problem Routing and scheduling decisions in the hierarchical hub location Computers and Operations Research

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Computers and Operations Research

journalhomepage:www.elsevier.com/locate/cor

Routing and scheduling decisions in the hierarchical hub location problem

Okan Dukkanci, Bahar Y. Kara

^∗

Department of Industrial Engineering, Bilkent University, Bilkent, Ankara, 06800, Turkey

a rt i c l e i n f o

Article history:

Received 15 October 2015 Revised 15 March 2017 Accepted 27 March 2017 Available online 29 March 2017 Keywords:

Transportation Hub location

Hierarchical network design Time-definite delivery Hub covering

a b s t r a c t

Hubsarefacilitiesthatconsolidateanddisseminateflowinmany-to-manydistributionsystems.Thehub locationproblemconsidersdecisionsthatincludethelocationsofhubsinanetworkandtheallocations ofdemand(non-hub)nodestothesehubs.Weproposeahierarchicalmultimodalhubnetworkstructure, andbasedonthisnetwork,wedefineahubcoveringproblemwithaservicetimebound.Thehierarchi- calnetworkconsistsofthreelayersinwhichweconsideraring-star-star(RSS)network.Thismultimodal networkmayhavedifferenttypesofvehiclesineachlayer.Fortheproposedproblem,wepresent and strengthenamathematicalmodelwithsomevariablefixingrulesandvalidinequalities.Also,wedevelop aheuristicsolutionalgorithmbasedonthesubgradientapproachtosolvetheprobleminmorereason- abletimes.WeconductthecomputationalanalysisovertheTurkishnetworkandtheCABdatasets.

© 2017ElsevierLtd.Allrightsreserved.

1. Introduction

Hubs function asswitching, transshipment and sorting points inmany-to-manydistributionnetworks.Insteadofconnectingeach origin-destination(o-d)pairby adirect link,hubsprovidea con- nectionbetweeneachpairby usingfewerlinksandconcentrating demandflowstoalloweconomiesofscale.

Thehublocationproblemistodecideonthelocationsofhubs andtheallocationsofdemandnodestohubs.Versions ofthehub locationproblemaredefinedas‘singleallocation’and‘multipleal- location’. In a single-allocation hub network, each demand node is assignedto exactly one hub. Whereas, ina multiple-allocation hub network, demand nodes can be allocatedto more than one hub.Theclassichublocationproblemhasthreemainassumptions.

First, the hub network is assumedto have a complete structure, withalinkbetweeneachhubpair.Second,thereareeconomiesof scale betweenhubs. Third,direct transportationbetweendemand node pairs (without using any hubs) is not allowed (Campbell, 1994).

Main application areasof the hub location problemare cargo delivery, telecommunications network design and air transporta- tion.Inthisstudy,wemainlyfocusonacargodeliveryapplication inTurkeywithagivenservicetimepromise.Aclassiccargodeliv- erysystemconsistsofbranchofficesandoperationcenters.Branch offices collect and distribute cargoes from/to customers directly,

∗ Corresponding author.

E-mail address: bkara@bilkent.edu.tr (B.Y. Kara).

andoperationcenterscollectanddistributecargoesfrom/tobranch officesorsendcargoestoanotheroperationcenter.Althoughthere canbemorethanonebranchofficeinacity,operationcentersdo notexistinevery city.Thus,each branchofficemustbe assigned tooperationcenter(s).

Accordingtotheaboveexplanationofacargodeliverysystem, cargodeliverynetworksandhub locationnetworksarevery sim- ilar. Branch offices and operation centers in cargo delivery net- workscanbeconsideredasdemandnodesandhubs, respectively.

Also,there areeconomies ofscale dueto bulk transportationbe- tweenoperationcenters.Therefore,thecargodeliveryproblemcan beconsideredasahublocationproblem(AlumurandKara,2009;

Kara andTansel, 2001; Tan and Kara, 2007; Yaman et al., 2007;

2012andAlumuretal.,2012b).InKaraandTansel(2001),theau- thorsemphasizetheimportanceofsynchronizationincargodeliv- erysystems. Later,Yaman etal. (2012) combinethe releasetime scheduling andhub location problems in cargo delivery applica- tions.

Theclassic hub locationproblemproposed by O’Kelly(1986a), 1986b),1987)onlyconsidersminimizationofthetotaltransporta- tion cost. However, in real life, cargo companies pay similar at- tentiontocustomer satisfaction.Toattract morecustomers,cargo companies focuson servicelevels. Servicelevel in cargodelivery isusually measured bydelivery time Yaman etal.(2012).Reduc- ingdeliverytimeisgenerallyconsideredtoincreasecustomersat- isfaction, andthus cargo companies offer differentdelivery time promises. For instance, in Turkey, cargo companies aim to de- livercargoeswithin24hours(next-daydelivery).However,dueto http://dx.doi.org/10.1016/j.cor.2017.03.013

0305-0548/© 2017 Elsevier Ltd. All rights reserved.

Fig. 1. Representation of a Multimodal Hierarchical Hub Network Instance.

Turkey’sgeographicalstructure,deliverywithinthistimeframeus- inggroundtransportationisalmostimpossibleforsomecitypairs.

Thus,inordertokeep thenext-daydelivery promisebetweenall citypairs,cargocompaniesinTurkeyhavebeguntouseairplanes intheirdistributionnetworks.

Intheclassichublocationproblem,thehubnetwork(thesub- network that is induced by the hub nodes and links between them)isusuallyassumedtobecompletewithalinkbetweeneach hubpair.Whenusingairplanes,acompletehubnetworkresultsin manyflights,andoperatingaflightisverycostly.Customarily,the operationalcostofaflight consistsofa fixeddispatchcost anda variabletransportation cost, which dependson the length ofthe flight. In this research, we limit our study to consider only the fixed dispatchcost andto those cases wherethat cost is signifi- cantlymorecrucialthanthevariabletransportationcost.Wehave twomain motivations forthislimitation. First,the fixed dispatch cost consistsof crucialcost componentsof a flight such as taxi- ingandtake off/landingcosts, whichare commonforeach flight notdepending onthe length offlight. Second, since we consider arealcargodeliveryapplicationinTurkey,thevariabletransporta- tioncostisnotasimportantasthefixeddispatchcostduetothe geographical structure of Turkey in which the difference among distances betweenany two airport hub candidates can be negli- giblecomparedto other countriesthat hasbiggerlandarea such asUSA.Basedonthesemotivationsandreasons,wesetthemain goalasminimizingthenumberofairlinesegments(flights).Thus, wewanttoprovidethesameworst-caselevelofservicetoallo-d pairswithminimalnumberofflights.

Motivated by the cargo company that uses trucks (small and large)alongwithairplanesintheirnetwork,weconsiderahierar- chicalmultimodalnetworkwiththreelayersandtwotypesofhubs (groundandairport).Fig.1showssuch aninstancewith18hubs;

nodes0to4areairporthubs;nodes5to17aregroundhubsand thesmallcircleswithnonumbersrepresentthedemandpoints.In thisrepresentation,airline segments are illustrated asthick lines betweentheairporthubs;highwaysegmentsareillustratedasthin linesbetweendemandnodesandhubs(groundorairport),andas dashedlinesbetweengroundhubsandairporthubs.

