Stock Sheet Research Articles

A hybrid simulated annealing metaheuristic algorithm for the two-dimensional knapsack packing problem

The rectangle knapsack packing problem is to pack a number of rectangles into a larger stock sheet such that the total value of packed rectangles is maximized. The paper first presents a fitness strategy, which is used to determine which rectangle is to be first packed into a given position. Based on this fitness strategy, a constructive heuristic algorithm is developed to generate a solution, i.e. a given sequence of rectangles for packing. Then, a greedy strategy is used to search a better solution. At last, a simulated annealing algorithm is introduced to jump out of the local optimal trap of the greedy strategy, to find a further improved solution. Computational results on 221 rectangular packing instances show that the presented algorithm outperforms some previous algorithms on average.

A squeaky wheel optimisation methodology for two-dimensional strip packing

The two-dimensional strip packing problem occurs in industries such as metal, wood, glass, paper, and textiles. The problem involves cutting shapes from a larger stock sheet or roll of material, while minimising waste. This is a well studied problem for which many heuristic methodologies are available in the literature, ranging from the basic ‘one-pass’ best-fit heuristic, to the state of the art Reactive GRASP and SVC(SubKP) iterative procedures. The contribution of this paper is to present a much simpler but equally competitive iterative packing methodology based on squeaky wheel optimisation. After each complete packing (iteration), a penalty is applied to pieces that directly decreased the solution quality. These penalties inform the packing in the next iteration, so that the offending pieces are packed earlier. This methodology is deterministic and very easy to implement, and can obtain some best results on benchmark instances from the literature.

A Distributed Stock Cutting using Mean Field Annealing and Genetic Algorithm

The composite stock cutting problem is defined as allocating rectangular and irregular patterns onto a large composite stock sheet of finite dimensions in such a way that the resulting scrap will be minimized. In this paper, we introduce a novel approach to hybrid optimization algorithm called MGA in MPI (Message Passing Interface) environments. The proposed MGA combines the benefit of rapid convergence property of Mean Field Annealing and the effective genetic operations. This paper also proposes the efficient data structures for pattern related information.

Bidirectional best-fit heuristic for orthogonal rectangular strip packing

In a non-guillotinable rectangular strip packing problem (RF-SPP), the best orthogonal placement of given rectangular pieces on a strip of stock sheet having fixed width and infinite height are searched. The aim is to minimize the height of the strip while including all the pieces in appropriate orientations. In this study, a novel bidirectional best-fit heuristic (BBF) is introduced for solving RF-SPPs. The proposed heuristic as a new feature considers the gaps in both horizontal and vertical directions during the placement process. The performance of BBF is compared to some previous approaches, including one of the best heuristics from the literature. BBF achieves better results than the existing heuristics and delivers a better or matching performance as compared to the most of the previously proposed meta-heuristics for solving RF-SPPs.

Solving the two-dimensional irregular objects allocation problems by using a two-stage packing approach

Packing problems are combinatorial optimization problems that concern the allocation of multiple objects in a large containment region without overlap and exist almost everywhere in real world. Irregular objects packing problems are more complex than regular ones. In this study, a methodology that hybridizes a two-stage packing approach based on grid approximation with an integer representation based genetic algorithm (GA) is proposed to obtain an efficient allocation of irregular objects in a stock sheet of infinite length and fixed width without overlap. The effectiveness of the proposed methodology is validated by the experiments in the apparel industry, and the results demonstrate that the proposed method outperforms the commonly used bottom-left (BL) placement strategy in combination with random search (RS).

A tabu search algorithm for a two-dimensional non-guillotine cutting problem

In this paper we study a two-dimensional non-guillotine cutting problem, the problem of cutting rectangular pieces from a large stock rectangle so as to maximize the total value of the pieces cut. The problem has many industrial applications whenever small pieces have to be cut from or packed into a large stock sheet. We propose a tabu search algorithm. Several moves based on reducing and inserting blocks of pieces have been defined. Intensification and diversification procedures, based on long-term memory, have been included. The computational results on large sets of test instances show that the algorithm is very efficient for a wide range of packing and cutting problems.

T-shape homogenous block patterns for the two-dimensional cutting problem

This paper presents an algorithm for unconstrained T-shape homogenous block cutting patterns of rectangular pieces. A vertical cut divides the stock sheet into two segments. Each segment consists of sections that have the same length and direction. A section contains a row of homogenous blocks. A homogenous block consists of homogenous strips of the same piece type. Each cut on the block produces just one strip. The directions of two strips cut successively from a block are either parallel or orthogonal. The algorithm uses a dynamic programming recursion to generate optimal blocks, solves knapsack problems to obtain the block layouts on the sections and the section layout on segments of various lengths, and optimally selects two segments to compose the cutting pattern. The computational results indicate that the algorithm is efficient in improving material usage, and the computation time is reasonable.

