Cluster analysis in manufacturing cellular formation

O M E G A Int. J. of M g m t Sci.. Vol. 17, No. 3, pp. 289-295, 1989

0305-0483,89 S3.00 + 0.00

Printed in Great Britain. All rights reserved

Copyright ~ 1989 Pergamon Press plc

Cluster Analysis in Manufacturing Cellular Formation C-H CHU Iowa State University, USA (Received August 1988) Ouster analysis,an analyticalmethodused for dmifying objects, hu been developedover the last century,and the Hteratureon cluster analysishas explededduringthe past two decades.Thistechnique has been broadly used in the fle~ of biology,social science,and lmydmingy,but only very little in mmmfaeturing.In this paper, we provide a state-of-the-artreview on the use of duster analysis in cellular formation-one of the first and most important issues in designingeeUnlar manufacturing systems. The problems, research directions, and related literature are presented for further studies.

Key words--cluster analysis applications, group technology, manufacturing cellular formation

INTRODUCTION DURING THE PAST DECADE, there has been a major shift in the design of manufacturing planning and control systems using manufacturing resource planning ( M R P I I ) logic, just-in-time philosophy, and flexible (cellular) manufacturing concepts. Cellular manufacturing (CM), especially has received considerable interest from practitioners as well as from academics [43] because it allows small batchtype production to gain economic advantages similar to those of mass production and still retain the flexibility of job-shop production. One of the first and most important steps in CM, called cellular formation, is to group parts with similar design features or processing requirements into families and form associated machines into cells. Numerous heuristic or analytical clustering methods have been applied or developed for forming manufacturing cells, but very little effort has been put into synthetizing the related literature. Though frameworks for structuring the problem have been proposed [3, 42], these studies are only concentrated on classifying alternative approaches. This paper focuses on a state-of-the-art review on the use of clustering techniques in cellular formation.

APPROACHES TO CELLULAR FORMATION The objective of cellular formation is to group parts or machines with similar features into families. There has been a considerable amount of work done in this area [3, 20, 41, 42]. One of the simplest approaches, called the ocular or eye-bailing method, is to examine the information and perform the classification using the human eye. Though the approach is easy to understand, its success is highly dependent on human experience and preference and the number of parts it can handle is still limited. The classification and coding sytem is an improvement on the eye-bailing method. It examines the design features of parts by coding. Parts with similar codes are formed into the same family. Much time and effort goes into coding parts and creating an elaborate database; nevertheless, this technique produces a weak connection between design and process features. A production flow analysis, on the other hand, groups parts into families according to routing information or operation sequences. This approach consists of three continuous levels [4]. The first level, a factory-flow analysis, makes use of the processing requirements to 289

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Chu--Cluster Analysis in Manufacturing Cellular Formation

last step examines the operation sequences and layouts of the machines using flow line concepts. Recently, the use of clustering algorithms in cellular formation has received extensive attention. Table 1 summarizes the clustering techniques appearing in the literature. A detailed discussion of the table follows.

obtain an overall material flow between the group of machines. The second step, called group analysis, utilizes the information about the relationship between parts and machines; the designer then proceeds to form machine/part families by reordering the rows and columns of the part/machine matrix. The

Table I. Summary of the related literature Method

Cock

Set]. a

Com. b

Language

Reference

(!) DESIGN-ORIENTED APPROACH Multiobjective clustering analysis Matrix formulation Dutta et al. heuristic

S D . D

Y N .

. N

Y(M) Y(S) . Y( )

FORTRAN BASIC

[14] [I 5] [23]

--

[12]

---

[18] [19]

(!I) PRODUCTION-ORIENTED APPROACH

(,4)Army.&ued Rank order clustering method (ROC)

---

N N

Y( Y(

) )

ROC2 algorithm MODROC algorithm Direct clustering algorithm (DCA) Jacobs' algorithm Modified bond energy algorithm

-S --

N N N

Y( ) Y(M) Y( )

-FORTRAN --

[20] [9] [6]

-S

N Y

m

N

BASIC FORTRAN ~

[16] [14]

Occupancy value method

Y(S) Y(S) Y( )

[17]

S S, D

N N

-Y( )

-CLUSTAN

[25] [39]

(B) HieraecMcaIelmsterina teclmiq~s Single linkage method

S S

Average linkage method Complete linkage method Centroid method Median method algorithm Lance and Williams flexible

N

--

--

S S S S, D S S, D S, D

N Y N N N N N N

Y(S) Y( ) Y( ) -Y( ) -Y( ) Y( )

S

N

--

S, D S S, D S, D --

N N N N N

Y( -Y( Y( Y(

D

N

BASIC FORTRAN --CLUSTAN

[27]

CLUSTAN CLUSTAN

[39] [39]

[5] [34] [39]

--

CLUSTAN -CLUSTAN CLUSTAN --

[39]

