Introduction to Linear Goal Programmingzalamsyah.staff.unja.ac.id/wp-content/uploads/... · me to...
Transcript of Introduction to Linear Goal Programmingzalamsyah.staff.unja.ac.id/wp-content/uploads/... · me to...
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title:
IntroductiontoLinearGoalProgrammingSageUniversityPapersSeries.QuantitativeApplicationsintheSocialSciences;No.07-056
author: Ignizio,JamesP.publisher: SagePublications,Inc.
isbn10|asin: 0803925646printisbn13: 9780803925649ebookisbn13: 9780585216928
language: Englishsubject Linearprogramming.
publicationdate: 1985lcc: T57.74.I351985ebddc: 519.7/2
subject: Linearprogramming.
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IntroductiontoLinearGoalProgramming
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SAGEUNIVERSITYPAPERS
Series:QuantitativeApplicationsintheSocialSciences
SeriesEditor:MichaelS.Lewis-Beck,UniversityofIowa
EditorialConsultants
RichardA.Berk,Sociology,UniversityofCalifornia,LosAngelesWilliamD.Berry,PoliticalScience,FloridaStateUniversity
KennethA.Bollen,Sociology,UniversityofNorthCarolina,ChapelHill
LindaB.Bourque,PublicHealth,UniversityofCalifornia,LosAngeles
JacquesA.Hagenaars,SocialSciences,TilburgUniversitySallyJackson,Communications,UniversityofArizona
RichardM.Jaeger,Education,UniversityofNorthCarolina,Greensboro
GaryKing,DepartmentofGovernment,HarvardUniversityRogerE.Kirk,Psychology,BaylorUniversity
HelenaChmuraKraemer,PsychiatryandBehavioralSciences,StanfordUniversity
PeterMarsden,Sociology,HarvardUniversityHelmutNorpoth,PoliticalScience,SUNY,StonyBrook
FrankL.Schmidt,ManagementandOrganization,UniversityofIowaHerbertWeisberg,PoliticalScience,TheOhioStateUniversity
Publisher
SaraMillerMcCune,SagePublications,Inc.
INSTRUCTIONSTOPOTENTIALCONTRIBUTORS
Forguidelinesonsubmissionofamonographproposaltothisseries,pleasewrite
MichaelS.Lewis-Beck,Editor
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SageQASSSeriesDepartmentofPoliticalScience
UniversityofIowaIowaCity,IA52242
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Series/Number07-056
IntroductiontoLinearGoalProgramming
JamesP.Ignizio
PennsylvaniaStateUniversity
SAGEPUBLICATIONSTheInternationalProfessionalPublishers
NewburyParkLondonNewDelhi
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Copyright©1985bySagePublications,Inc.
PrintedintheUnitedStatesofAmerica
Allrightsreserved.Nopartofthisbookmaybereproducedorutilizedinanyformorbyanymeans,electronicormechanical,includingphotocopying,recording,orbyanyinformationstorageandretrievalsystem,withoutpermissioninwritingfromthepublisher.
Forinformationaddress:
SAGEPublications,Inc.2455TellerRoadNewburyPark,California91320E-mail:[email protected]
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LibraryofCongressCatalogCardNo.85-072574
97989900010203111098765
Whencitingaprofessionalpaper,pleaseusetheproperform.RemembertocitethecorrectSageUniversityPaperseriestitleandincludethepapernumber.Oneofthetwofollowingformatscanbe
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adapted(dependingonthestylemanualused):
(1)IVERSEN,GUDMUNDR.andNORPOTH,HELMUT(1976)"AnalysisofVariance."SageUniversityPaperseriesonQuantitativeApplicationsintheSocialSciences,07-001.BeverlyHills:SagePublications.
OR
(2)Iversen,GudmundR.andNorpoth,Helmut.1976.AnalysisofVariance.SageUniversityPaperseriesonQuantitativeApplicationsintheSocialSciences,seriesno.07-001.BeverlyHills:SagePublications.
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Contents
SeriesEditor'sIntroduction 5
Acknowledgments 7
1.Introduction 9
Purpose 9
WhatIsGoalProgramming? 10
OntheUseofMatrixNotation 10
2.HistoryandApplications 11
3.DevelopmentoftheLGPModel 15
Notation 16
TheBaselineModel 17
Terminology 18
AdditionalExamples 21
ConversionProcess:LinearProgramming 21
LGPConversionProcedure:PhaseOne 23
LGPConversionProcess:PhaseTwo 25
AnIllustration 26
GoodandPoorModelingPractices 30
4.AnAlgorithmforSolution 32
TheTransformedModel 33
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BasicFeasibleSolution 35
AssociatedConditions 36
AlgorithmforSolution:ANarrativeDescription 38
TheRevisedMultiphaseSimplexAlgorithm 39
ThePivotingProcedureinLGP 41
5.AlgorithmIllustration 43
TheTableau 44
StepsofSolutionProcedure 46
ListingtheResults 54
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AdditionalTableauInformation 55
SomeComputationalConsiderations 57
BoundedVariables 59
SolutionofLPandMinsumLGPModels 62
6.DualityandSensitivityAnalysis 63
FormulationoftheMultidimensionalDual 64
ANumericalExample 66
InterpretationoftheDualVariables 68
SolvingtheMultidimensionalDual 69
ASpecialMDDSimplexAlgorithm 72
DiscreteSensitivityAnalysis 75
ParametricLGP 77
7.Extensions 81
IntegerGP 81
NonlinearGP 87
InteractiveGP 89
Notes 91
References 91
AbouttheAuthor 96
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SeriesEditor'sIntroductionAsthisseriesofvolumesinquantitativeapplicationshasgrown,wehavebeguntoreachouttothoseineconomicsandbusinessinthesamewaythatsomeoftheearliervolumesappealedmostespeciallytothoseinpoliticalscience,sociology,andpsychology.Ourgoal,however,continuestobethesame:topublishreadable,up-to-dateintroductionstoquantitativemethodologyanditsapplicationtosubstantiveproblems.
Oneofthefastest-growingareaswithinthefieldsofoperationsresearchandmanagementscience,intermsofbothinterestaswellasactualimplementation,isthemethodologyknownasgoalprogramming.Fromitsinceptionintheearly1950s,thistoolhasrapidlyevolvedintoonethatnowencompassesnearlyallclassesofmultipleobjectiveprogrammingmodels.Ofcourse,ithasalsoundergoneasignificantevolutionduringthattime.
InAnIntroductiontoLinearGoalProgramming,JamesIgnizio(apioneerandmajorcontributortothefield,whosefirstapplicationofgoalprogrammingwasin1962inthedeploymentoftheantennasystemfortheSaturn/Apollomoonlandingmission)providesaconcise,lucid,andcurrentoverviewof(a)thelineargoalprogrammingmodel,(b)acomputationallyefficientalgorithmforsolution,(c)dualityandsensitivityanalysis,and(d)extensionsofthemethodologytointegeraswellasnonlinearmodels.Toaccomplishthisextentofcoverageinashortmonograph,Igniziousesamatrix-basedpresentation,aformatthatnotonlypermitsaconciseoverviewbutonethatisalsomostcompatiblewiththemannerinwhichreal-worldmathematicalprogrammmingproblemsaresolved.
Thetextisintendedforindividualsinthefieldsofoperations
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research,managementscience,industrialandsystemsengineering,computerscience,andappliedmathematicswhowishtobecomefamiliarwithlineargoalprogramminginitsmostrecentform.Prerequisitesforthe
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textarelimitedtosomebackgroundinlinearalgebraandknowledgeofthemoreelementaryoperationsinmatricesandvectors.
RICHARDG.NIEMISERIESCO-EDITOR
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AcknowledgmentsDuringmorethantwodecadesofresearchinandapplicationsofgoalprogramming,Ihavebeeninfluenced,motivated,andguidedbytheworksandwordsofnumerousindividuals.Severaloftheseindividuals,inparticular,havehadamajorimpact.TheseincludeAbrahamCharnesandWilliamCooper,theoriginatorsoftheconceptofgoalprogramming;VeikkoJääskeläinen,anindividualwhosesubstantialimpactonthepresent-daypopularityofgoalprogramminghasbeenalmosttotallyoverlooked;andPaulHuss,whointroducedmetogoalprogramming,influencedmydevelopmentofthefirstnonlineargoalprogrammingalgorithmandapplication(in1962),andwasmyco-developerofthefirstlarge-scalelineargoalprogrammingcode(in1967).Ialsowishtoacknowledgetheinfluenceofthetext,AdvancedLinearProgramming(McGraw-Hill,1981)byBruceMurtagh.Murtagh'soutstandingtextanditsconciseyetlucidstylehavehadparticularinfluenceonthepresentationfoundinChapter4ofthiswork.Finally,particularthanksaregiventoTomCavalierandLauraIgnizioforcommentsandcontributionstotheoriginaldraftofthismanuscript.
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1.IntroductionAlthoughgoalprogramming(GP)isitselfadevelopmentofthe1950s,ithasonlybeensincethemid-1970sthatGPhasfinallyreceivedtrulysubstantialandwidespreadattention.MuchofthereasonforsuchinterestisduetoGP'sdemonstratedabilitytoserveasanefficientandeffectivetoolforthemodeling,solution,andanalysisofmathematicalmodelsthatinvolvemultipleandconflictinggoalsandobjectivesthetypeofmodelsthatmostnaturallyrepresentreal-worldproblems.YetanotherreasonfortheinterestinGPisaresultofagrowingrecognitionthatconventional(i.e.,singleobjective)mathematicalprogrammingmethods(e.g.,linearprogramming)donotalwaysprovidereasonableanswers,nordotheytypicallyleadtoatrueunderstandingofandinsightintotheactualproblem.
Purpose
Itisthenthepurposeofthismonographtoprovideforthereaderabriefbutreasonablycomprehensiveintroductiontothemultiobjectivemathematicalprogrammingtechniqueknownasgoalprogramming,withspecificfocusontheuseofsuchanapproachindealingwithlinearsystems.Further,inprovidingsuchanintroduction,weshallattempttominimizeboththeamountandlevelofsophisticationoftheassociatedmathematics.Assuch,theonlyprerequisiteforthereaderissomeexposuretolinearalgebraandaknowledgeofthemoreelementaryoperationsonmatricesandvectors.Itshouldbeemphasizedthatafamiliaritywithlinearprogramminghasnotbeenassumed,althoughit
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isbelievedlikelythatmostreaderswillhavehadsomepreviousworkinthatarea.Ithasbeenmyattempttoprovideabriefandconcise,butreasonablyrigoroustreatmentoflineargoalprogramming.
WhatIsGoalProgramming?
Atthispoint,letuspauseandreflectuponsomeofthenotionsexpressedabove,inconjunctionwithafewnewideas.First,letusnotethatgoalprogramminghas,initself,nothingtodowithcomputerprogramming(e.g.,FORTRAN,Pascal,LISP,BASIC).Thatis,althoughanyGPproblemofmeaningfulsizewouldcertainlybesolvedonthecomputer,thenotionof''programming"inGP(or,forthatmatter,inthewholeofmathematicalprogramming)isassociatedwiththedevelopmentofsolutions,or"programs,"foraspecificproblem.Thus,thename"goalprogramming"isusedtoindicatethatweseektofindthe(optimal)program(i.e.,setofpoliciesthataretobeimplemented)foramathematicalmodelthatiscomposedsolelyofgoals.Lineargoalprogramming,orLGP,inturnisusedtodescribethemethodologyemployedtofindtheprogramforamodelconsistingsolelyoflineargoals.
WeshallwaituntilChapter3torigorouslydefinethenotionofa"goal."Here,wesimplynotethatanymathematicalprogrammingmodelmayfindanalternaterepresentationviaGP.Further,notonlydoesGPprovideanalternativerepresentation,italsooftenprovidesarepresentationthatisfarmoreeffectiveincapturingthenatureofreal-worldproblemsproblemsthatinvolvemultipleandconflictinggoalsandobjectives.
Finally,wenotethatconventional(i.e.,singleobjective)mathematicalprogrammingmaybeeasilyandeffectivelytreatedasasubset,orspecialclass,ofGP.Forexample,asweshallsee,linearprogrammingmodelsareeasilyandconvenientlytreatedas"GP"models.Infact,
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andalthoughtheideaisconsideredradicalbythetraditionalists,itisnotreallynecessarytostudylinearprogramming(LP)ifonehasathoroughbackgroundinLGP.
OntheUseofMatrixNotation
Asmentionedearlier,oneoftheprerequisitesofthistextisthatthereaderhashadsomepreviousexposuretomatricesandvectors,andtheassociatednotation,terminology,andbasicoperationsemployedinsuchareas.Althoughatfirstglancethematrix-basedapproachusedhereinmayappearabitformidabletosomeofthereaders,beassured
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thatitspurposeisnottocomplicatetheissue.Instead,bymeansofsuchanapproachweareableto:
(1)provideapresentationthatistypicallyclearer,moreconcise,andlessambiguousthanifanonmatrix-basedapproachwereemployed;and
(2)providealgorithmsinaformfarclosertothatactuallyemployedindevelopingefficientcomputerizedalgorithms.
Ofparticularimportanceistheconcisenessprovidedviaamatrix-basedapproach.Usingmatricesandmatrixnotationweareable,inthisslimvolume,tostillcovernearlyalloftheusefulfeaturesoflineargoalprogramming(e.g.,areasonablycomputationallyefficientversionofanalgorithmforlineargoalprogramming,acomprehensivepresentationofduality,anintroductiontosensitivityanalysis,andevendiscussionsofvariousextensionsofthemethodology).Withouttheuseofthematrix-basedapproach,therewouldhavebeennopossibilityofcoveringthisamountofmaterialineventwoorthreetimestheamountofpagesusedherein.
Forthosereaderswhoseexposuretomatricesandvectorshasbeenlimited,orisapartofthenowdistantpast,thereisnoreasonforapprehension.Thelevelofthematrix-basedpresentationemployedhasbeenkeptquiteelementary.
2.HistoryandApplicationsAlthoughthereexistnumerousrelatedearlierdevelopments,thefieldofmathematicalprogrammingtypicallyistracedtothedevelopmentofthegenerallinearprogrammingmodelanditsmostcommonmethodforsolution,designatedas"simplex."LPandsimplexwere,in
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turn,developedin1947byateamofscientists,ledbyGeorgeDantzig,underthesponsorshipoftheU.S.AirForceprojectSCOOP(ScientificComputationOfOptimumPrograms).TheLPmodeladdressedasingle,linearobjectivefunctionthatwastobeoptimizedsubjecttoasetofrigid,linearconstraints.OneofthebestdiscussionsofthisradicalnewconceptisgivenbyDantzighimself(Dantzig,1982).
Withinbutafewyears,LPhadreceivedsubstantialinternationalexposureandattention,andwashailedasoneofthemajordevelopmentsofappliedmathematics.Today,LPisprobablythemostwidely
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knownandcertainlyoneofthemostwidelyemployedofthemethodsusedbythoseinsuchfieldsasoperationsresearchandmanagementscience.However,aswithanyquantitativeapproachtothemodelingandsolutionofrealproblems,LPhasitsblemishes,drawbacks,andlimitations.Ofthese,ourinterestisfocusedontheinabilityoratleastlimitedabilityofLPtodirectlyandeffectivelyaddressproblemsinvolvingmultipleobjectivesandgoals,subjecttosoftaswellasrigid(orhard)constraints.
ThedevelopmentofGPoneapproachforeliminatingoratleastalleviatingtheabove-mentionedlimitationsofLPoriginatedintheearly1950s.Atthistime,CharnesandCooperaddressedaproblemseeminglyunrelatedtoLP(orGP):theproblemof(linear)regressionwithsideconditions.Tosolvethisproblem,CharnesandCooperemployedasomewhatmodifiedversionofLPandtermedtheapproach"constrainedregression"(Charnesetal.,1955;CharnesandCooper,1975).
Later,intheir1961text,CharnesandCooperdescribedamoregeneralversionofconstrainedregression,onethatwasintendedfordealingwithlinearmodelsinvolvingmultipleobjectivesorgoals.Thisrefinedapproachwasdesignatedasgoalprogrammingandistheconceptthatunderliesallpresent-dayworkandgeneralizationsofGP.
