# Approaches to real world two-dimensional cutting problems

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<ul><li><p>Applications</p><p>Approaches to real world two-dimensional cutting problems</p><p>Enrico Malaguti a,n, Rosa Medina Durn b, Paolo Toth a</p><p>a DEI, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italyb Departamento de Ingeniera Industrial, Facultad de Ingeniera, Universidad de Concepcin, Chile</p><p>a r t i c l e i n f o</p><p>Article history:Received 9 October 2012Accepted 29 August 2013</p><p>Keywords:2-Dimensional cutting stockColumn generationHeuristics</p><p>a b s t r a c t</p><p>We consider a real world generalization of the 2-Dimensional Guillotine Cutting Stock Problem arising inthe wooden board cutting industry. A set of rectangular items has to be cut from rectangular stock boards,available in multiple formats. In addition to the classical objective of trim loss minimization, the problemalso asks for the maximization of the cutting equipment productivity, which can be obtained by cuttingidentical boards in parallel. We present several heuristic algorithms for the problem, explicitly consideringthe optimization of both objectives. The proposed methods, including fast heuristic algorithms, IntegerLinear Programming models and a truncated Branch and Price algorithm, have increasing complexity andrequire increasing computational effort. Extensive computational experiments on a set of realistic instancesfrom the industry show that high productivity of the cutting equipment can be obtained with a minimalincrease in the total area of used boards. The experiments also show that the proposed algorithms performextremely well when compared with four commercial software tools available for the solution of theproblem.</p><p>& 2013 Elsevier Ltd. All rights reserved.</p><p>1. Introduction</p><p>Given a set of rectangular items and innitely many identicalrectangular stock boards, the Two-Dimensional Cutting Stock Pro-blem (2DCSP) asks for cutting all the items by using the minimumnumber of boards or, equivalently, by minimizing the area of theused boards. In this paper we consider a generalization of theproblem, which models with adequate detail real world situationsarising in the wooden board cutting industry. The problem is knownto be NP-hard since it generalizes the Bin Packing Problem [18].</p><p>In wooden board cutting, a set of rectangular items has to be cutfrom rectangular stock boards, which are usually available inmultiple formats. Some items may be rotated, while others mayhave a compulsory orientation, which is determined by the woodgrain. The main objective, as in classical cutting stock problems, isthe stock usage (or equivalently, the trim loss) minimization,which is evaluated in terms of area (or cost) of used boards. Asecond objective is the maximization of machine productivity, acommon objective in many industrial settings. Machine produc-tivity is usually maximized by reducing the number of setups in aproduction process (see Allahverdi et al. [1]), as it is done, e.g.,when scheduling the sequence of operations on computer numer-ical control systems (see Zamiri Marvizadeh and Choobineh [24]),</p><p>or by dening optimal sequences of operations when the setuptimes are sequence-dependent (see, e.g., Pan and Ruiz [19]). Inwooden board cutting, instead, productivity can be improved byprocessing several items at the same time, as explained in thefollowing.</p><p>A few approaches are available for multi-objective optimizationproblems like the one we consider (see e.g., [16,4]), and the waythe two objectives are jointly optimized in practice depends on thespecic industrial setting. However, in the commercial softwaretools used in the wooden board cutting industry, it is usual tooptimize as an objective function a weighted combination of usedstock area and machine productivity.</p><p>Automatic wood cutting machines are equipped with a sawwhich can perform guillotine cuts (i.e., cuts that are parallel to thesides of the board and cross the board from one side to the other).A wood board is loaded on the machine, which performs a rststage of cuts, thus obtaining a set of strips. The rst stage of cutscan be either parallel to the horizontal side of the board (horizontaldirection), or parallel to the vertical side (vertical direction). Afterthe rst stage, the strips can be rotated inside the machine, and asecond stage of cuts is performed so as to obtain rectangular items.Generally, the items can be rotated again and a third (or further)stage of cuts can be performed, so as to obtain smaller items or toremove a waste part from an item. In Fig. 1, where we report anexample of guillotine cuts, the rst stage cuts (horizontal) arerepresented with a thick line, the second stage cuts (vertical) witha thin line, and the third stage cuts (horizontal) with a dotted line,while the waste area is represented by a dashed surface. However,</p><p>Contents lists available at ScienceDirect</p><p>journal homepage: www.elsevier.com/locate/omega</p><p>Omega</p><p>0305-0483/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.omega.2013.08.007</p><p>n Corresponding author. Tel.: 39 0512093151.E-mail addresses: enrico.malaguti@unibo.it (E. Malaguti),</p><p>rosmedina@udec.cl (R. Medina Durn), paolo.toth@unibo.it (P. Toth).