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Network Models For Disaster Management

Permanent Link: http://ufdc.ufl.edu/UFE0024855/00001

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Title: Network Models For Disaster Management
Physical Description: 1 online resource (179 p.)
Language: english
Creator: Arulselvan, Ashwin
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: connectivity, contraflow, disaster, evacuation, networks, routing
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We propose network models to study applications related to risk control and disaster management. We also perform a detailed complexity analysis and provide solution approaches for these problems. More specifically, we examine three different problems in detail, study their complexity and provide techniques to solve them. The first problem studies the connectivity properties of a node deleted subgraph. The objective is to identify important nodes of a graph responsible for the connectivity of the graph. This problem finds applications in studying robustness of a network, identifying important individuals of a social network and critical nodes of a telecommunication or transportation network. We study the complexity of the problem and develop heuristic procedures to solve the problem. We then study path planning problems that arises in military applications. The problem involves in routing unmanned vehicles in a network in order to visit targets in the most efficient way. We look at the complexity of these problems and provide heuristic solutions to solve the problems. We then provide a comparison of the heuristic solution with the optimal solution. Finally, we examine the evacuation problem that arises in an emergency situation. We provide complexity analysis for the contraflow problem, where in we allow arc reversals. We then develop a branch and price mechanism for establishing optimal evacuation routes for a bimodal multicommodity flow problem. We provide computational results for planar and grid graphs.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ashwin Arulselvan.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Pardalos, Panagote M.
Local: Co-adviser: Elefteriadou, Ageliki L.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-02-28

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024855:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024855/00001

Material Information

Title: Network Models For Disaster Management
Physical Description: 1 online resource (179 p.)
Language: english
Creator: Arulselvan, Ashwin
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: connectivity, contraflow, disaster, evacuation, networks, routing
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We propose network models to study applications related to risk control and disaster management. We also perform a detailed complexity analysis and provide solution approaches for these problems. More specifically, we examine three different problems in detail, study their complexity and provide techniques to solve them. The first problem studies the connectivity properties of a node deleted subgraph. The objective is to identify important nodes of a graph responsible for the connectivity of the graph. This problem finds applications in studying robustness of a network, identifying important individuals of a social network and critical nodes of a telecommunication or transportation network. We study the complexity of the problem and develop heuristic procedures to solve the problem. We then study path planning problems that arises in military applications. The problem involves in routing unmanned vehicles in a network in order to visit targets in the most efficient way. We look at the complexity of these problems and provide heuristic solutions to solve the problems. We then provide a comparison of the heuristic solution with the optimal solution. Finally, we examine the evacuation problem that arises in an emergency situation. We provide complexity analysis for the contraflow problem, where in we allow arc reversals. We then develop a branch and price mechanism for establishing optimal evacuation routes for a bimodal multicommodity flow problem. We provide computational results for planar and grid graphs.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ashwin Arulselvan.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Pardalos, Panagote M.
Local: Co-adviser: Elefteriadou, Ageliki L.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-02-28

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024855:00001


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NETWORKMODELFORDISASTERMANAGEMENT By ASHWINARULSELVAN ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2009 1

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c r 2009AshwinArulselvan 2

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Tomyfamilyandfriends 3

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ACKNOWLEDGMENTS ImustthankmyadvisorDr.PanosM.Pardalosforconstantsup portandencouragement forthelastveyearsofmygraduatestudies.Hehasbeenagoo dguide,friendand colloborator.Hewasresponsibleformytransformationfro mastudenttoaresearcher.He helpedmesustainmyinterestsingraduatestudiesasaconst antsourceofmotivation. Iwouldliketoexpressmygratitudemyco-advisorDr.LilyEl efteriadou.Shehas beenagreatsupportandgavemesomuchleverageinhelpingme nishthedisseration. Shehasalsobeenagoodcolloboratorandgreatsupervisorby beingrexibleandgivingme independenceinresearch. Igreatlyappreciatetheeortsofmycommitteemembers,Dr. JonathanC.Smith, Dr.JosephGeunes,Dr.MyT.Thaifortheirtimeandconstruct ivecriticismsthathelped inimprovingthequalityofdissertation.Iwouldalsoliket othankthemfortheirvaluable insightsthatprovideddirectionstome. Iamgreatlyindebtedtomyparentsandmybrotherfortheirim mensemoralsupport. Thisaccomplishmentwouldnothavebeenpossiblewithoutth em. Iwishtothanktomyfriends:AltannarChinchuluun,Bharath Badrinarayanan, ClaytonCommander,MichaelJ.Hirsch,SteenRebennack,Sr iramSethuraman, RaagavendraHareesh,AnandSrinivasan,VeraTomaino,Bala chandranVaidyanathan, AshishNemani,SuchandanGuha,ManishaGoswami,ErhunKund akcioglu,Petros Xanthopoulos,MehmetOnal,Gu~objortGylfadottir,Arni Jonsson,IngridaRadziukyniene, NikitaBoyko,SibelSonuc,AndyFan,AlexGrassas,AlexeySo rokin,VladimirBoginski, ErikaShort,OlegShyloandQipengZheng.Theyhavebeenmyco mpanyinthelastve yearsandtheywerepresentduringthegoodandbadtimes.The ywerealwaysproudof meandIamoverwhelmedbytheircompanionship. Finally,Iwouldliketothankallmycolloborators:Vladimi rBoginski,Clayton Commander,MichaelJ.Hirsch,AllaKammaradiner,SteenRe bennack,OlegShylo, 4

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MaruccioResende,PetrosXanthopoulos,QipengZhengandPi larMendoza.Theytaught metobeagoodresearcherandhavebeenterricworkingwitht hem. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 9 LISTOFFIGURES .................................... 10 ABSTRACT ........................................ 12 CHAPTER 1INTRODUCTION .................................. 13 2CRITICALNODEDETECTIONPROBLEMS .................. 16 Introduction ...................................... 16 CriticalNodeDetectionProblem .......................... 18 ProblemDenition ............................... 18 ComputationalComplexity ........................... 19 IntegerProgrammingFormulations ...................... 24 HeuristicforDetectingCriticalNodes ..................... 27 ComputationalResults ............................. 30 CardinalityConstrainedProblem .......................... 33 CC-CNPHeuristic ............................... 35 GeneticAlgorithmfortheCC-CNP ...................... 37 ComputationalResults ............................. 39 CC-CNPResults ................................ 40 ConcludingRemarks ................................. 42 3PATHPLANNINGPROBLEMS .......................... 48 TargetVisitationProblem .............................. 48 Introduction ................................... 48 ProblemDescription .............................. 52 Relatedproblems ............................ 52 Targetvisitationproblem .................... 56 GeneticAlgorithm ............................... 61 Evolutionarymechanisms ........................ 63 Localsearch ............................... 65 ComputationalResults ............................. 66 Numericalresults ............................ 68 Time-to-targetplots .......................... 70 CommunicationModelsforaCooperativeNetworkofAutonomo usAgents ... 77 ProblemFormulation .............................. 77 PreviousWork ................................. 80 ContinuousFormulations ............................ 85 6

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Formulation1:AcontinuousanalogofCCPM-D ........... 85 Formulation2:Acontinuousformulationensuringlocation visitations 89 CaseStudies ................................... 93 Conclusion .................................... 99 4REVIEWOFEVACUATIONPROBLEMS .................... 101 Introduction ...................................... 101 OptimizationTechniques ............................... 102 MaximumDynamicFlow ............................ 102 DynamicTracAssignment .......................... 106 Non-DeterministicMethods .......................... 111 SignicantFeaturesinOptimizationTechniques .................. 112 ContraFlows .................................. 112 EvacueeBehavior ................................ 113 DynamicOrigin-DestinationDemands .................... 115 MultimodalTransportation .......................... 117 MiscellaneousFactors .............................. 118 SimulationTechniques ................................ 119 MicroscopicSimulationTechniques ...................... 120 MacroscopicSimulationTechniques ...................... 123 Meso-SimulationTechniques .......................... 124 IntegratedTechniques ............................. 125 SignicantFeaturesinSimulationTechniques ................... 127 ReviewedFeatures ............................... 127 LaneReversalsandTracControl ...................... 128 DynamicDemandEstimation ......................... 129 MiscellaneousFactors .............................. 130 Conclusions ...................................... 131 5NETWORKFLOWPROBLEMSWITHLANEREVERSALS .......... 132 Introduction ...................................... 132 Background ...................................... 133 MaximumStaticContrarowProblems ....................... 135 MaximumDynamicContrarowProblems ...................... 138 SingleSourceandSingleSink ......................... 139 MultipleSourcesandMultipleSinks ..................... 143 DTCFis NP -completeinthestrongsense ............... 144 WhatmakesDTCFsotoughtosolve? ................ 147 ContrarowProblemswithArcSwitchingCost ................... 149 Conclusions ...................................... 152 7

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6MULTIMODALSOLUTIONSFOREVACUATIONPROBLEMS ........ 153 Introduction ...................................... 153 MultimodalProblem ................................. 153 ProblemDenition ............................... 154 AssumptionsandRealization ......................... 154 FormulationandDiscussion ............................. 155 BranchandPriceMechanism ............................ 156 RestrictedMasterProblem(RMP) ...................... 156 People-PathSubproblem ............................ 157 Car-PathSubproblem ............................. 158 BranchingStrategy ............................... 158 ComputationalResults ................................ 160 Conclusion ....................................... 161 7CONCLUSIONS ................................... 162 REFERENCES ....................................... 164 BIOGRAPHICALSKETCH ................................ 179 8

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LISTOFTABLES Table page 2-1ResultsofIPmodelandheuristiconterroristnetworkda tafromKrebs. ..... 44 2-2ResultsofIPmodelandheuristiconrandomlygenerateds calefreegraphs. ... 45 2-3ResultsofIPmodelandheuristicsonterroristnetworkd atafrom[1]. ...... 45 2-4ResultsoftheIPmodelandgeneticalgorithmandthecomb inatorialheuristic onrandomlygeneratedscalefreegraphs. ...................... 46 2-5Comparativeresultsofthegeneticalgorithmandthecom binatorialheuristic whentestedonthelargerrandomgraphs.Duetothecomplexit y,wewereunable tocomputethecorrespondingoptimalsolutions. ................. 47 3-1Comparativeresultsoftheoptimalsolutionstothecorr esponding tsp lop and tvp foreachinstance.Theabsolutevalueofthe tsp solutionsarereported. 68 3-2Thecorrespondingobjectivefunctionvaluesofeachoft he lop tsp ,and tvp aregivenforeachinstance.Foreachcolumn,oneoftheobjec tivesisconsidered andtheproblemsolvedtooptimality.Thesolutionoftherem ainingtwoproblems isgivenwhenevaluatedwiththeoptimalfunctionvalue. ............. 69 3-3ParametersusedfortheGAandHGAheuristics. ................. 69 3-4Thistableprovidesthenumericalresultsforasetofran domlygeneratedinstances. Therstcolumnsprovideinformationabouttheinstance.Ne xt,theoptimal solutionandrequiredcomputationtimeislisted.BoththeH GAandthestandard GAwereran250timesoneachinstance,andweprovidethemaxi mum,minimum, andaveragesolutionscomputedbyeachforall250tests.The averagecomputation timerequiredbyeachheuristictocomputethebestsolution isalsolisted. ... 71 6-1Branch&Pricemodeltestedongridgraphs .................... 160 6-2Branch&Pricemodeltestedonplanargraphswithonebusc ommodity ..... 161 9

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LISTOFFIGURES Figure page 2-1Heuristicfordetectingcriticalnodes. ........................ 27 2-2Localsearchalgorithmforcriticalnodeheuristic. ................. 29 2-3Heuristicwithlocalsearchfordetectingcriticalnode s. .............. 30 2-4TerroristnetworkcompiledbyKrebs. ........................ 31 2-5Optimalsolutionwhen k =20. ........................... 32 2-6ConnectivityIndexofnodesA,B,C,Dis3.ConnectivityI ndexofE,F,Gis2. ConnectivityIndexofHis0. ............................ 33 2-7Heuristicforthe cardinalityconstrainedcriticalnodeproblem ... 36 2-8Pseudo-codeforagenericgeneticalgorithm. .................... 37 2-9Anexampleofthecrossoveroperation.Inthiscase, CrossProb =0 : 65. ..... 38 2-10TerroristnetworkcompiledbyKrebs[1]. ...................... 40 2-11Optimalsolutionwhen L =4. ............................ 41 3-1Thisexamplecomparestheoptimalsolutionforthe tvp instancewiththerelated tsp and lop solutions. ................................ 50 3-2Pseudo-codeforgenericgeneticalgorithm. ..................... 62 3-3Anexampleofthecrossoveroperation.Inthiscase, CrossProb =0 : 65. ..... 64 3-4Graphicalrepresentationofgenerationalevolution. ................ 64 3-52-exchangelocalsearch. ............................... 65 3-6Time-to-TargetplotcomparingtheHybridGAandstandar dGAforinstance rand12-1 .Thetargetvalueistheoptimalsolutionfortheproblem. ....... 74 3-7Time-to-Targetplotforinstance rand14-2 .Asabove,thetargetvaluesisthe optimalsolution. ................................... 75 3-8Time-to-Targetplotforinstances rand16-1 .Thetargetvalueis : 95timesthe optimalsolution. ................................... 76 3-9Pseudo-codefortheshortest-pathconstructionheuris tic. ............. 81 3-10Pseudo-codefortheHillClimbingintensicationproc edure. ........... 82 3-11Pseudo-codefortheone-passheuristic. ....................... 83 10

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3-12GRASPformaximization. .............................. 83 3-13TheHeavisidefunction, H 1 ............................. 86 3-14 H 2 ,continuousapproximationto H 1 ........................ 87 3-15 H 3 ,continuouslydierentiableapproximationof H 1 ............... 88 3-16Examplewith5agents. ................................ 93 3-17Examplewith7agents. ................................ 93 3-18Examplewith10agents. ............................... 94 3-19Examplederivedfrom10agentexample,withoneagentre moved. ........ 95 3-20Examplederivedfrom10agentexample,withtwoagentsr emoved. ....... 96 3-21Examplederivedfrom10agentexample,withthreeagent sremoved. ...... 97 3-22Examplederivedfrom10agentexample,withfouragents removed. ....... 97 3-23Examplederivedfrom10agentexample,withveagentsr emoved. ....... 98 3-24Examplederivedfrom10agentexample,withsixagentsr emoved. ........ 98 5-1Transformedgraph G 3 SAT correspondingto3SATinstance ............ 145 5-2AtoughinstanceofDTCF .............................. 148 5-3InstanceforDTCFwithtimebound T =2 L +2resultingfromPARTITION .. 149 11

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy NETWORKMODELFORDISASTERMANAGEMENT By AshwinArulselvan August2009 Chair:PanosM.PardalosMajor:IndustrialandSystemsEngineering Weproposenetworkmodelstostudyapplicationsrelatedtor iskcontrolanddisaster management.Wealsoperformadetailedcomplexityanalysis andprovidesolution approachesfortheseproblems.Morespecically,weexamin ethreedierentproblemsin detail,studytheircomplexityandprovidetechniquestoso lvethem.Therstproblem studiestheconnectivitypropertiesofanodedeletedsubgr aph.Theobjectiveistoidentify importantnodesofagraphresponsiblefortheconnectivity ofthegraph.Thisproblem ndsapplicationsinstudyingrobustnessofanetwork,iden tifyingimportantindividuals ofasocialnetworkandcriticalnodesofatelecommunicatio nortransportationnetwork. Westudythecomplexityoftheproblemanddevelopheuristic procedurestosolvethe problem.Wethenstudypathplanningproblemsthatarisesin militaryapplications.The probleminvolvesinroutingunmannedvehiclesinanetworki nordertovisittargetsin themostecientway.Welookatthecomplexityoftheseprobl emsandprovideheuristic solutionstosolvetheproblems.Wethenprovideacompariso noftheheuristicsolution withtheoptimalsolution.Finally,weexaminetheevacuati onproblemthatarisesinan emergencysituation.Weprovidecomplexityanalysisforth econtrarowproblem,where inweallowarcreversals.Wethendevelopabranchandpricem echanismforestablishing optimalevacuationroutesforabimodalmulticommodityrow problem.Weprovide computationalresultsforplanarandgridgraphs. 12

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CHAPTER1 INTRODUCTION Theneedforanalysisofdisastermanagementisnecessaryin manyrealworld applications.Manystudiesinthisareaunderdierentname ssuchasnetworksurvivability, resiliencestudies,vulnerabilityanalysisisalreadykno wn.Themanagementofdisasters couldeitherbeproactive,whereweprepareourselvesbette rforadisasterorareactive, whereweplantocopeupwiththehazardscausedbythedisaste r.Inthisdissertation, wepresentsomeproactivestrategiesinchapters 2 and 3 andsomereactivestrategiesin chapters 5 and 6 .Chapter 4 dealswithasurveyofevacuationstudiesasthesubsequent chapters 5 and 6 primarilydealwithemergencymanagement. Identifyingcriticalnodesinagraphisimportanttounders tandthestructural characteristicsandtheconnectivitypropertiesofthenet work.Inchapter 2 ,wefocus ondetectingcriticalnodes,ornodeswhosedeletionresult sintheminimumpair-wise connectivityamongtheremainingnodes.Thisproblem,know nasthe CRITICAL NODEPROBLEM ,hasapplicationsinseveraleldsincludingtransportati on, biomedicine,telecommunications,andmilitarystrategic planning.Weshowthatthe decisionversionoftheproblemis NP -completeandderiveamathematicalformulation basedonintegerlinearprogramming.Inaddition,wepropos eaheuristicforthe problemwhichisthenenhancedbytheapplicationofalocali mprovementmethod.A computationalstudyispresentedinwhichweapplytheinteg erprogrammingformulation andtheheuristictorealandrandomlygenerateddatasets.F orallinstancestested, theheuristicisabletoecientlyprovideoptimalsolution sinafractionofthetime requiredbyacommercialsoftwarepackage.Wealsostudyava riationofthisproblem, whichinvolvesinminimizingthenumberofnodesdeletedbyc onstrainingthesizeofthe connectedcomponentsinthenode-deletedsubgraph. Inchapter 3 ,weconsidertheproblemofdetermininganoptimalpathfora n unmannedaerialvehiclewhichneedstovisitmultipletarge ts.Theobjectiveisto 13

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minimizethetraveldistancewhilemaximizingtheutilityo fthevisitationorder.This isknownasthe TARGETVISITATIONPROBLEM andhasseveralapplications includingcombatsearchandrescue,environmentalassessm ent,anddisasterrelief. First,weprovideamathematicalmodelbasedonintegerline arprogrammingandprove thattheproblemis NP -complete.Thenwedescribetheimplementationofagenetic algorithmforndingapproximatesolutions.Theheuristic isthenhybridizedbythe implementationofalocalsearchprocedure.Numericalresu ltsarepresenteddemonstrating theeectivenessoftheproposedprocedure.Wealsoconside rtheproblemofmaximizing thetotalconnectivityforasetofwirelessagentsinamobil eadhocnetwork.Thatis, givenasetofwirelessunitseachhavingastartpointandade stinationpoint,ourgoal istodetermineasetofroutesfortheunitswhichmaximizest heoverallconnectiontime betweenthem.Knownasthe cooperativecommunicationprobleminmobile adhocnetworks(ccpm) ,thisproblemhasseveralmilitaryapplicationsincluding coordinationofrescuegroups,pathplanningforunmanneda irvehicles,andgeographical explorationandtargetrecognition.The ccpm is NP -hard;thereforeheuristicdevelopment hasbeenthemajorfocusofresearch.Inthiswork,wesurveyt he ccpm examiningrst someearlycombinatorialformulationsandsolutiontechni ques.Thenweintroducenew continuousformulationsandcomparetheresultsofseveral casestudies.Byremovingthe underlyinggraphstructure,weareabletocreateamorereal isticmodeloftheproblemas supportedbythenumericalevidence. Overtheyears,investigatorsfromvariouseldshavestudi edtheevacuationproblem, whichresultedinamultitudeofevacuationmodels.Inchapt er 4 ,weconsolidatethiswork andmakeanassayoftheexistingmodelsavailableinthelite rature.Themethodology orapproachesemployedinthesemodelswerebroadlyclassi edaseitheroptimization orsimulationmethodsandpresented.Further,weanalyzeth efeaturesconsideredby eithertechniquesbyidentifyingthemajorfactorsinruenc ingtheevacuationeciencyand briengtheshortcomingsinthesetechniques. 14

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Chapter 5 providesacomprehensivestudyonnetworkrowproblemswith arcreversal capabilities.Theproblemistoidentifythearcstoberever sedinordertoachievea maximumrowfromsource(s)tosink(s).Theproblemndsitsa pplicationsinemergency transportationmanagement,wherethelanesofaroadnetwor kcouldbereversedtoenable rowintheoppositedirection.Wepresentseveralnetworkro wproblemswiththearc reversalcapabilityanddiscusstheircomplexityinthisch apter.Morespecically,we discussthepolynomialtimealgorithmsforthemaximumdyna microwproblemwith arcreversalcapabilityhavingasinglesourceandasingles ink,andforthemaximum (static)rowproblem.Thepresentedalgorithmsarebasedon graphtransformationsand reductionstopolynomiallysolvablerowproblems.Inaddit ion,weshowthatthequickest transshipmentproblemwitharcreversalcapabilityandthe problemofminimizingthe totalcostresultingfromarcswitchingcostsare NP -hard. Finallyinchapter 6 ,weprovidesolutionstobimodaltransportationproblemin an emergencysituation,whichincorporatesmulticommodityr ows.Thisproblemissimilar toalineplanningproblem.Wehavetwomodesoftransportati on,namelyprivatecars andbuses.Weassumethatthedemandsateverynodeforbothca rsandbusesareknown andtheasetoffeasibletoursofbuseshasbeenestablished. Weneedtodeterminethe ecientpathsandoptimalsetofbusesroutestosatisfythed emand.Weprovideapath basedformulation,whichenablesustoemployabranchandpr iceapproachtosolvethe problem.Wediscusstheassumptions,formulations,subpro blemsandbranchingstrategies. Thecomputationalresultsforsomegridandplanargraphsar eprovidedforthebranch andpricemechanism. 15

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CHAPTER2 CRITICALNODEDETECTIONPROBLEMS 2.1Introduction Inthischapter,westudytwovariantsofthe criticalnodeproblem .Ingeneral, theobjectiveofthe criticalnodeproblem(cnp) istondasetof k nodesina graphwhosedeletionresultsinthemaximumnetworkfragmen tation.Bythiswemean, maximizethenumberofcomponentsinthe k -vertexdeletedsubgraph.Studiescarriedout inthislineincludethosebyBavelas[ 2 ]andFreeman[ 3 ]whichemphasizenodecentrality andprestige,bothofwhichareusuallyfunctionsofanodesd egree.However,theylacked applicationstoproblemswhichemphasizednetworkfragmen tationandconnectivity. Givenagraphandaninteger k ,theobjectiveofthe criticalnodeproblem (cnp) istondasetof k nodesinthegraphwhosedeletionresultsinthemaximum networkfragmentation.Bythiswemean,minimizethepair-w iseconnectivitybetween thenodesinthethe k -vertexdeletedsubgraph.Inthesecondversionoftheprobl em,we seektheminimumnumberofnodeswhosedeletionconstrainst hecardinalityofconnected component. The criticalnodeproblems hasseveralapplicationsintheeldofsocialnetwork analysis.Socialnetworkshaveattractedasignicantamou ntofattentioninrecent years.Thestudyofthesegraphsisimportanttobetterunder standseveralproperties whicharemostcommoninnetworkdepictionsofsocialintera ctionsincludingcohesion, transitivity,andcentralityofspecicactorsofthegraph [ 4 ].Thestudyofthevarious propertiesofsocialnetworkssuchasdiameter,radiality, andconnectivityareresponsible forsocialcontagionandprovidescopeforcontainmentofan epidemicoutbreak.These propertiesalsohelpindesigningstrategiesforcommunica tionbreakdownsinhumanand telecommunicationnetworks[ 5 ]. The cnp ndsapplicationsinnetworkimmunization[ 6 7 ]wheremassvaccinationis anexpensiveprocessandonlyaspecicnumberofpeople,mod eledasnodesofagraph, 16

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canbevaccinated.Theimmunizednodescannotpropagatethe virusandthegoalisto identifytheindividualstobevaccinatedinordertoreduce theoveralltransmissibilityof thevirus.Thereareseveralvaccinationstrategiesinthel iterature(seee.g.,[ 6 7 ])oering controlofepidemicoutbreaks;however,noneofthepropose dareoptimalstrategies. Thevaccinationstrategiessuggestemphasizingthe centrality ofnodesasamajorfactor ratherthan critical nodeswhosedeletionwillmaximizethedisconnectivityoft hegraph. Deletionofcentralnodesmaynotguaranteeafragmentation ofthenetworkoreven disconnectivity,inwhichcasediseasetransmissioncanno tbeprevented.Becausesocial networksmodelthepatternsofhumans,theyverygreatlyove rtime.Therelationships betweenpeople,representedbyedgesinthesocialnetwork, aretransientandthereisa constantrewiringbetweenthenodesasnewrelationshipsar eestablished.Theproposed criticalnodetechniqueminimizesthetransmissionofthed iseaseoveraninstanceofthe dynamicnetwork. Furthermore,the cnp canbeappliedtothestudyofcovertterroristnetworks,whe re acertainnumberofindividualshavetobeidentiedwhosede letionwillresultinthe maximumbreakdownofcommunicationbetweenindividualsin thenetwork[ 1 ].Likewise inordertostopthespreadingofavirusoveratelecommunica tionnetwork,onecan identifythecriticalnodesofthegraphandtakethemoine. Similarly,ifone'sultimate goalistopreventcommunicationonawiredtelecommunicati onnetwork,anecientway ofdoingsowouldbetojamthecriticalnodes.Thishasbeenst udiedinthecontextof wirelessnetworksbyCommanderetal.in[ 8 ]. Beforeproceeding,wementiononenalareainwhichthe criticalnodeproblem ndsseveralapplications,andthatisintheeldoftranspo rtationengineering[ 9 ].Two particularexamplesareasfollows.Ingeneral,fortranspo rtationnetworks,itisimportant toidentifycriticalnodesinordertoensuretheyoperatere liablyfortransportingpeople andgoodsthroughoutthenetwork.Further,inplanningfore mergencyevacuations, identifyingthecriticalnodesofthetransportationnetwo rkiscrucial.Thereasonis 17

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two-fold.First,knowledgeofthecriticalnodeswillhelpi nplanningtheallocationof resourcesduringtheevacuation.Secondly,intheaftermat hofadisastertheywillhelpin re-establishingcriticaltracroutes. Borgatti[ 10 ]hasstudiedasimilarproblem,focusingonnodedetectionr esulting inmaximumnetworkdisconnectivity.Otherstudiesinthear eaofnodedetection suchascentrality[ 2 3 ]focusontheprominenceandreachabilitytoandfromthe centralnodes.However,littleemphasisisplacedontheimp ortanceoftheirrolein thenetworkconnectivityanddiameter.Perhapsonereasonf orthisisbecauseallofthe aforementionedreferencesreliedonsimulationtoconduct theirstudies.Althoughthe simulationshavebeensuccessful,amathematicalformulat ionisessentialforproviding insightandhelpingtorevealsomeofthefundamentalproper tiesoftheproblem[ 11 ].In thenextsection,wepresentamathematicalmodelbasedonin tegerlinearprogramming whichprovidesoptimalsolutionsforthe criticalnodeproblem Weorganizethischapterbyrstformallydeningtheproble manddiscussingits computationalcomplexity.Next,weprovideanintegerprog ramming(IP)formulation forthecorrespondingoptimizationproblem.InSection 2 weintroduceaheuristicto quicklyprovidesolutionsforinstancesoftheproblem.Wep resentacomputationalstudy inSection 2 ,inwhichwecomparetheperformanceoftheheuristicagains ttheoptimal solutionswhichwerecomputedusingacommercialsoftwarep ackage.Weperformthe samestudyonthecardinalityconstrainedprobleminsectio n 2 .Someconcludingremarks aregiveninSection 2 2.2CriticalNodeDetectionProblem ProblemDenition Theformaldenitionoftheproblemisgivenby: C RITICAL N ODE P ROBLEM (CNP) INPUT:Anundirectedgraph G =( V;E )andaninteger k 18

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OUTPUT: A =argmin P i;j 2 ( V n A ) u ij G ( V n A ) : j A j k; where u ij := 8>><>>: 1 ; if i and j areinthesamecomponentof G ( V n A ) ; 0 ; otherwise. Theobjectiveistondasubset A V ofnodessuchthat j A j k ,whosedeletion minimizesthepair-wiseconnectivityamongthenodesinthe inducedsubgraph G ( V n A ). Thisproblemissimilarto minimum k -vertexsharing [ 12 ],wheretheobjective istominimizethenumberofnodesdeletedtoachievea k -waypartition.Hereweare consideringthecomplementaryproblem,whereweknowthenu mberofverticestobe deletedandwetrytomaximizethenumberofcomponentsforme dandimplicitlylimitthe sizesofthecomponents.Borgatti[ 10 ]hasgivenacomprehensiveillustrationtofacilitate theunderstandingoftheobjectivefunctionanditsnon-tri viality. ComputationalComplexity Wenowprovethattherecognitionversionofthe cnp is NP -complete.Considerthe recognitionversionofthe cnp : K-C RITICAL N ODE P ROBLEM (K-CNP) INPUT:Anundirectedgraph G =( V;E )andaninteger k QUESTION:Isthereaset M ,where M isthesetofallmaximalconnected componentsof G obtainedbydeleting k nodesorless,suchthat P 8 i 2 M i ( i 1) 2 K where i isthecardinalityofcomponent i ,foreach i 2 M ? Inordertoprovethatthe K-cnp is NP -complete,wemakeuseofthefollowing lemmata.Inparticular,weprovethatoptimizingtheobject ivefunctionnotonly maximizesthepair-wisedisconnectivityamongthenodes,b utalsominimizesthevariance inthecardinalitiesofthecomponents.Particularly,inLe mma 2.1 weshowthatforany twosolutionsresultinginthesamenumberofcomponents,if thecardinalitiesofthe componentsareequalinonesolution,andnotequalintheoth er,thentheobjectivevalue ofthelatterwillalwaysbeworsethanthatoftheformer. 19

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Lemma2.1. Let M beapartitionof G =( V;E ) into L componentsobtainedby deletingaset D ofnodes,where j D j = k .Thentheobjectivefunction P 8 i 2 M i ( i 1) 2 ( j V j k ) j V j k L 1 2 ,withequalityholdingifandonlyif i = j ; 8 i;j 2 M ,where i isthesize of i th componentof M Proof. Case1: i 6 = j ; 8 i ; j 2 M : Notethat P i 2 M i = j V j k .Thengivenasolutionforaninstanceofthe cnp ,we havethat 1 L X i 2 M i 1 L X i 2 M i 2 = 1 L X i 2 M 2 i 1 L X i 2 M i 2 (2{1) 0 (2{2) = j V j k L 2 j V j k L 2 : (2{3) Thisimpliesthat 1 L X i 2 M 2 i j V j k L 2 : (2{4) Therefore, 1 L X i 2 M 2 i 1 L X i 2 M i j V j k L 2 1 L X i 2 M i (2{5) = j V j k L 2 j V j k L : (2{6) Thus,wehavethat X i 2 M i ( i 1) 2 ( j V j k ) j V j k L 1 2 : (2{7) Case2: i = j ; 8 i ; j 2 M : 20

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Inthiscase,eachcomponentof M willbeofsize j V j k L ,whichisobviouslytheaverage sizeofacomponentof M .Thus X 8 i 2 M i ( i 1) 2 = ( j V j k ) j V j k L 1 2 : (2{8) Conversely,if( 2{8 )holds,buteachcomponentof M isnotthesamesize,itfollowsthat notallcomponentswillbeofaveragesizeandhence 1 L X i 2 M 2 i 1 L X i 2 M i 2 > 0 (2{9) = j V j k L 2 j V j k L 2 : (2{10) Similartotheaboveresult,weseethat 1 L X i 2 M 2 i > j V j k L 2 : (2{11) Thus, 1 L X i 2 M 2 i 1 L X i 2 M i > j V j k L 2 j V j k L : (2{12) ) X i 2 M i ( i 1) 2 > ( j V j k ) j V j k L 1 2 : (2{13) Thisisacontradictionandwehavetheproof. Thefollowinglemmaprovidesasimilarresultasabove.Howe verinthiscase,the numberofcomponentsinducedbyeachsolutionarenotassume dtobeequal. Lemma2.2. Let M 1 and M 2 betwosetsofpartitionsobtainedbydeleting D 1 and D 2 sets ofnodesrespectivelyfromgraph G =( V;E ) ,where j D 1 j = j D 2 j = k .Let L 1 and L 2 bethe numbercomponentsin M 1 and M 2 respectivelyand L 1 L 2 .If i = j ; 8 i;j 2 M 1 ,then weobtainabetterobjectivefunctionvaluebydeletingthes et D 1 Proof. Let f ( M 1 )and f ( M 2 )betheobjectivefunctionvaluesobtainedbydeleting D 1 and D 2 respectively.Letusassumethat f ( M 1 ) >f ( M 2 ).Let u = ( j V j k ) L 2 .FromLemma 2.1 ,we 21

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havethat L 2 ( u )( u 1) 2 f ( M 2 ).Thenwehave L 2 ( u )( u 1) 2 f ( M 2 ) L 1 : (2{17) Thiscontradictsthehypothesisthat L 1 L 2 ,andwehavetheresult. Wecannowprovethefollowingtheoremregardingthecomplex ityofthe cnp Theorem2.1. The K-criticalnodeproblem is NP -complete. Proof. Toshowthis,wemustprovethat(1) K-cnp 2 NP ;(2)Some NP -completeproblem reducesto K-cnp inpolynomialtime. K-cnp 2 NP sincegivenanygraph G =( V;E ),anddeletinganysetofatmost k nodes,wecandeterminetheobjectivevaluein O ( j E j )timeusingadepth-rstsearch[ 13 ]. Tocompletetheproof,weshowareductionfromthe independentsetproblem (isp) [ 14 ],whichiswell-knowntobe NP -complete[ 15 ].Givenagraph G =( V;E ),the isp seekstodetermineif G containsanindependentsetofsize k .Thisisequivalentto determiningifthereexistsanemptysubgraphof G ofsize k bydeleting j V j k nodesand theiradjacentedges. Let G =( V; E )bethegraphobtainedbyreplacingeachnode u 2 G bya T -clique withoneofthecliquenodescoincidingwith u .Notethatif T =1,thenwehavethe originalgraph G .Considerthe K-cnp on G whichasksifthereexistsapartition M of G obtainedbydeleting j V j k nodessuchthat X 8 i 2 M i ( i 1) 2 kT ( T 1) 2 + ( j V j k )( T 1)( T 2) 2 : 22

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Weclaimthereisaone-to-onecorrespondencebetweenthe isp on G andthe cnp on G Ifthereisanindependentsetofsize k in G ,itisclearlyasolutiontothe cnp ,aswewill have k componentsofsize T and j V j k componentsofsize T 1.Thiswouldresultin therequiredobjectivefunctionvalueforthe cnp .Conversely,ifthereisapartitionof M satisfyingtheobjectivebydeleting j V j k nodes,thenwehaveanindependentsetofsize k Wegiveaconstructiveprooftoshowthis.Therstpartinvol vesshowingthatthe objectivevalueofthe cnp on G isalwaysbetterwhenthenodesoftheoriginalgraph G aredeletedfrom G asopposedtodeletingcliquenodes.Foragiven cnp solution,letus assumethatinaclique,anon-coincidingnodeisdeletedand thecoincidingnodefrom thiscliqueisnotdeleted.Ifweswapthesenodesinthesolut ion,thatis,ifwedeletethe coincidingnodeandreplacethenon-coincidingnode,thent heobjectivefunctionvalue eitherremainsthesame,ordecreasesifthenumberofcompon entsincreases. Nowletusassumethatanon-coincidingcliquenodeanditsco incidingnodeare deletedfrom G .Inthiscase,ifweswapfromthesolutionsetthenon-coinci dingnodewith anundeletedcoincidingnode,thenagaintheobjectivevalu ewilleitherdecreaseorremain unchanged.Toseethis,letusassumethecomponentcorrespo ndingtothecliquewith bothitscoincidingandnon-coincidingnodesdeletedisofs ize( T b ),where b 2,as theremaybeothernon-coincidingnodesdeletedfromthiscl ique.Also,letthecoinciding nodethatwasnotdeletedbeapartofsomecomponentofsize T + a .Nowtheobjective functionbeforetheswapwillbe Z 1 = S + ( T + a )( T + a 1) 2 + ( T b )( T b 1) 2 ; (2{18) where S isthecontributionfromothercomponentspresentinthegra ph.Afterswapping thesetwonodes,theobjectivefunctionvaluewouldbe Z 2 = S + ( T 1)( T 2) 2 + ( a )( a 1) 2 + ( T b )( T b +1) 2 : (2{19) 23

