This study develops a simulation-based dynamic traffic assignment, or dynamic user equilibrium (DUE), model for dynamic road pricing applications. This proposed model is considered as the bi-criterion DUE (BDUE) model, because it explicitly considers heterogeneous users with different values of time (VOT) choose paths that minimize the two path attributes: travel time and out-of-pocket cost. This study assumed trip-makers would select their respective least generalized cost paths, the generalized cost being the sum of travel cost and travel time weighted by the trip-maker's VOT. The VOT is modeled as a continuous random variable distributed across all users in a network.

The BDUE problem is formulated as an infinite dimensional variational inequality (VI), and solved by a column generation-based algorithmic framework which embeds (i) a parametric analysis (PAM) to obtain the VOT breakpoints which determine multiple user classes, and find the set of extreme non-dominated paths, (ii) a simulator to determine experienced travel times, and (iii) a multi-class path flow equilibrating scheme to update path assignments. The idea of finding and assigning heterogeneous trips to the set of extreme non-dominated paths is based on the assumption that in the disutility minimization path choice model with convex utility functions, all trips would choose only among the set of extreme non-dominated paths. Moreover, to circumvent the difficulty of storing the grand path set and assignment results for large-scale network applications, a vehicle-based implementation technique is proposed. This BDUE model is generalized to the multi-criterion DUE (MDUE) model, in which heterogeneous users with different VOT and values of reliability (VOR) make path choices so as to minimize their path travel cost, travel time, and travel time variability.

Another important extension of the BDUE model is the multi-criterion simultaneous route and departure time user equilibrium (MSRDUE) model, which considers heterogeneous trip-makers with different VOT and values of schedule delay (VOSD) making simultaneous route and departure time choices so as to minimize their respective trip costs, defined as the sum of travel cost, travel time weighted by VOT, and schedule delay weighted by VOSD. The MSRDUE problem is also solved by the column generation-based algorithmic framework. The Sequential Parametric Analysis Method (SPAM) is developed to find the VOT and VOSD breakpoints that define multiple user classes, and determine the least trip cost alternative (a combination of departure time and path) for each user class.