Guillotine Cutting Stock Problem Research Articles

A 2-dimensional guillotine cutting stock problem with variable-sized stock for the honeycomb cardboard industry

This paper introduces novel mathematical optimisation models for the 2-Dimensional guillotine Cutting Stock Problem with Variable-Sized Stock that appears in a Spanish company in the honeycomb cardboard industry. This problem mainly differs from the classical cutting stock problems in the stock, which is considered variable-sized, i.e. we have to decide the panel dimensions, width, and length. This approach is helpful in industries where the stock is produced simultaneously with the cutting process. The stock is then cut into smaller rectangular pieces that must meet the customers' requirements, such as the type of item, dimensions, demands, and technical specifications. Furthermore, in the problem tackled in this paper, the cuts are guillotine, performed side to side. The proposed mathematical models are validated using real data from the company, obtaining results that drastically reduce the produced material and leftovers, reducing operation times and economic costs.

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.

Linear Integer Optimization Model for Two-Stage Guillotine Cutting Stock Problem Using Branch and Bound Method in the Garment Industry

This paper discusses the Two-Stage Guillotine Cutting Stock Problem (2GCSP) in the garment industry, namely how to determine the two-stage guillotine pattern that is used to cut fabric stocks into several certain size t-shirt materials that are produced based on the demand for each size of the shirt. 2GCSP is modeled in the form of Linear Integer Optimization and finding solutions using the Branch and Bound method. In this paper also presented a Graphical User Interface with Maple software as an interactive tool to find the best fabric stock cutting patterns. The results show that the optimal solution can be determined by solving numerically using the Branch and Bound method and Maple optimization packages. The solution is shown with an illustration of the pattern and the amount of fabric cut based on the pattern.

Comparative analysis of pattern-based models for the two-dimensional two-stage guillotine cutting stock problem

This study presents a theoretical and computational analysis of integer programs based on the concept of the patterns, the so-called pattern-based models, for a two-dimensional two-stage guillotine cutting stock problem. The full-pattern models and the staged-pattern models with the corresponding column generation problems are analyzed and compared. Those pattern-based models have been actively used, however, their LP-relaxations have not been theoretically analyzed and compared. In this paper, we formally establish a hierarchy of the strength of lower bounds that can be obtained from their LP-relaxations. In addition, through computational tests with benchmark instances, we analyze the trade-off between the strength of the lower bounds and the required computation time to solve the LP-relaxations of the models. The quality of integer solutions obtained by an LP-based branch-and-bound algorithm with the columns generated during the process of solving the LP-relaxations of the models are also compared. The results show that one of the staged-pattern models, which has not been well-studied, shows competitive theoretical and computational performance.

Balanced approach for the two-dimensional rectangular guillotine cutting stock problem with setup cost

The main optimization objective of two-dimensional rectangular guillotine cutting stock problem (2DRGCSP) is minimizing stock cost. However, setup cost is also needed to be considered as an important factor in the real manufacturing process. A sequential pattern generation algorithm with rectangular homogenous strips (SPGA-RHSs) is presented to generate a cutting plan based on each single stock. A better cutting pattern considering both stock cost and setup cost is obtained with the cost balance utilization (CBU). The balanced approach combining SPGA-RHSs and CBU is used to achieve a current optimal cutting plan for 2DRGCSP with the minimum sum of stock and setup cost. The computational results prove that the proposed balanced approach has good optimization performance for 2DRGCSP with setup cost.

