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# New Algorithm for the Triangulation of Input-Output Tables and the Linear Ordering Problem

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page ACKNOWLEDGMENTS ............................. ii LISTOFTABLES ................................. iv LISTOFFIGURES ................................ v ABSTRACT .................................... vi 1INTRODUCTION .............................. 1 2LINEARORDERINGPROBLEM ..................... 5 2.1Applications .............................. 6 2.2ProblemFormulations ........................ 7 2.3PreviousWork ............................. 9 2.3.1ExactMethods ......................... 9 2.3.2HeuristicMethods ....................... 9 3AGRASPWITHPATH-RELINKINGALGORITHM .......... 11 3.1IntroductiontoGRASPandPath-Relinking ............ 11 3.2ProposedAlgorithm .......................... 13 3.2.1ConstructionPhase ...................... 15 3.2.2LocalSearch .......................... 16 3.2.3PathRelinking ......................... 16 4COMPUTATIONALRESULTS ....................... 18 5CONCLUDINGREMARKS ......................... 24 REFERENCES ................................... 26 BIOGRAPHICALSKETCH ............................ 28 iii

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Table page 4{1ResultsfortheLOLIBInstancesafter200and5000iterations. .... 20 4{2ResultsfortheMitchellInstancesafter200and5000iterations. .... 21 4{3SummaryofresultsobtainedfortheLOLIBandMitchelltestsetsafter200and5000iterations. ...................... 21 iv

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Figure page 3{1GenericGRASPpseudo-code ....................... 12 3{2Visualizationofapath-relinkingprocedure ............... 13 3{3GRASPpseudo-code ........................... 14 3{4GRASPconstructionphasepseudo-code ................ 15 3{5GRASPpath-relinkingpseudo-code ................... 17 4{1TheLOLIBInstances:gapfromoptimalsolutionasapercentageisshownasafunctionofthenumberofiterations. ........... 22 4{2TheLOLIBInstances:gapfromoptimalsolutionasapercentageisshownasafunctionofthetimeelapsed(seconds). .......... 22 4{3MitchellInstances:gapfromoptimalsolutionasapercentageisshownasafunctionofthenumberofiterations. ........... 23 4{4MitchellInstances:gapfromoptimalsolutionasapercentageisshownasafunctionofthetimeelapsed(seconds). .......... 23 v

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Theimpactofchangesofaneconomicvariablecanonlybeanalyzedbyunderstandingthecomplexseriesoftransactionstakingplaceamongthesectorsofaneconomy.FirstintroducedbyLeontiefintheearly1930s,input-outputmodelshavebecomeanindispensabletoolforeconomistsandpolicy-makersintheiranalysis,providingasystematicdescriptionofsuchinterrelationsamongthesectors[ 1 ]. Aninput-outputmodelbeginsbydividingtheeconomyofacountry(orregion)intoaspeciednumberofsectors.Thenatableisconstructed,wheretheentriesarethetotaltransactionsbetweeneverypairofsectors.Thetotaloutput(input)ofasectorcanbeobtainedbysummingtheentriesonthecorrespondingrow(column).Theresultingtablethussummarizestheinterdependenceamongtheeconomicsectors. Structuralpropertiesoftheinput-outputtablesmaynotbeapparent.Aparticularchoiceintheorderofthesectorsusedinconstructingthetablemightconcealanotherwiseevidentstructure.Thesefeaturesarerevealedbyaprocesscalledtriangulation,whoseobjectiveistondahierarchyofthesectorssuchthatthosewhoarepredominantlyproducerswillappearrst,whilethosewhoaremainlyconsumerswillappearlast. Theeconomicsignicanceisthatitshowshowtheeectsofchangesinnaldemandpropagatethroughthesectors.Note,however,thatinusingahierarchicordering,thereisanunderlyingassumptionthatnoowexistsfromlowertouppersectors.Infact,everyeconomyexhibitsacertaincircularityintheowofgoods|e.g.,themetallurgyindustrysuppliesthevehicleindustrywithrawmetalproducts, 1

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describingtheGRASPandpath-relinkingframework.ThecomputationalexperimentationisshowninChapter 4 .ThethesisconcludeswithChapter 5 ,wheresomepracticalissuesarediscussed.

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TheLOPisanNP-hardcombinatorialoptimizationproblemwithawiderangeofapplicationsineconomics,archaeology,andscheduling.Ithas,however,drawnlittleattentioncomparedtoothercloselyrelatedproblemssuchasthequadraticassignmentproblemandthetravellingsalesmanproblem. TheLOPcanbestatedasfollows.ConsiderasetNofnobjectsandapermutation:N!N.Eachpermutation=((1);(2);:::;(n))correspondsone-to-onetoalinearorderingoftheobjects.Leteij;i;j=1;2;:::;n,bethecostofhavingibeforejintheordering,andEbethen-squarematrixofcosts.Thenthelinearorderingproblemistondapermutationthatmaximizesthetotalcost Clearly,Equation 2.1 isthesumoftheelementsabovethediagonalofamatrixA,whoseelementsaijarethoseresultingfromapermutationoftherowsandcolumnsofmatrixE|i.e.,A=XEXT,whereXisthepermutationmatrixassociatedwiththepermutation[ 3 ].Inthecontextofitsapplicationineconomics,wecanrestateEquation 1.1 as whereisthesetofallpermutations,andKisapositiveconstantrepresentingthesumofalltheentriesinthematrix. TheLOPcanalsobeinterpretedasaproblemingraphs.LetG(N;A)beacompletedirectedgraphwithnodesetNandarcsetA=f(i;j):i;j2N^i6=jg. 5

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Leteijbetheweightofarc(i;j).AspanningacyclictournamentinGinducesauniquelinearorderingofthenodesetN[ 4 ].Atournamentisdenedasadirectedgraphinwhicheachpairofnodesisconnectedbyexactlyonearc,whichisclearlynecessarysinceeitheriisbeforejorjisbeforei. ThecomplexityofthemaximumLOPcanbeeasilyproventobeNP-hardbynoticingthatitisequivalenttotheminimumweightedfeedbackarcsetproblemonG,whichisknowntobeNP-hard[ 5 ]. TheLOPhasaninterestingsymmetryproperty.Ifapermutation=((1);(2);:::;(n))isanoptimalsolutiontothemaximizationversion,thenthereversepermutation 2-approximationalgorithm[ 4 ].Letbeanarbitrarypermutationand 2 whereisanoptimalpermutationandZ()>0.Nootherapproximationalgorithmexists[ 4 ].ItfollowsthatanypermutationisoptimalintheunweightedversionoftheLOP. 3 ]givesanextensivesurvey. Considertheproblemofhavingagroupofpeopleranknobjects.Eachindividualinthegroupisaskedtoexpressapreferencewithrespecttoeverypossiblepair.Ifweleteijbethenumberofpeoplewhopreferreditoj,thesolutiontothecorrespondingLOPistherankingthatmostlikelyreectsthepreferencesofthegroup.

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Asimilarapplicationcanbefoundinthecontextofsports.Forexample,consideratournamentofnteamsinwhicheveryteamplaysagainsteveryotherteam.Leteijbethescoreofthematchbetweeniandjifiwins,and0otherwise.TherankingobtainedbytheLOPisconsideredtobetheonethatmostcloselyreectsthe\true"performanceoftheteams.Still,ithasnotgainedsupportforitsimplementation,probablybecausetheoutcomeofaparticularmatchisnotcloselyrelatedtotheresultintheranking. Inarchaeology,theLOPisusedtodeterminethe\mostprobable"chronologicalorderingofasetofartifactsrecoveredfromdierentsites.Samplesbelongingtovarioustimeperiodsaregivenavaluebasedontheirdistancetothesurface.Theobjectiveistoaggregatethedataanddetermineanorderingoftheartifacts. Finally,itiseasytoseethattheLOPcanbeusedtodeterminetheoptimalsequenceofjobsinasingleserver,wherethecostofeachjobdependsuponitspositionwithrespecttotheentireschedule.

