Optimization for Decision Making: Linear and Quadratic Models

501 Pages·2010·5.757 MB·English

by  Katta G. Murty (auth.)

International Series in Operations Research
& Management Science
Volume 137
SeriesEditor
FrederickS.Hillier
StanfordUniversity,CA,USA
SpecialEditorialConsultant
CamillePrice
StephenF.AustinStateUniversity,TX,USA
Forfurthervolumes:
http://www.springer.com/series/6161
Katta G. Murty
Optimization for
Decision Making
Linear and Quadratic Models
ABC
KattaG.Murty
UniversityofMichigan
Dept.IndustrialandOperationsEngineering
1205BealAvenue
AnnArborMI48109-2117
USA
[email protected]
and
Professor,SystemsEngineeringDepartment
KingFahdUniversityofPetroleumandMinerals
Dhahran
SaudiArabia
ISSN0884-8289
ISBN978-1-4419-1290-9 e-ISBN978-1-4419-1291-6
DOI10.1007/978-1-4419-1291-6
SpringerNewYorkDordrechtHeidelbergLondon
LibraryofCongressControlNumber:2009932413
(cid:176)c SpringerScience+BusinessMedia,LLC2010
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten
permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,
NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Usein
connectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,
orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare
notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject
toproprietaryrights.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
To thememoryof mythesisadvisorDavid
Gale(my lastteacherintheformaleducation
system)who inspiredmeintodoingresearch
inOptimization,and tomymotherKatta
Adilakshmi(myfirstteacher inchildhood
athome)
Preface
Iwasfortunatetogetmyfirstexposuretolinearprogramminginacoursetaughtby
thefatherofthesubject,GeorgeDantzig,attheUniversityofCalifornia,Berkeley,
in the fall of 1965. It was love at first sight! I fell in love with linear program-
ming(LP),optimizationmodels,andalgorithmsingeneralrightthen,andbecame
inspired to work on them. Another of my fortunes was to have as my thesis ad-
visor David Gale, who along with Harold Kuhn and Albert Tucker contributedto
the developmentof optimality conditions.Eventually,I started teaching optimiza-
tionintheIOEDepartmentattheUniversityofMichigan,AnnArbor,andusingit
in applicationsmyself, and I wouldnow like to share thisbackgroundwith future
generationsofstudents.
Level oftheBook andBackground Needed
Thisisafirst-yeargraduate(Master’s)leveltextbookonoptimizationmodels,linear
and quadratic, for decision making, how to formulate real-world problems using
thesemodels,useefficientalgorithms(botholdandnew)forsolvingthesemodels,
andhowtodrawusefulconclusions,andderiveusefulplanninginformation,from
theoutputofthesealgorithms.
Itbuildsontheundergraduate(Junior)levelbook OptimizationModelsforDe-
cision Making Volume 1 on the same subject (Murty (2005)of Chap.1), which I
postedatthepublicaccesswebsite:
http://ioe.engin.umich.edu/people/fac/books/murty/optimodel/,
fromwhichyoucandownloadthewholebookforasmallcontribution.Readerswho
arenewtothesubjectshouldreadthisJunior-levelbooktoacquirethebackground
forreadingthisgraduate-levelbook.
vii
viii Preface
WhyAnother Book onLinear Programming
WhenfriendslearnedthatIwasworkingonthisbook,theyaskedme,“Whyanother
bookonlinearprogramming(LP)?”Therearetworeasons:
1. Almost all the best-known books on LP are mathematics books, with little
discussion on how to formulatereal-world problemsas LPs and with verysimple
modelingexamples.Withinashorttimeofbeginningworkonapplications,Ireal-
izedthatmodelingcouldactuallybeascomplexasprovingmathematicalresultsand
requiresveryspecialskills.Togetgoodresults,itisimportanttomodelreal-world
problemsintelligently.Tohelpthereaderdevelopthisskill,Idiscussseveralillus-
trativeexamplesfrommyexperience,andincludemanyexercisesfromavarietyof
applicationareas.
