an exact model for airline ﬂight network opmizaon based on

TRANSPORTES | ISSN: 2237-1346 14

An exact model for airline flight network op"miza"on

based on transport momentum and aircra' load

factor Daniel Jorge Caetano1, Nicolau Dionísio Fares Gualda2

1Escola Politécnica da Universidade de São Paulo, [email protected] 2Escola Politécnica da Universidade de São Paulo, [email protected]

Recebido:

23 de maio de 2017

Aceito para publicação:

03 de setembro de 2017

Publicado:

30 de dezembro de 2017

Editor de área:

Li Weigang

ABSTRACT

The problem of airline flight network op"miza"on can be split into subproblems such as

Schedule Genera"on (SG) and Fleet Assignment (FA), solved in consecu"ve steps or in

an integrated way, usually based on monetary costs and revenue forecasts. A linear pro-

gramming model to solve SG and FA in an integrated way is presented, but with an al-

terna"ve approach based on transport momentum and aircra' load factor. This alterna-

"ve approach relies on demand forecast and allows obtaining solu"ons considering min-

imum average load factors. Results of the proposed model applica"ons to instances of

a regional Brazilian airline are presented. The comparison of the schedules generated

by the proposed approach against those obtained by applying a model based on mone-

tary costs and revenue forecasts demonstrates the validity of this alterna"ve approach

for airlines network planning.

RESUMO

O problema da o"mização da malha de uma empresa aérea pode ser dividido em

subproblemas como a Programação de Voos (PV) e a Alocação de Frotas (AF), resolvidos

em etapas ou de maneira integrada, normalmente com base em previsões de custos e

receitas. Um modelo de programação linear é apresentado para resolver a PV e a AF de

maneira integrada, porém adotando uma abordagem alterna"va, baseada no momento

de transporte e na taxa de ocupação das aeronaves. Tal abordagem depende apenas de

previsões de demanda e permite considerar taxas mínimas de ocupação das aeronaves.

São apresentados resultados da aplicação do modelo a instâncias associadas a uma

empresa áerea regional brasileira. A comparação das programações de voo ob"das pela

abordagem proposta em relação às ob"das por um modelo baseado em custos e

receitas demonstra a validade dessa abordagem alterna"va para o planejamento das

malhas das empresas aéreas.

Keywords:

Air transporta"on,

Schedule genera"on,

Fleet assignment,

Linear programming.

Palavras-chaves:

Transporte aéreo,

Programação de voos,

Alocação de frotas,

Programação linear.

DOI:10.14295/transportes.v25i4.1383

1. INTRODUCTION

Themaingoalofairlinesstrategicplanningistheincreaseinef�iciencyandpro�itability.Determiningtheoptimal�lightnetworkisanimportantcomponentofthisstrategicplan,whichcomprehendswhatmarketsandhowmany�lightstoserveandwhichresources–aircraftsandcrew–toallocatetoeach�light(GU! RKANetal.,2016;SHERALIetal.,2013).

Asinglemodeltodeterminetheoptimal�lightnetwork,albeitdesirable,generallyleadstolarge-scaleproblemsoftheNP-Hardclass(HANEetal.1995;KLABJAN,2004).ItisacommonpracticetodividetheproblemintosmallerproblemssuchasScheduleGeneration,FleetAssignmentandCrewAssignment,solvedinconsecutivesteps,seekingtocomputeasolutionatfeasibletimes(RABETANETY,2006).Thispractice,however,doesnotguaranteetheoptimalglobalsolution.

Nomatterhowmanystepsareused tosolve theproblem, theusualapproachreliesonobjective

Caetano, D. J.; Gualda, N. D. F. Volume 25 | Número 4 | 2017


functions based on monetary costs and revenue forecasts, which, in turn, are based on passengerforecastsandcanthussuffersigni�icant�luctuationswhenchangesaremadetothe�lightnetwork.

Theobjectiveof thispaper is topresent an alternativeobjective function that allows solving theintegrated Schedule Generation and Fleet Assignment problem relying only on demand forecasts,avoidingtheneedofrevenuesand/orcostsforecasts.

Theobjectivefunctionproposedencompassesvariablesbasedontransportmomentumasaproxyofoperationalcostsandrevenues.Moreover,themodelproposedallowsimposingaminimumoverallloadfactorforsolvingtheproblem.

Thepapersequencebeginswithabriefreviewoftheairlineoperationalplanningconcepts,includingdetailsontheapproachwiththeobjectivefunctionrelatedtocostsand/orrevenues.Thisisfollowedbyapresentationofthealternativeobjectivefunctionproposed.Finally,theresultsofthemodelapplicationto instancesofaregionalBrazilianairlinearepresentedandcompared toresults fromtherevenue-relatedmodel,alongwithabriefanalysisconcerningtheimpositionofaminimumoverallloadfactor.

2. AIRLINE OPERATIONAL PLANNING

Airlinesoperationalplanningencompassesthede�initionofthe�lightstobeoffered,oftheaircrafttobeused for each �light, and of the crew to perform each of these �lights. These decisions are usuallyassociated to results of three interrelated processes that, in turn, may be divided into smallersubproblems,usuallysolvedsequentialy,asshowninFigure1.

Figure 1: Airline operational planning stages (Based on: CAETANO & GUALDA, 2010)

Flightde�initionisthe�irstprocess,whichde�inesthe�lightschedule.Thisprocesscanbedividedinto two subproblems: Route Development and Schedule Generation. In Route Development, thedemandsbetweencitypairsareidenti�iedandthuspotential�lightsbetweenairportpairsarede�ined.Itiscrucialtoaddressairportsrestrictionsatthisstep–suchasoperatingtimerestrictions–,topreventtheinclusionofoperationallyimpossible�lightsinthesolution.InScheduleGeneration,the�lightsthatwillactualybeon�lightscheduleareselected.(KLABJAN,2004;RABETANETY,2006).

