Springer handbook of transportation science 2003 ISBN1402072465

2 DISCRETE CHOICE MODELSWITH APPLICATIONS TO DEPARTURE TIME AND ROUTE We then present the alternative discrete choice model forms such as Logit, NestedLogit, Generalized Extreme Value an

TRANSPORTATION SCIENCE

Second Edition

Weyant, J / ENERGY AND ENVIRONMENTAL POLICY MODELING

Shanthikumar, J.G & Sumita, U / APPLIED PROBABILITY AND STOCHASTIC PROCESSES Liu, B & Esogbue, A.O / DECISION CRITERIA AND OPTIMAL INVENTORY PROCESSES Gal, T., Stewart, T.J., Hanne, T / MULTICRITERIA DECISION MAKING: Advances in

MCDM Models, Algorithms, Theory, and Applications

Fox, B.L / STRATEGIES FOR QUASI-MONTE CARLO

Hall, R.W./ HANDBOOK OF TRANSPORTATION SCIENCE

Grassman, W.K / COMPUTATIONAL PROBABILITY

Pomerol, J-C & Barba-Romero, S / MULTICRITERION DECISION IN MANAGEMENT

Axsäter, S / INVENTORY CONTROL

Wolkowicz, H., Saigal, R., & Vandenberghe, L / HANDBOOK OF SEMI-DEFINITE

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Hobbs, B.F & Meier, P / ENERGY DECISIONS AND THE ENVIRONMENT: A Guide

to the Use of Multicriteria Methods

Dar-El, E / HUMAN LEARNING: From Learning Curves to Learning Organizations

Armstrong, J.S / PRINCIPLES OF FORECASTING: A Handbook for Researchers and

Practitioners

Balsamo, S., Personé, V., & Onvural, R./ ANALYSIS OF QUEUEING NETWORKS WITH

BLOCKING

Bouyssou, D et al / EVALUATION AND DECISION MODELS: A Critical Perspective

Hanne, T / INTELLIGENT STRATEGIES FOR META MULTIPLE CRITERIA DECISION MAKING Saaty, T & Vargas, L / MODELS, METHODS, CONCEPTS and APPLICATIONS OF THE

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Chatterjee, K & Samuelson, W / GAME THEORY AND BUSINESS APPLICATIONS

Hobbs, B et al / THE NEXT GENERATION OF ELECTRIC POWER UNIT COMMITMENT

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Vanderbei, R.J / LINEAR PROGRAMMING: Foundations and Extensions, 2nd Ed.

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Feinberg, E & Shwartz, A / HANDBOOK OF MARKOV DECISION PROCESSES: Methods

and Applications

Ramík, J & Vlach, M / GENERALIZED CONCAVITY IN FUZZY OPTIMIZATION

AND DECISION ANALYSIS

Song, J & Yao, D / SUPPLY CHAIN STRUCTURES: Coordination, Information and

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of Stochastic Theory, Methods, and Applications

Dokuchaev, N / DYNAMIC PORTFOLIO STRATEGIES: Quantitative Methods and Empirical Rules

for Incomplete Information

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Zhu, J / QUANTITATIVE MODELS FOR PERFORMANCE EVALUATION AND BENCHMARKING Ehrgott, M & Gandibleux, X / MULTIPLE CRITERIA OPTIMIZATION: State of the Art Annotated

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DECISIONS

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Frederick S Hillier, Series Editor Stanford University

TRANSPORTATION SCIENCE

Second Edition

edited by

Randolph W Hall

University of Southern California

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Dordrecht

vii Preface to the Second Edition

Moshe Ben-Akiva and Michel Bierlaire

3 Activity-Based Modeling of Travel Demand

Chandra R Bhat and Frank S Koppelman

Automated Vehicle Control

Petros loannou and Arnab Bose

Spatial Models

9

10

Continuous Space Modeling

Tönu Puu and Martin Beckmann

Location Models in Transportation

Mark S Daskin and Susan H Owen

Network Equilibrium and Pricing

Michael Florian and Donald Hearn

Street Routing and Scheduling Problems

Lawrence Bodin, Vittorio Maniezzo and Aristide Mingozzi

Long-haul Freight Transportation

Teodor Gabriel Crainic

Crew Scheduling

Cynthia Barnhart, Ellis L Johnson, George L Nemhauser

and Pamela Vance

Economic Models

16 Revenue Management

Garrett van Ryzin and Kalyan Talluri

17 Spatial Interaction Models

Piet Rietveld and Peter Nijkamp

18 Transport Economics

Richard Arnott and Marvin Kraus

Biographies

Index

The Second Edition of the Handbook of Transportation Science is a compendium of

the fundamental concepts, methods and principles underlying transportation It hasbeen expanded from the first edition through the addition of four chapters Chapter

15 extends the networks section of the book by addressing supply chains, distributionnetworks and logistics While the emphasis is on freight transportation, theprinciples for network design extend to other applications, such as publictransportation Chapters 16 through 18 fall in a new section on transportationeconomics Chapter 16 addresses revenue management, a relatively recent topic intransportation, that has had substantial impact on the airline industry in particular.Chapter 17 presents spatial interaction models, which provides a mechanism foranalyzing patterns of development Lastly, Chapter 18 provides the principles oftransportation economics, with emphasis on pricing and public policy In addition tothe new chapters, several of the original chapters have been updated and revised Wehope that the Second Edition continues to inspire research into the science oftransportation

David Boyce, Carlos Daganzo, Michael Florian, and Nigel Wilson provided valuableguidance in suggesting chapter topics and authors Cenk Caliskan and Georgia Lumprovided a tremendous help in formatting chapters The work was supported, inpart, by a grant from the National Science Foundation (DMI-9732878), and supportfrom the United States Department of Transportation for the METRANS UniversityTransportation Center.

guidepost for defining what science is and how it is conducted.

