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Mathematics and Computers in Simulation 51 (2000) 257271

Grouping genetic algorithms: an efficient method to solve

the cell formation problem

P. De Lit , E. Falkenauer, A. DelchambreUniversit Libre de Bruxelles (ULB), Department of Applied Mechanics, Brussels, Belgium

Abstract

The layout problem arises in a production plant during the study of a new production system, but also during a

possible restructuring. The main aim of layout design is to reduce transportation and maintenance, which simplifiesmanagement, shortens lead time, improves product quality and speeds up the response to market fluctuations. A

principle of Group Technology (GT) advocates the division of a unity into small groups or cells. As it is most of the

time impossible to design totally independent cells, the problem is to minimise traffic of items between the cells, for

a fixed maximum cell size. This problem is known as cell formation problem (CFP). We propose here an original

approach to solve this NP-hard problem. It is based on a Grouping Genetic Algorithm (GGA), a special class of

genetic algorithms, heavily modified to suit the structure of grouping problems. The crucial advantage of this GGA

is that it is able to deal with large instancesof theproblem thus becoming a powerfultool foran engineer determining

a plant layout, allowing him or her to try several plant options, without the limitation of huge computation times.

2000 IMACS/Elsevier Science B.V. All rights reserved.

Keywords: Grouping genetic algorithms; Cell formation and decomposition; Group technology

1. Introduction

Layout problems arise with the study of a new production system, or during reorganisation due tointroduction of new resources or product design modification. Not too long ago, the layout of productionsystems was done according to two conceptual schemes, namely the jobshop (typical low volume, highproduct variety environments) and the transfer line or flowshop (typical high volume, low product varietyenvironments).

In the 1960s, J. L. Burbidge [4] developed a systematic planning approach on the concept accordingto which parts with similar features could be manufactured together with standardised processes. Today

Corresponding author. Tel.: +32-2-650-47-66; fax: +32-2-650-27-10

E-mail address: [email protected] (P. De Lit)

0378-4754/00/$20.00 2000 IMACS/Elsevier Science B.V. All rights reserved.

PII: S0378-4754(99)00122-6


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258 P. De Lit et al. / Mathematics and Computers in Simulation 51 (2000) 257271

large facilities regroup small independent units acting themselves as little factories. Creation of these unitsis based on the concept of Group Technology (GT) a theory of management based on the principlethat similar things should be done similarly. Products needing similar operations and a common set ofresources are grouped into families, the resources being regrouped into production subsystems. ThisGT has revealed itself as a key in production control and optimisation, as well as in material transport.This cellular manufacturing concept was developed to compromise the flexibility of the jobshop while

retaining the production management simplicity associated with the flowshop layout. GT aims to reducecarriage and handling, which leads to simplifications of the management, lessening of lead times, andindirectly, to an increasing product quality and a quicker response to market fluctuations.

One major problem to tackle in GT is the design of the manufacturing system. In an ideal cellularmanufacturing environment, products should be manufactured completely within a cell, and then possiblyassembled on an assembly system. This supposes that the product can be produced exclusively on a singlecell, and that no inter-cell transfers of parts is required. As it is most of the time illusory in industrialapplications to get totally independent cells, several approaches have been proposed to group machines.Researchers used matrix formulation, mathematical programming formulation, and graph partitioningmethods (we considered the problem as pertaining to the latter category). This cell formation problem(CFP) [5] (and most related ones) is known to be a NP-hard grouping problem, i.e., no algorithm ofpolynomial complexity to solve it seems to exist. Hence enumerative methods, while guaranteeing the

global optimum, break down on difficult instances of the problem. Heuristics have been developed toavoid the doom of enumerative methods, but they are subject to trapping in local extrema of the costfunction associated with the problem, sometimes giving poor results.

This paper is organised as follows. We briefly mention work related to ours in Section 2. We thendescribe in Section 3 the philosophy and principles of Grouping Genetic Algorithms (GGA), a class ofalgorithms well suited to Grouping problems. Section 4 is devoted to the description of the problem tobe solved, and the description of heuristics used in the GGA. Implementation details are explained inSection 5. Results of our algorithm will be given at Section 6 together with our conclusions.

