Contraflow Transportation Network Reconfiguration for - PDF document

ContraflowTransportationNetworkReconfigurationforEvacuationRoutePlanningSanghoKim,ShashiShekhar,,andMankiMin,—Givenatransportationnetworkhavingsourcenodeswithevacueesanddestinationnodes,wewanttofindacontraflownetworkconfiguration(thatis,idealdirectionforeachedge)tominimizetheevacuationtime.Contraflowlanereversalisconsideredapotentialremedytoreducecongestionduringevacuationsinthecontextofhomelandsecurityandnaturaldisasters(forexample,hurricanes).Thisproblemiscomputationallychallengingbecauseoftheverylargesearchspaceandtheexpenseofcalculatingtheevacuationtimeonagivennetwork.Toourknowledge,thispaperpresentsthefirstmacroscopicapproachesforthesolutionofacontraflownetworkreconfigurationincorporatingroadcapacityconstraints,multiplesources,congestion,andscalability.Weformallydefinethecontraflowproblembasedongraphtheoryandprovideaframeworkofcomputationalstructuretoclassifyourapproaches.AGreedyheuristicisdesignedtoproducehigh-qualitysolutionswithsignificantperformance.ABottleneckReliefheuristicisdevelopedtodealwithlargenumbersofevacuees.Weevaluatetheproposedapproachesbothanalyticallyandexperimentallyusingreal-worlddatasets.Experimentalresultsshowthatourcontraflowapproachescanreducetheevacuationtimeby40percentormore.IndexTerms—Contraflow,optimization,graphalgorithms,heuristicsdesign. Ç FFICIENTevacuationrouteplanningiscurrentlyanissueofmajorimportanceduetotheincreasingrisksfrombothterroristattacksandnaturaldisasters.Fortransporta-tionsystemplanners,themainissuehasbeentheseveretrafficjamsduringtheevacuationprocess.IntheaftermathofHurricanesKatrinaandRitain2005,thetransportationcommunityobservedtheneedforincreasedevacuationroutecapacity,aswellasamoreaccurateestimateoftheevacuationtime[29].Contraflow,orlanereversals,hasbeendiscussedasapotentialremedytosolvesuchatremendouscongestionbyincreasingtheoutboundevacuationroutecapacity[41],[42].Althoughcontraflowisprimarilyimportantforevacuations,itsapplicationsarenotlimitedtoemergencies.OtherexamplesofcontraflowincludethereversalsofthetwocenterlanesofthehighwaysysteminWashington,D.C.,duringmorningandeveningrushhours[7],[32]androadnetworkreconfigurationafterfootballThecontraflowproblemforevacuationcanbedefinedasfollows:Given1)atransportationnetworkwithedges,eachhavingacapacityandatraveltime,and2)sourceanddestinationjunctions,wefindareconfigurednetworkidentifyingtheidealdirectionforeachedgetominimizetheevacuationtimebyreallocatingtheavailablecapacity.Findingtheoptimalcontraflownetworkconfigurationisconsiderablychallengingbecausewemayhavetoenumer-atecombinationsofedge(thatis,roadsegment)directionsandcomparethosecombinationsbycalculatingtheevacua-tiontime.ThetaskisNP-complete,anditsproofisshowninthispaper.Inaddition,ittakesconsiderabletimetoevaluateeachcontraflowconfigurationcandidate,takingthedynamicsoftrafficflowintoaccount.Thus,weneedtoconsiderthebalancebetweenmodelrealismandprohibi-tivecomputingrequirementsengenderedbytheexhaustivesearchspaceandthedemandsoftherealisticmodelingoftrafficflow.Hamza-Lupetal.[22]proposedalgorithmstotacklethecontraflowproblem.Theirapproachisbasedonevacuationmodelingwithasinglesource,thusleadingtofindingtheoptimalpathstodestinationsandoverlayingthem.Thisplanningapproachdoesnottaketheoverallcapacityoftheroadnetworkintoaccount.TuydesandZiliaskopoulosdesignedamesoscopiccontraflownetworkmodel[39]basedonadynamictrafficassignmentmethod.Theirapproachissubjecttotheproblemofmathematicaloptimization,however,andthus,theyhavenotshownscalableexperiments.Inaddition,theirTabu-basedheur-isticapproach[40]isasearch-basediterativeoptimizationtechnique.AlthoughtheTabusearchsignificantlyreducesthesearchspacetobeexplored,thesearchspacemaystillbetoolargeincasesofverylargespatialnetworksduetothecombinatoriallyincreasingnumberofcandidatenet-worksreconfiguredbycontraflow.Toaddressthechallengesofevacuationrouteplanning,weintroducetheparameternamedOverloadDegree,whichclassifiesthecomputationalstructureofthecontra-flowreconfigurationproblembytheratioofthenumberoftravelingunits(forexample,evacuees)tothebottleneckcapacity(thatis,minimumcutingraphtheory)ofthegivennetwork.WeproposeheuristicstodeterminetheidealIEEETRANSACTIONSONKNOWLEDGEANDDATAENGINEERING,VOL.20,NO.8,AUGUST20081 S.KimiswiththeGeodatabaseteam,ESRI,380NewYorkStreet,Redlans,CA92373.E-mail:skim@esri.com.S.ShekhariswiththeUniversityofMinnesota,4-192,EE/CSBldg.,200UnionStreetSE,Minneapolis,MN55455.E-mail:shekhar@cs.umn.edu.M.MiniswiththeElectricalEngineeringandComputerScienceDepartment,SouthDakotaUniversity,133AdministrationBldg.,Brookings,SD57007.E-mail:manki.min@sdstate.edu.Manuscriptreceived1May2006;revised2Feb.2007;accepted23Oct.2007;publishedonline6Nov.2007.Forinformationonobtainingreprintsofthisarticle,pleasesende-mailto:tkde@computer.org,andreferenceIEEECSLogNumberTKDESI-0220-0406.DigitalObjectIdentifierno.10.1109/TKDE.2007.190722.1041-4347/08/$25.002008IEEEPublishedbytheIEEEComputerSociety directionofedgesintransportationnetworksforevacua-tion.TheGreedyheuristicrunsanevacuationrouteplannertodeterminetheconditionofcongestiononagivenoriginalconfigurationandflipshighlycongestedroadsegmentsinagreedymanner.TheBottleneckReliefheuristicidentifiesthebottleneckofagivennetworkandincreasesthebottleneckcapacitybycontraflow.Weevaluatedourapproachesusinganalyticalandexperimentalvalidationmethodologies.Intheexperimentalevaluation,wepreparedreal-worlddatasetstotesttheperformanceandscalabilityoftheapproaches.Experimen-talresultsandcasestudiesshowthattheproposedapproachescanreducetheevacuationtimeby40percentormore.Inaddition,wepresentfindingswithimportantimplicationsforplannersandfirstrespondersastheypreparecontraflowevacuationschemes.1.1MotivationEvacuationrouteplanninghasbecomeatopicofcriticalimportanceduetotheSeptember11terroristattacksandrecentcatastrophichurricanesthatrequiredlarge-scaleevacuationsacrosstheUS.In2005,twomajorhurricanes,KatrinaandRita,hitthesoutheasternpartoftheUSandcausedseveredamageacrossseveralcoastalstates[36].EspeciallyduringtheRitaevacuation,agreaternumberofevacueesthanexpectedfollowedtheevacuationorderwiththeirpersonalvehicles.Thefollowingarequotedobserva-tions[29]ofthetrafficproblemsthatoccurredduringtheRitaevacuation:Congestionproblem.“AnestimatedthreemillionpeopleevacuatedtheTexascoast,creatingcolossal,100-mile-longtrafficjamsthatleftmanystrandedandoutoffuel.DriversheedingthecalltoevacuateGalvestonIslandandotherlow-lyingareastookfourtofivehourstocoverthe50milestoHouston,andfromthereroadwayconditionswereevenworse,withtrafficcrawlingatjustafewmilesperhour....Aftercrawlingonly10or20milesinninehours,somedriversturnedaroundtotaketheirchancesathomeratherthanriskbeingcaughtintheopenwhenthehurricanestruck.”Contraflowproblem.“High-occupancyvehicle(HOV)laneswentunused,asdidmanyinboundlanesofhighways,becauseauthoritiesinexplicablywaiteduntillateThursdaytoopensomeup....Ascongestionworsened,stateofficialsannouncedthatcontraflowlaneswouldbeestablishedonInterstateHighway45(Fig.1b),USHighway290,andInterstateHighway10.Butbymidafternoon,withtrafficimmobileonUSHighway290,theplanwasdropped,strandingmanyandpromptingotherstoreversecourse.