An exact algorithm for constrained two-dimensional two-staged cutting problems

The constrained two-dimensional cutting (C_TDC) problem consists of determining a cutting pattern of a set of n small rectangular piece types on a rectangular stock plate S with length L and width W, to maximize the sum of the profits of the pieces to be cut. Each piece type i, i = 1, . . . , n, is characterized by a length l_i, a width w_i, a profit (or weight) c_i, and an upper demand value b_i. The upper demand value is the maximum number of pieces of type i that can be cut on S. In this paper, we study the two-staged C_TDC problem, noted C_2TDC. It is a classical variant of the C_TDC where each piece is produced, in the final cutting pattern, by at most two cuts. We solve the C_2TDC problem using an exact algorithm that is mainly based on a bottom-up strategy. We introduce new lower and upper bounds and propose new strategies that eliminate several duplicate patterns. We evaluate the performance of the proposed exact algorithm on problem instances extracted from the literature and compare it to the performance of an existing exact algorithm.