group technology based adaptive cell formation using predator–prey genetic algorithm
TRANSCRIPT
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Applied Soft Computing 12 (2012) 559–572
Contents lists available at ScienceDirect
Applied Soft Computing
j ourna l ho mepage: www.elsev ier .com/ locate /asoc
roup technology based adaptive cell formation using predator–prey geneticlgorithm
ndranil Banerjeea,∗, Prasun Dasb
Videonetics Technology Pvt. Ltd., Kolkata 700 091, IndiaSQC and OR Division, Indian Statistical Institute, Kolkata 700 108, India
r t i c l e i n f o
rticle history:eceived 29 June 2010eceived in revised form 24 July 2011ccepted 27 July 2011vailable online 11 August 2011
eywords:ellular manufacturingell formation problemredator–prey genetic algorithm
a b s t r a c t
In recent years, there has been a considerable growth of application of group technology in cellularmanufacturing. This has led to investigation of the primary cell formation problem (CFP), both in classicaland soft-computing domain. Compared to more well-known and analytical techniques like mathematicalprogramming which have been used rigorously to solve CFPs, heuristic approaches have yet gained thesame level of acceptance. In the last decade we have seen some fruitful attempts to use evolutionarytechniques like genetic algorithm (GA) and Ant Colony Optimization to find solutions of the CFP. Theprimary aim of this study is to investigate the applicability of a fine grain variant of the predator–prey GA(PPGA) in CFPs. The algorithm has been adapted to emphasize local selection strategy and to maintaina reasonable balance between prey and predator population, while avoiding premature convergence.
iagonal clusterottleneck
ntra-cluster densityransitional costrouping efficacy
The results show that the algorithm is competitive in identifying machine-part clusters from the initialCFP matrix with significantly less number of iterations. The algorithm scaled efficiently for large sizeproblems with competitive performance. Optimal cluster identification is then followed by removal ofthe bottleneck elements to give a final solution with minimum inter-cluster transition cost. The resultsgive considerable impetus to study similar NP-complete combinatorial problems using fine-grain GAs infuture.
. Introduction
Cellular manufacturing (CM) arises from the idea of decompos-ng a manufacturing process into sub-processes or groups (cells),n order to allocate similar resources to process a task family. Iteads to an improved workflow with reduced material cost, reduc-ion in machinery cost, better quality control, reduced set-up timend more efficient labor utilization to say a few. Group technol-gy (GT) based clustering has been widely used in the area ofM. The three-fold objectives of GT are to (i) perform similar jobsogether, (ii) standardize similar tasks, and (iii) effectively store andetrieve information about recurring problem [1–4]. In the simplestase, the objective is to group machines with similar character-stics together which can process a family of parts. The grouped
achines form cell which can process a family of parts within itsogistical boundary. Thus, the family of parts can be wholly pro-
essed within this logistical boundary, minimizing the need foroving parts to distance processing units (machines), hence reduc-ng the inter-cell movement of parts. In this way, transitional cost
∗ Corresponding author. Tel.: +91 9830407670.E-mail addresses: [email protected] (I. Banerjee), [email protected]
P. Das).
568-4946/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.asoc.2011.07.021
© 2011 Elsevier B.V. All rights reserved.
and overall production cost can be reduced. In the ideal case, eachcell of a machine is completely dedicated to a part family. Since forreal world manufacturing, the workflow of processes is complex,hence it usually does not lead to an independent cell formation. Theproblem, therefore, becomes optimizing the cell allocation so thatinter-cell transition is minimized while maintaining good cell den-sity. By cell density, we mean that the group of machines, whichforms the cell, is utilized by most of the parts in the parts family,thereby minimizing redundancy.
Predator–prey based approaches have served as a good alter-native to conventional GA, one reason being able to maintain localdiversity (through predator–prey selection strategy) in populationwithout compromising on global minima. Deb and Uday [5]; Li [6]showed two parallel approaches in applying predator prey strategyin multi-objective environment. In this study, a modified version ofPPGA has been developed in order to function it on a single objectiveenvironment, without curbing its ability to maintain good solutiondiversity. Solutions to CFPs have been proposed using this modifiedPPGA on binary incidence matrices. The use of binary incidencematrix for the CFP means that the complexity of problem space
gets reduced; hence giving more focus to developing the PPGA andstudy its applicability in this domain. So, the authors of this paperrestrict the study to binary incidence matrix based CFPs and donot incorporate alternate routings and other modeling parameters.
560 I. Banerjee, P. Das / Applied Soft Co
p1 p2 p3 p4 p5 p6 p7
m1 1 1
m2 1 1
m3 1 1 1 1
m4 1 1
m5 1 1 1
m6 1 1 1
m7 1 1
Iffhd
cmgcpS
2
puiCibs
a
zomawitembdoonnmmabtdsp
Fig. 1. A typical machine-part binary incidence matrix.
n this respect, a new grouping efficacy has also been developedor the binary incidence matrix based CFPs and used as the fitnessunction for the PPGA. After finding the optimal solution, an effortas been made in identifying bottleneck parts or machines, if any,epending on the cost measure developed.
The rest of paper is organized as follows. Section 2 defines theell formation problem and explains the proposed performanceeasure in relation to other studies in this domain. Section 3
ives a detailed description of the modified PPGA and its asso-iated nomenclature. In Section 4, results with fourteen testroblems have been analyzed, and the paper is concluded inection 5.
. Cell formation problem and performance metric
The machine-part cell formation problem (CFP) is NP com-lete [7], and has been studied extensively in classical domainsing matrix based, hierarchical, graph-theoretic clustering and
nteger programming. The main objective of group technology inFP is to identify machine cells and part family, while minimiz-
ng intercellular movement of parts of part families. In Fig. 1, ainary incidence matrix with seven machines processing six parts ishown.
