EFFICIENTSEQUENTIALDECISION‐MAKINGALGORITHMSFORCONTAINERINSPECTION
OPERATIONS
SushilMi;alandFredRobertsRutgersUniversity&DIMACS
DavidMadiganColumbiaUniversity&DIMACS
•CurrentlyinspecLngonlysmall%ofcontainersarrivingatports
PortofEntryInspecLonAlgorithms
•Goal:FindwaystointerceptillicitnuclearmaterialsandweaponsdesLnedfortheU.S.viathemariLmetransportaLonsystem
PortofEntryInspecLonAlgorithms
Aim:Developdecisionsupportalgorithmsthatwillhelpusto“opLmally”interceptillicitmaterialsandweaponssubjecttolimitsondelays,manpower,andequipment
Findinspec*onschemesthatminimizetotalcostincludingcostoffalseposi*vesandfalsenega*ves
MobileVACIS:truck‐mountedgammarayimagingsystem
SequenLalDecisionMakingProblem• Containersarrivingareclassifiedintocategories• Simplecase:0=“ok”,1=“suspicious”• Containershavea;ributes,eitherinstate0or1• Samplea)ributes:
– Doestheship’smanifestsetoffanalarm?– IstheneutronorGammaemissioncountabovecertainthreshold?
– DoesaradiographimagereturnaposiLveresult?– DoesaninducedfissiontestreturnaposiLveresult?
• Inspec3onscheme:– specifieswhichinspec*onsaretobemadebasedonpreviousobserva*ons
• Different“sensors”detectpresenceorabsenceofvariousa;ributes
•SimplestCase:A;ributesareinstate0or1
•Then:Containerisabinarystringlike011001
•So:ClassificaLonisadecisionfunc*onF thatassignseachbinarystringtoacategory.
F011001 0 or 1
Ifa;ributes2,3,and6arepresent,assigncontainertocategoryF(011001).
SequenLalDecisionMakingProblem
•Iftherearetwocategories,0and1,decisionfuncLonFisa Booleanfunc*on.
•Example:
•ThisfuncLonclassifiesacontainerasposiLveiffithasatleasttwoofthea;ributes.
abcF(abc)00000010010001111000101111011111
SequenLalDecisionMakingProblem
BinaryDecisionTreeApproach•BinaryDecisionTree:
–Nodesaresensorsorcategories(0or1)–Twoarcsexitfromeachsensornode,labeledlegandright.–Taketherightarcwhensensorsaysthea;ributeispresent,legarcotherwise
abcF(abc)00000010010001111000101111011111
CostofaBDT• CostofaBDTcomprisesof:– CostofuLlizaLonofthetreeand– CostofmisclassificaLon
0 0|0 0|0 1|0 1|0
1 0|1 0|1 1|1 1|1
0 0|0 1|0 1|0 1|0 1|0
1 0|1 0|1 0|1 1|1 0|1 1|1 0|1
( ) ( )
( )
( )
( )
a a b a b c a c
a a b a b c a c
a b c a c FP
a b a b c a c FN
f P C P C P P C P C
P C P C P P C P C
P P P P P P C
P P P P P P P P C
! = + + +
+ + + +
+ +
+ + +
ABDT, τwithn=3
P1ispriorprobabilityofoccurrenceofabadcontainer
Pi|j isthecondiLonalprobabilitythatgiventhecontainerwasinstatej,itwasclassifiedas i
SensorThresholds
Ps=0|0 + Ps=1|0 = 1Ps=1|1 + Ps=0|1 = 1
•Tscanbeadjustedforminimumcost
•Anandet.al.reportedthecheapesttreesobtainedfromanextensivesearchoverarangeofsensorthresholds.Forexample:forn=4,194,481testswereperformedwiththresholdsvaryingbetween[‐4,4]withastepsizeof0.4
• Approach:– BuildsonideasofStroudandSaeger1atLosAlamosNaLonalLaboratory
– InspecLonschemesareimplementedasBinaryDecisionTreeswhichareobtainedfromvariousBooleanfuncLonsofdifferenta;ributes
– Only“Complete”and“Monotonic”BooleanfuncLonsgivepotenLallyacceptableBinarydecisiontrees