AircraftAssignment,thesecondprocess,definesthesequenceofscheduledflightseachaircraftwillperform.Thisprocesscanalsobedividedintotwosubproblems:FleetAssignmentandMaintenanceRouting.InFleetAssignment,thebestaircrafttypeisassignedtoeachscheduledflight,aimingatprofitmaximization–usuallyrelatedtoaircraftcapacityanddemandforecasts.InMaintenanceRouting,theoperationalrestrictionsofeachaircraft–suchasmaintenanceschedule–areconsideredtodefineitsexactflightsequence.Thescheduleofeachaircraftshalldefinewhereandwheneachonewillbeoutofduty,inordertobeserviced(BARNHARTetal.2003;KLABJAN,2004,SHERALIetal.,2013).

CrewAssignment,thethirdprocess,de�inesthecrewmembersthatwillbeassignedtoeach�light.This process can be divided into two subproblems also: Crew Pairing and CrewRostering. In CrewPairing,�lightsaregroupedintosequencescalledpairings,whichmustrespectlaborlawsandtechnicalcriteria. InCrewRostering, thepairings–and, thus, the �lights–areassigned to thecrewmembers(BARNHARTetal.,2003;GOMES,2014;GOMES&GUALDA,2011;GOMES&GUALDA,2015;KLABJAN,2004;RABETANETYetal.,2006).

Flight Definition Aircraft Assignment Crew Assignment

Route

Development

Schedule

Generation

Fleet

Assignment

Maintenance

Routing

Crew

Pairing

Crew

Rostering



Thecompetitionamongairlinesandthecomplexityofaircraft,crewandpassengermanagementleadto large-scale models for airlines operational planning. Even when each planning subproblem isaddressedbyaspeci�icmodel,itisusuallyalsooftheNP-hardclass(HANEetal.,1995).Seekingglobaloptimization,modelsareaddressedtosolvetwoormoresubproblemssimultaneously,increasingthecomputational complexity (KLABJAN, 2004; SHERALI et al. 2013; RABETANETY et al., 2006).Consequently, heuristics and metaheuristics approaches have been proposed, such as the one byCaetanoandGualda (2011), addressed to solve theScheduleGenerationandFleetAssignment inte-gratedproblemwithanAntColonyMetaheuristicapproach,andtheonesbyGomesandGualda(2011;2015),forsolvingtheintegratedAirlineCrewAssignmentproblem.

Themostcommonapproachtosolvetheairlineplanningprocessistodivideitintoseveralsteps,asmentionedearlier.Eachstepisoptimizedbasedondifferentobjectivefunctions:theFlightDe�initionstepisusuallysolvedusingmarketing,revenue,aircraftavailabilityandotheroperationalrestrictionsas optimization criteria (KLABJAN, 2004; RABETANETY et al., 2006); the Aircraft Assignment stepusuallyinvolvesmainlytheconsiderationofcostsandrevenuestomaximizepro�itability(HANEetal.,2004;KLABJAN,2004;SHERALIetal.,2006);and, �inally, theCrewAssignmentstep isbasedon theavailabilityofhumanresources,legalmattersandcosts(BARNHARTetal.,2003;GOMES,2014;GOMES&GUALDA,2011;GOMES&GUALDA,2015;KLABJAN,2004).

Noticethatcostsandrevenuesaredirectlyorindirectlyrelatedtoeachstepoftheairlineplanningprocess.This is the reasonwhymodelsproposed to solvemore thanonestep in an integratedwayusually rely on costs and/or revenues as the main parameters of their objective functions. Othercharacteristics are usually addressed by model constraints (LOHATEPANONT & BARNHART, 2004;SALAZAR-GONZAF LEZ,2014).

Amongalltheairlineplanningstepsdescribedpreviously,ScheduleGenerationandFleetAssignmentproblemsareconsideredthemostimportantonesrelatedtotheairlinepro�itabilityandserviceleveland,therefore,goodcandidatestobesolvedinanintegratedway(DONGetal.,2016).Theseproblemshave been solved in an integrated way by Caetano & Gualda (2010; 2011) using a model thatencompasses some activities that can be considered part of Route Development – consideration ofalternative�lightstobeoffered–andsomethatcanbeconsideredpartofMaintenanceRouting–suchastheregularmaintenancetimeaftereach�light.ItisrelevanttoremarkthatthismodeldoesnotincludealltheaspectscoveredbyRouteDevelopmentandMaintenanceRoutingsubproblems.

ThemodelproposedbyCaetano&Gualda(2011)isbasedonpreviousmodelsbyBerge&Hopperstad(1993), Sherali et al. (2006) and Lohatepanont & Barnhart (2004). It is structured as a space-timenetwork,includingtheelementsshowninFigure2.Sincethearrivalslottimeconstraintsarebasedonthe�lightarrivaltime,thespace-timenetworkincludesexplicitarcsformaintenanceafter�light–duringwhichtheaircraftisunavailable–sothatthe�lightarcsendsatthecorrecttime,eveniftheaircraftisunavailableforalongerperiodoftime.

Figure 2: Space-time network (Based on CAETANO & GUALDA, 2011)



Notethatifone�leetcannotexecuteadirect�lightbetweentwoairports–becauseofitsrangelimi-tation–, thenetwork forthat �leetshallnot include thearcsrepresenting that �light–different �leetnetworksmayincludedifferentsetsofarcs.Thesamerationaleapplieswhenoneaircraftcannotoperateatanairport:no�lightarcshouldconnectthat�leettotherestrictedairport.