2. Related works

The cell formation problem has been studied using different formulations. A survey of approaches forconfiguring the groups is given in [15].

In the matrixformulation, where a binarymachine-part incidencematrix [aij] is constructed. An elementaij will be equal to 1 if part i is processed on machine j, 0 otherwise. There are several proceduresto solve this matrix formulation of the GT problem, like Production Flow Analysis [3,5], the use ofSimilarity Coefficients (SC) (like the Single Linkage Cluster Analysis (SLCA) [16] or Average LinkageClustering (ALC) [20]), matrix rearrangement procedures (Rank Order Clustering (ROC) [13], DirectCluster Algorithm (DCA) [6], Bond-Energy Algorithm (BEA) [17]). A comparison of these algorithmsor their variations is given in [18]. Several graph decomposition techniques were applied to solve theproblem, e.g., a variation of the Kernighan and Lins heuristic [12] developed by Askin and Chiu [1], orHaralakis [10] heuristic.

Simulated annealing [14] was applied to the cell formation problems ([19]), using the formalism

described at Section 4.1. The neighbourhood of a partition Cof the set of machines Mis defined as theset of partitions derived from Cby switch of two machines in two different cells, creation of a new cell


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P. De Lit et al. / Mathematics and Computers in Simulation 51 (2000) 257271 259

by extracting a machine from a given one, or reattribution of a machine from a cell to another. Thissimulated annealing will more rarely fall in local extrema than heuristics, but has the major drawbackto be extremely slow. However the approach vehicles the interesting idea to accept bad attributions orswitches of machines to escape from local optima.

Several programming formulations have also appeared in the literature e.g., [2], but computationalefficiency is most of the time a prohibiting factor for exact methods.

3. The grouping genetic algorithm

3.1. The grouping problems

The grouping problems constitute a large family of problems, many of them naturallyarising in practice,which consist in partitioning a set Uof items into a collection of mutually disjoint subsets Ui ofU, i.e.,such that:

i Ui = U

Ui Uj = i = j.

One can also see these problems as ones where the aim is to group the members of the set Uinto one ormore (at most card(U)) groups of items, with each item in exactly one group, i.e., to find a grouping ofthose items.

In most of these problems, notall possible groupings are allowed: a solution of the problem must complywith various hard constraints, the solution being otherwise invalid. That is, usually an item cannot begrouped with all possible subsets of the remaining ones.

The objective of the grouping is to optimise a cost function defined over the set of all valid groupings.This cost function depends on the composition of the groups, that is, where one item taken separately haslittle or no meaning.

3.2. The method

Introduced by J. Holland [11], the Genetic Algorithm (GA) is an optimisation technique inspired bythe process of evolution of living organisms. The basic idea is to maintain a population of chromosomes,each chromosome being the encoding (a description or genotype) of a solution (or phenotype) to theproblem being solved. The worth of each chromosome is measured by its fitness, which is often simplythe value of the objective function in the point of the search space defined by the (decoded) chromosome(in a maximisation problem).

Starting with an initial population generated mostly at random, the GA proceeds in quite the samemanner as Nature in evolving ever better solutions: chromosomes with high fitness are crossed over,producing progeny that replaces chromosomes with low fitness. A low rate of mutation, a small randommodification of a chromosome, is applied to prevent a premature convergence to a local optimum. A goodintroduction to GAs is given in [9].

E. Falkenauer [7] pointed out the weaknesses of standard GAs when applied to grouping problems

and introduced the Grouping Genetic Algorithm (GGA) [8], a GA heavily modified to match the struc-ture of grouping problems. The GGA differs from the classic GA in two important aspects. First, a


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special encoding scheme is used in order to make the relevant structures of grouping problems becomegenes in chromosomes. Second, given the encoding, special genetic operators are used, suitable for thechromosomes.

