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Mathematical Models and Decomposition Algorithms for Cutting and Packing Problems

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Alma Mater Studiorum - Universit`adi Bologna Dottorato di Ricerca in Automatica e Ricerca Operativa Ciclo XXIX Settore concorsuale di afferenza: 01/A6 - RICERCA OPERATIVA Settore scientifico disciplinare: MAT/09 - RICERCA OPERATIVA Mathematical Models and Decomposition Algorithms for Cutting and Packing Problems Presentata da Maxence Delorme Coordinatore Dottorato Relatore Prof. Daniele Vigo Prof. Silvano Martello Co-relatore Prof. Manuel Iori Esame finale anno 2017 Contents Acknowledgments v 1 Introduction 1 2 BPP and CSP: Mathematical Models and Exact Algorithms 7 2.1 Introduction.................................... 7 2.2 Formalstatement................................. 10 2.3 Upperandlowerbounds............................. 11 2.3.1 Approximation algorithms . 12 2.3.2 Lowerbounds............................... 13 2.3.3 Heuristics and metaheuristics . 15 2.4 Pseudo-polynomial formulations . ...... 17 2.4.1 ConsiderationsonthebasicILPmodel. 17 2.4.2 One-cutformulation ........................... 19 2.4.3 DP-flowformulation ........................... 21 2.4.4 Arc-flowformulations . .. .. .. .. .. .. .. 23 2.5 Enumerationalgorithms . 24 2.5.1 Branch-and-bound ............................ 24 2.5.2 Constraint programming approaches . 26 2.6 Branch-and-price ................................ 26 2.6.1 Set covering formulation and column generation . ...... 26 2.6.2 Integerround-upproperty. 29 2.6.3 Branch(-and-cut)-and-price algorithms . ....... 30 2.7 Experimentalevaluation . 33 2.7.1 Benchmarks................................ 33 2.7.2 Computercodes ............................. 35 2.7.3 Experiments ............................... 37 2.8 Conclusions .................................... 46 i ii CONTENTS 3 BPPLIB: A Library for Bin Packing and Cutting Stock Problems 47 3.1 Introduction.................................... 47 3.2 Computercodes ................................. 49 3.3 Benchmarks.................................... 53 3.4 Computationalexperiments . 54 3.4.1 GIinstances................................ 56 4 Enhanced PP Formulations for Bin Packing and Cutting Stock Problems 59 4.1 Introduction.................................... 60 4.2 The BPP, the CSP, and their well-known formulations . ........ 62 4.2.1 Problem description and notation . 63 4.2.2 Pattern-based formulations . 63 4.2.3 Pseudo-polynomial formulations . 65 4.3 Relationsamongmodels . .. .. .. .. .. .. .. .. 67 4.4 Reflect, an improved arc-flow formulation . ...... 72 4.4.1 Adapting reflect to solve large size instances: Reflect+........ 75 4.5 Generalizations ................................. 78 4.5.1 VariablesizedBPP............................ 78 4.5.2 BPP with item fragmentation . 80 4.6 Computationalresults . 82 4.6.1 ResultsonBPPandCSP .. .. .. .. .. .. .. 82 4.6.2 ResultsontheVSBPP. .. .. .. .. .. .. .. 86 4.6.3 ResultsontheBPPIF .......................... 88 4.7 Conclusion .................................... 89 Supplementary material 4.A Details for Lemma 1 . ...... 91 Supplementarymaterial4.B ProofofLemma2 . 91 Supplementary material 4.C Proof of Theorem 1 . ...... 93 Supplementary material 4.D Proof of Theorem 2 . 101 Supplementary material 4.E Proof of Theorem 3 . 102 Supplementary material 4.F Proof of Theorem 4 . 104 Supplementary material 4.G Algorithms for reflect . ......... 104 Supplementary material 4.H Proof of Theorem 5 . 108 CONTENTS iii 5 Logic Based Benders’ Decomposition for Orthogonal Stock Cutting Prob- lems 109 5.1 Introduction.................................... 109 5.2 Literaturereview ................................ 110 5.3 Mathematicalmodel ............................... 112 5.4 Preprocessing................................... 114 5.5 Decomposition algorithm . 115 5.6 Masterproblem.................................. 117 5.7 Slave problem and cut generation . 120 5.8 The case of identical item copies . 122 5.9 Computationalexperiments . 123 5.9.1 SCPinstances .............................. 124 5.9.2 Rectanglepackings. .. .. .. .. .. .. .. .. 126 5.9.3 Palletloading............................... 127 5.10Conclusion .................................... 129 6 A Training Software for Orthogonal Packing Problems 131 6.1 Introduction.................................... 131 6.2 Orthogonalpackingproblems . 134 6.3 Software...................................... 137 6.4 Experiments.................................... 140 6.4.1 Setup ................................... 141 6.4.2 Results .................................. 142 6.5 Conclusions .................................... 146 7 Mathematical Models and Decomposition Algorithms for the PRPP 147 7.1 Introduction.................................... 148 7.2 ProblemDescription .............................. 