Properties of dynamic user equilibrium solution: existence, uniqueness, stability, and robust solution methodology

Abstract

This paper reviews studies that deal with properties of dynamic user equilibrium (DUE) solutions to understand the extent to which they have been resolved in past studies and indicate theoretical problems that remain to be addressed in future research. DUE assignment is an equilibrium-based methodology for dynamic traffic assignment. In conventional equilibriumbased static traffic assignment, the four desirable properties of solutions -existence, uniqueness, stability of solutions, and the existence of a robust solution methodology -are all guaranteed. These properties must be ensured for planners to obtain a reliable solution for equilibriumbased assignment. Meanwhile, for DUE assignment, although some of these properties have been proved, many issues regarding these properties remain unresolved. This paper reviews studies that deal with these properties to understand how far they have been investigated and discusses what is required from future research to improve the robustness of DUE assignment.

Modeling Day-to-day Flow Dynamics on Degradable Transport Network

“…FP uniqueness is an attractive character of traffic network model, it has been extensively studied in existing literatures ([22,27,48]). For the dynamic evolution process as considered in this paper, this issue can be analyzed by investigating the monotonicity of the self-map about .…”

“…Stability is an important property of a dynamical model for its applicability in practice ([1,4,7,10,22,23,25,31,48–52]). A FP is (asymptotically) stable if from any (sufficiently close) starting state the system state tends to the fixed-point as t tends to infinity.…”

The Corridor Problem with Discrete Multiple Bottlenecks

“…Although various mathematical formulations of DUE have been proposed (see, Szeto and Wong, 2011), they are considerably complex ways to analyze properties of an equilibrium because they must handle the complicated nested structure between link and path travel times in a network. Consequently, many issues remain regarding DUE properties, such as their existence, uniqueness, and stability (Iryo, 2013). Moreover, there is no algorithm that ensures convergence to a DUE solution, although many algorithms have been proposed.…”

Investigating undesired spatial and temporal boundary effects of congestion charging

“…variational inequality formulation equivalent to dynamic user equilibrium (Friesz et al 1993, Ran andBoyce 1996), system dynamics model (Jin 2011), differential variational inequality (DVI) formulation for within-day dynamic user equilibrium (DUE) traffic assignment (Friesz et al 2011), etc. A review on the dynamic traffic assignment (DTA) problem is given in Peeta and Ziliaskopoulos (2001) and Iryo (2013) further discusses some desirable and non-desirable properties of DUE solutions. Different from these DUE models which require all used paths between an origin-destination (OD) pair be identical and minimized, Szeto and Lo (2006) and Ge and Zhou (2012) respectively proposed a class of dynamic user optimal (DUO) states which allows the existence of differences (fixed or variable tolerances) in travel costs among the used paths connecting an OD pair.…”