Corresponding Author Duke University Department of Electrical and Computer Engineering Durham, North Carolina , U.S.A.

Transcription

1 ForecastingUncertainHotelRoomDemand MihirRajopadhye,MounirBenGhaliayandPaulP.Wang DepartmentofElectricalandComputerEngineering Durham,NorthCarolina DukeUniversity TimothyBakerandCraigV.Eister Atlanta,Georgia BassHotelsandResorts Thisincreasinguncertaintyislikelytoincurbaddecisionsthatcanbecostlyinnancial Economicsystemsarecharacterizedbyincreasinguncertaintyintheirdynamics. Abstract offorecastingusingtheholt-wintersmethod.theproblemhereistoforecastthe uncertaindemandforroomsatahotelforeacharrivalday.forecastingispartofhotel terms.thismakesforecastingofuncertaineconomicvariablesaninstrumentalactivity inanyorganization.thispapertakesthehotelindustryasapracticalapplication revenuemanagementsystemwhoseobjectiveistomaximizetherevenuebymaking decisionsregardingwhentomakeroomsavailableforcustomersandatwhatprice. thathumanjudgmentisimportantwhendealingwithexternaleventsthatmayaect satisfactoryforcertaindays,thisisnotthecaseforotherarrivaldays.itisbelieved notincorporatemanagementexpertise.eventhough,forecastresultsarefoundtobe Theforecastapproachdiscussedinthispaperisbasedonquantitativemodelsanddoes thevariablesbeingforecasted.actualdatafromahotelareusedtoillustratethe Hotelindustry;Holt-Wintersapproach. KeyWords:Forecasting;Uncertainvariables;Economicrevenuemanagementsystems; forecastingmechanism. Corresponding Author Duke University Department of Electrical and Computer Engineering Durham, North Carolina , U.S.A. TEL: +1 (919) FAX: +1 (919)

2 Forecastingroomdemandisaveryimportantpartofmoderndayhotelrevenuemanagement 1systems.Theobjectiveofthesesystemsistomaximizetherevenueundertheconstraint Introduction optimizationandcontrols.theseoptimizationroutinesarecarriedoutoverseveraldays ofxedroomcapacity.tothisend,mosthotelshaveimplementedsomeformofinventory dayisrequiredtocarryouttheoptimization.thispaperdealswiththeproblemofforecastingunconstrainedhotelroomdemand.unconstrainedroomdemandisthenumberof andoptimizationareseparateissuesandarenotaddressedinthispaper. Optimizationoftheinventoryisveryimportanttotheyieldmanagementsystem.Theop- roomsthatcanberentediftherearenocapacityorpricingconstraints.roomallocation priortothearrivalday,soanestimateofthedemandforroomsforthatparticulartarget requestingmultiplenightstay.thismakesforecastinganimportantissue,sinceabetter ingthattherearemultipleroomtypeswhichcanbesoldatdierentratestocustomers price,withtheobjectiveofmaximizingtherevenue.thisisnotasimpletask,considertimizationprobleminvolvessellingtherighttypeofroomtotherightcustomerattheright forecastwouldresultinimprovedinventoryoptimization,andconsequently,increasedrevenue.indeed,forecastingandoptimizationareamongtheprimarycomponentsoftheyield managementsystem[1],andbothcomponentsarevitalfortheperformanceofthesystem. problem[2,3,4].however,tothebestofourknowledge,therehasbeenlittleornopublishedworkontheroomdemandforecastingaspect.inthispaper,weshowhowaparticular Alotoftheworkdoneonhotelrevenuemanagementsystemsdealswiththeoptimization Severalmethodshavebeenusedforthepurposeofforecastingdatainavarietyofbusiness forecastingprocedurecanbeappliedtothehotelroomdemandproblem. themodelrelatesthedatatotheinputswithasetofcoecients[5].anapplicationofthis applications[5].dierentmethodsvaryinthemannerinwhichthehistoricaldataismodeled.regressionalmethodsseektoexplainthedatawithoneormoreinputvariables,and methodisfoundin[5],whichusesalinearregressionmodeltotthemonthlymaintenance interpretation[6].averypopularforecastmethodisthebox-jenkinsapproachtotimeseries modelingandforecasting[7].thisapproachdoesnotassumethatsuccessiveobservations seriesmodeltothedata.suchamodelissetupintermsofcomponentswhichhaveadirect expensedataofamanufacturingplant.anothermethodinvolvesttingastructuraltime verycomplicatedanddiculttoimplement.smoothingmethods,ontheotherhand,are productinachemicalprocessismodeledusinganar(2)model.however,thesemodelsare anderrors.anexampleofsuchamodelcanbefoundin[5],inwhichtheviscosityofa anderrorsareindependent.futurevaluesareforecastedasafunctionofpastobservations simpleandgiveequivalentperformancewiththerightchoiceofmodel.smoothingproceduresdiscountpastobservationsinpredictingfuturedata,butthemannerinwhichpast dataisdiscountedisadhoc[6]. 2

