This is a repository copy of Marginal valuations of travel time and scheduling, and the reliability premium.

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1 Ths s a epostoy copy of Magal valuatos of tavel tme ad schedulg, ad the elablty pemum. Whte Rose Reseach Ole URL fo ths pape: Atcle: Batley, R.P. (2007) Magal valuatos of tavel tme ad schedulg, ad the elablty pemum. Taspotato Reseach E, 43 (4). pp ISSN Reuse See Attached Takedow If you cosde cotet Whte Rose Reseach Ole to be beach of UK law, please otfy us by emalg epts@whteose.ac.uk cludg the URL of the ecod ad the easo fo the wthdawal euest. epts@whteose.ac.uk

2 Whte Rose Reseach Ole Isttute of Taspot Studes Uvesty of Leeds Ths s a autho poduced veso of a pape ogally publshed Taspotato Reseach Pat E. It has bee pee evewed, but does ot clude the fal publshe s pagato ad fomattg. Whte Rose Repostoy URL fo ths pape: Publshed pape Batley, R.P. (2007) Magal valuatos of tavel tme ad schedulg, ad the elablty pemum - Taspotato Reseach Pat E: Logstcs ad Taspotato Revew 43(4) pp Whte Rose Cosotum epts Repostoy epts@whteose.ac.uk

3 Magalvaluatosoftaveltmeadschedulg,adthe elabltypemum RchadBatley IsttutefoTaspotStudes,UvestyofLeeds,LeedsLS29JT,UK Telephoe+44(0) Fax+44(0) Abstact Pevouswokshaveestablshedsyoymtybetweetheotosof ucetatyaduelablty,explotgthsdevgmagalvaluatos oftaveltmeadschedulgudeucetaty.whlstvaldfofoecastg demad,suchvaluatosfaltollumatethecostsofbeaguelablty heeefeedtoasthe elabltypemum.thepapedevesmagal valuatosoftaveltmeadschedulgatthecetatyeuvalet,showg thesetodvegefomthoseudeucetaty.thatdvegece,whch epesetsthemagalvaluatoofelablty,asesthepossbltyofbas shouldthecostsofuelabltyotbecludedappasal. 1

4 Keywods Relablty,Valuato,ExpectedUtlty,RskAveso,RskPemum 2

5 1.Itoducto Althoughapecseudestadghasofteseemedelusve,tswdely acceptedthattheelabltyoftaspotsystemsmaympactupothechoces oftavelles.pevouseseachhasllumatedsevealfacetsofths poposto,butoftewthouttheauthotyofcompehesveevdeceo thevalueofelabltytotavelles.thatsuchevdeceslackgcapehaps, tu,beattbutedtothedffcultyoffomulatgaeseachappaatusthat caestheoetcalvaldty,ssghtful,butemaspactcable.these aspatosaethecoceofthepesetpape. Thetaskofevewgelevatlteatueswellsevedbytheecet cotbutosofnoladadpolak(2002)addejogetal.(2004),adt wouldseemuecessaytooffefuthecommetaythsegad.suffce tosay,dejogetal.dstgushbetweetheeappoachestothevaluatoof elablty,specfcally:i)meavs.vaaceofthetaveltmedstbuto,ii) pecetlesofthetaveltmedstbuto,adiii)schedulgmodels.the pesetpapeexplotsthethdappoachpatcula,whchsfoudedo thehypothessthattavellesmayaccommodateexpectatosofuelablty thoughthetpschedulg. 3

6 Itheaalyssoftpschedulg,Small s(1982)appoachhaseceved cosdeablesuppot.smallextedsthemcoecoomctheoyoftme allocato(becke,1965;desepa,1971),supplemetgtheusualobectve poblemofutltymaxmsatosubecttomoeyadtmecostatswtha tpschedulgcostatdevgfomvckey(1969).implctsmall s appoach,howeve,stheassumptothatschedulgchocesaemade udecetaty,adthswouldseemtomposecosdeableestctoots applcablty.theusualaccommodatoofucetatyatleasttemsof mcoecoomctheoystoefomulatetheobectvepoblemfomthe maxmsatoofutltytooeofmaxmsgexpectedutlty,asfst poposedbyvoneuma&mogeste(1947).thelattesdeed explotedbynolad&small(1995),whoestablshmegebetweethewoks ofvoneuma&mogesteadsmall. Twopatcula,butelated,popetesofNolad&Small saalyssmghtbe oted.fst,boththechoce(.e.depatuetme)adpayoff(.e.avaltme) dmesosaeepesetedcotuously;thscaestheattactoof pemttgeasycalculatooftheoptmaldepatuetme.secod,teests estctedtothemogcommuteofcatavelles.cotutydepatue tmewouldappeamoeeasoablefocatavellesthafousesofpublc taspotsevces,scethelatteaetypcallycostaedbyfxedsevce tevals.batesetal.(2001)developnolad&smallfuthe,fstcosdeg 4

7 tsameabltytopublctaspotuses,adtheapplygtheaalysstoa teestmagalvaluatosoftaveltmeadschedulgude ucetaty.thesemagalvaluatosdevefomchocesbetweetwo publctaspotsevces,wheeeachsevceoffesaageofdepatue tmes,adthecoseuetavaltmesaechaactesedbyucetaty. ThepesetpapefollowsthebascthessofNolad&SmalladBatesetal., butpusuesaumbeofextesos.tmesepesetedasadscete vaablebothdepatueadavaldmesos.notolysthsmoe fathfultovoneuma&mogeste(1947),buttpemtseady accommodatoofpublctaspotuses.thedsceteepesetato, futhemoe,sameabletooeofthepcpalaalytcaltoolsoftavel demadaalyss,amelytheradomutltymodel(rum).asdefomths pesetatoaldstcto,thesubstatvecotbutoofthepapesto applythewokgsofnolad&smalladbatesetal.toateest tavelles atttudestosk.specfcally,thepapeecoclesthetaspot plae sotoofuelabltywththemcoecoomst sotoofsk(e.g. Patt,1964;Aow,1970),adsodogevealstheteatmetofskwth magalvaluatosoftaveltmeadschedulgudeucetaty.ths povokesthepopostoofafuthemetcfovalugelablty,efeedto asthe elabltypemum.theelabltypemum,whchsdawfom aalogywthpatt s skpemum,solatesthosecostsasgspecfcally 5

8 fomuelabltyad,euvaletly,thebeeftsthatwouldtaspeshould uelabltybeelmated.thepapeexplotstheelabltypemum devgafuthesetofmagalvaluatosoftaveltmeadschedulg, thstmeatthe cetatyeuvalet.thelattevaluatosaethecompaed wththosedevedudeucetaty,aexecsethatsevestoyeld magalvaluatosofelabltypese.thepapecosdestheelevaceof magalvaluatosofelabltyfobothfoecastgadecoomcappasal. 2.Theoyofdvdualchoceudeucetaty Mcoecoomctheoyofdvdualchoceudeucetatysfoudedo thepopostothattheeexstssomeelatobetweeadvdual schoces udeskoucetatyadadstbutoofoutcomes.thshasbee explotedtheelabltylteatue;thepesetpapeweshallexame thepatculapopostothattavelleschooseatmeofdepatueothe bassofadstbutoofthecoseuetavaltmes.thetepetatoof thepobabltydstbutohasbeethesouceofsomecotetothe mcoecoomclteatue,scetsemboledwththedchotomybetwee skaducetaty.keyes(1921,1936)adkght(1921)aehelpfulths egad,chaactesgskasstuatoswheepobabltesofoutcomesae kow(okowable),aducetatyasstuatoswheesuchpobabltes 6

9 maybeethekowableodefable.thoughtoffesaappealgclaty, twouldotseemcucaltothesubseuetaalyssthatoecommtstoths oayothetypology.rathethetemsskaducetatywll,what follows,beusedtechageablywthoutmplcato. Moecetaltoouteeststhepecseatueoftheelatobetweea dvdual schocesudeucetatyadthedstbutoofoutcomes;ths maybefomalsedthefollowgtems: Let E beafteadexhaustvesetofmutuallyexclusve evets : { e } E =,..., 1 e K Let E beassocatedwthavecto w,whchsefeedtoasa pospect,ad gvesthepobablty payoff w k pk thateachevet e k E wlloccu,togethewththe tothedvdualshouldthatevetdeedoccu,thus: w = ( w w ; p,..., ) 1,..., 1 K p K Followgtheusualulesofpobablty,tsecessalythecasethat K p k k = 1 = 1,mplygthat 0 p 1fo k = 1,..., K. k 7

