Modeling and Optimization Techniques for Ensuring Resilience in Heterogeneous Networked Systems

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Modeling and Optimization Techniques for Ensuring Resilience in Heterogeneous Networked Systems
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Sorokin, Alexey V
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Doctorate ( Ph.D.)
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University of Florida
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Industrial and Systems Engineering
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Pardalos, Panagote M
Committee Co-Chair:
Boginski, Vladimir L.
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Geunes, Joseph P
Hager, William W

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energy -- modeling -- networks -- optimization -- resilience -- robustness -- stochastic -- telecommunication -- transportation
Industrial and Systems Engineering -- Dissertations, Academic -- UF
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This dissertationpresents modeling and optimization approaches for addressing some aspects ofensuring resilience of various types of networked systems with respect topotential failures of network components. In the context of the consideredproblems, resilience of a networked system can be broadly defined as theability of a system to maintain its functionality after intentional attacks orunintentional (random) failures of system components (i.e., nodes and/orlinks). Many real-world systems and infrastructures can be modeled asheterogeneous networks, which can be considered either separately, or as partsof larger interdependent networked systems, which may include power,communication, transportation and other types of networks that may mutuallyinteract and affect each other’s performance. After a brief introduction to theresilience of heterogeneous networked systems in the first part of thisdissertation, the second part studies a transmission expansion planning problemfor interdependent electricity and natural gas systems. Electricity generationis one of the largest natural gas consumption. Natural gas is playing anincreasingly important role in the global energy market because of itsenvironment friendly properties, especially for electricity generation. Westudy a stochastic transmission expansion planning model, which considersexpansion of electricity generation and transmission, as well as natural gastransmission and LNG terminals location planning. The third part of thedissertation addresses optimization models for network topology design thatenable end-to-end dual-path support in a distributed wireless sensor network.We consider the case of a stationary sensor network with isotropic antennas, wherethe control variable for topology management is the transmission power on networknodes. For optimization modeling, the network metrics of relevance arecoverage, robustness and power utilization. The forth part of the dissertationconsiders a formulation for the fixed charge network flow (FCNF) problemsubject to multiple uncertain arc failures, which aims to provide a robustoptimal flow assignment in the sense of restricting potential losses usingConditional Value-at-Risk. We show that a heuristic algorithm referred to asAdaptive Dynamic Cost Updating Procedure previously developed for the deterministicFCNF problem can be extended to the considered problem under uncertainty andproduce high-quality heuristic solutions for large problem instances. Thereported computational experiments demonstrate that the described procedure cansuccessfully tackle both the uncertainty considerations and the large size ofthe networks. Finally, the fifth part of this dissertation considers a specificaspect of resilience analysis of a system of two interdependent networks interms of finding critical components, which removal minimizes pairwise connectivityof the remaining networks, as well as identifying an exact minimum-cardinalityset of nodes, whose removal completely disconnects both networks. Our resultsgeneralize some of the well-known properties of the critical nodes and minimumvertex cover problems. We study two types of interdependencies that are common,for instance, in telecommunication and energy supply contexts.
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by Alexey V Sorokin.
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Thesis (Ph.D.)--University of Florida, 2012.
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Adviser: Pardalos, Panagote M.
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Co-adviser: Boginski, Vladimir L.
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MODELINGANDOPTIMIZATIONTECHNIQUESFORENSURINGRESILIENCEINHETEROGENEOUSNETWORKEDSYSTEMSByALEXEYV.SOROKINADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012AlexeyV.Sorokin 2

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Tomyfamily 3

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ACKNOWLEDGMENTS IwanttosayaspecialthankyoutomythesisadvisorsDr.PanosM.PardalosandDr.VladimirL.BoginskifortheirgreatsupportandanexcellentguidanceduringmystudyattheUniversityofFlorida.Theirvaluableknowledgeandusefulsuggestionsalwayshelpedmethroughoutmyresearchworkandacademiceducation.Ihighlyappreciatealltheirhelpandeffortstowardsmyacademicexcellenceandsuccess.IamalsoverygratefultothedissertationcommitteemembersDr.JosephGeunesandDr.WilliamHagerfortheirgreatassistancewithmyresearchworkatUniversityofFloridaandreviewingthisdissertation.Additionally,Iwashighlyfortunatetocollaboratewithoutstandingknowledgeablescholars,duringthefouryearsofmystudyattheUniversityofFlorida.IwouldliketothankArtyomNahapetyan,MarcoCarvalho,BaskiBalasundaram,EduardoL.Pasiliao,AlexanderVeremyev,QipengPhilZheng,NikitaBoyko,andJosephPortellafortheirassistancewithbothpracticalandtheoreticalaspectsofmywork.Theyhavealwaysbeenofgreatassistance,whichsignicantlyimprovedthequalityofthisdissertationandmyoverallworkatUniversityofFlorida.IwanttothankmywifeEkaterinaandmyparentsfortheirendlesssupportandencouragementsofallmybeginnings. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 LISTOFSYMBOLS .................................... 9 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 2STOCHASTICEXPANSIONPLANNINGMODELFORCOMBINEDPOWERANDNATURALGASSYSTEMSWITHRISKMANAGEMENTCONSTRAINTS 18 2.1TheStochasticExpansionModelforCombinedElectricityandNaturalGasSystems .................................. 19 2.2RiskManagementConstraintsfortheExpansionPlanningProblem .... 26 2.3ComputationalExperiments .......................... 28 3TOPOLOGYDESIGNFORON-DEMANDDUAL-PATHROUTINGINWIRELESSNETWORKS ..................................... 31 3.1TopologyControlinWirelessSensorNetworks ............... 32 3.2OptimizationModels .............................. 35 3.2.1Arc-disjointODDPNDformulation ................... 36 3.2.2Node-disjointODDPNDformulation .................. 38 3.2.3Cutcoveringformulation ........................ 38 3.3ComputationalExperiments .......................... 39 4COMPUTATIONALRISKMANAGEMENTTECHNIQUESFORFIXEDCHARGENETWORKFLOWPROBLEMSWITHUNCERTAINARCFAILURES ...... 44 4.1GeneralSetupofFixedChargeNetworkFlowProblems .......... 44 4.2ComputationalChallengesandHeuristicAlgorithms ............ 46 4.2.1Exactalgorithms ............................ 46 4.2.2Heuristicalgorithms .......................... 47 4.3FixedChargeNetworkFlowProblemsunderUncertainArcFailures ... 48 4.3.1Uncertainarcdisruptions ....................... 48 4.3.2Quantitativeriskmeasures ....................... 50 5

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4.3.3MathematicalprogrammingformulationforFCNFunderuncertaintywithCVaRconstraints ......................... 53 4.4AContinuousApproximationTechniqueforSolvingRFCNF ........ 55 4.4.1ConcavepiecewiselinearapproximationofRFCNF ......... 56 4.4.2AnalgorithmtosolveRFCNF--R ................... 60 4.4.3Dynamicscenarioupdate ....................... 63 4.5ComputationalExperiments .......................... 64 5CRITICALNODESANDVERTEXCOVERPROBLEMSININTERDEPENDENTNETWORKSWITHCASCADINGFAILURES ................... 70 5.1LiteratureOverview .............................. 71 5.1.1Reliabilityanalysisofinterdependentinfrastructures ........ 71 5.1.2Criticalnodesproblem ......................... 74 5.1.3Vertexcoverproblem .......................... 76 5.2CascadingFailuresinInterdependentNetworks ............... 76 5.3CriticalNodesProbleminInterdependentInfrastructures .......... 78 5.3.1CNPwithoutcascadingfailures .................... 78 5.3.1.1One-to-onecorrespondencebetweeninfrastructures ... 79 5.3.1.2One-to-multiplecorrespondencebetweeninfrastructures 80 5.3.2CNPwithcascadingfailures ...................... 82 5.3.2.1Type1interdependence ................... 82 5.3.2.2Type2interdependence ................... 84 5.4MathematicalProgrammingFormulationsforVertexCoverProbleminInterdependentNetworks ........................... 85 5.4.1Type1interdependence ........................ 85 5.4.2Compactproblemformulation ..................... 87 5.4.3ALPrelaxationapproximationalgorithm ............... 90 5.4.4DepthofcascadeS .......................... 91 5.4.5Type2interdependence ........................ 92 5.5ComputationalExperiments .......................... 93 6CONCLUSION .................................... 99 REFERENCES ....................................... 103 BIOGRAPHICALSKETCH ................................ 114 6

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LISTOFTABLES Table page 2-1Setsandindices ................................... 21 2-2Parameters ...................................... 22 2-3Decisionvariables .................................. 22 3-1Runningtime(inseconds)forsolvingtooptimality. ................ 40 3-2Averagetimetotherstfeasiblesolution ...................... 41 3-3Averagerelativegaptotherstfeasiblesolution ................. 41 4-1Setofproblems. ................................... 65 4-2AveragecomputationalresultsforgroupsG1,G2,andG3. ........... 66 4-3DetailedcomputationalresultsforgroupsG1,G2,andG3. ........... 67 4-4ComputationalresultsforgroupsG4,G5andG6. ................. 69 4-5AveragerelativedifferenceofADCUPresultswithLPrelaxationresultsforproblemsets7-9. ................................... 69 5-1ComputationalresultsforproblemformulationsF1andF1c,n=200(n1=n2=100). ....................................... 94 7

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LISTOFFIGURES Figure page 1-1Conceptualviewofawirelesssensornetworkandanexampleofinformationow. .......................................... 13 1-2Multi-pathroutinginwirelesssensornetworks. .................. 14 2-1AgraphicaldepictionofVaRandCVaR. ...................... 27 2-2MinimalcostandCVaRrestrictions ......................... 29 2-3ObjectivevaluevsgasandelectricityCVaRrestrictionsGandEL ....... 30 3-1Symmetricnetworktopology. ............................ 33 3-2Asymmetricnetworktopology. ............................ 34 3-3Illustrativeexamplesofarc-disjointandnode-disjointODDPNDsolutions. ... 42 3-4Illustrationofpossiblearc-disjointandnode-disjointdualdatapaths. ...... 43 4-1AnexampleofoptimalsolutionstructureforRFCNPwithCVaRconstraints .. 55 4-2Approximationoffunctionfij(xij). .......................... 57 4-3ObjectivefunctionterminRFCNF-problem. ................... 59 5-1Anexampleofa2-layerinterdependentnetwork ................. 77 5-2One-to-onecorrespondenceina2-layernetwork.Connectionsbetweenthelayersarepresentedbybluecoloredarcs. ..................... 79 5-3One-to-multipledirectconnectionina2-layernetwork. .............. 81 5-4DepthofcascadesSfordifferentgraphsizesandnumberofinterdependencyedges. ......................................... 95 5-5Theminimumvertexcoversizeforuniformandpower-lawgraphsforType1interdependence,n=200(n1=n2=100) ..................... 96 5-6TheminimumvertexcoversizeforType2interdependence,n=50(n1=n2=25) ........................................ 97 5-7Minimumvertexcoversizesfortwonetworks((a)-uniform,(b)-powerlaw)withedgedensities0.5fortwotypesofinterdependencies,n=50(n1=n2=25),S=3. ......................................... 98 8

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LISTOFSYMBOLS,NOMENCLATURE,ORABBREVIATIONS ADCUP AdaptivedynamiccostupdatingprocedureCNP CriticalnodesproblemCVaR Conditionalvalue-at-riskDCUP DynamiccostupdatingprocedureE SetofedgesFCNF FixedchargenetworkowLNG LiqueednaturalgasLP Linearprogrammax() Maximummin() MinimumMIP MixedintegerprogramODDPND On-demanddual-pathnetworkdesigns.t. subjecttoNP NondeterministicpolynomialV SetofverticesVaR Value-at-riskWSN Wirelesssensornetworks 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyMODELINGANDOPTIMIZATIONTECHNIQUESFORENSURINGRESILIENCEINHETEROGENEOUSNETWORKEDSYSTEMSByAlexeyV.SorokinAugust2012Chair:PanosM.PardalosCochair:VladimirL.BoginskiMajor:IndustrialandSystemsEngineering Thisdissertationpresentsmodelingandoptimizationapproachesforaddressingsomeaspectsofensuringresilienceofvarioustypesofnetworkedsystemswithrespecttopotentialfailuresofnetworkcomponents.Inthecontextoftheconsideredproblems,resilienceofanetworkedsystemcanbebroadlydenedastheabilityofasystemtomaintainitsfunctionalityafterintentionalattacksorunintentional(random)failuresofsystemcomponents(i.e.,nodesand/orlinks).Manyreal-worldsystemsandinfrastructurescanbemodelledasheterogeneousnetworks,whichcanbeconsideredeitherseparately,oraspartsoflargerinterdependentnetworkedsystems,whichmayincludepower,communication,transportationandothertypesofnetworksthatmaymutuallyinteractandaffecteachother'sperformance. Afterabriefintroductiontotheresilienceofheterogeneousnetworkedsystemsintherstpartofthisdissertation,thesecondpartstudiesatransmissionexpansionplanningproblemforinterdependentelectricityandnaturalgassystems.Electricitygenerationisoneofthelargestnaturalgasconsumption.Naturalgasisplayinganincreasinglyimportantroleintheglobalenergymarketbecauseofitsenvironmentfriendlyproperties,especiallyforelectricitygeneration.Westudyastochastictransmission 10

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expansionplanningmodel,whichconsidersexpansionofelectricitygenerationandtransmission,aswellasnaturalgastransmissionandLNGterminalslocationplanning. Thethirdpartofthedissertationaddressesoptimizationmodelsfornetworktopologydesignthatenableend-to-enddual-pathsupportinadistributedwirelesssensornetwork.Weconsiderthecaseofastationarysensornetworkwithisotropicantennas,wherethecontrolvariablefortopologymanagementisthetransmissionpoweronnetworknodes.Foroptimizationmodeling,thenetworkmetricsofrelevancearecoverage,robustnessandpowerutilization. Theforthpartofthedissertationconsidersaformulationforthexedchargenetworkow(FCNF)problemsubjecttomultipleuncertainarcfailures,whichaimstoprovidearobustoptimalowassignmentinthesenseofrestrictingpotentiallossesusingConditionalValue-at-Risk.WeshowthataheuristicalgorithmreferredtoasAdaptiveDynamicCostUpdatingProcedurepreviouslydevelopedforthedeterministicFCNFproblemcanbeextendedtotheconsideredproblemunderuncertaintyandproducehigh-qualityheuristicsolutionsforlargeprobleminstances.Thereportedcomputationalexperimentsdemonstratethatthedescribedprocedurecansuccessfullytackleboththeuncertaintyconsiderationsandthelargesizeofthenetworks. Finally,thefthpartofthisdissertationconsidersaspecicaspectofresilienceanalysisofasystemoftwointerdependentnetworksintermsofndingcriticalcomponents,whichremovalminimizespairwiseconnectivityoftheremainingnetworks,aswellasidentifyinganexactminimum-cardinalitysetofnodes,whoseremovalcompletelydisconnectsbothnetworks.Ourresultsgeneralizesomeofthewell-knownpropertiesofthecriticalnodesandminimumvertexcoverproblems.Westudytwotypesofinterdependenciesthatarecommon,forinstance,intelecommunicationandenergysupplycontexts. 11

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CHAPTER1INTRODUCTION Thisdissertationdescribesnovelapproachestomodelingandoptimizationofheterogeneousnetworkedsystems,withspecialemphasisontheirresiliencytonetworkcomponentfailures.Networkedsystemsariseindiverseapplicationareasnowadays;therefore,inthisdissertationwewillconsiderdifferenttypesofnetworks,includingenergy,communication,andtransportationnetworks,aswellasinterdependentcombinationsofsuchnetworks. Chapter 2 considersastochasticoptimizationmodelforexpansionplanningofinterdependentnaturalgasandelectricitysystems.Concernedaboutglobalwarmingandshortageofcrudeoil,peoplebecomemoreinterestedinnaturalgas,whichisacleanerenergysourceandabundantinmanyplaces.Naturalgasmainlyconsistsofmethane,andwhenburntitreleasesalargeamountofenergyandlessgreenhouseemissionsthanoilandcoal.Asthemodernizationandindustrializationarecoveringthewholeglobe,peoplearemorethaneverdependentonenergysupply.Combinedcyclegasturbinesplayaveryimportantroletosupplypeoplegreenerelectricity,sincethepowergeneratorsreleaselessgreenhouseemissionsthantheelectricitypowergeneratorsthatusecoaloroil.AccordingtotheU.S.EnergyInformationAgency(EIA)AnnualEnergyOutlook2011themajorityofnewgenerationcapacityadditionsin2010willusenaturalgasforelectricityproduction,signicantlyincreasingthedemandfornaturalgasduringtheseyears[ 47 ]. Thischapterproposesastochasticprogrammingmodelforgeneralizedelectricityandgassystemsexpansionproblem,aswellasdescribesamethodologyforcopingwiththedemanduncertaintyforelectricityandnaturalgasinfuture.Becauseoftheimbalanceinnaturalgasbetweencountries,itisimportanttostudybothelectricity 12

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andnaturalgastransmissionnetworksexpansionplanning,aswellasLNGterminallocationplanningtogether.Thechapterdescribesthestochasticoptimizationmodelforexpansionplanningofbothnaturalgasandelectricitysystemsthatinsteadoftakingintoconsiderationallthescenarios,eventhosethatareunlikelytohappen,utilizesConditionalValue-at-Riskriskmeasure,whichallowsonetotakeintoaccounttheaverageofacertainpercentageofworst-casescenarios. Chapter 3 proposesoptimizationmodelsforon-demanddual-pathnetworkdesignforwirelesssensornetworks(WSN),thatarekeytechnologyenablersforvarioussystemsandcriticaloperationalsettingsincludingenvironmentalmonitoring,militaryoperationssupport,disasterrelief,andindustrialplantmonitoring[ 11 ].WSNsareoftenconguredasmeshwirelessnetworks,whereindividualsensornodesinteractonlywithitsimmediatepeers(i.e.othernodeswithinradiorange)andrelyonmulti-hopcommunicationsforend-to-endinformationsharingandcoordination.AsillustratedinFigure 1-1 ,informationcollectedbyindividualsensorsisgenerallypropagatedthroughthenetworktoagatewaynode,whichisresponsibleforbridgingtheWSNwitharemotemonitoringandcontrolinfrastructure. Figure1-1. Conceptualviewofawirelesssensornetworkandanexampleofinformationow. 13

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Themulti-hoppathillustratedinFigure 1-1 describeshowsensordataisrelayedtothegatewaynode.Thepathisdynamicallyconstructedandmaintainedbydistributedroutingalgorithmsrunningoneverynode.Thegoaloftheroutingalgorithmistoalloweachnodetodeterminethebestnext-hop(i.ethebestneighbor)todeliveradatapacket,givennaldestination.InFigure 1-1 asequenceordatapackets(e.g.images)isgeneratedbythesensor,andeachpacketisforwardedthroughthenodesinthehighlightedpath,tothegatewaynode. Routingalgorithmsaredesignedtoadapttotheunderlyingconnectivitybetweennodes.Mostalgorithmsrelyonlocalizedlinksensingtoestimatethetopologyandidentifyoptimummulti-pathroutesfordifferentdestinations.Adhocprotocols,inparticular,aredesignedtoprovidethatfunctionwithoutdependenciesonxedorcentralizedinfrastructures,theyoftenrelyonpeer-to-peerstatesharingandcoordination.Themulti-hoppathcanbeestablishedwithdifferentcriteriasuchasminimumdistance(aperformancemetric),minimumpowerconsumption,orgeographicalconstraints.Basedonspecicgoalsandrequirements,thealgorithmsweighlinkinformationtocalculatethepathaccordingly. Figure1-2. Multi-pathroutinginwirelesssensornetworks. 14

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ThemodelsproposedinChapter 3 considerpowerconsumptionofsensorsasaperformancemetricanddesignthetopologytocontainatleasttwoarc-disjointorinternallynode-disjointpathsfromthedesignatedtransmittertothedesignatedreceiver.Suchapproachissuitableforwirelessnetworkwithcentralizedcontrol,wherethecommandscanbetransmittedfromacentralnodetoallthenodesinthewirelessnetwork.Theproposedmodelemploysaquadraticobjectivefunctionofpowerconsumptionfortransmissionradius,whichisarealisticassumption,andconsidersdependenceofcommunicationlinkscreationontransmissionradius. Chapter 4 considersthexedchargenetworkowproblemwithuncertainfailuresofnetworkcomponents(arcs),whichcanbeutilized,forinstance,inthecontextofresilienceanalysisoftransportationorcommunicationnetworks.Fixedchargenetworkowproblemattractsalotofattentionamongresearchers,sinceavarietyofproblems,suchassinglesourceproblem,lot-sizeproblem,networkdesign,investmentdecision,pricingpolicy,scheduling,andmanyotherpracticalproblemsthatariseinthesupplychain,logistics,transportationscience,andinformationnetworkscanbemodeledasaFCNFproblem[ 50 92 ].Uncertaintiesinxedchargenetworkowproblemshavebeenmainlyaddressedinthepreviousliteraturefromtheviewpointofuncertaindemandsandcosts[ 8 51 54 81 102 119 120 ].Althoughtheseparametersareclearlyimportant,anothercrucialissuethatneedstobeaddressedinmanyreal-worldsituationsisthepresenceofpotentialdisruptions/failuresofnetworkarcs,whichareuncertainbynatureandmaybecausedbynaturalorman-madefactors.Inthecontextoftransportationnetworkinfrastructuretheuncertaindisruptionsareoftencausedbyweatherconditions(includingnaturaldisasters)alongwithmanyotherpossiblefactors.Similarissuesalsooftenoccurinthedesignandoperationofcommunicationnetworkinfrastructure,powergrid,andotherrelatedapplications[ 3 70 ].Themainconceptual 15

