Artificial intelligence
Dense packings of congruent circles in a circle
Discrete Mathematics
Artificial Intelligence: Structures and Strategies for Complex Problem Solving
Artificial Intelligence: Structures and Strategies for Complex Problem Solving
Approximate algorithms for constrained circular cutting problems
Computers and Operations Research
An effective hybrid algorithm for the problem of packing circles into a larger containing circle
Computers and Operations Research
Reformulation descent applied to circle packing problems
Computers and Operations Research
New heuristics for packing unequal circles into a circular container
Computers and Operations Research
A beam search algorithm for the circular packing problem
Computers and Operations Research
An Effective Hybrid Algorithm for the Circles and Spheres Packing Problems
COCOA '09 Proceedings of the 3rd International Conference on Combinatorial Optimization and Applications
Computers and Industrial Engineering
Solving the problem of packing equal and unequal circles in a circular container
Journal of Global Optimization
Two-dimensional equilibrium constraint layout using simulated annealing
Computers and Industrial Engineering
A heuristic approach for packing identical rectangles in convex regions
Computers and Operations Research
Packing unequal circles using formulation space search
Computers and Operations Research
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In this paper, we study the circular packing problem (CPP) which consists of packing a set of non-identical circles of known radii into the smallest circle with no overlap of any pair of circles. To solve CPP, we propose a three-phase approximate algorithm. During its first phase, the algorithm successively packs the ordered set of circles. It searches for each circle's "best" position given the positions of the already packed circles where the best position minimizes the radius of the current containing circle. During its second phase, the algorithm tries to reduce the radius of the containing circle by applying (i) an intensified search, based on a reduction search interval, and (ii) a diversified search, based on the application of a number of layout techniques. Finally, during its third phase, the algorithm introduces a restarting procedure that explores the neighborhood of the current solution in search for a better ordering of the circles. The performance of the proposed algorithm is evaluated on several problem instances taken from the literature.