An exact algorithm for generating homogenous T-shape cutting patterns

Both the material usage and the complexity of the cutting process should be considered in generating cutting patterns. This paper presents an exact algorithm for constrained two-dimensional guillotine-cutting problems of rectangles. It uses homogenous T-shape patterns to simplify the cutting process. Only homogenous strips are allowed, each of which contains rectangular blanks of the same size and direction. The sheet is divided into two segments. Each segment consists of strips of the same length and direction. The strip directions of the two segments are perpendicular to each other. The algorithm is based on branch and bound procedure combined with dynamic programming techniques. It is a bottom-up tree-search approach that searches the solution tree from the branches to the root. Tighter bounds are established to shorten the searching space. The computational results indicate that the algorithm is efficient both in computation time and in material usage.     

生成矩形毛坯最优两段排样方式的确定型算法

排样价值、切割工艺和计算时间是排样问题主要考虑的3个因素.文中提出一个新的基于排样模式的确定型排样算法——同质块两段排样算法,此算法适合剪冲下料工艺,在实现工艺简化的同时提高了排样价值时间比.首先通过动态规划算法生成最优同质块,然后求解一维背包问题生成块在级中的最优排样方式和级在段中的最优排样方式,最后选择两个段生成最优的两段排样方式.通过3组经典测题对该文算法进行了测试,将算法与4种著名算法进行了比较.实验结果表明,该文算法的优化结果好于以上4种著名算法,有效地提高了板材利用率,并且计算时间合理.     

二维剪切排样的束搜索启发式算法

针对约束二维矩形剪切排样问题,提出了一种基于束搜索的三阶段剪切排样算法。其切割过程包括三个阶段：板材剪切成段,段剪切成条带,条带切割成准确尺寸毛坯。采用动态规划确定段的价值,复杂度低的拼接递推不同长度子板的初始价值和板材的初始可行解,束搜索优化板材的排样方式。束搜索的节点用矩形对表示,分别是段组合而成的局部方式和未填充的剩余子板。以局部方式价值与剩余子板的初始价值之和作为节点的估计值。按估计值选择精英节点继续分支,其他节点直接删除不再回溯。实验结果表明该算法可缩短三阶段同质排样的计算时间,且所获得的余料大,利于余料的回收管理和再利用。     

同尺寸矩形毛坯排样的连分数分支定界算法

在确定同尺寸矩形毛坯最优排样方式的算法中,连分数算法的时间效率最高,但所生成排样方式的切割工艺复杂．提出连分数分支定界算法,该算法应用连分数法确定毛坯数最优值,采用贴切的上界估计方法;在搜索过程中只保留上界不小于最优值的分支,遇到下界等于最优值的分支时结束搜索．实验结果表明,该算法的时间效率和连分数算法接近,并可以有效地简化切割工艺,生成切割工艺最简单的排样方式．最后,通过实例分析说明该算法的节约材料潜力。     

冲裁条带最优多段排样方式的动态规划算法

讨论冲裁件无约束剪冲排样问题,用动态规划算法生成冲裁条带多段排样方式。采用一组相互平行的分割线将板材分成多个段,每段含一组方向和长度都相同的条带。通过动态规划算法确定所有可能尺寸段的最优价值以及板材中段的最优组合,使整张板材价值达到最大。实验结果表明该算法能够提高材料利用率,计算时间能满足实际应用的需要。     

面向可加工性的矩形件优化下料算法

针对目前矩形件优化下料算法侧重追求高材料利用率,而对实际切割成本考虑不足的现状,提出一种既维持高材料利用率,又使下料方案具有较低切割成本的矩形件优化下料算法。算法采用SVC框架和同质条带多级规范方式求解矩形件下料问题。利用条带共边排样的路径优化设计进行切割路径长度的计算,以生产成本（材料成本与切割成本之和）为优化目标得到高材料利用率、低切割成本的下料方案,最后通过实验证实该算法的可行性与有效性。     

Strips minimization in two-dimensional cutting stock of circular items

Circular items are often produced from stock plates using the cutting and stamping process that consists of two stages. A guillotine machine divides the plate into strips at the cutting stage, and then a press punches out the items from the strips at the stamping stage. The cutting cost at the first stage often increases with the number of strips in the cutting plan. An approach is presented for the two-dimensional cutting stock problem of the strips at the cutting stage. The objective is to minimize the sum of the material and the cutting costs. The approach formulates the problem as an integer linear programming, and uses a column generation method for generating the cutting patterns. The cutting patterns have the feature that each cut on the plate produces just one strip. The computational results indicate that the approach can greatly reduce the number of strips in the cutting plan.     

