A memetic algorithm to pack unequal circles into a square

摘要

A circle packing problem involves packing given circles into a container. The objective is to minimize the size of the container without causing any overlap. This paper focuses on a representative circle packing problem where the circles are unequal and the container is a square, called Packing Unequal Circles into a Square (PUCS). It proposes a memetic algorithm to solve the problem. The memetic algorithm can be regarded as a combination of a genetic algorithm (GA) and an iterated local search (ILS). It is composed of a local search phase and a global transformation phase. The global transformation phase evolves a population; the local search phase optimizes the offsprings generated by the global transformation phase. The proposed approach exhibits several novel features in its global transformation phase, such as the z-crossover operator based on the symmetry of the container and the complementarity of the configuration, a perturbation operator inspired by gene-fragment-insertion, a reproduction selector based on the genetic relationships of the individuals, and a hybrid population updating strategy based on the diversity of the reproduction operators. Experimental results show that the memetic algorithm is effective for the problem. (C) 2017 Elsevier Ltd. All rights reserved.