A mathematical approach for the formation of manufacturing cells

Computers & Industrial Engineering 48 (2005) 3–21 www.elsevier.com/locate/dsw

A mathematical approach for the formation of manufacturing cells Zahir Albadawia,1, Hamdi A. Bashirb,*, Mingyuan Chenc,2 a

Department of Mechanical Engineering, McGill University, 817 Sherbrooke St. West, Montreal, Que., Canada H3A 2K6 b Department of Mechanical and Industrial Engineering, Sultan Qaboos University, Alkhod, 33, Muscat, Oman 123 c Department of Mechanical and Industrial Engineering, Concordia University 1455 de Maisonneuve Blvd West, Montreal, Que., Canada H3G 1M8 Received 13 October 2003; accepted 21 June 2004

Abstract One of the major steps in designing cellular manufacturing systems is to form cells. This involves identification of machine cells and part families. This paper proposes a new mathematical approach for forming manufacturing cells. The proposed approach involves two phases. In the first phase, machine cells are identified by applying factor analysis to the matrix of similarity coefficients. In the second phase, an integer-programming model is used to assign parts to the identified machine cells. To evaluate its performance, the proposed approach was applied to sample problems from the literature and a real life problem from a manufacturing plant. The results indicate that the proposed approach performs very well in terms of a number of criteria and compares favorably to well-know existing cell formation methods. In addition to its good performance, the proposed approach has the flexibility to allow the cell designer to either identify the required number of cells in advance, or consider it as a dependent variable. Using algorithms which are available in many commercial software packages is the other advantage of the proposed approach. q 2004 Elsevier Ltd. All rights reserved. Keywords: Cellular manufacturing; Cell formation; Factor analysis; Integer programming

* Corresponding author. Tel.: C1 968 513316. E-mail addresses: [email protected] (Z. Albadawi), [email protected] (H.A. Bashir), mychen@ me.concordia.ca (M. Chen). 1 Tel.: C1 514 398 6296; fax: C1 514 398 7365. 2 Tel.: C1 514 848 2424x3134; fax: C1 514 848 3175. 0360-8352/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2004.06.008

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1. Introduction Cellular manufacturing has been recognized as one of the most recent technological innovations in job-shop or batch-type production to gain economic advantages similar to those of mass production (Chu & Tsai, 1990). Many firms have recently started to adopt cellular manufacturing systems in order to achieve flexibility and efficiency, which are crucial for survival in today’s competitive environment. Cell formation, which involves identification of machine cells and part families, is the first major step in designing a cellular manufacturing system. During the last three decades of research, a large number of cell formation methods have been developed. These methods are classified in the review papers by Jonies, Culbreth, and King (1996) and Selim, Askin, and Vakharia (1998) into array-based methods, heuristic methods, hierarchical methods, graph partition methods, artificial intelligence methods, and mathematical programming methods. Array-based clustering is one of the simplest classes of production-oriented cell formation methods. It requires a visual inspection of the output to determine the composition of the manufacturing cells. Examples of array-based clustering methods include the bond energy algorithm, BEA of McCormick, Schweitzer, and White (1972), the rank order clustering method, ROC of King (1980), and the direct clustering algorithm, RCA of Chan and Milner (1982). BEA maximizes the total ‘bond energy’ of the machine-part incidence matrix. The first step of this algorithm is determined by intuition; therefore, many possible solutions can be generated. ROC rearranges the machine-part incidence matrix based on the ‘binary rank orders’ of its rows and columns. Chandrasekharan and Rajagopalan (1986) have shown that this method is strongly dependent on the initial disposition of the machine-part incidence matrix; therefore, identification of the exceptional elements and bottleneck machines is somewhat arbitrary. RCA forms part and machine families by rearranging the rows and the columns of the incidence matrix based on the number of non-zero elements in each row and column. According to Wemmerlov and Hyer (1986), this algorithm may not produce viable or acceptable solutions because it redirects the diagonal in each iteration. Hierarchical clustering methods operate on an input data set described in terms of a similarity or distance function and produce a hierarchy of clusters or partitions. McAuley (1972) used the single linkage clustering (SLC) dendogram to identify machine groups from the distance matrix. The major drawback of SLC is the ‘chaining’ problem, which may be caused by two clusters joining together. To help reduce the chaining problem, Seifoddini (1986) applied the average linkage clustering (ALC) algorithm. Recently, artificial intelligence (AI) methods have been increasingly applied to the cellular manufacturing problem. Artificial neural networks have been applied to the production-oriented problems to determine the machine cells and part families. Malave and Ramachandran (1991) applied a modified version of the Hebbian learning rule to the cell formation problem. While some parts definitely belong to certain part families, it is not always clear which family is appropriate. Li and Ding (1988) and Xu and Wang (1989) applied fuzzy mathematics to this problem. Genetic algorithms and simulated annealing are also efficient stochastic search algorithms that solve a wide range of optimization problems, especially combinatorial problems. Boctor (1991) and Venugopal and Narendran (1992a) used simulated annealing to solve integer programming formulations of the cell formation problem. Joines (1993) developed a genetic algorithm to solve integer-programming formulations of the cell design problem, allowing multi-criteria objective functions and constraints on the number of permissible cells. Venugopal and Narendran (1992b) also used genetic algorithms to solve a multi-objective integer programming formulation of the cell formation problem.