Inthesame1961text,CharnesandCooperalsoaddressedthenotsoinsignificantproblemofattemptingtomeasurethe"goodness"ofasolutionforamultipleobjectivemodel.Theyproposedthreeapproaches,allofwhicharestillwidelyemployedtoday.Theseapproacheswereeachbasedonthetransformationofallobjectivesintogoalsbymeansoftheestablishmentofan"aspirationlevel,"or"target.''Forexample,anobjectivesuchas"maximizeprofit"mightberestatedasthegoal:"Obtainxormoreunitsofprofit."Obviously,anysolutiontotheconvertedmodelwilleitherbeunder,over,or
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exactlysatisfytheprofitaspiration.Further,anyprofitunderthedesiredxunitsrepresentsanundesirableorunwanteddeviationfromthegoal.Consequently,CharnesandCooperproposedthatwefocusonthe"minimizationofunwanteddeviations,"aconceptessentiallyidenticaltothenotionof"satisficing"asproposedbyMarchandSimon(Morris,1964).Usingthisconcept,CharnesandCooperspecifiedthefollowingthreeformsofGP:
(1)ArchimedeanGP(alsoknownas"minsum"or"weighted"GP):Hereweseektominimizethe(weighted)sumofallunwanted,absolutedeviationsfromthegoals;
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(2)ChebyshevGP(alsoknownas"minimax"GP):Ourpurposeistominimizetheworst,ormaximumoftheunwantedgoaldeviations;and
(3)non-ArchimedeanGP(alsoknownas"preemptivepriority"GPor"lexicographic"GP):Hereweseektheminimum(moreprecisely,thelexicographicminimum)ofanorderedvectoroftheunwantedgoaldeviations.
ItisofparticularinterestthatLP(oranysingle-objectivemethodology)aswellasArchimedeanandChebyshevGPmayallbeconsideredasspecialcasesofnon-ArchimedeanGPandthustreatedbythesamegeneralmodelandalgorithm(Ignizio,forthcoming).Asaresult,inthisworkwefocusourattentiononnon-ArchimedeanGPor,morespecifically,onlexicographiclineargoalprogramming.
InadditiontodescribingthelinearGPconceptandproposingtheabovethreemeasuresforevaluation,CharnesandCooperalsooutlined(again,intheir1961text)algorithmsforsolution.Evidently,however,actualsoftwarefortheimplementationofsuchalgorithmswasnotdevelopeduntilthelate1960s.Infact,totheauthor'sknowledge,thefirstcomputercodeforGPwastheonethatIdevelopedin1962(Ignizio,1963,1976b,1979b,1981b)forthesolutionofnonlinearGPmodelsmorespecifically,forthedesignoftheantennasystemsfortheSaturn/Apollomoonlandingprogram.
AsaresultofthesuccessofthealgorithmandsoftwarefornonlinearGP,orNLGP,IgainedaconsiderableappreciationofandinterestinGP.Asaconsequence,in1967,whenfacedwitharelativelylarge-scaleLGPmodel(onethatincludedthelexicographicminimum,orpreemptiveprioritynotions),IdevelopedacomputercodeforlexicographicLGPasbasedonasuggestionbyPaulHuss(personalcommunication,1967).Inatelephoneconversationwithme,HussproposedthatonesolvethelexicographicLGPmodelasasequenceof
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conventionalLPmodels.Thissuggestionwasrefinedandsoftwarefortheprocedurewasdevelopedbythesummerof1967.Thisspecificapproach,whichIdesignateassequentialgoalprogramming(orSGP;orSLGPinthelinearcase),althoughunsophisticated,
1resultedinacomputerprogramcapableofsolvinganLGPmodelofsizesequivalenttothosesolvedviaLP(Ignizio,1967,1982a;IgnizioandPerlis,1979).Infact,untilquiterecently,SLGP(alsoknownasiterativeLGPor"decomposed"LGP)evidentlyhasofferedthebestperformanceofanypackageforLGP(havingnowbeensupplantedbytheMULTIPLEXcodesforGP;Ignizio,1983a,1983e,1985a,1985b,forthcoming).
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Later,in1968through1969,VeikkoJääskeläinenalsoaddressedthedevelopmentofsoftwareforLGP.
2RatherthanemployingthecruderSLGPapproach(Jääskeläinenwasunawareofourworkaswereweofhis),Jääskeläinenemployedthealgorithmforlexicographic(i.e.,non-Archimedean)LGPasoriginallyoutlinedbyCharnesandCooper.Toimplementthisalgorithm,hemodifiedthesmallandextremelyelementaryLPcodeaspublishedinthetextbyFrazer(1968).Theresultwasasimplecode(e.g.,itrequiredafulltableau,employedelementarytextbookpivotingoperations,andlackedprovisionsforreinversion)capableofefficientlysolvingonlyproblemsofperhaps30to50variablesandalikenumberofrows.However,inasmuchasJääskeläinen'sintentwassimplytousethecodeonsmallproblemsaspartofhisinvestigationoftheapplicationofLGPtovariousareas,theelementarycodeprovedsufficient(Jääskeläinen,1969,1976).(AcompletediscussionofthiseffortmaybefoundinJääskeläinen's1969work.)
Oneofthemoreintriguingaspects(andonethatisbothfrustratingandembarrassingtotheserious,knowledgeableadvocatesofGP)oftheJääskeläinencodeforLGPisthattoday,thiscodeisthemostwidelyknownandemployedofallLGPsoftware.ThissituationisparticularlytrueinthecaseofmanyU.S.businessschoolswheresomeinvestigators,eventoday,areundertheillusionthatthiscoderepresentsthestateoftheartinLGPsoftware.Tocompoundthematterfurther,credittoJääskeläinenforthedevelopmentofthecodeisseldomifevergiven.TheonepositiveaspectofthesituationisthattheeasyavailabilityoftheJääskeläinencode(mostotherLGPcodesparticularlythosefortrulylarge-scalemodelsareproprietary)helpedencourageasubstantiallyincreasedinterestinGP.
Inthelate1960sandearly1970sIcontinuedtodevelopGP
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algorithmsandsoftware,includingthoseforintegerandnonlinearGPmodels(Drausetal.,1977;HarnettandIgnizio,1973;Ignizio,1963,1967,1976a,1976b,1976c,1977a,1978a,1979a,1979b,1979c;IgnizioandSatterfield,1977;Palmeretal.,1982;WilsonandIgnizio,1977).However,amuchmoreimportantcontributionresultedasaconsequenceofmyinterestindualityinLGP.Bytheearly1970s,arelativelycompleteandformalexpositionofthistopichadresulted(Ignizio,1974a,1974b).ThedualoftheLGPmodel,termedthe"multidimensionaldual,"rapidlyledtothedevelopmentofacompletemethodologyforsensitivityanalysisinLGPmodelsandinthedevelopmentofsubstantiallyimprovedalgorithmsandsoftware.Asaconsequence,todayonehasavailableafairlywideselectionofcomputationallyefficient
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softwareforbothlinearaswellasintegerandnonlinearGPmodels(CharnesandCooper,1961,1977;Charnesetal.,1975,1976,1979;Drausetal.,1977;GarrodandMoores,1978;HarnettandIgnizio,1973;Ignizio,1963,1967,1974b,1976b,1976c,1977a,1978b,1979a,1979b,1980b,1981a,1981b,1981c,1981d,1982a,1983b,1983c,1983d,1983e,1983f,1984,1985a,1985b,forthcoming;Ignizioetal.,1982;KeownandTaylor,1980;MasudandHwang,1981;McCammonandThompson,1980;Mooreetal,1980;MurphyandIgnizio,1984;PerlisandIgnizio,1980;Price,1978;Tayloretal.,1982;WilsonandIgnizio,1977).Onemaystate,infact,thattheperformanceofmodernGPsoftwareisequivalenttothatoftheverybestofthesoftwareusedinthesolutionofconventionalsingleobjectivemodels.
SpacedoesnotpermitadiscussionofGPapplications.However,wedoprovideanumberofreferencesthatdescribeavarietyofimplementationsofthemethodology(AndersonandEarle,1983;Bresetal.,1980;CampbellandIgnizio,1972;Charnesetal.,1955,1976;CharnesandCooper,1961,1975,1979;Cook1984;DeKluyver,1978,1979;Drausetal.,1977;FreedandGlover,1981;HarnettandIgnizio,1973;Ignizio,1963,1976a,1976c,1977,1978a,1979a,1979c,1980b,1981b,1981c,1981d,1983b,1983d,1983f,1984;Ignizioetal.,1982;IgnizioandDaniels,1983;Ijiri,1965;Jääskeläinen,1969,1976;KeownandTaylor,1980;McCammonandThompson,1980;Mooreetal.,1978;Ng,1981;Palmeretal.,1982;PerlisandIgnizio,1980;Pouraghabagher,1983;Price,1978;Sutcliffeetal.,1984;Tayloretal.,1982;WilsonandIgnizio,1977;ZanakisandMaret,1981).Obviously,anyproblemthatmaybeapproachedviamathematicalprogramming(optimization)isacandidateforGP.
3.
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DevelopmentoftheLGPModelInthischapter,weaddressthemostimportantaspect,byfar,intheGPmethodology.Specifically,wedescribeastraightforward,rational,andsystematicapproachtotheconstructionofthemathematicalmodelthatisdesignatedastheLGPmodel.
3
Thepurposeofanymathematicalprogrammingmethodisoratleastshouldbetogainincreasedinsightandunderstandingofthereal-worldproblemunderconsideration.Wehopetoaccomplishthisbyformingand"solving"aquantitativerepresentation(i.e.,themathe-
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maticalmodel)oftheproblem.Whatistoooftenforgotten,however,isthatthenumberssoderivedaresimplysolutionstotheabstractmodelandnot,necessarily,solutionstotherealproblem.Thepurposeoftheproceduretobedescribedistoattempttoprovideamathematicalmodelthatasaccuratelyaspossiblereflectstheproblem.Inthisway,weshouldbeabletominimizethediscrepanciesbetweenmodelandproblem.However,priortodescribingthemodelingprocess,wefirstprovideasummaryofsomeofthenotationthatshallbeusedthroughouttheremainderofthemonograph.
Notation
Ourmathematicalmodelsshallbe,forthemostpart,expressedintermsofmatrixnotation.
Matrices
Amatrixisarectangulararrayofrealnumbers.Werepresentthematrixviaboldface,capitalletterssuchasA,D,I.However,inthecaseof,say,amatrixcomposedsolelyofzeros,wedenotethisasboldfacezero,or0.
ElementsandOrderofaMatrix
Thei,jthelementofmatrixAisdesignatedasai,j.Thatis,ai,jistheelementinrowiandcolumnjofA.Theorderofamatrixisgivenas(m×n)wheremisthenumberofrowsandnisthenumberofcolumns.
SpecialMatrixTypes
Inthemonograph,weutilizeseveralspecialmatrixtypes,whichinclude:
·AT =thetransposeofA;·B1 =theinverseofB(where,ofcourse,Bmustbenonsingular);·I =theidentitymatrix;and·(A1:A2)=thepartitionofsomematrix,sayA.
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Vectors
Avectoriseitheranorderedcolumnorrowofrealnumbers.Inthistext,weshallassumethatallvectorsarecolumnvectors.Thus,intheeventoftheneedtodesignatearowvector,wewilldenotethisbythetransposeoperator.Typically,weshalluseboldface,lowercaselettersforavector,suchas:a,b,x.Asmentioned,thesearecolumnvectors.Thus,a,TbTandxTwouldberowvectors.
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SpecialVectorTypes
Someofthespecialtypesofvectorsusedhereinare:
·aj=thejthcolumnofmatrixA;
=thepartitionofsomevector,sayv;
·c(k)=avectorassociatedwiththekthsetofobjectsofc;and
·x³0;thisindicatesthatthecolumnvectorxisnonnegative.
ElementsofaVector
Typically,xjshallrepresentthejthelementofthevectorx.Thatis,thesubscriptshallindicatetheelement'spositionwithinthevector.
TheBaselineModel
Thefirstphaseofthemodelingprocessistogainasmuchappreciationoftheactualproblemaspossible.Typically,thisisaccomplishedbyobserving(ifpossible)theproblemsituation,discussingtheproblemwiththosemostfamiliarwithit,andsimplyspendingagreatdealoftimethinkingabouttheproblemanditspossiblereasonsforexistenceandpotentialalternatives.Onceonehasgainedsomedegreeoffamiliaritywiththeproblem,thenextstepistoattempttodevelopanaccuratemathematicalmodelforproblemrepresentation.
Itisintheinitialdevelopmentofthis(preliminary)mathematicalmodelthatourapproachdiffersfromthetraditionalprocedure.Thatis,ratherthanimmediatelydevelopingaspecific(andconventional)mathematicalmodel(e.g.,alinearprogrammingmodel),weshallfirstdevelopanextremelygeneral,aswellasuseful,problemrepresentation:the"baselinemodel"(Ignizio,1982a).
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Thegeneralformofthebaselinemodelisgivenbelow:
FindxT=(x1,x2,,xn)soasto
4
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Thecomponentsofthismodeltheninclude:(a)variables(specifically,structuralvariables,alsoknownascontrolordecisionvariables),(b)objectives(ofthemaximizingandminimizingform),and(c)goals(either"hard"or"soft").Further,insomecases(includingthecaseofeitherLPorLGPmodelsasderivedfromthebaselinemodel),anadditionalrestrictiontypicallyexistsinregardtothestructuralvariables.Specifically,thestructuralvariablesmayberestrictedtoonlynonnegativevalues;andthisiswrittenas:
Terminology
Beforeproceedingfurther,letusfirstmorepreciselydefinetheterminologyassociatedwiththebaselinemodel.Thisterminology,aswellasitsdifferencesfromthatusedinconventionalmathematicalprogramming,playsanimportantpartintheappreciationofLGP,ormulti-objectivemathematicalprogrammingingeneral.
Structuralvariable:Typicallydenotedasxj,thestructuralvariablesarethoseproblemvariablesoverwhichonecanexercisesomecontrol.Consequently,theyarealsoknownascontrolordecisionvariables.
Objective:Inmathematicalprogramming,anobjectiveisafunctionthatweseektooptimize,viachangesinthestructuralvariables.Thetwomostcommon(butnottheonly)formsofobjectivesarethosethatweseektomaximizeandthosewewishtominimize(i.e.,maximizeorminimizetheirrespectivevalues).Thefunctionsin(3.1)aremaximizingobjectiveswhereasthosein(3.2)areminimizingobjectives.
Goal:Thefunctionsof(3.3)aregoalfunctions.Specifically,theyappearasobjectivefunctionsinconjunctionwitharight-handside.Thisright-handside(e.g.,bt)isthe"targetvalue"or"aspirationlevel"associatedwiththegoal.
Tofurtherclarifythelastdefinition,thatofa"goal,"notefirsttherelationshipbetweenagoalandanobjective.Forexample,ifwe
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"wishtomaximizeprofit,"wearediscussinganobjective.However,ifwesaythatwe"wishtoachieveaprofitof$1000ormore,"thenwehavestatedagoal.Obviously,then,wemaytransformanyobjectiveintoagoalbymeansofcitingaspecifictargetvalue($1000inthepreviousexample).
Next,notethatgoalsmaybefurtherclassifiedaseither"hard"(i.e.,rigidorinflexible)or"soft"(i.e.,flexible)dependinguponjusthowfirm
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ourdesireistoachievethetargetvalue.Examine,forexample,theprofitgoallistedasfollows:
Now,ifweabsolutelymustachieveaprofitof$1000(e.g.,ifthefirmwillnotsurviveotherwise),thenthisfunctionisahard,orrigid,orinflexiblegoal.Or,usingtheterminologyofconventionalmathematicalprogramming,theexpressionrepresentsarigidconstraint.
Morelikely,suchagoalwouldnotbeinflexible.Thatis,thecompanymaywellwanttohaveaprofitof$1000butwillstillsurviveifitis$999or$990orperhapsevenless.Inthiscase,thegoalwouldbeconsideredsoftorflexible.
Basedontheseconceptsandterminology,letusnowconsiderthedevelopmentofasmall,simplifiednumericalexampleofabaselinemodel.Theproblemweconsiderinvolvestheconstructionofaground-waterpumpingstationtoprovidepotablewaterforasmallcountrytown.Thesiteofthestationisfixed,becauseoftheavailabilityofwellwater,andtheonlyquestionsremaining(thatweshallconsider)are:
(1)Whichoftwotypesofmonitoringstationshouldbeused?
