Introduction to Cutting and Packing Optimization. Problems, Modeling Approaches, Solution Methods

426 Pages·2018·8.45 MB·english

by  Guntram Scheithauer

Guntram Scheithauer
Introduction to Cutting
and Packing Optimization
Problems, Modeling Approaches,
Solution Methods
123
GuntramScheithauer
InstituteforNumericalMathematics
TUDresden
Dresden,Germany
ISSN0884-8289 ISSN2214-7934 (electronic)
InternationalSeriesinOperationsResearch&ManagementScience
ISBN978-3-319-64402-8 ISBN978-3-319-64403-5 (eBook)
DOI10.1007/978-3-319-64403-5
LibraryofCongressControlNumber:2017951441
©SpringerInternationalPublishingAG2018
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringerInternationalPublishingAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
The terms allocation, packing, and cutting problem comprise a wide variety of
theoretical and practical problems. Although various formulations with different
denotations appear frequently, there exist strong relations between them so that
solutionmethodsforcuttingproblemscanalsobeappliedforallocationandpacking
problems,andviceversa.
For example, on the one hand, any (two-dimensional) jigsaw puzzle can be
consideredaspackingorallocationproblem:Isitpossibletopackallgivenpuzzle
pieces into a predefinedregion? And,if yes, find a correspondingpattern. On the
otherhand,therelatedcuttingproblemcouldbe:Howtodissect(cut)apredefined
regionsothatasetofdesiredpiecesisobtained?
Inthisbook,weinparticulardealwithallocation,packing,andcuttingproblems
inwhichan optimizationcomponentisofimportance.Ingeneral,we differentiate
betweentwo kindsof problemstatementswhichare formulatedhere asallocation
problems:
• Computationofanoptimal(allocation)pattern
Inthisfirstproblemtype,aregion(container)andasetofsmallerobjects(pieces,
items)aregiven,andaprofitcoefficientisassignedtoeachobject.Thenthetask
istofindasubsetoftheobjectswithmaximaltotalprofitsuchthatallitemsof
itcanbeplacedsimultaneouslywithintheregionwhilemeetingsomeallocation
constraints.
• Computationofanoptimalcombinationof(allocation)patterns
In this case, a sufficiently large number of containers, each having a cost
coefficient,andasetofsmallerobjectsaregiven.Findasubsetofthecontainers
with minimal total costs such that all objects can be assigned to the chosen
containersregardingrelatedallocationconstraints.
Problemsofthefirsttypeaim,ingeneral,tomaximizematerialutilizationwhereas
thoseofthesecondtypehavetheminimizationofmaterialusageasobjective.
Within the last decades, a very large number of investigations have been pre-
sented concerningmostdiversified applicationsand correspondingresearch work.
Infact,itisimpossibletoencompassthewidevarietyofproblemstatementswhich
can be formulated as allocation, packing, or cutting problem. Therefore, attempts
toclassifyallthesetopicsareusefultoobtainabetteroverviewonrelatedproblem
formulationsandrelationshipsbetweenthem.Importantcontributionsinthisrespect
are donein Dyckhoff[H. Dyckhoff,A typologyof cuttingandpackingproblems.
Eur. J. Oper. Res. 44(2), 145–159 (1990)],Dyckhoff and Finke [H. Dyckhoff, U.
Finke, Cutting and Packing in Production and Distribution (Physica, Heidelberg,
1992)],andWäscherandH.Haußner[G.Wäscher,H.Haußner,H.Schumann,An
improvedtypologyofcuttingandpackingproblems.Eur.J.Oper.Res.183,1109–
1130(2007)].
However, the concern of this book consists in giving an introduction to the
treatment of cutting and packing problems. Therefore, we will consider several
one-, two-, and three-dimensional basic problems, some of their formulations,
relatedmodelingissues,andsolutionapproaches.Inparticular,wewilladdressthe
followingoptimizationproblemsarisinginthefieldofcuttingandpacking:
• knapsackproblems,
• cuttingstockandbinpackingproblems,
• feasibilityproblems,
• optimalguillotinepatternsofrectangularpieces,
• non-guillotinepatternsofrectangularpieces,
• packingofrectanglesintoastrip,
• considerationofqualityconstraints,
• palletandmulti-palletloadingproblems,
• containerandmulti-containerloadingproblems,
• allocationofpolygonalobjects,
• circleandspherepacking.