“Weneedthatroutesoresourcescanstillgetintothecity,”explainedanagencyspokeswoman.”DuringtheRitaevacuation,transportationanalysts[29]wereabletoobservetheinefficientuseofroadcapacityandtheeffectsfromtheill-plannedcontraflow,whichresultedindisorganizedmovementofpeople.TheylistedthefailuretousecontraflowlanesandroadshouldersforevacuationtrafficasoneoftheplanningproblemlessonslearnedfromKatrinaandRita.Althoughitisasubjectofrecentdramaticinterest,contraflowhasotherroutinebutimportantapplications.Oneapplicationistheuseofreversiblelanestodealwithmorningandeveningpeakcommutertime.WashingtonStatehasbeenoperatingreversibletwo-laneroadwaysforpeakperiodHOV-3vehicles[7],[32].Thereversiblelanesystemhasbeenreportedtoprovidesignificantsavingsintraveltime.Asecondapplicationofcontraflowiscommonduringspecialeventswhenalllanesarereversedtoaccommodateoutboundtrafficattheendofasportingeventorconcert.Thisisaspecialcaseofcontraflow,havingasinglesourcewithmultipledestinations.1.2RelatedWorkandOurContributionThematerialandliteratureonevacuationsingeneralandthecontraflowprobleminparticularhavebeenpublishedinvariousdomainsincludingsocialandbehavioralsciences,transportation,andmathematics[11],[12].Asurvey[42]ofevacuationissuesandcontraflowrevealedthattransportationplannershavenorecognizedstandardsorguidelinesforthedesign,operation,andlocationofcontraflowsegments.Manystatesthreatenedbyhurricanesandconsideringcontraflowplansaredependentonpastevacuationexperiences.Litman[29]identifiedtheplanningproblemsofHurricanesKatrinaandRitaandspecificallycriticizedunplannedcontraflowordersandthefailuretousecontraflowlanes.2IEEETRANSACTIONSONKNOWLEDGEANDDATAENGINEERING,VOL.20,NO.8,AUGUST2008 Fig.1.Beforecontraflow,thenorthboundlanesofInterstateHighway45inHoustonwerejammedduringtheHurricaneRitaevacuation.Aftercontraflow,thenorthboundtrafficissmootheronbothsidesofInterstateHighway45.(Source:wfaa.com).(a)Beforecontraflow.(b)Aftercontraf PastpapersandDepartmentofTransportationreports[15],[37],[41],[42]havemainlytackledthemanagerialandoperationalaspectsofcontraflow,suchassignalcontrol,merging,andcost.Whenplannersdesignanetworkconfigurationforevacuationscenarios,theymainlydependonempiricalguesses.Suchhandcraftedcontraflowplanshaverevealedthattheyareneitherefficientforfindingcriticalroadsegmentsofcontraflownorflexibleforaccommodatingvariousvariables[13],[14].Hamza-Lupetal.[22]introducedtwodifferentcontra-flowalgorithmsfromacomputerscienceperspective:onebasedonamulticastroutingproblemandtheotherbasedonbreadth-firstgraphtraversal.Thesealgorithmscanhandleonlyasinglecoordinatedincidentduetotheconflictsofmultipleoptimalpathsthatoccurinmultiple-sourceandmultiple-destinationevacuationmodels.Theauthorsdidnotclearlydescribetheuseofdifferentlinkcapacities.Thus,theirapproachisnoteffectivewhenthenumberoftravelingunitsisfinite,roadcapacitiesareconstrained,specificdestinationnodesareprescribed,orevacueesarespreadovermanylocations.TuydesandZiliaskopoulos[39]proposedamesoscopiccontraflownetworkmodelbasedonadynamictrafficassignmentmethod.Theyformulatedcapacityreversibilityusingamathematicalprogrammingmethod.Thediscre-tizedhypotheticalnetworkrequiredtosolvethetrafficassignmentproblem,however,hinderedlarge-scalenet-workscenariosfromrunningintheirframework.TheyalsotriedaTabu-basedheuristicapproach[40]toaddresscapacityreversibilityoptimization.Theirsolutionsrequiredaconsiderablenumberofiterations,thuslimitingtheirinputtoasmallnetwork.TheodoulouandWolshon[38]usedCORSIMmicro-scopictrafficsimulationtomodelthefreewaycontraflowevacuationaroundNewOrleans.Withthehelpofthemicroscaletrafficsimulator,theywereabletosuggestalternativecontraflowconfigurationsatadetailedlevel.However,themicroscopicsimulationmodelrequireslabor-intensivenetworkcodingandsignificantruntimeforeachscenario,makingitdifficulttotakeadvantageofspatialdatabasesoreasilycomparealternativeconfigurations.Evacuationrouteplanningwithothermicroscopictrafficsimulation(forexample,MITSIMLab[25])hasshownsimilarlimitations.Ourcontributions.Previously,KimandShekhar[28]proposedtwoheuristicsforcontraflowplanning.Oneheuristic,namedFlipHighFlowEdge(FHFE),isbasedonagreedyalgorithmwithaflowhistoryofedges.TheFHFEgeneratesasuboptimalcontraflowplanwithoutiteration.Theotherheuristicisbasedonasimulatedannealingoptimizationtechnique.Duetothesearchingproperty(thatis,globaloptimization)ofsimulatedannealing,itcangenerateslightlybettersolutionsthanFHFE,althoughthegainfromthesimulatedannealingmethodisrelativelysmalldespiteitslongruntimebyiterativesearch.Inthispaper,wepresentcapacity-awareglobalcontra-flowheuristicsthataredesignedtohandlemultiplesourceanddestinationnodes.WeclassifythecontraflowproblemusingOverloadDegreeandpresenttwoheuristics.OurGreedyheuristicisdesignedtohandlescenarioswithasignificantlylargepopulationandnetworksize.Italsohasaflexiblealgorithmstructurebyusinganevacuationrouteplannerasaplug-inmodule,thusleavingroomforimprovementswithfasterevacuationrouteplanners.TheotherproposedheuristicisaBottleneckReliefheuristic,whichtacklestheinherentcongestionproblemofcontra-flowbyidentifyingbottlenecksinthenetworkusingaminimumcut.Analyticalandexperimentalevaluationsareprovidedtovalidatetheusefulnessoftheproposedapproaches.Experi-mentalresultsshowthatlessthan30percentofthetotaledgesforcontraflowisenoughtoreduceevacuationtimebymorethan40percent.Wealsoprovideacomparisonofsolutionqualitybetweentheproposedheuristicsandintegerprogramming(IP)(optimalcontraflownetwork1.3ScopeandOutlineofthePaperOurevacuationmodelingisbasedongraphtheorywithflowanalysisonamacroscopicflowmodel.Ourmodelingdoesnotincludethesocialbehavioroftheevacuees,theoperationalcost/policyofcontraflow,ortrafficsignals.Ourfocusistodesignscalablecontraflowheuristicstoaddresslarge-scaletransportationnetworksandaccuratelycomparetheperformancebetweenagivennetworkandacontraflow-reconfigurednetworkwithinourcomputationalframework.Therestofthepaperisorganizedasfollows:Section2presentsthemodelingandhardnessoftheevacuationproblem.Section3providesacomputationalframeworkofthecontraflowproblemandpresentsourproposedap-proachestosolvingthecontraflowproblem.InSection4,wedescribedesigndecisionsandpresenttheiranalyticalevaluations.Section5givestheexperimentalsetupandevaluationoftheapproaches.Finally,Section6summarizesandconcludeswithadiscussionoffuturework.ODELINGAND2.1ProblemFormulationEvacuationisasituationwhereresidentsinadangerousareaareremovedtosafeplacesasquicklyaspossible.Manyapproacheshavebeenproposedtomodeltheevacuationsituationusingmicroscopicsimulation[8],[3],[31],[34],mesoscopicmodels[39],andmacroscopicnetworkflowmodels[2],[10],[21].Microscopicsimulationmodelstrafficflowatasingle-vehiclelevel.Thebehaviorofindividualdriversisundertheinfluenceofvehiclesintheirproximity.Thismodelisusuallyaccompaniedbycar-followingmodels.Mesoscopicsimulationmodelstrafficflowbygroupsoftrafficentities.Vehiclesarenotdescribedindividuallybutinmoreaggregatetermsusingprobability.Macroscopicmodelsdescribetrafficflowatahighlevelofaggregation.Althoughresearchershavedebatedthesuit-abilityofthesevariousapproachesfordescribingtrafficflow,manyfavormacroscopicmodelingduetotheincreasedpublicattention,improvedtechniques,andthecomputationalcapacitythisapproachoffers[24].Inthisresearch,weselectamacroscopicnetworkflowmodelusingamathematicalgraphtorepresenttheevacuationsituation.Themovementofevacueesisrepre-sentedasaflowonthegraph.Althoughamacroscopicmodellosesthepropertiesofsingle-vehicletrafficflow(forexample,congestionpropagation),itisstillapowerfultoolforevacuationplanningbecauseiteffectivelydealswiththeroaddensity,theweightedmeanspeed,and,mostim-portantly,thecapacityofagiventransportationnetwork. KIMETAL.:CONTRAFLOWTRANSPORTATIONNETWORKRECONFIGURATIONFOREVACUATIONROUTEPLANNING