Train Routing and Track Allocation Optimization Model of Multi-Station High-Speed Railway Hub

Abstract

A multi-station high-speed railway hub is the interaction of multiple high-speed railway lines, and its operation determines the efficiency of the whole network. In this study, we focus on the train routing problem for a multi-station railway hub and propose a flexible scheme to improve the capacity utilization. Based on the flexible scheme, a mixed-integer programming node-arc model is formulated to minimize the total cost of train and passenger routes. Specifically, for diversifying the train routes, we consider the individual train rather than the train flow as the basic unit, which differs from the approach in previous studies. For each train route, in addition to the macro-scale route between stations, the micro-scale track allocation inside stations is also considered. Afterward, the optimization solver Gurobi is used to solve the model and obtain the optimization scheme. A case study based on real data from the Zhengzhou railway hub in China is implemented to evaluate the effectiveness of the optimization model. By comparing the optimization scheme with the fixed scheme and the sequential scheme, the results show our method could reduce the total cost by 29.35% and 22.58%, and the line and track capacity utilization of the optimization scheme is more reasonable. We provide some suggestions to help railway managers improve the operation efficiency and service quality of multi-station high-speed railway hubs.

1. Introduction

With the continuous expansion of high-speed railway networks, many new high-speed railway lines are being constructed in the railway hubs [1,2]. Owing to the limited capacity of a single station, most hubs have built multiple passenger stations to accommodate more new trains, thus forming multi-station high-speed railway hubs. A multi-station high-speed railway hub is an urban area containing multiple high-speed railway stations and lines, such as the Zhengzhou railway hub in China and the Berlin railway hub in Germany.

At present, the stations in multi-station hubs independently operate trains, and railway operators usually set the connecting direction of each station. For example, at the Zhengzhou railway hub, Zhengzhou Station only deals with the Taiyuan and Xi’an trains; Zhengzhou East Station operates the Beijing, Xuzhou, Wuhan, and Jinan trains; and the newly built Zhengzhou South Station accommodates the Hefei and Chongqing trains. Although this fixed scheme is clear and easy to implement, it still has shortcomings. First, since the number of trains running on different railway lines varies, the operations of the stations are unbalanced. For example, Zhengzhou and Zhengzhou East stations are too busy to operate new trains, but Zhengzhou South station is idle and has surplus capacity. Second, this fixed scheme cannot meet the travel demands of all passengers in the city. Since sometimes no train travels to the target direction at nearby stations, passengers have to board at distant stations.

To address these weaknesses of the fixed scheme, in this study, we construct a flexible scheme to improve the overall utilization of all stations in multi-station hubs. In this scheme, the connecting directions of all stations are flexible, which means that all trains could stop at any station at the hub. Thus, some workload can be transferred from busy to idle stations. Moreover, owing to the diversity of train routes, passengers can board at their nearest station to reach their destination. Based on this flexible scheme, the optimization of the multi-station railway hub can be regarded as a train routing problem (TRP) [3,4,5], in which the aim is to optimize the train routes from origins to destinations.

We focus on the TRP at a multi-station high-speed railway hub, based on which we create a mathematical formulation to minimize the total cost of the train routes and passenger routes. In view of the effectiveness in solving the network flow problems, the optimization solver Gurobi is used to solve the model and generate the optimization scheme. The main contributions of this study are as follows:

(1)

We construct a flexible scheme that improves the overall utilization of all stations. Different from the current fixed scheme, this scheme allows all trains to stop at any station of the multi-station hub, which balances the operation of stations and improves the whole efficiency of the hub.

(2)

Based on the proposed flexible scheme, we formulate a mixed-integer programming node-arc model to route trains at multi-station high-speed railway hubs. In the formulation, the individual train, rather the train flow, is taken as the basic unit, which benefits the determination of various train routes. Each train route is divided into the route between stations and the track allocation inside stations, which is a combination of macroscopic and microscopic TRP.

(3)

Considering the passenger demand, we set the passenger traffic zones and define passenger decision variables. Then, the passenger cost is introduced into the objective function to optimize the passenger routes.

(4)

A case study of the Zhengzhou railway hub in China is carried out. We analyze the optimized results and compare them with the results of two other scenarios. Scenario 1 is a fixed scheme, and Scenario 2 is the sequential optimization of train routes and passenger routes. The comparison results verify the effectiveness and the benefits of the optimized results produced by the proposed method.

The remainder of this paper is organized as follows: Section 2 summarizes the related literature on TRP. Section 3 gives the detailed problem description. Section 4 formulates the mixed-integer programming node-arc model. Section 5 outlines a case study consisting of several experiments to examine the performance of the optimization model. Section 6 provides our concluding remarks for railway hub management and further research.

2. Literature Review

As the core problem of railway optimization, TRP has been widely studied. This problem can be divided into macroscopic and microscopic TRPs.

The macroscopic TRP involves the TRP on the railway corridors or networks and is also called the line planning problem [6,7]. These studies mainly aimed to optimize train routes, stop patterns, and service frequencies by constructing mathematical models. Generally, the objective function is to minimize total train travel cost, and the capacities of lines and stations are considered in the constraints [8,9,10,11]. The common models of macroscopic TRP include the node-arc and arc-route models [12,13,14]. In the node-arc model, whether the train occupies arcs or nodes is the decision variable, in which all possible routes can be considered. However, the huge number of variables in this model also increases the complexity of the solution, so this kind of formulation is more suitable for small-scale problems. Carey and Crawford [11] focused on the railway networks of busy complex stations and designed heuristic algorithms based on the experience of operators to avoid timetable conflicts. Lee and Chen [15] proposed a four-step heuristic algorithm to assign the trains to railway lines and generate feasible train timetables according a threshold accepting rule. Wang et al. [7] proposed a two-layer formulation within a simulation framework to optimize the stop patterns and service frequencies of trains. Zhang et al. [16,17] focused on a railway corridor with parallel lines and proposed integrated models considering the pattern of switch-line trains to minimize the sum of travel time and maintenance costs. In comparison, the decision variable of the arc-route model is whether the train occupies the alternative route, which can reduce the number of variables by constructing a set of alternative routes, which is more suitable for large-scale problems. Zhou et al. [18] constructed an extended space–time–speed network to simultaneously optimize the train routes and timetables. They introduced a Lagrangian relaxation solution with a dynamic programming algorithm to decompose the formulation. Zhou et al. [3] integrated the microscopic TRP with timetabling and rolling stock allocation and proposed an integrated model and a metaheuristic method to optimize the train operation. Stojadinović et al. [19] proposed a hybrid auction to allocate the capacity of congested railway lines and proved the feasibility of the method. Bi et al. [13] proposed a mixed-integer programming model to optimize the train routes between express distribution centers in a high-speed railway network. Meng et al. [2] introduced train speed control into the microscopic TRP and proposed a two-stage programming model to minimize the sum of train running costs and energy consumption. Dedík et al. [20] considered the impact of COVID-19 on the passenger frequencies and attempted to adjust the timetables of long-distance trains to improve the service quality.