Themodelbasedonsuchspace-timenetworkispresentedbelow(Caetano&Gualda,2011):

[ ]( )

( ),

. . . .∈ ∈

− + −

∑ ∑ f f

ij ij ij ij ij ij ij

i j Lf f F

min R C x R pa R d pa (1)

Subjectto:

∑ ��

�∈� ≤ 1 ∀� , �� ∈ ��, ∀� ∈ � (2)

∑ ��

�|��,��∈� − ∑ ��

�|��,��∈� = 0 ∀� ∈ �� , ∀� ∈ (3)

∑ ��

��,��∈�! ≤ "� ∀� ∈ (4)

∑ ∑ ��

�|��,��∈�#≤ 1 ∀ ∈ �$%�∈� (5)

∑ ∑ ��

�|��,��∈�#≤ 1 ∀� ∈ �$&�∈� (6)

∑ '� . ��

�∈� − )&�� ≥ 0 ∀� , �� ∈ �� (7)

%�� − )&�� ≥ 0 ∀� , �� ∈ �� (8) ∑ %��,��∈��+

− ,� = 0 ∀� ∈ � (9) Binaries:

��

∈ -0,1. ∀� , �� ∈ ��, ∀� ∈ � (10) Integers:

��

≥ 0 ∀� , �� ∈ �\��, ∀� ∈ � (11)

%�� ≥ 0 ∀� , �� ∈ �� (12)

)&�� ≥ 0 ∀� , �� ∈ �� (13) Where: M: set of allmarkets, indexed bym; eachmarketde�ines a demand and a timewindow

thatlimitswhich�lightscanservethisdemand.

Nf:setofallnodesforaircraftf,indexedbyi,j,o,dork,representinganairportataspeci�ictime.

Nrd:setofnodeswithdeparturerestrictions.

Nra:setofnodeswithlandingrestrictions.

F:setofalltypesofaircraft,indexedbyf.

L:setofarcsthatrepresentthemovementofaircraft,minimummaintenanceafter�light–turn-aroundtime–,waitingonthegroundorwrap,indexedby(i,j),iisthesourcenodeandjisthedestinationnodeofthemovement.

Lf:setofarcsthatrepresent�lightmovements.

Lfm:setofarcsrepresenting�lightsassignedtoamarketm.

Lt:setofarcswhoseorigintimeisequaltoorlessthantanddestinationtimeisaftert.Timetissettoavalidtimeaccordingtotheproblem.

Dm:unrestrictedpassengerdemandformarketm.

Cf:numberofseatsoftypefaircraft.

Rij:unitaryrevenueforapassengeronthe�lightfromnodeitonodej.Since(i,j)representaspeci�ic�light–includingdayandtime–each�lightmaybeassociatedwithaunitaryreve-nue.

Af:numberofaircraftoftypefavailable.

xfij:numberofaircraftoftypef�lowingthrougharc(i,j).



dij:numberofpotentialpassengers(demand)associatedtothe�lightfromnodeitonodej.

paij:numberofpassengersassociatedtothe�lightfromnodeitonodej.

Theobjective function–Equation1– seeks tominimize thesumof lost revenues.The �irst termrepresentsthedifferencebetweenmaximumrevenuefortheassignedaircraftandtherevenuereceivedfromassignedpassengers.Thesecondtermisassociatedtothelostrevenueduetolostdemand.

Equations2to4representtheusualcover,balanceandnumberofaircraftrestrictions(BERGE&HOPERSTEAD,1993;SHERALIetal.,2006;HANEetal.,1995).

Equations5and6representslotconstraints,assuringthatonlyoneaircraftwilldepartorlandonthosenodes,respectively.Equations7to9assurethateachmarketdemandwillbeassociatedtoeach�lightandthatthepassengersofa�lightwillneverbegreaterthantheassociatedaircraftcapacity.

Thevariablesrepresentingdemanded�lightarcsarebinary,andarespeci�iedinequation10.Alltheotherarcvariablesareintegersgreaterthanorequaltozero,asstatedinEquations11,12and13.

However,theuseofrevenuesasinputdataonplanningmodelspresentssomepracticalproblems.Firstofall,revenuesarenotonlybasedondemand–whichisestimated–butalsoonticketprice,whichcan�luctuateaccordingtotheyieldmanagementstrategy(MAYO,1999).Theformer,however,maynotexistorevennotre�lecttheoptimumforaspeci�icsolution–thecurrentairline�lights–and,therefore,itmayintroduceundesiredbiasinthesolutionprocess.

Modifying any model to rely on costs instead of revenues brings another set of problems todiscussion.Ifonlyoperationalcostsaretakenintoaccount,the“optimal”solutionmayleadtoalargepassengerspill–passengersnotserved.Thesolutionwouldbetoforceallthedemandtobeserved–whichisnotalwaysapracticalapproach–ortoadda“cost”foreachpassengernotserved.Thede�initionofthelattercostisaprobleminitself(KLABJAN,2004;SHERALIetal.,2006).

Theseproblemsarepresentnotonlyinthemodeldescribedbyequations1to13.Mostrecentmodelspresented in the literature to solve this integrated problem – such as the ones by Lohatepanon &Barnhart(2004),Sheraliet.al.(2013),DiWangetal.(2014)andSalazar-Gonzalez(2014),andDongetal.(2016)–relyonestimatedfares,revenuesand/orcostsand,therefore,mayincludesomekindofbias.

Also,whenanalyzingthewholeplanningprocess,thecostofa�lightisnotonlydependentonwhichaircraftisusedforeach�lightleg,butalsoontheorderinwhichtheselegsarecovered,onthechoiceofthecrewmembersineach�lightleg,andsoon(CAETANO&GUALDA,2011;GOMES,2014;GOMES&GUALDA, 2011; SHERALI et al., 2006; KLABJAN, 2004). In otherwords, since the �light cost is alsodependentonthemodeloutput,itisnotdesirabletouseitasamodelinput.

3. ALTERNATIVE OBJECTIVE FUNCTION

Themostcommonobjectiveofairlinesplanningis tomaximizepro�it.Therefore,objectivefunctionsignoringcostsandrevenuesarenotrealisticinmostcases.However,parametersthatareeasiertode-terminewithintheplanningtimemayactascostorrevenueproxiesandpreventtheintroductionofbiasesintheprocess.

