9 introduction to linear goal programming quantitative applications in the social sciences james p ignizio

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IntroductiontoLinearGoalProgramming
SageUniversityPapersSeries.Quantitative
ApplicationsintheSocialSciences;No.
07-056
Ignizio,JamesP.
SagePublications,Inc.
0803925646
9780803925649
9780585216928
English
Linearprogramming.
1985
T57.74.I351985eb
519.7/2

Linearprogramming.


IntroductiontoLinearGoalProgramming



SAGEUNIVERSITYPAPERS
Series:QuantitativeApplicationsintheSocialSciences
SeriesEditor:MichaelS.Lewis-Beck,UniversityofIowa
EditorialConsultants
RichardA.Berk,Sociology,UniversityofCalifornia,LosAngeles
WilliamD.Berry,PoliticalScience,FloridaStateUniversity
KennethA.Bollen,Sociology,UniversityofNorthCarolina,Chapel
Hill
LindaB.Bourque,PublicHealth,UniversityofCalifornia,Los
Angeles
JacquesA.Hagenaars,SocialSciences,TilburgUniversity
SallyJackson,Communications,UniversityofArizona
RichardM.Jaeger,Education,UniversityofNorthCarolina,
Greensboro
GaryKing,DepartmentofGovernment,HarvardUniversity
RogerE.Kirk,Psychology,BaylorUniversity
HelenaChmuraKraemer,PsychiatryandBehavioralSciences,
StanfordUniversity
PeterMarsden,Sociology,HarvardUniversity
HelmutNorpoth,PoliticalScience,SUNY,StonyBrook
FrankL.Schmidt,ManagementandOrganization,UniversityofIowa
HerbertWeisberg,PoliticalScience,TheOhioStateUniversity
Publisher

SaraMillerMcCune,SagePublications,Inc.
INSTRUCTIONSTOPOTENTIALCONTRIBUTORS
Forguidelinesonsubmissionofamonographproposaltothisseries,
pleasewrite
MichaelS.Lewis-Beck,Editor


SageQASSSeries
DepartmentofPoliticalScience
UniversityofIowa
IowaCity,IA52242




Page1

Series/Number07-056

IntroductiontoLinearGoalProgramming
JamesP.Ignizio
PennsylvaniaStateUniversity

SAGEPUBLICATIONS
TheInternationalProfessionalPublishers
NewburyParkLondonNewDelhi



Page2

Copyright©1985bySagePublications,Inc.
PrintedintheUnitedStatesofAmerica
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inanyformorbyanymeans,electronicormechanical,including
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Forinformationaddress:

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Whencitingaprofessionalpaper,pleaseusetheproperform.
RemembertocitethecorrectSageUniversityPaperseriestitleand
includethepapernumber.Oneofthetwofollowingformatscanbe


adapted(dependingonthestylemanualused):

(1)IVERSEN,GUDMUNDR.andNORPOTH,HELMUT(1976)
"AnalysisofVariance."SageUniversityPaperseriesonQuantitative
ApplicationsintheSocialSciences,07-001.BeverlyHills:Sage
Publications.
OR
(2)Iversen,GudmundR.andNorpoth,Helmut.1976.Analysisof
Variance.SageUniversityPaperseriesonQuantitativeApplications
intheSocialSciences,seriesno.07-001.BeverlyHills:Sage
Publications.




Page3

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

Page5

SeriesEditor'sIntroduction
Asthisseriesofvolumesinquantitativeapplicationshasgrown,we
havebeguntoreachouttothoseineconomicsandbusinessinthe
samewaythatsomeoftheearliervolumesappealedmostespecially
tothoseinpoliticalscience,sociology,andpsychology.Ourgoal,
however,continuestobethesame:topublishreadable,up-to-date
introductionstoquantitativemethodologyanditsapplicationto
substantiveproblems.
Oneofthefastest-growingareaswithinthefieldsofoperations
researchandmanagementscience,intermsofbothinterestaswellas
actualimplementation,isthemethodologyknownasgoal
programming.Fromitsinceptionintheearly1950s,thistoolhas
rapidlyevolvedintoonethatnowencompassesnearlyallclassesof
multipleobjectiveprogrammingmodels.Ofcourse,ithasalso
undergoneasignificantevolutionduringthattime.
InAnIntroductiontoLinearGoalProgramming,JamesIgnizio(a
pioneerandmajorcontributortothefield,whosefirstapplicationof
goalprogrammingwasin1962inthedeploymentoftheantenna
systemfortheSaturn/Apollomoonlandingmission)providesa
concise,lucid,andcurrentoverviewof(a)thelineargoal
programmingmodel,(b)acomputationallyefficientalgorithmfor
solution,(c)dualityandsensitivityanalysis,and(d)extensionsofthe

methodologytointegeraswellasnonlinearmodels.Toaccomplish
thisextentofcoverageinashortmonograph,Igniziousesamatrixbasedpresentation,aformatthatnotonlypermitsaconciseoverview
butonethatisalsomostcompatiblewiththemannerinwhichrealworldmathematicalprogrammmingproblemsaresolved.
Thetextisintendedforindividualsinthefieldsofoperations


research,managementscience,industrialandsystemsengineering,
computerscience,andappliedmathematicswhowishtobecome
familiarwithlineargoalprogramminginitsmostrecentform.
Prerequisitesforthe




