the effectiveness and superiority of the method. Majumder [24] delved into the realm of network optimization and employed a range of solution methodologies, including classical methods and the varying population genetic algorithm.

It can be seen from these studies that train timetabling and train stop planning are important aspects of the optimization of railway transportation. At present, the actual supply of freight volume cannot meet the demand, and the aims of optimization are always to reduce time and cost for a given capacity. In this paper, we therefore focus on solving for these two elements of a heavy-haul railway in GTO by aiming for maximum capacity. We also show that GTO is more effective in increasing the capacity of heavy-haul railway. The novel contributions of this research are threefold:

• We analyze the disparities between GTO and TO for the organization of transportation;
• We propose an optimization model and a meta-heuristic algorithm based on these identified disparities;
• We solve the mixed group train operation plan and demonstrate the benefits in terms of time and capacity through a case study in China.

3. Problem Statement

Freight transportation demand has several elements, including the initial station, the terminal station, the distance between the two stations, the committed arrival time, the theoretical demand, the actual supply and the actual gain. Since the operation plan for the heavy-haul train studied here is based on the background of organizing through trains at the initial technical station, we only consider the initial station and the terminal station for each freight transportation demand [3]. 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, ensuring that the freight volume is no lower than the minimum ratio and that the committed arrival time at each terminal station is met. The line and freight demand are shown in Figure 2. We note that there are differences between GTO and TO, which are mainly reflected in the following aspects.

At the initial marshaling station, a group control train is organized from unit trains under group operation, while for a conventional train, the car is used as the unit. Figure 3 and Figure 4 show the different marshaling regulations for the two modes. The order of the trains in the group should be based on the distance of the terminal; for example, the leading train in the group should have the farthest terminal station, while the tailing train should have the nearest.

When the tailing train is about to reach a terminal station, it will receive the “ungroup” command. It will then slow down to a certain speed while the remaining trains stay at the normal speed, which allows the tailing train to leave the train group. The tailing train will arrive at the corresponding terminal station, and the remaining group will continue to move on without stopping. If there is a train at the station that needs to go to a terminal station ahead, it will receive the “group” command; it will then leave the station and join the train group at a speed greater than the normal speed of the train group. The processes described above are illustrated in Figure 5.

4. Model Structure

4.1. Model Assumptions

Considering the actual production situation of special coal transport lines of heavy-haul railways in China and the generality and rigor of the research question, the following assumptions are made:

(1)

The cargo is homogenous throughout the route, and there is a substantial quantity of cargo available at the initial technical station;

(2)

The unit trains are dispatched in clusters of identical weight;

(3)

The average speed of a unit train within a group in the interval is consistent;

(4)

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

4.2. Symbols and Variables

Let $i\in A$ represent the stations along the line, and $M$ represents the number of groups in the operation plan. $m\in M$ represents the serial number of a group, $N_m$ is the number of trains in group $m$, and $n\in N_m$ is the serial number of a train. $c$ represents the capacity of a unit train, and $s$ the section between two adjacent stations. $l_s\in L$ represents the length of section $s,$ and ${\tau }_i$ is the adjacent section before arriving at station $i$. $V_{normal}$ is the speed of the trains when running normally and $V_{ungroup}$ is the speed when the trains are unmarshaling automatically. ${TS}_{(m,n)}$ represents the departure time of train $n$ in group $m$ from the initial technical station, and ${TE}_{(m,n)}$ represents its arrival time at the terminal station. $t_{group}$ represents the interval time between the departure of two adjacent groups, meaning the interval between the last train in the previous group and the first train in the next group.

Let $t_{unit}$ represent the interval time between two adjacent trains’ departure in a group, and $t_{tra}$ the interval time between the departures of two adjacent trains in TO. ${TW}_{start}$ represents the starting time of the window on the heavy-haul railway line, and ${TW}_{end}$ the ending time. $F_i$ represents the volume of theoretical demand at station $i$; $S_i$ represents the actual freight volume at station $i$; $T_i$ is the committed arrival time at station $i$; and ${\sigma }_i$ is the minimum proportion of total actual freight volume at each station. $T_{(m,n)}^{start}$ represents the departure time of train $n$ in group $m$ from the initial technical station. $T_m^r$ is the running time of the train group if all trains in group $m$ have the same terminal station $i$. $T_{(m,n)}^r$ represents the running time of train $n$ in group $m$.

