an empirical investigation of meta-heuristic and heuristic ... · (sa), na ıve evolution (ne)) and...

Theory and Methodology

An empirical investigation of meta-heuristic and heuristicalgorithms for a 2D packing problem

E. Hopper, B.C.H. Turton *

School of Engineering, Cardi� University, Queens Buildings, The Parade, P.O. Box 689, Cardi� CF2 3TF, UK

Received 7 January 1999; accepted 1 July 1999

Abstract

In this paper we consider the two-dimensional (2D) rectangular packing problem, where a ®xed set of items have to

be allocated on a single object. Two heuristics, which belong to the class of packing procedures that preserve bottom-

left (BL) stability, are hybridised with three meta-heuristic algorithms (genetic algorithms (GA), simulated annealing

(SA), na�õve evolution (NE)) and local search heuristic (hill-climbing). This study compares the hybrid algorithms in

terms of solution quality and computation time on a number of packing problems of di�erent size. In order to show the

e�ectiveness of the design of the di�erent algorithms, their performance is compared to random search (RS) and

heuristic packing routines. Ó 2001 Elsevier Science B.V. All rights reserved.

Keywords: Cutting; Packing; Genetic algorithms; Simulated annealing; Optimisation

1. Introduction

Cutting and packing problems are encounteredin many industries, with di�erent industries in-corporating di�erent constraints and objectives.The wood-, glass- and paper industry are mainlyconcerned with the cutting of regular ®gures,whereas in the ship building, textile and leatherindustry irregular, arbitrary shaped items are to bepacked.

Packing problems are optimisation problemsthat are concerned with ®nding a good arrange-ment of multiple items in larger containing regions(objects). The usual objective of the allocationprocess is to maximise the material utilisation andhence to minimise the ``wasted'' area. This is ofparticular interest to industries involved withmass-production as small improvements in thelayout can result in savings of material and aconsiderable reduction in production costs.

Our work is concerned with a 2D rectangularpacking problem. The problem consists of packinga collection of items onto a rectangular objectwhile minimising the used object space. Thepacking process has to ensure that there is nooverlap between the items. The speci®c problem

European Journal of Operational Research 128 (2001) 34±57www.elsevier.com/locate/dsw

* Corresponding author. Tel.: +44-1222-874425; fax: +44-

1222-874716.

E-mail address: [email protected] (B.C.H. Turton).

0377-2217/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved.

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that is addressed in this paper has the followingcharacteristics:· a set of items, which may contain identical

items;· one single object of ®xed width and in®nite

height;· all pieces are of rectangular shape;· items can be rotated by 90°;· non-guillotineable (Dyckho�, 1990).

The 2D stock cutting problem occurs as anindustrial problem where a number of rectangularitems need to be cut from a roll of material. Sincethe order (set of items) is small compared to thestock material, the height of the object can be re-garded as in®nite. Processor allocation can also betreated as a 2D packing problem (Hwang, 1997).

2. A review of meta-heuristic packing algorithms

Many heuristic packing algorithms have beensuggested in the literature. Surveys on solutionmethodologies for various types of the 2D rect-angle packing problem can be found in Hinxman(1980), Sarin (1983) and H�assler and Sweeney(1991). In comparison to the great quantity ofliterature on heuristic algorithms to the packingproblem, only a few researchers have experimentedwith meta-heuristic algorithms.

2.1. Packing and genetic algorithms

Genetic algorithms (GA) for packing problemsmainly concentrate on guillotineable layouts asfound in the wood, glass and paper industry(Kr�oger, 1995; Andr�as et al., 1996) and 1D bin-packing (Falkenauer and Delachambre, 1992). Areview of the application of genetic algorithms topacking problems is given in Hopper and Turton(1997). With respect to the speci®c packing prob-lem described in Section 1 three types of solutionapproaches involving GAs can be distinguished.

The majority of literature concentrates on hy-brid algorithms, where a GA is combined with aheuristic placement routine. In this two-stage ap-proach a GA is used to determine the sequence, inwhich the items are to be packed. A second algo-

rithm is then needed, which describes how thissequence is allocated onto the object. One of the®rst researchers who implemented GAs in thedomain of packing is Smith (1985). He experi-mented with two heuristic packing routines, one ofwhich implements backtracking and producesdenser layouts, however, is computationally moreexpensive. Comparisons between the two hybridapproaches show that the combination with themore sophisticated heuristic generates betterpacking patterns. The packing problem Smithstudied is special in that the orientation of therectangles is ®xed.

In the hybrid approach by Hwang et al. (1994)a GA is combined with a well-known heuristicfrom bin packing, the so-called First-Fit-Decreas-ing-Height algorithm (FFDH). Although thistechnique produces guillotineable layouts, this isnot treated as an additional constraint in this workand suggested for the general case of the rectanglepacking problem. Comparisons with another GAtechnique, which will be described below, and theFFDH heuristic itself show that this hybrid tech-nique performs best.

Jakobs (1996) uses a heuristic, which belongsto the class of bottom-left (BL) packing heuris-tics to hybridise an order-based GA. In order toreduce computational complexity the heuristicdoes not necessarily place an item at the lowestavailable BL position. However, it preserves BLstability in the layout (see Section 3.1). In orderto determine the in¯uence of the heuristic algo-rithm on the solution quality, Hopper and Tur-ton (1998) implement two hybrid GAs, which arecombined with di�erent placement heuristics (seeSection 5.1).

Lai and Chan (1997) use an evolutionary al-gorithm, which is combined with a heuristic rou-tine. This routine is similar to the BL-heuristic andplaces items in the position that is closest to thelower-left corner of the object. Comparisons witha mathematical programming algorithm show thatthe evolutionary approach is computationallymore e�cient, however, it generates patterns withslightly higher trim loss.

Dagli and Poshyanonda (1997) also used theGA to generate an input sequence for the place-ment algorithm, which is based on a sliding

E. Hopper, B.C.H. Turton / European Journal of Operational Research 128 (2001) 34±57 35

method combined with an arti®cial neural net-work. Every incoming item is placed next to thepartial layout and all scrap areas generated arerecorded. If there is a match between an incomingitem and one of the scrap areas, the neural net-work selects the best match.

A second category of solution approaches withGAs aims at incorporating some of the layoutinformation into the data structure of the GA.However, some additional rules are still needed to®x the position in the layout.

The GA by Kr�oger et al. (1991) is based on adirected binary tree to encode the problem. Thisrepresentation ®xes one dimension of the positionof an item in the layout. The second dimension isdetermined by the BL condition. Since its perfor-mance is compared to well-known packing heu-ristics, a relative comparison with our work ispossible.

