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 01 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.
Packing problems generally consist of packing of a certain number of items of known dimensions into one or more large objects or containers so as to minimize a certain objective (e.g., the unused part of the objects or waste). The shape of items and containers may vary from a circle, a square, a rectangle, and so forth.
This problem has been applied in different areas, such as the coverage of a geographical area with cell transmitters, storage of a cylindrical drums into containers or stocking them into an open area, packaging bottles or cans into the smallest box, planting trees in a given region so as to maximize the forest density and the distance between the trees, and so forth [
In this paper we address the problem of packing a set of circular items in a rectangular container. There are two principal types of objectives that have been used in the literature: (a) regard the circles (not necessary equal) as being of fixed size and the container as being of variable size and (b) regard the circles and the container as being of fixed size and minimize “waste.”
Examples of the first approach include the following [
For the square container minimize the length of the side and hence minimize the perimeter and area of the square.
Minimize the perimeter of the rectangle.
Minimize the area of the rectangle.
Considering one dimension of the rectangle as fixed, minimize the other dimension. Problems of this type are often referred to as strip packing problems (or as circular open dimension problems).
For the second approach various definitions of the waste can be used. The waste can be defined in relation to circles not packed (e.g., the number of unpacked circles or the perimeter/area of unpacked circles) or introducing a value associated with each circle that is packed (e.g., area of the circles packed), and so forth.
Many variants of packing circular objects in the plane have been formulated as nonconvex (continuous) optimization problems with decision variables being coordinates of the centres. The nonconvexity is mainly provided by no overlapping conditions between circles. These conditions typically state that the Euclidean distance separating the centres of the circles is greater than a sum of their radii. The nonconvex problems can be tackled by available nonlinear programming (NLP) solvers; however, most NLP solvers fail to identify global optima. Thus, the nonconvex formulation of circular packing problem requires algorithms which mix local searches with heuristic procedures in order to widely explore the search space. It is impossible to give a detailed overview on the existing solution strategies and numerical results within the framework of a single short paper. We will refer the reader to review papers presenting the scope of techniques and applications for the circle packing problem (see, e.g., [
In this paper we propose a new formulation for approximate solution of packing problems based on using a regular grid to approximate the container. 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 01 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. To the best of our knowledge, the idea to use a grid was first implemented by Beasley [
Suppose we have nonidentical circles
In what follows we will distinguish two cases of circle packing, depending on whether nesting circles inside one another is permitted or not. To the best of our knowledge, nesting problem was first mentioned in [
Consider first the problem without nesting. In order to let the circle
Similar to plant location problems [
Constraints (
If (
Thus the problem (
Since constraints of
This point can be constructed as follows. By the definition,
As follows from Proposition
By the definition,
To consider nesting circles inside one another, we only need to modify the nonoverlapping constraints. In order to let the circle
A rectangular uniform grid was used in the numerical experiments, such that all grid points are defined by the grid points on its edges. Let
All optimization problems were solved by the system CPLEX 12.5. The runs were executed on a DELL Power Edge T410, Intel Xeon 2.53 Ghz and 16 Gb RAM.
First, we compare different formulations for the case of packing equal circles. The same set of 9 instances as in [
Results of the numerical experiments.
Circle radius 

Problem dimension  Circle number  Complete  Half  Compact  Compact half 

0.625  0.15625  1403  10  6.4  125.5  144.8  110.6 
0.5625  0.0703125  2449  13  50.4  647.0  5034.5  1890.6 
0.5  0.125  697  18  2.6  3.2  5.2  24.1 
0.4375  0.0546875  3666  21  849.2  310.8  7459.9  3690.5 
0.375  0.046875  1425  32  50.0  21.4  3873.5  847.7 
0.3125  0.078125  2139  45  403.8  183.9  6514.3  4451.8 
0.275  0.06875  2880  61  1032.4  415.3  *  6985.4 
0.25  0.0625  3649  74  1234.8  535.5  14645.9  7840.6 
0.1875  0.046875  6897  140  1427.3  725.9  *  8765.2 
As we can see from Table
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Different integer formulations were proposed for approximate solution of the circle packing problem. We demonstrate that the pairwise formulation is stronger than the compact one obtained by summing up the nonoverlapping constraints. The presented approach can be easily generalized to the three (and more) dimensional case and to different shapes of the container, including irregulars. We also proposed a formulation permitting nesting circles inside one another. This problem was mentioned in [
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was partially supported by Grants from RFBR, Russia (12_01_00893_a), and CONACYT, Mexico (167019). The authors would like to thank anonymous referees for their constructive comments and suggestions.