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Integrated Assignment and Path Planning

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Integrated Assignment and Path Planning
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MURPHEY, ROBERT A. ( Author, Primary )
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2008

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Algorithms ( jstor )
Cost allocation ( jstor )
Customers ( jstor )
Integers ( jstor )
Linear programming ( jstor )
Marginal costs ( jstor )
Trajectories ( jstor )
Travel time ( jstor )
Vehicles ( jstor )
Weapons ( jstor )

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University of Florida
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University of Florida
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Copyright Robert A. Murphey. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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5/31/2007
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436098738 ( OCLC )

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INTEGRATEDASSIGNMENT ANDPATHPLANNING By ROBERTA.MURPHEY ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2005

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TABLEOFCONTENTS page LISTOFTABLES. ...................... ..........iv LISTOFFIGURES ...................... ..........v ABSTRACT............... .....................vii CHAPTER 1INTRODUCTION... ...........................1 1.1AMotivatingExample........................1 1.2Organization .................... ..........3 2WEAPONTARGETASSIGNMENTPROBLEMS............5 2.1IntroductionToWeaponTargetAssignment............5 2.2StaticAssignmentModels......................7 2.2.1UniformWeapons.......................9 2.2.2RangeRestrictedWeapons..................10 2.3TheDynamicAssignmentModel..................11 2.3.1Shoot-Look-ShootProblems.................12 2.3.2StochasticDemandProblems.................14 2.4AnAlternativeSPFormulationforTheDynamicWTA......21 2.5DiscussionOfWeaponTargetAssignment.............22 3PATHPLANNINGPROBLEMS. .....................24 3.1IntroductionToPathPlanning....................24 3.2TrajectoryDesign...........................24 3.3WaypointPlanning..........................27 3.3.1CapacitatedVRP(CVRP)..................29 3.3.2SplitDeliveryVRP(SDVRP)................30 3.3.3VRPWithTimeWindows(VRPTW)............30 3.3.4StochasticVRP(SVRP)...................30 3.4TheCaseForIntegratedResourceAssignmentandPathPlanning30 4MIXEDINTEGERPROGRAMS......................33 4.1TransshipmentProblems.......................33 4.2MultipurposeBatchPlants......................36 4.3UnmannedAirVehiclePathPlanning................43 ii

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5NEWPRODUCTDEVELOPMENTPROBLEMS............50 5.1IntroductionToNewProductDevelopment.............50 5.2ControlTheoreticModel.......................52 5.3MultidimensionalAssignmentProblemModel...........55 5.3.1ValueOfAnAssignment...................60 5.3.2UpperBoundOn3-IndexProblem..............62 5.3.3ExtensionsToArbitraryDimension.............65 5.4DiscussionOfMultidimensionalAssignmentProblems.......67 6INTEGRATEDASSIGNMENTANDPATHPLANNING ........69 6.1ModelingApproach..........................69 6.1.1GeneratingTheCostVector.................72 6.1.2AxialConstraints.......................85 6.1.3TravelTimeConstraints...................89 6.1.4PrecedenceConstraints ....................91 6.2TestProblems. .................. ..........94 6.2.1RandomScenario.......................94 6.2.2MilitaryScenario.. .....................96 7SOLUTIONMETHODS...........................104 7.1BranchandBound..........................104 7.2ConstructionHeuristics........................106 7.2.1PhaseOne:Construction...................107 7.2.2PhaseTwo:LocalSearch...................108 7.3ComputationalResultsOnTestProblems.............111 7.4Observations .................... ..........113 8CONCLUSIONSANDEXTENSIONS...................118 REFERENCES........ ...........................121 BIOGRAPHICALSKE TCH............. ...............126 iii

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LISTOFTABLES Table page 31De“ningTraitsOfTheObjectiveModel.................32 6AttributeScoresForIntegratedModels.. ...............70 62ExampleofALinearEquivalentIndexInColumnMajorOrder....86 63RandomTestProblems..........................95 6MilitaryScenarioObjectiveClasses ...................99 65ScenarioActivityTypeInformation..................99 6MilitaryScenarioCostsVehicleClassversusActivityType.. ....101 6MilitaryScenarioTestProblemStatistics ...............102 6MilitaryScenarioUniqueObjectiveDataForTestProblems ....102 6MilitaryScenarioUniqueVehicleDataForTestProblems.. ....103 6MilitaryScenarioUniqueTimeStepDataForTestProblems ....103 7ComputationalResultsUpperBoundPerformanceonRandomTest Problems...... ...........................112 7ComputationalResultsSide-by-SideComparisonof3Algorithms..113 7ComputationalResultsComparisonofGRASPLWithandWithout LocalSearch... ...........................114 iv

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LISTOFFIGURES Figure page 11ExampleOfIntegratedAssignmentandPathPlanning........2 12ExampleOfCorrelatedTasking.....................2 21ExampleStaticWeaponTargetAssignment..............6 22MinimumMarginalReturn(MMR)...................10 31ExampleVoronoiDiagram........................26 32ExampleDelaunayTriangulation....................27 41ExampleOfMultipurposeBatchPlant.................37 4ExampleTimelineandEventlineForAMultipurposeBatchPlant.38 51ExampleOfNDPActivities........................51 52ExampleOfNDPWithConstantDurationActivities..........57 53ExampleOf3-IndexAxialConstraints..................58 61Two-WayIntersectionCostsPseudo-code................79 62Three-WayIntersectionCostsPseudo-code...............80 63AssignmentCostsPseudo-code......................81 64Pseudo-codeForGeneratingAxialConstraints.............88 65Pseudo-codeForGeneratingTravelTimeConstraints.........91 6Pseudo-codeForGeneratingPrecedenceConstraints ..........93 67CostDistributionForRandomProblemM4N10T...........96 6CostDistributionForRandomProblemM4N10TWithWeights=3.97 6CostDistributionForRandomProblemM4N10TWithWeights=6.97 6CostDistributionForRandomProblemWithRandomWeights...98 71Pseudo-codeforagenericGRASP....................106 72GRASPconstructionpseudo-code....................108 v

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7GRASPLocalSearchMultidimensionalNeighborhoodPseudo-code110 7GRASPLocalSearchLinearEquivalentNeighborhoodPseudo-code111 7ComparisonOfExactAssignmentsWithandWithoutCorrelation..115 7ComparisonOfExactandGRASPLFor7663TPAW(10.10).. ....116 7ComparisonOfExactandGRASPLFor9994TPAW(5.10.5). ....117 vi

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFul“llmentofthe RequirementsfortheDegreeofDoctorofPhilosophy INTEGRATEDASSIGNMENT ANDPATHPLANNING By RobertA.Murphey May2005 Chair:ProfessorPanosPardalos MajorDepartment:IndustrialandSystemsEngineering Asurgeofinterestinunmannedsystemshasexposedmanynewandchallengingresearchproblemsacrossmany“eldsofengineeringandmathematics.These systemshavethepotentialoftransformingoursocietybyreplacingdangerousand dirtyjobswithnetworksofmovingmachines.Thisvisionisfundamentallyseparate fromthemodernviewofroboticsinthatsophisticatedbehaviorisrealizablenotby increasingindividualvehiclecomplexity, butinsteadthroughcollaborativeteaming thatreliesoncollectiveperception,abstraction,decisionmaking,andmanipulation. Obviousexampleswherecollectiveroboticswillmakeanimpactincludeplanetary exploration,spacestructureassembly,remoteandunderseamining,hazardous materialhandlingandclean-up,andsearc handrescue.Nonetheless,thephenomenondrivingthistechnologytrendistheincreasingrelianceoftheUSmilitaryon unmannedvehicles,speci“cally,aircraft.Onlyafewyearsago,followingyearsof resistancetotheuseofunmannedsystems,themilitaryandcivilianleadership intheUnitedStatesreverseditselfandhaverecentlydemonstratedsurprisingly broadacceptanceofincreasinglypervasiveuseofunmannedplatformsindefense surveillance,andevenattack.However,asrapidlyasunmannedsystemshave vii

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gainedacceptance,thedefenseresearchcommunityhasdiscoveredthetechnical pitfallsthatlieahead,especiallyforo peratingcollectivegroupsofunmanned platforms.Agreatdealoftalentandenergyhasbeendevotedtosolvingthese technicalproblems,whichtendtofallintotwocategories:resourceallocation ofvehiclestoobjectives,andpathplanningofvehicletrajectories.Anextensive amountofresearchhasbeenconductedineachdirection,yet,surprisingly,very littleworkhasconsideredtheintegratedproblemofassignmentandpathplanning. Thisdissertationpresentsaframeworkforstudyingintegratedassignmentandpath planningandthenmovesontosuggestanexactmathematicalmodelandsolution techniques.Theapproachadoptedisbaseduponthevery”exibleNewProduct Developmentmodelbutalsoblendsmanyfeaturesfromotherapproaches.Solutionmethodsusingbranchandboundandconstructionheuristicsaredeveloped andtestedonseveralexampleproblems,includingamilitaryscenariofeaturing unmannedairvehicles. viii

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CHAPTER1 INTRODUCTION Asurgeofinterestinunmannedsystemshasexposedmanynewandchallengingresearchproblemsacrossmany“eldsofengineeringandmathematics.The UnitedStatesDepartmentofDefensehasledthisinterestthroughtheincreasing useofunmannedplatformsindefensesurveillanceandevenattack.However, asrapidlyasunmannedsystemshavegainedacceptance,thedefenseresearch communityhasdiscoveredthetechnicalpitfallsthatlieahead,especiallyforoperatingcollectivegroupsofunmannedplatf orms.Thefundamentalnatureofthese problems,resourceallocationofvehiclestonon-homogeneousobjectivesandpath planningofvehicletrajectories,retainsthestructureofclassicallyrecognized KnapsackproblemsandTravelingSalesmanVehicleRoutingProblems(VRP) respectively.Hence,variouselaborationsonbothoftheseclassicalproblemswill bestudied.Inthischapter,theproblemof integratedresourceallocationandpath planning willbeintroducedandtheorganizationoftheremainderofthisworkwill bepresented.Agoodplacetobeginiswithanexample. 1.1 A Motivating Example ConsidertheproblempresentedinFigure1.1.Herethereare3unmanned vehicles,eachwithsomeuniqueability,eitherinsensing,strike,orcommunications, or,moregenerally,withvaryingabilities(e.g.,eciencies)inallthree.Thereare also6objectives,indicatedbytrianglesinthediagram.Theseobjectiveshave uniquerequirements,oractivities,indicatedbytheirletters,A,B,andC,sothat airvehiclesthatareecientatanactivityrequiredataparticularobjectivewill haveabetterchanceofsuccessthanavehiclenotadeptatthatactivity.Not apparentinthediagramisthefactthatprecedenceexistsamongtheobjectives. 1

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2 Figure1…1:ExampleOfIntegratedAssignmentandPathPlanning Figure1…2:ExampleOfCorrelatedTasking Therearetwoformsofprecedence.First,multipleactivitiesmayberequiredata singleobjectiveandthoseactivitiesmustbeperformedinsomeorder.Forexample, atargetmustbesurveyedbeforeitisattacked.Thesecondformofprecedenceis betweentwodisparateobjectives.Forexample,anair-to-surfaceweaponmustbe destroyedbeforethehighvaluetargetitwascoveringisattacked.Anexampleof thislatterformofprecedenceisshowninFigure1.1.Here,a“rstaircraftmust performanactivityatobjectiveAattime t beforeasecondaircraftcanperforman activityattheobjectiveBattime t + k .

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3 1.2 Organization Ingeneral,itcanbeseenthattheobjectivesareacollectionof activities and theaircraftare resources withuniqueecienciesatperformingthoseactivities. Precedencelinksactivities,betheyatthesamesite(objective)oratdierent sites.Thisexamplewillservetomotivatethedirectionofresearchintherest ofthiswork.Alogicalplacetobeginisbyconsideringtheobjectivesastargets andtheresourcesasweapons.Thisistheweapontargetassignmentproblem presentedinChapter2.Ofcoursetheexamplesmakeitquiteclearthatspace-time dependenciesarecriticaltothisproblem.ThereforeChapter3willexaminepath planningandtheconstructionoftrajectoriesforkinematicallyconstrainedvehicles. Itwillbecomeclearthatneitherweapontargetassignmentorpathplanning aresucientontheirown,orevenwhencombinedinahierarchicalmanner,for theintegratedproblem.Again,thisissynonymouswiththeVRPandKnapsack problems,suchthatpathplanningiscloselyrelatedtoVRPandassignmentto Knapsack.Quicklyreviewingthelessonsle arnedinChapters2and3,theessential aspectsoftheintegratedassignmentandpathplanningproblemwillbecome apparentandareprovidedattheconclusionofChapter3.Withthesede“ning characteristicsinmind,severalfairlygeneralintegerandmixedintegermodels (developedforvariousapplications,notnecessarilycollaboratingairvehicles) willbestudiedinChapters4and5.Eachmodelcontainssomeaspectofboth VRPandKnapsack,andfurthermore,adistinguishingfeaturethatiscriticalto theframeworkoftheintegratedproblem.Chapter6suggestsaframeworkand presentsawaytogeneratetestproblemsfortheintegratedproblem.Theapproach isbasedontheMultidimensionalAssignmentProblem(MAP)formulationofNew ProductDevelopmentwithmodi“cationsmadeaccordingtoelementsofother modelsstudied.Solutionmethodsareofparticularimportance,astheintegrated problemishighlycomplexandcomputationallychallenging.Therefore,good

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4 structureisneededtoobtainsolutions,oneofthereasonstheMAPformulation issuggested.Chapter7presents2basicapproachesto“ndingsolutionstothe integratedproblem.AlthoughtheMAPstructureallowsareasonableboundwith veryfastcomputationtime,aswillbeseen,itisnotfastenoughtomakebranch andbounduseful!Consequently,themajorityofcomputationalresultsarebased onconstructiveheuristics.ExtensionsoftheintegratedproblemsofChapter6are introducedinChapter8.

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CHAPTER2 WEAPONTARGETASSIGNMENTPROBLEMS 2.1 Introduction To Weapon Target Assignment Inthischapterweconsideraclassofnon-linearassignmentproblemscollectivelyreferredtoasTarget-basedWeaponTargetAssignment(WTA).The Target-basedWeaponTargetAssignment problemconsidersoptimallyassigning M weaponsto N targetssothatthetotalexpecteddamagetothetargetsismaximized.Weusethetermtarget-basedtodistinguishtheseproblemsfromthosethat areasset-based,thatisproblemswhereweaponsareassignedtotargetssuchthat thevalueofagroupofassetsismaximizedsupposingthatthetargetsthemselves aremissilesengagingtheassets.Theasset-basedproblemismostpertinentto strategicballisticmissile defense problemswhereasthetarget-basedproblems applyto oensive conventionalwarfaretypesofproblems.Previoussurveysof WTAproblemsweremadebyMatlin[27]andEcklerandBurr[11],the“rston oensive,targetbasedproblemsandthesecondprimarilyaboutdefensive,asset basedproblems. Ifatsometime t thenumbersandlocationsofweaponsandtargetsareknown withcertainty,thenasingleassignmentmaybemadeattime t suchthatallofthe weaponsarecommitted.Thisformulationisdenoted static WTA.Itisdepictedin Figure2.1. Initsmostgeneralform,staticWTAisknowntobe NP -complete.Amore dicultproblemresultswhenassignmentsmaybemadeatanyofseveraldiscrete pointsintime.Thesetypesofproblemsareknownas dynamic WTAandare formulatedin2ways.Inthe“rsttype,thenumbersandlocationsoftargetsareall knownapriori.However,sincethesurvivalofatargetwithweaponsassignedto 5

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6 Figure2…1:ExampleStaticWeaponTargetAssignment itisprobabilistic,theoutcomeofanytimeinterval(i.e.,whichtargetssurvive)is stochasticandsoin”uencesallfutureassignments.Theseproblemsareknownas shoot-look-shootsincesomeobservationofthetargetsisrequiredtoconditionthe problematthefollowinginterval.Shoot-look-shootproblemsarereadilyadapted toadynamicprogrammingformulation.InthesecondtypeofdynamicWTA,we assumethatthetargetsarenotobservab le;hencetheoutcomeofeachinterval isdeterministic.However,wecannotassumethatthenumbersandlocationsof targetsareknownapriori.Thatis,ateachinterval,asubsetofthe N targets isknownwithcertaintyandtheremainderareeithernotknownorknownonly stochastically.ThisdynamicWTAformulationmaybemodeledasstochastic programs.

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7 2.2 Static Assignment Models Considerthatthereare M weaponsand N independenttargets.De“nethe decisionvariable xij,i =1 , 2 ,...M,j =1 , 2 ,...,N as xij= 1:weapon i assignedtotarget j 0:else (2.1) Giventhatweapon i engagestarget j ;thatis xij=1,theoutcomeisrandomand assumesaBernoullidistribution: P (target j isdestroyedbyweapon i )= PijP (target j isnotdestroyedbyweapon i )=1 Š Pij(2.2) Assumethateachweaponengagementisindependentofeveryotherengagement. ThentheoutcomesoftheengagementsareindependentBernoullidistributed. Leteachtargetbeassignedapositiverealnumber Vjtoindicatepreference betweentargets.Assumethattargetsmaybepartitionedintoclassesandeach classhasauniquevaluetothedecisionmaker.Letthesetofallclassvaluesbe containedin V suchthat V isofcardinality K .Ourobjectiveistomaximizethe expecteddamagetothetargetswhichisequivalenttominimizingtheexpected targetvalue.With qij=(1 Š Pij),theresultingintegerprogrammingformulationis SWTA minimizeNj =1Vj Mi =1qxijijsubjecttoNj =1xij=1 ,i =1 , 2 ,...,M Vj V RK +,j =1 , 2 ,...,N x BM × N

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8 Notethat R denotestherealnumberspace,and Z and B denotetheintegerand binarysubspacesrespectively.TheequalityconstraintsofSWTAensurethat eachweaponisassignedtoexactlyonetarget.Noticethatnothingpreventsall weaponsfrombeingassignedtoasingletarget.Thisproblemformulationwas“rst suggestedbyManne[26]. Althoughthenonlinearassignmentpr oblemdevelopedinSWTAwasshownto be NP -complete[25],ausefullowerboundmaybedevelopedandthenappliedina branchandboundorotherglobalintegeroptimizationscheme.Alowerboundfor SWTAmaybeobtainedbyrelaxingintegralityof x RSWTA minimizeNj =1Vj Mi =1qxijijsubjecttoNj =1xij=1 ,i =1 , 2 ,...,M Vj V RK +,j =1 , 2 ,...,N 0 xij 1 ,i =1 , 2 ,...,M,j =1 , 2 ,...,N TheobjectiveofRSTWAisclearlyconvexwhiletheconstraintsarelinear,resultinginaconvexprogrammingproblem.Anumberofmethodsmaybeappliedto solveRSWTA;however,adualmethodwasshown[1,15]tobeveryfast.Thedual ofRSWTAissimply d ( )=minimizexL ( x, ) wheretheLagrangianisde“nedas L ( x, )=Nj =1Vj Mi =1qxijij+Mi =1iNj =1xijŠ 1 .

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9 Nowitcanbeseenthat d ( )hastheseparableform d ( )=Nj =1minx Vj Mi =1qxijij+Mi =1ixij ŠMi =1iandbyStrongDuality,thereisnodualitygap. 2.2.1 Uniform Weapons Iftheweaponsareallassumedtobe uniform (thatis,theyareallthesame), thenthe uniformstaticWTA problemmaybewrittenas SWTA Š U minimizeNj =1VjqxjjsubjecttoNj =1xj= M, Vj V RK +,j =1 , 2 ,...,N x ZNwherenow xjisthenumberofweaponsassignedtotarget j andtheindex i has beendroppedfrom q since,giventhattheweaponsareallthesame,theprobability ofdestroyingtarget j dependsonlyonthetarget. denBroeder etal. [9]presentedanalgorithmforsolvingSWTAoptimallyin O( N + M log( N ))timetermedtheminimummarginalreturn(MMR).Essentially, MMRassignsweaponsoneatatimetothetargetwhichrealizesthegreatest decreaseinvalue,wheretargetvalueisde“nedtobe qxjj.denBroedershowedthat thisgreedystrategyisoptimalfortheuni formweaponassumption.Thealgorithm isshowninFigure2…2. Hosein[15]latershowedthatavarian tofiterated1-optlocalsearchonany feasibleassignmentisalsooptimalforSWTA-U.Theiterated1-optvariantiseasily

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10 Step0 .Foreach j =1 ,...N ,let xj=0,anddenoteprobabilityof survivaloftarget j by Sj= Vjqxjj.Initializeweaponindex i =1. While i M DO Step1 .Findtarget k forwhichmunition i hasgreatesteect k =argmaxj{ Sj(1 Š qj) } . Step2 .Addmunition i totarget kxk= xk+1andreviseprobability ofsurvivaloftarget kSk= Skqk. i = i +1. Figure2…2:MinimumMarginalReturn(MMR) implementedforthisproblembyremovingoneweaponfromtarget j andreplacing itontarget k giventhatthemovedecreasestheobjectivevalue. Notethatifalltargetsarealsouniform, qjisreplacedby q and Vjisdropped. Itiseasilyshownthattheoptimalstrategyinsuchcasesistosimplyspreadthe weaponsevenlyacrossalltargets(see[25]). 2.2.2 Range Restricted Weapons RealisticWTAmodelsdictatethatnotallweaponswillbereachablebyall targets.Consequently,the rangerestricteduniformSWTA problemispresented SWTA Š URR minimizeNj =1Vjqi R jxijjsubjecttoNj =1xij=1 ,i =1 , 2 ,...,M Vj V RK +,j =1 , 2 ,...,N x BN

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11 where Rjisthesetofweaponindicesthatarewithinreachablerangeoftarget j . Byrepresentingeachweaponasanodeandeachtargetasanodeinabipartite graphpartitionedonweaponsandtarg ets,Hosein[15]showedthatSWTA-URR wasinfactsolvableasanetwork”owproblemwhenanedgeisaddedbetween weapon i andtarget j onlyif i Rjfor xij.Asupplyofoneisprovidedateach weaponnodeandthetargetnodesareallsunktoasinglesinknodewithdemand M .Thecapacityofallweapontotargetarcsisonewhilethecostofthesearcsis alwayszero.Thecapacityofallarcsfromthetargetstothesinknodeareeach M whilethecostsofthesearcsare Fjs( x )= Vj q x j+( x Š x ) q x jŠ q x j where”ow x onarc { j,s } iscontinuousbetween0and M .Since Fjs( x )isaconvex combination,theresultingnetwork”owproblemisclearlyconvex.Furthermore, therewillalwaysexistanintegervalued solutionthatisoptimalfortheconvex minimumcostnetwork”owproblemthatisalsooptimalforSWTA-URR. 2.3 The Dynamic Assignment Model InstaticWTAproblemstheassignmentismadeatasinglepointintime. Nowsupposethatassignmentsmaybemadeatanyof T discreteintervalsintime. Thereare2fundamentaltypesofdynamicWTAproblemsintheliterature.The moststudiedisthecaseinwhichthenumbersandlocationsoftargetsareall knownapriori.However,sincethesurvivalofatargetwithweaponsassignedto itisprobabilistic,theoutcomeofanytimeinterval(i.e.,whichtargetssurvive)is stochasticandsoin”uencesallfutureassignments.Theseproblemsareknownas shoot-look-shoot sincesomeobservationofthetargetsisrequiredpriortomaking anyfutureassignments.Shoot-look-shootproblemsarereadilyadaptedtoa dynamicprogramyetproveverydiculttosolveinthemostgeneralcase.In thesecondtypeofdynamicWTA,weassumethatthetargetsarenotobservable;

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12 hencetheoutcomeofeachintervalisessentiallydeterministic.However,weno longerassumethatthenumbersandlocationsoftargetsareknownapriori.That is,ateachinterval,asubsetofthe N targetsareknownwithcertaintyandthe remainderareeithernotknownorknownonlystochastically.ThesedynamicWTA formulationsmaybemodeledasstochasticprograms. 2.3.1 Shoot-Look-Shoot Problems Inshoot-look-shootproblems,theassignmentof M weaponsto N targets ismadeover T discreteintervalsintime.Therefore,thedecisionvariablemust beindexedbytimeasin xij( t ) ,t =1 , 2 ,...,T .Theassumptionisthatafteran assignment,theoutcomeofeachengagementcanbeobservedwithouterror.As aresult,wediscusstheevolutionofthe targetstate asafunctionoftimeandthe previousassignment.Atanytime t ,thisstateissimply Uj= 1:target j survivesstage t Š 1 0:else Sincethesurvivaloftarget j dependsonthevaluesof xij( t Š 1)and qij,thetarget stateisstochasticandevolvesas P ( Uj= u )= uMi =1qxij( t Š 1) ij+(1 Š u ) 1 ŠMi =1qxij( t Š 1) ij Similarly,the weaponstate isde“nedas wi= 1:weapon i notusedinstage t Š 1 0:else whichevolvesas wi=1 ŠNj =1xij( t Š 1)

