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Discrete and Geometric Approaches to Lifetime Maximization in Wireless Sensor Networks

Permanent Link: http://ufdc.ufl.edu/UFE0043321/00001

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Title: Discrete and Geometric Approaches to Lifetime Maximization in Wireless Sensor Networks
Physical Description: 1 online resource (143 p.)
Language: english
Creator: Behdani, Behnam
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: lifetime -- networks -- optimization -- sensor -- tsp -- wireless
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 address several problems related to the efficient use of energy resources in wireless sensor networks (WSNs). We first consider the problem of maximizing the lifetime of a WSN with energy-constrained sensors and a mobile sink. In this setting, the sink travels among discrete locations to gather information from all sensors. Data can be relayed among sensors and then to the sink, as long as the sensors and the sink are within a certain threshold distance of each other. However, sending information along a data link consumes energy at both the sender and the receiver nodes. We describe linear programming and column generation approaches for the problem of maximizing the lifetime of a WSN. We also propose decomposition and distributed algorithms for the case in which the underlying application can tolerate some degree of delay in delivering data to the sink. Our distributed algorithm can be embedded into a network protocol so that the senor nodes and the sink can run it directly as part of the network operation. We also address a variant of the traveling salesman problem known as the close-enough traveling salesman problem that is useful in devising energy-aware trajectories for the sink in a WSN. We formulate a mixed-integer programming model based on a discretization scheme for the problem. Both lower and upper bounds on the optimal tour length can be derived from the solution of this model, and the quality of the bounds obtained depends on the fidelity of the discretization scheme. Finally, we propose an extension to the lifetime maximization problem by examining the case in which the sink periodically travels at finite speed among a subset of sink locations to collect data from sensor nodes. In addition to the decisions regarding data routing and the sink's dwelling times, the problem finds a tour for the sink among a subset of sink locations. We provide mixed-integer programming formulations to model this problem, along with cutting planes, preprocessing techniques, and a Benders decomposition algorithm to improve its solvability. Computational results demonstrate the efficiency of the proposed algorithms.
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 Behnam Behdani.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Smith, Jonathan.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0043321:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043321/00001

Material Information

Title: Discrete and Geometric Approaches to Lifetime Maximization in Wireless Sensor Networks
Physical Description: 1 online resource (143 p.)
Language: english
Creator: Behdani, Behnam
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: lifetime -- networks -- optimization -- sensor -- tsp -- wireless
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 address several problems related to the efficient use of energy resources in wireless sensor networks (WSNs). We first consider the problem of maximizing the lifetime of a WSN with energy-constrained sensors and a mobile sink. In this setting, the sink travels among discrete locations to gather information from all sensors. Data can be relayed among sensors and then to the sink, as long as the sensors and the sink are within a certain threshold distance of each other. However, sending information along a data link consumes energy at both the sender and the receiver nodes. We describe linear programming and column generation approaches for the problem of maximizing the lifetime of a WSN. We also propose decomposition and distributed algorithms for the case in which the underlying application can tolerate some degree of delay in delivering data to the sink. Our distributed algorithm can be embedded into a network protocol so that the senor nodes and the sink can run it directly as part of the network operation. We also address a variant of the traveling salesman problem known as the close-enough traveling salesman problem that is useful in devising energy-aware trajectories for the sink in a WSN. We formulate a mixed-integer programming model based on a discretization scheme for the problem. Both lower and upper bounds on the optimal tour length can be derived from the solution of this model, and the quality of the bounds obtained depends on the fidelity of the discretization scheme. Finally, we propose an extension to the lifetime maximization problem by examining the case in which the sink periodically travels at finite speed among a subset of sink locations to collect data from sensor nodes. In addition to the decisions regarding data routing and the sink's dwelling times, the problem finds a tour for the sink among a subset of sink locations. We provide mixed-integer programming formulations to model this problem, along with cutting planes, preprocessing techniques, and a Benders decomposition algorithm to improve its solvability. Computational results demonstrate the efficiency of the proposed algorithms.
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 Behnam Behdani.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Smith, Jonathan.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0043321:00001


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DISCRETEANDGEOMETRICAPPROACHESTOLIFETIMEMAXIMIZATIONINWIRELESSSENSORNETWORKSByBEHNAMBEHDANIADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012BehnamBehdani 2

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Tomydearestparents,AbolghasemandFatemeh 3

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ACKNOWLEDGMENTS Iwouldliketoexpressmydeepestgratitudetomyadvisor,Dr.J.ColeSmith,forhisguidance,patience,andconstantsupportthroughoutmyyearsattheUniversityofFlorida.HehasbeenagreatmentorformeandIlearntsomuchfromhim,notonlyaboutmathbutalsolifelessons.Thisaccomplishmentwouldnothavebeenpossiblewithouthishelpandsupport.IwouldalsoliketothankDr.JosephGeunes,Dr.YongpeiGuan,andDr.YeXiaforservingonmyPh.D.committee.Theirvaluablesuggestionsandcommentscertainlyhelpedtoimprovethequalityofthisdissertation.ImustparticularlythankDr.Xiaforbeingagreatcoauthorandinstructorforme.DuringmytimeattheUniversityofFlorida,Ihadtheluxuryofmeetingmanyneindividualswhobecamegreatfriendsofmine.ImustspeciallythankmygreatfriendsEhsanSalariandSoheilHemmatiforallthehelpandsupportthattheyprovidedmewithduringmyyearsinGainesville.IwouldalsoliketothankmydearfriendsVeraTomaino,SomarKoria,DonatellaGranata,GudbjortGylfadottir,ArniOlafurJonsson,SaedAlizamir,MasuadZarepisheh,SibelBilgeSonuc,PetrosXanthopoulos,DmytroKorenkevych,ClayKoschnick,MikePrince,SaeedGhadimi,RezaSkandari,AshwinArulselvan,OlegShylo,AltannarChinchuluun,SiqianShen,MehmetOnal,andAlexeySorokinforallthewonderfulmomentsandmemoriesthatIwillalwaysrememberthemby.Andlastbutcertainlynottheleast,myspecialthanksgoestomydearestparents,AbolghasemandFatemeh,andmylovelybrothersandsisters,Behzad,Bahareh,Behrouz,andBahereh,fortheirendlessloveandconstantsupport.IowethemeverythingthatIhave,andamforevergratefultothem. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 11 2DECOMPOSITIONALGORITHMSFORMAXIMIZINGTHELIFETIMEOFWIRELESSSENSORNETWORKSWITHMOBILESINKS ........... 14 2.1LiteratureSurvey ................................ 14 2.2NetworkandEnergyConsumptionModel .................. 17 2.3MobileSinkModelwithoutDelayTolerance ................. 19 2.4MaximizingLifetimeinDelay-TolerantApplications ............. 25 2.5ComputationalResults ............................. 31 3ADISTRIBUTEDALGORITHMFORTHELIFETIMEMAXIMIZATIONPROBLEMINDELAY-TOLERANTAPPLICATIONS ...................... 38 3.1MotivationandLiteratureSurvey ....................... 38 3.2ProblemFormulation .............................. 40 3.3ALagrangianRelaxationApproach ...................... 43 3.3.1DistributedAlgorithmsforSubproblems ............... 45 3.3.2TheMainAlgorithm ........................... 48 3.4ConvergenceResults ............................. 49 3.5ComputationalExperiments .......................... 55 4ANEXACTAPPROACHTOTHECLOSE-ENOUGHTRAVELINGSALESMANPROBLEM ...................................... 58 4.1ProblemDenition ............................... 58 4.2Preliminaries .................................. 60 4.2.1DenitionsandNotation ........................ 61 4.2.2PartitioningSchemes .......................... 65 4.3LowerBoundingModel ............................ 68 4.3.1FormulationandBounds ........................ 68 4.3.2Cutting-PlaneGeneration ....................... 74 4.4AlternativeModelFormulationsandAlgorithms ............... 78 4.4.1ExpandedFormulation ......................... 79 4.4.2SubtourElimination ........................... 81 5

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4.4.3AlternativeFormulation ......................... 84 4.5CalculationofDistanceValues ........................ 89 4.6ComputationalExperiments .......................... 91 5EXTENSIONSTOTHELIFETIMEMAXIMIZATIONPROBLEM ......... 98 5.1TheMobileSinkModelwithFiniteSinkSpeed ............... 98 5.2ProblemStatement ............................... 101 5.2.1Preliminaries .............................. 101 5.2.2ProblemFormulation .......................... 104 5.2.3AlternativeDelayModel ........................ 107 5.2.4ComparisonofMSM,MSM-FS1,andMSM-FS2 .......... 116 5.3Cutting-planeAlgorithmforMSM-FS1 .................... 118 5.4BendersDecomposition ............................ 123 5.5ComputationalResults ............................. 125 6CONCLUSIONSandFUTURERESEARCHDIRECTIONS ........... 134 REFERENCES ....................................... 138 BIOGRAPHICALSKETCH ................................ 143 6

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LISTOFTABLES Table page 2-1Experimentalparametersandtheirvalues ..................... 31 2-2Performanceofthetwoalgorithmsforthemobilesinkmodel .......... 32 2-3ComparingtheresultsofthecolumngenerationalgorithmforthemobilesinkmodelwithdifferentlinearprogrammingsolversofCPLEX ............ 33 2-4Comparingthecolumngenerationandthesubgradientalgorithmsforthemobilesinkmodel ...................................... 34 2-5Performanceofthetwoalgorithmsforthedelay-tolerantmobilesinkmodel .. 36 2-6Comparingtheresultsofthecolumngenerationalgorithmforthedelay-tolerantmobilesinkmodelwithdifferentlinearprogrammingsolversofCPLEX ..... 37 4-1Comparinglowerboundformulationsonsix-nodeinstances ........... 93 4-2Lowerandupperboundsobtainedonsix-nodeinstances ............ 94 4-3ComparingformulationsLB1andLB3 ....................... 95 4-4PerformanceofBendersDecomposition ...................... 97 5-1Comparisonofthedirectsolveandcutting-planealgorithmsforMSM-FS1 ... 129 5-2Performanceofthecutting-planealgorithmforMSM-FS1underincreaseddelaytolerance .................................... 130 5-3ComparisonofthedirectsolveandBendersdecompositionalgorithmsforMSM-FS1 ....................................... 131 5-4PerformanceoftheBendersdecompositionalgorithmunderincreaseddelaytolerance ....................................... 132 5-5ComparisonofsolutiontimesforMSM-FS2models( 5 )and( 5 ) .... 133 7

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LISTOFFIGURES Figure page 2-1Illustrationofqueue-baseddelay-tolerantmobilesinkmodel ........... 30 2-2Convergenceofcolumngenerationalgorithm ................... 35 3-1Expandedgraphofdelay-tolerantmobilesinkmodel ............... 41 3-2Convergencetotheoptimalvalue ......................... 56 3-3Timeaverageoftotalvirtualqueuesizeovertime ................. 57 4-1AfeasibletourwithjMj=4 ............................. 61 4-2Illustrationofboundarypointoptimality ....................... 63 4-3Illustrationoftheconvexhulloftargetpoints .................... 65 4-4IllustrationoftheconvexhullargumentinproofofProposition 4.2 ........ 66 4-5Illustrationofgrid-basedpartitioning ........................ 67 4-6Illustrationofarc-basedpartitioning ......................... 68 4-7Aninstanceofthelowerboundproblem( 5 )anditsoptimalsolution .... 72 4-8Illustrationoftheobjectivefunctioncoefcientseijkin( 4 ) ........... 80 4-9Obtainingalowerboundonthedistancebetweentwoarcs ........... 91 4-10Resultsfortheiterativesolutionmethod ...................... 96 5-1Awirelesssensornetworkwithvesinklocations ................. 103 5-2IllustrationofG0fortheexampleinSection 5.2.1 ................. 117 5-3ConstructionofH0intheproofofProposition 5.5 ................. 122 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyDISCRETEANDGEOMETRICAPPROACHESTOLIFETIMEMAXIMIZATIONINWIRELESSSENSORNETWORKSByBehnamBehdaniAugust2012Chair:JonathanColeSmithMajor:IndustrialandSystemsEngineeringWeaddressseveralproblemsrelatedtotheefcientuseofenergyresourcesinwirelesssensornetworks(WSNs).WerstconsidertheproblemofmaximizingthelifetimeofaWSNwithenergy-constrainedsensorsandamobilesink.Inthissetting,thesinktravelsamongdiscretelocationstogatherinformationfromallsensors.Datacanberelayedamongsensorsandthentothesink,aslongasthesensorsandthesinkarewithinacertainthresholddistanceofeachother.However,sendinginformationalongadatalinkconsumesenergyatboththesenderandthereceivernodes.WedescribelinearprogrammingandcolumngenerationapproachesfortheproblemofmaximizingthelifetimeofaWSN.Wealsoproposedecompositionanddistributedalgorithmsforthecaseinwhichtheunderlyingapplicationcantoleratesomedegreeofdelayindeliveringdatatothesink.Ourdistributedalgorithmcanbeembeddedintoanetworkprotocolsothatthesenornodesandthesinkcanrunitdirectlyaspartofthenetworkoperation.Wealsoaddressavariantofthetravelingsalesmanproblemknownastheclose-enoughtravelingsalesmanproblemthatisusefulindevisingenergy-awaretrajectoriesforthesinkinaWSN.Weformulateamixed-integerprogrammingmodelbasedonadiscretizationschemefortheproblem.Bothlowerandupperboundsontheoptimaltourlengthcanbederivedfromthesolutionofthismodel,andthequalityoftheboundsobtaineddependsonthedelityofthediscretizationscheme.Finally,weproposeanextensiontothelifetimemaximizationproblembyexaminingthecase 9

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inwhichthesinkperiodicallytravelsatnitespeedamongasubsetofsinklocationstocollectdatafromsensornodes.Inadditiontothedecisionsregardingdataroutingandthesink'sdwellingtimes,theproblemndsatourforthesinkamongasubsetofsinklocations.Weprovidemixed-integerprogrammingformulationstomodelthisproblem,alongwithcuttingplanes,preprocessingtechniques,andaBendersdecompositionalgorithmtoimproveitssolvability.Computationalresultsdemonstratetheefciencyoftheproposedalgorithms. 10

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CHAPTER1INTRODUCTIONAdvancesinlow-powerwirelesscommunicationsandmicroprocessortechnologiesduringthepastdecadehavetriggeredtherapiddevelopmentandcommercializationoflow-costwirelesssensorsthatcanconstantlycollectinformationfromtheirsurroundings.Awirelesssensornetwork(WSN)consistsofalargenumberofwirelesssensorsthataredenselydeployedinanareaformonitoringpurposes.Sensorsusuallymeasureaphysicalattributeoftheirsurroundingarea(suchasthetemperature)orsensethemovementofobjectsintheirvicinity.Eachsensorcontainsamicroprocessorwithlimitedprocessingabilities,alimitedamountofmemory,alimitedpowersource,andoneormoresensingdevicesthatcontinuouslycollectinformationfromthesensor'ssurroundings.Thecollectedinformationneedstobetransmittedforfurtherprocessingtoadatacollectingnodecalledthesink.Thesensorsareusuallydeployedrandomly,butinhighquantitiestopreventcoveragebreach.Thisdensedeploymentofthesensornodesallowsmultihopdeliveryofthecollectedinformation:Eachsensorcancommunicatewithafewneighborsandusethemtorelayitscollectedinformationtothesink.TheneedforenergyconservationandfrequencyreusemakesmultihoproutinganessentialcomponentofWSNs[ 1 2 ].Despitetheirsimplestructure,WSNsprovideausefulmonitoringtoolinmanyreal-lifesettingssuchasenvironmentalmonitoring,homeautomationandhealth-careoperations.Itiswellknownthattheenergyrequiredforprocessingaunitofdataismuchlessthantheenergyrequiredforitstransmission[ 3 ].Eachsensornodeconstantlyconsumesenergytosenditsowndataaswellastorelay(possiblysomeof)itsadjacentnodes'datatothesinkorotherintermediatenodes.However,sensorbatteriesareoftenirreplaceableduetothelargenumberofsensorsinthenetworkandthefactthatsensorsmaybedeployedinhard-to-reachorhostileenvironments.Therefore,devisingenergy-awareroutingschemesforsensornodesisofcrucialimportanceinthedesign 11

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andoperationofWSNs[ 4 ].Networklifetimeisusuallydenedasthetimeuntiltherstsensornodecompletelyexhaustsitsenergy[ 5 6 ].Energy-awareoperationofWSNshasbeenthefocusofmanyresearcherslately[ 7 10 ].Inthisdissertation,wefocusonmathematical-programming-basedapproachestoenergy-awareoperationofWSNs.InChapter 2 ,weconsidertheproblemofmaximizingthelifetimeofaWSNwithamobilesinkthatmovesoverasetofpre-speciedsinklocationstocollectdatafromsensornodes.Thedecisionstobemadeinthismodelincludethesink'sdwellingtimeateachsinklocation,aswellasdatatransmissionpatternsfromsensornodestothesinkatdifferentsinklocations.Wedevelopacolumngenerationalgorithmforthelifetimemaximizationprobleminthissetting,andalsoforthecaseinwhichtheunderlyingapplicationcantoleratesomeamountofdelayindatatransmission.Exploitingtheunderlyingnetwork-owsstructureallowsustofurthersimplifythesubproblemintoseparableshortest-pathproblems.Columngenerationalgorithmsareimportantnotonlybecauseoftheirtheoreticalandcomputationalcontribution,butalsobecausetheyprovideafoundationforthedevelopmentofsemi-distributedalgorithmsforthelifetimemaximizationproblem.Distributedalgorithmsaregenerallymorewidelyusedandvaluedcomparedtotheircentralizedcounterpartsbecausetheycanbeincorporatedintothenetworkprotocolandbeusedasnetworkcontrolmechanisms.InChapter 3 ,weprovideafully-distributedalgorithmforthelifetimemaximizationprobleminadelay-tolerantWSNwithamobilesink.Ouralgorithmisproventoconvergetoanoptimallifetimeinthelong-run.Wealsoaddresstheproblemofndinganenergy-efcienttrajectoryforamobilesinkinWSNswherethesinkneedstotraveltodifferentareasofthesensoreldtocollectinformationfromsensornodes.IthasbeenshownbyCiulloetal.[ 11 ]thatanenergy-efcienttrajectoryforthesinkgetsclosertosensornodeswithhigherdatagenerationrates.Weusethisfacttomodelthisproblemasaninstanceoftheclose-enoughtravelingsalesmanproblem(CETSP)[ 12 13 ].TheCETSPisageometric 12

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generalizationofthewellknowntravelingsalesmanproblem(TSP)[ 14 ].WhileintheTSPthesalesmanhastovisitallcustomerlocations,intheCETSP,eachcustomerhasacompactneighborhoodsetarounditslocationwithinwhichthecustomeriswillingtotraveltomeetwiththesalesman.Therefore,eachcustomerisconsideredtobevisitedifthesalesman'stourvisitsatleastonepointinsideitsneighborhoodset.Weproposeaninteger-programming-basedapproachthatcansolvetheCETSPwithanylevelofaccuracyinChapter 4 .Finally,wepresentanimportantextensiontothelifetimemaximizationprobleminWSNswithamobilesinkbyliftingthecommonly-usedassumptionthatthesink'straveltimesarenegligible.InmanyenvironmentalmonitoringapplicationsofWSNs,assumingzerotraveltimesforthesinkisnotrealisticduetothelargeareaspannedbythenetwork.Weformulatethisproblemasamixed-integerprogramandprovidecutting-planeanddecompositionalgorithmstosolveitinChapter 5 13

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CHAPTER2DECOMPOSITIONALGORITHMSFORMAXIMIZINGTHELIFETIMEOFWIRELESSSENSORNETWORKSWITHMOBILESINKS 2.1LiteratureSurveyAwirelesssensornetwork(WSN)typicallyconsistsofalargenumberofsensornodesthatgatherinformationfromtheirneighborhoods.Thesensorscancommunicatewitheachotherviawirelesslinksiftheyarewithintheirmutualcommunicationrange.Thecollecteddatawillbedeliveredtothesinkforfurtherprocessing.Thedeliveryisoftenthroughmulti-hopcommunicationwiththesensorsrelayingthedatatowardthesink.Thesensornodesareusuallyexpectedtooperatewithbatteriesandmaybedeployedinnot-easily-accessibleorhostileenvironmentsinlargequantities.Therefore,itmaybedifcultorimpossibletoreplaceorrechargethebatteries.Ontheotherhand,thesinkistypicallyrichinenergy.BecausethesensorenergyisascarceresourceinaWSN,efcientenergyutilizationtoprolongtheWSNlifetimehasbeenanimportantresearchsubject.Recently,howtoimproveWSNlifetimebytakingadvantageofmobilityhasattractedsignicantattentionfromresearchers[ 7 10 15 19 ].Inthischapter,wewillfocusonsinkmobility.Followingtheconvention,wedenethelifetimeofaWSNasthetimeuntiltherstsensornodeexhaustsitsenergy[ 5 6 ].Ifthesinkisimmobile,theenergyconsumptionratecanbeseverelyimbalancedwiththenodesnearthesinkexhaustingtheirenergymuchsooner[ 20 22 ].Thisisknownastheenergyholeproblem.Thereasonisthatthecollecteddataisdirectedtowardthesinkandthenodesnearthesinkcanbeburdenedwithrelayingalargeamountoftrafcfromothernodes[ 23 ].Bymovingthesinkinthesensoreld,onecanmitigatetheenergyholeproblemandexpectanincreasednetworklifetime.Variouslifetime-maximizationproblemsareroutinelyformulatedaslinearorintegeroptimizationproblems[ 5 10 24 26 ].Thischapterfocusesontworelativelyrecentlinearprogrammingformulationsthatinvolveamobilesink.Theywillbecalledthe 14

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MobileSinkModel(MSM)andtheDelay-TolerantMobileSinkModel(DT-MSM).ThedifferencebetweenthetwoisthatDT-MSMisrelevanttoapplicationsthattoleratedelayedinformationdelivery.Ourmaincontributionsarecolumngenerationalgorithmsfortheseproblems.Thesealgorithmscanhelptoovercometwochallenges.First,solvingtheseproblemscanbeverytime-consumingusing(centralized)standardlinearprogramming(LP)solvers.WewilldemonstratethatouralgorithmfortheMSMisoftensignicantlyfasterthanhighly-tunedlinearprogrammingsolversevenwhenitrunsinacentralizedfashion.AlthoughouralgorithmfortheDT-MSMisoftenslowerthanthoseLPsolvers,itenjoysotherpotentialadvantagesassociatedwithbeingadecompositionalgorithm.1Second,therearefewknowndecompositionalgorithmsforsolvingthesetwoproblemsthatarealsofast.Decompositionalgorithmsareoftenmoreusefulandmoresought-afterthanmonolithiconesinthenetworkingareasinceportionsofthealgorithmscanbebuiltintonetworkprotocolsandbecomenetworkcontrolalgorithms.OuralgorithmfortheMSMinvolvessolvingseparateshortest-pathsubproblems.Thisstructuralpropertycanbeakeytoadaptingthealgorithmintodistributednetworkcontrolalgorithms,similartohowDijkstra'salgorithmisimplementeddistributedly[ 27 ].Moreover,ifthegoalisalgorithmspeedinsteadofnetwork-widedistributedimplementation,thesesubproblemscanbesolvedinparallelonmultipleCPUstofurtherspeeduptheprocessofndinganoptimalsolution.ThealgorithmfortheDT-MSM,afterdecomposition,involvessolvingasingleshortest-pathsubproblemonaspecialnetworkwiththesamesetofnodesastheoriginalnetworkandamoderatelylargerarc 1Findinganoptimalsolutionwithacentralizedsolvercanbeuseful.Insomeapplications,itispossibletomanagethenetworkwithacentralcontroller,whichcancontrolthenetworkelementsbasedontheknowledgeoftheoptimalsolution.Inaddition,evenwhendecentralizedalgorithmsmustbeappliedinpractice,anoptimalcentralizedsolutionprovidesaperformancebenchmarkandshowshowmuchlifetimeimprovementopportunitymayexist. 15

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set.Hence,distributedimplementationoftheshortest-pathalgorithmappearspossibleintheDT-MSMcaseaswell.Wenextbrieyreviewrelatedwork.GatzianasandGeorgiadis[ 9 ]formulateasimilarlifetimemaximizationproblemtoourrstproblem,theMSM,andprovideadistributedsubgradientalgorithmtosolveit.However,theydonotexploittheuncapacitatednatureoftheowbetweenthesensornodesandthesink.Asaresult,theysolveminimumcostowsubproblemsinsteadofshortestpathsubproblems.Anotherdrawbackoftheiralgorithmisthatitisverysensitivetothechoiceofstartingdualmultipliers,whilethecolumngenerationalgorithmsareknowntoberelativelyrobusttothechoiceofthestartingpoint.SankarandLiu[ 25 ]formulateastatic-sinklifetime-maximizationproblemandgiveadistributed,localalgorithmbasedonageneralnetwork-owalgorithm.Unlikethesubgradientalgorithm,itrelaxestheow-conservationconstraintsinsteadoftheenergyconstraint.YunandXia[ 26 ]proposeaframeworkofextendingthenetworklifetimebytakingadvantageofbothsinkmobilityandapplicationdelaytolerance.Forsituationswheretheunderlyingapplicationstoleratedelayedinformationdelivery(see[ 16 ]),eachsensornodecanstorethedatatemporarilyforuptoaprescribedmaximumdelay,andtransmititwhenthemobilesinkisatthelocationmostfavorableforachievingthelongestnetworklifetime.Theproposedmodelin[ 26 ]isexactlytheDT-MSMinthischapter.InthischapterwegiveacolumngenerationalgorithmfortheDT-MSM,whereasndingefcientalgorithmsisnotafocusof[ 26 ].Wangetal.[ 8 ]formulatealinearprogrammingproblem(LP)fordetermininghowtomovethemobilesinkandhowlongthesinkshouldstayateachstopalongthemovementpathinordertomaximizethenetworklifetime.Unlikethemodelsweconsiderinthischapter,theydonotincorporatetheroutingdecisionsintotheiroptimizationproblem.Instead,theyuseshortestpathalgorithmstodeterminehowtoroutethedatafromeachsensornodetothesink.Althoughshortestpathroutingprotocolsare 16

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commonchoicesinpractice(seeforexample[ 28 ]),inthiscase,itcausesanimbalanceinenergyconsumptionwhichresultsinshorterlifetime[ 29 ].Theproblemsofselectinganoptimalsetofsinkstopsand/orndinganoptimaltourofthelocationsareveryhardingeneral.ShiandHou[ 10 ]investigatehowtondtheoptimalsinkstopsandcomputethelengthofstayateachofthestops.Whenthecandidatelocationswherethesinkmaystoparenotspecied,theproblemisNP-hard,andtheauthorsproposeanapproximationalgorithmfortheproblem.Incontrast,werestrictthesetofsinkstopstobefromasetofgivencandidatelocationsinthischapter.Therestofthechapterisorganizedasfollows.Section 2.2 describesthenetworkandenergyconsumptionconceptsemployedthroughoutthechapter.Section 2.3 presentsacolumngenerationalgorithmfortheMSM,whileSection 2.4 describesacolumngenerationalgorithmfortheDT-MSM.WeevaluatethecomputationtimeofouralgorithmsinSection 2.5 2.2NetworkandEnergyConsumptionModelWebeginbybrieydiscussingsomekeyconceptsandnotationrelevanttothemodelspresentedinthischapter.LetNbethesetofsensornodes.Eachnodeistartswithaninitialenergyvalue,denotedbyEi,and,duringitslifespan,generatesdataataconstantrateofdi.LetLbethesetofpossiblelocationswherethesinkcanstop(alsoknownassinkstops).Thesinkcanmovewithinthesensoreld,stopatoneoftheselocationstogatherdatafromthesensornodesforaperiodoftime,andthenmoveontoanotherlocation.Thesinkdoesnotnecessarilystop(i.e.,stayforapositiveduration)atalllocationsinLintheinterestofmaximizingthenetworklifetime[ 8 10 ].Throughoutthischapter,weassumethatthesink'stravelingtimebetweenlocationsisnegligible.Thisisappropriatewhenthesink'stravelingtimeismuchsmallerthanthetimespentbythesinktocollectdataineachlocation.Withthisassumption,theorderofvisitingthesinkstopsdoesnotaffectthenetworklifetime. 17

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Similarto[ 30 31 ],theenergyrequiredperunitoftimetotransmitdataattherateofxijonthewirelesslink(i,j)canbemodeledas2 Etij=Ctijxij,(2)whereCtijistherequiredenergyfortransmittingoneunitofdataonthelink(i,j).Thisvalueisgivenby Ctij=+eij,(2)whereijistheEuclideandistancebetweennodesiandj,andarenon-negativeconstants,andeisthepathlossexponent.Typically,eisintherangeof2to6,dependingontheenvironment.NotethatCtijdoesnotdependonthelinkrate,and,inthelowrateregime,thisisavalidassumption.Hence,weneedtoassumethatthetrafcratexijissufcientlysmallcomparedtothecapacityofthewirelesslink.Theenergyspentatnodeiperunitoftimeforreceivingdatafromnodekisgivenby Erki=xki,(2)whereisagivenconstant.ThischapterdoesnotconsiderMAC-layercontention.ItisassumedthatcontentionisresolvedbysomeMAC-layerprotocol.TheoperationoftheMAC-layerprotocoldeterminesthelinkrates,whichareassumedtobelargeenoughsothattheydonotimposeaconstraintonthedatarates. 2Notethatwhenwerefertoawirelesslink(i,j),iisasensornode,andjmaybeasensornodeorthesinkinitscurrentlocation.Thiswillbemademoreprecisewhenneeded. 18

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2.3MobileSinkModelwithoutDelayToleranceInthissection,weformulatethelifetime-maximizationprobleminaWSNwithamobilesinkbutwithoutdelaytoleranceandprovideacolumngenerationalgorithmforitssolution.Thesinksojourntimeatalocationl2L,representedbydecisionvariablestl,isthetimethatthesinkspendsatltocollectdatafromthesensornodes.TheoverallnetworklifetimeisZ=Pl2Ltl.Notethatifthesinkdoesnotstopatlocationl2L,thentl=0.Whenthesinkisatstopl,thesetofneighborsofnodeiisdenedasSli=fj2N[flgjijg,whereisthemaximumtransmissionrangeofanynode.Notethatdependingonthedistancetol,Slicanincludethesinkitself.Tondtheoptimalnetworklifetime,weneedtoconsidertheroutingofthedata(includingrate)aswellasthesink'ssojourntimeateachstop(alsosee[ 7 10 ]).Letxlijbetheowonlink(i,j)whilethesinkisatstopl.Thelifetimemaximizationproblemcanbeformulatedasfollows[ 7 ].maxXl2Ltl (2)s.t.Xl2Ltl0@Xj2SliCtijxlij+Xj:i2Sljxlji1AEi8i2N (2)Xj:i2Sljxlji+di=Xj2Slixlij8i2N,l2L (2)tl08l2L (2)xlij08i2N,l2L,j2Sli. (2)Constraints( 2 )statethatthetotalenergyconsumedatanynodeicannotexceedtheinitialenergyEi.Constraints( 2 )ensureowconservationforallnodeswhenthesinkisatl.ToconverttheaboveformulationintoanLP,onecanintroducevariablesylijrepresentingthevolumeofdatatransmittedfromnodeitothenodejthroughthe 19