10 Let S beafteadexhaustvesetofsuchpospects,fomwhchthe dvdualsvtedtochoosehsohepefeedpospect: { w,..., } S = 1 w N ThesemalexpostoofvoNeuma&Mogeste(1947)establsheda setofecessayadsuffcetaxomsotheabovedeftossuchthata dvdualcouldbeepesetedasfchoosgthepospect w S thatyelds maxmumexpectedutlty.theseaxomshavesubseuetlybeeadapted vaousways;ofpatculaotethsegadaethecotbutosof Maschak(1950),HesteadMlo(1953)adFshbu(1970).Iwhat follows,howeve,weemafathfultotheogalexposto,whch povdesabassfothefollowgpoposto: Foaypaofpospects w, w S: w f ff Y ( ) Y ( w ) w w ( wheey w ) stheexpectedutltyofpospect w,adstselfgveby: 8

11 K ( ) = p U ( Y w w )foall w S (1) k = 1 k k ( k wheeu w ) stheutltydevgfompayoff wk. Havgsummasedthetheoy,letuscocludethssectowthdscussoof sevealssuesthatfollow.afstpottomakesthatvoneuma& Mogeste s(1947)axomsaeotsuffcettosuppotacotuous epesetatoofthepayoffdmeso,asadoptedbynolad&small(1995) adbatesetal.(2001),fowhchtheexpectatosopeato(1)would euetegato.acotuousepesetatosfactfeasble,butmust besuppotedbyasetofaxomsthatdvegesfomvoneuma& Mogeste s;seefshbu(1970)fofuthestucto. Asecodpotsthatadsceteepesetatoofthechocedmeso pomoteseadyapplcatotorum,asdemostatedbymaschaketal. (1963).Thscotastswththeheetlyambguouspocessof dscetsg acotuousdmeso(e.g.jotsakasaetal.,2004).iexplotgmaschak etal. sdemostato,tsusefultoatculatethebassfoadoptga pobablstcepesetato,whchsasfollows.cosdeadvdual facedwthaepeatedchocetaskudeucetaty.oaygveepetto ofthechocetask,heoshesabletoodeasetofpospectstemsof 9

12 expectedutlty,butosuccessveepettosthsodegmayshow vaablty.moefomally,thepobabltyofchoosgaypospect w S cabeexpessedasrum,suchthat: P ( w S) { Y ( w ) Y( w )} = P foall w S, (2) Athdpotcocesthecompaabltyofexpectedutlty Y ( w ) acossthe pospects w S.Baumol(1958)dsmssesthecovetoalwsdomthat choceudeucetatypemtsthemutatoofutltyfomaodal metctoacadaloe;thatstosay(1)elesdeedothepopostothat U ( wk ) scadal,butthsdoesotdstactfomtheeuemetthaty( w ) saetelyodalcostuct.moeove,theesobassfotepetg dffeecesexpectedutlty(oayatotheeof)acosspospects.whlst thedeftoofrum(2)adheestothseuemet,tsmpotatto esuethatapplcatosofrumtofoecastgoappasaldoot advetetlystaytocadalty.thelattepossbltyscosdeedby Batley(2006),whodemostatesthat logsum measuemetsofcosume suplusdevedfomrumcayafeeceofcadalty. Fally,tmghtbeackowledgedthatvoNeuma&Mogeste s(1947) aalysshasbeethesubectofsustaedassaultalmostevescetwas 10

13 coceved.ithecotempoaylteatue,expemetalecoomstshave beeactvecotbutos,pesetgstacesofdvdualchocesthat appaetlyvolateexpectedutltymaxmsato(kahema&tvesky s (2000)compedumcludessevealsuchwoks).Thshaspompted geealsatosofvoneuma&mogeste saalyss;foexample, sevealofthedetfedvolatosmaybeaccommodatedthougholea fomsotheevetpobabltes p k (1);seeEdwads(1955)adKahema &Tvesky(1979).Whlstmpotattoackowledge,thsctuehasotto datesucceededdeposgthepaadgmofexpectedutltymaxmsato, adtwouldtheefoeseemetelyeasoabletoadheetovoneuma& Mogestewhatfollows. 3.Thetheoyappledtotpschedulg ThecotbutosofNolad&Small(1995)adBatesetal.(2001)establsh pecedeceapplygthetheoyofdvdualchoceudeucetatyto thecotextoftpschedulg.thepuposeofthepesetsectosto explcatethsapplcato;thesubseuetaalyssfollowsthesepecedet wokseasoablyfathfully,savefotheepesetatooftmeasadscete vaable. 11

14 3.1.Pelmaydeftos Iapplygthetheoyoutledsecto2totpschedulg,twouldseem easoabletoepesetchocetemsofdepatuetmeadevetstems ofavaltme,adtopostulatethatucetatydevesfomthedstbuto ofavaltmesfoaygvedepatuetme.wthefeecetonolad& Small stypology,thsdstbutocouldfeasblydevefomecuetdelay, fomcdetelateddelay,ofombothoftheafoemetoed.moe fomally: Let A beafteadexhaustvesetofavaltmes: { a } A =,..., 1 a K Let D beafteadexhaustvesetofdepatuetmes: { d } D =,..., 1 d N Thelattecoespodstothechoceset S,asdefedsecto2: 12

15 { w,..., } S = 1 w N LettheexpectedutltyY ofaypospect w S begveby: Y K = p k = 1 k U k (3) wheeu stheutltydevgfomtheavaltme a,havgdepatedat d,ad k pk 3.2Utltyfucto stheassocatedevetpobablty. Itemastoeuputlty U k wthmoepecsefom.whlstthetheoyof k secto2wouldseemtomposefewestctososuchfom,thetp schedulglteatuedemostatescosdeablesuppotfosmall s(1982) utltyfucto,adthsfuctosstaghtfowadlyadoptedbynolad& Small(1995)adBatesetal.(2001).Iwhatfollows,weshallouselvesfollow thscoveto,butsodogdemostatethatsmall sutltyfucto mplespatculapopeteswthespecttotavelles atttudesto uelabltyavaltme. WthefeecetoSmall(1982),defeutlty: 13

16 U k = α T + βsde + γsdl + δl (4) k k k k whee: T staveltme SDE sscheduledelayealy SDLsscheduledelaylate L sadummyvaablethatsutaythecaseoflateaval,adzeo othewse HeceSmall sutltyfucto(4)cabeseetobealeafuctooffou attbutestaveltme,scheduledelayealy,scheduledelaylateada lateessdummywheethelattetheeaecodtoedbythe pefeed avaltme ( PAT )ofthetavelle,whchwetakeasgve.othsbass,let useexpesstheattbutesof(4)temsofoudmesosofteest avaltmeaddepatuetmefogve PAT : T k = a d k SDE SDL k k [( PAT a ),0] = max [( a PAT ),0] = max k k L = 1f ( a k PAT ) > 0, = 0othewse k 14

17 Nowcompletgouapplcatoofthetheoyofdvdualchoceude ucetatytothecotextoftpschedulg,letusexplcatethedeftoof thepospectvecto,thus: w = [( T SDE, SDL, L ),...,( T, SDE, SDL, L ); p,..., p ] 1, K K K K 1 K MoeovetheadoptoofSmall sutltyfuctomplesthepopostothat payoffsaegvebyaamalgamofthefouattbutesofthatfucto,thee ofwhchaemeasuedtmeuts,adthefouthasadummyvaable. ThuswhlepevousapplcatosofvoNeuma&Mogeste(1947) havefocussedheavlyomoetaypayoffs,twouldthepesetcotext seematual,adotoutsdetheemtofthetheoy,toepesetpayoffs tmeuts.iaycase,tavelcostsdootoutelyvaybyavaltme,such thattheavaltmedmesomayotbeeadlyameabletomoetsato. Thatdoesotpecludethepossbltythatavaltme(patcula,lateess) maycucdetalcosts.ratheaysuchcostswllaseasafuctoof avaltme,aswouldseemtutvethecaseoflateess,adtheatttudes oftavellestosuchcostswllmafesttheatttudestotheucetatyof avaltme. 15