Y(L)

FORTRAN

[8]

--

N Y

Y( ) . Y(L) Y( )

LINDO --

[28] [23] [22] [37]

S S

N Y

Y(L) --

FORTRAN --

---

N N

Y(L) --

FORTRAN --

[31] [1 I] [30] [40] [7]

--

S

N N

Y( ) Y(L)

-FORTRAN

[3] [38]

ODC

D

N

Y(

--

[121

Selvam et al. heuristic MACE

S S

Y N

-Y(M)

-FORTRAN

[35] [41]

C o s t - b a s e d heuristic

S

--

--

Cluster identification algorithm Cost analysis algorithm Polyhedral dynamics Mathematical classification

-~ $ S

N N N N

Y(L) Y(L) Y( ) Y( )

Ward's

McQuitty's similarity analysis A divisive procedure

) ) ) )

[39] [39] [36]

(C) NoR-MoarcMeal Modified MacQueen's method (D) Mat~mat/ea/proframm/gf Linear programming Zero-one integer programming Dynamic programming

S . S S

N .

.

(E) GrapMe tl~or,Me qpro=ek R&B algorithm De Witte's algorithm Pureheck V&R algorithm C&R algorithm (F) Hem'tn~ ~ WUBC ICRMA

otis

"Does it consider operating sequences (Y/N)? bL: Mainframe, M: minicomputer; S: microcomputar.

)

--

FORTRAN FORTRAN ~ --

[2]

[24]

[24] [33] [29]

Omega, Vol. 17, No. 3

STATE-OF-THE-ART REVIEW

Cluster algorithms As shown in Table 1, diverse clustering algorithms have been employed as effective tools in formatting manufacturing cells. Basically, they can be classified into two major classes. A design-oriented approach relies on the design features of parts to perform the necessary analyses. The production-oriented approach, on the other hand, is based on routing information to group parts or machines. The latter can be further divided into several categories with respect to the differences in clustering logic. An array-based clustering technique tries to rearrange the columns and rows of a machine/part matrix according to an index, until some diagonal blocks are formed. The method first represents routing information through a binary matrix. A 'i' means that the part is processed by the machine and a '0' means that the part is not processed by the machine. According to certain algorithms, such as the ranked order clustering (ROC), the direct clustering analysis (DCA), or the bond energy analysis (BEA), etc. the rows and columns are alternatively rearranged. Among the arraybased methods, the ROC has been used most frequently. A hierarchical clustering method must compute the similarity or dissimilarity between each pair of parts of machines in order to produce a dendogram for final judgement. Some methods used agglomerate philosophy while others use divisive philosophy for clustering hierarchically. A comprehensive review of these methods can be found in [1, 13]. Among the hierarchical clustering methods, the single linkage method has been used most frequently. Unlike hierarchical clustering methods, nonhierarchical clustering algorithms are iterative procedures. Thus far, only one non-hierarchical method, called ideal-seed method, has been used in cellular formation. The ideal-seed method is an expanded and improved version of the MacQueen's method [7]. Some mathematical programming techniques, such as linear programming, zero-one integer programming, and dynamic programming, etc. have also been used for formatting manufacturing cells. These methods are formulated to meet the objective functions such as minimum total cost, optimum throughput

291

time, and maximum total bond strength. They can use the routing information or operation sequences to find the optimal solution. The graph-theoretic approach, on the other hand, applies graph and network theory to perform the analysis. The method begins with an analysis of the relations between machine types and setups, the machine-graph and its cliques. It then uses the graphic positioning approach to form the machine cells and allocate parts to cells. The multitude of clustering approaches makes it difficult to select an appropriate method. Nevertheless, the single linkage method of hierarchical clustering has been used more often than others.

Clustering criteria To classify parts or machines into families or cells, a cluster algorithm must either rely on a built-in clustering logic or assign a clustering criterion as an objective function in order to optimize system performance. The criterion could be a similarity or a dissimilarity index derived from binary or numeric data. A similarity coefficient is used to measure the degree of similarity. The larger the coefficient, the higher the degree of similarity between each pair of parts or machines. A dissimilarity coefficient conversely, measures the degree of dissimilarity. The dissimilarity can also be defined as the distance between two clusters [7, 39]. Most clustering methods, as can be seen in Table 2, use a similarity coefficient, among which Jaccard's was used most frequently although no study testifies to its optimality. The effect of clustering criterion on cellular formation also remains uncertain.

Measures of performance Evaluating the relative performance of clustering algorithms is a very important, hut difficult task. Though various measures have been proposed, no single measuring criterion has received sufficient attention. This is due mainly to the fact that the performance of clustering techniques is data dependent. Basically, the performance of cluster algorithms can be measured from two directions:one approach is to examine its computational efficiency;the other is to evaluate the grouping effectiveness. Regarding the former approach, one can take into account the complexity of the algorithm,

Chu--Cluster Analysis in Manufacturing Cellular Formation

292

Table 2. Summary of the clustering criteria Similarity coeff. Dissimilarity coeff.