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link(i,j),whilethesinkisatlocationl.Thatis,ylijisequivalenttoxlijtl.maxXl2Ltl (2)s.t.Xl2L0@Xj2SliCtijylij+Xj:i2Sljylji1AEi8i2N (2)Xj:i2Sljylji+ditl=Xj2Sliylij8i2N,l2L (2)tl08l2L (2)ylij08i2N,j2Sli,l2L. (2)Notethatconstraints( 2 )canbeobtainedbymultiplyingconstraints( 2 )bytlandsubstitutingbyylij.Wenowdescribeastandardcolumngenerationalgorithmforthelinearprogram( 2 )( 2 ),andthenpresentmodicationsthatallowustoexploittheunderlyingnetworkstructure.Considerthesetoffeasiblesolutionsto( 2 )( 2 ).Thewellknownrepresentationtheorem(see,e.g.,[ 32 ])statesthatonecanrepresentapolyhedralsetasaconvexcombinationofitsextremepointsplusanon-negativecombinationofitsextremerays.Because( 2 )( 2 )isaconicset,ithasasingleextremepointthatliesattheorigin,andsoallofitspointscanberepresentedasanon-negativecombinationofitsextremerays.Now,letKbethesetofallextremeraysof( 2 )( 2 ),indexedbyk2K.Denoteby ylkijand tklthecomponentsofvectork2K.Thenndingafeasiblesolutionto( 2 )( 2 )isequivalenttonding 20

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non-negativescalarsk,k2K,suchthat:ylij=Xk2K ylkijk8i2N,j2Sli,l2Ltl=Xk2K tklk8l2LXl2L0@Xj2SliCtijylij+Xj:i2Sljylji1AEi8i2Nk08k2K. (2)Hence,wecanreformulateourproblemasfollows.maxXl2LXk2K tklk (2)s.t.Xl2LXk2K0@Xj2SliCtij ylkij+Xj:i2Slj ylkji1AkEi8i2N (2)k08k2K. (2)Thisproblemisnormallyreferredtoasthemasterproblemandeachofthevariableskiscalledacolumn.Asanalternativetosolvingthismasterproblem,wecaniterativelysolvearestrictedmasterproblemwithasubsetMofallcolumnsKandaddcolumnstoMasnecessary.Ineachiteration,arestrictedmasterproblemissolvedandanewsetofdualvariablesi,8i2N,associatedwithconstraints( 2 )isobtained.Usingthesedualvariables,wecancalculatetheminimumreducedcostoverallcolumnsinK.Iftheminimumreducedcostiszero,thecurrentrestrictedmasterproblemsolutionisoptimaltotheoriginalLP.Otherwise,thereexistsacolumnhavinganegativereducedcost,sowecanpotentiallyimproveoursolutionbyaddingthatcolumntoMandre-solvingthereducedmasterproblem.Tondacolumnhavingtheminimumreducedcost,constrainedby( 2 )( 2 ),wecansolvethefollowingsubproblem.minXl2Ltl)]TJ /F11 11.955 Tf 11.96 11.36 Td[(Xl2LXi2N0@Xj2SliiCtijylij+Xj:i2Sljiylji1A (2) 21

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s.t.Xj:i2Sljylji+ditl)]TJ /F11 11.955 Tf 11.95 11.36 Td[(Xj2Sliylij=08i2N,l2L (2)tl08l2L (2)ylij08i2N,j2Sli,l2L. (2)Iftheoptimalobjectivefunctionvaluetothissubproblemiszero,wecanstopandconcludethatthereisnoimprovingcolumn.Otherwise,thesubproblemmustbeunbounded,notingthattheright-hand-sidesofallconstraintsarezero,andthatthefeasibleregionisconic.Inthelattercase,wewilladdthecorrespondingdirectionofunboundednessasanewcolumnandre-solvethemasterproblem.Thecolumngenerationalgorithmdescribedabovecanbeimprovedinseveralways.Here,wedescribeamodiedversionofthealgorithm,whichleadstoasubproblemthatismucheasiertosolve.First,noticethatthesubproblem( 2 )( 2 )isseparableintojLjlinearprograms,becausetheobjectivefunctionislinearandtheconstraintsgoverningowsforeachsinklocationaredisjoint.Therefore,itismoreefcienttogenerateonecolumn(inthelowerdimension)fromeachoftheseconicsetsineachiteration.Notethatanoptimalsolutionexiststo( 2 )( 2 )inwhichtherearenocyclicowsforanysinklocationl.Hence,inourcolumngenerationalgorithm,wewillonlyconsidersolutionswithacyclicowpatterns.Forconvenience,wewillrefertothesetoffeasiblesolutionsto( 2 )( 2 )correspondingtosinklocationlasthelthconicset.LetCl=f(yl,tl):(yl,tl)isfeasibletothelthconicset,ylisacyclicg.Thisisthesetfromwhichwewillextractthenecessarycolumnsforthemasterproblem.Onceagain,allfeasiblesolutionstoClcanbewrittenasnon-negativelinearcombinationsofitsextremerays.Bysettingtl=1in( 2 )( 2 ),wegetthefollowingnormalizedsetforeach 22

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l2L:Xj:i2Sljylji+di=Xj2Sliylij8i2Nylij08i2N,j2Sli. (2)Now,letDl=fyl:ylisfeasibleto( 2 ),ylisacyclicg.Then,thefollowinglemmaholds. Lemma2.1. ThesetofallextremeraysofClisinone-to-onecorrespondencewiththesetofallextremepointsofDl. Proof. Firstnoticethateachfeasiblevectoryl2Dlcorrespondstoaray(yl,1)ofCl.Thatis,thisrayhasthesameylijcomponentsasyl,andtl=1.Moreover,ifylisanextremepointofDl,thenthevector(yl,1)isanextremerayofthecorrespondingconicsetCl.Toseethis,considertwodistinctrays(yl0,t0l)and(yl00,t00l)inClsuchthat(yl,1)=1(yl0,t0l)+2(yl00,t00l),where1,2>0and1+2=1.Notethatsinceyl0andyl00areassumedtobeacyclicows,t0landt00larenon-zeroscalars.Therefore,(yl,1)=(1t0l)((1=t0l)yl0,1)+(2t00l)((1=t00l)yl00,1)where1t0l+2t00l=1.Thismeansthat(yl,1)canberepresentedasaconvexcombinationoftwonormalizedrays((1=t0l)yl0,1)and((1=t00l)yl00,1),andsoylitselfisaconvexcombinationof(1=t0l)yl0and(1=t00l)yl00withthesamecoefcients.Thefactthat(1=t0l)yl0and(1=t00l)yl00belongtoDlimpliesthat(1=t0l)yl0=(1=t00l)yl00,whichinturngivesusthat(yl,1)isanextremerayofCl.Toprovethereverse,supposethatthenormalizeddirection(yl,1)isanextremerayofCl.WeclaimthatylisanextremepointofDl.Ifnot,thenwecanndyl1andyl2feasibleinDlsuchthatyl=yl1+(1)]TJ /F10 11.955 Tf 12.58 0 Td[()yl2where0<<1.Therefore,(yl,1)=(yl1,1)+(1)]TJ /F10 11.955 Tf 12.2 0 Td[()(yl2,1).Thisisacontradictionbecause(yl1,1)and(yl2,1)areraysofCl,whichwouldthenindicatethat(yl,1)isnotanextremerayofCl. 23

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SincealltheextremeraystoClhaveaone-to-onecorrespondencetotheextremepointsofDl,wecangeneratecolumnsfromDlinsteadofCl.WewillshowthatthisallowsthesubproblemtobesolvablebyDijkstra'salgorithm.Now,supposethatKlisthesetofallcolumnsgeneratedfromDl.Wedenotethecomponentsofsuchacolumnby ylkijwherek2Kl.Then,afeasiblesolutiontoClcanbewrittenintheformylij=Pk2Kl ylkijtlkandtl=Pk2Kltlk,whereeachtlk0.Pluggingthissolutionintotheenergyconstraint( 2 ),wegetthefollowingmasterproblem.maxXl2LXk2Kltlk (2)s.t.Xl2LXk2Kl0@Xj2SliCtij ylkij+Xj:i2Slj ylkji1AtlkEi8i2N (2)tlk08l2L,k2Kl. (2)Observethatforeachsinklocationl2L,thereisasetofKldifferentshortestpathtrees,eachwithaweightoftlk0inthemasterproblem.Wecaninterprettlkasthelengthoftimethatthesensorscommunicatewiththesink(whileatlocationl2L)usingtheshortestpathtreek2Kl.Leti0bethedualvariableassociatedwiththeithconstraintin( 2 ).Then,thereducedcostforanewcolumncorrespondingtovariabletlkandhavingcomponentseylijis:1)]TJ /F11 11.955 Tf 11.96 11.36 Td[(Xi2N0@Xj2SliiCtijeylij+Xj:i2Sljieylji1A. (2)Thelthsubproblemiswrittenasfollows.minXi2N0@Xj2SliiCtijeylij+Xj:i2Sljieylji1A (2)s.t.Xj:i2Sljeylji+di=Xj2Slieylij8i2N (2) 24

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eylij08i2N,j2Sli. (2)Theunderlyingnetworkforsubprobleml2Listhesameasthecommunicationgraphbetweenthesensorsandthesinkwhileatlocationl,withanexcessofdiateachsensornodeiandadecitofPi2Ndiatthesink.ThecostofthearcbetweeniandjisiCtij+j.Eachsubproblemhastheformofaminimumcostowproblem,butwithinnitecapacityonthearcsoftheunderlyingnetwork.Thus,eachsensornodesendsitsexcessowalongashortestpathtothesink.Theseshortestpathscanbeobtainedbysolvingoneshortestpathproblemthatcomputesashortestpathtreefromallsensornodestothesink.Notethatbecauseeachi0,thecostsonthearcsofthenewnetworkareallnon-negativeandwecanuseDijkstra'salgorithmtondtheshortestpaths.TheoptimalowcanbeobtainedbyaggregatingtheowsonalljNjpathsintheoriginalnetwork.Alsonoticethatbecausewesolvethisminimumcostowproblembyaggregatingowsalongtheshortestpathsfromsensornodestothesink,itwillnotgeneratecyclicows.Lettheoptimalobjectivefunctionvaluetothelthsubproblembezl.Notethatzl1sinceanycolumncorrespondingtoabasicvariableinthemasterproblemhaszeroreducedcost,andthecomponentsofsuchacolumnformafeasiblesolutionto( 2 )( 2 )withanobjectivefunctionvalueof1.Ifzl<1forsomel2L,thenthereducedcostofthegeneratedcolumnispositive,andweaddthecorrespondingcolumntoKlandre-solvethemasterproblemwithupdatedsetsKl,l2L.Ifzl=1,8l2L,thentheoptimalsolutiontothemasterproblemisalsooptimaltotheoriginalproblem( 2 )( 2 ). 2.4MaximizingLifetimeinDelay-TolerantApplicationsThissectiondescribestheDT-MSMgivenin[ 26 ]andhowcolumngenerationcanbeusedtosolveit.Recallthatinthiscase,eachnodecandeferthetransmissionofdatauntilthesinkisatthelocationmostfavorableforextendingthenetworklifetime.When 25

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anodeisnottransmittingthegenerateddatatothesink,itstoresthedataforfuturetransmission.Therefore,thenettransmissionrateatnodeiduringthecollectiontimecouldpossiblydifferfromtheconstantdatagenerationratedi.LetDbethemaximumtolerabledelay,orthedelaytolerancelevel.WeassumethatthesinkcompletesoneroundofvisitstoallstopsinDtimeunits,andthenrepeatswithanotherround.Therefore,twoconsecutivevisitstothesamestoptakesDtimeunits.Thisassumptionisnotrestrictivewhenthesink'stravelingtimesarenegligiblecomparingtothelifetimeofthenetwork.Moreover,theorderofvisittothelocationsisnotrelevanttothelifetimeofthenetwork;hence,wearbitrarilyassumethatthesinkvisitsthelocationsintheorder1,2,...,jLj,1,....NotethatthetotalstoreddataateachnodeiwithinacycleofDtimeunitsisatmostDdi.Forthisproblem,wedenethenetworklifetimeTtobethenumberofvisitingcyclesthatthesinkmakesuntiltherstnodeexhaustsitsenergy.TheactuallifetimeisTD.TwovariationsoftheDT-MSMareproposedin[ 26 ]whichcorrespondtotwotrafcbufferingschemes.Inthesub-ow-basedmodel,thesensornodescannotstoretheincomingtrafcfromothernodesandmustrelayothernodes'trafcimmediately.Therefore,eachnodecanstoreonlytheself-generatedtrafc.Ontheotherhand,inthequeue-basedmodel,sensornodescanstoretheincomingtrafc,aswellastheirowntrafc,potentiallybuildingupaqueueofmixeddataateachnode.Inthischapter,wefocusonthequeue-basedmodelandprovideacolumngenerationalgorithmforit.Wehavealsodevelopedacolumngenerationalgorithmforthesub-ow-basedmodel,butforbrevitywillomititsdescription.Thequeue-basedproblemcanbeformulatedasthefollowingnonlinearprogram.maxT (2)s.t.24Xl2Ltl0@Xj2SliCtijxlij+Xk:i2Slkxlki1A35TEi8i2N (2) 26

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tl0@Xk:i2Slkxlki1A+ql)]TJ /F13 7.97 Tf 6.59 0 Td[(1i)]TJ /F3 11.955 Tf 11.95 0 Td[(tl0@Xj2Slixlij1A=qli8i2N,l2L (2)q0i=Ddi8i2N (2)qjLji=08i2N (2)xlij08i2N,j2Sli,l2L (2)qli08i2N,l2L (2)T0. (2)Theenergyconstraints( 2 )aresimilartothemobilesinkmodel,notingthatthetotaltimespentbythesinkatlocationlistlT.TheyensurethattheenergyconsumedbyeachnodecannotbemorethanitsoriginalenergyendowmentEi.Theowconservationconstraintsareexpressedthroughthequeuelengthdynamicsin( 2 ).Here,variableqlidenotesthelengthofthequeueatnodei2Nrightafterthesinkdepartsfromlocationl.Constraints( 2 )and( 2 )ensurethatthedatageneratedinthepreviouscyclewillbetransmittedtothesinkduringthecurrentcycle.Lettingu=1=Tandylij=tlxlij,theaboveproblemcanbewrittenasthefollowinglinearprogrammingproblem:minu (2)s.t.Xl2L0@Xj2SliCtijylij+Xk:i2Slkylki1AEiu8i2N (2)Xk:i2Slkylki+ql)]TJ /F13 7.97 Tf 6.59 0 Td[(1i)]TJ /F11 11.955 Tf 11.96 11.36 Td[(Xj2Sliylij=qli8i2N,l2L (2)q0i=Ddi8i2N (2)qjLji=08i2N (2)ylij08i2N,j2Sli,l2L (2)qli08i2N,l2L (2) 27

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u0. (2)Weagainplacetheenergyconstraints( 2 )inthemasterproblemandobtaincolumnsfromconstraints( 2 )( 2 ).Similartothepreviousmodel,problem( 2 )( 2 )hasanoptimalsolutioninwhichtheowvariablesylijcorrespondtoacyclicowpatterns.Asaresult,wewillrestrictourselvestogenerateonlysuchacycliccolumns.Notealsothatallpossibleextremeraysofthepolyhedralset( 2 )( 2 )containcyclicows.Therefore,wecanignoretheextremeraysof( 2 )( 2 )informulatingthemasterproblem.DenotebyKthesetofgeneratedcolumns,wherethecomponentsofthekthgeneratedcolumnareoftheform ylkijand qlki.Wecanwritethefollowingmasterproblem:minu (2)s.t.Xk2KXl2L0@Xj2SliCtij ylkij+Xk:i2Slk ylkki1ArkEiu8i2N (2)Xk2Krk=1 (2)rk08k2K (2)u0. (2)Now,supposethat)]TJ /F10 11.955 Tf 9.3 0 Td[(iandarethedualvariablesassociatedwithconstraints( 2 )and( 2 ),respectively.Thereducedcostforanewcolumnis:Xl2LXi2N0@Xj2SliiCtijylij+Xk:i2Slkiylki1A)]TJ /F10 11.955 Tf 11.95 0 Td[(. (2)Therefore,thesubproblemiswrittenasfollows.minXl2LXi2N0@Xj2SliiCtijylij+Xk:i2Slkiylki1A (2) 28

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s.t.Xk:i2Slkylki+ql)]TJ /F13 7.97 Tf 6.59 0 Td[(1i)]TJ /F11 11.955 Tf 11.95 11.36 Td[(Xj2Sliylij=qli8i2N,8l2L (2)q0i=Ddi8i2N (2)qjLji=08i2N (2)ylij08i2N,j2Sli,l2L (2)qli08i2N,l2L. (2)Letzbetheoptimalobjectivefunctionvalueoftheabovesubproblem.Ifz>,weaddanothercolumn(extremepoint)tosetofcolumnsKandre-solvethemasterproblem.Otherwise,thesolutiontothemasterproblemisoptimaltotheproblem( 2 )( 2 )becausethereducedcostforallcolumnsmustbenon-positive.Subproblem( 2 )( 2 )isaminimumcostowproblemonaspecialnetwork.Figure 2-1 illustrateshowwecanbuildthisnewnetworkfromjLjcopiesoftheoriginalnetwork. TherearejNjjLj+1nodesdividedintojLjlayers,plusonenoderepresentingthesink. Thereisexactlyonenodecorrespondingtoeachsensori2NineachofthejLjlayers. ThereisanarcfromsensornodeitosensornodejinalljLjlayersofthenetworkifandonlyifthereisanarcfromitojintheoriginalnetwork. Ifthereisanarcfromasensoritothesinkatlocationl2L,thenthereisanarcfromthecorrespondingnodeiinthelthlayertothesinknodeinthenewnetwork. Foreachsensori2Nandeachlayerl2L,thereisanarcfromthenoderepresentingiinlayerltothecorrespondingnode(representingi)inlayerl+1.Wedenotetheseasqueuearcs. Thecostofeachqueuearciszero.ThecostofallotherarcsisoftheformofiCtij+j.Notethati0,8i2N.Hence,eachcostisnon-negative. NodeiintherstlayerhasanexcessofDdiandthesinkhasadecitofDPidi.Allothernodeshavenoowexcessordecit. Allarcshaveunlimitedcapacity. 29

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Figure2-1. Illustrationofqueue-baseddelay-tolerantmobilesinkmodel Similartothepreviouscase,herewealsohaveaminimumcostowproblemonanuncapacitatednetworkandtherefore,thesubproblemreducestondingashortestpathfromeachofthenodesintherstlayertothesink.Thisisbecauseeachnodesendsallitsexcessowusingtheshortestpathfromthatnodetothesink.Therefore,thesubproblemcanbesolvedbyoneapplicationofDijkstra'salgorithm.Whenusingtheenergyconsumptionmodeldenedby( 2 )( 2 ),theabovesubproblembecomeseveneasier.NotethatthecostofanyarcbetweentwosensornodesisthesameinalljLjlayersofthenetworksincethesensornodesdonotmove.Theonlydifferencebetweenthelayersisthearcsfromthesensornodestothesink.Thecostofsucharcscandifferbetweenthelayers.Therefore,thereisalwaysanoptimalsolutiontotheabovesubproblemusingonlythearcsintherstlayer,followedbya(possiblyempty)successionofthequeuearcsandendingwithanarctothesink 30

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Table2-1. Experimentalparametersandtheirvalues Numberofsensornodesf60,75,80,100,120gNumberofpossiblesinklocationsf20,25,30,35,40gpathlossexponent(e)f4.0gmaximumtransmissionrangef6.5,7.5,8.0gDatagenerationrate(di)0.05bps=10pJ/bit/m20.0013pJ/bit/m4InitialEnergy(Ei)500J (possiblyinanotherlayer).Therefore,foreachsensornodei2N,wecanaddanarc(i,s)withtheminimumcostoveralllayers(ifany)totherstlayerandsolvetheshortestpathproblemonthissmallernetwork.Theoptimalsolutiontotheexpandednetworkcanbeeasilyretrievedbykeepingtrackoftheassociatedlayerforeachaddedarc. 2.5ComputationalResultsWecomparethecomputationalefciencyofthealgorithmsthatweproposedinthischapteragainstthedirectsolutionofLPformulations.Foreachparticularinstancesize,werandomlygenerateseveralnetworksandcomparetheresultsofrunningthetwoalgorithmsonthesenetworks.AllalgorithmsareimplementedinC++callingCPLEXversion11.0viaILOGConcertTechnology2.5tosolvethelinearprogrammingproblems,andcompiledusingGNUg++version4.1.2.AllexperimentsareperformedonaDellPowerEdge2600machinewithtwoPentium43.2Ghz/1Mcacheprocessorsand6gigabytesofmemory,runningRedhatversion5.InTable 2-1 ,weshowthesystemparametersandtheirvaluesusedtogeneratethecongurationfortheexperiments.Thelastthreelinesinthetablearefrom[ 33 ].Thepositionsofthesensornodes(N)andpossiblesinkstops(L)arerandomlygeneratedaccordingtoauniformdistributionoveracircularareawithradiusof25.Themaximumtransmissionrangeiscarefullychosensothatresultinggraphisstronglyconnectedwithhighprobability.Table 2-2 showstheaverageexecutiontimes(inCPUseconds)forsolvingboththecolumngenerationalgorithmandtheoriginallinearprogram(usingCPLEX)forthe 31

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Table2-2. Performanceofthetwoalgorithmsforthemobilesinkmodel InstanceSizeColumnGenerationLPComparison (jNj,jLj)Time(s)NumberofColumnsTime(s)C-GWins (60,20)0.272100.4996%(70,25)0.543171.33100%(80,30)1.626404.24100%(100,35)2.927377.59100%(120,40)9.17145626.66100% MSM.Foreachproblemsize,wegenerate25randomnetworkinstancesandsolvethemusingbothmethods.Thesecondcolumn(Time(s))reportstheaveragetimeforthecolumngenerationalgorithmbasedonthe25instances,whilethethirdcolumn(NumberofColumns)representstheaveragenumberofcolumnsgeneratedfortheseinstances.Thefourthcolumnrepresentstheaveragetimeofsolvingthe25linearprogramsdirectlywiththeprimalsimplexsolverofCPLEX.Wealsoreportinthefthcolumn(C-GWins)thepercentageofalltheinstancesforwhichthecolumngenerationperformsbetterthanCPLEXintermsofCPUtime.ItisclearfromtheseresultsthatfortheMSM,thecolumngenerationalgorithmoutperformsCPLEXintermsofaverageexecutiontime,andalsoforthemajorityoftheindividualinstancesthataresolved.Table 2-3 containsacomparisonbetweentheperformanceofthecolumngenerationalgorithmfortheMSMandfourotherLPsolversofCPLEX.TheseresultsareforthesameinstancesbasedonwhichTable 2-2 isbuilt.Thesecondcolumnprovidestheaverageexecutiontimeofthecolumngenerationalgorithm.ForeachLPsolver,wereporttheaveragesolutiontimeofthe25instancesaswellasthefractionofalltheinstanceswherethecolumngenerationalgorithmsolvestheproblemfasterthanthecorrespondingCPLEXsolver.WecanconcludefromtheseresultsthatthecolumngenerationalgorithmsoutperformsalltheimplementedLPalgorithmsinCPLEXforsmallerinstances.Astheinstancesizegrows,theperformancegapbetweenthebarrieroptimizerofCPLEXandthecolumngenerationalgorithmbecomessmaller. 32

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Table2-3. ComparingtheresultsofthecolumngenerationalgorithmforthemobilesinkmodelwithdifferentlinearprogrammingsolversofCPLEX C-GDualBarrierSiftingConcurrent (jNj,jLj)TimeTimeC-GWinsTimeC-GWinsTimeC-GWinsTimeC-GWins (60,20)0.271.56100%0.7496%5.17100%1.57100%(70,25)0.545.37100%1.2796%10.05100%5.40100%(80,30)1.6219.43100%2.4688%19.42100%19.37100%(100,35)2.9234.59100%4.0980%32.33100%33.92100%(120,40)9.17140.96100%8.5752%56.86100%142.08100% 33

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Table2-4. Comparingthecolumngenerationandthesubgradientalgorithmsforthemobilesinkmodel ColumnGenerationSubgradient ProblemTime(s)NumberofColumnsTime(s)NumberofIterations TP10.01543.4347689TP20.0167>1000>13450599TP30+3674.881026989TP40.0153388.965094048TP50.0375683.638937513TP60.0150.213254TP70.0126>1000>13633391TP80.0162485.136419664TP90.0265>1000>13454004TP100.0279726.89918074 Table 2-4 providesacomparisonbetweentheperformanceofourcolumngenerationalgorithmfortheMSMandavariationofthesubgradientalgorithmpresentedin[ 9 ].Thereaderisreferredto[ 9 ]fordetailsonthedistributedimplementationofthecorrespondingsubgradientalgorithm.Ourimplementationisthesameasthatof[ 9 ]withsomeminormodications.Morespecically,wesolvetheminimumcostowsubproblemsusingDijkstra'salgorithminsteadofthescaled-relaxationalgorithmthattheauthorsusein[ 9 ].Also,weuserandomstartingvaluesforthedualvariablessincethechoiceofthestartingpointhadlittleimpactontheconvergenceofthealgorithminourexperiments.Forthesteplengthweusesk=1 k,assuggestedbytheauthors.TeninstancesofsizejNj=12andjLj=3havebeengeneratedandsolvedusingbothalgorithms.Ingeneratingthesesparseinstances,wesetthemaximumtransmissionrangeto35toensureconnectivity.Forthecolumngenerationalgorithm,wereporttherunningtimeandthenumberofcolumnsgenerated.Forthesubgradientalgorithm,wereportthetimeuntiltherstsolutionwithlessthan5%optimalityviolationisgeneratedaswellasthenumberofiterationsrequireduntilsuchasolutionisgenerated.A1000-secondtimelimitisimposedforeachrun.Itisclearfromtheseresultsthatthecolumngenerationalgorithmismuchfasterthanourimplementationofthesubgradientalgorithm. 34

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Figure2-2. ConvergenceofcolumngenerationalgorithmtotheoptimalsolutionwithjNj=120andjLj=40 InFigure 2-2 ,weillustratetheconvergenceofthecolumngenerationmethodtotheoptimalobjectivevalueonarandomMSMinstancewithjNj=120andjLj=40.TheverticallineT1representsthetimewhenourcolumngenerationalgorithmterminates,andtheverticallineT2representsthetimeatwhichtheLPterminates.Thegraphshowsthattheintermediatesolutionsofthemasterprobleminthecolumngenerationalgorithmrapidlyapproachtheoptimalvalue.Therefore,inthecasethatanearoptimalsolutionissufcientfortheapplication,thecolumngenerationalgorithmmayndonesuchsolutionquickly.Table 2-5 containstheresultsofrunningthecolumngenerationalgorithmfortheDT-MSMversussolvingtheoriginallinearprogramwiththeprimalsimplexsolverofCPLEX.TheseresultsareforthesameinstancesasthoseintheMSMcase,withthemaximumtolerabledelaybeingD=5000forallnetworks.Theresultssuggestthat 35

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Table2-5. Performanceofthetwoalgorithmsforthedelay-tolerantmobilesinkmodel InstanceSizeColumnGenerationLPComparison (jNj,jLj)Time(s)NumberofColumnsTime(s)C-GWins (60,20)0.23350.3076%(70,25)0.56370.6760%(80,30)1.58561.1160%(100,35)4.67632.4240%(120,40)19.101086.4528% thecolumngenerationalgorithmworksbetteronthesmallerDT-MSMinstances,whilesolvingtheoriginalLPisgenerallyfasterforinstancesoflargersize.However,aswementionedbefore,onemightstillpreferthecolumngenerationalgorithmsincethesubproblemcanbebuiltintonetworkprotocolsandbesolvedinadecentralizedmanner.Table 2-6 comparestheperformanceofthecolumngenerationalgorithmwithfourotherLPalgorithmsimplementedinCPLEX,andveriesthatwhilecolumngenerationisthemosteffectivealgorithmonsmallerinstances,theCPLEXprimalsimplexsolveristhemostefcientalgorithmonlargerinstances. 36

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Table2-6. Comparingtheresultsofthecolumngenerationalgorithmforthedelay-tolerantmobilesinkmodelwithdifferentlinearprogrammingsolversofCPLEX C-GDualBarrierSiftingConcurrent (jNj,jLj)TimeTimeC-GWinsTimeC-GWinsTimeC-GWinsTimeC-GWins (60,20)0.230.6884%0.6196%2.19100%0.6988%(70,25)0.562.2596%1.0996%4.75100%2.2396%(80,30)1.581.7860%1.9272%8.49100%1.8060%(100,35)4.675.9972%3.3556%15.7096%6.0372%(120,40)19.1029.7972%7.3432%41.3688%30.4372% 37

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CHAPTER3ADISTRIBUTEDALGORITHMFORTHELIFETIMEMAXIMIZATIONPROBLEMINDELAY-TOLERANTAPPLICATIONS 3.1MotivationandLiteratureSurveyThelifetimeofawirelesssensornetwork(WSN)isoftendenedasthetimeuntiltherstsensornodefailsbecauseofenergydepletion[ 5 6 ].Duetothemulti-hopdataroutingfromsensornodestothesink,thesensornodesthatareclosertothesinkareusuallythemostburdenedwithrelayingdatafromdistantnodes.Thistrafcimbalancecancauseearlyenergydepletionfornodesnearthesink,whichcreatesanenergyholearoundthesink.Theresultmaybeanearlydisconnectionofthesinkfromthesensornodesthatmaystillhaveplentyofenergy[ 20 21 34 ].Recently,exploitingmobility(particularlythesink'smobility)toimprovenetworklifetimehasattractedtheinterestofresearchers[ 7 10 15 19 26 ].AsexplainedinChapter 2 ,amobilesinkcanbetterbalancethetrafcloadacrossthesensoreldbymakingstopsatdifferentlocationsinthenetworktoreceivedatafromsensornodes.Therefore,sinkmobilitycanmitigatetheenergy-holeproblemandincreasethenetworklifetime.InChapter 2 ,wepresentedacolumngenerationalgorithmfortheDelayTolerantMobileSinkModel(DT-MSM).DT-MSMissuitableforapplicationswheresomeamountofdelayindatadeliverytothesinkispermitted[ 16 ].Inthissetting,sensornodesmaydelaythetransmissionofthecollecteddataforawhileandwaituntilthesinkarrivesatalocationthatismostfavorableforimprovingthenetworklifetime.ThegoalofthischapteristondadistributedalgorithmthatsolvesthelifetimemaximizationproblemassociatedwithDT-MSM.Thedecisionstobemadeincludehowlongthesinkshouldstayateachpotentialstop,andhowtoroutethedatatothesinkwhenitstops.Wepresenttwomaincontributionsinthischapter.First,ouralgorithmisbothdistributedandmostlylocal.Theoverallsolutionisdecomposedintosmallerdecisionproblemsandeachdecisioncanbedonelocallyinasensornode.Forthe 38

