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Packing in Two and Three Dimensions View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Calhoun, Institutional Archive of the Naval Postgraduate School Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 2003-06 Packing in two and three dimensions Martins, Gustavo H. A. http://hdl.handle.net/10945/9859 NAVAL POSTGRADUATE SCHOOL Monterey, California DISSERTATION PACKING IN TWO AND THREE DIMENSIONS by Gustavo H. A. Martins June 2003 Dissertation Supervisor: Robert F. Dell Approved for public release; distribution is unlimited THIS PAGE INTENTIONALLY LEFT BLANK REPORT DOCUMENTATION PAGE Form Approved OMB No. 0704-0188 Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instruction, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188) Washington DC 20503. 1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED June 2003 Dissertation 4. TITLE AND SUBTITLE: Title (Mix case letters) 5. FUNDING NUMBERS Packing in Two and Three Dimensions 6. AUTHOR(S) Martins, Gustavo H. A. 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING Naval Postgraduate School ORGANIZATION REPORT Monterey, CA 93943-5000 NUMBER 9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING / MONITORING N/A AGENCY REPORT NUMBER 11. SUPPLEMENTARY NOTES The views expressed in this thesis are those of the author and do not reflect the official policy or position of the Department of Defense or the U.S. Government. 12a. DISTRIBUTION / AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE Approved for public release; distribution is unlimited. 13. ABSTRACT (maximum 200 words) This dissertation investigates Multidimensional Packing Problems (MD-PPs): the Pallet Loading Problem (PLP), the Multidimensional Knapsack Problem (MD-KP), and the Multidimensional Bin Packing Problem (MD-BPP). In these problems, there is a set of items, with rectangular dimensions, and a set of large containers, or bins, also with rectangular dimensions. Items cannot overlap (share the same region in space), and, when packed, must be completely located within the bin. We develop new theory for PLP, and apply it to the construction of new bounds, heuristics, and an exact algorithm. The bounds verify that the heuristics optimally solve 99.999% of PLP instances with up to 100 items; in the instances that the heuristics fail to solve optimally, their best solution differs from the optimum by only one item. Using our new PLP theory, we implement algorithms to solve orthogonal non-guillotine MD-KP instances and are the first to obtain exact solutions for some instances from the literature. Using these MD-KP algorithms, we develop the first exact algorithm for the orthogonal non- guillotine MD-BPP and are the first to obtain exact solutions to several instances from the literature. 14. SUBJECT TERMS 15. NUMBER OF PAGES 176 Optimization, Combinatorics, Multidimensional Bin Packing Problem, Multidimensional Knapsack Problem, Pallet Loading Problem, Container Loading Problem, Cutting Stock. 16. PRICE CODE 17. SECURITY 18. SECURITY 19. SECURITY 20. LIMITATION CLASSIFICATION OF CLASSIFICATION OF THIS CLASSIFICATION OF OF ABSTRACT REPORT PAGE ABSTRACT Unclassified Unclassified Unclassified UL NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89) Prescribed by ANSI Std. 239-18 i THIS PAGE INTENTIONALLY LEFT BLANK ii Approved for public release; distribution is unlimited PACKING IN TWO AND THREE DIMENSIONS Gustavo H. A. Martins Commander, Brazilian Navy B.S., Escola Naval - Brazil, 1980 M.S., Naval Postgraduate School, 1993 Submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY IN OPERATIONS RESEARCH from the NAVAL POSTGRADUATE SCHOOL June 2003 Author: __________________________________________________ Gustavo H. A. Martins Approved by: ______________________ Robert F. Dell Associate Professor of Operations Research Dissertation Supervisor ______________________ _______________________ Gerald G. Brown Craig W. Rasmussen Distinguished Professor of Associate Professor of Mathematics Operations Research ______________________ ______________________ Alan R. Washburn R. Kevin Wood Professor of Operations Research Professor of Operations Research Approved by: __________________________________________________ James N. Eagle, Chair, Department of Operations Research Approved by: __________________________________________________ Carson K. Eoyang, Associate Provost for Academic Affairs iii THIS PAGE INTENTIONALLY LEFT BLANK iv ABSTRACT This dissertation investigates