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New Tools For Large-Scale Combinatorial Optimization Problems

Permanent Link: http://ufdc.ufl.edu/UFE0024719/00001

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Title: New Tools For Large-Scale Combinatorial Optimization Problems
Physical Description: 1 online resource (96 p.)
Language: english
Creator: Shylo, Oleg
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: optimization, parallel, portfolio, quadratic, restart, scheduling
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Many traditional algorithmic techniques used in combinatorial optimization have reached the computational limits of their scope. A relatively low cost and availability of parallel multi-core clusters offer a potential opportunity to shift these limits. Unfortunately, most existing serial algorithms are not easily adaptable to parallel computing systems. The objective of this work is to provide new algorithmic methods that can be easily and effectively scaled to parallel systems with large numbers of processing units. We concentrate on a scalability that comes from running a set of independent algorithms in parallel. We investigate the relationship between the parallel acceleration and properties of serial algorithms. In this dissertation, we present the algorithms based on Global Equilibrium Search method and Tabu Search method for Unconstrained Binary Quadratic Problem, Weighted MAX-SAT problem and Job Shop Scheduling Problem. These algorithm demonstrate the state-of-the-art performance comparing to the latest published work in the field.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Oleg Shylo.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Pardalos, Panagote M.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-02-28

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024719:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024719/00001

Material Information

Title: New Tools For Large-Scale Combinatorial Optimization Problems
Physical Description: 1 online resource (96 p.)
Language: english
Creator: Shylo, Oleg
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: optimization, parallel, portfolio, quadratic, restart, scheduling
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Many traditional algorithmic techniques used in combinatorial optimization have reached the computational limits of their scope. A relatively low cost and availability of parallel multi-core clusters offer a potential opportunity to shift these limits. Unfortunately, most existing serial algorithms are not easily adaptable to parallel computing systems. The objective of this work is to provide new algorithmic methods that can be easily and effectively scaled to parallel systems with large numbers of processing units. We concentrate on a scalability that comes from running a set of independent algorithms in parallel. We investigate the relationship between the parallel acceleration and properties of serial algorithms. In this dissertation, we present the algorithms based on Global Equilibrium Search method and Tabu Search method for Unconstrained Binary Quadratic Problem, Weighted MAX-SAT problem and Job Shop Scheduling Problem. These algorithm demonstrate the state-of-the-art performance comparing to the latest published work in the field.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Oleg Shylo.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Pardalos, Panagote M.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-02-28

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024719:00001


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Iwouldliketothankmyadviser,Dr.PanosM.Pardalos,forallofhisencouragementandsupport.Hisknowledge,experienceandsenseofhumorwerecrucialforthiswork.Also,IwouldliketoacknowledgecollaboratorsClaytonCommander,ValeriyRyabchenko,AlkisVazacopoulos,OlegProkopyev,AshwinAruselvan,IliasKotsireasandTimothyMiddelkoop,whohavebeenapleasuretoworkwith.Iamgratefultomyfriends,myfamilyandmygirlfriendErikafortheirlove,moralsupportandallthefunwehadtogether. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1RESTARTSTRATEGIESINOPTIMIZATION:PARALLELANDSERIALCASES ........................................ 10 1.1Introduction ................................... 10 1.2Motivation .................................... 11 1.3LiteratureOverview .............................. 12 1.4OptimalRestartStrategies ........................... 13 1.4.1SerialAlgorithm ............................. 13 1.4.2ParallelAlgorithm ............................ 16 1.5AlgorithmPortfolioandRestartStrategies .................. 20 2GLOBALEQUILIBRIUMSEARCH ........................ 25 2.1AlgorithmDescription ............................. 25 2.2ApproximationFormula ............................ 29 3UNCONSTRAINEDBINARYQUADRATICPROBLEM ............ 32 3.1Introduction ................................... 32 3.2DescriptionoftheAlgorithm .......................... 34 3.2.1SearchProcedures ............................ 35 3.2.2GenerationProcedure .......................... 35 3.2.3ComputationalExperiments ...................... 35 3.3ConcludingRemarks .............................. 41 4PERIODICCOMPLEMENTARYBINARYSEQUENCES ............ 42 4.1Introduction ................................... 42 4.2PAFandQuadraticForms ........................... 42 4.3ACombinatorialOptimizationFormalismforPCS .............. 45 4.4ApplicationsofPCSandRelatedSequences ................. 46 4.5OptimizationAlgorithm ............................ 47 4.6Results ...................................... 48 4.6.1PCS(50;2) ................................ 48 4.6.2Consequences .............................. 49 4.7Conclusion .................................... 49 5

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........................ 58 5.1Introduction ................................... 58 5.2AlgorithmfortheWeightedMAX-SAT .................... 59 5.2.1GenerationProcedure .......................... 59 5.2.2LocalSearchfortheWeightedMAX-SAT 59 5.3ComputationalExperiments .......................... 61 5.3.1Benchmarks ............................... 61 5.3.2AlgorithmsandSoftware ........................ 65 5.3.3ResultsandConclusions ........................ 65 6JOBSHOPSCHEDULINGPROBLEM ...................... 67 6.1Introduction ................................... 67 6.2MathematicalModelandNotations ...................... 68 6.3AlgorithmDescription ............................. 69 6.3.1EncodingandGenerationProcedure .................. 72 6.3.2LocalSearch ............................... 74 6.3.3AccelerationofMoveCostEstimation ................. 76 6.4ComputationalResults ............................. 77 6.5ConcludingRemarks .............................. 80 6.6BenchmarkProblems .............................. 82 REFERENCES ....................................... 89 BIOGRAPHICALSKETCH ................................ 96 6

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Table page 3-1ResultsforBeasleyinstances[ 1 ] ........................... 39 3-2ResultsforPalubeckis'instances[ 2 ] ......................... 40 3-3ResultsfortheMaximumIndependentSetinstancesarisingincodingtheory[ 3 ] 41 5-1Resultsforjnhbenchmarks ............................. 64 5-2Resultsforrndw1000bbenchmarks ......................... 64 5-3Resultsforrndw1000cbenchmarks ......................... 64 6-1ComputationalresultsbyGESforORB1-ORB10 ................. 82 6-2ComputationalresultsforLA01-LA40 ....................... 83 6-3ResultsforTA1-TA80 ................................ 84 6-4ResultsforTA1-TA80(continued) .......................... 85 6-5ComparisonwithTSSB ............................... 86 6-6Comparisonwithotheralgorithms ......................... 86 6-7ResultsforDMU1-DMU80 .............................. 87 6-8ResultsforDMU1-DMU80(continued) ....................... 88 7

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Figure page 1-1Tabualgorithmandjobshopschedulingproblem.Averagerunningtimeasafunctionofrestartparameter ............................ 11 1-2TheaveragerunningtimeasafunctionofrestartparameterR,haslog-normalldistributionwith=1and=2;h(x)=f(x) 1Prfxgishazardratefunction ... 15 1-3Thealgorithmicspeed-upasafunctionofthenumberofprocessors;haslog-normaldistributionwith=1and=2;theactualspeed-upisobtainedbycomparingtheoptimalserialrestartversiontotheoptimalparallelrestartversion;theplotforthelinearspeed-upisforcomparisonpurpposesonly. ............. 19 1-4Theeciencyofparallelization(linearspeed-uphasecencykoecientequalto1),haslog-normalldistributionwith=1and=2; ............ 20 1-5Exampleofimprovedperformancewhencombiningtwoalgorithmswithlog-normallydistributedrunningtime. ............................... 21 1-6Anotherexampleofimprovedperformancewhencombiningtwoalgorithmswithlog-normallydistributedrunningtime. ....................... 22 2-1GESmethod(generalscheme) ............................ 28 3-1LocalSearchTABU ................................. 36 3-2Generationprocedure ................................. 37 4-1GraphicalrepresentationsofthefoursymmetricmatricesM1;M2;M3;M4 44 5-1Generationprocedure ................................. 60 5-2LocalSearch1-opt .................................. 61 5-3LocalSearchk-opt .................................. 62 5-4LocalSearchTABU ................................. 63 6-1GESforJSP ..................................... 70 6-2Procedurethatgeneratessolution ......................... 73 6-3Improvementprocedure ............................... 76 6-4Estimationprocedure ................................ 78 8

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1.3 forexamples).Inaddition,therandomnatureofLasVegasalgorithmsallowsforasimpleparallelization,whichconsistsinrunningseveralindependentcopiesofthesamealgorithminparallelandhaltingtheexecutionwheneverthecorrectanswerisfound.ThispaperaddressestheproblemofminimizingtheaveragerunningtimeofLasVegasalgorithm,bothinserialandparallelsetups.Theconditionsthatarenecessaryfortheexistenceofeectiverestartstrategyarepresented(i.e.allowingforaccelerationofthealgorithm).Weshowthattheoptimalrestartstrategyfortheparallelcasemightbedierentfromtheoptimalrestartstrategyoftheserialalgorithm.Thisresultassertsthatthemosteectiveserialalgorithmisnotnecessarilythebestchoiceforparallelization.Ourresultscanbeusedtocreatetheapproximationmodelsdescribingthebehaviourofcomplexalgortihmsallowingforeectiverestartstrategies.AbriefoverviewoftherelatedworkispresentedinSection 1.3 .OurmainresultsandtheirproofsaredescribedinSection 1.4 10

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4 ].Thisalgorithmincorporatesthesimplerestartstrategy:thetabualgorithmisrestartedafterevery200000iterations.However,thereisnoexplanationordiscussionforsuchchoice.Thisisacommonsituationfortheresearchpapersonoptimizationalgorithmsandthelackofclarityinvolvingtheconceptofrestartstrategiesneedstobeaddressed.InFigure 1.2 ,thecomputationalexperimentsinvolvingtabusearchwithdierentrestartparametersarepresentedforasinglebenchmarkprobleminstance.Itisclear,thatthechoiceofrestartparameterprovidedin[ 4 ]isnear-optimalandwasbasedoncarefultuning.Additionally,thechoiceoftherestartparameteriscrucialforthealgorithm'sperformance. Figure1-1. Tabualgorithmandjobshopschedulingproblem.Averagerunningtimeasafunctionofrestartparameter 11