The lowest layer ofthe network consists ofthe allocations of thedemandpointstothegroundhubsandairporthubs(thinsolid lines)asnecessarytomeetthesingleallocation.Inthislayer,astar structureisusedtoallocatethedemandpoints.Eachdemandnode isconnectedtoexactlyonehub(groundorairport)withahighway link.Inreallife,smalltrucksareusedonthesehighwaysegments.

Themiddlelayerincludestheallocationofgroundhubstoair- porthubs(dashedlines),andweconsiderastarstructuretoallo- catethegroundhubshereaswell.Eachgroundhubisconnected toexactlyoneairporthubwithahighwaylink.Largetrucks,which arefasterandhavemorecapacitythansmalltrucks,areassumed to be used on these highway segments, and thus economies of scaleareconsidered.

Fig. 1 depicts a mesh structure in the first (top) layer (thick lines), where airport hubs are connected witheach other via an airline segment. However, to accomplish the fundamental goal, thatis,todecreasethenumberofflights,weproposeusingaring structure inthetop layerinstead ofa meshstructure,which can cause more flights (Fig. 2). We call this type of network a ring- star-star(RSS).Weassumeeachringwillbeservedbyseparateair- planes.Tocopewithsynchronizationissues,routing andschedul- ingdecisionsmustbeconsideredtogether.

Withthe ring structure in thetop layer, the airplane route is decided while covering all o-d pairs within a giventime bound.

In this study, motivated by the cargo company’s application, we adopta“pickup,thendeliver” typeofservice,whichmeansthere aretwo separate tours;apick-uptouranda deliverytour. Inthe pick-uptour,alldemandsarecollectedfromtheiroriginsandsent to a specific airport hub. After all demands arrive at thisairport hub, theyare sent totheir final destinationsinthe delivery tour.

We needa specificairport hub tocollect allthe demands atone point,andwecallthisthecentralairporthub.InFig.2,wedenote thecentralairporthubwithabigcircle.Ifoneairplaneinaringis notenoughtocoverallo-dpairswithinthetimelimitations,there canbemorethanonering,asshowninFig.2.Ineachring,exactly oneairplanecantravelbecausethereisnocapacityrestriction.

Pick-ups fromtheoriginsto thecentralairporthub anddeliv- eriesfromthecentralairporthubtothedestinationsareassumed

Fig. 2. Representation of a ring-star-star network instance.

to be symmetric. Therefore, theroute ofthe delivery tour is the reverserouteofthepick-uptour.

“Pickup,thendeliver” serviceiscommontomostcargocompa- nies.Themainissueistoreachtheconsigneewithinthepromised delivery time.Inthepick-uptour,theairplanemustcompletethe flightsinhalfthetimeboundsoitcancompletethedeliverytour intheotherhalf.

Based on the proposed hierarchical multimodal network, the problemcanbedefinedgivenasetofdemandnodes,asetofpos- siblelocationsforgroundandairporthubs,thelocationofthecen- tralairporthub,thenumberofhubstobelocated,thetimebound andtraveltimeparameters.Ourproposedproblemdeterminesthe locationofgroundhubsandairporthubs,theallocationofdemand nodestohubs(groundorairport),theallocationofgroundhubsto airporthubsandthelocationofairlinesegments,allwhileensur- ing thatall o-dpairscanbe servedwithin thegiventimebound.

In theobjectivefunction, thenumberof totalflights (airlineseg- ments)isminimized.

In Section 2, we review the related literature, and Section 3 presents the mathematical model and some valid inequalities. In Section 4, we develop a heuristic solution algo- rithmbasedonthesubgradientapproach.InSection5,weconduct a detailed computational analysis over two data sets, one from Turkey (TR) and one from the US Civil Aeronautics Board (CAB).

Concluding remarks and future research directions are given in Section6.

2. Relatedliterature

The hub location problem was first introduced by O’Kelly (1986a); 1986b); 1987). In these studies, the author defines the problem andproposesthe first mathematical model, which hap- penstobequadratic.Campbell(1994)categorizesthehublocation probleminto four problemsbased on theobjective function: the p-hubmedianproblem,thehublocationproblemwithfixedcosts, thep-hubcenterproblemandthehubcoveringproblem.Foreach problem, he presents linear formulations. After these pioneering studies, differentversionsofthe hub locationproblems arestud-

iedbyrelaxingorchangingthemainassumptionsofthehubloca- tionproblem,whicharehighlightedintheprevioussection.Next, weanalyze studiesintheliterature that relaxtheseassumptions, namely;the hub location problemwithring structures, the mul- timodalhub location problems andthe hierarchical hub location problems.

Relaxingtheassumptionwhichdoesnotallowanydirectlinks between two demand points, the ring structure concept is con- sideredbetweendemandnodes.The ringstructure,aspartofthe hublocationproblem,isfirstpresentedbyNagyandSalhi(1998), in a many-to-many hub location routing problem. The authors state that the many-to-many location routing problem(LRP) can bereducedto theclassic hublocation problemwhentherouting problemisnotconsidered.Theypresentamixed-integerprogram- mingformulationandproposesomesolutiontechniques.Theyalso presenta hierarchical heuristic, in which hub location is consid- ered as a master problem and routing problems are considered as sub-problems. Routing problems are solved via the neighbor- hoodsearchheuristicproposedbyNagyandSalhi(1996).Liuetal.

(2003)presentamixed-truckdeliverysystemthatallowshub-and- spokeshipmentsanddirectshipments.Aheuristicisdevelopedto determinethemode ofdelivery (hub-and-spokeordirect) andto performvehicleroutinginbothdeliverymodes.WasnerandZäpfel (2004) presenta multi-depot hub-locationvehicle-routing model fora networkdesignofparcelservicesinAustria.Thismodelcan be considered as LRP, determining the location of hubs and de- pots,theroutesbetweenhubs/depotsandtheir allocateddemand points.Thehublocationpartofthisproblemdiffersfromtheclas- sichublocationproblemintwoaspects.First,therecanbeadirect shipmentbetweentwodemandpoints.Second,thetransportation costbetweentwohubsdependsonthenumberoftransportsbe- tween those two hubs. The authors presenta mixed-integer op- timizationmodel;however, duetoits complexity,they developa heuristicbasedonalocalsearchprocedure.ForTurkey’spostalde- liverysystem, Çetineretal.(2010) proposea combinedhub loca- tion routing problem, whichincludes hub location decisions and routingdecisionsbetweendemandpoints.Inthatstudy,multiple- allocationisallowedanditisassumedthatthehubsandthevehi-

clesareuncapacitated. Theauthorsdevelopaniterativetwo-stage heuristictosolvethisproblem.DeCamargoetal.(2013)presenta newformulationforthemany-to-manyhub-locationroutingprob- lem,consideringsingle-allocationanduncapacitated hubsandve- hicles.Thecompletionofatourisboundedbyaserviceleveland eachcustomer is visitedexactlyonce.Using Bender’s decomposi- tion,they solve the problem forup to 100 nodes. In additionto theringstructure,atreetopologyonthesub-networkinducedby thehubs hasbeen alsostudied by severalresearchers (Contreras etal.,2009;2010;deSá etal.,2013).

