1. Introduction

With the continuous construction of railway lines, many cities have established multiple railway stations, and railway hubs with multiple stations have become typical [1]. Multi-station hubs not only alleviate the operational pressure of existing passenger stations but also provide more travel options for residents within the city. However, as the number of trains gradually increased after the line was opened, the disparity in workloads among the various stations within the hub also gradually became apparent, leading to an uneven distribution of capacity. This imbalance restricts the overall transportation efficiency of the hub, increases the time for trains to pass through railway hubs and also wastes the capacity of facilities, which is not conducive to reducing the operational costs of railway transportation.

Meanwhile, with the continuous expansion of urban scale, the travel demand of residents has grown rapidly and shown a diversified development trend of multi-regions, high efficiency and high quality [2]. However, affected by urban space and division of labor schemes, some passengers cannot go to the nearest station in large cities, they even have to travel a long distance to catch the train to their destination, which limits the quality of railway passenger transport services and causes the loss of passenger flow.

Therefore, how to optimize the division of labor among stations based on the configuration of hub facilities and the travel demand of urban residents has become a core issue in enhancing the transportation efficiency of railway hubs. This research abstracts the division of labor problem in multi-station railway hubs into a train routing problem, we optimize the division of labor in multi-station railway hubs by optimizing the train routes and stop schemes within the hub. Additionally, we further introduce travel zones to quantify the travel demand of residents within the city, ensuring that the division of labor scheme can take into account the travel demand of residents in different areas. On the one hand, it can provide optimization strategies for railway transportation departments to enhance the transportation efficiency of hubs, and activate the potential of the railway network. On the other hand, it can also offer development suggestions for passenger transport departments to improve the level of transport services, which is conducive to expanding the market share of railway transport.

The remainder of this research is organized as follows. Section 2 summarizes the related literature. Section 3 proposes the problem description. Section 4 formulates the mathematical model. Section 5 gives a case study consisting of several experiments to test the optimization model. Section 6 provides concluding remarks for railway hubs and further research.

2. Literature review

In the railway transportation, the optimization of trains travel path from origin to destination is typically formulated as the Train Routing Problem (TRP). The core objective of this problem is to minimize both travel time and operational cost. Depending on the path type and its hierarchical level, TRP could be further categorized into macroscopic and microscopic levels. As one of the most prevalent optimization problems in this field, TRP has generated a large number of studies.

At the macroscopic level, the railway network is abstracted as a graph comprising stations as nodes and railway lines as edges, with the primary focus on determining travel paths for trains. This level is mainly applied to line planning and timetable optimization at the network level. The optimization objective typically involves minimizing the total travel time and operating costs across all trains, subject primarily to line capacity constraints. While, at the microscopic level, the detailed topology within stations is further refined, including tracks, switches, and signaling equipment, with emphasis on arrival and departure routes. This level is mainly applied to track allocation and station operations optimization, where the objective is to minimize travel time and route occupation time, subject primarily to platform track capacity constraints and throat area operational constraints [3,4]. As this study focuses on railway passenger hubs comprising multiple stations, specifically addressing train routing and stopping patterns within such hubs. Accordingly, the literature review is structured around two main parts: macroscopic train routing problem and division of the hub.

