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On-Line Algorithms for Bin-Covering Problems with Known Item Distributions

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On-Line Algorithms for Bin-Covering Problems with Known Item Distributions ON-LINE ALGORITHMS FOR BIN-COVERING PROBLEMS WITH KNOWN ITEM DISTRIBUTIONS A Dissertation Presented to The Academic Faculty by Agni Ásgeirsson In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Industrial and Systems Engineering Georgia Institute of Technology May 2014 Copyright ¥c 2014 by Agni Ásgeirsson ON-LINE ALGORITHMS FOR BIN-COVERING PROBLEMS WITH KNOWN ITEM DISTRIBUTIONS Approved by: Sigrún Andradóttir, Advisor Pinar Keskinocak School of Industrial and Systems School of Industrial and Systems Engineering Engineering Georgia Institute of Technology Georgia Institute of Technology David Goldsman, Advisor Páll Jensson School of Industrial and Systems School of Science and Engineering Engineering Reykjavík University Georgia Institute of Technology Anton J. Kleywegt Date Approved: 24 January 2014 School of Industrial and Systems Engineering Georgia Institute of Technology Perseverance, secret of all triumphs. Victor Hugo iii ACKNOWLEDGEMENTS The gestation period for this thesis was unusually long, as the time between the thesis proposal defense and the thesis defense was more than thirteen years. A large group of people deserve thanks for supporting and encouraging me during this time. I would first like to thank my thesis advisors, Professors Sigrún Andradóttir and Dave Goldsman. Without Dave’s support and encouragement, I would probably have given up. Sigrún’s detailed feedback has made the thesis much better. The rest of the thesis committee deserve praise; Professor Anton Kleywegt gave very good guidance on Markov decision processes, Professor Pinar Keskinocak was very encouraging, and last, but not least, Professor Páll Jensson provided the insight that got me started on this journey more than twenty years ago. My mom, Albína Thordarson, was absolutely tireless in reminding me that I would never be happy with an unfinished thesis. My siblings, Páll Ágúst Ásgeirsson and Áslaug Ásgeirsdóttir, were always very supportive. My love, Hildur Pétursdóttir, de- serves praise for her encouragement during the last push to finish. Lára Jóhannsdóttir provided project management for a year or so, and my stepfather Ólafur Sigurðsson cheered me on as well. My dear daughters, Védís and Iðunn, have been an inspiration to finish before they go to university themselves, and their mother, Eyja Brynjarsdót- tir, was always supportive. Cousin Höskuldur and his wife Hilda provided working space on their farm, thank you. It was encouraging to know that my uncle Jón Thor- darson was always waiting to go to the defense. Jóhannes Pálsson and Hye Young were very hospitable in letting me stay at their flat in Atlanta. Finally, thanks to Geir Gunnlaugsson, Hörður Arnarson, Jón Þór Ólafsson, and Marel Hf, for their support. iv TABLE OF CONTENTS DEDICATION .................................. iii ACKNOWLEDGEMENTS .......................... iv LIST OF TABLES ............................... viii LIST OF FIGURES .............................. xi SUMMARY .................................... xvii 1 INTRODUCTION ............................. 1 2 BACKGROUND .............................. 4 2.1 BasicDefinitions............................. 4 2.2 LiteratureReview ............................ 7 3 THE PROSPECT ALGORITHM .................... 23 3.1 Introduction ............................... 23 3.2 TheProspectFunction ......................... 24 3.3 TheProspectAlgorithm ........................ 28 3.4 Performance Analysis of the Prospect Algorithm . 33 3.5 Comparison oftheNext-Fit, PR, andPD Algorithms . 38 3.6 ProspectFunctionforBin-Packing . 40 3.7 ProspectAlgorithmforBin-Packing . 46 4 EXTENSIONS TO THE PROSPECT ALGORITHM ....... 50 4.1 Bin-CoveringwithReject . 50 4.2 ConstraintsontheNumbersofItemsperBin . 54 4.3 ManyTypesofItems .......................... 55 4.4 Multiple Bin Capacities . 57 5 PERFECT PACKING ANALYSIS ................... 61 5.1 Introduction ............................... 61 5.2 TheSum-of-Squaresalgorithm . 61 v 5.3 ProspectAlgorithmsforPerfectPacking. 64 5.4 PerformanceComparison . 67 5.5 ImprovedProspect+Algorithm . 68 6 MARKOV MODELING OF ON-LINE BIN-COVERING . 77 6.1 Introduction ............................... 77 6.2 BinState-SpaceModel ......................... 78 6.3 MarkovDecisionProcessModel . 84 6.4 Markov Decision Process Model for the Prospect Function . .. 99 6.5 Improving the Prospect Algorithm Using Insights from the Optimal Strategy ................................. 100 6.6 MDPApproximationsforBiggerProblems . 114 6.7 RolloutProspectAlgorithm. 127 7 ON-LINE BIN-COVERING WITH LOOKAHEAD ........ 134 7.1 Introduction ............................... 134 7.2 Binary Knapsack and Minimization Binary Knapsack Problems . 