In thehub location problem,another importantassumptionis that between hubs and between hubs and demand nodes, only one transportation mode is used. Different transportation modes arenot widelystudied.Inthe literature,some researchersextend theirstudies by consideringtransportation mode decisionsinad- ditiontolocation andallocationdecisions,andalsoby increasing thenumberoftransportationmodes.Inmultimodaltransportation, transport modeshavedifferent cost structures. The firststudy of thehublocation problemthat includesa choiceoftransportation modeisproposed byO’KellyandLao(1991),wheretherearetwo fixedhublocations,calledaminihubandamasterhub.Thisprob- lemissolvedbyaddressingtwosub-problems.Thefirstproblemis thedecisionoftransportationmode(airortruck)whilesatisfying giventime limitations.The secondproblemistheallocationdeci- sionof cities to the minihub. The multimodal hub location and hub networkdesign problemis first introduced by Alumur etal.

(2012a),who,inadditiontothedecisions oftheclassic hubloca- tion problem, consider the decision of transportation mode. The authors present a linear mixed-integer programming model and considerdifferentvariantsofthisproblem.

Additionally, in the standard hub location problem the net- workconsistsoftwolayers;onebetweenhubsanddemandnodes and the other among hubs. However, real-life networks require morethantwo levelsduetotheir complexity.Thistype ofstruc- tures is called a hierarchical hub network. Smilowitz and Da- ganzo (2007) focus on the design of integrated package distri- butionsystems formultipletransportation modes andamultiple service-level delivery network. They consider separate networks foreachmode;forgroundandairtransportationmodes,theypro- posering-ring-complete andring-ring-treenetworks, respectively.

Theyusea continuum-approximationapproach tominimize cost.

Yaman(2009)proposesathree-levelhubnetwork,whichconsists of a complete network on the top level and star networks on thesecond andthird levels. Based on its objective,this problem canbe considered asa hierarchical hub medianproblem. Yaman (2009) also studies a different version of this problem by con- sideringservice-level quality,proposing amixed-integerprogram- mingmodel.SahraeianandKorani(2010)considerthesamethree- levelhub network structure under a maximal covering objective.

Finally, Alumur et al. (2012b) present a hierarchical multimodal hublocation problemwithtime-definitedeliveries.Theyconsider a star-incomplete-star network with air and ground transporta- tion modes. They propose mixed-integer programming anda set ofvalidinequalities.Inthemathematicalmodel,theyminimizethe totaltransportationandoperationalcosts.

Forcomprehensivesurveysonhublocation,wereferthereader toAlumurandKara(2008);Campbelletal.(2002);KaraandTaner (2011), and Campbell and O’Kelly(2012); and to Nagy and Salhi (2007)foralocation-routingsurvey.

3. Problemdefinitionandformulation

TheproblemisdefinedonacompletedirectedgraphG=(^N,A) whereN=

{

⁰,1,...,n

}

^{denotes the set of nodes and A}=

{

(ⁱ,j)^: i,j∈N&i=j

}

^{isthesetofarcs.DemandpointsetisN.Thepos-}

siblehubandairporthub location setsare denotedby HandAH

(H⊆ N &AH ⊆ H),respectively and0bethe centralairport hub (0∈AH).Afleetoftrucksandairplanesservesthecustomersfrom pgroundorairport hub(s)(thatneed tobe opened) within time boundTforeachorigin-destinationpair.Thetraveltimefromnode itonodejbysmalltruckandairplaneisdenotedbyt_ijandt^air_{i j} ,re- spectively.Thetotalloading-unloadingtimeatairportlisdenoted bym_l.

α

^{representsthediscountfactoroftimeforlargetrucksrel-}

ativetosmalltrucks.Maximumtraveltimebetweenagroundhub andan airporthub plusloading-unloading time atthat airportis denotedbyM.

Forthisproblem,wefirstproposealinearmixed-integermath- ematicalmodel.

Thedecisionvariables aredefinedasfollows: x_ij equalsto1if demand point i ∈N is allocated to hub j ∈H, and0 otherwise.

x_jj equalsto1ifahub isopenedatnodej ∈H,and0otherwise.

Ifhub j ∈Hisallocatedto airporthub l∈AH,then y_jl equals to 1,and0 otherwise.y_ll equals to 1ifan airport hub isopenedat node l ∈AH,and0 otherwise.u_kl equals to 1if thereis adirect link betweenairport hub k ∈AH andairport hub l ∈AH, and0 otherwise.Theearliesttimethat allsmalltrucks arriveathubj ∈ Hisdenotedbyr_jandr^{air plane}_l representstheearliesttimethatthe airplanedepartsfromairporthubl∈AHtootherairporthubs.

It is assumedthat travel time datais symmetric andsatisfies triangularinequality.Also, theloading andunloading time atair- portsisassumed tobe independent oftheloadsize. The mixed- integer programming formulation of the proposed problemis as follows:

Minimize 

k∈AH



l∈AH\{^k}

ukl (1)

subjectto



k∈H

x_ik=1

∀

ⁱ∈N (2)



k∈H

x_kk=p (3)

x_ik≤ xkk

∀

^{i∈N,k∈H (4)}



l∈AH

y_jl=x_{j j}

∀

^{j∈H (5)}

y_jl≤ yll

∀

^{j∈H,l∈AH (6)}

y₀₀=1 (7)



l∈AH\{^k}

ukl=ykk

∀

^k∈AH:k=0 (8)



l∈AH\{^k}

u_lk=y_kk

∀

^{k∈AH:k=0 (9)}



l∈AH\{⁰}

u_0l = 

l∈AH\{⁰}

u_l0 (10)

rj≥ ti j· xi j

∀

^{i∈N,j∈H (11)}

r^{air plane}_l ≥ rj+

( α

· tjl+ml

)

· yjl− M·

(

¹− yjl

)

∀

^j∈H,l∈AH:l=j (12)

r_k^{air plane}− r_l^{air plane}+T· ukl≤ T−

(

^tkl^air+ml

)

· ukl

∀

^k∈AH:k=0,l∈AH:l=k (13)

r_k^{air plane}≥

(

^t0^airk+m_k

)

· u0k

∀

^{k∈AH:k=0 (14)}

2· r^{air plane}₀ ≤ T (15)

x_ik∈

{

⁰,1

} ∀

ⁱ∈N,k∈H (16)

yjl∈

{

⁰,1

} ∀

^j∈H,l∈AH (17)

u_kl∈

{

^0,1

} ∀

^{k∈AH,l∈AH:l=k (18)}

r_j≥ 0

∀

^{j∈H (19)}

r_l^{air plane}≥ 0

∀

^l∈AH (20)

Theobjectivefunction(1)minimizesthenumberofairlinelinks between airport hubs. Constraint (2) ensures that every demand node isallocatedto exactlyone hub.ByConstraint (3), thenum- ber ofhubstobe locatedis p.WithConstraint (4),weguarantee thatnodemandnodeisallocatedtoanon-hubnode.ByConstraint (5),ahub isallocatedtoexactlyone airporthub.Also,withCon- straint(5),y_kk =1impliesthatx_kk = 1.Constraint(6)guarantees that no hub is allocated to a non-airport hub. Constraint (7) es- tablishesthecentralairporthub.Constraints(8)and(9)construct theringstructurefortheairporthubs(toplayerofthehierarchical network). Constraint (10) allows a structure withmore than one ring,whichmeanstherecanbe morethanoneairplane,ifneces- sary. Also, withConstraint(10),ifno airporthub isopened,then thecentralairporthubisusedasacentralgroundhub.

