Two-dimensional Guillotine Cutting Stock Problem Research Articles

Models for two- and three-stage two-dimensional cutting stock problems with a limited number of open stacks

We address three variants of the two-dimensional cutting stock problem in which the guillotine cutting of large objects produces a set of demanded items. The characteristics of the variants are the rectangular shape of the objects and items; the number of two or three orthogonal guillotine stages; and a sequencing constraint that limits the number of open stacks to a scalar associated with the number of automatic compartments or available space near the cutting machine. These problems arise in manufacturing environments that seek minimum waste solutions with limited levels of work-in-process. Despite their practical relevance, we are not aware of mathematical models for them. In this paper, we propose an integer linear programming (ILP) formulation for each of these variants based on modelling strategies for the two-dimensional guillotine cutting stock problem and the minimisation of the open stacks problem. The first two variants deal with exact and non-exact 2-stage patterns, and the third with a specific type of 3-stage patterns. Using a general-purpose ILP solver, we performed computational experiments to evaluate these approaches with benchmark instances. The results show that several equivalent solutions of the cutting problem allow obtaining satisfactory solutions with a reduced number of open stacks.

A simple approach to the two-dimensional guillotine cutting stock problem

Abstract Cutting stock problems are within knapsack optimization problems and are considered as a non-deterministic polynomial-time (NP)-hard problem. In this paper, two-dimensional cutting stock problems were presented in which items and stocks were rectangular and cuttings were guillotine. First, a new, practical, rapid, and heuristic method was proposed for such problems. Then, the software implementation and architecture specifications were explained in order to solve guillotine cutting stock problems. This software was implemented by C++ language in a way that, while running the program, the operation report of all the functions was recorded and, at the end, the user had access to all the information related to cutting which included order, dimension and number of cutting pieces, dimension and number of waste pieces, and waste percentage. Finally, the proposed method was evaluated using examples and methods available in the literature. The results showed that the calculation speed of the proposed method was better than that of the other methods and, in some cases, it was much faster. Moreover, it was observed that increasing the size of problems did not cause a considerable increase in calculation time. In another section of the paper, the matter of selecting the appropriate size of sheets was investigated; this subject has been less considered by far. In the solved example, it was observed that incorrect selection from among the available options increased the amount of waste by more than four times. Therefore, it can be concluded that correct selection of stocks for a set of received orders plays a significant role in reducing waste.

Special issue on cutting, packing and related problems

The purpose of this special issue is to encourage research on cutting, packing and related problems. During the last five decades this subject has received the attention of many academic researchers and industrial practitioners, including topics such as bin packing, knapsack problems, pallet and container loading, nesting, pattern sequencing, layout problems, multi-processor scheduling, and integrated problems as cutting and sequencing, lot sizing and cutting, routing and packing, among others. This issue features nine among the 32 submitted papers. Umetani et al. deal with the irregular strip packing problem and develop a guided local search algorithm for a core sub-problem to solve it: the overlap minimization problem. Bortfeldt and Winter propose a genetic algorithm to address guillotine and non-guillotine cases of the two-dimensional knapsack problem with rectangular pieces. Koch, Konig and Wascher present a case study of a special cutting process from the wood-processing industry: a decision support tool is developed, which incorporates a linear programming model, and experience from the application of this tool is reported. Christensen and Rousoe cope with the container loading problem with multi-drop constraints and present a heuristic to solve it based on a tree search framework: the algorithm is tested using data from a Danish company distributing construction products. Belov, Kartak and Scheithauer consider the feasibility problem in multi-dimensional orthogonal packing problem and examine one-dimensional relaxations and linear programming bounds for the problem. Kaluzny and Shaw propose a mixed integer programming formulation to solve a problem of determining the arrangement of items in a cargo hold that optimizes the load balance in aircrafts of the Canadian Forces. Egeblad presents a heuristic based on guided local search to the problem of placing irregular shapes in two or three dimensions within a container, such that the placement of the shapes is optimized for balance and inertia moment. Yi, Chen and Zhou introduce the pinwheel pattern as an alternative solution to the manufacturer’s pallet loading problem, in which a single pallet has to be loaded with a maximum number of identical boxes. Finally, Vasko and Bartkowski present in a short communication an iterative approach that judiciously uses Wang’s algorithm to obtain optimal cutting patterns for two-dimensional guillotine cutting stock problems in reasonable computer runtimes. We are indebted to 71 competent reviewers for their time and effort. Without their help this edition on cutting, packing, and related problems would not have been possible. We also wish to express our deepest appreciation to the authors of all 32 papers submitted. Special thanks for the support of Celso C. Ribeiro, the editor-in-chief of ITOR, who invited us to participate as guest editors of this special issue.