– n=4
1Stroud,P.D.andSaegerK.J.,“Enumera*onofIncreasingBooleanExpressionsandAlterna*veDigraphImplementa*onsforDiagnos*cApplica*ons”,ProceedingsVolumeIV,Computer,Communica*onandControlTechnologies,(2003),328‐333
Previouswork:Aquickoverview
OpLmumThresholdComputaLon• ExtensivesearchoverarangeofthresholdshassomepracLcaldrawbacks:– Largenumberofthresholdvaluesforeverysensor
– Largestepsize– GrowsexponenLallywiththenumberofsensors(computaLonallyinfeasibleforn>4)
• Therefore,weuLlizenon‐linearopLmizaLontechniqueslike:
– Gradientdescentmethod
– Newton’smethod
SearchingthroughaGeneralizedTreeSpace
• WeexpandthespaceoftreesfromStroudandSaeger’s“Complete”and“Monotonic”BooleanFuncLonstoCompleteandMonotonicBDTs,because…
• UnlikeBooleanfuncLons,BDTsmaynotconsiderallsensoroutputstogiveafinaldecision
• Advantages:– Allowsmore,potenLallyusefultreestoparLcipateintheanalysis
– HelpsdefininganirreducibletreespaceforsearchoperaLons
– MovesfocusfromBooleanFuncLonstoBinaryDecisionTrees
RevisiLngMonotonicity• MonotonicDecisionTrees– Abinarydecisiontreewillbecalledmonotonicifall
thelegleafsareclass“0”andalltherightleafsareclass“1”.
• Example:
abcF(abc)00000010010101111000101111001111
RevisiLngCompleteness• CompleteDecisionTrees– Abinarydecisiontreewillbecalledcompleteifeverysensoroccursatleastonceinthetreeandatanynon‐leafnodeinthetree,itslegandrightsub‐treesarenotidenLcal.
• Example:
abcF(abc)00000011010101111000101111011111
TheCMTreeSpace
No.ofaPributes
Dis*nctBDTsTreesFromCMBooleanFunc*ons
CompleteandMonotonicBDTs
2 74 4 4
3 16,430 60 114
4 1,079,779,602 11,808 66,000
TreeSpaceTraversal• GreedySearch
1. RandomlystartatanytreeintheCMtreespace2. FinditsneighboringtreesusingneighborhoodoperaLons3. Movetotheneighborwiththelowestcost4. IterateLllthesoluLonconverges
– TheCMTreespacehasalotoflocalminima.Forexample:9inthespaceof114treesfor3sensorsand193inthespaceof66,000treesfor4sensors.
• ProposedSoluLons• StochasLcSearchMethodwithSimulatedAnnealing• GeneLcAlgorithmsbasedSearchMethod
TreeSpaceIrreducibility
• WehaveprovedthattheCMtreespaceisirreducibleundertheneighborhoodoperaLons
• SimpleTree:– AsimpletreeisdefinedasaCMtreeinwhicheverysensoroccursexactlyonceinsuchawaythatthereisexactlyonepathinthetreewithallsensorsinit.
ToProve:Givenanytwotreesτ1,τ2inCMtreespace,τn,τ2canbereachedfromτ1byanarbitrarysequenceofneighborhoodoperaLons
Weprovethisinthreedifferentsteps:1. Anytreeτ1canbeconvertedtoasimpletreeτs12. Anysimpletreeτs1canbeconvertedtoanyothersimple
treeτs23. Anysimpletreeτs2canbeconvertedtoanytreeτ2
CMTreespace,τn
Simpletreesτ1
τs1 τs2
τ2
Results
• SignificantcomputaLonalsavingsoverpreviousmethods
• Haverunexperimentswithupto10sensors
• GeneLcalgorithmsespeciallyusefulforlargerscaleproblems
CurrentWork
• Treeequivalence
• TreereducLonandirreducibletrees
• CanonicalformrepresentaLonoftheequivalenceclassoftrees
• RevisiLngcompletenessandmonotonicity
ThankYou!
