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

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: enumerative 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 enumerative 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

Human-Centred Feasibility Restoration

Decision systems for solving real-world combinatorial problems must be able to report infeasibility in such a way that users can understand the reasons behind it, and understand how to modify the problem to restore feasibility. Current methods mainly focus on reporting one or more subsets of the problem constraints that cause infeasibility. Methods that also show users how to restore feasibility tend to be less flexible and/or problem-dependent. We describe a problem-independent approach to feasibility restoration that combines existing techniques from the literature in novel ways to yield meaningful, useful, practical and flexible user support. We evaluate the resulting framework on two real-world applications

Effectiveness and Economy of Hydrogen Production by Natural Gas Decomposition in the Thermal Plasma

Hydrogen produced by the thermal decomposition of natural gas in a low-temperature plasma reactor is the subject of this study. The main advantage of plasma pyrolysis of natural gas is that a high yield of hydrogen can be obtained without the emission of carbon monoxide or carbon dioxide since the main products of the process are carbon in the solid state and hydrogen. The use of plasma allows the decomposition of natural gas without the use of catalysts, which is one of the main problems of current technologies for hydrogen production from this feedstock. In this paper, an analysis of the process is presented using a thermodynamic equilibrium model based on the minimum of the Gibbs function in the temperature range of 500-2500 K. Undesirable components in the system such as carbon dioxide, hydrogen cyanide and nitrogen compounds such as ammonia and nitric oxide are evaluated. The analysis showed the useful energy of the system per kilogram of feedstock and the efficiency of the high-temperature plasma decomposition process in terms of hydrogen produced. The results of the numerical analysis showed an optimal temperature for the process evaluation of about 1200 K, at which an efficiency of about 50 % is achieved.18th Conference on Sustainable Development of Energy, Water and Environment Systems (SDEWES), 24-29 September, Dubrovnik, CroatiaScientific advisory board: [https://web.archive.org/web/20231215125756/https://www.dubrovnik2023.sdewes.org/scientific-advisory-board]Programme: [https://web.archive.org/web/20231215130738/https://www.dubrovnik2023.sdewes.org/programme

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

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: enumerative 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 enumerative 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

A Modified Algorithm for Convex Decomposition of 3D Polyhedra

This paper presents implementation details of a simple algorithm to compute a convex decomposition of a non-convex polyhedron without shells (internal voids). For such a polyhedron S with n edges and r notches (features causing non-convexity in polyhedra), the algorithm produces a worst-case optimal O(r²) number of convex polyhedra S i with [ i S i = S in O(nr ) time and O(nr²+ r ) space. The algorithm repeatedly cuts and splits polyhedra along planes that resolve notches. It works also for certain classes of non-manifold polyhedra. The algorithm is constructed for the finite precision input information. Thus, questions of numerical stability must be considered. Making geometric decisions in terms of whole facets, we obtain a robust and simple version of the algorithm