Though “Transportation Science” did not exist as a discipline in the time of

Jevons, his insights provide a motivation for The Handbook of Transportation Science The premise for our book is that transportation can be defined as a

scientific discipline that transcends transportation technology and methods Whether

by car, truck, airplane or by a mode of transportation that has not yet beenconceived transportation obeys fundamental properties The science oftransportation defines these properties, and demonstrates how our knowledge of onemode of transportation can be used to explain the behavior of another

Like any of the natural sciences, transportation science as a discipline arose out

of human curiosity, and the desire for explanations for how the world around usbehaves In the words of famed physicist Max Planck,

“The beginning of every act of knowing, and therefore the starting point of every science, must be in our own personal experience … They form the first and most real hook on which

we fasten the thought-chain of science.” (Planck, 1932, p 66)

And so is the case for transportation science When one looks back to the earliestpublications on the subject from the 1950s and early 1960s, we first see a desire tounderstand the dynamics of roadway traffic Then and now, there is hardly a person

in the profession who does not view a trip on the highway as a scientific experiment,seeking to understand why traffic flows as it does, how bottlenecks appear anddisappear, and what causes the myriad of driving behaviors Many of the early

pioneers were, in fact, trained in natural sciences, such as physics, and cleverlycombined knowledge of natural phenomena, such as thermodynamics and fluidmechanics, with their observations on traffic flow.

Transportation scientists are motivated by the desire to explain spatialinteractions that result in movement of people or objects from place to place Itsheritage includes research in the fields of geography, economics and location theory,dating over several centuries Its methodologies draw from physics, operationsresearch, probability and control theory It is fundamentally a quantitative discipline,relying on mathematical models and optimization algorithms to explain thephenomena of transportation

Publications in transportation science appear in many places, but they are most

concentrated in the journals Transportation Research B and Transportation Science,

and also in the proceedings of the International Symposium on Transportation andTraffic Flow Theory Transportation scientists perform both empirical andtheoretical work (many do both), and use real transportation systems as theirlaboratories They interact frequently with practitioners, with scientific findingsresulting from examination of real problems

Fundamentally, transportation science recognizes that all modes of transportationhave the same essential elements: vehicles, guideways, and terminals, operatingunder some control policy Vehicles comprise mobile resources that accompanypersons or shipments (P/S) as they travel from place to place They provide themotive power to propel P/Ss on their trip, and provide the carrying space to ensure asafe and/or comfortable journey Guideways are stationary resources that definefeasible paths of travel and provide the physical infrastructure to support vehicles andP/Ss They add to safety by restricting movements to defined paths, and provide anefficient surface for movement Terminals are stationary resources that reside atdiscrete location They offer the capability to sort vehicles, persons and objectsamong incoming and outgoing transportation routes Lastly, control represents therules, regulations and algorithms that determine movements and trajectories withintransportation systems

Many years ago, transportation occurred by human, animal and natural (e.g.,wind, currents, gravity) power, in simple vehicles (or none at all), on guideways thatrequired little in the way of construction Terminals, if they could be called that,were the market towns, caravansaries or trading posts, and control was executedthrough the minds of individual travelers By contrast, today most movementdepends on propulsion by motors or engines, built guideways and terminals, and, tosome degree, computer control So in many respects, one might say thattransportation modes of the late century have little in common with theirancestors

Nevertheless, similarities abound For any given mode of transportation,vehicles, guideways, terminals and control are configured to perform several basicfunctions All modes of transportation provide the capability to propel, brake andsteer Most (even animal and human) provide mechanisms to store energy forpropulsion, to sort persons and objects at terminals, to couple shipments together intoefficient loads, and to contain these shipments as they travel from place to place Theway that a mode of transportation accomplishes these functions may be unique, butthe basic tasks are the same (Hall, 1995).

As mentioned, this book is concerned with the properties and characteristics thattranscend individual modes of transportation, and collectively define a science oftransportation The chapters and structure of this book are intended to elucidatethese properties on a subject-by-subject basis, and not by mode We begin with thehuman element of transportation On a day-to-day basis, individuals are presentedwith a plethora of transportation choices, some of which are determined by ingrainedhabits and circumstances; others of which result from deliberation The routefollowed, the time of travel and, to some degree, the choice of destination and mode,are all daily decisions, and constitute short-term traveler behavior (Chapter 2) Thesedecisions are imbedded within the broader context of how we plan and organize ouractivities, the subject of Chapter 3 And the way we operate our vehicles is the maindeterminant of Transportation Safety, as covered in Chapter 4

Another important property of most transportation networks is that travel timedepends on traffic flows, as well as system design and system control Chapters 5(queueing) and Chapter 6 (traffic flow theory) show how congestion originates ontransportation networks, and how vehicles, travelers and shipments interact as theytravel across the network Chapter 5 addresses congestion and delay in a broadcontext, spanning all types of transportation, whereas Chapter 6 focuses onmovement along links of a network where vehicles interact with each other

At a more microscopic level, vehicle flows and trajectories depend on how theirspeed and direction are controlled Until fairly recently, this was a human task, butincreasingly vehicle control is automated, though electronic sensing and computerprocessors Chapter 7 covers automated techniques for controlling trajectories,whereas Chapter 8 covers control at the more macroscopic scale of regulating flows.Macroscopic control is typically executed by conveying messages to vehicleoperators (e.g., visible signals at intersections, network entrances and along lanes oftravel.) In the future, it is not hard to imagine a coalescence of vehicle and trafficcontrol within a single automated system

The next two topics are at the historical core of transportation science:continuous-space models (Chapter 9) and transportation location (Chapter 10) Thecontinuous-space approach has been used extensively as an explanatory tool foroptimal network design, both with respect to physically constructed networks(roadways, railroads, etc.) and operational networks (vehicle routes) It draws from

spatial economic theory and continuum models in physics Transportation locationalso addresses system design, largely from the perspective of placing discretefacilities, such as terminals and points of production It is the first of five chaptersthat include transportation optimization.

