Solving the Real-Time Train Dispatching Problem by Column Generation

TL;DR

The paper tackles real-time train dispatching by formulating the problem as a path-based MILP (TDP) and solving it with a column generation framework that uses maximal cliques to model conflicts. A two-stage approach separates speed-profile preprocessing from online optimization: speed-profiles are generated and then connected into train paths, while PMC-CG iteratively augments a restricted master problem with new paths via a MIP subproblem that accounts for clique shadow prices. Numerical experiments on a Bavarian dispatching area show that the method achieves meaningful delay reductions within tight time limits and yields a high percentage of integer solutions, indicating strong practical performance for real-time operation. The approach supports rolling-horizon deployment and can incorporate more complex, outside-dispatching-area effects through cost-function integrations, offering a promising route for scalable, real-time railway optimization.

Abstract

Disruptions in the operational flow of rail traffic can lead to conflicts between train movements, such that a scheduled timetable can no longer be realised. This is where dispatching is applied, existing conflicts are resolved and a dispatching timetable is provided. In the process, train paths are varied in their spatio-temporal course. This is called the train dispatching problem (TDP), which consists of selecting conflict-free train paths with minimum delay. Starting from a path-oriented formulation of the TDP, a binary linear decision model is introduced. For each possible train path, a binary decision variable indicates whether the train path is used by the request or not. Such a train path is constructed from a set of predefined path parts (speed-profiles) within a time-space network. Instead of modelling pairwise conflicts, stronger MIP formulation are achieved by a cliques formulated over the complete train path. The combinatorics of speed-profiles and different departure times results in a large number of possible train paths, so that the column generation method is used here. Within the subproblem, the shadow prices of conflict cliques must be taken into account. When constructing a new train path, it must be determined whether this train path belongs to a clique or not. This problem is tackled by a MIP. The methodology is tested on instances from a dispatching area in Germany. Numerical results show that the presented method achieves acceptable computation times with good solution quality while meeting the requirements for real-time dispatching.

Figures (10)

• Figure 1: Two-part approach for the TPD with an independent real-time train dispatching algorithm decoupled from the train path generation.
• Figure 2: Possible train paths for a train service $r$ from $S_1$ via $S_2$ to $S_3$. (a) Block sections utilized by the color-highlighted speed-profiles $v_1, ..., v_5$. (b) Space-time curve of the speed-profiles and connection of these to train paths $a_1 = \left( \left( v_1, t_{v_1} \right), \left( v_2, t_{v_2} \right) \right)$, $a_2 = \left( \left( v_1, t_{v_1} \right), \left( v_3, t_{v_3} \right) \right)$ and $a_3 = \left( \left( v_4, t_{v_4} \right), \left( v_5, t_{v_5} \right) \right)$.
• Figure 3: Conflict interval for a speed-profile $w$ defined by a speed-profile $v$ through the minimum headway times $l \left( v, w \right)$ and $u \left( v, w \right)$.
• Figure 4: (a) A halting conflict at platform $b_3$ between a crossing and halting train path with parts $\left( v,t_v \right)$ and $\left( w,t_w \right), \left( w^\prime,t_{w^\prime} \right)$, respectively. (b) A halting conflict at platform $b_3$ between two halting train paths with parts $\left( v,t_v \right), \left( v^\prime,t_{v^\prime} \right)$ and $\left( w,t_w \right), \left( w^\prime,t_{w^\prime} \right)$, respectively.
• Figure 5: Overview of the PMC-CG