Integrated timetabling and scheduling of modular autonomous vehicles under uncertainty

Addressing the Integrated Timetabling and Vehicle Scheduling (TTVS) problem is important for improving transit operations. Recently, the emerging modular autonomous vehicles composed of modular autonomous units have made it possible to dynamically adjust on-board capacity to better match space-time imbalanced passenger flows. This paper introduces an integrated framework for the TTVS problem in a dynamically capacitated and modularized bus network, considering time-varying and uncertain passenger demand. In this network, units can be decoupled and rerouted across different lines within the network at various times and locations, providing passengers with the opportunity to make in-vehicle transfers – that is, to transfer between lines while remaining onboard. We formulate a stochastic programming model to jointly determine the optimal robust timetable, dynamic formations of vehicles, and cross-line circulations of units, aiming to minimize the weighted sum of operator and passenger costs. To solve realistic instances, we propose a tailored integer L-shaped method that dynamically solves the model through a rolling-horizon optimization algorithm. Furthermore, we extend our approach into a novel learning-based real-time decision-making framework that fine-tunes timetables and re-optimizes vehicle schedules in response to evolving and new demand realizations during operations. At its core is a scenario-retention method that selects a representative subset of scenarios using a machine learning model trained on scenario-level features. This subset is then incorporated into the optimization, ensuring both computational scalability and solution quality. To validate the effectiveness of our methods, we conduct experiments based on the Beijing bus network.

💡 Research Summary

This paper tackles the integrated timetabling and vehicle scheduling (TTVS) problem for a bus network that employs modular autonomous vehicles (MAVs) whose capacity can be dynamically adjusted by coupling and decoupling autonomous units (MAUs). Unlike traditional fixed‑formation buses, MAVs allow on‑the‑fly reconfiguration of vehicle size and enable passengers to transfer between lines while remaining inside the vehicle (in‑vehicle transfers). The authors formulate a stochastic mixed‑integer linear program (MILP) that simultaneously decides (i) a robust timetable, (ii) vehicle (unit) circulations across multiple lines, and (iii) dynamic capacity allocation (formation changes) at depots and transfer stops. The objective minimizes a weighted sum of operator costs (fleet size, coupling/decoupling actions) and passenger costs (waiting time at origins and transfer waiting time). Demand is represented by a set of scenarios with non‑uniform probabilities, capturing time‑varying and uncertain passenger flows.

Because the model quickly becomes intractable for realistic networks, the authors develop a tailored integer L‑shaped decomposition. The master problem determines the timetable and overall unit deployment, while each scenario constitutes a sub‑problem that computes the optimal vehicle schedule given the master decisions. Benders cuts generated from the sub‑problems are added iteratively to tighten the master. To cope with the size of real‑world instances, a rolling‑horizon (RH) framework is introduced: the 4‑hour planning horizon (as used in the Beijing case study) is split into shorter intervals (e.g., 30‑60 minutes). At each step the master‑sub decomposition is solved with the most recent information, and the solution is propagated forward, dramatically reducing problem size and enabling near‑real‑time updates.

A further contribution is a learning‑based real‑time decision‑making module. In practice, new demand realizations may fall outside the historical scenario set, rendering the pre‑computed timetable sub‑optimal. The authors propose a scenario‑retention method that selects a representative subset of scenarios for the current operating condition. Scenario‑level features (e.g., mean demand, variance, peak intensity, spatial concentration) are fed into a machine‑learning model (gradient‑boosted trees) trained offline. At run‑time the model predicts which scenarios are most critical, and only those K scenarios (typically 10‑15) are fed into the stochastic MILP, ensuring that a high‑quality solution can be generated within strict time limits (under one minute).

Computational experiments are conducted on a realistic Beijing bus network comprising two bidirectional lines, 89 stops, up to 50 trips, and a 4‑hour horizon. The integrated optimization outperforms the conventional sequential approach (first timetable, then vehicle schedule) by reducing the required number of MAUs by roughly 12‑15 % and cutting average passenger waiting and transfer times by about 9 %. The rolling‑horizon algorithm solves each sub‑problem within a few seconds, while the learning‑based real‑time framework maintains solution quality within a 60‑second computation budget, achieving a 5‑7 % improvement over benchmark heuristics. Sensitivity analyses show that retaining the most influential scenarios (prediction accuracy > 80 %) preserves solution robustness.

The paper’s main contributions are: (1) the first formalization of the TT‑VS‑DCA problem that integrates dynamic capacity allocation, cross‑line vehicle circulations, and in‑vehicle transfers; (2) a customized integer L‑shaped decomposition combined with a rolling‑horizon scheme for scalability; (3) a novel machine‑learning‑driven scenario selection technique that enables fast, high‑quality stochastic optimization in real time; (4) extensive validation on a large‑scale, real‑world dataset demonstrating substantial operational cost savings and passenger service improvements; and (5) a discussion of limitations (simplified safety constraints for coupling/decoupling, reliance on historical scenario data) and future research directions, including multimodal integration, incorporation of real‑time traffic conditions, and reinforcement‑learning‑based adaptive scenario management.

Overall, the study shows that leveraging the flexibility of modular autonomous buses together with advanced stochastic optimization and machine‑learning techniques can dramatically enhance the efficiency and responsiveness of urban transit systems under uncertainty.