A comparison of existing grouping efficiency measures and a new weighted grouping efficiency...
Grouping efficiency is an evaluative measure of the machine-part groups in cellular manufacturing systems. This paper presents briefly both a critical review and a comparative study of different measures of grouping efficiency. Special emphasis is given to the evaluation of the goodness of clustering
1. Introduction
Due to its various advantages over other manufacturing techniques, the Cellular Manufacturing System (CMS) is currently a popular manufacturing technique to improve productivity in batch production systems. In a machine shop, a large variety of parts with different functional relationships are produced which can usually be partitioned into a homogenous cluster together with the group of machines. Thus, machine-part matrix clustering is the central issue in cellular manufacturing. In Srinivasan and Narendran [1] parts are partitioned into part-families and the machines are partitioned into cells depending on their similarity so that every part of each part-family undergoes its subsequent operations in the assigned machine group. The machine-part incidence matrix [[a.sub.ij]] is composed of binary digits where element [a.sub.ij] = 1 indicates that part j visits machine i and [a.sub.ij] = 0 means that part j does not visit machine i. Block-diagonalization is considered as the best approach to form part-families and machine-cells. In an ideal solution, all the ones will remain in the diagonal blocks of the matrix and all the zeros in the off-diagonal blocks, but an ideal case is rarely obtainable in practice.
1.1. Different clustering methods
Over the last few decades, different methods have been proposed for the block-diagonalization of a machine-part incidence matrix to achieve better cell formation [2-4]. Some of these methods are discussed below.
In array-dependent methods for block-diagonalization of the machine-part incidence matrix, rows and columns are alternately modified after ranking the rows and columns alternately and thus is continued until all the machine-parts cluster into blocks along the diagonal of the solved matrix. The Rank Order Clustering (ROC) algorithm [5], the ROC2 algorithm [6], the Modified ROC (MODROC) algorithm [7], the Direct Clustering Algorithm (DCA) [8], and the Cluster Identification Algorithm (CIA) [9] are a few examples of array-dependent methods. In rank order, in the case of ranking rows or columns, higher weights are assigned to the entries in the left upper corner of the matrix, but a problem, may occur in this case, concerning the ranking order of rows and columns, in which parts that should have been clustered with machines of higher weights may instead get exception operation positions. Also bottleneck machines may get the highest weights. So this algorithm is highly sensitive to form block-diagonalization of th e machine-part incidence matrix. Srinivasan and Narendran [1] note that Kusiak and Chow [9] have proposed a Cluster Identification Algorithm (CIA) which is more effective than the rank order algorithm for well-structured matrices. Some early surveys on this area are these of Greene and Sadowski [10], Miltenburg and Zhang [11], Shargal et al. [12], Sarker [13], Singh and Rajamani [14].
The clustering analysis method has been classified as being either hierarchical or non-hierarchical by Srinivasan and Narendran [1]. In the hierarchical clustering algorithm of McAuley [15], each machine or part is initially considered as being a single cluster and these single clusters are progressively grouped to form a cell. Chandrasekharan and Rajagopalan [16] have developed a non-hierarchical clustering algorithm that employed a graph-theoretic formulation to determine the number of groups in a machine-part incidence matrix. An ideal seed method called ZODIAC was used to cluster machines and parts. ZODIAC is the most improved ideal seed algorithm and was the first to mention a measure for the goodness of solutions [17].
Shargal et al. [12] used array-independent methods for the initial matrix in order to compute a distance or similarity matrix for rows and columns and thus they computed a similarity-coefficient between the machine and parts; subsequently groups are formed from this matrix. This method is also known as the similarity-coefficient based method. In the similarity-coefficient based method, similarity-coefficients are determined between the machines and parts and then they are grouped using either clustering analysis or other methods. In some cases this method uses hierarchical clustering of the similarity matrix to obtain part-families and machine-cells. In this method, machine and part groups are found separately and are mutually assigned to each other [11,13, 18-21].
In the graph theoretical method, a spanning tree with nodes and arcs is developed where the machines and parts is represented as nodes and the processing of parts represented by the arcs connecting the nodes. Srinivasan and Narendran [1] have noted that these models aim at obtaining disconnected sub-graphs from the machine-part graph to identify part-families and machine-cells. They also mentioned that Rajagopalan and Batra [22] have proposed a graph partitioning approach which used cliques of machine graphs as a means of grouping machines. Other works performed in this area are those of Kandiller [23], and Chen and Cheng [24].
1.2. The problem
Many algorithms have been developed to create block-diagonalization of machine-part incidence matrices and these include production flow analysis [25,26] and the GRAFICS search algorithm [1]. Most of the algorithms mentioned are suitable for the block-diagonalization of well-structured matrices, but in the case of ill-structured incidence matrices it is not clear of whether or not they give the best solution. Again a wide range of heuristics have been developed to solve the problem which may or may not give optimum results; but whatever the method used, one should choose a method that is based on some criteria that indicates the goodness of the solution.
None of the algorithms developed for Cellular Manufacturing Systems (CMSs) give a generalized formula for the solution of different practical problems even though they are the best for the solution of incidence matrices. So, in cellular manufacturing, the evaluation of the block-diagonal machine-part matrix is important, and the evaluation includes a grouping measure, performance measures, and simulated models [27]. The best way of evaluating the solutions obtained from different algorithms is to choose an absolute quantitative scale. If such an absolute scale is generated, then it becomes easier to compare the individual results objectively and in the case of forming the block-diagonalization of machine-part incidence matrices, such quantitative measure can be considered as the objective function which is to be maximized.
The objective of this research is to classify and generalize different measures for the determination of the goodness of clustering solutions and to compare them with respect to each other considering their advantages and limitations, and their applicability in different manufacturing systems.
2. Different measures of grouping efficiency
In this section, an attempt is made to present most of the notable measures of grouping or clustering efficiency considering that whenever a Group Technology (GT) problem is solved as a binary-digit block-diagonal matrix, an absolute scale is the best way to evaluate the goodness of a solution. Some definitions that will be frequently mentioned are listed here.
2.1. Definitions
Block: a submatrix of a solved machine-part incidence matrix where the positions of elements in the rows and columns are altered to form a cluster of part-families and machine-cells.
Void: a void at an element (i,j) inside a block means a part j is not assigned to machine i after the formation of the block-diagonalization of the machine-part incidence matrix.
Exceptional element: the elements in the off-diagonal blocks are the exceptional elements. The greater the number of exceptional elements, the greater the number of exceptional operations to be performed, and in that case, either the duplication of machines in cells is required or more inter-cell travel must be scheduled.
Perfect block-diagonal form: this is a block-diagonal form of a machine-part incidence matrix with a 100% efficiency where all the diagonal blocks are full of 'ones' with no voids, and no 'ones' fall outside the diagonal blocks, indicating that all the machines in a cell process all parts in that part family and no bottleneck machine exists. In practice, such a situation rarely exists. This perfect theoretical situation entails no inter-cell (block) flow of materials since bottleneck machines are not encountered and all parts in a family are to be scheduled to all machines in that cell.
Zero efficiency: this is the opposite to 100% efficiency where in a block-diagonalized incidence matrix, diagonal blocks are occupied by voids leaving the off-diagonal blocks full of exceptional elements.
Void density: this is the total number of voids with respect to the total number of elements in the diagonal blocks of a solved incidence matrix. A lower value of the void density is expected as it affects the overall efficiency of a solution.
Intensity of exceptions: this is the total number of exceptional elements with respect to the total number of elements or space in the diagonal blocks of a solved matrix. Like void density, a lower value of the intensity of exceptions is expected to obtain a better solution.
2.2. Grouping efficiency and grouping efficacy
Chandrasekharan and Rajagopalan [7,16], published the first quantitative measure of goodness of solution termed grouping efficiency which is the weighted average of two functions and defined as:
[eta] = q[[eta].sub.1] + (1 - q)[[eta].sub.2], (1)
where [[eta].sub.1] is the ratio of the number of ones in the diagonal blocks to the total number of elements (both zeros and ones) in the diagonal blocks, [[eta].sub.2] is the ratio of number of zeros in the off-diagonal blocks to the total number of elements (both zeros and ones) in the off-diagonal blocks, and q is a weighting factor (0 [less than or equal to] q [less than or equal to] 1).
This is a measure of the goodness of a solution which has established that ZODIAC [17] can elicit a better block-diagonal form from a given binary matrix than any other algorithm. This efficiency function has a non-negativity property and 0 [less than or equal to] [eta] [less than or equal to] 1, but it has a weaker discriminating power. The weighted average function in this measure gives enough freedom to choose on the relative importance between the number of voids in the diagonal blocks and the number of inter-cell movements (bottleneck machines). If q = 0.5, both get the same importance. Kumar and Chandrasekharan [28] found from an analysis of 100 randomly chosen data sets that, even though an algorithm produces the best possible block diagonal form in all the cases, the range of grouping efficiency varies practically from about 75 to 100%. So it is difficult to assign a value of q to correctly weigh voids and exceptional elements in a block diagonal matrix. The main problem is encountered with large ma trices where a lower value of q should be assigned instead of one with an equal impact on the number of voids and exceptional elements. Otherwise the second term (1 - q) will obtain a diminishing form and be less effective, and thus the overall measure will be less effective.
