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An Equilibrium Model for Schedule-Based Transit Networks with Hard Vehicle Capacities

Tobias Harks, Sven Jäger, Michael Markl, Philine Schiewe

TL;DR

The paper develops a rigorous framework for schedule-based transit networks with hard vehicle capacities by defining side-constrained user equilibria and characterizing them via a quasi-variational inequality. It shows existence for fixed departure times under a weak-regularity condition and provides both an efficient algorithm for single-commodity networks and an exponential-time exact approach for multi-commodity networks with departure-time choice, supplemented by a scalable heuristic. NP-hardness results for the multi-commodity, departure-time setting justify the need for practical algorithms and heuristics. Empirical tests on real-world networks demonstrate that the heuristic achieves flows close to equilibrium and near-optimal social costs in many cases, highlighting the approach’s potential for planning and policy analysis in urban transit systems.

Abstract

Modelling passenger assignments in public transport networks is a fundamental task for city planners, especially when deliberating network infrastructure decisions. A key aspect of a realistic model is to integrate passengers' selfish routing behaviour under limited vehicle capacities. We formulate a side-constrained user equilibrium model in a schedule-based transit network, where passengers are modelled via a continuum of non-atomic agents that travel from their origin to their destination. An agent's route may comprise several rides along given lines, each using vehicles with hard loading capacities. We give a characterization of (side-constrained) user equilibria via a quasi-variational inequality and prove their existence for fixed departure times by generalizing a well-known result of Bernstein and Smith (Transp. Sci., 1994). We further derive a polynomial time algorithm for single-commodity instances with fixed departure times. For the multi-commodity case with departure time choice, we show that deciding whether an equilibrium exists is NP-hard, and we devise an exponential-time algorithm that computes an equilibrium if it exists, and signals non-existence otherwise. Using our quasi-variational characterization, we formulate a heuristic for computing multi-commodity user equilibria in practice, which is tested on multiple real-world instances. In terms of social cost, the computed user-equilibria are quite efficient compared to a system optimum.

An Equilibrium Model for Schedule-Based Transit Networks with Hard Vehicle Capacities

TL;DR

The paper develops a rigorous framework for schedule-based transit networks with hard vehicle capacities by defining side-constrained user equilibria and characterizing them via a quasi-variational inequality. It shows existence for fixed departure times under a weak-regularity condition and provides both an efficient algorithm for single-commodity networks and an exponential-time exact approach for multi-commodity networks with departure-time choice, supplemented by a scalable heuristic. NP-hardness results for the multi-commodity, departure-time setting justify the need for practical algorithms and heuristics. Empirical tests on real-world networks demonstrate that the heuristic achieves flows close to equilibrium and near-optimal social costs in many cases, highlighting the approach’s potential for planning and policy analysis in urban transit systems.

Abstract

Modelling passenger assignments in public transport networks is a fundamental task for city planners, especially when deliberating network infrastructure decisions. A key aspect of a realistic model is to integrate passengers' selfish routing behaviour under limited vehicle capacities. We formulate a side-constrained user equilibrium model in a schedule-based transit network, where passengers are modelled via a continuum of non-atomic agents that travel from their origin to their destination. An agent's route may comprise several rides along given lines, each using vehicles with hard loading capacities. We give a characterization of (side-constrained) user equilibria via a quasi-variational inequality and prove their existence for fixed departure times by generalizing a well-known result of Bernstein and Smith (Transp. Sci., 1994). We further derive a polynomial time algorithm for single-commodity instances with fixed departure times. For the multi-commodity case with departure time choice, we show that deciding whether an equilibrium exists is NP-hard, and we devise an exponential-time algorithm that computes an equilibrium if it exists, and signals non-existence otherwise. Using our quasi-variational characterization, we formulate a heuristic for computing multi-commodity user equilibria in practice, which is tested on multiple real-world instances. In terms of social cost, the computed user-equilibria are quite efficient compared to a system optimum.

Paper Structure

This paper contains 26 sections, 22 theorems, 36 equations, 13 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Comparison to the Conference Version.
  4. Related Work
  5. Side-Constrained Equilibria for Schedule-Based Transit Networks
  6. The Time-Expanded Transit Network
  7. Side-Constrained User Equilibrium
  8. Characterization, Existence, and Price of Stability
  9. Quasi-Variational Inequalities
  10. Equilibria for Discontinuous Cost Functions
  11. Fixed Departure Times
  12. Generalization of Bernstein and Smith's Existence Result
  13. Existence of Equilibria in Schedule-Based Transit Networks
  14. Price of Stability
  15. Computation of Equilibria
  16. ...and 11 more sections

Key Result

Theorem 2

A feasible flow $f^*$ is a user equilibrium if and only if it is a solution to the quasi-variational inequality eq:QVI-SCDE.

Figures (13)

  • Figure 1: Two scheduled vehicle trips in the physical network (left) and their representation in the time-expanded transit network (right).
  • Figure 2: Visualization of the time-expanded transit network from \ref{['fig:intro']}. Each station is represented by a vertical timeline. The driving and dwelling edges are coloured according to their vehicle trip.
  • Figure 3: Compact representation of the time-expanded transit network from \ref{['fig:intro-lens']}.
  • Figure 4: $Q^{\mathrm{el}}_i(\pi)$ is the volume of particles of group $i$ that are willing to travel at a cost of at most $\pi$.
  • Figure 5: An example for non-existence when considering departure time choice.
  • ...and 8 more figures

Theorems & Definitions (57)

  • Definition 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Lemma 4
  • proof
  • Definition 5
  • Definition 6
  • Remark 7
  • ...and 47 more