Thus,macroscopicmodelsprovideevacuationplannerswiththemeanstoevaluatesystemwideevacuationstrate-gies.Inaddition,macroscopicmodelsaremoresuitablethanmicroscopicmodelsforlarge-sizenetworksduetotheirscalability.Inmacroscopicevacuationmodels,itisnecessarytorepresentthesituationwithamathematicalgraphstructure.LetG(N,E)beadirectednetwork,whereNisthesetofnodes,andEisthesetofedges.Eachnodehasaninitialoccupancyvalue—thatis,thenumberofresidentstoevacuate—andanodecapacity.Theuseofnodecapacitydependsontheevacuationmodel.Forexample,abuildingevacuationmaymodelaroomasanode.Inthiscase,roomsizeisanodecapacity.Ifanodeismodeledasanintersectioninatransportationnetwork,thenodecapacitycanbesettoinfinity.Inthecaseoflimitednodecapacity,suchasasmallstairwellinabuildingevacuationoratollplazainatransportationnetwork,suchanodewillbeachokingjunctionandnegativelyaffecttheedgeflowaroundthenode.Eachedgealsohasanedgecapacity,atraveltime,andaninitialdirection.Theedgecapacityisdefinedasthenumberoftravelingunits(forexample,vehiclesorpedestrians)pergivenunitperiod.Forexample,ahighwayedgesegmentmayhaveacapacityof1,800vehiclesperhourunderanormaloperation.Theevacuationsituationhasmultiplesourcenodesanddestinationnodes.Evacuationtimeisdefinedastheperiodfromthemomentwhenthefirstevacueeleavesasourcenodetothemomentwhenthelastevacueearrivesatadestinationnode.Itisworthwhiletonotethattherearetwodifferentwaysofmodelingedgecapacityinmacroscopicmethodology.Thefirstmethodcanbecalled“continuousentering.”Inthisapproach,itisassumedthattheevacueesequivalenttotheedgecapacitykeepenteringanedgeeveryunitoftimeaslongastheedgeisavailable.Thesecondmethodcanbecalled“occupyandempty.”Evacueesequivalenttotheedgecapacityoccupytheedgefortheedgetraveltime.Duringtheoccupyingperiod,theedgeisnotavailabletootherevacueesstayingata“from”node.Inourevacuationmodel,wechoose“continuousentering”tomodeltheedgecapacitybecauseitisamorerealisticrepresentationoftheevacuees’movement.Withthegivennetworksetup,wewanttofindanetworkreconfiguredbycontraflowwiththeobjectiveofminimiz-ingtheevacuationtime.Therearetwoconstraints.First,thecapacityneedstobeconstant.Ourproblemformulationdoesnotincorporatethedynamicnatureofnetworkproperties(forexample,bridgecollapseinthemiddleofanevacuationprocedure).Ontheotherhand,thetraveltimeofanedgecanbeeitherconstantorchangeable,dependingonthecharacteristicsoftheevacuationrouteplanner.Second,partialreversalisnotallowed.Thisconstraintwillkeeptheproblemsizeatareasonablelevel.Inourmodeling,wefollowatypicalnetworkmodel,whichiseasilypersistedinrelationaltables,takingfutureenhancementsintoaccount[23].Thefollowingisaformalsummaryofourcontraflowproblem:1.Atransportationnetwork,adirectedgraphG(N,E).2.Eachnodehasinitialoccupancyandcapacity.3.Eachdirectededgehasacapacity,atraveltime,andaninitialdirection.4.Sourceanddestinationnodes.Acontraflownetworkconfiguration(thatis,desireddirectionforeachedge).Minimizeevacuationtime.1.Capacityisconstant.2.Partialreversal(forexample,partialnumberoflanes)isnotallowed.Fig.2illustratesasimpleevacuationsituationbasedontheproblemformulation.NodesAandCaremodeledassourcenodes,whereasnodeEisadestinationnode(forexample,evacuationshelter).NodesBandDhavenoinitialoccupancyandonlyserveastransshipmentnodes.Theunittime(forexample,minute)isdefinedaccordingtothemodelscale.TheevacuationtimeoftheoriginalnetworkinFig.2ais22(detailsofhowtomeasureevacuationtimearediscussedinSection4.2).Figs.2band2cillustratetwopossiblecontraflowconfigurations.Allthetwo-wayedgesusedintheoriginalconfigurationaremergedbycapacityanddirectedinfavorofincreasingtheoutboundevacuationcapacity.TherearetwocandidateconfigurationsthatdifferinthedirectionofedgesbetweennodesBandD.The4IEEETRANSACTIONSONKNOWLEDGEANDDATAENGINEERING,VOL.20,NO.8,AUGUST2008 Fig.2.Graphrepresentationofasimpleevacuationsituationandtwofollowingcontraflowconfigurationcandidates.(a)Graphrepresentationofaevacuationsituation.(b)Contraflowconfiguration1.(c)Contraflowconfiguration2. networkinFig.2breducestheevacuationtimeto11(thatis,50percentoftheoriginalevacuationtime),whereasthenetworkinFig.2creducesevacuationtimeto14(64percent).Thisexampleillustratestheimportanceofchoiceamongpossiblecontraflownetworkconfigurations.Moreover,wehavetoknowthattherearecriticaledgesaffectingtheevacuationtimesuchasedge(B-D)inFig.2.2.2ProofofNP-CompletenessInthissection,weprovetheNP-completenessofthecontraflowproblem.Ingeneral,theprocessofdevisinganNP-completenessproofforadecisionproblemconsistsofthefollowingfoursteps[17]:showingthatisinNP,selectingaknownNP-completeproblemconstructingatransformation,andprovingthatisa(polynomial)transformation.Toconductourproof,weselectthe3-SATISFIABILITY(3SAT)problemasourknownNP-completeproblem.ThisproblemisconsideredtherootofmostotherNP-completeproblemsandisderivedfromtheSATISFIABILITYpro-blemwhoseNP-completenesswasprovenbyCook[17].The3SATproblemisspecifiedasfollows:ofclausesonafinitesetUofvariablessuchthatIsthereatruthassignmentforUthatsatisfiesalltheclausesinC?usedinthefollowingdefinitionisapolynomialfunctionthatcancalculatetheevacuationtimeofagivengraph.Forsimplicity,eachedgeinanundirectedgraphGcouldbeflippedineitherdirection.Anundirectedgraphwithinitialdenotesthepositiveintegers)forsome,destinationverticesforsome,capacityandtraveltimeforeach,adirectedgraph,andevacuationtimeIsthereafunction!½fisadirectededgeinsuchthatEVACTIMELemma1.CONTRAFLOWisNP-complete.Itiseasytoseethat,sinceanondeterministicalgorithmneedonlyguessanewdirectedgraphbyflippingalledgesrandomlyandcheckinpolynomialtimethathasevacuationtimeBorless.Wetransform3SATtoCONTRAFLOW.Letbeanyinstanceof3SAT.WemustconstructagraphandsetapositiveintegerBsuchthathasevacuationtimeBorlessifandonlyifCissatisfiable.Theconstructionconsistsofasourcecomponent,adestinationcomponent,andaflippingcomponentbetweenthesourceanddestinationcomponents.Thesourcecomponentconsistsofvertices.Thedestinationcomponentconsistsoftwolayers.ThefirstlayerconsistsofeachliteralandtheirnegatedliteralsinU(thatis, u1;u2; u2;un; ).Thesecondlayerconsistsoftheofeachpairofliterals(thatis, u1;u2 u2;un ).ThisXORlayerservesasadestinationnodesetintheCONTRAFLOWproblem.Thetwonodesinapair( )inthefirstlayerareconnectedtoeachnode( )inthesecondlayerwithedges,eachofwhichhas.Finally,aflippingcomponentconsistsofedgeswiththefollowingdefinition:Foreachclause,letthethreeliteralsinbedenotedby,and.Then,theedgesare,eachofwhichhas.Fig.3showsanexampleofthecontraflowgraphobtainedwhen¼ff u3; u4g;f u1;u2; Itiseasytoseehowtheconstructioncanbeaccomplishedinpolynomialtime.AllthatremainstobeshownisthatCissatisfiableifandonlyifEVACTIMEbyflippingedgesinGtoprovethattheaboveconstructionisindeedatransformation.:SupposethatCissatisfiable.WedefinetheifvisTRUEorifvisFALSE(thatis,drawanarrowheadontheTRUEnodeandanarrowtailontheFALSEnode).WeassumethatBisequaltothenumberofsourcenodes.IfCissatisfiable,atleastoneedgefromeachsourcenodewillbedirectedtowardthedestinationcomponent.Thisguaranteesthatoneoccupantineachsourcenodecanevacuatetothedestinationnodes(secondlayerinthedestinationcomponent)withatmostBevacuationtime.Theworst-caseevacuationtimeBhappenswhenallthesourcenodesarepointedtoonenodeinthefirstlayerofthedestinationcomponent.:SupposethatEVACTIMEbyusingthesameflippingfunctiondescribedabove.Foreachoccupantineachsourcenodetoevacuatetoadestinationnode,atleastoneedgefromthesourcenodeshouldbedirectedtowardthefirstlayerofthedestinationcomponent.ThisguaranteesthatCissatisfiable.RAMEWORKAND3.1ComputationalFrameworkandApproachOverviewInthissection,weintroducethecomputationalstructureofthecontraflowproblemusingOverloadDegreeandpresentappropriateapproachesaccordingtoeachworkloadzone. KIMETAL.:CONTRAFLOWTRANSPORTATIONNETWORKRECONFIGURATIONFOREVACUATIONROUTEPLANNING Fig.3.CONTRAFLOWinstanceresultingfrom3SATinstanceinwhich¼ff u3; u4g;f u1;u2; u4gg. Aswillbeshown,OverloadDegreeisakeydeterminantoftheoverallevacuationtimeandneedforcontraflow.Overloaddegree.Duetothecombinatorialnatureofthecontraflowproblem,acquiringtheoptimalsolutionbe-comesconsiderablychallengingasthesizeofthenetworkincreases.Toaddressthechallengesinproblemsize,weneedtodefineaparametertoclassifythecomputationalstructureoftheproblem.Thenumberoftravelingunitsandbottleneckcapacityofagivennetworkaretwocriticalfactorsaffectingthecomputationalstructure.Theevacuees’movementisanalogoustotheflowmovementthroughabottleneckofagourdbottle.Iftheamountofflowislargeorbottlenecksizeissmall,ittakesalongtimetofinishtheflowmovement.Withthisanalogy,wedefinetheterm“OverloadDegree”asfollows:Definition1(OverloadDegree).OverloadDegree=NumberofTravelingUnits/BottleneckCapacityWithoutContraflow.