Packing Circles and Irregular Polygons using Separation Lines

In this paper we propose a nonlinear mathematical model for the problem of packing circles, convex and nonconvex irregular polygons, within a rectangular envelope to be produced, obeying containment constraints and non-overlapping constraints; the objective of the problem is to minimize the area of the rectangular envelope. We consider free rotations of the polygons and use separation lines to ensure non-overlapping. Computational tests were run using instances presented in the literature that deal with circles and polygons simultaneously; different solutions, in which the area of the rectangular envelope is smaller than or equal to the ones found in the literature were found in most cases, and the execution time is very low. This indicates that our model is computationally efficient.