The microscopic TRP is the TRP inside the stations, which is also called the track allocation or train platforming problem [6,21]. Compared with the macroscopic TRP, more detailed train route information can be obtained with the microscopic TRP by considering the station layout and track capacity [22,23,24,25]. The common method used to solve the microscopic TRP is the conflict graph. Zwaneveld et al. [24,25] proposed a node packing model to overcome the conflicts between train routes and presented a branch-and-cut algorithm to solve the model. However, owing to the complexity of model construction, the conflict graph approach cannot be used to quickly obtain a satisfactory and feasible solution. Inspired by the manual methods of train planners, Carey and Carville [26] designed a scheduling heuristic algorithm for routing and scheduling trains at busy complex stations. Rodriguez [27] and Corman et al. [28] proposed a constraint programming model and an alternative graph programming model, respectively, to route trains through stations. Cacchiani et al. [29] constructed a tutorial and introduced common models and algorithms using a small-scale example. Recently, researchers have used the multicommodity flow method to tackle this problem. Meng and Zhou [30] focused on a network consisting of single- and double-track railway lines. They detailed the layout of stations, constructed a space–time network and proposed an integer programming model using cumulative flow variables to minimize the total deviation time of trains. Garrisi and Cervelló-Pastor [1] optimized the train routes between a series of busy stations and developed a heuristic algorithm to obtain high-quality solutions. Samà et al. [31] proposed an integer linear programming model to select the best alternative subsets from all possible routes for each train and solved the formulation through an ant colony algorithm. Xu et al. [32] focused on a double-track railway line and introduced three switchable scheduling rules to avoid conflicts. Luan et al. [33] focused on the problem of simultaneously scheduling trains and planning preventive maintenance time slots on a railway network. Through regarding each preventive maintenance as a virtual train, a mixed-integer integrated programming model based on the cumulative flow variable was proposed to minimize deviations of trains. Zhang et al. [34] used the big-M method to minimize the train travel time and the maintenance tardiness; they added speed restriction constraints to the model. Xu et al. [35] detailed the train formation process in high-speed railway stations and proposed a multi-objective programming model to minimize the train route cost and balance track allocation. Liao et al. [36] integrated the TRP with the vehicle circulation problem, an integer programming model was proposed to maximize the overall transportation efficiency, and a Lagrangian relaxation algorithm was developed to solve the integrated model. Gao and Niu [37] focused on the TRP of multi-type trains by considering all kinds of headways and relaxing the train time windows, a flexible framework was constructed to optimize train timetables, and an algorithm based on an alternating direction method of multipliers (ADMM) was developed. Zhang et al. [21] simultaneously reoptimized the train timetable and track allocation under scheduled track maintenance, aiming to minimize the number of train cancellations and the weighted train travel times.

With the intensification of transportation market competition, high-speed railway transportation quality will imminently be required [20,38]. Given this context, some researchers have begun to consider the impact of passenger demand on train routes [39,40]. Li et al. [41] considered passenger assignment in the TRP and proposed a bi-level multi-objective integer programming model to simultaneously optimize train routing and passenger assignment. Dong et al. [41] integrated train stop planning and timetabling under time-dependent passenger demand for commuter railways. Zhan et al. [42] proposed an integrated train scheduling and passenger routing programming model to minimize the total delay cost under disruption and designed a decomposition algorithm based on the ADMM to solve the formulation. Nguyen et al. [43] proposed a multi-objective integer programming model to minimize the travel time of passengers and number of trains, and then a cross-entropy method was developed to solve the model and generate the optimized service plans. Zhao et al. [44] considered passenger demand as a basic train formation factor and integrated formation with rolling stock allocation on an intercity railway line.

In summary, although the TRP has been well-studied, the approaches of previous research are not fully applicable to the multi-station hub, and the limitations are presented as follows: (1) Previous research mainly focused on one aspect of the macroscopic and microscopic level. The macroscopic research aimed to optimize the routes between stations and usually ignored the track layout of stations, while the microscopic research usually just concentrated on the track allocation of a single station or several adjacent stations due to the problem scale. However, the multi-station hub contains multiple railway lines and stations, and the detailed layout of stations should also be considered to optimize track allocation; thus, we need to combine the approaches of the macroscopic and microscopic TRP. (2) Most previous research still aimed to minimize the railway operation cost and did not consider passenger cost, which was not conducive to the competitiveness of railway companies. Since the main service target of the railway hub is urban residents, their travel demand should be taken into account in the optimized scheme. (3) In the modeling method of previous research, the train flow was usually taken as the basic unit, and although this method could simplify the formulation and reduce the problem scale, it may lead to centralized routes and congestion on some trunk lines. Therefore, more diversified train routes should be planned to avoid the unbalanced distribution of trains in the multi-station hub.

Thus, addressing the characters of the multi-station hub, this paper attempts to propose more applicable approaches. First, we focus on the TRP on macroscopic and microscopic levels, and both of the train routes between stations and the detailed track allocation are optimized simultaneously. Second, in order to ensure the service quality of the optimization scheme, the passenger demand is also introduced into the optimization. Third, to diversify the train routes and track allocation, we regard the individual train rather than the train flow as the basic unit.

3. Problem Description

Inspired by the network structure considered in previous studies [6,13,21,26,30], we construct a multi-station high-speed railway hub network, as shown in Figure 1. The railway hub network is expressed as $G=(V,E)$, where $V$ is the set of nodes, which represents the railway stations, connecting directions and traffic zones; $E$ is the set of arcs, which represents the railway lines. The notations are listed in

Table 1. Notations used in this paper.

