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Lower Bounds for Integer Programming Problems

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Lower Bounds for Integer Programming Problems LOWER BOUNDS FOR INTEGER PROGRAMMING PROBLEMS A Thesis Presented to The Academic Faculty by Yaxian Li In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Industrial and Systems Engineering Georgia Institute of Technology August 2013 Copyright c 2013 by Yaxian Li LOWER BOUNDS FOR INTEGER PROGRAMMING PROBLEMS Approved by: George L. Nemhauser, Ozlem Ergun, Ellis Johnson Committee Chair School of Industrial and Systems School of Industrial and Systems Engineering Engineering Georgia Institute of Technology Georgia Institute of Technology George L. Nemhauser, Ozlem Ergun, Maria-Florina Balcan Advisor School of Computer Science School of Industrial and Systems Georgia Institute of Technology Engineering Georgia Institute of Technology Martin Savelsbergh Date Approved: May 9th 2013 School of Mathematical and Physical Sciences University of Newcastle To my beloved grandfathers. iii ACKNOWLEDGEMENTS I would like to give my deepest gratitude to my advisors, Prof. George L. Nemhauser and Prof. Ozlem Ergun. This work could never be done without their generous support, invaluable guidance, and greatest patience with me. I am honored to work with both of them and have learned a great deal from them. I truly thank them for having belief in me which made me have more faith in myself. I would also like to thank Prof. Savelsbergh for his guidance in my early stage of research. Working with him was truly an enjoyable and inspiring experience. It is an honor to have Prof. Johnson on my committee, who has given me advice not limited to this thesis. I also want to thank Prof. Balcan for her encouragement and for exposing me a new direction in my research. I want to thank my friends and colleagues at ISYE, Huizhu, Qie, Linkan, Pengyi, Luyi, Kale, Feng, Juan Pablo, Fatma, who have made my days here happy and fulfilled. I cannot think of finishing the work without the encouragement and support from my parents. Their unconditional love has always been the biggest support for me. Finally, my most sincere thanks go to my husband, Nan, who has always been there for me. His encouragement, suggestions and love has made me who I am today, thank you. iv TABLE OF CONTENTS DEDICATION ...................................... iii ACKNOWLEDGEMENTS .............................. iv LIST OF TABLES ................................... vii LIST OF FIGURES .................................. viii SUMMARY ........................................ xi I INTRODUCTION ................................. 1 II PRICING FOR PRODUCTION AND DELIVERY FLEXIBILITY .. 14 2.1 Introduction . 14 2.2 The uncapacitated lot-sizing problem with pricing for delivery flexibility . 16 2.2.1 Solving the problem for a given discount factor α . 18 2.2.2 Analysis of the objective function . 20 2.3 Solution approach . 22 2.4 A special case . 25 2.5 Generalizations . 30 2.5.1 Per-period discounts . 30 2.5.2 Early and late delivery . 34 2.5.3 An example . 35 2.6 A computational study . 37 2.6.1 Pricing for delivery flexibility . 38 2.6.2 Delivery flexibility . 40 2.6.3 A single discount vs. per-period discounts . 42 III CORNER RELAXATION FOR MKP .................... 45 3.1 Introduction . 45 3.2 The periodic property and corner relaxation for KP . 48 3.3 Periodic property and corner relaxation for 2-KP . 52 3.3.1 Preliminaries . 53 3.3.2 The optimal basis fx1; s1g ...................... 56 3.3.3 The optimal basis fx1; x2g ....................... 60 v 3.4 The periodic property of the MKP . 64 IV MULTI-DIMENSIONAL KNAPSACK INSTANCES ........... 66 4.1 Introduction . 66 4.2 Instance generation and computational experiments . 68 V RELAXATION ALGORITHMS ........................ 78 5.1 Introduction . 78 5.1.1 Choosing Active Constraints . 82 5.1.2 Comparing Lower Bounds of Relaxations . 86 5.1.3 A Dual Heuristic Algorithm . 92 5.1.4 Modifying Parameters . 118 5.1.5 Conclusions and future research . 152 VI A SUBADDITIVE ALGORITHM AND SHORTEST PATH ALGORITHM FOR MKP ...................................... 154 6.1 Introduction . 154 6.2 Subadditive Dual and Shortest Path Algorithms . 