Table Of ContentS. A. MirHassani · F. Hooshmand
Methods and
Models in
Mathematical
Programming
Methods and Models in Mathematical Programming
(cid:129)
S. A. MirHassani F. Hooshmand
Methods and Models in
Mathematical Programming
S.A.MirHassani F.Hooshmand
DepartmentofMathematics DepartmentofMathematics
andComputerScience andComputerScience
AmirkabirUniversityofTechnology AmirkabirUniversityofTechnology
(TehranPolytechnic) (TehranPolytechnic)
Tehran,Iran Tehran,Iran
ISBN978-3-030-27044-5 ISBN978-3-030-27045-2 (eBook)
https://doi.org/10.1007/978-3-030-27045-2
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Abbreviations and Acronyms
The following table summarizes all abbreviations and acronyms, used throughout
thisbook,andthereadercanturntothislistatanytime.
Abbreviation Definition
B&B Branch-and-bound
B&C Branch-and-cut
BIP Binaryintegerprogramming
CLP Constraintlogicprogramming
CSP Cuttingstockproblem
CTP Coursetimetablingproblem
DCP Diseasecontrolproblem
FSSP Flowshopschedulingproblem
IFP Innerfitpolygon
IIS Irreducibleinfeasiblesubsystem
IP Integerprogramming
IPM Interiorpointmethod
KKT KarushKuhnTucker
LHS Lefthandside
LP Linearprogramming
LPR Linearprogrammingrelaxation
(mu) Monetaryunit
MCER McCormickenveloperelaxation
MILP Mixedintegerlinearprogramming
MINLP Mixedintegernonlinearprogramming
MIP Mixedintegerprogramming
MIS Minimalinfeasiblesubsystem
MPDP Multi-parametricdisaggregationtechnique
NDP Networkdesignproblem
NFP Nofitpolygon
NLP Nonlinearprogramming
(continued)
v
vi AbbreviationsandAcronyms
Abbreviation Definition
OR Operationsresearch
PDIP Pre-disasterinvestmentproblem
PIP Pureintegerprogramming
PLF Piecewiselinearfunction
PPSP Petroleumpipelineschedulingproblem
PSP Portfolioselectionproblem
QP Quadraticprogramming
RHS Right-handside
RLT Reformulationandlinearizationtechnique
SHP Sellorholdproblem
SNP Socialnetworkproblem
SOS1 Specialorderedsetoftype1
SOS2 Specialorderedsetoftype2
SSP Sportsschedulingproblem
TSP Travelingsalesmanproblem
VRP Vehicleroutingproblem
VRPTW Vehicleroutingproblemwithtimewindow
WRMP Waterresourcemanagementproblem
Preface
Operations research (OR) is a branch of mathematical sciences that aims to utilize
mathematicalmodeling,statisticalanalysis,andmathematicaloptimizationsystem-
atically to identify the best choice among available options. The source of optimi-
zation problems is the need for industrial, economic, and social decision-makers,
who are looking for the best possible way to solve the problems in their area of
expertise. In most cases, the number of options discussed is exceptionally high, so
thatitisnotpossibletoexpressallofthemexplicitly.Therefore,theseproblemsare
initially described by managers (problem description). Then, in order to identify
acceptable alternatives, possible choices are implicitly introduced in the form of
mathematical relations, so that any solution satisfying these relations is a valid
choice. In this way, mathematical models are formulated as a combination of vari-
ablesandparametersintheformofmathematicalequalitiesandinequalities(model
construction). Finally, methods of solving mathematical models are utilized to
determine the optimal choice from a range of possible options (model solving).
Trackingandreplicatingthisprocessinaninteractiveanddynamicenvironmentcan
leadtothedesiredoutcome,i.e.,theselectionofthebestoption.
