DISCRETE CHOICE METHODS AND THEIR APPLICATIONS TO SHORT TERM TRAVEL DECISIONS Moshe Ben-Akiva and Michel Bierlaire

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1 AND THEIR APPLICATIONS TO SHORT TERM TRAVEL DECISIONS Mosh B-Akiva ad Michl Birlair Chaptr for th Trasportatio Scic Hadbook MIT, April 1999 Itroductio Modlig travl bhavior is a ky aspct of dmad aalysis, whr aggrgat dmad is th accumulatio of idividuals dcisios. I this chaptr, w focus o short-trm travl dcisios. Th most importat short-trm travl dcisios iclud choic of dstiatio for a o-work trip, choic of travl mod, choic of dpartur tim ad choic of rout. It is importat to ot that short-trm dcisios ar coditioal o log-trm travl ad mobility dcisios such as car owrship ad rsidtial ad work locatios. Th aalysis of travl bhavior is typically disaggrgat, maig that th modls rprst th choic bhavior of idividual travlrs. Discrt choic aalysis is th mthodology usd to 1

2 2 M. B-Akiva ad M. Birlair aalyz ad prdict travl dcisios. Thrfor, w bgi this chaptr with a rviw of th thortical ad practical aspcts of discrt choic modls. Aftr a brif discussio of gral assumptios, w itroduc th radom utility modl, which is th most commo thortical basis of discrt choic modls. W th prst th altrativ discrt choic modl forms such as Logit, Nstd Logit, Gralizd Extrm Valu ad Probit, as wll as mor rct dvlopmts such as Hybrid Logit ad th Latt Class choic modl. Fially, w laborat o th applicatios of ths modls to two spcific short trm travl dcisios: rout choic ad dpartur tim choic. Discrt Choic Modls W provid hr a brif ovrviw of th gral framwork of discrt choic modls. W rfr th radr to B-Akiva ad Lrma (1985) for th dtaild dvlopmts. Gral modlig assumptios Th framwork for a discrt choic modl ca b prstd by a st of gral assumptios. W distiguish amog assumptios about th: 1. dcisio-makr -- dfiig th dcisio-makig tity ad its charactristics; 2. altrativs -- dtrmiig th optios availabl to th dcisiomakr; 3. attributs -- masurig th bfits ad costs of a altrativ to th dcisio-makr; ad 4. dcisio rul -- dscribig th procss usd by th dcisiomakr to choos a altrativ.

3 M. B-Akiva ad M. Birlair 3 Dcisio-makr Discrt choic modls ar also rfrrd to as disaggrgat modls, maig that th dcisio-makr is assumd to b a idividual. Th idividual dcisio-makig tity dpds o th particular applicatio. For istac, w may cosidr that a group of prsos (a houshold or a orgaizatio, for xampl) is th dcisio-makr. I doig so, w may igor all itral itractios withi th group, ad cosidr oly th dcisios of th group as a whol. W rfr to dcisio-makr ad idividual itrchagably throughout this chaptr. To xplai th htrogity of prfrcs amog dcisio-makrs, a disaggrgat modl must iclud thir charactristics such as th socio-coomic variabls of ag, gdr, ducatio ad icom. Altrativs Aalyzig idividual dcisio makig rquirs ot oly kowldg of what has b chos, but also of what has ot b chos. Thrfor, assumptios must b mad about availabl optios, or altrativs, that a idividual cosidrs durig a choic procss. Th st of cosidrd altrativs is calld th choic st. A discrt choic st cotais a fiit umbr of altrativs that ca b xplicitly listd. Th choic of a travl mod is a typical xampl of a choic from a discrt choic st. Th idtificatio of th list of altrativs is a complx procss usually rfrrd to as choic st gratio. Th most widly usd mthod for choic st gratio uss dtrmiistic critria of altrativ availability. For xampl, th possssio of a drivr s lics dtrmis th availability of th auto driv optio. Th uivrsal choic st cotais all pottial altrativs i th applicatio s cotxt. Th choic st is th subst of th uivrsal choic st cosidrd by, or availabl to, a particular idividual.

4 4 M. B-Akiva ad M. Birlair Altrativs i th uivrsal choic st that ar ot availabl to th idividual ar thrfor xcludd from th choic st. I additio to availability, th dcisio-makr s awarss of th altrativ could also affct th choic st. Th bhavioral aspcts of awarss itroduc ucrtaity i modlig th choic st gratio procss ad motivat th us of probabilistic choic st gratio modls that prdict th probability of ach fasibl choic st withi th uivrsal st. A discrt choic modl with a probabilistic choic st gratio modl is dscribd latr i this chaptr as a spcial cas of th latt class choic modl. Attributs Each altrativ i th choic st is charactrizd by a st of attributs. Not that som attributs may b gric to all altrativs, ad som may b altrativ-spcific. A attribut is ot cssarily a dirctly masurabl quatity. It ca b ay fuctio of availabl data. For xampl, istad of cosidrig travl tim as a attribut of a trasportatio mod, th logarithm of th travl tim may b usd, or th ffct of out-ofpockt cost may b rprstd by th ratio btw th out-ofpockt cost ad th icom of th idividual. Altrativ dfiitios of attributs as fuctios of availabl data must usually b tstd to idtify th most appropriat. Dcisio rul Th dcisio rul is th procss usd by th dcisio-makr to valuat th attributs of th altrativs i th choic st ad dtrmi a choic. Most modls usd for travl bhavior applicatios ar basd o utility thory, which assums that th dcisio-makr s prfrc for a altrativ is capturd by a valu, calld utility, ad th dcisio-makr slcts th altrativ i th choic st with th highst utility.

