Optimal bivariate clustering and a genetic algorithm with an application in cellular manufacturing

European Journal of Operational Research 160 (2005) 423–444 www.elsevier.com/locate/dsw

Discrete Optimization

Optimal bivariate clustering and a genetic algorithm with an application in cellular manufacturing David F. Rogers

a,*

, Shailesh S. Kulkarni

b

a

b

Department of Quantitative Analysis and Operations Management, College of Business, University of Cincinnati, 531 Lindner Hall, Cincinnati, OH 45221-0130, USA Department of Business Computer Information Systems, College of Business Administration, University of North Texas, P.O. Box 305249, Denton, TX 76203-5249, USA Received 27 July 2001; accepted 23 July 2003 Available online 18 November 2003

Abstract The problem of bivariate clustering for the simultaneous grouping of rows and columns of matrices was addressed with a mixed-integer linear programming model. The model was solved using conventional methodology for very small problems but solving even small to moderate-sized problems was a formidable challenge. Because of the NP-complete nature of this class of problems, a genetic algorithm was developed to solve realistically sized problems of larger dimensions. A commonly encountered example is the simultaneous clustering of parts into part families and machines into machine cells in a cellular manufacturing context for group technology. The attractiveness of employing the optimization model (with objective function being a sum of dissimilarity measures) to provide simultaneous grouping of part types and machine types was compared to solutions found by employing the commonly used grouping efficacy measure. For cellular manufacturing problem instances from the literature, the intrinsic differences between the objective of the proposed model and grouping efficacy is highlighted. The solution to the general model found by employing a genetic algorithm solution technique and applying a simple heuristic was shown to perform as well as other algorithms to find the commonly accepted best known solutions for grouping efficacy. Further examples in industrial purchasing behavior and market segmentation help reveal the general applicability of the model for obtaining natural clusters.
2003 Elsevier B.V. All rights reserved. Keywords: Clustering; Mathematical modeling; Integer programming; Genetic algorithms; Manufacturing; Cellular manufacturing

1. Introduction and problem description The basic aim of classical cluster analysis is to assign n objects to K mutually exclusive groups * Corresponding author. Tel.: +1-513-556-7143; fax: +1-513556-5499. E-mail addresses: [email protected] (D.F. Rogers), [email protected] (S.S. Kulkarni).

while minimizing some measure of distance or dissimilarity (usually a distance defined on a metric). Clustering of rows or columns in matrices has been considered by many researchers in applied mathematics. Statistical approaches (e.g., Anderberg, 1973), mathematical programming approaches (e.g., Rao, 1971; Klein and Aronson, 1991), and permutation approaches (e.g., McCormick et al., 1972) are examples of popular methods

0377-2217/$ - see front matter
2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2003.07.005

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for clustering matrices. The bivariate cluster analysis problem considered in this article is to determine groups for two separate but related populations represented by the rows and columns of a matrix. Rather than interest being in only finding clusters of rows or, separately, clusters of columns, the interest here is in finding clusters with both rows and columns as members of the cluster. The focus of this article is upon the formation of an integer-linear mathematical programming model for this general bivariate cluster analysis problem and a genetic algorithm (GA) for solving this general model. Researchers have applied mathematical programming techniques to this and other various cluster analysis problems, but not to the extent as other less computationally demanding heuristic approaches because of the tendency of apparently easily solvable problems to become combinatorially explosive as the problem size increases. Mathematical programming models often become an important step in specifically defining the problem and a heuristic technique, such as a GA as employed in this article, can provide very good solutions. Several applications for this model exist and are readily apparent. For example, determining clusters of purchasing behavior scripts and purchasing agents in market research, or determining groups of complaints and customer types––do certain types of complaints come from a particular profile of customers? Tracking website visitors and the particular areas of the website they visit may yield quite interesting results. A specific and commonly encountered problem with a compatible data set and objective is found in the cellular manufacturing (CM) application of finding families of part types grouped with cells of machine types. Larger problems were solved in the CM setting and the results obtained from the solution of this general integer model by a GA, coupled with a simple heuristic, were shown to find results as good as other approaches for classical problems in the literature. The remainder of this paper is organized as follows. Section 2 is a formal statement of the bivariate clustering model. In Section 3 solution attempts to some small sample problems using conventional software and techniques are presented as motivation for Section 4, which contains the basics of a

GA-based solution procedure for the bivariate clustering model. Section 5 contains an introduction to CM and a brief review of the literature related to mathematical programming formulations for the cell formation problem and other GA approaches. In Section 6, a heuristic is coupled with the GA to solve several problems from the CM literature, and Section 7 contains additional examples in industrial purchasing and market segmentation. Section 8 is a summary of the findings and suggested directions for future research.

2. The bivariate clustering model The input data necessary for the bivariate clustering problem considered here is any general matrix, A, with elements air , i ¼ 1; . . . ; m and r ¼ 1; . . . ; n. Thus row Ai is a function of the column data and column Ar is a function of the row data. The overall objective is to determine clusters of combinations of both rows and columns (and thus clusters of only rows and clusters of only columns may be deduced) such that some measure of dissimilarity or distance is minimized (or some measure of similarity or closeness is maximized). To find clusters of columns and/or clusters of rows, a mathematical programming model is considered and the point of departure is the mixedinteger linear programming model developed by Klein and Aronson (1991) and enhanced by Aronson and Klein (1989). Their model is a univariate optimal clustering model that can be used sequentially to cluster rows or columns based upon scaled distance or dissimilarity measures, e.g., any from the family of Minkowski metrics. For determining clusters of rows, their model is Minimize

n
1 X n X

dij Yij

i¼1 j¼iþ1

subject to Yij P Xik þ Xjk
1; 8i ¼ 1; . . . ; m
1; j ¼ i þ 1; . . . ; m; k ¼ 1; . . . ; K K X Xik ¼ 1; 8i ¼ 1; . . . ; m k¼1