One of the ways that traveler behavior is revealed is in the flow of traffic alonglinks in the transportation network And one of the most studied, and mostchallenging, areas of research in transportation science is network assignment, or theestimation and prediction of these flows (Chapter 11) Network assignment usesoptimization methods to predict the consequences of traveler behavior

The next four chapters 12, 13, 14 and 15 describe different aspects ofrouting and networks, represented by the assignment of persons/shipments to vehiclesand terminals, and the sequencing of stops along routes The emphasis of Chapter 12

is local routing, represented by vehicle tours that can be accomplished within thespan of a single day The emphasis of Chapter 13 is routing freight over longhaulnetworks, represented by tours that travel from city to city, and last more than oneday Chapter 14 is concerned with routing the crews that operate vehicles onlonghaul networks, with focus on the personnel constraints that dictate feasible tours.Chapter 15 addresses the design of transportation networks and supply chains,including the use of vehicles and terminals for shipment consolidation

The final section of the book – Chapters 16, 17 and 18 – address transportationeconomics Chapter 16 focuses on the recent topic of revenue management, or howtransportation companies can use pricing to maximize their returns on investment InChapter 17, research is presented on spatial interaction, which provides a frameworkfor predicting patterns of development, in light of transportation services andinfrastructure The final chapter covers transportation economics in general, withemphasis on pricing, markets, and public policy

We wrote this book with the intention of documenting the core knowledge oftransportation science As would be the case for any other science, this book cannotprovide ultimate conclusions But it can record the methods and issues that definethe discipline of transportation science as it exists at the end of the century Inthe words of the noted philosopher Karl Popper (1959, p 281):

“Science never pursues the illusory aim of making its answers final, or even probable Its advance is, rather, towards the infinite yet attainable aim of ever discovering new, deeper and more general problems, and of subjecting its ever tentative answers to ever renewed and ever more rigorous tests ”

We hope that the Handbook of Transportation Science provides this inspiration

for the transportation scientists of the future

Hall, R.W (1995) The architecture of transportation systems, Transportation Research C, 3, 129-142.

Jevons, W.S (1958) The Principles of Science, Dover Publishing, New York.

Planck, M (1932) Where is Science Going, W.W Norton and Company, New York.

Popper, K.R (1959) The Logic of Scientific Discovery, Basic Books, New York.

2 DISCRETE CHOICE MODELS

WITH APPLICATIONS TO DEPARTURE TIME AND ROUTE

We then present the alternative discrete choice model forms such as Logit, NestedLogit, Generalized Extreme Value and Probit, as well as more recent developmentssuch as Hybrid Logit and the Latent Class choice model Finally, we elaborate on theapplications of these models to two specific short-term travel decisions: route choiceand departure time choice

2.2 Discrete Choice Models

We provide here a brief overview of the general framework of discrete choicemodels We refer the reader to Ben-Akiva and Lerman (1985) for detailed discussion.General Modeling Assumptions

The framework for a discrete choice model can be presented by a set of generalassumptions We distinguish among assumptions regarding the:

decision-maker defining the decision-making entity and its characteristics;alternatives determining the options available to the decision-maker;

1

2

attributes measuring the benefits and costs of an alternative to the maker; and

decision-decision rule describing the process used by the decision-decision-maker to choose analternative

3

4

Decision-maker Discrete choice models are also referred to as disaggregate models,

meaning that the decision-maker is assumed to be an individual The “individual”decision-making entity depends on the particular application For instance, we mayconsider that a group of persons (a household or an organization, for example) is thedecision-maker In doing so, we may ignore all internal interactions within thegroup, and consider only the decisions of the group as a whole We refer to

“decision-maker” and “individual” interchangeably throughout this chapter Toexplain the heterogeneity of preferences among decision-makers, a disaggregatemodel must include their characteristics such as the socio-economic variables of age,gender, education and income

Alternatives Analyzing individual decision making requires not only knowledge of

what has been chosen, but also of what has not been chosen Therefore, assumptionsmust be made about available options, or alternatives, that an individual considersduring a choice process The set of considered alternatives is called the choice set

A discrete choice set contains a finite number of alternatives that can be explicitlylisted The choice of a travel mode is a typical example of a choice from a discretechoice set The identification of the list of alternatives is a complex process usually

referred to as choice set generation The most widely used method for choice set

generation uses deterministic criteria of alternative availability For example, thepossession of a driver’s license determines the availability of the auto drive option.The universal choice set contains all potential alternatives in the application’scontext The choice set is the subset of the universal choice set considered by, oravailable to, a particular individual Alternatives in the universal choice set that arenot available to the individual are therefore excluded from the choice set

In addition to availability, the decision-maker’s awareness of the alternative couldalso affect the choice set The behavioral aspects of awareness introduce uncertainty

in modeling the choice set generation process and motivate the use of probabilisticchoice set generation models that predict the probability of each feasible choice setwithin the universal set A discrete choice model with a probabilistic choice setgeneration model is described later in this chapter as a special case of the latent classchoice model

Attributes Each alternative in the choice set is characterized by a set of attributes.Note that some attributes may be generic to all alternatives, and some may bealternative-specific

An attribute is not necessarily a directly measurable quantity It can be anyfunction of available data For example, instead of considering travel time as anattribute of a transportation mode, the logarithm of the travel time may be used, orthe effect of out-of-pocket cost may be represented by the ratio between the out-of-

pocket cost and the income of the individual Alternative definitions of attributes asfunctions of available data must usually be tested to identify the most appropriate.