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differenceoftheproblemsetupwithmultipleuncertainarcfailuresfromthepreviouslyconsideredproblemformulationswithuncertaindemandsand/orcostsisthefactthatthestructure(topology)ofthenetworkitselfisnotdeterministicanymore,whichcreateschallengesintermsofoptimalrobustowassignmentsinthesenetworks,sincetheobtainedoptimalsolutionsneedtobenotonlycost-efcient,butalsorobustwithrespecttopotentialmultiplearcfailures.Thesefailurescancausethelossofowthroughanetwork,andtheimportantquestionsthatneedtoberigorouslyaddressedare:1)Howtoquantifytherisksofcollaterallossesofowassociatedwithuncertainarcfailures?2)Howtoefcientlyminimizeorrestricttheselossesintheframeworkoflarge-scalexedchargenetworkowproblems?ThemodelproposedinChapter 4 addressesthesequestionsandutilizesConditionalValue-at-Risktomeasurecollaterallossesassociatedwithuncertainarcfailures,whichresultsinanetworkowassignment,whichisbothcost-efcientandrobust.Thechapteralsodescribesaheuristicalgorithmfortheproposedproblemthatprovideshigh-qualitysolutionsinareasonablecomputationaltime. Chapter 5 describessomeaspectsofresilienceanalysisofinterdependentnetworkinfrastructuresinthecontextofthegeneralizedcriticalnodesandvertexcoverproblems.Interdependentinfrastructuresaremodeledasnetworkswitharcsbetweennetworksrepresentingprovisionofservicesbetweeninfrastructures.Reliabilityanalysisforsuchinterdependentinfrastructurescanbedrasticallydifferentthanforseparatenetworksandmayleadtonon-trivialresults[ 64 ].Analyzingresilienceandinterdependenciesinsuchnetworksmayimposesignicantchallenges.Thesolutiontothecriticalnodesproblemininterdependentnetworksgivesanexactsetofnodesthatshouldberemovedfromthenetworkstominimizepairwiseconnectivityoftheremainingnetworks.Vertexcoverproblemcanbeconsideredasaspecialcaseofthecriticalnodesproblem,the 16

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solutionforthevertexcoverproblemininterdependentnetworksgivesanexactsetofnodesthatshouldberemovedfromthenetworkstomakethesystemofinterdependentinfrastructurescompletelydisconnected.Thesectiondescribesnewoptimizationmodels,aswellasanapproximationalgorithmforthevertexcoverproblem,whichgeneralizetheresultsknownfortheclassicalvertexcoverproblemonasinglenetwork. Finally,Chapter 6 concludesthedissertationandoutlinespotentialfurtherresearchdirections. 17

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CHAPTER2STOCHASTICEXPANSIONPLANNINGMODELFORCOMBINEDPOWERANDNATURALGASSYSTEMSWITHRISKMANAGEMENTCONSTRAINTS AccordingtoU.S.EnergyInformationAgency(EIA),electricitygenerationisoneofthelargestpartofnaturalgasconsumptionintheU.S.Suchinterdependencebetweenelectricityandgassystemsshowstheimportanceofconsideringthesetwosystemstogether.AccordingtoEIAInternationalEnergyOutlook2010,worldaveragereserves-to-productionisestimatedtobe60years.However,thehighestratiosbyregionare46yearsforCentralandSouthAmerica,68yearsforAfrica,72yearsforRussia,andmorethan100yearsforMiddleEast.Duetosuchlargeimbalanceingasreserves,theintercontinentaltransportationhastobeconsidered.Sofar,themainintercontinentaltransportationisLNGshipment,wherenaturalgasisliquiedandtransportedbyLNGcarriersbetweencountries.Theincreaseofdemandforbothelectricityandnaturalgasduringthefollowingfewdecadesshowsthatitisimportanttoconsiderelectricityandgasnetworks,aswellasLNGlocationstogether. Zhengetal.[ 125 ]provideacomprehensivesurveyofoptimizationtechniquesappliedinnaturalgasindustry.Inthispaper,theauthorsalsoproposedastochasticmodel,whichconsiderstransmissionpipelinenetworkexpansion,aswellasLNGterminallocationplanningproblems.InvestmentdecisionsweremodeledbyusingbinaryvariablesforchoosingapredeterminedexpansionprojectforapipelineorLNGterminal.Formodelingthegassystem,theauthorsusedageneralizednetworkowmodelasdescribedbyBrooksandNeil[ 20 ].Anextensivesurveyaboutoptimizationmethodsappliedtoelectricitytransmissionnetworkexpansionproblemcanbefoundin[ 114 ].Inthischaptertheauthorsbrieydescribemostofthealgorithmsandmodelsusedintherecentliteraturefortransmissionexpansionproblemandprovideacomprehensivelistofreferences.Latorreetal.[ 76 ]presentanexcellentbibliographicalsurvey,whichprovides 18

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classicationofpapersontransmissionexpansionplanningpublishedbefore2003.Atwo-partpaper[ 103 ]and[ 104 ]describesamulti-periodgeneralizednetworkowmodelwithenergyowsfromcoalandgassupplierstoelectricityloads.Therstpartofthispaperdescribesmodelingtechniquesandproposesamodelformulationwherefuelsupplyandelectricitydemandnodesareconnectedbytransportationnetwork.Besideseconomicalaspects,themodelalsoconsidersenvironmentalimpacts.Thesecondpartofthepaperpresentstheimplementationproceduresaswellaspresentscasestudies,reportscomputationalresults,andhighlightsresearchdirectionsforfuturework.MoraisandMarangonLima[ 87 ]proposedapricingtechniqueforbothelectricitytransmissionandgaspipelinesystems.Theauthorsshowedhowfourdifferentchargingmethodscanaffectthelocationofnewgenerators.Ahmed[ 1 ]describesthedifcultyofstochasticprogrammingandintegerprogramming,andprovidesanoverviewofresearchprogressandassociatedchallenges.ForstochasticprogramswithbinaryvariablespresentonlyattherststageL-shapedmethod[ 113 ]orBendersdecompositioncanprovidesignicantimprovementsincomputationaltime.BirgeandLouveaux[ 18 ]provideanextensivedescriptionofstochasticprogramming,aswellasappropriatesolutionsalgorithms. 2.1TheStochasticExpansionModelforCombinedElectricityandNaturalGasSystems Theobjectiveoftheproposedoptimizationproblemistobothmeetthedemandsandmaximizethepresentworth.Inthismodel,weassumetheexpansionsarediscrete,whichisactuallywhatishappeningnow.Forexample,thediametersofthegaspipelinesarediscretewhenonetriestobuythemfrommanufacturers.ThecapacityexpansionplanofaLNGterminaloragas-redpowerplantisselectedfromasetofproposedprojects. 19

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Withbinaryvariableskijdenotingwhetherakijexpansionismadeforarc(i,j),thetotalcostofpipelineexpansioncanbemodeledasCostARC=X(i,j)2AGXk2Kijckijkij. Similarly,theLNGterminalopeningcostismodeledusingbinaryvariableskiasfollows:CostLNG=Xi2NLNGXk2Kickiki. Binaryvariablexijdenoteswhetherconstructionofaproposedelectricityline(i,j)shouldbeperformedwiththetotalcostofelectricitylinesexpansiondenedasfollows:CostEL=X(i,j)2AELPpijxij. Besidesthetransmissioncapacityexpansion,wealsoconsidergenerationcapacityexpansionofpowerplants.Fromasetofpossibleexpansionprojectsforeverypowerplant,wemustchosetheonewhichsatisestheelectricitydemandatminimumcost.Thetotalcostofgeneratorsexpansionisthefollowing:CostGEN=Xi2NGENXk2Kirkiyki. Inthemodelformulation,weconsidertwointerdependentnetworksforelectricityandgassystems.Everynodecanbelongtooneormoreofthefollowingsets:NGthesetofnodesinthegasnetwork,i.e.anodeicanbeusedforproduction,consumptionortransportationofnaturalgas.AnodecanproducegasifithasaLNGterminalorsomeothersourceofgaswhichweconsiderasaselfsupply.Similarly,NEListhesetofnodesintheelectricitynetwork.Forthegas-redpowerplantswedeneasetNGEN,whichisasubsetofNG\NEL.Theanalogousconceptappliestoarcsofgasandelectricity 20

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networks.AGisthesetofallarcs(pipelines)inthegasnetwork,AEListhesetofexistingelectricallinesandAELCisthesetofcandidateelectricallinesthatareconsideredforexpansion.ThecompletelistofsetsusedinthemodelispresentedintheTable 2-1 Table2-1. Setsandindices NGNodesinthegasnetworkNELNodesintheelectricitynetworkNLNGPossibleLNGterminals,NLNGNGNGENGas-redpowerplants,NGEN(NG\NEL)AGArcs(pipelines)inthegasnetworkA+iOutgoingarcsfromnodeiinthegasnetworkA)]TJ /F8 7.97 Tf 0 -8.28 Td[(iIncomingarcstonodeiinthegasnetworkAELExistingelectricallinesAELCCandidateelectricallinesKijExpansionsizesforanarc(i,j)KiExpansionsizesforanodei TheparametersofthemodelarepresentedintheTable 2-2 andareusedasaninputdataforthemodel.Amongthemkijisthesizeofkthexpansionprojectforagaspipeline,and,similarlykiisthesizeofLNGterminalorelectricitygeneratori.Thenewelectricitylinescanbeselectedfromasetofproposedprojectsinadditiontoexistinglines.TheotherparametersusedinthemodelaresummarizedintheTable 2-2 .ThecontinuousandbinarydecisionvariablesusedinthemodelareintroducedintheTable 2-3 WiththenotationsdenedintheTables 2-1 2-2 and 2-3 ,wecannowdescribetheproposedstochasticprogram.Theobjectivefunctionofthemodelisthetotalcostofexpansioninvestment,gastransportationandelectricitygeneration.TheexpansioncostconsistsofLNGexpansioncost,gaspipelinesexpansioncost,electricitygenerationcapacityexpansioncastandthecostofconstructingnewelectricallines.Alloftheobjectivefunctiontermsweredescribedabove.Theconstraintsthatmodelnaturalgassystemarepresentedin( 2 )( 2 ).Constraint( 2 )denesgasowbalance 21

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Table2-2. Parameters kijThekthexpansionsizeofgaspipeline(i,j)2AGkiThekthexpansionsizeofLNGportorgeneratori2NLNG[NGENckijExpansioncostsofsizekijforarc(i,j)2AGckiCostoftheexpansionofsizekiforLNGportorgeneratori2NLNG[NGENhijUnittransportationcostofgaspipeline(i,j)2AGlijTransmissionlossrateforarc(i,j)2AGSFiSelfsupplylimitofnodei2NGSLiLNGsupplylimitofnodei2NLNGu ijCurrentcapacityofgaspipeline(i,j)2AGv iCurrentcapacityofLNGporti2NLNGdGiDemandforgasatnodei(notforpowerplants),i2NGdELiDemandforelectricityatnodei2NELg maxiCurrentlimitofgeneratori2NGENfELmaxijCapacityofelectricalline(i,j)2AELiEfciencyofpowerplanti2NGENijSusceptanceofelectricalline(i,j)2AELmaxMaximumvoltageangledifferenceintheelectricitynetworkpijCostofconstructinganewelectricalline(i,j)2AELCriExpansioncostsoftheprojectkiongeneratori2NGENpriUnitgenerationcostforpowerplanti2NGENfELmaxijPowerlimitforelectricalline(i,j)2AEL[AELP Table2-3. Decisionvariables kijDenoteskijexpansionforgaspipeline(i,j)2AGkiDenoteskiexpansionforLNGporti2NLNGxijDenoteswhetheranewelectricalline(i,j)2AELCshouldbeconstructedykiDenoteswhetherakiexpansionismadeforgeneratori2NGENfijGasowofarc(i,j)2AGsiTotalsupplyfromnodei2NGziFutureLNGsupplyfromnodei2NLNGuijThecapacityofpipeline(i,j)2AGviThecapacityofLNGporti2NLNGdGPiConsumptionofgasbypowerplanti2NGENgiElectricitygeneratedatapowerplanti2NGENfELijPowerowatelectricalline(i,j)2AEL[AELCiVoltageangleatabusi2NEL 22

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constraint,whereincomingowminusoutgoingowmultipliedbyoneminuslossrateareequaltogassupplyminusgasdemandforcustomersorindustrialuse,andminusdemandforgasforelectricitygenerationatthatnodeifthereisagas-redpowerplantthere.Thelossrateimposedbythefactthatgaspumpsconsumegasinordertotransportitoverthenetworkofpipelines.Constraint( 2 )denearccapacityand( 2 )connectsarccapacitywiththeexpansionprojects.( 2 )limitsthethroughputcapacityofLNGterminalsand( 2 )showsthatLNGportcapacitycannotexceeditsownLNGsupplylimit.Constraint( 2 )requiresthatLNGterminalsupplycannotexceeditsownself-supplylimitplusLNGsupply.LNGterminalsexpansionprojectsareconsideredinconstraint( 2 ),andgassupplyfromeverynodeislimitedbyitsself-supplyinequation( 2 ).X(i,j)2A+ifij())]TJ /F11 11.955 Tf 19.62 11.36 Td[(X(j,i)2A)]TJ /F12 5.978 Tf 0 -6.41 Td[(i(1)]TJ /F4 11.955 Tf 11.95 0 Td[(lji)fji()+si()=dGi()+dGPi(),8i2NG,2, (2)fij()+fji()uij,8(i,j)2AG,2, (2)uij=u ij+Xk2Kijkijkij,8(i,j)2AG, (2)zi()vi,8i2NLNG,2, (2)zi()SLi(),8i2NLNG,2, (2)si()zi()+SFi(),8i2NLNG,2, (2)vi=v i+Xk2Kikiki,8i2NLNG, (2)si()SFi(),8i2NGnNLNG,2, (2)si(),fij(),uij(),vi(),zi()0,8(i,j)2AG,i2NG,2, (2)kij2f0,1g,8k2Kij,(i,j)2AG, (2)ki2f0,1g,8k2Ki,i2NLNG, (2) 23

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Electricityexpansionandtransmissionconstraintsareimposedbyequations( 2 )( 2 )below.Constraint( 2 )representstheefciencyofagas-redpowerplantandconnectselectricityandgassystemstogether.Powerbalanceateachnodeisenforcedbyconstraint( 2 ),wherepowerthoughexistingelectricallinespluspowerthroughproposedlinespluselectricitygenerationatthatnodeareequaltoelectricitydemand.SecondKirchhoff'slawsareimplementedinconstraints( 2 )forexistingelectricallinesand( 2 )forproposedlines.Constraints( 2 )and( 2 )imposetransmissioncapacityofexistingandproposedelectricallines.( 2 )limitstheoutputofageneratortoitsphysicalcapabilitiesincludingexpansionprojectsin( 2 ).Voltageanglelimitsareimposedby( 2 ).Ageneratorihasanupperlimitg maxi+Pk2Kikiykifortheelectricityproduced.gi()=idGPi(),8i2NGEN,2, (2)X(i,j)2AELfELij()+X(i,j)2AELCfELij()+gi()=dELi(),8i2NEL,2, (2)fELij())]TJ /F3 11.955 Tf 11.95 0 Td[(ij(i())]TJ /F3 11.955 Tf 11.96 0 Td[(j())=0,8(i,j)2AEL,2, (2)fELij())]TJ /F3 11.955 Tf 11.95 0 Td[(ijxij(i())]TJ /F3 11.955 Tf 11.96 0 Td[(j())=0,8(i,j)2AELC, (2))]TJ /F4 11.955 Tf 11.96 0 Td[(fELmaxijfELij()fELmaxij,8(i,j)2AEL,2, (2))]TJ /F4 11.955 Tf 11.96 0 Td[(fELmaxijxijfELij()fELmaxijxij,8(i,j)2AELC,2, (2)0.4Gigi()Gi,8i2NGEN,2, (2)Gi=g maxi+Xk2Kikiyki,8i2NGEN, (2))]TJ /F6 11.955 Tf 11.96 0 Td[(maxi())]TJ /F3 11.955 Tf 11.95 0 Td[(j()max,8(i,j)2AEL,2, (2)xij2f0,1g,8(i,j)2AELC, (2)yki2f0,1g,8k2Ki,i2NGEN (2) 24

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dGPi()0,8i2NGEN,2, (2) Usingtheconstraintsfornaturalgas( 2 )( 2 )andelectricity( 2 )( 2 )systemswecanmodelthestochasticexpansionplanningmodelas [GE-St]: MinX(i,j)2AGXk2Kijckijkij+Xi2NLNGXk2Kickiki+X2Prob()X(i,j)2AGhij()fij() (2)+X(i,j)2AELCpijxij+Xi2NGENXk2Kirkiykisubjectto( 2 )( 2 ). TheproposedmodelisnonlinearinthesecondKirchhoff'slawforproposedlinesinequations( 2 ),wherethedecisionvariablexijismultipliedbythevoltageangle.Welinearizethisequationsbyusingthefollowingcommontechnique:weintroducenewvariablesthatdenotethenon-linearmultiplicationofxij(i())]TJ /F3 11.955 Tf 12.51 0 Td[(j()):qij=xij(i())]TJ /F3 11.955 Tf -441.29 -23.91 Td[(j()).Then,theequation( 2 )canbeequivalentlyreplacedbythefollowingsetofconstraints:fELij())]TJ /F3 11.955 Tf 11.96 0 Td[(ijqij()=0,8(i,j)2AELC,2, (2))]TJ /F4 11.955 Tf 11.95 0 Td[(xijmaxqij()xijmax,8(i,j)2AELC,2, (2)qij()i())]TJ /F3 11.955 Tf 11.96 0 Td[(j()+max(1)]TJ /F4 11.955 Tf 11.95 0 Td[(xij),8(i,j)2AELC,2, (2)qij()i())]TJ /F3 11.955 Tf 11.96 0 Td[(j())]TJ /F6 11.955 Tf 11.95 0 Td[(max(1)]TJ /F4 11.955 Tf 11.95 0 Td[(xij),8(i,j)2AELC,2, (2) 25

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2.2RiskManagementConstraintsfortheExpansionPlanningProblem Thestochasticprogrampresentedintheprevioussectiontakesintoconsiderationallthescenarios,satisfyingtheexpansionrequirementsevenforthescenariosthatareunlikelytohappen,forexample,largescaledisasters.WeintroduceConditionalValueatRiskrestrictionstotheoriginalproblemthatwouldallowsomeshortageofnaturalgasand/orelectricityforasubsetofnodesforworstcasescenarios. Oneofthemostwell-knownriskmeasuresusedinoptimizationunderuncertaintyisValue-at-Risk(VaR):given2(0,1],thecorresponding-VaRvalueisthelowestvaluesuchthatwithprobability,thelossdoesnotexceed[ 73 ].OneofthedisadvantagesofusingVaRinoptimizationmodelsthatVaRisnon-convexriskmeasureandinclusionofVaRconstraintsincreasesthenumberofintegervariablesinaproblem. CVaRistheconditionalexpectationofthelossundertheconditionthatVaRisexceeded.Clearly,CVaRisamoreconservativemeasureofriskthanVaR;however,itisconvexanddoesnotintroduceanyadditionalbinaryvariablesintotheoptimizationproblem.AsimpleillustrativegraphicalrepresentationoftherelationshipbetweenCVaRandVaRisshowninFigure 2-1 .IntheproposedexpansionmodelweintroducenewvariablesELi()andGi()thatdenoteshortageofelectricityandgasfornodeiunderscenariorespectively.Theowbalanceconstraintsfornaturalgasandelectricity( 2 )and( 2 )planningwillbemodiedasX(i,j)2A+ifij())]TJ /F11 11.955 Tf 19.62 11.36 Td[(X(j,i)2A)]TJ /F12 5.978 Tf 0 -6.42 Td[(i(1)]TJ /F4 11.955 Tf 11.96 0 Td[(lji)fji()+si()=dGi()+dGPi())]TJ /F3 11.955 Tf 11.96 0 Td[(Gi(),8i2NG,2. (2)X(i,j)2AELfELij()+X(i,j)2AELCfELij()+gi()=dELi())]TJ /F3 11.955 Tf 11.95 0 Td[(ELi(),8i2NEL,2, (2) 26

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Figure2-1. AgraphicaldepictionofVaRandCVaR. Note,thatelectricityandgasshortagescanberestrictedfromabovebygivenparametersELi()andGi()respectively,whichcanbesettozeroforcriticalnodes.Withthatnotation,theCVaRriskconstraintscanbedenedasXi2NGGi()G+wG(),2, (2)G+X2Pr() 1)]TJ /F3 11.955 Tf 11.95 0 Td[(wG()G, (2)Xi2NELELi()EL+wEL(),2, (2)EL+X2Pr() 1)]TJ /F3 11.955 Tf 11.96 0 Td[(wEL()EL, (2)0Gi()Gi(),2,i2NG (2)0ELi()ELi(),2,i2NEL (2) 27