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Kusiak and Chow (1988) represented the part-machine matrix as a graph. Depending on the representation of nodes and edges three types of graphs can be formed (Singh, 1996): bipartite graph that formulates the problem as a k-decomposition problem in graph theoretic terms; transition graph which represents a part (machine) by a node and a machine (part) by an edge to determine the number of cells and cell sizes; and boundary graph that is a hierarchy of bipartite graph method. Purcheck (1975) was among the first to apply mathematical programming methods to the GT problem. Srinivasan, Narendran, and Mahadevan (1990) proposed an assignment model for the part families and machine grouping problem. The objective of the assignment model is to maximize the similarity. Since parts within the family interact with each other, it becomes important to account for the total family group interaction. Kumar, Kusiak, and Vannelli (1986) proposed a quadratic programming model for this purpose. Adil, Ragamani, and Strong (1993) also proposed an assignment allocation algorithm, a non-linear mathematical model, to identify part families and machine groups simultaneously without manual intervention. The objective of the model is to explicitly minimize the weighted sum of the exceptional elements and voids. Other cell formation methods rely on heuristics. Examples of these methods include the cost-based heuristic of Askin and Subramanian (1987), the inter-class traffic minimization method of Ballakur and Steudel (1987), and the identification, clustering, refinement, merging and allocation heuristic (ICRMA) of Tabucanon and Ojha (1987), which aim at reducing the inter-cell material flow in the system, belong to this class of cell formation methods. However, most of the existing cell formation methods suffer from one or more drawbacks. Their major common drawbacks include: † the deterioration of performance as the problem under consideration becomes larger, † the inflexibility in determining the number of cells (i.e. in some methods, the number of cells is a dependent variable, while in others it has to be identified in advance), and † the limited industrial application due to the unavailability of software programs supporting them (Selim, Askin, & Vakharia, 1998). New cell formation approaches that overcome these limitations are clearly needed. In response to this need, this paper proposes a new mathematical approach for the manufacturing cell formation. The proposed approach attempts to: † perform very well in terms of a number of well known criteria, † compare favorably to well-known existing methods, † have the flexibility in allowing the user either to identify the required number of cells in advance, or consider it as a dependent variable, and † be supported by available commercial software programs in order to facilitate industrial applications.

2. The proposed approach The proposed approach is implemented in two phases. In the first phase, the machine cells are identified using factor analysis. In the second phase, the parts are assigned to the machine cells using an integer-programming model. These two phases are described below through an example.