Xik ¼ 0; 1; 8i ¼ 1; . . . ; m; k ¼ 1; . . . ; K Yij P 0 8i ¼ 1; . . . ; m
1; j ¼ i þ 1; . . . ; m

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where dij ¼ non-negative distance (or dissimilarity measure) between row i and j. Xik ¼ 1, if row i is in group k; 0, otherwise. Yij ¼ 1, if rows i and j are in the same group; 0, otherwise. K ¼ number of clusters (groups) to be determined. m ¼ number of rows. To solve this model, an extension of the implicit enumeration algorithm of Balas (1965) was developed and special implicit constraints were applied to efficiently solve problems of reasonably practical size. An analogous model may be developed for determining the clusters of columns. To get a final solution to the bivariate problem of interest in this article, an additional procedure would then be required to match the columns with the appropriate rows. The Klein and Aronson (1991) model may be used to determine clusters of row elements and column elements sequentially, but the model is not for determining row and column clusters simultaneously and thus the relationship between the rows and columns are not considered. To enhance the model to further consider these relationships, the following new bivariate clustering model is introduced. The problem of clustering rows and/or columns individually and sequentially is extended to the simultaneous clustering of rows with columns by the following mixed-integer linear programming model: m
1 X m n
1 X n X X Minimize dij Yij þ drs0 Yrs0 i¼1 j¼iþ1

þ

n X m X r¼1

r¼1 s¼rþ1

bir Zir

i¼1

subject to Yij P Xik þ Xjk
1;

8i ¼ 1; . . . ; m
1;

j ¼ i þ 1; . . . ; m; k ¼ 1; . . . ; K Yrs0 P Xrk0 þ Xsk0
1; 8r ¼ 1; . . . ; n
1; s ¼ r þ 1; . . . ; n; k ¼ 1; . . . ; K

K X

Xik ¼ 1;

8i ¼ 1; . . . ; m

Xrk0 ¼ 1;

8r ¼ 1; . . . ; n

425

k¼1 K X k¼1

Zir P Xik þ Xrk0
1;

8i ¼ 1; . . . ; m;

r ¼ 1; . . . ; n; k ¼ 1; . . . ; K Xik ; Xrk0 ¼ 0; 1;

8i ¼ 1; . . . ; m;

r ¼ 1; . . . ; n; k ¼ 1; . . . ; K Zir P 0;

8i ¼ 1; . . . ; m; r ¼ 1; . . . ; n

Yij P 0;

8i ¼ 1; . . . ; m
1; j ¼ i þ 1; . . . ; m

Yrs0 P 0;

8 ¼ 1; . . . ; n
1; s ¼ r þ 1; . . . ; n

where, additionally, drs0 ¼ non-negative distance (or dissimilarity measure) between columns r and s. Xrk0 ¼ 1, if column r belongs to group k; 0, otherwise. Yrs0 ¼ 1, if columns r and s belong to the same group; 0, otherwise. Zir ¼ 1, if row i and column r belong to the same group; 0, otherwise. bir ¼ 0, if element in row i, column r > 0; 1, otherwise. n ¼ number of columns. K ¼ number of groups into which both columns and rows are to be clustered. The bivariate model is significantly larger in dimension as compared to the model of Klein and Aronson (1991). The additional clustering of rows with columns further complicates this cluster analysis problem. A comparison of the relative sizes of the two models is in Table 1 where the number of columns is assumed equal to the number of rows. Even for small-sized problems the bivariate model is significantly larger and likely to be much more difficult to solve as compared to the Klein and Aronson (1991) model. One reason for this is the m  K increase in the number of binary variables. The increase in difficulty is also because the bivariate model does not exhibit any special

Table 1 Comparison of model dimensions m rows, n columns, K groups

Klein and Aronson (1991) model

Bivariate clustering model

Difference (assuming m ¼ n)

Variables Constraints

nðK þ n
1 Þ 2 Kðnðn
1Þ Þþn 2

Kðm þ nÞ þ mn þ 12 ðnðn
1Þ þ mðm
1ÞÞ ðm þ nÞ þ mnK þ 12 Kðnðn
1Þ þ mðm
1ÞÞ

nK þ 32 n2
n2 n þ n2 K þ 12 ðKnÞðn
1Þ

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structure, cannot be decomposed, and cannot be represented as a network. Implicit enumeration techniques or intelligent heuristics must often be relied upon to obtain optimal or near optimal solutions for problem settings such as these.

3. Preliminary computational results Some preliminary attempts were made to solve three problem instances of the bivariate clustering model upon pure 0/1 matrices using traditional integer programming solution techniques. Squared Euclidean distance measures were employed for all problems considered in the article and resulted in rather dense upper diagonal distance matrices that contain elements on the range ½0; 1. The first problem was an m ¼ n ¼ 5 and K ¼ 3 problem solved with the generalized algebraic modeling system (GAMS) on a SPARC 2.0 UNIX workstation. The optimal solution was found after 11,454 iterations and the resource usage was 130.26 seconds. Subsequent problems were approached by modeling them with AMPL and solving them with either CPLEX 6.5 on a Pentium III enabled 700 MHz workstation with 256 MB of RAM or CPLEX 8.1 on a Pentium 2 GHz machine with 512 MB RAM. The m ¼ 12 and n ¼ 19 CM problem of Irani and Ramakrishnan (1995) was solved next and the optimal solutions for this problem with K ¼ 3 (optimal objective value ¼ 33.51) and K ¼ 4 (optimal objective value ¼ 60.14) are shown in Fig. 1. For CM problems, a commonly employed measure of performance is the grouping efficacy (GE) measure: GE ¼ 1
ðe0 þ ev Þ=ðe þ ev Þ, where e0 ¼ number of exceptional elements (ones not in a cluster or block diagonal), e ¼ total number of operations (ones) in the data matrix, and ev ¼ number of voids (zeroes) in the cells (clusters or blocks). The GE measure is not the same as the objective for the bivariate clustering model, but it is introduced so that results from the literature, for which GE is often employed, can be compared. See Rogers and Shafer (1995) for a complete survey for GE and other performance measures applied specifically to CM.