3.3. The encoding

The standard genetic operators are not suitable for grouping problems. The reason is that the structureof the simple chromosomes (which the above operators work with) is item oriented, instead of being grouporiented. In short, the encoding in standard GAs are not adapted to the cost function to optimise. Indeed,the cost function of a grouping problem depends on the groups, but there is no structural counterpart forthem in the chromosomes of the standard GAs.

The GGA uses an specific encoding scheme: the standard chromosome is augmented with a group part,encoding the groups on a one gene for one group basis. More concretely, let us consider a chromosome ofa standard GA. Numbering the items from 0 through 5, the item part of the chromosome can be explicitlywritten

0

A1

D2

B3

C4

E5

B : . . .

meaning the item 0 is in the group labelled (named) A, 1 in the group D, 2 and 5 in B, 3 in C, and 4 in E.

The group part of the chromosome represents only the groups. Thus:

. . . : BECDA

expresses the fact that there are five groups in the solution. Of course, what names are used for each ofthe bins is irrelevant in our grouping problem: only the contents of each group counts in this problem.We thus come to the raison dtre of the item part. Indeed, by a lookup there, we can establish what thenames stand for. Namely,

A = {0}, B = {2, 5}, C = {3}, D = {1} and E = {4}.

In fact, the chromosome could also be written

{0} {2, 5} {3} {1} {4}.

The important point is that the genetic operators will work with the group part of the chromosomes,the standard item part of the chromosomes merely serving to identify which items actually form whichgroup (note that this implies that the operators will have to handle chromosomes of variable length). Therationale is that in grouping problems it is the groups which are the meaningful building blocks, i.e., thesmallest piece of a solution that can convey information on the expected quality of the solution they arepart of. Note finally that the order of the groups in the chromosome is irrelevant in the GGA.

3.4. The crossover

Given the fact that the hard constraints and the cost function vary among different grouping problems,the ways groups can be combined without producing invalid or too bad individuals are not the same for

all those problems. Thus, the crossover used will not be the same for all of them. However, it will fit thefollowing pattern, illustrated in Fig. 1:


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Fig. 1. The GGA crossover.

1. Randomly select two crossing sites, delimiting a crossing section, in each of the two parents.2. Inject the contents of the crossing section of the first parent at the first crossing site of the second

parent. This means injecting some of the groups from the first parent into the second.3. Eliminate all items now occurring twice from the groups they were members of in the second parent,

so that the old membership of these items gives way to the membership specified by the newinjected groups. Consequently, some of the old groups coming from the second parent are altered:they do not contain all the same items anymore, since some of those items had to be eliminated.

4. If necessary, adapt the resulting groups, according to the hard constraints and the cost function tooptimise. At this stage, local problem-dependent heuristics can be applied.

5. Apply points 24 inverting the roles of the two parents in order to generate the second child.

As can easily be seen, the idea behind the GGA crossover is to promote promising groups by inheritance.We describe in Section 5.4.1 the adaptation of the crossover operator to our grouping problem.

3.5. The mutation

A mutation operator for grouping problems must work with groups rather than items. As for thecrossover, the implementation details of the operator depend on the particular grouping problem on hand.Our mutation operator is described at Section 5.4.2.

3.6. The inversion

The inversion operator serves to shorten good schemata in order to facilitate their transmission from

parents to offspring, thus ensuring an increased rate of sampling of the above-average ones ([11]). In aGrouping GA, it is applied to the group part of the chromosome.


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Thus for instance, the chromosome

ADBCEB : BECDA

could be inverted into

ADBCEB : CEBDA.

The example illustrates the utility of this operator: should B and D be promising genes (i.e., well-performing groups), the probability of transmitting both of them during the next crossover is improvedafter the inversion, since they are now closer together, i.e., safer against disruption.

4. Description of the cell formation problem (CFP)

The CFP involves the grouping of parts into families and the grouping of machines into cells, and theassignment of part families to machine cells.

