A Novel Multi-Agent-Based Approach for Train Rescheduling in Large-Scale Railway Networks

Abstract

Real-time train rescheduling is a widely used strategy to minimize knock-on delays in railway networks. While recent research has introduced intelligent solutions to railway traffic management, the tight interdependence of train timetables and the intrinsic complexity of railway networks have hindered the scalability of these approaches to large-scale systems. This paper proposes a multi-agent system (MAS) that addresses these challenges by decomposing the network into single-junction levels, significantly reducing the search space for real-time rescheduling. The MAS employs a Condorcet voting-based collaborative approach to ensure global feasibility and prevent overly localized optimization by individual junction agents. This decentralized approach enhances both the quality and scalability of train rescheduling solutions. We tested the MAS on a railway network in the UK and compared its performance with the First-Come-First-Served (FCFS) and Timetable Order Enforced (TTOE) routing methods. The computational results show that the MAS significantly outperforms FCFS and TTOE in the tested scenarios, yielding up to a 34.11% increase in network capacity as measured by the defined objective function, thus improving network line capacity.

1. Introduction

The rapidly increasing demand for railway transport has led to the intensive use of railway infrastructure. As more services are added to already busy networks, trains must operate at higher speeds and with shorter headways. Consequently, even a small initial delay can cascade into a series of knock-on delays affecting trains across the entire network due to interconnected schedules and network conflicts [1]. Traditionally, train monitoring and real-time rescheduling are traditionally performed by local dispatchers and real-time rescheduling decisions are made when significant delays occur, often using rule-based approaches such as

• Timetable Order Enforced (TTOE): Dispatch trains in the same sequence as their operational timetables.
• First Come First Served (FCFS): Prioritize the first train arriving at the last signal before a junction.
• Fast Line First (FLF): Prioritize trains on the fast line over those on the local line.
• Busy Line First (BLF): Dispatch trains from the line with a higher volume of traffic.

Once a dispatching decision is made, the signaling and interlocking systems translate it into future actions. The primary goal of generating a train rescheduling solution is to minimize knock-on delays among conflicting trains within the local area. However, dispatchers may not always find the optimal solution for the problem at hand [2]. Consequently, a series of studies have introduced advanced algorithms and decision-support systems to address train rescheduling challenges. Typically, these problems are modeled as mixed-integer linear programming (MILP) problems, allowing for the generation of optimal solutions within a reasonable computation time.

However, these algorithms face computational bottlenecks when dealing with highly complex problems. As the controlled network expands, search algorithms may struggle to find promising solutions, sometimes even performing less efficiently than traditional rule-based strategies. Additionally, the search process can become extremely time-consuming, making it unsuitable for the real-time demands of dispatching systems [3]. Moreover, because train timetables are strongly interdependent, adjusting the timetable for a single train can trigger a domino effect across the entire network, which local dispatchers cannot anticipate. As a result, a solution that appears optimal for a local network may not be globally feasible. In the context of railway networks, each local area is highly self-governing and aims to solve its traffic management problems independently. However, these local issues are often interconnected with other areas. Therefore, the train rescheduling problem should be approached from a global perspective, but in a decentralized manner.

This paper presents a novel multi-agent-based approach to solving the train rescheduling problem. The primary advantages of applying a multi-agent system (MAS) to railway traffic management are its inherently decentralized architecture and its ability to handle complex, large-scale problems. The proposed MAS generates real-time train rescheduling solutions by adjusting train passing sequences and times at junctions and stations, thereby minimizing knock-on delays caused by conflicts with the original timetable and ultimately improving rail service across the entire network. The key contributions of this paper are as follows:

(1)

Hierarchical Multi-Agent System for Large-Scale Networks: We apply a hierarchical MAS to address train rescheduling challenges in large-scale railway networks. This includes detailed descriptions of the methods for network decomposition and the allocation of optimization tasks across the entire MAS.

(2)

Enhancements in the Searching Algorithm: We introduce two additional operators in the searching algorithm: modification and classification. The modification operator acts as a post-correction mechanism to ensure the feasibility of solutions under local operational constraints. The classification operator groups the final generation based on train sequence indicators, configuring the voting entities for the subsequent voting process.

(3)

Condorcet Voting Mechanism for Collaboration: A Condorcet voting mechanism is employed as a collaborative approach within the MAS, enabling the selection of the most acceptable solution across the entire network and ensuring global feasibility.

(4)

Case Study and Comparative Analysis: We conduct a case study that demonstrates the benefits of the MAS-based approach compared to First-Come-First-Served (FCFS) and Timetable Order Enforced (TTOE) routing rules. The analysis highlights the strengths and weaknesses of the MAS approach in the context of train rescheduling.

The remainder of this paper is organized as follows: Section 2 provides a review of the state of the art in railway traffic management, focusing on formulation approaches and search algorithms, along with a brief introduction to MAS and its recent applications in the transportation industry. Section 3 describes the train rescheduling problem, which is modeled as a mixed-integer linear programming (MILP) problem. In Section 4, we propose a hierarchical MAS for train rescheduling, discussing network decomposition and the allocation of the optimization problem within the MAS. The section also details the local searching algorithm and introduces the Condorcet voting mechanism. Section 5 presents a performance evaluation of the MAS using a British railway network, comparing the MAS-generated solutions with those from FCFS and TTOE approaches and discussing the advantages and disadvantages of applying the MAS to train rescheduling.

2. Literature Review

Train operations are subject to a variety of disturbances, which can lead to delays of varying severity. Real-time decisions must be made based on the movement status of trains within the network, all within specific time horizons. In the context of the ICONIS decision support system for railway traffic management, developed by Alstom in 2013, train delays are categorized into three levels: small delays (within 15 min), medium delays (between 15 and 30 min), and large delays (up to 60 min). The computational time horizons for resolving these delays are set at 2 s, 5 s, and 10 s, respectively [4]. Despite advancements, generating an optimized solution within these brief timeframes remains a significant challenge for modern MILP techniques, although Alstom’s computational criteria are not universal standards.

To enhance the performance of decision support systems, numerous studies have introduced innovations in railway rescheduling, with approaches varying in terms of their modeling strategies and search algorithms. In this section, building on previous review papers [5,6], we briefly outline the state-of-the-art research in railway traffic management. Additionally, to connect with the approach proposed in this paper, recent applications of the MAS are also discussed.

2.1. Railway Traffic Management Approaches

Due to the nature of railway systems, a rail service can be conceptualized as the occupation of a series of track circuits. Consequently, the train rescheduling problem can be modeled as a job-shop scheduling problem with additional constraints, which can be solved using mathematical programming or similar techniques [7]. One notable system, RECIFE-MILP, proposed by Pellegrini et al. [8,9], represents the railway network at a microscopic level. In this system, the optimized variables include train timetables at specific junctions and station points within the network. The MILP problem is addressed using rule-based dispatching methods, advanced search algorithms, or CPLEX software, all aimed at minimizing the propagation of delays to other trains in the network. Samà et al. introduced an ant colony optimization algorithm designed for the train routing selection problem [10]. This approach, tested on two practical networks in France, generates near-optimal solutions that minimize consecutive train delays. Moreover, the optimal timing for solving the train routing selection problem is discussed in [11], which proposes search algorithms tailored for tactical, operational, and real-time traffic management levels. In another study, Chen et al. applied a differential evolution algorithm to solve train rescheduling problems in railway bottleneck sections. This algorithm outperformed First-Come-First-Served (FCFS) and automatic route setting, generating optimized solutions within 2–3 s in a British network [12].

Another common method for modeling railway infrastructure involves graph-based approaches. The concept of an alternative graph was first introduced by Mascis & Pacciarelli [13]. In an alternative graph, train movements are represented as the occupation of a track circuit over a certain period, with the entire journey of a train structured as a branch within the graph. This branch comprises nodes in the railway network and journey times for each track element. Building on this model, several resolution approaches have been developed for different levels of railway traffic problems. For instance, the ROMA system employs a branch-and-bound algorithm, as introduced by D’Ariano et al. [7]. Rule-based routing approaches and various resolution algorithms are compared in [14]. Additionally, algorithms optimizing train speed profiles, rescheduling, and rerouting solutions are explored in [15,16,17]. Van Thielen et al. introduced the concept of a dynamic impact zone for services within the network and proposed a conflict prevention strategy that explores rerouting options, as well as retiming and reordering plans [18]. Wang et al. proposed a C-DAS system for freight trains in mixed-service networks, which outputs advisory time and speed profiles to help freight trains find feasible time windows to join other rail corridors based on stairway predictions [19]. Additionally, there is also literature discussing cross-disciplinary approaches that bridge infrastructure design and operational efficiency. For example, the strategy for lightweight designing of a railway vehicle car body, including composite material and dynamic structural optimization, is discussed in [20], which also provides a comparative analysis and dynamic size optimization of aluminum and carbon fiber thin-walled structures of railway vehicle car body. Also, the vehicle aeromechanical performance of improved bogie design and its influence on train dynamics is discussed [21].

