Optimization of Multi-Day Flexible EMU Routing Plan for High-Speed Rail Networks

Abstract

With the continuous expansion and increasing operational complexity of high-speed railway networks, there is a growing need for more flexible and efficient EMU (Electric Multiple Unit) routing strategies. To address these challenges, in this paper, we propose a multi-day flexible circulation model that minimizes total connection time and deadheading mileage. A multi-commodity network flow model is formulated, incorporating constraints such as first-level maintenance intervals, storage capacity, train coupling/decoupling operations, and train types, with across-day consistency. To solve this complex model efficiently, a heuristic decomposition algorithm is designed to separate the problem into daily service chain generation and EMU assignment. A real-world case study in the Beijing–Baotou high-speed corridor demonstrates the effectiveness of the proposed approach. Compared to a fixed strategy, the flexible strategy reduces EMU usage by one unit, lowers deadheading mileage by up to 16.4%, and improves maintenance workload balance. These results highlight the practical value of flexible EMU deployment for large-scale, multi-day railway operations.

1. Introduction

The rapid expansion and increasing complexity of China’s high-speed railway (HSR) network present significant challenges in optimizing Electric Multiple Unit (EMU) routing plans. Traditional optimization methods have predominantly focused on single-day planning, addressing operational requirements within a fixed 24 h timeframe. However, the considerable variability in passenger demand—particularly evident between weekday commuter patterns and intensified weekend travel—highlights the inadequacy of these conventional single-day optimization methods. Therefore, there is an urgent need to develop multi-day EMU routing optimization strategies capable of dynamically responding to varying operational demands within China’s extensive HSR network.

Recent advancements in standardized EMUs, such as China’s “Fuxing” series, have significantly enhanced operational flexibility through standardized critical components and interfaces [1]. These improvements enable more adaptable routing optimization strategies, including multi-day flexible storage and maintenance planning, as well as the dynamic coupling and decoupling of trains at intermediate stations. Exploiting this versatility can substantially enhance EMU utilization efficiency and reduce overall operational expenditures.

In this study, we focus on multi-day EMU routing optimization within China’s high-speed railway network, incorporating flexible storage, maintenance planning, and station-based dynamic coupling and decoupling operations. By considering multi-day fluctuations in passenger demand, we propose a connection-based optimization model designed to effectively align EMU assignments with varying operational requirements. The model explicitly incorporates the flexible selection of inspection locations and the operational feasibility of coupling and decoupling at stations. The objective is to minimize connection times and empty-running distances, thereby significantly enhancing the adaptability and overall efficiency of EMU routing plans.

1.1. Research on Operation Optimization in EMU Routing Plans

In existing research, EMU routing plans are usually abstracted as vehicle routing problems with capacity constraints. The goal of optimization is to use fewer EMUs and build models based on multi-commodity networks.

Canca et al. [2] developed an arc-based network flow model, using graph theory to represent connection decisions and constraints, and proposed heuristic methods including local search and Lagrangian relaxation, validated with real-world data. Cacchiani et al. [3] proposed a fast heuristic for the train unit assignment problem, which seeks to allocate rolling stock to scheduled trips with known seat demands. The method is based on the Lagrangian relaxation of a natural integer programming formulation and solves a sequence of assignment problems. Compared to existing approaches, their heuristic offers significantly faster computation while maintaining high-quality solutions, making it well-suited for real-time applications and integration with other planning phases. Hong et al. [4] addressed EMU operations on Korea’s high-speed rail by ensuring the coverage of all services and minimizing empty movements and then reassigning EMUs under repositioning constraints to better utilize network-wide loops, solved via a two-stage approach. Haahr et al. [5] compared two exact methods for rolling stock (re)scheduling—one flow-based and one path-based—demonstrating that explicitly modeling unit order improves computational efficiency and practical applicability under real-world conditions. Wang et al. [6] addressed the high-speed train unit routing problem by incorporating both time- and mileage-based maintenance constraints. They developed a general connection-based routing model and adapted it to China’s two-day circulation structure. To improve tractability, a network reduction strategy was introduced. The study also extended the model to multi-depot scenarios and validated its effectiveness using real-world data from the Chinese high-speed railway. Lin et al. [7] investigated EMU routing optimization under practical operational requirements by considering three distinct objectives: minimizing the number of circulations, minimizing connection time between trips, and minimizing the number of train-sets used. They analyzed the performance and trade-offs under each objective to evaluate their impact on circulation planning outcomes.

With increasingly refined EMU management standards, operators impose stage-specific maintenance requirements to ensure functional safety. As a result, EMU routing optimization research has expanded to incorporate maintenance planning. Maróti et al. [8,9,10] addressed this by developing integer programming models that minimize the total cost of maintenance-related turnarounds. They designed heuristic algorithms combined with the CPLEX solver and validated their approach using real-world data from the Dutch railway network. Canca et al. [11] developed a mixed-integer programming model to minimize both the number of train-sets used and the number of empty movements, while incorporating maintenance requirements and depot locations. A branch-and-bound algorithm was designed to solve the model, which was validated using case studies from Serbian railways. Nishi et al. [12] introduced maintenance job duration constraints to ensure on-time inspections and proposed an integer programming model aimed at minimizing operating costs. They solved it using a column generation approach combined with Lagrangian relaxation, achieving tight bounds within reasonable computation time. Borndörfer et al. [13] formulated a highly scalable mixed-integer programming model based on hyper-graphs that considers train compositions, maintenance needs, and depot capacity limits. A local search heuristic was implemented and applied to practical cases on German railways. In the context of disruptions, Lusby et al. [14] proposed a rescheduling model to minimize connection costs, depot storage, and empty mileage. A branch-and-price algorithm was developed to handle large-scale instances, and the model was further extended by integrating maintenance constraints into subproblems. Zhang et al. [15] proposed a two-stage heuristic for rolling stock scheduling with maintenance constraints for China’s high-speed railway. Candidate schedules are first generated via MIP without maintenance and then evaluated for feasibility using an assignment approach. Tested on Zhengzhou Group data, the method outperformed manual planning and reduced operating costs by 10.5%. Niu et al. [16] developed a mixed-integer linear programming model to minimize the number of EMUs used, incorporating constraints such as overnight storage locations, platform track assignments, and both mileage- and time-based maintenance requirements. A two-stage branch-and-bound algorithm was designed to solve the model efficiently.

