NOISY DEMAND AND MODE CHOICE DAVID KAHN Trânsponânon SyûensCe.t€i, Caûbridee.MA 02142.U.S A. ANDRÉ Dri PaLM^ McMaslerUnive6ny,Hmilton, OnGno,Canada and JE^N Louls DENEUBoURG Unire6nél-ibÈ de Bruxelles,Bruxelle!,Belgique (Re.eired 21 FebruaLr 1981.in rer^ed lom 1a Ma, 1941) Abstract-Modc choiceunde.stæhaiicallyvaryinsdenandis nudiedvia a dynanicnalbematical nrodelwhich descnb€sthe bebaviouralinte@tons berweentoprlation eoups The norièl s dêleloped by assuningcompetingltltuctivny fùnciiônsbr autonobilù md pubhc tansû wh(h motilat rhù use wbcn rhr dcna.d is auotrcdro vârt stæbaiicalll, a sublecrro d olerrll demândi.r rranspodarion. sèt or stæhâsti. dihcrcnrial cqualionsdcsÙibing lhe nodel ùe oblai.en. Theseæ solved for their Qâdy sûre valùes.Il n foù.d rhal noisy d€nand can slructurelhe syslen qualir.tively difleFntly ùan wh€n the d€nrandis fixed. The noiseis iound lo generllly Educe the level ot public l(nsit nde^bip. at shich ncw Êgincs dcùr a.d, mo$ i.tcrcslingl!,it bu! ù d$ chanees thè vâlùesofrbc rhtcshold indùccsne* scady-sraresoluiionslor rideAhip al critical valuesol rhe !ùiance ol denrand.In rhe later cas, rcne b€comesa eu.ce ol new possibilitiesin the syn€n hy dÊgenng r {eady sùre ntùtion not presentin lhe noise f@ enlnonn.nt

INTRODUCTION modclsàrc dynâmic, dcsc.ibing,forcxamplc,lhe !ime evolution A la.ge classof traDsportation of thc velocily pâtlem in trâffic flow problems (Hâight, 1961). Another exâInpleof dynamic reprcscnlationwhich will be oùr concemhere is the individuâl's demândfor trânsponation,as in thc modc choice problem(Hirngen. 1974tVr'ihon. 1979;Dôneubourger al. , I 979; de Palma and Lefevre, I 983i Kahn €r dl. , I 981). Àn inponant reasonfor using a dynamic rcprcsentation is to capturethe effec! ofchanging rclativc attractivitiesof the modesas the mode choicesare being made. This cannot be adequatelydone wirh a static representation.The values of the parameteruintroducedinto thesemodelsare detemined cxogencouslyor are estimarcd,assum' ing them to be constânl-In rnâny situâtions,however, il is well recognizedthat this is not lrue. For example, lrip times will vâry frorn hour to hour and frorn day to day dep€nding upon such random factorsas weâlherconditions,the percentâgeof vehicle rnix at a given rnomentor the occunenceol an accident. It is rhereforc wonhwhile 1o cxaminc rhc effect of fluctuationsin thc pârùmetersof a transportationmodel on its b€havior.The fluctuationsaffectingthe parameterscould tÂke into accountseâsonalvâriâtioDsand vârialionsâmong individualsin thcir rcsistânccto mode change (Goodwin. 1977). as weii as random fluctuations. l'or linear systems (Arnold. 1974), if the fluctuations are rapid enough with regârd to the deterministic tiûre scale of the model. then the average values of the parameters are sufficient to determine the qualitative behavior of the systen. For nonlinear systems, howev€r, (he fluctuations may induce quaiitative changes in the system: Such phenomena are câlled noiseinducedtransitions(L€lèver ?r al.. 1979i Honthemte and lÊfèver, 1980). ln Section2 we recall a dynamic choice model for transpo(alion modes.In Section3 wc derivethe stochasticversionofthis model using somebasicresultsfrom the theôry of stochastic diftèrential equalioos;a simple version of this model wilt be analyzed.Finally, in Section4, we presen.the results of numerical solutions of a more complex versioDoi thc model and discusssome extensions. t,13

