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Measuring cellular manufacturing performance
Planning, Design, and Analysis of Cellular Manufacturing Systems A.K. Kamrani, H.R. Parsaei and D.H. Liles (Editors) 9 1995 Elsevier Science B.V. All rights reserved.
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Measuring cellular manufacturing performance D.F. Rogers" and S.M. S h a f e r b
aDepartment of Quantitative Analysis and Operations Management, College of Business Administration, University of Cincinnati, 531 Carl H. Lindner Hall, Cincinnati, Ohio 45221-0130, USA" bDepartment of Management, College of Business, Auburn University, 415 West Magnolia, Auburn, Alabama 36849-5241, USA
A large amount of research has focused on developing new cell formation procedures. Only recently, however, has research focused on comparing and evaluating alternative cell formation procedures. Perhaps the most significant hurdle associated with conducting studies for comparing and evaluating cell formation procedures is the absence of meaningful performance measures. A performance measure is considered meaningful when it is related to one or more of the design objectives associated with cellular manufacturing. In this paper several design objectives associated with cellular manufacturing are identified. Then, based upon these design objectives, appropriate performance measures are discussed and compared. Also included is a review and critique of performance measures used in previous studies for comparing cell formation procedures.
1. I n t r o d u c t i o n
It is widely acknowledged that the environment confronting manufacturers is becoming increasingly competitive as markets become more globalized. As a result, producers of goods are under constant and intense pressure to quickly and continuously improve their operations. Areas often targeted for improvement are productivity, quality, and responsiveness. In recent decades, Cellular Manufacturing (CM) has emerged as a promising approach for improving operations in batch and job shop environments, particularly in situations for which the divisions of the production processes are distinct. For CM, parts with similar processing requirements are identified and grouped together
*Dr. Rogers acknowledges Dr. J.E. Aronson and the Department of Management, College of Business Administration, University of Georgia for the support provided during his sabbatical leave from the University of Cincinnati.
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to form part families. The equipment requirements for the part families are simultaneously or subsequently determined and usually located together in close proximity. Thus, equipment is located together based upon the processing requirements of part families in CM and thereby differs from traditional layouts where similarly functioning equipment is often located together. Groups of mostly dissimilar machine types that are utilized for CM and located together are referred to as machine cells. The use of CM allows for an increase of local decision making in the factory and this also often results in quality and productivity improvements. There is a wealth of empirical evidence supporting the potential superiority of a cellular layout versus the more traditional functional layout. Schonberger [1] stated that moving the machines into cells was a basic step in the transformation of the General Electric Company's Louisville, Kentucky dishwasher plant into a world class manufacturing showcase. He noted that large numbers of manufacturers are utilizing CM in an effort to become world class competitors. There is also evidence to indicate that CM is not always the appropriate approach to take for certain scenarios. Results from the simulation models of Flynn and Jacobs [2 and 3] indicated that a well-organized job shop can outperform a CM configuration with respect to several criteria such as work-in-process inventory levels and average flow times. In these simulations the machine-component matrices were quite dense and forming distinct separations of the production processes to accommodate CM was not possible. Morris and Tersine [4] further revealed that an ideal environment for a cellular layout is characterized with a high ratio of setup to process time, stable demand, unidirectional work flow within a cell, and a considerable level of material movement times between process departments. In response to the considerable interest in CM, there has been much current research devoted to the development of cell formation procedures, i.e., techniques that facilitate the identification of part families and machine cells. WemmerlSv and Hyer [5] summarized research issues in CM that is also relevant today. Chu [6] provided a summary of general cluster analysis techniques and algorithms for forming manufacturing cells. Rogers, et al. [7] cast clustering into a larger framework for aggregation and disaggregation techniques. Shafer [8] identified over 80 contributions found in the literature associated with the development and comparison of cell formation procedures. Unfortunately, the vast majority of these contributions focused on developing new cell formation procedures. Very little research to date has focused on comparing these techniques. As a consequence, there is little or no guidance available concerning the appropriateness and/or usefulness of cell formation techniques. It may be argued that only profit, revenue, or cost-based measures, i.e., valuebased measures that reflect real improvement to investors, should actually be employed for judging the desirability of CM. This is ultimately true for almost any significant change in a firm, especially changes in production functions. Askin and Chiu [9] recognized this and incorporated an objective function of cost minimization for their linear integer programming problem to form machine cells and part families. The cost components of this objective function were 1) machine overhead - the cost of placing a particular number of a machine type in a cell, 2) group overhead - a fixed cost for using a particular grouping, 3) family tooling costs, and 4) intergroup
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material handling costs. However, it is often extremely difficult to accurately assess actual profits, revenues, or costs to a manufacturing system prior to actual implementation of the particular CM configuration selected. How, for example, may one reasonably assess the costs of intergroup material handling costs prior to knowing the cell configurations and relative locations? Choobineh [10] also formulated a linear integer programming problem with an objective function of minimizing the total average annual cost of producing all part families in all cells and the cost of providing the appropriate number of machines for each cell. These costs may often be exceedingly difficult to determine. In reality it may take months or even several years before the actual impacts of converting to a CM system may be assessed. It is also often extremely difficult to ascertain the costs of lost flexibility due to employing CM rather than a job shop, to assess the sociological implications of the factory employees as well as their management (see Huber and Hyer [11]), and to appropriately measure the quality increase in the end product that is often accompanied by employing CM. Because value-based methods are not easily implementable for determining part family and/or machine cell configurations we must typically judge the quality of a CM solution with measures that are not based upon value but nonetheless are quite good surrogates for value-based objectives. These surrogate measurements for CM evaluation should ideally coincide with the basic objectives of CM which were originally developed to coincide with value-based objectives. Therefore, the emphasis in this article will be to use measures for judging various CM configurations that appropriately match the objectives by which these cells are to be formed. An objective in this paper is to offer a framework for comparing alternate cell formation procedures. In the next section a review of studies for comparing various cell formation procedures is provided. In the following section is a closer examination of the performance measures employed in these studies and other performance measures that may prove to be interesting for future studies. Before useful comparison studies can be conducted, meaningful performance measures must be developed and a comparison of current performance measures is presented. Subsequently, a framework for comparing alternative cell formation solutions is offered. Finally, we provide a summary and discuss avenues for future research.
2. Literature Review of Cell Formation Technique Comparisons In this section studies that have focused on comparing alternate cell formation procedures will be reviewed. Many of these studies involve simple row and column manipulation and/or hierarchical and nonhierarchical statistical clustering of the rows and columns of the machine-component matrix X, where x~j = 1, if machine type i, i=l,...,M, is required for production of component j, j=I,...,N, and x~j=0, otherwise. In Table 1 is a summary of the cell formation solution quality performance measures that are included in these studies and other various gauges of performance that
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Table 1 Summary of cell formation solution quality performance measures. Performance Measure
Reference
Simple Matching Generalized Matching Product Moment Correlation Coefficient
Anderberg[22] Klastorin[34] Anderberg[22]
M osier[ 12]
Weighted Intercellular Transfers
Mosier[13]
M osier[ 12]
Used In
Avg. & Maximum WIP Shafer & Avg. & Max. Flow Time Meredith[14] Part Travel Distances Extra-Cellular Operations Longest Average Queue Total Bond Energy McCormick, Schweitzer & White[24] Clustering Measure Miltenburg & Zhang[27]
Chu & Tsai[23] Shafer & Rogers[31] Miltenburg & Zhang[27]
Proportion Exceptional Elements Machine Utilization
Chu & Tsai[23] Chandrasehkaran &Rajagopalan[26]
Chu & Tsai[23] Shafer & Rogers[31]
Grouping Efficiency
Chandrasehkaran &Rajagopalan[26]
Chu & Tsai[23]
Global Efficiency Group Efficiency Group Technology Efficiency
Harhalakis, Nagi & Proth[36]
Grouping Efficacy
Kumar & Chandrasehkaran [32] Miltenburg & Zhang[27]
Grouping Measure
Shafer & Meredith[14]
Shafer & Rogers[31] Miltenburg & Zhang[27]
Comments
9Solution quality defined in terms of how similar it is to original matrix. 9Not directly related to goals of CM. =Does not consider part sequencing or volumes. 9Relatively easy to calculate. 9Surrogate for intercellular transfers. 9Does not consider part sequencing but does consider part volumes. 9Relatively easy to calculate. 9Directly related to many CM goals. 9Generally not computationally efficient (requires simulation). 9Does consider part sequencing and part volumes, ,Not directly related to goals of CM. ,Does not consider part sequencing or part volumes. ,Relatively easy to calculate. ,Insensitive to family & cell definition. ,Surrogate measures related to CM goals. ,Does not consider part sequencing or volumes. ,Relatively easy to calculate. 9Must select arbitrary weight. 9Surrogate measure related to CM goals. 9Weak discriminating power. 9Does not consider part sequencing or volumes. 9Relatively easy to calculate. 9Surrogate measure related to CM goals. 9Considers part sequencing but not volumes. 9Relatively easy to calculate. 9Better discriminating power than grouping efficiency. 9No need to select an arbitrary weight. 9Does not consider part sequencing or volumes. 9Surrogate measure related to CM goals. 9Relatively easy to calculate.