(2)Whichofthreetypesofpumpingmachineryshouldbepurchased?Thetownwishes,ofcourse,tominimizethetotalinitialcost.However,asthereisahighlevelofunemploymentinthearea,theyalsowishtomaximizethenumberofworkersgainfullyemployed.Thedataassociatedwiththisparticularexampleisgivenbelow:
StationType PumpingMachineryTypeA B I II III
Initialcosts(inmillions)
2 1.5 5 4 3.5
Numberofpersonnel 4 6 6 10 15
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per8-hourshift
Toformthebaselinemodelforthisproblemwefirstlet
j=1,2,3,4,and5representingthesubscriptsassociatedwithstationtypeA,stationtypeB,machinerytypeI.machinerytypeII,andmachinerytypeIII,respectively;
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thenletting
Wearereadytoformtheobjectives,goals,andrigidconstraints.Now,exactlyoneofthemonitoringstationsandexactlyoneofthepumpingmachinerytypesmustbepurchased.Thismaybeexpressedas
and
Next,considertheinitialcoststhataretobeminimized.Fromthedatatablewecanimmediatelyconstructthecostobjectiveas
Further,itisdesiredtomaximizethenumberofworkersgainfullyemployed.Again,fromthedatathisobjectiveiswrittenas
Inaddition,wemaywritethenonnegativeconditionsas
Thus,reviewingthemodelweseethatwehavetwoobjectives(relationships3.8and3.9),twogoals(relationships3.6and3.7),andasetofnonnegativityconditions(3.10).However,noticethatforthismodelthenonnegativityconditionsareredundantbecausein(3.5)wehavealreadyrestrictedthestructuralvariablestononnegativevalues
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specifically,thevaluesofeither0or1.Assuch,wemayrewritethismodelinthestandardbaselineformshownbelow:
wherexj=0or1forallj
Ourfinalstepinbaselinemodeldevelopmentistoindicatewhichofthegoalsaretobeconsideredrigid.Asthestationmust,evidently,bebuilt,wemayconcludethatbothgoalsintheabovemodelaretobeconsideredasrigidconstraints.
Inreviewingthismodelitshouldbeobviousthat,eventhoughithasbeensimplified(e.g.,yearlyoperatingcostsandsalarieshavebeenignored),theproblemstillhastwoobjectivesandtheseobjectivesareinconflict.Thatis,theminimizationofinitialcostsadverselyaffectsthedesiretomaximizethenumberofworkersemployed,andviceversa.Further,weshouldnotethatthisparticularmodelisknownasa"zero-oneprogramming"modelbecauseoftherestrictionsonthestructuralvariablevalues.Inthistextweshallmainlyfocusonmodelswithstrictlycontinuousvariables.However,therearemethodstosolvethezero-onemodelasisbrieflydiscussedinChapter7.
AdditionalExamples
Becauseofthe(rigid)constraintsonthelengthofthismonograph,weshallnotaddressanyfurtherbaselinemodelexamples.However,thereaderdesiringfurtherdetailsandexamplesmayreviewChapter2ofmybookLinearProgramminginSingleandMultipleObjectiveSystems(Prentice-Hall,1982).
ConversionProcess:LinearProgramming
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Thebaselinemodelof(3.1)-(3.4)representsthequantitativemodelthat,ifproperlyandcarefullydeveloped,isclosesttorepresentingthe
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significantfeaturesofthecorrespondingreal-worldproblem.Unfortunately,inthegeneralcaseitisusuallynotfeasibletoattackdirectlysuchamodel.Thisisbecausetheavailablemethodsofsolutionaresimplynotyetadequatefordealingwithsuchageneralrepresentation.Asaconsequence,ournextstepinthemodelingprocessistotransformthebaselinemodelintoa''workingmodel,"bymeansofcertainassumptions.
AsmanyreadersmaybefamiliarwithLP,letusfirstdescribetheconversionofthebaselinemodelintoanLPmodel.First,wecitethegeneralformoftheLPmodel:
Findxsoasto
Forpurposeofillustration,wehaveselectedaformthathasaminimizationobjective(i.e.,function3.11).However,ifweinsteadwishedtomaximizethesingleobjective,wewouldsimplymultiplyitbynegativeoneandthenminimizetheresultantobjective.Thatis,
maximizez'=cTx
isequivalentto
minimizez=cTx
Now,comparingtheLPmodelof(3.11)-(3.13)withthebaselinemodelof(3.1)-(3.4),itshouldberelativelyapparentastohowthelatterwasconvertedintotheformer.Theprocessitselfmaybesummarizedasfollows:
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Step1:Selectoneobjectivefrom(3.1)or(3.2)andtreatitasthesingleLPobjective.Typically,thisistheobjectivethatisperceivedtobeof"mostsignificance."
Step2:Convertallremainingobjectivesintogoalsbymeansofestablishingassociatedaspirationlevels.Thatis,
maxfr(x)
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becomes
fr(x)³br
and
minfs(x)
becomes
wherebrandbsaretherespectiveaspirationlevelsforthetwoobjectivetypes.
Step3:Treatallgoals(i.e.,includingthoseasformedinstep2)asrigidconstraints.
Step4:Converteachinequalitygoaltoanequation(bymeansof"slack"or"surplus"variables.(See,forexample,Chapter6ofLinearProgramminginSingleandMultipleObjectiveSystems,1982.)
AlthoughfewanalystsareevertrainedtoproceedthroughtheabovefourstepsinformingtheLPmodel(i.e.,theytypicallyproceeddirectlytotheLPmodel),westronglybelievethattheassumptionsunderlyinganyLPmodelaremadefarmoreapparentviathisprocess.
LGPConversionProcedure:PhaseOne
Wenowconsidertheconversionofthebaselinemodelintothelexicographicform(i.e.,non-Archimedeanorpreemptivepriorityform)oftheGPmodel.Thismodelmostspecificallyinitslinearformis,ofcourse,thefocusofthismonograph.Thisconversionprocessproceedsthroughtwophases.Thefirstphaseconsistsofthefollowingsteps:
Step1:Allobjectivesaretransformedintogoals(inthesamemannerastreatedaboveforLP).Thus,thebaselinemodelof(3.1)-(3.4)
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becomes:
6
Notethateachgoalin(3.14),unlikeLP,maybeeitherhardorsoft,asdeemedappropriateforthemostaccuraterepresentationoftheproblembeingconsidered.
Step2:Eachgoalin(3.14)isthenrankorderedaccordingtoimportance.Asaresult,thesetofhardgoals(i.e.,rigidconstraints)is
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TABLE3.1InclusionofDeviationVariables
OriginalGoalForm ConvertedForm"Unwanted"DeviationVariable(theVariabletobeMinimized)
fi(x)£bi fi(x)+hiri=bi rifi(x)³bi fi(x)+hiri=bi hifi(x)=bi fi(x)+hiri=bi hi+ri
alwaysassignedthetoppriorityorrank(designatedtypicallyasP1).Allremaininggoalsarethenranked,inorderoftheirperceivedimportance,belowtherigidconstraintset.Notefurtherthatcommensurablegoalsmaybe(andshouldbe)groupedintoasinglerank(Ignizio,1976b;Ignizio,1982a;KnollandEnglebert,1978).
Step3:GiventhatthesolutionprocedureusedinsolvingLGPmodelsrequiresasetofsimultaneouslinearequations(asdoesLP),allofthegoalsof(3.14)mustbeconvertedintoequationsthroughtheadditionoflogicalvariables.
InLP,suchlogicalvariablesareknownasslackandsurplusvariables(and,whenneeded,artificialvariables).InGP,theselogicalvariablesaretermeddeviationvariablesorgoaldeviationvariables.WesummarizethisstepinTable3.1.
HavingconcludedthefirstphaseintheconversionofthebaselinemodelintotheGPmodel,wenowemphasizethatweseekasolution(i.e.,x)thatservesto"minimize"allunwanteddeviations.ThemannerbywhichwemeasuretheachievementoftheminimizationoftheundesirablegoaldeviationsiswhatdifferentiatesthevarioustypesofGP.Here,weshallusethelexicographicminimumconceptanapproachthat,asmentionedearlier,willalsopermitustoconsider,asspecialcases,Archimedean(i.e.,minsum)LGPaswellasconventionalLP.Beforeproceedingfurther,letusdefinethelexicographicminimum,or
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"lexmin,"ofanorderedvector.
LexicographicMinimum:Givenanorderedarray,saya,ofnonnegativeelements(aks),thesolutiongivenbya(1)ispreferredtoa(2)if andallhigherorderterms(i.e.,a1,a2,,ak1)areequal.Ifnoothersolutionispreferredtoa,thenaisthelexicographicminimum.
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Thus,ifwehavetwoarrays,saya(r)anda(s),where
a(r)=(0,17,500,77)T
a(s)=(0,18,2,9)T
thena(r)ispreferredtoa(s).
LGPConversionProcess:PhaseTwo
Wenowaddressthecompletionoftheconversionprocess,aprocessthatwillleadustothefollowinggeneralformofthelexicographicLGPmodel:
Findvsoasto
Ifweobservethat
thenwemaynotethat(3.16)issimplytherepresentationofallthegoals,includingtheirdeviationvariables(i.e.,seeTable3.1)fortheproblem.Itisnowleftonlytoexplainthemeaningandformationof(3.15).TheorderedvectoruT,istermedthe"achievement"functioninGP.Actually,itcouldbearguedthatamoreappropriatenameisthe"unachievement"functionasitreallyrepresentsameasureoftheunachievementencounteredinattemptingtominimizetherank-orderedsetofgoaldeviations.Thus:
uT=theGPachievementfunction,orvector,
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uk=thekthtermofuT;thetermassociatedwiththeminimizationofallunwanteddeviationsassociatedwiththesetofgoalsatrank,orpriorityk,and
c(k)T=the(rowvector)ofweightsassociatedwiththeunwanteddeviationvariablesatrankk.
NoteinParticulartheNotationUsedinc(k)T
Thatis,the"T"superscriptsimplydesignatesthetransposeofthecolumnvectorc(k).Thesuperscript(k)refers,however,totheprioritylevel,orrank,associatedwiththerespectivesetofweights.Forthereaderstilluncertainastotheprocedure,wenowdescribetheconversionviaasmallnumericalexample.
AnIllustration
Inordertobothclarifyandreinforcetheconceptofthedevelopmentofthebaselinemodel,weshallnowdescribeaspecific,numericalexample.Althoughfarsimplerandlesscomplexthanwouldbemostreal-worldproblems,themodelingprocessshouldstillindicatethetypicalprocedureused.
Weshallassumethatweareconcernedwiththeproblemsofaspecific,high-techfirm.Althoughthisfirmproducesnumerousitems,theirparticularproblemisinregardtothemanufactureofjusttwooftheseproducts.Theseproducts,designatedforsecurityas"x1"and"x2,"areproducedinoneisolatedsectoroftheplant,viaanextremelycomplexprocessascarriedoutonanexceptionallydelicatepieceofmachinery.Onceanitemisproduced,wehavejust24hours,atthemaximum,toshipandinstalltheitemataremotegovernmentinstallation.Thatis,unlessthefinishedunitofeitherproductx1orx2isinstalledwithin24hoursofitsmanufacture,theproductcannotbeenhancedchemicallyandmustbescrappedviaanextraordinarily
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expensiveandtime-consumingprocess;aprocessthatwould,infact,driveourcompanyoutofbusiness.
Thefirmhasacontractwiththegovernmenttosupplyupto30unitsperdayofproductx1andupto15unitsperdayofproductx2.However,thegovernmentinstallation,recognizingthedelicatenatureofthemanufacturingprocess,realizesthatreceiptofexactly30and15unitsofx1andx2,respectively,isunlikely.
Thefirmmakesanestimatedprofitperunitof$800forx1and$1200forx2.Theystatethattheycertainlywishtomaximizetheirdailyprofit.
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Onthesingle,speciallydesignedprocessor,ittakesjustoneminutetoproduceeachunitofx1andtwominutesforeachx2.However,duetothedelicatenatureofthemachine,thefirmwouldliketorunitnomorethan40minutesperevery24-hourperiod.Inthetimeduringwhichthemachineisnotrunning,itmaybeadjustedandfinetunedsoastosatisfythealmostcriticalmanufacturingrequirements.Thus,althoughthemachinecouldconceivablyberunformorethan40minutesperday,thiswouldnotbehighlydesirabletothefirm.
Tomodelthisproblem,inbaselineform,weshallfirstdefineourstructuralvariables:
x1=numberofunitsofproductx1producedperday;
x2=numberofunitsofproductx2producedperday.
Wenextformourobjectivesandgoals,asafunctionofthestructuralvariables.
Ourfirstsetofgoalswillbethatof"marketdemand,"thedaily(upper)requirementsofthegovernmentinstallation.Thus:
Notecarefullythatthegovernment,althoughwantingtheupperlimits,willacceptsomewhatfewerunits.Further,recallthevirtualdisasterthatwouldbeassociatedwithproducingmorethanthedailydemands.
Theprofitobjectivemaybewrittenasfollows:
maximize800x1+1200x2(dailyprofit)
However,thiswouldbepoormodelingpractice.Thatis,inmathematicalprogrammingoneshouldalwaysattempttoscaleallcoefficientssothatthedifferencebetweenthelargestandsmallestcoefficientisminimized.Thus,amoredesirableformoftheprofit
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objectiveis
Next,wenotethatwewouldliketolimitproductiontimeperdayto40minutestotal,althoughthefirmdoesindicatesomeflexibilityaboutthislimit.Thus,wewritethisgoalas
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Finally,althoughnotexplicitlymentionedinourproblemdescription,thefirmwouldobviouslyliketoproduceascloseto30unitsperdayofx1and15unitsperdayofx2.Indoingso,theynotonlyincreasetheirprofitsbutalsokeepthecustomerhappy.Weshallwaitforamomentbeforeactuallyformulatingtheselastgoals.However,notethattheyareassociatedwith(3.19)and(3.20).
Ournextstepistoconvertanyobjectivesintogoals.Theonlyobjectivelistedin(3.19)-(3.22)isthatofmaximizingprofit.Assumingthatthefirm'saspireddailyprofitfromthesetwoproductsis$100,000,weconvert
Wearenowreadytorankorderallgoals,inconjunctionwithdiscussionswiththefirm'sdecisionmakers.Forpurposeofdiscussion,weshallassumethattheorderofpresentationcoincideswiththeorderofpreference.Further,itisobviousthatthefirsttwogoals(dailyrequirements)aretheonlyonesthatarerigidinthisproblem.Thus,lettingPkrefertothekthpriorityorrank:
P1:producenomoreitemsperdayofeachitemthandemanded.
P2:achieveaprofitof$100,000perday,ormore.
P3:attempttokeepprocessingtimeto40minutesorlessperday.
P4:attempttosupplyascloseto30unitsand15unitsofx1andx2,respectively,perday.Further,weshallassumethatthefirmconsiderssupplyx2tobeoneandahalftimesmoreimportantthanx1.
Afterincludingthenecessarygoaldeviationvariables(i.e.,niandri)andformingtheachievementfunction,wedevelopthefinalformofthelexicographicLGPmodelasshownbelow:
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Notice,inparticular,thattwo(ormore)goalsmaybecombinedinonepriorityleveliftheyarecommensurable(i.e.,credibleweightsmaybeassignedtoeachgoalsothattheymaybeexpressedinasingleperformancemeasure).ThisoccursinP4(thefourthtermofuT).Further,allrigidconstraintsarealwayscombinedinP1,evenifnotinthesameunits,astheymustbeachievediftheprogramistobeimplementable.
Themodelof(3.24)-(3.26)mayberewritten,inamoregeneralform,as
s.t.
Av=b
v³0
where
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AsweshallseeinChapter5,theoptimalsolutiontothismodelis
v*=(3015:005800:00020)T
u*=(0580200)T
Fromv*,wenotethat and andthusweproduceexactlythedailylimits.Observingu*,wecandeterminehowwellourgoalsweremet.
thusallrigidconstraintsaresatisfied.theresultis580unitsbelowthegoalof1000.Thus,weachieveadailyprofitof$42,000ratherthan$100,000.theresultis20unitsovertheaspiredgoalof40.Consequently,ourmachinemustberunfor60minutesperdayratherthan40.thelastsetofgoalsarecompletelysatisfied.