134 7.3 TheSubset-SumProblem . 136 7.4 The Multiple Subset-Sum Problem . 142 8 PRACTICAL APPLICATION OF THE PROSPECT ALGORITHM ................................ 152 8.1 RealProductDistributions . 152 8.2 DoubleEffect .............................. 155 8.3 On-Line Item Distribution Estimation and Prospect Function Calcu- lation................................... 171 8.4 Setting Parameters for On-Line Item Distribution Estimation and ProspectFunctionCalculation . 187 8.5 Comparison of Different Versions of the Prospect Algorithm . 197 9 SUMMARY, CONCLUSIONS, AND FURTHER RESEARCH . 204 9.1 SummaryandConclusions . 204 9.2 FurtherResearch............................. 206 vi APPENDIX A — SIMULATION METHOD AND DATA DISTRI- BUTIONS .................................. 207 APPENDIX B — DETAILED COMPARISON OF THE PR AND PD ALGORITHMS FOR BIN-COVERING ............. 228 APPENDIX C — DETAILED COMPARISON OF THE PR AND PR+ ALGORITHMS FOR BIN-COVERING ............ 231 APPENDIX D — CONFIDENCE INTERVALS FOR SIMULATIONS FROM CHAPTER 5 ........................... 234 APPENDIX E — CONFIDENCE INTERVALS FOR SIMULATIONS IN CHAPTER 7 .............................. 239 APPENDIX F — DERIVATION OF EQUATIONS (8.6), (8.7), AND (8.9) ...................................... 241 REFERENCES .................................. 248 vii LIST OF TABLES 3.1 “Statistical winners” summary of the PR and PD Algorithms. 43 3.2 Summary of average overfill for the PR and PD Algorithms . 43 4.1 Summaryofthechickendistribution. 58 5.1 Comparison of average number of open bins using different prospect modificationfunctionsfortheSSalgorithm. 67 5.2 “Statistical winners” comparison of one- and two-sided zones. .... 71 5.3 “Statistical winners” summary of the PR and PR+ Algorithms. 72 6.1 Number of bin states as a function of nub , nfb , and k.......... 80 6.2 Sample item distribution and single bin state space. Bin size is 20. 81 6.3 Samplestatespacefortwobins. 82 6.4 Number of value iterations needed to get maximum error below 10 −12 . Boldfaced entries indicate that half updates were employed. 92 6.5 Overfill comparison I of the MDP and Prospect Algorithms. 97 6.6 Overfill comparison II of the MDP and Prospect Algorithms. 98 6.7 Optimal values for the parameter r ofthePREalgorithm. 109 6.8 Average overfill for all bin sizes using constant r values in PRE. 110 6.9 OptimalparametersforPRandPRE . 110 6.10 Summaryoftheaverage overfills ofMDP2+and PR. 123 8.1 Relative variance ( σ/µ ) for N D(100,20)anditsgrades. 155 8.2 Tally Matrix for FIFO queue 2 , 3 , 4 , 3 , 2 , 3 ............ 180 { 1 2 3 4 5 6} 8.3 Maximum FIFO queue size N for Tally Matrix calculations. 187 8.4 Approx. number of loops needed to recalculate/update a Prospect. 188 8.5 Comparison of non-Gaussian and Gaussian distribution statistics. 189 8.6 95% confidence intervals of average overfill as a function of Prospect function type and item distribution with µ = 100 and σ =10. 191 8.7 95% confidence intervals of average overfill as a function of Prospect function type and item distribution with µ = 100 and σ =15. 192 viii 8.8 95% confidence intervals of average overfill as a function of Prospect function type and item distribution with µ = 100 and σ =20. 193 8.9 Results of pre-selection simulations, PR algorithm and N D(100,15) itemdistribution. ............................. 199 8.10 Simulations as in Table 8.9 repeated for PR+ and PRE. 200 8.11 Effect of suboptimal settings on PR+ and PRE algorithms. 202 A.1 List of distributions used in simulations. 208 A.2 Histogram for the N D(100,10) distribution. 209 A.3 Histogram for the N D(100,15) distribution. 210 A.4 Histogram for the N D(100,20) distribution. 211 A.5 Histogram for the N D(50,10) distribution. 212 A.6 Histogram for the N D(10,1.0) distribution. 212 A.7 Histogram for the N D(10,1.5) distribution. 213 A.8 Histogram for the N D(10,2) distribution. 213 A.9 Histogram for the U D(75,125) distribution. 214 A.10 Histogram for the U D(8,12) distribution. 214 A.11 Histogram for the LS(100,10) distribution. 215 A.12 Histogram for the LS(100,15) distribution. 215 A.13 Histogram for the LS(100,20) distribution. 216 A.14 Histogram for the RS(100,10) distribution. 217 A.15 Histogram for the RS(100,15) distribution. 217 A.16 Histogram for the RS(100,20) distribution. 218 A.17 Histogram for the TP(100,10) distribution. 219 A.18 Histogram for the TP(100,15) distribution. 219 A.19 Histogram for the TP(100,20) distribution. 220 A.20 Histogram for the FS(653,43) distribution. 221 A.21 Histogram for the Earlybird breasts distribution. 222 A.22 Histogram for the Earlybird drums distribution. 222 A.23 Histogram for the Earlybird thighs distribution. 223 ix A.24 Histogram for the Earlybird wing distribution. 223 A.25 Cross checking data for random number generation. 227 B.1 Comparison of 95% confidence intervals for the average overfill ob- tained
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