Today, with the efforts of the scientific community, efficient algorithms have
been developed and embedded in mathematical software to satisfy the need of OR
practitioners. These software can receive the mathematical models, compile the
collected information in a specific form, and call an appropriate solver. Then, the
modelissolvedviamathematicalalgorithms,andtheresultisreturnedtotheuser.In
this way, theuser will beable todesign,formulate, andthen solve theproblemby
himselforherselfwithoutdealingwiththemathematicalcomplexityofalgorithms.
Thesuccessinthisareaisreliantontheconstructionofanaccurateandappropriate
model that not only observes the problem conditions but also considers the limita-
tionsofalgorithmsasaneffectivefactor.
Theinitialdescriptionofapracticalproblemisoftenbasedontheexpressionofa
collection of logical statements that should be used in the form of mathematical
equations and inequalities. Although describing a problem in the form of a mathe-
maticalmodelisanessentialstepinthereal-worldapplicationofoptimization,little
attention has been paid in the literature to the challenges and delicacies of the
vii
viii Preface
modelingprocess.Thistextbookfocusesonmathematicalmodelingandintroduces
its principles and inherent weaknesses. It attempts to describe the process of
constructing and evaluating models and explicitly outlines the required rules and
regulations so that the reader is able to generalize and reuse concepts in other
problemsbyrelyingonmathematicallogic.
This textbook is organized into six chapters: General principles of modeling,
differenttypesofmodelsandtheirimportance,themainstepsofmodeling,andthe
general structure of an optimization model are described in Chap. 1. Chapter 2
introduces the main components of an optimization model, provides an accurate
classification of models’ types, and presents a short review of solution methods.
Chapter3expresseshowtoincorporatelogicalpropositionsaslinearconstraintsinto
optimization models. Chapter 4 refers to some nonlinear functions frequently
appearing in optimization problems (e.g., multiplication of two or more variables,
piecewiselinearandseparablenonlinearfunctions,minimaxandmaximinfunctions,
absolute value and fractional functions) and explains some techniques for their
linearization.Inthecasesinwhichlinearizationisnotpossible,someapproximation
techniques are introduced. Chapter 5 addresses the qualitative comparison of
models, discusses the impact of the number of constraints and variables on the
quality of models, and introduces the ideal formulation. Furthermore, some tech-
niques of improving formulations are presented, and at the end of this chapter,
constraintlogicprogrammingisintroduced.Chapter6developsasetofreal-world
applications of mathematical programming, which are closely related to the meth-
odologypresentedinthepreviouschapters.Ineachcase,theproblemisdefined,the
model’scomponentsareintroduced,andthen,themodelispresentedinthegeneral
form. Afterward, the model is solved for a given set of data with the aid of a
computer, and the results are analyzed. Each chapter contains different exercises;
somearerelativelysimple,andothersaremorechallenging.
Undergraduate and postgraduate students of different academic disciplines,
including applied mathematics, computer science, operations research, industrial
engineering, and management science would find this textbook a suitable option
preparingthemforjobsandresearchesrequiringmodelingtechniques.Furthermore,
this textbook can be used as a reference for experts requiring advanced skills of
modelbuildingintheirjobs.
ThefirstversionofthisbookwaspublishedinPersianbyAmirkabirUniversity
ofTechnologyPress.Here,itisthesecondversionwritteninEnglish.
Thisbookistheresultofseveralyearsofteaching,andideasbroughtforwardby
readerswillbeappreciatedforimprovingpossibleshortcomingsinthefuture.