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have been developed. Mosier [12] used a mixture model experimental approach and compared seven similarity coefficients and four statistically-based hierarchical clustering procedures. The degree of cluster definition, i.e., the ratio of the non-zero density outside the clusters to the non-zero density inside the clusters, and the density of the non-zero entries within the clusters of the machine-component matrix were additional factors included in the study. The performance measures used were a simple and a generalized matching measure, a product moment correlation coefficient measure, and an intercellular transfer measure (Mosier [13]). Statistics were obtained for the results of the solved problems and the Jaccard similarity coefficient showed promise for these particular problems in spite of being one of the simplest similarity coefficients to calculate. None of the hierarchical clustering procedures appeared to uniformly dominate any of the others in the study. Shafer and Meredith [14] utilized computer simulation to compare cell formation procedures from all three categories of the taxonomy for cell formation procedures proposed by Ballakur and Steudel [15]: 1) part grouping, 2) machine grouping, and 3) simultaneous machine-part grouping. The clustering algorithms that were employed were the rank order clustering algorithm (King [16]), the direct clustering algorithm (Chan and Milner [17] and Wemmerl6v [18]), the cluster identification algorithm (Kusiak and Chow [19 and 20]), a technique for considering the sequence of operations developed by Vakharia and Wemmerl5v [21], and both single and average linkage hierarchical clustering routines (Anderberg [22]). These algorithms were applied to data collected from three companies. Computer simulation models were then developed based upon the solutions generated with the cell formation procedures. The performance measures included in this study were average and maximum work-in-process levels, average and maximum flow times, part travel distances, extra-cellular operations, and the longest average queue. It was found that using clustering algorithms to first form part families and then assigning machines to cells to accommodate the families provided more flexibility and worked best for these scenarios. Chu and Tsai [23] compared the rank order clustering algorithm, the direct clustering algorithm, and the bond energy analysis algorithm (McCormick, Schweitzer, and White [24] and Lenstra [25]) using the following performance measures: 1) total bond energy, 2) proportion exceptional elements, i.e., the proportion of the part types that must be transferred to other cells, 3) machine utilization, and 4) grouping efficiency (Chandrasekharan and Rajagopalan [26]). Eleven data sets from the literature were used for their study and the bond energy algorithm performed best for these problems. However, most of these data sets were extremely small example problems and not enough of the remaining problems were large enough to viably conclude the dominance of any one particular method. Miltenburg and Zhang [27] have performed one of the more extensive studies to compare CM algorithms upon 0/1 machine-part matrices. They tested nine different algorithmic approaches that employed different combinations of the rank order clustering algorithm, the modified rank order clustering algorithm (Chandrasekharan and Rajagopalan [28]), single and average linkage clustering, the bond energy algorithm, and an ideal seed non-hierarchical clustering algorithm
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(Chandrasekharan and Rajagopalan [26]). The objectives of their study are to form cells such that there is a high usage of machines by parts in each cell (a high cell density) and to form cells that do not tend to allow exceptional elements, i.e., parts that leave the cell and travel to another cell. The primary performance measure that they used to evaluate the clustering algorithms attempted to combine the influence of both of these objectives and may actually be attributed to Stanfel [29]. Two secondary performance measures employed were a measure of closeness of nonzero elements to the diagonal and average bond energy. Results were obtained for 544 solved problems and no significant differences were detected among most of the algorithms. The ideal seed non-hierarchical clustering algorithm showed the most promise for these particular randomly generated problems and performance measures. However, these approaches impose a significant computational burden for larger problems as compared to the other algorithms. Shafer and Rogers [30 and 31] investigated 16 measures of similarity or distance and four hierarchical clustering procedures that may be used for CM problems. The same performance measures as used by Chu and Tsai [23] were employed except that the grouping efficiency criterion was replaced by the improved grouping efficacy criterion developed by Kumar and Chandrasekharan [32]. A total of 704 solutions were derived and it was found that little difference typically existed among the results from different similarity coefficients. Single linkage clustering was, in general, found to be statistically inferior to average linkage, complete linkage, and Ward's clustering methods for the bond energy, machine utilization, and grouping efficacy criteria. However, single linkage clustering was statistically superior when considering the proportion of exceptional elements.
3. Comparison of Performance Measures
A fundamental criterion for evaluating performance measures is how closely related the performance measures are to the basic objectives associated with adopting CM. Shafer and Rogers [33] identified the following four key design objectives associated with CM: 1) setup time reduction, 2) to produce parts cell complete, i.e., minimize intercellular transfers, 3) to minimize investment in new equipment, and 4) to maintain acceptable machine utilization levels. Other important objectives associated with the adoption of CM include improving product quality, reducing inventory levels, and shortening lead times. In the remainder of this section the performance measures introduced in the previous section and summarized in Table 1 will be further discussed and compared. Mosier [12] employed four performance measures, a simple matching measure, a generalized matching measure, a product moment correlation coefficient measure, and an intercellular transfer measure. The first three performance measures (Anderberg [22] and Klastorin [34]) are for gauging how well the clustered solution matches the original machine-component matrix and thus do not allow for the possibility that the clustered solution obtained may be in some sense better than the configuration for the original matrix. The original configuration may not even be a good choice. Although these three measures are relatively easy to calculate, they
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are not related to the design goals associated with adopting CM, and part volumes or operation sequences are not considered with them. The last measure employed, the intercellular transfer measure, is the volume weighted number of parts that require processing in more than one cell. With this measure one may consider the possibility that the solutions generated by the cell formation procedures were better than the original randomly generated machine-component matrices. This measure does indirectly addresses the CM design objective of minimizing intercellular transfers. However, because operation sequences are not considered, it is only a surrogate measure for the actual number of intercellular transfers. All seven performance measures used by Shafer and Meredith [14], average and maximum work-in-process levels, average and maximum flow times, part travel distances, extra-cellular operations, and the longest average queue, require the development of computer simulation models. Deriving these measures is computationally more complex but they are all directly related to CM design objectives and are measures used for considering both part volumes and operation sequences. Four performance measures were employed by Chu and Tsai [23] and Shafer and Rogers [31]. The first performance measure, Total Bond Energy (TBE), was proposed by McCormick, Schweitzer, and White [24] as the performance measure to assist an algorithmic approach for identifying a permutation of the rows and columns such that the sum of the products of adjacent elements is maximized. This algorithm reveals a block diagonal matrix if one exists but its performance is unpredictable otherwise. TBE is defined as: M N TBE = I; I~ xij(xij+l .i- xij.1 -i- Xi+l J 4" Xi.l,j)12 i=1 j=l
(1)
where Xi0----X0j=XM+l,j=Xi,N+l=0. However, TBE is not directly related to the design goals associated with CM. The other three measures, proportion of exceptional elements, machine utilization, and grouping efficiency (grouping efficacy in Sharer and Rogers [31]), are surrogate measures for the CM design objectives of minimizing intercellular transfers and maintaining acceptable machine utilization levels. Proportion Exceptional Elements (PE) is the ratio of the number of nonzero elements not in the block diagonals (machine-component cells) in the final configuration of the clustered machine-cell matrix to the total number of nonzero elements in X given by: N M N M P E = I; I~ eij I T~ ~ Xi j i=1 j=l i=1 j=l
(2)