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5 ],theideaofrestartstrategieswasformulated,allowingforaccelerationofgeneralLasVegas-typealgorithms.TherestartstrategywasdenedasaninnitesetoftimeintervalsS=ft1;t2;:::gafterwhichaLasVegasalgorithmisre-initiated(i.e.restartedwithanewrandomseed).Thedomainofrestartstrategieswasextendedtothecasewhenti'sarerandomvariables,andrunsmaybesuspendedandrestartedatalatertime.Theauthorshaveshownthatfortheoptimalrestartstrategyti'sareconstant:t1=t2=:::=t(uniformstrategy).Toobtaintheoptimalrestartperiodt,onerequiresthefullknowledgeoftherunningtimedistribution.Theuniversalrestartstrategywasproposedtoovercomethisissue.SuchstrategyisonlyalogarithmicfactorslowerthantheoptimalrestartstrategyanddoesnotrequireanypriorknowledgeofLasVegasalgorithm.In[ 6 ],theseresultswereextendedtotheparallelenvironment.Similartotheserialcase,auniversalparallelrestartstrategywasproposedthatisonlyalogarithmicfactorslowerthantheoptimalparallelrestartstrategy.Theexampleswereprovidedforwhichnon-uniformparallelrestartscheduleperformsbetterthantheuniformstrategy.Theauthorsshowedthattheoptimaluniformparallelrestartstrategy(i.e.wheneachprocessorrunsindependentcopyofLasVegasalgorithmandisrestartedafterxedtimeperiod)iswithinaconstantfactorfromanoptimalparallelrestartstrategy.Theconceptofrestartstrategiesisalsoapplicableinthedomainofdeterministicalgorithms.In[ 7 ],itwasshownthatbyintroducingrandomnessandrestartstrategiesintodeterministictechniques(completesearchalgorithms),onecanachievealgorithmicspeedupforpracticalinstancesofconstraintsatisfactionproblemsandpropositionalsatisability.Theexistenceofeectiverestartstrategiesforenumerationalgorithmswasformallyexplainedandjustiedin[ 8 ].Theauthorsintroducedtheconceptofrestart-distribution,whichisthedistributionofrunningtime,suchthatthereexistsaneectiveoptimalrestartstrategy(i.e.allowingaccelerationofLasVegasalgorithm).Asearchtreemodelisprovidedin[ 9 ]allowingforaformalproofofheavy-tailedbehaviour 12

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10 ].TheloadingprocessofInternetpagecanbeacceleratedbyapplyingrestartstrategy(forexample,bysimplyclickingrefreshbuttoninthebrowser).TheauthorsprovidealgorithmstocalculatethemomentsofcompletiontimeforLasVegasalgorithmsundertheoptimalrestartstrategy.Thepresenceofecientrestartstrategiesforthetabusearchalgorithmwasreportedin[ 11 ].Itisworthnotingthattheconceptofrestartstrategiesiswidelyusedinthealgorithmicdesignofmetaheuristics(seeforexample[ 12 ],[ 13 ],[ 14 ]).However,itsimpactontheperformanceofthealgorithmisoftenunderestimatedandmerelymentioned. 13

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8 ],theauthorsintroducedthenotionofarestart-distribution,which,inotherwords,isthedistributionthathaspotentialforacceleration(byimplementingarestartversionofunderlyingalgorithm): (PrfRg)2ZR0xf(x)dx==1PrfRg 14

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1.4.1 Figure1-2. TheaveragerunningtimeasafunctionofrestartparameterR,haslog-normalldistributionwith=1and=2;h(x)=f(x) 1Prfxgishazardratefunction Proof. dRf(R)

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Proof. dR1PrfRg 1 ,atoptimalpointR:1PrfRg 2 16

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1(1PrfRg)n(1{5) 1(1PrfRg)nn(1PrfRg)n(1PrfRg)n1f(R) (1(1PrfRg)n)2+n(1PrfRg)n1f(R) (1(1PrfRg)n)2ZR0xfnmin(x)dx=n(1PrfRg)n1f(R) 1(1PrfRg)nR(1PrfRg)n 1(1PrfRg)n+Rfnmin(R) 1(1PrfRg)n+(1PrfRg)n 1(1PrfRg)n=(1PrfRg)n 1(1PrfRg)n 4 issimilartotheproofofProposition 2 andislefttothereader. Proof. 17

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5 ],whichstatesthattheoptimalrestartstrategyisuniform(i.e.onlyonecopyisrunningandrestartintervalisconstant). Proof. 1 andCorrollary 2 :T(R) 1PrfRng
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Thealgorithmicspeed-upasafunctionofthenumberofprocessors;haslog-normaldistributionwith=1and=2;theactualspeed-upisobtainedbycomparingtheoptimalserialrestartversiontotheoptimalparallelrestartversion;theplotforthelinearspeed-upisforcomparisonpurpposesonly. super-linearspeedup,ifitutilizesuptonprocessorsandprovidesthespeed-upthatislargerthann.Intheliterature,onecanndthepapersthatreport\super-linear"speed-upwhenusingsuchtrivialparallelization.Theorem 2 showsthatifthesuper-linearspeed-upisachieved,thentheunderlyingdistributionoftherunningtimeisarestart-distribution.Insomesense,thetrivialparallelizationtakesadvantageoftherestart-distributionpropertieswithoutimplementinganyrestartstrategy.Furthermore,theperformanceofthealgorithmcanbeimprovedbyapplyinganeectiverestartstrategyexplicitly.Inotherwords,thesuper-linearityismainlyduetoineciencesoftheserialalgorithm.Since,theeciencyofparallelizationquicklydegradeswithincreasingnumberofprocessors(seeFigure 1.4.2 ),inordertoachievethebestutilizationofavailableresources,oneshouldcombinethetrivialparallelization(runningindependentcopiesinparallel)withothertypesofparallelizing:bit-levelparallelism,instruction-levelparallelism,dataparallelism,taskparallelism. 19

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Theeciencyofparallelization(linearspeed-uphasecencykoecientequalto1),haslog-normalldistributionwith=1and=2; Intuitively,onecanassumethatthebestserialalgorithmisthebestcandidateforparallelizing.However,Theorem2togetherwithexamplepresentedaboveshowsthattheoptimalrestartparameterfortheoptimalserialalgorithmcanbenon-optimalinparallelenvironment.Inotherwords,thestableperformanceintheserialcasedonotguaranteetheecientparallelperformance.Ontheconrary,analgorithm,whichdoesnotperformveryecientinaserialcase,canoutperformmoreeectiveserialsearchprocedureswhenruninparallel. 20

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Exampleofimprovedperformancewhencombiningtwoalgorithmswithlog-normallydistributedrunningtime. Consequently,weinvestigatedthecombinationofalgorithmswiththerunningtimesfollowingrestart-distributions.Thelog-normaldistributionwasusedasabasisforourstudiessinceitbelongstothefamilyofrestartdistributions.Weconsideredapairsofalgorithmswithlog-normallydistributedrunningtimestooptimalsolution(dierentmeansandvariances).InFigure 1.5 ,wepresenttheresultsforthemixoftwoalgorithms(log-normalrunningtimes).Threedatasetscorrespondtothefollowingsetups: 1.5 ,wepresentanotherexampleofeectivecombination.Inthissectionwewilladdressthelimitationsofthisapproachwithrespecttoalgorithmsimplementingrestartstrategies. 21

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Anotherexampleofimprovedperformancewhencombiningtwoalgorithmswithlog-normallydistributedrunningtime. LetusconsidertwoalgorithmsA1andA2withrestartparametersR1andR2andrandomrunningtimes1and2respectively.AssumethatR1=kR2,wherekisinteger.Therunningtimeofthesealgorithmsortheircombinationwillbeconsideredtobeamultipleoftheirrestartparameters.Inotherwords,thealgorithmscanbeterminatedonlyattheirrestarttimepoints(afterthesolutionwasfound)andthelargestterminationtimeisthetotalrunningtimeofalgorithmportfolio.Theseassumptionsaremadetosimplifythediscussionandthegeneralcasewillbeaddressedlater.Letp1=Prf1R1gandp2=Prf2R2g.Firstly,letusshowthatundertheassumptionsdescribedabove,thebestalgorithmportfolioconsistsofasinglealgorithm,i.e.mixingalgorithmsisnotaprotablestrategy.LetNbeatotalnumberofavailableparallelprocessors,andtheoptimalmixedportfolioconsistsofn2copiesofA2.Supposethatthemixedportfolioisbetterthenasinglealgorithmportfolio.Then,therunningtimeofthemixshouldbebetterthentherunningtimeofthesinglealgorithmportfolioconsistingofsolelyofA1: 22

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1(1p1)Nn2(1p2)n2k
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1(1p1)N
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15 ],simulatedannealing(SA)[ 16 ],etc.)havegainedconsiderableamountsofattentionfromthescienticcommunityforbeingtheonlypracticaltoolthatcanbeappliedtoawiderangeofdicultproblems.GlobalEquilibriumSearchoersanotherhighlyeectivetoolforsolvinglargescaleoptimizationproblems[ 17 ].TheGESmethodsharesideassimilartothosethatinspiredtheSAtechnique,whileproviding,inpractice,fasterasymptoticconvergencetotheoptimalsolutiononawideclassofoptimizationproblems.Moreover,theGESmethodcanbeusedinanensemblewithothertechniques,whichmakesitmoreversatilethanmostofitspredecessors.Consideradiscreteoptimizationproblemofthefollowingform: minff(x):x2S:Sf0;1gng(2{1)wherefissomequalityfunction.Letusintroducearandombinaryvector,thattakesavaluesfromafeasiblesetSaccordingtotheBoltzmann'sdistributionwith0beingthetemperatureparameter: 25

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2{1 ).Itcanbeshowneasilythatundercertainconditions(i.e.symmetricneighbourhoodstructure)thestationaryprobabilitiesoftheMarkovchainassociatedwiththeSAmethodaregivenby( 2{2 ).ThesetScanbesplitintotwosubsetsinsuchawaythatoneofthemcontainsthefeasiblesolutionsforwhichthej-thcomponentis1,andanothersetwillcontainthesolutionwiththejthcomponentequalto0.LetusnamethesetwosetsS1jandS0j.Obviously,S1j[S0j=S.Thentheprobabilityofthejthcomponentofbeing1canbeexpressedas: 2{3 )(oritsapproximation)canbeusedforsuchageneration(substitutingSwithbSintheformula): 2{3 )controlsthelevelofsimilarityofthenewlygeneratedsolutionstohighqualitysolutionsinbS.TheuniquenessofthebestsolutionxinbSmentionedaboveshouldbemaintainedatallstagesofthealgorithm.Oneofthelimitationsofthestrategydescribedaboveisthatinordertoimplementit,thereshouldexistaneasywayofgeneratingrandomsolutionsfromSwiththedistributiongivenby( 2{4 ).Unfortunately,forsomeproblems,thestructureofthesetSwouldmakethishardtoachieve.Forsuchcases,thelocalsearchbasedtechniques(i.e.SAmethod,Tabumethod,GESmethod)arenoteasilyapplicable. 26

