A Value-Correction Construction Heuristic for the Two-Dimensional Cutting Stock Problem with Variable Sheet Size

Cutting and packing problems occur in many real-world applications such as industrial glass, paper or steel cutting, container loading and VLSI design [1]. In this work we consider in particular the two-dimensional cutting stock problem with variable sheet size (2CSV) in which we are given a set of nE rectangular element types E = {1, . . . , nE}, each i ∈ E specified by a height hi ∈ N, a width wi ∈ N and a demand di ∈ N. Furthermore, we have a set of nT stock sheet types T = {1, . . . , nT }, each t ∈ T specified by a height Ht ∈ N, a width Wt ∈ N, an available quantity qt ∈ N and a cost factor ct ∈ N. Both elements and sheets can be rotated by 90◦. The objective is to find a set of cutting patterns P = {P1, . . . , Pn}, i.e. an arrangement of the elements specified by E on the available stock sheets specified by T without overlap and using only up to a given number K of stages of guillotine cuts, s.t. the sum over the cost factors of all used sheets is minimal. A classical solution approach to the 2CSV has been proposed by Gilmore and Gomory [2] who employ column generation solving the pricing problem by dynamic programming (DP). Column generation as well as DP are still important components in many advanced algorithms for the 2CSV, see e.g. Cintra et al. [3]. More recently, several approaches using DP as their main framework have been proposed, however none of them considered sheets of variable size. In general, DP is not able to efficiently compute proven optimal patterns when demand constraints must be respected, and hence all these approaches are of heuristic nature. For example, Morabito and Pureza [4] compute a pattern for a single sheet by iteratively running a DP algorithm where after each iteration weights for the element types are updated. Similarly, Cui et al. [5] apply a valuecorrection heuristic to solve the cutting stock problem for a single sheet type and without imposing a stage limit.