where e~j is set to one if element x U in the rearranged X is an exceptional element, zero otherwise. Machine Utilization (MU), also known as the density of the machinepart cells, is the ratio of the number of nonzero elements in the machine-cell clusters
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to the total number of elements in these cells. Neither of these measure are for evaluating the actual achievement of the CM objectives largely because operation sequences, part volumes, and operation times are not considered. Chandrasekharan and Rajagopalan [26] developed the Grouping Efficiency (GE) measure, a weighted average defined as: GE = q MU + (l-q) ODV
(3)
where the Off-Diagonal Voids (ODV) is the ratio of the number of zero elements in the off-diagonal blocks of X over the total number of elements in the off-diagonal blocks and q is a selected weighting on [0,1] designed to reflect the relative importance the analyst desires to place on MU versus ODV. GE will also always take on values on [0,1] provided that there is more than one machine-cell cluster, GE=I for a perfect block diagonal form and GE=0 for the diametric case. Freedom to select the appropriate weighting q allows the analyst to decide the relative importance between intercellular moves and voids in the cells. Ng [35] scrutinized GE as a performance measure. He noted that nonzero elements inside a diagonal block correspond to work processed in a machine cell with smaller material handling and setup costs. Alternatively, exceptional elements correspond to work that must be processed outside of the cells and thus has larger material handling and setup costs. For a typical example with q=.5 it was shown that the rate of change of GE with respect to a nonzero entry inside a diagonal block was three times that of an exceptional element which is contrary to their relative costs. By adjusting the weighting to q=.2 the same rate of change would decrease from three to 3/4, a more reasonable value but still maybe too large in practice to appropriately reflect the relative costs. It was also revealed for a large and sparse X and using q=.l that this rate of change can still be much larger than 1.0. Furthermore, it was shown that any matrix X with two completely decomposable diagonal blocks will have a grouping efficiency of at least (1-q)+q(MU). If q is to be small to overcome the above-mentioned problems then this minimum value for GE will always be quite large. Smaller values of q should probably be used if calculating GE but the value of its usage is questionable. Kumar and Chandrasekharan [32] also noted that the use of GE as a measure of performance has several shortcomings. They revealed that even quite poor solutions could yield a GE of .75 and thus the efficiency function GE has weak discriminating power. Also, the requirement that a weight be selected, possibly quite arbitrarily, to determine the relative importance of MU and ODV may be a difficulty. They proposed the Grouping eFficacy (GF) measure to overcome these shortcomings while retaining the desirable properties associated with the use of GE: GF = ( 1-PE)I(1 .DV)
(4)
where the Diagonal Voids (DV) is the number of zero elements in the block diagonals in the final configuration of the clustered machine-cell matrix to the total number of nonzero elements in X. GF will also take on values on [0,1].
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Ng [35] noticed with an example that the value of GF may be larger for a problem with five clusters than an apparently better four-cluster solution for the same problem. He proposed a modification to remedy this problem with GF called the Weighted Grouping eFficacy (WGF) which was derived by placing a weight of q on each entry inside the diagonal blocks and a weight of 1-q on exceptional elements given by: W G F = r(1 -PE)I[PE+r(1 + D V - P E ) ]
(s)
where r=ql(1-q) and q is a weighting just as for GE. If q=.5 then r=l, WGF=GF, and thus WGF is a generalization of GF. WGF will likewise have values on [0,1]. Miltenburg and Zhang [27] utilized one primary and two secondary performance measures. Their primary performance measure, called the Grouping Measure (GM), is also an attempt to overcome some similar supposed weaknesses with using the GE measure and is given by: G M = M U - DV
(6)
The value of GM is bounded by negative and positive unity, i.e., -I_
(7)
where dh(Xij)is the horizontal distance between a nonzero element and the diagonal and dv(X~j)is the vertical distance between a nonzero element and the diagonal. CLM is an overall indicator of the closeness of the nonzero elements to the diagonal, regardless of whether the nonzero elements are in a diagonal block or not. It is the average squared Euclidean distance of a nonzero element from the diagonal. CLM is not directly related to any of the CM design objectives but it may be a good gauge of cohesiveness for some circumstances just as TBE may be. Harhalakis, Nagi, and Proth [36] proposed the following three performance measures which have not yet been used in a study to compare cell formation procedures: 1) global efficiency - the ratio of the number of operations performed within the primary cells to the total number of operations, 2) group efficiencycalculated by subtracting from one the ratio of the actual number of external cells visited to the maximum number of cells that could be visited and 3) group technology efficiency- calculated by subtracting from one the ratio of actual number of intercellular moves required to the maximum number of intercellular moves possible.
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These measures do include the consideration of operation sequences but do not regard any data for part volumes. Compared to performing computer simulations, they are relatively easy to calculate. Numerous other authors have implemented performance measures implicitly through the objective functions of the mathematical models that they have developed for which there are many possibilities. A representative group of this work includes Stanfel [37, 38, and 39] who formed a linear integer programming problem with the objective of minimizing the total number of intercell transitions made by part groups. Kusiak [40] and Gunasingh and Lashkari [41] both proposed the maximization of total similarity for their linear integer programming formulations and this is a quite reasonable approach, especially if the similarities were developed with volume and/or routing data. Gunasingh and Lashkari [41] additionally used another objective function to minimize the total fixed cost of all machines in all cells less the savings in intercell movement costs that resulted form parts belonging to a particular machine cell. This latter savings appears to be quite difficult to estimate and the resulting performance measure is directly related to the CM objective of minimizing investment in new equipment but only slightly related to the objective of producing parts cell complete. Wei and Gaither [42] suggested several potential objective functions for their linear integer programming problems, including the minimization of intercell capacity imbalances, intercell shipments, and capacity underutilization. Shafer and Rogers [33] utilized goal programming to minimize deviations from several goals that included 1) minimize the amount setup times exceed zero, 2) maximize or minimize capacity utilization in each cell, 3) maximize or minimize the funds available to purchase new equipment, and 4) minimize the number of intercellular moves. In Table 2 is a categorization of the performance measures listed in Table 1 based upon the information the measures required for their calculation. Performance measures are categorized on the basis of whether or not they require part volume data and/or operation sequence data. Table 2 Categorization of p e r f o r m a n c e m e a s u r e s Part Volumes and Sequencing Not Considered
Part Volumes Considered
Simple Matching GeneralizedMatching
Weighted Intercelluar GlobalEfficiency Transfers Group Efficiency
Product Moment Correlation
Coefficient Total Bond Energy Proportion ExceptionalElements Machine Utilization Grouping Efficiency Grouping Efficacy Grouping Measure .Clustering Measure
Part Sequencing Considered
Both Part Volumes and Sequencing Considered
AverageWIP MaximumWIP Group Technology Average FlowTime MaximumFlowTime Efficiency Part Travel Distances Extra-Cellular Operations Longest Average
Queue
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The example machine-component matrices shown in Figure 1 were developed to help illustrate the limitations of some of the performance measures listed in the first column of Table 2. The simple matching, generalized matching, and the product moment correlation coefficient performance measures were not included in this comparison because only the final configuration of the machine-component matrix was developed and the value of calculating it is questionable unless the analyst has prior knowledge of an ideal configuration. In addition, the GE measure was not included because of its weak discriminating power and other evident problems associated with its use. Rather, the dominant measures of GF, WGF, and GM will be considered. Matrix B
Matrix A 1
2 3
4
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Matrix C
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5
2 ii~iiii~iii~iiiiiiii~iiii~iii~ili!ii!~iii~iiii 3
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4
o 0
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3
o 0
o !iii~ililililililili!~!ili!ii!