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wG()0,wEL()0,2. (2) Withtheriskconstraintdenedabove,theresultingoptimizationproblemcanbeformulatedas [GE-ST-R]:Min( 2 ) (2)subjectto( 2 ),( 2 ),( 2 )( 2 ),( 2 ),( 2 )( 2 ),( 2 )( 2 ),( 2 )( 2 ). 2.3ComputationalExperiments Thissectionsummarizesthecomputationalexperimentsfortheproposedexpansionplanningproblem.ThemodelwasimplementedinC++andsolvedbyCPLEXr12.1onacomputerwithIntelrCorei72.8GHzprocessorand2GBofRAMavailableforcomputations.WeconductedthenumericalexperimentsfortheriskmodelGE-Rtondouthowriskconstraintsaffectthemodelperformanceandvalueoftheobjectivefunction.Figure 2-2 showsthesummaryofcomputations,wherewevariedthecondenceintervalandthevalueofCVaRrighthandsideconstraintsfornaturalgasGtoshowhowCVaRrestrictionschangethevalueoftheobjectivefunction.WithG=0themodeldoesnotallowgasshortageforanyofthescenariosleadingtohighvaluesoftheobjectivefunction.Also,withG=0theobjectivevalueshouldbethesamefor 28

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Figure2-2. MinimalcostandCVaRrestrictions allcondenceintervalsbecausenoshortageisallowedunderanyofthescenarios.Withincreasing,moreshortageofgasisallowedundersomeofthescenarios,whichleadstolowerobjectivevalues.AtapointaroundofG=1400andlargertheCVaRconstraintsbecomeinactiveduetolargeshortageallowancesanddonotaffecttheobjectivefunctionvalue.TheelectricityshortagerestrictionELsimilarlyaffectstheobjectivefunctionvalues.Note,thattheobjectivefunctionvaluechangesdiscreetlywithchangestoGandELbecauseoftheassumptionthattheexpansionisdiscreet.Figure 2-3 showsthechangeoftheobjectivefunctionwiththechangeofCVaRallowancesfornaturalgasGandelectricityEL. 29

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Figure2-3. ObjectivevaluevsgasandelectricityCVaRrestrictionsGandEL 30

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CHAPTER3TOPOLOGYDESIGNFORON-DEMANDDUAL-PATHROUTINGINWIRELESSNETWORKS Oneoftheimportancecriteriainsensornetworksisthereliabilityofthesignal,forexample,inmonitoringinfrastructures,wherethelossofacriticalevent(suchasanintruder,orafailureinaprotectionperimeter)canhaveverydamagingconsequences.Forsuchapplications,end-to-endreliabilityistypicallyprovidedthroughreliabletransportprotocols(operatingabovetheroutinglayer)including,forexample,reliableUDP[ 59 100 122 ],TCP[ 101 ].Morerecently,however,reliableroutingprotocolsarestartingtobeconsideredtohelpprovidehigherlevelsofreliability,withouttheimplicitlatenciesinducedbyend-to-endreliabletransportprotocols. Routingreliabilityisoftenimplementedthroughconstrainedmulti-pathroutingalgorithms,whichattempttobuildsimultaneousandnonself-interferingdatapathscarryingduplicatesofeachdatapacket(Figure 1-2 ).Eventuallossesoneitherpatharelikelytobecompensatedbythearrivalofduplicates,tradingoffcapacitywithreliability.Amajorchallenge,however,inmulti-pathroutingisthatglobaloptimalityunderanyperformancemetricisdifculttoachievevialocalizedalgorithms.Furthermore,individualnodesmustnotonlymaketheirroutingdecision,buttheymustalsocoordinatewithremotenodesthatareroutingduplicatepacketstoensureseparationandnon-interference. Whileadhocdistributedalgorithmshavebeenproposed[ 33 124 ],thealgorithmsareoftendifculttoimplement,especiallygivenarbitraryandoftendynamicphysicalconstraintsoftheradiofrequency(RF)topology.Aspreviouslyreported[ 88 ],thedynamicsoftheunderlyingphysicaltopologycancompromisetheconvergenceandstabilityofadhocroutingprotocols,andtomitigatesucheffectscross-layerstrategiesfornetworkmanagementhaverecentlybeenproposed[ 5 26 27 52 ]. 31

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3.1TopologyControlinWirelessSensorNetworks Topologycontrolintroducesthenotionofdrivingthelower-levelRFtopology,forinstancethoughexplicitnodepositionordifferentiatedpowercontrol,toimprovethebehaviorofhigh-levelprotocolsandapplicationperformance.Forexample,topologycontrolcanbeusedtocreateoptimizedsensornetworksforminimalpowerusage(toincreaselifespan),minimumlatency,orrobustoperations.Asidefromenvironmentalandinterferenceeffects,thetopologyofwirelessnetworkswithxednodesisdenedbythecombinedallocationoftransmissionpowerandsignalfrequency/coding.Inthiswork,weconsiderthesimpliedcasewheretheonlycontrolvariablefortopologymanagementisthetransmissionpoweroneachnode,whichconstrainstheproblemoftopologycontrolinWSNstondingatransmissionpowerallocationP=fp1,p2,...,pngforallnodesinthenetworksuchthatitgloballyminimizes(ormaximizes)someperformancemetric,whilemaintainingpre-denedtopologicalpropertiesinthedesignednetwork.Theperformancemetricmaybecontextdependentandcouldincludetheminimizationofoverallpowerutilizationfortransmitters,interferencereduction,throughputmaximization,etc.Inthispaper,wewillconsiderthecaseswherethedual-pathsbetweenadesignatedtransmitterandreceiverarearc-disjointaswellasinternallynode-disjointinthedesignednetwork.Eachconditioncanbeassociatedwithadifferentoperationalcontextandcanbedynamicallyenabledinthenetworksatrun-time. ThephysicalwirelessnetworktopologyisrepresentedbyasimpledirectednetworkG=(V,E)whereVrepresentsthenodes,andEVVrepresentsasetofdirectedarcseij.Adirectedarceijfromnodei2Vtonodej2Vimpliesthat,undernointerferenceconditions,thetailnodei(transmitter)cansuccessfullytransmitdatatotheheadnodej(receiver).Theactualtransmissionpatternaroundthetransmitterisafunctionofitsantenna,thetransmissionpower,andotherenvironmentaleffects.For 32

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example,foranidealizedisotropicantenna,thesignalpropagationfromthetransmitterdecaysequallyinalldirections.Inpractice,however,thetransmissionpatternofantennasisnotuniformandmayvarysignicantly.However,forthepurposesofthisworkwewillconsideromnidirectionaltransmitters,withidealandisotropicantennas. Figure3-1. Symmetricnetworktopology. Inthetypesofnetworksunderconsideration,foragivenuniformpowersetting(i.e.,p1=p2=p3...=pn),theRFtopologyissolelydenedbytherelativepositionbetweennodesasillustratedinFigure 3-1 .Thisapproximatemodeliswidelyusedinmobileadhocnetwork(MANET)applications,wherethetopologyisprimarilyaffectedbytherelativemobilityofnodes[ 11 ].Conversely,inWSNs,nodestendtobexed,andtopologyisprimarilymanagedthroughchangesinpowerorfrequencyallocation.Auniformchangeintransmissionpowerforallnodeswillincrease(ordecrease)thedensityofthenetwork.Higherpowerlevelswillincreasetheaveragedegreeofthenodesandinterference,whilereducedpowerlevelswillreducetheaveragedegreeandpossiblytheinterference,butwilllikelyincreasethepathlengthsinthenetworkandpossiblycausenetworksegmentation.Uniformpowercontrolisgenerallynot 33

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veryeffectiveasittreatsdifferentpartsofthenetwork(denseorsparseareas)withacommonstrategy. Amoreexibleapproachtotheproblemistheuseofdifferentiatedpowersettingsforeachnode.Inthiscase,eachnodemaybeassignedadifferenttransmissionpower,anddifferentareasofthenetworkcanbeseparatelyandmoreeffectivelyaddressed.Geographicallydenseareasofthenetworkmayuselesspowerfordatatransmissionthansparseregions,reducingpowerrequirementsandself-inducedinterference. However,differentialallocationofpowerbetweentransmittersmayleadtoasymmetricdatalinks,asillustratedinFigure 3-2 .Whileasymmetricdatanetworksaretheoreticallypossible,inpracticetheyarenotcompatiblewithmostcommoncommunicationprotocols.Layertwoprotocols,forinstance,relyonpacketacknowledgmentatthelinklevelforcoordination,andmostcommonroutingprotocolsignoreasymmetriclinkstomitigatecomplexityandtimingissuestotransportprotocols. Figure3-2. Asymmetricnetworktopology. So,directedarcsinthenetworkareoftenignoredbyhigherlevelprotocols,butstillplayaveryimportantroleinthetopologicalpropertiesandinterferencemetricsforthenetwork.Forthepurposesofthiswork,wewillconsiderthepowerallocation 34

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strategiesthatresultinasymmetricdatalinksmodeledbydirectedarcsinthenetwork.Itisimportanttohighlightthatbi-directionallinksattheRF-leveldonotnecessaryimplythesamedifferentiatedpowerallocation. Inthispaperweproposenewglobaltopologydesignmodelstosupporton-demanddual-pathroutinginWSNs.Weemploypowerconsumptionasourperformancemetricanddesignthetopologytocontainatleasttwoarc-disjointorinternallynode-disjointpathsfromthedesignatedtransmittertothedesignatedreceiver.Suchdesignscouldbeimplementedon-demandwhereacentralizedalgorithmmonitorsthetopology(forexamplethroughalinkstateroutingalgorithm)andcomputesoptimal,separatepathsfordatatransmissionfromagivensourceandtoagivedestination.Itshouldbenotedthatweuseaquadraticobjectiveintheoptimizationmodelssincepowerconsumptioncanbeassumedtoincreaseasaquadraticfunctionofthetransmissionradiusrequired,whichismorerealisticthanassuminglinearity.Theproposedmodelsalsocapturedependenceincommunicationlinkcreationinourmodels.Thatis,aspowersettingisincreasedatatransmitter,linksareestablishedtoallreceiversinthetransmissionradius.Whilethesechoicescomplicatemodeldevelopmentandsolution,theyremovesometypicalbutunrealisticassumptionsmadeinsuchapplications.InSection 3.2 ,weintroducethedifferentmixedintegernonlinearoptimizationformulationsstudied.Section 3.3 presentstheresultsofourpreliminarynumericalexperimentsinsolvingtheproblemsusingacommercialsolver. 3.2OptimizationModels Theon-demanddual-pathnetworkdesign(ODDPND)problemcanbestatedasfollows.GivenasetofwirelessnodesdenotedbyV=f1,...,ng,locatedintheEuclideanplane,determinethetransmissionradiusofeachnodei2Vthatminimizestotalenergyconsumption(assumedtobeadditiveandproportionaltothesquareof 35

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thetransmissionradius),andensuresthattheresultingtopologysatisesthefollowingconditions: 1. Thereisadirectedcommunicationlinkfromitojifandonlyifthetransmissionradiusofiisatleastdij,theEuclideandistancebetweennodesiandj; 2. Foradistinguishedpairofnodess,t2Vwheresistheoriginandtisthedestination,thereexistatleast2arc-orinternallynode-disjointdirectedpathsfromstot; 3. Foreverydistinctpairofnodesi,j2V,thereexistsadirectedpathfromitoj. TheODDPNDproblemisverysimilartothesurvivablenetworkdesign(SND)problemswithprescribednodeconnectivity[ 29 86 115 ],orprescribedarc-connectivity([ 82 ]andreferencestherein),aswellasnetworkaugmentation(byarcaddition)problemstoproduceanetworkwithdesiredarc[ 12 ]ornodeconnectivity[ 79 ].TheODDPNDproblemdiffersfromSNDbecauseoftheassumptionswerelaxinthisparticularapplicationsetting.Consequently,thekeydistinctionsaretheuseofaquadraticobjectiveandthedependenceinarccreation(asprescribedbythecondition 1 ).Bycontrast,SNDproblemsemployalinearobjectiveandthearcsareassumedbeindependentofeachotherandanysubsetofarcscanbecreated/added.Despitethesedifferences,thebasicformulationideasdevelopedforSNDcanbeadaptedaswedemonstrateinthefollowingsections.Theassumptionsrelaxedclearlyresultinmorecomplicatedmodels,butareconsideredaworthwhiletrade-offgiventheimprovedabilitytoimplementsuchdesigns. 3.2.1Arc-disjointODDPNDformulation Thetransmissionradiusofnodei2Visdenotedbythedecisionvariableyi.ParameterMiiseitherthemaximumtransmissionradiusthatnodei2Vcansupport,ormaxj2Vdij,whicheverissmaller.Constraint( 3 )enforcesthisboundonyi.Ifthebinaryvariablexij=1,constraint( 3 )ensuresthatyidijanditisredundantotherwise. 36

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Constraint( 3 )enforcesthatyidijifxij=0.Togethertheycapturethedependenceinlinkcreationasthenodepowerisincreased. Weneedtoensurethatthewholenetworkisstronglyconnectedandfromnodestonodet,thereare2arc-disjointpaths.Thisisaccomplishedviaowbalanceconstraintssimilartoamulti-commoditynetworkowproblem[ 2 ].Nodek2Vwillhaveasupplyofn)]TJ /F6 11.955 Tf 12.24 0 Td[(1unitsofcommodityk2Vwhereasallothernodeswillhaveademandofoneunitforcommodityk.Byimposingtheseconstraintsweensurethatthereisadirectedpathfromeverynodetoeveryothernodeinthenetwork.Inconstraints( 3 ),fork2V,bk(i)=n)]TJ /F6 11.955 Tf 12.1 0 Td[(1ifi=k,andbk(i)=)]TJ /F6 11.955 Tf 9.3 0 Td[(1fori2Vnfkg.Therequirementthatthereshouldbeatleasttwopathsbetweennodessandtisimposedbyconstraints( 3 ),wherebn+1(s)=2,bn+1(t)=)]TJ /F6 11.955 Tf 9.3 0 Td[(2,andbn+1(i)=0fori2Vnfs,tg.Constraints( 3 )and( 3 )ensurethatowscanonlygothougharcscreatedinthenetwork.MinimizeXi2Vciy2i (3)subjectto:Xj2Vnfigfkij)]TJ /F11 11.955 Tf 18.82 11.35 Td[(Xj2Vnfigfkji=bk(i),8i2V,k2V (3)Xj2Vnfigfn+1ij)]TJ /F11 11.955 Tf 18.81 11.36 Td[(Xj2Vnfigfn+1ji=bn+1(i),8i2V (3)fkij(n)]TJ /F6 11.955 Tf 11.96 0 Td[(1)xij,8i,j2V:i6=j,k2V (3)fn+1ijxij,8i,j2V:i6=j (3)dijxijyi,8i,j2V:i6=j (3)Mixijyi)]TJ /F4 11.955 Tf 11.95 0 Td[(dij,8i,j2V:i6=j (3)0yiMi,8i2V (3)fkij08i,j2V:i6=j,k2V[fn+1g (3) 37

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xij2f0,1g8i,j2V:i6=j (3) 3.2.2Node-disjointODDPNDformulation Formulation( 3 )( 3 )canbemodiedtoensuretwointernallynode-disjointdirectedpathsfromnodestonodet.Forthatpurposetheincomingowforcommodityn+1mustbelimitedtooneunitateachnodeinthenetwork.ThiscanbedonebyaddingthefollowingconstraintstoFormulation( 3 )( 3 ) Xj2Vnfigfn+1ji1,8i2Vnfs,tg(3) Constraint( 3 )ensuresthateverynodeexcepttheoriginandthedestinationmayhavenomorethanoneunitofowofcommodityn+1incoming.Thatis,thereisonlyonepathfromstottraversedbycommodityn+1comingthroughanyparticularnode. 3.2.3Cutcoveringformulation Thesocalledcutcoveringformulation[ 16 ]usesanimplicationofthetheoremsofMenger[ 83 ]andFord-Fulkerson[ 66 ],thatfordistinctverticess,t2VinadirectednetworkG=(V,E),theminimumsizeofans,t-cutequalsthemaximumnumberofpairwisearc-disjoints,t-dipaths.Wedevelopacutcoveringformulationonlyforthearc-disjointcaseoftheODDPNDproblemasitrequiresanexponentialnumberofconstraints.Notethatthisformulationisbestsolvedwithaspecializedrow-generation/cutting-planeapproachwhereviolatedconstraintsaredetectedandaddedduringthecourseofthesolutionprocessandnotinadvance.Asourintentioninthispreliminarypaperistosolvetheformulationusingbasicalgorithmsavailableinacommercialsolver,wedonotconsidertheinternallynode-disjointcaseoftheODDPNDproblem.Welimitourselvesexploringsuchaformulationforjustoneofthetwocasesfor 38

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thesakeofcomparison.MinimizeXi2Vciy2i (3)subjectto:Xi2S,j2VnSxij8>><>>:2ifs2S,t2VnS,8;6=SV1otherwise (3)dijxijyi,8i,j2V:i6=j (3)Mixijyi)]TJ /F4 11.955 Tf 11.96 0 Td[(dij,8i,j2V:i6=j (3)0yiMi,8i2V (3)xij2f0,1g8i,j2V:i6=j (3) 3.3ComputationalExperiments Thissectionsummarizescomputationalexperimentsforcutcoveringarc-disjoint(CCAD),multi-commodityarc-disjoint(MCAD)andmulti-commoditynode-disjoint(MCND)problemformulationspresented.TheformulationswereimplementedinC++andsolvedbyCPLEXr12.1onacomputerwithIntelrCorei72.8GHzprocessorand2GBofRAMavailableforcomputations.Althoughtheprocessorhad4cores,welimitedthenumberofthreadsCPLEXcanuseupto1forcomparisonpurposes.Foreveryprobleminstanceweuseda2-dimensionalplanewithcoordinatesfrom0to100forbothoftheaxes.Thecoordinatesofthenodesweregeneratedrandomlyandtherstnodewasassignedtobes-node,whereasthelastnodewast-node.Thecostcoefcientsciintheobjectivefunctionwerethesameforallthenodesfromtheassumptionthatallthesensorsinthenetworkwereidentical.ParameterMi,whichcorrespondstomaximumtransmissionradiuswasalsothesameforallthenodesandequalto35.Computational 39

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timepresentedisinsecondsandrelativegapisinpercents.Everygroupconsistsof30randomlygeneratedprobleminstances.Table 3-1 presentsaveragecomputationaltimesaswellasminimumcomputationaltime,standarddeviation(SD)andmaximumtimeforeachgroupofinstances. Table3-1. Runningtime(inseconds)forsolvingtooptimality. nCCADTimeMCADTimeMCNDTimeAverageAverageAverage(Min,SD,Max)(Min,SD,Max)(Min,SD,Max) 100.10.30.3(0.1,0.1,0.3)(0.1,0.2,0.9)(0.1,0.2,0.9)154.12.72.9(0.4,7.1,32.8)(0.3,2.8,13.9)(0.4,3.7,20.0)2085.662.4()(3.7,133.6,635.1)(3.4,85.3,371.6)252035.32152.9()(116.0,2242.3,9570.0)(127.7,2400.5,11231.0) Oneofthepotentialproblemswiththeproposedapproachofcentralizedtopologycontrolarisesfromthefactthatafterasensorsendsarequestforatopologychangeithastowaituntilanoptimalsolutioniscomputedandtransmittedbacktosensorsfromthecomputationalcluster.Toovercomethatproblemthecomputationalclustercantransmitagoodfeasiblesolutionassoonasitisobtained.Forthatpurposewecalculatedaveragetimeofgettingrstfeasiblesolutionsforthesameprobleminstances.WithFoptdenotingtheoptimalobjectivefunctionvalueandFfeasdenotingtheobjectivefunctionvalueattherstfeasiblesolution,therelativegapRGoftheobjectivefunctionvalueattherstfeasiblesolutionwithrespecttotheoptimalsolutioniscalculatedaccordingtothefollowingformula: RG=Ffeas)]TJ /F4 11.955 Tf 11.95 0 Td[(Fopt Ffeas.(3) TheaveragetimetotherstfeasiblesolutionandtheaveragerelativegapRGarereportedinTables 3-2 and 3-3 40

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Table3-2. Averagetimetotherstfeasiblesolution nCutCoveringArcDisjointNodeDisjointAverageAverageAverage(Min,SD,Max)(Min,SD,Max)(Min,SD,Max) 100.10.10.1(0.0,0.0,0.1)(0.1,0.1,0.3)(0.1,0.1,0.4)150.60.80.8(0.3,0.3,1.37)(0.1,0.7,2.6)(0.2,0.7,3.0)206.66.0()(0.5,4.1,14.5)(0.5,3.8,14.1)2537.035.1()(8.9,18.6,81.0)(11.0,16.9,73.9) Table3-3. Averagerelativegaptotherstfeasiblesolution nCutCoveringArcDisjointNodeDisjointAverageAverageAverage(Min,SD,Max)(Min,SD,Max)(Min,SD,Max) 106.110.49.7(0.0,5.9,20.0)(0.0,9.9,35.6)(0.0,9.9,34.3)158.714.913.2(0.0,6.3,23.0)(0.0,13.9,55.8)(0.0,13.6,47.9)2026.220.6()(5.0,22.3,70.3)(3.4,18.2,65.3)2561.550.9()(6.9,21.6,80.0)(3.9,28.1,78.3) IntheTables 3-1 3-2 ,and 3-3 correspondstothecasesthatCPLEXcouldnotsolvetooptimalityduetoalargenumberofconstraintsinthecutcoveringformulationandconsequentlylargememoryrequirements.AsonecanseefromtheTable 3-1 ,node-disjointsolutionstakeapproximatelyasmuchtimeasarc-disjointsolutions,consequentlytherequirementforthepathstobenode-disjointdoesnotmaketheproblemmoredifcultsolveforthisconguration.Table 3-1 alsoshowsthatasexpected,thecutcoveringformulationisnotsuitableforexplicituseonproblemswithmorethan15nodes.Therelativegapbetweenrstfeasiblesolutionandoptimalsolutionfornode-disjointformulationappearstobesmallerthanacorrespondinggapforarc-disjointformulation. 41

