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THEGREENLANDAIRTRANSPORTMODELSYSTEM-ACOMBINEDTRANSPORTMODELLINGANDOPTIMISATIONPROBLEM

OttoAnkerNielsen1,a,Professor,Ph.D.JeppeRich,a,AssociateProfessor,Ph.D.

MetteAagaardKnudsen,a,Ph.D.student,M.Sc.RasmusDyhrFrederiksen,b,M.Sc.Partner,Rapidisltd.

aCenterforTrafficandTransport(CTT),TechnicalUniversityofDenmark(DTU)

Building115,st.tv.,2800Lyngby,Denmark

bRapidisLtd.,Jgersborgall4,2920Charlottenlund,Denmark

SHORTABSTRACTThe paper describes a combined demand and optimisation model for air transport inGreenland.Demandand routechoice forpassengerand freight isaddressedat the lowerlevelproblem,whereastheservicenetwork,flightschedules(time-tableoptimization),de-cision on airplane types (considering cost and capacity), and finally creation of airplaneschedules are dealt with in the upper level problem. Due to special capacity restrictionscombinedwiththeneedforaschedulebasedassignmentmodel,thisbi-levelproblemcan-not be solved analytically. The lower level model is a non-analytical non-linear non-continuousmappingofthesolutionoftheupperlevelproblem.ThemodelisnowusedbytheGreenlandHomeRuletodecideuponanewairportstruc-tureinGreenland(changeofmainairportsandcreationofnewairports),policiestoobtainmorecompetitionandtoliberalisethemarket,andtodesignthemainairtransportsystem.Inthepaper,wepresentthemodelaswellasastudyinwhichthelocationofanewAtlan-ticAirportisanalysed.Thecost-benefitanalysisindicatesthatthepresentairportstructureishighlyin-optimalandthatsignificantbenefitswillresultifthelocationofthenewair-portismovedtoNuuk,theCapitalofGreenland.

1. INTRODUCTIONBeingtheworldslargest island,butwithonly56,000inhabitants inamountainousarcticareawithnopossibilitiesforinterurbanroadtransport,airtransportplaysacentralroleforthesociety.However,thepresenttransportnetworkneedslargepublicsubsidiesyethavingveryhighusercostsandofteninconvenienttransfersfortheusers.

1CorrespondingAuthor.E-mailaddress:oan@ctt.dtu.dk;Tel.:+4545251514;Fax:+4545936412.

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1.1 BackgroundThe main airport of Greenland is not located at the Capital (Nuuk), but at the locationSdr.StrmfjordatthebottomofthefjordSnderstrmfjord150km.fromthecost-lineandonly10km.fromtheicecap.Thislocationwasoriginallychosenduetohistoricalrea-sons, since the US built the runway and airfield during the Second World War. At thattime,themainconsiderationwastominimizeaccessibilityforpotentialGermansubma-rines. Clearly, today this objective has changed and with increasing air transport inGreenland,theefficiencyofthesystemhasbeenquestioned.Threemajorissueshavebeendiscussed; Thereisa feeder-trafficstructurebetweenNuukandSdr.Strmfjord.Thelinkhasto

beservicedbyDASH7turbo-propairplanes(50seatmax.),whichisthelargestair-planethatcanlandandtake-offatthe899meterrunwayinNuuk.Tocompare,the4hour flights fromSdr.Strmfjord toCopenhagenarepresently servedbya245seatsAirbuses-200withadditionalcargofacilities.

TherehavetobeanArtificialcommunityinSdr.Strmfjordtosupporttheoperationoftheairportandthereislargeoperationcostsassociatedwiththis.Asecondproblemisthat,asaresultofglobalwarming,thepermafrostonwhichtheSdr.Strmfjordrun-wayisbasedhasstartedtothaw,makingitcostlytomaintain.

Fordomesticflights,aDash7airnetworkhasNuukashub.Therefore,theIslandop-erateswithtwohubs,eventhoughthepopulationisonly56,854inhabitants.

EventhoughitmightseemveryobvioustobuildanAtlanticairportinNuuk,thedeci-sionisnot trivialduetotheconstructioncostsrelatedtothis.Realisinganeedforbetterdecision tools, the Greenland Home Rule asked the Technical University of Denmark(DTU)todevelopanintegratedsystemofatransportmodel,anairnetworkoptimizationmodel,andasocio-economicdecisionsupportmodel.

1.2 Overviewofthemodelsystem

Whilethewholemodelsystemincludesdemandmodels,andsocio-economicimpactmod-els,thepresentpaperfocusontheinteractionbetweentheassignmentmodelsandtheop-timisationmodels.Theroutechoicemodelassignspassengersontothe(air)network.Themodeloperateswithexacttimetables,andtakesintoaccountcapacityconstraints(seataswellasweightrestric-tions).Themodelisformulatedasastochastic(mixedprobit)multi-classuserequilibriummodel. A special consideration is that post (letters) have the highest priority, and thatpostal sacks literally can replace passengers. Passengers then have higher priority thanfreight.Thechoiceofthepassengersdependsonthenetwork,i.e.theleg-structure,thedeparturetimes(andfrequencies)andtheaircraftsbeingused(fastjet-plainsmaymakeadifferencecomparedtolesscomfortableslowpropelledairplanes).Theoptimizationmodeldesigntheoverallairtransportationnetwork,legstructure(depar-turetime),choosesplaintypes,andcalculatesthecostsofoperationsincludingtheneces-sarynumberofaircrafts.