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2.1. Phase 1: identification of machine cells using factor analysis Since cell formation can be considered as a dimension reduction problem in which a large number of interrelated machines are grouped into a smaller set of independent cells, the proposed approach applies factor analysis, a dimension reduction technique, to the initial part-machine matrix to identify machine cells. Factor analysis is a powerful multivariate analysis tool used to analyze interrelationships among a large number of variables to reduce them into a smaller set of independent variables called factors. Factor analysis was developed in 1904 by Spearman in a study of human ability using mathematical models (Rummel, 1988). Since then, most of the applications of factor analysis have been in the psychological field. Recently, its applications have expanded to other fields such as mineralogy, economics, agriculture and engineering. Factor analysis requires having data in form of correlations, and uses different methods for extracting a small number of factors from a sample correlation matrix. These methods include: common factor analysis, principal component analysis, image factor analysis, and canonical factor analysis. Principal component analysis (PCA) is the most widely used. PCA is a straightforward and quantitatively rigorous method, which transforms a given set of interrelated variables into a new set of variables called the principal components (corresponding to factors in factor analysis). The set of principal components generated are uncorrelated linear combinations of the original variables and account for the total variance of the original data. In this method all the principal components are generated in such a way that they are orthogonal to each other so the correlation between them is zero. The principal components are generated in a sequentially ordered manner with decreasing contributions to the variance, i.e. the first principal component explains most of the variation present in the original data, and successive principal components account for decreasing proportions of the variance. Accordingly, the first principal component can be viewed as the best summary of the linear relationships existing in the original data set. The second principal component can be viewed as the second best representative of the linear relationships among the variables under the condition that the second principal component is orthogonal to the first. In order to be orthogonal to the first principal component, the second principal component must account for that proportion of the variance, which is not accounted for by the first principal component. Thus, the second principal component can be defined as the linear combination of variables, which accounts for the maximum variance after the effect of the first principal component is removed from the data. The remaining principal components are defined similarly until all the variance in the data is explained. The full set of the principal components is as large as the original set of variables, but usually the variances of the first few principal components exceed 80% of the total variance of the original data. This property means that the data points can be rigorously separated into distinct clusters when projected into a space spanned by the first few principal components, which are called factors. This achieves the dimensionality reduction objective of factor analysis. The factors obtained by the PCA method usually have a complex structure and are difficult to interpret since they are significantly correlated with many of the original variables. For solving this problem another method from statistical data analysis is used. Rotation methods transform the factors to simpler and more interpretable constructs. After rotation, each variable will be only related to one of the factors and each factor will have high correlation with only a small set of variables. The brief description of factor analysis presented above only highlights the most important features of this approach. Detailed description of this approach can be found in the relevant literature such as Kleinbaum, Kupper, and Muller (1988) and Rummel (1988).

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Fig. 1. Part-machine matrix.

To apply factor analysis to a cell formation problem, the following major steps are carried out: (1) generation of a similarity coefficient matrix of the machines by considering similarity between machines as the measure of distance between every pair of machines; (2) extraction of the initial cells using the PCA method; (3) optimization of the initial cell formation using a rotation technique. These steps are explained below through a demonstrative example of a simple machine cell formation problem provided by Singh and Rajamani (1996). As shown in Fig. 1, the initial machine-part matrix of this problem consists of six machines (labeled 1–6) and eight parts (labeled 1–8). It should be noted, however, that this simple problem can be easily solved using any other cell formation method and was selected only for the purpose of illustration. 2.1.1. Generation of the matrix of similarity coefficients The implementation of factor analysis requires the correlation matrix of the initial data set. This correlation matrix can be obtained in the form of a similarity coefficient matrix. Each element of the correlation matrix of a set of variables can be represented as a scalar product of two vectors representing two of the variables. A scalar product which represents the relationship between two vectors can be estimated using one of the similarity measures proposed in the literature such as the Jacard similarity measure which is defined by Eq. (1). PN kZ1 Xijk (1) Sij Z PN kZ1 ðYijk C Zijk K Xijk Þ where Sij Xijk Yijk Zijk

similarity coefficient between machines i and j, 1 if operation on part k is performed on both machine i and j and 0 otherwise, 1 if operation on part k is performed only on machine i and not on j and 0 otherwise, 1 if operation on part k is performed only on machine j and not on i and 0 otherwise.

This coefficient indicates maximum similarity when the two machines process the same part type, SijZ1, and maximum dissimilarity when the two machines do not process the same part type, SijZ0.

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Fig. 2. Similarity matrix.