A More Efficient Method for Clustering Sheet Metal Shapes

The nesting of two-dimensional irregular shapes is a common problem which is frequently encountered by a number of industries where raw material has to be, as economically as possible, cut from a given stock sheet. A frequently recurring problem as far as cutting stock is concerned, is how to obtain the best nesting of some pieces of flat patterns which occupy minimalarea convex enclosure. The area of convex enclosure is related to the convex hull of the union of patterns which can be imagined as a large rubber band surrounding the set of all polygons. Our goal is to automatically obtain the smallest area convex shape containing all the patterns. As a matter of fact, Cheng and Rao have proposed an heuristic “stringy effect” procedure for clustering which follows a descending order of area of patterns. The “stringy effect” is able to put each new piece in a position which minimises the value of the distance between the centroid of each added piece and the centroid of the already formed cluster. The procedures till now shown in literature are quite complex. They make use of sliding techniques, and are not able to effectively work with relatively multiply-connected figures. In particular, the different procedures proposed are based on the No Fit Polygon computation of non-convex polygons, which often generates holes. This work is a proposal for a more efficient method, which can be used in heuristic procedure. In this paper a new procedure for the calculation of “No Fit Polygon” (NFP) of non-convex polygons is presented. Given two non-convex polygons, the algorithm is able to calculate their NFP very quickly and without any approximation by a polygon clipping method. By iterating this procedure with every polygon of our set, and positioning them using the “stringy effect” technique, it is so possible to obtain a convex shape that contains all the patterns, having the minimal area.

Two-dimensional Minkowski sum optimization of ganged stamping blank layouts for use on pre-cut sheet metal for convex and concave parts

With the increasing number of parts that manufacturers need to place on a piece of material such as sheet metal, the need increases for more sophisticated algorithms for part orientation and spacing. With greater part shape complexity, the ability of a skilled worker is challenged to minimize waste. Building on the previous work of Nye, this paper presents a Minkowski sum method for maximizing the number of parts within gangs on a rectangular sheet of material. The example provided uses a simply shaped part to illustrate the presented method, yielding a packing efficiency of 62% that is identical to the efficiency that a skilled worker would produce without the algorithm. The paper also provides results for laying out a more complex part in ganged sections, demonstrating a result that would be difficult for a human to reproduce. This work extends that of Nye by adding practical constraints such as the number of parts that can be blanked at once as well as the amount of horizontal and vertical spacing between ganged blanking sets. Additionally, the paper adds an algorithm for laying out polygons with concave geometries by separating the part into a set of convex polygons. Two examples for optimization, one of a chevron-shaped part and one of a complex shape previously used by Nye [Nye TJ. Stamping strip layout for optimal raw material utilization. Journal of Manufacturing Systems 2000;19(4):239–46] and Choi et al. [Choi JC, Kim BM, Cho HY, Kim C. A compact and practical CAD system for blanking or piercing of irregular-shaped sheet metal products and stator and rotor parts. International Journal of Machine Tools & Manufacture 1998;38:931–63], are provided, demonstrating the existence of a local maximum number of parts that may be stamped within a single ganged blank. The algorithm is extendable to a program that may provide stamping manufacturers with a tool that can maximize the total number of parts stamped on stock sheet metal, or for other tiling problems.

Recursive algorithm for the two‐dimensional cutting problem of punched strips

PurposeTo provide an algorithm for the two‐dimensional guillotine‐cutting problem of punched strips.Design/methodology/approachIt is assumed that the stock sheet is cut into blanks in two stages. First a guillotine shear cuts the sheet into strips, and then a stamping press punches out the blanks from the strips. To generate good strip layout, the sheet is divided into two segments with an orthogonal cut. Each segment consists of strips in the same direction. The strip directions of the two segments are perpendicular to each other. A recursion function is established to determine the optimal strip layouts on segments of different lengths. All possible segment lengths are considered either explicitly or implicitly. Two segments of different strip directions are selected optimally to compose the final cutting pattern.FindingsA strip can be taken as consisting of rectangular pieces, where the length of the first piece may be longer than that of the others. Normal lengths and widths can be defined according to the properties of punched strips. Considering only normal segment lengths and using lower bound in the recursion function can reduce the computation time drastically.Practical implicationsBased on the algorithm, practitioners may develop applications to solve real world two‐dimensional cutting problem of punched strips.Originality/valueThe two‐segment cutting patterns for punched strips are proposed. They are simple to cut and may be welcomed in practice.