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mostpart,onlylocalinformationatanodeitselfandatitsneighborsisneeded.Ingeneral,distributedalgorithmsaremoreusefulfornetworkingproblemsbecausetheycanbereadilybuiltintonetworkprotocolsandbecomenetworkcontrolalgorithms.Localalgorithmshavetheadditionalbenetofrestrictingthecontroltrafctobeamonglocallyinteractingnodes.Second,weanalyzeouralgorithmandshowthatinthelong-run,itconvergestotheoptimalobjectivefunctionvalueofthelifetimemaximizationproblem,andthatthevirtualqueuesizesareboundedinthelong-runaveragesense.Wenextbrieysurveyrelatedworks.GatzianasandGeorgiadis[ 9 ]studythelifetimemaximizationprobleminaWSNwithamobilesinkandproposeadistributedsubgradientalgorithmforsolvingit.Unliketheiralgorithm,standardconvergenceresultsforsubgradientalgorithmsdonotapplytoouralgorithmandtheanalysisaboutconvergenceandalgorithmperformanceinourcasereliesonadifferentframework.Inaddition,theirapproachrelaxestheenergyconstraints,whereaswerelaxtheowconservationconstraints.OurconvergenceandstabilityanalysisisbasedonanalyzingaLyapunovdrift.TheLyapunovdrifttechniqueiswidelyusedforstudyingthestabilityofcontrolandoptimizationalgorithmsfornetworksofqueues.Arepresentativeworkis[ 35 ]wheretheauthorsapplythistechniquetothestudyofalinkschedulingalgorithminmulti-hoppacketradionetworks.Ourmethodiscloselyrelatedtothatof[ 36 ],whichalsoanalyzesadynamiccontrolalgorithminwirelessnetworks.SimilartoourassumptionsinChapter 2 ,weassumethatthesink'straveltimesbetweendifferentstopsarenegligibleinthischapter.Thisassumptioniswidelyusedintheliteratureofsimilarmobilesinkproblems[ 7 8 10 17 19 26 ].Moreover,werestrictthesetofpossiblesinklocations(wherethesinkcanstopat)toagivenniteset.Therestofthischapterisorganizedasfollows.Section 3.2 describestheunderlyingsystemmodelandtheproblemformulation.Thedistributedalgorithmfor 39

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theproblemisdiscussedinSection 3.3 .InSection 3.4 ,weanalyzetheoptimalityandconvergenceofouralgorithmandSection 3.5 containstheexperimentalresults. 3.2ProblemFormulationWemodelawirelesssensornetworkasadirectedgraphG0=(N[L,A),whereN=f1,...,Ngisthesetofsensornodes,L=f1,...,Lgisthesetofthesinklocations,andAisthesetofwirelesslinksinterconnectingthesensornodesandthesinklocations.LetijbetheEuclideandistancebetweennodesiandj.WedenethesetofdownstreamneighborsofiasN(i)=fj2N[Lj(i,j)2Ag=fj2N[Ljijg,whereisthemaximumtransmissionrangeofthenodesinthenetwork.EachsensornodeigeneratesdataataconstantratediandhasaninitialenergyendowmentEi.Letc:A!R+beacostfunctiononthearcsetsuchthatc(i,j)denestherequiredenergyforthetransmissionofoneunitofdataalongarc(i,j)2A.AnimportantparameterinDT-MSMisthemaximumdelayDthattheunderlyingapplicationcantolerate.Thatis,thesensorscanlocallystoreanddelaythetransmissionoftheircollecteddataforuptoDtimeunits.WealsoassumethatthesinkmustcompleteoneofitstourswithinDtimeunits(see[ 26 ]).Therefore,maximizingnetworklifetimeisequivalenttomaximizingthetotalnumberofsinktours,denotedbyT.Throughoutthischapter,wewillassumethattheorderinwhichthesinkvisitsthesinklocationsisgivenas1!2!!L.Decisionvariablesaretheamountoftimethatthesinkstaysateachsinklocationl2Lwithineverytour(denotedbytl)andthedatatransmissionratefromitojwhenthesinkisatl(denotedbya(l)ij).Wedenex(l)ij=tla(l)ijasthetrafcvolumeonlink(i,j)whenthesinkisatl.LetGl=(N[flg,Al)betheunderlyinggraphwhenthesinkisatl,whereAl=f(i,j)2Aji2N,j2N[flgg.Whenthesinkisatlocationl,theneighborhoodsetofiisdenotedbyNl(i)=fjj(i,j)2Alg.Weassumethatforeachsensornodei2Nandeverysinklocationl2L,thereexistsatleastonepathfromitol 40

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Figure3-1. Expandedgraphofdelay-tolerantmobilesinkmodel inGl.Thisassumptionisrequiredtoensurethepossibilityofdeliveringdatafromeverysensornodetothesinkatanypointintime.SimilartotheconstructioninSection 2.4 ,wecreateanexpandedgraphfromthegraphsGl,l2L,suchthatthemaximumnetworklifetimecanbeobtainedbysolvingaspecialnetwork-owsproblemonthisexpandedgraph.Figure 3-1 illustratesthisexpandedgraph.Thecostofeachverticalarc(i(l),j(l))isassignedasfollows: e(l)ij=8>>>>>><>>>>>>:c(i,j),ifi,j2Nc(i,l),ifi2N,j=l1,otherwise.(3) 41

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Thecostofeachhorizontalarc(i(l),i(l+1))is0.Theowconservationconstraintsforthesensornodescanbestatedasfollows. 8>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>:Xj:i2N1(j)x(1)ji)]TJ /F11 11.955 Tf 17.3 11.36 Td[(Xj2N1(i)x(1)ij)]TJ /F3 11.955 Tf 11.95 0 Td[(y(1)i=)]TJ /F3 11.955 Tf 9.3 0 Td[(Ddi,forl=1,i2NXj:i2Nl(j)x(l)ji)]TJ /F11 11.955 Tf 16.64 11.36 Td[(Xj2Nl(i)x(l)ij+y(l)]TJ /F13 7.97 Tf 6.58 0 Td[(1)i)]TJ /F3 11.955 Tf 11.95 0 Td[(y(l)i=0,forl2f2,...,L)]TJ /F4 11.955 Tf 11.95 0 Td[(1g,i2NXj:i2NL(j)x(L)ji)]TJ /F11 11.955 Tf 17.45 11.36 Td[(Xj2NL(i)x(L)ij+y(L)]TJ /F13 7.97 Tf 6.59 0 Td[(1)i=0,forl=L,i2N(3)AtthebeginningofeachcycleoflengthDtimeunits,nodeihasaccumulatedDdiunitsofdatafromthepreviouscycle,whichmustbedeliveredtothesinkduringthecurrentcycle.Variablex(l)ijrepresentstheamountofdatasentonarc(i,j)whenthesinkisatlocationl,whiley(l)idenotestheamountofbuffereddata(queuesize)atnodeirightafterthesinkleaveslocationl.Thus,y(l)]TJ /F13 7.97 Tf 6.59 0 Td[(1)i)]TJ /F3 11.955 Tf 12.42 0 Td[(y(l)iisthechangeinthebuffereddataatnodeiwhilethesinkisatlocationl.Inaddition,atthesink(nodes),allarrivaltrafcmustbedrained.Thus,wemusthave LXl=1Xj:s2Nl(j)x(l)js)]TJ /F5 7.97 Tf 17.29 14.94 Td[(NXi=1Ddi=0.(3)Theproblemthatweaddressinthischapteristomaximizethenumberofsinktours(orcycles),T,whilemaintainingtheowconservation( 3 )and( 3 ),subjecttotheenergyconstraintsatthesensornodes.Moreprecisely,theproblemcanbewrittenasmaxT (3)s.t.( 3 )and( 3 ) (3) 42

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0@LXl=1Xj2Nl(i)e(l)ijx(l)ij1ATEi,8i2N (3)x(l)ij0,8l2L,i2N,j2Nl(i) (3)y(l)i0,8l2L,i2N (3)T0. (3)Constraints( 3 )ensurethatthetotalenergyexpenditureateverysensornodeduringTcyclesdoesnotexceedthenode'sinitialenergyendowment. 3.3ALagrangianRelaxationApproachInthissection,weproposeadistributedalgorithmtosolvethelifetimemaximizationproblemdescribedinSection 3.2 .Westartbypresentinganequivalentlinearprogramthatisobtainedfromproblem( 3 )( 3 )byreplacing1=Twithz.Forconvenience,wedeneM=PNi=1Ddi.minz (3)s.t.LXl=1Xj2Nl(i)e(l)ijx(l)ijzEi,8i2N (3)8>>>>>>>>><>>>>>>>>>:Xj:i2Nl(j)x(l)ji)]TJ /F11 11.955 Tf 16.64 11.36 Td[(Xj2Nl(i)x(l)ij+y(l)]TJ /F13 7.97 Tf 6.59 0 Td[(1)i)]TJ /F3 11.955 Tf 11.95 0 Td[(y(l)i=0,8l2L,i2NLXl=1Xj:s2Nl(j)x(l)js)]TJ /F3 11.955 Tf 11.96 0 Td[(M=0 (3)y(0)i=Ddi,y(L)i=0,8i2N (3)x(l)ij0,8l2L,i2i2N,j2Nl(i) (3)y(l)i0,8l2L,i2N (3)z0. (3) 43

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NotethatMisanupperboundonthetrafcvolumeofanyarcinthenetwork(includingbuffereddata).Intheremainderofthischapter,wewillusethetermsowandvol-umeinterchangeably.Hence,thenewformulationminimizesthemaximumenergyconsumption(normalizedwithrespecttoEi)amongallnodesinonecycle,whilesatisfyingtheowconservationconstraints.Let(l)iandsbethedualvariablesassociatedwiththeconstraintsin( 3 ).TheLagrangianfunctionof( 3 )is L(z,x,y,)=z+s(LXl=1Xj:s2Nl(j)x(l)js)]TJ /F3 11.955 Tf 11.95 0 Td[(M)+LXl=1NXi=1(l)i(Xj:i2Nl(j)x(l)ji)]TJ /F11 11.955 Tf 16.64 11.36 Td[(Xj2Nl(i)x(l)ij+y(l)]TJ /F13 7.97 Tf 6.59 0 Td[(1)i)]TJ /F3 11.955 Tf 11.95 0 Td[(y(l)i),where=((l)i,s).Rearrangingtheaboveequality,weobtain L(z,x,y,)=z+LXl=1X(i,j)2Al((l)j)]TJ /F10 11.955 Tf 11.95 0 Td[((l)i)x(l)ij+NXi=1L)]TJ /F13 7.97 Tf 6.59 0 Td[(1Xl=1((l+1)i)]TJ /F10 11.955 Tf 11.96 0 Td[((l)i)y(l)i)]TJ /F3 11.955 Tf 11.96 0 Td[(DNXi=1(s)]TJ /F10 11.955 Tf 11.96 0 Td[((1)i)di.(3)Forconvenience,wedene(1)s=(2)s=...=(L)s=sinthesecondtermofequation( 3 ).Notethatthelasttermof( 3 )isaconstantforagiven.Therefore,theLagrangiandualfunction()isgivenby()=minL(z,x,y,) (3)s.t.LXl=1Xj2Nl(i)x(l)ije(l)ij)]TJ /F3 11.955 Tf 11.95 0 Td[(zEi0,8i2N (3)x(l)ij0,8l2L,i2N,j2Nl(i) (3)0y(l)iM,8l2f1,...,L)]TJ /F4 11.955 Tf 11.96 0 Td[(1g,i2N (3)z0. (3) 44

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Wecandecomposetheproblem( 3 )( 3 )intothefollowingtwosubproblems.S1:minNXi=1L)]TJ /F13 7.97 Tf 6.58 0 Td[(1Xl=1((l+1)i)]TJ /F10 11.955 Tf 11.96 0 Td[((l)i)y(l)is.t.0y(l)iM,8l2f1,...,L)]TJ /F4 11.955 Tf 11.96 0 Td[(1g,i2N. (3)S2:min8<:z+LXl=1X(i,j)2Al((l)j)]TJ /F10 11.955 Tf 11.96 0 Td[((l)i)x(l)ij9=;s.t.0x(l)ijM,8l2L,i2N,j2Nl(i)LXl=1Xj2Nl(i)e(l)ijx(l)ij)]TJ /F3 11.955 Tf 11.95 0 Td[(zEi0,8i2Nz0. (3) 3.3.1DistributedAlgorithmsforSubproblemsThesolutionmethodforS1isstraightforward:If((l+1)i)]TJ /F10 11.955 Tf 12.36 0 Td[((l)i)isnegative,thenwesetthecorrespondingvariabley(l)itoitsupperboundM.Otherwise,y(l)iequalszero.Clearly,thisalgorithmcanbeimplementedinadistributedandlocalmanner.WenowturnourattentiontosubproblemS2.Considerrstthecaseinwhichzisxedatsomenonnegativevaluezanddenef(z)asfollows.f(z)=max8<:)]TJ /F4 11.955 Tf 9.45 0 Td[(z+NXi=1LXl=1Xj:j2Nl(i)((l)i)]TJ /F10 11.955 Tf 11.96 0 Td[((l)j)x(l)ij9=;s.t.0x(l)ijM,8l2L,i2N,j2Nl(i)LXl=1Xj2Nl(i)e(l)ijx(l)ijzEi,8i2N.Fori2N,letfi(z)=maxLXl=1Xj:j2Nl(i)((l)i)]TJ /F10 11.955 Tf 11.95 0 Td[((l)j)x(l)ijs.t.0x(l)ijM,8l2L,j2Nl(i) 45

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LXl=1Xj2Nl(i)e(l)ijx(l)ijzEi.Then,f(z)=)]TJ /F4 11.955 Tf 9.44 0 Td[(z+PNi=1fi(z).Therefore,maximizingf(z)canbedecomposedintosmallerproblemsinwhicheachnodeiwillndfi(z).Findingthevalueoffi(z)ateachnodeiisequivalenttosolvinganinstanceofthefractionalknapsackproblem,whichispolynomiallysolvable[ 37 ].DetailsarelistedinAlgorithm 1 .Again,thisalgorithmcanalsobeimplementedinadistributedandlocalmanner. Algorithm1FractionalKnapsack(z)forFindingfi(z) sort(i,l,j)inthenonincreasingorderof((l)i)]TJ /F10 11.955 Tf 11.96 0 Td[((l)j)=e(l)ij U(zEi foreachof(i,l,j)inthesortedlistdo if((l)i)]TJ /F10 11.955 Tf 11.95 0 Td[((l)j0)then break elseif(U)]TJ /F3 11.955 Tf 11.95 0 Td[(Me(l)ij)<0then x(l)ij(U=e(l)ij break else x(l)ij(M U(U)]TJ /F3 11.955 Tf 11.95 0 Td[(Me(l)ij endif endfor Algorithm 1 solvessubproblemS2foragivenvalueofz.Tocompleteourargument,weneedtosuggestawayofndinganoptimalvalueforz,sothattheoverallobjectivefunctionf(z)ismaximized.Notethateachfi(z)isanondecreasing,concave,andpiecewise-linearfunctionofz.Hence,f(z)isalsoconcaveandpiecewise-linear.Supposewewanttosearchforanoptimalsolutionbystartingfrom0andincrementallyincreasingzuntilwendanoptimalvalue.Notethatweonlyneedtoconsiderz-valuesthatmarkthebeginningorendofalinearsegmentoff(z).Letzbetherstoptimalsolutionencounteredinthesearch.Then,f(z)mustbeincreasingforzz(exceptforthetrivialcasewheref(z)equalszeroatoptimality).Forzz,f(z)mustbenonincreasing.Hence,wearelookingforavaluez0suchthatf0(z)>0forz
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andf0(z)0forz>z.Alternatively, 8>><>>:Pi2Nf0i(z)>1,zz.(3)Alsonotethatf0(z)changesonlywhenoneofthef0i(z)changes.FromAlgorithm 1 ,weobservethatforeachi,f0i(z)changesonlywhenweselectthemostprotableiteminthelistoftheremainingitems.Supposeitem(i,j,l)isselected.Thenewf0i(z)isgivenbyf0i(z)=((l)i)]TJ /F10 11.955 Tf 12.6 0 Td[((l)j)=e(l)ij.Furthermore,thenexttimewhenf0i(z)changesagainiswhenzisincrementedbyMe(l)ij=Ei. Algorithm2SolutionforS2 foreachi2Ndo sort(i,l,j)innonincreasingorderof((l)i)]TJ /F10 11.955 Tf 11.96 0 Td[((l)j)=e(l)ij discardanyitem(i,l,j)if((l)i)]TJ /F10 11.955 Tf 11.96 0 Td[((l)j)0 k(0;zk(0 foreachof(i,l,j)inthesortedlistdo zk(zk+Me(l)ij=Ei Pi[k]((zk,((l)i)]TJ /F10 11.955 Tf 11.96 0 Td[((l)j)=e(l)ij) k(k+1 endfor endfor ndzwhichsatises( 3 )bysearching(Pi)i2N eachnodeiappliesAlgorithm 1 withz Tosummarize,theprocedureforsearchingzistokeeptrackofthesequenceofpointswheref0(z)changes(orthesequenceofpointswheref0i(z)changes,foreachi).Foraxedi,supposethat((l)i)]TJ /F10 11.955 Tf 12.41 0 Td[((l)j)=e(l)ijissortedinnonincreasingorderwhereanyitem(i,l,j)forwhich((l)i)]TJ /F10 11.955 Tf 12.02 0 Td[((l)j)0isdiscarded.Startingwithz0=0,wecangenerateasequencezk=zk)]TJ /F13 7.97 Tf 6.58 0 Td[(1+Me(l)ij=Eiiteratively,where(i,j,l)usedintheupdatetogetzkisthekthiteminthelist.Then,f0i(z)canchangeonlyateachofthepointszk.Algorithm 2 formalizesthisidea.ThearrayPi[]recordsthesequenceofzkandf0i(zk). 47

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3.3.2TheMainAlgorithmWenowassumethatthenetworkoperatesinatime-slottedfashion.TheLagrangiandualproblemof( 3 )( 3 )is Dual:max()(3)Hence,onecanproposeasubgradientprojectionmethodfor( 3 )whereupdated-valuesateachiterationcanbeobtainedusingthefollowingequations.(l)i(k+1)=[(l)i(k))]TJ /F10 11.955 Tf 11.95 0 Td[((Xj2Nl(i)x(l)ij(k)+y(l)i(k))]TJ /F11 11.955 Tf -221.89 -26.88 Td[(Xj:i2Nl(j)x(l)ji(k))]TJ /F3 11.955 Tf 11.95 0 Td[(y(l)]TJ /F13 7.97 Tf 6.59 0 Td[(1)i(k))]+,8i2N,l2L (3)s(k+1)=[s(k))]TJ /F10 11.955 Tf 11.96 0 Td[((M)]TJ /F5 7.97 Tf 18.17 14.95 Td[(LXl=1Xj:(j,s)2Alx(l)js(k))]+. (3)Here,[b]+=maxf0,bgand(>0)isasufcientlysmallstepsize.Lettingq(l)i(k)=(l)i(k),weproposethefollowingmodicationoftheabovealgorithmforthelifetimemaximizationproblemgivenby( 3 )( 3 ).Fortechnicalreasons,herewehavechangedtheupperboundoftheowvariablesfromMtoM(),M+NL,whereisasmallpositivevalue.MainAlgorithm y(k)=argminy(l)i2[0,M()],l2f1,...,L)]TJ /F13 7.97 Tf 6.59 0 Td[(1g,i2NfNXi=1L)]TJ /F13 7.97 Tf 6.59 0 Td[(1Xl=1(q(l+1)i(k))]TJ /F3 11.955 Tf 11.96 0 Td[(q(l)i(k))y(l)ig (3)(z(k),x(k))=argminx(l)ij2[0,M()],l2L,i2N,j2Nl(i)PLl=1Pj2Nl(i)e(l)ijx(l)ijzEi,i2Nz0fz +LXl=1Xi2NXj2Nl(i)(q(l)j(k))]TJ /F3 11.955 Tf 11.96 0 Td[(q(l)i(k))x(l)ijg (3) 48

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q(l)i(k+1)=[q(l)i(k))]TJ /F4 11.955 Tf 11.95 0 Td[((Xj2Nl(i)x(l)ij(k)+y(l)i(k))]TJ /F11 11.955 Tf -210.77 -26.87 Td[(Xj:i2Nl(j)x(l)ji(k))]TJ /F3 11.955 Tf 11.95 0 Td[(y(l)]TJ /F13 7.97 Tf 6.59 0 Td[(1)i(k))]+,8l2L,i2N (3)qs(k+1)=0. (3)Problems( 3 )and( 3 )aresolvedusingthealgorithmspresentedinSection 3.3.1 withasuitablemodicationinthenotation.Onecanthinkofq(l)iasthevirtualqueueatnodei(l)inFigure 3-1 ,and( 3 )canbeunderstoodasthequeuedynamicsinthetwoconsecutiveslots.Sinceq(k+1)s=0,allowreachingthesinkmustbedrainedout.Notethattheabovealgorithmisnotexactlyasubgradientoptimizationonebecauseoftheprojectionthatweuseinupdatingq-valuesandalsobecauseqs(k+1)=0.Therefore,standardsubgradientoptimizationresultscannotbeappliedtoprovethecorrectnessofouralgorithm.WewilluseaLyapunovdriftanalysismethodinthenextsectiontoshowconvergenceandstabilitypropertiesofthisalgorithm. 3.4ConvergenceResultsInthissection,weshowthatouralgorithmconvergestoanoptimalsolutionandthevirtualqueuesarebounded(bothinthelong-runaveragesense).Theanalyticaltechniquethatweuseissimilartothatof[ 36 ].Werstdenean-perturbedproblem,whichwillbeusedlater.Here,isasmallpositivevalue.minz (3)s.t.LXl=1Xj2Nl(i)e(l)ijx(l)ijzEi,8i2N (3) 49

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8>>>>>>>>><>>>>>>>>>:Xj:i2Nl(j)x(l)ji)]TJ /F11 11.955 Tf 16.64 11.36 Td[(Xj2Nl(i)x(l)ij+y(l)]TJ /F13 7.97 Tf 6.58 0 Td[(1)i)]TJ /F3 11.955 Tf 11.96 0 Td[(y(l)i=)]TJ /F10 11.955 Tf 9.3 0 Td[(,8l2L;i2NLXl=1Xj:s2Nl(j)x(l)js=M+NL (3)y(0)i=Ddi,y(L)i=0,8i2N. (3)Theusualnonnegativityconstraintsarestillrequired.Intheaboveproblem,weinjectanextrasupplyofmagnitudeateachnodei(l),i2N,l2L.ThedemandatthedestinationnodesisalsomodiedtoM+NLtoensuretheexistenceofafeasibleow.Werstdiscussthepropertiesoftheoptimalobjectivevaluefunctionoftheabove-perturbedproblem.Forbrevity,wereplaceourproblemwithagenerallinearprogrammingproblemasfollows.(P)mincTxs.t.Ax=dx0. (3)Inproblem(P())below,therighthandsideof(P)isperturbedbyalongthedirectiond.(P())mincTxs.t.Ax=d+dx0. (3)Letfandf()betheoptimalobjectivevaluesforproblem(P)and(P()),respectively. Lemma3.1. f()iscontinuous,convex,andpiecewise-linearin. 50

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Proof. Thedualof(P())isdenedas(D())max(d+d)Ts.t.ATcLetSdenotethefeasibleregionof(D()).Bythestrongdualitytheoreminlinearprogramming[ 32 ],weknowthatf()=maxf(d+d)Tj2Sg.Also,weknowthatforevery,theoptimalobjectivefunctionvalueof(D())canbeobtainedbyexaminingallextremepointsofS.Let^SbethesetofextremepointsofS.Hence,f()=maxfdT+dTj2^Sg.Thismeansthatf()isobtainedbytakingthemaximumofanitenumberoflinearfunctionsof.Thereforef()isacontinuous,convex,andpiecewise-linearfunctionof[ 38 ]. Theorem3.1. Letz()betheoptimalobjectivefunctionvalueofproblem( 3 )( 336 )andzbetheoptimalobjectivefunctionvalueoftheunperturbedproblem.Then,z()!zas!0. Proof. If=0,thenthe-perturbedproblemisidenticaltotheunperturbedproblemdenedasin( 3 )( 3 ).Therefore,theresultholdsusingLemma 3.1 andthefactthatz()isacontinuousfunctionof. Next,wewanttoproveouralgorithmconvergestotheoptimalobjectivefunctionvalueinthetimeaveragesense.LetusdeneaLyapunovfunctionofthequeuesbyV(q)=Pi2NPl2L(q(l)i)2.Let(k),V(q(k+1)))]TJ /F3 11.955 Tf 11.95 0 Td[(V(q(k)). Lemma3.2. ThereexistpositiveB,,andsuchthatforalltimeslotkandforallq(k)thefollowingconditionholds. (k)+2 z(k)B+2 ^z())]TJ /F4 11.955 Tf 11.96 0 Td[(2Xl2LXi2Nq(l)i,(3)where^z()istheoptimalsolutionofthe-perturbedproblem. 51

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Proof. Bysquaring( 3 )andarrangingit,weget(q(l)i(k+1))2)]TJ /F4 11.955 Tf 11.95 0 Td[((q(l)i(k))2g2(i,l,k))]TJ /F4 11.955 Tf 11.95 0 Td[(2q(l)i(k)g(i,l,k),whereg(i,l,k)=Pj2Nl(i)x(l)ij(k))]TJ /F11 11.955 Tf 12.83 8.97 Td[(Pj:i2Nl(j)x(l)ji(k)+y(l)i(k))]TJ /F3 11.955 Tf 12.83 0 Td[(y(l)]TJ /F13 7.97 Tf 6.59 0 Td[(1)i(k).Notethatg(i,l,k)NMbecausePj2Nl(i)x(l)ij(k)(N)]TJ /F4 11.955 Tf 13.14 0 Td[(1)Mandy(l)i(k)Mforallk.Aggregatingtheaboveinequalityoveralli2Nandl2L,wehaveLXl=1NXi=1q(l)i(k+1)2)]TJ /F11 11.955 Tf 11.95 13.27 Td[(q(l)i(k)2LXl=1NXi=1g2(i,l,k))]TJ /F4 11.955 Tf 11.96 0 Td[(2LXl=1NXi=1q(l)i(k)g(i,l,k)LN3M2+2LXl=1NXi=1q(l)i(k)0@)]TJ /F11 11.955 Tf 15.97 11.36 Td[(Xj2Nl(i)x(l)ij(k)+Xj:i2Nl(j)x(l)ji(k))]TJ /F3 11.955 Tf 11.96 0 Td[(y(l)i(k)+y(l)]TJ /F13 7.97 Tf 6.58 0 Td[(1)i(k)1A=B)]TJ /F4 11.955 Tf 11.95 0 Td[(2LXl=1X(j,s)2Alq(l)j(k)x(l)js(k)+2LXl=1X(i,j)2Al;j6=sq(l)j(k))]TJ /F3 11.955 Tf 11.95 0 Td[(q(l)i(k)x(l)ij(k)+2NXi=1L)]TJ /F13 7.97 Tf 6.59 0 Td[(1Xl=1q(l+1)i(k))]TJ /F3 11.955 Tf 11.96 0 Td[(q(l)i(k)y(l)i(k)+2NXi=1q(1)i(k)y(0)i(k), (3)whereB,LN3M2.Equation( 3 )canbeobtainedbyregroupingthetermsbasedonvariablesxandy.Notethattheforthterminthelastequalityexcludeslinkstothesink. 52

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Adding2qs(k)(PlP(j,s)2Alx(l)js(k))=0to( 3 ),wehave (k)B+2NXi=1q(1)i(k)Ddi+2XlX(i,j)2Alq(l)j(k))]TJ /F3 11.955 Tf 11.95 0 Td[(q(l)i(k)x(l)ij(k)+2NXi=1L)]TJ /F13 7.97 Tf 6.59 0 Td[(1Xl=1q(l+1)i(k))]TJ /F3 11.955 Tf 11.95 0 Td[(q(l)i(k)y(l)i(k).(3)Notethatthethirdtermnowincludesthelinkstothesink.Wealsousedthefacty(0)i(k)=Ddi.Adding(2=)z(k)tobothsidesofinequality( 3 ),weget(k)+2 z(k)B+2NXi=1q(1)i(k)Ddi+28<:z(k) +XlX(i,j)2Al(q(l)j(k))]TJ /F3 11.955 Tf 11.96 0 Td[(q(l)i(k))x(l)ij(k)9=;+2NXi=1L)]TJ /F13 7.97 Tf 6.59 0 Td[(1Xl=1q(l+1)i(k))]TJ /F3 11.955 Tf 11.96 0 Td[(q(l)i(k)y(l)i(k)B+2NXi=1q(1)i(k)Ddi+28<:^z() +XlX(i,j)2Al(q(l)j(k))]TJ /F3 11.955 Tf 11.95 0 Td[(q(l)i(k))^x(l)ij()9=;+2NXi=1L)]TJ /F13 7.97 Tf 6.59 0 Td[(1Xl=1q(l+1)i(k))]TJ /F3 11.955 Tf 11.96 0 Td[(q(l)i(k)^y(l)i(), (3)where(^z(),^x(),^y())isanoptimalsolutionof-perturbedproblemdenedas( 3 )-( 3 ).Notethatthelastinequalityholdsbecause(z(k),x(k),y(k))isaminimizeroftheobjectivefunctioninsubproblemsS1andS2.Afterregroupinginequality( 3 )accordingtothedualmultipliers,wehave(k)+2 z(k)B+2 ^z()+2LXl=1NXi=1q(l)i(k)(Xj:i2Nl(j)^x(l)ji())]TJ /F11 11.955 Tf 16.64 11.36 Td[(Xj2Nl(i)^x(l)ij())]TJ /F4 11.955 Tf 12.24 0 Td[(^y(l)i()+^y(l)]TJ /F13 7.97 Tf 6.59 0 Td[(1)i()) 53

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=B+2 ^z())]TJ /F4 11.955 Tf 11.95 0 Td[(2LXl=1NXi=1q(l)i(k) (3)Intherstinequality,2NXi=1q(1)i(k)Ddi=2NXi=1q(1)i(k)^y(0)i(),whichisthesecondtermin( 3 )andisnowcombinedwiththethirdterm.Also,recallthat(^z(),^x(),^y())satisestheowconservationconstraint( 3 ).Therefore,Pj^x(l)ji())]TJ /F11 11.955 Tf 12.64 8.97 Td[(Pj^x(l)ij())]TJ /F4 11.955 Tf -434.42 -23.9 Td[(^y(l)i()+^y(l)]TJ /F13 7.97 Tf 6.58 0 Td[(1)i()=)]TJ /F10 11.955 Tf 9.3 0 Td[(andtherstequalityissatised.Thisconcludestheproof. Weshowinthetheorem 3.2 thatthetimeaveragedsolutionof(z(k),x(k),y(k))convergestotheoptimalsolutionoftheprimalproblem.DeneQ(k)=PLl=1PNi=1q(l)i(k).NotethatQ(k)canbeinterpretedasthesumofallvirtualqueuesizesinthesystematthetimeslotk. Theorem3.2. Thefollowinginequalitieshold: limsupT!1,!0z(T)B 2+z,(3) limsupT!11 TT)]TJ /F13 7.97 Tf 6.59 0 Td[(1Xk=0Q(k)B 2+1 ^z()(3) Proof. Aggregatinginequalities( 3 )overk=0,1,...,T)]TJ /F4 11.955 Tf 11.95 0 Td[(1,wehave V(q(T)))]TJ /F3 11.955 Tf 11.95 0 Td[(V(q(0))+2 T)]TJ /F13 7.97 Tf 6.58 0 Td[(1Xk=0z(k)BT+2 T^z())]TJ /F4 11.955 Tf 9.3 0 Td[(2T)]TJ /F13 7.97 Tf 6.58 0 Td[(1Xk=0Q(k)(3)Multiplying=(2T)andarrangingappropriately,wegetz(T),1 TTXk=0z(k)B 2+^z())]TJ /F10 11.955 Tf 13.15 8.09 Td[( TT)]TJ /F13 7.97 Tf 6.59 0 Td[(1Xk=0Q(k))]TJ /F10 11.955 Tf 13.15 8.09 Td[(V(q(T)) 2T+V(q(0)) 2TB 2+^z()+V(q(0)) 2T. (3) 54