“BottleneckCapacityWithoutContraflow”referstoaminimumcutvalue(ormaximumflowvalue)ofagivennetworkwithoutcontraflow.Inthecalculationofthebottleneckcapacity,nodecapacityisnotconsidered.ThehardnessofthecontraflowproblemisafunctionoftheOverloadDegree.ForasmallOverloadDegree,wecanconsidermathematicalprogramming,search-basedap-proaches,ormicroscopicsimulationtoproduceoptimalresults.Forexample,supposethatthereare500evacueesonanetworkwhoseminimumcutis100.TheOverloadDegreeisonly5.Insuchacase,thenetworkhasenoughbottleneckcapacitytoevacuatethe500evacuees.Thus,thecomputa-tionalworkloadisrelativelylow.ForamediumOverloadDegree,wedefinitelyneedheuristicapproachestoachieveabalancebetweentheresultqualityandthereasonablecomputationalworkload.ForalargeOverloadDegree,weneedamorecomputationallyefficientapproach.Fig.4isthesameexampleevacuationnetworkasinFig.2butwithdifferentinitialoccupancy.Thedottedlineisabottleneckofthisnetwork,separatingthesourcenodesandthedestinationnodes.Thevalueofthebottleneckcapacityisfoundtobe3byaddingthecapacityofedge(B-E)(thatis,fromBtoE)and(D-E).SupposenodeAhasoccupancy2andChas1.Wedonotneedcontraflowbecausethecurrentbottleneckcapacityisenoughtohandlethesmallnumberoftravelingunits(thatis,evacuees).Inthiscase,theOverloadDegreeis1.Asthenumberofinitialoccupancyincreases(for),thecurrentbottleneckcapacitybecomesinsufficient.Westartthinkingofcontraflowtoreducetheevacuationtime.Thecomputationalworkloadaccordinglybecomesheavytocalculatetheschedulingofthelargenumberoftravelingunits.SupposenodesAandChave2,000and1,000evacueeseach.Then,thesituationbecomesclosetoaninfinitesourceproblemaswecanignorethetransitionalstartingandendingperiodsofevacuation.Asshowninthisexample,OverloadDegreeisacriticalparameterfordeterminingtheproblemsizeanditscon-gruentsolutionforagivennetwork.Intheabsenceofoverload(forexample,OverloadDegreeislessthan1),contraflowoffersfewornobenefitsbecausetheoriginalnetworkhasenoughcapacityforthecurrentevacueestopassthrough.IftheOverloadDegreeissmall(forexample,aone-digitnumber),itiscomputation-allyfeasibletoidentify“optimal”contraflowconfigurationsbyusingoptimizationtechniquessuchasmathematicalprogramming,search-basedoptimization,ormicroscopicsimulation.OurIPformulationbelongstothesmallOver-loadDegreecategory.ResultsfromtheIPformulationareusefultoassessthequalityofsolutionsobtainedbyourIftheOverloadDegreeismedium,wehavetoconsiderheuristicsduetotheheavycomputationalworkload.Atthislevelofworkload,itisimpossibletouseaniterativelearningprocess,whichisonlyfeasibleinasmallOverloadDegree.Wesuggestanoniterativeheuristicbasedonagreedyapproach.Last,iftheOverloadDegreeislarge,itisclosetothecasewherethenetworkhasaninfinitesourceofevacuees.Here,itisnecessarytosimplifytheevacuationmodelingtoaddresssuchaheavycomputationalworkload.Wehavedesignedaminimumcutandmaximumflow[16]basedBottleneckReliefheuristicthatignorestheamountofthepopulationconstraint.Useofanevacuationrouteplanner.Theroleofanevacuationrouteplannerinourframeworkistocalculatetheflowhistoryandevacuationtimeofagivennetwork.Theflowhistoryofanedgeisequivalenttothetotalnumberoftravelingunitsthatpassthroughtheedgeduringanevacuationtime.Ourcontraflowsolutionframeworkseparatestheevacuationrouteplannerfromthecontraflowreconfigurationalgorithm.Thus,theevacuationrouteplannertobepluggedincantakeeitheramicroscopicoramacroscopicsimulationapproachaslongastheplannerconformstotheinput/outputrules.Fig.5showshowtheevacuationrouteplannersfunctionwithinourproposedframework.Theinputtothesystemisanoriginalevacuationnetworkwithpredefinedsource/destinationnodesandedgeswithcapacityandtraveltime.Therearethreealgorithmiccomponents:IP,aGreedyheuristic,andaBottleneckReliefheuristic.FortheIPapproach,theevacuationrouteplanneriscombinedwithamathematicaloptimizertoevaluatethenetworksgeneratedbyiterativeenumerationandserveasanobjectivefunction.TheGreedyheuristicusestheflowhistoryoftheoriginalnetworkasinputandproducesareconfiguredcontraflownetwork.TheBottleneckReliefheuristicusestheoriginalnetworkasinputanddirectlyproducesareconfiguredcontraflownetwork.Thecostofrunninganevacuationrouteplannerincreaseswiththesizeofthenetwork.Thus,howtheevacuationrouteplannerisusediscriticalintheframe-work.TheGreedyheuristicusestheevacuationroute6IEEETRANSACTIONSONKNOWLEDGEANDDATAENGINEERING,VOL.20,NO.8,AUGUST2008 Fig.4.Simpleevacuationnetworkwithbottleneckcapacity3. planneroncetogenerateacontraflownetwork.TheBottle-neckReliefheuristicdoesnotuseanevacuationrouteplanner.Ontheotherhand,theIPapproachusestheevacuationrouteplanneriteratively.3.2GreedyHeuristicThebasicassumptionoftheGreedyheuristicisthatwhenwerunanevacuationrouteplanneroveranoriginalnetworkconfigurationwithoutcontraflow,theedgeshav-ingmorecongestionhistoryaremoreinfluentialinthedecisionofedgeflips.Therefore,itisnecessarytoquantifythecongestionhistoryoneachedgewiththedatafromtheevacuationrouteplanner.WedefinetheFlowHistoryCongestionIndexofanedgeinthefollowingway:Definition2(flowhistory).FlowHistory=TotalnumberoftravelingunitsgoingthroughedgeEvacuationTimeDefinition3(congestionindex).CongestionIndexFlowHistoryCapacityEvacuationTimeFlowHistoryisacquiredfromtheresultoftheevacua-tionrouteplanner.ThedenominatorinDefinition3referstothemaximallypossibleamountofflowofedgeEvacuationTime.Thus,CongestionIndexindicatesthepercentageofedgeutilizationduringEvacuationTimeCongestionIndexvaluemeansthattheedgebeenmorecongestedduringtheevacuationprocess.Thethirddefinitionusedinthegreedyapproachisthe“DegreeofContraflow.”WecandefinetheDegreeofContraflowinthereconfigurednetworkasfollows:Definition4(DegreeofContraflow).DegreeofContra-flow(DoC)=NumberofFlippedEdges/TotalNumberofThispercentageparameterindicateshowmanyedgesareflippedamongalledgesinthereconfigurednetwork.OurGreedyheuristichastheabilitytocontrolthisparameter,whichisimportantinthecontextofevacuationbecauseunnecessaryflipsleadtothewasteofresources.Thatis,moreemergencyprofessionalsareneededasthenumberofreversedroadsegmentsincreases.Inaddition,theunflippededges(thatis,in-boundroadsegments)canbeusedascapacityforincomingemergencyvehicles(forexample,ambulancesandfiretrucks).Algorithm1.1:runanevacuationrouteplannertoproduceFlowHistoryEvacuationTimeforallCongestionIndexFlowHistoryCapacityEvacuationTimeendfor5:sortedgesbyCongestionIndexindescendingorder;reconfiguredforalli;jinthefirstedgesinthesortededgesreconfigured:flipi;jendforreconfiguredTheGreedyalgorithmshowninAlgorithm1worksinthefollowingway:First,werunanyevacuationrouteplannertogeneratetheflowhistoryandevacuationtimeofagivenoriginalnetwork.Second,weassignacongestionindexvaluetoeachedge.Third,theedgesaresortedbycongestionindexindescendingorder.Finally,weflipedgesinfavorofahighercongestionindexvalueamongthefirstofthesortededgeset.Theevacuationrouteplannermustberunagainoverthereconfigurednetworktogettheevacuationtimeofthereconfigurednetwork.Thisnoniterativealgorithmstructuremayresultinanetworkdisconnectionproblembecausethealgorithmsuggestsareversalthatdisconnectstwosubnetworks.Wecanshowthatsuchadisconnectionproblemdoesnothappen.Supposethatwehavetwonetworks,andtheyareconnectedbytwobidirectionaledges,)and(source)and(source).Assumetothecontraryisdisconnectedbyreversingtheedge.Thedisconnectednetworkmeansthatthereisatleastoneroutegeneratedbytheevacuationrouteplanner.Sucharoutedoesnotexistbecausedoesnothaveanydestination,contradictingourassumption.Fig.6showsaseriesofstepsusingtheGreedyalgorithmtogenerateacontraflownetworkfromagivenoriginalnetwork.WeassumethatthegivenDegreeofContraflowis60percent.Thenetworkinstep1isthegivenoriginalnetwork.Ifwerunanevacuationrouteplanneronthenetwork,weacquiretheflowhistory,aswellastheevacuationtime.Anoptimalrouteplannerproducesevacuationtime22.Thenetworkinstep2showstheflowhistoryvalue.Forexample,thevalue17overedgemeansthat17evacueespassthroughtheedgeduringevacuationtime22.Instep3,thecongestionindexvaluesaregeneratedfromtheinforma-tionofstep2usingtheformulationinthecongestionindexdefinition.Thecongestionindexvaluesaresorted