Hyperheuristic Codification for theMulti-Objective 2D Guillotine Strip Packing Problem

Gara Miranda, Jesica de Armas, Carlos Segura, and Coromoto Leon

Abstract— Most research on Strip Packing Problems is focusedon the single-objective formulation of the problem. However, inthis work we deal with a more general and practical variant ofthe problem, which not only seeks to optimize the usage of theraw material, but also the production process. For the problemsolution, we have applied some of the most-known multi-objectiveevolutionary algorithms, since they have shown a promisingbehavior when affording multi-objective real-world problems.For an initial implementation, we proposed a solution codificationwhich is based on a complete representation of the patternlayouts. Such an approach was promising but wasn’t suitableto afford large instances. For this reason, we have focused onthe design of a codification which can be much more competitivewhen compared to some tailor-made methods. In this sense, wepresent a hyperheuristic-based codification as an alternative tocombine heuristics in such a way that a heuristic’s strengthsmake up for the drawbacks of another. Results demonstrate theadvantage of using multi-objective approaches, hyperheuristic-based representations, and of course, the importance on theselection of appropriate solution codifications.

I. INTRODUCTION

Strip Packing Problems (SPPs) arise in many productionindustries where the raw materials are in the form of rolls(paper, steel, textiles, etc.). For all practical purposes, therolls may be considered to be of infinite length, so in theseproblems, the stock sheet is often known as a “strip”. Inparticular, this work is focused on the 2D Guillotine StripPacking Problem (2DSPP) which is one of the most interestingvariants of the SPP. The 2DSPP involves the cutting of aset of n demanded pieces from a large roll or stock sheetof raw material. The stock sheet has fixed width W andunlimited length L. Each rectangular piece i demanded hasfixed dimensions (li, wi), although they can be rotated 90degrees, thus allowing for the dimensions (wi, li) as well.All demanded pieces must be orthogonally arranged on thematerial in such a way that they do not overlap and that onlyvertical or horizontal builds of pieces are generated [1]. Thislast constraint always gives the possibility of producing thepacking pattern through guillotine cuttings, i.e. the pieces haveto be cut with their edges parallel to the edges of the stocksheet, going from one border straight to the opposite side [1].The production of non-guillotinable cuts may entail a morecomplex machinery operation. For this reason, and in order toencompass a wider range of industrial cutting machines, wehave focused on the guillotinable formulation of the problem.

Dpto. Estadıstica, I. O. y Computacion, Universidad de La Laguna, Avda.Astrofısico Fco. Sanchez s/n, 38271 La Laguna, Santa Cruz de Tenerife,Spain (phone: +34 922 318180; email: gmiranda@ull.es, jdearmas@ull.es,csegura@ull.es, cleon@ull.es).

Following related classifications in the literature [2], theproblem posed here belongs to a subclass of packing andcutting problems denoted as Open Dimension Problems. Inthese problems, one of the dimensions of the given largeobjects is considered as a variable, such that the probleminvolves a decision on fixing the extension of this dimension;i.e. the goal is to minimize the strip length required to cut thewhole set of demanded pieces. However, in some industrialfields, the raw material is either very cheap or can be easilyrecycled, so in such cases, a more important criterion forthe pattern generation may be the speed at which the piecescan be obtained, thus minimizing the production times andmaximizing the usage of the cutting equipment. The cuttingspeed is specifically limited by the features of the machineryavailable but, in general, it is determined by the number of cutsinvolved in the packing pattern. Moreover, the number of cutsrequired for the cutting process is also crucial to the life of theindustrial machines. Since the number of cuts is an importantaspect in determining the cost and efficiency of the productionprocess, a comprehensive optimization methodology shouldtake also this criterion into consideration. Therefore, in thisstudy, the number of cuts is taken as a second design objective.This way, the problem can be posed as a multi-objectiveoptimization problem for optimizing the layout of rectangularparts so as to minimize the required sheet length (trim loss)as well as the number of cuts to achieve the final demandedpieces. This formulation of the problem is denoted as Multi-Objective Strip Packing Problem (MOSPP).