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1.3 DecisioncontextThemodelcontextmaybeconsideredtorepresentatthreeplayergameconsistingofandfindinganequilibriumbetween;

1. Home rule decisions (which are feed into the system manually as exogenous as-sumptions)

Airportstructure(location,lengthofrunways) Subsidiesand/orpackagesofroutes MinimumservicefrequenciesdefinedbytheHomerule

2. Aircompanydecisions Legs(andgeneralstructureofnetwork) Schedule Airplaneallocation Non-Greenlandflights Fare

3. Passengerdecisions Number of trips, destination, mode choice (domestic to some limited ex-

tend), route choice, timeof travel (partlypart of route choice), fare level,company.

1.4 ObjectfunctionsintheoptimisationmodelTheobjectfunctionoftheairmodelincludesthefollowingelements;

1passenger(dis)utility+2costs_of_operation+3fare_revenueThefunctionismaximised,wherebythemodelcreatestheairnetwork.Thepassengerdis(utility)isafunctionoffrequency(dis-utilityoffewdepartures),waitingtime,traveltime,transfertime,numberoftransfers(theextradisutilityoftransfering),andfare.Thecostsofoperationsoftake-off(includinghandlingattheairport),adistancedependentcost and a turn-arround cost (the costs of not using the airplain between arrival anddeparture, i.e. lost returnof investment).Thesecostsdependon the typeofaircraft.Thetotalcostsusually increaseswith thesizeof theaircraft,however, theunitcostperPAX(passenger)isusuallylowerforlargeraircrafts.InGreenland,helicoptershavetobeusedto locations without runways, however, helicopters have a much higher unit cost thanairplanes.Thetwooveralloptimizationcriteriafortheservicenetworkdesignarepartlyconflicting;

Tominimizetheoperationscostsandmaximizerevenue(companyobjectives) Tomaximizetheutilityforpassengersandsociety

Theobjectfunctioncanbeconfiguredaspurecompanyprofitmaximization,i.e. Max[carerevenue-costsofoperations]

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Normallyitisreasonabletoassumethatairlinecompaniestriestomaximizethisobjectiveandthattheroutenetworkwouldreflectsuchfunctioninacompletelyfreemarket2.

Thefunctioncanalsobeconfiguredasapuresocioeconomicoptimization,i.e. Min[passenger(dis)utility+costofoperation]Inotherwords, thepassengersshouldobtainminimumgeneralised travelcost taking theoperationcostsintoaccount.Thissocio-economicoptimizationresultsinadifferentroutenetworkthantheoperationaloptimization. Incaseof substantialdifferencesbetween the twosolutions itcouldfavourregulations. I.e. a change in the operational conditions through subsidizes or taxes andduties

Acombinedfunctioneveninasocio-economicvaluationcouldalsobeasolution,i.e.Min[1passenger(dis)utility+2operationcost+3ticketrevenue]

Itcanbeargued,thatsinceashareoftheticketrevenuesissubmittedtotheHomerules,thisobjectivemightbereasonable.Thiscouldaswellbethecase,especiallyiftherevenuecomes fromforeigners.Also, itmightbe lessdistorting than the taxon income (less taxdistortion).3 is between 0 and -1. Isolated from Greenlands point of view it could beargued that theoperation costshas apartly increasing affect on employment and that2shouldbesetbelow1.

Hence,inpoliticalanalysisitispossibletoimplementsensitivitycalculationsandevaluatethe optimized structure of the route network given different assumptions and politicalpriorities. Within the model tests so far there are however used a purely operationalconfigurationasitispresumedthatAirGreenlandprimarydesignstheroutenetworkfromprofitableconsiderations.

1.5 PassengerbehaviourpredictionEachpassengerwilltypicallymaximizehis/herownutility.However,inaddition,itcanbeassumedthatpassengershaveincompleteknowledgeofalternativesandaswelldifferentpreferences.Asa result, themodeloperateswith twodifferentobjectivefunctions in thetrafficmodelandtheoptimisationmodelrespectively;

Theobjectivefunctionforthetrafficmodeldescribesthepreferencesofthepassen-gers (e.g. the value-of-time) and includes a random term and stochastic coeffi-cients.

Theobjectivefunctionoftheoptimisationmodelisweightedfunctionofsystemop-timalcriteriasandoperationalcriterias,whichbothisdeterministic.

The different models operates by trip purpose (business, tourists, and natives), each one

23may,however,belessthan1,iftheticketpriceincludestaxes.

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withaseparateutilityfunction

2. MAINSTRUCTUREOFTHEMODELThemodelframeworkconsistsoftwoseparatemodules;1. Demand models, which describe number of trips and destination choice. The models

have been stratified according to trip purpose (business, tourists, natives, post, andfreight). In addition, most of the demand models have been formulated with separatemodulesfornationalandinternationaltrips.

2. Supplymodels,whichdefinetheleg-structure(planeconnections),stockofplanes(typeandnumberofplanes), and thepassengerschoiceof routeasa functionof trafficde-mand.Theroutechoicemodelsdistinguishbetweenmodes,butevaluatetheassignmentsimultaneouslyduetocapacityconstraints,whichisaffectedbyallpurposes.