Although Chandrasekharan and Rajagopalan [17] contend that an equal weight leads to equal weights for voids and exceptional elements, a further study reveals that this claim is not true. They observed that when large matrices are block diagonalized for more than two blocks, the quantity q[[eta].sub.1] has a much smaller denominator compared to the quantity (1 - q)[[eta].sub.2] while the numerator is approximately of the same order. As the matrix dimension increases, this disparity between these quantities widens, and for large and/or sparse matrices (many jobs having fewer operations rarely on many machines) the quantity (1 - q)[[eta].sub.2] becomes less and less effective. In fact, if q and (1 - q) are of the same order, the effect of inter-cell movements (exceptional elements) is hardly reflected in the efficiency values for large and sparse matrices. Therefore, Kumar and Chandrasekharan [28] suggested that it is necessary to choose a very low value of q to have equal weights on the voids and exceptional e lements. It is also desirable to have a common denominator for both quantities in order to choose a good weight factor q.
Kumar and Chandrasekharan [28] proposed another measure named grouping efficacy which has overcome the weaker discriminating power of the grouping efficiency measure by putting equal weights on the number of voids and the number of exceptional elements. This measure is defined as:
[tau] = (1 - [psi])/(1 + [varphi]) = e - [e.sub.0]/e + [e.sub.v], (2)
where [psi] is the ratio of the number of exceptional elements to the total number of operations, [varphi] is the ratio of the number of voids in the diagonal blocks to the total number of operations, e is the total number of ones (i.e., the total number of operations) in the matrix, [e.sub.0] is the number of exceptional elements, and [e.sub.v] is the number of voids.
Although it overcomes the previously noted limitations an analysis of some matrices' results highlighted the following problems: (a) the built-in-mechanism assigns weights to voids and exceptional elements which are unknown to the user; (b) grouping efficacy is more sensitive to changes in the number of voids than that of the exceptional elements; and (c) in the case of grouping efficacy, the user has no freedom to assign the weights for voids and exceptional elements.
The block-diagonal space has a great impact on the efficiency measure and so to obtain an accurate and meaningful result, grouping efficacy is expressed as:
[[tau].sub.1] = (B - [e.sub.0])/(B + [e.sub.v]). (3)
Here, B is the sparsity of the solved matrix or in other words, it is the number of elements within the diagonal blocks of the solved matrix. Again to produce an impact on the number of voids and number of exceptional elements, a weighted factor q has to be introduced and thus the modified expression for the grouping efficacy is:
[[tau].sub.2] = (1 - q[e.sub.v] + (1 - q)[e.sub.0]/B)/(1 + q[e.sub.v] + (1 - q)[e.sub.0]/B). (4)
The modified grouping efficacy is comparatively better than the grouping efficacy as it simultaneously affects the block-diagonal space, number of voids and exceptional elements, and the weighting factor. The problem is, though this measure ensures an almost equal weighting to the number of voids and exceptional elements, it can take a negative value when the diagonal blocks are full of voids and the off-diagonal blocks have more ones.
Ng [29] noticed that
[partial][tau]/[partial][psi]/[partial][tau]/[partial][varphi] = (1 + [varphi])/(1 - [psi]) [greater than or equal to] 1,
which means that the penalty imposed on an exceptional element is always greater than the penalty imposed on a void inside the diagonal blocks. He proposed a new measure, [gamma], the weighted grouping efficacy for a partitioned incidence matrix:
[gamma] q(e - [e.sub.0])/q(e + [e.sub.v] - [e.sub.0]) + (1 - q)[e.sub.0]. (5)
The purpose of introducing the weight q is to enable the ratio:
\([partial][gamma]/[partial][psi])\/\([partial][gamma]/[partial][varphi])\,
to correctly represent how [gamma] changes with respect to the number of exceptional elements and the number of voids in a practical situation by suitably choosing the value of the weighing factor.
2.3. Index measures
In addition to the grouping efficiency, grouping efficacy measures and their variant modified or weighted grouping efficacy measures a few other measures have been recently introduced. These measures have either incorporated some additional factors or have been defined with different perspectives on the problems. Some of these measures are: (1) grouping index; (ii) grouping capability index; and (iii) quality index.
(i) The Grouping Index (GI): Nair and Narendran [30] introduced a measure to overcome the previously mentioned drawbacks in the modified or weighted grouping efficacy. They added a correction factor to [[tau].sub.2], which is introduced along with the space B and the weighting factor q by:
[[tau].sub.3] = (1 - q[e.sub.v] + (1 - q)([e.sub.0] - A)/B)/(1 + q[e.sub.v] + (1 - q)([e.sub.0] - A)/B), (6)
where A is a correction factor, A = 0 if [e.sub.0] [less than or equal to] B, and A = [e.sub.0] - B if [e.sub.0] [greater than] B. The grouping index is a measure which is to some extent successful in overcoming the limitations of two measures, grouping efficiency, [eta] and grouping efficacy, [tau] It is quite reasonable to refer to the block-diagonal space to define efficiency which ensures equal weights to the voids and exceptional elements, and a good discriminating power under all circumstances. So the choice of q would be linked with the size and sparsity of the matrix.
(ii) Hsu [31] proposed the Grouping Capability Index (GCI) as a measure of efficiency which is contrary to the grouping efficiency and grouping efficacy measures that include zero entries from the matrix calculation. Seifoddini and Djassemi [32] report this measure as:
GCI = 1 - ([e.sub.0]/e), (7)
where [e.sub.0] and e have the same meanings, as previously.
(iii) Quality Index: Seifoddini and Djassemi [27] tried to convert a job shop to a cellular manufacturing system (CMS). To measure the effectiveness of independency of machine-component group, they proposed a new measure called quality index (QJ) and developed the threshold value of this measure beyond which a CMS will outperform the corresponding job shop system. Their quality index, QI, is defined as
QI = 1 - ICW/PW
where the total plant workload (P W) is calculated as PW = [[[sigma].sup.M].sub.m=1] [[[sigma].sup.P].sub.p=1] [V.sub.p][T.sub.mp][X.sub.mp] and the total intercellular workload (ICW) is defined as ICW = [[[sigma].sup.C].sub.c=1] [[[sigma].sup.M].sub.m=1] [Y.sub.mc] [[[sigma].sup.P].sub.p=1] [V.sub.p][T.sub.mp][X.sub.mp] (1 - [Z.sub.pc]) in which [V.sub.p] = volume of part p (p = l,2,...,P), and = processing time of part p on machine m (m = 1,2,...,M), [X.sub.mp] = 1 if part p has an operation on machine m or 0 otherwise, [Y.sub.mc] = 1 if machine m is assigned to cell c (c = 1,2,...,C) or 0 otherwise, and [Z.sub.pc] = 1 if part p is assigned to cell c or 0 otherwise.
Seifoddini and Djassemi [27] pointed that it has the potential for significantly improving the evaluation of block diagonal forms if production volume and processing times are incorporated in the calculation of grouping measure QI. The production volume and the processing time are the two important factors that affect the performance of a CMS. As a result, QI is more closely related to the performance of a CMS that all other grouping measures that solely use the binary data in the incidence matrix.
2.4. Primary and secondary measures
Miltenburg and Zhang [11] proposed the grouping measure as their primary measure and, clustering measure and bond-energy measure as secondary measures. Grouping measure compares the performance of a number of algorithms and it is also a direct measure of the effectiveness of an algorithm to obtain a final grouped matrix.
(i) The Grouping Measure (GM) gives higher values if both the number of voids and exceptional elements is small and it can be defined as:
[[eta].sub.g] = [[eta].sub.u] - [[eta].sub.m], (8)
where [[eta].sub.u] is the ratio of the number of ones to the number of total elements in the diagonal blocks and [[eta].sub.m] is the ratio of exceptional elements to the total number of ones in the matrix. Miltenburg and Zhang [11] observed that, when a final partmachine matrix is displayed in standard block-diagonal form, then often the resulting machine-part cells will be better arranged when the non-zero elements are clustered more tightly around the diagonal.
(ii) The Clustering measure is an efficiency measure that examines how close the cluster is around the diagonal of the solved matrix. This measure is defined for m (m 1,...,M) machines and p (p= 1,...,P) parts as:
[[eta].sub.c] = [[sigma].sub.All] [a.sub.mp] = 1 [square root][[[delta].sup.2].sub.h]([a.sub.mp]) + [[[delta].sup.2].sub.v] ([a.sub.mp])/[[[sigma].sup.M].sub.m=1] [[[sigma].sup.P].sub.p=1] [a.sub.mp], (9)
where:
[[delta].sub.h] = m - P(M - 1)/(P - 1) - (P - M)/(P - 1),
and
[[delta].sub.v] = p - m(P - 1)/(M - 1)+(P - M)/(M - 1),
[[delta].sub.h] is the horizontal distance between a non-zero element [a.sub.mp] and the diagonal, [[delta].sub.v] is the vertical distance between a non-zero element [a.sub.mp] and the diagonal, and [a.sub.mp] is an element of the machine-part incidence matrix.
