Traffic Network Equilibrium Problems with Capacity Constraints of Arcs and Linear Scalarization Methods

X Q Tian,Y D Xu

doi:10.1155/2012/612142

Abstract

Traffic network equilibrium problems with capacity constraints of arcs are studied. A (weak) vector equilibrium principle with vector-valued cost functions, which are different from the ones in the work of Lin (2010), and three kinds of parametric equilibrium flows are introduced. Some necessary and sufficient conditions for a (weak) vector equilibrium flow to be a parametric equilibrium flow are derived. Relationships between a parametric equilibrium flow and a solution of a scalar variational inequality problem are also discussed. Some examples are given to illustrate our results.

Highlights

The earliest traffic network equilibrium model was proposed by Wardrop 1 for a transportation network
A vector equilibrium should be sought based on the principle that the flow of traffic along a path joining an O-D pair is positive only if the vector cost of this path is the minimum possible among all the paths joining the same O-D pair
We introduce a weak vector equilibrium principle with vector-valued cost functions, which are more reasonable from practical point of view than the ones in 18, 19

Summary

Introduction

The earliest traffic network equilibrium model was proposed by Wardrop 1 for a transportation network. H2 2 > 0 and path 1 is nonsaturated path of h∗ It follows from Definition 3.3 that the flow h∗ is not in φ-parametric equilibrium. If the traffic network equilibrium problem with capacity constraints of arcs, the converse of Theorem 3.13 may not hold. The following theorem shows that the converse of Theorem 3.13 is valid when the traffic equilibrium problem with capacity constraints of paths. Let. K : ⎩h | λ ≤ h ≤ μ, hp dw, ∀w ∈ W⎭, 3.27 p∈Pw be the feasible set of traffic network equilibrium problem with capacity constraints of paths, where λ λ1, λ2, . The φ-parametric equilibrium principle of traffic equilibrium problem with capacity constraints of paths is as follows. A path h ∈ K is in φ-parametric equilibrium if and only if the flow h solves the following scalar variational inequality:. It follows from the definition of the φ-parametric equilibrium flow that φ Tu h ≥ φ Tv h , ∀u ∈ Bw, v ∈ Aw

Tu h

Conclusions