<%BANNER%>

Decomposition Algorithms for Two Stage Stochastic Integer Programming

Permanent Link: http://ufdc.ufl.edu/UFE0024769/00001

Material Information

Title: Decomposition Algorithms for Two Stage Stochastic Integer Programming
Physical Description: 1 online resource (106 p.)
Language: english
Creator: Penuel, John
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: algorithms, cutting, integer, plane, programming
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Stochastic programming seeks to optimize decision making in uncertain conditions. This type of work is typically amenable to decomposition into first- and second-stage decisions. First-stage decisions must be made now, while second-stage decisions are made after realizing certain future conditions and are typically constrained by first-stage decisions. This work focuses on two stochastic integer programming applications. In Chapter 2, we investigate a two-stage facility location problem with integer recourse. In Chapter 3, we investigate the graph decontamination problem with mobile agents. In both problems, we develop cutting-plane algorithms that iteratively solve the first-stage problem, then solve the second-stage problem and glean information from the second-stage solution with which we refine first-stage decisions. This process is repeated until optimality is reached. If the second-stage problems are linear programs, then duality can be exploited in order to refine first-stage decisions. If the second-stage problems are mixed-integer programs, then we resort to other methods to extract information from the second-stage problem. The applications discussed in this work have mixed-integer second-stage problems, and accordingly we develop specialized cutting-plane algorithms and demonstrate the efficacy of our solution methods.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by John Penuel.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Smith, Jonathan.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024769:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024769/00001

Material Information

Title: Decomposition Algorithms for Two Stage Stochastic Integer Programming
Physical Description: 1 online resource (106 p.)
Language: english
Creator: Penuel, John
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: algorithms, cutting, integer, plane, programming
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Stochastic programming seeks to optimize decision making in uncertain conditions. This type of work is typically amenable to decomposition into first- and second-stage decisions. First-stage decisions must be made now, while second-stage decisions are made after realizing certain future conditions and are typically constrained by first-stage decisions. This work focuses on two stochastic integer programming applications. In Chapter 2, we investigate a two-stage facility location problem with integer recourse. In Chapter 3, we investigate the graph decontamination problem with mobile agents. In both problems, we develop cutting-plane algorithms that iteratively solve the first-stage problem, then solve the second-stage problem and glean information from the second-stage solution with which we refine first-stage decisions. This process is repeated until optimality is reached. If the second-stage problems are linear programs, then duality can be exploited in order to refine first-stage decisions. If the second-stage problems are mixed-integer programs, then we resort to other methods to extract information from the second-stage problem. The applications discussed in this work have mixed-integer second-stage problems, and accordingly we develop specialized cutting-plane algorithms and demonstrate the efficacy of our solution methods.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by John Penuel.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Smith, Jonathan.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024769:00001


This item has the following downloads:


Full Text

PAGE 1

1

PAGE 2

2

PAGE 3

3

PAGE 4

Thisworkwouldnotbepossiblewithoutthetirelessenthusiasm,support,andguidanceofmyadvisorandfriend,Dr.J.ColeSmith.IwouldalsoliketothanktheIndustrialandSystemsEngineeringDepartmentoftheUniversityofFloridaforgivingmeallthetoolsandsupportnecessarytosucceed. 4

PAGE 5

page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 6 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 8 CHAPTER 1INTRODUCTION .................................. 9 2TWO-STAGEFACILITYLOCATION ....................... 13 2.1ProblemDescription .............................. 13 2.2MathematicalProgrammingFormulation ................... 14 2.3ACutting-PlaneAlgorithmfortheTwo-StageFacilityLocationProblem 17 2.4BendersCuttingPlanes ............................ 19 2.5Laporte/LouveauxCuttingPlanes ....................... 20 2.6ResidualPathCuttingPlanes ......................... 21 2.7ComputationalTesting ............................. 32 3DECONTAMINATINGAGRAPHWITHMOBILEAGENTS .......... 37 3.1Introduction ................................... 37 3.2BasicDescription ................................ 40 3.3TheNodeCleaningProblem .......................... 41 3.4TheMinimumNumberofAgentsRequiredtoCleanaGraph ........ 54 3.5PositioningAgentstoMinimizeExpectedCleaningTime .......... 61 3.6IntegerProgrammingComputationalTesting ................. 63 3.7FastestCleaningStrategyHeuristic ...................... 72 3.8ADecompositionApproachforMINAGENT ................. 78 3.9FeasibilityCuttingPlanes ........................... 81 3.10MINAGENTDecompositionProcedureComputationalTesting ....... 87 3.11ADecompositionApproachforAGENTPOS ................. 91 3.12OptimalityCuttingPlanes ........................... 93 3.13AGENTPOSDecompositionProcedureComputationalTesting ....... 95 4CONCLUSIONANDFUTURERESEARCH ................... 100 REFERENCES ....................................... 103 BIOGRAPHICALSKETCH ................................ 106 5

PAGE 6

Table page 2-1Possiblevaluesfortwo-stagefacilitylocationtestproblemdata ......... 34 2-2Summaryofcutting-planealgorithmsandextensiveformulationperformancesfortwo-stagefacilitylocationprobleminstances. .................. 35 3-1CLEANMIPgiventheentiregraphiscontaminated. ............... 65 3-2TightnessofvariousCLEANMIPrelaxationsgiventheentiregraphisinitiallycontaminated. ..................................... 66 3-3MINAGENTMIPgiventheentiregraphiscontaminated. ............ 67 3-4AGENTPOSMIPgiventheentiregraphiscontaminated. ............ 68 3-5CLEANgivencontaminationbeginsonasinglenode. ............... 69 3-6TightnessofvariousCLEANMIPrelaxationsgivencontaminationbeginsonasinglenode. ...................................... 70 3-7MINAGENTgiventhecontaminationbeginsonasinglenode. .......... 71 3-8AGENTPOSMIPgiventheentiregraphiscontaminated. ............ 72 3-9CLEANheuristicwithvariousmetrics. ...................... 76 3-10CLEANheuristicwithvariousconstructionparameters. .............. 77 3-11TunedCLEANheuristicforinstanceswheretheentiregraphiscontaminated. 79 3-12TunedCLEANheuristicforinstanceswherethecontaminationstartsonasinglenode. .......................................... 79 3-13ReviewofMINAGENTextensiveformulationtesting. ............... 88 3-14MINAGENTdecompositionprocedure. ....................... 89 3-15MINAGENTcutting-planealgorithm. ....................... 90 3-16ReviewofAGENTPOSextensiveformulationtesting. ............... 96 3-17AGENTPOSdecompositionprocedure. ....................... 97 3-18AGENTPOScutting-planealgorithm. ........................ 98 4-1Problemcomplexitysummary. ............................ 101 6

PAGE 7

Figure page 2-1Facility1doesnotneedtobeactivatediffacility3islocated. .......... 23 2-2Asituationwherenoprimaryintersectionexists.Adashedlineindicateszeroow.Asolidlineindicatesoneunitofow. .................... 28 2-3FlowispushedonarcsinfPjfromtheoriginoffPjtovlastj,andthenowisreversed(eectivelyremoved)fromarcsinf1jfromvlastjtotheoriginoff1jforj=1;2;3.Theow-cycleisidentiedwiththedottedline. .................. 29 2-4Remainingowafteroneunitofowonthecyclehasbeenremoved. ...... 30 3-1Diamondstructure. .................................. 43 3-2ACLEANbinstancetransformedfromthetheexampleX3Cinstance. ...... 44 3-3Ahexagonalstructurewithasix-cliqueforthemiddlenodes. ........... 47 3-4ACLEANpinstancetransformedfromthetheexampleX3CinstancedescribedintheproofofTheorem 3 .............................. 48 3-5Avertexcoverinstance(left)transformedtoaMINAGENTsgraph(right). ... 56 3-6Avertexcoverinstance(left)transformedtoaMINAGENTbgraph(right). .. 57 7

PAGE 8

Stochasticprogrammingseekstooptimizedecisionmakinginuncertainconditions.Thistypeofworkistypicallyamenabletodecompositionintorst-andsecond-stagedecisions.First-stagedecisionsmustbemadenow,whilesecond-stagedecisionsaremadeafterrealizingcertainfutureconditionsandaretypicallyconstrainedbyrst-stagedecisions.Thisworkfocusesontwostochasticintegerprogrammingapplications.InChapter 2 ,weinvestigateatwo-stagefacilitylocationproblemwithintegerrecourse.InChapter 3 ,weinvestigatethegraphdecontaminationproblemwithmobileagents.Inbothproblems,wedevelopcutting-planealgorithmsthatiterativelysolvetherst-stageproblem,thensolvethesecond-stageproblemandgleaninformationfromthesecond-stagesolutionwithwhichwerenerst-stagedecisions.Thisprocessisrepeateduntiloptimalityisreached.Ifthesecond-stageproblemsarelinearprograms,thendualitycanbeexploitedinordertorenerst-stagedecisions.Ifthesecond-stageproblemsaremixed-integerprograms,thenweresorttoothermethodstoextractinformationfromthesecond-stageproblem.Theapplicationsdiscussedinthisworkhavemixed-integersecond-stageproblems,andaccordinglywedevelopspecializedcutting-planealgorithmsanddemonstratetheecacyofoursolutionmethods. 8

PAGE 9

Thisworkfocusesondecompositionmethodsforsolvingacertainclassofstochasticintegerprograms(SIP).Decision-makingproblemsunderuncertaintycanoftenbebrokenupinto\here-and-now"or\rst-stage"decisions,and\recourse"or\second-stage"decisions,whicharedependentontherealizationofsomerandomfuturedata.Forexample,consideringadistributionnetwork,itisunclearhowdemandwilluctuateinthefuture,butweneedtobuilddistributionfacilitiesnow.Wemustrstbuilddistributionfacilities,andinthefuturewhendemandisrealized,wearelimitedtothepreviouslylocatedfacilitiestodistributeowthroughthenetwork.Whenrst-orsecond-stagedecisionsinvolvebinary\yes-or-no"ordiscretedecisions,wecalltheseproblemstwo-stageSIPs.Afterrst-stagedecisionsaremade,wesolveasecond-stageproblemtoassessthequalityofrst-stagedecisions.Fortheproblemsconsideredinthisdissertation,stochasticityismodeledbyconsideringanitebutpotentiallylargesetoffuturescenarios.Thus,givenrst-stagedecisions,wewillsolvethesecond-stageproblemassociatedwitheachscenario.Thedicultyisgleaningusefulinformationfromthesolutionofthesecond-stageproblemssothatwecaneectivelyrenerst-stagedecisions. Two-stagestochasticlinearprograms(LP)withrandomsecond-stagedatawererstconsideredbyDantzig[ 14 ]andWalkup[ 41 ].ThetechniqueofBendersdecomposition[ 7 ](sometimesreferredtoastheL-shapedmethodasdescribedbyVanSlykeandWets[ 40 ]andWollmer[ 42 ])servesasthefoundationforourapproach.Thistechniqueidentiesasplitbetween\complicating"and\separable"variables,wherebytheresolutionofcomplicatingvariablevaluespermitsthesolutionofseveralindependentproblemsinvolvingtheseparablevariables.Withrespecttoourdiscussion,rst-stagedecisionsrepresentthecomplicatingvariables,andrecoursedecisionsforaspecicscenariorepresenttheseparablevariables.Bendersdecompositioniterativelysolvesarst-stagemasterproblemintermsofthecomplicatingvariables,whileoptimisticallyestimatingthe 9

PAGE 10

BendersdecompositionassumesthattherecoursesubproblemsareLPs,andexploitsthedualityofthesubproblemstoderivevalidinequalitiesthatstatearelationshipbetweenrst-stagedecisionvariablesandtheestimatedsecond-stageobjectivevalue.However,theproblemsdiscussedinthisworkdonothavecontinuousrecourseproblems,butrathermixed-integerprogramming(MIP)recourseproblemsthatincludecontinuous,binary,orinteger-valuedvariables.Withasecond-stageMIP,nodualinformationexists,andthusBendersdecompositionisnotdirectlyapplicabletotheseproblems.Instead,welookatthestructureofproblemsandattempttoderiveanerelationshipsbetweenrst-stagevariablesandsecond-stageobjectives.Forexample,wetrytoestimatehowsecond-stageobjectivescanbeimprovedwhenadditionalresourcesarededicatedintherststage.Themajorcontributionthisdiscussionisthederivationofcuttingplanesspeciallytailoredtoourproblems. LouveauxandvanderVlerk[ 30 ]studiedsimpleintegerrecoursemodels,wherethesecond-stageproblemminimizesthepenaltychargeforshortageorsurplusfromrst-stagedecisions.Later,LaporteandLouveaux[ 26 ]introducedtheintegerL-shapedmethodfortwo-stageSIPswithbinaryrst-stagevariables,whichisequippedtodealwithanyMIPrecourseproblem.CareandTind[ 11 ]generalizedtheintegerL-shapedmethodviageneraldualitytheory.However,themasterproblemobtainedfromtheirmethodshasnonlinearconstraints,andisnotgenerallycomputationallyattractive.SenandHigle[ 34 ]introducedthedisjunctivedecomposition(D2)algorithm,whichusesdisjunctiveprogrammingtoconvexifyMIPsubproblemsandderivevalidcuttingplanes.SenandSherali[ 35 ]puttheD2algorithmintoabranch-and-cutframework. 10

PAGE 11

27 28 ].Theirmethodisverygeneral,andsothereisroomforimprovementgivenspecicassumptionsonproblemstructure.InChapter2,wedemonstrateanintegerdecompositionmethodtailoredtoaspecicapplication,whichwedemonstratetobemorecomputationallyecientthantheL-shapedmethodofLaporteandLouveaux. Inthenextpartofthisdissertation,weconsideravariationofthescenario-basedstochasticfacilitylocationproblemmodeledonanetwork.Infuturescenarios,problemdataregardingdemandpoints,arccapacities,andotheraspectsmayvary.Inatypicaltwo-stagestochasticfacilitylocationproblem,wewouldlocatefacilitiesintherststageandthenforeachsecond-stagerecoursesubproblem,establishowsfromlocatedfacilitiestosatisfydemandinthenetwork.Intheproblemweconsider,wemustlocateaprioriasetoffacilitiesthatmaypossiblybeactivatedinafuturescenario;such\facilitylocation"inthiscontextdoesnotnecessarilyimplythatthefacilityisreadyforuseinafuturescenario.Theproblemcanthenbedecomposedintosubproblemsassociatedwitheachscenario,giventherst-stagefacilitylocations.Thecomplicatingissueinourproblemisthatineachsubproblem,wemustactivateasubsetoflocatedfacilities,incurringaxed-chargeactivationfeeinadditiontoitslocationcost.Theactivationoffacilitiesismodeledasasetofbinaryvariables,andthusthesubproblemsthatweencounterinatypicaldecompositionapproachareMIPs.InSection 2.1 ,weintroducethetwo-stagefacilitylocationproblemgivealiteraturereviewoffacilitylocationunderuncertainty.Section 2.2 presentsadetaileddescriptionoftheproblemandmodelitasamixed-integerprogram.Section 2.3 describesclassicalcutting-planealgorithmsfortwo-stageoptimizationproblems,alongwithanovelapproachthatweintroduceforthetwo-stagefacilitylocationproblem.Weperformcomputationaltestsonthegivenmethodsforthetwo-stagefacilitylocationprobleminSection 2.7 InChapter3,weconsidertheproblemofusingmobileagentstodecontaminatethenodesofagraphgivenaspreadingcontamination.Section 3.1 providesanintroduction 11

PAGE 12

3.3 ),(2)theminimumnumberofagentsrequiredtodecontaminatethegraphandtheirinitialpositions(Section 3.4 ),and(3)thebestpositionsforsomepredeterminednumberofagentstominimizeexpectedcleaningtime(Section 3.5 ).Weperformcomplexityanalysisonallvariantsoftheseproblemsforgeneralgraphs.Weapproachtheseproblemsfromanintegerprogrammingperspective,provideformulationsforallproblemsdiscussed,andpresentbasiccomputationaltestingoftheMIPmodels(Section 3.6 ).Wethenprovideadvancedtechniquesforsolvingthedierentgraphcleaningproblemsandcomputationaltesting.Tondthefastestcleaningstrategyweprovideagreedyrandomconstructionheuristic(Section 3.7 ).Tosolvetheminimumnumberofagentsandndthebestpositions,wedevelopcutting-planealgorithmsthatutilizeBenders-decomposition-styleinequalitiesandothercombinatorialcuttingplanes(Sections 3.8 { 3.13 ). Finally,weconcludeinChapter 4 anddescribefutureresearchdirections. 12

PAGE 13

Asanotherexample,considerthelocationofhurricanereliefsheltersinhurricane-proneregions.Ifahurricanewastostrikeacertainregion,somesheltersintheregionwouldbeopenedandstaed,incurringaxedactivationcost.Intherecourseproblem,onceahurricanehasstruckaregion,wearelimitedtothefacilitiesthathavebeenlocatedintherststagetoaddresstheneedsofthehurricanevictims.Thevariableowcostswouldbeproportionaltothenumberofpeopletheshelterservesinagivenscenario,whichcouldbemodeledbysendingowsfromthesheltertothosecommunitiesseekingaid. Theoptimizationchallengeposedinthisdocumentisalsoinherentinmanymilitaryplanningsituations.Along-rangestrategicplanmayinitiallycapturecertainvitalareasinsometheater,whichcouldthenbepursuedasforwardbasesforfutureactioninthecampaign.Therst-stage\facilitylocation"herewouldrepresentthevitalareastobesecuredearlyintheconict.Thesecond-stagedecisionswouldrepresentthe\activation"ofthefacilities(bymobilizingtroopsandestablishingadvancedcontrolcenters),plusthecostsassociatedwithdeployingtroopsfromtheseestablishedbases.(Notethatadvancedbasescannotbeestablishedinvitalareasthatwerenotpreviouslysecured.) Facilitylocationisanexpensiveandlong-termplanningdecision,andthereforerequiressignicantconsideration.Compensatingforuncertaintyisanaturalextensionof 13