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2{4 )arestoredseparately.Therefore,thereisnoneedtostorethewholesetbSinthememory!ThenotionofbSisusedbelowmainlyforthesimplicityofdiscussion.TheperformanceofanyGESbasedalgorithmisdependantonthechoiceofthetemperatureschedule.AswiththeSAmethod,thereisnobasicrecipetoprovideanoptimalschedulefortheGES.Thegeneraladvicehereistochoosethesequenceofincreasingvalues0=0;1;2=1;:::;K=K1,(Kisanumberoftemperaturestagesand>0)insuchamannerthatthealgorithmwillndthebestsolutionfromthesetbSalmostforsurewhengeneratingsolutionswithtemperatureparameterK.However,thereisnoneedtoprovideaseparatecoolingscheduleforeachproblemsolved.Simplescalingofthecostfunction(f0(x)=Cf(x),C>0)canmakeonetemperatureschedulesuitableforawiderangeofproblemsfromthesameclass.Thechoiceofscalingfactorcanbemade,forexample,intheinitialstageofalgorithm,when=0.Additionally,ifwemultiplythedenominatorandnumeratorof( 2{4 )byexp(f(x)),wherexisthebestsolutionfrombS,thentheconvergencetothebestsolutionfrombSislessdependantontheabsolutevaluesofsolutioncosts.ThegeneralschemeoftheGESmethodispresentedinFigure 2-1 .Therearesomeelementsthatareincludedinthescheme,butwerenotdiscussedabove:elitesolutionsset,prohibitionofcertainsolutionsandrestartingthesearch.TheseelementsarenotnecessaryforsuccessoftheGESmethodandcanbeeasilyexcluded.However,forsomeclassesofproblemstheycanprovideasignicantperformanceimprovement. 27

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elite set(E,R) GESmethod(generalscheme) 28

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2{4 )inthebeginningofeachtemperaturestage(line12).Foreachprobabilityvector,ngensolutionsaregenerated(lines13-22).Thesesolutionsareusedasinitialsolutionsforthelocalsearchprocedure(line15).ThesubsetofencounteredsolutionsRisusedtoupdatethesetbS(line16).Somesetofthesolutionscanbestoredinmemory,inordertoprovideafastinitializationofthealgorithm'smemorystructures(lines27,34).Suchsetisreferredtoasanelitesetinthealgorithmpseudocode.Certainsolutionscanbeexcludedfromthissettoavoidsearchingthesameareasmultipletimes.Inlines29and30,thesolutionsforwhichtheHammingdistancetotoxbestislessthanparameterdpareexcludedfromtheeliteset. 2{4 forgenerationprobalitiesisverysensitivewheneverthesetofknowsolutionsissmall.BySuchsituationistypicalfortheinitialstage,sincethesetbSisinitiallyempty.Inordertoovercomethisinstability,thealternativeapproximationforformula 2{3 isusuallyusedincomputations. 29

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2{3 ,weobtainthefollowingexpression:@pj (Px2Sexp(f(x)))2Px2S1jf(x)exp(f(x)) 2{5 )givesusthefollowingformulaforpj(),whichcanbeveriedbysimplesubstitutioninto( 2{5 ): 1+1pj(0) 30

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2{7 ): ~pj(k)=1 1+1~pj(0) ~pj(0)expf1 2Pk1i=0(i+1i)(E0ij+E0i+1jE1ijE1i+1j)g(2{9)GEScaneithergeneratesolutionsaccordingtothedistributiondenedby( 2{9 ),orcanapplysomenumberofperturbationstothegivensolutionwhichasymptoticallyleadtothisdistribution.Weshouldalsopointoutherethatformula( 2{9 )isjustoneofthepossiblewaystoestimate( 2{7 ).Anothervariantofestimationprobabilities( 2{7 )forJSPisgiveninChapter 2 31

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maxx2f0;1gnf(x)=nXi=1nXj=1qijxixj;(3{1)whereqijareelementsofannnsymmetricrealmatrixQ2Rnn.Thisproblemisreferredtoasanunconstrainedbinaryquadraticprogrammingproblem(UBQP).Sincex2i=xiforall0{1variables,linearfunctionofxcanbemovedintothequadraticpartoftheobjectivefunctionin( 3{1 ).Manyfundamentalproblemsinscience,engineering,nance,medicineandotherdiverseareascanbeformulatedasquadraticbinaryprogrammingproblems.Quadraticfunctionswithbinaryvariablesnaturallyariseinmodelingselectionsandinteractions.Forexample,considerasetofnobjectsf1;:::;ng,eachofwhichisselectedornot.Foreachpair(i;j)ofobjectsweassociateaweightqijmeasuringtheinteractionbetweenpointsiandj.Letxi=1iftheobjectisselected,andxi=0otherwise.Iftheglobalinteractionisthesumofallinteractionsbetweentheselectedpoints,thentheirglobalinteractioncanberepresentedasaquadraticbinaryfunctionPni=1Pnj=1qijxixj.Aclassofsuchproblemshasbeenstudiedinsolid-statephysics[ 18 19 ].Furthermore,applicationsofconstrainedandunconstrainedversionsoftheproblem( 3{1 )canbefoundinnumerousareasincludingbutnotlimitedtomedicine[ 20 ],computer-aideddesign[ 19 21 22 ],scheduling[ 23 ],modelsofmessagemanagement[ 24 ]andchemistry[ 25 ].Manygraphtheoreticalproblemscanbenaturallyrepresentedintermsofquadraticbinaryoptimization,includingwell-studiedmaximumcliqueandmaximumindependentsetproblems[ 26 { 28 ]. 32

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29 ].Thereisonlyalimitednumberofclassesknowntobepolynomiallysolvable[ 30 { 33 ].ItisalsoknownthattheproblemofcheckingifanunconstrainedbinaryquadraticproblemhasauniquesolutionisNP-hard[ 34 ].Furthermore,quadraticbinaryprogrammingproblemremainsNP-hardevenifweknowthattheglobaloptimumisunique[ 34 ].OneoftheclassicalexamplesofanNP-hardproblem[ 35 ],reformulatedintermsofquadratic0{1optimization,istheaforementionedmaximumcliqueproblem[ 27 ].Approximationoflargecliquesisalsodicult,sinceasitisshownbyHastad[ 36 ]thatunlessNP=ZPPnopolynomialtimealgorithmcanapproximatethecliquenumberwithinafactorofn1forany>0.Khotimprovedthisboundton=2(logn)1[ 37 ].RegardingapproximabilityweshouldalsomentionanimportantresultbyNesterov[ 38 ],whichisageneralizationofideabyGoemansandWilliams,whodevelopedanapproximationalgorithmforthemaximumcutproblem[ 39 ].Nesterovprovedthatbooleanquadraticprogramming,maxfq(x)=xTQxjx2f1gngcanbeapproximatedbysemideniteprogrammingwithaccuracy4=7.ExtensionsofthisresultcanbefoundinNesterov[ 40 ]andYe[ 41 ].Unfortunately,duetotheintrinsiccomplexityoftheproblem( 3{1 ),exactalgorithms(see,forexample,[ 26 42 43 ])canhandleinstancesonlyofsizeofseveralhundredvariablesforreasonablysparsematricesQ.Forlargerinstanceswithdensematricesonlyheuristicapproachesareapplicable.Amongvariousproposedheuristicmethodsforsolvingtheunconstrainedbinaryquadraticproblemweshouldmentionalgorithmsbasedontabusearch[ 12 44 45 ],evolutionary[ 46 ].Someofthealgorithmiccodes,testproblemsgeneratorsandbenchmarkinstancesarepubliclyavailableandcanbefoundat[ 1 2 47 48 ]. 33

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17 ](inRussian).AbriefdescriptionofGES(inEnglish)isgiveninasurvey[ 49 ].RecentapplicationofGESforsolvingjobshopschedulingproblem(GESJSP2006)withexcellentresultsisreportedin[ 13 ].TheresultsofourcomputationalexperimentsindicatethatthedevelopedGES-basedheuristicalgorithmishighlyecient(itoutperformsthebestavailablesolver[ 12 ]intheliterature)andcanbeusedforsolvinghardinstances(withseveralthousandsofzero-onevariables)oftheunconstrainedquadraticbinaryprogrammingproblem.Anotherimportantandpromisingconclusionofthestudyisasfollows.Tabusearchisoneofthemostpowerfulheuristicmethodssuccessfullyappliedforsolvingavarietyofproblems[ 15 ].Furthermore,thisapproachmaybecomeevenmoreecientifitisusedwithinsomemulti-startframework(see,forexample,[ 12 ]).TheresultsofourcomputationalstudiesindicatethatthecombinationoftheGESmethodandtabusearchcanprovideanotherveryattractivealternativeforsolvinglarge-scalecombinatorialoptimizationproblems. 1 wasusedtoconstructthealgortihmfortheUBQP(seeFigure 2-1 ).Theonlysolutionsthatarememorizedbythealgorithmaresocalledelitesolutions(setE),whichareusedtoformtheinitialsetofknownsolutions(lines27,34,Figure 2-1 ).ThesetEcontainsEsizebestsolutionsfromtheset^S(line21,Figure 2-1 ).Inline29-30(Figure 2-1 )weremovefromthesetEthebestknownsolutionandallsolutionswithintheHammingdistanceofdpfromit.Theideabehindthisprohibitionistoforcethealgorithmtoexplorethesolutionswithdierentstructurethenthosethathavebeenalreadyfound.Thelines29-31(Figure 2-1 )areusedtorestartthesearchprocess. 34

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2-1 ,line15)usedinourstudyarebasedonthe1-ipneighborhood.Asolutionybelongto1-ipneighborhoodofbinaryvectorxiftheHammingdistanced(x;y)isequalto1.Inotherwords,ify2N1(x)thenyj=1xjforsomeindexjandyk=xkfork6=j.Inordertoaccelerate1-ipmoveevaluations,thespecialdatastructureismaintainedthroughoutthesearchprocess.Letgains(x)denoteavectorinwhichthej-thelementrepresentsagaininthecostfunctionresultingfromapplying1-ipoperatortothevariablexj.Thiscanbecalculatedinlineartime:gainsj(x)=qjj(12xj)+2nXi=1;i6=jqij(12xj)xi:Basedonthisneighborhood,thesearchmethodbasedontabusearch[ 15 ]wasimplemented(seeFigure 3-1 ). 1 .Thepseudo-codeofgenerationprocedureisshowninFigure 3-2 .Giventhevectorx,thegenerationprocedureappliesthenumberof1-ipmoveoperators(lines2-16).Theuniformlydistributedvariableintheinterval(0,1)isusedtodeterminewhethertoapplyperturbationtoagivenvariableornot(lines4,8).Theprocedureterminates,whenevertheHammingdistancebetweenxandaperturbedsolutionbecomesequaltotheparameterdistmax. 35

