Constrained two-dimensional cutting stock problems a best-first branch-and-bound algorithm

In this paper, we develop a new version of the algorithm proposed in Hifi (Computers and Operations Research 24/8 (1997) 727–736) for solving exactly some variants of (un)weighted constrained two-dimensional cutting stock problems. Performance of branch-and-bound procedure depends highly on particular implementation of that algorithm. Programs of this kind are often accelerated drastically by employing sophisticated techniques. In the new version of the algorithm, we start by enhancing the initial lower bound to limit initially the space search. This initial lower bound has already been used in Fayard et al. 1998 (Journal of the Operational Research Society, 49, 1270–1277), as a heuristic for solving the constrained and unconstrained cutting stock problems. Also, we try to improve the upper bound at each internal node of the developed tree, by applying some simple and effcient combinations. Finally, we introduce some new symmetric-strategies used for neglecting some unnecessary duplicate patterns . The performance of our algorithm is evaluated on some problem instances of the literature and other hard-randomly generated problem instances.     

A New Algorithm for Two-Dimensional Line Clipping via Geometric Transformation

Line segment clipping is a basic operation of the visualization process in computer graphics.So far there exist four computational models for clipping a line segment against a window,(1)the encoding,(2)the parametric,(3)the geometric transforma tion,and (4)the parallel cutting.This paper presents an algorithm that is based on the third method.By making use of symmetric properties of a window and transformation operations,both endpoints of a line segment are transformed,so that the basic cases are reduced into two that can be easily handled,thus the problems in NLN and AS where there are too many sub-procedure calls and basic cases that are difficult to deal with are tackled.Both analytical and experimental results from random input data show that the algorithm is better than other developed ones,in view of the speed and the number of operations.     

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.     

凸多面体线消隐算法的研究与改进

为使三维形体有较强的立体感,物体因自身遮挡和物体间的相互遮挡产生的线段就必须被消除.在研究了三维几何形体消隐算法中的线消隐算法之后,针对传统的凸多面体线消隐算法存在计算量大、消隐时间长、效率低的缺点进行改进,在原来线消隐算法的基础上加入包围盒的最大最小测试方法和深度优先排序方法.算法使用C++编程实现,实验证明算法的时间复杂度由原来的N2降低为N,大大提高了消隐效率.     

A hidden-line algorithm for graphical reconstruction of serially sectioned objects

An algorithm for hidden-line removal in graphical reconstructions of serially sectioned biological objects is developed for PC-like computer configurations. The aim is to handle complicated structures defined by vast numbers of coordinate pairs and intersections between contours, without requiring excessive computing time. The procedure depends upon a fast searching routine for the localization of intersections and visibility.     

A linear-time algorithm for solving the strong hidden-line problem in a simple polygon

A linear-time algorithm is presented for solving the strong hidden-line problem in a simple polygon P, or alternately, determining the region in P weakly visible from a specified edge of P. The algorithm combines results from visibility and shortest paths with the linear-time polygon triangulation algorithm discovered recently by Tarjan and Van Wyk. Previous published algorithms for the strong hidden-line problem require O(n logn) steps even after triangulation, where n is the cardinality of P.     

A parallel algorithm for constrained two-staged two-dimensional cutting problems

In this paper, we solve the two-staged two-dimensional cutting problem using a parallel algorithm. The proposed approach combines two main features: beam search (BS) and strip generation solution procedures (SGSP). BS employs a truncated tree-search, where a selected subset of generated nodes are retuned for further search. SGSP, a constructive procedure, combines a (sub)set of strips for providing both partial lower and complementary upper bounds. The algorithm explores in parallel a subset of selected nodes following the master-slave paradigm. The master processor serves to guide the search-resolution and each slave processor develops its proper way, trying a global convergence. The aim of such an approach is to show how the parallelism is able to efficiently solve large-scale instances, by providing new solutions within a consistently reduced runtime. Extensive computational testing on instances, taken from the literature, shows the effectiveness of the proposed approach.     

用启发算法和神经网络法解决二维不规则零件排样问题

本文提出一种用启发算法和神经网络法相结合的算法解决二维不规则零件的排料问题。此算法具有优化效果好、自动化程度高、并且速度快等特点。     

Efficient parallel solutions to some geometric problems

This paper presents new algorithms for solving some geometric problems on a shared memory parallel computer, where concurrent reads are allowed but no two processors can simultaneously attempt to write in the same memory location. The algorithms are quite different from known sequential algorithms, and are based on the use of a new parallel divide-and-conquer technique. One of our results is an O(log n) time, O(n) processor algorithm for the convex hull problem. Another result is an O(log n log log n) time, O(n) processor algorithm for the problem of selecting a closest pair of points among n input points.     

Decomposition algorithm for geometric objects in 2D packing and cutting problems

A class of basic 2D objects is introduced whose Φ-functions are known, and the decomposition theorem is proved for arbitrary φ-objects whose boundaries are formed by circular arcs and line segments. A step-by-step decomposition algorithm is proposed for arbitrary two-dimensional φ-objects. The algorithm efficiently constructs Φ-functions of arbitrary φ-objects in mathematical and computer modeling of packing and cutting problems. Results of numerical experiments are presented.     