5 M. B-Akiva ad M. Birlair 5 This cocpt, mployd by cosumr thory of micro-coomics, prsts strog limitatios for practical applicatios. Th udrlyig assumptios of this approach ar oft violatd i dcisio-makig xprimts. Th complxity of huma bhavior suggsts that th dcisio rul should iclud a probabilistic dimsio. Som modls assum that th dcisio rul is itrisically probabilistic, ad v complt kowldg of th problm would ot ovrcom th ucrtaity. Othrs cosidr th idividuals dcisio ruls as dtrmiistic, ad motivat th ucrtaity from th limitd capability of th aalyst to obsrv ad captur all th dimsios of th choic procss, du to its complxity. Spcific familis of modls ca b drivd dpdig o th assumptios about th sourc of ucrtaity. Modls with probabilistic dcisio ruls, lik th modl proposd by Luc (1959), or th limiatio by aspcts approach proposd by Tvrsky (1972), assum a dtrmiistic utility ad a probabilistic dcisio procss. Radom utility modls, usd itsivly i coomtrics ad i travl bhavior aalysis, ar basd o dtrmiistic dcisio ruls, whr utilitis ar rprstd by radom variabls. Radom Utility thory Radom utility modls assum, as dos th coomic cosumr thory, that th dcisio-makr has a prfct discrimiatio capability. Howvr, th aalyst is assumd to hav icomplt iformatio ad, thrfor, ucrtaity must b tak ito accout. Maski (1977) idtifis four diffrt sourcs of ucrtaity: uobsrvd altrativ attributs; uobsrvd idividual charactristics (also calld uobsrvd tast variatios ); masurmt rrors; ad proxy, or istrumtal, variabls.

6 6 M. B-Akiva ad M. Birlair Th utility is modld as a radom variabl i ordr to rflct this ucrtaity. Mor spcifically, th utility that idividual associats with altrativ i i th choic st C is giv by U i = V i + ε i, whr V i is th dtrmiistic (or systmatic) part of th utility, ad ε i is th radom trm, capturig th ucrtaity. Th altrativ with th highst utility is chos. Thrfor, th probability that altrativ i is chos by dcisio-makr from choic st C is P(i C ) = P[ U i U j j C ] = P[U i = max j C U j ]. I th followig w itroduc th assumptios cssary to mak a radom utility modl opratioal. Locatio ad scal paramtrs Cosidrig two arbitrary ral umbrs α ad, whr > 0, w hav that P[ U i U j j C ] = P[U i +α U j +α j C ] = P[ U i -U j 0 j C ]. Th abov illustrats th fact that oly th sigs of th diffrcs btw utilitis ar rlvat hr, ad ot utilitis thmslvs. Th cocpt of ordial utility is rlativ ad ot absolut. I ordr to stimat ad us a spcific modl arbitrary valus hav to b slctd for α ad. Th slctio of th scal paramtr is usually basd o a covit ormalizatio of o of th variacs of th radom trms. Th locatio paramtr α is usually st to zro. S also th discussio blow of Altrativ Spcific Costats.

7 M. B-Akiva ad M. Birlair 7 Altrativ Spcific Costats Th mas of th radom trms ca b assumd to b qual to ay covit valu c (usually zro, or th Eulr costat γ for Logit modls). This is ot a rstrictiv assumptio. If w dot th ma of th rror trm of altrativ i by m i = E[ε i ], w ca dfi a w radom variabl i = ε i - m i +c such that E[ i ]=c. W hav P[ U i U j j C ] = P[ V i + m i + i V j + m j + j j C ], a modl i which th dtrmiistic part of th utilitis ar V i + m i ad th radom trms ar i (with ma c). Th trms m i ar th icludd as Altrativ Spcific Costats (ASC) that captur th mas of th radom trms. Thrfor, w may assum without loss of grality that th rror trms of radom utility modls hav a costat ma c by icludig altrativ spcific costats i th dtrmiistic part of th utility fuctios. As oly diffrcs btw utilitis ar rlvat, oly diffrcs btw ASCs ar rlvat as wll. It is commo practic to dfi th locatio paramtr α as th gativ of o of th ASCs. This is quivalt to costraiig that ASC to zro. From a modlig viwpoit, th choic of th particular altrativ whos ASC is costraid is arbitrary. Howvr, Birlair, Lota ad Toit (1997) hav show that th stimatio procss may b affctd by this choic. I th cotxt of th Multiomial Logit Modl, thy show that costraiig th sum of ASCs to 1 is optimal for th spd of covrgc of th stimatio procss. This rsult is also gralizd for th Nstd Logit Modl. Th dtrmiistic trm of th utility Th dtrmiistic trm V i of ach altrativ is a fuctio of th attributs of th altrativ itslf ad th charactristics of th dcisio-makr. That is

8 8 M. B-Akiva ad M. Birlair V i = V(z i, S ) whr z i is th vctor of attributs as prcivd by idividual for altrativ i, ad S is th vctor of charactristics of idividual. This formulatio is simplifid usig ay appropriat vctor valud fuctio h that dfis a w vctor of attributs from both z i ad S, that is x i = h(z i, S ). Th choic of h is vry gral, ad svral forms may b tstd to idtify th bst rprstatio i a spcific applicatio. It is usually assumd to b cotiuous ad mootoic i z i. For a liar i th paramtrs utility spcificatio, h must b a fully dtrmid fuctio (maig that is dos ot cotai ukow paramtrs). Th w hav V i = V(x i ). A liar i th paramtrs fuctio is dotd as follows V = β x. i k ik k Th dtrmiistic trm of th utility is thrfor fully spcifid by th vctor of paramtrs β. Th radom part of th utility Amog th may pottial modls that ca b drivd for th radom parts of th utility fuctios, w dscrib blow th most popular. Th modls withi th Logit family ar basd o a probability distributio fuctio of th maximum of a sris of radom variabls, itroducd by Gumbl (1958). Probit ad Probitlik modls ar basd o th Normal distributio motivatd by th Ctral Limit Thorm.

9 M. B-Akiva ad M. Birlair 9 Th mai advatag of th Probit modl is its ability to captur all corrlatios amog altrativs. Howvr, du to th high complxity of its formulatio, vry fw applicatios hav b dvlopd. Th Logit modl has b much mor popular, bcaus of its tractability, but it imposs rstrictios o th covariac structur. Thy may b uralistic i som cotxts. Th drivatio of othr modls i th Logit family is aimd at rlaxig rstrictios, whil maitaiig tractability. W discuss hr th spcificatio ad proprtis of th modls from th Logit family (th Multiomial Logit modl, th Nstd Logit modl, th Cross-Nstd modl ad th Gralizd Extrm Valu modl). Aftr prstig th Probit modl, w itroduc mor advacd modls. Th Gralizd Factor Aalytical Rprstatio ad th Hybrid Logit modls ar dsigd to bridg th gap btw Logit ad Probit modls. Th Latt Class Choic modl is a furthr xtsio dsigd to xplicitly iclud i th modl discrt uobsrvd factors. Th LOGIT family Logit-basd modls hav b widly usd for travl dmad aalysis. Practitiors ad rsarchrs hav usd, rfid ad xtdd th origial Biary Logit Modl to obtai a class of modls basd o similar assumptios. W rfr to this class as th Logit-family. Multiomial Logit Modl Th Logistic Probability Uit, or th Logit Modl, was first itroducd i th cotxt of biary choic whr th logistic distributio is usd. Its gralizatio to mor tha two altrativs is rfrrd to as th Multiomial Logit Modl. Th Multiomial Logit Modl is drivd from th assumptio that th rror trms of th utility fuctios ar idpdt ad idtically Gumbl