Applying Eq. (1) to the initial machine-part matrix given in Fig. 1 yields the matrix of similarity coefficients given in Fig. 2. For example, the similarity coefficient between machines 1 and 6, S16 is calculated as follows: S16 Z

2 Z 0:40 4 C3 K2

(2)

2.1.2. Extraction of the initial cells Assuming the machines as the original set of variables, and the similarity coefficient matrix as an estimate of the correlation matrix explaining the correlations between each pair of machines, we proceed to use the PCA framework for grouping the machines into separate independent clusters, which form the initial cells. In the PCA method, the initial cells are extracted out by the eigenvalue–eigenvector analysis of the similarity coefficient matrix as presented in Eq. (3) ðS K Ili ÞYi Z 0;

i Z 1; 2; .; P

(3)

where S is an P!P similarity coefficient matrix, I is the identity matrix, li are the characteristic roots (eigenvalues), and Yi are the corresponding eigenvectors. Eq. (3) is an eigenvalue–eigenvector equation where the terms l1Rl2R.Rlp are the real, nonnegative roots of the determinant polynomial of degree P given as: jS K li Ij Z 0

(4)

This equation is solved for li, and then Yi can be calculated, using the values of li in Eq. (3). It is proven that the eigenvectors thus computed represent the unique set of P independent principal components (factors) of the data set, which maximize the variance (Basilevsky, 1994). According to the PCA method each of the P independent principal components (factors) can be written as a linear combination of the original variables (machines), with the elements of the P eigenvectors as the coefficients of these linear combinations. Furthermore, the elements of these eigenvectors reflect the degree of association between each principal component (factor) and the machine, and are called the ‘factor loadings’ of the machines on the ith principal component in factor analysis terminology. Each of the P independent principal components represents a cell. There should be low similarities among machines that are associated with different cells, and high similarities among machines strongly associated with the same cell. The corresponding eigenvalues and eigenvectors for the similarity matrix given in Fig. 2 are illustrated in Figs. 3 and 4.

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Fig. 3. The eigenvalue matrix.

In order to determine the number of cells needed to group the machines, the user has two options, either to identify the required number of cells in advance, or to consider it as a dependent variable. In both cases, the cells have to be ranked in a descending order according to the percentage of the total variance explained by each cell. The total variance of each cell is the sum of the variances of all machines in the cell, or the eigenvalue corresponding to that cell (Basilevsky, 1994). If the number of cells is determined by the user, then the cells with the highest eigenvalues are to be selected. Otherwise, cells with eigenvalues greater than or equal to one should be selected (Kaiser, 1960). Both criteria ensure that a high percentage of the variance is accounted for. The computed eigenvalues for the matrix given in Fig. 2 are listed and ranked in a descending order in Table 1. According to Kaiser’s criterion, only the first two cells are needed to group the machines. Table 1 illustrates the initial statistics for each cell. The total variance explained by each cell is listed in the column labeled eigenvalues. The next column contains the percentage of the total variance attributable to each cell. The percentage of the total variance explained by each factor is used to decide on the number of cells. The last column indicates the cumulative percentage, which is the percentage of variance attributable to each cell and the cells that precede it in the table. Table 1 shows that almost 70% of the total variance is attributable to the first two cells. The remaining four cells together, account for only 30% of the variance. One of the best advantages of this method is the possibility of obtaining the optimum number of cells by considering the cells with the greater percentages of the total variance. Table 2 displays the initial machine-cell matrix produced by the PCA. This table simply contains the elements of the two selected eigenvectors corresponding to the highest eigenvalues. The absolute values of the elements of the eigenvectors (the factor loadings) reflect the associations between the cells and the machines. For example, machine 1, m1 can be expressed as: m1Z0.36F1C (K0.47)F2, so the absolute value of the loadings that associate machine 1 to the cells 1 and 2 are,

Fig. 4. The eigenvector matrix.

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Table 1 Percentage of variance associated with each cell Cells

Eigenvalues

% of total variance

Cumulative percentage (%)