2.2. Multi-Agent System (MAS)

A MAS consists of a set of agents embedded in a shared environment interacting with each other to achieve a common goal. In this context, an “agent” refers to an intelligent entity characterized by autonomy, activity, reactivity, and mobility [22]. In detail, an agent operates autonomously, making decisions without direct human intervention, while also exhibiting proactiveness by taking goal-driven actions rather than just reacting. It dynamically responds to changes in its environment through reactivity, adapting its behavior in real time. Additionally, some agents possess mobility, allowing them to move across systems or physical spaces to perform tasks efficiently. Together, these features—autonomy, proactiveness, reactivity, and mobility—enable intelligent agents to function effectively in dynamic and complex environments. Due to the inherent nature of the MAS, it is well-suited to tackle large-scale and/or complex problems. Similarly, public transportation systems, such as buses, urban trains, and airplanes, operate within extensive networks with strong internal coupling, making the MAS an attractive approach for researchers in the railway sector.

For instance, Castro & Oliveira proposed a MAS to solve airline operation problems, specifically addressing crew recovery [23]. Their approach specialized the traditional Airline Operations Control Centre as an agent responsible for crew allocation, considering associated costs like hotel expenses. Another application of the MAS is found in the work of Shibghatullah et al., who introduced a MAS for bus assignment that adapts to changes in passenger flow, highlighting the potential advantages of the MAS in transportation research [24]. Similarly, Abbink et al. presented a systematic approach for the allocation and reallocation of human resources—such as train drivers, train managers, and service staff—in passenger railways [25]. Their MAS emphasized a negotiation protocol for crew assignment and multi-layer structural control within railway operations. Liu et al. also demonstrated a MAS framework for real-time train dispatching in a bottleneck section in the UK [1].

In previous research, the MAS has typically been applied as a conceptual modeling technique, with limited discussion on the interactions among agents within the system. Additionally, most recent railway traffic management research has focused on developing algorithms specific to particular network layouts, limiting their scalability and applicability to larger networks. To the best of the authors’ knowledge, there is no existing work that discusses the application of the MAS to the train rescheduling problem, particularly with regard to the scalability of conflict resolution approaches. This paper introduces a novel MAS to address the large-scale train rescheduling problem, offering improved scalability and potential for extension across entire railway networks.

3. Problem Statement

The mixed-integer linear programming (MILP) model is used to present the train rescheduling problem. The journey of a train can be described as the process of passing through specific nodes in a network and occupying corresponding track circuits. An operational conflict arises in the railway network when multiple trains request access to the same track circuit simultaneously. To resolve this, real-time decisions must be made to determine the order in which the conflicted trains pass through the track circuit. The most common conflict areas within a railway network are junctions, where trains typically merge onto a single line heading in the same direction. However, once a decision is made regarding the order of trains at a junction, it becomes impossible to alter that order unless there is a looping area ahead. Therefore, real-time decisions must consider the entire network rather than focusing solely on the local area. In this paper, the passing times and sequences of the trains at junction areas are modeled as optimization variables. The optimization process takes into account the different types of services, recognizing that not all services are scheduled to dwell at stations. To optimize overall service performance across the network, both junctions and station areas are considered in the model. The abbreviations provide a summary of the notations used in the MILP model.

Building on prior research on the train rescheduling problem [26], the MILP formulation for rescheduling in large-scale railway networks could be as follows:

Minimize

$$TD={\displaystyle \sum _{J_m\in \mathbb{J}}}{\displaystyle \sum _{{\tau }_m\in \mathbb{T}}}\left|t_{{\tau }_m}^{J_m}-\left.{t_{{\tau }_m}^{J_m}}^{\mathcal{O}}\right|\right.$$

(1)

Subject to

$$H_{{\tau }_m,{\tau }_n}^{J_m}=f_{headway}\left({\tau }_m,{\tau }_n, J_m\right), {\tau }_m,{\tau }_n\in \mathbb{T},J_m\in \mathbb{J}$$

(2)

$$r_m=f_{routing}\left({\tau }_m,J_m\right), {\tau }_m\in \mathbb{T},J_m\in \mathbb{J},r_m\in {\mathbb{R}}^{J_m}$$

(3)

$$t_{{\tau }_m}^{S_m,dep}=\mathit{max}\left(t_{{\tau }_m}^{S_m,arr}+D_{{\tau }_m}^{S_m}{i}_{{\tau }_m}^{S_m}, { t_{{\tau }_m}^{S_m,dep}}^{\mathcal{O}}\right), S_m\in \mathbb{S},{\tau }_m\in \mathbb{T}$$

(4)

$$t_{{\tau }_m}^{S_m,dep}=\mathit{max}\left(t_{{\tau }_n}^{S_m,arr}+C_{{\tau }_m,{\tau }_n}^{S_m}{i}_{{\tau }_m,{\tau }_n}^{S_m}, { t_{{\tau }_m}^{S_m,dep}}^{\mathcal{O}}\right), S_m\in \mathbb{S},{\tau }_m\in \mathbb{T}$$

(5)

$$i_{{\tau }_m,{\tau }_n}^{J_m}=i_{{\tau }_m,{\tau }_n}^{J_n}\ast i_{{\tau }_m,{\tau }_n}^{J_m,J_n} , IFF i_{{\tau }_m,{\tau }_n}^{J_m,J_n}=1, {\tau }_m,{\tau }_n\in \mathbb{T},J_m,J_n\in \mathbb{J}$$

(6)

$$t_{{\tau }_m}^{J_m}\ge {t_{{\tau }_m}^{J_m}}^{\mathcal{O}}, J_m\in \mathbb{J},{\tau }_m\in \mathbb{T}$$

(7)

$$t_{{\tau }_m}^{J_m}\ge t_{{\tau }_m}^b+t_{{\tau }_m}^{b,J_m}, J_m\in \mathbb{J},{\tau }_m\in \mathbb{T}$$

(8)

$$t_{{\tau }_m}^{J_n}\ge t_{{\tau }_m}^{J_m}+t_{{\tau }_m}^{J_m,J_n} , J_m,J_n\in \mathbb{J},{\tau }_m\in \mathbb{T}$$

(9)

$$t_{{\tau }_n}^{J_m}\ge t_{{\tau }_m}^{J_m}+H_{{\tau }_m,{\tau }_n}^{J_m}{i}_{{\tau }_m,{\tau }_n}^{J_m} , {\tau }_m,{\tau }_n\in \mathbb{T}$$

(10)

$$t_{{\tau }_n}^{J_m}\ge t_{{\tau }_m}^{J_m}+H_{{\tau }_m,{\tau }_n}^{J_m}{i}_{{\tau }_m,{\tau }_n}^{J_m} i_{r_m,r_n}^{J_m}, {\tau }_m,{\tau }_n\in \mathbb{T}$$

(11)

$$i_{{\tau }_m,{\tau }_n}^{J_m},i_{r_m,r_n}^{J_m},i_{{\tau }_m}^{S_m},i_{{\tau }_m,{\tau }_n}^{S_m},i_{{\tau }_m,{\tau }_n}^{J_m,J_n}\in \left\{\mathrm{0,1}\right\}$$

(12)

The total delay (TD) is the objective function in the MILP formulation, as shown in Equation (1), and it is minimized during the optimization process. Equation (2) defines the headway between two neighboring trains at a junction area, while Equation (3) specifies that a routing action is required for train ${\tau }_m$ as it approaches junction $J_m$. Train dwell times and connection constraints are managed by Equations (4) and (5), respectively. The strong coupling of train passing sequences at interconnected junctions is ensured by Equation (6). Additionally, operational constraints are enforced through Equations (7)–(11). Specifically, Equation (7) provides the essential range for the variable $t_{{\tau }_m}^{J_m}$, Equation (8) ensures the minimum travel time for train ${\tau }_m$ between the control boundary and junction $J_m$, and Equation (9) guarantees the minimum travel time between two neighboring junctions. Equations (10) and (11) further ensure the required headway between trains on the same route and those on conflicting routes, respectively.

In this MILP model, the optimization process adjusts the variable $t_{{\tau }_m}^{J_m}$ and the binary variable $i_{{\tau }_m,{\tau }_n}^{J_m}$. The variables ${t_{{\tau }_m}^{J_m}}^{\mathcal{O}}$, ${t_{{\tau }_m}^{S_m,dep}}^{\mathcal{O}}$, $D_{{\tau }_m}^{S_m}$, and $C_{{\tau }_m,{\tau }_n}^{S_m}$ are obtained from the train working timetable, which is provided in advance by the rail operators. The indicators $i_{{\tau }_m}^{S_m}$ and $i_{{\tau }_m,{\tau }_n}^{S_m}$ are derived from the planned journey of each train at every station within the control area. The coupling indicator $i_{{\tau }_m,{\tau }_n}^{J_m,J_n}$ is mainly identified from the layout of each junction and is determined by the routes requested by the approaching trains ${\tau }_m$ and ${\tau }_n$, with its value provided online by each junction. The variable $H_{{\tau }_m,{\tau }_n}^{J_m}$ and the generation of the corresponding routing $r_m$ are decided by each junction based on its database and the approaching train. The passing time of train ${\tau }_m$ at the control boundary $t_{{\tau }_m}^b$ is monitored by the railway traffic management system as the train crosses its control boundary. All the travel times for train ${\tau }_m$ between nodes in the network, $t_{{\tau }_m}^{b,J_m}$ and $t_{{\tau }_m}^{J_m,J_n}$, can be supported by the train kinematic model, using data from the traffic management system’s database.