t14

D. KÂHN.A. DEparMÀmd J. L DETTUBoLko THE DETERMINISTIC MODEL

Recently.systemsof nonlineûrdifferentialequâtions (Deneubourg ?l dl., i9?9; Kahner dl. , 198I . 1983)bavebeendeveloped ro describethe tempomlevotutionof differentmodes of ransportationin comp€ririonwith eachother for riders. Thesemodelsffe rreat€das choice rnodelsamongthe differenr transpondionmodes. We considerhere(seeDeneubourg er dl., 19?9)the câseof a choiceberweena privâ(e node. the câr, and publicone,the bus. Iæl X and y be the nurnbe.of individuêls who chooserhe car and lhe bus. resDectivetv. D repre'cnr\ùe rorâlnumbe'X - y ol 'ndi\iduatsuho are tacingrhe modechorc;p,obtem ând rs assumedto b€ consianrfor the presenttime (unetasricilemand).Iæl y.6, y) a;d Vt(X, D be the observâbleutility functionsof rhe car and the bus. The ftêcrion of individuals usins the cù and Lhebu, i' a-umed ro be denlabje from lhe jogir modet(McFadden.t981, ds

4= D

" ' "" evtx. tt + e\tx. t).

! = ,' _ \

D

(l)

D.

The derivation of this equationin â dynamicâlcontext bâs been presenredin de palrnâ and teferre r 1981,.Theseaurhorrhavepropos€d â dynarnrLat adiusrminrproce- whichde.cribes huq rndi!iduaj. \elecrâ modechorce rherea trdnspondtion mode/d. a function ot rrme{i.e. lrom dây to day). As â function of time, the equationsof evotutionof the car and bus users are(BenAkiva er a/., 1984:de Pâlmaandlæfèvre.1983)

dx

dv

dx

dt

(2)

Tbe dymmic modechoicernodel, inrroduc€din Deneubourg?r rl. (19?9) is surnmârize.d below.Fu' the car it i\ a\\umfd lhal rheulilrr) i\ con\lanr: y. = ln d,

(3)

For the bus, the utility is âssumed to be a funclionof tbe ..publicity..usedto plomotebus "imitâtive" usage(factor6:) andofthe behâviour effecr)resultingin bususage lor bandwagon (seede Palrnaandl€fèvre, 1983): ys = ln (@,) + o:_),).

(.4)

Tbe lariÂblesX and y in eqn(2) may approachstarionâryvalueswhererheywilt no tonger . chângewith turther changesin time_ The srâtionarysolulions of eqns (2), defined by dy/ dt = 0 anddyldr = 0, ale preciselythe forrnulasgivenin eqn(t)_ For rhefuncrionatforrns (3) ând14)selected,we obtain,for rhesrâtionary srare, X D

al + Ozy + nrf '

Y = D - X .

(s)

The solutionsof eqn (5) are given by Y = O ;X = D

d,Y' + (6, - a,D)Y + (.1t - D@) :

(6)

O.

(7)

Thereâre qualiiâtively rwo differenl bifurcariondiagramsdisplayingthe srârjonarystatesas a functionof, for the y solutionsof eqn (7). The biturcationdiâgarnswilt yield informarion

Noisy demand and nodc choice

---

Stâble Stationary State -

Unstable Stationary State

Y

û,'t dz'l

Oz'z

Y'

T----------+ t

-

Stabl€ Stationary State

----

UnstableStationaryState

D

q2' 1

0z'os

ôc Fig. I BituEaliondiaSraùsfor ôc casewhen(a) 6: > V.r,, a and(b) O: < \4" dl

on which of the possiblesolutionsare acceptedby the sysGn. as well âs informalion on rheii stabilny propenies when subjected to pcnurbations in X and y. The diâgârns âre shown in Fig. 1. In Fig. I we have introducedcritical densitiesdefinedby

D ,l - o- a 1

l4!tq,\\'

t8)

- O2

(e)

146

D. l(^HN, A. DE pÀda a.d J L. DENEûBoIRC

For Fig. l(a), the publicityfacror@, hâsa higb vàluel@, > (d, dJ1r,l_In this casewe see thatno busservicecanbe iniiiâledif rhedemandD is lessrhanthe criticaldernând D:. lf the publicityis low IFig. l(b)l l@, < (il1lr:)1'11, a lârg€rdernandis necessary before (D > ,. > D!). However,evenin rhiscase,ifr. < D < D!', bui service serviceis possibte may not be observed, sincetwo stabteregimescoexisr,onewiih zeroridership(yi = 0) and onewjth positiveridership(f+ > 0).

