A Comparison of Models for Rolling Stock Scheduling

TL;DR

This paper compares two mathematical models for rolling stock scheduling in passenger railways, analyzing their theoretical strengths and practical performance on real-world instances.

Contribution

It provides a detailed comparison of the Composition and Hypergraph models, including variants, and evaluates their effectiveness for a real-world railway scheduling problem.

Findings

The Hypergraph model's LP bounds are as strong as the Composition model's when sufficiently expressive.

The Composition model is more compact and finds solutions faster in practice.

Both models have equally strong LP bounds under certain conditions.

Abstract

A major step in the planning process of passenger railway operators is the assignment of rolling stock, i.e., train units, to the trips of the timetable. A wide variety of mathematical optimization models have been proposed to support this task, which we discuss and argue to be justified in order to deal with operational differences between railway operators, and hence different planning requirements, in the best possible way. Our investigation focuses on two commonly used models, the Composition model and the Hypergraph model, that were developed for Netherlands Railways (NS) and DB Fernverkehr AG (DB), respectively. We compare these models in a rolling stock scheduling setting similar to that of NS, which we show to be strongly NP-hard, and propose different variants of the Hypergraph model to tune the model to the NS setting. We prove that, in this setting, the linear programming bounds of both models are equally strong as long as a Hypergraph model variant is chosen that is sufficiently expressive. However, through a numerical evaluation on NS instances, we show that the Composition model is generally more compact in practice and can find optimal solutions in the shortest running time.