3 Theexponentialsmoothingprocedureisasimplemethodtoforecastfuturedatabasedon pastobservations[8].inthismethod,previousobservationsarediscountedsuchthatrecentobservationsaregivenmoreweightsandobservationsfurtherinthepastaregivenless weight.theweightsdecreasebyaconstantratio,andthuslieonanexponentialcurve. tothis,certainadaptationsarerequiredinordertouseitfortimeseriesthatariseinreal Thismethod,however,canbeusedonlyfornon-seasonaltimeseriesshowingnotrend.Due problems. AmoregeneralvariationofthesimpleexponentialsmoothingprocedureistheHolt-Winters method[9].thelatterconsidersthelocallineartrendandseasonalityinthedata.thetrend representsthedirectioninwhichthetimeseriesismoving,whiletheseasonalityexplains theeectsofdierentseasonsinthedata.thismethodowesitspopularitytothefact thatitisverysimpletoimplementandiscomparablewithanyotherunivariateforecasting procedureintermsofaccuracy[10].also,thecomponentsoftheforecast(viz.mean,trend andseasonality)lendthemselvestoaneasyinterpretation.thesecomponentsarediscussed indetailinthenextsection.inthispaper,weapplytheholt-wintersproceduretoforecast unconstrainedroomdemandforanactualhotel.datacollectedfromanactualhotelisused TheobjectiveofthispaperistoapplyandevaluatetheHolt-Wintersproceduretothe intheinitializationoftheforecastcomponents. forecastofhotelroomdemand.theforecastgeneratedisbasedonlyontheharddatain theformofhistoricaldataandcurrentbookingactivity.inthispaper,nohumaninputis accountedforintheforecastmechanism.thecurrentstudyispartofanongoingresearch aimingatdevelopingarobustforecastsystemwherebothharddataandhumaninputare combined.themotivationforthisresearchcomesfromthefactthatagoodforecastwould greatlyenhancemanagementdecisionmaking. modelselectioncriteriaarediscussed.section3dealswiththeroomdemandforecasting. castingprocedureinsection2.both,additiveandmultiplicativemodelsarepresented,and Theremainderofthepaperisorganizedasfollows:WerstpresenttheHolt-Wintersfore- ofhistoricaldataandsimulationofreservationrequestsarealsoincludedinthissection, followedbytheforecastalgorithm.simulationresultsarepresentedinsection4.finally,in Section5,wepresenttheconclusionsandsuggestionsforfurtherresearchwork. Wediscussthereservationcharacterizationandformulatetheforecastingproblem.Analysis TheHolt-Wintersmethodisanextensionoftheexponentiallyweightedmovingaverage 2(EWMA)procedure[6].TheEWMAalgorithmforecastsfuturevaluesbasedonpastobser- vations,andplacesmoreweightonrecentobservations.intheholt-wintersmethod,the values.thedistinctivefeatureoftheholt-wintersprocedureisthatitincorporateslinear Holt-WintersForecastingMethod forecastcomponentsareupdatedinasimilarfashion,i.e.moreweightsareplacedonrecent 3

4 trendandseasonalityintothesimpleexponentialsmoothingalgorithm[6]. TheHolt-Wintersmodelsthetimeserieswiththreecomponents:mean,localtrendand seasonality.dependingonhowtheseasonalvariationisincludedinthemodel,therearetwo versionsoftheholt-wintersforecastprocedure:theadditivemodelandthemultiplicative model.theadditivemodelassumesthattheforecastattimet,yt,isgivenby whilethemultiplicativemodelassumestheforecastisgivenby yt=(meanatt?1+localtrend)+(seasonality)+error (1) Eachcomponentoftheforecastisdescribedbelow. yt=(meanatt?1+localtrend)(seasonality)+error (2) 2.1 Themeancomponent,denotedbymt,ofthemodelgivesthelevelofthetimeseriesattime instantt.itisthebasecomponentofthetimeserieswhichismodiedbythetrendand Mean takenastheforecastoffutureobservations. seasonalityeectstogivethenalvalue.foraconstant,non-seasonalprocess,themeanis Amajorityofthetimeseriesdonotuctuateaboutaconstantlevel,butexhibitshiftsin 2.2 eithertheupwardordownwarddirections.thiseectismodeledbythetrendcomponent, Trend denotedbybt,anditgivesthegeneraldirectioninwhichtheseriesisprogressing.thetrend canbeclassiedasglobalorlocal,linearornon-linear,etc.theholt-wintersprocedure modelsalocal,lineartrend. Timeseriesgenerallyshowseasonalvariationsi.e.thereisaperiodicshiftinthelevelofthe series.thisisespeciallytrueincaseofhotelroomdemand,whichhavedistinctperiodsof 2.3 Seasonality highandlowdemanddependingonthetypeofhotelproperty,timeofyearetc.seasonal eectsarecyclici.e.itrepeatsitselfafteraxedintervaloftime.theholt-wintersmethod modelsanitenumberofseasonalvariations,andtheseasonalcomponentisdenotedbyct. 4