4. Multi-Agent System for Train Rescheduling Problems

This section presents a systematic MAS designed to address train rescheduling challenges in large-scale railway networks. The section is organized as follows: First, we introduce the approach used to decompose the railway network and the allocation of the optimization process across different local areas. Next, we detail the structure of the proposed MAS, defining the roles and functions of each agent within the system. Following this, we explain the searching algorithm employed by the MAS. Finally, we describe the Condorcet-voting-based collaborative approach, which ensures effective coordination among agents.

4.1. Problem Decomposition

Typically, the MILP model for train rescheduling can be addressed using global search techniques in a centralized manner. However, as the size of the network increases, the search space expands significantly, potentially leading to degraded solutions and extensive computational times—both of which are unsuitable for real-time decision-making. To overcome this challenge, this paper proposes a method to divide large-scale networks into manageable segments. By decoupling the train rescheduling problem into localized decision-making areas, we can solve the problem more efficiently within each area.

In the proposed MAS, the railway infrastructure is decomposed down to the single-junction level. Figure 1 illustrates an example of this decomposition. A single junction is defined as the area starting from the signal before the junction and includes the entire control area around the junction, encompassing all associated signals. Additionally, platform areas can also be segmented at the junction level. The junctions surrounding the station areas are capable of adjusting train arrival and departure times at the station by determining the train’s passing time through the junction. To balance computational complexity and minimize the geographical coverage of each agent, the control boundary of an agent is defined by all preceding signals before entering the junction, ensuring that it encompasses the associated routing and interlocking decisions.

In each junction area, the local decision-making process is solely responsible for generating dispatching solutions within that specific junction area. Therefore, the variables under optimization for a single junction area $J_m$ include only the train passing time $t_{{\tau }_m,{\tau }_n}^{J_m}$ and the indicator $i_{{\tau }_m,{\tau }_n}^{J_m}$. However, these variables are strongly coupled with the dispatching decisions of neighboring junctions, influenced by the value of $i_{{\tau }_m,{\tau }_n}^{J_m,J_n}$. Consequently, the decision-making process for a local area must also consider a global view of the entire network. To address this, a collaborative approach has been designed, which involves post-correcting the values of $t_{{\tau }_m,{\tau }_n}^{J_m}$ and $i_{{\tau }_m,{\tau }_n}^{J_m}$, as detailed in Section 4.4. The objective function (Equation (1)) can then be allocated to each junction area as follows:

$${TD}^{J_m}=\sum _{{\tau }_m\in \mathbb{T}}\left|t_{{\tau }_m}^{J_m}-{t_{{\tau }_m}^{J_m}}^{\mathcal{O}}\right|$$

(13)

The optimization within a local area takes into account all the constraints outlined in Section 3, with the exception of Equation (6). This selective consideration reduces the computational complexity of each local area, ensuring that the decision-making process can meet the time constraints required for real-time traffic control.

4.2. Structure of MAS for Train Rescheduling Problem

To reduce computational complexity, a large-scale railway network is decomposed into single junctions. Each junction is represented by a junction agent, and these agents are overseen by a collaborative agent in the system’s upper layer. The collaborative agent supervises the decision-making process of each junction agent, ensuring that local decisions align with the global network goals. Below are the detailed definitions of these agents, with Figure 2 illustrating the overall system structure and data flow.

Definition 1.

A junction agent is an intelligent entity that gathers train movement information from the signaling and interlocking systems of the network. It makes decisions regarding the future actions of the local junction based on its local knowledge. In the optimization process of the train rescheduling problem, the junction agent will only consider the operational constraints in the local area.

Definition 2.

A collaborative agent monitors the junction agents within its control area, ensuring that the decisions made by each junction agent are feasible from a global perspective. The collaborative agent is responsible for maintaining a balance between local and global optimality while ensuring solution feasibility throughout the optimization process.

In the proposed system, the collaborative agent and junction agents are organized in a hierarchical structure to address the train dispatching problem in a decentralized manner. This hierarchical multi-agent system (MAS) establishes a closed-loop framework for dynamic railway traffic management, ensuring both local responsiveness and global coordination. At the local level, junction agents continuously monitor their designated segments of the rail network, employing real-time optimization algorithms to generate efficient train rescheduling solutions in response to disruptions, such as delays or track blockages. These agents operate autonomously, leveraging localized data—including train positions, speeds, and track occupancy—to make time-critical decisions that minimize immediate congestion and maintain operational fluidity within their respective areas. Once local solutions are formulated, they are transmitted to the collaborative agent, which serves as the system’s global decision-making hub. The collaborative agent aggregates and evaluates these proposals using advanced conflict-detection algorithms and multi-objective optimization techniques, weighing factors such as network-wide throughput, energy efficiency, and adherence to timetable priorities. By synthesizing these inputs, the collaborative agent resolves interdependencies and conflicts between junction-level solutions, producing a harmonized, globally optimal dispatching plan that balances competing demands across the entire rail network. The finalized dispatching decisions are then disseminated back to the junction agents, enabling them to adjust their local operations in alignment with the global strategy. This feedback loop ensures continuous adaptation and coordination, fostering system-wide resilience and efficiency. Furthermore, the hierarchical architecture of the MAS is deliberately designed to interface seamlessly with existing railway traffic management systems, which are inherently decentralized and hierarchical. By mirroring the organizational and operational structure of conventional systems, the MAS enhances interoperability, simplifies deployment, and facilitates incremental upgrades without requiring extensive overhauls of legacy infrastructure. Currently, real-time dispatching decisions are centralized in a traffic control center that disseminates instructions to local areas for signaling and interlocking. By adopting the MAS, we can potentially enhance this system, addressing the growing complexity as traffic volume increases.

The advantage of the hierarchical MAS lies in its ability to manage complexity through decomposition. By breaking down the rescheduling problem into manageable local areas handled by individual junction agents and allowing the collaborative agent to coordinate these local solutions, the overall computational complexity is significantly reduced. Additionally, the parallel processing of local solutions by multiple junction agents can further decrease the total computational time, making the system more efficient and scalable.

4.3. Algorithm Development

A genetic algorithm (GA) is employed for local searching in the proposed system. This algorithm determines the passing sequence of trains through a junction. Unlike traditional GAs, our approach incorporates two additional operators: modification and classification.

• Modification Operator: This post-correction operator ensures that the solutions generated by the GA are feasible within the local context of each junction. It addresses any inconsistencies or infeasibilities that may arise in the solution space, ensuring that the results adhere to operational constraints.
• Classification Operator: This operator classifies the final generation of solutions based on the decision variable $i_{{\tau }_m,{\tau }_n}^{J_m}$. Classification helps to organize and manage solutions, facilitating a more structured and efficient evaluation process.

The detailed implementation of these operators is discussed in the following sections. The pseudo-code for the designed GA is provided below, illustrating how these operators are integrated into Algorithm 1.

Algorithm 1: GA for local searching in MAS

1. Initialization: generate  $N_{gen}$ of individuals from $\left[{t_{{\tau }_m}^{J_m}}^{\mathcal{O}},\right.\left.{t_{{\tau }_m}^{J_m}}^{\mathcal{O}}+t_{slot}\right]$

2. While (! The local terminating condition is satisfied)

3.     Crossover: two-point crossover;

4.     Mutation: extended-line mutation;

5.     Modification: ensures all the operational constraints;

6.     Sorting;

7.     Selection: Maximum keeps $2N_{gen}$ individuals;

8. End;

9. Classification;

10.   Output the solution table to collaborative agent.

4.3.1. Initialization

The algorithm starts its local searching from initialization, which generates $N_{gen}$ individuals for the first generation of the GA. Each individual encodes the sequence and timing of train passages through the local junction, and the passing time for train ${\tau }_m$ is randomly chosen from $\left[{t_{{\tau }_m}^{J_m}}^{\mathcal{O}},\right.\left.{t_{{\tau }_m}^{J_m}}^{\mathcal{O}}+t_{slot}\right], {\tau }_m\in \mathbb{T}$, where $t_{slot}$ is the delay to train ${\tau }_m$ in minutes against its original passing time in its operational timetable, ${t_{{\tau }_m}^{J_m}}^{\mathcal{O}}$. Furthermore, the generated individual is stated as ${\mathit{X}}_k^{J_m}$, where $k$ is an index variable, $k\in \left\{\mathrm{1,2},3\dots ,N_{gen}\right\}$, and $N_{gen}$ stands for the number of individuals in the initial generation. A generation in the searching algorithm is defined as $\left\{{\mathit{X}}_1^{J_m},{\mathit{X}}_2^{J_m},{\mathit{X}}_3^{J_m},\dots ,{\mathit{X}}_{N_{gen}}^{J_m}\right\},J_i\in \mathbb{J}$. To ensure that the final solution is not worse than FCFS in terms of objective function, FCFS is also added as a seed in the first generation.