As formulation variant of a considered problem we will optionally choose a
description of it as an allocation, packing, or cutting problem. A translation into
anotherformulationisobvious,ingeneral.Frequently,thepresentationofmodeling
issues and solution approachesis supplemented by some examples and exercises.
Correspondingsolutionsareprovided,too.
This bookis addressedto everyonewho is dealingwith allocation,packing,or
cutting of regular or irregular objects, and to those who intend to deal with such
problems. In particular, this monograph is directed at students of mathematics,
optimization,businessortechnomathematics,operationsresearch,andengineering.
Moreover,our aim is to provide fundamentalknowledge to researchers and prac-
titioners which have to solve real-world allocation, packing, or cutting problems.
Based on the presentedmodelingmethodsand the proposedsolution strategiesas
wellasnumerousreferencestorelatedproblems,themanuscriptshouldenablethe
reader to successfully tackle further new and more complex cutting and packing
problems.
This book is a result of a long-time work on the field of cutting and packing,
andof severallectureson this topic.Itconstitutesa majorextensionofthe earlier
German version [G. Scheithauer, Zuschnitt- und Packungsoptimierung (Vieweg +
Teubner, Wiesbaden, 2008)]. The ingredients of the book are based on many
theoreticallyorientedinvestigationsandexperienceacquiredinnumerouspractical
projects.
I am deeply grateful to Johannes Terno who introduced me to the fields of
optimization and cutting and packing. My sincere thanks go to many researchers
and practitionerswho were and/orare workingtogether with me on a pluralityof
very interesting topics within that area. In particular, I would like to thank again
JürgenRietz for his help to finish the German predecessor of this manuscript, for
whichJohnMartinovicgaveveryvaluablesupport.Notleast,Ithankmydearwife
Monikaforherunderstandingandactivehelp.
Dresden,Germany GuntramScheithauer
May2017
Contents
1 Modeling..................................................................... 1
1.1 Set-TheoreticalModels .............................................. 1
1.2 RepresentationofObjects............................................ 5
1.3 RepresentationofPatterns........................................... 7
1.4 ApproximationandInternalDescriptionofObjects................ 8
1.5 ANonlinearOptimizationModel,ˆ-Functions.................... 9
1.6 IntegerLinearProgrammingModels................................ 12
1.7 FurtherOpportunitiesofModeling.................................. 14
1.8 Exercises.............................................................. 14
1.9 Solutions.............................................................. 15
References.................................................................... 17
2 KnapsackProblems ........................................................ 19
2.1 ProblemStatement................................................... 19
2.2 AlgorithmofGilmoreandGomory ................................. 20
2.3 LongestPathMethod................................................. 23
2.4 Branch-and-BoundAlgorithms...................................... 25
2.5 PeriodicityandDominanceofSolutions............................ 29
2.6 KnapsackProblemswithUpperBounds............................ 32
2.7 SetsofPotentialAllocationPoints .................................. 34
2.8 Exercises.............................................................. 38
2.9 Solutions.............................................................. 40
References.................................................................... 45
3 One-DimensionalBinPacking ............................................ 47
3.1 ProblemStatementandModeling................................... 47
3.2 LowerBounds........................................................ 49
3.2.1 NaturalBounds............................................. 49
3.2.2 CombinatorialBounds ..................................... 52
3.2.3 DualFeasibleFunctions.................................... 54
3.2.4 LP-BasedLowerBound.................................... 58
3.3 ReductionMethodsandEquivalenceofInstances.................. 58
3.4 Heuristics............................................................. 60
3.5 PerformanceResultsfor1BPPHeuristics........................... 62
3.5.1 Worst-CasePerformance................................... 63
3.5.2 Average-CasePerformance ................................ 67
3.6 ExactApproachesandExtensions................................... 67
3.7 Exercises.............................................................. 68
3.8 Solutions.............................................................. 69
References.................................................................... 71
4 One-DimensionalCuttingStock .......................................... 73
4.1 ProblemStatementandNotations................................... 74
4.2 TheGilmore/GomoryModel ........................................ 75
4.3 SolvingtheContinuousRelaxation ................................. 78
4.3.1 TheRevisedSimplexMethod.............................. 78
4.3.2 TheFarleyBound........................................... 82
4.3.3 ColumnGeneration......................................... 83
4.4 GettingIntegerSolutions,Heuristics................................ 88
4.5 ABranch-and-BoundApproach..................................... 90
4.6 LowerBounds........................................................ 92
