Byung-Wook Wie

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.

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.

The existence, uniqueness and computation of an arc-based dynamic network user equilibrium formulation

Pad devices disposed between a lift sling and a load and permitting relative sliding motion between the load and the sling without the load and sling being in direct engagement and thereby avoid and/or minimize damage to the sling and to the load.

The Augmented Lagrangian Method for Solving Dynamic Network Traffic Assignment Models in Discrete Time

We develop and test an augmented Lagrangian method for solving dynamic traffic assignment models formulated as optimal control problems. Our presentation is in terms of the discrete time, system optimal traffic assignment problem. However, the basic ideas presented here are readily applied to continuous time models and to other behavioral assumptions regarding traffic assignment which may be expressed as optimal control problems. The proposed algorithm obviates the need for path enumeration and exploits the natural decomposition of the traffic assignment problem by time period which is possible when an optimal control formulation is employed.

A comparison of system optimum and user equilibrium dynamic traffic assignments with schedule delays

Abstract One way to estimate the potential benefits of new traffic control and management systems is to compare the total cost incurred in equilibrium with the system optimized total cost. To do this, we formulate the dynamic traffic assignment models with schedule delays under the system optimum and user equilibrium principles and solve them using numerical methods. System optimum and user equilibrium dynamic assignments on an 18-arc test network are then compared in terms of total travel times and schedule delays at different levels of traffic congestion. This comparison provides important implications for the success of the intelligent transportation systems (ITS) in reducing traffic congestion.

A differential game approach to the dynamic mixed behavior traffic network equilibrium problem

Abstract The dynamic mixed behavior traffic network equilibrium model is formulated as a noncooperative N -person nonzero-sum differential game under the open-loop information structure. A simple network is considered where one origin-destination pair is connected by parallel arcs and two types of players - User Equilibrium (UE) and Cournot-Nash (C-N) - interact through the congestion phenomenon. Each of UE and C-N players attempts to achieve its own prescribed objective by making a continuum of simultaneous decisions of departure time, route, and departure flow rate over a fixed time interval. The necessary and sufficient conditions are derived and given economic interpretation as a dynamic game theoretic generalization of the mixed behavior traffic network equilibrium principle which requires equilibration of average costs for UE players and equilibration of marginal costs for C-N players. An approximate iterative algorithm is proposed for solving the model in discrete time, which makes use of the augmented Lagrangian method and the gradient method. A numerical example is presented and future extensions of the model and the algorithm are also discussed.

A differential game model of Nash equilibrium on a congested traffic network

This paper considers the problem of the competition among a finite number of players who must transport the fixed volume of traffic on a simple network over a prescribed planning horizon. Each player attempts to minimize his total transportation cost by making simultaneous decisions of departure time, route, and flow rate over time. The problem is modeled as a N-person nonzero-sum differential game. Two solution concepts are applied: [1] the open-loop Nash equilibrium solution and [2] the feedback Nash equilibrium solution. Optimality conditions are derived and given an economic interpretation as a dynamic game theoretic generalization of Wardrops second principle. Future extensions of the model are also discussed. The model promises potential applications to Intelligent Vehicle Highway Systems (IVHS) and air traffic control systems.