GoodandPoorModelingPractices
Notetheachievementfunctionof(3.24),orthegeneralformof(3.15).Thisfunctionisanorderedvector,witheachelementcorrespondingtothemeasureoftheminimizationofcertainunwantedgoaldeviations.Whenonerefersbacktotheearlierdefinitionofthelexicographicminimum,wenotethatthisdefinitionis,infact,baseduponthenotionofanorderedvector.Despitethis,someemployaGPachievementfunction(whichtheytypicallytermasan"objectivefunction")thatindicatesasummationoftheindividualelementsofuT.Thatis,theywillwriteuTin(3.24)asfollows:
minimizeP1(r1)+r2)+P2(h3)+P3(r4)+P4(h1+1.5h2)
wherePkindicatestherank,orpriority,oftheterminparentheses.Althoughincommonuse,thisnotationisexceptionallypoorpracticeasittotallycontradictstheverydefinitionofthelexicographicminimum(orof"preemptivepriorities").Therealproblemappears,however,
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whenthoseusingsuchnotationattempttodevelopanyextensionsofGPorsupportingproofs.Thatis,theinvalidsummationsserveto,quiteoften,totallyconfusethosepursuingsuchextensions.Consequently,it
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ismybeliefthat,despitethe"tradition"ofsuchaform,itismathematicalnonsenseandshouldbeavoided.
Anothermodelingpracticesometimesadvocatedbyothersistoconstructgoalswithouttheinclusionofanystructuralvariables.Toillustratethis,considerthefollowingexample;
Here,thecircleddeviationvariablesarethosewewishtominimize.Thus,G1is whereinG2isanattempttostatethatinG1thenegativedeviationfrom40shouldbe10unitsorless.Iclaimthatthefollowingrepresentationismoreeffective:
Thereadershouldnotethatbothrepresentationsareequivalent.However,thefirstrepresentationwillrequire,whensolvedbyanyLGPalgorithm,morevariablesthanthesecond.ThereasonforthiswillbecomeclearinChapter4.Here,wesimplynotethattheso-calledinitialbasicfeasiblesolutioninLGPalwaysconsistsofthenegativedeviationvariables(nis).Assuch,eachhimustappearinexactlyonegoal.SuchisnotthecaseinG1andG2.Consequently,toalleviatethis,atleastonenewvariablemustbeaddedtotheformulation.Thus,althoughG1andG2meanthesamethingasG1'andG2',thelatterisamorecomputationallyefficientrepresentation.Toconclude,wesimplynotethatitisnevergoodpracticetoformagoalconsistingsolelyofdeviationvariablesandanalternate,properrepresentationmayalwaysbefound.
Wealsonotethattherigidconstraintsshouldalwaysappear,
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separately,inprioritylevelone(i.e.,asu1).Allremaininggoalsshouldberankedbelowthembutmayappearinasingleprioritylevel(orasgroupedsets)ifreasonableweightsmaybefoundsoastoachievecommensurability.Further,weshouldalwaysrealizetheimplicationofseparateprioritylevels.Thatis,theachievementofPralwayspreemptsthatofPsifs>randthusthegoalsatPscanbeachievedonlytothepoint
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thattheydonotdegradeanyhigher-ordergoals.Itisforthisreasonthatitispoorpracticetohavemorethanabout5or6prioritylevels.Thatis,thelikelihoodofbeingabletodealwithanygoalsataprioritylevelof,say,tenisvirtuallynil.
AgoodGPmodelwillmakegoodsense.Thatis,itshouldbelogicalandexpresstheproblemaccurately.Ifitdoesnot,oneshouldseektoimprovethemodelsothatitdoesmakesense.Somechecksthatshouldalwaysbemadeinclude:
(1)Arealltherigidconstraints(andonlytherigidconstraints)atpriorityone?
(2)Aretheunwanteddeviationsthosethatappearintheachievementfunction?
(3)Aretheremorethan5or6prioritylevels(real-worldproblemstypicallyhavenomorethan2to5prioritylevels)?
(4)Areallsetsofcommensurablegoalsgroupedwithinthesameprioritylevel?
(5)Doanygoalsconsistsolelyofdeviationvariables?Ifso,replacethemasdiscussedearlier.
4.AnAlgorithmforSolutionAsdiscussedinChapter2,anumberofdifferentalgorithms(andassociatedcomputersoftware)havebeendevelopedforthesolutionofthelexicographicLGPmodel.Further,thebestofthesealgorithms(e.g.,MULTIPLEX;Ignizio,1983e,1985a,forthcoming)arecapableofsolvingmodelsofcomparablesizes(i.e.,severalthousandsofrowsbytensofthousandsofvariables)andwithequivalentcomputational
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efficiencyasthatfoundincommercialsimplexsoftware(i.e.,forconventionalLPmodels;Ignizio,1983e,1984,1985a,forthcoming).
InthischapterweshalladdressjustoneversionofthemanyLGPalgorithms,aversionusingmultiphasesimplex.ForthosewithafamiliaritywithLP,wenotethatmultiphasesimplexissimplyastraightforwardandrathertransparentextensionofthewell-known''twophase"simplexprocedure(CharnesandCooper,1961;Ignizio,1982a;Lasdon,1970;Murtagh,1981)forconventionalLP.
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TheTransformedModel
InChapter3,wepresentedthegeneralformofthelexicographicLGPmodelin(3.15)-(3.18).Wehaverewrittenthismodelbelow:
Findvsoasto
where
Notethathandrarethelogical(orgoaldeviation)variableswhereasxisthestructuralvariable.
Further,as representstheweightgiventovariablej,atpriorityorrankk,thenall are,inLGP,nonnegative.Thatis,
AlthoughthepreviousmodelrepresentsandconvenientlysummarizesthelexicographicLGPmodel,weshallworkwithatransformationofthismodel.This"transformedmodel"isalsoknownasthe"tabular"modelor''reducedform"model.Theadvantagesofthetransformedmodelincludethefactthat,fromit,variousLGPconditionsmaybeeasilyderived.WenowproceedwiththedevelopmentofthetransformedLGPmodel.
Wefirstnotethatthesetofgoalsisgivenas:
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However,themxnmatrix,A,maybepartitionedinto:
A=(B:N)
where:
B=amxmnonsingularmatrix,designatedasthebasismatrix,and
N=amx(nm)matrix
Further,thevariableset,v,maybesimilarlypartitionedinto:
where:
vB=thesetofbasicvariablesthoseassociatedwithBand
vN=thesetofnonbasicvariablesthoseassociatedwithN
Consequently,wemayrewrite(4.6)as
and,asBisnonsingular(andthushasaninverse),wemaypremultiplyeachtermin(4.8)byB1toobtain
B1BvB+B1NvN=B1b
or
vB+B1NvN=B1b
and,solvingforvB:
Next,examinetheLGPachievementfunctionasgivenin(4.1).Thegeneral,orkthelementofuisgivenas
c(k)Tv
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However,recallthatvwaspartitionedaccordingto(4.7)andthustheabovetermmayberewrittenas
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whereinthesubscriptsforcreflectthosecoefficientsassociatedwiththesetofbasicvariablesorthoseassociatedwiththenonbasicvariables"B"orN",respectively.
Wemaynow,using(4.9),substituteforvBin(4.10)toobtain
Further,let
whereajisthejthcolumnofA.
Usingtheabove,wemaywritethegeneralformoftheLGPmodelfrom(4.1)-(4.3)inthe"transformed"or"reduced"formasgivenbelow.
Findvsoasto
Analternativeandquiteconvenientwayinwhichwemaysummarize(4.15)-(4.17)isbymeansofatableau,asshowninTable4.1.
BasicFeasibleSolution
Wemaydefineabasicsolutionasoneinwhichallnonbasicvariablesaresetattheirbound.Forourpurposes,thisboundshallbezero.Thus,
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ifvN=0,abasicsolutionresults.Morespecifically,ifvN=0,thenB1NvN=0andthus
Further,abasicfeasiblesolutionisonewhereinalltermsin(4.18)arenonnegative.Thus,abasicfeasiblesolutionis:
InLGP,asinLP,theoptimalsolutionmayalwaysbefoundasabasicfeasiblesolution(CharnesandCooper,1961).
AssociatedConditions
ThethreeprimaryconditionsassociatedwiththereducedformoftheLGPmodelarefeasibility,implementability,andoptimality.GiventhatvN=0,thesetermsaredefinedasfollows:
Feasibility:Ifb=B1b³0,theresultantsolution,orprogram,isdenotedasbeingfeasible.
Implementability:If thentheresultantprogram(vB)isdesignatedasbeinganimplementablesolution.Thatis,thetoprankedsetofgoals(i.e.,thesetofrigidconstraints)areallsatisfied.
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Now,beforedefiningtheconditionsfortheoptimalityofagivenprogramforanLGPmodel,letusfirstnotethat,in(4.15),theterms
aredesignatedasthevectorofincreasedcostsasthisindicatesjusthowmuchthekthtermoftheachievementfunctionwillincreaseasvNchanges.
Weshallletthejthelementof(4.20)bedesignatedas
wherein
Consequently,associatedwitheachnonbasicvariableisacolumnvectorof elements.Thisvectoristermed,inLGP,asthevectorofmultidimensionalshadowpricesorassimplytheshadowpricevector,dj.
Optimality:Ifeveryshadowpricevector,dj,islexicographicallynonpositive,theassociatedbasicfeasiblesolution(vB)isoptimalforthegivenLGPmodel.Thisoptimalprogramisdesignatedas
Alexicographicallynonpositivevectorisone,inturn,inwhichthefirstnonzeroelementisnegative.Ofcourse,avectorofsolelyzerosisalsolexicographicallynonpositive.
Wemaynotethat"implementability"isaconditionthatisuniquetoGP.Further,unlikeLP,thereisnoconditionofunboundednessinGP.Thismaybeobservedbysimplyexaminingtheachievementfunctionformgivenoriginallyin(4.1)andrepeatedbelow:
lexmin
Now,uTcouldonlybeunbounded(inthecaseofseekingthe
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lexicographicminimumofuT)ifoneormoreelementsofuTcoulddecreasetominusinfinity.Thisisobviouslyimpossiblebecause
v³0andc(k)³0forallk
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Thus,theabsoluteminimumvalueforanyukiszero.
7Despitethis,othermaterialortextsonGPsometimesdiscusstheunboundedconditionforLGPasifitwereactuallypossibleand,infact,evenproposecheckswithintheirproposedalgorithmsforsolutions.Theimplementationofsuchunnecessarycheckssimplyincreasescomputationtime.Evidently,thoseproposingsuchconditionsandchecksaresimplycopying,withoutthoughttotherationale,thechecksforunboundednessthatexistinLP.Thissituation,aswellasnumerousothers,providesamplereasontonotblindlytreatGP,orLGP,assimply"anextensionofLP."Infact,amorelogicalviewwouldbetoconsiderLPasbutaspecialsubclassofLGP.
AlgorithmforSolution:ANarrativeDescription
BeforeproceedingtoalistingofthespecificstepsofouralgorithmforsolutionoftheLGPmodel,letusfirstattempttodescribetheoverallnatureofthisalgorithm.Aswithanyalgorithmformathematicalprogramming,ourassumptionisthatacorrectmathematicalmodelhasbeendeveloped.
Initially,wefocusourattentiononprioritylevelone,theachievementofthecompletesetofrigidconstraints.Thus,ourinitialmotivationistodevelopabasicsolutionthat(ifpossible)simultaneouslysatisfiesalltherigidconstraints.Thisisaccomplishedwhenevera1=0.Inourattempttoachievethis,weinitiallysetallstructuralvariablesandpositivedeviationvariablesnonbasic.Consequently,ourfirstbasisconsistssolelyofthesetofnegativedeviationvariables.Typically,thisbasis(which,inessence,isthe"donothing"solution)willnotsatisfyallrigidconstraintsandthusweinitiatethesimplexpivoting
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procedure.Specifically,weexchangeonebasicvariableforonenonbasicvariableifsuchanexchangewill:(a)retainfeasibility,and(b)leadtoanimprovedsolutiononethatmorecloselysatisfiesthesetofrigidconstraints.Thepivotingstepisthenrepeateduntilallrigidconstraintsaresatisfiedascloselyasispossible(i.e.,untila1reachesitsminimumvalueavalueofzero,itishoped).
Havingobtainedtheminimumvaluefora1,wenextattempttominimizea2withoutdegradingthevaluepreviouslyobtainedforourhigherprioritylevel(i.e.,a1).Minimizationofa2isaccomplished,onceagain,viathesimplexpivotingprocess.However,inadditiontoconsideringthefeasibilityandimprovementofanyproposedexchange,wemustalsonotpermitanexchangethatwoulddegradethevalueofa1(i.e.,increaseitsvalue).
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Theprocedurecontinuesinthismanneruntilthelexicographicminimumofaisfinallyobtained.Thosereaderswithapreviousexposuretoconventionallinearprogrammingwillrecognizethattheprocedureusedisbutaslightlymodifiedversionofthewell-knownsimplexalgorithm;specificallythetwo-phasesimplexprocedureusedinmostcommercialsimplexsoftwarepackages.
TheRevisedMultiphaseSimplexAlgorithm
ThealgorithmforthesolutionoftheLGPmodel,asisdescribednext,istermedtherevisedmultiphasesimplexalgorithm.Assuch,itisbasicallyastraightforwardmodificationofrevisedsimplexforLPwhereintheso-calledtwo-phasesimplexprocessisutilized.Themodificationitselfpermitsmultiple"phases,"ratherthanjusttwoasinconventionalLP.
Undertheassumptionthatthealgorithmistobeultimatelyimplementedviaacomputer,theinformationmaintainedincomputerstoragemustconsistofsomerepresentationoftheoriginalLGPmodelasgivenin(4.1)-(4.3).Specifically,westoreA,b,andc(k)Tforallk.
Webeginthealgorithmbyassumingthatwehaveaninitialbasicfeasiblesolution,somerepresentationofB1andtheassociatedprogram:b=B1b(withthelatterdesignatedasthe"currentrighthandside").WhenemployingtheLGPmodel,thesearetrivialrequirementsbecause,initially,h=bandx,r=0willalwaysprovideabasicfeasiblesolution(weassumethatallgoalsarewrittenwithnonnegativerighthandsides).Further,thebasisassociatedwithvB=histheidentitymatrix,I,andthusB1=Iinitially.
Wemaythengeneratethemultidimensionalshadowpricevectors,dj,forallnonbasicvariablesanddeterminewhetherornotthepresentbasicfeasiblesolutionisoptimal.
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8Ifso,wemaystop.Otherwise,wemustproceedtoapivotingoperation.Pivotinginvolvestheexchangeofanonbasicvariableforabasicvariableinamannersuchthat:
(1)thenewsolutionisstillabasicfeasiblesolution,and
(2)theresultantvalueofuTisimproved,orisatleastnoworsethanbeforethepivot.
Oncewehavepivoted,wesimplyupdateB1andbandrepeattheprocess.Inactualpractice,wemayaugmenttheproceduredescribedabovebynumerousrefinementsandshortcutssoastosubstantially
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improvecomputationalperformance.WenowlistthestepsoftherevisedsimplexprocedureforLGP.
Step1.Initialization.LetvB=h.Thus,B=I,B1=Iandb=b.Setk1.Initially,allvariablesareunchecked.
Step2.Developthepricingvector.Determine:
Step3.PriceoutallUNCHECKED,nonbasiccolumns.Compute:
where isthesetofnonbasicanduncheckedvariables.
Step4.Selectionofenteringnonbasicvariables.Examinethose ascomputedinstep3.Ifnonearepositive,proceedtostep8.Otherwise,selectthenonbasicvariablewiththemostpositive (tiesmaybebrokenarbitrarily)astheenteringvariable.Designatethisvariableasvq.
Step5.Updatetheenteringcolumn.Evaluate:
Step6.Determinetheleavingbasicvariable.Weshalldesignatetheleavingvariablerowasi=p.Usingthepresentrepresentationofbandthevaluesofaq,asderivedinstep5,wedetermine:
Again,tiesmaybebrokenarbitrarily.Thebasicvariableassociatedwithrowi=pistheleavingvariable,vB,p.
Step7.Pivot.
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9WereplaceapinBbyaqandcomputethenewbasisinverse,B1.Returntostep2.
Step8.Convergencecheck.Ifeitherone(orboth)ofthefollowingconditionsholds,STOPaswehavefoundtheoptimalsolution:
(a)ifall ascomputedinstep3arenegative,or
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(b)ifk=K(whereK=numberofprioritylevels,ortermsinuT).Otherwise,"check"allnonbasicvariablesassociatedwithanegative,setk=k+1andreturntostep2.
TheaboveeightstepsrepresenttheprimaryelementsoftherevisedmultiphasesimplexalgorithmforLGP.However,itmustberealizedthattrulyefficientcomputersoftwarefortheimplementationofsuchanalgorithmwilltypicallyinvolvenumerousmodificationsandrefinementstailoredaboutthespecificadvantagesaswellaslimitationsofthedigitalcomputer.