Parameters	Definition
$V$	Set of nodes, indexed by $v$
$A$	Set of stations in the hub, indexed by $a$
$B$	Set of connecting directions, indexed by $b$
$Bh,Bu,Bi$	$Bh$ refers to high-speed directions; $Bu$ refers to normal-speed directions; $Bi$ refers to intercity directions
$Q$	Set of branch nodes, indexed by $q$
$K$	Set of trains, indexed by $k$, $(b,k)$ indicates the $k$th train of direction $b$
$K_{bd},K_{ba},K_{mn}$	$K_{bd}$ is the set of departure trains in connecting direction $b$, $K_{ba}$ is the set of arrival trains in connecting direction $b$, $K_{mn}$ is the set of passing trains from direction $m$ to direction $n$
$G$	Set of tracks, indexed by $g$
$G_a$	$G_a$ is the set of tracks of station $a$
$O$	Set of traffic zones, indexed by $o$
$El$	Set of arcs, including $E_2,E_3,E_4,E_5$, indexed by $(i,j)$
$Et$	Set of arcs $E_1$, indexed by $(o,a)$
$n{f}_b$	Number of departure trains moving from the stations in the hub to direction $b$ (trains/day)
$n{d}_b$	Number of arrival trains moving from direction $b$ to the stations in the hub (trains/day)
$n_{mn}$	Number of passing trains moving from direction $m$ to direction $n$ (trains/day)
$c_{ij}$	Capacity of arc $(i,j)$ (trains/day)
$d_{ij}$	Length of arc $(i,j)$ (km)
$w_{ag}$	Capacity of track $g$ in station $a$ (trains/day)
$p_{ob}$	Number of passengers moving from traffic zone $o$ to direction $b$ (persons/day)
$d_{oa}$	Distance from traffic zone $o$ to station $a$ (km)
$u_l$	Per cost of train running on lines (thousand RMB/train·km)
$u_g$	Cost of train operation on tracks (thousand RMB/train)
$u_p$	Per travel cost of passengers (thousand RMB/person·km)
$p_h$	Passenger capacity of high-speed train (persons/train)
$p_u$	Passenger capacity of normal-speed train (persons/train)
$p_c$	Passenger capacity of intercity train (persons/train)

, with the related concepts presented below.

(1)

Set of nodes

The set of nodes $V$ includes (a) stations $A$, indexed by $a$; (b) connecting directions $B$, indexed by $b$; (c) traffic zones $O$, indexed by $o$; (d) branch nodes $Q$, indexed by $q$. Specifically, each station $a$ includes a set of tracks $G_a$, which represents the track layout of the station, indexed by g. The connecting direction $b$ indicates the direction of the railway line. To express the passenger demand, we construct traffic zone $o$ to represent a set of local passengers traveling to a certain direction. In addition, to represent different types of trains on the same railway lines, we set branch node $q$ to indicate the intersection of the high-speed and normal-speed connecting directions.

(2)

Set of arcs

The set of arcs $E$ includes (a) the arcs $E_1$ between traffic zones $O$ and stations $A$, as shown by the red arcs, which indicates the travel route of passengers; (b) the arcs $E_2$ between stations $A$, as shown by the green arcs, which refers to a railway line connecting stations; (c) the arcs $E_3$ between stations $A$ and branch nodes $Q$; (d) the arcs $E_4$ between stations $A$ and directions $B$; (e) the arcs $E_5$ between branch nodes $Q$ and directions $B$. Arcs $E_3,E_4,E_5$ refer to the double-track railway lines and are represented by blue arcs.

(3)

Arc capacity

The capacity of arcs $E_2,E_3,E_4,E_5$ refers to the number of trains that can travel the railway lines within a certain period of time [13,45,46]. Owing to the unpredictability of passenger behavior, we define the capacity of arcs $E_1$ as infinity. Specifially for arc $E_5$, the sum of the capacity of arc-linked directions is equal to the capacity of arc-linked stations.

(4)

Track capacity

Similar to the arcs, the track capacity refers to the number of trains that can operate on a track within a certain period of time.

The train types include departure, arrival, and passing trains. Departure trains are those departing from the hub, arrival trains are those arriving at the hub, and passing trains are those passing through the hub without stopping.

Based on the railway hub network $G=(V,E)$, the TRP of a multi-station railway hub can be stated as follows: Given (a) the sets of departure trains $K_{bd}$ and arrival trains $K_{ba}$ of each direction $b$, and the sets of passing trains $K_{mn}$ from direction $m$ to $n$; (b) the number of passengers $p_{ob}$ in traffic zone $o$ heading in direction $b$; (c) the length of the railway lines and passenger routes; and (d) the capacity of arcs and tracks, by taking the individual train as the basic unit, and we aim to minimize the total cost of train and passenger routes.

4. Mathematical Model

Here, we introduce the method used for modeling the TRP of a multi-station high-speed railway hub. By using the multicommodity flow modeling method, we design a mixed-integer programming node-arc model to minimize the total train and passenger costs.

4.1. Assumptions

Without loss of generality, the following assumptions are given to simplify the formulation:

(1)

For each connecting direction, the number of trains is predetermined, including departure, arrival, and passing trains. As the capacity occupied by the non-stopping trains is relatively small, we ignore the number of non-stopping trains.

(2)

Based on the flexible scheme, trains do not have preferred routes, and they can stop at any station in the hub.

(3)

The capacity of arcs and tracks is calculated after deducting the occupation of freight trains, and the occupation of high-speed and normal-speed trains is the same.

(4)

The track capacities are different, but their operational cost is the same, which ensures that the trains have no preferred tracks.

(5)

For traffic zones, only the passengers departing from the stations are considered. Since the arriving and transit passengers are coming from outside the cities and their amounts are less than that of departing passengers, these parts of passenger demand are ignored.

(6)

The scale of each station is enough to accommodate passengers boarding at this station.

(7)

The capacity of other hub facilities is sufficient, such as the train maintenance depot. It is because the train operation volume of these facilities is relatively small and does not affect the train routes.

4.2. Variable Definition

To consider all routes and detail the track allocation, we choose the node-arc rather than the arc-route model, so the decision variables focus on the occupation of the arcs and tracks. The definitions of the decision variables are provided in

Table 2. Definitions of the decision variables.

Notations	Definition
$x_{ij}^{bk}$	Binary {0,1}, equals 1 if the kth departure train of direction $b$ occupies arc $(i,j)$, and 0 otherwise, $\forall b\in B,\forall k\in K_{bd},\forall (i,j)\in El$
$y_{ij}^{bk}$	Binary {0,1}, equals 1 if the $k$th arrival train of direction $b$ occupies arc $(i,j)$, and 0 otherwise, $\forall b\in B,\forall k\in K_{ba},\forall (i,j)\in El$
$z_{ij}^{mnk}$	Binary {0,1}, equals 1 if the $k$th passing train from direction $m$ to direction $n$ occupies arc $(i,j)$, and 0 otherwise, $\forall m,n\in B,\forall k\in K_{mn}$, $\forall (i,j)\in El$
$r_{ag}^{bk}$	Binary {0,1}, equals 1 if the $k$th departure train of direction $b$ stops at track $(a,g)$, and 0 otherwise, $\forall b\in B,\forall k\in K_{bd},\forall a\in A,\forall g\in G_a$
$s_{ag}^{bk}$	Binary {0,1}, equals 1 if the $k$th arrival train of direction $b$ stops at track $(a,g)$, and 0 otherwise, $\forall b\in B,\forall k\in K_{ba},\forall a\in A,\forall g\in G_a$
$t_{ag}^{mnk}$	Binary {0,1}, equals 1 if the $k$th passing train from direction $m$ to direction $n$ stops at track (a, g), and 0 otherwise, $\forall m,n\in B,\forall k\in K_{mn}$, $\forall a\in A,\forall g\in G_a$
$p_{oba}$	Integer representing the passenger number of traffic zone $o$ boarding at station $a$ to travel in direction $b$, $\forall o\in O,\forall b\in B,\forall a\in A$

.

