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An Equilibrium Dynamic Traffic Assignment Model with Linear Programming Formulation

Victoria Guseva, Ilya Sklonin, Irina Podlipnova, Demyan Yarmoshik, Alexander Gasnikov

Abstract

In this paper, we consider a dynamic equilibrium transportation problem. There is a fixed number of cars moving from origin to destination areas. Preferences for arrival times are expressed as a cost of arriving before or after the preferred time at the destination. Each driver aims to minimize the time spent during the trip, making the time spent a measure of cost. The chosen routes and departure times impact the network loading. The goal is to find an equilibrium distribution across departure times and routes. For a relatively simplified transportation model we show that an equilibrium traffic distribution can be found as a solution to a linear program. In earlier works linear programming formulations were only obtained for social optimum dynamic traffic assignment problems. We also discuss algorithmic approaches for solving the equilibrium problem using time-expanded networks.

An Equilibrium Dynamic Traffic Assignment Model with Linear Programming Formulation

Abstract

In this paper, we consider a dynamic equilibrium transportation problem. There is a fixed number of cars moving from origin to destination areas. Preferences for arrival times are expressed as a cost of arriving before or after the preferred time at the destination. Each driver aims to minimize the time spent during the trip, making the time spent a measure of cost. The chosen routes and departure times impact the network loading. The goal is to find an equilibrium distribution across departure times and routes. For a relatively simplified transportation model we show that an equilibrium traffic distribution can be found as a solution to a linear program. In earlier works linear programming formulations were only obtained for social optimum dynamic traffic assignment problems. We also discuss algorithmic approaches for solving the equilibrium problem using time-expanded networks.
Paper Structure (14 sections, 2 theorems, 20 equations, 3 figures)

This paper contains 14 sections, 2 theorems, 20 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Paper Structure
  4. Problem Statement in Continuous Time
  5. Transportation Model
  6. Toy Example
  7. Mathematical Formulation in Discrete Time
  8. Mathematical Formulations
  9. Algorithmic Approaches
  10. Link-Based
  11. Column Generation
  12. Dual Subgradient Methods
  13. Conclusion
  14. Acknowledgments.

Key Result

lemma thmcounterlemma

Any solution to VI eq:vi is an equilibrium assignment eq:compl-eq:demand

Figures (3)

  • Figure 1: Scheme of processing junctions. a) Schematic picture of a junction with indicated allowed directions of movement. b) Graph representation of the junction. c) Processed graph. Nodes inside the dashed circle are artificial nodes which correspond to node $v_2$. Blue links belong to subset $E_J$, while black links belong to subset $E_R$.
  • Figure 2: Let consider some part of a city divided on zones. For example, zones $i \in O, j \in D$ are showed in the figure. There are several paths $p_{ij} \in P_{ij}$ between these nodes, where $P_{ij}$ is a set of all possible paths with origin $i$ and destination $j$.
  • Figure 3: Example of time-expanded graph. Node $i$ is a source node with different departure times. Node $j'$ is a destination zone witth different arrival times. Node $j$ shows aimed arrival time. Edges between $i$ and $j'$ are physical edges and have some travel costs, however edges between $j'$ and $j$ does not exists in reality and used to introduce arrival costs.

Theorems & Definitions (3)

  • lemma thmcounterlemma
  • proof
  • theorem thmcountertheorem