PAGE 14

37 ]presentsathoroughreviewofstochasticfacilitylocationmodelsandsolutionapproaches.Wehaveformulatedourproblemasatwo-stagestochasticintegerprogramcontainingbinarysecond-stagevariables,withtheobjectiveofminimizingfacilitylocationcostsplusaweightedaverageofsecond-stageactivationandowcosts. Inourapplicationitwouldbenaturaltoconsiderthefacilitylocationvariablesasourcomplicatingmasterproblemvariables,andthefacilityactivationandmaterialroutingvariablesasrecoursesubproblemvariables.However,thesubproblems'activationvariablesarerestrictedtobebinary,andthussubproblemdualitycannotbeexploitedtoderivecuttingplanes.Asanalternative,binarysubproblemvariablescouldbemovedtotherststage,leavingonlycontinuousvariablesinthesubproblems,thusmakingtheproblemamenabletodirectsolutionbyBendersdecomposition.BienstockandShapiro[ 9 ]solvedanapplicationforanelectricutilitydealingwithfuelcontractsandplantconstructionwiththisstrategy.However,thisapproachresultsinarst-stageproblemhavingamuchlargernumberofbinaryvariables.Inordertoavoidthissharpincreaseinthenumberofmasterproblembinaryvariables,weinvestigateamethodthatkeepsintegerrecoursevariablesinthesubproblems. LetSbethesetofscenariosthatwewishtoconsider.Eachscenarios2Sisassociatedwithsomepriorityps.Theseprioritiesweighttherecourseobjectivecomponentassociatedwiththescenariosbytakingintoaccounttheprobabilitythateachscenariooccursaswellastheimportanceofaddressingeachscenario.Inlocatinghurricaneshelters,wecouldconsiderthepossibilityofahurricanemakinglandfalloncertaincoastalregionswithroughlyuniformprobability.However,scenariosinwhichahurricanemakes 14

PAGE 15

Theremainingdatainourproblemdependsonthescenarios2S.Theactivationcostforeachfacilityisgivenbygsiforalli2V.Iffacilityi2Visactivatedinscenarios2S,thenvertexihasasupplyofbsi0.Also,letthecostofshippingonarc(i;j)2Ainscenarios2Sbegivenbycsij0,andthecapacityofarc(i;j)inscenariosbegivenbyusij0.Finally,thedemandatvertexi2Visgivenbydsi0.Thisscenario-specicdataallowsustomodelinstancesinwhichsupplyandtransportationcosts,aswellasarccapacitiesandnodaldemands,areaectedbythescenariosthemselves. Inthecontextofhurricaneshelterlocation,facilitiesarerstlocated,andthensitidleuntilahurricanemakeslandfallinthearea.Thesheltermustthenbestaedandequippedinahurricaneemergency,denotedbyscenarios2S,incurringthexedactivationcostgsi.Inaddition,emergencyvehiclestravelingalongarcstovictimsincursomeowcost.Road(arc)capacitiescanmodiedappropriatelyineachscenariotomodelthedisaster'spredictedimpactoninfrastructure. Wecanmodelthisproblemasalarge-scalemixed-integerformulation.Thedecisiontolocateafacilityonsomevertexi2Visrepresentedbythebinaryvariablexi,wherexi=1ifafacilityislocatedati2Vandzerootherwise,foralli2V.Theremainingvariablesdependonscenarios2S.Wehavebinaryvariablesysi,foralli2V,whichareequalto1ifthefacilityonvertexi2Visactivatedinscenarios2S,andzerootherwise.Thecontinuousvariableswsij,forall(i;j)2A,representtheowvaluesonarcsinscenarios2S. 15

PAGE 16

Theobjectivefunction( 2{1a )minimizesthefacilitylocationcostplusapriority-weightedcombinationoffacilityactivationandowcostsovereachscenario.Constraints( 2{1b )imposetheconditionthatafacilityonvertexi2Vcannotbeactivatedunlessafacilityhasbeenlocatedonvertexi2V.Constraints( 2{1c )ensurethatdemandismetbyrequiringthattheowintonodei2Vplustheinitialsupplyatnodei,minustheowoutofnodei,isnolessthanthedemandfornodeiineachscenario.Constraints( 2{1d )and( 2{1e )stateboundsandlogicalrestrictionsonthedecisionvariables.(Notethatifmodel( 2{1 )issolveddirectly,thex-variablescanequivalentlyberelaxedascontinuousvariables,withbinarinessenforcedonlyonthey-variables.) TheprimarydicultywithsolvingEXTdirectlyisitslargescale.ThereareO(jVjjSj)structuralconstraintsandbinaryvariables,plusO(jAjjSj)continuousvariableshavingsimplebounds.Especiallywhenthenumberofscenariosislarge,weanticipatethatEXTwillrequiresubstantialcomputationalresourcestosolve.Astandardtechniqueforsolvingscenario-basedproblemsisbydecomposition,althoughtherearerelativelyfeweectivetechniquescapableofsolvingtwo-stageproblemsinwhichintegervariablesappearinbothstages.Wedevelopsomecandidateintegerdecompositionapproachestosolvingthisprobleminthefollowingsection. 16

PAGE 17

WethensolveeachMIPsubproblemanddetermineifafeasiblesolutionexistsgiven^x.Ifnot,thena\feasibility"cutisaddedtoRMPthatenforcesavalidinequalityonfacilitylocationsinorderforthecurrentlyinfeasiblesubproblemtobecomefeasible.Else,iftheactualsecond-stagecostforsomescenariosisgreaterthanthevalue^zsobtainedfromsolvingtheRMP,thenan\optimality"cutisaddedtoRMPthatstatesalowerboundonazsvariableasananefunctionofx-variables. Therelaxedmasterproblemweconsideris 17

PAGE 18

Initially,wemaytakeopt=feas=;.Ifthisisthecase,ontherstiterationwewouldobtaintheinfeasibletrivialsolutionxi=0,foralli2V,zs=0,foralls2S.Asanalternative,\warmingconstraints"canbeaddedtoRMPtohelpproducemorerealisticsolutionsinearlyiterationswhenthesetsoptandfeasareemptyorsmall.Warmingconstraintsaresimple,necessaryconstraintsthatcanbeenumeratedwithoutsolvinganysubproblem.Forinstance,wemaychoosetoaddwarmingconstraintstoRMPoftheform: foreachs2S,whichstatethatwemustlocateenoughfacilitiessothatthesumofavailablesupplyinthemissionareaisatleastasmuchasthetotaldemandinthemissionareaforscenarios.(Constraints( 2{3 )arenotsucientforfeasibilitysincetheyignorescenario-specicarccapacities.) GivenatentativeRMPsolution^x,regardingasetoflocatedfacilities,recallthateachscenariosubproblemhastheformofacapacitatedfacilitylocationproblem.Forsomescenarios2S,thesubproblemisgivenby 18

PAGE 19

2{4 )asSUBs(^x),andthelinearrelaxationof( 2{4 )as SUBs(^x).IfSUBs(^x)isinfeasible,thenwetakes(^x)=1.Similarly,welet SUBs(^x),setting SUBs(^x)isinfeasible. ThechallengeistoaddcutstofeasifSUBs(^x)isinfeasible,ortooptifthevalueof^zsobtainedfromRMPislessthantheoptimalobjectivevalues(^x)foranys2S.Themethodsandjusticationforthreecutting-planetechniquesaregiveninthefollowingsections.Thersttwofollowstandarddecompositiontheory,whilethethirddescribesanewclassofcuttingplanestailoredforthisapplication. SUBs(^x)andSUBs(^x),butnottoforceconvergencetoanoptimalsolution.For SUBs(^x),associates-variableswithconstraint( 2{4b ),s-variableswith( 2{4c )ands-variableswiththeupperboundinginequalitiesin( 2{4d ).Thedualof SUBs(^x)is Problem( 2{5 )isalwaysfeasiblesincecsij;gsi0forall(i;j)2A;i2V,andsothetrivial(all-zero)solutionisfeasible.Ifproblem( 2{5 )isunbounded,thenweobtainanextreme 19

PAGE 20

Otherwise,ifSUBs(^x)isfeasible,but^zs< s(^x),weobtainanoptimalextremepointsolution(^s;^s;^s)tothedualof SUBs(^x)andaddthefollowingoptimalitycuttoopt: Notethatsincetheboundsonys-variablesarebetween0and1,afeasiblesolutionwithfractionalys-variablescanbemodiedtoabinaryfeasible(notnecessarilyoptimal)solutionbysettingallfractionalys-variablesto1.Thusthesetofconstraints( 2{6 )forallextremeraystothepolyhedrongivenby( 2{5b ){( 2{5e )issucienttoguaranteefeasibilityforboth SUBs(^x)andSUBs(^x). Oncealldualsubproblemsaresolved,ifnoinequalitiesareaddedtofeasoropt(i.e.,^zs 2{7 )arenot(ingeneral)sucienttoforceconvergencetoanoptimalsolutionofourproblem,themethodofLaporteandLouveaux[ 26 ]canbeusedtogenerateL-shapedcuttingplanesfromsubproblemshavingbinaryandcontinuousvariables,whichwillforceconvergencetoanoptimalsolution.AccordingtotheLaporte/Louveaux(LL)method,theMIPsubproblemSUBs(^x)associatedwithsgiven^xwouldbesolveddirectly,andthefollowinginequalitywouldbeaddedtotheRMP: 20

PAGE 21

LLcutsareverygeneral,andcanbeappliedtoanystochasticintegerprogramwhoserst-stagevariablesareallbinary-valued.Duetothegeneralityofthecuts,however,theinequalitiesoftheform( 2{8 )tendtobeweak.Forthecapacitatedfacilitylocationproblemwearestudying,ifanyx-variablechangesfromitspriorvalue,thentheLLcutallowsthevalue-functionvariablezstotakeonitslowerboundLs.LLcutscanbemodiedforourapplicationto: whichstatesthatthereisnodecreaseinthevaluefunctionvariablezsifsomelocatedfacilityati2^Xwereremoved.Thisstatementisjustiedbythefactthats(x1)s(x2)wheneverx1ix2i,foralli2V,sinceSUBs(x1)isarelaxationofSUBs(x2).Still,inequality( 2{9 )statesthatifsomefacilitywerelocatedati2Vn^Xintherststage,andthatfacilityhaspositivesupplyinscenarios,thevaluefunctionvariablezsisallowedtoachieveitslowerboundLs.Forlargeprobleminstances,weanticipatethattheseinequalitieswillbetooweaktoforceouralgorithmtoconvergewithinpracticalcomputationallimits. 21

PAGE 22

FromtheoptimalsolutiontoSUBs(^x),weoptimisticallyestimatesi(^x)0asthebenettotheobjectivevalueofsubproblemswhenanadditionalfacilityi2Vn^Xislocatedintherststage.IncontrasttoLLcuts,RPcutsestimatethedecreaseinsecond-stageobjectivevaluewithrespecttochangesinrst-stagedecisions,ratherthanautomaticallyallowingthesecond-stageobjectivevaluetoachieveitslowerbound. SimilartomodiedLLcuts,theremovalofalocatedfacilitywillnotdecreasethesubproblemobjectivevalues.However,duetothenonconvexityoftheoptimalsubproblemobjectivevaluesasafunctionofx,ingeneralitisimpracticaltoassesstheminimumobjectiveincreaseresultingfromtheremovalofafacility.Hence,weignorexi-termscorrespondingtoi2^Xin( 2{10 ). Wefocusondeterminingthemaximumamountbywhichasubproblemobjectivecandecreasewithadditionallocatedfacilities.Givenfacilitylocations^x,weidentifyanoptimalsolution(^ys;^ws)tosubproblems:WedenetheresidualnetworkRs(^x;^ys;^ws)havingnodesVanddirectedarcs(i;j)ifandonlyifeither(a)(i;j)2Aand^wsij0(inwhichcasewecall(i;j)a\backwardarc").Forwardarcs(i;j)haveowcostscsij,andbackwardarcs(i;j)havecostsofcsji.Todeterminethemaximumbenetoflocatingafacilityatnodei2Vn^X,weconsiderthepotentialimpactthateachunitshippedfromthislocationcouldhaveontheoptimalrecoursesolutiontoSUBs(^x).Ifafacilityislocated(andthenactivated)ati,theowfromiwouldeectivelyaugment(andreplace)owemanatingfromcurrentlyactivatedfacilities.Additionally,someactivatedfacilitiescouldbedeactivated,addingtothecostsavings. Tomodelthepossibilityoffacilitydeactivation,wemodifytheresidualnetworkbyaddingdummydestinationnodet,anduncapacitatedarticialarcsfromeachactivated 22

PAGE 23

Evenifactivatedfacilityjemittedmultipleunitsintheprevioussolution,wemustconsiderthepossibilityofdeactivatingfacilityjwhentheamountshippedfromjisreducedbyasingleunit.ConsidertheexampledepictedinFigure 2-1 ,whichdepictsanoptimalsolutioninwhichafacilitylocatedatnode1suppliesthreeunitsofdemandtonodeA.However,whenafacilityatnode3islocatedandactivated,itsuppliesoneunitofdemandtonodeA.Thefacilityatnode1cannowbedeactivated,reducingcostsbyg11=10,despitethefactthatnode1originallysentthreeunitsofdemandtonodeA,andonlyonenewunitofowisavailablefromnode3. Figure2-1. Facility1doesnotneedtobeactivatediffacility3islocated. Thereforelocatingafacilityatnodeiforcesustoactivatethefacilityatacostofgsi,butallowsustosaveowandactivationcostsasfollowsinthebestcase.Therstunitofowfromnodeichangesowcostsontheaugmentingpathbyincreasingcostsonforward 23

PAGE 24

1 ,accordingtotheprinciplesestablishedabove. Procedure 1 iterativelycomputesashortestpath,andwhileitisstillpotentiallyprotabletodoso,sendsonemoreunitfromnodeitonodet.Thisroutinestopseitherwhenbtemp,theremainingsupplyatnodei,equalszero,orwhenadditionalowisnolongerprotable,orwhens(^x)
PAGE 25

1 toterminateappropriatelyandultimatelyreturnsi(^x)=0. Astheroutineiterates,weupdatethecostofarticialarcsfromactivatedfacilitiestodummynodetinordertoenforcetheconditionthatcostsavingsgainedfromfacilitydeactivationcannotbeattainedmorethanonceforeachactivefacility.Sinceweupdatearticialarccosts,weneedtorecomputetheshortestpathfromitotateveryiterationinwhichthesecostsaremodiedonline8ofProcedure 1 .Atsomeiteration,theshortestpathfromitotmightuseanarc(j;t)thathasacostcjt=0.Inthiscase,thispathwillbetheshortestpathfromitotinallfutureiterations,andsoweaddthecostofpushingallremainingbtempunitsalongthisshortestpathtosi(^x)inasingleiterationofthewhile-loop(lines11and12ofProcedure 1 ). Lines15{17ofProcedure 1 ensurethatsi(^x)Lss(^x),whichisvalidbecausethelocationofoneadditionalfacilitycannotreducethescenarioobjectivebelowitslowerboundLs.(NotethatProcedure 1 couldcomputeaverynegativevalueforsi(^x),forinstance,byaccruingpotentialcostsavingsfromdeactivatingmultiplefacilities.)Byplacinglowerboundsonthe-valuesofLss(^x),weguaranteethatourinequalitiesareatleastasstrongasthemodiedLLcuts( 2{9 ).Next,lines18{20guaranteethatsi(^x)0,whichisjustiedbythefactthatintheworstcase,wehavetheoptionofsimplynotactivatingfacilityiinscenarios. FindingashortestpathfromitotinRs(^x;^ys;^ws)canbeperformedinpolynomialtimebecausetherearenonegative-costcyclesinRs(^x;^ys;^ws).Theresidualgraphcorrespondstoanoptimalactivationschemeandowvalues,andsoifanegative-costcyclewerepresent,owcouldbesentaroundthecycleandtheobjectivevaluecouldbeimproved,whichcontradictstheoptimalityof(^ys;^ws)given^x.Furthermore,Rs(^x;^ys;^ws)doesnothaveanydirectedcycles(ofanycost)thatcontainarticialarcssincetheout-degreeofnodetiszero.Procedure 1 willeitherperformatmostO(jVj)iterations(deactivatingeachactivefacilityateachiteration),orbsiiterations(pushingeachunit 25

PAGE 26

1 willperformatmostminfjVj;bsigiterationsoftheinnerwhile-loop.Usingaconservativeshortest-pathcomplexity,eachsuchiterationrequiresO(jVj2)operations.HencetheprocedurecanbeperformedinO(jVj3)time. Theorem 1 assertsthevalidityofusingsi(^x)inthederivationofvalidinequalitiesoftheform( 2{10 ).Inthefollowingdiscussion,westatethatvectorxcorrespondstoasubsetXVifxi=1wheni2X,andxi=0otherwise,8i2V. andthereforeaninequalityoftheform: isvalidtotherelaxedmasterproblem. Throughouttheproof,wewillrefertothreesolutions.Wewillcalloneoptimalsolution(^ys;^ws)toSUBs(^x)the\previoussolution,"withobjectivevalues(^x),fromwhichtheresidualnetworkwillbecreated.WerefertosomeoptimalsolutiontoSUBs(^x+x1)asthe\currentsolution."Wewillthenconstructa\modiedsolution,"whichonlyactivatesfacilitiesin^X.Using( 2{13 )togetherwiththeconstructionofthemodiedsolution,wewillshowthatthemodiedsolutionhasanobjectivevaluestrictlylessthans(^x).Becausethemodiedsolutiononlyactivatesfacilitiesin^X,thefactthatitsobjective 26

PAGE 27

2{13 )isfalse. First,wepartition^XintosetsX2andX3.Intheprevioussolution,allfacilitiesinX2^Xareactivatedandemanateows,andnofacilitiesinX3areactivated.FlowsfromfacilitiesinX2canbedecomposedinto(possiblyoverlapping)unit-owpaths;werefertothissetofactiveowpathsfromtheprevioussolutionasFP. ItissucienttoconsiderthecaseinwhicheveryfacilityinX1isactivatedinthecurrentsolution.(IfothernodesinX1arenotactivated,thefactthatsi(^x)0;8i2VnX1,ensuresthat( 2{12 )remainsvalid.)Inthecurrentsolution,wewillhaveowcomingfromeachfacilityinX1;otherwise,facilitiesemittingnoowswouldbedeactivatedinanoptimalsolution.WemayormaynothaveowemanatingfromfacilitiesinX2andX3(andthesefacilitiesmayormaynotbeactivated).DeneFiasthesetofunit-owpathsemanatingfromXifori=1;2;3inthecurrentsolution. Wenowprovideanalgorithmtoconstructthemodiedsolutionfromthecurrentsolution.Theobjectiveofthisalgorithmistoaugmentallowfrom,anddeactivatefacilitiesin,X1whilemaintainingafeasiblesolution. WeseekanodevbetweenapathfP2FPandapathf12F1suchthatfPdoesnotintersectanynodecontainedinanypathinF1beforev,andf1doesnotintersectanynodecontainedinanypathinFPbeforev.WecallvaprimaryintersectionoffPandf1.Ifthereexistsaprimaryintersectionv2VbetweentwopathsfP2FPandf12F1,thenwemodifythecurrentsolutionbyaddingaunitofowalongpathfPfromtheoriginoffPtov,andthendeletingaunitofowalongpathf1fromvtotheoriginoff1.Tworemarksareimportanttonoteregardingthismodication.One,thesolutionremainsfeasible,sinceowcurrentlyexistsonthearcsinthesegmentoff1leadingtonodev,andcapacitymustbeavailableonthesegmentoffPleadingtonodev;theowbalancesateachnodeareclearlyunaected.Two,theaugmentingpathstartingonf1andgoingtonodev,thenmovingbackwardonfPtoitsstartingvertexinX2,musthavebeenpresent 27