10 10 M. B-Akiva ad M. Birlair distributd (or Typ I xtrm valu). That is, ε i for all i, is distributd as: ( ε η) [ ] F( ε ) = xp, > 0 [ ] f ( ε ) = ( ε η ) xp ( ε η) whr η is a locatio paramtr ad is a strictly positiv scal paramtr. Th ma of this distributio is whr γ = lim k k i= 1 η + γ / 1 l( k) i is th Eulr costat. Th variac of th distributio is π 2 /6 2. Th probability that a giv idividual chooss altrativ i withi th choic st C is giv by Vi P ( i C ) =. V j j C A importat proprty of th Multiomial Logit Modl is Idpdc from Irrlvat Altrativs (IIA). This proprty ca b statd as follows: Th ratio of th probabilitis of ay two altrativs is idpdt of th choic st. That is, for ay choic sts C 1 ad C 2 such that C 1 C ad C 2 C, ad for ay altrativs i ad j i both C 1 ad C 2, w hav

11 M. B-Akiva ad M. Birlair 11 ( i C1) ( j C ) P P 1 ( i C2) ( j C ) P =. P A quivalt dfiitio of th IIA proprty is: Th ratio of th choic probabilitis of ay two altrativs is uaffctd by th systmatic utilitis of ay othr altrativs. Th IIA proprty of Multiomial Logit Modls is a limitatio for som practical applicatios. This limitatio is oft illustratd by th rd bus/blu bus paradox i th modal choic cotxt. W us hr istad th followig path choic xampl. Cosidr a commutr travlig from origi O to dstiatio D. H/sh is cofrotd with th path choic problm dscribd i Figur 1 blow, whr th choic st is {1,2a,2b} ad th oly attribut cosidrd for th choic is travl tim. W assum furthrmor that th travl tim for ay altrativ is th sam, that is V(1) = V(2a) = V(2b) = T, ad that th travl tim o th small sctios a ad b is δ. 2 O Path 2 a D b Path 1 Figur 1 Th probability of ach altrativ providd by th Multiomial Logit Modl for this xampl is

12 12 M. B-Akiva ad M. Birlair T 1 P( 1 { 1, 2a, 2b}) = P( 2a { 1, 2a, 2b}) = P( 2b { 1, 2a, 2b}) = T = 3 j { 1, 2a, 2b} Clarly, this rsult is idpdt of th valu of δ. Howvr, wh δ is sigificatly smallr tha th total travl tim T, w xpct th probabilitis to b clos to 50%/25%/25%. Th Multiomial Logit Modl is ot cosistt with this ituitiv rsult. This situatio appars i choic problms with sigificatly corrlatd radom utilitis, as it is clarly th cas i th path choic xampl. Idd, altrativs 2a ad 2b ar so similar that thir utilitis shar may uobsrvd attributs of th path ad, thrfor, th assumptio of idpdc of th radom parts is ot valid i this cotxt. Nstd Logit Modl Th Nstd Logit Modl, first proposd by B-Akiva (1973 ad 1974), is a xtsio of th Multiomial Logit Modl dsigd to captur som corrlatios amog altrativs. It is basd o th partitioig of th choic st C ito M sts C m such that ad C = M m=1 C m C m C m = m m. Th utility fuctio of ach altrativ is composd of a trm spcific to th altrativ ad a trm associatd with th st. If i is a altrativ from st C m, w hav ~ U = V + ~ ~ ε + V + ~ ε. i i i Cm Cm

13 M. B-Akiva ad M. Birlair 13 Th rror trms ~ ε i ad ~ ε Cm ar supposd to b idpdt. As i th Multiomial Logit Modl, th rror trms ~ ε i ar assumd to b idpdt ad idtically Gumbl distributd, with scal paramtr m (it ca b diffrt for ach st). Th distributio of ~ ε Cm is such that th radom variabl maxu j is Gumbl j C m distributd with scal paramtr. Each st withi th choic st is associatd with a composit utility V Cm = ~ V Cm 1 + l m j C m Th scod trm is calld xpctd maximum utility, LOGSUM, iclusiv valu or accssibility i th litratur. Th probability for idividual to choos altrativ i withi st C m is giv by whr ~ mv j P( i C ) = P( C C ) P( i C ) m m. ad P( C m P( i C ) = m C ) j C m = M l= 1 VC m ~ mvi ~ mv. j VCl, Paramtrs ad m rflct th corrlatio amog altrativs withi th st C m. Th covariac btw th utility of two altrativs i ad j i st C m is

14 14 M. B-Akiva ad M. Birlair Cov( U i, U j var( ~ ε ) = 0 Cm ad th corrlatio is 2 1 Corr( U, ) = 2 i U j m 0 ) if i ad j C othrwis if i ad j C othrwis Thrfor, as th corrlatio is o gativ, w hav 0 1, m m m. ad = 1 corr( U, U ) = 0. m i j Th paramtrs ad m ar closly rlatd i th modl. Actually, oly thir ratio is maigful. It is ot possibl to idtify thm sparatly. A commo practic is to arbitrarily costrai o of thm to a spcific valu (usually 1). As a xampl, w apply ow th Nstd Logit Modl to th rout choic problm dscribd i Figur 1. W partitio th choic st C ={1,2a,2b} ito C 1 ={1} ad C 2 ={2a,2b}. Th probability of choosig path 1 is giv by 1 P (1 {1,2a,2b}) =, 1+ 2 whr 2 is th scal paramtr of th radom trm associatd with C 2, ad is th scal paramtr of th choic btw C 1 ad C 2. 2

