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Integrated timetabling and scheduling of modular autonomous vehicles under uncertainty

Dongyang Xia, Jihui Ma, Shadi Sharif Azadeh

TL;DR

This paper tackles the integrated timetabling, vehicle scheduling, and dynamic capacity allocation problem for modular autonomous vehicles across a network with cross-line circulations and in-vehicle transfers under time-varying uncertainty. It develops a stochastic MILP formulation for TT-VS-DCA, and proposes a tailored integer L-shaped method augmented by a rolling-horizon framework to solve realistic network-scale problems. To address real-time operational needs, it introduces a learning-based scenario-retention framework that selects representative demand scenarios for rapid re-optimization, ensuring solution quality within tight time limits. The approach is validated on Beijing-subnetwork data and a virtual network, demonstrating significant reductions in fleet size and operational costs, improved passenger transfer convenience, and robust performance under demand perturbations, with superior scalability and real-time applicability compared to benchmarks.

Abstract

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.

Integrated timetabling and scheduling of modular autonomous vehicles under uncertainty

TL;DR

This paper tackles the integrated timetabling, vehicle scheduling, and dynamic capacity allocation problem for modular autonomous vehicles across a network with cross-line circulations and in-vehicle transfers under time-varying uncertainty. It develops a stochastic MILP formulation for TT-VS-DCA, and proposes a tailored integer L-shaped method augmented by a rolling-horizon framework to solve realistic network-scale problems. To address real-time operational needs, it introduces a learning-based scenario-retention framework that selects representative demand scenarios for rapid re-optimization, ensuring solution quality within tight time limits. The approach is validated on Beijing-subnetwork data and a virtual network, demonstrating significant reductions in fleet size and operational costs, improved passenger transfer convenience, and robust performance under demand perturbations, with superior scalability and real-time applicability compared to benchmarks.

Abstract

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.

Paper Structure

This paper contains 59 sections, 5 theorems, 37 equations, 10 figures, 15 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature review
  3. Integrated timetabling and vehicle scheduling with fixed-formation vehicles
  4. Timetabling of MAVs with flexible formations
  5. Real-time optimization under uncertainty
  6. Contributions against the state-of-the-art studies
  7. Problem description
  8. Network structure and operational framework
  9. Network structure.
  10. Trips and MAVs.
  11. Timetabling, vehicle scheduling, and dynamic capacity allocation.
  12. Time-dependent and uncertain path-based passenger demand
  13. Decisions and costs of operators
  14. Timetabling decisions.
  15. Vehicle scheduling and dynamic-capacity allocations decisions.
  16. ...and 44 more sections

Key Result

Proposition 1

(Time window-based search space reduction) Constraints (cons_ve_timetable) are valid for the TT-VS-DCA model (stochastic_problem). where $\underline{\varsigma}^l_{k,i}$ and $\overline{\varsigma}^l_{k,i}$ represent the earliest and latest departure times of the MAV assigned to trip $k$ leaving stop $i$ of line $l$, respectively. The specific values of $\underline{\varsigma}^l_{k,i}$ and $\overline

Figures (10)

  • Figure 1: The optimized timetable, formations of MAVs assigned to trips, and cross-line schedules of MAUs of the instance$\_$rp$\_$6. Note: Shaded area indicates the transfer stop.
  • Figure 2: Results of passengers’ and operational costs when the weighting coefficient related to passengers' costs in the objective function increases.
  • Figure 3: Results of the proportion of in-vehicle transfers among all transfers and the used MAUs when the weighting coefficient related to passengers' costs in the objective function increases.
  • Figure 4: Effect of overload allowance $\lambda$ on violations of the nominal capacity.
  • Figure 5: Approximate Pareto frontier of the TT-VS-DCA model.
  • ...and 5 more figures

Theorems & Definitions (7)

  • Definition 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Lemma 1
  • Example 1