18 Letusdeveaschematcepesetatooftheutltyfucto(4)foay depatuetme d D,butestctattetotoabaysubsetofavaltmes ~ A A.Ipatcula,let A ~ = { a, a },wheethefollowgelatoshold: a a < PAT < m a.thatstosay,let a betheealestfeasbleavaltme (.e.feeflowcodtos),adlet a ad a betwofutheavaltmes thataedefedabtalysavefotheeuemetthat m a fallsbefoethe PAT ad a aftethe PAT. Fgue1:Small sutltyfucto,fogvedepatuetme,wthα < β (ABOUT HERE) ThssllustatedbyFgue1,whchepesetsavaltmeothehozotal axs,adutltyothevetcal.allattbutesoftheutltyfuctoae bad,.e.α, β, γ, δ < 0 (4),adU stheefoedawtheloweghtuadat ofthefgue;otealsothatu ogatesattheealestfeasbleavaltme am.nowcosdetheutltydevedatavaltmes a ad a, espectvely: At a = a :U = α( a d ) + β ( PAT a ) At : a a = U = α ( a d ) + γ ( a PAT ) + δ 16

19 Cosdealsotheutltydevedastheavaltmeappoachesthe PAT fom a ad a,espectvely: As ( PAT a ) 0:U α ( PAT d ) α( a d ) + α( PAT ) As ( ) 0 a a PAT : α ( PAT d ) + δ α a d α a PAT + U ( ) ( ) δ WecaowestablshtheslopeoftheutltyfuctoU,whchfoaval tmesbefoethe PAT mustbe ( α β ),adfoavaltmesaftethe PAT mustbe ( α + γ ).Fopuposesofllustato,Fgue1explotsSmall s(1982) empcalfdgthatα < β,.e.thattaveltmemposesgeatedsutlty thascheduledelayealy.thsegedesthepopetythatu sstctly deceasgutlty,adsteepeaftethe PAT thabefoe.cotawse,f tsteadheldthat β < α,thethetwopotosofu wouldshowopposg slopes,wthavaltmesbefoethe PAT chaactesedbypostveslope,ad avaltmesaftethe PAT byegatveslope.ithslattecase,theabsolute slopeofthefstpotoofu maybegeatethaolessthathesecod poto,depedgotheelatvedsutltyofscheduledelayealy.fgue2 embodestheelatos β < α ad ( α β ) < ( α + γ ),suchthatthetwo potosofu showopposgslopes,wththepotobefoethe PAT caygloweabsoluteslopethathepotoaftethe PAT. 17

20 Fgue2:Small sutltyfucto,fogvedepatuetme,wth β < α (ABOUT HERE) Fally,tmghtbeobsevedthatFgues1ad2dvegefomthemoeusual pesetatoofsmall sutltyfucto,aspehapsllustatedbyfgue1of Batesetal.(2001).ThssbecauseFgues1ad2ofthepesetpape cosdebothtaveltmeadschedulgpaametes,wheeasfgue1of Batesetal.cosdesolyschedulgpaametes.Thepesetatoadopted thepesetpapesmotvatedbyadesetoclealyatculatetheelato ofthecompleteutltyfuctototheexpectedutltyfucto,ateestthat weshallpusuethefollowgsectos.moeove,tspehapshelpfulfo puposesofdstctotoefetofgue1ofbatesetal.assmall s schedulgfucto,adtofgues1ad2ofthepesetpapeassmall s utltyfucto. 3.3Expectedutltyfucto Foeasosofbevty,thsadsubseuetsectoswllfocusattetoo Small sutltyfuctowththepopetyα < β,.e.asfgue1, patculalyasthspopetycaestheempcalsuppotofsmall(1982) hmself.whlsttshouldbeeassuedthatthegeealpcplesofthe 18

21 aalyssextedalsotofuctoswththepopety β < α,.e.asfgue2, somedffeecesaalyssdoase,adtheseaeemakedupowhee elevat. Iodetopomoteexpostoalclaty,letusoceagaestct cosdeatotoabaysubsetofavaltmes ~ A A ~ = a a,,whee A {, } thstmemposg(atleastfothemomet)theestctoolythat a < Foeachdepatuetme d D,defethepospect: a. [( T, SDE, SDL, L ), ( T, SDE, SDL, L ); p,( p )] w = 1 Nowpopagatgexpectedutlty(3)wththeutltyfucto(4),letuswte theexpectedutltydevgfomtheabovepospect: Y = { p [ αt + βsde + γsdl + δl ]} + {( 1 p )[ α T + βsde + γsdl + δl ]} (5) Omoegeeally: Y ( T ) + βe( SDE ) + γe( SDL ) δe( L ) = α E + (6) 19

22 Fgue3:Utltyadexpectedutltyfuctos,fogvedepatuetme,wthα < β (ABOUTHERE) Thebaysubsetofavaltmes(5)pemtseasytaslatotothe dagammatcaalyssoffgue3,wheetheecasesmghtusefullybe detfeddepedgotheelatooftheavaltmestothe PAT,thus: Case1: a < a PAT Case2: PAT a < a Case3: a < PAT < a Letusdawsomeobsevatoselatgtoeachofthetheecases,asfollows: Case1: a < a PAT ExpectedutltylabelledY Case1 thefguewllthscasefallsomewhee othesectooftheutltyfuctou pecedgthe PAT ;pecselywhee wlldepedothepobablty p thatavaltme a actuallyoccus. Case2: PAT a < a Bycotasttothepevouscase,expectedutltyY Case 2 wllowfall somewheeothesectooftheutltyfuctou thatcomesaftethe PAT. Sceutltysstctlydeceasgtaveltme,tmustholdthat Case 2 Case1 Y < Y ;hececase1domatescase2. 20

23 Case3: a < PAT < a AgawthefeecetoFgue3,cosdethepaofavaltmes Case 3 a ad Case3 a,whchsatsfytheeuemetsofcase3,butaeothewsedefed Case 3 Case3 abtaly.ifwedetfythepotatwhch ad tesectthe a a utltyfuctou,theexpectedutltyy Case 3 wllfallsomewheeothe staghtleogthesetwopots.thedscotutyofu toduces somecomplextes,howeve,asweshallseeduecouse. 3.4Chocebetweepospects Havgdevedexpectedutltyfoaydepatuetme d D,letus cosdethechocebetweeapaofpospects w, w tothedepatuetmes d S,whchcoespod, d D.Futhemoe,let d < d,suchthat d s theealeofthetwodepatues.scevoneuma&mogeste s(1947) axomsefeetelytothepopetesofpas,abaychocewouldot seemueasoablyestctve.itmghtalsobeemakedthatbaychoce wouldseempatculalyameabletomplemetatowthstated Pefeece(SP). Followgsecto2,tmustholdthat: 21

24 w f ffy Y 0 w w f ffy Y 0 w w ~ ffbothy Y 0 ady Y 0 w Scethesetheepefeeceelatosaedctatedbythedffeece expectedutlty,letusdeve: Y Y ( d d ) + [( p p )( α a + β SDE + γ SDL + δ L) ] = α (7) whee: a = SDE = SDL = ( a a ) > 0 ( SDE SDE ) 0 ( SDL SDL ) 0 L = 1f a < PAT <, = 0othewse a Itmaytheefoebesee,wthefeeceto(7),thatthedffeeceexpected utltysafuctoofthedffeecedepatuetme,aswellasdffeeces expectedavaltme,expectedscheduledelayealy,expectedschedule delaylateadtheexpectedlateessdummy(wheethelattemayothewse beefeedtoasthepobabltyoflateess). 22

25 NowdectgattetotoFgue4,letustaslatethepecedgalgebac aalysstoadagammatcoe.theeade sattetosfstdawtothe utltyfuctosu ad U,whchpetatothedepatuetmes d ad d espectvely.wthefeecetothehozotalaxs,otethatthetwo depatueswll,assumgcostatfeeflowtaveltme,havedffeettmes ( ) ( ) fotheealestfeasbleavals(.e. am d < am d ).Thetwoutlty fuctosogate,theefoe,fomdffeetpotsotheavaltmeaxs. Wthefeecetothevetcalaxs,thetwofuctosaesepaatedbythe costat α ( d ) d,whchepesetstheutltydffeeceasgfomthe dffeecetheespectvedepatuetmes.othewsethetwofuctos caythesamepopetes,wthcommoschedulgpaametesad PAT. Fgue4:Chocebetweedepatuetmes,wthα < β (ABOUTHERE) Havgdefedtheelevatevets,wecaowdevetheexpectedutlty fuctoselatgtothepospects w ad w ;thesefuctosaelabelledy ad Y espectvelythefgue.pecselywheeexpectedutltyfallso thesefuctoswllbedetemedbytheexpectedavaltmes E E( a ).Ifhowevetholdsthat ( ) a ad a a,.e.theealeavaltmewththe m pospectsfallsatoaftetheealestfeasbleavaltme,twouldseem ucotovesaltomaketheassetothat p > p,.e.theealedepatue 23