Waghodekar and Sahu's

[5,9,25] [27, 32. 34] [38, 39, 411 [421

M ultiplicative Additive

[27]

"Jaccard's

Binary

Yule's

Squared Euclidean distance

[7] [39]

Dutta's et at.

[121

[7] [8] [I 5]

Hamann's

Baroni-Urbani and Buser's Sorensen's

Kulcznski's Un-Name (Sokal and Sheath's) Dot Product

[39] [39] [391 [39l

~"De Witte's ~ Multiplieative weighted

Ill]

[27]

Numerical "~ Additive weighted | Modified muhiplicative (. weighted

Minkowski metric

[27]

function

the program's execution time, and the memory needed for storage [24]. The effectiveness measure, on the other hand, requires some vehicles to compare the clustering results with an original matrix, a standard result, or a result from other methods [10, 21]. For example, one may simply count the number of points misclassified from a standard result, or compute the similarity between two clusterings of the same data

[321. Numerous measures, either individual or aggregate, have been proposed in the literature. (See Table 3.) The major difference between them is that aggregate measures evaluate the clustering results by more than one factor. In

[271

manufacturing cases, the number of exceptional elements, the cost of inter-cell or intra-ceU movement, material handling cost, machine idle time, work-in-process, setup time, etc. may serve as a measure. The number of exceptional elements, which evaluates the number of elements remaining outside the diagonal blocks, has been most popularly used. Grouping parts, macldnes, or both? Though most algorithms deal with the problem of clustering parts into families [5,7,8, 12, 14, 15,39] or machines into cells [2, 9, 11, 25, 27, 31, 34, 38, 411, it is also possible

Table 3. Summary of the measures of performance Reference

Measure of performance

(I) AGGREGATE MEASURE

(A) Cost based

Minimum costs of inter-cell and intra-cell movement Minimum costs of material handling and machine idle time Minimum system costs of work-in-process, cycle inventory, material handling, set-up,

[25] [35] [2]

fixed machine cost

(n) NoR-cost based Group efficiency (Weighted inter-cell movement and machine utilization) Weighted inter-cell and extraneous machine transitions

[8, 10] [36]

(H) INDIVIDUAL MEASURE Number of exceptional elements (Inter-cell movement)

Machine utilization Number of extra duplicated machines Simple matching measure Production movement correlation coefficient Generalized matching measure Total bona energy

Distance Sum of similarity Sum of bond strength Overall dissimilarity coefficient (ODC) Occupancy value

[3.6. 19, 27. 31, 38, 40. 41] [39] [39]

[26]

[16, 231 [23] [37] [12] [17]

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to use an iterative approach in which the clustering is alternated between parts and machines until mutually harmonious clusters are achieved for both. Production flow analysis and arraybased clustering methods [6, 16, 18, 19, 20, 26] follow this logic. ZODIAC [8] can also simultaneously form part families and machine cells.

The determination of number of cells The decision to find an optimal number of cells is a controversial issue. To some algorithms, such as non-hierarchical clustering methods, mathematical programming approach and graphic-theoretical methods, the determination of the number of groups is a dependent variable; that is, it must be decided before starting the clustering procedures. According to graph theory, Chandrasekharan and Rajagopalan [77] proposed a method to estimate the number of groups. To some others, such as hierarchical clustering techniques, the number of families depends where a dendogram is cut. Though a big-jump principle, which selects the largest changes between fusions of dendrogram, has been used informally, the principle may not be applicable to some cases [1, 13]. Since the array-based clustering techniques can automatically determine the number of cells through the procedures, the solution to the number of groups is not a problem. How to handle bottleneck machines?

According to Everitt [13], clustering algorithms are very sensitive to the data to be clustered, especially if they contain outliers, i.e. there exist bottleneck machines or exceptional parts. Examining the examples in the literature, we quickly conclude that if the tested problems could be well blocked, most methods would generate good clusters. However, the classification results are quite different for those problems with exceptional or bottleneck entities, and they normally entail subjective judgements. Therefore, the problem of how to handle bottleneck machines and exceptional elements has gained researchers' and practitioners' attention. Two approaches have been developed. One is to relax the bottlenecks [19], i.e. to duplicate one machine for each exceptional element, and thereafter compose these machines together; the other alternative is to eliminate the bottlenecks first and then duplicate the machines after clustering [6, 20]. The latter is more efficient [20].