The decision binary variable ${\rho }_s^j$ represents whether section $s$ is contained in the previous section of station $j$(${\rho }_s^j$ = 1 means it is contained; otherwise, it is not contained). The decision binary variable ${\epsilon }_s^j$ represents whether the destination of section $s$ is station $j$ (${\epsilon }_s^j$ = 1 means the destination is station $j$; otherwise, the destination is not station $j$). The decision binary variable ${\mu }_m$ represents whether all trains in group $m$ have the same terminal station (${\mu }_m$ = 1 means that they do; otherwise, they do not). The decision binary variable ${\theta }_{(m,n)}^j$ represents whether the terminal station of train $n$ in group $m$ is station $j$ (${\theta }_{(m,n)}^j$ = 1 means the terminal station is $j$; otherwise, the terminal station is not $j$).

4.3. Mathematical Model

4.3.1. The Goal of Model

The goal of this scheme is to maximize the capacity of a heavy-haul railway under limited circumstances under GTO.

$$Max:C={\sum }_{m\in M}{N}_m·c,$$

(1)

$$t_{unit}\le t_{group}=t_{tra},$$

(2)

$$M,N_m\in Z^{+}$$

Equation (1) is the objective function, in which the term on the right is the capacity of the train groups. Equation (2) shows the advantage of GTO in shortening the train interval, which is shown in more detail in Figure 6.

For TO, the goal can be shown as Equation (3), where $m$ represents the number of trains:

$$Max:C={\sum }_{m\in M}c,$$

(3)

As the interval time between the departure of two adjacent trains operating in a group is less than in TO, there can be more trains or groups running within the same period in GTO, as shown in Figure 7. The number of trains in the group is no less than two, and as the number of trains in the group increases, $∆t_{extra}$ shown in Figure 7 will become larger. This proof only takes into account the time interval relationships.

4.3.2. Model Constraints

To further explore the use of GTO in optimizing the operational plan, we design nine constraints to reflect a realistic situation.

In addition to Equations (1)–(3), the other formulations necessary to solve the problem are as follows:

$${TS}_{(m,n)}\ge {TW}_{end},$$

(4)

$${TE}_{(m,n)}\le {TW}_{start},$$

(5)

$$S_i={\sum }_m{\sum }_n{\theta }_{(m,n)}^i·c{\le F}_i,$$

(6)

$$S_i/{\sum }_i{S}_i\ge {\sigma }_i,$$

(7)

$$T_{(m,n)}^{start}+{{\mu }_m·T}_m^r+\left(1-{\mu }_m\right){·T}_{\left(m,n\right)}^r\le T_i,$$

(8)

$$T_m^r={\sum }_{s\in S}{l}_s·{\rho }_s^j·{\theta }_{\left(m,n\right)}^j/V_{normal},$$

(9)

$$T_{\left(m,n\right)}^r={\sum }_{s\in \left(S-{\tau }_i\right)}\left(l_s·{\rho }_s^i·{\theta }_{\left(m,n\right)}^i/V_{normal}\right)+l_{{\tau }_i}/V_{ungroup},$$

(10)

$${TE}_{(m,n)}={TS}_{(m,n)}+T_{\left(m,n\right)}^r,$$

(11)

$$N_m\le N_m^{max},$$

(12)