Afavorableproxyfortheoperationalcostisthepotentialtransportmomentum(seats.milesorseats.kilometers),orPTM,oncethereisagoodcorrelationbetweenthem(SWAN&ADLER,2006).Sincethedemandforecastisarequiredinputparameterfortheairlineplanningprocess,andsincethe�lightdis-tancecanbedetermined,PTMisreadilyavailableattheplanningtime.Since�lightdistancesand�lighttimesarealsocorrelated,analternativecouldbetodeterminePTMintermsofseats.hours.

Forsimilarreasons,consideringauniformyieldmanagementstrategy,theeffectivetransportmo-mentum(passengers.miles,passengers.kilometersorpassengers.hours),orETM,maybeafavorableproxyforoperationalrevenues.

However,givenPTMandETMasproxiesofcostsandrevenues,thepro�itabilitycannotbeobtainedbysimplysubtractingPTMfromETM:theresultwouldalwaysbenegative.Hence,additionalconsider-ationsareneededtoestimatewhethera�lightwouldbepro�itableornot.



Analyzingthesituationfromanotherperspective,itispossibletoconsideremptyseatsona�lightaswastedpotentialtransportmomentum,orWPTM.Theseseatscouldhavegeneratedsomerevenue,buttheyhavenot.Then,WPTMcorrespondstothedifferencebetweenPTMandETM.

Inamaximumcoveragemodel–all�lightsmustbeassigned–,theobjectivefunctioncanbesimplystatedtominimizeWPTM.However,ifthemodelde�ineswhich�lightsshouldbeperformed–inasched-ulegenerationprocess,forinstance–minimizationofWPTMwillalwaysleadtoanemptyschedule–thereisnopenaltyforunmetdemandandnoemptyseatsareaccountedwhenthereareno�lights.Thesolutionforthislimitationcanbeachievedbyconsideringthespilleddemandaslostrevenue,and,like-wise,thewastedeffectivetransportmomentum,orWETM.

The�lightwastedtransportmomentum,WTM,canbeobtainedbyaddingWPTMtoWETM,andtheobjectivefunctioncouldbesimplytominimizethesumoftheWTMforeach�lightleg,aspresentedinEquation14.

[ ]( )

( ), ∈

− + ∑ ij ij ij

i j Lf

Min PTM ETM WETM (14)

MinimizingthesumofWPTMandWETMenforcestheneedto�indabalancebetweenemptyseatsandunmetdemand.Theformulation,however,impliesthatnotcarryingapassengerhasthesamecostasanemptyseatonaspeci�ic�light.Inotherwords,thereisanimplicitassumptionthattheoperationalbreak-evenwillbemetwitha50%loadfactor,orLF.Thisoccupationratioisnotrealisticinmostcases,andprobablymostairlineswouldprefertosetaspeci�icbreak-evenloadfactor,BELF,theoccupationratiothatmakesa�lightattractive,accordingtotheirindividualexpectations.

Foragiven�lightdemand,theaircrafttypeselectedhasdirectimpactontheloadfactor.Theoppositeisalsotrue:variationsintheminimumacceptableloadfactorwillimpacttheviableaircrafttypechoicesforaspeci�ic�light.Inpractice,therelationshipbetweenloadfactorandaircraftsize,associatedtode-mandand�lightdistance,canbeusedtoforecastaircraftmovementsina�lightleg(KO! LKERetal.,2016)oranaircraftroute.Theserelationshipssuggestthat�lightdemand,�lightlengthandloadfactorshouldallbeconsideredforselectingwhich�leetwillperformeach�light.

Asmentionedearlier,anewmodelconstraintcouldbeusedtoovercometheimplicit50%BELF.Theminimumloadfactorona�light,δ,couldbelimitedbytheconstraintpresentedinEquation15.

0.

ij

f f

ijf F

pa

C xδ

∈

− ≥ ∑

(15)

However,themodelshouldallowlowerloadfactorsonselected�lightsonceproventhatthose�lightsarebeingusedtorepositionanaircrafttoperformahigh-occupancy�light(or�lights).Therefore,abetterapproachwouldbetoallowloadfactorcompensationamong�lights.The�lightlengthshouldalsobeconsidered,sincelowoccupationratioonshorter�lightsislessharmfulthanonlonger�lights.Adoptingthe�lighttimeTijasaweight,thepreviousconstraintcanberewrittentomeettheserequirementstolimittheminimumtime-weightedloadfactor,TWLF,asshowinEquation16.

( ),

0.

δ∈

∈

− ≥

∑∑f

ij

ij f fi j L ijf F

paT

C x (16)

Unfortunately,thisconstraintisnotlinearwhenthenumberofpassengersper�light,paij,isadecisionvariable.Nonetheless,itispossibletoachieveBELFcontrolbyusingcostweightsintheobjectivefunc-tionwhilekeepingthelinearnatureofthemodel:ifthecostofanemptyseatisthesameasthecostofalostpassenger,theBELFwillbe50%.Ontheotherhand,ifanemptyseatcostsmorethanalostpas-senger,toperforma�lightat50%occupationwouldbemoreexpensivethannotperformingthat�lightatall,andBELFwillbehigherthan50%.

TheobjectivefunctionpresentedinEquation17allowsthiskindofcontrolthroughweightsαandβ.



[ ]( )

( ),

. .α β∈

− + ∑ ij ij ij

i j Lf

Min PTM ETM WETM (17)

ToachieveaBELFofδ,theα/βratioshouldbecalculatedaspresentedinEquation18.

1

α δβ δ

=−

(18)

Asanexample,ifα=3andβ=2,δis60%.Fixingβas1wouldsimplifythemodel,butkeepingthembothallowstheuseofintegralvaluesinmostcases,whichmaybedesirabletoavoidroundingerrors.