1 2 3 4 5 6

2.38 1.81 0.78 0.53 0.29 0.20

39.70 30.00 13.00 9.00 4.80 3.50

39.70 69.70 82.70 91.70 96.50 100.00

respectively, 0.36 and 0.47. This means that machine 1 has a stronger relationship with cell 2 than cell 1. However, it is difficult to decide whether machine 5 is more closely related to cell 1 or 2. Thus, to achieve more interpretability, the initial cells corresponding to the resulting eigenvectors have to be optimized. 2.1.3. Optimization of the initial cells Although the matrix obtained in the extraction stage indicates the relationship between the cells and the individual machines, it is usually difficult to identify meaningful cells based only on this matrix. In other words, in many cases, it is not easy to determine which machine corresponds to which cell. This is because the machines and the cells do not appear correlated in any interpretable pattern and most cells are correlated with many machines. To overcome this problem, the structure of the initial cells needs to be simplified to achieve more interpretability. This is usually done by using one of the rotation methods such as the varimax algorithm (Rummel, 1988). Fig. 5 illustrates graphically how rotation helps to achieve a simpler structure. As shown, the purpose of rotation is to rotate the axes I (cell 1) and II (cell 2) to new positions so that each machine is close to only one of the two rotated axes, I 0 (cell 1) and II 0 (cell 1). After rotation each machine is clustered with only one of the rotated axes. Mathematically, the varimax algorithm finds the optimal position of the component axes by maximizing Eq. (5) " !2 # p p Q X 1X 2 2 1 X 2 (5) ðb Þ K 2 bij p iZ1 ij p jZ1 iZ1 where Q represents the number of cells, p represents number of machines, and bij denotes the loading of the ith variable on the jth principal component. This equation represents the variance of the loadings bij Table 2 Initial machine-cell matrix Machines

Cell 1

Cell 2

1 2 3 4 5 6

0.36 0.47 0.45 0.38 0.44 0.33

K0.47 0.37 K0.41 0.31 0.46 K0.41

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Fig. 5. Graphical illustration of rotation.

across all cells and the varimax algorithm seeks to maximize this variance in order to obtain the cells with high correlation for one machine and no correlation at all with other machines. Since most statistical packages such as the Statistical Package for the Social Sciences (SPSS, 1999) can perform the varimax algorithm, there is no need to describe the computation involved in further detail. SPSS has been used to apply the varimax algorithm to the initial cells shown in Table 2, which yields the matrix shown in Table 3. As shown in Table 3, machines 2, 4 and 5 have the highest loadings on cell 1, while machines 1, 3 and 6 have the highest loadings on cells 2. Therefore, the best grouping for the six machines is to group them into two cells: cell 1 consists of machines 2, 4, and 5, while cell 2 consists of machines 1, 3, and 6. 2.2. Phase 2: assigning parts to cells To complete the cell formation, the parts need to be allocated to the machine cells. Different criteria could be used to assign parts to cells. For example, if the goal is to minimize the number of exceptional elements, then the following simple integer-programming model can be used. XX nkc rkc (6) Maximize k

c

Table 3 Final machine-cell matrix (after rotation) Machines 1 2 3 4 5 6

Cells 1

2 K2

K4.02!10 0.596 7.26!10K2 0.488 0.633 K2.30!10K2

0.594 3.37!10K2 0.607 2.11!10K2 K6.16!10K2 0.523

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Fig. 6. Final cell formation.

Subject to X

rkc Z 1 c k

(7)

c

where rkcZ1 if part k is assigned to cell c and rkcZ0 otherwise, and nkc is the number of visits to cell c by part k. Applying the model to the problem under consideration yields the cell design in Fig. 6. As shown, the solution is to group components 3, 4, and 7 in one family and the rest of the components in another family. The first family belongs to cell 1, which consists of machines 2, 4, and 5, while the second family belongs to cell 2, which consists of machines 1, 3, and 6.

3. Evaluation of performance Three objective criteria widely used in the literature are selected to evaluate the performance of the proposed approach (McCormick et al., 1972). These criteria are the percentage of exceptional elements, machine utilization, and the grouping efficiency. 3.1. Percentage of exceptional elements The quality of a clustering can be measured by the number of machine-part cells that remain outside the diagonal blocks (Chan & Milner, 1982; King, 1980). These off-diagonal machine-part cells are called exceptional elements. Percentage of exceptional elements (PE) is calculated by dividing the number of exceptional elements by the total number of elements with 1 as the entry. This measure is normalized with respect to the problem size. Better clustering algorithms result in a smaller percentage of exceptional elements. PE Z

number of exceptional elements !100 total number of operations

(8)

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3.2. Machine utilization Machine utilization (MU) indicates the percentage of time the machines within the clusters are used in production. MU is defined by Chandrasekharan and Rajagopalan (1986) as MU Z

