Problems, Models and Algorithms in One- and Two-Dimensional Cutting

Preface Within such disciplines as Management Science, Information and Computer Science , Engineering, Mathematics and Operations Research, problems of cutting and packing (C&P) of concrete and abstract objects appear under various specifications (cutting problems, knapsack problems, container and vehicle loading problems, pallet loading, bin packing, assembly line balancing, capital budgeting, changing coins, etc.), although they all have essentially the same logical structure. In cutting problems, a large object must be divided into smaller pieces; in packing problems, small items must be combined to large objects. Most of these problems are NP-hard. Since the pioneer work of L.V. Kantorovich in 1939, which first appeared in the West in 1960, there has been a steadily growing number of contributions in this research area. In 1961, P. Gilmore and R. Gomory presented a linear programming relaxation of the one-dimensional cutting stock problem. The best-performing algorithms today are based on their relaxation. It was, however, more than three decades before the first 'optimum' algorithms appeared in the literature and they even proved to perform better than heuristics. They were of two main kinds: enu-merative algorithms working by separation of the feasible set and cutting plane algorithms which cut off infeasible solutions. For many other combinatorial problems , these two approaches have been successfully combined. In this thesis we do it for one-dimensional stock cutting and two-dimensional two-stage constrained cutting. For the two-dimensional problem, the combined scheme provides mostly better solutions than other methods, especially on large-scale instances, in little time. For the one-dimensional problem, the integration of cuts into the enumera-tive scheme improves the results of the latter only in exceptional cases. While the main optimization goal is to minimize material input or trim loss (waste), in a real-life cutting process there are some further criteria, e.g., the number of different cutting patterns (setups) and open stacks. Some new methods and models are proposed. Then, an approach combining both objectives will be presented, to our knowledge, for the first time. We believe this approach will be highly relevant for industry. The presented investigation is a result of a three-year study at Dresden University , concluded in August 2003. I would like to express my gratitude to all the people who contributed to the completion of my dissertation. Dr. Guntram Scheithauer has been a very attentive and resourceful supervisor, not only in the research. He and Professor Johannes Terno, who died in 2000 after a …