A hybrid heuristic to reduce the number of different patterns in cutting stock problems

We propose a hybrid procedure to obtain a reduced number of different patterns in cutting stock problems. Initially, we generate patterns with limited waste that fulfill the demands of at least two items when the patterns are repeatedly cut as much as possible but without overproducing any of the items. The problem is reduced and the residual problem is solved. Then, pattern reduction techniques (local search) are applied starting with the generated solution. The scheme is straightforward and can be used in cutting stock problems of any dimension. Variations of the procedure are also indicated. Computational tests performed indicated that the proposed scheme provides alternative solutions to the pattern reduction problem which are not dominated by other solutions obtained using procedures previously suggested in the literature.     

基于两阶段的分段单一矩形优化排样

为有效解决分段单一矩形优化排样问题,给出一个求解分段单一矩形优化排样问题的两阶段方法。第一阶段完成标准子段最佳排样方式求解,并将二维排样问题转化为一维下料问题,第二阶段使用适合于一维下料问题求解的算法完成板材最佳排样方式求解。使用该方法开发了一个单一矩形优化排样系统,该系统既可以解决分段单一矩形排样问题也可以解决其他类型的单一矩形优化排样问题。企业应用实例表明该方法是求解分段单一矩形优化排样问题的一个较为有效的方法。     

Heuristic for constrained T-shape cutting patterns of rectangular pieces

T-shape patterns are often used in dividing stock plates into rectangular pieces, because they make good balance between plate cost and cutting complexity. A dividing cut separates the plate into two segments, each of which contains parallel strips, and the strip orientations of the two segments are perpendicular to each other. This paper presents a heuristic algorithm for constrained T-shape patterns, where the optimization objective is to maximize the pattern value, and the frequency of each piece type does not exceed the demand. The algorithm considers many dividing-cut positions, determines the pattern value associated to each position using a layout-generation procedure, and selects the one with the maximum pattern value as the solution. Pseudo upper bounds are used to skip some non-promising positions. The computational results show that the algorithm is fast and able to get solutions better than those of the optimal two-staged patterns in terms of material utilization.     

生成最优单毛坯条带T型布局方式的精确算法

为解决大规模矩形件布局问题,提出一个生成单毛坯条带T型布局方式的精确算法。该算法不仅可在合理时间内取得好的优化结果,而且在满足实际下料工艺的同时化简了切割工艺。该算法首先确定最优单毛坯条带,然后通过求解一维背包问题确定单毛坯条带在级中的布局方式和级在段中的最优布局方式,最后选择两个最优段生成布局方式。通过文献中的63道基准测题,将该算法与5种著名算法(经典两阶段、普通T型、同质块两阶段、普通布局算法和启发式算法TABU500)进行了比较。实验结果表明,该算法在计算时间和材料利用率两方面都有效。     

应用精确两阶段排样图的板材下料算法

求解基于精确两阶段排样图的二维下料问题,用最小的板材成本,生产出所需要的全部毛坯。将顺序启发式算法和排样图生成算法相结合,顺序生成排样方案中的各个排样图;采用顺序价值修正策略,在生成每个排样图后修正其中所含各种毛坯的价值。经过多次迭代生成多个排样方案,从中选择最好者。实验计算时与商业软件和文献算法相比较,结果表明所述算法可以更为有效地减少板材消耗。     

Genotype–phenotype heuristic approaches for a cutting stock problem with circular patterns

