Numerical optimization method for packing regular convex polygons

An algorithm is presented for the approximate solution of the problem of packing regular convex polygons in a given closed bounded domain $G$ so as to maximize the total area of the packed figures. On $G$ a grid is constructed whose nodes generate a finite set $W$ on $G$, and the centers of the figures to be packed can be placed only at some points of $W$. The problem of packing these figures with centers in $W$ is reduced to a $0$–$1$ linear programming problem. A two-stage algorithm for solving the resulting problems is proposed. The algorithm finds packings of the indicated figures in an arbitrary closed bounded domain on the plane. Numerical results are presented that demonstrate the effectiveness of the method.