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whereA0AisthearcsetofthespanningacyclictournamentonG.ThentheproblemofndingthemaximumofEquation 2.1 becomesmaxX(i;j)2Aeijxij 2.4 )denethetournamentpolytope.Itcanbeproventhatthe3-dicycleinequalities(Equations 2.5 )aresucienttopreventanycycles[ 3 ].Togethertheydenethelinearorderingpolytope.Thereare2n2variablesandn2+2n3constraintsinthisformulation. Thetournamentconstraints(Equations 2.4 )motivatetheuseofasinglevariabletorepresentthetwopossiblewaysinwhicheverypairofnodescanbeconnected.Letussubstitutexji=1xij,foreveryi;j2N;j>i,thenanequivalentintegerprogrammingformulationismaxXf(i;j)2A:i
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2.3.1ExactMethods 3 7 ].Thegoalistoobtainanapproximatedescriptionoftheconvexhullofthesolutionsetbyintroducingvalidinequalitiesthatareviolatedbycurrentfractionalsolutions,whichareaddedtothesetofinequalitiesofthecurrentlinearprogrammingproblem.Reinelt[ 3 ]introducedfacetsinducedbysubgraphs.Bolotashvilietal.[ 8 ]extendedReinelt'sresultsintroducingageneralizedmethodtogeneratenewfacets.Acompletecharacterizationhasonlybeenobtainedforverysmallproblems(n=7)[ 9 ].Infact,weknowthatunlessP=NPthereexistanexponentialnumberofsuchfacets.However,researchinthisareahasresultedinvalidinequalitiesthatimprovetheperformanceofbranch-and-cutalgorithms[ 3 ]. TherstauthorstoconsideraninteriorpointalgorithmfortheLOPwereMitchellandBorchers[ 10 11 ].Thesolutiongivenbyainteriorpointalgorithmisusedasastartingpointforasimplexcuttingplanealgorithm[ 11 ]. Heuristics,asopposedtoapproximationalgorithms,donotgiveaguaranteedqualityoftheobtainedsolutions.Nevertheless,theexibilitywehaveindevelopingheuristicsallowsustoexploitthespecialstructureoftheproblem,tailoringtheexistingmethods,andresultinginverywellperformingalgorithms.Thequalityofaheuristic,however,mustbevalidatedbyextensivetesting. HeuristicmethodssuchasGRASP,tabusearch,simulatedannealing,geneticsearch,andevolutionstrategieshaveshowntobeabletoecientlyndhigh

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qualitysolutionstomanycombinatorialandglobaloptimizationproblemsbythoroughlyexploringthesolutionsspace.Arecentsurveyonmulti-startheuristicalgorithmsforglobaloptimizationisgivenbyMart[ 12 ]. OneoftheearliestheuristicsfortheLOPwasproposedbyCheneryandWatanabe[ 13 ]inthecontextofthetriangulationofinput-outputtables.Givenasectori,theratiooftotalinputtothetotaloutput isusedtoarrangethesectorsintheorderofdecreasingui.TheuseofEquation 2.9 givesafairlygoodorderingconsideringitssimplicity. BasedonthesymmetrypropertymentionedinChapter 2 ,ChanasandKobylanski[ 14 ]developedaheuristicthatperformsasequenceofoptimalinsertionsandreversals. Lagunaetal.[ 15 ]developedanalgorithmbasedontabusearch.Theyanalyzedseveralintensicationanddiversicationtechniques,andcomparedtheiralgorithmwiththatofChanasandKobylanski. Camposetal.[ 16 ]usedascattersearchapproach.AcorrectiontermbasedonthefrequencybywhichanobjectiappearsinaparticularpositionintheorderingisaddedtoEquation 2.9 toreectprevioussolutions. GRASP,whichisaniterativerestartapproach,hasproventobeoneofthemosteectiveheuristicstodate.Inthisthesis,wedevelopedaGRASP-basedalgorithmfortheLOP,oeringasignicantimprovementonthecomputationaltimeandqualityofsolutioncomparedtopreviousheuristics.Inthenextsection,wediscussthebasicprinciplesforimplementinganewlocalsearchschemeandPath-RelinkinginGRASPframework.

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17 ].Festaetal.[ 18 ]havecompiledanextensiveannotatedbibliographyofGRASPapplications. EachiterationofGRASPconsistsoftwophases:aconstructionphaseinwhichweseektoobtainafeasiblesolution,andalocalsearchphasethatattemptstoimprovethesolution.Figure 3{1 showsthepseudo-codeofagenericGRASPalgorithm. Duringtheconstructionphase,weiterativelybuildasolutionbyrandomlyselectingobjectsfromarestrictedcandidatelist(RCL).Ateachstep,weformtheRCLchoosingthoseobjectswiththehighestmeasureofattractiveness,weselectarandomobjectfromtheRCL,andadaptthegreedyfunctiontoreecttheadditiontothesolution.Figure 3{4 showstheconstructionphaseforourimplementation. Thesizeofthelististypicallyrestrictedinoneoftwoways:byquality,whenwechoosetheelementsbasedonathresholdonthegreedyfunction,orbycardinality.Intheliterature,theyareoftenreferredtoasand,respectively.ThesizeoftheRCLcontrolsthedegreeofgreedinessandrandomnessoftheconstructionphase.AnullRCL|i.e.,ofsize1|resultsinapurelygreedysolutionwhereasaRCLsizeequaltothesizeoftheproblemyieldsapurelyrandomsolution. Afterasolutionisconstructed,weattempttoimproveitbyperformingalocalsearchinanappropriatelydenedneighborhood.Givenasolutionx,we 11

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01BestSolutionFound=;; 02EliteList=;; 03nNonImprovingIt=0; 04fork=1,2,...,MaxIteration 05x=ConstructGreedyRandomizedSolution(); 06LocalSearch(x); 07ifxisbetterthanworsesolutioninEliteList 08DoPathRelinking(EliteList,x); 09nNonImprovingIt=0; 10elseifnNonImprovingIt> 12DoPathRelinking(EliteList,x); 13nNonImprovingIt=0; 14elsenNonImprovingIt=nNonImprovingIt+1; 13UpdateSolution(BestSolutionFound,x); 14end; 15returnBestSolutionFound; Figure3{3: GRASPpseudo-code Inthecontextoftheinput-outputtables,Equation 3.1 representsthenetowofasector.InearlierexperimentationswithaGRASPalgorithm,wecomparedtheuseofEquation 3.1 withtheratiosasdenedinEquation 2.9 .Althoughwehavenotobservedsignicantdierencesinthequalityofthesolutions,adaptingthegreedyfunctionwhenratiosareusediscomputationallymoreexpensive.TheuseofEquation 2.9 demandsO(n2)timetoupdatethegreedyfunctionwhereasEquation 3.1 requiresO(n)time. Linearorderingproblemsusuallyhavemanyalternativesolutions|optimalandsuboptimal|withthesameobjectivefunctionvalue.Therefore,itmayoccuratsomepointinthealgorithmthattheelitelistbecomesmostlypopulatedbyalternativesolutions.Furthermore,itisincreasinglydiculttoenterapath-relinkingasthebestsolutionfoundapproachestheoptimal.Weattempttoavoidsuchsituationsbyexpandingthesizeoftheelitelistandforcingapath-relinkingprocedureafteracertainnumberofnon-improvingiterations.Ifanimprovementis

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1Solution=;,RCL=;; 2whilejSolutionj
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breakingtiesarbitrarily.FirstintroducedbyChanasandKobylanski[ 14 ]aspartoftheirheuristic,itcanbeconsideredasaveryecientlocalsearchprocedureinarelativelysmallneighborhood.Infact,itcanbeimplementedinO(k). Astepofthecontructionphasenalizeswiththetaskofadaptingthegreedyfunction.Therowandcolumncorrespondingtotheobjectsareremovedfromthematrix,andtheattractivenessoftheobjects(Equation 3.1 )isupdated.Wesetds=M,whereMisalargepositivevalue,andre-sortthetopnkobjectsthathaveyettobeselected.Theprocedurecontinuesuntilasolutionisconstructed.TheoverallcomplexityoftheconstructionphaseisO(n2). Z(;i;j)=e0(i)(j)j1Xk=i+1(e0(i)(k)+e0(k)(j))(3.3) wheree0ij=eijeji.Atcompletion,thelocalsearchwouldhaveexchangedthepairofobjectsthatmaximizesEquation 3.3 .TheprocedureofexploringtheneighborhoodandperformingtheexchangecanbeimplementedinO(n2).

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Herewediscussthecomputationalresultsweobtainedwhenapplyingouralgorithmtotwosetsofproblems: 21 ].Theyconsistof49input-outputtablesforsomeEuropeancountries,withsizesupto60objects. 10 ],withsizesrangingfrom100to250objects.Threedierentpercentagesofzeroentrieswereused:0,10,and20%,denotedbythelastdigitontheproblemname.Thegeneratoraswellastheinstancesareavailableattheauthor'swebsite[ 22 ]. Theoptimalvaluesforallinstancesareknown.ThegeneratedinstancesaresimilartothosefromLOLIBexceptforthenumericalrangeoftheentries|considerablylargerforthelatter.Despiteattemptstoreplicatethecharacteristicsofreal-worldinstancessuchasthosefoundinLOLIB,Mitchell'stestsetissignicantlyhardertosolve.AllpreviousworkonthelinearorderingproblemthatincludedcomputationalresultspredatestheMitchellinstances,hencefeaturingonlytheLOLIBproblems. ThealgorithmwaswritteninC++andexecutedonaPentium4,2.7GHz,with512MBofmemory.Empirically,wedetermined=0:25and=0:35asthebestvaluesfortheparameters(e.g.,foraproblemofsizen,thesizeoftheRCLandtheelitelistareatmost0:25nand0:35nrespectively).Thealgorithmwasexecutedvetimesforeachprobleminstance,withalimitof5000GRASPiterationsonitsrunningtime.Wereporttherunningtime,numberofiterations, 18

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andthegapbetweenthebestsolutionandtheoptimalsolution.Alltimesarereportedinsecondsandgapsaspercentages. Tables 4{1 and 4{2 showtheelapsedrunningtimeandgapvaluesafter200and5000iterations.Theresultsreportedaretheaveragesof5runsforeachprobleminstance.AsummaryoftheresultsispresentedinTable 4{3 .Figures 4{1 and 4{2 showtheevolutionofthegapasafunctionofthenumberofiterationsandtherunningtimefortheLOLIBinstances,respectively.CorrespondingresultsfortheMitchellinstancesareshowninFigures 4{3 and 4{4 .Notethattheunitsontheordinatesarepercentagepoints. Thealgorithmfoundoptimalsolutionsfor47of49LOLIBinstances,17ofwhichwereconsistentlysolvedtooptimality.TheaveragegapfortheremainingLOLIBinstanceswas0.0127%.Theaveragerunningtimeforthesereal-worldinstanceswas0.87seconds.AlthoughnoneoftheMitchellinstancesweresolvedtooptimality,theaveragegapafter5000iterationswas0.0173%withanaveragerunningtimeof21.98seconds.

