The two-dimensional bin packing problem with variable bin sizes and costs

The primary objective in cutting and packing problems is trim loss or material input minimization (in stock cutting) or value maximization (in knapsack-type problems). However, in real-life production we usually have many other objectives (costs) and constraints. Probably the most complex auxiliary criteria in one-dimensional stock cutting are the number of different cutting patterns (setups) and the maximum number of open stacks during the cutting process. There are applications where the number of stacks is restricted to two. We design a sequential heuristic to minimize material input and show its high effectiveness for this purpose. Then we extend it to restrict the number of open stacks to any given limit. Then, the heuristic is simplified and integrated into a setup-minimization approach in order to combine setup and open-stacks minimization. To get a smaller number of open stacks, we may split up the problem into several parts of smaller sizes. Different solutions are evaluated in relation to the multiple objectives using the Pareto criterion.

Setup and Open-Stacks Minimization in One-Dimensional Stock 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 …

/pdf/problems-models-and-algorithms-in-one-and-two-dimensional-4xtul5m5uc.pdf

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

Two algorithms for the one-dimensional cutting problem, namely, a modified branch-and-bound method (exact method) and a heuristic sequential value correction method are suggested. In order to obtain a reliable assessment of the efficiency of the algorithms, hard instances of the problem were considered and from the computational experiment it seems that the efficiency of the heuristic method appears to be superior to that of the exact one, taking into account the computing time of the latter. A detailed description of the two methods is given along with suggestions for their improvements.

/pdf/linear-one-dimensional-cutting-packing-problems-numerical-168w1p0hur.pdf

Linear one-dimensional cutting-packing problems: numerical experiments with the sequential value correction method (SVC) and a modified branch-and-bound method (MBB)

Consideration was given to the problem of packing the rectangular half-infinite strip. It was suggested to solve it using a one-point evolutionary algorithm with search of the best solution in the neighborhood of the local lower bound. Proposed was an algorithm to construct this neighborhood including the packings obtained by solving a special problem of one-dimensional cutting. Its solution was shown to be the local lower bound in the considered neighborhood. An improved global lower bound was proposed. The results of numerical modeling were presented. The record value obtained was compared with the global bounds.

Local search of orthogonal packings using the lower bounds

Different optimal cutting problems are considered in this paper. Among them are cutting forming problems (closed packing problems) and problems of cutting totality planning with intensities of their application. For solving these planning problems, linear or integer programming is used. Furthermore, different cutting technological and organizational situations are considered. Different optimal criteria and a compromise solution choice procedure are presented. All the statements are illustrated by numerical examples from a guillotine cutting area. The possibility of linear cutting approximation for a non-guillotine (closed packing) cutting stock problem is shown. Rectangular packing algorithms based on this approximation can be built. These algorithms form the basis of special software. Their characteristics are presented in the conclusion of the paper.

Linear programming cutting problems

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Linear programming cutting problems