Hwang et al. (1994) also use a directed binarytree, which combines two rectangles to a largerrectangle by either placing them horizontally orvertically next to each other. The position withinthe containing larger rectangle is left justi®ed. Asmentioned before, comparisons with a hybrid GAtechnique show that this method is less e�cient interms of packing height.

The third group of GA solution approachesattempts to solve the problem in 2D space. Her-bert and Dowsland (1996) developed a 2D codingtechnique for a pallet-loading problem of identicalrectangles. The layout is represented by 2D matrixindicating available positions for vertical andhorizontal placement, whereby the horizontal onehas priority. This technique works well for smallproblems. In order to improve the outcome formedium sized problems additional repair and en-hance operators have been introduced.

The method developed by Ratanapan and Da-gli (1997) is di�erent from the other approachesdescribed so far, since it does not make use of adata structure to represent the problem. The itemsare represented as 2D pieces with their true geo-metric dimensions. After the initialisation process,which places all items into non-overlapping posi-tions on the object, a series of genetic operators isapplied, which consist of moving, relocation andrecombination operations.

2.2. Packing and simulated annealing

Only few researchers have applied simulatedannealing (SA) to 2D rectangular packing prob-lems. K�ampke (1988) applies SA to 1D bin pack-ing. Dowsland (1993) has experimented with SAon pallet loading problems involving identical aswell as non-identical boxes. In the identical casethe number of feasible positions for the placementof one item is reduced to the co-ordinates, whichare a multiple of its length and width away fromthe container edge. The neighbourhood has beende®ned as the set of solutions, which is obtained,when each item is moved to any other positionwith some restrictions. Since these movements leadto overlapping patterns, this constraint has beendealt with in the objective function. Extending thismethod to non-identical pieces, the condition forthe feasible positions is that it has to be at a validcombination of lengths and widths of the otheritem types from the container edge.

2.3. Comparison of meta-heuristic methods

As can be seen from the literature overviewgiven above GAs and SA have been successfullyapplied to the 2D rectangle packing problem.However, none of the researchers have comparedthe performance of these meta-heuristic algorithmsusing the same packing problems. Burke andKendall (1998) have carried out the only researchin this area on the clustering of rectangles. Their®ndings indicate that tabu search and SA outper-form GAs for this speci®c problem.

In this paper our main objective is to comparethe performance of GAs, na�õve evolution (NE) andSA with each other for small to large packingproblems. For all methods a two-stage approachhas been chosen, where the meta-heuristic algo-rithm is combined with a heuristic packing policy.As far as we are aware SA has not been imple-mented in a two-stage approach for packingproblems such as GAs (see Section 2.1).

Two di�erent heuristic routines are used tohybridise the meta-heuristic algorithms. The ®rsttechnique is the BL heuristic, which has been usedby Jakobs (1996) in a hybrid GA. The second

36 E. Hopper, B.C.H. Turton / European Journal of Operational Research 128 (2001) 34±57

heuristic algorithm also creates BL justi®ed pat-terns. However, it is more sophisticated and hencecomputationally more expensive (Chazelle, 1983).The outcome of meta-heuristic algorithms iscompared with the two packing heuristics as wellas with other search techniques such as hill-climbing and random search (RS). In order toshow the e�ectiveness of the design of the GAstheir performance is compared with an NE algo-rithm. Results indicate that the performance of thehybrid algorithms is strongly dependent on thenature of the placement routine and the problemsize. In particular in industrial applications notonly the solution quality has to be considered, butalso the computational cost of the various packingmethods.

This paper is organised as follows: Section 3gives a brief introduction into the heuristic andmeta-heuristic algorithms used in our research. InSection 4 the experiments and the test problemsare outlined. Sections 5 and 6 contain an overviewand a discussion of the results and Section 7summarises the ®ndings of this research.

3. Heuristic placement algorithms

In this paper we turn our attention to the classof BL heuristics (Baker et al., 1980). These packingprocedures preserve BL stability in the layout. Anitem is allocated in a BL stable position if it cannotbe moved any further to the left or downwards.Two implementations of a BL heuristic are com-bined in our study with meta-heuristic algorithms.

3.1. BL algorithm

The BL algorithm described below has beenused by Jakobs (1996) in a hybrid GA. Startingfrom the top-right corner each item is slid as far aspossible to the bottom and then as far as possibleto the left of the object. These successive verticaland horizontal movement operations are repeateduntil the item locks in a stable position. A validposition is found when the rectangle collides withthe partial layout at its lower and left sides. Fig. 1shows the placement of a sequence of rectangles,

which is described by the permutation (2, 6, 4, 3, 0,1, 5).

The major disadvantage of this routine consistsof the creation of empty areas in the layout, whenlarger items block the movement of successiveones. On the other hand its time complexity is onlyO(N2), when N is the number of items to bepacked. Due its low complexity this heuristic isfavourable in a hybrid combination with a meta-heuristic, since the decoding routine has to be ex-ecuted every time the quality of a solution isevaluated and hence contributes to a great extentto the run time of the hybrid algorithm.

3.2. BLF-algorithm

Since the BL-routine described above tends togenerate layouts with relatively large empty areas,a second more sophisticated BL heuristic has beenconsidered for hybridisation with meta-heuristics.The strategy here consists of placing a rectangleinto the lowest available position of the object andleft-justifying it. Fig. 2 demonstrates the placement

Fig. 1. BL routine (Jakobs, 1996).

Fig. 2. Bottom-left-®ll heuristic.


policy using the same permutation example as inFig. 1.

Since the generation of the layout is based onthe allocation of the lowest su�ciently large regionin the partial layout rather than on a series of BLmoves, it is capable of ®lling existing gaps inpacking pattern. In order to distinguish it from theBL-algorithm described in Section 3.1 it is referredto as the BL-Fill (BLF) heuristic. Compared to theBL-routine this method results in denser packingpatterns. The major disadvantage, however, lies inits time complexity, which is O(N3) (Chazelle,1983).

4. Heuristic search techniques

The quality of the layout which is constructedusing the above placement algorithms depends onthe sequence in which the rectangles are presentedto the routine. Since the number of combinationsis too large to be explored exhaustively in a rea-sonable amount of time, meta-heuristic algorithmsare used as a more e�cient search strategy. In thefollowing hybrid approaches the task of the meta-heuristic is to search for a good ordering of theitems. A placement routine is then needed to in-terpret the permutation and evaluate its quality.Heuristic search techniques will result in a good,however, not necessarily optimal solution withinreasonable computing time.