1.2. Research on Flexible Train Formation

In recent years, the increasing convenience of high-speed rail has led to growing passenger demand, with significant variations in flow across stations along high-speed lines. In response, flexible EMU composition strategies are being explored to dynamically match these demand fluctuations. Consequently, a number of researchers have begun to investigate how passenger demand patterns and train formation configurations influence EMU routing plans.

Yu et al. [17] investigated the energy cost implications of flexible train formations on high-speed rail. They developed an optimization model to minimize energy consumption, considering passenger demand and train composition constraints. Case studies revealed that flexible formations significantly reduce energy costs under low-demand conditions, providing theoretical support for the broader adoption of flexible EMU operation strategies. Fioole et al. [18] developed an integer programming model to minimize operational costs, allowing train coupling and decoupling at specific stations based on Dutch railway practices. The model incorporated composition constraints and was solved using CPLEX. Peeters et al. [19] proposed a mixed-integer programming model under a fixed timetable and set seat demand, also accounting for coupling and decoupling operations. They designed an exact branch-and-bound algorithm and validated their approach with Dutch railway data. Cadarso et al. [20] extended this line of research by formulating an integer program that integrates train composition and passenger demand, aiming to minimize total operational costs. The model was solved using a Benders decomposition method. Gao et al. [21] proposed a trip sequence graph to model EMU movements and coupling/splitting operations in high-speed rail networks, incorporating maintenance constraints and the periodicity of trip sequences. Based on this graph, they formulated both path-based and arc-based integer linear programming models. A tailored branch-and-price algorithm was developed for the path-based model. Application to the eastern China high-speed rail network demonstrated the models’ effectiveness and practical relevance through numerical experiments. Wu et al. [22] developed an optimization model for high-speed rail operations with variable train formations, aiming to maximize ticket revenue while accounting for passenger demand and flexible EMU configurations. Their results showed that, under sufficient rolling stock availability, the optimized variable formation strategy significantly outperformed fixed formation plans in terms of revenue generation.

Studies integrating passenger demand, train formation, and EMU circulation planning are also common in urban rail transit. Zhao et al. [23] jointly optimized train formation and rolling stock scheduling via a multi-objective mixed-integer nonlinear programming (MINLP) model based on a time–space network, which was linearized into single-objective MILP and solved using CPLEX. Zhu et al. [24] proposed a “shadow train” model to optimize coupling-based circulation plans, aiming to minimize both total connection costs and the variance in rolling stock utilization, solved via a multi-objective chaotic particle swarm algorithm. Zhou et al. [25] developed a two-stage stochastic programming model to address the dynamic and stochastic nature of commuting demand, incorporating both flexible train formations and robust passenger flow control. The model was reformulated into MILP for solving using CPLEX. Wang et al. [26] tackled uneven urban passenger flow using a path-based MILP model with flexible train formations, solved using a branch-and-price algorithm. However, due to differences in demand patterns and operational characteristics, these urban transit models—typically ignoring empty runs—require significant adaptation before being applied to high-speed rail.

1.3. Summary of Literature Review

In summary, extensive research—both domestic and international—has explored EMU routing plans from multiple perspectives, establishing a relatively mature theoretical foundation in areas such as circulation frameworks, optimization models, and algorithmic strategies. These studies have offered valuable references for EMU routing planning in China’s high-speed railway network. However, limited attention has been paid to multi-day EMU routing plans that incorporate flexible maintenance arrangements and dynamic coupling/decoupling operations. The relevant literature is summarized in

Table 1. Comparison of selected literature on EMU routing plans.

References	Model Type *	Background *	Objective *	Solution Methods	Variable Maintenance Depot	Coupling/ Decoupling Operation	Number of Days
[6]	BIP	R	TUA	CPLEX	✓		two days
[11]	MIP	R	RSCP	Branch-and-bound			one day
[13]	MIP	R	RSCP	Local search heuristic			one day
[15]	MILP	R	RSS	Two-stage iterative solution		✓	two days
[16]	MILP	R	EMURP	Two-stage branch-and-bound	✓		two days
[21]	BIP	R	RSCP	Branch-and-price		✓	one day
[23]	MINLP	URT	TFP + RSCP	Cplex		✓	one day
[24]	BIP	URT	TFP + RSCP	Multiobjective Chaos Particle Swarm		✓	one day
[25]	MILP	URT	RSCP	Cplex		✓	one day
[26]	MILP	URT	TFP + RSCP	Branch-and-price		✓	one day
Our paper	MILP	R	EMURP	Gurobi and heuristic	✓	✓	six days

* R: rail; URT: urban rail transit; BIP: binary integer programming; MILP: mixed-integer linear programming; MIP: mixed-integer programming; TFP: train formation plan; RSS: rolling stock scheduling; RSCP: rolling stock circulation plan; TUA: train unit assignment; EMURP: EMU routing plan.