Constraint(11)calculatestheearliesttimethatsmalltrucksar- riveattheirallocatedhub. Constraint(12)guaranteesthat anair- planecannotleaveanairporthubbeforeallthevehicles(smalland largetrucks)fromtheothernodes(demandpointorgroundhub) allocated to that airport hub have arrived. With Constraint (13), weensurethattheearliesttimean airplanedeparts fromanyair- porthubiswithinapredeterminedtimebound.Constraint(13)is an adaptationofthewell knownMiller-Tucker-Zemlin(MTZ)con- straint proposedby Miller etal.(1960).Constraint (14)calculates theearliest time anairplane departsfromtheairport hubwitha directlink tothecentral airporthub. Wecompute thistime sep- arately to complete the ring structure for the top layer. By Con- straint (15), we guarantee that all o-d pairs are covered within a giventime bound. Because we assume symmetricaltravel time data,we considerpick up anddeliverythe same,sowe multiply by2.Finally,Constraints(16)-(20)arethedomainconstraints.

This mathematical model is a mixed binary programming modelwithO(n²)binaryvariables,O(n)non-negativevariablesand O(n²)constraintswherenisthenumberofnodes.

3.1. Pre-processingandvalidinequalities

We nowproposesome variablefixing rules andvalidinequal- ities. First, we present two variable fixing rules, which are pre- specifiedasparametersofthemodel:

VariableFixingRule1:Fori ∈N andj ∈H\{i},ift_ij >T/2,then demandnodeicannotbe allocatedtothehub(groundorairport) j,becausetraveltimebetweencityi andcityjexceedshalfofthe

timeboundT.WithVariableFixingRule1,ift_ij >T/2,thenx_ijwill beequalto0.

Variable FixingRule 2: For j ∈ Hand l ∈A\{j}, if

α

· tjl+m_l>

T/2,then hubj cannot be allocatedto theairport hubl,because the reduced travel time between city j and city l and the load- ing/unloadingtimeatcitylexceedshalfofthetimeboundT.With VariableFixingRule2,if

α

· tjl+m_l>T/2,theny_jlwillbeequalto 0.

Wenowproposetwovalidinequalities:

Fori∈N,j∈H\{i}andl∈A\{i,j},ift_{i j}+

α

· tjl+m_l>T/2,then demandnode i cannot be allocatedto groundhub j,andground hubjcannotbeallocatedtoairporthublatthesametime.There- fore,theinequality

xi j+yjl≤ 1

∀

ⁱ∈N,j∈H

\{

ⁱ

}

,l∈A

\{

i,j

}

,ifti j+

α

· tjl+ml>T/2 (21) isvalid.

Fori ∈N andj ∈H\{i},ift_{i j}+

α

· tjl+m_l>T/2forairporthub l andthereis a hub atcity j, then demandnode i cannot be al- locatedtogroundhubj,andgroundhub j cannotbe allocatedto airporthublatthesametime.Therefore,theinequality

x_{i j}+ 

l∈A\{i, j}^{:ti j}+α·tjl+ml>T/2

y_jl≤ xj j

∀

^i∈N,j∈H

\{

ⁱ

}

⁽²²⁾

isvalid.Noteherethatvalidinequality(22)isthestrongerversion ofvalidinequality(21).

Weanalyzetheperformancesofthesetwovalidinequalitiesin detailin Section 5. Based onthese analysis, we include validin- equality(22)intothemathematicalmodel.

4. Lagrangianrelaxationbasedsolutionapproach

Sincegettingtheoptimalsolutiontakestoomuchtime despite ofallvariablefixingrulesandvalidinequalities,weproposeanal- ternative solution approach forthe problemunder consideration.

In order to find the optimal or near optimal solutions in a rea- sonabletime, a differentsolutionmethodbasedon thesubgradi- entalgorithmisdevelopedtosolvethoseprobleminstances,which cannotbe solvedinafew seconds.So,we proposethisalgorithm fortheprobleminstanceswithtightertimebounds.

Subgradientalgorithm is one of the well-known solution ap- proachesforthecombinatorialoptimizationproblemsbasedonthe Lagrangian relaxation. The algorithm consists of two main com- ponents. First, the problem is relaxed by removing some sets of constraintsfromtheformulationandaddingthemtotheobjective function after multiplying them with Lagrange multipliers. Solv- ingtherelaxedproblem,incaseofminimization,providesalower bound forthe original problem. Second, by utilizing the solution of the relaxedproblem, a feasible solution is obtained. Thisfea- siblesolution gives an upperbound for the original problem. By using these two components, the subgradient algorithm tries to strengthenthelowerandupperboundsinordertofillthegapbe- tweenthemandreachtheoptimalsolution.

Now,we explain the proposed subgradientalgorithm in three parts.Initially,wedescribetherelaxedproblem.Secondly,wegive adetailedexplanationonhowtofindafeasiblesolutionbyusing the solutionobtained forthe relaxedproblem. Finally,we define thesubgradientalgorithmitself.

4.1. Relaxedproblem

In order to apply this Lagrangian relaxation approach, Con- straints(13)arerelaxedfromtheoriginalformulation.Constraints (13)are versions of the Miller-Tucker-Zemlinsubtour elimination constraintthatkeepsthedeparturetimesoftheairplanesinatour.

Itisabig-Mtypeconstraint.Theformulationoftherelaxedprob- lemwithLagrangemultipliers

π

kl isasfollows:

Minimize 

k∈A



l∈A\{^k} u_kl

+ 

k∈A\{⁰}



l∈A\{^k}

π

kl·

(

^{rair plane}_k − r_l^{air plane}+

(

^T+t_kl^air+m_l

)

· ukl− T

)

subjectto

(2-12),(14-20),(22)

Solving theabove formulationwillgive alower bound forthe originalproblem.