The works dealing with such a multi-objective problem arealmost inexistent [3], [4], [5] and they all are based on theusage of Multi-Objective Evolutionary Algorithms (MOEAs).Moreover, the approaches tackling guillotine problems arebased on representations where the pattern layouts are ex-plicitly encoded [3], [5]. In this work, we propose a new non-direct codification which involves the usage of widely knownlow-level heuristics and the principles of hyperheuristics. Theremaining content of this paper is organized as follows.Section II gives a general state-of-the-art on the solutionof the widely studied single-objective formulation of theproblem. The number of works dealing with multi-objectiveformulations of the problem are almost inexistent as shownin section III. In section IV, we present the new proposal -based on hyperheuristics concepts - which tries to improveexisting approximations. The experimental comparison of thenew proposal and the already existing one is presented insection V. Finally, the conclusions and some lines of futurework are given in section VI.

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II. SINGLE-OBJECTIVE APPROACHES

As occurs with most cutting and packing problems, theproblem of finding an optimal solution for the 2DSPP isNP-hard, so research has focused mainly on developing ap-proximation algorithms. Approximation algorithms find nearoptimal solutions but do not guarantee finding the optimalpacking for every data set. Exact algorithms, on the contrary,ensure the achievement of optimal solutions but cannot dealwith large and real instances of the problem. This is why awide variety of heuristic strategies have been formulated, soas to obtain good quality solutions (although not necessarilythe optimal ones) in an acceptable computational time.

The existence of different choices on how to sort the list ofrectangles, how to position them inside a level, etc., yields ahigh number of possible heuristic algorithms for the 2DSPP.Many of these alternatives have been studied, though dueto the extensive number of published papers, it is difficultto ascertain the level of current research in this area [6],[7]. Some of the best-known level-oriented algorithms whichare able to deal with guillotine cuttings are [8]: First-FitDecreasing Height algorithm [9], Next-Fit Decreasing Heightalgorithm [9], Best-Fit Decreasing Height algorithm [7], andSPLIT algorithm [7]. Many papers propose improvements tothese level-based placement heuristics, or combine them withother approaches, such as genetic algorithms [10].

In order to overcome the main disadvantage of local searchalgorithms, whose weakness lies in the inability to escapefrom local minima, more sophisticated heuristic search strate-gies have been designed to avoid such a situation. This impliesthe temporary acceptance of a lower quality state. Hencemetaheuristic algorithms can be considered to a certain extentas local search strategies, which include a means to escapefrom local minima. In this sense, as a more general andsophisticated method, different types of hybrid algorithms andmetaheuristics have been considered [10], [11], [12], [13].These algorithms are guided through the problem search spaceby previous attempts and involve a set of parameters thatmust be tuned in order to improve the performance achieved.In general, when designing metaheuristic approaches for thesolution of a given problem, they are many design decisionswhich are decisive for the impact of the solution method. Inthis sense, one of the main decisions concerns the represen-tation of an individual solution in the population. Usually,metaheuristics for the 2DSPP generate a number of differentsolutions or input sequences that are usually interpreted byplacement heuristics. According to [14], different groups ofmetaheuristic solution approaches for the 2DSPP can bespecified, depending on the type of codification chosen forthe solutions:• Most metaheuristic proposals in the literature use a cod-

ing of solutions [6], [7]. Typically, an encoded solutionstipulates a placement sequence for the pieces. Themetaheuristic search is carried out within the encodedsolution space and usually involves problem-independentoperators. A placement or decoding routine transformsencoded solutions into complete layouts.

• An intermediate alternative combines the solution encod-ing with the usage of decoding heuristics [15]. Whileencoded solutions already contain, to a certain extent,layout or geometrical information, an additional place-ment routine is also required for the final positioning. Thecoding of solutions is usually problem-specific, whichis often based on graphs, and the use of correspondingproblem-specific operators.