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used(j)=j2M used(j)>tabuORf(x)+gj(x)>f(xbest)then used(j)=step;xj=1xj used(j)>tabuORf(x)+gj(x)>f(xbest)then used(indbest)=step;xindbest=1xindbest LocalSearchTABU 36

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Generationprocedure fortemperaturestages1-5anddistmax=n=5fortemperaturestages6-K,thenumberofsolutionsgeneratedngen=27,thetabutenuretabu=21,prohibitionparameterdp=200,nfail=1.Initialprobabilitiesatthebeginningofeachtemperaturecycleareallinitiallysetto0.5.Forthetemperatureschedulethefollowingvalueswereused:0=0,1=107,k=k1log10log1 1 ]andPalubeckis[ 2 ])andTRUEforthemaximumindependentsetprobleminstances(Sloane[ 3 ]).ThebestalgorithmsfortheUBQPproblemarebasedontabusearchbasedtechniques[ 12 15 45 ].AmongthemwehaveselectedtheimplementationofPalubeckis[ 12 ]toprovideacomparativeexperimentalstudyoftheGESperformance.Therearemultiplereasonsforthischoice.Firstofall,toourbestknowledgetheMST2 37

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12 ]iscurrentlythebestheuristicforsolvingtheUBQPproblem.Itshowsexcellentandrobustperformanceonavarioussetofbenchmarkinstances.ThecomparativestudyofMST2withotherwell-knownapproachesincludingtabusearch,simulatedannealingandgeneticlocalsearchisgivenin[ 12 ].Furthermore,thesourcecodeforthealgorithmisavailableonline[ 2 ].Therefore,itallowedustoorganizethecomputationalexperiments,inwhichthesamepersonalcomputerwasusedtorunthealgorithms.Hence,thecomputationaltimesprovidedbelowcanbeeasilyusedasoneofthemeasuresforcomparisonofthealgorithms.Wedesignedourexperimentsinthefollowingmanner.First,theMST2algorithmofPalubeckiswasbeingrun10timesoneachproblemfromthebenchmarksset.Foreachinstance,werecordthebestsolutionfoundbyMST2in10runs(fbest),theaverageofthebestsolutionvaluefound(favr),theaveragetimeuntilthebestsolutionwasfound(tbest)andtheaveragetotalrunningtimettotal(themaximumnumberofiterationwasusedasterminationcriterionforMST2).WealsorecordhowmanytimesMST2wasabletoobtainthesolutionvaluegivenbyfbestforagivenproblem(#found).ThesimilarexperimentwasconductedfortheGESalgorithm.TheGESalgorithmterminatedwheneverthesolutionofthesameorbetterqualitythanfbestprovidedbyMST2wasfound(i.e.thebestsolutionfoundbyMST2wasusedastargetsolutionfortheGES).ThemeaningofthedatacolumnsforGESisthesameasforMST2,exceptforthecolumn'#found',whichcontainsthenumberoftimestheGESalgorithmwasabletondthesolutionofthesameorbetterqualitythenthetargetsolutionprovidedbyMST2(fbest).Suchdesignofthecomputationalexperimentsallowedustoobtainquitemeaningfuldataforthecomparisonofalgorithmicperformance.TherstsetofbenchmarkinstancesusedinourstudywasoriginallyintroducedbyBeasley[ 44 ].Theexperimentswerelimitedtothelargestinstancesofthisset,sincethesmalleronesareextremelyeasyandthereforedonotprovideagoodbasisforthe 38

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ResultsforBeasleyinstances[ 1 ] Algorithm MST2 1515944 10 3.1 465.6 1515944 10 2.6 MST2 1471392 10 19.4 460.4 1471392 10 47.40 MST2 1414192 10 17.5 469.6 1414192 10 8.71 MST2 1507701 10 2.1 467 1507701 10 1.5 MST2 1491816 10 2.2 462 1491816 10 2.77 MST2 1469162 10 4.8 473 1469162 10 5.46 MST2 1479040 10 8.7 469 1479040 10 24.5 MST2 1484199 10 7.8 460 1484199 10 2.6 MST2 1482413 10 14.9 462 1482413 10 7.5 MST2 1483355 10 26.4 458.4 1483355 10 24.7 12 ]torefertotheseinstances:b2500-1,b2500-2,...,b2500-10(thesizeoftheproblemsis2500variables).ThemaximalnumberofiterationsforMST2(stoppingcriterion)wassetto100.TheresultsoftheexperimentaltestingsarepresentedinTable1.BasedonthedatapresentedinTable1,onecanseethatbothtechniquesareabletoobtainthesolutionofthesamequality.Thecomputationaltimesdoesnotdiertoomuch.ForsomeoftheinstancesMST2outperformsGES,fortheothersGESoutperformsMST2.OnaverageMST2isperformingslightlybetter,howeverthedierenceisnotsignicantenough.Theseresultsassertstheneedformoredicultinstancesinordertocomeupwithmoremeaningfulconclusionsregardingthecomparativeeectivenessofthesealgorithms.Forthesecondsetofexperiments,theproblemsofhigherdimensionalityandhigherdensitywereused.TheseproblemswereintroducedbyPalubeckisandusedinhisstudies[ 12 ].Inourexperiments,welimitedtestingtothelargestproblemsoftheset(p5000-1,...,p5000-5,p6000-1,p6000-2,p6000-3,p7000-1,p7000-3).ThemaximalnumberofiterationsforMST2(stoppingcriterion)wassetto1000.ThecomputationalresultsarepresentedinTable2. 39

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ResultsforPalubeckis'instances[ 2 ] Algorithm MST2 8566497.1 2 2280.5 4567.7 8566602 10 2269.8 MST2 10833518 10 710.6 7431.1 10833518 10 442.26 MST2 10482962 10 1721.7 7610.3 10482962 10 1271.3 MST2 12256474 10 2387.9 9486.9 12256474 10 2081.6 MST2 12727352.3 9 2871.9 9729.1 12727442 10 1345.5 MST2 11385007 10 1958.1 6443.3 11385007 10 3141.7 MST2 14328050.5 5 6144.8 10507.4 14328191 10 1950.3 MST2 16138848.6 8 2540.7 13605.6 16138892 10 3549.9 MST2 14473796.4 2 4545.5 8567 14474315 10 2231.1 MST2 20570091.2 1 6351.9 17985.6 20571136 10 5641.1 28 ].ForthethirdsetofexperimentsweusedalgorithmsMST2andGEStosolvemaximumindependentsetproblemonvegraphsarisingincodingtheory(1zc1024,1zc2048,1zc4096,1zc8192and1zc16384)[ 3 ].ThemaximalnumberofiterationsforMST2(stoppingcriterion)wassetto1000.TheresultsofcomputationalexperimentsforthemaximumindependentsetproblemsaregiveninTable 3-3 40

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ResultsfortheMaximumIndependentSetinstancesarisingincodingtheory[ 3 ] Algorithm MST2 109 10 18.9 95.4 109 10 4.2 MST2 178 1 33.9 206.2 181.3 10 45.1 MST2 311.5 2 157.9 529.7 313.9 10 4.0 MST2 564.5 4 3.3 1519.9 580.7 10 10.7 MST2 1049.5 4 10.3 5646.4 1065.9 10 39.8 12 ]onasetofwell-knownpubliclyavailablebenchmarkinstances[ 1 { 3 ].Theideasintroducedbytheglobalequilibriumsearchmethodprovideaverypromisingapproachforconstructionofalgorithmsforsolvingdiscreteoptimizationproblems. 41

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50 ],inthestudyofsequenceswithPeriodicAutocorrelationFunctionzero.WebeginwiththedenitionofthePeriodicAutocorrelationFunction,PAF,from[ 51 ]: 51 ]. 1 liesinthefactthatforasequenceoflengthnoneneedstoconsideronlytherstn

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2;whenajak2PA(i);j;k2f1;:::;ng0;otherwise;i=1;:::;mThefollowinglemmacanbeprovedbystraightforwardcomputation. 2PA(m): 2PA(4)M1=26666666666666666666666666666666401 2000001 21 201 20000001 201 20000001 201 20000001 201 20000001 201 20000001 201 21 2000001 20377777777777777777777777777777775;M2=266666666666666666666666666666664001 20001 200001 20001 21 20001 200001 20001 200001 20001 200001 20001 21 20001 200001 20001 200377777777777777777777777777777775;M3=2666666666666666666666666666666640001 201 20000001 201 20000001 201 21 2000001 2001 2000001 21 201 20000001 201 20000001 201 2000377777777777777777777777777777775;M4=26666666666666666666666666666666400001 2000000001 2000000001 2000000001 21 2000000001 2000000001 2000000001 20000377777777777777777777777777777775

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GraphicalrepresentationsofthefoursymmetricmatricesM1;M2;M3;M4 2 wecanreformulatethisproblemasfollows:ProblemIIFindtwobinarysequencesa,b,(viewedasn1columnvectors)suchthataTMia+bTMib=0;i=1;:::;m:wherea=[a1;:::;an]andb=[b1;:::;bn]andai;bi2f1;+1g.Weareinterestedin1valuesofthe2nvariablesai;bisuchthatPA(i)+PB(i)=0;i=1;:::;morequivalentlyaTMia+bTMib=0;i=1;:::;m:

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2 toprovideaCombinatorialOptimizationformalismforthegeneralproblemofndingPCS(n;p). 52 ].TheygavenecessaryexistenceconditionsandproducedadiagramsummarizingtheirresultsfortheexistenceandnonexistenceofPCSforallvalues2p12and2n50.TheyalsodescribesynthesismethodsforPCS,usingtheconceptsofmates,interleaving,matriceswithorthogonalcolumns,perfectarraysandperiodicproducts.Vialemma 2 ,thesearchforPCS(n;p)canbeexpressedasaCombinatorialOptimizationproblem:Findasetofbinarysequencesa1,a2,:::,ap,(viewedasn1columnvectors)suchthataT1Mia1++aTpMiap=0;fori=1;:::;m:ThennmatricesM1;:::;MmthatappearintheCombinatorialOptimizationformalismofthePCS(n;p)problem,satisfyasimpleadditiveproperty,giveninthefollowinglemma. 2MwhereMisannnmatrixwithelementsmij=1ijwhereijdenotestheusualKronecker'sdelta.Proof