Elements of the matrix are given by,
ij ={
1, if machine i processes the part j0, otherwise
The elements of the matrix are also called operations; hereeros are not shown for the sake of clarity. In the binary versionf the CFP, the main objective is to form completely independentachine cells, associated with independent part families, such that
part belonging to a particular part family is exclusively associatedith one machine cell only. In matrix or array based clustering,
nitial incidence matrix is transformed through iterative transposi-ion of rows and columns until clusters are identified. McCormickt al. [8] used heuristic Bond Energy Algorithm (BEA) to maxi-ize the sum of bond energies of elements of the incidence matrix
y taking some permutation of rows and columns. This methodid not give optimal result under existence of bottleneck partsr machines. Rank order clustering (ROC) [9] is another examplef such re-ordering algorithm of rows and columns. These tech-iques usually suffer due to the non-deterministic and overlappingature of machine cells and part families and existence of one orore bottleneck elements. Bottleneck elements are residual ele-ents, which cannot be further grouped in cells. In graph theoretic
pproach, the incidence matrix is represented by a graph (usuallyipartite graph), where a machine and part has an edge between
hem if the part is processed by the machine in the initial inci-ence matrix. In one such application, Kumar et al. [10] used a twotage approach: initial clusters are formed first by solving a trans-ortation problem and then it is further optimized. A mixed integermputing 12 (2012) 559–572
programming (IP) model with alternative process plans have beenproposed by Rajamani et al. [11]. Shafer et al. [12] proposed anovel mathematical programming based approach to model thecost associated in dealing with exceptional elements, which formbottlenecks in CFPs. Recently, an integrated approach consider-ing CFP, process planning, and production planning together issolved using a combination of IP and heuristic based approach[13].
In the past decade and a half one has witnessed a consid-erable acceleration in application of the heuristic methodologiesto solve CFPs. These soft-computing techniques encompass fuzzytheory, artificial neural networks (ANNs), evolutionary strategiesand other related meta-heuristic tools. Kusiak and Chung [14]applied a 2-layer Carpenter-Grossberg neural network to solveCFP. Here, the input layer acts as comparison layer and the outputlayer as the recognition layer. In more recent work, Miljkovic andBabic [15] developed adaptive resonance theory (ART) and ANNbased clustering to investigate manufacturing similarity and cre-ate machine-part cells. Yang and Yang [16] proposed a modifiedART based approach which adaptively sets the vigilance param-eter for the ART network. There has been parallel developmentto solve CFPs using fuzzy theory. Most of these approaches try toovercome the overlapping nature of machine-part cluster by usingfuzzy modeling. Gindy et al. [17] assesses the applicability of fuzzymodeling with a new efficacy measure, which tries to incorpo-rate both intra-cluster compactness and inter-cluster overlapping.In a more constrained environment considering various process-ing requirements like intra part dependency and processing times,Narayanaswamy et al. [18] applied fuzzy modeling to solve multi-criteria CFPs.
GA based techniques are inherently more robust since thecluster formation is done by considering machine and partssimultaneously. In recent years there have been some noticeableattempts to solve different types of CFPs using GA and other similartechniques like simulated annealing (SA) and differential evolution.Joines et al. [19] explored the CFP using integer programming inconjunction with GA. They also showed that there is only a lin-ear scaling in complexity in a GA based search algorithm, withrespect to increase in dimension of CFP incidence matrix and max-imum number of cells that can be formed. Boulif and Atif [20]used a binary coded branch and bound genetic algorithm, whichthey showed it to be quite effective for constrained CFPs of mod-erate complexity. Grouping based GAs also found to be applicablein CFP domain [21,22]. Grouping genetic algorithms (GGAs) havebecome quite popular in the niche domain of partitioning prob-lems, clustering problem. A good example would be the NP-hard‘bin picking’ problem where one has to place a number of items indifferent bins without exceeding its capacity, while using optimalnumber of bins. James et al. [21] applied GGA with the additionalintroduction of a local search mechanism to enhance the diver-sity of population. In another application of local search heuristic,Conc alves and Resende [23] used it in a feedback construction toassist the GA based search. Using simulated annealing (SA), Wuet al. [24] proposed another heuristic algorithm for CFP. In a morerecent development, Noktehdan et al. [25] explored CFPs usingdifferential evolution. They extended the concept of GGAs in dif-ferential evolution, and showed their strategy to be quite effectivein dealing with different types of problems found in the litera-ture.
Two solutions of the CFP (ref. Fig. 1) are shown in Fig. 2. For mostof the real world problems in CM, it is very difficult to find separableclusters, and in some cases they do not exist. The incidence matrix
of Fig. 1 is one such case. A cell formation (here ‘cell’ means thecluster formed by machines cell and parts families, and it is differentfrom machine cells) solution of the incidence matrix is given inFig. 2.