• The last type of algorithms do not use coding. Thesearch is carried out directly in the space of fully definedlayouts, which are therefore manipulated as such byspecific operators [10], [16].

III. MULTI-OBJECTIVE APPROACHES

Multi-objective or multi-criteria optimization problems(MOPs) [17], [18] arise in most real-world disciplines. Themultiple objectives are usually conflicting or competing, butmust be simultaneously optimized. In practice, it is oftenimpossible to find a single optimum that dominates all othersolutions. Therefore, a solution optimizing every single objec-tive might not exist. Instead there is rather a set of alternativetrade-offs, known as a Pareto-optimal. These solutions areoptimal in the sense that no other solutions in the search spaceare superior to them when all objectives are considered.

In order to simplify the MOPs solution, the original problemcan be converted into a single-objective optimization problemby weighting the different objectives or by translating some ofthe objectives into constraints. This combination of multipleobjectives into one optimization criterion has the advantagethat the classical single-objective optimization strategies canbe applied without any further modifications. However, theapplication of such approaches requires a prior and in-depthknowledge of the problem that is not usually available.Therefore, a MOP solution calls for efficient and alterna-tive approaches. Usually, the most appropriate approximationinvolves the application of techniques that can specificallyhandle multiple objectives and MOPs intrinsic complexity(very large search spaces, uncertainty, noise, disjoint Paretocurves, etc.). As with single-objective optimization, large andcomplex search spaces can make the search difficult andpreclude the use of exact optimization methods [19]. For thisreason, a wide variety of approximate algorithms have beendesigned. The ideal approach must be capable of samplingintractably large and highly complex search spaces, and alsoof generating the exact Pareto-optimal set or approximationsof it. From the resulting final solution set, a human decisionmaker will be able to select a suitable compromise solution.

EAs have shown great promise for calculating solutionsto large and difficult optimization problems and have beensuccessfully used over a wide variety of real-world appli-cations [20]. In fact, when applied to MOPs, EAs seem toperform better than other blind search strategies [21], [22].Although this statement must be qualified with regard to theno free lunch theorems for optimization [23], to date there arefew, if any, alternatives to EA-based multi-objective optimiza-tion [24]. The numerous applications of MOPs have led to agrowing interest in the area of multi-objective optimization,

Fig. 1. Layout on the mother sheet for the gene ‘1 3 H 2 V 010’

whose most novel and efficient algorithms are primarily basedon evolutionary approaches [25]. The use of EAs to solveproblems of this special nature has been driven mainly bythe fact that they are able to capture multiple Pareto-optimalsolutions in a single simulation run - which is possible thanksto their population-based feature - and to exploit similarities ofsolutions by recombination. EAs that are specifically designedto deal with multiple objective functions are known as Multi-Objective Evolutionary Algorithms (MOEAs) [26].

Many proposals appear in the literature considering thesingle-objective formulation of the SPP, but only a fewdeal with real-world multi-objective constraints [3], [4], [27].In [27] several secondary optimization criteria are analyzed.The multi-objective formulation of the 2DSPP involving theminimization of the strip length and the number of cuts istackled in [3], [4], [5]. These existing approaches for thismulti-objective formulation of the 2DSPP are all based onthe application of MOEAs. The cutting patterns provided bythe approach presented in [3], [5] always allow for guillo-tinable cuts, while those achieved by [4] can only be non-guillotinable. Works dealing with guillotine cuttings [3], [5]are based on a codification of solutions where the cutting lay-outs are directly represented through a post-fix notation. Theapproach we proposed in [5] clearly improved the previousproposal presented in [3]. In such a initial approach [5], wedidn’t use coding of solutions; i.e., the representation useddefines a fully defined guillotine layout (see Figure 1, lastbits define the rotation of pieces). This way, the search iscarried out directly in the space of all the possible layoutsand the variation operators must be specifically designed tomanipulate such layout representations. Although the resultsobtained were competitive, not only compared to the otherexisting multi-objective approach [3], but also when comparedto some single-objective algorithms [7], we realized thatthe search space defined by the chosen codification becameintractable for large problem instances. For this reason, in thefollowing section we propose a new codification which triesto reduce the search space in order to be more competitive forlarge problem instances. Such a codification introduces codingof solutions, incorporating low-level placement heuristics todecode the solutions.