2. All the available books on LP discuss only the simplex method (developed
basedonthestudyofLPusingthesimplex,oneofthesolidsinclassicalgeometry)
and perhaps existing interior point methods (developed based on the study of LP
using the ellipsoid). All these methods are based on matrix inversion operations
involving every constraint in the model in every step, and work well for LPs in
which the coefficient matrix is very sparse. We discuss also a new method being
developedbasedonthestudyofLPusingthesphere,whichusesmatrixinversion
operationssparinglyandseemswellsuitedtosolvelarge-scaleLPs,andthosethat
maynothavethepropertyofbeingverysparse.
Contents oftheBook
Chapter 1 contains a brief account of the history of mathematical modeling, the
Gasuss–Jordaneliminationmethodforsolvinglinearequations;thesimplexmethod
for solving LPs and systems of linear constraints including inequalities; and the
importanceofLPmodelsindecisionmaking.
Chapter 2 discusses methods for formulating real-world problems, including
thosein whichthe objectivefunctiontobe optimizedisa piecewise linearconvex
functionandmultiobjectiveproblems,aslinearprograms.Thechapterisillustrated
withmanyexamplesandexercisesfromavarietyofapplications.
Chapter3explainstheneedforintelligentmodelinginordertogetgoodresults,
illustrated with three case studies: one from a container terminal, the second at a
bus-rentalcompany,andthethirdataninternationalairport.
Chapter 4 discusses the portion of the classical theory of polyhedralgeometry
thatplaysan importantrolein the studyoflinearprogrammingandin developing
algorithmsforsolvinglinearprograms,illustratedwithmanynumericalexamples.
Chapter5treatsdualitytheory,optimalityconditionsforLP,andmarginalanaly-
sis;andChap.6discussesthevariantsoftherevisedsimplexmethod.Bothchapters
dealwithtraditionaltopicsinlinearprogramming.InChap.5wediscussalsoopti-
malityconditionsforcontinuousvariablenonlinearprogramsandtheirrelationship
tooptimalityconditionsforLP.
Preface ix
Chapter 7 discusses interior point methods (IPMs) for LP, including brief
descriptionsoftheaffinescalingmethod,whichisthefirstIPMtobedeveloped,and
theprimal-dualIPM,whichismostcommonlyusedinsoftwareimplementations.
Chapter 8 discusses the sphere methods, new IPMs that have the advantage of
using matrix inversion operations sparingly, and thus are the next generation of
methodsforsolvinglarge-scaleLPs.
Chapter9discussesextensionsofthespheremethods–toconvexandnonconvex
quadraticprograms,andto0–1integerprogramsthroughquadraticformulations.
Additional Exercises
Exercises offer students a great opportunity to gain a deeper understanding of
the subject. Modeling exercises open the student’s mind to a variety of applica-
tions of the theory developed in the book and to a variety of settings where such
useful applications have been carried out. This helps them to develop modeling
skills that are essential for a successful career as a practitioner. Mathematical ex-
ercises help train the student in skills that are essential for a career in research
or a career as a higher-levelpractitioner who can tackle very challenging applied
problems.
Becauseofthelimitationsonthelengthofthebook,notallexercisescouldbein-
cludedinit.Theseadditionalexerciseswillbeincludedinthewebsiteforthebookat
springer.cominthenearfuture,andevenmoreaddedovertime.Someoftheformu-
lationexercisesatthewebsitedealwithmedium-sizeapplications;theseproblems
canbeusedascomputationalprojectproblemsforgroupsoftwoorthreestudents.
FormulatingandactuallysolvingsuchproblemsusinganLPsoftwarepackagegives
thestudentatasteofreal-worlddecisionmaking.
Citing References inthe Text
Attheendofeachchapter,welistonlyreferencesthatarecitedinthetext.Thusthe
list of referencesis actually small; it does not provideextensive bibliographiesof
thesubjects.Forreaderswhoareinterested,wereferthemtootherbooksavailable
thathaveextensivebibliographies.
We use the following style for citing references: A citation such as “Wolfram
(2002)”referstothepaperorbookofWolframofyear2002listedamongreferences
attheendofthecurrentchapterwherethiscitationappears.Alternately,areference
suchas“(Dikin(1967)ofChap.1)referstothedocumentofDikinofyear1967in
thelistofreferencesattheendofChap.1.
x Preface
Solutions Manual
Springer will host the solutions manual at springer.com,allowing token access to
registeredadoptingfaculty.