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3 .ThetermsPA(1);:::;PA(m)aremadeupfromn(n1) 2dierentmonomialsoftheformajak(withj6=k).Thenmonomialsoftheformajak(withj=k)i.e.thesquares,donotappearinthetermsPA(1);:::;PA(m). 2266666666666666666666666666401111111101111111101111111101111111101111111101111111101111111103777777777777777777777777775 52 ],PCSareusedtoconstructsequenceswithdesirablepropertiesforradarapplications,asdescribedin[ 53 ].Againasnotedin[ 52 ],PCSinterveneincodedapertureimaging[ 54 ]andhigher-dimensionalsignalprocessingapplicationssuchastime-frequency-coding[ 55 ]orspatialcorrelation[ 56 ].Thebook[ 57 ]containsachapteronapplicationsofcombinatorialdesignsingeneral,inauthenticationcodes,thresholdschemesandgrouptestingalgorithms. 46

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58 ]isarichsourceofinformationonapplicationsofsequenceswithlowautocorrelationfunctionpropertiesingeneral,insignaldesignforcommunications,radar,andcryptographyapplications.SomecategoriesofPCS,e.g.PCS(n;2),areusedasrstrowsofcirculantmatricestoconstructHadamardmatricesfromtwocirculantsubmatrices,see[ 59 ]andreferencestherein.TheseHadamardmatricesarethenusedforCodingTheorypurposes,i.e.toconstructbinarylinearcodeswithdesirableproperties. 60 ]withslightmodications.Firstly,thesubstitutionofaiwith2xi1,wherexi2f0;1g,transformedtheproblemto0{1domain: 2qjk=0;ifmjk=0Letdi(x)=xTQix+ndenoteanerrorcorrespondingtotheithequation(positive,negativeorzero).Thesumoftheabsoluteerrorsforeachequationisusedasthecostfunctiontominimize:f(x)=mXi=1abs(xTQix+n)=mXi=1abs(di(x))Searchmethodsusedin[ 60 ]arebasedonthe1-ipneighborhood.Asolutionbbelongsto1-ipneighborhoodofbinaryvectoraifbj=1ajforsomeindexjandbk=akfork6=j.Inordertoaccelerate1-ipmoveevaluations,thespecialdatastructureismaintainedthroughoutthesearchprocess.Letgains(x)denoteavectorinwhichthe 47

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15 ]wasimplementedbasedon1-ipneighbourhood(see[ 60 ]forimplementationdetails).Ateachiteration,thetabualgorithmexplores1-ipneighbourhoodofthecurrentsolution.Thebestsolutionfromtheneighbourhoodissettobeacurrentsolutionandtheprocedureisrepeated.Aprohibitionrulesareusedtoescapefromlocallyoptimalsolutions.Ifthevariablexjwasinvertedduringthemove,thenitisisprohibitedtoinvertxjforaxednumberofiterations,referredtoasatabutenure.Arandombinarysequencewasusedtoinitializethetabusearch.Thealgorithmisrestartedafterevery10000iterationswithanewrandominitialsolution.Thealgorithmisstoppedwheneverthesolutionxisobtained,suchthatf(x)=0.Thetabutenureparameterofthealgorithmwassetto21iterations. 61 ].Weusethestandardnotation,+standsfor+1andstandsfor1.

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52 ])thattheexistenceofPCS(n;p)impliestheexistenceofPCS(n;pk)forallpositiveintegersk1. Theprevioustheoremhasanimportantcorollary,that(basedonourresults)settlessixmoreopencasesinthetableofopencaseslistedonpage319ofthesecondeditionoftheHandbookofCombinatorialDesigns[ 61 ]. 49

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61 ].Therefore,theprogramstartedbyBomerandM.Antweilerin1990(see[ 52 ])cannowbedeclaredcompleted.WepresentthenewPCS(n;p)thatwefoundusingournewtechnique.Inconjunctionwithcorollary1,thissettlesallopencasesintheupdatedBomerandAntweilerdiagramstatedonpage319of[ 61 ].

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=m^i=10@jCij_j=1`ij1A;(5{1)wherejCijisthenumberofliteralsinaclauseCi,and`ijisaliteral,i.e.,abooleanvariablexk,oritsnegationxk,1kn.Aclauseissatisedifatleastoneofitsliteralsistrue.IntheMAXIMUMSATISFIABILITY(MAX-SAT)problemweneedtondanassignmentofvaluestothevariablesthatsatisesasmanyclausesaspossible.AnaturalgeneralizationoftheaboveproblemistodeneapositiveweightwiforeachclauseCiandsearchforanassignment,whichmaximizesthetotalweightofthesatisedclauses.TheMAX-SATaswellasitsweightedversionremainNP-hardevenifeachclausehasatmosttwoliterals(MAX-2SATproblem)[ 35 ].AmongvariousheuristicapproachesforsolvingtheMAX-SATproblemweshouldmentionalgorithmsbasedonreactivetabusearch[ 62 ],simulatedannealing[ 63 ],GRASP[ 64 65 ],iteratedlocalsearch[ 66 ]andguidedlocalsearch[ 67 ].DetailedsurveysonMAX-SAT,relatedapplicationsandsolutionapproachescanbefoundin[ 68 ].Someofthealgorithmiccodesandbenchmarkinstancesarepubliclyavailableandcanbefoundat[ 69 { 72 ].InthisnoteweconsideraheuristicapproachforsolvingweightedMAX-SATbasedontheglobalequilibriumsearch(GES)frameworksuggestedin[ 17 49 ].Weinvestigateperformanceofthreevariantsoftheproposedmethodandcomparethemwithotherexistingalgorithmsonpubliclyavailablebenchmarksinstances.Thecomputationalresults 58

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60 ]. 1 .TheframeworkofthealgortihmissimilartothepseudocodepresentedinFigure 2-1 .However,ourimplementationfortheweightedMAX-SATproblemisslightlydierentfromthegeneralframework.Thesedierencesareduetoproblemspecicsandcanbesummarizedinafollowinglist: 1. Thesetofelitesolutionisnotused,E=;throughoutthealgorithmstages. 2. ThesetofRofsolutionsthatareusedtoupdatethememoryofthealgorithm(Figure 2-1 ,line15)consistsofthebestsolutionfoundbythesearchprocedure. 3. ThesetPofprohibitedsolutionisnotused,P=;throughoutthealgorithmstages. 4. RESTART-criterion=TRUE(Figure 2-1 ,line31).Inotherwords,thealgorithmdeletesalltheinformationaftertheunsuccesfultemperaturecycle(completerestart). 5-1 )scansthroughthecomponentsofxinarandomsequenceandthevaluesofthesolutioncomponentsarechangedbasedonthe~pkj(i.e.thej-thcomponentofsolutionxissetto1withprobability~pkj,k-isthenumberofthetemperaturestageatwhichgenerationprocedureisapplied).TheperturbationscyclestopswhenevertheHammingdistancebetweenxandperturbedsolutionbecomesequaltotheparameterdistmax. 2-1 ,line15)wereimplementedandtested.Allofthemarebasedona1-ipmoveoperator,whichchangesthevalueofasingle 59

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Generationprocedure variablexitoitsnegation:xi,i.e.,fromxito1xi(intermsof0{1variables),forsomei2 60

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LocalSearch1-opt non-improvingmoveisappliedtosomevariable,itisremovedfromthesetM.IfthesetMbecomesempty,thesearchisrestartedfromthebestsolutionfoundduringthelocalsearch.Thelocalsearchreturnsthebestsolutionfoundiftherewasnoimprovementinthelastnbaditerations.Thethirdsearchmethodinvestigatedinthisnoteisbasedonthetabumethod[ 15 ].methodusedinouralgorithmispresentedinFig. 5-4 5.3.1Benchmarks 70 ](\easy"instances).Thenumberofvariablesisn=100.Thenumberofclausesrangesfromm=800tom=900.Theseinstanceswereusedforcomputationalexperimentsin[ 65 { 67 73 ]. 48 ](\harder"instances).Thenumberofvariablesandclausesisn=1000andm=11050,respectively.Thissetwasutilizedin[ 66 73 ] 70 ]). 61

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LocalSearchk-opt 62

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used(j)=,j2M used(j)>tabuORf(x)+gj(x)>f(xbest)then used(j)=step;xj=1xj used(j)>tabuORf(x)+gj(x)>f(xbest)then used(indbest)=step;xindbest=1xindbest LocalSearchTABU 63

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Resultsforjnhbenchmarks sa ils tabu ges 1-opt ges k-opt ges tabu gls1 gls2 jnh04 35.09 6.57 67.83 10.10 2.24 0.70 1.72 0.84 jnh05 1.24 0.29 41.63 0.29 0.57 0.10 1.11 0.10 jnh06 0.07 0.12 0.44 0.13 0.09 0.06 0.09 0.13 jnh08 2.06 3.64 1.28 0.33 0.53 0.13 0.02 0.04 jnh09 5.34 37.73 4.68 2.75 1.14 0.12 2.00 0.77 jnh10 8.95 4.17 30.57 0.58 0.49 0.12 0.04 0.04 jnh11 21.63 18.55 27.25 4.28 3.19 0.60 0.48 0.57 jnh13 0.51 0.26 0.76 0.17 0.29 0.03 0.03 0.02 jnh14 0.54 1.39 4.12 0.64 0.90 0.20 0.03 0.02 jnh15 1.87 0.65 27.50 0.81 0.41 0.08 0.07 0.11 jnh16 47.74 0.12 25.80 5.94 2.93 0.36 3.21 0.57 jnh18 5.65 7.74 13.22 0.89 0.51 0.08 1.14 1.01 jnh19 3.21 73.31 28.34 27.70 20.43 2.69 0.07 0.18 jnh202 32.22 34.33 63.62 4.34 1.86 0.36 0.04 0.05 jnh203 16.02 18.15 2.20 0.48 0.65 0.78 0.04 0.07 jnh208 1.35 2.98 0.99 0.38 0.45 0.23 0.13 0.07 jnh211 0.32 0.50 0.74 0.76 0.46 0.04 0.02 0.05 jnh214 2.70 1.44 6.42 0.24 0.34 0.18 0.08 0.04 jnh215 11.27 0.69 5.04 1.30 0.48 0.17 0.05 0.05 jnh216 0.50 1.45 10.70 0.44 0.38 0.10 0.30 0.27 jnh219 1.99 34.26 13.98 8.15 2.58 0.14 0.12 0.14 jnh302 0.07 0.23 1.23 0.23 0.18 0.11 0.13 0.19 jnh303 2.39 0.70 7.81 0.81 1.08 0.13 0.35 0.21 jnh304 0.71 1.75 7.87 1.52 0.19 0.03 0.08 0.10 jnh305 1.90 3.88 9.08 3.67 1.91 0.25 5.43 1.18 jnh306 0.27 0.31 1.32 0.29 0.21 0.04 0.03 0.04 jnh307 0.32 0.79 3.10 0.26 0.47 0.10 0.22 0.28 jnh308 48.85 66.23 36.63 8.49 2.58 0.19 0.24 0.22 jnh309 0.17 2.03 0.21 0.20 0.16 0.05 0.14 0.10 jnh310 0.42 1.88 209.53 1.26 2.04 0.07 0.29 0.38 Totaltime(sec): 255.38 326.16 653.89 87.44 49.73 8.21 17.68 7.81 Resultsforrndw1000bbenchmarks ges tabu gls1 gls2 rndw1000 meant meanf bestf meant meanf bestf meant meanf bestf 420 5552786 5553057 2381 5551629 5552657 2533 5551695 5552065 b02 319 5519999 5520245 2412 5519335 5519829 2526 5519337 5519605 b03 484 5579425 5579673 2300 5578475 5579159 2518 5578744 5578208 b04 1258 5504266.9 5504348 2322 5502842 5503341 2531 5503249 5504246 b05 624 5527403 5527445 2392 5525988 5527357 2528 5526290 5527059 b06 333 5523689 5523832 2329 5522259 5523423 2538 5523143 5523559 b07 247 5513115 5513393 2364 5511917 5512232 2525 5512292 5512983 b08 829 5491782 5491999 2313 5490155 5490566 2523 5490915 5491696 b09 451 5543211 5543353 2498 5542028 5543114 2557 5542167 5542606 b10 1270 5554784 5554855 2276 5553519 5554709 2543 5553444 5553745 Resultsforrndw1000cbenchmarks ges tabu gls1 gls2 rndw1000 meant meanf bestf meant meanf bestf meant meanf bestf 393 5520297 5520841 1962 5519435.5 5520066 2041 5519233 5519983 c02 834 5497278 5497582 1946 5495597 5496645 2018 5496246 5497066 c03 4117 5541418 5541452 1954 5539629.3 5540355 2028 5540140 5541399 c04 491 5522087 5522372 1958 5521264.2 5521925 2032 5521756 5521947 c05 593 5501340 5501664 1961 5500468.1 5501213 2040 5500578 5501249 c06 543 5502583 5502903 1959 5501467.5 5502483 2040 5501556 5502016 c07 409 5505078 5505222 1955 5503706.8 5504510 2038 5504234 5504984 c08 676 5533540 5533641 1958 5532340.6 5533259 2038 5532846 5533450 c09 482 5544435 5544702 1958 5542906.7 5544191 2038 5543638 5544265 c10 422 5524889 5525283 1963 5523939.1 5524726 2039 5524047 5524505