Multidimensional Packing Problems (MD-PPs): the Pallet Loading Problem (PLP), the Multidimensional Knapsack Problem (MD-KP), and the Multidimensional Bin Packing Problem (MD-BPP). In these problems, there is a set of items, with rectangular dimensions, and a set of large containers, or bins, also with rectangular dimensions. Items cannot overlap (share the same region in space), and, when packed, must be completely located within the bin. We develop new theory for PLP, and apply it to the construction of new bounds, heuristics, and an exact algorithm. The bounds verify that the heuristics optimally solve 99.999% of PLP instances with up to 100 items; in the instances that the heuristics fail to solve optimally, their best solution differs from the optimum by only one item. Using our new PLP theory, we implement algorithms to solve orthogonal non-guillotine MD-KP instances and are the first to obtain exact solutions for some instances from the literature. Using these MD-KP algorithms, we develop the first exact algorithm for the orthogonal non-guillotine MD-BPP and are the first to obtain exact solutions to several instances from the literature. v THIS PAGE INTENTIONALLY LEFT BLANK vi EXECUTIVE SUMMARY This dissertation investigates Multidimensional Packing Problems (MD-PPs): the Pallet Loading Problem (PLP), the Multidimensional Knapsack Problem (MD-KP), and the Multidimensional Bin Packing Problem (MD-BPP). In these problems, there is a set of items, with rectangular dimensions, and a set of large containers, or bins, also with rectangular dimensions. Items cannot overlap (share the same region in space), and, when packed, must be completely located within the bin. In some cases, we want to maximize the value of items packed in a single bin (PLP and MD-KP). In others, we want to minimize the number of bins used to pack all items (MD-BPP). MD-PPs arise when preparing a shipment of goods in rectangular boxes inside rectangular containers, or cutting smaller pieces from a larger stock (also known as the One-, Two-, or Three-Dimensional Cutting Stock Problem); allocating resources over time; planning the layout of newspaper pages; designing new computer chips; assigning processors in parallel computers; and many others. MD-PPs arise in the obvious military logistic applications and in others, e.g., campaign planning, where forces available and time are the two dimensions involved. Most MD-PPs are known to be NP-complete and the related literature is rich in heuristics and approximation algorithms. There are efficient algorithms that yield optimal solutions for special cases of the general problem. The complexity of some of these special cases is still unknown. In this dissertation, we implement new techniques, both exact and heuristic, to solve instances of PLP, MD-KP and MD-BPP. These techniques are based primarily on new theory developed for PLP. Initially, we analyze the effect of restricting rotation and the effects of other common pattern restrictions on the number of bins required to solve instances of Two- Dimensional Bin Packing Problems (2D-BPP). We define the Minimum Size Instance (MSI) of an equivalence class of PLP, and show that every class has one and only one MSI. We also develop bounds on the dimensions of item and pallet in the MSI of a class, and show that a bound used for almost 15 years is incorrect. Applying the newly developed bounds on the MSI dimensions, we enumerate all 3,080,730 equivalence vii classes with an area ratio (AR) bound no greater than 100 boxes. Previous work in this area considered only a subset of these classes. We also present new bounding procedures for PLP. Some of these bounds rely on new relations among distinct equivalence classes of PLP (restricted and relaxed classes). We develop new heuristics for PLP. The G5-heuristic computes optimal solutions to all instances of PLP with an AR bound of up to 51 boxes, and 99.999% of all possible instances with an AR bound of up to 100 boxes, differing from the optimum solution by at most one box in the 0.001% remaining instances. We develop three other heuristics to solve some instances not solved by the G5-heuristic or by heuristics from the literature. After applying the new bounding procedures and heuristics, 6,952 equivalence classes of PLP with an AR bound of up to 100 boxes remain without a proven optimal solution. We develop an exact algorithm, the HVZ algorithm, to solve these remaining instances. With this
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