Inthefollowingchapter,weillustratetheimplementationofthedescribedalgorithmonanumericalexample.However,beforeproceedingtothatdiscussion,wenextexamine,inmoredetail,certainstepsoftheabovealgorithm.
ThePivotingProcedureinLGP
TheheartofthesolutionalgorithmfortheLGPmodelisaproceduredenotedaspivoting.Inessence,pivotinginvolvesthemodificationofapriorbasicfeasiblesolution.Associatedwitheverybasicfeasiblesolution,vB,isabasis,B.ThisbasisiscomposedofasetofmlinearlyindependentcolumnvectorsfromA,thematrixof"technologicalcoefficients"thatappearsinthestatementoftheLGPgoals(i.e.,Av=b).
Initsmostelementaryform,pivotinginvolvestheexchangeofacolumnvectorofB(asassociatedwithapresentbasicvariable)foranonbasiccolumnvectorfromN.Thatis,anonbasicvariableissaidto"enter"thebasiswhileabasicvariable"leaves"thebasis.SuchanexchangeismadeifitresultsinanimprovementinthelexicographicminimumofuT;or,atworst,ifuTisnotdegraded.
Thechoiceofthenonbasicvectortoenterthebasisismadeviaan
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operationtermedthe"priceout"procedure.Simplyput,thepriceoutprocedureevaluatesthepotentialimprovementinuTforeachcandidatenonbasicvariableandselectstheonethatappearstoprovidethegreatestreduction.Thedeterminationofthebasicvariablethatistoleavethebasisisslightlymoreinvolved.Giventhatanewvariable(i.e.,apresentlynonbasicvariable)istoenterthebasis,abasicvariablemustobviouslyleaveandthechoiceofthebasicvariablethatdepartsisnotarbitrary.Specifically,thechoiceismadesothatthenewbasisisassociatedwithanewbasicfeasiblesolution.Thismaybeillustratedasfollows.First,observethetransformedformoftheLGPmodelthatreflectsthebasicfeasiblesolution,forexample:
vB=B1bB1NvN
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withb=B1bandrealizingthatonlyasinglenonbasicvariableistoenterthebasis,wemayrewritetheaboveexpressionas
vB=bB1aqvq
whereqisthesubscriptassociatedwiththeenteringnonbasicvariable.However,wemayreplaceB1aqwithaq.Thus,
vB=baqvq
RewritingvBintermsofeachofitscomponents,wehave
Thus,asvqincreases,bidecreasesifai,qispositive.Thatis,
where isthenewvalueofbi.InorderthatvBremainfeasible,eachbimustremainnonnegative.Assuch,ifanyai,qispositive,thecorrespondingvalueofbiwilldecrease,eventuallypassingthroughzero.Thus,thefirstsuchbithatwouldreachzerodeterminestheso-calledblockingbasicvariablethebasicvariablethatmustleavethebasisifvqenters.Thisleadsdirectlytothedepartingbasicvariablerulegivenas(4.26).inthepreviousalgorithm.
Theoperationofthepivotingprocedure,summarizedsobrieflyinstep7ofthealgorithm,istocomputethenewinverseasaresultofthepivot.Nearlyallpresentdaycommercialsoftwarefortheconventionalsimplexmethod(i.e.,forLP)orthatforLGPutilizetheso-calledproductformoftheinverse(CharnesandCooper,1961;Ignizio,1982a;Lasdon,1970;Murtagh,1981)toaccomplishthisevaluation.Thereare,however,severalalternateapproachesand,in
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fact,weshallpresentoneoftheseinthechaptertofollow(inthediscussionofthetabularsimplexprocessforLGP).Thismethodistermedthe"explicitformoftheinverse"andit,aswellasitsvariations,providesareasonablycomputationallyefficientapproachtoLGP,whetherperformedbyhandoronthecomputer.
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5.AlgorithmIllustrationToillustratetheimplementationofthealgorithmaslistedinChapter4,weapplythatproceduretothesameLGPmodeloriginallyformedinChapter3andgivenin(3.24)-(3.26).Specifically,theproblemaddressedisasfollows:
Findxsoasto
s.t.
However,theaboveformofthemodelisnotconvenienttoworkwithandthuswereplaceitbythemoregeneralform:
Findvsoasto
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and
TheTableau
InChapter4,Table4.1,wepresentedoneparticularformoftheLGPtableau.Specifically,thatversionistermedthe''full"or"extended"tableauform.Actually,wemighttermitthe"toofull"tableauasitcontainsfarmoreinformationthanisactuallyneededtoperformthealgorithm.Althoughthefulltableauisoftenusedintextbookpresentations,itissimplynotsuitableforrealisticcomputerimplementation.Thus,hereweuseanalternativeandconsiderablymoreconvenienttableauform.Weadvocatetheuseofthistableauwhetheroneissolvingtheproblembyhandordevelopingacomputerprogramforreasonablyefficientsolution.ThistableauisshowninTable5.1.
Notethatthistableaucontains
B1theinverseofthepresentbasis;b thepresentright-handside(vB=B1b),orprogram;
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u theachievementvector;andP
thematrixof elements.
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TABLE5.1LGPTableau,ExplicitFormofInverse
vB,1...vB,i B1 b...vB,mp(1)T.. P u.p(k)T
RemarkingonP,wenotethateachrowinP,givenasp(k)Tingeneral,isthepricingvectorasspecifiedinstep2ofthealgorithmofChapter4.
Actually,thesectionofthistableauthatcontainsuisnotreallyneededinperformingthealgorithm.Wekeepitmerelytonotethecontinuingimprovementofuineachiteration.
Wemaynowwritetheassociatedinitialtableaufortheproblemgivenin(5.4)-(5.6).
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Notecarefullythatv3throughv6(whichcorrespondtoh1throughh4)aretheinitialsetofbasicvariables,asspecifiedinstep1ofthealgorithm.Thatis,
vB,1=v3=h1
vB,2=v4=h2
vB,3=v5=h3
vB,4=v6=h4
Further,theaj'scorrespondingtothissetofvariablesformthecolumnsofthebasisinversefortheintialbasisaswellasforanyotherbasicfeasiblesolution.
BecausetheinitialbasisisequaltoI,theassociatedright-handside(rhs),orb,is
EachelementofPisthengivenby(4.23),thatis,
TotherightofthePmatrixaretheachievementvectorvaluesasgivenby(4.15).Specifically,
Inpractice,thereisnoneedtolistalltherowsofP.Thatis,thealgorithmpresentedinChapter4permitsustolistonlytherowsofPcorrespondingtothespecificprioritylevelunderconsideration.Thus,theinitialtableauusedinthesolutionprocessneedonlycontainthefirstrowofP,orp(1)T.Wearenowreadytoproceedtothediscussion
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ofthesolutionprocedure,asadaptedtoourspecifictableau.
StepsofSolutionProcedure
Ourfirststepoftheactualsolutionprocedureforthepreviousexamplecombinesbothsteps1and2ofthe8-stepalogrithmofChapter4.
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Thatis,weconstructaninitialtableauwhereinourinitialsetofbasicvariablesarethenegativedeviationvariablesoftheLGPmodel(i.e.,h1,h2,h3andh4whichcorrespondtov3,v4,v5,andv6ofthegeneralform).Correspondingtothebasisisabasisinversethatistheidentitymatrix.Wethencomputeb,p(1)Tandu1as
Asaresult,ourinitialtableauisgivenas
v3 1 0 0 0 30v4 0 1 0 0 15v5 0 0 1 0 1000v6 0 0 0 1 40p(1)T 0 0 0 0 0
Wearenowreadytoproceedtostep3andcompute forv1,v2,v7,v8,v9,andv10(i.e.,forx1,x2,r1,r2,r3,andr4).Thesevaluesareasfollows:
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Proceedingtostep4,wenotethattherearenopositivevaluedelements(i.e.,forthesetofnonbasicanduncheckedvariables).Thus,wemovetostep8.
Thereaderisnowadvisedtopayparticularattentiontohowstep8iscarriedout,asitisespeciallytailoredforourspecifictableau.Fromstep8,wefirstnotethatneitherstoppingconditionholds.Thus,we"check"variablesv7andv8.
)
Checkedvariablesshallneverbecandidatestoenteranysubsequentbasis(astheirintroductionwouldonlyservetodegradetheachievementofhigher-levelgoals).Next,andalthoughnotspecificallyspelledoutinthealgorithm,wecrossouttheentiretableaurowassociatedwithp(k)T(i.e.,p(1)Tatthisstep).Finally,andstillasapartofstep8,weset
k=k+1=2
andthencomputetheentirenewbottomrowofthetableauasassociatedwithrankorprioritytwo.Thatis,wecomputep(2)Tandu2fromtheformulaspreviouslyspecified.
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Returningtostep2,thetableaunowassociatedwiththisstepisasshownbelow.
(Ö)v3 1 0 0 0 30 v7v4 0 1 0 0 15 v8v5 0 0 1 0 1000v6 0 0 0 1 40p(2)T 0 0 1 0 1000
Thecheckedvariablesarenowlistedtotherightofthetableau.
Movingtostep3,wecomputethevaluesof forv1,v2,v9,andv10(i.e.,wedonotevaluatethesevaluesforthecheckedvariables).Thus:
Proceedingtostep4,wenotethatv2isselectedastheenteringvariable.Thatis,
q=2
Wenextupdatetheenteringcolumn,a2,instep5:
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Movingtostep6,theleavingbasicvariableisdetermined.Thatis,
Asaresult,weseethatp=2.Thatis,thesecondbasicvariable(vB,2)isthedepartingbasicvariable.Fromthetableau,wenotethatvB,2correspondstov4.Thus:
vB,2=v4,thedepartingvariable
vq=v2,theenteringvariable
Beforeproceedingfurther,notethattheprevioussteps4though6maybemoreconvenientlycarriedoutdirectlyinconjunctionwiththetableau.Specifically,fromstep5,weentertheupdatedaq(a2inthiscase)directlytotherightofthematrix,asshownbelow.
Notethatthe"12"undera2,andintheverybottomposition,isthevalueof ascomputedinstep3.Directlytotherightofthea2columnisthecolumnheadedby"Q,"where
Thatis,Qissimplythesetofratiosfrom(4.26)andusedtodeterminethedepartingbasicvariable.TheQiwiththeminimumratioisdenotedbyanasterisk.
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Wearenowreadyforstep7,thepivotingoperation.Toaccomplishthis,weusethelasttableauaslistedabovewhereinq=2,p=2.Ourso-calledpivotelement,a2,2=1,iscircledinthistableau.Thepurelymechanicalprocedurebywhichanewbasisinversemaybeformedisnowdescribed.
First,wedefineB1asthe"old"basisinverseandbi,jastheelementofB1intheithrow,jthcolumn.Correspondingly, and arethe"new"basisinverseandnewelements,respectively.Toderivethenewbasisinverse,weusethefollowingformulas:
Usingtheseformulasplusthoseforcomputinguk,b,andp(k)T,wedeveloptheresultantnewtableau,asshownbelow,andreturntostep2.
BasisInverse b (Ö)v3 1 0 0 0 30 v7v2 0 1 0 0 15 v8v5 0 12 1 0 820v6 0 2 0 1 10p(2)T 0 12 1 0 820
Beforeproceedingtostep3,notecarefullythatv2hasreplacedv4(inthepositionofthesecondbasicvariable,vB,2)intheabovetableau.
Step3.
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Step4.v1istheenteringvariableandq=1.
Steps5and6arethensummarizedinthefollowingtableau:
Consequently,v6(thesmallestQratio)isthedepartingvariableandv1enters.Thisleadstothenexttableau,viathepivotingprocess:
(Ö)v3 1 2 0 1 20 v7v2 0 1 0 0 15 v8v5 0 4 1 8 740v1 0 2 0 1 10p(2)T 0 4 1 8 740
Step3.Wedeterminethat
Step4.Theenteringvariableisv10,thusq=10.
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Steps5and6areagainsummarizedinthetableau:
Thus,v10entersandv3departs.Thepivotingprocessthenleadstothefollowingtableau:
(Ö)v10 1 2 0 1 20 v7v2 0 1 0 0 15 v8v5 -8 -12 1 0 580v1 1 0 0 0 30p(2)T -8 -12 1 0 580
Step3.Wedeterminethat
Step4.No arepositivesowegotostep8.
Step8.Neitherstoppingruleismetsowecheckthevariablesv3,v4,andv9.Wealsocrossoutthep(2)Trowandsetk=k+1=3.
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Step2.Thetableaunowassociatedwiththestepisshownbelow:
(Ö)v10 1 2 0 1 20 v7v2 0 1 0 0 15 v8v5 8 12 1 0 580 v3v1 1 0 0 0 30 v4
v9p(3)T 1 2 0 1 20
Step3.Wenextevaluate forallnonbasic,uncheckedvariables.Thatis,
Step4.Gotostep8.
Step8.Thefirststoppingconditionofstep8issatisfied.Thus,westopwiththeoptimalsolution.
ListingtheResults
Havingfollowedthealgorithmthroughtoconvergence,wearenow,ofcourse,concernedwithdeterminingthespecificvaluesassociatedwiththesolution.Fromthelasttableaudevelopedwemayimmediatelyreadofftheoptimalprogram.Thatis,
Thus,intheoriginalmodel
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andallothervariablesarenonbasic,orzerovalued.Alternatively,wecouldhavecomputed as
vB=B1b=b
andvN=0bydefinition.
Theachievementvectoristhusdeterminedby
uk=c(k)TB1b''k
andis
And,infact,allotherinformationofinterest(e.g.,theshadowpricevectors)maybesimilarlyderivedviaaknowledgeofB1.
AdditionalTableauInformation
InChapter4,wediscussedcertainconditionsassociatedwithabasicsolution.Wenowdescribehowtheseconditionsmaybedetectedviaanexaminationofthetableauxusedinalgorithmicimplementation.
Feasibility
Thebasisisfeasibleifandonlyifb³0.WithLGP(exceptincertainspecialcasessuchasintegerLGPandsensitivityanalysis),wealwaysstartwithabasicfeasiblesolution,andviatheexaminationoftheqratiosneverpermitthebasistobecomeinfeasible.Consequently,iftheright-handside(i.e.,b)ofthetableaueverincludesanegative
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element,anerrorincomputationisindicated.Note,however,thatwhenusingadigitalcomputer,abivalueofzero(whichisacceptable),could
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bedeliveredassomeverysmallnegativevalue.Consequently,certaintolerancesaboutzeroarealwaysmaintainedandanyvaluewithinthesetolerancesistreatedasazero.
Implementability
Aslongas =0,thebasicfeasiblesolutionisimplementable.However,withLGPandquiteunlikeLPthesolutionprocesswillcontinueevenifu1>0,andasolutionasclosetoimplementableaspossiblewillbedeveloped.
Optimality
InChapter4,wenotedthatabasicfeasiblesolutionisoptimalwheneverallshadowpricevectors(dj),fornonbasicvariables,arelexicographicallynonpositive.Althoughthisistrue,wedonotexplicitlyexaminethesevectorsinourconvergencecheck.Rather,thisexaminationisachievedimplicitlyviastep8.Assuch,theoptimalityconditioncannotdirectlybeobservedbysimplyexaminingthetableauweuse.
AlternateOptimalSolutions
Analternateoptimalsolutionisindicatedwhen,inthelastimplementationofstep3,thereisatleastoneshadowpriceelementhavingavalueofzero.Thatis,ifsomedj(k)=0foranuncheckedandnonbasicvariableinthelastcyclethroughstep3,thenthereexistsanalternateoptimalsolution(actually,aninfinitenumberofalternateoptimalsolutionsandafinitenumberofalternateoptimalbasicfeasiblesolutions).Inourpreviousexample,therewerenoalternateoptimalsolutions.Givenanalternateoptimalsolution,thismeansthatthesetofoptimalsolutionstotheLGPmodelmaybeencompassedwithinaregionratherthanstrictlyontheboundaryofaregionasisthecaseinconventionalLP.SucharesultisuniquetoGPand(despitecertainprotestsbytheoreticians)is,infact,anadvantageinthe
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practicalsense.Thatis,ifaregionexistswhereinallsolutionshavethesameu*,wehaveavarietyofacceptablesolutionstopresenttoourdecisionmaker.Further,asiswellestablished,interiorsolutionsareinvariablymorestable(i.e.,lessaffectedbyvariationsinthemodelparameters)thenarethoseontheboundaries.
