Emerging reservation-based travel technologies offer a promising solution to mitigate supply-demand mismatches in metro systems. This paper presents a framework to support metro operators by optimizing time-varying reservation slot allocation plans, passenger flow control strategies, and train schedules. The proposed approach ensures that passengers with reservations can directly access platforms and board the first available train services, while those without reservations are managed through effective passenger flow control strategies to optimize train capacity utilization. To address this, an integer nonlinear programming model is formulated, incorporating constraints that capture interactions between passengers with and without reservations, with the objective of minimizing passengers’ waiting time and line congestion. A hybrid algorithm is developed to improve computational efficiency, combining the adaptive large neighborhood search method with a commercial solver and incorporating valid inequalities tailored to the properties of the model. The effectiveness of the proposed approaches is demonstrated through numerical experiments using real-world operational data from the Beijing metro Batong line. Computational results indicate that the integrated optimization approach reduces the objective value by 6.19 % compared to a step-by-step optimization method, achieving better alignment of capacity with dynamic passenger flows. In addition, the extreme unfairness between reserved and unreserved passengers, where passengers with reservations have a 100 % service ratio compared to less than 20 % for unreserved passengers, is mitigated by increasing passenger waiting times by 3.51 % and line congestion by 0.51 %. Furthermore, the proposed algorithm efficiently solves large-scale and real-world instances, outperforming the state-of-the-art commercial solver.

Integrating emerging shared mobility with traditional fixed-line public transport is a promising solution to the mismatch between supply and demand in urban transportation systems. The advent of modular vehicles (MVs) provides opportunities for more flexible and seamless intermodal transit. The MVs, which have been implemented, are comprised of automated modular units (MUs), and can dynamically change the number of MUs comprising them at different times and stops. However, this innovative intermodal urban transit brings with it a new level of dynamism and uncertainty. In this paper, we study the problem of jointly optimizing the timetable and the vehicle schedule within an intermodal urban transit network utilizing MVs within the context of distributionally robust optimization (DRO), which allows MVs to dynamically (de)couple at each stop and permits flexible circulations of MUs across different transportation modes. We propose a DRO formulation to explore the trade-off between operators and passengers, with the objective of minimizing the worst-case expectation of the weighted sum of passengers’ and operating costs. Furthermore, to address the computational intractability of the proposed DRO model, we design a discrepancy-based ambiguity set to reformulate it into a mixed-integer linear programming model. In order to obtain high-quality solutionss of realistic instances, we develop a customized decomposition-based algorithm. Extensive numerical experiments demonstrate the effectiveness of the proposed approach. The computational results of real-world case studies based on the operational data of Beijing Bus Line illustrate that the proposed integrated timetabling and vehicle scheduling method reduces the expected value of passengers’ and operating costs by about 6% in comparison with the practical timetable and fixed-capacity vehicles typically used in the Beijing bus system.

The collaborative design of the timetable and dynamic-capacity allocation plan of emerging modular vehicles (MVs) is a promising solution to the mismatch between supply and demand in public transportation studies; however, such efforts are subject to high-level dynamics and uncertainty inherent in operating environments. In this study, we focus on the timetabling and dynamic-capacity allocation problem of MVs within the context of distributionally robust optimization under time-dependent demand uncertainty. The dynamic capacity refers to the number of modular units (MUs) comprising an MV can be potentially changed at different times and stops. A Wasserstein distance-based ambiguity set with a time-dependent and station-wise perturbation parameter is adopted to incorporate all potential distributions within a 1-Wasserstein distance for addressing the uncertainty of passenger demand. Further, a data-driven distributionally robust optimization model that considers time-varying capacity is formulated to minimize passenger waiting costs and dispatching costs of operators over all possible demand distributions within the ambiguity set. Subsequently, an expansion that allows for flexible formations of MVs assigned to each trip at each stop is proposed, and this results in more customized operational plans driven by the passenger demand. To improve the computational efficiency of realistic problems, we design a customized integer L-shaped method to exactly solve the models, which incorporates a class of valid equalities to further speed up the computation. The effectiveness of the proposed approaches in reducing the costs for both passengers and operators compared with the practical fixed-capacity operations is verified by real-world case studies based on the operating data of Beijing Bus Line 468. Furthermore, the superiority of the distributionally robust optimization method in comparison to the stochastic programming and the robust optimization approaches is demonstrated.