Exact and approximate Mixed-Integer Programming models for the nesting problem

Several industrial problems involve placing objects into a container without overlap, with the goal of minimizing a certain objective function. This problems arise in many industrial fields such as apparel manufacturing, sheet metal layout, shoe manufacturing, VLSI layout, furniture layout, etc., and are known by a variety of names: layout, packing, nesting, loading, placement, marker making, etc. When the 2-dimensional objects to be packed are non-rectangular the problem is known as the (irregular) nesting problem. The nesting problem is strongly NP-hard. Furthermore, the geometrical aspects of this problem make it really hard to solve in practice. In this paper we define a Mixed-Integer Programming (MIP) model for the nesting problem, based on the an earlier proposal of Daniels, Li and Milenkovic, and analyze it computationally. We also introduce a new MIP model for a subproblem arising in the construction of optimal nesting solutions, called the multiple containment problem, and show its potentials in finding improved solutions.