5 TheseasonalitycomponentdeterminestheversionoftheHolt-Wintersforecastprocedure. Theadditiveversion,inwhichtheseasonalcomponentisaddedtothebaseandtrendcomponents,isusedwhentheamplitudeoftheseasonalvariationisindependentofthelevelof version,theseasonaleectismultipliedtothebaseandtrendcomponents. theseasonalvariationisproportionaltothelevelofthetimeseries.inthemultiplicative AdditiveModel Thetimeseriesisrepresentedbythemodel thetimeseries.theuseofthemultiplicativeversionisappropriatewhentheamplitudeof wheretistherandomerrorcomponentwithmean0andvariance2.weassumethelength ofaseasontobesperiods.theequationsforupdatingthecorrespondingcomponentsare yt=mt+btt+ct+t (3) ^mt=(yt?^ct?s)+(1?)(^mt?1+^bt?1) ^bt=(^mt?^mt?1)+(1?)^bt?1 (4) where,andarethesmoothingconstantsforthebase,trendandseasonalcomponents ^ct=(yt?^mt)+(1?)^ct?s (5) respectively,and^mt,^btand^ctaretheestimatesofthebase,trendandseasonalcomponents (6) respectivelyattimet.theforecastforanyfuturetime=1;2;:::isgivenby Intheaboveequation,weusetheestimateoftheseasonalcomponentattimet+computed speriodsago. ^yt+=^mt+^bt+^ct+?s (7) MultiplicativeModel Themultiplicativemodelassumesthetimeseriestobeoftheform where,asabove,tistherandomerrorcomponent.theupdateequationsare yt=(mt+btt)ct+t (8) ^mt=yt=^ct?s+(1?)(^mt?1+^bt?1) ^bt=(^mt?^mt?1)+(1?)^bt?1 ^ct=yt=^mt+(1?)^ct?s (10) (11) (9) 5

6 where,andhavethesamemeaningasintheadditivemodel.theforecastforany futuretime=1;2;:::isgivenby^y Asseenfromequations1through10,itisquitestraightforwardtoimplementtheHolt- Wintersmethod(eitherversion)onadigitalcomputer.Wehaveusedthemultiplicative t+=(^mt+^bt)^ct+?s (12) versiontoforecastroomdemand,basedontheassumptionthattheseasonaleectsare proportionalinsizetothelocalmean.initially,thesmoothingconstantsareassignedvalues arbitrarily,andisoptimizedbeforeeachforecastruntominimizethemostrecentforecast error. 33.1ForecastingUnconstrainedRoomDemand andthelengthofstay.arequestforroomreservationalwaysspeciesaparticulararrival Areservationrequestischaracterizedbythreequantities:thearrivalday,marketsegment CharacterizingReservationRequests date.availabilityand/orpriceconsiderationsmaycausethecustomertochangethearrival isnochangeintherequestedarrivaldayofeachreservation.also,hotelshavedierenttypes day,butsinceweareinterestedinforecastingunconstraineddemand,weassumethatthere ofrooms(e.g.suites,doublerooms,economyetc.)andeachoftheseissoldatdierentrates. Additionally,thesametypeofroommaybesoldatdierentratestodierentcustomers, eachreservationrequestischaracterizedbythemarketsegmentorratecategoryrequested. dependingonpromotionalpackages,concessionalratesforgovernmentemployees,etc.thus, Thelengthofstayisanimportantfactorasithasadirectimpactonthedailyrevenue, capacity,etc. isrequestedfor.areservationrequestmaybeforonenight,twonights,orseveralnights. Lastly,thedemandforaroomisalsocharacterizedbythenumberofnightsthereservation 3.2 ilarlines.however,onemustrealizetwoimportantdierencesbetweenthetwo.firstly,only Likereservations,cancelationsofsomeexistingreservationsarealsocharacterizedalongsim- CharacterizingCancelationRequests reservations.theotherdierenceisthatitisimportanttoknowwhenthecancelationoccurred,i.e.howfarbeforethearrivaldatewasthereservationcanceled.asseenlaterin anexistingreservationcancancel.inthissense,cancelationsdependontheoutstanding Theforecastproblemcanthenbedenedasestimatingthenumberofnetreservations(reservations-cancelations)thatwillbereceivedforeachfuturearrivaldateperrateclassforeach section3.4,thisaectstheshorttermdemandforecast. 6

7 Demand > Figure1:Plotofactualunconstrainednetdemandofrooms 20 dierentstaydurationsaccordingtohistoricaldata,andwefocusourattentiononforecastingthenetdemandforeachfuturearrivaldayforeachmarketsegment.thesimulations possiblelengthofstay.inthispaper,weassumethatthereservationsaredistributedover Days > arecarriedoutforonerateclass.extendingthisworktoforecastunconstraineddemandfor severalrateclassesisstraightforward. 3.3 Wehaveuseddatafromanactualhotelfortheinitializationandtestingoftheforecastalgorithm.Reservationdatafor58weeks(406days)wasusedforthispurpose.Theproperty AnalysisofHistoricalData fromwhichthedatawasobtainedisabusiness/conventioncenterproperty.themethodologydevelopedhereisgeneral,andcanbeappliedtoanytypeofhotel(eg.businesstravelestrainednetdemandforthehotelproperty. Fromthegraph,weobservethefollowingpoints: -Adistinctseasonaleectisseenintheplot.Theaverageleveloftheseriesdecreases -Whilenodistinctglobaltrendisseenintheplot,itispossibletoidentifylocallinear between75and200daysascomparedtotheotherportionsofthetimeseries. trendsinthedata. property,leisuretravelerproperty).figure1showsthetimeseriesplotoftheactualuncon- 7