For the macroscopic train route between the stations, the binary variable $x_{ij}^{bk}$ refers to whether the $k$th departure train in direction $b$ occupies arc $(i,j)$ ($x_{ij}^{bk}=1$) or not ($x_{ij}^{bk}=0$). Similarly, binary variables $y_{ij}^{bk}$ and $z_{ij}^{mnk}$ represent the arrival and passing trains, respectively. For the microscopic track allocation inside the stations, the binary variable $r_{ag}^{bk}$ refers to whether the $k$th departure train in direction $b$ stops at track $(a,g)$ ($r_{ag}^{bk}=1$) or not ($r_{ag}^{bk}=0$). Binary variables $s_{ag}^{bk}$ and $t_{ag}^{mnk}$ correspond to the arrival and passing trains, respectively. For the passengers, the integer variable $p_{oba}$ refers to the passenger numbers in traffic zone $o$ boarding at station $a$ to direction $b$.

4.3. Formulation

Based on the decision variables, a mixed-integer programing node-arc model is constructed to minimize the total cost of trains and passengers. Specifically, the objective function is represented as follows:

$$\begin{array}{c}\min F={\displaystyle \sum _{(i,j)\in E}[{\displaystyle \sum _{b\in B}({\displaystyle \sum _{k\in K_{bd}}{x}_{ij}^{bk}}+}}{\displaystyle \sum _{k\in K_{ba}}{y}_{ij}^{bk}})+{\displaystyle \sum _{m,n\in B}{\displaystyle \sum _{k\in K_{mn}}{z}_{ij}^{mnk}}}]\cdot d_{ij}\cdot u_l+\\ [{\displaystyle \sum _{b\in B}({\displaystyle \sum _{k\in K_{bd}}{\displaystyle \sum _{a\in A}{\displaystyle \sum _{g\in G_a}{r}_{ag}^{bk}}}}+{\displaystyle \sum _{k\in K_{ba}}{\displaystyle \sum _{a\in A}{\displaystyle \sum _{g\in G_a}{s}_{ag}^{bk}}}})+{\displaystyle \sum _{m,n\in B}{\displaystyle \sum _{k\in K_{mn}}{\displaystyle \sum _{a\in A}{\displaystyle \sum _{g\in G_a}{t}_{ag}^{mnk}}}}}]\cdot u_g}]+{\displaystyle \sum _{o\in O}{\displaystyle \sum _{a\in A}({\displaystyle \sum _{b\in B}{p}_{oba}^{}\cdot d_{oa}\cdot u_p})}}\end{array}$$

(1)

where the objective function includes two parts: the train cost and the passenger cost. For the trains, we aim to obtain the shortest routes, and the train cost is further divided into the train running and waiting costs. The train running cost is the total length of all train routes multiplied by the running cost per kilometer, and the train waiting cost is the operational cost of all trains on tracks. As for the passengers, we aim to ensure all passengers board at their nearest stations, so the passenger cost is the total length of passenger routes multiplied by the travel cost per kilometer.

Then, the constraints of the optimization model are represented as follows:

Constraints (2)–(4) indicate the flow-balancing constraints for nodes $A$, $B$, and $Q$, which are divided into three groups: departure, arrival, and passing trains.

$${\displaystyle \sum _{j\in V,j\ne i}{x}_{ij}^{bk}}-{\displaystyle \sum _{j\in V,j\ne i}{x}_{ji}^{bk}}=\{\begin{array}{l}{\displaystyle \sum _{g\in G_i}{r}_{ig}^{bk}}, i\in a\hfill \\ 0\hspace{1em}\hspace{1em},i\in B\cup Q,i\ne b\hspace{1em}\hspace{1em}\hspace{1em}\forall b\in B, k\in K_{bd}\hfill \\ -1\hspace{1em},i=b\hfill \end{array}$$

(2)

$${\displaystyle \sum _{j\in V,j\ne i}{y}_{ij}^{bk}}-{\displaystyle \sum _{j\in V,j\ne i}{y}_{ji}^{bk}}=\{\begin{array}{l}-{\displaystyle \sum _{g\in G_i}{s}_{ig}^{bk}}, i\in a\hfill \\ 0\hspace{1em}\hspace{1em}\hspace{1em},i\in B\cup Q,i\ne b\hspace{1em}\hspace{1em}\hspace{1em}\forall b\in B, k\in K_{ba}\hfill \\ 1\hspace{1em}\hspace{1em}\hspace{1em}, i=b\hfill \end{array}$$

(3)

$${\displaystyle \sum _{j\in V,j\ne i}{z}_{ij}^{mnk}}-{\displaystyle \sum _{j\in V,j\ne i}{z}_{ji}^{mnk}}=\{\begin{array}{l}1,i\in b \mathrm{and} i=m\hfill \\ 0, i\in V \mathrm{and} i\ne m,i\ne n\hspace{1em}\forall b\in B, k\in K_{mn}\hfill \\ -1,i=b\hfill \end{array}$$

(4)

Constraints (5)–(7) are the coupled constraints of both arc and track decision variables, which ensure that departure, arrival, and passing trains can only stop at the stations on their routes.

$${\displaystyle \sum _{j\in V}{x}_{ij}^{bk}}+{\displaystyle \sum _{j\in V}{x}_{ji}^{bk}}\ge {\displaystyle \sum _{g\in G_i}{r}_{ig}^{bk}} ,\forall i\in A,b\in B,k\in K_{bd}$$

(5)

$${\displaystyle \sum _{j\in V}{y}_{ij}^{bk}}+{\displaystyle \sum _{j\in V}{y}_{ji}^{bk}}\ge {\displaystyle \sum _{g\in G_i}{s}_{ig}^{bk}} ,\forall i\in A,b\in B,k\in K_{ba}$$

(6)

$${\displaystyle \sum _{j\in V}{z}_{ij}^{mnk}}+{\displaystyle \sum _{j\in V}{z}_{ji}^{mnk}}\ge {\displaystyle \sum _{g\in G_i}{t}_{ig}^{mnk}} ,\forall i\in A,(m,n)\in M,k\in K_{mn}$$

(7)

Constraints (8)–(10) are the no-circle constraints of the train routes, which guarantee that departure, arrival, and passing trains do not circle back on their routes.

$${\displaystyle \sum _{j\in V}{x}_{ij}^{bk}}\le 1 ,{\displaystyle \sum _{j\in V}{x}_{ji}^{bk}}\le 1\hspace{1em}\hspace{1em}\forall i\in V,b\in B,k\in K_{bd}$$