To highlight the intrinsic differences between GE and the dissimilarity objective for the bivariate clustering model, consider part types 15 and 16 for the three-cluster solution in Fig. 1. Both of these part types could be placed in the cluster with machine types 1, 7, and 10 resulting in an improved (higher) GE, but the objective value for the bivariate clustering model would increase from 60.14 to 62.09. Also note part type 11 in the three-cluster solution––it might naturally be considered for being processed in the cluster with part types 1, 7, 8, and 9, but this would result in both a lower GE because of more voids in the clusters, and a higher value for the objective value for the bivariate model. Similar observations can be made for the four-cluster solution of Fig. 1 where moving machine type 7 to the cluster with machine types 11 and 12 would increase GE but also results in the objective for the bivariate model increasing from 33.51 to 35.91. The objective for the bivariate clustering model is naturally more complicated and additional information––the amount of dissimilarity between different elements and clusters––is utilized, whereas the GE measure only consists of a simple counting and ratio of the number of operations in the matrix, the number of exceptional elements, and the number voids in the clusters, and does not take into account the distances. An attempt was made to solve a CM problem from Burbidge (1969) with m ¼ 20, n ¼ 35, density ¼ 0.193, and K ¼ 4 resulting in 1705 variables (220 binary) and 5995 constraints. After some experimentation with allowing the solver to run for up to 5000 branch-and-bound seconds, the solution process was terminated prematurely because the size of the branch-and-bound tree became excessively large. The clustering solution obtained upon termination after 100 branch-andbound seconds is shown in Fig. 2. Traditional branch-and-bound techniques did not prove efficient for solving the problem of moderate size to optimality. However, we observe from Figs. 1 and 2 that the solutions obtained in a very short duration as measured by branch-andbound seconds, prove to be quite good if not optimal. But much larger problems need to be solved in practice and more efficient solution techniques

D.F. Rogers, S.S. Kulkarni / European Journal of Operational Research 160 (2005) 423–444 P M a c h i n e T y p e s

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r 2

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s 9 12 13 14 15 16 17 19 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

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Fig. 1. Optimal solutions for the Irani and Ramakrishnan (1995) problem (K ¼ 3 and 4, respectively).

need to be explored. One approach is to use heuristic procedures that provide good incumbents for commercially available solvers or vice versa. It would be even more beneficial if the heuristic(s) could provide near optimal solutions. In the next section, employing a GA as a solution technique will be explored and how the GA was tailored to fit the requirements of the bivariate clustering model will be described.

4. A genetic algorithm for the bivariate clustering model The bivariate clustering model suffers from the curse of dimensionality for which integer programming problems typically display. The increase

in problem difficulty for the bivariate model as compared to the univariate model is displayed in Table 1. Since the initial attempts to solve small versions of the bivariate clustering problem with traditional optimal approaches were not always successful, a GA was developed to provide reasonable solutions. 4.1. Genetic algorithms: basic principles GAs have found increasing acceptance and have gained in popularity among members of the optimization community in recent years. There are a number of reasons for this. Michalewicz (1996) described a GA as a stochastic search algorithm for modeling some natural phenomena such as genetic inheritance and Darwinian strife for

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Fig. 2. Solution for the Burbidge (1969) problem terminated after 100 branch-and-bound seconds.

survival. GAs are a class of general purpose (domain independent) search methods. They strike a remarkable balance between exploration and exploitation of the search space. One of the key properties of a GA is the ability to maintain a pool of potential solutions at any instance unlike other methods, which maintain a single solution. Thus, a GA effectively performs a multi-directional search by exchanging information between these solutions and manipulating them. A GA is initiated by randomly creating (or seeding by the user) a population of solutions. This is the first generation of the GA. Each solution is called a chromosome. The chromosome consists of genes. Each of these solutions is evaluated for their ‘‘fitness’’ according to a user defined fitness function. Then, a new population is selected based upon the relative fitness of the individual chromosomes. Some chromosomes from the new population undergo changes to their structure by means of a crossover (a swapping of part of the genetic information between chromosomes) and

mutation (alteration of a single gene). The chromosomes from an earlier generation are called parent chromosomes and those from the immediate next generation are called child chromosomes. After crossover and mutation the child population then becomes the parent population for the next generation and the process is repeated until it exceeds the user-specified limit upon the maximum number of generations or satisfies some other stopping criterion. Thus a GA mimics the evolutionary process by implementing a survival of the fittest strategy wherein the fittest individuals of any population tend to reproduce and pass their genes to the next generation, thus improving successive generations (Joines et al., 1996). 4.2. Details of the genetic algorithm for the bivariate clustering model Michalewicz (1996) claimed that a GA for a particular problem must have the following five components:

D.F. Rogers, S.S. Kulkarni / European Journal of Operational Research 160 (2005) 423–444

