# A Novel Multi-Agent-Based Approach for Train Rescheduling in Large-Scale 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**  
\[TD=\sum_{J_{m}\in\mathbb{J}}\sum_{\tau_{m}\in\mathbb{T}}\left|t_{\tau_{m}}^{J _{m}}-t_{\tau_{m}}^{J_{m}}\right| \tag{1}\]  
**Subject to**  
\[H_{\tau_{m},\tau_{n}}^{J_{m}}=f_{headway}(\tau_{m},\tau_{n},\;J_{m}),\;\tau_{ m},\tau_{n}\in\mathbb{T},J_{m}\in\mathbb{J} \tag{2}\]  
\[r_{m}=f_{routing}(\tau_{m},J_{m}),\;\tau_{m}\in\mathbb{T}  
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 \(t_{\tau_{m}}^{\text{J}_{m}}\), 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 \(t_{\tau_{m}}^{\text{J}_{m}}\) and the binary variable \(i_{\tau_{m},\tau_{m}}^{\text{J}_{m}}\). The variables \(t_{\tau_{m}}^{\text{J}_{m}}\), \(t_{\tau_{m}}^{\text{S}_{m},\text{L}\text{P}}\), \(D_{\tau_{m}}^{\text{S}_{m}}\), and \(C_{\tau_{m},\tau_{m}}^{\text{S}_{m}}\) are obtained from the train working timetable, which is provided in advance by the rail operators. The indicators \(i_{\tau_{m}}^{\text{S}_{m}}\) and \(i_{\tau_{m},\tau_{m}}^{\text{S}_{m}}\) are derived from the planned journey of each train at every station within the control area. The coupling indicator \(i_{\tau_{m},\tau_{m}}^{\text{J}_{m},\tau_{m}}\) is mainly identified from the layout of each junction and is determined by the routes requested by the approaching trains \(\tau_{m}\) and \(\tau_{m}\), with its value provided online by each junction. The variable \(H_{\tau_{m},\tau_{m}}^{\text{J}_{m}}\) 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 \(t_{\tau_{m}}^{\text{b}}\) is monitored by the railway traffic management system as the train crosses its control boundary.