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AArc-disjointODDPNDsolution BNode-disjointODDPNDsolution Figure3-3. Illustrativeexamplesofarc-disjointandnode-disjointODDPNDsolutions. Anillustrationthatdemonstratesthedifferencebetweenarc-disjointdualpathsandnode-disjointdualpathsispresentedinFigure 3-3 .Inthegure,thesizeofeachnoderepresentsitsrelativetransmissionpower:largernodesaretransmittingwithhigherpowerthansmallerones.Thedifferentialpowerallocationcreatesdirectedlinksbetweeneachnodethatcreateanoptimal,directedtopologytosupportarc-disjointdualpaths(Figure 3-3A )andnode-disjointdualpaths(Figure 3-3B ). Figure 3-4 ,showsanexampleofadualdatapathcreatedbyaroutingprotocolovertheprovidedtopologiesforthearc-disjoint(Figure 3-4A ),andnode-disjoint(Figure 3-4B )cases,foragivenoriginanddestinationnodes.Itisimportanttonotethatthismaynotbetheonlypossibledual-pathcombinationforeachtopology.Theproposedapproachensuresthattheminimumoverallpowerallocationwillresultinatleastonedual-pathcombination.Itispossible,duetothewirelessconnectivityconstraintsthatmorethanonepathiscreatedfortheminimumpowerallocation.Itisalsoimportanttohighlightthatouralgorithmdoesnotoptimizethelengthofthepathinnumberofhops,buttheoverallpowerutilizationforagivensourceanddestinationpair.Additional 42

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APossiblearc-disjointdualdatapaths BPossiblenode-disjointdualdatapaths Figure3-4. Illustrationofpossiblearc-disjointandnode-disjointdualdatapaths. constraintssuchasthemaximumnumberofhopscouldalsobeaddedtoformulationtoaddressotherprotocollevelconstraints. 43

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CHAPTER4COMPUTATIONALRISKMANAGEMENTTECHNIQUESFORFIXEDCHARGENETWORKFLOWPROBLEMSWITHUNCERTAINARCFAILURES Inthischapter,weaddressbothconceptualandcomputationalissuesassociatedwiththesequestions.Therestofthechapterisstructuredasfollows.Section 4.1 reviewsamathematicalprogrammingformulationforFCNFproblemandSection 4.2 providesabriefoverviewofrelatedliteratureforthedeterministicFCNFproblems.Section 4.3 describesthequantitativeriskmeasuresandshowshowuncertainarcfailuresareincorporatedintothemodel.Section 4.4 discussesanefcientheuristictechniqueforsolvingFCNFproblemswithuncertainarcfailuresusingConditionalValue-at-Risk(CVaR)asariskmeasureandintroducesanapproachthatcanbeutilizedforreducingthesizeoflarge-scaleproblemsandimprovingthecomputationalefciencyoftheconsideredtechniques.ComputationalresultsarepresentedinSection 4.5 4.1GeneralSetupofFixedChargeNetworkFlowProblems Ingeneral,aninstanceoftheMINIMUMCOSTFLOWPROBLEM(MCF)isrepresentedbyadirectedgraphG=(V,E),whereeachdirectededge(arc)(i,j)2Ehastheassociatedcostcijperunitofowalongthisedge,aswellasthecapacityijdenotingthemaximumamountofowxijthatcantraversetheedge(i,j).Foreachnodei2V,didenotesthedemand(supply)ofnodei.ThentheMCFcanbeformulatedasthefollowingwell-knownlinearprogram: (MCF)minXf(i,j)2Egcijxij (4) s.t.Xfj:(i,j)2Egxij)]TJ /F11 11.955 Tf 23.66 11.36 Td[(Xfj:(j,i)2Egxji=di,8i2V, (4) 0xijij,8(i,j)2E. (4) 44

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Inpractice,itisoftenrequiredtopayacertainxedsetupcost(xedcharge)inordertohaveanonzeroowthroughacertainarc.Letsijdenotethexedsetupcostforarc(i,j).Thenifxijistheowthroughthisarc,thecostassociatedwiththisarccanbedenedasfij(xij)=8><>:cijxij+sij,xij2(0,ij]0,xij=0. ThestandarddeterministicFIXEDCHARGENETWORKFLOWPROBLEM(FCNF)canthenbeformulatedas (FCNF)minXf(i,j)2Egfij(xij) (4) s.t.Xfj:(i,j)2Egxij)]TJ /F11 11.955 Tf 23.66 11.36 Td[(Xfj:(j,i)2Egxji=di,8i2V, (4) 0xijij,8(i,j)2E. (4) Itisalsoeasytorepresentthisformulationasalinearmixedintegerprogrammingproblembyintroducingbinaryvariableszij,whichdenotewhetherthereisanon-zeroowthrougharc(i,j)asfollows: (FCNF-IP)minXf(i,j)2Eg(cijxij+sijzij) (4) s.t.Xfj:(i,j)2Egxij)]TJ /F11 11.955 Tf 23.66 11.35 Td[(Xfj:(j,i)2Egxji=di,8i2V, (4) 0xijijzij,8(i,j)2E, (4) zij2f0,1g,8(i,j)2E. (4) 45

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4.2ComputationalChallengesandHeuristicAlgorithms ThexedchargenetworkowproblemhasadiscontinuousobjectivefunctionandisNP-hard.Therefore,theexactmethodsforFCNFrequirelargecomputationaleffortsandbecomeimpracticalforlarge-scaleproblems.BelowweprovideabriefreviewofexactmethodsaswellasheuristictechniquesforsolvingdeterministicFCNFproblems. 4.2.1Exactalgorithms Asmentionedabove,thedeterministicFCNFproblemcanbemodeledasa0-1mixedintegerproblem[ 63 ].Theproblembelongstoaclassofconcaveminimizationproblemsandhasasolutiononavertexofthefeasibleregion.Basedonthatfact,Murty[ 90 ]proposedasolutionalgorithmbasedonrankingtheextremepoints.Thealgorithmworkswellonproblemswithrelativelyhighvariablecostandsmallxedcost.ManyoftheexacttechniquesfortheFCNFproblemutilizeabranch-and-boundapproach.Gray[ 53 ]proposedabranch-and-boundalgorithmwhichperformswellwhenthexedcostislarge.CombiningthismethodandMurtyapproach,acompositealgorithmwasalsodevelopedin[ 53 ].KenningtonandUnger[ 69 ]proposedamoreefcientalgorithmwhichusesthetransportationstructureoftheproblemandcalculatespenaltiesforintegervariablesusingalinearprogrammingrelaxation.Barretal.[ 14 ]developedabranch-and-boundalgorithmforsparsenetworksjustifyingitbythefactthatmostofreal-worldproblemshavesparsestructure;moreover,theauthorsproposedanewmethodforcalculatingupanddownpenaltiesforthecorresponding0-1variables.CabotandErenguc[ 25 ]usedaLagrangianrelaxationforcalculatingstrongerupanddownpenaltiesandreportedimprovedboundsonsmallbranch-and-boundtrees.Cruzetal.[ 36 ]proposedabranch-and-boundmethodforuncapacitatedxedchargenetworkowproblem.InsteadofusingaLPrelaxationforcomputinglowerbounds,theauthorsusedaLagrangianrelaxation,whichdoesnotprovideabetterlowerbound 46

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thanLP,butisusedforcalculatingverytightupperbounds.OrtegaandWolsey[ 95 ]proposedabranch-and-cutmethodforuncapacitatedxedchargenetworkowproblems.Themethodemploystwoheuristictechniques:KimandPardalos[ 72 ]forndingagoodfeasiblesolutionatthetopnodeandagreedymin-costowheuristicforgeneratingagoodsubsetofnodes.Besidesthebranch-and-boundapproaches,Bendersdecompositionhasbeenappliedtothexedchargenetworkdesignproblems.Costa[ 81 ]presentsanextensivesurveyofthistopic. 4.2.2Heuristicalgorithms Severalheuristicalgorithmsarebasedonthesolutionoftherelaxedversionoftheproblemwiththexedcostignored.FromtheoptimalsolutionofsuchLPtheheuristicsconstructasolutiontotheoriginalFCNFproblem([ 35 ],[ 38 ]and[ 116 ]).Adual-ascentbasedheuristicalgorithmhasbeenproposedforuncapacitatedproblemsin[ 9 ].AftersolvingthedualLPproblem,adrop-addprocedureisemployedinordertoimproveexistingsolution.Thealgorithmproducesasolutionfrom1%to4%fromoptimality,butdoesnotworkwelloncapacitatedproblems.KimandPardalos[ 72 ]proposedaDynamicSlopeScalingProcedure(DSSP)forFCNFproblem.Thealgorithmbasedonanapproximationofaconcaveobjectivefunctionbyalinearone,i.e.thevariablecostandthexedcostareapproximatedbyonelinearfactor.NahapetyanandPardalos[ 91 ]developedanAdaptiveDynamicCostUpdatingProcedurewhichapproximatestheoriginalobjectivefunctionbyapiecewiselinearonewithtwolinearpieces.Theauthorsprovedthatthesetwoproblemsareequivalent.However,ndingparametersofthetwolinearpiecesisdifcultandtheauthorssolvedasequenceofbilinearrelaxationmodifyingtheparameterseachtimeuntiltherearenochangesandtheapproximatesolutionisfound.ThecomputationalresultsshowedthattheproposedmethodoutperformsallmodicationsofDSSPinbothsolutionqualityandcomputational 47

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time.Laterinthispaperchapter,wewillshowthatthistechniquecanbeextendedtothecaseofxedchargenetworkowproblemswithuncertainarcfailures,duetothefactthatextraconstraintsgeneratedfromriskconsiderationsdonotcontain0-1variables.Recently,Hewittetat.[ 62 ]developedanalgorithmformulti-commodityFCNFproblemwhichcombinesheuristicandexactmethodsforLPandIP.TheauthorscomparethesolutionsreturnedbyalgorithmandbyCPLEX9.1andshowthattheproposedalgorithmcanproducenear-optimalheuristicsolutionsfasterthanCPLEX. Finally,itshouldbenotedthatthesizeofthetestinstancesforthestandarddeterministicFCNFproblemconsideredintheliteratureisrathermoderateeveninthecaseofheuristicalgorithms,thatis,thelargestnetworksconsideredinthepreviouslyreportedcomputationalstudiescontainnomorethanafewthousandarcs.Inthischapter,besidesincorporatinguncertainarcfailuresandriskconstraintsintotheclassicalFCNFproblemsetup,wedemonstratethattheconsideredheuristicapproachescantackleverylargeprobleminstances(e.g.,upto200,000arcs)andstillproducehigh-qualitynear-optimalsolutions.Thefollowingsectionaddressestheconsideredproblemsetupinmoredetail. 4.3FixedChargeNetworkFlowProblemsunderUncertainArcFailures Inthissectionwedescribetheuncertainarcfailuresconsiderations,provideabriefoverviewofquantitativeriskmeasuresandproposeamathematicalprogrammingformulationforFCNFproblemunderuncertaintywithCVaRconstraints. 4.3.1Uncertainarcdisruptions Duetothefactthatmultiplenetworkarcsmayfailundertheassumptionsofourmodel,withthesefailuresbeinguncertain,weneedtouserigoroustechniquestodescribetheseuncertaintiesandexplicitlyquantifythelossesassociatedwithpotentialarcfailures.Specically,weintroduceafunctionL(x,Y)thatdependsonboththe 48

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decisionvectorxandandthevectorofuncertainparametersY.Thisfunctionisusuallyreferredtoasthelossfunction.NotethatL(x,Y)isarandomvariablebyitself,althoughitsdistributionmaynotnecessarilyhaveanexplicitanalyticalexpression. Inthecontextofanetworkowproblem,thevectorxconsistsofthevariablesxijrepresentingtheamountofowthrougharc(i,j).Further,letYbearandomvectorrepresentingtheuncertainarcfailuresinthenetwork.Inthemodelsbelow,weassumethateacharccaneitherbefullyfunctionalorcompletelyfail.Inthiscase,eachcomponentYijofthevectorYisaBernoullirandomvariabledenedasfollows:Yij:=8>><>>:1,withprobabilitypij,0,withprobability1)]TJ /F4 11.955 Tf 11.95 0 Td[(pij, wherepijistheprobabilityoffailureforthearc(i,j).Furthermore,basedonthedistributionsofYij,onecangenerateasetoffailurescenarioss=1,2,...,S,wherethevaluesysijaredenedforeachscenarioasfollows: ysij:=8>><>>:1,ifarc(i,j)failsunderscenarios,0,otherwise. (4) Then,inthesimplestcasetherandomvariableL(x,Y)representingthetotallosscanbedenedasthefollowinglinearexpression L(x,Y)=Xf(i,j)2EgijxijYij.(4) Notethatinthisdenitionweintroducethecoefcientsijcorrespondingtoeacharc(i,j),whichcanbeviewedaseitherarcimportancecoefcientsorreroutingcostcoefcients.Thevaluesofarcimportancecoefcientsmaydifferdependingonspecic 49

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applicationsandprovideacertaindegreeofexibilityforthequantitativeexpressionofthecollaterallossesassociatedwiththefailuresofdifferentarcs.Thisinterpretationisappropriate,forinstance,inthecaseoftheclassicaltransportationproblemsetupwithxedcostsanduncertainarcfailures;however,inamoregeneralcase,thesecoefcientscanalsobeviewedasreroutingcostcoefcientsthatshowcostofowreroutingtoanalternativepath.Exactcomputationofthesecoefcientsforspecicnetworktopologiesandallfailurescenariosmightbechallenging;however,someestimationsofthesecoefcientscanbeobtainedfromhistoricaldata.Inthiscaseonecanassumethatreroutingcostmaybedifferentfordifferentarcsandlinearlyproportionaltotheoriginalowxij.Inthefurtherdiscussion,withoutlossofgenerality,wesetij=1,8(i,j)2E;however,alltheformulasderivedbelowarevalidforarbitraryvaluesofij.Further,usingtheabovenotations,wedenetherealizationoftherandomvariableL(x,Y)foreachscenarios=1,...,S: L(x,ys)=Xf(i,j)2Egxijysij.(4) Equation( 4 )denesthelossofowinthewholenetworkforoneparticularscenarios.OnecanusetheexpressionforL(x,ys)tomeasuretheoveralllossoccurredinthenetworkforeveryscenario. 4.3.2Quantitativeriskmeasures Tofacilitatetheupcomingdiscussion,webrieyreviewsomeofthebasicstatisticalconcepts(commonlyreferredtoasquantitativeriskmeasures),whichweutilizeinthisstudy. Oneofthemostwell-knownriskmeasuresusedinoptimizationunderuncertainty(especiallyinnancialapplications)isknownasValue-at-Risk(VaR).Given2(0,1],thecorresponding-VaRvalueisthelowestvaluesuchthatwithprobability,the 50

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lossdoesnotexceed[ 73 ].Ineconomicterms,VaRissimplythemaximumamountatrisktobelostfromaninvestment.VaRisthemostwidelyappliedriskmeasureinprobabilisticsettingsprimarilybecauseitisconceptuallysimpleandeasytoincorporateintoamathematicalmodel[ 34 ].Howeverwiththiseaseofuse,itisaccompaniedbyseveralcomplicatingfactors.SomedisadvantagesarethattheinclusionofVaRconstraintsincreasesthenumberofintegervariablesinaproblem.Also,VaRisnotaso-calledcoherentriskmeasure,implyingamongotherpropertiesthatitisnon-convexandnotsub-additive. AnotherriskmeasurecloselyrelatedtoVaRistheso-calledConditionalValue-at-Risk(CVaR)(alsoknowninthenanceliteratureasExpectedTailLoss,ExpectedShortfall,andWorstConditionalExpectation).Bydenition,CVaRistheconditionalexpectationofthelossundertheconditionthatVaRisexceeded.Clearly,CVaRisamoreconservativemeasureofriskthanVaR.RockafellarandUryasev[ 108 ]provedseveralimportantresultsregardingoptimizationofCVaR,whichmakethisriskmeasureratherattractivefromtheoptimizationviewpoint.Inparticular,CVaRhasbeenshowntopossessthepropertiesthatVaRlacks;inparticular,itiscoherent(whichincludesconvexityamongotherproperties).Thismakesthisstatisticalmeasuremuchmoreconvenienttohandleinoptimizationmodels.InordertodeneVaRandCVaRmoreformally,weintroducethefollowingnotations.SupposethatarandomvariableL(x,Y)(referredtoasalossfunctionanddenedabove)representsatotalcollaterallossassociatedwithadecisionvectorx2XRn,andarandomvectorY2Rmwhichrepresentstheuncertainparametersthatmayaffecttheperformanceofasystemunderconsideration. ThelossL(x,Y)foreachx2XisarandomvariablehavingadistributioninRinducedbythatofY.ThereforetheprobabilityofL(x,Y)notexceedingsomevalueis 51

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denedas (x,):=PfL(x,Y)g.(4) Byxingx,thecumulativedistributionfunctionofthelossassociatedwiththedecisionxisthusgivenby (x,).GiventhelossrandomvariableL(x,Y)andany2(0,1),wecanuseequation( 4 )todene-VaRas (x):=minf2R: (x,)g.(4) FromthisweseethattheprobabilitythatthelossL(x,Y)exceeds(x)is1)]TJ /F3 11.955 Tf 12.18 0 Td[(.Usingthedenitionabove,CVaRistheconditionalexpectationthatthelossdeterminedbythedecisionvectorxexceeds(x)[ 108 ].Thuswehave-CVaRdenotedas(x)denedas (x):=EfL(x,Y)jL(x,Y)(x)g.(4) InordertoincludeCVaRandVaRconstraintsinoptimizationmodels,onecancharacterize(x)and(x)intermsofafunctionF:XR7!Rdenedby F(x,):=+1 1)]TJ /F3 11.955 Tf 11.95 0 Td[(EfmaxfL(x,Y))]TJ /F3 11.955 Tf 11.95 0 Td[(,0gg.(4) Itcanbeshownthatasafunctionof,F(x,)isconvexandcontinuouslydifferentiable.Moreover,foranyx2X,(x)=min2RF(x,),forall2R.Furthermore,ifA(x):=argmin2RF(x,)isthesetconsistingofthevaluesofforwhichFisminimized,thenA(x)isanon-empty,closedandboundedintervaland(x)istheleftendpointofA(x).Inparticular,itisalwaysthecasethat(x)2argmin2RF(x,)and (x)=F(x,(x))[ 108 ].IthasalsobeenshownthatforanyprobabilitythresholdandlosstoleranceC,constraining(x)CisequivalenttoconstrainingF(x,)C[ 109 ]. 52

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4.3.3MathematicalprogrammingformulationforFCNFunderuncertaintywithCVaRconstraints Withthelossfunctiondenedinsubsection 4.3.1 ,thecharacteristicfunctionforCVaRconstraintscanbeapproximatedby F(x,)=+1 1)]TJ /F3 11.955 Tf 11.95 0 Td[(SXs=1smaxXf(i,j)2Egxijysij)]TJ /F3 11.955 Tf 11.95 0 Td[(,0,(4) wheresistheprobabilityofscenarioysfors=1,2,...,S.SinceL(x,Y)islinearwithrespecttox,F(x,)isconvexandpiecewiselinear[ 73 ].Thenwecanmodeltherobustxedchargenetworkow(RFCNF)problemwithCVaRconstraintsasshownbelow.Notethatthetermrobustcanhavedifferentmeanings,andinthecontextofthemodelsusedinthischapter,wespecicallymeanthatarobustsolutiontoFCNFistheonethatisfeasiblefortheoriginaldeterministicproblemandalsosatisestheCVaRconstraint(thatis,theCVaRcorrespondingtothevaluesofxijdoesnotexceedapre-denedupperthresholdC).ThenotationRFCNFwillbeusedinthisspeciccontext. (RFCNF)minXf(i,j)2Eg(cijxij+sijzij) (4) s.t.Xfj:(i,j)2Egxij)]TJ /F11 11.955 Tf 23.66 11.36 Td[(Xfj:(j,i)2Egxji=di,8i2V, (4) 0xijijzij,8(i,j)2E,+1 1)]TJ /F3 11.955 Tf 11.96 0 Td[(SXs=1smaxXf(i,j)2Egxijysij)]TJ /F3 11.955 Tf 11.96 0 Td[(,0C, (4) zij2f0,1g,8(i,j)2E, (4) 2R. (4) WecanlinearizeF(x,)byusingasetofextravariablests,s=1,2,...,S,andreplacingF(x,)bythelinearfunction+1 1)]TJ /F15 7.97 Tf 6.59 0 Td[(PSs=1stsandaddingthefollowingsetof 53

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linearconstraints: tsL(x,ys))]TJ /F3 11.955 Tf 11.95 0 Td[(,8s=1,2,...,S, (4) ts0,8s=1,2,...,S. (4) Forthepurposesoffurtherdiscussion,thefullylinearizedmathematicalprogrammingformulationoftheROBUSTFIXEDCHARGENETWORKFLOWPROBLEMwithConditionalValue-at-Riskconstraintscanequivalentlybewritteninthefollowingnotations: (RFCNF-f)minXf(i,j)2Egfij(xij) (4) s.t.Xfj:(i,j)2Egxij)]TJ /F11 11.955 Tf 23.67 11.36 Td[(Xfj:(j,i)2Egxji=di,8i2V, (4) +1 1)]TJ /F3 11.955 Tf 11.96 0 Td[(SXs=1stsC, (4) tsXf(i,j)2Egxijysij)]TJ /F3 11.955 Tf 11.96 0 Td[(,8s=1,2,...,S, (4) ts0,8s=1,2,...,S, (4) 2R, (4) wherefij(xij)=8><>:cijxij+sij,xij2(0,ij]0,xij=0. AnillustrationthatdemonstratesthedifferencebetweentheoptimalsolutionstructuresforthedeterministicFCNFandtheproblemsetupunderuncertaintywithCVaRconstraintsisprovidedinFigure 4-1 .Figure 5-7A showstheoriginalnetwork,Figure 5-7B showstheoptimalsolutionforthisnetworkindeterministicsettingsand 54

