Fast pattern-based algorithms for cutting stock

The conventional assignment-based first/best fit decreasing algorithms (FFD/BFD) are not polynomial in the one-dimensional cutting stock input size in its most common format. Therefore, even for small instances with large demands, it is difficult to compute FFD/BFD solutions. We present pattern-based methods that overcome the main problems of conventional heuristics in cutting stock problems by representing the solution in a much more compact format. Using our pattern-based heuristics, FFD/BFD solutions for extremely large cutting stock instances, with billions of items, can be found in a very short amount of time.     

Efficient indexing algorithms for one-dimensional discretely-scaled strings

The discretely-scaled string indexing problem asks one to preprocess a given text string T, so that for a queried pattern P, the matched positions in T that P appears with some discrete scale can be reported efficiently. For solving this problem, Wang et al. first show that with an O(|T|log|T|)-time preprocessing on T, all matched positions can be reported in O(|P|+Ud) time, where |T|, |P|, and Ud denote the length of T, the length of P, and the number of matched positions for discretely-scaled P in T, respectively. In this paper, for fixed alphabets we propose the first-known optimal indexing algorithm that takes O(|T|) and O(|P|+Ud) time in its preprocessing and query phases, respectively. For integer and unbounded alphabets, our new algorithm can also be extended to obtain the best-known results.     

一种有效的求解一维下料问题的启发式算法

使用了一种改进的顺序启发式算法,在排样方式的生成过程中不断修正当前排入毛坯的价值,使之趋于合理,依次选取求解背包函数获得的最大单位价值的排样方式组成当前排样方案,迭代调用该过程多次,最终选取最优的排样方案。在保证较高材料利用率的同时考虑减少排样方式,增加最后一根材料余料长度等多个优化目标。通过多组实验结果比较,证实了算法的有效性。     

Discrete gradient algorithms of high order for one-dimensional systems

We show how to increase the order of one-dimensional discrete gradient numerical integrator without losing its advantages, such as exceptional stability, exact conservation of the energy integral and exact preservation of the trajectories in the phase space. The accuracy of our integrators is higher by several orders of magnitude as compared with the standard discrete gradient scheme (modified midpoint rule) and, what is more, our schemes have very high accuracy even for large time steps.     

Analysis of distributed genetic algorithms for solving cutting problems

In this paper, a solution to the three‐stage two‐dimensional cutting problem is presented by using sequential and parallel genetic algorithms (GAs). More specifically, an analysis of including distributed population ideas and parallelism in the basic GA are carried out to solve the problem more accurately and efficiently than with ordinary sequential techniques. Publicly available test problems have been used to illustrate the computational performance of the resulting metaheuristics. Experimental evidence in this work will show that the proposed algorithms outperform their sequential counterparts in time (high speedup with multiprocessors) and numerically (lower number of visited points during the search to find the solutions).     

Two algorithms for a reachability problem in one-dimensional space

Two algorithms are proposed to solve a reachability problem among time-dependent obstacles in 1D space. In the first approach, the motion planning problem is reduced to a path existence problem in a directed graph. The algorithm is very simple, with running time O(n²), where n is the complexity of obstacles in space-time. The second algorithm uses a sweep-line technique and has running time O(n log₂ n). Besides, the latter algorithm can be easily modified to compute a collision-free trajectory, if such trajectories exist     

Near-optimal solutions to one-dimensional cutting stock problems

This paper describes a set of heuristic procedures for efficiently generating good solutions to one-dimensional cutting stock problems in which (i) there are multiple stock lengths with constraints on their availability, (ii) it is desirable to cut the trim into as few pieces as possible and (iii) it is difficult because of the problem's structure, to round fractional, LP solutions to good feasible cutting plans. The point of departure for the procedures is the column generation technique of Gilmore and Gomory. The computational experience reported here suggests that the heuristics are both effective and efficient.     

基于交货期的多目标一维优化下料模型研究

根据目前对优化下料问题新特性的研究,指出优化下料问题呈现多目标和复杂约束状态,建立了以原料消耗最少和不同交货期的惩罚最小为目标函数的优化下料问题的数学模型.利用多目标智能优化算法求解模型,并结合算例验证了模型的有效性和实用性.     

Heuristics for the one-dimensional cutting stock problem with limited multiple stock lengths

This paper deals with the classical one-dimensional integer cutting stock problem, which consists of cutting a set of available stock lengths in order to produce smaller ordered items. This process is carried out in order to optimize a given objective function (e.g., minimizing waste). Our study deals with a case in which there are several stock lengths available in limited quantities. Moreover, we have focused on problems of low demand. Some heuristic methods are proposed in order to obtain an integer solution and compared with others. The heuristic methods are empirically analyzed by solving a set of randomly generated instances and a set of instances from the literature. Concerning the latter, most of the optimal solutions of these instances are known, therefore it was possible to compare the solutions. The proposed methods presented very small objective function value gaps.     

An efficient implementation of one-dimensional discrete wavelet transform algorithms for GPU architectures

The Journal of Supercomputing - In this paper, the authors present several self-developed implementation variants of the Discrete Wavelet Transform (DWT) computation algorithms and compare their...     