Decision Rule The decision rule is the process used by the decision-maker to

evaluate the alternatives in the choice set and determine a choice Most models used

for travel behavior applications are based on utility theory, which assumes that the

decision-maker’s preference for an alternative is captured by a value, called utility,and the decision-maker selects the alternative in the choice set with the highestutility

This concept, employed by consumer theory of micro-economics, presents stronglimitations for practical applications The underlying assumptions of this approachare often violated in decision-making experiments The complexity of humanbehavior suggests that the decision rule should include a probabilistic dimension.Some models assume that the decision rule is intrinsically probabilistic, and evencomplete knowledge of the problem would not overcome the uncertainty Othersconsider the individuals’ decision rules as deterministic, and motivate the uncertaintyfrom the limited capability of the analyst to observe and capture all the dimensions ofthe choice process, due to its complexity

Specific families of models can be derived depending on the assumptions aboutthe source of uncertainty Models with probabilistic decision rules, like the modelproposed by Luce (1959), or the “elimination by aspects” approach proposed byTversky (1972), assume a deterministic utility and a probabilistic decision process.Random utility models, used intensively in econometrics and in travel behavioranalysis, are based on deterministic decision rules, where utilities are represented byrandom variables

Random Utility Theory

Random utility models assume, as does the economic consumer theory, that thedecision-maker has a perfect discrimination capability However, the analyst isassumed to have incomplete information and, therefore, uncertainty must be takeninto account Manski (1977) identifies four different sources of uncertainty:unobserved alternative attributes; unobserved individual characteristics (also called

“unobserved taste variations”); measurement errors; and proxy, or instrumental,variables

The utility is modeled as a random variable in order to reflect this uncertainty

More specifically, the utility that individual n associates with alternative i in the

where is the deterministic (or systematic) part of the utility, and is the randomterm, capturing the uncertainty The alternative with the highest utility is chosen

Therefore, the probability that alternative i is chosen by decision-maker n from

choice set is

choice set is given by

In the following we introduce the assumptions necessary to make a random utilitymodel operational.

where we have that

The above illustrates the fact that only the signs of the differences between utilities

are relevant here, and not utilities themselves The concept of ordinal utility isrelative and not absolute In order to estimate and use a specific model arbitraryvalues have to be selected for and The selection of the scale parameter isusually based on a convenient normalization of one of the variances of the randomterms The location parameter is usually set to zero See also the discussion below

of Alternative Specific Constants

Alternative specific constants The means of the random terms can be assumed to

be equal to any convenient value c (usually zero, or the Euler constant for Logit

models) This is not a restrictive assumption If we denote the mean of the error term

of alternative i by we can define a new random variable

such that We have

a model in which the deterministic part of the utilities are and the randomterms are (with mean c) The terms are then included as Alternative Specific

Constants (ASC) that capture the means of the random terms Therefore, we mayassume without loss of generality that the error terms of random utility models have a

constant mean c by including alternative specific constants in the deterministic part of

the utility functions

As only differences between utilities are relevant, only differences between ASCsare relevant as well It is common practice to define the location parameter as thenegative of one of the ASCs This is equivalent to constraining that ASC equal zero.From a modeling viewpoint, the choice of the particular alternative whose ASC isconstrained is arbitrary However, Bierlaire, Lotan and Toint (1997) have shown thatthe speed of convergence of the estimation process may be improved by imposingdifferent constraints

Location and scale parameters Considering two arbitrary real numbers and

The deterministic term of the utility The deterministic term of each alternative

is a function of the attributes of the alternative itself and the characteristics of thedecision-maker That is

where is the vector of attributes as perceived by individual n for alternative i, and

is the vector of characteristics of individual n.

This formulation is simplified using any appropriate vector valued function h that

defines a new vector of attributes from both and that is

Then we have

The choice of h is very general, and several forms may be tested to identify the best

representation in a specific application It is usually assumed to be continuous andmonotonic in For a linear in the parameters utility specification, h must be a fully

determined function (meaning that is does not contain unknown parameters) A linear

in the parameters function is denoted as follows

or in vector form

The deterministic term of the utility is therefore fully specified by the vector ofparameters

The random part of the utility Among the many potential models that can be

derived for the random parts of the utility functions, we describe below the mostpopular The models within the Logit family are based on a probability distributionfunction of the maximum of a series of random variables, introduced by Gumbel(1958) Probit and Probit-like models are based on the Normal distribution motivated

by the Central Limit Theorem

The main advantage of the Probit model is its ability to capture all correlationsamong alternatives However, due to the high complexity of its formulation,relatively few applications have been developed The Logit model has been muchmore popular, because of its tractability However, Logit imposes restrictions on thecovariance structure that may be unrealistic in some contexts Other models in the

“Logit family” are aimed at relaxing restrictions, while maintaining tractability

We present first the Generalized Extreme Value Models, a class of random utilitymodel that includes Logit and Nested Logit Next we present the Probit model andother advanced models including the Generalized Factor Analytical Representation

and the Hybrid Logit models (designed to bridge the gap between Logit and Probitmodels) and the Latent Class Choice model (designed to explicitly include discreteunobserved factors in the model).

The Generalized Extreme Value Models Family

The Generalized Extreme Value (GEV) model has been derived from the randomutility model by McFadden (1978) This general model consists of a large family of

models The probability of choosing alternative i within is

is the number of alternatives in and G is a non-negative differentiable function

defined on with the following properties:

The Multinomial Logit Model is an instance of the GEV family, with

McFadden’s original formulation with was generalized to by Ben-Akivaand François (1983)

1,

1

yielding to the following probability model :

An important property of the Multinomial Logit Model is Independence from

Irrelevant Alternatives (IIA) This property can be stated as follows: The ratio of the

probabilities of any two alternatives is independent of the choice set That is, for any

choice sets and such that and and for any alternatives i and j

in both and we have

An equivalent definition of the IIA property is: The ratio of the choice probabilities

of any two alternatives is unaffected by the systematic utilities of any other alternatives.