158 6.2.1 The Subadditive Lifting Method . 158 6.2.2 A Subadditive Dual Algorithm . 159 6.2.3 A Shortest Path Algorithm . 167 6.3 Computational Results . 170 6.4 Conclusion . 173 VII CONCLUSIONS AND FUTURE RESEARCH ............... 174 REFERENCES ..................................... 177 vi LIST OF TABLES 1 Instances with n = 50; m = 100 . 70 2 Instances with n = 50; m = 500 . 71 3 Instances with n = 50; m = 1000 . 71 4 Instances with n = 100; m = 100 . 71 5 Instances with n = 100; m = 500 . 72 6 Instances with n = 100; m = 1000 . 72 7 Choosing active constraints: Lower bounds of 50 500 0 0.25 instances . 83 8 Choosing active constraints: Lower bounds of 50 500 500 1 instances . 84 9 Choosing active constraints: Lower bounds of 100 500 0 0.25 instances . 84 10 Choosing active constraints: Lower bounds of 100 500 500 1 instances . 85 11 Lower bounds obtained from various relaxations: 50 500 0 0.25 instances . 88 12 Lower bounds obtained from various relaxations: 50 500 500 1 instances . 89 13 Upper bounds obtained from the lazy relaxations: 50 500 0 0.25 instances . 89 14 Upper bounds obtained from the lazy relaxations: 50 500 500 1 instances . 90 15 LB using Lazy Constraints: 50 1000 0 0.25 . 91 16 LB using Lazy Constraints: 50 5000 0 0.25 . 91 17 Lower Bounds of the Shortest path Algorithm: 50 500 . 171 18 Lower Bounds of the Shortest path Algorithm: 50 1000 . 172 vii LIST OF FIGURES 1 Lot-sizing model . 2 2 Branch and bound algorithm . 4 3 Cutting plane algorithm . 5 4 Piecewise Quadratic Objective Function . 22 5 Recursive Partitioning . 25 6 Lower Bound for Relative Controllable Cost Decrease . 26 7 Demand of Instance . 36 8 Example . 36 9 Relative cost decrease for varying holding cost . 38 10 Relative cost decrease for varying delivery window ∆ . 39 11 Delivery flexibility for varying holding cost . 41 12 Delivery flexibility for varying delivery window size . 41 13 Discount factors for varying holding cost . 43 14 Discount factors for varying delivery window size . 43 15 Cones and basis . 57 16 Gomory { Conditions for Tight Corner Relaxation . 60 17 Periodicity of f(y1; y2).............................. 63 18 Relative Gap at Node 10,000,000 . 74 19 Number of Nodes Processed at optimality . 75 20 Lower bounds . 96 21 Lower bounds . 97 22 Lower bounds . 98 23 Lower bounds . 99 24 Lower bounds . 100 25 Relative gaps . 101 26 Relative gaps . 102 27 Relative gaps . 103 28 Relative gaps . 104 29 Relative gaps . 105 viii 30 # Remaining nodes in B&B tree compared with CPLEX . 106 31 # Remaining nodes in B&B tree compared with CPLEX . 107 32 # Remaining nodes in B&B tree compared with CPLEX . 108 33 # Remaining nodes in B&B tree compared with CPLEX . 109 34 # Remaining nodes in B&B tree compared with CPLEX . 110 35 Average time spent on each node in seconds . 112 36 Average time spent on each node in seconds . 113 37 Average time spent on each node in seconds . 114 38 Average time spent on each node in seconds . 115 39 Average time spent on each node in seconds . 116 40 Frequency of constraints in relaxations . 117 41 Lower bounds for modifying Slack . 120 42 Lower bounds for modifying Slack . 121 43 Lower bounds for modifying Slack . 122 44 Lower bounds for modifying Slack . 123 45 Lower bounds for modifying Slack . 124 46 Average time per node for modifying Slack . 125 47 Average time per node for modifying Slack . 126 48 Average time per node for modifying Slack . 127 49 Average time per node for modifying Slack . 128 50 Average time per node for modifying Slack . 129 51 Lower bounds for modifying percentage of special nodes . 131 52 Lower bounds for modifying percentage of special nodes . 132 53 Lower bounds for modifying percentage of special nodes . 133 54 Lower bounds for modifying percentage of special nodes . 134 55 Lower bounds for modifying percentage of special nodes . 135 56 Check P ercent for modifying percentage of special nodes . 136 57 Check P ercent for modifying percentage of special nodes . 137 58 Check P ercent for modifying percentage of special nodes . 138 59 Check P ercent for modifying percentage of special nodes . 139 60 Check P ercent for modifying percentage of special nodes . 140 ix 61 Lower bounds for modifying the relative gap.
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