Optimal tool partitioning rules for numerically controlled punch press operations

The purpose of this paper is to formulate and solve an optimization problem arising in the operations of a numerical control machine such as a punch press. Holes of n different types must be punched by a press on a linear object such as a metallic bar. Each type of hole requires a different tool and tools are mounted on a fixed rotating carousel. Holes are punched in several passes on the bar, using each time a contiguous subset of tools. The problem is to determine a partition of the tools into subsets that will minimize expected completion time. The problem is first formulated as a shortest path problem on an acyclic graph. It is shown how this problem can be solved in O(n²) or in O(n³) time and that a closed form solution can sometimes be obtained in constant time.