PAGE 28

Asituationwherenoprimaryintersectionexists.Adashedlineindicateszeroow.Asolidlineindicatesoneunitofow. intheresidualgraphoftheprevioussolutionsincecapacityexistedonthearcsinf1andowexistedonthearcsinfP.Afterthismodication,removef1fromF1andremovefPfromFP.StopifF1=;,andotherwiseseekanotherprimaryintersection. Next,supposethatnoprimaryintersectionexists(Figure 2-2 ).Inthiscase,wemusthaveasetofcycle-inducingpathsfP1;:::;fPKinFPandf11;:::;f1KinF1,whichcanbeidentiedasfollows.Tracethepathofanyf112F1untilitrstintersectsapathfP12FP.Sincethepointofintersectionisnotataprimaryintersection,pathfP1mustrstbeintersectingsomeotherpathf122F1beforeintersectingf112F1.However,pathf12mustbeintersectinganotherpathfP22FPbeforefP1(orelseaprimaryintersectionwouldexist).Now,fP2mustintersectsomeotherpathinF1beforef12.Ifthatpathisf11,thenwehaveidentiedourcycle-inducingpaths(fP1;fP2;f11;f12).Else,fP2rstintersectsf132F1,butf13rstintersectsapathfP32FP.IffP3intersectsf11beforeanyotherpathinF1(asdepictedinFigure 2-2 ),thenacardinality-6setofcycle-inducingpathshasbeenidentied(fP1;fP2;fP3;f11;f12;f13),andifitrstintersectsf12,thenacardinality-4setofcycle-inducingpathshasbeenidentied(fP2;fP3;f12;f13).Otherwise,wecontinueasbeforeuntilsuchacycleofpathsisfound.ThenitenessofFP,andthelackofaprimaryintersection,guaranteesthatsuchasetofcycle-inducingpathsmusteventuallybefound. 28

PAGE 29

FlowispushedonarcsinfPjfromtheoriginoffPjtovlastj,andthenowisreversed(eectivelyremoved)fromarcsinf1jfromvlastjtotheoriginoff1jforj=1;2;3.Theow-cycleisidentiedwiththedottedline. Now,modifyowsalongtheseKcycle-inducingpathsasfollows.First,foreachj=1;:::;K,sendaunitofowfromfPjuntilitsrstintersectionwithf1j;removeowfromthebeginningoff1jtothispoint(Figure 2-3 ).Asbefore,eachoftheKaugmentingpathsstartingonthef1jpathandendingatthefPjoriginispresentintheresidualgraphoftheprevioussolution.Moreover,theseoperationscontinuetosatisfytheowbalanceconstraints. However,thereisnoguaranteethattheadditionalowonfP-pathssatisfycapacityrestrictionsontheirarcs.TheremustexistsparecapacityonfPjarcsencounteredbeforetherstpointofintersectionwithowsinF1,whichwecallvrstj.Afterthispoint,fPjmayintersectotherarcsusedinF1pathsbeforearrivingtothepointofintersectionwithf1j,whichwecallvlastj.Toovercomethisproblem,observethatourmodiedgraphnowcontainsatleastoneunitofowbetweenvrstjandvlastj,foreachj=1;:::;K.Also,aftervlastj,pathf1jcontinuestotransmitowtoitsintersectionwithfPj1atnodevrstj1(wherefP0fPKandvrst0vrstK),becausethissegmentofowwasnotaectedbyourmodication.Therefore,ownowexistsinacycleinthisnetwork,proceedingfromvrstjtovlastj,thenfromvlastjtovrstj1,forj=K;:::;1,loopingbacktovrstK.Removingaunitof 29

PAGE 30

Remainingowafteroneunitofowonthecyclehasbeenremoved. owfromthiscycle,weensurethatcapacityrestrictionsonthearcsonthecyclefromvrstjtovlastj,foreachj=1;:::;K,aresatised,andthattheestablishedowpatternremainsfeasible(Figure 2-4 ).Followingthetwomodications(owaugmentationandcyclicowremoval),weremovef1jfromF1andremovefPjfromFPforeachj=1;:::;K.Wethenreturntolookingforaprimaryintersectionorasetofcycle-inducingpaths,andterminatewhenF1=;. Procedure 2 formalizestheseconceptsandreturnstheappropriatemodicationsnecessarytotransformthecurrentsolutionintothemodiedsolution. AfterProcedure 2 hasterminated,wehavecollectedasetofaugmentingpathsandcycles.Foreach2,dene1()astheendpointofinX1,and2()astheendpointofinX2.Partitionintoi,i2X1,where2isplacediniifandonlyif1()=i. Now,totransformthecurrentsolutiontothemodiedsolution,processiforeachi2X1inanyorderaccordingtothefollowingprocedure.Foreach2i,letcostdenotetheaggregatecostofaugmentingowusingpath.Activatefacility2()2X2ifthereiscurrentlynoowfrom2()inthemodiedsolutionandaddgs2()tocost.Pushowalong,reversingowemanatingfromi.(Leticost=P2icostandletcost=Pi2X1icost.) 30

PAGE 31

Notethatthesamepath,withcostatleastassmallastheaggregatecostcomputedabove,existsinRs(^x;^ys;^ws).Thecostswillbeequalifbothpathseitherdo,ordonot,subtractgs2()fromtheaggregatecost.However,therstsuchpathfromitotusingarticialarc(2();t)inRs(^x;^ys;^ws)willalwaysaccountforthedeactivationcostsavingsgs2(),whilethisinitialsubtractionisnotnecessarilyavailableinourschemeabove,becausefacility2()mayhavebeenactivatedinourpriorconsiderationofadierentsubsetj,j2X1. Afterprocessingallowpathsinandremovingowfromcyclesin(wherethecostofcycle2iscostandcost=P2cost),wehaveconstructedthemodied 31

PAGE 32

Itisimportanttonotethat foralli2X1,wheresi(^x)iscomputedviaProcedure1,since(a)boththeleft-handandright-handtermsof( 2{15 )includegsi,(b)eachpath-costincludedinthecomputationoficostcorrespondstoanidenticalpathinRs(^x;^ys;^ws)ofidenticalorlessercost(moreover,positive-costpathsinRs(^x;^ys;^ws)areessentiallysupplantedwithzerocostsinthecomputationofsi(^x),duetothe\cSP<0"conditioninline3ofthewhile-loopinProcedure1),and(c)theoverallcostofsi(^x)isboundedabovebyzero. Noting( 2{13 )andsubstitutingfors(^x+x1)asin( 2{14 ),wehave However,sincecost0and(gsiicosti(^x))0foreachi2X1dueto( 2{15 ),( 2{16 )impliesthatsmodLss(^x)forseverali2V,s2S,andthustheRPcutsaretighterthanLLcuts.Ontheother 32

PAGE 33

Accordingly,wedevelopedsixdataprolestocomparetheperformanceofthesethreeapproaches.Therearetentestinstancesperprole,foratotalofsixtyinstances.AlltestinstanceshadjVj=40andjSj=50.OurprolesaredesignatedA1,A2,B1,B2,C1,andC2.InstancesinproleA1havelargelocationandactivationcostsandfacilitieshavelargesuppliessothatasinglefacilitycanalmostalwaysaccountforalldemandineachscenario.InstancesfromproleB1havemuchhighershippingcoststhanactivationcosts,andwhile,asinproleA1,facilitiesusuallyhaveenoughsupplytosatisfyalldemandineachscenario,multiplefacilitiestypicallyneedbeactivatedduetoshippingcostsandconnectivity/capacityrestrictions.InstancesfromproleC1haveanequalratiooftotalcosts,andfacilitiesthatarenotcapableofsatisfyingalldemand.ProlesA1{C1haveanarcdensityof10%,whileprolesA2{C2havethesamecharacteristicsrespectivetoA1{C1,buthaveanarcdensityof80%. Table 2-1 givesthesetofpossibledatavaluesforeachprole,fromwhichasinglevalueischosenwithequalprobability.Dene0qasasetofqzeros.Notethatmultiplecopiesofidenticalvaluesincreasetheprobabilityofselectingthatvalue.Forinstance,inprolesB,ineachscenario,eachnodehasa25%chanceofreceivingaunitofdemand,anda75%chanceofhavingnodemand.Also,ifanodeisassignedpositivedemandinsomescenario,thenthesupplyatthatnodeissettozerointhatscenario.Finally,wegenerateprioritiesbyrstassigningeitherps1orps2foreachscenario,andthennormalizingtheprioritiessuchthatPs2Sps=1. AllcomputationswereperformedonaDellPowerEdge2600computerwithtwoPentium43.2Ghzprocessorsand6Gofmemory.LinearandintegerprogrammingproblemsweresolvedusingCPLEX10.2.Weimposedaone-hour(3600second)timelimitforsolvingeachinstance. 33

PAGE 34

Possiblevaluesfortwo-stagefacilitylocationtestproblemdata ABC LocationCost(fi)f100;:::;250gf50;60;70gf4;5gActivationCost(gsi)f100;:::;250gf40;50gf4;5gSupply(bsi)f100;:::;200gf20;30gf200;:::;400gDemand(dsi)075[f1;:::;10g03[f1g0150[f100;:::;150gArcCapacity(usij)f25;:::;75gf2;:::;5g050[f150;:::;250gArcCost(csij)f1;:::;10gf10;15;20gf0:01;0:02;0:03g 2{7 )derivedfromthelinearprogrammingrelaxationsofthesubproblems.Specically,weimplementthecutting-planealgorithmsasfollows.AftereachsolutionoftheRMP,werstsolvethelinearprogrammingrelaxationforeachsubproblem.Iftherelaxationisinfeasible,weaddafeasibilitycut( 2{6 )totheRMP.Iftherelaxationisfeasibleand^zs< s(^x),weaddanoptimalitycut( 2{7 )totheRMP.ThenwesolvetheMIPsubproblemandif^zs
PAGE 35

Summaryofcutting-planealgorithmsandextensiveformulationperformancesfortwo-stagefacilitylocationprobleminstances. A1A2B1B2C1C2 InstancessolvedRP101071042LL101071042EXT110000 AverageCPUsecondsRP1852031698?1832575?3023?LL4316031907?1412633?3243?EXT3392?3485?3594?3593?3593?3594? Eachpassthroughthewhile-loopresultsinarealisticaugmentationoftheowfromactivefacilitiesandinamodestdecreaseinthe-value.ProlesA1andA2producedthehighestpercentageof-valuesthatwerenotboundedfrombelowby(Lss(^x)):15.9%forA1and15.8%forA2.ThestrongerRPcuttingplanesleadtofewersolutionsoftheRMPandareductionintotalCPUtime. InstancesgeneratedaccordingtoprolesB1andB2havefairlylargearccosts,andthereforetotalshippingcostsaresignicantlylargerthantotalactivationcosts.However,thesuppliesatfacilitieswerelargeenoughtosatisfymostofthedemand,andonaveragelessthan1.5facilitieswereactivatedinallscenariosovertheB1andB2testinstances.Onaverage,forinstancesinB1,8.4%of-valueswerenotbounded,and4:6%forinstancesinB2.RPcutshadaslightadvantageintheB1instances,leadingtoalowernalaverageoptimalitygapaswellasareductioninaverageCPUtime.InB2,withthehigherarcdensity,Procedure 1 wasabletondlargerpotentialreductions,leadingtomore-valuesbeingbounded,andweakeningofthecuts.WhileProcedure 1 terminatesinapolynomialnumberofsteps,thecoecientsfortheLLcutsaregeneratedinstantly,andwhenthe 35

PAGE 36

ProbleminstancesfromproleC1andC2provedtobethemostchallengingforbothcuttingplanealgorithms.Theseinstanceshaveanevenratiooftotallocation/activationtototalshippingcosts,buthavelargedemandandsupplygures.Asaresult,Procedure 1 performsmoreiterationsofthewhile-loop,thusdecreasingthe-valuessubstantially.Thispromptedthehighestrateof-valuesbeingboundedfrombelowby(Lss(^x)).ForinstancesfromC1thatterminatedunderthetimelimit,onaverageonly5.6%of-valueswerenotboundedfrombelow.ForC2,onaverage0.6%of-valueswerenotboundedfrombelow.TheRPcutsstillprovideaslightadvantageovertheLLcutsfortheC1case,andhavenodiscernablebenetinsolvingtheC2instances. 36

PAGE 37

Whilethisproblemcanbeusedinthecontextofcombattingthespreadofinfectiousdisease,itisalsoapplicableinseveralothercontexts.Forinstance,considerthesearchforatargetedperson(e.g.,afugitiveorkidnappedperson).Thecontaminationoriginatesatthepointwherethetargetisisdetected.(Inoutmodel,weassumethatinitialdetectionisautomatic,andfocusondecontaminationstrategies.)Thespreadofcontaminationrepresentsthecandidatenodestowhichthetargetcouldmoveasthesearchevolves.Inthissense,itdoesnotmatterwhetherthetargetishiding[ 2 13 24 ]orattemptingto 37

PAGE 38

1 3 4 29 ].Weassumeonlythatthereisnocommunicationthathelpstheagentsinseekingthetarget,andexaminesearchstrategiesthatwouldguaranteethetarget'scaptureasquicklyaspossible.Anothercontemporaryapplicationarisesindeployingmobilesoftwareagents,whichareusedtodetectanintruderoranomaly[ 22 23 ],orremoveharmfulsoftwarefromcompromisedcomputersonacomputernetwork[ 17 { 20 ]. Parson[ 32 33 ]andBreisch[ 10 ]introducedtheoriginalgraphdecontaminationproblem.Intheirpapers,thegraphrepresentsasystemoftunnels,andagentssearchthegraphbyasequenceofmovements(includingteleporting).FindingtheminimumnumberofagentstosearchthegraphwasshowntobeNP-Completeforgeneralgraphs[ 8 25 31 ],andlinearlycomputablefortrees[ 31 ].Classicresultshavealsoestablishedrelationshipsbetweentheminimumnumberofsearchersrequiredforthesesearchproblemsandgraphtheoreticmetricssuchasvertexseparation[ 16 ],treewidth[ 36 ],andpathwidth[ 15 ].Forcasesinwhichagentscannotteleport,asassumedinthispaper,Barriereetal.[ 5 6 ]introduceda\connectednodesearch"model,whichconstrainsthemovementsofagentstoedgesonthegraph. Inourmodels,theagentsandcontaminationmoveandspreadatthesameunitspeed,andallcontaminationandcleaningisperformedinstantly.Mostpreviousliteratureallowsthecontamination(ortarget)tospreadarbitrarilyfastalonganyopenpath.However,intheapplicationslistedabove,itisasensibleassumptionthatagents,fugitives,etc.,movealongthenetworkedgesatthesamespeed. Inourmodel,weseektodecontaminateonlythenodesofagraph,andnotitsedges.SimilarproblemsareconsideredbyFlocchinietal.[ 17 { 20 ],wheretheagentsarenotallowedtoteleport.Specictopologiesconsideredincludehypercubes[ 17 ],chordalringsandtori[ 18 ],hexagonalmeshes[ 20 ],andarbitraryregularnetworks[ 19 ].Ineachofthesepapers,theadvantagesofdierentagentcapabilitiesareevaluated.Thesecapabilitiesincludevisibility,abilitytoclonethemselves,communicationwithotheragents,and 38

PAGE 39

Thenotableassumptionsofourmodelanddierencesbetweenpreviousworkare: 1. Edgesareneutral,andweonlyseektodecontaminatenodes. 2. Weconsidersynchronousmovements,whereagentsandthecontaminationmoveandspreadatthesamespeed.Everyedgetakesonetimesteptotraverse.Allcontamination/cleaningisperformedinstantly.Itisworthnotingthatinmostpreviousliterature,thecontamination(orintruder)canspreadarbitrarilyfastalonganyopenpath.Inourmodel,whenthecontaminationandagentsrepresentsoftware,itisasensibleassumptionthatanytypeofsoftware(goodorbad)movesaboutthenetworkatthesamespeedbasedonthecapabilitiesofthenetwork.Forphysicalsearchproblems,suchasapprehendingafugitive,theequivalent-speedassumptionisalsoreasonable. 3. Ourmodelcontainsperfectinformation,i.e.,theagentscaninferwhichnodesarecontaminatedduetoknowledgeofthegraphtopology. 4. Wedonotrestrictagentcleaningstrategiestobemonotone.Acleaningstrategyismonotoneifafteranodeisdecontaminated,theagentsmoveinsuchawaythatthenodeneverbecomesrecontaminated.Weseekanymovementstrategythatcleansallnodesasfastaspossible. 5. Weseekafastestpossiblecleaningstrategyfortheagents,withoutconstrainingthemtofollowsimpleprotocols. 6. Wedonotconsideraspecicnetworktopology.Theformulationsandsolutionsmethodsweproposeareapplicabletoanynitegraph. Wepresentmoresophisticatedsolutiontechniquesfortheseproblems.Tondamovementstrategyforagentstodecontaminatethegraphasfastaspossible,wepresentagreedyrandomconstructionheuristicthatusestherelaxationsofthemathematicalprogrammingformulations.Fortheagentdeploymentproblems,wepresentatwo-stage 39

