A New Class of Instantaneous Dynamic User-Optimal Traffic Assignment Models

Abstract

The instantaneous dynamic user-optimal (DUO) traffic assignment problem is to determine vehicle flows on each link at each instant of time resulting from drivers using instantaneous minimal-time routes. Instantaneous route time is the travel time incurred if traffic conditions remain unchanged while driving along the route. In this paper, we introduce a different definition of an instantaneous DUO state. Using the optimal control theory approach, we formulate two new DUO traffic assignment models for a congested transportation network. These models include new formulations of the objective function and flow propagation constraints, and are dynamic generalizations of the static user-optimal model. The equivalence of the solutions of the two optimal control programs with DUO traffic flows is demonstrated by proving the equivalence of the first-order necessary conditions of the two programs with the instantaneous DUO conditions. Since these optimal control problems are convex programs with linear constraints, they have unique solutions. A numerical example is presented indicating that this class of models yields realistic results.

User-equilibrium properties of fixed points in dynamic traffic assignment

“…This definition takes into account the actual path durations experienced by vehicles which enter the links of the path at various times, like the simultaneous route-departure equilibrium of Friesz et al (1993), rather than the instantaneous equilibrium of Ran et al (1993) which only considers path durations consisting of the link travel times prevailing at the time the trip is begun.…”

“…The model is typically given variational constraints and/or an objective function intended to force the trip durations to satisfy user equilibrium. Models proposed in the literature include Janson (1991), Ghali and Smith (1995), and Wie et al (1995) in discrete time and Friesz et al (1993), Ran et al (1993), Smith and Wisten (1995), and Lam and Huang (1995) in continuous time. However, these models suffer from restrictive definitions of equilibrium and conditions on traffic flow dynamics and/or failure to establish existence of equilibrium solutions.…”

Some general results on the optimal ramp control problem

“…This theory has been applied to the ramp control problem (Papageorgiou, 1983), and has recently gained popularity in dynamic network assignment (e.g. Ran et al, 1993). In this section, we present the standard form of the discrete optimal control problem, and state its necessary conditions for optimality, which will be used to analyze the ramp control problem.…”

Discrete-time dynamic traffic assignment models with periodic planning horizon: system optimum

“…The First-In-First-Out (FIFO) condition requires that cars entering an arc at time t must leave the arc before those entering after time t. In the literature, many (see, e.g., [35], [47], and [34]) assume that the travel cost function satisfied certain conditions to ensure FIFO. To avoid making additional assumptions, we ensure FIFO by adding the following constraints to DTDTA instead.…”