Heuristics for Two-Dimensional Rectangular Guillotine Cutting Stock

Two-Dimensional Rectangular Guillotine Cutting Stock Problem (2DRGCSP) is one of the most significant problems in the manufacturing industries. A set of small rectangular size of paper, aluminum rolls, glasses, fiber glasses, or plastic are required to cut from a set of big size rectangular sheet of its raw materials. A guillotine cut is used, where the sheet is cut from one side to another side without changing the direction of the blade to produce the strips. Then each strip is cut again to produce the small rectangular size of panels that match with the required size, and the number of the panels must satisfy with the demand. This paper presents the heuristic techniques to solve the cutting problems. The heuristic techniques create the good cutting patterns such that the waste of the sheets is minimized and demands for each panel are satisfied. There are five proposed heuristics, developed to solve this problem. They are 2D simple heuristic cutting (2DSHC), 2D horizontal construction (2DHC), 2D vertical construction (2DVC), 2D horizontal improvement (2DHI), and 2D vertical improvement (2DVI). The testing instances are created from the real problems in the Printed Circuit Board (PCB) Company. The results are presented and compared with column generation (CG) method. The proposed heuristics provide good cutting patterns with a comparable waste to the column generation method. Moreover, the heuristics can produce solutions in a short computational time even in the large size instances, where the column generation method cannot find the solution. Therefore, these proposed heuristics techniques are applicable to solve the 2DRGCSP in the industry where the solutions from the exact algorithms cannot be found.

An adapted particle swarm optimization approach for a 2D guillotine cutting stock problem

This paper focuses on the two dimensional rectangular non-oriented guillotine cutting stock problem in which pieces with different dimensions need to be cut with different quantities in order to satisfy customers orders. In order to optimize the raw material utilization, make-to-order and make-to-stock production strategies are combined by considering forecast plans with inventory constraints in addition to firm orders. An adapted version of Particle Swarm Optimization approach is proposed and combined to a hybrid heuristic. Results show that the proposed algorithm outperforms the hybrid heuristic in a reasonable computational time.

A new heuristic algorithm for two-dimensional defective stock guillotine cutting stock problem with multiple stock sizes

A new heuristic algorithm for two-dimensional defective stock guillotine cutting stock problem with multiple stock sizes

3D overlapped grouping Ga for optimum 2D guillotine cutting stock problem

The cutting stock problem (CSP) is one of the significant optimization problems in operations research and has gained a lot of attention for increasing efficiency in industrial engineering, logistics and manufacturing. In this paper, new methodologies for optimally solving the cutting stock problem are presented. A modification is proposed to the existing heuristic methods with a hybrid new 3-D overlapped grouping Genetic Algorithm (GA) for nesting of two-dimensional rectangular shapes. The objective is the minimization of the wastage of the sheet material which leads to maximizing material utilization and the minimization of the setup time. The model and its results are compared with real life case study from a steel workshop in a bus manufacturing factory. The effectiveness of the proposed approach is shown by comparing and shop testing of the optimized cutting schedules. The results reveal its superiority in terms of waste minimization comparing to the current cutting schedules. The whole procedure can be completed in a reasonable amount of time by the developed optimization program.

Two‐stage two‐dimensional guillotine cutting stock problems with usable leftover

AbstractIn this study, we solve the nonexact two‐stage two‐dimensional guillotine cutting problem considering usable leftovers, in which stock plates remainders of the cutting patterns (nonused material or trim loss) can be used in the future, if they are large enough to fulfill future demands for items (ordered smaller plates). This cutting problem can be characterized as a residual bin‐packing problem because of the possibility of putting back into stock residual pieces, as the trim loss of each cutting/packing pattern does not necessarily represent waste of material depending on its size. Two bilevel mathematical programming models to represent this nonexact two‐stage two‐dimensional residual bin‐packing problem are presented. The models basically consist of cutting/packing the ordered items using a set of plates of minimum cost and, among all possible solutions of minimum cost, choosing one that maximizes the value of the generated usable leftovers. Because of special characteristics of these bilevel models, they can be reformulated as one‐level mixed integer programming models. Results of some numerical experiments are presented to show that the models represent appropriately the problem and illustrate their performance.