IV. HYPERHEURISTIC-BASED CODIFICATION

As shown in section II, in the literature we can find awide range of placement heuristics which allow to derivehigh quality solutions in acceptable computational times. Theexistence of such a wide variety of heuristics and types ofproblem instances makes difficult the choice of the mostsuitable algorithm. In this sense, hyperheuristics appear as

Fig. 2. Heuristics

Fig. 3. Representation based on hyperheuristics

a more general procedure for optimization which deals withthe process to choose/combine the right heuristics for solvingthe problem at hand. A hyperheuristic can be viewed as aheuristic that iteratively chooses between a set of given low-level (meta)-heuristics in order to solve an optimization pro-blem [28]. The motivation behind the approach is that, ideally,once a hyperheuristic algorithm has been developed, severalproblem domains and instances could be tackled by onlyreplacing the low-level (meta)-heuristics. In fact, the search ison a (meta)-heuristic search space rather than a search spaceof potential problem solutions. The hyperheuristic solves theproblem indirectly by recommending which solution methodto apply at which stage of the solution process. Generally,the goal of raising the level of generality is achieved at theexpense of reduced - but still acceptable - solution qualitywhen compared to tailor-made (meta)-heuristic approaches.

For the 2DSPP, there exists some works applying theprinciples of hyperheuristics in order to determine how tocombine single low-level heuristics [4], [29]. In this work, wepropose a non-direct codification which determines the typeof low-level placement heuristic to apply and the number ofpieces that each heuristic must arrange.

A. Low-level heuristics

The first decision to implement the hyperheuristic-based ap-proach lies on the definition of the set of placement heuristicsto be combined. In the literature we can find a huge amountof heuristics for non-guillotine 2DSPPs, but the number ofproposals for the guillotine case is not so extensive. From theexisting heuristics we have selected four of the most-known.They all fix the orientation of the objects such that their widthis not lower than their height. Then, objects are ordered fromhighest to lowest length. Once the order of the pieces to bepacked is established, the heuristics must decide where toarrange a given object at a given moment, considering the openand still unfilled levels (see Figure 2). The selected heuristics

Fig. 4. Two-point crossover

are briefly described next:• Next Fit Decreasing Height (NFDH) [9]: rectangles are

packed left justified on a level until the next rectanglewill not fit, in which case it is used to start a new levelabove the previous one, on which packing proceeds.

• First Fit Decreasing Height (FFDH) [9]: places eachrectangle left justified on the first (i.e., lowest) level inwhich it will fit. If none of the current levels has room,then a new level is started.

• Best Fit Decreasing Height (BFDH) [7]: packs the nextrectangle left justified on that level, among those that canaccommodate it, for which the residual horizontal spaceis a minimum. If no level can accommodate it, a newone is created.

• Best Fit Decreasing Height* (BFDH*) [10]: seeks toimprove BFDH heuristic by allowing object rotations, sothat when the algorithm searches to include the currentobject into a sub-area it tests both orientations.

B. Representation

In this case, an individual is represented by a sequenceof pairs (PH, p), where PH is the identifier of the low-levelplacement heuristic to be applied and p is the number of piecesthat such a heuristic must arrange on the available strip (seeFigure 3). Chromosomes have a variable length j, which goesfrom j = 1 (there is one single pair (PH,n) so that the sameheuristic arranges all the available pieces) to j = n (the navailable objects are arranged on an independent way, i.e.,∀i ∈ [1, n], pi = 1). Note that in a valid representation allthe demanded objects must be arranged by the heuristics, i.e.,(∑j

i=1 pi) = n. For the generation of the initial individuals,pairs (PH, p) are generated until there are no more pendingpieces to be arranged. For each pair, PH is randomly selectedfrom the set of four used placement heuristics and p israndomly selected from the interval [1, ap], where ap is thenumber of remaining or still available pieces.