A STOCHASTIC VERSION O! THE MODEL

In the preceding anaiysis,rhe rotâtdemandD wasâssumed to b€ consrant. In facr,for a numberof reâsons D vâriesas â functionof time. D wilt vârybecause of seÀlonalvariarions, pnce modificâtionswhich affe€t rhe elasticity of demêndfor transponadonând orher exr€mal factors.Theseincludeemployment conditions,weatherconditio;sand sirnilarextematiries influencing the demândfor ûânsponation.The summarioî of rheseeffecrs implies thar the demand, appears to be fluctuating. Lel u. no$ dellne! llocha\ricdemandlunctrùn D,=D+é,

(10)

whereD is the averagedemaDdând (, is rhe fluctuaringcomponenrof the dernand.We will assumeùat +.

(le)

of theprobability,dP(y)/dv, followsthesignofthe functiono: In thiscasethederivative whercoi = (z.it/O, \DO, - o.\).

p0)

D probabiliiydnmbution;D = a,/O,. Fis. 2. Stationâry

D. K^HN.

, DI P^LM,\ and J. L. IÈNEÙBoTRC

r(v) o:

o

"' - 9 0"

Y

Fi! ]\ Display orrùnction r(x)t D.l k tO)).' @roll The€ are rwo qualitatively difièrenl situationsin ihis câse: (1)lf Dêl a'lo. t o'lOJ. ihen F(v) is a monotonically decreasing functionof v with F(0) : (zrlt@r@O, - ar) andF(D d,/Ot = 0 (seeFig. 3). Thusihe probabilityP(D (wheno: > oi). will eitherexhibila maximum(for dr < d:) or be nonotonicâllydecreasing This is shownin Fig. 4(â) and (b)- Wc thusseethat the introductionof fluctuarions in the dcmândhirsthe effecrof shiftingthe maximumof P(}J from $e deterministicvalu€Y : D - all Otfor o' = 0) to zerofor or > o: (seeFigs. 3 anda). (ii) The secondsituâtionis wheDD > i d,/O:. Thena(n is an increasing functionof y for y 2Dl3 .rr/@2.This situarionis depictedin Fig. 5. Thustheprobabiliry P(y) hasa uniquemaxirnumwhenor < ol exhjbitsbotha mininum ândâ rnâximumwhen o? < 01 < (2D13)3O2la1 and is a monotonically decreasing functionof y \|hen 01 > l2Dl These 3)r@1o,. threedifferentcasesare shownin Fig. 6. in the casesdepict€din Fig. 6(a) and6(c).fluctuarions in thedemandhaveintroduced similârphenomenê al.€adydiscussed with reference to Fig. 4(â) ând(b). However,ir the casedepictediD Fig. 6(b) a qualitâtivelynew

0

D

Y

Sktionary probabihy distnbùtion..Dtl @tr6.- t ta I O)l

Noisv d€nand and n.dc choicc

r(v) , 2 D , s0 2

tsro,

2J-4t

s 0"