1 ~!i!i:ti~i~i~i~i!i~i~i~;t~i~i~i~!]
Matrix E
Matrix D 1
5 0 1 o
4
5
iii~iiiiiiii~iiiiiii~iii~o !.i.!~!.!.i.i.il.ii~.i!i!ii!~.iii~i~ii o
o o
!ii..liii~ii~i~i~ii!ii!i!i!i~i!iii1 0 0 1 1 -iiii:li!i!i!i!i!i!iiii~ii!iiiiii 0 0 1 -i!!!~ii!iliiiiiiiiii~l!~ii!i!i . . . . . . . . . . . . . . . . . . .
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Performance Measure Machine Utilization (MU) Proportion Exceptional Elements (PE) Total Bond Energy (TBE) Grouping Efficacy (GF) Diagonal Voids (DV) Grouping Measure (GM)
Matrix A
Matrix B
Matrix C
Matrix D
Matrix E
1.00 0.64 12.00 0.36 0.00 1.00
0.56 0.00 12.00 0.56 0.79 -0.23
0.85 0.21 12.00 0.69 0.14 0.71
0.77 0.29 18.00 0.59 0.21 0.56
1.00 0.07 18.00 0.93 0.00 1.00
Figure 1. Example machine-component matrices and performance measures The machine-component Matrices A-C in Figure 1 all have the same configuration; however, the part families and machine cells are defined differently as is illustrated with the shaded regions. In Matrix A five small machine cells and part families have been defined, in Matrix B one large machine cell and part family has been defined, and in Matrix C two machine cells and part families have been defined. As is illustrated in Figure 1 for Matrices A-C, MU is often maximized by creating a larger number of small cells while PE may be minimized by creating a fewer number of larger cells. Also, note that the TBE measure is indifferent among the cell assignments shown in Matrices A-C. In fact, TBE is only sensitive to the configuration of the entire machine-component matrix and is indifferent to the actual
158
assignment of part types to families and machine types to cells. This would also be true for CLM. The GF measure is greatest for the more reasonable assignment shown in Matrix C and the GM is greatest and at its upper bound for Matrix A which is probably not a good configuration for CM. Note that GM will always be equal to 1.0 whenever the cells consist of all nonzero elements regardless of the number of exceptional elements. Matrices D and E were developed to illustrate another weakness associated with the TBE measure. Both machine-component Matrices D and E have the same number of rows, columns, and nonzero entries. However, while the machinecomponent Matrix E has more clearly defined clusters, TBE is the same for both matrices. It is interesting to note that the other five performance measures all appear to indicate that the layout in Matrix E is superior. The results for using different values of q for calculating WGF are listed in Table 3. Note that WGF is constant for Matrix B over all selected values of q. This is because WGF reduces to WGF=II(I+DV) whenever PE=0 and thus WGF is insensitive to q. A similar property is not found when DV=0 as seen for the results for Matrices A and E. For the case of DV=0, WGF=r(1-PE)I[PE+r(1-PE)] and is dependent upon r=ql(1-q) and thus remains sensitive to different values for q. Note that for the four matrices with PE>0, the ranking with respect to WGF is consistent over all values of q and thus consistent decisions regarding the relative performance of different cell formation procedures may be made.
Table 3 Weighted grouping efficacy for the matrices from Figure 1. for various values of q q
Matrix A
Matrix B
Matrix C
Matrix D
Matrix E
.5 .4 .3 .2 .1
0.36 0.27 0.19 0.12 0.06
0.56 0.56 0.56 0.56 0.56
0.69 0.63 0.56 0.45 0.28
0.59 0.52 0.45 0.33 0.20
0.93 0.90 0.85 0.78 0.60
In summary, of the performance measures that are not for considering operation sequences and part volumes, the family of WGF measures often appear to be one of the better choices for capturing both features of PE and DV. The value of q can be set to reflect managerial preferences for the relative costs of DV and PE but the relative ranking with respect to WGF for different configurations will usually not be altered. The simple matching, generalized matching, and product moment correlation coefficient are not related to any of the design objectives associated with CM and are only for considering how closely the clustered machine-component matrix matches the original randomly generated machine-component matrix. TBE and CLM are only for considering the final configuration of the machine-component matrix and do not consider how part families and machine cells are defined. In
159
addition, TBE may be considerably insensitive to the degree of cluster definition in the rearranged machine-component matrix. The PE performance measure may have a bias toward a small number of large cells while the MU measure may have a bias toward a large number of small cells. 4. A Framework for Comparing Cell Formation Solutions
In this section a framework for comparing alternate cell formation procedures is suggested. A basic prerequisite for a performance measure employed to compare alternate cell formation procedures should be that it is directly related to specific CM design objectives. The design objectives included in this framework are the following: 1) minimize intercellular transfers, 2) minimize machine setup time, 3) minimize the investment in new equipment, 4) maintain acceptable machine utilization levels, 5) improve quality, 6) reduce work-in-process levels, 7) reduce production lead times, 8) maintain on-time delivery performance, and 9) increase job satisfaction. Also, static and dynamic performance measures will be distinguished in the framework. Static performance measures are analytical or formula based. Dynamic performance measures require computer simulation models and analysis. Finally, performance measures used to specifically assess the achievement of the design goals (i.e., direct performance measures) and measures that approximate or indirectly assess the achievement of the design goals (i.e., surrogate performance measures) will also be distinguished in the framework. In the remainder of this section static, dynamic, direct, and surrogate performance measures for each of the CM design objectives will be discussed. The CM design objective most frequently assessed is the number of intercellular transfers required. Usually it is measured indirectly as the proportion of elements in the final rearranged machine-component matrix that are exceptional elements as defined with PE. Mosier [12] modified PE by including part volume data and derived the Volume Weighted Proportion of Exceptional Elements (VWPE): N M N M VWPE= T. T~V~e~kl3 3V~x~k i=1 k=l i=1 k=l
(8)
where V~ is the annual demand for part i. Likewise, a surrogate measure for assessing the proportion of operations that require intercellular transfers can be developed based upon operation sequence information. The Operations Sequences Proportion of Exceptional Elements (OSPE) is: N 0i-1 OSPE = 3 3 r i=lj=l
N
(9) i=1
where (z~j is set to one if operation j and operation j+l on part i are performed in different cells, zero otherwise, and O~ is the total number of operations required by
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part type i. Finally, a measure for incorporating both operation sequence and part volume data to assess the actual proportion of operations requiring intercellular transfers may be developed. The Actual Proportion of Intercellular Operations (APIO) is:
N Ocl N APIO = Z T_,Via~j I T_,V,O~ i=1 j=l i=1
(10)
The example data shown in Figure 2 was developed to illustrate the four intercellular transfer measures of PE, VWPE, OSPE, and APIO. In Figure 2, machine routings, annual part type volume data, and three alternative machinecomponent assignments are provided. The results for the PE measure, which does not incorporate operation sequence or part type volume data, yields an equal ranking for all three machine-component assignments shown in Matrices A-C. When employing VWPE, the machine-component assignments shown in Matrices A and C are ranked as best. Employing the OSPE measure reveals that the machinepart assignment shown in Matrix B is best. The APIO measure ranks the machinecomponent assignment given in Matrix A as best. We thus note that when employing the three surrogate performance measures for the actual number of intercellular transfers, i.e., PE, VWPE, and OSPE, none of them consistently indicated the machine-component assignment shown in Matrix A as did APIO. Part T y p e
cl c2 c3 c4 c5 c6 Matrix A
Volume
10 20 20 10 20 100
Machine Routing
ml-m2-m3-m4 ml-m2-m4 m3-ml-m3-m2-m3 m6-m5-m4-m3 m6-m5-m3 m4-m5-m6 Matrix C
Matrix B
cl c2 c3 c4 c5 c6 cl c2 c3 c4 c5 c6 cl c2 c3 c4 c5 c6 0 0 ml i~i~i.lii~ii~i~i~i~i~i~:l~i~ii~i~i~i~ii~i~l~0ii~ii~i~ 0 ml iiiii~iiiiiiiiiiiiiiii!~iiiiiiiiiii!!iiiii~iiiiiiiiiil0 0 0 ml ::::::::?::?::::::::::::::::::::::::0::::::::::::0 iiiiiililiiiii~iiiiiiiiiiii filialiiiiiii] 0 0 0 m2 ilili~iiiiiiiiiiiliiiiii"..iiiiilililiiiiiiiili~iiiiiiiiilo o o m2 iilii~ m2 i!ili]iiiiiiiiiiiiiiiii~iiiiiiiiiiiiiiiiii~iiiiiiiiiii 0 0 0 .:...:.:.:.:................................................, o ~ Ii~i~i~i~i~i~i~i~i~i1:::::::::::::::::::::::::::::: m3 .iiiii~::.~i~iiii~i::i::iiii~i!~iiiii::i::i::i::i~iii::i~i~ii m3 iiiiiiiliiiiiiiiiiiiii~iiiiiiiili!!iiii!~iliiiiiii!i 1 1 0 1 1 0 m3 o :::::::::::::::::::::::::::::::::::::::::::::::::::: : 0 1 m4 , m4 1 1 0 i!;i;~i;i;i;i;i;!;i;i;i~;i;!;!;!;i;!;i;!;~;i;i;i;!;i;i;ilm4 :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0 0 0 :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: m5 o o o l~i~i~i~i~i~i~!~i~i~i~i~i~i~i~i~i~! mS 0 0 0 |iiii;~iiiiiiiiiiii!iiiii]iliiiiiiiiiiiiiiiii]iiiiiiii;iiiii I m5 mS o o o !!!!!~!ii!i!i!!ii!!iii~ii!i!i!!!!iiiii~i!i!!iiili!] m6 o o o ~ili~iii!i!ilili!iii!i!!i!i!!ii!iii:~i!!!!ii!iilme o o o ~!~i~iii!i!i!i!i~iii!ii!i!i!!!!i!i:ti!iii~
Performance Measure
ProportionExceptionalElements(PE) Volume Weighted ProportionExceptionalElements(VWPE) OperationsSequence Proportionof ExceptionalElements(OSPE) Actual Proportionof IntercellularOperations(APIO)
Matrix A
Matrix B
Matrix C
0.20 0.11 0.18 0.10
0.20 0.25 0.14 0.22
0.20 0.11 0.27 0.18
Figure 2. Example part type routings, volumes & machine-component matrices.