8 -Largeandsuddenincreasesordecreasesindemandareseenatdierentpartsofthe plot.thesemaycorrespondtocertaineventstakingplaceinthehotelorcity,and generallyhavedistincthighdemandandlowdemandseasons,anddierentroomallocation Theseasonaleectisspeciallyimportantinthehotelrevenuemanagementproblem.Hotels theseareclassiedasnon-randomeects. withexcessrooms.consequently,itisveryimportanttobeabletoestimatetheroomdemandbasedonthecurrentseason. partoftheroominventoryduringlowdemandseasontoreducetheoverheadsassociated strategiesareusedinthesedierentseasons.asanexample,hotelsmayworkwithonlya Toevaluatetheperformanceoftheforecast,weneedtosimulatetheprocessofreceiptof 3.4 requestsforhotelrooms.theoutputoftheforecastisthenumberofcustomersineach SimulatingReservationRequests randomreservationandcancelationrequestsbasedonhistoricaldata. mentandeacharrivaldayinthesimulationperiod.thus,theproblemisthatofgenerating marketsegmentthatwillactuallyshow-uponanyparticulararrivalday.thus,weneedto simulatethebuild-upofnetdemand(reservationsminuscancelations)foreachmarketseg- Poissondistributionandbinomialdistributionrespectivelyareverycommonintheliteratureandhavebeenusedbyseveralresearchers[2,3,4].In[2],aPoissondistributedrandotionisusedtomodelcancelationrequests.Modelingreservationsandcancelationsusing variableisusedtosimulatedierenttypesofreservationrequests,andabinomialdistributionformodelingno-showsandcancelations.optimalstrategiesforrentinghotelrooms theobjectiveofmaximizationofprotaredevelopedin[4],andabinomialdistributionis takingmultipledaysstayintoaccountarestudiedin[3],andatruncatedpoissonprocess isusedtodescribethearrivalprocess.dynamicoperatingrulesformotelreservationswith usedforcancelationsandno-showsinthesimulation. APoissondistributionisintuitivelyappealingsincetherequestsarereceivedprimarilyfor individualroomsratherthangroups.themeanofthepoissondistribution,whichrepresents therateatwhichreservationsarerequestedvarydependingonhowcloseorfarthearrival therateatwhichreservationrequestsarereceived,isobtainedfromhistoricaldata.since dayisfromthesimulationday,wemustusetime-varyingreservationratestomodelthe processmoreaccurately.toaccomplishthis,wedividethebookinghorizonintoanumber ofbookingperiods,andeachbookingperiodhasitsreservationrequestandcancelationrate. ThisconceptofbookingperiodsisillustratedinFigure2below.Inthegure,riandcirepresentthereservationandcancelationrates,respectively,inbookingperiodt=i.Notethat dierentbookingperiodscontaindierentnumberofdays.thisisbecausebookingperiods closertothearrivaldayreceivegreaternumberofrequests,andsotheyhavefewerdaysin ReservationrequestsaregeneratedusingaPoissondistribution,whileabinomialdistribu- 8

9 theperiod.conversely,bookingperiodsfurtherawayfromthearrivaldayhavemorenumber ment.thisimplementationgivesamorerealisticapproximationtotheactualbuild-upcurve. ratesarealsoobtainedfromhistoricaldataforthatparticulararrivaldayandmarketseg- ofdayssincetheyrecordsmallnumberofrequests.thedierentreservationandcancelation Reservation and cancelation rates in different periods Booking r1,c1 r2,c2 r3,c3 rn,cn periods t=1 t=2 t=3 t=n TheprobabilityfunctionforgeneratingrandomreservationrequestsaccordingtothePoisson lawwithtimevaryingratesisgivenby Figure2:BookingHorizonandBookingPeriods Arrival Approaching the arrival day Start of booking Day horizon f(x)=e?t(t)x where, x! forx=0;1;2;:::; Intheaboveequations,f(x)istheprobabilitythattherandomvariable(numberofrequests t=numberofrequestsreceivedinbookingperiodt receivedperday)takesonvaluesfrom0,1,2,:::.thet'sarethedierentrequestratesin periodst=1;:::;n,wheret=1isthebookingintervalclosesttothearrivalday,t=nis theintervalfarthestfromthearrivaldayandnisthetotalnumberofbookingperiods. Areservationmayalsobecanceledpriortothearrivalday,andweseektomodelthese cancelationsaswell.cancelationshavetobetakenintoaccounttobeabletocompensate forthelossinpotentialrevenueduetocanceledreservations.wemodelthecancelationwith obtainedfromhistoricaldata,andvariesaccordingtohowcloseorfarthearrivaldayis withparameterequaltotheprobabilityofthereservationcancelling.thisprobabilityis abinomialrandomvariable.foreachreservation,abinomialrandomvariateisgenerated, fromthesimulationday.abinomialdistributionwaschosenasthereareonlytwopossible outcomes:eitherthereservationwillcanceloritwillnot.bygeneratingsucharandom day/marketsegmentcombination.wecanthencalculatethepercentageofthesenetreservationsthatactuallyshow-uponthearrivaldaybymultiplyingthenetreservationsandthe cancelationvariateforeachreservation,wecancalculatethenetreservationsforeacharrival 9 0 otherwise (13) numberofdaysinbookingperiodt (14)