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13 thatis,weapon i isavailableforuseinstage t onlyifitwasnotusedinstage t Š 1. Impliedinthismodelisthefactthatweapon i isavailableforuseinstage t only ifitwasnotusedin anyprevious stage.Thisisbasedonaninductiveargument thatassumesthatasolutionofstage t Š 1considerswhatweaponswerenotusedin stage t Š 2andsoon. De“nethecostofastage T assignmenttobe F1( u,w ),thecostofastage T Š 1assignmenttobe F2( u,w ),andsoon.Nowthedynamicshoot-look-shoot WTAproblem(DWTA-SLS)isde“nedasthe followingrecursiveoptimization problem DWTA Š SLS min F1=minxu ZNP ( U = u ) F 2( u,w ) subjectto x BM × N Nj =1xij=1 Š wi,i =1 ,...,M wheretheoptimalsolutiontothestage T Š 1problem,denotedby F 2( u,w ),is obtainedbysolvingDWTA-SLSwiththestage T Š 1solutionnowde“nedby F1.Inalikemanner, F 3( u,w )isobtainedandsoon.Notehowever,thatweneed onlyusea2-stagede“nitionatanytime.Thisformulationisclearlyamenableto DynamicProgramming(DP)methods.Unfortunately,evenfor T =2,thenumber ofcomputationsrequiredforDPisunrealisticallylarge,because2Massignments arepossibleinstage1,eachforwhichthereare Nm2possibleassignmentsinstage 2,where m2isthenumberofweaponsleftforstage2. Hosein[15]studiedDWTA-SLSindetailandfoundusefulsolutionsfora simplercasewhentheprobabilitiesofsurvival qijwerenolongerdependenton weaponortargetbutinsteadonlydependentonthestage t .Whenthisassumption

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14 ismadeourdecisionvariableisnow mt,thenumberofweaponsusedinstage t . Hoseinshowsthattheseproblemsshareanoptimalstrategy. Optimal strategy of stage dependent probabilities. For t =1 ,...,T ,the mtweaponsusedatstage t shouldbespreadevenlyoverallthe N targets. Ofcoursewemuststilldetermineoptimalvaluesfor mt,t =1 ,...T .Hosein examines3specialcases. Case 1. If M N thenassignallweaponsinthestagewiththelowest q ( t ) . ThisisessentiallytheSWTAproblem. Case 2. If M N and q ( t ) q ( t +1) ,t =1 ,...,T Š 1 ,then m 1 N . Case2meansthatfor q ( t )nondecreasing,theoptimal“rststageassignmentis touseatleastasmanyweaponsastherearetargets. Case 3. If T> 1+M Š N 2for M>N and q ( t )= q forall t ,then m 1= N . Thissimplymeansthatforlargenumbersofintervalsandaconstant q ,the optimalstrategyistoassignexactly N weapons,andsince mtweaponsusedat stage t shouldbespreadevenlyoverallthe N targets,wesurmisethateachtarget shouldgetexactlyoneweaponinstage1. Finally,HoseinexploredDWTA-SLSforverylargenumbersoftargetsby developingalimiton F1as N . 2.3.2 Stochastic Demand Problems InthestaticWTAformulations,thedecisionmakerknowsatoneinstantin timethetotalnumberoftargets N andtheirlocations.ConsideranotherWTA problem,whereatsometime t ,onlyasubsetofthetotaltargetsisknowntothe decisionmaker.Letthenumberoftargetsknownattime t bedenotedby n ( t )and n ( t ) N forany t .Astimeprogresses,additionaltargetsarediscovered;hence n ( t )isnondecreasingwithtime.Since N isunknown,anassignmentofweapons canonlybemadetothe n ( t )weaponsthatareknownorelsereservedfortargets

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15 thatareexpectedinthefuture.ThesetypesofDynamicWTAproblemsessentially haveastochasticdemandandwere“rstintroducedbyMurphey[30]. Ifthetargetsarerankedaccordingtotheirvalue,asinthestaticWTA problems,thenitispossiblethatattime t onlylowvaluetargetshavebeenfound, inwhichcaseitisdesirabletowaitbeforemakinganassignment.If,afterwaiting time ,allofthetargetsarediscovered,(thatis n ( )= N ),thenastaticWTA problemresults.Assumehoweverthatthereisacostassociatedwithwaitingto makeanassignment.Thiscostisduetothefactthateachweaponhas“nitefuel, soonceitisdeployed,ithasa“nitetimetodetectandengagetargets.Indeed,the targetmaypassoutoftheweapons“eldofviewaltogether. Therearetwoapproachesforastochasticmodel:two-stageandmultistage. Wewillonlydevelopthetwo-stagemodelheresinceitisdiculttopredictexactly when anunknowntargetmightappearaswouldberequiredinamultistage model.Itisfairlystraightforwardtomodelhowmanytargetsofeachclassremain undetectedwithina“nitegeographicregionatanyinstantintime.Asaresult, the2-stagemodelmustbesolvedrecursivelyateachtimeinterval,eachsolution consistingoftwosubproblems:(1)a“rststageproblemwhichincludesalltargets detectedduringthecurrentintervalandallpreviousintervals,and(2)asecond stageorrecourseproblemthatincludesalltargetsnotyetdetected.The“rststage demandisdeterministicwhilethesecond stagedemandisstochasticandthetrick istobalanceassigningweaponstoknowntargetsagainstreservingweaponsfor possible,butnotyetdetermined,futuretargets. Letarandomvector denotethenumberoftargetsineachclass k,k =1 ,...,K thathaveyettobedetected(e.g.,targetsinstage2).Ifweare giventheprobabilitiesofsurvivalandthetargetvaluesforeachtargetclass,then foranyinstanceofrandomvector ,thesecondstagevaluesoftargetvalue V2, probabilityofsurvival q2andthetotalnumberofyettobedetected(stage2)

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16 targets n2areeasilydetermined.The2-stagestochasticprogrammingformulation isasfollows SP Z1( x )=minxf1( x )+ E [ Z2( x,j)](2.3) subjectton1i =1xi M, (2.4) x b (2.5) x Zn1(2.6) where f1( x )isthestageonecostfunctionofthe“rststageassignment x andis integer-convex;thatis,thecontinuousrelaxationof f1( x )isconvex. E [ Z2( x,j)] istheexpectationoperatorwithrespectto .Theconstraintin(2.5)isused tolimitthenumberofweaponsassignedtoanysingletarget,preventingthe assignmentoflargenumbersofweaponstohighvaluetargets.Therighthandside of(2.5)isdeterminedbysettingathresholdjonthereducedvalueofeachtarget asin qxii jwhichcanalwaysbewrittenintheformof(2.5)byapplyingalogarithm. Z2( x,j) isthesolutiontoasecondstageproblemthatclearlyshowsitsdependenceon thestochasticparameter andthe“rststagedecision x ascapturedinthe followingprogram

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17 S2P Z2( x,j)=minyfj 2( y )(2.7) subjectton1i =1xi+n2( j)i =1yi= M, (2.8) y Zn2(2.9) where fj 2( y )isthestagetwocostfunctionofthenumberofweapons y assignedin thesecondstage.Since y dependsontheoutcomeof , fj 2( y )dependson and furthermoreisinteger-convex.Speci“cally, f1( x )and fj 2( y )arethefollowing f1( x )=n1i =1Vi 1( qi 1)xi(2.10) fj 2( y )=n2( )i =1Vi 2( ) qi 2( )yi(2.11) Duetothediscretesupportof ,SPmaybereplacedwiththefollowingsocalled deterministicequivalent program DESP Z1( x )=minxf1( x )+sj =1pjZ2( x,j)(2.12) subjectton1i =1xi M, (2.13) x b (2.14) x Zn1(2.15) where s isthetotalnumberofscenarios(outcomes)fortherandomvector .

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18 EachstageofDESP/SP2hasaninteger-convexobjectiveandintegerane constraints.Theclassicapproachtosolvingastochasticprogrammingproblem withdiscreterandomsupportisbythedecompositionmethod.Decomposition methodsdecouplethestage1andstage2problemsby“rstsolvingavariantof thestage1problem,oftencalledthe currentproblem (CP)withascalarvariable takingtheplaceofthestage2solutionsothat sj =1pjZ2( x,j) . Thestage2problemissolvedforeach j,j =1 ,...s .Informationfromthestage2 problemsisusedtodevelopasupportinghyperplaneconstraintwhichboundsthe “rststageresourceto ( optimalitycut ).ThisconstraintisaddedtoCPwhichis solvedonceagain.IterationsbetweentheCPandstage2problemscontinueuntil thenewsolutionis -closetothelatestoptimalitycut. Unfortunately,theintegralityoftheobjectiveinDESP/SP2nolongerallows supportinghyperplanesforthestage2problem(seeforexample[3,5,20,42]). Nonetheless,alowerboundontheoptimalsolutioncanbeobtainedbyapplyinga decompositionmethodonarelaxedversionofDESP/SP2.Thealgorithmpresented “ndsalowerboundbysolvingasimilarproblemto -optimality.Theproblem solvedreplacestheintegralityof y inS2Pwithacontinuousversion CS2P Z2( x,j)=minyfj 2( y )(2.16) subjectton1i =1xi+n2( j)i =1yi M, (2.17) y 0 ,y Rn2(2.18)

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19 DetailsofthemethodareinMurphey[30]whereitisshownthatitterminatesin a“nitenumberofiterationswithasolutionthatis -optimalforDESP/CS2Pand consequentlyyieldsalowerboundsolutionforDESP/S2P. Step0 .Set =0, L =0, k =1. Step1 .Solvethecurrentproblem ¯ Z1( x )=minx,f1( x )+ (2.19) subjectton1i =1xi M, (2.20) x b (2.21) Akx + dk k =1 ,...,L (2.22) x 0 ,x Zn1(2.23) Ifinfeasible,STOP:problemisinfeasible.Elsedenotesolutionby( xk,k). Step2 .For j =1 ,...,s solvesecondstageprimal Z2( xk,j)=minyfj 2( y ) subjectton1i =1xk i+n2( j)i =1yi= M, y 0 ,y Rn2anditsdual maxminy 0fj 2( y )+ jn1i =1xk i+n2( j)i =1yiŠ M Letthesolutionstotheseproblemsbe( yj k,j k). Step3. De“ne

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20 A =sj =1pjj kd =sj =1pj fj 2( yk)+ j k( eTyj kŠ M ) where e isavectorofonesofcomparablesizeto yj k.If k 0wherehere e isavectorofonesofcomparablesizeto x ,thenaddcutto currentproblem AkeTx + dk .Let k = k +1.Returntostep1. ElseSTOP:( xk,k)is -optimalsolutionforDESP/CS2P. Analgorithmforthecurrentprobleminstep1ofthelowerboundingalgorithmisneededsincenon-Booleanintegralityandnon-linearitypreventusfrom usingMMRasforSWTA-U.WhereastheSWTAconstrainsthesumofthedecisionvariablestoequalitywith M ,thecurrentproblemhasinequalityconstraints onthesumofthedecisionsasinequatio ns(2.20)and(2.22).Anaturalapproach forhandlingtheconstraintsistoiterativelyapplyMMRtothecurrentproblem wheretheconstraints(2.20)and(2.22)arereplacedwiththeconstraintNj =1xj= m where m = Š d A and m isiteratedfrom0 , 1 ,...M .Thenforeachvalueof m ,thevalueof is uniquelydeterminedandhence“xedintheobjective.

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21 2.4 An Alternative SP Formulation for The Dynamic WTA Anotherwaytospecifythe2-stagestochasticprogramforWTAisasfollows min Kk =1 nkj =1Vkqxkjk+ E[ Q ( x , )] subjecttoKk =1 nkj =1xkj M, xkj log( kjŠ qk) , k=1,2, ...,K,j =1 , 2 ,...,nk(reducedvaluebound, qxkjk kj) xkj Z ,k =1 ,...,K,j =1 ,...,nkwiththesecondstagesolutionfrom Q ( x , )=min Kk =1 s kj =1Vkqys kjk subjecttoKk =1 nkj =1xk+Kk =1 s kj =1ys kj= M, ys kj Z ,k =1 ,...,K,s =1 ,...,S,i =1 ,...,s kUsingthisform,itcanbedeterminedthatthe“rststageobjectiveisalinear combinationofunivariatenon-linearsummands qxkjkwhichcanbereplacedbya piecewiselinearfunction j( xkj)suchthat xkj Z+: j( xkj)= qxkjj,j =1 ,...,N whichmeansthatallverticesofthelinearfunctionarelocatedatintegerpoints andforintegervaluesoftheargument,thelinearfunctionequalsthenonlinear summands.Nowbyrelaxingtheintegralityofthedecisionvariables xkjand representingthepiecewiseconvexfunctions j( xkj)bythemaximumof nklinear functions,eachbeingalinesegment( qk)m[(1 Š qk)( m Š xki)+1] ,k =1 ,...,K,i = 1 ,...,nk,m =0 ,...,M Š 1,alinearized“rststageisobtained.Thesamecanbe

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22 doneforthesecondstagevariables ykj.ThelinearizedSPisthen min Kk =1 nki =1Vkzki+ 1 SSs =1 Kk =1 s ki =1Vkus ki subjectto zki ( qk)m[(1 Š qk)( m Š xki)+1] , k =1 ,...,K,i =1 ,...,nk,m =0 ,...,M Š 1 ,Kk =1 nki =1xki M, us ki ( qk)m[(1 Š qk)( m Š ys ki)+1] , k =1 ,...,K,i =1 ,...,s k,m =0 ,...,M Š 1 ,s =1 ,...,S,Kk =1 nki =1xki+Kk =1 s ki =1ys ki= M,s =1 ,...,S, xki,ys ki,zki,us ki 0 . Notetheaddedconstraints,with zkjand ukjreplacing xkjand ykjsuchthatthey areboundedabove(maximum)bythelinesegments. 2.5 Discussion Of Weapon Target Assignment WeaponTargetAssignmentproblemslikethosediscussedinthischapterhave beenstudiedformanyyearsandhavecapturedmanyaspectsoftheproblemposed intheintroductorychapter.However,theWTAmodelsalsoleavemuchtobe desired.ThefundamentalworkinWTAwasdevelopedfordirect“reweaponsand somissesmuchoftheintelligentandloiteringaspectsoftheunmannedaircraftof interest. First,allthemodelsexaminedassumethatweaponsoraircraftareindependent,which,forthetypesofintelligent,sensingunmannedaircraftandmunitions ofinterest,issimplynotthecase.Asanexample,warheadeectsshouldbecorrelatedwhenthedamagecausedbythe“rststrikechangesthesecondweapons probabilityofdestroyingthetarget.Clearlythisismoreoftenthecasethannot.

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23 Further,consider2weaponsthatareabletoachieveahighlevelofeectivenessby sequentiallystriking2distinctaimpoints-the“rstaimpointdesignedtoŽsoftenŽ thetargetorstopitfrommovingsothatthesecondstrikeismoreeective.Extendingthislineofthought,correlationappearsinmanyofthetaskingsexpected oftheseaircraft.Forinstance,sensorsoneachplatformdonotmakeindependent observationsofacommontarget-theyareactuallycorrelatedsincethesceneat eachplatformismostlythesame. Second,usingtheWTAmodels,taskingoftheaircraftislimitedtoengagementanddestructionofthetargets.Theproblemsofinterestclearlyhaveother, lessdestructivemodesofoperation,forexample,search,classify,actasadecoy, andsoon. Finally,thekinematicscapturedbytheWTAmodelsarecoarseatbest.These kinematicaspectsleadtotheconsiderationofpathplanningproblems.

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CHAPTER3 PATHPLANNINGPROBLEMS 3.1 Introduction To Path Planning Inpathplanningproblems,thegoalistodeveloptrajectoriesfor M vehiclesto N objectives,whichcouldbetargetsbutmoregenerallyaresimplypointsinspace or waypoints .Someexampleobjectivesareasfollows. € Minimizesearchtimeto“ndallobjectiveswithunknownlocations € Maximizecoverageordetectionrateofobjectiveswithunknownlocations € Minimizeexposuretothreatswhileenroutetoobjectives,whereobjectives andthreatsknown € Minimizesurvivingvalueoftargetobjectives € Minimizetimetocoverallobjectives Fundamentally,pathplanningproblemsmaybedividedinto2categories: smoothtrajectorydesign,andwaypointplanning,thelatterbeingcloselyrelatedto theVehicleRoutingProblemvariantoftheTravelingSalesmanproblem. 3.2 Trajectory Design Considertheproblemofplottingatime-spacetrajectoryin2ormoredegrees offreedomforasingleairvehicle.Theremaybemanyconsiderationsinvolved, includingmaintenanceoftrim”ightcondi tionsofairspeed,altitude,angle-ofattack,andthelikethatareadvantageousforgoodsensorperformance,avoiding obstacles,andavoidingthreatslikeantiaircraftgunsormissiles.Likewise,the objectiveofthevehiclemaybeto“ndandreportonobjectsofinterest,survey featuresoftheterrainandhumaninfrastructure,oravoidthreatswhileenrouteto knownobjects. 24

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25 Nowconsiderthegeneralproblemoftrajectorydesignfor M vehicles.The considerationsforsinglevehiclesapplyhereaswell,however,therearesomeunique aspectsaswell.Collisionavoidancebetweenvehicles,ensuringecientsearchof theregionofinterest,andpossiblyprecisecontrolofrelativepositionbetween vehicles.Inanycase,trajectorydesignproblemsshareacommonproduct-smooth curvesintime-spacethatthevehiclesareexpectedtofollowquiteclosely. Tothisend,therearetwoapproachestodevelopingsmoothtrajectories: CalculusofVariationsthatoptimizeaperformancecriterionsuchasminimum risk(exposuretothreats)anddevelopcontinuoussolutionsforthetrajectory andsplinedconstrainedpathplanningwherediscretepointsinspace-timeare smoothedintocontinuoustrajectoriesbyusingsplinesorsomeothergeometrical approximationlikeVoronoiDiagrams. Agreatdealofworkhasbeendoneonminimumriskpaths[33,48].The majorcontributionofvariationalmethodsisthatagreatdealofinsightisobtained thatcanbeusedtodevelopon-lineheuristics.Variationalmethodsarenotuseful foron-lineimplementationsincetheydonothandleevenminorexcursionsfrom precisemodels.theexample[48]makesaverygoodcase-in-point:nomorethan asinglethreatcanbeconsideredwithoutsubstantiallossof“delity.Morethan2 threatsiscomputationallyintractable.Hencetheauthorsturntodiscretemethods, likeconstrainedshortestpath,wherebysplinesorVoronoidiagramscanbeusedto smooththetrajectories. ToseehowVoronoidiagramscanbeusedtocreatesmoothtrajectories, consideraset S of n sitesinEuclideanspaceofdimension d .Foreachsite p S , theVoronoicell V ( p )isthesetofpointsthatarecloserto p thantoanyother pointin S .Therefore,Voronoidiagram, V ( S ),isthespacepartitioninducedby Voronoicellsforall p S .Inpracticalterms,theedgesoftheVoronoidiagramare equi-distantfromadjacentpointsin S .AnexampleisinFigure3.2.

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26 Figure3Â…1:ExampleVoronoiDiagram ThepracticaluseofVoronoidiagramsistheconstructionoftheirgeometric dual,theDelaunaytriangulation;thatis,twopointsin S arelinkedbyanedge intheDelaunaytriangulationifandonlyiftheircellsareincidentintheVoronoi diagramof S . AnexampleofaDelaunaytriangulationisshowninFigure3.2.Observethat asthenumberofpointsin S areincreased,theedgesintheDelaunaytriangulation aresmoothed.So,thegoalofpathplanningbyusingVoronoidiagramsreducesto thatofconstructingaset S thatyieldsasucientlysmoothtrajectoryconsisting offollowingedgesintheDelaunaytriangulation.Theedgesthatmakeupthe trajectorymaybeobtainedusingashortestpathalgorithm. Inthecaseofrendezvous,Voronoidiagramshavebeenusedtogreatsuccess [28].Heretwovehiclesarerequiredtorendezvousatsomepointinspace.The objectiveistosimultaneouslyobserveorattackanobjectortarget. Withtheexceptionofrendezvous,whichisanarrowinterpretationofresource assignment,theintentofweapontargetassignmentislostinallcasesoftrajectory design.Thereisnoattempttoassignindividualvehiclestoobjectives.Instead,

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27 Figure3…2:ExampleDelaunayTriangulation thegoalistosimplybuildtrajectoriestogetthere.Thisisagoodtimetoconsider thatpursuingtheintegratedproblemofresourceassignmentandtrajectorydesign maybeafoolserrand.Afterall,resourcescanbeallocated,asinweapontarget assignmentproblems,afterwhichtraject oriesinspace-timemaybeconstructed. However,aswillbediscussedintheSection3.4,thispositionignoresafundamental couplingbetweentheproblems;thatfeasibleresourceassignmentmustbeconstrainedbytrajectorykinematicsandthatkinematicsmustrespondtoresource assignment,evendynamically.Adduncertaintyintothemixanditisclearthat somemeansoftreatingtheintegratedproblemmustbefound. 3.3 Waypoint Planning BeginwiththeTravelingSalesmanProblem(TSP).Whiletherearemany variants,thegoalofthebasicTSPisto“ndatourof N citiesoflowestcostthat beginsandendsatthesamepoint.TheVehicleRoutingProblem(VRP)isa variantthatattemptsto“nd M suchtours,oneforeachof M vehicles(salesmen), suchthateachcityappearsinexactlyon etour.Thegeneralformulationofthis

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28 problemisasfollows[7].De“nedecisionvariables xijk= { 1 , ifvehicle k visitscustomer j immediatelyaftercustomer i ;0else yik= { 1 , ifvehicle k visitscustomer i ;0else andcost cijcorrespondingtotheeortordistancefromcustomer i to j .Let Qkbe thecapacityofvehicle k .ThentheVRPisthefollowingintegerprogram Min i Ij Icijk KxijkSubjectto k Kyik= m,i =1 k Kyik=1 ,i =2 , 3 ,...N i Iqiyik Qk, k K j Ixijk= j Ixjik= yik, i I, k K i Ij Iyik| S |Š 1, S { 2 , 3 ,...N } , k K wheretheobjectiveisto“nd M toursofminimumtotaleortwhere M = | K | . The“rstconstraintensuresthatallvehiclesvisitthedepot-acommonlaunching point.Thesecondconstraintensuresthateachcustomerisvisitedbyasingle vehicleexactlyonce.Thethirdconstraintisacapacityconstraint-assumingthat eachvehicledelivers1unitofproducttoeachcustomerandeachcustomerhasa unitydemand, Qkissimplythenumberofcustomersvehicle k canvisit.Theforth constraintensuresthateverycustomerisadescendentandaprecedent,thusa cycleisguaranteed.The“fthconstraintistoeliminatesubtours. VRPsarehardtosolve.Whilebranchandboundmethodshavebeenattempted,theyarenotsuccessfulforevenmoderatesizedproblems.Asaresult, constructiveheuristics,thatseparatetheproblemintothefundamentalassignment androutingcomponents,tendtotakeprecedenceinpracticalliterature.Thetwo

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29 basicapproachesarecluster“rst,routesecondŽandroute“rst,clustersecond.Ž Themostfamousexamplesofthe“rstarethePetalalgorithm[41]andtheSweep algorithm[37].Thesecondmethodisdominatedbymethodsthatsimplybuildone largeTravelingSalesmanrouteandthenuseashortestpathalgorithmtodivide thesingletourintomultipletoursforeachvehicle. Anotherapproachtowaypointpathplanningthatmayseemabitsurprising istheMultidimensionalMultichoiceKnapsackProblem.Theknapsackofinterest has N sizes(features)andthereare M elementstoplaceintotheknapsack,the elementscanbepartitionedinto G groups.Eachelement i hasasinglevalue vibut N weights, wij,eachcorrespondingtoasizeoftheknapsack.Theobjectiveisto placeoneelementfromeachgroupintotheknapsacksuchthatthesumofweights foreachsizeisboundedbysomecapacity cj. Thismodeliscraftilyadapted[29]todescribeaVRPvariantdeveloped[8] whereallthepossibletoursareenumeratedapriori.Thealgorithmthenconsiders thewaypointplanwhichhas N waypointsand M feasibleroutes,previously enumerated.Theroutesaregroupedinto G vehicles.Theweight wij{ 0 , 1 } is thecapabilityofroute i tovisitwaypoint j , viisthevalueorgoodnessofroute i . Theobjectiveisthentochoose1routeforeachvehiclesuchthatthesum(over vehicles)ofthecapabilitiestovisitawaypointisconstrainedto cjvisits. TherearemanyextensionstotheVRP.Someoftheextensionsofinterestare nowdescribed. 3.3.1 Capacitated VRP (CVRP) Here,eachcustomerhasademand Diwhichonemightrelatetotheweapon targetassignmentproblemasbeingthevalueofthetarget(customer).If Qk= Qk= Q k = k ,thentheCVRPhasatractablesolutionbasedon“nding minimumK-trees[12].