4.3.2. Crossover

Crossover is an operator for generating offspring based on two parents. Firstly, two of the individuals from the last generation will be randomly selected as the parents, and based on the length of the individual, a crossover point is chosen to exchange the elements within the two parents. Finally, the parents will generate two offspring, which are organized by a new series of passing times for the trains considered. The crossover will repeat $N_{gen}/2$ times and generate $N_{gen}$ new individuals for the new generation.

4.3.3. Mutation

Mutation is an operator that inserts random changes into the new individuals generated by crossover. The mutation generates a random array of variables and inserts them into the passing times of the trains for an individual. In this paper, we introduce the extended-line mutation, which can generate the individual with the best diversity. Equation (14) states the extended-line mutation:

$$\left\{\begin{array}{c}{\mathit{X}}_{mut}={\mathit{X}}_{ori}+F\ast \left({\mathit{X}}_{best}-{\mathit{X}}_{ori}\right), \mathbf{I}\mathbf{F} 0<P_{xline}<{\displaystyle \frac{1}{3}}\\ {\mathit{X}}_{mut}={\mathit{X}}_{ori}+F\ast \left({\mathit{X}}_{ori}-{\mathit{X}}_{best}\right), \mathbf{I}\mathbf{F} {\displaystyle \frac{1}{3}}<P_{xline}<{\displaystyle \frac{2}{3}}\\ {\mathit{X}}_{mut}={\mathit{X}}_{best}+F\ast \left({\mathit{X}}_{best}-{\mathit{X}}_{ori}\right), \mathbf{I}\mathbf{F} {\displaystyle \frac{2}{3}}<P_{xline}<1\end{array} \right.$$

(14)

where ${\mathit{X}}_{ori}$ is the original individual made by crossover, and ${\mathit{X}}_{mut}$ is the individual generated by the mutation operator. ${\mathit{X}}_{best}$ is the best individual from the last generation, and $\mathit{F}$ is a scaling array with the same size as individual ${\mathit{X}}_{ori}$, and the elements in $\mathit{F}$ are chosen from $\left(0,\left.1\right]\right.$. $P_{xline}$ is a probability variable, which is chosen from $\left(0,\left.1\right]\right.$.

4.3.4. Modification

As the individual generated by crossover and mutation operators may not be feasible with its local operational constraints, a modification operator is applied to ensure the feasibility of the solutions with respect to its network layout and its local knowledge of train operation. The following equations are applied to the modification operator:

$$\left\{\begin{array}{c}\mathbf{I}\mathbf{F} t_{{\tau }_m}^{J_m}<t_{{\tau }_m}^b+t_{{\tau }_m}^{b,J_m}\\ \mathbf{THEN} t_{{\tau }_m}^{J_m}=t_{{\tau }_m}^b+t_{{\tau }_m}^{b,J_m}, J_m\in \mathbb{J}, {\tau }_m\in \mathbb{T}\end{array}\right.$$

(15)

$$\left\{\begin{array}{c}\mathbf{I}\mathbf{F} t_{{\tau }_n}^{J_m}<t_{{\tau }_m}^{J_m}+H_{{\tau }_m,{\tau }_n}\ast i_{{\tau }_m,{\tau }_n}^{J_m} \\ \mathbf{THEN} t_{{\tau }_n}^{J_m}=t_{{\tau }_m}^{J_m}+H_{{\tau }_m,{\tau }_n}\ast i_{{\tau }_m,{\tau }_n}^{J_m}, J_m\in \mathbb{J}, {\tau }_m,{\tau }_n\in \mathbb{T}\end{array}\right.$$

(16)

$$\left\{\begin{array}{c}\mathbf{I}\mathbf{F} t_{{\tau }_m}^{S_m,dep}<\mathit{max}\left(t_{{\tau }_m}^{S_m,arr}+D_{{\tau }_m}^{S_m}\ast i_{{\tau }_m}^{S_m}, { t_{{\tau }_m}^{S_m,dep}}^{\mathcal{O}}\right)\\ \begin{array}{c}\mathbf{T}\mathbf{H}\mathbf{E}\mathbf{N} t_{{\tau }_m}^{S_m,dep}=\mathit{max}\left(t_{{\tau }_m}^{S_m,arr}+D_{{\tau }_m}^{S_m}\ast i_{{\tau }_m}^{S_m}, { t_{{\tau }_m}^{S_m,dep}}^{\mathcal{O}}\right), \\ S_m\in \mathbb{S},{\tau }_m\in \mathbb{T}\end{array}\end{array}\right.$$

(17)

$$\left\{\begin{array}{c}\mathbf{I}\mathbf{F} t_{{\tau }_m}^{S_m,dep}<\mathit{max}\left(t_{{\tau }_n}^{S_m,arr}+C_{{\tau }_m,{\tau }_n}^{S_m}\ast i_{{\tau }_m,{\tau }_n}^{S_m}, { t_{{\tau }_m}^{S_m,dep}}^{\mathcal{O}}\right) \\ \mathbf{THEN} t_{{\tau }_m}^{S_m,dep}=\mathit{max}\left(t_{{\tau }_n}^{S_m,arr}+C_{{\tau }_m,{\tau }_n}^{S_m}\ast i_{{\tau }_m,{\tau }_n}^{S_m}, { t_{{\tau }_m}^{S_m,dep}}^{\mathcal{O}}\right),\\ S_i\in \mathbb{S}, {\tau }_i\in \mathbb{T}\end{array}\right.$$

(18)

$$\left\{\begin{array}{c}\mathbf{I}\mathbf{F} t_{{\tau }_m}^{J_n}<t_{{\tau }_m}^{J_m}+t_{{\tau }_m}^{J_m,J_n}\\ \mathbf{THEN} t_{{\tau }_m}^{J_n}=t_{{\tau }_m}^{J_m}+t_{{\tau }_m}^{J_m,J_n}, J_m,J_n\in \mathbb{J}, {\tau }_m\in \mathbb{T}\end{array}\right.$$

(19)

In the operators above, (15) ensures the minimum operational travel time for train ${\tau }_m$ from the control boundary and junction area. And in (16), the decision variable $i_{{\tau }_m,{\tau }_n}^{J_m}$ will be adjusted, and the operational headway between trains ${\tau }_m$ and ${\tau }_n$ is satisfied based on the decision variable $i_{{\tau }_m,{\tau }_n}^{J_m}$. Here, the decision variable will be decided without the decision from other junctions; that is, (6) is not considered in this section, and $i_{{\tau }_m,{\tau }_n}^{J_m}$ has to keep the same sequence for trains ${\tau }_m$ and ${\tau }_n$ if they come from the same route and there is no looping in the route to the junction area. Furthermore, train dwelling time and connection time are checked by (17) and (18), respectively, if the train is supposed to stop in the local control area of the junction. The departure time is chosen using the same logic as in (4) and (5), where the departure time is decided as the maximum value between the departure time in the timetable and the delayed arrival time plus minimum dwelling time or connection time of train ${\tau }_m$. Finally, the minimum traveling time between two junctions is guaranteed based on (19). Here, in (16) and (19), the indicator $i_{{\tau }_m,{\tau }_n}^{J_m,J_n}$ is not considered; in other words, the junction will decide $i_{{\tau }_m,{\tau }_n}^{J_m}$ by itself, though it is coupled with the indicator $i_{{\tau }_m,{\tau }_n}^{J_m,J_n}$. Couplings among the decisions of junction agents are ensured in the collaborative strategy illustrated in Section 4.4.

4.3.5. Sorting and Selection

In the sorting operator, all the new individuals and their parents (last generation) will be organized into a generation and arranged with respect to their values for the objective function, i.e., the individual that has the lowest value for the objective function will be placed at the top of the generation, and the highest-valued individual will be put at the bottom. In the selection operator, a number of top individuals in the group generated from the sorting operator will be kept, and the rest of the individuals will be discarded. The following rule is applied in the selection:

$$\left\{\begin{array}{c}N=P_{sel}\ast 2N_{gen} \mathit{I}\mathit{F} N\le 2N_{gen}\\ N=2N_{gen} \mathit{I}\mathit{F} N>2N_{gen}\end{array}\right.$$

(20)

where $N$ is the number of individuals left after selection, and $P_{sel}$ is a percentage parameter that is applied to cut the individuals when the size of a generation is less than $2N_{gen}$.

