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On Path-based Marginal Cost of Heterogeneous Traffic Flow for General Networks

Jiachao Liu, Sean Qian

TL;DR

This work tackles the challenge of computing Path Marginal Cost (PMC) in networks with heterogeneous vehicle classes by proposing a simple, approximate decomposition of PMC into intra-class and inter-class terms, where inter-class effects are captured via conversion factors derived from multi-class DNL dynamics. It also addresses non-differentiability near capacity through sub-gradient formulations and integrates these PMC approximations into a PMC-based dynamic system-optimum traffic assignment solved with a standard MSA algorithm. Through experiments on a small corridor and a large real-world network, the method consistently improves system-wide travel costs by accounting for cross-class externalities, with lower-bound sub-gradients further reducing costs. The approach supports broader uses in heterogeneous travel-time estimation, multi-class OD estimation, control, and resilience analysis, offering a tractable pathway to more accurate and robust policy analysis in mixed-traffic networks.

Abstract

Path marginal cost (PMC) is a crucial component in solving path-based system-optimal dynamic traffic assignment (SO-DTA), dynamic origin-destination demand estimation (DODE), and network resilience analysis. However, accurately evaluating PMC in heterogeneous traffic conditions poses significant challenges. Previous studies often focus on homogeneous traffic flow of single vehicle class and do not well address the interactive effect of heterogeneous traffic flows and the resultant computational issues. This study proposes a novel but simple method for approximately evaluating PMC in complex heterogeneous traffic condition. The method decomposes PMC into intra-class and inter-class terms and uses conversion factor derived from heterogeneous link dynamics to explicitly model the intricate relationships between vehicle classes. Additionally, the method considers the non-differentiable issue that arises when mixed traffic flow approaches system optimum conditions. The proposed method is tested on a small corridor network with synthetic demand and a large-scale network with calibrated demand from real-world data. Results demonstrated that our method exhibits superior performance in solving bi-class SO-DTA problems, yielding lower total travel cost and capturing the multi-class flow competition at the system optimum state.

On Path-based Marginal Cost of Heterogeneous Traffic Flow for General Networks

TL;DR

This work tackles the challenge of computing Path Marginal Cost (PMC) in networks with heterogeneous vehicle classes by proposing a simple, approximate decomposition of PMC into intra-class and inter-class terms, where inter-class effects are captured via conversion factors derived from multi-class DNL dynamics. It also addresses non-differentiability near capacity through sub-gradient formulations and integrates these PMC approximations into a PMC-based dynamic system-optimum traffic assignment solved with a standard MSA algorithm. Through experiments on a small corridor and a large real-world network, the method consistently improves system-wide travel costs by accounting for cross-class externalities, with lower-bound sub-gradients further reducing costs. The approach supports broader uses in heterogeneous travel-time estimation, multi-class OD estimation, control, and resilience analysis, offering a tractable pathway to more accurate and robust policy analysis in mixed-traffic networks.

Abstract

Path marginal cost (PMC) is a crucial component in solving path-based system-optimal dynamic traffic assignment (SO-DTA), dynamic origin-destination demand estimation (DODE), and network resilience analysis. However, accurately evaluating PMC in heterogeneous traffic conditions poses significant challenges. Previous studies often focus on homogeneous traffic flow of single vehicle class and do not well address the interactive effect of heterogeneous traffic flows and the resultant computational issues. This study proposes a novel but simple method for approximately evaluating PMC in complex heterogeneous traffic condition. The method decomposes PMC into intra-class and inter-class terms and uses conversion factor derived from heterogeneous link dynamics to explicitly model the intricate relationships between vehicle classes. Additionally, the method considers the non-differentiable issue that arises when mixed traffic flow approaches system optimum conditions. The proposed method is tested on a small corridor network with synthetic demand and a large-scale network with calibrated demand from real-world data. Results demonstrated that our method exhibits superior performance in solving bi-class SO-DTA problems, yielding lower total travel cost and capturing the multi-class flow competition at the system optimum state.
Paper Structure (16 sections, 2 theorems, 36 equations, 13 figures, 3 tables, 1 algorithm)

This paper contains 16 sections, 2 theorems, 36 equations, 13 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Explaining PMC in Multi-class System Optimum
  4. Revisiting Multi-class Cell Transmission Model by qian2017modeling
  5. Fundamental Diagram of Two Vehicle Classes
  6. Road space splits and perceived densities
  7. Flux between cells
  8. Approximating PMC of Heterogeneous Traffic and Application in SO-DTA
  9. Decomposing PMC for bi-class traffic
  10. Intra-class terms
  11. Inter-class terms
  12. Solution Algorithm for PMC-based SO-DTA
  13. Numerical Experiments
  14. A small network
  15. A large network
  16. ...and 1 more sections

Key Result

Proposition 1

The path-based SO-DTA can be transformed into an equivalent Variational Inequality Problem (VIP), which is to find $\bf{f}^{*} = \{f^{rs,*}_{p,t,i}\}_{rs,p,t,i}\in \varOmega$ such that where the PMC can be represented in a sub-gradient form as follows without loss of generalization

Figures (13)

  • Figure 1: Fundamental diagram for the bi-class traffic stream
  • Figure 2: Single Bottleneck
  • Figure 3: Cumulative curves of two vehicle classes
  • Figure 4: Small network
  • Figure 5: Convergence of two DSO runs (with/without inter-class terms) on small network
  • ...and 8 more figures

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 2