4.2.Findingafeasiblesolution

In ordertoapply a subgradientalgorithm, weneed tofindan upperboundonthe originalproblem. One approachisto change thesolutionobtainedfromthe relaxedproblemtomake itfeasi- blefortherelaxedconstraint.Inourproblem,itishardtochange thesolutiontomakeitfeasiblebecauseweneedtoadjustthede- parturetimeofthe airplanesandalsowe needto eliminatesub- toursifthereare any.Thus, weproposetosolveanotherproblem whichisarestrictedversionoftheoriginalproblem.Aftersolving therelaxedproblem, wegive the valuesofthedecisionvariables associatedwiththe location ofhubs(x_jj) tothe original problem andthen wesolve theresultingnewprobleminordertofindan upperbound.Theformulationofthenewrestrictedproblemisas follows:

Minimize 

k∈A



l∈A\{^k}

u_kl (1)

subjectto (2-20),(22)

xj j=x _{j j}

∀

^j∈J:

|

^J

|

^{=p & J⊆ H (23)}

wherex _{j j}isobtainedfromtherelaxedproblem.

However, byusingConstraint(23)wecannot ensurethefeasi- bilityoftherestrictedproblem.Duetothetimeboundconstraint, itis possiblethat theopened facilities cannot coverall o-dpairs withinthegiventimelimit.Therefore,iftheproblembecomesin- feasiblebecauseofConstraint(23),weneedtodecreasethenum- ber of fixed locations obtainedfrom the relaxed problem. While decreasingthenumberoffixedlocations, werandomly selectthe fixedlocationtoremove.Thereductiononthenumberoffixedlo- cationscontinuesuntilonefixed locationthatisthegivencentral airporthubremains.So,wecanadoptConstraint(23)asfollows:

x_{j j}=x _{j j}

∀

^j∈J:

|

^J

|

^{≤ p & J⊆ H (23*)}

whereset|J|dependsonthefeasibilityoftheproblem.

By solving this restricted problem, we can obtain an upper boundontheoriginalproblem.

4.3.Subgradientalgorithm

By using lower andupper boundsobtainedfrom thesolution techniquesexplainedpreviously,wecanapplyasubgradientalgo- rithm to the proposed problem. The algorithm is constructed as follows:

Step0:ChooseaninitialLagrangemultiplier

π

_kl^{0 andsett}=0. Step 1: Let

π

kl=

π

_kl^{t andsolve the relaxedproblem withthe}

optimalvaluez(

π

kl^t)^{andupdatethelowerboundasfollows:}

LB←max

{

^LB,z

( π

kl^t

) }

Step 2: Giventhe location of hubsfrom the relaxedproblem, solvetherestrictedproblemwithz(^xt_{j j})^{.Iftherestrictedproblem} yieldaninfeasiblesolution,thenumberoffixedlocationsobtained

fromtherelaxedproblemisdecreaseduntiltherestrictedproblem providesa feasiblesolution.Then,theupperbound isupdatedas follows:

UB←min

{

UB,z

(

^xtj j

) }

Step3:UpdatetheLagrangemultipliersasfollows:

π

_kl^t+1←max

{ π

^t+

μ

t·

(

^{rair plane}_k − r^{air plane}_l +

(

^T+t^air_kl +m_l

)

· ukl− T

)

,0

}

where

μ

^t= f· ^{UB−z(πklt)}

||^r_k^{air plane}−r^{air plane}_l +(T+t_kl^air+m_l)·u_kl−T||² andbeingfanum- bertakenbetween0and2that isdecreased afteracertain num- berofiterationswithoutimprovement.

Step4:t←t+1and

Step 5: If t reaches the maximum number ofiterations, then Stop.Otherwise,gotoStep1.

In the proposed subgradientalgorithm, initially, an initial La- grange multiplier (

π

kl⁰) is chosen. Then, based on the chosen La- grangemultiplier,therelaxedproblemissolved andifthe objec- tivefunctionvalueoftherelaxedsolutionisgreaterthanthelower bound, thelower bound isupdated accordingly.After that, based on the location of hubs obtained from the relaxed solution, the restrictedproblemis solved tofind a feasiblesolution andifthe objective function value ofthis feasible solution is less than the upperbound,thentheupperbound isupdatedaccordingly.Next, theLagrangemultipliersareupdatedbasedonthegivenformulain Step3.Afterupdating theLagrange multipliers,the relaxedprob- lem is solved again with the new Lagrange multipliers and the algorithm continues until the maximum number of iterations is reached.Thevaluesofthedifferentparametersofthesubgradient algorithmaregiveninthecomputationalanalysissection.

5. Computationalstudy 5.1. Datasets

Inthe computationalstudies, sincewe consider acargodeliv- eryapplicationinTurkey,weuseTurkishnetwork(TR)dataset.In 2007,Tan andKara(2007)introducedtheTRdatasettotheliter- ature,anditconsistsof81citiesinTurkey.

These cities are illustrated in Fig. 3; the numbers represent Turkey’svehicle licenseplatenumbers,which areunique to each city. Eachcity is considered as a demand point, so there are 81 demand points (|N| =81). There are 22 potential hub nodes (|H|

= 22),representedasred circlesinthe map(Fig.3).Since Afyon (3),Aksaray(68)andDuzce(81)donothaveairports,19ofthese potentialhubnodesareconsideredaspotential airporthubs(|AH|

= 19).Ankara(6)isconsidered thecentral airporthub dueto its geographical andgeopolitical advantages:it is nearthe centerof Turkey,ithasthecountry’s secondbiggest amountofflowandit isthecapitalcity.

The time discount factor

α

^{is taken as 0.9. Distance data is}

taken from Tan and Kara (2007). Travel times are calculated by assuming that the trucks travel ata speed of70 km/hrand that theairplanestravelataspeedof700km/hr.Theloading/unloading timeatanairportistakenas30minutes.

Inaddition to theTR dataset, we alsoconsider theCAB data set, whichis based on airline passengerinteractions between25 UScitiesin 1970,andwasintroducedto theliterature byO’Kelly (1987).TheCABdatasetisillustratedinFig.4.Twenty-fivenodes representthedemandnodesandthepotentialgroundandairport hubs (

|

^N

|

=

|

^H

|

=

|

^AH

|

=25). The central airport hub is assigned toNewYork(17)becauseitisthebiggestcity intheUSinterms ofpopulation.ThedistancedataistakenfromO’Kelly(1987).The othersettingsarethesameasintheTurkishnetwork.

Fig. 3. Map of Turkey with cities and potential hub sets.

Fig. 4. Map of the US with 25 Cities.

FormoreinformationontheCABandTRdatasets,wereferthe readertoBeasley(2012).

Thesubgradientalgorithmrelatedparametersare takenasthe sameforbothdatasets. TheinitialLagrange multipliers(

π

_kl⁰)^are setto0.fistakenas1anditishalvedafter5iterationswithout improvement.Theinitiallowerandupperboundsofthealgorithm are0andthehighestnumberofairlinesegmentspossible,respec- tively.Whileapplyingthesubgradientalgorithm,we usedifferent maximumnumberofiterationssuchas1,3,5,10,20,50&100.

Computationalstudieswerecarriedoutonaserverwith4AMD OpteronInterlagos6282SEsand96GBofRAM.Theproposedfor- mulation andthe proposed subgradientalgorithm were coded in Java via NetBeansIDE 8.0.2. As solver, we usedGurobi optimiza- tionsoftware,version6.0.3.Thetimelimitwassetas6hours.