Two-stage general block patterns for the two-dimensional cutting problem

This paper presents an algorithm for the unconstrained two-dimensional cutting problem of rectangular pieces. It proposes the two-stage general block pattern that can be cut into general blocks in two stages, with another two or more stages being required to cut the blocks into pieces. Firstly vertical cuts divide the stock sheet into segments, and then horizontal cuts divide the segments into general blocks. A general block contains general strips, and each cut on the block produces just one strip. The directions of two strips cut successively from a block are either parallel or perpendicular to each other. The algorithm uses a dynamic programming recursion to determine the strip layout on each block, solves knapsack problems to obtain the block layout on each segment and the segment layout on the sheet. The computational results indicate that the algorithm is highly efficient in improving material usage, and the computation time is reasonable.

Generating optimal multiple-segment cutting patterns for rectangular blanks

This paper considers the unconstrained two-dimensional guillotine-cutting problem of rectangular blanks. It suggests the application of multiple-segment cutting patterns. A multiple-segment pattern consists of one or more segments that have the same width as that of the stock sheet. The lengths of the segments may be different. Strips in each segment are of the same length and direction. An algorithm based on dynamic programming is presented for generating the optimal multiple-segment patterns. Computations are performed both on benchmark problems and on random problems. The algorithm can solve most small-scale problems to optimality. For problems of larger scale, it can give solutions very close to optimal, and the time efficiency is much more higher than that of the exact algorithm for non-staged cutting. Furthermore, the algorithm can consider the restriction of the cutting process and generate cutting patterns that are easy to cut.

Dynamic Programming and Hill-Climbing Techniques for Constrained Two-Dimensional Cutting Stock Problems

In this paper we propose an algorithm for the constrained two-dimensional cutting stock problem (TDC) in which a single stock sheet has to be cut into a set of small pieces, while maximizing the value of the pieces cut. The TDC problem is NP-hard in the strong sense and finds many practical applications in the cutting and packing area. The proposed algorithm is a hybrid approach in which a depth-first search using hill-climbing strategies and dynamic programming techniques are combined. The algorithm starts with an initial (feasible) lower bound computed by solving a series of single bounded knapsack problems. In order to enhance the first-level lower bound, we introduce an incremental procedure which is used within a top-down branch-and-bound procedure. We also propose some hill-climbing strategies in order to produce a good trade-off between the computational time and the solution quality. Extensive computational testing on problem instances from the literature shows the effectiveness of the proposed approach. The obtained results are compared to the results published by Alvarez-Valdes et al. (2002).

A hybrid algorithm for the two‐dimensional layout problem: the cases of regular and irregular shapes

The two‐dimensional layout (TDL) optimization problem consists of finding the minimal length layout of a set of both regular and irregular two‐dimensional shapes on a stock sheet of finite width but infinite length. The layout should not contain any overlap. In this paper, we solve the TDL problem by proposing two approximate algorithms. The first one is a new heuristic based on a constructive approach specifically designed for irregular shapes, but that works well with regular ones. The second is a hybrid method in which the constructive approach displays the layouts corresponding to the chromosomes yielded by the genetic algorithm. The resulting algorithm yields satisfactory results in relatively short computational times as shown by the extensive computational testing on problems from the literature.

Approximate algorithms for constrained circular cutting problems

In this paper, we study the problem of cutting a rectangular plate R of dimensions ( L, W) into as many circular pieces as possible. The circular pieces are of n different types with radii r i , i=1,…, n. We solve the constrained circular problem, where d i the maximum demand for piece type i is specified, using two heuristics: a constructive procedure-based heuristic and a genetic algorithm-based heuristic. Both of these approaches search for a good ordering of the pieces and use an adaptation of the best local position procedure (Studia. Inform. Univ. 2 (1) (2002) 33) to find the “best” layout of this ordered set. This positioning procedure is specifically tailored to circular cutting problems. It acts, for constrained problems, as one of the mutation operators of the genetic algorithm. We compare the performance of both proposed approaches to that of existing approximate and exact algorithms on several problem instances taken from the literature. The computational results show that the proposed approaches produce high-quality solutions within reasonable computational times. The genetic algorithm-based heuristic is easily parallelizable; one of its important features to be investigated in the near future. Scope and purpose In many industrial sectors, minimizing waste is a critical issue. This is particularly the case for the packing, textile, naval, and aerospace industries, where minimizing the waste of packing and cutting is a frequent problem. Our paper studies the constrained circular packing problem where a set of circles needs to be cut on a rectangular stock sheet of fixed width and length. The objective is to maximize the usage of the rectangular sheet while respecting the upper demand value for each circle type. We propose two approximate algorithms to solve this problem and prove their efficiency.