I. Banerjee, P. Das / Applied Soft Computing 12 (2012) 559–572 561
p3 p5 p6 p1 p4 p2 p7
m3 1 1 1 1
m5 1 1 1
m6 1 1 1
m1 1 1
m7 1 1
m2 1 1
m4 1 1
p2 p7 p1 p3 p4 p5 p6
m2 1 1
m4 1 1
m1 1 1
m3 1 1 1 1
m5 1 1 1
m6 1 1 1
m7 1 1
g. 1 an
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s
a
s
iw(ocwpcttSeetcsaot
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(a) Fig. 2. (a) Rearranged incidence matrix of Fi
In Fig. 2(a), 3 machine cells and corresponding part families aredentified as,
ol1 =∣∣∣∣∣{m3, m5, m6|p3, p5, p6}
{m1, m7|p1, p4}{m2, m4|p2, p7}
And, in Fig. 2(b), 2 machine cells and corresponding part familiesre identified as,
ol2 =∣∣∣∣ {m2, m4|p2, p7}{m1, m3m5, m6, m7|p1, p3, p4, p5, p6}
The sol1 of Fig. 2(a) is an optimal solution, if the only criterions to maximize intra-cluster density and utilization of machines
ithin a cell. In sol1, part 4 is processed by machines of both cell-1m3) and cell-2 (m1, m7). The elements which lie outside of the diag-nal blocks are called exceptional elements. When a part becomesommon to two independent groups of machines, and cannot beholly processed within one machine’s cell and thus forcing theart to ‘travel’ in the job-shop environment. This results in inter-ellular movements of parts. One of the goals of CM is to minimizehe inter-cellular movements of parts, also referred to as transi-ional cost or carrying cost of parts from one machine to another.o, the objective must include minimization of these exceptionallements as much as possible. In sol2, it is seen that there is noxceptional element, but there has been a trade-off in intra clus-er density, i.e., there are more number of parts in a machine-partluster which are not processed by machines. Though, both theolutions are optimal with respect to their objectives, in practice,
compromise solution need to be found with respect to both thebjectives simultaneously. A common formula [26] for measuringhis degree of cluster formation (called grouping efficacy) is,
0 = e − ee
e + ev(1)
here,
e = total number of operation of incidence matrixee = total number of diagonal entries outside the diagonal clusters(exceptional elements)ev = total number of zero entries inside the diagonal elements(voids)
In this study, a new grouping efficacy measure is proposed. Inq. (1), if both ee and ev are zero, i.e., perfectly formed clusters witho exceptional elements then ˝0 = 1. This efficacy measure has two
(b) d (b) rearranged incidence matrix of Fig. 1.
main deficiencies: (i) the two objectives; maintaining good opera-tion density within a cluster and, secondly, to minimize the numberof inter-cellular movements of parts, are combined in non-linearway. It is difficult to determine the degree of influence of each sep-arate objective in measuring the efficacy of a solution; (ii) sincethe two objectives are non-linearly combined there is no possibleway of guiding the GA search in favour of a particular objective.In this study, a linear combination of the two objectives is used asgrouping efficacy. It is given as,
= (1 − (ed/D))2︸︷︷︸f1
+ ((e/ed) − 1)2︸︷︷︸f2
(2)
where,
ed = Total number of 1’s (i.e., operations) inside diagonal clustersD = Total size of diagonal clusters
Here, the first term inside the braces represents the first objec-tive (f1). If there are no voids inside the diagonal blocks, then ed = D,and f1 becomes zero. Similarly, if there are no exceptional ele-ments, i.e., all the elements contained inside the diagonal blocksthen ed = e and f2 becomes zero, thus = 0. It is found from simula-tion study that this representation of grouping efficacy gives moredegree of freedom in GA search space, and controls over type ofoptimal solutions to be searched.
In most of the practical problems, the incidence matrix is suchthat no optimal solution can be found with respect to independentcells. Some exceptional elements will always remain. In such cases,some parts or machines cannot be grouped into any of the cellsformed in order to remove the exceptional elements, and are knownas bottleneck elements. If the bottleneck is avoided by removingparts or machines containing exceptional elements, the bottleneckcondition is known as part or machine bottleneck, respectively.After clusters are found (corresponding to the minimum value of˝) having exceptional elements, a cost function Cbtnk is used todetermine which type of bottleneck condition may give the bestresult in terms of maintaining good cluster size and intra-clusterelement density. It is defined as,
Cbtnk = eem/Nm
eep/Np(3)
where,
eem = number of machine having exceptional elementseep = number of parts having exceptional elements
562 I. Banerjee, P. Das / Applied Soft Co
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wvsme
3
sptbfiptphgbmdpicntGukpptbtgepo
ig. 3. Prey and predators in a two-dimensional grid, in each iteration a predatorries to kill the least fit prey in its 8 – connected (Moore) neighborhood, this structurean also be represented by a un-directed graph.
Nm = total number of machinesNp = total number of parts
If Cbtnk ≥ 1, then column or part bottleneck is considered, other-ise machine bottleneck is assumed. In case if bottleneck exists, a
alue of Cbtnk, close to either side of 1 will give relatively low gradeolution with respect to both types of bottleneck. After this, parts orachines (depending on the type of bottleneck) having exceptional
lements are removed from the formed clusters.
. Predator–prey evolutionary model
The basic predator–prey strategy comes from ecology, achematic model which is shown in Fig. 3. This model was firstroposed by Li [6]. In this model, an individual in prey popula-ion represents a solution. Individual preys in the population canreed and expand or get killed by predators according to its rank ortness. The preys represent the evolutionary active agents in thisredator–prey model. On the other hand, predators help to controlhe prey population while creating selection pressure in the preyopulation. A predator tries to kill the least fit prey in its neighbor-ood, thus removing (local) bad solutions and preserving (local)ood solutions in the population. This does not affect the globalest individuals, as they are also selected as local best solutions forating. This unique local selection helps to develop good solution
iversity and reduces the chance of premature convergence of therey population. After predator prey interaction and local selection
s employed, spatial distribution of prey and predator changes. Newhild solutions (prey) are placed at random location in the grid, notecessarily near its parents. In this way, uniform population overime will lead to local clusters with unique solution sets. ClassicalA only uses global selection operator based on fitness of individ-al solution with respect to population average. This is one of theey differences of PPGA from conventional GA. The local selectionressure, based on predator–prey interaction where the least fitrey in the neighborhood of a predator is killed in each move ofhe predator, leads to a more robust selection strategy. It has alsoeen found that, in some cases, this may lead to good local solu-ion but may not give the optimum solution with respect to the
lobal minimum. So, in this study, a more adaptive selection strat-gy has been used, which considers both local fitness (based onredator interaction pressure) and also takes into account the rankr standing of the prey solution with respect to entire population.mputing 12 (2012) 559–572
In the following subsections, a general outline of the PPGA algo-rithm is given first, followed by a more detailed explanation of theproposed PPGA with respect to the above mentioned strategy. Thedetails of chromosome formation and genetic operator selection,for the PPGA, are given in the subsequent section.