The IIA property of Multinomial Logit Models is a limitation for some practicalapplications This limitation is often illustrated by the red bus/blue bus paradox in themodal choice context We use here instead the following path choice example.Consider a commuter traveling from origin O to destination D He/she isconfronted with the path choice problem described in Figure 2-1, where the choiceset is {1,2a,2b} and the only attribute considered for the choice is travel time Weassume furthermore that the travel time for any alternative is the same, that is

and that the travel time on the small sections a and b is

The probability of each alternative provided by the Multinomial Logit Model forthis example is

Clearly, this result is independent of the value of However, when is

significantly smaller than the total travel time T, we expect the probabilities to be

close to 50%/25%/25% The Multinomial Logit Model is not consistent with thisintuitive result This situation appears in choice problems with significantlycorrelated random utilities, as it is clearly the case in the path choice example.Indeed, alternatives 2a and 2b are so similar that their utilities share manyunobserved attributes of the path and, therefore, the assumption of independence ofthe random parts is not valid in this context

The Nested Logit Model, first proposed by Ben-Akiva (1973 and 1974) and

derived as a random utility model and a special case of GEV by McFadden (1978), is

an extension of the Multinomial Logit Model designed to capture some correlationsamong alternatives It is based on the partitioning of the choice set into M nests

such that

and It is also an instance of the GEV family, with

where and Each nest within the choice set is associated with acomposite utility

where denotes the partial utility common to all alternatives in the nest The secondterm is called expected maximum utility, LOGSUM, inclusive value or accessibility

in the literature The probability for individual n to choose alternative i within nest

is given by

where

and

Parameters and reflect the correlation among alternatives within the nest

The correlation between the utility of two alternatives i and j in nest can bederived (see Ben-Akiva and Lerman, 1985) as

Therefore,

The parameters and are closely related in the model Actually, only their ratio ismeaningful It is not possible to identify them separately A common practice is toarbitrarily constrain one of them to a specific value (usually 1) If the NestedLogit Model collapses to a Multinomial Logit Model

This is illustrated by the following example (Bierlaire, 1998) We apply theNested Logit Model to the route choice problem described in Figure 1 We partition

choosing path 1 is given by

where is the scale parameter of the random term associated with and is thescale parameter of the choice between and Note that we require

The probability of the two other paths is

In this example, we need to normalize either or to 1 In the latter case wehave

and

and we require that Note that for we obtain the MNL result Forapproaching zero, we obtain the expected result when paths 2a and 2b fully overlap.

A model where the scale parameter is normalized to 1 is said to be “normalizedfrom the top.” A model where one of the parameters is normalized to 1 is said to

be “normalized from the bottom.” The latter may produce a simpler formulation ofthe model We illustrate it using the following example

model with two nests: contains the public transportation modes and

contains the private transportation modes For the example’s sake, weconsider the following deterministic terms of the utility functions:

where is the travel time using mode i and and are parameters to be estimated.Note that we have one parameter for private and one for public transportation, and

we have not included the alternative specific constants in order to keep the examplesimple

Applying the Nested Logit Model, we obtain

The normalization from the bottom is obtained by defining

(1998) and Hensher and Greene (2002) for further discussion Note that the newpackage BIOGEME (Bierlaire, 2001b) for GEV model estimation does not impose aspecific normalization for the Nested Logit model and therefore does not requiresuch techniques.

A direct extension of the Nested Logit Model consists in partitioning some or allnests into sub-nests, which can in turn, be divided into sub-nests The modeldescribed above is valid at every layer of the nesting, and the whole model isgenerated recursively Therefore, a tree structure is a convenient representation ofNested Logit models Clearly, the number of potential structures reflecting thecorrelation among alternatives can be very large No technique has been proposedthus far to identify the most appropriate correlation structure directly from the data.The Nested Logit Model is designed to capture choice problems wherealternatives within each nest are correlated No correlation across nests can becaptured by the Nested Logit Model When alternatives cannot be partitioned intowell separated nests to reflect their correlation, the Nested Logit Model is notappropriate

The Cross-Nested Logit Model is a direct extension of the Nested Logit Model,

where each alternative may belong to more than one nest It is also an instance of theGEV family, with

This model was first presented by McFadden (1978) as a special case of the GEVmodel It was applied by Small (1987) for departure time choice, by Vovsha (1997)for mode choice, and by Vovsha and Bekhor (1998) for route choice Swait (2001)proposes a Cross-Nested formulation for a model including choice set generation.The general formulation proposed above has been introduced by Ben-Akiva andBierlaire (1999) The proof that it is indeed a GEV model is detailed by Bierlaire(200la) Wen and Koppelman (2001) provide an analysis of the model elasticities.They use the name “Generalized Nested Logit” model for Cross-Nested Papola(2000) describes a technique to design a specific Cross-Nested logit model for anygiven homoscedastic variance-covariance structure

The parameter is usually interpreted as the degree at which alternative j belongs to nest m Therefore, a common normalization of the model imposes that

We emphasize that this condition is a convenient normalizationcondition, but is not necessary for the model to comply with random utility theory.The Recursive Nested Extreme Value Model (RNEV), proposed by Daly (2001),generalizes the Cross-Nested model by allowing several levels of nests in theformulation

The Network GEV model is a class of models within the GEV family proposed

by Bierlaire (2002) and based on the same idea as Daly’s RNEV Each instance isdefined by a network where each edge (m,k) is associated with a non-negativeparameter The network must have the following properties

1 It does not contain any circuit

2 It has one special node with no predecessor, called the root

3 It has J special nodes with no successor, called the alternatives

4 For each alternative i, there exists a path between the root and i such thatall parameters on the path are non-zero

Each node m of the network is associated with an homogeneous function withhomogeneity parameter such that

If each alternative i is associated with the trivial function

then the G function associated with each node of the network generates a GEVmodel In general, only the GEV model associated with the root is considered Thisresult, formally proven by Bierlaire (2001c), provides an intuitive and general way ofgenerating new GEV models Namely, all GEV models mentioned above fit in thatframework

Multinomial Probit Model

The Probability Unit (or Probit) model is derived from the assumption that the errorterms of the utility functions are normally distributed The Probit model capturesexplicitly the correlation among all alternatives Therefore, we adopt a vectornotation for the utility functions:

multivariate normal distributed with a vector of means 0 and a covariance matrix

variance-The probability that a given individual n chooses alternative i from the choice set

is given by

Denoting the matrix such that

are

The matrix is such that the column contains -1 everywhere If the column isremoved, the remaining matrix is the identity matrix For example, in thecase of a trinomial choice model, we have

Given this transformation, we have that

The density function is given by

where

and

We note that the multifold integral becomes intractable even for a relatively lownumber of alternatives Moreover, the number of unknown parameters in thevariance-covariance matrix grows with the square of the number of alternatives Werefer the reader to McFadden (1989) for a detailed discussion of multinomial Probitmodels The complexity of Probit models can be reduced using a Factor Analyticform of the model, as described in the next section