^

C{r Az

Fig 5 Display oltunctiona(y)rD> 1(t,rO,) behavioris exhibitedin thc syslernwhich now admitstwo maxina. This is an examplewhercby the noise has introducedâ peâk-splittingtransition.This would correspondin thc macroscopic descriptionto a new observableslâie of the system. N U M E R I C A LR E S U L T S ln this section rle presentnumericàlt€snlis for the more complex nodel (l: I 0 treâled in the deleministic casein Deneubourget al. 11979).This will cûable us to comparedirecdy the dererminisric ctse in which the denand D is coûsta with the stochâstic câse treâted here where the demandfluctuates.The equationfor the extremais given by

l +

D(q+ a,Y\ al + O,Y + ull

t)trt(6,+ ti{)(t9t + zah (ut+ O,Y + a,f)l 2

(20)

whichreducesto eqn(15) whend: = 0. \Ve remind rhe readerùal the extremav of the stationarypobâbiliry distribution P(v) macroscopic statesof thesynem.Themaximâ eqû Lsee (l3)l âreidentifiedwith theobservable nost ofits timearethestablesteadystates,whilethemimima spends ofP(v) whererheprocess whererelativelylitlle time is spentarethe unstâblestâtes.ln the limit a: + 0, the extrema (1979)1. eqùâlionlâlreâdyobtainedin Deneubourg of thedeterminist;c aresimplythesolotions differentialeqt we now comparethedeleministicsolutionswith solutionsfromthe stochastic (12) âsgivenby e.qn(20).This equationis solvedfo. r asa functionof thedemandD andas â func.ion of the variancet'r for the samenunerical valucs oi the pâramele.sâs in the deter' ministiccasô.Theresultsareshownin Figs.7 and8. In thesefigures,thesolidlinesreprcsent the maximaof the probability funclion P(v) correspondingto observablemacroscopicstates, and the dashedlines represenlthe minima correspondingto unobseflablemâcroscopicstates. Figure7, in which r is plottedas a tunctionof, for differentvâluesof d:, exhibitsall rkee phenomenainducedby a îuctuating demandas discussedin the previous section. we peak danpins, forcing the maxinum value of P(u to seethat the noisy demandhas caldsed becomedepressedfor all finite vâluesof rr comparedto the deternilislic câsewherea' = 0. The figure further showsthe phenomenonofp?dt sl,ri"8 by which the loca.ion of the transilionpoinrsareshifted. The prâcticâlsignificanceof this is that oneneedsa higherdemand

150

D. KN,

A DEPaLMÀ md J. L. DÊNr1ue@c

P(v)

Y Fie 6. Sbtionary ptubabihy dntribuiiont D > , (a /dJ.

D befbre bus serviceis initiated. The value ofrhe demandnecessùf to initiate bus servicefor difièrentvâluesof d'is cornpùled by crlculalingthe tunctiondDlôy = 0 and solvinCir for Y to obtâinthe cnticalvaluesof D. Finally the phenomenonof padl rplirir,g is Âlso seer ro occur, wheieby scopicallyobservâbles1âteappearsin the sys!,en.For example,at D : 5 onty one obsenâble stale âppeârsin the deterrninisticcased, = 0, whereasrwo observablesratesappearwhen a' - 10 (seeFig. 7).

Noisy dcmandand node .hoice

151

I ".'.\

Fig. ?. Ride^hip rs,sus ucmano.

Figure 8(a) and (b) show v as â fuûction of d' directly. Figurc 8(a) con€spondsto the previousFig. 5. \Ve note thal beyond â critical noise l€vel dl no bus serviceis possiblefor âbove. the givenlevelof demand,asdiscussed Figure 8(b) shows the appearanceof â new obse able macroscopicstate I I 0 for a given value of dle varianceo:. This, of course,is qualitativelynew, not possiblein lhe deterministiccâsewhe.€ o' = 0. Thus. here two non-zeroroots can co€xist; one root is the continuation of thedeterministic root(upperbrânch),whiletheotherdid not existin th€noiseftee deterministiccase. CONCLUSIONS

