The Traffic Assignment Problem: Models and Methods

The Traffic Assignment Problem

Models & Methods

The Traffic Assignment Problem

Models & Methods

Michael Patriksson

Department of Mathematical Sciences

Chalmers University of Technology

and Gothenburg University

Dover Publications, Inc.

Mineola, New York

Copyright

Copyright © 1994, 2015 by Michael Patriksson All rights reserved.

Bibliographical Note

This Dover edition, first published in 2015, is an unabridged republication of the work originally published in the Topics in Transportation Series by VSP Publishers, Utrecht, The Netherlands, in 1994. An errata list has been added to page xiii for the Dover edition.

Library of Congress Cataloging-in-Publication Data

Patriksson, Michael.

The traffic assignment problem: models and methods / Michael Patriksson.

p. cm.

A n unabridged republication of the work originally published in the Topics in Transportation Series by VSP Publishers, Utrecht, The Netherlands, in 1994.

Includes bibliographical references and index.

eISBN-13: 978-0-486-80227-5

1. Traffic assignment—Mathematical models. I. Title.

HE336.T68P37 2015

388.3’140151—dc23

2014033576

Manufactured in the United States by Courier Corporation

78790701         2015

www.doverpublications.com

Contents

Preface

Some notations

I Models

1Urban traffic planning

1.1Introduction

1.2The transportation planning process

1.3Organization and goal definition

1.4Base year inventory

1.5Model analysis

1.5.1Trip generation

1.5.2Trip distribution

1.5.3Modal split

1.5.4Traffic assignment

1.6Travel forecast

1.7Network evaluation

1.8Discussion

2The basic equilibrium model and extensions

2.1The Wardrop conditions

2.1.1The fixed demand case

2.1.2The variable demand case

2.1.3Discussion

2.2The mathematical program for user equilibrium

2.2.1The fixed demand case

2.2.2Network representations

2.2.3The elastic demand case

2.2.4Equivalent fixed demand reformulations

2.2.5Discussion

2.3Properties of equilibrium solutions

2.3.1Existence of equilibrium solutions

2.3.2Uniqueness of equilibrium solutions

2.3.3Further properties of equilibrium solutions

2.3.4Stability and sensitivity of equilibrium solutions

2.4User equilibrium versus system optimum

2.5Nonseparable costs and multiclass-user transportation networks

2.6Related network problem

2.6.1Traffic equilibria and network games

2.6.2Discrete traffic equilibrium models

2.6.3Traffic equilibria and electrical networks

2.6.4Spatial price equilibria

2.6.5Optimal message routing in computer communication networks

2.7Discussion

2.8Some extension

2.8.1Stochastic assignment models

2.8.2Side constrained assignment models

3General traffic equilibrium models

3.1Introduction

3.1.1Alternative definitions of equilibria

3.1.2Variational inequality problems

3.1.3Nonlinear complementarity problems

3.1.4Fixed point problems

3.1.5Mathematical programming reformulations

3.2Traffic equilibrium models

3.2.1Variational inequality models

3.2.2Nonlinear complementarity models

3.2.3Fixed point models

3.3Properties of equilibrium solutions

3.3.1Existence of equilibrium solutions

3.3.2Uniqueness of equilibrium solutions

3.3.3Further properties of equilibrium solutions

3.3.4Stability and sensitivity of equilibrium solutions

II Methods

4Algorithms for the basic model and its extensions

4.1The Frank-Wolfe algorithm and its extensions

4.1.1The Frank-Wolfe algorithm

4.1.2Termination criteria

4.1.3The use of the Frank-Wolfe approach for the solution of [TAP] .