N Q X

(9)

mk pk

kZ1

where N is the total number of ones within the part family-machine cells, Q is the number of cells, mk is the number of machines in the kth cell, and pk is the number of parts in the kth cell. Generally, the higher the value, the better the machines are being utilized. 3.3. Grouping efficiency Grouping efficiency (GE) is an aggregate measure, which takes both the number of exceptional elements and machine utilization into consideration. GE is defined by Chandrasekharan and Rajagopalan (1986) as: 1 1 0 0 C C B B C C B N B NE C C ð1 K aÞB1 K C GE Z aB C C BX B Q Q X A A @ @ MN K mk p k mk pk kZ1

(10)

kZ1

The weight a is assigned to reveal the relative importance of each term, though a value of 0.5 is commonly used. MN is the size of machines-part matrix and NE is the number of exceptional elements. 3.4. Computational results In order to evaluate the proposed approach and to compare its performance with other cell formation methods, six sets of data (problems) have been chosen from the literature. Table 4 summarizes the special features and the sources of these data sets. Applying the proposed approach to these data sets yields the solutions shown in Figs. 7–12. The performance evaluation results of these solutions are summarized in Table 5. It can be seen that the percentage of the exceptional elements ranges from 0 to 14.8%, the percentage of machine utilization ranges from 61.8 to 100%, and the percentage of group Table 4 Features and sources of cell formation problems Test problem

Size

No. of cells

References

1 2 3 4 5 6

5!7 12!10 15!10 8!20 14!24 16!43

2 3 3 3 4 5

Waghodekar and Sahu (1984) McAuley (1972) Chan and Milner (1982) Chandrasekharan and Rajagopalan (1986) King (1980) King and Nakornchai (1982)

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Fig. 7. Result of applying the proposed approach to problem 1.

efficiency ranges from 80.7 to 96%. The best clustering performance results that are obtained from the literature are also included for comparison purposes (Chen, 1996; Chu & Tsai, 1990). As shown in Table 5, the proposed approach compares favorably on average to neural network clustering (Chen, 1996), rank order clustering, ROC (King, 1980), direct clustering algorithm, DCA (Chan & Milner, 1982), and bond energy algorithm, BEA (McCormick et al., 1972). Since BEA has the second best performance after the proposed approach, we decided to compare its performance with the proposed approach using a real problem from a manufacturing plant. This plant has 34 machines and manufactures 97 different parts. The initial machine-part matrix produced has a density of 0.10. The density of this matrix is varied randomly to produce six more problems. The proposed approach and BEA are then applied to a total of seven problems of the same size, but with different densities. The performance evaluation results are shown graphically in Figs. 13–15. A visual assessment of these graphs indicates that the performance of the proposed approach has a lower

Fig. 8. Result of applying the proposed approach to problem 2.

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Fig. 9. Result of applying the proposed approach to problem 3.

sensitivity to the problem density compared to BEA and that the proposed approach performs better than BEA in all criteria. This visual assessment is supported by a ‘significance testing’ technique. Because the distribution of the data is unknown, a distribution-free test known as Wilcoxon Rank–Sum test (Montgomery & Runger, 2003) is adopted. For each criterion, the null hypothesis tested is H0 denoting that no significant difference exists between the performance of the proposed approach and that of BEA, and the alternative hypothesis tested is H1 denoting that the performance of the proposed approach is better than that of BEA. Table 6 contains a summary of the performed tests. As shown, all the null hypotheses are rejected, at 0.005 level of significance. In other words, the proposed approach performs better than BEA in all three criteria.

Fig. 10. Result of applying the proposed approach to problem 4.

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Fig. 11. Result of applying the proposed approach to problem 5.

Fig. 12. Result of applying the proposed approach to problem 6.

Problem

1 2 3 4 5 6 Average

The proposed approach

Neural network clustering

Rank order clustering, ROC

Direct clustering algorithm, DCA

PE (%)

MU (%)

GE (%)

PE (%)

MU (%)

GE (%)

PE (%)

MU (%)

GE (%)

PE (%)

MU (%)

GE (%)

PE (%)

MU (%)

GE (%)