The imponanceof this work, we believe,lies in the new behâviorexhibitedby â transpoflâtion systen subjectto environmentalnoise uflder very broâd conditions. As long âs the noiseis multiplicaiive andthe systen non linear, whicb is the usuâlcâsefor real trânsportâtion âswell âsqualitâtive effèctsnotpresent in thenoisesystems, tbenoisemayinducequântitative This is trueevenif thesystemis in equilibriumin the sensethattherândom lreeenvironmentUuctuations or noisecanceloul on lhe average. The effectsinducedby the noiseareof the following tJ?es:first, thereis a generaldecreas€ in bus ridershipas the fluctuâtionsincreâse;secondly,ihe rninimumdemandnecessaryto have bus serviceincrea,.ies with increasingfluctualionsin demând;finally, the presenceof noisecan shift an ini.ially high level of bus ridershipoperationinùoâ low-tevel ridershipregime. Behâviourally,we interprettheseresultsto signily peoples rclu€tanceto frequenta service fmm dayto dây. which,thoughon average maintains a givenlevelofsewice,is unprediclable Thereis a mode switch to the car undertheseconditionsof variâbilily- Malhernâticâlly,fmrn the assumptions of the modeltseeeqns(2Ha)1, the fractionof individualstâling the bus increases with increasingdemandin a concavefâshion(seeFig. 7)- Thus, the downward fluctuârionsin total demad are more detrimentalthan the symmeiric upwârd fluctuationsin From the poin( of view of â bus conpany which wishes to intoduc€ a new tine, lhe presenceof â fluctuâting demandincreasesthe marketnecessaryto supponthe line. Thus the companymay wish to develop methodsto reducethe impact of t'luctuationsâs well as the level,the compânyrnâywish fluctuations themselves. On a long'rangeor overâllmanagerial to consider poticies and infrasructural changeswhich would help ao reduce the impact of

t52

D, K^H\, A Dtr PALMÀand J, L DlNtsU8oURc

Y

- ---

Stable unstable

D =2 . 5

F,e 8 Ride6hiD lesus vanance of demnd

fluctuâtions in demand.Oneway ro do thiswouldb€ to makethevelociryor frequency ofbus servicelesssensidve 10the demand. On anothernoæ operâlionallevel, the compânymây reducethe fluctuarionsdirectty by suchstratagems asthe useof back-upbuses.A lesscapital-intensive strategyfor reducingrhe fluctuationsin demandwould be to provide transportarionusels with information conceming tie-upsor bottlenecks.The idea here is ro relocarerhe locâl demandfrom one line ro ânorhcr as traffic conditionschange.ln fâct thereis a programwhich is now atremptingto determine the benefitsof providing such information to potential users(TransponadonSysremsCenter. l98l). This progam is calledConpuienzedRiderlnfbrrnarionSystcm(CRIS)and is being tesledin Pilrsburgh, E.ie, Albany,SanDiego,SaltLakeCity andColurnbus. Whilethedetails of the Plogrâm differ fmn city to city, the essentiâlcommon feature is rhe provision of informationto â telephonecaller on the time of anivai of â busâr stopsspecifiedby rhâtcâller. This informâ.ionis expectedio reducethe amountandthe vâriabiliry of wait time ai bus sroFj. It is hop€dthâtthis, in tum, will leadto increased ridership.To the extenrrhata reductionin

Non! demand mdnodcchoi.c 153 wait tine vânâbilitywill resultin an associated reductiorin dle variabilityof totaldemand, this papcr would predicran increasein lrânsit ridership. projectin Minneêpolis-St. Paulcouldprovide Thetlansirservicereliâbilitydemonstration dataneededto testthe model.The readeris referredro theMinneâpoliFst. PaulTrânsnSeflice Reliâbility Demonstrationreport (Multiplications, 1983)for detailedinfonnarionon tbe actual deûonstration resùlts.This typeof demonstration may be usedfor modeltestingby obtaining "before" datâ on the demândfor aansportationbetweentwo areasservicedby a bus route (theirroute5, say).Ihe daily variÂnce of this demandwouldbe obtainedas an averageover severâlwe€ks.Bus ridershiplevelswould also bc obtaincdfor tbis p€nod for this route. Following thesebeforedarâcollectionâctivities, r€al-timeholding strategieswould be usedto increasebus reliâbitily tkough increasedadhererceto scheduleddepanuretimes from the vdriousstopsalonglhe busroute.lf ii is assunedthal improvemcnt actuallyoccurs(asit did "âfter" in the demonstrâtion), dâtâwill be obtÂinedon demand.on the daily vârianceof demandand on bus .idelship. After accountingfor ridenhip changesdue to other faclors (the extemalitiet, the dâtâwould be analyzedfor relatiorshipsberwe€nrideruhipchangesand with rnodelpedictions,such changesin demandanddemandvaiability ând thencompàred ;n Figs. 7 atd 8. as shown.fo. exanple. The âùrhon would like ro ùmk Moshe Ben Akiva, R Litèler dnd R Aûôlt tnt ustul sùg A.horLd8e,ena gestions We xl$ kkc plcâsùE in acknowledgingI PriCogirctnr sue8èsingthè âppli.arion of rbc scha$r method ior lbe sûrdvot sdial Dhenonena.