3.1. The basic predator–prey strategy
The following algorithm gives a stepwise description of the gen-eral predator–prey strategy applied to genetic algorithm.
Step 1: Set control parameters, i.e., maximum number of genera-tion, predator–prey ratio, fixed mutation rate, etc.
Step 2: Initialize prey population, and place them at random pointsor cells in the grid. Also place predators at randomlythroughout the grid.
Step 3: Evaluate the fitness of prey population and for each preda-tor, search its local neighborhood (cells or vertices directlyconnected with a predator) for the worst fit prey, if a preyis found then kill it.
Step 4: Chose random prey in the neighborhood of the worst fitprey which was chosen by the predator. Generate offspringby crossover and mutation. These offspring will replace thepreys killed by the predators.
Step 5: Move each predator one cell in random direction, which isempty.
Step 6: If termination criterion is not met, then go to step 3.
3.2. Outline of the proposed PPGA
Below is the proposed PPGA with adaptive local selection strat-egy.
Step 1: The predator–prey population grid is initialized first. Inthe initial population, prey to predator ratio is kept around 1/5th to1/10th. This ratio is an important parameter of any PPGA algorithm.Let Pi
rat be defined as,
Pirat = initial prey to predator ratio
prey to predator ratio of ith generation(4)
Taking constant predator size, Pirat becomes equivalent to,
Pirat = initial prey size
prey size at ith generation(5)
The number of moves a prey can make in each iteration isdirectly proportional to this parameter (no. of moves by prey =max moves × Pi
rat). Since in each move a prey has the chance tomate and reproduce, hence it is quite evident that this will makeprey population more active in breeding, if Pi
rat increases, i.e., if preypopulation decreases. This way predator to prey ratio is adaptivelymaintained more or less to the initial value throughout the PPGArun.
Step 2: After calculating Pirat for a particular generation, another
parameter �ixy, which gives the size (in radius) of selection neigh-
borhood (Moore neighborhood) for each prey at position (x, y) inthe population, is determined as follows:
�ixy =
∣∣∣∣�max(f ixy − f i
max)
f imin − f i
max
∣∣∣∣ (6)
where
�max = maximum radius of selection neighborhood
f imax = maximum fitness of any individual in the population at ithgenerationf imin = minimum fitness of any individual in the population at ith
generation
oft Computing 12 (2012) 559–572 563
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I. Banerjee, P. Das / Applied S
This simple transformation maps the fitness f ixy ∈ (f i
min, f imax) of a
rey at point (x, y) on the grid to �ixy ∈ (�max, 0). This gives a prey with
ower (better) fitness a larger selection neighborhood, hence theres better chance of finding good prey solution for mating. Since thearameter �i
xy is calculated globally for each prey and determineshe local selection, the algorithm tries to maintain a good balanceetween searching for global minimum, while maintaining localiversity.
Step 3: When a prey finds a suitable mate, crossover and muta-ion operators are applied to generate one child solution. Then, thehild is placed at any random location in the grid, only if the loca-ion is empty or has a prey which has a higher fitness than the newhild solution. There may be multiple attempts till an offspring isuccessfully placed into a particular cell. In this study, the maxi-um number of tries to determine a suitable location for the child is
aken to be max (1 + �ixy, 10). This gives the offspring’s of preys with
etter (lower) fitness value a better chance to get placed (survived)n the grid.
Step 4: After all preys in the population have made their moves,ach predator is given a chance to move. The number of moves aredator can make (NMpred), in one iteration, is determined fromhe rate of convergence of prey population, and given as
ipred =
max1≤x≤i
∣∣f xmin − f x−1
min
∣∣∣∣f imin − f i−1
min
∣∣ (7)
The value of �ipred
is linearly mapped between 1 and upperound of NMpred. It is apparent that if there is no change in the min-
mum fitness form (i − 1)th to ith generation, then each predatoran make NMpred moves. This reduces the premature convergencen population. In each move, a predator searches an 8-connectedeighborhood or Moore neighborhood, to determine the least fitrey. If the worst fit prey is found, the predator moves to the posi-ion of the prey in the grid and subsequently the prey is killed.fter all predators have made their moves, the grid is updated. In
his study, the value of NMpred is taken to be 10.Step 5: The stopping criteria is chosen as the maximum num-
er of generations (Gm). If current generation i < Gm, the sequenceoves to Step 2.
.3. Chromosome formation and genetic operator for PPGA
The cell formation problem is easier to represent using integeroded chromosome as shown below.
m1, m2, m3, . . . , mNm |p1, p2, p3, . . . , pNp }As can be seen from the above representation, the whole chro-
osome is divided into two parts; in the first part genes, mi ∈ (1,max) is the cell or cluster number the ith machine belongs to. Cmax
s the maximum number of cells. In the later half, pi ∈ (1, Cmax) ishe cell number the ith part belongs to. The minimum value of Cmax
s determined from the following empirical formula,
max =[√
max(Nm, Np)]
(8)
If we choose a higher value of Cmax, then we have more combi-ations or choices for the cells formed. Since, the range of each gene
s proportional to Cmax, the search space increases with higher val-es of Cmax. So, this leads to a wider but less controlled search andxperimentally shown to be rather unpredictable. One reason beinghe highly non-linear and discontinuous nature of the search space
f CFPs and also the efficacy measure is a many-to-one mappingrom the genotype to phenotype. So, it is quite important to limithis value, while avoiding such lower values that actually restricthe PPGA of ever finding the optimal solution for the problem. TheFig. 4. The two types of mutation operator used in this study.
above empirical formulae give good balance, at least for the set ofproblems chosen for this study. It is needed to be mention that suchresult is, however, preliminary and therefore needs more investi-gation on the value of this parameter for CFPs. Here, minimum cellsize is assumed to be 2 × 2.