PAGE 40

7 ]andsupplementedwithintegercuttingplanes.Forthecutting-planealgorithm,werstestimatethenumberofagentsandtheirpositions,andinthesecondstage,weverifythateverypossiblecontaminationcanbecleanedwithintheimposedtimelimit. Theremainderofthischapterisorganizedasfollows.InSection 3.2 ,wediscussthespecicdynamicsofhowthecontaminationspreadsandagentsmoveaboutthegraph.InSections 3.3 3.4 ,and 3.5 ,weperformcomplexityanalysisandgivemathematicalprogrammingformulationsforthegraphdecontaminationproblem,theproblemofdeployingtheminimumnumberofagentsrequiredtocleanthegraph,andtheproblemofdeployingagentstominimizeexpectedcleaningtime,respectively.InSection 3.6 ,wetestthemathematicalprogrammingformulationsdeveloped.InSection 3.7 ,wegiveagreedy,randomconstructionheuristictondthefastestmovementstrategyforagentstofollowwhencleaningagraphandpresentcomputationaltestingontheheuristic.InSections 3.8 { 3.10 and 3.11 { 3.13 ,wedevelopandtestdecompositionsolutionmethodsforndingtheminimumnumberofagentstocleanthegraphandthedeploymentstrategyforagentsthatminimizesexpectedcleaningtime,respectively. ThereexistKjVjagentsthatcleanthespreadingcontamination.Anodeoccupiedbyanagentiscleaned,butinsomecasescanberecontaminatedatafuturetime.Agentsareidenticalintheircapabilitiesandmultipleagentscansimultaneouslyoccupyanode. 40

PAGE 41

WeassumethatthecontaminationstartsonsomesubsetofnodesLV.Ifanagentisinitiallypositionedonanode`2L,thenthatnodeisnotinitiallycontaminated.IfagentsarepositionedoneverynodeinL,thencleaningtimeiszero. Inthefollowingdiscussion,weconsidertwovariationsregardingthespreadofcontamination:withblockingandwithpassing.Passingpermitsthespreadingcontaminationtotraverseanedgeintheoppositedirectionofanagentonanedgewithoutbeingcleaned.Consideranedge(i;j),inwhichnodenodejiscontaminated.Thennodeibecomescontaminatedatthenexttimestepifnoagentispresentatnodeiatthenexttimestep.Theversionoftheseproblemswithblockingassumesthatthecontaminationwillnotpassthroughanagentmovingalonganedge.Hence,intheaboveexample,thecontaminationatjwillspreadtonodeiunlessanagentmovestonodeiinthenexttimestep,orifanagenttraversesedge(i;j)immediatelybeforethenexttimestep. Wealsoexaminetwovariationsoftheproblemwhereagentscan\clone"themselves.Inonevariation,theagentleavesastationarycloneateachnodevisited,thuspreventingrecontaminationofthevisitednode.Thisversionalsocapturesthecaseinwhichnocloningactuallytakesplace,butanagentbothcleansandfortiesanynodeitvisits.Inthecloningversion,agentssendcopiesofthemselvestoalladjacentnodesateachtimestep,andeectivelyspreadaboutthegraphinthesamewayacontaminationspreadsaboutthegraph.(Notethatthenotionofblockingorpassingisirrelevantforcloned-agentproblems,becausenoderecontaminationisnotpossible.) 3.2 .Weknowtheinitialpositionsoftheagents,andwealsoknowthesetofnodesfromwhichthecontaminationoriginates.Theversionthatallowsforagents 41

PAGE 42

WeleaveopenthequestionofwhethertheseproblemsbelongtoNP.Verifyingwhetheragiveninstanceisfeasibleappearstorequireastep-by-stepanalysisofagentmovementandcontaminationspread,whichis(nomorethan)O(KjEjtmax)incomplexity.However,whileKisO(jVj)andjEjisO(jVj2),tmaxisnotknowntobeboundedbyapolynomialfunctionofjVj.IfitcouldbeshownthattmaxisO(p(jVj)),wherep(jVj)isapolynomialfunctionofjVj,thenwecouldconcludethattheCLEANs,CLEANb,andCLEANpproblemsarestronglyNP-Complete,asinallofthefollowingproofsourtransformationsareallfromstronglyNP-Completeproblems,andwecreatenonumericaldatainourtransformations. 21 ].ThedecisionversionoftheHamiltonianPath(HP)problemseeksapathstartingonsomenodesthatvisitsallothernodesexactlyonce.WetransformanyinstanceoftheHPproblemtoaCLEANsinstancebycopyingtheHPgraph,positioningoneagentonnodes.Lettheentiregraphbecontaminated,andsetthetimelimittmax=jVj1.Sincetheagentmustvisiteverynode,wecancleanthegraphinjVj1stepsifandonlyiftheHPinstanceisfeasible.ThedecisionversionofCLEANsisatleastasdicultasHP,andthereforeCLEANsisNP-Hard.2

PAGE 43

Diamondstructure. 21 ].AnX3CinstancecontainsasetofelementsX=f1;:::;3qg,andasetofclausesC=fc1;:::;crg,whereq
PAGE 44

ACLEANbinstancetransformedfromthetheexampleX3Cinstance. strategy.)LetL=V,i.e.,theentiregraphisinitiallycontaminated.Weclaimthatthegraphcanbecleanedintmax=4timestepsifandonlyiftheX3Cinstanceisfeasible. Adiamondstructure(Figure 3-1 )canbecleanedbythreeagentsusingStrategy1,describedasfollows.Placeallthreeagentsonthetailnode.Inonestep,sendallthreeagentstooccupyallmiddlenodes,andinthenextstep,sendallagentstooccupytheheadnode,atwhichpointallnodesareclean.Inthefollowingtwoclaims,werefertothefour-cliquecomposedofallmiddlenodesandtheheadnodeastheheadfour-clique. Supposethatonlyonenode`inthiscliqueiscontaminated.Sincethedegreeofnode`isatleastthree,iftherearefewerthanthreeagentsintheclique,thereisanadjacentnodektowhichthecontaminationwillspreadinthenexttimestep.Lettingktaketheroleof`inthepreviousstep,wehavethatatleastonenodeinthediamondstructurewillalwaysbecontaminatedateachtimestepunlessthreeormoreagentscleantheclique.2

PAGE 45

Case(1)issatisedbysendingoneagenttotheheadnodeofthediamondstructuresothatitarrivesattimet+2,withthetrailingagentsarrivingonallmiddlenodesattimet+1.Sincewehavethreeormoretrailingagentsarrivingonthemiddlenodes,theyblocktheedgesonwhichtheytravelandpreventrecontaminationofthetailnode.Case(2)canbesatisedbydelayingoneoftheagentsatthetailnodebetweentimestandt+1,andrepeatingtheabovestrategyforCase(1).Strategy1showsthatCase(3)issatised. Toshowtheconverseoftheclaim,rstnotethatifwedonothaveanyagentsarriveonthetailnodeattimet,thennoagentscanreachtheheadnodebytimet+2.Ifwehaveonlyoneagentarriveonthetailnodeattimet,itmustpassoveronemiddlenodeattimet+1beforearrivingattheheadnodeattimet+2.Attimet+1,theothermiddlenodesremaincontaminated.Iffewerthanthreeagentsenterthetailnodeattimet+1,onemiddlenodewillbecomerecontaminated(orremaincontaminated)attimet+2.Next,ifwehavetwoagentsarriveonthetailnodeattimet,followedbyonereserveagentattimet+1,thenbythesameargument,atleastonemiddlenodeiscontaminatedattimet+2.2 Intherststep,allreserveagentsmovefromnodes1tos2,andallforwardagentsmoveingroupsofthreetothetailnodesofdiamondstructurescorrespondingtoclausesinC.Inthenextstep,reserveagentswillmoveingroupsofthreetothetailnodesofremainingdiamondstructures.AgentscleanthediamondstructuresaccordingtoStrategy 45

PAGE 46

Next,weshowthatifnosolutionexiststotheX3Cinstance,thenwecannotcleanthegraphinveorfewertimesteps. Claim 1 statesweneedatleastthreeagentstocleaneveryheadfour-clique.Claim 2 statesthatiffewerthanthreeagentsenterthediamondstructureatthesametime,theheadfour-cliquecannotbecleanintwostepsunlessatotaloffourormoreagentsvisittheheadfour-clique.Notethatnoagentcanvisittwodierentheadfour-cliquesinfoursteps.Therefore,everyheadfour-cliqueisvisitedbyexactlythreeagentswhogatheratthediamond'stailnodeatthesametime.Inorderfortheforwardagentstoreachtheelementnodesbytimefour,theycannotwaitforatimesteptobepairedwithareserveagent.Therefore,theforwardagentsmustmoveingroupsofthreetoqdierentdiamondstructures.However,sincenoX3Cexists,nosetofqdiamondstructureswillallowtheseforwardagentstoreachall3qelementnodes.2 3 46

PAGE 47

Ahexagonalstructurewithasix-cliqueforthemiddlenodes. ThetransformedCLEANpgraphisdescribedasfollows.Foreachclause,wecreateahexagonalstructureasdepictedinFigure 3-3 .Createatotalof3qelementnodes.Foreachclauseci,connecttheheadnodeofthehexagonalstructurecorrespondingtocitothethreeelementnodesassociatedwiththeelementsinclauseci.Createnodess1,s2,ands3.Connectnodes1tos2,s2tos3,andconnects3tothetailnodeofeachhexagonalstructure(Figure 3-4 ). TheinitialconditionsfortheCLEANpinstancearedescribedasfollows.Wehaveatotalof3(rq)reserveagentsonnodes1and2.Wehave3qforwardagentsonnodes2and3.Thus,wehave3(rq),3r,and3qagentspositionedonnodes1,2,and3,respectively,foratotalof6ragents.Theentiregraphisinitiallycontaminated. 3 holds,notethatiftherearefewerthansixagents,andifthereisatleastonecontaminatednodeinthesix-clique,thenonenodemustremaincontaminatedateachstep.2 47

PAGE 48

ACLEANpinstancetransformedfromthetheexampleX3CinstancedescribedintheproofofTheorem 3 contaminationfrompassingbacktoanycleanednodes.Attimet+2,movetherstgroupofagentstothethreemiddlenodesconnectedtotheheadnode.Attimet+3,therstgroupofagentsarriveontheheadnode,andthehexagonalstructureisclean. First,wehavethatat1sincetheshortestpathlengthfromthetailtotheheadnodeisthree(andhence,thestructurecannotbecleanbeforetimet+3).Ifat=1,thenwemusthaveat+13sothatthreeagentscanarriveonthemiddlenodesconnectedtotheheadnodeattimet+3.Else,thecontaminationwouldremainononeofthemiddlenodesconnectedtotheheadnodewithoutanagent(bythesamelogicgiveninClaim 2 ). 48

PAGE 49

Ifat=2,thenwemusthaveat+12sothattwoagentscanarriveontheremainingcontaminatedmiddlenodesconnectedtotheheadnodeattimet+3.Again,thethreemiddlenodesconnectedtothetailnodewillstillbecontaminated,andsowemusthaveat+23.Thusat+at+1+at+27.Ifat3,thenwithat+1=6at,wecandelayat3agentsatthetailnodeattimetandemployStrategy3.2 Distributeforwardagentsingroupsofsix(withthreeleadagentsandthreebackupagents)tohexagonalstructuresassociatedwithclausesinC,anddistributereserveagentsingroupsofsix(againwiththreeleadagentsandthreebackupagents)totheremaininghexagonalstructures.UseStrategy3tocleanallhexagonalstructures.ThosehexagonalstructuresassociatedwithclausesinCwillbecleanandwillhavethreeagentsontheirheadnodesattimefour.Attimeve,cleanallelementnodeswithagentsatheadnodescorrespondingtoC.Letthesecondgroupofforwardagentsarriveontheheadnodesattimeve,thuspreventingrecontaminationinthehexagonalstructures.Attimeve,thereserveagentswillarriveattheheadnodesoftheremaininghexagonalstructures(correspondingtoCnC),preventingthecontaminationfromspreadingtoremaininghexagonalstructuresandcompletingthegraphcleaninginvesteps. Next,supposethatnosolutionexiststotheX3Cinstance.Claims 3 and 4 demonstratethatwemustcleaneachhexagonalstructureusingexactlysixagentsinordertohavethegraphcleanthegraphinvesteps.Again,notethatnoagentcanreachthemiddlenodesofmorethanonehexagonalstructureinvesteps,thus,wehavethatallfeasiblecleaning 49

PAGE 50

Denebinarycontaminationvariablesvti,whichtakeonavalueof1ifnodeiiscontaminatedattimetand0otherwise,foralli2V,t=0;:::;tmax,wheretmaxisthegiventimelimit.Letbinaryvariablesutiequal1ifaclonedagenthasappearedonnodeiattimetorbefore,and0otherwise,fori2V,t=0;:::;tmax.Finally,thebinaryvariableytequals0ifallnodesarecleanattimet,andequals1otherwise,fort=0;:::;tmax.Giventhatagentsarepositionedonnodesaccordingtox,wherexiistheintegralnumberofagentslocatedonnodei2V,andacontaminationbeginsonnodesLV,thecleaningtimeQcL(x)is Theobjectivefunction( 3{1a )seekstominimizethenumberoftimestepsuntilthegraphisdecontaminated.Notethat0QcL(x)tmax+1.Ifnofeasiblesolutioncanbe 50

PAGE 51

3{1b )and( 3{1c )aretheinitialconditionsfortherecourseproblemthatstatewhichnodesarerstcontaminatedandwheretheagentsarepositioned,respectively.Constraint( 3{1d )statesthatanagentcannotappearonnodeiattimetunlesstherewasanagentonsomeadjacentnodehattimet1.Itisworthnotingthattheagentcloningconstraint( 3{1d )alonedoesnotforceaclonedagenttoappearonadjacentnodes,butanoptimalsolutionwillexistinwhichagentscloneandspreadtoalladjacentnodesateachtimestep.Constraint( 3{1e )modelsthespreadofcontamination.Itstatesthatifnodejiscontaminatedattimet1,thenthecontaminationwillspreadtoadjacentnodeiattimet,unlessanagenthasappearedonnodeiatorbeforetimet.Constraint( 3{1f )statesthatifanynodesarecontaminatedattimet,thenyt=1.Constraint( 3{1g )statesthatv-variablesarenonnegative.Nonnegativityofthey-variablesisimpliedby( 3{1f ).Constraint( 3{1h )statesthatu-variablesarenonnegativeandnomorethan1. Wenowshowthatthereexistsanoptimalsolutioninwhichallu-,v-,andy-variablestakeonbinaryvalues.First,notethatoptimalitywillforcey-variablestobeassmallaspossibleforeacht=0;:::;tmax.Sincethelowerboundsonthey-variablesarestatedby( 3{1f ),optimalityforcesv-variablestobeassmallaspossible.Theinitialconditionsstatethatv0`1unlessagentsarepositionedonnode`(forall`2L).Thespreadingconstraints( 3{1e )willpropagatebinaryvaluesforvt-variablessincevt1-variableswillbebinary.Bythesametoken,optimalitypushesu-variablestotheirlargestpossiblevalues(tominimizev-values),andwillreachtheirlimitofmaxfPh2N[i]uti;1gfori2Vandt=0;:::;tmax.Sinceoptimalityforcesyt=maxi2Vfvtig,theny-variablesmustalsobebinary-valued. FortheCLEANsformulation,weintroduceadditionalagent-owvariableswtij,whichgivetheintegralnumberofagentsthatmovefromnodeitojattimet,8(i;j)2E,t=0;:::;tmax1.Inthisformulation,utiisequaltothetotalnumberofstationaryclones 51

PAGE 52

3{1b );( 3{1c );( 3{1e )( 3{1g )Xj2N(i)w0ij=xi8i2V (3{2c)Xh2N[i]wthi+uti=ut+1i8i2V;t=0;:::;tmax1 (3{2d)wtij0andintegral8i2V;j2N[i];t=0;:::;tmax1: Constraint( 3{2b )setstheinitialpositionsofmobileagents.Constraint( 3{2c )actsasaowbalanceconstraintforthemovementofmobileagents.Notethatconstraints( 3{2b )and( 3{2c )preventmobileagentsfromtraversingself-loops,sincetherewillalwaysexistanoptimalstrategyforthestationarycloningvariantinwhichnoagentstraverseself-loops.Constraint( 3{2d )increasesthevalueofuti-variablesbythenumberofagentsvisitingnodei2Vattimet=1;:::;tmax.Constraints( 3{1c )and( 3{2d )enforcetheconditionthattheu-variablesarenonnegativeintegers.Integralityoftheu-,v-andy-variablesisguaranteedbytheargumentgivenformodel( 3{1 ). 52

PAGE 53

3{1b );( 3{1f );( 3{1g );( 3{2e )Xj2N[i]w0ij=xi8i2V (3{3c)vtivt1iXh2N[i]wt1hi8i2V;t=1;:::;tmax Constraint( 3{3b )setstheinitialpositionsofmobileagentsandconstraint( 3{3c )actsasaowbalanceconstraintforthemovementofmobileagentsandinthiscase,allowsforagentstotraverseself-loops.Constraints( 3{3d )and( 3{3e )modelthespreadofcontamination.Constraint( 3{3d )statesthatifnodeiiscontaminatedattimet1,thenitwillremaincontaminatedattimetunlessoneormoreagentsarriveonnodeiattimet.Constraint( 3{3e )statesthatifnodej2N(i)iscontaminatedattimet1,thenthecontaminationwillspreadtonodeiattimetunlessoneormoreagentsarriveatnodeiattimet,oranagentblocksthespreadingofthecontaminationbytraversingedge(i;j)attimet1. TheformulationoftheCLEANpproblemisnearlyidenticalto( 3{3 ) 3{1b );( 3{1f );( 3{1g );( 3{3b );( 3{3c );( 3{2e )vtivt1jXh2N[i]wt1hi8i2V;j2N[i];t=1;:::;tmax: 53

PAGE 54

3{3b )and( 3{3c ),whichmodelthespreadofcontamination,arereplacedwithsingleconstraint( 3{4b ).Constraint( 3{4b )statesthatacontaminationwillspreadfromnodejattimet1toadjacentnodei(includingself-loops)attimetunlessagentsarriveonnodei. GiventhesameinitialagentpositionsxandcontaminationstartingonnodesinL,wehavethatQcL(x)QsL(x)QbL(x)QpL(x),duetotherelativeeectivenessofthevarioustypesofagents. 21 ].TheDSdecisionproblemseeksadominatingsetDsuchthatjDjK. 54