(iii) The Bond-energy measure is a secondary measure, which is also named as the, bond-energy method by McCormick et al. [33], in which the non-zero elements of the machine-part matrix are located close to each other. For m (m = 1,2,...,M), the row index for a machine in a binary matrix, and p (p = 1,2,...,P), the column index for a part, [a.sub.mp] is a binary entry in row m and column p of the binary matrix [[a.sub.mp]], the effectiveness of the bond energy measure of McCormick et al. [33] is normalized by Miltenburg and Zhang [11] as:
[[eta].sub.BE]
= [[[sigma].sup.M].sub.m=1] [[[sigma].sup.P-1].sub.p=1] [a.sub.mp][a.sub.m,p+1] + [[[sigma].sup.M-1].sub.m=1] [[[sigma].sup.P].sub.p=1] [a.sub.mp][a.sub.m+1,p]/[[[sigma].sup.M].sub.m=1] [[[sigma].sup.P].sub.p=1] [a.sub.mp] (10)
which is basically the expression of McCormick et al. [33] for the bond energy of a matrix, divided by the total number of non-zero elements in the matrix. Thus [[eta].sub.BE] is therefore, the average bond energy for a machine-part operation (i.e., a non-zero [a.sub.ij]). Normalizing the measures helps to compare other measures. Miltenburg and Zhang [11] mentioned that large values of [[eta].sub.BE] were an indicator of preference although it was sometimes difficult to interpret a high value of the bond energy. However, as this study evaluates and compares the quality of solutions, it enriches the analysis.
There are many other measures of the efficiency measure of a clustering solution but we continue to give our attention to those measures that determine the goodness of a clustering solution, i.e., those measures which evaluate the efficiency of clustering solutions.
2.5. Clustering efficiency measures
Shargal et al. [12] have proposed other clustering efficiency measures that give two values for any incidence matrix since the block-diagonal formation is simply a combination of a row and column permutation. Four different measures of this group, Neighbor Clustering Efficiency (NCE), Ones Clustering Efficiency (OCE), Ones-Zero Clustering Efficiency (OZCE), and the Jaccard Clustering Efficiency (JCE) are listed in Table 1.
The Neighbor Clustering Efficiency (NCE) is an extension of the bond-energy method which gives two values for any matrix because of row and column permutation. The Ones Clustering Efficiency (OCE) is a measure that belongs to the line-properties family which checks to verify that in each row or column all ones group together with no zeros between. The Ones Zero Clustering Efficiency (OZCE) is another measure which is an extension of the ones clustering efficiency in which weight is given to the distribution of zeros in each row or column. In this case importance must be given on the way the zeros are being distributed. The Jaccard Clustering Efficiency (JCE) is a measure that is based on the similarity between rows and columns and the distance between them. The algorithm ZODIAC has placed an emphasis on the similarity and distance between rows and columns in order to obtain block-diagonal forms of matrices.
Shargal et al. [12] observed from specific studies that all these measures give similar profiles in that they give similar values of efficiency with respect to the row-cost and CPU time. These clustering efficiency measures have been experimentally tested on 14 different data sets ranging from eight to 43 machines and 10 to 38 parts from both the literature and a manufacturing company. They used an ideal algorithm for the machine-part matrix to maximize each of these clustering efficiency measures without taking a large amount of CPU time. They also indicate that the choice of algorithm alone does not guarantee high values for any clustering efficiency measures. Most efficiency measures given in the literature are computation intensive and are thus time consuming, but these are specialized in their realms in measuring efficiency of different clustering solutions.
2.6. A doubly weighted grouping efficiency measure
Considering the factors involved in structural construction and predictive evaluation of a block-diagonal form of an incidence matrix, a doubly weighted grouping efficiency measure, [[eta].sub.Q], has been suggested by Sarker [34,35] to capture those important characteristics as follows:
[[eta].sub.Q] = ([q.sub.1][e.sub.1] + (1 - [q.sub.1])[e.sub.v]/[e.sub.1] + [e.sub.v]) ([q.sub.2][e.sub.1] + (1 - [q.sub.2])[e.sub.0]/[e.sub.1] + [e.sub.0]), (11)
where
[q.sub.1], [q.sub.2] = a weighting factor which varies from 0 to 1, i.e., 0 [less than or equal to] ([q.sub.1], [q.sub.2]) [less than or equal to] 1;
[e.sub.1] = the number of ones in the diagonal blocks of the solved machine-part incidence matrix;
[e.sub.v] = the number of voids in the diagonal blocks in the machine-part incidence matrix; and
[e.sub.0] = the number of exceptional elements in machine-part incidence matrix.
In this measure, two distinct weighted terms of efficiencies, the weighted intra-block efficiency ([[eta].sub.b]) of the diagonal blocks (first term), and the weighted relative efficiency ([[eta].sub.0]) of the off-diagonal blocks (second term) of any solved machine-part incidence matrix are combined together. The weighted intra-block efficiency of diagonal blocks is the ratio of the weighted average of diagonal elements to the total number of diagonal elements, and the weighted relative efficiency of off-diagonal blocks is the ratio of weighted exceptional elements to the total number of operations. The ultimate purpose of this measure is to eliminate the effect of the number of voids and the number of exceptional elements in a goodness of clustering solution, and it is obtained by assigning weights to these factors individually. In this measure, the number of exceptional elements has more effect on the clustering solution than does the number of voids, and that is why, a lower value for the exceptional eleme nts is always expected. The higher the number of exceptional elements, the lower the efficiency, and the same situation arises in the case of a higher number of voids in a lower degree. Thus the block-diagonal efficiency can be increased by giving higher weights to 'ones' in the diagonal blocks and lower weights to the voids and exceptional elements.
This new measure is dependent on the value of the weighting factor and the number of exceptional elements, and its value always varies from zero to one. For equal weights on both terms (i.e., [q.sub.1] = [q.sub.2] = q), the efficiency in (11) converts to
[[eta].sub.Q] = (q[e.sub.1] + (1 - q)[e.sub.v]/[e.sub.1] + [e.sub.v]) (q[e.sub.1] + (1 - q)[e.sub.0]/[e.sub.1] + [e.sub.0]). (12)
For example, in [12] if [e.sub.1] = 16, [e.sub.0] = 2, and [e.sub.v] = 6 the new doubly weighted grouping efficiency yields 0.0303 [less than or equal to] [[eta].sub.Q] [less than or equal to] 0.6464 for any value of q on the range 0 [less than or equal to] q [less than or equal to] 1.
3. Comparative studies of different measures
It is evident from the previous discussion that all the grouping measures evaluate their performances as to their cohesiveness of block-diagonalization of operations in the machine-part incidence matrix though the measures are different quantitatively. Each grouping efficiency measure is based on various assumptions and concepts, and their varied degrees of structural definitions lead to different conclusions. In this section, a comparative study is pursued so as to enable the assessment of the performance of the different measures on the same scale.
3.1. An empirical investigation
Some grouping efficiency measures are applied here to a solved machine-part incidence matrix to show their relative performance. All the necessary data are taken from the solution matrix (Fig. 1), given by Kusiak and Chow [21].
The solution matrix in Fig. 1 has e = 23, [e.sub.0] = 7, [e.sub.v] = 12, and B = 28. Thus the grouping efficiency, [[eta].sub.1] = 16/28 and [[eta].sub.2] = 42/49, and the grouping efficiency [eta] is 72% for q = 0.5. Also, [psi] = 7/23 and [varphi] = 12/23; thus the grouping efficacy [tau] is 46%. In the case of modified or weighted grouping efficacy for the same value of q, the efficiency [[tau].sub.1] and [[tau].sub.2] are 52 and 49% respectively which are a little bit higher than the value of the grouping efficacy since these measures incorporate the value of space B = 28. The Grouping Capability Index (GCI) is 69%. From this solution matrix, the total number of operations is 23, the number of ones in the diagonal blocks is 16, and the number of voids is 12. Thus, the grouping measure [[eta].sub.g] is 27%.
From the computation of different measures, a comparative higher value is obtained by grouping efficiency [eta] and a lower value is obtained by grouping measure [[eta].sub.g]. In all of these measures, the grouping efficacy is comparatively giving a reasonable result as it considers the value of the weighting factor, the number of voids and exceptional elements, and the block-diagonal space.
3.2. Comparative studies of different measures
A detailed analysis of the comparative performance of different measures of grouping efficiency is provided here to help understand which measure gives the best goodness of different clustering solutions. For this purpose, three more matrices B, C and D, shown in Figs. 2-4, are introduced in addition to matrix A.
Table 2 shows a comparison of grouping efficiency, grouping efficacy, weighted grouping efficacy, grouping index, grouping capability index, and grouping measure whose values are calculated using the data from the matrices A, B, C and D. From this table, it is obvious that for the same value of weighting factor q, grouping efficacy, weighted grouping efficacy, grouping index, and grouping measure have nearly similar values of efficiency. In the grouping measure case more impact is given on the usage of parts in the cell, and relatively less impact on inter-cell movement.
Table 3 shows a comparison between the grouping efficiency and the grouping efficacy with respect to four different algorithms: ZODIAC by Chandrasekharan and Rajagopalan [17], GRAFICS by Srinivasan and Narendran [1], the Assignment Allocation Algorithm (AAA) by Adil et al. [36] and the Simulated Annealing Algorithm (SAA) by Adil et al. [36]. In this table, two effective measures: grouping efficiency and grouping efficacy have been used to evaluate the performance of these four algorithms for seven data sets ranging from well structured to the most ill-structured incidence matrices taken from Chandrasekharan and Rajagopalan [37] by the authors. It is observed from the table that AAA and SAA compare favorably with ZODIAC and GRAFICS for both measures for the same matrices. From the table, it can be concluded that for the same set of exceptional elements and voids, AAA and SAA provide a better partition with a higher value of the grouping measures than those for ZODIAC and GRAFICS.