Arc-flow model for the two-dimensional guillotine cutting stock problem

We describe an exact model for the two-dimensional cutting stock problem with two stages and the guillotine constraint. It is an integer linear programming (ILP) arc-flow model, formulated as a minimum flow problem, which is an extension of a model proposed by Valério de Carvalho for the one dimensional case. In this paper, we explore the behavior of this model when it is solved with a commercial software, explicitly considering all its variables and constraints. We also derive a new family of cutting planes and a new lower bound, and consider some variants of the original problem. The model was tested on a set of real instances from the wood industry, with very good results. Furthermore the lower bounds provided by the linear programming relaxation of the model compare favorably with the lower bounds provided by models based on assignment variables.

Algorithms for two-dimensional cutting stock and strip packing problems using dynamic programming and column generation

We investigate several two-dimensional guillotine cutting stock problems and their variants in which orthogonal rotations are allowed. We first present two dynamic programming based algorithms for the Rectangular Knapsack (RK) problem and its variants in which the patterns must be staged. The first algorithm solves the recurrence formula proposed by Beasley; the second algorithm – for staged patterns – also uses a recurrence formula. We show that if the items are not so small compared to the dimensions of the bin, then these algorithms require polynomial time. Using these algorithms we solved all instances of the RK problem found at the OR-LIBRARY, including one for which no optimal solution was known. We also consider the Two-dimensional Cutting Stock problem. We present a column generation based algorithm for this problem that uses the first algorithm above mentioned to generate the columns. We propose two strategies to tackle the residual instances. We also investigate a variant of this problem where the bins have different sizes. At last, we study the Two-dimensional Strip Packing problem. We also present a column generation based algorithm for this problem that uses the second algorithm above mentioned where staged patterns are imposed. In this case we solve instances for two-, three- and four-staged patterns. We report on some computational experiments with the various algorithms we propose in this paper. The results indicate that these algorithms seem to be suitable for solving real-world instances. We give a detailed description (a pseudo-code) of all the algorithms presented here, so that the reader may easily implement these algorithms.

Remote service to solve the two-dimensional cutting stock problem: an application to the Canary Islands costume

This paper presents the Graphical User Interface (GUI) for a remote service used to solve the two-dimensional guillotine cutting stock problem as applied to the textile industry. This interface allows for the patterns in the regional woman's dress of the city of La Orotava, in the Canary Islands, to be defined and manipulated. The user must choose the garment, its quantity and its sizes, as well as, the amount of material available. This information is then used by the system to arrange the patterns. Individual patterns may also be selected. Internally, the system implements a variant of the Viswanathan-Bagchi algorithm to solve the two-dimensional cutting stock problem. The solution is calculated remotely in a dedicated server. The user may, however, keep track of all those possibilities that were rejected by the algorithm in its search for the final solution. He/She can also modify the final solution by dragging the patterns with the mouse into the desired position on the surface. The GUI and the remote service were implemented using Java, while the solving algorithm was coded using C/C++.