12.50 12.80 0 14.80 3.30 2.40 7.63

82.35 85.00 92.00 100.00 68.60 61.80 81.62

85.62 89.40 96.00 95.80 83.9 80.70 88.57

15.00 12.82 9.61 14.80 3.30 2.40 9.65

72.27 80.95 94.00 100.00 66.28 61.80 79.21

77.10 82.27 94.50 95.80 82.74 80.70 85.51

18.80 13.20 0 21.30 6.60 2.40 10.38

72.22 78.60 92.00 86.00 66.30 61.80 76.15

77.29 86.10 96.00 86.60 82.30 80.70 84.83

18.80 18.40 0 * 6.60 2.40 12.46

72.22 81.60 92.00 * 66.30 59.90 74.40

77.29 86.50 96.00 * 82.30 79.80 84.37

12.50 13.20 0 14.80 3.30 2.40 7.70

82.35 82.50 92.00 100.00 68.60 61.80 81.20

85.62 88.10 96.00 95.80 83.90 80.70 88.35

GE, grouping efficiency; PE, exceptional elements; MU, machine utilization.

Bound energy algorithm, BEA

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Table 5 Summary of performance evaluation results

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Fig. 13. Density versus percentage of exceptional elements.

Fig. 14. Density versus machine utilization.

Fig. 15. Density versus grouping efficiency.

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Table 6 Wilcoxon rank-sum for testing significance Parameters of interest

m1 and m2, the mean of parentage of exceptional elements using the proposed approach and BEA, respectively m1 and m2, the mean of machine utilization using the proposed approach and BEA, respectively m1 and m2, the mean of grouping efficiency using the proposed approach and BEA, respectively

H0

H1

Test statistic w1

w2

Critical value, w0:005

Rejection region

Decision

m1Zm2

m1!m2

31

74

32

Reject H0 if w1%w0.005

Reject H0

m1Zm2

m1!m2

77

28

32

Reject H0 if W2%w0.005

Reject H0

m1Zm2

m1!m2

77

28

32

Reject H0 if W2%w0.005

Reject H0

Clustering and data reorganization methods such as BEA do not always give a solution matrix with a desirable structure. For some simple problems, the user may manually interchange some rows and columns of a solution matrix so as to obtain non-overlapping blocks. However, for large problems, this is not possible. There are no known systematic procedures to replace the manual method for this purpose (Boe & Cheng, 1991). Array sorting methods such ROC have fast convergence and a low computation time. However, they do not always produce good grouping solutions and must rely on the user to identify the exceptional elements and the bottleneck machines that prevent the formation of a block structure in a solution matrix. DCA cannot produce exact diagonal matrices (Boe & Cheng, 1991). The proposed approach performs better than the mentioned algorithms in that it gives the best diagonal block solution matrices while converging fast.

4. Conclusions During the last three decades of research, numerous algorithms have been developed to solve cell formation problems and this research still remains of interest to this day. Designing appropriate cells is the first step towards configuring a cellular manufacturing system. This paper presents a new mathematical approach for the manufacturing cell formation. The principal objective of the proposed approach is to formulate a multivariate analysis model to generate optimal machine cells and part families. The approach has the capability of providing good solutions compared to other existing methods. Using six test problems from the literature, the proposed approach is found to perform very well in terms of a number of objective criteria, and compares favorably to neural networks clustering, ROC, DCA, and BEA. Furthermore, because of its mathematical foundation, the performance of the proposed approach does not deteriorate as the problem under consideration becomes larger. Real life problems are typically large, and solving such problems requires special solution procedures. This research provides a suggestion of a solution for this purpose. When applied to a real life problem from a manufacturing plant and to other six randomly generated problems, the proposed approach was found

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not only to perform significantly better than bond energy, but also to have lower sensitivity to the density of machine-part matrix. This approach has the flexibility to allow the cell designer to either identify the required number of cells in advance, or consider it as a dependent variable. Another aspect of this research, which makes it easily portable into practice, is that it uses algorithms, which are available in many commercial software packages. For example, factor analysis can be performed on most statistical packages including SAS (1985) and SPSS (1999). The proposed approach has been developed to address some deficiencies in the existing cell formation methods. It remains to be seen how this approach can be extended to address other issues highlighted in the literature.

Acknowledgements This research is supported by Research Grant from Natural Sciences and Engineering Research Council (NSERC) of Canada and Faculty of Engineering and Computer Science Research Support Fund, Concordia University. The authors would like to thank Ms T. Ehtiati at McGill University, Canada, for her inputs in this paper and the anonymous referees for their useful comments on an earlier version of this paper.

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