Generating optimal cutting patterns for rectangular blanks of a single size

This paper deals with the problem of minimizing trim loss in cutting rectangular blanks of a single size from a rectangular sheet using orthogonal guillotine cuts. First we prove that we can obtain the unconstrained optimal layout by searching among normal multi-section layouts. Next we present an unconstrained algorithm to search for it. The unconstrained algorithm uses a branch-and-bound method with a tight upper bound. Later we discuss the algorithm for the constrained problem where the blank demand must be met exactly. Finally, the unconstrained algorithm is extended to cope with the blade length constraint. Experimental computations show that the algorithms are extremely efficient.     

A hybrid evolutionary algorithm for the two-dimensional packing problem

This paper presents a hybrid evolutionary algorithm for the two-dimensional non-guillotine packing problem. The problem consists of packing many rectangular pieces into a single rectangular sheet in order to maximize the total area of the pieces packed. Moreover, there is a constraint on the maximum number of times that a piece may be used in a packing pattern. The set of packing patterns is processed by an evolutionary algorithm. Three mutation operators and two types of quality functions are used in the algorithm. The best solution obtained by the evolutionary algorithm is used as the initial solution in a tree search improvement procedure. This approach is tested on a set of benchmark problems taken from the literature and compared with the results published by other authors.     

Descomposición en pistones circulares para la obtención de la radiación acústica de un pistón rectangular

Obtaining pressure radiated by flat surfaces is not a new problem. This problem has studied by the complexity of the topic and its application to design flat speakers. These kind of speakers are the speakers that we use in televisions, ceilings, cinema screens, panels, etc. in this cases usually we have rectangular speakers.The single source model is the simplest model to convert the surface vibration to radiated pressure in a point. This is an easy model but it is very slow, especially when we want calculate at high frequencies. For rectangular surfaces there are models that use relatively complex auxiliary functions. In this case the calculation is accelerated but its implementation is more complicated and is necessary to particularize each situation.This paper presents the decomposition of a rectangular surface in several circular surfaces, by means of area associations, seeking a rapid method based on circular pistons whose behavior is known with an acceptable error in the allocation.     

Heuristic for the rectangular two-dimensional single stock size cutting stock problem with two-staged patterns

Two-staged patterns are often used in manufacturing industries to divide stock plates into rectangular items. A heuristic algorithm is presented to solve the rectangular two-dimensional single stock size cutting stock problem with two-staged patterns. It uses the column-generation method to solve the residual problems repeatedly, until the demands of all items are satisfied. Each pattern is generated using a procedure for the constrained single large object placement problem to guarantee the convergence of the algorithm. The computational results of benchmark and practical instances indicate the following: (1) the algorithm can solve most instances to optimality, with the gap to optimality being at most one plate for those solutions whose optimality is not proven and (2) for the instances tested, the algorithm is more efficient (on average) in reducing the number of plates used than a published algorithm and a commercial stock cutting software package.     

复数平方根z~(1/2)的有效算法

本文为复数平方根之主值z~(1/2)提供一适用于计算机的有效算法.该算法除已克服有效数字消失、从而能确保计算结果的精度之外,还有处理简单统一、适用范围广的优点.在附录中给出了极坐标与直角坐标互换的实用算法.     

A distributed global optimization method for packing problems

Packing optimization problems aim to seek the best way of placing a given set of rectangular cartons within a minimum volume rectangular container. Currently, packing optimization methods either have difficulty in finding a globally optimal solution or are computationally inefficient, because models involve too many 0–1 variables and because use of just a single computer. This study proposes a distributed computation method for solving a packing problem by a set of personal computers via the Internet. First, the traditional packing optimization model is converted into an equivalent model containing many fewer 0–1 variables. Then the model is decomposed into several sub-problems by dividing the objective value into many intervals. Each of these sub-problems is a linearized logarithmic program expressed as a linear mixed 0–1 problem. The whole problem is solvable and reaches a globally optimal solution. The numerical examples demonstrate that the proposed method can obtain the global optimum of a packing problem effectively.     

