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Transportation Network Analysis Transportation Network Analysis Volume I: Static and Dynamic Traffic Assignment Stephen D. Boyles Civil, Architectural, and Environmental Engineering The University of Texas at Austin Nicholas E. Lownes Civil and Environmental Engineering University of Connecticut Avinash Unnikrishnan Civil and Environmental Engineering Portland State University Version 0.8 (Public beta) Updated August 25, 2019 ii Preface This book is the product of ten years of teaching transportation network anal- ysis, at the The University of Texas at Austin, the University of Wyoming, the University Connecticut, and Portland State University. This version is being released as a public beta, with a final first edition hopefully to be released within a year. In particular, we aim to improve the quality of the figures, add some additional examples and exercises, and add historical and bibliographic notes to each chapter. A second volume, covering transit, freight, and logistics, is also under preparation. Any help you can offer to improve this text would be greatly appreciated, whether spotting typos, math or logic errors, inconsistent terminology, or any other suggestions about how the content can be better explained, better orga- nized, or better presented. We gratefully acknowledge the support of the National Science Foundation under Grants 1069141/1157294, 1254921, 1562109/1562291, 1636154, 1739964, and 1826230. Travis Waller (University of New South Wales) and Chris Tamp`ere (Katholieke Universiteit Leuven) hosted visits by the first author to their re- spective institutions, and provided wonderful work environments where much of this writing was done. Difficulty Scale for Exercises Inspired by Donald Knuth's The Art of Computer Programming, the exercises are marked with estimates of their difficulty. The key reference points are: 0 : A nearly trivial problem that you should be able to answer without any pencil-and-paper work. 20 : A straightforward problem that may require a few minutes of effort, but nothing too difficult if you have given the chapter a good read. 40 : A typical problem requiring some thought, or a few attempts at solution in different ways, but the answer should yield after some dedicated effort. 60 : A problem of above-average difficulty, where the correct approach is not obvious. You may require a bit of scratch paper or computer work as you try out different approaches before settling on a solution. 80 : A highly challenging or involved problem, which may be appropriate as a course project or other long-term study. 100 : An open problem in the research literature, whose solution would be a substantial advance in the understanding of transportation networks. The tens digit indicates the intellectual difficulty of the exercise, while the ones digit indicates the amount of calculation required. An exercise rated at 50 may require more cleverness and insight than one ranked at 49, but the ultimate solution is shorter. Of course, each student will find different problems more challenging than others. Contents I Preliminaries 1 1 Introduction to Transportation Networks 3 1.1 Transportation Networks . .3 1.2 Examples of Networks . .5 1.3 The Notion of Equilibrium . .6 1.4 Traffic Assignment . 11 1.5 Static Traffic Assignment . 13 1.5.1 Overview . 13 1.5.2 Critique . 15 1.6 Dynamic Traffic Assignment . 16 1.6.1 Overview . 17 1.6.2 Critique . 18 1.7 Target Audience . 20 1.8 Exercises . 20 2 Network Representations and Algorithms 23 2.1 Terminology . 23 2.2 Acyclic Networks and Trees . 26 2.3 Data Structures . 28 2.4 Shortest Paths . 31 2.4.1 Shortest paths on acyclic networks . 34 2.4.2 Shortest paths on cyclic networks: label correcting . 36 2.4.3 Shortest paths on general networks: label setting . 