Using Valid Inequalities and Different Grids in LP-Based Heuristic for Packing Circular Objects

Abstract

Using a regular grid to approximate a container, packing is reduced to assigning objects to the nodes of the grid subject to non-overlapping constraints. The packing problem is then stated as a large scale linear 0–1 optimization problem. A problem of packing unequal circles in a fixed size rectangular container is considered. The circle is considered as a set of points that are all the same distance (not necessary Euclidean) from a given point. Different shapes, such as ellipses, rhombuses, rectangles, octagons, etc. can be treated similarly by simply changing the definition of the norm used to define the distance. Valid inequalities are used to strengthening the LP-relaxation. An LP-based heuristic is proposed. Numerical results on packing circles and octagons are presented to demonstrate the efficiency of the proposed approach.