OD-matrices are feed from demand models to supply models, describing the number oftravelersbetweenthedifferentairports.Thesupplymodelcalculatestraveltimeandcostsat the OD level, which is used to re-calculate demand. As a result, the two models williterateuntilequilibriumisreached.BecausetherearegreatseasonalvariationindemandandsupplyinGreenland,themodelissplitinto4models,oneforeachquarter.Thefigurebelowillustratestheoverallmodelstructure(notethefeedbackfromall5demandmodels).Figure1:Overallmodelstructure.

Turismmodel.WorldtoGreenlandand

domesticdistributionofturism

Privatetravelermodel

Businestravelermodel Airmailmodel Airfreightmodel

RoutechoicemodelServicenetworkdesign

SchedulingAircraftallocation

Trafficvolumesfromthedemandmodelsto

thenetworkmodels

Feedbackoftraveltimes

andtransportnetwork

2.1 MainflowThemainflowinthemodelis;1. Run the demand models. These models calculate the transport demand on a weekly

basis,i.e.numberoftripstoandfromeachairportandthedistributionoftripsbetweentheairports.Therebythismodelstepproduces5ODtripmatrices(Origin-Destinationmatrices)

2. Thetripmatricesareatfirstdividedintothe7weekdayswhichisthensubdividedinto7timeintervals.Thesplit-functionsvarybetweenthe5trippurposes.

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3. Thenextstepistherunofascenariogeneratorthatgeneratesthepossibleairlinecon-nectionsbetweenall theairports (incl. theheliports)given theavailableplane types,thelengthoftherunways,andthedistancebetweentheairports.

4. Theplaneoptimizationmodeloptimizesthelegstructure(airlineconnectionsbetweentheairports),numberofdepartures,andtheusedplanetypeswithiterationswiththeassignmentmodelasaniterativeprocesswithanassignmentmodel

5. Thesupplydatatraveltimes,costsetc.calledLoS(LevelofService)isaggregatedtoweeklybasisforeachtrippurposeasaweightedaverageofdaysandtimeperiods.

6. Thedemandmodels(tourism,business,visiting,mailandcargo)arerepeatedwiththenewsupplydata

7. Thematricesaresubdividedintoweekdaysandtimeperiodsasinstep2.

8. Thecalculationswiththeplaneoptimizationmodelisrepeated

9. Aplanebalancingmodel isstarted,whichcalculatesandbalancestheuseofaircraftmaterial.Duringtheprocesseachlegisupgradedandnewlegscouldbeadded.

10. Thefinalassignmentmodel

It couldbe argued that thedemandmodels and the supplymodels should iterate severaltimesbut there isanupgradefunctionbuilt in theassignmentmodelwhichupgrades thematerialafterthedemandcalculation.Thisissueisthereforelesscritical.

3. ROUTECHOICEMODELThepassengerroutechoicemodelfindstheuser-equilibrium,e.g.thesituationwherenopassengersperceivedutilitycanbeimprovedbyhe/herunitarilychangingrouteatthede-siredtimeofdeparture.Routechoiceisdependentontrippurpose,preferencesofpassengers,andtheuseoftime.Theutilityisstronglyflowdependent,sincemostairportsinGreenlandcanonlybeservedbysmallairplains, themaintypebeingtheDASH7withmaximum50seats.Onlythreeinternationalcivilairportsexist3;

Sdr.Strmfjord,whichisservedbya245seatAirbus-200 Nasarsuaq,whichisservedbya180seatBoing-757 Kulusuk,which isservedbyaanother turbo-propairplane(Fokker50which is

quitesimilartoDASH7)

3TheMilitaryairportlocatedatThuleisnotopenforcivilairtraffic.

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Thestrictcapacityrestrictionscombinedwiththelowfrequencyledtothedecisiontoim-plement a Stochastic User Equilibrium timetable-based assignment model (Nielsen &Frederiksen,2006)thatisrunonaweeklyschedule(sincesomelegareonlyservedonceortwiceaweek).ThismodelwasthenmodifiedtoreflectanumberofspecialissuesinGreenland;

Passengercapacityisastronglylimitingfactor;passengersaresimplyrejected,notonlydelayedase.g.inclassicalroutechoicemodels.Thiscausessomedifficultiesinthemodelformulationandsolutionalgorithm.

Airmailhashigherprioritythanpassengers, i.e.aneedforasequentialapproachwithregardtothiscomparedtotraditionalmulti-classequilibriummethods.

Payloadrestrictionsmayrestrictthenumberofpassengersandtheamountofcargo. Airplane(schedules)hastobepartofthemodel. Tripsmayevenhavetowaittothenextweekortheymayoriginfromtheweekbe-

fore.Firstacyclicgraphapproachwasexploredforthis,whilstabeforeandafterdemandandnetworkperiodwasfinallydecided.

3.1 UtilityfunctionsThe model has the following explanatory variables that are optimized in a linear-in-parameterutilityfunctionforeachclassofpassengers(andbysimulatingstatisticaldistri-butionsofeachpassengeraswell);SupposethattherearekpassengerclassesandiIroutealternatives,thentheutilityfunc-tionsisgivenas;

Uk,i=k,edEDk,i+k,ldLDk,i+k,tpTPk,i+k,ttTTk,i+k,opOPk,i+k,itITk,i+cCi+k,iWhere;EDk,i:Earlydeparturepenalty(sortofhiddenwaitingtime)LDk,i:Latedeparturepenalty(traditionalhiddenwaitingtime)ITk,i:Traveltime(traditionalInvehicletimespendintheairplane)TTk,i:Transfertime.

