363 research outputs found
Computer tool for maximizing the placement of congruent polyhedra
Given multiple identical polyhedral objects and a parallelepiped container, how should one place the
objects so that the largest number fits inside the container? This simple question is important in many applications,
yet the answer is elusive. In fact, we know of no published solution for this very general formulation. Still, in many
circumstances, further restrictions apply, resulting in a large number of variations requiring different algorithmic
strategies. This paper is the continuation of [12] and focus on the fundamental concepts and tools that are used for
this kind of problem, such as the no-fit polygon. We also present some of its many variations, giving in particular one
that applies to the stereolithographic rapid prototyping technology
Recommended from our members
Compaction Algorithms for Non-Convex Polygons and Their Applications
Given a two-dimensional, non-overlapping layout of convex and non-convex polygons, compaction refers to a simultaneous motion of the polygons that generates a more densely packed layout. In industrial two-dimensional packing applications, compaction can improve the material utilization of already tightly packed layouts. Efficient algorithms for compacting a layout of non-convex polygons are not previously known. This dissertation offers the first systematic study of compaction of non-convex polygons. We start by formalizing the compaction problem as that of planning a motion that minimizes some linear objective function of the positions. Based on this formalization, we study the complexity of compaction and show it to be PSPACE-hard. The major contribution of this dissertation is a position-based optimization model that allows us to calculate directly new polygon positions that constitute a locally optimum solution of the objective via linear programming. This model yields the first practically efficient algorithm for translational compaction-compaction in which the polygons can only translate. This compaction algorithm runs in almost real time and improves the material utilization of production quality human-generated layouts from the apparel industry. Several algorithms are derived directly from the position-based optimization model to solve related problems arising from manual or automatic layout generation. In particular, the model yields an algorithm for separating overlapping polygons using a minimal amount of motion. This separation algorithm together with a database of human-generated markers can automatically generate markers that approach human performance. Additionally, we provide several extensions to the position-based optimization model. These extensions enables the model to handle small rotations, to offer the flexible control of the distances between polygons and to find optimal solution to two-dimensional packing of non-convex polygons. This dissertation also includes a compaction algorithm based on existing physical simulation approaches. Although our experimental results showed that it is not practical for compacting tightly packed layouts, this algorithm is of interest because it shows that the simulation can speed up significantly if we use geometrical constraints to replace physical constraints. It also reveals the inherent limitations of physical simulation algorithms in compacting tightly packed layouts. Most of the algorithms presented in this dissertation have been implemented on a SUN SparcStationTM and have been included in a software package licensed to a CAD company.Engineering and Applied Science
A scanline-based algorithm for the 2D free-form bin packing problem
Abstract This paper describes a heuristic algorithm for the two-dimensional free-form bin packing (2D-FBP) problem, which is also called the irregular cutting and packing, or nesting problem. Given a set of 2D free-form bins, which in practice may be plate materials, and a set of 2D free-form items, which in practice may be plate parts to be cut out of the materials, the 2D-FBP problem is to lay out items inside one or more bins in such a way that the number of bins used is minimized, and for each bin, the yield is maximized. The proposed algorithm handles the problem as a variant of the one-dimensional bin-packing problem; i.e., items and bins are approximated as sets of scanlines, and scanlines are packed. The details of the algorithm are given, and its application to a nesting problem in a shipbuilding company is reported. The proposed algorithm consists of the basic and the group placement algorithms. The basic placement algorithm is a variant of the first-fit decreasing algorithm which is simply extended from the one-dimensional case to the two-dimensional case by a novel scanline approximation. The group placement algorithm is an extension of the basic placement algorithm with recombination of input items. A numerical study with real instances shows that the basic placement algorithm has sufficient performance for most of the instances, however, the group placement algorithm is required when items must be aligned in columns. The qualities of the resulting layouts are good enough for practical use, and the processing times required for both algorithms are much faster than those by manual nesting. 1