Exact solutions to two-dimensional homing problems

Two-dimensional controlled Wiener and Bessel diffusion processes are considered in circles. Both the infinitesimal means and variances of the controlled processes depend on the control variables. The processes are controlled until they hit the circles for the first time. The objective is to minimize the expected value of the time spent by the controlled processes inside the circles, while taking the control costs into account. Explicit and exact expressions are obtained for the optimal values of the control variables as well as for the value functions. The method of similarity solutions is used to solve the appropriate dynamic programming equation.     

一种抗几何失真的彩色图像数字水印算法*

提出了一种基于SIFT(尺度特征变换)特征点校正几何参数的小波域彩色图像水印算法.该算法利用RGB彩色图像中蓝色与绿色分量的小波系数之间的关系,在蓝色水平细节和垂直细节中嵌入相同的水印;水印检测前,利用红色分量中的SIFT特征点估计几何参数,校正失真水印图像,从而可以无失真地提取水印.实验结果表明,该算法对压缩、噪声、剪裁、旋转、缩放等各类几何攻击具有很好的鲁棒性.     

An exact dynamic programming algorithm for large-scale unconstrained two-dimensional guillotine cutting problems

In the unconstrained two-dimensional cutting problems (U2DCP) small rectangular objects have to be extracted from a large rectangular sheet, with no limits on the number of small objects.The exact U2DCP solving approaches present in literature show some limits in tackling very large size instances, due to the high memory requirements.In this work we propose five improvements, three original and two derived from the literature, in order to overcome these limits and to reduce the computational burden of the knapsack function based U2DCP solving approaches. These improvements, based on proofed theoretical results, allow to reduce the search space and to avoid redundant solutions without loss of the feasible ones.The presented improvements, together with several computational refinements, are integrated in a new dynamic programming algorithm, which modifies the one by Russo et al. (2013 [16]). The proposed algorithm has been experienced on test instances present in literature and compared with the best U2DCP solving approaches. The obtained results show that it significantly outperforms them and it determines the optimal solution of unsolved very large size instances.     

Hybrid genetic algorithm and simulated annealing for two-dimensional non-guillotine rectangular packing problems

In this paper, genetic algorithm (GA) and simulated annealing (SA) with improved bottom left (BL) algorithm were applied to two-dimensional non-guillotine rectangular packing problems. The performance and efficiency of these algorithms on several test problems [Hopper, E., Turton, B.C.H., 2000. An empirical investigation of meta-heuristic and heuristic algorithms for two-dimensional packing problem. European Journal of Operational Research 128 (1), 34–57] were compared. These test problems consist of 17 and 29 individual rectangular pieces to place in main object limited with a size of 200×200 units. Both solution approaches were compared, based on the trim losses of the test problems. Also, the influences of the GA parameters (population sizes, mutation rates, crossover techniques) and of the SA parameters (cooling schedules, neighborhood moves, the number of inner loop, different temperature values) on the solution of these problems were examined. For considering all solutions of the test problems, the hybrid GA produces much better results than the hybrid SA.     

On the parallel-decomposability of geometric problems

There is a large and growing body of literature concerning the solutions of geometric problems on mesh-connected arrays of processors. Most of these algorithms are optimal (i.e., run in timeO(n ^1/d ) on ad-dimensionaln-processor array), and they all assume that the parallel machine is trying to solve a problem of sizen on ann-processor array. Here we investigate the situation where we have a mesh of sizep and we are interested in using it to solve a problem of sizen >p. The goal we seek is to achieve, when solving a problem of sizen >p, the same speed up as when solving a problem of sizep. We show that for many geometric problems, the same speedup can be achieved when solving a problem of sizen >p as when solving a problem of sizep.     

On the parallel-decomposability of geometric problems

There is a large and growing body of literature concerning the solutions of geometric problems on mesh-connected arrays of processors. Most of these algorithms are optimal (i.e., run in timeO(n ^1/d ) on ad-dimensionaln-processor array), and they all assume that the parallel machine is trying to solve a problem of sizen on ann-processor array. Here we investigate the situation where we have a mesh of sizep and we are interested in using it to solve a problem of sizen >p. The goal we seek is to achieve, when solving a problem of sizen >p, the same speed up as when solving a problem of sizep. We show that for many geometric problems, the same speedup can be achieved when solving a problem of sizen >p as when solving a problem of sizep.The research of M. J. Atallah was supported by the Office of Naval Research under Contracts N00014-84-K-0502 and N00014-86-K-0689, the Air Force Office of Scientific Research under Grant AFOSR-90-0107, the National Science Foundation under Grant DCR-8451393, and the National Library of Medicine under Grant R01-LM05118. Jyh-Jong Tsay's research was partially supported by the Office of Naval Research under Contract N00014-84-K-0502, the Air Force Office of Scientific Research under Grant AFOSR-90-0107, and the National Science Foundation under Grant DCR-8451393.     

Non-negative matrix factorization: Ill-posedness and a geometric algorithm

Non-negative matrix factorization (NMF) has been proposed as a mathematical tool for identifying the components of a dataset. However, popular NMF algorithms tend to operate slowly and do not always identify the components which are most representative of the data. In this paper, an alternative algorithm for performing NMF is developed using the geometry of the problem. The computational costs of the algorithm are explored, and it is shown to successfully identify the components of a simulated dataset. The development of the geometric algorithm framework illustrates the ill-posedness of the NMF problem and suggests that NMF is not sufficiently constrained to be applied successfully outside of a particular class of problems.