A memetic algorithm to pack unequal circles into a square

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.