• A genetic representation for potential solutions to the problem. • A way to create an initial population of potential solutions. • An evaluation function that plays the role of the environment, rating solutions in terms of their fitness. • Genetic operators for altering the composition of children. • Values for various parameters employed for the GA, e.g., population size, probabilities of applying genetic operators, random number seed, and maximum number of generations. Each potential solution for the bivariate clustering model can be represented as a two-dimensional binary array. The initial population was randomly generated. The evaluation function consists of the objective function of the clustering model plus a penalty function for penalizing the violation of constraints from the constraint set. Michalewicz et al. (1996) showed that a penalty function with a variable penalty, i.e., a penalty based upon the relative magnitude of violation of the constraints, performs much better than a penalty function which is based only upon the number of constraints violated and this type of penalty function was employed. In the GA, fit individuals in each generation were selected based upon standard proportional selection incorporating the elitist model, and then a single point crossover is performed using a simple crossover operator. A random uniform mutation operator was used. For further information on the operators and selection strategies used see Michalewicz (1996). The population size at each generation was maintained between 65 and 100. The GA was coded in C and the detailed procedures are in Kulkarni (1998).

5. An application in cellular manufacturing Manufacturing companies in todayÕs competitive environment are faced with the challenge of providing a high degree of customization, low response times and increased flexibility and quality to an increasingly aware customer. To face this

429

challenge, manufacturers have adopted a number of innovative techniques such as optimized production technology, just-in-time, flexible manufacturing systems, and group technology. CM is but one of the primary applications of GT principles to manufacturing. Conversion from, for example, a functional layout to a cellular layout involves, first, the identification of families of parts with similar machining/processing requirements. Subsequently, the machines necessary to produce the part families are determined and placed together with the part families in a cell. Thus, a typical CM setup has cells, each of which contains functionally dissimilar machines processing a particular part family. Employing CM can provide job shops with many of the advantages of mass production without severely compromising the advantages of a functional layout and also provide benefits such as reductions in work-in-process inventory, reduced lead times, simplified shop floor control, simplified product changeovers, shortened manufacturing throughput time, and improved quality (Shafer and Rogers, 1993a,b; Hyer, 1984). CM has also been found to be an important constituent for the successful adaptation of a just-in-time setup (Inman and Mehra, 1993). Manufacturers who produce a variety of low-volume products benefit most from a CM type of layout (Miltenburg and Zhang, 1991). However, there exist some disadvantages to a CM layout such as lower machine and labor utilization rates and higher capital investment due to machine and tool duplication (Singh, 1993). A number of analytical techniques have been utilized for the cell formation problem. These have been broadly classified by Chu (1995) as: classification and coding systems which are used only for identifying part families (Hyer and Wemmerl€ ov, 1985); array-based clustering methods which include the direct clustering method of Chan and Milner (1982), the rank order clustering algorithms of King (1980a,b) and King and Nakornchai (1982), and the bond energy algorithm developed by McCormick et al. (1972); statistical clustering approaches such as those developed by Seifoddini and Wolfe (1986) and Chu (1989); graph-theoretic approaches such as that of Askin and Chiu (1990); knowledge-based and pattern-recognition

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methods were developed by Chu and Shih (1992); fuzzy clustering and modeling approaches were considered by Chu and Hayya (1991); and neural network approaches were examined by Chu (1994). The most widely applied approach for the cell formation problem is specially tailored mathematical programming models and some notable references are Kusiak (1985), Gunasingh and Lashkari (1991), Joines et al. (1996), Shafer and Rogers (1991), Boctor (1991), Wei and Gaither (1990), Sankaran (1990), Sankaran and Kasilingam (1993), Vakharia et al. (1992), Kusiak and Cheng (1990), and Zhu et al. (1995). A great deal of variety can be found in the models developed, based upon model complexity and the particular objective functions and constraints utilized. Chu (1995) classified the complexity of these models as limited (linear programming and 0–1 integer programming), modest (mixed-integer and 0–1 non-linear programming), and very great (mixedinteger non-linear programming and 0–1 non-linear fractional programming). A review of the various objective functions and constraints used in mathematical programming models for CM can also be found in Chu (1995) where 45 different types of constraints used in cell formation problems to attempt to capture manufacturing realities were identified. Constraints were categorized as either (1) logical, (2) for determining cell size, (3) physical, or (4) for modeling needs. The objective for most mathematical programming models for CM have been based either upon cost, operation, or similarity/dissimilarity (s/d) coefficients. For the case of s/d coefficient based objective functions, the aim is to minimize the dissimilarity or maximize the similarity between data units. The s/d coefficients for this setting can be computed using data for the design features of parts, processing sequences, tooling requirements between parts and machines, demand, and/or processing times. Although it might appear that the s/d coefficients based upon the above factors may not reflect cost and operational factors adequately, they still capture a large portion of manufacturing reality. Hence, an objective function that is based upon s/d coefficients and not designed to include objectives such as minimizing intercel-

lular movement and maximizing machine utilization does implicitly capture the essence of some of these other objectives. For example, when a mathematical programming model is used to identify part families or machine cells based upon the s/d coefficients derived from such data, the tendency is to also approach minimizing the exceptional parts and duplicate machines and hence minimizing intercellular movement is approached as well. Mathematical programming models for CM can be formulated with objectives and constraints that capture many more of the aspects of manufacturing reality. However, as with many large integer linear mathematical programming models, the ability to solve realistically sized problems to optimality is quickly hindered as more complex features are considered. Thus, the pure bivariate clustering model from Section 2 may be promising for solving larger problems. Only pure 0/1 machine-component matrices will be considered here, but it may be modified by incorporating information such as sequencing, capacity, or volume and the methods then applied. For the CM framework, part type distances are based upon the set of machines that they must visit and distances for machine types are based upon the set of part types that they process. Derived for this research was a Euclidean distance standardized with respect to the largest distance obtained resulting in all distances belonging to the interval ½0; 1. Several GA approaches have been specifically designed for the cell formation problem. Various production data and heuristics have been incorporated into the construction of them and they have not been generalized to solve the general bivariate clustering model. Joines et al. (1996) developed a GA to form part type families and machine type cells for CM to directly maximize the often employed and non-linear grouping efficacy measure. Lee et al. (1997) created a GA for the CM setting by maximizing similarity coefficients developed by including production volume, alternate routings, and process sequences. Moon and Gen (1999) incorporated volume, capacity, and processing time in their formulation for the CM setting and GA solution. Lee-Post (2000) developed a simple and specially tailored GA to