However,notethatBELFdoesnotrepresentthe“desiredoccupationratio”,butthereferenceaverageoccupationratiotobeacceptedbythemodelasaviablesolution.Asadirectconsequence,ifanairlinesetsthisvaluetoohigh–suchas85%,whichisacommonoccupationratiotarget–,severalpro�itablelong�lightsmaybeeliminatedfromthescheduleduetotheneedofsomelowoccupation,shortreposi-tioning�lights.

Thealternativemodelformulation,incorporatingtheproposedobjectivefunction,becomes:

[ ]( )

( ),

. . . . ∈ ∈

− + − ∑ ∑ f f

ij ij ij ij ij

i j Lf f F

min T α C x pa β d pa (19)

Subjectto:

( )1 , , ∈

≤ ∀ ∈ ∀ ∈∑ f

ij m

f F

x i j Lf m M (20)

( ) ( )| , | ,

0 , ∈ ∈

− = ∀ ∈ ∀ ∈∑ ∑f f

ok kd f

o o k L d k d L

x x k N f F (21)

( ),

f

ij f

i j Lt

x A f F∈

≤ ∀ ∈∑ (22)

( )| ,

1 ∈ ∈

≤ ∀ ∈∑ ∑f

f

ij

f F j i j L

x i Nrd (23)

( )| ,

1 ∈ ∈

≤ ∀ ∈∑ ∑f

f

ij

f F i i j L

x j Nra (24)

( ). 0 ,∈

− ≥ ∀ ∈∑ f f

ij ij

f F

C x pa i j Lf (25)

( )0 ,− ≥ ∀ ∈ij ijd pa i j Lf (26)

( ),

0m

ij m

i j Lf

d D m M∈

− = ∀ ∈∑ (27)

Binaries:

{ } ( )0,1 , , ∈ ∀ ∈ ∀ ∈f

ij mx i j Lf m M (28)

Integers:

( )0 , \ , ≥ ∀ ∈ ∀ ∈f

ij mx i j L Lf m M (29)

( )0 ,≥ ∀ ∈ijd i j Lf (30)

( )0 ,≥ ∀ ∈ijpa i j Lf (31)

ThemodeldescribedbyEquations19to31isverysimilartothatpresentedbyEquations1to13.Infact,theonlychangeisthenewobjectivefunctionshowninEquation19,whichisthefullformofEqua-tion17:PTMijisrepresentedbythe�lighttimeTijmultipliedbythecapacityoftheusedaircraftCf.xfij;ETMijisrepresentedbythe�lighttimeTijmultipliedbythepassengersallocatedtothat�lightpaij;and,�inally,WETMijisrepresentedbythe�lighttimeTijmultipliedbythespilleddemandassociatedtothat�light(dij-paij).



Sincenootherconstraintswerechanged,thismodelcansolvethesameinstancesfromthemodelpresentedbyCaetano&Gualda(2011),allowingforcomparingtheresultsprovidedbybothmodels.TheonlyadditionalrequireddataistheBELFspeci�icationintheformofαandβvalues.

4. METHODOLOGY

This study proposes an alternative objetive function to amodel previously designed for solving theintegratedScheduleGenerationandFleetAssignmentproblem,whichhasbeenappliedandtestedwithinstancesbasedondata fromaBrazilianregionalairline. Inorder toallowdirectcomparisonof theresults,allthecharacteristicsoftheoriginalmodelandinstanceswillbekept:

• Theschedulemustcoveranentireoperationweek.

• Theschedulemustbecyclical–the�leetthatstartstheweekatoneairportshould�inishtheweekatthatsameairport.

• Flighttimeswillbetheaverage�lighttimebetweentwoairports,thesameforallaircrafts.

• Performancedifferencesamongaircraftswillnotbeconsidered.

• Onlydirect�lightsareconsidered,therearenohubs.

• Anyaircraftmayperformany�light.

Besidesthedataprovidedbytheairline,additionaldatawereobtainedfrompublicsources–suchasthe Brazilian Civil Aviation regulatory agency website and annual reports. Since the demand wasprovidedaggregatedintheformofannualtotalsforeachorigin/destinationpair–andtheonlyservingairlinebetweenthoseairportswastheoneconsideredinthisstudy–,thedemandwasestimatedintwodifferent ways: an average per �light demand – based on the number of �lights between eachorigin/destinationperweek–andanaverageperdayperiod–morningandafternoon.Sincedemandshouldalwaysbeaninteger,theaveragedemandforeach�lightorperiodwasrounded.

Sincethemodelisnotdesignedtocreatenew�lightsbyitself,theoriginalproposedscheduled�lightsare complemented with additional alternative �lights, allowing the possibility of alternativerepositioning�lights.Theadditional�lightsarecreatedasthe“earliest”�lightpossibleattheendofeachproposed�lightandthe“latest”possible�lightarrivingjustbeforeeach�light.Sincetheairlinedoesnotperform�lightsbetweensomepairsofairports,thementionednetworkexpansionprocessisperformedtwicetoassurethatanyaircraftmayberepositionedtoanyotherairport.

Initially, the model incorporating the alternative objective function will be used to solve someinstancesalreadysolvedbytheoriginalmodel.Theresultswillbecomparedintermsofselected�lights,demandmet,averageloadfactorandweightedloadfactorfortwodifferentBELFs:oneverylow–50%–andamoreusualone–75%.Sincenootherchangeswereintroducedinthemodelbesidestheobjectivefunction,acomparisonoftheresultsmaybeusedtovalidatethemodel:ifthenewobjectivefunctionworksasexpected,andprovidedthattherevenuesareoptimallyde�inedfortherevenue-basedmodel,thesameinstancessolvedbybothmodels,withusualandalternativeobjectivefunctions,shouldleadtosimilar�lightschedules.

Oncethemodelhasbeenvalidated,severalinstanceswillbetested,withdifferentBELFvaluesandtheresultswillbecomparedintermsofselected�lights,demandmet,averageloadfactorandweightedloadfactor.