Degeneracy
Weobserveadegeneratebasicsolutionwheneveranyvariableinthebasistakesonavalueofzero.Thus,inthetableau,wesimplyneedtoobserveb.Examiningallthetableauxassociatedwithourpreviousexample,weseethatdegeneracywasneverencountered.Theimportanceofdegeneracywas,untilrelativelyrecently,felttobemainlyoftheoreticalinterest.Specifically,afewpathologicalexampleswereconstructedwheredegeneracyledtoacyclingphenomenoninthe
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pivotingprocess.However,analystsnowhaveobservedsuchcyclinginsomereal-world,large-scaleproblemsand,inparticular,inintegerprogramming.Fortunately,therearepracticalwaystoavoidoratleastalleviatetheproblemsassociatedwiththiscondition(CharnesandCooper,1961;Lasdon,1970;Murtagh,1981).
UnboundedPrograms
AsdiscussedinChapter4,theLGPachievementvector,uT,canneverbeunbounded.However,incertaincasestheprogrammaybe(i.e.,thevalueofsomevjthatis,nonbasicanduncheckedmayapproachinfinity).Thisinstancemaybedetectedinthefinaltableau.If,forsomevj(wherevjisnonbasicandunchecked),bothdj=0(i.e.,thedjcomputedinthelastimplementationofstep3)andaj£0,thenanunboundedprogramexists.Thismaymeanthattheright-handsideofsomegoalcouldbeincreased(ordecreased)withoutbound.Ifso,thissituationmaybeexaminedviaLGPsensitivityanalysis.
SomeComputationalConsiderations
Thedifferencebetweenthemannerinwhichanalgorithmisimplementedbyhand(i.e.,asinthepreviousexampleorasisshowninvirtuallyalltextbookdiscussions)andhowitisimplementedonthecomputercanbequitevast.Onlythemostnaiveindividualwouldexpecttodevelopanefficientcomputercodebysimplyemployingthestepsoutlinedinatextbookalgorithm.Further,thedevelopmentoftrulyefficientcomputersoftwareforLGP(orLP)implementationcombinesagreatdealofartandexperience,alongwithscience.Inthissectionwemerelypointoutbutafewfactorstoconsiderincomputerimplementation.
Oneoftheguidelinestypicallyusedinmodeldevelopment(andwhichwasemployedinthepreviousexample)isto"scale"themodel.Thatis,thedifferencebetweenthelargestandsmallestelements(i.e.,
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c(k),A,b)shouldbekeptassmallaspossible.Thisservestoreducecomputerroundingerrors.
Theactualgrowthofroundingerrorsshouldbemonitored.Onewayistosimply,onsomeperiodicbasis,checktherelation
Av=b
Wecomputethevalueoftheleft-handside(rowbyrow)byusingtheoriginalvaluesofAinconjunctionwiththemostrecentlycomputedvaluesofv.Theseresultsarethencomparedwiththeoriginalsetofb.If
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differencesare"significant"(thatis,theaccumulationofroundingerrorsexceedscertainprescribedlimits),thenthebasisinverse,B1,mustbe"cleanedup."Thiscleaning-upprocessisaccomplishedbyatechniqueknownas"reinversion"(Lasdon,1970;Murtagh,1981).
Thepivotingprocess,particularlyanunsophisticatedonesuchasdepictedinourexample(seeformulas5.8and5.9),cancontributetoroundingerrors.Thatis,ifap,q(thepivotelement)isverysmall,itsdivisionintoarelativelylargenumeratorcancreateanextremelylargenumberwhoseactualvaluemustbesubstantiallyroundedoffinthecomputer'srepresentationofthenumber.Assuch,givenatieinenteringordepartingvariables,onewaytobreakthetiewouldbetofavorthepivothavingthelargestpivotelement.
Yetanotherconsiderationinthepivotingprocedureisthechoiceofenteringvariables.Herewearenotreferringtoacaseinwhichtiesforthemostpositive exist.Rather,theenteringvariablecouldbeselectedasonehavingavalueof thatisactuallynotthelargestvalue.Further,suchachoicecouldleadtoconvergenceinfeweriterationsand/orwithlessroundingerror.InLP,numerous"pricingmethods"(Murtagh,1981)havebeendevelopedtoaccomplishsuchimprovementsandtheycouldbeused(and,insomecases,arebeingused)inLGPaswell.
Asonefurthercommentinourbriefsurveyofcomputationalrefinements,wenotethatthesometimesdazzlingperformanceofLPorLGPcodesisbasedonacriticalassumption,thatthematrixAissparse(i.e.,mostelementsarezeros).Infact,folkloreamongsttheLPanalystswouldhaveoneconcludethatsuchdensitiesrarelyexceed5%andare,inmostcases,less(orwellless)than1%.Thismythcontinuestobeperpetuatedbecausemostmathematicalprogramminganalysts(specificallythoseinthefieldsofoperationsresearchandmanagementscience)seemtoreligiouslyavoidproblemsin
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engineeringdesign(wheredensitiesof20%toeven100%mayexist).However,ifonedoesindeedhaveasparseproblem(orchoosestoconfinehisorherintereststosuchproblems),thentremendousadvancesincomputationalperformancearetobehad.Themostimmediateofthese,andmostobvious,isintherealmofdatastorage.CommercialLPcodesroutinelyusesophisticateddata"packing"and"unpacking"algorithmstotakeadvantageofsparsity.Takingamorebruteforcepointofview,wenotethatthestorageoftheAmatrixisusuallybestaccomplished(andalwaysbestaccomplishedifthematrixissparse)whenAisstoredcolumnbycolumnandonlynonzerocolumnentriesareactuallyrecorded.
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Examiningthestepsofthealgorithm,aslistedinChapter4(andascarriedoutintheexampleofthischapter),wecannotethatthebulkofoperationsarethoseinsteps3and7.However,inasmuchasinstep3weperformthefollowing:
andajisassumedsparse,thenumberofoperations(i.e.,ofanelementinp(k)Twithanelementinaj)canbekeptquitesmalliftheoperationsarerestrictedtoonlythosethatinvolvenonzeroelementsinaj.Thissimpleobservationalonecanprovidetremendousreductionincomputertime.
BoundedVariables
Onerelativelysimplecomputationalrefinement,andonenotdiscussedintheprevioussection,maybehadifoneisdealingwithboundedvariables(and,inparticular,iftherearealargenumberofsuchvariables).Specifically,ifsomeorallofthestructuralvariablesintheLGPmodelareboundedfromaboveforexample,if
whererjistheupperboundon thenonemaytakeadvantageofaslightlymodifiedversionofthe8-stepalgorithmandpossiblyachievesubstantialsavingsintimeandstorage.
However,ifonedecidesnottotakespecialnoteoftheboundedvariablesthen mustbeincludedin SuchinclusionincreasesthenumberofrowsinA,whichhasadirectimpactonbothcomputationtimeandcomputerstorage.Thus,inthissectionweshalldealdirectlywiththeboundedvariablesituation.
Priortodescribingtherevisionsnecessarytothealgorithm,letusfirstexaminetheeffectsofboundedvariablesonthebasicsolutionifsuchboundsarenotincludedinA.Recallthatthesetofgoalsmaybe
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writtenas
BvB+NvN=b
Letusthendefinethefollowing:
s=thesetofnonbasicvariablesatzero,and
s'=thesetofnonbasicvariablesattheirupperbound.
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Thus:
Ns=thecolumnsofAassociatedwithvNeS
Ns'=thecolumnsofAassociatedwithvNeS'
Then:
BvB+Ns'vN(s)+NsvN(s)=b
and,lettingvN(s)=0
vB=B1bB1Ns'VN(s')
Thatis,vBisnolongersimplyequaltoB=B1bifboundedvariablesareexplicitlyconsidered.
NotonlyisthereachangeinthewaywecomputevB,theremustalsobeachangeinthewayenteringanddepartingvariablesaredetermined.Ratherthanderivingthesenewrules,weshallsimplylisttheminourmodifiedalgorithmsteps.
Theproceduretofollowinthecaseofexplicitconsiderationofboundedvariablesisasfollows:
(1)Donotincludetheboundedconditioninthegoalset.Thatis,donotincludex'£rinAv=b.
(2)Modifystep4ofthe8-stepalgorithmofChapter4toreadasfollows:
Step4(revisedforboundedvariables).Examinethose ascomputedinstep3.If:
(a)vj=0,thenvjisacandidatetoenterthebasisif
(b)vj=rj,thenvjisacandidatetoenterthebasisif .Iftherearenocandidates,proceedtostep8.Otherwise,selectthecondidatewiththelargestabsolutevalueof toenterthebasis.Designatethe
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enteringvariablesasvq.
(3)Finally,wemodifystep6ofthealgorithmaslistedbelow:
Step6(revisedforboundedvariables).Ifvq=0thengoto(a),below.Otherwise,ifvq=rq,goto(b).
(a)Determinetheleavingbasicvariable(vB,p)bycomputingthefollowingratioforeachrow:
Ifai,q<0thenQi=(rB,i)vB,i)/ai,q;
10
Ifai,q³0thenQi=vB,i/ai,q.
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LetrowpbetherowwiththeminimumQratio.IfQp³rq,thenvqdoesnotenterthebasisbutissettoitsupperbound.Gotostep6(c).Otherwise,replacevB,pbyvqinthebasiswhere
vq=Qp
andvBisadjustedaccordingto
Nowgotostep7.
(b)Determinetheleavingbasicvariablebycomputingthefollowingratioforeachrow:
Ifai,j>0thenqi=(rB,ivB,i)/ai,q;
Ifai,q<0thenqi=vB,i/ai,q;
Ifai,q=0thenqi=¥.
LetrowpbetherowwiththeminimumQratio.IfQp³rq,thenvqdoesnotenterthebasisbutissettozeroandweproceedtostep6(c).Otherwise,replacevB,pbyvqinthebasiswhere
vq=rqQp
andvBisadjustedaccordingto
Nowgotostep7.
(c)RecomputevBwhere
vB=B1bB1Ns'vN(s')
andreturntostep2.
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Inadditiontofollowingtheprocedurelistedintheabovethreesteps,itshouldbeobviousthatwemustalwayskeeparecordofthosevariablesthatareattheirupperbound.Withtheseconsiderationsinmind,onemayemploythemodifiedalgorithmonLGPmodelshavingboundedvariables.
SolutionofLPandMinsumLGPModels
AlthoughthefocusofthisworkisonthelexicographicLGPmodel,itisstressedthatthealgorithmpresentedmayalsobeusedtosolvenumerousothermultiobjectivemodelsaswellasconventionalLPmodels(Ignizio,1976b,1982,1983c,forthcoming).Assuch,itprovidesasingleapproachforahostofmodels.
Touseouralgorithmforthesolutionofalternatemodels(e.g.,LPandminsumLGP)requiresthatsuchmodelsfirstbeplacedintheformatgivenby(4.1)-(4.4).Forexample,thegeneralformoftheLPmodel,asshownbelow:
s.t.
isconvertedtotheformof(4.1)-(4.4)viathefollowingsteps:
(1)Eachrigidconstraint(of5.11)istransformedintotheGPformatviatheproceduresummarizedinTable3.1(i.e.,negativeandpositivedeviationvariablesareaugmentedtoeachconstraint).
(2)Theachievementvectorisformedwiththerigidconstraintshavingfirstpriorityandthesingleobjectivehavingsecondpriority.
Asaresult(5.10)-(5.12)becomes
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Findvsoasto
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where
=oneifhiistobeminimized,andzerootherwise.
=oneifr8istobeminimized,andzerootherwise.
Ratherobviously,(5.13)-(5.15)isnowintheformspecifiedvia(4.1)-(4.4).
Wemaythenapplythe8-stepalgorithmofChapter4tothemodelbysimplymodifyingasinglestep.Specifically,step6ischangedbynotingthatiftherearenoQiratioswhereinai,q>0,westopwithanunboundedsolution.Thatis,dTxisthenunbounded.
Solutionoftheminsum(orArchimedean)LGPmodelisevenmorestraightforward.Specifically,ifthelexicographicLGPmodelhasbuttwotermsinuT,itisconsideredaminsumLGPmodel.Thatis:
u1=thetermassociatedwithallrigidconstraints,
u2=thetermassociatedwithallsoftgoals,whereintheyareweightedaccordingtoimportance.
6.DualityandSensitivityAnalysisWhenwesolveamodelwhatevertypeofmodelandwithwhateveralgorithmwearetypicallyonlyroughlymidphaseinouroverallprocessofanalysis.Thatis,thesolutionderivedforourlexicographicLGPmodelisonlyguaranteedtobevalidforthespecific,deterministicrepresentationused.However,intherealworld,thedata
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collectedtorepresentthemodelcoefficientsaretypicallyonlyestimates.Further,therecouldbeerrorsinthemodelingprocessor,oncethemodelhasbeenbuilt,thesystemitrepresentsmaychange.Assuch,itisvitaltoatleastexaminetheimpactofsuchchanges,errors,and/orestimatesonthesolutionasderivedviaouralgorithm.
Inconventionallinearprogramming,suchimpact,orsensitivityanalysis,maybeconductedinastraightforwardmannerviaasystematic
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procedure.ThisabilityistheresultoftwopropertiesofLP:thefactthatthemodelislinearandtheexistenceoftheLPdual.Further,thisabilityisofsuchpowerandimportancethatitalonecanexplainmuchofthereasonforthepopularityofLP.Infact,inmanycaseswetransformnonlinearmodelstoLPmodels(e.g.,viavariousapproximations)soastotakefulladvantageoftheabilitiesofLP,andinparticularitsabilitytoprovideafullanalysisofsensitivity.
AlloftheabilitiesinherentinconventionalLParealsoinherenttolexicographicLGP,includingtheabilitytoperformacompleteandcomprehensivesensitivityanalysis.Further,asisthecasewithLP,theexistenceofsuchsensitivityanalysisislargelybasedupontheexistenceandexploitationofthedualofthelexicographicLGPmodel.Consequently,beforedescribingsensitivityanalysisinLGP,weshallfirstdiscussthedevelopmentoftheLGPdualthemultidimensionaldual.
FormulationoftheMultidimensionalDual
Bythelate1960s,IhaddevelopedapartialsetoftoolsforsensitivityanalysisinLGP.However,tocompletetheapproach,itwasnecessarytoconstructarepresentationofthedualoftheinitial,orprimalLGPmodel.Bytheearly1970s,thisdualwhichIdenotedasthe"multidimensionaldual"wasestablished(Ignizio,1974a,1974b).However,mostexistingpapersandtextbookdiscussionsofthemultidimensionaldualhavebeenataratherelementarylevel.Asaconsequence,inthisworkwenowprovideaconcisebutmorecompleteandsomewhatmorerigorousdevelopment;onethatlendsitselftoawiderangeofboththeoreticalandpracticalextensions(Ignizio,1974a,1974b,1979b,1982a,1983b,1985a;MarkowskiandIgnizio,1983a,1983b).ThisdevelopmentisbasedonthetransformedformofthelexicographicLGPmodel(i.e.,theprimal)asgivenin(4.15)-(4.17)andrepeatedbelow.
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LGPprimal:Findvsoasto
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Ifwerecallthatwemayrewrite(6.2)as
thenthedualof(6.1),(6.4),and(6.3)mayimmediatelybewrittenasfollows:
LGPMultidimensionalDual:FindYsoasto
s.t.
ThosereaderswithafamiliaritywithLPwillrecognizethatthedevelopmentofthemultidimensionaldual,orMDD,fromtheLGPprimalfollowsasetofrulessimilartothoseusedtoformaconventionalLPdual.However,thereareseveralratherunusualfeaturesoftheMDDthatweshallnowcommenton.
First,notethatthesetofdualvariables,Y,isamatrixratherthansimplyavector.Further,eachelementofY,designatedas isunrestrictedinsign.Wethusdefine asfollows:
=theithdualvariableforthekthright-handside.
Thatis,thereisaseparateset(orvector)ofsuchdualvariablesforeachright-handsideof(6.6).
Second,weseethat(6.5),theMDD''achievementfunction,"isanorderedvectorforwhichweseekthelexicographicmaximum.
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Further,the"goalset"of(6.6)hasmultipleandprioritizedright-handsides.This
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lastfeatureisalsoreflectedintheuseofthesymbol forthelexicographicinequality,in(6.6).