.

However, most existing studies focus on single-day EMU routing and often assume fixed depot assignments without considering the flexibility of inspection site selection and dynamic coupling/decoupling operations. This limits the adaptability of these models to real-world high-speed railway networks, where multi-day demand fluctuations and operational constraints are significant. Furthermore, although the urban rail transit domain has explored flexible train formations and multi-day scheduling, these approaches are not directly applicable to high-speed railway systems due to fundamental differences in service patterns, turnaround constraints, and maintenance requirements.

To bridge this gap, this study proposes a novel flexible multi-day EMU routing optimization model that explicitly incorporates flexible inspection site selection and dynamic coupling/decoupling operations. The proposed approach aims to enhance practical applicability, improve operational efficiency, and better support large-scale high-speed railway networks with variable multi-day demand.

2. Model Description

2.1. Problem Description

2.1.1. Problem Analysis

The EMU routing plan connects all train services in the timetable through feasible service connections, generating an ordered sequence that reflects the logical continuity of train operations. This sequence serves as the operational framework for daily EMU operation. To efficiently fulfill the scheduled train services, EMUs typically depart from their assigned depots, perform a sequence of connected trips, and then return to the depot for scheduled inspection and the completion of the circulation cycle.

To illustrate the operational logic of EMU routing under a multi-day planning horizon, we consider the representative multi-day train timetable shown in Figure 1. In this example, stations A, M, and B are intermediate or terminal, and two EMU depots are connected to stations A and C, respectively. Days 1 to 4 represent weekday service patterns, consisting of eight short-formation and two long-formation train services (depicted by thin and thick red slanted lines, respectively). Due to increased passenger demand on weekends (Days 5 to 7), two additional short-train services are added (shown by green slanted lines).

To illustrate the characteristics of the EMU routing problem under a multi-day timetable, two operation strategies are compared in Figure 2 and Figure 3. Figure 2 shows a fixed operation scheme, while Figure 3 presents a flexible operation scheme, both designed to fulfill the service requirements in Figure 1.

In the fixed operation scheme (Figure 2), five EMU routing plans are required, involving five short-formation EMUs. EMUs are not allowed to couple or decouple and can only connect with services of the same formation type. In addition, each EMU must return to its assigned depot for inspection at the end of each day, leading to an average of two empty trips per day.

In contrast, the flexible operation scheme (Figure 3) requires only four EMU routing plans. Under this approach, EMUs are permitted to couple and decouple at designated stations, and maintenance can be performed at any available depot according to operational conditions, rather than being limited to the originally assigned depot. As a result, only one empty trip is needed to initiate the first-day service, and the total number of required EMUs and the empty-running mileage are significantly reduced.

Based on this, the problem addressed in this study is formulated as follows: Given a set of stations, EMU depots, and a multi-day train timetable, the goal is to construct feasible EMU routing plans that incorporate in-station coupling and decoupling, along with flexible depot selection for inspection. The objective is to satisfy all maintenance constraints while minimizing total empty-running mileage and overall operational costs.

2.1.2. Basic Assumptions

To support the formulation of the EMU routing optimization model, the following assumptions are made:

(1)

The multi-day train timetable is known and fixed, including all train numbers, service schedules, and formation types. The model performs routing optimization based on this predetermined timetable without altering train paths or schedules.

(2)

No additional empty train movements are introduced. Optimization is limited to the given timetable and the deadhead trips implied by it.

(3)

Only first-level maintenance is considered. Each EMU is required to return to a depot for overnight inspection upon completing its daily assignments.

(4)

EMUs of the same type are allowed to couple and decouple at designated stations.

(5)

Coupling and decoupling capacity constraints at stations are not explicitly modeled.

2.2. Model Construction

2.2.1. Construction of the EMU Circulation Connection Network

To support the formulation of the multi-day EMU routing plan model, we constructed a directed connection network that systematically represents the movement and operational status of EMUs. This network abstracts key operational elements—such as depots, stabling yards, and scheduled train services—into vertices and establishes a set of directed arcs to represent feasible transitions under operational and technical constraints. Maintenance requirements are embedded through dedicated maintenance arcs. Figure 4 illustrates a simplified version of the connection network based on the routing scenario shown in Figure 3. For clarity, only a subset of feasible arcs is displayed.

Network vertices:

(1) Source/sink vertices: These are virtual vertices representing the start and end points of every feasible circulation path.

(2) Depot vertices: Each EMU depot is modeled as a vertex with associated parameters, including the maximum number of daily inspections, the storage capacity, and the set of associated stations.

(3) Stabling yard vertices: These represent stations equipped with overnight stabling capabilities. Each vertex is defined by its maximum capacity and the corresponding set of stations.

(4) Train service vertices: Each scheduled train operation is modeled as a vertex. These are divided into short-formation and long-formation categories. Each vertex includes attributes such as departure/arrival station and time, formation type, distance, and duration.

Network Arcs:

(1) Virtual entry and exit arcs: Connect the source vertex with depot or stabling vertices, indicating the initiation and termination of EMU movement paths.

(2) Depot service arcs: Represent direct connections between depots and train service vertices, corresponding to EMUs starting or ending service at a depot.

(3) Deadhead arcs: Denote non-revenue-generating movements (empty runs) between train service terminals and depot-connected stations.

