Construction heuristics for two-dimensional irregular shape bin packing with guillotine constraints

The paper examines a new problem in the irregular packing literature that has many applications in industry: two-dimensional irregular (convex) bin packing with guillotine constraints. Due to the cutting process of certain materials, cuts are restricted to extend from one edge of the stock-sheet to another, called guillotine cutting. This constraint is common place in glass cutting and is an important constraint in two-dimensional cutting and packing problems. In the literature, various exact and approximate algorithms exist for finding the two dimensional cutting patterns that satisfy the guillotine cutting constraint. However, to the best of our knowledge, all of the algorithms are designed for solving rectangular cutting where cuts are orthogonal with the edges of the stock-sheet. In order to satisfy the guillotine cutting constraint using these approaches, when the pieces are non-rectangular, practitioners implement a two stage approach. First, pieces are enclosed within rectangle shapes and then the rectangles are packed. Clearly, imposing this condition is likely to lead to additional waste. This paper aims to generate guillotine-cutting layouts of irregular shapes using a number of strategies. The investigation compares three two-stage approaches: one approximates pieces by rectangles, the other two approximate pairs of pieces by rectangles using a cluster heuristic or phi-functions for optimal clustering. All three approaches use a competitive algorithm for rectangle bin packing with guillotine constraints. Further, we design and implement a one-stage approach using an adaptive forest search algorithm. Experimental results show the one-stage strategy produces good solutions in less time over the two-stage approach.     

Column generation and sequential heuristic procedure for solving an irregular shape cutting stock problem

The research addressing two-dimensional (2D) irregular shape packing has largely focused on the strip packing variant of the problem. However, it can be argued that this is a simplification. The materials from which pieces are required to be cut will ultimately have a fixed length either due to the physical dimensions of the material or through constraints on the cutting machinery. Hence, in order to cut all the pieces, multiple sheets may be required. From this scenario arises the 2D irregular shape cutting stock problem. In this paper, we will present implementations of cutting stock approaches adapted to handle irregular shapes, including two approaches based on column generation (CG) and a sequential heuristic procedure. In many applications, setup costs can be reduced if the same pattern layout is cut from multiple sheets; hence there is a trade-off between material waste and number of patterns. Therefore, we describe the formulation and implementation of an adaptation of the CG method to control the number of different patterns. CG is a common method for the cutting stock problem; however, when the pieces are irregular the sub-problem cannot be solved optimally. Hence we implement CG and solve the sub-problem using the beam search heuristic. Further, we introduce a version of CG for instances where the number of rows is less than the number of columns.     

A fully general,exact algorithm for nesting irregular shapes

This paper introduces a fully general, exact algorithm for nesting irregular shapes. Both the shapes and material resource can be arbitrary nonconvex polygons. Moreover, the shapes can have holes and the material can have defective areas. Finally, the shapes can be arranged using both translations and arbitrary rotations (as opposed to a finite set of rotation angles, such as 0 \(^\circ \) and 180 \(^\circ \) ). The insight that has made all this possible is a novel way to relax the constraint that the shapes not overlap. The key idea is to inscribe a few circles in each irregular shape and then relax the non-overlap constraints for the shapes by replacing them with non-overlap constraints for the inscribed circles. These relaxed problems have the form of quadratic programming problems (QPs) and can be solved to optimality to provide valid lower bounds. Valid upper bounds can be found via local search with strict non-overlap constraints. If the shapes overlap in the solution to the relaxed problem, new circles are inscribed in the shapes to prevent this overlapping configuration from recurring, and the QP is then resolved to obtain improved lower bounds. Convergence to any fixed tolerance is guaranteed in a finite number of iterations. A specialized branch-and-bound algorithm, together with some heuristics, are introduced to find the initial inscribed circles that approximate the shapes. The new approach, called “QP-Nest,” is applied to three problems as proof of concept. The most complicated of these is a problem due to Milenkovic that has four nonconvex polygons with 94, 72, 84, and 74 vertices, respectively. QP-Nest is able prove global optimality when nesting the first two or the first three of these shapes. When all four shapes are considered, QP-Nest finds the best known solution, but cannot prove optimality.     

How to solve a design centering problem

This work considers the problem of design centering. Geometrically, this can be thought of as inscribing one shape in another. Theoretical approaches and reformulations from the literature are reviewed; many of these are inspired by the literature on generalized semi-infinite programming, a generalization of design centering. However, the motivation for this work relates more to engineering applications of robust design. Consequently, the focus is on specific forms of design spaces (inscribed shapes) and the case when the constraints of the problem may be implicitly defined, such as by the solution of a system of differential equations. This causes issues for many existing approaches, and so this work proposes two restriction-based approaches for solving robust design problems that are applicable to engineering problems. Another feasible-point method from the literature is investigated as well. The details of the numerical implementations of all these methods are discussed. The discussion of these implementations in the particular setting of robust design in engineering problems is new.     