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Table4{1: ResultsfortheLOLIBInstancesafter200and5000iterations. Gap(%)Time(s)Gap(%)Time(s) be75eec500:01990:02820:00420:4062be75np500:00570:01880:00181:0406be75oi500:04590:02180:01250:7062be75tot500:00560:02180:00020:5156stabu1600:01240:04060:00241:3030stabu2600:00590:04680:00512:1624stabu3600:03290:05620:00901:9280t59b11xx440:00120:01560:00120:8250t59d11xx440:00820:01560:00000:5720t59f11xx440:02020:01240:00000:6562t59i11xx440:00000:01560:00000:7686t59n11xx440:05150:01860:00790:6592t65b11xx440:02230:01240:01330:7624t65d11xx440:00030:01560:00000:5876t65f11xx440:00900:01560:00060:5842t65i11xx440:01370:01560:00070:5938t65l11xx440:00000:01560:00000:8032t65n11xx440:01240:01560:00000:4624t65w11xx440:00140:01560:00000:5594t69r11xx440:01220:01560:00020:6844t70b11xx440:00800:01240:00800:7280t70d11xn440:02880:07500:00732:7406t70d11xx440:02320:01560:01200:4938t70f11xx440:00300:01260:00120:4282t70i11xx440:00220:01260:00090:4970t70k11xx440:00000:01560:00000:7218t70l11xx440:00070:02500:00000:6124t70n11xx440:02600:01260:00190:3970t70u11xx440:00230:01260:00000:7938t70w11xx440:00120:01560:00000:3874t70x11xx440:00000:01260:00000:6906t74d11xx440:01250:01880:00420:5032t75d11xx440:00280:02180:00200:7936t75e11xx440:00480:01260:00000:4688t75i11xx440:00170:01260:00000:6220t75k11xx440:00000:01560:00000:7156t75n11xx440:07100:01560:00000:5062t75u11xx440:00070:01560:00030:6530tiw56n54560:01700:03740:00440:8436tiw56n58560:01270:03120:00001:0124tiw56n62560:01870:02820:00071:0970tiw56n66560:01790:03420:00430:5218tiw56n67560:00630:03440:00062:0312tiw56n72560:00780:03760:00131:0470tiw56r54560:01570:03120:00200:7062tiw56r58560:00460:04380:00201:2720tiw56r66560:03530:04060:00610:8688tiw56r67560:00860:03120:00071:9250tiw56r72560:00060:03120:00061:8970

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Table4{2: ResultsfortheMitchellInstancesafter200and5000iterations. Gap(%)Time(s)Gap(%)Time(s) r100a20:02680:23740:00472:5374r100b20:04620:25940:02072:6032r100c20:04160:30940:03022:6376r100d20:03890:22200:03432:4906r100e20:02700:20300:01656:9312r150a00:01370:65940:00708:0562r150a10:02630:71260:02268:3282r150b00:02300:97200:01028:1970r150b10:02241:32200:01949:1000r150c00:02071:28440:016727:9032r150c10:03690:92820:01919:0032r150d00:01461:19700:01228:5656r150d10:02121:08740:019134:6218r150e00:01480:81260:005633:1438r150e10:03881:48420:02899:5812r200a00:01812:60620:008021:2470r200a10:04092:89060:030522:4062r200b00:01962:01560:015121:4780r200b10:03333:65320:027123:6062r200c00:00963:19380:008420:4376r200c10:02442:73120:021921:5844r200d00:01592:77200:008932:5562r200d10:03451:98440:026222:5906r200e00:02122:96560:015822:8470r200e10:02972:37800:024435:9750r250a00:02335:47500:015647:1220r250b00:01893:81880:010243:5406r250c00:015910:13120:013252:2126r250d00:01848:37500:011849:1438r250e00:02415:48440:014648:9250 SummaryofresultsobtainedfortheLOLIBandMitchelltestsetsafter200and5000iterations. Gap(%)Time(s)Gap(%)Time(s) LOLIBAverage0:01250:02350:00240:8685Std.Dev.0:01460:01310:00350:5188Maximum0:07100:07500:01332:7406 MitchellAverage0:02542:40550:017321:9791Std.Dev.0:00962:36380:008115:6266Maximum0:046210:13120:034352:2126

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Figure4{1: TheLOLIBInstances:gapfromoptimalsolutionasapercentageisshownasafunctionofthenumberofiterations. Figure4{2: TheLOLIBInstances:gapfromoptimalsolutionasapercentageisshownasafunctionofthetimeelapsed(seconds).

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Figure4{3: MitchellInstances:gapfromoptimalsolutionasapercentageisshownasafunctionofthenumberofiterations. Figure4{4: MitchellInstances:gapfromoptimalsolutionasapercentageisshownasafunctionofthetimeelapsed(seconds).

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[1] WassilyLeontief,Input-OutputEconomics,OxfordUniversityPress,NewYork,USA,1986. [2] GeorgeB.Dantzig,\Linearprogramming,"OperationsResearch,vol.50,no.1,pp.42{47,January2002. [3] GerhardReinelt,Thelinearorderingproblem:algorithmsandapplications,Number8inResearchandExpositioninMathematics.HeldermannVerlag,Berlin,Germany,1985. [4] MichaelJunger,PolyhedralCombinatoricsandtheAcyclicSubdigraphProblem,Number7inResearchandExpositioninMathematics.HeldermannVerlag,Berlin,1985. [5] MichaelR.GareyandDavidS.Johnson,ComputersandIntractability:AGuidetotheTheoryofNP-Completeness,W.H.FreemanandCo.,NewYork,USA,1979. [6] R.E.Burkard,E.Cela,P.M.Pardalos,andL.S.Pitsoulis,\Thequadraticassignmentproblem,"inHandbookofCombinatorialOptimization,P.M.PardalosandD.-Z.Du,Eds.,pp.241{338.KluwerAcademicPublishers,1998. [7] MartinGrotschel,MichaelJunger,andGerhardReinelt,\Acuttingplanealgorithmforthelinearorderingproblem,"OperationsResearch,vol.2,no.6,pp.1195{1220,1984. [8] G.Bolotashvili,M.Kovalev,andE.Girlich,\Newfacetsofthelinearorderingpolytope,"SIAMJournalonDiscreteMathematics,vol.12,no.3,pp.326{336,1999. [9] ThomasChristof,Low-Dimensional0/1-PolytopesandBranch-and-CutinCombinatorialOptimization,Ph.D.dissertation,UniversityofHeidelberg,Heidelberg,Germany,1997. [10] JohnE.Mitchell,\Computationalexperiencewithaninteriorpointcuttingplanealgorithm,"Tech.Rep.,MathematicalSciences,RensellaerPolytechnicIntitute,Troy,NY12180-3590,USA,1997. [11] JohnE.MitchellandBrianBorchers,\Solvinglinearorderingproblemswithacombinedinteriorpoint/simplexcuttingplanealgorithm,"inHighPerformanceOptimization,H.Frenketal.,Ed.,chapter14,pp.345{366.KluwerAcademicPublishers,Dordrecht,TheNetherlands,2000. 26

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[12] RafaelMart,\Multi-startmethods,"inHandbookofMetaheuristics,FredGloverandGaryA.Kochenberger,Eds.,InternationalSeriesinOperationsResearch&ManagementSciences,chapter12,pp.355{368.KluwerAcademicPublishers,Dordrecht,TheNetherlands,2003. [13] HollisB.CheneryandTsunehikoWatanabe,\Internationalcomparisonsofthestructureofproduction,"Econometrica,vol.26,no.4,pp.487{521,October1958. [14] StefanChanasandPrzemyslawKobylanski,\Anewheuristicalgorithmsolvingthelinearorderingproblem,"ComputationalOptimizationandApplications,vol.6,pp.191{205,1996. [15] ManuelLaguna,RafaelMart,andVicenteCampos,\Intensicationanddiversicationwithelitetabusearchsolutionsforthelinearorderingproblem,"Computers&OperationsResearch,vol.26,pp.1217{1230,1999. [16] VicenteCampos,FredGlover,ManuelLaguna,andRafaelMart,\Anexperimentalevaluationofascattersearchforthelinearorderingproblem,"JournalofGlobalOptimization,vol.21,no.4,pp.397{414,December2001. [17] ThomasA.FeoandMauricioG.C.Resende,\Greedyrandomizedadaptivesearchprocedures,"JournalofGlobalOptimization,vol.2,pp.1{27,1995. [18] PaolaFesta,MauricioG.C.Resende,andGeraldoVeiga,\AnnotatedbibliographyofGRASP,"AT&TResearchLabs,Jan.2004, http://www.research.att.com/mgcr/grasp/annotated. [19] FredGloverandManuelLaguna,TabuSearch,KluwerAcademicPublishers,Boston,USA,1997. [20] ManuelLagunaandRafaelMart,\GRASPandpathrelinkingfor2-layerstraightlinecrossingminimization,"INFORMSJournalonComputing,vol.11,no.1,pp.44{52,1998. [21] GerhardReinelt,\Linearorderinglibrary(LOLIB),"UniversityofHeidelberg,Dec.2002, http://www.iwr.uni-heidelberg.de/groups/comopt/software/LOLIB/. [22] JohnE.Mitchell,\Generatinglinearorderingproblems,"RensellaerPolytech-nicIntitute,Dec.2002,http://www.rpi.edu/mitchj/generators/linord.

