ALGORITHM FOR THE CUTTING STOCK PROBLEM WITH MULTIPLE RAWS AND LIMITED NUMBER OF CUTTING KNIVES by PITJAYA TANGTATSWAS Submitted in partial fulfillment of the requirements For the Master of Science Advisor: Vira Chankong CASE WESTERN RESERVE UNIVERSITY May, 2017 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the thesis/dissertation of Pitjaya Tangtatswas Candidate for the degree of Master of Science Committee Chair Vira Chankong, Ph.D. Committee Member Mingguo Hong, Ph.D. Committee Member Evren Gurkan-Cavusoglu, Ph.D. Date of Defense December 22, 2016 *We also certify that written approval has been obtained for any proprietary material contained therein. i Table of Contents Chapter 1 .................................................................................................................................................... 1 Introduction .............................................................................................................................................. 1 1.1 Background and Significance ................................................................................................. 1 1.2 Research Objective and Contributions ............................................................................... 2 1.3 Outline of the thesis ................................................................................................................... 3 Chapter 2 .................................................................................................................................................... 4 Literature Review ................................................................................................................................... 4 2.1 Cutting Stock Problem .............................................................................................................. 4 2.2 Column Generation .................................................................................................................... 6 2.2.1 Dynamic Programming .................................................................................................... 9 2.2.2 Branch and Bound Method .......................................................................................... 14 2.2.3 Dynamic Programming versus Branch and Bound ............................................ 19 Chapter 3 ................................................................................................................................................. 22 Solution Methods for Solving CSPs ............................................................................................... 22 3.1 Cutting Stock Problem with Multiple Raws ................................................................... 22 3.1.1 Find the Initial Solution ................................................................................................ 23 3.1.2 Knapsack Problem .......................................................................................................... 25 3.1.3 Further additional algorithm ...................................................................................... 26 3.1.4 Example ............................................................................................................................... 26 ii 3.2 Cutting Stock with Limited Number of Rolls of Raw Material ............................... 33 3.2.1 Example ............................................................................................................................... 35 3.3 Cutting Stock Problem with Limited Number of available Cutting Knives ....... 41 3.3.1 Lagrangian Relaxation Review ....................................................................................... 42 3.3.1 Example ............................................................................................................................... 45 Chapter 4 ................................................................................................................................................. 52 Computational Results ....................................................................................................................... 52 4.1 Cutting Stock with Multiple Raw Material ..................................................................... 52 4.2 Cutting Stock with Multiple Limited Number of Rolls of Raw Material ............. 55 4.3 Cutting Stock with Limited Number of Available Cutting Knives ......................... 56 Chapter 5 ................................................................................................................................................. 62 Conclusions ............................................................................................................................................ 62 Bibliography .......................................................................................................................................... 64 iii List of Tables Table 1 ...................................................................................................................................................... 13 Table 2 ...................................................................................................................................................... 14 Table 3 ...................................................................................................................................................... 21 Table 4 ...................................................................................................................................................... 52 Table 5 ...................................................................................................................................................... 53 Table 6 ...................................................................................................................................................... 53 Table 7 ...................................................................................................................................................... 53 Table 8 ...................................................................................................................................................... 54 Table 9 ...................................................................................................................................................... 54 Table 10 ................................................................................................................................................... 55 Table 11 ................................................................................................................................................... 55 Table 12 ................................................................................................................................................... 55 Table 13 ................................................................................................................................................... 55 Table 14 ................................................................................................................................................... 57 Table 15 ................................................................................................................................................... 57 Table 16 ................................................................................................................................................... 58 Table 17 ................................................................................................................................................... 59 Table 18 ................................................................................................................................................... 59 Table 19 ................................................................................................................................................... 61 iv List of Figure Figure 1 ....................................................................................................................................................... 4 Figure 2 .................................................................................................................................................... 13 Figure 3 .................................................................................................................................................... 51 Figure 4 .................................................................................................................................................... 56 Figure 5 .................................................................................................................................................... 57 Figure 6 .................................................................................................................................................... 58 Figure 7 .................................................................................................................................................... 60 Figure 8 .................................................................................................................................................... 60 v Acknowledgements My deepest gratitude to my advisor, Prof. Vira Chankong. He has helped me throughout my research with great patience. He has shared his wisdom and knowledge without reserve. He cares for his students deeply.