Approximate packing: Integer programming models, valid inequalities and nesting

Abstract

Using a regular grid to approximate a container, packing objects 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. Different formulations for non-overlapping constraints are presented and compared. Valid inequalities are proposed to strengthening formulations. This approach is applied for packing circular and L-shaped objects. Circular object is considered in a general sense 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., are treated similarly by simply changing the definition of the norm used to define the distance. Nesting objects inside one another is also considered. Numerical results are presented to demonstrate the efficiency of the proposed approach.