15 M. B-Akiva ad M. Birlair 15 Not that w rquir 0 / 2 1. Th probability of th two othr paths is P (2a C ) = P(2b C ) = I this xampl, w d to ormaliz ithr or 2 to 1. I th lattr cas w hav ad 1 P( 1 { 1, 2a, 2b}) = P( 2a C ) = P( 2b C ) = ad w rquir that 0 1. Not that for =1 w obtai th MNL rsult. For approachig zro, w obtai th xpctd rsult wh paths 2a ad 2b fully ovrlap. A modl whr th scal paramtr is ormalizd to 1 is said to b ormalizd from th top. A modl whr o of th paramtrs m is ormalizd to 1 is said to b ormalizd from th bottom. Th lattr may produc a simplr formulatio of th modl. W illustrat it usig th followig xampl. I th cotxt of a mod choic with C ={bus, mtro, car, bik}, w cosidr a modl with two sts: C 1 ={bus,mtro} cotais th public trasportatio mods ad C 2 ={car,bik} cotais th privat trasportatio mods. For th xampl s sak, w cosidr th followig dtrmiistic trms of th utility fuctios: V bus =β 1 t bus ; V mtro =β 1 t mtro ; V car =β 2 t car ; V bik =β 2 t bik

16 16 M. B-Akiva ad M. Birlair whr t i is th travl tim usig mod i ad β 1 ad β 2 ar paramtrs to b stimatd. Not that w hav o paramtr for privat ad o for public trasportatio, ad w hav ot icludd th altrativ spcific costats i ordr to kp th xampl simpl. Applyig th Nstd Logit Modl, w obtai t t t ( 1β1 bus 1β 1 mtro 1β1 bus l + ) P ( 1 bus) = 1β1t bus 1β1tmtro t t t 2 t l( 1β1 bus 1β 1 mtro ) l( 2β2 car + + β 2 bik ) Dfi θ 1 = / 1, θ 2 = / 2, β 1 * = 1 β 1 ad β 2 * = 2 β 2 to obtai P(bus) = * β1tbus * β1 tbus + * β1 tmtro β * t * t 1 bus β + 1 mtro 1l θ β* t * t 1 bus β + 1 mtro 1l θ + θ β * t * t 2 car β + 2 bik 2 l with 0 θ 1,θ 2 1. This formulatio simplifis th stimatio procss. For this raso, it has b adoptd by th B-Akiva ad Lrma (1985) txtbook ad i stimatio packags lik ALOGIT (Daly, 1987) ad HiLoW (Birlair, 1995, Birlair ad Vadvyvr, 1995). W mphasiz hr that ths packags should b usd with cautio wh th sam paramtrs ar prst i mor tha o st. Spcific tchiqus ispird from artificial trs proposd by Bradly ad Daly (1991) must b usd to obtai a corrct spcificatio of th modl. I th abov xampl, if 1 = 2, th imposig th rstrictio β 1 =β 2 is straightforward. Howvr, for th cas of 1 2 ad β 1 =β 2 =β, w dfi β*= 1 2 β ad crat artificial ods blow ach altrativ, with a scal 2 for th first st ad scal 1 for th scod. W rfr th radr to Kopplma ad Ch (1998) for furthr discussio. A dirct xtsio of th Nstd Logit Modl cosists i partitioig som or all sts ito sub-sts which ca i tur, b dividd ito sub-sts. Th modl dscribd abov is valid at vry,

17 M. B-Akiva ad M. Birlair 17 layr of th stig, ad th whol modl is gratd rcursivly. Bcaus of th complxity of ths modls, thir structur is usually rprstd as a tr. Clarly, th umbr of pottial structurs rflctig th corrlatio amog altrativs ca b vry larg. No tchiqu has b proposd thus far to idtify th most appropriat corrlatio structur dirctly from th data. Th Nstd Logit Modl is dsigd to captur choic problms whr altrativs withi ach st ar corrlatd. No corrlatio across sts ca b capturd by th Nstd Logit Modl. Wh altrativs caot b partitiod ito wll sparatd sts to rflct thir corrlatio, th Nstd Logit Modl is ot appropriat. Cross-Nstd Logit Modl Th Cross-Nstd Logit Modl is a dirct xtsio of th Nstd Logit Modl, whr ach altrativ may blog to mor tha o st. Similar to th Nstd Logit Modl, th choic st C is partitiod ito M sts C m. Morovr, for ach altrativ i ad ach st m, paramtrs α im (0 α im 1) rprstig th dgr of mmbrship of altrativ i i st m ar dfid. Th utility of altrativ i is giv by U = ~ V + ~ + ~ ε V + ~ ε + lα. Th rror trms ~ ε i ad im i i Cm Cm im ~ ε C m ar idpdt. Th rror trms ar idpdt ad idtically Gumbl distributd, with uit ~ ε i scal paramtr (this assumptio is ot th most gral, but simplifis th drivatio of th modl). Th distributio of such that th radom variabl maxu j C m jm ε ~ C m is is Gumbl distributd with scal paramtr. Th probability for idividual to choos altrativ i is giv by

18 18 M. B-Akiva ad M. Birlair whr M P( i C ) = P( C C ) P ( i C ) m m m= 1 P( C m C ) = M l= 1 VC m VCl, ad V Cm ~ = V Cm + l Vi αim P ( i Cm ) = ~ V α j C m ~ V j jm α. j C m This modl was first prstd by McFadd (1978) as a spcial cas of th GEV modl that is prstd blow. It was applid by Small (1987) for dpartur tim choic ad by Vovsha (1998) for rout choic. Gralizd Extrm Valu Th Gralizd Extrm Valu (GEV) modl has b drivd from th radom utility modl by McFadd (1978). This gral modl cosists of a larg family of modls that iclud th Multiomial Logit ad th Nstd Logit modls. Th probability of choosig altrativ i withi C is jm ~ j G V V 1 J (,..., ) Vi V V 1 J (,..., ) V i P( i C ) =. G,

19 M. B-Akiva ad M. Birlair 19 J is th umbr of altrativs i C ad G is a o-gativ diffrtiabl fuctio dfid o IR + J with th followig proprtis: 1. G is homogous of dgr > 0 1, 2. lim G( x1,..., xi,..., x J ) =, i = 1,..., J xi 3. th kth partial drivativ with rspct to k distict x i is ogativ if k is odd, ad o-positiv if k is v, that is, for ay distict i 1, i k {1, J } w hav k k G J ( 1) ( x) 0 x IR x... x +. i1 ik Th Multiomial Logit Modl, th Nstd Logit Modl ad th Cross-Nstd Logit Modl ar GEV modls, with for th Logit modl, G( x) = J x i i = 1 G( x) = for th Nstd Logit modl ad M m= 1 m x i i C m m 1 McFadd s origial formulatio with =1 was gralizd to >0 by B-Akiva ad Fraçois (1983).