26 d smoelkelytoaveat a thathelatedepatue d.thoughtsleft pactcetoempcalvestgatotodetfytheexpectedavaltmes ( ) a ( ) E ( ) ( ) a E ad,thefguellustatesastuatowhee E a < PAT < E, a wthaoutcomethat Y >.Ithsstuato,theefoe,pospect w would bechoseovepospect Y w. 4.Magalvaluatosoftaveltmeadschedulg,adtheelablty pemum Thepevoussectoappledthetheoyofdvdualchoceude ucetatytotpschedulg.thseupsuswththeecessaytheoyto owdevelopthepcpalteestofthepape,whchsthevalueof elabltytoadvdualtavelle.whlstackowledggthatthevalueof elabltyhasbeevaouslydefed(seedejogetal.,2004),thefollowg aalysswllexplotadextedbatesetal. s(2001)defto,whchs goudedthetheoyofsecto3.letusfstsummasebatesetal. s defto,befoecosdegtsexteso. 4.1Batesetal. smagalvaluatosoftaveltmeadschedulgudeucetaty 24

27 FollowgBatesetal.,theelabltyofavaltmemaybeepeseted temsoftheevetpobabltesofvoneuma&mogeste s(1947) aalyss;wthefeeceto(5),foexample,aychageelablty(.e. p ad p )wllmpactupoexpectedavaltmead,bymplcato,expected taveltme,expectedscheduledelayealy,expectedscheduledelaylate,ad theexpectedlateessdummy.applygthsepesetatoofelabltyto empcalvestgato,batesetal.devseadmplemetabaychocesp expemet,adfomthsdatafemagalvaluatosofexpectedtavel tme,expectedscheduledelayealy,expectedscheduledelaylateadthe expectedlateessdummy. LetusllustatetheubofBatesetal. sempcalvestgato,butwththe cotextofouowwokg.webegbysupplemetgtheexpectedutlty fucto(5)wthavaableepesetgtavelcost,otgmpotatlythat costsdexedbydepatuetmebutotbyavaltme.moefomally,fo aypospect w S let: Yˆ = Y + φc (8) whee c sthetavelcostofpospect w.adoptg(8),letusetutothe chocebetweethepa w, w S.Ctcaltothschocesthepotat 25

28 whchthedvdualsdffeetbetweethepospects;ewokg(7)we caestablshthat: IfYˆ ˆ Y = 0 the: ( c c ) = α ( d d ) + [( p p )( α a + β SDE + γ SDL + δ L) ] φ (9) Thepopostoowemegesthatadvdualtavellemghtbewllgto exchageaadustmetcostdffeece,.e.thelefthadsdeof(9),fo adustmetsoeomoeofthetaveltmeadschedulgdffeeces,.e. theghthadsdeof(9).acceptgthspopostoofexchage,twould seemaelatvelysmallextesotosolatetheateofexchagebetwee tavelcostadeachcosttuetofexpectedutlty,adtheebydeve magalvaluatosofexpectedtaveltme,expectedscheduledelayealy, expectedscheduledelaylate,adtheexpectedlateessdummy.fo example,letusdevemagalvaluatoofexpectedtaveltme Vo E T thus: IfYˆ ˆ = 0 the: φ Y β, γ, δ = 0 ( c c ) = α{ [ E( a ) E( a )] + ( d d )} Vo ( E( T )) α = = φ ( c c ) [ E( a ) E( a )] + ( d d ) ( ( )) (10), 26

29 Magalvaluatosofexpectedscheduledelayealy Vo ( E( SDE) ),expected scheduledelaylate Vo ( E( SDL)),adtheexpectedlateessdummyVo( E( L) ) cabedevedaalogously,asshowappedxa. 4.2Atttudestouelablty Theexpostoofsecto3would,applygthetheoyofdvdualchoce udeucetatytotheelabltyofavaltme,seemtoestablsh syoymtybetweethemcoecoomst sotoofucetaty(osk)ad thetaspotplae sotoofuelablty.thstupovokesa teestecoclgbatesetal. smagalvaluatosoftaveltmead schedulgudeucetatywthestablshedmcoecoomcmethodsfo valugtheskheetpospects.ipusugthsteestoemght, ekdlgtheealedscussosecto3.2,dstgushbetweeatttudes toskymoetayoutcomesadatttudestoskytmeoutcomes.whlstthe fomehaveattactedcosdeableeseachatteto,thelattehaveot, hecetheoppotutyfofuthevestgato. Tothsed,letuscosdethepatculacoceptofthe skpemum (Patt, 1964),whchcaesaappealgtutoadcommadscosdeable 27

30 suppotacossthemcoecoomccommuty.theskpemumasesfom theelatobetweetheexpectedutltyfuctoadtsudelygutlty fucto,adscucallydctatedbythedvdual satttudetowadssk. Befoepoceedgtotheskpemum,letusthecosdeatttudestoskas theyapplytoouteesttheelabltyofavaltme.webegby devgtheutltyoftheexpectedavaltme,thus: U ( E( a )) = α [ E( a ) d ] + β max[ ( PAT E( a )),0] + γ max[ ( E( a ) PAT ), 0] + δl( E( a )) whee: E ( a ) p a + ( p ) a = 1 (11) L( E( )) = 1f ( ( a ) PAT ) > 0 a E, = 0othewse Thecompagtheutltyoftheexpectedavaltmetoexpectedutlty, Jese seualty(seefoexamplejohasso,1991)povdesabassfothe followgfeeces: IfY = U E( a )) thethetavelles skeutalavaltme IfY > U IfY < U ( ( E ( a )) ( E ( a )) thethetavelles skpefeedavaltme thethetavelles skaveseavaltme Itemastoestablshthepevaleceofthesetheeelatos,whchaga vokesthesametheecasesassecto3.3: 28

31 Case1: a < a PAT Thsalwaysyeldstheeualty Y = U ( E( )) skeutalty. Case2: PAT a < a a,suchthatthetavelleexhbts TheeualtyY = U E( a )) agaholds,adafeeceofskeutaltymay theefoebedaw. ( Case3: a < PAT < a Thscasesmoeambguous,adtsstuctvetotoducethefuthe dchotomy: ( a ) PAT Case3.1: E < WheeCase3holdsadtheexpectedavaltmesealethathe PAT,the dffeecebetweeexpectedutltyadutltyoftheexpectedavaltmes gvebytheuatty: Y [ β + δl ] ( E( a )) = ( p ) ( a PAT ) + γ ( a PAT ) U 1 Sce β, γ, δ < 0 ad 0 1bydefto,ad p ( PAT ) > 0 assumpto,tmustbethecasethat Y < U ( E( )) a a by.thetavelletheefoe 29

32 exhbtsskaveso.thscabecofmedthoughefeecetofgue3, wheetheutltyfuctou lesabovetheexpectedutltyfuctoy thoughouttheteval[ a, PAT ]. Case3.2 PAT < E( ) a WheeCase3cotuestoholdbuttheexpectedavaltmesowlatetha the PAT,thedffeecebetweeexpectedutltyadutltyoftheexpected avaltmebecomes: Y U ( E( a )) = p [ β ( PAT a ) + γ ( PAT a ) δl] whee L L = L( E( )) = a Iasmlamaetobefoe, β, γ, δ < 0 ad 0 p 1bydefto,ad ( PAT a ) > 0 byassumpto.ulkecase3.1,howeve,tcaotbe detemedapowhchofy adu ( E( )) wllbethegeate.ratheoe mustdefetotheempcaloutcome,whchwllbedctatedbythepoxmty oftheavaltmes a ad a tothe PAT.WthefeecetoFgue3,fo example,theutltyfuctou lesabovetheexpectedutltyfuctoy fo ayavaltmetheteval [ PAT, ] a a ;.e.skaveso.cotastthswth Fgue5,wheetheexpectedutltyfuctotesectsthevetcalsegmet 30