4.3.6. Terminating Condition

In the designed GA, the local search will be terminated when the solution with the lowest value for the objective function does not change in $N_{ter}$ iterations.

4.3.7. Classification

At the end of the local search, a local best solution is found. However, (6) is not considered in local searching; the coupling of train timetables among coupled junctions is not considered, which may lead to the best solution from local searching not being feasible globally. To ensure global feasibility, we propose a voting mechanism to decide the coupling indicator $i_{{\tau }_m,{\tau }_n}^{J_m,J_n}$ in the next section, and to ease the voting among junction agents, a classification operator is applied.

The local search will end with a generation whose seeds consist of their decision by the decision variable $i_{{\tau }_m,{\tau }_n}^{J_m}$ and $t_{{\tau }_m}^{J_m}$. And with their different decision for $i_{{\tau }_m,{\tau }_n}^{J_m}$ for all the trains passing the junction area, $t_{{\tau }_m}^{J_m}$ may change due to different operational constraints raised by decision variable $i_{{\tau }_m,{\tau }_n}^{J_m}$. In the classification operator, all the individuals in the final generation will be classified into several pairing groups to indicate their different decisions for the decision variable $i_{{\tau }_m,{\tau }_n}^{J_m}$, which is shown in the following equations:

$$\begin{array}{c}\stackrel{⃡}{{\tau }_m,{\tau }_n}=\left\{\begin{array}{c}{i}_{{\tau }_m,{\tau }_n}^{J_m}=1, \left\{{\mathit{X}}_1^{J_m},{\mathit{X}}_2^{J_m},\dots ,\left.{\mathit{X}}_{N^{i=1}}^{J_m}\right\}\right.\\ i_{{\tau }_m,{\tau }_n}^{J_m}=0, \left\{{\mathit{X}}_1^{J_m},{\mathit{X}}_2^{J_m},\dots ,\left.{\mathit{X}}_{N^{i=0}}^{J_m}\right\}\right.\end{array}\right.\\ N^{i=1}+N^{i=0}=2N_{gen}\end{array}$$

(21)

where $\stackrel{⃡}{{\tau }_m,{\tau }_n}$ states the relationship in ordering between train ${\tau }_m$ and train ${\tau }_n$ when passing the junction, and it consists of the individuals that launch the train ${\tau }_m$ before train ${\tau }_n$, where $i_{{\tau }_m,{\tau }_n}^{J_m}=1$. Similarly, the rest of the individuals are classified into a group whose solutions decide to dispatch train ${\tau }_n$ before train ${\tau }_m$. The subscript of the individual ${\mathit{X}}^{J_m}$ stands for the index of the individual in each group, and the total number of individuals in each group is $2N_{gen}$, which should be the same as the number of individuals in the final generation. In each sub-group, the individuals are still sorted by their ascending numerical value for the objective function. Moreover, the pairing group will not be generated if all the individuals in the final generation have the same order for trains ${\tau }_m$ and ${\tau }_n$. Finally, the final generation is organized into a solution table that is indexed by $\stackrel{⃡}{{\tau }_m,{\tau }_n}$, and the solution table is delivered to the collaborative agent.

4.4. Collaborative Strategy: Condorcet Voting Method

To ensure that the final dispatching solutions generated by each junction agent are feasible from a global perspective, a Condorcet voting-based collaborative strategy is proposed. Each junction agent optimizes local solutions without considering the coupling effects on train timetables across the network. This often results in conflicts and infeasibilities in the operational plan, as the solutions generated by different agents may not align with each other. Additionally, due to the local preferences of each agent, the top solutions—those with the lowest objective function values—may form a cyclic pattern, making it difficult to reach a consensus among the junction agents.

Example of Cyclic Preference:

• Junction Agent A prefers the passing sequence of conflict trains (S1, S2, S3) to S1, S2, S3.
• Junction Agent B prefers S2, S3, S1.
• Junction Agent C prefers S3, S1, S2.

In such cases, it becomes challenging to identify a majority preference that is acceptable to all junction agents. An appropriate strategy must be devised to consider the preferences of all agents and identify the most acceptable majority solution for each junction.

The Condorcet voting method, originally proposed by Marquis de Condorcet, addresses the social choice problem through a series of voting actions on cyclic preferences [27]. In the context of train rescheduling, the conflicting decisions made by junction agents can be seen as a manifestation of the Condorcet paradox. This paradox is resolved by breaking down cyclic preferences into sequential pairwise elections. The final decision is made based on the statistical results of these pairwise comparisons [28]. In this section, the detailed approach to adjusting the local solution from each junction agent with the Condorcet voting strategy is illustrated. The Condorcet voting starts from generating a voting sequence for all conflict decision pairs, and then the detailed voting approach is illustrated and the solution selection method is discussed.

After all junction agents complete their local searches (discussed in Section 4.3), the collaborative agent receives solution tables from each junction agent and begins the voting process to make the final decision for the entire network. As previously described, a passing solution in a junction area is represented by the decision variable $i_{{\tau }_m,{\tau }_n}^{J_m}$. Therefore, voting pairs are organized by this decision variable. The collaborative agent will examine all indicators $i_{{\tau }_m,{\tau }_n}^{J_{m,}{J}_n}$ and the associated decision variables, $i_{{\tau }_m,{\tau }_n}^{J_m}$ and $i_{{\tau }_m,{\tau }_n}^{J_n}$. A voting pair will be created if junctions $J_m$ and $J_n$ are strongly coupled and the decisions made by each junction regarding the passing sequence of trains ${\tau }_m$ and ${\tau }_n$ differ, as stated below:

$$\begin{array}{c}\mathcal{V}\left(J_m,J_n|{\tau }_m,{\tau }_n\right)=1 ,\mathit{I}\mathit{F}\mathit{F} { i}_{{\tau }_m,{\tau }_n}^{J_m,J_n}=1, i_{{\tau }_m,{\tau }_n}^{J_m}\ne i_{{\tau }_m,{\tau }_n}^{J_n}, \\ J_m,J_n\in \mathbb{J},{\tau }_m,{\tau }_n\in \mathbb{T}\end{array}$$

(22)

where $\mathcal{V}\left(J_m,J_n|{\tau }_m,{\tau }_n\right)$ denotes a voting section that will decide the passing sequence between trains ${\tau }_m$ and ${\tau }_n$ at junctions $J_m$ and $J_n$.

Definition 3.

In this paper, the first two parameters, $J_m$ and $J_n$, are called the position variables in a voting section, and the second two parameters, ${\tau }_m$ and ${\tau }_n$, are defined as voting seeds in a voting section. The solutions generated by the junction agent $J_m$ with different values for the decision variable $i_{{\tau }_m,{\tau }_n}^{J_m}$ will be the voting candidates in this voting section. $\mathcal{V}\left(J_m,J_n|{\tau }_m,{\tau }_n\right)=1$ stands for the voting between the passing sequences of trains ${\tau }_m$ and ${\tau }_n$.

In the proposed MAS for train rescheduling, the collaborative agent plays a crucial role in ensuring that the decisions made by each junction agent are globally feasible. The collaborative agent achieves this by checking all coupling indicators among junctions and their respective decisions on train passing sequences. It then compiles a voting list that includes all the voting sections to be executed. To ensure an organized and effective voting process, the voting sections are ordered as follows:

(1)

Voting sections covering multiple position variables (Priority 1):

• Voting sections that involve the same voting seeds (i.e., common trains or decision points) and cover the most position variables are prioritized.
• These sections are voted on first because their outcomes have the broadest impact on the network and can significantly influence other decisions.
• Resolving these high-impact voting sections early helps minimize conflicts and dependencies in subsequent votes.

(2)

Voting sections affecting other voting sections (Priority 2):

• Next, priority is given to voting sections that include the same one or two voting seeds, especially those in which the voting results could affect other sections.
• These sections are critical, as they may create cascading effects across the network, impacting decisions in other junctions.
• Voting on these sections second ensures that the most interconnected decisions are resolved before addressing more isolated conflicts.

(3)

Voting sections involving only two junction agents (Priority 3):

• Finally, voting sections that involve only two junction agents are addressed.
• These sections are voted on last because they are typically more localized and have a limited impact on the broader network.
• Resolving these simpler, less interconnected conflicts after the more complex sections helps streamline the decision-making process.

By applying this structured approach, the collaborative agent ensures that the voting process is both efficient and effective, resolving the most critical and interconnected conflicts first. This hierarchical approach helps optimize the overall train rescheduling process, leading to more globally feasible and robust solutions.