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In addition to directly measuring the number of intercellular transfers statically using APIO, it can also be measured dynamically using computer simulation. For example, in the computer simulation models developed by Shafer and Meredith [14], a variable was updated each time a part traveled between two cells. Since both direct static and dynamic measures are available, a question that naturally arises is whether one method in some sense outperforms the other. Since computer simulation would tend to require more effort, APIO might be preferred. However, when using APIO one must implicitly assume that the demand for the parts is known and is relatively stable. If these assumptions are met then APIO may perform well for assessing the actual proportion of operations requiring an intercellular transfer. Alternatively, if these assumptions are not realistic, then computer simulation may be a more attractive choice to more accurately assess intercellular transfers. Another design goal associated with CM is to reduce setup times. Assessing setup times statically via an equation is very difficult and to do this a great deal of information such as part volumes, part routings, part processing sequences, and sequence dependent setup times for all parts at all machines is needed (for an example, see Shafer and Rogers [33]). In addition, a number of assumptions must be made. For example, it must usually be assumed that all parts are produced on each production cycle and that the processing sequence of the parts does not vary. As a result, setup times are probably best measured dynamically via computer solution. Measuring setup times directly using computer simulation also requires a great deal of data, most importantly sequence dependent setup times for each machine for all part type pairs processed by a given machine. Because it is often not practical to collect this data, setup times in the near future will most likely be measured indirectly. For example, assumptions about setup time reductions can be built into computer simulation models (see Flynn and Jacobs [3]). Assessing the investment in new equipment is perhaps the easiest CM design objective to measure. A direct static measure is to simply sum over all machine types the number of machines of the type needed times the cost of the machine. Assessing machine utilization levels can be accomplished statically or dynamically. Statically, ratios of capacity needed to capacity available have long been used. Likewise, resource utilization measures are among the most common measures employed in simulation analysis. Caution should be exercised, however, to not overly emphasize the machine utilization objective. For example, assuming adequate machine capacity, a shop's performance should be considered improved if its lead times and work-in-process levels are reduced regardless of the level of machine utilization. Likewise, a shop's performance should be considered improved whenever quality improves and setup times are reduced regardless of the level of machine utilization. Another important consideration related to machine utilization is how the machines are weighted. Average machine utilization with all machines weighted equally may frequently not be very appropriate because in most plants only a few machines are critical. Quality is perhaps the most difficult CM design objective to measure. Presently, only potential improvements in quality can be indirectly measured by considering the
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relationship of quality to other CM design objectives. For example, reductions in the number of intercellular transfers permits holding workers more accountable for product quality. Similarly, ensuring that machines are not overutilized facilitates higher quality because there is time available for preventive maintenance. Finally, reductions in work-in-process levels and lead times facilitate uncovering hidden problems in the production process and permits the discovery of quality problems sooner so that corrective action can be initiated. Thus, assessing the extent of quality improvements can be approximated by assessing the achievement of other CM design objectives. Finally, assessing work-in-process levels and lead times is best accomplished dynamically via computer simulation. Work-in-process levels can be measured as a time persistent variable and lead times as a simple average of individual lead times over all parts. Analytical models that rely upon statistically-based queueing theory require assumptions about the frequency distribution of arrivals and service times and are too often not easily applicable or impossible to apply to complex situations.
5. Summary and Future Research Choosing a measure of performance is part of the modeling of any production process and is ultimately up to the management's stated objectives. If appropriate value-based objectives of profit, revenue, or cost are available to judge the quality of CM configurations then they should be utilized. However, accurate data for such objectives are difficult to obtain at any point in time and especially difficult to obtain prior to implementing CM. In order to compare competing CM configuration proposals we must often judge them by how aligned they are with the basic objectives of CM. Most of the performance measures in this article may be rationally utilized to gauge a CM configuration depending upon the situation encountered and the managerially-stated objectives and in this respect there is not a truly dominant measure. If, however, no guidance is provided regarding the desired outcomes of converting to CM, other than the basic tenets of CM, then the WGF measure appears to be one of the better choices to incorporate for making static decisions because a tradeoff is made between having exceptional elements outside of the cells and the voids in the cells. WGF is even a better measure if in the formula PE can be replaced by APIO. An analyst may also want to separately consider MU, DV, APIO (or OSPE, VWPE, or PE if the data for APIO is not available), TBE, CLM, GE, and GM but the properties and weaknesses of several of these measures should not be overlooked. A large amount of research has been devoted to the development of cell formation procedures. Recently, a few studies have focused on comparing these cell formation procedures. The purpose of this paper is to review solution quality performance measures available and to offer a framework for comparing alternate cell formation procedures. Before useful comparisons of cell formation procedures can be made, meaningful performance measures must be developed. For a performance to be meaningful it should be related to one or more of the design
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objectives associated with CM. The following nine design objectives associated with CM were identified: 1) minimize intercellular transfers, 2) minimize machine setup time, 3) minimize the investment in new equipment, 4) maintain acceptable machine utilization levels, 5) improve quality, 6) reduce work-in-process, 7) reduce production lead times, 8) maintain on-time delivery performance, and 9) increase job satisfaction. Additionally, performance measures were classified as direct or surrogate, and as static or dynamic. In terms of assessing intercellular transfers, static and dynamic direct measures are available. A static measure may be more appropriate when product volumes are known and the product mix is relatively stable. Otherwise, computer simulation should be used to directly measure intercellular transfers. For assessing setup times, machine utilization, work-in-process levels, and lead times dynamic measures may often perform best. Because of the large amount of data required, setup times probably can only be practically measured indirectly, while the other three measures can be easily assessed directly. Measuring the investment required for new equipment can be measured directly via static measures. Finally, at present, the recommended approach for assessing potential quality improvements is to combine several performance measures such as machine utilization, work-in-process levels, lead times, and intercellular transfers. Combining these measures may often provide an indication of how much the manufacturing environment will support and enhance quality improvement activities.