RCC8的一致分割及其算法

区域连接演算（Region Connection Calculus,RCC）是一种用于空间定性表示和推理的形式化模型,如RCC5,RCC8等,其一致性检查被证明是一个NP问题。幸运的是,在其可处理子集上,路径一致性和一致性等价,即便这样也有[O(n3)]的时间复杂度和[O(n2)]的空间复杂度。为了提高一致性检查的效率,提出了一致分割的概念,给出了其定义和成立的充分必要条件,用来将RCC8的约束图在保持一致性的前提下分割成若干个子图,分而求解各个子图的一致性;并随后给出了几种一致分割的充分条件,和相应的高效分割算法。在随机生成的大型、稀疏约束图上的实验表明了一致分割的有效性。     

Heuristic and exact algorithms for generating homogenous constrained three-staged cutting patterns

An approach is proposed for generating homogenous three-staged cutting patterns for the constrained two-dimensional guillotine-cutting problems of rectangles. It is based on branch-and-bound procedure combined with dynamic programming techniques. The stock plate is divided into segments. Each segment consists of strips with the same direction. Only homogenous strips are allowed, each of which contains rectangles of the same size. The approach uses a tree-search procedure. It starts from an initial lower bound, implicitly generates all possible segments through the builds of strips, and constructs all possible patterns through the builds of segments. Tighter bounds are established to discard non-promising segments and patterns. Both heuristic and exact algorithms are proposed. The computational results indicate that the algorithms are capable of dealing with problems of larger scale. Finally, the solution to a cutting problem taken from a factory that makes passenger cars is given.     

蜂群遗传算法在一维下料问题中的应用

针对一维下料优化问题,根据企业的实际生产情况,考虑能够满足和不满足生产两种情况,建立一个新的优化模型,并使用蜂群遗传算法求解方案。用各零件长度的一个排列作为一个染色体,每个零件的长度作为染色体的一个基因,根据蜂群原理设置两个不同的种群,种群1用于全局搜索,种群2用于局部搜索。实验结果表明,该模型具有一定的实用价值。     

一维下料问题的AB分类法

为了解决大规模的一维下料问题的计算困难, 根据一维下料问题的特点,把贪心算法和随机搜索技术有机地结合起来,利用随机搜索技术对贪心算法进行了有效的改进,提出了一种简单实用的AB分类法。 实验表明,该算法对规模较大的问题也能较快地获得问题最优解或精度较高的近似最优解。     

Mathematical models and routing algorithms for CAD technological preparation of cutting processes

Resource-conscious technologies for cutting sheet material include the ICP and ECP technologies that allow for aligning fragments of the contours of cutouts. In this work, we show the mathematical model for the problem of cutting out parts with these technologies and algorithms for finding cutting tool routes that satisfy technological constraints. We give a solution for the problem of representing a cutting plan as a plane graph G = (V,F,E), which is a homeomorphic image of the cutting plan. This has let us formalize technological constraints on the trajectory of cutting the parts according to the cutting plan and propose a series of algorithms for constructing a route in the graph G = (V,F,E), which is an image of an admissible trajectory. Using known coordinates of the preimages of vertices of graph G = (V,F,E) and the locations of fragments of the cutting plan that are preimages of edges of graph G = (V,F,E), the resulting route in the graph G = (V,E) can be interpreted as the cutting tool’s trajectory.The proposed algorithms for finding routes in a connected graph G have polynomial computational complexity. To find the optimal route in an unconnected graph G, we need to solve, for every dividing face f of graph G, a travelling salesman problem on the set of faces incident to f.     

Evaluation of BIC-based algorithms for audio segmentation

The Bayesian Information Criterion (BIC) is a widely adopted method for audio segmentation, and has inspired a number of dominant algorithms for this application. At present, however, literature lacks in analytical and experimental studies on these algorithms. This paper tries to partially cover this gap.Typically, BIC is applied within a sliding variable-size analysis window where single changes in the nature of the audio are locally searched. Three different implementations of the algorithm are described and compared: (i) the first keeps updated a pair of sums, that of input vectors and that of square input vectors, in order to save computations in estimating covariance matrices on partially shared data; (ii) the second implementation, recently proposed in literature, is based on the encoding of the input signal with cumulative statistics for an efficient estimation of covariance matrices; (iii) the third implementation consists of a novel approach, and is characterized by the encoding of the input stream with the cumulative pair of sums of the first approach.Furthermore, a dynamic programming algorithm is presented that, within the BIC model, finds a globally optimal segmentation of the input audio stream.All algorithms are analyzed in detail from the viewpoint of the computational cost, experimentally evaluated on proper tasks, and compared.     

基于EPFF算法的下料问题模型

研究一维单一原料下料问题,将最优化模型和EPFF算法相结合,建立了混合型模型,即先采用EPFF算法得到下料方式阵,再将其代入线性规划模型中,加上了加工时间以及最大加工能力的限制;最后确定了满足要求的实用下料方案。     

Evaluation of projection algorithms

A number of linear and nonlinear mapping algorithms for the projection of patterns from a high-dimensional space to two dimensions are available. These two-dimensional representations allow quick visual observation of a data set. A combination of two popular mapping algorithms-Sammon's mean-square error technique and the triangulation method-is proposed to overcome the limitations in the individual algorithms. Some factors which describe the goodness of a projection are described, and a comparison is made of six of these algorithms by running them on four data sets. The results obtained support the use of the proposed algorithm.     

An improved tabu search approach with mixed objective function for one-dimensional cutting stock problems

In this paper, a methodology with both efficiency and effectiveness of using improved tabu search approach is applied successfully to solve one-dimensional cutting stock problem (1D-CSP). By introducing a varying mixed objective function, the chance of obtaining optimum solution in solving 1D-CSP, which is treated as a sequence problem, has increased. The total incentive trim loss, which functions as an incentive term in this paper, makes it possible to obtain good 1D-CSP results through the use of the new proposed TS approach. After testing against actual sample cases from shipyards and literatures, the solutions obtained from the proposed algorithm are superior. Computational results indicate that the methodology proposed outperforms the ones from other meta-heuristic methods.