The performance of the doubly weighted grouping efficiency measure with respect to other commonly used measures are also reported in Table 4. It is observed that the efficiency [[eta].sub.q] is robust in the sense that it performs in very much the same manner as the other measures. The main reason for its insensitivity is due to the weights on all parameters such void, cell operation (ones) and inter-cell movements (exceptional elements).
Chandrasekharan and Rajagopalan [37] have identified the relation between the average similarity and the ultimate grouping efficiency. They found that the final grouping efficiency is strongly related to the standard deviation of the data as well as to the average similarity, although the relation with the former appears to be more pronounced in terms of absolute values. With the available information, they came to the conclusions that standard deviation values over 0.3 give excellent results and those below 0.15 very poor groupability, with a smooth variation and a saturation tendency in between. They also noted that the value of the ultimate grouping efficiency, [[eta].sub.u], for rows and columns begins to become steady at a certain value of average similarity, but below 0.13, the ultimate grouping efficiency is always higher than that for columns.
Seifoddini and Djassemi [27] have developed a simulated model for performance evaluation which is the basis for the comparison of five different measures: bond-energy measure, grouping efficiency, grouping efficacy, grouping capability measure, and, quality index. It is used to determine which measure most accurately predicts the performance of the machine-part incidence matrix solution where the average flowtime and in-process inventories are used as variables to show the comparison of the measures graphically. They found that, for a certain value of the mean flow time, the efficiency of all these measures is 100% and then as the value of the mean flow time increases, the efficiency of all these measures begins to fall, but the steepest fall occurs to the Quality Index (QI) measure. The grouping capability index and the grouping efficiency measures give similar values of efficiency and grouping efficacy and the bond- energy measures give similar values for the efficiency. Of these five measures the quality index gives the highest efficiency for a range of mean flow times and then falls beyond any other measure noted here. Seifoddini and Djassemi [27] claimed that the QI is the measure of independence of machine-part groups. Since independent machine-cells are ideal for the formation of CMSs, a high value of QI is expected to lead to a high performance level in the corresponding CMS.
4. A new weighted grouping efficiency
In the previous sections, different grouping efficiency measures in cellular manufacturing systems have been discussed in detail with respect to their structural and formational characteristics, advantages, and limitations. It is clear that the grouping efficiency of machine-part incidence matrices depends on the block-diagonalization form of the instances. Different heuristics have been developed to obtain the block-diagonalization form of instances but they are not the subject matter of this paper, what is important now is to choose the algorithm that would be perfectly suitable for block-diagonalization in order to obtain a better grouping efficiency. Grouping efficiency has been defined in several ways by different researchers. These definitions often ignore important factors which if considered could strengthen their attempts to get better grouping efficiency. These factors are mentioned below:
(i) The number of parts to be produced within the machine-cells, or in other words, the number of ones in the diagonal blocks of the solution matrix should be as high as possible with respect to `ones' in the off-diagonal blocks. It is always desirable to produce such a machine-cell that can provide as many operations of parts as possible because it gives better utilization of machines within the cells and also provides more parts to be produced within the cell reducing inter-cell movement, duplication of machines, set-up time, etc. In other words, it is preferable to have machines with fewer operations outside the diagonal blocks as bottleneck machines if the case arises at all.
(ii) A weighting factor must be considered to show how much weight should be given to each machine within the cells to obtain better efficiency. This purpose is served by varying the weights on the machines.
(iii) The number of voids in the diagonal blocks is important because the grouping efficiency is sensitive to the number of voids; a high value of this number leads to under-utilization of machines and causes less production of parts. So a low number is always expected in the cells in order to obtain better grouping efficiency in a solution matrix.
(iv) The number of exceptional elements in a solution is an influential factor for grouping efficiency and a low value is usually desirable as it gives rise to either duplication of machines or inter-cell movement of parts.
4.1. A new weighted grouping measure
Considering all these factors individually, a new weighted grouping efficiency measure is proposed here as
[[eta].sub.q] = q[e.sub.1] + (1 - q)[e.sub.v]/[e.sub.1] + [e.sub.v] - (1 - q)[e.sub.0]/[e.sub.1] + [e.sub.0], (13)
where
q = a weighting factor for exceptional and block-diagonal elements;
[e.sub.1] = number of ones in the diagonal blocks of a solved machine-part incidence matrix;
[e.sub.v] = number of voids in the diagonal blocks in the solved matrix; and
[e.sub.0] = number of exceptional (off-diagonal) elements in the solved matrix.
In this new measure, there are two efficiency terms where the first one is the efficiency of the diagonal blocks and the second one is the inefficiency of the off-diagonal blocks of a solution for a machine-part incidence matrix. The efficiency of diagonal blocks is the ratio of the weighted average of diagonal elements, q[e.sub.1] + (1 - q)[e.sub.v], to the total number of diagonal elements [e.sub.1] + [e.sub.v], and the inefficiency due to off-diagonal (exceptional) elements is the ratio of weighted exceptional (off-diagonal) elements (1 - q)[e.sub.0] to the total number of operations, [e.sub.l] + [e.sub.0]. The ultimate purpose of this measure is to eliminate the effect of the number of voids in the diagonal blocks and the number of exceptional elements in the off-diagonal blocks for the goodness in the clustering solution. This objective is achieved by assigning equal weights to these factors individually. Thus the efficiency of solution matrix is under-weighted by the inefficiency (1 - q)[e.sub.0]/([e.sub.1] + [e.sub.0]) in order to discourage exceptional elements.
In this efficiency measure, the number of exceptional elements is a more effective factor than the number of voids, and hence, a lower value of exceptional element [e.sub.0] is always expected to enhance the grouping efficiency. The higher the number of exceptional elements, the lower the efficiency, and the same situations arises in the case of a higher number of voids but to a lower degree. So the group efficiency in a solution can be increased by giving more weight to 'ones' in the diagonal blocks and less weight to the number of voids [e.sub.v] and exceptional elements [e.sub.0].
This new measure is dependent on the value of the weighting factor and the number of exceptional elements. For example, in Fig. 4, [e.sub.1] = 16, [e.sub.0] = 2, and [e.sub.v] = 6. Thus, for q = 0.5, the new weighted grouping efficiency in Equation (13) yields [[eta].sub.q] = 0.4444.
4.2. Characteristics of the new weighted grouping efficiency measure, [[eta].sub.q]
Since the second part of the weighted grouping efficiency [[eta].sub.q] is a negative function of [e.sub.0], the following characteristics are observed:
If
[[e.sup.1].sub.0] [less than or equal to] [[e.sup.2].sub.0] [less than or equal to] [[e.sup.3].sub.0] [less than or equal to] ... [less than or equal to] [[e.sup.E-1].sub.0] [less than or equal to] [[e.sup.E].sub.0],
then
[[[eta].sup.1].sub.q] [greater than or equal to] [[[eta].sup.2].sub.q] [greater than or equal to] [[[eta].sup.3].sub.q] [greater than or equal to] ... [greater than or equal to] [[[eta].sup.E-1].sub.q] [greater than or equal to] [[[eta].sup.E].sub.Q],
where [[e.sup.k].sub.0] (k = 1, 2,...,E) is an arbitrary value of the off-diagonal elements for an instance k, and [[e.sup.E].sub.0] is the maximum number of exceptional elements of a particular machine-part incidence matrix. The effect of the weights and the role of exceptional elements in this efficiency measure are discussed below.
4.2.1. Effect of weights
For a perfect block-diagonalized machine-part incidence matrix (i.e., [e.sub.v] = 0 and [e.sub.0] = 0), the weighted grouping efficiency, [[eta].sub.q] = q(0 [less than or equal to] q [less than or equal to] 1). So, the new measure largely depends on the value of the weighting factor:
(i) If q = 0, [[eta].sub.q] = [e.sub.v]/([e.sub.1] + [e.sub.v]) - [e.sub.0]/([e.sub.1] + [e.sub.0]). For perfect block-diagonalization, [e.sub.v] = 0 and [[eta].sub.q] = 1/[e.sub.1] - [e.sub.0]/([e.sub.1] + [e.sub.0]) [greater than] 0. If the solution matrix has a structure such that at the worst case, diagonalization, is not formed, i.e., [e.sub.1] = 0 and [[eta].sub.q] = 0. When the bottleneck problem does not exist, then [e.sub.0] = 0 and [[eta].sub.q] = [e.sub.v]/([e.sub.1] + [e.sub.v]).
(ii) If q = 1, [[eta].sub.q] = [e.sub.1]/([e.sub.1] + [e.sub.v]) in which case a bottleneck problem does not exist. For perfect block-diagonalization, [e.sub.v] = 0 and [[eta].sub.q] = 1 and at the worst case where block diagonals are not formed, [e.sub.1] = 0 and [[eta].sub.q] = 0.
Thus, the overall range of this efficiency measure is 0 [less than or equal to] [[eta].sub.q] [less than or equal to] 1. Theorem 1 provides information on the effective bounds of the weight q.
Theorem 1. If [[[eta].sup.P].sub.q] and [[[eta].sup.N].sub.q] are the weighted grouping efficiencies in a perfect and imperfect block-diagonalized solution, respectively, then [[[eta].sup.P].sub.q] [greater than or equal to] [[[eta].sup.N].sub.q] if q [greater than or equal to] 1/2 or vice-versa.