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Hence limsupT!1,!0z(T)B 2+z,(3)wherezistheoptimalobjectivefunctionvalueof( 3 )( 3 ).Notethat^z()!zas!0bytheTheorem 3.1 .From( 3 ),wehave2T)]TJ /F13 7.97 Tf 6.58 0 Td[(1Xk=0Q(k)BT+2T ^z()+V(q(0)))]TJ /F3 11.955 Tf 11.96 0 Td[(V(q(T)))]TJ /F4 11.955 Tf 13.15 8.09 Td[(2 T)]TJ /F13 7.97 Tf 6.58 0 Td[(1Xk=0z(k)BT+2T ^z()+V(q(0)) (3)Multiplyingthebothsideoftheaboveinequalityby1=(2T),weget 1 TT)]TJ /F13 7.97 Tf 6.58 0 Td[(1Xk=0Q(k)B 2+1 ^z()+V(q(0)) 2T.(3)Hence, limsupT!11 TT)]TJ /F13 7.97 Tf 6.59 0 Td[(1Xk=0Q(k)B 2+1 ^z(),(3)whichconcludestheproof. 3.5ComputationalExperimentsInthissection,wepresenttwonumericalexperimentstoprovethevalidityofouralgorithm.Morespecically,weshowthatouralgorithmconvergestotheoptimalobjectivefunctionvalueofproblem( 3 )( 3 ).WealsoshowhowtheLyapunovdriftandthequeuesizesevolve.Fortheseexperiments,werandomlyplace50sensorsinacircularregionwitharadius25mandselect6sinklocationsinthesameregionforthesinktovisit.Thetransmissioncostbetweentwonodesdependsonthedistancebetweenthem[ 30 ].Datagenerationrateofnodeiisselectedfrom[0,500]bpsandeachnodehas500Jofinitialenergy.Inthesubgradientprojectionmethod,weuseaconstantstepsizeof=10)]TJ /F13 7.97 Tf 6.58 0 Td[(8.Inallexperiments,theperturbationparameteris=10)]TJ /F13 7.97 Tf 6.59 0 Td[(8. 55

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Figure3-2. Convergencetotheoptimalvalue Figure 3-2 showstheconvergenceofouralgorithmtotheoptimalobjectivefunctionvalue.Asareference,theoptimalsolutionoftheprimalproblem( 3 )( 3 )isobtainedusingCPLEX.Thecurvelabeledasz(k)istheaveragevalueofz(k)atiterationk.Figure 3-2 veriesthevalidityoftherstinequalityinTheorem 3.2 .Finally,Figure 3-3 showsthelong-runaveragevalueofthetotalqueuesize,PlPiq(l)i.BythesecondpartofTheorem 3.2 ,thisvalueisboundedfromabove,whichisveriedhere. 56

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Figure3-3. Timeaverageoftotalvirtualqueuesizeovertime 57

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CHAPTER4ANINTEGER-PROGRAMMING-BASEDAPPROACHTOTHECLOSE-ENOUGHTRAVELINGSALESMANPROBLEM 4.1ProblemDenitionGivenacollectionofnodesandthesetofdistancesbetweeneachpairofnodes,thetravelingsalesmanproblem(TSP)seekstondashortesttourthatvisitseachnodeexactlyonce.ThebroadapplicabilityoftheTSP,orslightvariationsthereof,haveresultedinanimpressiveslateofresearchonTSP-relatedproblems.SeveralTSPextensionscanbefoundintheliterature,suchasthecoveringsalesmanproblem[ 39 ],prizecollectingTSP[ 40 41 ],andthecoveringtourproblem(CTP)[ 42 ].Foracomprehensivesurveyonthehistory,algorithms,andapplicationsoftheTSP,thereaderisreferredto[ 14 ].Theclose-enoughtravelingsalesmanproblem(CETSP)isageneralizationoftheTSPinwhichthesalesmandoesnotneedtovisittheexactlocationofeachcustomer.Instead,acompactregionoftheplanecontainingeachnodeisspeciedasitsneigh-borhoodset,andthegoalistondashortesttourthatstartsfromaspecieddepotlocationandintersectsalloftheseneighborhoodsets.Intuitivelyspeaking,intheCETSP,eachofthesalesman'sclientsiswillingtotraveltoanypointinsideitsparticularneighborhoodtomeetwiththesalesman.AtypicalapplicationoftheCETSPariseswhenanairbornevehicleistryingtondastationarytargetonatwo-dimensionaleldbyscanningacollectionofcandidatelocations.Supposethatthedetectionoccurswhentheaircraftisnomorethanrunitsawayfromthetarget.Thentheneighborhoodsetofatargetisadiscofradiusratthetarget'slocation.Theproblemofndingtheshortesttrajectoryfortheaircraftisequivalenttondingatourofminimumlengththatintersectseachneighborhoodset.AnotherimportantapplicationfortheCETSParisesinwirelesssensornetworkoperations.AswediscussedinChapters 2 and 3 ,inthesenetworkssensornodesperiodicallyrelaydatathattheyhaveaccumulatedtoamobiledata-collectingdevice 58

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calledasink.Energyminimizationisanimportantcomponentofwirelesssensornetworks.Ciulloetal.[ 11 ]showthatbylettingthesinkcomeclosertosensorswithhigherdatagenerationrates,onecansignicantlyreducetheconsumedenergyforthetransmissionofdata.Therefore,itisreasonabletoassignasensingradiusritoeachsensornodeibasedonitsdatagenerationrate,andrequirethatthesinkmustvisitapointwithintheri-neighborhoodofnodeitocollectitsdata.Again,theproblemofndingashortesttrajectoryofthesinktoretrievedatafromallsensorsisaninstanceoftheCETSP(seealso[ 43 ]).Gulczynskietal.[ 44 ]proposeseveralheuristicsforacommonspecialcaseoftheCETSPwheretheneighborhoodsetsofallnodesarediscsofthesameradius.Theproblemarisesinsituationswhereradiofrequencyidentication(RFID)tagsareusedthatcanremotelyprovideamobiledatacollectorwiththerequireddata.Asanexample,manyutilitycompaniesarenowusingRFID-basedautomatedmeterreadersthatcanreadtheusageofeachcustomerremotely[ 12 ].Mennelletal.[ 13 ]proposeaheuristicalgorithmbasedonSteinerzones,whicharenonemptyintersectionsoftheneighborhoodsets.Theirapproachconsistsofthreephases:(1)identifyingacollectionofSteinerzonesthatcovereveryneighborhoodset;(2)representingeachSteinerzonewithoneofitspoints;and(3)ndingaTSPtourovertheserepresentativepoints.ArkinandHassin[ 45 ]proposepolynomial-timeapproximationalgorithmsforseveralspecialcasesoftheproblem.Inparticular,theyprovidealgorithmsthatyielderrorboundsfortheCETSPinwhichtheneighborhoodsetseithertaketheformofparallelunitsegments,translatesofpolygonalregions,ordiscs.OtherapproximationalgorithmsincludetheworkofMataandMitchellin[ 46 ]andDumitrescuandMitchell[ 47 ].TheheuristicalgorithmofMennell[ 13 ]andthepolynomial-timeapproximationalgorithmsin[ 45 47 ]areabletoefcientlyndagoodfeasibleCETSPtour.OnedifcultyinevaluatingthequalityofsuchfeasiblesolutionsisthelackofexactalgorithmsfortheCETSPintheliterature.Hence,developingtightlowerboundsfortheCETSPis 59

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ofcrucialimportance.Suchlowerboundswouldenableoneto(conservatively)evaluatethequalityofafeasibletourobtainedbyanon-exactalgorithm.WhilethereexistseveralmethodsofefcientlyobtainingsuchlowerboundsfortheTSP,developinglowerboundsfortheCETSPisconsideredtobeanon-trivialtask.OurcontributionistargetedtowardndingarbitrarilytightlowerandupperboundsontheoptimalCETSPtourlengthviamixed-integerprogrammingmodels.SomeofthemostsuccessfulexactalgorithmsforsolvingthesymmetricTSPcombinetheuseofcuttingplanesandefcientheuristicsinabranch-and-cutscheme.Examplesofsuchbranch-and-cutalgorithmsaretheworkofPadbergandRinaldi[ 48 ]andthewellknownConcordeTSPsolver[ 49 ].TheapproachthatweproposeinthischapterrequiressolvingintegerprogrammingproblemstoobtainaseriesofupperandlowerboundsthatconvergetoanoptimalCETSPtour.ThediscretenatureofthisalgorithmprovidesaframeworkforusingcuttingplanesthathavebeenprovedtobemostsuccessfulinsolvinglargesymmetricTSPinstances.Therestofthischapterisorganizedasfollows.Section 4.2 establishesfoundationalconceptsrelatedtooptimalCETSPtoursandthediscretizationschemeusedinthischapter.Section 4.3 providesamathematicalprogrammingformulationthatyieldsalowerboundontheoptimalCETSPtourlength.Basedontheanticipateddifcultiesofsolvingthismodel,weproposeanalternativemodelandassociatedtwo-stageoptimizationalgorithminSection 4.4 .WepresentcomputationalresultsthatdemonstratetheefciencyofourapproachinSection 4.6 4.2PreliminariesWebeginbystatingtheformaldenitionoftheCETSPanddescribingthenotationusedinthischapterinSection 4.2.1 ,andthenpresentourcorediscretizationschemeinSection 4.2.2 60

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4.2.1DenitionsandNotationLetMbeasetofpointsinatwo-dimensionalplanealongwithadepotpointp0.Foreachpointm2M,letSmbeacompactsetthatcontainsm.TheCETSPseekstondashortesttour(withrespecttoEuclideandistance)thatstartsfromp0,intersectseverysetS1,...,SjMj(inanyorder),andterminatesatp0. Figure4-1. AfeasibletourwithjMj=4 Figure 5-1A illustratesaCETSPtourthatintersectsfourcompactsets.NotethattheCETSPisclearlyNP-hardasitreducestotheTSPwheneachsetSmconsistsofasinglepoint.Withoutlossofgenerality,weassumethatp0=2Sm,8m2M,orelsewecansimplyignoreallm2Mforwhichp02Sm.WewillrefertotheelementsofMasthetargetlocationsandtheirassociatedcompactsetsastheneighborhoodsets.Theseneighborhoodsetsareoftendiscs,e.g.,fortheairmonitoringandmeterreadingapplicationsmentionedpreviously.However, 61

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theapproachthatwepresentinthischapterforsolvingtheCETSPingeneraldoesnotdependontheshapeoftheneighborhoodsets.Thefollowingpropositionisusefulinthedevelopmentofouralgorithm. Proposition4.1. AlloptimalsolutionstotheCETSPconsistofanitesetofconnectedlinesegments(p0p1),(p1p2),...,(pk)]TJ /F13 7.97 Tf 6.59 0 Td[(1pk),(pkp0),wherekjMj.Moreover,foreachpointpi,i=1,...,k,thereexistsatleastonem2MsuchthatpiisontheboundaryofSm. Proof. LetTbeanyfeasibleCETSPtour.Wecanselectanitenumberofpointsp1,...,pkonTsuchthatkjMjand)]TJ /F2 11.955 Tf 5.48 -9.69 Td[([kj=1pj\Sm6=;,8m2M.Notethatthelinesegmentbetweenanypairofthesepointsistheuniqueminimizerofdistancebetweenthem.Therefore,theclassofsolutionscomposedoflinesegmentsconnectingthesepointsdominatesallotherclassesofsolutions,andsoanoptimalCETSPtourmustconsistofsomekjMjlinesegments.Hence,wenowconsideranoptimaltourconsistingoflinesegments(p0p1),(p1p2),...,(pk)]TJ /F13 7.97 Tf 6.59 0 Td[(1pk),(pkp0),where1kjMj.Foreachi=1,...,k,deneViasthesetoftargetneighborhoodsvisitedbypi,butnotbyp1,...,pi)]TJ /F13 7.97 Tf 6.58 0 Td[(1,i.e.,Vi=fm2M:pi2Smandm=2Vjforj
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traversesfromp0idirectlytop00iandbypassespiisnolongerthanT,butstillintersectsRi.Therefore,theremustexistanoptimalsolutioninwhicheachpointpi,i=1,...,k,liesontheboundaryofatleastonesetSmforsomem2M.Thiscompletestheproof. Figure4-2. Illustrationofboundarypointoptimality Proposition 5.4 impliesthatanyoptimalsolutiontotheCETSPcanbecharacterizedbyadiscretesetofpointsintheplane,whereeachpointbelongstotheboundaryofatleastoneneighborhoodset.Intherestofthechapter,wewillcallthesetheturnpointsofthecorrespondingtour.Proposition 4.2 furthercharacterizesthosesectionsoftheboundarysetsthatmaycontainaturnpointofanoptimaltour. Proposition4.2. SupposethateveryneighborhoodsetSmisadisccenteredatm2M.LetTbeanoptimalCETSPtourthatischaracterizedbyasetofturnpointsfp0,...,pkg.Then,pi2conv(M[fp0g),fori=1,...,k,whereconv(M[fp0g)denotestheconvexhullofthecorrespondingpoints. Proof. First,deneD=conv(M[fp0g).ThedimensionofDmustbeatleastone(orelseD=fp0gandthusjMj=0).IfthedimensionofDisone,thepropositionfollowsbynotingthatanoptimaltourtraversesthelengthofDfromp0towardeachofitstwoendpoints,stoppingonceallneighborhoodsetsarevisitedneareachendofDandthenbacktop0.NowconsiderthecaseinwhichDistwo-dimensional,andbycontradiction,supposethatTisanoptimaltourthatcontainspointsoutsideD.Letq0beapointon 63

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TthatintersectstheboundaryofD,suchthatthetrajectoryofTimmediatelyleavesD(i.e.,8"suchthat0<" ",pointq0+"disonTbutisnotinD,forsometrajectoryvectordandapositive ").Becauseq0isontheboundaryofD,itmustbeaconvexcombinationoftwopointsc1andc2inM[fp0g.Considerthe(innite)lineintersectingc1andc2anddeneHasthehalfspacewithrespecttothislinethatcontainsD;also,dene HasthehalfspaceconsistingofthissamelineandpointsontheoppositesideofD(sothatH\ Hisjustthelineintersectingc1andc2).FollowingthetrajectoryofTafteritleavesDatq0,supposethatTre-entersHatq00.Thispointmustexistbecausethetourmustreturntop02H.Nowconsideranalternativetour^T,whichisthesameasTexceptthatthesegmentofTbetweenq0andq00(allofwhichliesoutsideofH)isreplacedwithasegmentthatliescompletelyonthelinespanningc1andc2.Let M=fm2M:Tvisitstheneighborhoodsetofmonthesegmentbetweenq0andq00thatliesin Hg.If M=;,thenreplacingthesegmentofTwiththestraightsegmentbetweenq0andq00retainsthefeasibilityofthetourwhilereducingitslength;hence,Tcouldnotbeoptimal(Figure 4-3 ).Otherwise,supposethat M6=;.Becausethelinesegmentintersectingc1andc2inducesafacetofD,alltargetsin Mcanbevisitedbytraversingthissegment(orelse,somem2 MwouldnotbelongtoD).Supposethat[min,max]isthesmallestintervalsuchthatthesetofallpointsq0+(q00)]TJ /F3 11.955 Tf 12.15 0 Td[(q0)for2[min,max]intersectsneighborhoodsetforeachm2 M.Dene min=minf0,mingand max=maxf1,maxg.Theuniqueshortesttoursegmentthatiscompletelycontainedin H,startsatq0,intersectstheneighborhoodsetofeachm2 M,andendsatq00isdescribedasfollows.Segmentstartsatthepointonthelinesegmentwhere=0,movestothepointwhere= min,thentothepointwhere= max,andreturnstothepointwhere=1(Figure 4-4 ).Now,supposethatwecreateanewtour^T,whichisthesameasTexceptwherereplacesthesegmentfromq0toq00inT.BecausethesegmentinTjoiningq0andq00iscontainedin Handisnotidenticalto,itsdistanceislongerthanthatof.Thus,tour^Tisstill 64

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Figure4-3. Illustrationoftheconvexhulloftargetpoints feasiblebutisshorterthanT,whichcontradictstheoptimalityofT.Thiscompletestheproof. 4.2.2PartitioningSchemesSinceanoptimalCETSPtourcanberepresentedbyanitenumberofturnpoints,anaturalwayofobtainingafeasibletouristoapproximatethesolutionspacebyadiscretesetofpoints.However,suchanapproachresultsonlyinanupperboundontheoptimalCETSPtourlength.OurapproachforobtaininglowerboundsontheoptimalCETSPtourlengthinthischapterisbasedonpartitioningthecontinuoussolutionspaceintosmallersetsandidentifyingthosepartitionsthatpossiblycontainaturnpointofanoptimalCETSPtour. Denition4.1.AsetC=fC0,...,CngiscalledaCETSP-partitioningofthetwodimensionalplaneif: 1. C0=fp0g. 2. Ciisanonemptycompactset,forall1in. 65

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Figure4-4. IllustrationoftheconvexhullargumentinproofofProposition 4.2 3. conv(Ci\Cj)hasanemptyinterior,forall1i,jn. 4. ([ni=1Ci)([i2MBm),whereBmistheboundaryofSm.WheneveryneighborhoodsetSmisadiscthatiscenteredatitscorrespondingtargetlocation,BmintheabovedenitioncanberestrictedtoincludethosepointsontheboundaryofSmthatlieinsideconv(M[fp0g).2WerefertotheelementsofaCETSP-partitioningCascells.ForanytwocellsCiandCj,lijisdenedastheshortestlinesegmentlengththatconnectstheboundariesofCiandCj.SeeSection 4.5 fordetailsoncomputinglij-values.Inthischapter,weconsidertwowaysofpartitioningtheplane:grid-basedandarc-based.Inagrid-basedpartitioning,eachgridcellCiisarectanglethatintersectsatleastoneneighborhoodset(seeFigure 4-5 ).Thegridcellsarechosensothattheycollectivelycontainalltheneighborhoodsetsandthedepot.Moreover,wewillrestrictourselvestogrid-basedpartitioningsinwhichthereexistsnoneighborhoodsetthatisentirelycontainedinoneoftherectangles.WedeneasetPofgridpointsasthecollectionofp0andanyotherpointthatisavertexofatleastonegridcell.Throughout 66

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thechapter,wewillrepresentgridcellibyatuple(ai,bi,w1i,w2i),where(ai,bi)denotesthelower-leftvertex,andw1iandw2iarethetwosidelengthsoftherectangle,sothat(ai+w1i,bi+w2i)denotestheupper-rightvertex. Figure4-5. Illustrationofgrid-basedpartitioning Asopposedtothegrid-basedscheme,arc-basedpartitioningisprimarilyintendedforcircularneighborhoodsets.Figure 4-6 illustratesanexampleofanarc-basedpartitioningforaCETSPinstance.Here,acellisdenedasanarc(speciedbyitstwoendpoints)ontheboundaryofasetSmthatliesonorinsidetheconvexhullofthetargetlocationsandthedepot.NotethatbyProposition 4.2 ,everyturnpointofanoptimalCETSPtourbelongstoatleastonearcinthispartitioning.Wewillspecifyanarcwith(ai,bi,ri,1i,2i)where(ai,bi)andriarerespectivelythecenterandradiusofthecorrespondingdisc,and1iand2irespectivelydenotethestartandnishanglesofthecorrespondingarc.Thecentralangleofanarcisdenedasi=2i)]TJ /F10 11.955 Tf 12.16 0 Td[(1i.Throughoutthechapter,wewillassumethati
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Figure4-6. Illustrationofarc-basedpartitioning 4.3LowerBoundingModelWerstformulateamixed-integerprogram(MIP)forobtaininglowerboundstotheoptimalCETSPtourlengthinSection 4.3.1 ,alongwithaclosed-formexpressionforanaccompanyingupperbound.Wethendescribetwocutting-planestrategiestoaidinthesolutionofthismodelinSection 4.3.2 ,andcommentonthecomplexityoftheirseparationroutines. 4.3.1FormulationandBoundsLetC=fC0,...,CngbeaCETSP-partitioningoftheplanewithapairwisedistancematrixL=flijg.Foreachm2MdenethesetofcellsintersectingSmasN(m)=f1in:Ci\Sm6=;g.NotebyDenition 4.1 wehaveN(m)6=;forallm2M.ConsiderthefollowingMIP. MinnXi=0nXj=0lijxij (4a)s.t.nXj=0xij=nXj=0xji,8i=0,...,n (4b) 68

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yi=nXj=0xij,8i=0,...,n (4c)Xi2N(m)yi1,8m2M (4d)Xi2SXj=2Sxijyv,8Sf1,...,ng:2jSjjCj)]TJ /F4 11.955 Tf 17.93 0 Td[(2andv2S (4e)xij2f0,1g,8i=0,...,n,j=0,...,n (4f)0yi1,8i=1,...,n;y0=1. (4g)Anoptimalsolutionto( 5 )isinfactashortestTSPtourwithrespecttodistancematrixL,overasubsetofCthatvisitsatleastoneelementinN(m)foreachm2M.DecisionvariablexijindicateswhetherthecorrespondingtourmovesfromCitoCj,whileauxiliarybinaryvariableyiindicateswhetherCiisvisitedonthetour.Theobjectivefunction( 4a )minimizesthetotaldistancetraveledinthetour.Constraints( 4b )ensurethatforeachnodei,thenumberofincomingtourarcsequalsthenumberofoutgoingtourarcs.Constraints( 4c )deney-variablesintermsofx-variables.(Infact,theformulationcanbegivenwithoutthey-variables;theyareincludedonlyforconvenienceinpresentation.)Constraints( 4d )ensurethatforeachm2M,atleastoneelementofCisvisitedthatcoversm.Constraints( 4e )aresubtoureliminationconstraints(see[ 50 ]).Finally,( 4f )and( 4g )statelogicalrestrictionsandboundsonthevariables.Problem( 5 )isaspecialcaseoftheCTP[ 42 ].TheCTPisdenedonagraphG=(V[W,E)whereVisthesetofverticesthatcanbevisitedandWisthesetoftargetsthatmustbecoveredbyverticesinV.Foreveryw2W,thereexistsanonemptysubsetVwVofverticesthatcoverw.Thereisalsoasubset VVcontainingtheverticesthatmustbevisitedbyanyfeasibletour.ThegoalintheCTPistondashortesttouronasubsetofVthatvisitsallverticesof Vaswellasatleastone 69

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vertexofVw,foreveryw2W.Therefore,problem( 5 )isaspecialcaseoftheCTPonacompletedirectedgraphwithV=C[fp0g,W=M, V=fp0g,andVm=N(m).Thefollowingpropositionestablishesarelationshipbetweentheoptimalobjectivefunctionvalueofproblem( 5 )andtheoptimalCETSPtourlength. Proposition4.3. SupposethattheoptimalCETSPtourlengthisl,andconsideranoptimalsolution(x,y)to( 5 )withobjectivefunctionvaluezLB1.DeneI=fi:yi=1g,whichindicatesthesetofturncellsassociatedwiththisoptimalsolution,andletNi=jfm2M:i2N(m)gjbethenumberofneighborhoodsetsintersectedbyCi.ThenzLB1lzLB1+Xi2Ihi, (4)wherehi=8><>:2Nirisin)]TJ /F17 7.97 Tf 6.68 -4.57 Td[(i 2ifNi2,2ri2sin)]TJ /F17 7.97 Tf 6.67 -4.57 Td[(i 4+(Ni)]TJ /F4 11.955 Tf 11.95 0 Td[(1)sini 2(Ni)]TJ /F13 7.97 Tf 6.59 0 Td[(1)ifNi3, (4)foranarc-basedCETSP-partitioning,andhi=8>>>>>>><>>>>>>>:2p w1i2+w2i2ifNi=1,w1i+w2i+p w1i2+w2i2ifNi=2,2(w1i+w2i)ifNi=3,2(w1i+w2i)+p w1i2+w2i2ifNi4, (4)foragrid-basedCETSP-partitioning. Proof. SupposethatanoptimalCETSPtourTischaracterizedbyanorderedsetofturnpoints(p0,...,pk).LetC0t=Cit2CbeanelementoftheCETSP-partitioningthatcontainspt,fort=1,...,kanddeneC0k+1=C00=C0andik+1=i0=0.Becausecellsmightcontainmultipleturnpoints,someelementsofC0=fC01,...,C0kgmaybeidentical.Supposethat yRit=1,fort=0,...,k, 70

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xR0,i1=1,and xRit,is=1,fort=1,...,kands>t,ifandonlyifxRiu,it=1forsomeu
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Figure4-7. Aninstanceofthelowerboundproblem( 5 )anditsoptimalsolution consistsofatmostNichordsandtherefore( 4 )holds.ForNi3,werstcomputeanupperboundontheshortesttotallengthofNi)]TJ /F4 11.955 Tf 12.45 0 Td[(1chordsthatinterconnectpi1andtheremainingmiddlepoints.Thisboundisgivenbytheoptimalobjectivefunctionvalueofthefollowingproblem.MaxNi)]TJ /F13 7.97 Tf 6.59 0 Td[(1Xk=12risink 2 (4)s.t.Ni)]TJ /F13 7.97 Tf 6.59 0 Td[(1Xk=1ki (4)k0,8k=1,...,Ni)]TJ /F4 11.955 Tf 11.96 .01 Td[(1. (4)Sincei<,theobjectivefunctionisconcaveandsotheuniqueoptimalsolutiontotheaboveproblemis1==Ni)]TJ /F13 7.97 Tf 6.59 0 Td[(1=i=(Ni)]TJ /F4 11.955 Tf 12.82 0 Td[(1),andthemaximumlengthoftheseNi)]TJ /F4 11.955 Tf 12.76 0 Td[(1chordsisgivenby2ri(Ni)]TJ /F4 11.955 Tf 12.76 0 Td[(1)sin(i=2(Ni)]TJ /F4 11.955 Tf 11.95 0 Td[(1)).However,wepossiblyneedadditionalchordstocompletethisconnectingtrajectory.Whenpi1isanendpointoftheunderlyingarc,theconnectingsectionstartsfrompi1,visitsallthemiddlepoints,andthentraversestopi2.Anupperboundonthelengthofthetrajectoryinthiscaseisgivenby2risin(i=2)+2ri(Ni)]TJ /F4 11.955 Tf 12.01 0 Td[(1)sin(i=2(Ni)]TJ /F4 11.955 Tf 11.96 0 Td[(1)).Nowsupposepi1isnotanendpointoftheunderlyingarc,andletP1betheendpointthatisclosertopi1(comparedtopi2) 72

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andP2betheotherendpoint.Onepossiblewaytoconnectpi1andpi2istostartfrompi1,visitallthemiddlepointsonthewaytoP1,comebacktopi1,visittheremainingmiddlepoints,andnallymovetopi2.Inthiscase,twochords(P1pi1,andP2pi2)areaddedtothepreviousNi)]TJ /F4 11.955 Tf 12.83 0 Td[(1chords.Let1and2bethecentralanglescorrespondingtothetwonewchords.Because1+2i,thetotallengthofthesetwonewchordscannotexceed4risin(i=4)(whichisgreaterthan2risin(i=2)forallvaluesofi).Hence,anupperboundonthelengthoftheshortestconnectingtrajectoryisgivenby4risin(i=4)+2ri(Ni)]TJ /F4 11.955 Tf 11.96 0 Td[(1)sin(i=2(Ni)]TJ /F4 11.955 Tf 11.95 0 Td[(1)).Nowconsideragrid-basedCETSP-partitioning.Theworst-casescenarioforeachvalueofNioccurswhenpi1existsatavertexofthegridcell.WhenNi=1,thelengthofaminimalconnectingsegmentisclearlynomorethan2p w1i2+w2i2,whichhappenswhenpi1=pi2butadiagonalofthegridcellneedstobetraversedoncetocoveratarget,andoncetoreturntopi2.WhenNi=2,theconnectingtrajectoryconsistsofthreeconnectedlinesegments.Itcanbeveriedthatthemaximumlengthoftheminimaltrajectoryintheunderlyingrectangleisgivenbyw1i+w2i+p w1i2+w2i2,whichhappenswhenpi1=pi2andtwooftheothergridverticesneedtobevisitedtocoverthecorrespondingtargets.WhenNi=3,themaximumlengthhappenswhenpi1=pi2andtheotherthreegridverticesoftherectangleneedtobevisitedtocoverallthetargets.(ThedetailsshowingthatthesituationsaboveareindeedtheworstcaseswhenNi=2or3areintuitiveandareomittedforbrevity.)Finally,whenNi4,onecanensurethatallNitargetsarecoveredbystartingfrompi1,traversingtheperimeteroftherectangleandthenmovingtopi2.(NointeriorpointofthegridcellneedstobevisitedbyourassumptionthatnoSmiscontainedwithintheinteriorofanycell.)Thelengthoftheconnectingtrajectoryinthiscaseisboundedby2(w1i+w2i)+p w1i2+w2i2. Remark4.1.Notethattheupperboundsin( 4 )areobtainedbyperformingaworst-caseanalysis.Inpractice,onecandesignapostprocessingsubroutinethatusesthesolutionof( 5 )aswellastheproblem'sgeometrytobuildafeasibleCETSPtour, 73

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whichwouldtypicallyyieldasmallerupperboundthantheonegiveninProposition 4.3 .Aheuristicalgorithm(suchasthosein[ 51 ]and[ 13 ])canalsobeusedforthispurpose.Alternatively,onecanobtainanupperboundontheoptimalCETSPtourlengthbysolving( 5 )overasetPofgridpoints,wherep02PandP\Sm6=;,8m2M,withrespecttotheEuclideandistancebetweenthecorrespondingpoints.2Proposition 4.3 impliesthatthesetoflinksinanoptimalsolutionto( 5 )canbeusedtoformafeasibleCETSPtourwhoselengthcanbecomearbitrarilyclosetotheoptimalCETSPtourlength.Tothatend,onecanrenetheunderlyingpartitioninginsuchawaythatthegridcellsizesbelongingtoIaresmallenoughtoyieldanacceptablysmalloptimalitygap.Onepossiblestrategyistosolve( 5 )andobtainI,subdivideallcellsinIintosmallercells,andsolvetherevisedinstanceof( 5 )inarepeatedfashion. 4.3.2Cutting-PlaneGenerationSubtourEliminationConstraints.Becausethereareanexponentialnumberofsubtoureliminationconstraints( 4e ),weinitiallyrelaxtheseconstraintsandaddthosethatareviolated(withrespecttoapre-determinedviolationthreshold)ateachnodeofthebranch-and-boundtree.Here,wediscussthecorrespondingseparationprocedures.Tondasubtoureliminationconstraint( 4e )thatisviolatedbyasolution( x, y),weuseawellknownseparationprocedure(see[ 52 53 ])asfollows.Deneacompletedirectedgraphwiththenodesetf0,...,nginwhichthecapacityofanarc(i,j)equals xij.Foreachv2f1,...,ngwith yv>0,wendthemaximumowfrom0tov,whichwillgenerateaminimumcut( S,S)where02 Sandv2S.Ifthecapacityofthiscutislessthan yv,thatis,ifPi2 SPj2S xij< yv,thenSandvdeneaviolatedinequality( 4e ).Whenthisproceduredoesnotndaviolatedsubtourinequality,( x, y)satisesallthesubtoureliminationconstraints( 4e ).ModelTighteningInequalities.Recallthattheobjectivefunctioncoefcientsin( 5 )arepairwiseminimumdistancesofthecorrespondingcells.Ingeneral,the 74