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Now,ifwetakethedierenceweseethat Z 1 Z 2 = aT + b 1 0 ; 8 T 0 : (2{20) Sincewearedeletingonly j V j k nodes,wehaveapartition M with X 8 i 2 M i ( i 1) 2 kT ( T 1) 2 + ( j V j k )( T 1)( T 2) 2 ; bydeletingonlythenodesoftheoriginalgraph.Sincenoneo fthenewnodes(i.e.,the nodesfromthe T -cliques)aredeletedfrom G ,thedeletionofthe j V j k nodesresultsin exactly j V j k componentsofsize T 1.Thiscontributesexactly ( j V j k )( T 1)( T 2) 2 towards theobjectivefunction.Theremaining kT nodesformatmost k components.Hencefrom Lemma 2.2 ,thiscontributesatleast kT ( T 1) 2 totheobjectivefunction.FromLemma 2.1 the kT nodesinvolveexactly k componentsofsize T ,representingthe T -cliquesof G ,with onenodeineach T -cliquepresentintheoriginalgraph G ,andnoneofthemconnectedto eachother.Hencedeletionof j V j k nodesfrom G resultsin k independentnodesinthe originalgraph G .Thiscompletestheproof. IntegerProgrammingFormulations Whenstudyingcombinatorialproblems,integerprogrammin gmodelsareusually quitehelpfulforprovidingsomeoftheformalpropertiesof theproblem[ 11 ].Withthisin mindwenowdevelopalinearintegerprogrammingformulatio nforthe cnp Tobeginwith,denethesurjection u : V V 7!f 0 ; 1 g asabove.Further,we introduceasurjection v : V 7!f 0 ; 1 g denedby v i := 8>><>>: 1,ifnode i isdeletedintheoptimalsolution, 0,otherwise. (2{21) 24

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Thenthe criticalnodeproblem admitsthefollowingintegerprogramming formulation (CNP-1) Minimize X i;j 2 V u ij (2{22) s.t. u ij + v i + v j 1 ; 8 ( i;j ) 2 E; (2{23) u ij + u jk u ki 1 ; 8 ( i;j;k ) 2 V; (2{24) u ij u jk + u ki 1 ; 8 ( i;j;k ) 2 V; (2{25) u ij + u jk + u ki 1 ; 8 ( i;j;k ) 2 V; (2{26) X i 2 V v i k; (2{27) u ij 2f 0 ; 1 g ; 8 i;j 2 V; (2{28) v i 2f 0 ; 1 g ; 8 i 2 V: (2{29) Notetheobjectiveistondthesetof k nodeswhoseremovalresultsinagraph whichhastheminimumpari-wiseconnectivitybetweenthere mainingnodes.Thisis accomplishedbytheobjectivefunction.Therstsetofcons traintsin( 2{23 )impliesthat ifnodes i and j areindierentcomponentsandifthereisanedgebetweenthe m,then oneofthemmustbedeleted.Furthermore,constraints( 2{24 )-( 2{26 )togetherimplythat foralltripletsofnodes i;j;k; thatif i and j areinsamecomponentand j and k arein samecomponent,thennecessarily k and i mustbeinthesamecomponent.Constraint ( 2{27 )ensuresthatthetotalnumberofdeletednodesislessthano requalto k .Finally, ( 2{28 )and( 2{29 )denetheproperdomainsforthevariablesused.Thus,asol utionto theintegerprogrammingformulation CNP-1 characterizesafeasiblesolutiontothe cnp Ontheotherhand,itisclearthatafeasiblesolutiontothe cnp willdeneatleastone feasiblesolutionto CNP-1 .Therefore, CNP-1 isacorrectformulationforthe cnp Wenoteherethatinalllikelihood,thereexistalternative mathematicalprogramming formulationsforthe cnp .Forexample,noticethattheconditionswhichsatisfythe 25

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circularconstraints( 2{24 ),( 2{25 ),and( 2{26 )in CNP-1 canbesatisedbythesingle constraint u ij + u jk + u ki 6 =2 ; 8 ( i;j;k ) 2 V: Byappropriatelydeninganewsetof binaryvariables,thisconstraintsetcouldbeincorporate dintothemodel.Thismightbe usefulforbreakingdownthesymmetryoftheproblemifonewa sattemptingtoexploit thepolyhedralstructureofthemodel. Noticethatiftheobjectivewasafunctionofthenumberofco mponents,then anapproximationforthe maximum K -cutproblem [ 15 16 ]couldbeemployedby modifyingthecostfunctionoftheGomory-Hutree[ 17 ].Anevensimplerapproachwould betoidentifythecutverticesinthegraph,ifanyexist.Stu diestoassessthevulnerability ofanetworkwithsimilarobjectivefunctionsarestudiedin [ 18 19 ].Theobjective functionin[ 19 ]istomaximizetracrowwhiledeletingasetof k edges[ 19 ]fromthe graph.In[ 18 ],averysimilarobjectivetotheoneproposedinthisworkis presented. Whereasourobjectiveistominimizethepair-wiseconnecti vitybetweenthenodesafter deleting k nodes,theobjectivein[ 18 ]maximizesthenodedisconnectivitybetweenasetof sourcenodesandsinknodesbydeletingasetof k arcs. Recallthat P i;j 2 V u ij isameasureofthetotalpair-wiseconnectivityofthegraph Noticethatsince u ij isbinaryandequalto1ifandonlyif i and j areinthesame componentintheoptimalsolution,theobjectivefunctionc ouldberewrittenas X j 2 S j ( j 1) 2 ; (2{30) where S issetofallcomponentsand j isthesizeofthe j -thcomponent,whichcan beeasilyidentiedbyfastalgorithmslikebreadthordepth rstsearchalgorithmsin O ( j E j )timeusinganadjacencylistrepresentationofthenetwork [ 13 20 ].Wewilluse ( 2{30 )astheobjectivefunctionoptimizedbytheheuristicinthe followingsection.In additiontotherelativeeaseofcalculatingthecardinalit yofthecomponentsofagraph, thereisanintuitiveexplanationforthechoiceof( 2{30 )asourobjectivefunction.Aswe provedinLemma 2.1 andLemma 2.2 above,optimizing( 2{30 )maximizesthenumberof 26

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procedure CriticalNode ( G;k ) 1 MIS MaximalIndepSet ( G ) 2 while ( j MIS j6 = j V j k ) do 3 i argmin P j 2 S j ( j 1) 2 : S 2 G (MIS [f j g ) ;j 2 V n MIS 4 MIS MIS [f i g 5 endwhile 6 return V n MIS/ setof k nodestodelete / endprocedure CriticalNode Figure2-1.Heuristicfordetectingcriticalnodes. connectedcomponentswhilesimultaneouslyminimizingthe varianceinthecomponent sizes.Forexample,consideranarbitraryunweightedgraph with150nodes.Accordingto ourobjective,itismorepreferabletohaveapartitionwith 3componentseachwith50 nodes,asopposedtoapartitionwith5componentswithoneha ving146nodesandthe restofthemhavingasinglenode.HeuristicforDetectingCriticalNodes Pseudo-codefortheproposedheuristicisprovidedinFigur e 2-1 .Tobeginwith, thealgorithmndsamaximalindependentset(MIS).Thisset isinitiallyempty,and iscomputedsequentiallyasfollows.First,asinglevertex isaddedtotheset.Nextby iteratingthroughthevertices,anodethatisnotadjacentt othestartingnodeisaddedto theMIS.Thenavertexadjacenttoneitheroftheseisadded,a ndsoon.Thisiscontinued untilwecanndnomoreverticestoinclude,andthustheseti smaximalindependent. AftertheinitialMISiscomputed,intheloopfromlines2-5, theheuristicgreedily selectsthenode i 2 V notcurrentlyintheMISwhichreturnstheminimumobjective functionforthegraph G (MIS [f i g ).ThesetMISisaugmentedtoincludenode i ,andthe processrepeatsuntil j MIS j = j V j k .Atthistime,themethodterminatesandthesetof criticalnodestobedeletedisgivenasthosenodes j 2 V suchthat j 2 V n MIS. Theintuitionbehindusinganindependentsetisthatthesub graphinducedbythis setisempty.Statedotherwise,thedeletionofthosenodest hatare not intheindependent setwillresultinanemptysubgraph.Noticethatthiswillpr ovidetheoptimalsolution 27

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foraninstanceofthe cnp if j MIS jj V j k .However,ifthesizeoftheMISisless than j V j k ,wesimplykeepaddingnodeswhichprovidethebestobjectiv evaluetothe setuntilitreachesthedesiredsize.Theheuristiciscompu tationallyecientandthe complexityisgiveninthefollowingtheorem. Theorem2.2. Theproposedalgorithmhascomplexity O ( j V j 2 j E j ) Proof. Tobeginwith,ndingtheMISusingthesequentialmethoddes cribedabove requireslineartime.Next,the while loopfromlines2-5williterateatmost O ( j V j k ) times.Ineachiteration,thenumberofsearchoperationsde creasesfrom j V j 1to j V j ( j V j k )= k .Notethatweareperformingthesearchofasparsegraph,whi chis initiallyempty.Therewillbeonecomparisonstepforevery searchperformedinorderto determinethenodethatprovidestheminimumincreaseinthe objectivefunction.This willinturnbedominatedbythecomplexityofthesearchproc edurewhichrequires O ( j E j ) time.Hence,thetotalnumberofiterationswillbeO ( j V j 1+ j V j 2+ + j V jj V j + k )= O 0@ j V j 1 X i =1 i k 1 X i =1 i 1A = O ( j V j 2 k 2 )= O ( j V j 2 ) : Thustheoverallcomplexityis O ( j V j 2 j E j ),andtheproofiscomplete. Theproposedalgorithmndsafeasiblesolutiontothe criticalnodeproblem ; however,thesolutionisnotguaranteedtobegloballyorloc allyoptimal.Therefore, wecanenhancetheheuristicwiththeapplicationofalocals earchroutineasfollows. Considerthepseudo-codepresentedinFigure 2-2 .Theroutinereceivesasinputthe solutionfromthe CriticalNode heuristicandperformsa2-exchangelocalsearch.Let f : V 7! Z beafunctionreturningtheobjectivefunctionvalueforagi vensetinthe senseof( 2{30 )above.Thatis,considerapairofnodes i and j suchthat i 2 MISand j 62 MIS.Thenforallsuchpairs,weset j 2 MISand i 62 MISandexaminethechangein theobjectivefunction.Ifitimproves,thentheswapiskept ;otherwise,weundotheswap andcontinuetothenextnodepair.Noticethattheloopfroml ines3-16repeatswhile 28

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procedure LocalSearch ( V n MIS) 1 X MIS 2 local improvement : TRUE : 3 while local improvement do 4 local improvement : FALSE : 5 if i 2 MIS and j 62 MIS then 6 MIS MIS n i 7 MIS MIS [ j 8 if f (MIS)
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procedure CriticalNodeLS ( G;k ) 1 X ; 2 f ( X ) 1 3 for j =1to MaxIter do 4 X CriticalNode ( G;k ) 5 X LocalSearch (X) 6 if f ( X )
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Figure2-4.TerroristnetworkcompiledbyKrebs. Asabasisofcomparison,wehaveimplementedtheintegerpro grammingmodelfor the criticalnodeproblem usingtheCPLEX TM version9optimizationsuitefrom ILOG[ 21 ].CPLEXcontainsanimplementationofthesimplexmethod[ 22 ],andusesa branchandboundalgorithm[ 23 ]togetherwithadvancedcutting-planetechniques[ 24 25 ]. WetestedtheIPmodelandheuristiconasetofrandomlygener atedgraphsranging insizefrom75to150nodeswithvaryingdensities.Thegraph sweregeneratedwith version1 : 4ofthepubliclyavailableBarabasigraphgeneratorbyDre ier[ 26 ].Foreach randominstance,wereportsolutionsfor3valuesof k ,thenumberofnodestobedeleted. Inaddition,wehavetestedthealgorithmsontheterroristn etworkcompiledbyKrebs [ 1 ]showninFigure 2-10 .Thisnetworkdepictstherelationshipsbetweentheterror ists involvedinthehorricattacksofSeptember11,2001.Thegr aphwasconstructedafter theattackswithdatawhichwerepubliclyavailablebefore9 = 11. 31

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Figure2-5.Optimalsolutionwhen k =20. Webeginbyprovidingtheresultsfromtheterroristnetwork [ 1 ]showninFigure 2-10 Thisgraphhas62nodesand153edges.Noticethatnode38isth ecentralnodewith degree22.WeappliedtheIPformulationandtheheuristicto thisnetworkwith6values of k .TheresultsareprovidedinTable 2-1 .Noticethatforallvaluesof k ,theheuristic computedtheoptimalsolutionrequiringonaverage0 : 013secondsofcomputationtime. TheaveragetimetocomputetheoptimalsolutionusingCPLEX was5387 : 31seconds. Clearlyevenforthisrelativelysmallnetwork,theheurist icisthemethodofchoice. Figure 2-5 showstheresultinggraphoftheterroristnetworkaccordin gtotheoptimal solutiontothe cnp fortheinstanceof k =20. Inordertodeterminethescalabilityandrobustness,thepr oposedheuristicwas testedonasetofrandomlygeneratedscale-freegraphs.Tab le 3-4 presentstheresults oftheheuristicandtheoptimalsolverwhenappliedtothera ndominstances.Foreach 32

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instance,wereportthenumberofnodesandarcs,valueof k beingconsidered,theoptimal solutionandcomputationtime,andnallytheheuristicsol utionandthecorresponding computationtime.Foreachgraph,wereportsolutionsfor3d ierentvaluesof k .The graphsweregeneratedwithversion1 : 4ofthepubliclyavailableBarabasigraphgenerator byDreier[ 26 ]. Noticethatforallinstancestested,ourmethodwasabletoc omputetheoptimal solution.Furthermore,therequiredtimetocomputetheopt imalsolutionwaslessthan onesecondforallbutoneinstance,averagingonly0 : 33secondsforall27instances.On theotherhand,CPLEXrequired289 : 44secondsonaveragetocomputetheoptimal solution,requiringover5000secondsintheworstcase.Our computationalexperiments indicatethattheproposedheuristicisabletoecientlypr ovideexcellentsolutionsfor large-scaleinstancesofthe cnp 1 2.3CardinalityConstrainedProblem Wenowprovidetheformulationforaslightlymodiedversio nofthe cnp based onconstrainingtheconnectivityindexofthenodesinthegr aph.Givenagraph G = Figure2-6.ConnectivityIndexofnodesA,B,C,Dis3.Connec tivityIndexofE,F,Gis2. ConnectivityIndexofHis0. ( V;E ),the connectivityindex ofanodeisdenedasthenumberofnodesreachable 1 Theexperimentaldataincludingtheinstancestestedareav ailableatthefollowingurl: http : == plaza : ufl : edu = clayton8 = cnp = 33

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fromthatvertex(seeFigure 2-6 forexamples).Toconstrainthenetworkconnectivityin optimizationmodels,wecanimposeconstraintsontheconne ctivityindices. Thisleadstoacardinalityconstrainedversionofthe cnp whichweaptlyrefertoas the cardinalityconstrainedcriticalnodedetectionproblem(cccnp) .The objectiveistodetectasetofnodes A V suchthattheconnectivityindicesofthenodes inthevertexdeletedsubgraph G ( V n A )islessthansomethresholdvalue,say L .Using thesamedenitionofthevariablesasintheprevioussubsec tion,wecanformulatethe cc-cnp asthefollowingintegerlinearprogrammingproblem. (CC-CNP-1) Minimize X i 2 V v i (2{31) s.t. u ij + v i + v j 1 ; 8 ( i;j ) 2 E; (2{32) u ij + u jk + u ki 6 =2 ; 8 ( i;j;k ) 2 V; (2{33) X i;j 2 V u ij L; (2{34) u ij 2f 0 ; 1 g ; 8 i;j 2 V; (2{35) v i 2f 0 ; 1 g ; 8 i 2 V; (2{36) where L isthemaximumallowableconnectivityindexforanynodein V First,weseethattheobjectivefunctiongivenclearlymini mizesthenumberofnodes deleted.Constraints( 2{32 )and( 2{33 )followexactlyasinthe cnp formulation.The onlydierenceisnowwemustconstraintheconnectivityind exofeachnode.Thisis accomplishedbyconstraint( 2{34 ).Finallyconstraints( 2{35 )and( 2{36 )denethe domainsofthedecisionvariables. TheproofofNP-completenessisobtainedfromtheresultpro vedbyKrishnamoorthy andDeo[ 27 ]foraclassofnodedeletionproblems. Lemma2.3. [ 27 ]Let beaspeciedgraphpropertythatisdeterminedbyitscompon ents, andsupposethereisagraphFwithanode\s"suchthatthefoll owinghold: 34

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(1) ThegraphFandthesubgraphresultingafterdeletingnode\s "fromFsatisfy. (2) IfanodexisaddedtothegraphFandnodesxand\s"arejoinedb yanedge,then theresultinggraphisaforbiddeninducedsubgraphforprop erty. thenthenode-coverproblemispolynomiallytransformable tothenodedeletionproblemfor property Letusconsiderthegraphswiththepropertythatthesizeofa nyconnected componentinthegraphislessthanorequalto L ,where L> 0.ThenCC-CNPis anodedeletionproblem[ 27 ]forthisproperty.Additionally,thepropertysatisesth e conditionsstatedinLemma1(onecantakeanyconnectedcomp onentofsize L asgraph F ).ThuswehaveapolynomialtimereductionfromtheNODECOVE RPROBLEM whichiswell-knowntobeNP-complete[ 28 ].Aprecisegeneralizationoftheproperty isprovidedin[ 29 ]ashereditaryandnon-trivial.Aproperty ishereditaryifagraph satises ,theneverynodeandedgeinducedsubgraphofthegraphsatis es and itisnon-trivialifthereareinntielymanygraphsthatsat isfytheproperty.Thenode deletionproblemforhereditarypropertieswaslaterprove dtobemax-SNPhardasthe transformationprovidedwasapproximationpreserving[ 30 ].Thisimpliesthatthereisno polynomialtimeapproximationschemetosolvetheproblemu nlessP=NP. CC-CNPHeuristic Wecanappropriatelymodifytheheuristicsfor cnp tosolveinstancesof cc-cnp Todothis,noticethatnowweareonlyconcernedwiththeconn ectivityindicesofthe nodes.Stateddierently,weareonlyconcernedwiththesiz esofthecomponentsin thevertexdeletedsubgraph.Unlikebefore,thereisnolimi tonthenumberofcritical nodeswechoose,solongastheconnectivityconstraintsare satised.Wecouldgenerate pathologicalinstancestodemonstrateitsineciency,sow eprovideageneticalgorithmin alatersection. Pseudo-codefortheproposedalgorithmisprovidedinFigur e 2-7 .Theheuristicstarts othesameasbeforebyidentifyingamaximalindependentse t(MIS).Then,theboolean 35

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procedure ConstrainedCriticalNode ( G;L ) 1 MIS MaximalIndepSet ( G ) 2 OPT FALSE 3 NoAdd 0 4 while (OPT : NOT : TRUE ) do 5 for ( i =1to j V j ) do 6 if j s j ( j s j 1) 2 L 8 s 2 S G (MIS [f i g ): i 2 V n MIS then 7 MIS MIS [f i g 8 else 9 NoAdd NoAdd+1 10 endif 11 if (NoAdd= j V jj MIS j ) then 12 OPT TRUE 13 BREAK 14 endif 15 endfor 16 endwhile 17 return V n MIS/ setofnodestodelete / endprocedure ConstrainedCriticalNode Figure2-7.Heuristicforthe cardinalityconstrainedcriticalnodeproblem variableOPTissetto FALSE .Finallyinline3,avariableNoAddisinitializedto0.This variabledetermineswhentoexitthemainloopfromlines4-1 6.Afterthisloopisentered, theprocedureiteratesthroughtheverticesanddetermines whichcanbeaddedbackto thegraphwhilestillmaintainingfeasibility.Ifvertex i canbeadded,MISisaugmentedto include i instep7,otherwiseNoAddisincremented.IfNoAddisevereq ualto j V jj MIS j thennonodescanbereturnedtothegraphandOPTissetto TRUE .Thenloopisthen exitedandthealgorithmreturnsthesetofnodestobedelete d,i.e. V n MIS. Theorem2.5. Theworst-casecomplexityofthe ConstrainedCriticalNode heuristicis O ( j V j 2 + j V jj E j ) Proof. ThisproofissimilartotheproofofTheorem 2.2 above.Theloopfromlines4-16 williterateatmost O ( j V j )times.Eachlooprequiresatmost O ( j V j + j E j )timetoverify theifasolutionwillremainfeasibleafteranodeisre-incl udedinthegraph.Thuswehave theresult. 36

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GeneticAlgorithmfortheCC-CNP procedure GeneticAlgorithm 1 Generatepopulation P k 2 Evaluatepopulation P k 3 while terminatingconditionnotmet do 4 Selectindividualsfrom P k andcopyto P k +1 5 Crossoverindividualsfrom P k andputin P k +1 6 Mutateindividualsfrom P k andputin P k +1 7 Evaluatepopulation P k +1 8 P k P k +1 9 P k +1 ; 10 endwhile 11 return bestindividualin P k endprocedure GeneticAlgorithm Figure2-8.Pseudo-codeforagenericgeneticalgorithm. Geneticalgorithms(GAs)mimicthebiologicalprocessofev olution.Inthissubsection, wedescribetheimplementationofaGAforthe cc-cnp .Recallthegeneralstructureof aGAasoutlinedinFigure 2-8 .Whendesigningageneticalgorithmforanoptimization problem,onemustprovideameanstoencodethepopulation,d enethecrossover operator,anddenethemutationoperatorwhichallowsforr andomchangesinospring tohelppreventthealgorithmfromconvergingprematurely[ 31 ]. Forourimplementation,weusebinaryvectorsasanencoding schemeforindividuals withinthepopulationofsolutions.Whenthepopulationisg enerated,(Figure 2-8 ,line1), arandomdeviatefromadistributionwhichisuniformonto(0 ; 1) 2 R isgeneratedfor eachnode.Ifthedeviateexceedssomespeciedvalue,theco rrespondingalleleisassigned value1,indicatingthisnodeshouldbedeleted.Otherwise, thealleleisgivena0,implying itisnotdeleted.Inordertoevaluatethetnessofthepopul ation,perline2,wemust determinewhethereachindividualsolutionisfeasibleorn ot.Determiningfeasibilityisa relativelystraightforwardtaskandcanaccomplishedin O ( j V j + j E j )usingadepth-rst search[ 13 ]. 37

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Inordertoevolvethepopulationoversuccessivegeneratio ns,weuseareproduction schemeinwhichtheparentschosentoproducetheospringar eselectedusingthebinary tournamentmethod[ 32 33 ].Usingthismethod,twochromosomesarechosenatrandom fromthepopulationandtheonehavingthebesttness,i.e.t helowestobjectivefunction value,iskeptasaparent.Theprocessisthenrepeatedtosel ectthesecondparent.The twoparentsarethencombinedusingacrossoveroperatortop roduceanospring[ 34 ]. Tobreednewsolutions,weimplementastrategyknownas parameterizeduniform crossover [ 35 ].Thismethodworksasfollows.Aftertheselectionofthepa rents,refer totheparenthavingthebesttnessas MOM .Foreachofthenodes(alleles),abiased coinistossed.Iftheresultisheads,thentheallelefromth e MOM chromosomeischosen. Otherwise,theallelefromtheleasttparent,callit DAD ,isselected.Theprobability thatthecoinlandsonheadsisknownas CrossProb ,andisdeterminedempirically. Figure 3-3 providesanexampleofapotentialcrossoverwhenthenumber ofnodesis5and CrossProb =0 : 65[ 31 ]. CoinToss T H H T H MOM 0 : 56 0 : 81 0 : 22 0 : 7 0 : 86 DAD 0 : 29 0 : 49 0 : 98 0 : 12 0 : 32 Ospring 0 : 29 0 : 81 0 : 22 0 : 12 0 : 86 Figure2-9.Anexampleofthecrossoveroperation.Inthisca se, CrossProb =0 : 65. Afterthechildisproduced,themutationoperatorisapplie d.Mutationisa randomizingagentwhichhelpspreventtheGAfromconvergin gprematurelyand escapetolocaloptima.Thisprocessworksbyrippingabiase dcoinforeachalleleof thechromosome.Theprobabilityofthecoinlandingheads,k nownasthemutationrate ( MutRate )istypicallyaverysmalluserdenedvalue.Iftheresultis heads,thenthevalue ofthecorrespondingalleleisreversed.Forourimplementa tion, MutRate =0 : 03. Afterthecrossoverandmutationoperatorscreatethenewo spring,itreplacesa currentmemberofthepopulationusingtheso-called steady-state model[ 32 34 36 ]. Usingthismethodology,thechildreplacestheleasttmemb erofthepopulation,provided 38

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thatacloneofthechildisnotanexistingmemberinthepopul ation.Thismethod ensuresthattheworstelementofthepopulationismonotoni callyimprovinginevery generation.Inthesubsequentiteration,thechildbecomes eligibletobeaparentandthe processrepeats.ThoughtheGAdoesconvergeinprobability totheoptimalsolution, itiscommontostoptheprocedureaftersome\terminatingco ndition"(seeFigure 2-8 line3)issatised.Thisconditioncouldbeoneofseveralth ingsincluding,amaximum runningtime,atargetobjectivevalue,oralimitonthenumb erofgenerations.Forour implementation,weusethelatteroptionandthebestsoluti onafter MaxGen generationsis returned.ComputationalResults Alloftheproposedheuristicswereimplementedinthe C ++programminglanguage andcompliedusingGNU g ++version3 : 4 : 4,usingoptimizationragsO2 .Itwastestedon a PC equippedwitha1700MHzIntel R r Pentium R r Mprocessorand1 : 0gigabytesofRAM operatingundertheMicrosoft R r Windows R r XPProfessionalenvironment. Inordertodeterminethescalabilityandrobustness,thepr oposedheuristicwastested onasetofrandomlygeneratedscale-freegraphs.Table 3-4 presentstheresultsofthe heuristicandtheoptimalsolverwhenappliedtotherandomi nstances.Foreachinstance, wereportthenumberofnodesandarcs,thevalueof k beingconsidered,theoptimal solutionandcomputationtimerequiredbyCPLEX,andnally theheuristicsolutionand thecorrespondingcomputationtime.Foreachgraph,werepo rtsolutionsfor3dierent valuesof k Noticethatforallinstancestested,ourmethodwasabletoc omputetheoptimal solution.Furthermore,therequiredtimetocomputetheopt imalsolutionwaslessthan onesecondforallbutoneinstance,averagingonly0 : 33secondsforall27instances.On theotherhand,CPLEXrequired289 : 44secondsonaveragetocomputetheoptimal solution,requiringover5000secondsintheworstcase.Our computationalexperiments 39

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Figure2-10.TerroristnetworkcompiledbyKrebs[ 1 ]. indicatethattheproposedheuristicisabletoecientlypr ovideexcellentsolutionsfor large-scaleinstancesofthe cnp CC-CNPResults Wecontinuewiththeresultsofthetwoalgorithmsdeveloped forthe cc-cnp ,namely thecombinatorialalgorithmandthegeneticalgorithm.Asa bove,wetestedtheIPmodel andbothheuristicsontheterroristnetwork[ 1 ]andasetofrandomlygeneratedgraphs. Foreachinstancetested,wereportsolutionsfor3valuesof L ,theconnectivityindex threshold.Finally,wehaveimplementedtheintegerprogra mmingmodelforthe cc-cnp usingCPLEX TM Table 2-3 presentscomputationalresultsoftheIPmodelandheuristi csolutionswhen testedontheterroristnetworkdata.Noticethatforall5va luesof L tested,thegenetic 40

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algorithmandthecombinatorialalgorithmwithlocalsearc h(ComAlg+LS)computed optimalsolutions.Figure 2-11 showstheoptimalsolutionforthecasewhen L =4. Figure2-11.Optimalsolutionwhen L =4. Wenowconsidertheperformanceofthealgorithmswhenteste dontherandomly generateddatasetscontainingupto50nodestakenfrom[ 37 ].Theresultsareshown inTable 2-4 .Fortheserelativelysmallinstances,wewereabletocompu tetheoptimal solutionsusingCPLEX.Foreachinstance,weprovidesoluti onsfor3valuesof L ,the maximumconnectivityindex.Noticethatfortheseproblems ,thegeneticalgorithm computedoptimalsolutionsforeachinstancetestedinafra ctionofthetimerequiredby CPLEX.Thecombinatorialheuristicfoundoptimalsolution sforallbut3casesrequiring approximatelyhalfofthetimeoftheGA. Table 2-5 presentsthesolutionsfortherandominstancesfrom75to15 0nodes [ 37 38 ].Again,inordertodemonstratetherobustnessoftheheuri stics,weprovide 41

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solutionsfor3valuesof L foreachinstance.Inthistable,weprovidetheresultsfor thegeneticalgorithmandcombinatorialheuristicwithand withoutthelocalsearch enhancement.CPLEXwasunabletocomputeoptimalsolutions withinreasonabletime limitsforanyoftheinstancesrepresentedinthistable. Weseefromthistablethattheintermsofsolutionqualityth eGAisthebest performingmethod.The ComAlg + LS alsofavorswell,butrequiresmorecomputation timethantheGAandrequiresmorecomputingtimeonaverage. Thecombinatorial algorithmwithoutthelocalsearchprocedureproducessolu tionwhicharearguably reasonablegiventhattherequiredcomputationtimeisover 36timesfasterthantheGA, whilethesolutionsareonly1 : 2timesworsethanthosecomputedbytheGA.Nevertheless, thegeneticalgorithmrequiredonly5 : 748secondsonaveragetocomputethebestsolution. Thetrade-oofsolutionqualityversuscomputationtimeis adecisionthatwouldbemade byanoperatordependingonthesizeofthenetworkandthetim econstraintsimposedon detectingthecriticalnodesofagivengraph. 2.4ConcludingRemarks Inthischapter,weproposedseveralmethodsofforthethede tectionofthecritical nodeswhosedeletionresultsinthemaximumnetworkdisconn ectivity.Ingeneral,the problemofdetectingcriticalnodeshasawidevarietyofapp licationsfromjamming communicationnetworksandotheranti-terrorismapplicat ions,toepidemiologyand transportationscience[ 37 38 ]. Inparticular,weexaminedtwoproblems,namelythe criticalnodeproblem (cnp) aswellasthe cardinalityconstrainedcnp(cc-cnp) .Givenagraphand aninteger k ,theobjectiveofthe cnp istodetectasetof k criticalnodeswhosedeletion resultsinthemaximumnumberofdisconnectedcomponentswh osecardinalitieshavethe minimumvariance.Thedenitionofthe cc-cnp isslightlydierentinthatinsteadof given k 2 Z ,themaximumnumberofnodestodelete,wearegivensomevalu e L 2 Z whichrepresentsthemaximumconnectivityindexanodemayh ave.Theobjectiveinthis 42

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caseistodeletetheminimumnumberofnodeswhileensuringt hattheconnectivityindex ofeachnodedoesnotexceed L Theproposedproblemsweremodeledasintegerlinearprogra mmingproblems.Then weprovedthatthecorrespondingdecisionproblemsare NP -complete.Furthermore,we proposedaseveralheuristicsforecientlycomputingqual itysolutionstolarge-scale instances.Theheuristicproposedforthe cnp wasacombinatorialalgorithmwhich exploitedpropertiesofthegraphinordertocomputebasicf easiblesolutions.Themethod wasfurtherintensiedbytheapplicationofalocalsearchm echanism.Byusingthe integerprogrammingformulationwewereabletodeterminet heprecisionofourheuristic bycomparingtheirrelativesolutionsandcomputationtime sforseveralnetworks.The computationalexperimentsindicatedthattheheuristicfo undoptimalsolutionsforall instancestestedinafractionofthetimerequiredbythecom mercialIPsolverCPLEX. Forthe cc-cnp weproposedtwoalgorithms,namelyamodiedversionofthe combinatorialalgorithmdescribedaboveandageneticalgo rithm[ 39 ].Onceagain,the computationalexperimentsindicatedthatbothmethodsare robustandareableto ecientlycomputeapproximatesolutionsforinstancesupt o150nodes. Wealsoconcludewithafewwordsonthepossibilityoffuture expansionofthis work.Aheuristicexplorationofcuttingplanealgorithmso ntheIPformulationwould beaninterestingalternative.Otherheuristicapproaches worthyofinvestigationinclude hybridizingthegeneticalgorithmwiththeadditionofaloc alsearchorpath-relinking enhancementprocedure[ 40 ].Finally,thelocalsearchusedinthecombinatorialalgor ithm wasasimple2-exchangemethod,whichwasthecauseofasigni cantslowdown incomputationasnotedinTable 2-5 .Amoresophisticatedlocalsearchsuchasa modicationoftheoneproposedbyResendeandWerneck[ 41 42 ]shouldbeamajor focusofattention. Furthermore,anodeweightedversionwillbeaninteresting study.Asitisrationalto perceiveapplicationscontainingweightednetworksinwhi chthecostofdeletingonenode 43

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isdierentfromanother.Also,pertainingtoapplications outsidethescopeofjamming networks,astudyofepidemicthresholdvariationwithresp ecttotheheuristicresultswill helpdeterminetheimpactsoncontagionsuppressioninbiol ogicalandsocialnetworks. Table2-1.ResultsofIPmodelandheuristiconterroristnet workdatafromKrebs. InstanceIPModelHeuristic NodesObjectiveExecutionObjectiveExecution Deleted( k )ValueTime(s)ValueTime(s) 202012 : 69200 : 01 1561277 : 77610 : 01 101693337 : 061690 : 02 92142792 : 332140 : 02 828215111 : 942820 : 01 732710792 : 083270 : 01 44

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Table2-2.ResultsofIPmodelandheuristiconrandomlygene ratedscalefreegraphs. InstanceIPModelHeuristicHeuristic+LS NodesArcsDeletedObjCompObjCompObjComp Nodes( k )ValueTime(s)ValueTime(s)ValueTime(s) 75140203666 : 7920 : 12360 : 03 75140251833 : 28390 : 28180 : 03 751403074 : 23180 : 0270 : 04 75210252693 : 71780 : 1260 : 04 752103083 : 57310 : 0580 : 05 752103524 : 36160 : 1820 : 04 752803326749 : 19540 : 00260 : 04 752803520164 : 34380 : 09200 : 06 75280371383 : 98240 : 39130 : 11 1001942544151 : 141420 : 731440 : 09 100194302059 : 66720 : 56200 : 11 10019435108 : 51330 : 66100 : 12 1002854023136 : 47481 : 151230 : 11 1002854217263 : 82380 : 4170 : 17 100285451116 : 78290 : 53110 : 23 1003804522128 : 13580 : 58220 : 15 1003804716243 : 07421 : 191160 : 16 1003805010228 : 72230 : 31100 : 11 12524033625047 : 51970 : 721620 : 30 1252404029118 : 92491 : 562290 : 24 125240451617 : 09320 : 14160 : 39 150290404041 : 61251 : 832400 : 47 150290501226 : 29642 : 773120 : 831 15029060124 : 92351 : 09110 : 851 150435611929 : 55532 : 313190 : 741 150435651331 : 45370 : 991131 : 952 150435671137 : 91310 : 52110 : 801 Table2-3.ResultsofIPmodelandheuristicsonterroristne tworkdatafrom[ 1 ]. InstanceIPModelGeneticAlgComAlgComAlg+LS MaxConn.ObjCompObjCompObjCompObjComp Index( L )ValTime(s)ValTime(s)ValTime(s)ValTime(s) 321188 : 98210 : 25220 : 01210 : 1 417886 : 09170 : 741190 : 01170 : 45 51530051 : 09150 : 871200 : 18251 : 331 8 130 : 39140 : 05130 : 07 10 110 : 741120 : 07110 : 05 45