Tehran,Iran S.A.MirHassani
F.Hooshmand
Contents
1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 ModelandItsTypes. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 ValueofMathematicalModels. . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 StepsofModelBuilding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4.1 Step1:IdentifyandDefinetheProblem. . . . . . . . . . . . 4
1.4.2 Step2:CollectDataandPresenttheModel. . . . . . . . . . 4
1.4.3 Step3:SolvetheModel. . . . . . . . . . . . . . . . . . . . . . . . 4
1.4.4 Step4:ValidatetheModel. . . . . . . . . . . . . . . . . . . . . . 5
1.4.5 Step5:ProvideResultstotheOrganization. . . . . . . . . . 5
1.5 StructureofaMathematicalModel. . . . . . . . . . . . . . . . . . . . . . 5
1.5.1 DecisionVariables. . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5.2 Constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5.3 ObjectiveFunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5.4 Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5.5 SolutionTypes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5.6 ModelAssumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 NotesandReferences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.7 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 MainComponentsofMathematicalModels. . . . . . . . . . . . . . . . . . . 13
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 FreeVariables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.2 FiniteDomainVariables. . . . . . . . . . . . . . . . . . . . . . . 14
2.2.3 Semi-continuousVariables. . . . . . . . . . . . . . . . . . . . . . 15
2.2.4 PartialIntegerVariable. . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 SimpleBoundandGeneralBound. . . . . . . . . . . . . . . . 16
2.3.2 SoftandHardConstraints. . . . . . . . . . . . . . . . . . . . . . 17
ix
x Contents
2.3.3 RedundantandBindingConstraints. . . . . . . . . . . . . . . 18
2.3.4 CutsandLazyConstraints. . . . . . . . . . . . . . . . . . . . . . 19
2.3.5 ConstraintsandSpecialOrderedSets. . . . . . . . . . . . . . 21
2.3.6 ChanceConstraints. . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.7 ConvertingConstraints. . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 ObjectiveFunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 MathematicalModels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5.1 ContinuousandDiscreteModels. . . . . . . . . . . . . . . . . 26
2.5.2 LinearandNonlinearModels. . . . . . . . . . . . . . . . . . . . 26
2.5.3 SingleandMulti-ObjectiveModels. . . .. . . . . . .. . . . . 30
2.5.4 ConstrainedandUnconstrainedModels. . . . . . . . . . . . 31
2.5.5 StaticandDynamicModels. . . . . . . . . . . . . . . . . . . . . 31
2.5.6 DeterministicandNon-deterministicModels. . . . . . . . . 32
2.5.7 LPModelswithIntervalCoefficients. . . . . . . . . . . . . . 32
2.5.8 SingleandMulti-stageModels. . . . . . . . . . . . . . . . . . . 34
2.5.9 SingleandMulti-levelModels. . . . . . . . . . . . . . . . . . . 36
2.6 SolutionMethods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.6.1 SolvingLPProblems. . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.6.2 SolvingMIPProblems. . . . . . . . . . . . . . . . . . . . . . . . . 43
2.6.3 SolvingNLPProblems. . . . . . . . . . . . . . . . . . . . . . . . 44
2.6.4 SolvingMulti-objectiveProblems. . . . . . . . . . . . . . . . . 50
2.6.5 DynamicProgrammingtoSolveOptimization
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.6.6 SolvingBi-levelProblems. . . . . . . . . . . . . . . . . . . . . . 52
2.6.7 SolvingProblemswithIntervalCoefficients. . . . . . . . . 55
2.7 NotesandReferences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.8 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 ModelsandMathematicalLogic. . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2 AtomicPropositions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3 CompositePropositions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4 PropositionalFunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.5 PropositionsandBinaryVariables. . . . . . . . . . . . . . . . . . . . . . . 70
3.6 ConstraintsandIndicators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.6.1 ConditionalIndicators. . . . . . . . . . . . . . . . . . . . . . . . . 75
3.6.2 ComplementarityRule. . . . . . . . . . . . . . . . . . . . . . . . . 79
3.6.3 Bi-ConditionalIndicators. . . . . . . . . . . . . . . . . . . . . . . 80
3.6.4 Either-orConstraints. . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.6.5 ConstraintSelection. . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.6.6 Not-EqualityConstraints. . . . . . . . . . . . . . . . . . . . . . . 94
3.6.7 If-ThenConstraints. . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.6.8 If-Then-ElseConstraints. . . . . . . . . . . . . . . . . . . . . . . 100
3.7 MiscellaneousExamples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.8 NotesandReferences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.9 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