The formulation we propose will lead to the decomposition of the set of machines into cells, thisdecomposition fixing the product families and their attribution to cells. In our formalism, the hard task

is to propose the cell decomposition. The forming of part families is immediate once cells have beenformed (we supposed that a family was attributed to a cell, but the generalisation to group of cells isstraightforward). In the following section, we formally describe the decomposition of the set of resourcesinto cells and the subsequent grouping of parts into families.

4.1. Mathematical formulation of the CFP

Let us consider the sets P={(p0,u0),. . . ,(pi ,ui ),. . . (pn1,un1)} and M={m0,...,mj,...,mm1} withcard(P) = n, card(M) = m, and ui R

Let {rk(pk )} be a suite of elements mkj M, and C= {C0,...,C1} a partition ofMwith card(C) = .

Let us define for pk,xklm as the number of times m

kj Cm is immediately preceded by m

ki Cl in {rk},

mki , mkj with Cl, Cm Cand m = l. We call traffic between the two partitions:

Tlm =

n1k=0

uk(xklm + x

kml).

Note that one can represent these traffics between the subsets with an matrix, (Tlm). Let us finallyintroduce Nmax .

The problem is to find the partition C = {C0,... ,C1

} minimising

2i=0

1j=i+1

Tij with card(Ck C

) Nmax.

The traffic between two partitions Cl , Cm Ccan also be defined as follows. Let ykij be for pk the numberof times mkj is immediate successor ofm

ki in rk (pk). We call traffic between two elements m

ki and m

kj:


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tij =

n1k=0

uk (ykij + y

kji ).

The traffics between elements are independent from partition C and can be represented as an m mmatrix. For a given partition C={C0,...,C1}, the traffic between the subsets is given by:

Tij =

mk Ci

ml Cl

tkl , k = l.

An application to production systems of the above presented formalism is the following:P is the set of couples (product, product weight), ui being a production volume or a cost factor;

Mis the set of machines on the workshop;rkis the manufacturing sequence of product pk;Cis the partition of the set of machines, each Ci representing a cell;xkij is the number of times a machine in cell Cj is immediate successor of a machine in cell Ci during

the manufacturing sequence of product pk;ykij is the number of times a machine m

kj is immediately preceded by m

ki in the manufacturing sequence

of product pk;

Tij is the inter-cell traffic, and tij the inter-machine traffic;Nmax is a maximum number of machines allowed in a cell.

The problem can be seen as the partition of an undirected weighted graph, the nodes representing themachines and the weighted edges the traffic between these resources. Each subgraph will represent agroup. The aim is to find the minimal cut, restraining the size of each subgraph to Nmax nodes. So our cellformation problem is more a decomposition problem, sometimes named workshop cell decompositionproblem.

Note that this formulation takes the following aspects into account: maximal size of a cell, productionvolume of the different products, possibleloops in the manufacturingsequence(e.g., 1 2 4 1 5).

Until now, the only constraint we considered was the size of the cells. Several constraints can be takeninto account: machines that should or must be grouped together or resources that must not or should notbe allocated to the same cell. User preferences can be considered as a supplementary traffic tij between

machines (which may be negative if one prefers not to group some machines together):

tij = tij + tij.

Hard associative constraints can be expressed by considering each set of associated machines S as aunique resource with a size s = card(S). The traffic between the fictive machine created and the other onesbecomes:

tkl =mS

tml.

Hard dissociative constraints are satisfied thanks to a check when we try to allocate a machine to a group.This test does not influences the methods proposed to tackle the generic problem.

As these constraints do not change the way to tackle the problem, nor the algorithms used, we will nottake them into account into the further description of our algorithm for sake of clarity.


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Table 1

Manufacturing sequences and production volume for six products

Product Sequence Volume

p0 0, 3, 2, 3 2

p1 7,8 2

p2 2, 4, 1, 6 3

p3 5, 4, 6 1

p4 5, 7, 8 4

p5 0, 2, 1, 4 2

4.2. Allocating parts to cells and forming part families

The part families will easily be determined after machine clustering. Suppose we allocate a part to asingle cell. It will be attributed to the cell in which it provokes the most important traffic (the extensionfrom a cell to several ones in straightforward). Parts assigned to the same cell will form a family.