3.4. Crossover operator
In this study, two types of crossover operators are chosen: onebeing uniform crossover and other two-point crossover. In uniformcrossover, a child solution is chosen as shown below,
Ch(x)i+1 =
⎧⎨⎩
⌊�Ch(x)i
1 + (1 − �)Ch(x)i3
⌋if Ch(x)i
1 < Ch(x)i2⌈
�Ch(x)i1 + (1 − �)Ch(x)i
2
⌉otherwaise
(9)
where, Ch(x)i+1 is child solution of (i + 1)th generation, x being theelements within the chromosome. Here Ch(x)i
1 and Ch(x)i2 are the
two parent solutions. Here, � is a random number between 0 and1. This is similar to the BLX- crossover with = 0. Since, this isan integer coded GA, ceiling or floor function is applied beforetaking the child solution. Uniform crossover, however, tends tofavour solution to the mid-range rather than the extreme pointsof the variable range, so this crossover is used in combinationwith the two-point crossover. In two-point crossover, child solu-tion is formed by interchanging the machine and part portionsof the parents. In this study, both types of crossover are usedwith crossover probability between 0.1 and 0.5, the probability ofchoosing two-point crossover over uniform crossover. This pro-duces roughly an equal number of offspring based on both theoperators. For larger problems it was found that a smaller valueof crossover probability helped the PPGA convergence rate signifi-cantly.
3.5. Mutation operator
Like the crossover operators, the mutation operator is also acombination of two chosen types, with equal probability of use.In the first operator (Fig. 4(a)), one gene site is chosen from thechromosome and is replaced by a random value between (1, Cmax).In the alternate operator (Fig. 4(b)), using a mutation probability
parameter (which is kept between 0.05 and 0.075), each site isgiven a chance to mutate. With respect to the problem domain ofthe study, it is found that the second operator has more influencein population convergence.
564 I. Banerjee, P. Das / Applied Soft Computing 12 (2012) 559–572
Table 1Simulation parameters for the proposed PPGA.
PPGA parameters Set-2 Problems (6–12, 14) Set-2 Problems (6–12)
Stopping criteria Maximum generation count 150 200Successive stagnated generations 5 10
Mutation rate 0.05 0.075Crossover 0.5 0.1Grid size 25 × 25 30 × 30P0
rat 1/5 1/10Number of trial per problem 30 30
Table 2Problem description and their references.
Problem Reference Size (Nm × Np) Number of clusters Beast known value of ˝0c
1 Fig. 5 7 × 8 2 0.852 Fig. 6 7 × 11 3 NA3 Fig. 7 11 × 7 3 NA4 Fig. 8 6 × 7 2 NA5 Nagi et al. [28] 17 × 20 5 0.73076 Burbidge [29] 20 × 35 4 0.75717 Lee et al. [30] 30 × 40 6 0.77958 Ch-1989 Data-1a 24 × 40 7 1.09 Ch-1989 Data-2a 24 × 40 7 0.851010 Ch-1989 Data-3a 24 × 40 7 0.748311 Ch-1989 Data-4a 24 × 40 7 0.735112 Ch-1989 Data-5a 24 × 40 7 � = 77.31d
13 Irani and Ramakrishnanb [31] 12 × 19 3, 4 0.5, 0.52514 Ch-1987 [33] 40 × 100 10 � = 90.33
a Chandrasekharan and Rajagopalan [27] optimal results given are those found by the original authors.b As per results given in Rogers and Kulkarni [32].c Higher the value the better for both ˝0 and �.d Value of � is in percentage.
Table 3Results of simulation with PPGA on the above test problems.
Problem Number of clusters ˝0 � Cbtnk ˝btnk
1 2 0.075* 0.8500 92.5 NA –2 3 0.320* 0.5483 77.8 1.1786 0.13883 3 0.1726* 0.7037 86.1 0.6364 0.14294 2 0.256* 0.5454 76.2 1.1667 0.22225 5 0.1686* 0.7307 90.1 0.2353 0.09176 4 0.1209* 0.7571 88.1 1.7500 0.11517 6 0.1101* 0.8031 89.3 1.3333 0.09638 7 0.0* 1.0 100.0 NA –9a 7 0.0837* 0.8510 95.20 – –
10a 7 0.1756 0.7279 90.0 – –11a 7 0.1664 0.7351 91.2 – –
a
4
htcos
TA
clustering every problem poses some unique challenge to the PPGAsearch due its characteristic search space structure. Parameters ofthe algorithm like mutation probability, crossover rate and initialseed population are all affect the evolutionary process in different
12 7 0.4422
14a 10 0.1399
a Not considered for bottleneck
. Implementation and results and performance analysis
In this section, fourteen CFPs’ have been studied and solutionas been proposed based on the grouping efficacy measure given in
his paper. The problems, which are dealt with here, have differentharacteristics attributed by their size, groupablity [27] and naturef bottlenecks. They also emphasize the points made in the earlierection on the two-fold objectives of cost minimization. In groupable 4verage search space coverage in terms of number of fitness function evaluations.
Problem Average searchcomplexity
Size of the search space(approx.)