Approximate and exact algorithms for the double-constrained two-dimensional guillotine cutting stock problem

In this paper, we propose approximate and exact algorithms for the double constrained two-dimensional guillotine cutting stock problem (DCTDC). The approximate algorithm is a two-stage procedure. The first stage attempts to produce a starting feasible solution to DCTDC by solving a single constrained two dimensional cutting problem, CDTC. If the solution to CTDC is not feasible to DCTDC, the second stage is used to eliminate non-feasibility. The exact algorithm is a branch-and-bound that uses efficient lower and upper bounding schemes. It starts with a lower bound reached by the approximate two-stage algorithm. At each internal node of the branching tree, a tailored upper bound is obtained by solving (relaxed) knapsack problems. To speed up the branch and bound, we implement, in addition to ordered data structures of lists, symmetry, duplicate, and non-feasibility detection strategies which fathom some unnecessary branches. We evaluate the performance of the algorithm on different problem instances which can become benchmark problems for the cutting and packing literature.

A computational study of LP-based heuristic algorithms for two-dimensional guillotine cutting stock problems

In this paper we develop and compare several heuristic methods for solving the general two-dimensional cutting stock problem. We follow the Gilmore-Gomory column generation scheme in which at each iteration a new cutting pattern is obtained as the solution of a subproblem on one stock sheet. For solving this subproblem, in addition to classical dynamic programming, we have developed three heuristic procedures of increasing complexity, based on GRASP and Tabu Search techniques, producing solutions differing in quality and in time requirements. In order to obtain integer solutions from the fractional solutions of the Gilmore-Gomory process, we compare three rounding procedures, rounding up, truncated branch and bound and the solution of a residual problem. We have coded and tested all the combinations of algorithms and rounding procedures. The computational results obtained on a set of randomly generated test problems show their relative efficiency and allow the potential user to choose from among them, according to the available computing time.

An efficient approach for large-scale two-dimensional guillotine cutting stock problems

An efficient approach for large-scale two-dimensional guillotine cutting stock problems

An efficient approach for large-scale two-dimensional guillotine cutting stock problems

We present an efficient method for solving approximately both constrained and unconstrained two-dimensional cutting stock problems. The algorithm guarantees a constant approximation ratio for some versions of the problem. The performance of the proposed algorithm is evaluated on several large-scale randomly generated problem instances and on many instances of the literature. Computational results show that our algorithm produces high-quality solutions within reasonable computational times.

A Polynomial Time Algorithm For The Guillotine Pallet Loading Problem

A polynomial time algorithm is presented for solving the two-dimensional guillotine cutting stock problem where all small rectangles are of the same dimensions (generating pallet loading patterns)....

An algorithm for the 2D guillotine cutting stock problem

The two-dimensional guillotine cutting stock problem seeks to apply a sequence of guillotine (edge to edge) cuts to extract a collection of smaller rectangles from a stock piece. In this paper an O( n 3) approximation algorithm for the two-dimensional guillotine cutting stock problem is presented. The algorithm considers each rectangle, in turn, and attempts to locate a guillotine feasible position on the stock piece. The algorithm has the property that a position will be located for the n-th rectangle if and only if such a feasible placement exists. This paper establishes the efficiency and effectiveness of the new algorithm by presenting computational results.

A computational improvement to Wang's two-dimensional cutting stock algorithm

Wang developed algorithms [ Opns Res. 31, 573–586 (1983)] for the constrained two-dimensional guillotine cutting stock problem. Given a bound on trim waste based on a feasible cutting pattern, her algorithms are guaranteed to generate the optimal guillotine cutting pattern. In this paper, we discuss two algorithms: (1) an algorithm, SPAM, which quickly generates solutions to the constrained two-stage two-dimensional guillotine cutting stock problem, and (2) an enhanced version of Wang's Algorithm One [1] which significantly improves its computational performance. SPAM is used to generate an initial upper bound for the minimum trim waste of the more general (non-staged) constrained guillotine cutting stock problem. Then, this bound is used in both Wang's Algorithm One and the enhanced version of it to solve 120 cutting stock problems. On average, the enhanced version was more than 25 times faster than the original algorithm, and the computational benefits of the enhancements increased as the problem complexity increased.