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30 3.3.2 Split Delivery VRP (SDVRP) ThisisageneralizationoftheVRPwhereeachcustomersmaybeservedby morethanonevehicle.Ifthecustomerdemandsareclosetoorlargerthanthe vehiclecapacities,thenthisformulationbecomesquiteimportant.Totransform aVRPintoaSDVRP,spliteachcustomerorderintoanumberofsmallerorders [4].Oneofthedownsidestodoingthisisthatthenumberofvariablesisincreased greatly.Notsurprisingly,[10]itisconc ludedthattheSDVRPismuchharderthan VRP. 3.3.3 VRP With Time Windows (VRPTW) Supposeatimewindowisassociatedwitheachcustomer,de“ningatime intervalduringwhichthecustomerhastobesupplied.Thevehiclesareallassumed tobethesameandtheobjectiveistominimizethenumberofvehiclesneededor thetotaltravelingtime.Severalauthors[22,23,39,43]havelookedatthisdicult problem.Solutionmethodstendtofocusonmetaheuristics. 3.3.4 Stochastic VRP (SVRP) ConsideraVRPwhereoneorseveralcomponentsoftheproblemarerandom. ThreedierentkindsofSVRPareconsideredintheliterature. 1.Stochasticcustomers:Eachcustomerispresentwithsomeprobability. 2.Stochasticdemands:Thedemandofeachcustomerisarandomvariable. 3.Stochastictimes:Servicetimesandtraveltimesbetweencustomersare randomvariables. SolutionmethodsforSVRPimplyapplicationofstochasticprogramming[21]. 3.4 The Case For Integrated Resource Assignment and Path Planning TheVehicleRoutingProblemandallitsextensionsjustdiscussedlooklikea promisingwaytomodelintegratedresourceassignmentandpathplanning.Indeed, theextensionsaddingcapacities,splitcustomerdemands(delivery),timewindows,

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31 andstochasticdemandsareexactlythesortoffeaturesdescribedinthemotivating exampleinSection1.1. Aspromisingasthismaybe,VRPsstillfailtocapturethefundamental knapsackstructureofWTAproblems,inthatVRPsforcetourstoallcustomers, evenifskippingonecustomerinfavorofmaking2ormoredeliveriestoanother isabettersolution.TheonlywayaVRPcanallowassignmentofmorethan onevehicletoacustomeristosplitthedemandatthatcustomerintosmaller pieces,eachpiecebeingsuppliedbyaseparatevehicle.Unfortunately,thisnot onlyaddsvariablestotheproblem,italsofailstosolvetheprobleminthatnow allthesubdemandsmustbesatis“ed.Inotherwords,theVRPdoesnotcontain theknapsack-like”exibilityinnon-homogeneousassignmentoptionsthatWTA possesses.Asanotherexample,considerthatVRPsignoretheuniquenessofeach vehicleasde“nedforthegeneralWTA.Thatis,acustomerdoesnotcarewhich vehiclemeetshisdemand,justsolongasthedemandismet.InWTAthisis clearlynotthecaseassomeweaponsaremuchmoreeectiveatdestroyingcertain targetsthanotherweapons. Nowthatthegoodandbadfeaturesofnon-homogeneousassignment(knapsack)andVehicleRoutingProblemshavebeenuncovered,itistimetoexplicitly setforththede“ningtraitsoftheintegratedresourceassignmentandpathplanningproblem.TheyareasinTable3…1. Inthefollowingtwochaptersmoregeneralmethodsofresourceassignment andpathplanningarestudied.Somemethodshaveafairlyhighlevelofkinematic integration,whileothersincludeagreatdealofresource”exibility,including precedenceintasks,cooperativeresourcesharing,resourcedivisibilityamong taskings,andresourceexpertise.Anobviousplacetostartisintegerprogramming. So,inChapter4integerprogrammingmethodswillbeexplored.

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32 Table3…1:De“ningTraitsOfTheObjectiveModel 1. Non-HomogeneousTargets/Objectives. Sometargetsorobjectivesmaybeworthmorethanothers. 2. KinematicConstraints. Vehiclesmustobeydynamicallawsof motion. 3. MultidisciplinaryPrecedence. Asequenceofmultidisciplinary tasks(“nd,classify,track,engage,assess,spoof,jam,...)isdesiredfor eachtarget/objective.Ifthechainisnotcompleted,theremaystillbe valueorderinchainisnotstrict,i.e.,generalprecedence. 4. CorrelatedResources,ActivitySharing. Thechainmayexhibit dependenciesandcorrelationintimeandspace-onthesametarget/objectiveoracrossmultipletargetsorvehicles.Activitiesmaybe sharedbymorethanoneresource. 5. Expertise. Ecient,robustsolutionsmayfavornon-homogeneous vehicles:possessingdistinctlevelsofexpertise(e.g.,sensorspectrum, processing,comm,warheadeect,endurance,speed,...)foreachpotentialtasking.Lessthanperfectandincompleteexecutionoftasks maystillhavevalue. 6. TerminatingVehicles. Vehiclesmaybemunitionsthemselvesand thereforehaveanintentionalterminalassignmentstate(detonateor crash). 7. Objective:Eciency,Accuracy,andCompletenessofCollectiveMission. Vehicleshave“xedmissiontimes(duration).Therefore,minimumcompletiontimemaynotbearealisticgoal. 8. Uncertainty. Quantity,type,andvalueoftargets/objectivesare uncertain.Theuncertaintymaynotbemodeled(risk).

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CHAPTER4 MIXEDINTEGERPROGRAMS Beginninginthischapterandcontinuingintoallthefollowing,asmuchas possible,acommonnotationwillbeused.Thisismainlytomakeiteasierto comparemodelsbutisalsogoodpreparationforborrowingvariouscomponentsof eachmodelforbuildingtheobjectivemodelinChapter6. 4.1 Transshipment Problems The“rstmodeltobeexaminedattemptstocombineattributesofVRPand Knapsackassignment.Consideracollectionofairvehiclesthathave2con”icting goals,searchfornew(unknown)targetsorprosecuteknowntargetsasmembersof multidisciplinaryteams.Teamshavenonhomogeneousmembers;thatis,[13]there existmultipleteamrolesincludingsearchŽ,classifyŽ,strikeŽ,assess,Žandsome vehiclesarebetteratcertainrolesthanothersasiscapturedintheparameter cgrjwhichisthevalueofassigningvehicle j toteam g withrole r . Vehiclesalsohavevariablepro“cienciesatsearchingfornewtargetsasis determinedbyvariable vjwhichisthevalueofvehicle j beingassignedtosearch forunknowntargets. Sovehiclesarepartitionedintoteamsandlonesearchers.Butsimultaneously, theteamrolesaredecidedandthetargetsareassignedtothedierentteams.So, thereisaverygoodrepresentationoftheresourceexpertisetraitsoftheobjective model.Theproblemdatamaybesummarizedasfollows 33

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34 Indexsets: Activities: i I Alltargets Resources: j J Allvehicles Team:g G Allteamsformedfromindividualvehicles TeamRoles:r R Allroletypes:search,classify,strike,assess Parameters: vj:Valueofvehicle j searchingforunknowntargets wg:Targetassignedtoteam g (determinedo-line) ng:Numberofrolesonteam g cgrj:Valueofassigningvehicle j toteam g withrole r DiscreteVariables: xj= { 1 , ifvehicle j assignedtosearch;0else eg= { 1 , ifteam g exists;0else ygrj= { 1 , ifvehicle j isassignedrole r onteam g ;0else Clearlyanintegerprogramwillresult.Interestingly,thisIPtakestheformofa so-calledcapacitatedtransshipmentproblemwhere,inthegeneralcase,”owsare shippedthroughintermediatenodes.Inth iscase,thenodesaretargetsorsearch requirements.Theedgesonthistransshipmentproblemgrapharecapacitatedfor teamcomposition.Theproblemformulationisasfollows

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35 Max j Jvjxj+ g Gr Rj JcgrjygrjSubjectto g Geg 1 , i I, j Jygrj= eg, g G, r R g Gr Rygrj+ xj=1 , j J wheretheobjectiveismaximizingalinearmixtureofthebene“tof“ndingnew targetsandprosecutingexistingones.The“rstconstraintensuresthateverytarget hasatmostoneteamassignedtoit.Theabilityofthatteamtoprosecutethe targetisassumedapriori.Thesecondconstraintensuresthatforeveryvehicle assignedtoateamhasaroletoplay.Thethirdconstraintsrequireseveryvehicle toeitherjoinateamorsearchforunknowntargets. With egpredeterminedor“xed,thebasismatrix;thatistheconstraintmatrix A intheexpression Ax b ,istotalunimodularwhichguaranteesanintegral solution.Withthisbeingthecase, xjand ygrjmayberelaxedtopositiverealand abranchandboundappliedon egtosolvetheIPandthesolutionwillbeintegral. Clearlythetransshipmentmodelhassomenicefeatures.Itdoesagoodjob ofresourceassignmenttocon”ictingobjectives.Itmodels,tosomeeect,resource collaborationandresourceexpertise,and recognizestheneedformultidisciplinary typesofactivitiesoneachtarget.Aframeworkforuncertaintyisaddressed,ina limitedsense,byposingsearchversusteamassignment.Finally,themodelcanbe computationallytractable,especiallyforsmallnumberofteams. Therearesomeshortcomingsaswell.First,theemphasisisonassignment tothecompleteexclusionofkinematics.Thatisevidentfromthefactthatthe resourcesandtasksare“xedinspaceandtheassignmentisaone-timedecision withoneactivityperresource/teamandnoattempttobepredictive.Thetargets

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36 areclearlymultidisciplinary,butthetasksareallassumedtooccursimultaneously andthereisnoprecedenceoccuring.Thisislikelypushedoutfortheteamtode con”ictasaseparateoptimizationproblem.Essentially,thismodelisverymuch liketheWTAmodelinthateverytargetisterminating.Thusthereisnoconcept ofschedulingtrajectoriesfromoneassignmenttothenext. Whileresourcescooperateonactivities,thereisnowaytomodelcorrelated resourceeciency.Thatwouldmeanthattheteamcompositionwouldbeaected toensureinclusionofresourcesthatareabletocollaborate.Theuncertainty frameworkmodeledbyposingsearchfornewtargetsversusassigntoateamto prosecuteanexistingoneisidenticalinspirittothatoftheStochasticWTA problem.However,herethereisnoexplicitmeasureofuncertaintyregardingthe beliefthatnewtargetsexistonnot.Thatuncertaintyislumpedintothecost vjwhichisnotquantitativelyrelatedtothenumberoftargetsorthetypeof targetsorthetrajectoryoftargets,allofwhichmaybemadestochasticwithsome operationalmeaning.Finally,theform ulationiscomputationallynicebutthe totalunimodularitystructurewillmostcertainlyfallapartwhentheadditional constraintsandvariablesneededtomodeltheseadditionalfeaturesjustmentioned areadded. 4.2 Multipurpose Batch Plants MultipurposeBatchPlantsorMPBsareaproductofthelatestadvancesin combiningresourcesharingandschedulingformultiproductproductionandplant design.Assuch,thesemodelsmayoersomeideasfortheintegratedassignment andpathplanningproblem.ThefeaturesoftheMBP[16,17,24]arethatthereare avarietyofproductsproduced,eachwithauniquerecipe.Theuniquenessofeach productimpliesthattheywillallrequireschedulingofresourcesindierentorders andfordierenttimes,needuniqueprocesseswiththoseresources,andrequire timephasedhandlingofuniquematerialrequirements.Schedulingasingleplant

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37 Figure4…1:ExampleOfMultipurposeBatchPlant toco-manufacturemultiple,uniqueproductsinatimeeectivemannerimplies con”ictsandthesharingofresources.AnotherfeatureofMBPsisthatresource schedulingiscompletelywithnotonlymaterial”ow,butalso,synthesisofthe resourcesthemselves.Thus,theMBPissimultaneouslyoptimizingthequantities andtypesofresourcesavailablewiththeactualassignmentofthoseresourcesand material”owstoproductlines.Thisalllooksquitepromisingfortheintegrated assignmentandpathplanningproblem. AnexampleofaMBPisprovidedinFigure4.2.Hereweseethreeresources, amixer,reactor,andpuri“cator,eachcapableofmultipletypesofactivities.For example,thepuri“catorcanpurify,separa te,orstore,dependinguponthedemands oftheparticularproduct.Thisisanalogou stotheintegratedassignmentandpath planningproblemsneedformultidisciplinaryprecedence. WhiletherehasbeensomeworkgoingonforseveralyearsinMPBs,the modelshavebeenhamperedbyverylargenumbersofintegralvariableswhichmade solutionsimpossibletocomeby.Therealbreakthroughinmodelingwasmadeby IerapetritouandFloudas[16]whichoersauniqueformulationthatdecouples theactivityassignmentfromtheresourceassignment.Thisdirectlyleadstoa signi“cantreductioninthenumberofbinaryvariables.Forinstance,ifthereisa threeindexintegervariable xi,j,kwithdimensions ni,nj,nkrespectively,thenvia aseparationof( i,k )and( j,k ),therearenow nink+ njnkvariablesinsteadof

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38 Figure4…2:ExampleTimelineandEventlineForAMultipurposeBatchPlant ninjnkvariables.For ni= nj= nk=10 , thismeansadierenceof200versus1000 variables. InFigure4.2,isatimelinethatillustratesthebasisofthevariableseparation. Here,acontinuousvariable Ts ijkisusedtode“nethestarttimeofactivity i by resource j ateventtime k .Sothediscrete,Booleanvariables wvikand yvjk,are usedtodecideifactivity i startsateventtime k andifresource j alsobeginsat eventtime k respectively.Therefore,thereisaseparationofwhatwouldhavebeen a3-indexdecisionofresource i workingonactivity j attime k intoamixedinteger program,wherethenumberofdiscretevariableshasbeenreducedthroughthe introductionofcontinuousvariables.Thistypeofseparationmightalsobeuseful inreducingthecomplexityofthemultidimensionalassignmentproblem,discussed inSection5.3,whichisultimatelythemathprogramstructureadoptedforthe integratedassignmentandpathplanningproblem. Inadditiontothe3variablesjustintroduced,consideralsothe“nishtime Tf ijkofactivity i byresource j ateventtime k and4othercontinuousvariablesusedto settheresourcecapacitiesandmaterial”ows. bijkistheamountofmaterialthat resource j canprocessonactivity i ateventtime k , sjisthedesignedcapacityof resource j , dskistheamountofmaterialtype s thatisdeliveredateventtime k , and stskistheamountofmaterialtype s availableateventtime k .

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39 Thereisalsooneadditionaldiscretevariable, ej,whichissetto1ifresource j ischosentobeapartofthedesignandzerootherwise. TheMBPhasseveralparametersthatmaybesettocontroltherelative proportionofstateproducedandconsumedbyproduct s .Yes,aproductline productionmayactuallyconsumeanotherproductthatisproducedevenifthat otherproductisasaleableendproductitself.Thesellingpriceofproductsissetas arethecapitalcostsontheresourcesthatmayormaynotbechosentobeapart oftheplant.Insummary,theproblemdatafortheMBPis Indexsets: Activities: i I Allactivities, Ij I activitiesthatcanbeperformedbyresource j Resources: j J Allresources, Ji J resourcesthatcanperformactivity i States:s SAllstate(material)types EventTime: k K Alleventspoints, T istimehorizon Parameters: ps:Sellingpriceofstate(product) s j, j, j:Capital(designed)costofresource j (constant,coecient,exponent) Vmin j,Vmax j:Designcapacityconstraintsonresource j rs:Marketdemandforstate(product) s attime T p si:Proportionofstate(product) s producedbyactivity i c si:Proportionofstate(product) s consumedbyactivity i DiscreteVariables: ej= { 1:ifresource j exists;0else } wvik= { 1:ifactivity i startsateventtime k ;0else } yvjk= { 1:ifresource j startsateventtime k ;0else }

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40 ContinuousVariables: bijk=Amountofmaterialresource j processesonactivity i ateventtime k sj=Designedcapacityofresource j dsk=Amountofmaterialtype s deliveredateventtime k stsk=Amountofmaterialtype s availableateventtime k Ts ijk=Starttimeofactivity i byresource j ateventtime k Tf ijk=Finishtimeofactivity i byresource j ateventtime k TheMBPisset-upasamixedintegernon-linearprogram(thenon-linearityisin theobjectivefunction)asfollows min j J jej+ js jj Š s Sk KpsdskSubjectto yvjk ej, j J, k K Vmin jej sj Vmax jej, j J i Ijwvik= yvjk, j J, k K bijk sj, i I, j Ji, k K stsk= stsk Š 1Š dsk+ i Ip sij Jibijk Š 1+ i Ic sij Jibijk, s S, k K k Kdsk rs, s S Sequenceconstraints: Ts ijk +1 Tf ijk, i I, j Ji, k K Ts ijk +1 Tf ijkŠ T (1 Š wvik) ,i = i, i Ij, i Ij, j J, k K Ts ijk +1 Tf ijkŠ T (1 Š wvik) ,i = i,j = j, i I, i I, j Ji, j Ji, k K Ts ijk T ,Tf ijk T , i I, j Ji, k K

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41 The“rstconstraintistoestablishifaresourceischosenfortheplantornot.The secondthenconstrainsthechosenresourcestowithinspeci“edcapacities.The thirdconstraintsimplyrequiresthatifanyactivityisstarted,thentheremustbe aresourceavailabletoworkonit.Theforthconstraintmaintainsalimitonthe materialcapacities.Constraint5statesthatifmaterialisneededforanactivity itmustbetheamountnecessarytodothejob.Thenext3constraintsensure precedenceinthemodel.The“rst2sequenceconstraintsensureprecedencefor thesameactivitybeingexecutedbyadierentresource.Soforexample,themixer andreactorperformingthestoreŽactivitymustdosoatdierenttimes.This isanimportantdistinction,forintheintegratedassignmentandpathplanning problem,itmaybeadvantageoustohave2ormoreresourcesworkontheactivity atthesametimewhereasthisisnotsofortheMBP.Thethirdsequenceconstraint statesthatasingleresourcemustsequence2dierentactivities;thatis,itcannot workon2thingsatonce.Thelastconstraintsimplyensuresthatallassignments arecompletedbeforetheendofthe“nitetimehorizon, T .Theobjectiveisto minimizethecapitalcostsofinstallingtheplantminusthepro“tsduetosalesof theproductsproduced. Certainlythisisavery“nemodelwithlotstorecommendit.Theintroduction ofatimedependencymeansthatthereiscausetosequenceresourcesandactivities soastomeetmultipleobjectives(products).Thenprecedenceisintroducedthat ispreciselythesortof”exibilityneededinservicingmulti-activityjobs.The“nite horizonisexactlythesamesortof“xedtimeoptimizationthattheintegrated assignmentandpathplanningproblemrequire.And“nally,theformulationis certainlyniceforreducingthenumberofdiscretevariablesandgivesoneaccessto wellknowmixedintegersolverslikeCPLEXandMINOPTthatactuallyperform quirewellforratherlargeproblems.

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42 Still,therearesomeissueswiththemodel.First,theresourcedonotneedto travel,so,whilethereispathplanninginthesenseofroutingmaterialsthrough resources,therearenokinematicsassociatedwiththis-itisassumedthatthe productmovesinstantly,whichisproperforamanufacturingplant,butnotfor ”yingvehicles!Furthermore,asalludedtoearlier,theresourcescannotcooperate onanactivity, i Ijwvik= yvjkimpliesonlyoneresourceperactivity.Ofcoursethis constraintcouldbechanged,butthatwoulddonogoodforthereisnoroominthe costmodeltoaccountforcorrelatedresourceeciencywhen2ormoreresources workthesameactivityatthesametime. Theplantdesignfeatureofmodel,whileinterestinginitsownright,isnot needed,noristhenonlinearcapitalcostfeature. Theproblemofintegratedassignmentandpathplanningdoesnotrequireany sortofmaterial”ow.However,material”owiswhattriggerdemandforactivities intheMBPmodelandsomustbekeptforthatpurposeonly.Tryingtomodel acontinuousmaterial”owforthekindsofproblemsbeingconsideredhere,while notoutofthequestion,isapoor“tandintroducesmorecomplexthannecessary. Imaginerepresentinga”eeinggroundtargetasaproductlineandtheactivities are“ndingthetarget,identifyingit,trackingit,andsoon.Butwhatdoesproduct volumeandcapacitymeaninthislatersense?WouldmoreproductŽmeanmore target?ŽTargetsdonotsimplyscaleinvolumeasmaterialdoes,sohavingmore targetŽtoforcelargercapacityresourcesdoesnotreallyhavemeaninginthe integratedassignmentandpathplanningproblem. Alsoofinterestinintegratedassignmentandpathplanningareterminating activities.FortheMBP,thiswouldrequirethatresourcesbedeactivatedafter aperforminganspeci“cactivitythatisterminating.ŽItseemsasthoughthis sortofconstraintcouldbeaddedtothismodel.Finally,uncertaintyismodeled incapitalcost,whichisreallyofnovaluetotheintegratedassignmentandpath

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43 planningproblem.Uncertaintyisneededontheresourceecienciesoractivity types,quantities,andtime-trajectories.Unfortunately,thisisnotpossiblesincethe modelstrictlyde“nesallactivitiesaprioribysettingproductiongoalsonproduct andestablishesrecipesthatdesignatespeci“cactivityuse.Theproductionlevels forproductscouldbemadestochastic,butthentheproblemde“neintheprevious paragraphreturns. 4.3 Unmanned Air Vehicle Path Planning Abodyofwork[2,40]directlycitethegoalofdevelopingaframeworkfor integratedassignmentandpathplanningproblems.Inthiswork,acollectionof airvehiclesisconsideredthatarerequiredtoover”yacollectionofobjectiveseachobjectivebeingasinglewaypointlocationinspace.Theobjectiveisthento developtrajectoriesforeachoftheairvehiclessothatallwaypointsarevisited inminimumtimeandeort.WhilethissoundsquitesimilartotheVRP,andin manyrespectsitis,akeydierenceisthatinthisworkthevehiclesmusttravelon smoothtime-spacetrajectoriesfromonelocationtothenext.Hencethemodels fromthiseortarethe“rsttotrulyincorporatekinematicsintopathplanningas discussedinSection3.2formultiplevehicles. Obstacleavoidanceisalsoasigni“cantpartofthemodelinthatthevehicles must”yaroundobstaclethatlieintheirpathswhiletraversingfromonewaypoint tothenext.Thisissigni“cantbecauseitmakesthekinematicsindispensablefor determininghowwaypointswillbescheduled.However,forbrevity,theobstacle avoidancepartofthemodelwillnotbeincludedhere. MuchliketheMBP,thismodelisbaseduponasystemofinteractingcontinuousanddiscretedecisionvariables.Thecontinuousvariablesrepresentthe continuousmotionoftheaircraftsuchthat¯ zkj=[ xvkj,yvkj, xvkj, yvkj]Tisthestate (positionandvelocities)vectorofvehicle j attime k and ¯ fkj=[ fkj1,fkj2]Tisthe forcevectorofvehicle j attime k .ThediscretevariablesareBooleansothat bijkis

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44 1ifvehicle i visitswaypoint j attime k andzerootherwise.Twoothercontinuous variablesareaddedtominimizethetimevehiclesspend”yingthetrajectories. Theyare tT,whichistheoverallmissioncompletiontime,and tjwhichisthe missioncompletiontimeofjustvehicle j .Thislatervariableisaddedtoensure thateachvehiclemakesabesteorttobeecientanddoesnotwastetimesimply becausesomeothervehicleisessentiallycriticalpathfor tT. Thecontinuousstatesofthevehiclesaremodeledbyadierenceequation typicalofthatfoundintheestimatio nandcontroloflineardynamicalsystems.Thus A , B are“xedlinearsystemdynamicmatrices, vmaxjand jare themaximumvelocityandminimumturnrateofvehicle j andthevehiclestate evolvesas¯ z( k +1) j= A ¯ zkj+ B ¯ fkj.Theinitialvehiclestatesarespeci“edby Sj, 1:4=[ xv0 j,yv0 j, xv0 j, yv0 j].So,byspecifyingforce ¯ fkj,acompletetimetrajectory ofthestate¯ zkjforvehicle j maybedescribed. Vehiclesexhibitstrictexpertiseintheirabilitiestoprocesswaypoints-they areeitherabletoserviceawaypointornotwhichiscapturedinarray Qjiwhichis 1ifvehicle j canvisitwaypoint i andiszerootherwise. Precedenceisalsoafeatureofthemodel.Theindexset D containstheset ofwaypointindicesthatdisplaysomeformofdependency.Amatrixdjde“nes theactualprecedencerelationshipsbetweenwaypoints.So, diis Š 1ifwaypoint i isaprecedentfordependency d D andis+1ifwaypoint i isadescendantfor dependency d D .Clearly,istotalunimodular.Thedatafortheunmannedair vehiclepathplanningproblemissummarizedinthefollowing

