Approximate Packing: Integer Programming Models, Valid Inequalities and Nesting (original) (raw)
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Integer Programming Formulations for Approximate Packing Circles in a Rectangular Container
Mathematical Problems in Engineering, 2014
A problem of packing a limited number of unequal circles in a fixed size rectangular container is considered. The aim is to maximize the (weighted) number of circles placed into the container or minimize the waste. This problem has numerous applications in logistics, including production and packing for the textile, apparel, naval, automobile, aerospace, and food industries. Frequently the problem is formulated as a nonconvex continuous optimization problem which is solved by heuristic techniques combined with local search procedures. New formulations are proposed for approximate solution of packing problem. The container is approximated by a regular grid and the nodes of the grid are considered as potential positions for assigning centers of the circles. The packing problem is then stated as a large scale linear 0-1 optimization problem. The binary variables represent the assignment of centers to the nodes of the grid. Nesting circles inside one another is also considered. The resulting binary problem is then solved by commercial software. Numerical results are presented to demonstrate the efficiency of the proposed approach and compared with known results.
Lecture Notes in Computer Science, 2014
A problem of packing a limited number of unequal circles in a fixed size rectangular container is considered. The aim is to maximize the (weighted) number of circles placed into the container or minimize the waste. Frequently, the problem is formulated as a nonconvex continuous optimization problem which is solved by heuristic techniques combined with local search procedures. A new formulation is proposed using a regular grid approximating the container and considering the nodes of the grid as potential positions for assigning centers of the circles. The packing problem is then stated as a large scale linear 0-1 optimization problem. The binary variables represent the assignment of centers to the nodes of the grid. The resulting binary problem is then solved by commercial software. Two families of valid inequalities are proposed to strengthen the formulation. Numerical results are presented to demonstrate the efficiency of the proposed approach.
Packing circular-like objects in a rectangular container
Известия Российской академии наук. Теория и системы управления, 2015
Packing problems generally consist of packing a set of items of known dimensions into one or more large objects in order to minimize a certain objective (e.g. the unused part of the objects or waste).
Computational Aspects of Packing Problems
2016
Packing problems have been investigated in mathematics since centuries. For polygons, circles, or other objects bounded by algebraic curves or surfaces it can be argued that packing problems are computable. However, even some of the simplest versions in the plane turn out to be NP-hard unless the number of objects to be packed is bounded. This article is a survey on results achieved about computability and complexity of packing problems, about approximation algorithms, and about very natural packing problems whose computational complexity is unknown. Packing objects is a quite natural problem and has been investigated in mathematics and operations research for a long time. Applications concern the physical nonoverlapping packing of concrete objects during storage or transportation but also in two dimensions how to eciently cut prescribed pieces from cloth or sheet metal while minimizing waste. Even more abstract problems like, e.g., ecient scheduling with respect to time and space c...
Using Different Norms in Packing Circular Objects
Lecture Notes in Computer Science, 2015
A problem of packing unequal circles in a fixed size rectangular container is considered. The circle is considered in a general sense, as a set of points that are all the same distance (not necessary Euclidean) from a given point. An integer formulation is proposed using a grid approximating the container and considering the nodes of the grid as potential positions for assigning centers of the circles. The packing problem is then stated as a large scale linear 0-1 optimization problem. Valid inequalities are proposed to strengthening the original formulation. Nesting circles inside one another is considered tacking into account the thickness of the circles. Numerical results on packing circles, ellipses, rhombuses and octagons are presented to demonstrate the efficiency of the proposed approach.
Approximate Packing Circles in a Rectangular Container: Valid Inequalities and Nesting
Journal of Applied Research and Technology, 2014
A problem of packing a limited number of unequal circles in a fixed size rectangular container is considered. The aim is to maximize the (weighted) number of circles placed into the container or minimize the waste. This problem has numerous applications in logistics, including production and packing for the textile, apparel, naval, automobile, aerospace and food industries. Frequently the problem is formulated as a nonconvex continuous optimization problem which is solved by heuristic techniques combined with the local search procedures. A new formulation is proposed based on using a regular grid approximated the container and considering the nodes of the grid as potential positions for assigning centers of the circles. The packing problem is then stated as a large scale linear 0-1 optimization problem. The binary variables represent the assignment of centers to the nodes of the grid. The resulting binary problem is then solved by the commercial software. Two families of valid inequalities are proposed to strengthening the formulation. Nesting circles inside one another is also considered. Numerical results are presented to demonstrate the efficiency of the proposed approach.
A particular approach for the Three-dimensional Packing Problem with additional constraints
Computers & Operations Research, 2010
This paper focuses on a restrained concretization of the general NP-hard Container Loading Problem that arises from a real world application. This particular problem can be informally described as: given different sets of bins and boxes, find the packing of the boxes into the smallest number of bins obeying some additional restrictions on the placement. The mathematical programming formulation that appears to better model this application is the Three-dimensional Bin-Packing Problem (3D-BPP) which is no more than an extension of the classic Bin-Packing Problem to the orthogonal packing of solid objects.
A Literature Review on Circle and Sphere Packing Problems: Models and Methodologies
Advances in Operations Research, 2009
This paper reviews the most relevant literature on efficient models and methods for packing circular objects/items into Euclidean plane regions where the objects/items and regions are either two-or three-dimensional. These packing problems are NP hard optimization problems with a wide variety of applications. They have been tackled using various approaches-based algorithms ranging from computer-aided optimality proofs, to branch-and-bound procedures, to constructive approaches, to multi-start nonconvex minimization, to billiard simulation, to multiphase heuristics, and metaheuristics.
A global optimization method for packing problems
Engineering Optimization, 2006
The packing optimization problem is to seek the best way of placing a given set of rectangular boxes within a rectangular container with minimal volume. Current packing optimization methods (Chen et al. [2], Li and Chang [6]) are either difficult to find an optimal solution or required to use too many extra 0-1 variables to solve the problems. This paper proposes a new method for finding the global optimum of the packing problem within tolerable error based on piecewise linearization techniques, which is more computationally efficient than current methods.