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

Abstract

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.

Jostle heuristics for the 2D-irregular shapes bin packing problems with free rotation

“…For the latter, the position of the pieces in the allocated bin is determined by the Bottom-Left (BL) algorithm developed by Jakobs (1996) or an improved bottom left (BLLT) version proposed by Liu and Teng (1999) or a variant of the constructive approach presented by Hifi and M'Hallah (2003). Song and Bennell (2014) solve the irregular shape cutting stock problem.…”

A beam search approach to solve the convex irregular bin packing problem with guillotine cuts

“…In both cases, they only deal with the packing of one bin, and the beam search will find the best sequencing of pieces based on a placement procedure. To the best of our knowledge, within the field of cutting and packing problems, only Song and Bennell (2014) use beam search when there are multiple bins. They tackle a 2D cutting stock problem.…”

Matheuristics for the irregular bin packing problem with free rotations

“…Another study that considers multiple bins and irregular pieces is that of Song and Bennell (2013), who propose a column generation procedure for solving the irregular shape cutting stock problem. The authors use the beam search algorithm proposed in Bennell and Song (2010) for generating the patterns and study three solution approaches: column generation, adapted column generation and a sequential heuristic procedure.…”