4.1.4Shortest route algorithms

4.1.5Convergence characteristics of the Frank-Wolfe method

4.1.6Improvements and extensions

4.2Algorithm concepts

4.2.1Partial linearization algorithms

4.2.2Decomposition algorithms

4.2.3Column generation algorithms

4.2.4Discussion

4.2.5A taxonomy of algorithms for [TAP]

4.3Algorithms for the basic model

4.3.1Decomposition algorithms

4.3.2Sequential decomposition algorithms

4.3.3Parallel decomposition algorithms

4.3.4Aggregate simplicial decomposition algorithms

4.3.5Disaggregate simplicial decomposition algorithms

4.3.6Comparisons between aggregated and disaggregated representations

4.3.7Dual algorithms

4.3.8Network aggregation algorithms

4.3.9Other algorithms

4.4Algorithms for elastic demand problems

4.5Algorithms for stochastic assignment models

4.5.1Stochastic network loading

4.5.2Stochastic user equilibrium

4.6Algorithms for side constrained assignment models

4.6.1Algorithms for capacity side constrained assignment models

4.7Discussion

5Algorithms for general traffic equilibria

5.1Introduction

5.2Algorithm concepts

5.2.1Cost approximation algorithms

5.2.2Decomposition algorithms

5.2.3Column generation algorithms

5.2.4Algorithmic equivalence results

5.2.5Descent algorithms for variational inequalities

5.3Algorithms for general traffic equilibria

5.3.1Linear approximation algorithms

5.3.2Sequential decomposition algorithms

5.3.3Parallel decomposition algorithms

5.3.4Algorithms based on the primal and dual gap functions

5.3.5Column generation algorithms

5.3.6Dual algorithms

5.3.7Other algorithms

5.4Discussion

A Definitions

References

Index

Preface

This book is the result of several years of research into the modelling and efficient solution of problems in transportation planning and related areas. A previous version appeared as a long survey in my licentiate thesis ([743]) presented at the Department of Mathematics, Linköping Institute of Technology, and received a positive response from some leading researchers in the field of transportation research. Their positive criticism inspired me to further develop the survey into what has become the present book.

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.

I can envisage two main uses for this book. The first is by 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 second use is 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; I personally recommend using Bazaraa et al. [43].

A work of this type would be impossible without the help of many people. I especially thank my former tutor Prof. T. Larsson for guiding me through the optimization landscape, and for his collaboration in research upon which parts of this book is based, and Prof. A. Migdalas for introducing me to the area of transportation research. The assistance given by the library staff over the years in gathering many of the references has been invaluable. Pamela Vang helped in improving the English of the text. The book was sponsored in part by grants from the Swedish Transport and Communications Research Board (KFB), Swedish Institute, and the Royal Swedish Academy of Sciences.

Linköping, June 1994

Michael Patriksson

Some notations

Errata and comments list for The Traffic Assignment Problem—Models and Methods

Michael Patriksson

9 September, 2004

Part I

Models

Chapter 1

Urban traffic planning

1.1 Introduction

A significant amount of the activity in an urban area concerns the movement of people and goods between different locations in the transportation infrastructure, and a smooth and efficient transportation system is essential for the economic health and the quality of life within the urban region. When analyzing the present infrastructure for future investments and operating policies, a careful study of the transportation system is therefore among the most important components of the planning process.

The decades following World War II have seen an enormous increase in the demand for transportation. A vast majority of this increase is accounted for by the development of personal transport, which has its roots in the urbanization and the rising standards of living.¹ The increase of mobility has, however, also brought many serious problems into urban regions, such as pollution, increased accident rates, unwanted social effects on urban life due to highway expansion, and an inefficient use of the transportation system because of high congestion.

In transportation planning studies alterations of the existing transportation systems are evaluated with the objective of alleviating the above mentioned problems (among others), while also utilizing the full range of transport modes available.

Urban transportation planning has been an evolutionary process. Its beginnings may be traced to the home-interview studies conducted in more than 100 cities in the United States during the decade following the end of World War II. The concept of small sample interviews was then combined with cordon line surveys in order to derive patterns of urban travel. Future traffic usage of urban highway projects was predicted by manually assigning selected origin-destination (O-D) movements to the routes being planned. In the early 1950s there were studies investigating land use and traffic relationships because better estimating methods were needed in order to forecast the travel in the design year. Methods of forecasting future population and its distribution, trip generation analysis relating travel to underlying household characteristics (car ownerships, etc.), and planning for networks instead of single routes were introduced at this time. Improved procedures were facilitated by the growing use of punch card data processing systems and later by the increasing capabilities of electronic computers. The latter permitted greater sophistication in transportation planning because they permitted the examination of more alternatives. The modelling of future land-use plans and future highway and transit systems was combined with more elegant methods of evaluation. Criteria for determining if plans met community objectives (a concept itself not generally introduced until the mid-1960s) could be increasingly quantified.