(8)

$${\displaystyle \sum _{j\in V}{y}_{ij}^{bk}}\le 1 ,{\displaystyle \sum _{j\in V}{y}_{ji}^{bk}}\le 1\hspace{1em}\hspace{1em}\forall i\in V,b\in B,k\in K_{ba}$$

(9)

$${\displaystyle \sum _{j\in V}{z}_{ij}^{mnk}}\le 1 ,{\displaystyle \sum _{j\in V}{z}_{ji}^{mnk}}\le 1,\hspace{1em}\hspace{1em}\forall i\in V,(m,n)\in M,k\in K_{mn}$$

(10)

Constraints (11)–(13) are the track-only train constraints, which ensure that each train can only stop on one track at one station.

$${\displaystyle \sum _{a\in A}{\displaystyle \sum _{g\in G_a}{r}_{ag}^{bk}}}=1,\hspace{1em}\forall b\in B,k\in K_{bd}$$

(11)

$${\displaystyle \sum _{a\in A}{\displaystyle \sum _{g\in G_a}{s}_{ag}^{bk}}}=1,\hspace{1em}\forall b\in B,k\in K_{ba}$$

(12)

$${\displaystyle \sum _{a\in A}{\displaystyle \sum _{g\in G_a}{t}_{ag}^{mnk}}}=1,\hspace{1em}\forall (m,n)\in M,k\in K_{mn}$$

(13)

Constraint (14) is the line-capacity constraints for arcs El, which ensures the total number of trains passing through arc $(i,j)$ is less than the capacity of arc $(i,j)$.

$${\displaystyle \sum _{b\in B}({\displaystyle \sum _{k\in K_{bd}}{x}_{ij}^{bk}+{\displaystyle \sum _{k\in K_{ba}}{y}_{ij}^{bk}})}}+{\displaystyle \sum _{(m,n)\in M}{\displaystyle \sum _{k\in K_{mn}}{z}_{ij}^{mnk}}} \le c_{ij},\forall (i,j)\in El$$

(14)

Constraint (15) is the branch node constraint, which indicates that the sum of the capacities of arc-linked directions is equal to the capacities of arc-linked stations.

$${\displaystyle \sum _{j\in V}{c}_{ij}=}{\displaystyle \sum _{j\in V}{c}_{ji}},\hspace{1em}\forall i\in Q$$

(15)

Constraint (16) is the track-capacity constraint for all tracks, which ensures that the total number of trains stopping on the track is less than the track capacity.

$${\displaystyle \sum _{b\in B}({\displaystyle \sum _{k\in K_{bd}}{r}_{ag}^{bk}+{\displaystyle \sum _{k\in K_{ba}}{s}_{ag}^{bk}})}}+{\displaystyle \sum _{(m,n)\in M}{\displaystyle \sum _{k\in K_{mn}}{t}_{ag}^{mnk}}} \le w_{ag},\hspace{1em}\forall a\in A,g\in G_a$$

(16)

Constraint (17) shows the distribution of the passengers in the traffic zones, which indicates that the sum of passengers boarding at all stations is equal to the total number of passengers in traffic zone $o$ traveling in direction $b$.

$${\displaystyle \sum _{a\in A}{p}_{oba}=p_{ob}},\hspace{1em}\forall o\in O,b\in B$$

(17)

Constraints (18)–(20) ensure that the passenger-carrying capacity of departure and passing trains stopping at each station meets the demand of boarding passengers. The above constraints correspond to high-speed, normal-speed, and intercity railways, respectively.

$$({\displaystyle \sum _{k\in K_{bd}}{\displaystyle \sum _{g\in G_a}{r}_{ag}^{bk}}})\cdot P_n+({\displaystyle \sum _{k\in K_{mn}}{\displaystyle \sum _{g\in G_a}{t}_{ag}^{mnk}}})\cdot P_p\ge {\displaystyle \sum _{o\in O}{p}_{oba}},\hspace{1em}\forall b,m,n\in B_h,a\in A$$

(18)

$$({\displaystyle \sum _{k\in K_{bd}}{\displaystyle \sum _{g\in G_a}{r}_{ag}^{bk}}})\cdot P_U+({\displaystyle \sum _{k\in K_{mn}}{\displaystyle \sum _{g\in G_a}{t}_{ag}^{mnk}}})\cdot P_p\ge {\displaystyle \sum _{o\in O}{p}_{oba}},\hspace{1em}\forall b,m,n\in B_u,a\in A$$

(19)

$$({\displaystyle \sum _{k\in K_{bd}}{\displaystyle \sum _{g\in G_a}{r}_{ag}^{bk}}})\cdot P_n+({\displaystyle \sum _{k\in K_{mn}}{\displaystyle \sum _{g\in G_a}{t}_{ag}^{mnk}}})\cdot P_p\ge {\displaystyle \sum _{o\in O}{p}_{oba}},\hspace{1em}\forall b,m,n\in B_i,a\in A$$

(20)

Constraints (21)–(23) represent the distribution of departure, arrival, and passing trains, respectively. For each type of train, these constraints ensure that the sum of trains stopping at all stations is equal to the total number of trains.

$${\displaystyle \sum _{k\in K_{bd}}{\displaystyle \sum _{a\in A}{\displaystyle \sum _{g\in G_a}{r}_{ag}^{bk}}}}=n{f}_b,\hspace{1em}b\in B$$

(21)

$${\displaystyle \sum _{k\in K_{ba}}{\displaystyle \sum _{a\in A}{\displaystyle \sum _{g\in G_a}{s}_{ag}^{bk}}}}=n{d}_b,\hspace{1em}b\in B$$

(22)

$${\displaystyle \sum _{k\in K_{mn}}{\displaystyle \sum _{a\in A}{\displaystyle \sum _{g\in G_a}{t}_{ag}^{mnk}}}}=n_{mn},\hspace{1em}(m,n)\in M$$

(23)

Constraint (24) indicates the domains of the decision variables.

$$x_{ij}^{bk},y_{ij}^{bk},z_{ij}^{mnk},r_{ag}^{bk},s_{ag}^{bk},t_{ag}^{mnk}\in \{0,1\},\hspace{1em}{p}_{oba}\in N$$

(24)

According to the model structure, it can be conducted that the model is a mixed-integer linear programming model. The objective function and constraints are all linear, and there are no constraints involved in a huge number of variables, so it can be solved by the commercial solver directly [13,16]. Gurobi is a kind of commercial solver that includes many kinds of precise and heuristic algorithms. Owing to its high-efficiency solving, Gurobi is usually used to solve the multicommodity flow problem and other railway optimization problems. Since our proposed optimization model can be regarded as a combination of train and passenger shortest path problems with capacity constraints, it can be solved by the branch-and-bound algorithm inserted in Gurobi. Therefore, we use the Python language to call Gurobi in PyCharm to solve the formulation and evaluate its performance in different scenarios.