4.1. Hill-climbing

Hill-climbing is a local search technique, whichmoves from one solution to another one in theneighbourhood. If the quality of the new solutionis better than the previous one, this move is ac-cepted and the search continues from here. If theneighbouring state does not result in an improve-ment, the move is rejected and the search contin-ues from the current state. The main disadvantageof this method is that the search process might gettrapped in a local minimum, which is not equal tothe global one. A useful variation on simple hill-climbing considers a series of moves from thecurrent state and selects the best one as the next

state. This method is known as gradient search orsteepest-ascent hill-climbing.

4.2. Meta-heuristic algorithms

In order to overcome the main disadvantage oflocal search algorithms such as hill-climbing,whose weakness lies in the inability to escape fromlocal minima, more sophisticated heuristic searchstrategies are designed to avoid such a situation.This implies the temporary acceptance of a state oflower quality. Hence meta-heuristic algorithmscan be considered to some extent as local searchstrategies, however, they include a means to escapefrom local minima.

4.2.1. Genetic algorithmsGAs are search and optimisation procedures

that operate in a similar way to the evolutionaryprocesses observed in nature. The search is guidedtowards improvement using the `survival of the®ttest principle'. This is achieved by extracting themost desirable features from a generation of so-lutions and combining them to form the nextgeneration. The quality of each solution is evalu-ated and the `®tter' individuals are selected for thereproduction process. Continuation of this processthrough a number of generations will result inoptimal or near-optimal solutions. The main dif-ference between GAs and other meta-heuristicapproaches such as SA and tabu search is that theydeal with populations of solutions rather than asingle solution and therefore explore the neigh-bourhood of the whole population. Operatorssuch as selection, crossover and mutation are usedto explore the neighbourhood and generate a newgeneration. Further theoretical and practical de-tails can be found in (Davis, 1991; Goldberg,1989).

4.2.2. Na�õve evolution algorithmThe basic idea behind NE is the same as for

the GA. However, no crossover operator is ap-plied to manipulate the search space. Only themutation operator is used for the generation ofthe next population. An NE algorithm canbe used to test the e�ciency of the crossover


operator in a GA. Falkenauer (1998) applied thistechnique in experiments on 1D bin-packingproblems.

4.2.3. Simulated annealingEglese (1990) has investigated the application of

SA as a tool for Operations Research. SA wasintroduced as an optimisation tool in the 1980swhen the concept of physical annealing ®rst wasapplied in combinatorial optimisation. Transfer-ring this model to combinatorial problems theenergy states in a system correspond to the variousfeasible solutions for a problem and the energy ofthe system to the cost function to be minimised.

SA can be seen as a variant of the hill-climbingmethod, however, it attempts to avoid gettingtrapped in a local minimum. Instead of only ac-cepting neighbouring solutions that result in animprovement, also solutions, which are worse,may be accepted randomly with a certain proba-bility. This probability depends on the increase incost and a control parameter, i.e., temperature inphysical annealing. The smaller the increase in thecost and the higher the temperature the more likelyuphill moves will be accepted. During the anneal-ing process the temperature is gradually loweredaccording to the cooling schedule. This means thealgorithm becomes more and more selective inaccepting new solutions. At the end of the processonly moves, which result in an improvement areaccepted in practice. The search process terminateswhen it arrives at a lower bound for temperatureor cost.

5. Search techniques and test data

5.1. Implementation of the search algorithms

A number of problem-speci®c and generic de-cisions have to be made for the implementation ofthe meta-heuristic search algorithms. The prob-lem-speci®c choices concern the objective function,initial solution, representation scheme as well asthe operators applied to manipulate the searchspace. Generic decisions include the probabilitiesat which the search space manipulators such as

crossover and mutation are applied, the coolingschedule in the case of SA and the population andgeneration sizes for the GA as well as stoppingcriteria for the search algorithms.

In our work the packing problem is tackledwith a two-stage approach, where the meta-heu-ristic search methods (GA and SA) and the localsearch algorithm search the solution space forgood permutations. The permutation representsthe order, in which the ®nite set of items is packed.

The heuristic placement routine is then used todecode and evaluate the quality of the permutationaccording to the ®tness function. The quality of apacking pattern is ®rst of all determined by itsheight, since the unused rectangular area can be re-used. However, this variable is not su�cient toexpress how tightly the items are packed. For the®tness function a weighted sum has been used sothe packing height is weighted at 70% and thepacking density at 30%.

The search space of this problem is extended bythe orientation of the items, which can rotate by90°. In order to allow the meta-heuristic and localsearch algorithms to explore the orientation of theitems, an operator is used which ¯ips the orienta-tion of each rectangle in the sequence with a cer-tain probability.

5.1.1. Genetic algorithmSince an order-based encoding is used for this

problem, care has to be taken that valid chromo-somes are generated during the crossover andmutation operations. Partially matched crossover(PMX) (Goldberg, 1989) and order-based muta-tion (Syswerda, 1991) are suitable for this type ofencoding and have been used in this case. Pro-portional selection and generational replacementhave been applied. The orientation of the rectan-gles is considered in the GA in the form of muta-tion. In the case of orthogonal packing only twoorientations for an item are possible. The rotationoperator is applied to every item in the chromo-some and changes the orientation with a certainprobability.

Further techniques that have been implementedinclude elitism and seeding (Goldberg, 1989). Sinceheuristic placements with pre-ordered input se-quences can yield packing patterns, whose quality


can lie above average (Co�man et al., 1984), theinitial population has been seeded with the per-mutation, which describes the rectangles sortedaccording to decreasing height. For the seeding thebest out of 50 evaluations of the placement heu-ristic using randomly generated but height-sortedinput sequences has been taken. The initial popu-lation has been generated randomly and containsthe seeded individual. The GAs we implementeduse a population size of 50 and a generation size of1000. The probability for crossover is 60% and theone for the two types of mutations is 3%.

5.1.2. Na�õve evolution algorithmThe problem-speci®c and generic decisions for

the NE algorithm are the same as for the GA withthe di�erence that no crossover operator is used tomanipulate the solution space.

5.1.3. Simulated annealing algorithmThe packing problem is represented by a per-

mutation that is interpreted as the order in whichthe rectangles are packed. The neighbourhoodstructure of the current solution is de®ned by theset of solutions that can be reached applying thefollowing two manipulation operations. The ®rstone is analogous to the order-based mutationoperator used in the GA and swaps two ran-domly selected items in the permutation. Thesecond operator considers only the orientationand ¯ips the rotation variable of one randomlyselected item. In the translation to the next so-lution only one of the operators is applied with a50% chance. The initial solution is randomlygenerated.