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Figure 4-1C presentsthesolutionoftheproblemwithCVaRconstraints.Althoughtheguresdonotshowindetailallthenodeandarcparameters(nodesupplies/demands,arccostsandprobabilitiesoffailure),thegeneralobservationisthatthesolutionusesarcswithlowerprobabilityoffailureacceptingthehighercostfortheobjectivefunction.Bydiversifyingtheowacrossmanyarcs,onecanensurethatthesolutionismorerobustwithrespectto(multiple)arcfailures. AOriginalnetwork BOptimalsolutionstructurewithoutCVaRconstraints(noarcfailures) COptimalsolutionstructurewithCVaRconstraints(withuncertainarcfailures) Figure4-1. AnexampleofoptimalsolutionstructureforRFCNPwithCVaRconstraints Inthenextsections,wediscussheuristictechniquesforsolvingRFCNFproblemswithmultipleuncertainarcfailuresforlargenetworkswithinafewpercentfromoptimality. 4.4AContinuousApproximationTechniqueforSolvingRFCNF RFCNF-fisaFCNFproblemwithadditionalCVaRconstraintsimposedforageneratedsetofscenarios.Asmentionedabove,thedeterministicFCNFisanNP-hardproblembyitself,anditispracticallyimpossibletondanoptimalsolutionforlargenetworks.Uncertaintymakestheproblemevenmorecomputationallychallenging,sinceitrequiresaddingmanyCVaRconstraintstotheoriginalproblem.Althoughadditionalconstraintshaverathersimplestructureanddonotrequireadditionalintegervariables,in 55

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practiceuncertaintyrequiresconsideringalargenumberofscenarios,andaddingCVaRconstraintsforeachscenariomakestheproblemtoolargetobesolvedexactly. NahapetyanandPardalos[ 91 ]developedanapproximationtechniquewheretheytransformedthedeterministicFCNFproblemintoanotherequivalentcontinuousbilinearformulation.Furthermore,aheuristicmethodAdaptiveDynamicCostUpdatingProcedure(ADCUP)wasdeveloped,whichcouldtacklerelativelylargedeterministicproblemsandgenerateasolutionthatisveryclosetoanoptimaloneinthereportedinstances.SincetheconsideredCVaRconstraintsarelinearanddonotintroduceadditionalintegervariables,anextendedtechniquecanbeappliedtoRFCNF-fproblemtondaclose-to-optimalsolution.Inthissection,wedescribehowthemethodcanbeadaptedfortheRFCNF-fproblemwithuncertainarcfailures.FormoredetailsontheoremproofsandpropertiesofthealgorithmutilizedforthedeterministicFCNF,wereferthereaderto[ 91 ]. 4.4.1ConcavepiecewiselinearapproximationofRFCNF ObservethatRFCNF-fandFCNFhavethesamecostfunctionf(x)=Pf(i,j)2Egfij(xij),where fij(xij)=8><>:cijxij+sij,xij2(0,ij]0,xij=0. Note,thatthefunctionisdiscontinuousintheoriginwherefij(0)=0andfij(xij)>sijforallxij2(0,ij].Thisfunctioncanbeapproximatedbythefollowingpiecewiselinearconcavefunction: "ijij(xij)=8><>:cijxij+sij,xij2["ij,ij]c"ijijxij,xij2[0,"ij), 56

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Figure4-2. Approximationoffunctionfij(xij). where"ij2(0,ij](Figure 4-2 ).Observethat"ijij(xij)isalowerapproximationoffij(xij),i.e.,"ijij(xij)fij(xij),8"ij2(0,ij]and8x2[0,].Furthermore,notethat"ijij(xij)=fij(xij),8xij2f0gS["ij,ij]andgiven"ij2(0,ij]. InRFCNF-f,letusreplacefij(xij)by"ijij(xij)intheobjective,andconsiderthefollowingapproximationoftheproblem: (RFCNF-)min"(x)=Xf(i,j)2Eg"ijij(xij) (4) s.t.Xfj:(i,j)2Egxij)]TJ /F11 11.955 Tf 23.66 11.35 Td[(Xfj:(j,i)2Egxji=di,8i2V, (4) +1 1)]TJ /F3 11.955 Tf 11.95 0 Td[(SXs=1stsC, (4) tsXf(i,j)2Egxijysij)]TJ /F3 11.955 Tf 11.95 0 Td[(,8s=1,2,...,S, (4) ts0,8s=1,2,...,S, (4) 2R, (4) xij2[0,ij],8(i,j)2E. (4) 57

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NotethatRFCNF-isapiecewiselinearconcaveminimizationproblemwithlinearconstraints.LetPrepresentthesetofextremepointsofthefeasibleregion,and=minfxpijjxp2P,(i,j)2E,xpij>0g.Inotherwords,wetakethevectorsrepresentingtheextremepointsofthefeasibleregion,considerpositiveelementsofthosevectors,anddenotetheminimumofthosevaluesby.Bydenition,>0.In[ 91 ],theauthorsappliedthistechniquetoFCNFproblemandshowedthatbychoosing"ij2(0,],8(i,j)2E,FCNFanditsapproximationproblemhavethesameoptimalsolutionset.RFCNF-fisanFCNFproblemwithadditionallinearconstraintsandcontinuousvariables.However,theseconstraintsdonotcreateanydifcultiesduetothefactthattheydonotcontainintegervariables,andthesameresultholdsforRFCNF-fanditsapproximationRFCNF-,i.e.,bychoosingsufcientlysmall"ij2(0,],8(i,j)2E,ndinganoptimalsolutioninRFCNF-isequivalenttondinganoptimalsolutioninRFCNF-f. Letusdenote1ij(xij)=cijxij+sijand2ij(xij)=c"ijijxij.Figure 4-3 depicts1ij(xij)and2ij(xij)functions,anditiseasytoseethattheobjectivefunctionofRFCNF-canberewritteninthefollowingform "(x)=Xf(i,j)2Eg"ij(xij)=8><>:1ij(xij),xij2[0,"ij)2ij(xij),xij2["ij,ij]=minf1ij(xij),2ij(xij)g. Byintroducingtwoadditionalvariablesforeacharc,RFCNF-canbetransformedintothefollowingrelaxedproblem(RFCNF--R)withbilinearobjectivefunction,linearconstraints,andcontinuousvariables. (RFCNF--R)minx,zXf(i,j)2Eg1ij(xij)z1ij+2ij(xij)z2ij (4) s.t. 58

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Figure4-3. ObjectivefunctionterminRFCNF-problem. Xfj:(i,j)2Egxij)]TJ /F11 11.955 Tf 23.67 11.35 Td[(Xfj:(j,i)2Egxji=di,8i2V, (4) +1 1)]TJ /F3 11.955 Tf 11.96 0 Td[(SXs=1stsC, (4) tsXf(i,j)2Egxijysij)]TJ /F3 11.955 Tf 11.96 0 Td[(,8s=1,2,...,S, (4) ts0,8s=1,2,...,S, (4) 2R, (4) xij0,8(i,j)2E, (4) z1ij+z2ij=18(i,j)2E, (4) z1ij,z2ij0,8(i,j)2E. (4) SuchtransformationissuitableinasensethatanoptimalsolutionofRFCNF--RisanoptimalsolutionofRFCNF-problemorasolutionofRFCNF-problemcanbeveryeasilyconstructedfromthesolutionofRFCNF--Rproblem.IntuitivelythiscanbeshownusingFigure 4-3 ,andfordetailedproofwereferto[ 91 ]. Letusassumethat(x,z)isasolutionofRFCNF--R.ObservethatxisfeasibletoRFCNF-.Ifxij2[0,"ij),than1ij(xij)<2ij(xij)(Figure 4-3 );therefore,itisbenecial 59

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toincreasethevalueofz1ijanddecreasethevalueofz2ijwithinrestrictionsenforcedbyconstraints( 4 )and( 4 ).Sincepair(z1ij,z2ij)=(1,0)isfeasible,wecanconcludethatthisistheoptimalchoiceforz1ijandz2ijvariables.Similarwaywecanconcludethatifxij2("ij,ij],thenatoptimalityz1ij=0andz2ij=1.Incaseofxij="ij,1ij(xij)=2ij(xij);therefore,allvaluesofz1ijandz2ijsatisfyingconstraints( 4 )and( 4 )areoptimal.However,sinceweareinterestedinasolutionwithintegervaluesforzvariables,wecantakeeitherpair(z1ij,z2ij)=(1,0)or(z1ij,z2ij)=(0,1).Usingthisanalysis,oneconcludesthatRFCNF--RandRFCNF-havethesameobjectivefunctionvalueat(x,z)andx,respectively.Fromthelatter,iteasilyfollowsthatxisanoptimalsolutionofRFCNF-. 4.4.2AnalgorithmtosolveRFCNF--R Inthissection,wediscussanalgorithmforndinganear-optimalheuristicsolutionfortherobustxedchargenetworkowproblemwithCVaRconstraints.Intheprevioussection,wetransformedtheRFCNF-fproblemintoanalternativeonewithbilinearobjectivefunctionRFCNF--Rbychoosing"ij2(0,],8(i,j)2E.However,itiscomputationallyexpensivetondthevalueof,whichrequiresenumeratingallverticesofthefeasibleregion.In[ 91 ]theAdaptiveDynamicCostUpdatingProcedure(ADCUP)forndinganear-optimalsolutiontotheFCNFproblemwasdeveloped.SincethestructureofRFCNF-fisverysimilartoFCNFproblem,weapplythealgorithmtondanear-optimalsolutionfortheRFCNF-f. TheADCUPrequiressolvingasequenceofbilinearproblemsRFCNF--Rbygraduallydecreasingthevalueof"(Algorithm 1 ).Specically,inthebeginningitsetsthevaluesof"ijtothecorrespondingarccapacitiesij.Then,ateachiteration,itconstructsaninstanceofRFCNF--Rusingtheselected".SinceRFCNF--Risabilinearproblem,itiscomputationallyexpensivetondaglobaloptimumoftheproblem.Instead,weconsiderndingalocalsolutionsusingDynamicCostUpdatingProcedure(DCUP).If 60

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DCUPgeneratesasolutionwherexij=0orxij"ij,8(i,j)2E,thentheprocedurestops;otherwise,itupdatesthevalueof"ijandproceedsagaintoStep2tondalocalminimumofupdatedRFCNF--RproblembyrunningDCUP.ItisimportanttonotethatateachiterationDCUPislookingforalocalminimumstartingfromthesolutiongeneratedinthepreviousiteration. Algorithm1:AdaptiveDynamicCostUpdatingProcedure(ADCUP) Step1:Let"(1)ij ij,x(1)ij=0,z(1)ij=(z1ij,z2ij)=(1,0).m 1. Step2:RunDCUPtosolvetheRFCNF--Rproblemwithinitialvector(x(m)]TJ /F7 7.97 Tf 6.59 0 Td[(1)ij,z(m)]TJ /F7 7.97 Tf 6.58 0 Td[(1)ij).Let(x(m)ij,z(m)ij)bethesolutionthatisreturnedbyDCUP. Step3:If9(i,j)2Esuchthatx(m)ij2(0,"(m)ij)then"(m+1)ij "(m)ij,2(0,1),m m+1,andgotostep2.Otherwise,stop. DCUPisanotheriterativeprocess,whichsolvesasequenceoftwoLPproblemstondalocalminimumoftheRFCNF--R(Algorithm 2 ).Specically,itxesthevaluesofzvariablesinRFCNF--RandsolvestheresultingLP(z)linearproblemtondthecorrespondingoptimalsolutionx.Then,inRFCNF--Ritxesthevalueofxvariablestotheoptimalvaluesx,andsolvestheresultingLP(x)linearproblem.ThisiterativeprocesscontinuesuntilitconvergestoalocalminimumofRFCNF--Rwithstoppingcriteriaz(k)]TJ /F7 7.97 Tf 6.59 0 Td[(1)=zk,i.e.,vectorzwasnotupdatedbetweentwoconsecutiveiterations. LP(z)minxXf(i,j)2Egc1ijz1ij+c2ijz2ijxij (4) s.t.Xfj:(i,j)2Egxij)]TJ /F11 11.955 Tf 23.67 11.36 Td[(Xfj:(j,i)2Egxji=di,8i2V, (4) +1 1)]TJ /F3 11.955 Tf 11.95 0 Td[(SXs=1stsC, (4) 61

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tsXf(i,j)2Egxijysij)]TJ /F3 11.955 Tf 11.95 0 Td[(,8s=1,2,...,S, (4) ts0,8s=1,2,...,S, (4) 2R, (4) xij2[0,ij],8(i,j)2E. (4) LP(x)minzXf(i,j)2Egc1ijxijz1ij+(s2ij+c2ijxij)z2ij (4) s.t."z2ijxij"z1ij+ijz2ij,8(i,j)2E, (4) z1ij+z2ij=18(i,j)2E, (4) z1ij,z2ij0,8(i,j)2E. (4) Algorithm2:DynamicCostUpdatingProcedure(DCUP) Step1:Setz(1)ij=z(m)]TJ /F7 7.97 Tf 6.58 0 Td[(1)ij.k 1 Step2:Letx(k)=argminfLP(z(k)]TJ /F7 7.97 Tf 6.59 0 Td[(1)g,andz(k)=argminfLP(x(k))g. Step3:Ifz(k)=z(k)]TJ /F7 7.97 Tf 6.59 0 Td[(1)thenstop.Otherwise,k k+1andgotostep2. Notethatthechoiceoftheparameter2(0,1)inAlgorithm 1 hasadirectinuenceontheCPUtimeoftheprocedureaswellasthequalityofthesolution.Inparticular,ifthevalueoftheisclosetoone,"decreasesslowlyandtheprocedurerequiresalargenumberofiterationstostop.Ontheotherhandsmallvaluesoftheparametercanworsenthequalityofthesolution.Inourcomputationalexperiments,weuse=0.5sinceitproducesafairlyhigh-qualitysolutionsusinglittleCPUtimeformostoftheconsideredtestproblems(Section 4.5 ). 62

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4.4.3Dynamicscenarioupdate Asmentionedbefore,wesimulatealargenumberofscenariosinordertocopewithuncertainarcfailures.In[ 19 ]theauthorshaveshowedthatinordertoaccuratelyapproximatetheCVaRcorrespondingtolinearlossfunctionswithanyxedprecisionlevel,thenumberofrequiredscenariosissO(m2=2),wheremisthenumberofarcsinthenetwork.Althoughtherequirednumberofscenarios(andadditionalvariablesandconstraints)ispolynomial,itcanstillbeverylarge,makingaproblemcomputationallychallenginginlarge-scalenetworks. Inordertocopewiththisissue,weproposetheDynamicScenarioUpdateprocedure,whichcandecreasethecomputationaltimeandstillproducehigh-qualityheuristicsolutions.TheoutlineoftheprocedureissummarizedinAlgorithm 3 .ThistechniquecanbeappliedtoRFCNF-faswellasRFCNF--Rformulations.Itrequiresre-solvingtheoptimizationproblemsbygraduallyaddingmorescenariosintoCVaRconstraintsateachiteration.Specically,inthebeginning,wegeneratealargepoolofscenarios,butinsteadofincorporatingallofthemintotheoptimizationproblem,westoretheminthememoryandinitiallysolvetheproblemwithoutCVaRconstraints.Onceweobtainasolutionoftheproblem,weevaluatethecorrespondingCVaRwithrespecttoallthescenariosfromthepoolwehavegenerated.IfthesolutionexceedstheallowanceCforCVaR,weincorporatesomescenariosfromtheoriginalpoolintotheproblemandsolveitagainwiththeCVaRconstraintsthatuseonlythesescenarios.Thisway,wearetryingtoapproximatetherealCVaRbyusingasmallernumberofscenarios.Afterobtainingasolutionoftheresultingproblem,weagaincheckifthesolutionsatisestheCVaRallowanceCwiththeoriginallygeneratedlargepoolofscenarios,andtheniterativelyaddmorescenariostotheoptimizationproblemifnecessary. 63

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Algorithm3DynamicScenarioUpdate Step1:Generatealargepoolofscenariosys,s=1,...,S. Step2:SolveproblemRFCNF--R(orRFCNF-f)withouttheCVaRconstraints. Step3:Usingallgeneratedscenarios,computeCVaRforthesolution.IfCVaR>C,gotoStep4;otherwise,stop. Step4:ConstructCVaRconstraintsbyincorporatingsomeadditionalscenarios(fromthelargepoolofscenarios)intotheoptimizationproblem. Step5:SolvetheresultingRFCNF--R(orRFCNF-f)problem,andgotoStep3. 4.5ComputationalExperiments Thissectionsummarizesthecomputationalresultsfortheconsideredtechniques.Therearesixgroupsoftestproblemsbasedonthesizeofthenetwork,thenumberofsupply/demandnodesandthenumberofscenariosusedtoapproximateCVaR.Foreachgroupofproblemsweconstructeddifferenttypesoftheobjectivefunctionsfij(xij)wherethexedandvariablecostsweregeneratedfromuniformdistributionwithparametersspeciedinTable 4-1 .Eachgroupconsistsof9setofproblemswithdifferentobjectivefunction.Everyproblemsetconsistsof30randomlygeneratedproblems.Thecomponentsofthesupply/demandvectoraregenerateduniformlybetween30and50units.ThemodelswereimplementedinC++andsolvedbyCPLEX11.2.ComputationswereperformedonadesktopwithPentium43.4GHzprocessorand2GBofmemory.Computationaltimepresentedisinseconds.InordertocomparethequalityofADCUPsolutionaswellascomputationaltimeswesolvedproblemsingroupsG1-G3byformulatingthemasintegerprograms( 4 )( 4 )andusingMIPsolverofCPLEX.Therelativeerrorsforthesesetsofproblemswerecomputedaccordingtothefollowingformula:RE=fADCUP)]TJ /F4 11.955 Tf 11.95 0 Td[(fexact fexact100%. 64

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Table4-1. Setofproblems. GrpSetSpp/DemVar.FixedGrpSetSpp/DemVar.Fixed##jNj=jEjNodesjSjCostCost##jNj=jEjNodesjSjCostCost G1130/2003/3200U[1,5]U[50,100]G2140/3004/4200U[1,5]U[50,100]2200U[100,200]2200U[100,200]3200U[200,400]3200U[200,400]4200U[10,20]U[50,100]4200U[10,20]U[50,100]5200U[100,200]5200U[100,200]6200U[200,400]6200U[200,400]7200U[30,40]U[50,100]7200U[30,40]U[50,100]8200U[100,200]8200U[100,200]9200U[200,400]9200U[200,400]G3150/5005/5200U[1,5]U[50,100]G41100/1,25010/10200U[1,5]U[50,100]2200U[100,200]2200U[100,200]3200U[200,400]3200U[200,400]4200U[10,20]U[50,100]4200U[10,20]U[50,100]5200U[100,200]5200U[100,200]6200U[200,400]6200U[200,400]7200U[30,40]U[50,100]7200U[30,40]U[50,100]8200U[100,200]8200U[100,200]9200U[200,400]9200U[200,400]G51200/5,00020/20300U[1,5]U[50,100]G61500/50,00050/50500U[1,5]U[50,100]2300U[100,200]2500U[100,200]3300U[200,400]3500U[200,400]4300U[10,20]U[50,100]4500U[10,20]U[50,100]5300U[100,200]5500U[100,200]6300U[200,400]6500U[200,400]7300U[30,40]U[50,100]7500U[30,40]U[50,100]8300U[100,200]8500U[100,200]9300U[200,400]9500U[200,400] 65

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AveragerelativeerrorsandsolutiontimesfortheseproblemsaresummarizedinTable 4-2 belowandpresentedinTable 4-3 inmoredetail. Table4-2. AveragecomputationalresultsforgroupsG1,G2,andG3. Group#RelativeError,%IPtimeADCUPtime G11.384.470.39G21.97133.790.53G32.623155.410.87 AsonecanseetheCPUtimeforIPincreasesveryfastwiththesizeofthenetwork.Incontrast,ADCUPprovidesthesolutionintimelymannerwithinonlyafewpercentfromoptimality.AreadercannotethatthemaximumcomputationaltimeforIPingroupG3was68246.1secondsorabout19hours.Forlargeproblems,suchassetsG4,G5andG6,CPLEXwasunabletondanoptimalsolutionforintegerprograms.InordertoevaluateperformanceofADCUPonlarge-scalenetworks,wecomparedthesolutionfoundbyACDUPwithLPrelaxationofIPformulation( 4 )( 4 ),whichcanserveasanupperboundfortheoptimalitygap. ItshouldbenotedthatthesolutionsoftheLPrelaxationsoftheconsideredRFCNFproblemsdonothaveanypracticalsignicanceotherthanestablishingthelowerboundsontheoptimalobjectivevaluesofthecorrespondingIPproblems.However,sincendingexactsolutionsofIPproblemsforlargenetworkinstancesiscomputationallychallenging,comparingADCUPsolutionswithLPrelaxationsolutionsistheonlywaytojudgeabouttheperformanceoftheADCUPprocedureinlarge-scalenetworks.Clearly,iftherelativegapbetweenADCUPandLPrelaxationobjectivevalueissmall,thentheADCUPprocedureperformswell,sincetheactualgapwiththeexactIPsolutionwillbeevensmaller.However,ifthegapbetweenADCUPandLPrelaxationsolutionsislarge,thenonecannotdrawanyconclusionsaboutthequalityoftheADCUPsolution,sincethegapbetweenIPandLPrelaxationsolutioncanbelargeaswell.Therefore,inevaluating 66