Proof. For a perfect block-diagonalized solution, [e.sub.v] = 0 and [e.sub.0] = 0 while in an imperfect block-diagonalized solution, [e.sub.v] [neq] 0. Assume [e.sub.v]/[e.sub.1] = c. Now
[[eta].sub.q] = q[e.sub.1] + (1 - q)[e.sub.v]/[e.sub.1] + [e.sub.v] - (1 - q)[e.sub.0]/[e.sub.1] + [e.sub.0]
= q + (1 - q)[e.sub.v]/[e.sub.1]/1 + [e.sub.v]/[e.sub.1] - (1 - q)[e.sub.0]/[e.sub.1] + [e.sub.0]
= q + (1 - q)[e.sub.v]/[e.sub.1]/1 + [e.sub.v]/[e.sub.1] (assuming [e.sub.0] = 0 for both cases).
Now, [[[eta].sup.P].sub.q] = q (since [e.sub.v] = 0) and
[[[eta].sup.N].sub.q] = q + (1 - q)[e.sub.v]/[e.sub.1]/1 + [e.sub.v]/[e.sub.1] = q + (1 - q)c/1 + c.
Hence, the condition [[[eta].sup.P].sub.q] [greater than or equal to] [[[eta].sup.N].sub.q] leads to that conclusion that
q [greater than or equal to] q + (1 - q)c/1 + c
which further simplifies to q [greater than or equal to] 1/2 for c [greater than or equal to] 0 (which is always true).
Corollary 1. For [e.sub.0] = 0, [[[eta].sup.P].sub.q] [right arrow] 1/2 and [[[eta].sup.N].sub.q] [right arrow] [[[eta].sup.P].sub.q] as [e.sub.v] [right arrow] [e.sub.1].
Proof.
[[[eta].sup.N].sub.q] = [Lim.sub.[e.sub.v][right arrow][e.sub.1]] (q + (1 - q)[e.sub.v]/[e.sub.1]/1 + [e.sub.v]/[e.sub.1]) = 1/2.
At the limiting condition [e.sub.v] = [e.sub.1]. Hence, by simple argument, [[[eta].sup.P].sub.q] = [e.sub.1]/([e.sub.1] + [e.sub.v]) = 1/2.
To illustrate this result and the role of q in the weighted grouping efficiency for perfect and imperfect block-diagonalized solutions, an example is chosen. Assume two solutions, a perfect block-diagonal matrix P (with [e.sub.v] = 0 and [e.sub.0] = 0) and an imperfect block-diagonal matrix N (with [e.sub.0] = 0 and [e.sub.v] = c[e.sub.1]). So for q = 0.6 and c = 0.1, [[[eta].sup.P].sub.q] = q = 0.6, and [[[eta].sup.N].sub.q] = (q + (1 - q)c)/(1 + c) = 0.5818 which is an acceptable result with respect to perfect and imperfect block-diagonalization because solution P is supposed to more efficient than solution N for [e.sub.v] [neq] 0. On the contrary, for q = 0.4 and c = 0.1, [[[eta].sup.P].sub.q] = q = 0.4, and [[[eta].sup.N].sub.q] = (q + (1 - q)c)/(1 + c) = 0.4182 which numerically proves the theoretical result, but in reality, [[[eta].sup.N].sub.q] of an imperfect block-diagonalized solution cannot be greater than [[[eta].sup.P].sub.q] for a perfect block-diagonalized solution. Therefore, the weighting fac tor, q, plays an important role, as indicated in Theorem 1, in evaluating a solution.
4.2.2. Role of exceptional elements
The efficiency of the diagonal blocks, (q[e.sub.1] + (1 - q)[e.sub.v])/([e.sub.1] + [e.sub.v]), can be written as q - (2q - 1) ([e.sub.v]/([e.sub.1] + [e.sub.v])). So, for a particular value of the weighting factor q, the efficiency of the block-diagonalized portion varies with [e.sub.1]/([e.sub.1] + [e.sub.v]) while the inefficiency contributed by the number of exceptional elements, [e.sub.0], varies with [e.sub.0]/ ([e.sub.1] + [e.sub.0]).
4.3. Analysis of the new measure
To test the relative efficiency of the new measure we investigate, three different machine-part incidence matrices. Although equal weights are given to the number of voids and the exceptional elements, the efficiency measure varies with the weight factor and the variation of exceptional elements in the solved matrix. Four different incidence matrices A (Fig. 1), B (Fig. 2), C (Fig. 3) and D (Fig. 4) are used here to show the performance of weighted grouping efficiency [[eta].sub.q] with respect to exceptional elements (Table 5). The incidence matrices A, B and C are chosen from the existing literature and incidence matrix D is generated randomly. In order to capture different perspectives of the problem, matrix A is a bad solution in terms of both good block-diagonalization and exceptional elements (bottleneck parts), matrix B is chosen with a perfect block-diagonal structure, and both B and C have bottleneck machines. Also, matrix C has more workload (relatively more incidence in the block-diagonals) than matrix D.
Table 6 provides insights on the functional behavior of the weighted group efficiency [[eta].sub.q] with respect to [e.sub.0] and at different values of weights at q = 0.3, 0.5 and 0.7 over four different data sets A, B, C, and D. The relative performance of the weighted group efficiency [[eta].sub.q] with respect to other commonly used efficiency measures is reported in Table 7. It may be noted that the grouping efficiency decreases with an increase in the exceptional element [e.sub.0] but increases with the weighting factor q.
5. Relative performance of the measures
Some of the comparison criteria for evaluating grouping efficiency measures that are available in the literature were discussed earlier. It is difficult to compare one efficiency measure with another since their structural characteristics and formational purposes are unique in most cases and different in perspective. A new measure called the weighted group efficiency, [[eta].sub.q] was presented in the previous section. This measure is evaluated here to assess its performance with respect to other measures.
5.1. Comparison with existing models
In this section, an attempt is made to compare the new grouping efficiency with other grouping efficiencies such as grouping efficiency, grouping efficacy, grouping index, grouping capability index, and grouping measure. These measures are dependent on the number of voids, number of exceptional elements, weighting factor, and the sparsity of the incidence matrix. For the purpose of comparison, four different machine-part incidence matrices of (24 x 40) elements in each are adopted from Figs. 1-4 of Nair and Narendran [30]. The values of the efficiencies of the problems defined by these figures (renamed NN-l, NN-2, NN-3 and NN-4, respectively) are reported in Table S in which the weighting factor is chosen as q 0.5. It may be noted that the new weighted grouping efficiency [[eta].sub.q] is a more general measure of grouping capability for different weights. For example, [[eta].sub.q] reduces to half the Hsu [31] Grouping Capability Index (GCI), when q = 0.5. This result is obvious in Table 8. Because each meas ure differs to some extent structurally and objectively, it is difficulty to say that one measure is better than another [12]. Nevertheless, Sarker and Li [38] have shown that grouping efficiency and efficacy measures have some drawbacks in computing the efficiency value on the full range of the measure. Since 0 [less than or equal to] [[eta].sub.q] [less than or equal to] 1, new measure can have different scales of comparison with other measures depending on 0 [less than or equal to] q [less than or equal to] 1; this is a major advantage of this measure with a good discriminating capability. Such characteristics have been discussed in details earlier and its performance at different weights is given later.
5.2. Relative performances
A new approach is introduced here to compare different grouping efficiency measures. Let [[g.sup.k].sub.ij] be the absolute value of the difference in the grouping efficiency calculated from two different grouping efficiency measures i and j over a data set k, that is,
[[g.sup.k].sub.ij] = \[[[eta].sup.k].sub.i] - [[[eta].sup.k].sub.j]\, i,j = 1,2,...,N, (14)
where the weighted grouping efficiency [[eta].sub.q] is redefined as [[[eta].sup.k].sub.i] for measure i(i = 1,2,...,N) over data set k(k = 1,2,...,K). Any [[g.sup.k].sub.ij] is the discriminating amount of value between the corresponding grouping measures i and j among N measures.
Let [G.sub.k] = [[[g.sup.k].sub.ij], = 1,2,...N]. Since [[g.sup.k].sub.ij] = [[g.sup.k].sub.ji], it will be appropriate to redefine [G.sub.k] as
[G.sub.k] = [[[g.sup.k].sub.ij], i,j = 1,2,...N; i [less than or equal to]j], (15)
which will make it an upper triangular matrix. Let G be a (N x N) matrix such that G = [G.sub.1] + [G.sub.2] + ... + [G.sub.k]. Thus, each element of G represents the total difference between the two measures corresponding to the rows and columns over a total of K data sets. The element in the measure of G is a quantity that represents the degree by which other measures are discriminated. The average difference matrix, G is simply given by
G = 1/K [[[sigma].sup.K].sub.k=1] [G.sub.k]. (16)
Since G is a function of the data set, a higher number of machine-part incidence matrices will result in more steady results and thus any conclusion drawn on the grouping efficiency measures will be more reliable. The computation of the G values are written here in algorithmic form.
Algorithm G: Relative performance
Step 0. Choose a set of measures {1, 2,...,N} for evaluation of their performance. Assume an machine-part incidence matrix, [M.sub.k] (k = 1,2,...,K). Set k = 1.
Step 1. Using matrix [M.sub.k], computer [[[eta].sup.k].sub.i] for i = 1,2,...,N.