C. Evaluation of Objectives

In this non-direct codification, MOEAs evolve populationsof individuals which represent hyperheuristics, so that, thechromosome must be interpreted to obtain the problem solu-tion. The representation is analyzed from left to right, applying

for each pair (PH, p) the heuristic PH in order to locatethe next p pieces. When pieces are located on the top ofexisting levels, verticals constructions are created. When anew level is opened at the right of the existing ones, ahorizontal combination of the adjacent levels is generated.The decodification of the gen provides a problem solutionrepresented by a post-fix notation of vertical and horizontalcuts. Then, such layout of the pieces over the available rawmaterial can be evaluated - as shown in [5] - to obtain thenecessary length and number of cuts.

D. Operators

Several mutation and crossover operators were designedand tested with this representation. Some of them weremore general and others more specific to deal with certainconstraints of the representation. Here, the operators whichshowed a better behavior are described.

The selected mutation operator applies three different typeof movements inside the gen. Each of the following move-ments is applied upon the algorithm mutation probability pm:• add: randomly selects a pair (PHi, pi) inside the repre-

sentation. Then it generates a new pair (PH, p) wherePH is a random selected heuristic and p is a randomnumber within the interval [1, pi]. Then, pi is updatedwith the value pi−p. If after the update, pi 6= 0, pairs inpositions i, . . . , j are displaced one position to the right,increasing the total length of the individual (j = j + 1).Finally, the new pair is introduced in position i.

• remove: randomly selects a pair (PHi, pi) inside therepresentation. If the selected pair is the last one inthe representation (i = j), then pi−1 = pi−1 + pi. Inother case, pi+1 = pi+1 + pi and the pairs in positionsi+1, . . . , j are displaced one position to the left. In bothcases, the length is updated (j = j − 1). This operationcan be applied only if initially j > 1.

• replace: randomly selects a pair (PHi, pi) inside therepresentation. PHi is randomly fixed to one of thedefined low-level heuristics.

For the crossover, we have selected a two-point crossoverwhich considers the number of accumulated pieces withinthe representation. Two number of pieces np1 and np2 arerandomly generated, such that 1 ≤ np1 < np2 < n. For each

TABLE ICONFIGURATION FOR DIRECT AND HYPERHEURISTIC CODIFICATIONS

Codification Algorithm Crossover Mutation PopulationDirect NSGA-II 0.7 0.3 50

Hyperheuristic NSGA-II 0.6 0.4 50

parent, it is necessary to find the gene positions where thenumber of arranged pieces sums np1 and np2. When the sumof the pieces do not coincide with the generated value np,the first pair (PHi, pi) satisfying

∑im=1 pm > np should be

split into two different pairs (PHj , pj) and (PHk, pk), suchthat PHj = PHk = PHi, pk = (

∑im=1 pm) − np, and

pj = pi − pk (see Figure 4). Once the two cross points aredetermined inside both parents, the central pairs between themare exchanged among the two parents in order to generate thenew offsprings.

V. EXPERIMENTAL EVALUATION

In order to avoid the implementation of the most widelyused MOEAs, we made usage of METCO [30], a ParallelPlugin-Based Framework for Multi-Objective Optimization.The framework provides implementations of MOEAs such asNSGA-II [31], SPEA2 [32], IBEA [33], etc. It allows theusers to simply specify the details related to the problem(representation, evaluation of objectives, operators, ...) withouthaving to worry about the internal details of the algorithmimplementation. The framework also provides a simple, fle-xible, and efficient interface to setup and tune the parametersbeing used inside the algorithms. In order to perform moreaccurate parametrizations, the tool is able to run on parallelenvironments. The experimental evaluation was performedon a dedicated Debian GNU/Linux cluster of 20 dual-corenodes. Each node consists of two Intel R© Xeon 2.66 Ghzand has 1Gb RAM and a Gigabit Ethernet interconnectionnetwork. The framework and both approaches for the problemwere implemented in C++ and compiled with gcc 4.1.3 andMPICH 1.2.7. For the computational study, some test instancesavailable in the literature [34] have been used.