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431

just as the originating authors found with single linkage cluster analysis and King (1980b) found with the rank order clustering algorithm. The optimal for the Chandrasekharan and Rajagopalan (1986a) problem with K ¼ 3 was determined in 47 seconds on CPLEX 6.5 and the same solution with nine exceptional elements was found, just as the original authors found with the ideal seed nonhierarchical clustering algorithm, a modified kmeans algorithm. For K ¼ 4, the bivariate model optimum was found in 1639 seconds on CPLEX 6.5 resulting in 18 exceptional elements, but previous results are unavailable for benchmarking purposes. The K ¼ 3 optimum for the Chandrasekharan and Rajagopalan (1986b) problem was determined with CPLEX 6.5 in 6926 seconds and resulted in 43 exceptional elements and objective value of 40.31. For K ¼ 4, CPLEX was suboptimally terminated after 5000 branch-and-bound seconds resulting in the solution in Fig. 3 with GE ¼ 0.4396 and objective value of 24.61. The optimal solution is in Fig. 4 and also has GE ¼ 0.4396 but an optimal objective value of 24.15. This again reflects the nature of the differences of the two performance measures, GE and that of the bivariate clustering model to minimize a sum of dissimilarities. Minimizing the sum of dissimilarities will indicate not only the membership of the cells, but additionally which cells should be physically placed next to each other to minimize distances between the cells for which exceptional parts are transferred, an inherent advantage of employing the bivariate clustering model. Chandrasekharan and Rajagopalan (1986b) noted ‘‘that the given data may not render itself amenable to a GT solution’’. Indeed, the relatively large density of this data is atypical of CM settings (Srinivasan, 1995,

maximize part type similarity to form part families. Zhao and Wu (2000) considered a GA for a multiple objectives approach to CM with routing, volume, and workload data. Onwubolu and Mutingi (2001) minimized workload variation among cells and the number of exceptional elements in their GA approach. Arzi et al. (2001) considered a bivariate objective of grouping inefficiency and machine costs (purchase and variable costs) specifically for an environment where capacity is affected by demand variability, and employed a GA to find the efficient frontier for a non-linear mixed-integer program. Dimopoulos and Mort (2001) considered only the pure binary matrix for CM to maximize grouping efficacy and developed a GA to search for the maximum set of similarity coefficients to be used as inputs to a single linkage clustering procedure. Brown and Sumichrast (2001) developed a GA for the binary CM matrix with several specific heuristics designed to increase machine utilization and decrease intercell traffic.

6. Computational results for cellular manufacturing data Several additional problems from the machinery manufacturing industry were employed to test the initial performance and applicability of the bivariate clustering model specifically in the CM arena. The density, m, n, and K for these problems are provided in Table 2. The first three problems were solved to optimality on CPLEX. The Ham et al. (1985) problem solved to optimality in 3.5 seconds on CPLEX 6.5 and the same solution for the part families and machine cells with two exceptional elements was determined, Table 2 Test problems from the CM literature Test problem source

Density

mn

K

Ham et al. (1985) Chandrasekharan and Rajagopalan (1986a) Chandrasekharan and Rajagopalan (1986b) Burbidge (1969) Burbidge (1975) Lee et al. (1997)

0.3250 0.3813 0.5688 0.1943 0.1831 0.1267

10 · 8 8 · 20 8 · 20 20 · 35 16 · 43 30 · 40

3 3, 4 3, 4 4 4 6

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Part Types 2 3 4

Machine Types

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reported densities in the range ½0:034; 0:139) and the number of exceptional elements is naturally larger. At some point of density increase, a general job shop may often become the preferred solution. A comparison to the originating authorsÕ solution with the modified rank order clustering algorithm was not attempted because the modified rank order clustering method includes machine duplication that can result in markedly different nonoverlapping part families. Since the GE measure is so pervasive in the CM literature, the ability of the GA to be modified to recover acceptable GE results was considered. After the GA has been run for a desired number of generations and the potentially suboptimal solution has been found, the following heuristic was employed to decide the final clustering. Consider the set consisting of only machine types that are in a cluster with at least n=K part types. If any part

type has not been assigned to a cluster with one of the machines of this set, then assign that part type to the group that will maximize the increase in GE. This heuristic is but one of numerous methods that could be adopted to form the final solution. The heuristic operates by beginning with the GA solution, considering the machine types with the most frequency, and then greedily assigning part types and has thus far performed quite well in the trials. The exploration and comparison of heuristics for the purpose of taking the GA solution and obtaining the final groupings lies in future research. Kulkarni (1998) appended the heuristic to the GA and the combined pseudo-code for the optimization, GA, and heuristic procedure for the bivariate clustering model applied to CM is given in Fig. 5. The Burbidge (1969) example for which the attempt to solve to optimality using a commercial