The generic choices for the implementation ofan SA algorithm are summarised in the annealingor cooling schedule. The schedule presented inPress et al. (1995) is used in this study with a fewmodi®cations. The temperature function is geo-metric and decreased by 10%. The initial valuefor the temperature has been determined as 19.23applying the method described in Press et al.(1995). The number of iterations at each tem-perature has been modi®ed in order to reduce thetotal simulation time. The temperature is heldconstant for 50N total moves or 5N successful

moves at each step with N representing thenumber of items.

5.1.4. Hill-climbingThe neighbourhood structure as well as the

manipulators for the search space used in the localsearch procedure is the same as in SA. The initialsolution is generated randomly. The search processis stopped after N unsuccessful moves in the searchspace.

5.1.5. Random searchIn order to demonstrate the e�ectiveness of the

design the di�erent meta-heuristic search tech-niques are compared to RS. In RS permutationsare generated randomly. The search process is runover the same number of function evaluations asthe GA.

5.2. Test problems

Performance of the meta-heuristic and heuristicalgorithms has been tested with seven di�erentsized packing tasks ranging from 17 to 197 items.Three instances have been generated for eachproblem category. The dimensions of the rectan-gles are produced randomly with a maximumaspect ratio of 7. The problems have been con-structed such that the optimal solution is known(see Table 1). The ratio of the two dimensions ofthe object varies between 1 and 3. Three instancesof each problem have been simulated. Detailedinformation about the various item sets is given inAppendix A.

Table 1

Test problems

Problem

category

Number of

items

Optimal

height

Object

dimensions

C1 16 or 17 20 20 ´ 20

C2 25 15 40 ´ 15

C3 28 or 29 30 60 ´ 30

C4 49 60 60 ´ 60

C5 72 or 73 90 60 ´ 90

C6 97 120 80 ´ 120

C7 196 or 197 240 160 ´ 240


5.3. Simulation

The GA and the NE algorithms have both beensimulated over 1000 generations using a populationsize of 50. The stopping criterion for the SA and thehill-climbing algorithm is based on the number ofunsuccessful moves. Hence both search processeshave been run until termination by this criterion. Inorder to establish the e�ciency of the optimisationprocesses RS has been applied over the sameamount of iterations as the GA (i.e., 50 000). Theresults presented below are the average of 10 sim-ulations. The outcome of the meta-heuristicmethods is compared on the basis of the number ofiterations. The heuristic methods have been run 50times using random input sequences as well as se-quences which have been sorted by decreasingheight (DH) or width (DW). The simulations havebeen run on a PC with a Pentium Pro 200 MHzprocessor and 65MB of RAM under WindowsNT4.0. The algorithms have been implemented inC++ using LEDA (Library of E�cient Data typesand Algorithms), version 3.7.1 (Mehlhorn et al.,1998), to generate the graphic output.

6. Results

6.1. Comparison between the heuristic algorithmsBL and BLF

The comparison of the two heuristic packingalgorithms shows that the more sophisticatedplacement routine (BLF) achieves better layouts.Table 2. summarises the relative distances betweenthe lowest packing height found and the height ofthe optimal solution. Using random input se-quences the layouts generated by the BLF algo-

rithm are between 10 and 30% better than the onesobtained with the BL rule.

Pre-ordering the input sequences according todecreasing width (DW) or height (DH) of theitems improves the outcome of both packingheuristics by 5±10% compared to the performanceon random input sequences. Comparing the bestsolutions achieved with both methods for eachproblem shows that the BLF heuristic outperformsthe BL routine by up to 25% with the performancegain being higher for the larger problems.

As already mentioned in Section 3.1 the com-putational complexity of the BLF algorithm ishigher than the one of the BL routine. The averagerun time increases exponentially with the problemsize. This di�erence is particularly noticeable forlarger packing problems (Fig. 3). The average timeneeded to place one item is higher for the BLFalgorithm (Table 3). The number of available BLstable positions, which are tested on average, be-fore a suitable position is found, is higher than thenumber of movements carried out using the slidingmethod of the BL algorithm. The number of free

Table 2

Relative distance of best solution to optimum height (%) for heuristic methods with and without pre-ordered input sequences

C1 C2 C3 C4 C5 C6 C7

BL 25 39 33 33 31 34 41

BL±DH 17 68 27 21 18 19 31

BL±DW 18 31 24 18 22 21 29

BLF 14 20 17 15 11 12 10

BLF±DH 11 42 12 6 5 5 4

BLF±DW 11 12 12 5 5 5 5

Fig. 3. Average elapsed time per 100 iterations for heuristics.


BL-positions in the layout increases exponentiallywith the problem size.

6.2. Comparison between the heuristic algorithmsBL and BLF

In the following the performance of the meta-heuristic algorithms is investigated. First of all,the meta-heuristics methods, which use the BLdecoder, are compared. Then the same comparisonis made for the meta-heuristic algorithms, whichare hybridised with the more sophisticated BLFroutine. Finally, the performance of the two typesof hybrid combinations is analysed. In order tostudy the e�ciency of the meta-heuristic methodsover the simple heuristics their outcomes havebeen compared to RS, which evaluates the packingroutines over the same number of iterations as theGA (i.e., 50 000) using random input sequences.

The meta-heuristic search methods using theBL decoder achieve layouts of higher quality thanthe simple packing heuristic (BL). The packingheights achieved by the hybrid are up to 24%better than the ones by the BL (Table 4). Theperformance of the RS lies between the one of themeta-heuristics and the packing heuristic. Hencesome of the performance gain achieved by themeta-heuristics is due to the higher number of it-

erations. The outcomes of the two evolutionarymethods (GA and NE) are very similar with theNE algorithm performing slightly better for someproblems (up to 2%). Hill-climbing performs bet-ter than RS for most problems or at least equallywell and outperforms the simple BL heuristic byup to 5%. For all techniques, heuristics as well asmeta-heuristics, the di�erence between the packingheights achieved and the optimal height becomeslarger with increasing problem size.

The best layouts for the hybrids with the BLdecoder have been obtained with SA in all prob-lem categories. The di�erence between SA and thetwo evolutionary techniques (GA and NE) liesbetween 1% and 8% and is higher for the largerproblems. However, the number of iterationsneeded by the SA is up to 5 times higher on av-erage. Fig. 4 shows that the SA converges veryslowly, whereas the GA reaches the ®nal packingheight earlier.