四边简支正交各向异性波纹型夹心矩形夹层板的固有频率

给出了一种把具有波纹型夹心的正交各向异性夹层板的控制方程组化为仅包含一个位移函数的单一方程的简单方法,从而获得了四边简支条件下其自由振动固有频率的精确解,同时还对两种具有重要实际意义的特殊情况进行了讨论。     

Anordnung in Rechtseitebenen und Integritätssystemen

H. Karzel [6] has shown that rectangular planes can be algebraically represented by metric integral systems. Here, rectangular planes, which are simultaneously ordered planes, are considered. Necessary and sufficient conditions on the integral system are given for the corresponding rectangular plane to be ordered.Moreover, it is shown that the maximal multiplicative system of an ordered rectangular plane is a valuation ring.     

State Reconstruction in Higher Dimensional PDEs with Spatially Varying Parameters

The state estimation problem is considered for a diffusion-reaction system with spatially varying parameters defined on a 3-dimensional rectangular domain with the measured output being restricted to a single surface. For this, a backstepping-based observer design is applied, which enables to obtain the observer gains such that the observer error dynamics decays exponentially in the L₂-norm. At first, an idealized system output restricted to a single surface is assumed as an available measurement. Secondly, in view of a practical realization of the proposed observer, the idealized system output is reconstructed from a set of finite-dimensional measurements. The observer error convergence and the applicability of the proposed approach are evaluated by means of numerical simulations. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

中心裂纹夹紧矩形板的拉伸*

本文利用单裂纹基本解及无限板条的Fourier变换解,将含有中心裂纹的夹紧矩形板的拉伸问题,化归为解一组奇异积分方程,进而使用Gauss-Jacobi求积公式,计算了中心裂纹的应力强度因子及夹紧边的法向应力,在应力强度因子表中还作了数值结果比较.     

关于环上长方矩阵的加权群可逆性

研究任意环上长方矩阵的加权群逆和加权（1,5）-逆。利用矩阵分解,得到了长方矩阵积的加权群逆存在的一些等价条件和计算方法及任意环上长方矩阵的加权（1,5）-逆的刻画表达式。得到的定理推广了有关方阵群逆和（1,5）-逆的相关结果。结果还可适合应用于加法范畴中的态射。     

Dipole Interactions in Doubly Periodic Domains

We consider the interactions of finite dipoles in a doubly periodic domain. A finite dipole is a pair of equal and opposite strength point vortices separated by a finite distance. The dynamics of multiple finite dipoles in an unbounded inviscid fluid was first proposed by Tchieu, Kanso, and Newton in Tchieu et al. (Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 468(2146):3006–3026, 2012) as a model that captures the “far-field” hydrodynamic interactions in fish schools. In this paper, we formulate the equations of motion governing the dynamics of finite dipoles in a doubly periodic domain. We show that a single dipole in a doubly periodic domain exhibits periodic and aperiodic behavior, in contrast to a single dipole in an unbounded domain. In the case of two dipoles in a doubly periodic domain, we identify a number of interesting trajectories including collision, collision avoidance, and passive synchronization of the dipoles. We then examine two types of dipole lattices: rectangular and diamond. We verify that these lattices are in a state of relative equilibrium and show that the rectangular lattice is unstable, while the diamond lattice is linearly stable for a range of perturbations. We conclude by commenting on the insight these models provide in the context of fish schooling.     