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48 72 ].ThewebsiteofYagiura[ 48 ]providesthecodesforasetofalgorithmsfortheweightedMAX-SATproblem.Amongthose,thetabusearchalgorithm(tabu),simulatedannealingalgorithm(sa)anditeratedlocalsearch(ils)dominatetherestofthealgorithmsgivenat[ 48 ]intermsoftheircomputationaleciency.Anothertechniqueusedinourstudiesisguidedlocalsearch(GLS).In[ 67 72 74 ]theauthorsprovideanimplementationofguidedlocalsearchfortheweightedMAX-SATproblem.TheauthorsprovidetwoversionsofGLSreferredtoasgls1andgls2(herethesamenotationisusedasin[ 72 74 ]).Thedemoversionsoftwareoftheirimplementationisavailableat[ 72 ]. 65

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tabualgorithm,i.e.,ges tabuterminatedassoonasthesolutionofthesameorbetterqualitythanthetargetsolutionwasobtained.Wepresentthedatafortheaveragesolutionquality(meanf),averagesolvingtime(meant),thebestsolutionfound(bestf).Thesevaluesarecalculatedbasedon10independentrunsofeachalgorithm.TheresultsarepresentedinTables2and3.TheperformancedatainTables2and3indicatesthat,overall,on\harder"instancesges tabuperformsrobustlyanddominatesbothgls1andgls2.Foreveryinstanceandeveryrunofges tabu,thealgorithmwasabletondthesolutionofthesameorbetterqualityasthebestsolutionsfoundbygls1andgls2in10runs.Besidesndingbettersolutionsintermsofquality,ges tabuwasspendinglessCPUrunningtimetoobtainthem(exceptrndw1000c03). 66

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75 ],butevenwithinthisclassofproblemsitisconsideredasoneofthemostchallenging.Exactmethodsweresuccessfullyappliedtoproblemsofsmalldimensions,butforjobshopproblemswithmorethan15machinesand15jobstheyarenotabletondahigh-qualitysolutionwithreasonablecomputationaleort.Forlargedimensionsthereisaneedforgoodapproximatealgorithms.Thelocalsearchbasedmethodsweresuccessfullyappliedtosuchproblems:tabumethod[ 76 ],[ 77 ],greedyrandomizedadaptivesearch[ 78 ],andsimulatedannealing[ 79 ].ThedetailedreviewofthemethodsfortheJSPcanbefoundin[ 80 ]and[ 81 ].Anapproximatealgorithmforthejob-shopschedulingproblembasedontheGlobalEquilibriumSearch(GES)method[ 17 ]isproposedinthischapter.Themainideaofthisapproachistousethehistoryofthesearchforguidingitintotheareaswithpotentiallygoodsolutions.Atrst,theminimummakespanproblemofjob-shopschedulingisformalizedintermsofamathematicalmodel,andthentheproposedheuristicalgorithmbasedontheGESmethodforthisproblemisdescribed.Finally,theresultsofthecomputationalexperimentsonthewell-knownbenchmarkinstancesareshown,andourapproachiscomparedwithsomeofthebest-performingalgorithmsforjob-shopscheduling. 67

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max2O[s()+p()](6{1)isminimizedsubjectto Thevaluegivenin( 6{1 )isoftenreferencedtoasthemakespanoftheschedulerepresentedbystartingtimesofoperationsfromO.ThisproblemcanbedescribedusingthedisjunctivegraphmodelofRoyandSussmann[ 82 ].LetG=(O;A;E)denotethedisjunctivegraph,whereOisasetofnodes(anodeforeachoperation),Aisasetofdirectedarcs((0;)2Aand(;N+1)2Aforall2O),andE=mSk=1Ekisasetofundirectedarcs.Theweightofeachnodeisgivenbytheprocessingtimeofthecorrespondingoperation.LetS(selection)denoteasetofdirectedarcswhichisobtainedfromEbychoosingthedirectionforeachofitsarcs.Then 68

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17 ]hassomecommonfeatureswiththesimulatedannealingmethod.Ithasbeenappliedtomanydiscreteoptimizationproblems,andverypromisingresultswereobtained[ 17 ].ThemainideaoftheGESmethodistocollectinformationaboutthesolutionspaceforitsfurtheruseinthenextstagesofthesearch.Thesearchisorganizedasaseriesoftemperaturecycles.Duringeachstageofthetemperaturecyclesomesetofsolutionsisrandomlygenerated.Ontheearlystagesofeachtemperaturecyclethealgorithmgeneratesabsolutelyrandomsolutions,butonthefurtherstagesthesolutionsaregeneratedinsuchawaythattheyaremoreandmorelikelytohavesomecommonfeatureswiththebestsolutionsfoundinthepreviousstages 69

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for JSP initializealgorithmparameters(iter;maxiter;ngen;maxnfail) bestxx FbestxfF{setofalreadyknownsolutionsg generation probabilities(prk;F) generation0 generation
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generation probabilities)aredescribedinthenextsection.Ateverystagewegeneratengenrandomfeasiblesolutions,andthesetFisupdatedwithnewsolutions.Afterthatwechangethecurrenttemperature,andthenewstagebeginswithrecalculationofgenerationprobabilities.Inourimplementationafterthetemperaturecycleisnished,we"forget"allsolutionsfoundduringthecycleexceptthebestsolutionbysettingF=fbestxg.Furthermore,ifthealgorithmrunsformaxnfailcycleswithoutanimprovementofthebestsolutionthenwesetF=;;inotherwords,thesearchprocessrestarts.Thereareseveralbasicelementswhichhavetobedenedmoreproperly.Themainpartofthisframeworkistheprocedurethatgeneratesrandomsolutions.Italmostalwaysappearsthatrandomlygeneratedsolutionscanbeimprovedbyapplyingsomelocalsearch 71

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72

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solution(pr,x)fpr{probabilities,x{initialsolutiong tabuSettabuSet[(!;) endif endwhile returntransformedsolutionx Procedurethatgeneratessolution 2.Addingarcs(!;),(MP();!),(;MS(!).AveryusefulpropertyofsuchtransformationisthatitcanneverleadtoacyclicgraphDS0[ 79 ].EmpiricalanalysisrevealsthatitismuchmoreecienttoapplyaseriesofrandommoveswithintheN5-neighborhoodtoescapefromthelocalminimathantoconstructatotallynewsolution.In[ 83 ]itisshownthattheaveragenumberofmovesthatarerequiredtoescapefromalocalminimumisusuallyverysmallformostjobshopschedulingproblems.TheprocedurethatgeneratessolutionsisoutlinedinFigure 6-2 .Wehavealreadyintroducedtheprobabilitiesforgenerationoftheinitialsolutions.Nowweshalldescribehowweusethesolutionhistorytoupdatetheseprobabilities.ThetemperaturecyclefortheGESalgorithmisdeterminedbyasetofK+1temperatures0;:::;K,suchthat0<1<:::
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84 ](weusetheneighborhoodnotationintroducedin[ 85 ]).TheN4-neighborhoodconsistsofsolutionsobtainedbymovingacriticaloperationeithertothebeginningortotheendofitscriticalblock.InmanyapplicationsthepriorityhasbeengiventothehighlyrestrictedN5moveoperator.ThesolutionsfromtheN5-neighborhoodareobtainedbychangingtheprocessingorderoftwoconsecutivecriticaloperationssuchthatatleastoneofthemiseithertherstorthelastoperationofitscriticalblock.Themaindrawbackofsucharestrictionisthattheresultingsearchspacebecomesdisconnected,i.e.theexistenceofthepath,withrespecttotheN5,fromanyarbitrarysolutiontotheoptimumisnotguaranteed.Onthecontrary,theN4moveoperatorinducestheconnectedsearchspace.Additionally,astheN5moveoperatorisjustarestrictionoftheN4moveoperator,itisclearthatthelatterprovidesamorethoroughsearch.NowickiandSmutnickiproposedaneectivemethodofthemakespancalculationfortheN5moveoperator[ 4 ],whichsignicantlydecreasestheoveralltimespentonmoveevaluations.Basedontheirapproach,weusethesimilaracceleratorfortheN4move 74