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Table2-4.ResultsoftheIPmodelandgeneticalgorithmandt hecombinatorialheuristic onrandomlygeneratedscalefreegraphs. InstanceIPModelGeneticAlgComAlg+LS NodesArcsMaxConn.ObjCompObjCompObjComp Index( L )ValueTime(s)ValueTime(s)ValueTime(s) 2045290 : 0490 : 0290 : 03 2045460 : 1360 : 0460 : 862 2045850 : 3950 : 0451 : 482 25602110 : 07110 : 49110 : 08 25604914 : 192 : 113100 : 01 25608726 : 6470 : 0580 : 06 30502110 : 07110 : 06110 : 01 3050480 : 180 : 0580 3050861152 : 1560 : 0960 307541018 : 77100 : 14100 : 02 307569442 : 4190 : 0990 : 04 307510764 : 9470 : 1880 35602120 : 13120 : 14120 : 14 35604829 : 8980 : 71180 35606731 : 6170 : 3170 : 01 40702150 : 17150 : 1150 : 101 4070411341 : 97110 : 06110 40706878 : 9480 : 280 : 04 45802160 : 24160 : 06160 : 1 458041148 : 17110 : 05110 : 02 458068118 : 2380 : 0980 : 071 501352190 : 36190 : 27190 : 05 50135415165 : 18150 : 63150 : 291 501356145722 : 88140 : 721140 : 03 Total(Sum)248257 : 58246 : 705273 : 417 46

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Table2-5.Comparativeresultsofthegeneticalgorithmand thecombinatorialheuristic whentestedonthelargerrandomgraphs.Duetothecomplexit y,wewere unabletocomputethecorrespondingoptimalsolutions. InstanceGeneticAlgorithmComAlgComAlg+LS NodesArcsMaxConn.ObjCompObjCompObjComp Index( L )ValueTime(s)ValueTime(s)ValueTime(s) 751405181 : 622210181 : 502 751408141 : 442200 : 02141 : 181 7514010121 : 231200 : 12123 : 364 752105231 : 532290 : 012318 : 476 752108212 : 443230 : 01222 : 934 7521010202 : 794240 : 092021 : 17 752805313 : 464350 : 101313 : 144 752808292 : 874310 : 05293 : 746 7528010283 : 775300 : 13284 : 787 1001945225 : 317330 : 02222 : 774 10019410173 : 224220 : 241176 : 499 10019415152 : 954220 : 021150 : 44 1002855335 : 08380 : 02331 : 262 10028510284 : 376310 : 052811 : 076 10028515275 : 728280 : 16271 : 142 1003805409 : 052470 : 051425 : 739 100380103611 : 506410 : 02373 : 866 10038015356 : 198400 : 39363 : 034 1252405297 : 951370 : 251311 : 472 12524010249 : 984290 : 07241 : 993 12524015225 : 888260 : 18229 : 233 1502905317 : 981400 : 421305 : 798 15029010264 : 967320 : 2255 : 107 15029015235 : 457291 : 1012319 : 889 1504355499 : 143570 : 06496 : 459 150435104019 : 407500 : 44415 : 518 15043515389 : 703450 : 073813 : 699 Total(Sum)731155 : 1838804 : 297737165 : 304 47

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CHAPTER3 PATHPLANNINGPROBLEMS 3.1TargetVisitationProblem Introduction Pathplanningproblemsareamongthemoststudiedtopicsino perationsresearch [ 43 { 46 ].Infact,the travelingsalesmanproblem(TSP) ,arguablythemostfamous ofalloptimizationproblemsfallsintothisclass[ 47 ].Inthischapter,weconsiderthe problemofdeterminingtheoptimalrouteforanunmannedaer ialvehicle(UAV)which needstovisitmultipletargetsandreturntoitspointofori gin.Theobjectiveisto minimizethetotaldistancetraveledandmaximizetheutili tyofthevisitationsequence. Thisisknownasthe targetvisitationproblem(TVP) andhasseveralapplications includingcombatsearchandrescue,disasterrelief,anden vironmentalassessment[ 48 ]. Workonthe tvp islimited.ItwasrstposedinapaperbyGrundelandJecoat datingfrom2004[ 48 ].Inthiswork,theauthorsdescribetheproblemandtheimpl ementation ofagreedyrandomizedadaptivesearchprocedure(GRASP)fo rcomputingapproximate solutions.Theaforementionedpaperwasintendedtoprovid eanintroductiontothe problem,notanextensivecomputationalanalysis.Instead ,theauthorsprovidea combinatorialformulationandexaminethesimilaritiesbe tweenthe tvp andtwoother well-knownproblems,namelythe travelingsalesmanproblem [ 47 ]andthe linear orderingproblem [ 15 ].Weprovideasimilaranalysisinalatersubsection. Westudyanothercloselyrelatedprobleminthischapter.Th eproblemdealswith maximizingcommunicationamongagentswhileroutingthemi nacooperativenetwork. Researchintheareaofcooperativenetworkshassurgedinre centyears[ 43 { 46 ].This particularbranchoftelecommunicationsisleadingtheway forfuturetechnologiesand thedevelopmentofnewnetworkorganizations[ 5 ].Inparticular,so-called mobileadhoc networks (MANETs)areattheforefrontoftheworkinautonomouscoope rativenetworks [ 49 ].MANETsarecomprisedofasetoflooselycoupledagentswhi chcommunicatevia 48

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asharedradiochannelwithotheragentswithinaspeciedra nge.Theuniquefeature ofMANETsthatseparatesthemfromtraditionalcellularnet worksisthefactthatthe topologyofMANETsisdynamic.Thatis,witheachmovementof theagents,anew topologyisestablished. Thelackofapre-establishedinfrastructureisanattracti vefeatureofMANETs. Theyareparticularlyusefulinsituationswherecommunica tionisrequired,butnoxed telecommunicationsystemexists.MANETsarealsohelpfulw henasetofmobileusers needtobeinconstantcontactwitheachother.Specicexamp lesincludecombatsearch andrescueteams,andmedicalteams.Inthewakeofdisasters suchastheterrorist attacksofSeptember11,2001,andHurricaneKatrina,thena tionsawrsthandthat communicationamongtheemergencyresponderswascritical tothesuccessoftherescue operations. Inthischapter,wedescribetheimplementationofagenetic algorithmfornding approximatesolutionsforthe tvp .Theencodingschemeisbasedonrandomkeys[ 50 ]. Theheuristicisthenhybridizedbytheimplementationofal ocalsearchprocedure. Numericalresultsarepresenteddemonstratingtheeectiv enessoftheproposedprocedure. Theremainderofthechapterisorganizedasfollows.Next,w eprovideamathematical modelforthe tvp andprovethatndinganoptimalsolutionis NP -complete.Then,in Section 3 wedescribeahybridgeneticalgorithmforsolvinglargeins tances.Computational resultsareprovidedinSection 3 comparingtheproposedheuristictoastandardgenetic algorithmandtheoptimalsolutionsascomputedbyacommerc ialintegerprogramming package.InSection 3 ,westudytheCommunicationModelsforaCooperativeNetwor kof AutonomousAgents( ccpm ).Section 3 ,weprovidetheproblemformulationforthe ccpm TheninSection 3 wepresentareviewofthepreviousworkintheareaofcommuni cation incooperativenetworks.InSection 3 ,wederivetwomixedintegerformulationsforthe ccpm usingthecombinatorialproblemasaguide.Weprovidesomep reliminarynumerical resultsinSection 3 anddiscussconclusionsanddirectionsoffutureresearchi nSection 3 49

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Origin A B C D AThewaypointstobevis-ited. Origin A B C D K ) r s BTheminimumdistancesolution. Origin A B C D k Y 1 ? CThebestsequencesolution. Origin A B C D 1 ? DTheoptimal tvp solution. Figure3-1.Thisexamplecomparestheoptimalsolutionfort he tvp instancewiththe related tsp and lop solutions. Astechnologiesadvance,theuseofunmannedaerialvehicle s(UAVs)forcivilianand militaryapplicationsisincreasing.Civilianapplicatio nsincludeenvironmentalassessment andsearchandrescue.Moreover,UAVshavebeenusedinmilit aryapplicationsfor decadesandhelptoensurethatcoalitionforcesmaintainac ompetitiveadvantagein theglobalwaronterrorism.Oftentimes,pathplanningfort heparticularmission,be itcivilianormilitary,isanon-trivialprocess.Twoimpor tant,oftencompetingfactors aretheoveralldistancetraveledbytheUAVandthesequence inwhichvarious\targets" or\pointsofinterest"arevisited[ 48 ].Beforemovingontotherigorousmathematical formulation,weprovideasimpleexampledemonstratingthe ideabehindthe tvp and motivatingitsuseinseveralpracticalmilitaryandcivili anapplications.In[ 48 ],the authorsprovideasimilarexampleinthecontextofaUAVsurv eillinganenvironmental mishap,whichwefollowwithamodicationofthetheme. 50

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SupposethatthesetofpointsinFigure 3-1 (a)representacollectionofvillagesin whichasoughtafterterroristissuspectedofhidingout.Th epointlabeled\Origin"isthe locationofthecoalitionforce.Moreoverassumetheavaila bleintelligencedatasuggests thatthemostlikelyhidingspotoftheterroristispointD,a ndthesecondmostlikely locationispointA.Inanapplicationsuchastheonedescrib ed,itiswell-knownthatthe personofinterestmovesfrequently,andthattheolderthei ntelligencedateis,theless accurateitbecomes.Supposethecoalitionforcehastheabi litytolaunchaminiature UAVfromtheOriginandhaveitpassthroughapre-establishe dsetofwaypointsbefore returningtothestartingpoint.Duringitsright,theUAVis capableoftelemeteringdata backtothecoalitionforcehelpingtoestablishtheknownlo cationoftheterroristthey seek. Theremainingsubguresdemonstratetheoptimalsolutions whenvariousobjectives areconsidered.InFigure 3-1 (b),onlytheoveralldistancetraveledbytheUAVistakenin toaccount.NoticeherethatpointD,themostprobablelocat ionoftheterroristisvisited last.Clearlythisisnotadesirablevisitationsequence.I nFigure 3-1 (c),maximizing thepreferencesinwhichthevillagesarevisitedistheonly objectiveconsidered.This sequence,whilebetterthanthepreviousisstilllongandco uldperhapsbeshorteneda bitwithoutsacricingtoomuchtimebetweenvisitingthehi ghlyprobablywaypoints. TheroutegiveninFigure 3-1 (d)doespreciselythis.Here,acombinationofdistanceand preferenceisconsideredinthepathplanning.Weseethatth isrouteprovidestheoptimal mixtureofvisiting\highchance"waypointsquickly,sotha tthecoalitionforcemayact ontheintelligencetheyreceive.Thissimpleexampledemon stratestheimportanceof consideringbothdistanceandvisitationsequencewhensol vingapathplanningproblem foraUAV.Aswewillseeinthenextsection,whenconsideredi ndividually,thesetwo objectivesgeneralizetwowell-studiedproblemsincombin atorialoptimization.Asitturns out,bothare NP -hardwhichprovidessomeindicationastothecomplexityof the tvp 51

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ProblemDescription Inthissectionweprovideaformaldescriptionofthe targetvisitationproblem anddiscusscomplexityissues.Wealsodiscussthesimilari tiesbetweenthe tvp andother combinatorialproblems.Whatfollowsisabriefsurveyofth isnature. Relatedproblems Herewebrieryexaminethesimilaritiesbetweenthe targetvisitationproblem andtwowell-knownproblemsindiscreteoptimization,name lythe travelingsalesman problem andthe linearorderingproblem TravelingSalesmanProblem .The travelingsalesmanproblem(tsp) isthe moststudiedandwidelyrecognizedprobleminoperationsre search[ 15 23 47 51 { 54 ].It hasbeenthefocusofresearchforover50yearsandremainsac hallengeeventoday.Given agraph G =( V;E ),asubsetofedges T E issaidtobea travelingsalesmantour ifitis asimplecycleof G havinglength j V j .Inthiscontext(andour's)atourisaHamiltonian cycle,butthiseasilygeneralizesdependingonone'sdeni tionofatour. Supposenowthatthegraphisweighted( G;c ),where c ij representsthecostof traversingarc( i;j ) 2 E andthat j V j = n .Theobjectiveisnowtondatourofminimum cost.Thenifwerepresentatourasapermutation ofthenodes,thenthegoalistond thatminimizes Z ( )= n 1 X i =1 c ( i ) ; ( i +1) + c ( n ) ; (1) : (3{1) Theoriginalintegerprogrammingformulationofthe tsp isduetoDantzig,Fulkerson, andJohnson(DFJ)[ 52 ]andcontinuestobeapopularformulationtoday.Let x : E 7! f 0 ; 1 g beadecisionvariableassociatedwitheacharc.ThentheDFJ formulationofthe 52

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tsp isgivenasthefollowing0-1integerprogram[ 52 ]. DFJ :max n X i =1 n X j =1 c ij x ij (3{2) subjectto n X i =1 i 6 = j x ij =1 ; 8 j 2 V; (3{3) n X j =1 j 6 = i x ij =1 ; 8 i 2 V; (3{4) X i 2 S X j 2 S x ij j S j 1 ; 8 S V; 2 j S j n 1 ; (3{5) x ij 2f 0 ; 1 g ; 8 i;j 2 V: (3{6) TheDFJformulationcontains n ( n 1)integervariablesand2 n constraints.While minimizingthetotaltourcost,thesetsofconstraintsin( 3{3 )and( 3{4 )ensurethateach nodeisvisitedexactlyonce.Constraintset( 3{5 )preventssubtours,byensuringthat feasiblesolutionsarebiconnected[ 23 ].ThemajordrawbackoftheDFJformulationisthe exponentialnumberofsubtoureliminationconstraints. Anotherformulationofthe tsp thatiswidelyusedinpracticeisduetoMiller, Tucker,andZemlin(MTZ),andwasrstpublishedin1960[ 55 ].TheMTZformulation reducesthenumberofsubtoureliminationconstraintstobe polynomiallybounded,atthe expenseofincreasingthenumberofdecisionvariables.Let s : V 7! R beabijectionwhere s i :=therelativepositionofnode i inthetour.(3{7) Forexample,ifnode2isvisitedthirdinthetour,then s 2 =3.The s i variablesare commonlyreferredtoas sequencingvariables andcanhavemanyinterpretations[ 56 ]. Withoutthelossofgeneralitysupposethatthedepot,orpoi ntoforiginisdenedtobe node1.Usingthedecisionvariablesdenedaboveandtheseq uencingvariables,wecan 53

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formulatetheMTZ tsp asfollows: MTZ :max n X i =1 n X j =1 c ij x ij (3{8) subjectto n X i =1 i 6 = j x ij =1 ; 8 j 2 V; (3{9) n X j =1 j 6 = i x ij =1 ; 8 i 2 V; (3{10) y i y j + n x ij n 1 ; 8 i;j 2 V nf 1 g ;i 6 = j; (3{11) x ij 2f 0 ; 1 g ; 8 i;j 2 V; (3{12) y i 2 R ; 8 i 2 V nf 1 g : (3{13) TheMTZformulationcontains n 2 decisionvariables,and O ( n 2 )constraints.The sequencingvariablesusedinthismodelallowconsiderable rexibilityandtheabilityto modelrelatedproblemseasily.Wetakeadvantageofthisfor ourformulationofthe tvp in thefollowingsection.IthasbeenshownhoweverthattheMTZ formulationisweakerthan theDFJformulation[ 57 ];however,themodelcanbestrengthenedbyliftingadditio nal edgesasshownbyDesrochersandLaporte[ 56 ]. Theserepresentjustafewoftheformulationswhichhavebee nproposedfor solving travelingsalesmanproblems .Forextensivereviewandanalysisofother formulations,thereaderisreferredto[ 58 ]and[ 57 ].Asforthecomputationalcomplexity ofthe tsp ,itiswell-knowntobe NP -hard.Further,nonmetricinstancesofthe tsp cannotbeapproximatedwithinaconstantfactorunless P = NP [ 15 ].Athoroughreview oftheproblemisavailablein[ 59 ]. LinearOrderingProblem .The linearorderingproblem(lop) isanother optimalsequencingproblem.Givenaset N of n itemsandacorrespondingmatrix D = f d i;j g n n ,whichrepresentsthepreferencesfororderingitem i beforeitem j ,theobjective istondanorderingoftheitemswhichmaximizestheprefere nces[ 60 ].Applications 54

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aboundincludingrankingathletesorsportsteams,ranking preferencestoobtainancestry relationships,andineconomics[ 61 ].Intermsofthematrix D ,theoptimalsolutionisa permutation oftherowsandcolumnsof D suchthatthesumofthevaluesintheupper trianglearemaximized.The lop canrepresentedasacombinatorialoptimizationproblem asfollows: max Z ( )= n 1 X i =1 n X j = i +1 d ( i ) ; ( j ) ; where ( i )representstheiteminposition i ofthepermutation[ 48 ].Anequivalentgraph theoreticalproblemistondanacyclictournamentinacomp leteweightedgraphin whichthesumofthearcweightsismaximal[ 62 ]. Anintegerprogrammingformulationforthe tvp canbecomputedasfollows.Let x : N N 7!f 0 ; 1 g bedenedas x ij := 8>><>>: 1 ; if i isorderedbefore j; 0 ; otherwise. (3{14) Thenthe lop admitsthefollowingintegerprogrammingformulation: LOP :max X ( i;j ) 2 N d ij x ij (3{15) subjectto x ij + x ji =1 ; 8 i;j 2 N; (3{16) x ij + x jk + x ki 2 ; 8 i;j;k 2 N;i 6 = j;i 6 = k;j 6 = k; (3{17) x i;j 2f 0 ; 1 g ; 8 i;j 2 N: (3{18) The lop formulationconsistsof n 2 decisionvariablesand O ( n 3 )constraints.Constraint set( 3{16 )ensuresthateachitemisconsideredonlyonce,thusenforc ingthestrict precedencerelation.Theconstraintsin( 3{17 )ensurethatthesolutionisacyclic. Notsurprisingly,thedecisionversionofthe lop isknowntobe NP -complete,andthe correspondingoptimizationproblemis NP -hard[ 63 64 ]. 55

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Aswecansee,the targetvisitationproblem combinesattributesofboththe travelingsalesmanproblem andthe linearorderingproblem .Wearenow readytoformallydenethe tvp andexamineseveralmathprogrammingformulations. Targetvisitationproblem Aninstanceofthe targetvisitationproblem consistsofaset N = f 1 ; 2 ;:::;n g oftargetslocatedatdistinctpoints.Thereisalsoanassoc iateddistancematrix D = f d i;j g m m ,where m := n +1.The d i;j entriesrepresentthedistancesbetweennodes i;j 2 N .Notethatthedistancesmaybeasymmetric,i.e. d i;j 6 = d j;i necessarily.Also, foralltargets i ,thereisavalue d 0 ;i whichrepresentsthedistancefromtheUAVspointof origintotarget i .Furthermore,amatrix R = f i;j g n n isprovidedwhere i;j represents thepreferenceorutilityofvisitingtarget i beforetarget j .Thiscanbeinterpretedasthe assumed\threatlevel"orrelativeimportanceofvisitingo netargetbeforeanother.The intuitionisthattargetswithhigherprioritiesshouldbev isitedearlierinthesequence. Asmentionedin[ 48 ],obtainingthevaluesof d i;j isusuallyaneasytasksinceliteral distancemeasuresorothermetricssuchastraveltimeareav ailableoraretrivialto calculate.However,derivingthevaluesof i;j ,thevalueofvisitingtarget i beforetarget j isnotalwayssosimple.Thereareseveralmethodsusedbymil itaryplannerswhen developingroutesforthe tvp .Themostcommonmethod,andtheoneweadoptin thischapter,isknownas\targetvaluereconciliation"[ 48 ].Inthismethodagroupof expertsoerasetofpair-wiserankingsforthetargetsfrom whichthepreferencematrixis derived.Morespecically,foralltargets i and j ,eachexpertistospecifyapreferenceof visitingtarget i before j [ 65 ].Thevalueof i;j issimplythecumulativenumberofexperts whoprefertovisit i before j Afeasiblesolutionforthe targetvisitationproblem isoneinwhichtheUAV leavesitspointoforigin,visitsalltargetsexactlyonce, andreturnstotheorigin.The objectiveistominimizethedistancetraveledwhilemaximi zingtheutilityofthevisitation sequence[ 48 ].Let beapermutationofthesetofintegers[1 ;:::;n +1) \ Z ,suchthat 56

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j =: ( i )impliesthattarget j isthe i th positionofthevisitationsequence.Withthis,we canformulatethe tvp asthefollowingcombinatorialoptimizationproblemrstg ivenin [ 48 ]: Maximize Z ( )= h n 1 X i =1 n X j = i +1 ( i ) ; ( j ) i h d 0 ; (1) + n 1 X k =1 d ( k ) ; ( k +1) + d ( n ) ; 0 i (3{19) Permutationbasedmodelsofcombinatorialproblemsareoft enusefulforgainingan intuitiveunderstandingoftheproblem.However,integerp rogrammingmodelsareusually themosthelpfulforprovidingsomeoftheformalproperties oftheproblem[ 11 ].Withthis inmindwenowdevelopalinearintegerprogrammingformulat ionforthe tvp The targetvisitationproblem canbeconvenientlydescribedasacombinatorial problemonagraph.Consideradoublyweighteddirectedgrap h( G;d; ),where V = f 0 ; 1 ; 2 ;:::;n g isthesetofnodes.Supposethat V representsthesetoftargets,hence n = j N j .Weincludeanextranodewhichrepresentstheorigin.Also, assumewithout thelossofgeneralitythat G isacompletegraph.Foreachedge( i;j ) 2 E ,thereisan associatedweight d i;j whichrepresentsthedistancefromtarget i totarget j .Furthermore, foreachedge( i;j ) 2 E; thereisanassociatedvalue i;j whichisthepreferenceforthe correspondingtargetpairasdescribedabove.Now,let x : V V 7!f 0 ; 1 g beasurjection denedby x i;j := 8>><>>: 1 ; i i = ( k ) ) j = ( k +1) ; for k 2 Z n ; 0 ; otherwise, (3{20) where isdenedasabove.Saiddierently, x i;j =1impliesthat( i;j ) 2 E isalinkinthe tour.Next,weintroduceanothersurjectivefunction w : V V 7!f 0 ; 1 g denedby w i;j := 8>><>>: 1 ; i i = ( k ) ) j = ( l ) ; for k;l 2 Z n suchthat k
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Finally,wehaveabijection y : V 7! R ,where y i :=sequenceinwhichtarget i isvisited.(3{22) Withthiswecanformulatethe targetvisitationproblem asthefollowinginteger linearprogram. TVP 1:max n X i =1 n X j =1 i 6 = j i;j w i;j n X i =0 n X j =0 i 6 = j d i;j x i;j (3{23) subjectto n X i =0 i 6 = j x i;j =1 ; 8 j 2 V; (3{24) n X j =0 j 6 = i x i;j =1 ; 8 i 2 V; (3{25) y i y j + n x i;j n 1 ; 8 i;j 2 V nf 0 g ;i 6 = j; (3{26) w i;j + w j;i 1 ; 8 i;j 2 V;i 6 = j; (3{27) y i y j + n w i;j 0 ; 8 i;j 2 V nf 0 g ;i 6 = j; (3{28) x i;j 2f 0 ; 1 g ; 8 i;j 2 V; (3{29) w i;j 2f 0 ; 1 g ; 8 i;j 2 V; (3{30) y i 2 R ; 8 i 2 V nf 0 g : (3{31) Thisformulationhasatotalof3 n 2 5 n +4constraintsand2 n 2 n 1integervariables. Noticethatingraphtheoreticalterms,theobjectiveofthe tvp istondaHamiltonian cyclewhichisofminimumweight,butwhichalsomaximizesth evisitationpreferences. Thisisaccomplishedbytheobjectivefunctionin( 3{32 ).The2 n assignmentconstraintsin ( 3{33 )and( 3{34 )ensurethateachtargetisvisitedonlyonceinthetour.The n 2 3 n +2 constraintsin( 3{35 )aresubtoureliminationconstraintsandhencepreventdis jointcycles fromoccurringinthetour. 58

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Theconstraintsin( 3{27 )ensurethatonlyoneof w i;j or w j;i isnonzeroforall( i;j ) pairs.Inordertoensurethat w i;j =1onlywhen i isvisitedbefore j ,wehavethe O ( n 2 ) constraintsin( 3{28 ).Theyinsistthat w i;j isnonzeroonlywhen y i
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and( 3{36 )togetherensuresthattherecannotbesubtours,asthisisj usttheMTZbased formulationforTSP.Inordertoprovethat w ij 6 = w ji ,letusconsiderconstraintset( 3{35 ). If i isvisitedbefore j ,thenwehave y i y j < 0andmaximizationoftheobjectivefunction ensuresthat w ij isone.If j isvisitedbefore i ,thenwehave y i y j > 0andhence w ij has tobezero. The tvp representsasetofcombinatorialdecisionsthatmustbemad e[ 11 ].Clearly, foranyasymmetricinstanceconsistingof n targetsthereare n !possibleroutestoconsider. Nowthatwehaveanintegerprogrammingmodelforthe tvp ,wecanexaminethe computationalcomplexity.Itisnotsurprisingthatnding anoptimalsolutionis NP -hardaswewillnowshowbyprovingthattherecognitionvers ionoftheproblem is NP -complete.Therecognitionversionofthe tvp canbestatedasfollows:( K-tvp ) Givenaninstanceofthe targetvisitationproblem ,doesthereexistatourofcost lessthanorequalto K ? Theorem1. Thedecisionversionofthe targetvisitationproblem ( K-tvp )is NP -complete. Proof. Toshowthis,wemustprovethat(1) K-tvp 2 NP ;(2)Some NP -completeproblem reducesto K-tvp inpolynomialtime. Clearly K-tvp 2 NP sinceanysolutioncanbeveriedinpolynomialtimetobefea sibleor not.Tocompletetheproof,weshowasimplereductionfromthe Hamiltoniancycle problem whichiswell-knowntobe NP -complete[ 15 ].Let G =( V;E )beagraphin whichaHamiltoniancyclehastobedetermined.Constructac ompletegraph G =( V; E ) witharcdistance1if( i;j ) 62 E and0otherwise.Furthermore,constructthepreference matrixsuchthat i;j = k; where k 2 R ,forall( i;j )pairs.Theobjectiveofthedecision problem K-tvp istodetermineifasolutionexistswithcost K k n .A`yes'instanceof the Hamiltoniancycleproblem on G correspondstoa`yes'instanceforthe K-tvp 60

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on G .Noticethatthecostofthe K-tvp tourissimplythesumofthepreferencecosts, i.e. k n ,asthedistancecomponentoftheobjectivefunctioniszero Toprovetheconverse,observethatthecostofany K-tvp tourisatleast k n .Thus a`yes'instanceof K-tvp wouldmeanthatallofthearcsinthetourhavezerocost.This impliesallofthearcsinthe K-tvp tourarepresentinthegraph G andalsoformavalid tourin G .Thisresultsina`yes'instanceforthe Hamiltoniancycleproblem .Thus theproofiscomplete. Asnotedin[ 48 ],itmightbethecasethatforaparticularinstanceofthe tvp ,the termsofoneofthematricesintheobjectivefunctionmaydom inatetheother.However, bothdistanceandutilityareimportantfactorsandshouldb egivenequalattentionina solution.Therefore,weuseasimplebalancingheuristicr stgivenbyGrundelandJecoat in[ 48 ].Let r bearandompermutationofthetargetstobevisited.Further ,dene r 2 R suchthat ~ R := r R .Inordertonormalizethe D and R matrices,weadjusttheparticular valueof r sothat P n 1 i =1 P nj = i +1 ~ r ( i ) ; r ( j ) d 0 ; r (1)+ P n 1 i =1 d r ( i ) ; r ( i +1) + d r ( n ) ; 0 1 : (3{40) Thenwithoutthelossofgenerality,theparameter r canbeusedtoweighteithermatrixif itisdeterminedthattheoneofthedistanceorpreferenceco mponentsismoreimportant thantheother.Henceincreasingthevalueof r placesmoreimportanceontheutilityof thevisitationsequencerelativetothetotaldistancetrav eled[ 48 ]. Thecomplexityofthe tvp motivatestheneedforecientheuristicssincending optimalsolutionsforlargeinstancesisimpractical.Ther efore,inthenextsectionwe proposetheuseofgeneticalgorithmforndingnearoptimal solutionsforthe tvp GeneticAlgorithm Geneticalgorithms(GAs)gettheirnamefromthebiological processwhichthey mimic.MotivatedbyDarwin'sTheoryofNaturalSelection[ 66 ],thesealgorithmsevolvea population ofsolutions,called individuals ,overseveral generations untilthebestsolution iseventuallyreached.Eachcomponentofanindividualisca lleda allele .Individualsin 61

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thepopulationmatethroughaprocesscalled crossover ,andnewsolutionshavingtraits, i.e.allelesofbothparents,areproduced.Insuccessivege nerations,onlythosesolutions havingthebest tness arecarriedtothenextgenerationinaprocesswhichmimics thefundamentalprincipleofnaturalselection, survivalofthettest [ 39 ].Figure 3-2 providespseudo-codeforastandardgeneticalgorithm.Tho ughtheGAdoesconverge inprobabilitytotheoptimalsolution,itiscommontostopt heprocedureaftersome \terminatingcondition"(seeline3)issatised.Thiscond itioncouldbeoneofseveral thingsincluding,amaximumrunningtime,atargetobjectiv evalue,oralimitonthe numberofgenerations.Forourimplementation,weusethela tteroptionandthebest solutionafter MaxGen generationsisreturned. procedure GeneticAlgorithm 1 Generatepopulation P k 2 Evaluatepopulation P k 3 while terminatingconditionnotmet do 4 Selectindividualsfrom P k andcopyto P k +1 5 Crossoverindividualsfrom P k andputin P k +1 6 Mutateindividualsfrom P k andputin P k +1 7 Evaluatepopulation P k +1 8 P k P k +1 9 P k +1 ; 10 endwhile 11 return bestindividualin P k endprocedure GeneticAlgorithm Figure3-2.Pseudo-codeforgenericgeneticalgorithm. Whendesigningageneticalgorithmforanoptimizationprob lem,onemustprovide ameanstoencodethepopulation,denethecrossoveroperat or,anddenethe mutation operatorwhichallowsforrandomchangesinospringtohelp preventthealgorithmfrom convergingprematurely.Theencodingschemeweproposefor ourGAisbasedonrandom keysandfollowsexactlyasdescribedbyBean[ 50 ].Asmentionedin[ 50 ],GAsoftenhave adiculttimemaintainingfeasibilityofsolutionsinsucc essivegenerations.Thisproblem isovercomebytheuseofrandomkeysasanencodingmechanism forthepopulation. 62

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Randomkeysworkbyencodingthesolutionvectorusingrando mnumbers.Thefeasibility issueisthenmovedintotheobjectivefunction,andsubsequ entlyallospringproduced areguaranteedtobefeasiblesolutions. FortheGAimplementationforthe tvp ,wehavethefollowingdenitions.As mentionedabove,solutionsarerepresentedbyarandomvect or.Todeterminethe visitationsequence,arandomdeviatefromadistributionw hichisuniformonto(0 ; 1) 2 R isgeneratedforeachtarget.Thetourisdeterminedbysorti ngtherandomnumbersand sequencingthetargetsindescendingorderofthesort.Fore xample,supposethereare n =3targetstovisit.Thenachromosomesuchas ( : 34 ;: 71 ;: 28) wouldcorrespondtothevisitationsequence 2 1 3 : Theobjectivevalueofthesequencecanbeevaluated,thusde terminingthetnessofthe chromosome.Evolutionarymechanisms Inordertoevolvethepopulationoversuccessivegeneratio ns,weuseareproduction methodwhichcopiesthebestindividualsinthecurrentgene rationtothenext.We aptlyrefertothissetthe BEST set.Thistechniqueensuresthatthebestsolutionis monotonicallyimprovingineverygeneration[ 50 ].Tobreednewsolutions,weimplementa strategyknownas parameterizeduniformcrossover [ 35 ].Thismethodworksbyselecting twosolutionstoserveasparents.Inourimplementation,on eparentischosenatrandom fromthe BEST set,andtheotherischosenfromtheentirepopulation(incl uding BEST ). Then,foreachtargettobevisited,abiasedcoinistossed.I ftheresultisheads,thenthe alleleofthe BEST parentischosen,otherwisethealleleistakenfromtheothe rparent. Theprobabilitythatthecoinlandsonheadsisknownas CrossProb ,andisdetermined 63

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empirically.Figure 3-3 providesanexampleofapotentialcrossoverwhenthenumber of targetsis5and CrossProb =0 : 65. CoinToss T H H T H Parent1 0 : 56 0 : 81 0 : 22 0 : 7 0 : 86 Parent2 0 : 29 0 : 49 0 : 98 0 : 12 0 : 32 Ospring 0 : 29 0 : 81 0 : 22 0 : 12 0 : 86 Figure3-3.Anexampleofthecrossoveroperation.Inthisca se, CrossProb =0 : 65. Finally,themutationoperatorisdenedasfollows.Instea dofintroducingrandom perturbationstoselectedospring,weinsteadreplacease tofindividualshavingtheworst tnesswithnewsolutionsgeneratedatrandomfromthesamed istributionastheoriginal population.Thisreplacementsetisreferredtoasthe WORST set.Usingthismethod,we areabletoensurethattheGAdoesnotconvergeprematurely. Thisisacommonmethod, sometimesreferredtoas immigration andappearsthroughouttheliterature[ 50 67 ].An overallpictorialviewofthegenerationalevolutionofthe proposedGAisprovidedin Figure 3-4 Figure3-4.Graphicalrepresentationofgenerationalevol ution. 64

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Localsearch InadditiontothestandardGA,weproposeahybridizationte chniquetoproduce bettersolutions.Inparticularweimplementa2-exchangel ocalsearchoneachofthe ospringproducedbycrossoveroperator.Pseudo-codefort hisheuristicisprovided inFigure 3-5 .A2-exchangelocalsearchisahill-climbingprocedurewhi chexamines pairsofallelesandperformsaswap.Iftheresultingswapin creasesthetnessofthe individual,theswapiskept.Otherwise,itisundoneananot herpairisexamined.Such localimprovementmethodsaboundintheliteratureandareu sedtoenhancemethods suchasgreedyrandomizedadaptivesearchprocedures(GRAS P)[ 68 ],tabusearch[ 69 ], andothercombinatorialoptimizationheuristics[ 70 ]. procedure LocalSearch ( X ) 1 X X 2 f ( X ) f ( X ) 3 temp 0 4 while X isnotlocallyoptimal do 5 for i =1to j X j do 6 for j =1to j X j do 7 temp X ( i ) 8 X ( i ) X ( j ) 9 X ( j ) temp 10 if f ( X ) >f ( X ) then 11 X X 12 else 13 temp X ( i )/ undoswap / 14 X ( i ) X ( j ) 15 X ( j ) temp 16 endif 17 endfor 18 endfor 19 endwhile 20 return ( X ) endprocedure LocalSearch Figure3-5.2-exchangelocalsearch. 65