KIMETAL.:CONTRAFLOWTRANSPORTATIONNETWORKRECONFIGURATIONFOREVACUATIONROUTEPLANNING Fig.5.Useofanevacuationrouteplannerinthecontraflowsolution indescendingorder,andthefirst60percentofthem(underlinededges)aregreedilyselected,asshowninTable1.Eachselectededgeiscomparedwithitsoppositeedge,andtheoppositeedgeisflippediftheselectededgewins.Thefinalreconfigurednetworkisshowninstep4,aftertheflippingprocessisfinished.Theflowonedgecanbegeneratedinstep2becausesomeamountofflowoscillatesbetweenthetwonodes.Thismaynothappeninanactualevacuationscenariobutmayhappeninaflowgraph.Theoscillationdoesnotaffectthefinalevacuationtime.Thefinaldecisionbetweennodesisedgebecausethedirectionshowsmorecongestion,asseeninstep3inFig.6.3.3BottleneckReliefHeuristicforaLargeOverloadTheBottleneckReliefapproachstartsfromthewell-knowntheorembyFordandFulkerson[16],whichstatesthat“Thevalueofthemax-flowinacapacitatednetworkisequaltothevalueofthemin-cut.”Inthecontextoftransportationnetworks,themin-cutisabottleneckorchokecapacity.Theideabehindthisapproachistoidentifythebottleneckandincreaseitscapacitybycontraflow.Algorithm2.maxflow�maxflow2:find3:flipedgesacrosstowarddestination;maxflowmaxflowmaxflowendwhileIfthegivengraphhasmultiplesourcesandmultipledestinations,wehavetoplaceasupersourceconnectingtothesourceswithinfinitecapacityandasuperdestinationconnectingtothedestinationwithinfinitecapacitybeforethealgorithmBottleneckReliefisapplied.ThealgorithmBottleneckRelief,showninAlgorithm2,findsamin-cutofthegivengraphandflipsedgesacrossthemin-cut.Then,thelocationofthemin-cutwillchange.Thealgorithmkeepsfindingthemin-cutuntilthemaximumflowdoesnotimprove.ThisalgorithmissuitableforanetworkhavingalargeOverloadDegreebecausethemaximumflowisbasedontheinfiniteflowfromsourcestosinks.Evacuationscenariosoverheavilycrowdedareasandreversiblehigh-waysystemsforspecifiedperiodsoftimeareexamplestowhichwecanapplythisalgorithm.Supposethattheoriginalnetworkhasnumberofoccupancy,edges.Aproposedrandomizedalgorithm[26]findsaminimumcutwithhighprobabilityin.Intheworstcase,ourBottleneckReliefheuristicrunswhichleadstoruntime.Thenetworkdiscon-nectionproblemdoesnothappenbytheBottleneckReliefheuristic.TheproofisthesameasfortheGreedyheuristic.Fig.7illustratestheapplicationoftheBottleneckReliefheuristictooursimplegraph.NodesAandCarestillsourcenodes,whereasnodeEisadestinationnode.Thesourcenodesareconnectedfromasupersourceasshowninstep1.Themin-cut(ormax-flow)intheoriginalgraphisrepresentedasadottedlineinstep1andhasvalue3.Instep2,weflipedgesacrossthefirstmin-cutinfavorofincreasingcapacitytothedestination.Then,thepreviousmin-cutisnolongeramin-cutduetoitsincreasedcapacitybycontraflow.Asecondmin-cutisalsoshownasadottedlineinstep2.Wecontinuethesestepsuntilthemax-flowdoesnotincrease.Step4showsthefinalnetworkreconfiguration.8IEEETRANSACTIONSONKNOWLEDGEANDDATAENGINEERING,VOL.20,NO.8,AUGUST2008 Fig.6.ExampleoftheGreedyalgorithm.TABLE1SortedCongestionIndexesfromStep3inFig.6 Fig.7.ExampleoftheBottleneckReliefheuristic. 3.4IntegerProgrammingFormulationTheIPapproachproducesanoptimalcontraflowplanthatcanminimizetheevacuationtime.TheIPapproachisusefulincomparingthesolutionqualitybetweentheproposedheuristicsandoptimalplan.Duetolimitedspace,weintroduceonlythemostimportantformulationsusedintheIPexperiment.Table2showsselectedformulationsoftheIPapproach.Equation(1)inTable2definestheobjectivefunctionsuchisthetotalamountoftimetofinishtheevacuation,assumingthatislargeenough.Ifissettolessthantheminimumvalue,thentheformulationbecomesinfeasible.Equation(2)describestheflowconservationconstraints,meaningthatinflowequalsoutflow.Equa-tion(3)isacontraflowconstraint,anditrestrictstheselectionofcontraflowasfollows:Whenthereisonlyoneedgebetweentwonodes,wehaveonlytwooptions:normalfloworcontraflow.Whentherearetwoedgesbetweentwonodes,wehavethreeoptions:twonormalflowsoronecontraflow.Wedonotconsiderthecaseofreversingthetwoedgesatthesametime.Equation(4)isusedtoensuretheproperallowedamountofflowoneachedgebasedonthevalueoftheedgecapacity.ECISIONSAND4.1OverloadDegreeandResultQualityInthissection,weexplaintherelationshipbetweentheOverloadDegreeandtheresultqualityoftheproposedapproaches.Wecanclassifythequalityofresultsintotwolevels:optimalandheuristic.Anoptimalresultmeansthattheevacuationtimeisminimal.Optimalresultsareobtainedfromahugenumberofcombinatorialnetworkcandidates.Theheavycomputationalloadfromcombina-torialoptimizationlimitstheIPapproachtocasesofsmallOverloadDegree,asshowninFig.8.Theprohibitivecomputationalworkloadofachievingoptimalresultsledustoexploreeffectiveheuristics.WedesignedtheGreedyheuristictomeettheneedsinmedium-overloadscenarios.AsdetailedinSection5,theresultqualityoftheGreedyheuristiccomparesreasonablywellwiththeoptimalresult.TheBottleneckReliefheuristicistailoredtoaddresscasesoflargeOverloadDegree.4.2ChoiceofRoutePlannerInthecontraflowcomputationalframework,anevacuationrouteplannerplaysanimportantroleinbothestimatingtheevacuationtimeofagivennetworkandprovidingtheinformationofthetotalnumberoftravelingunitspassingthrougheachedgeinthenetwork.Theestimatedevacua-tiontimeisusedtomeasurethequalityofthenetworkreconfiguredbycontraflow.Whenweselectanevacuationrouteplanner,thereisatrade-offbetweentheresultqualityandruntime.Anoptimalevacuationrouteplannercangenerateanoptimalevacuationtimebyperformingthefollowingthreesteps:creatingatime-expandednetwork,applyingaminimum-costflowalgorithm,andextractinganevacuationtime.Theexistingminimum-costflowsolvers(forexample,NETFLO[27],RELAX[6],RNET[20],andCostScaling(CS)[18])arealloptimalevacuationrouteplanners.Themajordrawbacksofoptimalevacuationrouteplannersaretheirpoorscalabilityandtherequirementofpriorknowledgeoftheupperboundoftheevacuationtime.Theselinearmethodshaveanexponentialruntimeproportionaltoagivennetworksize.Aheuristicevacuationrouteplanner,bycontrast,avoidstheseissues,oftenproducingaclose-to-optimalevacuationtimewithgoodscalability.TheCCRP[30]algorithmistheonlyheuristicevacuationrouteplanneravailableinthisdomain.Thealgorithmdividesevacueesfromeachsourceintomultiplegroupsandassignsarouteandtimeschedule KIMETAL.:CONTRAFLOWTRANSPORTATIONNETWORKRECONFIGURATIONFOREVACUATIONROUTEPLANNING TABLE2ContraflowIPFormulation