20 20 M. B-Akiva ad M. Birlair M G( x) = α m= 1 j C for th Cross-std Logit modl. m jm x j m Multiomial Probit Modl Th Probability Uit (or Probit) modl should hav b calld Normit, for Normal Probability Uit modl. It is drivd from th assumptio that th rror trms of th utility fuctios ar ormally distributd. Th Probit modl capturs xplicitly th corrlatio amog all altrativs. Thrfor, w adopt a vctor otatio for th utility fuctios: U = V + ε, whr U, V ad ε ar (J 1) vctors. Th vctor of rror trms ε =[ε 1,ε 2,...,ε J ] T is multivariat ormal distributd with a vctor of mas 0 ad a J xj variac-covariac matrix Σ. Th probability that a giv idividual chooss altrativ i from th choic st C is giv by P( i C ) = P( U U 0 j C ). j i Dotig i th (J -1 J ) matrix such that i U =[U 1 -U i,,u (i-1) -U i,u (i+1) -U i,, U -U i] T, w hav that Th dsity fuctio is giv by i U ~ N( i V, i Σ i T ). J

21 M. B-Akiva ad M. Birlair 21 whr fi ( x) = λ xp 1 ( 2 x V ) i T ( Σ i T i ) 1 ( x V i ) ad λ = (2π ) J 1 2 Σ i T i 1/ ( i C ) = P( i 0) =... fi ( x) dx1... dxi 1dxi dxj P U. Th matrix i is such that th ith colum cotais -1 vrywhr. If th ith colum is rmovd, th rmaiig (J 1 J 1) matrix is th idtity matrix. For xampl, i th cas of a triomial choic modl, w hav = W ot that h multifold itgral bcoms itractabl v for a rlativly low umbr of altrativs. Morovr, th umbr of ukow paramtrs i th variac-covariac matrix grows with th squar of th umbr of altrativs. W rfr th radr to McFadd (1989) for a dtaild discussio of multiomial Probit modls. W prst blow th Gralizd Factor Aalytic formulatio dsigd to dcras th dgr of complxity of Probit modls. Gralizd Factor Aalytic Spcificatio of th Radom Utility Th gral formulatio of th factor aalytic formulatio is

22 22 M. B-Akiva ad M. Birlair U = V + ε = V + F ζ, whr U is a (J 1) vctor of utilitis, V is a (J 1) vctor of dtrmiistic utilitis, ε is a (J 1) vctor of radom trms, ζ is a (M 1) vctor of factors which ar IID stadard ormal distributd, ad F is a J M matrix of loadigs that map th factors to th radom utility vctor. This spcificatio is vry gral. If M = J, th umbr of altrativs i th uivrsal st, w ca dfi th matrix F as th Cholsky factor of th variac-covariac matrix Σ, that is Σ=F F T. F is th obtaid by rmovig th rows associatd with uavailabl altrativs. W dscrib hr spcial cass of factor aalytical rprstatios. Thy ar discussd i mor dtails by B-Akiva ad Bolduc (1996). Htroscdasticity A htroscdastic 2 modl is obtaid wh F is a J J diagoal matrix. Lt T b a diagoal matrix cotaiig th altrativ spcific stadard dviatios σ i. F is obtaid by rmovig th rows ad colums of th uavailabl altrativs. W obtai th followig modl, i scalar form: Factor Aalytic U i = V i + σ i ζ i. I this modl, th gral matrix F is dividd ito a matrix of loadigs Q ad a diagoal matrix T cotaiig th factor spcific stadard dviatios. W obtai th followig modl, 2 Htroscdasticity hr rfrs to diffrt variacs amog th altrativs. W us it i this cotxt to rfr to a diagoal variac-covariac matrix with pottially diffrt trms o th diagoal.

23 M. B-Akiva ad M. Birlair 23 Or, i scalar form: U =V + Q T ζ. M U = V + q σ ζ i i im m m m= 1 whr q im ar th lmts of Q ad σ m ar th diagoal lmts of T. Th matrix Q is ormalizd so that = 1. 2 q im i, m Wh th matrix Q is kow ad dos ot d to b stimatd th modl is rfrrd to as th Error Compot Formulatio. Gral Autorgrssiv Procss W cosidr th cas whr th rror trm ε is gratd from a first-ordr autorgrssiv procss: ε = ρw ε + T ζ, whr W is a (J J ) matrix of wights dscribig th ifluc of ach compot of th rror trms o th othrs, ad ζ ~N(0,I j ), as abov. Th w hav ε = (I-ρW ) -1 T ζ, which is a spcial cas of th factor aalytic rprstatio with Q = (I-ρW ) -1.,

24 24 M. B-Akiva ad M. Birlair Hybrid Logit Modl Th Multiomial Probit with a Logit krl, or Hybrid Logit 3, modl has b itroducd by B-Akiva ad Bolduc (1996). It is itdd to bridg th gap btw Logit ad Probit modls by combiig th advatags of both of thm. It is basd o th followig utility fuctios: U i = V i + ξ i + υ i, whr ξ i ar ormally distributd ad captur corrlatio btw altrativs, ad υ i ar idpdt ad idtically distributd Gumbl variabls. If th ξ i ar giv, th modl corrspods to a Multiomial Logit formulatio: P( i C, ξ ) = j C Vi + ξi V j + ξi whr ξ =[ξ 1,..., ξ J ] T is th vctor of uobsrvd radom trms. Thrfor, th probability to choos altrativ i is giv by P ( i C ) = ξ P( i C, ξ ) f ( ξ ) dξ whr f(ξ ) is th probability dsity fuctio of ξ. This modl is a gralizatio of th Multiomial Probit Modl wh th distributio f(ξ ) is a multivariat ormal. Othr distributios may also b usd. Th arlist applicatio of this modl to captur radom cofficits i th Logit Modl (s blow) was by Cardll ad Dubar (1980). Mor rct rsults highlightd th robustss of Hybrid Logit (s McFadd ad Trai, 1997)., 3 Somtims calld Mixd Logit

25 M. B-Akiva ad M. Birlair 25 Hybrid Logit with factor aalytical rprstatio. Th Hybrid Logit Modl ca b combid with th factor aalytical rprstatio prstd abov to allow practical stimatio usig a simulatd maximum liklihood procdur (s B-Akiva ad Bolduc, 1996). Th Probit rror trm is trasformd usig ay appropriat factor aalytical rprstatio to obtai th followig choic probability: P( i C ) = P( i C, ζ ) N( 0, I ) dζ. M ζ This formulatio of th multiomial Probit is spcially usful wh th umbr of altrativs is so high that th us of probability simulators is rquird. Radom cofficits W coclud our discussio of th Hybrid Logit modl with a formulatio of th Multiomial Logit Modl with radomly distributd cofficits: U =V +υ = X β +υ. Assum that β ~N(β,Ω). If Γ is th Cholsky factor of Ω such that ΓΓ T =Ω, w rplac β by β+γζ to obtai U = X β+ X Γζ +υ. It is a Hybrid Logit modl with a factor aalytic rprstatio with F = X Γ. Latt class choic modl Latt class choic modls ar also dsigd to captur uobsrvd htrogity. Th udrlyig assumptio is that th htrogity is gratd by discrt costructs. Ths costructs ar ot dirctly obsrvabl ad thrfor ar rprstd by latt classs.