Dynamic stability of a porous rectangular plate

The study is devoted to a axial compressed porous-cellular rectangular plate. Mechanical properties of the plate vary across is its thickness which is defined by the non-linear function with dimensionless variable and coefficient of porosity. The material model used in the current paper is as described by Magnucki, Stasiewicz papers. The middle plane of the plate is the symmetry plane. First of all, a displacement field of any cross section of the plane was defined. The geometric and physical (according to Hook's law) relationships are linear. Afterwards, the components of strain and stress states in the plate were found. The Hamilton's principle to the problem of dynamic stability is used. This principle was allowed to formulate a system of five differential equations of dynamic stability of the plate satisfying boundary conditions. This basic system of differential equations was approximately solved with the use of Galerkin's method. The forms of unknown functions were assumed and the system of equations was reduced to a single ordinary differential equation of motion. The critical load determined used numerically processed was solved. Results of solution shown in the Figures for a family of isotropic porous-cellular plates. The effect of porosity on the critical loads is presented. In the particular case of a rectangular plate made of an isotropic homogeneous material, the elasticity coefficients do not depend on the coordinate (thickness direction), giving a classical plate. The results obtained for porous plates are compared to a homogeneous isotropic rectangular plate. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A goal-driven prototype column generation strategy for the multiple container loading cost minimization problem

In the multiple container loading cost minimization problem (MCLCMP), rectangular boxes of various dimensions are loaded into rectangular containers of various sizes so as to minimize the total shipping cost. The MCLCMP can be naturally modeled as a set cover problem. We generalize the set cover formulation by introducing a new parameter to model the gross volume utilization of containers in a solution. The state-of-the-art algorithm tackles the MCLCMP using the prototype column generation (PCG) technique. PCG is an effective technique for speeding up the column generation technique for extremely hard optimization problems where their corresponding pricing subproblems are NP-hard. We propose a new approach to the MCLCMP that combines the PCG technique with a goal-driven search. Our goal-driven prototype column generation (GD-PCG) algorithm improves the original PCG approach in three respects. Computational experiments suggest that all three enhancements are effective. Our GD-PCG algorithm produces significantly better solutions for the 350 existing benchmark instances than all other approaches in the literature using less computation time. We also generate two new set instances based on industrial data and the classical single container loading instances.     

ON A FAMILY OF MULTIVARIATE LEAST-SQUARES ORTHOGONAL POLYNOMIALS

In this paper the new notion of multivariate least-squares orthogonal polynomials from the rectangular form is introduced. Their existence and uniqueness is studied and some methods for their recursive computation are given. As an application, a new family of multivariate Pade-type approximants from the rectangular form is constructed.     

A hybrid genetic algorithm for the two-dimensional single large object placement problem

In the two-dimensional single large object placement problem, we are given a rectangular master surface which has to be cut into a set of smaller rectangular items, with the aim of maximizing the total value of the pieces cut. We consider the special case in which the items cannot be rotated and must be cut with their edges always parallel to the edges of the surface. We present new greedy algorithms and a hybrid genetic approach with elitist theory, immigration rate, heuristics on-line and tailored crossover operators. Extensive computational results for a large number of small and large benchmark test problems are presented. The results show that our approach outperforms existing heuristic algorithms.     

弹性地基上矩形板弯曲的CC型级数解

本文利用双变量函数的Stockes变换,用CC型级数求弹性地基上矩形板弯曲问题的解析解.以弹性地基上四边自由矩形板中点作用一集中力为例给出数字计算结果.     

An integer programming model for two- and three-stage two-dimensional cutting stock problems

In this paper, an integer programming model for two-dimensional cutting stock problems is proposed. In the problems addressed, it is intended to cut a set of small rectangular items of given sizes from a set of larger rectangular plates in such a way that the total number of used plates is minimized.     

An exact algorithm for the pallet loading problem

The two-dimensional packing problem of finding optimal layouts for identical rectangular boxes on a rectangular pallet has interested OR practitioners for many years. The problem is NP-complete and solution methods to date tend to be heuristic. This paper discusses the development of an exact tree search algorithm based on a graph-theoretic model of the problem.     

矩形槽水中非传播孤波[英文]

本文主要研究在外部驱动下浅水槽内部的非传播孤波，用渐进方法中的多重尺度法较详细讨论和导出波动振幅所满足的非线性薛定谔方程及其非传播单孤波解。采用一些近似条件，又可以由非线性薛定谔方程得到两个独立的线性拉普拉斯方程。