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Table4-3. DetailedcomputationalresultsforgroupsG1,G2,andG3. AverageAverageCPUTime,sec.GroupSetVar.FixedRelativeError%ADCUPIP##costcost(min,st.dev,max,)(min,st.dev,max,)(min,st.dev,max,) G11U[1,5]U[50,100]1.60.36.2(0,2.4,7.9)(0.2,0.1,0.4)(0.5,13.6,82.2)2U[100,200]2.90.310.7(0,3.6,12.8)(0.2,0.1,0.4)(0.6,19.0,113.9)3U[200,400]3.80.310.9(0,2.8,10.4)(0.2,0.1,0.4)(0.2,13.5,76.5)4U[10,20]U[50,100]0.20.51.3(0,0.4,1.8)(0.3,0.1,0.7)(0.2,1.5,7)5U[100,200]1.00.52.0(0,1.6,6.5)(0.3,0.1,0.7)(0.3,2.2,7.9)6U[200,400]1.40.33.3(0,2.0,7.4)(0.1,0.1,0.4)(0.3,4.6,22)7U[30,40]U[50,100]0.20.40.6(0,0.4,2.1)(0.3,0.1,0.6)(0.1,0.7,3.3)8U[100,200]0.50.41.2(0,0.8,2.7)(0.3,0.1,0.7)(0.2,1.3,6.8)9U[200,400]0.80.42.7(0,1.2,4.7)(0.2,0.1,0.7)(0.3,4.2,19.5)G21U[1,5]U[50,100]3.50.599.6(0,2.4,7.9)(0.4,0.1,0.7)(7.1,184.6,1037.5)2U[100,200]3.00.5232.2(0,3.1,9.6)(0.4,0.1,0.8)(10.0,324.4,1690.8)3U[200,400]4.10.5767.0(0,2.7,8.4)(0.4,0.1,0.8)(15.6,1493.4,8474.7)4U[10,20]U[50,100]1.00.510.3(0,0.7,2.2)(0.4,0.1,0.8)(0.4,25.8,151.8)5U[100,200]1.20.518.5(0,0.9,3.6)(0.4,0.1,0.8)(0.8,19.0,77.7)6U[200,400]1.90.532.2(0,1.6,6.3)(0.4,0.1,0.8)(2.3,32.6,126)7U[30,40]U[50,100]0.20.54.7(0,0.5,1.2)(0.3,0.1,0.7)(0.2,15.1,91.8)8U[100,200]0.40.59.7(0,0.6,1.3)(0.3,0.1,0.7)(0.5,17.6,102.5)9U[200,400]0.60.527.4(0,0.8,2.7)(0.3,0.1,0.7)(0.5,65.4,337.7)G31U[1,5]U[50,100]3.70.83600.0(0,2.0,8.2)(0.7,0.1,1.0)(23.6,7319.4,28405.2)2U[100,200]5.20.96980.5(0,2.9,10.5)(0.7,0.1,1.2)(49.4,9590.7,43648.0)3U[200,400]6.40.916785.5(0,4.1,14.9)(0.7,0.1,1.2)(404.8,15190.4,68246.1)4U[10,20]U[50,100]0.60.959.8(0,0.9,3.6)(0.6,0.1,1.1)(1.1,71.9,271.4)5U[100,200]1.90.9137.0(0,1.2,4.6)(0.5,0.2,1.4)(14.0,129.4,469.9)6U[200,400]2.10.8707.1(0,1.4,4.8)(0.5,0.1,1.1)(20.8,1083.5,5629.0)7U[30,40]U[50,100]0.10.912.9(0,0.2,0.7)(0.6,0.2,1.3)(0.4,16.9,71.2)8U[100,200]0.50.854.2(0,0.5,1.8)(0.6,0.2,1.3)(0.5,90.6,378.3)9U[200,400]1.10.871.8(0,1.1,3.6)(0.6,0.1,1.0)(9.1,92.3,369.5) 67

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thequalityofADCUPonlargeinstances,weonlyconsideredthosesetsofproblems(withspeciccombinationsofxedcostandvariablecostparameters)thatexhibitedsmallIP-LPrelaxationgapsinmoderate-sizenetworkinstances.WepresumedthatlargeprobleminstanceswiththesamecombinationsofxedandvariablecostswouldalsohavesmallgapsbetweenIPandLPrelaxationsolutions.AlthoughwecouldnotverifythisbysolvingthecorrespondinglargeIPproblemsduetoobviouscomputationalchallenges,theexperimentswithADCUPprocedurefortheseproblemsets(presentedbelow)conrmedthisassumptionanddemonstratedthattheprocedureproduceshigh-qualitysolutionsintheselargeinstances. Table 4-4 presentstherelativedifferencesbetweenADCUPandLPsolutionsforthetestproblems.WecomputedarelativedifferencebetweenADCUPsolutionandLPrelaxationaccordingtothefollowingformula:RDADCUP=fADCUP)]TJ /F4 11.955 Tf 11.95 0 Td[(fLP fLP100%. Thecomputationalexperimentsshowedthatthisrelativedifferenceisreasonablysmallforproblemsets7,8and9withlargervariablecosts.WeusedthesesetsofproblemsinordertoestimatethequalityofADCUPsolutionforlargeproblems,suchasingroupsG4,G5andG6.AsshowninTable 4-5 ,therelativedifferenceslightlyincreasesasthesizeofthenetworkincreases.ItshowsthatthequalityofADCUPsolutionforthelargeproblemsisaboutthesameasforthegroupsG1-G3(forwhichtheexactsolutionshavebeenfound).ThelargestproblemthatwewereabletosolvebyusingADCUPalgorithmwithDynamicScenarioUpdatecontained1,000nodesandapproximately200,000arcs(withobjectivefunctionparameterscorrespondingtoset#9),whichtook3788secondsofCPUtime. 68

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Table4-4. ComputationalresultsforgroupsG4,G5andG6. AverageAverageCPUTime,sec.RelativeDifference%GroupSetVar.FixedwithLPRelaxationADCUPLPRelaxation##costcost(min,st.dev,max,)(min,st.dev,max,)(min,st.dev,max,) G47U[30,40]U[50,100]12.25.60.2(10.7,1.1,15.1)(4.1,0.7,7.6)(0.1,0.0,0.2)8U[100,200]24.16.00.2(19.9,2.1,28.4)(5.0,0.8,7.9)(0.1,0.0,0.2)9U[200,400]46.66.10.2(41.4,2.8,53.0)(4.1,1.1,8.8)(0.1,0.0,0.3)G57U[30,40]U[50,100]12.949.31.0(11.9,0.8,15.0)(41.6,5.2,59.7)(0.9,0.1,1.1)8U[100,200]24.247.51.0(20.9,1.4,26.6)(39.6,5.0,59.9)(0.8,0.1,1.1)9U[200,400]47.847.81.0(42.0,3.0,53.4)(38.1,6.0,62.2)(0.9,0.1,1.1)G67U[30,40]U[50,100]12.81041.416.2(11.9,0.5,13.8)(891.0,266.5,2436.8)(15.3,1.0,19.1)8U[100,200]24.6968.715.7(21.9,1.1,27.3)(858.0,69.9,1169.9)(14.6,0.9,18.7)9U[200,400]48.298116.7(44.5,3.1,49.9)(878.0,87.4,1131.2)(15.6,0.8,18.1) Table4-5. AveragerelativedifferenceofADCUPresultswithLPrelaxationresultsforproblemsets7-9. G1G2G3G4G5G6SetjEj=200jEj=300jEj=500jEj=1,250jEj=5,000jEj=50,000#jNj=30jNj=40,jNj=50jNj=100jNj=200jNj=500jSj=200jSj=200jSj=200jSj=200jSj=300jSj=500 710.610.511.512.212.912.8821.121.322.224.124.224.6940.240.042.346.647.848.2 69

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CHAPTER5CRITICALNODESANDVERTEXCOVERPROBLEMSININTERDEPENDENTNETWORKSWITHCASCADINGFAILURES Moderninfrastructuresprovideservicestopeopleonaregularbasis,theyaredifferentbynature,suchaselectricity,telecommunication,andtransportation;however,manyofthemdependoneachother(i.e.interdependent)andprovideservicesonlyifunderlyinginfrastructuresareoperating.Electricityistypicallyconsideredtobeoneoftheunderlyinginfrastructuresthatisrequiredformanyotherinfrastructurestoprovideservices.Anoverwhelminghumandependenceonelectricitypresentsthechallengeofdeterminingareliableandsecureenergysupply.Withinfrastructuresoperatingneartheirphysicallimitsthereliabilityofthesystemsmaydecrease.Recentpoweroutagecausedbyashort-circuitinNorthGila,ArizonaaffectedmillionspeopleinCalifornia,ArizonaandMexicoonSeptember8th,2011.Thetransportationinfrastructurewasdramaticallyaffectedinurbanareasduetothelackofpower.Energysystemsappeartobeveryvulnerableduetomutualdependencieswithotherinfrastructures.OneoftherecentexamplesofcascadingfailuresininterdependentinfrastructuresoccurredinItalyonSeptember28th,2003,whereshutdownofapowerstationledtofailureofInternetcommunicationnetworkcomponents,whichwereusedforcontrolofelectricitynetworkandledtodevelopmentoffurtherfailuresinpowernetwork[ 21 ]. Thischapterconsidersareliabilityanalysisforinterdependentnetworksandstudiestwotypesofinterdependenciesthatarecommonintelecommunicationandenergycontexts.Wepresentmathematicalprogrammingmodelsforthecriticalnodesproblemandminimumvertexcoverproblemintheinterdependentnetworkenvironmentunderdifferentassumptionsonsurvivabilityofthedependentnodes,consideringcascadingfailures.Criticalnodesproblem(CNP)helpstounderstandthestructuralcharacteristicsandconnectivitypropertiesofthenetwork.CNPidentiesasetofnodes,whichdeletion 70

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minimizesthepairwiseconnectivityoftheremainingnetwork.Ingeneralcase,thevertexcoverproblemcanbestatedastoidentifytheminimumnumberofnodes,whichfailuresmakethesetofaffectednodestobecompletelydisconnected,i.e.avertexcover.Thefollowingtwosubsectionsprovideabriefoverviewoftheliteratureonreliabilityanalysisofinterdependentinfrastructures,aswellasonthecriticalnodesandvertexcoverproblems. 5.1LiteratureOverview 5.1.1Reliabilityanalysisofinterdependentinfrastructures Interdependentinfrastructuresattractalotofattentionfromtheresearchersduringtherecentyears.Rinaldietal.[ 107 ]describedinfrastructurecharacteristics,typesofinfrastructureinterdependencies,examplesofnth-orderinterdependencies,aswellasstudiedinterdependentinfrastructuresassociatedwithelectricpower.Thepaperalsoemphasizeschallengesassociatedwithmodelingandsimulationofinterdependentinfrastructures.Leontiefmodelforeconomiccommodityowshasbeenappliedtotheanalysisofriskincomplexinfrastructuresin[ 57 65 ].Theauthorsconsideredtime-independentanalysisofgeneration,transshipmentandconsumptionofcommoditiesindifferentinterconnectedinfrastructures.Twopapers[ 56 ]and[ 55 ]extendedtheproposedmodelandprovidedextensivecase-studies.Amin[ 4 ]identiedsecuritychallengesinelectricity,telecommunicationandtransportationinfrastructures.Theauthorpointsouttheneedforresearchanddevelopmentinefciencyandrobustnessofinterdependentinfrastructures.Little[ 78 ]describedcausesandconsequencesofinterdependentinfrastructurefailures.Rihaldi[ 106 ]emphasizedfactorsthataffectinterdependenciesanalysisaswellassummarizedmodelingandsimulationtechniquesusedforanalisysofinterdependentinfrastructures.Duenasetal.[ 44 ]summarizedtopologicpropertiesofthenetworks,reviewedrandom,small-world 71

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andscale-freenetworks,andprovidedsimulationresultsforresilienceofeachofthesenetworkswithrespecttoverticesandedgesremoval.Also,theauthorsproposedaninterdependentinfrastructuremodelwithprobabilisticfailuresofdependentcomponents.Themodelwastestedonasmallexampleofwater,transportation,gas,andelectricitynetworks.Thisworkhasbeenextendedin[ 45 46 ],wheretheauthorsmodelednetworkinterdependenciesutilizingaconditionalprobabilityoffailurebetweentwoelementsofpowerandwaternetworksandconsideredpossiblemitigationactions.Thisparameterwasadjustedaccordingtothegeographicalproximityofnetworkcomponents.Newmanetal.[ 93 ]consideredriskassessmentininterdependentinfrastructuresandappliedprobabilisticanddynamicalmodelsthathadbeenusedtostudyblackoutdynamicsinpowersystems.Ouyangetal.[ 96 ]presentvulnerabilityanalysisandsimulationmodelofinterdependentelectricityandgasnetworks.ThebookchapterofHealetat.[ 60 ]discussessecurityofinterdependentnetworks,aswellasapplicationsforsupplychainmanagementandcomputersecurity.SvendsenandWolthusen[ 118 ]proposedtwoapproachesformodelingnetworkinterdependencies.Bothofthemarebasedonageneralizednetworkowmodelandutilizeanunbufferedowmodelforcommoditiesthatcannotbestored,suchaselectricity,aswellasabufferedapproachforcommoditiesthatcanbestored,suchasnaturalgasandoil.Minetal.[ 84 ]proposedmodelingframeworkforinterdependentinfrastructuresusingsystemdynamics,functionalmodelingandnonlinearoptimizationmethods.Restorationmodelforinterdependentinfrastructureshasbeenproposedin[ 77 123 ].Casalicchioetal.[ 28 ]describedafederatedagent-basedmodelingandsimulationapproachforinterdependentITinfrastructures.Luiijfetal.summarizedin[ 80 ]thecascadingfailuresofcriticalinfrastructuresoccurredinEuropeanUnion,emphasizingbothcascadeinitiatingandaffectedinfrastructures.HelsethandHolen[ 61 ]proposedamodelfor 72

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studyingvulnerabilityofinterdependentelectricanddistrictheatingsystems.Reedetal.[ 105 ]presentedamethodologyforassessingtheresilienceofinterdependentinfrastructuresfornaturalhazardevents.Buldyrevetal.[ 21 ]introducedasimulationmodelforcascadesoffailuresininterdependentnetworks.Forthecase-studiestheauthorsusedthereal-worlddatafromblackoutthathappenedinItalyin2003andconsideredpowergridandtheInternetastwointerdependentnetworks,wherefailureinpowernetworkcancausefailureofInternetserver,whichinturncanaffectthepowergrid.Theauthorsdevelopedananalyticalsolutionforthefractionofnodestobedeletedthatwouldleadtoaseriesofcascadesandcompletefragmentationoftwonetworks.Theextensionsofthismodelhasbeenstudiedin[ 49 ],wheretheauthorsintroducedaunidirectionaldependencybetweenthenetworksandstudiedthecasesofmorethattwointerdependentnetworks.Theauthorsalsopresentedanalyticalsolutionsforthesizeofgiantconnectedcomponentforthreetypesofinterdependentnetworks:nfullydependentErdos-Renyinetworks,star-likenetworkofnpartiallyinterdependentErdos-Renyinetworksandloop-likenetworkofpartiallyinterdependentErdos-Renyinetworks.Barrettetal.[ 15 ]studiedcascadingfailuresintransportationandtelecommunicationnetworksduringevacuations.Inparticulartheauthorsstudiedhowextendedroadcongestioncanleadtoafailureofcommunicationinfrastructureinurbanregions.Parshanietal.[ 98 ]suggestedthatrealinterdependentnetworksarecoupledbynodeswithsomesimilarity.Asacase-studytheauthorsusedworldportandairportnetworks.Theyintroducedtwocoefcientsthatcanmeasurenetworksimilarityandbyusingasimulationmodelshowedthatinterdependentnetworkswithinter-similardependenciesaremorerobustthanthenetworkswithrandomdependencies.SimilarresultwasobtainedbyBuldyrevetal.in[ 23 ],wheretheauthorsstudiedtwointerdependentnetworkswithbidirectionaldependencylinksandone-to-one 73

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correspondencebetweenthenodesoftwonetworks.Theauthorsconstructedthenetworksinsuchawaythatdependentnodeshadthesamedegreeandstudiedthesizeofgiantmutuallyconnectedcomponentafterremovingafractionofnodesfromthenetworks.Amongotherresultstheauthorsshowedthatthenetworkscoupledbythenodeswithpositivelycorrelateddegreesaremorerobustthanrandom-couplednetworks.Shaoetal.[ 111 ]studiedadifferenttypeofdependencybetweentwonetworksandanalyticallyestimatedthesizeofgiantmutuallyconnectedcomponent.Incontrasttopreviouswork,whichfocusedonrandomnoderemovalininterdependentnetworks,Huangetal.[ 64 ]studiedvulnerabilityofinterdependentnetworksundertargetedattacks.Theauthorstransformedthetargeted-attackproblemtoarandom-attackproblemandshowedthatinterdependentnetworksincontrasttosingle-layernetworksarevulnerableevenifhigh-degreenodesareprotected.ZioandSansavini[ 126 ]proposedasimulationmodelforidenticationofoperatingmarginsforcascade-safeoperationofinterdependentinfrastructures.Parshanietal.[ 99 ]showedbothanalyticallyandnumericallythatreducingthedependencybetweentwointerdependentnetworkschangesthepercolationtransitionfromtherstordertothesecondone.Schneideretal.[ 110 ]showedasimilarresultthatmakingasmalloptimalfractionofthenodesininterdependentnetworksautonomouscouldleadtothechangeofpercolationtransitionfromdiscontinuoustocontinuous,signicantlyimprovingtheresilienceofthenetworks. 5.1.2Criticalnodesproblem Theproblemofndingcriticalcomponentsinnetworkedsystemshasbeenextensivelystudiedintherecentresearchliterature.Arulselvanetal.[ 6 ]describedamodelforCNPthatidentiescriticalnodes,whichremovalresultsintheminimumpair-wiseconnectivityamongtheremainingnodesinthenetwork.TheauthorsprovedthattheproblemisNP-completeandproposedaheuristicalgorithm,whichfound 74

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optimalsolutionsformostofthecasestudieswithlittlecomputationaltime.In[ 7 ]theauthorsextendedtheCNPbyincorporatingcardinalityconstraintsforremainingdisconnectedcomponents.Dingetal.[ 42 ]consideredtheproblemofminimizingpairwiseconnectivitycalled-disruptor,aswellasprovedNP-CompletenessofthisproblemandinapproximabilitywithinafactorofO(lognloglogn).DinhandThaiproposedabranch-and-cutalgorithmforthe-disruptorintegerprogrammingformulationin[ 40 ]andanapproximationalgorithmin[ 41 ].Kuhlmanetal.[ 75 ]studiedtheproblemofinhibitingdiffusionofrumorsinsocialnetworksbyidentifyingasmallfractionsofthenodestoberemovedformthenetwork.FanandPardalos[ 48 ]proposedamixedintegerprogrammingformulationfortheproblemofgraphpartitioningandcriticalnodesdetection.Theauthorsintroducedrobustoptimizationmodelsandpresentedasolutionalgorithmfortheresultingproblembasedonadecompositionmethodononevariable.DiSummaetal.[ 117 ]studiedtheCNPovertreesandprovedthattheproblemisstillNP-completewithgeneralconnectioncosts.Forthecasewithnonnegativeconnectioncosts,theauthorsproposedanalgorithmwithtimecomplexityO(1.618034n).ShenandSmith[ 112 ]proposedpolynomialtimealgorithmsforasubclassofCNPfortreesandseries-parallelgraphs,wheretheobjectivesweretomaximizethenumberofdisconnectedcomponentsandtominimizethelargestcomponentinthegraph.DasguptaandBiswas[ 37 ]studiedtheproblemofcriticalnodesinthecontextofunevennetworktrafcdistributionandidentiedcriticalringsaroundcriticalnodestorelieveuneventrafcdistribution.DiSummaetal.[ 39 ]proposedanewformulationfortheCNP,studiedvalidinequalitiesofthemodelandprovidedtheoreticalresultsaboutthem.Furthermore,theauthorsproposedanalternativequadraticmodelforCNPandpresentedextensivecomputationalresults.Murray[ 89 ]providedanoverviewofthemodelsfornetworkreliability. 75