26 26 M. B-Akiva ad M. Birlair For xampl, htrogity may b producd by tast variatios across sgmts of th populatio, or wh choic sts cosidrd by idividuals vary (latt choic st). Th latt class choic modl is giv by: S P( i X ) = P( i X ; βs, Cs ) P( s X ; θ) s= 1 whr S is th umbr of latt classs, X is th vctor of attributs of altrativs ad charactristics of dcisio-makr, β s ar th choic modl paramtrs spcific to class s, C s is th choic st spcific to class s, ad θ is a ukow paramtr vctor. Th modl P( s X ; θ) is th class mmbrship modl, ad is th class-spcific choic modl. P( i X ; β, C ) A spcial cas: latt choic sts s s A spcial cas is th choic modl with latt choic sts: P( i, ) = P( i C ) P( C ) C G whr G is th st of all o-mpty substs of th uivrsal choic st M, ad P(i C ) is a choic modl. W ot hr that th siz of G grows xpotially with th siz of th uivrsal choic st. Th latt choic st ca b modld usig th cocpt of altrativ availability. Th, a list of costraits or critria ar usd to charactriz th availability of altrativs. For ach altrativ i, a biary radom variabl A i is dfid such that A i =1 if altrativ

27 M. B-Akiva ad M. Birlair 27 i is availabl to idividual, ad 0 othrwis. A list of K i costraits is dfid as follows: A i = 1 if H ik 0, k=1,,k i. For xampl, i a path choic cotxt, o may cosidr that a path is ot availabl is th ratio btw its lgth ad th shortst path lgth is abov som thrshold, rprstd by a radom variabl. Th associatd costrait for path i would th b: L i / L * 2+ε whr L * is th lgth of th shortst path, L i is th lgth of path i ad ε a radom variabl with zro ma. It mas that, o avrag, paths logr tha twic th lgth of th shortst path ar rjctd. Th probability for a altrativ to b availabl is giv by P(A i = 1) = P(H ik 0 k=1,,k i ). Th latt choic st probability is th: P( C P( Ai = 1, i C ad Aj = 0, j C ) ) =. 1 P( A = 0, l M ) l If th availability critria ar assumd to b idpdt, w hav i C P( A i = 1) l M j C = 0) P( C ) =. 1 P( A = 0) l P( A Swait ad B-Akiva (1987) stimat a latt choic st modl of mod choic i a Brazilia city. i

28 28 M. B-Akiva ad M. Birlair Rout choic applicatios Th rout choic problm ca b statd as follows. Giv a trasportatio twork composd of ods, liks, origis ad dstiatios; ad giv a origi o, a dstiatio d ad a trasportatio mod m, what is th chos rout btw o ad d o mod m. This discrt choic problm has spcific charactristics. First, th uivrsal choic st is usually vry larg. Scod, ot all physically fasibl altrativs ar cosidrd by th dcisio-makr. Third, th altrativs ar usually corrlatd, du to ovrlappig paths. W ow dscrib typical assumptios associatd with rout choic modls. Dcisio-Makr Th travlr s charactristics most oft usd for rout choic applicatios ar: Valu-of-tim. Obviously, travl tim is a ky attribut of altrativ routs. Its ifluc o bhavior, howvr, may vary across idividuals. A Wall Strt brokr is likly to prciv ad valuat travl tim diffrtly from a rtird Floridia. Th ssitivity of a idividual to travl tim is usually rfrrd to as th valu-of-tim. It ca b rprstd by a cotiuous variabl (.g., th dollar-valu quivalt of a miut spt travlig) or by a discrt variabl idtifyig th dcisio-makr s valu-of-tim as low, mdium or high. Accss to iformatio. Iformatio about twork coditios may sigificatly ifluc rout choic bhavior. Thrfor, it may b importat that a rout choic modl xplicitly diffrtiats travlrs with accss to such iformatio from thos without accss. It may b modld by a sigl biary

29 M. B-Akiva ad M. Birlair 29 attribut (accss/o accss) or by svral biary variabls idtifyig th typ of iformatio availabl to th travlr (pr-trip iformatio, o-board computr, tc.) Trip purpos. Th purpos of th trip may sigificatly ifluc th rout choic bhavior. For xampl, a trip to work may b associatd with a palty for lat arrival, whil a shoppig trip would usually hav o such palty. Howvr, ot that th trip purpos may b highly corrlatd with th valu-of-tim. Altrativs Idtifyig th choic st i a rout choic cotxt is a difficult task. Two mai approachs ca b cosidrd. First, it may b assumd that ach idividual ca pottially choos ay path btw hr/his origi ad dstiatio. Th choic st is asy to idtify, but th umbr of altrativs ca b vry larg, causig opratioal problms i stimatig ad applyig th modl. Morovr, this assumptio is bhaviorally uralistic. Scod, a rstrictd umbr of paths may b cosidrd i th choic st. Th choic st gratio ca b dtrmiistic or stochastic, dpdig o th aalyst s kowldg of th problm. Dial (1971) proposs to iclud i th choic st rasoabl paths composd of liks that would ot mov th travlr farthr away from hr/his dstiatio. Th lablig approach (proposd by B-Akiva t al., 1984) icluds paths mtig spcific critria, such as shortst paths, fastst paths, most scic paths, paths with fwst stop lights, paths with last cogstio, paths with gratst portio of frways, paths with o lft turs, tc. A applicatio of a implicit probabilistic choic st gratio modl has b proposd by Casctta ad Papola (1998), whr th utility fuctio associatd with path i by idividual is dfid as