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45 Indexsets: Waypoints(activities): i I Allwaypoints Vehicles(resources): j J Allvehicles, Ji J vehiclesthatcanvisitwaypoint i DynamicStates(vehicleorobject): s { 1 , 2 , 3 , 4 } ,s =1: x,s =2: y,s =3: x,s =4: y Time: k K Alldiscretetimepoints Precendence: d D Allwaypoint(activity)dependancies Parameters: vmaxj:Maxvelocityofvehicle j j:Minturnratesofvehicle j Sjs:Intitialstate s ofvehicle j : Sj, 1:4=[ xv0 j,yv0 j, xv0 j, yv0 j] Wis:Coordinate s ofwaypoint(object) i : Wi 1= xwi,Wi 2= ywiQji:Vehicle j capabilityw.r . t . waypoint i : Qj,i= 1:If i Ji0:else dj:Timedependency d forwaypoint i di= Š 1:Ifwaypoint i isprecedentfordependency d +1:Ifwaypoint i isdecendantfordependency d 0:else d:Timeintervalbetweenwaypointswithdependency d A , B :Linearsystemdynamicmatrices 1 ,2:Positiveweightsonrelativevalueoftime-vs-eort R:Alargepositivenumber

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46 ContinuousVariables: ¯ zkj:Statevectorofvehicle j attime k :¯ zkj=[ xvkj,yvkj, xvkj, yvkj]T¯ fkj:Forcevectorofvehicle j attime k : ¯ fkj=[ fkj1,fkj2]Ttj:Missioncompletiontimeofvehicle j tT:Overallmissioncompletiontime DiscreteVariables: bijk= { 1:ifvehicle i visitswaypoint j attime k ;0else } Themathprogrammingmodelisamixedintegerlinearprogramthatminimizesthe“nalcompletiontimeoftheentirecollectionofaircraftplussomelinear combinationoftheindividualcompletiontimesofeachaircraftandtheamountof forceexpendedinmovingtheaircraftalongthetrajectories.Thelinearcombinationoftheselatertwoobjectivesisbecausetheytendtobeoppositegoals-one couldgetafastcompletiontimewithalargeexpenseofenergyorconversely,an energyconservingpolicycouldbedevelopedthatwouldtakealongtimetocomplete.Themultipliers, 1,and 2aredesignedtoallowtheanalysttodetermine therelativeimportanceoftotal“nishingtimeversusamountofforce(ergofuel) expended.Themathprogramisasfollows

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47 min¯ f, ¯ z, ¯ btT+ 1j J tj+ 2k K( | fkj1| + | fkj2| ) Subjectto ¯ z( k +1) j= A ¯ zkj+ B ¯ fkj, j Ji, k K xvkjŠ Wi 1 R (1 Š bijk) , i I, j Ji, k K xvkjŠ Wi 1Š R (1 Š bijk) , i I, j Ji, k K yvkjŠ Wi 2 R (1 Š bijk) , i I, j Ji, k K yvkjŠ Wi 2Š R (1 Š bijk) , i I, j Ji, k K k Kj JQjibijk=1 , i I i Idik Kj Jkbijk d, d D tj j Jkbijk, i I, j J tT tj, j J The“rstconstraintiskinematic,toensurethevehiclestravelsmoothtrajectories.Thenext4constraintssimplydetectwhenavehiclehasvisitedŽawaypoint. Thenextconstraintensuresthateachwaypointisvisitedexactlyonce,constraint6 ensuresprecedenceisobeyedforeachwaypoint,andthelasttwoconstraintssetthe completiontimesforeachvehicleandthetotalschedule. Aspreviouslystated,thismodelisthemostsincereattempttointegrate kinematicsdirectlyintoanintegratedpathplanningandresourceassignmenttask. Italsomodelsprecedencebetweenactivities,andaddressesresourceexpertise, thoughinasomewhatlimitedfashion.Theweaknessinresourceexpertiseisthat thevehiclecapabilitiesareBooleanforeachwaypoint,whichdoesnotpermitless thanperfectperformanceforawaypoint-thevehicleeitherknowshowtoservice thatactivityoritdoesnt.Ofcourse Qjicouldbemadenon-Boolean,butthe optimizationframeworkstillhasnowaytodierentiatebetweenvariousexpertise levels.

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48 Themodelalsotreatseachobjectiveasasingleactivityormultipleactivities aresimplylumpedtogetheratasinglewaypoint.Thiscouldbe“xedfairlyeasily bymakingadditionalvirtualwaypointsatthesamelocationforeachadditional activity.Eachobjectiveisalsoofequalvalueinthismodel-thereisnowayto showpreferencebetweenthedierentwaypoints. Theobjectivetominimizetotaltimeandindividualvehiclecompletiontimes isad-hocandnotreallyanissueforvehi clesthathave“xedmissiontimes.The assumptionthatminimumtimecompletionisimportantassumesthattherewill beverylittleuncertaintyinthescenarioandthatthereareenoughvehiclesto serviceallthewaypointsasspeci“ed.Iftherearenotenoughvehicles,eitherdue touncertaintyorsimplysupplyissues,thentheproblemwillbeinfeasible.This doesnotallowtheanalysttoperformmissionsthatmeetmostoftheobjectives,Ž orperformthebestvalueeort.ŽRelativetothis,themodelalsodoesnotallow anyformofactivitysharingbetweenresourcesandsohasnoroomforcorrelated eciencybetweenmultipleresources. Toensuretrulysmoothpaths,timemustbediscretizedintosucientlysmall stepstoallowthedierenceequationtomeetsomeleveloferror.Thistimestep sizeisdirectlysetbythedynamicsofthevehicles,sothataveryfastmoving,fast turningvehiclewouldrequireshortertimeintervalsthanaslowervehiclewould. Evenso,forarealisticUAV,thenumberoftimestepstendstogrowquitelarge ratherquicklyandsothedimensionoftheproblemexplodesandwithit,the likelihoodofsolvingtheproblemdiminishes.Theauthorshavelookedatways to“xthisproblembyeitherrestrictingthetimehorizonandessentiallysliding ittotherightwithasolutionstepeverysooftentoaddthenewtimeintervals, orbyrelaxingtheproblemviaaseparationofthetrajectorygenerationandthe assignmenttoresources.

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49 Overall,thismodelisaverygoodVRPwithkinematicconstraintsbutfalls shortontheresourceassignmenttraits.

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CHAPTER5 NEWPRODUCTDEVELOPMENTPROBLEMS 5.1 Introduction To New Product Development NewProductDevelopment(NPD)concernstheallocationoflimitedresources totheactivitiesofaportfolioofnewdevelopmentprojects.Speci“cally,the resourcesareoftenpeopleandtheprojectsmayexhibitoverlappingneedsfor resources.Theresourcesareassumedtobe”exibleinthattheymayperformmore thanoneactivityortaskwithvaryingdegreesofeciencyorsuccess.Oftenin NPDtheresourcesareconsideredtobedivisiblewhichmaybeappropriateata macrolevel.However,inthispapereachresourceismodeledtoagranularitythat requirestheresourcesattentiontobefocusedonasingletaskatanymoment intime.ActivitysharingisanotherhallmarkoftheNPDinthatmorethan oneresourcemaybeassignedtoworkagivenactivityconcurrently.Activity sharingincorporatesthedynamicsofteamwork,wherebyworkersaremore productiveingroupsthanindividually.Indeed,someactivitiesmayrequirea teamofresourcestoachieveevenminimalsuccess( e.g., liftingheavyobjects, followingsafetyconstraints).Eachprojectisassumedtoconsistofacollection ofactivities.Anexampleofactivitiesfor3projectsisgiveninFigure5.1.The activitiesmaybeindependent,however,inmanycasestheyarecoupledthrough precedencerelations.Theserelationstypicallyinhibitworkonadescendentactivity untilitsprecedentshavebeencompleted.Finally,sinceNPDsmayspanmany diverseproducts,theywillalsotypicallyspangeographicboundaries.Soitisthat activitiesmaybeaccomplishedatmultipleworksitessuchthatifanygivenproject issplitbetween2sites,theneithertheprojectortheresourcesmusttravelbetween 50

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51 Figure5…1:ExampleOfNDPActivities. thesiteswithsome“nite,measurabledelay.See[38,44,45]formoreinformation andbackgroundonmodelsandalgorithmsforNPDs. De“nethefollowingindexsets Projects: p P Allprojects Activities: i I Allactivitiesinallprojects, Ip I allactivitiesinproject p Di I allactivitiesthatprecedeactivity i Resources: j J Allresources InSection5.2,acontroltheoreticmodelwillbeintroducedthattreatsthe amountofworkremainingforanactivity asacontinuous”owwithstateevolution dictatedbya“rstorderdierentialconstraint.Itwillbeseenthattheresulting optimalcontrolproblemisintractableforthegeneralcaseofactivityprecedence. Furthermore,sinceresourcesaredivisible,theauthorsdidnotconsiderresource traveltimefromoneactivitytothenext.Forthesereasons,analternatemodelis introducedinSection5.3.ThismodelisaMultiDimensionalAssignment(MAP) formulationforNPDsandpromisestoc apturethedynamicsofresourcesand

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52 includegeneralprecedenceofactivitiesandcorrelatedresourceeciencies.In thissectionthevalueofanassignmentwillbedevelopedfromjointprobability distributionsthatdescribethesuccessofresourcesprocessingactivities.Using thisde“nitionofcost,anupperboundwillbeconstructedonthemaximization formulationoftheMAPforthethreeindexassignmentproblem(3AP).Thisresult willbeextendedtoMAPsofarbitrarydimension. 5.2 Control Theoretic Model WangandPerkins[45]recognizedthatacontinuous”owmodelcouldbe developedthatwouldtreatthedecisionvariableasacontinuoustimefunction specifyingtheamountofworkperformedbyaresourceonagivenactivity.The evolutionoftheworkloadisadierentialconstraintandtheoptimaldecisionis obtainedthroughanoptimalcontrolformulation.Themainadvantageofthis approachisthatitcapturesdivisibilityofresourcesandactivitysharing.Thatis,a singleresourcemayworkmorethanoneactivityatonceandanyactivitymaybe processedbymorethanoneresource.Thislatterfeatureisofparticularinterestas theapproachesconsideredsofarhavenotconsideredthis. Considerthefollowingmodelwithparameters µij:Maxratethatresource j canprocessactivity i Di I Allactivitiesthatprecedeactivity i andcontinuousvariables t :Time ij( t ):Fractionofresource j dedicatedtoactivity i attime t wi( t ):Amountofactivity i remainingattime t i( t ):Lagrangemultiplierforactivity i stateevolutionconstraint anddiscretevariables ei( t )= { 0if wi( t ) > 0forany i Di;1else }

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53 Ofthevariables, ij( t )istakenastheonlycontrolordecisionvariable, and ei( t )issimplyanindicatorfunctionthattakesthevalueoneonlywhenall activitiesthatprecedeactivity i arecompleted.Since ij( t )iscontinuous,thenwe knowthattheamountofactivity i remainingattime t is wi( t )= wi(0) Š j Jµij t0ij( ) d where wi(0)istheamountofworkonactivity i attimezero.Dierentiatingthis expressionyieldsadierentialequationdescribingthedynamicconstraintsonthe activitycompletionrate wi( t )= Š j Jµijij( t ) . (5.1) Theprecedenceconstraintsdonotallowanyfractionofresource j beallotted toactivity i aslongasalltheprecedentsofactivity i , Diareincompletedas determinedby ei( t )=1.So,foreachresource, i Iij( t ) ei( t ) 1 , j J. (5.2) UsingPontryaginsMinimumPrinciple[18],anoptimalcontrolisthesolutionthat minimizestheHamiltonianof min H ( w ( t ) , ( t ) , ( t ))=min { g ( w ( t ) , ( t ))+ ( t ) w ( t ):s . t . i Iij( t ) ei( t ) 1 , j J } =min { g ( w ( t ) , ( t )) Š i Ij Ji( t ) µijij( t ):s . t . i Iij( t ) ei( t ) 1 , j J } where g ( w ( t ) , ( t ))issomecostfunctionintheperformancecriterionof =T0g ( w ( t ) , ( t ))

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54 and i( t )areLagrangemultipliersofthedynamicconstraintsonactivity i obtained fromEquation5.1.TheMinimumPrincipledictatesthat (t)= Šw ( t )g ( w ( t ) , ( t )) . Nowsupposethat g ( w ( t ) , ( t ))=1.Thisistheresultofusingtotalcompletiontime, T astheperformancecriterionsinceT0dt = T. Then ( t )=argmin ( t ){ 1 Š i Ij Ji( t ) µijij( t ) } (5.3) Subjectto: i Iij( t ) ei( t ) 1 , j J (5.4) Thisisaverydicultproblemtosolvebecauseoftheprecedenceconstraints inEquation5.4.Thatis,determining ei( t )explicitlyasafunctionofprecedent activitycompletion.Togetatractableformulation,theauthorsassumedarelaxed formofprecedence,where µij= µij, i I, j J, and j> 0 wherenow µiisanactivityspeci“cprocessingrate,independentoftheresource, and jistheeciencyofresource j regardlessoftheactivity.Essentially,ifa resourceisslowatprocessingoneactivity,itisslowatallactivities.Thesame holdstrueforfastresources.De“netheaverageeciencyofallresourcesas R= j Jj.Thentheminimumcompletiontimeisproven[45]tobe T = 1 R i Iwi(0) µi

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55 oressentially,theaverageactivityworkload,weightedbyanactivityspeci“c processingrate,andnormalizedbyaverageeciencyofresources, R .Theproof howeverdoesnotspecifytheoptimalcontrol ( t )whichinpointoffact,not unique.Theauthorswereunabletoachieveanysolutionsonanyproblemswith strongerprecedenceconditions.Furthermore,tryingtoaddkinematicsofresources andnon-trivialdistancesbetweenactivi tiestothemodelwouldmaketheproblem evenmoredicult.Whiletheactivitysharingfeatureofthemodelisattractive, itdoesnotgofarenoughbyconsideringcorrelatedbehavior.Asaresult,anew formulationfortheNPDproblemneedstobeconsideredandispresentedinthe nextsection. 5.3 Multidimensional Assignment Problem Model Inthefollowing,theunionofsets A and B is A B andtheirintersectionis A B . Consideraportfolioof p =1 , 2 ,...,P projectseachconsistingofsome“nite numberofactivities.Thereexist j J resourcesand k K timestepswith whichtocompletealloftheprojects.Assumeeachactivityrequiresonetime steptocomplete.Wemaydothiswithoutlossofgeneralitybyequatingthetime stepduration, t ,withthedurationoftheshortestactivityandensuringthat allotheractivitiesaredividedintosubactivitieseachof t duration.Notethat theactivitiesofunmannedaircraftarealreadyfairlysimilarinduration,sothis assumptionisnotcomplicatingforintegratedassignmentandpathplanning.An exampleofthisactivityframeworkissh owninFigure5.3.Thetotaltimealloted, T ,istheplanninghorizonandconsidered“niteand“xed.Let I betheunique setofallactivitiesintheportfolio.Assumethateachresourcepossessesaskill set, Ij I possiblynotunique,thatthetotalcollectionofskillsissucientto accomplisheachproject,andthateachactivitymaybeaccomplishedatonlya limitednumberofapprovedworksites.Let Ji J betheallresourcesthatcan

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56 performactivity i .Usingthisframework,thenotionthatanactivitymayreceive serviceatmorethanoneworksiteisrealizedbycreatingaresourcethatrepresents eachworksite.Theresourcesrequiredtoperformtheworkonanactivity,typically personnel,mustbejointlyassignedtotheactivitywiththeworksiteactivity.They mustalsomovebetweentheworksiteresourceswithsomeconstanttimedelay. De“ne Tiijtobethetraveltimedelayforresource j fromactivity i toactivity i.AsmentionedinSection5.1,precedenceofactivitiesisanimportantaspect ofNewProductDevelopmentproblems.Hence,let Di I beallactivitiesthat immediatelyprecedeactivity i .De“neaBooleandecisionvariablethatschedules resourceassignmentsasfollows xk,i1,i2, ··· ,iM= 1 , Ifresource1assignedtoactivity i1andresource2assignedto activity i2and ··· resource M assignedtoactivity iMattime k, 0,Else suchthat i1 I1,i2 I2,...,iM IM.Eachdecisionhassomevalue ck,i1,i2, ··· ,iMthatwillbedescribedinthefollowingsection.Insummary,theproblemdatais Indexsets: Projects : p P Allprojects Activities : i I Allactivitiesinallprojects, I = N,ij Ij I all activitiesinskillsetofresource j, Ij = njResources : j J Allresources, J = M,Ji J resourcesthatcanperform activity i Time : k K Discretetime

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57 Figure5…2:ExampleOfNDPWithConstantDurationActivities. Parameters: Di I :Allactivitiesthatimmediatelyprecedeactivity i Tiij:Traveltimebetweenactivity i and iforresource j ck,i1,i2, ··· ,iM:Valueofassignment xk,i1,i2, ··· ,iMDiscreteVariables: xk,i1,i2, ··· ,iM= 1 , Ifresource1assignedtoactivity i1andresource2assignedto activity i2and ··· resource M assignedtoactivity iMattime k, 0,Else Eachresourcemustbecommittedtoanactivityateachtimestep.To accommodaterest,repair,andtravelbetweensites,activitiessodesignated maybecreatedor,ifnotstrictlyrequired,dummyactivitiesmaybeused.In Figure5.3theactivitieslabeled D denotedummyactivities.Resource j maybe assignedanyoralloftheactivitiesin Ij.Let N = || I || ,/M = || J || ,/nj= || Ij|| withaspecialcase nkreservedforthenumberoftimesteps.Dummyactivities maybeaddedtotheskillsetforeachactivitysothat nj= n forall j where

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58 Figure5…3:ExampleOf3-IndexAxialConstraints. n =max { mK,n1,n2,...,nM} .Whilethisisnotstrictlynecessary,itcanbe convienientforanalysis.Theaboveassumptionsallowstheformationof axial constraints as i1 I1i2 I2··· iM IMxk,i1,i2, ··· ,iM=1 , k K (5.5) k Ki1 I1··· ij Š 1 Ij Š 1ij +1 Ij +1··· iM IMxk,i1,i2, ··· ,iM 1 , ij Ij, j J (5.6) A3-indexexampleoftheaxialconstraintsisshowninFigure5.3.Herethere are2projects:AandB,withactivities { A 1 ,A 2 A 3 } ,and { B 1 B 2 } ,2 resourceswithskillsets I1= { A 1 ,A 2 ,A 3 ,B 1 ,D } ,I2= { A 1 ,A 2 ,B 2 ,D } ,and 4timessteps.Thenotation A B indicates A precedes B andthesymbol D denotesadummyactivity.Theaxialconstraintsallowthechoiceof3-tuplesof { time,activity (Resource1) ,activity (Resource2) } .Theexampleshowsahintof thetypesofcomplexbehaviortheaxialcon straintscancapture.Forexample,the sizeofskillsetsforResource1and2neednotbethesame.Activity A 1isshared bybothresourcesattimestep2whichdemonstratesteaming,and“nally,thereis adummyactivityassignedtoResource1attimestep3representingatransition timeasforinstancewhen TA1 , B1 , 1=1 ,TB1 , A3 , 1=2 ,TA1 , A2 , 2=1 ,TA1 , B2 , 2=1.

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59 AsstatedintheIntroduction,anactivitywithaprecedentcannotbestarted untilitsprecedentshavebeencompleted.Aconstraintthatguaranteesprecedence isobeyedisthefollowing i1 I1··· ij Š 1 Ij Š 1ij +1 Ij +1··· iM IMxk,i1...ij Š 1,i,ij +1, ··· ,iMŠ k K : k
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60 Collectingconstraints,thefollowingMultiDimensionalAssignmentproblem representstheNPD max k Ki1 I1i2 I2··· iM IMck,i1,i2, ··· ,iMxk,i1,i2, ··· ,iM(5.7) Subjectto i1 I1i2 I2··· iM IMxk,i1,i2, ··· ,iM=1 , k K (5.8) k Ki1 I1··· ij Š 1 Ij Š 1ij +1 Ij +1··· iM IMxk,i1,i2, ··· ,iM 1 , ij Ij, j J (5.9) i1 I1··· ij Š 1 Ij Š 1ij +1 Ij +1··· iM IMxk,i1...ij Š 1,i,ij +1, ··· ,iMŠ k K : k
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61 value.Let ck,i1,i2,...,iM=ProbabilityofSuccess(resource1 i1,..., resource M iMattime k ) = P ( k i1··· iM) . Thevalueofatupleoftime-assignmentsisjusttheunionprobabilityofsuccess. Themarginalprobabilitiesareeasytospecify. P ( ij)istheprobabilityofresource j successfullycompletingactivity ijindependentofanyotherresourceoractivity. P ( k )istheprobabilityofsuccessattime k ,regardlessoftheresourceoractivity. Theunionprobabilityisperhapsalittlelessintuitive.Thereare2waystothink abouttheproblem.First,wemayrequirethatallresourcesmustsuccessfullycompletetheirtaskstoclaimsuccessforthatstepintime.ThisistheMultiplicative ProbabilityLaw.Thesecondapproachclaimsthatsomemeasureofsuccessata timestepmaybedeclaredevenifonlyoneresourcehassuccess,butthatmore resourceshavingsuccessesshouldaddtotheoverallmeasureofsuccess.Thismodel istheAdditiveProbabilityLawandistheapproachtakeninthispapersince itmoretypicallyresemblestherealityofprojectdevelopment.Forthe3-index problemthisleadsto P ( k i1 i2)= Pk+ Pi1+ Pi2Š Pki1Š Pi1i2Š Pki2+ Pki1i2andforthe4-indexproblem P ( k i1 i2 i3)= Pk+ Pi1+ Pi2+ Pi3Š Pki1Š Pki2Š Pki3Š Pi1i2Š Pi1i3Š Pi2i3+ + Pki1i2+ Pki1i3+ Pki2i3+ Pi1i2i3Š Pki1i2i3andsoon.Here Pijisshorthandnotationfor P ( ij)and Pijij isthejointprobabilityoftheintersection j ij j ij.Notethatifresourcesareindependent, then Pijij = PijPij .Ifresourcesaremutuallyexclusive,then Pijij =0.Otherwise,

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62 Pijij = PijPij \ ijwith Pij \ ijtheconditionalprobabilityofsuccessofresource jcompletingactivity ijgiventhatresource j completedactivity ij. Clearlytheprobabilitymodelforcostadmitsarichdescriptionofteaming resources,timedependencies,andsoon.Ifresourcesaremutuallyexclusivein alltheiractions,thecostssimplifytosumsofmarginalprobabilitiesandthe MAPeasilyseparatesinto M +1integerlinearprogramswithlinkedconstraints. However,thisdoesignoresigni“cantaspectsoftheNPDproblem. 5.3.2 Upper Bound On 3-Index Problem PreviousworkonalgorithmsfortheMAPhavefocusedonheuristicsthat performwithoutbounds[31,32].Thisismainlyduetothedicultyindeveloping reasonableboundsontheproblem.To“ndanupperboundonthe3-IndexAssignmentProblem(3AP)arelaxationisrequired.Theonlyrelaxationsavailablefor MAPsareeitheracontinuousrelaxationofintegralityoraLagrangianrelaxation oftheconstraints.Theformerapproachisnotoriouslyverypoorasabound,the secondapproachmayprovidegoodresultsbutiscomputationallyexpensivefor large R [35,36].Inthispaperaseparationofthecostissoughtwhichmaynotbe feasiblebutwillprovideabestcasesolutionwhencertainaspectsoftheproblem arerelaxed.Withthisapproach,itishopedthatthestructureofthesolutionwill bepreservedtoalargedegreeandthattheboundwillbegood.Fortheremainder ofthepaper,theprecedenceandtraveltimeconstraintswillberelaxed.Forthe purposesofthisdevelopment,itissucienttorealizethatwithoutthem,theupper boundisstillvalid.Thederivationrequires2Lemmas. Lemma1. M i =1P ( i ) P ( M i =1i ) . Proof. TheprooffollowsdirectlybyapplicationofBoolesInequality. Lemma1couldbeusedasabounddirectly,butasitturnsouttheboundis trivialsincenoconstraintsareactive.Thereasonswillbeexplainedattheendof thissection.AbetterboundmaybeobtainedfromthefollowingLemma.