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textarelimitedtosomebackgroundinlinearalgebraandknowledge
ofthemoreelementaryoperationsinmatricesandvectors.
RICHARDG.NIEMI
SERIESCO-EDITOR



Page7

Acknowledgments
Duringmorethantwodecadesofresearchinandapplicationsofgoal
programming,Ihavebeeninfluenced,motivated,andguidedbythe
worksandwordsofnumerousindividuals.Severalofthese
individuals,inparticular,havehadamajorimpact.Theseinclude

AbrahamCharnesandWilliamCooper,theoriginatorsoftheconcept
ofgoalprogramming;VeikkoJääskeläinen,anindividualwhose
substantialimpactonthepresent-daypopularityofgoalprogramming
hasbeenalmosttotallyoverlooked;andPaulHuss,whointroduced
metogoalprogramming,influencedmydevelopmentofthefirst
nonlineargoalprogrammingalgorithmandapplication(in1962),and
wasmyco-developerofthefirstlarge-scalelineargoalprogramming
code(in1967).Ialsowishtoacknowledgetheinfluenceofthetext,
AdvancedLinearProgramming(McGraw-Hill,1981)byBruce
Murtagh.Murtagh'soutstandingtextanditsconciseyetlucidstyle
havehadparticularinfluenceonthepresentationfoundinChapter4
ofthiswork.Finally,particularthanksaregiventoTomCavalierand
LauraIgnizioforcommentsandcontributionstotheoriginaldraftof
thismanuscript.




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1.
Introduction
Althoughgoalprogramming(GP)isitselfadevelopmentofthe
1950s,ithasonlybeensincethemid-1970sthatGPhasfinally
receivedtrulysubstantialandwidespreadattention.Muchofthe
reasonforsuchinterestisduetoGP'sdemonstratedabilitytoserveas
anefficientandeffectivetoolforthemodeling,solution,andanalysis
ofmathematicalmodelsthatinvolvemultipleandconflictinggoals
andobjectivesthetypeofmodelsthatmostnaturallyrepresentrealworldproblems.YetanotherreasonfortheinterestinGPisaresultof
agrowingrecognitionthatconventional(i.e.,singleobjective)

mathematicalprogrammingmethods(e.g.,linearprogramming)do
notalwaysprovidereasonableanswers,nordotheytypicallyleadtoa
trueunderstandingofandinsightintotheactualproblem.
Purpose
Itisthenthepurposeofthismonographtoprovideforthereadera
briefbutreasonablycomprehensiveintroductiontothemultiobjective
mathematicalprogrammingtechniqueknownasgoalprogramming,
withspecificfocusontheuseofsuchanapproachindealingwith
linearsystems.Further,inprovidingsuchanintroduction,weshall
attempttominimizeboththeamountandlevelofsophisticationofthe
associatedmathematics.Assuch,theonlyprerequisiteforthereader
issomeexposuretolinearalgebraandaknowledgeofthemore
elementaryoperationsonmatricesandvectors.Itshouldbe
emphasizedthatafamiliaritywithlinearprogramminghasnotbeen
assumed,althoughit






Page10

isbelievedlikelythatmostreaderswillhavehadsomepreviouswork
inthatarea.Ithasbeenmyattempttoprovideabriefandconcise,but
reasonablyrigoroustreatmentoflineargoalprogramming.
WhatIsGoalProgramming?
Atthispoint,letuspauseandreflectuponsomeofthenotions
expressedabove,inconjunctionwithafewnewideas.First,letus
notethatgoalprogramminghas,initself,nothingtodowithcomputer