Approaches to real world two-dimensional cutting problems

We consider a real world generalization of the 2-Dimensional Guillotine Cutting Stock Problem arising in the wooden board cutting industry. A set of rectangular items has to be cut from rectangular stock boards, available in multiple formats. In addition to the classical objective of trim loss minimization, the problem also asks for the maximization of the cutting equipment productivity, which can be obtained by cutting identical boards in parallel. We present several heuristic algorithms for the problem, explicitly considering the optimization of both objectives. The proposed methods, including fast heuristic algorithms, Integer Linear Programming models and a truncated Branch and Price algorithm, have increasing complexity and require increasing computational effort. Extensive computational experiments on a set of realistic instances from the industry show that high productivity of the cutting equipment can be obtained with a minimal increase in the total area of used boards. The experiments also show that the proposed algorithms perform extremely well when compared with four commercial software tools available for the solution of the problem.

A branch-and-price algorithm for the two-stage guillotine cutting stock problem

We investigate the two-stage guillotine two-dimensional cutting stock problem. This problem commonly arises in the industry when small rectangular items need to be cut out of large stock sheets. We propose an integer programming formulation that extends the well-known Gilmore and Gomory model by explicitly considering solutions that are obtained by both slitting some stock sheets down their widths and others down their heights. To solve this model, we propose an exact branch-and-price algorithm. To the best of our knowledge, this is the first contribution with regard to obtaining integer optimal solutions to Gilmore and Gomory model. Extensive results, on a set of real-world problems, indicate that the proposed algorithm delivers optimal solutions for instances with up to 809 items and that the hybrid cutting strategy often yields improved solutions. Furthermore, our computational study reveals that the proposed modelling and algorithmic strategy outperforms a recently proposed arc-flow model-based solution strategy.

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.

A new formulation of the two-dimensional cutting-stock problem with consideration of demand planning

This paper focuses on the two dimensional rectangular non-oriented guillotine cutting stock problem (TDRCSP) in which many pieces with different dimensions need to be cut with different quantities in order to satisfy customers' orders. In order to maximise the use of raw materials, make-to-order 'MTO' and make-to-stock 'MTS' production strategies are combined; in addition to the (firm) orders that need to be fulfilled, other quantities of pieces are considered when designing cutting patterns in order to satisfy the forecast plan over a fixed time horizon. A new formulation of the optimisation problem is proposed in this paper considering multi-period demand planning as well as inventory management constraints. To solve this new problem, a hybrid heuristic, based on the combination of the Bottom Left and Shelf algorithms, is considered. It is shown that formulating the problem combining the original non-oriented guillotine TDRCSP with demand and inventory planning optimises the raw material use and yields good solutions in very short computational times. Results show that integrating forecast and inventory constraints are much more of an additional way to improve the raw material use than a real constraint. Finally, a discussion on the hybrid heuristic sensitivity and robustness is reported.

A column generation heuristic for the two-dimensional two-staged guillotine cutting stock problem with multiple stock size

We consider a two-dimensional cutting stock problem where stock of different sizes is available, and a set of rectangular items has to be obtained through two-staged guillotine cuts. We propose a heuristic algorithm, based on column generation, which requires as its subproblem the solution of a two-dimensional knapsack problem with two-staged guillotines cuts. A further contribution of the paper consists in the definition of a mixed integer linear programming model for the solution of this knapsack problem, as well as a heuristic procedure based on dynamic programming. Computational experiments show the effectiveness of the proposed approach, which obtains very small optimality gaps and outperforms the heuristic algorithm proposed by Cintra et al. [3].

Using Wang's two‐dimensional cutting stock algorithm to optimally solve difficult problems

AbstractP. Y. Wang's classic bottom‐up two‐dimensional cutting stock algorithm generates cutting patterns by building rectangles both horizontally and vertically. This algorithm uses a parameter β1 to tradeoff the number of rectangles generated by the algorithm and hence the quality of the cutting pattern solution obtained versus the amount of computer resources required. Several researchers have made relatively straightforward modifications to Wang's basic algorithm resulting in improved computational times. However, even with these modifications, Wang's approach tends to require large amounts of computer resources in order to optimally or near‐optimally solve difficult two‐dimensional guillotine cutting stock problems. In this paper, we present an iterative approach that judiciously uses Wang's basic algorithm (with some previously defined modifications) to obtain optimal cutting patterns to difficult two‐dimensional cutting stock problems in reasonable computer run times.

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++.