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This book is the result of several years of research into the modelling and efficient solution of problems in transportation planning and related areas.

The aim of this book is to provide a unified account of the development of models and methods for the problem of estimating equilibrium traffic flows in urban areas, from the early days of transportation planning heuristics to today's advanced equilibrium models and methods. Also, the aim is to show the scope and—just as important—the limitations of present traffic models. The development is described and analyzed using the powerful instruments of nonlinear optimization and mathematical programming within the field of operations research. The book includes historical references as well as many recent developments, and aims to clarify the close relationships between several lines of development by placing them in a new, unifying framework.

The first part of the book is devoted to mathematical models for the analysis of transportation network equilibria. Chapter 1 describes the traditional transportation planning process of which traffic assignment is a central part. The development of traffic assignment heuristics is described. Chapter 2 analyzes the basic models of traffic assignment, based on the principles of Wardrop. Existence, uniqueness and stability results are given.

Extensions of the basic models, including non-deterministic travel cost perceptions and additional flow relationships modelled through the introduction of side constraints, are discussed. Chapter 3 analyzes traffic equilibrium models for general travel cost functions such as variational inequality, nonlinear complementarity, and fixed point problems. The recent development of optimization reformulations of asymmetric variational inequalities is accounted for in detail.

The second part of the book is devoted to methods for traffic equilibrium problems.

Chapter 4 gives a uniform description of methods for the basic traffic assignment models and their extensions discussed in Chapter 2. Important concepts, such as partial linearization, decomposition, and column generation, are described in detail for general convex programs, and are subsequently used to describe and interrelate traffic assignment methods. Chapter 5 gives the corresponding treatment of the general traffic equilibrium models described in Chapter 3, based on the concepts of cost approximation, decomposition, and column generation. Optimization reformulations of general traffic equilibrium problems are utilized to derive a new class of traffic equilibrium methods which requires mild assumptions on the models.

An appendix summarizes the definitions of the concepts most frequently used.

The scope of the material is limited to static models of traffic equilibrium; neither dynamic nor combined traffic models are dealt with in detail. The results obtained in this book can, however, be applied to the analysis and solution of such models also.

In order to economize with the space available, the reader is often directed to other works for more details. The resulting reference list is extensive—it contains more than 1,000 entries—and serves the additional purpose of being a source for anyone interested in acquiring deeper knowledge in the field.

The book will be of interest to researchers in transportation, operations research, and quantitative economics—and those entering these areas of research—who wish to extend their knowledge of equilibrium modelling and analysis, and of the foundations of efficient optimization methods adapted for the solution of large-scale models. The book can also be used in advanced graduate courses in the areas just mentioned. This book could provide the basic material for a course in transportation research. A course in structured mathematical programming, with application to traffic equilibrium problems, is defined by Chapters 2 and 4, or by Chapters 2-5, the latter including the foundations of variational inequality models and methods. A course in equilibrium modelling is defined by Chapters 2 and 3. The text assumes some familiarity with nonlinear programming theory and techniques: it would

therefore be preferable to combine material from this book with that of a modern textbook in nonlinear programming.

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