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63 Lemma2. P ( i j k ) P ( i j )+ P ( i k )+ P ( j k ) . Proof. Let A = i j .ThenfromLemma1, P ( i j k )= P ( A k ) P ( A )+ P ( k )= P ( i j )+ P ( k ).Since P ( k ) P ( i k )and P ( j k ) 0, P ( i j k ) P (i j )+ P ( i k )+ P ( j k ). ByapplicationofLemma2,aseparationofthe3APmaybeachievedas follows. ck,i1,i2= P ( k i1, i2) ,ck,i1= P ( k i1) ,ck,i2= P ( k i2) ,ci1,i2= P ( i1 i2) So, ck,i1,i2 ck,i1+ ck,i2+ ci1,i2.Inthecostfunctionofthe3AP max k Ki1 I1i2 I2ck,i1,i2xk,i1,i2 max k Ki1 I1i2 I2( ck,i1+ ck,i2+ ci1,i2) xk,i1,i2Therighthandsideistherelaxedobjectivefunctionandisequivalentto max k Ki1 I1ck,i1yk,i1+ k Ki2 I2ck,i2yk,i2+ i1 I1i2 I2ci1,i2yi1,i2where yk,i1= i2 I2xki1i2,yk,i2= i1 I1xki1i2,yi1,i2= k Kxki1i2. Substitutethenew2-indexdecisionvariables, y ,intothe3APconstraintstoget6 newconstraints k Ki1 I1xk,i1,i2 1 , i2 I2 k Kyk,i2 1 i1 I1yi1,i2 1 i1 I1i2 I2xk,i1,i2 1 , k K i1 I1yk,i1 1 i2 I2yk,i2 1 k Ki2 I2xk,i1,i2 1 , i1 I1 i2 I2yi1,i2 1 k Kyk,i1 1

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64 Combiningtherelaxedobjectiveandthenewconstraintstheupperboundproblem is max k Ki1 I1ck,i1yk,i1s.t. k Kyk,i1 1 , i1i1 I1yk,i1 1 , k + max k Ki2 I2ck,i2yk,i2s.t. k Kyk,i2 1 , i2i2 I2yk,i2 1 , k + max i1 I1i2 I2ci1,i2yi1,i2s.t. i1 I1yi1,i2 1 , i2i2 I2yi1,i2 1 , i1 (5.12)Thisisasigni“cantreductionincom putation.Whiletheoriginal3APhas complexityof O ( n !2),where n =max { T,n1,n2} ,thecomplexityofthebound subproblemsis O ( n3). ItshouldalsonowbeclearwhytheinequalityofLemma1wouldyieldavery poorbound.Suppose ck= P ( k ) ,ci1= P ( i1) ,ci2= P ( i2).TheLemma1bound wouldbe ck,i1,i2 ck+ ci1+ ci2.Thenthenewvariableswouldbe yk= i1 I1i2 I2xki1i2,yi1= k Ki2 I2xki1i2,yi2= k Ki1 I1xki1i2andtheresultingboundproblemis max k Kckyks.t. yk 1 , k + max i1 I1ci1yi1s.t. yi1 1 , i1 + max i2 I2ci2yi2s.t. yi2 1 , i2 (5.13) whichhasatrivial(andratheruseless)solution yk=1 , k,yi1=1 , i1,yi2=1 , i2. ThisdoesunderlinethefactthatLemma1,andindeed,themoregeneralBonferroniInequalities,maybeusedtodevelopmanydierentboundsforthisvalue structure.Forexample,the4indexprobabilitieshavethefollowingbound P ( k i1 i2 i3) P ( k i1)+ P ( i2 i3) .

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65 Whileat“rstblushthisseemslikeareasonablepossibility,ittoosuersfroma lackofadequateconstraints.Notethatthereisnodependencybetweentime( k ) andeitherresource2or3.Thisistrueofresource1aswell.Theboundsuggested inLemma2doesretainallpairwisedepende nciesandthereforeshouldyieldmore reasonableresults. 5.3.3 Extensions To Arbitrary Dimension Therelaxationresultsintheprevioussectionwillnowbeextendedtoan M-dimensionalAssignmentProblem(MAP)ofarbitrarydimension. Theorem1. P ( i1 i2··· iM) Mr =1 Ms = r +1P ( ir is) . Proof. Consider2cases,when M isevenandodd. M even. Let A1= i1 i2,A2= i3 i4,...,AM 2= iM Š 1 iM. ThenbyLemma1 P ( i1 i2··· iM)= P ( A1 A2··· AM 2) P ( A1)+ P ( A2)+ ··· + P ( AM 2) andsince P ( A1)+ P ( A2)+ ··· + P ( AM 2)= P ( i1 i2)+ P ( i3 i4)+ ··· + P ( iM Š 1 iM) Mr =1 Ms = r +1P ( ir is) , theTheoremholds. M odd. Let A1= i1 i2,A2= i3 i4,...,AM Š 1 2= iM Š 2 iM Š 1. ThenbyLemma1 P ( i1 i2··· iM)= P ( A1 A2··· AM Š 1 2 iM) P ( A1)+ P ( A2)+ ··· + P ( AM Š 1 2)+ P ( iM) .

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66 Since P ( A1)+ P ( A2)+ ··· + P ( AM Š 1 2)+ P ( iM)= P ( i1 i2)+ P ( i3 i4)+ ··· + P ( iM Š 2 iM Š 1)+ P ( iM)) Mr =1 Ms = r +1P ( ir is) againtheTheoremholds. Nowthemainresult.Withoutlossof generality,letthe“rstindexnowbe i1insteadof k andthedimension M includesthetimeindex.Thatis,relabelthe M indices k,i1,i2,...,iM Š 1as i1,i2,i3,...,iM.Thisissimplytoeasethenotationused intheproof.Ofcourse,theoriginalindicesmayalwaysbeusedinapplications. Theorem2. AnupperboundontheM-dimensionalAssignmentProblemwith axialconstraintsandcostsdeterminedbytheAdditiveProbabilityLawmayalways beobtainedbysolving M 2 decoupledLinearAssignmentProblemsoftheform UB =Mr =1 Ms = r +1 max ir Iris Iscir,isyir,iss.t. ir Iryir,is 1 , is Isis Isyir,is 1 , ir Ir . Proof. Use ci1,i2,...,iM= P ( i1 i2··· iM)andapplyTheorem1 ci1,i2,...,iMMr =1 Ms = r +1ciris. ThensubstituteintoEquations5.7,5.8,and5.9withallindicesconvertedto i1,...iM: MAP max i1 I1i2 I2··· iM IMMr =1 Ms = r +1cir,is xi1,i2,...,iMs.t. i1 I1··· ij Š 1 Ij Š 1ij +1 Ij +1··· iM IM 1 , ij Ij, j J,

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67 Let yir,is= i1 I1··· ir Š 1 Ir Š 1ir +1 Ir +1··· is Š 1 Is Š 1is +1 Is +1··· iM IMxi1,i2,...,iMandsubstitutethisintotheupperboundproblem.Thereare M2Š M new constraintsonvariables y ,eachsummedoverasingleindexas ir Iryir,is 1 , is Isor is Isyir,is 1 , ir Ir. Thereare M 2 newobjectivessoeachobjectivegetsM ( M Š 1) (M 2) =2newconstraints. Thetransformedandseparatedproblemis MAP maxMr =1 Ms = r +1 max ir Iris Iscir,isyir,iss.t. ir Iryir,is 1 , is Isis Isyir,is 1 , ir Ir So,theoriginalproblemwithcomplexity O ( n !M Š 1)wasreplacedwithabound problemofcomplexity O ( M 2 n3),asigni“cantcomputationalsavings. 5.4 Discussion Of Multidimensional Assignment Problems ThemultidimensionalassignmentformulationfortheNewProductDevelopment(NPD)problemholdsagreatdealofpromiseforincorporatingmostof thefeaturesofintegratedassignmentandpathplanning.ThevalueofanassignmentisrealisticallyrepresentedbytheAdditiveProbabilityLawwhichpermits resource”exibilityanddependencyconsistentwiththerealworld.Anupperbound problem,thatexploitsaninequalityfeatu reofthevaluefunctionensuresthatthe verycomplexmultidimensionalassignmentproblemwithaxialconstraintsmaybe decomposedinto M 2 linearintegerassignmentproblems.Thisboundpromisesto

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68 befairlystrongsincetheoverallstructureoftheproblemispreserved,unlikerelaxationsofintegrality.However,therearestillsomeshortcomingsofthemethods. First,theboundneedstobecheckedwithinanexperimentalframeworkincluding abranchandboundimplementationanduseofaglobalheuristiclikeGreedy RandomizedAdaptiveSearchProcedure(GRASP).Second,aclosedformsolution fortheoptimalitygapofthisboundmightbepossiblethroughtheapplicationof alternatingBonferroniInequalitieswhichcouldproveusefulindevelopingafast algorithm.Theprecedenceandtraveltimeconstraintsneedtobeaddedbackinto theboundingproblem,fortheoptimalitygapintheexistingformulationisonly validwithoutconsiderationofthesecomplic atingconstraints.Explicitlyrepresentingthepathstateevolutionmaymakethismucheasier,asdemonstratedforUAV pathplanninginChapter4.3.Aterminatingconstraintstillneedstobeaddedinto themodel.Finally,uncertaintyintheactivitystatesisnotyetmodeledbutthe methodusedinKrokhmalet.al.,[19]isdirectlyapplicabletothismodel.

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CHAPTER6 INTEGRATEDASSIGNMENTANDPATHPLANNING InTable6…1,eachoftheattributeslistedinSection3.4asthede“ningtraits oftheintegratedassignmentandpathplanningproblemareontheleft.Acrossthe topthe“vemodelsexaminedinChapters4and5arelisted.Entriesinthetable indicateweatherornot,andifsohowwell,eachmodelmeetsthecorresponding attributes.AGentryindicatestheattributeiswellmodeled,Yindicatesonly partiallymodeledandnoentrymeansthattheattributeisnotmodeledatall. Itisfairlyeasytoseethatthelastcolumn,MAPNPD,hasthebestcoverage oftheattributes.Theweaknessesarethelackofterminatingconstraints,inability tohandleuncertainty(ofwhichnomodelscored)andweakmodelingofkinematics. So,itislogicaltobegindevelopinganintegratedassignmentandpathplanning modelfromtheNDPMAP.Firstandforemost,theMAPNPDmustbeclearly extendedtointegratedassignmentandpathplanningdynamics.Thismeansthat correlatedsensorandothervehicleeectsmodelsmustbedevelopedandtest problemsdeveloped.Thetestproblemswillrequiredynamicstateevolutionof objectivesandtheabilitytorepresentvehicledynamics.Oncetestproblemsare built,numericalapproachesforobtainingsolutionsareneeded.Thiswillbedonein thefollowingchapter. 6.1 Modeling Approach AsintroducedinChapter1andrecalledmanytimeafter,theauthoris motivatedtostudytheproblemofIntegratedPathPlanningandAssignment bytheneedtocontrolmultipleairvehiclesinspaceandtime.So,asapointof departurefromtheNPDmodel,theair vehicles areequivalenttotheresourcesin theNDPmodelandtheprojectsinNDParenowtermed objectives .Justaseach 69

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70 Table6…1:AttributeScoresForIntegratedModels projecthasmultipleactivitiesassociatedwithit,sotoodoobjectives.Similarly, theseactivitieshaveprecedence-dependencerelationshipsthatcanbede“nedas constraintsasinEquation5.10.Itisimporta nttorealizethatwhileitisdesirable toplanoncompletingalltheactivitiesforeachobjective,thereisstillsomevalue incompletingonlysomeoftheactivities,solongasprecedenceisobeyed. Thevalueofaccomplishinganactivityisafunctionofitsrelativemerit andthedegreeofsuccesswithwhichanygivenresourceisabletocomplete thatactivity.JustaswiththeNDP,aresourcesmeasureofsuccessissimplya probabilityofsuccess,giventheactivityisengaged,andisuniquetothatresourceactivitypairing.ThereforeallofSection5.3.1applieshereaswell.Themethodfor generatingtheseprobabilitieswillbediscussedlaterinthissection. Itwillbeassumedthateachobjectivehasa“xedlocationinCartesian space.Inastrictsense,thisisnotalwaystruesinceanobjectivecouldbea

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71 vehiclemovingontheground.Nonetheless,thisassumptionwillholdforamoving objectivesolongthefollowingconditionsaremet. 1. Theobjective p movesataconstantspeed, v . 2. Thespeedoftheobjectivedescribesacircularregionofuncertaintythatis acontinuousfunctionoftime;thatis RU ( t )= ( vt )2.Thelocationofthe objectiveisattachedtothecenterof RU ( t ) . 3. Theregionofuncertaintyattheendoftimeinterval k ,termed RUk,issimply RUk= ( vk t )2,where t isthelengthofonetimeinterval. 4. Theregionsearchablebyaresourceinonetimeintervalis RS . 5. Forsometimeinterval k, RUk RS .Forall k>k, RUk>RS .Interval kis termedthe“escapetime”forthatobjective-resourcepair. 6. TheprobabilitiesofsuccessforallassignmentsthatincludeanyactivityassociatedwiththemovingobjectiveANDanytimeintervalgreaterthan karesetto zero.Thatis, P ( k i1 i2 ,..., iM)=0 , k>k, Subset { i1,i2,...,iM} Ipwhere Ipareallactivitiesofobjective p . JustaswithNDP,resourcesarekinematicallyconstrained.Thatis,each vehiclerequiresanonzero,“nitetimetoreachanobjective.Oncethere,itis assumedthatthevehiclerequiresexactlyonetimeintervaltoperformasingle activity.ThereforeEquation5.11holds. Toensurethataresourcealwayshasthe abilitytotravelfromoneobjectivetoanother,the nullactivity isintroduced.To beconsistent,thenullactivityindexwillalwaysbethe“rstactivityindexforeach vehicle.Therefore im=1correspondstovehicle m beingassignedtothenull activityandistrueforall m =1 , 2 ,...,M .Thenullactivityisnotrelatedtoany particularobjective.Allresourcesmayutilizethenullactivityasmanytimesas theywish.Therefore,thenullactivitymustbeassignableasmanytimesasthere

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72 aretimesteps.Thatis,ifthereare nktimeintervals,thentherighthandsideofall axialconstraintsthatinvolvethenullactivitymustbe nk.Thenullactivityisnot apreferredassignment.Thisistobesurethevehiclesdonotspendaninordinate amountoftimesimplytravelingaround.Thereforeallresourcesmusthaveavery smallprobabilityofsuccessforchoosingthenullactivity.Thenullactivityalsohas noprecedentsordescendants-itmaybeassignedforanytimeinterval. Activitiesinheritrelativemeritfromtheobjectivestheyareassociatedwith. So,unliketheNDPmodel,eachobjectivehasaunique preferenceweight associated withitthatisareal,positivescalar.Thisweightissimilarinnaturetothe preferenceweightusedintheWeaponTargetAssignmentmodel, Vj,and,aswas thecaseforWTA,isusedtoindicatepreferenceamongtheobjectives.Forthesake ofsimplicity,itwasdecidedthattheminimumweightanobjectivemayhaveisone andthemaximum10.Thisspanofweightswasfoundtoclearlyseparatecostsand forceanobservablebiasinthefavorofactivitiesassociatedwithobjectiveswith higherweights.Thenullactivityhasaweightof1. Inthefollowing4sections,thespeci“cdatastructuresforgeneratingtest problemswillbediscussed. 6.1.1 Generating The Cost Vector Inmodelingthisproblem,the“rsttaskwillbetogeneratethecostvector.An outlineoftheapproachisasfollows. 1. Capturethecostsofallpossiblesingle(marginal)vehicle-to-activityassignments 2. Buildanarraycontainingmarginalcosts 3. Buildanarraythatmarksactivitiesthatarecorrelatedifperformedatthesame time 4. Buildarrayscontainingthe2-wayand3-wayintersectionprobabilities 5. Buildthenalcostarray

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73 Ofcoursethemeasureofperformanceisactuallyavaluenotacost,butit ismathprogrammingconventiontousecostsŽineitherminimizationormaximizationproblems.AsrelatedinSection5.3.1thecost(value)ofanassignmentis simply, ck,i1,i2,...,iM=ProbabilityofSuccess(resource1 i1,..., resource M iMattime k ) = P ( k i1··· iM) whichcanbealwaysbedecomposedas P ( k i1 i2··· iM)=Mm = k, 1P ( im) ŠMr = k, 1 Ms = r +1P ( ir is)+Mr = k, 1 Ms = r +1 Mt = s +1P ( ir is it) Š ... =marginals-2-wayintersections+3-wayintersections... which,forourcostnotationisequivalentto ck,i1,i2,...,iM=Mm = k, 1cimŠMr = k, 1 Ms = r +1cir is+Mr = k, 1 Ms = r +1 Mt = s +1cir is itŠ ... (6.1) Thesecondandthirdorderintersectioncostsare cir is= ciscir\ isand cir is it= cis itcir\ is itrespectively. Computingthecostofanassignmentisasimpletaskoncethemarginal costsandconditionalcostsbetweeneventsarecapturedforaspeci“cscenario. Marginalcostsandconditionalcostsarespeci“ctothevignette,andassuch,could becomequiteaburdenontheanalysttosupply,especiallyiftheproblemchanges evenslightly.Fortunately,thisprocesscanbesimpli“edbyplacingallpotential objectives,activities,andvehiclesonemightencounterintoseveralcategories typicalforabroadlyde“nedscenario.Oncethemarginalsarecapturedforthese categoriesofobjectives,activities,andvehicles,itisasimplemattertogetallthe

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74 marginalcostsforanyvignette,nomatterhowmanyorwhattypesofobjectivesor vehiclesthereare. Assignmentsforeachobjectivemaybegroupedintoacommonlistofactivity types .Forinstance, resupplyrescueteam and findrefugees are2distinct objectivesin2distinctlocations,butbothrequireactivitytype findobjective inuncertainregion .Thusitisasimplematterforananalysttospecifywhat typesofactivitiesareassociatedwitheachobjective.Tomakeiteveneasieronthe analyst,partitionallobjectivesintoasetof classes ,suchthateachclassrequires theexactsameactivitiestypes.Forinstance, resupplyrescueteamA and resupplyrescueteamB haveidenticaltypesofactivities,eventhoughtheyare 2dierentrescueteamsat2distinctlocationsandthereforeare2objectives.This classassignmentiscapturedinthematrix Bobj( p,l )= 1ifobjective p isinclass l 0else Hence,theanalystneedonlyspecifyactivitytypesforcommonclassesofobjectives ascapturedinthematrix Itypes( i,l )= 1ifactivitytype i existsinobjectiveclass l 0else Thefollowingmatrixproductisanincidencematrixsuchthatavalueof1indicatesactivitytype i existsforobjective p Iobj= Itypes× BT objTheuniqueactivitylistisobtainedby“ndingall1valuesin Iobj.Thiswillbedone incolumnmajororder Iunique=“ndcolmajorder( Iobj==1) [ rowindex , columnindex ]

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75 sothe rowindex correspondstotheactivitytypeandthe columnindex tothe objective. Iuniqueisa numact × 2array,where numact isthenumberofunique activities. Aswithobjectives,vehiclesmayalsobeaggregatedintoa“nitesetofclasses. Forinstance, unmannedsensorcraftA and unmannedsensorcraftB are identicalintheircapabilitieseventhoughtheyaretwodistinctaircraft.This aggregationismadesuchthatvehicleswithinthesameclasswillalwayshave identicalsuccessatperformingthesameactivity.So,forinstance, unmanned sensorcraftA and unmannedsensorcraftB areexpectedtohaveidentical probabilitiesofsuccessforactivity findrefugeesatA .Furthermore,itwill alwaysbeassumedthat2vehiclesinthesameclasswillalwayshaveidentical probabilitiesofsuccessforthesameactivitytype.Thus unmannedsensorcraft A and unmannedsensorcraftB haveidenticalprobabilitiesofsuccessforeither activity findrefugeesatA or findrefugeesatB .So,theanalystneedonly specifythemarginalprobabilitiesofsuccessforeachvehicleclassperformingeach activitytype.Thisiscapturedinthearray Cclass( i,g )=ProbabilityofSuccess(vehicleclass g resourcetype i ) whiletheassignmentofvehiclestovehicleclassesisdoneinthearray Bveh( j,g )= 1ifvehicle j isinclass g 0else Thecrossproductofthetwoprovidetheprobabilitiesofsuccessforeachvehicle versuseachactivitytype Ctypes= Cclass× BT veh

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76 andtheprobabilitiesofsuccessforeachvehicleversuseachuniqueactivityisthen Cunique= Ctypes(:, Iunique( rowindex )) where Iobj( rowindex )aretheactivitytypeindicesforeachuniqueactivity.This matrix Cuniquecontainsthemarginalprobabilitiesofsuccessforeachunique vehicle-uniqueactivitypairing.Itisthefoundationofthemarginalcostmatrix. Thenextstepinpreparingthemarginalsmatrixistoaddarowtoaccountfor nullactivityassignmentsandacolumnforthetimeindices.Firstarowof10Š 4isaugmentedtothetopof Cunique.Thesenear-zerovaluescorrespondtothe probabilityofsuccessforeachvehiclebeingassignedthenullactivity.Thevalueis setverylowtoforcethevehiclestodoproductiveworkandtravelnomorethan necessary.Theyarenotexactlyzerotokeeptheassignmentviable,aswillbe seenlater,zerovalueswillbecompressedoutinaneorttoreducethenumberof variables.Ofcoursetheymaybesettoanevenlowervalue,butitwasfoundin simulationsthat10Š 4wassucienttopreventunnecessarytraveling. Next,an nklengthcolumn,withelements ckcorrespondingtotheprobabilities ofsuccessattime k, k K regardlessoftheresourceoractivity,isaugmented totheleftof Cunique.Thesemarginalcostsrepresentthecostofdoingbusiness atparticulartimes.So,forexample,ifnoactivityattime k isdesired,itcan beestablishedbysettingthecorrespondingcost cktonearlyzero.Or,ifthere isanincreasingriskoffailureasafunctionoftime,thiscouldbemodeledas ckdecreasingas k increases.Asthe“rstindexin ck,i1,i2,...,iMisreservedforthetime index,the“rstcolumninthemarginalcostarrayisalsoreservedforthetime index.Ingeneral,the“rstrowwillnothavethesamenumberofrowsas Cunique. ThisisnotaproblemthoughsinceintheNDP, ni = nj, i = j .Whenthisisthe case,asitusuallywillbe,itisasimplemattertopadeitherthe“rstcolumnor theremainingcolumnswithzeros,dependinguponwhichcontainsmorerows.The

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77 resultingmarginalsmatrixisthen Cmarg= ck(1) . . . ck( nk) 10Š 4, 10Š 4..., 10Š 4 Cunique The“nalstepinpreparingthemarginalsistore-labelthematrixtomakeitmore amenabletoprogrammingdatastructuresandtocompressthecolumnsof Cmargbysqueezingoutanyzeroentries.Tore-labelthematrix,(aswasdoneinSection 5.3.3),re-labelthe M indices k,i1,i2,...,iM Š 1as i1,i2,i3,...,iM.Thatis,letthe timeindexnowbe i1insteadof k andthedimension M includesthetimeindex. Toeliminatethezeroentriesinthemarginalmatrix,itmustbeputinto compressedform . De“nition1. Ifarectangularmatrixmaybeputintotrapezoidalformbypermutingcolumns,thenitissaidtobecompressed.Ifnot,thentheremustbeatleastone zero-valuedelementsuchthatanon-zeroelementisinarowbelowandwithinthe samecolumn. Thematrixissaidtobecompressedbecauseonlythe“rst nmrowsofcolumn m havenon-zerovalues;henceanyindexingintothematrixshouldbelimitedto the“rst nmrows.Aswillbeseen,thismovecanseverelyreducethenumberof variables.Clearlyanymarginalmatrixnotincompressedformshouldbemade so.Toputanon-compressedmatrixintocompressedform,permuteanon-zero valuefromalowerrowandwithinthesamecolumnintothepositionofthezero. Continuethisrowpermutationoperationuntilthematrixisincompressedform.