The results obtained by the methods using thesophisticated packing heuristic, i.e., BLF, show asimilar ranking of the various methods (Table 5).For problems up to category C3 RS and hill-climbing perform better than the simple heuristic.Hill-climbing, however, is outperformed by RS,especially for the smaller problems. The packingheights achieved by the evolutionary algorithms,GA and NE, are very similar and outperform RS

Table 4

Relative distance of best solution to optimal height (%) for heuristic and meta-heuristic methods combined with BL routine

C1 C2 C3 C4 C5 C6 C7

GA + BL 6 10 8 9 11 15 21

NE + BL 6 8 8 8 11 13 19

SA + BL 4 7 7 6 6 7 13

HC + BL 9 18 11 14 14 20 25

RS + BL 6 14 14 16 18 20 28

BL 17 31 24 18 18 21 29

Table 3

Average elapsed time per placement of one item for heuristic methods (ls)

Problem C1 C2 C3 C4 C5 C6 C7

Size 17 25 29 49 73 97 197

BL 46 61 56 95 119 158 324

BLF 72 87 114 252 470 794 3234


and hill-climbing. SA yields the best results in eachproblem category.

Summarising Table 5, the meta-heuristicmethods, which use the BLF decoder, achievepacking heights, which are very close to the op-timum height (between 3% and 7%). Even theBLF heuristic on its own leads to very lowpacking heights. Especially, for larger problems(C2±C7) the di�erence to the meta-heuristics ismaximally 2%. This ®nding is di�erent from theBL case, where the solution qualities obtained bythe simple heuristic and the meta-heuristic di�erparticularly for the larger problems (Table 4).Using the BLF decoder the di�erence to the op-

timal height does not increase with the problemsize as is the case with the BL decoder (see Tables4 and 5). The solution quality remains at thesame level for each problem category (up to 7%from the optimum).

The comparison of the hybrid algorithms showsthat the combinations with the BLF placementroutine produce better layouts than the combina-tions with the BL routine. The hybrids with theBLF routine generate layouts that are up to 16%better. The di�erence is especially high for thelarge problems. The same is true for the NE hy-brids. In Table 6 the results obtained with the twodi�erent decoding algorithms are compared. The

Fig. 4. GA and SA+BL for large problem.

Table 5

Relative distance of best solution to optimum height [%] for heuristic and meta-heuristic methods combined with BLF decoder

C1 C2 C3 C4 C5 C6 C7

GA + BLF 4 7 5 3 4 4 5

NE + BLF 5 7 4 4 4 4 5

SA + BLF 4 6 5 3 3 3 4

HC + BLF 7 10 7 7 6 7 7

RS + BLF 5 8 7 7 6 7 7

BLF 11 16 12 5 5 5 5


di�erence between the best solutions found usingthe meta-heuristic with the BL and the BLF de-coder are stated.

Fig. 5. demonstrates the performance of the twoGAs. Both algorithms achieve the highest perfor-mance gain within the ®rst 10 000 iterations, i.e.,2000 generations.

Although the meta-heuristics perform better interms of solution quality, the combinations withthe BLF decoder have longer run times (Table 7).Run times become extremely long for large prob-lems (C5±C7) due to its higher computationalcomplexity. The BL algorithm o�ers an advantagein that respect. Especially, SA, which achieves thebest layouts, has high execution times. However,

further adjustments to the annealing schedule willmost likely reduce the run time of this meta-heu-ristic.

7. Discussion

7.1. Packing heuristics

The BLF packing algorithm achieves betterpacking patterns than the BL heuristic for ourexample problems. Since the BLF routine ®rst at-tempts to ®ll the gaps in the layout, the majority ofthe small items will be `absorbed' within the ex-isting partial layout and does not contribute

Fig. 5. Comparison of the two GAs combined with BL and BLF routines.

Table 6

Di�erence between the best solutions of the hybrids with the BLF routine and the ones with the BL routine (%)

C1 C2 C3 C4 C5 C6 C7

GA 2 3 3 6 7 11 16

NE 1 1 4 4 7 9 14

SA 0 1 2 3 3 3 9

HC 2 8 4 7 8 13 18

RS 1 6 7 9 12 13 21

Heuristic 6 15 12 13 13 16 24


further to the packing height, which is mainly de-termined by the larger items (Fig. 6). With the BL-rule, however, unused regions in layout cannot beaccessed and smaller rectangles also contribute tothe height. The results in Section 6.1 show that thedi�erence to optimum solution gets smaller withincreasing problem size. This is due to the fact thatlarger problems contain a larger number of smallitems, which are allocated in the empty areascontained in the partial layout. In particular pre-ordered input sequences (height or width) achievedense layouts (Fig. 6). Although the executiontime for the BLF algorithm is considerably larger,the performance gain especially for large packingproblem justi®es the application of the BLF rou-

tine. In a combination with a meta-heuristic,however, the time complexity plays a more im-portant role.

7.2. Meta-heuristics and local search

Looking at the results stated in Section 6.2 itcan be seen that the meta-heuristic methods out-perform the hill-climbing algorithm due to theirability to escape from local minima. Hence meta-heuristics o�er a clear advantage over the localsearch algorithm in that respect. Since the hill-climbing algorithm terminates in a local mini-mum its run time is shorter than that of SA. The

Fig. 6. Best layouts for a large problem (C6) with BL (left) and BLF (right); height-sorted sequence.

Table 7

Average elapsed time for heuristics and meta-heuristics per run in (min); for BL and BLF in (ms)

C1 C2 C3 C4 C5 C6 C7

GA + BL 0.5 0.8 0.9 2.4 4.0 6.7 23

NE + BL 0.4 0.6 0.7 1.6 2.6 4.1 30

SA + BL 0.4 1.4 1.8 7.5 17 31 117

GA + BLF 1.0 2.0 3.0 13 36 86 777

NE + BLF 0.7 1.3 2.1 8.3 23 55 483

SA + BLF 0.7 2.4 4.0 33 115 382 4181

BL (ms) 0.8 1.3 1.6 4.7 8.7 15.3 63.7

BLF (ms) 1.2 2.2 3.3 12 34 77.0 636


hill-climbing technique only allows exploration ofa limited area of the search space and is outper-formed by the meta-heuristic algorithms from thebeginning of the search process (Fig. 7). Althoughthe ®nal outcome of hill-climbing is slightly betterthan the RS method, a random walk through thesearch space results in better solutions during mostof the search process. Whereas hill-climbing onlyexplores the search space locally, SA can exploitthe space more e�ectively and concentrate onpromising areas.