PAGE 55

IfthereexistsadominatingsetDofcardinalityKforsomeDSinstance,thenwecansolvethecorrespondingMINAGENTcproblembypositioningKagentsonthenodesinD:agentssendclonesofthemselvestoalladjacentnodes,andinonetimestep,thereisanagentoneverynode.Now,assumethattheMINAGENTcinstanceisfeasible.Allnodeseitherhostanagent,orareadjacenttoanodehavinganagent,orelseatleastonenodewillremaincontaminatedinonestep.Thepositionsofagentsthusformadominatingset.2 21 ]foragraphG=(V;E),whereweselectacoversetSVsuchthatforeveryedge(i;j)2E,eithernodeiorjisinS.TheVCdecisionproblemseeksacoverSsuchthatjSjq. WetransformGtoMINAGENTsgraphG0=(V0;E0)withthefollowingmodications.FirstsetV0=VandE0=;.Foreverynodei2V,addtwoextranodesi1andi2toV0,andconnectnodeitoi1andi2,byaddingedges(i;i1)and(i1;i2)toE0.Next,foreachoriginaledge(i;j)2E,createanewnodevij.AddnodevijtoV0andaddnewedges(i;vij);(j;vij)toE0.Forconvenience,werefertonodesinVas\originalnodes,"allnodesi1connectedtoi2V0as\1-chainnodes,"allnodesi2connectedtoi12V0as\2-chainnodes,"andallnewnodesvij2V0addedforeachoriginaledge(i;j)2Eas\splitnodes."SeeFigure 3-5 foranexampletransformation. Let=ffig:i2V0g.WeshowthatwecanpositionK=jVj+qagentsonthegraphG0andcleananycontaminationinonesteporlessifandonlyiftheVCinstanceisfeasible. 55

PAGE 56

Avertexcoverinstance(left)transformedtoaMINAGENTsgraph(right). AnyfeasibleMINAGENTssolutionmustpositionanagentonnodei1ori2,foreveryoriginalnodei2V0;otherwise,wewouldnotbeabletocleanacontaminationstartingfromsome2-chainnodei2inonestep.Ifafeasiblesolutionexists,onesolutionplacesexactlyoneagentonnodei1andnoagentsoni2:thereisnobenetinpositioninganagentonbothi1andi2,andplacinganagentoni1wouldbenolesseectivethenplacingitonnodei2. Theremainingqagentsareonoriginalnodesorsplitnodes.Ifsomecontaminationstartsonoriginalnodei,onwhichnoagentisinitiallypositioned,thenanagenton1-chainnodei1canshiftupandcleannodeiinonestep.However,thecontaminationwilltrytospreadtoadjacentsplitnodesvij,foralljadjacenttoiinG.Therefore,foreachedge(i;j)incidenttoiinG,wemusthavepositionedanagentonsplitnodevijand/ororiginalnodej.Ifwehaveanagentpositionedonnodej,theagentcanshifttosplitnodevijandinterceptthespreadingcontaminationinonestep.Andclearlythecontaminationwillnotspreadtonodevijifthereisanagentonvij.ThepositionsofqagentsonoriginalandsplitnodescannowbeusedtoformacoversetfortheVCinstance.Foreachoriginalnodej2V0withanagentonit,addjtoS.Foreachsplitnodevij2V0withanagentonit,addeitheriorjtoS.Thus,foreveryoriginaledge(i;j)2E,eitheriorjbelongstoS. 56

PAGE 57

Avertexcoverinstance(left)transformedtoaMINAGENTbgraph(right). IftheVCinstanceisfeasiblewithcoversetS,thenwecanpositionagentsonalloriginalnodesinthecoverset,andoneagentonevery1-chainnodeinG0,andbythesameargumentasbefore,wecancleananycontaminationinonestep.2 WetransformGtoG0=(V0;E0)exactlyasprescribedintheproofofTheorem 6 ,butwiththeadditionofincludingalledges(i;j)2EinE0aswell.Hence,foreachedge(i;j)2E,wehaveatriangle(i;j;vij)inG0,asillustratedinFigure 3-6 First,supposethatwehaveasolutionStotheVCinstance.WecopytheVCsolutionontotheoriginalnodesinV0,positioningoneagentoneachnodeinS.Wealsopositionanagentonevery1-chainnode.Anycontaminationstartingata2-chainnodeiscleanedinonestep,andthecontaminationcannotstartata1-chainnode. Assumesomecontaminationstartsatanoriginalnodei2Vonwhichnoagentisinitiallypositioned.SinceSisavertexcover,therewillbeanagentateachnodejsuchthat(i;j)2E.ForallnodesjadjacenttonodeiinG,moveagentsfromnodejtosplitnodevijandmoveallagentspositionedon1-chainnodestotheircorrespondingoriginal 57

PAGE 58

Ifacontaminationstartsatasplitnodevij,thensinceSisavertexcover,adjacentnodeiorjwillhaveanagentpositionedonit.Withoutlossofgenerality,supposethereisanagentonnodei.Tocleanthegraph,movetheagentfromnodeitosplitnodevijandmoveagentspositionedon1-chainnodesi1andj1tooriginalnodesiandj,respectively.Thus,graphG0iscleaninonestep. Now,supposethatthereexistsasolutiontotheMINAGENTbproblem.BythesamelogicintheproofofTheorem 6 ,ifMINAGENTbisfeasible,thereisafeasiblesolutionthathasanagentonevery1-chainnode,andnoagentson2-chainnodes.Consideronesuchsolution. Notethateachtriangle(i;j;vij)containsatleastoneagent.(Ifnot,acontaminationthatstartsatsplitnodevijcannotbecleanedinonestepsincevijisonlyadjacenttonodeiandj.)Bymovingallagentsatsplitnodesvijtoeitheroftheiradjacentoriginalnodes,wehavethatforeachedge(i;j)2E,thereexistsanagenteitheratnodeiorj,i.e.,wehaveavertexcoverontheoriginalgraph.SincejVjagentsexistat1-chainnodes,nomorethanqagentsarenowpositionedonoriginalnodes,andhencethepositionsofagentsformavertexcoversetofcardinalitynomorethanq.2 7 .Then,foreachnodei2V,create3-chainnodei3andconnecti2toi3.Addi3toV0and(i2;i3)toE0.WewillshowthatwecanpositionK=2jVj+qagentsonnodesinG0andcleananycontaminationin=ffig:i2V0gintmax=1timesteporless(giventhatthespreadingcontaminationcanpassanagentwhiletraversinganedge)ifandonlyifwecanndavertexcoveronGofcardinalityq. 58

PAGE 59

Assumethatwehaveavertexcover,S,forG.ThenwecanpositionqagentsonthenodesinG0correspondingtoS,inadditionto2jVjagentspositionedonevery1-and2-chainnode.Ifacontaminationstartsonany3-chainnode,thenwecancleanitbymovingtheagentfromi2toi3andpreventi2frombecomingcontaminatedbymovingtheagentfromi1toi2.Foreachoriginalnodeinotinthevertexcover,wewillhaveanagentpositionedonalloriginalnodesjadjacenttoiinG.Ifnodeiiscontaminated,wecanmoveallagentsatadjacentoriginalnodestocoverallsplitnodesvij.Topreventcontaminationofnodeiandi1,wemoveagentsati1andi2toiandi1,respectively.Topreventcontaminationatoriginalnodesjadjacenttoi,wemoveagentsfromj1toj. Now,assumethatwehaveafeasibleMINAGENTpsolution.Considerasolutionhavingagentspositionedonevery1-and2-chainnodeandnoagentsonany3-chainnode.Theremainingagentswillbepositionedonoriginalorsplitnodes.ByasimilarreasoningasintheproofofTheorem 7 ,therewillbeanagentoneverytriangle(i;j;vij).Aftermovinganyagentonasplitnodevijtoeitheriorj,theagentpositionsontheoriginalnodesformavertexcoverontheoriginalgraph.Sincewehave2jVjagentson1-and2-chainnodes,thereareatmostqagentsonoriginalnodes,andhencetheagentpositionsonoriginalnodesformavertexcoverofcardinalitynomorethanq.2

PAGE 60

3.3 ,QcL(x)istheminimumcleaningtime,givenagentpositionsxandinitialcontaminationatnode`2V.TheMINAGENTcproblemcanbeformulatedsuccinctlyas minXi2Vxi (3{5b)xi0andintegral8i2V: Toobtainasinglemixed-integerformulation,wecanincorporatealltheconstraintsnecessarytodescribeQcL(x),forallL2.Theobjectivefunction( 3{5a )seekstominimizethenumberofagentsavailableinthesystem.Constraint( 3{5b )statesthecleaningtime,givenagentpositionsxandcontaminationstartingatnodesL2,mustbecleanedintmaxtimestepsorfewer.Constraint( 3{5c )givesboundandintegralityrestrictionsonx-variables. TheMINAGENTs,MINAGENTb,andMINAGENTpMIPsarethesameas( 3{5 ),exceptthatconstraints( 3{5b )arereplacedwithreplacedwithQsL(x)tmax;8L2,QbL(x)tmax;8L2,orQpL(x)tmax;8L2,respectively. AnalternativeformulationcanbeobtainedfortheMINAGENTcproblemifeachnodecanbecomecontaminatedinatleastonecontaminationsetin,e.g.,=fVg.Observethatthegraphcanbecleanedintmaxorfewertimestepsifandonlyifeverynodeiswithintmaxhopsofsomeagent(i.e.,apathoflengthtmaxexiststoeverynodefromsomenodeonwhichanagentislocated).Thisconditionisnecessary,becauseifsomenodei2Vdoesnotliewithintmaxhopsofanagentposition,itcannotbecleanedintmaxsteps.Theconditionissucientbecauseallnodeswillhaveagentsintmaxstepsiftheconditionholds.DeneNk[i]=fj2V:SP(i;j)kgasthek-hopneighborhoodofnodei,whereSP(i;j)istheshortestpathlength(intermsofedges)betweennodeiandnodej. 60

PAGE 61

minXi2Vxi 1 ,whichwillbeusedtoshowthatthedierentvariantsoftheAGENTPOSproblemareNP-Hard. 61

PAGE 62

5 6 7 ,and 8 ,respectively.Let=ffig:i2V0g,andassignpL=1 1 showsthat,foreachvariant,wecanachievetheminimumexpectedcleaningtimeof 5 { 8 )andcleananycontaminationinonesteporless.Furthermore,asshowninTheorems 5 { 8 ,wecanpositionKagentsonG0andcleananycontaminationinonesteporlessifandonlyifthetransformedNP-Completeprobleminstanceisfeasible.2 3{5 ),theintegralvariablexiisthenumberofagentspositionedonnodei2V.ThefunctionQcL(x)isdescribedinSection 3.3 .TheAGENTPOScMIPis minXL2pLQcL(x) (3{7a)s.t.Xi2Vxi=K (3{7c)xi0andintegral8i2V: Theobjectivefunction( 3{7a )minimizesexpectedcleaningtime.Constraint( 3{7b )statesthatwemustpositionexactlyKagentsonnodesofthegraph.RecallthatiftheCLEANcproblemforpositionsxandcontaminationLisinfeasible,thenformulation( 3{1 )givesanobjectivevalueoftmax+1.Constraint( 3{7c )enforcesthatagentpositionsxaresucient 62

PAGE 63

3{7d )statestheboundsandintegralityrestrictionsonx-variables. TheMIPformulationsoftheAGENTPOSs,AGENTPOSb,andAGENTPOSpareidentical,butsubstituteQsL(x),QbL(x),orQpL(x),respectively,forthetermQcL(x)intheobjectivefunctionandinconstraint( 3{7c ). 3.7 3.10 ,and 3.13 .)Inthefollowingtests,theexecutionofeachMIPwaslimitedto10CPUminutes. Fifty20-nodetestinstancegraphsweregeneratedaccordingtothefollowingprocedure.Foreachpairofnodes(i;j),weaddedge(i;j)withprobabilityd,forsomespecieddensityparameter0
PAGE 64

Foreachtestinstance,thefourvariantsaresolvedinthefollowingorder:cloning(c),stationarycloning(s),blocking(b),andthenpassing(p).AsmentionedinSection 3.3 ,QcL(x)QsL(x)QbL(x)QpL(x),andsoafterwesolveavariantforsomeinstance,weusethebestlowerboundobtainedforthatinstanceasavalidlowerboundinoptimizingthenextvariant. RecallingthatCLEANcistriviallysolvable,wefocusonexaminingthesolvabilityofCLEANs,CLEANb,andCLEANp.Sincethecleaningtimeobjectiveisinteger-valued,weadjusttheCPLEXstoppingcriteriontoterminatewhenevertheupperandlowerbounds(UBandLB)oncleaningtimeislessthan1103=0:999.Ofthe150instances,131/150,88/150,and71/150terminatedwithinoneCPUsecondforCLEANs,CLEANb,andCLEANp,respectively.Also,nostationarycloninginstancerequirednomorethanvesecondstooptimize.However,thesolutionofCLEANbandCLEANpinstancesposedamoredicultchallenge,andseveralinstancescouldnotbesolvedtooptimalitywithin10minutes.Table 3-1 summarizesthecomputationalresultsfortheblockingandpassingvariantsoftheCLEANproblem.Thecolumnlabeled\CleaningTime"presentstheminimumlowerboundandthemaximumupperboundoncleaningtimeforthe25instanceswiththegivenparametersfoundafter600seconds.Thecolumnlabeled\Solved"indicatesthenumberofinstancesthatcouldbesolvedtooptimalitywithinthe10-minutetimelimit.Fortheseinstances,anyfractionallowerboundisroundedup.Relativeoptimalitygap(orsimply\Gap"inTable 3-1 )isapercentagedenedas100(UBLB)=LB.Notethatifaninstancedoesnotsolvein600seconds,wecounttheexecutiontimeas600secondswhencomputingtheaverageCPUtimes.Hence,theaverageCPUtimesreportedareunderestimatesofthetruetimerequiredtosolvetheseinstancestooptimality.Table 3-1 showsthatinstanceshavingalargenumberofagentsareeasiertosolvebecausethecleaningstrategiesemployedonlyrequireafewsteps.If 64

PAGE 65

CLEANMIPgiventheentiregraphiscontaminated. VariantdKSolved%GapCPUSecondsCleaningTimeminavgmaxminavgmax[min,max] thereisasmallnumberofagents,theformulationsseekamorecomplexcleaningstrategythatrequiresmoretimesteps. Weexaminethetightnessofvariousrelaxationsoftheseproblemsbasedonrelaxingintegralityrestrictionsonw-variables.Ifweweretoenforcethebinarinessofthey-variablesinsteadofinsistingonintegralw-variables,thenitispossiblethata(unique)optimalsolutiontoCLEANproblemswillnothavehaveintegralu-,v-,andw-variables.Similarly,ifweholdholdv-variablesbinary,thenbythesameargumentgivenformodel( 3{1 ),y-variableswillbebinaryatoptimality;however,wecouldstillhavethatu-orw-variablesarefractionalinuniqueoptimalCLEANsolutions.AsdescribedinTable 3-1 ,25CLEANbinstancesand32CLEANpinstanceswerenotsolvedtooptimality,andhenceweomitthosecasesinTable 3-2 .Table 3-2 summarizestheempiricallyobservedtightnessoftheserelaxationscomparedtotheoptimalobjectivevaluesfoundinthepreviousexperiment.Lowerboundsareobtainedfromtheoptimalsolutionoftherelaxedproblems.Alllinearprogrammingandbinary-y-variablerelaxationsterminatedinunderoneCPUsecond.Also,allbinary-v-variablerelaxationsforstationarycloningterminatedwithinoneCPUsecond,andforblockingandpassing,97/150and106/150binary-v-variablerelaxationsterminatedwithinoneCPUsecond,respectively.Forbinary-v-variable 65

PAGE 66

TightnessofvariousCLEANMIPrelaxationsgiventheentiregraphisinitiallycontaminated. 10LPrelaxation22/2515/253/2331543binaryy-vars.25/2522/258/230522binaryv-vars.25/2525/2523/23000 15LPrelaxation24/2521/2515/253717binaryy-vars.25/2525/2522/25006binaryv-vars.25/2525/2525/25000 20%5LPrelaxation0/250/20/13558980binaryy-vars.4/250/20/13283520binaryv-vars.24/252/213/13100 10LPrelaxation22/252/250/744477binaryy-vars.25/2510/250/703033binaryv-vars.25/2525/257/7000 15LPrelaxation25/2519/2515/2501924binaryy-vars.25/2525/2525/25000binaryv-vars.25/2525/2525/25000 problems,thelongestpassinginstancetook324seconds,andfourblockinginstancesdidnotterminatewithin10minutes. Next,fortheMINAGENTproblem,wegeneratedtestinstancesgiventimelimitsoftmax=1and3steps.Again,wearesolvingtheoptimizationversionoftheMINAGENTproblemandthusweseekthefewestnumberofagentsrequiredtohavethegraphcleanratherthanjustdeterminingifaxednumberofagentsKissucient.Table 3-3 summarizestheresults.Sincetheseinstanceshaveintegraloptimalobjectivevalues,weagainallowCPLEXtoterminatetheabsoluteoptimalitygapisnomorethan0.999.Thecolumnlabeled\OptimalK"givestheminimumlowerboundandmaximumupperboundfortheminimumnumberofagentsrequiredforthe25instanceswiththegivenparametersfoundafter600seconds.Thecloningandstationarycloningvariantsoftheproblemsallterminatedquickly.Theblockingandpassingversionstooksubstantiallylongertosolve.Only10of25MINAGENTbinstancessolvedtooptimalitywithinthe 66

PAGE 67

MINAGENTMIPgiventheentiregraphiscontaminated. 10%1c25/25000000[5,8]s25/25000000[10,13]b25/25000001[11,13]p25/25000001[14,15] 3c25/25000001[1,3]s25/25000016[5,7]b25/25000375235[6,7]p9/250184381499600[7,10] 20%1c25/25000000[4,6]s25/25000000[10,11]b25/25000012[11,12]p25/250000420[14,14] 3c25/25000001[1,2]s25/250001616[5,6]b10/250225048455600[6,9]p0/25134183600600600[6,11] 10-minutetimelimit,giventmax=3andd=20%.Also,nineMINAGENTpinstancessolvedtooptimalitywithinthetimelimitfortmax=3andd=10%,whilenoinstancesweresolvedtooptimalityforthepassingvariationwhentmax=3andd=20%. Onceagain,settingtmax=3notonlyincreasesthenumberofintegervariablesintheformulation,butalsoincreasesthecomplexityofthecleaningstrategiesfound.ItisthusintuitivethatincreasingtmaxresultsinlongerCPUtimes.Similarly,increasingdincreasesproblemsizeandmakestheseproblemshardertosolve. TotesttheAGENTPOSformulations,wevarythenumberofagentsKavailableto5or10,andweimposeahardtimelimitoftmax=5.Forthecasewhere=fVg,wewillalsohaveanintegralexpectedcleaningtime,andhencewecanagainsetCPLEXtoterminatewhentheabsoluteoptimalitygapfallswithin0.999.Table 3-4 summarizestheresults.Thecolumnlabeled\Exp.CleanTime"givestheminimumlowerboundandmaximumupperboundonexpectedcleaningtimesforthe25instancesforthegivenparametersfoundafter600seconds.Ifthereisatleastoneinstancethatdoesnotidentifyafeasiblesolutionandupperbound,thentheupperboundgivenis1:

