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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:[email protected];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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15
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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17
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