Fig.8.Dominancezoneoftheproposedapproaches. toeachgroupbasedonitsdestinationarrivaltime.Intermsoftrafficassignment,CCRPisneithersystemoptimalnoruserequilibrium.However,CCRPdoesuseacloseapprox-imationofasystem-optimalapproach.Theheuristicgivesprioritytotheroutewiththeearliestdestinationarrivaltime.EventhoughCCRPdoesnotproduceoptimalsolutionsinallevacuationscenarios,experimentresultsshowthatmostsolutionsarelessthan10percentlongerthantheoptimalevacuationtime.Inaddition,theplannerdoesnotrequirethepreprocessingofagivennetworkortheupperboundoftheevacuationtime.Thefollowingsectionspresentananalyticalevaluationofevacuationrouteplanners.Weusethefollowingnotationstodescribetheoriginalnetwork:numberoftravelingunits,vertices,and4.2.1OptimalRoutePlanner,RELAX[6]RELAXisasoftwarecodetosolvetheclassicminimumcostflowproblemwithintegerdata.Itisbasedontherelaxationmethod,whichisdesignedtosolvesimultaneousequationsbyguessingasolutionandreducingtheerrorsthatresultbysuccessiveapproximationsuntiltheerrorsarelessthansomespecifiedamount.Bertsekas,whocreatedthemini-mumflowsolverRELAX,notedthatthereisnoknownpolynomialcomplexityboundfortherelaxationmethod[5].Thus,wehavetodependonexperimentalevaluationtomeasuretheperformanceoftheRELAXevacuationrouteplannerincombinationwiththeGreedyheuristic.4.2.2OptimalRoutePlanner,CostScaling[18]TheCSminimumcostflowalgorithmcombinestheideasofcostscaling,thepush-relabelmaximumflowmethod,andtherelaxationmethod.Goldbergincorporatedseveralheuristics(forexample,priceupdate,pricerefinement,arcfixing,andpushlook-ahead)toimprovethepracticalperformanceoftheCSalgorithm.However,wewillusetheasymptoticworst-casetimeboundouranalysis,whichisnotaffectedbytheheuristics(biggestcost).Asdescribedpreviously,anoptimalevacuationrouteplannerrunsthreestepstogenerateanevacuationplan.First,itgeneratesatime-expandedgraphfromagivennetwork.Second,aminimumcostflowmethodisappliedonthetime-expandedgraph.Third,thepostprocessingoftheflowhistoryresultretrievestheevacuationtime.Thesecondstepisdominantintermsofruntime.isthetime-expandedgraphbuiltfromtheoriginalnetworkwithupperbound.Theupperboundnumberofnodes,andtheupperboundnumberofedgesis,wherenotestraveltimeofedgei;j[21].Ifweassume,asisgenerallytrue,thatthetransportationnetworkissparse,withanaveragedegreeofvertices3,wecanassumethat.Wecanalsoassumethatthemaximumevacua-tiontimeisproportionaltotheoccupancyvalueisproportionalto,andisalsoproportionalwithoutlossofgenerality.ThetimeboundofCSis.Ifwecombineourassumptionswiththeupperbound,wecanacquirethefollowingÞÞ¼.Thatis,thecombinationoftheGreedyheuristicwithCSrunssuperlinearlyproportionaltothenumberofnodesandtravelingunits.4.2.3HeuristicRoutePlanner,CCRPThealgebraiccostmodelofCCRPispresentedin[30].TheCCRPevacuationrouteplannerusesaniterativeapproach.Ateachiteration,therouteforonegroupofpeopleischosen,andthecapacitiesalongtheroutearereserved.Thetotalnumberofiterationsisequivalenttothenumberofgroupsgenerated.ThecomputationofroutesforeachgroupisperformedbyrunningthegeneralizedDijkstra’sshortestpathsearch.Theimplementationfollowingdoublebucketdatastructuresleadstoanalgebraiccostmodelof,whereisthemaximumedgeweight.Thatis,thecombinationoftheGreedyheuristicwithCCRPrunslinearlyproportionaltothenumberofnodesandevacuees.Lemma1.TheGreedyheuristicwithaheuristicevacuationrouteplannerisfasterthantheGreedyheuristicwithanoptimalrouteevacuationplanner.IntheGreedyalgorithm,step1runstheevacuationrouteplanneronetimeandisdominantintermsofruntime.Thus,adirectcomparisonbetweentheoptimalandheuristicevacuationrouteplannerruntimescanprovethelemma.Theoptimalevacuationrouteplanner(CS)runsinandtheheuristicevacua-tionrouteplanner(CCRP)runsin.Bycomparingthetworuntimes,wecanconcludethattheGreedyalgorithmwithaheuristicevacuationrouteplannerisfasterthanthatwithanoptimalevacuationrouteplannerwhenthe relationholds,whichisalwaystrueintransportationnetworks.Lemma2.TheBottleneckReliefheuristicisfasterthantheGreedyonewithCCRPif TheBottleneckReliefheuristicrunsinTheruntimeofCCRPis.Bycompar-ingthetworuntimeswiththeassumptionofasparsetransportationnetwork,wecanconcludethattheBottleneckReliefheuristicisfasterthantheGreedyonewithCCRPif .Wecanverifytheformulausingthemetropolitanscenariousedinourexperimentswithatwo-mileradiuszone,whichhas269,635occupancy,562nodes,andedges.Thisdatasetfitsthesparsenetwork.Iftheparametervaluesarepluggedin,wecanobservethattheformulasatisfiesthefollowing: 5.1ExperimentSetupWeimplementedandevaluatedthealgorithmsusingreal-worlddatasets.ThelanguageusedwasC++,andtheexperimentswereperformedonadual-CPUPentiumIII650-MHzworkstationwith2Gbytesofmemoryrunning5.1.1SystemDesignWeimplementedtheIPformulationonCPLEX,amathe-maticalprogrammingoptimizer.CPLEXisawell-knowncommercialoptimizationtooltosolveIP.TheGreedyheuristicusesanevacuationrouteplannerasaplug-in10IEEETRANSACTIONSONKNOWLEDGEANDDATAENGINEERING,VOL.20,NO.8,AUGUST2008 externalmodule.Thetwocommunicateviatextfilestoexchangeevacuationtimeandflowhistoryinformation.Thisimplementationframeworkgivesaflexiblestructuretoaccommodatenewevacuationrouteplannersorfutureenhancementsofexistingplanners.Evaluatingthereconfi-guredcontraflownetworkisoptionalandthereforenotincludedinmeasuringtheperformanceoftheapproaches.InthecaseoftheBottleneckReliefheuristic,theoriginalnetworkisdirectlyusedasaninputbecausetheheuristicrequiresonlythecapacityinformationofagivennetwork.5.1.2DataSetDescriptionWepreparedtwodatasets.ThefirstisavirtualscenarioofanuclearpowerplantfailureinMonticello,Minnesota.Thereare12citiesdirectlyaffectedbythefailurewithin10milesofthefacilityandonedestinationshelter.Thisscenarioisaspecialtypeofevacuationhavingasingledestinationbecauseallevacueesshouldhavearadioactiveclearancecheckatthedesignatedfacility.ThedemographicdataisbasedonCensus2000populationdata.Thetotalnumberofevacueesisabout42,000.Ifthegivensituationisrepresentedasagraph,itcontains47nodeswith148edges.Thegraphstructureisbasedonlargeedgegranularitywithaninterstatehighway(I-94)andmajorarterialroads.Thus,thesizeofthenetworkisrelativelysmall.Theinterstatehighwayhasalargercapacitythanotheredges.Theevacuationtimewiththeoriginalnetworkconfigurationis272minutes(4hoursand32minutes).Wecreatedthissmall-sizecasefile(intermsofnumberofnodes)tocomparethequalityofourheuristicswiththeoptimalnetworkconfigurationfromIP.Theseconddatasetwaspreparedwithourevacuationscenariogenerationsoftware.Thesoftwarehasthecap-abilityofspecifyingthesizeoftheevacuationzone,adjustingtheamountofpopulation,changingthemodeofevacuationbetweendrivingandwalking,andgloballyadjustingthecapacityofedges.Withthesefunctionalities,wewereabletogeneratescenarioswithvarioussizesaroundtheMinneapolis-SaintPaul,Minnesota,metropoli-tanarea.Thedatausedinthesoftwareisgivenasfollows:Mapdata.ThisconsistsofTP+(planningpurpose)andMinnesotaDepartmentofTransportationbase-mapdata(detailedgeometryrepresentation).TheTP+containsroadtype,roadcapacity,traveltime,numberoflanes,etc.Italsocontainsvirtualnodesaspopulationcentroidsforeachtrafficanalysiszone.Populationdata.ThisconsistsofCensus2000data(nighttimeestimation)andemploymentdata(day-timeestimation)butnotincludingtravelers(forexample,shoppers).Weselectedthreedifferentlocations,asmandatedbytheDepartmentofHomelandSecurity,withthreedifferentnetworksizes(thatis,half-,one-,andtwo-mileradii).Forsecurityreasons,thespecificnamesofthelocationshavebeenremovedinthispaper.Theprimarypurposeoftheseconddatasetistocomparetheresultsfromheuristicapproachesandtotestthescalabilityduetotherelativelylargernetworksizecomparedtothefirstdataset.5.2OverloadDegreeandResultQualityAsexplainedpreviously,theOverloadDegreeisanimportantparameterinclassifyingthecomputationalstructureofthecontraflowproblemandtheproposedheuristicsaccordingtothedegreeofcomputationalwork-load.Inthissection,weexaminetherelationshipbetweentheOverloadDegreeandotherfactorssuchastheevacuationtimeandruntimeofourheuristicsusingtheMonticellodataset.Fig.9ashowstheeffectsoftheOverloadDegreeonevacuationtime.Weperformedthistestbychangingthenumberoftravelingunitsoversourcenodes.WecanobservethattheevacuationtimeislinearlyproportionaltotheOverloadDegreeforallmethods.Mostheuristicsshowedareductionofevacuationtimebyabout30percentregardlessoftheOverloadDegree.TheGreedyheuristicwithanoptimalevacuationrouteplanner(RE-LAX)alwaysshowedminimumevacuationtime.ThecombinationoftheGreedyheuristicwithaheuristicevacuationrouteplanner(CCRP)placedinthemiddle.TheBottleneckReliefheuristicalsoshowedcomparableresultqualitywiththeGreedyheuristic.Fig.9bshowstherelationshipbetweentheOverloadDegreeandruntime.TheGreedyheuristicwithRELAX(optimalevacuationrouteplanner)hasasteeply,perhapssuperlinearly,increasingruntime.GreedywithCS(optimalevacuationrouteplanner)showedaremarkablyfasterruntimethanGreedywithRELAXastheOverloadDegreeincreases.However,GreedywithCCRP(heuristicevacua-tionrouteplanner)wasthefastest(almost-zeroruntime,alongwiththeBottleneckReliefheuristicinFig.9b)amongthecombinationsofGreedyheuristicswithvariousevacua-tionrouteplanners.Theseresultsindicatethattheselectionofevacuationrouteplannerisacriticaldesigndecisionforscalability.TheBottleneckReliefheuristichadaconstant