(4) Overnight stabling arcs: Represent connections from terminal train services to yard vertices when overnight storage is possible and servicing facilities are available.

(5) Connection arcs: Link two train service vertices, allowed only when the terminal station of the first matches the departure station of the second and the time gap meets the minimum turnaround requirements. Long-formation services may be split into equivalent short-formation services to allow for flexible coupling or decoupling.

(6) Maintenance arcs: Represent train-to-depot connections for scheduled inspection. These arcs are feasible when the train’s terminal station is associated with the depot and sufficient time is allowed for inspection.

Each complete circulation of an EMU corresponds to a path starting and ending at the source vertex, traversing a series of arcs that represent service execution, stabling, and maintenance. This network structure allows the routing problem to be formulated as an arc-based multi-commodity flow model.

2.2.2. Symbol Definition

The sets, indexes, parameters, intermediate variables, and decision variables defined in the model are shown in

Table 2. Index and set descriptions.

Symbol	Definition
$i,j$	Vertex index
$(i,j)$	Arc index
s	Station index
k	Days in the planning horizon index
v	EMU index
V	Set of EMUs; $v\in V$
$K/\mathrm{day}$	Set of days in the planning horizon; $K=\{1,2,\dots ,|K\left|\right\}$
O	Set of virtual source and sink vertices
D	Set of EMU depot vertices
S	Set of station stabling yard vertices
$\mathcal{C}$	Set of all train service vertices in the timetable; $\mathcal{C}={\mathcal{C}}_1^1\cup {\mathcal{C}}_2^1\cup \dots \cup {\mathcal{C}}^k$
${\mathcal{C}}^k$	Set of all train service vertices on day k, including both high-speed and conventional fast trains; ${\mathcal{C}}^k={\mathcal{C}}_1^k\cup {\mathcal{C}}_2^k$
${\mathcal{C}}_1$, ${\mathcal{C}}_2$	${\mathcal{C}}_1$ is the set of high-speed train service vertices; ${\mathcal{C}}_2$ is the set of conventional fast train service vertices
${\mathcal{C}}_{1,\mathrm{long}}^k$, ${\mathcal{C}}_{1,\mathrm{short}}^k$	Long- and short-formation high-speed train service vertices on day k
${\mathcal{C}}_{2,\mathrm{long}}^k$, ${\mathcal{C}}_{2,\mathrm{short}}^k$	Long- and short-formation conventional fast train service vertices on day k
$\mathcal{N}$	Set of all vertices in the connection network; $N=O\cup D\cup S\cup \mathcal{C},i,j\in \mathcal{N}$

,

Table 3. Parameter descriptions.

Symbol	Definition
$P_i^{\mathrm{store}}$	Nighttime stabling capacity at vertex i (station or depot); $i\in \mathcal{S}\cup \mathcal{D}$
$P_i^{\mathrm{maint}}$	Maintenance capacity at depot i; $i\in \mathcal{D}$
$L_{\mathrm{cycle}}$/km	Maximum cumulative mileage for EMU first-level maintenance cycle
$T_{\mathrm{cycle}}$/km	Maximum cumulative time for EMU first-level maintenance cycle
$T_{\mathrm{con}}$	Minimum train connection time
$T_{\mathrm{cou}},T_{\mathrm{decou}}$	Minimum technical operation time required for coupling and decoupling
$l_i,l_j$	Distance between train service vertices i and j; $i,j\in \mathcal{C}$
$t_i,t_j$	Travel duration between train service vertices i and j; $i,j\in \mathcal{C}$
$t_i^a,t_j^a$	Arrival time at train service vertices i and j
$t_i^d,t_j^d$	Departure time at train service vertices i and j
$t_{ij}$	Connection time between vertices i and j
$s_i^a,s_j^a$	Arrival station of train service vertices i and j
$s_i^d,s_j^d$	Departure station of train service vertices i and j
$\zeta$	EMU idle cost coefficient
$F_{ij}$	Empty run cost from i to j

and

Table 4. Decision variable definitions.

Variable	Definition
$L_{i,j}^{v,k}$	Continuous variable; cumulative operating distance of EMU v on arc $(i,j)$ on day k
$T_{i,j}^{v,k}$	Continuous variable; cumulative operating time of EMU v on arc $(i,j)$ on day k
$x_{i,j}^{v,k}$	Binary variable; equals 1 if v passes arc $(i,j)$ on day k and is 0 otherwise
$y_{i,j}^{v,k}$	Binary variable; equals 1 if v completes first-level maintenance while traversing arc $(i,j)$ on day k and is 0 otherwise
$S{D}_{v,k}$	Binary variable; equals 1 if v selects conventional fast train service on day k and is 0 otherwise
$S{G}_{v,k}$	Binary variable; equals 1 if v selects high-speed train service on day k and is 0 otherwise

.

2.2.3. Mathematic Model

Objective Function: In this study, the optimization objective is formulated as a weighted sum of total connection time and empty-running mileage, as shown in Equation (1). This formulation can be interpreted as a scalarization of a multi-objective optimization problem, where the weights ${\alpha }_1$ and ${\alpha }_2$ represent the relative importance assigned to each objective.

Specifically, minimizing the total connection time reduces the number of EMUs and lowers overall operational costs. In contrast, minimizing empty-running mileage improves EMU utilization efficiency and helps achieve a more balanced distribution of operating mileage among different EMUs.