Inside METCO we have defined two different individualsfor the solution of the 2DSPP: one direct codification basedon post-fix representation of pattern layouts and other witha hyperheuristic-based codification. For each approach, wedefined the corresponding representation and implementedthe evaluation, generation, and operator methods involved.Both approaches were evolved using three different MOEAs:NSGA-II, SPEA2, and an adaptive version of IBEA. More-over, every pair algorithm-codification was tested also usingdifferent parameter configurations. After performing an initialtuning, we selected the algorithms and parameters betterperforming for each of the codifications (see Table I). Notethat, as demonstrated in previous works [3], [5], NSGA-IIshowed a better behavior than the other algorithm alternatives.

The application of MOEAs that generate solutions to the2DSPP according to two different optimization criteria hasa major advantage for potential customers: such approachesprovide a set of solutions offering a range of trade-offsbetween the two objectives, from which clients can choose

TABLE IICOMPARISON OF MULTIPLE AND SINGLE OBJECTIVE APPROACHES

Problem NFDH FFDH SPLIT GA NSGA-IINice.25 133 118 138 108 106.12Nice.50 120 119 134 108 109.41

Nice.100 112 111 137 111 110.84Nice.200 110 108 138 - 113.33Nice.500 108 107 139 - 126.13Path.25 132 120 136 109 101.43Path.50 143 131 154 108 103.42

Path.100 120 109 137 112 108.39Path.200 128 116 138 123 120.69Path.500 112 105 141 - 147.16

according to their needs, e.g. cost associated with the rawmaterial or even times imposed for the production process.However, dealing with more than one optimization objectivedoes not necessarily imply a reduced solution quality atthe expense of possibly optimizing multiple objectives. Onthe contrary, in the problem studied, we initially checkedthe direct codification approach and we realized that, byconsidering both, the number of cuts and the length, theapproach derived solutions with wastage levels similar tomost previous approximations which just seek to optimizethe overall length. In fact, initial studies presented in [5]and here summarized in Table II demonstrated the validityof the 2DSPP multi-objective approach. The table shows, forsome 2DSPP test instances available in the literature [34],the length achieved by three different low-level heuristics anda genetic algorithm proposed in [7]. All these approachesdeal with the single-objective formulation of the 2DSPP. Thelast column of the table represents the solutions achieved bythe direct codification approach. The approach was executedusing the parameters presented in Table I. For small problemsthe stop criterion was fixed to 10000000 evaluations whilefor larger instances it was fixed to 25000000 evaluations.Execution times ranged from two minutes up to two hours forlarger problem instances. Thirty repetitions were performedfor each test problem. For each repetition, the best valuefor the “overall length” objective is considered. Then, thesevalues are used for the calculation of the average best value ofthe executions. For smaller test instances, the multi-objectiveapproach provides better solutions than those obtained by thesingle-objective strategies. Only for larger problem instancesis our multi-objective approach not able to provide betteror similar solutions than the ones achieved by the single-objective approaches. In general, we can state that when thesize of the instances increases, the specific single-objectiveapproximations tend to behave better, thanks to their simplicityand to their single optimization considerations. However, themulti-objective approach applies an evolutionary algorithm toevolve individuals which, in this case, where initially codedby a direct representation of the cutting patterns. Such directcodification allows to represent any of the possible solutions,thus obtaining an approach which is able to deal with the com-plete search space of solutions. The size of the codifications,2∗n−1+n, is proportional to the number of available piecesn. For smaller problems - with a reduced number of pieces- the search space of solutions is much more reduced and