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NEW ALGORITHM FOR THE TRIANGULATION OF INPITT-OUTPITT
TABLES AND THE LINEAR ORDERING PROBLEM

By
BRUJNO H. CHIARINI

A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FITLFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2004

ACKNOWLEDGMENTS

I would like to thank all the people who phali- d a role in my career at the

University of Florida (Elif Akgahl, Donald Hearn, and Edwin Romeijn, among

others). I specially thank my committee members (Stanislay Uryasey and Panos

Pardalos) for their time and candid review of my thesis.

There is somebody who will forever have a special place in my heart: Panos

M. Pardalos. I thank him for all his support and advice throughout my career. I

will miss him, but I am leaving with the hope of having other opportunities to work

together in the future.

The last mention is reserved for some very special people, without whom this

would have been an impossible journey. They have endured ups and downs, and

grown with me to form a stronger nucleus. They are the source from which I pull

strength and wisdom, and the recipients of my work. This work is dedicated to

them. I thank Diane and Gianluca for their eternal love, patience, and support.

pagfe

ACKNOWLEDGMENTS .........i

LIST OF TABLES ......... ... .. iv

LIST OF FIGURES ......... .. .. v

ABSTRACT ......... .... .. v

1 INTRODUCTION . ...... .... 1

2 LINEAR ORDERING PROBLEM ...... .. 5

2.1 Applications ......... .. .. 6
2.2 Problem Formulations . ...... .. 7
2.3 Previous Work ......... .. .. 9
2.3.1 Exact Methods . ...... .. 9
2.3.2 Heuristic Methods . ..... .. 9

:3 A GRASP WITH PATH-RELINK(ING ALGORITHM .. .. .. 11

:3.1 Introduction to GR ASP and Path-Relinking ... .. .. 11
:3.2 Proposed Algforithm ........ ... .. 1:3
:3.2.1 Construction Phase . ..... .. 15
:3.2.2 Local Search ........ .. 16
:3.2.3 Path Relinkingf ...... .. .. 16

4 COMPUTATIONAL RESULTS . ..... .. 18

5 CONCLUDING REMARK(S ...... .. .. 24

REFERENCES ......._._.. ....... ... 26

BIOGRAPHICAL SK(ETCH ....._._. .. .. 28

LIST OF TABLES
Table pagfe

4-1 Results for the LOLIB Instances after 200 and 5000 iterations. .. 20

4-2 Results for the Mitchell Instances after 200 and 5000 iterations.. .. 21

4-3 Summary of results obtained for the LOLIB and Mitchell test sets
after 200 and 5000 iterations. ...... .... 21

LIST OF FIGURES
Figure page

3-1 Generic GRASP pseudo-code ...... .. 12

3-2 Visualization of a path-relinking procedure ... .. .. .. 13

3-3 GRASP pseudo-code .. ... .. 14

3-4 GRASP construction phase pseudo-code ... ... .. 15

3-5 GRASP path-relinkingf pseudo-code ... .. .. .. 17

4-1 The LOLIB Instances: gap from optimal solution as a percentage is
shown as a function of the number of iterations. .. .. .. 22

4-2 The LOLIB Instances: gap from optimal solution as a percentage is
shown as a function of the time elapsed (seconds). .. .. .. 22

4-3 Mitchell Instances: gap from optimal solution as a percentage is
shown as a function of the number of iterations. .. .. .. 23

4-4 Mitchell Instances: gap from optimal solution as a percentage is
shown as a function of the time elapsed (seconds). .. .. .. 23

Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

NEW ALGORITHM FOR THE TRIANGULATION OF INPITT-OUTPITT
TABLES AND THE LINEAR ORDERING PROBLEM

By

Bruno H. Chiarini

Al li- 2004

C'I ny~: Panos 31. Pardalos
Major Department: Industrial and Systems Engineering

Developed by Leontief in the 1930s, input-output models have become an

indispensable tool for economists and policy-makers. They provide a framework

on which researchers can systematically analyze interrelations among sectors of

an economy. In an input-output model, a table is constructed where each entry

represents the flow of goods between each pair of sectors. Special features of the

structure of this matrix are revealed by a technique called ',.:.<*.l; I.J.;i..n.

Triangulation is shown to be equivalent to the linear ordering problem

(LOP), which is an HiP-hard combinatorial optimization problem. Because of

the complexity of the triangulation procedure, it is essential in practice to search

for quick approximate (heuristic) algorithms for the linear ordering problem. In

addition to the triangulation of input-output tables, the LOP has a wide range of

applications in practice, including single-server scheduling and ranking of objects

by pairwise comparisons. However, a higher emphasis is placed on the triangulation

of input-output tables, the original motivation for this work.

Our study developed a new heuristic procedure to find high-quality solutions

for the LOP. The proposed algorithm is based on a Greedy Randomized Adaptive

Search Procedure (GR ASP), which is one of the most effective heuristics for solving

combinatorial and global optimization problems to date. We propose an improved

solution technique by using a new local search scheme and integrating a path-

relinkingf procedure for intensification. We tested our implementation on a set of 49

real-world instances of input-output tables in LOLIB. In addition, we tested a set

of 30 large random instances generated by Mitchell. Most of the LOLIB instances

were solved to optimality within 0.87 seconds on average. The average gap for the

Mitchell instances was 0.01'7:' with an average running time of 21.98 seconds.

Results prove the efficiency and high-quality of the algorithm, and its potential to

be applied to large problems.

CHAPTER 1
INTRODUCTION

The impact of changes of an economic variable can only be an~ II,. I1 by

understanding the complex series of transactions taking place among the sectors

of an economy. First introduced by Leontief in the early 1930s, input-output

models have become an indispensable tool for economists and policy-makers in

their analysis, providing a systematic description of such interrelations among the

sectors [1].

An input-output model begins by dividing the economy of a country (or

region) into a specified number of sectors. Then a table is constructed, where the

entries are the total transactions between every pair of sectors. The total output

(input) of a sector can be obtained by summing the entries on the corresponding

row (column). The resulting table thus summarizes the interdependence among the

economic sectors.

Structural properties of the input-output tables may not be apparent. A

particular choice in the order of the sectors used in constructing the table might

conceal an otherwise evident structure. These features are revealed by a process

called te.:r,:l,, I.J.;i.. n, whose objective is to find a i 7 I /I of the sectors such that

those who are predominantly producers will appear first, while those who are

mainly consumers will appear last.

The economic significance is that it shows how the effects of changes in final

demand propagate through the sectors. Note, however, that in using a hierarchic

ordering, there is an underlying assumption that no flow exists from lower to upper

sectors. In fact, every economy exhibits a certain J.~iil, h;in the flow of goods-

e.g., the metallurgy industry supplies the vehicle industry with raw metal products,

while the metallurgy sector needs vehicles as part of the cost of doing business.

Obviously, the flow between any two sectors is hardly symmetric.

The degree to which an economic structure Igees" with a hierarchy of

the sectors is called 'ii,.w9i;, In a perfectly linear economy, the flow of goods

"cascades" from the upper sectors to the lower sectors of the hierarchic ordering.

If we arrange the rows and columns of the input-output matrix according to the

hierarchy, such a situation would be reflected by a matrix that has an upper

triangular structure (that is, all entries below the diagonal would be zero). On the

other hand, if there is flow of goods back to the upper sectors, then there would be

positive values on the entries below the diagonal. This leads to the definition of a

quantitative measure of linearity. Let a denote the number of sectors and E = {es; }

be the n-square matrix representing the input-output table. Assume that the rows

and columns have been arranged according to the hierarchy. Then, the linearity of

an economy is given by

That is, linearity is the ratio of the sum of the elements above the diagonal to the

sum of all elements (except the diagonal). It follows that A = 1 for a perfectly

linear economy. Researchers have observed that large and highly developed

economies tend to have a low degree of linearity-i.e., there is a high circulation

in the flow of goods among sectors-whereas underdeveloped economies tend to

exhibit a clearer hierarchy. Typical linearity values are 711I' for a highly developed

economy, and 911' for an underdeveloped economy [1].