Archetypal shapes based on landmarks and extension to handle missing data

Archetype and archetypoid analysis are extended to shapes. The objective is to find representative shapes. Archetypal shapes are pure (extreme) shapes. We focus on the case where the shape of an object is represented by a configuration matrix of landmarks. As shape space is not a vectorial space, we work in the tangent space, the linearized space about the mean shape. Then, each observation is approximated by a convex combination of actual observations (archetypoids) or archetypes, which are a convex combination of observations in the data set. These tools can contribute to the understanding of shapes, as in the usual multivariate case, since they lie somewhere between clustering and matrix factorization methods. A new simplex visualization tool is also proposed to provide a picture of the archetypal analysis results. We also propose new algorithms for performing archetypal analysis with missing data and its extension to incomplete shapes. A well-known data set is used to illustrate the methodologies developed. The proposed methodology is applied to an apparel design problem in children.     

Three-Dimensional Bodies of Minimum Total Drag in Hypersonic Flow

The problem of constructing three-dimensional bodies of minimum total drag is studied within the framework of a local interaction model. Under certain assumptions, this model can be adopted to describe the distributions of both pressure and skin friction on the body during its high-speed motion through gases and dense media. Without any constraints on the possible drag law within the scope of the accepted model, the optimum shapes providing the minimum drag are found without any simplifying assumptions regarding their geometry. It is shown that, for a given base area and specified limitations on the body size, one can construct an infinite number of optimum forebody shapes. It is proved that the desired shapes are formed by combinations of surface parts whose normal makes a certain constant angle with the direction of motion. The optimum angle is determined by the velocity and medium characteristics in terms of the constants of the drag law. A method of optimum shape design is proposed; in particular, it allows one to construct optimum bodies like missiles with aft feather and optimum bodies with a circular base. All the bodies constructed have the same minimal total drag for the given base area. Even for asymmetrical bodies, the acting force has no component in a plane perpendicular to the direction of motion. Special attention is paid to the particular case of the minimum drag body design in hypersonic flow, when the pressure on the body is specified by the Newton formula. A comparative study of the results obtained for Newtonian flow shows that the proposed shapes are more effective in providing a drag reduction than bodies found to be optimum in earlier studies under special simplifying assumptions.     

Packing tripods: Narrowing the density gap

In 1984, Stein and his co-authors posed a problem concerning simple three-dimensional shapes, known as semicrosses, or tripods. By definition, a tripod of order n is formed by a corner and the three adjacent edges of an integer n×n×n cube. How densely can one fill the space with non-overlapping tripods of a given order? In particular, is it possible to fill a constant fraction of the space as tripod order tends to infinity? In this paper, we settle the second question in the negative: the fraction of the space that can be filled with tripods must be infinitely small as the order grows. We also make a step towards the solution of the first question, by improving the currently known asymptotic lower bound on tripod packing density, and by presenting some computational results on low-order packings.     

Mixed-integer programming models for nesting problems

Several industrial problems involve placing objects into a container without overlap, with the goal of minimizing a certain objective function. These problems arise in many industrial fields such as apparel manufacturing, sheet metal layout, shoe manufacturing, VLSI layout, furniture layout, etc., and are known by a variety of names: layout, packing, nesting, loading, placement, marker making, etc. When the 2-dimensional objects to be packed are non-rectangular the problem is known as the nesting problem. The nesting problem is strongly NP-hard. Furthermore, the geometrical aspects of this problem make it really hard to solve in practice. In this paper we describe a Mixed-Integer Programming (MIP) model for the nesting problem based on an earlier proposal of Daniels, Li and Milenkovic, and analyze it computationally. We also introduce a new MIP model for a subproblem arising in the construction of nesting solutions, called the multiple containment problem, and show its potentials in finding improved solutions.     

The Design of axisymmetric ducts for incompressible flow with vorticity.