By adjusting the weights ${\alpha }_1$ and ${\alpha }_2$, the proposed method can be flexibly adapted to various operational scenarios, thereby enhancing its practical applicability in large-scale high-speed railway networks.

$$minZ={\alpha }_1{Z}_1+{\alpha }_2{Z}_2$$

(1)

(1)

Total Connection Time

The total connection time $Z_1$ is used as a proxy to reflect the number of EMUs required. Since the train operation schedule is fixed, a circulation plan with a shorter total connection time implies more efficient vehicle turnover and fewer required EMUs. The objective $Z_1$ is given by

$$Z_1=\sum _{v\in V}\sum _{k\in K}{t}_{ij}·x_{ij}^{v,k},\forall i,j\in \mathcal{D}\cup \mathcal{C},i\ne j$$

(2)

The connection time $t_{ij}$ between vertices i and j is computed as follows:

$$t_{ij}=\left\{\begin{array}{cc}{t}_j^d-t_i^a\hfill & \mathrm{if}{t}_j^d>t_i^a\mathrm{and}{s}_i^a=s_j^d\hfill \\ t_j^d-t_i^a+1440\hfill & \mathrm{if}{t}_j^d<t_i^a\mathrm{and}{s}_i^a=s_j^d\hfill \\ +\infty \hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(3)

(2)

Empty-Running Mileage

To improve the utilization of EMUs, it is essential to minimize total empty-running mileage. A penalty coefficient, $\xi$, is used to convert empty mileage into the same unit as that of connection time. The corresponding objective $Z_2$ is defined in Equation (4):

$$Z_2=\xi \left(\sum _{v\in V}\sum _{k\in K}\sum _{i\in D}\sum _{j\in C^k}{x}_{ij}^{v,k}·F_{ij}+\sum _{v\in V}\sum _{k\in K}\sum _{i\in C^k}\sum _{j\in D}{x}_{ij}^{v,k}·F_{ij}\right)$$

(4)

Here, $F_{ij}$ represents the empty-running distance between vertices i and j, and $x_{ij}^{v,k}$ is the binary decision variable indicating whether EMU v performs the connection from i to j on day k.

Constraints: According to the rules of multi-day EMU flexible routing and maintenance requirements, the following constraints are established:

(1)

Flow Balance Constraints.

These constraints ensure that for each EMU, the number of inbound and outbound arcs at each node in the connection network is balanced. They also ensure that each EMU starts and ends at the virtual depot node.

$$\sum _{j\in N}{x}_{ij}^{v,k}-\sum _{j\in N}{x}_{ji}^{v,k}=0,\forall i\in S\cup D\cup C,v\in V,k\in K$$

(5)

$$\sum _{i\in O}{x}_{ij}^{v,k}=1,\forall v\in V,k\in K$$

(6)

$$\sum _{j\in O}{x}_{ij}^{v,k}=1,\forall v\in V,k\in K$$

(7)

(2)

Train Assignment Constraints.

Each train task in the operation diagram must be assigned to exactly one EMU of the appropriate type:

$$\sum _{v\in V}\sum _{i\in N}{x}_{ij}^{v,k}=1,\forall k\in K,j\in C^k,i\ne j$$

(8)

(3)

Depot Departure and Return Constraints.

Each EMU must depart from a depot on the first day:

$$\sum _{j\in C^1}{x}_{ij}^{v,1}=1,\forall i\in D,v\in V$$

(9)

Each EMU must return to its previous departure point on the following day:

$$\sum _{j\in C^k}{x}_{ij}^{v,k}=\sum _{i\in C^{k-1}}{x}_{ji}^{v,k-1},\forall k<\left|K\right|,i\in D\cup S,v\in V$$

(10)

Finally, each EMU must return to a depot at the end of the planning period:

$$\sum _{i\in N}{x}_{ij}^{v,\left|K\right|}=1,\forall j\in D,v\in V$$

(11)

(4)

Cumulative Mileage for Maintenance.

Each EMU is required to undergo Level I maintenance either after accumulating a certain mileage or after a fixed operational time. Let $L_{i,j}^{v,k}$ be the accumulated mileage of EMU v on arc $(i,j)$ on day k. Then we define the following:

$$L_{i,j}^{v,k}=\sum _{i\in C^k}\sum _{j\in C^k\cup D}{x}_{i,j}^{v,k}·l_i+L_{i,j}^{v,k-1}·\left(1-\sum _{i\in C^k}\sum _{j\in D}{y}_{i,j}^{v,k}\right),\forall k>1,v\in V$$

(12)

For the first day, the initial running mileage of the EMU needs to be considered:

$$L_{i,j}^{v,k}=\sum _{i\in C^k}\sum _{j\in C^k\cup D}{x}_{i,j}^{v,k}·l_i+L_v^0,\forall k=1,v\in V$$

(13)

(5)

Cumulative Operation Time for Maintenance.

Similarly, let $T_{i,j}^{v,k}$ denote the cumulative operation time of v on arc $(i,j)$ on day k:

$$T_{i,j}^{v,k}=\sum _{i\in C^k}\sum _{j\in C^k\cup D}{x}_{i,j}^{v,k}·t_i+T_{i,j}^{v,k-1}·\left(1-\sum _{i\in C^k}\sum _{j\in D}{y}_{i,j}^{v,k}\right),\forall k>1,v\in V$$

(14)

On the first day, the initial state of the EMU needs to be considered:

$$T_{i,j}^{v,k}=\sum _{i\in C^k}\sum _{j\in C^k\cup D}{x}_{i,j}^{v,k}·t_i+T_v^0,\forall k=1,v\in V$$

(15)

Moreover, it is worth noting that flexibility in the allocation of inspection sites is explicitly incorporated in the proposed framework. Instead of requiring EMUs to return to a fixed depot for Level I maintenance, the model allows for the selection of different eligible inspection locations along the network. This flexibility reduces empty-running mileage caused by mandatory returns, facilitates a more balanced distribution of maintenance workloads across different sites, and enhances operational robustness and scheduling adaptability. As a result, it contributes to improved EMU utilization efficiency and lower overall operational costs.

