Solving a multi-resolution model of the train platforming problem using Lagrangian Relaxation with dynamic multiplier aggregation

Abstract

High-speed railway stations are crucial junctions in high-speed railway networks. Compared to operations on the tracks between stations, trains have more routing possibilities within stations. As a result, track allocation at a station is relatively complicated. In this study, we aim to solve the train platforming problem for a busy high-speed railway station by considering comprehensive track resources and interlocking configurations. A multi-resolution space–time network is constructed to capture infrastructure information from a macroscopic and a microscopic perspective. Additionally, we propose a nonlinear programming model that minimizes a weighted sum of total travel time and total deviation time for trains at the station. We apply Lagrangian Relaxation combined with dynamic multiplier aggregation to a linearized version of the model and demonstrate how this induces a decomposable, macroscopic train-specific path choice problem that is guided by aggregated Lagrange multipliers, which are dynamically generated based on microscopic resource capacity violations. As case studies, the proposed model and solution approach are applied to a small virtual railway station and two high-speed railway hub stations located on two of the busiest high-speed railway lines in China. Through a comparison of other approaches that include Logic-based Benders Decomposition, we highlight the superiority of the proposed method; on realistic instances, the proposed method finds solution that are, on average, approximately 2% from optimality for one station and less than 5% from optimality for the other.