10 80 70 Actual Build up Simulated Build up Cum. Bkgs Figure3:Actualandsimulatedbuild-upcurves 10 obtainedfromhistoricaldata. show-uprateforthatarrivalday/marketsegmentcombination.theshow-uprateisalso Days Before Arrival Theprobabilityfunctionofabinomialdistributionusedtogeneraterandomcancelationsis thatreservationwillcancel.heren,whichrepresentseachreservation,equals1,andf(x)is wherexistherandomvariable,nisthenumberofbernoullitrialsandpistheprobability theprobabilitythattherandomvariablextakesvalues0or1.therandomvariableitself representstheeventthatthereservationwillcancel,withx=1meaningthatthereservation correspondtothecumulativebookingsthatshoweduponthearrivalday.asseenfromthe Actualandsimulatedbuild-upcurvesareshowninFigure3.Theordinateinthetwogures willcancelandx=0meaningitwillnot. twogures,weareabletoobtainarealisticapproximationofthebuild-upprocessusingthe givenby f(x)=nxpx(1?p)n?x (15) approachdescribedabove. 10

11 Theforecastedvalueofdemandiscomprisedoftwocomponents:thelongtermandtheshort 3.5 termforecasts.asthenameindicates,thelongtermforecastestimatesthenaldemandfor LongTermandShortTermForecasts requestsarereceivedduringthe60daysbeforethearrivalday;hencethenameshortterm propertystartsreceivingbookingforanarrivalday.typically,mostoftheadvancebooking Theshorttermforecastontheotherhand,estimatesthenaldemandonlyafterthehotel thedierentarrivaldates/marketsegmentcombinationswellinadvanceofthearrivaldates. forecasts. Thelongtermforecastisthedominantcomponentofthenalforecastwhenthearrivalday forecast.thenalforecastisaweightedcombinationofthelongtermandtheshortterm termforecastbecomesthedominantcomponent,sincetheshorttermforecastdependson alsochangeasthearrivaldayapproaches.initially,whenthearrivaldayisfaraway,the theactualbookingrate.thecorrespondingweightsassociatedwitheachofthetwoforecasts isfarawayfromtheprocessingday.conversely,asthearrivaldayapproaches,theshort longtermforecastweightisnearunity,whiletheshorttermforecastweightisalmostzero. forecastweightdecreases.eventually,whenthearrivaldayisveryclose,theshortterm Asthearrivaldaygetscloser,theshorttermforecastweightincreasesandthelongterm Wewishtopointoutthatthisapproachofusingtwoforecasts(shorttermandlongterm) Eachcomponentoftheforecastandtheforecastweightsaredescribedbelow. forecastweightwillapproachunitywhilethelongtermforecastweightwillbenearzero. andthencalculatingacombinedestimateiswellknownintheliterature[11].notably,estimationofloadonelectricpowersystemsusethismethodologyveryfrequently.procedures models-astochasticloadmodelbasedonhistoricaldataandaweatherloadmodelwhich forelectricloadestimationusingtwoforecastedarediscussedin[12].thispaperusestwo takesintoaccounttheeectofweathervariablesontheloadpattern.theforecastsfrom bothmodelsarethencombinedoptimallytogivethenalforecast LongTermForecast Thelongtermforecastestimatesthedemandforfuturearrivaldatesbasedonhistoricaldata. longtermforecast.weusetheholt-wintersprocess(describedinsection2)forthisforecast. Theforecastmaybemadeasmuchasayearaheadofthearrivaldate,andsoitiscalledthe Theshorttermforecastisanestimateofunconstrainednetdemandforfuturedatesbased 3.5.2ShortTermForecast ontheactualadvancebookingactivity.thisisincontrasttothelongtermforecast,which thedemand,theshorttermforecastusesthecurrentreservationheld,thecancelationrates estimatesthedemandentirelyonthebasisofthehistoricaldemandpattern.inestimating 11