For train routing problem, early studies on train routing optimization primarily focused on single-track railways, and subsequently expanded to double-track railways before further extending to complex railway networks [5]. This progression reflects a transition in research objects from simplified line structures toward more realistic and complicated network topologies, meeting the growing complexity of modern railway systems [6]. Carey et al. [7,8] developed a model, algorithms, and strategy for routing trains with varying speeds and stopping patterns on a unidirectional double-track railway in a previous paper, and then extended to more general and complex rail networks encompassing line choices, and station platforms. Lee et al. [9] proposed an optimization heuristic algorithm that integrates train pathing and train timetabling, with the capability to solve practically problems. A distinctive feature of this heuristic algorithm is its modeling flexibility, which improved the conventional assumption of fixed parameters, thereby better reflecting real-world operational conditions. Sun et al. [10] proposed a multi-objective optimization model, which considered the average travel time, the energy consumption, and the user satisfaction. And an improved genetic algorithm was designed to solve the train routing problem. Li et al. [11] proposed a bi-level model for simultaneously obtaining the optimal plan of passenger assignment and train routing considering the total train travel time, the average capacity utilization rate, the total travel time of passengers, and the average train attendance rate for all sections. And a particle swarm approach was adopted to solve the problem. Zhou et al. [12] constructed a directed graph to represent the composite travel network encompassing both train services and passenger flows, through which train operational choices, passenger travel choices, and their interdependent constraints could be simultaneously captured. And a mixed-integer linear programming approach was developed to jointly solve the line planning and timetabling problem. Wang et al. [13] addressed the challenge of long-distance cross-line operations under resource constraints by proposing an integrated train line planning and timetabling approach that modified train operation zones. A bi-objective model was formulated to minimize deviations of same-line trains from ideal timetables while maximizing direct cross-line origin-destination services. Zhou et al. [14] addressed complex railway networks comprising both unidirectional and bidirectional tracks, and proposed a train-based Lagrangian relaxation decomposition method for the simultaneous optimization of passenger train routing and timetabling. Zhang et al. [15] proposed a unified integer linear programming (ILP) model with two types of binary decision variables coupled through cross-resolution consistency constraints. The line planning component satisfied passenger demand via ILP, while cyclic timetabling was formulated as a multi-commodity network flow model with track capacity constraints. Wang et al. [16] proposed a mixed integer programming model based on an event-activity network framework, and the model objective was set to minimize the deviation of main-line trains from their ideal timetables while maximizing the frequency of direct services for cross-line passengers.

For division of the hub problem, it is defined as the rational planning of the types and numbers of passenger trains to be received and dispatched at each station within a railway hub, based on the coordinated utilization of all transportation resources in the hub, with the objective of minimizing train operational costs under the capacity constraints of various equipment and facilities. Caimi et al. [17] proposed a model predictive control framework as a dispatching assistant for complex central station areas. Liao et al. [18] advanced railway capacity estimation beyond conventional infrastructure-centric approaches by integrating both infrastructure and vehicle resources. Capacity was defined through a timetable saturation method, formulated as an integer programming model on a hybrid time–space network to maximize overall transportation performance. Shi et al. [19] formulated a multi-commodity flow model on a two-layer time–space network to capture the joint optimization of train platforming and shunting with service scheduling in railway hubs. The model incorporated the spatiotemporal coupling between train arrival, departure operations and shunting services, aiming to mitigate operational conflicts, reduce shunting workload, and achieve balanced resource utilization across stations and depots. Park et al. [20] investigated line planning problems with multiple patterns, analyzed the computational complexity, and developed a column generation algorithm for solution. The effectiveness of the approach was validated, offering novel modeling insights for improving passenger service efficiency and network coverage. Pellegrini et al. [21] addressed the real-time railway traffic management problem by proposing a mixed-integer linear programming with fine-grained infrastructure representation. Khaled et al. [22] proposed an optimization model under disruptive scenarios, aiming to minimize system-wide total cost, including classification time at yards and travel time along links. The optimization was to determine the optimal number of trains, their routes, and associated blocks, subject to various capacity and operational constraints at rail links and yards. Sama et al. [23] compared two selection strategies: a tactical-level approach that pre-screened alternative routings offline after timetabling using historical data, and an operational-level approach that selected routings online before real-time dispatching based on up-to-date perturbation information. Szymula et al. [24] proposed a Railway Network Vulnerability Model (RNVM) from the dual perspective of passenger flows and train operations, assessing system vulnerability by identifying critical link combinations. An integrated solution framework was developed that synergized column generation, row generation, and mixed integer linear programming, enabling the simultaneous generation of critical links, reassigned passenger flows, rerouted train paths, and adjusted timetables. Yu et al. [25] addressed the coordination of heavy and empty vehicle flows in railway hubs by developing an integrated optimization model and designing a logistic chaos map-enhanced artificial bee colony algorithm. Zhou et al. [26] addressed flow assignment for stage plan formulation across multiple marshalling stations. An optimization model was constructed with the objective of minimizing total dwell time, taking into account disassembly and assembly capacity at marshalling stations, transfer operation capacity between stations, connection time limits for inbound and outbound trains, and heterogeneous full workload constraints.