4.7 EquivalenceofInstances............................................. 94
4.8 AlternativeModels................................................... 96
4.8.1 ANonlinearModel......................................... 96
4.8.2 Kantorovich-TypeModels ................................. 97
4.8.3 TheOne-CutModel........................................ 100
4.8.4 TheArcflowModel......................................... 101
4.9 NeighboringProblems............................................... 103
4.10 TheCuttingStockProblemwithMultipleStockLengths ......... 105
4.11 IntegerRoundUpandModifiedIntegerRoundUpProperty...... 107
4.11.1 Definitions .................................................. 107
4.11.2 Transformations ............................................ 109
4.11.3 SubproblemsPossessingtheMIRUP...................... 109
4.11.4 FurtherRelaxations......................................... 113
4.11.5 MIRUPandHigher-DimensionalCuttingStock
Problems.................................................... 115
4.11.6 Resume...................................................... 115
4.12 Exercises.............................................................. 116
4.13 Solutions.............................................................. 117
References.................................................................... 121
5 OrthogonalPackingFeasibility,Two-DimensionalKnapsack
Problems .................................................................... 123
5.1 NonlinearBasicModels ............................................. 124
5.2 IntegerLinearProgrammingModels................................ 125
5.2.1 TheBeasley-TypeModel................................... 125
5.2.2 ThePadberg-TypeModel .................................. 127
5.3 NecessaryConditionsforFeasibility................................ 129
5.3.1 NaturalCriteriaforFeasibility............................. 129
5.3.2 TheBarRelaxation......................................... 130
5.3.3 TheContiguousRelaxation ................................ 134
5.4 TheInterval-GraphApproach ....................................... 138
5.4.1 TheInterval-GraphModelandaSimplification .......... 138
5.4.2 TheAlgorithmofFeketeandSchepers.................... 140
5.5 SufficientConditionsforFeasibility ................................ 143
5.5.1 RectangularPieces.......................................... 143
5.5.2 QuadraticPieces............................................ 144
5.6 AConstructiveBranch-and-BoundAlgorithm ..................... 145
5.6.1 TheContourConcept....................................... 145
5.6.2 ABranch-and-BoundAlgorithm........................... 147
5.6.3 UpperBounds............................................... 148
5.6.4 EquivalenceandDominance............................... 151
5.7 FurtherApproachesfortheOPP..................................... 152
5.8 Exercises.............................................................. 153
5.9 Solutions.............................................................. 154
References.................................................................... 155
6 OptimalGuillotineCutting................................................ 157
6.1 ProblemStatements.................................................. 157
6.2 GeneralGuillotineCutting........................................... 158
6.3 Two-StageGuillotineCutting........................................ 164
6.4 Three-StageGuillotineCutting ..................................... 165
6.5 FurtherPatternTypes ................................................ 170
6.6 BoundedGuillotineCutting ......................................... 171
6.6.1 TheAlgorithmofWang.................................... 171
6.6.2 ILPModelsforBounded2-StageCutting................. 172
6.7 GuillotineCuttingStockProblems.................................. 175
6.7.1 ColumnGenerationApproach............................. 175
6.7.2 One-CutILPModelofthe2-and3-Stage2CSP ......... 176
6.7.3 Extensions .................................................. 177
6.8 Three-DimensionalGuillotineCutting.............................. 177
6.9 Exercises.............................................................. 179
6.10 Solutions.............................................................. 179
References.................................................................... 180
7 PackingRectanglesintoaStrip........................................... 183
7.1 IntegerLinearProgrammingModels................................ 184
7.1.1 ABeasley-TypeModel..................................... 184
7.1.2 ALinearMixed-IntegerModel ............................ 185
7.2 LowerBounds........................................................ 186
7.2.1 NaturalLowerBounds ..................................... 186
7.2.2 CombinatorialBounds ..................................... 189
7.2.3 LPBasedBounds........................................... 190
7.3 HeuristicsfortheStripPackingProblem ........................... 195
7.3.1 HeuristicsfortheOfflineStripPackingProblem......... 195
7.3.2 PerformanceResults........................................ 198
7.3.3 ShelfAlgorithmsforOnlineProblems.................... 203
7.4 LocalSearchandMetaheuristics.................................... 207
7.4.1 LocalSearch................................................ 207
7.4.2 Metaheuristics .............................................. 208
7.5 ABranch-and-BoundAlgorithm.................................... 211
7.6 GuillotineStripPacking ............................................. 214
