Column generation and sequential heuristic procedure for solving an irregular shape cutting stock problem

Research addressing two-dimensional irregular shape packing has largely focused on the strip packing variant of the problem. However, it can be argued that this is a simplification. Materials from which pieces are required to be cut will ultimately have a fixed length either due to the physical dimensions of the material or through constraints on the cutting machinery. Hence, in order to cut all the pieces, multiple sheets may be required. From this scenario arises the two-dimensional irregular shape cutting stock problem. In this paper, we will present implementations of cutting stock approaches adapted to handle irregular shapes, including two approaches based on column generation and a sequential heuristic procedure. In many applications, set up costs can be reduced if the same pattern layout is cut from multiple sheets; hence there is a trade-off between material waste and number of patterns. Therefore, we describe the formulation and implementation of an adaptation of the column generation method to control the number of different patterns. Column generation is a common method for the cutting stock problem, however when the pieces are irregular the sub problem cannot be solved optimally. Hence we implement column generation and solve the sub problem using the beam search heuristic. Further, we introduce a version of column generation for instances where the number of rows is less than the number of columns.