Generalized Factor Analytic Specification of the Random Utility

The general formulation of the factor analytic formulation is

where is a vector of utilities, is a vector of deterministic utilities,

is a vector of random terms, is an (M×1) vector of factors which are IID

standard normal distributed, is a matrix of loadings that map the factors to

the random utility vector and T is a MxM lower triangular matrix, capturing the

Cholesky factor of the variance-covariance matrix This specification is very general,and allows explicitly specifying some structure in the model and, consequently,decreasing the complexity We describe here special cases of factor analyticrepresentations They are discussed in more detail by Ben-Akiva, Bolduc, andWalker (2001) and Walker (2001)

Heteroscedasticity A heteroscedastic2 model is obtained when is the identitymatrix, and T is a diagonal matrix containing the alternative specific standarddeviations

Error components The error component formulation is based on fixed factor

loadings equal to 0 or 1 Entry in row i and column j of is 1 if error term

applies to alternative i, and 0 otherwise A typical specification of is based on anested structure, where each alternative belongs to exactly one next In that case, afactor is associated with each nest, and the entry (i,j) of is 1 if alternative i belongs

to nest j A cross-nested specification is also possible, by allowing an alternative to

belong to more than one nest Finally, we note that the matrix T is usually diagonaland must be estimated

Factor analytic The term “factor analytic” usually refers to the formulation where

the loading factor is not imposed a priori and must be estimated In general, thematrix T is usually diagonal in that case

General autoregressive process Assuming that the disturbances follow an

autoregressive process allows decreasing the model complexity while keeping areasonable level of generality Interestingly, such an assumption fits in theGeneralized Factor Analytic Specification We consider the case where the error term

is generated from a first-order autoregressive process:

where is a matrix of weights describing the influence of each component

of the error terms on the others, and is an (M×1) vector of error terms which are

IID standard normal distributed We can write the process as

which is a special case of the Generalized Factor Analytic Specification with

Random parameters We consider a utility function and we assume that the

parameters are normally distributed with mean and variance-covariance matrixTT’ Therefore, where are IID normal distribution The utility functioncan be written as

2Heteroscedasticity here refers to different variances among the alternatives We use

it in this context to refer to a diagonal variance-covariance matrix with potentiallydifferent terms on the diagonal

which is a Generalized Factor Analytic formulation, with Using such a utilityfunction in a probit model does not cause any difficulty, as all random terms arenormally distributed If a Multinomial Logit model is preferred, the formulationcontains both Gumbel and normal error terms, and the model becomes a HybridLogit model, or Mixed Logit model, which is described in the next section.

Hybrid Logit Model

The Multinomial Probit with a Logit kernel model, called Hybrid Logit or MixedLogit, has been introduced by Bolduc and Ben-Akiva (1991) It is intended to bridgethe gap between Logit and Probit models by combining the advantages of both It isbased on the following utility functions:

where are normally distributed and capture correlation between alternatives, andare independent and identically distributed Gumbel variables If the are given,the model corresponds to a Multinomial Logit formulation:

where is the vector of unobserved random terms Therefore, the

probability to choose alternative i is given by

where is the probability density function of This model is a generalization ofthe Multinomial Probit Model when the distribution is a multivariate normal.Other distributions may also be used The earliest application of this model to capturerandom coefficients in the Logit Model (see below) was by Cardell and Dunbar(1980) More recent results highlighted the robustness of Hybrid Logit (seeMcFadden and Train, 2000) We note that the Hybrid Logit model can be combinedwith any Generalized Factor Analytic formulation The random parameters modelpresented above is an example of such a combination We refer the reader to Ben-Akiva et al (2002) for a review of Hybrid Logit models

Latent Class Choice Model

Latent class choice models are also designed to capture unobserved heterogeneity(see Everitt, 1984, for an introduction to latent variable models) The underlyingassumption is that the heterogeneity is generated by discrete constructs Theseconstructs are not directly observable and therefore are represented by latent classes.For example, heterogeneity may be produced by taste variations across segments ofthe population, or when choice sets considered by individuals vary (latent choice set)

The latent class choice model is given by:

where S is the number of latent classes, is the vector of attributes of alternatives

and characteristics of decision-maker n, are the choice model parameters specific

to class s, is the choice set specific to class s, and is an unknown parametervector

The model

is the class membership model, and

is the class-specific choice model (Kamakura and Russell, 1989, Gopinath, 1995)

Special case: latent choice sets A special case is the choice model with latent choice

sets:

where G is the set of all non-empty subsets of the universal choice set M, and

is a choice model We note here that the size of G grows exponentially

with the size of the universal choice set

The latent choice set can be modeled using the concept of alternative availability.For such a model, a list of constraints or criteria is used to characterize the

availability of alternatives For each alternative i, a binary random variable isdefined such that if alternative i is available to individual n, and 0 otherwise A

list of constraints is defined as follows:

For example, in a path choice context, one may consider that a path is not available ifthe ratio between its length and the shortest path length is above some threshold,

represented by a random variable For example, the associated constraint for path i

could be:

where is the length of the shortest path, is the length of path i and a random

variable with zero mean It means that, on average, paths longer than twice the length

of the shortest path are rejected

The probability for an alternative to be available is given by

The latent choice set probability is then:

If the availability criteria are assumed to be independent, we have

Swait and Ben-Akiva (1987) estimate a latent choice set model of mode choice in aBrazilian city See also Ben-Akiva and Boccara (1995) for a more detail analysis ofdiscrete choice models with latent choice sets

2.3 Model estimation

The estimation of discrete choice models from sample data is a difficult andimportant task Most statistical packages provide estimation capabilities for simplemodels, like the Multinomial Logit models Dedicated commercial software packagesare available for the estimation of Nested-Logit models Free software for estimation

of Hybrid, or Mixed, Logit models (emlab.berkeley.edu/users/train/software.html)and GEV models (rosowww.epfl.ch/mbi/biogeme) is also available However, thesepackages do not cover the entire range of models, and a specific implementation of

an estimation procedure is sometimes necessary We discuss here some issues related

to such implementation

Maximum likelihood estimation is the most widely used technique for discretechoice model estimation (see statistical textbooks, such as Sprott, 2000, and Severini,2000) It aims at identifying the set of parameters maximizing the probability that agiven model perfectly reproduces the observations It is a nonlinear programmingproblem The nature of the objective function and of the constraints determines thetype of solution algorithm that must be used