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ComputationalResults Theproposedheuristicwasimplementedinthe C ++programminglanguageand compliedusingGNU g ++version3 : 4 : 4,usingoptimizationragsO2 .Itwastestedona PC equippedwitha1700MHzIntel R r Pentium R r Mprocessorand1 : 0gigabytesofRAM operatingundertheMicrosoft R r Windows R r XPProfessionalenvironment. Asabasisforcomparison,weexaminetheresultsforthehybr idgeneticalgorithm (HGA)withthestandardgeneticalgorithm(GA).Inaddition ,wehaveimplementedthe integerprogrammingmodelforthe targetvisitationproblem usingtheCPLEX TM version10optimizationsuitefromILOG[ 21 ].CPLEXcontainsanimplementationofthe simplexmethod[ 22 ],andusesabranchandboundalgorithm[ 23 ]togetherwithadvanced cutting-planetechniques[ 24 25 ].TheinstancesweretestedusingtheCPLEXdefault settings.Thealgorithmsweretestedonasetofrandomlygen eratedinstancesvarying insizefrom8-16targets 1 .Duetothecomplexityoftheproblem,CPLEXwasunable toobtainoptimalsolutionsforinstanceswith j N j > 16.Foreachinstance,thenumber of\experts"usedtoderivetheutilitymatrixis10.Also,th ematriceswerebalanced usingtheheuristicdescribedinEquation( 3{40 )above.Foreachinstance,themaximum distancebetweenthetargetsvariedfrom20to150units.The distancematrixforeach instancewasgenerateduniformlyatrandomwithsomeuserde nedupperandlower bounds.Theinstanceswerecreatedusingasimpleproblemge neratorwrittenin C ++. Theinstancesusedinthischapterarenon-symmetricandnon -metric;however,this optionisbuiltintothegenerator.Itisareasonableassump tiontohavenon-symmetric instances,sinceinareal-worldscenariothematrix D mightrepresentotherfactorsthan simplythedistancebetweentwotargetssuchasanextracost orriskassociatedwith visitingaparticulartarget.Inthiscase d ij 6 = d ji necessarily.Furthermore,itisassumed thattheUAViscapableoftravelingallcyclesinthegraph.D ependingontheindividual 1 Thetestproblemsmaybedownloadedfrom http://plaza.ufl.edu/clayton8/tvp.tar.gz 66

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applicationandfactorssuchasthenumberoftargetsandthe sizeofthebattlespace,it maybereasonabletoimposeahardconstraintonthetotaldis tancetraveledinthe tvp tour.Thiswouldmodelproblemscenarioswhenbatteryconsu mptionandfuelcapacity arecritical.Intheinstancestestedwedonotimposesuchco nstraints;howeverdoingso wouldmostlikelyreducetheoverallCPLEXrunningtime. Wementionherethatintheinstancesconsidered,thepriori tyfunctionshavebeen normalizedbythedistancefunctioninordertoavoidthedom inationofonecostfunction bytheother.Thisjustiesthecostofobjectivefunction.F orabetterrealizationof thisfact,wemadetestrunsontheinstancessolvingboththe tsp and lop problems separately.TheseresultsareincludedinTable 3-1 .Tofurtherillustratethis,Table 3-2 providestheobjectivefunctionvalueofthe lop tvp ,and tsp whenoneobjective isconsideredandsolvedtooptimality.The lop solution(non-optimalcomponent) correspondingtothe tsp optimalroute(optimalcomponent)andthe tsp solution (non-optimalcomponent)correspondingtotheoptimal lop route(optimalcomponent) havebeenpresented.Thenon-optimalcomponentsareweaker whencomparedtoeach componentintheoptimal tvp solution.Thisnumericalevidencesupportstheclaimthat inordertondhighquality tvp routes,itisnotadvisabletodecoupletheprobleminto simplya travelingsalesmanproblem or linearorderingproblem ,butratherto considerbothobjectivesinconcert. Thealternativewayofsolvingabiobjectiveoptimizationp roblemisbydetermining asetofecientsolutions.Ansolutionisecientifboththe costfunctionsaredominated byanyothersolution[ 71 ].Thisisachievedbytreatingtheproblemwithasingleobje ctive functionandrecursivelysolvingtheproblembyrecomputin gthenewcostsbasedonthe costobtainedfromthepreviousiteration.Thishoweverass umesthattheboththecost functionsinvolveswiththesameproblem.Inourcase,wehav eonecostfunctiontobe employedfora tsp problemandanothercostfunctionfora lop problemandanecient 67

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algorithmforsolvingboththeproblemshastobedesignedfo rthispurpose,whichis beyondthescopeofthischapter. Table3-1.Comparativeresultsoftheoptimalsolutionstot hecorresponding tsp lop and tvp foreachinstance.Theabsolutevalueofthe tsp solutionsare reported. Instance loptvptsp NameTargetsOptimalOptimalOptimal SolutionSolutionSolution rand 8-18110 : 08560 : 276631 rand 8-28197 : 139115 : 94455 rand 8-38335 : 74195 : 33376 rand 8-4859 : 222229 : 007420 rand 8-58646 : 16314221 rand 10-110259 : 926157 : 40474 rand 10-21024720821 rand 10-310734 : 569520 : 679149 rand 10-410720 : 167532 : 597 rand 10-510565 : 781365 : 125105 rand 12-112208 : 29124 : 17956 rand 12-212491 : 677318 : 38104 rand 12-312646 : 772420 : 959142 rand 12-412944 : 093594 : 546169 rand 12-512640 : 055472 : 35489 rand 14-114204 : 414137 : 60939 rand 14-214549 : 708405 : 77472 rand 14-314897 : 804631 : 711153 rand 14-414292 : 389176 : 63165 rand 14-514976 : 921679 : 625131 rand 16-116518 : 62381 : 93463 rand 16-216706 : 98431 : 531164 rand 16-316571 : 735415 : 33999 rand 16-416707 : 22421 : 658162 rand 16-516364 : 152249 : 93968 Numericalresults Webeginbypresentingsomecomparativeresultsoftheheuri stics.AsGoncalveset al.correctlyindicate,despitethemassiveamountsoflite ratureongeneticalgorithms, thereislittleknowledgeofhowbesttotunetheparametersf oragivenapplication[ 72 ]. Foralloftheinstancestested,theparametersusedfortheg eneticalgorithm(GA)andthe 68

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Table3-2.Thecorrespondingobjectivefunctionvaluesofe achofthe lop tsp ,and tvp aregivenforeachinstance.Foreachcolumn,oneoftheobjec tivesisconsidered andtheproblemsolvedtooptimality.Thesolutionoftherem ainingtwo problemsisgivenwhenevaluatedwiththeoptimalfunctionv alue. Instance lop opt tvp opt tsp opt NameTargets tsploptsploptsplop rand 8-18-100110 : 085-3191 : 2766-3191 : 2766 rand 8-28-174197 : 139-77192 : 944-55161 : 486 rand 8-38-242335 : 74-115310 : 333-76219 : 592 rand 8-48-3959 : 222-2958 : 0075-2044 : 6444 rand 8-58-502646 : 16-221574-221501 : 84 rand 10-110-139259 : 926-94251 : 404-74203 : 68 rand 10-210-55247-37245-21195 rand 10-310-648734 : 569-199719 : 679-149575 : 743 rand 10-410-401720 : 167-120652 : 5-97580 rand 10-510-344565 : 781-123488 : 125-105391 : 979 rand 12-112-184208 : 29-64188 : 179-56170 : 941 rand 12-212-328491 : 677-110428 : 38-104392 : 211 rand 12-312-379646 : 772-149569 : 959-142514 : 653 rand 12-412-721944 : 093-279873 : 546-169674 : 352 rand 12-512-513640 : 055-138610 : 354-89512 : 341 rand 14-114-150204 : 414-48185 : 609-39170 : 287 rand 14-214-336549 : 708-84489 : 774-72427 : 967 rand 14-314-707897 : 804-157788 : 711-153670 : 773 rand 14-414-280292 : 389-93269 : 631-65195 : 421 rand 14-514-774976 : 921-176855 : 625-131708 : 104 rand 16-116-399518 : 62-92473 : 934-63421 : 125 rand 16-216-671706 : 98-185616 : 934-164564 : 846 rand 16-316-392571 : 735-110525 : 338-99457 : 987 rand 16-416-693707 : 22-178599 : 658-162565 : 597 rand 16-516-290364 : 152-83332 : 939-68288 : 484 hybridgeneticalgorithm(HGA)aregiveninTable 3 .Ithasbeenshownintheliterature thatparameterssimilartothoseweimplementedhavebeene ectivewhenimplementing ahybridGA[ 67 72 { 74 ].Table 3-4 presentsthecomparativeresultsoftheHGAand Table3-3.ParametersusedfortheGAandHGAheuristics. CrossProb =0 : 7PopulationSize( PopSize )=2 j N j MaxGen =10000 j BEST j = : 1 PopSize j WORST j = : 2 PopSize standardGAappliedto25randomlygeneratedinstances.The numberoftargetsranges 69

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from8to16.Fiveinstancesweretestedforeachvalueof j N j .Thetableprovidesthe instancenameandthecorrespondingnumberoftargets.Next theoptimalsolutions areprovidedalongwiththecorrespondingcomputationtime requiredbyCPLEX.We testedeachoftheheuristics250timesoneachinstance,and weprovidethemaximum, minimum,andaveragesolutionscomputedforeach.Finallyf orbothheuristics,weprovide theaveragecomputationtimeforthe250runsaswellastheav eragedeviationfromthe optimum. Noticethatforall6250experiments,thehybridGAcomputed optimalsolutions 99 : 93%ofthetimerequiring2 : 687secondsofcomputationtimeonaverage.Thecompares favorablywiththeaveragetimerequiredbyCPLEXtocompute theoptimalsolutions whichwas601 : 428seconds.Wenoteherethataheuristicsolutionwasuseda sastarting pointforthecomputationoftheoptimalsolutionsforthein stancescontaining16targets. Withoutthis,therunningtimeforCPLEXfortheseproblemsw asontheorderof75000 seconds.Weseealsothatthestandardgeneticalgorithmper formedreasonablywell,with anaverageoptimalitygapof4 : 429%.Inaddition,thestandardGAscaledwellaveraging lessthanonehalfsecondofcomputationtimeforallinstanc es.However,weseethat ultimatelythehybridizedalgorithmwasthemostrobustoft hetwomethods.Forthe HGA,theincreaseistheaveragesolutiontimeforthelargei nstancesisarguablyoset byitsstellarperformance.Tothecontrary,onemightargue thatgivenmoretimethe performanceofthestandardGAwouldmatchthatofthehybrid method.However,inthe nextsubsectionweperformaprobabilisticanalysisonthet imerequiredforeachheuristic tocomputeatargetvalue,andweshallseethatthisargument isultimatelyuntrue. Time-to-targetplots Inthissubsection,weinvestigatetheempiricaldistribut ionsoftheheuristicrunning times.Ithasbeenobserved[ 75 { 77 ]thatsolutiontimesforstochasticheuristicssuchas 70

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Table3-4.Thistableprovidesthenumericalresultsforase tofrandomlygeneratedinstances.Therstcolumnsprovide informationabouttheinstance.Next,theoptimalsolution andrequiredcomputationtimeislisted.Boththe HGAandthestandardGAwereran250timesoneachinstance,an dweprovidethemaximum,minimum,and averagesolutionscomputedbyeachforall250tests.Theave ragecomputationtimerequiredbyeachheuristicto computethebestsolutionisalsolisted. InstanceIPModelHybridGA StandardGA NameTargetsOptimalExecutionMaxMinAvg.Avg.Avg.MaxMin Avg.Avg.Avg. SolutionTime(s)SolnSolnSolnTime(s)Dev(%)SolnSolnSol nTime(s)Dev(%) rand 8-1860 : 27660 : 0160 : 276660 : 276660 : 27660 : 005-60 : 276656 : 340459 : 88950 : 0540 : 642 rand 8-28115 : 9440 : 02115 : 944115 : 944115 : 9440 : 047-115.944112.653115.6810 : 0440 : 27 rand 8-38195 : 3330 : 01195 : 333195 : 333195 : 3330 : 011-195 : 333194 : 96188 : 5550 : 0325 : 006 rand 8-4829 : 00740 : 0229 : 007429 : 007429 : 00740 : 027-29 : 007425 : 859228 : 8880 : 0520 : 412 rand 8-583140 : 033143143140 : 001-3143143140 : 008rand 10-110157 : 4043 : 01157 : 404157 : 404157 : 4040 : 111-157 : 404140 : 133140 : 1330 : 12310 : 972 rand 10-2102082 : 87208204207 : 9840 : 3070 : 008208200207 : 360 : 0440 : 308 rand 10-310520 : 6790 : 01520 : 679520 : 679520 : 6790 : 092-520 : 679437 : 12518 : 6980 : 0490 : 380 rand 10-410532 : 52 : 45532 : 5532 : 5532 : 50 : 059-532 : 5489 : 667529 : 8910 : 1070 : 49 rand 10-510365 : 1254 : 87365 : 125365 : 125365 : 1250 : 034-365 : 125303 : 615349 : 4570 : 0684 : 291 rand 12-112124 : 17945 : 37124 : 179124 : 179124 : 1790 : 129-124 : 179106 : 645121 : 0220 : 1482 : 54 rand 12-212318 : 3861 : 17318 : 38318 : 38318 : 380 : 039-318 : 38266 : 641308 : 6680 : 1283 : 050 rand 12-312420 : 95951 : 89420 : 959420 : 959420 : 9590 : 191-420 : 959341 : 866403 : 310 : 09514 : 193 rand 12-412594 : 54616 : 44594 : 546594 : 546594 : 5460 : 427-594 : 546487 : 099580 : 9560 : 1372 : 286 rand 12-512472 : 35414 : 68472 : 354472 : 354472 : 3540 : 305-472 : 354409 : 102456 : 7350 : 1313 : 307 rand 14-114137 : 609303 : 55137 : 609137 : 609137 : 6090 : 792-137 : 609110 : 948128 : 2080 : 2256 : 832 rand 14-214405 : 774370 : 01405 : 774397 : 503405 : 7412 : 1280 : 008405 : 774334 : 807383 : 210 : 295 : 561 rand 14-314631 : 711184 : 82631 : 711614 : 917631 : 6442 : 7950 : 011631 : 711508 : 412594 : 7650 : 2675 : 849 rand 14-414176 : 631301 : 71176 : 631172 : 789176 : 6033 : 1480 : 016176 : 631146 : 979164 : 3770 : 2506 : 938 rand 14-514679 : 6252700 : 78679 : 625679 : 625679 : 6251 : 546-679 : 625530 : 617638 : 1610 : 2256 : 101 rand 16-116381 : 9341353 : 77381 : 934376 : 039381 : 628 : 9540 : 082381 : 934298 : 207351 : 2750 : 3518 : 027 rand 16-216431 : 5312556 : 77431 : 531414 : 606428 : 7510 : 0820 : 645431 : 531333387 : 6590 : 32210 : 167 rand 16-316415 : 338462 : 5415 : 338408 : 324415 : 07413 : 3580 : 064415 : 338332 : 868380 : 3190 : 3298 : 431 rand 16-416421 : 6582810 : 45421 : 658409 : 171419 : 30512 : 9030 : 558417 : 109314 : 109386 : 3550 : 3978 : 372 rand 16-516249 : 9393788 : 49249 : 939243 : 534249 : 0999 : 6760 : 336249 : 939187 : 592234 : 1920 : 3266 : 3 71

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geneticalgorithms,tabusearch,andGRASPtatwo-paramet ershiftedexponential distribution[ 78 ].Morespecically,let P : R 7! [0 ; 1]beaprobabilitymeasureona Borelset.Thentheprobabilityof not ndingatargetsolutionin t timeunitsisgivenby P ( t ):= e ( t ) = ,where 2 R + and 2 R Foreachinstance,wemake100runsofboththehybridGAandth estandardGA. Therunsareassumedtobeindependentsinceeachwasdoneusi ngadierentseedforthe randomnumbergenerator.Foreachinstance/heuristicpair ,thealgorithmsranuntilthey calculatedatargetsolutionandtherequiredcomputationt imewasrecorded.Thenfor eachinstance,therunningtimesforeachheuristicwassort edindescendingorder.The i th sortedrunningtime, t i isassociatedwiththeprobability p i :=( i 1 2 ) = 100andthepoint z i =( p i ;t i ),forall i =1 ;:::; 100[ 77 ].The z i pointswerethenplottedinwhatisknownas a Time-to-TargetPlot (TTTplot)usingthepubliclyavailable perl software tttplots 2 by Aiex,Resende,andRibeiro[ 78 ]. SinceitisunreasonabletoprovideaTTTplotforeachinstan cetested,instead weprovidearepresentativesubsetalltheinstancesinFigu res 3-6 3-8 .Noticethat forallcasesthehybridGAconvergesfasterthanthestandar dGAwhichisvisualized bythefactthattheHGAcurvesarecompletelytotheleftofth estandardGAcurves inalltheTTTplots.TheTTTplotsimplythatforaxedamount oftime,thehybrid methodhasahigherprobabilityofreachingthetargetsolut ion.Forexample,consider theplotforinstance rand12-1 showninFigure 3-6 .Weseethatgivenonesecondof computingtime,theprobabilitythatthehybridGAwillcomp utetheoptimalsolution is0 : 926,comparedwithaprobabilityof0 : 443forthestandardGA.Likewise,foraxed probability,theplotindicatesthatthehybridmethodwill ndthetargetsolutionquicker 2 Availableat http://www.research.att.com/ ~ mgcr/tttplots/ 72

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thanthesimplegeneticalgorithm.ConsidertheTTTplotfor instance rand14-2 in Figure 3-7 .InorderfortheGAtocomputethetargetsolutionforaxedp robabilityof 0 : 6,approximately10 : 6secondsofcomputationtimearerequired.ThehybridGAreq uires 2 : 32secondstondthetargetsolutionwith60%success.TheTT Tplotsparticularly highlightthescalabilityandrobustnessofthehybridgene ticalgorithmwhentested onlargerinstancesasindicatedbythenearverticalplotso ftheHGAprobabilitiesin Figure 3-8 .Themainpointtobemadehereisnottoarguethatthehybridg enetic algorithmoutperformsthestandardmetaheuristicinterms ofobjectionfunctionvalue. Ofcourse,ithasbeenshownthatthestandardgeneticalgori thmwillconvergetothe optimalsolutionwithprobability1.Whatwehavedemonstra tedisthatbyenhancingthe metaheuristicwiththeapplicationofalocalsearch,weare abletodramaticallydecrease thecomputationtimerequiredtoconvergetotheoptimalsol ution.Wecanconcludethat forverylarge-scaleinstancesofcombinatorialproblemss uchasthe tvp ,theadvantageof usingthehybridGAenablesustondhighqualitysolutionsm uchfasterthanthebasic geneticalgorithm.Forproblemsinvolvingmilitaryapplic ationssuchasthe tvp timeis usuallycritical.Timespentsearchingforagoodsolutionc anleadtothelossofequipment orthedeathofpersonnelinthebattleeld.Inthesecases,t headvantageofthehybrid methodisclear.Thequickerasolutioncanbecomputed,thef asterthesystemcanbe deployedandthecompetitiveadvantageisretainedontheba ttleeld. 73

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0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 14 16 18 20 cumulative probabilitytime to target solution rand12-1: target value = 124.179 GA HGA 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 cumulative probabilitytime to target solution rand12-2: target solution = 318.38 GA HGA Figure3-6.Time-to-TargetplotcomparingtheHybridGAand standardGAforinstance rand12-1 .Thetargetvalueistheoptimalsolutionfortheproblem. 74

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0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 140 160 cumulative probabilitytime to target solution rand14-1: target value = 137.609 GA HGA 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 cumulative probabilitytime to target solution rand14-2: target solution = 405.774 GA HGA Figure3-7.Time-to-Targetplotforinstance rand14-2 .Asabove,thetargetvaluesisthe optimalsolution. 75

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0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 140 160 cumulative probabilitytime to target solution rand16-1: target solution = 362.8373 GA HGA 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 140 cumulative probabilitytime to target solution rand16-4: target solution = 527.1199 GA HGA Figure3-8.Time-to-Targetplotforinstances rand16-1 .Thetargetvalueis : 95timesthe optimalsolution. 76

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3.2CommunicationModelsforaCooperativeNetworkofAuton omous Agents Mostcooperativenetworksrequirecoordinationamongtheg roupofusersin ordertoaccomplishtheobjective.Thecoordinationofthes ystemusuallydepends oncommunicationbeingguaranteedamongsttheagents.Inty picaladhocnetworks, bandwidthandcommunicationtimeareverylimitedresource s.Therefore,weseethat thelackofacentralcommandcenterforMANETs,whileappeal ingfromadistributed perspective,doesleadtoseveralproblemsintermsofrouti ng,communication,and path-planning[ 79 80 ].Perhapsthemostimportantamongthese,andthefocusofth is chapter,isthestudyofcommunicationmodelsinthenetwork .Inparticular,westudy theproblemofcoordinatingasetofwirelessagentsinvolve dinataskthatrequires themtotravelfromasourcelocationtoadestination.Theob jectiveistodetermine thepaths,ortrajectoriesfortheagentswhichmaximizesth econnectivitybetweenthem subjecttoconstraintsontheinitialandnalconguration s,andseverallimitationsonthe movementsoftheagents[ 81 ].Thisproblemisknownasthe cooperativecommunicationprobleminmobileadhocnetworks(ccpm) ,andisknowntobe NP -hard [ 15 ].Inthenextsection,wereviewthecurrentworkonthe ccpm ,whichisprimarily focusedonheuristicsfortheproblemposedasadiscreteopt imizationproblem. ProblemFormulation Consideranundirectedgraph G =( V;E ),where V = f v 1 ;v 2 ;:::;v n g represents thesetofavailablepositionsforthewirelessagents.Each nodein V isassumedtobe connectedonlytonodesthatcanbereachedinonetimestep.A lso,dene N ( v ) 2 V for v 2 V ,torepresentthesetofneighbors,ornodes,whichareadjac enttonode v .Let U representthesetofagents, S = f s 1 ;s 2 ;:::;s j U j g V thesetofinitialpositions,and D = f d 1 ;d 2 ;:::;d j U j g V thesetofdestinationpositionsfortheagents.Givenatime 77

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horizon T ,theobjectiveoftheproblemistodetermineasetofroutesf ortheagents,such thateachagent u i 2 U startsatsourcenode s i andnishesatitsrespectivedestination node d i afteratmost T unitsoftime[ 49 ]. Foreachagent u 2 U ,thefunction p t : U V returnsthepositionoftheagentat time t 2f 1 ; 2 ;:::;T g .Thenateachtimeinstant t ,anagent u 2 U caneitherremainin itscurrentlocation,i.e. p t 1 ( u ),ormovetoanodein N ( p t 1 ( u )). Wecanrepresentarouteforanagent u 2 U asapath P = f v 1 ;v 2 ;:::;v k g V where v 1 = s u v k = d u ,and,for i 2f 2 ;:::;k g v i 2 N ( v i 1 ) [f v i g .Finally,if fP i g j U j i =1 isthe setoftrajectoriesfortheagents,wearegivenacorrespond ingvector L suchthat L i isa thresholdonthesizeofpath P i .Thisvalueistypicallydeterminedbyfuelorbatterylife constraintsonthewirelessagents. Weassumethattheagentshaveomnidirectionalantennasand thattwoagentsin thenetworkareconnectedifthedistancebetweenthemisles sthansomeradius r 2 R Theparticularvalueof r isdeterminedbythecapabilitiesofthewirelessequipment suchastheantennastrengthandpoweramplier.Morespeci cally,let : V V R representtheEuclideandistancebetweenapairofnodesint hegraph.Then,wecan deneafunction c : V V !f 0 ; 1 g suchthat c ( p t ( u i ) ;p t ( u j ))= 8>>><>>>: 1 ; if ( p t ( u i ) ;p t ( u j )) r 0 ; otherwise : (3{41) 78

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Withthis,wecandenethe ccpm asthefollowingoptimizationproblemasgivenby Commanderetal.[ 49 ]: max T X t =1 X u;v 2 U c ( p t ( u ) ;p t ( v )) (3{42) s.t. n i X j =2 ( v j 1 ;v j ) L i ; 8P i = f v 1 ;v 2 ;:::;v n i g (3{43) p 1 ( u )= s u 8 u 2 U (3{44) p T ( u )= d u 8 u 2 U; (3{45) whereconstraint( 3{43 )ensuresthatthelengthofeachpath P i islessthanorequaltoits maximumallowedlength L i Ithasbeendetermined[ 81 ]thattheproblemdescribedaboveis NP -hard.Thiscan beshownbyareductionfromthewellknown3 sat problem.Moreover,itis NP -hardeven tondanoptimalsolutionforonestageoftheproblematagiv entime t .Toseethis, consideranalgorithmthatmaximizesthenumberofconnecti onsattime t ,bydeningthe positionsformembersofthenetwork;clearlythealgorithm justdescribedcomputesthe valueofthemaximumcliqueontheunderlyingunitgraph[ 82 ].Runningthisalgorithm fordierentsets U i ,with j U i j = i and i varyingfrom1to T ,thealgorithmstopswhenthe numberofconnectionsislessthen i 2 ,andthevaluereturnedis i 1.Computingoptimal solutionsforthe maximumcliqueproblem onaunitgraphisknowntobe NP -hard [ 83 ]. Duetothecomputationalcomplexityoftheproblem,real-wo rldinstancescannot besolvedexactly.Therefore,weturnourattentiontothede signandimplementationof ecientheuristicstosolvelarge-scaleinstanceswithinr easonablecomputingtimes.Inthe followingsection,wereviewtherecentworkinthisareaand describetheimplementation 79

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oftherstadvancedmetaheuristicforthe ccpm basedontheGreedyRandomized AdaptiveSearchProcedure[ 68 ]. PreviousWork Sincetheintroductionofthe ccpm byOliveiraandPardalos[ 81 ],heuristicdesignhas beenamajorfocus[ 49 82 ].In[ 49 ],theauthorsintroducedaconstructionheuristicfor the ccpm basedonshortestpaths[ 13 ].Thegoalwastoprovideawaytoquicklycalculate setsoffeasibletrajectoriesfortheagents.Pseudo-codef orthisalgorithmisprovidedin Figure 3-9 .Theproceduretakesasinputaninstanceofthe ccpm consistingofthegraph G ,thesetofagents U ,sourcenodes S ,destinationnodes D ,andamaximumtraveltime T .Thetotalnumberofconnections( c )representsthevalueoftheobjectivefunctionfrom equation( 3{42 )andisinitializedtozero.Thesetoftrajectoriesforthea gents( solution ) isinitializedtotheemptyset.Inline3,wecomputetheshor testsource-destinationpath foreachagentusingtheFloyd-Warshalalgorithm[ 84 85 ].Foreachagent i 2 U ,the correspondingshortestpathisassignedasthetrajectory P i fortheagent.Thetrajectory isfeasibleifagent i isabletoreachitsdestination d i inatmost T timeunits.Anyagent whichreachesitstargetlocationinlessthan T timestepswillremainthereuntilall otheragentsreachtheirrespectivedestinations.Ifanypa thisinfeasible,thealgorithm terminates.Otherwise,thenumberofconnectionsisupdate dandtheprocessrepeatsuntil allagentshavebeenconsidered. Theaforementionedalgorithmprovidesfeasiblesolutions forinstancesofthe ccpm in O ( j V j 3 )time.However,thetrajectoriescalculatedarenotguaran teedtobelocally optimal,letalonegloballyoptimal.Therefore,alocalnei ghborhoodsearchenhancement wasapplied.Ingeneral,alocalsearchmethodreceivesafea siblesolutionasinput andreturnsasolutionthatisnoworsethantheoneinput,wit hrespecttothegiven neighborhoodstructure.Findingalocallyoptimalsolutio ninthelocalsearchphase 80

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procedure ShortestPath ( G;U;S;D;T ) 1 c 0 2 solution ; 3 Computeallshortestpaths SP ( s i ;d i )foreachpair( s i ;d i ) 2 S D 4 for i =1to j U j do 5 P i SP( s i ;d i ) 6 if lengthof P i >T then 7 return ; 8 else 9 solution solution [P i 10 c c +newconnectionsgeneratedby P i 11 end 12 end 13 return ( c,solution ) endprocedure ShortestPath Figure3-9.Pseudo-codefortheshortest-pathconstructio nheuristic. depends,amongotherthings,ontheactualstructureandden sityoftheneighborhood. Let S bethesetoffeasiblesolutionsforaninstanceofthe ccpm .Thenforsome s 2S theneighborhoodof s ,denoted N ( s ),canbedenedasthesetofallsolutions s 2S that dierfrom s inexactlyoneroute.Noticethatthenumberoffeasiblepath sbetweenany source-destinationpairisexponential,andcouldleadtou nreasonablecomputationtimes. Thereforeinsteadofexhaustivelysearchingtheentirenei ghborhoodtheauthorsprobe only j U j neighborsateachiteration(oneforeachsource-destinati onpair).Also,because oftheexponentialsizeoftheneighborhood,themaximumnum berofiterationsperformed waslimitedtoaconstant MaxIter Pseudo-codeforthelocalimprovementheuristiccanbeseen inFigure 3-10 .Let f representtheobjectivefunctionforthe ccpm asgiveninequation( 3{42 )above. Newroutesarecomputedusingarandomizedversionofthesta ndarddepth-rst search(DFS)[ 13 ].Asmentionedin[ 49 ],ateachstepoftherandomizedDFS,thenode selectedtoexploreisuniformlychosenamongtheavailable childrenofthecurrentnode. Randomizationhelpstondaroutethatmayimprovethesolut ion,whileavoidingbeing 81

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trappedatalocaloptimumafteronlyafewiterations.Thelo calsearchisastandard hill-climbingmethod[ 24 ].Beginningwiththefeasiblesolutionfromtheshortestpa th constructor,thelocalsearchbeginscomputingnewtraject oriesfortheagentsbyusingthe randomizedDFStoexploretheneighborhoodasdescribedabo ve.Themethoditerates overalltheagentsandrepeatsatotalof MaxIter iterationsafterwhichthecurrentbest solutionisdeemedlocallyoptimalandreturned. procedure HillClimb ( solution ) 1 c f ( solution ) 2 while solution notlocallyoptimal and iter < MaxIter do 3 for i =1to j U j do 4 solution solution nfP i g 5 P i DFS ( s i ;d i ) 6 c' f ( solution [ P i ) 7 if lengthof P i c then 8 c c' 9 iter 0 10 else 11 Restorepath P i 12 end 13 endfor 14 iter iter +1 15 endwhile 16 return ( solution ) endprocedure HillClimb Figure3-10.Pseudo-codefortheHillClimbingintensicat ionprocedure. Wenowdescribetheimplementationofamoreadvancedrandom izedmulti-start heuristicforthe ccpm basedontheGreedyRandomizedAdaptiveSearchProcedure (GRASP)[ 68 ]framework.GRASPisatwo-phasemetaheuristicforcombina torial optimizationthataimstondverygoodsolutionsthoughthe controlleduseofrandom sampling,greedyselection,andlocalsearch.GRASPhasbee nusedextensivelyinthelast decadeonnumerousoptimizationproblemsandproducesexce llentresultsinpractice[ 86 ]. Let F bethesetoffeasiblesolutionsfortheaproblem,whereeac hsolution S 2 F 82

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procedure OnePass ( G;U;S;D;T ) 1 solution ShortestPath ( G;U;S;D;T ) 2 solution HillClimb ( solution ) 3 return ( solution ) endprocedure OnePass Figure3-11.Pseudo-codefortheone-passheuristic. iscomposedof k discretecomponents a 1 ;:::;a k .GRASPconstructsasequence f S g i of solutionsfor,suchthateach S i isfeasiblefor.Attheend,thealgorithmreturnsthe bestsolutionfound. procedure GRASP ( MaxIter ) 1 X ; 2 for i =1to MaxIter do 3 X ConstructionSolution ( G;g;X ) 4 X LocalSearch ( X; MaxIterLS ) 5 if f ( X ) f ( X ) then 6 X X 7 end 8 end 9 return X endprocedure GRASP Figure3-12.GRASPformaximization. Pseudo-codefortheGRASPisprovidedinFigure 3-12 .NoticethateachGRASP solutionisbuiltintwostages,called greedyrandomizedconstruction and intensication phases.Theconstructionphasereceivesasparametersanin stanceoftheproblem,a rankingfunction g : A ( S ) R (where A ( S )isthedomainoffeasiblecomponents a 1 ;:::;a k forapartialsolution S ),andaconstant0 << 1.Itstartswithanempty partialsolution S .Assumingthat j A ( S ) j = l ,thealgorithmcreatesalistofthebest ranked l componentsin A ( S ),andreturnsauniformlychosenelement x fromthislist. Thecurrentpartialsolutionisaugmentedtoinclude x ,andtheprocedureisrepeateduntil thesolutionisfeasible,i.e.until S 2 F 83

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Theintensicationphaseconsistsoftheimplementationof ahill-climbingprocedure. Givenasolution S 2 F ,let N ( S )bethesetofsolutionsthatcanfoundfrom S by changingoneofthecomponents a 2 S .Then, N ( S )iscalledaneighborhoodof S .The improvementalgorithmconsistsofnding,ateachstep,the element S suchthat S =argmax s 2 N ( S ) f ( s ) ; where f : F R istheobjectivefunctionoftheproblem.Attheendofeachst ep,we assign S S if f ( S ) >f ( S ).Thealgorithmwillconvergetoalocaloptimum,inwhich casetheprocedureabovewillgenerateasolution S suchthat f ( S ) f ( S )foreach S 2 N ( S ). ToapplyGRASPtothe ccpm ,weneedtospecifytheset A ,thegreedyfunction g ,theparameter ,andtheneighborhood N ( S ),for S 2 F .Thecomponentsofeach solution S arefeasiblemovesofamemberoftheadhocnetworkfromanode v toa node w 2 N ( v ) [f v g .Thecompletesolutionisconstructedaccordingtothefoll owing procedure.Startwitharandom u 2 U andndtheshortestpath P from s u to d u .If thetotaldistanceof P isgreaterthan D u ,thentheinstanceisclearlyinfeasible,andthe algorithmends.Otherwise,thealgorithmconsiderseachfe asiblemove.Afeasiblemove connectsthenalnodeofasub-path P v ,for v 2 U nf u g ,toanothernode w ,suchthatthe shortestpathfrom w to d v hasdistanceatmost D v P e 2 P v dist( e ).Thesetofallfeasible movesinasolutionisdenedas A ( S ). Thegreedyfunction g returnsforeachmovein A ( S )thenumberofadditional connectionscreatedbythatmove.Asdescribedabove,theco nstructionprocedurewill ranktheelementsof A ( S )accordingto g ,andreturnoneofthebest j A ( S ) j elements. Thisisrepeateduntilacompletesolutionfortheproblemis obtained. 84

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Theimprovementphaseisdenedbytheperturbationfunctio n,whichconsistsof selectingawirelessagent u 2 U andreroutingit,i.e.ndingacompletepathusingthe proceduredescribedaboveforeachtimestep1to T .Thesetofallperturbationsofa solution S isitsneighborhood N ( S ).Ateachstep,allelements u 2 U aretested,andthe procedurestopswhennosuchelement u thatimprovesthecurrentsolutioncanbefound [ 82 ]. ContinuousFormulations Inthissection,wepresentcontinuousformulationsofthe ccpm [ 87 ].These formulationswillprovideamorerealisticscenariothanth ediscreteformulationprovided above,inthatmovementisnotrestrictedtoadiscretesetof positions.Wewillassume thattheagentsareoperatinginabattlespace Q R d ,where Q isacompact,convex setwithunitvolumeandtheEuclideannorm jjjj 2 in R d .Forourpurposes,weare goingtoconsidertheplanarcase,i.e. d =2,withtheunderstandingthatextensions tohigherdimensionsareeasilyachieved.Supposethereare M wirelessagentsinthead hocnetwork.The M agentsareassumedtobeomnidirectionalandaremodeledasp oint masses.Weallowtheagentstomovefreelywithin Q atsomeboundedvelocity. Formulation1:AcontinuousanalogofCCPM-D Inordertoderiveacontinuousformulation,weneedtotode neanobjectivefunction thatisconsistentwiththatofthediscreteformulation.Le t R ij bethecommunication constantforagents i and j .Thatis, R ij istheradiusofcommunicationforthetwoagents. Onepossibleobjectiveistomaximizethe Heavisidefunction ,denedas H 1 R ij jj ~x ( t ) i ~x ( t ) j jj 2 = 8>>><>>>: 1 ; if jj ~x ( t ) i ~x ( t ) j jj 2 R ij 0 ; if jj ~x ( t ) i ~x ( t ) j jj 2 >R ij : (3{46) 85