In summary, the existing researches of train routing problem mainly focus on either macroscopic line planning or microscopic track allocation, which are difficult to be directly applied to division of the hub problem. In addition, the existing researches on hub division mainly took the perspective of railway transportation enterprises, with relatively insufficient consideration of passenger travel demand. Based on this, the main contribution of this paper is stated as follows: According to the layout of lines and stations in railway hubs, we combine macroscopic line planning with microscopic track allocation to detail the train routes in railway hubs. Meanwhile, we also set up travel zones to quantify the travel demand of urban residents, and incorporate them into the optimization model to improve the quality of the division plan.

3. Problem description

The issue of division of labor among passenger stations within a railway hub essentially involves the rational allocation of the types and quantities of trains. Furthermore, this study takes a single train as the basic object to make the scheme more flexible, unlike previous studies that took the flow of trains with the same origin and destination as the basic object. Therefore, the division of labor problem can be regarded as the train routing problem in the hub, where the train route indicates the running path and the operation track, and the railway hub network is constructed as shown in Fig 1.

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Fig 1. The network structure of the railway hub.

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The details of the railway hub are presented as follows:

1. (1). Set of nodes

The set of nodes includes: (a) railway stations , indexed by , (b) connecting directions , indexed by , (c) traffic zones P, indexed by . Specifically, the connecting direction indicates the direction of a railway line, and the traffic zone indicates the travel demand of residents going to certain directions.

1. (2). Set of arcs

The set of arcs includes: (a) the arcs between railway stations and connecting directions , as shown by the blue arcs and the dotted arcs, which refer to the double-track railway lines, (b) the arcs between railway stations and traffic zones P, indicate the passenger travel routes and are shown by the yellow arcs.

1. (3). Set of tracks

The set of tracks G refers to the tracks of all stations, indexed by g. The tracks of station are represented as .

1. (4). Capacity of railway lines and tracks

The capacity of arc refers to the number of trains that can pass through the railway line within a certain period of time. The capacity of track refers to the number of trains that can be operated on the track within a certain period of time.

Based on the railway hub network, the division of labor can be represented as the train routing problem as follows: given (a) the network structure of the railway hub, including the layout of the railway lines and the details of the stations, (b) the capacity and time of the railway lines and tracks, (c) the number of various types of trains, (d) the travel demand of passengers, (e) the passing time of railway lines, tracks and passenger travel routes, (f) the unit time cost of trains and passengers, we aim to minimize the total cost of trains and passengers.

4. Modelling

4.2 Notations

The definition of the sets and parameters mentioned in the railway hub network are summed in Tables 1 and 2.

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Table 1. Notations of the sets.

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Table 2. Notations of the parameters.

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Notably, in order to indicate that departure and arrival trains can be operated at any station within the hub, virtual nodes and arcs are set. For the departure train , the virtual origin is built to connect all stations, and the arc refers to the train can be operated in station . For the arrival train , the virtual destination is also built to connect all stations, and the arc refers to the train can stop in station .

4.3 Formulation

The common approaches of the train routing problem include the Node-Arc and the Arc-Route formulation. The Node-Arc formulation focuses on the occupancy of nodes and arcs, which can detail the train routes. The Arc-Route formulation needs to construct candidate sets of train routes, which can simplify the solving methods. In this study, to comprehensively represent all possible routes and track allocation, the Node-Arc formulation is adopted instead of the Arc-Route formulation. Therefore, the decision variables are defined to represent arc usage and track occupancy. The formal definitions of these variables are listed in Table 3.

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Table 3. Notations of decision variables.

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Drawing on the relevant theories of multi-commodity network flow and combining the above decision variables, the optimization model for train routing problem in the railway hub considering multi-station collaboration and passenger demand is constructed. The objective function and constraints are stated as follows:

1. (1). The objective function

To minimize the total cost of trains and passengers, the objective function is represented as follows:

(1)

In Eq. (1), the objective function includes two parts: the train cost and the passenger cost. For the first part, we aim to obtain the shortest train routes, so the train cost is calculated by the train running time on the railway lines and the operation time at tracks within stations multiplied by per time cost of trains. For the second part, we aim to make all passengers board at their nearest stations, so the passenger cost is calculated by the passenger travel time on the travel routes multiplied by per time cost of passengers.

