Proportional Packing of Circles in a Circular Container

The authors consider a problem of packing circles of given types in a circular container. Circles are allowed to cross the container boundary in a predefined neighborhood that depends on the circle type (pseudo-inclusion condition). A family of circles is placed in the container under the conditions of their non-intersection, pseudo-inclusion, and compliance with the given proportions of circle types (proportionality condition) in order to maximize the total number of circles. A mathematical model is constructed as a problem of mixed integer nonlinear programming. A heuristic algorithm is proposed that applies a nonlinear programming problem to pack a given number of circles in a circular container, maximizing the variable radii of the circles. The results of the computational experiments are given.