(6)

Overnight Storage Capacity Constraints.

EMUs must be stored overnight at stations or depots with sufficient capacity. Let $P_j^{store}$ be the capacity of station j for overnight parking on day k; the constraint is

$$\sum _{v\in V}\sum _{i\in C^k}{x}_{i,j}^{v,k}\le P_j^{store},\forall 1<k<\left|K\right|-1,j\in S\cup D$$

(16)

(7)

Coupling and Decoupling Operation Constraints.

During the scheduling process, EMUs must satisfy service continuity requirements, meaning that adjacent train services operated by the same EMU must meet minimum connection time constraints. Specifically, short-formation trains must connect to short-formation trains and long-formation trains to long-formation trains:

$$T_{\mathrm{con}}-M(1-x_{ij}^{v,k})\le t_{ij},\forall v\in V,k\in K,i,j\in C_{1,\mathrm{long}}^k\mathrm{or}i,j\in C_{1,\mathrm{short}}^k$$

(17)

$$T_{\mathrm{con}}-M(1-x_{ij}^{v,k})\le t_{ij},\forall v\in V,k\in K,i,j\in C_{2,\mathrm{long}}^k\mathrm{or}i,j\in C_{2,\mathrm{short}}^k$$

(18)

When short-formation trains are coupled to form a long train, the coupling time must be considered in addition to the minimum connection time:

$$T_{\mathrm{con}}+T_{\mathrm{cou}}-M(1-x_{ij}^{v,k})\le t_{ij},\forall v\in V,k\in K,i\in C_{1,\mathrm{short}}^k,j\in C_{1,\mathrm{long}}^k$$

(19)

$$T_{\mathrm{con}}+T_{\mathrm{cou}}-M(1-x_{ij}^{v,k})\le t_{ij},\forall v\in V,k\in K,i\in C_{2,\mathrm{short}}^k,j\in C_{2,\mathrm{long}}^k$$

(20)

Similarly, when long-formation trains are decoupled into two short-formation trains, the decoupling time must be considered:

$$T_{\mathrm{con}}+T_{\mathrm{decou}}-M(1-x_{ij}^{v,k})\le t_{ij},\forall v\in V,k\in K,i\in C_{1,\mathrm{long}}^k,j\in C_{1,\mathrm{short}}^k$$

(21)

$$T_{\mathrm{con}}+T_{\mathrm{decou}}-M(1-x_{ij}^{v,k})\le t_{ij},\forall v\in V,k\in K,i\in C_{2,\mathrm{long}}^k,j\in C_{2,\mathrm{short}}^k$$

(22)

(8)

First-Level Maintenance Requirement.

After completing all scheduled train services during the planning horizon, each EMU must undergo first-level maintenance if its cumulative operating mileage or time exceeds the predefined threshold. The following constraints ensure that if such thresholds are reached, the EMU returns to a depot for maintenance:

$$L_{i,j}^{v,k}\le L_{\mathrm{cycle}}+M\left(1-\sum _{i\in C^k}\sum _{j\in D}{x}_{ij}^{v,k}\right),\forall v\in V,k\in K$$

(23)

$$T_{i,j}^{v,k}\le T_{\mathrm{cycle}}+M\left(1-\sum _{i\in C^k}\sum _{j\in D}{x}_{ij}^{v,k}\right),\forall v\in V,k\in K$$

(24)

(9)

First-Level Maintenance Location and Capacity Constraint.

If an EMU is scheduled for first-level maintenance, it must return to a depot equipped with corresponding facilities. Each EMU can only be assigned to one such location for inspection, and the number of EMUs undergoing maintenance at any depot must not exceed its capacity. The constraint is defined as follows:

$$\sum _{v\in V}\sum _{i\in C^k}{y}_{ij}^{v,k}\le P_j^{\mathrm{maint}},\forall j\in D,k\in K$$

(25)

(10)

Consistency Constraint on Train Formation Types Across Days.

To ensure the operational consistency of train formation types throughout the planning horizon, if an EMU selects a specific type of train formation service, the number of EMUs using that type should remain consistent across all days. The constraint is formulated as follows:

$$S{D}_{v,k}=\sum _{i\in N}\sum _{j\in C_1^k}{x}_{ij}^{v,k}\forall v\in V,k\in K$$

(26)

$$S{G}_{v,k}=\sum _{i\in N}\sum _{j\in C_2^k}{x}_{ij}^{v,k}\forall v\in V,k\in K$$

(27)

$$S{D}_{v,k-1}=S{D}_{v,k}\forall v\in V,k\in K,k>1$$

(28)

$$S{G}_{v,k-1}=S{G}_{v,k}\forall v\in V,k\in K,k>1$$

(29)

Here, $S{D}_{v,k}$ and $S{G}_{v,k}$ denote the number of short-formation and long-formation train services assigned to EMU v on day k, respectively. These constraints ensure the consistency in formation type used by EMUs across consecutive days.

