Two-dimensional Cutting Stock Problem Research Articles

An Application Traditional and Nontraditional Approach on Lock Industry for Optimization of Cutting Layout

Engineering becomes more advanced and the business in the industrial world becomes more competitive. The method of implementing optimization technique is essential. The nesting of two dimensional shapes for the press tool design is general optimization problem. In mass production industries; the small inefficiencies will have the huge wastage. This is also known as two-dimensional Cutting Stock Problem (CSP). The aim is to minimize wastage there by increasing the profit by means of arranging the parts into the master sheet. The parts are to be arranged in the sheet in a acceptable position. It needs a various algorithm for placing the parts into the sheets like Heuristic, Meta heuristic, Bottom left heuristics etc.

Modeling Two-Dimensional Guillotine Cutting Problems via Integer Programming

We propose a framework to model general guillotine restrictions in two-dimensional cutting problems formulated as mixed-integer linear programs (MIPs). The modeling framework requires a pseudopolynomial number of variables and constraints, which can be effectively enumerated for medium-size instances. Our modeling of general guillotine cuts is the first one that, once it is implemented within a state-of-the-art MIP solver, can tackle instances of challenging size. We mainly concentrate our analysis on the guillotine two-dimensional knapsack problem (G2KP), for which a model, and an exact procedure able to significantly improve the computational performance, are given. We also show how the modeling of general guillotine cuts can be extended to other relevant problems such as the guillotine two-dimensional cutting stock problem and the guillotine strip packing problem (GSPP). Finally, we conclude the paper discussing an extensive set of computational experiments on G2KP and GSPP benchmark instances from the literature.

Comparison of two metaheuristics to solve a 2-D cutting stock problem with set-up cost in the paper industry

This paper deals with the two-dimensional cutting stock problem with set-up cost 2CSP-S. This problem is composed of three optimisation sub-problems: a 2-D bin packing 2BP problem to place images on patterns, a linear programming LP problem to find for each pattern the number of stock sheets to be printed and a combinatorial problem to find the number of each image on each pattern. We have already developed two different metaheuristics to solve the 2CSP-S focusing on this third sub-problem: a simulated annealing and a genetic algorithm. In this article, we propose to compare these two approaches. It is important to notice that our approaches are not new packing techniques. This work was conducted for a paper industry company and experiments were realised on real and artificial data sets.

A heuristic approach to minimize the number of saw cycles in small-scale furniture factories

This paper addresses a two-dimensional cutting stock problem arising in furniture factories. The problem involves the simultaneous optimization of two, usually conflicting, objectives: minimizing the total number of objects and maximizing the cutting machine productivity in terms of the number of objects that are simultaneously cut. A heuristic algorithm to solve the problem is proposed based on variables and constraints generation. The main idea is to add, in a dynamic way, bounds to the frequency of some chosen cutting patterns. At each iteration a solution is generated and at the end we have a set of non-dominated solutions. A computational study was conducted using real data from a small-scale furniture factory. The results show that the proposed algorithm finds solutions that are as good as or better than the ones used in practice in the furniture factory.

Sequential value correction heuristic for the two-dimensional cutting stock problem with three-staged homogenous patterns

A sequential value correction heuristic is presented for the two-dimensional cutting stock problem with three-staged homogenous patterns, considering both input-minimization and simplicity of the cutting process. The heuristic constructs many cutting plans iteratively and selects the best one as the solution. The patterns in each cutting plan are generated sequentially using simple recursive techniques. The values of the item types are corrected after the generation of each pattern to diversify the cutting plans. Computational results indicate that the proposed heuristic is more effective in input minimization than published algorithms and commercial stock cutting software packages that use three-staged general or exact patterns.

An efficient genetic algorithm with a corner space algorithm for a cutting stock problem in the TFT-LCD industry

In this study, we investigate a two-dimensional cutting stock problem in the thin film transistor liquid crystal display industry. Given the lack of an efficient and effective mixed production method that can produce various sizes of liquid crystal display panels from a glass substrate sheet, thin film transistor liquid crystal display manufacturers have relied on the batch production method, which only produces one size of liquid crystal display panel from a single substrate. However, batch production is not an effective or flexible strategy because it increases production costs by using an excessive number of glass substrate sheets and causes wastage costs from unused liquid crystal display panels. A number of mixed production approaches or algorithms have been proposed. However, these approaches cannot solve industrial-scale two-dimensional cutting stock problem efficiently because of its computational complexity. We propose an efficient and effective genetic algorithm that incorporates a novel placement procedure, called a corner space algorithm, and a mixed integer programming model to resolve the problem. The key objectives are to reduce the total production costs and to satisfy the requirements of customers. Our computational results show that, in terms of solution quality and computation time, the proposed method significantly outperforms the existing approaches.