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triangleinequalitydoesnotholdforthesecostcoefcients,whichmayresultinaweaklowerbound.Toillustratethispoint,notethatthereexistsanoptimalCETSPtourinwhichnotwoturnpointscoveranidenticalsetoftargets(orotherwise,bypassingeitherofthemresultsinaCETSPtourthatisnolongerthantheoriginalone).However,itispossiblethattwocellsonthetourcoverthesamesetoftargetsinauniqueoptimalsolutionto( 5 ).ConsidertheCETSPinstanceinFigure 4-7 ,whereC1isthearcfrompoint1to2,andC2isthearcfrompoint2to3.Here,x12=1atoptimality.ThisoptimalsolutioncontainsbothC1andC2,whichcoverthesametarget,andallowsthetourtogofrom1to3atadistanceofzero(notingthatl12=0becausepoint2belongstoC1andC2).Thefollowingproposition,whichisageneralizationofsimilarinequalitiesfortheCTP[ 42 ],canbeaddedto( 5 )tostrengthentheobtainedbound. Proposition4.4. ConsidertwononemptysubsetsC1andC2ofCsuchthatC1\C2=;.DeneV(Cp)=[Ci2Cpfm2M:i2N(m)g, (4)forp=1,2.IfV(C1)V(C2),thenthefollowinginequalityisvalid:Xi2C1[C2yiminfjC1j+jC2j)]TJ /F4 11.955 Tf 17.93 0 Td[(1,jV(C2)jg, (4)inthesensethatsolvingmodel( 5 )augmentedwith( 4 )stillyieldsavalidlowerboundontheoptimalCETSPtourlength. Proof. TocoverallthetargetlocationsinV(C2),ashortestCETSPtourneedsatmostjV(C2)jturnpoints.Moreover,anoptimalCETSPtourwillnotcontainaturnpointineverycellofC1[C2becauseinthatcase,ashorterfeasibletourcanbeconstructedbyexcludinganyoftheturnpointsinC1.Asaresult,(xR,yR)intheproofofProposition 4.4 satises( 4 ),whichcompletestheproof. 75

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Remark4.2.InProposition 4.4 ,( 4 )canbestrengthenedinthefollowingformwhenjC1j2,jC2j2,andV(C1)=V(C2):Xi2C1[C2yiminfjC1j+jC2j)]TJ /F4 11.955 Tf 17.94 0 Td[(2,jV(C2)jg. (4)ThereductioninthersttermisduetothefactthatifanoptimalCETSPtourcontainsaturnpointineverycellinC1(orC2),thennocellsinC2(orC1)arevisitedbythisoptimaltourorotherwise,omittingthoseturnpointsinC2(orC1)resultsinashortertour.SincejC1jandjC2jarebothatleasttwo,(xR,yR)intheproofofProposition 4.4 satises( 4 )andsotheinequalityobtainsavalidboundinthiscase.Else,thereexistsatleastonecellinbothC1andC2thatdoesnotcontainanyturnpointofanoptimalCETSPtour,and( 4 )holdsinthiscaseaswell.2Givena(possiblyfractional)vector y,aCETSP-partitioningC,asetMoftargets,andthecoveringsetVi=V(fCig)foreachCi2C,theseparationproblemof( 4 )seekstondnonemptydisjointsubsetsC1andC2ofCsuchthatV(C1)V(C2)andPi2C1[C2 yi>jC1j+jC2j)]TJ /F4 11.955 Tf 18.51 0 Td[(1.LetthisdecisionproblembedenotedbySP1.WeprovethatSP1isstronglyNP-completebyareductionfromtheexactcoverby3-setsproblem(X3C),denedasfollows[ 54 ]. Denition4.2.ProblemX3C:LetG=fg1,...,g3qgandconsideracollectionF=fF1,...,Fpgof3-elementsubsetsofG.IsthereasubsetF0FsuchthateveryelementofGoccursinexactlyonememberofF0? Theorem4.1. ProblemSP1isstronglyNP-complete. Proof. SP1clearlybelongstoNP,becauseaninputguesscanbeveriedtosatisfytheSP1conditionsinO(n)time.ToshowthatSP1isNP-complete,wereduceanyarbitraryinstance(G,F)ofX3CintoaninstanceofSP1asfollows.LetM=fm1,...,m3q+pgandC=fC0,...,Cp+1g.ForeachCi,i=1,...,p,deneVi=fmj:gj2Fig[fm3q+igandletVp+1=fm1,...,m3qg.Also,let yi=q+ q+1fori=1,...,p+1,where0<<1 q+2.We 76

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assertthatthereexistsanexactcoverby3-setsofGifandonlyifthereexistsubsetsC1andC2ofCthatsolvetheaboveinstanceofSP1.=)SupposethereexistsasubsetF0FsuchthatjF0j=qandeveryelementofGoccursinexactlyonememberofF0.LetC1=fCp+1gandC2=fCi2C:Fi2F0g.Then,Xi2C1[C2 yi)-222(jC1j)-223(jC2j=(q+1)q+ q+1)]TJ /F3 11.955 Tf 11.96 0 Td[(q)]TJ /F4 11.955 Tf 11.95 0 Td[(1 (4)=)]TJ /F4 11.955 Tf 11.96 0 Td[(1>)]TJ /F4 11.955 Tf 9.3 0 Td[(1. (4)(=SupposethatthereexistdisjointsubsetsC1andC2thatsolveSP1.ThenwehavethatPi2C1[C2 yi>jC1j+jC2j)]TJ /F4 11.955 Tf 19.32 0 Td[(1,i.e.Pi2C1[C21)]TJ /F17 7.97 Tf 6.59 0 Td[( q+1<1.NotethatC1cannotcontainanyofC1,...,CporotherwiseV(C1)*V(C2).Therefore,C1=fCp+1gandC2mustcoverallelementsinfm1,...,m3qg,whichrequiresjC2jtobeatleastq.Hencewehavethat(jC2j+1)1)]TJ /F17 7.97 Tf 6.59 0 Td[( q+1<1,orequivalently,jC2j
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Also,let yi=q+1+ q+3fori=1,...,p+3,where0<<2 q+4.Thereexistsanexactcoverby3-setsofGifandonlyifthereexisttwonon-emptydisjointsubsetsC1andC2ofCthatsolvetheaboveinstanceofSP2,whereC1=fCp+1,Cp+3gandC2consistsofCp+2andtheelementsofCcorrespondingtoanelementFiinanexactcoverby3-setsforG.=)SupposethereexistsasubsetF0FsuchthatjF0j=qandeveryelementofGoccursinexactlyonememberofF0.LetC1=fCp+1,Cp+3gandC2=fCi2C:Fi2F0g[fCp+2g.Then,Xi2C1[C2 yi)-222(jC1j)-222(jC2j=(q+3))]TJ /F4 11.955 Tf 9.29 0 Td[(2+ q+3>)]TJ /F4 11.955 Tf 9.3 0 Td[(2. (4)(=SupposethereexistdisjointsubsetsC1andC2ofCsuchthatjC1j2,jC2j2,V(C1)=V(C2)andPi2C1[C2 yi>jC1j+jC2j)]TJ /F4 11.955 Tf 18.44 0 Td[(2.ThelatterconditionisequivalenttoPi2C1[C22)]TJ /F17 7.97 Tf 6.59 0 Td[( q+3<2.OnepossibilityforsetsC1andC2thatsatisesjC1j2,jC2j2,andV(C1)=V(C2)isgivenbyC1=fCp+1,Cp+2gandC2=fC1,...,Cpg[fCp+3g.However,thissolutiondoesnotviolate( 4 )unlesswhenp=q,inwhichcasetheanswertotheX3Cinstanceistrivial.TheonlyotherchoicesforC1andC2thatsatisfytheaboveconditionsarethoseinwhichC1=fCp+1,Cp+3gandCp+22C2(orviceversa).Therefore,C2nfCp+2gcoversallelementsoffm1,...,m3qg,andsojC2jq+1.Hence,(jC2j+2)2)]TJ /F17 7.97 Tf 6.59 0 Td[( q+3<2,orequivalently,jC2j<2q+2+2 2)]TJ /F17 7.97 Tf 6.58 0 Td[(.Because<2 q+4,wehavethatjC2j
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modelinSection 4.4.2 .Wethenformulateathirdlower-boundingmodelinSection 4.4.3 ,anddemonstratethatthismodelissolvablebyaBendersdecompositionstrategy. 4.4.1ExpandedFormulationAnotherwayofreducingtheconservativenessofformulation( 5 )istoredenethedecisionvariablessothattheycapturesomeofthetraveldistancesinsidethecells,andhence,contributetoatighterlowerbound.Tothatend,weneedtoaddsequence-relatedbinaryvariablestotheproblem.Letsijk=xijxjkbeabinaryvariablethatequalsoneifandonlyifthesolutiontothelowerboundproblemconsecutivelyvisitsCi,Cj,andCk.Moreover,leteijk(lij+ljk)denotethelengthofashortestpaththatgoesfromapointinCitosomepointinCj,andthenfromthesamepointinCjtoapointinCk.(Section 4.5 discussesdetailspertainingtothecalculationoftheseeijk-values.)Figure 4-8 illustratesthedenitionofeijkforanarc-basedpartitioningscheme.Considerthefollowingintegerprogrammingproblem. Min1 2nXi=0nXj=0nXk=0eijksijk (4a)s.t.nXl=0slij=nXk=0sijk,8i=0,...,n,j=0,...,n (4b)yj=nXi=0nXk=0sijk,8j=0,...,n (4c)Xi2N(m)yi1,8m2M (4d)Xi2SXj=2SnXk=0sijkyv,8Sf1,...,ng:3jSjjCj)]TJ /F4 11.955 Tf 17.93 0 Td[(3andv2S (4e)sijk2f0,1g,8i=0,...,n,j=0,...,n,k=0,...,n (4f)0yi1,8i=1,...,n;y0=1. (4g) Proposition4.5. LetlbetheoptimalCETSPtourlengthandsupposezLB1andzLB2aretheoptimalobjectivefunctionvaluesofproblems( 5 )and( 4 ),respectively. 79

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Figure4-8. Illustrationoftheobjectivefunctioncoefcientseijkin( 4 ) Then,zLB1zLB2l. (4) Proof. SupposeagainthatanoptimalCETSPtourTvisits(inorder)thesetofturnpoints(p0,...,pk).SimilartotheproofofProposition 4.3 ,letC0t=Cit2CbeanelementoftheCETSP-partitioningthatcontainspt,fort=1,...,kanddeneC0k+1=C00=C0andik+1=i0=0.Consider(xR,yR)asdenedintheproofofProposition 4.3 andletsRijk=xRijxRjk,forall0i,j,kn.Notethat(sR,yR)denesafeasiblesolutionto( 4 ).LetzRLB1andzRLB2denotetheobjectivefunctionvalueof(xR,yR)and(sR,yR),respectively.Bydenition,ifsRit,iu,iv=1,wehavelit,iu+liu,iveit,iu,ivv)]TJ /F13 7.97 Tf 6.59 0 Td[(1Xj=tjpjpj+1j, (4)wherepk+1=p0.Aggregatingalloftheaboveinequalitiesyields2zRLB12zRLB22l.Therefore,zLB2zRLB2l.Nowlet(^s,^y)beanyfeasiblesolutionto( 4 ).Correspondingto(^s,^y),wedeneauniquefeasiblesolution(^x,^y)to( 5 ),where^xij=Pnk=1^sijk,forall0i,jn.Usinglij+ljkeijk,itisstraightforwardtoshowthat 80

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^zLB1^zLB2,where^zLB1and^zLB2denotetheobjectivefunctionvalueof(^x,^y)and(^s,^y),respectively.Therefore,zLB1zLB2,whichconcludestheproof. Proposition4.6. DenotebyP1andP2theLPrelaxationpolytopesofformulation( 5 )andformulation( 4 ),respectively.Then,thereexistsamappingtsuchthatP1=projt(P2),whereprojt(P2)=f(x,y):(x,y)=t(x,s)forsome(x,s)2P2g. Proof. Supposetisdenedtomapanypoint(s,y)inP2ontoapoint(x,y)suchthatxij=Pnk=0sijk.Itisthenstraightforwardtocheckthatanypoint(x,y)inthecodomainoftsatisesalltheconstraintsof( 5 ).Therefore,projt(P2)P1.WenextprovethatP1projt(P2).Let(x,y)representasolutioninP1.Toshowthattheabovesolutionisinprojt(P2),webuildavectorssuchthat(s,y)isfeasibletotheLPrelaxationof( 4 )andxij=Pnk=0sijkholdtrueforeachpairiandj.ForeachCj2C,deneAj=fi:xij>0gandBj=fk:xjk>0g.Takei2Ajandk2Bjandletsijk=minfxij,xjkg.Thenrevisethevaluesofxijandxjkbysubtractingsijkfromthem.Notethatafterthisstep,atleastoneofxijandxjkwillbecomezero,inwhichcaseweeliminateeitheriorkfromthecorrespondingsetAjorBjandperformtheaboveprocedureagain.WecancontinuethisprocedureuntilbothsetsAjandBjareempty,whichwilleventuallyhappensincePixij=Pkxjk.Thisprocedureresultsinvectorssandythatdeneafeasiblesolutionto( 4 ).Thisistruebecause Pnl=0slij=xij=Pnk=0sijk,byconstruction; yj=Pi2Cxij=Pi2CPk2Csijk;and BecausePnk=0sijk=xij,sandysatisfythesubtoureliminationconstraints( 4e ).Thiscompletestheproof. 4.4.2SubtourEliminationToseparate( 4e ),wecanusethesameprocedurespeciedfortheseparationof( 4e ),withthemodicationofsettingthecapacityofarc(i,j)asPnk=0 sijk.Forthe 81

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expandedmodel,though,wealsoconsideranalternativeformofsubtoureliminationconstraintsasfollows. Proposition4.7. Considerthefollowingsetofsubtoureliminationconstraintsfor( 4 ).Xi2SXj2SXk2Ssijk)]TJ /F11 11.955 Tf 11.95 11.35 Td[(Xi=2SXk=2SsivkXj2Snfvgyj)]TJ /F3 11.955 Tf 11.96 0 Td[(yv,8Sf1,...,ng:3jSjjCj)]TJ /F4 11.955 Tf 17.94 0 Td[(3andv2S. (4)LetY1=f(s,y):(s,y)isfeasibleto( 4 )g,Y2=f(s,y):(s,y)isfeasibleto( 4 )with( 4e )substitutedby( 4 )g.Moreover,let Y1betheLPrelaxationpolytopeof( 4 )anddene Y2astheLPrelaxationpolytopeof( 4 )with( 4 )insteadof( 4e ).Then,Y1=Y2and Y2 Y1. Proof. Werstprovethat Y2 Y1byshowingthatevery(possiblyfractional)solutionto Y2satises( 4e )aswell.Byrearranging( 4 )weobtainXi2SXj2SXk2Ssijk+2yvXj2Syj+Xi=2SXk=2Ssivk. (4)NotethatPi=2SPk=2SsivkPi=2SPj2SPk=2SsijkandPj2Syj=Pni=0Pj2SPnk=0sijk.Therefore,wehave2yvnXi=0Xj2SnXk=0sijk+Xi=2SXj2SXk=2Ssijk)]TJ /F11 11.955 Tf 11.96 11.36 Td[(Xi2SXj2SXk2Ssijk. (4)Theabovecanberestatedas 2yvXi=2SXj2SXk2Ssijk+Xi2SXj2SXk=2Ssijk+2Xi=2SXj2SXk=2Ssijk (4a)=Xi=2SXj2SnXk=0sijk+nXi=0Xj2SXk=2Ssijk. (4b) 82

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Notethatthersttermof( 4b )representsthenumberoftimesa(possiblyfractional)tourenterssetS,andthesecondtermrepresentsthenumberoftimesatourexitssetS.By( 4b ),thesevaluesmustbeequal,andso( 4b )reducesto2Pni=0Pj2SPk=2Ssijk.ButsincenXi=0Xj2SXk=2Ssijk=Xi2SXj=2SnXk=0sijk, (4)wehavethat( 4 )implies( 4e ).WenowprovethatY1=Y2.Weshowedabovethatifasolutionsatises( 4 ),italsosatises( 4e ).Therefore,Y2Y1.Nowconsideranintegersolution( s, y)inY1,whichrepresentsatouroverasubsetofCthatintersectsp0.Weshowthat( s, y)satises( 4 )aswell.SupposeSCischosensuchthat0=2Sandv2S.Considerthefollowingcases. Pi2SPj2SPk2S sijk1:NotethatXj2S yj=Xi2SXj2SXk2S sijk+Xi=2SXj2SXk2S sijk+Xi2SXj2SXk=2S sijk+Xi=2SXj2SXk=2S sijk. (4)Moreover,wehaveXi=2SXj2SXk2S sijk1 (4)Xi2SXj2SXk=2S sijk1 (4)Xi=2SXj2SXk=2S sijk0. (4)Therefore,Xi2SXj2SXk2S sijkXj2S yj)]TJ /F4 11.955 Tf 11.96 0 Td[(2Xj2S yj)]TJ /F4 11.955 Tf 11.96 0 Td[(2yv. (4)and( 4 )holds. Pi2SPj2SPk2S sijk=0and yv=0:( 4 )clearlyholdsinthiscase. Pi2SPj2SPk2S sijk=0and yv=1:Inthiscase,ifPj2Snfvgyj1then( 4 )isvalid.IfPj2Snfvgyj=0,thenPi=2SPk=2Ssivk=1,and( 4 )remainsvalid. 83

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NotethatsinceY1=Y2,thesameseparationprocedurethatgenerates( 4e )canalsobeusedtogenerate( 4 ).Weconcludethissectionbypresentingstrongersubtoureliminationconstraintsfor( 4 ).SupposeSf1,...,ngandlet S=f0,...,ngnS.DeneeM(S)=Mnfm2M: S\N(m)6=;g,i.e.,eM(S)isthesetoftargetsnotcoveredbyacellin S.LeteC(S)beasmallestcardinalitysubsetofSthatcoversalltargetsineM(S).ThenXi2SyijeC(S)j (4)isvalidfor( 4 ).IfjeC(S)j1,then( 4 )impliesthevalidityofthefollowingstrongersubtoureliminationconstraintfor( 4 ):Xi=2SXj2SnXk=0sijk1. (4)Notethat( 4 )isamodiedversionofsimilarconnectivityinequalitiesfortheCTP[ 42 ].WhenjeC(S)j2,( 4 )canbefurtherstrengthenedasfollows.Xi=2SXj2SXk2Ssijk+Xi2SXj2SXk=2Ssijk+Xi=2SXj2SXk=2Ssijk2. (4)Therefore,whenaninequality( 4 )isgeneratedinourseparationroutine,werstchecktoseeifcanbestrengthenedintotheform( 4 )or( 4 )beforeaddingthecuttothemodel. 4.4.3AlternativeFormulationFormulations( 5 )and( 4 )containO(n2)andO(n3)binaryvariables,respectively,whichpossiblymakesthemintractableforCETSP-partitioningshavingalargenumberofcells.WedevelopanalternativeMIPformulationwhosenumberofbinaryvariablesdoesnotdependonn. 84

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Notethatanyfeasiblesolutionto( 5 )canbecharacterizedbytwosetsofdecisions:(a)theorderinwhichtheneighborhoodsetsSmarevisited,and(b)foreachSm,whichcellrepresentstargetmonthetour.Givenanyparticularorderofvisitingtheneighborhoodsets,theproblemofoptimallyidentifyingarepresentativecellforeachtargetcanbesolvedbysolvingashortestpathproblemasfollows.ForaCETSP-partitioningC=fC0,...,CngdenesetQasQ=f(i,m):Ci2C,m2M,i2N(m)g[f ,g, (4)where andrespectivelyserveassourceanddestinationnodesinthegraph.Anorderedpair(i,m)belongstoQforeachcellCi2Candeverym2MsuchthatCi\Sm6=;.Forconvenience,wealsodescribetheelementsofQbyGreekletters,withcomponents^i()and^m(),where^i( )=^i()=0and^m( )=^m()=p0.ThedistancebetweenanytwoelementsandinQisdenedasd=l^i(),^i().Letulmbeabinaryvariablethatindicateswhethertargetlisvisitedimmediatelybefore(orsimultaneouslywith)targetm.Weseekashortesttourthatstartsfromthedepot,visitsacellineveryneighborhoodset(intheorderdeterminedbytheu-variables),andreturnstothedepot.Thisisequivalenttondingashortestpath(withrespecttoarccostsd)from toinagraphG(u)=(Q,A(u)),whereA(u)=f(,):,2Q,andu^m(),^m()=1g.Tomodelthisshortestpath,weintroducenonnegativeowvariablesfalongwiththenecessaryow-balanceconstraints.Thecorrespondingformulationisstatedbelow,whereM0=M[fp0g. MinX2QX2Qdf (4a)s.t.Xm2M0uml=Xm2M0ulm=18l2M0 (4b)Xl2SXm2SulmjSj)]TJ /F4 11.955 Tf 42.01 0 Td[(1,8SM0,2jSjjM0j)]TJ /F4 11.955 Tf 17.94 0 Td[(2 (4c)fu^m(),^m()82Q,2Q (4d) 85

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X2Qf)]TJ /F11 11.955 Tf 12.31 11.36 Td[(X2Qf=0,82Qnf ,g (4e)X2Qf=1, (4f)f0,82Q,2Q (4g)ulm2f0,1g,8l2M0,m2M0. (4h)Intheaboveformulation,constraints( 4b ),( 4c ),and( 4h )imposeaTSPtouroversetM0,while( 4d )( 4g )modelthecorrespondingCETSPtourasashortestpathfrom toinG(u).NotethatwhenjC1j=jC2j=1,anequivalentformof( 4 )canbeappliedtoformulation( 4 )byaddingf=f=0tothemodel,where^i()2C1and^i()2C2. Proposition4.8. SupposethattheoptimalCETSPtourlengthisl,andconsideranoptimalsolution(u,f)to( 4 )withobjectivefunctionvaluezLB3.DenethesetofturncellsasI0=f2Qnf ,g:P2Qf=1g.Then,zLB3lzLB3+X2I0h0, (4)whereh0=8><>:2r^i()sin^i() 2ifC^i()S^m(),4r^i()sin^i() 2ifC^i()*S^m(), (4)foranarc-basedCETSP-partitioning,andh0=2q w1,^i()2+w2,^i()2, (4)foragrid-basedCETSP-partitioning. Proof. ConsideranoptimalCETSPtourcharacterizedbyanorderedsetofturnpoints(p0,...,pk)withacorrespondingsolution(xR,yR)asconstructedintheproofofProposition 4.3 .SupposethatthissolutioncorrespondstoatourC0!...! 86

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Cr!C0.Foreachi=1,...,rdeneMi=fm2M:m=2[i)]TJ /F13 7.97 Tf 6.59 0 Td[(1h=1Mhand9j2f1,...,kgsuchthatpj2Sm\Cig.NotethatMi6=;foralli=1,...,rorotherwise,ashorterCETSPtourcanbeobtainedbyexcludingallturnpointsinCifromT.NowsupposeMi=fmi1,...,mijMijg,8i=1,...,r,andconsiderthefollowingfeasiblesolutionto( 4 ). f ,(1,m11)=1, f(i,mij),(i,mij+1)=1forall1ir,j=1,...,jMij)]TJ /F4 11.955 Tf 17.93 0 Td[(1, f(i,mijMij),(i+1,mi+11)=1for1ir)]TJ /F4 11.955 Tf 11.95 0 Td[(1, f(r,mrjMrj),=1, ulm=1forl,m2M0,ifandonlyifl6=mandthereexistandinQsuchthat^m()=l,^m()=m,andf=1.Theobjectivefunctionvalueofthissolutionequalsthatof(xR,yR).Therefore,zLB3zRl.ValidityoftheupperboundfollowsfromasimilardiscussiontothatofProposition 4.3 ,notingthatinthiscaseeveryvisittoCirequiresavisittoatmostonemiddlepoint. Proposition 4.8 impliesthattheoptimalobjectivefunctionvalueof( 4 )canbeusedasalowerboundontheoptimalCETSPtourlength.NotethatregardlessofthesizenoftheunderlyingCETSP-partitioning,problem( 4 )containsO(jMj2)binaryvariables.WenextdescribeaBendersdecompositionalgorithmforsolving( 4 ).Tothatend,suppose dlmequalstheminimumdistancebetweentwoneighborhoodsetsSlandSm,forl,m2M0.Formulation( 4 )canbeequivalentlystatedas MinXl2M0Xm2M0 dlmulm+X2QX2Q(d)]TJ ET q .478 w 302.43 -560.98 m 309.76 -560.98 l S Q BT /F3 11.955 Tf 302.43 -570.95 Td[(d^m(),^m())f (4a)s.t.Constraints( 4b )( 4h ). (4b) 87

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Wereformulate( 4 )as: MinXl2M0Xm2M0 dlmulm+(u) (4a)s.t.Constraints( 4b ),( 4c ),and( 4h ), (4b)where (u)=MinX2QX2Q(d)]TJ ET q .478 w 266.92 -146.15 m 274.25 -146.15 l S Q BT /F3 11.955 Tf 266.92 -156.12 Td[(d^m(),^m())f (4a)s.t.Constraints( 4d )( 4g ). (4b)IntheBendersdecompositionframework,welet( 4 )serveasamasterprobleminwhich(u)isreplacedbyavaluefunctionvariable.Then,givenaxedvalueofu,( 4 )servesasthesubproblem.Solving( 4 )eitherprovestheoptimalityofu,oryieldsaBenders(optimality)inequality.Given u,deneasajQj-vectorthatcontainsthedualvaluesassociatedwiththisshortestpathproblem,i.e.,denotesthedualvalueassociatedwithconstraint( 4e )for2Qnf ,g,isthedualvalueassociatedwithconstraint( 4f ),and 0.Suchdualvaluesindicatethelengthofashortestpathfrom toanyothernodeinG( u).Forany2Qand2Q,dene)]TJ /F10 11.955 Tf 9.3 0 Td[(asthedualvalueassociatedwithconstraint( 4d ).Hence,=maxf0,)]TJ /F11 11.955 Tf 11.95 9.68 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(d)]TJ ET q .478 w 133.25 -434.08 m 140.58 -434.08 l S Q BT /F3 11.955 Tf 133.25 -444.06 Td[(d^m(),^m())]TJ /F10 11.955 Tf 11.95 0 Td[(g,andletlm=X2Q:^m()=lX2Q:^m()=m. (4)Notethatlmisnonnegativeandcantakepositivevaluesonlyfor(l,m)2U0=f(l,m):l2M0,m2M0,and ulm=0g.Hence,thecorrespondingBendersinequalitycanbewrittenas+X(l,m)2U0 lmulm, (4) 88

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where lm=minflm,g.Thesereducedcoefcientsarevalidduetothenonnegativityoflm,8(l,m)2U0,andthefactthat0.Thisconceptcanbeusedtofurthertighten( 4 ).Notingthat( 4 )isvalidatu= u,regardlessofthecoefcientsoftheu-variablesinU0,weneedtoensurethat( 4 )isvalidforu06= u.Notethatu0containsatleasttwovariablesinU0thatequalone.Let=min(l,m)2U0f lmg.SupposeU01denotesasubsetofU0whoseelementssatisfy lm())]TJ /F10 11.955 Tf 13.28 0 Td[(,anddeneU02=U0nU01.If 2,then( 4 )canbestrengthenedas+X(l,m)2U01()]TJ /F10 11.955 Tf 11.96 0 Td[()ulm+X(l,m)2U02lmulm; (4)otherwise,thefollowinginequalityisvalidfor( 4 ).+X(l,m)2U0 2ulm. (4) 4.5CalculationofDistanceValuesWedescribehowtocalculatethecostcoefcientslijandeijkforthecellsofagivenCETSP-partitioning.TwocellsCiandCjaresaidtoberegularlyplacedwithrespecttoeachotherifbothofthefollowingconditionshold: 1. ai+w1iajoraiaj+w1j,and 2. bi+w2ibjorbibj+w2j.IfCiandCjarenotregularlyplacedwithrespecttoeachotherandaj)]TJ /F3 11.955 Tf 9.44 0 Td[(w1i
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Nowlet(v1v2)betheshortestlinesegmentconnectingtworegularlyplacedcellsCiandCjand(v3v4)betheshortestlinesegmentconnectingcellsCjandCk,whicharealsoregularlyplacedwithrespecttoeachother.Hence,lij=jv1v2jandljk=jv3v4j.Toevaluateeijk,weconsiderthefollowingthreecases. Case1.:v2andv3arethesamevertexofCj.Inthiscase,eijk=lij+ljk. Case2.:v2andv3areonthesameedgeofCj.Letvi=(!1i,!2i)fori=1,...,4.Sincev2andv3areonthesamesideofcellCj,either!12=!13or!22=!23.Withoutlossofgenerality,suppose!22=!23= !.TheminimumdistancefromanypointinCitoanypointinCjandthentoanypointinCkisobtainedbygoingfromv1tosomepoint(, !)onthelinesegmentbetweenv2andv3andfromtheretov4.Therefore,thecorrespondingoptimizationproblemisasfollows. min!12!13)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(()]TJ /F10 11.955 Tf 11.96 0 Td[(!11)2+( !)]TJ /F10 11.955 Tf 11.96 0 Td[(!21)21 2+)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(()]TJ /F10 11.955 Tf 11.96 0 Td[(!14)2+( !)]TJ /F10 11.955 Tf 11.96 0 Td[(!24)21 2(4)Sincetheobjectivefunctionisstrictlyconvex,theoptimalvalueofiseither!12,!13,orwhereistheunconstrainedminimizerof( 4 ),suchthat )]TJ /F10 11.955 Tf 11.96 0 Td[(!11 !14)]TJ /F10 11.955 Tf 11.95 0 Td[(= !)]TJ /F10 11.955 Tf 11.96 0 Td[(!21 !)]TJ /F10 11.955 Tf 11.96 0 Td[(!24,assuming !6=!24. Case3.:v2andv3arediagonallyoppositeeachotherinCj.Inthiscase,ifthelinesegment(v1,v4)passesthroughCj,theneijk=jv1v4j.Otherwise,e=jv1v0j+jv0v4j,wherev0isoneoftwoverticesofCjotherthanv2andv3(whicheverisaminimizerofdistance),becausethispathdominatesallotherpathsthatdonotpassthroughavertexofCj.Therefore,calculatingeijk-coefcientscanbedoneinO(1)time.Unlikethegrid-basedCETSP-partitioning,computingobjectivefunctioncoefcientsforanarc-basedCETSP-partitioningdoesnotappeartobeeasy.Thisisbecausethecorrespondingdistanceminimizationproblemingeneralisnotconvex.Onewaytodealwiththisdifcultyistonumericallycalculatealltheminimumdistancesbetweenanytwoarcspriortosolvingthemathematicalprogrammingproblems.Wecanalso 90

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Figure4-9. Obtainingalowerboundonthedistancebetweentwoarcs useaneasily-computablevalidlowerboundontheminimumdistance.Toobtainthislowerbound,onecancalculatetheminimumdistancebetweentwominimaltriangles(outer-approximations)thatcontainthetwoarcs.Thisisequivalenttosolvingaconvexoptimizationproblem.Thecorrespondingtrianglescanbeobtainedusingthetangentlinesatthetwoendpoints,plusthearc'schord.Moreover,wecandivideeachofthetwoarcsintoseveralsmallerportionstoobtainacollectionofsmallertriangles,asillustratedinFigure 4-9 .Theminimumdistancebetweenanypairoftrianglesrepresentingdifferentarcsprovidesavalidlowerboundontheminimumdistancebetweenthetwoarcs.Thislowerboundcanbecomearbitrarilytightbyincreasingthenumberoftriangles. 4.6ComputationalExperimentsToexaminetheefciencyoftheproposedformulationsandalgorithms,wegeneraterandomCETSPinstancesonwhichweobtainupperandlowerboundsfortheoptimalCETSPtourlength.AllalgorithmsareimplementedinC++programminglanguageandcompiledusingMicrosoftVisualStudio2008.IntegerprogramminginstancesaresolvedusingCPLEXversion12.2viaILOGConcertTechnology2.9onaPCwithanIntelCore2QuadprocessorQ9500and4GBofmemory,runningWindows7.AllCETSPinstancesaregeneratedaccordingtothefollowingprocedure.ThejMjtargetlocationsandthedepotarechosenrandomlyonarectangleoflength16andwidth10.Theneighborhoodsetforalltargetsarediscsofidenticalradiusr(whichwevaryinourcomputationaltests).Foreachdisc,wespecifytheintersectionofitsboundaryandtheconvexhullofthetargetlocationsandthedepot,anddividethis 91