Due to the manipulation technique used in SA,which either only changes the rotation of one ele-ment at a time or the position of two elements inthe permutation, the search process with the SA isslow at the beginning and even RS results in betterlayouts. RS, however, is quickly outperformed,when the SA algorithm starts exploiting promisingareas in the solution space and ®nds solutionswhich would not have been found on a randombasis.

Looking at Fig. 7 the most successful searchstrategy in the beginning of the search process areGAs. Solutions found by the GA improve rapidly

over the number of evaluations and only get out-performed by the SA towards the end of the search(Fig. 4). Hence the technique which GAs use toexplore the space is more successful than the oneused by SA. One of the di�erences between thealgorithms is that a crossover operator is used inthe GA to manipulate the current best solutions.This obviously creates larger changes in the se-quences than with the mutation operator used inSA and hence could explain the rapid progress theGA makes in the beginning.

Comparisons with an NE algorithm, which isonly based on mutation and has no crossover,show, however, that both strategies are equallysuccessful (Fig. 8). Hence the crossover operatorused in this implementation is not the reason forthe better exploration of the search space in thebeginning of the search process. If crossover wasthe main contributor, then the di�erence betweenGA and NE would be higher. However, the mu-tation operator implemented is obviously su�cientfor this task. Di�erent from the SA method theevolutionary techniques work on a population ofsolutions, which they explore simultaneously.

Fig. 7. Comparison between local search and meta-heuristic methods (+BL).


While the SA operates only on one solution at atime, the recombination method in evolutionaryalgorithms guarantees that the most successful so-lutions are utilised in the following generation.Hence, they allow exploration of the solution spacein parallel. The GA, however, is outperformed bythe SA technique towards the end of the searchprocess, when the population has converged. Onlythe SA technique, where solutions of minor qualitycan be accepted over a series of moves, can thenlead the search into promising regions.

In order to establish how well the meta-heu-ristic algorithms explore the search space, an RSprocess has been applied to the packing problems.Whereas the RS only èxplores' the solutionspace, the in-built search mechanisms allow themeta-heuristic strategies to èxploit' good regions.The di�erence between the GA and RS for theBLF case is smaller, which indicates that theèxploitation' of the solution space is limited(Fig. 9).

The comparison between the meta-heuristics,hill-climbing and RS shows that the improvementover the BLF heuristic is largest for smallerpacking problems, i.e., problems with less than 50items (C1±C3). Fig. 10 indicates the improvement

the ®ve di�erent methods have achieved over thebest solution obtained with the BLF algorithm.For problems consisting of a higher number ofitems (C4±C7) only meta-heuristic methods aresuccessful and result in better layouts than theBLF heuristic. RS and hill-climbing cannotexplore the enormous search space su�ciently andare easily outperformed by the simple BLF heu-ristic. Unlike RS the simple BLF method used inthis comparison also includes pre-sorted sequenc-es, where the items are sorted according to theirheight or width. As it can be seen in Table 2sorting almost always is better than random input.Since RS only stands a theoretical chance of®nding a sorted sequence it can be outperformedby the simple BLF method as indicated in Fig. 10.All meta-heuristic methods manage to improve theheuristic solution, however, only by a few percent.The most successful method on large problems isSA.

7.3. Hybrid methods

The results summarised in Tables 4 and 5 showthat the combination between the meta-heuristics

Fig. 8. Evolutionary algorithms (GA and NE) and random search for BL for C3.


and the BLF packing routine achieve better out-comes compared to the combinations with the BLroutine. The di�erence is higher for the largerproblems (Table 6).

Fig. 6 shows the performance of the two heu-ristics in combination with a GA. Since the pack-ing heights achieved with the BLF on its own arealready very close to the optimum height (less than

Fig. 10. Improvement of the meta-heuristics + BLF in comparison to the best of the heuristic solutions (BLF including sorted se-

quences) for each problem category.

Fig. 9. Random search and GAs for both decoders (BL and BLF).


8% from the optimum on average), the meta-heuristic cannot improve the performance of theheuristic as much as in the BL case. In other wordsby using the `poorer' BL-decoder, the meta-heu-ristic is needed to ®nd a good input sequence,

whereas applying the better BLF-heuristic goodlayouts are achieved in a smaller number of iter-ations. Figs. 11 and 12 show initial and best lay-outs obtained by both GAs for the largestproblem.

Fig. 11. Initial and best layout for GA + BL.

Fig. 12. Initial and best layout for GA + BLF.


This questions the use of a hybrid combina-tion between a meta-heuristic and a `poor' de-coder for this type of packing problem. In orderto achieve high quality layouts it may often besu�cient to apply the BLF routine over a smallnumber of iterations. This is especially true forlarger problems (>30 items, C4±C7), where thehybrid methods only manage to improve theheuristic solution by less than 2% (Fig. 10),however, at a very high computational cost (Ta-ble 7). For smaller problems on the other hand,where the execution times are acceptable, themeta-heuristic with the BLF rule outperforms thesimple heuristic by about 7%. In this case theapplication of a meta-heuristic algorithm o�ersadvantages.

In order to reduce the computation time the BLheuristic could be considered in connection withthe meta-heuristic. In spite of the reduction of theexecution time the solution qualities achieved withthis approach for large packing tasks are up to15% worse than those obtained with the simpleBLF heuristic (Fig. 13). For smaller problemswhere combination with the BLF routine achievesbetter layouts, the computation time is low any-way. Summarising the hybrid approach with thepoorer decoder, which is more e�cient in terms ofcomputation time, cannot be justi®ed, since it iseasily outperformed by the BLF-heuristic on largeproblems.

An implementation of the BLF algorithm,which has a lower computational complexity, is of

Fig. 13. Improvement of the meta-heuristics + BL in comparison to the best heuristic solution (BLF) for each problem category.

Table 8

Method for best results within speci®ed time limit (hybrids are with BLF decoder)

<1 min <10 min <1 h <10 h <24 h

C1 GA, SA GA, SA GA, SA GA, SA GA, SA

C2 BLF SA SA SA SA

C3 BLF NE NE NE NE

C4 BLF NE SA SA SA

C5 BLF GA, NE GA, NE SA SA

C6 BLF BLF NE SA SA

C7 BLF BLF BLF NE NE


great bene®t for the hybrid approach using theBLF rule. Chazelle (1983) developed an imple-mentation of this algorithm that has a complexityof O(N2). Using a more time e�cient implemen-tation the execution times of both heuristic pack-ing rules become comparable, and will only di�erby a factor rather than an order of magnitude.This means the hybrid algorithms with the BL rulelose their major advantage over the ones using theBLF decoder.