programming(e.g.,FORTRAN,Pascal,LISP,BASIC).Thatis,
althoughanyGPproblemofmeaningfulsizewouldcertainlybe
solvedonthecomputer,thenotionof''programming"inGP(or,for
thatmatter,inthewholeofmathematicalprogramming)isassociated
withthedevelopmentofsolutions,or"programs,"foraspecific
problem.Thus,thename"goalprogramming"isusedtoindicatethat
weseektofindthe(optimal)program(i.e.,setofpoliciesthatareto
beimplemented)foramathematicalmodelthatiscomposedsolelyof
goals.Lineargoalprogramming,orLGP,inturnisusedtodescribe
themethodologyemployedtofindtheprogramforamodelconsisting
solelyoflineargoals.
WeshallwaituntilChapter3torigorouslydefinethenotionofa
"goal."Here,wesimplynotethatanymathematicalprogramming
modelmayfindanalternaterepresentationviaGP.Further,notonly
doesGPprovideanalternativerepresentation,italsooftenprovidesa
representationthatisfarmoreeffectiveincapturingthenatureofrealworldproblemsproblemsthatinvolvemultipleandconflictinggoals
andobjectives.
Finally,wenotethatconventional(i.e.,singleobjective)mathematical
programmingmaybeeasilyandeffectivelytreatedasasubset,or
specialclass,ofGP.Forexample,asweshallsee,linearprogramming
modelsareeasilyandconvenientlytreatedas"GP"models.Infact,


andalthoughtheideaisconsideredradicalbythetraditionalists,itis
notreallynecessarytostudylinearprogramming(LP)ifonehasa
thoroughbackgroundinLGP.
OntheUseofMatrixNotation
Asmentionedearlier,oneoftheprerequisitesofthistextisthatthe
readerhashadsomepreviousexposuretomatricesandvectors,and
theassociatednotation,terminology,andbasicoperationsemployed

insuchareas.Althoughatfirstglancethematrix-basedapproachused
hereinmayappearabitformidabletosomeofthereaders,beassured




Page11

thatitspurposeisnottocomplicatetheissue.Instead,bymeansof
suchanapproachweareableto:
(1)provideapresentationthatistypicallyclearer,moreconcise,and
lessambiguousthanifanonmatrix-basedapproachwereemployed;
and
(2)providealgorithmsinaformfarclosertothatactuallyemployed
indevelopingefficientcomputerizedalgorithms.
Ofparticularimportanceistheconcisenessprovidedviaamatrixbasedapproach.Usingmatricesandmatrixnotationweareable,in
thisslimvolume,tostillcovernearlyalloftheusefulfeaturesof
lineargoalprogramming(e.g.,areasonablycomputationallyefficient
versionofanalgorithmforlineargoalprogramming,acomprehensive
presentationofduality,anintroductiontosensitivityanalysis,and
evendiscussionsofvariousextensionsofthemethodology).Without
theuseofthematrix-basedapproach,therewouldhavebeenno
possibilityofcoveringthisamountofmaterialineventwoorthree
timestheamountofpagesusedherein.
Forthosereaderswhoseexposuretomatricesandvectorshasbeen
limited,orisapartofthenowdistantpast,thereisnoreasonfor
apprehension.Thelevelofthematrix-basedpresentationemployed
hasbeenkeptquiteelementary.

2.

HistoryandApplications
Althoughthereexistnumerousrelatedearlierdevelopments,thefield
ofmathematicalprogrammingtypicallyistracedtothedevelopment
ofthegenerallinearprogrammingmodelanditsmostcommon
methodforsolution,designatedas"simplex."LPandsimplexwere,in


turn,developedin1947byateamofscientists,ledbyGeorge
Dantzig,underthesponsorshipoftheU.S.AirForceprojectSCOOP
(ScientificComputationOfOptimumPrograms).TheLPmodel
addressedasingle,linearobjectivefunctionthatwastobeoptimized
subjecttoasetofrigid,linearconstraints.Oneofthebestdiscussions
ofthisradicalnewconceptisgivenbyDantzighimself(Dantzig,
1982).
Withinbutafewyears,LPhadreceivedsubstantialinternational
exposureandattention,andwashailedasoneofthemajor
developmentsofappliedmathematics.Today,LPisprobablythemost
widely




Page12

knownandcertainlyoneofthemostwidelyemployedofthemethods
usedbythoseinsuchfieldsasoperationsresearchandmanagement
science.However,aswithanyquantitativeapproachtothemodeling
andsolutionofrealproblems,LPhasitsblemishes,drawbacks,and
limitations.Ofthese,ourinterestisfocusedontheinabilityoratleast
limitedabilityofLPtodirectlyandeffectivelyaddressproblems