5.2. Performanceofthevalidinequalities

Wefirsttestedtheperformancethetwovalidinequalitiespro- posed in Section 3 for the mathematical model on the Turkish network data set. We did not consider Valid Inequality (21) and Valid Inequality(22)together becauseValid Inequality(22)isthe strongerversionofValidInequality(21).Theresultsaredepictedin Table1.ThefirsttwocolumnsofTable1representtheparameters:

thetimebound(T)inhoursandthenumberofhubstobeopened (p). The“NoValid Inequalities”,“ValidInequality(21)” and“Valid Inequality(22)” columnsrepresentthe solutions ofthe appropri- ate models. The columnsindicated by “LPGap” and “CPU” show

thegapin linearprogrammingrelaxationfromtheoptimalvalue inpercentagesandtheCPU timerequirementinseconds,respec- tively.Finally,thecolumnwithlabel“Nodes” presentsthenumber ofnodesthatwereevaluatedinabranch-and-boundtree.

As evident from Table 1, including only Valid Inequality (21)andonly Valid Inequality(22)generallydecreasesCPU time.

When we comparethe resultsin termsof CPU time, thehighest improvementwasobservedintheValidInequality(21)columnfor threeinstances,intheValidInequality(22)columnforremaining seveninstances.Basedontheseresults,andbecauseValidInequal- ity(22)isthestrongerversionofValidInequality(21),weincluded onlyValidInequality(22)intothemathematicalmodel.

5.3.Computationalanalysisofthemathematicalmodel

Wenextobservedthe effectofT (timebound)andp(number of hubs to be opened) on the optimal network configuration by varyingthem. Wevaried the timebound andnumberof hubsto beestablishedonbothdatasets.FortheTRdataset,weevaluated Tbetween24and14hours,anddeterminedthattheRSSmodelis infeasiblewhenthetimeboundis12hoursorless.BecausetheUS ismuchlargerthanTurkeyintermsofarea,theservicetimelevel must be higher forthat case. Therefore, we varied T from 60 to 32hours, anddetermined that the RSSmodel isinfeasible when the time bound is 22 hours or less. Foreach T bound, we used threedifferentvalues ofpstartingfromthe first feasiblepvalue forthecorresponding Tvalue aslongastheCPU time allowed it

Table 1

Performance of Valid Inequalities, TR Data Set.

T p No Valid Inequalities Valid Inequality (21) Valid Inequality (22)

Lp Gap CPU Nodes Lp Gap CPU Nodes Lp Gap CPU Nodes

18 6 95.83 321.27 202700 82.92 85.41 127476 82.92 59.65 145235 18 7 95.83 888.57 1910129 82.92 32.5 166974 82.92 123.52 190014 18 8 95.83 398.09 1001490 82.92 103.96 296374 82.92 85.16 177697 17 6 92.86 24.08 98732 78.57 25.51 95806 78.57 23.83 81900 17 7 92.86 641.98 1584395 78.57 303.15 1289257 78.57 172.77 873851 17 8 92.86 1021.43 4340362 78.57 899.56 8071544 78.57 494.23 2339322 16 6 94.44 60.4 288113 78.89 16.92 83396 78.89 74.22 226549 16 7 94.44 885.92 4408706 78.89 450.47 2890848 78.89 822.47 4992912 16 8 93.75 6 864.4 9 19089335 76.25 2404.16 10474880 76.25 1731.65 8296941 15 8 95 7765.05 39483982 70 1577.5 10597225 70 1210.64 8305331

Table 2

Results on TR data set.

T p # Flights # Airplanes A. Hubs G. Hubs (allocated A. Hubs) CPU

24 2 2 1 6,23 – 0.09

24 3 2 1 6,23 21(23) 0.26

24 4 2 1 6,25 21(25), 65(25) 0.41

23 2 2 1 6,23 – 0.09

23 3 2 1 6,21 61(21) 0.28

23 4 2 1 6,25 21(25), 61(25) 0.29

22 2 2 1 6,23 – 0.09

22 3 2 1 6,23 21(23) 0.3

22 4 2 1 6,23 20(6), 61(23) 0.26

21 3 3 1 6,21,25 – 0.41

21 4 2 1 6,25 1(6), 65(25) 1.04

21 5 2 1 6,25 1(6), 21(25), 65(25) 0.81

20 4 3 1 6,21,25 34(6) 0.9

20 5 3 1 6,23,65 34(6), 61(23) 1.27

20 6 3 1 6,21,25 23(21), 65(21), 81(6) 1.36

19 5 5 2 6,21,25,26 27(21) 2.08

19 6 4 1 6, 21, 25, 34 16(6), 27(21) 5.73

19 7 4 1 6, 21, 25, 34 16(6), 27(21), 42(6) 1.3

18 6 6 2 6,27,34,61,65 20(6) 59.65

18 7 6 2 6,25,34,44,65 20(6), 55(6) 123.52

18 8 6 2 6,26,44,61,65 23(61), 34(26), 81(6) 85.16

17 6 7 2 6,25,27,34,35,65 – 23.83

17 7 7 2 6,1,16,20,25,65 27(1) 172.77

17 8 7 2 6,16,20,25,27,65 21(65), 23(25) 494.23

16 6 9 4 6,20,34,44,61,65 – 74.22

16 7 9 4 6,16,20,44,61,65 21(44) 822.47

16 8 8 3 6,7,16,21,25,27,65 55(6) 1731.65

15 8 10 3 6,20,21,25,27,34,61,65 – 1210.64

15 9 10 3 6,1,20,21,25,34,61,65 68(6) 14987.67

14 8 11 4 6,1,20,21,25,34,61,65 – 990.34

14 9 10 3 6,1,20,21,25,34,61,65 16(34) 18659.45

(the CPU time requirement of the modelincreases exponentially withp).Thus, forsome Tvalues,weonlyreport resultswithtwo differentpvalues.TheresultscanbeseeninTables2and3.

ThefirsttwocolumnsinTable2representthetwoparameters:

Tandp.Thethirdandfourthcolumnsinthetablepresenttheop- timalobjectivefunctionvalue(thenumberofairlinelinks)andthe numberof airplanes (number of rings),respectively. We can de- ducethe numberofairplanes from thesolution byobserving for howmanyi nodes,thevalue ofu_0i equalsto 1.Thefifth column lists the location of airport hubs. The sixth column presentsthe locationofgroundhubsandtheirallocatedairporthubsinparen- thesis.Finally,the lastcolumnindicates the CPU timein seconds tosolvetheinstancetooptimality.