PAGE 68

AGENTPOSMIPgiventheentiregraphiscontaminated. 10%5c525/25000001[1,2]]s16/250123315414600[3,4]b3/250{1405591600[3,1]p19/250{1143360600[4,1] 10c525/25000001[1,1]s25/250000414[1,2]b25/250001957[2,2]p0/255054100600600600[2,4] 20%5c525/25000001[1,2]s23/250333867600[3,4]b0/251{1600600600[3,1]p1/250{1432595600[3,1] 10c525/25000001[1,1]s25/25000014[1,2]b23/250122005215600[1,3]p0/2550{1600600600[1,1] WesolvetheoptimizationversionoftheAGENTPOSproblem,andthusweseektheinitialpositionsfortheKagentsaswellasthefastestexpectedcleaningstrategyfortheagents.(Althoughsinceweonlyhaveonecontaminationscenario,theexpectedcleaningtimewilljustbethetimerequiredtocleantheentiregraph,ratherthananaverageofpossiblecontaminationscenarios.)Notethatwealsoincludeconstraint( 3{7c ),whichrequiresthegraphbecleaninatmosttmaxsteps.EventhoughwearesolvingtheoptimizationversionoftheAGENTPOSproblem,includingconstraint( 3{7c )canmakethemathematicalprogrammingformulationinfeasible.Provingthataninstanceisinfeasiblewithin10minutesisconsideredsolvingtheproblem.ForAGENTPOSp,withK=5,andd=10%,19ofthe25instanceswereproventobeinfeasiblewithin10minutes.Theremainingsixinstancesfoundavalidlowerbound,butdidnotndafeasiblesolutionwithcorrespondingupperboundandthushaveaninniteoptimalitygap.Ofthe25passingvariantinstanceswithK=5,d=20%,oneinstancewasproventobeinfeasiblein10minutes. 68

PAGE 69

CLEANgivencontaminationbeginsonasinglenode. VariantdKSolved%GapCPUSecondsCleaningTimeminavgmaxminavgmax[min,max] 3-5 summarizestheresultsfromsolvingtheoptimizationMIPformulationsoftheinstancesfortheblockingandpassingvariants. ComparingtheresultstheCLEANinstancesfromtheprevioussectioninwhichtheentiregraphwasinitiallycontaminated,itappearsthatinstanceswithonenodeinitiallycontaminatedsolvemuchquicker.Thecasesthatposedthemostdicultywereinstanceshavinghighedgedensityandlownumberofagents.Inthesecases,thecontaminationcanspreadfasterthantheagentscanmovetocleanit,andamoresophisticatedcleaningstrategymustbederived. WesolvedthevariousCLEANrelaxations(linearprogramming,binaryy-variables,andbinaryv-variables)againtotestthetightnessoflowerboundsthattheyprovided.Alllinearprogrammingandbinary-y-variablerelaxationsterminatedwithinonesecond.Forthebinary-v-variablerelaxations,allstationarycloninginstancesterminatedwithin 69

PAGE 70

TightnessofvariousCLEANMIPrelaxationsgivencontaminationbeginsonasinglenode. 20%5LPrelaxation18/2512/227/1592434binaryy-vars.25/2519/2212/15066binaryv-vars.25/2522/2215/1500010LPrelaxation24/2524/2520/25119binaryy-vars.25/2525/2525/25000binaryv-vars.25/2525/2525/2500015LPrelaxation25/2525/2525/25000binaryy-vars.25/2525/2525/25000binaryv-vars.25/2525/2525/25000 oneCPUsecond.Fortheblockingandpassingvariants,143/150and138/150instancesterminatedwithinoneCPUsecond,respectively,andthelongestinstancestook257and210seconds,respectively.ThetightnessoftheserelaxationsissummarizedinTable 3-6 Giventhecontaminationcanstartonanysinglenode,weneedtoconsiderjVjcontaminationsthatmustbecleanedintmaxstepsorfewerwhenwearedeterminingtheminimumnumberofagentsrequiredtocleanthegraphandtheirpositions.WetestedtheMINAGENTformulationsanalogouslywithtmax=1ortmax=3.TheresultsarepresentedinTable 3-7 .Comparedtothecasewhenweonlyhaveonecontaminationscenariotoaddress(theentiregraphiscontaminated),thecasewherethecontaminationcanoriginateonanysinglenodeappearstobemarginallymorecomputationallyintensive.Notably,thestationarycloning,blockingandpassingvariantsonlysolvedfourinstanceseachtooptimalitywithd=20%andtmax=3. 70

PAGE 71

MINAGENTgiventhecontaminationbeginsonasinglenode. 10%1c25/25000001[5,8]s25/25000001[6,8]b25/25000001[7,9]p25/25000001[7,11] 3c25/25000013[1,3]s24/2501333164600[3,4]b22/2504330157600[3,4]p17/2508254349600[3,6] 20%1c25/25000001[4,6]s25/25000013[5,7]b25/25000017[6,7]p25/2500005100[6,8] 3c25/25000001[1,2]s4/25063246539600[2,6]b4/2502767381593600[3,6]p4/2502275136560600[4,7] WetestedtheAGENTPOSformulationsgiventhatanysinglenodecanbecomecontaminated.Wesettheprobabilitythatthecontaminationstartsonsomenodeas1 3-8 summarizestheresults.Thesituationwherecontaminationcanoriginateonanysinglenodeappearstobecomputationallyeasier.Thisisduetotheeaseinwhichcleaningstrategiessingle-nodecontaminationsarederived.Thecauseofdicultyinalloftheseproblemsappearstobederivingsophisticatedcleaningstrategies. Again,notallAGENTPOSinstancesarefeasible.Withd=10%;K=5,twoAGENTPOSpinstancesdidnotndafeasiblesolutioninthetimelimit.Withd=20%;K=5,veAGENTPOSbinstancesand20AGENTPOSpinstancesfailedtoidentify 71

PAGE 72

AGENTPOSMIPgiventheentiregraphiscontaminated. 10%5c525/25000029[0.75,1.1]s25/25000318100[0.85,1.15]b25/25000247226[0.85,1.35]p19/250{121264600[1.05,1] 20%5c525/25000013[0.75,0.85]s25/250005113242[0.75,0.95]b11/250{137416600[0.77,1]p1/250{1490597600[0.81,1] 5c1025/25000001[0.5,0.5]s25/250000311[0.5,0.5]b25/250000324[0.5,0.5]p25/250000318[0.5,0.5] afeasiblesolutioninthetimelimit.Givenmoreagents,theproblemsndfeasible(andoptimal)solutionsmuchquicker. Procedure 3 describesaconstructionheuristicforndingsolutionstotheCLEANproblem.Procedure 3 computesSstrategiesandreturnsastrategyhavingthefastestcleaningtime.Thealgorithmconstructsacleaningstrategystep-by-step,ateacht=0;:::;tmax1.Givenagentpositionsxtiforalli2V,werandomlygenerateMcandidatemovesets.Wethenscorethequalityofeachcandidatemovesetaccordingtooneofthreemetrics,wherealowermetricvalueisbetter.Letlmbethevalueofmovesetm,andlet 72

PAGE 73

73

PAGE 74

NotethateachmovesetcanbecreatedbyrstcomputingjN[i]j(orjN(i)jforthestationarycloningcase),foralli2V.Foreachagentatnodei2V,wethencomputearandomintegermodulojN[i]j,andmovetheagentalongthe+1stedgeincidenttonodei.ThisprocessrequiresO(K)operations. Tocomputethemetricvalueforeachmoveset,weneedtodeterminewhichnodeswillbecontaminatedattimet+1,giventhesetofnodescontaminatedattimet.Assumingthatwehavealreadymarkedtheedgesonwhichagentsaretraveling,thiswillrequireO(jEj)operations. Ifweusethecountingmetrictoevaluatethevalueofamoveset,aftercomputingthemovesetanddeterminingwhichnodesarecontaminatedattimet+1,evaluatingthemetricvalueofthemovesetsimplyrequiresO(jEj)operations.Thus,thecomplexityoftheentireheuristicisO(SMtmaxjEj). IfweusetheLP-relaxationmetric,thenthecomplexityisO(SMtmaxjEjLjVj;jEj;tmax),whereLjVj;jEj;tmaxisthecomplexityofsolvingalinearprogramwithjVjnodes,jEjedges,andtimelimittmax.NotethatwecansolvetheCLEANbinary-yrelaxationbyincrementingandxingy-variablesforeachcleaningtimeandattemptingtondfeasiblevaluesforothercontinuousvariables.Forexample,onecanxy0==yt0=1andyt0+1==ytmax=0,andattempttondafeasiblesolutiontotheresultingLP.Ifitisfeasible,weknowthatthebinary-yrelaxationobjectiveisnomorethant0.PerformingabinarysearchtondtheoptimalobjectivevaluerequiresO(log(tmax))operations.Hence,ifweusethebinary-yrelaxationmetric,thecomplexityoftheheuristic 74

PAGE 75

Wealsoaddadditionalstoppingcriteriabasedontheboundsgleanedfromourrelaxations.Beforestartingtheheuristic,wecanobtainalowerboundontheminimumcleaningtimebysolvingthebinary-vCLEANrelaxation.Thenumberofoperationsrequiredtondasolutiontothebinary-vrelaxationisnotpolynomiallybounded,butempiricallyspeaking,thebinary-vrelaxationsolvesquicklyinmostcases.Tokeepthecomplexityoftheentireheuristicprocedurepolynomial,thenumberofnodesinthebranch-and-boundtreeofthebinary-vrelaxationcanbeboundedbysomepolynomialfunctionofthenumberofnodes,orwecouldsimplyrestricttheCPUtimeallocatedtosolvetheproblem.Attheendofthisprocess,ourLBistheceilingofthelowerboundontheoptimalobjectivefunctionvalueforthebinary-vrelaxation. Initiallywestartwithanupperbound(UB)oftmax+1.IfanyoftheSstrategiesfoundbytheheuristichaveafastercleaningtime,thenweupdateUBaccordingly.IfoneoftheSstrategiesconsideredhasacleaningtimethatmatchesLB,thenwecanstoptheprocedureandconcludethatwehavefoundanoptimalcleaningstrategy.Also,ifweareusingametricbasedonsomerelaxationoftheCLEANMIP,thenwhenweareexaminingacandidatemovesetattimet,weonlyconsideramovesetmwithmetriclmsuchthatt+lmUB1.Ifnoneofthemovesetsgeneratedsatisfythiscondition,wecanabandonthecurrentstrategyunderconstructionsinceitwillnothaveafastercleaningtimethanUB. WeperformcomputationaltestingonthesamegraphsandCLEANinstancestestedinSection 3.6 75

PAGE 76

CLEANheuristicwithvariousmetrics. binary-vSkipped55267CPU043 CountingmetricSolved205180155CPU043 LPmetricSolved206179153CPU43026 Binary-ymetricSolved204177161CPU74041 3-9 presentstheresultsfromtheheuristicwithvariousmetricsusedaswellastheresultsfromtestingthefullMIPformulationsinSection 3.6 .Thesectionofthetablelabeled\MIP"givesthenumberofinstancessolved(outof300)andtheaverageCPUtimes(roundedtothenearestseconds)foreachvariant.TheaverageCPUtimesforMIPsweretruncatedsincea10-minutetimelimitwasimposed.Thesectionofthetablelabeled\binary-v"givestheresultsfromrunningthebinary-vrelaxationtogetalowerboundoncleaningtimebeforetheheuristicisstarted.Therowlabeled\Skipped"showsthenumberofinstancesthatwerefoundtobehavealowerboundofmorethantmax.Thenextsectionsofthetablegivethenumberofinstancessolved(whichincludesthoseskippedduetoaLB>tmaxsuggestedbytheCLEANbinary-vrelaxationandtheaverageCPUtimefortheheuristic(whichincludesthetimespentsolvingtheCLEANbinary-vrelaxationtogetalowerbound). FirstnotethatthestationarycloninginstancessolvedviatheMIPformulationallterminatedwithinoneCPUsecond,andthustheheuristicisunnecessaryforthisvariant.Fortheblockingandpassingvariants,theheuristicprovidesafast,eectiveboundingmoduleforMIPsolutions. 76

PAGE 77

CLEANheuristicwithvariousconstructionparameters. binary-vskipped55267CPU043 RandomStrategysolved128131109 Theheuristicprocedurewasabletosolvemorethanhalfofallinstancestooptimalitywitheverymetric.TheperformanceoftheconstructionproceduredidnotseemtobesignicantlyimprovedusingtheLP-relaxationorbinary-y-relaxationmetricoverthesimplisticandcomputationallycheapcountingmetric.Ineveryinstance,constructingO(S)cleaningstrategieswiththecountingmetriccontributedlessthanonesecondtothetotalrunningtimeoftheheuristic.BasedonthenumberofinstancessolvedandthenominalCPUtime,weproceedtotunetheheuristicusingonlythecountingmetric. 3-10 summarizestheresults.NotethatCPUtimeisomittedforthestrategyconstructionprocedureforeachparametersetbecauseitisdeterministicallyboundedandempiricallyobservedtobelessthanoneCPUsecondforallparameterchoices.Theresultsshowthatincreasingthenumberofmovesetsevaluatedgivesanimprovementinthenumberofinstancessolved.Varyingthe-parameterfrom0to0:2doesnothaveanimpactontheeectivenessoftheconstructionprocedure.Thisis 77

PAGE 78

Tables 3-11 and 3-12 summarizetheperformanceofthetunedheuristic(withS=50,M=100,thecountingmetric,and=0)fortheinstanceswheretheentiregraphisinitiallycontaminatedandwhenthecontaminationstartsonasinglenode,respectively.Thecolumnslabeled\Gap"givetheaveragerelativeoptimalitywhichisdenedas(UBLB)/LB100.Thecolumnslabeled\CPU"givetheaverageCPUtimeroundedtothenearestsecond(includingsolvingthebinary-vrelaxationtogetaLB).Notsurprisingly,theblockingandpassinginstanceswheretheentiregraphiscontaminatedprovedextremelychallengingtosolve.TheseresultsareconsistentwithsolvabilityoftheCLEANMIPproblemsbyintegerprogramming.Overall,theheuristicprocedureiscapableofsolvingalmost2/3oftheCLEANinstancesinsubstantiallylesstimethandirectlysolvingthefullMIPformulationswouldtake. 3.9 ,weoutlinethedierenttypesofcuttingplanesimplementedinthealgorithm.InSection 3.10 ,weprovidecomputationaltuningofthealgorithmandadetailedsummaryoftheperformanceofthetunedcutting-planealgorithm. Wewillalsoutilizeabinary-vrelaxationof( 3{5 ).SimilartotheCLEANbinary-vrelaxations,weenforcethatv-variablesarebinaryandrelaxagent-oww-variables.Wecontinuetoenforcethatagent-deploymentx-variablesareintegralandpositive.Theoptimalobjectivevalueofthebinary-vrelaxationwillgivealowerboundKLBontheminimumnumberofagentsrequiredtocleanthegraph.Beforethecutting-planealgorithmisstarted,wesolvethebinary-vrelaxationof( 3{5 )andtestthefeasibility 78

PAGE 79

TunedCLEANheuristicforinstanceswheretheentiregraphiscontaminated. HeuristicMIPVariantKdSkippedSolved%GapCPUSolved%GapCPU Table3-12. TunedCLEANheuristicforinstanceswherethecontaminationstartsonasinglenode. HeuristicMIPVariantKdSkippedSolved%GapCPUSolved%GapCPU 79

PAGE 80

Thecutting-planealgorithmapproachseparatesthedecisionsintorst-andsecond-stageproblems.Intherst-stageproblem,wedeterminethenumberofagentsandwheretheyaredeployedonthegraph.Inthesecond-stageproblem,weverifythateachpossiblecontaminationincanbecleanedintmaxsteps. Therst-stageproblemisformulatedasarelaxationofthefullMINAGENTMIP( 3{5 ), MINAGENT.Weseparatethex-variablesintoauxiliarybinaryvariablesik,whereik=1iftherearekKagentspositionedonvertexi,andzerootherwise.The MINAGENTMIPformulationis minXi2Vxi Constraint( 3{8b )imposesalowerboundonthetotalnumberofagentsdeployed.ThelowerboundKLBisinitiallyseededbythebinary-vrelaxationof( 3{5 ),andasthecutting-planealgorithmprogresses,weupdateKLBtobetheoptimalobjectivefunctionfoundatthepreviousiterationofthealgorithm.The MINAGENTformulationdoesnotincludeconstraint( 3{5b ).Inordertocapturetheserestrictions,weincludeconstraints 80

PAGE 81

3{8f ),whichrepresentasetofvalidinequalitiesthathavebeenaddedtotherelaxationateachiterationofthecutting-planealgorithm.Thepurposeoftheseconstraintsistoeliminateagentdeploymentpositionsthatcannotcleanallcontaminationswithintmaxstepsfromthesolutionspace.WediscussthederivationoftheseinequalitiesinSections 3.9 .Constraint( 3{8c )givestherelationshipbetween-andx-variables.Constraint( 3{8d )forcesustochoosesomenumberofagentsfornodei.Constraints( 3{8e )stateintegralityrestrictionsonthe-variables.(Notethatx-variablesarethereforeconstrainedtobenonnegativeintegersdueto( 3{8c ).) Ourcutting-planealgorithmiteratesasfollows.First,wesolve MINAGENTandgetanoptimalsolution^x.Next,wesolvetheLPrelaxationoftheCLEANproblemwithagentpositions^xforeachcontaminationL2.ForeachCLEANLPrelaxationthathasanobjectivevaluegreaterthantmax,weaddacuttingplaneto.IfnocuttingplanesaresuggestedbytheCLEANLPrelaxations,webeginsolvingtheCLEANMIPrecourseproblemsforeachcontaminationL2.(NotethatweonlyseekfeasiblesolutionstoCLEANMIPproblems.)IfoneCLEANMIPcannotndafeasiblesolutionthatterminatesintmaxstepsorfewer,weaddanothercuttingplaneto,anddiscontinuesolvingtheremainingCLEANMIPproblems.Ifnocutsareadded,wecanconcludethatthenumberofagentsdeployedistheminimumnumberofagentsrequiredtocleanallcontaminationsL2intmaxstepsorfewer.Ifoneormorecutsareadded,wewillrepeattheprocess.NotethatweavoidsolvingtheCLEANMIPrecourseproblemsuntilwehavegleanedasmuchinformationaspossiblefromtheCLEANLPrelaxations. 7 ]to,asdescribedbelow. First,weassociatedualvariable`0,forall`2Lwithconstraint( 3{1b ).Dualvariablesiandi,i2Vareassociatedwith( 3{1c )and( 3{2b ),respectively,andnote 81

