An exact algorithm for two-dimensional cutting problems based on multi-level pattern

A multi-level pattern is proposed for the unconstrained two-dimensional cutting problems of rectangular items, and an exact generation algorithm is constructed. The arrangement of rectangular items with the same type in multiple rows and columns is referred to as a 0-level pattern. An n-level pattern is the horizontal or vertical combination of an n-1 level pattern with a pattern whose level will not exceed n-1. The generation algorithm of multi-level pattern is constructed on the base of dynamic programming, and the multi-level patterns with various sizes are generated with increased level order. The normal size is chosen to reduce unnecessary computation in the algorithm. Three sets of benchmark instances and one set of random production instance from the literatures are used for comparison. Comparing to the exact algorithm in the literature, the results in this paper are equivalent, but the computation time is shorter. Comparing to heuristic algorithms in literatures, the results in this paper are better and the computation time is still good enough for practical applications.