Exact and Heuristic Approaches for Assignment in Multiple-Container Packing

This paper deals with cutting/packing problems in which there is a set of pieces to be allocated and arranged in a set of "containers." In an apparel manufacturing application, the containers might be unused areas of the fabric after large pieces have been placed, and the pieces of interest might be the smaller pieces. In a sheet metal application, the containers could be the sheets themselves, and the pieces the entire set of pieces to be arranged. The specific problem addressed takes as input a set of groups (of pieces), and mappings from pieces to groups and groups to containers. The method in which the groups are generated and the particular geometric constraints (e.g., translation only, or translation plus rotation) is not critical for the methods developed here. This paper presents an integer programming formulation of the multiple-container group assignment problem (MCGAP). Based on long and/or highly variable solution times for some problem instances, a Lagrangian heuristic procedure is developed along with a local improvement heuristic. When combined together these methods yield a powerful solution technique, reaching the optimal solution in all but one problem instance tested, while using relatively little CPU time. Moreover, solution times are more predictable than when standard branch & bound methods are used. Cases when all pieces are distinct and when certain pieces form equivalence classes are treated. key words: packing problems, cutting stock, integer programming, Lagrangian relaxation, heuristic, covering problems