The cutting stock problem has been studied in the context of different industrial applications inducing NP-hard problems in most instances. However, the application in sawmill has not received the same attention. In this paper, we deal with the problem of determining the number of logs to cut over a period of several days and the geometry of sawmill patterns in order to satisfy the demand while minimizing the loss of material. First, the problem is formulated as an integer programming problem of the form of a constrained set covering problem where the knowledge of a priori cutting patterns is necessary to generate its columns. In our implementation, these patterns are obtained using a genetic algorithm (GA) or a simulated annealing method (SA). Then, two different approaches are introduced to solve the problem. The first approach includes two methods that combine a metaheuristic to generate the number of logs and a constructive heuristic to generate the cutting patterns for each of the logs. In the second approach, we use an exact procedure CPLEX to solve the integer programming model where the cutting patterns are generated with the GA method (GA+CPLEX) or the SA method (SA+CPLEX). These four methods are compared numerically on 11 semi-randomly generated problems similar to those found in real life. The best results for the loss are obtained with the two-stage GA+CPLEX approach that finds the best values for 7 problems.     

改进的矩形毛坯三块排样方式及其算法

致力于改进矩形毛坯三块排样方式的生成算法,采用三种策略缩小解的搜索范围,并将该算法与线性规划相结合形成排样方案生成算法,用于求解大规模矩形毛坯排样问题.通过实验证明,与二阶段、T形、两段、三阶段排样算法相比,排样方案生成算法生成的排样方案虽然板材利用率稍低,但排样方案简单,能够简化切割工艺.     

Generating optimal two-section cutting patterns for rectangular blanks

This paper presents an algorithm for generating unconstrained guillotine-cutting patterns for rectangular blanks. A pattern includes at most two sections, each of which consists of strips of the same length and direction. The sizes and strip directions of the sections must be determined optimally to maximize the value of the blanks cut. The algorithm uses an implicit enumeration method to consider all possible section sizes, from which the optimal sizes are selected. It may solve all the benchmark problems listed in the OR-Library to optimality. The computational results indicate that the algorithm is efficient both in computation time and in material utilization. Finally, solutions to some problems are given.     

A genetic algorithm approach for the cutting stock problem

In this paper, a genetic algorithm approach is developed for solving the rectangular cutting stock problem. The performance measure is the minimization of the waste. Simulation results obtained from the genetic algorithm-based approach are compared with one heuristic based on partial enumeration of all feasible patterns, and another heuristic based on a genetic neuro-nesting approach. Some test problems taken from the literature were used for the experimentation. Finally, the genetic algorithm approach was applied to test problems generated randomly. The simulation results of the proposed approach in terms of solution quality are encouraging when compared to the partial enumeration-based heuristic and the genetic neuro-nesting approach.     

双排多段排样方式及其生成算法

为解决大规模矩形毛坯无约束的二维剪切排样问题,提出双排多段排样方式及其 生成算法。排样时采用一条剪切线将板材切分为两段,用一组剪切线将每段切分成一系列的块, 每个块由一组水平方向的同质条带构成。采用枚举法确定两段分界线的最优位置,通过求解背 包模型确定所有可能尺寸的块的最大价值和块在段中的最优布局。利用文献中的2 组基准测题 对所述算法进行测试,实验结果表明,该算法能在合理的计算时间内取得较好的优化结果。     

Optimization of cutting process by GA approach

The paper proposes a new optimization technique based on genetic algorithms (GA) for the determination of the cutting parameters in machining operations. In metal cutting processes, cutting conditions have an influence on reducing the production cost and time and deciding the quality of a final product. This paper presents a new methodology for continual improvement of cutting conditions with GA. It performs the following: the modification of recommended cutting conditions obtained from a machining data, learning of obtained cutting conditions using neural networks and the substitution of better cutting conditions for those learned previously by a proposed GA. Experimental results show that the proposed genetic algorithm-based procedure for solving the optimization problem is both effective and efficient, and can be integrated into an intelligent manufacturing system for solving complex machining optimization problems.     

A Bayesian method for fitting parametric and nonparametric modelsto noisy data

We present a simple paradigm for fitting models, parametric and nonparametric, to noisy data, which resolves some of the problems associated with classical MSE algorithms. This is done by considering each point on the model as a possible source for each data point. The paradigm can be used to solve problems which are ill-posed in the classical MSE approach, such as fitting a segment (as opposed to a line). It is shown to be nonbiased and to achieve excellent results for general curves, even in the presence of strong discontinuities. Results are shown for a number of fitting problems, including lines, circles, elliptic arcs, segments, rectangles, and general curves, contaminated by Gaussian and uniform noise     

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

P. 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 β ₁ 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.