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13 ]forthedetails.Asaresultofsuchacceleration,themoveevaluationsareperformedupto3timesfaster.Inordertoprovideanevenmorethoroughsearch,weintroduceaprocedure,whichhassomecommonfeatureswiththeiteratedlocalsearchtechnique[ 86 ],[ 83 ].Wheneveranoperationismovedeithertothebeginningortheendoftheblock,thereasonthatsuchmovedoesnotimprovethesolutionisbecauseofaconstraintcomingfromthejobpredecessorofsomeoperationwhichbelongedtothesamecriticalblockasthemovedoperation.Therefore,afteridentifyingsuchanoperation,wepushitsjobpredecessorsbackintheprocessingordertoremovethisspecicconstraint.Afterthetransformationisnished,thelocalsearchstartsfromtheobtainedsolution(seeFigure 6-3 ).Thisprocedurewasdiscussedin[ 13 ]andisusedinourstudiesasanenhancementofthelocalsearch.Theinvestigationofitsimpactonthealgorithmperformancewasprovidedin[ 13 ].Whenevertwosolutionshavethesamemakespanvalue,weconsiderthetotalprocessingtimeoftheircriticaloperations.Thesolutionwithsmallertotalprocessingtimeisconsideredanimprovingsolution.Thisruleallowsustotakeintoaccountcaseswhereweencountermultiplecriticalpaths(Figure 6-3 :line27). 75

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LetM=N4(x)(Grabowski'sneigborhood) M=My ndanoperation2Bsuchthat ifhead(JS(!))6=end(!)then break endif else break endif endwhile endif RETURNyfImprovement!g endwhile RETURNxfNoimprovementg Improvementprocedure 76

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6-4 theprocedureforaquickmoveestimationisoutlined,whichusesthisproperty.ThisprocedurereturnsthelowerboundofthelongestpathinDS0.Heres(S;)denotesthelengthofthelongestpathfrom0tointhegraphDS=(O;A[S);q(S;)denotesthelengthofthelongestpathfromtoN+1inthegraphDS=(O;A[S),f1S(i)denotestheoperationsuchthatfS(f1S(i))=i.Thisprocedurecanbefurtherimprovedsothatitcanreturnanexactvalueofs(S0;N+1),asitisdonein[ 87 ].Onlyslightmodicationsforthisprocedurearenecessarywhenisthelastoperationofitscriticalblock. 1 ].Weusedseveralwell-knownproblemclassesfromtheliterature: 77

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move(S,FS,(;!))f{rstoperationincriticalblockg IfS0(resultingselection)isacyclicselectionreturn1 if(est>s(S;N+1)then returnest Estimationprocedure 1)10problemsdenotedas(ORB1-ORB10)duetoApplegateandCook[ 88 ];2)40problemsdenotedas(LA01-LA40)duetoLaurence[ 89 ];3)80problemsdenotedby(TA1-TA80)duetoTaillard[ 77 ].Optimalsolutionsareknownfor47problemsfromthisclass.4)80problemsdenotedby(DMU1-DMU80)duetoDemircoletal.[ 90 ].Thecomputationalresultsaregiveninthetablesbelow.Fortheproblemswithanunknownoptimalsolution,weprovideitslowerbound(LB)andupperbound(UB).WecompareourresultswiththoseobtainedbyBalasandVazacopoulos(1995a)(BV), 78

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6-2 ,whereallcomponentsofvectorpraresetequalto1 2.InourimplementationofTSABalgorithmweusetheaccelerationproposedin[ 87 ].EachrunofTSABusesthefollowingparameters:maxt=8,maxl=5,maxc=2,max=100,andmaxiter=250000,see[ 76 ]fordetails.Wehavealsoinvestigatedthebehaviorofouralgorithmwhenthegenerationprobabilitiesareallxedandareequalto1 2foralliterations.Hence,inthiscasethealgorithmdoesn'tuseanyofGESfeatures.WerefertothisalgorithmasRIMP;itstestingresultsprovideagoodillustrationoftheperformanceofthesimplecombinationofthelocalsearchandtheimprovementheuristicdescribedaboveandallowtorevealtheenhancementprovidedbyGESstrategy.IncomparativetestsweuseanotationRI,whichisapercentagebywhichthebestfoundsolution(withmakespanCMax)isabovethebest-knownupperbound:RI=100%CMaxUB UBAllresultsforotherapproachesweretakenfromtheliteratureandsomeresultsweretakenfromTaillard'shomepage(http://www.eivd.ch/ina/collaborateurs/etd/default.htm).TheresultsforproblemsLa01-La40andOrb01-Orb10areshowninTables 6-1 6-2 .Alloftheseinstancesweresolvedoptimallybyourapproach.ThecomparisonwithalgorithmTSSB[ 91 ]isgiveninTable 6-5 79

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6-3 showstheresultsfortheseproblems.Despitethefactthattherewasalotoftimeconsumingexperimentsonsolvingtheseproblemswithmanydierentalgorithms,ourapproachwasabletondseveralnewupperbounds(TA11,TA15,TA19,TA20,TA32,TA46).Ingeneraltheproposedalgorithmisperformingquitewellonallsetsofproblemscomparativelywithshiftingbottleneckprocedure[ 92 ]andtabusearchalgorithm[ 91 ].ThecomputationalresultsforRTSABrevealthatGESoutperformsitontheprevailingmajorityofinstances.ComparisonwithRIMPshowsthatGESisperformingsignicantlybetter.Foreasyproblems,theuseofGESmethodologyallowsustodecreasesolvingtime.Ontheotherhand,thisstrategyallowsustoobtainabettersolutionforhardinstances.TheaverageresultsforallalgorithmsaregiveninTable 6-6 .TheDMUclassofproblemsconsistsof80instances.InstancesDMU41-80areconsideredasparticularyhard[ 90 ],butcomputationalexperimentsshowedthatourapproachallowsustoobtainhigh-qualitysolutionsevenforthesefragileproblemsandformanyofthemthebestupperboundswereimproved.Wetestedeachprobleminsuchawaythatthealgorithmstoppedifeitherthebestknownupperboundwasimprovedoroverallrunningtimeexceeded30000seconds.ThecomputationalexperimentsfortheseproblemsaregiveninTable 6-7 80

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81

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91 ]Time(sec)GEScomputingtimeinsecondsTavaveragecomputingtimeRTSABbestsolutionbyRTSABalgorithmrunfor5000secondsRIMPRbestsolutionbyRIMPRalgorithmrunfor5000secondsBVbestsolutionamongthoseprovided[ 93 ]ARIaverageRIforthebestsolution ComputationalresultsbyGESforORB1-ORB10 jMj UB GES1 Time(sec) orb01 10 10 1059 1059 1059 11 orb02 10 10 888 888 888 1 orb03 10 10 1005 1005 1005 6 orb04 10 10 1005 1005 1005 50 orb05 10 10 887 887 887 4 orb06 10 10 1010 1010 1010 2 orb07 10 10 397 397 397 2 orb08 10 10 899 899 899 3 orb09 10 10 934 934 934 14 orb10 10 10 944 944 944 1

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ComputationalresultsforLA01-LA40 GES1 GES2 TSSB la01 10x5 666 666 666 666 la02 10x5 655 655 655 655 la03 10x5 597 597 597 597 la04 10x5 590 590 590 590 la05 10x5 593 593 593 593 la06 15x5 926 926 926 926 la07 15x5 890 890 890 890 la08 15x5 863 863 863 863 la09 15x5 951 951 951 951 la10 15x5 958 958 958 958 la11 20x5 1222 1222 1222 1222 la12 20x5 1039 1039 1039 1039 la13 20x5 1150 1150 1150 1150 la14 20x5 1292 1292 1292 1292 la15 20x5 1207 1207 1207 1207 la16 10x10 945 945 945 945 la17 10x10 784 784 784 784 la18 10x10 848 848 848 848 la19 10x10 842 842 842 842 la20 10x10 902 902 902 902 la21 15x10 1046 1046 1046 1046 la22 15x10 927 927 927 927 la23 15x10 1032 1032 1032 1032 la24 15x10 935 935 935 938 la25 15x10 977 977 977 979 la26 20x10 1218 1218 1218 1218 la27 20x10 1235 1235 1235 1235 la28 20x10 1216 1216 1216 1216 la29 20x10 1152 1153 1152 1168 la30 20x10 1355 1355 1355 1355 la31 30x10 1784 1784 1784 1784 la32 30x10 1850 1850 1850 1850 la33 30x10 1719 1719 1719 1719 la34 30x10 1721 1721 1721 1721 la35 30x10 1888 1888 1888 1888 la36 15x15 1268 1268 1268 1268 la37 15x15 1397 1397 1397 1411 la38 15x15 1196 1196 1196 1201 la39 15x15 1233 1233 1233 1240 la40 15x15 1222 1226 1222 1233

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ResultsforTA1-TA80 UB GES2 GES1 RTSAB RIMP TSSB BV ta01 15x15 1231 1231 1231 1231 1231 1231 1241 1231 ta02 15x15 1244 1244 1244 1244 1244 1244 1244 1244 ta03 15x15 1218 1218 1218 1218 1220 1218 1222 1218 ta04 15x15 1175 1175 1175 1175 1175 1175 1175 1181 ta05 15x15 1224 1224 1224 1224 1229 1224 1229 1233 ta06 15x15 1238 1238 1238 1238 1238 1238 1245 1243 ta07 15x15 1227 1227 1228 1228 1228 1228 1228 1228 ta08 15x15 1217 1217 1217 1217 1217 1217 1220 1217 ta09 15x15 1274 1274 1274 1274 1280 1274 1291 1274 ta10 15x15 1241 1241 1241 1241 1241 1241 1250 1241 ta11 20x15 1323 1357 1357 1357 1367 1365 1371 1392 (1358) ta12 20x15 1351 1367 1367 1375 1377 1377 1379 1367 ta13 20x15 1282 1342 1344 1344 1350 1350 1362 1350 ta14 20x15 1345 1345 1345 1345 1345 1345 1345 1345 ta15 20x15 1304 1339 1339 1339 1347 1345 1360 1353 (1340) ta16 20x15 1302 1360 1360 1360 1361 1365 1370 1369 ta17 20x15 1462 1462 1469 1473 1476 1473 1481 1478 ta18 20x15 1369 1396 1401 1401 1408 1411 1426 1396 ta19 20x15 1297 1332 1332 1332 1338 1342 1351 1341 (1335) ta20 20x15 1318 1348 1348 1352 1355 1357 1366 1359 (1351) ta21 20x20 1539 1644 1647 1647 1648 1652 1659 1659 ta22 20x20 1511 1600 1602 1602 1607 1606 1623 1603 ta23 20x20 1472 1557 1558 1558 1560 1563 1573 1558 ta24 20x20 1602 1647 1653 1653 1654 1656 1659 1659 ta25 20x20 1504 1595 1596 1596 1597 1598 1606 1615 ta26 20x20 1539 1645 1647 1647 1654 1655 1666 1659 ta27 20x20 1616 1680 1685 1685 1691 1687 1697 1689 ta28 20x20 1591 1614 1614 1616 1619 1619 1622 1615 ta29 20x20 1514 1625 1625 1625 1627 1625 1635 1629 ta30 20x20 1472 1584 1584 1584 1585 1585 1614 1604 ta31 30x15 1764 1764 1764 1766 1764 1766 1771 1766 ta32 30x15 1774 1793 1793 1801 1818 1816 1840 1803 (1796) ta33 30x15 1778 1793 1799 1802 1805 1817 1833 1796 ta34 30x15 1828 1829 1832 1832 1832 1835 1846 1832 ta35 30x15 2007 2007 2007 2007 2007 2007 2007 2007 ta36 30x15 1819 1819 1819 1819 1821 1824 1825 1823 ta37 30x15 1771 1778 1779 1779 1789 1793 1813 1784 ta38 30x15 1673 1673 1673 1673 1678 1682 1697 1681