REFERENCES 1. R.J. Schonberger, World Class Manufacturing, Free Press, New York, 1986. 2. B.B. Flynn and F.R. Jacobs, "A simulation comparison of group technology with traditional job shop manufacturing", Int. J. of Prod. Res., 24 (1986) 1171. 3. B.B. Flynn and F.R. Jacobs, "An experimental comparison of cellular (group technology) layout with process layout.", Dec. Sci., 18 (1987) 562. 4. J.S. Morris and R.J. Tersine, "A simulation analysis of factors influencing the attractiveness of group technology cellular layouts", Man. Sci., 36 (1990) 1567. 5. U. WemmerlOv and N.L. Hyer, "Research issues in cellular manufacturing", Int. J. Prod. Res., 25 (1987) 413. 6. C-H. Chu, "Cluster analysis in manufacturing cellular formation", OMEGA Int. J. Man. Sci., 17 (1989) 289. 7. D.F. Rogers, R.D. Plante, R.T. Wong, and J.R. Evans, "Aggregation and disaggregation techniques and methodology for optimization", Oper. Res., 39 (1991) 553. 8. S.M. Shafer, "Cellular manufacturing: a selected bibliography", Working Paper (1994) Department of Management, College of Business, Auburn University, Auburn, Alabama, USA.
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9. R.G. Askin and K.S. Chiu, "A graph partitioning procedure for machine assignment and cell formation in group technology", Int. J. Prod. Res., 28 (1990) 1555. 10. F. Choobineh, "A framework for the design of cellular manufacturing systems", Int. J. Prod. Res., 26 (1988) 1161. 11. V.L. Huber and N.L. Hyer, "The human factor in cellular manufacturing", J. of Oper. Man., 5 (1985) 213. 12. C. Mosier, "An experiment investigating the application of clustering procedures and similarity coefficients to the GT machine cell formation problem", Int. J. Prod. Res., 27 (1989) 1811. 13. C. Mosier, "Weighted similarity measure heuristics for the group technology machine clustering problem", OMEGA Int. J. Man. Sci., 13 (1985) 577. 14. S.M. Shafer and J.R. Meredith, "A comparison of selected manufacturing cell formation techniques", Int. J. Prod. Res., 28 (1990) 661. 15. A. Ballakur and H.J. Steudel, "A within-cell utilization based heuristic for designing cellular manufacturing systems", Int. J. Prod. Res., 25 (1987) 639. 16. J.R. King, "Machine-part grouping in production flow analysis: an approach using a rank order clustering algorithm", Int. J. Prod. Res., 18 (1980) 213. 17. H.M. Chan and D.A. Milner, "Direct clustering algorithm for group formation in cellular manufacture", J. Man. Sys., 1 (1982) 65. 18. U. Wemmerl(~v, "Comments on direct clustering algorithm for group formation in cellular manufacture", J. Man. Sys., 3 (1984) vii. 19. A. Kusiak and W.S. Chow, "Efficient solving of the group technology problem, J__. Man. Sys., 6 (1987) 117. 20. A. Kusiak and W.S. Chow, "An efficient cluster identification algorithm", IEEE. Trans. Sys., Man and Cyb., SMC-17 (1987). 21. A.J. Vakharia and U. WemmerlOv, "Designing a cellular manufacturing system: a materials flow approach based on operation sequences", liE Trans., 22 (1990) 84. 22. M.R. Anderberg, Cluster Analysis for Application, Academic Press, New York, NY, 1973. 23. C-H. Chu and M. Tsai, "A comparison of three array-based clustering techniques for manufacturing cell formation", Int. J. Prod. Res., 28 (1990) 1417. 24. W.T. McCormick, P.J. Schweitzer, and T.W. White, "Problem decomposition and data recognition by a clustering technique", Oper. Res., 20 (1972) 993. 25. J.K. Lenstra, "Clustering a data array and the traveling salesman problem", Oper. Res., 22 (1974) 413. 26. M.P. Chandrasekharan and R. Rajagopalan, "An ideal seed non-hierarchical clustering algorithm for cellular manufacturing", Int. J. Prod. Res., 24 (1986) 451. 27. J. Miltenburg and W. Zhang, "A comparative evaluation of nine well-known algorithms for solving the cell formation problem in group technology", J. Oper. Man., Special Issue on Group Technology and Cellular Manufacturing, 10 (1991)44.
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28. M.P. Chandrasekharan and R. Rajagopalan, "MODROC: an extension of rank order clustering for group technoiogy", Int. J. Prod. Res., 24 (1986) 1221. 29. L.E. Stanfel, "Machine clustering for economic production", Eng. Costs and Prod. Econ., 9 (1985) 73. 30. S.M. Shafer and D.F. Rogers, "Similarity and distance measures for cellular manufacturing part I: a survey", Int. J. Prod. Res., 31 (1993) 1133. 31. S.M. Shafer and D.F. Rogers, "Similarity and distance measures for cellular manufacturing part I1: an extension and comparison", Int. J. Prod. Res., 31 (1993) 1315. 32. C.S. Kumar and M.P. Chandrasekharan, "Grouping efficacy: a quantitative criterion for goodness of block diagonal forms of binary matrices in group technology", Int. J. Prod Res., 26(1990) 233. 33. S.M. Shafer and D.F. Rogers, "A goal programming approach to the cell formation problem", J. Oper. Man., Special Issue on Group Technology and Cellular Manufacturing, 10 (1991 ) 28. 34. T.D. Klastorin, "The p-median problem for cluster analysis: a comparative test using the mixture model approach", Man. Sci., 31 (1985) 84. 35. S.M. Ng, "Worst-case analysis of an algorithm for cellular manufacturing", Eur. J. Oper. Res., 69 (1993) 384. 36. G. Harhalakis, R. Nagi, and J.M. Proth, "An efficient heuristic in manufacturing cell formation for group technology applications", Int. J. Prod. Res., 28 (1990) 185. 37. L.E. Stanfel, "A successive approximations method for a cellular manufacturing problem", Ann. Oper. Res., 17 (1989) 13. 38. L.E. Stanfel, "Successive approximations procedures for a cellular manufacturing problem with machine loading constraints", Eng. Costs and Prod. Econ., 17 (1989) 135. 39. L.E. Stanfel, "lterative determination and matching of part groups and machine cells: optimization and successive approximations", Int. J. Prod. Econ., 23 (1991) 213. 40. A. Kusiak, "The generalized group technology concept", Int. J. Prod. Res., 25 (1987) 561. 41. K.R. Gunasingh and R.S. Lashkari, "Machine grouping problem in cellular manufacturing systems - an integer programming approach", Int. J. Prod. Res.,27 (1989) 1465. 42. J.C. Wei and N. Gaither, "An optimal model for cell formation decisions", Dec. Sci., 21 (1990) 416.
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Measuring cellular manufacturing performance D.F. Rogers" and S.M. S h a f e r b
aDepartment of Quantitative Analysis and Operations Management, College of Business Administration, University of Cincinnati, 531 Carl H. Lindner Hall, Cincinnati, Ohio 45221-0130, USA" bDepartment of Management, College of Business, Auburn University, 415 West Magnolia, Auburn, Alabama 36849-5241, USA
A large amount of research has focused on developing new cell formation procedures. Only recently, however, has research focused on comparing and evaluating alternative cell formation procedures. Perhaps the most significant hurdle associated with conducting studies for comparing and evaluating cell formation procedures is the absence of meaningful performance measures. A performance measure is considered meaningful when it is related to one or more of the design objectives associated with cellular manufacturing. In this paper several design objectives associated with cellular manufacturing are identified. Then, based upon these design objectives, appropriate performance measures are discussed and compared. Also included is a review and critique of performance measures used in previous studies for comparing cell formation procedures.