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Figure3-13.TheHeavisidefunction, H 1 Agraphicalrepresentationof H 1 isdisplayedinFigure 3-13 .Whilethisfunctionwillwork asanobjective,itisveryextremeinthesensethatthereisa largejumpfromperfect communicationatdistanceslessthanorequalto R ij tonocommunicationassoonasthe distancebecomeslargerthan R ij .Amoredesirablefunctionisonethatapproximates H 1 butdegradesinacontinuousfashionfromperfecttonocommu nication. Weconsidertwoalternativesto H 1 .Therstisapiecewisecontinuous,linear functiondenedby H 2 R ij jj ~x it ~x jt jj 2 = 8>>>>>>><>>>>>>>: 1 ; if jj ~x it ~x jt jj 2 R ij 2 jj ~x it ~x jt jj 2 R ij ; if R ij < jj ~x it ~x jt jj 2 2 R ij 0 ; if jj ~x it ~x jt jj 2 2 R ij : (3{47) Thisfunction,whosegraphisprovidedinFigure 3-14 ,hasavalueequaltooneifagents i and j arewithinthecommunicationradius R ij ofoneanother.Thefunctionthen decreasesconstantlyuntiltheagentshavedistance2 R ij ,atwhichtimetheyareunableto communicate. 86

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Figure3-14. H 2 ,continuousapproximationto H 1 Thethirdandnalobjectivefunctionwewillconsiderisaco ntinuouslydierentiable decreasingfunctionofthedistancebetweenagents i and j .Thisfunction,displayedin Figure 3-15 ,isdenedby H 3 jj ~x it ~x jt jj 2 ;R ij = e jj ~x it ~x jt jj 2 R ij 2 : (3{48) Thisisperhapsthebestapproximationof H 1 inthatitcanbeinterpretedasthe probabilityofagents i and j directlycommunicatingasafunctionofthedistancebetwee n them. Nowthatwehavefoundasuitableobjectivefunctionwecande netheremaining parametersandconstraintsoftheproblem.Let ~x i ( t )bethepositionofagent i attime t Similarly,let ~v i ( t )bethevelocityofagent i attime t .Therelationshipbetweenvelocity andpositionisthestandardone,givenby ~v i ( t )= dx i ( t ) dt .Inordertoformulatethe continuoustimeanalogofthe ccpm ,wemustconstrainthemaximumspeedofeach agent.Thisisthecontinuoustimeanalogoftheconstraints onthemaximumdistance traveledinthediscreteformulation,betweenanytwotimes teps.If s i 2 R 2 isthestarting 87

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Figure3-15. H 3 ,continuouslydierentiableapproximationof H 1 positionofagent i ,and d i 2 R 2 isthedestinationpointofagent i ,thenwecanformulate the continuouscooperativecommunicationproblemonmobileadh oc networks(ccpm-c) asfollows. max Z T 0 X i
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veryinteresting.Alternatively,considerasetofUAVsinv olvedinasearch-and-rescueor reconnaissancemission.Forobviousreasons,missionsoft hissortgenerallyrequirethe UAVstotraversealargeportionofthebattlespacebeforear rivingattheirdestinations.In thiscase,theaboveformulationisnothelpful.Withthisin mind,wemoveontodevelop asecondcontinuousformulationwhichnotonlymaximizesth ecommunicationbetween theagents,butalsomaximizesthecoverageofpredenedreg ionsofthebattlespace. Formulation2:Acontinuousformulationensuringlocation visitations Inthefollowingparagraphs,wederiveasecondcontinuousf ormulationwhich guaranteesthatcertainlocationswillbevisitedbytheUAV sastheytraversethe battlespacefromtheirsourcestotheirrespectivedestina tions.Previousworkontarget visitationproblemsappearin[ 88 ].Onceagain,weareconsideringasetof M UAVs.We keeptheassumptionthatthe i thUAVstartsataposition s i =( s ix ;s iy ),attime0,and endsatposition d i =( d ix ;d iy ),attime T .The i thUAV,attime t 2 [0 ;T ],hasposition ~x i ( t )=( x i ( t ) ;y i ( t )).Assumethatthefollowingholds: ~x i ( t ) 2 [ x low ;x high ] [ y low ;y high ] 8 i =1 ;:::;M;t 2 [0 ;T ] : Furthermore,assumethatthereexists J positionsinthedomain,eachofwhichmust bevisitedbyatleastoneUAVinthetimeinterval[0 ;T ].Thesepositionsaregivenby Q j =( x j ; y j ),foreach j =1 ;:::;J .Lastly,the i thUAVhasamaximumspeedgivenby maxi foreach i =1 ;:::;M andweassumetheminimumspeedtobezero. Inordertoimplementasolutiontechniqueinadigitalcompu ter,wemakeuseofthe L 1 -normasameasureofthedistancebetweentwopointsanddisc retizethetimedomain into equaltimesteps, t = T= ( 1).Let t k = k t; foreach k =0 ;:::; 1.Thus thepositionofthe i thUAVattimestep k isgivenby ~x i ( t k )=( x i ( t k ) ;y i ( t k )),foreach i =1 ;:::;M ,andforeach k =0 ;:::; 1. 89

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Thentheproblem,whichisdenotedas CCPM-C ,canbewrittenas: min X i 1
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Wecanlinearizethemixedintegerprogrammingformulation in( 3{54 )-( 3{61 )as follows.Tobeginwith,replacetheobjectivefunction( 3{54 )with X i 1
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Finally,wereplace( 3{58 )with ijk + ijk =0(3{72) andaddtheconstraints ijk x i ( t k ) x j ; 8 i; 8 j; 8 k (3{73) ijk x i ( t k ) x j ; 8 i; 8 j; 8 k (3{74) ijk y i ( t k ) y j ; 8 i 8 j 8 k (3{75) ijk y i ( t k ) y j ; 8 i; 8 j; 8 k (3{76) ijk ijk ( x high x low ) ; 8 i; 8 j; 8 k (3{77) ijk 0 ; 8 i; 8 j; 8 k (3{78) ijk ijk ; 8 i; 8 j; 8 k (3{79) ijk ijk 1 ijk x high x low ; 8 i; 8 j; 8 k (3{80) ijk ijk ( y high y low ) ; 8 i; 8 j; 8 k (3{81) ijk 0 ; 8 i; 8 j; 8 k (3{82) ijk ijk ; 8 i; 8 j; 8 k (3{83) ijk ijk 1 ijk y high y low ; 8 i; 8 j; 8 k: (3{84) Thustheproblembecomesoneofminimizing( 3{62 )subjecttotheconstraints( 3{55 ), ( 3{56 ),( 3{59 )-( 3{61 ),( 3{63 )-( 3{84 ).Theresultingformulationisamixedintegerlinear program(MILP)andcanbesolvedusinganumberofcommercial softwarepackages.In thefollowingsection,wepresentsomepreliminaryresults fromonesuchpackage,aswell asprovidingadiscussionoftheexperiments. 92

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0 2 4 6 8 10 0 1 2 3 4 5 6 7 8 9 UAV1 UAV2 UAV3 UAV4 UAV5 J Figure3-16.Examplewith5agents. 0 2 4 6 8 10 0 1 2 3 4 5 6 7 8 9 10 UAV1 UAV2 UAV3 UAV4 UAV5 UAV6 UAV7 J Figure3-17.Examplewith7agents. CaseStudies WehaveimplementedtheMILPformulationofthe ccpm usingtheCPLEX TM optimizationsuitefromILOG[ 21 ].CPLEXcontainsanimplementationofthesimplex 93

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method[ 22 ],andusesabranchandboundalgorithm[ 23 ]togetherwithadvanced cutting-planetechniques[ 24 25 ]. 0 2 4 6 8 10 0 1 2 3 4 5 6 7 8 9 10 UAV1 UAV2 UAV3 UAV4 UAV5 UAV6 UAV7 UAV8 UAV9 UAV10 J Figure3-18.Examplewith10agents. Theinstancesweretestedongridsofsize10.Thesetofcoord inates, J ,tobevisited weregenerateduniformlyatrandom.Threesetsofcoordinat esweregeneratedandeach visitedbythreedierentsetsofUAVs,numbering5,7,and10 .The y -coordinatesofthe startingandendingpositionswerealsorandomlygenerated usingauniformdistribution andthe x -coordinateswereassumedtobe0and10respectively.Thesc enarioswere solvedmakinguseoftheMILPformulationderivedabove.The optimalsolutionswere obtainedfortheinstanceswith5UAVs.Theinstanceswith7U AVsand10UAVswere stoppedatoptimalitytolerancesof10%and25%respectivel y.Atime-frameof10units wasprovidedasaninputandtheminimumandmaximumspeedoft heUAVswere0and 2unitsrespectively. Wehaveprovidedthreegraphicalrepresentationsofthetra jectoriesoftheagents fromeachproblemset.Figure 3-16 showsthepathstraversedinoneofthescenarios 94

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containing5agents.Themovementsoftheagentsarefromlef ttorightinthegure.The pointswhichmustbevisitedaredenotedasstars.Weseethat fromtheirstartingpoints, theagentstendtoconvergeintoatightformation.Noticeth atUAV2separatesfrom thegrouparoundpoint(7 ; 6)inordertovisitthree\mustvisit"pointsandarriveatit s destination.Theremainingagentstraveltogetheruntilth eymustdivergetoreachtheir destinations. InFigure 3-17 wehaveanexamplescenariocontaining7agents.Asbefore,t heagents quicklyconvergetoatightlycoupledformationwithagents leavingthegrouponlyto visitthepointsin J orarriveattheirdestinations.A10agentscenarioisdepic tedin Figure 3-18 andsimilarbehaviorsoftheagentscanbeobserved.Figures 3-19 3-24 show howthepathsoftheagentschange(fromthe10agentscenario inFigure 3-18 )asagents arerandomlyremovedfromthescenario.Asexpected,therem ainingagentsareforcedto spreadoutinordertoensurevisitingthe J targetpoints. 0 2 4 6 8 10 0 1 2 3 4 5 6 7 8 9 10 UAV1 UAV2 UAV3 UAV4 UAV6 UAV7 UAV8 UAV9 UAV10 J Figure3-19.Examplederivedfrom10agentexample,withone agentremoved. 95

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0 2 4 6 8 10 0 1 2 3 4 5 6 7 8 9 10 UAV1 UAV2 UAV3 UAV4 UAV6 UAV7 UAV8 UAV10 J Figure3-20.Examplederivedfrom10agentexample,withtwo agentsremoved. Theresultspresentedareverypromising.Indeed,theagent sexhibittheexact behaviorwewouldexpecttoseegiventhenatureofthe ccpm .Ascommunication strengthisinverselyproportionaltodistance,theconver gencetoacommonpathclearly indicatesthattheUAVsareattemptingtomaximizethecommu nicationamongstthe group.ConsiderthescenarioinFigure 3-16 .Noticethatmidwaythroughthemissiona veryclearclusteringeectcanbeseenastwodistinctgroup sofagentsmaketheirway towardsthe\mustvisit"pointsandultimately,theirdesti nations. 96

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0 2 4 6 8 10 0 1 2 3 4 5 6 7 8 9 10 UAV1 UAV3 UAV4 UAV6 UAV7 UAV8 UAV10 J Figure3-21.Examplederivedfrom10agentexample,withthr eeagentsremoved. 0 2 4 6 8 10 0 1 2 3 4 5 6 7 8 9 10 UAV1 UAV3 UAV4 UAV7 UAV8 UAV10 J Figure3-22.Examplederivedfrom10agentexample,withfou ragentsremoved. 97

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0 2 4 6 8 10 0 1 2 3 4 5 6 7 8 9 10 UAV3 UAV4 UAV7 UAV8 UAV10 J Figure3-23.Examplederivedfrom10agentexample,withve agentsremoved. 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 UAV3 UAV4 UAV7 UAV8 J Figure3-24.Examplederivedfrom10agentexample,withsix agentsremoved. 98

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Conclusion Inthischapter,westudiedtheso-called targetvisitationproblem whose objectiveistoplanthesequenceforanunmannedaerialvehi cletovisitasetoftargets whichminimizesthetotaldistancetraveledandmaximizest heutilityofthesequence. Untilnow,theliteratureonthisproblemhasbeenslight[ 48 ].Thischapterpresentsthe rstextensivecomputationalanalysisfortheproblem.Fir stweprovidedamathematical modelforthe tvp basedonintegerlinearprogrammingandprovedthatndinga n optimalsolutionis NP -complete.Toovercomethecomputationalcomplexity,wede scribed theimplementationofarandomkeysbasedgeneticalgorithm forndingnearoptimal solutions[ 50 ].Theheuristicwasthenhybridizedbytheimplementationo falocalsearch procedure.Thenumericalresultspresenteddemonstratedt heeectivenessoftheproposed procedure.Outof6250experiments,thehybridheuristicca lculatedoptimalsolutions forover99 : 9%ofthetrialsinafractionofthetimerequiredbythecomme rcialinteger programmingsolverCPLEX. Sincethe tvp isarelativelynewproblemintheliterature,thereareseve raldirections forfutureresearch.Clearlyothermetaheuristicscanbeim plementedandcomparedwith GAapproach.However,duetothecomputationalcomplexity, decompositiontechniques shouldmostlikelytobethefocusinordertodetermineoptim alsolutionsforlarge-scale instancesoftheproblem.Extensionstothemodelproposeda realsopossiblesuchas imposingaconstraintonthetotaldistancetraveledandgen eralizingthemodeltoinclude multiplevehicles.Visitingmobiletargetswouldpresento therinterestingchallenges beyondthemodelpresentedinthischapter. Wealsoprovidedareviewofrecentworkintheareaofcoopera tivecommunicationin mobileadhocnetworks.Whileinherentlyaproblemofpathpl anning,ourformulations incorporatedcommunicationasameasureofthetnessofagi vensolution.Wepresented 99

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somediscreteversionsoftheproblemandderivedtwocontin uousformulations,therst timethishasbeenconsidered.Theadvantageofthenewmodel sisthattheyensurethata speciedamountofthebattlespaceisexploredbytheagents .Thisadditionisimportant inreal-worldapplicationsparticularlyintheareasofsur veillance,reconnaissance,and rescueoperations.Thepreliminarynumericalresultsdemo nstratetheeectivenessofthe proposedmodels. Duetotheinherentcomplexityoftheproblem,futureresear chwillfocuson continuousheuristictechniquesforthenewlyproposedmod els,similartothosefoundin [ 89 90 ].Percentileriskconstraintswillbeincorporatedintoth eformulation.Commonly appliedinnancialapplications,riskmeasuressuchasVal ue-at-Risk(VaR)and ConditionalValue-at-Riskhaveproventobeeectivetools formilitaryapplicationsas well[ 8 80 91 ].Wealsoplantoprovidesometheoreticalresultsregardin gfeasibilityof probleminstances. 100

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CHAPTER4 REVIEWOFEVACUATIONPROBLEMS 4.1Introduction Hurricanes,earthquakes,industrialaccidents,nucleara ccidents,terroristattacks andothersuchemergencysituationsposeagreatdangertoth elivesofthepopulace. Evacuationduringthesesituationsisonewaytoincreasesa fetyandavoidescalationof damages.Evacuationproblemsarebeinggivenincreasedatt entionoverthelastveyears. Thetechniquesthatarecurrentlyemployedcouldbebroadly categorizedintooptimization orsimulationmethods.Ineithercase,theevacuationprobl emisdealtoveranetwork wherearcsoredgesaretheroadslinkingtwoplacesorthenod esofthenetwork.The typicalfactorsusuallytakenintoconsiderationbythesem odelsareorigin-destination assignment,responsetimeoftheevacuees,modesoftranspo rtation,contrarowsetc. Factorssuchasorigin-destinationassignmentandarccapa citiescouldbesetasstatic ordynamicandthesedecisionsheavilyinruencetheevacuat ioneciency.Mostofthe recentsurveysandreviewsintheeldofevacuationweremad eforspecicinstances [ 92 93 ].Thesurveyscatertoaspecictechniqueanddiscussindet ailontheirimpacton emergencyevacuation.Thischapterismoreacrosstheboard .Theobjectiveistopresent acomprehensivereportonthetechniquesthatareavailable intheliteratureandbroadly classifythem.Aninitialclassicationwasmadedepending ontheapproachesusedinthe evacuationmodels,namelyoptimization-basedorsimulati on-basedapproaches.Further, asubclassicationwasmadebasedonsomevitalfeaturescon sideredbythemodel.These arefeaturesthatareexpectedtohaveasignicantimpacton evacuationeciency.The solutionmethodologies,foreverymodelundertheseclassi cations,werediscussedindetail andwerealsoassessedbasedontheircomputationalperform ance,scalability,extensibility, realizabilityandthemajorcomponentsconsidered.Comput ationaleciencyofthemodel isimportantincaseofunforeseenevents.Themodelsneedst obeexecutedquicklyto generatealternativeplansandprepareforthedynamicscen arios.Thecomplexityof 101

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theevacuationproblemsmakestheresearchersresorttofor heuristicproceduresand simulationbasedapproaches.Thusthesolutionstestedand providedneedstobescalable, orinotherwords,realizableonlargerinstances.Mostofth emodelsarecustomized tospecicsituationsbutwhenneededtoaccommodateadditi onalfeaturesandhandle moreparameterstheyneedtobeextensible.Therealizabili tyofthemodelisachieved whentheyaretestedonrealtimenetworksandmostofthetheo reticalguaranteesare accomplished.Thechapternallyconcludeswithfewwordso ntheshortcomingsofthe availablemodelsthatmayrequireattentionandlaysdownvi talfeaturesthatanew designerhastoseekinamodel. 4.2OptimizationTechniques Alargenumberofoptimizationmodels,sometimesreferredt oasanalyticalmodels intheliterature,havebeendevelopedforevacuationstudi es.Whilesomeofthesemodels arediscrete,therestareextensionsfromthesebasemodels .However,thesemodelsare essentiallyasimplenetworkrowproblemtryingobtainamin imumcostrowfromsource todestination.Adetaileddiscussionoftherowproblemsis carriedoutinthefollowing section.MaximumDynamicFlow Anelementaryevacuationrowproblemscouldbeformulateda salinearinteger programusingavariantofthemaximumdynamicrowproblem[ 94 95 ].Themaximum dynamicrowproblemistodeterminethemaximumamountrowfr omoriginto destinationwithinaspecictimeT.FordandFulkersonform ulatedthisproblemon atimeexpandedstaticnetwork,whereeachnodeandedgeisre placedbyTcopies correspondingtoeachtimeinstance[ 94 ].Givenadigraph G ( V;E )andtimeintervalT, let c ( u;v )and t ( u;v )bethecapacityandtraversaltimeofarc( uv ) 2 E .Let x ( u;v; )be theamountofrowleavingnode u alongarc( u;v )attime .Lettheoriginnodebe s and destinationnodebe t .Assumingholdingoverofrowoveranynodeisallowed,wehav e thefollowingintegerlinearprogrammingproblemthatsolv esthemaximumdynamicrow 102

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problem: Maximize T X =0 X ( su ) 2 E x ( s;u; ) T X =0 X ( us ) 2 E x ( u;s; t ( u;s )) (4{1) s.t. X ( uv ) 2 E x ( u;v; ) X ( vu ) 2 E x ( v;u; t ( v;u ))=0 ; 8 u 6 = s;t;u 2 V; =0 ; 1 ;:::;T (4{2) [ T X =0 X ( su ) 2 E x ( s;u; ) T X =0 X ( us ) 2 E x ( u;s; t ( u;s ))] +[ T X =0 X ( tu ) 2 E x ( t;u; ) T X =0 X ( ut ) 2 E x ( u;t; t ( u;t ))]=0 (4{3) 0 x ( u;v; ) c ( u;v ) ; 8 ( uv ) 2 E; =1 :::T (4{4) Theobjectivefunctionthathastobemaximizedgivesthenet amountofrowleavingthe originnodesbytimeperiodT.Theconstraintset( 4{2 )ensuresconservationofrowat everynode,wheretheamountofrowthatentersanodeisexact lyequaltotheamount ofrowthatleavesthenodeatanytimeperiod.Theconstraint set( 4{3 )ensuresthat attheendofTtimeperiodstheamountofrowthatleavestheor iginsisequaltothe amountofrowthatentersthedestinationt.Theaboveformul ationsolvesamaximum rowproblemoveratimeexpandedgraph.Theproblembecomese xtremelydicultto solveforlargergraphswithabiggertimeframe.However,in [ 94 ],theauthorssuggesteda strictlypolynomialalgorithmforthemaximumdynamicrow. Theysolvedtheminimum costrowproblemontheoriginalgraph,withouttimeexpansi on,anddecomposedthe rowintoasetofpaths.Thentheyobtainedthemaximumdynami crowbytemporally repeatingtherowalongthepaths. Mostoftheevacuationproblemsramifyfromthemaximumdyna microwproblem. Thequickestrowproblem,sometimesreferredtoasevacuati onproblem,istodetermine theminimumtimerequiredtosendagivenamountofrowfromth eorigintothe destination.Thisproblemisasimplevariationofthemaxim umdynamicrowproblem 103

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andcouldbesolvedthroughabinarysearch.In[ 96 ],theauthorsprovedthisreduction fromthemaximumdynamicrowproblemtothequickestrowprob lemandprovideda stronglypolynomialtimealgorithmthroughaparametricse archonthequickesttimeby repeatedlysolvingthemaximumdynamicrowproblem.Manyot herworkhavebeendone inthislinegeneralizingthisconcept.In[ 97 ],aworkonmulticommodityshipmentofrows wasprovided.Amulticommodityrowproblemonastaticgraph withoutatimeboundis: Givenagraphwithaarctraveltimeandcapacityoneacharcan dasetofcommodities K =1 ;:::;k witheachcommodityhavingspecicorigin s i anddestination t i ,itis requiredtosendaspecicamountofrowfrom s i to t i ofthecorrespondingcommodityin theminimumamountoftime.Theproblemisformulatedinthef ollowingmanner. Minimize k X i =1 X ( uv ) 2 E c ( u;v;i ) x ( u;v;i ) (4{5) s.t. X ( uv ) 2 E x ( u;v;i ) X ( vw ) 2 E x ( v;w;i )=0 ; 8 v 2 V;i =1 ;:::;k (4{6) X ( ut i ) 2 E x ( u;t i ;i ) X ( t i v ) 2 E x ( t i ;v;i ) d i ; 8 i =1 ;:::;k (4{7) 0 x ( u;v;i ) 8 uv 2 E;i =1 ;:::;k (4{8) k X i =1 x ( u;v;i ) ; c ( u;v ) 8 ( uv ) 2 E;i =1 ;:::;k (4{9) x ( u;v;i )istheamountofrowonarc( uv )ofcommodity i and c ( u;v;i )isthecostofunit rowonarc( uv )ofcommodity i .Theconstraintset( 4{6 )impliestherowconservation atanodeforaparticularcommodityandtheconstraintset( 4{7 )ensuresthatthesinks node t i receives d i amountofrow.Finally,( 4{8 )and( 4{9 )restraintstherowcapacity oneacharc.Theaboveisamulticommodityrowproblemforast aticgraph.Inorder tosolvethequickestmulticommodityproblemwehavetoexte ndtheformulationtoa timeexpandedgraphsimilartothepreviousproblemformula tion.However,thisresults innumerousconstraintsandvariablesifthetimehorizon,T ,consideredinlarge.Each nodeandarcofthegraphisreplacedbyTcopies,eachcorresp ondingtothespecic 104

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timeinstant.Themulticommodityrowproblemisknowntobe NP -hardevenfora staticgraph[ 98 ].Toovercomethetimeexpansiondicultyascalingalgorit hmcouldbe employed,whereineachnodeandarcisreplacedby T= copiesinsteadT,thusreducing theproblemsizeconsiderablyandstrikingabalancebetwee ntheprecisionoftheresult andtherunningtimeofthealgorithm[ 97 ].In[ 99 ],theauthorsprovidedsolutionsfor threevariationsofmaximumdynamicrowproblem.Theyprovi dedapolynomialtime approximationfortheearliestarrivalrowproblem,whichw asstudiedby[ 100 ]and [ 101 ].Theearliestarrivalrowproblemrequirestherowtobemax imizedateachtime stepofthegivenhorizon,unlikethemaximumdynamicrowpro blem.Theproposed algorithmisbasedonsuccessiveshortestpathalgorithm,w heretherowisaugmented alongtheshortestpath,quickestpathinourcase,andthech aindecompositionisgiven byaugmentationsperformedinasequenceofstaticgraphs.H owever,successiveshortest pathisapseudo-polynomialalgorithm.Thisdicultyisusu allyhandledbyscaling. Unliketraditionalscaling,theproposedalgorithmperfor msanupwardcapacityscaling. Adynamicrowquickestpathwithasmallcapacitycouldberep eatedtemporallyto obtainamaximumrow.Wereferto[ 99 ]foradetailedaccountofthealgorithmandproof ofapproximation.Thesecondproblemstudiedwaslexicogra phicmaximumdynamic row.Givenasetofsourcesandtheirpriorityofevacuationt helexicographicmaximum dynamicrowmaximizestherowsleavingthesourcesinthespe ciedorder.Finally, theystudiedthequickestrowproblemwithxednumberofsou rcesanddestinations withequalprioritieshavingspecicsupplydemands.In[ 102 ],theauthorspointedout thattheproblemofdeterminingquickestrowfromanysubset ofnodestoanyother subsetofnodesisequivalenttoasinglesourceshortestpat hproblem.Thusforaxed numberofsourcesanddestinationsthisproblemispolynomi allysolvable,ifweconsiderall possiblesubsetsofthesourcesanddestinations.Inthesam efashion,foraxednumber ofsourceanddestination,thelexicographicmaximumdynam icrowcouldbesolvedforall possibleorderingofthesourcestosolvetheaboveproblem. Adetailedandmoreecient 105

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algorithmwasgivenin[ 103 ]forthisquickesttransshipmentproblemwithmorethanone originanddestination.Thesemodelsbasedonmaximumdynam icrowproblemhavetheir practicallimitationsastheyareoblivioustofactorssuch ascongestioncontrol,dynamic originanddestinationdemandandmultimodaltransportati on,whicharequiteconceivable inarealtimesituation.Therowproblemsareratherasimpli edversionsofrealtime model.Themodelsarecomputationallyecientwithpolynom ialtimesolutionsforexact orapproximatesolutions.Theproblemsthuscouldhelpinpr ovidingquicksolutionsto largeproblemsizesunderasimpliedsetting.Therealizab ilityofthemodelislimited buttheircomputationaleciencycouldbetakenadvantageo finpreprocessingstagesof analyticaltechniquesandheuristicdevelopments.DynamicTracAssignment In[ 104 ],thedynamictracassignment(DTA)problemwasintroduce dand formulatedasanon-linearprogramwithanon-convexmathem aticalprogram.The problemistondanassignmentofrowsonthelinksoptimally .Likebefore,letusassume theplanninghorizontobeT.And G ( V;E )bethedigraphunderconsideration.Let x ( u;v; )betheamountofrowonarc( u;v )attime .Let F ( u; )betheexternalinput innodeuattime and h uv; ( x ( u;v; ))bethecostfunction.Also,let g uv ( x ( u;v; ))be theexitfunctiondenotingtheamountofrowthatexitsfroma rcu,vduringperiod Finally, d ( u;v; )betheamountofrowenteringarc( uv )atnodeu.TheDTAproblemis formulatedasfollows: 106

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Minimize T X =0 X ( uv ) 2 E h uv; ( x ( u;v; )) (4{10) s.t. x ( u;v; +1) x ( u;v; )+ g uv ( x ( u;v; )) d ( u;v; )=0 ; 8 ( uv ) 2 E; =1 ; 2 ;:::;T (4{11) X 8 uv 2 E d ( u;v; ) F ( u; ) X 8 pu 2 E g pu ( x ( p;u; ))=0 ; 8 u 2 V n t; =1 ; 2 ;:::;T (4{12) 0 x ( u;v; ) ; 8 ( uv ) 2 E; =0 :::T (4{13) 0 d ( u;v; ) 8 ( uv ) 2 E; =0 :::T (4{14) ThisworkbyMerchantandNemhauserwastheopeningtodynami ctrac assignment.Theyformulateddiscretepiecewiselinearize dversionofthetracassignment problem.Theproblemassumesthatthedemandsareavailable andithasasingle destination.Inmostoftheemergencysituationstheseassu mptionsarenotpreserved.For instance,ofamulticommodityrowwheretheitisrequiredto maintaintheorigin-destination pairasingledestinationmodelmaynotbesuitable.Also,dy namicdemandinplaceof astaticdemandestimateismoreecientandmorepreciseina real-timesituation.In [ 105 ],themodelwasvalidatedbyprovingthattheconstraintssa tisfylinearindependence constraintqualicationastheexitfunctioniscontinuous lydierentiable.In[ 106 ],the authorslinearizedtheexitandcostfunctionandobtainedt heglobaloptimumforthe nonlinear,non-convexoptimizationproblembysolvingase riesofT+1optimization problems.Alinkrownon-linearmixedintegerprogrammingf ormulationandaconvergent dynamicalgorithmtosolvethedynamicuserequilibriumpro blemwasprovidedin [ 107 ].Itexplicitlyseeksequilibriumintermsofpathtravelti mesunlikeMerchant andNemhauser'smodel.Themodeldependsonstaticuseequil ibriumfunctionswith additionalconstraintstoensuretemporallycontinuousro w.Thetechniqueitselfisnot 107

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pertinenttoevacuationstudiesandhencewerefertothewor kin[ 108 ]whomadea detailedstudyondynamictracassignmentintheirsurvey. Wearemoreinterestedinits applicabilityinemergencysituation,howeverthelittleb ackgroundprovidedisrequired. Thecellbaseddynamictracassignmentthatsegmentedtheh ighwaylinksinto equalsizedcells,suchthateachcellcouldbetraversedina nunittimewasprovided in[ 109 ].Eachcellhasaspeciccapacityandthecongestionisexpl icitlyhandledby restrainingtheamountofrowfromonecelltoanother.In[ 110 ],Daganzo'sformulation wasrelaxedbyholdingofrowatnodesintherowconservation equations.In[ 111 ],the authorsimplementedadynamictracmodelingtechniquebas edonthecelltransmission model[ 109 ]foranoptimalno-noticemassevacuation.Theyalsopropos edanetwork transformationtoasingledestinationnetworkandthenace llnetwork,fortheabove implementation.Theirobjectiveinno-noticeevacuationa lsoincludesidenticationof destinations.Themodelingisdonethroughagraphtransfor mation.Themodelemploys aggregationofzonesandhencecouldbescaledappropriatel ytohandlelargegraphs.The modelassumespriorknowledgeoforiginatingdemandsandzo nalinformation,whichin mostcasesareavailable.Also,theoptimizationformulati onwasprovidedforacell-based transmissionforatimeexpandedgraph.Themodeloptimizat ionmodelassuchmightbe timeconsumingwhileappliedoverrealtimegraphs.Theauth orspointedoutthatwithout userequilibriumconstraintsthemodelmaynotbeofpractic alinterest.Themodelmight beusefulduringno-noticemassevacuation,assumingthatt hereisanecienttechnique tosolveit,buttheassumptionsmakeitrigidtoextendittoo theremergencysituations. Anotherworkemployingcelltransmissionmodelisin[ 112 ].Theydiscussthestaged evacuationprocedure,whereinazonalclassicationofthe nodesisdonebasedonthe severityofimpactstheysuerandstartingtimeforeacheva cuationzoneisdetermined, takingtheresponsetimeofevacueesintoconsideration.Th esemodelsbasedoncell basedtransmissioninvolvesinaformulationforatimeexpa ndedstaticgraphwitheach linkreplacedbyagroupofcells.Thismethodologymaynotyi eldquickerresultsforan 108

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evacuationproblemconsideredoveralargerspaceandtimea ndthismaybeanecessity inanemergencysituation.Thecomplexityofthemodelswill furtherincreasewhen theyconsidermorecomplexfeaturessuchasuserequilibriu mconstraints,congestions management,dynamicdemandetc.Thiscomputationaldicul tycouldbeovercomethe aggregatingthenodesofthenetworkunderconsiderationan dsolvingtheproblemin asmallergraph.Theeciencycouldalsobeimprovedbyincre asingthecoarsenessof thediscretizationoftimedependingontheaccuracyofther esultsdesired.Themodels suggestedanevacuationprocedurewithsingledestination orsink,whichcannothandle multicommodityrowstoidentifyoptimaldestinations.In[ 113 ],theauthorssuggested aheuristictothecell-basedtransmissionmodelwithmulti pledestinationstoovercome thisimplementationdiculty.However,theaccuracyofthe heuristicanditsconvergence whentheproblemsizegrowsisaquestionofconsideration.I n[ 114 ],theconcernsina timeexpandedevacuationproblemareprovidedindetail.Ti meexpandednetworks,while havingcomputationallimitationsinalargescaleevacuati ons,mightbenetthesmall scalenetworkssuchasthebuildingevacuationasthenetwor kunderconsiderationis quitesmallcomparedtolargescaleevacuationnetworks.Ap seudopolynomialalgorithm forsolvingthemaximumdynamicrowandquickestrowproblem swithatime-varying traveltime,nodeandarccapacitywasprovidedin[ 115 ].Wereferto[ 116 ]formoredetails onsmall-scaleevacuation,wheretheymadeacomprehensive reviewonthebuilding evacuationmodels. Thevariationalinequalityapproachisanotherwayofformu latingthetheDTA problem[ 108 117 { 119 ].VariationalInequalityprovidesaconvenientformulati ontechnique fornetworkequilibriumproblemsarisingineconomics,na nceandtransportation.It wasputforthby[ 120 ]tostudypartialdierentialequationsproblems.In[ 121 ],the authorsstudiedtheequilibriumproblemsintransportatio nnetworksapplyingvariational inequalityforastatictraveldemand.Anitedimensionalv ariationalinequalityproblem 109

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couldbestatedas:Givenasubset k R n ndavector x 2 k suchthat h F ( x ) ;x x i 0 ; 8 x 2 k (4{15) where F : k R n isacontinuousfunction.Incaseofthetransportationprob lem the F isthecostfunctionand x istherowonaarcandnbeingthenumberofarcsin thenetwork.TheequilibriumconditionscouldbeviewedasK uhn-Tuckerconditions foraconvexnonlinearproblem.In[ 122 ],adetailedaccountonvariationalinequalities andtheirapplicationstoseveralequilibriumproblemsisg iven.Amuchwiderrange ofproblemsincludingmultimodaltransportationandelast icdemandswerediscussed alongwithadiscussionofalgorithmstonetworkequilibriu mproblemsformulatedwith variationalinequalities.In[ 123 ],avariationalinequalityformulationforacontinuous timeproblemwithdynamicroutechoicesanddeparturetimed ecisions,adynamicversion ofthestaticWardropianuserequilibrium,wasprovided.It considersthepathcosts realizedthroughtraveltimeandpenaltyforearlyorlatear rivaltimeandtheoptimalrow patternissimultaneousroutedepartureequilibrium.Asth emodelisdealtinacontinuous timedomainitmightlackagoodalgorithmtoecientlysolve complexintegralsonthe pathcostspresentedinthemodel.Anothercontinuoustimem odelfortransportation networkswasprovidedin[ 124 ].Theyconsideredasinglecommodityproblemwithelastic demandandatime-dependentequilibriumandexpressedthec orrespondingequilibrium problemasquasi-variationalinequality.Thesubset k R n inequation( 4{15 )isreplaced byapoint-to-setmap K : E R n .[ 125 ]presentedanothervariationalinequality basedequilibriumproblemforamulticommodityrowproblem wheretheuser'sdesired origin-destinationpairanddepartureandarrivaltimewin dowisprovided.Adetailed sectionofvariationalinequalitymodelsinthetransporta tionframeworkwasprovided [ 118 ]. 110