- Strongly non-linear, as long transfer times can be used constructively, e.g. onmuskoxtrips,orvisitingtheinlandiceatSdr.Strmfjordusinglongtransfertimesfore.g.tourism

TPk,i:Transferpenalty(disutilitybynon-directtravels)OPk,i:Overnightpenalty(disutilityandcostwhenneedforovernighttransfers/stays)Ci:Cost(Ticketcosts)k,i:IndependentidenticalGumbelerror.

Inthemodel,thebehaviouralparametersk,ed,k,ld,k,tp,k,tt,k,op,k,it,callfollowedlog-normal distributions (over the population), with the relative size between passengerclassesandtimecomponentsbasedonexperiencefromCopenhagen.

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3.2 ChosenparametersSinceitisdifficultandcomputationalhardtoimplementnon-linearutilityfunctionstheassumednonlinearityinthetotaltraveltimeswasapproximatedbyusingahightransferpenalty(asproxyfornon-linearity)andalowvalueoftransfertime.Differentvaluesweretestedwhere40%ofthefreetraveltimegavethebestresult.

Thescalingofthedifferenttimevaluesforeachpassengertrippurposesisingeneralsimi-lar.Howeverthehiddenwaitingtimeforvisitingtripsisassumedtohaveasmallerimpactontheutilitythanbusinesstripsandanevensmallerimpactfortourismtrippurposes.ForairmailandcargoitisassumedthattraveltimefromOtoDandthepriceofthetrans-portationistheonlyparametersinfluencingtheutility.Whichtimecomponentsthetripcanbesubdividedintoisconsideredwithoutinfluenceforcustomers.Theaccesstimeishow-evergivenahighertimevalueasthisisregardedasadiscomfortforcustomersandthefreightcompanies.

TheticketpricesareingeneralpricedinDKr.BecausetherestofthevariablesarescaledintoDKrusingtimevaluesthecoefficientsforticketpricesisnormallysetto1.Fortheo-reticalreasonsthiscoefficientisfixed.Inthemodelitispossibletodefineamaximumwaittimeandamaximumearlydeparturetimetoimplementhowlongtimethepassengersarewillingtowaitandleavebeforetheplaneddeparture.Table1belowshowsthefinalparametersintheroutechoicemodel.

Concerningtherandomvariation(theerrorterm)itisassumedtobeadditivenonnegativedistributed(gammadistributed)andthatthepassengersarefamiliarwiththeroutenetworkwhythevarianceisratedlow(5%).The last definitions describe the average weight per passenger inclusive luggage andweight units for mail and cargo. For some plane types the passenger seats can beexchangedwithmailsacks.Finally there is a specified rangingof the travellers,mails and cargo in themodel.ThisresultsfrominterviewsinGreenlandwheremailshavehigherprioritythanpassengersandpassengershavehigherprioritythancargo.

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Table1Parametersintheroutechoicemodel

Parametername Business Tourism Visiting Mail Cargo

BASISValueoftime 2,29 1,62 1,62 1 0,5

Accesstime(WalkTimeW) 3,435 2,430 2,430 1,500 0,750Variance(WalkTWVar) 0,3435 0,2430 0,2430 0,1500 0,0750Changetime(WaitTimeW) 0,916 0,648 0,648 1 0,5Variance(WaitTWVar) 0,0916 0,0648 0,0648 0,1000 0,0500Timetoairport(ConTimeW) 2,748 1,944 1,944 1 0,5Variance(ConTWVar) 0,2748 0,1944 0,1944 0,1000 0,0500Hiddenwaittimeinzone(WaitIZoneW) 0,687 0,162 0,486 1 0,5

Variance(WaitZWVar) 0,0687 0,0162 0,0486 0,1000 0,0500Changepenalty(ChangePen) 137,4 97,2 97,2 0 0Variance(ChPenVar) 6,87 4,86 4,86 0 0Earlydeparture(EarlyDW) 0,687 0,162 0,486 1,000 0,500Variance(EarlyDVar) 0,069 0,016 0,049 0,100 0,050Ticketprice(CostW) 1 1 1 1 1Variance(CostVar) 0 0 0 0 0Distributiontype(DistAll) 7 7 7 7 7Variance(StocCofAll) 0,05 0,05 0,05 0,05 0,05Weightperseat(SeatUnitWeight) 100 100 100 100 100

Seatsinplane(Seat) 1 1 1 1 0Cargoroom(Cargo) 0 0 0 1 1RankingLoad(PreLoad) 1 1 1 0 2

3.3 CapacityrestrictionsIneachofthethreeassignmentstepsdescribedabovethecapacityrestraintsarehandled.Andforeachpassengerclassloadedintheiterationthefollowingprocedurearefollowed:

Foreachdepartureintheroutenetwork:o Ifthepassengerclassisfreightwhichcannotbetransportedonpassenger

seatsandtheplanetypedoesnothaveacargoroom,thedepartureisclosedfortravellers

o Ifthepassengerclassonlycantravelonpassengerseatsandtheplanetypeonlyhasacargoroomthedepartureisclosedfortravellers

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o Otherwisethedepartureisexaminedforexceededcapacity(seebelow).Ifcapacityisexceededthedepartureisclosedformoretravellers

o Ifdepartureisnotclosedfornewpassengersorfreightitispossibleforthespecificpassengerclasstousethisdepartureinthisspecificiteration.