3. Solution Algorithm

The multi-day flexible EMU routing plan for railway networks must be generated based on train timetables and connection rules. Subsequently, EMUs are assigned according to the initial train in each circulation. Given the complexity of decision variables and the abundance of constraints, directly solving the model is computationally challenging. Therefore, in this paper, we propose a heuristic decomposition algorithm based on the model structure, enabling efficient solution acquisition within a reasonable time.

The algorithm decomposes the original problem in three main phases. Firstly, it focuses on minimizing the total connection time for daily EMU routing planning, considering intra-day connection rules. Secondly, it constructs an assignment network based on task chains, network layout, and maintenance resources. Lastly, it solves a multi-day integrated circulation plan with flexible inspection requirements and connection constraints. The detailed steps are as follows:

Step 1: Generation of Daily Train Service Task Chains using Gurobi

Each EMU is required to perform a sequence of train services on a given day, which collectively form a daily task chain. In generating these chains, the model does not consider first-level maintenance capacity constraints. Instead, it focuses exclusively on intra-day train connection constraints. This simplification is intended to ensure feasibility in later stages by avoiding infeasibility caused by the number of task chains exceeding the available storage or inspection capacity at intermediate stations or depots.

Moreover, the inter-day continuity between task chains on day k and day $k+1$ is not considered in this stage. The primary objective of this subproblem is to minimize the total connection time of EMUs on each day. The constraints include only same-day connection feasibility rules, excluding multi-day dependencies and maintenance requirements. The detailed procedure is illustrated in Figure 5, and we solve this subproblem using Gurobi [27]. The output of this stage provides the foundation for subsequent EMU assignment and routing decisions.

Step 2: Construction of EMU Assignment Network

Using the daily train task chains from Step 1, a bipartite network is constructed to assign rolling stock to each task chain. Each task chain is treated as a demand node with attributes such as origin, destination, operation time, and mileage, while supply nodes represent available depot and yard resources. Feasible arcs between demand and supply nodes are defined based on network layout, the initial departure station and time, and depot–station connectivity (see Figure 6).

Step 3: Generation of Multi-day EMU Routing Plan

Finally, based on the assignment network, the EMU routing plan is optimized over the entire planning horizon. The objective is to minimize total deadhead mileage, while satisfying cross-day connection continuity and inspection constraints. This final stage is modeled similarly to the mathematical formulation in Section 2, with task chains replacing individual train services as basic planning units (see Figure 7).

Based on the above three-step decomposition, the proposed heuristic algorithm transforms the complex multi-day EMU routing optimization problem into a tractable solution framework. Through first determining feasible task chains for each day and then constructing a structured assignment network to match available EMUs with task chains, the method integrates operational logic and maintenance feasibility. The final routing plan is generated through global coordination, ensuring consistency across days.

4. Experiments and Computational Results

4.1. Network Operation Data

To evaluate the effectiveness of the proposed method, a case study was conducted on a network composed of the Beijing–Zhangjiakou, Zhangjiakou–Hohhot, and Zhangjiakou–Datong high-speed railways. The network includes major stations such as Beijing North, Qinghe, Yanqing, Taizicheng, Zhangjiakou, Ulanqab, Datong South, Hohhot East, and Baotou. Among these, Yanqing, Taizicheng, and Ulanqab stations are not equipped with overnight stabling facilities. The network structure is shown in Figure 8.

Three depots are connected to Qinghe, Datong South, and Hohhot East stations, respectively, which provide multi-point Level I maintenance services for EMUs. According to the operational timetable, the rolling stock required includes two EMU types: CRH3A and CR400BF. Based on national maintenance standards, the Level I inspection threshold for CR400BF (350 km/h) is 7700 km or 48 h, and for CRH3A (250 km/h), it is 6600 km or 72 h.

Train connection rules stipulated a minimum of 15 min between consecutive services, and 20 min was required for coupling or decoupling operations. All computations were performed using a machine with an Intel Core i7-11800H CPU at 2.3 GHz and 32 GB of RAM. Using the proposed approach, EMU routing plans for a six-day operation period were obtained within 30 min, demonstrating the computational efficiency and practical viability of the method.

4.2. Comparative Analysis of EMU Utilization Under Fixed and Flexible Routing Plans

To demonstrate the practical relevance and effectiveness of the proposed multi-day flexible EMU routing optimization model, a case study was conducted on the Beijing–Zhangjiakou high-speed railway network. The input data consisted of 130 scheduled train services on weekdays and 140 on weekends. Two planning scenarios were defined for comparison:

• Scenario 1: A fixed routing plan with constrained EMU assignments;
• Scenario 2: A flexible routing plan based on the proposed optimization method.

Analysis of EMU Utilization

Figure 9 illustrates the Gantt chart for the fixed routing scenario (Scenario 1), while Figure 10 presents the results under the flexible routing scenario (Scenario 2). Both scenarios were applied to the same six-day high-speed railway timetable covering weekday and weekend services.

Under Scenario 1, a total of 43 EMUs were required to complete all train operations. This included 22 high-speed CR400BF and 21 medium-speed CRH3A units.

In contrast, Scenario 2 accomplished the same operational tasks using only 42 EMUs, comprising 22 CR400BF and 20 CRH3A units. This represents a net reduction of one CRH3A unit while maintaining full timetable coverage, thereby demonstrating the proposed model’s effectiveness in reducing the required fleet size and operational costs.

To address this issue, we compared the fixed and flexible multi-day circulation plans in terms of operational performance across different EMU types. As shown in

Table 5. Comparison of EMU circulation metrics under fixed and flexible operation.