As an example let us consider the following problem. We try to group nine machines into cells con-taining at most three machines. Eight products have to be manufactured, with the following sequence andproduction volume, presented in Table 1. The weighted graph corresponding to this problem is shown in

Fig. 2.The optimal solution yields the groups and part families given in Table 2. Note that part p5 could either

be allocated to cell C0 or C2. We obtain three cells and three part families:

P F0 = {p0, p5}, P F1 = {p1, p4}, P F2 = {p2, p3}.

In the following, we will focus on the allocation of machines to cells.

Fig. 2. Weighted traffic graph for the example.

Table 2

Results of the CFP

Cell Machines Parts

C0 0, 2, 3 p0, p5

C1 5, 7, 8 p1, p4C2 1, 4, 6 p2, p3


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5. Algorithm implementation

5.1. Pitfalls for heuristics

Two heuristics influenced our work. G. Haralakis et al. [10] have proposed a simple heuristic tominimise the inter-cell traffic, which is divided into two phases: an aggregation and a local refinement.

At the beginning of the aggregation each machine is in a cell. The possible aggregations are the onenot exceeding the maximum cell size. The two cells between which there exists the highest traffic aregrouped. After this aggregation, a refinement phase tries to convert the inter-cell traffic into intra-celltraffic. Each machine is considered as a separate entity, and its traffic with each cell is computed. A givenmachine is attributed to the group it has the most important interaction with. Note that most of the timea machine will be reattributed to its cell, but some changes may occur.

This algorithm is simple, but is a heuristic and does not always yield the optimal solution. The smallestproblem which the algorithm is deceived by is illustrated in Fig. 3. The optimal solution yields groups{1,3,4} and {2} and an inter-cell traffic of 7.

Another popular heuristic for graph partitioning is Kernighan and Lins heuristic ([6]), which can beadapted to multiple groups and variable group sizes (e.g., [7]). This procedure starts from a given partitionand first tries to find the best possible swap between the groups (swap may concern subsets of the two

partitions). Once the best swap has been performed, the process restarts. This heuristic yields good results(for example, it will solve the deceptive problem illustrated in Fig. 3), but is inefficient if the improvementof a partition needs a swap between more than two groups. On the following example, showed in Fig. 4, the

Fig. 3. Minimal deceiver for Haralakis heuristic (Nmax =3).

Fig. 4. Two cells swap deceptive problem.


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heuristic will not be able to lead to the optimal solution, no swap improving the proposed decomposition(this first decomposition was obtained using Harhalakis aggregation procedure). The optimal solutionasks for a swap of three machines together (1 from G0 to G2, 4 from G2 to G1, and 7 from G1 to G0).

The two above described heuristics were adapted in our algorithm, to add randomness in the localoptimisations our GGA performs.

5.2. Hard constraints

We fixed two conditions for an individual to be valid: the size of the cells may not exceed Nmax, andthe subgraphs associated to the different cells must be connected. Note that this constraint influencesthe number of groups proposed in the optimal solution (we could otherwise propose to group machineswithout traffic between them), but has no influence on the quality of the solution according to our costfunction.

5.3. Cost function

The intracell traffic for a cell Ci is:

Ti =kCi

lCi

tlk

2 .

As the total traffic between machines stays constant (Ttotal), the inter-cell traffic is given by:

Tinter = Ttotal Tintra.

So minimising the inter-cell traffic is equivalent to maximising the total intracell traffic, given for q cellsby:

max (Tintra) = max

q1i=0

Ti

.

This cost function is well adapted to our problem, the evolution of the traffic only depending on affectedcells during a perturbation.