Once the voting sequence is decided, the voting for each voting section regarding the decision variables with coupling is started. The voting for each pair of trains is inherently a trade-off for passing indicators that have conflicts with each other. From the solution tables for each junction agent, the classified solutions can be indexed with their decision for $i_{{\tau }_m,{\tau }_n}^{J_m}$, and the voting value is chosen from the values for objective function of top solutions of $i_{{\tau }_m,{\tau }_n}^{J_m}=1$ and $i_{{\tau }_m,{\tau }_n}^{J_m}=0$ among all the junctions that have a conflict in the train passing sequence, which is illustrated in (23):

$$\begin{array}{c}\mathcal{V}\left(J_1,J_2,\dots , J_{N_v}|{\tau }_m,{\tau }_n\right)\stackrel{\scriptscriptstyle\mathrm{def}}{=}\left\{\begin{array}{c}{i}_{{\tau }_m,{\tau }_n}^{J_1,J_2,\dots ,J_{N_v}}=1, ⟨{\tau }_m|{\tau }_n⟩={\displaystyle \sum _{J_1,J_2,\dots ,J_{N_v}}}TD\left({\mathit{X}}_{1^{i=1}}^{J_m}\right)\\ i_{{\tau }_m,{\tau }_n}^{J_1,J_2,\dots ,J_{N_v}}=0, ⟨{\tau }_n|{\tau }_m⟩={\displaystyle \sum _{J_1,J_2,\dots ,J_{N_v}}}TD\left({\mathit{X}}_{1^{i=0}}^{J_m}\right)\end{array}\right.\\ \mathbf{I}\mathbf{F} \mathcal{V}\left(J_1,J_2|{\tau }_m,{\tau }_n\right)=\mathcal{V}\left(J_2,J_3|{\tau }_m,{\tau }_n\right)=\dots =\mathcal{V}\left(J_{N_v-1},J_{N_v}|{\tau }_m,{\tau }_n\right)=\mathcal{V}\left(J_{N_v},J_1|{\tau }_m,{\tau }_n\right)=1\end{array}$$

(23)

where $\mathcal{V}\left(J_1,J_2,\dots , J_{N_v}|{\tau }_m,{\tau }_n\right)$ denotes the voting on the passing sequence of trains ${\tau }_m$ and ${\tau }_n$ among junctions $J_1,J_2,\dots , J_{N_v}$, and the subscript $N_v$ stands for the number of conflict junctions. The voting values $⟨{\tau }_m|{\tau }_n⟩$ and $⟨{\tau }_n|{\tau }_m⟩$ are calculated by the sum of the values for the objective function (total delay) of the best solution in a pair of voting sections in every conflict junction agent. Furthermore, a decision that a junction does not include in its solution table is viewed as extremely bad locally, so a very large integer is applied as the value of its objective function. The final decision on the train passing sequence depends on the numerical values of $⟨{\tau }_m|{\tau }_n⟩$ and $⟨{\tau }_n|{\tau }_m⟩$ as shown below.

$$\begin{array}{c}{i}_{{\tau }_m,{\tau }_n}^{J_1}=i_{{\tau }_m,{\tau }_n}^{J_2}=\dots =i_{{\tau }_m,{\tau }_n}^{J_{N_v}}=i_{{\tau }_m,{\tau }_n}^{J_1,J_2,\dots ,J_{N_v}}\\ =\left\{\begin{array}{c}\begin{array}{cc}1& \mathit{I}\mathit{F} ⟨{\tau }_m|{\tau }_n⟩<⟨{\tau }_n|{\tau }_m⟩ \end{array}\\ \begin{array}{cc}0& , otherwise\end{array}\end{array}\right.\end{array}$$

(24)

Once a decision on the passing sequence of trains ${\tau }_m$ and ${\tau }_n$ is made, the collaborative agent saves this part of the solution as the final solution and discards all the individuals that include a different value for the decision variable in the solution table. It then moves to the next voting section until all voting sections are completed.

The Condorcet voting mechanism was selected over simpler consensus-based approaches due to its unique ability to resolve cyclic conflicts inherent in decentralized railway rescheduling. When junction agents generate locally optimal but globally incompatible solutions (e.g., conflicting train sequences at coupled junctions), traditional averaging methods fail to guarantee operational feasibility. The Condorcet method addresses this by (1) decomposing cyclic preferences into pairwise comparisons of alternative sequences (Equations (22) and (23)), (2) evaluating each option’s network-wide impact through summed delay penalties (Equation (24)), and (3) selecting the sequence with minimal total disruption. This approach maintains the MAS’s distributed architecture while ensuring strict adherence to precedence constraints that averaging would violate through fractional solutions. Furthermore, the hierarchical voting process—where junction agents propose alternatives and the collaborative agent mediates conflicts—provides computational scalability that is unattainable with centralized resolution methods. The mechanism’s effectiveness is particularly evident in scenarios with strong timetable coupling, where it prevents locally optimal but globally detrimental decisions by incorporating downstream delay propagation into the voting metric. This property aligns with real-world dispatching practices while overcoming the combinatorial limitations of alternative conflict-resolution approaches.

5. Case Study

In this study, the area surrounding Derby station in the UK was selected to test the MAS-based approach. The test area includes five key junctions: Ambergate Junction, Derby Station North Junction, Derby London Road Junction, Derby LNW Junction, and North Stafford Junction. These junctions encompass the primary conflict areas within the network. The layout of these junctions is depicted in Figure 3. Derby station itself is surrounded by three junctions: Derby LNW Junction, Derby London Road Junction, and Derby Station North Junction. These junctions play a critical role in managing train movements, as they can adjust the arrival and departure times of trains at Derby station by determining the passing sequence within their respective local junction areas. To evaluate the effectiveness of the MAS-based approach, a simulation was conducted involving 49 services operating within this network. The simulation covered both normal and peak hours, specifically from 6 a.m. to 10 a.m., on a typical workday. The services, denoted as Si (where i is a number), included a mix of local and long-distance passenger trains on the Willington and Matlock branch and Clay Cross North Junction, as well as local passenger trains and freight trains from Spondon. Additionally, the simulation included long-distance passenger trains dispatched from Derby station.

All the simulations were executed on a PC with an Intel Core i7-4790 CPU @ 3.60 GHz processor with 12 GB RAM, and the proposed approaches were commanded in Java.

5.1. Computational Complexity Analysis

In the following tables,

Table 1. Asymptotic complexity of the whole problem and each junction agent in MAS.

Variable or Constraints	Asymptotic Complexity of the Whole Problem	Asymptotic Complexity (in MAS)
Continuous variable	$\left|\mathbb{T}\right|·\left|\mathbb{J}\right|·x$	$\left|\mathbb{T}\right|·x$
Binary variable	${\displaystyle \frac{1}{2}}\left|\mathbb{T}\right|·(\left|\mathbb{T}\right|-1)·\left|\mathbb{J}\right|·x$	${\displaystyle \frac{1}{2}}\left|\mathbb{T}\right|·(\left|\mathbb{T}\right|-1)·x$
Railway operational constraints	$\left|\mathbb{T}\right|+\left|\mathbb{T}\right|·\left|\mathbb{J}\right|·x+{\displaystyle \frac{1}{2}}\left|\mathbb{T}\right|·(\left|\mathbb{T}\right|+1)·\left|\mathbb{S}\right|+{\displaystyle \frac{1}{2}}\left|\mathbb{T}\right|·\left(\left|\mathbb{T}\right|-1\right)·\left|\mathbb{J}\right|·x+{\displaystyle \frac{1}{2}}\left|\mathbb{T}\right|·\left|\mathbb{J}\right|·(\left|\mathbb{J}\right|-1)·x$	$\left|\mathbb{T}\right|+\left|\mathbb{T}\right|·x+{\displaystyle \frac{1}{2}}\left|\mathbb{T}\right|·\left(\left|\mathbb{T}\right|+1\right)·\left|\mathbb{S}\right|+{\displaystyle \frac{1}{2}}\left|\mathbb{T}\right|·\left(\left|\mathbb{T}\right|-1\right)·x+{\displaystyle \frac{1}{2}}\left|\mathbb{T}\right|·(\left|\mathbb{J}\right|-1)·x$
Timetable coupling constraints	${\displaystyle \frac{1}{4}}\left|\mathbb{T}\right|·\left(\left|\mathbb{T}\right|-1\right)·\left|\mathbb{J}\right|·(\left|\mathbb{J}\right|-1)·x$	${\displaystyle \frac{1}{4}}\left|\mathbb{T}\right|·\left(\left|\mathbb{T}\right|+1\right)·\left|\mathbb{J}\right|·(\left|\mathbb{J}\right|-1)·x$

N.B. x is the number of train passes at junction areas.

and

Table 2. The exact value for computational complexity analysis in the case study.

Variable or Constraints	Whole Problem	North Strafford Jn	Derby LNW Jn	Derby London Road Jn	Derby Station North Jn	Ambergate Jn	Collaborative Agent
Continuous variable	149	27	27	4	5	26	0
Binary variable	2273	351	351	946	300	325	0
Operational constraints	453	459	837	2104	724	429	0
Timetable coupling constraints	16,820	0	0	0	0	0	16,820

illustrate the computational complexity in a train rescheduling problem and that in a junction agent in the proposed MAS. The computational complexity analysis includes the total number of continuous variables $t_{{\tau }_m}^{J_m}$, binary variables $i_{{\tau }_m,{\tau }_n}^{J_m}$, railway operational constraints, and timetable coupling constraints. And the second table illustrates the exact number of variables and constraints in this case study. In this study, we use asymptotic complexity, indicating the worst case counts in terms of optimization.