KIMETAL.:CONTRAFLOWTRANSPORTATIONNETWORKRECONFIGURATIONFOREVACUATIONROUTEPLANNING Fig.9.Evacuationtime/runtimewithregardtotheOverloadDegreeusingtheMonticelloscenario. runtimebecauseitdidnotinvolveoccupancydata(thatis,thenumberofevacuees)aspartoftheinput.Fig.10ashowsthequalityofGreedyheuristicsbycomparingtheresultsfromIP.First,Greedyheuristics,regardlessofevacuationrouteplanner,showedabouta40percentreductionofevacuationtime.Greedywithanoptimalevacuationrouteplanner(RELAXorCS)resultedinonlyaslightlybetterevacuationtimethanthatwiththeheuristicevacuationrouteplanner(CCRP).Second,weobservethatagap(14minutes)existsbetweenGreedyheuristicsandoptimalresults.Fig.10bshowsaruntimecomparison.IPresultedinamuchhigherruntime(205sec-onds)becausetheIPformulationtook130,109iterationstoproduceanoptimalcontraflownetwork,whereastheGreedyheuristicstookonlyoneiteration.5.3ChoiceofRoutePlannerandScalabilityFig.11showstheconvergencepatternwithregardtotheDegreeofContraflowusingRELAXandCCRP.AlthoughGreedywithRELAXalwaysproducedbetterresultsthanthatwithCCRP,CCRPprovidedasimilarresultqualityasRELAX,showingonlya4-minutegapinevacuationtime(RELAX:170minutes,CCRP:174minutes).BothplannersalsoshowedsimilarconvergencepatternswithregardtotheDegreeofContraflow.IntheMonticellocase,lessthan10percentofthetotaledgescontributetotheconstantlyreducedevacuationtime.Wealsoperformedexperimentsonmetropolitanscenar-iostoexaminetheconvergencepatternsoftheDegreeofContraflow.Mostevacuationtimesconvergedwithin30percentDegreeofContraflow.Thismeansthatthelimitedresourcesrequiredtoimplementcontraflow,suchasbarricadetrucksandpolicecars,canbeeffectivelydispatchedtotheappropriatelocationsbasedontheDegreeofContraflowparameter.ThemaximumgapinevacuationtimeobservedbetweenRELAXandCCRPwas32minutes,andtheminimumgapwas0minutes.Ontheaverage,themetropolitandatasetsshoweda45percentreductioninevacuationtimebycontraflowfromtheoriginaltothereconfigurednetwork.Fig.12showsthescalabilityoftheGreedyheuristicwithdifferentevacuationrouteplannersusingmetropolitanscenarios.TheevacuationrouteplannerRELAXshowedasteepruntimeincrease.TheevacuationrouteplannerCSshowedbetterscalabilityeventhoughitproducesthesameresultqualityasRELAX.Asshowninthegraph,CCRPprovidedthebestperformancescalabilitywithregardtothenetworksize.Nowadays,evacuationatthemetropolitanscaleisoftentheissueofinterest.Insuchcases,CCRPwillplayanimportantroleinscalingourapproachestotacklinghugenetworks.5.4MonticelloScenarioResultsandImplicationsforPlanningInthissection,wedescribetwofindingsfromtheMonticelloscenariothathaveespeciallyimportantimplica-tionsforevacuationrouteplanning.Thefirstfindingistheefficiencyofthecomputerizedevacuationrouteplanning.Fig.13comparesahandcraftedplanwithroutessuggestedbytransportationanalystsoftheDepartmentofTransporta-tionandaplangeneratedbytheheuristicCCRPevacuationrouteplannerandtheGreedyheuristic.Thehandcraftedversion(Fig.13a)resultsinanevacuationtimewithoutcontraflowthatistwiceaslong(554minutes)asthatgeneratedbyCCRP(276minutes)(Fig.13b).Themainreasonforthereductioninevacuationtimeachievedbytherouteplannerisitsabilitytocorrectlyselectthedirectionofedges,aswellasitsextensiveuseofvariousroutesaroundthedestination.Asecondfindingistheefficiencyofthecomputerizedcontraflowreconfiguration.OnthenetworkshowninFig.13c,weobservethat10percentoftheedgesarechosenforcontraflowbytheGreedyheuristic.Theresultingreconfigurednetworkcanfurtherreducetheevacuationtimeto180minutes,whichis32percentofthetimerequiredbytheoriginalhandcraftedversion.The10percentDegreeofContraflowismeaningfulinthatwecanapplylimitedresourcestothemostcongestedroadsegmentforcontraflowandreservetheremainingcapacityforincomingemergencytraffic.Inthecontextoftransportationplanning,mostedgesselectedforcontraflowinourexperimentscorrespondtomajorhighwayswithlargecapacityandlocalarterialroadsaroundthedestination.Thisselectionscheme12IEEETRANSACTIONSONKNOWLEDGEANDDATAENGINEERING,VOL.20,NO.8,AUGUST2008 Fig.10.ResultqualityandruntimecomparisonbetweentheGreedyheuristicandIPformulationusingtheMonticelloscenario. Fig.11.ConvergencepatternoftheevacuationtimewithregardtotheDegreeofContraflowusingtheMonticelloscenario. willhelpplannerstoidentifyandrefinemoreefficientroutesforcontraflow.ONCLUSIONANDCurrentevacuationproceduresdependheavilyontheuseofsurfacetrafficthroughthelimitedcapacityofroadnetworks.Fromthisperspective,contraflowmustbeseenasoneofthekeyelementsinanyevacuationplannedontheexistingtransportationinfrastructure.Takingintoaccountthenatureoftransportationnetworks,wemodeledandanalyzedevacuationsituationsusinggraphtheory.Inourmodel,oneormoresourcenodescanbeadded,whereasexistingalgorithmsonlycoversituationswithasinglesourceduetoconflictsofoptimalpathsfromdifferentsourcenodes.Themultiple-sourceandmultiple-destinationcontraflowproblembelongstoacategoryofNP-completeproblems.Ourmaincontributionliesinthefactthatweaddresssuchachallengingcontraflowproblemwithcomputationalstructureanalysisandprovidescalableheuristicswithhigh-qualitysolutions.Wealsopresentedanalyticalandexperimentalevaluations.Thefollowingsummarizesthetwocontraflowheuristicswedeveloped:Greedyheuristic.Thisguaranteesapromisingsolutionqualityinspiteofitsfastruntime.Theevacuationplanningsoftwareneedstobeinteractiveduetovariouscombinationsofinputparametersandevolvingdatasets.Thus,runtimeisacriticalfactorwhenweimplementacomputerizedcontra-flowplanner.Ourwell-designedapproach,basedonaGreedyimplementationthatistailoredtocontra-flowproblems,hassomeadvantagesovergeneraliterativemethods.WithourGreedyheuristic,thenumberofcontraflowededgesisadjustable.Thescalabilityissuperiortothatofmathematicalprogrammingorsimulation-basedapproaches.BottleneckReliefheuristic.Thisissuitableforacontraflowsituationwithlargenumbersofevacuees.AlthoughwewereabletoobservecomparableresultqualitywiththeGreedyapproach,theruntimeoftheBottleneckReliefheuristicisfastestregardlessofthenumberoftravelingunits. KIMETAL.:CONTRAFLOWTRANSPORTATIONNETWORKRECONFIGURATIONFOREVACUATIONROUTEPLANNING Fig.12.Scalabilitywithregardtothenetworksizeusingmetropolitanscenarios.(a)ScenarioA–Runtime.(b)ScenarioB–Runtime.(c)ScenarioC–Runtime. Fig.13.Handcraftedversuscomputerizedplans.(a)Handcraftedplanevacuationtime:554min,100percent.(b)Planbyheuristicrouteplanner(CCRP)evacuationtime:276min,50percent.