The first transportation studies made concerned only highway traffic, and saw the problem as being that of providing enough capacity for the estimated future demand for personal transport. Since the 1950s, however, it has been realized that transportation is not an isolated activity; indeed, the demand for travel facilities is a function of human land use activity and, conversely, the provision of transport facilities stimulates land use activity. This development can also be seen in the Federal-Aid Highway Act of 1962, which states that federally assisted highway projects must be ... based on a continuing comprehensive transportation planning process carried on cooperatively by states and local communities .... As a result of these findings, recent transportation studies form integrated parts of the overall planning process, and the so called 3C philosophy of continuing, comprehensive, and cooperative urban transportation planning characterizes the current status of the process. Transport planners focus more on improving public transport, as an alternative to the auto mode, in order to reduce highway congestion.

The transportation system is very complex, and its performance depends on decisions made on many levels of society (the goals and purposes of which may be in conflict with each other). The process of evaluating, designing and managing such a system can therefore not be carried out without the aid of properly formulated models.

Depending on the purpose of the transportation study, models may concern different components of the transportation system (land use patterns, control policies, trip generation and distribution, etc.), different levels of aggregation of the physical reality (macroscopic or microscopic models), different planning horizons (from the use in realtime traffic management systems up to 20 year forecasts), and be based on different modelling principles (statistical models, optimization models, simulation models).

As the understanding of the transportation system has grown, together with the increase in availability of computational tools for its analysis, the planning problem has become more complex. The costs have also increased, due partly to the increase in costs for the inventory stage, and also because several more alternatives are tested.² However, viewing these costs against the scale of the plans they produce, the planning costs are less than one percent of the total ([37]).

1.2 The transportation planning process

The basis of the modelling of transportation problems is a set of assumptions, the most important ones being that travel patterns are tangible, stable, and predictable, and that the demand for transportation is directly related to the distribution and intensity of land uses, which are capable of being accurately determined for some future date ([130]).

Domencich and McFadden [264, Chap. 1] provide one list of criteria which a demand-based transportation planning model should meet in order to be a practical tool for policy analysis: it should be sensitive to transportation policy, so that the effects of policy alternatives can be forecast; it should be causal, establishing the behavioural link between the attributes of the transportation system and the decisions of the individual. This leads to the investigation of behavioural models of individual travel demand. Further, it should be flexible, allowing application to a wide variety of planning problems without major data collection and calibration costs; it should be transferable from one urban setting to another, allowing reuse without expensive reestimation in each new setting; finally, it should be efficient, in terms of providing maximum forecasting accuracy per monetary unit spent on data collection.

The traditional approach to transportation planning is to identify a number of simple submodels of the whole system, which are then analyzed separately, and most often in sequence. This transportation planning process can be divided into the following steps:

Step 1 (Organization and goal definition) The first stage of the process includes obtaining agreement on the funding, participation, and organizational form, setting up the committee structure, and arranging for staffing the study. Statements of goals and objectives of the study are also made.

Step 2 (Base year inventory) At this stage the data that may be relevant to the analysis of the transportation system is collected. It includes an inventory of existing transportation facilities and their characteristics, existing travel patterns determined through origin-destination surveys and traffic measurements, and planning factors, such as land use, income distribution, neighbourhood structure, and types of employment. It also includes the collection of historical data for trend analyses, such as population growth and car ownership.

Step 3 (Model analysis) The purpose of this phase is to establish relations among various quantities measured in Step 2,and to calibrate these relations for the base year. The relations are usually determined through the use of the following mathematical models, which are considered in sequence, and where the output from one model is input to the next.

(a) (Trip generation) This model is used to determine the number of trips originating and terminating in different zones of the study area. These numbers, which are sometimes called production and attraction numbers, are usually defined as functions of socio-economic, locational and land use characteristics of the zone in question, and are divided into different categories of purpose, such as work and recreational trips.