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5.1.3Vertexcoverproblem Vertexcoverproblem(givenagraphG=(V,E),ndasubsetofverticesthattoucheseveryedgeinthisgraph)isoneoftheKarp's21NP-completeproblems[ 68 ].ItisknowntobeAPX-complete[ 97 ]andisNP-hardtoapproximatewithinafactorof1.3606[ 43 ].Theproblempossessesahalf-integralityproperty(eachsolutionofthelinearprogrammingrelaxationhasonly0,1,or1/2coordinates),whichimmediatelyleadstoasimplefactor2approximationalgorithmfortheproblem:solveaLPrelaxationandroundall1/2coordinatesto1[ 121 ].However,ndingabetterapproximationalgorithmisachallengingproblemthathasbeenstudiedin[ 13 58 85 ].ThealgorithmthatusesaSDPrelaxationandso-calledtriangleinequalitieshasbeendevelopedrecentlybyKarakostas[ 67 ]andhasthebestsofarapproximationfactorof1)]TJ /F6 11.955 Tf 11.56 0 Td[(1 p logn.KhotandRegev[ 71 ]showedthatvertexcoverproblemishardtoapproximatewithinaconstantfactorsmallerthan2.Parameterizedvertexcoverproblem(givenagraphGandanumberk,showthatthevertexcoverforthisgraphisatmostkvertices)hasbeenextensivelystudiedintheliterature[ 10 24 30 31 94 ].ThemostrecentpaperthatstudiesthatproblembyChenetal.[ 32 ]describesanO(1.2738k+kn)-timepolynomialspacealgorithm,improvingpreviouscorrespondingresults. 5.2CascadingFailuresinInterdependentNetworks Inthisworkwemodeltwointerconnectedinfrastructuresasatwo-layerinterdependentnetworkwithG1=(V1,E1)beingasetofnodesandedgesforonelayerandG2=(V2,E2)fortheother.Note,thatthenumberofnodesinthenetworkscanbedifferent(n1=jV1jfortherstlayerandn2=jV2jforthesecondone)allowingone-to-manyrelationshipsbetweenlayers.WithE12wewilldenoteasetofarcsbetweentwonetworkswithnodesV2beingdependentonnodesV1,i.e.afailureofanodeinnetwork1mayleadtoafailureofnodesinnetwork2;E21willdenotethesamedependenceofV1onV2. 76

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ForanexamplenetworkdepictedinFigure 5-1 :E12=f(2,1),(2,2),(3,3),(4,3)gandE21=f(4,4),(3,2)g. 1 4 2 3 1 4 2 3 Figure5-1. Anexampleofa2-layerinterdependentnetwork AnimportantdetailthatisintroducedinthisworkisthenumberoffailurestagesSaparameterthatdeneshowmanyfailurestagesarebeingconsideredinthemodel.Themainmotivationforintroducingthisparameteristhefactthatinmanycasesthecascadingfailuresdonotspreadinstantaneouslybutratherwithsomedelaybetweencascades[ 17 ].Incommunicationcontextthisisassociatedwithoverloadingsomeoftheequipmentthatmayleadtoachangeofnetworktopologyandreroutingoftrafc,whichconsequentlymayoverloadothernodesandmakethemnonfunctional.Intheproposedmodelweconsiderthatanodebelongstoazerofailurestageonlyifitwasattacked.Consequently,anodebelongstotherstfailurestageifiteitherbelongstoazerofailurestage,orfailedtoworkduetoitsdependenceonnodesthatfailedatstagezero.Inotherwords,stageoneconsistsofthenodesthatarenotfunctionalafteronecascadeoffailures.Thisprocedurecanbeeasilyextendedtothesthfailurestage,meaningthatthenodesthatbelongtothisfailurestageareeitherfailedonstage(s)]TJ /F6 11.955 Tf 12.37 0 Td[(1)orfailedduetotheirdependenceonnodesfrom(s)]TJ /F6 11.955 Tf 12.03 0 Td[(1)thfailurestage.Cascadingfailuresmaycontinue 77

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uptostageS.Subsection 5.4.4 analysesthemaximumpossiblenumberofcascadesthatcanhappeninagiventwo-layerinterdependentnetwork. ThenextsectionswilldescribeindetailthetwodifferenttypesofinterdependenciesintroducedinDenitions 1 and 2 Denition1. Type1interdependenceassumesthatafteranodefailureinonenetworkallthedependentnodesintheothernetworkwillalsofail. Denition2. Type2interdependenceassumesthatanodeinonenetworkcanfailonlyifallthenodesintheothernetworkfromwhichthisnodeisdependentfail. Foranexamplenetworkdepictedingure 5-1 ,forType1interdependence,node3innetwork2willfailifeithernode3ornode4innetwork1fail.ForType2interdependence,node3innetwork2willfailonlyifbothnode3andnode4innetwork1fail.Therstcasecanbeappliedtotheinformationtransmissioncase,wherethedependentarcsbetweenthenetworklayersrepresenttheinformationowfromonenetworktoanother.Thesecondcaseisapplicabletothepowersupplycase,wherethearcsofdependenciesrepresentthepowersupplygoingfromonelayertoanother. 5.3CriticalNodesProbleminInterdependentInfrastructures 5.3.1CNPwithoutcascadingfailures ThissectionpresentsseveralformulationsforCNPininterdependentnetworksbasedondifferentrelationsbetweentheinfrastructures.Inthesimplestcase,presentedin[ 22 ]thereisone-to-onerelationshipsbetweenthenodesindifferentnetworks.Insuchinfrastructuresafailureofanodeinonenetworkcancauseonlyafailureofonenodeinanothernetwork;Section 5.3.1.1 introducesamodelformulationforsuchproblemsetup.Section 5.3.1.2 describesamoregeneralproblem,wherethenodesbetweenthelayerscouldbeconnectedwithdirectedarcsandafailureofonenodecancauseafailureofmultiplenodesinanothernetwork. 78

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5.3.1.1One-to-onecorrespondencebetweeninfrastructures Thissubsectiondescribesasimplecaseoftwointerdependentnetworks(V1,E1)and(V2,E2)withthesamenumberofnodesineachlayerandanynodeinthelayer1isconnectedtoonlyonenodeinthelayer2(Figure 5-2 ). 1 4 2 3 1 4 2 3 Figure5-2. One-to-onecorrespondenceina2-layernetwork.Connectionsbetweenthelayersarepresentedbybluecoloredarcs. Suchproblemscanbesimpliedtotheoneshavingonlyonesetofvertices(V)andtwosetsofedges(E1,E2).Furtherinthepaperwewillusevariablesu1ijtodenotewhethernodesiandjareconnectedinlayer1ofthenetwork(u1ij=1)ornot(u1ij=0).Similarly,variablesu2ijwillbeusedforthesecondlayerofthenetwork.Variablesviwilldenoteifnodeiisremovedfromthenetwork(vi=1)andthemaximumnumberofnodesthatcanbedeletedwillberestrictedbyaparameterK.Withthatnotationwecandescribetheproposedcriticalnodesproblemininterdependentnetworks.Theobjectiveofthemodelistominimizethenumberofnodesconnectedtoeachother.Constraints( 5 )and( 5 )ensurethatifnodesiandjareintheseparatecomponents,thanoneofthemmustbedeleted,otherwisetheyremainconnected.Constraint( 5 )restrictthe 79

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numberofnodesthatcanberemovedfromthenetwork(attacked).Constraints( 5 )ensurethatifnodesiandjaredisconnectedintheremainingnetwork,therewillbenocaseswhenbothfi,kgandfk,jgareconnected.Constraints( 5 )statethesamerestrictionforthesecondnetwork.Finally,constraint( 5 )ensuresthatmaximumKnodescanbedeleted.minimizeXi2VXj2V(u1ij+u2ij) (5)subjecttou1ij+vi+vj1,8(i,j)2E1, (5)8>>>>>><>>>>>>:u1ij+u1jk)]TJ /F4 11.955 Tf 11.95 0 Td[(u1ki1,u1ij)]TJ /F4 11.955 Tf 11.95 0 Td[(u1jk+u1ki1,)]TJ /F4 11.955 Tf 9.3 0 Td[(u1ij+u1jk)]TJ /F4 11.955 Tf 11.95 0 Td[(u1ki1,8(i,j,k)2V, (5)u2ij+vi+vj1,8(i,j)2E2, (5)8>>>>>><>>>>>>:u2ij+u2jk)]TJ /F4 11.955 Tf 11.95 0 Td[(u2ki1,u2ij)]TJ /F4 11.955 Tf 11.95 0 Td[(u2jk+u2ki1,)]TJ /F4 11.955 Tf 9.3 0 Td[(u2ij+u2jk)]TJ /F4 11.955 Tf 11.95 0 Td[(u2ki1,8(i,j,k)2V, (5)Xi2VviK. (5) 5.3.1.2One-to-multiplecorrespondencebetweeninfrastructures InthissubsectionwedescribeamoregeneralcaseofnetworkinterdependencewiththenodesbetweenlayersbeingconnectedbytwosetsofdirectedarcsE12andE21(Figure 5-3 ).Ifonenodeisdestroyed,thecorrespondingnode(s)towhichitwasdirectlyconnectedandtheiradjacentedgesarealsodestroyed.Forexample,ifnode2inFigure 80

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5-3 isdestroyedinthelayer1,nodes1and2inthelayer2andtheiradjacentarcsarealsodestroyed.Inthisproblemwedonotconsiderpossiblecascades,i.e.theattackonnode4inthelayer1doesnotcausethesubsequentfailureofnode2inthelayer1duetofailureofnode3inthelayer2. 1 4 2 3 1 4 2 3 Figure5-3. One-to-multipledirectconnectionina2-layernetwork. Inthissectionweintroducethefollowingvariables:wewillusevariablesy1itodenotewhetheranodeiisattackednlayer1ofthenetwork(y1i=1)andy2iforthesecondlayer.Parametersn1andn2willdenotethenumberofnodesinlayer1and2ofthenetwork.Finally,a1ij=1if(i,j)2E12,adsimilarlya2ij=1if(i,j)2E21Theproblemminimizesthenumberofconnectedpairsofnodesinbothlayers,subjecttothemaximumnumberofnodesthatcanbeattackedbeingK.Inadditiontotheconnectivityconstraints( 5 )( 5 )introducedintheprevioussection,equations( 5 )and( 5 )imposeinterdependentnetworkfailurerequirements,andequations( 5 )and( 5 )arenodesurvivalconditionsforbothofthenetworklayers.minimizeXi2V1Xj2V1u1ij+Xi2V2Xj2V2u2ij (5) 81

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subjecttou1ij+v1i+v1j1,8(i,j)2E1, (5)8>>>>>><>>>>>>:u1ij+u1jk)]TJ /F4 11.955 Tf 11.96 0 Td[(u1ki1,u1ij)]TJ /F4 11.955 Tf 11.96 0 Td[(u1jk+u1ki1,)]TJ /F4 11.955 Tf 9.3 0 Td[(u1ij+u1jk)]TJ /F4 11.955 Tf 11.96 0 Td[(u1ki1,8(i,j,k)2V1, (5)u2ij+v2i+v2j1,8(i,j)2E2, (5)8>>>>>><>>>>>>:u2ij+u2jk)]TJ /F4 11.955 Tf 11.96 0 Td[(u2ki1,u2ij)]TJ /F4 11.955 Tf 11.96 0 Td[(u2jk+u2ki1,)]TJ /F4 11.955 Tf 9.3 0 Td[(u2ij+u2jk)]TJ /F4 11.955 Tf 11.96 0 Td[(u2ki1,8(i,j,k)2V2, (5)v1i1 n1+1 y1i+Xj2V2a2jiy2i!,8i2V1, (5)v2i1 n2+1 y2i+Xj2V1a1jiy1i!,8i2V2, (5)v1iy1i+Xj2V2a2jiy2i,8i2V1, (5)v2iy2i+Xj2V1a1jiy1i,8i2V2, (5)Xi2V1y1i+Xi2V1y2iK. (5) 5.3.2CNPwithcascadingfailures 5.3.2.1Type1interdependence Thissectionextendstheproblemconsideredintheprevioussectionforageneralfailurecasewithcascades.Theproblemsettingassumesthattherearetwolayers:(V1,E1)and(V2,E2)(Figure 5-3 ).Nodesbetweenthelayersareconnectedbytwosets 82

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ofdirectedarcsE12andE21.Themodelminimizesthenumberofconnectedpairofnodesinbothlayers,subjecttoacertainnumberofnodesthatcanbeattacked.Withtheproblemintroducedintheprevioussectiontheattackonnode4inthelayer1,forexample,destroysonlynode3inthelayer2.Intheconsideredcase,node3inthelayer2isdirectlyconnectedtonode2inlayer1,whichwasnotaffected.Inthiscaseanydestroyednodecausesthefailureofallthenodesthatitwasconnectedtointheotherlayer.Hence,ifnode4isdestroyedinthelayer1,itwillcausethedestroyingofnodes1,2,3inthelayer2andnode2inthelayer1.Toensurethatsuchcascadeshappen,thesubsequentnodedestroyingconstraints( 5 )( 5 )shouldbechangedtov1i1 n1+1 y1i+Xj2V2a2jiv2i!,8i2V1, (5)v2i1 n2+1 y2i+Xj2V1a1jiv1i!,8i2V2, (5)v1iy1i+Xj2V2a2jiv2i,8i2V1, (5)v2iy2i+Xj2V1a1jiv1i,8i2V2. (5) However,itshouldbenotedthattheseconstraintscanbeintroducedonlyiftherearenocyclesintheinterconnectionedgesE12,andE21.Alternatively,theproblemcanbemodiedtoconsiderthecascadingfailureswithoutsuchnocycleassumption.Inadditiontothevariablesintroducedintheprevioussectionswewillusevsti=1todenotewhethernodeifailsatstagesinthelayertofthenetwork.Theparameterntwilldenotenumberofnodesinthelayertoftheinterdependentnetwork.Constraints( 5 )and( 5 )describeinterdependentnodefailures,andconstraints( 5 )and( 5 )imposenodesurvivalrequirementsduringcascadingfailures. 83

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minimizeXt=1,2Xi2V1Xj2V1utij (5)subjecttoutij+vSti+vStj1,8(i,j)2Et,t=1,2, (5)8>>>>>><>>>>>>:utij+utjs)]TJ /F4 11.955 Tf 11.95 0 Td[(utsi1,utij)]TJ /F4 11.955 Tf 11.96 0 Td[(utjs+utsi1,)]TJ /F4 11.955 Tf 9.3 0 Td[(utij+utjs)]TJ /F4 11.955 Tf 11.96 0 Td[(utsi1,8(i,j,k)2Vt,t=1,2, (5)vs1i1 n2+1 vs)]TJ /F7 7.97 Tf 6.59 0 Td[(11i+Xj2V2a2jivs)]TJ /F7 7.97 Tf 6.59 0 Td[(12i!,8i2V1,s=1,...,S, (5)vs2i1 n1+1 vs)]TJ /F7 7.97 Tf 6.59 0 Td[(12i+Xj2V1a1jivs)]TJ /F7 7.97 Tf 6.59 0 Td[(11i!,8i2V2,s=1,...,S, (5)vs1ivs)]TJ /F7 7.97 Tf 6.58 0 Td[(11i+Xj2V2a2jivs)]TJ /F7 7.97 Tf 6.59 0 Td[(12i,8i2V1,s=1,...,S, (5)vs2ivs)]TJ /F7 7.97 Tf 6.58 0 Td[(12i+Xj2V1a1jivs)]TJ /F7 7.97 Tf 6.59 0 Td[(11i,8i2V2,s=1,...,S, (5)Xt=1,2Xi2V1v0tiK. (5) 5.3.2.2Type2interdependence Inthissectionweconsiderthecriticalnodesproblemsetupwhenanodebelongstothesthfailurestageifandonlyifitbelongsto(s)]TJ /F6 11.955 Tf 12.22 0 Td[(1)thfailurestageorallthenodesitisconnectedtofromotherlayersbelongto(s)]TJ /F6 11.955 Tf 12.15 0 Td[(1)thfailurestage.Thisassumptioncanbeintroducedbychangingtheconstraintsonnodesremoval( 5 )( 5 )tovstivs)]TJ /F7 7.97 Tf 6.59 0 Td[(1ti,8i2Vt,t=1,2,s=1,...,S, (5) 84

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1)]TJ /F4 11.955 Tf 11.96 0 Td[(vs1i1 n2+1 Xj2V2a2ji(1)]TJ /F4 11.955 Tf 11.95 0 Td[(vs)]TJ /F7 7.97 Tf 6.59 0 Td[(12i)!)]TJ /F4 11.955 Tf 11.96 0 Td[(vs)]TJ /F7 7.97 Tf 6.59 0 Td[(11i,8i2V1,s=1,...,S, (5)1)]TJ /F4 11.955 Tf 11.96 0 Td[(vs2i1 n1+1 Xj2V1a1ji(1)]TJ /F4 11.955 Tf 11.95 0 Td[(vs)]TJ /F7 7.97 Tf 6.59 0 Td[(11i)!)]TJ /F4 11.955 Tf 11.96 0 Td[(vs)]TJ /F7 7.97 Tf 6.59 0 Td[(12i,8i2V2,s=1,...,S, (5)1)]TJ /F4 11.955 Tf 11.96 0 Td[(vs1iXj2V2a2ji(1)]TJ /F4 11.955 Tf 11.96 0 Td[(vs)]TJ /F7 7.97 Tf 6.59 0 Td[(12i),8i2V1,s=1,...,S, (5)1)]TJ /F4 11.955 Tf 11.96 0 Td[(vs2iXj2V1a1ji(1)]TJ /F4 11.955 Tf 11.96 0 Td[(vs)]TJ /F7 7.97 Tf 6.59 0 Td[(11i),8i2V2,s=1,...,S. (5) 5.4MathematicalProgrammingFormulationsforVertexCoverProbleminInterdependentNetworks Minimumvertexcoverproblemcanbeconsideredasaspecialcaseofcriticalnodesproblemwithallthenodesbeingdisconnectedfromeachotheraftercascadingfailures.ThissectionstudiestheminimumvertexcoverproblemininterdependentnetworksforthetwotypesofinterdependencethatwereintroducedinSection 5.2 5.4.1Type1interdependence ThissectionintroducesamathematicalprogrammingformulationforthevertexcoverproblemininterdependentnetworkswithType1interdependencethatmaybeappropriateinthecontextofinformationexchangenetworks.WeassumethatafailureprocessinthecouplednetworkscontinuesuptoacertainstageSjV1j+jV2j)]TJ /F6 11.955 Tf 18.06 0 Td[(1.Themodelidentiestheminimumnumberofnodes,whosefailuresmakethesetofnodescompletelydisconnected.Thisproblemsettingassumesthatifanodefails,thenallofitsdependentnodesalsofail.Forexample,ifnode4inlayer1isattacked(Figure 5-3 ),node3inthelayer2andtheiradjacentarcsfailintherstfailurestage.Then,node2inlayer1failsatthesecondstage,andnodes1,2inthelayer2failinthethirdstage.Afterthat,noadditionalnodeswillfailatnextstages. 85

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ForthesakeoffurtherdiscussionwedenoteA11,A22,A12,A21tobeadjacencymatricesforthesetsofedgesE1,E2andthesetsofdependencyarcsE12,E21respectively,withcorrespondingelementsa11ij,a22ij,a12ij,a21ij.Binaryvariablesvsliwillshowwhetheranodeiinnetworklfailsatstages(vsli=1)ornot(vsli=0).Usingthesevariables,theminimumvertexcoverproblemforinterdependentnetworkscanbedescribedasfollows.Theobjectiveoftheproblem( 5 )istondaminimumsetofnodestoberemovedfrombothofthenetworksinordertocompletelydisconnectthenetworks.Asitismentionedabove,thenodesthatareattackedfailatstagezero.Constraints( 5 )imposecoveringofeveryedgeinnetworks1and2.Cascadingfailuresofnodesaremodeledwiththeconstraints( 5 )and( 5 ):equations( 5 )statethatnodeiinnetworkonefailsatstages(vsli=1)ifithasfailedatstages)]TJ /F6 11.955 Tf 12.29 0 Td[(1oranyofthenodesinthesecondnetworkthatnodeidependsonhasfailed.Constraints( 5 )imposethesamerequirementsbutforthenodesinnetwork2thatdependonnetwork1.Equations( 5 )provideconditionsfornodesinnetwork1tobeoperatingiftheyhavenotfailedatpreviousstagesandthenodesfromwhichtheyaredependentoninnetwork2areoperating.Equations( 5 )imposethesameconditionsforthenodesinthesecondnetwork. Problem1(F1). minimizeXl=1,2Xi2Vlv0li (5)subjecttovSli+vSlj1,8(i,j)2El,l=1,2, (5)vs1i1 n2+1 vs)]TJ /F7 7.97 Tf 6.59 0 Td[(11i+Xj2V2a21jivs)]TJ /F7 7.97 Tf 6.59 0 Td[(12j!,8i2V1,s=1,...,S, (5) 86

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vs2i1 n1+1 vs)]TJ /F7 7.97 Tf 6.59 0 Td[(12i+Xj2V1a12jivs)]TJ /F7 7.97 Tf 6.59 0 Td[(11j!,8i2V2,s=1,...,S, (5)vs1ivs)]TJ /F7 7.97 Tf 6.58 0 Td[(11i+Xj2V2a21jivs)]TJ /F7 7.97 Tf 6.59 0 Td[(12j,8i2V1,s=1,...,S, (5)vs2ivs)]TJ /F7 7.97 Tf 6.58 0 Td[(12i+Xj2V1a12jivs)]TJ /F7 7.97 Tf 6.59 0 Td[(11j,8i2V2,s=1,...,S, (5)vs1i,vs2j2f0,1g,8i2V1,8j2V2,s=1,...,S. (5) Theproblem( 5 )( 5 )isalinear0-1programmingproblemwith(S+1)(n1+n2)binaryvariables.Thefollowingsectiondescribesamorecompactformulationforthesameproblemofminimumvertexcoverininterdependentnetworks. 5.4.2Compactproblemformulation Inthissectionwedevelopacompactmodelformulation,whichutilizesonly(n1+n2)binaryvariablesincomparisonto(S+1)(n1+n2)variablesusedinthepreviousformulation.WealsoconsideraLPrelaxationoftheconsideredproblemandprovesomefactsaboutanapproximationalgorithmbaseonLPrelaxationfortheproposed0-1problem. ForanynodeiinlayerrdeneasetDrm(i)asthesetofnodesinlayermsuchthatfailureofanynodeinDrm(i)causesnodeitofail.Formally,itcanbedenedas Drm(i)=fk2Vm:ifkfails,theni2Vrfailsg.(5) Tosimplifyfurthernotation,wedeneforanypairofnodes(i,j)inlayerrDrm(i,j)=Drm(i)[Drm(j).1 1WeusenotationDrm(i,j,S),orDrm(i,S)ifweneedtoemphasizeitsdependencyonthenumberofconsideredcascadingstagesS 87

