Quantum and classical algorithms for daily railway rolling stock circulation plans

TL;DR

The paper tackles daily rolling stock circulation for EMUs with predefined coupling and new bicycle-capacity constraints, framed around a real regional operator’s needs. It develops a flexible acyclic ILP/hyper-graph model and a QUBO reformulation for quantum optimization, comparing classical and quantum-inspired solutions on real-world data. Findings show that the ILP approach scales to hundreds of trips and dozens of EMU types within practical times, while current quantum and quantum-inspired solvers are limited to substantially smaller subproblems due to embedding and QUBO size growth; these results quantify the current frontier of QUBO methods in this domain. The work advocates a hybrid architecture where classical optimization handles end-to-end planning, and quantum/quantum-inspired solvers address targeted local subproblems, with potential extensions to multi-day planning leveraging column-generation ideas.

Abstract

We study daily rolling stock circulation planning for electric multiple units (EMUs) on a regional passenger network, focusing on services where identical EMUs may be coupled in pairs on selected routes. Motivated by the operational needs of the regional operator Silesian Railways in Poland, we formulate an acyclic mixed-integer linear program on a one-day horizon that incorporates depot balance constraints, demand-driven seat and bicycle capacity limits (which is a new aspect requested by the regional operator and local society of passengers), and simple crew availability constraints. The model is designed to support both baseline planning and disruption management under increased passenger demand. Using a graph-hypergraph representation of trips and single or coupled EMU movements, we first solve the problem with a classical ILP solver. We then derive a Quadratic Unconstrained Binary Optimization (QUBO) reformulation - which is frequently used as the input for quantum optimization - and evaluate its solution by quantum annealing on D-Wave Advantage systems and by the classical quantum-inspired VeloxQ solver. Computational experiments on real-world instances from the Silesian network, with up to 404 train trips and 11 EMU types, show that the ILP approach can obtain high-quality daily circulation plans within at most about 40 minutes, whereas current quantum and quantum-inspired solvers are restricted to substantially smaller sub-instances (up to 51 and 78 train trips, respectively) due to embedding and QUBO size limitations. These results quantify the present frontier of QUBO-based methods for rolling stock circulation and point towards hybrid decision-support architectures in which quantum or quantum-inspired optimizers address only local subproblems within a broader classical planning framework.

Figures (9)

• Figure 1: The workflow of the paper
• Figure 2: Illustration of the three hyper-arc types: (a) coupling of two EMUs into a multiple-unit service, (b) transfer of a coupled composition represented by a hyper-arc composed of two parallel arcs, and (c) decoupling into two single-EMU services.
• Figure 3: Illustration of the network structure of the optimal solution of the toy problem. Black arrows represent arcs, while green arrows represent the single hyper-arc.
• Figure 4: Schematic topology of the network, for the more detailed one see https://www.kolejeslaskie.com/rozklad_jazdy/schemat-linii-komunikacyjnych/
• Figure 5: Train diagrams of ILP solutions of the following instances 2 (left) and 2a (right). The green line means two rolling pieces of stock coupled. We use $\alpha = 0.01$ in Eqs. \ref{['eq::obj']}–\ref{['eq::binary_req']}.