Roger L. Tobin

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.

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.

Dynamic user optimal traffic assignment on congested multidestination networks

An equivalent continuous time optimal control problem is formulated to predict the temporal evolution of traffic flow pattern on a congested multiple origin-destination network, corresponding to a dynamic generalization of Wardropian user equilibrium. Optimality conditions are derived using the Pontryagin minimum principle and given economic interpretations, which are generalizations of similar results previously reported for single-destination networks. Analyses of sufficient conditions for optimality and of singular controls are also given. Under the steady-state assumptions, the model is shown to be a proper dynamic extension of Beckmanns mathematical programming problem for a static user equilibrium traffic assignment.

A Discrete Time, Nested Cost Operator Approach to the Dynamic Network User Equilibrium Problem

In this paper we formulate the dynamic network user equilibrium problem as a variational inequality problem in discrete time in terms of unit path cost functions. We then show how arc exit flow functions and nested cost operators can be used to calculate unit path costs given the departure time and route choices of network users. We also demonstrate that, assuming certain regularity conditions hold, a discrete time dynamic network user equilibrium is guaranteed to exist. Finally, a heuristic algorithm and numerical results are presented.

Dynamic congestion pricing models for general traffic networks

In this paper we develop two types of dynamic congestion pricing model, based on the theory of marginal cost pricing. The first model is appropriate for situations where commuters have the ability to learn the best route choices through day-to-day explorations on a network with arc capacities and travel demands that are stable from day to day. The second model is appropriate for situations where commuters optimize their routing decisions each day on a network with arc capacities and travel demands that fluctuate significantly from day to day. We show that two types of time-varying congestion tolls can be determined by solving a convex control formulation of the dynamic system optimal traffic assignment problem on a network with many origins and many destinations. We also show that the dynamic system optimal traffic assignment is an equilibrium for commuters under the tolls in both cases.

Heuristic algorithms for delivered price spatially competitive network facility location problems

We review previous formulations of models for locating a firms production facilities while simultaneously determining production levels at those facilities so as to maximize the firms profit. We enhance these formulations by adding explicit variables to represent the firms shipping activities and discuss the implications of this revised approach. In these formulations, existing firms, as well as new entrants, are assumed to act in accordance with an appropriate model of spatial equilibrium. The firm locating new production facilities is assumed to be a large manufacturer entering an industry composed of a large number of small firms. Our previously reported proof of existence of a solution to the combined location-equilibrium problem is briefly reviewed. A heuristic algorithm based on sensitivity analysis methods which presume the existence of a solution and which locally approximate price changes as linear functions of production perturbations resulting from newly established facilities is presented. We provide several numerical tests to illustrate the contrasting locational solutions which this papers revised delivered price formulation generates relative to those of previous formulations. An exact, although computationally burdensome, method is also presented and employed to check the reliability of the heuristic algorithm.