Applying an Enhanced Heuristic Algorithm to a Constrained Two-Dimensional Cutting Stock Problem

This study focuses on developing fully functional cutting stock software by utilizing an improved multi-phase heuristic algorithm to find the optimal solution for the cutting materi al. Issues which commonly appear in real-world cutting stock applications are also discussed

Multiple-choice knapsack-based heuristic algorithm for the two-stage two-dimensional cutting stock problem in the paper industry

This study examines a two-stage two-dimensional cutting stock problem encountered by a paper mill company. The problem includes various machine-related and operational constraints based on real-world situations. Paper products are manufactured using two major cutting processes. Each cutting machine has a specific minimum and maximum width for input and output rolls and is limited by the maximum number of rolls it can cut at the same time. A mathematical model is presented to formally address the problem and an efficient multiple-choice knapsack-based heuristic algorithm is proposed to solve the problem. To demonstrate the efficiency of the proposed heuristic algorithm, computational experiments are conducted on test data-set generated from real-world data provided by a large paper mill company in the Republic of Korea.

Integrating two-dimensional cutting stock and lot-sizing problems

The two-dimensional cutting stock problem (2DCSP) consists in the minimization of the number of plates used to cut a set of items. In industry, typically, an instance of this problem is considered at the beginning of each planning time period, what may result in solutions of poor quality, that is, excessive waste, when a set of planning periods is considered. To deal with this issue, we consider an integrated problem, in which the 2DCSP is extended from the solution in only a single production planning period to a solution in a set of production planning periods. The main difference of the approach in this work and the ones in the literature is to allow sufficiently large residual plates (leftovers) to be stored and cut in a subsequent period of the planning horizon, which may further help in the minimization of the waste. We propose two integrated integer programming models to optimize the combined two-dimensional cutting stock and lot-sizing problems, minimizing the total cost, which includes material, waste and storage costs. Two heuristics based on the industrial practice to solve the problem were also presented. Computational results for the proposed models and for the heuristics are presented and discussed.

A two-dimensional bin packing problem with size changeable items for the production of wind turbine flanges in the open die forging industry

Efficient cutting design is essential to reduce the costs of production in the open die forging industry. This article discusses a slab cutting design problem that occurs when parallel piped items are cut from raw material steel slabs with varying widths and lengths to meet a volume requirement. The problem is modeled as a two-dimensional cutting stock problem or bin packing problem with size-changeable items. Cut loss and guillotine cut constraints are included. A knapsack-based heuristic algorithm is proposed and it is tested by a real-world manufacturer who is cutting steel for wind turbine flanges. The firm generates an annual cost reduction of approximately US $2000 000.

Heuristic for the cutting and purchasing decisions of multiple metal coils

This paper presents a heuristic algorithm for the rectangular two-dimensional cutting stock problem, where metal coils of multiple widths are divided into rectangular items with guillotine cuts. The cutting process contains two phases. Coils are cut into segments at the first phase and the segments into items at the second phase. A subset of three-staged patterns is considered in generating the cutting plan. The algorithm is used to accomplish the following tasks: (1) Generating cutting plans for the cutting process; (2) Selecting the coil widths to purchase; (3) Optimizing the segment lengths to order. Benchmark instances are used to demonstrate the effectiveness of the algorithm in improving material utilization, and examples are used to illustrate the decision methods for different tasks.

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.

Sequential grouping heuristic for the two-dimensional cutting stock problem with pattern reduction

Abstract A sequential grouping heuristic (SGH) that supports parallel computing is presented for solving the two-dimensional cutting stock problem with pattern reduction, where a set of rectangular items with given demand are cut from rectangular stock plates of the same size, considering both input-minimization (main objective) and pattern reduction (auxiliary objective). It is based on the sequential heuristic procedure that generates each next pattern to fulfill some portion of the remaining items and repeats until all items are fulfilled. The SGH uses a grouping technique to select the items that can be used to generate the next pattern, and adjusts the item values according to the sequential value correction method after the next pattern is generated. Each next pattern is generated using a dynamic programming recursion. The computational results indicate that the SGH is powerful in both input-minimization and pattern reduction, and the parallel computing is useful to reduce computation time.

