Building and Solving Mathematical Programming Models: 50 Practical Examples (International Series in Operations Research & Management Science, 329)

408 Pages·2022·8.355 MB·English

by  José Manuel García Sánchez

International Series in
Operations Research & Management Science
José Manuel García Sánchez
Building
and Solving
Mathematical
Programming
Models
50 Practical Examples
International Series in Operations Research
& Management Science
FoundingEditor
FrederickS.Hillier,StanfordUniversity,Stanford,CA,USA
Volume 329
SeriesEditor
Camille C. Price, Department of Computer Science, Stephen F. Austin State
University,Nacogdoches,TX,USA
EditorialBoardMembers
Emanuele Borgonovo, Department of Decision Sciences, Bocconi University,
Milan,Italy
BarryL.Nelson,DepartmentofIndustrialEngineering&ManagementSciences,
NorthwesternUniversity,Evanston,IL,USA
BruceW.Patty,VeritecSolutions,MillValley,CA,USA
MichaelPinedo,SternSchoolofBusiness,NewYorkUniversity,NewYork,NY,
USA
RobertJ.Vanderbei,PrincetonUniversity,Princeton,NJ,USA
AssociateEditor
JoeZhu,TheBusinessSchool,WorcesterPolytechnicInstitute,Worcester,MA,USA
The book series International Series in Operations Researchand Management
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(cid:129)Advancedexpositorybooksthatextendandunifyourunderstandingofpartic-
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Theseriesemphasizesthefollowingfourareas:
MathematicalProgramming: Including linearprogramming, integerprogram-
ming, nonlinear programming, interior point methods, game theory, network opti-
mization models, combinatorics, equilibrium programming, complementarity
theory, multiobjective optimization, dynamic programming, stochastic program-
ming,complexitytheory,etc.
Applied Probability: Including queuing theory, simulation, renewal theory,
Brownian motion and diffusion processes, decision analysis, Markov decision
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applications,etc.
ProductionandOperationsManagement:Includinginventorytheory,produc-
tion scheduling, capacity planning, facility location, supply chain management,
distributionsystems,materialsrequirementsplanning,just-in-timesystems,flexible
manufacturing systems, design of production lines, logistical planning, strategic
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ing, public policy, military operations research, humanitarian relief and disaster
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ThisbookseriesisindexedinScopus.
á
José Manuel García S nchez
Building and Solving
Mathematical Programming
Models
50 Practical Examples
JoséManuelGarcíaSánchez
HigherTechnicalSchoolofEngineering
UniversityofSeville
Seville,Spain
ISSN0884-8289 ISSN2214-7934 (electronic)
InternationalSeriesinOperationsResearch&ManagementScience
ISBN978-3-030-97625-5 ISBN978-3-030-97626-2 (eBook)
https://doi.org/10.1007/978-3-030-97626-2
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To my students
Preface
The book presents the construction and resolution of 50 optimization problems,
divided into five levels of difficulty, although assigning a level of difficulty is
subjective in some cases. The first levels are dedicated toward problems oriented
to learning the model construction methodology. The difficulty increases with the
levels, up to the complex optimization environments presented in the last level.
Despite this, understanding all the problems, including the last levels, is feasible
after knowing the modeling methodology and progressing with analysis of the
problemsfromthebeginning.
A wide range of problems are presented, from problems associated with games
(UnblockMe,sudokus)tologisticalproblems,plantdistribution,production,oper-
ations scheduling, management, and resource allocation. Many of them with an
innovative nature, since they had not been modeled until now, and even others
presentscenarioswithnewcharacteristics.
Besides all this, another fundamental characteristic of the book is that all the
modelsareimplementedinanoptimizationlibrary,LINGO,andaresolvedinorder
to analyze the correct construction of the model. The book incorporates a simple
guidetobuildandsolvemodelsinLINGO,easilyandinanyformatfortheinputand
output data (text files, spreadsheets, or databases). I have preferred to include the
LINGO code for the building of each model within the proper section to each
problem, instead of in an annex, so that the reader can more easily visualize the
wayinwhicheachcomponentofthemodelisimplementedinLINGO.
Somethingveryimportantinthebookisthefirstchapter.Itcontainsasummary
ofthemaincharacteristicsofthemodelingmethodology.Allthereferencesusedfor
themodelingoflogicalpropositionsarepresented,addingnewdetailsandexplana-
tions. Obviously, not all of the details of the methodology are included in this
chapter. Having access to the book Modelling in Mathematical Programming:
Methodology and Techniques (Springer, 2021) is an important aid to understand
themodelingtechniquesused.
At the beginning of each chapter, the characteristics of all the problems are
presented in a structured way, with the use of tables, according to the modeling
vii
viii Preface