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78 The“nalmatrixhasthefollowingform Cmarg= ci1(1) ci2(1) ci3(1) ··· ciM(1) ci1(2) ci2(2) ci3(2) ··· . . . . . . . . . . . . ··· ciM( nM) ci1( n1) . . . ci3( n3) ci2( n2) (6.2) The2-way,3-way,andhigherorderintersectioncostsmayalwaysbespeci“edas afunctionofthemarginalsandconditionalprobabilities.Intheexperiments,it wasalwaysobservedthatall4-wayandhigherintersectionvalueshadanegligible impactontheassignmentvalues,nomatterthelevelofconditioning.So,onlythe 2-wayand3-wayintersectionvalueswillbecomputed. Asexplainedinitem3ofthesectionoutline,theonlytypeofcorrelatedcosts ofinterestareforsingleactivitiesthatareeitherenhancedormadeworseiftwo vehiclesareassignedtothem atthesametime .Thismaybemodeledfairlysimply byhavingtheanalyst“rst mark thoseactivities,andthenspecify,asapercentage, theincreased(decreased)costaboveandbeyondthatoftwovehiclesworking independently.Onlythe3-wayintersectioncostsareaectedbecausetheeventsto bemarkedare(vehicle r irandvehicle s is, attime i1). Forthemodelingassumptionsmade,all2-wayinteractionsareindependent sothatthe2-wayintersectioncostsare cir is= ciscir\ is= ciscirwhichareeasily computedfromthemarginalsandplacedintothearray C2-way( r,s,ir,is)= circis. ThealgorithmissimplebutprovidedinFigure6…1. WhilethemathematicsoftheMAPdonotrestrictthemodelto3-way interactions,itjustseemsratherpracticaltomakethisassumption.Itisdicultto

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79 procedure 2WayCosts ( solution ) 1 FOR r =1 ,...,M,s = r +1 ,...,M, ir Ir, is Is2 C2-way( r,s,ir,is)= circis3 ENDFOR 4 return ( solution ); end 2WayCosts ; Figure6…1:Two-WayIntersectionCostsPseudo-code conceptualizeofmanyreal-lifeexamplesof2-wayinteractions.Theyarelimitedto twoforms 1. ci1 ir,r =1:activitiesthataretimedependentwhileotheractivitiesarenot 2. cir isr =1 ,s =1 ,r = s :activitiesthatareco-dependentindependentoftime. The“rstcaseiscertainlyusefulbutnotforthescenariosmodeledbytheauthor. Thesecondcaseiseasilymodeledusingjust3-wayintersections. The3-wayintersectioncostswithoneoftheindices, r beingthetimeindexare cir is it= cis itci1\ is it= ciscitci1\ isitIfanactivityisnotmarked,thentheconditionaltermbecomes ci1\ isit= ci1andhence cir is it= ci1ciscit.Ifanactivityismarked,thenwecanassumethe correlationiseasilyaccountedfor byaddingtheeciency,denotedby ,tothe marginaltimecostbutsaturatingtheproductatavalueof1toensureitremainsa probability.Theresultis cir is it=max { 1 , ( ci1+ ) ciscit} . Thiswilleectivelyincrease(decrease)the3-wayintersectioncostforpositive (negative)eciencywhichwill,in-turn,increase(decrease)theassignmentcost because3-waycostsarepositiveinEquation6.1.Thisisexactlywhatonewould expect.If =0,then cir is it= ci1ciscit;thatis,thereisnocorrelationandthe termsareindependent,alsoasonewouldexpect.Finally,thevaluesof cir is itare

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80 procedure 3WayCosts ( solution ) 1 Create activity ( ij,j ) uniqueactivityof ij; 2 Create ( activity ) percentincreasedeciencyif activity assignedto 2vehiclessimultaneously 3 FOR r =1 ,s =1 ,...,M,t = s +1 ,...,M, ir Ir, is Is, it It4 WHILE activity ( is,s )== activity ( it,t ) 5 C3-way( r,s,t,ir,is,it)=max { 1 , ( cir+ ( activity )) ciscit} 6 ENDWHILE 7 ENDFOR 8 return ( solution ); end 3WayCosts ; Figure6…2:Three-WayIntersectionCostsPseudo-code intheinterval[0 , 1],asexpectedofaprobability.Analgorithm,baseduponthis process,isprovidedinFigure6…2.Afunction activity ( ij,j )isintroducedthat returnstheuniqueactivityIDforanyactivity ijofvehicle j . Notethatthereisoneotherpossibletypeofinterdependencynotyetdiscussed andthatisthe3-wayinteractionwhen2dierentactivitiesareassignedatthe sametime.Again,thiscaseisnotofinteresttotheauthorbutisnotdicultto capturewithinthismodelingframework. Theassignmentcostsmaynowbecomputed.ThisalgorithmfollowstheapproachindicatedbyEquation6.1exceptthatpreferenceweightsarealsoincluded. Theweightsarespeci“edasavectorofvalues suchthat ( i )istheweightattachedtoobjective i .Thepseudocodethenusesafunction objectype ( ij,j )to returntheobjectivetypeforanyindex ij,muchinthesamemannerthat activity worksinFigure6…2.Theweightsareelement-wise(notationbeing . )multiplied withthemarginalarray.Therootmeansquare(rms)oftheweightscorresponding to irand isaremultipliedelement-wisewiththetwo-waycosts.Thermsisperformedbecausetheintersectionoftwouniqueactivitiesin irand ismostoftenwill leadtotwouniqueweightsyetonlyoneterm: C2-way.Thelogicalwaytohandle thisistoaveragethetwoweightsintoasingleterm.Thesamelogicappliesto applyingweightstothe3-waycosts.

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81 procedure AssignCosts ( solution ) 1 Capture Bobj,Itypes,Cclass,Bveh,, 2 DO a Iobj Itypes× BT objb Iunique “ndcolmajorder( Iobj==1) [ rowindex , columnindex ] c Ctypes Cclass× BT vehd Cunique Ctypes(:, Iunique( rowindex )) e Cmarg ck(1) . . . ck( nk) 10Š 4, 10Š 4..., 10Š 4 Cunique f W ( objectype ) 3 WHILE columnperm ( Cmarg) NOTtrapezoidal 4( Cmarg) Compress ( Cmarg) 5 ENDWHILE 6 C2-way 2WayCosts ( I1,...,IM,Cmarg) 7 C3-way 3WayCosts ( I1,...,IM,Cmarg, ) 8 FOR r =1 ,...,M,s = r +1 ,...,M, ir Ir, is Is9 C2-wayTotal C2-wayTotal+ C2-way( r,s,ir,is) rms([ ( r,ir) , ( s,is)]) 10 FOR t = s +1 ,...,M, it It11 C3-wayTotal C3-wayTotal+ C2-way( r,s,t,ir,is,it) rms([ ( r,ir) , ( s,is) , ( t,it)]) 12 ENDFOR 13 ENDFOR 14 Cassign rows,col( . Cmarg)+ C2-wayTotal+ C3-wayTotal15 return ( solution ); end AssignCosts ; Figure6…3:AssignmentCostsPseudo-code ThemarginalvaluesareobtainedfromEquation6.2,the2-wayintersection costsfromFigure6…1,andthe3-wayintersectionsobtainedfromFigure6…2.As mentionedpreviously,4-wayandhigherintersectionswereignored,nomatterthe dimensionoftheproblem,simplybecauseintheexperiments,theynevermadeany signi“cantchangetotheoutcome.ThealgorithmissummarizedinFigure6…3. Numerical Example. Theanalystbeginstheprocessbyspecifyingthetypesofobjectivesand vehiclestobeexpectedinthescenario.Thisdataisenteredin Bobj,Itypes,Bveh,and

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82 Cclass.Inthisexample,thereare2objectives,oneeachoftype1and3 objectiveclass123 Bobj= 100 001 objective1 objective2 Objectivetype1isspeci“edwithactivitytypes1and2,objectivetypewith activitytype2,andobjectivetype3withactivitytypes1and3as objectiveclass123 Itypes= 101 110 001 activitytype1 activitytype2 activitytype3 Thereare2vehicles,oneeachinvehicleclasses1and2 vehicleclass123 Bveh= 100 010 vehicle1 vehicle2

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83 andthecostsofeachvehicleclassforeachactivitytypeareinthematrix vehicleclass123 Cclass= 0 . 90 . 60 . 3 0 . 00 . 40 . 0 0 . 30 . 50 . 9 activitytype1 activitytype2 activitytype3 Theinstrinsiccostofeachtimestepisspeci“edbytheanalyst.Herean attritionmodelisadopted,wherebyastimeincreases,thelikelihoodofsuccessfalls olinearly.Thisisaccomplishedbysettingthetimecostasdecreasingwith k . ck= 10 . 90 . 80 . 7 TFromhereontheprocessisautomated.The Iobjmatrixiscomputedtoalign speci“cobjectiveswithactivitytypes,thenthe Iuniquearrayiscomputedwhich speci“estheuniqueactivities. objective12 Iobj= 11 10 01 activitytype1 activitytype2 activitytype3 Iunique= activitytype 1at objective 1 activitytype 2at objective 1 activitytype 1at objective 2 activitytype 3at objective 2 uniqueactivity1 uniqueactivity2 uniqueactivity3 uniqueactivity4

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84 Usingtherowindicesof Iunique,thecostsforactivitytypesofeachuniquevehicle areindexedinto CtypesIunique( rowindex )= 1213 Tvehicle12 Ctypes= 0 . 90 . 6 0 . 00 . 4 0 . 30 . 5 activitytype1 activitytype2 activitytype3 Thenthosecostsaremappedtotheuniqueactivitiesas vehicle12 Cunique= Ctypes(: , [1213]T)= 0 . 90 . 6 0 . 00 . 4 0 . 90 . 6 0 . 30 . 5 uniqueactivity1 uniqueactivity2 uniqueactivity3 uniqueactivity4

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85 Finally,themarginalcostsareintegratedintoasinglematrixthatincludes instrinsictime,nullactivities,andvehicles timeveh1veh2 Cmarg= 1 . 010Š 410Š 40 . 90 . 90 . 6 0 . 80 . 00 . 4 0 . 70 . 90 . 6 0 . 30 . 5 nullactivity uniqueactivity1 uniqueactivity2 uniqueactivity3 uniqueactivity4 The“nalstepistocompressoutthezerossothat Cmargisassmallaspossible. Thisleaves timeveh1veh2 Cmarg= 1 . 010Š 410Š 40 . 90 . 90 . 6 0 . 80 . 90 . 4 0 . 70 . 30 . 6 0 . 5 6.1.2 Axial Constraints Therearethreetypesofconstraintsintheproblem,andgeneratingthese constraintsisstraightforwardifthemultidimensionalcostanddecisionvariables arraysare“rstconvertedintoalinearequivalentindex.Thatis,fortheindices i1,i2,...,iMthereisasingleindex l inalinearequivalentproblem.Adheringto thestandardintroducedintheprevioussectionwhen“ndingtheuniqueactivity list,thelinearequivalentindexwillbegeneratedfromthemultidimensionalindex

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86 Table6…2:ExampleofALinearEquivalentIndexInColumnMajorOrderFor M =3, n1= n2= n3=2 i1 i2 i3 l 1 1 1 1 2 1 1 2 1 2 1 3 2 2 1 4 1 1 2 5 2 1 2 6 1 2 2 7 2 2 2 8 incolumnmajororder.Forexample,if M =3and n1= n2= n3=2,Table6…2 showsthemultidimensionalindicesinthe3 leftcolumnsandthelinearequivalent indexincolumnmajororderintherightmostcolumn.Thefunction Multi2Lin is introducedthatconvertsanymultidimensionalindices i =[ i1,i2,...,iM]Twhere thelimitsare n =[ n1,n2,...,nM]T,intoalinearequivalentindex l .Thefunction iscalledas l = Multi2Lin ( n,i ).So,forexample,4= Multi2Lin ([222]T, [221]) whichiscon“rmedfromTable6…2.Thistypeoffunctioniswidelyavailablein computerlanguagelibrariesformanytypesoflanguages.Nowde“nethelinear indexset L = Multi2Lin ( n,i ) , i .Therefore L isthelinearequivalentto I1× I2×···× IM.InTable6…2, L = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 } .Moreimportantly, Lim= i= Multi2Lin ( n,i ) , i : im= iisthesubsetof L with i1“xedat i. Thus Li1= iisthelinearequivalentto i1= i× I2× I3×···× IM.InTable6…2, Li1=1= { 1 , 3 , 5 , 7 } and Li1=2= { 2 , 4 , 6 , 8 } .Withthesede“nitionsin-hand,the

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87 MAP max i1 I1i2 I2··· iM IMci1,i2, ··· ,iMxi1,i2, ··· ,iM(6.3) Subjectto i2 I2i3 I3··· iM IMxi1,i2, ··· ,iM=1 , i1 I1(6.4) i1 I1··· ij Š 1 Ij Š 1ij +1 Ij +1··· iM IMxi1,i2, ··· ,iM 1 , ij Ij, j =1 J (6.5) isequivalenttothelinearassignmentproblem max l Lcle( l ) xle( l )(6.6) Subjectto l Li 1 = i xle( l )=1 ,i=1 , 2 ,...,n1(6.7) l Li j = i xle( l ) 1 ,i=1 , 2 ,...,nj, j =1 J (6.8) The“rst n1constraintsareequalitytoensurethateveryvehicleisassignedat eachtimestep.Onemodi“cationisnownecessary-asstatedinSection6.1,all vehiclesmayutilizethenullactivityasmanytimesastheywish.Therefore,the nullactivitymustbeassignableasmanytimesastherearetimesteps.Thatis,if thereare n1timeintervals,thentherighthandsideofallaxialconstraintsthat involvethenullactivitymustbe n1.Duetocolumnmajorgenerationofthelinear indices,theserighthandsideswilloccurat l = { n1+1 ,n1+ n2+1 ,n1+ n2+ n3+1 ,..., 1+ M Š 1 i =1ni} .Analgorithmforcreatingtheaxialconstraintsisprovided inFigure6…4.TheArrays Aaand bade“netheaxialconstraintsinthelinear

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88 procedure Axial ( solution ) 1 numac M i =1ni,numvars M i =1ni2 l =0 ,Aa= zeros ( numc,numvars ); 3 FOR m =1 ,...,M 4 FOR j =1 ,...,nm5 l = l +1; 6 Lim= j= Multi2Lin ( n, [ i1,...,im Š 1,j,im +1,iM]); 7 Aa( l,Lim= j)=1; 8 ENDFOR 9 ENDFOR 10 ba= ones ( numac, 1); 11 FOR j =1 ,...,M Š 1 12 cumsum j i =1ni13 ba( cumsum +1) n114 ENDFOR 15 return ( solution ); end Axial ; Figure6…4:Pseudo-codeForGeneratingAxialConstraints assignmentproblem max cT lexle(6.9) Subjectto Aaxle= ba,l =1 ,...,n1(6.10) Aaxle ba,l = n1+1 ,n1+2 ,...,numac (6.11) (6.12) where numac isthenumberofaxialconstraintsand numvars isthenumberof variablesintheproblem,eachcomputedatline1ofthealgorithm.Line7returns row l of Aacorrespondingtoconstraint l ,andline10returnsalltherighthand sidesofones.Line13returnstherighthandsidesforthenullactivityconstraints.

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89 6.1.3 Travel Time Constraints GeneratingthetraveltimeconstraintsofEquation5.11isdoneinalike mannertobuildingtheaxialconstraints.Thatis,thefunction Multi2Lin willbe usedto“ndallthelinearindicesactiveintheconstraint. First,theanalystspeci“esthelocationofeachobjective.Again,itisassumed thattheobjectiveisata“xedlocation,eventhoughthismaynotbeexactlyso (seeexplanationinSection6.1).Thelocationisan x and y coordinatespeci“ed onaCartesian,”atplanthatislocallyleveltosomeprede“nedoriginandaxes orientation.ThisreferenceframeistermedlocallylevelŽorsometimesnortheast-downŽornorth-east-upŽindicatingthedirectionstheaxesarepointed. Inthiswork,onlytwodirectionswillbeconsidered,thosethatareintheplane paralleltotheground,thusnorthŽandeast.ŽTheverticalcoordinatecouldbe addedveryeasily.Thetwocoordinatesareenteredasarowina P × 2array X , wherethe pthrowcontainsthelocationofobjective p andthe“rstandsecond columnsarethenorthandeastcoordinatesrespectively. Next,amatrixisbuiltcontainingallthedistancesfromeachuniqueactivity toalltheothers.RecallfromSection6.1.1thatthesecondcolumnofarray Iuniquepointstotheobjectivescorrespondingtoeachuniqueactivity.Therefore X ( Iunique( i, 2) , :)isa1 × 2vectorholdingthenorthandeastcoordinatesfor uniqueactivity i .Thedistancebetweeneachuniqueactivityiscomputedas d ( i,i)= || X ( Iunique( i, 2) , :) Š X ( Iunique( i, 2) , :) || where ||·|| isthe2-normoperation thatreturnsthedistancebetweenthetwopointscorrespondingtouniqueactivities i and i. Thenextstepistocomputethetraveltimefromeachuniqueactivitytoall theothersforeachvehicle.Thiswillbecontainedinthe numact × numact × M Š 1 array T andiscomputedasthedistancedividedbythevelocityofvehicle m .Thus

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90 thetimefromactivity i to iforvehicle j is T ( i,i,j )= d ( i,i) v ( j ) , wherethevelocitiesofallvehiclesarestoredinthe( M Š 1) × 1vector v .Itis assumedthatthevehiclestravelataconstant(oraverage)velocity.Notethatthe numberofuniqueactivitiesisincreasedbyonewhenthenullactivityisaddedinto themodel.Clearlythedistancefromanyactivitytothenullactivitymustbezero. Recallthatthetraveltimeconstraintis i2 I1··· ij Š 1 Ij Š 1ij +1 Ij +1··· iM IM[ xi1,i2...ij Š 1,i,ij +1, ··· ,iM+ k I1: k = i1max { 1 , 1 Š Tii j} kk< min { n1, 1+ Tii j}xi1,i2,...ij Š 1,i,ij +1, ··· ,iM] 1 , i Ij, i Ij: i = i, j J, i1 I1Thisconstraintmaynowbeconstructedbysimply“ndingtheproperlinearindicesand“xingthemintheconstraintmatrixaswasdoneinthe Axial algorithm.Forthe“rstterm, { Li1= i1 Lij= i} returnsalltheindiceswherethe timestepis i1andvehicle j isbeingassignedtoactivity i.Forthesecondterm, { Li1= k Lij= i} returnsallindiceswherethetimeis kandvehicle j isassigned activity i .Thekeytosuccessissettingupindexarraysfor kand i suchthat k{ I1: k = i1andmax { 1 , 1 Š T ( i,i,j ) } k< min { n1, 1+ T ( i,i,j ) }} and i { Ij: i = i} .ThealgorithmisinFigure6…5.The“rsttermoftheconstraint issetinlines6and7,thesecondhalfinlines10and11.Theindexsetfor kis line8.Inlines12and14,theactiveindicesfor“rstandsecondtermsoftheconstraintareconcatenatedontopoftheotherandthensortedtoreturntheindices intheproperorder.Theprogramreturns Atand btinlines15and21whichare numtc × numvars and numtc × 1sizearraysrespectively,where numtc isthe numbertravelconstraintsconstructed.Thesearraysaresimplyaugmentedtothe

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91 procedure Travel ( solution ) 1 numvars M i =1ni2 l =0; 3 FOR j =2 ,...,M 4 FOR i=1 ,...,nj5 FOR i = i+1 ,...,nj6 FOR k =1 ,...,n15 l = l +1; 6 Li1= k,ij= i= Multi2Lin ( n, [ k,i2,...,ij Š 1,i,ij +1,iM]); 7 S1 Li1= k,ij= i; 8 Ik= { I1: k = k andmax { 1 , 1 Š T ( i,i,j ) } k< min { n1, 1+ T ( i,i,j ) }} ; 9 FOR all k Ik10 Li1= k,ij= i= Multi2Lin ( n, [ k,i2,...,ij Š 1,i,ij +1,iM]); 11 S2 Li1= k,ij= i; 12 S S1... S2 13 ENDFOR 14 S sortlow2high( S ) 15 At( l,S )=1; 16 ENDFOR 17 ENDFOR 18 ENDFOR 19 ENDFOR 20 numtc l 21 bt= ones ( numTc, 1); 22 return ( solution ); end Travel ; Figure6…5:Pseudo-codeForGeneratingTravelTimeConstraints bottomofthearrays Aaand bainthelinearassignmentprograminEquations6.6, 6.7,6.8. 6.1.4 Precedence Constraints The“nalsystemofconstraints,whichwillalsobeaugmentedtothebottomof theconstraintarraysbuiltthusfararetoenforceprecedence.Firstanarraythat capturesprecedencebetweenactivitiesmustbeassembled. Theanalystenterstheprecedentsforeachactivitytype.Indoingso,the analystonlyconsiderstheprecedentsthatcome immediately beforetheactivity

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92 typeofinterest.Forthesakeofsimplicity,thenumberofimmediateprecedentsfor anyactivitytypemustbelessthanorequaltoone.Thiscouldeasilybechanged toanynumberbyexpandingthevectorde“nedbelowtoamatrix.Itwasnot necessaryfortheproblemsstudiedhere.Theprecedentinformationisentered inthe numtypes × 1vector D where numtypes isthenumberofactivitytypes. Hence D ( i )istheindexoftheactivitytypethatisprecedentforactivitytype i . If D ( i )=0thenactivitytype i hasnoprecedent.Asanexample, D =[0112]Tde“nesthatactivitytype1hasnoprecedent,theprecedentofactivitytypes2and 3areactivitytype1,andtheprecedentofactivitytype4isactivitytype2.The nullactivitywillhavenoprecedents. Now,justaswiththeaxialandtravelt imeconstraintspreceding,thefunction Multi2Lin willbeusedto“ndallthelinearindicesactiveintheconstraint.The precedenceconstraintis i1 I1··· ij Š 1 Ij Š 1ij +1 Ij +1··· iM IMxk,i1...ij Š 1,i,ij +1, ··· ,iMŠ k I1: k
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93 procedure Precedence ( solution ) 1 numvars M i =1ni2 l =0; 3 FOR j=2 ,...,M 4 FOR i=1 ,...,nj5 i= D ( activity ( i,j); 6 IF i = DO 7 FOR k=1 ,...,n18 l = l +1; 9 Li1= k,ij = i= Multi2Lin ( n, [ k,i2,...,ijŠ 1,i,ij+1,iM]); 10 S1 Li1= k,ij = i; 11 FOR j =2 ,...,M 12 FOR i =1 ,...,nj13 IF activity ( i,j )== i14 FOR k =1 ,...,kŠ 1 15 Li1= k,ij= i= Multi2Lin ( n, [ k,i2,...,ij Š 1,i,ij +1,iM]); 16 S2 Li1= k,ij= i; 17 S S1... S2 18 ENDFOR 19 ENDIF 20 ENDFOR 21 ENDFOR 22 S sortlow2high( S ) 23 Ap( l,S )=1; 24 ENDFOR 25 ENDDOENDIF 26 ENDFOR 27 ENDFOR 28 numpc l 29 bp= ones ( numpc, 1); 30 return ( solution ); end Precedence ; Figure6…6:Pseudo-codeForGeneratingPrecedenceConstraints

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94 6.2 Test Problems 6.2.1 Random Scenario Therandomtestproblemsweredesigne dsimplytocheckthequalityofbounds introducedinSection5.3.3.Allofthetestproblemshaverandomvaluesforthe costsofeachactivity-vehiclepairingandthetraveltimesbetweenactivities.For allthreeproblems,therandomcostsweregenerateddirectlyfor Cmargcompletely bypassingobjectiveclassesandvehicleclassesandleavingoutthenullactivity. Todoso,aproblemdimensionwasspeci“edbysettingthevaluesof M and N , where M Š 1isthenumberofvehiclesand N isthenumberofactivitiesforeach vehicleandthenumberoftimesteps.Thecostarrayisthereforea N × M matrix ofuniformlyrandomnumbersontheinterval[0 , 1]givenby Cmarg= uniformN × M(0 , 1) . Thetraveltimeswerealsorandomlygenerateddirectlyforthearray T ( i,i,j ), bypassingspeci“cationsoflocationsin X ,vehiclevelocities,andcomputationsof distances d .Thematrix T ( i,i,j )issymmetricaboutthemaindiagonalforeach valueof j ,sotherandomvaluesweregeneratedsothat T retainedthissymmetric property.Furthermore,therandomvalueswereuniformontheinterval[0 ,N ]so thattherewouldbesomevaluesnear N therebyexcludingtravelfrom i to ifor thatentry.Thematrixwasbuiltas FOR j =1 ,...,M Š 1 T ( i,i,j )= uniformN × N(0 ,N ) ENDFOR Therewasalsonocorrelationbetweenactivitiesforthe“rsttwotestproblems, nopreferenceweights,andnospeci“cationofprecedence.Thestatisticsforthe“rst twotestproblems,M4N10TandM10N5T,areinthe“rst2rowsofTable6.2.1. ThecolumnlabeledInst,Žspeci“esthenumberoftestinstancesbuild,eachwitha

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95 Table6…3:TestProblemsWithUniformlyRandomData Test Constraints Problem Inst M N Corr Variables Axial Travel Preced M4N10T 10 4 10 n/a 10,000 40 1350 0 M10N5T 10 10 5 n/a 59,049 30 81 0 M4N9CTP 1 4 9 50% 2,268 29 546 91 dierentrandomseed.Thecolumns M and N specifythenumberofcolumnsand rowsinthecostmatrix.ThecolumnVariablesŽspeci“esthenumberofvariables intheproblems.Therearethreecolumnswithentriesfromlefttorightwiththe numberofaxial,traveltime,andprecedenceconstraints. Thethirdrandomtestproblem,M4N9CTP,wasdesignedtoseewhatimpact, ifany,theadditionofcorrelationandprecedencewouldhave.Correlationwas speci“edbymarking2activitiesatrandomandspecifyingtheeciencyofboth as as0.9.Precedencewasaddedto50%oftheactivitiesatrandom.Sofor N activities,N 2werechosenatrandomandforeach i chosen, D ( i )= uniform (0 ,N ) \ i, \ D ( D ( i )) where \ i eliminates i frombeingprecedenttoitself,and \ D ( D ( i ))eliminatessingle precedencecycles,suchaswouldoccurif D ( i )= iand D ( i)= i .Ensuringthere werenolongercycleswasdonebyvisualinspection.Thestatisticsforthistest problemareinthethirdrowofTable6.2.1. Itisinterestingtoseehowthecostsaredistributedfortheseproblems.Figure 6.2.1showsthedistributionforoneinstanceofproblemM4N10T.Thissymmetric shapeisquitetypicalforeachoftestinstances.Themeanisatone,andthecosts areveryclosetooneanother.Thiswouldindicatemanylocaloptimameaning thiswilllikelybeaverydicultproblemtosolve.InFigures6.2.1and6.2.1the eectsofmultiplyingwithpreferenceweightsof3and6onallactivitiesisseen.