PAGE 82

3{1f ).Dualvariabletji0,fort=1;:::;tmax,(i;j)2E,isassociatedwithspreadingconstraint( 3{1e ),( 3{3d )and( 3{3e ),or( 3{4b ),dependingonthevariant.Aspreadingconstraintmodelsthespreadofcontaminationsfromnodejtoibytimet.Weassociateti,fort=1;:::;tmax1,i2V,withagent-owbalanceconstraint( 3{2c ).Andnally,forthestationarycloningvariantonly,weassociatedualvariablesti,fort=1;:::;tmax,i2V,withconstraint( 3{2d ).Restrictionsonvariablesaresummarizedasfollows. (3{9a)`08`2L (3{9g)tiunrestricted8i2V;t=1;:::;tmax: 82

PAGE 83

maxX`2L(1x`)`+Xi2Vxii+Xi2Vxii 3{9a )( 3{9h )Xi2Vti=18t=0;:::;tmax1 (3{10b)+Xi2Vtmaxi=1 (3{10c)`Xj2N[`]1`j0`08`2L (3{10i)tmaxi+Xj2N[i]tmaxji=08i2V (3{10l)tmaxj+tmax1i08(i;j)2EnE0: Constraint( 3{10b )isassociatedwithy-variablesexceptforytmax,whichisrelatedtodualconstraint( 3{10c ).Constraint( 3{10d )isassociatedwithprimalvariablesv0`for`2Landconstraint( 3{10e )isassociatedwiththeremainingv0-variablesfornodesi2VnL.Constraint( 3{10f )isassociatedwithvt-variables,fort=1;:::;tmax1,andconstraint 83

PAGE 84

3{10g )isassociatedwithvtmax-variables.Constraints( 3{10k ),( 3{10l ),and( 3{10m )areassociatedwithw0-,wt,(fort=1;:::;tmax2),andwtmax1variables,respectively. ThedualoftheCLEANbLPrelaxationis maxX`2L(1x`)`+Xi2Vxii 3{10b )( 3{10g );( 3{9a );( 3{9b );( 3{9d )( 3{9g )i+Xh2N[i]1hi1i08(i;i)2E0 (3{11d)Xh2N[i]t+1hj+t+1ji+tit+1j08(i;j)2EnE0;t=1;:::;tmax2 (3{11e)Xh2N[i]tmaxhi+tmax1i08(i;i)2E0 Dualconstraints( 3{11b ),( 3{11d ),and( 3{11f )areassociatedwithself-loopw-variableswithtimeindext=0,t=1;:::;tmax2,andt=tmax,respectively.Dualconstraints( 3{11c ),( 3{11e ),and( 3{11g )areassociatedwithwij-variablesfortheremainingedges(i;j)2EnE0,withtimeindicest=0,t=1;:::;tmax2,andt=tmax1,respectively. 84

PAGE 85

maxX`2L(1x`)`+Xi2Vxii 3{10b )( 3{10g );( 3{9a );( 3{9b );( 3{9d )( 3{9g )i+Xh2N[i]1hj1j08(i;j)2E (3{12c)Xh2N[i]tmaxhj+tmax1i08(i;j)2E: Dualconstraints( 3{12b ),( 3{12c ),and( 3{12d )areassociatedwithw-variables,withtimeindext=0,t=1;:::;tmax2,andt=tmax1,respectively. Ifthedualisunboundedwithextremeraycomponents^,^,and^,thetraditionalBendersfeasibilitycutforthestationarycloningvariantis 0X`2L^`(1x`)+Xi2V(^i+^i)xi:(3{13) Dene^i=0fori2VnL.Notethati0foralli2V.Also,sincetheprimalconstraintsassociatedwithiandicanequivalentlybewrittenasu0ixiandPj2N(i)w0ijxiforalli2V,all-and-variableswillbenonpositive.Thus(^i^i^i)0.Thereforecut( 3{13 )canbestrengthenedto Next,observethatthe-variablesdonotappearinthedualsoftheblockingandpassingformulations.TheBendersfeasibilitycutforthesetwovariantsisofthesameformas( 3{14 ),where-variablesareassociatedwith( 3{3b )insteadof( 3{2b ),andwhere 85

PAGE 86

3{14 )forthiscasewith^i=0foralli2V. SincetheBendersfeasibilitycuts( 3{14 )arederivedfromtheLPrelaxationsofCLEANsubproblems,theyaresucienttoguaranteethatQcL(x)tmax,sinceCLEANcisanLP.However,theyarenotsucienttoguaranteethattheotherCLEANvariants,whichcontainintegervariables,arefeasible.Forthesevariants,weincorporateadditionalcuttingplanes. Forthestationarycloning,blocking,andpassingvariants,ifwecannotobtainanycutsoftheform( 3{14 ),webeginsolvingtheCLEANMIPrecourseproblemforeachcontaminationL2.IfsomeCLEANMIPisinfeasiblegiven MINAGENTsolution^,weaddacombinatorialfeasibilitycuttingplaneto MINAGENToftheform: anddiscontinuesolvinganyremainingCLEANMIPrecourseproblems.Constraint( 3{15 )statesthatthenumberofagentsonsomenodeimustbeincreased.Toseethat( 3{15 )isvalid,notethatifx0^x,thenQvL(x0)QvL(^x),foranyvariantv.SinceQvL(^x)>tmax,anyfeasiblesolutionxmusthavexi>^xiforsomei,whichisexactlytherestrictionstatedby( 3{15 ).(See[ 12 38 39 ]forrelatedstrategiesoncombinatorialinequalitiessuchas( 3{15 ).) Ifnode`willbecomecontaminatedinatleastonecontaminationscenario,thenitisnecessarytohaveatleastoneagentpositionednomorethantmaxstepsawayfromnode`.Formally,letN1[`]=N[`],andletNh[`]=fi2V:(i;j)2Eforsomej2Nh1[`]gforall 86

PAGE 87

3.6 .Again,weconsidertwopossiblecontaminationsituations:=fVg,wherethereisonlyonepossiblecontamination,inwhichtheentiregraphisinitiallycontaminated,or=ffig:i2Vg,wherethecontaminationcanstartonanysinglenode.SincetheMINAGENTcformulationcanbedirectlysolvedwithinoneCPUsecondforallinstances,wefocusonsolvingthestationarycloning,blocking,andpassingvariations. Werstdiscussthetuningofthecutting-planealgorithmwithasubsetofinstances.Weselecteightinstances(oneforeachparameterset)foreachvariant.Thebinary-vlower-boundingprocedureisnotimplementedinthistuningandKLBisstartedatone. Preliminarytestingsuggeststhatwarmingconstraintsoftheform( 3{16 )donotimprovethenumberofiterationsrequiredtondfeasiblesolutions.Thisisduetotherandomnessofthegraphsweconsider,whicharetypicallynotplanarorgrid-like.Givenlarger,sparser,ormoreorganizedgraphs,thesewarmingconstraintsmayprovemoreuseful.Warmingconstraints( 3{16 )areomittedinallfurtherexperiments. Next,wecomparesolvingtheCLEANMIPrecourseproblemateachiterationregardlessofwhetherornotBendersfeasibilitycuts( 3{14 )areaddedto MINAGENT,orwaitingtosolvetheCLEANMIPrecourseproblemsuntiltheCLEANLPrelaxationsdonotproduceanycuts.FortheimplementationinwhichwesolveCLEANMIPsonlyif( 3{14 )failstocuto^x,weperformanaverageof70extraiterationsbeforethe10minutetimelimitisreached,and17/24MINAGENTinstancesaresolvedcomparedto16/24MINAGENTinstances.TheCLEANLPrelaxationssolveveryquicklyandweareabletogeneratemorecutsfaster,thusrapidlyrestrictingthe MINAGENTsolutionspace.Thus 87

PAGE 88

ReviewofMINAGENTextensiveformulationtesting. inallfurthertesting,wewaittosolveCLEANMIPsuntilwecannolongergenerateanycutsfromCLEANLPs. Finally,wetestthesolutiontimeof MINAGENTwithandwithoutexplicitlyenforcingintegralityonx-variables.The MINAGENTrequiresonaverage0.1secondslesstosolvewithcontinuousx-variables,andsox-variablesarerelaxedinallfurthertesting. WhenrunningtheMINAGENTdecompositionprocedure,wesolvethebinary-vrelaxationof( 3{5 )and( 3{8 )toanabsoluteoptimalitygapof0:999.WhensolvingaCLEANrecourseproblem,werstemploythecleaningstrategyconstructionheuristic(butdonotsolvetheCLEANbinary-vrelaxationtogetalowerbound)andifafeasiblestrategyisfound,weskipsolvingtheCLEANMIP.SolvingtheMINAGENTbinary-vrelaxationisgivenaoneCPU-minutetimelimitandtheentireprocedureisgivena10CPU-minutetimelimit. Table 3-13 reviewsthepreviouscomputationaltestingonthefullMINAGENTMIPformulation.Therowslabeled\ExtensiveMIPsolved"givethenumberofinstancesoutof100thatweresolvedtooptimality.Therowslabeled\ExtensiveMIPCPU"givetheaverageCPUtimeinsecondsforthe100instancesdescribed.Whentmax=1,theextensiveformulationcanbesolvedquickly.Whentmax=3,morecomplicatedcleaningstrategiesmustbeimplementedandtheMINAGENTMIPislargerandmorechallengingtosolve. Table 3-14 summarizestheperformanceoftheMINAGENTdecompositionsolutionprocedure.Thetablepresentsthenumberofinstancessolvedandthetruncatedaverage 88

PAGE 89

MINAGENTdecompositionprocedure. MIPBin.-vC.-P.Alg.OverallvtmaxdSolvedCPUSolvedCPUSolvedCPUCPU Total178/20089166/2001415/3425771 Total161/200160156/2001833/4419666 Total130/200252147/2002626/53363112 CPUtimefortheAGENTPOSbinary-vrelaxationboundingprocedure(\Bin.-v")andthecutting-plane(\C.-P.Alg."),respectively,aswellastheoverallCPUtime. Comparedtothedirectsolutionof( 3{5 ),whichsolved486/600instances,thedecompositionproceduresolved543/600instancesinhalfthetime.Asbefore,allinstanceswithtmax=1solvedquickly.Comparingtheinstanceswithtmax=3,wehave81,89,and73(outof100)instancessolvedbythedecompositionprocedureforthestationarycloning,blockingandpassingvariants,respectively.Thisisespeciallysignicantforthepassingvariantinstances,ofwhichtheMIPwasonlyabletosolve30/100in 89

PAGE 90

MINAGENTcutting-planealgorithm. 3{14 )( 3{15 ) tenminutes.Ofthe25,393CLEANrecourseproblems,theheuristicreturnedafeasiblesolutiontoapproximately58%. Table 3-15 presentsaveragestatisticsforthoseinstancesinwhichthecutting-planealgorithmwasimplemented.Thetableillustratestheaveragenumberofiterations,Bendersfeasibilitycuttingplanes( 3{14 )added,andcombinatorialfeasibilitycuttingplanes( 3{15 )added. Thecutting-planeprocedurewasespeciallyeectivefortheblockingandpassingvariantswithcontaminationset=ffig:i2Vg,solving30/31and26/37instances,respectively.AsshowninSection 3.6 ,theCLEANLPrelaxationsproduceanoptimalobjectivevaluethatisequaltotheCLEANMIPoptimalobjectivevaluefarmore 90

PAGE 91

3{14 ). 3.12 .WethenprovidecomputationaltestingofthedecompositionmethodinSection 3.13 SimilartothepreviousdiscussionontheMINAGENTdecompositionmethod,wecansolveabinary-vrelaxationof( 3{7 ),wherev-variablesareheldbinary,andagent-deploymentx-variablesareheldintegralandallothervariablesarerelaxed.AnoptimalsolutiontotheAGENTPOSbinary-vrelaxationwillprescribeagentpositionsxandalowerboundontheoptimalobjectivevalueof( 3{7 ).Givensuggestedagentpositionsx,wesolveeachindividualCLEANMIPrecourseproblemtooptimalityandcomputetheactualexpectedcleaningtime.Iftheactualexpectedtimematchesthelowerboundsuggestedbythebinary-vrelaxation,weconcludewehavefoundtheoptimalagentdeploymentpositions.Otherwise,weinitiatethecutting-planealgorithm. TheAGENTPOScutting-planealgorithmagainseparatesthedecisionsintorst-andsecond-stageproblems.Intherst-stageproblem,wedeterminethepositionsofagentsandestimatecleaningtimesforeachcontaminationL2.Inthesecond-stageproblem,weverifythattheestimatedcleaningtimesareaccurateandthatallcontaminationscanbecleanedintmaxstepsorfewer. Therst-stageproblem,whichwerefertoas AGENTPOS,isformulatedasarelaxationofthefullAGENTPOSMIP( 3{7 ).WedenezLLLastherst-stageestimateofcleaningtimeforacontaminationthatbeginsonnodesinL2,withlower 91

PAGE 92

3.8 ,wealsodecomposethex-variablesintoauxiliarybinary-variables.The AGENTPOSMIPformulationis AGENTPOS:minXL2pLzL 3{8c )( 3{8e );( 3{7b )XL2LzL+Xi2VKXk=0ikik82 (3{17b)zLLL8L2: Theobjectivefunction( 3{17a )seekstominimize(estimated)expectedcleaningtime.Constraint( 3{17b )representsasetofvalidinequalitiesthathavebeenaddedtotherelaxationateachiterationofthecutting-planealgorithmtoforceestimatesofcleaningtimezLtoconvergetoactualcleaningtimevalues.Finally,( 3{17c )enforceslowerboundsoncleaningtimegivenaninitialcontaminationatnodesinL.(WediscusssimplestrategiesforcomputingLLforourtestinstancesinSection( 3.13 ).)Likethe MINAGENTformulation,the AGENTPOSformulationdoesnotincludeconstraint( 3{5b ),andwemustguaranteethatzLtmaxviaasetofcuttingplanes. Thecutting-planealgorithmrstsolves AGENTPOS,getsanoptimalsolutionoftheform(^x;^z),andthensetsLBequaltotheobjectiveof AGENTPOS.Notethattheoptimalobjectiveof AGENTPOSisnondecreasingandalwaysequalstheLB.IfUBLB,forsomeabsoluteterminationconstant>0,weterminate.Otherwise,werstsolvetheLPrelaxationoftheCLEANproblemforeverypossiblecontamination.IfoneCLEANLPrelaxationinfeasible,wewilladdaBendersfeasibilitycut( 3{14 )to.IftheCLEANLPrelaxationisfeasible,denetheoptimalobjectivevalueas 3.12 ).IfeachCLEANLPrelaxationisfeasible,and^zL 3{15 )toanddiscontinuesolvinganyremaining 92

PAGE 93

3.12 ).IfallCLEANMIPproblemsarefeasible,wecancomputeactualexpectedcleaningtimegivenpositions^xandupdateUB.Again,ifUBLB,weterminate.Ifatanypoint AGENTPOSisinfeasible,thenwecanconcludethattheAGENTPOSinstanceisinfeasible. 7 ]canbederivedfromthefulllinearprogrammingrelaxationofthevariousCLEANproblems.Givenanoptimalsolutiontothe AGENTPOSproblem(^x;^z),iftheLPrelaxationoftheCLEANproblemisfeasiblewithoptimaldualsolutioncomponents^,^,and^,but AGENTPOS.TheBendersoptimalitycutforthestationarycloningvariantisgivenby where^i=0fori2VnL. WecanmodifytheBendersoptimaltycutto ThemodiedBendersoptimalitycutstatesthatifthenumberofagentsonsomenodeiisincreasedfromktok0>k,cleaningtimewilldecreasebytheamountsuggestedbyeitherthetraditionalBendersoptimalitycut( 3{18 )orthemaximumpossibledecreaseoncleaningtime(whicheverisless). Ingeneral,inequalities( 3{18 )or( 3{19 )donotdominateoneanother.Consideringnodei2Vwithkagentspositionedonit,( 3{18 )wouldimplythatifthenumberofagentsonnodeiwerereducedthenestimatedcleaningtimemayincrease.Constraint( 3{19 )wouldnotimposeanysuchpenaltyonz-variablesfordecreasingx-variables.However,constraint( 3{19 )willprovideamoreaccurateestimateofhowcleaningtimeis 93

PAGE 94

TheBendersoptimalityconstraintfortheblockingorpassingvariantisthesame,but^i=0foralli2V.TheBendersoptimalitycutforthecloningvariantisalsothesame,but^i=0foralli2V. IfallCLEANLPrelaxationsarefeasibleandzL AGENTPOSforallLsuchthatQL(^x)>^zL.WeutilizeamodiedversionofthecombinatorialoptimalitycutssuggestedbyLaporteandLouveaux[ 26 ].IfQL(^x)>^zL,wecanaddaconstraintto AGENTPOSoftheform: Acombinatorialoptimalitycutstatesthatifagentpositionsarekeptasis,theestimatedcleaningtimezLforacontaminationthatstartsonnodesinLisforcedtobeatleastQL(^x).However,ifagentpositionsarechangedinanyway,thenthezL-variableisreleasedandallowedtoachieveitslowerboundofLL.ThisconstraintissucienttoforceallvariantsoftheAGENTPOScutting-planealgorithmtoconverge. Althoughtheseintegeroptimalitycutsaresucient,theyareweakinthesensethatswitchinganyagent'spositionallowszL=LLwithrespecttothemostrecentlygeneratedconstraint( 3{20 ).Ifonlythesecombinatorialoptimalitycutsareimplementedinthecutting-planealgorithm,thenthealgorithmmayenumerateallpossiblen+K1Kchoicesforagentpositionsbeforesettlingonanoptimalsetofpositions. AGENTPOSbeforethecutting-planealgorithmisstarted.Notethatifnoagentispositionedonnode`2L,thencleaningtimeforcontaminationLwillbeatleastone.Denebinaryvariable!`=1ifnoagentsarepositionedonanynodesadjacentto`,i.e.,xh=0forallh2N[`],else!`=0.Ifnoagentsarepositionedon`oranynodesadjacentto`,thenitwilltakeatleasttwosteps 94