Equations (4) and (5) are the constraints on the window time for the heavy-haul railway line, meaning that the running time of train $n$ in group $m$ on the line should be outside of the window time. Equation (6) means that the volume of the actual gain for station $i$ should be less than that of the theoretical demand. Equation (7) shows that the ratio of the actual freight volume of each station to the total actual value should not be less than ${\sigma }_i$. Equation (8) is the constraint on the arrival time at each terminal station $i$, which means that the arrival time of each train to the terminal station $i$ cannot be later than this time. Equation (9) shows the formula of $T_m^r$, which is the ratio of the group’s running distance to its normal running speed. Equation (10) shows the formula of $T_{\left(m,n\right)}^r$, which is the sum of the ratio of the distance of preceding sections to the normal running speed and the ratio of the distance of the last section to the unmarshaling speed. In this case, there are two or more different terminal stations for the trains in group $m$. Equation (11) shows the formula for ${TE}_{(m,n)}$. Equation (12) shows that the number of unit trains in a group should not exceed the limiting value arising from current technical constraints.

4.4. Model Solution

We consider the departure time of the first train as the end time of maintenance, for both forms of operation. Based on the constraints on the intervals, we can obtain the total number of groups or trains under the two modes of operation.

Despite the evolution of sophisticated metaheuristic techniques, the unique strengths of simulated annealing warrant exploration. Firstly, in handling the complexity inherent in Mixed-Integer Nonlinear Programming (MINLP), where both continuous and discrete variables are involved, its adaptability to explore diverse solution spaces stands out. Furthermore, it has the ability to globally search by strategically accepting suboptimal solutions, thereby avoiding premature convergence to local optima. Notably, its simplicity and the flexibility to fine-tune parameters such as temperature decay rates and acceptance probabilities allow it to adapt to specific problem structures, a significant advantage. Lastly, when dealing with smaller-scale problems or limited computational resources, simulated annealing demonstrates higher efficiency in resource utilization, making it a particularly economical choice [25]. Therefore, we designed an algorithm based on the simulated annealing (SA) algorithm to solve this problem. The steps are as follows:

Step 0—Input the distance matrix, cargo demand, committed arrival time at each station and other parameters of the heavy-haul railway line;

Step 1—Set the initial solution.

Set $T=T_0$. Randomly generate the value $M$ in the given range and the matrix $N_m$ based on $M$. Each value in matrix $N_m$ should also be within the given range. Search for the corresponding routes for each value in matrix $N$. Randomly set the initial sequence of stations as the route for each group, and calculate the corresponding running time schedule for each group based on these routes. The content above is regarded as the initial solution ${\vartheta }_0$. Judge whether ${\vartheta }_0$ meets the constraints we have set. If it does not meet them, the initial solution ${\vartheta }_0$ needs to be updated. Calculate the value of the objective function, which is the corresponding transportation volume $S_0$;

Step 2—Create a new solution.

Set $T_n=\rho ·T_{n-1}$. Randomly perturb the solution ${\vartheta }_{n-1}$ to get a new solution ${\vartheta }_n$. Calculate the value of the objective function $S_n$ (we preferentially update the solution by perturbing the values of $M$ and $N_m$, just like in Step 1, because they usually affect the value of transportation volume directly. In this step, there is no specific updating direction, in order to avoid getting stuck in the local optima);

Step 3—Decide whether to accept the new solution.

Calculate the gap ${∆S}_n$ between $S_{n-1}$ and $S_n$. If $∆S<0$, accept the new solution ${\vartheta }_n$. Otherwise, calculate $P_n=exp(-{∆S}_n/T)$. If $P_n>P_{random}$, accept the new solution ${\vartheta }_n$;

Step 4—Decide whether to stop.

If the termination condition is met ($T\le T_{min}$), the current best solution is output and the program is terminated.

5. Case Study

5.1. Study Area and Data

To reflect the current situation with regard to heavy-haul railways and to make the case study more concise, we created a simplified model of a coal heavy-haul railway in China, as shown in Figure 8 [26]. In Figure 8, letters $A-F$ represent the terminal stations. The values of some variables are shown in

Table 2. Values of variables.

Variable	Value
$t_{group}$ (minutes)	12
$t_{tra}$ (minutes)	12
$t_{unit}$ (minutes)	6
${TW}_{start}$	23:00
${TW}_{end}$	02:00 (the next day)
$V_{group}$ (km·h−1)	100
$V_{ungroup}$ (km·h−1)	80

.

