A three-stage stochastic optimization approach for robust train timetabling and rolling stock planning with virtual (de)coupling

The emergence of suburban clusters has significantly reshaped urban mobility patterns, intensifying the reliance on central business districts for work and commerce. This dependency has led to an asymmetric and time-dependent demand for passengers, characterized by peak-hour congestion in one direction and under-utilization in the opposite direction. Existing studies often address passenger-dependent and time-dependent uncertainties in isolation, neglecting their combined impact on transit operations. To bridge this gap, this study proposes a novel three-stage stochastic optimization model that integrates robust train timetabling and rolling stock planning under virtual (de)coupling, explicitly considering both time-dependent and asymmetric passenger-dependent uncertainties. Passenger-dependent uncertainty is modeled through stochastic variations in arrival rates, while time-dependent uncertainty captures operational delays in running and dwell times. The model reformulation into a tractable mixed-integer program leverages the superposition principle and the mean conditional-value-at-risk criterion. This approach simultaneously optimizes nominal and robust scheduling aspects, including passenger loading, train timetables, rolling stock assignments, and virtual (de)coupling decisions. The primary objective is to develop an integrated robust train timetable and rolling stock plan that minimizes schedule deviations, unserved passengers, and operational costs. To enhance computational efficiency, a branch-and-price algorithm based on Dantzig-Wolfe decomposition is introduced, decomposing the problem into a master problem for rolling stock planning and sub-problems for robust train timetable and passenger loading process. The proposed methodology is validated through extensive computational experiments, including small- and medium-scale examples, as well as a real-world case study of the Beijing Batong metro line using historical data. The results demonstrate the effectiveness of the approach in generating robust train schedules that mitigate operational disruptions while optimizing rolling stock utilization.