Hybrid combinations between meta-heuristicsand a heuristic packing rule as investigated in thisstudy not only achieve high quality layouts. Theirmain advantage lies in simplicity of the imple-mentation. The meta-heuristic search is hybridisedwith a heuristic and acts as tuner of the packing

Table 11

Category 3: 28 or 29 items; object: 60 ´ 30

P1 P2 P3

w h w h w h

7 5 18 6 24 9

14 5 12 2 8 9

14 8 7 10 11 9

4 8 23 4 17 9

21 13 1 4 24 4

7 11 7 7 8 4

14 11 4 11 6 1

14 5 5 6 5 1

4 5 7 2 17 4

18 3 11 6 6 3

21 3 19 10 5 3

17 11 5 11 5 12

4 11 2 4 13 12

7 4 5 7 14 14

5 4 2 4 14 2

6 7 12 7 2 2

18 5 13 7 3 8

3 5 6 3 9 8

7 3 10 6 14 12

5 3 16 9 2 12

18 4 4 1 3 6

3 4 10 4 9 6

12 2 24 6 5 2

6 2 9 9 13 2

18 5 1 2 18 3

21 5 5 8 14 3

17 3 5 3 16 3

4 3 25 7 12 3

21 5

Table 10

Category 2: 25 items; object: 40 ´ 15

P1 P2 P3

w h w h w h

11 3 11 2 12 7

13 3 2 3 7 7

9 2 10 7 7 1

7 2 8 4 5 1

9 3 9 5 3 2

7 3 7 2 6 2

11 2 4 1 7 2

13 2 6 1 5 2

11 4 4 5 3 1

13 4 8 3 6 1

3 5 1 3 12 6

11 2 5 5 9 6

2 2 3 1 12 2

11 3 12 4 7 2

2 3 6 2 10 3

5 4 2 4 4 1

6 4 11 4 5 1

12 2 10 2 16 3

1 2 3 2 5 3

3 5 11 2 4 2

13 5 3 4 5 2

12 4 26 4 10 3

1 4 8 4 9 3

5 2 3 2 16 3

6 2 6 2 5 3

Table 9


P1 P2 P3

w h w h w h

2 12 4 1 4 14

7 12 4 5 5 2

8 6 9 4 2 2

3 6 3 5 9 7

3 5 3 9 5 5

5 5 1 4 2 5

3 12 5 3 7 7

3 7 4 1 3 5

5 7 5 5 6 5

2 6 7 2 3 2

3 2 9 3 6 2

4 2 3 13 4 6

3 4 2 8 6 3

4 4 15 4 10 3

9 2 5 4 6 3

11 2 10 6 10 3

7 2


routine. The representation as a sequencingproblem allows the use of well-known manipula-tion techniques for the search space, e.g., order-based crossover operators, rather than developingproblem-speci®c operators that only can be used inone speci®c context.

On the other side it can be argued that thegeometric information concerning the layout ishidden in the heuristic decoder. Hence it cannot beexploited by meta-heuristic search processes to thesame extent as if a representation was used thatincludes more geometric information in the de®-nition of the neighbourhood structure and chro-mosomes, respectively.

Comparisons with the approach by Ratanapanand Dagli (1997), where the items have been rep-resented as true geometric objects in an evolu-tionary search algorithm show that this does notnecessarily generate better layouts. The packingdensities achieved by Ratanapan and Dagli's

technique for two medium-sized problems (21 and31 items) are about 92%. Using the same data setsall of the hybrid meta-heuristic techniques (SA,GA and NE) in combination with either of thepacking routines were able to outperform this re-sult. The achieved packing densities range between92% and 98%, with the SA obtaining the bestoutcomes. For the smaller problem the SA evenfound the optimum solution in 3 runs out of 10.Unfortunately, Ratanapan and Dagli do not statethe computational e�ort of the rearrangements inthe layout after application of the mutation oper-ators.

Dagli and Poshyanonda (1997) developed anapproach involving a hybrid between GAs and aneural network, which achieves packing densitiesbetween 95% and 97%. Applying the meta-heu-ristics in combination with the BLF routinedensities between 95% and 96% are obtained.However, the test problem Dagli and Poshyan-

Table 12

Category 4: 49 items; object: 60 ´ 60

P1 P2 P3

w h w h w h w h w h w h

2 7 3 3 10 14 2 4 10 4 12 7

24 7 8 3 3 13 2 7 12 4 9 4

16 4 5 20 28 5 3 4 13 5 4 4

18 4 3 17 5 8 5 30 3 5 9 9

16 7 3 7 14 9 5 3 7 22 2 5

18 7 5 7 12 14 10 26 6 22 20 5

2 4 3 7 13 10 6 5 9 23 9 5

24 4 4 7 3 17 4 9 10 19 4 5

4 28 4 21 1 5 1 4 3 15 4 2

6 18 10 19 4 1 9 2 5 13 12 2

14 12 4 17 18 4 4 17 2 10 3 15

2 12 8 17 1 1 5 2 2 10 21 11

18 19 3 10 2 6 4 4 13 18 11 3

9 8 5 10 4 14 6 2 3 18 3 3

7 8 7 6 3 18 4 10 2 3 11 23

9 11 8 6 4 14 2 4 2 3 11 23

7 11 15 12 8 17 3 12 5 2 11 8

14 6 3 12 11 5 6 5 4 2 3 8

2 6 11 10 9 12 3 9 3 4 21 4

6 10 5 10 4 7 7 18 9 4 14 4

16 10 4 2 25 8 6 6 7 1 3 13

3 5 8 2 7 5 18 7 6 1 35 13

4 5 10 2 24 9 13 9 2 4 11 5

8 12 12 2 9 14 25 7 20 4 11 5

3 18 12 19 4 7


onda use is very special in the sense that theratio between the width and the height of theobject is very large. Even simple heuristics as theheight and width sorted BLF routines achieve apacking density of 94%. Hence, the higher e�ortof the implementation in Dagli and Poshyan-onda's approach is not re¯ected in a much betteroutcome.

The 2D matrix representation Herbert andDowsland (1996) developed for a GA for thepallet loading problem of identical boxes does notachieve better outcomes as the 1D binary en-coding as the authors conclude. Applying thehybrid meta heuristics combined with BL andBLF routines to the same problems shows, thatthey ®nd optimal solutions more often thanHerbert and Dowsland's approach for the smallproblems (6 16 items). For the larger problemsonly near-optimal solutions have been found.Only with the implementation of `enhance' and

`improvement' operators Herbert and Dowslandmanage to improve the performance for largerproblems.