Indicator	Fixed Operation		Flexible Operation
	CRH3A	CR400BF	CRH3A	CR400BF
Avg. cumulative mileage per EMU (km)	3172	3241	3276	3404
Avg. deadheading mileage per EMU (km)	278	276	237	231
Number of EMUs stored at intermediate station at night/group	2	3	8	5
No. of EMUs per depot (Beijing North/Datong South/Hohhot East)	8/2/5	11/4/6	6/5/5	8/7/6

, we analyzed several key indicators including average cumulative mileage, average deadheading mileage, and the average number of first-level maintenance cycles per EMU over the six-day horizon.

Under the flexible circulation plan, both CRH3A and CR400BF EMUs exhibited higher average cumulative mileage compared to those in the fixed plan. Moreover, the average deadheading mileage was significantly reduced by 14.8% for CRH3A units and 16.4% for CR400BF units. This improvement is primarily attributed to better depot allocation flexibility and the more balanced distribution of inspection tasks.

Additionally, under the flexible plan, the maintenance assignments were more evenly distributed among EMUs, indicating higher resource utilization and better alignment of maintenance cycles with actual train usage patterns. This balanced distribution helps improve operational realism and robustness, ensuring that maintenance schedules match operational demands. Overall, these results confirm that the proposed model effectively reduces the required number of EMUs and non-revenue mileage while enhancing fleet utilization, operational robustness, and overall efficiency in large-scale, real-world high-speed railway networks.

4.3. Sensitivity Analysis of Weight Coefficients

To further evaluate how the minimum train connection time and the minimum coupling/decoupling time affect the model’s two main objectives—minimizing total connection time and deadheading mileage—we conducted a sensitivity analysis by adjusting the weight coefficients ${\alpha }_1$ and ${\alpha }_2$.

Specifically, we selected four representative parameter settings and solved the model under each one. The performance was assessed based on total objective value, total connection time, the number of EMUs used, deadheading mileage, and solution time. The results are summarized in

Table 6. Sensitivity analysis of objective weights.

Weight Setting	Objective Value	Total Connection Time (min)	EMUs	Deadheading Mileage (km)	Solving Time (s)
${\alpha }_1=0.8,{\alpha }_2=0.2$	51694	55157	42	37846	1452
${\alpha }_1=0.7,{\alpha }_2=0.3$	49666	55589	42	35846	1375
${\alpha }_1=0.6,{\alpha }_2=0.4$	47081	54641	42	35741	1248
${\alpha }_1=0.5,{\alpha }_2=0.5$	48583	57628	43	39538	1669

.

As shown in the table, when ${\alpha }_1=0.6$ and ${\alpha }_2=0.4$, the model achieved a balanced trade-off between the number of EMUs (42 units), total train connection time (54,641 min), and deadheading mileage (35,741 km), with a relatively fast solving time (1248 s). In contrast, other parameter settings either slightly increased the number of EMUs or led to longer connection time or deadhead mileage.

Figure 11 illustrates the convergence trajectories of the objective function under four representative weight settings. The trends observed validate the effectiveness and robustness of the proposed optimization algorithm. In all cases, the objective values exhibit a steep decline within the first few iterations, followed by convergence toward a stable solution, indicating that the algorithm efficiently reaches a near-optimal region in a relatively short time.

More specifically, when the weight coefficients are set as ${\alpha }_1=0.8,{\alpha }_2=0.2$, the objective function stabilizes after approximately 20 iterations, albeit at a relatively higher value compared to the other scenarios. As the weight assigned to deadheading mileage ${\alpha }_2$ increases, the final objective value decreases, reflecting an improved trade-off between total train connection time and empty-running mileage.

Among the four configurations, the setting ${\alpha }_1=0.6,{\alpha }_2=0.4$ achieves the lowest final objective value, which is consistent with the quantitative results summarized in Table 6. This outcome supports the selection of this particular weight configuration as the baseline for subsequent analysis, as it yields superior overall performance across key metrics, including EMU utilization, connection efficiency, and computational efficiency.

5. Conclusions

In this study, we propose a flexible multi-day EMU routing optimization model that accounts for dynamic coupling/decoupling operations and flexible inspection location assignment. The presented model integrates connection time and empty-run mileage into a unified objective, enabling the dynamic assignment of EMUs across planning days while respecting maintenance constraints. To efficiently solve this complex optimization problem, a heuristic decomposition algorithm is developed, which involves the following: (i) the generation of daily train connection task chains; (ii) the construction of the EMU assignment network; (iii) multi-day routing plan optimization with inspection and connection continuity constraints.

We validated the proposed method using a real-world case study on the Beijing–Baotou high-speed railway corridor. The results demonstrate that compared to the fixed routing strategy, the flexible scheme reduces EMU fleet requirements (from 43 to 42 units), decreases total deadheading mileage, and achieves more balanced inspection scheduling. Moreover, it maintains high operational feasibility, generating solutions within 30 min on standard computational hardware.

In summary, the presented model enhances EMU operational efficiency by comprehensively leveraging the flexibility of modern high-speed EMUs and station infrastructure. It offers a scalable and practical framework for EMU routing plans in dense and dynamic railway networks, particularly under scenarios with variable demand and constrained maintenance resources.

Future research could further extend the model to incorporate energy efficiency and passenger flow dynamics. Additionally, to improve practical applicability, future extensions may include explicit coupling/decoupling capacity constraints and multi-level maintenance requirements. These enhancements will enable the model to more accurately reflect complex operational realities and provide even more robust decision support for large-scale railway networks.