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intersectionintoN0smallerarcs,whichareinitialelementsofthepartitioningsetC.Therefore,theinitialCETSP-partitioningcontainsjMjN0+1cells.WerstprovideacomparisonbetweentheperformanceofdifferentmethodsofobtainingalowerboundontheoptimalCETSPtourlength.Forbrevity,wewillrefertoformulations( 5 ),( 4 )(withsubtoureliminationconstraints( 4 )),and( 4 )asLB1,LB2,andLB3,respectively.Thesubtoureliminationconstraintsforallthreemodelsareinitiallyrelaxed.Ateachnodeofthebranch-and-boundtree,weusetheseparationprocedureexplainedinSection 4.3 tondviolatedsubtourinequalities.Iftheviolationmagnitudeisgreaterthanorequalto0.5forasubtoureliminationconstraint,weaddthatinequalitytothelinearprogrammingrelaxationofthecorrespondingnode.TheresultsofrunningallthreemodelsoverasetoftensmalltestinstancesarereportedinTables 4-1 and 4-2 .Theseinstancescontainsixtargetlocationswiththeneighborhoodsetofeachlocationdenedasadiscofradiusr=0.25,andeacharcispartitionedintoN0=4smallerones.A1500-secondlimitisimposedontherunningtimeofallmethods.TheLowerBoundcolumnsreportthelowerboundthatisobtainedviaeachmethod,whiletheUpperBoundcolumnsreporttheupperboundprovidedby( 4 )(forLB1andLB2)and( 4 )(forLB3).Wealsoreporttheaverageabsolutegap(difference)betweenthelowerandupperboundsofallmethodsintheAvg.Gaprow.WeobservethatwhileLB2alwaysprovidesthebestlowerbound,itgenerallyrequiresasignicantlylongertimetosolvethantheothertwomethods.NotethatLB1consumesroughly30timestheCPUtimerequiredtosolveLB3,anditprovidesalowerboundthatisdominatedbythatofLB3.ThefastrunningtimeofLB3isevidentlydueatleastinparttothelownumberofsubtourcutsthatarerequiredtosolvetheproblem.Table 4-3 comparestherunningtimeofformulationsLB1andLB3forten8-and10-nodeinstancesforwhichLB2istoolargetobepracticallyuseful.Overall,theseresultsdemonstratethatLB3issignicantlyeasiertosolvethanLB1,andonceagain,thereappearstobedirectcorrelationbetweenthenumberofsubtourcutsgenerated 92

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(manyfewerforLB3thanforLB1).Thelowerboundsobtainedbythesealgorithmsarenotdisplayedinthetable,becausealllowerboundswerethesameexceptforthreeinstancesforr=0.25andsixforr=0.5.ForeightoutofthesenineinstanceswefoundthatLB3providesaslightlybetterlowerboundthanLB1.Inoneinstance,thelowerboundobtainedfromLB1wasbetterthanthatofLB3withadifferenceof0.052. Table4-1. Comparinglowerboundformulationsonsix-nodeinstances LB1LB2LB3 InstanceSubtourCutsTime(s)SubtourCutsTime(s)SubtourCutsTime(s) CETSP-6-012614.6145263.3270.8CETSP-6-0235722.7205442.3210.8CETSP-6-0344826.3195378.4141.6CETSP-6-0412715.4021.6140.5CETSP-6-0515617.5022.6121.2CETSP-6-0617320.2022.570.8CETSP-6-0723323.7216364.7100.7CETSP-6-0826921.2226429.0231.5CETSP-6-0911512.5025.9190.8CETSP-6-1049628.8372767.2280.9 Averages263.519.2135.9273.717.50.9 Table 4-4 demonstratestheefciencyoftheBendersdecompositionalgorithmforLB3insolving80randominstances.ForeachvalueofjMj,wesolvetwoinstancesassociatedwithsettingr=0.25andr=0.5.ThenumberofarcsperneighborhoodsetissettobeN0=4foreachinstance.Wepresenttherunningtime(inCPUseconds)requiredbytheBendersdecompositionalgorithm,andforsolvingformulation( 4 )directlyusingCPLEX.Wereportthenumberofaddedsubtoureliminationconstraintsforbothmethods.FortheBendersdecompositionalgorithm,wealsoreportthenumberofBendersinequalitiesneededtosolvethemasterproblemtooptimality.(NotethattheaverageCPUtimesfactorin1500secondsforthecaseinwhichaprobleminstanceisnotsolvedwithinthetimelimit.)TheresultsdemonstratethatusingBendersdecompositiondramaticallyimprovesthemodel'ssolvability.Using( 4 ),wecanalsoobtainupperboundsontheoptimalCETSPtourlengthforallinstances.Theaveragedifferencebetweentheupperandlowerboundsfordifferentinstancesizesarespeciedbelow,wheretherst,second,andthirdcomponentsdenotejMj,randtheaverage 93

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Table4-2. Lowerandupperboundsobtainedonsix-nodeinstances LB1LB2LB3 InstanceLowerBoundUpperBoundLowerBoundUpperBoundLowerBoundUpperBound CETSP-6-0133.160434.276833.499434.615833.160434.2768CETSP-6-0227.061127.856827.352128.147427.061527.8568CETSP-6-0324.704825.643925.050925.9924.704825.6439CETSP-6-0436.100937.083836.429537.412436.100937.0838CETSP-6-0521.386522.559821.597822.77121.386522.9133CETSP-6-0627.119528.138727.511428.530627.119528.1387CETSP-6-0733.945335.010234.370735.432933.94835.0102CETSP-6-0823.470924.51423.813324.856423.470924.514CETSP-6-0927.823129.024328.13729.338127.823129.0243CETSP-6-1033.674234.953334.195235.474333.674234.9533 Avg.Gap1.06141.06111.0965 94

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Table4-3. ComparingformulationsLB1andLB3 r=0.25r=0.5 LB1LB3LB1LB3 InstancejMjSubtourCutsTime(s)SubtourCutsTime(s)SubtourCutsTime(s)SubtourCutsTime(s) CETSP-8-01860325.9351.234614.7381.9CETSP-8-02870037.1492.576031.9836.5CETSP-8-0382108.1150.531411.4150.7CETSP-8-04861919.4462.71893.5343.0CETSP-8-058159664.4732.566325.0674.6CETSP-8-0682329.0331.034112.9653.7CETSP-8-07824911.7541.01965.6611.2CETSP-8-08838722.3521.291735.2601.7CETSP-8-09825412.7420.81376.1441.2CETSP-8-10837917.3362.137421.3628.3Averages522.922.843.51.5423.716.852.93.3CETSP-10-0110123192.2301.61906192.3413.7CETSP-10-021097297.1935.71698215.312212.5CETSP-10-031093852.5685.641945.07912.7CETSP-10-04101812231.5504.1100878.2514.8CETSP-10-051089073.7891.963853.316619.6CETSP-10-061068756.7591.61887227.06813.2CETSP-10-071041524.5382.556463.3386.3CETSP-10-08102598196.3507.253053.510418.4CETSP-10-091027915.4341.438524.6774.7CETSP-10-101050952.3241.230923.79018.2Averages1033.189.253.53.3934.497.683.611.4 95

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absolutegap,respectively:(14,0.25,3.80),(14,0.5,7.78),(16,0.25,4.38),(16,0.5,9.20),(18,0.25,5.20),(18,0.5,11.04),(20,0.25,5.79),and(20,0.5,12.68).Notethatafewinstanceswerenotsolvedwithinthetimelimitbyeitheralgorithm.ThelowerboundthatwereportfortheseinstancesisgivenbythelowerboundonzLB3after1500seconds.TheupperboundfortheseinstancesemploysProposition 4.8 ,withzLB3givenbytheobjectivecorrespondingtothebestfeasiblesolutionfoundforLB3after1500seconds.Figure 4-10 illustratestheconvergenceoflowerandupperboundsforaninstanceofsizejMj=12andr=0.25.Ineachiteration,weobtainlowerandupperboundsontheoptimalCETSPtourlengthusingtheBendersdecompositionalgorithmthatwasproposedinSection 4.4.3 .Figure 4-10 alsoshowsthesolutiontimeofeachiterationfortheupperandlowerboundproblems. Figure4-10. Resultsfortheiterativesolutionmethod 96

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Table4-4. PerformanceofBendersDecomposition r=0.25r=0.5 MonolithicBendersMonolithicBenders InstancejMjTime(s)GapSubtourCutsTime(s)BendersCutsSubtourCutsTime(s)GapSubtourCutsTime(s)GapBendersCutsSubtourCuts CETSP-14-011449.70.0%1482.588150353.70.0%2165.30.0%257202CETSP-14-021410.70.0%801.5219224.10.0%1602.20.0%4988CETSP-14-031410.50.0%2096.44814076.40.0%19911.70.0%223213CETSP-14-041410.00.0%1392.628154103.90.0%3604.50.0%137243CETSP-14-05145.60.0%1302.1209870.30.0%2113.30.0%69174CETSP-14-061417.20.0%1153.7714177.40.0%1832.50.0%52213CETSP-14-07145.30.0%662.486571.00.0%2512.80.0%37106CETSP-14-081421.30.0%1106.32616771.00.0%2777.40.0%242196CETSP-14-091411.80.0%942.9149721.10.0%1061.90.0%42111CETSP-14-10147.90.0%1683.42114827.10.0%1776.00.0%154174Averages15.00.0%125.93.428.1125.289.60.0%214.04.80.0%126.2172.0CETSP-16-011689.30.0%3555.0166231709.40.0%3238.30.0%294239CETSP-16-021622.70.0%2281.8181931060.80.0%6528.70.0%238302CETSP-16-031611.70.0%1536.712175111.20.0%1989.70.0%168244CETSP-16-041623.50.0%2583.025289250.30.0%46013.80.0%384460CETSP-16-051661.00.0%3113.247209577.40.0%5548.40.0%202255CETSP-16-061617.10.0%1734.047251802.60.0%48014.20.0%337334CETSP-16-071640.20.0%2727.530172213.30.0%29715.60.0%450248CETSP-16-08167.80.0%1492.0814529.30.0%1767.20.0%50196CETSP-16-0916218.00.0%26211.62032461500.973.7%1046520.10.0%5365521CETSP-16-101652.00.0%1625.271158331.40.0%35715.80.0%339199Averages54.30.0%232.35.062.7206.9558.77.4%454.362.20.0%782.7299.8CETSP-18-0118296.00.0%5568.11454581500.815.2%74535.60.0%790545CETSP-18-0218269.20.0%35312.83212651500.84.2%467202.00.0%2597413CETSP-18-031820.70.0%1829.1662301500.817.4%622102.90.0%1536550CETSP-18-041849.90.0%2012.715127382.50.0%29111.00.0%132146CETSP-18-0518121.70.0%2054.765165862.70.0%42728.00.0%483230CETSP-18-06181500.910.0%59613.42594441501.580.1%884173.90.0%2142718CETSP-18-071855.70.0%3014.239292140.40.0%28311.40.0%108211CETSP-18-081875.70.0%29322.05332951501.096.1%9121501.06.7%7068693CETSP-18-091885.00.0%2525.761198442.10.0%29438.40.0%680274CETSP-18-101831.80.0%2724.768306176.50.0%39236.20.0%506307Averages250.71.0%321.18.8157.2278.0950.921.3%531.7214.00.7%1604.2408.7CETSP-20-0120625.00.0%28611.72212301500.97.4%367556.40.0%4397656CETSP-20-022036.80.0%20911.81753201500.94.6%493135.40.0%1211609CETSP-20-032018.40.0%3398.3987721501.017.7%94841.50.0%586814CETSP-20-042036.90.0%2549.21222521501.413.6%52884.60.0%992311CETSP-20-0520709.70.0%3098.2953091501.781.4%842716.90.0%4628987CETSP-20-0620366.60.0%46415.01203061501.23.1%567177.30.0%1734707CETSP-20-072073.00.0%23815.82472511501.276.9%10711501.33.3%6872741CETSP-20-08201501.42.5%41739.46324811500.810.0%7191501.04.3%61001233CETSP-20-092091.00.0%32217.4169441476.50.0%51557.00.0%746387CETSP-20-102020.40.0%29910.133292359.40.0%62321.40.0%289346Averages347.90.2%313.714.7191.2365.41284.521.5%667.3479.30.8%2755.5679.1 97

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CHAPTER5EXTENSIONSTOTHELIFETIMEMAXIMIZATIONPROBLEM 5.1TheMobileSinkModelwithFiniteSinkSpeedInthischapter,weaddressanimportantextensiontothelifetimemaximizationprobleminwirelesssensornetworks(WSNs).SimilartothemodelspresentedinChapter 2 ,hereweexamineaWSNconsistingofasetoflimited-powersensorsthataredeployedinanareafordatacollectionpurposes.Thecollectedinformationneedstobetransmittedforfurtherprocessingtoadata-collectingnodecalledthesink.Sensorsareoftendeployedrandomly,butinhighquantitiestopreventcoveragebreach.Thisdensedeploymentofthesensornodesallowsmultihopdeliveryofthecollectedinformation:Eachsensorcanremotelycommunicatewithothernearbysensorsviawirelesslinksandusethemtorelayitscollectedinformationtothesink.Aswediscussedinpreviouschapters,sensorenergyisascarceresourceinWSNs.However,ifsensorsaredeployedinnot-easily-accessibleorhostileenvironments,thetaskofreplacingtheirbatteriesbecomesimpractical.AnimportantoptimizationproblemistodevisebalancedcommunicationschemesbetweenthesensorsandthesinktoprolongWSNlifetime.VariouslifetimemaximizationproblemsforWSNshavebeenstudiedrecently[ 5 6 24 26 ],especiallywithrespecttothetaskofexploitingsinkmobility[ 7 10 15 19 26 55 56 ].Whenthesinkisstatic(i.e.,locatedatonexedposition),thesensornodesclosertothesinkareburdenedwithtrafcaggregationbecausetheyneedtorelayothernodes'trafc.Therefore,thesenodesexhausttheirbatteryenergysoonerthantherestofthesensornodes,disconnectingtherestofthenetworkfromthesink.Thisphenomenonisknownastheenergyholeproblem[ 20 22 ].Movingthesinkinthesensoreldcanmitigatetheenergyholeproblem,whichresultsinanextendednetworklifetime.Inthischapter,wefocusonsituationsinwhichthereexistsonesinkthatmovesoverasubsetofpre-speciedsinklocationsinthesensoreld. 98

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InWSNshavingastaticsink,themaindecisionaffectingnetworklifetimeishowtoroutethedatafromeachsensornodetothesink.Ontheotherhand,whenthesinkismobile,onemustalsodeterminehowlongthesinkshouldstayateachsinklocation.Wewillrefertothesedecisionsasschedulingdecisions.Whenthesinkcanmovearbitrarilyfastbetweenitspossiblelocations,schedulingonlyreferstotheamountoftimethatthesinkstaysateachlocation.(Laterinthischapter,wewillalsoincludetheorderinwhichthoselocationsarevisited.)Theproblemisthentondasetofroutingandschedulingdecisionsthatmaximizethenetworklifetime,wherethelifetimeisdenedasthetimeuntiltherstsensornodeexpendsallofitsbatterypower[ 5 6 ].Athigherdischargerates,batteriesaretypicallylessefcientatconvertingtheirchemicallystoredenergyintoavailableelectricalenergy(see,forexample,[ 57 ]).Therefore,itisenergy-efcienttouseroutingandschedulingschemesthatrequirefrequentchangesinthesensors'energyconsumptionrates.Inthischapter,weconsideracyclicmodelinwhichthesinkisrequiredtonishsome(positiveinteger)Ccyclesduringthenetworklifetime(see[ 26 58 ]forsimilarcyclicmodels).Ourlifetimemaximizationproblemisthenfocusedonndinganoptimaltrajectoryforthesinkaswellasoptimalroutingschemesforthesensornodesduringonecycle.AsmentionedinChapter 2 ,PapadimitriouandGeorgiadis[ 7 ]proposealinearprogrammingformulationfortheproblemofmaximizingthelifetimeofawirelesssensornetworkwithonemobilesink.Theirformulationintegratesbothroutingandschedulingdecisions.(Seealso[ 55 ]forrelatedworkonthisproblemthatemployscolumn-generationtechniques.)Amajorassumptioninthesepapersisthatthesink'straveltimefromonelocationtoanotherisnegligible,orequivalently,thattherearemanystationarysinks,oneateachpossiblelocation.However,whenthesink'straveltimesaresignicantcomparedtoitsdwellingtimesateachsinklocation,theorderinwhichthesinkvisitstheselocationsbecomesimportant. 99

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Keskinetal.[ 58 ]formulatealifetimemaximizationproblemthatassumesnonzerosinktraveltimes,andproposeheuristicalgorithmstosolveit.Theirformulationrequiresashortestpathroutingofthecollecteddatafromsensornodestothesink,whichmayresultinasuboptimalnetworklifetimeasdiscussedin[ 7 ].Unlike[ 58 ],themodelsweproposeinthischapterincorporatetheroutingofdatafromsensornodestothesink.WeformulatetheWSNlifetimemaximizationproblemasamixed-integerprogram(MIP)tocapturetheorderinwhichthesinklocationsarevisitedduringeachcycle.Weassumethatthesinkcannotgatherinformationfromthesensornodeswhiletravelingbetweendifferentsinklocations.Asaresult,theunderlyingapplicationmustbedelay-tolerantwithamaximumtolerabledelayD,i.e.,datacanbedeliveredtothesinkwithamaximumdelayofDtimeunits.Insomedelay-tolerantWSNapplications,sensornodesarecapableofpostponingdatatransmissiontothesinkbystoringthedatalocally.Alternatively,ifthesensornodes'computationalpowerandstoragecapabilitiesarelimited,thesensorsmayinsteadcontinuedatatransmissiontotheirtargetsinklocations,wherethetransmitteddataisstoreduntilasubsequentvisitbythesink.Whileourmodelsandsolutionmethodsinthischaptercanbeappliedtobothcases,wetaketheperspectiveofthelattercase,wheresinklocationsstoredata.Inadditiontoanoptimalsetofroutingschemesfromthesensornodestoeachsinklocation,ourformulationsndanoptimaltrajectoryforthesinkthatsatisesthemaximumtolerabledelayrestrictionfortheunderlyingapplication.Therestofthischapterisorganizedasfollows.InSection 5.2 wepresentMIPformulationsforthelifetimemaximizationproblemsexaminedinthischapter.InSection 5.3 ,wedescribeacutting-planeapproachformoreeffectivelysolvingonevariantoftheproblem.Next,werecasttheproblemasatwo-stageoptimizationprobleminSection 5.4 ,andprescribeaBendersdecompositionalgorithmforitssolution.WeconcludethechapterbypresentingsomecomputationalresultsinSection 5.5 100

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5.2ProblemStatementWebegininSection 5.2.1 byintroducingnotationanddiscussingseveralkeycharacteristicsofthelifetimemaximizationproblem.WethenformulatethelifetimemaximizationprobleminSection 5.2.2 andpresentanalternativeproblemvariationinSection 5.2.3 .WeprovideacomparisonofthepresentedmodelsinSection 5.2.4 5.2.1PreliminariesLetNbethesetofsensornodes(nodesforshort)andLbethesetofsinklocationsofaWSN.ThesinkperiodicallyvisitssinklocationsinasubsetofLandcollectsdatafromthenodes.Whenthesinkisatlocationl2L,nodeicansenditsdata(orrelayothernodes'data)onlytoasetofdownstreamneighborsSli.ThissetisoftengivenasSli=fj2N[flg:ij g,whereijistheEuclideandistancebetweeniandjand isthemaximumtransmissionrangeofanynode.DenotebyEitheinitialenergylevelofnodei,andsupposethatthesinkisrequiredtonishCcyclesduringthenetworklifetime.Therefore,maximizingthenetworklifetimeisequivalenttomaximizingthecycleduration,whileensuringthateachsensornode'senergyexpenditureinacycledoesnotexceed Ei=Ei=C.Therequiredenergyfortransmittingoneunitofdatafromnodeitoapointj2Sliwhilethesinkisatlocationl2Lisdenotedbyelij>0.Similarly,>0istherequiredenergyforreceivingoneunitofdataateverynode.Finally,di>0representsthedatagenerationrateofnodei.Underthecommonly-usedassumptionofzerosinktraveltimes[ 7 55 ],thelifetimemaximizationprobleminWSNswithamobilesinkcanbeformulatedasalinearprogram.Letvariablezldeterminehowlongthesinkstaysatlocationl2Lduringeachcycle,andylijrepresentthevolumeofdatatransmittedfromsensornodeitoj2Sliwhilethesinkisatlocationl.Thelifetimemaximizationproblemcanbeformulatedasfollows[ 7 ]. MaxXl2Lzl (5a) 101

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s.t.Xl2L0@Xj2Slielijylij+Xj:i2Sljylji1A Ei,8i2N (5b)Xj2Sliylij)]TJ /F11 11.955 Tf 14.23 11.36 Td[(Xj:i2Sljylji=dizl,8i2N,l2L (5c)zl0,8l2L (5d)ylij0,8i2N,j2Sli,l2L. (5e)TheobjectivefunctionoptimizestheWSNlifetimebymaximizingthesumoftimethatthesinkspendsstayingoverallpossiblesinklocations.(TheactuallifetimewouldequalthisvaluemultipliedbyC.)Constraints( 5b )ensurethatthetotalenergyexpenditureofeachsensornodeiduringeachcycledoesnotexceed Ei.Constraints( 5c )areowconservationconstraintsandensurethatforeachsinklocationl,everysensornode'sout-owequalsitsin-owplusthevolumeofdatageneratedatthesensornodeduringthesink'sstayatl.Constraints( 5d )and( 5e )statethenonnegativityrequirementsforz-andy-variables.WerefertothismodelastheMobileSinkModel(MSM).Observethatthismodelimplicitlyassumesthatdataisroutedtothesinklocationsinthesamemannereachtimethesinktraversesacycle.(Here,routingincludesbothpathselectionandrateallocation.)Thisassumptionismadeforthesakeofsimplifyingroutinganddatacollection,andallmodelswepresenthereemploythiscyclicroutingassumption.Whenthesinkmoveswithanitespeed,Formulation( 5 )maynotnecessarilyyieldafeasibletrajectoryforthesink.ConsidertheWSNinFigure 5-1A wherethesinkvisitssinklocationsfl1,l2,l3,l4g.Figure 5-1B containsthedistancesbetweeneachpairofsinklocations.Supposethatinsomesolutionto( 5 ),thesinkstaysatlocationl1forz1=5timeunits,duringwhichtimethesensornodesroutedatatothesinkaccordingtoacommunicationpatternspeciedbyvectory1.Also,supposethatthesolutionrequiresaz2=7,z3=8,andz4=6time-unitstayatsinklocationsl2,l3,andl4withcommunicationschemesy2,y3,andy4,respectively.Ifthesinkmovesarbitrarilyfast, 102

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AIllustrationofthenetwork l1l2l3l4l5 l1 041285l2 40994l3 129076l4 89703l5 54630 BPairwisedistancesbetweenthesinklocations Figure5-1. Awirelesssensornetworkwithvesinklocations asassumedinMSM,thenthissolutionneednotprescribeaspecicorderofvisittothesinklocations.Supposeinsteadthatthesinktravelswithaspeedof2distanceunitspertimeunit.Wecanbuildafeasibletrajectoryforthesinkbasedontheforegoingoptimalz-andy-valuesasfollows: Thesinkleavesl1towardl2attimet=0.Duringthesink'stravelfroml1tol2,thesensornodestransmittheircollecteddatatol2accordingtothecommunicationpatternspeciedbyy2.Thetransmitteddataisstoredatl2untilthesinkarrivesthereattimet=2. Thesinkstaysatl2untilt=7andcollectsdatafromthesensornodes(accordingtoy2),andthenheadstol3. Thesinkreachesl3attimet=11.5,collectsallstoreddataimmediately,continuescollectingdatauntilt=15(accordingtoy3),andthenleavesl3togotol4. Thesinkthenreachesl4att=18.5,andleavesthereatt=21. 103

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Finally,thesinkreachesl1att=25,andstaysthereuntiltheendofthecycleatt=26.Clearly,thesinkcanonlytravelonanarcifitstraveltimeisnomorethanthedwellingtimeatthedestination.Now,supposethatthesinkonlymoves1distanceunitpertimeunit.Thereexistsnofeasibletrajectoryforthesinkthatspansfl1,l2,l3,l4g,givenz1,...,z4:Thefactthatz1++z4=26impliesthatthesink'stourcannottakelongerthan26timeunits,buttheshortesttourspanningthesefournodesis28timeunits. 5.2.2ProblemFormulationInthissection,weconsidertheWSNlifetimemaximizationproblemwithanitesinkspeed.Supposethatittakestlm>0timeunitsforthesinktogofromlocationl2Ltolocationm2L.Datageneratedwhilethesinktravelsfromltomistransmittedandstoredatlocationmandiscollectedbythesinkonceitreachesm.ThemaximumtolerabledelayisassumedtobeD,sothesinkcanonlytravelonarcsthatsatisfytlmD.WewillrefertothedirectedgraphG=(L,A)witharcsetA=f(l,m):tlmDgasthesinkgraph.InthesinkgraphdeneN+l=fm2L:(l,m)2AgandN)]TJ /F5 7.97 Tf -.93 -8.28 Td[(l=fm2L:(m,l)2Ag.OurgoalistomaximizetheWSNlifetimebyndingatourforthesinkoverasubsetofL,aswellasroutingpatternsfromeachsensornodetoeachvisitedsinklocation.WerefertothisproblemasMSM-FS1.ToformulatetheproblemasaMIP,weintroducebinaryvariablesulmthatequal1ifandonlyifthesinktravelsalongarc(l,m)intheunderlyingsinkgraph.Recallthatthemainconstraintinthismodelrequiresthesink'sdwellingtimeatltobeatleastaslargeastmlifuml=1.Ourformulationsatisesthisrequirementbecausethetransmissiontimetoeachsinklocationisnowgivenbyzl+Pm2N)]TJ /F12 5.978 Tf -.58 -6.42 Td[(ltmluml,wherezlisdenedasthesink'sstoplengthatl.Wealsointroduceauxiliarybinaryvariablesvlthatequal1ifandonlyiflocationlisvisitedbythesink.ProblemMSM-FS1canbeformulatedasfollows, 104

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whereconstantsMl,8l2L,arelargevaluesthatwespecifylaterinthischapter. MaxXl2L0@zl+Xm2N)]TJ /F12 5.978 Tf -.57 -6.42 Td[(ltmluml1A (5a)s.t.Xl2L0@Xj2Slielijylij+Xj:i2Sljylji1A Ei,8i2N (5b)Xj2Sliylij)]TJ /F11 11.955 Tf 14.22 11.36 Td[(Xj:i2Sljylji=di0@zl+Xm2N)]TJ /F12 5.978 Tf -.58 -6.42 Td[(ltmluml1A,8i2N,l2L (5c)zlMlvl,8l2L (5d)Xm2N+lulm=vl,8l2L (5e)Xm2N)]TJ /F12 5.978 Tf -.58 -6.42 Td[(luml=Xm2N+lulm,8l2L (5f)Xl2SXm2 Sulmvk+vr)]TJ /F4 11.955 Tf 11.95 0 Td[(1,8SL:2jSjjLj)]TJ /F4 11.955 Tf 17.94 0 Td[(2,k2S,r2 S (5g)vl2f0,1g,8l2L (5h)ulm2f0,1g,8l2L,m2N+l (5i)zl0,8l2L (5j)ylij0,8i2N,j2Sli,l2L. (5k)Theobjectivemaximizesthetimethatthesinkdwellsatsinklocations,plusthetimeitspendstravelingbetweentheselocations.Constraints( 5b )and( 5c )aretheenergyandowconservationconstraints,respectively,similarto( 5b )and( 5c ).Constraints( 5d )forcevl=1wheneverzl>0,8l2L.Constraints( 5e )denetherelationshipbetweenu-andv-variablesandensurethatatmostonearcenterseachsinklocation,whileconstraints( 5f )ensurethatforeachl2L,thenumberofincomingandoutgoingarcsinafeasibletourofsinklocationsareequal.Finally,constraints( 5g )arethegeneralizedsubtoureliminationconstraints[ 40 ].Theseconstraintsensurethatforany 105

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subsetSLsuchthat2jSjjLj)]TJ /F4 11.955 Tf 18.29 0 Td[(2,thereexistsanarcgoingfromSto SifbothSand Scontainasinklocationthatisonthetour.Remark1.NotethatwhenC=1,thesinkdoesnotneedtoreturntoitsinitiallocationandtherefore,oneseeksafeasiblepathforthesinkinsteadofacycle.Inthiscase,letbinaryvariablel(l)indicatewhetherthesink'spathstarts(ends)atsinklocationl.Wewouldthensubstituteconstraints( 5e )and( 5f )withthefollowingconstraints. l+Xm2N+lulm=vl,8l2L (5a)Xm2N)]TJ /F12 5.978 Tf -.58 -6.41 Td[(luml)]TJ /F11 11.955 Tf 15.67 11.36 Td[(Xm2N+lulm=l)]TJ /F10 11.955 Tf 11.95 0 Td[(l,8l2L (5b)Xl2Ll=1 (5c)l2f0,1g,8l2L (5d)l2f0,1g,8l2L. (5e)Moreover,becauseweseekadirectedpath,theleft-hand-sideofsubtoureliminationconstraints( 5g )becomesPl2SPm2 S(ulm+uml).NotethattheequalityPl2Ll=1isimpliedbytheaggregationof( 5b )and( 5c ).2Forsimplicity,wepreprocessthecasesinwhichthesinkvisitsonlyasinglesinklocationl2L.Toaccomplishthis,wesolvetheMSMinwhichLisrestrictedtothesingletonflg.LetM0lbetheoptimalMSMobjectivetothisproblem.Fromthispointforward,weassumethatthesinkvisitsatleasttwolocationsinL.Ourultimatesolutionisonehavingthelargestobjectiveamongthesingle-sink-locationsolutionsandthemultiple-sink-locationsolutionweobtain.WenextaddressthederivationoftheMl-valuesusedin( 5d ).NotingthatMlmustbeatleastaslargeasthemaximumdwellingtimeatanylocationl2L,weconsiderthescenarioinwhichthesinkspends(almost)theentirecycledurationatlocationl.Recall 106

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thatM0listhemaximumamountoftimethesinkcanspendatl2L,andthesinkmustspendsometimeintransittol,duetotheassumptionthatthesinkvisitsatleasttwolocations.Hence,M0lisanupperboundonzl+tml,wheremiswhicheversinklocationvisitedimmediatelybeforel.Therefore,Ml=M0l)]TJ /F4 11.955 Tf 15.45 0 Td[(minm2N)]TJ /F12 5.978 Tf -.58 -6.42 Td[(ltml (5)canbeusedin( 5d ).Alternatively,wecandeterminetheminimumamountofenergythatanodemustexpendwhilethesinkisatl.Dividingthenode'senergybythisminimumexpenditureyieldsavalidboundforM0l,andenablesustouseMl=mini2N( Ei diminj2Slielij))]TJ /F4 11.955 Tf 15.45 0 Td[(minm2N)]TJ /F12 5.978 Tf -.58 -6.42 Td[(ltml (5)asavalidupperboundonzl.AlthoughcalculatingMlvia( 5 )ismorecomputationallyexpensivethancalculating( 5 ),itgenerallyresultsinastrongerbound,andhencewecomputeMlvia( 5 )intheremainderofthischapter. 5.2.3AlternativeDelayModelProblemMSM-FS1assumesreal-timetransmissionofdatawheneverpossible,i.e.,asinklocationstoresdataonlywhenthesinkisenroutetothelocation,anddatatransmissionhappensinrealtimewhilethesinkdwellsatasinklocation.Thismodelisappropriateforapplicationssuchassurveillanceandreal-timemonitoringwhereunnecessarydelaymustbeavoided,except,e.g.,whenthesinkistravelingbetweenthelocations.Inothermoredelay-tolerantWSNapplications,itmaybemoreappropriatetoallowsensorstotransmitdatatosomelocationl,evenwhenthesinkisnotcurrentlypresentat,orenrouteto,locationl.However,westillhonorthemaximumdelayrestrictionstatingthatonceinformationbeginstransmissiontolocationl,thesinkmustarriveatnodelnomorethanDtimeunitslater.Toillustratethisalternativedelaymodel,consideragainthewirelesssensornetworkinFigure 5-1A andsupposethatthesink'strajectoryisl1!l2!l3!l1.Thecycle 107