30 30 M. B-Akiva ad M. Birlair U i = V i + l q i + ε i, whr q i is a radom variabl with ma q i 1 = k 1+ A γ k X ik A X ik ar th attributs for availability ad prcptio of th path ad γ k ar paramtrs to b stimatd. Som rct modls (Nguy ad Pallottio, 1987, Nguy, Pallottio ad Gdrau, 1988) cosidr hyprpaths istad of paths as altrativs. A hyprpath is a collctio of paths with associatd stratgis at dcisio ods. This tchiqu is particularly appropriat for a public trasportatio twork. Attributs I dscribig th attributs of th altrativs to b icludd i th utility fuctio, w d to distiguish btw lik-additiv ad o-lik-additiv attributs. If i is a path composd of liks a Γ i, x i is a lik-additiv attribut of i if x =, i x a a Γi whr x a is th corrspodig attribut of lik a. For xampl, th travl tim o a path is th sum of th travl tims o liks composig th path. Qualitativ attributs ar i gral o-likadditiv. For xampl, a biary variabl x i qual to o if th path is a habitual path ad 0 othrwis, is o-additiv. I th cotxt of public trasportatio, variabls lik trasfrs ad fars ar usually ot lik-additiv. Th distictio is importat bcaus som modls,.

31 M. B-Akiva ad M. Birlair 31 dsigd to avoid path umratio, us lik attributs ad ot path attributs. Amog th may attributs that ca pottially b icludd i a utility fuctio, travl tim is probably th most importat. But what dos travl tim ma for th dcisio-makr? How dos sh/h prciv travl tim? May modls ar basd o th assumptio that most travlrs ar sufficitly xpricd ad kowldgabl about usual twork coditios ad, thrfor, ar abl to stimat travl tims accuratly. This assumptio may b satisfactory for plaig applicatios usig static modls. With th mrgc of Itlligt Trasportatio Systms, modls that ar abl to prdict th impact of ral-tim iformatio hav b dvlopd. I this cotxt, th "prfct kowldg" assumptio is cotradictory with th ITS srvics that provid iformatio. Svral approachs ca b usd to captur prcptios of travl tims. O approach rprsts travl tim as a radom variabl i th utility fuctio. This ida was itroducd by Burrll (1968) ad is capturd by a radom utility modl. Also, th ucrtaity or th variability of travl tim alog a giv path ca b xplicitly icludd as a attribut of th path. I additio to travl tim, th followig attributs ar usually icludd. Path lgth. Th lgth of th path is likly to ifluc th dcisio makr s choic. Also, this attribut is asy to masur. Not that it may b highly corrlatd with travl tim, spcially i ucogstd tworks. Travl cost. I additio to th obvious bhavioral motivatio, icludig travl cost i th utility fuctio is cssary to forcast th impact of tolls ad cogstio pricig, for xampl. It is commo practic to distiguish th so-calld out-of-pockt costs (lik tolls), which ar dirctly associatd

32 32 M. B-Akiva ad M. Birlair with a spcific trip, from othr gral costs (lik car opratig costs). Trasit spcific. Attributs spcific to rout choic i trasit tworks iclud umbr of trasfrs, waitig ad walkig tim ad srvic frqucy. Othrs. Traffic coditios (.g. lvl of cogstio, volum of coflictig traffic strams or pdstria movmts), obstacls (.g. umbr of stop sigs, umbr of traffic lights, umbr of lft turs agaist traffic), road typs (.g. dummy variabl capturig prfrc for frways) ad road coditio (.g. surfac quality, umbr of las, safty, scry) ar som of th othr attributs that may b cosidrd. Whthr to iclud thm i th utility fuctio dpds o thir bhavioral prtic i a spcific cotxt, ad o data availability. Dcisio Ruls Shortst path Th simplst possibl dcisio rul i th rout choic cotxt assums that ach idividual chooss th path with th highst utility. Modls basd o dtrmiistic utility maximizatio ar supportd by fficit algorithms to comput shortst paths i a graph (.g. Dijkstra, 1959, ad Dial, 1969). Howvr, th bhavioral limitatios of this approach hav motivatd th dvlopmt of stochastic modls basd o th radom utility modl.

33 M. B-Akiva ad M. Birlair 33 Logit rout choic A Multiomial Logit Modl with a fficit algorithm for rout choic has b proposd by Dial (1971). Usig th cocpt of rasoabl paths to dfi th choic st ad assumig th paths attributs to b lik-additiv, this algorithm avoids xplicit path umratio. As dscribd arlir, th IIA proprty of th Multiomial Logit Modl is th major wakss of Dial s algorithm i th cotxt of highly ovrlappig routs. Thrfor, its us is limitd to tworks with spcific topologis. A Logit modl may also b usd with a choic st gratio modl, such as th Lablig approach, that rsults i a small siz choic st with limitd ovrlap. Probit rout choic Giv th shortcomigs of th Logit rout choic modl, Probit modls hav b proposd i th cotxt of stochastic twork loadig by Burrll (1968) ad Dagazo ad Shffi (1977). Th two problms i this cas ar (i) th complxity of th variaccovariac matrix ad (ii) th lack of a aalytical formulatio for th probabilitis. Th covariac structur ca b simplifid wh path utilitis ar lik-additiv, th variac of lik utility is proportioal to th utility itslf, ad th covariac of utilitis of two diffrt liks is zro. A Mot-Carlo simulatio is oft usd to circumvt th absc of a closd aalytical form. C-Logit Th C-Logit modl, proposd by Casctta t al. (1996) i th cotxt of rout choic, is a Multiomial Logit Modl which capturs th corrlatio amog altrativs i a dtrmiistic way. Thy add to th dtrmiistic part of th utility fuctio a trm,

34 34 M. B-Akiva ad M. Birlair calld commoality factor, that capturs th dgr of similarity btw th altrativ ad all othr altrativs i th choic st. ( Vi CFi ) P ( i C ) = ( V j CF j ) j C Casctta t al. (1996) propos th followig spcificatio for th commoality factor CF i = β CF l j C L i ij L L j γ whr L ij is th lgth 4 of liks commo to paths i ad j, ad L i ad L j ar th ovrall lgth of paths i ad j, rspctivly. β CF is a cofficit to b stimatd. Th paramtr γ may b stimatd or costraid to a covit valu, oft 1 or 2. Cosidrig th path choic xampl i Figur 1, th commoality factor for path 1 is zro bcaus it dos ot ovrlap with ay othr path. Th commoality factor for paths 2a ad 2b is β CF l(1 + [(T-δ)/T] γ ). Not that th commoality factor of a altrativ is ot o of its attributs. It ca b viwd as a masur of how th altrativ is prcivd withi a choic st. PS-Logit Path-Siz Logit is a applicatio of th otio of lmtal altrativs ad siz variabls. S B-Akiva ad Lrma (Chaptr 9) for dtails about modls with lmtal ad aggrgat altrativs. I th rout choic cotxt, w assum that a 4 or ay othr lik-additiv attribut