151 7.3 A Compact Mathematical Formulation . 153 7.4 Lower Bounds based on a Decomposition Method . 155 7.4.1 Cuttingcomponent(CC) . 156 7.4.2 Schedulingcomponent(SC) . 159 7.5 UpperBoundingProcedures. 161 7.5.1 A Heuristic Based on a Generalized Assignment Problem . 162 iv CONTENTS 7.5.2 An Iterated Local Search Algorithm . 163 7.6 ComputationalExperiments. 165 7.6.1 Instances ................................. 165 7.6.2 Algorithmperformance . 167 7.7 ConcludingRemarks ............................... 169 8 ILP and CP for Project Scheduling Problems 171 8.1 Introduction.................................... 171 8.2 Literaturereview ................................ 172 8.3 Mathematicalmodels.. .. .. .. .. .. .. .. .. 173 8.3.1 RCPSPformulations. 174 8.3.2 DTCTPformulations . 175 8.3.3 MMRCPSPformulations . 176 8.4 Proposedapproaches.............................. 176 8.4.1 RCPSPimprovedalgorithm. 177 8.4.2 DTCTPimprovedalgorithm . 179 8.4.3 MMRCPSP improved algorithm . 180 8.5 Computationalexperiments . 180 8.5.1 Computational experiments for the RCPSP . 181 8.5.2 Computational experiments for the DTCTP . 182 8.5.3 Computational experiments for the MMRCPSP . 183 8.6 Conclusion .................................... 184 Bibliography 185 Acknowledgments First of all, I would like to thank my tutors Prof. Silvano Martello and Prof. Manuel Iori for the help they provided during three years of Ph.D. studies, for giving me the opportunity to work on many different and interesting research projects, and for teaching me a lot about Operations Research and Combinatorial Optimization. I would also like to thank all the other components of the group of Operations Research of the Department of Electrical, Electronic and Information Engineering ”Guglielmo Mar- coni” (DEI) of the University of Bologna, namely Prof. Paolo Toth, Prof. Daniele Vigo, Prof. Andrea Lodi, Prof. Enrico Malaguti, Prof. Michele Monaci, Dr. Valentina Cacchiani, Dr. Paolo Tubertini, Ph.D. students Carlos Emilio Contreras Bolton, Alberto Santini, and the former members Dr. Claudo Gambella, Dr. Tiziano Parriani, Dr. Dimitri Thomopulos, and Dr. Sven Wiese. I am grateful to my co-authors, who helped me in writing the reports that form part of this thesis: Prof. Jean Fran¸cois Cˆot´efrom the Universit´eLaval, and Prof. Anand Subra- manian from the Federal University of Para´ıba. Thanks are also due to the undergraduate and master students I collaborated with during these years, in particular Gianluca Costa and Vitor Nesello. I would also like to thank the Ministero dell’Istruzione, dell’Universit`ae della Ricerca (MIUR) for the financial support given to my Ph.D. course Special thanks are addressed to Prof. Fran¸cois Clautiaux, Prof. Joshua D. Habiger, and Prof. Alexandru-Adrian Tantar for encouraging me to pursue a Ph.D. and to Prof. Clarisse Dhaenens, who made me discover and enjoy Combinatorial Optimization. Finally, I would like to thank my friends Damien and Julien, my mother, my father, and all my family for always supporting me in my decisions. Bologna, March 27, 2017 Maxence Delorme v List of Figures 1.1 Number of papers dealing with bin packing and cutting stock problems, 1991-2016 ..................................... 2 2.1 DP-flow graph construction for Example 1 . ..... 22 2.2 Arc-flow representation of the graph of Figure 2.1 . ......... 23 3.1 The interactive visual solver . ..... 52 4.1 L(FAF ) solution and path decomposition for Example 2 . 68 4.2 An L(FOC ) solution of Example 2 represented as a set of trees (¯zt = value of tree t). ..................................... 69 4.3 An L(FDP ) solution of Example 2 (selected arcs in bold, values taken by the selected variables on the arcs) . 69 4.4 Graphical representation of relations among CSP formulations. 70 4.5 Set of arcs required by the standard arc-flow (above) and by reflect (below) for Example 2 (item arcs are depicted in straight lines, loss arcs in dotted lines)........................................ 73 4.6 Solution of L(FRE ) for Example 2 (selected item arcs are depicted in straight lines, selected loss arcs in dotted lines, variable values onthearcs) . 74 4.7 Set of arcs required by reflect for Example 3 (item arcs are depicted in straightlines)................................... 79 4.8 Set of arcs built by arc-flow for the BPPIF Example 4 (item arcs are depicted instraightlines) ................................. 80 4.9 Construction of an L(FAF ) solution for Example 2 through Algorithm 5. Each iteration, from top to bottom, processes the cut associated with they ¯ variablegivenontheleft. 100 4.10 An invalid L(FAF ) solution of Example 2 with a negative flow (cycle), ob- tained by executing Algorithm 5 on an input L(FOC ) solution that does not satisfy(4.51). ................................... 101 4.11 Example 2 shows that FAF is not included in FDP ............... 102 vii
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