MonotonicBooleanFunc3ons:•Giventwostringsx1x2…xn, y1y2…yn
•Fismonotoniciffxi ≥ yi foralliimpliesthatF(x1x2…xn) ≥ F(y1y2…yn).
CompleteBooleanFunc3ons:•BooleanfuncLonFisincompleteifFcanbecalculatedbyfindingatmostn-1a;ributesandknowingthevalueoftheinputstringonthosea;ributes•Inotherwords,Fiscompleteifallthea;ributescontributetowardstheoutput
Previouswork:Aquickoverview
Previouswork:Aquickoverview• StroudandSaeger:“bruteforce”algorithmforenumeraLng
binarydecisiontreesimplemenLngcomplete,monotonicBooleanfuncLonsandchoosingleastcostBDT.
263,515,92068945x10185
11,8081141,079,779,6024
60916,4303
42742
BDTsfromCMBooleanFunc*ons
CMBooleanExpressions
Dis*nctBDTsNo.ofaPributes
Infeasiblebeyondn>4!
ProblemswithStandardApproaches• GradientDescentMethod:SetngthevalueofthestepsizeheurisLcally,since:– Toosmallstepsize:longLmetoconverge
– Toobigstepsize:mightskiptheminimum
• Newton’sMethod:
– TheconvergencedependslargelyonthestarLngpoint– OccasionallydrigsinthewrongdirecLonandhencefailstoconverge.
• SoluLon:combina*onofgradientdescentandNewton’smethods
TheCombinedMethod1. IniLalizeTasvectorofrandomsensorthresholdvalues
2. Compute∂f ,Hf(τ)3. IfHf(τ)isnotposiLvedefinite,thenfindacloseapproximaLon
4. IfHf(τ)isnotwell‐condiLoned,thentakeafewstepsusinggradientdescentunLlitbecomeswell‐condiLoned
5. TakeastepusingNewton’smethod6. Repeatsteps1‐5unLlthesoluLonconverges7. Repeatsteps1‐6afewLmesandchoosetheoverallminimumcost
TreeNeighborhoodandTreeSpace• Structurebasedmethods
• ClassificaLonbasedmethods
• Wechoosestructurebasedneighborhoodmethodsbecause:
• Smallchangesintreestructuredonoteffectthecostsignificantly,and…
• AllBDTswithsameBooleanfuncLonmaydifferalotincost
TreeNeighborhoodandTreeSpace
• DefinetreeneighborhoodsuchthattheCompleteandMonotonic(CM)treespaceisirreducible
• Irreducibility– AnytreeintheCMtreespacecanbereachedfromanyothertreebyusingtheneighborhoodoperaLonsrepeLLvely
– AnirreducibleCMtreespacehelps“search”forthecheapesttreesusingneighborhoodoperaLons
SearchOperaLons
• SplitPickaleafnodeandreplaceitwithasensorthatisnotalreadypresentinthatbranch,andtheninsertarcsfromthatsensorto0andto1.
SearchOperaLons
• SwapPickanon‐leafnodeinthetreeandswapitwithitsparentnodesuchthatthenewtreeissLllmonotonicandcompleteandnosensoroccursmorethanonceinanybranch.
SearchOperaLons
• MergePickaparentnodeoftwoleafnodesandmakeitaleafnodebycollapsingthetwoleafnodesbelowit,orpickaparentnodewithoneleafnode,collapsebothofthemandshigthesub‐treeupinthetreebyonelevel.
SearchOperaLons
• ReplacePickanodewithasensoroccurringmorethanonceinthetreeandreplaceitwithanyothersensorsuchthatnosensoroccursmorethanonceinanybranch.