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Fig. 5. Pseudo-code for the GA and heuristic for the bivariate clustering model applied to CM.

solver was unsuccessful was solved solely from the GA without the heuristic and the results are shown in Fig. 6. In Fig. 7 is the final solution after implementation of the heuristic and it is identical to the best-known solution available in the literature. In Table 3 are the details of the parameters employed for the GA. For this problem, penalizing each constraint on the basis of degree of violation seemed to work better than assigning each constraint with a fixed penalty. Note from Table 3 that the termination criterion was the number of generations required to get to the solution, a function of the problem characteristics and the performance of the operators used in the GA, and is a rather large number. The stopping crite-

rion could also have been the standard deviation of the population fitness (stop if less than some small n > 0), rather than the maximum number of generations. Also in Table 3 are the values for GE before and after employing the GA. The GA by itself resulted in a GE of 0.8999 of the GE for the final solution after employing the heuristic; thus the heuristic resulted in the additional 0.1001. The Lee et al. (1997) problem resulted in 2835 variables (420 binary) and 14,560 constraints. Fig. 8 is the solution provided solely from the GA and Fig. 9 is the best solution for the problem derived from the GA-based solution after incorporating the heuristic and also as given by the original

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authors. In Table 3 the details of the GA run are provided. The GA by itself accounted for 0.8326 of the GE found after employing the heuristic and the heuristic accounted for the remaining 0.1674 of the value of the best-known GE. The Burbidge (1969, 1975) problems were solved with the initial solution to the GA being a terminated (suboptimal) solution found from employing CPLEX 6.5 for the bivariate optimal clustering model, rather than the original problem being the input for the GA. In Table 4 are the details of the GA parameters and the GE of the input solution, GA only solution, and the final solution after the heuristic. For the Burbidge (1969) problem, the suboptimal initial solution was determined after 100 seconds and accounted for 0.63 of the final GE; after the GA was implemented 0.85 of the final GE was obtained; and the heuristic accounted for the remaining 0.15 of the GE. The GA before the heuristic accounted for less of the final GE than by employing the original

problems as the initial start, but only 3000 generations were employed rather than 5000 previously, and different mutation rates and crossover probabilities were explored. The same final solution was determined. For the Burbidge (1975) problem, the suboptimal solution was determined after 183 seconds and accounted for 0.74 of the final GE; after the GA, 0.96 of the final GE was determined; the heuristic accounted for the final 0.04 of GE. The Burbidge (1975) solution to the problem could not be compared since he duplicated machines in his method. The GA and heuristic was quite successful in providing solutions to these cell formation problems for recovering acceptable GE measures. The attractiveness of the procedure includes a relatively easy implementation and moderate solution times for large problems, as compared to commercial optimal solvers that may terminate prematurely. A number of additional implementation issues inherent to employing a GA may be of interest.

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Table 3 Details for the problems of Burbidge (1969) and Lee et al. (1997) GE of GA before the heuristic GE after applying the heuristic # of generations for GA solution CPU resource usage for the GA GA mutation rate GA crossover probability

The solutions given by the GA depend heavily upon the operator probabilities. The best combination was identified by trial-and-error, however it may not always be possible to do so. It is important that the representation of the fitness function be studied carefully to ascertain the best operator probabilities. Also, different random number streams give different solutions and it is very important to conduct a formal experimental design for each problem. It is necessary to keep track

Burbidge (1969)

Lee et al. (1997)

0.6813 0.7571 5000 2 h 0.001 0.87

0.6490 0.7795 5000 5 h 0.001 0.87

of the seeds used, and store them for future replications. The best solution for the GA typically improved fairly rapidly in the first 1000–2000 generations, but after that the improvement was very marginal. Thus, running the GA for 5000 or more generations did not prove very beneficial. To reduce the number of generations required to get within close range of a good solution depends upon the operators used, the representation scheme, and the

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feasibility maintenance mechanism. Finally, instead of penalizing the objective function for vio-

lated constraints, other strategies can be employed such as (1) repairing infeasible individuals or (2)

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Table 4 Details for the problems of Burbidge (1969, 1975) with CPLEX 6.5 start GE of suboptimal CPLEX 6.5 solution GE of GA before the heuristic GE after applying the heuristic # of generations for GA solution GA mutation rate GA crossover probability

generating feasible solutions and making sure that the offspring are always feasible.

7. Additional applications and computational results Although the model has been illustrated for use in CM applications, it is general enough to be applied to a variety of clustering problems. Unlike many previous solution techniques that are specifically designed to solve the cell formation problem, the GA is for solving the mixed-integer programming model, and the solution technique can be applied to any clustering situation cast in the form of the bivariate clustering optimization model. Although the model and solution technique work at least as well as established approaches for solving the cell formation problem, the generality and wide application of the model and solution technique makes them attractive for use and further research across a variety of functional areas. This is now illustrated with two additional examples. 7.1. Industrial purchasing behavior The first example is a data set consisting of m ¼ 30 responses from purchasing agents indicating if they performed each of n ¼ 35 particular activities or not when making a purchase of industrial equipment from the author of Ghingold (1985). The activities are in Table 5 and the matrix has a density of 0.139. When applying the bivariate clustering model to this data set with K P 3, the solutions can be best described as (1) having two very large clusters and several very small clusters or (2) clusters with a large number of purchasing agents and few of the responses or vice

Burbidge (1969)

Burbidge (1975)