REFERENCES Amold L. (1914) Sro.rasti. D'fcrcûiat Equnio \ wil.t Inren.ience, New Yôik. Ben Alira M . Cynâ M and dc Pdna A. (1984) Dymnic nodel ol peal penql consesrion ?ra,v, R4 l8B. 319 355 D€neùbouCJ L . de Palna A. a.d Kahn D. (1979) Dynamic mod€lsol conp€ilion bêtsècn tuspona.o. nodes. Eùircnnenl Ptûnn.lL, 665 673. dePalmaA. ( 198r)Mod€lsstæbaniqùes desconpartenenccolledilsdûs lessy$.hes.oûplexesPh.D DnserlarionUnileôilé LibE de Bruxeues.Bruielles. (1933)Indivdùâldê.is n nakinein dynmic collætiresy$€ms.J Marr. Jr. ,9, l{ll dè PâlnâA. ddktèveC t21 CoodwinP B (19??)Urr,, S.!d 14, 95 98 H.rght FFFFF A. (1963)Mzrr?tuital Th?orie: ofhqlir Ftov academic Press,New York Hdecn D T. (197,1)Irdap,rdr,on 3- 45 58 Kann D. , D€neùboùreJ. L. md de Palna A. ( l98l ) Tûnsponadonnole choice. ,]."ùr"rn.t Pla,". A 13, I 163 t1?4. KahnD., DeneubourC L L. dd de PalmaA ( I 983) Transl)(nâlon node choiceand.ny sùbuibantùbl ic iûnspondtion \èNice.Tranvn,RPs l?a. 25,13 Hosthemkc w dd kÈvcrR (1S80)A p€rtuôalion expatrsionfor exlemalwide bandMdtovian noise Applicatio.s Moher 10, 1.11 211 b trdsitions inducedby Onsrein Uhlenb€ckNoi€, z Pnt\'ik a londùed klcvcr R. ùd Houlheùlc W ( l9?9)Multiplelrdsinonsindùccdbyligbtinrensny flucrurionsinillumiÉl€dchenical syslens. Pr@. Nat. A.al. J.t. U.S.A. 16(6), 2190 2491. McFâdd€nD ( I98I ) Ecùnonetic h.xleh of piobabllsiic chôi.e In Manski C F md McFaddenD. (eds),Sû!.r!fal Atultsis ôl Dirr.L Datu \|ùh Eùnaneti. Appliûti.Â. Mll Pt.*. Cadbridge, Ma* Mùhiprications 11983)Mi,n?dpolk.St.Paul lrdnsù Serric. Rèliabi|i1\D.noatrûti.n, Fi.al Repon.Clmbddse, Masschùselts,for the Urbm MassT.osit Adninislration thrcùgbconlract nùnber DOI-TSC-1756. TÉnsponrtiônsy$ensCc.rr (1981)Scruiæùd MdbodsDcnônshùonPùgrah Reporl.UMTA MA,06-ùM9-8112. Conpùtcriæd Rider lnlbrn.tion Sysens. tsoslon. Mas. wirson A. C. (r9t9) Arp..ts o/ Catalroph? îhear! ond Bilur.olion Th.or- ih Rqiotut s.i.h... wP 2.19.Sch@l of Geogr.phy.Univc6riyof rÆds, kcds. E.Etmd.