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Queue Replacement Approach to Dynamic User Equilibrium Assignment with Route and Departure Time Choice

Takara Sakai, Takashi Akamatsu, Koki Satsukawa

TL;DR

The paper tackles dynamic user equilibrium with route and departure-time choice on general networks by introducing the Generalized Queue Replacement Principle (GQRP), which ties equilibrium queueing delays to a shadow-price solution of an auxiliary LP. A two-step LP framework—cost determination (yielding a candidate queueing pattern) followed by flow determination (verifying the GQRP and computing the equilibrium flows)—enables exact DUE solutions when the GQRP holds, and provides high-quality initialization when it does not. The approach is validated on Braess, Sioux Falls, and Eastern Massachusetts networks, achieving near-zero objective values and scalable computation times, while also illustrating a no-GQRP case where the method remains a valuable heuristic. Overall, the work offers a practical, scalable decomposition of the DUE-RDTC problem, links DUE to first-best dynamics under certain conditions, and suggests policy levers to influence the GQRP in real networks.

Abstract

This study develops a hybrid analytical and numerical approach for dynamic user equilibrium (DUE) assignment with simultaneous route and departure time choice (RDTC) for homogeneous users. The core concept of the proposed approach is the generalized queue replacement principle (GQRP), which establishes an equivalence between the equilibrium queueing-delay pattern and the solution to a linear programming (LP) problem obtained by relaxing some conditions in the original DUE-RDTC problem. We first present a method for determining whether the GQRP holds. Based on the GQRP, we then develop a systematic procedure to obtain an exact DUE solution by sequentially solving two LPs: one for the equilibrium cost pattern, including queueing delays, and the other for the corresponding equilibrium flow pattern. Computational results on networks of varying scales confirm the effectiveness of the proposed method.

Queue Replacement Approach to Dynamic User Equilibrium Assignment with Route and Departure Time Choice

TL;DR

The paper tackles dynamic user equilibrium with route and departure-time choice on general networks by introducing the Generalized Queue Replacement Principle (GQRP), which ties equilibrium queueing delays to a shadow-price solution of an auxiliary LP. A two-step LP framework—cost determination (yielding a candidate queueing pattern) followed by flow determination (verifying the GQRP and computing the equilibrium flows)—enables exact DUE solutions when the GQRP holds, and provides high-quality initialization when it does not. The approach is validated on Braess, Sioux Falls, and Eastern Massachusetts networks, achieving near-zero objective values and scalable computation times, while also illustrating a no-GQRP case where the method remains a valuable heuristic. Overall, the work offers a practical, scalable decomposition of the DUE-RDTC problem, links DUE to first-best dynamics under certain conditions, and suggests policy levers to influence the GQRP in real networks.

Abstract

This study develops a hybrid analytical and numerical approach for dynamic user equilibrium (DUE) assignment with simultaneous route and departure time choice (RDTC) for homogeneous users. The core concept of the proposed approach is the generalized queue replacement principle (GQRP), which establishes an equivalence between the equilibrium queueing-delay pattern and the solution to a linear programming (LP) problem obtained by relaxing some conditions in the original DUE-RDTC problem. We first present a method for determining whether the GQRP holds. Based on the GQRP, we then develop a systematic procedure to obtain an exact DUE solution by sequentially solving two LPs: one for the equilibrium cost pattern, including queueing delays, and the other for the corresponding equilibrium flow pattern. Computational results on networks of varying scales confirm the effectiveness of the proposed method.

Paper Structure

This paper contains 29 sections, 13 theorems, 56 equations, 18 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model
  3. Network and user
  4. Dynamic user equilibrium
  5. Queue Replacement Approach
  6. Overview of the approach
  7. Step 1: Cost Determination
  8. Step 2: Flow Determination and the GQRP Verification
  9. Cost determination step
  10. Flow determination step
  11. Generalized queue replacement principle and numerical flow determination
  12. Sufficient condition for the GQRP
  13. Summary of the Procedure
  14. Discussion
  15. Policy implications
  16. ...and 14 more sections

Key Result

Lemma 2.1

Consider a following LCP and QP: Let $\bm{x}^{\ast\ast}$ be a solution to the [QP]. The solution to the [LCP] is given by $\bm{x}^{\ast} = \bm{x}^{\ast\ast}$ and the optimal value of the [QP] is zero, i.e., $f(\bm{x}^{\ast\ast})=0$.

Figures (18)

  • Figure 1: Framework of the proposed GQRP-based approach
  • Figure 2: Many-to-one network
  • Figure 3: Schedule delay cost function
  • Figure 4: Definition of variables in the Lagrangian-like coordinate system
  • Figure 5: Flowchart of the hierarchical numerical algorithm
  • ...and 13 more figures

Theorems & Definitions (24)

  • Definition 2.1
  • Lemma 2.1: Cottle2009-xz
  • Proposition 2.1
  • Proposition 3.1
  • Proposition 3.2
  • Definition 3.1: Generalized Queue Replacement Principle
  • Lemma 3.1
  • Proof
  • Theorem 3.1
  • Corollary 3.1
  • ...and 14 more