12 fortheseheldreservationsandthenumberofreservationsthatwereturneddown.inan equationform,itisexpressedas ThetermsontherightsideofEqn.16areselfexplanatory.Thenetreservationsheldcorrespondtothenumberofreservationsforanarrivaldatethatareoutstandingatthetimeof netreservationsturneddown (16) S.T.forecast=(netreservationsheldcancelationrate)+ calculatingtheshorttermforecast.thisquantity(netreservationsheld)whenmultipliedby thecancelationrategivesthenumberofreservationsthatwillactuallyshowuponthearrival amountisincludedintheshorttermforecastsincewewanttoforecasttheunconstrained date.thesecondtermontherightsideofeqn.16representsanestimateofthenumberof requestswereaccepted,thenthedemandwouldbehigherbyanequivalentamount.this reservationsthatwereturneddownordeniedduetocapacityorpricingconstraints.ifthese demand ForecastWeights Thenalforecastconsistsofcombiningthelongtermandtheshorttermforecaststoproduce asinglecompositeforecast.thisisachievedbytakingaweightedsumofthetwoforecasts. Theobjectiveistogivethelongtermforecastahigherweightthantheshorttermforecast whentheprocessingdayisfarawayfromthearrivalday.alternately,theshorttermforecast isgivenahigherweightwhentheprocessingdayisclosetothearrivalday.theweightsare normalized,i.e.,theirsumisunity. Initially,theweightsaresetaccordingtothenumberofbookingsandnumberofforecasted Theupdatefactordependsonthemeansquareerror(MSE)betweeneachoftheforecast demand.duringtheexecutionoftheforecastprogram,theweightsarecontinuallyupdated. andtheactualvalue.thenewweightsaregivenby Theweightsarethemselvesupdatedbytakingaweightedaverageofthenewweightsand newweight= STforecastMSE+LTforecastMSE theoldweights.theupdatedweightsaregivenby (17) togivetothenewweights.weuse=0:9inthesimulations. Theparameter0<<1isxedarbitrarily,dependingonhowmuchimportancewewish updatedweight=oldweight+(1?)newweight (18) 12

13 3.5.4CombinedForecast Afterhavingcalculatedtheshorttermandthelongtermforecastsandtheforecastsweights, thenalcombinedforecastcanbecalculatedas where nalforecast=lt-weightltforecast+st-weightstforecast (19) Thenalforecastiscalculatedforeacharrivalday. st-weight=1?lt-weight lt-weight=updatedweight Havingseenthedierentcomponentsoftheforecast,wecannowstudytheactualprocedure used.theforecastalgorithmcanbeeasilyunderstoodwiththehelpoftheowchartshown 3.6 ForecastAlgorithm infigure4. Theowchartshowshowthetwoforecastcomponentsarecalculatedandcombinedtogive thenalforecast.initializationofthelongtermcomponentinvolvessettingthevaluesof theforecastprocess.oneyear(52weeks)worthofdataisusedfortheinitialization.all themean,trendandseasonalcomponents,andisperformedonlyonce,atthebeginningof theothercomponentsofthelongtermforecastareupdatednightly.thenextstepinthe longtermforecastistondtheoptimalvalueofthesmoothingparameter(refersection betweentheforecasteddemandforthecurrentdayandtheactualdemandforthesame 2)thatminimizesthecurrentforecasterror.Thecurrentforecasterroristhedierence day.wealsousetherootsquareforecasterrortodetectnon-randomeectsinthedata. (0.5and0.025respectively)areimposedonthevalueof.Westartwithinitialvaluesof Dependingonwhethersucheectsarepresent,theincrementswithwhichischangedis decided.toensurethestabilityoftheholt-wintersprocedure,upperandlowerbounds =0:2;=0:05and=0:1.Oncethevaluesofthesmoothingconstantsarexed,the Theinitializationfortheshorttermforecastcomprisesofdeterminingthecancelationrates Theseindividualcomponentsarethenusedtogeneratethenallongtermforecast. individualcomponentsofthelongtermforecastarecomputedforallfuturearrivaldays. cancelationrateandtheturndownstoestimatethenetdemand.eachday,thenumberof ratesdonotchangefromyeartoyear.theshorttermforecastusesthereservationsheld, daywillbeknownonlyafterthearrivaldayhaspassed.weassumethatthecancelation fromthehistoricaldata.thisisrequiredsincetheactualcancelationrateforagivenarrival usedinthegenerationoftheshorttermforecast. reservationsandturndownsareobservedandthesetogetherwiththecancelationrateare 13

14 Short Term Component Long Term Component Initialization Initialization Find optimal α Observe bookings, cancelations and turndowns Actual demand Calculate mean, trend & seasonal components Generate short term forecast Generate long term forecast Compute short term forecast MSE Compute long term forecast MSE Set forecast weights Weighted short Weighted long Figure4:FlowchartofForecastAlgorithm term forecast term forecast Calculate combined forecast 14