Acknowledgments
Ireceivedcomments,encouragement,andotherhelpfrommanypeopleinpreparing
this book,includingRichard Chen, Jose Dula, Stein-Erik Fleten, Santosh Kabadi,
ShantisriKatta,JustinLin,MohammadOskoorouchi,A.RaviRavindran,Romesh
Saigal,ArvindSharma,EricSvaan,VolodymyrBabich.Iamgratefultoallofthem.
Iamalsogratefultomyeditorsandsupporters,FredHillier,CamillePrice,andthe
Springerteamforconstantencouragement.FinallyIthankmywifeVijayaKattafor
beingmycompanioninalltheseyears.
Conclusion
Optimum decision making is all about improving lives. As the Sanskrit proverb
(jiivaassamastaassukhinoobhavamtu)showninTeluguscriptsays:
Ihopereaderswillusethesemethodstoimprovethelivesofalllivingbeings!
9April2009 KattaGopalakrishnaMurty
OtherTextbooks by Katta G.Murty
Linear and Combinatorial Programming, first published in 1976, available from
R.E.Krieger,Inc.,P.O.Box9542,Melbourne,FL32901.
Linear Programming, published in 1983, available from John Wiley & Sons, 111
RiverStreet,Hoboken,NJ07030-5774.
Linear Complementarity, Linear and Nonlinear Programming, published in
1988 by HeldermannVerlag, Germany;now available as a downloadfor a volun-
tary contribution at: http://ioe.engin.umich.edu/people/fac/books/murty/linear
complementarity webbook/
Preface xi
Operations Research: Deterministic Optimization Models, published in 1995,
availablefromPrentice-HallInc.EnglewoodCliffs,NJ07632.
Network Programming, published in 1992, available from Prentice-Hall Inc.
Englewood Cliffs, NJ 07632; also as a download for a voluntary contribution at:
http://ioe.engin.umich.edu/people/fac/books/murty/network programming/
Optimization Models for Decision Making: Volume 1, Junior Level, available as
a download for a voluntary contribution at: http://ioe.engin.umich.edu/people/fac/
books/murty/opti model/
Computational and Algorithmic Linear Algebra and n-Dimensional Geometry,
Sophomorelevel,availableasadownloadforavoluntarycontributionat:http://ioe.
engin.umich.edu/people/fac/books/murty/algorithmic linear algebra/
Contents
1 Linear Equations, Inequalities, LinearProgramming:
ABriefHistoricalOverview................................................. 1
1.1 MathematicalModeling,Algebra,SystemsofLinear
Equations,andLinearAlgebra ........................................ 1
1.1.1 EliminationMethodforSolvingLinearEquations ......... 2
1.2 ReviewoftheGJMethodforSolvingLinearEquations:
RevisedGJMethod .................................................... 5
1.2.1 GJ Method Using the Memory Matrix
toGeneratetheBasisInverse ................................ 8
1.2.2 TheRevisedGJMethodwithExplicitBasisInverse....... 11
1.3 LackofaMethodtoSolveLinearInequalitiesUntilModernTimes 14
1.3.1 The Importance of Linear Inequality
ConstraintsandTheirRelationtoLinearPrograms ........ 15
1.4 FourierEliminationMethodforLinearInequalities.................. 17
1.5 HistoryoftheSimplexMethodforLP................................ 18
1.6 The Simplex MethodforSolvingLPs and Linear
InequalitiesViewedasanExtensionoftheGJMethod.............. 19
1.6.1 GeneratingthePhaseIProblemifNoFeasible
SolutionAvailablefortheOriginalProblem................ 19
1.7 TheImportanceofLP.................................................. 21
1.7.1 MarginalValuesandOtherPlanningTools
thatcanbeDerivedfromtheLPModel..................... 22
1.8 Dantzig’sContributionstoLinearAlgebra,Convex
Polyhedra,OR,ComputerScience.................................... 27
1.8.1 ContributionstoOR.......................................... 27
1.8.2 ContributionstoLinearAlgebraandComputerScience ... 27
1.8.3 Contributionsto the MathematicalStudy
ofConvexPolyhedra ......................................... 28
1.9 InteriorPointMethodsforLP ......................................... 29
1.10 NewerMethods......................................................... 30
1.11 Conclusions............................................................. 30
1.12 HowtoBeaSuccessfulDecisionMaker? ............................ 30
1.13 Exercises................................................................ 31
References...................................................................... 37
xiii