Late-night and early-morning train scheduling with non-traffic hour maintenance window in urban rail transit systems

Regular maintenance during non-traffic hours (NTH) is vital for the resilience of urban rail transit (URT) systems, yet an insufficient NTH maintenance window poses a challenge for URT systems in various cities. For instance, the Hong Kong MTR Corporation has noted that the required NTH maintenance time often exceeds the available window, prompting service adjustments such as earlier late-night closures and/or later early-morning starts. To address this challenge, this study develops an optimal scheduling framework that links late-night and early-morning URT services through the NTH maintenance window requirement to maximize public welfare. A Decoupled Optimization Model (DOM) first derives closed-form solutions for the number of train services and each train headway in both periods by relaxing the NTH maintenance window constraint. These results determine the last and first train departure times, service duration, and the available NTH maintenance window, accordingly. Building on the DOM, a Coupled Adjustment Model (CAM) addresses cases where the required NTH maintenance window exceeds the available NTH maintenance window by adjusting train services through four schemes: canceling services or reducing headways in either period. Analytical results and numerical experiments for the Chengdu Metro show that optimal headways increase (decrease) over time with declining (rising) passenger demand during the late-night (early-morning) period, and this pattern persists after service duration adjustment. Service cancellation (headway reduction) is more effective when the NTH shortage gap is large (small). The framework generates optimal schedules for any feasible exogenous NTH maintenance window requirement, offering practical guidance for URT operations and integrating the timetabling of the last-train and first-train services.