(c)10percentcontraflowbygreedyheuristicevacuationtime:180min,32percent. Eventhoughacontraflowoperationonurbanarterialroadwaysandlongsectionsofinterstatefreewaysforevacuationsisaccompaniedbycomplicatedissuesofsafety,accessibility,andcost,ourproposedalgorithmsforsimpli-fiedsituationsshouldbeconsiderablyhelpfultoplannersdesigningcontraflowplansbecausetheobjectiveofourresearchistominimizetheevacuationtime,whichisanessentialpartofplanning.Futurework.Morein-depthresearchisrequiredforcontraflowalgorithms.Otherpossiblemethodsshouldbeexaminedsuchasthepossibilityofflippingapathinsteadofanedge.Inaddition,weneedtoexploretheapplicationofqueuingtheory[1],[4],[9]andsearchtechniquesintheartificialintelligencefield[19],[33],[35]tothecontraflowproblem.Wewilluseanevacuationrouteplannerbasedonmicroscopicsimulationtoseehowdetailedcongestionphenomenaaffectthechoiceofedgestobereversed.Inboundtrafficdemandshouldbeconsidered.Networkcapacityshouldbepreemptedforemergencyvehicles,trafficofficers,orfirefighters.Partiallanereversalandtime-dependentcapacity-varyingedgesneedtobeincorporatedinthemodeling.ThequantitativevaluesoftheOverloadDegree—forexample,thepracticalrangeofmediumOver-loadDegree—needtobeestablishedandrefined.Wewilldevelopamorerealisticcongestionindexformulausingthefundamentaldiagrambetweenflow,density,andspeedfrequentlyusedinthetrafficoperationsarea.Finally,thetime-dependentnatureoftrafficflowcanbeaddressedintheevacuationrouteplanner.TheauthorsareparticularlygratefultothemembersoftheSpatialDatabaseResearchGroupattheUniversityofMinnesotafortheirhelpfulcommentsandvaluablediscus-sions.Theauthors’evacuationresearchmembers,includingQingsongLuandBetsyGeorge,gavethemespeciallyvaluablecomments.TheArcGISNetworkAnalystandgeodatabaseteamsatESRIhavegiventheauthorsachancetoimplementtheideasinthispaperontheirGISdevelopmentframework.KimKoffoltandESRIhelpedtheauthorsimprovethereadabilityofthepaper.TheyarethankfulfortheuseoftheprogramRelaxIV,developedbytheCRIFORresearchgroup,andCS,developedbyA.Goldberg.Thisworkwassupportedbyagrant(Contract81655)fromtheMinnesotaDepartmentofTransportation.[1]I.Adan,QueueingTheory.EindhovenUniv.ofTechnology,2001.2001.R.K.Ahuja,T.L.Magnanti,andJ.B.Orlin,NetworkFlows:Theory,Algorithms,andApplications.PrenticeHall,1993.1993.M.Ben-Akiva,“DevelopmentofaDeployableReal-TimeDynamicTrafficAssignmentSystem:DynaMITandDynaMIT-PUser’sGuide,”technicalreport,MassachusettsInst.Technology,2002.2002.A.W.Berger,L.M.Bregman,andY.Kogan,“BottleneckAnalysisinMulticlassClosedQueueingNetworksandItsApplication,”QueueingSystems,vol.31,no.3-4,pp.217-237,1999.1999.D.Bertsekas,PersonalCommunicationbyE-mail,AvailableuponRequest,2006.2006.D.BertsekasandP.Tseng,“Relax-IV:AFasterVersionoftheRelaxCodeforSolvingMinimumCostFlowProblems,”TechnicalReportP-2276,LaboratoryforInformationandDecisionSystems,MassachusettsInst.Technology,Cambridge,Mass.,1994.1994.R.J.CaudillandN.M.Kuo,“DevelopmentofanInteractivePlanningModelforContraflowLaneEvaluation,”TransportationResearchRecord,UrbanTrafficSystems,vol.906,no.7,pp.47-54,47-54,X.Chen,J.Meaker,andF.Zhan,“Agent-BasedModelingandAnalysisofHurricaneEvacuationProceduresfortheFloridaNaturalHazards,vol.38,pp.321-338,2006.2006.W.C.ChengandR.R.Muntz,“OptimalRoutingforClosedQueueingNetworks,”Proc.14thIFIPWG7.3Int’lSymp.ComputerPerformanceModelling,MeasurementandEvaluation(Performancepp.3-17,1990.1990.T.J.CovaandJ.P.Johnson,“ANetworkFlowModelforLane-BasedEvacuationRouting,”TransportationResearchPartA:PolicyandPractice,vol.37,pp.579-604,2003.2003.J.G.DohenyandJ.L.Fraser,“Mobedic—ADecisionModelingToolforEmergencySituations,”ExpertSystemswithApplications,vol.10,pp.17-27,1996.1996.M.Ebihara,A.Ohtsuki,andH.Iwaki,“AModelforSimulatingHumanBehaviorduringEmergencyEvacuationBasedonClassificatoryReasoningandCertaintyValueHandling,”computersinCivilEng.,vol.7,pp.63-71,1992.1992.FederalHighwayAdministration,CatastrophicHurricaneEvacua-tionPlanEvaluation,http://www.onewayflorida.org/,2007.2007.FloridaDepartmentTransportation,StateofFloridaContraflowhttp://www.onewayflorida.org/,2007.2007.G.Ford,R.Henk,andP.Barricklow,“InterstateHighway37ReverseFlowAnalysis—TechnicalMemorandum,”technicalreport,TexasTransportationInst.,2000.2000.L.R.FordandD.R.Fulkerson,FlowsinNetworks.PrincetonUniv.Press,1962.1962.M.GareyandD.Johnson,ComputersandIntractability:AGuidetotheTheoryofNPCompleteness.W.H.Freeman,1979.1979.A.V.Goldberg,“AnEfficientImplementationofaScalingMinimum-CostFlowAlgorithm,”J.Algorithms,vol.22,pp.1-29,1-29,R.Greiner,“ProbabilisticHill-Climbing:TheoryandApplica-Proc.NinthCanadianConf.ArtificialIntelligence,pp.60-67,60-67,M.D.Grigoriadis,“AnEfficientImplementationoftheNetworkSimplexMethod,”Math.ProgrammingStudy,vol.26,pp.83-111,83-111,H.W.HamacherandS.A.Tjandra,“MathematicalModelingofEvacuationProblems:StateoftheArt,”PedestrianandEvacuationpp.227-266,2001.2001.G.Hamza-Lup,K.A.Hua,M.Le,andR.Peng,“EnhancingIntelligentTransportationSystemstoImproveandSupportHomelandSecurity,”Proc.SeventhIEEEInt’lConf.IntelligentTransportationSystems(ITSC),(ITSC),E.Hoel,W.Heng,andD.Honeycutt,“HighPerformanceMultimodalNetwork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“1998PerformanceofRegionalHigh-OccupancyVehicleFacilitiesonFreewaysintheWashingtonRegion,”AnalysisofPersonandVehicleVolumes,MetropolitanWashingtonCouncilofGov’t,1999.1999.S.Nahar,S.Sahni,andE.Shragowitz,“SimulatedAnnealingandCombinatorialOptimization,”Proc.23rdACM/IEEEDesignAuto-mationConf.(DAC’86),pp.293-299,1986.1986.M.Pidd,F.Silva,andR.Eglese,“SimulationModelforEmergencyEuropeanJ.OperationalResearch,vol.90,no.3,pp.413-419,1996.1996.S.RussellandP.Norvig,ArtificialIntelligence:AModernApproach,seconded.PrenticeHall,2003.2003.FederalResponsetoHurricaneKatrina:LessonsLearned,www.whitehouse.gov/reports/katrina-lessons-learned/,TheWhiteHouse,2006.2006.G.Theodoulou,“ContraflowEvacuationontheWestboundI-10outoftheCityofNewOrleans,”master’sthesis,LouisianaStateUniv.,2003.2003.G.TheodoulouandB.Wolshon,“AlternativeMethodstoIncreasetheEffectivenessofFreewayContraflowEvacuation,”tionResearchRecord:J.TransportationResearchBoard,vol.1865,pp.48-56,2004.2004.H.TuydesandA.Ziliaskopoulos,“NetworkRe-DesigntoOptimizeEvacuationContraflow,”TechnicalReport04-4715,Proc.83rdAnn.MeetingoftheTransportationResearchBoard,Board,H.TuydesandA.Ziliaskopoulos,“Tabu-BasedHeuristicforOptimizationofNetworkEvacuationContraflow,”TechnicalReport06-2022,Proc.85thAnn.MeetingoftheTransportationResearchBoard,2006.2006.B.Wolshon,“One-Way-Out:ContraflowFreewayOperationforHurricaneEvacuation,”NaturalHazardsRev.,vol.2,no.3,pp.105-112,2001.2001.B.Wolshon,E.Urbina,andM.Levitan,“NationalReviewofHurricaneEvacuationPlansandPolicies,”technicalreport,HurricaneCenter,LouisianaStateUniv.,BatonRouge,Louisiana,SanghoKimreceivedthemaster’sdegreeandPhDdegree(in2007)incomputersciencefromtheUniversityofMinnesota.HeiscurrentlyasoftwaredeveloperwiththeGeodatabaseteamatESRI.Hisresearchinterestsincludeissuesingeodatabases,spatialnetworkproblems,eva-cuationroutingproblems,optimizations,andgeographicinformationsystems.ShashiShekharisaMcKnightdistinguisheduniversityprofessorattheUniversityofMinne-sota.HeisthecoauthorofapopulartextbookonspatialdatabasesandisservingontheNAS/NRCMappingScienceCommitteeandasacoeditorinchiefofGeoInformatica.HewaselectedanIEEEfellowforcontributionstospatialdatabasestoragemethods,datamining,andgeographicinformationsystems.MankiMinreceivedthePhDdegreeincompu-terandinformationsciencesfromtheUniversityofMinnesotain2004.HeiscurrentlyanassistantprofessorintheElectricalEngineeringandComputerScienceDepartment,SouthDakotaStateUniversity(SDSU).BeforejoiningSDSU,hewasinapostdoctoralresearchassociateattheUniversityofFlorida,workingwithDistinguishedProfessorDr.PanosM.Pardalos.Hiscurrentresearchinterestsincludethedesignandanalysisofoptimizationtechniquesandheuristicswithmainapplicationstovariousnetworktopologies.HeisamemberoftheFormoreinformationonthisoranyothercomputingtopic,pleasevisitourDigitalLibraryatwww.computer.org/publications/dlib. KIMETAL.:CONTRAFLOWTRANSPORTATIONNETWORKRECONFIGURATIONFOREVACUATIONROUTEPLANNING