15 Thelongtermandshorttermforecastsgeneratedaretheneachcomparedwiththeactual demandforthecurrentday,andthecurrentforecastmeansquareerrorsarecalculatedfor eachforecast.theforecastweights,whicharegiveninitialvaluesduringthelongterm Theweightedshorttermandlongtermforecastsarethensummeduptogivethenal forecastinitialization,areupdatedbasedontheseerrorsasgiveninequations17and18. compositeforecast. initializationandsimulationpurposes.datafromtherst52weeksareusedforinitialization 4AsdiscussedinSection3.2,58weeksofdatafromanactualhotelpropertywereusedfor SimulationResults cancelationrequests. reservationandcancelationrequestratestosimulaterandomreservationrequestsaswellas oftheforecastparameters,anddatafromthefollowing6weeksareusedinobtainingthe Toseetheeectsofthelongtermandtheshorttermforecastsforaparticulararrivalday distinctly,weshouldstartforecastingthedemandforthatarrivaldaywellinadvance.since thesimulationperiodisonly6weeks,wecanexpecttoobservebothcomponentsofthe choicewasmadebecausethehotelpropertyfromwhichthedatawasobtainedcatersprimarilytobusinesstravelers,andsodierentbookingpatternscanbeexpecteddepending forecastforthelasttwoweeksofthesimulationperiod.wechooseoneweekdayandone weekenddayinthelastweekofthesimulationperiodtotesttheforecastalgorithm.this foraweekday(testday1)inthelastweekofthesimulationperiod.theforecastofdemand onthedayoftheweek. foraparticulararrivaldayisdoneeverynightpriortothearrivalday.forthisexample, theforecaststarts38nightsbeforethearrivalday. Figure5showstheactualbuildupofreservations,thecombinedforecastanditscomponents InFigure5,thecombinedforecastanditstwocomponentsgiveestimatesofthedemand uptoonedaybeforethearrivalday.thearrivaldaycorrespondstothezerocoordinateon thex-axis.thenalforecastedvalueslieontheverticallinedrawnatonedaybeforethe arrivalday.thepointatwhichtheactualdemandcurvemeetsthey-axisgivesthevalueof thenumberofreservationsthatshoweduponthearrivalday. Asseenfromthegure,thecombinedforecastfollowsthelongtermforecastcloselywhen thearrivaldayisfarfromthedaytheforecastismade.thenumberofactualreservations forecasts,liesinbetweenthetwoestimates.eventually,thecombinedforecastgivesasatisfactoryestimateofthenaldemand. Asthearrivaldayapproaches,therateofreservationrequestsincreases,andtheshortterm forecastgivesbetterestimates.thecombinedforecast,beingaweightedsumofthetwo initiallyisverysmall,andsotheshorttermforecastdoesnotgiveaveryaccurateestimate. 15

16 Actual Demand buildup Combined Forecast LT Forecast ST Forecast Final Actual Demand Demand Figure5:ActualandForecastedUnconstrainedDemandforTestDay Figure6showsasimilargraphforTestDay2inthelastweekofthesimulationperiod Similartothepreviouscase,theforecastforthisarrivaldaystarts40nightsbeforethe Days Before Arrival arrivalday. Theresultsobtainedinthesecondtestcasearesimilartothoseobtainedintherst.The combinedforecastfollowsthelongtermforecastinitially,andasthearrivaldayapproaches, ittakesintoaccounttheeectoftheshorttermforecast.weobtainagoodestimateofthe unconstrainedroomdemandinthiscaseaswell. Figure7showsasimilarplotforanotherdayinthelastweek.Inthiscasehowever,thereis alargedeviationoftheforecastedvaluefromtheactualvalue.thereasonsforthisbehavior oftheforecastarediscussedlaterinthesection.theforecsatforthisdaystarted36nights priortothearrivalday. Fromtheaboveresults,weseethatneitherthelongtermnortheshorttermforecastcan giveareasonableestimateofthedemandindividually,butaweightedsumappearstobe eectiveinsomecases.thisshouldnotbesurprising,sincethelongtermforecastgives fortheuseofalongterm-shorttermforecastmethodology. whenthearrivaldayisveryfarawayfromtheprocessingday.thisprovidesthemotivation anestimateonthebasisofhistoricaldemand.theshorttermforecast,ontheotherhand, dependonactualreservationsandturndownsandthismakesthetaskofforecastingdicult 16

17 Actual Demand buildup Combined Forecast LT Forecast ST Forecast 150 Final Actual Demand Demand 100 Figure6:ActualandForecastedDemandforTestDay Days Before Arrival Actual Demand buildup Combined Forecast LT Forecast ST Forecast 140 Demand Final Actual Demand 80 Figure7:ActualandForecastedDemandforTestDay Days Before Arrival

18 obtainarealisticapproximationtothereservation/cancelationprocess.inthispaper,the randomly,anditisessentialthatweaverageoverseveralrealizationsoftheserequeststo MonteCarlosimulationshavebeenusedtocomparetheactualandforecasteddemand.This actualandforecasteddemandhavebeenaveragedover100runsforeacharrivalday. wasnecessitatedbythefactthatthereservationandcancelationrequestsaregenerated absolutedeviationisgivenby algorithm.thisiscalculatedoverthelast2weeksofthesimulationperiod.themean Themeanabsolutedeviation(MAD)isusedasameasureofperformanceoftheforecast AplotofthemeanabsolutedeviationisshowninFigure8. MAD=Pjactualdemand?forecasteddemandj numberofdays (20) Mean Absolute Deviation Anothermeasureofperformanceisthemeanabsolutepercentageerror(MAPE).MAPEis 5 computedbyaveragingtheabsoluteerrorbetweentheactualandforecasteddemandexpressedasapercentageofactualdemandovertime.figure9showsaplotofthemean Figure8:MeanAbsoluteDeviationfornal2weeksofsimulationperiod absolutepercentageerror. MAPE= Pjactualdemand?forecasteddemandj Fromthevariousgraphs,wenoticethatwhileagoodforecastisobtainedfromsomedays, thereareoccasionsonwhichthealgorithmdoesnotperformsatisfactorily.theprimary 18 numberofdays (21)