</p><p>Please cite this article as: Malaguti E, et al. Approaches to real world two-dimensional cutting problems. Omega (2013), http://dx.doi.org/10.1016/j.omega.2013.08.007i</p><p>Omega () </p></li><li><p>while the rst and second stages can be automatically managed bythe machine, the third (and higher) stage cuts usually require themanual intervention of an operator. Thus, if on one hand cuts ofthird (and higher) stage increase the opportunities of the stockusage minimization, on the other hand their use reduces themachine productivity. We only consider cuts up to the third stage.</p><p>A peculiarity of wood cutting consists of the possibility ofstacking in a pile several identical boards, say h, which can be cutwith a single cutting cycle, thus obtaining h copies of each itemcontained in a single board. This possibility leads to a dramaticproductivity increase. Clearly, the geometry of the cuts to beperformed, i.e., the so-called cutting pattern, must be the samefor all the stacked boards, and thus the parallel cut of stackedboards limits the opportunities of stock usage minimization. Givena pattern to be repeated r times and a maximum number ofboards for a board pile, we have to perform c dr= cuttingcycles.</p><p>As anticipated, we are considering a multi-objective optimizationproblem, for which it is not possible to dene a unique approach,valid for any industrial setting. The two objectives of stock usageminimization and machine productivity are partially in contrast, andin theory it would be interesting, although impractical, to enumerateall the Pareto-efcient solutions to the problem. In practice commer-cial software tools produce one solution, and have some exibility ongiving more importance to trim loss minimization or productivity.Thus, in this paper we use a scalarization technique, and evaluate asolution to the cutting problem according to a generic weightedobjective function, including three different terms: the area of usedboards, the number of cutting cycles and the number of third stagecuts, where the last two terms are strongly correlated with machineproductivity.</p><p>Other approaches for multi-objective optimization problemsinclude the -constraints method, where one objective is mini-mized, and the remaining objectives are constrained to be lessthan or equal to given target values; the goal programming, whichcan be seen as the search for a solution satisfying specic goalvalues of the objectives, or the solution closest to the speciedgoals if none exists satisfying the goal values; the lexicographicmethod, where the objective functions are arranged in order ofimportance, then the minimizers of the rst objective function arefound; secondly, the minimizers of the second objective functionare searched for, and so on, until all the objective functions havebeen optimized on successively smaller sets. For a deeper analysisof multi-objective optimization we refer the reader to the survey[16] and the book [4] and the references therein.</p><p>Although commercial software tools are designed to implicitlytake into account the minimization of cutting cycles and thirdstage cuts, quite often these two objectives, and their relativeimportance with respect to the minimization of the area of usedboards, are not stated explicitly, and are obtained as a side-productof the optimization. In our research, we show instead thatexplicitly considering cutting cycles and third stage cuts allowone to obtain solutions which are much more efcient withrespect to these two dimensions, at a very small cost in terms ofthe area of used boards.</p><p>More formally, we consider a real world 2DCSP with thefollowing additional features:</p><p> the items must be obtained from stock boards through at mostthree-staged guillotine cuts;</p><p> some items have a xed orientation with respect to the boardsides, while other items may be orthogonally rotated;</p><p> board rotation is allowed (i.e., the rst stage can be eitherhorizontal or vertical with respect to the board side);</p><p> boards of multiple formats are available; up to boards can be stacked in a pile and cut with a single</p><p>cutting cycle; the area of used stock A, the number of cutting cycles C, and the</p><p>number of third stage cuts Z have to be globally minimized,according to the weighted objective function:</p><p>min w1Aw2Cw3Z 1where w1, w2 and w3 are given non-negative weights.