TABLE IIISINGLE-OBJECTIVE LOW-LEVEL HEURISTICS AND MULTI-OBJECTIVE APPROACHES

Solution nice200 40 path200 40 nice500 5 path500 5Approach length cuts length cuts length cuts length cuts

NFDH 110.90 371 127.30 350 108.80 943 119.61 896FFDH 107.01 374 114.32 364 106.17 946 112.84 904BFDH 107.01 374 114.25 392 106.17 946 112.84 904BFDH* 106.96 373 114.10 354 106.08 946 111.17 905Direct 123.51 337.80 128.89 331.86 148.84 908.03 187.19 886.70

Codification 154.67 306.80 174.70 297.80 186.46 861.40 235.51 849.93Hyperheuristic 104.83 373.60 103.94 353.13 103.36 944.36 103.65 904.63Codification 107.92 368.93 107.68 334.00 104.76 938.06 106.06 883.13

Fig. 5. Attainment surfaces for direct and hyperheuristic codifications

so, the MOEA is able to perform an exhaustive explorationof the space, thus obtaining high quality solutions which inmany cases improve the ones obtained by the single-objectiveapproaches. For larger problems, where a high amount ofpieces must be arranged, the search space of solutions is toolarge, and so, the single-objective approaches which seek tooptimize the overall length obtain better results.

For this reason, the main aim of this work was focusedon the design of a codification which can deal with a morereduced search space, so that solutions with quality compa-rable to those achieved by the single-objective approachescan be obtained. Following such a goal, we have designeda hyperheuristic-based codification which combines differentexisting low-level heuristics in order to improve the solutionsgiven when the heuristic methods are individually applied. Ta-

ble III shows the values of both optimization objectives whenthe low-level heuristics are individually applied and when thetwo proposed multi-objective approaches are applied to solvefour large problem instances. For each codification, thirtyexecutions are performed, using the configuration parametersgiven in Table I. The stop criterion is fixed to 30 minutes.For each repetition, two solution points are selected: the onewith lower length and the one with lower number of cuts. Theaverage values for both objectives are shown for the case ofthe lowest-length solution (first row of the codification) andthe lowest-cuts solution (second row). Analyzing the resultsgiven in the table, we realize that the direct codification imple-mentation achieves cutting values considerably lower than theones obtained by the heuristics but it is not able to reach theirlength values. However, the designed hyperheuristic approach

is able to improve both objectives values when compared tothe single-objective low-level heuristics. In this sense, by theapplication of hyperheuristics principles and multi-objectiveapproaches, we have highly improved the solutions obtainedby tailor-made algorithms for the single-objective 2DSPP.

The new designed codification has been able to improvethe low-level heuristics results, but if we compare the twocodifications applied by the MOEA, we realize that noneof them is superior to the other when both objectives areconsidered. The direct codification allows for a considerablyreduction of the number of cuts objective, while the hy-perheuristic codification achieves the lowest values for thestock sheet length. The different codifications are analyzingdifferent regions of the solution space. However, we have justanalyzed the solutions obtaining the minimum lengths or cuts.Now, it is necessary to compare the complete set of solutionsobtained by the codifications. For comparing two differentmulti-objective approaches [35] we could use some multi-objective metric such as hypervolume [36] or ε-indicator [37].But in this case, we would like to clearly identify the searchspace areas being explored. Directly plotting Pareto frontscould be rather messy since we are dealing with the results ofthirty executions, so as an alternative we have used attainmentsurfaces [38]. An attainment surface is the family of tightestgoals that has been attained by the approximation set definingit. Importantly, by exchanging the plot of the (approximationset) points only, with the plot of the attainment surface,it is much easier to identify “gaps” in the distribution ofpoints. Another advantage of attainment surfaces over simplyplotting points comes when we want to display the outcomeof multiple runs of one or more optimizers. In this case,instead of plotting one front for each of the executions, wecan make use of the summary attainment surfaces [39]. Notethat if we have performed n different runs, the summaryattainment surface s weakly dominates summary attainmentsurfaces s+ 1, s+ 2, . . . n. Summary attainment surface plotsare easier to interpret than plots of many result surfaces sincesummary surfaces never cross each other.