Step 2. (a) Find [[g.sup.k].sub.ij] = \[[[eta].sup.k].sub.i] - [[[eta].sup.k].sub.j]\, i,j = 1,2,...,N.
(b) Construct the upper triangular matrix, [G.sub.k] = [[[g.sup.k].sub.ij], i,j = 1,2,...,N; i [less than or equal to] j].
(c) If k [less than] K, Set k = k + 1 and repeat Steps 1 and 2.
Step 3. Compute the average difference matrix for K data sets,
G = 1/K [[[sigma].sup.K].sub.k=1] [G.sub.k].
Step 4. Stop.
5.3. An illustrative example and comparison
Some commonly known grouping efficiency measures mentioned previously are compared here to assess their relative average differences with respect to each other. These grouping efficiency measures which are candidates for comparison are arranged sequentially from one to seven in Table 9, and they will be identified by the notation [[[eta].sup.k].sub.i] (i = 1,2,...,7) as indicated in Table 9.
Computations are performed on 20 machine-part incidence matrices to obtain the final average-difference matrix of the absolute differences of different grouping efficiency measures. In orderly form the grouping efficiency measures are shown in Table 9 wherein [e.sub.0],[e.sub.1],[e.sub.v], e, q, B bear the usual meaning and et stands for the total number of elements in the off-diagonal blocks of thesolved machine-part incidence matrix. Twenty (K = 20) different machine-part incidence matrices are used here.
Algorithm G: Finding the relative performance
From the incidence matrix, [M.sub.1] in Fig. 5, [e.sub.1] = 15, [e.sub.v] = 3, [e.sub.0] = 2, e = 17, and the sparsity B = 18. Therefore, the efficiency in percentage and the absolute value elements for this matrix are given below in the following forms:
Step 1. Using matrix [M.sub.1], the respective efficiency [[[eta].sup.1].sub.i] and their absolute differences [[g.sup.1].sub.ij] (i,j = 1,2,...,7) for all seven measures are computed as:
[[[eta].sup.1].sub.1] = 86.11% [[g.sup.1].sub.11] = \[[[eta].sup.1].sub.1] - [[[eta].sup.1].sub.1]\ = \86.11 - 86.11\ = 0,
[[[eta].sup.1].sub.2] = 75.00% [[g.sup.1].sub.12] = \[[[eta].sup.1].sub.1] - [[[eta].sup.1].sub.2]\ = \86.11 - 75.00\ = 11.11,
[[[eta].sup.1].sub.3] = 75.61% [[g.sup.1].sub.13] = \[[[eta].sup.1].sub.1] - [[[eta].sup.1].sub.3]\ = \86.11 - 75.61\ = 10.5,
[[[eta].sup.1].sub.4] = 88.24% [[g.sup.1].sub.14] = \[[[eta].sup.1].sub.1] - [[[eta].sup.1].sub.4]\ = \86.11 - 88.24\ = 2.13,
[[[eta].sup.1].sub.5] = 71.57% [[g.sup.1].sub.15] = \[[[eta].sup.1].sub.1] - [[[eta].sup.1].sub.5]\ = \86.11 - 71.57\ = 14.54,
[[[eta].sup.1].sub.6] = 75.00% [[g.sup.1].sub.16] = \[[[eta].sup.1].sub.1] - [[[eta].sup.1].sub.6]\ = \86.11 - 75.00\ = 11.11,
[[[eta].sup.1].sub.7] = 44.12% [[g.sup.1].sub.17] = \[[[eta].sup.1].sub.1] - [[[eta].sup.1].sub.7]\ = \86.11 - 44.12\ = 41.99.
Similarly, [[g.sup.1].sub.24] = \[[[eta].sup.1].sub.2] - [[[eta].sup.1].sub.4]\ = \75.00 - 88.24\ = 13.24 and so on.
Step 2. Computing in this fashion, for the matrix [M.sub.1] in Fig. 5, the difference matrix is obtained as
[G.sub.1] = [[[g.sup.1].sub.ij], i,j = 1,2,...,7; i [less than or equal to] j]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Since k [less than] K = 20, this process is repeated for all 20 machine-part matrices (see Table Al in the Appendix). This process of computation for 20 matrices is very time consuming and extensive. Thus the summary computations are provided in Table 10 in which the difference matrices [G.sub.k] (k = 2, 3,...,20) are to be computed from the given values of [[[eta].sup.k].sub.i] (i = 1,2,...,7; k = 1,2,...,20). Information on the input data and its references are provided in the Appendix.
Step 3. Upon finding [G.sub.k] (k = 1,2,...,20), the final average-difference matrix G is obtained as:
G = 1/K [[[sigma].sup.K].sub.k=1] [G.sub.k]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Step 4. Stop.
From the results of the final average matrix, some important information can be gathered about these seven grouping efficiencies. It is usually thought that a higher value of grouping efficiency is always better but, from the comparison, it is observed that the grouping efficiency measure [[eta].sub.q] gives the highest percentage of efficiency but this higher value also represents its drawback in that it ignores some important parameters such as the number of exceptional elements ([e.sub.0]), total operations (e), etc. The same behavior is observed for the Grouping Capability Index (GCI) but in the present study, although it gives the lowest value of efficiency it has a higher discriminating power as it considered almost all the parameters individually. Again, it is also observed that the weighted grouping efficiency [[eta].sub.q] is always half the GCI for a value of q = 0.5 although the grouping capability index is independent of weighting factor q.
The importance of the weighting factor q on the weighted grouping efficiency can be easily understood when this formula is applied to a perfect diagonal-block matrix where both the number of exceptional elements [e.sub.0] and the number of voids [e.sub.v] is zero. In this case all the grouping efficiency measures give 100% efficiency except for the weighted grouping efficiency if the value of weighting factor q is less than one which means that 100% for a perfect diagonal matrix can be obtained by weighted grouping efficiency [[eta].sub.q] if a 100% weighting is given on the machines within the cells. From this comparison, it is found that grouping efficacy and weighted grouping efficacy measures always give an equal value of efficiency for q = 0.5 for the same solved incidence matrix. In this analysis, the grouping measure gives the second lowest value of efficiency since the number of exceptional elements have a mere negative effect on this measure than the number of voids in the solved machine-part incide nce matrix.
The final average matrix of absolute values of the difference of seven grouping efficiency measures will vary for the same solved machine-part incidence matrices if the value of the weighting factor changes which is obvious from the information in Table 11. Thus it shows the importance of individually considering the effect of the weighting factor, q in the grouping efficiency measures.
6. Conclusions
It is clear in cellular manufacturing systems that the goodness of solution of a machine-part incidence matrix mainly depends on the perfection in producing a standard block-diagonal matrix. Many other factors contribute to the goodness such as: production volume, CPU time, machining requirements of parts, processing time, etc. which are taken into consideration almost in all the measures mentioned here. It is notable that, an important factor, cost, is not introduced effectively in any of the mentioned efficiency measures. Here the computational results of the measures of the effectiveness are reported. A comparative study shows their relative performance and goodness for computing efficiency. Because of their limitations, they have been restricted to use in the generalized form for any practical problem for the determination of goodness of clustering solution.
From the analysis of the new weighted grouping efficiency measure it is obvious that the new measure has a higher discriminating capability and it is sensitive to the sparsity of the incidence matrix. The main problem with this grouping efficiency is that it is highly dependent on the weighting factor and it is really difficult to assign the value of this factor. It gives higher emphasis on the number of exceptional elements and thus considers inter-cell travel for parts to different machines. It also gives a comparatively reasonable value of efficiency of solved machine-part incidence matrices than those of other grouping efficiencies mentioned earlier.
Acknowledgement
The authors are thankful to the anonymous referees for their constructive comments and suggestions. This research was supported by National Science Foundation grant DMII: 96-22306.
Biographies
Bhaba R. Sarker is currently a Professor at the Department of Industrial and Manufacturing Systems Engineering, Louisiana State University, Baton Rouge, LA 70803. He obtained a B.A. (Mathematics) from Dhaka University, a B.S.M.E. from BUET (Dhaka), an M.Tech (IE&OR) from Indian Institute of Technology (Kharagpur), M.S. (Eng. Admin) and I.Engr degrees from Syracuse University and a Ph.D. in Industrial Engineering from Texas A&M University, College Station, TX. He has taught at several other schools including The University of Texas at Austin and Texas A&M University at College Station. He also worked with the World Bank and US Army Corps of Engineers besides his other appointments abroad. Bhaba Sarker was the recipient of Best Dissertation Awards from DSI (1990), lIE (1991), and POMS (1990). He has published more than 60 papers in refereed journals and 50 papers in conference proceedings, and in addition, presented more than 50 papers in national and international conferences. Bhaba Sarker is currently serving on the Editorial Boards of IIE Transactions, International Journal of Production Economics, International Journal of Production Research, and Production and Operations Management. He also served as President of the IIE (Institute of Industrial Engineers) Chapter at Baton Rouge, LA. Bhaba Sarker is a member of DSI, lIE, IEEE, INFORMS, POMS, SME and New York Academy of Science. He is currently working in the area of production and manufacturing systems, and supply chain management.