(b) (Trip distribution) At this step, formulas are derived to describe the allocation of trips from a point of origin to the destination zones. These formulas are typically defined as functions of the production and attraction numbers of the different zones, produced in step (a), and of the travel costs between them. In some models, traffic counts are used when determining the trip matrix.

(c) (Modal split) This model determines the portion of the total number of trips made between an origin and destination using different transport modes, the two most commonly considered being cars and public transit. The portions of trips in an origin-destination relation is normally derived from relative travel times and costs between modes, and also, in some cases, from the socio-economic and land use characteristics of the origin and destination, respectively.

(d) (Traffic assignment) In this model, the origin-destination trips are allocated to routes in the transportation network, in order to estimate the traffic volumes and travel times on the roads as functions of the network characteristics. The underlying behavioural principle in the choice of route is normally that travellers try to minimize their own travel costs.

Step 4 (Travel forecast) Based on the data collected in Step 1 and trend analyses, future land use, population distribution, etc., are predicted for a design year. The models developed and calibrated in Step 3 are then used to estimate the generation and distribution of trips on the future transportation network.

Step 5 (Network evaluation) If alternative future transportation networks and facilities are proposed, in this step costs and benefits are compared between their predicted flow patterns, in order to provide a basis for an economic evaluation of the proposed new facilities.

In order to achieve a consistent output the steps of the planning process must be repeated. Indeed, the travel costs of the future transportation network given by Step 4 influence the trip distribution, and even the projected land use and trip generation! This inconsistency problem can (at least partially) be alleviated by considering parts of the process simultaneously. Recent research efforts are being made in this direction.

In the sequel, we shall study the different parts of the transportation planning process in more detail, and outline the most common methods employed for their solution. We will here concentrate on the models and methods developed within transportation planning studies, and describe those developed through academic research in subsequent parts of the book.

1.3 Organization and goal definition

It is important for the result of the transportation study to establish goals and objectives early in the process, since these will guide the evaluations towards conformity with the desire of the community ([893, 130]). Traditionally, as already mentioned, the main objective of the transportation study has been to evaluate alternative highway constructions for increased personal transport capacity ([862]). Other goals considered have also mainly been orientated toward traffic functional aspects, such as an increased safety, a saving of travel time, a reduction of operating costs, and an increase in efficiency and mobility. It is only during the last 25 years that environmental aspects and the transit alternative have been considered essential elements of the transportation study. See [984, 37] for a more detailed description of the goal setting.

The topology of the study area, the population distribution and many other socioeconomic factors vary from study to study. The form of the study may therefore differ significantly among different countries and regions.

Studies may be of long-range type, in which case the most important questions to be answered deals with the density and configuration of the future transportation system. Short-term plans may include immediate-action programs for arterial improvements. The scale of the study may also differ; some plans include proposals for new facilities, such as parking, terminals, and transit lines, while others may describe highway locations with ramp connections pinpointed, or only deal with single corridors.

The personnel organization of the study can also have several different forms. The Transportation and Traffic Engineering Handbook [37, pp. 517-518] lists the following alternatives: A centralized state staff may be an existing agency or a new department incorporating the necessary multidisciplinary talents. The Chicago Area Transportation Study [170], established as an ad hoc joint effort and responsive to a multiagency board, illustrates the use of a semi-independent organization. A council of governments is a study organization which may be created under a council made up of elected representatives of communities within the region. Established planning bodies for metropolitan regions are sometimes the organizations housing the transportation planning staff. In a contract study organization consultants under the supervision and monitoring of either a state representative or local study director perform all or some of the stages in the planning process. The procedure has been used extensively in the U.S. ([882]).