0.4783 0.6455 0.7571 3000 0.0050 0.75

0.3426 0.4429 0.4630 3000 0.0080 0.55

versa. The best solution obtained upon applying CPLEX for 1340 branch-and-bound seconds and the GA to the bivariate clustering model for K ¼ 2 appears in Fig. 10. This solution would be the one utilized if only larger clusters are sought. Thus a nice implicit and important feature of the model is the ability for the researcher to experiment with different K to search for a desired type of solution, if that is known apriori. The solution in Fig. 10 has a smaller cluster of 11 purchasing agents clustered with 10 of the activities. These activities are all associated with either (1) involving various other individuals or (2) the lesser details of working with the vendors. The larger cluster with 19 purchasing agents has 24 of the activities and can best be described as ‘‘core activities’’, i.e., activities that must be perfunctorily done, the more important constituents to involve, and the larger issues of working with vendors. Thus for this data there is a tendency for a smaller number of purchasing agents to be more social and/or political by seeking more inputs from several individuals and involving themselves with details of working with the vendors. Another group of purchasers tends to be more involved with the practical aspects of the core purchasing activities. 7.2. Segmentation strategies for the automobile industry For all previous examples, values in the data matrix were binary. Robustness of the model for solving problems with a more general data format will be addressed in the following application. In Table 6 is data for automobiles from 34 manufacturers––columns correspond to the automobile names and rows correspond to performance

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Table 5 Activities for an industrial purchase 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

Identify equipment purchase need Determine equipment currently on-site Discuss requirements with potential vendors Gather information from personnel who operate the equipment Make final decision on equipment order Prepare for receipt of the equipment––location and personnel Visit sites that use various equipment Consult experts Revise/refine/review/approve equipment specifications Install equipment Trial run of equipment Receive and inspect equipment Seek vendor instructions and/or support List and contact known suppliers Seek vendor demonstrations, tests, and sales visits Prepare evaluation criteria––cost/feature analyses Postpone the purchase––assess the purchase later Pay the invoice Discuss and negotiate with vendors Reduce the number of potential vendors Evaluate the vendorsÕ financial and legal status Review existing equipment contracts Verify funding commitment for the purchase Obtain cost estimates Get plant engineers to evaluate the proposed equipment Establish a purchasing committee Present a report to top management Top management becomes involved Discuss and get approval from others concerned with the purchase Track and expedite the order Assign a project manager Obtain recommendations from suppliers Match user needs to software capabilities Purchase department contacted and involved Compare vendorsÕ past performances

measures/criteria. Johnson (1998) used multidimensional scaling, a non-linear optimization method to reduce the standardized distances between points in a p-dimensional data set to q dimensions, 2 6 q < p (see Young and Hammer, 1987, for a detailed exposition). Distances between the automobiles in Table 6 from the ten-dimensional space of the performance measures/criteria to a lower two-dimensional space were mapped. Unlike JohnsonÕs (1998) clustering of the automobiles on two ‘‘unknown’’ dimensions using multidimensional scaling, the bivariate clustering model simultaneously clusters the automobiles with the performance measures/criteria thereby providing

an indication of why some of the automobiles may belong to the same cluster, i.e., the attributes that the automobiles have in common. The bivariate clustering model is linear and when the model is solved, a global optimal solution is guaranteed–– that type of property may not be indicated for multidimensional scaling, which also suffers from the issue that as the number of dimensions increase the convergence criteria may not be adequately satisfied. The data on performance measures/criteria in Table 6 are on different scales, so the automobiles were ranked from 1 to 34 for each of the 10 performance measures/criteria. These ranks were then

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utilized to calculate the standardized distances between automobiles and between the performance measures/criteria. The resulting bivariate clustering model with K ¼ 3 solved to optimality in 1230 branch-and-bound seconds and the solution is in Table 7. The mixed-integer program solved to optimality in this case, but the GA is not confined to a specific application such as CM and it could have been utilized if an optimal solution had not been obtained. Johnson (1998) provided a limited report of the clustering that he obtained using multidimensional scaling and mentioned that the Volvo 850 Turbo SportsWagon, the Honda Prelude VTEC, and the Volkswagen Corrado SLC are very similar; the Mercedes Benz 600SEC and the Bentley Turbo R are quite similar; and the Buick Riviera and Oldsmobile Aurora are very similar. He also noted that the two dimen-

sions for clustering may not be easily identifiable in a meaningful way except perhaps by someone who is an expert on automobiles. The results of the bivariate clustering model validate the findings of Johnson (1998) but go beyond that by clustering the performance measures/criteria with the clustering for the automobiles. This could provide vital information for a marketing analyst who wants to decide on various segmentation strategies for sales and marketing of the automobiles from different manufacturers. The grouping of the performance measures/criteria achieved by the bivariate clustering model can also be utilized to combine them to represent latent dimensions such as those used in structural equation modeling. This example is an indication that the bivariate clustering model and solution technique are general enough to be applied across a broader class of

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Table 6 Automobile and attribute data from Johnson (1998) Automobile Acura NSX Alfa Romeo 164 Quad Audi S4 Bentley Turbo R BMW 850 CSi Bugatti EB110 Buick Riviera Cadillac Eldorado Touring Coupe Chevrolet Corvette ZR1 Chrysler LHS Dinan M5 Dodge Viper RT/10 Ford Mustang Cobra Honda Prelude VTEC Infiniti Q45 Jaguar XJR-S Coupe Lincoln Mark VIII Lotus Esprit Turbo Mazda Millenia S Mercedes 600SEC Mercury Villager Mitsubishi 3000 GTVR-4 Nissan 300ZX Turbo Oldsmobile Aurora Plymouth Grand Voyager LE Pontiac Firebird Formula Porsche 911 Turbo 3.6 Saab 9000 Aero Saleen Mustang S-351 (Ford) Subaru SVX Toyota Supra Turbo Vector W8 Twin Turbo Volkswagen Corrado SLC Volvo 850 Turbo Sportswagon