3.3.1 ExaminationofexceededcapacityForeachdepartureexaminedforexceededcapacitythefollowingprocedurearefollowed.Incaseofprioriterationstepsthenumberofoccupiedpassengerseatsandthetotalpre-loadedweightis:

Seatspreload,WeightpreloadThesumsofoccupiedseatsandweightinthecurrentiterationarecalculatedbyworkingthrougheachpassengerclassesloadedonthespecificdepartureinthecurrentassignmentcalculation.Inthisprocedurethethreerunningtotalsiscalculated:

PassengerSeatstotal,Weighttotal,PostSeatstotal

Thisisdoneasfollows:Foreachpassengerclassithetrafficinseatunitsisti:

Iftionlycanusepassengerseats(passengers):o PassengerSeatstotal=PassengerSeatstotal+tio Weighttotal=Weighttotal+ti*seatunitweighti

Iftionlycanusethecargoroom(cargo)o Weighttotal=Weighttotal+ti*seatunitweighti

Ifticanusebothcargoroomandseats(airmail):o Iftheplanetypehascargoroom:

Weighttotal=Weighttota+ti*seatunitweightio Iftheplanetypehasnocargoroom:

PostSeatstotal=PostSeattotal+ti Weighttotal=Weighttotal+ti*seatunitweighti

Forsomeplanetypesgroupsofseatsareremovedtomakeroomformaile.g.groupsof4,8or12seats.Ifthisisthecase,thePostSeatstotalisroundeduptothenearestmultiple.Afterwardsitistestedwhetherthecapacityrestraintsareexceededwhichisdescribedby:

PlaneMaxSeats,PlaneMaxWeight,PlaneMaxPostSeatsThethefollowingistested:If

(PassagerSeatstotal+PostSeatstotal)>(PlaneMaxSeats+Seatspreload)or

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PostSeatstotal>PlaneMaxPostSeatsor

Weighttotal>(PlaneMaxWeight-Weightpreload)Ifoneofthesecriteriaistruethecapacityisexceededandtheflightisclosedforthepre-sentiterationintheassignmentcalculation.

4. NETWORKDESIGNMODELThenetworkdesignmodeldesign/forecastthe

Airnetwork(servicenetwork),i.e.betweenwhichairportsthereisadirectleg. Schedule,inthemodelformulatedasdiscretedeparturetimesina30.min.segmen-

tation Useofairplanes(typesandnumbers)

Thisiscalculatedconditionalontheexpectedpassengerflows,whicharetheresultsoftheroutechoicemodelontheoutputofthenetworkdesign(i.e.abi-leveloptimisationprob-lem4).In the solution algorithm, the air model (the set of optimisation models) and the routechoicemodelisruniterativelyinatabu-searchalgorithm,whichturnedouttobethemostefficientsolutionalgorithm.Themodeltakesthefollowingdecisions;

Passengerschoiceofroutes,andpossibleupgradingofairplanetypeduetothis Downgradingofairplanetypeduetosmallpassengervolumes Airplaneallocationmodel Removing or adding direct legs (conditional to the economics of operation and

someminimumservicerestriction) Time-scheduledesign

4.1 Mainsolutionalgorithm

Themainsolutionapproachwasdecidedasfollows; Step1;Agrossnetworkisgenerated

30minutesdeparturesbetweenallrelevantdestinationsFeasibleairplanesforeachlegisset

Dependsprimarilyofthelengthsoftherunway Secondarilyfromthisbyhowlongitispossibletogowiththemost

farreachingairplanethatcanoperate

4Wherethesub-problemsarenon-linear,discreteandwithanon-closedformformulation,andis

solvedinternallybyiterativeprocedures.

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Step2;Calculationsaremadebasedonweeklyscheduleswith3daysadditionallybeforeandafter(somedestinationsmayhavedownto1connectioneachweek)

Each leg (flight between airports) can be operated by a list of possible air-planetypeswithdifferentsizesandoperationscosts

Basedonmodelledpassengerpotentials,alllegsaredowngradedtotheopti-maltypeofairplane

Step3;Routechoicewithnocapacityconstraints,i.e.theoptimalscheduleises-timatedinthepassengersviewpoint

Step4; Linkswithvery fewpassengers aredeleted (However, only to the extentthis does not violate minimum restraints). Since the graph was enormously, thisheuristicwasaddedinordertoreducetheoptimisationproblemtoasizethatcouldbesolvedbymorerigidmethodsinthelaterphasesoftheoveralloptimisational-gorithm.

Step5;Downgradingofairplanestofitdemandbasedonpurebusinesseconomicdecision.

Step6;Iterativenetworkimprovement6.1Closingoflegswithleastutility(givenasetofminimalcriteriaforopera-

tionsandwhethertheleghasnotbeeninvestigatedbefore)6.2Anewassignmentismade.After5iterations,airplanesmaybeupgraded

during assignment. The network is redesigned then due to possible in-creasedspeed

6.3 Airplane disposition / scheduling (estimation of turn-arround costs andbalancingoflegsandairplanesonairports)usingaheuristicmethod.

6.4Newstochasticcapacitydependentroutechoice(legscanbeclosed,up-gradedandusingbiggerandfasterplains),whichinfluenceroutechoices.

