Irregular packing problems: A review of mathematical models

Packing of concave polyhedra with continuous rotations using nonlinear optimisation

In this paper, continuous and differentiable nonlinear programming models and algorithms for packing ellipsoids in the n-dimensional space are introduced. Two different models for the non-overlapping and models for the inclusion of ellipsoids within half-spaces and ellipsoids are presented. By applying a simple multi-start strategy combined with a clever choice of starting guesses and a nonlinear programming local solver, illustrative numerical experiments are presented.

/pdf/packing-ellipsoids-by-nonlinear-optimization-10q33r6ica.pdf

Packing ellipsoids by nonlinear optimization

Hyper-heuristics aim at interchanging different solvers while solving a problem. The idea is to determine the best approach for solving a problem at its current state. This way, every time we make a move it gets us closer to a solution. The problem changes; so does its state. As a consequence, for the next move, a different solver may be invoked. Hyper-heuristics have been around for almost 20 years. However, combinatorial optimization problems date from way back. Thus, it is paramount to determine whether the efforts revolving around hyper-heuristic research have been targeted at the problems of the highest interest for the combinatorial optimization community. In this work, we tackle such an endeavor. We begin by determining the most relevant combinatorial optimization problems, and then we analyze them in the context of hyper-heuristics. The idea is to verify whether they remain as relevant when considering exclusively works related to hyper-heuristics. We find that some of the most relevant problem domains have also been popular for hyper-heuristics research. Alas, others have not and few efforts have been directed towards solving them. We identify the following problem domains, which may help in furthering the impact of hyper-heuristics: Shortest Path, Set Cover, Longest Path, and Minimum Spanning Tree. We believe that focusing research on ways for solving them may lead to an increase in the relevance and impact that hyper-heuristics have on combinatorial optimization problems.

/pdf/a-systematic-review-of-hyper-heuristics-on-combinatorial-2bflxp45o3.pdf

A Systematic Review of Hyper-Heuristics on Combinatorial Optimization Problems

The concept of configuration space of geometric objects is introduced. Its generalized variables are metric parameters of the spatial form and parameters of the location of objects. The properties of configuration spaces of complex geometric objects are considered. The structures of configuration spaces for various classes of geometric object placement problems, including packing and covering problems, are analyzed. The concept of Φ-function of geometric objects with variable metrical parameters is generalized.

Configuration Space of Geometric Objects

We describe our methodology for solving NP-hard irregular placement problems. We deal with an accurate representation of objects bounded by circular arcs and line segments and allow their free rotations within a container. We formulate a basic irregular placement problem (IRPP), which covers a wide spectrum of practical packing, cutting, nesting, clustering, and layout problems. We provide a nonlinear programming (NLP) model of the problem, employing the phi-function technique. Our model involves a large number of inequalities with nonsmooth functions. We describe a solution tree for our placement problem and evaluate the number of its terminal nodes. We reduce IRPP problem to a sequence of NLP-subproblems with smooth functions.

Placement Problems for Irregular Objects: Mathematical Modeling, Optimization and Applications

Packing optimization problems have a wide spectrum of real-word applications, including transportation, logistics, chemical/civil/mechanical/power/aerospace engineering, shipbuilding, robotics, additive manufacturing, materials science, mineralogy, molecular geometry, nanotechnology, electronic design automation, very large system integration, pattern recognition, biology, and medicine. In space engineering, ever more challenging packing optimization problems have to be solved, requiring dedicated cutting-edge approaches.

Optimized Packings in Space Engineering Applications: Part I

A packing (layout) problem for a number of clusters (groups) composed of convex objects (e.g., circles, ellipses, or convex polygons) is considered. The clusters have to be packed into a given rectangular container subject to nonoverlapping between objects within a cluster. Each cluster is represented by the convex hull of objects that form the cluster. Two clusters are said to be nonoverlapping if their convex hulls do not overlap. A cluster is said to be entirely in the container if so is its convex hull. All objects in the cluster have the same shape (different sizes are allowed) and can be continuously translated and rotated. The objective of optimized packing is constructing a maximum sparse layout for clusters subject to nonoverlapping and containment conditions for clusters and objects. Here the term sparse means that clusters are sufficiently distant one from another. New quasi-phi-functions and phi-functions to describe analytically nonoverlapping, containment and distance constraints for clusters are introduced. The layout problem is then formulated as a nonlinear nonconvex continuous problem. A novel algorithm to search for locally optimal solutions is developed. Computational results are provided to demonstrate the efficiency of our approach. This research is motivated by a container-loading problem; however similar problems arise naturally in many other packing/cutting/clustering issues.

/pdf/optimized-packing-clusters-of-objects-in-a-rectangular-4ytrzm1kkd.pdf

Optimized Packing Clusters of Objects in a Rectangular Container

We consider the problem of packing convex polytopes in a cuboid of minimum volume. To describe analytically the non-overlapping constraints for convex polytopes that allow continuous translations and rotations, we use phi-functions and quasi-phi-functions. We provide an exact mathematical model in the form of an NLP-problem and analyze its characteristics. Based on the general solution strategy, we propose two approaches that take into account peculiarities of phi-functions and quasi-phi-functions. Computational results to compare the efficiency of our approaches are given with respect to both the value of the objective function and runtime.

Alexandr Pankratov

Papers

Placement Problems for Irregular Objects: Mathematical Modeling, Optimization and Applications

Optimized Packings in Space Engineering Applications: Part I

Optimized Packing Clusters of Objects in a Rectangular Container

Two Approaches to Modeling and Solving the Packing Problem for Convex Polytopes

3D Irregular Packing in an Optimized Cuboid Container