5. APPLICATION AND RESULTS

Theproposedmodelwasappliedtoinstancesbasedonadomesticregionalairlinethatcarries104weeklyflightsandoperatesinfiveairports–Network1–andinstancesthatrepresentexpansionstothisbasenetwork,includinganewdestinationwithatotalof164weeklyflights–Network2.Theairport in thisnewdestinationhasoperationrestrictions–arrivalanddeparture timeslots– thathinderthepossibilityofall60flightsbeingselected.ThesenetworkswillbetestedconsideringseveralfleetswiththreetypesofaircraftcommonlyusedbyBrazilianregionalairlines:ATR-42/300(AT42,for



50passengers),Embraer120(E120,for30passengers),andEmbraer170(E170,for70passengers)–allofthemrequiringa15-minuteturn-aroundtime.

Thefollowingcombinationsof�leettypeswereselected:

• TypeI:3xAT42(originalairline�leet).

• TypeII:3xE170.

• TypeIII:2xE120,2xAT42and1xE170.

• TypeIV:2xAT42and1xE170.

Asmentionedbefore, inordertoallowthedirectcomparisonofresultswiththosefrompreviousstudies(Caetano&Gualda,2011),thedemandisestimatedonannualpassengertotalsasprovidedbytheBrazilianCivilAviationregulatoryagency,ANAC(2007).Thedemanddistributionforeachinstancecanbeofthreedifferenttypes:

• Fixed:thedemandassociatedtoeach�lightis�ixedat50passengers.

• Flight:thedemandisassociatedtoeach�lightandistheaveragedemandper�light,basedonvaluesprovidedbyANAC.

• Period:thedemandbetweentwoairportsassociatedtoaperiodofday–morningorevening–istheaveragedemandbydayperiod,basedonvaluesprovidedbyANAC.

Thetotaldemandwillbedifferentineachcase:whenconsidering�ixeddemand,thetotalwillbe50multipliedbythenumberoftotal�lights.The�lightandperiodtotaldemandsshouldbedependentontheoriginanddestinationpairsservedonly;however,thevaluesmaybeslightlydifferentbecauseofroundingerrors.

Since there is interest in the direct comparison of results between the proposed and the usualobjective functions, instances 1-I-Fixed, 2-I-Flight and 2-I-Period are exactly the same instancespresented by Caetano& Guada (2011) under number 1, 4 and 7, respectively. These instanceswillhereafterbereferredtoas“referencegroup”.

Table 1: Optimization Results

Network Fleet Demand BELF Flights Demand WTM

(pax.h) LF (%)

TWLF

(%)

MLF

(%) Total Selected Total Met

1 I Fixed 50 104 104 5,200 5,200 0 100 100 100

75 104 104 5,200 5,200 0 100 100 100

1 II Fixed 50 104 104 5,200 5,200 1,537 71.4 71.4 71.4

75 104 0 5,200 0 3,842 - - -

2 I Fixed 50 164 94 8,200 4,700 2,917 100 100 100

75 164 94 8,200 4,700 2,917 100 100 100

2 I Flight 50 164 80 13,697 3,555 14,956 88.9 92.3 0

75 164 76 13,697 3,503 15,263 92.2 96.8 0

2 III Flight 50 164 127 13,697 4,993 13,489 83.8 88.7 0

75 164 101 13,697 4,264 13,664 94.9 95.5 66

2 I Period 50 164 84 13,680 3,930 14,502 93.6 95.6 88

75 164 84 13,680 3,930 14,502 93.6 95.6 88

2 IV Period 50 164 84 13,680 4,330 13,569 93.6 95.6 88

75 164 84 13,680 4,330 13,569 93.6 95.6 88

2 III Period

50 164 143 13,680 5,215 13,143 84.4 88.9 46.7

60 164 110 13,680 4,862 13,135 89.9 92.3 46.7

75 164 109 13,680 4,745 13,233 90.6 93.6 56.7

Table1showstheresultsfromusingdifferentweightvaluesforαandβ,withatleasttwobreak-evenloadfactors–50%and75%.TheWTMvaluesareproportionaltotheobjectivefunctionvaluesand,thus,the lowest values correspond to the best solutions. All the instanceswere solved by integer linearprogrammingtechniquesusingtheGurobiOptimizer©softwareversion7onaquad-corei7processorrunningat3.6GHz.ProcessingtimesfortheinstancespresentedinTable1variedfromafewseconds



(mostinstances)toabout4hours(instance2-III-Flightand2-III-Period).

The�irstrelevantobservationisthatthe�lightscheduleobtainedbyprocessingeachinstanceofthe“referencegroup”,whenconsideringaBELFof50%,waspracticallythesameobtainedbyCaetano&Gualda (2011), with some �lights being performed at slightly different departure times. This resultimpliesthat,forthosecases,bothobjectivefunctionsleadtothesameoptimalsolutionand,thus,thealternative function is as good as the revenue function. On the other hand, a BELF of 50% can beconsideredlow,suggestingthatthereissomeroomforimprovingthesolution.

Thealternativeapproachpresentedinthispaper,however,allowsforseveralexperimentswiththebreak-evenloadfactor.Furthermore,severalaircraftcombinationsmaybetestedwithouttheneedtomakeassumptionsaboutfaresandoperationalcosts.Onecombinationofspeci�icinterestisthesolutionfor instance2-I-Flight–partof the“referencegroup”–withaBELFof75%,whencompared to thesolutionforaBELFof50%.WhentheBELFwassetat75%,four�lightswereexcludedfromtheschedule,reducingthedemandmetby52passengers,butincreasingtheTWLFfrom92.3%to96.8%andthussuggestingaslightlymoreef�icient�lightschedulethanthatobtainedwiththeoriginalmodel–whileavoidingtheneedforadditionalassumptionsregardingtheyieldmanagement.