The introduction of input-output tables and other quantitative economic

models originated a profusion of research in many areas. For instance, Dantzig's

early work in the Air Force (before his development of the simplex algorithm

for linear programming) consisted of investigating methods to efficiently solve

large systems of linear equations, motivated by the applications to input-output

tables [1, 2]. However, the triangulation problem described next has not been given

much attention.

Triangulation. We have assumed so far the knowledge of a hierarchy

of sectors. Tr.17ttall;lrl..> n of an input-output table is the process of finding such

hierarchy among all possible orderings. It is clear from the discussion above

that such ordering most closely resembles an upper triangular matrix, and thus

has the maximum value of A. Note that every ordering is a permutation of the

sectors, and it is applied to both the rows and columns of the input-output matrix.

Additionally, the denominator of Equation 1.1 is constant for all permutations.

Therefore, we can state the te.:~,:l,,~r.: ad./.- problem as that of finding a permutation

of the rows and columns, such that the sum of the elements above the diagonal

is maximum. Clearly, this is equivalent to a combinatorial optimization problem,

known as the linear ordering problem (LOP).

Finding such permutation is not an easy task. In fact, the linear ordering

problem is an AP~P-hard problem, and as such we can only aspire to obtain

approximate solutions. Furthermore, the extent to which input-output methods are

useful depends on the efficiency of the computations. Limited by computational

power, practitioners often must recur to .I__-oegfation, with the consequent loss of

information and accuracy, to find optimal solutions within an acceptable time.

Therefore, it is essential in practice to search for quick approximate (heuristic)

algorithms .

In this thesis we propose a new algorithm based on a greedy randomized

adaptive search procedure (GRASP) to efficiently solve the LOP. The algorithm is

integrated with a path-relinkingf procedure and a new local search scheme.

The remainder of this work is organized as follows. In C'!s Ilter 2 we introduce

the LOP, give other applications, and discuss some previous work. In C'!s Ilter 3 we

give a detailed implementation of our algorithm, preceded by an introduction

describing the GRASP and path-relinking framework. The computational

experimentation is shown in C'! Ilpter 4. The thesis concludes with C'!s Ilter 5,

where some practical issues are discussed.

CHAPTER 2
LINEAR ORDERING PROBLEM

The LOP is an HiP-hard combinatorial optimization problem with a wide

range of applications in economics, archaeology, and scheduling. It has, however,

drawn little attention compared to other closely related problems such as the

quadratic assignment problem and the travelling salesman problem.

The LOP can be stated as follows. Consider a set NV of a objects and a

permutation xr : NV N Each permutation xr = (x (1), x (2), (u)) corresponds

one-to-one to a linear ordering of the objects. Let eij, i, j = 1, 2,. ., n, be the cost

of having i before j in the ordering, and E be the n-square matrix of costs. Then

the linear ordering problem is to find a permutation xr that maximizes the total

cost
n-1 n

i=1 j=i+1
Clearly, Equation 2.1 is the sum of the elements above the diagonal of a

matrix A, whose elements aij are those resulting from a permutation xr of the

rows and columns of matrix E-i.e., A = XEXT, where X is the permutation

matrix associated with the permutation xr [3]. In the context of its application in

economics, we can restate Equation 1.1 as

A = max {Z(x) } (2.2)
K ren

where II is the set of all permutations, and K is a positive constant representing

the sum of all the entries in the matrix.

The LOP can also be interpreted as a problem in graphs. Let G(NV, A) be a

complete directed graph with node set NV and arc set A = {(i, j) : i, 1 e A i / j}.

Let eij be the weight of arc (i, j). A spanning .l i-ll 1 tournament in G(~. indce a

unique linear ordering of the node set NV [4]. A tournament is defined as a directed

graph in which each pair of nodes is connected by exactly one arc, which is clearly

necessary since either i is before j or j is before i.

The complexity of the maximum LOP can be easily proven to be HiP-hard by

noticing that it is equivalent to the minimum weighted feedback arc set problem on

G, which is known to be HiP-hard [5].

The LOP has an interesting symmetry property. If a permutation xr =

("(1),wr(2), ..., x(u)) is an optimal solution to the maximization version, then

the reverse permutation xr = (xr(u),wr(n 1),...,xr(1)) is an optimal solution

to the minimization version. In fact, the LOP accepts a trivial ~-approximation

algorithm [4]. Let xr be an arbitrary permutation and Wr its reverse. It is easy to see

that Z(xr) + Z(Wr) is a constant. Ch.. .. -- & such that Z(ii) = max{Z(r), Z(Wr)}, then

we get
Z(r*) Z(-i) 1
Z(xr*) 2
where xr* is an optimal permutation and Z(xr*) > 0. No other approximation

algorithm exists [4]. It follows that any permutation is optimal in the unweighted
version of the LOP.

2.1 Applications

Next, we discuss a few applications of the LOP besides that in economics,

which are of particular relevance to the present volume. Reinelt [3] gives an

extensive survey.

Consider the problem of having a group of people rank n objects. Each

individual in the group is asked to express a preference with respect to every

possible pair. If we let eij be the number of people who preferred i to j, the

solution to the corresponding LOP is the ranking that most likely reflects the

preferences of the group.

A similar application can be found in the context of sports. For example,

consider a tournament of a teams in which every team ptil- against every other

team. Let eij be the score of the match between i and j if i wins, and 0 otherwise.

The ranking obtained by the LOP is considered to be the one that most closely

reflects the "true" performance of the teams. Still, it has not gained support for its

implementation, probably because the outcome of a particular match is not closely

related to the result in the ranking.

In archaeology, the LOP is used to determine the inn~-I pro' .I I!.

chronological ordering of a set of artifacts recovered from different sites. Samples

belonging to various time periods are given a value based on their distance to the

surface. The objective is to .I_ egate the data and determine an ordering of the

artifacts.

Finally, it is easy to see that the LOP can be used to determine the optimal

sequence of jobs in a single server, where the cost of each job depends upon its

position with respect to the entire schedule.

2.2 Problem Formulations

As with most combinatorial optimization problems, the linear ordering

problem has many alternative formulations. The LOP can be expressed as an

integer programming problem as follows. Let G(NV, A) be the complete directed

graph associated with the LOP as shown in the previous section. Define

1i if (i, j) E A'

otherwise

where A' CA is the arc set of' the- spanning ... i:-- lic toraen nG Then the

problem of findings the maximum of Equation 2.1 becomes

max ) eqxey, (2.3)
(i, j) EA

s.t. xij + xji = 1 ij Ni< (2.4)

xij + Zjk k xi < 2 i,,k Ni ,i kj k (2.5)

Xij E {0, 1}.

The constraints (Equations 2.4) define the tournament polytope. It can be

proven that the 3-dicycle inequalities (Equations 2.5) are sufficient to prevent

anly cycles [3]. Together the~y defineI the~ linear~ ordering polytope. There~ are~ 2 (
variables,,,, an ") + ')ln ~ontrai~nts in? this formula~ltion.

The tournament constraints (Equations 2.4) motivate the use of a single

variable to represent the two possible v- .--s in which every pair of nodes can be

connected. Let us substitute xji = 1 xij, for every i, je N ,j > i, then an

equivalent integer programming formulation is

s.t. xs j k<1Vj ,i
xij + Zjk xik > 0 i ,keN (2.8)

Xij E {0, 1}

whree = e; -- eji. Tis formIulaion~V hasI (") varIIiable andI 2(")/ contrintsIII.

Finally, the linear ordering problem can be formulated as a quadratic

assignment problem (QAP). The distance matrix of the QAP is the matrix of

weights E and the flow matrix F = { fgy is constructed as follows, fij = -1 if i < j

and fij = 0, otherwise [6].

2.3 Previous Work

2.3.1 Exact Methods

A common approach in the LOP literature is the use of cutting plane

algorithms [:3, 7]. The goal is to obtain an approximate description of the convex

hull of the solution set by introducing valid inequalities that are violated by current

fractional solutions, which are added to the set of inequalities of the current

linear programming problem. Reinelt [:3] introduced facets induced by subgraphs.

Bolotashvili et al. [8] extended Reinelt's results introducing a generalized method

to generate new facets. A complete characterization has only been obtained for

very small problems (n = 7) [9]. In fact, we know that unless P = HiP there exist

an exponential number of such facets. However, research in this area has resulted in

valid inequalities that improve the performance of branch-and-cut algorithms [:3].

The first authors to consider an interior point algorithm for the LOP were

Mitchell and Borchers [10, 11]. The solution given by a interior point algorithm is

used as a starting point for a simplex cutting plane algorithm [11].

2.3.2 Heuristic Methods

Most hard problems in combinatorial optimization require the use of heuristics

to obtain approximate solutions due to their inherent intractability. In this case, we

are interested in findings solutions that are close "enowl,h to the optimal value at a

low computational cost.

Heuristics, as opposed to approximation algorithms, do not give a guaranteed

quality of the obtained solutions. Nevertheless, the flexibility we have in developing

heuristics allows us to exploit the special structure of the problem, tailoring the

existing methods, and resulting in very well performing algorithms. The quality of

a heuristic, however, must he validated by extensive testing.