6.5Theoverallutilityfunctioniscalculated.Ifthisisimproved,thechangestakesplace,otherwise,theyareregrettedandmarkedastabu

6.6Stopcriterion Step4-6isrepeatedninetimeswithgraduallyincreasedrestrictions. Step7;detailedoptimizationoftheairplaneschedulingusingalinearprogramfor-

mulatedintheMOSELsoftware. Step7aseparatemodulecalculatestheoperationcostincludingturnaroundcost. Step8;Finalroutechoicemodel

Basically,thisheuristicfirstuseaverysimplepassenger-basedapproachtoreducethepos-sible solutionspace (step4). Inpractice thisapproach reduces theproblemsizewith re-specttonumberoflegsfrom58,000to5,000.

Thenaheuristicisusedforthecalculationoftheoptimalairplanedispositionandschedul-ing.(Step6.3).Themethodforthisproblem,whichconsistoftwocoreelements,calcula-tionofturn-aroundtimeasalowerlevelproblemandbalancingofairplanesastheupperlevel problem, where the cost of balancing is given by the lower level calculation. This

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heuristicprovidesamuchmorepreciseestimateoftheobjectfunctionvalue,thanjustthepassengerflows.However,itwasfirstfoundfeasiblefornetworksizesbelow5000legs.

Finallyamorerigidoptimisationwasattemptedfortheairplanedispositionasthiswasfor-mulatedasamathematicalproblemandsolvedbyMOSEL.This,however,wasonlyabletosolveproblemsupto600legs,andwastooslowtoberuniterativelywiththetime-tableandlegoptimisationmodels.Itwasalsonecessarywithsimplificationstosolveitasapuremathematical.Inthefinalmodel,itwasthereforedecidednottousestep7,andtousetheheuristic(refertosection4.2)forthelaststepaswell.

All-in-all using the above combination of heuristics made it possible to optimise the airnetwork inGreenland conditional to thepassengers choiceof route anddeparture time.Theoverallcalculationtimeisabout84hours,whichmaybeconsideredhigh,butwhichhoweverindeedwasabletopin-pointpossibilitiesofrestructuringtheairtransportnetworkwithconsiderablybenefitsforthesociety.

4.2 LegbalancingThegeneralprincipleintheleg-balancingmodelistoensurebalanceforeachairportoneby one. By starting with the airport with the smallest number of legs, the balancingproblem with the fewest degrees of freedom is solved first. In this way, the algorithmalways finish in the largest hups (Nuuk, Sdr. Strmfjord), where it should be easier toensureabalance.Evenif thereisa lackofbalancehere(e.g.from45to44legs) it is inrelativetermslessproblematicthanifasmallairportareassigned1fromlegand2to-legs.The balance is ensured by successively removing legs if balance is not meet initially.Subsequentlybalanceisensuredbyplanetypesbyupgradinglegstolargerplanetypes.

4.2.1 SolutionalgorithmSumBalOK:=0;SumBalEjOK:=0(Thetwobalancingvariablesareinitialised)Thesimplifiedturnaroundcostmodelisruninordertogenerateaninitialturnaroundcost(refertosection3).Airports are sorted according to the total number of trips in and out of the airport.LufthavneneAsorteresefterhvormangelegsdersamletgrtilogfralufthavnenForeachairportAinthesortedsequence(thesmallestairportfirst),legsarebalancedasfollows5{

Legs,which isequal to theminimumservicecriteria ismarkedastabu.ThenumberoflegsLf(legs,whichdepartfromtheairport)andnumberoflegsLt(legs,whicharrivetotheairport)arecounted.IfLf>Lt{

Ifatleastonefrom-legLf,whichisnotmarkedastabu{The from-leg Lf, which is not tabu and have the lowest

5Thetrickisthatthelargertheairportthemorechanceofbeingabletobalance.Ifthereisbalancein

thesmallairports,theriskoflargerairportsnotbeingbalancedissmaller.

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object value (e.g., revenue operation costs turn-aroundcosts)6areremoved.BalA:=OK(theairportisbalanced)SumBalOK:=SumBalOK+1(thestatusvariablesisupdated)

}Or(theairportcouldnotbebalanced)7BalA:=NOTOK(Theairportcouldnotbebalanced)SumBalEjOK:= SumBalEjOK + 1 (the status variables isupdated)

}IfLt>Lf thesameprocedureasforfrom-legsarecarriedout,butwithto-legsremoved.Itshouldbenoted,thateventhoughbalanceisnotensuredfortheairportasawhole,thebalancingwillcontinueinthatitmightbepossibletoensureabetterbalanceonplanetypes.If there are at least two different plane types in an airport, legs for eachplanetypesareinvestigated.Thelargestplanetypeisinvestigatedfirstandtheprocedurecontinuesuntilthesecondsmallestairplanetype.8{

ThenumberoflegsfromL(X)f(legs,whichdepartfromtheairportofplanetypeX)andthenumberoflegsL(X)t(legs,whicharrivetotheairport)arecounted.IfL(X)f>L(X)t{

All to-legsofplane typeL(X-1)t,andwhicharrivefromanairportthatallowsplanetypeXexists9

Ifthereisatleastonleginthisset10 Cturnarround:=null,Lopt:=null

Foreachoftheselegs{The turnarroundcost is calculated for the set{all L(X) + the actual L(X-1)} by sameprinciple as for the simplified turnaroundmodel11.

6Thisisthereasonwhythesimpleturn-aroundmodelneedstoberunbeforethesolutionalgorithm.

