A BRANCH-AND-PRICE ALGORITHM FOR BIN PACKING PROBLEM MASOUD ATAEI A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE GRADUATE PROGRAM IN APPLIED AND INDUSTRIAL MATHEMATICS YORK UNIVERSITY TORONTO, ONTARIO August 2015 © Masoud Ataei, 2015 Abstract Bin Packing Problem examines the minimum number of identical bins needed to pack a set of items of various sizes. Employing branch-and-bound and column generation usually requires designation of the problem-specific branching rules compatible with the nature of the pricing sub-problem of column generation, or alternatively it requires determination of the k-best solutions of knapsack problem at level kth of the tree. Instead, we present a new approach to deal with the pricing sub-problem of column generation which handles two- dimensional knapsack problems. Furthermore, a set of new upper bounds for Bin Packing Problem is introduced in this work which employs solutions of the continuous relaxation of the set-covering formulation of Bin Packing Problem. These high quality upper bounds are computed inexpensively and dominate the ones generated by state-of-the-art methods. Keywords: Bin Packing Problem, Branch-and-Bound, Column Generation. ii To my parents iii Acknowledgments I would like to express my deep sense of appreciation to my supervisor, Dr. Michael Chen. Had it not been for his invaluable guidance and tremendous support, I would not be able to surmount all obstacles laid in the path of research. His flexibility and dedication allowed me to explore different areas of Operational Research that caught my interest while keeping in sight my road map and final goals. Not only had I the opportunity to pursue my interests in mathematics during the time of working with Dr. Chen, but also I learned how to follow my heart when making vital decisions in life and free myself from those pressures which mechanize the mind and make for routine thinking as founding president of York University, Murray G. Ross once envisaged. Furthermore, I would like to thank Dr. Murat Kristal for taking time, to be part of the supervisory committee, and for advising me on my research. I am also grateful to Professor Jeff Edmonds for agreeing to be on my examining committee. Throughout my study at York University, I benefited a lot from professors and friends who were always helpful. I especially want to thank Dr. Vladimir Vinogradov for his unconditional generosity and help. My sincere thanks also to Mourad Amara who applied his editorial talents to my thesis and made it look better. I am also thankful to Primrose Miranda for her administrative support during the whole period of my study. Last but not least, I want to thank my parents from whom I learned the alphabet of love and to whom I am indebted for all the achievements I have in life. Even though the words are not enough when it comes to expressing appreciations for love, I want to give my special thanks to my mother who made lots of sacrifices to be with me in Canada and support me thoroughly during the time I was working on my thesis. Also, my deepest gratitude to my father both as a person I always admired in my life and as a university professor who unfailingly helped and encouraged me to realize the true nature of doing research. My thanks finally go to my beloved sister, Minoo, for her whole-hearted love and encouragement during the years of my schooling. iv Table of Contents Abstract ………………………………………………………………………………………………………....ii Dedication……………………………………………………………………...…….………………………….iii Acknowledgments …………………………………………………………………………………..…….……iv Table of Contents………………..…………………………………………….……………………………..….v List of Tables…………………………………………………………………………...………………………vii List of Figures……..……………………………………………….………………………..……...…………viii 1 Introduction ...................................................................................................................................................... 1 1.1 Overview .................................................................................................................................................... 1 1.2 Thesis outline .............................................................................................................................................. 3 2 Literature Review ............................................................................................................................................ 5 2.1 Definitions .................................................................................................................................................. 5 2.2 Problem formulation ................................................................................................................................... 7 2.2.1 Compact formulation of BPP .............................................................................................................. 7 2.2.2 Set-covering formulation of BPP ........................................................................................................ 8 2.3 Column generation.................................................................................................................................... 11 2.4 Round up properties of set-covering formulation of BPP ......................................................................... 14 2.4.1 Round-up property (RUP) ................................................................................................................. 14 2.4.2 Integer round up property (IRUP) ..................................................................................................... 14 2.4.3 Modified integer round up property (MIRUP) .................................................................................. 14 2.5 Round up solutions ................................................................................................................................... 15 2.6 Branch-and-price ...................................................................................................................................... 16 2.6.1 Branching Strategies .......................................................................................................................... 16 2.6.1.1 Finding the k-best solutions of knapsack problem ..................................................................... 16 2.6.1.2 Implicit methods in exclusion of the forbidden patterns from search domain ........................... 17 2.6.2 Search strategies ................................................................................................................................ 19 2.6.3. Termination conditions ..................................................................................................................... 21 2.6.4 Primal approximations, heuristics, and meta-heuristics .................................................................... 21 2.6.4.1 Approximations .......................................................................................................................... 22 2.6.4.2 Heuristics and meta-heuristics ................................................................................................... 24 2.6.5 Algorithms for solving pricing sub-problem (one-dimensional knapsack problem) ......................... 26 2.7 Benchmark instances ................................................................................................................................ 28 3 Solving the master problem by column generation ..................................................................................... 30 3.1 Restricted master problem (RMP) ............................................................................................................ 31 3.2 Revised simplex method ........................................................................................................................... 38 3.3 Computational results ............................................................................................................................... 42 4 An upper bound for BPP using solutions of the continuous relaxation of set-covering formulation of BPP (SCCUB) .................................................................................................................................................... 46 v 4.1. SCCUB procedure ................................................................................................................................... 46 4.2 Computational results ............................................................................................................................... 50 5 Upper bounds for BPP using solutions of the continuous relaxation of set-covering formulation with an elimination operator -p (SCCUB-p) ................................................................................................................ 65 5.1. Procedure SCCUB-p ................................................................................................................................ 65 5.2 Computational results ............................................................................................................................... 68 6 Upper bounding technique for BPP using solutions of the continuous relaxation of set-covering formulation with an elimination operator- p and a dual BPP approach (SCCUB-p-d) ............................. 80 6.1 Maximum cardinality bin packing problem .............................................................................................