1. (2). Constraints

The constraints focus on the capacity of the railway hub and passenger demand, which include flow balance constraints, no circle constraints, stop frequency constraints, the relationship between stop stations and train routes constraints, capacity of arcs constraints, capacity of tracks constraints, capacity of trains constraints, distribution of passengers constraints and decision variables constraints. The specific formula is represented as follows:

Constraints (2) are flow balance constraints, which are the conventional constraints for multi-commodity network flow. For any train , the total occupation of all arcs departing from its origin minus the total occupation of arcs arriving at it equals 1. The total occupation of arcs departing from its intermediate nodes equals that of arcs arriving at them. The total occupation of arcs departing from its destination minus the total occupation of arcs arriving at it equals −1.

(2)

Constraints (3) and (4) are no circle constraints, which ensure that there are no circles in the train routes.

(3)(4)

Constraints (5) and (6) are stop frequency constraints. Considering excessive stops would lead to frequent starts and stops of the train, not only increasing energy consumption but also occupying capacity and prolonging the time to pass through the hub. Therefore, we stipulate that a train can make at most two stops within the hub, and each stop can only be carried out on one track for departing and arriving operations.

(5)(6)

Constraints (7) indicate the relationship between stop stations and train routes, which ensure that the stop stations of the train must be located on its route.

(7)

Constraints (8) are capacity of arcs constraints, which belong to the capacity constraint in multi-commodity network flow. This set of constraints ensures that the number of trains passing through any railway line within the decision-making period is less than or equal to the capacity of arcs.

(8)

Similar to the previous constraints, constraints (9) are capacity of tracks constraints, which represent the receiving and dispatching capacity constraint of tracks at the station. It ensures that the number of trains departing and arriving on a certain track within the decision-making period is less than or equal to its receiving and dispatching capacity.

(9)

Constraints (10) are capacity of trains constraints, which are coupling constraints between train decision variables and passenger decision variables, indicating that the total number of passengers from all traffic zones boarding at station to direction is less than the total passenger capacity of all trains departing from station to direction .

(10)

Constraints (11) are distribution of passengers constraints, which indicate the distribution of passengers from the traffic zone to direction among stations.

(11)

Constraints (12) are decision variables constraints, respectively represent the domains of the train and passenger decision variables.

(12)(13)

As for the objective function (1) and constraints (2–13) are linear, the model M1 can be regarded as a mixed integer linear programming model (MILP). The proposed MILP belongs to the class of NP-hard problems, the worst-case computational time grows exponentially with problem size in theory. However, as the model structure is similar to the classical Vehicle Routing Problem(VRP), which has been solved efficiently by many mature algorithms, when the problem scale is not too large, M1 can be solved by using commercial solvers. The GUROBI solver is equipped with various exact and heuristic algorithms, offering fast solution speed and high solution quality. Therefore, the Python programming language is used to call the GUROBI solver within the PyCharm Community software platform to solve model M1. The running environment is a laptop with an Intel(R) Core(TM) i7-9750H CPU at 2.60 GHz and 16GB of memory.

5. Case study

5.1 Data source

Based on the current development status of railway hubs in China, it can be observed that the number of stations in a hub is generally no more than 3, and the number of connecting directions is generally no more than 10. For example, as one of the largest hubs in China, Zhengzhou High-Speed Railway Hub consists of three passenger stations: Zhengzhou Station, Zhengzhou East Station and Zhengzhou South Station, which connect 8 directions including Beijing, Jinan, Xuzhou, Hefei, Wuhan, Chongqing, Xi’an and Taiyuan railway lines.Therefore, this paper designs a calculation example based on the actual railway network scale. The example railway hub network designed in this section includes three passenger stations, seven directions and three traffic zones. The specific network structure is shown in Fig 2. All railway lines are single-direction double-track railways, and the line conditions of up and down directions are the same. The running time and capacity of each railway line, as well as the travel time of passengers on each travel route, are marked on the corresponding arcs in the figure. The operation time and capacity of tracks in each station are shown in Table 4. The operation time is directly proportional to the distance between the track and the main line. The train types and quantities of each direction within the hub are shown in Table 5, and the number of passengers traveling from each traffic zone to each direction is shown in Table 6. Based on actual railway operational data, the procurement cost of each train is approximately 150–200 million yuan. With an annual utilization of 140 days and factoring in depreciation expenses, the resulting operating cost amounts to roughly 1,000 yuan/minute [15,29]. In addition, passenger costs include direct time value and indirect costs, according to the average annual disposable income of urban residents, the travel time cost of passengers is approximately 10–15 yuan/minute [30,31]. Furthermore, in order to ensure that the influence of train cost and passenger cost on the objective function is at the same order of magnitude, after several sets of test experiments, per time cost of trains is set to 1,000 yuan/minute, and per time cost of passengers is 10 yuan/minute. The passenger capacity of all trains is set to 2,500 people per train.