1 0.6264 224
3 0.2309 236
5 0.1572 286
8–12 0.1008–0.1102 2180
7 0.0910 2197
14 0.0588 2465
0.4728 78.4 – –0.8401 90.9 – –
1 2 3 4 5 6 7 81 1 1 1
1 1 2 11 3
1 1 4 11 1 5
1 6 11 1 1 7
1 6 2 3 5 8 4 7 2 1 1 4 1 1 1 1 1 1 5 1 1 1 7 1 1 1 1 3 1 1 6 1
(b)(a)
Fig. 5. Matrices are shown conventionally with machines are in rows and parts arein columns (a) original machine-part incidence matrix and (b) incidence matrix aftercell formation.
oft Co
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Fi
Fi
I. Banerjee, P. Das / Applied S
ays for different problems. So, a robust algorithm must be able
o perform well in a general setting without resorting to problempecific biasing.In most of the solutions given in the current literature and inimilar studies in the past, the results obtained are optimal with
1 2 3 4 5 6 7 8 9 10 11 1 1 1 1
1 1 2 1 1 3
1 1 1 4 1 1 5
1 1 1 1 1 6
1 1 1 1 1 7
2 5 3 6 1 4 7
5 10 11 2 3 6 2 1 5 1 3 1 1 6 1 1 1 1 4 1 1 7 1 1
(a)
(c)
ig. 6. (a) Original machine-part incidence matrix, (b) cluster matrix obtained after PPdentifying bottleneck parts.
1 2 3 4 5 6 7 1 1 1
1 1 2 1 1 1 3 1 1 4
1 1 5 1 1 6
1 1 7 1 8
1 9 1 1 10
1 1 11
4 7 1 4 1 1 5 1 8 1 1 10 1 3 1 7 1 11 1 2 6 1 9 1
(a)
(c) ig. 7. (a) The original machine-part incidence matrix, (b) clustered matrix obtained afdentifying bottleneck machines.
mputing 12 (2012) 559–572 565
respect to the efficacy measure for the particular CFP. All prob-
lems, which are used in simulation, are given with optimal resultsobtained both with respect to our proposed efficacy and the con-ventional ˝0 measure. However, we expectedly observed that ifwe change the relative weight of importance between the two1 5 8 4 9 10 11 2 3 6 7 1 1
1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
7 1 8 4 9
1 1
1 1 1 1 1
1 1 1 1
(b)
GA run, before identification of bottleneck parts, and (c) the final solution after
4 7 1 5 6 2 3 4 1 1 5 1 1 8 1 10 1 1 3 1 1 1 7 1 1 11 1 1 1 1 1 2 1 1 6 1 1 9 1
5 6 2 3
1 1 1
1 1 1 1 1
1 1
1
(b)
ter PPGA run, before identification of bottleneck parts, and (c) final solution after
5 oft Co
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4
mopbmftwoi
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Fi
66 I. Banerjee, P. Das / Applied S
arts (inter-cell movement and intra-cell utilization) of the pro-osed efficacy measure then we get a set of solutions which areiased accordingly. We study this effect on a particular problemith non-removable exceptional elements. The grouping results in
ig. 12 give five instances of solutions of Problem 12 with differenteights.
The section is dived into four sub-sections. The Section 4.1 illus-rates in detail the simulation parameters for the PPGA, detail ofhosen problems and results of the simulation. The results givenere are with respect to the fitness measure in the PPGA run,owever, we give both the ˝0 value and another similar mea-ure � = q(ed/D) + (1 − q)(1 − (e − ed)/(NmNp − D)), where q = 0.5 byefault) which can be found in [27] for comparative analysis withtate-of-art the literature. Section 4.2 discusses the effect of theariation of the weights of the objective values in the efficacy mea-ure in finding biased solutions with the proposed PPGA. Section.3 discusses comparative performance of our algorithm with otherimilar approaches with respect to number function evaluations.ection 4.4 gives a general review of the obtained results and itscope. Due to space constraints, only the obtained optimal solutions given for well cited problems.
.1. Simulation results and algorithm parameters
In the PPGA algorithm, maximum generation count or maxi-um successive stagnated generation (where the minimum fitness
f the population does not improve further) is taken as the stop-ing criteria. The selected CFPs are coarsely dived into two groupsased on the parameters chosen for them to give the best perfor-ance; this, however, does not mean that the set of parameters
or a particular group cannot be used for the other. In fact, forhe bigger problems, we needed more prey individuals, in general,hich may not be necessary for smaller size problems. The details
f the parameters and their problem specific variations are givenn Table 1.
In our simulation we found that it is not necessary to increase
he grid size linearly according to the problem size, which alsolows down the computation considerably (O(N2) slowdown, Neing the grid size). In fact the problems, which are highly regularwhere there exists perfect clusters and has little or no exceptional1 2 3 4 5 6 7 1 1 1 1
1 1 2 1 1 3
1 1 4 1 1 5
1 1 6
4 5 6 1 1 2 1 1 3 1 4 5
6
(a)
(c)
ig. 8. (a) The original machine part incidence matrix, (b) cluster matrix obtained aftedentifying bottleneck parts.
mputing 12 (2012) 559–572
elements in the optimal case), are found to be quite easier to solvewith respect to premature convergence as our PPGA was able toget the global optimum in very regular fashion (>95% of the time)and it did not matter much for the problem size. However, forsome irregular problems having exceptional elements the proposedPPGA sometime gets trapped in local optimum which was over-come by increasing the stagnated generation count in the stoppingcriteria although, for these problems (problem no. 6, 11, 12 and 13),the average optimal convergence dropped to about 77%.
For each problem, 30 trials are made and the result given belowis the most probable outcome, or the solution (i.e., the fitness value)that occurs with highest frequency. Also, it is necessary to mentionthat for a single value of the grouping efficacy, there may be morethan one optimal arrangement for the cells formed. But, they willbe equivalent with respect to the measure proposed. In such cases(like Problem 3), only one such solution is shown for convenience.Only the order of machines and parts inside a cell, and inter cellu-lar order will be different for such cases. Table 2 gives the detailsof the problems chosen along with reference to their origin, wherepossible, number of clusters for the optimal grouping (if any) andtheir best known values in ˝0 (for the referenced problems). InTable 3, the simulation results of the selected problems are givenwhich include optimal efficacy values found by the proposed PPGA,and the type of bottleneck (if any) found. Also, the equivalent effi-cacy in terms of ˝0 and � is given for comparative analysis withprevious work. We also mention whether or not these results areoptimal with respect to the measure (highlighted by an asterisk).The Cbtnk value, where applicable, gives the type of bottleneck con-sidered for the final rearranging, and we also present the valueafter removing bottleneck (˝btnk). In Appendix, the actual CFP datamatrices are given along with their optimal solution found by theproposed PPGA.