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ResultsforTA1-TA80(continued) UB GES2 GES1 RTSAB RIMP TSSB BV ta39 30x15 1795 1795 1795 1795 1806 1798 1815 1798 ta40 30x15 1631 1674 1680 1683 1685 1695 1725 1686 ta41 30x20 1859 2014 2022 2029 2035 2047 2045 2026 ta42 30x20 1867 1956 1956 1957 1968 1969 1979 1967 ta43 30x20 1809 1859 1870 1875 1875 1887 1898 1881 ta44 30x20 1927 1984 1991 1992 2000 2006 2036 2004 ta45 30x20 1997 2000 2004 2004 2013 2016 2021 2008 ta46 30x20 1940 2011 2011 2011 2024 2036 2047 2040 (2016) ta47 30x20 1789 1903 1903 1911 1922 1931 1938 1921 ta48 30x20 1912 1952 1962 1975 1966 1978 1996 1982 ta49 30x20 1915 1968 1969 1974 1980 1988 2013 1994 ta50 30x20 1807 1926 1931 1940 1942 1945 1975 1967 ta51 50x15 2760 2760 2760 2760 2760 2760 2760 2760 ta52 50x15 2756 2756 2756 2756 2756 2756 2756 2756 ta53 50x15 2717 2717 2717 2717 2717 2717 2717 2717 ta54 50x15 2839 2839 2839 2839 2839 2839 2839 2839 ta55 50x15 2679 2679 2679 2679 2679 2679 2684 2679 ta56 50x15 2781 2781 2781 2781 2781 2781 2781 2781 ta57 50x15 2943 2943 2943 2943 2943 2943 2943 2943 ta58 50x15 2885 2885 2885 2885 2885 2885 2885 2885 ta59 50x15 2655 2655 2655 2655 2655 2655 2655 2655 ta60 50x15 2723 2723 2723 2723 2723 2723 2723 2723 ta61 50x20 2868 2868 2868 2868 2868 2868 2868 2868 ta62 50x20 2869 2869 2872 2872 2877 2875 2942 2900 ta63 50x20 2755 2755 2755 2755 2755 2755 2755 2755 ta64 50x20 2702 2702 2702 2702 2702 2702 2702 2702 ta65 50x20 2725 2725 2725 2725 2734 2725 2725 2725 ta66 50x20 2845 2845 2845 2845 2881 2845 2845 2845 ta67 50x20 2825 2825 2825 2825 2825 2825 2865 2826 ta68 50x20 2784 2784 2784 2784 2784 2784 2784 2784 ta69 50x20 3071 3071 3071 3071 3071 3071 3071 3071 ta70 50x20 2995 2995 2995 2995 2995 2995 2995 2995 ta71 100x20 5464 5464 5464 5464 5464 5464 5464 5464 ta72 100x20 5181 5181 5181 5181 5181 5181 5181 5181 ta73 100x20 5568 5568 5568 5568 5568 5568 5568 5568 ta74 100x20 5339 5339 5339 5339 5339 5339 5339 5339 ta75 100x20 5392 5392 5392 5392 5392 5392 5392 5392 ta76 100x20 5342 5342 5342 5342 5342 5342 5342 5342 ta77 100x20 5436 5436 5436 5436 5436 5436 5436 5436 ta78 100x20 5394 5394 5394 5394 5394 5394 5394 5394 ta79 100x20 5358 5358 5358 5358 5358 5358 5358 5358 ta80 100x20 5183 5183 5183 5183 5183 5183 5183 5183

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ComparisonwithTSSB TSSB ARI Tav ARI Tav la01-la05 10x5 0.00 0.00 9,8 la06-la10 15x5 0.00 0.00 la11-la15 20x5 0.00 0.00 la16-la20 10x10 0.00 0.00 61,5 la21-la25 15x10 0.00 17.2 0.11 115 la26-la30 20x10 0.02 15.4 0.28 105 la31-la35 30x10 0.00 0.00 la36-la40 15x15 0.07 109.2 0.58 141 Comparisonwithotheralgorithms GES1 RTSAB RIMP TSSB BV ARI Tav ARI Tav ARI Tav ARI Tav ARI Tav ta01-ta10 0.01 84.5 0.11 566.5 0.01 190.0 0.45 2175 0.17 1498 ta11-ta20 0.21 1154.6 0.55 1612.5 0.60 2044.3 1.19 2526 0.75 4559 ta21-ta30 0.13 1668.8 0.31 2241.7 0.34 1561.6 1.01 34910 0.61 6850 ta31-ta40 0.19 1460 0.46 1171.4 0.62 2052.2 1.41 14133 0.30 8491 ta41-ta50 0.49 2111.4 0.78 3229.9 1.18 2205.6 1.92 11512 1.11 16018 ta51-ta60 0.00 6.4 0.00 238.8 0.00 7.5 0.02 421 0.00 196 ta61-ta70 0.01 185.2 0.19 1155.2 0.02 188.6 0.40 6342 0.11 2689 ta71-ta80 0.00 38.1 0.00 452.5 0.00 38.2 0.00 231 0.00 851

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ResultsforDMU1-DMU80 n m LB UB GES2 DMU1 20 15 2501 2563 2566 DMU2 20 15 2651 2706 2706 DMU3 20 15 2731 2731 2731 DMU4 20 15 2601 2669 2669 DMU5 20 15 2749 2749 2749 DMU6 20 20 2834 3250(3252) 3250 DMU7 20 20 2677 3053(3063) 3053 DMU8 20 20 2901 3197(3199) 3197 DMU9 20 20 2739 3092 3092 DMU10 20 20 2716 2984(2985) 2984 DMU11 30 15 3395 3453(3457) 3453 DMU12 30 15 3481 3518(3519) 3518 DMU13 30 15 3681 3697(3698) 3697 DMU14 30 15 3394 3394 3394 DMU15 30 15 3332 3343 3343 DMU16 30 20 3726 3781(3787) 3781 DMU17 30 20 3697 3848(3854) 3848 DMU18 30 20 3844 3849(3852) 3849 DMU19 30 20 3650 3807(3814) 3807 DMU20 30 20 3604 3739(3740) 3739 DMU21 40 15 4380 4380 4380 DMU22 40 15 4725 4725 4725 DMU23 40 15 4668 4668 4668 DMU24 40 15 4648 4648 4648 DMU25 40 15 4164 4164 4164 DMU26 40 20 4647 4667(4670) 4667 DMU27 40 20 4848 4848 4848 DMU28 40 20 4692 4692 4692 DMU29 40 20 4691 4691 4691 DMU30 40 20 4732 4732 4732 DMU31 50 15 5640 5640 5640 DMU32 50 15 5927 5927 5927 DMU33 50 15 5728 5728 5728 DMU34 50 15 5385 5385 5385 DMU35 50 15 5635 5635 5635 DMU36 50 20 5621 5621 5621 DMU37 50 20 5851 5851 5851 DMU38 50 20 5713 5713 5713 DMU39 50 20 5747 5747 5747 DMU40 50 20 5577 5577 5577 DMU41 20 15 2839 3267(3275) 3267 DMU42 20 15 3066 3401(3416) 3401 DMU43 20 15 3121 3443(3455) 3443

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ResultsforDMU1-DMU80(continued) n m LB UB GES2 DMU44 20 15 3112 3489(3501) 3489 DMU45 20 15 2930 3273 3273 DMU46 20 20 3425 4099(4101) 4099 DMU47 20 20 3353 3972(3973) 3972 DMU48 20 20 3317 3810(3834) 3810 DMU49 20 20 3369 3754(3765) 3754 DMU50 20 20 3379 3768(3772) 3768 DMU51 30 15 3839 4247(4252) 4247 DMU52 30 15 4012 4380(4383) 4380 DMU53 30 15 4108 4450(4454) 4450 DMU54 30 15 4165 4424(4425) 4424 DMU55 30 15 4099 4331(4332) 4331 DMU56 30 20 4366 5049 5051 DMU57 30 20 4182 4779(4781) 4779 DMU58 30 20 4214 4829(4834) 4829 DMU59 30 20 4199 4694(4696) 4694 DMU60 30 20 4259 4888(4890) 4888 DMU61 40 15 4886 5293(5294) 5293 DMU62 40 15 5004 5342 5354 DMU63 40 15 5049 5437 5439 DMU64 40 15 5130 5367 5388 DMU65 40 15 5072 5269(5271) 5269 DMU66 40 20 5357 5902(5910) 5902 DMU67 40 20 5484 6012(6016) 6012 DMU68 40 20 5423 5934(5936) 5934 DMU69 40 20 5419 5891 6002 DMU70 40 20 5492 6072(6076) 6072 DMU71 50 15 6050 6302 6333 DMU72 50 15 6223 6571 6589 DMU73 50 15 5935 6283 6291 DMU74 50 15 6015 6376 6376 DMU75 50 15 6010 6380(6384) 6380 DMU76 50 20 6329 6974(6975) 6974 DMU77 50 20 6399 6930 7006 DMU78 50 20 6508 6962 6988 DMU79 50 20 6593 7158(7164) 7158 DMU80 50 20 6435 6824 6843

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OlegShylowasbornonSeptember03,1981,inKiev,Ukraine.HereceivedhisbachelorsandspecialistsdegreesinappliedmathematicsfromtheNationalTechnicalUniversityofUkraine(\KPI")inKiev,Ukraine,in2003and2004,respectively.InJanuary2005,hebeganhisdoctoralstudiesintheIndustrialandSystemsEngineeringDepartmentattheUniversityofFlorida.HeearnedhisPh.D.inAugust2009. 96