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Non-DeterministicMethods Signicantworkhasbeendoneonevacuationproblemsfromno n-deterministic perspective.Smithet.alhavedonesubstantialworkonstoc hasticevacuationnetwork [ 126 { 128 ].Theyprovidenon-inferiorsolutionstobi-objectiverou tingprobleminqueueing networks.Thetwoobjectivesmodeledwerethedistancetrav eledandtheclearancetime. Theproblemalsoconsideredthemulticommodityrowproblem .Theyprovidedapath basedbi-objectiveformulationwithrowconservationandc apacityconstraintsforallthe routingalternatives.Theroutesaregeneratedandfedtoth eaboveformulationbyan algorithmthatiterativelygeneratescandidatepathstoas sesscongestion.In[ 129 ],gave anoverviewofdeterministic,stochasticandhybridmethod sforevacuationproblemsand comparedaanalyticalqueueingnetworkwithasimulationmo delforahospitalevacuation consideringevacuationtime,congestionandoptimalroute s.In[ 130 ],twosolutionswere provided,onewithoptimalevacuationrouteandanotherwit hsetofstrategiesfromwhich theevacueecanchoosethebestarcateachstep.Theydealtth eevacuationproblem onnetworkwithstochastic,time-varyingtraveltime.In[ 131 ],theauthorsconsidereda stochasticdynamicnetworkinwhichtheorigindestination demandarerandomvariables withknownprobabilitydistributions.Alinearprogrammin gformulationbasedonthe systemoptimaldynamictracassignmentpropagatingtrac bycelltransmissionwas provided.Theresultsarerobustastheybuildacondencele velwhichrequiresthe solutionstomeettheexpecteddemands.In[ 132 ],theauthorsincorporatedcapacity constraintsincelltransmissionmodelforsystemoptimald ynamictracassignment. Thecapacityconstraintsareprobabilisticinnaturedueto theimpactsofthedisaster. Theyalsocomparedthemodeltoonewithadeterministiccapa city.In[ 133 ],theresource allocationproblemwasstudiedintheemergencyevacuation network.Theyemployed queueingtheorytomodelthevaryingcapacitiescirculatio nspaces,suchasnitesized corridors,staircasesetc.andstudytheireectsonthroug hput.Theevacueesinthis 111

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modelexperienceaserviceattheevacuationnodesandthiss erviceratedecayswiththe increasingamountoftrac. 4.3SignicantFeaturesinOptimizationTechniques Thissectionwilldiscussthefactorsthatarewidelyregard edtoinruencethe evacuationeciency.Whilesomeofthemareonlygermanetoo ptimizationtechniques, therestwillberevisitedwhenwediscussthesimulationtec hniques. ContraFlows Acontraroworreversiblerowisastrategytoimprovetheec iencyofevacuation byallowingtracintheoppositedirectionoftheroadsduri nganemergency.Mostofthe recentevacuationplansproposedincorporatescontrarowi nthem.Thecontrarowswhile beingecientinevacuationtheycomewithafewpitfallslik eincreaseinaccidentsand operationcost,whichmightbeaconsiderationfortheirimp lementationintheevacuation plans[ 134 ]. Considerthefollowinglanereversalproblem:Givenadirec tedgraph G ( V;E )with non-negativearccapacitiesandtraveltimes,asubsetofno des S assourcesandplanning periodorhorizonT,determinethequickesttimetoevacuate alltheoccupantsfrom sourcetodestinationbyallowinglanereversals.Asingles ourcesinglesinkcontrarow problemispolynomiallysolvable,butamultiplesourceand sinkproblembecomes stronglymathsfNP-hard[ 135 ].Thus,evensimplecasesoftheproblemishardtosolveand researchershavetoresorttoheuristic.Simulationproced urescouldbeperformedfora givencongurationsofnetworksandhencemaynotbeasuitab letooltoidentifyoptimal laneidentication,buttheanalyticalproceduresarenotc omputationallyecientevenfor simplemodelswithoutcomplexfeatures[ 136 ].f In[ 137 ],theauthorsproposedtwoheuristicstosolvetheevacuati onproblemwith contrarowmorethanonesourceanddestination.Therstheu risticwasproposed foratimeexpandedstaticgraph G T ( V T ;E T ).Theheuristicrunsonatimeexpanded graph,makingitlessattractiveforlargescaleevacuation withlargenumberofnodesand 112

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withevacuationplannedoverabiggertimeframe.Thesecond heuristicproposedwas asimulatedannealingprocedure.Thealgorithmbeginswith solutionandperturbsita littletoobtainabettersolutionandthisiterativeproced ureiscarriedoutuntilastopping conditionissatised.Thisisastandardtechniqueemploye dtoescapelocalminimum andobtainvaluescloseglobalminimum.Asimilartabu-base dheuristicalgorithmtosolve lanereversibleevacuationproblemwasproposedin[ 138 ].Itisalocalsearchprocedure attemptingtogetthebestsolutioninagivenneighborhood. Thesystemoptimaldynamic tracassignmentmodelforthecellbasedtransmissionwhic hwasdescribedinsection 2.2isconsideredfortheimplementationofcontrarowsandi tpermitspartialcapacity reversal,unliketheformerapproach.Inanotherwork[ 139 ],theyformulatedthedynamic tracassignmentmodelwithlanereversiblecapabilities. Thisinformationcouldbeused asanecientinitialsolutionforthetabubasedsearch.The modelsdiscussedaresimple rowproblemswithlanereversalcapabilityandisfarfrombe ingpracticallyrealizable. Themodelscouldhoweverserveasagoodpreprocessingtechn iquetoidentifythelanesto bereversedandthenamorecomplexmodelcouldbeemployedon arecongurednetwork. Theheuristicsdevelopedtosolvethem,thoughcomputation allyecient,doesnothave anytheoreticalguaranteeinthequalityofthesolutions.T hustheissueofscalabilityis aconcern,asthequalityofthesolutioncouldbecompromise dwiththeproblemsize. Approximatesolutionstothecontrarowproblemsforsuchgu aranteesisstillanopenarea ofstudy.EvacueeBehavior Likecontrarows,anotherfactorthatisincorporatedinthe evacuationmodelsinthe recentyearsisthetheresponsetimeoftheevacuees.Theres ponsetimeisthetimetaken foradecisionbytheevacueetoevacuate.Thislatencycould beduetotheineciency intheinformationdissemination,panicinevacueesorthee mergencysituationitself. Conventionalmodelsdisregardthis,assumingzeroloading andunloadingtime.In[ 140 ], theauthorsformulatedaNuclearpowerplantevacuationthr oughbilinearprogramming 113

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andincorporatedtheresponsetimeoftheevacuees.Theyuse dtheformulationin[ 118 ] fordynamictracassignment.Theyproposedadeparturemod elwiththeobjectiveof quickestevacuationtimewithonemodeldeparturetimefrom originandanothermodel arrivaltimeatthedestination.In[ 141 ],theauthorssimulatedtheescapepanicsituation inwhichtheclogging,jammingatwidening,panicinitiatio nandimpatiencearecaptured. Thiswouldhelpinvalidatinganevacuationmodeldeveloped withoutconsideringhuman factors.However,thestudywasaimedtomodelthepedestria nbehavior,injuriescaused duetocongestion,uncoordinatedpassingofbottlenecks,p hysicalinteractionsamong people.Thuscarefulscalingofparametersisrequiredifwe needtorealizethisinlarger networkwithvehicularmotions.Themodelwouldmoreservea satoolforcalibration thanforestablishingoptimalroutesofevacuation.Thesur veyabouthumanbehavior inre[ 142 ]givesagistoftheinruenceofevacueebehavioronmodels.T hestudy catersspecicallytothesituationcausedbyreandcertai nconcernsraised,suchas smokeeects,realarms,cannotbeextendedtoagenericsit uation.Thiswouldhelp inasimilarfashionasthepreviousmodelinsettingstandar dsforanevacuationmodel andfeaturestobeidentiedinamodel.Amodelwasdeveloped in[ 143 ]topredictthe responsebehaviorofhurricaneAndrewinLouisianaandthey werefoundtobesimilarto hurricaneFloydinSouthCarolina.Theresponsecurvecould beusedtomodelresponse behaviorinevacuationmodels.Sincethestudywasperforme dandtestedforbehavior duringhurricane,responsebehaviorsduringsituationssu chaseventmanagement,nuclear accidents,rehazards,andno-noticeevacuationsneedsto beanalyzedinordertoassure itsgenericapplicability.[ 144 ]providedthemeasuresforeciencyofanevacuation plan.Amongthevariousfactorstheyconsideredtheloading curveofthetimetaken forevacueestodecidetoevacuatewasconsideredasamajorf actorasthisiscrucialin determiningthetraveltimeoftheindividualevacuees.In[ 145 ],theauthorssuggested thatthepanicsituationasresultofadangerresultsinthee vacueestomoveatrandom beforerestoringorder,especiallywhentheywanttondthe familymembersbeforetrying 114

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toreachsafety.Theyproposedatwostageformulationestab lishhouseholdtripchain sequence.Intherstphasetheoptimizationmodeldetermin esameetinglocationforthe familyandinthesecondphasecomprisesofthetripassignme nttoevacuatepeopleto safety.Thestrategiesarelatersuppliedtoasimulatortos tudytheeectsofreassignment. Itprovidesadierentperspectivetobehaviormodelingbut nottheonlyfeaturetobe captured. Incaseofalargescaleevacuationsystem,wherethetravelt imeismuchlarger comparedtotheresponsetimeoftheevacueestheerrorinthe caseofexcludingthe responsetimewouldbeveryminimal.Howeverincorporating itmightincreasethe complexityoftheoptimizationmodelandhencedecreasethe computationaleciency. Theimplementationintwodierentphases,likethatofcont rarows,inordertoovercome thisproblemcouldalsobedicultasthebehaviormodelingd ynamicallychangesthe tracrow.Thisisshouldbeaanotherfactortakenintoconsi derationwhiledeveloping amodel.Sometimesbehaviorialresponseisrelatedclosely tothesituationandmaybe dierentatdierenttimes.DynamicOrigin-DestinationDemands Anobservationonthemostdicultobstaclebeforedeployin gDTAisestimatingthe timedependentdemandwasmadein[ 108 ].Thedemandbetweenorigin-anddestination inanevacuationproblemistypicallysubjecttotheenviron mentalchanges.Forinstance, anevacuationscenariowhichjustrequirestheevacueestor eachsafetymightnotroute passengerstotheinitiallydesignateddestination,asthi swouldcompromisetheevacuation time.Thiscouldbebecauseofanyunforeseencircumstances suchascongestion,road blockage,lanereversalsorunavailabilityofshelteretc. Mostresearchershaveassumed tohaveorigindestinationdataavailableaprioriandtheyd on'tchangeovertime.Later, theanalysiswascarriedoutonastaticdemand.In[ 146 ],theauthorsformulatedan evacuationproblemwheretheorigindestinationpairsaren otxed.Givenasetof originordestination,theobjectiveistoshipallthecommo ditiesfromtheoriginset 115

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toanysubsetofdestinationnodes.Thisismoreintuitivein anevacuationprocess, wherethesafetyofindividualsisthepriority.Thiscouldb eachievedthroughasimple transformationinwhichadummysuper-sinknodeisaddedwit hzerocostandinnite capacityarcsfromallotherdestinations.Givenandirecte dgraph G ( V;A )andsetof originsIanddestinationsJwitharccapacities c ( a ) ; 8 a 2 A andcostfunction h ( x ( a )).Let D ij bethedemandmatrixbetweenODpair.Nowwehavethefollowin gstaticdemand evacuationproblem. Minimize X 8 a 2 A h ( x ( a )) (4{16) s.t. x ( a )= X 8 r 2 R ar p r ; 8 a 2 A (4{17) X 8 r 2 R ij p r = D ij ; 8 i 2 I;j 2 J (4{18) 0 x ( a ) c ( a ) ; 8 a 2 A (4{19) R isthesetofallpathsbetweenalloriginsanddestinations, p r istherowalongthe r t h pathand ar isindicatorvariablewithvalue1isarcaisinpathrand0oth erwise. Constraintset( 4{18 )willbereplacedby X 8 r 2 R i p r = D i ; 8 i 2 I (4{20) forthesingledestination. Thisisanotherformulationforatimeexpandedstaticgraph ,havingallthe drawbacks,withregardstocomputationaleciency,mentio nedearlier.Themodel whenassumingindependentevacueesasrowvariables,assum ethatthedierentfamily memberscouldreachdierentdestinations.Asapathbasedm odelhasbeenpresented thescalabilityofthemodelneedsmoreexaminationasthean alyticaltechniquesusing pathorroutebasedmodelingneedstoconsiderexponentiall ymanypathsavailable. Thesequentiallogitmodelwasappliedtothemodelthedynam ictracdemandfor 116

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evacuationproblemswiththeunderlyingassumptionthatth edemandvarieswiththe evacueebehavior[ 147 ].Thetimehorizonisdiscretizedandthedecisiontoevacua teat eachtimeintervalismodeledasasequenceofbinarydecisio ns.Latertheprobabilitythat ahouseholdwillevacuateataspecictimeiscalculatedusi ngrandomutilitytheory, whichinturnisusedtoestimatetimevaryingdemand.Thepro blemofupdating origin-destinationmatrixbasedonthetraclinkcounthas alsobeenstudiedin [ 148 149 ].Abi-levelprogramming,wheretheODmatrixisupdatedatt heupperlevel andtheupdatedmatrixislaterusedforDTAatthelowerlevel .Theformulationallows tracrowaccordingtothetracconditionsandtheyarebrou ghtunderanentropy maximizationframework.Anotherusefulstudyinthislinew asin[ 150 ],wherethey comparedtwomodelsofcomplianceandnon-compliancetopre determinetherouteand destinationassignmentsforevacuationusingsimulations oftware.Itwasproposedin[ 151 ], ageneticalgorithmtoestimatethedynamicdemandbetweent heorigindestinationpair. ThemethodisbasedonthefactthatdynamicdemandODmatrixd ependsonspatial andtemporalvariationsofcongestionorinotherwordsthev ariationindemandbased onthetraccountonalink.Thecongestionoflinkandhencea specicpathfroman origintodestinationwouldcausetheoptimalsolutiontose ekanalternatepathcausingan alterationindemand.Thegeneticalgorithmstriestondth eoptimalODdemandmatrix fromtheseedODmatrixprovidedasinput.MultimodalTransportation Multimodaltransportationarerealisticinanemergencysi tuationanditalso considerablyaectstheevacuationeciency.Itistherefo reessentialtoincorporate modechoicesinthemodelsinordertodevelopprecisemodels .Aformulationbasedon celltransmissionwasprovidedin[ 152 ]permitsintermodaltransportationwithcarsand busesforasingledestinationnetwork.Theformulationwas asystemoptimalinteger linearprogramming.Thusthecorrespondingrowproblemisa bimodaltransportationfor asinglecommodity.Themodelassumesnopedestrianmovemen ts,buttransitcellsare 117

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consideredwhereevacueescanmovetoanothervehicle.Them odelisaverybasicmodel andhaveseveralassumptionsthatmakesitpracticallyinfe asible.Forinstance,themodel assumesthatthereisanunlimitedsupplyofbusesandcars.A lso,themodelscontains enormousamountintegervariablesandconstraints.Additi onofmulticommodityrow constraintswouldmakethemodelevenmorecomplex.In[ 153 ],theauthorsstudiedthe inruenceofmultimodaltransportationontheevacuatione ciencyinbuildingcomplexes. Theyextendedthecellularautomatamodel,wherespaceisdi videdintocellsandtime isdiscretizedsuchthatvehiclescanmovefromonecelltoan otherinonetimestep, tocapturetheinteractionbetweeninpedestriansandvehic lebehavior.Toovercome thecomputationalcomplexityposedbytheformulationsint heoptimizationmodels investigatorsexploredheuristicalgorithmstosolvethee vacuationproblem.In[ 154 ],the authorsproposedaheuristicalgorithm,HASTE,whichprovi dedacloseapproximationto theoptimalsolutionthatcouldbeachievedthroughalinear programforthecellbased dynamictracassignmentproblem.Thisovercomesthedicu ltyofthecomputational timespentontheoverwhelmingsizeoftheproblematapriceo fapproximatingthe optimalsolution.Theaccommodationofmultimodaltranspo rtationusuallyincreasesthe precisionofthemodelasitismorerealisticandalsotheeva cuationeciency.However, itincreasesthecomplexityofthemodelasawholeanddecrea sesthecomputational eciency.MiscellaneousFactors Thekeylogisticalissuesduringtheaftermathofadisaster havebeencarefully analyzedin[ 155 ].Theresponseandrecoveryoperationssuchassupplyoffoo dand medicineonreducedcapacitynetworkarediscussed.Theres earchwasbasedonpublic accountsandinterviewsmadeduringtheeldvisitstoKatri naimpactedarea.Another interestingstudyinthecontextofevacuationistoexamine thevulnerabilityofthe networkandidentifytheinsecurelinksinthenetwork.Itis importantbecauseit accountsfortheconnectivitybetweentheoriginanddestin ationduringevacuation. 118

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Thismayallowustoestimatethearcorlinkcapacitiesofthe networkforevacuation.In [ 156 ],thevulnerabilityofanetworkusingavulnerabilityinde xwasstudiedanditwas aggregatedthemoverallorigindestinationpairs.Atopolo gicalindextodeterminethe depressiveness/concentration,whichhelpsinidentifyin gisolateddistrictswasprovided [ 157 ]andestimatetherobustnessoftheroadnetworkinanemerge ncysituation.The travelchoiceistiedtothevulnerabilityofthenetwork.Th echangeinthedemandsand rowsowningtodisruptionsinthenetworkwasalsostudiedin [ 158 ].Theyformulateda traveldemandmodeltoderiveameasuretoassessthevulnera bilityofatransportation network.Themeasuresarecapableofvaluatingthechangesi nbothdemandandsupply. In[ 144 ],theauthorsusedagentbasedsimulationtechnique,PARAM ICS,tocomparethe stagedandsimultaneousevacuation.Theyconcludedthatfo rageneralnetworktopology thereisnospecicstratgeythatresultsinbetterevacuati oneciency.Howeverahigh densitypopulationcouldbeevacuatedquickeroveragridst ructurednetworkbystaged evacuation. 4.4SimulationTechniques Thelargesizeandtimeoverwhichtheoptimizationmodelsha vetobeimplemented makethemlesssuitableforimmediaterealizability.Simul ationbasedapproachesfor evacuationstrategiesarewidelyadoptedtoovercomethisp racticaldiculty.Also, simulationbasedmodelscouldbeatooltostudythecurrentp lanswithoutactually executingit.Theevacuationplansgeneratedbytheanalyti calmodelsdiscussedabove couldbesimulatedtoidentifyinconsistenciesinthemodel ,thusservingasagood instrumentforvalidation.Thesimulationmodelspermitst hedesignersandresearchersto visualizetheevacuationandhenceitismoreincisivecompa redtoanalyticalmodels. Onelevelofbreakdownofthesimulation-basedtechniquewi llbeasmacro,microor mesotracsimulations.Amicroscopictracmodelcaptures thelineamentsofindividual vehicles,whereasamacroscopictracmodelprovidesacoll ectivevehicledynamics. Macroscopicsimulationsaresimilartoruidrowstudy,wher etheestimatesarebasedona 119

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groupofvehiclesasawhole.Macroscopicsimulationsareno tcomputationallyexpensive, butfailstoadapttorandomandrapidchangesintheenvironm ent.Microsimulations isadiametricsimulationtechnique,wherethecharacteris ticsofindividualvehiclesare capturedandthusabletopredictandadapttothechangesint hemodelmoreeciently thanmacroscopicsimulations.Thedisadvantageofthistec hniqueisthecomputational expenseencounteredasthesystemsizegrowsandmorevehicl esareaddedtothesystem. However,themicro-simulatorsaregettingpopularinthere centyearswiththeincreasein thecomputationalcapacity.Adetailednoteaboutmacroand microsimulationsandtheir relativeadvantagesanddisadvantagesisdiscussedin[ 159 ]. MicroscopicSimulationTechniques Agentbasedsimulationsarealsocalledasmicro-simulator s,astheagentsordrivers involvedinthesystemcouldbestudiedindividually.Inmos ttracsimulationstudies thereisaneedforthesimulationtobeperformedwithmorepr ecision.Forinstance, vehicularinteractionsduringcongestionlanechangeortr acsignalsatintersectionscould bedealtonlyinamicroscopicmodel.In[ 160 ],theauthorsperformedanagentbased neighborhoodevacuationstudy.Themodelhasanicepropert ytostudythedisaggregate outputssuchasthevehiclesafetyandtraveltimeswithinzo nesratherthanaveragetravel timeofentiresystem.Howeversomeoftheassumptionsareri gidsuchasdeparturetimes wereassumedtobeavailableandthedestinationswerepreas signed.Theseassumptions maynotbepreservedinrealtimesituation.Thetopologyoft hetestednetworkwas relativelysimplecomparedtorealtimenetworks.Itwillbe interestingtocomparethe computationaleciencywhentestedrealtimewithfeatures suchqueues,spillovers andcongestion.ThePARAMICSisamicro-tracsimulationso ftwarewidelyusedin simulation.Manysimulationbasedevacuationstudiesdepl oyedPARAMICSfortrac simulationstudies.In[ 161 ],theauthorsdevelopedasimulationmodelusingPARAMICS tosimulatetheevacuationofahighriskareainSantaBarbar acalledMissionCanyon Neighborhood.Themodelisusedtoanalyzethepossibleevac uationscenarioschanging 120

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traccontrol,numberofvehiclesperhousehold,openingof alternateexitandcritical linksinthenetwork.Althoughthesemaynotbetheoptimalon es,theyarevalidated stronglybasedonempiricalobservations.Thesimulationp rovidesthelowerboundon theactualevacuationtimeasevacueebehaviorisnotconsid eredanddemandandarrival timeestimatesarenotaccurate.Butthesimulationresults providingthebestscenario intermsoftheevacuationtimemightbeindicativeofthenec essaryfactorstokeepin mindinanemergencysituation.Inanotherstudy,In[ 162 ],PARAMICSwasemployed tocomparetheecienciesofstagedandsimultaneousevacua tionstudiedunderfree-row andcongestionsituation.Thestudywasperformedfordier entnetworktopologies.They madesomesimplisticassumptionslikeavailabilityofrout echoices,destinationinformation thatpreventsitfromimmediaterealization.Thendings,w hichincludestagedevacuation duringcongestion,wouldguidethedesignofnewmodels.The aggregationofareasinto zonesisatechniqueemployedtogaincomputationalecienc y,butwithacompromisein theprecision.ThedefaultrulesofPARAMICSfortripgenera tionanddestinationand routechoiceswereusedforthesimulation.Theyalsoprovid edareviewofagentbased simulationmodelsandexplaineditsadvantage.Theyperfor medthesimulationoverthree typesofnetworknamelyring,gridandanactualroadnetwork .Thestudyindicated thatstagedevacuationsbasedonzonesareecientthansimu ltaneousevacuation.The modelsassumethatthedriversareassumedtotaketheshorte stpathwhichleadsto frequentcongestionandhencetheresultsmaynotrerectthe bestsimultaneousevacuation time.CORSIMisanothermicro-simulationsoftwareusedfor tracsimulation.CORSIM isacombinationoftwoothermicrosimulationmodelsnamely NETSIMandFRESIM, whichareusedfortracsimulationinsurfacestreetsandfr eewaysrespectively.In[ 163 ], CORSIMwasusedtocreateatransportationmodelforBirming ham.Thegenerated modelwasthenusedtestseveralemergencyscenarios.Theya lsomadeabriefreviewof micro-simulatedevacuationmodelsanddiscussedthevital featuresofCORSIM.Several responseactionssuchastracdiversions,alteringsignal timings,roadwayclearanceand 121

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accessrestrictionwereincorporatedinthetesting.Thesi mulationmodelwasthentested ondierentscenariosnamely,adiscretetracevent,evacu ationsituation,andsimulation ofavailableresponseplans.Theperformancemeasures,whi chincludesspeed,queueing lengthandqueueingtime,werenotjustiedtobethebestset andtheydonotguarantee thatthesimulatedresultscorrespondtothemostecientso lutioncorrespondingto theassumptionsmade.Theextensibilityofthemodelisacon cernfornetworksbesides thenetworksexaminedunderthesamesettings.Themodelwil lnothaveimmediate realizability,butisusefulinvalidatingandcalibrating modelsandtherecommendations withrespectthemeasuresthatwerefocussedinthestudy.CO RSIMwasalsousedto testtheeciencyofcontrarowsorlanereversalundervario usevacuationscenarios[ 164 ]. Thestudywascarriedoutinveryplainsettings,justtotest theeectivenessoflane reversalofthelinkinanetwork.Thestudycouldserveonlya saguidetoevaluatethe lanereversaleciencyratherthanstandalonemodeltoesta blishevacuationroutes.The simplicityofthemodelmakesitcomputationallyecientto obtainquickresults.We discussthedemeritsofestablishinglanereversalstrateg iesusingsimulationtechniques insection 4 .Anotherpopularmicro-tracsimulationsoftware,VISSIM ,isalsousedin evacuationstudies.In[ 165 ],VISSIMwasusedfornuclearpowerplantaccidentswith dynamictracassignmentandmostdesirabledestinations. In[ 166 ],acomparison wasprovidedofVISSIMandCORSIMwiththedemandestimatesp rovidedbyFederal EmergencyManagementAgency(FEMA)tostudytheeciencyof lanereversals.Both thesemodelsindicatedincreasedthroughputanddecreased queuewithlanereversal implementation.Throughputatkeynodes,queuelengthsand averagespeedswere consideredtobetheeciencymeasuresoftheevacuationpla n.Theseeciencymeasures werecomparedbetweenalanereversalplanandado-nothingp lan.Theevacuation scenarioswereconsideredessentiallyforareversingonly twovitallinksofthenetwork. Thelanereversalswereconsideredforonlythecriticallin ksofthenetwork,buttherest ofthemodelisheavilydependentonthesimulatorsusedbyth emodel.Asmicroscopic 122

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simulationskeepstrackofindividualentitiesinthesyste mitiscomputationallyexpensive anditwasoftenusedinsmallscaleevacuationsystems.Also ,thedetailskeptbythese modelsmakesthemmorecomplex.Theirfunctionlogicgetsco mplicatedforoperation andtheyoftenresultinbefuddlingnumberofparameterstha taretoughtokeeptrack. However,thislimitationisbeingslowlysubduedwiththead ventoffastercomputers. MacroscopicSimulationTechniques Asmentionedearliermacroscopicmodelscapturesthevehic lesandtheiractivities morecoarselycomparedtoitscounterpart.Tracisaggrega tedandtheaggregatedrow willbestudiedthroughmodelvariablessuchasspeed,densi tyandrowrate.NETVAC [ 167 ]andMASSVAC[ 168 ]aretwopopularmacro-tracsimulationsoftware.NETVAC modelsradialevacuationfromriskarea,similartoevacuat ionduringanuclearaccident. Thetraveldemandisincidentalinthemodel,sincealltheho useholdswithintheriskarea requireevacuation.MASSVACisamacroscopicsimulationmo deldevelopedforrural networksandwastestedonasmallruralnetworkinVirginiaf orseveralloadingcurves. Itcomprisesofacommunitymoduletodenetheboundaryofth ehazard,apopulation characteristicsmoduletodeterminethepopulationalloca tionspatiallyandanevacuation modulethatperformstheactualtracassignment.Ithasasi mpleinputstructure andtripdistribution.LaterversionsofMASSVAChavebeend evelopedtoincorporate complexfeaturessuchasuserequilibriumassignments.Oak RidgeEvacuationModeling System(OREMS)isapopularmacro-simulationevacuationsi mulator.Identifying bottlenecksandfeasibilityofevacuation,establishinge vacuationroutesandidentifying alternateconrodstrategiesaresomeoftheimportantfeatu resofOREMS.GISinterface, whichisausefuladd-onintherecentyearsisalsoavailable inOREMS.Moriartyetal., comparedmajormacro-simulationsoftwaresincludingOREM S,DYNEVandETISwhere thefactorsinruencingtheevacuationresponsewereidenti edandseveralenhancements weresuggestedtoimprovetheevacuationeciency[ 169 ].Mostofthesemodelswere providedwiththeevacuationdemandasaninputtothemodela ndthemodelsarea 123

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coarseapproximationofreality.Theincreaseincomputati onalcapacityintherecent yearsaremakingmicrosimulatorsmoreattractiveastheyco uldalsoprovideprecisionin performinglocalizedstudiesandnetworksofreasonablesi zes.Howevermacrosimulators couldbeintegratedwithmicrosimulatorsoranalyticalmod els.Theycouldserveasquick validatorsinaevaluatingadevelopedmodel.Also,thevery largescalenetworkarestill alittlefarfetchedforanalyticalmodelsandmicrosimulat orsandaremostlyhandledby macrosimulators.Macroscopicmodelsaremuchsimplercomp aredtothemicroscopic models,whenitcomestotocalibrationandcomputation.Thi sisonlyachievedthrough someshortcomings.Theycannotbeappliedtoinstanceswhic hrequiresindividual vehiculardynamicstobetracked.Asthevehiclepositionsa renotknowninthesemodels, themoderntraccontrolsystemssuchasrampmetering,lane changemaneuversand afewotherfeaturesofintelligenttransportationsystems cannotbecapturedbythese models.Also,evacuationthatrequiresmodelingofhumanbe haviorcannotemploy macroscopicsimulationstudiesforthesamereason.Theacc uracyiscompromisedbythe aggregationofthetracandnetwork.Theyaremoreapplicab leinlargescaleevacuation withrespecttotimeandspace,whenevertheseshortcomings arelessrealized. Meso-SimulationTechniques Anewgenerationofsimulatorscalledmeso-simulatorsarep opularintherecentdays. Theycombinetheprosandconsofthemicroandmacrosimulato rs.Themeso-simulators clubsthevehiclesintopacketsorplatoonswhicharethensi mulatedasseparateentities. In[ 170 ],CEMPS,ameso-simulator,wasemployedtodevelopaspacia ldecisionsupport system.Thisisverypracticalmodelinthesensethatthedec isionsupportsystem integratestherealtimedataobtainedfromgeographicalin formationsystem(GIS)to maketracdecisionsandaobjectorientedsimulationcompr isesofdecisionmodelingand dynamicanalysiscomponents.CEMPSprovidesaconvenientf rameworkthatpermitsthis communication.Themodeldoesnotsucientlyclarifytheco mputationalandeconomic concernsthatmayariseduetotheincreaseinthesizeofevac uationnetwork.In[ 171 ], 124

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theauthorsprovidedameso-microscalesimulationstudyus ingSmartCAPthatallows monitoringandstudyingofaggregatetracrowbehaviorsuc hasdensity,row,and velocityandintegrateditwithamicro-simulator,SmartAH S,whichisusedrecordthe individualdetails.Theintegratedsimulatorconsistsof\ window"ofthemicro-simulator thatcommunicateswiththemeso-simulator.Essentiallyth evehiclesaremicro-simulated andthemeso-tracrowcharacteristicssuchasvelocity,ro wrate,etc.arepreserved.The claimedboostinthecomputationalspeedispertinenttospe cicsituationsbytheuseof meso-simulators.Moreprecisely,thegainwillbelinearin termsofthelevelofaggregation ofvehiclesintopackets.Thisstillcannotjustifythephas ingoutofmacrosimulatorsor microsimulators.However,thelevelofaggregationwillhe lpinachievingthecapabilityof amicrosimulatortothedesiredaccuracy.DYNEMO[ 172 ]proposedanothermesoscopic tracrowmodelthatwasdeveloped,whereunitofatracrowi sanindividualvehicle unlikepackets.Themotionofthevehiclesisdeterminedbyt heirlink'stracdensity.The functionthatgivestherelationbetweentracdensityands peedisprovidedasaninput tothemodel.IntegratedTechniques Simulationtechniquescouldbeusedinconjunctionwithana lyticalmodelsinorder togainthebestoutoftheoptimizationandsimulationtechn iques.Theevacuationplans thataregeneratedthroughtheoptimizationtechniquescou ldbeagoodlowerbound onactualevacuationtime,thustheseplanscouldserveasac andidateforsimulators toidentifythediscrepanciesinthemodel,foreseetherequ irementsandcapturethe featuresthatwerenotincorporatedintheoptimizationtec hnique.In[ 173 ],theauthors providedabi-levelsimulationbasedapproachtosolvethee vacuationproblem.Atone levelthetimedependentrouteassignmentsaredetermineda ndatanotherleveladynamic loadingproblemissolvedandtheoutputislateraggregated .Thetimedependentroute assignmentissolvedusingthemethodofsuccessiveaverage sandthetracdemand simulationisdoneusingDYNASMART-Ptoestimatethevehicl etriptimesandlink 125

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traveltimes,thusachievingasystemoptimalschedule.The dynamictracassignment isimplicityhandledbyDYNASMART-P,relievingtheoptimiz ationmodel.Themodel iscomputationallyecientbutthegaininspeedcouldbesof tenedbythedegreeof accuracyoftheheuristicwiththeincreaseinproblemsizea ndthecomplexityduetonew featureswhenextendedtoothersituations.In[ 174 ],anevacuationsystemwaspresented foroceancityintegratingoptimizationandsimulationtec hniques.Anevacuationplanis generatedbytheoptimizationmoduleandtheplanisrevised bytheresultsofsimulation evaluation.Theprosandconsoftheplansestablishedwered iscussed.Themodelprovides routechoiceoptionsthatpermitsuserstodeviatefromthea vailableorpretestedplans, butthemodelassumestheavailabilityofpreestimatedorfo recasteddemands.The formulationisbasedoncelltransmissionmodelproposedin [ 109 ]withnecessaryrow conservationanddemandconstraints.Theresultofthistec hniqueisasetofcandidate plans,whichcouldlaterbenalizedusingcalibratedsimul ators.Theresultswerethen evaluatedwithamicroscopicsimulationprogram.Thiscell -basedformulationmakesthe modelmorecomplicatedintermsofthesizeandcomplexitywh endealtoveralarger systemandsoasthenumberofplansthathavetobestoredandt estedinthenextphase. Themethodiscomprehensivecomparedtoanintegratedtechn iquethatproducesone singlesolution,butmakesitcomputationallyunattractiv e.Thecandidatesetcouldbe alimitedsettoovercomethisundesiredeect.Asimilartwo leveloptimizationmethod forevacuationoftheoceancitywasproposedin[ 175 ].Atabroadlevel,theymaximize thethroughputatthehigherlevelintermsofthenumberofve hiclesbeingevacuated andminimizethetraveltimeatthelowerlevel.Thecell-bas edformulationwasmade byaccommodatingcellsofvaryingsizesinordertodecrease thecomplexityofthe optimizationmodel.In[ 176 ],anotherintegratedtechniquewasproposedinwhichthe cellbasedoptimizationmodelwasusedtoformulatethedema ndconstraintsandtherow andstoragecapacityconstraints.Theresultoftheoptimiz ationmoduleisthenfedas theinputtothemicroscopicsimulator,CORSIM,whichmodel srealtimeoperational 126

