New heuristics for packing unequal circles into a circular container

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Abstract

We propose two new heuristics to pack unequal circles into a two-dimensional circular container. The first one, denoted by A1.0, is a basic heuristic which selects the next circle to place according to the maximal hole degree rule. The second one, denoted by A1.5, uses a self look-ahead strategy to improve A1.0. We evaluate A1.0 and A1.5 on a series of instances up to 100 circles from the literature and compare them with existing approaches. We also study the behaviour of our approach for packing equal circles comparing with a specified approach in the literature. Experimental results show that our approach has a good performance in terms of solution quality and computational time for packing unequal circles.

Introduction

The two-dimensional (2D) circle packing problem is a famous packing problem [1], encountered in some industries (textile, glass, wood, paper, etc.). It consists in placing a set of circles in a container without overlap. The usual objective is to maximize the material utilization and hence to minimize the wasted area. There are various options in the problem, e.g. the container can be circle, rectangle or polygon, the radii of circles to be placed can be equal or unequal. In this paper, we investigate the problem of packing unequal circles into a circular container. The problem is known to be NP-hard [2].

Most published research on circle packing focused on packing equal circles into a container [3], [4], [5], [6]. The proposed approaches were heavily influenced by the congruence of the circles.

Several authors addressed the problem of packing unequal circles into a rectangular or a strip container. George et al. [7] proposed a set of heuristic rules and different combinations of these rules to pack unequal circles into a rectangle and the best ones were a quasi-random algorithm and a genetic algorithm. Stoyan and Yaskov [8], [9] proposed a mathematical model and a solution method based on a combination of the branch-and-bound algorithm and the reduced gradient method to pack unequal circles into a strip. Hifi and M’Hallah [10] presented a Bottom-Left (BL) based genetic algorithm to solve a dual of circle packing problem: cutting unequal circles on a rectangular plate.

But to our knowledge, no significant published research addresses the problem of packing unequal circles into a circular container, except a simulation approach based on an elasticity physics model, proposed by Huang et al. [11], [12] and Wang et al. [13]. This approach can obtain good enough solutions for packing unequal circles and very good solutions for packing equal circles, but the computational time greatly increases for large instances.

In this paper, we propose two new heuristics to solve the problem. A preliminary version of this work appeared in [14], [15], and then extended to solve the problem of packing unequal circles into a rectangular container [16]. The basic idea of our approach is a quantified measure, called hole degree, to evaluate the benefit of placing a circle in the container, which gives the first algorithm, denoted by A1.0. Another feature of our approach is the use of a self look-ahead strategy to improve A1.0 and gives the second algorithm, denoted by A1.5. We evaluate our approach on a series of instances up to 100 circles from the literature [8], [9], [12], [13], [14], [15], [17] and compare it with existing approaches. We also study the behaviour of our approach when it is applied to pack equal circles. Experimental results show that our approach has a good performance in terms of solution quality and computational time for packing unequal circles.

The paper is organized as follows. In Section 2, we give a formal definition of the circle packing problem. In Section 3, we present the two heuristics A1.0 and A1.5. In Section 4, we present and analyze the experimental results. Section 5 concludes the paper.

Section snippets

Circle packing problem

We consider the following circle packing optimization problem: given a set of unequal circles, find the minimal radius of a circular container so that all the circles can be packed into the container without overlap. The associated decision problem is formally stated as follows.

Given a circular container of radius r0 and a set of n circles c1,,cn of radii r1,,rn. There may be several circles with the same radius. Let (0,0) be the coordinate of the container centre, and (xi,yi) the coordinate

Algorithms A1.0 and A1.5

Inspired from human experiences in packing, we propose a quantified measure, called hole degree, to evaluate the benefit of placing a circle in the container. The selection of the next circle to pack is guided by the maximal hole degree (MHD) rule, which gives our basic packing algorithm A1.0.

Following the same line, we propose a self look-ahead strategy to improve A1.0, which consists in using A1.0 itself to evaluate the benefit of placing a circle in the container, which gives our second

Experimental results and analysis

In this section, we evaluate our approach on a series of instances up to 100 circles from the literature [8], [9], [12], [13], [14], [15], [17], and compare our approach with the simulation approach proposed by Huang et al. and Wang et al. [11], [12], [13] in terms of solution quality and running time. We also study the behaviours of our approach when it is applied to pack equal circles.

The simulation approach in [11], [12], [13] is practically a Local Search based on an elasticity physics

Conclusion

We have presented two new heuristics A1.0 and A1.5 for packing unequal circles into a 2D circular container. The main features of our approach are the use of the MHD rule to select a next circle to place and obtain A1.0, and the use of a self look-ahead strategy to improve A1.0 and obtain A1.5.

We evaluate our approach on a series of instances from the literature and compare with existing approaches. Experimental results show that our approach works a good performance for packing unequal circles

Acknowledgements

The authors thank anonymous referees for their helpful comments and suggestions which improved the quality of the paper.

References (19)

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This work is supported by “The National Natural Science Foundation of China under grant No. 10471051” and “Programme de Recherches Avancées de Coopérations Franco-Chinoises (PRA SI02-04)”.

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