The objective function of the maximum likelihood estimation problem for GEVmodels is a nonlinear analytical function, as the probability density function has aclosed form In general, the function is not concave (except for the Multinomial LogitModel) and, therefore, significantly complicates the identification of a (global)maximum Most nonlinear programming algorithms (see Dennis and Schabel, 1983,

or Bertsekas, 1995) are designed to identify local optima of the objective function.There exists some meta-heuristics designed to identify global optima (like geneticalgorithms, and simulated annealing) but none of them can guarantee that theprovided solution is a global optimum Therefore, whatever algorithm is preferred,starting it from different initial solutions is a good practice

For the Probit or Hybrid-logit models, the objective function does not have ananalytical form and must be evaluated based on Monte Carlo (Metropolis and Ulam,1949) or Quasi-Monte Carlo methods (Morokoff and Caflish, 1995) Contrarily toMonteCarlo, Quasi-Monte Carlo techniques are deterministic They require fewer

“draws” than Monte-Carlo simulation to reach the same level of accuracy (seeSpanier and Maize, 1994).

Not all parameters of a model can be identified from the data Parameteridentification and model normalization issues are important to analyze beforeperforming an actual estimation We refer the reader to Ben-Akiva and Lerman(1985) for a general discussion on such issues Bunch (1991) and Bolduc (1992)address the case of the Probit model Walker (2001) provides a detailed analysis ofidentification issues for the Hybrid Logit model

The parameters to be estimated must verify some constraints First, most of themmust lie within bounds in order for the model to be consistent with the theory (e.g.the homogeneous parameters of GEV functions must be non-negative) or with theirintuitive interpretation (e.g the coefficient for cost or travel time in a utility function

is usually non-positive) Moreover, some constraints have to be verified in order forthe model to be estimable (e.g the sum of parameters must sum up to one in aCross-Nested Logit model) In the past, it was usually advised to ignore the boundconstraints, to eliminate other constraints by incorporating them in the objectivefunction, and to use unconstrained optimization algorithms The increasingcomplexity of the models, combined with the availability of efficient softwarepackages for constrained optimization motivate now the explicit management ofconstraints in the estimation process

2.4 Route Choice Applications

The route choice problem plays an important role in many transportation relatedapplications In this section, we analyze its specific assumptions, and present somemodels designed to capture this complex behavioral problem

Given a transportation network composed of nodes, links, origins and

destinations; and given an origin o, a destination d and a transportation mode m, what

is the chosen route between o and d on mode m This discrete choice problem has

specific characteristics First, the universal choice set is usually very large Second,the decision-maker considers not all physically feasible alternatives Third, thealternatives are usually correlated, due to overlapping paths

We now describe typical assumptions associated with route choice models

Decision-Maker

The traveler’s characteristics most often used for route choice applications are:Value-of-time Obviously, travel time is a key attribute of alternative routes.Its influence on behavior, however, may vary across individuals Thesensitivity of an individual to travel time is usually referred to as the value-of-time It can be represented by a continuous variable (e.g., the dollar-valueequivalent of a minute spent traveling) or by a discrete variable identifyingthe decision-maker’s value-of-time as low, medium or high

Access to information Information about network conditions maysignificantly influence route choice behavior Therefore, it may be importantthat a route choice model explicitly differentiates travelers with access to suchinformation from those without access It is also an important policy variable.

It may be modeled by a single binary attribute (access/no access) or byseveral binary variables identifying the type of information available to thetraveler (pre-trip information, on-board computer, etc.)

Trip purpose The purpose of the trip may significantly influence the routechoice behavior For example, a trip to work may be associated with a penaltyfor late arrival, while a shopping trip would usually have no such penalty

Dial (1971) proposes to include in the choice set “reasonable” paths composed oflinks that would not move the traveler farther away from her/his destination The

labeling approach (proposed by Ben-Akiva et al., 1984) includes paths meeting

specific criteria, such as shortest paths, fastest paths, most scenic paths, paths withfewest stop lights, paths with least congestion, paths with greatest portion offreeways, paths with no left turns, etc

Azevedo et al (1993) propose the link elimination approach, where the shortest

path (according to a given impedance) is first calculated and introduced in the choiceset Then, some links belonging to the shortest path are removed, and a the shortestpath in the modified network is computed and introduced in the choice set

Cascetta and Papola (1998) propose an implicit probabilistic choice set generation

model, where the utility function associated with path i by individual n is defined as

where is a random variable with mean

are the attributes for availability and perception of the path and areparameters to be estimated

Swait (2001) combines the probabilistic choice set generation with the routechoice model within a Cross-Nested structure.

Some recent models (Nguyen and Pallottino, 1987, Nguyen, Pallottino andGendreau, 1988) consider hyperpaths instead of paths as alternatives A hyperpath is

a collection of paths with associated strategies at decision nodes This technique isparticularly appropriate for a public transportation network

Attributes

In describing the attributes of the alternatives to be included in the utility function,

we need to distinguish between link-additive and non-link-additive attributes

If i is a path composed of links is a link-additive attribute of i if

where is the corresponding attribute of link a For example, the travel time on a

path is the sum of the travel times on links composing the path Qualitative attributesare in general non-link-additive For example, a binary variable equal to one if thepath is a habitual path and 0 otherwise, is non link-additive In the context of publictransportation, variables like transfers and fares are usually not link-additive Thedistinction is important because some models, designed to avoid path enumeration,use link attributes and not path attributes