38 2.4.4 Shortest paths from one origin to one destination . 40 2.5 Exercises . 42 3 Mathematical Concepts 51 3.1 Basic Definitions . 51 3.1.1 Indices and summation . 52 3.1.2 Vectors and matrices . 55 3.1.3 Sets . 59 3.1.4 Functions . 63 3.2 Tools for Modeling Equilibrium . 71 3.2.1 Fixed-point problems . 72 iii iv CONTENTS 3.2.2 Variational inequalities . 73 3.3 Exercises . 77 4 Optimization 81 4.1 Introduction to Optimization . 81 4.2 Examples of Optimization Problems . 84 4.3 Convex Optimization . 91 4.4 Basic Optimization Techniques . 94 4.4.1 One-dimensional problems . 95 4.4.2 Bisection method . 97 4.4.3 Multidimensional problems with nonnegativity constraints 98 4.4.4 Constrained problems . 99 4.5 Metaheuristics (*) . 101 4.5.1 Simulated annealing . 103 4.5.2 Genetic algorithms . 109 4.6 Exercises . 114 II Static Traffic Assignment 121 5 Introduction to Static Assignment 123 5.1 Notation and Concepts . 123 5.1.1 Commentary . 127 5.2 Principle of User Equilibrium . 128 5.2.1 A trial-and-error solution method . 131 5.3 Three Motivating Examples . 134 5.4 Exercises . 141 6 The Traffic Assignment Problem 145 6.1 Mathematical Formulations . 145 6.1.1 Multifunctions and application to network equilibrium (*) 149 6.2 Properties of User Equilibrium . 152 6.2.1 Existence and link flow uniqueness . 153 6.2.2 Path flow nonuniqueness, entropy, and proportionality . 155 6.2.3 Aggregation by origins or destinations . 162 6.3 Alternative Assignment Rules . 165 6.3.1 System optimal assignment . 166 6.3.2 Bounded rationality (*) . 167 6.4 User Equilibrium and System Optimal . 170 6.4.1 Externalities . 171 6.4.2 Mathematical equivalence . 173 6.4.3 Price of anarchy . 174 6.5 Exercises . 177 CONTENTS v 7 Algorithms for Traffic Assignment 181 7.1 Introduction to Assignment Algorithms . 182 7.1.1 Why do different algorithms matter? . 182 7.1.2 Framework . 183 7.1.3 Convergence criteria . 185 7.2 Link-Based Algorithms . 186 7.2.1 Method of successive averages . 187 7.2.2 Frank-Wolfe . 193 7.2.3 Small network example . 195 7.2.4 Large network example . 197 7.2.5 Conjugate Frank-Wolfe . 199 7.3 Path-Based Algorithms . 206 7.3.1 Projected gradient . 208 7.3.2 Gradient projection . 210 7.4 Bush-Based Algorithms . 214 7.4.1 Bush labels . 217 7.4.2 Shifting flows on a bush . 221 7.4.3 Improving bushes . 228 7.5 Likely Path Flow Algorithms . 230 7.5.1 Primal method . 235 7.5.2 Dual method . 238 7.5.3 Traffic assignment by paired alternative segments . 242 7.6 Exercises . 256 8 Sensitivity Analysis and Applications 267 8.1 Sensitivity Analysis Preliminaries . 267 8.2 Calculating Sensitivities . 271 8.2.1 Changes to the OD matrix . 272 8.2.2 Changes to a link performance function . 276 8.3 Network Design Problem . 277 8.4 OD Matrix Estimation . 283 8.5 Exercises . 289 9 Extensions of Static Assignment 293 9.1 Elastic Demand . 294 9.1.1 Demand functions . 294 9.1.2 Network transformation . 296 9.1.3 Variational inequality formulation . 297 9.1.4 Optimization formulation . 297 9.1.5 Solution method . 299 9.1.6 Example . 301 9.2 Link Interactions . 301 9.2.1 Formulation . 303 9.2.2 Properties . 304 9.2.3 Algorithms . 309 9.3 Stochastic User Equilibrium . 314 vi CONTENTS 9.3.1 Discrete choice modeling . 315 9.3.2 Logit route choice . 317 9.3.3 Stochastic network loading . 324 9.3.4 Stochastic user equilibrium . 330 9.3.5 Relationships between logit loading and most likely path flows (*) . 337 9.3.6 Alternatives to logit loading . 339 9.4 Exercises . 340 III Dynamic Traffic Assignment 347 10 Network Loading 349 10.1 Link Model Concepts . 350 10.1.1 Sending and receiving flow . 351 10.1.2 Point queue . 353 10.1.3 Spatial queue . 355 10.2 Node Model Concepts . 357 10.2.1 Links in series . 360 10.2.2 Merges . 362 10.2.3 Diverges . ..
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