5. Case Study

In this section, we apply the proposed model to the Zhengzhou high-speed railway hub in China as a case study. Then, we analyze the optimized results and compare them with those results obtained in different scenarios to evaluate the effectiveness of the optimization model.

5.1. Data Source

5.1.1. Description of Zhengzhou High-Speed Railway Hub

Located at the intersection of multiple high-speed railway lines, the Zhengzhou high-speed railway hub is a typical multi-station high-speed railway hub; its operation volume is huge. To use the station resources more effectively, we apply the proposed model to the Zhengzhou railway hub to optimize the train routes and track allocation.

Figure 2 illustrates the detailed layout of the Zhengzhou railway hub. Three passenger stations connect to eight directions at this hub. Among these directions, Beijing, Xuzhou, Wu Han, and Xi’an are connected with both high-speed and normal-speed railway lines. Notably, Zhengzhou East Station has two independent yards, Jing-Guang and Xu-Lan. Because these two yards have independent groups of tracks, we divide them into two station nodes. Similarly, we divide Zhengzhou South Station into Zheng-He and Zheng-Wan yards.

5.1.2. Related Parameters

Based on the relevant technical data provided by the China National Railway Group Zhengzhou Railway Bureau Co., Ltd., the values of the related parameters are predetermined. The lengths and capacity of arcs in the Zhengzhou railway hub are listed in

Table 3. Arc lengths and capacities.

Arcs	Length (km)	Capacity (Trains/Day)	Arcs	Length (km)	Capacity (Trains/Day)
Railway lines
(q1, a2)	179	263	(q4, a1)	331	150
(q1, a3)	179	263	(q4, a2)	201	120
(b2, a2)	237	263	(q4, a3)	201	221
(b2, a3)	237	263	(b8, a1)	111	287
(q2, a2)	250	164	(b1a, q1)	0	263
(q2, a3)	250	164	(b1b, q1)	0	263
(b4, a4)	212	287	(b3a, q2)	0	164
(b4, a5)	212	287	(b3b, q2)	0	164
(q3, a1)	102	150	(b5a, q3)	0	263
(q3, a2)	91	263	(b5b, q3)	0	150
(b6, a4)	282	273	(b7a, q4)	0	221
(b6, a5)	282	273	(b7b, q4)	0	200
Connecting lines
(a2, a4)	36	331	(a1, a4)	43	331
(a2, a5)	36	331	(a1, a5)	43	331
(a3, a4)	36	331	(a1, a2)	15	331
(a3, a5)	36	331
Travel routes
(o1, a1)	5	-	(o2, a4)	10	-
(o1, a2)	10	-	(o2, a5)	10	-
(o1, a3)	10	-	(o3, a1)	10	-
(o1, a4)	15	-	(o3, a2)	15	-
(o1, a5)	15	-	(o3, a3)	15	-
(o2, a1)	15	-	(o3, a4)	5	-
(o2, a2)	5	-	(o3, a5)	5	-
(o2, a3)	5	-

-, the travel routes had no capacity.

. As all the railway lines are double-track, the lengths and capacities of the opposite tracks are similar. The track capacities of the stations are listed in

Table 4. Track capacities in stations (trains/day).

Track Number	1	2	3	4	5	6	7	8	9	10	11	12	13
a1	36	38	40	45	40	38	38	38	36	36	36	30	30
a2	30	36	40	42	45	38	38	45	45	40	36	30	30
a3	30	45	34	36	45	45	38	36	44	42	40
a4	30	36	38	45	44	40	36	38	34
a5	30	38	40	45	36	38	38	34

. As for the information on trains, since there are up to 1198 trains stopping at the hub within one day, it is too long to list them all here, so we present the detailed data of their origins and destinations in Appendix A,

Table A1. Train numbers of connecting directions (trains/day).

	D	b1a	b1b	b2	b3a	b3b	b4	b5a	b5b	b6	b7a	b7b	b8	Z
O
b1a		0	0	0	3	0	0	26	0	0	6	0	0	8
b1b		0	0	0	0	8	4	0	64	20	0	20	0	19
b2		0	0	0	0	0	0	0	19	12	0	17	0	29
b3a		3	0	0	0	0	0	2	0	0	24	0	0	10
b3b		0	8	0	0	0	0	0	20	13	0	36	4	24
b4		0	4	0	0	0	0	0	0	0	0	12	14	22
b5a		26	0	0	2	0	0	0	0	0	4	0	0	7
b5b		0	64	19	0	20	0	0	0	0	0	15	10	25
b6		0	20	12	0	13	0	0	0	0	0	0	20	31
b7a		6	0	0	24	0	0	4	0	0	0	0	0	6
b7b		0	20	17	0	36	12	0	15	0	0	0	0	26
b8		0	0	0	0	4	14	0	10	20	0	0	0	19
Z		8	19	29	10	24	22	7	25	31	6	26	19

. Similarly, the number of passengers is also listed in Appendix A,

Table A2. Passenger numbers of traffic zones (thousand persons/day).

	D	b1a	b1b	b2	b3a	b3b	b4	b5a	b5b	b6	b7a	b7b	b8	Total
O
o1		10	30	12	10	20	7	10	30	15	10	30	17	201
o2		10	30	12	10	20	7	10	30	15	10	30	17	201
o3		10	30	12	10	20	7	10	30	15	10	30	17	201
Total		30	90	36	30	60	21	30	90	45	30	90	51

. Moreover, to ensure that the railway operational and passenger travel costs have the same level of effect on the objective function, we set the values of basic parameters as listed in

Table 5. Values of other basic parameters.

Parameters	Definition	Value
$u_l$	Per cost of train running on lines	0.3 (thousand RMB/train·km)
$u_g$	Cost of train operation on tracks	50 (thousand RMB/train)
$u_p$	Per travel cost of passengers	0.04 (thousand RMB/person·km)
$p_h$	Passenger capacity of the high-speed train	1000 (persons/train)
$p_u$	Passenger capacity of the normal-speed train	1460 (persons/train)
$p_p$	Passenger capacity of the passing train	800 (persons/train)

.

5.2. Optimized Results

Using the commercial solver Gurobi on a computer with Intel^® Core™ [email protected] GHz CPU and 16.00 GB RAM, we substituted the above real data into the proposed model to obtain the optimized results. The computational time is 369.76 s, which proves that the model can be solved efficiently within a short time.

For the optimized results, the objective function is RMB 295,109,000, in which the cost of trains running on lines is RMB 114,609,000, the cost of trains waiting on tracks is RMB 59,900,000, and the passenger travel cost is RMB 120,600,000. Since listing all train routes would be challenging, we only report the detailed departure and arrival routes of trains in directions b_1a, b_1b, b₂, b_3a and b_3b. The routes of other directions can be seen in Appendix B. In

Table 6. Detailed departure and arrival routes of trains in directions b_1a, b_1b, b₂, b_3a and b_3b.