Among the many attributes that can potentially be included in a utility function,travel time is probably the most important But what does travel time mean for thedecision-maker? How does she/he perceive travel time? Many models are based onthe assumption that most travelers are sufficiently experienced and knowledgeableabout usual network conditions and, therefore, are able to estimate travel timesaccurately This assumption may be satisfactory for planning applications using staticmodels With the emergence of Intelligent Transportation Systems, models that areable to predict the impact of real-time information have been developed In thiscontext, the “perfect knowledge” assumption is contradictory with the ITS servicesthat provide information Several approaches can be used to capture perceptions oftravel times One approach represents travel time as a random variable in the utilityfunction This idea was introduced by Burrell (1968) and is captured by a randomutility model Also, the uncertainty or the variability of travel time along a given pathcan be explicitly included as an attribute of the path

In addition to travel time, the following attributes are usually included

Path length The length of the path is likely to influence the decision maker’schoice Also, this attribute is easy to measure Note that it may be highlycorrelated with travel time, especially in uncongested networks

Travel cost In addition to the obvious behavioral motivation, including travelcost in the utility function is necessary to forecast the impact of tolls andcongestion pricing, for example It is common practice to distinguish the so-

called out-of-pocket costs (like tolls), which are directly associated with aspecific trip, from other general costs (like car operating costs).

Transit specific Attributes specific to route choice in transit networks includenumber of transfers, waiting and walking time and service frequency

Others Traffic conditions (e.g level of congestion, volume of conflictingtraffic streams or pedestrian movements), obstacles (e.g number of stopsigns, number of traffic lights, number of left turns against traffic), road types(e.g dummy variable capturing preference for freeways) and road condition(e.g surface quality, number of lanes, safety, scenery) are some of the otherattributes that may be considered Whether to include them in the utilityfunction depends on their behavioral pertinence in a specific context, and ondata availability

Finally, the level of path overlapping can also be included in the utility function of

a path It is not one of its attributes per se It is more a measure of how thealternative is perceived within a choice set Several formulations have been proposed

in the literature

Commonality Factor. Cascetta et al (1996) propose the following specification for

the commonality factor

where is the length3 of links common to paths i and j, and and are the overall

length of paths i and j, respectively is a coefficient to be estimated Theparameter may be estimated or constrained to a convenient value, often 1 or 2.Considering the path choice example in Figure 1, the commonality factor for path

1 is zero because it does not overlap with any other path The commonality factor forpaths 2a and 2b is

Path Size The Path Size model, first proposed by Ben-Akiva and Bierlaire (1999),

is an application of the notion of elemental alternatives and size variables In theroute choice context, we assume that an overlapping path may not be perceived as adistinct alternative Indeed, a path contains links that may be shared by several paths.Hence, the size of a path with one or more shared links may be less than one We

include in the utility function of path i for individual n a size variable defined by

3 or any other link-additive attribute

and is the set of links in path i; and are the length of link a and path i,

respectively; is the link-path incidence variable that is one if link a is on path j and

0 otherwise; and is the length of the shortest path in Considering again thepath choice problem from Figure 1, the size of path 1 is 1, and the size of paths 2a

and 2b is

Generalized Path Size Ramming (2001) proposed a generalized formulation of the

Path Size where

and G is a function with paratemer g The Exponential Path-Size formulation isobtained with Note that corresponds to a normalized version ofthe original Path Size model Based on experiments on a case study in Boston,Ramming (2001) observes a better behavior of the Path Size correction in terms of itscapability of reproducing observed data, compared to the Commonality Factor Also,

it is observed that low values of may lead to counter-intuitive results, motivating thegeneralized version

Logit route choice A Multinomial Logit Model with an efficient algorithm for

route choice has been proposed by Dial (1971) Using the concept of “reasonablepaths” to define the choice set and assuming the paths attributes to be link-additive,this algorithm avoids explicit path enumeration

As described earlier, the IIA property of the Multinomial Logit Model is the majorweakness of Dial’s algorithm in the context of highly overlapping routes Therefore,its use is limited to networks with specific topologies A Logit model may also beused with a choice set generation model, such as the Labeling approach, that results

in a small size choice set with limited overlap

Probit route choice Given the shortcomings of the Logit route choice model,Probit models have been proposed in the context of stochastic network loading by

Burrell (1968), Daganzo and Sheffi (1977) and Yai et al (1997) The two problems

in this case are the complexity of the variance-covariance matrix and the lack of ananalytical formulation for the probabilities The covariance structure can besimplified when path utilities are link-additive, the variance of link utility isproportional to the utility itself, and the covariance of utilities of two different links iszero The use of a factor analytic formulation, where the matrix is the link-pathincidence matrix, enables to reduce the complexity of the model from the number ofpaths in the network down to the number of links A Monte Carlo or Quasi-MonteCarlo simulation is often used to circumvent the absence of a closed analytical form

Cross-Nested Logit route choice Vovsha and Bekhor (1998) have proposed an

interesting Cross-Nested formulation, where each link of the network corresponds to

a nest, and each path to an alternative The parameters of the Cross-Nested logitare not estimated They capture the network topology and, consequently, the pathoverlapping Note that this approach can be combined with attribute-based pathoverlapping measures, like the commonality factor and the path-size

Hybrid Logit route choice Ramming (2001) has estimated a route choice model

based on Hybrid Logit Although there are some issues regarding the number ofdraws for the Simulated Maximum Likelihood Estimation, experiments on a casestudy in Boston report a good behavior of the Hybrid Logit approach, especiallywhen it is combined with the Path Size overlapping attribute

2.5 Departure Choice Applications

Modeling the choice of departure time appears in the context of dynamic trafficassignment as an extension of the route choice problem It is important to distinguishthe departure time choice itself and the choice of changing departure time The latterappears usually in the context of Traveler Information Systems, where individualsmay revisit a previous choice using additional information We now describe typicalmodeling assumptions associated with the departure time choice model

Decision-Maker

The central traveler’s characteristic of departure time choice models is thepreferred arrival time at the destination It is often presented as a time interval orwindow with variable length reflecting schedule flexibility Other relevant traveler’scharacteristics are the (monetary and psychological) penalties for early and latearrivals In the context of a departure time change, the individual’s “habitual”departure time must also be known