Ifnot,therewouldnotbeanestimatefortheturn-aroundcosts.7Thepurposeofthisupdatingismainlyinternalvalidation.8Thesmallestairplaneisaresidualoftheotherupgrades.9Allto-legs,whichcanbeupgradedisdetected,however,onlywithinthepresentplanetype.10

IftherearenotrelevantlegsoftypeX-1,thealgorithmproceedtocheckX-2and,etc.11

Itwillgiveabiasedestimateoftheturn-aroundcostifonlythecalculationiscarriedoutforthe

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If Cturnarround:=null or the calculatedturnarround cost for L(X-1); C(L(X-1)) L(X)f thesameprocedureiscarriedout,butwherefromlegsisupgraded.

}}Thesimplifiedturnarroundcostmodelisrun13Thefollowinginformationisloadedtotheplaneoptimizationmodel{ Deletedlegs Upgradedlegs Newturnaroundcosts}

4.2.2 DiscussionThe general plane optimisation model delete links one by one (or several links in oneprocess, but without considering balancing). Initially, up to 50% of all airports areunbalanced.However,becausemanyairportsmeet theminimumserviceconstraints, it isexpectedtobeless.Afterrunningtheheuristicfortheleg-balancing,mostifnotallairportswillbebalanced.Onlyinveryspecialcasesbalancingmaynotbemet.After the balancing on legs the balancing on plane types are carried out. It must beassumedthat,inmostcases,itispossibletoupgradeplanetypes.All-in-all the heuristic will ensure a much better balancing of the final solution. This

actualL(X-1).However,thereisalimitedlistofalternatives,whichischeckedaccordingtotheFIFOprinciple(purelinearoperation).Asaresult,thecompleteisinvestigated.12

TheactuallegisthebestcandidatesofarforupgradingofplanetypefromX-1toX.13

Theleg-structureischanges,whichiswhyweneedtocalculatenewturnaroundcosts.

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ensuresthattheinputtotheoptimisationmodelisbetter.However,itshouldbeunderlinedthateventhoughbalanceisensured,thefinalplaneoptimizationmodelmaystillupgradelegs inorder to reduce turn-aroundcosts.Also, itmay in rareoccasionsbebeneficial tointroduceemptylegs.

5. MODEL APPLICATION: LOCATION CHOICE AND SIZE OF ATLANTICAIRPORT

During2007,TheGreenlandHomeRulewill take themost important transportdecisionever,namely,thelocationandthesizeofanewAtlanticairport.Toillustratetheimpor-tanceof thedecision, theconstructioncostalonemayequate10-20%oftheannualGNPdependingon the specific location.Tohelp thisdecision, themodelhasbeenapplied tothreedifferentscenarios,whichvarieswiththelengthof therun-wayinNuukaswellasthe location in the Nuuk area14. The alternatives are compared in a social cost-benefitanalysisandbenchmarkedtoabasesituation,inwhichthelocationoftheAtlanticairportremainsinSdr.Strmfjord.Thethreescenariosareoutlinedbelow;

Nuuk1799meter:Therunway inNuuk isextended to1799meterenablingforaBoeing-757toland.

Nuuk2200meter:TherunwayinNuukisextendedto2200meterenablingforAir-bus-200toland.

Nuuk3000meter:TherunwayinNuukisextendedto3000meterenablingforAir-bus-200toland.

ApartfromthedifferenceinthelengthoftherunwayinNuukthereisconsiderablediffer-ence inconstructionandmaintenance costsandalso in thedemandestimates, especiallyfortourists.

Theservicenetworkandthetrafficflowaremodeledinthereferenceyearwithandwith-outchangestothenetwork.Afterwards,theservicenetworkandthetrafficflowaremod-eled with the network changes and a new OD matrix is calculated on the basis of thechangeinthelevelofservice.

Thetrafficflowof todayhasthreemainroutes;fromCopenhagentoSdr.Strmfjordandfrom Sdr.Strmfjord to either Nuuk or Illulissat. With the extensions of the runway inNuuk,thesemainrouteschangeinthatpassengerwilltraveldirectlybetweenCopenhagentoNuukandfromNuuktoeitherIllulissatorNarsarsuaq.Thechangestothenetworkre-sultsindirectroutesfromDenmarktoNuukanddirectroutefromIsland.Ingeneral,the

14Thepresentrun-waycanonlybeextendedto2200meterandtwoalternativelocationshavebeen

selectedforthe3000meteralternative,whichatpresentisassumedtohavethesameconstructioncosts.

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numberof routes is reducedconsiderablywhenSdr.Strmfjord isclosedbecause agreatdealofthefeeder-trafficistakencareofbydirectroutes.Figure1:IllustrationofhowthenetworkstructurechangesasaresultofanewAtlanticairportinNuuk.

Trafficflowinreferenceyear TrafficflowinscenarioNuuk2200

Theresultsfromthethreescenariospointsinthesamedirection.TheservicenetworkhasalmostthesameroutesbuttheplanetypesdifferbetweenCopenhagenandNuuk(refertofigure4below).Thismakesaslightlydifferenceinlevelofservicewhichgiverisetodif-ferentdemandformations(ODmatrices).Also,sincetheuseofplanetypesdifferbetweenthescenarios,theproductioncostsdifferaswell.

Figure2:B/CratiosfortheNuukscenarios.