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96 Figure6…7:CostDistributionForRandomProblemM4N10T Thedistributionwidensastheweightsincrease,eectivelyscalingthedata.This scalingmayormaynothelpsolvers,asitisstrictlyascalingphenomenon.In Figure6.2.1theweightsareassignedtoactivitiesatrandomandareuniformon theinterval[1 , 6].Interestingly,thereisevenmoredispersionthanthatofFigure 6.2.1.Itisexpectedthattheuseofweightsinrealdatathatlookmorerandom thanuniformshouldhelpsolversarriveatgoodsolutions. 6.2.2 Military Scenario Themilitaryscenariopitsanynumberof”yingvehiclesagainstanynumber ofground-basedmilitaryobjectives.Thereare4typesofobjectives-armored vehicles,enemyairdefenseunits(SEAD),commandcenters(C2),andcommunicationscenters(Comm).These4typesofobjectiveseachhavemultipleactivity typesassociatedwiththem,asshowninTable6.2.2andcapturedinthearray ITypes.ThelastrowofTable6.2.2showsthedefaultpreferenceweightsassociated witheachobjectiveclass.TheCommandŽobjectiveclassisthemostpreferred, theCommunicationsŽclasstheleast.Ofcoursethenullactivityhasapreference weightofone.Thesevaluesareenteredbytheanalystintovector .Insomeof

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97 Figure6Â…8:CostDistributionForRandomProblemM4N10TWithWeights=3 Figure6Â…9:CostDistributionForRandomProblemM4N10TWithWeights=6

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98 Figure6…10:CostDistributionForRandomProblemM4N10TWithRandom Weights theexperiments,thesevalueswillbechangedtoseetheeectoftheweightsonthe solutions. Precedenceinthisscenarioisexhibitedintheactivitytypesasshowninthe “rstcolumnofTable6.2.2.So,forexamp le,amobiletargetcannotbeclassi“ed (e.g.,ClassifyMobileŽ)untilitis“rstf ound(e.g.,searchŽ).Thisdataiscapturedinthe D array.Thesecondcolumnmarksalltheactivitytypesthatexhibit correlation.AsstatedinSection6.1,anactivitywithanon-zeroeciency, ,experiencesanincreasedlikelihoodofsuccess,aboveandbeyondthatofthe2vehicles actingindependently,whentwovehiclesserviceitatsametime.Thetableshows thatonlyactivitytype1,search,Žismarkedwithaneciencyof90%.Allother activitytypeshavenodependencies. Thevehiclesmaybeamixturedrawnfrom4classes:LargeUnmannedAir Vehicles(UAVs),SmallUAVs,CruiseMissiles,andGlideBombs.Eachvehicle classhasaknowncostforeachactivitytype.AsrelatedinSection6.1,thesecosts arecapturedbytheanalystinthearray Cclassanddisplayedinthe“rst8rows

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99 Table6…4:MilitaryScenario„ActivitiesandPreferencesForEachObjectiveClass ObjectiveClass ActivityType Armor SEAD Command Communications Search yes yes no no ClassifyFixed no yes no no ClassifyMobile yes no no no StrikeFixed no yes yes yes StrikeMobile yes no no no DetectRadio no no no yes JamRadio no no yes yes TrackMobile yes no no no PreferenceWeight 5 10 7 3 Table6…5:MilitaryScenarioActivityTypeInformation„PrecedenceandEciency ActivityType ImmediatePredecessor Eciency Search None 0.9 ClassifyFixed Search 0.0 ClassifyMobile Search 0.0 StrikeFixed ClassifyFixed 0.0 StrikeMobile ClassifyMobile 0.0 DetectRadio None 0.0 JamRadio DetectRadio 0.0 TrackMobile ClassifyMobile 0.0

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100 ofTable6.2.2.Thelastrowindicatesthespeedofeachvehicleclass.Ascanbe seen,thevelocitieswereallspeci“edtobeidentical,which,at“rstblushseemsto makenosense.Anexplanationisorder.Thetypesofvehiclesinthisscenariohave widelydierenttypesofsensors-theLargeUAVhasaverywide“eldofview,on theorderof10to100kilometers,whereasthecruisemissilehasaverysmall“eld ofviewofseveralhundredmetersatmost.Theglidebombhasnosensoratall. So,thenotionofhowlongittakesforavehicletoarriveatanobjectiveisrather confusingasitdependsnotonlyonthevehiclespeedbutalsothevehiclesaltitude andsensor“eldofregard.Whileacruisemissilemay”yordersofmagnitude fasterthanalargeUAV,ithasamuchsmallersensorscan.TheUAVcouldswing itssensoraroundand,after”yingaveryshortdistance,“ndtheobjective.The cruisemissilewouldhavetoover”ytheobjectiveitselftoseeit.Therefore,for thepurposesofthissimulation,allthevehiclescanbeassumedtotravelatabout thesamespeed.Thisassumptionobviou slyhas”awswhentheactivitychanges tostrike,Žsincethenthevehiclemusttrulytraveltheentiredistance,butbear in-mindthattheonlyvehiclescapableofstrikearethecruisemissileandbomb whichtravelatsimilarhighspeeds.Finally,thesamespeedassumptionisabout thebestthatcanbedonewiththemodelathand.Obviously,somechangesneed tobemadeinthemodelthatallowtraveltimestobeafunctionoftheactivity type,sensorspeci“cations,andperhapsotherparametersinadditiontojustthe vehiclevelocity. Forthisscenario,thereare7testproblems.Thestatisticsofthesetest problemsaresummarizedinTable6.2.2.Inthistable,thenumberofobjectives andnumberofvehiclesareprovidedincolumns3and4respectively.Thecolumn CorrŽsimplyindicateswhetherornottheeciencyspeci“edinTable6.2.2 wasused.TheremainingcolumnsareidenticaltothoseofTable6.2.1.Thetest problemnamingnomenclatureisasfollows:

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101 Table6…6:MilitaryScenarioCosts„VehicleClassversusEachActivityType VehicleClass ActivityType LargeUAV SmallUAV CruiseMissile GlideBomb Search 0.7 0.9 0.9 0.0 ClassifyFixed 0.4 0.7 0.2 0.0 ClassifyMobile 0.5 0.9 0.9 0.0 StrikeFixed 0.0 0.0 0.7 0.9 StrikeMobile 0.0 0.0 0.9 0.5 DetectRadio 0.9 0.0 0.0 0.0 JamRadio 0.0 0.9 0.0 0.0 TrackMobile 0.9 0.8 0.7 0.0 Velocity(m/sec) 75 75 75 75 1.Theleadingnumbersindicatethedimensionof n .Thusthe“rsttestproblem has n =[7663]. 2.Thelettersthatfollowindicatethetypeofconstraintsordatathatispresent. Iftheletterisnotthere,thecorrespondingconstraintordataisnotused. Thelettersare: € C Presenceofcorrelation € T Presenceoftraveltimeconstraints € P Presenceofprecedenceconstraints € A Presenceofattrition-Ifthevalueof ckisdecreasingwith k . € W Presenceofnon-unityweights 3.Thedotseparatednumbersinparenthesesthatfollowa W arethepreference weightvalues.Iftherearenonumbers,thenthepreferenceweightswerenot used,inotherwords,allsetto1. QuicklyexaminingTable6.2.2revealsthatthetestproblemsbuildfromsimple tocomplex.Hence,testproblemsnumbered1through5havethesameobjectives andvehicles.However,the“rsttestproblemhasonlytraveltimeconstraints.As theproblemnumbersincreaseto5,featuresaresuccessivelyaddedon.Inthisway, theeectofeachfeaturemaybeexplicitlydetermined.Testproblems6and7were

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102 Table6…7:MilitaryScenario„TestProblemStatistics Test Constraints Num Problem Obj Veh Corr Vars Axial Trvl Prec 1 7663T 2 3 no 756 22 231 0 2 7663TP 2 3 no 756 22 231 56 3 7663TPAW(10.10) 2 3 no 756 22 231 56 4 7663TPAW(5.10) 2 3 no 756 22 231 56 5 7663CTPAW(5.10) 2 3 yes 756 22 231 56 6 9994CTPAW(5.10.5) 3 3 yes 2,916 31 702 117 7 99944CTPAW(5.10.5) 3 4 yes 11,664 35 756 144 Table6…8:MilitaryScenario„UniqueObjectiveDataForTestProblems ObjectiveType Location ForProbNum Obj Armor SEAD C2 Comm N E 1,2 3 4,5 6 1 1 0 0 0 5000 4000 1 10 5 5 2 0 1 0 0 10000 5000 1 10 10 10 3 1 0 0 0 5000 10000 n/a n/a n/a 5 designedtopushthelimitsofthesolvers.Theyincreasethenumbersofobjectives andvehiclesandkeepalltheintricatefeaturesandconstraints. Withthegeneralscenarioinformationcaptured,theanalystmayenter thedataspeci“ctothenumberandtypesofeachobjectiveandvehicle.These valuesarespeci“ctotheexperiment.Thereare4speci“carraystopopulate: theobjectiveclassesofeachuniqueobjective, Bobj,thepositionofeachunique objective, X ,thevehicleclassofeachuniquevehicle, Bveh,andtheintrinsiccost andweightofeachtimestep, ck,and krespectively.ThisdataisinTables6.2.2, 6.2.2,and6.2.2.

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103 Table6…9:MilitaryScenario„UniqueVehicleDataForTestProblems VehicleType Veh LgUAV SmUAV CruiseMiss GuidBomb 1 1 0 0 0 2 0 1 0 0 3 0 0 0 1 4 0 0 0 1 Table6…10:MilitaryScenario„UniqueTimeStepDataForTestProblems ckForProbNum TimeStep 1,2 3,4,5 6 k 1 1 0.9 0.9 1 2 1 0.8 0.8 1 3 1 0.8 0.8 1 4 1 0.7 0.7 1 5 1 0.7 0.7 1 6 1 0.6 0.6 1 7 1 0.6 0.6 1 8 n/a n/a 0.5 1 9 n/a n/a 0.5 1

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CHAPTER7 SOLUTIONMETHODS Numericalapproachesforobtainingsolutionstothetestproblemsinthe previouschapterareneeded.Theapproachesofinteresthereare,notsurprisingly, branchandbound,branchandcut,andconstructiveheuristics.Thesuccess ofbranchandbound/cutmethodsusuallyhingesongoodupperbounds(for maximizationproblems).However,aswillbeseen,evenveryfastboundsdont helpoutmuch.Whentheproblemsbecomelarge,theexactmethodstakean enormousamountoftimetocompute. 7.1 Branch and Bound Branchandboundisaverywellestablishedapproachforsolvinginteger,0-1 integer,andmixedintegerlinearprograms.Itisbasedontheideaofdivideand conquer.Itdividestheproblemintopiecesby“xingvariablesatbudnodesŽ, oneatatimetoanintegervalue.Theproblemsofinterestherecontainsonly0-1 integers,sothevaluesselectedareeither0or1.Thealgorithmthenconquersby computinganupperbound(lowerforminim izationproblems)andcomparingthis totheincumbent,whichisthebestknownsolutionsofar.Theupperboundis alwaysinfeasiblebutoptimistic.Iftheboundonthebudnodeislessthanthe valueoftheincumbent,thenthebudnode,andallnodesthatdescendfromit,are prunedŽfromfurtherconsideration.Branchandboundfor0-1integerprogramsis verywelldiscussedintheliteratureandsowillnotbediscussedhereinanygreat depth.Thereferences[6,34,46]areexcellentsourcesforfurtherinformation.An overviewofthetermsisnowpresented. Node: Anypartialorcompletesolution. Leaf: Acompletesolution(node)whereallvariablevaluesare“xed. 104

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105 Bud: Apartialsolution(node)whichmayormaynotbefeasible. Searchtree: Agraphicaltreestructurewhereeachnodecorrespondstoa particularrelaxedproblemandeacharcrepresentsa“xedvalueononeofthe integervariables. Bounding: amethodofoptimisticallyover-estimatingthevalueoftheobjective functionforanybud. Fathoming: Determinationthatitisnotnecessarytoexplorethedescendantsofa particularnodeinthesearchtree. Incumbent: Bestobjectivefunctionvaluefoundsofarintheenumeration process,andcorrespondingfeasiblesolution. Partitioning: Dividingtherelaxedfeasibleregionintomutuallyexclusivepartsby either“xingavariableatsomeintegervalueorrestrictingittosomeupperor lowerbound. Branching: Selectingabudnodeforpartitioning. Backtracking: Movingfromafathomednodetoalivenodeinsuchamanner thattheentiresearchtreeissystematicallyexploredsoastoguaranteethat theoptimalsolutionisnotoverlooked. Thereare2majorissuesforbranchandbound,thebranchingstrategyand thefunctionusedtodevelopboundsonbudnodes.InSection5.3.3averyecient upperboundontheNPDMAPwasintroduced.Thatboundwasinfactintegrated intothestructureofabranchandboundalgorithmandresultsgenerated.A simplelinearprogrammingcontinuousre laxationwasalsoused.Thebranching strategyforbothalgorithmswasbest“rst,Žwherethenodewiththebestsolution anywhereonthetreeisselectedtobranchfrom.Theresultsofthisalgorithmare inSection7.3.

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106 7.2 Construction Heuristics Theclassofheuristicsthatdevelopasolutiononepieceatatimearecalled constructionheuristics.TheseareusedquiteofteninVRPsinattempttoeither routeonevehicleatatime,orassignonedemandatatimetoavehicle,andthen createroutesforallthevehicles.Theweaknessofconstructionheuristicsisthat theyoftenaretrappedatlocalminima.TheGreedyRandomizedAdaptiveSearch Procedure(GRASP)heuristicwasintroducedtocircumventthisproblem.Indeed, theGRASPhasbeenshowntoproduceverygoodsolutionsquiteecientlyforthe QuadraticAssignmentProblem(QAP)[47] andthemultidimensionalassignment problemsencounteredindatafusionanddataassociationproblems[31,32].Since thebasicstructureoftheintegratedassignmentandpathplanningproblemisa multidimensionalassignmentproblem,itmakessensetotryithereaswell. GRASPisaniterativealgorithmthatdevelopsafeasiblesolutionatevery iteration.TheGRASPiteratesapre-speci“ednumberoftimesandthebest solutionamongalliterationsiskeptastheapproximateglobaloptimum.Each GRASPiterationproceedsin2phases:aconstructionphaseandalocalsearch phase.Intheconstructionphase,ara ndomadaptiveruleisusedtobuilda feasiblesolutiononecomponentatatime.Alocalsearchisthenappliedtothis constructedsolutiontomakefurtherimprovements.GRASPisillustratedin pseudo-codeinFigure7…1. procedure GRASP ( ListSize,MaxIter,Rando mSeed ) 1 InputInstance() ; 2 DO k =1 ,..., MaxIter 3 ConstructGreedyRandomizedSolut ion(ListSize, RandomSeed) ; 4 LocalSearch(BestSolutionFound) ; 5 UpdateSolution(BestSolutionF ound) ; 6 ENDDO ; 7 return BestSolutionFound end GRASP ; Figure7…1:Pseudo-codeforagenericGRASP

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107 7.2.1 Phase One: Construction Theobjectiveofphase1istogenerateagoodŽfeasiblesolutionthatisa partitionofvehiclestoactivitiesforeachtimestep.Foranyprobleminstance therearealargenumberofpossibleM-tuplesofpossibleassignments,ofwhich onlyasmallnumberneedtobeselectedtoconstructapartition.Eachtupleis modeledasthevariable xi1,i2,...,iM.Therefore,the“rststepofphase1istocreate arestrictedcandidatelistofthe %bestŽvariables-thosethathavethehighest cost ci1,i2,...,iM.Thealgorithmthenselectsoneoftheseatrandomtoinitialize thesolution.Sincechoosingatuplemeansthatthosecorrespondingassignments arenowineligibleforfurtherconsideration,therestrictedcandidatelistwillnow likelycontaininfeasibletuples.Hence,itisadaptedtodisqualifyfromfurther considerationallthosevariablesnowrenderedinfeasible,andupdatedtoinclude thenew highestcostcandidates.Asecondtupleisnowrandomlyselectedfrom thislisttoaddtotheconstruction.Thisprocessofrandomselectionandcandidate listadaptationrepeatsuntilasolutionhasbeencompleted,whichisindicated bythelackofanyfurthertimeslotsforassignments,oralackofanyfeasible candidates. Thiscompletephaseisrepeated s1times,eachiterationusingadierent randomseedandyieldinganinitialfeasible solution.Intuitively,weexpectthatthe larger s1becomes,themorelikelyitwillbethatweobtainanear-optimalsolution. However,alarge s1alsoimpliesalongercomputationtime.Theconstructionphase pseudo-codeisillustratedinFigure7…2. Forthisproblemobservethatacompleteassignment,asconstructedinphase 1,isamatrixwith n (1)rowsand M columnssuchthateachrowcorrespondstoa tupleofassignmentsatsomedistincttime.Callthismatrix, P .Theentryinrow i , column j ofmatrix P istheactivityassignedtovehicle j Š 1forassignment i .If P issortedbythe“rstcolumn,whichcontainsthetimeindicesforeachassignment,

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108 procedure Construction ( CostArray,solution ) 1 CostVector=SortCosts(Cost Array) ; 2 createRCL ; 3 do k =1 ,..., MaxIter 4 s=selectRandomElement(RCL) ; 5 solution=solution s ; 6 adaptRCL(Costarray,s,RCL) ; 7 od ; 8 LastAssignment(RCL,solution) ; 9 return ( solution ); end Construction ; Figure7…2:GRASPconstructionpseudo-code thentheentryinrow i ,column j correspondstoactivityassignedtovehicle j Š 1 fortimestep i .Inthisform,itiseasytovisualizethesolutions. 7.2.2 Phase Two: Local Search Inthelocalsearchphase,eachphase1solutionisimprovedbysearchingits neighborhoodforabettersolution.Theneighborhoodmaybede“nedinmany dierentways.Inthiswork,asintherelatedworkforthedataassociationproblem [31,32],theneighborhoodwillbeall2-exchangepermutations. Let p beavectorof n elements.De“nethe dierence between p andapermutationof p , pnew,tobe ( p,pnew)= ! i | p ( i ) = pnew( i ) " De“nethe distance betweenthemas d ( p,pnew)= | ( p,pnew) | A k-exchangeneighborhood ofvector p isthesetofallpermutations pnewsuch that d ( p,pnew) k ,i.e.,

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109 Nk( p )= ! pnew| d ( p,pnew) k, 2 k n " Nowconsiderthecasewhen k =2,whichoccurswhenany2elementsin p areinterchanged.Recallthesolutionisamatrix P ,with M columnsand N rows. Noticethatthe pthcolumnof P isapermutationoftimelocationsforthe pthvehicle.Thusonepossible2-exchangeneighborhoodwouldsimplylookforall N 2 ineachrow.Theentire2-exchangeneighborhoodof P is N2( P )= N2( p1,p2,...pM)= N2( p1) N2( p2) ··· N2( pM)(7.1) Then N2( P )has N 2Mmembers.Obviously,thissearchneedstolimitedtosome subsetoftheentireneighborhoodsince,intheworstcase,analgorithmbased onthisneighborhoodde“nitionwouldruninexponentialtime.Therefore,de“ne N2( P )tobeallofthe2-exchangesof p1 p2··· pM,where pmisthe mthcolumn of P .Then N2( P )= # N2( pj) # $i =1 ...M i = jpj% j =1 ...M % (7.2) Thesizeofthisneighborhoodis | N2( p1) | + | N2( p2) | + ··· + | N2( pM) | = M N 2 . Ofcourse, N2( P ) N2( P ). Thelocalsearchpseudo-codeisgiveninFigure7…3andproceedsasfollows. € foreachphase1iterationwithpartition P ,comparethecostof P with thatofitsneighbors, N2( P ).Ifalowercostneighborisfound,checkthe feasibilityoftheneighbor.Iffeasible, N isreplacedwiththeneighborandthe localsearchbeginsalloveragainwithacorrespondingnewneighborhood. € Iftheneighborisnotfeasiblethesearchoftheneighborhoodcontinues untileitheralowercostfeasibleneighborisfoundortheneighborhoodis

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110 exhausted.Ifexhausted,thecurrentpartitionisthelocalminimumandthe searchterminates. procedure LocalSearch ( solution ) 1 QUIT=false ; 2 while notQUIT do 3 if [ searchfinished ], QUIT=true ; 4 STOP=false ; 5 counter=0 ; 6 while notSTOP do 7 counter=counter+1 ; 8 ProduceMatrixPermutation ( counter, P , Pnew); 9 if LowerCost ( counter, P , Pnew) 10 P = Pnew; 11 STOP=true ; 12 “ ; 13 if ( notSTOP ) and ( counter=|neighborhood| ) QUIT=true 14 od ; 15 od ; 16 return ( solution ); end LocalSearch ; Figure7…3:GRASPLocalSearch„MultidimensionalNeighborhoodPseudo-code Noticethatthereissome”exibilityinhowthesolutionisupdated:2-exchange permutationscouldbegeneratedone-by-oneandthe“rstlowercostfeasible partitionchosentoreplacetheincumbent;theentireneighborhoodcouldbe searchedandthelowestcostfeasibleneighborchosentobethereplacement;orthe “rstmethodcouldbemodi“edbyrandomlyselectingthenextneighbortotest.In thisworkthe“rstmethodwaschosen. A Second Local Search Method Insteadofde“ningthelocalsearchonpermutationswithinthecolumns,the localsearchcouldalsobeconducteddirectlyonthelinearequivalentindexsolution vector xleintroducedinequation6.6.Inthisvector n (1)elementsinthesolution thathaveavalueof1.Theother n Š n (1)arenotinthesolutionandhavea valueofzero.The2-exchangeneighborhoodonthisvectorisde“nedasfollows.