In this paper a numerical algorithm is described for solving the boundary value problem associated with axisymmetric, inviscid, incompressible, rotational (and irrotational) flow in order to obtain duct wall shapes from prescribed wall velocity distributions. The governing equations are formulated in terms of the stream function and the function as independent variables where for irrotational flow can be recognized as the velocity potential function, for rotational flow ceases being the velocity potential function but does remain orthogonal to the stream lines. A numerical method based on finite differences on a uniform mesh is employed. The technique described is capable of tackling the so–called inverse problem where the velocity wall distributions are prescribed from which the duct wall shape is calculated, as well as the direct problem where the velocity distribution on the duct walls are calculated from prescribed duct wall shapes. The two different cases as outlined in this paper are in fact boundary value problems with Neumann and Dirichlet boundary conditions respectively. Even though both approaches are discussed, only numerical results for the case of the Dirichlet boundary conditions are given. A downstream condition is prescribed such that cylindrical flow, that is flow which is independent of the axial coordinate, exists. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimization for automated assembly of puzzles

The puzzle-assembly problem has many application areas such as restoration and reconstruction of archeological findings, repairing of broken objects, solving jigsaw type puzzles, molecular docking problem, etc. The puzzle pieces usually include not only geometrical shape information but also visual information such as texture, color, and continuity of lines. This paper presents a new approach to the puzzle-assembly problem that is based on using textural features and geometrical constraints. The texture of a band outside the border of pieces is predicted by inpainting and texture synthesis methods. Feature values are derived from these original and predicted images of pieces. An affinity measure of corresponding pieces is defined and alignment of the puzzle pieces is formulated as an optimization problem where the optimum assembly of the pieces is achieved by maximizing the total affinity measure. A Fast Fourier Transform based image registration technique is used to speed up the alignment of the pieces. Experimental results are presented on real and artificial data sets.     

Fuzzy bin packing problem

This paper deals with the fuzzy bin packing problem that is a packing problem of non-rigid rectangles into an open rectangular bin. This problem is different from the conventional bin packing problem, which considers only rigid rectangles. The goal of the fuzzy bin packing problem is to minimize both the height of a packing and the extra cost due to the reduction of each piece. The total cost of the problem is represented as the sum of the height cost and the extra cost due to reductions of the pieces, which is called reduction cost. Because the conventional bin packing problem itself is an NP-hard problem, the presented optimization method assumes that an initial packing for non-reduced pieces has already been found. A closed form solution is presented for fuzzy bin packing problems, in which fuzzy numbers are triangular and the reduction cost is given by a quadratic function.     

Control the Fibre Orientation Distribution at the Outlet of Contraction

A two dimensional model of the orientation distribution of fibres in a paper machine headbox is studied. The goal is to control the fibre orientation distribution at the outlet of contraction by changing its shape. The mathematical formulation leads to an optimization problem with control in coefficients of a linear convection-diffusion equation as the state problem. Then, the problem is expressed as an optimal control problem governed by variational forms. By using an embedding method, the class of admissible shapes is replaced by a class of positive Radon measures. The optimization problem in measure space is then approximated by a linear programming problem. The optimal measure representing optimal shape is approximated by the solution of this linear programming problem. In this paper, we have shown that the embedding method (embedding the admissible set into a subset of measures), successfully can be applied to shape variation design to a one dimensional headbox. The usefulness of this idea is that the method is not iterative and it does not need any initial guess of the solution.      

Complete and robust no-fit polygon generation for the irregular stock cutting problem

The no-fit polygon is a construct that can be used between pairs of shapes for fast and efficient handling of geometry within irregular two-dimensional stock cutting problems. Previously, the no-fit polygon (NFP) has not been widely applied because of the perception that it is difficult to implement and because of the lack of generic approaches that can cope with all problem cases without specific case-by-case handling. This paper introduces a robust orbital method for the creation of no-fit polygons which does not suffer from the typical problem cases found in the other approaches from the literature. Furthermore, the algorithm only involves two simple geometric stages so it is easily understood and implemented. We demonstrate how the approach handles known degenerate cases such as holes, interlocking concavities and jigsaw type pieces and we give generation times for 32 irregular packing benchmark problems from the literature, including real world datasets, to allow further comparison with existing and future approaches.     

The k‐piece packing problem

A k‐piece of a graph G is a connected subgraph of G all of whose nodes have degree at most k and at least one node has degree equal to k. We consider the problem of covering the maximum number of nodes of a graph by node disjoint k‐pieces. When k = 1 this is the maximum matching problem, and when k = 2 this is the problem, recently studied by Kaneko [ 19 [, of covering the maximum number of nodes by disjoint paths of length greater than 1. We present a polynomial time algorithm for the problem as well as a Tutte‐type existence theorem and a Berge‐type min‐max formula. We also solve the problem in the more general situation where the “pieces” are defined in terms of lower and upper bounds on the degrees. © 2006 Wiley Periodicals, Inc. J Graph Theory     