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Table 4. Operation time and capacity of tracks in the example stations.

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Table 5. Number of trains of all directions in the example hub.

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Table 6. Number of passengers of all traffic zones in the example hub.

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Fig 2. The network structure of the example railway hub.

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5.2 Optimized results

By inputting the basic data from the previous section into the optimization model M1. The Python language is used to call the GUROBI solver to obtain the optimization solution, with a solution time of 3.26 seconds.

The objective function value is calculated to be 21,396 thousand yuan, of which the train cost is 396 thousand yuan, including 332 thousand yuan for the train running cost on the railway lines and 64 thousand yuan for the train operation cost on the tracks. The passenger cost is 21,000 thousand yuan. The train routes and the operation tracks they stop at are shown in Table 7, and the passenger travel routes are shown in Table 8.

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Table 7. Optimized train routes of the example hub.

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Table 8. Optimized passenger travel routes of the example hub.

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Based on the train routes and passenger travel routes in Tables 7 and 8, the following conclusions can be drawn: ① The routes of most trains are the shortest routes in the railway hub network, demonstrating the effectiveness of the optimization model M1 in limiting train operation costs; ② All passengers can reach the nearest stations to travel to their target directions, indicating that the optimization model M1 can not only optimize train routes but also better meet passenger demand; ③ Trains with the same origin-destination may have different routes and stop stations. For instance, No. 49 and 50 trains are both passing trains from b7 to b2, with various routes b7-a3-a2-b2 and b7-a3-a1-a2-b2, and stops at a2, a3 and a1, a3 respectively. This shows that the optimization model M1 can flexibly arrange the division of labor plan based on the track utilization rate of stations and the travel demand of nearby passengers; ④ Eight trains stop twice within the hub, proving that the optimization model M1 can take into account the passenger demands of multiple traffic zones; ⑤ The three passenger stations in the hub each undertake 20 trains, which indicates that the division of labor plan can effectively coordinate the capacity of all stations within the hub and balance the operation burden among stations.

5.3 Discussion

To further illustrate the effectiveness of the optimization method, this section will compare the optimization scheme with two other schemes, namely the fixed scheme and the stop-once scheme.

1. (1). Comparison with the fixed scheme.

The fixed scheme is a commonly used division mode in current railway hubs. Its characteristic is that each passenger station only receives and dispatches the trains of the nearest connecting direction. This division plan is relatively clear and conducive to train scheduling for the railway department.

The train routes are allocated based on the principle of the shortest path, and the travel routes of passengers are allocated under the passenger capacity constraints. The costs of each item in the objective function under the fixed scheme are calculated and compared with those in the optimized scheme. The results are shown in Table 9.

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Table 9. Comparison between the cost of fixed scheme and optimization scheme (thousand yuan).

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The comparison results in Table 9 lead to the following conclusions: ① Compared with the fixed scheme, the total cost of the optimization scheme decreased by 0.1%, among which the train cost decreased by 5.3%, while the passenger cost remained unchanged. This indicates that the optimization scheme can effectively reduce the railway operation cost on the basis of meeting the passenger demand; ② Compared with the fixed scheme, the train running cost decreased by 4.6% and the track operation cost decreased by 8.6%. This shows that the optimization scheme can simultaneously optimize the train routes and stop plans.

To further refine the analysis of the optimization effect, focusing on the railway lines, the analysis is conducted from the perspectives of line capacity utilization rate and track capacity utilization rate, as shown in Figs 3 and 4 and Table 10.

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Table 10. Track capacity utilization rates of fixed and optimization scheme.

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Fig 3. Comparison between arc capacity utilization rates of fixed and optimization scheme.

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Fig 4. Comparison between track capacity utilization rates of fixed and optimization scheme.