Muslema Khan obtained a B.S. degree in Mechanical Engineering from Bangladesh University of Engineering and Technology, Dhaka in 1996 and an M.S. in Management Information Systems from the University of Texas at Dallas, Richardson, TX in 1999. Previously she studied at the Department of Industrial and Manufacturing Systems Engineering, Louisiana State University at Baton Rouge where she performed this research work. Ms. Khan also presented a part of this research at the INFORMS (Institute For Operations Research and Management Science) National Meeting at Dallas, TX on October 26- 29, 1997. Her current research interests are in manufacturing and management information systems.
Contributed by the Manufacturing Systems Modeling Department
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(27.) Seifoddini, H. and Djassemi, M. (1996) The threshold value of a quality index formation of cellular manufacturing systems. International Journal of Production Research, 34(12), 3401-3416.
(28.) Kumar, C.S. and Chandrasekharan, M.P. (1990) Grouping efficacy: a quantitative criterion for goodness of block diagonal forms of binary matrices in group technology. International Journal of Production Research, 28(2), 233-243.
(29.) Ng, S.M. (1993) Worst-case analysis of an algorithm for cellular manufacturing. European Journal of Operational Research, 69(2), 384-398.
(30.) Nair, G.J.K. and Narendran, T.T. (1996) Grouping index: a new quantitative criterion for goodness of block diagonal forms in group technology. International Journal of Production Research, 34(10), 2767-2782.
(31.) Hsu, C.P. (1990) Similarity coefficient approaches to machine component cell formation in cellular manufacturing: a comparative study. Ph.D. Dissertation, Department of industrial and Systems Engineering, University of Wisconsin, Milwaukee, WI.
(32.) Seifoddini, H. and Djassemi, M. (1994) Analysis of efficiency measures for block diagonal machine-component charts. Computers and Industrial Engineering, 27(1-4), 237-240.
(33.) McCormick, W.T., Schweitzer, P.J. and White, T.W. (1972) Problem decomposition and data reorganization by clustering technique. Operations Research, 20(5), 993-1009.
(34.) Sarker, B.R. (1997) A doubly weighted grouping efficiency measure for cellular manufacturing systems. Working Paper BRS-97-5, Department of Industrial and Manufacturing Systems Engineering, Louisiana State University, Baton Rouge, LA 70803-6409.
(35.) Sarker, B.R. (2000) Measures of grouping efficiency in cellular manufacturing systems. European Journal of Operational Research, (in press).
(36.) Adil, G.K., Rajamani, D. and Strong, D. (1997) Assignment allocation and simulated annealing algorithms for cell formation. IIE Transactions, 29(1), 53-67.
(37.) Chandrasekharan, M.P. and Rajagopalan, R. (1989) GROUPABILITY: an analysis of the properties of binary data matrices for group technology. International Journal of Production Research, 27(6), 1035-1052.
(38.) Sarker, B.R. and Li, Z. (1998) Measuring matrix-based cell formation with alternative routings. Journal of the Operational Research Society, 49(9), 953-965.
Clustering efficiency measures
Measure Row
NCE [[[sigma].sup.max rows].sub.i=1] M[i,j]
{M[i - 1,j] + M[i + 1,j]}
OCE [[[sigma].sup.max rows].sub.i=1]
[L.sub.i]/[C.sub.i][N.sub.i]
OZCE [[[sigma].sup.max rows].sub.i=1]
[(1 - S)[L.sub.i]/[C.sub.i][N.sub.i] +
S(n - [N.sub.i] - [Z.sub.i])/
(n - [N.sub.i])]
JCE [[[sigma].sup.max rows-1].sub.i-1]
[[[sigma].sup.max rows].sub.j=j+1]
JC[i,j]/D[i,j]
Measure Column
NCE [[[sigma].sup.max columns].sub.j=1] M[i,j]
{M[i,j + 1] + M[i,j - 1]}
OCE [[[sigma].sup.max columns].sub.j=1]
[L.sub.j]/[C.sub.j][N.sub.j]
OZCE [[[sigma].sup.max columns].sub.j=1]
[(1 - S)[L.sub.j]/[C.sub.j][N.sub.j] +
S(m - [N.sub.j] - [Z.sub.j])/
(m - [N.sub.j])]
JCE [[[sigma].sup.max columns-1].sub.i=1]
[[[sigma].sup.max columns].sub.j=j+1]
JC[i,j]/D[i,j]
[C.sub.i] ([C.sub.j]) = number of groups of consecutive ones in row i (column j); D[i,j] = distance between rows (or columns) i or j; JC[i,j] = Jaccard similarity coefficient between rows (columns) i or j; [L.sub.i] ([L.sub.j]) = largest number of consecutive ones in row i (column j); m = number of columns in the matrix; M[i,j] = element in row i and column j in matrix M; n = number of rows in the matrix; [N.sub.i] ([N.sub.j]) = total number of ones in row i (column j); S = scale factor, 0 [less than or equal to] S [less than or equal to] 1; and [Z.sub.i] ([Z.sub.j]) = number of zeros in between ones in row i (column j).
Grouping efficiency in percent (q = 0.5)
Problem Grouping Grouping Modified
efficiency, [eta] efficacy, [tau] grouping
efficacy, [[tau].sub.1]
A 71.43 45.71 52.50
B 82.50 65.00 70.21
C 92.32 81.48 83.10
D 83.98 72.73 69.23
Problem Weighted Grouping Grouping
grouping index, [[tau].sub.3] capability
efficacy, [[tau].sub.2] index, GCI
A 49.33 49.33 69.56
B 65.00 70.21 100.00
C 82.76 83.10 92.31
D 66.67 69.23 88.89
Problem Grouping
measure, [[eta].sub.g]
A 26.71
B 65.00
C 81.19
D 61.62
Comparison of grouping efficiency and
efficacy (adopted from Adil et al. [36])
Problem AAA SAA Grouping efficiency, [eta]
[e.sub.0] [e.sub.v] [e.sub.0] [e.sub.v] ZODIAC
1 0 0 0 0 1.00
2 23 17 23 11 0.92
3 56 17 54 14 0.77
4 70 7 63 15 0.69
Problem Grouping efficacy, [tau]
GRAFICS AAA SAA ZODIAC GRAFICS AAA SAA
1 1.00 1.00 1.00 1.00 1.00 1.00 1.00
2 0.92 0.92 0.93 0.38 0.74 0.73 0.73
3 0.79 0.88 0.90 0.20 0.43 0.51 0.53
4 0.79 0.91 0.88 0.18 0.42 0.44 0.46
Comparative study of some weighted
grouping measures in percent (q = 0.85)
Problem [*] [e.sub.v] [e.sub.0] [e.sub.1] e [eta] [tau] [[tau].sub.3]
NN-1 7 19 124 143 96.2 82.7 87.41
NN-2 19 7 112 119 92.3 81.2 76.79
NN-3 20 30 111 141 90.6 68.9 71.80
NN-4 30 20 101 121 87.3 66.9 64.26
Problem [*] [[eta].sub.g] GCI [[eta].sub.Q]
NN-1 81.4 86.7 61.51
NN-2 79.6 94.1 60.05
NN-3 63.4 78.7 52.09
NN-4 60.6 83.5 50.64
(*.)NN = Nair and Narendran
(The problem numbers corresponds to their figure numbers)
Performance of weighted grouping
efficiency, [[eta].sub.q] (q = 0.8)
[e.sub.0] A (Fig. 1) B (Fig. 2) C (Fig. 3)
[e.sub.1] = 16 and [e.sub.1] = 13 and [e.sub.1] = 24 and
[e.sub.v] = 12 [e.sub.v] = 7 [e.sub.v] = 3
0 -- 59.00 --
2 -- 55.92 71.79
4 -- 52.85 70.26
6 -- 49.77 68.72
8 45.62 46.69 67.18
10 44.59 43.62 65.64
12 43.71 40.54 64.10
14 42.95 37.46 62.56
16 42.27 34.38 61.03
18 41.70 31.31 59.49
20 41.17 28.23 57.95
22 40.71 25.15 56.41
24 40.28 -- 54.87
[e.sub.0] D (Fig. 4)