Figure 5 shows the summary attainment surfaces 1, 15, and30, for four different large problems. On each graphic, sum-mary attainment surfaces for both, direct and hyperheuristic-based codifications, are attached. As clearly shown in thefigures, the hyperheuristic codification covers the upper-leftpart of the solution space (high values for the cuttings - lowvalues for the length). Meanwhile, the direct codification isfocused on the middle/bottom-right region (low values for thecuttings - medium/high values for the length). There is onlya small region for three problems, where the lowest points ofthe hyperheuristic approach completely dominate the uppestpoints of the direct codifications. Anyway, we can see that,in general, the different codifications are covering differentregions of the solution space. Note that the gap between thetwo covered regions tends to increase for larger problems.Definitely, some partial optimal solutions can’t be obtainedby the hyperheuristic, because the low-level heuristics fixthe orientation and order of the objects, thus avoiding the

generation of some of the feasible solutions. However, theadvantage of having efficient different codifications coveringdifferent areas of the solution space lies in the fact thatdecision makers can choose the type of codification dependingon the area of the solution space which is more interesting forthe problem at hand.

VI. CONCLUSIONS

A real-world multi-objective formulation of the 2DSPP hasbeen presented. The two objectives considered were: minimizethe overall length of the raw material and minimize the totalnumber of cuts needed to obtain the complete set of de-manded pieces. Many approximations to the single-objectiveformulation of the problem appear in the literature, but thenumber of multiple-objective approaches is very reduced. Dueto the inherent complexity of most MOPs, their solution is bestobtained by those techniques that are specifically tailored tohandle multiple objectives. Since evolutionary algorithms havebeen successfully applied to many real-world MOPs and tomany single-objective approaches of the problem, they werechosen to deal with the multi-objective 2DSPP.

The results obtained with some of the best-known MOEAsfor an initial codification based on the representation ofcutting patterns, demonstrated the validity of evolutionarystrategies in this kind of multi-objective real-world problems.For smaller problem instances, the proposal derived solutionswith wastage levels similar to most previous approximationswhich just seek to optimize the overall length. For largeproblems, the approach has to deal with very large searchspaces, and so, it is not able to produce competitive solutionswhen seeking to optimize the overall length. For this reason,we have proposed a new type of codification based onthe combination of different low-level heuristics which havebeen designed to optimize just the length objective. Resultsdemonstrate that such a new codification is able to improvethe solutions obtained by the single heuristics, even when bothobjectives are together considered. However, as an effect ofthe more reduced search space of the hyperheuristic and theusage of low-level heuristics focused on the optimization ofthe stock sheet length, both codifications are covering differentregions of the search space.

As we have demonstrated, the design of a codificationfor the representation of solutions inside a MOEA is a veryimportant issue. The definition of large search spaces makesdifficult to obtain higher quality solutions, but when thesearch spaces are reduced some (or even important) regionsof the solution space can be missed. For this reason, wethink that an important line of future research lies on thedesign of new codifications for the 2DSPP. In particular,we are interested on codifications that can seek on regionswhich better balance the optimization of both objectives. Inthis sense, the hyperheuristic codification can be extended toexplicitly contemplate the combination of different low-levelheuristics, sorting of pieces, orientation of pieces, selection ofholes, etc. Moreover, the MOEAs can be hybridized with theintroduction of local searches which can try to better expandthe search at certain regions of the solution space.

ACKNOWLEDGMENT

This work was funded by the EC (FEDER) and the SpanishMinistry of Science and Technology as part of the ‘PlanNacional de I+D+i’ (TIN2008-06491-C04-02). The Canary Gov-ernment has also funded this work through the PI2007/015research project. The work of Jesica de Armas was fundedby grant FPU-AP2007-02414.

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