Price

HP

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TIM2

TS

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BRAK2

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SS

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68,600 36,690 43,750 174,900 98,500 35,000 27,632 41,535

270 230 227 315 372 611 225 300

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14 15.8 15 15.5 14.4 12.5 16 15.3

168 150 130 128 155 207 114 150

120 150 147 147 135 112 139 138

200 268 246 253 220 209 254 249

0.87 0.77 0.83 0.73 0.89 0.99 0.75 0.78

62.3 59.3 60 56.8 62 64.3 57.9 61.7

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65,318 28,254 60,700 50,000 23,535 24,500 44,000 73,000 38,800 67,345 31,400 132,000 21,798 40,900 39,500 31,995 22,883

375 214 382 400 240 190 278 318 280 264 210 389 151 320 300 250 162

5.6 9.1 5.6 4.8 6.9 7.1 7.5 7 7.6 5.3 8 6.6 13.4 5.7 6 8.6 10.7

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178 112 160 160 140 139 150 167 130 167 142 155 115 159 155 135 112

142 153 122 156 130 137 159 137 137 121 151 125 178 122 124 131 161

256 275 216 261 236 242 272 225 248 225 269 240 293 218 279 244 285

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Notes: Price ¼ base price of the automobile in US dollars, HP ¼ engine horsepower, TIM1 ¼ time to reach a speed of 60 mph, TIM2 ¼ time to reach a speed of 80 mph, TS ¼ top speed, BRAK1 ¼ braking distance from 60 mph, BRAK2 ¼ Braking distance from 80 mph, SP ¼ steady speed cornering grip, SS ¼ speed through a slalom course, MPG ¼ average miles per gallon (fuel efficiency).

clustering problems. Future research into how the model and the GA may be modified to include multivariate clustering is being planned.

8. Summary and future research In this paper, a new model was developed for bivariate clustering of a matrix, complexity

of the problem was explored, and a GA-based solution procedure for solving realistically sized problems with a direct application to the clustering of part types and machine types in a CM environment was developed. Problems from the CM literature were solved using the GA-based procedure and the results provided the best solutions known to be available for these problems. Additional problems in industrial

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Table 7 Clustering of JohnsonÕs (1998) automobile/attribute data into three groups Automobile

Performance measure

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Acura NSX Ford Mustang Cobra Honda Prelude VTEC Nissan 300ZX Turbo Plymouth Grand Voyager LE Pontiac Firebird Formula Saab 9000 Aero Subaru SVX Volkswagen Corrado SLC Volvo 850 Turbo Sportswagon

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Price HP TS SP

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Alfa Romeo 164 Quad Audi S4 Buick Riviera Cadillac Eldorado Touring Coupe Chrysler LHS Infiniti Q45 Lincoln Mark VIII Mazda Millenia S Mercury Villager Oldsmobile Aurora

TIM1 TIM2

purchasing and market segmentation were also explored. There are a number of avenues for further research. A general study is being prepared to determine the performance of the optimal bivariate model and the GA approach w.r.t. terminating at the optimal solution and computer time requirements. Control variables include the nature of the specific elements of the input matrix (such as binary, non-negative, or unbounded), the matrix density, and the number of rows and columns. For GA applications, experimenting with different GA operators and with different strategies for generating feasible solutions are being considered. For

example, experiments may be conducted with different operators such as non-uniform mutation, Gaussian mutation, two-point crossover, arithmetic crossover, and also with selection strategies such as tournament selection. Using one of these operators might improve the performance of the GA resulting in the GA being more robust to solve problems of larger dimensions and possibly capable of providing the best solution directly. Other potential future research includes comparing the effectiveness of the GA-based procedure to other meta-heuristic solution techniques such as simulated annealing or tabu search. Incorporating implicit constraints directly in the bivariate

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clustering model to derive a special structure that can then be exploited by using Lagrangian relaxation techniques to get good bounds and/or optimal solutions for very large problems may prove interesting. Indicated for the CM area is a close examination of the performance of the bivariate model by comparing the results obtained to those from many other approaches designed specifically for CM. Adapting the bivariate clustering model and/or the GA to CM by including options such as alternate part routings, limits on cell size, part processing times, machine costs, and part volumes may be fruitful. Also, a comparison of the effectiveness of the GA versus the effectiveness of the heuristic in the application for CM would be interesting. How might the heuristic alone perform? At what point should the optimization model or the GA be terminated and the heuristic employed? Acknowledgements We are grateful to W. David Kelton, Professor, Department of Quantitative Analysis and Operations Management, College of Business, University of Cincinnati, Cincinnati, OH, USA, and Dr. E. Jack Chen, BASF Corporation, Mt Olive, NJ, USA, for allowing us to use their random number generation software. References Anderberg, M.R., 1973. Cluster Analysis for Applications. Academic Press, New York. Aronson, J.E., Klein, G., 1989. A clustering algorithm for computer-assisted process organization. Decision Sciences 20 (4), 730–745. Arzi, Y., Bukchin, J., Masin, M., 2001. An efficiency frontier approach for the design of cellular manufacturing systems in a lumpy demand environment. European Journal of Operational Research 134, 346–364. Askin, R.G., Chiu, K.S., 1990. A graph partitioning procedure for machine assignment and cell formation in group technology. International Journal of Production Research 28, 1555–1572. Balas, E., 1965. An additive algorithm for solving linear programs with zero–one variables. Operations Research 4 (13), 517–549. Boctor, F.F., 1991. A linear formulation of the machine cell formation problem. International Journal of Production Research, 343–356.

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