From the complexity analysis presented earlier, it is evident that the train rescheduling problem in the case study involves a vast number of variables that must be optimized under a substantial set of constraints. Solving this mixed-integer linear programming (MILP) problem with conventional tools like CPLEX typically requires around two hours of computation time. Furthermore, the authors’ experimentation has shown that centralized solution approaches—such as genetic algorithms (GAs) using parameter configurations identical to this study—often encounter excessively large search spaces. In some cases, this can even yield inferior solutions compared to simple first-come-first-served (FCFS) dispatching rules.

The multi-agent system (MAS) approach offers a crucial advantage for train rescheduling problems through its distributed optimization framework. By decomposing the overall problem and assigning specific optimization tasks to individual junction agents, the MAS approach achieves three key benefits:

• It dramatically reduces the problem’s search space to computationally tractable levels.
• It enables truly real-time operation for traffic control applications.
• It facilitates parallel computation through distributed junction agents, yielding significantly faster solutions than centralized methods.

While the junction agents’ localized optimization might appear more limited in scope compared to global optimization approaches, the integrated Condorcet voting mechanism effectively resolves any resulting infeasibilities, ensuring system-wide solution quality. Furthermore, timetable coupling constraints, which ensure the synchronization of train schedules across different junctions, are managed by the collaborative agent. Although the voting process may require adjustments to the train sequence variables $i_{{\tau }_m,{\tau }_n}^{J_m}$ and the corresponding indicators $i_{{\tau }_m,{\tau }_n}^{J_m,J_n}$, not all decisions made by the junction agents will conflict with one another. As a result, the number of timetable coupling constraints may be significantly smaller than initially anticipated, depending on the test scenario. This reduction is supported by the classification operator within the proposed GA in the MAS. In contrast, a centralized system must account for all timetable coupling constraints, which further extend the computational time. Therefore, solving large-scale railway network train rescheduling problems using the MAS can not only significantly reduce computational time but also potentially improve the quality of the final solution.

5.2. Scenario Design

Most of the past research used historical delay data as the test scenario in case studies; however, it would be challenging to reflect all potential delay scenarios in practice. In that case, we use a sensitive testing approach to design the delay scenarios in this paper, as such, to evaluate the effectiveness of the proposed algorithm.

In this paper, three classes of test scenarios were designed: individual train entrance delays, multiple train entrance delays, and minor perturbations in station areas [29].

• Class I: Individual Entry Delay

In this class, a delay is introduced at the point where a train enters the simulation. This could occur either at the edge of the network, where a train enters the simulation area, or at a station within the network, where a scheduled journey begins. The purpose of this scenario is to observe how a single train’s delay propagates across the entire network under different dispatching strategies. These scenarios are labeled I_Si, where Si represents the delayed train. Each scenario includes 25 cases, with input delays ranging from 0 to 1440 s, in increments of 60 s.

• Class II: Multiple Entry Delays

In Class II, delays are introduced to two individual services at their entry points into the simulation. These scenarios are labeled M_Si_Sj, where Si and Sj represent the delayed services. Each scenario includes 156 cases, testing various dispatching rules. Si is delayed from 0 to 1440 s in increments of 120 s, while Sj is delayed from 60 to 1380 s, also in increments of 120 s.

• Class III: Minor Perturbation Delays

In this class, delays are introduced at each scheduled stopping point for every service, affecting the departure time. The delay is added as follows:

(1)

the scheduled departure time and

(2)

the arrival time + the required dwell time.

The value of delays is drawn from the Weibull distribution stated in Equation (25):

$$f\left(x;\lambda ,k\right)=\left\{\begin{array}{c}\hfill 0, x<0\\ {{\displaystyle \frac{k}{\lambda }}\left({\displaystyle \frac{x}{\lambda }}\right)}^{k-1} e^{-{\left(x/\lambda \right)}^k}, x\ge 0\end{array}\right.$$

(25)

where $\lambda$ and $k$ are the scale and shape parameters, respectively. We developed three kinds of cases with different parameter settings, as shown in

Table 3. Summary of minor perturbation delay scenarios.

Index	Train ID	Delay Distribution
Class III_a	All trains potentially delayed on departure	$k=0.45, \lambda =4$
Class III_b		$k=0.5, \lambda =6$
Class III_c		$k=0.55, \lambda =8$

.

5.3. Computational Experiments and Discussion

The final evaluation is based on the total value of the objective function for the five junction dispatching agents. Train cancellations and replatforming actions are not considered. The TTOE strategy, which follows the original train schedule sequence, is included for comparison. However, TTOE often performs worse than FCFS and the proposed MAS, and is sometimes not feasible. FCFS is used as the baseline to assess improvements achieved by the MAS over traditional human dispatching. The specific data settings for each local dispatching agent are detailed in the following sections, providing a clear framework for evaluating the MAS approach’s effectiveness in optimizing train dispatching in a complex railway network.

• $t_{slot}=30$ min;
• $N_{gen}=100$;
• $P_{sel}=0.8$;
• $N_{ter}=50$.

To show the advantages of the proposed MAS approach, an example scenario (M_S28_S4) is detailed and discussed below. In

Table 4. Summary of total delay (TD) in second in scenarios M_S28_S4.

	S4_0060s	S4_0180s	S4_0300s	S4_0420s	S4_0540s	S4_0660s	S4_0780s	S4_0900s	S4_1020s	S4_1140s	S4_1260s	S4_1380s
S28_0000s	60	80	1458/ 1836	1836	1842	1897	2145	2535	2735	3027	3206	3516
S28_0120s	80	300	1583/ 1956	1956	1962	2017	2265	2655	2855	3147	3326	3636
S28_0240s	303	423	1706/ 2079	2079	2085	2140	2388	2778	2978	3270	3449	3759
S28_0360s	1085	1205	2382/ 2861	2861	2867	2922	3170	3560	3760	4052	4231	4541
S28_0480s	2022	2142	3377/ 3824	3824	3825	3859	4107	4495	4697	4989	5168	5478
S28_0600s	3009	3129	4292/ 4830	4830	4839	4922	5094	5482	5684	5976	6155	6465
S28_0720s	4338	4338	5255/ 5613	5613	5619	5674	5922	6312	6512	6804	6983	7293
S28_0840s	3837	4338	5256/ 5613	5613	5619	5674	5922	6312	6512	6804	6983	7293
S28_0960s	3837	4020	5255/ 5613	5613	5619	5674	5922	6312	6512	6804	6983	7293
S28_1080s	5083/ 5225	5203/ 5345	7254/ 7618	6997/ 7127	6997/ 7127	7006/ 7127	7181/ 7310	7483/ 7649	7654/ 7822	8263	8371	8681
S28_1200s	5083/ 5225	5203/ 5345	7079/ 7618	7352/ 7618	6997/ 7127	7006/ 7127	7181/ 7310	7483/ 7649	7654/ 7822	8105/ 8263	8190/ 8371	8681
S28_1320s	5083/ 5793	5203/ 5913	7080/ 7618	7352/ 7618	7396/ 7611	6993/ 7798	7166/ 7878	7484/ 8217	7746/ 8213	8128/ 8775	8240/ 9090	8555/ 9249
S28_1440s	5203/ 5801	5323/ 5921	7074/ 7604	7352/ 7604	7406/ 7629	7633/ 7864	7286/ 7886	7604/ 8225	7866/ 8246	8262/ 8783	8360/ 9098	8675/ 9257

, each value states the TD in seconds across the whole network; numbers in black indicate that the decisions made from the FCFS and MAS approaches are the same in terms of TD, and in the scenario with two values, the numbers in red and green stand for FCFS and MAS, respectively.

To illustrate the difference between the FCFS and MAS approaches, we take the case of S28 with a 1320 s delay and S4 with a 300 s delay as an example. The detailed timetables of the representative trains for FCFS and MAS are shown in

Table 5. Rescheduled timetable for representative trains in FCFS.

	S6	S4	S28	S24	S12	S50	S14
North Stafford Jn	--	--	08:04:37	08:11:40	--	07:55:51	08:01:50
Derby LNW Jn	--	--	07:59:53	08:07:19	--	07:59:15	07:57:51
Derby London Road Jn	07:48:15	07:49:17	07:50:45/ 07:58:00	07:57:03/ 08:06:24	07:58:42	--	--
Derby Station	07:49:10–07:52:10	07:50:18–07:53:18	07:51:43–07:56:43	07:58:00–08:06:00	07:53:00	08:01:07–08:10:00	07:53:00–07:57:00
Derby North Jn	07:52:25	07:53:44	--	--	--	08:10:30	07:52:08
Ambergate Jn	08:07:28	08:08:51	--	--	--	--	07:30:10

and

Table 6. Rescheduled timetable for representative trains in MAS.