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Consideration of operating sequences Considering operating sequences at the formation stage could provide valuable information and thereafter save layout time in implementing the machine cells. According to Table 1, most clustering techniques classify the parts into families without considering the sequence of operations. Only a few studies concern this problem. One common approach is to develop a similarity index that reflects the sequencing requirements [11, 35, 37]. An alternative is to enter process sequences in the machine/part matrix and then rearrange the columns and rows of this matrix [26]. Gongaware and Ham [14] assigned operating sequences as one of the objectives in multiobjective clustering algorithm in order to optimize the solution. Level of computerization Because of the complexity of clustering algorithms, a computer is needed to perform the analyses. Though several generic statistical analysis packages such as SAS, SPSS, etc. and some special-purpose clustering packages such as CLUSTAN have been used widely in other clustering related researches [1, 13], they are not being used widely in manufacturing settings. Nevertheless, most analyses are performed by in-house programs written in high-level programming languages such as F O R T R A N and BASIC. Regarding the computer facilities used, most are run on mainframe or mini computers. Recently, some research has been done on microcomputers. The use of microcomputers to solve the cellular formation problem will become popular due to the rapidly changing technology. The database Since clustering algorithms are very sensitive to the data to be clustered, the selection of data sets for testing is very important. The testing data sets can be either generated from a random number generator via computer or collected from the literature. Table 4 lists the examples used in the literature. Several observations are worth noting. First, almost all of the examples tested on these articles are artificial or partial of the actual data from a plant [8, 39]. The most popular example is Burbidges's [4]. The largest sample tested included 40 machines and 100

Chu--Cluster Analysis in Manufacturing Cellular Formation

294

Table 4. Summary of the problem (data) sets Problem (data) Set

Problem size (M X p)a

Chandrasekharan and Rajagopalan Burbidge Tarsuslugil and Bloor Stanfel Tabucanon and Ojha Burbidge Burbidge Jacobs King De Witte Chandrasekharan and Rajagopalan Chan and Milner McAuley Tabucanon and Ojha Gongaware and Ham Slevan and Balasubramanian King and Nakornchai Modified King and Nakornchai's Mo$ier

40 x 100 36 x 90c 78 × 30 30 × 50 30 x 40 16 × 43 20 × 35 20 x 24 14 x 24 12 x 19 8 × 20 15 × 10 12 x 10 7 x 14 9x9 10 x 5 5x7 5x7 6x4

Matrix taypeb

Bottleneck machines? (Y/'N)

Exceptional elements? (Y/N)

N Y Y N N Y N N N Y N N N N N N N N N

Y Y N N Y Y Y Y Y N Y N Y Y Y N N Y N

I I I n I ! I I I I I II I [

I II 1 I II

References [8] [20, 40] [391 [36] [38] [3, 6, 17, 19, 34, 40, 41] [5, 9] [16] [18, 36] [I I] [7, 9] [3,6] [25] [38] [14, 37] [35] [I 7, 20, 31,37, 41] [41], [3] [27]

"Machines x parts. bMatrix type:

!Part (I) Machine I

]Machine (II) Part [

Cln this example, six machines are idle. Actually, it is a 30 x 90 example.

parts [8]. In fact, the actual number of parts a company has is usually much larger than the tested examples. Second, if the test problems could be well blocked, most of the methods would result in good clusters. For those problems that have exceptional elements, the classification results, however, are quite varied from algorithm to algorithm and require subjective judgement or human interaction. Finally, there seems to be no systematic study that involves a comparison of alternative methods using different sets of data, rather, most studies rely on a single set of tests. CONCLUSIONS In reviewing the cellular formation problem under discussion, we become immediately aware that a substantial number of problems still remain. First, the multitude of clustering algorithms, clustering criteria, and measures of performance make it difficult to evaluate and select an appropriate or better clustering method. Although a few studies [39] address this problem, the experimental setting is relatively small; therefore, the results are limited for reference. Second, although it is clear that different clustering criteria may produce different grouping results even if the same algorithm and data are used, a general contention regarding the effect

of the criterion on clustering is yet to be established. Third, although minimizing the number of exceptional elements has been widely used as a measure, the questions: "Is that measure appropriate?" or "Does any other better measure exist?" etc. remain to be answered. Particularly under a cellular manufacture setting, some measures, such as work-in-process and throughput time, may be very important to the system performance, but are not considered in cellular formation. Generally speaking, the problem is this: most clustering algorithms do not take processing time and production volume into consideration. Fourth, the quality of clustering depends highly on the quality of data. The questions: "Is there any method that is less sensitive to the data with outliers?"; "How to flawlessly detect bottleneck machines and exceptional elements?"; and "How to handle these bottlenecks?" become challenge tasks. Fifth, the determination of an optimal number of manufacturing cells is a controversial issue. Although a few principles can be applied, the decision is still subjective, and must rely on personal judgement by considering some constraints such as available spaces, production volume, setup time etc. Finally, though it would be much more convenient to consider the processing sequences during the clustering stage, the problem of integration remains a tough issue.

Omega, Vol. 17, No. 3

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