4.2. Analysis of variation of the weight of the two objectives in ˝
In Section 2 we explained how the variation in weights in the
two objective components of would result in different types ofoptimal groupings. Here we give an example based on the PPGA runfor the Irani, 1995 problem. This specific problem was so chosenbecause it has the significant number of bottleneck elements in the4 5 6 7 1 2 3 1 1 1 1 2 1 13 1 1 4 1 1 5 1 1
6 1 1
7 1 2 3 1 1
1 1 1
1 1
1 1
(b)
r PPGA run, before identification of bottleneck parts, and (c) final solution after
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4
t
I. Banerjee, P. Das / Applied S
ptimal grouping due to its highly irregular structure. Apart fromhe default (0.5, 0.5) weights as given in Eq. (2) we also used the fol-owing combinations of weights (1, 0), (0.75, 0.25), (0.25, 0.75) and0, 1) for the objectives f1 and f2, respectively. The results are givenn Fig. 12 of Appendix A. It is expectedly seen that as more emphasiss given to f1, the optimal results are obtained with tighter clustersaving significant number of exceptional elements. In fact, we willot be able to use the Cbtnk measure as they would totally destroyhe tight groupings in these cases ((1, 0), (0.75, 0.25)). However, forower f1 weights, we can clearly see loose clusters with significantumber of voids.
.3. Comparative performance study
From the context of simulation we study the performance ofhe proposed PPGA with respect to fitness function evaluation and
1 2 3 4 5 6 7 8 9 10 11
1 1 1 1 1 1 1 1 1 2
1 3 4
1 1 1 5 1 1 1 1 1 1 1 1 1 6
1 7 1 1 1 8
9 1 10
1 1 1 1 1 11 12 13 14
1 1 1 1 15 16
1 1 1 1 17
11 12 13 6 7 8 9 10 18 19
3 1 1 1 7 1 1 1 10 1 1 2 1 1 1 1 5 1 1 1 15 1 1 1 1 4 1 1 12 1 1 14 1 1 9 13 16 1 8 11 6 1 1 1 1 1 17 1 1 1 1 1
(a)
(b) Fig. 9. (a) Original incidence matrix of Problem 5 and (b)
mputing 12 (2012) 559–572 567
average computational time. It is quite noteworthy that the num-ber of fitness function evaluations did not scale linearly with thesize of the CFP as it was quite dependent on the grid size forthe PPGA run. If maximum number of groups or clusters takenfor the PPGA run is Cmax as given in equation 8, then the sizeof the search space will be bounded by 2(Nm + Np)
⌈log2Cmax
⌉.
The number of fitness function evaluations gives a rough ideaabout the fraction of the ‘volume’ of the search space searchedby the PPGA. Due to the huge difference in order between thenumber of fitness function evaluations and the size of the searchspace we give the logarithmic (base 2) ratio of former withrespect to the latter in terms of average search complexity for
some selected problems. This indicates an increasing searchingefficiency for larger problems as, in these cases, only a verysmall fraction of the decision search space is actually searched(Table 4).12 13 14 15 16 17 18 19 20
1 1 1 1 1
1 1
1 1 1 1 1
1 1 1 1 1
1 1 1
1 1 1 1 1 1 1 1
20 14 15 16 17 1 2 3 4 5
1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1
final solution after removing machine bottleneck.
568 I. Banerjee, P. Das / Applied Soft Computing 12 (2012) 559–572
31
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Fig. 10. Final solution after removing part bottleneck for Burbidge [29].
40
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1 1
3 4 5 23 25 27 29 1 2 22 24 10 12 13 17 8 9 15 16 19 20 6 11 14 18 7 26 28
30
21
Fig. 11. Final solution after removing machine bottleneck for Lee et al. [30].
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I. Banerjee, P. Das / Applied S
To give a rough idea regarding actual computational time, forarge problems (search space > 2100) are solved in about 5–12 minn average, although, for smaller problems, it took about 1–2 min.roblem 14 took 35 min, in average, to converge to the opti-um under simulation. The simulation was carried out on a
entium IV 1.8 GHz machine. To give a comparative idea abouthe efficiency of proposed PPGA, authors in [32] using a GA based
pproach took about 5 h to solve the Lee, 1997 problem and abouth for the Burbidge, 1969 problem on a Pentium III 700 MHzachine.