Heuristic methods such as GRASP. Labu .search. .simulated r,,:tr.:...lle genetic

.search. and evolution strategies have shown to be able to efficiently find high

quality solutions to many combinatorial and global optimization problems by

thoroughly exploring the solutions space. A recent survey on multi-start heuristic

algorithms for global optimization is given by Marti [12].

One of the earliest heuristics for the LOP was proposed by ('I!. in~ a y and

Watanabe [13] in the context of the triangulation of input-output tables. Given a

sector i, the ratio of total input to the total output

Ui = Ck(2.9)

is used to arrange the sectors in the order of decreasing ui. The use of Equation 2.9

gives a fairly good ordering considering its simplicity.

Based on the symmetry property mentioned in C'!s Ilter 2, ('1! I1. I-4 and

K~obylariski [14] developed a heuristic that performs a sequence of optimal

insertions and reversals.

Laguna et al. [15] developed an algorithm based on tabu search. They

analyzed several intensification and diversification techniques, and compared

their algorithm with that of ('I! I1. I-; and K~obylariski.

Campos et al. [16] used a scatter search approach. A correction term based on

the frequency by which an object i appears in a particular position in the ordering

is added to Equation 2.9 to reflect previous solutions.

GRASP, which is an iterative restart approach, has proven to be one of the

most effective heuristics to date. In this thesis, we developed a GRASP-based

algorithm for the LOP, offering a significant improvement on the computational

time and quality of solution compared to previous heuristics. In the next section,

we discuss the basic principles for implementing a new local search scheme and

Path-Relinkingf in GRASP framework.

CHAPTER 3
A GRASP WITH PATH-RELINK(ING ALGORITHM\

3.1 Introduction to GRASP and Path-Relinking

Since its inception by Feo and Resende in the late 1980s, GR ASP has been

successfully used in many applications. In 1995, the authors formally introduced

GR ASP as a framework for the development of new heuristics [17]. Festa et al. [18]

have compiled an extensive annotated bibliography of GR ASP applications.

Each iteration of GR ASP consists of two phases: a construction pherse in

which we seek to obtain a feasible solution, and a local .search phase that attempts

to improve the solution. Figure :31 shows the pseudo-code of a generic GR ASP

algorithm.

During the construction phase, we iteratively build a solution by randomly

selecting objects from a restricted candidate list (RCL). At each step, we form the

RCL choosing those objects with the highest measure of attractiveness, we select a

random object from the RCL, and adapt the greedy function to reflect the addition

to the solution. Figure :34 shows the construction phase for our implementation.

The size of the list is typically restricted in one of two Cw-i~; by quality,

when we choose the elements based on a threshold on the greedy function, or by

cardinality. In the literature, they are often referred to as a~ and /3, respectively.

The size of the RCL controls the degree of greediness and randomness of the

construction phase. A null R CL-i.e., of size 1-results in a purely greedy solution

whereas a RCL size equal to the size of the problem yields a purely random

solution.

After a solution is constructed, we attempt to improve it by performing a

local search in an appropriately defined neighborhood. Given a solution .r, we

Procedure GRASP(RCLSize, StoppingCondition)
1 BestSolutionFound = 0:;
2 while StoppingCondition not satisfied, do
:3 x = ConstructGreedyRandomizedSolution(RCLSize)
4 LocalSearch(x);
5 UpdateSolution (Best SolutionFound,x) ;
6 en d;
7 return BestSolutionFound;
en d;

Figure :31: Generic GR ASP pseudo-code

explore the neighborhood NV(.) aspiring to find a local (global) optimal solution.

Although larger neighborhoods increase the probability of finding the global

optimum, local search algorithms are often computationally expensive and thus

careful consideration must he given to the election of the neighborhood. It is in

this part of GR ASP where the particular properties of the problem in question can

he exploited to develop schemes that can provide an intensification of the search,

while not compromising its running time. Finally, the best solution is updated

if necessary with the newly found solution. The procedure is repeated until a

stopping condition is met-for example, number of iterations, running time, etc.

Path-relinking was introduced by Glover and Laguna [19] as a method to

integrate intensification and diversification to tabu search. It generates new

solutions by exploring routes that connect high-quality solutions by starting

from one of these solutions, so-called initiating .solution. and generating a path

in the neighborhood space that leads toward the other solutions, a so-called

guiding .solution (Figure :32). This is completed by selecting moves that introduce

attributes contained in the guiding solutions.

Path-relinking is a directly-focused instance of a strategy that seeks to fit

in features of high quality solutions. On the other hand, instead of using an

incentive that supports the inclusion of such attributes, the path relinkingf approach

subordinates all other considerations to the goal of choosing moves that initiate the

o

Solution

Figure 3-2: Visualization of a path-relinking procedure. Given the initial solution
x and the guiding solution g, we iteratively explore the trajectory linking these
solutions, checking for any improvements on the way.

attributes of the guiding solutions, in order to generate a good attribute composition

in the current solution. The composition at each step is determined by choosing the

best move, using customary choice criteria, from the restricted set of moves that

incorporate a maximum number of the attributes of guiding solutions.

Laguna and Marti [20] were the first to combine GRASP with a path-relinking

procedure, essentially adding memory to a procedure that would otherwise be a

multi-start algorithm.

3.2 Proposed Algorithm

In this section we propose a new heuristic that integrates GRASP with path-

relinking for the linear ordering problem. Figure 3-3 shows the pseudo-code for the

implementation.

The measure of attractiveness for each object i consists of the difference

between its row and column sums, given by

ds= (s y) i 12 ..n (3.1)
j=1

Procedure GRASP(P, p, Maxlteration)
01 BestSolutionFound = 0);
02 EliteList = 0);
03 nNonImprovingIt = 0;
04 for k = 1,2,... ,Maxlteration
05 x = ConstructGreedyRandomizedSolution(P);
06 LocalSearch(x);
07 if x is better than worse solution in EliteList
09 nNonImprovingfIt = 0;
10 else if nNonImprovingIt > y
11 ExpandEliteList(El; 1i. L-1 ,);
13 nNonImprovingIt = 0;
14 else nNonImprovingIt = nNonImprovingIt + 1;
13 Up date Solution(Best SolutionFound,x) ;
14 end;
15 return BestSolutionFound;
en d;

Figure 3-3: GRASP pseudo-code

In the context of the input-output tables, Equation 3.1 represents the net flow

of a sector. In earlier experimentation with a GRASP algorithm, we compared

the use of Equation 3.1 with the ratios as defined in Equation 2.9. Although we

have not observed significant differences in the quality of the solutions, adapting

the greedy function when ratios are used is computationally more expensive. The

use of Equation 2.9 demands O(n2) time tO update the greedy function whereas

Equation 3.1 requires O(u) time.

Linear ordering problems usually have many alternative solutions-optimal

and suboptimal--with the same objective function value. Therefore, it may occur

at some point in the algorithm that the elite list becomes mostly populated by

alternative solutions. Furthermore, it is increasingly difficult to enter a path-

relinkingf as the best solution found approaches the optimal. We attempt to avoid

such situations by expanding the size of the elite list and forcing a path-relinking

procedure after a certain number of non-improving iterations. If an improvement is

procedure ConstructGreedyRandomizedSolution(P)
1 Solution = 0, RCL = 0;
2 while |Solution| < NV
3 MakeRCL(RCL,P);
4 s = Select Element At Random( RC L);
5 Insert(Solution,s);
7 en d;
en d;

Figure 3-4: GRASP construction phase pseudo-code

still not obtained and the elite list size reaches its limit, the elite list is deleted and

a new one is constructed. Next we discuss in detail the different components of the

algorithm.

3.2.1 Construction Phase

We initiate the procedure by creating the restricted candidate list (RCL)

(Figure 3-4). The parameter P of GRASP determines the cardinality limit on

the RCL--i.e., the number of elements in RCL. Larger values of P achieve greater

diversity but at the cost of constructing many lower-quality solutions. The best

value of is usually determined by extensive testing.

After selecting an element at random from the RCL, we proceed to insert

it in the partial solution. A conventional GRASP implementation would simply

append the recently selected object a to the end of the partial solution. Instead,

we added a procedure named lI!- : t" (line 5 on Figure 3-4) that seeks to insert

the object in an optimal position. More precisely, let Tk, 1 t, 2, /k), k =

1,2,..., n, denote the current (partial) solution obtained after k steps. The Insert

operation intercalates the most recently selected object s in Tk, in the position r

that maximizes
r-1

breaking ties arbitrarily. First introduced by C'I I1. I-4 and K~obylariski [14] as part of

their heuristic, it can be considered as a very efficient local search procedure in a

relatively small neighborhood. In fact, it can be implemented in O(k).

A step of the construction phase finalizes with the task of adapting the greedy

function. The row and column corresponding to the object s are removed from the

matrix, and the attractiveness of the objects (Equation 3.1) is updated. We set

d, = -M~, where M~ is a large positive value, and re-sort the top n k objects that

have yet to be selected. The procedure continues until a solution is constructed.

The overall complexity of the construction phase is O(n2)

3.2.2 Local Search

We used a 2-exchange neighborhood for our local search. Given a solution

xr, its 2-exchange neighborhood NV(x) consists of all the solutions obtained by

permuting the position of two objects in the ordering--i.e., if xr = (3, 1, 2), then

N 1) {1,/3, 2), (2 13)(3, 2,1 ~ ,1)}. Clearly frapolem of size, |Nx) =i,\ .