ByanalyzingthedatapresentedinTable1,itispossibletonoticethatadjustingtheBELFdoesnotalwaysintroducesigni�icantchangesintheresults.Thiscanbeparticularlyobservedininstances1-I-Fixedand2-I-Fixed.Thisusuallymeansthatmost�lightshaveahighloadfactor,suggestingthatexisting�leetsareaverygood�itforthesupplieddemandforecasts.Thisisnotanunexpectedresult,sinceinthoseinstancesthedemandwaspurposefullyde�inedasexactlythesameaseachaircraftcapacity.

Instance 1-II-Fixed acts as a sanity check, using a 70-passenger aircraft for �lights with only 50passengers.WhenimposingaBELFof50%,every�lightisperformedwithanLFof71.4%.Ontheotherhand,imposingaBELFof75%willhindertheviabilityofall�lights.

Instances2-I-Flightand2-III-FlightshowhowthemanipulationoftheBELFcanimprovetheLFandTWLF.Thereductionofthenumberof�lights(5%inthe�irstcaseandmorethan20%inthesecond)comparedtothereductionofthenumberofpassengerstransported(justover1%inthe�irstcaseandabout15%inthesecond)impliesthatthereisabetterusageoftheequipment,whichisexpressedinthehigherLFandTWLFvalues.ThelowerWTMalsoshowsthatthe2-III-Flightcon�igurationbettermeetsthedemand–whichiscoherent,oncethedemandforeach�lightisverydiverseand�leettypeIIIiscomposedofaircraftofseveralcapacities.

TheTWLFobtainedishigherthantheBELFrequiredforeverytestedinstance.Thisresultisexpected,sinceBELFde�inesabaselineand,therefore,most�lightsincludedintheschedulewillhavealoadfactorhigherthanBELF.Thereare�lightswithloadfactorbelowtheBELF–asshownbythe“minimumloadfactor”column,MLF,theloadfactoroftheleastoccupied�lightinthesolution–,buttheyareactingasrepositioning�lights,inordertoallowother�lightswithhighoccupationratiostobeselected.Inotherwords,every�lightwithoccupationbelowBELFimpliesatleasta�lightwithoccupationaboveBELF.

The analysis of instances 2-I-Period and 2-IV-Period shows something similar to that, but it alsoshows that some �lightsselected tobe �lownbyATR-42/300donothaveenoughdemand to its fulloccupancy.TheexchangeofoneATR-42/300byanEmbraer170allowstransportingmorepassengers(4,330versus3,930),buttheoccupationremainsthesame.ItispossibletonoticethattheEmbraer170replacedtheATR-42/300justfor�lightsinwhichthedemandwashigherthan70passengers,and,forthisreason,theLFandTWLFhavenotchangedforbothinstances.

The instance 2-III-Period, however, shows that adding a smaller aircraft – the Embraer 120, forexample–tothemix,ledtoasigni�icantimprovementinthenumberofpassengerstransportedandinthenumberof�lights,reducingtheWTM.WhenadoptingtheBELFof50%,however,thereisasigni�icantloadfactordecrease.Also,inthiscon�iguration,oneATR-42/300aircraftwasnotused.TherewassomeimprovementinLFandTWLFwhenBELFwasincreasedto60%,butverylowoccupancyremainedwithaloadfactorofjust46.7%forsomeaircraft.Itwasveri�iedthatincreasingBELFto75%canimproveLFandTLWevenfurther,andtheleastoccupiedaircraft,inthiscase,hadaLFof56.7%.WithBELFof60%



and75%,all�iveaircraftareused,althoughtheuseofeachofthemislessfrequentthaninthecasewitha50%BELF.ThereasonforahigherBELF,whichcanleadtoanincreaseinthenumberofusedaircraft,isthatseveralrepositioning,lowoccupation�lights,arenotallowedinthosecon�igurations.

Itisworthmentioningthattheactualcon�igurationoftheairlinecouldnotbecomparedtopreviousresultsbecauseitincludessomeairlinerequired�lightsbetweentwospeci�icairports–theinequalityinEquation20mustberewrittenasanequalityforthose�lights.

Theresultsforthisnetwork1’,whichincludestherequired�lights,isshowninTable2.Thistablealsopresentsanew�leettypeV,composedof2xEmbraer120,2xATR-42/300and1xAirbus320–thelatestinthe156seatscon�igurationandrequiringatleast30minutesofturn-aroundtimebetween�lights.

Thesolutionobtainedforinstance1’-I-Flight–theactualairlinecon�iguration–,bothwithBELFof50%and75%,wasexactlytheactual�lightscheduleperformedbythatairline,implyingthattheairlinewasperformingtheoptimalschedulewhenconsideringits�leetconfrontedtotheactualdemand.Ontheotherhand,theMLFshowsthatsome�lightsareselectedwithaverylowoccupationratio–exactlythoserequiredbytheairline.Theseresultssuggestthattheairlinewouldhavebetterresultsutilizingsmalleraircrafts, suchasEmbraer120.There is somespilleddemandonsome �lights,whichwouldsuggest incorporating a larger aircraft, such as Embraer 170. Adding these �leets, the results arepresentedinthesameTable2,intheformofinstance1’-III-Flight.

Table 2: Optimization Results

Network Fleet Demand BELF Flights Demand WTM

(pax.h) LF (%)

TWLF

(%)

MLF

(%) Total Selected Total Met

1’ I Flight 50 104 104 4,177 3,673 1,604 70.6 69.0 28.0

75 104 104 4,177 3,673 1,604 70.6 69.0 28.0

1’ III Flight 50 104 104 4,177 3,645 1,023 80.1 81.0 44.0

75 104 104 4,177 3,414 1,032 84.1 83.3 44.0

1’ V Flight 50 104 104 4,177 3,572 1,055 80.6 81.4 44.0

75 104 104 4,177 3,362 1,063 84.2 83.4 44.0

Table 2 allows observing that the alternative �leet in 1’-III-Fleet reduces the number of totalpassengers,butalsoreducestheWTMandincreasestheloadfactorbyasigni�icantamountforbothBELF values. This means that, from an operational costs standpoint – and excluding maintenanceconsiderations–,itisconvenientforthisairlinetooperatewithaircraftsofmultiplesizes.