33 oftheutltyfucto,adtheutltyfuctotheefoelesbelowthe expectedutltyfuctothoughouttheteval[ PAT, ];.e.skpefeece. a Fgue5:Utltyadexpectedutltyfuctos,fogvedepatuetme,wthα < β (ABOUTHERE) Befoemovgo,tshouldbeackowledgedthatPolak(1987)pusues smlateeststotheabove,albetwthdffeetfocus.polakdevotes caefulattetotothefomofhsutltyfucto,postulatgthatthe fuctoshouldbemootocallydeceasgtavelcostaddemostatea costatdegeeofskavesototavelcost(seepatt(1964)adaow (1970)fodscussoofmeasuesofskaveso);cotastthswththe pesetpape,whchsfoudedoatttudestotmeskathethacostsk. Adoptgaexpoetalfomasawokgepesetatoofthepostulated popetes,polakseekstodetfy,asmlamaetosecto3.4above, thepefeeddepatuetmeofadvdualtavelle.thedevelopg deasfuthe,hetoducestheotoofthe safetymag,aspoposedby Gave(1968)adKght(1974),adestablshesaelatoshpbetweethe magtudeofthesafetymagadthedegeeofskaveso. 4.3Theelabltypemum 31

34 Retugtoouowaalyss,letusowdevelopPatt s(1964)coceptof theskpemumthecotextoftpschedulg.asapecuso,letusfst toducethecoceptofa cetatyeuvalet,whchthepesetcotext maybedefedasfollows.thecetatyeuvaletelatgtoaypospect w S stheavaltme a ~ thatyeldsthesameutltywthcetatyasthe expectedutltyofthepospect.icases1ad2ofthepevoussecto,both ofwhchmplyskeutalty,utltyadexpectedutltyaeoeadthe same,adtheotoofacetatyeuvaletstheefoesomewhat tautologcal. OfathemoeteestsCase3,scetcaesapotetalfoskaveso. Subseuetdscussowllevealatutothat,developgthe cetatyeuvaletfocase3,thecetatyeuvaletshouldbecostaed tofallwththetevalofavaltmesdefedbythepospect,.e. a a ~ a.letusfothemometsmplyacceptthseuemetad,by aalogytocases3.1ad3.2,dstgushbetweecaseswheethecetaty euvaletfallsbefoeadaftethe PAT,asfollows: Case3.3: a a ~ < PAT 32

35 Idevelopgthscase,letusestablshbywayofassetoeuvalece betweeexpectedutltyadtheutltydevgfomthecetaty euvalet,.e. Y α ~ ~ ),whee a a ~ PAT Let = ( a d ) + β ( PAT a < Reaagg,wecathedetfythecetatyeuvalet: Y + αd βpat a~ = (12) ( α β ) Case3.4: PAT < a ~ a Nowepeatgthesameexecse,butfoacetatyeuvaletlatethathe PAT : Y = ~ ~ δ ~ Let α ( a d ) + γ ( a PAT ) + L( a ) ~ = whee L( ) 1f ( PAT ) 0 a a~ >, = 0othewse,ad PAT < a ~ a Reaagg: 33

36 Y + αd + γpat δl( a~ ) a~ = ( α + γ ) Notethat,wthefeecetoealedscussosecto3,thedeomatoof thecetatyeuvaletfocases3.3ad3.4sgvebytheslopeofthe elevatsectooftheutltyfucto.whlsttheabovewokgcaesa appaetclaty,tmghthowevebecautoedthatthepopetesofthe fuctomoepatculaly,tsdscotutycaythemplcatothata exactcetatyeuvaletsotempcallyguaateed.nevetheless,letus poceedtothedeftooftheskpemum. Igeealtems,theskpemummeasuesthedvdual swllgessto pay,utsofthepayoff,toavodtheskofchoosgauceta pospect.omoesuccctly,theskpemummeasuesthe costofsk beag.applygthesedeftostoouteesttheelabltyofaval tme,theskpemumopesetpalace,the elabltypemum measues,foagvedepatuetme,thedelayavaltme(wthts coseuetmpactsotaveltme,scheduledelayealy,scheduledelaylate adthelateessdummy)thatthedvdualwouldbewllgtopay exchagefoelmatgtheuelablty.theelabltypemumthus measuesthecostsboebythetavellethatasespecfcallyfom uelabltyavaltme. 34

37 Nowadoptggeatefomalty,defetheelabltypemum: K [( a~ E( a )), 0] = max (13) Scethepayoffsthsstacedefedoabad(.e.avaltme),the expectedpayoffmustbesubtactedfomthecetatyeuvaletodeto yeldtheelabltypemum K.Notetheeuemetothesgof K, whchmplesthataozeoelabltypemumsepesetatveofsk avesebehavou.theelabltypemumsllustateddagammatcally Fgues6ad7,agawthα < β.fgue6cosdesaexpectedlateaval, addemostatesthatatavellewouldyeldeualutltyfomthepospect Y adthecetaavaltme a ~,theebydetfygtheelabltypemum tobethedstace a ~ E a ). Fgue7applesaalogouslytothecaseofa ( expectedealyaval,wthsmlaesult;thetavellewouldbedffeet betweethepospectadacetaavaltmeustealyofthe PAT.Whlst Fgues6ad7wouldseemeasoablyclea,thedscotutyofU toducessomecomplextes;fpatculay tesectsthevetcal segmetofu themaeoffgue5,thethecetatyeuvaletmay fallbefoetheexpectedavaltme,suchthattheskpemumbecomeszeo accodacewth(13). 35

38 Fgue6:Theelabltypemumofaexpectedlateaval,fogvedepatuetme, wthα < β (ABOUTHERE) Fgue7:Theelabltypemumofaexpectedealyaval,fogvedepatue tme,wthα < β (ABOUTHERE) Ifweowexteddscussooftheelabltypemumtocosdethe elato β < α,thetsappaetthatfuthecomplextesase.wth efeecetofgue8,whchomtssomelabellgthehopeofpomotg claty,theopposgslopesofthetwosegmetsofu yeldthepaof cetatyeuvalets a ~ ad a ~.Itmghtbeotedthat,wheeas a ~ falls wththeteval [ ] a a,, a ~ doesot.futhemoe,whlst a ~ fallsaftethe expectedavaltmeadtheefoeyeldsthepossbltyofaozeo elabltypemum(ackowledggagathepossbltyofazeoelablty pemumshouldy tesectthevetcalsegmetofu ), a ~ fallsbefoethe expectedavaltmeadwllevetheefoeyeldaozeoelablty pemum.hece,weaveatatutofothecostat a a ~ a toducedattheoutsetofthssecto. 36

39 Fgue8:Theelabltypemumofaexpectedealyaval,fogvedepatue tme,wth β < α Befoepoceedg,tsusefultodstgushtheotooftheelablty pemumfomtheafoemetoedotoofthesafetymag,thateach devesfomadffeetefeecepot.wheeastheelabltypemum devesfomtheefeecepotofthecetatyeuvalet,thesafetymag devesfomtheefeecepotofthe PAT.Thspovokesthempotat dstctothatthesafetymagpetasolytoealyavals,wheeasthe elabltypemummaypetatoetheealy(.e.fgue7)olate(.e. Fgue6)avals.Ideed,theecessalytheoetcaldscussoabovemght beembellshedwththeemakthattheelabltypemumoffesatoale foapactcethatswdelyappledthepublctaspotdusty,thus.a opeato,ffacedwththestuatopesetedfgue6,couldtoducea ceasedoueytme(.e. a~,suchthatthedvdualcusalate d aval)adstllmatamaketshae(.e.matathedvdual slevelof utltyat Y ),povdedtcouldesuefullelabltyofsevce(.e.movefom theexpectedutltyfuctoy totheutltyfucto U ). 4.4RecoclgtheelabltypemumwthBatesetal. 37

40 Havgtoducedtwoalteatveotosofthevalueofelabltythe elabltypemum(secto4.3)adbatesetal. s(2001)magalvaluatos oftaveltmeadschedulgudeucetaty(secto4.1)letusowseek toecoclethem.tothsed,adexplotgtheaalyssofthepevous secto,wecaeexpessexpectedutltyastheutltyatthecetaty euvalet: ( a ~ ) = α ( a~ d ) + β max[ ( PAT a~ ),0] + γ max[ ( a~ PAT ),0] + δ L( a ) Y = U ~ Ifthedvdualtavellesskavesethewecasubsttutefo a ~ usgthe skpemum K,thus: Y = α [ E( a ) + K d ] + β max[ ( PAT E( a ) K ),0] + γ max[ ( E( a ) + K PAT ),0] + δ L( E( a ) + K ) (14) Shouldweowtoducetavelcostto(14)themaeof(8),theweca deveafuthesetofmagalvaluatos,butthstmeatthecetaty euvalet.foexample,letusdevemagalvaluatooftaveltmeatthe cetatyeuvalet: IfYˆ ˆ = 0 the: Y β, γ, δ = 0 38