As evidentfromTable2,twoairlinesegments andtwo airport hubswithoneairplaneareenoughtocoverallo-dpairsinTurkey within24, 23or22hours.WhenwedecreaseTto21hours,three airporthubswiththreeairline segmentsarerequired.Then,ifwe increasepfrom3to4,twoairporthubswithtwoairlinesegments areenoughtocoverthecountrybecauseoftwoadditionalground hubs(whichshowstheimportanceofgroundhubs).WhenT=20

andp=3,theproblembecomesinfeasible.Ifwe increasepfrom 3 to 4,there can be a solution withthree airport hubsand one groundhubrather thanwithfourairporthubs. Whenwe further decrease thetime bound to 19hours with fivehubsare opened, two airplanes are required to cover all cities inTurkey withfive flights among four airport hubs. If we continue to decrease the time bound to 16, 15 and 14 hours (which are very tight time boundsfortheTurkish network),the numberofairlinesegments increases to eight, nine, 10 and11, andthe number of airplanes increasestothreeandfour.

On the other hand, one airplane and two airport hubs are enoughto coverthe USwhenthetimebound isbetween60and 56 hours and where the central airport hub is New York (17) (Table3).Ifthetimeboundisbetween52and36hours,thengen- erallyone airplaneandmorethan twoairport hubsarerequired.

Whenwe reduce Tto 32hoursorfewer, morethanone airplane isneeded.

Also,asevidentfromTable3,whenweincreasepforeachtime bound level, generally the additionalhub is opened asa ground hub because each additional airport hub can lead to an increase

Table 3

Results on CAB Data Set.

T p # Flights # Airplanes A. Hubs G. Hubs (allocated A. Hubs) CPU

60 2 2 1 17,8 – 0.06

60 3 2 1 17,8 18(17) 0.23

60 4 2 1 17,22 7(17), 10(17) 0.63

56 3 2 1 17,8 23(8) 0.16

56 4 2 1 17,8 3(17), 23(8) 0.41

56 5 2 1 17,8 10(8), 12(8), 23(8) 0.73

52 3 3 1 17,10,22 – 0.16

52 4 3 1 17,8,22 16(17) 1.76

52 5 2 1 17,8 1(17), 22(8), 23(8) 0.58

48 3 3 1 17,13,22 – 0.13

48 4 3 1 17,13,22 1(13) 3.46

48 5 3 1 17,13,22 11(13), 14(17) 2.71

44 5 4 1 17,13,19,23 14(13) 10.05

44 6 4 1 17,12,13,23 8(13), 24(13) 3.4

44 7 4 1 17,12,13,23 8(13), 16(13), 24(13) 6.64

40 5 5 1 17,1,11,12,23 – 2.33

40 6 5 1 17,1,12,15,23 8(15) 8.81

40 7 5 1 17,1,12,15,23 8(15),13(1) 11.73

36 6 5 1 17,1,11,12,23 10(1) 7.51

36 7 5 1 17,11,12,16,23 8(11), 14(16) 7.67

36 8 5 1 17,1,11,12,23 8(11), 10(1), 16(1) 43.54

32 7 7 2 17,11,12,13,14,23 8(11) 197.15

32 8 7 2 17,10,11,12,23,24 8(11), 15(11) 493.92

32 9 7 2 17,11,12,13,14,23 8(11), 18(17), 22(12) 224.24

Table 4

Results of different central airport hubs, TR Data Set.

T p Ankara (6) Istanbul (34) Izmir (35) Kayseri (38) Elazig (23)

# # CPU # # CPU # # CPU # # CPU # # CPU

Flights Airplanes Flights Airplanes Flights Airplanes Flights Airplanes Flights Airplanes

24 3 2 1 0.26 2 1 0.36 3 1 0.39 3 1 0.15 2 1 0.55

24 4 2 1 0.41 2 1 0.43 3 1 4.54 3 1 0.75 2 1 0.62

24 5 2 1 0.43 2 1 0.66 3 1 4.11 3 1 0.96 2 1 1.29

21 5 2 1 0.81 5 2 13.39 4 1 37 4 1 2.99 3 1 2

21 6 2 1 0.49 4 1 19.67 4 1 8.62 4 1 2.98 3 1 2.04

21 7 2 1 0.68 4 1 21.12 4 1 224.08 4 1 2.54 3 1 1.82

18 6 6 2 59.65 7 2 454.41 7 2 445.31 6 2 66.11 6 2 68.88

18 7 6 2 123.52 7 2 2662.24 7 3 2361.68 6 2 104.17 6 2 78.09

18 8 6 2 85.16 7 2 2122.34 7 3 2985.12 6 2 96.1 6 2 158.18

inthenumberofairlinesegmentsastheobjectivefunctionofthe mathematicalmodelminimizesit.

Decreasingthetime boundandincreasingthenumberofhubs tobeopenedgenerallyincreasestheCPUtime.FortheTRdataset, ifthetimeboundisbetween24and19hours,themodelissolved within afew secondsforall instances.Ifitisbetween18 and14 hours,theCPUtimerequirementiswithintwohoursexceptintwo instances, T = 15 and p = 9 andT = 14 and p = 9. When the time boundis between60and48 hoursinthe CABdataset,the problemissolvedwithinafewseconds.Ifitisbetween44and32 hours,allinstancesaresolvedwithin10minutes.

Wealsoanalyzedtheeffectofadifferentcentralairporthublo- cationontheresultsoftheRSSmodelforbothdatasets(Table4).

InTurkey,wechose Istanbul(34;northwesternTurkey)andIzmir (35; western Turkey) as central airport hubs due to the high amountofdemand;Istanbulisthecountry’slargestcityandIzmir is thethird largest.Wealsoselected Kayseri (38)andElazig(23) because of their locations (central and eastern Turkey, respec- tively).Wevariedthetimeboundsandthenumberofhubstobe locatedforeachpossiblecentralairporthublocation.Asexpected, thecities’geographicalpositionsdirectlyaffectstheresults.Forex- ample, for some p values,the problem becomesinfeasible when thecentral airporthubisnot Ankara.When T=24andp=2,if thecentralairporthubisAnkara,Istanbul orElazig,thereexistsa solution;ifitisIzmirorKayseri,theproblemisinfeasiblebecause thetime boundcannot be satisfiedwithtwo hubs. BecauseIzmir

is located inthe far west of Turkey,cities in the east cannot be coveredwithonlytwoairporthubs.Interestingly,althoughKayseri isin the centerof Turkey,two airporthubs are alsonot enough tocoverall o-dpairs inTurkeybecauseifasecond airporthubis openedinthe west,then citiesinthe eastcannot be reachedon time,andifit intheeast, thencities inthewest cannot becov- eredontime.Tosatisfythetime boundforKayseri,atleastthree hubsarerequired;therefore,havingthecentralairporthubinthe centerofthecountrymaynotbeasefficientasitmightseem.

Whenwecomparetheresults,weseethat theobjectivefunc- tionvalues(thenumberofairlinelinks)fortheIstanbul,Izmirand KaysericasesaregenerallyhigherthanforAnkaraandElazig.This findingindicates that ifthecentral airporthub islocated onone sideofthecountry(suchasinIstanbul orIzmir)orinthecenter ofthecountry (suchasinKayseri), more flights arenecessary to ensurecoverageofthewholecountry.Ontheotherhand,citieslo- catednearthecenter,butnotexactlyinthecenter(suchasAnkara andElazig)aremoreadvantageous forbeingacentralairporthub intermsofthenumberofflights.

