Note on: An improved algorithm for the packing of unequal circles within a larger containing circle

Basin filling algorithm for the circular packing problem with equilibrium behavioral constraints

“…It is noteworthy that refs. [13,15] and this paper use the same target energy function, but the potential energy of the resulting configuration X obtained by BF for each instance satisfies U (X) < 10 −20 , whereas IQPQH's only satisfies U (X) < 5 × 10 −6 , and ISS's satisfies U (X) < 10 −8 , i.e., we obtain the precise solutions of these two problems. For the other three instances, we find some new smaller radii (which are in boldface) of the circular container.…”

“…Among the five instances, Instances 4 and 5 are most challenging and have been tested by all six algorithms. To make intuitively a comparison between the results of six methods, we give the ratios of the results of other five methods to reference values (100%), shown in Figures 4 and 5, where reference values are the minimal radius of the outer enveloping circle obtained by ISS in [15] and the static non-equilibrium value by IQPQH in [13], respectively. Because of the differences in the performance of the running computers and the programming languages, we do not compare the speed of our algorithm with the other five methods.…”

“…The minimal radius R 0 (mm) of the circular container (i.e. the outer enveloping circle), the average static non-equilibrium value J (g·mm) and the average running time t(s) by BF over five independent runs are listed in Table 2, in comparison with those obtained by the human-computer interactive genetic algorithm (HCGA) [8], the mutation particle swarm optimization (MPSO) algorithm [10], the adaptive particle swarm optimizer (APSO) [12], the improved quasi-physical quasi-human (IQPQH) algorithm [13], and the improved scatter search (ISS) [15]. Table 2 shows the optimal radii for Instances 1 and 2.…”

“…Lei and Qiu [12] gave a novel adaptive particle swarm optimizer by modifying on the traditional PSO algorithm. Huang and Chen [13] proposed an improved quasi-physical quasi-human algorithm which introduces an efficient strategy of accelerating the search process into the quasi-physical quasi-human algorithm [14]. Wang et al [15] developed an improved scatter search algorithm which incorporates two types of local search methods, the gradient method and Nelder-Mead simplex algorithm, into the scatter search.…”

“…The absence of a powerful optimization method is the key obstacle to solving this type of problems. Different global search methods, including stochastic algorithms (for example, the genetic algorithm [7][8][9], and the particle swarm optimizer [10][11][12]) and heuristic algorithms (for example, the method of model-changing iteration [6], and the quasi-physical quasi-human algorithm [13]), have their respective limitations. On the one hand, stochastic algorithms have global search ability and broad adaptability; on the other hand they lack accuracy of search and speed of convergence.…”