![]() Specifically for the two-stage problem, Gilmore and Gomory (1965) also proposed an integer programming model with two sets of decision variables: a set of decision variables related with the definition of the height of the stripes and the other set related with the items included in each stripe. The subproblem of generating attractive patterns is also solved by column generation. Vanderbeck (2001) also used a column generation based approach for a three-stage 2DCS problem with additional constraints related to the cutting process. (2008), the subproblem of the column generation algorithm is solved by dynamic programming. The subproblem generates attractive patterns, which, for the two-stage problem, are obtained by solving a sequence of two integer knapsack problems. As in the one-dimensional case, in the restricted master problem, variables are associated with cutting patterns. Previous exact approaches have been proposed for 2DCS problems.įollowing their seminal work on the one-dimensional cutting stock problem, Gilmore and Gomory (1965) proposed a column generation algorithm for the two-stage 2DCS problem. 2 for an example of a non-exact three-stage cutting pattern. In both cases, the restricted version of the three-stage problem is considered, in which, in each stripe, there is always a stack with only one item defining its height. In the former, all the items in a stack must have the same width. In the latter, an additional set of vertical cuts is allowed for separating the items from the waste. As in the two-stage problem, the exact and non-exact cases are considered. In this problem, a third set of cuts is allowed: after the plate is cut in stripes (in the first stage), each stripe is then cut in stacks (second stage) and finally the items in each stack are separated by a third set of (horizontal) cuts (third stage). The three-stage problem is also considered in this paper. 1 shows a non-exact two-stage cutting pattern, where the white rectangles correspond to waste and the shaded rectangles to items. If an additional set of horizontal cuts is allowed in order to separate the waste from the items, then the problem is a non-exact two-stage problem (in opposition to the exact case where all the items in a given stripe have the same height). In a two-stage problem, the items are obtained by a set of horizontal cuts, dividing the plate in stripes, followed by a set of vertical cuts, separating the different items. Furthermore, the number of stages (set of cuts with the same orientation – horizontal or vertical) in which a plate can be cut is also frequently limited to two or three. In this industry, due to technological constraints, usually only cuts parallel to the sides of the plate (orthogonal cuts) and from one border to the opposite one (guillotine cuts) are allowed. The motivation behind this work lies in the woodcut industry where plates of wood must be cut in pieces (items) to satisfy customer orders. ![]() In Section 3, the extension to deal with multiple stock sizes (MSSCSP) is also described. (2007) the problems addressed in this paper are two-dimensional rectangular Single Stock Size Cutting Stock Problems (SSSCSP) with additional constraints related with the types of cuts allowed. Each item type is defined by a width, a height and a demand (corresponding to the number of items of the type to be cut).Īccording to the typology of Wäscher et al. ![]() ![]() ![]() A set of items to be cut, grouped by types according to their dimensions (width and height), is given. The plates are available in (a virtually) infinite number and all have the same dimensions, i.e., the same width and height. In a 2DCS problem it is intended to cut a set of rectangular items from a set of rectangular plates in such a way that the number of used plates is minimized. The approach is based on the definition of an integer programming model, with a pseudo-polynomial number of variables and constraints, to be optimized directly by a general integer programming solver. In this paper, an exact approach to two-dimensional cutting stock (2DCS) problems is proposed. ![]()
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