1 2 4 5 6 7 10 12 13 14
1 1 1 1 1 3 4 1 1 1 1 1 6 1 1 1 7 1 1 1 1 1 1 1 8 1 1 1 1 1 9 1 1 1 1 1 10 1 1 1 11 1 1 1 12 1 1 2 5
1 2 3 4 5 6 7 10 12 13
1 1 1 1 1 1 2 1 4 1 1 1 1 1 1 6 1 1 1 7 1 1 1 1 1 1 1 8 1 1 1 1 1 1 9 1 1 1 1 1 10 1 1 11 1 1 12 1 1 3 5
12 13 14 15 16 17 18 19 1
11 1 1 1 1 1 1 12 1 1 1 1 1 1 2 3 4 15 8 19 11 1 1 16 1 7 1 1 1 1 1 1 10 1 1 1 1
(a)
(b)
(c)
ig. 12. Solutions of Problem 13 based on different combinations of weights of the two objc) f1 = 0.5, f2 = 0.5 and � = 0.3304, (d) f1 = 0.75, f2 = 0.25 and = 0.3320, and (e) f1 = 1, f2 = 0
mputing 12 (2012) 559–572 569
4.4. Discussion
In this study, one of the main objectives was to find clus-ters with minimum inter-cellular movements of parts. Simulationresults show that considering machine and part bottleneck, theseproblems produce better solutions in which relatively large num-ber of exceptional elements will always remain inherent from the
structure of the original incidence matrix. The use of PPGA givesconsiderable leverage in determining suitable solutions with onlylimited number of iterations. Use of linear combination of two15 16 17 18 11 19 3 8 9
1 1 1 1 1
1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1
14 15 16 17 18 11 19 8 9
1 1 1 1 1
1 1 1 1 1 1 1 1 1
1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
2 3 4 5 6 7 8 9 10 11
1 1
1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1
ectives in Eq. (2). (a) f1 = 0, f2 = 1 and = 0.0896, (b) f1 = 0.25, f2 = 0.75 and = 0.2209, and = 0.1304.
570 I. Banerjee, P. Das / Applied Soft Computing 12 (2012) 559–572
11 12 13 17 19 2 3 4 5 6 10 14 15 16 18 1 7 8 9
11 1 1 1 1 1 1 12 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 7 1 1 1 1 1 1 1 1 1 1 10 1 1 1 1 1 1
1 2 1 3 1 1
1 1 1 1 4 1 1 1 1 5 1 1
1 1 1 1 6 1 1 1 1 1 1 1 8 1 1 1 1
1 1 1 9 1 1 1 1
1 2 3 4 7 10 5 6 11 14 15 16 18 8 9 12 13 17 19
1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 7 1 1 1 1 1 1 1 1 1 1 1 1 10 1 1 1 1 1 1 2 1 1 3 1 1 5 1 1 6 1 1 1 1 1 1 1 8 1 1 1 1 1 1 1 1 9 1 1 1 1 1 1 1 11 1 1 1 1 1 1 12 1 1 1 1 1 1
(d)
(e)
Fig. 12. continued
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2 5 11 19 4 16 7 14 23 24 3 13 20 22 6 8 12 15 18 1 21 9 10
17
Fig. 13. Final solution for Problem 12, bottleneck is not considered here and it is given here for comparative study with the optimal configuration found by authors in [27].
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F
TG
I. Banerjee, P. Das / Applied S
bjectives (Eq. (2)) gives the user flexibility in choosing a partic-lar type of solution. To illustrate this, the two solutions of the
ncidence matrix of Fig. 1(a) are found by setting f2 = 0 and f1 = 0,
espectively, in the PPGA run and Fig. 12, in Appendix A, illustratesimilar fact in actual simulation for Problem 13. The cost functionCbtnk) chosen here can be extended to determine suitable choiceig. 14. Final solution for Problem 14, bottleneck is not considered here and it is given he
able A.1ives the optimal grouping found in the simulation for the Problems 8–11 in terms of ma
Problem Machine cells Part family
8 9 10 17, 6 7
7 14 23 24, 3 25
1 13 21 22, 1 9
6 8 12 15 18, 4 5
3 20, 2 11
2 5 11 19, 10 13
4 16 8 19
9 4 16, 8 19
1 13 21 22, 1 9
7 14 23 24, 3 25
9 10 17, 6 7
6 8 12 15 18, 4 5
2 5 11 19, 10 13
3 20 2 11
10 9 10 17, 6 7
1 13 21 22, 1 9
2 5 11 19, 10 13
3 20, 2 11
7 14 23 24, 25 32,4 16, 8 19
6 8 12 15 18 4 5
11 2 5 11 19, 10 13
3 20, 2 11
4 16, 19 21
7 14 23 24, 3 25
1 13 21 22, 1 9
6 8 12 15 18, 4 5
9 10 17 6 7
mputing 12 (2012) 559–572 571
of machine and parts elements, when both types of bottlenecks areconsidered simultaneously. In this respect, Cbtnk can be rewrittenas,
Cbtnk = εem
Nm+ eep
Np
re for comparative study with the optimal configuration found by authors in [33].
chine cells and corresponding part families.
20 29 40,32,16 17 33,18 26 27 30,12 15 23 24 31 34,14 22 35 36,21 28 37 38 3921 28 37 38 39,16 17 33,32,20 29 40,18 26 27 30,14 22 35 36,12 15 23 24 31 3420 29 40,16 17 33,14 22 35 36,12 15 23 24 31 34, 3
21 28 37 38 39,18 26 27 3014 22 35 36,12 15 23 24 31 34, 828 37 38 39,32,16 17 33,18 26 27 30,20 29 40
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72 I. Banerjee, P. Das / Applied S
Using GA, this can be minimized, and corresponding machinesnd parts can be identified which will be part of the bottlenecklock, so that only minimum number of machines or parts remainutside the clustered cells.
. Concluding remarks
In this paper, a solution to the problem of cell formation istudied using modified PPGA method. The modifications made inhe PPGA regarding adaptive selection neighborhood size help inchieving good solution diversity, and optimal solution was foundith relatively a fewer number of iterations. This helps in explor-
ng large solution space with respect to the new grouping efficacyeasure proposed. The bottleneck cost function guides in selecting
he type of bottleneck to be considered. This work can be extendedo incorporate sequential operation and alternate routing in thencidence matrix. In such case, the size of the bottleneck block maye further reduced by considering both types of bottleneck simul-aneously. Also, the two-fold grouping efficacy can be extendeds a bi-objective problem allowing the use of multi-objectivePGA.
cknowledgements
The authors thank the significant contributions of the reviewersn providing valuable comments and fruitful suggestions towardsmprovement of this article.
ppendix A. Simulation results on CFP data-sets
Figs. 5–14 and Table A.1
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