Constructive procedures to solve 2-dimensional bin packing problems with irregular pieces and guillotine cuts
This paper presents an approach for solving a new real problem in cutting and packing. At its core is an innovative mixed integer programme model that places irregular pieces and defines guillotine cuts. The two-dimensional irregular shape bin packing problem with guillotine constraints arises in the glass cutting industry, for example, the cutting of glass for conservatories. Almost all cutting and packing problems that include guillotine cuts deal with rectangles only, where all cuts are orthogonal to the edges of the stock sheet and a maximum of two angles of rotation are permitted. The literature tackling packing problems with irregular shapes largely focuses on strip packing i.e. minimizing the length of a single fixed width stock sheet, and does not consider guillotine cuts. Hence, this problem combines the challenges of tackling the complexity of packing irregular pieces with free rotation, guaranteeing guillotine cuts that are not always orthogonal to the edges of the stock sheet, and allocating pieces to bins. To our knowledge only one other recent paper tackles this problem. We present a hybrid algorithm that is a constructive heuristic that determines the relative position of pieces in the bin and guillotine constraints via a mixed integer programme model. We investigate two approaches for allocating guillotine cuts at the same time as determining the placement of the piece, and a two phase approach that delays the allocation of cuts to provide flexibility in space usage. Finally we describe an improvement procedure that is applied to each bin before it is closed. This approach improves on the results of the only other publication on this problem, and gives competitive results for the classic rectangle bin packing problem with guillotine constraint
Packing of concave polyhedra with continuous rotations using nonlinear optimisation
We study the problem of packing a given collection of arbitrary, in general concave, polyhedra into a cuboid of minimal volume. Continuous rotations and translations of polyhedra are allowed. In addition, minimal allowable distances between polyhedra are taken into account. We derive an exact mathematical model using adjusted radical free quasi phi-functions for concave polyhedra to describe non-overlapping and distance constraints. The model is a nonlinear programming formulation. We develop an efficient solution algorithm, which employs a fast starting point algorithm and a new compaction procedure. The procedure reduces our problem to a sequence of nonlinear programming subproblems of considerably smaller dimension and a smaller number of nonlinear inequalities. The benefit of this approach is borne out by the computational results, which include a comparison with previously published instances and new instances
Learning Gradient Fields for Scalable and Generalizable Irregular Packing
The packing problem, also known as cutting or nesting, has diverse
applications in logistics, manufacturing, layout design, and atlas generation.
It involves arranging irregularly shaped pieces to minimize waste while
avoiding overlap. Recent advances in machine learning, particularly
reinforcement learning, have shown promise in addressing the packing problem.
In this work, we delve deeper into a novel machine learning-based approach that
formulates the packing problem as conditional generative modeling. To tackle
the challenges of irregular packing, including object validity constraints and
collision avoidance, our method employs the score-based diffusion model to
learn a series of gradient fields. These gradient fields encode the
correlations between constraint satisfaction and the spatial relationships of
polygons, learned from teacher examples. During the testing phase, packing
solutions are generated using a coarse-to-fine refinement mechanism guided by
the learned gradient fields. To enhance packing feasibility and optimality, we
introduce two key architectural designs: multi-scale feature extraction and
coarse-to-fine relation extraction. We conduct experiments on two typical
industrial packing domains, considering translations only. Empirically, our
approach demonstrates spatial utilization rates comparable to, or even
surpassing, those achieved by the teacher algorithm responsible for training
data generation. Additionally, it exhibits some level of generalization to
shape variations. We are hopeful that this method could pave the way for new
possibilities in solving the packing problem
Capturing points with a rotating polygon (and a 3D extension)
This is a post-peer-review, pre-copyedit version of an article published in Theory of computing systems: an international journal. The final authenticated version is available online at: http://dx.doi.org/10.1007/s00224-018-9885-yWe study the problem of rotating a simple polygon to contain the maximum number of elements from a given point set in the plane. We consider variations of this problem where the rotation center is a given point or lies on a segment or a line. We also solve an extension to 3D where we rotate a polyhedron around a given point to contain the maximum number of elements from a set of points in the space.Peer ReviewedPostprint (author's final draft
- …