Column Generation for Periodic Timetabling

TL;DR

This work addresses scalable periodic timetabling by reformulating the $PESP$ on a time-expanded graph and developing a path-based model, $cXPESP$, that embeds more problem structure than traditional arc-based formulations. The approach leverages column generation with pricing problems that are solvable as shortest-paths on DAGs, yielding a stronger LP relaxation and enabling exact IP solutions for limited instances. An extension, $cXPESP^{w}$, reduces the number of expanded transfer arcs to alleviate model size, while an integrated variant $cXTTP$ combines timetabling with passenger routing, offering improved dual bounds at the cost of heavier LP computations. Computational experiments show that the LP relaxations for $cXPESP$ and $cXTTP$ are notably tighter than those of classical $PESP$ formulations, but solving the full IP remains computationally challenging, highlighting theoretical gains and practical limitations. Future work may focus on exploiting problem symmetries to further prune the column space and improve scalability.

Abstract

Periodic timetabling for public transportation networks is typically modelled as a Periodic Event Scheduling Problem (PESP). Solving instances of the benchmark library PESPlib to optimality continues to pose a challenge. As a further approach towards this goal, we remodel the problem by a time discretization of the underlying graph and consider arc-based as well as path-based integer programming formulations. For the path-based case, we use cycles on the graph expansion of the operational lines as variables and, therefore, include more of the problem inherent structure into the model. A consequence is the validity of several known inequalities and a lower bound on the LP-relaxation, that is the best known to date. As an extension we integrate passenger routing into the new model. The proposed models have an advantage in the linear programming relaxation, on the one hand, but have an increased problem size, on the other hand. We define the corresponding pricing problems for the use of column generation to handle the size. Both models are practically tested on different problem instances.

Figures (9)

• Figure 1: Exemplary graph expansion. (a) The line network $G$ of a two-line example with driving activity bounds fixed to 1. (b) The corresponding event-activity network $N$ with exemplary bounds for waiting, transfer, and turnaround activities. (c) The corresponding expanded event-activity network $D$ for period $T=2$. For each event, the different time steps are marked by the numbers in the nodes.
• Figure 2: A cycle (blue) in an expanded line cycle. The closed walk colored in red does not fulfill the definition of a cycle due to the cardinality of its node set.
• Figure 3: Illustration of the coupling constraints \ref{['eq:c1new']} and \ref{['eq:c2new']} in cXPESP. The summed values of the identically colored arc variables equal the summed values of the correspondingly colored cycle variables.
• Figure 4: Illustration of counterexample in Remark \ref{['rem:incl']}.
• Figure 5: Counterexample for Theorem \ref{['thm:splitclosure']} for arbitrary cycles.

Theorems & Definitions (32)

• Definition 3.1
• Definition 4.1
• Lemma 4.2
• Remark 4.3
• Definition 5.1
• Definition 5.2
• Definition 5.3
• Theorem 5.4
• Theorem 5.5
• Corollary 5.6