5.4. Genetic operators

5.4.1. Crossover

Crossover is applied as described in Section 3, with the difference that affected groups are emptied.Re-injection of the machines uses the following heuristic. The traffic between a chosen (by to choose,we mean to draw lots) non attributed machine and the existing cells is computed, and this machine isattributed to a group with a probability pro rata of this traffic. Note that the complexity of this heuristicis O(x2). We create a new cell if none can accept a chosen machine trice in a row. Fig. 5 illustrates ourwords. The traffics between machine 4 and groups 1 and 2 amount to 24 and 36, respectively. The machine

will be injected in group 1 and with a probability of 0.4 for the former and 0.6 for the latter. The samereasoning goes for machine 3. This one having no connections with the existing cells, a new one is created


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Fig. 5. Machine reattribution heuristic.

to greet it. This ranking and selection of the greeting cells aims to help the GGA to leave local optima itcould get stuck in.

After having applied the heuristic described above, if the GGA seems to be stuck in a local optimum, wesearch for the best possible swap of two machines that reduces the inter-cell traffic. If there is none, a swapworsening the solution will sometimes be applied, to help the GGA to leave that local optimum. Notethat this probabilistic swap is crucial in case of problems requiring cyclic swaps like the one illustratedin Fig. 4. Without this heuristic, the algorithm gets stuck in local optima for about 1000 generations

for medium-sized instances of the problem (about 300 machines; these instances are in fact severalindependent elementary problems presented in Fig. 4).

Note that no reproduction is applied: half the population is crossed at each generation.

5.4.2. Mutation

The mutation operator is only applied if the crossover does not generate a new individual in thepopulation. The mutation removes one tenth of theobjects among thegroups andre-injects them accordingto the heuristic we described in Section 5.4.1.

5.5. Generating the first population

The first individual of the population is generated by Haralakis aggregation procedure, the others using

heuristic described at Section 5.4.1. These aggregations are followed by a swapping heuristic: if a grouphas external connections, the best swap between it and all others is made. The swap will occur onceper group (so a given group will not undergo two successive swaps). This first population generationprocedure insures us to find the global optimum at initialisation if a perfect decomposition is possible.The reader could object that the generation of an individual which could sometimes be highly better fitthan the others will lead to a broad spreading of schemata belonging to a suboptimal solution amongthe population. Tests on deceptive problems indicate that the proposed genetic operators make the GGAforget the effect of Haralakis aggregation at population initialisation.

6. Experimental results

For all simulations, performed on a 266 MHz Pentium, size of the population was 30 individuals.Results presented here are mean and standard deviation values of 20 runs.


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Fig. 6. Results for perfect decomposition without Haralakis heuristic. The left graphic presents the time to optimum. The right

one represents the generations to optimum.

6.1. Perfect decompositions

Those simple problems, for which Haralakis heuristic always converges to the optimum, allow the

evaluation of the algorithm on single mode functions. To avoid finding the optimal solution at evaluation,we suppressed Haralakis procedure and machine swaps at initialisation. The evolution of the computingtime and generations to the optimum according to the size of the problem are given in Fig. 6.

6.2. Deceptive problems

6.2.1. Haralakis minimal deceiver

We studied Haralakis minimal deceiver to see the effects of the swapping heuristic on the search ofthe optimum. We disabled Haralakis heuristic at initialisation, as it leads to the optimal solution whenassociated with initialisations swap heuristic. Results are reported in Fig. 7. One can see that that the

Fig. 7. Results for minimal deceivers with andwithout swap heuristic(Nmax = 3). The left graphic represents the time to optimum,

while the right one presents the generations to optimum.


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Fig. 8. Deceptive problem.

Fig. 9. Results for thecompleteproblem. (Nmax = 5). The left graphic represents thetime to optimum theright one thegenerations

to optimum.

swap heuristic enabled after 10 generations without improvement increases the speed to reach the optimalsolution.

6.2.2. Complete problem

The graph associated to the elementary deceptive problem we studied is given in Fig. 8. Optimal solution,for Nmax = 5 yields groups {0,3,4,5,6} and {1,2}. Note that the application of Haralakis followed by theswap heuristic is ineffective in this case, because the algorithm will swap machines 2 and 3. The instancesare composed of independent elementary deceivers. The results in Fig. 9 show that the GGA is able todeal with important instances of the problem in reasonable amount of time.