41 φ ( c c ) = α {( K K ) + [ E( a ) E( a )] + ( d d )} Vo ( T ) α = = φ ( c c ) ( K K ) + [ E( a ) E( a )] + ( d d ) (15) Magalvaluatosatthecetatyeuvaletofscheduledelayealy, scheduledelaylateadthelateessdummyaemoecomplcated.whlstt seasoabletoestctattetotocase3(.e. a < PAT < a ),emembeg thattheelabltypema K ad K aepetetolytocodtosofsk aveso,thsdoesotcostathecetatyeuvalets a ~ ad a ~ tofall ethesdeofthe PAT.Rathetsecessaytodefetheefuthe possbltes,asextesosofcases3.3ad3.4,thus: Case3.5: a~, a~ PAT Case3.6: PAT a~, a~ Case3.7: a ~ < PAT < a~ TableIdsplaysthecompletesetofmagalvaluatosatthecetaty euvalet,wththefsttheecolumsdscmatgbytheabovethee cases.ofpatculateestswhethethesemagalvaluatosatthe cetatyeuvaletaccodwththosepevouslydevedudeucetaty. Fopuposesofthscompaso,tsusefultocosdehow(10)ad(A1)to (A3)applytoCase3,adtheseaegvethefalcolumofTableI. 39

42 TableI:Magalvaluatosoftaveltmeadschedulgudeucetatyadat thecetatyeuvalet(abouthere) WthefeecetoTableI,compasoofthemagalvaluatosoftavel tmeudeucetatyadatthecetatyeuvalet,whchaegveby α φ (.e.fomeuato(15))adα φ (.e.fomeuato(10))espectvely, evealsthatthedeomatosofthetwovaluatosdffebytheuatty ( ) K K.Thesevaluatoswlltheefoebeeualolyf K =.Whlst K thelatteeualtymayfeasblyhold,theesoapobassfoecessaly expectgthsesult,adtwouldseemutepossblethatmagal valuatooftaveltmeudeucetatywlldffefomthatatthecetaty euvalet. Nowcosdemagalvaluatoofthelateessdummy.ICases3.5ad 3.6,themagalvaluatosudeucetatyadatthecetatyeuvalet (.e.δ φ adδ φ espectvely)wllbeeualf c = c.icase3.7,smla eualtywllasefboth p = 1ad p = 0.Notepassgthat,assecto 3.4hasaleadycosdeed,oewouldpactceexpecttheeualty p > p tohold,whchwouldbecosstetwtheualtybetweemagal 40

43 valuatosofthelateessdummyudeucetatyadatthecetaty euvalet., ad Fally,themagalvaluatosofscheduledelayealy ( β φ β φ) scheduledelaylate ( γ φ γ φ), showsmladffeecesthefomulae whecompagucetatywththecetatyeuvalet.itwouldhoweve seemmoedffculttopedcttheoutcomeapo,adoemuststeaddefe toempcalvestgato. Thatotwthstadg,thepecedgtheoetcalagumethasllumatedthe possbltythatmagalvaluatosoftaveltmeadschedulgatthe cetatyeuvaletwllshowdscepacyfomthesamevaluatosude ucetaty.whethetheoetcaldscepacyesultsempcaldscepacy emastobesee,adthspovokesacallfofutuewokaddessgsuch mattes.shouldthedscepacybecofmedempcally,thetcaes mpotattepetatoasthedvdualtavelle swllgesstopay(ow moetayuts)toelmateuelabltyavaltme.foexample,the uatty ( α φ α φ) epesetsthedvdual swllgesstopayto elmateuelabltyavaltme,specfcallyastmpactsupotavel tme;tmghttheefoebeefeedtoasthe magalvaluatoofelablty taveltme.smlatepetatosapplytotheothecosttuetsof 41

44 Small s(1982)utltyfucto;theuatty ( β φ β φ) mghtbeefeedto asthe magalvaluatoofelabltyscheduledelayealy,adsoo. 5.Thepevaleceaddstbutoofbeeftsfomtheelabltypemum Whlstmagalvaluatosoftaveltmeadschedulgudeucetaty aethemselvespefectlyadeuatefopuposesoffoecastgdvdual choceudeucetaty,tsolythoughthecompasotomagal valuatosoftaveltmeadschedulgatthecetatyeuvaletthatthe effectofuelabltyavaltmeochocecabeevealed.suchsght mghtusefullyfompublctaspotopeatosastotheeffectsofchages elabltyothedvdualtavelle schoceofdepatuetme.moe geeally,themmsatoofskmaybeampotatpolcyaspato,ad magalvaluatosofelabltycould,tothsed,offeausefulmeasof dscmatgbetweealteatvevestmetoptos. Thoughtsctomagalvaluatosofelablty,aguablythemoe sgfcatcotbutooftheelabltypemumstoecoomcappasal. AsPeaceadNash(1981)obseve,thepoectedbeeftsofaytaspot schemewhchmaycludebeeftsfommpovedelabltyshould mtgatefothecostofheetsk,adfaluetodosotoducesthe 42

45 possbltyofbas.hecetwouldseemcucaltomakeexplctstatemetof thecostofuelabltyavaltmetothedvdualtavelle(o euvaletly,thebeeftofelmatgthatuelablty),whchfoay pospect w S sgveby: Vo ( K ) = α φ *[ a~ E( a )] + β φ *{ max[ ( PAT a~ ),0] max[ ( PAT E( a )),0]} γ φ *{ max[ ( a~ PAT ),0] max[( E( a ) PAT ),0]} δ φ *[ L( a~ ) L( E( a ))] + (16) + Notgmpotatlythatthepotetalbeeftofelmatguelablty mafestsolyudeccumstacesofskaveso,letuscosdethe pevaleceofsuchbeeftelatotothesametheecasescosdeed eale. Case1: a < a PAT Ithscase,theewouldbelbeeft(.e. Vo( ) = 0 K (16))tothetavelle fomtheelmatoofuelablty;thssbecausethetavelleexhbtssk eutalty. Case2: PAT a < a ThswouldyeldthesameoutcomeasCase1. 43

46 Case3: a < PAT < a Thetavellemghtomghtotealsebeeftfomtheelmatoof uelablty,asfollows: ( a ) PAT Case3.1: E < Theewouldbeuambguousbeeft(.e. Vo ( ) > 0 K ),scethetavelle exhbtsskaveso. Case3.2 PAT < E( ) a Thepevaleceofbeeftwoulddepedothepoxmtyoftheavaltmes a ad a tothe PAT,adthecoseueteffectoatttudesto uelablty. Scetheaboveoutcomesaedctatedbytheelatooftheavaltmes a ad a tothe PAT,oecouldtheefoeexpectthepevaleceofbeeftto vaybythedepatuetmes d D.Thatstosay,oemghteasoably expectelatvelyealydepatuestopetatocase1,elatvelylate depatuestocase2,adtemedatedepatuestocase3. 44

47 Itshouldbeemphassed,howeve,thattheabovecoclusosapply specfcallytothecaseofadvdualtavelle.if,assmoecommo pactce,oesteestedasampleoftavelles, saytheusesofa patculapublctaspotdepatue,thethesamplewouldtypcally demostatesomeheteogeetyespectofthe PAT.Themplcato followsthat,foaympovemettheelabltyofavaltmefothat sevce,someusesmghtealseabeeftfomthatmpovemet,whlst othesmghtot.thetotalbeefttothesamplewouldtheefoedepedo theumbeofbeefcaes,adthedvdualvaluatosoftheelablty pemum.wthefeeceto(2),teestasampleofdvdualssofte developedthoughaetepetatoofrum,wth epettos becomg dvduals,adpobabltesofchocedevgfomdffeecesthe pefeecesofdvdualsacossthesample.iespectveofwhethet devesfomtadvdualotedvdualvaabltypefeeces, extesooftheelabltypemumtoaccommodaterumwouldgveseto dstbutedmagalvaluatosoftaveltmeadschedulg,aswellasto dstbutedmagalvaluatosofelabltytaveltmeadschedulg. Asdefomthematteofaggegato,afutheestctooftheabove aalyssstoabaysubsetofavaltmes,adtsappopateto ackowledgethattheclatyofcase3becomescompomsedocethesetof avaltmessextededbeyodthebay.isuchccumstaces,the 45