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constraintsanddriverbehaviorthatwerenotcapturedinth eoptimizationmodel.The evacuationsystemsproposedweremodeledwithstaticdeman ds.Theseintegration techniquesareveryusefulforanumberofreasons.Theevacu ationplancouldbevalidated throughsimulationandhencecanbereliable.Thecomputati onallyexpensivetaskcould besimulatedandotheroperationscouldbeoptimizedandthu stheyattempttoseekan optimalbalancebetweenprecisionandspeed. 4.5SignicantFeaturesinSimulationTechniques ReviewedFeatures Asurveyin[ 92 ]identied22evacuationmodelsandassessedthembygroupi ngthem basedonfourdierentperspectives,namelyenclosurei.et henenessorcoarsenessof anetwork,populationperspectivewhereactionsaretakenb asedonindividualsorby groups,behavioralperspectivebasedonhowtheoccupantsr eacttotheenvironmentand thenatureofmodelapplications.In[ 93 ],asurveyofsimulationmodelswaspresentedfor emergencyevacuation.Theymadeaclassicationbasedonmo delingapproach,namely rowbased,agent-basedandcellularautomataanddiscussed theirrelativeadvantages fromtheperspectiveofevacueebehavior.Arowbasedmodeli ngcomprisesofnodes representingthestructuresorplacestoandfromtheevacue eshavetobemovedand arcsmappingtothehallways,roads,etc.,thatlinksthesen odes.Sincetheindividual characteristicsarenotemphasizedinthismodelingthepra cticalrealizationofthese modelsareimpeded.However,ifthesemodelswereusedincon junctionwithanalytical modelscouldbeverypowerfultoolintermsofbothspeedandp recision.Incellular automatatheevacuationspaceisdiscretizedintocellsorg ridswithspeciccapacityand theentitiesrowfromonecelltoanother.Themodelsbasedon cellbasedtransmission couldbeveryaccurate.In[ 177 ],32dierentmicro-simulationmodelswereanalyzedand comparedthefeaturesavailableinthemodels.Scaleofappl ication,i.ethesizeofthe networkthatthemodelcanhandlewasoneofthefeaturesthat wasdiscussedinthe papers.Adetailedstatisticsoftheobjectsandphenomenon thatthemodelsincluded 127

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wasprovided.Queuespillbacks,weaving,incidentsandcom mercialvehicleswerebyand largemodeled.Indicatorsofobjectivesshowedthatthespe ed,traveltime,congestion andqueuelengthwerewidelyadoptedbythemodelsandindica torssuchascomfort andperformancewererarelyused.Analysisoftelematicfun ctionsandinterfacewere alsodiscussed.Acomprehensivesurveyonsimulationstudi eswasprovided[ 178 ]and analyzedvarioussimulationsoftwarethatareavailablean dpresentedallthecomponents consideredforevacuationbythesesoftware.Thecategorie sofevaluationofthemodels weremodeling,behavior,operationsandhazards.Mostofth esefeatureshavealready beendiscussed.Thisanalysisprovidedageneralizedframe workforsimulationstudiesof emergencyevacuation.LaneReversalsandTracControl Contrarowsorlanereversalsarebeingconsideredinthesim ulationmodelsjust likeinoptimizationmodels.Therecenttrendandstudiesha veindicatedthatthereisa signicantdecreaseinevacuationtimewiththeimplementa tionoflanereversals.Amore ecientwayofcontrollingthetracisbyemployingtracpe rsonnel.In[ 164 ],trac controlwasstudiedbycapturingthesescenariosandsimula tingthem.Thetraveltime wassignicantlylessfortheinstanceswithlanereversal. In[ 151 ],theauthorsundertook aresearchtostudytheeectofcontrarowimplementationsi nevacuationinNewOrleans. TheyemployedCORSIMtosimulatethefreewaycongurationa ndtwootherscenarios withcontrarowimplementation.Thestudycarriedoutvario usloadingcongurationsof majorhighwaypermittingcontrarows.Theexperimentswere usedtodemonstratethe benetsofcontrarow.Asignicantimprovementinthemeasu resofeectiveness,namely traveltimeandaveragespeedwasachievedforthecongurat ionspermittingcontrarow. Inmostcasesthecontrarowsareassumecompletelanerevers als,thatis,theentire capacityoftheroadisswitchedtowardsthedestinations.T hereareacoupleofreasons forholdingacapacityineitherdirections.Incaseofalarg enetwork,alinkofthenetwork ineitherdirectioncouldbeusedinaroutetoreachthedesti nation.Incaseofalink 128

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failure,wherealternatepathsarerequired,thereversedl inkscannolongerbepartofan alternatepathtothedestination.Ifthiswasnotthecasean devenifalinkcannotbea partofroutefromorigintodestination,thesearccapaciti escouldbeusedinarouting problemwhereemptybuseshavetoberoutedbacktopickuppoi nts.Mostofthepapers currentlyassumethattherewillbeenoughnumberofbusesto supporttheevacuationby makingjustonetrip.Thusstudiesthatidentifytheamounto fcapacitiesthathastobe reversedwithinhighwayratherthancompletereversalsare limited.Simulationstudies arenotthebestwaytostudytheeectsofcontrarowastheyne edainputplanwhich couldbecalibratedorvalidated,butcannotbeatooltoiden tifythelinksortheamount ofcapacitiesthathastobereversed.Analyticaltoolscoul dbeemployedforthispurpose andtheplanscouldlaterbetestedandveriedinsimulation s. DynamicDemandEstimation Inalargescaleevacuationtheevacuationdistanceislarge andassumingstatic tracconditionsisnotprecise.DTAallowsdierenttracc onditionstobeincluded inthesimulation.Itcapturesthecomplexdynamicdemandpa tternthatarisesdue tocongestion,queues,spillbackanddelays.Therewillbea signicantdierencein theevacuationeciencybetweenstaticanddynamicdemand. Thefactorsthatresults inthedierencehasbeendiscussedbefore.Thedynamicdema ndestimationthrough asimulatormaynotbeasdicultasthecontrarows.In[ 179 ],theauthorsprovided asimulationmodelwithademandpredictivecapabilitiesim plementedasapartof DynaMITsimulator.Themodelgeneratesapre-tripdemandba sedonthehistorical demandandtwosystematicdeviationsbasedondailydemandr uctuationsanddriver responsebehavior.Thepre-tripdemandisusedtoestablish adisaggregatepre-trip traveldecisionsforthedrivers.Thenabehavioralmodelis employedtoexploitthe availablerealtimedataandalterroutedecisions.Thenadi saggregateorigindestination matrixisgeneratedbasedonthetraccountonthelinksofth enetwork,whichisthen aggregatedtoestimatethenewdemand.Themodeltakesintoa ccountvariousfactors 129

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includingbehaviorpattern,dailydemandructuationsanda randomerrorcomponent, thusgivingmorereliabilitytothemodel.Themodelemploys atimediscretizationfor determiningintervalsbetweenwhichthedistanceupdatesa remade.Thismakesthe modelcomputationallyexpensivebasedonthelengthofthei nterval.Thequestionof considerationwillbethecapabilityofthemodeltohandleo therfeaturesinadditionto thedynamicdemandestimation.Thismodelwasjustanextens ionoftheworkin[ 180 ]. Thedemandestimationthroughupdateofhistoricaldemandb asedondriverbehaviorand wasfurtherenhancedbysystematicdeviationsinthenewerm odels.Asimilarmethod, QUEENSOD,[ 181 ]involvesaseedmatrixsimilartothepre-tripdemandmatri xbasedon historicaldemand.Theseedmatrixestablishestraccount sandamicro-simulatorisused toestablishthetracroutesbasedontheestimateandthent heseedmatrixisalteredto reducetheerrorbetweentheestimatedandobservedtracco unts. MiscellaneousFactors Thenumbervehiclesperhouseholdinmicro-tracsimulatio nmaybeimportant asthisincreasesthedemand.Theuseofcriticallinksisama jorfactorinevacuationas itisimportanttoensurethatthecriticallinksarenotcong ested,whichwouldresultin heavytracdisruption.Incidentmanagementisaminorfeat urethatmodelsfocuses ontheneedforalternateroutesincaseofaccidentsandesti mationoftracpersonnel fordiversionandlanecontrol.In[ 182 ],theimpactofstagedevacuationeciencywas studied,wheretheyconsideredsixdierentscenariosanda ndtheycarriedoutsimulation onarepresentativetracnetworkthatresultedinsuccessf ulstagedevacuations.The scenarioscomprisedofcombinationsofshiftingthedepart uretimesofevacuations.This resultedinincreaseinthetotalnumberoftrips,butpreven tedcongestionandqueues. Theyconcludedthatthedeparturetimeshiftingandtotalnu mberoftripsmadehada positiveimpactontheclearancetime. 130

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4.6Conclusions Themodelscurrentlyavailableintheliteratureareusuall ycustomizedforthe evacuationofspecicregionsingeographytailoredtoitsn eeds.Themodelshavetheir relativeadvantagesanddisadvantages,butprecisetothei rneeds.Itisrathertediousto generateauniedmodelthatcouldbeusedinallsituations. Inclusionofseveralfeatures impactsthecomplexityofthemodelandhencethecomputatio nalspeed.Ontheother hand,simplifyingamodelwouldcompromisetheprecisionof themodel.However,the modelshavesomecommonoverlappingfeaturesthatwehighli ghtedinthisreport.We sawthatthehybridmodelscouldbereasonablyaccurateandp recisebyexploitingthe relativeadvantagesofsimulationandoptimizationtechni ques.Thischapterprovided broadclassicationofevacuationmodelbasedonsimulatio noroptimizationmethods. Further,itidentiedthefactorsconsideredbytheseoptim izationmodelsandcommented ontheapproaches.Wehighlightedthefeaturesthatwillhav easignicanteectonthe traveltimenamelyintermodaltransportation,dynamictra cdemandestimationand contrarowsorlanereversalsincaseofawideareaevacuatio n.Modelsincorporating intermodaltransportationandcontrarowsdemonstratedth eimprovementinevacuation eciencycomparedtothetraditionalmodels.Also,staticd emandmodelhavebecome obsoleteanddynamicdemandisnecessaryforpracticalreal izationofthemodels.Some areasthatneedsattentionareoptimizationproblemstoest ablishalternateevacuation pathsforincidentmanagements.Criticalnodedetectionan dtracmanagementon criticallinksarestudiesthatmightimprovetheeciencyo ftheevacuationandalsomight giveanindicationofthenecessarylinksthatwecouldfocus oncontrarows.Alsoheuristic explorationofoptimizationtechniquescouldsignicantl yreducethecomputationalspeed. Researchintheeldofclusteringofnodesandzonaldivisio nofnetworkisverylimited. Thismightshedlightinperformingevacuationoverasmalle raggregatednetworkhelping incomputationaleciency. 131

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CHAPTER5 NETWORKFLOWPROBLEMSWITHLANEREVERSALS 5.1Introduction Westudycontrarownetworkproblems,whereinwetrytomaxim izerowinagraph whilepermittingdirectionreversalsofanarc,resultingi nacapacityincreaseinthe directionofswitch.Theapplicationsarerealizedinaneme rgencysituation,wherepeople havetobe`evacuated'fromaspecicarea; i.e. afootballstadiumafteragame,acity expectingaroodorhurricane,azonewhereanunexplodedord nancedevicehasbeen found,oraregionwhichhasbeenattackedbyterrorists.Inm ostofthesecases,the evacueesareexpectedtoleavetheareaofrisk,thesource(s ),towardsasaferplace,the sink(s).Arowtowardsthesourceisundesiredduringmostof thesescenariosandwedo notexpecttheevacueestogointhisdirection.Asadirectco nsequence,allthearcsthat arenotapartofanypathfromthesourcenode(s)tothesink(s )mightbeleftunused. Onecanevenencounteridlearcsduringcertainscenarios,s uchasmanagingafootball event,whereinwedohavesomeamountrowtowardsthesource. Theseidlearcscouldbe usedtoincreasetheeciencyofevacuationbyreversingthe irdirections.Thescenarios involvinginpartiallanereversalcapabilitycouldbecapt uredwithappropriategraph transformation.Wediscussseveralscenariosthatmayaris eduringthereconguration, whichincludespermittingonlyasubsetofarcstobereverse d,imposingaswitchingcost tothearcsinvolvedinthereversals. Thereareveryfewoptimizationtechniquesintheliteratur ehandlingarcreversals. Kim and Shekar [ 137 ]proposedasimulatedannealingprocedureforthisproblem and providedempiricalresults.Theyalsoprovideasketchofth eproofthattheproblemis NP -complete.Atabu-basedheuristicwasproposedby Tuydes and Ziliaskopolos [ 138 ] fortheproblem.Theyfocustheirstudyonaspecializedvers ion,wheretheypermitlane reversalswithpartialcapacities. Hamza-Lup etal.[ 183 ]proposedaheuristicforthis contrarowproblem.Thesetechniquesandtheirpitfallswer ediscussedin[ 137 ].Afew 132

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otherstudiesintheliteraturethatarenotanalyticalinna turewerealsoproposed.They relyonsimulation-basedmethodsanddecisionsupporttool s[ 151 164 ]. Inthischapter,weprovideadetailedstudyofthearcrevers al(orcontrarow) problemswithrespecttotheircomputationalcomplexity.T hemotivationistointroduce theproblemsformallytoprovideabasisforfurtherresearc hinthisarea.Asthe applicationsaremainlyrealizedduringemergencysituati ons,thedynamicrowproblems areofprincipalinterest,butwestudystaticcasesaspresu ppositionsandalsoforthe sakeofcompletenessofthestudy.Insection 5 ,weprovideabriefbackgroundofthe networkrowproblemsandexplaintheterminologyusedinthe restofthechapter.We thenprovideadiscussionofstaticrowproblemsinSec. 5 .Apolynomialtimealgorithm throughagraphtransformationisintroducedforthestatic maximumrowproblem witharcreversalcapability.Theresultisevidentanditis usefulinSec. 5 inshowing thatthedynamicmaximumcontrarowproblem,withsinglesou rceandsinglesink,is polynomiallysolvable.WeshowinSec. 5 thatthethedecisionversionofthemultiple sourcesandmultiplesinksversionoftheproblemis NP -completethroughareductionfrom 3-SATISFIABILITY(3SAT).InSec. 5 ,weshowthattheproblembecomes NP -complete byhavingjusttwosourcesorsinks.Inaddition,wediscusst heinabilityofthegraph transformationthatwasemployedearliertoprovidefeasib lesolutions.Wenallyshow inSec. 5 thattheproblemofndingtheminimumtotalcost,incurredd uetoanarc switchingcost,toidentifythearcstobereversedis NP -hard,eveninthestaticcase. 5.2Background Thebasicterminologiesanddenitionsthatarepredominan tlyusedinthenetwork rowsliteratureandthatareessentialfortherestofthecha pterareexplainedinthis section. Denition1 (Staticfeasiblerow) 133

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Givenisagraph G =( V;A ) withcapacities c e 2 Z + forallarcs e 2 A .Astaticrow, characterizedbythefunction f : A R + ,withvalue v ,from s 2 V to t 2 V isfeasible,if f e c e ; 8 e 2 A (5{1) P ( i;j ) 2 A f i;j P ( j;k ) 2 A f j;k = 8>>>>>><>>>>>>: v;j = s 0 ; 8 j 2 V nf s [ t g v;j = t (5{2) Wecallnode s asthe`source',node t asthe`sink'andrestofthenodesas`intermediate' or`transhipment'nodes. Equation 5{1 ensuresthattherow f e alongeacharc e 2 A meetsthecapacity constraints;asweassumealllowerboundsontherowtobe0.I nequation 5{2 ,thenet rowoutof s is v and t is v .Forallintermediatenodesitis0andisalsoreferredtoas rowconservation.Thedenitionofafeasiblerowgeneraliz esinanaturalwayforthecase ofmultiplesourcesandmultiplesinks. Asequenceofdistinctnodes x 1 ;x 2 ;:::;x n ofagraph G =( V;A )iscalledachainif ( x i ;x i +1 ) 2 A; 8 i =1 ;:::;n .Achainisalsoreferredtoasadirectedpath.Let P bethe setofallchainsfrom s to t .Wedeneanotherrowfunction, h : P R + ,intermsofthe rowalongthechainsfrom s to t .Afeasiblerow f withvalue v couldbedecomposedinto asetofchains, P ,from s to t ,suchthat v = j P j X i =1 h i : Theprocessofobtainingrowalongthechainsthiswayiscall edas`chaindecomposition'. Amoredetailedaccountoftheseterminologiescouldbefoun din[ 94 184 ]. Inadynamicgraphornetwork G =( V;A )eacharcisassociatedwithatraveltime, t : A R + ,besidesthecapacityfunction.Thegraphexpandedover T timeperiods, G T =( V T ;A T ),isobtainedbyreplacingeachnodeby T copiesandhavingnodes v l i and v l + t i;j j connectedin G T =( V T ;A T )if v i and v j areconnectedin G ,forall l =0 ;:::;T t i;j 134

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Thisconceptofafeasiblerowcanbedirectlyadoptedtothed ynamiccasebyensuring thatbothequations 5{1 and 5{2 aresatisedforalldiscretetimesteps.Hence,afeasible dynamicrowisafeasiblerowinthetimeexpandedgraphwitht hevalueequaltothe sumofthenetrowsoutofallthe T copiesof s .Formoredetailsabouttimeexpanded graphsrefer,forinstance,to[ 184 ,Chapter19.6]. 5.3MaximumStaticContrarowProblems Inthissection,weprovideapolynomialtimealgorithmsolv ingthemaximum contrarowprobleminastaticgraph.Theresultspresentedi nthissectionareverybasic andstraightforward.Nevertheless,wediscussthemindeta ilasthishelpsusindeveloping themainresultsinSection 5 Now,letusdenedenethemaximumrowproblemwitharcrever salcapability. Denition2 (MaximumContrarow(MCF)) Instance :Givenadirectedgraph G =( V;A ) withsource s + 2 V ,sink s 2 V and capacity c e 2 Z + oneacharc e 2 A Question :Whatisthemaximumrowfromnode s + tonode s ifthedirectionofthe arcscanbereversed? Thisproblemisalsocalled maximumrowproblemwitharcreversal .Considernow procedureP-MCF.Intherststep,anauxiliarygraph e G =( V; e A )isconstructed.The transformationfromtheoriginalgraph G isobtainedbysummingthecapacitiesofarcs ( i;j )and( j;i ).ThisallowsustoreducetheMCFproblemtothemaximumrowp roblem onthetransformedgraphinstep2.Step3removescyclerowsi nthetransformedgraph. ThisensuresthattheconstructedsolutionoftheMCFproble minstep4iswelldened. Wehavethepriorknowledgethatthereexistsanoptimalrowt othemaximumrow problemthatdoesnothavecycles.Thus,arcsoneitherdirec tionwillneverbeusedin thisrowforthemaximumrowproblem.Thisisthebasicideaof procedureP-MCFthat motivatesthegraphtransformationgiven.Thisresultisst raightforwardbutwecanrealize itsimpactinSec. 5 135

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Procedure MaximumContrarow(P-MCF) 1.Constructthetransformedgraph e G =( V; e A )wherethearcsetisdenedas ( i;j ) 2 e A ,if( i;j ) 2 A or( j;i ) 2 A; Thearccapacityfunction e c isgivenby e c i;j := c i;j + c j;i ; forallarcs( i;j ) 2 e A 2.Solvethemaximumrowproblemongraph e G withcapacity e c 3.Performrowdecompositionintopathandcyclerowsofthem aximumrowresulting fromstep2.Removethecyclerows. 4.Arc( j;i ) 2 A isreversed,ifandonlyiftherowalongarc( i;j )isgreaterthan c i;j orifthereisanon-negativerowalongarc( i;j ) = 2 A andtheresultingrowisthe maximumrowwitharcreversalforgraph G =( V;A ). Endprocedure Theorem1 (Proofofcorrectness) ProcedureP-MCFsolvesthemaximumrowproblem witharcreversalforgraph G =( V;A ) optimally. Proof. Theproofconsistsoutoftwosteps.First,weshowthatanyso lutionofthe procedureP-MCFisfeasiblefor G =( V;A ).Second,weshowitsoptimality. Forfeasibility,weonlyhavetoshowthatstep4inthealgori thmiswelldened; i.e. notbotharcs( i;j )and( j;i )havetobeswitched.However,thisisensuredbystep3. Theoptimalsolutionaftertherowdecompositionresultsin asetofpathsfromsourceto sinkandasetofcycleswithpositiverows.Aftertherowdeco mpositionwecouldcancel thepositiverowsalongallcyclesandensurethatthereisno rowalonganycycle.This ensuresthatthereiseitherarowalongarc( i;j )or( j;i ),butneveronbotharcs.Hence, theresultingrowfromstep4isafeasiblerowwitharcrevers alforgraph G =( V;A ). Now,weprovethattheresultingrowisalsooptimal.Notetha tanyoptimalsolution tothemaximumrowproblemwitharcreversalongraph G =( V;A )isalsoafeasible solutiontothemaximumrowproblemonthetransformedgraph e G =( V; e A ).Asthe 136

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amountofrowsendfrom s to t isnotchangedinsteps3and4,theresultingrowisan optimalsolutiontothemaximumrowproblemwitharcreversa longraph G =( V;A ). TherunningtimeofprocedureP-MCFisdominatedbysolvinga maximumrow probleminstep2andbytherowdecompositioninstep3;asste ps1and4canbedone in O ( j A j ).Letusdenotetherunningtimeforsolvingthemaximumrowp roblemby S 1 ( j V j ; j A j )andfortherowdecompositionproblemby S 2 ( j V j ; j A j ).Then,therunning timeofprocedureP-MCFisgivenby O ( S 1 ( j V j ; j A j )+ S 2 ( j V j ; j A j )).Usingthehighest-label prerow-pushalgorithmleadsto S 1 ( j V j ; j A j )= O ( j V j 2 p E ),[ 185 ].Therowdecomposition canbedone,forinstance,in O ( j V jj E j ),[ 184 ].Thisprovesthefollowingtheorem. Theorem2 (Runningtime) ProcedureP-MCFsolvesthemaximumcontrarowproblem instronglypolynomialtime. Wearenowabletoextendtheresultabovetothecaseofmultip lesourcesand multiplesinks.Thisproblemisalsocalled maximumtransshipmentcontrarow(MTCF) problem Corollary1. Thestaticversionofthemaximumcontrarowproblemwithmul tiplesources andmultiplesinksispolynomiallysolvable. Corollary 1 canberealizedthroughasimplereduction.Let S + and S bethesetof sourcesandthesetofsinks,respectively.Then,adda`supe r-source' u + anda`super-sink' v togetherwiththearcs( u + ;s + ),forall s + 2 S ,witharccapacitiesequaltheirrespective surplusand( s ;v )forall s 2 S withtheirarccapacitiesequaltheirrespectivedecits. Formoredetails,referto[ 94 ]. Recognizethatwebasicallyshowinthissectionthatthemax imumcontrarow problemisequivalenttoamaximumrowproblemonanundirect ed(modied)graph. Thiscouldbeseeninthegraphtransformationprovidedinst ep1ofprocedureP-MCF witharcshavingsamecapacitiesineitherdirections. 137

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5.4MaximumDynamicContrarowProblems Inthissection,wediscussthe maximumdynamiccontrarow(MDCF)problem .The maximumdynamicrowproblemwasstudiedby Ford and Fulkerson [ 94 ],wherethey trytomaximizetherowsentfromsourcetosink,withinagive ntimehorizon T .Unlike thestaticcase,inthedynamicnetworkrowproblemtherowov eranarccanberepeated overtime. Ford and Fulkerson provedthatthisproblemisequivalenttosolvinga minimumcostrowproblemwiththearccostsastraveltimeson thearcs.Thenthe optimalrowonthearcsfromsourcetosinkisdecomposedinto asetofpathsorchains. Thesechainsarethen temporallyrepeated overtimetoobtaintherequireddynamicrow. Inotherwords,thereisalwaysatemporallyrepeatedchainr owthatisequivalentto themaximumdynamicrow.Letusassumethereare P pathsobtainedfromthechain decompositionoftheoptimalminimumcostrow.Thenthemaxi mumdynamicrowis givenby X i 2 P ( T +1 t i ) h i ; where h i istherowalongthe i th pathand t i isthetimetakentotravelthe i th path. Inthissection,werststudythesinglesourceandsinglesi nkdynamicrowproblem havingarcreversalcapability.Weprovideanalgorithmemp loyingasimilarkindofgraph transformationasprocedureP-MCFanddiscussitsproofofc orrectnesstogetherwithits worstcaserunningtimeanalysis.Thisimpliesthatthe quickestcontrarow(QCF)problem isalsopolynomiallysolvable.Inthe quickestrowproblem ,thetimetosendagivenrow fromsourcetosinkisminimized. Burkard etal.[ 186 ]gaveastronglypolynomialtime algorithmforthisproblem. Hoppe [ 187 ]studiedthemultiplesourcesandmultiplesinksversionof thisproblem, alsocalledthe quickesttransshipmentproblem ,wheretheyminimizethetimetakento sendthesupplyatthesourcestothesinkssatisfyingtheird emands.Instaticnetwork rows,themultiplesourcesandmultiplesinksarehandledby addinga`super-source' anda`super-sink'.Thentheyareconnectedtothesourcesan dsinksrespectively,see 138

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Corollary 1 .However,thissolutionprocedureisnotapplicableinadyn amiccaseanymore. Forthesamereason,thedynamiccontrarowproblemwithmult iplesourcesandmultiple sinksis NP -complete.Weprovideanexampleillustratingthistogethe rwithaproofofits NP -completeness. SingleSourceandSingleSink LetusextendtheMCFproblemofSec. 5 tothedynamiccase. Denition3 (MaximumDynamicContrarow(MDCF)) INSTANCE :Givenadirectedgraph G =( V;A ) withsourcenode s + 2 V ,sinknode s 2 V ,capacity c e 2 Z + andtransmissiontime t e 2 Z + oneacharc e 2 A with t i;j = t j;i if ( i;j ) ; ( j;i ) 2 A ,andanoveralltimehorizon T 2 Z + QUESTION :Determinethemaximumamountofrowthatcanbesendin T units oftimefromsource s + tosink s ,ifthedirectionofthearcscanbereversedattime0. Note:Inthiscase,ifwechoosetoswitchanarc,itremainssw itchedfromtime 0 to T .Thecasewhereweallowswitchingofarcsbackandforthinti meistrivialasthe quickesttranshipmentcontrarowproblem,withthisassump tion,reducestothequickest transhipmentproblemthroughthegraphtransformationsug gestedinprocedureP-MDCF andhenceispolynomiallysolvable. Denition 3 statesthatinaMDCFproblem,thegraphisallowedtobeasymm etric withrespecttothearccapacities.However,wheneverbothd irectionsofanarcare includedinthegraph,thenthetravelingtimeofthesetwoar csmustbethesame.This assumptionimpliesthattheswitchingofanarconlychanges thecapacitiesofthearcsbut doesnotaltertheirtravelingtime. Theconceptoftemporallyrepeatedrowsisveryfundamental forthemaximumrow problemwithsinglesourceandsinglesink.Ouralgorithmfo rsolvingtheMDCFproblem ismainlybasedonthisconcept.Hence,letusrepeattheden itiongivenby Ford and Fulkerson ,[ 94 ,page147]. Denition4 (temporallyrepeated) 139

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Adynamicrowwhichcanbegeneratedbyrepeatingchainrowso fastaticrowin graph G iscalledtemporallyrepeatedrow. Thefollowingtheoremrevealstheusefulnessoftemporally repeatedrowsinthe contextofsinglesourceandsinglesinknetworkrowproblem s,[ 94 ,Theorem9.1]. Theorem3. Thereisatemporallyrepeateddynamicrowthatismaximalov erall dynamicrowsfor T periods. Therowtobetemporallyrepeatedcouldthenbedeterminedby justsolvinga minimumcostrowproblem.Letusdenoteitsrunningtimeby S 3 ( j V j ; j A j ).Using,for instance,theminimummeancycle-cancelingalgorithmlead stoastronglypolynomial runningtimeof O ( j V j 2 j E j 3 log( j V j )),[ 188 ]. Beforeweproceedtothenextlemma,weneedtoknowthatutili zingtheconceptof timeexpandedgraphsinasolutionalgorithmleadstoapseud o-polynomialrunningtime. Inthiscase,therunningtimedependson j T j ,ratherthanlog( j T j )whichwouldthenlead toaweaklypolynomialrunningtime.Nevertheless,weuseth econceptoftimeexpanded graphsinTheorem 4 ConsidernowprocedureP-MDCF.WeshowinTheorem 4 thatitsolvestheMDCF problemcorrectly.ThemaindierencesofprocedureP-MCFa ndP-MDCFisgivenin step2.Forthedynamicproblem,weneedtemporallyrepeated rows.Thisensuresthat onlyoneofthearcs( i;j )or( j;i )isusedintherow.Thisenablesustousethesame rippingruleforthearcsasinprocedureP-MCF. InordertoshowthecorrectnessofprocedureP-MDCF,weneed thefollowinglemma. Lemma1. Themaximumamountofrowinthesinglesourceandsinglesink maximum dynamiccontrarowproblemforgraph G =( V;A ) islessthantheoptimalrowinthe maximumcontrarowproblemforthecorrespondingtimeexpan dedgraph G T =( V T ;A T ) Proof. Theresultfollowsdirectlyfromtheobservationthatevery feasiblerowtothe maximumdynamiccontrarowproblemhasanequivalentfeasib lerowtothemaximum contrarowproblemofthetimeexpandedgraph. 140

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Procedure MaximumDynamicContrarow(P-MDCF) 1.Constructthetransformedgraph e G =( V; e A )wherethearcsetisdenedas ( i;j ) 2 e A ,if( i;j ) 2 A or( j;i ) 2 A: Thearccapacityfunction e c isgivenby e c i;j := c i;j + c j;i andthetravelingtimeis e t i;j = e t j;i := t i;j ; if( i;j ) 2 A t j;i ; otherwise ; forallarcs( i;j ) 2 e A 2.Generateadynamic,temporallyrepeatedrowongraph e G withcapacity e c and travelingtime e t 3.Performrowdecompositionintopathandcyclerowsofther owresultingfromstep 2.Removethecyclerows. 4.Arc( j;i ) 2 A isreversed,ifandonlyiftherowalongarc( i;j )isgreaterthan c i;j orifthereisanon-negativerowalongarc( i;j ) = 2 A andtheresultingrowisthe maximumrowwitharcreversalforgraph G =( V;A ). Endprocedure PleasenotethatLemma 1 holdsgoodformorethanonesourceandonesink. However,ingeneral,equalityholdsonlyforthecaseofasin glesourceandasinglesink, aswewillseeinthefollowingtheorem.Wearenowreadytopro vethecorrectnessof procedureP-MDCF. Theorem4 (Proofofcorrectness) ProcedureP-MDCFsolvesthemaximumdynamic contrarowproblemforgraph G =( V;A ) optimally. Proof. TheconceptofthisproofissimilartotheproofofTheorem 1 .First,weprove thatallthestepsinprocedureP-MDCFarewelldenedandres ultinafeasiblesolution. Second,weshowoptimality. Forfeasibility,theprooffollowsdirectlyfromthefactth attheconstructedrows aretemporallyrepeatedandhence,thereisonlyarowinoned irectionoftwonodes, andneverinbothdirectionsatthesametimeaswellasatdie renttimeperiods.After 141

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cancelingtherowsalongthecycles,wehaverowseitheronar c( i;j )oron( j;i )butnot onboth.Thisensuresthattherowislessthanthereversedca pacitiesonallthearcsat alltimeunits.Thisalsoensuresthefeasibility.Inotherw ords,wenowhaveestablished thefact [ G =( V;A )] MDCFopt [ e G =( V; e A )] MDFopt ; bytheargumentthateveryfeasiblerowofthedynamicrowpro bleminthetransformed graph e G =( V; e A )isfeasibletothemaximumdynamiccontrarowprobleminthe graph G =( V;A ).Ourproofiscompleteifweshowthat [ G =( V;A )] MDCFopt [ e G =( V; e A )] MDFopt : Toseethis,rstnotethatthemaximumcontrarowingraph G T =( V T ;A T ) maximumdynamiccontrarowingraph G =( V;A ),fromLemma 1 .Hencewehave, [ G =( V;A )] MDCFopt [ G T =( V T ;A T )] MCFopt : ByTheorem 1 wehavethatthemaximumcontrarowproblemingraph G T = ( V T ;A T )isequivalenttothemaximumrowprobleminthegraph e G T =( V T ; e A T ),where thearcset e A T isdenedas ( i;j ) 2 e A T ,if( i;j ) 2 A T or( j;i ) 2 A T ; andthearccapacityfunction e c isgivenby e c ti;j := c ti;j + c tj;i : Thus, [ G T =( V T ;A T )] MCFopt =[ e G T =( V T ; e A T )] MFopt : ByTheorem 3 ,themaximumrowinthetimeexpandedgraph e G T =( V T ; e A T )canbe obtainedbyatemporallyrepeatingachainrowofastaticgra ph e G =( V; e A ).Hencewe 142

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havethefact, [ e G T =( V T ; e A T )] MFopt =[ e G =( V; e A )] MDFopt : JustlikeprocedureP-MCF,runningtimedominatingarestep sare2and3for procedureP-MDCF.Thisresultsinaworstcaserunningtimeo f O ( S 2 ( j V j ; j A j )+ S 3 ( j V j ; j A j ));whichisstronglypolynomial. Theorem5 (Runningtime) ProcedureP-MDCFsolvesthemaximumrowproblemin stronglypolynomialtime. Forgivenexcess b ,the quickestcontrarowproblem determinestheminimumtime horizon T neededbyanyfeasiblerow. Corollary2. Thequickestcontrarowproblemcanbesolvedinastronglypo lynomialtime. OnewaytorealizeCorollary 2 isthroughtheworkby Burkard etal.forthe quickestrowproblem,[ 186 ].First,obtainanupperboundonthequickesttimeand second,performabinarysearchbyrepeatedlysolvingthemi nimumdynamiccontrarow problem.Suchaboundcanbeobtainedinpolynomialtime,for instance,bycomputing apathfromsourcetosinkandtemporallyrepeatingrowalong thepathuntilallsupply atthesourceissenttothesink.However,thisleadstoaweak lypolynomialalgorithm.A stronglypolynomialalgorithmcouldbeobtainedthroughap arametricsearchsuggested by Megiddo [ 186 189 ]. MultipleSourcesandMultipleSinks Letusstartwiththedenitionofthemultiplesourcesandmu ltiplesinksversionof theMDCFproblem. Denition5 (DynamicTransshipmentContrarow(DTCF)) INSTANCE: Adirectedgraph G =( V;A ) ,asetofsources S + V ,asetofsinks S V ,arccapacities c e 2 Z + andtransmissiontime t e 2 Z + foreacharc e 2 A with t i;j = t j;i if ( i;j ) ; ( j;i ) 2 A ,andanoverallpositiveintegertimebound T 143

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QUESTION: Isthereafeasibledynamicrowwithintimehorizon T ,allowingeach arctobereveredonceattime0? NotethattheDTCFproblemisadecisionproblemcorrespondi ngtothemaximum dynamiccontrarowproblemwithmultiplesourcesandmultip lesinks. DTCFis NP -completeinthestrongsense Inthissection,weproofthattheDTCFproblemis NP -complete.Asketchofthe proofoutlinewasgivenin[ 137 ].However,weprovidearigorousproof.Also,theproofhas somedierencesthoughweprovidethereductionfromthesam eproblem, 3SAT ,[ 28 ,page 46]: Denition6 (3SAT) INSTANCE: Collection C = f c 1 ;c 2 ;:::;c m g ofclausesonaniteset U ofvariables suchthat j c 1 j =3 for 1 i m QUESTION: Isthereatruthassignmentfor U thatsatisesalltheclausesinC? 3SATisknowntobe NP -completeinthestrongsense,see[ 28 ,Theorem3.1]. Foraninstanceof3SAT,constructagraph G 3 SAT =( V;A )forDTCFasfollows. Foreachclause c i wehaveonesourcenode c +i withasurplusof1.Eachvariable u j 2 U ispresentedbysixnodesinthegraph:twoforeachliteral,n amed u 1j u 2j u 1j and u 1j respectively,onesourcenodewithsurplus1, d +j ,andonesinknodewithdecit-1, d j Finally,thereisonenodewithdecit j C j ,named s .Thissumsupto j V j = j C j +6 j U j +1 nodes.Eachclausenode c +i isconnectedtothenodeswithsuperscript1representingit s literals,taking3timeunits.Foreach j ,thenode u 1j isconnectedtoitscopy, u 2j ,with transshipmenttimeof1.Nodes d +j areconnectedto u 2j and u 2j withtransshipmenttime1, whilenodes d j areconnectedto u 1j and u 1j havingatransshipmenttimeof1.Finally,each secondcopy(superscript2)oftheliteralsisconnectedtot hesink s takingatimeof1. Allarcshaveacapacityof j C j .Thisleadsto j A j =3 j C j +8 j U j arcsingraph G 3 SAT .One suchgraphtransformationisshowninFig. 5-1 144

