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Aims and ScopeOptimization has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in all areas of applied mathematics, engineering, medicine, economics and other sciences.The Springer Optimization and Its Applications series publishes undergraduate and graduate textbooks, monographs and state-of-the-art expository works that focus on algorithms for solving optimization problems and also study applications involving such problems. Some of the topics covered include nonlinear optimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control, discrete optimization, multi-objective programming, description of software packages, approximation techniques and heuristic approaches. © Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com IntroductionScience is made of facts just as a house is made of bricks, but a collection of facts is no more science than a pile of bricks is a house.Henri PoincaréThe aim of the disciplines of praxis is not theoretical knowledge. . . . It is to change the forms of action. . . . AristotleTransportation systems consist not only of the physical and organizational elements that interact with each other to produce transportation opportunities, but also of the demand that takes advantage of such opportunities to travel from one place to another. This travel demand, in turn, is the result of interactions among the various economic and social activities located in a given area. Mathematical models of transportation systems represent, for a real or hypothetical transportation system, the demand flows, the functioning of the physical and organizational elements, the interactions between them, and their effects on the external world. Mathematical models and the methods involved in their application to real, large-scale systems are thus fundamental tools for evaluating and/or designing ...

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Traditionally, traffic assignment models, both for within-day static and dynamic demand, have been formulated following an equilibrium approach in which a state ensuring internal consistency between demand (flows) and costs is sought. However, equilibrium analysis is significant under some assumptions on its “representativeness” (coincidence or closeness with the actual attractor of the system) and analytical properties, such as existence, uniqueness, and stability. Moreover, transients due to modifications of demand and/or supply cannot be simulated through equilibrium models, nor can a statistical description of the state of the system, i.e. means, modes, moments and, more generally, frequency distributions of flows over time be obtained. In this paper, interperiodic (day-to-day) dynamic modeling of transportation networks is addressed following two different approaches, namely deterministic and stochastic processes. In both cases several theoretical results are shown by making use of a formal framework covering most models discussed in the literature as well as some possible extensions. Most of the results reported can be extended to cover within-day dynamic models but these models are not explicitly dealt with. Within the framework of deterministic processes the relevance of day-to-day dynamic models for demand/supply interaction in comparison with the traditional user equilibrium approach is discussed, and conditions for coincidence of fixed-point attractors and equilibrium states are stated. Conditions for existence and uniqueness of fixed-point attractors are proposed, generalizing and extending those presented in the literature for user equilibrium. Conditions for stability of both fixed-points and equilibrium states were formulated by making use of results from non-linear dynamic system theory. Moreover, it is possible to devise a new family of “dynamic” algorithms which simulate the system convergence to a fixed-point in order to obtain an equivalent equilibrium state, as opposed to conventional “optimisation” algorithms. In this case the fixed-point stability analysis can be viewed as a convergence analysis for the algorithms specified this way. Conditions for stochastic process regularity are proposed ensuring, among other things, existence and uniqueness of a stationary probability distribution of system states. These conditions generalize and extend results presented in the literature to a wider class of possible dynamic models. Relationships between a deterministic process, together with corresponding fixed-points or equilibrium states, and stochastic probability distribution are also briefly addressed. Finally, some numerical examples confirming theoretical results are reported for a small test network.