48 elatobetweetheutltyoftheexpectedavaltmeadexpectedutlty emaspetetbutmustbeevealedempcally.thatsad,adwth efeecetotheaspatosoutledthetoductotothspape,tmght beaguedthatabaysubsetofavaltmeswouldbgappealg coveecetospaalyssoftheelabltypemum.ishot,thebay casesfafomabstact.icotasttocase3,theuambguousesultsof Cases1ad2eadlyextedtotomalolageavaltmesets. 6.Wokedexampleoftheelabltypemum Letusowllustatethetheoetcalexpostooftheelabltypemumby measofawokedexample,otgthatwewllestctattetotoasgle dvdualadtheefoeavodthecomplcatosofaggegatoust metoed.wthefeecetotableii,whchuatfesalltmesmutes aftemdght,cosdeaoewaycommutewthadepatuetmepofleof 420 (.e.7:00am)to 495 (.e.8:15am),cemetsof5mutes.aval tmesaesmlalydefedcemetsof5mutes,adevealammum oueytmeof30mutes(.e. a = m 450 ).Thscouldbeepesetatveofa hghfeuecyscheduledpublctaspotsevce;alteatvely,tcouldbea dsceteappoxmatotoacabasedouey.thebodyofthetabledsplays theevetpobabltesbydepatueadavaltmes.itmghtbeobseved 46

49 thatthesubsetofavaltmesvaesbydepatuetme,adcotas betweetwoadfvepossbleavals.scethepoblemsmoegeeal thathebaysubsetofavaltmescosdeedabove,aalyssofcase3 musttheefoedefetotheempcalesultsthatfollow. TableII:Payoffmatxfowokedexample(ABOUTHERE) Ifocussgotheelabltypemum,tsuecessaytoexplctlycosde tavelcost,adwetheefoepoceedwththefomulatosofutltyad expectedutltygveby (4)ad(6)espectvely. Letuspopulatethesewth theestmatesofα, β, γ adδ fommodel(1)ofsmall(1982),specfcally α = 0.106, β = ,γ = adδ = 0. 58,otgthatα < β.letus assumealsothat PAT = 525 (.e.8:45am).fgue9plotsthevaousattbutes ofexpectedutltyexpectedtaveltme,expectedscheduledelayealy, expectedscheduledelaylate,adtheexpectedlateessdummyagast expectedavaltme.thepopetesofthsfgueaccodwththoseof pevouspesetatosthelteatue,foexamplefgue2ofbatesetal.( 2001),adcofmthepostoofthe PAT at525. Fgue9:Expectedtaveltme,SDE,SDLadlateessdummyvs.expectedaval tme(abouthere) 47

50 Fgue10plots,foeachdepatuetme,expectedutltyY adutltyofthe expectedavaltme U ( E( a) ).Itmghtbeemakedthatmayofthe depatuetmesaefeo,thatthemaxmumutltyfallsshotofthe mmumutltyofothedepatues.othsbass,wecaestctatteto todepatuetmestheage450to475.ideedexpectedutltys maxmsedwththsage,specfcallyat465.compagtheplotsofy ad U ( E( a) ),tcabeobsevedthatexpectedutltyadutltyofthe expectedavaltmecocdefothevastmaotyofdepatuetmes;these depatuespetatocases1ad2(.e.theespectveavaltmesfall ethealwaysbefoeoalwaysaftethe PAT ).Bycotast,thetwoplots dvegefothe465,470ad475depatues,eachofwhchpetastocase3 (.e.theespectveavaltmesstaddlethe PAT ).Moespecfcally, expectedutltyslessthatheutltyoftheexpectedavaltmefothe465 ad470depatues,wheeastheeveseapplesfothe475depatue.hece wthefeecetothedscussoofsecto4.2,tmaybeseethatthe pefeeddepatuetmeof465selatvelyskycompasotoothe avalabledepatues. Fgue10:Expectedutltyadutltyofexpectedavaltme,bydepatuetme (ABOUTHERE) 48

51 Fally,letuscosdeaexampleoftheelabltypemum,takgthe patculacaseofthe465depatue(scethsdepatueschaactesedby skaveso).theempcalutltyfuctofothsdepatuesshow Fgue11;thsfollowsthechaactestcshapeofthetheoetcalutlty fuctosfgues1ad3to7,wthdstctsectosbefoeadaftethe PAT,adα < β.theempcalexpectedutltyfucto,bycotast,caot beshowthemaeofthetheoetcalexamples,scewehaveexpaded thesetofavaltmesbeyodthebay.suffcetosay,thea valtme wdowextedsfom510to530,hecethepotslabelledy. Fgue11:Utltyadexpectedutltyfuctosfo d = 465 (ABOUTHERE) Fothe465depatue,wecacalculatetheexpectedavaltme(11),gvg ( a 465 ) = E,adthecetatyeuvalet(12),gvg a ~465 = The applygthesetotheelabltypemum(13),wecacalculate K = a~ E a ( ) =,suchthatacetaavaltme2.65muteslate thatheexpectedavaltmewouldyeldthesameutltyastheexpected utltyofthepospect.fally,thevalueofthselabltypemums,wth efeeceto(16),gveby: 49

52 Vo α β φ ( K ) = 65 Cotastthswththe460depatue,whee U ( E( )) Y = ad Vo K 460 = a 460 ( ) Heceelmatoofuelabltyavaltmewouldfothe465depatue (.e.case3)yeldaaddtoalbeefttothedvdual,buttheeso possbltyofsmlabeeftfothe460depatue(.e.case1). 7.Summayadcocluso Uelabltysedemcmaytaspotsystems,adthsstmulates teestwhetheadhowuelabltympactsupothechocesof tavelles.thepapepusuedspecfcteesttheeffectofuelablty avaltmeoschedulgchoce.followgthepecedetofnolad& Small(1995)adBatesetal.(2001),thswasdevelopedthoughthemaage ofsmall s(1982)utltyfuctowthvoneuma&mogeste s(1947) theoyofdvdualchoceudeucetaty.asgfomthelatteuo sthepopostothatuelabltymposesdsutltyothetavelle.hece thepotetalfobeeftshoulduelabltybeeducedodeedelmated. 50

53 Icotasttothepecedetwoksoelablty,thepesetpapeadopteda dsceteepesetatooftme,motvatedpatculabyadesetopomote mplemetatowththeappaatusofrumadsp.thesubstatve cotbutoofthepape,howeve,wasthescopeofthetheoetcal exposto,whchoffeedsgfcatextesosbeyodtheextatelablty lteatue.wthefeecetothetheoetcallteatueoatttudestosk(e.g. Patt,1964;Aow,1970),thepapecosdeedthemplcatosofSmall s utltyfuctofotavelles atttudestouelabltyavaltme,ad patculadetfedccumstacesudewhchtavelleswouldbesk avese.iesposetothelatteobsevato,addawgaalogywth Patt s(1964)coceptoftheskpemum,thepapetoducedtheotoof the elabltypemum.thsmeasues,foagvedepatuetme,thedelay avaltmethatthedvdualwouldbewllgtopayexchagefo elmatguelabltyavaltme. Thepapethesoughttoecoclethscosdeatoofatttudestoskwth Batesetal. s(2001)magalvaluatosoftaveltmeadschedulgude ucetaty.thelatteasefomthepopostothatadvdualtavelle would,choosgbetweepospects,bewllgtoexchagetavelcostfo expectedtaveltme,expectedscheduledelayealy,expectedscheduledelay late,adthepobabltyoflateaval.explotgtheelabltypemum,the papeestablshedabassfocompasobetweemagalvaluatosof 51

54 taveltmeadschedulgudeucetaty,adaalogousvaluatos devedatthecetatyeuvalet.thscompasoevealedthetheoetcal possbltythatthetwosetsofvaluatosmghtshowdscepacy.should thstheoetcaldscepacymafestempcaldscepacywhch emastobeseethetcaesmpotattepetatoasthemagal valuatoofelabltyavaltme. Whlstmagalvaluatosoftaveltmeadschedulgudeucetaty aeadeuatefodemadfoecastg,ecoomcappasalshouldmtgatethe poectedbeeftsofaschemeagastthecostsofskbeag,adthss wheetheelabltypemumbecomespetet.itscucaltoackowledge thatelabltybeeftsaseolyudethepatculaccumstacesofsk aveso,adthatskavesos,thetemsoftheutltyfucto,dctated bytheelatoofthepossbleavaltmestothepefeedavaltme. Moeove,thepevaleceofbeeftwlllkelyvaybydepatuetmefo gvepefeedavaltme.thsassumesasgledvdualhoweve;oce theaalysssextededtoasampleofdvduals,theoutcomewllbe complcatedbyheteogeetythepefeedavaltmead,tfollows, heteogeetythepevaleceadmagtudeofelabltybeefts.thus foaypatculadepatuetme,ampovemettheelabltyofaval tmemghtyeldbeeftfosometavellesbutobeeftfoothes. 52