</p><p>Hence, the input data dening an instance of the problem are: alist of m rectangular item classes, each item class i with dimensionsli;wi, a demand di and an attribute specifying if the item can beorthogonally rotated, and a list of b rectangular board classes, eachboard class k with dimensions Lk;Wk and available in an innitenumber of copies. In the following, li and Lk will be denoted as lengthof an item class i and of a board class k, respectively; wi and Wk willbe denoted as width of an item class i and a board class k,respectively. The maximum number of stack boards in a pile andthe weights w1, w2 and w3 of the objective function conclude theinput data. For each item class i, the number of produced items canexceed the corresponding demand di (the overproduced items areconsidered as waste). Following the typology introduced by Wscheret al. [23] and by disregarding the component of the problem relatedto productivity, the problem can be classied as a Two-DimensionalMultiple Stock Size Cutting Stock Problem. Other real world general-izations of the 2DCSP, characterized by dimension, planning situa-tion, goal, restrictions and solution approach, are described byDyckhoff et al. [6].</p><p>According to the literature on cutting stock problems, the realworld problem we consider combines a generalization of the2DCSP with a variant of the Pattern Minimization Problem, whichwas mainly addressed in the one-dimensional case (see, e.g.,[13,22,21,3]). Note that pattern minimization goes in the directionof productivity but it is not precisely the same as minimization ofthe cutting cycles and of the third stage cuts.</p><p>The 2DCSP was introduced by Gilmore and Gomory [1012],who also considered the k-staged version of the problem and alsoconsidered boards of different dimensions. Riehme et al. [20]considered the two-staged version of the problem in the casewhere boards of different dimensions are available and the itemdemands differ in a large range. Cintra et al. [5] considered several2DCSPs with guillotine cuts and their variants in which orthogonalrotations are allowed and boards of different dimensions areavailable. They presented dynamic programming algorithms forthe Two-Dimensional Knapsack Problem (2DKP), which are thenused to generate patterns in approaches for the 2DCSP based oncolumn generation. Given a set of rectangular items with asso-ciated positive prots and one rectangular bin, the 2DKP is to cutthe subset of items of maximum prot which can t into the bin.Recently, Furini et al. [9] proposed a column-generation basedheuristic for the 2DCSP, where orthogonal rotations are allowedand boards of different dimensions are available, which improveson previously known results from the literature. Exact models forthe same problem (without rotation) were proposed and compu-tationally compared by Furini and Malaguti [8]. The problemaddressed in [9,8] does not consider the possibility of stacking</p><p>Fig. 1. Guillotine cuts.</p><p>E. Malaguti et al. / Omega () 2</p><p>Please cite this article as: Malaguti E, et al. Approaches to real world two-dimensional cutting problems. Omega (2013), http://dx.doi.org/10.1016/j.omega.2013.08.007i</p></li><li><p>several boards in a pile and of third-stage cuts, and thus does notconsider the optimization of cycles. In addition, the problemaddressed in [9,8] only considers two-stage cuts plus trimming,i.e., third stage cuts are only allowed for removing a waste area.A survey on several Two-Dimensional Packing Problems can befound in Lodi et al. [15].</p><p>The literature on the One-Dimensional Cutting Stock Problemwith Pattern Minimization considers the case of identical boards.An early heuristic was proposed by Haessler [13], where the cost ofusing a pattern is added as an additional cost term to the objectivefunction, and new patterns are iteratively added to the currentsolution by taking into account their trim loss and the possibilityof multiple usage. A metaheuristic approach to the problem wasproposed by Umetani et al. [21], where the number of differentcutting patterns is an input parameter and different solutions aregenerated by binary search. Belov and Scheithauer [3] proposed asequential heuristic for the combined minimization of the trimloss and of the number of different cutting patterns. Concerningexact approaches, Vanderbeck [22] presented an ILP (IntegerLinear Programming) formulation and a Branch-and-Cut-and-Price algorithm for minimizing the number of different patterns,once the maximum number of used boards is xed as a constraint.</p><p>The main contribution of this paper is to propose severalalgorithms for a real world 2DCSP explicitly considering theoptimization of the three objectives mentioned in Eq. (1). Sincethe problem considers the machine productivity maximization(PM) in a 2DCSP setting, we denote it as PM-2DCSP. The proposedalgorithms have increasing complexity and require increasingcomputational effort. Extensive computational experiments showthat high productivity of the cutting equipment can be obtainedwith a minimal increase in the total area of used boards.</p><p>More in detail, in Section 2 we describe three fast heuristicalgorithms for the PM-2DCSP, and a procedure for reducing thenumber of cutting cycles. The aim of these algorithms is to obtaingood quality solutions in short computing time. In Section 3 wepresent a straightforward integration of the three algorithmsthrough the solution of an optimization model which is able toproduce improved solutions. In Section 4 the best performingalgorithm among the previously described ones is embedded intoan overall metho...</p></li></ul>

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