PAGE 95

(3{21a)!`1Xh2N[`]x`8`2L Constraint( 3{21a )statesthatunlesswepositionagentoneverynode`2L,thencleaningtimeforcontaminationLwillbezero.Ifonenode`isnotcovered,thencleaningtimewillbeatleastone,andifnoagentisadjacenttonode`,thenconstraint( 3{21b )willforce!`1andcontributeatleastonemoretimeunittocleaningtime.Constraint( 3{21c )statesthat!-variablesarenonnegative.Optimalitywillforcez-variablesaslowaspossible,andthus!-variableswillbenomorethanone,andzerowheneverpossible. 3.6 .Weconsiderthesametwopossiblecontaminationsituations:=fVgand=ffig:i2Vg.WeomittheAGENTPOScvariantasallinstancesoftheextensiveformulationcanbesolveddirectlywithin10seconds. Werstdiscussthetuningofthecutting-planealgorithmwithasubsetofinstances(oneinstanceforeachparametersetforatotalofeight,testedforeachvariantforatotalof24).Thebinary-vlowerboundingprocedureisomittedinthefollowingdiscussion. First,notethatcleaningtimesarealwaysintegral,howeverwedonotnecessarilyrestrictcleaning-time-estimatez-variablestobeintegers.Ifz-variablesareheldintegral,thentwoextrainstanceswheretheentiregraphisinitiallycontaminatedcanbesolvedintenminutes.(All=ffig:i2Vginstancesaresolvedtooptimalityquicklyineithercase.) Second,notethatcleaningtimeforthecasewhere=fVgwillalwaysbeatleastone.Hence,warmingconstraints( 3{21 )canonlyhelptosuggestthatzL>1if 95

PAGE 96

ReviewofAGENTPOSextensiveformulationtesting. thedeploymentpositionsdonotformadominatingset.Indeed,warmingconstraints( 3{21 )donotimprovetheconvergenceofthecutting-planealgorithmforinstanceswith=fVg.However,warmingconstraints( 3{21 )dohelptheinstanceswhere=ffig:i2Vg.Withnowarmingconstraints,thetuninginstancessolvedin(onaverage)14.4iterationscomparedto7.3iterationsforinstanceswithwarmingconstraints.Infulltesting,thez-variablesareheldintegralwhensolving AGENTPOSandwarmingconstraints( 3{21 )areonlyaddedtoinstanceswith=ffig:i2Vg. SpecialconsiderationsareimplementedfortheAGENTPOSsolutionmethod.Beforethecutting-planealgorithmisstarted,thebinary-vrelaxationof( 3{7 )issolvedgivenatimelimitofoneminute.If=fVg,thenwesolvethebinary-vrelaxationof( 3{7 ) AGENTPOStoanabsoluteoptimalitygapof0.999,otherwisewesolvetooptimality.WerstattempttosolveCLEANrecourseproblemsusingtheheuristic.Ifthatfails,wesolvetheCLEANMIPwiththesuggestedboundstoa0.999absoluteoptimalitygap.Given=fVg,weletsetthecutting-planealgorithmtoterminatewhenUBLB0:999.Given=ffig:i2Vg,weterminatethealgorithmwhenUBLB0:001.Whenimplementingoptimalitycutsoftheform( 3{19 )and( 3{20 ),weneedtochooseanappropriatelowerboundLLforcleaningtime.If=fVg,thenclearlywewillneverbeabletocleanthegraphinstantly,henceLL=1.If=ffig:i2Vg,wewillbeabletoachievezerocleaningtimeforsomecontaminations,andhenceLL=0,foreachL2. Table 3-16 reviewshowtheextensiveAGENTPOSformulationsperformed.GivenK=5agents,theinstancesaremorediculttosolvebecausetheagentsmustpursueamoresophisticatedcleaningstrategyinordertocleanthegraph. 96

PAGE 97

AGENTPOSdecompositionprocedure. MIPBin.-vC.-P.Alg.OverallvKdSolvedCPUSolvedCPUSolvedCPUCPU Total189/20081189/20077/1134924 Total137/200235135/2002221/65261144 Total90/200379127/2003528/72209134 Table 3-17 summarizestheresultsfromtheAGENTPOSsolutionprocedure.Weomitaveragerelativeoptimalitygapbecausetheaverageforeachparametersetwasalwayseither0(allinstancesweresolvedorproventobeinfeasible),or1(duetoalackofavalidupperboundfromoneormoreinstances). TheAGENTPOSbinary-vrelaxationreturnedanoptimalsolutionin451/600instancesinanaverageof100seconds.Thecutting-planealgorithmsolved56ofthe149instancesforwhichitwasinvoked.TypicallywhenK=10,instancessolvedquickly.Givenfeweragents,moresophisticatedcleaningstrategiesmustbedeveloped,andthe 97

PAGE 98

AGENTPOScutting-planealgorithm. 3{14 )(( 3{18 ),( 3{19 ))( 3{15 )( 3{20 ) AGENTPOSinstancesbecomemorediculttosolve.Approximately95%ofthe7,448CLEANrecourseproblemsweresolvedwiththeheuristic. Table 3-18 summarizestheperformanceofthecutting-planealgorithmontheinstancesforwhichitwasinvoked.Thetablepresentstheaveragenumberofiterations,thenumberofBendersfeasibilitycuts( 3{14 ),Bendersoptimalitycuts( 3{18 )and( 3{19 ),combinatorialfeasibilitycuts( 3{15 ),andcombinatorialoptimalitycuts( 3{20 )added. Therewerefewerfeasibilitycutsaddedintheseinstancesbecauseofalargertmax=5asopposedtotmax=1or3inSection 3.10 .Exceptforthepassingvariant,theseinstancestypicallyhaveasucientnumberofagentspresentinthegraphtoensurefeasibility.Approximately7%oftheBendersfeasibilitycutcoecientsgeneratedhad 98

PAGE 99

3{14 ).TheAGENTPOScutting-planealgorithmwasnotassuccessfulastheMINAGENTcutting-planealgorithmduetoalargersolutionspace.Giventmax=5,therearemorefeasibledeploymentlocationsforagentstobepositioned,andthusBendersfeasibilitycuts( 3{14 )andevencombinatorialfeasibilitycuts( 3{15 )werenotgeneratedasfrequentlyperiteration. 99

PAGE 100

Wehavediscussedtwostochasticintegerprogrammingapplicationsinthisdissertation.First,wecoveredatwo-stagefacilitylocationproblemwithbinaryactivationvariablesinthesecondstage.Next,wecoveredavarietyofproblemsrelatedtoagraphdecontaminationproblemwithmobileagents.Foreachapplication,Bendersdecompositionwashelpfulbutnotsucient,anditwasnecessarytoderiveandprovethevalidityofaclassofcutting-planeinequalities. Fortheclassofstochasticfacilitylocationprogramshavingintegerrecoursevariables,wepresentedasolutionmethodbasedonaspecializedcutting-planealgorithm.WecomparedourapproachtoastrengthenedversionoftheLaporteandLouveaux[ 26 ]integerL-shapedalgorithmformoregeneral,binarystochasticprogrammingproblems.ThecuttingplanesprescribedinthemethodofLaporteandLouveauxblindlyestimatesecond-stageobjectiveimprovementwithrespecttochangesinrst-stagedecisionsbasedonlyonwhichfacilitieswerelocatedinthepreviousRMPsolution^x,andtheoptimalsecond-stageobjectivesgiven^x.Thus,thesecutsgivelittleinsightintowhichfacilitylocationscanactuallyeectareductioninsecond-stageobjectives.Bycontrast,wederiveRPcuts,whichestimatesecond-stageimprovementbasedonsubproblemactivationandarcowdata,thuspermittingthegenerationofstrongercuttingplanes.Wehavedemonstratedonasuiteoftestproblemsthatacutting-planealgorithmbasedonRPcutsperformsatleastaswellasonebasedonLLcuts,andthateithertechniqueispreferabletosolvingtheproblembyitsextensiveformrepresentation.Inparticular,theRPcutting-planeapproachismosteectiveforinstancesinwhichasinglefacilitycanservetheentireregion. Futureresearchonthetwo-stagefacilitylocationproblemmayincludeheuristicapproacheslinkedwiththelower-boundingprocedureoutlinedinthisdissertationinordertoidentifygood-qualitysolutionswithprovableoptimalityboundsfortheproblem.We 100

PAGE 101

Problemcomplexitysummary. VariantCLEANMINAGENT CloningTriviallysolvableNP-Hard(byDS)StationaryClonesNP-Hard(byHP)NP-Hard(byVC)BlockingNP-Hard(byX3C)NP-Hard(byVC)PassingNP-Hard(byX3C)NP-Hard(byVC) AGENTPOSvariantshavethesamecomplexityastheirrespectiveMINAGENTvariants. mayalsoconsiderotherformsofthestochasticfacilitylocation/activationprobleminwhichotherobjectivesandrestrictionsareconsidered.Finally,itmaybeappropriatetoinvestigatearobustversionofthecurrentprobleminwhichweminimizetheworst-casesecond-stageobjectiveratherthanaweightedaverageofsecond-stageobjectives. Wehavealsointroducedseveralvariantsofthesynchronousgraphdecontaminationproblemandanalyzedthecomplexityofeachvariant(Table 4-1 ).WealsoprovidedMIPformulationsforeveryproblemdiscussed.PreliminarycomputationaltestinghasshownthattheMIPformulationsprovidedcansolvemostinstancesswiftly;however,someinstancesrequiringmorecomplexcleaningstrategiesbecomediculttoderive. TheMINAGENTandAGENTPOSproblemsarealsoamenabletodecompositionmethodsbasedonthesetofpossiblecontaminations.Theproblemswereformulatedastwo-stagestochasticintegerprograms,whereagentsarepositionedintherststage,andcleaningstrategiesforpossiblecontaminationsaredevisedinthesecondstage.Wedescribedacutting-planealgorithmwithintegercutssupplementedwithstrengthenedBenderscuttingplanestosolvetheMINAGENTandAGENTPOSproblems. Tostrengthenthecutting-planealgorithm,wecouldfurtherexploittherelaxationsoftheCLEANformulationsthroughtheReformulation-LinearizationTechnique(RLT)thatwouldpartiallyconvexifythefeasibleregion,allowingustoexploitdualityinformationinderivingstrengthenedBendersfeasibilityandoptimalitycuttingplanes. AnotherareaoffutureresearchcouldalsoincludeanalysisoftheMINAGENTproblemonspecialgraphssuchastreesormeshes.Giventhoseresults,generalgraphs 101

PAGE 102

Heterogeneousagentmodelsmaybeconsidered,inwhichagentshavingdierentcapabilitiesareavailabletocleanthegraph.MIPformulationsforthistypeofproblemwouldbeafairlystraightforwardamalgamationofthevariousconstraintsandvariablesintheMIPformulationsoftheCLEANproblemsgiveninthispaper. Thegraphdecontaminationproblemcouldbeconsideredwithnotimelimit.Ifnotimelimitimposed,thenitisnecessarytofocusontheminimumnumberofagentsandtheirinitialpositionsrequiredtocleanthegraph.ObviouslytheMIPformulationsfortheCLEANproblempresentedarenotappropriateforthisassumption;however,theCLEANheuristiccouldbeimplementedinanattempttondasolution.WeconjecturethatdeterminingwhetheraspecicnumberofagentsandtheirinitialpositionsaresucienttocleanagraphgivennotimelimitisalsoanNP-Hardproblem. Finally,itisunclearnowwhetherornottheCLEANproblemisinNP.WehaveshowntheCLEANs,CLEANb,andCLEANpvariantsarestronglyNP-Hard;however,thenumberoftimeunitsrequiredtocleanthegraphisnotnecessarilyboundedbyapolynomialfunctionofthenumberofnodesinthegraph,andthusitisnotclearifwecanverifyasolutiontotheCLEANprobleminpolynomialtime.FutureresearchmayfocusonspecialgraphsandshowthatsolutionstoCLEANproblemsonthesegraphscanbeveriedinpolynomialtime. 102

PAGE 103

[1] S.Alpern,Therendezvoussearchproblem,SIAMJControlOptimizat33(1995),673{683. [2] S.Alpern,Hide-and-seekgamesonatreetowhichEuleriannetworksareattached,Networks52(2008),162{166. [3] S.Alpern,V.J.Baston,andS.Essegaier,Rendezvoussearchonagraph,JApplProbability36(1999),223{231. [4] S.AlpernandS.Gal,Theoryofsearchgamesandrendezvous,KluwerInternationalSeriesinOperationsResearchandManagementScience,Kluwer,Boston,2003. [5] L.Barriere,P.Flocchini,P.Fraigniaud,andN.Santoro,Captureofanintruderbymobileagents,Procfrom14thACMSympParallelAlgorithmsArchitectures(SPA,Winnipeg,Canada,2002,pp.200{209. [6] L.Barriere,P.Fraigniaud,N.Santoro,andD.M.Thilikos,Searchingisnotjumping,LectureNotesinComputSci2880(2003),31{45. [7] J.F.Benders,Partitioningproceduresforsolvingmixed-variableprogrammingproblems,NumerMathematik4(1962),238{252. [8] D.BienstockandP.Seymour,Monotonicityingraphsearching,JAlgorithms12(1991),239{245. [9] D.BienstockandJ.F.Shapiro,Optimizingresourceaquisitiondecisionsbystochasticprogramming,ManageSci34(1988),215{229. [10] R.Breisch,Anintuitiveapproachtospeleotopology,SouthwesternCavers6(1967),72{78. [11] C.C.CareandJ.Tind,L-shapeddecompositionoftwo-stagestochasticprogramswithintegerrecourse,MathProgram83(1997),451{464. [12] G.CodatoandM.Fischetti,CombinatorialBenders'cutsformixed-integerlinearprogramming,OperRes54(2006),756{766. [13] A.DaganandS.Gal,Networksearchgameswitharbitrarysearcherstartingpoint,Networks52(2008),156{161. [14] G.B.DantzigandA.Madansky,Linearprogrammingunderuncertainty,ManageSci1(1955),198{206. [15] N.D.Dendris,L.M.Kirousis,andD.M.Thilikos,Fugitive-searchgamesongraphsandrelatedparameters,TheoretComputSci172(1997),233{254. [16] J.Ellis,H.Sudborough,andJ.Turner,Thevertexseparationandsearchnumberofagraph,InformatComputation113(1994),50{79. 103

PAGE 104

P.Flocchini,M.J.Huang,andF.L.Luccio,Decontaminationofhypercubesbymobileagents,Networks52(2008),167{178. [18] P.Flocchini,F.L.Luccio,andM.J.Huang,Decontaminationofchordalringsandtoriusingmobileagents,IntJFoundationsComputSci18(2007),547{564. [19] P.Flocchini,A.Nayak,andA.Schulz,Cleaninganarbitraryregularnetworkwithmobileagents,LectureNotesinComputSci3816(2005),132{142. [20] P.Flocchini,N.Santoro,andL.X.Song,Onthecomplexityofdecontaminatingahexagonalmeshnetwork,ProcIntMulti-ConferenceComputinGlobalInformatTechnology,2007,pp.21. [21] M.R.GareyandD.S.Johnson,Computersandintractability:AguidetothetheoryofNP-completeness,W.H.FreemanandCo.,NewYork,NY,1979. [22] W.Jansen,Intrusiondetectionwithmobileagents,ComputCommun25(2002),1392{1401. [23] W.Jansen,P.Mell,T.Karygiannis,andD.Marks,Applyingmobileagentstointrusiondetectionandresponse,Interimreport6416,NationalInstituteofStandardsandTechnology,1999. [24] A.JotshiandR.Batta,Searchforanimmobileentityonanetwork,EurJOperRes191(2008),347{359. [25] A.Lapaugh,Recontaminationdoesnothelptosearchagraph,JACM40(1993),224{245. [26] G.LaporteandF.V.Louveaux,TheintegerL-shapedmethodforstochasticintegerprogramswithcompleterecourse,OperResLett13(1993),133{142. [27] G.Laporte,F.V.Louveaux,andH.Mercure,Thevehicleroutingproblemwithstochastictraveltimes,TransportationSci26(1992),161{170. [28] G.Laporte,F.V.Louveaux,andL.vanHamme,Exactsolutiontoalocationproblemwithstochasticdemands,TransportationSci28(1994),95{103. [29] W.S.LimandS.Alpern,Minimaxrendezvousontheline,SIAMJControlOptimizat34(1996),1650{1665. [30] F.V.LouveauxandM.H.vanderVlerk,Stochasticprogrammingwithsimpleintegerrecourse,MathProgram61(1993),301{325. [31] N.Megiddo,S.L.Hakimi,M.R.Garey,D.S.Johnson,andC.H.Papadimitriou,Thecomplexityofsearchingagraph,JACM35(1988),18{44. [32] T.D.Parson,\Pursuit-evasioninagraph,"TheoryandApplicationsofGraphs,Y.AlaviandD.R.Lick(Editors),Springer-Verlag,1976,pp.426{441. 104

PAGE 105

T.D.Parson,Thesearchnumberofaconnectedgraph,9thSoutheasternConferenceCombin,GraphTheory,Comput,1978,pp.549{554. [34] S.SenandJ.L.Higle,TheC3theoremandaD2algorithmforlargescalestochasticmixed-integerprogramming:Setconvexication,MathProgram104(2005),1{20. [35] S.SenandH.D.Sherali,Decompositionwithbranch-and-cutapproachesfortwo-stagestochasticmixed-integerprogramming,MathProgram106(2006),203{223. [36] P.SeymourandR.Thomas,Graphsearchingandamin-maxtheoremfortree-width,JounalCombinatorialTheory,SerB58(1993),22{33. [37] L.V.Snyder,Facilitylocationunderuncertainty:areview,IIETrans38(2006),537{554. [38] Z.C.TasknandJ.C.Smith,Cuttingplanealgorithmsforsolvingarobustedge-partitionproblem,Technicalreport,DepartmentofIndustrialandSystemsEngineering,UniversityofFlorida,Gainesville,FL,2007. [39] Z.C.TasknandJ.C.Smith,Mixed-integerprogrammingtechniquesfordecomposingimrtuencemapsusingrectangularapertures,Technicalreport,DepartmentofIndustrialandSystemsEngineering,UniversityofFlorida,Gainesville,FL,2007. [40] R.M.VanSlykeandR.Wets,L-shapedlinearprogramswithapplicationstooptimalcontrolandstochasticprogramming,SIAMJApplMath17(1969),638{663. [41] D.W.WalkupandR.J.B.Wets,Stochasticprogramswithrecourse,SIAMJApplMath15(1967),1299{1314. [42] R.M.Wollmer,Two-stagelinearprogrammingunderuncertaintywith0-1rststagevariables,MathProgram19(1980),279{288. 105

PAGE 106

IhavehadanexceptionaltimeattheUniversityofFlorida.Icaughtalotofsh,wenttoalotoffootballgames,andcompletedmydoctorate. 106