Wealso explored differentoptions forthe central airporthub for the CAB data set (Table 5). We first chose Los Angeles (12), whichisthesecond-biggestcityintheUSintermsofpopulation.

WealsochoseKansasCity(11),becauseitisvery nearthecenter ofthe US.Additionally,we considered Memphis(13) andCincin- nati(5)asthecentralairporthubbecauseseveralcargocompanies areheadquarteredthere.

Table 5

Results for different central airport hubs, CAB data set.

T p New York (17) Los Angeles (12) Kansas City (11) Memphis (13) Cincinnati (5)

# # CPU # # CPU # # CPU # # CPU # # CPU

Flights Airplanes Flights Airplanes Flights Airplanes Flights Airplanes Flights Airplanes

60 2 2 1 0.06 2 1 0.1 2 1 0.05 2 1 0.05 2 1 0.06

60 3 2 1 0.23 2 1 0.78 2 1 0.21 2 1 0.2 2 1 0.23

60 4 2 1 0.63 2 1 5.9 2 1 0.13 2 1 0.16 2 1 0.2

48 4 3 1 3.46 3 1 8.73 3 1 1.74 3 1 0.85 2 1 0.18

48 5 3 1 2.71 3 1 7 3 1 2.28 3 1 2.76 2 1 0.17

48 6 3 1 2.23 3 1 7.01 3 1 1 3 1 0.9 2 1 0.2

36 6 5 1 7.51 6 2 148.56 5 1 8.04 6 1 17.4 5 1 2.07

36 7 5 1 7.67 6 2 159.36 5 1 9.02 6 2 47.46 5 1 3.84

36 8 5 1 43.54 5 1 87.18 5 1 7.07 6 2 17.3 5 1 3.71

Table 6

Coverage Percentages of O-D Pairs in TR Data Set for T = 19, p = 6.

Ankara Istanbul Izmir Kayseri Elazig

# Flights 4 flights 6 flights 6 flights 5 flights 4 flights

# Airplanes 1 airplane 2 airplanes 2 airplanes 2 airplanes 1 airplane Service Time Percentage of Coverage

18 98.3 98.09 99.78 96.45 97.84

16 88.03 84.29 91.88 80.46 81.48

14 67.22 61.94 67.35 58.24 58.55

12 41.42 38.77 37.96 35.19 33.83

10 20.43 18.55 17.41 17.13 15.74

We varied the time bound and the number of hubs to be openedfor thesefive central airport hub locations, as we did in the New York case. As evident from the table, the results differ markedlyfromeachother,especiallyforthetighttimebounds,be- causeoftheirlocationsindifferentregions.NewYorkislocatedin thenortheastern US,Los Angelesis inthe southwestandKansas Cityinthecenter.MemphisandCincinnatiarelocatedinthecen- traleasternportionoftheUS.

Foraloosetimeboundsuchas60or48hours,theresultsare generallythesameforeachcase. Ifwe tightenthetimeboundto 36hours,againforNewYork,KansasCityandCincinnati,the re- sultsarethesame,butforLosAngeles,moreairplanesandflights are required because of how far west it is located and because mostofthecitiesintheCAB datasetare locatedintheeast.The Memphiscasealsorequiresmoreairplanesandflights,forasimi- larreasonastheKaysericaseintheTRdataset.

TheresultsoftheTRandCABdatasetsindicatethatifthecen- tralairporthub isintheeastorwestofthecountry, theairplane firsttravelstoanairporthublocatedontheothersideofthecoun- try,whichisgenerallyfarthestfromthecentralone.Ifthecentral airporthubisinthecentralpartofthecountry,theairplanefirst travelstooneside(westoreast)ofthecountry,thentotheother side and then returns to the central airport hub. The results for bothdatasetsalsoshowthatwhenthecentralairporthubismore centrallylocated,CPUtimegenerallydecreases.

While analyzing the outputs for both data sets, we observed thatalthoughallo-dpairsarecoveredwithinthefixedtimebound T, some are covered within a bound far shorter than the time boundT overthe proposed network.We now providea different analysiswherewecomparetheresultingnetworksbasedontheir

“percentageofcoverage” performance,whichwe define as the per- centageofthewholedemandservedwithindifferent(smallerthan T)timebounds.IntheTRdataset,weanalyzedoneinstance(T= 19,p=6) forfivecentralairport hublocations. Forthisanalysis, aftercalculatingtheservicetimeforeveryo-dpair,wecomputed thepercentageofcoveragebasedonservicetime(Table6).

The analysis indicates that actually more than halfof the o-d pairsarecoveredwithin 14hours,regardlessofwherethecentral airporthub islocated.When wecomparetheresults,weseethat

Izmir hasthehighestservicelevel percentageforT= 18,16and 14.Whenthetimeboundisequalto12and10hours,weseethat Ankarahasthehighestcoverage.Generally,the coveragepercent- agesforAnkara,IzmirandIstanbularemorethanforKayseriand Elazig.However, weshouldnote thattheIstanbul,IzmirandKay- sericaseshavetwoairplanesandtheAnkaraandElazigcasesjust one.Havingtwoairplanesdirectlyincreasesservicelevelpercent- ages; therefore,the Istanbul, Izmir andKayseri cases have more advantagescomparedtotheAnkaraandElazigcases.Nevertheless, Ankarahasthe second-highestservicelevelpercentagewhen the time boundisbetween18and14hours, andthehighestwhenT is between12 and10 hours. Therefore, in terms ofservice level percentage,Ankaraisthemostadvantageouscity(T=19,p=6).

Wealsocomparetheservicelevelpercentagesofthefivecen- tralairporthublocationsintheCABdatasetforT=48andp=4 (seeTable7).

Based on the results shownin Table 7, nearly 70% of all o-d pairsarecoveredwithin32hoursforeachcase.Whentheservice levelisbetween44and36hours, theCincinnatiandKansasCity cases have the highest coverage percentages. If the service level istightenedtofewerthan36hours,CincinnatiandNewYorkhave thehighestpercentages.AlthoughNewYorkislocatedontheeast- ernedgeofthecountry,whenwedecreasethetimebound,itsser- vicelevelpercentageisgenerallyhigherthantheother casesper- centages;thisfindingisrelatedtothehighnumberofcitiesclose toit.ThissituationisalsovalidforCincinnati,butnotforKansas City,Memphisor(andespecially)LosAngeles.TheCincinnaticase generallygivesthe highestcoveragepercentages forthisinstance (T=48,p=4).

5.4. Computationalanalysisofthesubgradientheuristicalgorithm

In this section, the solution obtained from the subgradient based heuristicalgorithm willbe compared withthe optimalso- lutioninordertoevaluatethequalityoftheproposedsolutionap- proachesintermsoftheoptimalitygapandCPUtime.

When we apply the proposed subgradient algorithm for both TR andCABdata set,formostof theinstances, thegapbetween lowerandupperboundsisreallyhigh.Thereasonforthiscanbe