Table 3. Constraints on freight volume at each station.

Station	Theoretical Demand	Minimum Proportion of Total Actual Coal Volume
A	100	15%
B	100	18%
C	100	13%
D	100	12%
E	100	12%
F	100	12%

shows that there is a large demand for freight volume at each station, and the aim of the experiment is to determine how much freight volume can actually be realized in GTO. We also set a minimum proportion of total freight volume for each station, in order to ensure that the coal is not delivered only to the first few stations.

5.2. Solution Results

The mixed group train operation plan is displayed in Figure 9, which includes the quantity of trains in different groups, the stopping plan and the duration of the journey. In Figure 9, the number and the duration of the journey of the train are, respectively, indicated above the line (Q1-1 represents the first train in the first group). Observations in Figure 9 reveal variations in the travel durations of trains arriving at the same terminal station, such as Q3-1 and Q4-2. These variations stem from differences in train types. For instance, Q3-1, being the leading train in a group, does not need to have an unmarshaling process. Conversely, Q4-2 requires an automatic unmarshaling in the last section at a lower speed. Therefore, despite the fact that they share the same terminal station, the variance in the trains’ running times stems from differences in the unmarshaling process and the required speed to reach the terminal station. Figure 10 shows the convergence process of the SA algorithm. As the object of the optimized model is the transportation capacity, the trend in the objective value is to become larger. Under the specific requirements for this case, the maximum capacity is 680,000 tons per day (02:00–23:00).

In the optimization period in this case, the time required to achieve the above volume on the line is 20 h, which is less than the difference value between the start and end times of the maintenance. A comparison between the actual and minimum proportions of volume at each station is shown in

Table 4. Comparison of freight volumes.

Station	A	B	C	D	E	F
Actual proportion	26.47%	20.58%	14.71%	12.5%	13.24%	12.5%
Minimum proportion	15.00%	18.00%	13.00%	12.00%	12.00%	12.00%
Difference value	11.47%	2.58%	1.71%	0.50%	1.24%	0.50%

. If the minimum proportion is met during the optimization period, a better result will be achieved by sending more cargo to closer terminal stations.

5.3. Benefit Analysis

On current heavy-haul railway lines, coal forms the main bulk of the transport. Hence, when developing a heavy-haul railway operation plan, we need to consider the characteristics of coal, combine the advantages of a heavy-haul railway with the characteristics of the line, pay attention to the needs of customers, and design integrated freight products suitable for railway development, in order to increase the diversity of railway freight and meet the needs of various customers.

Nowadays, customers have much higher requirements for the timeliness of freight products, and improvements to this aspect are mainly made by shortening the delivery times and increasing on-time deliveries. The timeliness of freight products is affected by existing equipment and facilities, the level of organization of the transportation system, and other factors. Different goods also have different requirements as regards timeliness. We therefore analyzed the time cost as part of the benefit evaluation.

In order to analyze this problem more comprehensively, we designed two contrastive experiments. The first used the same committed arrival times, minimum cargo proportion, maintenance time and other variables as in the core experiment, and we obtained a comparison of the optimized maximum transport capacity. The capacity of a single train was set to 5000 tons.

In

Table 5. Comparison of benefits in different experiments.

Situation	GTO		TO		Ratio (%)
Same constraints	680,000 tons	20 h	425,000 tons	20 h	37.5
Same capacity	680,000 tons	20 h	680,000 tons	28.9 h	30.6

, TO means traditional operation, and “Ratio” means the optimization ratio of GTO to TO.

With the same constraints, the transportation volume achieved in TO was 425,000 tons in the same period (20 h), which was less than the volume for GTO. The iteration diagram is shown in Figure 11a.

The second contrastive experiment was based on the freight volume achieved in the core experiment. We studied the minimum time needed to complete the same freight volume without limiting the committed arrival times and maintenance time. The iteration diagram is shown in Figure 11b. Figure 10 and Figure 11 display mean values of target objectives above each stationary stage corresponding to the iteration stages. In this experiment, the time taken to transport the same amount of cargo as in the core experiment was 28.9 h.