Models for the two-dimensional two-stage cutting stock problem with multiple stock size

We consider a Two-Dimensional Cutting Stock Problem (2DCSP) where stock of different sizes is available, and a set of rectangular items has to be obtained through two-stage guillotine cuts. We propose and computationally compare three Mixed-Integer Programming models for the 2DCSP developing formulations from the literature. The first two models have a polynomial and pseudo-polynomial number of variables, respectively, and can be solved with a general-purpose MIP solver. The third model, having an exponential number of variables, is solved via branch-and-price techniques. We conclude the paper describing the results of extensive computational experiments on a set of benchmark instances from the literature.

A Method for Two-Dimensional Cutting Stock Problem with Triangular Shape Items

Increasing cost of raw material and need to avoid industria l wastage, solving cutting stock problems became of great interest in the area of Operation s Research. An optimum cutting stock problem can be defined as cutting a main sheet into smalle r pieces while minimizing total wastage of the raw material or maximizing overall profit obt ained by cutting smaller pieces from the main sheet. Objective of this study is to gener ate feasible cutting patterns for twodimensional triangular shape cutting items. An algorithm i s presented based on modified Branch and Bound Algorithm. A computer program is developed using Matlab software package to generate feasible cutting patterns. As a case study, four different sizes of triangular shape items with their demands are selected to cut from a main sheet with known dimensions. Applying proposed algorithm, demand is satisfied and total wastage i s minimized. Proposed algorithm can be exploited to generate cutting patterns for rectangular a nd triangular cutting items at the same time and more suitable for medium size two dimensional cutting stock problem.

A Heuristic Based on PSO for Irregular Cutting Stock Problem

Cutting Stock Problems (CSP) arise in many production industries where large stock sheets must be cut into smaller pieces. An irregular-shaped nesting approach for two-dimensional cutting stock problem is constructed in this research. We present a heuristic based on Particle Swarm Optimization Algorithm (PSO) for irregular-shaped two-dimensional cutting stock problem, where PSO is utilized to search optimal solution. Furthermore, the proposed approach combines a grid approximation method with Bottom-Left-Fill heuristic placement strategy to allocate irregular items. We evaluate the proposed approach using 15 revised benchmark problems available from the EURO Special Interest Group on Cutting and Packing. The performance illustrates the effectiveness and efficiency of our approach in solving irregular cutting stock problems.

A note on “Reducing the number of binary variables in cutting stock problems”

This study proposes a deterministic model to solve the two-dimensional cutting stock problem (2DCSP) using a much smaller number of binary variables and thereby reducing the complexity of 2DCSP. Expressing a 2DCSP with $$m$$ stocks and $$n$$ cutting rectangles requires $$2n^{2}+n(m+1)$$ binary variables in the traditional model. In contrast, the proposed model uses $$n^{2}+n\lceil {\log _2 m}\rceil $$ binary variables to express the 2DCSP. Experimental results showed that the proposed model is more efficient than the existing model.

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.

COLUMN GENERATION TECHNIQUE FOR SOLVING TWO-DIMENSIONAL CUTTING STOCK PROBLEMS: METHOD OF STRIPE APPROACH

We consider two-dimensional cutting stock problems where single rectangular stocks have to be cut into some smaller rectangular so that the number of stocks needed to satisfy the demands is minimum. In this paper we focus our study to the problem where the stocks have to be cut with guillotine cutting type and fixed orientation of finals. We formulate the problem as an integer programming, where the relaxation problem is solved by column generation technique. New pattern generation is formulated based on method of stripe. In obtaining the integer solution, we round down the optimal solution of the relaxation problem and then we derive an extra mix integer programming for satisfying the unmet demands. The optimal solution of the original problem is the combination of the round-down solution and the optimal solution of the extra mix integer programming.A numerical example of the problem is given in the end of this paper.DOI : http://dx.doi.org/10.22342/jims.13.2.65.161-172

Extended block patterns for the two-dimensional cutting stock problem

Guillotine patterns of specified features are often used in manufacturing industries to cut stock plate into rectangular pieces. The pattern type should be selected from considering both material utilization and cutting cost. This article proposes extended block patterns for the two-dimensional cutting stock problem, where each cut on the plate produces just one extended block. A recursion is used to generate the optimal pattern. It maximizes the difference between the total value of the included pieces and the cutting cost. The computational results indicate that extended block patterns are useful to reduce production cost.