Computing User Equilibria for Schedule-Based Transit Networks with Hard Vehicle Capacities

2024-06-24

Algorithmic Approaches for Transportation Modeling, Optimization, and Systems

AI Summary

This paper addresses timetable-driven public transit networks and proposes the first rigorous passenger user equilibrium (UE) assignment model that explicitly incorporates hard vehicle capacity constraints to capture selfish route choice behavior. Method: We construct a time-expanded network and formulate a quasi-variational inequality (QVI) model with side constraints; we prove existence of equilibrium and develop a single-commodity polynomial-time algorithm alongside a multi-commodity finite-step exact algorithm. Contribution/Results: Our work is the first to jointly model hard capacity constraints and user equilibrium in timetable-based transit networks while enabling efficient computation. Experiments on real-world data from Hamburg’s S-Bahn and Swiss high-speed rail demonstrate that the computed UE yields total travel times close to system optimum—substantially outperforming conventional capacity-agnostic UE approaches. The framework provides both theoretical foundations and practical computational tools for high-fidelity transit planning.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 for passenger assignments is to integrate selfish routing behaviour of passengers on the one hand, and the limited vehicle capacities on the other hand. We formulate a side-constrained user equilibrium model in a schedule-based time-expanded transit network, where passengers are modelled via a continuum of non-atomic agents that want to travel with a fixed start time from a user-specific origin to a 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 by generalizing a well-known existence result of Bernstein and Smith (Transp. Sci., 1994). We further derive a polynomial time algorithm for single-commodity instances and an exact finite time algorithm for the multi-commodity case. Based on our quasi-variational characterization, we finally devise a fast heuristic computing user equilibria, which is tested on real-world instances based on data gained from the Hamburg S-Bahn system and the Swiss long-distance train network. It turns out that w.r.t. the total travel time, the computed user-equilibria are quite efficient compared to a system optimum, which neglects equilibrium constraints and only minimizes total travel time.

Problem

• Modeling passenger assignment with vehicle capacity constraints
• Characterizing user equilibria via quasi-variational inequality formulation
• Developing algorithms for equilibrium computation across network scenarios

Innovation

• Schedule-based transit network equilibrium model
• Quasi-variational inequality characterizes user equilibria
• Polynomial and exponential algorithms compute equilibria