Terry L. Friesz

A variational inequality formulation of the dynamic network user equilibrium problem

In the present paper we are concerned with developing more realistic dynamic models of route choice and departure time decisions of transportation network users than have been proposed in the literature heretofore. We briefly review one class of models that is a dynamic generalization of the static Wardropian user equilibrium, the so-called Boston traffic equilibrium. In contrast, we then propose a new class of models that is also a dynamic generalization of the static Wardropian user equilibrium. In particular, we show for the first time that there is a variational inequality formulation of dynamic user equilibrium with simultaneous route choice and departure time decisions which, when appropriate regularity conditions hold, preserves the first in, first out queue discipline.

Dynamic network traffic assignment considered as a continuous time optimal control problem

Two continuous time formulations of the dynamic traffic assignment problem are considered, one that corresponds to system optimization and the other to a version of user optimization on a single mode network using optimal control theory. Pontryagins necessary conditions are analyzed and given economic interpretations that correspond to intuitive notions regarding dynamic system optimized and dynamic user optimized traffic flow patterns. Notably, we offer the first dynamic generalization of Beckmanns equivalent optimization problem for static user optimized traffic assignment in the form of an optimal control problem. The analysis further establishes that a constraint qualification and convexity requirements for the Hamiltonian, which together ensure that the necessary conditions are also sufficient, are satisfied under commonly encountered regularity conditions.

Sensitivity Analysis for Equilibrium Network Flow

Direct application of existing sensitivity analysis methods for nonlinear programming problems or for variational inequalities to nonlinear programming or variational inequality formulations of the equilibrium traffic assignment problem is not feasible, since, in general, the solution to the equilibrium traffic assignment problem does not satisfy the uniqueness conditions required by the sensitivity analysis methods. This paper presents an approach for sensitivity analysis of equilibrium traffic assignment problems in which an equivalent restricted problem is developed which has the desired uniqueness properties; the existing methods are applied to this restricted problem to calculate the derivatives of the equilibrium arc flows with respect to perturbations of the cost functions and of the trip table. These derivatives are then shown to be equivalent to the derivatives of the original unrestricted equilibrium traffic assignment problem; therefore, the method yields the desired sensitivity analysis results.

Day-To-Day Dynamic Network Disequilibria and Idealized Traveler Information Systems

In this paper we present tatonnement models for calculating static Wardropian user equilibria on congested networks with fully general demand and cost structures. We present both a qualitative analysis of stability and numerical studies which show that such an approach provides a reliable means for determining static user equilibria. We also describe circumstances for which these models depict day-to-day adjustments from one realizable disequilibrium state to another and how these adjustment processes differ depending on the “quality” of the information being provided by (abstract) traveler information systems. Specifically, we demonstrate that such dynamic adjustment processes settle down to equilibria both when information is complete and when it is incomplete.

A Simulated Annealing Approach to the Network Design Problem with Variational Inequality Constraints

The equilibrium network design problem can be formulated as a mathematical program with variational inequality constraints. We know this problem is nonconvex; hence, it is difficult to solve for a globally optimal solution. In this paper we propose a simulated annealing algorithm for the equilibrium network design problem. We demonstrate the ability of this algorithm to determine a globally optimal solution for two different networks. One of these describes an actual city in the midwestern United States.

Transportation network equilibrium, design and aggregation: Key developments and research opportunities

This paper attempts: to provide a review of key developments in transportation network equilibrium modeling and the related fields of network design and network aggregation; and to suggest research opportunities in the fields of network equilibrium, design and aggregation which are critical to the scientific progress of the larger discipline of transportation.

The multiobjective equilibrium network design problem revisited: A simulated annealing approach

Abstract In this paper we present a model for continuous multiobjective optimal design of a transportation network. The model presented here explicitly incorporates user equilibrium constraints and takes the form of a difficult nonlinear, nonconvex mathematical program. The user equilibrium constraints form a finite set of nonlinear, albeit nonconvex, inequalities, and give rise to a single level mathematical program, as opposed to the now standard mathematical programming/variational inequality representation which leads to a bilevel formulation of the equilibrium network design problem. We show that simulated annealing is ideally suited for solving multiobjective versions of the equilibrium network design problem articulated in this fashion. We employ the ‘weighting’ method together with simulated annealing to generate the Pareto optimal set. A numerical example is provided to demonstrate the efficacy of this solution methodology.

Hierarchical optimization: an introduction

Decision problems involving multiple agents invariably lead to conflict and gaming. In recent years, multi-agent systems have been analyzed using approaches that explicitly assign to each agent a unique objective function and set of decision variables; the system is defined by a set of common constraints that affect all agents. The decisions made by each agent in these approaches affect the decisions made by the others and their objectives. When strategies are selected simultaneously, in a noncooperative manner, solutions are defined as equilibrium points [13,51] so that at optimality no player can do better by unilaterally altering his choice. There are other types of noncooperative decision problems, though, where there is a hierarchical ordering of the agents, and one set has the authority to strongly influence the preferences of the other agents. Such situations are analyzed using a concept known as a Stackelberg strategy [13, 14,46]. The hierarchical optimization problem [11, 16, 23] conceptually extends the open-loop Stackelberg model toK players. In this paper, we provide a brief introduction and survey of recent work in the literature, and summarize the contributions of this volume. It should be noted that the survey is not meant to be exhaustive, but rather to place recent papers in context.

Prediction of intercity freight flows, I: Theory

This paper presents the conceptual framework for a predictive network equilibrium model of a freight transportation system in which the generation, distribution, modal split and assignment of freight movements are performed simultaneously. A neoclassical profit maximization model is stated for the supply-side of the transportation market, the demand-side is represented by a spatial price equilibrium model, and the economic mechanism which integrates the supply and demand submodels is described. The theoretical limitations imposed on the model by the requirement that it be capable of solving large-scale problems are also addressed.

A Sequential Shipper-Carrier Network Model for Predicting Freight Flows

A sequential shipper-carrier predictive freight network model is presented. This model is applied to three detailed network data bases. The predicted arc loadings are compared to published railway arc density codes and goodness-of-fit measures are reported for the origin-destination flows of 15 commodities. A discussion of some sources of error in the model and recommendations for further research to lessen that error are given.