	S6	S4	S28	S24	S12	S50	S14
North Stafford Jn	--	--	08:02:21	08:11:40	--	07:55:51	08:03:42
Derby LNW Jn	--	--	07:58:02	08:07:19	--	08:00:45	07:59:25
Derby London Road Jn	07:48:15	07:49:17	07:50:45/ 07:57:06	07:58:12/ 08:06:24	07:59:09	--	--
Derby Station	07:49:10–07:52:10	07:50:18–07:53:18	07:51:43–07:56:43	07:59:17–08:06:00	07:53:00	08:02:54–08:10:00	07:53:00–07:57:00
Derby North Jn	07:53:58	07:53:44	--	--	--	08:10:30	07:52:08
Ambergate Jn	08:08:12	08:01:09	--	--	--	--	07:30:10

below, and their differences are highlighted in red color:

S28 and S4 are long-distance services with a single stop in Derby station, originating from Long Eaton station and terminating at Willington and Clay Cross North Junction, respectively. S6 is local service to Matlock station from Long Eaton station with stops at Spondon, Derby, Duffield, Belper, Ambergate, Whatstandwell, Cromford, Matlock Bath, and Matlock stations. S6, S4, and S28 enter the simulation at 7:38:00, 7:39:00, and 7:43:00, respectively; due to their same trajectory between Long Eaton and Derby stations, the movement of S6 blocks the movement of S4 and S28. S6 has a dwell at Spondon station, which means that S4 and S28 have to wait outside Spondon station and extend their delays. As they approach Derby station, S6, S4, and S28 dwell at platforms 6, 4, and 5, respectively. After dwelling at Derby station, S6 and S4 are dispatched to the Ambergate Junction side at 7:52:10 and 7:53:44, respectively. However, S6 and S4 will ask for the same infrastructure between Derby station and Ambergate Junction, so if Derby Station North Junction makes the decision according to FCFS, dispatching S6 before S4, the future movement of S4 will be blocked by S6 and its dwelling at Duffield and Belper stations, resulting in a significant delay of S4 at Ambergate Junction, though FCFS has a smaller value for its local objective function. In the MAS approach, Derby Station North Junction launches S4 first, followed by S6. Besides that, S24 is also a long-distance service having the same trajectory as S28 but with a dwell at Derby station at platform 3, and S12 is a high-speed train that enters the simulation from Derby station and is dispatched to the Long Eaton station side from the same platform as S28. In the context of FCFS, the train will be dispatched by Derby London Road Junction in the sequence S24, S28, S12; that decision will increase the delay of S28 and attach a larger knock-on delay to S12. However, the MAS decides to launch S28 first, with the sequence S28, S24, S12. Then, the delay of S28 will be smaller than under FCFS, and the delay to S24 caused by its conflict with S28 will be easily absorbed by the dwell time at Derby station. The detailed passing sequence in scenario M_S28(1320s)_S4(300s) is shown in

Table 7. Passing sequence of trains in all junctions considered in FCFS and MAS.

Passing sequences of trains in each junction in FCFS

The order for North Strafford Junction is: S5, S1, S51, S2, S19, S13, S22, S23, S15, S52, S50, S18, S14, S28, S31, S24, S35, S41, S37, S42, S33, S36, S40, S43, S32, S44, S54; The order for Derby LNW Junction is: S5, S1, S51, S2, S13, S19, S22, S15, S23, S52, S14, S50, S28, S18, S24, S31, S41, S35, S37, S42, S33, S43, S36, S40, S44, S32, S54; The order for Derby London Road Junction is: S53, S51, S47, S5, S49, S9, S51, S5, S57, S47, S2, S25, S49, S11, S10, S19, S15, S19, S6, S4, S52, S28, S24, S28, S12, S50, S26, S24, S50, S55, S16, S29, S42, S35, S42, S35, S38, S43, S36, S39, S30, S36, S44, S45, The order for Derby Station North Junction is: S53, S58, S9, S47, S53, S49, S47, S8, S25, S57, S27, S57,S49, S52, S6, S4, S26, S16, S29, S55, S43, S38, S30, S39, S45; The order for Ambergate Junction is: S58, S9, S8, S25, S13, S1, S14, S27, S52, S26, S22, S16, S6, S29, S4, S23, S41, S18, S55, S43, S30, S37, S45, S38, S39, S40.

Passing sequences of trains in each junction in MAS

The order for North Strafford Junction is: S5, S1, S51, S2, S19, S13, S22, S23, S15, S52, S50, S18, S28, S14, S31, S24, S35, S41, S37, S42, S33, S36, S40, S43, S32, S44, S54; The order for Derby LNW Junction is: S5, S1, S51, S2, S13, S19, S22, S15, S23, S52, S28, S14, S50, S18, S24, S31, S41, S35, S37, S42, S33, S43, S36, S40, S44, S32, S54; The order for Derby London Road Junction is: S53, S51, S47, S5, S49, S9, S51, S5, S57, S47, S2, S25, S49, S11, S10, S19, S15, S19, S6, S4, S52, S28, S28, S24, S12, S50, S26, S24, S50, S55, S16, S29, S42, S35, S42, S35, S38, S43, S36, S39, S30, S36, S44, S45; The order for Derby Station North Junction is: S53, S58, S9, S47, S53, S49, S47, S8, S25, S57, S27, S49, S57, S52, S4, S6, S26, S16, S29, S55, S43, S38, S30, S39, S45; The order for Ambergate Junction is: S58, S9, S8, S25, S13, S1, S14, S27, S52, S26, S22, S16, S4, S29, S6, S23, S41, S18, S55, S43, S30, S37, S45, S38, S39, S40.

below, and the difference in sequence is highlighted in the table.

In this single scenario, due to the global voting mechanism proposed in this paper, the delays of trains S4 and S28 are reduced and the delays of trains S6, S24, S12, S50 and S14 are enlarged against the FCFS dispatching rule. The improvements in delays for S4 and S28 are larger than the values of deterioration for S6, S24, S12, S50, and S14, so the value of total delay across the whole network is reduced. With the changes in the passing sequence between the solutions of FCFS and MAS, the maximum value of prolonged delay for trains S6, S24, S12, S50, and S14 is up to 3 min, and the mean value is about 1 min, and the reduced delays for S4 and S28 are up to 5 min and 2.5 min, respectively. Although improving the performance of S4 and S28 results in more delays for other trains, the prolonged delays are usually around 1 min, which is acceptable for daily operations. Furthermore, S4 and S28 are long-distance services that are more sensitive to punctuality, so it is worth changing the train passing sequence to the MAS version.

In brief, 11.8% and 20.64% of the solutions generated by the MAS are better than those by FCFS in class I and class II, respectively. The improvement based on FCFS in terms of the values of the objective function is up to 34.11%, 28.7%, and 21.27% in class I, class II, and class III, respectively. Figure 4 illustrates the detailed average values of the objective function in class III, and Figure 5 shows the computational time of each agent in the MAS.

In summary, the MAS effectively addresses the train dispatching problem during delays, achieving up to a 34.11% improvement in network-wide delay reduction. The computational times for the junction agent and collaborative agent are 13.49 s and 7.62 s, respectively. The overall computational time for the system, which depends on the slowest junction agent and the collaborative agent’s voting process, averages 27.39 s, with a maximum of 33.87 s. A key advantage of the MAS is its ability to offer a global perspective to each junction agent, enabling it to balance solutions across the network. This collaborative strategy ensures that overly greedy decisions made by individual junction agents can be corrected, thereby enhancing the overall feasibility and optimization of train dispatching solutions. The collaborative agent plays a crucial role in validating the global feasibility of the solutions and selecting the optimal approach from the junction agents under its control.

6. Conclusions

In this paper, we propose a multi-agent system (MAS) for addressing train rescheduling problems in large-scale railway networks. We present a mixed-integer linear programming (MILP) formulation for the large-scale train dispatching problem and describe our approach to network decomposition, the local search algorithm, and the collaborative strategy. By decomposing the railway network, the train rescheduling problem is broken down into smaller sub-problems, reducing computational complexity and enabling real-time traffic control. Simulation results demonstrate that the MAS outperforms the First-Come-First-Served (FCFS) routing approach in 19.264% of the tested scenarios, with improvements of up to 34.11% in key control objectives. In all remaining cases, the MAS performs at least as well as FCFS, maintaining solution quality without degradation. The decentralized nature of the MAS significantly reduces computational time, with simulations completed in approximately 0.5 min, making it highly suitable for real-time applications.

The proposed MAS offers a decentralized alternative to traditional centralized rescheduling methods, improving scalability and adaptability for larger networks. Future work will explore additional operational factors, including train cancellation, replatforming, and the influence of human factors such as crew scheduling and passenger flow dynamics. We will also examine the constraints imposed by existing infrastructure on railway operations and explore potential upgrades to enhance operational flexibility.