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Optimization Models and Methods for Equilibrium Traffic Assignment

Authors: Dr. Alexander Krylatov, Prof. Dr. Victor Zakharov, Prof. Tero Tuovinen

Publisher: Springer International Publishing

Book Series : Springer Tracts on Transportation and Traffic

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book is focused on the discussion of the traffic assignment problem, the mathematical and practical meaning of variables, functions and basic principles. This work gives information about new approaches, methods and algorithms based on original methodological technique, developed by authors in their publications for the past several years, as well as corresponding prospective implementations. The book may be of interest to a wide range of readers, such as civil engineering students, traffic engineers, developers of traffic assignment algorithms etc. The obtained results here are to be used in both practice and theory. This book is devoted to the traffic assignment problem, formulated in a form of nonlinear optimization program. The most efficient solution algorithms related to the problem are based on its structural features and practical meaning rather than on standard nonlinear optimization techniques or approaches. The authors have carefully considered the meaning of the traffic assignment problem for efficient algorithms development.

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Table of Contents

Frontmatter

Traffic Flow Issues

Frontmatter

Chapter 1. Introduction

Abstract

A road network is a large scale system with huge amounts of interdependent elements such as streets, junctions, traffic lights, traffic flows, and the like. Deep understanding of the nature of existing dependencies between the elements of the road network could offer decision makers and managers of different levels significant support in the transportation sphere. Thus, the Introduction chapter is devoted to a brief review on traffic theory development, and it illuminates important research directions in the field to be taken under additional careful investigation. The major attention is paid to problems concerning traffic assignment. The discussion tends to cover both practical and theoretical aspects. Essentially, the Introduction is committed to specify a general line of the book in a short way so as possible.

Alexander Krylatov, Victor Zakharov, Tero Tuovinen

Optimization Traffic Assignment Models

Frontmatter

Chapter 2. Principles of Wardrop for Traffic Assignment in a Road Network

Abstract

In this chapter is devoted to user equilibrium and system optimum of Wardrop. Discussion on the mathematical formulation of traffic assignment problems with regard to their meaning is available in the Sect. 2.1. The specification of necessary basic statements completes this discussion further. The dual traffic assignment problem with travel times between all origins and destinations as dual variables is considered in the Sect. 2.2. The practical significance of such dual formulation is shown to become evident due to the wide spread of online traffic services. The route-flow assignment problem and link-flow assignment problem are reduced to fixed-point problems with explicit operators in the Sect. 2.3 and Sect. 2.4 respectively. Proofs of corresponding theorems are fully given.

Alexander Krylatov, Victor Zakharov, Tero Tuovinen

Chapter 3. Nash Equilibrium in a Road Network with Many Groups of Users

Abstract

In this chapter concentrates on the relationships between individual and group behaviour of drivers in a road network. Such relationships are established by comparing the optimal routing of drivers (system optimum of Wardrop), the competitive drivers’ groups routing (Nash equilibrium), and the selfish drivers routing (user equilibrium of Wardrop). Thus, the boundary conditions for traffic assignment in a road network were recently obtained for the first time. Wide analytical discussion on the topic as well as a survey of relevant references are presented. Moreover, a new behavioural model of traffic assignment in case of simultaneous selfish and group behaviour of drivers in a road network is formulated in the last section. An explicit solution to a behavioural traffic assignment problem is offered for a single-commodity linear network with non-interfering routes.

Alexander Krylatov, Victor Zakharov, Tero Tuovinen

Optimization Traffic Assignment Methods

Frontmatter

Chapter 4. Methods for Traffic Flow Assignment in Road Networks

Abstract

In this chapter is devoted to approaches for solving traffic flow assignment problems. The most popular gradient descent method for solving traffic assignment problems is discussed in the first section. New projection algorithms based on the obtained, explicitly fixed-point operators for the route-flow assignment problem and link-route assignment problem are presented in the third and fourth sections respectively. Obtained operators is proved to be contractive that leads to the linear convergence of provided algorithms. Moreover, under some fairly natural conditions the algorithms converge quadratically. The technique for representing a linear route-flow assignment problem in the form of a system of linear equations is presented in the fourth section. A simple example demonstrates the evident usability of the developed technique for its implementation and further extensions.