超尺寸物品装箱问题

本文给出一类新的装箱问题，超尺寸物品装箱问题。就实际解决该问题所普遍彩的两步法，证明了当采用经典目标函数并且拆分次数不超过2时，第二步采用FFDLR的渐进最坏比为3/2。进而针对超尺寸物品装箱问题的算法提出了一个评价效率更高的目标函数。证明了在此目标函数下，当不限制物品的最大尺寸时，第二步采用最优装法两步法的渐近最坏比为2。最后，给出渐近最坏与拆分次数的关系。     

Unsupervised classification of children’s bodies using currents

Object classification according to their shape and size is of key importance in many scientific fields. This work focuses on the case where the size and shape of an object is characterized by a current. A current is a mathematical object which has been proved relevant to the modeling of geometrical data, like submanifolds, through integration of vector fields along them. As a consequence of the choice of a vector-valued reproducing kernel Hilbert space (RKHS) as a test space for integrating manifolds, it is possible to consider that shapes are embedded in this Hilbert Space. A vector-valued RKHS is a Hilbert space of vector fields; therefore, it is possible to compute a mean of shapes, or to calculate a distance between two manifolds. This embedding enables us to consider size-and-shape clustering algorithms. These algorithms are applied to a 3D database obtained from an anthropometric survey of the Spanish child population with a potential application to online sales of children’s wear.     

Creeping flow about a slightly deformed sphere

The problem of slow streaming flow of a viscous incompressible fluid past a spheroid which departs but little in shape from a sphere with mixed slip-stick boundary conditions, is investigated. The explicit expression for the stream function is obtained to the first order in the small parameter characterising the deformation. The case of an oblate spheroid is considered as a particular example and the force on this non-spherical body is evaluated. It is found that the parameter ₁, which arises in connection with the boundary condition, has significant effect upon the hydrodynamic force. In fact, it is shown that, the force is a quadratic function of this parameter up to the first order of deformation. Also, it is observed that the drag in the present case is less than that of the Stokes resistance for a slightly oblate spheroid. Some other special cases are also deduced from the present result. A brief discussion of the results to other body shapes is presented.     

Hypersonic shapes of maximum drag for a given heat-transfer rate

The problem of determining the shape of a blunt, axisymmetric body which maximizes the drag for a given heat-transfer rate and diameter is considered. For blunt bodies, pressure drag predominates and is estimated from modified Newtonian flow considerations. With regard to heat transfer, it is assumed that the body is operating in the range of hypersonic speeds where the radiative heating rate can be neglected with respect to the convective heating rate. The latter is estimated from the boundary-layer analysis due to Lees. Optimum power-law shapes as well as variational shapes are determined and are shown to yield almost identical results. For low-to-moderate values of the convective heat-transfer parameter, the optimum shape is very flat and is approximately a one-half power-law shape; in this range, spherical segments are approximately one-half power-law shapes and, hence, are nearly maximum drag shapes. There exists a maximum value of the convective heat-transfer parameter for which maximum drag shapes exist, and the corresponding optimum shape is a cone, or a power-law shape of exponent unity. This limiting shape is shown to be that which maximizes the convective heat-transfer rate for a given diameter.This research was supported in part by NASA-Manned Spacecraft Center under Contract No. NAS-6963. The authors are indebted to Dr. John J. Bertin for helpful discussions and suggestions concerning heattransfer aspects of this paper.     

Three-dimensional bin packing problem with variable bin height

This paper studies a variant of the three-dimensional bin packing problem (3D-BPP), where the bin height can be adjusted to the cartons it packs. The bins and cartons to be packed are assumed rectangular in shape. The cartons are allowed to be rotated into any one of the six positions that keep the carton edges parallel to the bin edges. This greatly increases the difficulty of finding a good solution since the search space expands significantly comparing to the 3D-BPP where the cartons have fixed orientations. A mathematical (mixed integer programming) approach is modified based on [Chen, C. S., Lee, S. M., Shen, Q. S., 1995. An analytical model for the container loading problem. European Journal of Operational Research 80 (1), 68–76] and numerical experiments indicate that the mathematical approach is not suitable for the variable bin height 3D-BPP. A special bin packing algorithm based on packing index is designed to utilize the special problem feature and is used as a building block for a genetic algorithm designed for the 3D-BPP. The paper also investigates the situation where more than one type of bin are used and provides a heuristic for packing a batch of cartons using the genetic algorithm. Numerical experiments show that our proposed method yields quick and satisfactory results when benchmarked against the actual packing practice and the MIP model with the latest version of CPLEX.