Regardless of the organization structure, an additional organization must be appointed to ensure that the activity of the planning staff agrees with the goals and objectives set up ([489]). This organization could comprise of the following committees ([37, pp. 518-520]):

The policy committee includes representatives of agencies participating financially in the study, as well as officials and executives of local and regional planning organizations. The function of this committee is to provide budget control, establish regulations for study personnel, supervise technical matters, establish objectives, assist in the plan development, and recommend a final plan ([692]). The technical committee includes technical personnel from agencies represented on the policy committee, and sometimes also from other local agencies. The function of this committee is to review and evaluate study methods, assist in developing alternative plans, perform technical evaluations, coordinate technical service contributions of participating agencies, and enlist the interest of local agencies in the planning process. The composition and function of the citizens advisory committee vary with the size of the study area, and the interest in the study objectives of the communications media. The committee provides the policy committee with information on public thinking, and can thus assist in the definition of planning goals and objectives, improve public understanding of the planning process, and build support for plan implementation.

1.4 Base year inventory

The inventory stage can be divided into four categories:

(1) (Transportation facilities) Here, the study area is defined, and divided into sectors, districts and zones. The physical network is represented by a graph, with streets and road sections represented by links (or arcs), and intersections and trip origins and destinations by nodes. The boundary of the study area, referred to as the external cordon, is chosen to approximate the commuter-shed associated with the urban centre. The zones represent aggregates of trips and socio-economic conditions; the choice of zones is very important, since the number of zones determines the complexity of the study, and the wrong choice of zone size and distribution would obscure a lot of the information in the data collected ([130, 264, 805]).³ The number of zones ranges typically from 10 to 1000, and their sizes from a few blocks to several square kilometers. Whenever zones are small, their locations may be defined by single points in the network description, the so called zone centroid nodes.

Next, the characteristics of the existing transportation network are colleted; data includes measurements of traffic flows, speeds, travel times (or delays), link lengths, capacities, and the quality of transit service. There are many techniques for measuring these performance characteristics; some of the data required is recorded automatically by many traffic control systems, other information can be obtained from census data ([130]).

(2) (Travel patterns) Data relating to the present-day movement between zones is collected at this stage. Traffic pattern data is required for all combinations of external and internal movements. The data may be divided into trip mode and purpose. The goal of this data collecting is to estimate the number of trips made between zones within the study area, and the number of trips passing through, into or out of the area.

Movements through the area and external-internal movements are surveyed at the external cordon, and possibly at an internal cordon or screen line; this is done by manual or automatic counts. Internal-external movements are surveyed in the home-interview study and at the external cordon, while internal movements are surveyed by home-interview studies and, sometimes in addition to check, by an internal cordon or screen line survey.

The size of the sample to be interviewed depends on the total population of the area, the degree of accuracy required, and sometimes on the density of the population. The recommended sample sizes for home interviews are between 4 and 25 percent of the total population ([130]). For roadside interviews, the sample can be based on time or volume clustering, or could vary among classes of vehicles ([52, Chap. 4]).

In home-interviews, the information gathered includes address and size of household, job information, income, number of vehicles, and information about all journeys made in a previous time period, usually 24 hours. The interview procedure is outlined in Behr [52, Chap. 4]. Additional information is collected by interviews at commercial premises. For further reading, see, e.g., [467].

Roadside interviews are made on the external cordon to cover trips passing through or into the area. The questions asked depend on the purpose of the study, and the type of vehicle. Alternatives to direct interviews which delay travellers, are to ask drivers to complete and return prepaid postcards, to record registration numbers, or to place coded tags on the vehicles; see [52, Chap. 4] for further details.

More economical methods for estimating the existing origin-destination flow pattern are made possible by the automatic counts provided by many traffic control and signal setting systems, and by optimization models, with which possible origin-destination flow matrices may be derived from the counts (see, e.g., [999] and the references cited therein).

(3) (Economic activity and population) This information, together with that of land use, form the basis for developing relationships between the movements of goods and people and the distribution and intensity of land use. Data typically collected includes: historic population patterns (past distribution, migrations, density, and trends in growth), present population (distribution by area, density, average income, car ownership), employment trends and present employment, economic activity (patterns of investments in manufacturing, services, redevelopment, and other real estate), and transportation resources (outlays for regional transportation facilities).