The GA approach developed by Kr�oger et al.(1991) is based on a graph structure, which allowsincluding some geometric information in the datastructure. Since the data sets used in the experi-ments are not published only an indirect compar-ison is possible. The packing heights achieved withthis technique are 1% and 7% better than thoseobtained by the BLF heuristic and lie in the sameregion as our ®ndings.

8. Conclusions

Two types of hybrid algorithms for the rect-angle packing problem have been implementedconsisting of a combination of a meta-heuristic

Table 13


P1 P2 P3

w h w h w h w h w h w h w h w h w h

6 34 10 2 2 3 3 5 1 2 30 10 6 37 5 11 4 2

3 13 6 6 9 6 14 3 5 1 14 19 10 15 4 5 3 5

5 13 5 6 9 6 9 27 1 1 4 26 4 7 5 5 4 5

12 10 7 14 1 6 6 24 3 3 3 3 12 7 1 3 4 24

12 10 6 14 2 6 21 7 5 20 5 23 4 18 6 3 15 12

7 6 3 16 7 5 7 10 6 23 5 20 10 8 1 4 13 12

15 6 5 16 18 5 1 2 7 2 15 4 5 8 6 4 19 7

7 25 6 14 9 3 13 19 11 21 10 6 4 25 4 2 9 7

15 25 5 14 9 3 4 17 8 7 6 3 5 25 5 2 5 4

12 21 13 14 18 9 4 13 6 15 5 2 4 8 19 25 2 4

7 16 2 3 5 6 17 3 2 1 4 2 12 8 5 9 12 5

5 16 7 3 2 6 24 10 13 14 3 1 10 10 4 9 9 22

3 21 2 11 12 2 5 4 3 14 2 3 5 10 3 6 5 1

5 21 7 11 9 2 2 2 5 26 14 3 3 4 3 6 2 1

7 5 7 6 3 8 6 1 9 14 9 2 7 4 6 13 15 12

5 5 6 6 9 8 11 9 10 3 7 8 7 10 20 13 13 12

1 4 14 33 9 10 4 26 4 13 32 6 2 10 3 18 2 5

10 4 4 12 5 3 2 1 1 3 6 2 7 15 3 18 5 5

13 6 3 12 2 3 4 7 14 11 26 5 4 18 5 16 12 17

13 12 18 16 18 3 7 38 7 10 1 2 15 18 4 16 2 12

9 12 3 12 7 3 3 2 14 12 6 5 3 18 6 11 5 12

6 23 18 12 3 2 3 1 18 3 13 3 7 18 13 4 4 5

3 7 4 4 9 2 4 2 7 4 10 3 7 5 7 4 28 5

5 7 3 4 4 6 2 7 2 5 13 7

1 2 1 3 2 1 7 28 4 11 3 2


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47

12

28

18

16

19

20

59

63

32

12

32

55

17

16

11

48

17

12

28

72

04

20

47

10

11

25

11

318

12

46

24

62

47

84

27

30

17

16

410

97

39

811

817

10

16

36

22

10

23

71

24

19

316

26

10

53

17

113

64

75

72

15

10

22

67

12

77

10

29

19

13

819

11

21

313

36

10

93

10

14

14

32

37

12

39

59

10

10

51

68

21

11

42

28

825

85

31

26

32

810

93

922

813

51

81

03

39

217

710

619

93

98

10

14

35

43

813

25

19

10

63

42

912

15

78

12

36

18

12

33

44

21

10

818

13

22

57

619

10

311

83

351

810

21

38

827

21

24

11

37

30

85

315

84

11

99

527

15

94

71

43

81

026

74


algorithm (GA, NE and SA) and heuristicpacking routine to allocate the items on theobject. The heuristic packing routine generatesthe layouts in a BL justi®ed manner. Whereasone of the techniques (BL) is a time-e�cientimplementation based on a sliding principle, thesecond one (BLF) is able to ®ll enclosures in thelayouts, however, at higher computational cost.The meta-heuristic hybrid algorithms have beentested on a number of packing problems andcompared with heuristic and local search meth-ods and also with approaches used by other re-searchers.

In terms of solution quality the meta-heuristicalgorithms outperform the heuristic packing rou-tines and the hill-climbing approach with SAperforming best. The combinations with the moresophisticated heuristic (BLF) achieve better lay-outs than the ones using the BL decoder. Forindustrial problems the question, which techniquein combination with the BLF routing to choose, isa trade-o� between material cost and simulationcost. In our study SA has achieved the best layoutquality over all problem categories. Its executiontime, however, becomes larger with increasingproblem size. The evolutionary algorithms, GAand NE, are better in terms of execution time andyield results, which are slightly worse than onesobtained by the SA. Hence if the time for solvinga packing task is limited, GA and NE are ap-propriate. For very small time margins, onlyheuristic packing algorithms will be able to meetthis criterion. Table 8 summarises which methodsare appropriate for di�erent sized packing tasksunder limited execution time. The information inTable 8 has to be seen in connection with thecomputational power of the simulation equipmentused. With the processing power constantly in-creasing, it will be possible to apply meta-heuris-tics e�ciently even in larger problems in thefuture.

Since the performance di�erence between thehybrid methods using the BL decoder and the oneusing the BLF decoder is only due to the improvedheuristic, the decoder has a larger e�ect on theoutcome of the hybrid technique than the meta-heuristic technique itself. This seems to suggestapproaches, where more layout speci®c knowledge

is incorporated in the meta-heuristic rather thanthe decoder. However, representation schemesstudied by other researchers that use more layoutinformation did not necessarily achieve higherpacking densities than hybrid techniques for theproblems tested.

Concerning the methodology, hybrid algo-rithms are well suited for industrial demands. Thelayouts achieved are of similar quality as othertechniques. The implementation of the hybrid al-gorithm is easier, since it is based on well-knowntechniques and does not require development ofproblem-speci®c algorithms. With heuristicsalready being applied in industry, the acceptanceof research methods such as meta-heuristics cer-tainly will be higher.

Appendix A

The packing tasks are classi®ed by the problemsize, i.e., the number of items. For each categorythree problem instances have been constructed.The dimensions of the items are given in Tables 9±15. The optimum solution for each problem isknown and is achieved by packing the rectangles inthe order they are stated in the tables using theBLF routine.

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