35 M. B-Akiva ad M. Birlair 35 ovrlappig path may ot b prcivd as a distict altrativ. Idd, a path cotais liks which may b shard by svral paths. Hc, th siz of a path with o or mor shard liks may b lss tha o. W iclud a siz variabl i th utility of a path to obtai th followig modl : whr th siz S i is dfid by ( Vi + l Si ) P ( i C ) = ( V j + l S j ) j C l 1 a S i = * a Γ L i i LC δ aj j C L j, ad Γ i is th st of liks i path i; l a ad L i ar th lgth of lik a ad path i, rspctivly; δ aj is th lik-path icidc variabl that is o if lik a is o path j ad 0 othrwis; ad L * C is th lgth of th shortst path i C. Cosidrig agai th path choic problm from Figur 1, th siz of path 1 is 1, ad th siz of paths 2a ad 2b is (T+δ)/2T. It is itrstig to ot that th siz variabl formulatio is quivalt to th commoality factor formulatio for th xtrm cass whr δ=0 or δ=t, assumig that β CF =1 ad for ay valu of γ. Howvr, th two modls ar diffrt for itrmdiary valus. Dpartur choic applicatios Modlig th choic of dpartur tim appars i th cotxt of dyamic traffic assigmt as a xtsio of th rout choic problm. It is importat to distiguish th dpartur tim choic

36 36 M. B-Akiva ad M. Birlair itslf ad th choic of chagig dpartur tim. Th lattr appars usually i th cotxt of Travlr Iformatio Systms, whr idividuals may rvisit a prvious choic usig additioal iformatio. W ow dscrib typical modlig assumptios associatd with th dpartur tim choic modl. Dcisio-Makr Th rlvat travlr s sociocoomic charactristics ar similar to thos of rout choic modls. Additioal charactristics importat for dpartur tim choic ar dsird arrival tim ad paltis for arly ad lat arrival. I th cotxt of dpartur tim chag, th idividual s habitual or historical dpartur tim must also b kow. Altrativs Th choic st spcificatio for dpartur tim modls is a itricat problm. First, th cotiuous tim must b discrtizd. A rasoabl compromis must b foud btw a fi tmporal rsolutio ad th modl complxity. Idd, thr is a pottially larg umbr of altrativs, particularly for ralistic dyamic traffic applicatios. Scod, th corrlatio amog altrativs caot b igord, spcially wh tim itrvals ar short. Choosig btw th 7:45-7:50 ad 7:50-7:55 tim itrvals diffrs from choosig btw 7:45-7:50 ad 8:45-8:50. I th first cas, th two altrativs ar likly to shar uobsrvd attributs. Third, th prcptio of th altrativs dpds o trip travl tim. Most idividuals roud tim ad th roudig may dpd o th travl tim ad travl tim variability. For short trips, 7:52 may b roudd to 7:50, whras for log trips it may b approximatd by 8:00.

37 M. B-Akiva ad M. Birlair 37 Th choic st gratio cosists of dfiig a accptabl rag of dpartur tim itrvals cosidrd by a idividual. A commo procdur is basd o th dsird arrival tim AT *. Lt [AT,mi ; AT,max ] b th fasibl arrival tim itrval, ad lt [TT,mi ; TT,max ] b th rag of travl tims. Th th itrval of accptabl dpartur tims is [DT,mi ; DT,max ] = [AT,mi -TT,max ; AT,max -TT,mi ]. Small (1987) aalyzd th impact of trucatig th dpartur tim choic st. H cocludd that thr is o problm if th tru modl is a Multiomial Logit Modl. Som adjustmts ar dd if a Cross-Nstd Logit with ordrd altrativs is assumd. I th cotxt of dpartur tim chag, th altrativs may b dscribd i a rlativ way. Atoiou t al. (1997) propos a choic st with fiv altrativs: do ot chag, switch to a arlir or a latr dpartur, by o or two tim itrvals. Attributs Travl tim is a ky attribut of dpartur tim altrativs. Othr importat attributs ar th arly ad lat schdul dlays. Giv a dsird arrival tim AT *, a palty-fr itrval is dfid: [AT *,mi;at *,max]. It is assumd that th idividual suffrs o palty if th arrival tims lis withi th itrval. Th actual arrival tim AT is qual to DT +TT(DT ),whr TT(DT ) is th travl tim if th trip starts at tim DT. Th arly schdul dlay is dfid as Max [AT *.mi AT, 0] ad th lat schdul dlay is dfid as Max [AT AT *.max, 0].

38 38 M. B-Akiva ad M. Birlair I th cotxt of dpartur tim chag, a palty ca also b associatd with dpartur tims vry diffrt from th habitual choic, capturig th irtia associatd with habits. Dcisio Ruls Small (1982) ad Casctta t al. (1992) us Multiomial Logit Modls for dpartur tim choic. Howvr, th itrisic aformtiod corrlatio amog altrativs is ot capturd by such modls. Small (1987) proposd a Ordrd Gralizd Extrm Valu modl. It is a Cross-Nstd Logit Modl, whr m adjact dpartur tim itrvals ar std togthr, capturig thir itrisic corrlatio. A sigl dpartur tim itrval blogs to m diffrt sts, sourc of th cross-std structur. I th cotxt of dpartur tim chag, Atoiou t al. (1997) propos a Nstd Logit Modl for joit choic of dpartur tim ad rout. Liu ad Mahmassai (1998) propos a Probit modl whr day-to-day corrlatio is assumd. Coclusio Discrt choic mthods ar costatly volvig to accommodat th rquirmts of spcific applicatios. This is a xcitig fild of rsarch, whr a dp udrstadig of th udrlyig thortical assumptios is cssary both to apply th modls ad dvlop w os. I this Chaptr, w hav summarizd th fudamtal aspcts of discrt choic thory, ad w hav itroducd rct modl dvlopmts, illustratig thir richss. A discussio o rout choic ad dpartur tim choic applicatios hav show how spcific aspcts of ral applicatios must b addrssd.

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