Approximate Circle Packing in a Rectangular Container: Integer Programming Formulations and Valid Inequalities

A problem of packing a limited number of unequal circles in a fixed size rectangular container is considered. The aim is to maximize the (weighted) number of circles placed into the container or minimize the waste. Frequently, the problem is formulated as a nonconvex continuous optimization problem which is solved by heuristic techniques combined with local search procedures. A new formulation is proposed using a regular grid approximating the container and considering the nodes of the grid as potential positions for assigning centers of the circles. The packing problem is then stated as a large scale linear 0-1 optimization problem. The binary variables represent the assignment of centers to the nodes of the grid. The resulting binary problem is then solved by commercial software. Two families of valid inequalities are proposed to strengthen the formulation. Numerical results are presented to demonstrate the efficiency of the proposed approach.