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From the comparison results of the capacity utilization rate in Figs 3 and 4 and Table 10, the following conclusions can be drawn: ① The arc capacity utilization rate of the fixed scheme and the optimization scheme is not significantly different. The number of tight sections (capacity utilization rate over 40%) is one for both schemes, and the number of busy sections (capacity utilization rate between 26% and 40%) is six and five respectively. This indicates that the optimization scheme has alleviated the busy situation of some sections, such as [a1, a3]; ② The fixed scheme can only obtain a feasible solution by relaxing the track capacity constraints, with the maximum track capacity utilization rate reaching 183%, which proves that the load on the busy tracks in the fixed scheme is relatively large. The maximum capacity utilization rate in the optimized scheme is 100%, which reduces the tight capacity situation of the busy tracks and avoids the overloading of stations; ③ The variances of the track operation capacity utilization rate for the fixed scheme and the optimization scheme are 0.36 and 0.17 respectively, which indicates that the optimization scheme makes more reasonable and balanced use of tracks, which is conducive to maintaining the flexibility of the station operation.

1. (2). Comparison with the stop-once scheme.

In practical railway operations, to reduce the train passing time within the hub, it is conventional to arrange trains to only stop once in the hub. However, this practice may be unable to fully accommodate the passenger demand of different urban areas. Specifically, the comparison of each cost item in the objective function is presented in Table 11, and the comparison of passenger travel routes is shown in Table 12 and Fig 5.

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Table 11. Comparison between the cost of stop-once scheme and optimization scheme(thousand yuan).

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Table 12. Passenger travel routes of the stop-once scheme and optimization scheme.

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Fig 5. Comparison between passenger travel routes of stop-once and optimization scheme.

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Based on the comparison results in Tables 11 and 12 and Fig 5, the following conclusions can be drawn: ① Compared with the stop-once scheme, the total cost of the optimization scheme decreased by 2.7%, among which the train cost increased by 4.2% while the passenger cost decreased by 2.8%. This indicates that although the optimization scheme slightly increased the train cost by stopping more times, it better balanced the passenger demand of multiple traffic zones, achieving the optimization effect of reducing the total cost; ② In the stop-once scheme, 71.4% of the passengers can board at the nearest stations to the target directions, while the optimization scheme enables all passengers to travel from the nearest stations, proving that the optimization scheme can better serve the passenger demand of different traffic zones within the city, which is conducive to improving the quality of railway passenger transport services.

6. Conclusions and future work

This paper takes railway hubs with multiple stations as the research subject, abstracting the division of labor problem among different stations as the train routing problem within the hub. To accommodate the travel demands of urban residents, multi-stop operations of trains are incorporated. By leveraging the multi-commodity network flow optimization theory and the Node-Arc modeling approach, a MILP optimization model for railway hubs is constructed. The objective function aims to minimize the sum of train cost within the hub and passenger cost travel from traffic zones to stations, with constraints including arc capacity, track capacity, and passenger capacity. Finally, a case study of a railway hub with three stations is designed to validate the effectiveness of the proposed model. The key conclusions are as follows:

1. (1). Compared with the fixed scheme, the optimization scheme reduces the total cost by 0.1%—with the total train cost decreasing by 5.3% while passenger remains unchanged. This demonstrates that the optimization scheme can effectively lower railway operation cost without compromising passenger demand and better balance the operational loads among different stations.
2. (2). Compared with the stop-once scheme, the optimization scheme cuts the total cost by 2.7%: train cost increases by 4.2% (attributed to additional stops), while passenger cost decreases by 2.8%. This indicates that despite a slight rise in track operation cost, the optimization scheme better addresses the passenger demand of multiple traffic zones.
3. (3). To ensure hub operational efficiency and service quality, railway departments should fully utilize the connecting lines within the hub to design more flexible division of labor schemes—thereby reducing train passing time and balancing operational loads. Additionally, appropriate multi-stop arrangements should be considered based on the distribution of passengers to better meet their mobility needs.

However, there are some limitations in this paper and the further work will focus on the following two points:

1. (1). This paper focuses on the outward travel demand of passengers within the hub but does not account for the transfer needs of transfer passengers. As the number of hub connecting directions increases in the future, rational accommodation of transfer passengers’ needs will become a critical optimization direction for division of labor.
2. (2). In the modeling process, some infrastructure conditions not central to this study (e.g., station throat capacity, EMU depot capacity) were simplified. Future work can refine these model details to make the optimization scheme more aligned with actual railway operations.