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.

IntroductiontoLinearGoalProgramming

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

SageQASSSeriesDepartmentofPoliticalScience

UniversityofIowaIowaCity,IA52242

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Series/Number07-056

IntroductiontoLinearGoalProgramming

JamesP.Ignizio

PennsylvaniaStateUniversity

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Copyright©1985bySagePublications,Inc.

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Whencitingaprofessionalpaper,pleaseusetheproperform.RemembertocitethecorrectSageUniversityPaperseriestitleandincludethepapernumber.Oneofthetwofollowingformatscanbe

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

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

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,

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

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

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

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

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.

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.

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).

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

"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

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

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:

Step1:Selectoneobjectivefrom(3.1)or(3.2)andtreatitasthesingleLPobjective.Typically,thisistheobjectivethatisperceivedtobeof"mostsignificance."

Step2:Convertallremainingobjectivesintogoalsbymeansofestablishingassociatedaspirationlevels.Thatis,

maxfr(x)

Page23

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)

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

"lexmin,"ofanorderedvector.

LexicographicMinimum:Givenanorderedarray,saya,ofnonnegativeelements(aks),thesolutiongivenbya(1)ispreferredtoa(2)if andallhigherorderterms(i.e.,a1,a2,,ak1)areequal.Ifnoothersolutionispreferredtoa,thenaisthelexicographicminimum.

Page25

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

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

objectiveis

Next,wenotethatwewouldliketolimitproductiontimeperdayto40minutestotal,althoughthefirmdoesindicatesomeflexibilityaboutthislimit.Thus,wewritethisgoalas

Page28

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:

Page29

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

Page30

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,

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,

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

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:

Page34

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

However,recallthatvwaspartitionedaccordingto(4.7)andthustheabovetermmayberewrittenas

Page35

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,

Page36TABLE4.1LGPTableau

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.

Page37

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

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

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).

Page39

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.

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.

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

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

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;

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

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

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

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.,

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

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

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

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

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

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.

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.

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).

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.

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.

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.

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

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

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<¥.

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

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.

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

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

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;

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,

thetechniqueknownas"augmentedGP"(Ignizio,1979a,1979b,1981a,1981d,1982a,1983c;Ignizioetal.,1982).

AugmentedGPproceedsasifonewere,atfirst,simplysolvingaGPmodel.Thatis,theinitialinputrequiredofthedecisionmakerisasfollows:

(1)estimatesastotheaspirationlevelsasrequiredtoconvertallobjectives(ofthebaselinemodel)intogoals,and

(2)estimatesastotheorderoftheimportanceofallgoals.

Page90

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).

Page91

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.

11.Note,however,thatthecolumncheckoperationofcontinuousLGPisnotpermittedhereifanyvariablesarerestrictedtobeintegers.

12.Where"È"denotestheunionoperator.

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