methodology. These characteristics are defined throughout Chap. 1. The level of
difficultyoftheproblemshasbeendefinedonthebasisofthesecharacteristics.
The format for modeling each problem is identical, which makes the models
easier to read and understand. For some problems, two versions are presented for
modeling. The versions differ in the configuration of the elements or the decision
activities. It is not the object of the book to carry out experimental analyses of the
differentmodelingalternatives.
Seville,Spain JoséManuelGarcíaSánchez
Contents
1 IntroductiontoModelingMethodology. . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2.1 AttributesofanElement. . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 TheQuantitativeNatureoftheElements. . . . . . . . . . . . 3
1.2.3 GroupingofElementsinSets. . . . . . . . . . . . . . . . . . . . 4
1.3 DecisionActivities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.1 AuxiliaryCalculations. . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.2 LogicalCalculations. . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.3 Lower/UpperBoundCalculations. . . . . . . . . . . . . . . . . 9
1.5 Specifications. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . .. . . . . . 10
1.5.1 SpecificationsStatedasSimplePropositions. . . . . . . . . 10
1.5.2 SpecificationsStatedasCompoundPropositions. . . . . . 12
1.5.3 DatainPropositions. . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.6 ObjectiveCriterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.6.1 CostAccordingtotheIntervalofValues. . . . .. . . . . . . 23
1.6.2 CostAccordingtotheValueofAnotherVariable. . . . . 24
1.6.3 CostsDependingontheDeviationoftheVariable. . . . . 25
1.6.4 BalanceFunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 BuildingandSolvingModelswithLINGO. . . . . . . . . . . . . . . . . . . 31
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 BasicFormat:TheExtendedFormat. . . . . . . . . . . . . . . . . . . . 34
2.3 CompactSet-BasedFormat. . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.1 SETSSection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3.2 DATASection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.3 CompactDefinitionofConstraintsandObjective
Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
ix
x Contents
2.3.4 IncompatibilitiesintheUseofPrimitiveSetsInstead
ofDerivedSets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.3.5 CALCSection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.4 ExecutionoftheSolvingProcess. . . . . . . . . . . . . . . . . . . . . . 58
2.5 LINGOSolutionReport. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.6 OutputDataFormat. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.6.1 OutputDatainExternalTextFiles. . . . . . . . . . . . . . . . 61
2.6.2 OutputDatainSpreadsheetFiles. . . . . . . . . . . . . . . . . 64
2.6.3 OutputDatainDatabaseFiles. . . . . . . . . . . . . . . . . . . 64
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3 Low-LevelProblems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2 Problem1:ProductManufacturing. . . . . . . . . . . . . . . . . . . . . 71
3.3 Problem2:FruitSale(Salazar2021). . . . . . . . . . . . . . . . . . . . 74
3.4 Problem3:SecuritiesPortfolio(Castilloetal.2002). . . . . . . . . 76
3.5 Problem4:CapitalInvestments(D’Auria2008). . . . . . . . . . . . 80
3.6 Problem5:NumberPartitioning(Korf1998). . . . . . . . . . . . . . 85
3.7 Problem6:InstallationofParkingSpacesforElectricCars. . . . 88
3.8 Problem7:IndependentEdgeSet(Alan1985). . . . . . . . . . . . . 92
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4 Low-MediumLevelProblems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.2 Problem8.PowerGenerationPlanning
(SarkerandNewton2007). . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.3 Problem9.PurchaseofProcessingMachines. . . . . . . . . . . . . . 106
4.4 Problem10.ProjectSelection(Castroetal.2011). . . . . . . . . . 112
4.5 Problem11.AssignmentofWorkerstoCenters. . . . . . . . . . . . 115
4.6 Problem12.WarehouseRental. . . . . . . . . . . . . . . . . . . . . . . . 120
4.7 Problem13.ArticleandPositionPlanning. . . . . . . . . . . . . . . . 126
4.8 Problem14:OnlineSale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.9 Problem15.PlotCultivation. . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.10 Problem16.ProductionandDistribution. . . . . . . . . . . . . . . . . 139
4.11 Problem17.ProductionofDivisibleTasks.. . . .. . . .. . . .. . . 148
4.12 Problem18.SelectionofNodesandEdges. . . . . . . . . . . . . . . 153
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5 Medium-LevelProblems. .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . 159
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.2 Problem19.ManagementofPollutingProducts
(Castroetal.2011). . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 161
5.3 Problem20.AssignmentofTaskstoOperators. . . . . . . . . . . . 168
5.4 Problem21.AssignmentofStudentstoGroups. . . . . . . . . . . . 173
5.5 Problem22.HomeBuilding. . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.6 Problem23.AssignmentofExtracurricularActivities. . . . . . . . 183