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Departure time choice user equilibrium for public transport demand management

Xia Zhou, Zhenliang Ma, Mark Wallace, Daniel D. Harabor

TL;DR

This study defines the Departure Time Choice User Equilibrium for Public Transport (DTUE-PT) in multi-line, schedule-based networks with hard train capacity and develops a nonlinear mathematical program that minimizes the system gap across departure-time options. An AdaGDD algorithm with two nested loops and adaptive step-size control solves the model, while a simulation-based transit assignment captures FCFS boarding and transfers. Validation on a synthetic four-line network shows AdaGDD substantially improves solution quality over MSA and Day-to-Day learning, and sensitivity analyses confirm robustness across initial conditions. Application to the Hong Kong MTR central network demonstrates design-level gains in system costs and reveals how added route options can shift congestion patterns, highlighting policy-relevant insights for demand management and planning. The framework extends naturally to incorporate route choice and policy levers such as incentives or fares, offering a practical tool for evaluating congestion alleviation in complex PT networks.

Abstract

Departure time management is an efficient way in addressing the peak-hour crowding in public transport by reducing the temporal imbalance between service supply and travel demand. From the demand management perspective, the problem is to determine an equilibrium distribution of departure times for which no user can reduce their generalized cost by changing their departure times unilaterally. This study introduces the departure time choice user equilibrium problem in public transport (DTUE-PT) for multi-line, schedule-based networks with hard train capacity constraints. We model the DTUE-PT problem as a Non-linear Mathematical Program problem (NMP) (minimizing the system gap) with a simulation model describing the complex system dynamics and passenger interactions. We develop an efficient, adaptive gap-based descent direction (AdaGDD) solution algorithm to solve the NMP problem. We validate the methodology on a multi-line public transport network with transfers by comparing with classical public transport assignment benchmark models, including Method of Successive Average (MSA) and day-to-day learning methods. The results show that the model can achieve a system gap ratio (the solution gap relative to the ideal least cost of an origin-destination option) of 0.1926, which significantly improves the solution performance from day-to-day learning (85%) and MSA (76%) algorithms. The sensitivity analysis highlights the solution stability of AdaGDD method over initial solution settings. The potential use of DTUE-PT model is demonstrated for evaluating the network design of Hong Kong mass transit railway network and can be easily extended to incorporate the route choice.

Departure time choice user equilibrium for public transport demand management

TL;DR

This study defines the Departure Time Choice User Equilibrium for Public Transport (DTUE-PT) in multi-line, schedule-based networks with hard train capacity and develops a nonlinear mathematical program that minimizes the system gap across departure-time options. An AdaGDD algorithm with two nested loops and adaptive step-size control solves the model, while a simulation-based transit assignment captures FCFS boarding and transfers. Validation on a synthetic four-line network shows AdaGDD substantially improves solution quality over MSA and Day-to-Day learning, and sensitivity analyses confirm robustness across initial conditions. Application to the Hong Kong MTR central network demonstrates design-level gains in system costs and reveals how added route options can shift congestion patterns, highlighting policy-relevant insights for demand management and planning. The framework extends naturally to incorporate route choice and policy levers such as incentives or fares, offering a practical tool for evaluating congestion alleviation in complex PT networks.

Abstract

Departure time management is an efficient way in addressing the peak-hour crowding in public transport by reducing the temporal imbalance between service supply and travel demand. From the demand management perspective, the problem is to determine an equilibrium distribution of departure times for which no user can reduce their generalized cost by changing their departure times unilaterally. This study introduces the departure time choice user equilibrium problem in public transport (DTUE-PT) for multi-line, schedule-based networks with hard train capacity constraints. We model the DTUE-PT problem as a Non-linear Mathematical Program problem (NMP) (minimizing the system gap) with a simulation model describing the complex system dynamics and passenger interactions. We develop an efficient, adaptive gap-based descent direction (AdaGDD) solution algorithm to solve the NMP problem. We validate the methodology on a multi-line public transport network with transfers by comparing with classical public transport assignment benchmark models, including Method of Successive Average (MSA) and day-to-day learning methods. The results show that the model can achieve a system gap ratio (the solution gap relative to the ideal least cost of an origin-destination option) of 0.1926, which significantly improves the solution performance from day-to-day learning (85%) and MSA (76%) algorithms. The sensitivity analysis highlights the solution stability of AdaGDD method over initial solution settings. The potential use of DTUE-PT model is demonstrated for evaluating the network design of Hong Kong mass transit railway network and can be easily extended to incorporate the route choice.

Paper Structure

This paper contains 16 sections, 64 equations, 11 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Generalized travel costs in PT
  4. Transit assignment and simulation
  5. User equilibrium of departure time choices
  6. Solution Algorithm
  7. Case study
  8. Synthetic network
  9. Model validation
  10. Sensitivity analysis
  11. Hong Kong MTR network
  12. Model Extension (incorporating route choices)
  13. Conclusion
  14. Picking up passengers at origin nodes
  15. Dropping off passengers and picking up passengers at transfer nodes
  16. ...and 1 more sections

Figures (11)

  • Figure 1: Flow conservation in a congested PT network for evaluating arrival times (see Appendix-A)
  • Figure 2: Test Public Transport Network
  • Figure 3: Comparison of model solution algorithms: Day-to-day learning, MSA, and AdaGDD
  • Figure 4: System relative gaps (SRG) under 5 initial model solution settings: Day-to-day learning , MSA, and AdaGDD
  • Figure 5: Extracted Hong Kong transit networks: (a) previous network (before 2020), (b) current network. Reproduced from web images (https://www.checkerboardhill.com/2020/05/mtr-rail-operating-divisions/) and (https://www.mtr.com.hk/en/customer/services/system_map.html)
  • ...and 6 more figures