55 Ackowledgemets TheauthowassuppotedbyaDepatmetalReseachFellowshpatthe IsttutefoTaspotStudes(ITS),UvestyofLeeds,adbythePlatfom GatGR/S46291/01 Towadsaufedtheoetcalfamewokfotaspot etwokadchocemodellg,whchsfudedbytheukegeegad PhyscalScecesReseachCoucladalsoheldatITS.Theautho gatefullyackowledgesthecommetsofpofessodavdwatlg,pofesso GeaddeJogadMJamesLadoealedafts,aswellasthecommets oftwoaoymousefeees.ayemageosaethesoleesposblty oftheautho. 53

56 AppedxA:Devatoof Vo ( E( SDE) ), Vo( E( SDL) ) ad Vo( E( L) ) IfYˆ ˆ = 0 the: (A1) Y α, γ, δ = 0 Vo ( E( SDE) ) β = = φ ( c c ) ( p p ) SDE IfYˆ ˆ = 0 the: (A2) Y α, β, δ = 0 Vo ( E( SDL) ) γ = = φ ( c c ) ( p p ) SDL IfYˆ ˆ = 0 the: (A3) Y α, β, γ = 0 Vo ( E( L) ) δ = = φ ( c c ) ( p p ) L 54

57 Refeeces Aow,K.J.(1970)EssaystheTheoyofRskBeag.NothHollad, Amstedam. Bates,J.,Polak,J.,Joes,P.adCook,A.(2001)Thevaluatoofelabltyfo pesoaltavel.taspotatoreseachpate,37(23), Batley,R.(2006)Oodalutlty,cadalutlty,adadomutlty. Wokgpape,IsttutefoTaspotStudes,UvestyofLeeds. Baumol,W.(1958)Thecadalutltywhchsodal.TheEcoomc Joual,68(272), Becke,G.S.(1965)Atheoyoftheallocatooftme.TheEcoomcJoual, 75(299), DeJog,G.,Koes,E.,Plasmee,R.,Sades,P.adWaffemus,P.(2004) Thevalueofelablty.PapepesetedattheEuopeaTaspot Cofeece,Stasboug,46Octobe

58 DeSepa,A.C.(1971)Atheoyoftheecoomcsoftme.TheEcoomc Joual,81(324), Edwads,W.(1955)Thepedctoofdecsosamogbets.Joualof ExpemetalPsychology,50(3), Fshbu,P.C.(1970)UtltyTheoyfoDecsoMakg.JohWley&Sos, NewYok. Gave,D.P.J.(1968)Headstatstategesfocombatgcogesto. TaspotatoScece,2(2), Heste,I.N.adMlo,J.(1953)Aaxomatcappoachtomeasuable utlty.ecoometca,21(2), Johasso,P.O.(1991)AItoductotoModeWelfaeEcoomcs. CambdgeUvestyPess,Cambdge. Jotsakasa,A.,Hess,S.adPolak,J.W.(2004)Otheeffectsofalteatve dscetsatostategesdepatuetmechocemodellg.papepeseted atthe36 th AualCofeeceoftheUvestes TaspotStudyGoup, UvestyofNewcastleupoTye,57Jauay

59 Kahema,D.adTvesky,A.(1979)Pospecttheoy:aaalyssofdecso udesk.ecoometca,47(2), Kahema,D.adTvesky,A.(2000)Choces,ValuesadFames. CambdgeUvestyPess,Cambdge. Keyes,J.M.(1921)ATeatseoPobablty.Macmlla,Lodo. Keyes,J.M.(1936)TheGeealTheoyofEmploymet,IteestadMoey. Macmlla,Lodo. Kght,F.H.(1921)Rsk,UcetatyadPoft.HoughtoMffl,Bosto. Kght,T.E.(1974)Aappoachtotheevaluatoofchagestavel uelablty:a safetymag hypothess.taspotato,3(4), Maschak,J.(1950)Ratoalbehavou,ucetapospectsadmeasuable utlty.ecoometca,18(2),pp

60 Maschak,J.,Becke,G.M.adDeGoot,M.H.(1963)Stochastcmodelsof chocebehavo.imaschak,j.(1974)ecoomcifomato,decsoad Pedcto:SelectedEssays(Volume1).D.Redel,Dodecht. Nolad,R.B.adPolak,J.W.(2002)Taveltmevaablty:aevewof theoetcaladempcalssues.taspotrevews,22(1),3954. Nolad,R.adSmall,K.(1995)Taveltmeucetaty,depatuetme choceadthecostofthemogcommute.wokgpapeuciitswp95 1,IsttuteofTaspotatoStudes,UvestyofCalfoa,Ive. Peace,D.W.adNash,C.A.(1981)TheSocalAppasalofPoects:AText CostBeeftAalyss.Macmlla,Lodo. Polak,J.(1987)Taveltmevaabltyaddepatuetmechoce.Taspot StudesGoupDscussoPapeNo.15,PolytechcofCetalLodo. Patt,J.W.(1964)Rskavesothesmalladthelage.Ecoometca,32 (12), Small,K.(1982)Theschedulgofcosumeactvtes:woktps.Ameca EcoomcRevew,72(3),

61 Vckey,W.S.(1969)Cogestotheoyadtaspotvestmet.Ameca EcoomcRevew,59(2), voneuma,j.admogeste,o.(1947)theoyofgamesadecoomc Behavo(SecodEdto).PcetoUvestyPess,Pceto. 59

62 Fgue1:Small sutltyfucto,fogvedepatuetme,wthα < β aval tme a m a PAT a Slope = α β utlty Slope = α + γ U 60

63 Fgue2:Small sutltyfucto,fogvedepatuetme,wth β < α aval tme a m a PAT a utlty Slope = α β Slope = α + γ U 61

64 Fgue3:Utltyadexpectedutltyfuctos,fogvedepatuetme,wthα < β aval tme Case 3 a PAT Case 3 a utlty Case 3 Y U, Y Cases1& 2 62

65 Fgue4:Chocebetweedepatuetmes,wthα < β aval tme ( ) d a m d a m ( ) E ( ) ( ) a a PAT E a a α ( d ) d utlty ( Y Y ) Y Y U U 63

66 Fgue5:Utltyadexpectedutltyfuctos,fogvedepatuetme,wthα < β aval tme PAT Case 3 a Case 3 a Case 3 Y utlty U 64

67 Fgue6:Theelabltypemumofaexpectedlateaval,fogvedepatuetme, wthα < β aval tme E a a a PAT ( ) a ~ K U utlty Y Y U U 65

68 Fgue7:Theelabltypemumofaexpectedealyaval,fogvedepatue tme,wthα < β aval tme ( ) E a a a a ~ PAT K U Y utlty Y U U 66

69 Fgue8:Theelabltypemumofaexpectedealyaval,fogvedepatue tme,wth β < α aval tme a ~ E( a ) a ~ K K utlty Y Y U 67

70 Fgue9:Expectedtaveltme,SDE,SDLadlatepealtyvs.expectedavaltme 80 E(T), E(SDE), E(SDL), E(L) E(a) E(T) E(SDE) E(SDL) E(L) 68

71 Fgue10:Expectedutltyadutltyofexpectedavaltme,bydepatuetme utlty U(E(a)) Y depatue tme 69

72 Fgue11:Utltyadexpectedutltyfuctosfo d = utlty U U Y aval tm e 70

73 TableI:Magalvaluatosoftaveltmeadschedulgudeucetatyadatthecetatyeuvalet CERTAINTYEQUIVALENT UNCERTAINTY Case3.5: Case3.6: Case3.7: Case3: a~, a~ PAT < PAT < a~, a~ a ~ < PAT < ~ a a < PAT < a α ( c c ) φ ( K K ) + [ E( a ) E( a )] + ( d d ) β ( c c ) φ ( K K ) + [ E( a ) E( a )] γ φ δ φ 0 0 PAT ( c c ) K E ( a ) ( c c ) ( c c ) ( K K ) + [ E( a ) E( a )] K + E a ( ) PAT 0 0 ( c c ) ( c c ) [ E( a ) E( a )] + ( d d ) ( c c ) [( p p )( PAT a )] ( c c ) [( p p )( a PAT )] ( c c ) ( p p ) α φ β φ γ φ δ φ 71

74 TableII:Payoffmatxfowokedexample a a a a a a a a a a a a a a a a a a a a a a d d d d d d d d d d d d d d d d