AssociatedwiththeMDDisasetofconditionsthatencompassallthoseexistingwithinconventionalLP.Forexample,thedualoftheLGPdualistheprimal.(Forthereaderdesiringfurtherdetails,werecommendthefollowingreferences:Ignizio,1976b,1982a,1985a,forthcoming;MarkowskiandIgnizio,1983a,1983b.)
ANumericalExample
ThemechanicsofthedevelopmentoftheMDDmaybemosteasilyillustratedviaanexample.WethuslistthefollowingprimalLGPmodel.
Findxsoasto
s.t.
wherein:
and,becausetheinitialbasisalwaysconsistsofthenegativedeviationvariables:
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Usingthis,wemaythusformtheMDDasfollows:
FindYsoasto
s.t.
Thereadershouldnoteinparticularthatthefollowingrelationshipswereusedtoconstructtheabovedualform:
UsinganyalgorithmforsolutiontotheMDD(Ignizio,1976b,1982a,1985a),wewouldfindthattheoptimalMDDprogramisgivenas
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Thatis,thesolutiontothemodelforthefirstright-handside(i.e.,priority)is
Forthesecondright-handside,wehave
Inasubsequentsection,weshallseehowsuchresultsmaybeobtained.
InterpretationoftheDualVariables
Thedualvariables,Y,areinterpretedinamanneressentiallythesameasemployedinconventionallinearprogramming.Thatis,thesolutiontotheMDDgivenintheprevioussectionwas
Ifweweretosolvetheprimaloftheaboveproblem,wewouldfindthattheshadowpricevectorsforeachoriginalbasicvariable(i.e.,thehtermsorv3throughv6)are
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Thus,thefirstrowofY*correspondstotheshadowpricevectorforh1,thesecondrowtotheshadowpricevectorforh2,andsoon.
Asaresult,weseethat istheperunitcontributionofresourcei(oftheprimal)tothekthtermoftheachievementfunction.Forexample,bynotingthat
weseethatanincreaseofoneunittob1(whereb1=12)in(6.9)willresultin
(a)noimpactonu1,orimplementability,and
(b)animprovement(i.e.,reduction)inu2of75/3foreveryunitthatb1isincreased.
Theseobservationsaretrueonlyaslongasthefinal(optimal)basisremainsunchanged.Weshalldiscusshowsuchrangesmaybedeterminedlaterinthechapter.
SolvingtheMultidimensionalDual
InsensitivityanalysisforLGP,itisnotabsolutelyessentialthatoneknowhowtosolvetheMDDtoobtainthesolutionsimplylistedintheprevioussection.However,forcompletenessinpresentationweprovide,inthissection,abriefdescriptionofonerelativelyrecentandparticularlyefficientwaytoobtainthesolutiontotheMDD(and,asaresult,alsoobtainthesolutiontotheprimal).IhavedesignatedthismethodasthesequentialMDDsimplexalgorithm(Ignizio,1985a).
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OneparticularlyinterestingfeatureofthisapproachisthattheMDDissolvedviathesolutionofasequenceofconventionalLPmodels.
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Further,eachLPmodelinthesequencetypicallyisconsiderablysmallerthanitspredecessor.Welistthestepsofthisalgorithmasfollows:
Step1.Establishthemultidimensionaldualasgivenin(6.5)-(6.7).Setk=1.
Step2.FormtheLPmodelfrom(6.5)-(6.7)thatincludesonlythekthright-handsidevectorof(6.6).Solveusinganyconventionalsimplexalgorithm.Ifk=K,gotostep4.Otherwise,gotostep3.
Step3.Forthelinearprogrammingmodelpreviouslysolved,removeallnonbindingconstraints(thisisanalogoustothenonbasicvariable"checking"procedureinthealgorithmfortheprimal).Ifthesubsequentmodelhasnoconstraints,gotostep4.Otherwise,setk=k+1andreturntostep2.
Step4.ThepresentsolutionisthatwhichisoptimalfortheMDDandthekthright-handside.Thecorrespondingoptimalsolutiontotheprimalmodelisgivenbytheshadowpricesasassociatedwiththeinitialsetofbasicvariablesforthekthdualmodel.
Toillustrate,weshallsolvetheLGPmodelgiveninprimalformin(6.8)-(6.10)andinMDDformin(6.11)-(6.13).ThefirstLPmodeltobesolvedisthus:
s.t.
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andy(1)unrestricted.
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WemaynotethatthefirstfourconstraintsinconjunctionwiththelastfoursimplydenoteupperandlowerboundsonY(1).SolvingthisproblemviaanyconventionalLPalgorithm,weobtain
Further,forthissolution,boththeseventhandeighthconstraintsarenonbindingandthusmaybedroppedfromtheLPmodelfork=2.Thenext,andfinalLPmodelinthesequenceisthus:
s.t.
andy(2)unrestricted.
Again,solvingviaanyLPsimplexalgorithmweobtain
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Further,fromtheshadowpricesforthefinalLPtableau,wemaydeterminethattheoptimalprimalprogramis
Inactualpractice,theSequentialMDDSimplexalgorithmmaybeenhancedbynumeroussimplifications(Ignizio,1983a,1985a,forthcoming)andthusthealgorithmprovidesexceptionallygoodcomputationalperformance.Infact,whencomparisonsweremadewiththeverylatestversionoftheSequentialLGP,orSLGPmethod(aprimalbasedmethoddiscussedbrieflyinChapter2),thedualbasedschemewassubstantiallysuperior.
ASpecialMDDSimplexAlgorithm
ThealgorithmdiscussedabovemaybeusedtosolveanyLGPmodel(i.e.,initsMDDform).Inthissectionwediscussafarmorerestricteddualbasedalgorithmthatwillbeofconsiderableuseincertain,veryspecialsituations(Ignizio,1974a,1974b,1976b,1982a).IncludedamongsuchspecialsituationsisthatofLGPsensitivityanalysis.
TousethisspecialalgorithmwemustsatisfythefollowingconditionswithregardtotheLGPprimal:
(1)atleastoneelementinvB(i.e.,b)mustbenegative,and
(2)allshadowpricecolumnvectors,dj,mustbelexicographicallynonpositive.
Giventheseconditions,thealgorithmlistedbelowmaybeemployedsoastoregainfeasibilitywhilemaintainingtheoptimalitycondition.
Step1.SelecttherowwiththemostnegativevB,ielement.Thebasicvariableassociatedwiththisrowisthedepartingvariable.Denotethisrowasi=p.Tiesmaybearbitrarilybroken.
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Step2.Developthepricingvectorsforallk:
Step3.Priceoutallnonbasiccolumns,foralllevelsofk:
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Step4.Computeap,jforall(nonbasic)j:
where
b'p=thepthrowofB1
aj=thejthcolumnofA
Step5.Determinethenonbasicvariableassociatedwiththelexicographicallyminimum"columnratio"wherethiscolumnratioisgivenby
Designatethenonbasicvariablewiththelexicographicallyminimumrjasbeingcolumnj=q.Tiesmaybearbitrarilybroken.
Step6.Usingthepivotingprocedure,exchangetheenteringvariableforthedepartingvariableanddevelopthenewtableau.
Step7.Repeatsteps1through6untilallvB,iarenonnegative.
Todemonstratetheemploymentoftheabovealgorithm,weshallusetheexamplegivenbelow:
lexmin
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Abasicsolutiontothismodelisshowninthetableaubelow.Notecarefullythat,althoughbasic,theprogramisinfeasible.
BasisInverse bv2 1 1 0 0 2v1 0 1 0 0 12v5 3 2 1 0 2v6 1 0 0 1 2p(1)T 0 0 0 0 0p(2)T 3 2 1 0 2p(3)T 0 0 0 0 0
Usingp(k)T,wemaycomputealldj(forjeN):
WethusnotethatthistableaudoessatisfythetwoconditionsfortheemploymentofthespecialMDDsimplexalgorithm.Thatis,vB,1=v2=2andalldjarelexicographicallynonpositive.
Proceedingthroughthestepsofthealgorithm,wenotethatvB,1(i.e.,b1=v2)isthedepartingvariableandthusi=p=1.Movingtostep4wemaythencomputeallap,jvia(6.14).Step4leadsto:
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Becausea1,4anda1,7aretheonlya-valuesthatarenegative,weneedonlycomputethecolumnratiosassociatedwithv4andv7.Theseareasfollows:
Thus,r4istheminimumcolumnratioandsov4(j=q=4)istheenteringvariable.LettingvB,1=v2departandv4enter,ournewtableaubecomes
BasisInverse bv4 1 1 0 0 2v1 1 0 0 0 10v5 5 0 1 0 6v6 1 0 0 1 2
Sinceb³0,thisnewsolutionisnowfeasibleandoptimal.
DiscreteSensitivityAnalysis
WearenowreadytoproceedtoourpresentationofsensitivityanalysisinLGP(Ignizio,1982a).WebeginthiswithadiscussionofhowdiscretechangesintheoriginalLGPmodelaredealtwith.Weconsider:
·achangeinsome
·achangeinsomebi,
·achangeinsomeai,j,
·theinclusionofanewstructuralvariable,and
·theinclusionofanewgoalorrigidconstraint.
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Weshallconsider,inturn,howeachoneofthesechangesmaybedealtwith.However,letusfirstnotethatweshallplaceacaretoverthenewparametersoastodistinguishitsnewvaluefromitsoriginalvalue.
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Achangeinsome
Todeterminetheimpactofachangeinthevalueofsome wemustfirstdetermineifxjispresentlyabasicornonbasicvariable.Thatis,ifxjisnonbasicand ischangedto then:
Thatis,theonlyresultofsuchachangeisitsimpactonasingleshadowpricevectorelement.However,thiscouldresultinasolutionthatisnownotoptimalandtheoptimizingalgorithmmustthenbecontinued.
If,however, isassociatedwithsomexjthatisbasic,weaffectanentiresetofshadowpriceelements(i.e.,allthoseatlevelk)pluswemaychangethevalueofuk.Thatis,ifxjisbasicand ischangedtothen:
Achangeinsomebi
Thechangeofsomeelementoftheoriginalright-handsidevectorisfeltonbandu.Thatis,ifbiischangedtobithen:
Asbcanchange,itcanactuallycontainoneormorenegativeelements.Inthiscase,ourspecialMDDsimplexalgorithmmaybeemployedtoregainfeasibility.
Achangeinsomeai,j
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Themannerinwhichachangeinai,j(the''technologicalcoefficients")isdealtwithdependsonwhetherxjisbasicornonbasic.Ifxjisbasicwecanproceedthroughalongandrathercumbersomeproceduretodeterminetheimpact.Someanalystsfeelthatitmaybebettersimplytoresolvetheproblemfromthebeginning.Consequently,weshallonlydescribetheprocessusedwhenxjisnonbasic.Thatis,ifxjisnonbasicandai,jischangedtoâi,jthen:
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Thus,ifsomeai,j(xjnonbasic)ischangedtheentireajvectormaychangeand,inaddition,anentireshadowpricevectorcouldalsochange.Thus,achangeinai,jmayaffecttheoptimalityofasolution.
AddingaNewStructuralVariable
Theadditionofsomenewstructuralvariable,xj,hasanimpactidenticaltothatofachangeinanai,jasassociatedwithanonbasicvariable.Thatis,wemaythinkoftheajvectorforthenewvariableashavingpreviouslybeen0.Wethencomputethenewajand valuesasnoteddirectlyabove.Theresultwillbethateitherthepresentbasisisstilloptimalorthatitisnot.Inthefirstcasethisindicatesthatthenewvariableshouldnotenterthebasiswhereasinthesecondwenotethatthenewvariablewillimprovethepresentsolution.
AddingaNewGoal
Theadditionofanewgoal(whetheritisflexibleorarigidconstraint)requiressomewhatmoreworkthanwasrequiredforthepreviouschanges.First,ifsomenewgoal,sayGr,isaddedtothelexicographicLGPmodelwemustmakesurethat(forotherthanthecaseofrigidconstraints)itiscommensurablewithallothergoalsattheprioritylevelinwhichitisincluded.Second,thenewgoalwillincreasethesizeofthebasisbyonerowandcolumn.Third,todeterminethenewbasis,wemustfirst"operate"onthenewgoalsoastoeliminatethecoefficientsofanybasicvariablefromthegoal.Thiscanbeaccomplishedviaordinaryrowoperations.Finally,theinclusionofthenewgoalcanaffectboththefeasibilityand/oroptimalityofthepresentsolution.
ParametricLGP
ThepreviousdiscussionfocusedsolelyondiscretemodificationstotheoriginalLGPmodel.Inthissectionweshallbrieflyexamine
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parametricLGP,ortheinvestigationofchangesoveracontinuousrange(Ignizio,1982a).Indoingso,weshallconfineourpresentationtojustparametricchangesinbi(i.e.,theoriginalright-hand-sidevalueofgoali)and (theoriginalweightorcoefficientofvariablejatthekthprioritylevel).Weshalldealwiththislattercasefirst.
AParameterintheAchievementFunction
Theeasiestwaytoexplaintheapproachusedisviaasimplenumericalexample.WeshallusethefollowingLGPmodel:
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Rewritingthismodelingeneralformwehave
where
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Wefirstdeterminetheoptimalsolutiontotheabovemodel.Theresultisgiveninthetableaubelow:
BasisInverse bv3 1 0 0 0 20v2 0 1 0 0 35v5 0 3 1 0 115v6 0 1 0 1 95p(1)T 0 0 0 0 0p(2)T 0 1 1 2 305p(3)T 1 0 0 0 20
Intheoriginalmodelabove,thecoefficientofv6(i.e.,h4)atprioritylevel2(k=2)was"2."Letusnowdeterminetherangeofvaluesforthiscoefficientoverwhichtheaboveprogramisstilloptimal.Thus,inplaceof weshalluseaparameter,say"t."Thisresultsinachangein fortheabovetableau.Thatis,
or
Usingthenewvalueof wemaycomputetheshadowpricecolumnvectorsforallnonbasicvariables.Thisresultsin
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Now,inorderforthepreviousprogramtobeoptimal,alldjmustbelexicographicallynonpositive,or5+t<03+5£0t£0
Andthesethreerelationshipsaresimultaneouslysatisfiedonlywhen
0£t£3
Thus,aslongasallotherparametersremainunchanged,theweightonh4atprioritylevel2mayvaryfrom0upto3andtheoriginalprogramwillstillbeoptimal.Usingthesameprocess,wecouldexamine,oneatatime,alloftheremainingachievementfunctionparameters.However,itshouldbeobvioustothereaderthattheonlyotherparameterofinterestinthismodelwouldbetheweightassociatedwithh3atprioritylevel2.
AParameterintheRight-HandSide
Aparameterintheachievementfunctionmustbeexaminedforthatrangeoverwhichtheoriginalprogramisstilloptimal.Aparameterintheright-handside,however,willbeexaminedfortherangeoverwhichtheoriginalprogramisstillfeasible.Todemonstrate,letusexaminetherangeofvaluesofb1forwhichtheoriginalprogramremainsfeasible.Thatis,wereplace20(i.e.,thevalueofb1)by"t".Usingtheapproachdiscussedearliertoinvestigateadiscretechangeinbi,wefind:
Andthisnewbisfeasibleaslongas0£t<¥.
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Next,examinetherangeonb3.Thatis,wereplaceb3bytanddeterminethenewright-handside:
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andthisisfeasibleaslongast105³0.Thus,therangeonb3is
105£t<¥
Again,wecanperformsuchananalysisonanyoralloftheright-handsideelements.
7.ExtensionsInthisvolume,ourattentionhasbeenfocused,atleastforthemostpart,onlexicographiclineargoalprogramming(i.e.,LGPwithapreemptiveprioritystructure,ornon-Archimedeanweights).However,evenwiththisseeminglynarrowperspective,wehaveseenthatthemethodologypresentedmayalsobedirectlyappliedto:
·lexicographiclineargoalprogramming,
·minsum(orArchimedean)lineargoalprogramming,and
·conventionallinearprogramming.
Inaddition,withbutminormodificationwemayextendourapproachtoencompassevenfurtheralternativemethodsformultiobjectiveoptimizationincludingfuzzyprogramming,fuzzygoalprogramming,andthegeneratingmethod(Ignizio,1979,1981a,1982b,1983b;IgnizioandDaniels,1983;IgnizioandThomas,1984;Yu,1977;Zimmermann,1978).Assuch,themethodologythusfarpresentedprovidesanapproachtoeithersingle-objectivelinearprogrammingormostclassesofmultipleobjectivelinearmathematicalprogramming.