(4) (Land use) The inclusion of land use studies into the transportation planning process was made during the 1950s; prior to this, the future demand for transportation was extrapolated, using simple growth factors, from counts on existing flows. Mitchell and Rapkin [681], however, demonstrated the close relationship between traffic flows and land use, and subsequent U.S. studies were orientated more towards the influence of land use activity on the generation of flows. While much of the attention still was focused on the traffic functional aspects, the consideration of future land uses was gradually incorporated during the 1960s. In Great Britain, a similar development can be traced, following the Traffic in Towns Report [133].

Typical data collected include: historic development trends such as patterns of urbanization, topography and physical constraints on development, classified measures of acres of land vacant or in urban use, location of major travel generators, identification of social neighbourhood and community boundaries, nature of existing land use controls, and identification of redevelopment areas ([650, 93]).

This part of the planning process has historically been the most costly one; as much as 49 percent of the total cost has been reported ([496]). With more advanced techniques for forecasting, and as knowledge of the urban transportation system improves, the availability of data from external sources or from automatic measurements increases and more alternatives are evaluated, and thus, the portion of the total cost decreases ([37, p. 521]).

1.5 Model analysis

In this step, relationships are sought between the land use and traffic characteristics of the present-day situation. These relationships are then used to estimate the future traffic situation, given the future estimated land use and proposed network facilities. An underlying assumption in this process is, of course, that these relationships will not alter significantly in the future.

These relationships are usually derived and calibrated through considering a sequence of models, rather than by a single analysis. The basic qualities of these models and their solution are outlined below.

1.5.1 Trip generation

The purpose of the trip generation step is to estimate the number of trips (typically per day) that originates or terminates in each of the zones previously defined, as a function of land use, socio-economic and locational characteristics of the zones. The most important dependent variables used are trip purpose, family income, vehicle ownership, land use activity at the zones defining the trip origin and destination, length and mode of trip, and time of day (see, e.g., [650, 837, 130, 805]).

The first transportation studies employed simple growth rates to estimate future trip generation ([249]). Subsequent studies analyzed the correlation between the above mentioned variables, using multiple regression analysis ([668, 265, 266, 130, 805]).

If it is assumed that trip generation characteristics remain stable with time, then future estimates can be made using the regression equations obtained. However, some modifications are usually necessary to reflect the estimated future conditions.

There are several sources of error in this use of regression analysis ([130, 264, 805]), and the underlying assumptions of independence and continuity among the variables in the regression analysis are not entirely correct. Because of the difficulties associated with the regression technique, household based disaggregate models, usually referred to as category analyses ([993, 754]), have been developed and used. The underlying assumption here is that the household is the fundamental unit in the trip generation process, and that the journeys generated depend on household characteristics and location. The main advantage of this method of analysis is that household categories may be estimated from census data using known relationships, such as distributions of income, car ownership and family structure; large scale home interviews can thus be avoided, resulting in a large saving compared to the regression approach. Furthermore, the analysis is computationally cheaper, and the disaggregated information may reflect individual behaviour more realistically than the zonal aggregated information. A disadvantage of this technique is that the distributions used may not be valid in the future planning period.

Domencich and McFadden [264, Chap. 2] argue that since transportation facilities do not enter the trip generation step trip frequency is independent of changes in the transportation system, making the trip generation both non-behavioural and non-causal, and also non-policy orientated.

For further reading on trip generation models and methods, see [922, 265, 93, 266].

1.5.2 Trip distribution

The purpose of this model is to estimate the number of trips performed from an origin zone to a destination zone, given aggregated trip numbers from the previous step.

The traditional techniques used for estimating the future origin-destination (O-D) flows can be divided into two categories: growth factor (or analogy) methods, and inter-area travel (or synthetic) methods.

Growth factor methods

The philosophy behind growth factor methods is that present travel patterns may be projected into the future on the basis of zonal growth rates, which may be obtained from the productions and attractions assessed in the previous stage; the future O-D flows are calculated by simply multiplying the present-day pattern by the growth rates.

, from zone p P to zone q Q, (p, q) C, through the general formula

, generated by zone p, attracted by zone q, defines the growth factors for the O-D pairs. This factor may be a single factor, or a combination of several factors, and it may be the same for all O-D pairs, or vary with the zone. The above formula may give results that are inconsistent with the estimated trip totals, dest; in this case, an iterative procedure must be adopted,