B	Arrival Trains	Track Allocation	N	Departure Trains	Track Allocation	N
b1a	b1a-q1-a2	a2 [5, 9, 10]	3	a1-a2-q1-b1a	a1 [5]	1
				a4-a2-q1-b1a	a4 [1, 2]	2
	b1a-q1-a3	a3 [5, 7, 7, 9,11]	5	a4-a3-q1-b1a	a4 [1, 2, 9]	3
				a5-a2-q1-b1a	a5 [3, 5]	2
b1b	b1b-q1-a2	a2 [4, 4, 5, 5, 8, 9, 9, 10, 11]	9	a1-a2-q1-b1b	a1 [1, 4, 5, 5, 5, 10, 11, 12]	8
				a4-a2-q1-b1b	a4 [2, 3, 4, 5, 5, 7, 9]	7
	b1b-q1-a3	a3 [6, 8, 8, 8, 9, 9, 10, 10, 10, 11]	10	a4-a3-q1-b1b	a4 [9]	1
				a5-a2-q1-b1b	a5 [1, 1]	2
				a5-a3-q1-b1b	a5 [1]	1
b2	b2-a2	a2 [4, 6, 6, 7, 8, 8, 9]	7	a1-a2-b2	a1 [3, 3, 4, 5, 6, 9, 10, 10, 10, 11, 12, 12]	12
				a2-b2	a2 [4, 5, 5, 10, 10, 13]	6
	b2-a3	a3 [1, 1, 2, 3, 4, 4, 4, 5, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11,11]	22	a3-b2	a3 [2, 5, 8, 9, 9, 9, 10, 11]	8
				a4-a3-b2	a4 [4, 4]	2
				a5-a3-b2	a5 [5]	1
b3a	b3a-q2-a2	a2 [5, 7, 10]	3	a1-a2-q2-b3a	a1 [1, 1, 2, 2, 2, 4, 4]	6
				a4-a2-q2-b3a	a4 [5, 6]	2
	b3a-q2-a3	a3 [3, 7, 8, 8, 10, 11]	7	a5-a2-q2-b3a	a5 [8]	1
				a5-a3-q2-b3a	a5 [8]	1
b3b	b3b-q2-a2	a2 [4, 4, 5, 5, 5, 6, 6, 8, 8, 8, 9, 13]	12	a1-a2-q2-b3b	a1 [3, 3, 3, 4, 4, 5, 6, 9, 10, 11, 11, 12, 12, 13]	14
				a4-a2-q2-b3b	a4 [2, 2, 5, 5, 9, 9]	6
	b3b-q2-a3	a3 [2, 6, 9, 9, 10, 10, 10, 11, 11, 11, 11, 11]	12	a4-a3-q2-b3b	a4 [4, 4, 5]	3
				a5-a2-q2-b3b	a5 [2]	1

, B refers to the direction, track allocation refers to the occupied tracks, and N refers to the number of trains with the same route. The origin traffic zone (O), the destination direction (B), and the number of passengers boarding at each station are presented in

Table 7. Passenger routes with boarding stations.

O/B	b1a	b1b	b2	b3a	b3b	b4	b5a	b5b	b6	b7a	b7b	b8
o1	a1, 10	a1, 30	a1, 12	a1, 10	a1, 20	a1, 7	a1, 10	a1, 30	a1, 15	a1, 10	a1, 30	a1, 17
o2	a2, 7.6 a3, 2.4	a2, 21.2 a3, 8.8	a2, 7.6 a3, 2.4	a2, 4 a3, 8	a3, 20	a2, 6.2 a3, 0.8	a2, 10	a2, 30	a2, 15	a2, 7.6 a3, 2.4	a2, 30	a2, 17
o3	a4, 7.08 a5, 2.92	a4, 26.2 a5, 3.8	a4, 8.6 a5, 3.4	a4, 5.48 a5, 4.52	a4, 19 a5, 1	a4, 0.2 a5, 6.8	a4, 8.54 a5, 1.46	a4, 19.8 a5, 10.2	a4, 7.4 a5, 7.6	a4, 9.2 a5, 0.8	a4, 22.6 a5, 7.4	a4, 8.2 a5, 8.8

a₁, 10, there are 10 thousand persons boarding at station a₁.

.

From the results in

Table 8. Connecting directions of stations in the fixed scheme.

Station	Yard	Directions
Zhengzhou	a1	b1a, b3a, b5a, b7a, b8
Zhengzhou East	a2	b1b, b5b
	a3	b2, b3b, b7b
Zhengzhou South	a4	b4
	a5	b6

and

Table 9. The results of the fixed scheme.

Scheme	Total Cost (Thousand RMB)	Train Cost (Thousand RMB)	Passenger Cost (Thousand RMB)	Line Capacity Utilization (%)	Track Capacity Utilization (%)
Scenario 1	417,709	176,509	241,200	28.54	58.74
OP	295,109	174,509	120,600	28.73	58.89

OP, optimized results based on the flexible scheme described in Section 4.2.

, the following conclusions can be drawn:

(1)

Most trains select the shortest route of all possible routes. For example, all the departure trains in the Chongqing (b6) direction depart from the Zhengzhou South Station (a4 and a5), and most trains to Taiyuan (b8) depart from the Zhengzhou Station (a1). This result proves the effectiveness of the train cost in the objective function, which makes trains prioritize the shortest routes.

(2)

The trains with the same destinations take various routes, and their track allocations are also different. For instance, the arrival trains to Beijing (b1b) have five routes, and the track allocation of each train route is also different. This is because the individual train is taken as the basic unit, which diversifies the routes between the same pair of origin and destination. This result illustrates that the proposed method can avoid centralized train routes and reduce the pressure on trunk lines.

(3)

All of the passengers board at their nearest stations, which means the passenger cost in the objective function works and the passenger route lengths are minimized. This result illustrates that the stops of the trains are convenient for the passengers, which benefits service quality.

In addition, from the perspective of capacity utilization, the utilization rates of all lines and tracks should be under 70% to reserve space for future trains. According to this rule, we represent the line and track capacity utilization rates as two scatter diagrams in Figure 3 and Figure 4 and demonstrate the distributions using two box diagrams in Figure 5.

Figure 5 shows that the capacity utilization rates of most lines are within [10%, 50%]. Specifically, as shown in Figure 3, the maximum line capacity utilization rate is 94.17%, the minimum is 0%, and the average is 28.73%. Furthermore, 88.71% of the railway lines have capacity utilization rates lower than 60%, and only three lines have rates above 80%. These results indicate that these railway lines have enough capacity for new trains, proving that the optimization model can enable the rational utilization of line capacity.