Goalprogramming(recallourdiscussioninChapter2)isnot,however,limitedtolinearsystems.Rather,powerfulextensionsoftheGPconceptexistandfindreal-worldapplicationinbothintegerand
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nonlinearmodels.InthissectionweshallverybrieflydiscussafewoftheseextensionssothatthereaderhasatleastsomefamiliaritywithotherthanstrictlylinearGPmodels.Inaddition,toconcludethischapter,weshalldescribeinteractiveapproachesviaGP,atopicofsomeconsiderablerecentinterest.
IntegerGP
SequentialIntegerGoalProgramming
AmongthefirstapproachestointegerGP,orIGP,wasthesocalledsequentialGPapproach.This
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sequentialapproachwasfirstproposedanddevelopedbyIgnizioandHussin1967forthesolutionofstrictlylinearGPmodels(Ignizio,1967,1982a,1983c;IgnizioandPerlis,1979;MarkowskiandIgnizio,1983b).However,thereisnoreasonwhytheconceptcannotbeusedinIGPoreveninnonlinearGP(NLGP)and,infact,itoftenfindssuchapplication.ThebasicthrustofsequentialGPistopartitiontheGPmodelintoarelatedsequenceofconventional,orsingle-objectivemodels.ThegeneralalgorithmforsequentialGPisgivenbelow.
Step1.EstablishtheGPformulationofthemodelandsetk=1(whereK=totalnumberofprioritylevelsinu).
Step2.Establishthemathematicalmodelforprioritylevel1only.Thatis,
Step3.Solvethesingle-objectiveproblemassociatedwithprioritylevelkviaanyappropriatealgorithm.
11Lettheoptimalsolutionbedesignatedas .
Step4.Setk=k+1.Ifk>K,gotostep6.Otherwise,gotostep5.
Step5.Establishtheequivalentsingle-objectivemodelforthenextprioritylevel(levelk).Thismodelisgivenas
minimizeuk=c(k)Tv
s.t.
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andthenproceedtostep3.
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Step6.Thesolutionvectorv*,asassociatedwiththelastsingle-objectivemodelistheoptimalsolutiontotheoriginalGPmodel.
NotecarefullythatthisalgorithmaspresentedisapplicabletoanytypeofGPmodel(i.e.,linear,integer,ornonlinear).AnumberoforganizationsutilizethisapproachindealingwithIGPmodelsandreportsuccessfulresults.Thepoweroftheapproachis,ofcourse,directlydependentuponthepowerofthespecificsingle-objectivealgorithm(orassociatedsoftware)asemployedinstep3.
ModificationstoClassicalApproaches
AnalternativeapproachtoIGPisavailablebytherelativelystraightforwardmeansofmodifying"classical"approachestointegerprogramming;suchasthecuttingplanemethod(Gomory,1958),branchandbound(LandandDoig,1960),ortheBalasalgorithm(Balas,1965).Idevelopedanumberofsuchalgorithmsinthelate1960sandearly1970s,someofwhichappearasChapter5ofGoalProgrammingandExtensions(Ignizio,1976b).Ingeneral,however,thesemodifiedalgorithmshavenotprovenveryeffective.Theyhaveallthelimitationsandproblemsassociatedwiththeirsingle-objectivecounterpartsandtypicallyonlyexhibitadequateperformanceonrelativelysmalltomodestsizemodels.Further,becausetheultimateperformanceofthemethodrestsprimariliyupontheefficiencyofthestructureofthealgorithmsanditscoding,itismyopinionthatoneisbetterofftakingadvantageofalreadydevelopedsingle-objectiveIPcodessuchasareavailableviasequentialIGPorthegoalaggregationmethod,asdescribednext.
GoalAggregation
Thegoalaggregationapproachproceedsasfollows(Ignizio,1985b).First,onesolvestherelaxedlinearIGPmodel(i.e.,theintegerrestrictionsareignored).Next,usingtheshadowpricecolumnvectors
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oftherelaxedmodel'sfinaltableau,weconstructasmallLPmodel.ThesolutiontothisLPmodelprovidesasetofweightsbywhichwemaychangethelinearIGPmodelintoaconventionalIPmodel,andthensolvebyconventionalIPsoftware.
Thestepsofthegoalaggregationalgorithmareasfollows:
Step1.FormthelinearIGPmodel.
Step2.Solve,viaanyLGPcode,therelaxedversionofthemodeldevelopedinstep1.Ifallintegervariablesareintegervalued,stop.Otherwise,deleteanydeviationvariableshaving andgotostep3.
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Step3.DeterminetheweightsforthemodifiedformbysolvingthefollowingLPmodel:
s.t.
Step4.Usetheweightsfoundinstep3toconvertthelinearIGPmodelintoanew,equivalent,linearIPmodel.Thatis,
Step5.Solvethemodeldevelopedinstep4byanyappropriateconventionallinearIPalgorithm.
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Toillustratetheapproach,considerthisexample:
x1+2x2+h1r1=1x2+h2r2=3
8x1+10x2+h3r3=8010x1+8x2+h4r4=80
andx1andx2mustbenonnegativeintegers.
Thefinaltableaufortherelaxedversionofthismodelisgiveninthefollowingtable.Alsolistedaretheshadowpricecolumnvectorsforeachnonbasicvariable.
BasisInverse bx2=v2 1/2 0 0 0 1/2h2=v4 1/2 1 0 0 5/2h3=v5 5 0 1 0 75h4=v6 4 0 0 1 76p(1)T 0 0 0 0 0p(2)T 1/2 1 0 0 5/2p(3)T 54 0 10 1 826
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Fromstep2weseewemustproceedasnotallintegervariableshaveintegervalues(i.e.,x2=1/2).Further,r1(orv7)mustbedeletedbecause
Movingtostep3,ourLPmodeltobesolvedisasfollows:
Althoughthestrictinequalityconstraintsareunusual,wemayaccommodatethembyaddingsomesmallnegativeamount,saye,totheright-handsides.Thus,wereplace<0by£e.Onesolutiontothismodelis
Consequently,theaggregatedIPmodelisgivenas
ThislastmodelmaybesolvedbyanyconventionalapproachtolinearIPandtheresultingprogram mustbeoptimalfortheoriginalIGPmodels.
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NetworkSimplex
Certainconventionalintegerprogrammingproblemsmayfindconvenientrepresentationasnetworks.Ifso,someverypowerfulapproachesbasedonnetworksimplexmaybeusedtoobtainasolution.Further,thetimerequiredtofindthatsolutionmaybeonlyafractionofthatrequiredbymore''conventional"approaches(Gloveretal.,1974).
ItisalsopossibletotreatcertainintegerGPproblemsasnetworksandthenapplyanextensionofconventionalnetworksimplextodevelopasolution.Justasintheconventional(i.e.,single-objective)case,whenthisispossiblethecomputationalefficiencymaybeconsiderablyenhanced.Assuch,forthoseIGPproblemsthatmayfindsuchrepresentation,thenetworksimplexapproachshouldcertainlybeconsidered(Ignizio,1983d,1983f;IgnizioandDaniels,1983;Price,1978).
HeuristicProgramming
DespiteyearsofsubstantialeffortinIP,itisstilltruethatmanyrealworldIPorIGPproblemsaresimplytoolargeand/orcomplexforsolution(oratleastsolutioninareasonableamountoftime)byexactmethods(Ignizio,1980a).Insuchcaseswetypicallyresorttospeciallytailoredheuristicmethods.Withthesemethodsweseekacceptablesolutionsinanacceptableamountoftime.ThereferencesprovideadiscussionofsomeoftheusesofheuristicprogramminginIGP(HarnettandIgnizio,1973;Ignizio,1976c,1979a,1981d,1984;Ignizioetal.,1982;MurphyandIgnizio,1984;Palmeretal.,1982).
NonlinearGP
SequentialNonlinearGoalProgramming
Usinganapproachanalogoustothatdescribedforsequentialinteger
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goalprogrammingwecould,ifwewished,solvenonlinearGPmodels.However,althoughthisispossibleandsometimesdone,itisbothinefficientandunnecessaryinthecaseofnonlinearGP.ThereasonforthisisthatconventionalnonlinearprogrammingalgorithmsandcodesmayeasilybemodifiedtohandledirectlythenonlinearGPcase.
ModificationstoClassicalApproaches
TheveryfirstapproachtononlinearGP(Ignizio,1963)wasbasedonthemodificationofexisting,conventionalnonlinearprogrammingmethods.Specifically,the"patternsearch"methodofHookeandJeeves(1961)wasconvertedintoanalgorithmandcodefornonlinearGP.Thesuccessofthisresultledmetoinvestigatesuchconversionsforvirtuallyallotherconventionalnonlinearprogrammingalgorithms.Inmostcases,thekeychangetotheconventionalcodeisthesimplereplacementofthescalarobjective
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function,z,withtheachievementvector,u.Forexample,atypicalsearchalgorithmforconventionalnonlinearprogrammingwillproceedasfollows:
Step1.Formulatethemodelandsett=1(wheretissimplyacounter).Determinesometrialsolutionanddenotetheprogramandsolutionasxtandzt.
Step2.Definearegion,or"neighborhood"aboutxtandthendetermineadirectionofimprovement,fromxt,withinthisneighborhood.
Step3.Determinea"steplength"alongthedirectionofimprovementfoundinstep2andmovealongthislengthtoanewprogram,xt+1.Evaluatezt+1.
Step4.Repeatsteps2and3untiloneconvergestotheoptimalsolution(aresultrarelyknownexceptfortrivialmodels)ormuststopaccordingtocertainstoppingrules(e.g.,toomanyiterations,lackofsignificantimprovement).
Now,tomodifythisapproachsoastohandleNLGPmodelswemaysimplyreplaceztbyutinsteps1and3.Modificationofmostsearchalgorithmstoaccommodatetheresultingevaluationofuistypicallyaminorprocedure.
Oftheclassicalalgorithmsconverted,thebestresults,byfar,havebeenachievedwithalgorithmsbasedupon:
·patternsearch(HookeandJeeves,1961),
·theGifffith/Stewarttechnique(1961),and
·generalizedreducedgradientmethods(Lasdon,1970).
TheresultsaccomplishedwiththemodifiedpatternsearchmethodforNLGP(Drausetal.,1977;Ignizio,1963,1976a,1979b,1981b;
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McCammonandThompson,1980;Ng,1981)havebeenparticularlyimpressive.Engineeringdesignproblems(e.g.,phasedarrays,transducerdesign)withthousandsofvariablesandhundredsofrowsareroutinelysolvedwiththelatestversionsofNLGP/PS(i.e.,nonlinearGPviamodifiedpatternsearch).Further,notonlyaresuchproblemsoflargesize,theyarealsotypicallyofhighdensity(infact,densitiesof100%arenotuncommon).Althoughsuchdensitiesalonetypicallydefeatsimplexbasednonlinearalgorithms,theNLGP/PScodesarerelativelyunaffected.
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InteractiveGP
Inthepastseveralyearstherehasbeen,insomequarters,anintenseinterestinso-calledinteractivemethodsfordecisionsupport.(GassandDror,1983;Ignizio,1979a,1979b,1981a,1981d,1982a,1983c;Ignizioetal.,1982;KhorramshahgolandIgnizio,1984;MasudandHwang,1981).Byinteractive,itismeantthatoneencouragesandutilizescertaindirectsupportofthedecisionmakerinactuallysolvingthedecisionmodel.Suchapproacheshavereceivedparticularlyfavorablereviewsbymanyofthoseinthefieldsofmulticriteriadecisionmaking(MCDM)andmultiobjectivemathematicalprogramming.
IncludedamongtheinteractiveapproachesproposedareanumberthatrequirethedecisionmakertositinfrontofaCRT(i.e.,monitor)andreacttoasetofalternatives.Thatis,heorsheindicatesthemost(or,perhaps,least)preferredalternativefromasmallgroupofalternatives.Usingthisinformation,theproceduremovestoanewgroupofalternativesandagainthedecisionmakerisaskedtorespond.Itishopedthatwiththeinputprovidedbythedecisionmaker,suchaprocedurewillleadtoeitherthe"optimal"resultoratleastonethatisacceptable.
Unfortunatelyatleastforthosewhowouldhopetousesuchmethodsonrealproblemswithrealdecisionmakersmanyoftheseinteractivemethodsarebasedonarathernaiveviewoftheworldand,inparticular,ofreal-worlddecisionmakers.Thatis,itis(atleastfrommyexperience)raretofindachiefexecutiveofficerwhoiswillingtoeventakethetimetobeshownhowsuchmethodswork,muchlessspendthetimerequiredtoprovidethenecessaryinteraction.Itisforthesereasonsthat(successful)interactiveversionsofGPtypicallyaredesignedtominimizethetimeandeffortrequiredofthebusydecisionmaker.Toclarify,weshallpresentjustoneinteractiveGPapproach,
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thetechniqueknownas"augmentedGP"(Ignizio,1979a,1979b,1981a,1981d,1982a,1983c;Ignizioetal.,1982).
AugmentedGPproceedsasifonewere,atfirst,simplysolvingaGPmodel.Thatis,theinitialinputrequiredofthedecisionmakerisasfollows:
(1)estimatesastotheaspirationlevelsasrequiredtoconvertallobjectives(ofthebaselinemodel)intogoals,and
(2)estimatesastotheorderoftheimportanceofallgoals.
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AsolutiontothisinitialGPmodelisthenderivedandpresentedtothedecisionmaker(s)asa"candidatesolution."Now,whendealingwithanynontrivialreal-worldproblemsubjecttomultiple(andconflicting)objectives,anysolutiondevelopedrepresentsacompromise.Consequently,somegoalsmaybecompletelyachievedwhereasothersarerelativelyfarfromachievement.Thus,ifthecandidatesolutionisconsideredunacceptable,thenextstepintheprocedureistoaskthatthedecisionmakerindicatejusthowmuchheorshewouldalloweachgoaltobedegradedifsuchdegradationwouldresultinsomesubstantialimprovementtoanothergoalorgoals.Theindicationofthesedegradationsthendefinesa"regionofacceptabledegradation."Wenextdevelopasubsetoftheefficient(i.e.,nondominated)solutionsinthisregionandpresentthissubsettothedecisionmaker.
Ifanymemberofthesubsetisacceptable,wemaystop.Otherwise,inexaminingthissubsetthedecisionmakercangetafairlygoodideaastohowmuchimpactthedegradationofonegoalwillhaveontheothers.Forexample,thedecisionmakermayfindthecostofacandidatesolutionacceptablebutmaynotbepleasedwith,say,theresultantsystemreliability.However,ifheorshenotesthatcostmustbedrasticallyincreasedtoproduceevenasmallincreaseinreliability,thisinformationmaywellresultinthedecisionmakeracceptingsomepreviouslyrejectedearliercandidatesolution.
Forthosereadersdesiringfurtherreferences(andexamplesofimplementation)onaugmentedGP,wesuggestthefollowingreferences:Ignizio(1979a,1979b,1981a,1981d,1982a,1983c)andIgnizioetal.(1982).
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Notes1.Huss,infact,consideredtheresultssotransparentastonotwarrantevenanattemptatpublication.However,inrecentyearssequentialGPhasbecomethefocusofsomeratherintense,althoughbelated,interest.
2.ForlexicographicLGP,tobeprecise.
3.ThisspecificmodelisalsoaspecialcaseofthemoregeneralformofmathematicalprogrammingmodelknownastheMULTIPLEXmodel(Ignizio,forthcoming).
4.Thesymbol denotes"forall."
5.Inequation3.3onlyoneoftherelations(£,=,or³)isassumedtoholdforeacht.
6.Where,again,onlyoneoftherelations(£,=,or³)holdsforeachi.
7.UsingthetransformedformoftheLGPmodel,thereadershouldbeabletoproveeasilythattheprogram(i.e.,v)foranLGPmodelcanitselfbeunboundedbut,ofcourse,uTwillbefinite.Further,standardpivotingrulesprecludeapivottoanunboundedprogram(i.e.,somevj®¥).
8.Inactualpractice,andinthealgorithmtofollow,thereisnoneedtogeneratebutoneelementofeachdjforeachiteration.
9.Numerousapproachesforaccomplishingstep7havebeendescribed.Ourapproach,tobedescribedintheexampletofollow,usesthe"explicitformoftheinverse."
10.NotethatrB,iistheupperboundontheithbasicvariable.Further,bi=vB,i.
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11.Note,however,thatthecolumncheckoperationofcontinuousLGPisnotpermittedhereifanyvariablesarerestrictedtobeintegers.
12.Where"È"denotestheunionoperator.
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