Exploringthistrendevenfurther,instance1’-V-FlightreplacestheEmbraer170byanAirbus320.Table2showsthattheexchangeoftheEmbraer170fortheAirbus320reducesthenumberoftotalpassengersandincreasestheWTM,despitemarginallyincreasingtheloadfactorswhencomparedtoinstance1’-III-Flight.Theexplanationforthisresultcanbeobtainedbyanalysingthescheduleforeachaircraft:whileall�iveaircraftunitsareusedininstance1’-III-Flight,onlyfourareusedininstance1’-V-Flight–twoATR-42/300andtwoEmbraer120.ThereisnotenoughactualdemandtojustifytheuseoftheAirbus320and,thus,theresultingscheduledoesnotincludeany�lighttobe�lownwiththisaircraft.

Note,however,thattheseresultswereassociatedtooptimalcon�igurationsfortheairlineoperationbasedontheprovideddata.Sinceitisusualthatcostsandrevenuesmatriceschangealongtheyears,aswellasoperationalrules,thetargetedloadfactormaychangeovertimeand,thus,theadequatevalueforBELFcannotbeconsideredstatic.Therefore,theactualairline�lightscheduleandtheadequateBELFmustbedeterminedfromadequatedemandprojectionsfromcurrentdata.

6. MODEL EXPANSIONS AND FUTURE STUDIES

Theproposedobjectivefunctiondoesnottakeintoaccountdifferencesinperformanceforeachaircrafttype.Ifseveral�leetsaresuitablefora�lightbutdifferinperformance–intermsofunitaryrevenue–,thisdifferencecouldbeaddressedwithanothersetofconstantsγfijintheobjectivefunction,asshowninEquation32.



[ ]( )

( ),

. . . . . ∈ ∈

− + − ∑ ∑ f f f

ij ij ij ij ij ij

i j Lf f F

min T α γ C x pa β d pa (32)

Theconstantγfijactsasamultiplierofthepotentialtransportmoment,changingthenumberofpas-sengersneededtoguaranteepro�itabilitywhen�leetf�liesfromitoj.Fleetswithγfijequalto1.0aretheef�iciencyreferencefor�light(i,j).Valuesofγfijgreaterthan1.0meanthat�leetfislessef�icientthanthereference�leet.Valuesofγfijsmallerthan1.0meantheopposite:that�leetfismoreef�icientthanthereference�leet.Thecalibrationoftheseconstantsisoutofthescopeofthisarticleandisaninterestingtopicforafuturestudy.

Moreover,themodeldoesnottakeintoaccountthedifferencesin�lighttimes.Eventhoughitispos-sibletomodifyittoallowdifferent�lighttimesforeach�leet,thisvariationofthemodelisoutofthescopeofthisstudy.

Regardingtheproposedobjectivefunction,futurestudiesshouldevaluatetheimpactofusingdiffer-enttypesofaircraft–turbopropversusturbofan–toincorporateandtocalibratetheγfijconstants.

Aboutthemodelasawhole,futurestudiesmayconsiderhubsandconnectionstoaddressthede-mand,aswellasdemandrecapturesuchasproposedbyothermodels(LOHATEPANONT&BARNHART,2004),andtheincorporationoflongtermmaintenanceconstraints(SALAZAR-GONZAF LEZ,2014).

Sincelargerinstancesthanthosepresentedherein, includingmore�leettypesandairports, ledtohugeprocessingtimes–Gurobiwasunabletoreachtheoptimalvalueafterrunningmorethan48hours–,itissuitabletodevelopaheuristicapproachtosolvetheproblem,asproposedbyCaetano&Gualda(2011).

7. CONCLUSION

Thispaperpresentedanexactmodelwithanalternativeobjectivefunctiontosolvetheintegrated�lightschedule and the �leet assignment problem. The proposed objective function uses the transportmomentumasaproxytotheoperationalcosts,avoidingtheuseofestimatedmonetaryparameters.

Theexactmodelproposedwas testedandvalidatedagainstapreviouslydevelopedmodelwhichreliedonairlinerevenues.Theschedulegeneratedforeachoftheinstancesreferredtoas“referencegroup”wasthesameforbothapproaches,butthemodelwiththealternativeobjectivefunctionreadilyallowedforgeneratinganimproved�lightschedulewithoutanyfurtherassumptions.Thisresultpermitstoconsiderthatthenewproposedobjectivefunctioncanreplacetherevenue-basedobjectivefunction,which,inpractice,permitstheairlineplannertooptfortheformulationthatbest�itshis/herreality.

The comparison of results for instances with similar proposed �lights, but considering severalcombinationsof�leetsanddemanddistributionalongtheday,allowedforacomprehensiveevaluationof theresults in termsofcoverage,wasted transportmomentum,numberofpassengersserved,andef�iciencyofaircraftusage.

Sincenobiaswasintroducedinthesolutionbyexogenouslyde�inedcosts,theresultsmayrepresentacleanslate,theyieldmanagementstrategycanbebuiltuponandencouragetheadoptionofthenewobjectivefunctioncharacteristicstomodelandtosolveotherairlineoptimizationproblems.

ACKNOWLEDGEMENTS

TheauthorsacknowledgeCNPq(ConselhoNacionaldeDesenvolvimentoCientı�icoeTecnologico–NationalCouncilforScien-ti�icandTechnologicalDevelopment)foraResearchProductivityFellowship(Process:307622/2015-0)andLPT/EPUSP(La-boratoriodePlanejamentoeOperaçaodeTransportesdaEscolaPolitecnicadaUniversidadedeSaoPaulo–TransportationPlanningandOperationLaboratoryofthePolytechnicSchooloftheUniversityofSaoPaulo)forthetechnicalsupport.

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an exact model for airline ﬂight network opmizaon based on

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