Note that the important dispersion on the results is due to the fact that the swap heuristic is triggered

when the GGA has not improved the solution for 30 generations. This heuristic being time consuming,it notably increases computation time when it is enabled.


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7. Conclusions

In this paper, we addressed the cell formation problem (CFP), which is an important aspect of GroupTechnology (GT). We represented the set of machines by an undirected weighted graph. The problemto tackle becomes finding the partition which gives the minimal cut for a given maximal size of eachsubgraph. Since this problem isNP-hard, enumerative methods will crash down for important instances of

it. We thus proposed a Grouping Genetic Algorithm which does not present this major drawback, makingit applicable to industrial cases, and offering the advantage not to be stuck in local optima like heuristics.Further research on the subject will deal with machine sizes, to deal with placement constraints on the

shop floor. Several routing will also be allowed for the products. The objective will then be a compromisebetween minimum inter-cell traffic and size of equipment in a cell. The maximum cell size constraintwill also be released.

Acknowledgements

This paper is based on results of the project Outils daide la conception interactive des produitset de leur ligne dassemblage. This project is made in collaboration with the Universit Catholique

de Louvain (UCL), the Facult Polytechnique de Mons (FPMs), and the Belgian Research Center forthe metalworking industry (CRIF). We particularly thank the Rgion Wallonne which has funded thisproject.

References

[1] R.G. Askin, K.S. Chiu, A graph partitioning procedure for machine assignment and cell formation in group technology,

Int. Jour. Prod. Res. 24(3) (1990) 471481.[2] F.F. Boctor, A linear formulation of the machine-part cell formation problem, Int. J. Prod. Res. 29(2) (1991) 343356.[3] J.L. Burbidge, Production flow analysis, Prod. Eng. (1971) 139152.[4] J.L. Burbidge, The Introduction of Group Technology, Wiley, New York, 1975.[5] J.L. Burbidge, Production Flow Analysis, Clarendon Press, Oxford, 1989.[6] H.M. Chan, D.A. Milner, Direct clustering algorithm for group formation in cellular manufacturing, J. Manuf. Syst. 1(1)

(1982) 6574.[7] E. Falkenauer, A hybrid grouping genetic algorithm for bin packing, J. Heuristics 2(1) (1996) 530.[8] E. Falkenauer, Genetic Algorithms and Grouping Problems, Wiley , 1998.[9] D.E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, 1989.

[10] G. Harhalakis, R. Nagi,J.M. Proth, An efficient heuristic in manufacturing cellformationfor group technology applications,

Int. J. Prod. Res. 28(1) (1990) 185198.[11] J.H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, 1975.[12] B.W. Kernighan, S. Lin, An efficient heuristic procedure for partitioning graphs, The Bell Syst. Tech. J. 49 (1970) 291307.[13] J.R. King, Machine-component group formulation in production flow analysis: an approach using a rank order clustering

algorithm, Int. J. Prod Res. 18(2) (1980) 213222.[14] S. Kirkpatrick, C.D. Gelatti, M.P. Vecchi, Optimization by simulated annealing, Science 220 (1983) 671680.[15] A. Kusiak, W.S. Chow, Decomposition of manufacturing systems, IEEE J. Robotics Automation 4(5) (1988) 457471.[16] J. McAuley, Machine grouping for efficient production, Prod. Eng. (1972) 5357.[17] W.T. McCormick, P.J. Schweitzer, T.W. White, Problem decomposition and data reorganization by cluster technique, Oper.

Res. 20(5) (1972) 9931009.[18] J. Miltenburg, W. Zhang, A comparative evaluation of nine well-known algorithms for solving the cell formation problem

in group technology, J. Op. Manage. 10(1) (1991) 4472.


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[19] B. Pommerenke, Choix optimal des cellules de production par un algorithme gntique de groupement, Travail de fin

dtudes prsent en vue de lobtention du grade dIngnieur Civil Physicien, Universit Libre de Bruxelles, 1995.

[20] H. Seifoddini, P.M. Wolfe, Application of the similarity coefficient method in group technology, IIE Trans. 18(3) (1986)

271277.

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