1. I n t r o d u c t i o n
It is widely acknowledged that the environment confronting manufacturers is becoming increasingly competitive as markets become more globalized. As a result, producers of goods are under constant and intense pressure to quickly and continuously improve their operations. Areas often targeted for improvement are productivity, quality, and responsiveness. In recent decades, Cellular Manufacturing (CM) has emerged as a promising approach for improving operations in batch and job shop environments, particularly in situations for which the divisions of the production processes are distinct. For CM, parts with similar processing requirements are identified and grouped together
*Dr. Rogers acknowledges Dr. J.E. Aronson and the Department of Management, College of Business Administration, University of Georgia for the support provided during his sabbatical leave from the University of Cincinnati.
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to form part families. The equipment requirements for the part families are simultaneously or subsequently determined and usually located together in close proximity. Thus, equipment is located together based upon the processing requirements of part families in CM and thereby differs from traditional layouts where similarly functioning equipment is often located together. Groups of mostly dissimilar machine types that are utilized for CM and located together are referred to as machine cells. The use of CM allows for an increase of local decision making in the factory and this also often results in quality and productivity improvements. There is a wealth of empirical evidence supporting the potential superiority of a cellular layout versus the more traditional functional layout. Schonberger [1] stated that moving the machines into cells was a basic step in the transformation of the General Electric Company's Louisville, Kentucky dishwasher plant into a world class manufacturing showcase. He noted that large numbers of manufacturers are utilizing CM in an effort to become world class competitors. There is also evidence to indicate that CM is not always the appropriate approach to take for certain scenarios. Results from the simulation models of Flynn and Jacobs [2 and 3] indicated that a well-organized job shop can outperform a CM configuration with respect to several criteria such as work-in-process inventory levels and average flow times. In these simulations the machine-component matrices were quite dense and forming distinct separations of the production processes to accommodate CM was not possible. Morris and Tersine [4] further revealed that an ideal environment for a cellular layout is characterized with a high ratio of setup to process time, stable demand, unidirectional work flow within a cell, and a considerable level of material movement times between process departments. In response to the considerable interest in CM, there has been much current research devoted to the development of cell formation procedures, i.e., techniques that facilitate the identification of part families and machine cells. WemmerlSv and Hyer [5] summarized research issues in CM that is also relevant today. Chu [6] provided a summary of general cluster analysis techniques and algorithms for forming manufacturing cells. Rogers, et al. [7] cast clustering into a larger framework for aggregation and disaggregation techniques. Shafer [8] identified over 80 contributions found in the literature associated with the development and comparison of cell formation procedures. Unfortunately, the vast majority of these contributions focused on developing new cell formation procedures. Very little research to date has focused on comparing these techniques. As a consequence, there is little or no guidance available concerning the appropriateness and/or usefulness of cell formation techniques. It may be argued that only profit, revenue, or cost-based measures, i.e., valuebased measures that reflect real improvement to investors, should actually be employed for judging the desirability of CM. This is ultimately true for almost any significant change in a firm, especially changes in production functions. Askin and Chiu [9] recognized this and incorporated an objective function of cost minimization for their linear integer programming problem to form machine cells and part families. The cost components of this objective function were 1) machine overhead - the cost of placing a particular number of a machine type in a cell, 2) group overhead - a fixed cost for using a particular grouping, 3) family tooling costs, and 4) intergroup
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material handling costs. However, it is often extremely difficult to accurately assess actual profits, revenues, or costs to a manufacturing system prior to actual implementation of the particular CM configuration selected. How, for example, may one reasonably assess the costs of intergroup material handling costs prior to knowing the cell configurations and relative locations? Choobineh [10] also formulated a linear integer programming problem with an objective function of minimizing the total average annual cost of producing all part families in all cells and the cost of providing the appropriate number of machines for each cell. These costs may often be exceedingly difficult to determine. In reality it may take months or even several years before the actual impacts of converting to a CM system may be assessed. It is also often extremely difficult to ascertain the costs of lost flexibility due to employing CM rather than a job shop, to assess the sociological implications of the factory employees as well as their management (see Huber and Hyer [11]), and to appropriately measure the quality increase in the end product that is often accompanied by employing CM. Because value-based methods are not easily implementable for determining part family and/or machine cell configurations we must typically judge the quality of a CM solution with measures that are not based upon value but nonetheless are quite good surrogates for value-based objectives. These surrogate measurements for CM evaluation should ideally coincide with the basic objectives of CM which were originally developed to coincide with value-based objectives. Therefore, the emphasis in this article will be to use measures for judging various CM configurations that appropriately match the objectives by which these cells are to be formed. An objective in this paper is to offer a framework for comparing alternate cell formation procedures. In the next section a review of studies for comparing various cell formation procedures is provided. In the following section is a closer examination of the performance measures employed in these studies and other performance measures that may prove to be interesting for future studies. Before useful comparison studies can be conducted, meaningful performance measures must be developed and a comparison of current performance measures is presented. Subsequently, a framework for comparing alternative cell formation solutions is offered. Finally, we provide a summary and discuss avenues for future research.
2. Literature Review of Cell Formation Technique Comparisons In this section studies that have focused on comparing alternate cell formation procedures will be reviewed. Many of these studies involve simple row and column manipulation and/or hierarchical and nonhierarchical statistical clustering of the rows and columns of the machine-component matrix X, where x~j = 1, if machine type i, i=l,...,M, is required for production of component j, j=I,...,N, and x~j=0, otherwise. In Table 1 is a summary of the cell formation solution quality performance measures that are included in these studies and other various gauges of performance that
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Table 1 Summary of cell formation solution quality performance measures. Performance Measure
Reference
Simple Matching Generalized Matching Product Moment Correlation Coefficient
Anderberg[22] Klastorin[34] Anderberg[22]
M osier[ 12]
Weighted Intercellular Transfers
Mosier[13]
M osier[ 12]
Used In
Avg. & Maximum WIP Shafer & Avg. & Max. Flow Time Meredith[14] Part Travel Distances Extra-Cellular Operations Longest Average Queue Total Bond Energy McCormick, Schweitzer & White[24] Clustering Measure Miltenburg & Zhang[27]
Chu & Tsai[23] Shafer & Rogers[31] Miltenburg & Zhang[27]
Proportion Exceptional Elements Machine Utilization
Chu & Tsai[23] Chandrasehkaran &Rajagopalan[26]
Chu & Tsai[23] Shafer & Rogers[31]
Grouping Efficiency
Chandrasehkaran &Rajagopalan[26]
Chu & Tsai[23]
Global Efficiency Group Efficiency Group Technology Efficiency
Harhalakis, Nagi & Proth[36]
Grouping Efficacy
Kumar & Chandrasehkaran [32] Miltenburg & Zhang[27]
Grouping Measure
Shafer & Meredith[14]
Shafer & Rogers[31] Miltenburg & Zhang[27]
Comments
9Solution quality defined in terms of how similar it is to original matrix. 9Not directly related to goals of CM. =Does not consider part sequencing or volumes. 9Relatively easy to calculate. 9Surrogate for intercellular transfers. 9Does not consider part sequencing but does consider part volumes. 9Relatively easy to calculate. 9Directly related to many CM goals. 9Generally not computationally efficient (requires simulation). 9Does consider part sequencing and part volumes, ,Not directly related to goals of CM. ,Does not consider part sequencing or part volumes. ,Relatively easy to calculate. ,Insensitive to family & cell definition. ,Surrogate measures related to CM goals. ,Does not consider part sequencing or volumes. ,Relatively easy to calculate. 9Must select arbitrary weight. 9Surrogate measure related to CM goals. 9Weak discriminating power. 9Does not consider part sequencing or volumes. 9Relatively easy to calculate. 9Surrogate measure related to CM goals. 9Considers part sequencing but not volumes. 9Relatively easy to calculate. 9Better discriminating power than grouping efficiency. 9No need to select an arbitrary weight. 9Does not consider part sequencing or volumes. 9Surrogate measure related to CM goals. 9Relatively easy to calculate.