StochasLcSearchMethod1. RandomlystartatanytreeinCMspace
2. Finditsneighboringtrees,andfindtheiropLmumcosts
3. Selectmoveaccordingtothefollowingprobability.Ifweareattheithtreeτi,thentheprobabilityofgoingtoitskthneighborτik,isgivenby
whereniisthenumberofneighboringtreesofτi4. IniLalizethetemperaturet = 1,andloweritindiscreteunequal
stepsagereverymhopsunLlthesoluLonconverges
5. Repeatsteps1‐4afewLmesandchoosetheoverallminimum
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TreeSpaceIrreducibility
1. τ1τs1:
•Repeatedsubtreemerger•Toremoveanodeatdepthk, atmostk-2 needtobecheckedforcompleteness•WeprovethatthereisatleastonenodeinasubtreeatanyLme,thatcanbemergedwithoutdisturbingtheoverallcompletenessconstraint
TreeSpaceIrreducibility
2. τs1τs2:
•Firstconvertτs1tohavesimilar“skeleton”asτs2•ThenuserepeatedSwapoperaLons
SPLIT SPLITMERGE
MERGE
SWAPSWAPSWAP
TreeSpaceIrreducibility
3. τs2τ2:
•TheprocessofgoingfromatreetoasimpletreeisenLrelyreversible.Forexample:
–anysplitoperaLoncanbereversedusingamergeoperaLonandvice‐versa
–swapandreplaceoperaLonscanbereversedbyoppositeswapandreplaceoperaLons,respecLvely
•Therefore,τ2τs2impliesτs2τ2
GeneLcAlgorithmsbasedSearch
• TheunderlyingideaistogetapopulaLonof“be;er”treesfromacurrentpopulaLonof“good”treesbyusingthebasicoperaLons:– SelecLon– Crossover–MutaLon
• “be;er”decisiontreescorrespondtotheonescheaperthanthecurrentones(“good”)
GeneLcAlgorithmsbasedSearch
• Selec*on:– Selectarandom,iniLalpopulaLonofNtreesfromCMtreespace
• Crossover:– PerformedkLmesbetweeneverypairoftreesinthecurrentbestpopulaLon,bestPop
GeneLcAlgorithmsbasedSearch
– ForeachcrossoveroperaLonbetweentwotreesτiandτj,werandomlyselectanodeineachtreeandexchangetheirsubtrees
– However,weimposecertainrestricLonontheselecLonofnodes,sothattheresultanttreessLlllieinCMtreespace
GeneLcAlgorithmsbasedSearch
• Muta*on:
– Performedagereverym generaLonsofthealgorithm
– WedotwotypesofmutaLons:
1. Generateallneighborsofthecurrentbest
populaLonandputthemintothegenepool
2. ReplaceafracLonofthetreesofbestPopwithrandomtreesfromtheCMtreespace
ResultsI‐ThresholdOpLmizaLon
• ManyLmestheminimumobtainedusingtheopLmizaLonmethodwasconsiderablylessthantheonefromtheextensivesearchtechnique.
0 20 40 60 80 100
100
150
200
250
300
350
400
450
500
Tree Number
Total Cost
Tree costs at optimum thresholds
Combined Optimization
Extensive search
ResultsII‐SearchingCMTreeSpace
• StochasLcSearchMethod:• Successfullyperformedexperimentsforupton =5• Forexample,for4sensors(66,000trees)
– 100differentexperimentswereperformed
– Eachexperimentwasstarted10LmesrandomlyatsometreeandchainswereformedbymakingstochasLcmovesintheneighborhood,unLlconvergence
– Only4890treeswereexaminedonaverageforeveryexperiment
– Globalminimumwasfound82Lmeswhilethesecondbesttreewasfound10Lmes
ResultsII‐SearchingCMTreeSpace
• GeneLcAlgorithmsbasedMethod:• Successfullyperformedexperimentsforupton =10• For4sensors(66,000trees)
– 100differentexperimentswereperformed
– EachexperimentwasstartedwitharandompopulaLonof20treesandwasconLnuedfor27generaLonseach;themutaLonsareperformedagerevery3generaLons
– Only1440treeswereexaminedonaverageforeveryexperiment
– Globalminimumwasfoundall100Lmes– ThealgorithmreturnsawholepopulaLonofgoodtreesmostofwhichbelongto50besttrees