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timeequalsP3m=1rm,wherermisthelengthoftheperiodduringwhichthesensornodessendtheirdatatosinklocationlmaccordingtothedatatransmissionschemegivenbyym.Dataistransmittedtothesesinklocationsinthesameorderinwhichtheyarelatervisitedbythesink.Unlike( 5 ),itisnowpossible,forinstance,thatwhilethesinkismovingfroml1tol2,thesensornodesstarttransmittingdatatol3.First,notethatthethesink'stourlengthcannotexceedthecycletime.Therefore,theremustexistatleastonesinklocationlonthesink'stour,suchthatsensorsarestillsendingdatatolwhenthesinkarrivesatl.(Thischaracterizationincludesthecaseinwhichsensorsstoptransmittingtolimmediatelywhenthesinkarrivesatl.)Wewillrefertoonesuchsinklocationasthetour'sorigin.Thisassumptionisnecessarytoensuretheuniformityofdataroutingandcollectionduringeachcycleperformedbythesink.Inourexample,ifl1servesastheorigin,delayrestrictionsimposethefollowingconstraints. t12D, (5a)maxft12)]TJ /F3 11.955 Tf 11.95 0 Td[(r2,0g+t23D, (5b)maxfmaxft12)]TJ /F3 11.955 Tf 11.96 0 Td[(r2,0g+t23)]TJ /F3 11.955 Tf 11.95 0 Td[(r3,0g+t31D. (5c)Hereinequality( 5a )ensuresthattransmissiondelayalongthepathll!l2doesnotexceedD,while( 5b )and( 5c )ensurethatthisrequirementismetalongll!l2!l3andll!l2!l3!l1,respectively.Moreover,becausel1servesastheorigin,thesensornodesmuststillbesendingdatatol1whenthesinkarrivesthere,andsothefollowingconstraintmustbesatised:r1maxfmaxft12)]TJ /F3 11.955 Tf 11.96 0 Td[(r2,0g+t23)]TJ /F3 11.955 Tf 11.96 0 Td[(r3,0g+t31.Toformulatetheseconstraintswhenthetour'soriginisnotdeterminedapriori,letvariablesulmandvldeterminethesink'strajectory(asmodeledby( 5e )( 5i )).Also,denebinaryvariablesl,forl2L,whichequals1iflocationlservesasthetour's 108

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origin.Weimposeconstraintsanalogousto( 5 )foreverycombinationoforiginnodel2Landsinklocationm2L.Theseconstraintsaremadeinactive(byweakeningtheright-hand-sidewithabig-Mterm)wheneversl=0orvm=0,andareotherwiseactive.Tomodelthesedelayconstraints,letvariableqlm,8l,m2L,representtheamountoftimethatsensorstransmitdatatombeforethesinkbeginsmovingtowardmfromitsprecedingsinklocation,giventhatlservesasthetour'sorigin.Therefore,dataistransmittedtomstartingfromqlm+Pk2N)]TJ /F12 5.978 Tf -.58 -5.09 Td[(mtkmukmtimeunitsbeforethesinkarrivesatm,andthefollowingsetofconstraintsmusthold:qlm+Xk2N)]TJ /F12 5.978 Tf -.58 -5.09 Td[(mtkmukmD,8l2L,m2L. (5)Constraints( 5a )and( 5b )belowensuretheexistenceofadesignatedoriginfromamongthosenodesvisitedbythetour. Xl2Lsl=1 (5a)slvl,8l2L. (5b)Withoutlossofgenerality,ifthesink'stourtraversessinklocations1,...,pinorderwithoriginnode1,thenq1,w+1=maxfq1w+tw)]TJ /F13 7.97 Tf 6.58 0 Td[(1,w)]TJ /F3 11.955 Tf 12.58 0 Td[(rw,0g,whereallindexadditionandsubtractionisperformedmodulop,foreachw=1,...,p.Moreover,becauselocation1istheorigin(andlocation2shouldnotcollectdatabeforethesinkleaves1),q12=0.Allotherq-variablesshouldequal0.Thefollowingconstraintsproperlydenetheq-variablesassuchintermsofther-,s-,andu-variables. qlmXk2N)]TJ /F12 5.978 Tf -.58 -5.09 Td[(m24qlk+Xh2N)]TJ /F12 5.978 Tf -.57 -6.42 Td[(k(thk)]TJ /F3 11.955 Tf 11.95 0 Td[(rk)uhk35ukm)]TJ /F3 11.955 Tf 11.96 0 Td[(D(1)]TJ /F3 11.955 Tf 11.96 0 Td[(sl),8l2L,m2L (5a)D(1)]TJ /F3 11.955 Tf 11.95 0 Td[(sl)+rlqll+Xm2N)]TJ /F12 5.978 Tf -.58 -6.41 Td[(ltmluml,8l2L. (5b) 109

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Toseethat( 5a )and( 5b ),alongwith( 5 ),correctlydenetheq-variables,werstclaimthatthereexistsasolutiontotheseconstraintsinwhicheachq-variabletakesthesmallestvalueallowedby( 5a )and( 5b )(duetoqlmbeingontheleft-hand-sideof( 5 ),whichisofthesense,andbecauseq-variablesarenotpresentelsewhereinthemodel).Ifsl=0,thennotethatsettingqlk=0,8k2L,isfeasible:Theright-hand-sideof( 5a )wouldbenomorethanthk)]TJ /F3 11.955 Tf 12.65 0 Td[(D0(foranyh2N)]TJ /F5 7.97 Tf -.93 -8.27 Td[(k),and( 5b )permitsqll=0duetothefactthattmlDforallm2N)]TJ /F5 7.97 Tf -.93 -8.28 Td[(l.Nowconsiderthel2Lsuchthatsl=1.Ifvm=0,thenPk2N)]TJ /F12 5.978 Tf -.57 -5.09 Td[(mukm=Pk2N+mumk=0,andthussettingqlm=0isfeasibleto( 5a ).Finally,considerm2L:vm=1.Reindexthelocationssothatthetourvisitslocations1,...,pinorder,withoriginl=1(wherep2).Tobegin,notethat( 5b )restrictsr1q11+tp1,andso( 5a )form=2reducestoq12q11+tp1)]TJ /F3 11.955 Tf 12.27 0 Td[(r1,whichisthusdominatedbyq120.Hence,q12=0asdesired.Byinduction,assumethatq12,...,q1warecorrectlydenedforsome2wp.Constraint( 5a )denesql,w+1qlw+tw)]TJ /F13 7.97 Tf 6.58 0 Td[(1,w)]TJ /F3 11.955 Tf 11.02 0 Td[(rw(wherew+11ifw=p,andw)]TJ /F4 11.955 Tf 11.02 0 Td[(1pifw=1),whichalongwiththenonnegativityoftheq-variablesforcesql,w+1maxfqlw+tw)]TJ /F13 7.97 Tf 6.59 0 Td[(1,w)]TJ /F3 11.955 Tf 12.52 0 Td[(rw,0gasdesired.Repeatingthisargumentshowsthatqlwiscorrectlydenedforallw=1,...,p.Tolinearize( 5a ),weintroducenonnegativevariableslkm=qlkukm,hkm=uhkukm,andhkm=rkuhkukm,andaddthefollowingconstraints:D(1)]TJ /F3 11.955 Tf 11.95 0 Td[(ukm)+lkmqlk,8l2L,m2L,k2N)]TJ /F5 7.97 Tf -.94 -7.89 Td[(m (5)hkmuhk+ukm)]TJ /F4 11.955 Tf 11.95 0 Td[(1,8h2L,k2N+h,m2N+k (5)hkmrk,8h2L,k2N+h,m2N+k (5)hkmDuhk,8h2L,k2N+h,m2N+k (5)hkmDukm,8h2L,k2N+h,m2N+k, (5) 110

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where( 5a )isnowrevisedasqlmXk2N)]TJ /F12 5.978 Tf -.57 -5.09 Td[(m24lkm+Xh2N)]TJ /F12 5.978 Tf -.58 -6.42 Td[(k(thkhkm)]TJ /F10 11.955 Tf 11.96 0 Td[(hkm)35)]TJ /F3 11.955 Tf 11.96 0 Td[(D(1)]TJ /F3 11.955 Tf 11.95 0 Td[(sl).Again,because-and-variablesappearontheright-hand-sideof( 5a )(a-constraint),therealwaysexistsanoptimalsolutioninwhichthesevariablestakeontheirsmallestpossiblevaluepermittedby( 5 )and( 5 ),andsotheupper-boundingconstraintsrequiredtolinearize-and-variablesareunnecessaryinthisformulation.Byasimilarargument,thelower-boundingconstraintsforlinearizing-variablesarenotrequiredintheformulation,becausetherealwaysexistsanoptimalsolutioninwhichthesevariablesassumetheirlargestpossiblevalues.Hence,problemMSM-FS2canbeformulatedasfollows. MaxXl2Lrl (5a)s.t.Xl2L0@Xj2Slielijylij+Xj:i2Sljylji1A Ei,8i2N (5b)Xj2Sliylij)]TJ /F11 11.955 Tf 14.23 11.36 Td[(Xj:i2Sljylji=dirl,8i2N,l2L (5c)rlM0lvl,8l2L (5d)Constraints( 5e )( 5i ) (5e)Constraints( 5 )( 5 ) (5f)qlm0,8l2L,m2L (5g)rl0,8l2L (5h)lkm,hkm,hkm0,8l2L,k2N+l,m2N+k. (5i)Formulation( 5 )containsanexponentialnumberofsubtoureliminationconstraints.Inpractice,onecanrelaxthesubtoureliminationconstraints,andaddthoseinequalitiesthatareviolatedateachnodeofthebranch-and-boundtreebya 111

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polynomiallysolvableseparationroutine[ 52 ].WenextformulateanalternativeMIPmodelforMSM-FS2thatispolynomiallysized.Tothatend,wedecomposeeachdwellingtimerlintovariablesr1landr2l,wherer1lrepresentstheamountoftimethatsensorssenddatatolbeforethesinkarrivesatl,andr2lrepresentstheamountoftimethatthesinkstaysatl.Next,denethestartofasink'scycleasthetimethesinkdepartstheorigin.Wealsointroducevariableslasthedurationfromthestartofeachcycleuntilthesensorsstarttransmittingdatatol,andlasthedurationfromthestartofeachcycleuntilthesinkreachesl.Giventhesenewvariabledenitions,considerthefollowingalternativeformulationforMSM-FS2,wherethe-and-variablesservethedualrolesofenforcingdelaylimitsandpreventingsubtours. MaxXl2Lr1l+Xl2Lr2l (5a)s.t.Xl2L0@Xj2Slielijylij+Xj:i2Sljylji1A Ei,8i2N (5b)Xj2Sliylij)]TJ /F11 11.955 Tf 14.23 11.36 Td[(Xj:i2Sljylji=di(r1l+r2l),8i2N,l2L (5c)r1l+r2lM0lvl,8l2L (5d)Constraints( 5e ),( 5f ),( 5h ),( 5i ),( 5a ),and( 5b ) (5e)mXl2N)]TJ /F12 5.978 Tf -.57 -5.08 Td[(mtlmulm,8m2L (5f)mM(2)]TJ /F3 11.955 Tf 11.95 0 Td[(sl)]TJ /F3 11.955 Tf 11.95 0 Td[(ulm),8l2L,m2L (5g)ml+r2l+tlmulm)]TJ /F3 11.955 Tf 11.95 0 Td[(M(1)]TJ /F3 11.955 Tf 11.96 0 Td[(ulm+sl),8l2L,m2L (5h)r1l+r2lm)]TJ /F10 11.955 Tf 11.95 0 Td[(l)]TJ /F3 11.955 Tf 11.95 0 Td[(M(1)]TJ /F3 11.955 Tf 11.96 0 Td[(ulm+sl),8l2L,m2L (5i)r1l+r2ll)]TJ /F10 11.955 Tf 11.95 0 Td[(l)]TJ /F3 11.955 Tf 11.96 0 Td[(D(1)]TJ /F3 11.955 Tf 11.95 0 Td[(sl),8l2L (5j)r1l+r2ltml(uml+sl)]TJ /F4 11.955 Tf 11.95 0 Td[(1),8l2L,m2L (5k)r1ll)]TJ /F10 11.955 Tf 11.96 0 Td[(lD,8l2L (5l) 112

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ylij0,8i2N,j2Sli,l2L (5m)r1l,r2l0,8l2L. (5n)Thebig-Mvaluesinconstraints( 5g ),( 5h ),and( 5i )canbesettotheoptimalobjectivefunctionvalueofthecorrespondingMSMinstance.Thenexttwopropositionsformallyestablishtheequivalenceofformulations( 5 )and( 5 ). Proposition5.1. Afeasiblesolutionto( 5 )doesnotcontainanysubtours. Proof. Let(r1,r2,y,u,s,v,,)denoteafeasiblesolutionto( 5 ).Supposethatthereexistsasubtour,T,thatdoesnotcontainthetour'sorigin.Withoutlossofgenerality,assumethatTcontainsarcsf(1,2),(2,3),...,(p,1)g,andsl=0,8l=1,...,p.Aggregatingconstraints( 5h )forallarcsinTyieldst12+t23++tp1+Ppl=1r2l0,whichisacontradictionbecauseallt-valuesarepositiveandallr2-variablesarenonnegative.Thiscompletestheproof. Proposition5.2. Formulations( 5 )and( 5 )areequivalent. Proof. Weprovethateachsolutiontoformulation( 5 )correspondstoasolutiontoformulation( 5 )havingthesameobjectivefunctionvalue,andviceversa.First,leta=(r1,r2,y,u,s,v,,)denoteafeasiblesolutionto( 5 ).Denerl=r1l+r2l,foralll2L.Forl,m2L,letqlm=maxf0,Pk2N)]TJ /F12 5.978 Tf -.58 -5.09 Td[(m(k)]TJ /F10 11.955 Tf 11.97 0 Td[(m)ukmgifslvm(1)]TJ /F3 11.955 Tf 11.97 0 Td[(ulm)=1,andqlm=0otherwise.Solutionb=(r,y,q,u,s,v)hasthesameobjectivefunctionvaluefor( 5 )asadoesfor( 5 ).Wenextshowthatbisfeasibleto( 5 ).UsingtheforegoingdenitionsandProposition 5.1 ,bsatisesconstraints( 5b )( 5e ),( 5g ),( 5h ),( 5a ),and( 5b ).Also,wecomputethe-,-,and-valuesaccordingtotheirdesiredlinearizations,andhencethesolutionsatises( 5 )( 5 )and( 5i ).Hence,wemustshowthatbsatises( 5 ),( 5a ),and( 5b ).Ifqlm=0forl,m2L,thenbsatises( 5 )becausetkmDforallk2N)]TJ /F5 7.97 Tf -.93 -7.3 Td[(m.Nowsupposethatqlm>0forl,m2L.Then,bydenition,wemusthavesl=vm=1and 113

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ulm=0.Letk2N)]TJ /F5 7.97 Tf -.93 -7.29 Td[(mnlbethelocationforwhichukm=1.NotethatDm)]TJ /F10 11.955 Tf 11.95 0 Td[(m(by( 5l ))k+r2k+tkm)]TJ /F10 11.955 Tf 11.95 0 Td[(m(by( 5h ),writtenfor(k,m))k+tkm)]TJ /F10 11.955 Tf 11.96 0 Td[(m.BecauseDtkmaswell,weconcludethatbsatises( 5 )becauseDmaxfk)]TJ /F10 11.955 Tf 11.96 0 Td[(m,0g+tkm=qlm+tkm.Wenextprovethatbsatises( 5a )forl,m2L.Notethatifvm=0,then( 5a )holdsbecauseukm=0forallk2N)]TJ /F5 7.97 Tf -.94 -7.29 Td[(m.Also,ifvm=1andsl=0,thenbecauseldoesnotserveastheorigin,wehavethatqlk=0,8k2L,and( 5a )holdsbecauserk0fork2N)]TJ /F5 7.97 Tf -.93 -7.29 Td[(mandthkDfork2Landh2N)]TJ /F5 7.97 Tf -.94 -8.27 Td[(k.Hence,wefocusonthecaseinwhichvm=sl=1.Supposethatk2N)]TJ /F5 7.97 Tf -.93 -7.3 Td[(m,andh2N)]TJ /F5 7.97 Tf -.93 -8.28 Td[(karechosensuchthatuhk=ukm=1.Weconsiderthreedifferentcasesforthesequenceh!k!m.Incase(i),assumethatk=l.Wehavethatqlm=0and( 5a )reducesto0qll+thl)]TJ /F3 11.955 Tf 12.28 0 Td[(rl.Ifqll=0,theinequalityissatisedbecauseof( 5k )writtenfor(l,h),andotherwiseifqll=h)]TJ /F10 11.955 Tf 11.96 0 Td[(l>0,thentheinequalityissatisedbecauserl=r1l+r2ll)]TJ /F10 11.955 Tf 11.96 0 Td[(l(h+r2h+thl))]TJ /F10 11.955 Tf 11.96 0 Td[(l=qll+r2h+thlqll+thl,wherethersttwoinequalitiesholdbecauseof( 5j )and( 5h )(writtenfor(h,l)),respectively.Incase(ii),assumethath=l.Hence,qlk=0and( 5a )reducestoqlmtlk)]TJ /F3 11.955 Tf 11.95 0 Td[(rk.Usingthedenitionofqlm,( 5i )(for(k,m)),and( 5f ),wehavethatqlmk)]TJ /F10 11.955 Tf 11.95 0 Td[(mk)]TJ /F3 11.955 Tf 11.95 0 Td[(rk)]TJ /F10 11.955 Tf 11.96 0 Td[(ktlk)]TJ /F10 11.955 Tf 11.95 0 Td[(k)]TJ /F3 11.955 Tf 11.95 0 Td[(rk. 114

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Also,( 5g )impliesthatk=0,andhence( 5a )holds.Finally,incase(iii),assumethath6=landk6=l.Thefollowinginequalitieshold:qlmk)]TJ /F10 11.955 Tf 11.96 0 Td[(m(denitionofqlm)h+r2h+thk)]TJ /F10 11.955 Tf 11.95 0 Td[(m( 5h )for(h,k)h+r2h+thk)]TJ /F4 11.955 Tf 11.95 0 Td[((k+r1k+r2k)( 5i )for(k,m)h)]TJ /F10 11.955 Tf 11.96 0 Td[(k+thk)]TJ /F3 11.955 Tf 11.96 0 Td[(rk, (5)Also,theinequalityqlmh+r2h+thk)]TJ /F10 11.955 Tf 11.96 0 Td[(mleadstothefollowing:qlmk)]TJ /F10 11.955 Tf 11.96 0 Td[(mh+r2h+thk)]TJ /F10 11.955 Tf 11.95 0 Td[(mh+r1h+r2h+thk)]TJ /F10 11.955 Tf 11.96 0 Td[(mk+thk)]TJ /F10 11.955 Tf 11.95 0 Td[(mm)]TJ /F3 11.955 Tf 11.95 0 Td[(rk+thk)]TJ /F10 11.955 Tf 11.96 0 Td[(m=thk)]TJ /F3 11.955 Tf 11.95 0 Td[(rk. (5)Using( 5 )and( 5 ),weobtainqlmmaxf0,h)]TJ /F10 11.955 Tf 11.95 0 Td[(kg+thk)]TJ /F3 11.955 Tf 11.95 0 Td[(rk=qlk+thk)]TJ /F3 11.955 Tf 11.95 0 Td[(rk.Thereforebsatises( 5a ).Wenextprovethatbsatises( 5b ).Ifsl=0,then( 5b )holdsbecauseqll=0,tmlD,forallm2N)]TJ /F5 7.97 Tf -.93 -8.28 Td[(l,andrl0.Nowsupposethatlandk2N)]TJ /F5 7.97 Tf -.94 -8.28 Td[(larechosensuchthatsl=ukl=1.Ifqll=k)]TJ /F10 11.955 Tf 11.95 0 Td[(l,thenusing( 5j )and( 5h )for(k,l)wehaverl=r1l+r2ll)]TJ /F10 11.955 Tf 11.95 0 Td[(lk+r2k+tkl)]TJ /F10 11.955 Tf 11.95 0 Td[(lk+tkl)]TJ /F10 11.955 Tf 11.95 0 Td[(l.Hence,rlqll+tkland( 5b )issatised.Otherwisewemusthaveqll=0,and( 5k )impliesthatrktkl,whichagainimpliesthat( 5b )issatised. 115

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Conversely,letb=(r,y,q,u,s,v)denoteafeasiblesolutionto( 5 ).Letlandmbetwosinklocationsforwhichsl=ulm=1.Dene: r1k=qlk+Ph2N)]TJ /F12 5.978 Tf -.57 -6.41 Td[(kthkuhkandr2k=rk)]TJ /F3 11.955 Tf 11.96 0 Td[(r1k,fork2L. m=0andm=tlm. k=h+rhifuhk=1andh6=l. k=h+r2h+thkifuhk=1andh6=l.Thena=(r1,r2,y,u,s,v,,)hasthesameobjectivefunctionvalueasdoesbin( 5 ),andsatises( 5b )( 5e )and( 5n )duetothefactthataisfeasibleto( 5 ).Solutionbsatisesallremainingconstraintsto( 5 )byconstruction(thedetailsofthisanalysisarestraightforwardandareomittedforbrevity). 5.2.4ComparisonofMSM,MSM-FS1,andMSM-FS2Weconcludethissectionbystatingtherelationshipbetweentheoptimalobjectivefunctionvaluesofthethreelifetimemaximizationproblems.Let( z, y)denoteanoptimalsolutionto( 5 ).Fromthissolution,webuildadirectedgraphG0=(L0,A0)withL0=fl2L: zl>0gasthenodesetandA0=f(l,m)2A: zmtlmgasthearcset.Figure 5-2 illustratesG0fortheexamplesdiscussedinSection 5.2.1 .Notethatwhenthesinkmovesslower,G0becomessparser(Figure 5-2A )andasweshowedearlier,thereexistsnofeasibletrajectoryforthesink.However,withafastersink,G0isHamiltonianandasweobservedearlier,afeasibletrajectoryforthesinkexists. Proposition5.3. LetZMSM,ZMSM-FS1,andZMSM-FS2denotetheoptimalobjectivefunctionvalueofproblemsMSM,MSM-FS1,andMSM-FS2,respectively.ThenZMSM-FS1ZMSM-FS2ZMSM. (5) Proof. Supposethat(z,y,u,v)representsanoptimalsolutiontoMSM-FS1andletrl=zl+Pm2N)]TJ /F12 5.978 Tf -.58 -6.42 Td[(ltmlumlandqlm=0foralll2Landm2L.Also,supposethatsl=1for 116

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AG0whensinkspeed=1 BG0whensinkspeed=2 Figure5-2. IllustrationofG0fortheexampleinSection 5.2.1 anarbitrarylonthetourthatisrepresentedby(u,v),andsm=0,8m2Lnflg.WeprovethatZMSM-FS1ZMSM-FS2byshowingthat(r,y,q,u,v,s)isfeasibletoMSM-FS2.(Thetwosolutionsclearlyprovidethesameobjectivefunctionvalue.)Notethatifuml=1,thentmlrlandtherefore( 5b )holdsforalll2L.ThesinkcanonlytravelonanarcwhosetraveltimeisnotmorethanD,andso( 5 )issatised.Finally,constraints( 5a ),( 5b ),and( 5a )holdbydenition.Also,becauseMSM-FS2isarestrictionofMSM,wehavethatZMSM-FS2ZMSM,whichcompletestheproof. Proposition 5.4 establishesasufcientconditionunderwhichZMSM-FS1=ZMSM-FS2=ZMSM. Proposition5.4. ZMSM-FS1=ZMSM-FS2=ZMSMifthereexistsanoptimalsolutiontoMSMforwhichthecorrespondinggraphG0isHamiltonian. Proof. BecauseZMSM-FS1ZMSM-FS2ZMSM,weprovetheclaimbyshowingthatanoptimalsolutiontoMSMsatisfyingtheassumptionsofthepropositioncorrespondstoafeasiblesolutiontoMSM-FS1havingthesameobjectivefunctionvalue.Consideranoptimalsolution( z, y)toMSMsatisfyingtheassumptionsoftheproposition,anddeneTasthesetofarcsinaHamiltoniancycleingraphG0inducedby z.Letulm=1if(l,m)existsinT,andsetzm= zm)]TJ /F11 11.955 Tf 12.55 8.97 Td[(Pl2N)]TJ /F12 5.978 Tf -.58 -5.09 Td[(mtlmulm.Byconstruction,(u,z, y)isfeasibleto 117

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( 5 ),andhasthesameobjectivefunctionvalueasdoestheoptimalMSMsolution.Thiscompletestheproof. 5.3Cutting-planeAlgorithmforMSM-FS1Inthissection,wefocusonthedevelopmentofacutting-planealgorithmforsolvingMSM-FS1.In( 5 ),werevisethedenitionofrlasrl=zl+Pm2N)]TJ /F12 5.978 Tf -.58 -6.42 Td[(ltmluml,i.e.,rldenotesthesink'sdwellingtimeatlocationlplusthetimethatittakesforthesinktoreachlfromitsprevioussinklocation,andthusrepresentsthetotalamountoftimedataissenttolocationl.Usingthisreviseddenitionofrl,wecanreformulateMSM-FS1asfollows. MaxXl2Lrl (5a)s.t.Xl2L0@Xj2Slielijylij+Xj:i2Sljylji1A Ei,8i2N (5b)Xj2Sliylij)]TJ /F11 11.955 Tf 14.23 11.36 Td[(Xj:i2Sljylji=dirl,8i2N,l2L (5c)rlMlvl,8l2L (5d)rmXl2N)]TJ /F12 5.978 Tf -.58 -6.41 Td[(ltlmulm,8l2L,m2N+l (5e)Constraints( 5e )( 5i ),and( 5k ) (5f)rl0,8l2L. (5g)Nowconsiderthefollowingmasterproblemrelaxationof( 5 ). MaxXl2Lrl (5a)s.t.Constraints( 5b )( 5d ),( 5g ),and( 5k ) (5b)tlmwlmrm,8l2L,m2N+l (5c)wlmvl,8l2L,m2N+l (5d) 118

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vl2f0,1g,8l2L (5e)wlm2f0,1g,8l2L,m2N+l. (5f)Here,auxiliarybinaryvariablewlmcanequaloneonlywhentlmrm(D)andvl=vm=1.BecausethedelayconstraintspresentinMSM-FS1areabsentin( 5 ),theoptimalobjectivefunctionvalueto( 5 )matchesthatofMSM,andthereforeprovidesanupperboundontheoptimalobjectivefunctionvalueof( 5 ).Moreover,anyfeasiblesolutionto( 5 )inducesadirectedgraphG0=(L0,A0)withL0=fl2L:vl=1gasitsnodesetandA0=f(l,m):wlm=1gasitsarcset.(Moreprecisely,after( 5 )issolved,oneshouldpostprocessthesolutionandsetwlm=1ifpermittedby( 5c ),foreachl2Landm2N+l.)ByProposition 5.4 ,ifG0isHamiltonian,thenanyoptimalsolutionto( 5 )canbeusedtoconstructanoptimalsolutionto( 5 ).Letusdenethefollowingsetswithrespecttoanoptimalsolution( r, y, v, w)to( 5 ).Vq=fl2L: vl=qg,forq=0,1; (5)W1=f(l,m):l2L,m2N+l,and wlm=1g (5)W0=f(l,m):l2L,m2N+l,and wlm=0g[f(l,m):l2L,m2LnN+lg. (5)ThefollowingsubproblemseeksaHamiltoniancycleoverV1,whereedge(l,m)hasanobjectivefunctioncostcoefcientofdlm=0if(l,m)2W1,andacostofdlm=1if(l,m)2W0(includingthecasewhenm=2N+l).Iftheoptimalobjectivefunctionvaluetosubproblem( 5 )belowequals0,thenanyofitsoptimalsolutionsrepresentsaHamiltoniancycleinG0,whichinturnindicatesthatthecurrentoptimalsolutiontothemasterproblemcorrespondstoanoptimalsolutiontoMSM-FS1. MinXl2V1Xm2V1dlmxlm (5a)s.t.Xm2V1xml=1,8l2V1 (5b) 119

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Xm2V1xlm=1,8l2V1 (5c)Xl2SXm2SxlmjSj)]TJ /F4 11.955 Tf 17.93 0 Td[(1,8SV1,2jSjjV1j)]TJ /F4 11.955 Tf 17.93 0 Td[(2 (5d)xlm2f0,1g,8l2V1,m2V1. (5e)Notethat( 5 )isaninstanceoftheTravelingSalesmanProblem(TSP)[ 14 ].Letndenotetheoptimalobjectivefunctionvalueof( 5 ).Ifnispositive,thenanyHamiltoniancycleinG0(ifoneexists)usesatleastnarcsfromW0,andwemustaddanappropriatecuttingplanetothemasterproblem( 5 )toensuretheexistenceofaHamiltoniancycleinG0eitherbymodifyingr-values(whichpossiblymodiesW0andW1)orthecompositionofV1.Wenextdescribeourmethodofgeneratingcuttingplanesforthemasterproblem.Let(r,y,v,u)beafeasiblesolutionto( 5 );thissolutionisalsofeasibleto( 5 )bylettingwlm=ulm.Thefollowingpropositionpresentsaclassofcuttingplanesforthemasterproblem( 5 )thatarevalidforallfeasiblesolutionsof( 5 ). Proposition5.5. Supposethattheoptimalobjectivefunctionvalueofthesubproblem( 5 )given vand wisn>0.Thefollowinginequalityfor( 5 )cutsoffthesolutioncontaining vand w,andisvalidforallfeasiblesolutionsto( 5 ):2Xl2V1(1)]TJ /F3 11.955 Tf 11.96 0 Td[(vl)+X(l,m)2W0wlmn+Xl2V0vl. (5) Proof. Firstnotethat( 5 )cutsoff( v, w)whenn>0,becausetheleft-hand-sideof( 5 )evaluatesto0forthissolutionwhiletheright-hand-sideequalsn.Wesaythatlocationlisactiveifvl=1,andisinactiveotherwise.Supposethatafeasiblesolution(br,by,bv,bu)to( 5 )inducesaHamiltoniancycleHinwhichthereexistsp0inactivesinklocationsinV1,andr0activesinklocationsinV0.Notethatifp=jV1j,then 120