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111 € foreachphase1iterationwithsolution xle,comparethecost clewiththatof itsneighbors, N2( xle).Thisisdonebyremovingoneofthe xle( l )variables fromthesolutionandreplacingitwithavariable xle( l) ,l = l thatisnot yetinthesolution.Ifalowercostneighborisfound,checkthefeasibilityof theneighbor.Iffeasible, N isreplacedwiththeneighborandthelocalsearch beginsalloveragainwithacorrespondingnewneighborhood. € Iftheneighborisnotfeasiblethesearchoftheneighborhoodcontinues untileitheralowercostfeasibleneighborisfoundortheneighborhoodis exhausted.Ifexhausted,thecurrentsolutionisthelocalminimumandthe searchterminates. ThisiscapturedinFigure7…4. procedure LocalSearch ( solution ) 1 QUIT=false ; 2 while notQUIT do 3 if [ searchfinished ], QUIT=true ; 4 STOP=false ; 5 counter=0 ; 6 while notSTOP do 7 counter=counter+1 ; 8 ProduceVectorPermutation ( counter, xle( l ) , xle( lnew)); 9 if LowerCost ( counter, xle( l ) , xle( lnew)) 10 xle= xle( lnew); 11 STOP=true ; 12 “ ; 13 if ( notSTOP ) and ( counter=|neighborhood| ) QUIT=true 14 od ; 15 od ; 16 return ( solution ); end LocalSearch ; Figure7…4:GRASPLocalSearch„LinearEquivalentNeighborhoodPseudo-code 7.3 Computational Results On Test Problems TherandomtestproblemsweredesignedtoseehowgoodtheBonferroni boundis.Sotheexperimentalresultsonthosetestproblemsisrestrictedto

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112 Table7…1:ComputationalResults„UpperBoundPerformanceonRandomTest Problems Bonferroni LPContinuous Problem Solt Runtime Solt Runtime M4N10T 39.04 0.38 10.64 68.25 M10N5T 91.87 0.68 34.19 57.24 M4N9CTP 29.91 0.37 3.70 34.11 obtainingupperboundsonthecompletelyunfathomedtree.Theseresultsare summarizedinTable7…1.Allruntimesareexpressedinseconds. ItcanbeseenfromTable7…1thattheupperbounddevelopedontheBonferroniinequalityisabout3timesaslargeastheLPboundbutisanorderof magnitudefaster.SoonewouldexpectthattheBonferronibound,notbeingvery tight,willnotbeaseectiveastheLPboundinpruningbuds.However,tomake upforthat,itisveryfastsothatBonferronibranchandboundisabletocheck10 nodesinthetimeittakestheLPbranchandboundtocheckone.Itseemsthatit isworthlookingintomoreclosely. Fouralgorithmsforsolvingtheintegratedassignmentandpathplanning problemhavebeenintroduced.Theyare: BBBB BranchandboundwiththeBonferroniinequalityupperboundand best-“rstbranching. BBLP BranchandboundwithLPRelaxationupperboundandbest-“rstbranching. GRASPM GRASPwithamultidimensionalneighborhoodlocalsearch. GRASPL GRASPwithalinearequivalentneighborhoodlocalsearch. Asitturnsout,localsearchonthemultidimensionalneighborhoodisnot eectiveatall.Thiswassomewhatofasurprisesinceitiswasshowntobeavery eectiveneighborhoodonotherMAPproblems.Infact,itwassounreliable,that theresultsarenotshownhere.Thedicultyappearstobewiththeprecedence

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113 Table7…2:ComputationalResults„Side-by-SideComparisonof3Algorithms BBBB BBLP GRASPL Problem Solt Runtime Solt Runtime Solt Runtime 7663T 8.05 1,115 8.05 1,433 8.04 55 7663TP 8.01 1,503 8.01 1,520 7.01 64 7663TPAW(10.10) 39.21 1,330 39.21 1,312 30.14 64 7663TPAW(5.10) 60.29 1,298 60.29 1,358 48.15 67 7663CTPAW(5.10) 60.73 1,140 60.73 399 48.70 69 9994TPAW(5.10.5) 61.98 32,321 61.98 26,525 50.28 2314 79944TPAW(5.10.5) ns > 43,000 ns > 43,000 ns ns andtraveltimeconstraints.Withonlyaxialconstraints,asinpreviousworks,a permutationwithinacolumnisquiteoftenfeasible.However,withtraveltime,and precedenceinparticular,beingabletopermutetimeslotsforavehiclesactivities assignmentsisarareeventandtheresultingneighborhoodisactuallyverysmall. Consequently,itrarelyoeredanimprovementontheconstructedsolution.A side-by-sidecomparisonofthethreeremainingalgorithmsissummarizedinTable 7…2.Allruntimesareexpressedinseconds.AllalgorithmswerecodedinMatlab 7.0.Unlessotherwisenoted,theGRASPalgorithmsalluseda =15%with5 constructioniterationsandalocalsearchoneachconstruction.Allofthebranch andboundsolutionsshownareoptimal. InTable7…3acomparisonofGRASPLwithandwithoutthelocalsearchis made. 7.4 Observations 1.Forthetraveltimeconstrainedproblems(7663TP),theruntimeoftheBonferronibranchandboundiscertainlyfasterthanthatoftheconventional LPbranchandbound.However,byaddingprecedenceconstraints(7663TP andbelow),thespeedbene“tislost,andinfact,forcorrelatedproblems (7663CTPAW(5.10)andbelow),theconventionalLPboundoutperformsthe Bonferronialgorithm.Whywouldthisbethecase?LookingatTable7…1

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114 Table7…3:ComputationalResults„ComparisonofGRASPLWithandWithout LocalSearch GRASPL GRASPL,noL.S. Problem Solt Runtime Solt Runtime 7663T 8.04 55 7.73 9 7663TP 7.01 64 7.01 9 7663TPAW(10.10) 30.14 64 18.58 9 7663TPAW(5.10) 48.15 67 26.76 9 7663CTPAW(5.10) 48.70 69 ns ns 9994TPAW(5.10.5) 50.28 2314 35.02 224 79944TPAW(5.10.5) ns ns 30.81 781 problemM4N9CTP,theboundqualityisnearly10timesworsethantheLP boundforthecorrelatedtestproblemwhereaselsewhereitisonly3times worse.Clearly,thefasterspeedofthebounddoesnotmake-upforthepoor boundqualityforthecorrelateddata.Whenthereisnocorrelationbutprecedence,asintestproblems7663TP,7663TPAW(10.10),and7663TPAW(5.10), theruntimestendtobeaboutequal.Apparentlyprecedenceconstraintsare makingheavierpruningmoreimportanttoecientlysearchingthetree. 2.Duetocomputationtime,thebranchandboundalgorithmislimitedto fairlysmallproblems.Indeed,neithersolverfoundasolutioninlessthan 24hoursforthelasttestproblem,sotherunswereterminated.GRASPis alsohamstrungbydimensionalitybutnotasbadlyasforbranchandbound. Unfortunately,runtimesexceeding1minutearenotpracticalforon-line algorithms.OfcourseplacingtheGRASPconstructionsinparallelwouldcut theGRASPruntimesdramatically. 3.CorrelationsinthecostdataseemstospeeduptheLPboundedbranch andboundsolutionquitedrastically.ThisisclearlyobservedinTable 7…2forproblem7663CTPAW(5.10)whichistheexactsameproblemas 7663TPAW(5.10)onlywithoutcorrelatedcosts.Theruntimedierence,

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115 PNoCorrelation= 1621 2731 3151 4114 5211 6371 7568 PCorrelation= 1661 2771 3111 4118 5221 6331 7554 Figure7…5:ComparisonOfExactAssign mentsWithandWithoutCorrelation 399secondsversus1358seconds,isanover4timesimprovement!This improvementisnotseenfortheBonferronibranchandboundhowever,likely forthereasonsdiscussedinthe“rstparagraph.And,evencorrelationinthe costdoesnothelptheLPboundasthedimensionoftheproblemincreases. CorrelationhasnoeectonruntimefortheGRASPsolver.Itisinstructive tolookatthematrixformsolutionswithandwithoutcorrelation.InFigure 7…5thematrixformoftheexactsolutionsfortestproblems7663TPAW(5.10) and7663CTPAW(5.10)areshown. The“rstcolumncontainsthetimeindex,columns2through4theindexof theuniqueactivityassignedtovehicles1through3respectively.Attimestep 1,thesolutionwithoutcorrelationassignsvehicle1totask6andvehicle2to task2.Inthesolutionwithcorrelatedcosts,tasks2and6aremarkedwith aneciencyof90%increase.Asaresult,vehicles1and2arebothassigned totask6astheygetadditionalbene“tbydoingso.Thesameistruefortask 2attimestep5.Theothersareallalignedsimplybycoincidence. 4.LookingatTable7…3,GRASPlocalsearchmostcertainlyhasaneect, especiallywhentheweightsareincludedinthedata(seetestproblems 7663TPAW(10.10)andbelow.However,thelocalsearchalsoslowsthesolver downbyafactorof8to10.Asfarasthesolutionqualitywithlocalsearch, theGRASPsolutionsarewithin12to20percentoftheoptimalsolutions

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116 PExact= 1261 2371 3511 4118 5121 6731 7654 PGRASP= 1161 2211 3311 4131 5151 6621 7711 Figure7Â…6:ComparisonOfExactandGRASPLFor7663TPAW(10.10) obtainedusingbranchandbound.Whatdoesthismeanfromapractical standpoint?InFigure7Â…6isaside-by-sidecomparisonoftwosolutionsfor testproblem7663TPAW(10.10). ThecostdierencebetweenexactandGRASPListhehighestofallthe problemstried-about23%.Butlookatthedierencesintheassignments. Fortheexactsolution,vehicles1and2incombinationmanagetocoverall theactivitiestheyareabletoserviceandvehicle3coversbothactivitiesitis solelycapableof.FortheGRASPLsolution,vehicles1and2incombination coveralltheactivitiestheyareabletoservice,justinadierentorderthan intheexactsolutionandvehicle3managestodonothingatall.So,isthis acceptablefromalogisticsstandpoint?Thevehicle1and2resultslikelyare, butthevehicle3resultwouldnotbe.Thisistheworsecaseexperienced. Lookingatanexamplewithcorrelation,Figure7Â…7comparestheexactand GRASPLsolutionsfortestproblem9994TPAW(5.10.5). ItappearsthattheGRASPalgorithmshaveahardtimewithcorrelation -theexactsolutionmanagestoassignmarkedactivities2,3,and7inthe sametimeslotsandonlymissesmarkedactivity5.TheGRASPonlygets activity5inthesametimeslotandasaresult,excludesanysimultaneousassignmentsfor2,3and7.Italsoendsupwithmoretravelingthan necessary.

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117 PExact= 1611 2161 3771 4111 5111 6228 7331 8511 9154 PGRASP= 1111 2551 3611 4161 5113 6211 7111 8311 9121 Figure7Â…7:ComparisonOfExactandGRASPLFor9994TPAW(5.10.5)

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CHAPTER8 CONCLUSIONSANDEXTENSIONS Operatingcollectivegroupsofunmannedplatformshasmanytechnical challenges,notleastofwhichiseectivelyroutingthevehiclestononhomogeneous objectives.Aftermuchstudy,thisproblemwasshowntopossessthefundamental natureof2typesofclassicaloperation sresearchproblems-Knapsackproblems andTravelingSalesmanVehicleRoutingProblems(VRP)representingResource allocationofvehiclestononhomogeneousobjectives,andpathplanningofvehicle trajectoriesor,inthediscretesense,thesequencingofwaypointsforeachvehicle respectively.ThedicultyisthatthestructureofknapsackandVRParecontrary; thatis,knapsackproblemsdonotreadilyadmitroutingandVRPsnotreadily admitknapsack-like”exibilityinnon-hom ogeneousassignmentoptionsandresource uniqueness.Neitherframeworkhaseveraddressedcorrelatedbehaviorofresources. Soaftermuchbackgroundstudyofknapsackproblems,vehiclerouting problems,andmanynewextensionsthatattempttocombinethetwo,aframework usingtheNewProductDevelopmentproblemstructurewithaMultidimensional AssignmentProblemasthecoreofthemathematicalmodelwasadoptedasthe objectivemodel forintegratedassignmentandpathplanning.TheNPDframework wasthe“rsttocapturecorrelatedbehaviorofresourcesandalsoisabletoadmit mostoftheknapsackandVRPfeaturesstudiedhere. AlgorithmsweredevelopedusingbranchandboundandGreedyRandomized AdaptiveSearchProcedure(GRASP)frameworksandappliedtoseveraltest problems,includingamilitaryscenariowithautonomousairvehiclesandground mobiletargets.Theresultsareinterestingbutbearfurtherstudy.Theintegral complexityoftheMAPindicatetheuseofheuristicslikeGRASP,butthesolution 118

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119 qualityand,forlargeproblems,eventheruntimesdonotlookallthatfavorably evenonGRASP.FutureeortsshouldfocusonmethodstomaketheGRASPmore ecient.Twopossibilitiesarenowdiscussed. 1.ThevariableseparationtrickintroducedforMultiPurposeBatchPlants wasshowntheretosigni“cantlyreducethecomputationalcomplexityofthe problem.There,acontinuousvariable Ts ijkwasusedtode“nethestarttime ofactivity i byresource j ateventtime k .Adiscrete,Booleanvariables wvikand yvjk,wasusedtodecideifactivity i startsateventtime k andifresource j alsobeginsateventtime k respectively.Theseparationofwhatwouldhave beena3-indexdecisionofresource i workingonactivity j attime k intoa mixedintegerprogramreducesthenumberofdiscretevariablesthroughthe introductionofcontinuousvariables.Thissameapproachcouldbeappliedto theMAPbycreatingcontinuousvariableson M dimensionsandthediscrete variablesononly2. 2.AsstatedattheendofSection7.4,theGRASPspendsagooddealoftime lookingforimprovementsthat,withcorrelateddata,willimplyassignments ofmorethanoneresourcetothesametimeslot.Theresultisthatalotof eortiswastedlookingforwhatshouldactuallybequiteobvious-those activitiesmarkedashavingcorrelatedeectwillalwaysturnupinpairs onthesametimeslot.Alloftheassignmentsthatcontainthesepatterns, foreverytimeslotandresourcepairing ,couldfairlyecientlybetriedout beforethecandidatelistislookedat.Inessence,thesepatternsshouldbe systematicallyseededintotheconstruction.This patternseeding routine wouldbefairlyeasytoimplementandtestandshouldmaketheGRASP solutionsmuchbetterqualitywithlesstravelingtime.

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120 Terminatingconstraintswerediscussedseveraltimesbutneveraddedinthe model.Thisisstillverynecessaryandonceaccomplished,itshouldbestraightforwardtoshowthattheobjectivemodelgeneralizestheWeaponTargetAssignment problemandseveralvariantsofVRPandknapsack. Stochasticextensionstothedeterministicmodelwerealsosuggestedbut neveraddedtothemodel.Uncertaintyonthequantity,typeof,andlocationof theobjectivesisthelikelyneed.Thestochasticprogrammingapproach,andrisk measuresoutlinedinChapter2[19]aregoodcandidateapproaches.

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REFERENCES [1]M.Athans,P.HoseinandJ.Walton,(1988),DynamicWeaponTarget AssignmentProblemsWithVulnerable C 3Nodes,ŽProceedingsofthe CommandandControlSymposium,Monterey,CA,pp.75-85. [2]J.S.Bellingham,M.Tillerson,M.Alighanbari,andJ.P.How,(2002),CooperativePathPlanningForMultipleUAVsinDynamicandUncertain Environments,ŽProceedingsoftheI EEEConferenceonDecisionandControl,LasVegas,NV,Dec.2002. [3]J.R.BirgeandC.H.Rosa(1996),ParallelDecompositionofLarge-scale StochasticNonlinearPrograms,Ž AnnalsofOperationsResearch ,vol.64,pp. 39-65,1996. [4]W.Burrows,(1988),TheVehicleRoutingProblemwithLoadsplitting:A HeuristicApproach,Ž24thAnnualConferenceoftheOperationalResearch SocietyofNewZealand,pp.33-38. [5]C.C.Carøe,andJ.Tind(1998),L-shapedDecompositionofTwo-stage StochasticProgramsWithIntegerRecourse,Ž MathematicalProgramming , vol.83,pp.451-464. [6]W.J.Cook,W.H.Cunningham,W.R.Pulleyblank,andA.Schrijver(1997), CombinatorialOptimization, Wiley-Interscience,JohnWileyandSonsInc., NewYork. [7]N.Christo“des(1985),VehicleRouting,ŽChapter12of TheTraveling SalesmanProblem ,E.L.Lawler,J.K.Lenstra,A.H.G.RinnooyKan,andD.B. Shmoys,Ed.s,JohnWileyandSonsLtd.,NewYork,pp.431-448. [8]N.Christo“des,A.Mingozzi,andP.Toth,(1980),ExactAlgorithmsFor TheVehicleRoutingProblemBasedonSpanningTreeAndShortestPath Relaxations,Ž Math.Programming 20,pp.255-282. [9]G.G.denBroeder,R.E.Ellison,andL.Emerling(1959),OnOptimum TargetAssignments,Ž OperationsResearch ,vol.7,pp.322-326. [10]M.Dror,G.Laporte,andP.Trudeau,(1994),VehicleRoutingWithSplit Deliveries, DiscreteAppl.Math.50, pp.239-254. [11]A.R.EcklerandS.A.Burr(1972), MathematicalModelsofTargetCoverageandMissileAllocation, MilitaryOperationsResearchSocietyPress, Alexandria,VA. 121

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122 [12]M.Fisher,(1994),OptimalSolutionofVehicleRoutingProblemsMinimum k-trees,Ž OperationsResearch, 37,pp.319-328. [13]M.Hennebry,A.Kamel,K.Nygard,(2003),AnalyticalAndDiscrete OptimizationApproachesInOptimalTrajectoryGeneration,Ž inRecent DevelopmentsInCooperativeControl ,R.MurpheyPanosPardalos,andS. Butenko,Eds.,KluwerAcademicPublishers,Boston,pp.167-174. [14]J.B.Hofman,W.A.MetlerandF.L.Preston(1990),ASuiteOfWeapon AssignmentAlgorithmsForAnSDIMidcourseBattleManager,Ž Naval ResearchLaboratoryReport 6713. [15]P.A.Hosein(1989),AClassofDynamicNonLinearResourceAllocation Problems,ŽPh.D.Thesis,DepartmentofElectricalEngineeringandComputerScience,MassachusettsInstituteofTechnology,MA. [16]M.G.IerapetritouandC.A.Floudas,(1998),EectiveContinuous-Time FormulationforShort-TermScheduling.1.MultipurposeBatchProcesses,Ž IndustrialandEngineeringChemistryResearch, 37,pp.4341-4359. [17]M.G.IerapetritouandC.A.Floudas,(1998),EectiveContinuous-Time FormulationforShort-TermScheduling.2.ContinuousandSemicontinuous Processes,Ž IndustrialandEngineeringChemistryResearch, 37,pp.4360-4374. [18]D.E.Kirk,(1970), OptimalControlTheory, Prentice-Hall,Inc.,Englewood, NJ. [19]P.Krokhmal,R.Murphey,P.Pardalos,S.UryasevandG.Zrazhevski, (2002),RobustDecisionMaking:AddressingUncertaintiesInDistributions,Ž CooperativeControl,Models,Algorithms,andApplications ,R. MurpheyandPanosPardalos,andS.Butenko,Eds.,KluwerAcademic Publishers,Boston,pp.165-186. [20]G.LaporteandF.V.Louveaux(1993) ,TheIntegerL-shapedMethodFor StochasticIntegerProgramsWithCompleteRecourse,Ž OperationsResearch Letters ,vol.13,pp.133-142. [21]G.Laporte,andF.V.Louveaux,(1998),SolvingStochasticRoutingProblemsWithTheIntegerL-shapedMethod,Ž FleetManagementandLogistics, T.G.CrainicandG.Laporte(eds.), KluwerAcademicPublishers,Boston, pp.159-167. [22]H.C.Lau,M.SimandK.M.Teo,(2003),VehicleRoutingProblemwith TimeWindowsAndaLimitedNumberOfVehicles,Ž EuropeanJournalof OperationalResearch 148,pp.559-569. [23]L.H.Lee,K.C.Tan,K.OuandY.H.Chew,(2003),VehicleCapacity PlanningSystem:acasestudyonVehicleRoutingProblemWithTime

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123 Windows,Ž IEEETransactionsonSystems,ManandCybernetics,PartA, 33(2),pp.169-178. [24]X.LinandC.Floudas,(2001),Design,SynthesisandSchedulingofMultipurposeBatchPlantsviaanEectiveContinuous-TimeFormulation,Ž ComputersandChemicalEngineering, 25,pp.665-674. [25]S.P.LloydandH.S.Witsenhausen(1986),WeaponsAllocationis NP complete,ŽIEEESummerSimulationConference,Reno,Nevada. [26]A.S.Manne(1958),ATargetAssignmentProblem,Ž OperationsResearch , vol.6,pp.346-351. [27]S.Matlin(1970),AReviewOfTheLiteratureOnTheMissileAllocation Problem,Ž OperationsResearch 18,pp.334-373. [28]T.McLain,P.Chandler,S.Rasmussen,andM.Pacter,(2001),CooperativeControlOfUAVRendezvous,ŽIEEEAmericanControlConference, Arlington,VAJune25-27,pp.23092314. [29]M.Moser,D.Jokanovic,N.Shiratori,(1997),AnAlgorithmForThe MultidimensionalMultiple-choiceKnapsackProblem,Ž IEICETrans.Fundamentals ,vol.E80-A,No.3March1997. [30]R.A.Murphey(1999)AnApproximateAlgorithmForAWeaponTarget AssignmentStochasticProgram,Ž ApproximationandComplexityinNumericalOptimization:ContinuousandDiscreteProblems ,P.MPardalos,(Ed.), KluwerAcademicPublishers,Boston. [31]R.A.Murphey,P.M.Pardalos,andL.Pitsoulis(1997),AGreedyRandomizedAdaptiveSearchProcedureForTheMultitargetMultisensorTracking Problem,Ž NetworkDesign:ConnectivityandFacilityLocation, Panos M.PardalosandDing-ZhuDu(eds.),AmericanMathematicalSocietypp. 277-301. [32]R.A.Murphey,P.M.Pardalos,andL.Pitsoulis(1998),AParallelGRASP forTheDataAssociationMultidimensionalAssignmentProblem,Ž Parallel ProcessingofDiscreteProblems ,P.M.Pardalos(Ed.),Springer,pp.159-179. [33]R.A.Murphey,S.Uryasev,andM.Zabarankin,(2005)TrajectoryOptimizationInaThreatEnvironment,Ž SubmittedtoJournalofGuidance,Control, andDynamics,AIAA . [34]G.L.NemhauserandL.A.Wolsey(1988), IntegerandCombinatorial Optimization, JohnWileyandSonsInc.,NewYork. [35]A.B.PooreandN.Rijavec(1993),ALagrangianRelaxationAlgorithmfor MultidimensionalAssignmentProblems arisingfromMultitargetTracking,Ž SIAMJournalonOptimization ,3,pp.544-563.

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124 [36]A.B.PooreandN.Rijavec(1994),MultidimensionalAssignmentFormulationofDataAssociationProblemsArisingfromMultitargetandMultisensor Tracking,Ž ComputationalOptimizationandApplications ,3,pp.37-54. [37]J.RenaudandF.F.Boctor,(2002),ASweep-basedAlgorithmForTheFleet SizeAndMixedVehicleRoutingProblem,Ž EuropeanJournalofOperational Research 140,Elsevier,pp.618-628. [38]N.Repenning,(2000)ADynamicModelofResourceAllocationinMultiProjectResearchandDevelopmentSystems,Ž SystemDynamicsReview ,16, 3,pp.173-212. [39]A.RhalibiandG.Kelleher,(2003),AnApproachToDynamicVehicleRouting,ReschedulingAndDisruptionMetricsŽ,Systems,ManandCybernetics Proceedings(SMCC03),WashingtonD.C.,U.S.A.,pp.3613-3618. [40]A.Richards,J.Bellingham,M.Tillerson,andJ.How(2002),Coordination AndControlOfMultipleUAVs,ŽAIAAGuidance,NavigationandControl Conference,Monterey,CA,AIAApaper2002-4588. [41]D.M.Ryan,C.HjorringandF.Glover(1993),ExtensionsofthePetal MethodForVehicleRouting,Ž JournaloftheOperationalResearchSociety ,44, pp.289-296. [42]R.Shultz,L.Stougie,andM.H.vanderVlerk(1998),SolvingStochastic ProgramsWithIntegerRecourseByEnumeration:aFrameworkUsing Gr¨ obnerBasisReductions,Ž MathematicalProgramming ,vol.83,pp.229-252. [43]M.Solomon,(1995),AlgorithmsForTheVehicleRoutingProblemWith TimeWindows,Ž TransportationScience,29(2), pp.156-166. [44]G.L.Urban,andJ.R.Hauser(1993), DesignandMarketingofNewProducts ,Prentice-Hall,UpperSaddleRiver,NewJersey,SecondEdition. [45]Y.WangandJ.R.Perkins(2002),OptimalResourceAllocationInNew ProductDevelopmentProjects:aControlTheoreticApproach,Ž IEEETrans. AutomaticControl ,vol.47,no.8,pp.1267-1276. [46]L.A.Wolsey(1988), IntegerProgramming, Wiley-Interscience,JohnWiley andSonsInc.,NewYork. [47]L.Yong,P.M.Pardalos,andM.G.C.Resende,(1994),AGreedyRandomizedAdaptiveSearchProcedureForTheQuadraticAssignmentProblem,Ž QuadraticAssignmentandRelatedProblems ,PanosM.PardalosandHenry Wolkowicz(eds.),16,AmericanMathematicalSociety,pp.237-262. [48]M.Zaberenkin,R.A.Murphey,andS.Uryasev,(2003),OptimalPath PlanningInaThreatEnvironment,Ž RecentDevelopmentsInCooperative

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125 Control ,R.Murphey,P.Pardalos,andS.Butenko,Eds.,KluwerAcademic Publishers,Boston,pp.349-406.

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BIOGRAPHICALSKETCH RobertMurpheyistheChiefoftheNavigationandControlBranchofthe MunitionsDirectorate,AirForceResearchLaboratory,atEglinAirForceBase, wherehehasworkedforover17years.Heischargedwithoversightofallbasicand appliedresearchinnavigation,guidanceandcontrolforair-launchedconventional weapons.Heiswidelypublishedintheoptimizationandcontrolliteraturewith 4booksandover30peerreviewedpublicationstohisnameandhasbeenrecognizedbytheAirForceforexcellenceinbasicresearch.Robertisco-founderofthe KluwerCooperativeSystemsbookseriesandisco-organizeroftheannualCooperativeControlandOptimizationConference,nowinits“fthyear.Heisanassociate fellowofAIAAandaseniormemberoftheIEEE.Robertisamemberofthe advisoryboardfortheUniversityofFlo ridaIndustrialandSystemsEngineering Departmentwherehereviewsthecurricu lumofthedepartmentsundergraduate andgraduateprogramsandmakesrecommendationsonteachingmethods,courses, andresearchinvestments.RobertismarriedtoCoriMurpheyandtheyhavetwo children,Erin,age7,andRyan,age5.TheycurrentlyresideinNiceville,Florida.