The Number of Setups (Different Patterns) in One-Dimensional Stock Cutting

The primary objective in cutting and packing problems is trim loss or material input minimization (in stock cutting) or value maximization (when packing into a knapsack). However, in real-life production we usually have many other objectives (costs) and constraints, for example, the number of different patterns. We propose a new simple model for setup minimization (in fact, an extension of the Gilmore-Gomory model for trim loss minimization) and develop a branch-and-price algorithm on its basis. The algorithm is tested on problems with industrially relevant sizes of up to 150 product types. The behavior is investigated on a broad range of problem classes and significant differences between instances of a class are found. Allowing even 0.2% more material input than the minimum significantly improves the results, this tradeoff has not been investigated in the earlier literature. Comparison to a state-of-the art heuristic KOMBI shows mostly better results; to a previous exact approach of Vanderbeck, slightly worse solutions and much worse LP bound, which is a consequence of the simplicity of the model.