Similarly, Figure 5 shows that most track capacity utilization rates are within [40%, 80%], which is a reasonable range. Figure 4 displays that the maximum track capacity utilization rate is 90.48%, the minimum is 15.56%, and the average is 58.89%. Of the tracks, 57.41% have utilization rates lower than 70%. The above results prove that new trains can be operated on the tracks. Moreover, by taking the station as the basic unit, the utilization rates of Zhengzhou, Zhengzhou East, and Zhengzhou South Stations are 53.44%, 60.83%, and 54.49%, respectively. The gaps between different stations are relatively small, which proves that the optimization model could balance the train operation among stations.

5.3. Discussion

To illustrate the effectiveness of the optimization model, we compare the optimized results with the results obtained in two different scenarios. In Scenario 1, as the optimized results are based on the flexible scheme, we compare them with the results of the fixed scheme, which is currently implemented for the Zhengzhou railway hub by railway operators. The comparison here is to test the validity and advantages of the optimization model. In Scenario 2, the train routes and the passenger routes are sequentially optimized to prove the importance of considering passenger demand in the optimization model.

5.3.1. Scenario 1: Fixed Scheme

At present, the Zhengzhou railway hub is operated on a fixed scheme, in which the connecting directions of stations are fixed. The connecting directions of different stations in this scheme are shown in Table 8.

The fixed scheme is described as follows:

(1) Zhengzhou Station handles normal-speed trains in all directions, and the high-speed trains of Taiyuan. (2) At Zhengzhou East Station, the Jing-Guang yard is responsible for all high-speed Beijing and Wuhan trains, whereas the Xu-Lan yard operates all the high-speed Jinan, Xuzhou, and Xi’an trains. (3) At Zhengzhou South station, the Zheng-He yard handles all the high-speed Hefei trains, and the Zheng-Wan yard operates the high-speed trains of Chongqing.

Then, we calculate the cost of the fixed scheme and compare the results with the optimized results based on the flexible scheme. The results are shown in Table 9, including the total train cost, the total passenger cost, the average line capacity utilization, and the average track capacity utilization. Figure 6 shows the capacity utilization rate of each line. As all railway lines are double-track, we mark two values for the two directions, and darker-colored lines indicate larger values. Figure 7 provides the average track utilization rates of the stations.

From Table 9 and Figure 6 and Figure 7, the following conclusions can be obtained:

(1)

The total cost of the flexible scheme is 29.35% lower than that of the fixed scheme, where the train and passenger costs are 1.13% and 50% lower, respectively. These results indicate that the optimization model identifies shorter train and passenger routes because the flexible scheme assigns most trains to the shortest routes without considering fixed directions. Additionally, it allows passengers to board at the nearest stations.

(2)

The line capacity utilization rates of these two schemes are almost similar. Notably, the partial objective of the optimization model is to minimize the train cost, which usually concentrates trains on the shortest routes and leads to an imbalance among lines. However, for the Zhengzhou railway hub, as the difference in route lengths between routes are not too large, the imbalance in the results produced by the flexible scheme is minor.

(3)

Figure 5 shows that the average track capacity utilization rate of the Zhengzhou East Station is much larger than the Zhengzhou South Station in the fixed scheme, and the track capacity utilization between stations is imbalanced. However, this problem does not occur in the flexible scheme, where the track capacity utilization of the stations is more balanced.

5.3.2. Scenario 2: Sequentially Optimize Train and Passenger Routes

Previously, researchers usually optimized train routes from the perspective of railway transportation enterprises and ignored passenger demand. To prove the importance of considering passenger demand in the optimization model, we design Scenario 2. In this scenario, the train routes are firstly optimized to minimize the total train cost, and then the passengers are allocated to different stations.

Table 10. Results of Scenario 2.

Scheme	Total Cost (Thousand RMB)	Train Cost (Thousand RMB)	Passenger Cost (Thousand RMB)	Line Capacity Utilization (%)	Track Capacity Utilization (%)
Scenario 2	381,206	171,686	209,520	26.01	58.56
OP	295,109	174,509	120,600	28.73	58.89

OP, optimized results based on the flexible scheme described in Section 4.2.

compares the results. Figure 8 shows the distributions of the line and track capacity utilization rates, and Figure 9 details the track capacity utilization rates using pie charts.

From Table 10, Figure 6 and Figure 7, we obtain the following conclusions:

(1)

The total cost of Scenario 2 is 22.58% higher than that of the optimized results. Although the train cost is slightly lower in the former, the passenger cost is higher by 73.73% because the train cost is considered first in the objective and passengers are passively assigned. This sequential optimization leads to centralized train routes and to passengers in some traffic zones traveling to distant stations, resulting in an increase in passenger cost.

(2)

The line capacity utilization in Scenario 2 is more imbalanced. Moreover, Figure 7 shows that, in Scenario 2, many tracks have utilization rates of more than 80%, 18 tracks have utilization rates of more than 90%, and 8 tracks have utilization rates of 100%. Due to the centralized train routes, the operation pressure on some tracks is too high to operate new trains, so this scheme cannot be applied in practice. The above results illustrate that passenger demand should be directly introduced into the objective function rather than the sequential optimization.

6. Conclusions and Further Works

This research proposes a mixed-integer programming node-arc model based on a flexible scheme to optimize train routes at a multi-station high-speed railway hub. Considering the individual train as the basic unit, the train routes in the hub are optimized, including the macroscopic routes between stations and the microscopic track allocation inside stations. Additionally, the passenger demand is introduced into the objective function to improve the service quality provided by railway transport. Through comparing with the fixed and sequential schemes based on the Zhengzhou railway hub, we conclude that the total cost of the optimization scheme is lower than that of schemes by 29.35% and 22.58%, respectively. The results illustrate the effectiveness and benefits of the optimization model based on the flexible scheme.

As the multi-station hub is a new type of hub, its operation scheme still needs to be optimized. Therefore, based on the method and results in this paper, we provide some suggestions:

(1)

For railway operators, when formulating the operation scheme, all stations should be considered together and regarded as a whole. This not only takes advantage of the scale effect of multi-station hubs but also balances the operations of the different stations.

(2)

Multiple routes should be planned for each origin–destination pair of trains to avoid congestion on railway lines, and the routes should be detailed in terms of track allocation inside stations.

(3)

With the intensification of competition in the transportation market, passenger demand should be considered in the planning of train routes to improve service quality.

However, this study has some limitations, and the further work should focus on the following two points:

(1)

The railway hub that we study is mostly relevant to an already constructed railway hub. For hubs still in the planning stage, the location of railway passenger stations can be combined with the optimization scheme to achieve collaborative optimization.

(2)

We only study the departure passenger flow and do not consider the transfer passenger flow. In the next stage, we would introduce transfer passenger flow into the model to construct a transfer scheme, which could increase the suitability of the optimization scheme for the transfer passengers.