19 Mean Absolute Percentage Error reasonforthisisthattheforecastisbasedentirelyonaquantitativemodelandharddata. Figure9:MeanAbsolutePercentageErrorfornal2weeksofsimulationperiod demand,butknowntothehotelmanager.also,thealgorithmisunabletodistinguishnonrecurringeventsbasedonlyonhistoricaldata.thisprovidesagoodmotivationforincluding humanknowledgeintheforecastprocedure. Itdoesnottakeintoaccountexternal/non-randomeectswhichmayhaveinuencedthe 2 ThispaperdiscussedtheHolt-Wintersforecastingprocedureanditsapplicationtoforecastinghotelroomdemand.TheparamatersoftheHolt-Wintersmodelareinitializedusing 5 DiscussionandFutureResearch tocomputethelongtermforecastofroomdemand.theshorttermforecastwascomputed historicaldataobtainedfromanactualhotel.theholt-wintersforecastapproachwasused basedonactualbookingactivity.thenalforecastisaweightedsumofthelongterm andtheshorttermforecasts,andtheforecastweightsaredecidedbythemeansquareerror aresimulatedusingpoissonandbinomialdistributions,respectively.theparametersfor betweentheforecastedandtheactualvalues.randomreservationandcancelationrequests thedistributionswereobtainedusinghistoricaldata.thesimulationsarecarriedoutfora singlerateclass.extendingthealgorithmformultiplerateclassesisstraightforward. Alotofthedicultiesencounteredintestingthealgorithmweredatarelated.Having58 weeksofdataisbarelysucienttoinitializeandtesttheforecastalgorithm.duetothis, itisnotpossibletoupdatetheseasonalcomponentandobserveitseect.additionally, 19

20 itispossiblethatthedatacontainednon-randomeects(whichmayaectthedemand) orunexplainedcrestsortroughsindemandandthiscausedthealgorithmtogiveerratic atrulyaccurateforecastalgorithm.oneofthemostimportantobjectivesofthisprojectis Thereareseveralissueswhicharestilltobeaddressedandaccountedforinordertoobtain demandvaluesonsomeoccasions. ofthearrivalday.thisisbecausetheyareabletosupplementtheirexperiencewiththeir thathotelmanagersareabletogiveveryaccurateforecastswithinaperiodof2-3weeks tobeabletoincorporateexpertknowledgeintothesystemforecast.ithasbeenobserved knowledgeofevents,demandpatterns,etc.weseektorepresentthisknowledgeintothe forecastalgorithm.theresultingalgorithmisnotintendedtoreplacethehotelmanagers; insteaditwouldtaketheexpertknowledgeasaninputandcombineitwithamathematical forecasttogivethebestpossibleforecast. Theroleoftheforecastshouldalsobeseenintheproperperspective.Asmentionedin thebeginningofthepaper,wefeelthatagoodforecastwillresultinbetterinventory optimizationandmanagement.thus,theobjectiveistobeabletoobtainagoodforecast days.thislogicallytakesustotheuseoffuzzylogicintheforecastalgorithm.theseideas Workalongthislineofthoughtwillbereportedinafuturepaper. arebeinginvestigatedandtheycertainlypresentanewapproachoftacklingtheproblem. consistentlyforallfuturedays,ratherthanhavinganexactpredictionofthedemandonsome References [1]Cross,R.G.,RevenueManagement,BroadwayBooks,NewYork,1997. [2]Bitran,G.R.,andGilbert,S.M.,ManagingHotelReservationswithUncertainArrivals, [3]Bitran,G.R.,andMondschein,S.V.,AnApplicationofYieldManagementtotheHotel Opns.Res.,Vol.44,No.1,(Jan.-Feb.1996)pp [4]Ladany,S.P.,DynamicOperatingRulesforMotelReservations,Dec.Sci.,Vol7,(1976) IndustryConsideringMultipleDayStays,Opns.Res.,Vol.43,No.3,(May-Jun.1995) pp [5]Montgomery,D.C.,andJohnson,L.A.,ForecastingandTimeSeriesAnalysis,McGraw- Hill,NewYork,1976. pp [6]Harvey,A.C.,TimeSeriesModels,MITPress,Cambridge,MA,2nded.,1993. [7]Box,G.E.P.,andJenkins,G.M.,TimeSeriesAnalysis,ForecastingandControl, Holden-Day,SanFrancisco,Rev.ed,

21 [8]Chateld,C.,TheAnalysisofTimeSeries:AnIntroductionChapmanandHall,London, [9]Winters,P.R.,ForecastingSalesbyExponentiallyWeightedMovingAverages,Mgmt. 2nded.,1980. [10]Chateld,C.,TheHolt-WintersForecastingProcedure,AppliedStatistics.,Vol27,No Sci.,Vol6,No3,(1960)pp [11]Sage,A.P.,OptimumSystemsControl,Prentice-Hall,EnglewoodClis,NJ, ,(1978)pp [12]Gupta,P.C.,andYamada,K.,AdaptiveShort-TermForecastingofHourlyLoadsusing [13]Harvey,A.C.,Forecasting,StructuralTimeSeriesModelsandtheKalmanlter,CambridgeUniversityPress,Cambridge,1989. WeatherInformation,IEEETran.PowerAppar.Syst.,Vol.PAS-91,Sept./Oct.1972, pp