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.

A tabu search algorithm for large-scale guillotine (un)constrained two-dimensional cutting problems

In this paper we develop several heuristic algorithms for the two-dimensional cutting problem (TDC) in which a single stock sheet has to be cut into a set of small pieces, while maximising the value of the pieces cut. They can be considered to be general purpose algorithms because they solve the four versions of the TDC: weighted and unweighted, constrained and unconstrained. We begin by proposing two constructive procedures based on simple bounds obtained by solving one-dimensional knapsack problems. We then use these constructive algorithms as building blocks for more complex procedures. We have developed a greedy randomised adaptive search procedure (GRASP) which is very fast and obtains good results for both constrained and unconstrained problems. We have also developed a more complex tabu search algorithm that obtains high quality results in moderate computing times. Finally, we have implemented a path relinking procedure to improve the final results of the above algorithms. For the computational results we have used the set of large-scale test problems collected and generated by Fayard et al. (J. Oper. Res. Soc. 49 (1998) 1270). Scope and purpose The two-dimensional cutting problem (TDC) consists of cutting a single rectangular stock sheet into a set of small rectangular pieces of given sizes and values to maximise the total value of the pieces cut. This problem has a wide range of commercial and industrial applications, whenever a sheet of wood, glass, paper or metal has to be cut. The TDC problem can also be considered as a subproblem of more general cutting problems, involving several available stock sheets of different sizes. In this paper we develop several heuristic algorithms for solving TDC problems. The computational results show that they produce high-quality results in short computing times.

The trim-loss problem in a wooden container manufacturing company

The principal raw material for making wooden containers is rectangular veneer sheets. These sheets are cut into smaller rectangular sheets according to the customer orders for various sizes of wooden containers. The resulting offcuts, which do not meet the minimum dimensional requirements, are considered as trim loss or waste. At the time of the study, the company making these containers experienced veneer waste of about 40%, the major contributing factor being trim loss. This problem has been modeled here as a two-dimensional trim-loss problem. Fourteen different heuristics for allocating stock sheets to order sheets have been considered and evaluated with respect to five criteria including trim loss. A heuristic that results in acceptable levels of trim loss but reduces the administration of stock dramatically has been suggested.

A decision-making approach for nesting scheduling: A textile case

In many industrial cases, the nesting problem and the scheduling problem have to be addressed at the same time. The complexity of the combined problem often prevents to take effectively into account both nesting efficiency and overall production objectives. This paper presents a scheduling approach for the combined problem of production scheduling and nesting. The aim of the proposed approach is to provide a good solution both for the nesting and scheduling problem. The approach involves the generation of scheduling alternatives, their transformation through a rule base mechanism into nesting solutions and finally their evaluation using different criteria that reflect the overall production objectives such as meeting due dates, minimizing of the cost and maximizing the machines and stock sheet utilisation. The proposed approach has been implemented in a software system for the purpose of solving a problem in the textile industry. Specifically, the scheduling of the carpet weaving processa problem of nesting rectangular patterns under complex production constraints-has been examined. A set of experiments has been conducted for producing realistic nesting schedules in order to evaluate the proposed system's performance. The results show that the proposed approach may be applied in real-life manufacturing processes under complex production constraints and multiple objectives.

Microstructure of Friction Stir Welded Joints in AA5182

Two 2.5 mm thick hot mill stock sheets of AA5182 aluminum alloy were pretreated differently before friction stir welding, one was cold rolled by 20% reduction in thickness whereas the other was 20% cold rolled then recovery annealed at 230 °C for 100 hours. After friction stir welding of these two sheets, a uniform dynamic recrystallized microstructure of fine grain size with no evidence of abnormal grain growth in the heat affected zone of the weld was obtained. X-ray pole figure study showed that the weld zone comprised of triclinic texture components which were recognizably of rotated cube texture and could be quantitatively analyzed. The predicted sheet formability of the weld zone is better than that of the start material.