B/CratiofortheNuukscenarios

0 1 2 3 4 5 6

Nuuk1799

Nuuk2200

Nuuk3000

B/Cratio

In figure 2, the B/C ratios are shown foreachscenario.TheNuuk1799scenariohasturnedout tobe themost cost-effectiveofthethreeprojects.Themainreasonsforthedifferences in the projects are the initialcosts, production cost, and ticket revenueforoperators.Thisdifferenceresolvesinanimportant difference in the calculated in-ternalratewhichis26.3%forNuuk1799,20.9%forNuuk2200andonly8.9%forNuuk3000.

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Onthefiguresbelowthedifferentbenefitandcostcomponentsisshown.Nuuk2200andNuuk3000haveequalbenefitsbecausethemodelsettingsallowsthesametypeofplanestoland,however,thereislargedifferencewhenlookingatcostcomponents.

Benefit

0 2000 4000 6000

Nuuk1799

Nuuk2200

Nuuk3000

Benefitmio.DKKChangeintotaloperat ioncost Changeintotalproduct ioncost

Ticket revenueoperator LeavingKangerlussuaq

Timesavings Ticket revenuepassenger

Other

Cost

0 1000 2000 3000 4000 5000 6000

Nuuk1799

Nuuk2200

Nuuk3000

Costmio.DKKInit ialcost

Changeintotaloperationrevenue

Regulat ionsloss

Other

Figure3:BenefitandcostcomponentsforthethreeNuukalternatives.

The model more than indicate that significant socio-economic benefits can be expected,when themainairportaremovedfromSdr.Strmfjord toNuuk.However,when judgingbetweenalarge(Nuuk2200orNuuk3000)andamedium(Nuuk1799)sizesolutioninNuuk, themediumsizesolution turnsout tobe theonewith thehighest internal rent,duetolowercosts.Clearly,theresultisbasedonthedemandestimatedfor2012.Ifasig-nificantincreaseinthenumberofpassengersbeyondthe2012levelisexpected,thecapac-itylimitsmaybemet,andthe2200maybepreferable.However,thesenumbersareverydifficulttoforecast,whichiswhytheGreenlandhome-rulehasdecidedtoforcastonlyto2012.

Theapplicationshowsthatithasbeenpossibletoimplementamodelsystem.Allthough,themodelsisbasedonnumerousassumptions,itprovideanimportantdecisiontoolanditgivesaclearindicationthataswitchoflocationfortheAtlanticairportisbeneficial.

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Theuseofdifferentplanetypesinreferenceyear

TheuseofdifferentplanetypesinscenarioNuuk2200

Figure4:IllustrationofhowthetypeofplaneschangesasaresultofanewAtlanticair-portinNuuk.

Thepracticalimplementationandthecalibrationofthemodelturnedouttobeverycom-plex.However,inthefinalversion,themodelwasabletoreplicatethenetworkintheref-erenceyearsufficientlywell.Theouterloopbetweenthedifferentmodelcomponentscon-verged,andthefinalsolutionwasnotonlyabletobesimilarlyasgoodastheexistingsys-tem,butalsosuggestimprovements.

6. CONCLUSIONInthepaper,acombinedtransportandoptimisationmodelforairtransportinGreenlandhasbeenpresented.Thestructureofthemodeliscomposedasabi-leveloptimisationproblem.Atthelowerleveloptimisation,demandandroutechoiceforpassengersandfreight isad-dressed,whereasat theupper leveloptimisation, the servicenetwork, theconfigurationofplanetypesandtheflightschedulingisdealtwith.

Thecentreoftheupperleveloptimisationisanobjectivefunction,consistingofpassengerutility,operationcosts,andrevenuesfromsaleoftickets.Thefunctionmaybeformulatedasa function that represent a socio-economic optimum or an operator-optimum (air compa-nies),whichwillrepresentpartlyconflictingpurposes.Thepresentversionofthemodelonlyoptimisestheobjectivesoftheoperator,whichreflectsthepresentdecisioncontext.

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The model is solved conditional on a number of constraints, e.g. capacity constraints forplanetypes(weightandseatconstraints),constraintsforvariouslegs(minimumservicelevelrequirements),mileageconstraintsforplanetypes,constraintsconcerningminimumrunwaylengths for certain airplane types, and finally certain airports (landing grounds) that canonlybeservedwithhelicopters.

Themodelsolveacomplicatedproblem,with28airports,upto1000legsinthefinalflightschedulesandallocationofmanytypesofairplanestothenetwork.Thetestingshowsthatthemodelindeedisabletosolvetheproblemandthatthesolutionisbetterthanthepresentsystem.Moreoverthemodelisabletosuggestservicenetworksinfuturescenarioswheretheconstraintsarechanged.Typicallythisischangesofairportlocationsandlengthsoftherunway.ThemainscenariotobuildanAtlanticAirportintheCapitalNuukturnedouttohaveverypositivesocio-economiccharacteristics.

1. Introduction1.1 Background1.2 Overview of the model system1.3 Decision context1.4 Object functions in the optimisation model1.5 Passenger behaviour prediction

2. Main structure of the model2.1 Main flow

3. Route choice model3.1 Utility functions3.2 Chosen parameters3.3 Capacity restrictions3.3.1 Examination of exceeded capacity

4. Network design model4.1 Main solution algorithm4.2 Leg balancing4.2.1 Solution algorithm4.2.2 Discussion

5. Model application: Location choice and size of atlantic airport6. Conclusion