Enforcing Priority in Schedule-based User Equilibrium Transit Assignment

TL;DR

This paper addresses the enforcement of boarding priority in schedule-based transit assignment by formalizing an implicit-priority framework that encodes continuance priority and FCFS boarding via available capacity on priority arcs. It develops a rigorous NCP formulation for the original implicit-priority model, proves equilibrium existence under mild conditions, and reveals potential non-uniqueness with behaviorally unrealistic equilibria. To remedy this, the authors introduce a refined arc-level NCP (R-UEIP), reformulate it as a continuously differentiable MPEC using the Fischer–Burmeister function, and propose two solution methods (implicit differentiation-based descent and NLP-based SQP). Numerical experiments on a Hong Kong case study, Nguyen’s benchmark, and the Sioux Falls network demonstrate that the model reproduces FCFS queuing and departure-time adjustments while offering computationally tractable solution methods. The work provides a principled, DNL-free approach to incorporating priority-driven passenger behavior into schedule-based transit analysis, with potential implications for transit planning and design.

Abstract

Denied boarding in congested transit systems induces queuing delays and departure-time shifts that can reshape passenger flows. Correctly modeling these responses in transit assignment hinges on the enforcement of two priority rules: continuance priority for onboard passengers and first-come-first-served (FCFS) boarding among waiting passengers. Existing schedule-based models typically enforce these rules through explicit dynamic loading and group-level expected costs, yet discrete vehicle runs can induce nontrivial within-group cost differences that undermine behavioral consistency. We revisit the implicit-priority framework of Nguyen et al. (2001), which, by encoding boarding priority through the notion of available capacity, characterizes route and departure choices based on realized personal (rather than group-averaged) travel experiences. However, the framework lacks an explicit mathematical formulation and exact computational methods for finding equilibria. Here, we derive an equivalent nonlinear complementarity problem (NCP) formulation and establish equilibrium existence under mild conditions. We also show that multiple equilibria may exist, including behaviorally questionable ones. To rule out these artifacts, we propose a refined arc-level NCP formulation that not only corresponds to a tighter, behaviorally consistent equilibrium concept but also is more computationally tractable. We reformulate the NCP as a continuously differentiable mathematical program with equilibrium constraints (MPEC) and propose two solution algorithms. Numerical studies on benchmark instances and a Hong Kong case study demonstrate that the model reproduces continuance priority and FCFS queuing and captures departure-time shifts driven by the competition for boarding priority.

Figures (12)

• Figure 1: An example transit network.
• Figure 2: Event–activity graph of the example transit network.
• Figure 3: Illustration of the implicit priority and UEIP.
• Figure 4: Visualization of a non-UEIP flow for the example transit network.
• Figure 5: An illustrative example of unreasonable UEIP states.

Theorems & Definitions (18)

• Definition 1: Route availability
• Definition 2: User equilibrium with implicit priority
• Proposition 1
• proof
• Theorem 1
• proof
• Proposition 2
• proof
• Definition 3: Relative route availability
• Definition 4: Refined user equilibrium with implicit priority