Consider a solution xr and two objects xr(i) and xr(j) located in positions i and j

respectively. For simplicity assume that i < j. The change in the objective function

for an exchange of objects xr(i) and xr(j) is

j-1

k=i+1

where egj = eij eji. At completion, the local search would have exchanged

the pair of objects that maximizes Equation 3.3. The procedure of exploring the

neighborhood and performing the exchange can be implemented in O(n2)

The solution provided by the local search procedure is used as the initial

solution for the path-relinking. We randomly select a solution from the elite list as

the guiding solution, determining the trajectory to be followed by the procedure.

1 TenipSolution = 0:;
2 g = Select SolutionAt Randon1(EliteList);
:3 while .r / g do
4 TenipSolution = MakeNextMove(.r, g);
5 LocalSearch (TenipSolution);
6 if TenipSolution is better than .r or y then
7 EliteList = EliteList U TenipSolution;
8 en d;
9 Adj ustEliteList (Elite LM- 1 ) ;
en d;

Figure :35: GR ASP path-relinkingf pseudo-code

Figure :35 shows the pseudo-code for the path relinking procedure. The parameter

p determines the size of the elite list as a fraction of the problem size.

With the trajectory defined by the two end solutions, we proceed to perform

a series of moves that will transform the initial solution into the guiding solution.

In each iteration the algorithm performs a single move, thus creating a sequence

of intermediate solutions (Figure :32). To add intensification to the process, we

search the 2-exchange neighborhood of the intermediate solutions. The solutions

obtained in this manner are added to the elite list if they are better than either the

initial or the guiding solutions.

It should be noted that the search on the 2-exchange neighborhood may yield

a previously examined solution in the path. However, this is not of concern in our

intplenientation since we do not use the local nxininia during the procedure.

The moving process terminates when the algorithm reaches the guiding

solution. At this point, the elite list size could have grown considerably due to the

added solutions. The procedure "AdjustEliteList" will discard the worst solutions,

keeping the best pn. The list is kept sorted at all times and therefore no sorting is

needed. The complexity of the path-relinking is O(n ).

CHAPTER 4
COMPUTATIONAL RESULTS

Here we discuss the computational results we obtained when applying our

algorithm to two sets of problems:

LOLIB. These are real-world instances of linear ordering problems that are
publicly available on the internet [21]. They consist of 49 input-output tables
for some European countries, with sizes up to 60 objects.

Mitchell. This is a set of 30 randomly-generated instances by Mitchell [10],
with sizes ranging from 100 to 250 objects. Three different percentages of
zero entries were used: 0, 10, and 211' denoted by the last digit on the
problem name. The generator as well as the instances are available at the
author's web site [22].

The optimal values for all instances are known. The generated instances

are similar to those from LOLIB except for the numerical range of the entries-

considerably larger for the latter. Despite attempts to replicate the characteristics

of real-world instances such as those found in LOLIB, Mitchell's test set is

significantly harder to solve. All previous work on the linear ordering problem

that included computational results predates the Mitchell instances, hence featuring

only the LOLIB problems.

The algorithm was written in C++ and executed on a Pentium 4, 2.7 GHz,

with 512 MB of memory. Empirically, we determined /3 = 0.25 and p = 0.35 as

the best values for the parameters (e.g., for a problem of size n, the size of the

RCL and the elite list are at most 0.25n and 0.:-+, respectively). The algorithm

was executed five times for each problem instance, with a limit of 5000 GR ASP

iterations on its running time. We report the running time, number of iterations,

and the gap between the best solution and the optimal solution. All times are

reported in seconds and gaps as percentages.

Tables 4-1 and 4-2 show the elapsed running time and gap values after 200

and 5000 iterations. The results reported are the averages of 5 runs for each

problem instance. A summary of the results is presented in Table 4-3. Figures 4-1

and 4-2 show the evolution of the gap as a function of the number of iterations and

the running time for the LOLIB instances, respectively. Corresponding results for

the Mitchell instances are shown in Figures 4-3 and 4-4. Note that the units on

the ordinates are percentage points.

The algorithm found optimal solutions for 47 of 49 LOLIB instances, 17 of

which were consistently solved to optimality. The average gap for the remaining

LOLIB instances was 0.01";' The average running time for these real-world

instances was 0.87 seconds. Although none of the Mitchell instances were solved

to optimality, the average gap after 5000 iterations was 0.017 :' with an average

running time of 21.98 seconds.

T~Iabl: 4-1: Results fo~r thle LOIIB TIlnstaces: alfl~ter c200 d ..:::1- iteraionlCI s.

Ins;a~nce Size 20 train ::: : Iterations
G:alp( ) TIime (s) C:a~pi )Trn(s
be75eec 50 0.0199 0.0282 0.0042 0.4062
bue75np 50 0.0057 0.0188 0.0018! 1.0406
be750oi 50) 0.04359 0.0218 0.0125, 0.706i2
be75tot 50) 0).0056i 01.0218 0.00012 0).j515
stab-ul 60 0.0124 0.0406 0.0024 1.3030)
: 60 0.0059 0.0468 1- : 'L 21624
stabu3il 60i 0.0329 i: :: -- 0.0090)i 1.9280)
1 1xx 44 0.00 12 0.15 .0012
(,59d11Ixx 44 0.0082 0.0156i 0.000)0 0.5720)
44 1:: ** 0.0124 0.0000 0.6562
t59ill~xx 414 0.0000 0.0156 0.0000 0.76i86
t59n11xx 414 0.0515 0.0186i 0.00)79 0.6592
t65b11lxx 44 0.23 .02 0.0133 03.76324
t65d11lxx 441 0.000:3 0.160.0000 0).5876
44 0.0090 0.0156 0.0006 0.5842
44 0.0137 0.0156 0.000)7 0.5938
44 0.0000 0.16 0.0000 0.8032
44 0.0124 0.0156 0.000)0 0.4624
11xx 44 0.0014 0.0156 0.0000 0.r5594
t69r 1xx 44 0.0122 0).0156 i : ::: : 0.6844
t70b1~,1 xx 44 0.0080 0.01234 0.0080 0.72380
t70d11lxn 442 0).0288 01.0750) 0.0073' 2?.7406~l
t70dii11xx 441 0).0232 05 0.012,0 0).4938
ti::: : 44 0.0030 0.0126 0.0012 0.4282
t70~illxx 44- 0.0022 0.0126i 0.00019 0.4970)
t70k11lxx 441 0).100001 0.0156 0.0000)( 0.7218
170)111xx 44 1* ***** 0.000)0 0.6124
t70n 11xx 44 1:: 0.0126 0.0019 0.3970
t70u.11xx 44 i::::: 0).0126 0.0000 0.7938
t70wl 1xx 44 0.00 12 0.160.0000 0.3874
44 0.0000 0.0 0.0000 0.6906
t74dii11xx 441 0).0125 0.0)188 0.0042% 0.5032
t75d11xxr 44 0.0028! 0.02>18 0.0020 0.7936
t73e11xx 44- 0.00418 0.0126i 0.00010 0.4688
t75ilxx 44 0.07 .02 0.0000( 0).6i220
t75k11xx 441 0.0000 005 0.0000 0).7156
t75n11xxr 44 0.0'710 0.0156 0.0000 0.5062
t75ull1xx 44 0.0007 0).0156 :
tiwi56n54, 56 0.01 70 0.0374 0.04 .8436
Liw5n5856 0.0127 0.0312 0.000)0 1.0124
tiwov6n62 56 0.0187 : 0.0007 1.0970
tiwb6n66 56 0.0179 0).0342 : ::: : I: : 8
ti --- 56i 0.006i3 0.0344- 0.0006 2.0:312
tiw56in72 56i 0).0078 01.0376 .01 1 -
tiw5F6r54i 56 0).0157 001 0.002,0 0).7062
tiw56r58 56 0.0046: 0.0438 0.0020 1.>72,0
tiw56ir66 56i 0.0:333 0.0406- 0.0061- 0.8688
tiw56r67 563 0.0086 0.0312 0).10007 1 *
tiw56ir72 56i 0.0006 0.0312 0.000)6

Ta~ble 4-2: Results fo~r the M~i i: i In~stances after : an~d 5000 iteration s.