Alexander Krylatov, Victor Zakharov, Tero Tuovinen

Chapter 5. Parallel Decomposition of a Road Network

Abstract

In this chapter a new road network decomposition approach is proposed. Necessary statements are proved in the first section, and the formal description of the approach as well as a brief review on relevant references are presented. The offered approach is completely based on the very features of the equilibrium traffic assignment and, consequently, forms the methodological basis for direct solving algorithms. The implementation of the developed approach to a single-commodity network can be found in the second section. A new decomposition technique for route-flow assignment in a general topology network is presented in the third section. The fourth section is reserved for a decomposition technique for link-flow assignment in a general topology network. Moreover, capabilities for route-flows aggregation and link-flows disaggregation which appeared by virtue of the new approach are discussed.

Alexander Krylatov, Victor Zakharov, Tero Tuovinen

Optimization Models and Methods for Network Design

Frontmatter

Chapter 6. Topology Optimization of Road Networks

Abstract

In this chapter is mainly devoted to the capacity allocation problem as one of the most significant for road network topology optimization. A brief review on problems concerning network design and relevant fields is given in the Sect. 6.1. Capacity allocation control for a general topology network in the form of a mathematical problem is formulated in the Sect. 6.2. The Sect. 6.3 is devoted to solving the capacity allocation problem for a single-commodity linear network of non-interfering routes. The solution is obtained explicitly that allows to make practically substantial conclusions. The Sect. 6.4 addresses the problem of optimal capacity allocation control under multi-modal traffic flows. The multi-modality influence on optimal control strategy for capacity allocation is also discussed.

Alexander Krylatov, Victor Zakharov, Tero Tuovinen

Chapter 7. Optimal Transit Network Design

Abstract

In this chapter, the optimal transit network design is under investigation. A special kind of transport is assumed to be given the best traffic conditions in a road network (the smallest amount of travel time between origins and destinations). Optimality criteria for transit network design in case of selfish routing are defined in the first section. The Sect. 7.2 is devoted to the estimation of selfish traffic assignment in a network with a transit subnetwork. Optimality criteria for transit network design in case of competitive drivers’ groups routing are defined in the Sect. 7.3. The Sect. 7.4 is devoted to traffic assignment in case of competitive drivers’ groups routing in a network with a transit subnetwork. Explicit illustrative solutions are presented for a single-commodity road network with non-interfering routes.

Alexander Krylatov, Victor Zakharov, Tero Tuovinen

Networking Issues

Frontmatter

Chapter 8. Transportation Processes Modelling in Congested Road Networks

Abstract

In this chapter, the models of different transportation processes in a congested road network are considered. The first section is devoted to a signal control problem formulated as a bi-level optimization program. An analytical solution for a two-commodity linear road network offers a practical and illustrative result to be taken into consideration by decision-makers in this sphere. A new algorithm for OD-matrix estimation based on the dual traffic assignment problem is described in the second section. The third section is devoted to the problem of emission reduction. The approaches presented in this book are shown to be well-implemented for coping with such problems. The time-depended vehicle routing problem in a congested road network is considered in the last section.

Alexander Krylatov, Victor Zakharov, Tero Tuovinen

Chapter 9. Load Flow Estimation in a Transmission Network

Abstract

This chapter includes an investigation on a power smart grid with multiple suppliers and consumers. The first section is devoted to the corresponding model description. The power load flow estimation is presented in a form equivalent to a link-flow traffic assignment problem. The second section concentrates on the competition of consumers, while the third section concentrates on the cooperation of suppliers. From mathematical perspectives, both defined models of economic interaction imply bi-level optimization problems. Thus efficient computational algorithms are required. The last section is devoted to pricing mechanisms for transmission networks with multiple suppliers and consumers. Possible techniques for the arising complex problem of transmission cost sharing are proposed.

Alexander Krylatov, Victor Zakharov, Tero Tuovinen

Title: Optimization Models and Methods for Equilibrium Traffic Assignment
Authors: Dr. Alexander Krylatov
Prof. Dr. Victor Zakharov
Prof. Tero Tuovinen
Copyright Year: 2020
Publisher: Springer International Publishing
Electronic ISBN: 978-3-030-34102-2
Print ISBN: 978-3-030-34101-5
DOI: https://doi.org/10.1007/978-3-030-34102-2

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