Joint optimization of train services and freight delivery in a metro-based underground logistics system

This study investigates a shared-use metro network in which passengers and freight are transported on different trains, enhancing resource utilization and promoting the sustainable growth of urban areas. From the perspective of public authorities seeking to balance the interests of all stakeholders, we address the tactical problem of determining the optimal number, stop pattern, and schedule of freight train services needed for freight delivery, while maintaining high passenger service quality by minor timetable adjustments. A non-convex mixed-integer quadratic programming model is developed to minimize freight transportation, train operational costs, and deviations in passenger train schedules. A Frank-Wolfe-based heuristic is developed with a novel gradient approximation that accounts for the hierarchy between binary and continuous decisions. The step size is defined as a discrete gap between disjoint subspaces determined by train operations, reflecting that partially shifting flow does not effectively impact the objective due to the indivisibility of train activation. Experiments are conducted based on instances generated from the Beijing Metro network. The results show that under high freight demand, dedicated freight trains cause less passenger delay than utilizing existing passenger trains. Prioritizing passenger schedules shifts freight to lower-priority lines, while incorporating a quadratic consolidation cost helps reduce fragmentation and excessive transhipments.