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( 5 )reducestoX(l,m)2W0wlmn)]TJ /F4 11.955 Tf 11.96 0 Td[(2jV1j+r,whichisvalidbecausenjV1jandP(l,m)2W0wlm=r.(TheequalityholdsbecauseinthiscaseHcontainsexactlyrarcswithbothendpointsinV0.)Therefore,wewillassumeintheremainderofthisproofthatpjV1j)]TJ /F4 11.955 Tf 19.04 0 Td[(1,i.e.,HvisitsatleastonesinklocationfromV1.SupposebycontradictionthatthesolutionassociatedwithHviolates( 5 ).Thenwemusthave2p+X(l,m)2W0bulmn)]TJ /F4 11.955 Tf 11.96 0 Td[(1+r. (5)Hence,Hcontainsatmostn)]TJ /F4 11.955 Tf 13.15 0 Td[(1+r)]TJ /F4 11.955 Tf 13.15 0 Td[(2parcsfromW0.Indexthesenodesasi1,j11,...,j1q1,...,it,jt1,...,jtqt,i1,wherei1,i2,...,itbelongtoV1andallothernodesbelongtoV0.(Ifqe=0forsomee=1,...,t,thenthetourproceedsfromietoie+1,whereit+1i1.)WebuildanewcycleH0byrstremovingallnodesinV0fromH,sothatH0initiallyconsistsofi1,i2,...,it,i1(asshowninFigure 5-3 ).Notethatinremovingnodesje1,...,jeqefromH,forsomee2f1,...,tgwithqe1,atotalofqe+1arcsinW0areremovedfromH0whileaddingatmostonearc(ie,ie+1)inW0toH0.Hence,H0nowcontainsatmostn)]TJ /F4 11.955 Tf 12.11 0 Td[(1)]TJ /F4 11.955 Tf 12.1 0 Td[(2parcsinW0.Next,supposethatweexpandH0toincludeallpnodesinV1thatwerenotcontainedinH.ThesepnodescanbeinsertedtoH0inanyarbitraryorder,witheachinsertionrequiringatmosttwoinactivearcs.Now,H0consistsofatmostn)]TJ /F4 11.955 Tf 12.02 0 Td[(1arcsinW0,whichcontradictstheassumptionthattheoptimalobjectivefunctionvalueof( 5 )isn.Thiscompletestheproof. Remark2.ThelaststepintheproofofProposition 5.5 canalternativelybestatedinthefollowingway:Ifp>0,includeallsinklocationsit+1,...,it+pinV1notpresentinH0(andH)bysimplyextendingH0tovisitalllocationsintheorderi1,i2,...,it+p,i1,which 121

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intheworstcaseaddsp+1morearcsinW0toH0.Then,H0spansV1andcontainsatmostn)]TJ /F3 11.955 Tf 12.46 0 Td[(parcsinW0whenp>0,andatmostn)]TJ /F4 11.955 Tf 12.46 0 Td[(1arcsinW0whenp=0.Thissuggeststhefollowingalternativevalidinequalityformasterproblem( 5 ):1+Xl2V1(1)]TJ /F3 11.955 Tf 11.96 0 Td[(vl)+X(l,m)2W0wlmn+Xl2V0vl. (5)Notethatwhile( 5 )isvalidandcutsoffsolutionsthatarenotcutoffby( 5 ),itdoesnotcutoffaninfeasiblemasterproblemsolutionwhenthesubproblemyieldsanoptimalobjectivefunctionvalueofn=1.However,wecanaddboth( 5 )and( 5 )tothemasterprobleminourcutting-planescheme,oradd( 5 )whenn=1and( 5 )otherwise.Abriefcomputationalstudyrevealsthattheuseof( 5 )byitselfinthecutting-planealgorithmisthemosteffectiveimplementation.2 Figure5-3. ConstructionofH0intheproofofProposition 5.5 Now,wepresentamethodthatpermitsustosetcertainw-valuesequaltozeroinapreprocessingphase.LetZLBdenotealowerboundontheoptimalobjectivefunctionvalueof( 5 ),e.g.,ascomputedviatheobjectivefunctionvalueofafeasiblesolutionto( 5 ).Todetermineifagivenarc(l,m)2Acouldbetraversedbythesinkinanoptimalsolutionto( 5 ),wecheckanecessaryconditionforittobeusedinatleastonefeasiblesolutionhavinganobjectivefunctionvaluethatisnolessthanZLB.Specically, 122

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if Xi2Ndi!mini:m2Smifemimgtlm+0@Xi2Nnfi:m2Smigdi1Atlm+(ZLB)]TJ /F3 11.955 Tf 11.96 0 Td[(tlm)Xi:m2Smidiminl2L,j2Slielij>Xi:m2Smi Ei, (5)then(l,m)cannotbeincludedinanyoptimalsolutionto( 5 )andhencewlm=0isvalidfor( 5 ).Thersttwotermsin( 5 )providealowerboundonthetotalenergyexpendedbysensornodesthatcancommunicatewiththesinkatlocationmduringthesink'stravelonarc(l,m),whilethethirdtermisalowerboundonthecombinedenergyexpendedbythesesensorsduringtherestofthenetworklifetime.Remark3.WecanalsousetheinformationfromanoptimalsolutiontoMSMtoaddvalidinequalitiestoMSM-FS1.LetL0LandsupposethatZTSP(L0)>ZMSM,whereZTSP(L0)istheminimumTSPtourlengthoverthesinklocationsinL0.Then,anyTSPtouroverasubsetofLthatvisitsallsinklocationsinL0isnotassociatedwithafeasibleto( 5 )becauseanysuchsolutionwouldhaveanobjectivefunctionvaluegreaterthanZMSM.Therefore,Xl2L0vljL0j)]TJ /F4 11.955 Tf 17.94 0 Td[(1 (5)isvalidfor( 5 ).Inparticular,iftlm>ZMSM 2,thenulm=0isvalidfor( 5 ).2 5.4BendersDecompositionThecutting-planealgorithmpresentedinSection 5.3 fortheMSM-FS1startsfromarelaxationoftheoriginallifetimemaximizationproblemwhoseoptimalsolutionmaynotadmitaHamiltoniancycleoftheactivesinklocations.ThealgorithmtheninducestheexistenceofaHamiltoniancyclebyiterativelyaddinginequalitiesofform( 5 )tothemasterproblem.Inthissection,weproposeadecompositionapproachthatstartsfromatouroverasubsetofthesinklocationsandreachesanoptimalsolutionbyaddingappropriateBenderscuts[ 59 ].Ourdecompositionapproachmakesthesink'srouting 123

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decisionsinthefollowingmasterproblem. MaxXl2LXm2N)]TJ /F12 5.978 Tf -.58 -6.41 Td[(ltmluml+ (5a)s.t.Xm2N)]TJ /F12 5.978 Tf -.58 -5.09 Td[(muml=vl,8l2L (5b)Xm2N)]TJ /F12 5.978 Tf -.58 -5.09 Td[(muml=Xm2N+lulm,8l2L (5c)Xl2SXm2 Sulmvk+vr)]TJ /F4 11.955 Tf 11.95 0 Td[(1,8SL,k2S,r2 S (5d)vl2f0,1g,8l2L (5e)ulm2f0,1g,8l2L,m2N+l, (5f)whereisdenedasfollows,givenvaluesu= uandv= v =MaxXl2Lzl (5a)s.t.Xl2L0@Xj2Slielijylij+Xj:i2Sljylji1A Ei,8i2N (5b)Xj2Sliylij)]TJ /F11 11.955 Tf 14.22 11.35 Td[(Xj:i2Sljylji=di0@zl+Xm2N)]TJ /F12 5.978 Tf -.58 -6.42 Td[(ltml uml1A,8i2N,l2L (5c)zlMl vl,8l2L (5d)zl0,8l2L (5e)ylij0,8i2N,j2Sli,l2L. (5f)Problems( 5 )and( 5 )togetherprovideanequivalentreformulationfor( 5 ).Letfi,gil,andhldenotethedualvariablesassociatedwithconstraints( 5b ),( 5c ),and( 5d ),respectively.Takingthedualof( 5 ),weget =MinXi2N Eifi+Xi2NXl2Ldi0@Xm2N)]TJ /F12 5.978 Tf -.58 -6.42 Td[(ltml uml1Agil+Xl2LMl vlhl (5a) 124

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s.t.Xi2N)]TJ /F3 11.955 Tf 9.29 0 Td[(digil+hl1,8l2L (5b)gil)]TJ /F3 11.955 Tf 11.96 0 Td[(gjl+elijfi+fj0,8i2N,j2Slinfsg,l2L (5c)gil+elisfi0,8i2N,s2Sli,l2L (5d)hl0,8l2L (5e)fi0,8i2N. (5f)Ifproblem( 5 )(whichservesastheBendersdualsubproblem)hasanunboundeddirection(bf,bg,bh),weaddthefollowingfeasibilitycuttotheBendersmasterproblem( 5 ).Xi2N Eibfi+Xi2NXl2LXm2N)]TJ /F12 5.978 Tf -.58 -6.41 Td[(l(ditmlbgil)uml+Xl2L(Mlbhl)vl0. (5)When( 5 )hasanoptimalsolution(f,g,h),weaddthefollowingoptimalitycuttothemasterproblem.Notethat( 5 )isneverinfeasible,becausesettinghl=1,foralll2L,andallothervariablesequaltozero,isalwaysfeasible.Xi2N Eifi+Xi2NXl2LXm2N)]TJ /F12 5.978 Tf -.57 -6.42 Td[(l(ditmlgil)uml+Xl2L(Mlhl)vl. (5) 5.5ComputationalResultsWeexaminethecomputationaleffectivenessoftheproposedalgorithmsinthischapteronacollectionofrandomlygeneratedinstances.Wegeneratedtheseinstancesbyrandomlyplacingsensornodesandsinklocationsinacircularareaofradius25.Asensornodecancommunicatewithsensornodesandsinklocationsinadiskofradius15aroundit.Weassumeaninitialenergyandxeddatagenerationrateof Ei=500anddi=50forallsensornodes.Tocalculatetherequiredenergyfortransmissionofaunitofdataalongthelink(i,j),weuseanenergymodelsimilartothatof[ 8 ].Morespecically,weuseelij=0.0005+0.000134ijand=0.0005,whereijdenotesthe 125

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distancebetweentwonodesiandjthatcancommunicatewithoneanotherwhilethesinkisatasinklocationl2L.Wehaveimplementedallmixed-integerprogramsinC++callingCPLEXversion11.0viaILOGConcertTechnology2.5.AllcodesarecompiledusingMicrosoftVisualC++2008andthecomputationalexperimentsarecarriedoutonaPChavinganIntelCore2QuadprocessorQ9500and4GBofmemory,runningWindows7.A1000-secondtimelimitisimposedonallrunningtimes.WerstprovideacomparisonbetweensolvingMSM-FS1bythecutting-planealgorithmofSection 5.3 andbysolvingproblem( 5 )directlyusingCPLEX.Forthisexperiment,wegeneraterandominstancesoftwodifferentsizes(jNj,jLj)=(50,15)and(jNj,jLj)=(60,20)aswellastwodifferentlevelsD=0.2andD=0.4forthemaximumtolerabledelay.Foreachcombinationofproblemsizeanddelaytolerancelevel,tenrandomnetworksaregenerated.WethengeneratetwoinstancesofMSM-FS1associatedwiththetwodifferentsinkspeeds,andsolvetheseinstancesbothdirectlyviaCPLEX(whichwecallCPLEXinourcomputationaltables),andalsobythecutting-planealgorithminSection 5.3 .OurCPLEXimplementationforMSM-FS1usesaspecialcallbackfunctioninsidethebranch-and-boundtreetoaddtheviolatedsubtoureliminationconstraintsateachnode.Thiscallbackfunctionisanimplementationofthefollowingseparationprocedure[ 52 ],where ulmand vldenotecurrentvaluesofvariablesulmandvl,respectively. BuildacompletedirectedgraphGhavingLasitsnodeset,andxthecapacityofeacharc(l,m)inGto ulm. Foreverypairofsinklocationslandmsuchthat vl+ vm>1,solveamaximumowproblemfromlandmonG. Ifthecapacityoftheresultingminimumcut(S, S)islessthan vl+ vm)]TJ /F4 11.955 Tf 12.58 0 Td[(1,thatis,ifPl2SPm2 S ulm< vk+ vr)]TJ /F4 11.955 Tf 12.49 0 Td[(1,thenaddtheviolatedinequalitytothelinearprogrammingrelaxationatthecurrentnode. 126

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Validinequalities( 5 )arealsoaddedtothemasterproblem( 5 )usingacallbackfunction.Weaddtheseinequalitiesonlyatbranch-and-boundnodeshavingintegersolutions.Wedonotincludeinequalitiesbasedon( 5 )inourpreprocessingstepbecauseefcientdeploymentoftheseinequalitiesrequiresaheuristicprocedureforobtaininggoodlower-boundsolutions,whichisbeyondthescopeofthischapter.Similarly,separationofinequalities( 5 )foragivensubsetofLrequiressolvinganinstanceoftheTSP.Therefore,wedonotaddtheseinequalitiestoourformulation.Table 5-1 containstheresultsofrunningthetwoalgorithmsonthegeneratedinstancesforD=0.2.Table 5-2 containstheresultsforD=0.4.Theseresultssuggestthatthecutting-planealgorithmisadvantageousonlyforinstancesinwhichafewcutsareneededtosolvetheproblemtooptimality.Table 5-3 presentstheresultsofrunningtheBendersdecompositionalgorithmofSection 5.4 forMSM-FS1versusadirectsolvebyCPLEXfortwodifferentsinkspeedsandtwoproblemsizes(jNj,jLj)=(50,15)and(jNj,jLj)=(60,20).Here,weassumeamaximumtolerabledelayofD=0.2timeunits.Allcuttingplanes,includingthesubtoureliminationandBendersfeasibilityandoptimalitycuts,areaddedusingacallbackroutine.TheBendersfeasibilityandoptimalitycutsareaddedonlyatbranch-and-boundnodeshavingintegersolutions.TheresultsclearlyfavortheuseofBendersdecompositionforbothsinkspeeds,astheaveragetimeforBendersdecompositionissignicantlyshorterthanthatofCPLEX.Table 5-4 providesacomparisonoftheperformanceofthetwoalgorithmsforthesameinstancesandthesamesinkspeedsbutwithanincreasedmaximumtolerabledelayofD=0.4timeunitsforallinstances.Inthiscase,theresultsforv=25stillsuggestusingtheBendersdecompositionalgorithm.However,resultsforinstanceswithv=50suggestusingadirectsolvebyCPLEXratherthantheBendersdecompositionalgorithm.Thecombinationoflowertraveltimesduetoincreasedsinkspeedandtheincreaseinmaximumtolerabledelaymakestheunderlyingsinkgraphsforv=50denserthan 127

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theircounterpartsforlowersinkspeedsandshortermaximumtolerabledelaylevels.Weconcludethatadirectsolveof( 5 )byCPLEXtendstooutperformBendersdecompositiononinstanceshavingrelativelydensesinkgraphs.OneexplanationfortheeffectivenessofBendersdecompositiononsparsesinkgraphsisthatsparsegraphscontainfewercycles,whichinturnrequiresfewerBendersdecompositioniterations.Weconcludethissectionbyprovidingacomparisonbetweentherunningtimesofformulations( 5 )and( 5 )insolvingMSM-FS2inTable 5-5 .TheseresultsareobtainedbysolvingMSM-FS2onthesamenetworksusedinTable 5-4 .ThemaximumtolerabledelayissettoD=0.4.Thesubtoureliminationconstraintsin( 5 )areinitiallyrelaxedandaddedinsideacallbackfunctionusingtheforegoingseparationprocedure.Inadditiontotherunningtimesforbothformulations(inCPUseconds),wereportthenumberofsubtoureliminationconstraintsaddedto( 5 )foreachinstance.Theseresultssuggestthatsolving( 5 )isgenerallyfaster,despitethefactthatitisnotpolynomiallysized.Also,theresultssuggestthatMSM-FS2tendstobemorechallengingandcomputationallycomplexthanMSM-FS1. 128

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Table5-1. Comparisonofthedirectsolveandcutting-planealgorithmsforMSM-FS1 v=25v=50 CPLEXCutting-PlaneCPLEXCutting-Plane TimeSubtoursTimeCutsTimeSubtoursTimeCuts MSM-FS1-50-15-0116.6814.516.44059.859MSM-FS1-50-15-027.0814.3112.57622.922MSM-FS1-50-15-030.600.801.202.80MSM-FS1-50-15-040.300.3027.272127.038MSM-FS1-50-15-055.6163.4233.07278.073MSM-FS1-50-15-060.900.907.3168.01MSM-FS1-50-15-074.785.2111.98436.758MSM-FS1-50-15-080.600.901.982.40MSM-FS1-50-15-090.500.507.33219.521MSM-FS1-50-15-103.883.1135.7156323.0211Average4.14.84.40.614.455.668.048.3MSM-FS1-60-20-0118.54466.51742.5216>1000.0>1274MSM-FS1-60-20-022.903.403.003.20MSM-FS1-60-20-0319.63224.4537.7160448.3918MSM-FS1-60-20-0431.23633.1864.2172919.7916MSM-FS1-60-20-052.408.2098.0104>1000.0>352MSM-FS1-60-20-0625.81623.4255.4100284.1256MSM-FS1-60-20-077.2020.7253.7306>1000.0>1320MSM-FS1-60-20-0847.44877.05925.152>1000.0>1070MSM-FS1-60-20-09223.552355.831415.0168>1000.0>95MSM-FS1-60-20-1025.35621.41470.3406>1000.0>654Average40.428.463.413.886.5168.4>765.6>685.5 129

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Table5-2. Performanceofthecutting-planealgorithmforMSM-FS1underincreaseddelaytolerance v=25v=50 CPLEXCutting-PlaneCPLEXCutting-Plane TimeSubtoursTimeCutsTimeSubtoursTimeCuts MSM-FS1-50-15-014.65225.0517.51213.542MSM-FS1-50-15-0212.36817.6195.4786.78MSM-FS1-50-15-031.502.601.802.00MSM-FS1-50-15-0423.78089.22711.1252>1000.0>1030MSM-FS1-50-15-0525.456137.25719.1210>1000.0>1352MSM-FS1-50-15-065.5167.451.682.20MSM-FS1-50-15-0716.75630.54223.5380>1000.0>2006MSM-FS1-50-15-081.601.3032.4108200.548MSM-FS1-50-15-095.03215.8915.7176>1000.0>783MSM-FS1-50-15-1022.894343.913420.8280>1000.0>218Average11.945.467.134.413.9150.4>522.5>548.7MSM-FS1-60-20-0144.1250>1000.0>103025.2208>1000.0>1021MSM-FS1-60-20-023.683.304.61410.40MSM-FS1-60-20-0320.392>1000.0>366133.1468>1000.0>366MSM-FS1-60-20-0452.4166>1000.0>28030.1278>1000.0>1046MSM-FS1-60-20-05154.0170>1000.0>12822.415046.76MSM-FS1-60-20-0647.4166292.710866.7688>1000.0>544MSM-FS1-60-20-0739.4276>1000.0>68237.7310138.8101MSM-FS1-60-20-0818.448>1000.0>4518.3112382.890MSM-FS1-60-20-09326.9168998.478355.3658>1000.0>362MSM-FS1-60-20-1073.5518>1000.0>46254.7820>1000.0>216Average78.0186.2>829.6>317.974.8370.6>657.9>375.2 130

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Table5-3. ComparisonofthedirectsolveandBendersdecompositionalgorithmsforMSM-FS1 v=25v=50 CPLEXBendersCPLEXBenders TimeSubtoursTimeBendersCutsSubtourCutsTimeSubtoursTimeBendersCutsSubtourCuts MSM-FS1-50-15-116.682.0286.4402.8596MSM-FS1-50-15-27.082.32812.5762.9580MSM-FS1-50-15-30.603.3281.204.79134MSM-FS1-50-15-40.301.71027.2723.28142MSM-FS1-50-15-55.6162.032433.0722.55126MSM-FS1-50-15-60.903.5107.3164.3548MSM-FS1-50-15-74.781.72811.9842.7790MSM-FS1-50-15-80.601.3101.982.3792MSM-FS1-50-15-90.501.7107.3322.4660MSM-FS1-50-15-103.881.92835.71563.06172Average4.14.82.11.76.414.455.63.16.3104.0MSM-FS1-60-20-118.5447.569642.521611.012406MSM-FS1-60-20-22.907.96743.007.44432MSM-FS1-60-20-319.6327.744837.71606.75392MSM-FS1-60-20-431.2366.045664.21728.510378MSM-FS1-60-20-52.405.52098.01047.78516MSM-FS1-60-20-625.8164.832455.41006.68312MSM-FS1-60-20-77.205.431653.73068.211424MSM-FS1-60-20-847.4486.868025.1527.16346MSM-FS1-60-20-9223.5527.67140415.01687.55248MSM-FS1-60-20-1025.3565.838070.34067.66514Average40.428.46.54.461.486.5168.47.87.5396.8 131

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Table5-4. PerformanceoftheBendersdecompositionalgorithmunderincreaseddelaytolerance v=25v=50 CPLEXBendersCPLEXBenders TimeSubtoursTimeBendersCutsSubtourCutsTimeSubtoursTimeBendersCutsSubtourCuts MSM-FS1-50-15-014.6523.0736967.51215.054391068MSM-FS1-50-15-0212.3683.0575805.47816.06841860MSM-FS1-50-15-031.504.35291481.805.72618396MSM-FS1-50-15-0423.7803.213814211.1252133.0972401834MSM-FS1-50-15-0525.4562.808712619.121032.199511562MSM-FS1-50-15-065.5164.3225481.688.03317758MSM-FS1-50-15-0716.7562.71479023.53806.45818754MSM-FS1-50-15-081.602.12169232.41085.38121756MSM-FS1-50-15-095.0322.66866015.717620.733541104MSM-FS1-50-15-1022.8943.5561018020.8280104.2851452366Average11.945.43.26.9106.213.9150.434.764.41145.8MSM-FS1-60-20-0144.12509.109941225.2208NANANAMSM-FS1-60-20-023.689.12794964.614133.3391643442MSM-FS1-60-20-0320.3927.126400133.146834.58561980MSM-FS1-60-20-0452.41668.631343830.1278NANANAMSM-FS1-60-20-05154.01709.0621451822.4150NANANAMSM-FS1-60-20-0647.41666.32631266.7688147.7471342798MSM-FS1-60-20-0739.42767.652942437.7310NANANAMSM-FS1-60-20-0818.4486.806734618.3112NANANAMSM-FS1-60-20-09326.91687.555236355.3658NANANAMSM-FS1-60-20-1073.551814.9471656654.782050.529621990Average78.0186.28.69.4414.874.8370.6NANANA NA:Instancenotsolvedduetomemorylimitations. 132

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Table5-5. ComparisonofsolutiontimesforMSM-FS2models( 5 )and( 5 ) v=25v=50 Formulation( 5 )Formulation( 5 )Formulation( 5 )Formulation( 5 ) TimeSubtoursTimeTimeSubtoursTime MSM-FS2-50-15-015.3508.87.168143.1MSM-FS2-50-15-028.98016.530.053413.6MSM-FS2-50-15-033.301.33.9846.4MSM-FS2-50-15-0419.810821.0>1000.0>3010>1000.0MSM-FS2-50-15-0521.57846.7>1000.0>2870>1000.0MSM-FS2-50-15-061.702.01.806.4MSM-FS2-50-15-0713.78424.535.7442>1000.0MSM-FS2-50-15-081.6181.621.1328397.6MSM-FS2-50-15-096.83216.7>1000.0>1932>1000.0MSM-FS2-50-15-1032.9120138.8>1000.0>2542>1000.0Average11.557.027.8>410.0>1181.0>556.8MSM-FS2-60-20-0161.6258815.8>1000.0>3574>1000.0MSM-FS2-60-20-024.903.32.904.1MSM-FS2-60-20-0335.7104493.3>1000.0>616>1000.0MSM-FS2-60-20-0460.8202522.8NANA>1000.0MSM-FS2-60-20-0596.1174>1000.0>1000.0>2256319.3MSM-FS2-60-20-0626.786117.7>1000.0>2004999.4MSM-FS2-60-20-0754.0398179.7>1000.0>4222>1000.0MSM-FS2-60-20-0818.794126.6>1000.0>2356>1000.0MSM-FS2-60-20-09408.2194634.5NANA>1000.0MSM-FS2-60-20-1045.0378>1000.0>1000.0>4308>1000.0Average81.2188.8>489.4NANA>832.4 133

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CHAPTER6CONCLUSIONSANDFUTURERESEARCHDIRECTIONSInthisdissertation,weconsideredseveralmathematicalprogrammingmodelsformaximizingthelifetimeofawirelesssensornetwork(WSN).InChapter 2 ,westudiedtwodifferentmodelsforthelifetimemaximizationinWSNswithmobilesinks.Onemodelprohibitsdelaytolerance(theMSM),andtheotherallowsit(thequeue-basedDT-MSM).Foreachofthesemodels,wedescribedlinearprogrammingformulationsandproposedacolumngenerationalgorithm.Weshowedthatinbothmodels,thesubproblemofthecorrespondingcolumngenerationalgorithmcanbesolvedbyDijkstra'salgorithm.Also,fortheMSM,itisshownthattheproblemisseparabletoseveralsmallerproblems.ThecomputationalresultsshowedtheefciencyoftheproposedcolumngenerationalgorithmfortheMSM,butrevealedthatLPisquiteefcientforthequeue-basedDT-MSMinstancesthatwegenerated.Oneofthekeystothesuccessfulapplicationofanoptimalroutingand/orrateallocationalgorithmtothereal-worldnetworks,includingourcolumngenerationalgorithms,iswhetherthealgorithmcanbeimplementedinadistributedmanner.Ingeneral,thesubproblemsarisingfromourcolumngenerationalgorithmsexhibitashortestpathstructure,whichareamenabletodecentralizedimplementation.However,themasterproblemsforthesealgorithmsarenotreadilysolvablebyadistributedalgorithm.Toapplythecolumngenerationalgorithms,atentativearrangementmaybetoletthesinksolvethemasterproblems.Despitethisshortcoming,ourcolumngenerationalgorithmsarestillmuchmoreopentodistributedimplementationandapplicationstoreal-worldnetworksthanastandard,monolithicLPalgorithm.Findingadistributedimplementationofourcolumngenerationalgorithmsisoneimportantfutureresearchdirection.Inaddition,wewillcontinuetoexploredecompositionapproachesforrelatedWSNlifetimeproblems,anddeterminecasesinwhichtheyoutperformsimplelinearprogrammingapproaches. 134

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InChapter 3 ,weproposedanalgorithmformaximizingtheWSNlifetimewhenthereisamobilesinkandtheunderlyingapplicationcantoleratesomedegreeofdelayindeliveringthedatatothesink.Ourmaincontributionwasthatthealgorithmisdistributed,andinaddition,mostlyuseslocalinformation.Suchanalgorithmcanbeimplementedbyparalleland/ordistributedexecutionandtheoverheadofmessagepassingislow.Itisalsopossibletoembedthealgorithmintoanetworkprotocolsothatthesenornodesandthesinkcanrunitdirectlyaspartofthenetworkoperation.Wealsogaveaproofofthealgorithm'soptimalityandtheboundednessofthequeuesizes.TheproofwasbasedonanalyzingaLyapunovdrift.Theresultsandtheanalyticaltechniquearebothsubstantiallydifferentfromthestandardoptimizationtheory.InChapter 4 ,wepresentedseveralmethodsofobtaininglowerandupperboundsontheoptimalobjectivefunctionvalueofanimportantgeometricvariantofthetravelingsalesmanproblem,calledtheclose-enoughtravelingsalesmanproblem(CETSP).TheabilitytoobtaintightlowerboundsontheoptimalCETSPtourlengthisvitalinevaluatingthequalityofanyheuristicsolutiontotheproblem.WeprovedseveralpropertiesofanoptimalCETSPtour,andusedthemtoestablishawayofpartitioningthecontinuoussolutionspace.ThispartitioningschemeisthenusedinformulatingthreedifferentintegerprogrammingproblemsthatyieldlowerandupperboundsontheoptimalCETSPtourlength.Inparticular,weformulatedalowerboundingproblemthatisaspecialinstanceofthecoveringtourproblem,anddescribedanalternativeformulationthatyieldsatighterlowerbound.WealsodescribedawayofreformulatingtheunderlyingintegerprogramthatmakesitamenabletoBendersdecomposition.Thesubprobleminthedecompositionschemendsashortestpathinaspecialdirectedgraphwithanexpandednodeset.Weobservedthatthisreformulationyieldsalowerboundthatisnotdominatedbythatoftheoriginalformulation,whilegreatlyenhancingtherunningtimeviaBendersdecomposition.Aniterativereningoftheunderlyingpartitioningschemeensuresthat 135

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thelowerboundsobtainedfromthethreemethodsconvergetotheoptimalCETSPtourlength.TheframeworkpresentedinChapter 4 canbeextendedtomanyotherproblemswithcontinuoussolutionspace.Candidatesincludethelawnmowingproblem[ 60 ]andthethepolygonexplorationproblem[ 61 ].Insomeapplications,aturningcostmaybepresentbasedontheanglethatisformedateachturnpointofthetour(see,e.g.,[ 62 ]).Formulation( 4 )may,inparticular,beextendedtoaddresspracticalinstancesofgeometrictourproblemswithturncosts.Anotherinterestinglineoffutureresearchwouldperformacomprehensivepolyhedralstudyontheproposedintegerprogrammingproblemsformulatedinthischapter.InChapter 5 ,weproposedexactalgorithmsforsolvingaversionofthelifetimemaximizationprobleminWSNswithamobilesink.Ourapproachdiffersfromthoseintheliteratureasittriestondanoptimalcyclictrajectoryforthesinkinthesensoreldwhenthesink'straveltimesbetweenitsdifferentlocationsarenotnegligible.Weformulatedseveralmixed-integerlinearprogramstomodeldifferentvariantsofthisproblem.Wethenfocusedononeoftheseformulationsandprovidedcutting-planeanddecompositionalgorithmstosolveit.Ourcomputationalexperimentsindicatedthattheproposedalgorithmscanpotentiallyimprovethesolvabilityoftheunderlyingmodel.AnimportantfutureresearchdirectionwouldfocusondevelopingefcientexactalgorithmsforMSM-FS2,whichprovestobeacomputationallychallengingproblem.MSM-FS2generallyyieldsahighernetworklifetimeandseemstobemoreappropriateforapplicationsinwhichreal-timetransmissionofdataislesscrucial.Anotherproblemmayinvestigatethecaseinwhichthesinktrajectorydoesnotneedtobeacycleandcantakeonmoregeneralforms,e.g.,agure-eighttrajectory.Furthermore,wealsosuggestthatourresearchcanbeextendedtothecaseofadelaytolerantmobilesinkmodelasformulatedin[ 26 56 ]. 136

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Fromapracticalpointofview,ourmodelsinChapter 5 areprimarilyintendedforndinganoptimalsinktrajectory.Givenaspecictrajectoryforthesink,anotherfutureresearchproblemwoulddeviseadistributedalgorithmforndingasetofoptimalroutingdecisionsforthesensornodesinadecentralizedmanner.Adistributedroutingalgorithmmightbemorevaluablethanacentralizedoneincertainapplications,e.g.,whentheroutingalgorithmcanbebuiltintothenetworkprotocol.WhilethereexistseveraldistributedalgorithmsforMSM(see,e.g.,[ 9 ]),noneofthemcanbedirectlyappliedtothemodelsproposedinChapter 5 .Hence,thedevelopmentofdistributedalgorithmsforsolvingthemodelspresentedinChapter 5 posesanotherfutureresearchchallenge. 137

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BIOGRAPHICALSKETCH BehnamBehdaniwasbornin1983inBirjand,Iran.Hereceivedhisbachelor'sandmaster'sdegreesinindustrialengineeringfromSharifUniversityofTechnology,Tehran,Iran.HethenjoinedtheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFloridainAugust2007.HereceivedhisDoctorofPhilosophydegreeinindustrialandsystemsengineeringfromtheUniversityofFloridainthesummerof2012.Hisresearchinterestslieintheoperationsresearcharea,includingintegerprogramming,networkoptimization,decompositionapproachestolarge-scaleoptimizationproblems,androbustoptimization. 143