ResultsII‐SearchingCMTreeSpace
• Similarly,forn=5,thetreespaceconsistsofmorethan22.5billiontrees,wealwaysobtainedoneofthefollowingbesttrees:
• Eachofthesetreescosts41.4668
ResultsII‐SearchingCMTreeSpace
• Forn=10,followingwerethebesttreesoverafewruns:
CurrentWork• TreeEquivalence
–DecisionEquivalence:TwoormoredecisiontreesarecalleddecisionequivalentiftheirunderlyingBooleanfuncLonissame–CostEquivalence:Twotreesarecalledcostequivalentifftheyare“transposes”ofeachother.Forexample:
–ThesizeoflargestequivalenceclassalsoincreasesmorethandoubleexponenLallywithn–Therefore,wedefineaspaceofequivalenceclassesofdecisiontrees,withaunique,canonicalrepresentaLonofeachclass
CurrentWork• TreeReducLonandIrreducibleTrees
–Atransposeofacompletetreecanbeincomplete.Forexample:
–IrreducibleTrees:Atreewillbecalledirreducible,ifallthetreesbelongingtoitsequivalenceclassarecomplete
CurrentWork• CanonicalFormRepresentaLon
– WechosealexicographicrepresentaLonoftheequivalenceclass– “Pull‐up”thelexicographicallysmallestsensorastheroot
nodeandrecursivelyrepeattheprocedureinthelegandrightsubtrees
– AcanonicalformrepresentaLonofanequivalenceclassenablesusto“shrink”thetreespace
– Everytreeisfirstconvertedtoitscanonicalform,beforecheckingforitscost,thereforecheckingthecostofonlyonetreefromanequivalenceclassissufficient
CurrentWork• CanonicalFormRepresentaLon:Example
CurrentWork• RevisiLngCompleteness:
1. Atanynodeinatree,thelegandrightsubtreesshouldnotbe
cost‐equivalent
2. Atanynodeinatree,thelegandrightsubtreesshouldnothave
idenLcalBooleanfuncLon
• 2covers1,therefore…
• Equi‐completeBDT:Abinarydecisiontreewillbecalledequi‐
completeifeverysensoroccursatleastonceinthetreeand,atany
non‐leafnode,thelegandrightsubtreesdonotcorrespondtosame
BooleanfuncLon.
CurrentWork• RevisiLngMonotonicity:
1. Acost‐equivalenttreeofamonotonictreecanbenon‐monotonic(‘0’asrightleaf,‘1’aslegleaforboth).
• Equi‐monotonicBDT:Abinarydecisiontreewillbecalledequi‐monotonic,ifallthetreesbelongingtoitsequivalenceclassaremonotonic.
Discussion
1. TheexhausLvesearchmethod,forfindingtheopLmumthresholdsforagiventree,becomepracLcallyinfeasiblebeyondaverysmallnumberofsensors.
2. ThethresholdopLmizaLontechniquediscussedinourworkprovidefasterandbe;erwaystocalculatetheopLmaltotalcostofatree.
3. TheexhausLvesearchmethod,forfindingthecheapesttreeintheenLrespaceoftreesisalsohardtoextendbeyondaverysmallnumberofsensors.
4. WedescribedacoupleofefficientsearchmethodstofindthebesttreesintheCMtreespace
Discussion
5. ExpandingtheideasofmonotonicityandcompletenessfromBDFstoBDTsisreasonablebecause:• certaintreesobtainedfromincomplete/non‐
monotonicBDFsarepotenLallyvalidBDTsand,• itfacilitatestreesearchalgorithms
6. WeprovedthattheproposedCMtreespaceisirreducibleunderthedefinedneighborhoodoperaLons.
7. WediscussedtheideasoftreeequivalenceandtreereducLonthathelpus“shrink”thetreespace
8. Wedescribewaytorepresentanequivalenceclasswithaunique,canonicalform.
FutureWork
• Amorebasicandrigorousanalysisofmonotonicityisrequired
• DifferentinstancesofasensorinatreecanbesettodifferentthresholdsforopLmumcost
• Sensormodels,otherthantheoneweusecouldbetried
• Dr.FredRoberts
• DIMACS,NSFandONR
• Dr.PeterMeerandOncelTuzel
• Dr.EndreBoros
Acknowledgements