7.6.1 QualityofGuillotinePatterns.............................. 215
7.6.2 ILPModelsfork-StageGuillotineStripPacking......... 218
7.7 Exercises.............................................................. 221
7.8 Solutions.............................................................. 222
References.................................................................... 225
8 Two-DimensionalBinPacking ............................................ 227
8.1 ProblemStatementandModeling................................... 227
8.1.1 Non-GuillotinePatterns.................................... 229
8.1.2 Two-StageGuillotinePatterns ............................. 230
8.1.3 Three-StageGuillotinePatterns............................ 232
8.2 BasicResults ......................................................... 234
8.3 LowerBounds........................................................ 237
8.4 SolutionApproaches................................................. 240
8.5 Exercises.............................................................. 241
8.6 Solutions.............................................................. 242
References.................................................................... 243
9 QualityRestrictions........................................................ 245
9.1 CuttingofVariableLengths.......................................... 245
9.1.1 ProblemFormulation....................................... 246
9.1.2 Modeling.................................................... 247
9.1.3 OptimalValueFunction.................................... 248
9.1.4 UpperBoundsandSolutionStrategy...................... 250
9.1.5 AllocatingaSinglePiece................................... 254
9.1.6 SolutionApproaches ....................................... 258
9.1.7 Example..................................................... 260
9.2 DefectiveorForbiddenRegions..................................... 264
9.2.1 ProblemFormulation....................................... 264
9.2.2 BasicRecursion............................................. 266
9.2.3 Improvements............................................... 267
9.2.4 ExamplesandExtensions.................................. 270
9.3 Exercises.............................................................. 272
9.4 Solutions.............................................................. 274
References.................................................................... 277
10 PalletLoading............................................................... 279
10.1 TheStandardPalletLoadingProblem............................... 279
10.1.1 ProblemStatement ......................................... 280
10.1.2 EquivalenceofPLPInstances.............................. 280
10.1.3 UpperBounds............................................... 282
10.2 TheG4Heuristic..................................................... 284
10.2.1 Notations,Block-Heuristics................................ 284
10.2.2 TheG4-Structure........................................... 286
10.2.3 TheBasicRecursion........................................ 287
10.2.4 Improvements............................................... 289
10.3 TheGuillotinePalletLoadingProblem............................. 291
10.3.1 BasicModelandAlgorithm................................ 291
10.3.2 DefinitionofSubproblems................................. 294
10.3.3 PropertiesofE0andE1..................................... 296
10.3.4 AlgorithmforSubproblem1............................... 299
10.4 ThePalletLoadingProblemwithSeveralItemTypes.............. 302
10.5 TheMultiPalletLoadingProblem .................................. 305
10.5.1 ProblemStatement ......................................... 305
10.5.2 SolutionStrategy ........................................... 306
10.5.3 TheMPLPAlgorithm...................................... 307
10.5.4 TheDistributingProcedure ................................ 308
10.5.5 TheLoadingProcedure..................................... 309
10.5.6 AlternativeLoadingProcedure ............................ 311
10.5.7 Examples.................................................... 311
10.6 Exercises.............................................................. 313
10.7 Solutions.............................................................. 313
References.................................................................... 315
11 ContainerLoading ......................................................... 317
11.1 ProblemStatements.................................................. 317
11.2 TheContainerLoadingProblem .................................... 318
11.2.1 IntegerLinearProgrammingModel....................... 318
11.2.2 TheBasicAlgorithm ....................................... 321
11.2.3 TheContourConcept....................................... 323
11.2.4 FurtherHeuristics........................................... 327
11.3 MultiContainerLoadingandThree-DimensionalBinPacking ... 328
11.4 Three-DimensionalStripPacking................................... 329
11.5 LPBounds............................................................ 330
11.5.1 TheBarRelaxationoftheCLP............................ 330
11.5.2 TheBarRelaxationoftheMCLP.......................... 336
11.5.3 ALayerRelaxationfortheContainerLoading
Problem ..................................................... 338
11.6 Exercises.............................................................. 340
11.7 Solutions.............................................................. 341
References.................................................................... 343