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c +2 +1 c +1 +1 u 13 u 23 u 13 u 23 u 12 u 22 u 12 u 22 u 11 u 21 u 11 u 21 d 3 1 d +3 +1 d 2 1 d +2 +1 d 1 1 d +1 +1 s 2 (2 ; 3) (2 ; 1) Figure5-1.Transformedgraph G 3 SAT correspondingto3SATinstance Theproofofthevalidityofthetransformationisbasedonth efollowingkey observation. Lemma2. Inanyfeasiblerow f inthegraph G 3 SAT withintime T =5 ,thereisarowof value1fromnode d +j tonode d j ,forall j Proof. Letusxindex j andassumethattherowtonode d +j isintegral.Iftherow tonode d j doesnotcomefromnode d +j ,thenitcanonlycomefromexactlyoneofthe nodes c i or d +k with k 6 = j .However,inbothcases,therowarrivesatnode d j earliestat time6,or7respectively.Thisproofsthelemmaforthecaseo fintegerrows.Thecaseof fractionalrowissimilar:Ifsomefractionoftherowtonode d j comesfromadierent nodethen d +j ,thentherowarrivesaftertime T =5. Lemma 2 impliesthatforafeasiblerow,atleastoneofthearcs( u 1j ;u 2j )or( u 1j ; u 2j ) hasbeenswitchedforall j {withotherwords,atmostoneofthetwoarcs( u 1j ;u 2j )and 145

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( u 1j ; u 2j )keeptheirdirectioninanyfeasiblerowwithtimebound T =5.Now,weareable toproofthefollowinglemma. Lemma3. Aninstanceof3SATisa`YES'instance,ifonlyifthetransfo rmedgraph G 3 SAT isa`YES'instanceforDTCFwithoveralltimebound T =5 Proof. \ ) "Let3SAThavethefeasibleassignment u j = a j forallvariables,with a j 2f 0 ; 1 g .Then,reversethearcs( u 1j ;u 2j )if a j =0,andreversearc( u 1j ; u 2j )otherwise. Now,forall j ,sendoneunitofrowfrom d +j to d j alongthereversedarc.Asonlyoneof thearcs( u 1j ;u 2j )or( u 1j ; u 2j )hasbeenswitched,wecansendrowfromanyofthenodes c +i throughanynon-switchedarc,dependentontheassignmento ftheliterals.Thisleadstoa feasiblerowforDTCFwithintime T =5. \ ( "Wehavetoshow,thatanyfeasiblerow f forDTCFneeding(atmost)5unitsof timeleadstoa`YES'instanceofthe3SAT.Weassignthefollo wingvaluetoeachvariable u j 2 U as u j := 8><>: 0 ; ifarc( u 1j ;u 2j )isreversedinrow f 1 ; otherwise : (5{3) Wehavetoshowthatthisisasatisfyingtruthassignmentfor the3SATinstance.Now, assumethatclause c i isnotatruthassignment.Oneunitofrowissendfromnode c +i to node s throughoneofthenodes u 1j or u 1j with u j 2 c i or u j 2 c i .Noticethatthisrow cannotgothroughanyothernode c +k with1 k m and k 6 = i .Lemma 2 impliesthat thecorrespondingvalueofvariable u j hasbeenset; i.e. u j =1iftherowpassesnode u 1j or u j =1ifitpassesthroughnode u j =1.Thisleadstoacontradiction. ThesecondpartoftheproofofLemma 3 togetherwithLemma 2 givetheideaofthe transformationfrom 3SAT .First,wehavetosendoneunitofrowfromeachofthenodes d +j to d j .Thisensuresthat(atleast)oneofthearcsbetweenthecopi esoftheliteralshas tobereversed.Thearcwhichhasnotbeenswitchedcanthenbe usedfortherowofthe 146

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nodes c +j ,allowingtheclausestohaveatruthassignment.Hence,the valueoftheliterals isrerectedbytheswitchingofthearcs. Theorem6. DTCFis NP -completeinthestrongsense. Proof. DTCF 2 NP ,asanon-deterministicalgorithmneedsonlyguesstheseto farcsto bereversedtogetherwitharow f andcheckiftherowisfeasiblewithtimebound T =5; whichcanallbedoneinpolynomialtime.Lemma 3 statesthatthegiventransformation G 3 SAT from 3SAT toDTCFisvalid.Asthecardinalityofthenodesetandthearc set oftheconstructedgraphis O ( j C j ),thetransformationispolynomialintheinputsizeof 3SAT Wewanttomentionthatthetransformationfrom 3SAT caneasilybechangedto thegeneralSATbyonlychangingtheappropriatearcsfromth eclausenodestothenodes representingtheliterals.WhatmakesDTCFsotoughtosolve? Ford and Fulkerson introducedtheideaoftemporallyrepeatedchainrowsof astaticrow.Thisenabledthemtosolvethemaximumdynamicr owproblemwithone sourceandonesink.Thefundamentalprincipleisthatthere isalwaysanoptimaldynamic rowwhichusesonlyonedirectionofanarc,butneverboth.Th eycallthisa standard chaindecomposition .Thispropertyallowsustosolvethemaximumdynamicrowpro blem instronglypolynomialtime.WeexploitthispropertyinSec tion 5 tosolvetheMDCF problem. Theconceptofstandardchaindecompositionisnotsucient forsomewellknown dynamicrowproblems[ 187 190 191 ].AnexampleisgiveninFig. 5-2 .Graph G =( V;A ) showninFig. 5-2 (a)hasafeasiblerowwithtime T =6asillustratedinFig. 5-2 (b).The dashedandgraylinesshowthetworowsfromnodes s +1 and s +2 tonode s ,respectively. Analyzingthegraphrevealsthatthereisnofeasiblerowwit hintimehorizon T =6using onlyoneofthearcs( n 1 ;n 2 )or( n 2 ;n 1 );thiscanbeseen,forinstance,byconsideringthe rowtroughthecutseparating s fromtherestofthegraph. 147

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However,itwasstillpossibletosolvethemaximumdynamicr owproblemwith multiplesourcesandmultiplesinksinthetime-expandedgr aph;resultinginapseudo-polynomial runningtimealgorithm.However, Hoppe wasabletoprovideapolynomialtime algorithmforthedynamictransshipmentproblem[ 187 ].Heintroducedtheconceptof non-standardchaindecomposition ,allowingrowineitherdirectionsofanarcatdierent timesteps{ifbothdirectionsofanarcarepresentinthegra ph. Looselyspeaking,theproceduresP-MCFandP-MDCFreverset hearcsonthery andtheyareblindwhethertheyreverseanarcornot.Thisdoe snotcauseanyproblems inthecontextofstaticrowsorsinglesourceandsinglesink dynamicrows,asina standardchaindecomposition,onecanalwaysderiveanopti malsolutionusingonlyone thearcsduringthewholetimehorizon.However,inthecaseo fmultiplesourcesand multiplesinks,thepotentialofusingbotharcsleadstothe problemthatwehavetoknow ifanarchasbeenreversedornot.Butexactlythismemoryand thetradeoofreversing thearcnoworatalatertime,makestheproblem NP -complete.ConsiderFig. 5-2 again. ApplyingtheideaofproceduresP-MDCFtothisproblemleads tothefollowingresult:At time1,wewouldswitcharc( n 2 ;n 1 )inordertoincreasethecapacityandattimepoint3, wewouldswitchitbackagain;resultinginarowneedingonly T =5timesteps. s +2 s +1 n 2 n 1 s (7,3) (10,1) (1,1) (1,1) (2,1) (2,1) 7 10 -17 AGraph G =( V;A ) s +2 s +1 n 2 n 1 s 7 10 1 1 1 6 9 1 BFeasiblerowwith T =6usingboth arcs( n 1 ;n 2 )and( n 2 ;n 1 ) Figure5-2.AtoughinstanceofDTCF InSec. 5 ,weshowedthatDTCFis NP -complete.Thereductionfrom 3SAT involves j C j + j U j sourcenodesand j U j +1sinknodes.Inthefollowing,weshowthatthereisno 148

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polynomialtimealgorithmfortheDTCFproblemhavingonlyt wosourcesandonesink (oronesourceandtwosinks),unless P = NP .Inotherwords,allowingonlyonemore sourceorsinktoDTCFmakestheproblem NP -complete. Wedonotgointofulldetailhere,butratherprovidetheidea ofareduction fromPARTITION,whichismotivatedbythekeyobservationof Lemma 2 andthe NP -completenessproofby Melkonian ,[ 192 ].Givenisaniteset A andasize a i 2 Z + foreach i 2 A .ThePARTITIONproblemdecideswhetherthereisasubset A A such that P i 2 A a i = P j 2 A n A a j ,ornot.PARTITIONisknowntobe NP -complete(intheweak sense),see[ 28 ,Theorem3.5,Chapter4.2].Let P i 2 A a i =2 L with L 2 Z + .Weconstruct aninstanceoftheDTCFwithtwosourcenodes s +1 s +2 andonesinknode s ,asshownin Fig. 5-3 .Theideaofthistransformationisthattherowatnode s +2 hastopassthrough node v 1 0 toreachnode s ,andoneunitofrowfromnode s +1 hastotravelthoughnode v 1 n tonode s .Thisisindeedtrueasotherwisethetotaltimeboundof T =2 L +2wouldbe exceeded.Therowthroughthenodes v 1 0 to v 1 n andbackgivestheassignmenttoset A ; i.e. i 2 A ifandonlyifarc( v 1 i 1 ;v 1 i )isnotreversedinthegraph. Figure5-3.InstanceforDTCFwithtimebound T =2 L +2resultingfromPARTITION 5.5ContrarowProblemswithArcSwitchingCost Toallowtheswitchingofanarcinordertoincreasethecapac ityinonedirection resultsfromtheapplicationinevacuationscenarios.Howe ver,inpractice,youmight notbeabletoswitchcertainarcs.Forinstance,inevacuati onscenarios,certainstreets 149

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arereservedforemergencyvehiclesbutcanalsobeusedby(l imitednumberof)other travelers; i.e. thiscanbemodeledbyreducingthecapacityofthisarcandbl ockingit frombeingreversed.Inaddition,theswitchingofanarcish ighlycostly; i.e. inorderto switchthedirectionofahighway,wehavetosetuppoliceblo cksoneachentrytothe highway.Hence,itisnaturaltoaskwhataretheminimumcost incurredinswitchingthe arcsallowingacertain(minimum)amountofrow.Thisleadst othefollowingproblem. Denition7 (FixedSwitchingCostContrarow(FSCF)) Instance :Adirectedgraph G =( V;A ) withasetofsources S + ,asetofsinks S excess b 2 Z j V j ,arccapacities c e andarc-switchingcost b fe foreacharc e 2 A Question :Findafeasiblerow f in G withminimaltotalcost,ifthedirectionofthe arcscanbereversedwith(xed)cost b f NotethatFSCFisastaticproblemwithmultiplesourcesandm ultiplesinks.The xedcost b fe occur,wheneverarc e isreversed.Thisdenitionallowstomodelthe situationdescribedabove:Wheneveranarccannotberevers ed,thenitscostcanbe assignedahighvalue; i.e. Big M .Asthecostofswitchingcandierforeacharc,wecan distinguishbetweentheeortofreversinganarc; i.e. reversingahighwayoranalleyway involvesdierentcostorresources. Thexedswitching-costcontrarowproblemhasthefollowin ginterestingvalue. OnecansolvetheMTCFproblemanddeterminetheoptimalrowi nthegraph,see Corollary 1 .Later,onecanapplytheFSCFproblemtodeterminetheminim alcost impliedbytheswitchingofarcs,whilestillpushingtheopt imalamountofrowtroughthe graph. NoticethattheFSCFproblemhasasimilarstructureasthe minimumconcave-cost networkrow problems.Theseproblemsasktondafeasiblerowwhilemini mizingthe totalcostwhichareinthiscasethesumofconcave-costsind ucedbyusingofthearcs.For anexactdenitionandanoverviewaboutthisproblem,pleas eseethesurveyby Guisewite and Pardalos ,[ 193 ].Wecanbasicallyassumetheconcave-costperarctoconsis t 150

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ofxedcost,occurringwheneverthisparticulararcisused ,andavariablecost,depending onhowmuchrowissendtroughthisarc,see[ 194 ].Fixingthevariablecosttozeroleads toaspecialproblemcalled minimumcostxedrow (MCFF)problem. Krukme etal. provethatthisproblemis NP -hardinthestrongsenseevenonseries-parallelgraphs, [ 195 ,Theorem14].Series-parallelgraphshaveaveryspecialst ructureandaredened recursively,see[ 196 197 ].Furthermore, Krukme etal.showthattheminimumcostxed rowproblemisequivalenttothefollowingproblem,[ 195 ,Theorem8]: Denition8 (0/1-MinimumImprovementFlow(MIF)) Instance: Agraph G =( V;A ) withsinknode s + ,sourcenode s ,excess b 2 Z j V j ,arc capacities c e 2 Z + ,maximumcapacities C e 2 Z + ;C e c e andcapacityimprovementcost b e 2 Z Question :Determineanimprovementstrategy d : A !f 0 ;C e c e g withminimum cost P e 2 A d e b e ,suchthatthegraphwiththeimprovedcapacity u e + d e ; 8 e 2 A ,allowsa feasiblerow f from s + to s Thedenitiongivenhereisslightlydierentthentheonein thepaperby Krukme etal.,[ 195 ,Denition7].Basically,weassumealldatatobepositivei ntegral.The improvementstrategyfunction d isa0-1decisionifadditionalcapacityisusedornot; independentofhowmuchadditionalcapacityisused.Thecos tforthisadditionalcapacity forarc e isxatvalue( C e c e ) b e .InordertoprovethatFSCFisstrongly NP -hard,we showthatitisequivalenttoMIF. Theorem7. Fixedswitching-costcontrarowisequivalentto0/1-minim umimprovement row. Proof. Withoutlossofgenerality,wecanassumetheFSCFproblemto havesinglesource andsinglesink.Recognizethatthegraphtransformationpr ovidedforCorollary 1 works here. \ ) "GivenaninstanceofFSCFforgraph G =( V;A )witharccapacity c e and arc-switchingcost b fe .ConstructaninstanceofMIFforgraph G =( V ; A )asfollows.If 151

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thereisanarc( i;j ) 2 A and( j;i ) = 2 A ,then( i;j ) ; ( j;i ) 2 A with i;j = C i;j = C j;i := c i;j i;j = j;i :=0,and j;i := b fi;j =c i;j respectively.Forthecasethat( i;j ) ; ( j;i ) 2 A ,wedene ( i;j ) ; ( j;i ) 2 A with i;j := c i;j C i;j = C j;i := c i;j + c j;i i;j := b fj;i = ( c i;j + c j;i ), j;i := c j;i and j;i := b fi;j = ( c i;j + c j;i )respectively.Byapplyingthecyclereductionprincipleu sedin Sec. 5 and 5 ,wecanseethatthistransformationisindeedvalid. \ ( "GivenaninstanceofMIFforgraph G =( V;A )with c e C e and b e ,construct aninstanceofFSCFforgraph G =( V ; A )asfollows.Foranyarc( i;j ) 2 A ,wehavethe threearcs( i;j ) ; ( i; i ) ; ( j; i ) 2 A .Dene i;j := c i;j fi;j = fi; i := M i; i = j; i = C i;j c i;j and fj; i := b i;j ( C i;j c i;j ),where M isabignumberpreventingtoswitchthecorrespondingarc inanoptimalsolution. Recognizethathavingxedcostforarcreversalsmakesthep roblem NP -hard,even inthestaticcase.Onereasonis,forinstance,theprevious lymentionedobservation,that theprocedureP-MCFis`blind'forthearcreversaldecision s.Addingatimecomponent toFSCFmakesitpracticallyevenmorediculttosolve.Thet imecomponentreveals alsothedierencesbetweenthe(dynamic)xedswitching-c ostcontrarowproblemand the(dynamic)0/1-minimumimprovementrowproblem:MIFae ctsonlyaparticulararc ( i;j ),whileinFSCFalsothereversearc( j;i )isaected,ifbotharcsarecontainedinthe graph. 5.6Conclusions Thischapterformallyintroducesthecontrarowproblemtha thasapplications inemergencytransportationmanagement.Severalclassicn etworkrowproblemsare studied,includingstaticanddynamicnetworks.Apolynomi altimealgorithmforthe dynamiccontrarowproblemwithsinglesourceandsinglesin kisgiven,togetherwithan NP -completenessproofforthedynamictranshipmentcontraro wproblem.Thehardnessof thecontrarowproblemwitharcreversalcostisalsoindicat ed. 152

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CHAPTER6 MULTIMODALSOLUTIONSFOREVACUATIONPROBLEMS 6.1Introduction Thesurveyonevacuationproblems[ 198 ]indicatesthatthereisashortageof analyticaltechniquesinmultimodalevacuationstudies.T hischapterfocuseson establishingecientevacuationrouteswithbimodaltrans portation.Weconsider emergencymanagementoreventmanagementsituationssucha sfootballgame,which assumestheabsenceofpanicsituationsbutstillcapturest heseveralaspectsofan evacuationsettings.Theseincludehighdemandsduringthe event,needtosatisfy demandsquicklyandcongestionduetohighdemands.Weassum eprivatecarsandbuses asthemodesoftransportation.Thecarsaretotakeapathfro msourcetodestination, whilethebusesarerouted.Weassumethattheroutesofthebu sesareknown.Weneed toestablishecientpathsforthecarsanddeterminethefre quencyofthebusesalong theroutes.Theproblemiscomparabletothelineplanningpr oblem[ 199 { 201 ],where multiplelinesormodesoftransportationareavailableand demandsofpeopleareavailable atspecictimewindows.Thelinesarepredenedpathsandth efrequencyofalineneeds tobedetermined.Columnsgenerationproceduresarequitep opularforthelineplanning problemandmuchresearchhavebeendoneinthisarea[ 201 202 ].Weemploythebranch andpriceapproachtosolvetheproblem. Inthenextsection,weformallyintroducetheproblemanddi scusstheinteger linearprogrammingformulation.Insection( 6 )wediscussthemultimodalrowproblem andintegerprogrammingformulation.Insection( 6 ),wediscussthebranchandprice procedureforthemultimodalrowproblem.Finallyinsectio n( 6 ),weprovidesome numericalresults. 6.2MultimodalProblem MultimodalrowproblemsareknowntobeNP-hard[ 178 ].Thus,theproblem, tailoredtotheneedsofevacuationstudies,requiresecie ntapproachestosolvethem 153

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eitherapproximatelyorexactly.Weprovidedapathbasedfo rmulationthatwouldenable ustoemployabranchandpriceproceduretosolvetheproblem .Weformallydenethe problemandstatetheassumptions.ProblemDenition Inthisproblem,wehavetwosetsofpeopledependingontheir modesoftransport. Werecognizethemodesoftransportationasprivatecarsand buses.Weknowthe demandsofcarsandpeopletravelingbybusforeverypairofn ode.Wealsohaveasetof bustoursthathasalreadybeenestablished.Thearcsofthen etworkunderconsideration issharedbybothcarsandbuses.Eachlinkhaveacapacityand acost.Weneedto determinethemostecientpathforthecarsbetweentheorig insanddestinationand theoptimalbusesroutesrequiredwithoutexceedingthecap acityofthearcs.Networks withtraveltimesontheirlinkscouldbeexpandedovertimew itheachlinkhavingacost dependingonthetimeinstanceoftheoriginatingnodeandit sdistancefromitsnearest sourcenode.Theoptimalbusroutesthenwouldprovideuswit hthefrequencywithwhich thebuseshavetoberouted. Weformallydenetheproblemas:Givenagraph G ( V;E )with c ij u ij asthecost andcapacityoneacharc ij 2 E ,asetofTbustours,andtwosetsoforigindestination pairs( OD ) 1 and( OD ) 2 correspondingtodemandsoforiginanddestinationsofcars and peopletravelingbybusrespectively,determinetheminimu mcostpathofthecarsandthe optimalsubsetofbusroutessatisfyingdemandsandcapacit y. AssumptionsandRealization Wemakesomesimplisticassumptions,whichdoesnothindert herealizationofthe model.Weassumethatthedemandsbetweentheorigin-destin ationpairsareknown andremainstatic.Weassumethatthebusroutesareestablis hedandweonlyneedto determinetheirfrequency.Wealsoassumezeroloadingandu nloadingtimeforthebuses. Thelastassumptioncouldhoweverbeovercomeinthecurrent modelbyappropriately 154

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changingthebusroutestoaccommodatexedloadingtimes.W enowprovideapathand routebasedformulationthatwillenableustoimplementabr anchandpricemechanism. 6.3FormulationandDiscussion Abimodalevacuationproblemisconsideredinwhichtwomode sofevacuation, namelyprivatecarsandbuses,areusedforevacuatingpeopl efromtheoriginnodes.The peoplehavetheirdestinationpreferencesfromtherespect iveorigins.Inanevacuation setting,thebusesareroutedtopickuppeoplefromtheirori ginsanddropthemat theirdestinations.Theprivatecarsaretakenbypeopledir ectlyfromtheorigintothe respectivedestinations.Thedemandsareknownintermsofn umberofcarsandnumberof peoplefortherespectiveorigin-destinationpair.Thebus esandthecarshavetosharethe capacityofthearcs.Weaimtoreachthedestinationswithth eleastpossiblecost.The objectivefunctionisalittlelooselydened,butwewillel aborateitshortly.Weprovide abranchandpriceframeworktosolvetheproblem.Wehavetwo subproblems,oneto generatethepathsoftheprivatecarsandtheothertogenera tepathsofpeople.Forthe timeexpandedformulation,weneedtodetermineanupperbou ndonthevalueof T .A looseboundonthisvaluecouldbeobtainedbyindividuallyb oundingthetimesforbuses andcarsseparatelyandaddingthemtogether.Theimplicati onisweseriallyroutethem oneaftertheotherandthisisstillafeasiblesolutiontoth eproblem.Thiswillalsohelp usobtainaninitialfeasiblesolutiontoourbranchandpric eprocedure.Wediscussthe proceduretoobtainindividualtimeboundslater. f p ( x p )isabinaryvariableindicatingwhetherpath p isusedtosatifythedemandof thecorrespondingorigindestinationpair. b t isbinaryvariablewithvalue1ifabustour t isusedand0otherwise. 1 ( ij;p )( 2 ( ij;p ))isanindicatorvariablewithvalue1ifarc ij is inbuspath(carpath) p and0otherwise. ( ij;t )isanindicatorvariablewithvalue1ifarc ij isintour t and0otherwise. r 1 ( st;p )( r 2 ( st;p ))isanindicatorvariablewithvalue1ifa buspath(carpath) p hasoriginanddestinationas s and t respectivelyand0otherwise. Let P 1 and P 2 bethesetsofallbuspathsandcarpathsrespectivelyandTbe thesetof 155

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allbustours. B isthecapacityofabus. OD 1 and OD 2 arethesetsoforigins-destination pairscorrespondingtobusesandcarsrespectively. d 1st and d 2st isthedemandofpeople usingbusesandcarsrespectivelyfromorigin s todestination t .Finally, b t isabinary variablewithvalue1ifatourispickedand0otherwise. Minimize X p 2 P 1 c p x p + X p 2 P 1 c p f p (6{1) s.t. X p 2 P 1 r 1 ( st;p ) x p =1 ; 8 st 2 OD 1 (6{2) X p 2 P 2 r 2 ( st;p ) f p =1 ; 8 st 2 OD 2 (6{3) X p 2 P 1 d 1st 1 ( ij;p ) x p X t 2 T ( ij;t ) Bb t 0 ; 8 ij 2 E (6{4) X p 2 P 2 d 2st 2 ( ij;p ) f p + X t 2 T ( ij;t ) b t u ij ; 8 ij 2 E (6{5) x p 2f 0 ; 1 g ; 8 p 2 P 1 (6{6) f p 2f 0 ; 1 g ; 8 p 2 P 2 (6{7) b t 2f 0 ; 1 g ; 8 t 2 T (6{8) 6.4BranchandPriceMechanism Wenotethattheproblemhasexponentiallymanypathsinterm softheinputsizeof thegraphandwewillbegeneratingthesevariablesinthesub problem.Thisisastandard approachinmostoftheroutingandschedulingproblems.RestrictedMasterProblem(RMP) Therestrictedmasterproblemisobtainedbyrelaxingconst raints( 6{6 )-( 6{7 )as continuousvariablesandreplacingthesets P 1 and P 2 bytherestrictedsets P 1 P 1 and P 2 P 2 respectively.Thisleadstotherestrictedmasterproblem. 156

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Minimize X p 2 P 1 c p x p + X p 2 P 1 c p f p (6{9) s.t. X p 2 P 1 r 1 ( st;p ) x p =1 ; 8 st 2 OD 1 (6{10) X p 2 P 2 r 2 ( st;p ) f p =1 ; 8 st 2 OD 2 (6{11) X p 2 P 1 d 1st 1 ( ij;p ) x p X t 2 T ( ij;t ) Bb t 0 ; 8 ij 2 E (6{12) X p 2 P 2 d 2st 2 ( ij;p ) f p + X t 2 T ( ij;t ) b t u ij ; 8 ij 2 E (6{13) 0 x p 1 ; 8 p 2 P 1 (6{14) 0 f p 1 ; 8 p 2 P 2 (6{15) 0 b t 1 ; 8 t 2 T (6{16) Let 2 R OD 1 betheunrestricteddualvariablecorrespondingtoconstra intset ( 6{10 ), 2 R OD 2 betheunrestricteddualvariablecorrespondingtoconstra intset( 6{11 ), 2 R E bethenon-positivedualvariablecorrespondingtothecons traintset( 6{12 ) andnally 2 R E bethenon-positivedualvariablecorrespondingtothecons traintset ( 6{13 ).Inthepricingproblem,weareinterestedinthereducedco stofthevariable x p and f p .Wedeterminetheminimumreducedcostofthepathrowvariab lesinthepricing subproblems.Iftheminimumreducedcostcorrespondingtoa origindestinationpair isnegativeweaddittotherestrictedset(correspondingto thecarsorbuses)andthe restrictedmasterproblemissolvedagain.People-PathSubproblem Inthepeople-pathsubproblem,wedeterminetheminimumneg ativereducedcost, x p ofapathrowvariable, x p ,forpeopletakingbusesbetweenagivenorigin-destinatio npair st 2 OD 1 .Thisisgivenby 157

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x p = c p ( st + X 8 ( ij ) 2 E 1 ( ij;p ) ij )= st + X 8 ( ij ) 2 E ( 1 ( ij;p ) c ij 1 ( ij;p ) ij )(6{17) Thecostofthepathisgivenbythesumofthecostonarcsinthe aboveequation. Now,theshortestpathproblemforallpairsof st 2 OD 1 withtheabovearccosts c ij ij issolved.If st ismorethanthelengthofapath,thenitisaddedtotherestri ctedpath set P 1 andtheRMPissolvedagain.Weobservedthatthedualvariabl e isnegativeand hencethecostoneacharcispositive.Thustheresultingsho rtestpathproblemissolvable inpolynomialtime.Car-PathSubproblem Inthecar-pathsubproblem,weareconcernedwiththenegati vereducedcost, f p ,of carrowvariables, f p ,forthepairsoforigin-destination st 2 OD 2 .Thisgivenby f p = c p ( st + X 8 ( ij ) 2 E 2 ( ij;p ) ij )= st + X 8 ( ij ) 2 E ( 2 ( ij;p ) c ij 2 ( ij;p ) ij )(6{18) Wesolvetheshortestpath,justasinpeople-pathsubproble m,butwitharccost c ij ij andifthecostofapathfrom s to t islessthan st ,weaddittotherestrictedset P 2 andtheRMPissolvedagain. BranchingStrategy Thebranchingrulesisimportantasthisdeterminesthecomp lexityofthepricing problem.Thebranchingalsoinducessomepracticaldicult iesthatneedstobeexplicitly handled.Weaddressthetwoimportantproblemsencountered whilebranching. Thedecisiontobranchoccursatanodeofthebranchandbound tree,whenwe cannotenteranymorecolumnstotherestrictedsetsfromeit hersubproblemsandthe relaxedLPsolutionatthecurrentnodeisinfeasibletothei ntegerprogram.Atthis juncture,ifanyofthe b t variablesarefractional,wedecidetobranchonthem.Itise asy toseethatthisbranchingwillnotcauseanydicultytothes ubproblemsasthearc 158

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costscorrespondingtotheshortestpathproblemsstillrem ainspositive.Ifnoneofthe b t variablesarefractionalandifarowvariableisfractional ,abranchingonthefractional pathrowvariablewouldrestrictthesubproblemstogenerat epathsotherthanthepath thatwasfractional.Forinstance,let f p bethefractionalpathwithvalue f p .Webranch byadding f p b f p c toonebranchand f p d f p e totheotherbranch.Thedicultynowinthesubproblemistha tacandidatepath generatedforthatorigin-destinationpairshouldnotbeth ebranchedvariable.Wecannot guaranteethattheshortestpathproblemcouldgeneratesuc hapath.Infactafter k branchings,wemighthavetosolvethe k th shortestpathsubproblem.Thisisacommon dicultythatarisesinbranchandpriceapproachesformult icommodityrowproblems. Thereareafewtechniquesintheliteraturethathandlesthi sissue[ 203 { 206 ].One techniqueistomakeanarcorasetofarcsofthepaththatwasb ranchedasforbidden arcsinthebranches.Thusthesubproblemwillnotregenerat ethepath[ 203 ].Another techniqueintheliterature[ 204 ]tosolvetheproblemistobranchonarcrowvariable insteadofpathrowvariable.Forinstance,theamountofcar sonanarc ij isgivenby P p 2 P 2 d 2st 2 ( ij;p ) f p andlet x ij bethefractionalrowonthearc.Sowecanaddthe constraint X p 2 P 2 d 2st 2 ( ij;p ) f p b x ij c toonebranchand X p 2 P 2 d 2st 2 ( ij;p ) f p d x ij e tootherbranch.Thishoweverdoesnotguaranteepositivear ccostsanymoreinthe subproblemsandtheybecome NP -hard.Thisproblemwasovercomebyaddingseparate variablesforrowsalongcyclesintheRMP.Thusthesubprobl emhastoreturnashortest 159

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pathifavailableoranegativecostcycle.Thiscouldbesolv edinpolynomialtime.Inthis problem,werevisitthetechniqueemployedby[ 205 ]forabandwidthpackingproblem, whichisverysimilartothemulticommodityrowproblem.Afr actionalpathvariable isdealtbycreatinganumberofbranches.Eachbranchcorres pondstoanarcofthe fractionalpath,whereitisforbiddenandthereisoneaddit ionalbranchingnodeinwhich thepathisxedtothesolution. Thenextproblemtobeaddressedoccurswhenwearriveatabra nchandbound nodewiththeLPrelaxationresultinginaninfeasiblesolut ion.Inanelementarybranch andboundprocedureweprunethesearchinthissituation.Ho wever,thisisnotpossible inbranchandpricemechanismaswehavenotyetconsideredth eentiresetofcolumns andhencetheremightexistapaththathasnotbeenenteredbu tcouldprovideafeasible solutioninthefuture.Wetakecareofthisissue,byaddingd ummypathsintheinitial solutionwithhighcostthatwillprovideuswiththefeasibl esolution. 6.5ComputationalResults Wetestedgridgraphsofsize25to400nodes.Table 6-1 enumeratestheinstanceswe tested.Thelargestinstancetestedwasgridgraphwith400n odesand1520edgeswith100 carsand40buses. Table6-1.Branch&Pricemodeltestedongridgraphs #NodesArcsCarsBusesToursCPUTime 125806630.01322580101040.06432580201040.17442580151540.08951003605540.1386100360101051.1617100360201051.0638100360202070.89225840101075.637 102258402020101.205114001520100401452.503 160

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Additionally,inordertotesttheroubustnessofthecodewe testedafewonline benchmarkinstancesformulticommodityrowproblems[ 207 ]forfourplanargraphs andresultsareprovidedintable 6-2 .Wegeneratedonedummybustourandonebus commodityforeachoftheinstances. Table6-2.Branch&Pricemodeltestedonplanargraphswitho nebuscommodity #NodesArcsCarsCPUTime 130150920.082502502670.1773804405430.509410053210851.605 6.6Conclusion Thischapterprovidesabranchandpriceframeworktosolvea bimodalmulticommodity rowproblem.Weconsidercarsandbusesastwomodesoftransp ortationandweobtained optimalpathsforcarsandidentiedthebusroutesneededfo rtransportation.Wetested themodelongridgraphsofsizesupto400nodes.Asafuturewo rk,weneedtoemploy heuristicmethodsforthesubproblems,developprocedures thatwouldprovidegoodlower boundforthebranchandbound.Wearecurrentlyintheproces sofimplementingabox stabilizationtechniqueinordertoacceleratetheconverg enceofthebranchandprice procedurethatwouldenableustotestmuchlargerinstances 161

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CHAPTER7 CONCLUSIONS Inthisdissertation,wefocusedonnetworkmodelshavingap plicationsindisaster management.Weexaminedthreekindsofapplications,inwhi chwestudieddierent problems.Therstapplicationdealtwiththeidenticatio nofcriticalnodesinagraph thatarecrucialforitsconnectivity.Weanalysedandprovi dedsolutiontechniquestotwo variationsoftheproblem.Wethenstudiedtwopathplanning problemsthatinvolvesin routingofagentsorvehiclesinanetworkinordertovisitta rgetsalongtheirroutes.We nallystudiedevacuationproblemsinordertoestablishev acuationroutesandcontrarow plans.Inalltheseproblems,weexaminedthecomplexityoft heproblem,providedinteger programmingformulationsandheuristicorexactsolutiont echniquestosolvethem. Inchapter 2 ,westudiedthevariationsofthe criticalnodedetectionproblem Theproblemidentiesthenodeswhosedeletionresultsinas ubgraphthathasmaximum fragmentation.Weprovidedcomplexityanalysisforthepro blems,integerprogramming formulationandheuristicsolutions.Wewouldliketostudy aweightedversionofthe problemandexploreapproximationalgorithmsthatwouldgu aranteeatheoreticalbound onthesolution. Inchapter 3 ,westudiedtwopathplanningproblems.Intherstproblem, the targetvisitationproblem ,weformulateditasanintegerprogramandprovideda geneticalgorithmfortheproblem.Wethencomparedthesolu tionsoftheheuristicwith theCPLEXsolutions.Wealsostudied communicationmodelsforcooperative network ,wherewehavetoroutemultipleagentstovisittargetsinac ooperative networkinordertomaximizetheircommunication.Weprovid edanintegerprogramming formulationandaGRASPbasedheuristicproceduretosolvet heproblem.Asafuture work,wewouldliketoparallelizetheheuristicandenhance itwithmoresophisticated searchprocedures. 162

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Inchapter 5 ,westudiedvariousnetworkrowproblemwitharcreversalca pabilities. Weperformedadetailedcomplexityanalysisfortheseprobl ems.Futureresearchinthese problemsinvolvesindevelopingecientsolutiontechniqu esforthe NP -hardproblems.In chapter 6 ,westudiedtheevacuationproblemforwhichweprovidedabr anchandprice approachtosolveabimodalmulticommodityrowproblem.Wea recurrentlyworkingon stabilizationtechniquestoacceleratetheconvergenceof thealgorithminordertotest largescaleinstances. 163

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BIOGRAPHICALSKETCH Theauthorofthisdissertation,Mr.AshwinArulselvan,isa graduatestudentatthe UniversityofFloridainindustrialandsystemsengineerin g.HeobtainedhisBachelor ofEngineeringfromtheCollegeofEngineering,Guindywith amajorinmechanical engineering.HeworkedasasoftwareengineeratInfosysTec hnologiesLtd.foranyear. HethenobtainedhisMasterofSciencewiththethesisoption fromUniversityofFlorida beforehestartedhisPhDwithDr.PanosM.Pardalos. 179