involvingmultipleobjectivesandgoals,subjecttosoftaswellasrigid
(orhard)constraints.
ThedevelopmentofGPoneapproachforeliminatingoratleast
alleviatingtheabove-mentionedlimitationsofLPoriginatedinthe
early1950s.Atthistime,CharnesandCooperaddressedaproblem
seeminglyunrelatedtoLP(orGP):theproblemof(linear)regression
withsideconditions.Tosolvethisproblem,CharnesandCooper
employedasomewhatmodifiedversionofLPandtermedthe
approach"constrainedregression"(Charnesetal.,1955;Charnesand
Cooper,1975).
Later,intheir1961text,CharnesandCooperdescribedamore
generalversionofconstrainedregression,onethatwasintendedfor
dealingwithlinearmodelsinvolvingmultipleobjectivesorgoals.
Thisrefinedapproachwasdesignatedasgoalprogrammingandisthe
conceptthatunderliesallpresent-dayworkandgeneralizationsofGP.
Inthesame1961text,CharnesandCooperalsoaddressedthenotso
insignificantproblemofattemptingtomeasurethe"goodness"ofa
solutionforamultipleobjectivemodel.Theyproposedthree
approaches,allofwhicharestillwidelyemployedtoday.These
approacheswereeachbasedonthetransformationofallobjectives
intogoalsbymeansoftheestablishmentofan"aspirationlevel,"or
"target.''Forexample,anobjectivesuchas"maximizeprofit"might
berestatedasthegoal:"Obtainxormoreunitsofprofit."Obviously,
anysolutiontotheconvertedmodelwilleitherbeunder,over,or


exactlysatisfytheprofitaspiration.Further,anyprofitunderthe
desiredxunitsrepresentsanundesirableorunwanteddeviationfrom
thegoal.Consequently,CharnesandCooperproposedthatwefocus
onthe"minimizationofunwanteddeviations,"aconceptessentially

identicaltothenotionof"satisficing"asproposedbyMarchand
Simon(Morris,1964).Usingthisconcept,CharnesandCooper
specifiedthefollowingthreeformsofGP:
(1)ArchimedeanGP(alsoknownas"minsum"or"weighted"GP):
Hereweseektominimizethe(weighted)sumofallunwanted,
absolutedeviationsfromthegoals;




Page13

(2)ChebyshevGP(alsoknownas"minimax"GP):Ourpurposeisto
minimizetheworst,ormaximumoftheunwantedgoaldeviations;
and
(3)non-ArchimedeanGP(alsoknownas"preemptivepriority"GPor
"lexicographic"GP):Hereweseektheminimum(moreprecisely,the
lexicographicminimum)ofanorderedvectoroftheunwantedgoal
deviations.
ItisofparticularinterestthatLP(oranysingle-objective
methodology)aswellasArchimedeanandChebyshevGPmayallbe
consideredasspecialcasesofnon-ArchimedeanGPandthustreated
bythesamegeneralmodelandalgorithm(Ignizio,forthcoming).Asa
result,inthisworkwefocusourattentiononnon-ArchimedeanGPor,
morespecifically,onlexicographiclineargoalprogramming.
InadditiontodescribingthelinearGPconceptandproposingthe
abovethreemeasuresforevaluation,CharnesandCooperalso
outlined(again,intheir1961text)algorithmsforsolution.Evidently,
however,actualsoftwarefortheimplementationofsuchalgorithms
wasnotdevelopeduntilthelate1960s.Infact,totheauthor's

knowledge,thefirstcomputercodeforGPwastheonethatI
developedin1962(Ignizio,1963,1976b,1979b,1981b)forthe
solutionofnonlinearGPmodelsmorespecifically,forthedesignof
theantennasystemsfortheSaturn/Apollomoonlandingprogram.
Asaresultofthesuccessofthealgorithmandsoftwarefornonlinear
GP,orNLGP,Igainedaconsiderableappreciationofandinterestin
GP.Asaconsequence,in1967,whenfacedwitharelativelylargescaleLGPmodel(onethatincludedthelexicographicminimum,or
preemptiveprioritynotions),Idevelopedacomputercodefor
lexicographicLGPasbasedonasuggestionbyPaulHuss(personal
communication,1967).Inatelephoneconversationwithme,Huss
proposedthatonesolvethelexicographicLGPmodelasasequenceof


conventionalLPmodels.Thissuggestionwasrefinedandsoftwarefor
theprocedurewasdevelopedbythesummerof1967.Thisspecific
approach,whichIdesignateassequentialgoalprogramming(orSGP;
orSLGPinthelinearcase),althoughunsophisticated,
1resultedinacomputerprogramcapableofsolvinganLGPmodelof

sizesequivalenttothosesolvedviaLP(Ignizio,1967,1982a;Ignizio
andPerlis,1979).Infact,untilquiterecently,SLGP(alsoknownas
iterativeLGPor"decomposed"LGP)evidentlyhasofferedthebest
performanceofanypackageforLGP(havingnowbeensupplantedby
theMULTIPLEXcodesforGP;Ignizio,1983a,1983e,1985a,1985b,
forthcoming).