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Consequently,allaffectednodesformavertexcoverifandonlyifforanyedge(i,j)inlayer1thereisatleastonefailed(attacked)nodeinsetD11(i,j)[D21(i,j)andforanyedge(i,j)inlayer2thereisatleastonefailed(attacked)nodeinsetD22(i,j)[D12(i,j).UsingthisobservationanddenitionofDrm(i),theproblemofminimumvertexcovercanbeformulatedas( 5 )( 5 ). minimizeXl=1,2Xi2Vlvli (5)subjecttoXk2D11(i,j)v1k+Xk2D21(i,j)v2k1,8(i,j)2E1, (5)Xk2D22(i,j)v2k+Xk2D12(i,j)v1k1,8(i,j)2E2, (5)v1i,v2j2f0,1g,8i2V1,8j2V2. ThisproblemformulationrequirestheidenticationofsetsDrm(i)foreverynodeiinthenetwork.Thepropositionbelowshowshowtondthesesets Proposition5.1. ForS1deneamatrixTrmas T11=bS 2cXi=0(A12A21)i,T22=bS 2cXi=0(A21A12)i,(5) T12=bS)]TJ /F13 5.978 Tf 5.75 0 Td[(1 2cXi=0A12(A21A12)i,T21=bS)]TJ /F13 5.978 Tf 5.75 0 Td[(1 2cXi=0A21(A12A21)i.(5) Then Drm(i)=fj2Vr:Trmji>0g.(5) 88

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Proof. Theproofoftheprepositioncanbeshownbyinduction.WeconsideracaseforD11(i)=fj2V1:T11ji>0g,theproofforothercasesissimilar.Observethati2D11(i)andT11ii>0sincetherstmatrixinthesummation( 5 )isanidentitymatrix.Thesecondterminthesummation( 5 )isthematrixA12A21.Theelement(i,j)ofthismatrixisequalto(A12A21)ij=n2Xk=1a12ika21kj. Thus,(A12A21)ij>0ifandonlyifthereisapathoflengthatmost2inthedependentarcsgoingfromnodeitonodej,or,inotherwords,afailureofnodeicausesfailureofnodejinlayer1withS2.Applyingthisconclusiontoothermatricesinthesummation( 5 )itfollowsthatanelementT11ijisequaltothenumberofpathsoflengthatmostSinthedependentarcsfromnodeitonodejinthelayer1.Therefore,ifnodejfails,thennodeifailsifandonlyifT11ji>0. Usingtheproposition 5.1 wecanrewritetheproblemformulationinthefollowingform: Problem2(F1c). minimizeXl=1,2Xi2Vlvli (5)subjectton1Xk=1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(T11ki+T11kjv1k+n2Xk=1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(T21ki+T21kjv2k1,8(i,j)2E1, (5)n2Xk=1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(T22ki+T22kjv2k+n1Xk=1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(T12ki+T12kjv1k1,8(i,j)2E2, (5)v1i,v2j2f0,1g,8i2V1,8j2V2. (5) 89

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5.4.3ALPrelaxationapproximationalgorithm Itisawell-knownfactthataLPrelaxationoftheminimumvertexcoverproblemprovidesa2-approximationalgorithm[ 121 ].InthissectionweanalyseanapproximationalgorithmforvertexcoverproblemininterdependentnetworksbasedonLPrelaxationandnditsapproximationfactor,whichgeneralizesthealgorithmforthestandardproblem. Proposition5.2. Dene1and2as 1=max(i,j)2E1jD11(i,j)j+jD21(i,j)j,(5) 2=max(i,j)2E2jD22(i,j)j+jD12(i,j)j,(5) and=max(1,2).Then,aLPrelaxationoftheminimumvertexcoverproblemininterdependentnetworks( 5 )( 5 )providesa-approximation. Proof. Letvli,i=1,...,nl,l=1,2bethesolutionoftheLPrelaxationoftheproblem( 5 )( 5 ),andletwLP,wIPbethecorrespondingobjectivevaluesoftheoptimalsolutionsofLPandIP,wLPwIP.Dene vli=8>><>>:0,vli<1 l,1,vli1 l.(5) Sincethenumberofvariablesineveryconstraintoftheproblem( 5 )( 5 )isnogreaterthan,vectorvli,i=1,...,nl,l=1,2isafeasiblesolutionoftheIP( 5 )( 5 ).LetwLPIPbethecorrespondingobjectivefunctionofthissolution, wIPwLPIP.(5) 90

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Also,bydenition( 5 ),iffollowsthat wLPIPwLP.(5) Combiningthelasttwoequations,weget wIPwLP.(5) Notethatwithnointerdependenciesbetweenthelayers,=2andtheproblem( 5 )( 5 )reducestothestandardminimumvertexcoverproblem. 5.4.4DepthofcascadeS IntheprevioussectionsweassumedthatthemaximumnumberoffailurestagesconsideredintheoptimizationmodelsaboveisS=jV1j+jV2j)]TJ /F6 11.955 Tf 18.7 0 Td[(1.Thisisthelongestpossiblefailureprocess,sinceateverystagecascadeseitherstoporcauseatleastonemorenodefailure.But,forcomputationalreasons,itmightnotbeefcienttousethehighestpossibleSintheminimumvertexcoverproblemformulations(F1,F1c)sinceitmayrequirealotofunnecessarycalculationsandusageofredundantvariables.InthissectionweshowthatthereexistssuchanumberoffailurestagesSjV1j+jV2j)]TJ /F6 11.955 Tf 18.64 0 Td[(1,sothattheproblemformulationsF1(S)forallSS,andF1c(S)forallSShavethesameoptimalsolution2.Moreover,wepresentanalgorithmforcalculationofSforagiveninterdependentnetwork. Proposition5.3(depthofcascade). LetG=(V1[V2,E12[E21),anddist(i,j)bethelengthoftheshortestdirectedpathfromnodeitonodejinG(dist(i,j)=1ifthereis 2Ifnecessary,weuseanargumentStorefertotheproblemformulationsF1,F1cwithvariablenumberofcascadingstages. 91

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nopathfromitoj),andlet S=max(i,j)2V1[V2(dist(i,j)jdist(i,j)<1),(5) thentheproblemformulationsF1(S)forallSS,andF1c(S)forallSShavethesameoptimalsolution. Proof. NotethatbydenitionoftheproblemformulationF1(S),thestatementinpropositionisvalidautomatically.FortheproblemformulationF1c(S)observethatbydenitionofthesetDrm(i,S),forallSSandanyxedr,m,ithesetsDrm(i,S)areequal.Indeed,ifweassumethatthereexistr,m,iandS2>S1S,suchthatDrm(i,S1)Drm(i,S2),itmeansthatthereexistsnodek2GsuchthatS1
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subjecttovSli+vSlj1,8(i,j)2El,l=1,2, (5)vstivs)]TJ /F7 7.97 Tf 6.58 0 Td[(1ti,8i2V1,V2,s=1,...,S, (5)1)]TJ /F4 11.955 Tf 11.95 0 Td[(vs1iPj2V2a2ji(1)]TJ /F4 11.955 Tf 11.96 0 Td[(vs)]TJ /F7 7.97 Tf 6.59 0 Td[(12i) n2+1)]TJ /F4 11.955 Tf 11.96 0 Td[(vs)]TJ /F7 7.97 Tf 6.59 0 Td[(11i,8i2V1:Xj2V2a2ji1,sS, (5)1)]TJ /F4 11.955 Tf 11.95 0 Td[(vs2iPj2V1a1ji(1)]TJ /F4 11.955 Tf 11.96 0 Td[(vs)]TJ /F7 7.97 Tf 6.59 0 Td[(11i) n1+1)]TJ /F4 11.955 Tf 11.95 0 Td[(vs)]TJ /F7 7.97 Tf 6.58 0 Td[(12i,8i2V2:Xj2V1a1ji1,sS, (5)1)]TJ /F4 11.955 Tf 11.95 0 Td[(vs1iXj2V2a2ji(1)]TJ /F4 11.955 Tf 11.95 0 Td[(vs)]TJ /F7 7.97 Tf 6.59 0 Td[(12i),8i2V1:Xj2V2a2ji1,sS, (5)1)]TJ /F4 11.955 Tf 11.95 0 Td[(vs2iXj2V1a1ji(1)]TJ /F4 11.955 Tf 11.95 0 Td[(vs)]TJ /F7 7.97 Tf 6.59 0 Td[(11i),8i2V2:Xj2V1a1ji1,sS, (5)vs1i,vs2j2f0,1g,8i2V1,8j2V2,sS. 5.5ComputationalExperiments Thissectionsummarizesthecomputationalexperimentsfortheproposedvertexcovermodelsandapproximationalgorithm.Tothebestknowledgeofour,therearenoreal-worldinstancesavailableinpublicdomain.Therefore,consistentlywithotherrelatedstudies,weconsiderrandomgraphswithcertainwell-knowndegreedistributions:power-lawanduniform.AlltheproposedmodelswereimplementedinXpressMPandsolvedonacomputerwithPentium43.4GHzprocessorand8GBofmemory.Computationaltimepresentedisinseconds.WedenotewithjEdepjthenumberofarcsbetweentwonetworksandjEdepj=jE12j+jE21j. Table 5-1 presentsoptimalobjectivevalues(Opt.Obj.),computationaltimeandLPrelaxationsfortheproblemformulationsF1( 5 )( 5 )andF1c( 5 )( 5 ).Fourinstancesofinterdependentnetworkswithtwolayersaregeneratedrandomlyforthisexperiment.Parameterpdenotestheprobabilitythatanyedgeexistsinnetworks 93

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G1andG2,andparameterqdenotestheprobabilitythatanygivennodeinonelayerisdependentonanygivennodeinotherlayer. Table5-1. ComputationalresultsforproblemformulationsF1andF1c,n=200(n1=n2=100). GraphSOpt.Obj.F1ctimeF1timeLPF1cLPF1 #10139105.563106.136100100p=0.1,1692.7913.3635353q=0.1,2500.7314.1074037.8S=15.5390.1760.8103531.615390.1610.7663531.2#2013829.35930.002100100p=0.1,1579.857.15442.542.5q=0.15,2331.1144.43327.523.6S=18.5210.1820.9601915.618190.1470.5931715.1#30158359.763355.426100100p=0.2,1818.5176.51354.554.5q=0.1,2631.5893.3284341.2S=15.5550.8212.96038.537.615520.1610.76637.537.5#401613137.093098.94100100p=0.2,168199.2758.96845.345.3q=0.15,2412.63110.37829.226.8S=20.5240.5322.11520.520.120220.4575.1392020 Figure 5-4 showsdepthofcascadesSthatwasobservedinthenetworkswiththenumberofnodes200,400,600,and1000.Thehorizontalaxisshowsthenumberofinterdependencylinksbetweenthenetworksdividedbythetotalnumberofnodesinbothofthenetworks.ThedepthofcascadesSissmallatthebeginning,butitincreaseswiththenumberofinterdependencylinksandreachesmaximumwithjEdepj=nbeing 94

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somewherebetweenoneandtwo.Furtherincreaseinthenumberofinterdependencylinksleadstothedecreaseofthenumberofcascadingfailures. Figure5-4. DepthofcascadesSfordifferentgraphsizesandnumberofinterdependencyedges. Figure 5-5 showstheminimumvertexcoversizeforthenetworkswithdifferentdensitiesandthenumberofcascadesconsideredforType1interdependence.Figures 5-5A and 5-5B showtheminimumvertexcowersizesfortheuniformgraphswithdensities0.1and0.2.Figures 5-5C and 5-5D showthesameinformationforpower-lawnetworkswithdensities0.1and0.2.ForallofthecasesdepictedinFigure 5-5 ,theminimumvertexcoversizedecreaseswiththeincreaseinthenumberofinterdependencyedges. 95

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ATwouniformnetworkswithdensities0.1 BTwouniformnetworkswithdensities0.2 CTwoscale-free(powerlaw)networkswithden-sities0.1 DTwoscale-free(powerlaw)networkswithden-sities0.2 Figure5-5. Theminimumvertexcoversizeforuniformandpower-lawgraphsforType1interdependence,n=200(n1=n2=100) Figure 5-6 showstheminimumvertexcoversizefortwouniformgraphsandtwopower-lawgraphswithdensities0.3and0.5forType2interdependence.WithjEdepj=nincreasing,theminimumvertexcoversizedecreasesatthebeginning,reachesitsminimumforjEdepj=nbeingsomewherebetweenoneandtwoandincreasesagain. Figure 5-7 comparestheminimumvertexcoversizesforuniformandpower-lawgraphswithdifferenttypesofinterdependence.TheminimumvertexcoversizeforType1interdependencewaslessthanorequaltotheminimumvertexcoversizeforType2interdependenceinourexperiments.SuchbehaviorcanbeexplainedbythefactthatinterdependentnetworkswithType2interdependencyaremoreresilientto 96

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ATwouniformnetworkswithedgedensities0.3 BTwouniformnetworkswithedgedensities0.5 CTwopower-lawnetworkswithedgedensities0.3 DTwopower-lawnetworkswithedgedensities0.5 Figure5-6. TheminimumvertexcoversizeforType2interdependence,n=50(n1=n2=25) cascadingnodefailuresthanthenetworkswithType1interdependence,sinceforType2interdependenceforanodeinonenetworkallthenodesthatareconnectedtothisnodeinanothernetworkmustfailtocauseacascadingfailureofthisnode. 97

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A B Figure5-7. Minimumvertexcoversizesfortwonetworks((a)-uniform,(b)-powerlaw)withedgedensities0.5fortwotypesofinterdependencies,n=50(n1=n2=25),S=3. 98

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CHAPTER6CONCLUSION Thisdissertationfocusesonensuringresilienceininterdependentnetworks,whichisacomplextaskthathasbeenstudiedforyears.First,thisdissertationstudiesthegeneralizedexpansionplanningproblemforinterdependentelectricityandnaturalgassystems.Duetothelong-termnatureoftheproblemanduncertaintiesinfuturedemandfornaturalgasandelectricityweutilizeConditionalValue-at-Risktocopewithuncertantyinsteadofoptimizationagainstworstcasescenario,whichisunlikelytohappen.Becauseoftheimbalanceofnaturalgasresourcesamongthecountries,wealsoconsidertheproblemofLNGterminalexpansionplanning.Oneofthepotentialfutureresearchdirectionforthisproblemisinclusionofintegervariablesintotheefciencyconstraintsforelectricitygenerators.Suchapproachwillallowabetterprecisionfortheoperationalmodeofpowerplants. Second,thisdissertationstudiestheproblemoftopologydesignforon-demanddual-pathroutinginwirelessnetworks.Missioncriticalandtimesensitiveapplicationsofsensornetworksmaybenetfrommulti-pathdatatransportfromasourcenode(sensor)toadestinationnode(gateway).Previousresearchonmulti-pathroutingandtransportalgorithmshavetraditionallystartedfromagivennetworktopologyatthephysicallevel,withlittleattentiontoacoordinatedruntimedesignofthenetworktopologyitself.Inthisdissertation,weintroducedacentralizedoptimalnetworkdesignapproachthatenablestheexistenceofdualpathsbetweenanygivensourceanddestinationnodesinthenetwork.Indesigningthenetworktopology,ourapproachtakesintoaccountpracticalconstraintsassociatedwiththephysicalcharacteristicsofwirelesslinksandreliessolelyontheuseoftransmissionpowercontrolfornetworkdesign.Whilecentralized,theproposedallocationtechniquesareprogressive,providingincreasinglybetterpower 99

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congurationateachiteration.Ateachstepsolution,anewcongurationcanbedeployedinthenetwork,enablinganewpartialmulti-pathsolutionthatwillbeimprovedinthesubsequentiteration,anditisguaranteednottodegradebetweeniterations.Intheproposedformulations,wehavefocusedontheidenticationofedgeconstructiontominimizetheoveralltransmissionpowerofallnodes,butotherobjectivesandconstraintscouldalsohavebeenconsideredinourformulation. Ourrstresults,stillpreliminaryatthispoint,showthepotentialforapracticalcalculationofoptimalnetworkdesignformulti-pathdatasupport.Possiblefutureresearchdirectionsincludeaddingtheconstraintsneededtoenablenon-interferingmulti-pathcalculationandthedesignofadistributedversionoftheproposedapproach.Non-interferingpathswillensurethatmulti-hoptransmissionstakingplaceatoneofthepaths,willnotaffect(throughinterference)themulti-hoptransmissionstakingplaceonthesecondpath.Forexample,referringbacktoFigure 3-4B ,anon-interferingdesignwouldpreventadirectarcbetweenanodeinf2,7,8g,andanodeinf3,4,5,6,11g.Also,someoftheconstraintsoftenadoptedinwirelesssensornetworkformaximumtransmissionlength,andasymmetricdatapathscanbeincorporatedintotheformulatedproblems. Therearebasicallythreelevelsoftopologydesignsthatcouldbeconstructedtoenabledisjointdatapathsbetweennodes.Thelessrestrictiveimplementationisanarc-disjointtopologydesign,whichhaslittlepracticaluse,sinceitreliesonthepossibleuseofcommonnodesforthedifferentdatapaths.Asecondlevelofdesigninvolvestheconstructioninternallynode-disjointdatapaths,ensuringthatthepathshavenocommonarcsornodesbetweenanygivensourceordestination.Athirdandevenmorerestrictivedesignapproachisfornon-interferingpaths,whichimpliestheconstructionofpathsthatarearc-disjoint,node-disjointandnon-interfering.Thedesignofnon-interferingpaths 100

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imposesastricterconstraintonthetopology,requiringnotonlythatpathsarearc-disjointandinternallynode-disjoint,butalsothatnoarcdirectlyconnectsanodefromonedatapathwithanodefromtheother,withtheexceptionofthesourceanddestinationnodes.Inthisdissertationwehavefocusedonthersttwotopologydesigns,leavingthemorerestrictivedesignaspartofourfuturework. Next,wehaveintroducedamathematicalprogrammingmodelforthexedchargenetworkowproblemwithuncertainmultiplearcfailures,whichutilizesConditionalValue-at-Riskasameansofcontrollingtherisksandrestrictingcollaterallosses.WehaveshownthattheAdaptiveDynamicCostUpdateProcedureisapplicablefortheconsideredproblemsetupunderuncertaintyandcanbeefcientlyutilizedforndinghigh-qualityheuristicsolutionstotheresultingproblem,eveninthecaseoflarge-scaleinstances.Inaddition,theconsideredmodelallowsadjustingthetradeoffsbetweensolutionrobustnessandcostefciency.Thenumericalexperimentsshowthattherelativeerrorincreasesfortheproblemswithhighxedcostduetothefactthattheresultingpiecewiselinearobjectivefunctiondoesnotapproximatetheoriginalfunctionasgoodasinthecaseswithhighvariablecost. Forthefutureresearch,onemayconsiderotherexpressionsoflossfunctions,aswellasreplacingCVaRconstraintswithmoregeneralmeasuresofrisk,referredtoasHigherMomentCoherentRiskmeasures(HMCR)describedin[ 74 ]whichcontainsCVaRasaspecialcase.HMCRmeasuresmaybesuitableforlossdistributionswithheavytails,forinstance,underlarge-scaledisasterscenarios. Finally,wehavestudiedthecriticalnodesproblemandvertexcoverproblemforinterdependentinfrastructures.Theproposedcriticalnodesmodelidentiesasetofnodes,whichdeletionminimizestheconnectivityoftheremainingnetwork.Thevertexcovermodelidentiesthenodes,whichremovalcompletelydisconnectstheremaining 101

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network.Weproposedmathematicalprogrammingformulationsfortheconsideredproblemsundertwodifferentassumptionsonsurvivabilityofthedependentnodesthatarecommonintelecommunicationandenergysupplycontextsduringthecascadingfailures.WehavealsodescribedanapproximationalgorithmbasedonaLPrelaxationofthevertexcoverproblem.Forthefutureresearchdirectionsweconsideramorein-depthstudyaboutthepropertiesofthecriticalnodesproblemininterdependentinfrastructures. 102

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BIOGRAPHICALSKETCH AlexeySorokinwasborninVoronezh,Russia.HegraduatedfromVoronezhStateTechnicalUniversityinJune2008withspecialistdegreeincomputerengineering.ThesameyearhejoinedthegraduateprogramatIndustrialandSystemsEngineeringDepartmentatUniversityofFlorida.InAugust2010,hereceivedhismasterofsciencedegreeinindustrialandsystemsengineering.InAugust2012,AlexeySorokinobtainedhismasterofsciencedegreeinmanagementfromHoughGraduateSchoolofBusiness.HegraduatedfromUniversityofFloridawithhisPh.D.degreeinindustrialandsystemsengineeringinAugust2012.AlexeySorokinistheeditorofthreebooksandtheauthorofseveralscienticpapers,surveys,andbookchapters. 114