The purpose of the first of these experiments was to examine the difference in freight volume between traditional trains and group operation under the same requirements, while that of the second was to examine the difference in the time required to complete the same cargo volume with GTO.

The results show that the freight volume for GTO reflected in the first experiment was 37.5% higher than for TO. The second experiment showed a 30.6% reduction in the time for which the line was occupied by trains under GTO.

In order to maintain the consistency of the experiment, we use MATLAB 9.13 (MathWorks, Natick, MA, USA) to program the core experiment and the comparison experiment. In these two experiments, we set the parameters of the simulated annealing algorithm, including the initial temperature ($T_0$) of 1 × 10³, the minimum temperature ($T_{min}$) of 1 × 10⁻⁶, and the number of internal iterations of 20.

The core experiment involves the group train planning, whose solution set is more complex, including multiple solutions of different dimensions, such as three-dimensional matrices and multiple two-dimensional matrices of different sizes. Because of the complexity of the solution, and the difficulty involved in the random generation of the initial solution and the neighborhood solution to meet the constraints, the solution time of the core experiment is longer. Specifically, through programming in MATLAB, the solution in the core experiment took 7.8 h. The contrastive experiments took into account TO and only took about 0.1 h. The core experiment did take longer than the computation in TO. This difference is mainly due to the relatively simple structure of the solution in the comparison experiment, which led to a significant reduction in the solving time of the simulated annealing algorithm.

6. Conclusions

GTO, a practice that involves the simultaneous operation of multiple trains within a single block, has emerged as a promising solution to some existing challenges.

Regarding its theoretical implications, the research delves into the disparities between GTO and TO within the transportation organization. This analysis contributes significantly to the theoretical understanding of operational logistics in the transportation sector. By identifying and scrutinizing these disparities, the study paves the way for a more comprehensive comprehension of the intricacies and distinctions between GTO and TO methodologies. It offers insights into how these organizational approaches differ, potentially leading to advancements in theoretical models of transportation logistics.

In practical terms, the research proposed a mathematical model to optimize the mixed group train operation plan for heavy-haul trains and solve it by applying a specific and effective algorithm. We then focused on a compelling case study in China and designed two contrastive experiments as a comparison with the core experiment, to verify the integrity and robustness of our results. The demonstrated improvements in terms of time efficiency and increased transportation capacity provide practical, implementable solutions that will benefit transportation organizations looking to optimize their operations.

Together, these results lead us to believe that this new form of operation can offer tangible benefits, and can provide valuable guidance to railway operators, policymakers, and stakeholders seeking to enhance the efficiency and sustainability of heavy-haul rail transportation in the 21st century.

The present research has some limitations, and leaves scope for further research. In future work, a more comprehensive optimization model that includes more policy parameters and more practical objectives could be implemented. The objectives in this future model should then be subdivided, as this means that they can be more detailed. We also suggest further studies that consider a more complex and realistic railway network.

From the point of view of optimization, stations that are closer are preferred for delivery, in order to maximize the capacity of a single trip. For this reason, the policy parameters set in this study were the committed arrival time and the minimum ratio of freight volume for each terminal station. After the addition of the policy parameters, the capacity in a single trip was made lower than in the above situation. It can be seen that the committed arrival time and the minimum ratio of freight volume to each terminal station are the constraints of the problem, and that these are also the aspects that need to be focused on by the railway enterprises.

This gives rise to several additional suggestions for railway enterprises. Firstly, if possible, a larger ratio of freight volume should be set for terminal stations that are closer to the original station, and a smaller ratio for terminal stations further away. Secondly, the committed arrival time is determined by the customer, and the railway enterprises need to ensure timely arrival. This can be achieved in several ways. High-value or time-sensitive freight shipments can be prioritized to ensure that the committed arrival times are met. Investments could also be made in real-time monitoring systems that track the movements of trains, so that adjustments can be made immediately if they deviate from the schedule, thereby ensuring the active management of the committed arrival times.