Instance 200 Itera~tions

5000 Iterations
Cap( ) 'Time (s)
**** i. 2.5374
0.0207 2.6032
0.0302 2.6376
0.0343 2.4906
0.0165 6.9312
0.0070 8.0562
0.0226 8.3282
0.0102 8.1970
0.0194 9.I000
0.0167 27 -
0.0191 9.00323
0.0122 :
0.019)1 1 6218
0.0056 33.1438
0.0289 9.5812
0.0080 2
0.0305 22.406i2
0.0151 21.4780
0.27 3.6i062
0.0084 20.4376
0.0219 21.5844

0.0262%
0.01 2
0.0244 35.9750
0.0156 47.1220
0.0102 43.5406
0.0132 52.2126
0.0118 49.1438
0.0146 48.9250

LO)L li i and Mitchell test sets after

Gapt )

0.0462
0.0416
0.0:389
0).0270
0).0137
0.0263
0.0230
0.0'224

0.0146
0.02312
0).0148
0).03X88
0.0181
0.04019
0).0196
0).0333

0.0244
0.0159

0.0297
0.0233
0.0189
0).0159
0.0184
0.0241

Thine (s)
0.2374
0.2L594
0.3094
0.2220

0.7126
0.9720
1.3230
1.2844
i: --
1.1970
1.0874

2.6062
2.8906

3.65r32

'2.7312
'2.7720

2.3'780
5.4750
3.8188

8.3'750
5.4844

r100a2
rl300b2
cl100c2
r100d2
r100e2
rl50~a0

rl5Ob0
rI 1
rl1 **
rl50ct
r S~l50d
r lI ==
rl50e0
rl5i0el
r200a0
r200al
r200b0 ~
r200b1
r2L00c0
r200~e1
r20010
r200di 1
r2L00e0
r200el
r250a0
r250b0 l
r25l0c0
r250d0

Table 4 3: Sur~nmary of results obtained flor the
200 andi 5000 iterations.

M~eas~re

Average
Dpv.
Maximum
Average
Dev.
M\1aximum r

5000 Iterations
G~apj ) Timet (s)
0.0024 .8685
0.0035 0.5188
0.0133 2.7406
0.0173 21.9791
0.0081 15.6266
0.0343 52 9126

20 i:

0.0710
0.0254
0).0096

nations
Timet (s)
0.023'5
0.01:31
0.0750
2.401;55

10.1312

U\LOLIB

0 .10

0.08

0.06

0.04

0.02

1000 2000 3000 4000

5000

Figure 4-1: The LOLIB Instances: gap from optimal solution as a percentage is
shown as a function of the number of iterations.

S. 10

0 .08

0 .06

0 .04

0 .02

1 2 3 4 5

Figure 4-2: The LOLIB Instances: gap from optimal solution as a percentage is
shown as a function of the time elapsed (seconds).

0 .10

0.08

0.06

0.04

1000

2000

3000

4000

5000

Figure 4-3: Mitchell Instances: gap from optimal solution as a percentage is shown
as a function of the number of iterations.

S. 10

0.08

0.06

Figure 4-4: Mitchell Instances: gap from optimal solution as a percentage is shown
as a function of the time elapsed (seconds).

0.04

0.02

CHAPTER 5
CONCLUDING REMARKS

Our study implemented a new heuristic algorithm for the linear ordering

problem, inspired by its application to the triangulation of input-output tables

in economics. The algorithm is based on a greedy randomized adaptive search

procedure (GRASP) with the addition of a path-relinking phase to further intensify

the search.

The algorithm was tested on two sets of problems, exhibiting a remarkably

robust performance as shown in Table 4-3. For all instances we obtained optimality

gaps of less than 0.05' within 200 iterations and times ranging from 0.02 to

2.40 seconds on the average. We found optimal solutions for most of the LOLIB

instances. No optimal solutions were obtained for the Mitchell instances, however,

the average gap at termination for this set was 0.01'7:' The results confirm the

benefit of embedding GRASP with a path-relinkingf procedure.

Researchers in economics often use simulations in which many triangulation

problems need to be solved in limited time. The efficiency and high-quality

performance of our algorithm makes it a superior candidate for such application.

Furthermore, since the algorithm is based upon the modelling of the triangulation

problem as a linear ordering problem (LOP), it can be used for any application

that accepts an LOP formulation such as those mentioned in Section 2.1.

We believe that further research is required for analyzing the relation of the

LOP with other known problems. In particular, could the LOP be thought of as a

travelling salesman problem where, in addition to the distances between locations,

we have priorities assigned each of these? Could we use the extensive research done

in the TSP as a basis for the LOP? Answers to these and other related questions

will be pursued in future research.

REFERENCES

I1 Wassily Leontief, Input-Output Economics, Oxford University Press, New
York, USA, 1986.

121 George B. Dantzig, "Linear programming," Op~erations Research, vol. 50, no.
1, pp. 42-47, January 2002.

13 Gerhard Reinelt, The linear ordering problem: rlly.>rithm~s and applications,
Number 8 in Research and Exposition in Mathematics. Heldermann \: ~11 I.
Berlin, Germany, 1985.

14] Michael Jiinger, Polyhedral C'ombinatoric~s and the Al to II.:- S"Io.ll..1/9'-r
Problem, Number 7 in Research and Exposition in Mathematics. Heldermann
\: 11 .-- Berlin, 1985.

[3 Michael R. Garey and David S. Johnson, C'omputer~s and Intracterbility: A
Guide to the Ti,*** *U of NP-G'omp~leteness, W.H. Freeman and Co., New York,
USA, 1979.

::jR.E. Burkard, E. Cela, P.M. Pardalos, and L.S. Pitsoulis, "The quadratic
assignment problem," in Handbook of C'ombinatorial Op~timization, P.M.
Pardalos and D.-Z. Du, Eds., pp. 241-338. K~luwer Academic Publishers, 1998.

17 Martin Groitschel, Michael Jiinger, and Gerhard Reinelt, "A cutting plane
algorithm for the linear ordering problem," Op~erations Research, vol. 2, no. 6,
pp. 1195-1220, 1984.

18] G. Bolotashvili, 31. K~ovalev, and E. Girlich, N. i.-- facets of the linear ordering
polytope," SIAM~ Journal on Discrete M~athematics, vol. 12, no. 3, pp.
326-336, 1999.

L: Thomas ('!!itI&*! Low-Dimensional 0/1-Polytopes and Branch-and-G'ut in
Combinatorial Op~timization, Ph.D. dissertation, University of Heidelberg,
Heidelberg, Germany, 1997.

[10] John E. Mitchell, "Computational experience with an interior point cutting
plane algorithm," Tech. Rep., Mathematical Sciences, Rensellaer Polytechnic
Intitute, Troy, NY 12180-3590, USA, 1997.

[11] John E. Mitchell and Brian Borchers, "Solving linear ordering problems
with a combined interior point/simplex cutting plane algorithm," in High
Performance Op~timization, H. Fr-enk et al., Ed., chapter 14, pp. 345-366.
K~luwer Academic Publishers, Dordrecht, The Netherlands, 2000.

[12] Rafael Marti, jl!Illi:-start methods," in Handbook of Ietakeuristics, Fred
Glover and Gary A. K~ochenberger, Eds., International Series in Operations
Research & Management Sciences, chapter 12, pp. 355-368. K~luwer Academic
Publishers, Dordrecht, The Netherlands, 2003.

[132] Hollis B. C'I..! 31. and Tsunehiko Watanabe, 11.1 ~ ii !. 1..i. i comparisons of the
structure of production," Econometrica, vol. 26, no. 4, pp. 487-521, October
1958.

[14] Stefan C'I I.. s- and P1.. In!i--1i.x-- K~obylariski, "A new heuristic algorithm
solving the linear ordering problem," Comp~utational Op~timization and
Applications, vol. 6, pp. 191-205, 1996.

[157] Manuel Laguna, Rafael Marti, and Vicente Campos, ls.!i..01 n
diversification with elite tabu search solutions for the linear ordering problem,"
Computers 8I Op~erations Research, vol. 26, pp. 1217-1230, 1999.

[16] Vicente Campos, Fred Glover, Manuel Laguna, and Rafael Marti, "An
experimental evaluation of a scatter search for the linear ordering problem,"
Journal of Global Op~timization, vol. 21, no. 4, pp. 397-414, December 2001.

[17] Thomas A. Feo and Mauricio G.C. Resende, "Greedy randomized adaptive
search procedures," Journal of Global Op~timization, vol. 2, pp. 1-27, 1995.

[18] Paola Festa, Mauricio G.C. Resende, and Geraldo Veiga, "Annotated
bibliography of GRASP," AT& T Research Labs, Jan. 2004,
http: //www.research.att.com/~mgcr/grasp/annotatd

[19)] Fred Glover and Manuel Laguna, Tabu Search, K~luwer Academic Publishers,
Boston, USA, 1997.

1: Manuel Laguna and Rafael Marti, "GRASP and path relinking for 2-1 .v. r
straight line crossing minimization," INFORM~S Journal on Companyl.::l vol.
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[21] Gerhard Reinelt, "Linear ordering library (LOLIB)," University of Heidelberg,
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http: //www.iwr.uni-heidelberg. de/groups/comopt/software/LOLIB/.

[22] John E. Mitchell, "Generating linear ordering problems," Rensellaer Polytech-
nic Intitute, Dec. 2002, http://www.rpi .edu/~-mitchj /generators/linord.

BIOGRAPHICAL SKETCH

Bruno H. Chiarini was born on December 21, 1974, in Buenos Aires,

Argentina. In 1996 he was admitted to the Instituto Tecnologico de Buenos Aires

(ITBA). He transferred to the University of Florida in January 2001, where he was

awarded a Bachelor of Science degree in Industrial and Systems Engineering in

August 2002. He then pursued graduate studies at the same institution, obtaining

a Master of Science degree in Industrial and Systems Engineering in May 2004.