[e.sub.1] = 16 and
[e.sub.v] = 6
0 --
2 61.41
4 59.19
6 56.97
8 54.75
10 52.53
12 50.30
14 48.10
16 45.86
18 43.64
20 41.41
22 39.19
24 36.97
Performance of weighted
group efficiency [[eta].sub.q]
with respect to [e.sub.0] and q
[e.sub.0] A (Fig. 1) B (Fig. 2)
[e.sub.1] = 16 [e.sub.1] = 13
and [e.sub.v] = 12 and [e.sub.v] = 7
q q
0.3 0.5 0.7 0.3 0.5 0.7
3 -- -- -- 27.85 38.46 49.08
5 -- -- -- 17.08 30.77 44.46
7 25.84 34.78 43.73 6.31 23.08 39.85
9 21.94 32.00 44.42 4.46 15.38 35.23
[e.sub.0] C (Fig. 3) D (Fig. 4)
[e.sub.1] = 24 [e.sub.1] = 16
and [e.sub.v] = 3 and [e.sub.v] = 6
q q
0.3 0.5 0.7 0.3 0.5 0.7
3 26.37 44.23 62.09 29.24 41.67 54.09
5 20.98 40.38 59.79 21.46 36.11 50.76
7 15.59 36.54 57.48 13.69 30.56 47.42
9 10.21 32.69 55.17 5.91 25.00 44.09
Comparison of new efficiency measure,
[[eta].sub.q], with other measures (q = 0.5)
Problem Grouping Grouping Modified
efficiency, [eta] efficacy, [tau] grouping
efficacy, [[tau].sub.1]
A 71.43 45.71 52.50
B 82.50 65.00 70.21
C 92.32 81.48 83.10
D 83.98 72.73 69.23
Problem Weighted Grouping Grouping
grouping index, [[tau].sub.3] measure, [[eta].sub.g]
efficacy, [[tau].sub.2]
A 65.00 70.21 26.71
B 65.00 70.21 65.00
C 82.76 83.10 81.19
D 66.67 69.23 61.62
Problem Weighted
grouping
efficiency, [[eta].sub.q]
A 34.78
B 50.00
C 46.15
D 44.44
Comparative of weighted grouping
efficiency in percent (q = 0.5) [*]
Figure [e.sub.v] [e.sub.0] [e.sub.1] e [eta] [tau] GI GCI [[eta].sub.g]
NN-1 7 19 124 143 96.2 82.7 81.9 86.7 81.4
NN-2 19 7 112 119 92.3 81.2 81.9 94.1 79.6
NN-3 20 30 111 141 90.6 68.9 67.9 78.7 63.4
NN-4 30 20 101 121 87.3 66.9 67.9 83.5 60.6
Figure [[eta].sub.q]
NN-1 43.4
NN-2 47.1
NN-3 39.4
NN-4 41.7
(*.)NN = Nair and Narendran
Commonly known measures
Efficiency i Name of the measures
1 Grouping effciency, [eta]
2 Grouping efficacy, [tau]
3 Grouping index, [[tau].sub.3]
4 Grouping Capability Index, (GCI)
5 Grouping measure, [[eta].sub.g]
6 Weighted grouping efficacy, [gamma]
7 Weighted grouping efficiency,
[[eta].sub.q]
Efficiency i Measure [[[eta].sup.k].sub.i]
1 q[e.sup.1]/([e.sup.1] + [e.sub.v])
+ [e.sub.t] - [e.sub.0]/[e.sub.t]
2 (e - [e.sub.0])/(e + [e.sub.v])
3 B - q[e.sub.v] - (1 - q)([e.sub.0] - A)/
B + q[e.sub.v] + (1 - q)([e.sub.0] - A),
A = {0, [e.sub.0][less than or equal to]B
[e.sub.0] - B, [e.sub.0] [greater than]B
4 1 - [e.sub.0]/e
5 [e.sub.1]/([e.sub.1] + [e.sub.v]) - [e.sub.0]/e
6 q(e - [e.sub.0]) q(e + [e.sub.v] - [e.sub.0])
+ (1 - q) [e.sub.0]
7 q[e.sub.1] + (1 - q) [e.sub.v]/
[e.sub.1] + [e.sub.v] - (1 - q) [e.sub.0]/
[e.sub.1] + [e.sub.0]
Efficiency i Reference
1 Chandrasekharan and
Rajagopalan [7, 16]
2 Kumar and Chandrasekharan [28]
3 Nair and Narendran [30]
4 Hsu [31]
5 Miltenburg and Zhang [11]
6 Ng [29]
7 Proposed measure
Values of efficiency measure for 20 problems (for q = 0.5)
Problem k [*] Data from 20 matrices
[e.sub.1] [e.sub.v] [e.sub.0] e B
1 15 3 2 17 18
2 10 4 1 11 14
3 20 4 3 23 24
4 9 1 0 9 10
5 16 12 6 22 28
6 24 3 2 26 27
7 9 6 1 10 15
8 52 0 9 61 52
9 44 6 5 49 50
10 14 1 12 26 15
11 12 4 2 15 16
12 14 4 7 21 18
13 9 1 0 9 10
14 20 4 2 22 24
15 10 3 3 13 13
16 11 2 1 12 13
17 19 6 0 19 25
18 18 0 0 18 18
19 23 2 3 26 25
20 16 4 0 16 20
Problem k [*] [[[eta].sup.k].sub.1] [[[eta].sup.k].sub.2]
1 86.11 75.00
2 83.93 66.67
3 85.42 74.07
4 95.00 90.00
5 72.45 47.06
6 92.56 82.76
7 76.67 56.25
8 95.58 85.25
9 91.50 80.00
10 84.17 51.85
11 76.79 63.16
12 75.43 56.00
13 95.00 90.00
14 87.50 76.92
15 81.64 62.50
16 89.37 78.57
17 88.00 76.00
18 100.00 100.00
19 93.12 82.14
20 90.00 80.00
Problem k [*] [[[eta].sup.k].sub.3] [[[eta].sup.k].sub.4]
1 75.61 88.24
2 69.70 90.91
3 74.55 86.96
4 90.48 100.00
5 51.35 72.72
6 83.05 92.31
7 62.16 90.00
8 84.01 85.25
9 80.18 89.80
10 39.53 53.85
11 64.10 80.00
12 53.19 66.67
13 90.48 100.00
14 77.78 90.91
15 62.50 76.92
16 79.31 91.67
17 78.57 100.00
18 100.00 100.00
19 81.82 88.46
20 81.82 100.00
Problem k [*] [[[eta].sup.k].sub.5] [[[eta].sup.k].sub.6]
1 71.57 75.00
2 62.34 66.67
3 70.29 74.07
4 90.00 90.00
5 29.87 47.06
6 81.19 82.76
7 50.00 56.25
8 85.25 85.25
9 77.80 80.00
10 47.18 51.85
11 55.00 63.16
12 44.44 56.00
13 90.00 90.00
14 74.24 76.92
15 53.85 62.50
16 76.28 78.57
17 76.00 76.00
18 100.00 100.00
19 80.46 82.14
20 80.00 80.00
Problem k [*] [[[eta].sup.k].sub.7] [G.sub.k]
1 44.12 [G.sub.1]
2 45.46 [G.sub.2]
3 43.48 [G.sub.3]
4 50.00 [G.sub.4]
5 36.36 [G.sub.5]
6 46.76 [G.sub.6]
7 45.00 [G.sub.7]
8 42.62 [G.sub.8]
9 44.90 [G.sub.9]
10 26.92 [G.sub.10]
11 40.00 [G.sub.11]
12 33.33 [G.sub.12]
13 50.00 [G.sub.13]
14 45.45 [G.sub.14]
15 38.46 [G.sub.15]
16 45.83 [G.sub.16]
17 50.00 [G.sub.17]
18 50.00 [G.sub.18]
19 44.23 [G.sub.19]
20 50.00 [G.sub.20]
(*.)Table A1 in Appendix
Performance of grouping efficiency measure
(in percent) [*]
Number q [eta] [tau] GI GCI [[eta].sub.g] [[tau].sub.3] [[eta].sub.g]
1 0.9 83.88 75.00 72.25 88.24 71.57 82.32 75.49
2 0.7 85.00 75.00 73.91 88.24 71.57 79.55 59.80
3 0.5 86.11 75.00 75.61 88.24 71.57 75.00 44.12
4 0.4 86.67 75.00 76.47 88.24 71.57 71.43 36.27
5 0.2 87.78 75.00 78.22 88.24 71.57 57.69 20.59
(*.) Based on [M.sub.1] ([e.sub.1] = 15, [e.sub.0] = 2,
[e.sub.v] = 3, e = 17, B = 18)
Appendix
References of 20 problems and their dimensions
Problem, k Reference of Data Matrix, [M.sub.k]
1 Adil et al. [36] [M.sub.1]
2 Generated randomly [M.sub.2]
3 Generated randomly [M.sub.3]
4 Kusiak and Chow [21] [M.sub.4]
5 Kusiak and Chow [21] [M.sub.5]
6 Miltenburg and Zhang [11] [M.sub.6]
7 Generated randomly [M.sub.7]
8 Shargal et al. [12] [M.sub.8]
9 Shargal et al. [12] [M.sub.9]
10 Generated randomly [M.sub.10]
11 Generated randomly [M.sub.11]
12 Kusiak and Chow [9] [M.sub.12]
13 Kusiak and Chow [9] [M.sub.13]
14 Kusiak and Chow [9] [M.sub.14]
15 Generated randomly [M.sub.15]
16 Generated randomly [M.sub.16]
17 Seifoddini and Djassemi [27] [M.sub.17]
18 Adil et al. [36] [M.sub.18]
19 Generated randomly [M.sub.19]
20 Kusiak and Chow [21] [M.sub.20]
Problem, k Dimension, m x n [*] Triangular Matrix, [G.sub.k]
1 6 x 6 [G.sub.1]
2 6 x 7 [G.sub.2]
3 6 x 8 [G.sub.3]
4 4 x 5 [G.sub.4]
5 7 x 11 [G.sub.5]
6 10 x 8 [G.sub.6]
7 6 x 5 [G.sub.7]
8 8 x 20 [G.sub.8]
9 10 x 15 [G.sub.9]
10 7 x 9 [G.sub.10]
11 5 x 6 [G.sub.11]
12 4 x 11 [G.sub.12]
13 4 x 5 [G.sub.13]
14 6 x 8 [G.sub.14]
15 5 x 7 [G.sub.15]
16 5 x 6 [G.sub.16]
17 7 x 11 [G.sub.17]
18 6 x 6 [G.sub.18]
19 7 x 11 [G.sub.19]
20 7 x 8 [G.sub.20]
(*.)m machines and n parts.
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