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have been developed. Mosier [12] used a mixture model experimental approach and compared seven similarity coefficients and four statistically-based hierarchical clustering procedures. The degree of cluster definition, i.e., the ratio of the non-zero density outside the clusters to the non-zero density inside the clusters, and the density of the non-zero entries within the clusters of the machine-component matrix were additional factors included in the study. The performance measures used were a simple and a generalized matching measure, a product moment correlation coefficient measure, and an intercellular transfer measure (Mosier [13]). Statistics were obtained for the results of the solved problems and the Jaccard similarity coefficient showed promise for these particular problems in spite of being one of the simplest similarity coefficients to calculate. None of the hierarchical clustering procedures appeared to uniformly dominate any of the others in the study. Shafer and Meredith [14] utilized computer simulation to compare cell formation procedures from all three categories of the taxonomy for cell formation procedures proposed by Ballakur and Steudel [15]: 1) part grouping, 2) machine grouping, and 3) simultaneous machine-part grouping. The clustering algorithms that were employed were the rank order clustering algorithm (King [16]), the direct clustering algorithm (Chan and Milner [17] and Wemmerl6v [18]), the cluster identification algorithm (Kusiak and Chow [19 and 20]), a technique for considering the sequence of operations developed by Vakharia and Wemmerl5v [21], and both single and average linkage hierarchical clustering routines (Anderberg [22]). These algorithms were applied to data collected from three companies. Computer simulation models were then developed based upon the solutions generated with the cell formation procedures. The performance measures included in this study were average and maximum work-in-process levels, average and maximum flow times, part travel distances, extra-cellular operations, and the longest average queue. It was found that using clustering algorithms to first form part families and then assigning machines to cells to accommodate the families provided more flexibility and worked best for these scenarios. Chu and Tsai [23] compared the rank order clustering algorithm, the direct clustering algorithm, and the bond energy analysis algorithm (McCormick, Schweitzer, and White [24] and Lenstra [25]) using the following performance measures: 1) total bond energy, 2) proportion exceptional elements, i.e., the proportion of the part types that must be transferred to other cells, 3) machine utilization, and 4) grouping efficiency (Chandrasekharan and Rajagopalan [26]). Eleven data sets from the literature were used for their study and the bond energy algorithm performed best for these problems. However, most of these data sets were extremely small example problems and not enough of the remaining problems were large enough to viably conclude the dominance of any one particular method. Miltenburg and Zhang [27] have performed one of the more extensive studies to compare CM algorithms upon 0/1 machine-part matrices. They tested nine different algorithmic approaches that employed different combinations of the rank order clustering algorithm, the modified rank order clustering algorithm (Chandrasekharan and Rajagopalan [28]), single and average linkage clustering, the bond energy algorithm, and an ideal seed non-hierarchical clustering algorithm
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(Chandrasekharan and Rajagopalan [26]). The objectives of their study are to form cells such that there is a high usage of machines by parts in each cell (a high cell density) and to form cells that do not tend to allow exceptional elements, i.e., parts that leave the cell and travel to another cell. The primary performance measure that they used to evaluate the clustering algorithms attempted to combine the influence of both of these objectives and may actually be attributed to Stanfel [29]. Two secondary performance measures employed were a measure of closeness of nonzero elements to the diagonal and average bond energy. Results were obtained for 544 solved problems and no significant differences were detected among most of the algorithms. The ideal seed non-hierarchical clustering algorithm showed the most promise for these particular randomly generated problems and performance measures. However, these approaches impose a significant computational burden for larger problems as compared to the other algorithms. Shafer and Rogers [30 and 31] investigated 16 measures of similarity or distance and four hierarchical clustering procedures that may be used for CM problems. The same performance measures as used by Chu and Tsai [23] were employed except that the grouping efficiency criterion was replaced by the improved grouping efficacy criterion developed by Kumar and Chandrasekharan [32]. A total of 704 solutions were derived and it was found that little difference typically existed among the results from different similarity coefficients. Single linkage clustering was, in general, found to be statistically inferior to average linkage, complete linkage, and Ward's clustering methods for the bond energy, machine utilization, and grouping efficacy criteria. However, single linkage clustering was statistically superior when considering the proportion of exceptional elements.
3. Comparison of Performance Measures
A fundamental criterion for evaluating performance measures is how closely related the performance measures are to the basic objectives associated with adopting CM. Shafer and Rogers [33] identified the following four key design objectives associated with CM: 1) setup time reduction, 2) to produce parts cell complete, i.e., minimize intercellular transfers, 3) to minimize investment in new equipment, and 4) to maintain acceptable machine utilization levels. Other important objectives associated with the adoption of CM include improving product quality, reducing inventory levels, and shortening lead times. In the remainder of this section the performance measures introduced in the previous section and summarized in Table 1 will be further discussed and compared. Mosier [12] employed four performance measures, a simple matching measure, a generalized matching measure, a product moment correlation coefficient measure, and an intercellular transfer measure. The first three performance measures (Anderberg [22] and Klastorin [34]) are for gauging how well the clustered solution matches the original machine-component matrix and thus do not allow for the possibility that the clustered solution obtained may be in some sense better than the configuration for the original matrix. The original configuration may not even be a good choice. Although these three measures are relatively easy to calculate, they
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are not related to the design goals associated with adopting CM, and part volumes or operation sequences are not considered with them. The last measure employed, the intercellular transfer measure, is the volume weighted number of parts that require processing in more than one cell. With this measure one may consider the possibility that the solutions generated by the cell formation procedures were better than the original randomly generated machine-component matrices. This measure does indirectly addresses the CM design objective of minimizing intercellular transfers. However, because operation sequences are not considered, it is only a surrogate measure for the actual number of intercellular transfers. All seven performance measures used by Shafer and Meredith [14], average and maximum work-in-process levels, average and maximum flow times, part travel distances, extra-cellular operations, and the longest average queue, require the development of computer simulation models. Deriving these measures is computationally more complex but they are all directly related to CM design objectives and are measures used for considering both part volumes and operation sequences. Four performance measures were employed by Chu and Tsai [23] and Shafer and Rogers [31]. The first performance measure, Total Bond Energy (TBE), was proposed by McCormick, Schweitzer, and White [24] as the performance measure to assist an algorithmic approach for identifying a permutation of the rows and columns such that the sum of the products of adjacent elements is maximized. This algorithm reveals a block diagonal matrix if one exists but its performance is unpredictable otherwise. TBE is defined as: M N TBE = I; I~ xij(xij+l .i- xij.1 -i- Xi+l J 4" Xi.l,j)12 i=1 j=l
(1)
where Xi0----X0j=XM+l,j=Xi,N+l=0. However, TBE is not directly related to the design goals associated with CM. The other three measures, proportion of exceptional elements, machine utilization, and grouping efficiency (grouping efficacy in Sharer and Rogers [31]), are surrogate measures for the CM design objectives of minimizing intercellular transfers and maintaining acceptable machine utilization levels. Proportion Exceptional Elements (PE) is the ratio of the number of nonzero elements not in the block diagonals (machine-component cells) in the final configuration of the clustered machine-cell matrix to the total number of nonzero elements in X given by: N M N M P E = I; I~ eij I T~ ~ Xi j i=1 j=l i=1 j=l
(2)
where e~j is set to one if element x U in the rearranged X is an exceptional element, zero otherwise. Machine Utilization (MU), also known as the density of the machinepart cells, is the ratio of the number of nonzero elements in the machine-cell clusters
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to the total number of elements in these cells. Neither of these measure are for evaluating the actual achievement of the CM objectives largely because operation sequences, part volumes, and operation times are not considered. Chandrasekharan and Rajagopalan [26] developed the Grouping Efficiency (GE) measure, a weighted average defined as: GE = q MU + (l-q) ODV
(3)
where the Off-Diagonal Voids (ODV) is the ratio of the number of zero elements in the off-diagonal blocks of X over the total number of elements in the off-diagonal blocks and q is a selected weighting on [0,1] designed to reflect the relative importance the analyst desires to place on MU versus ODV. GE will also always take on values on [0,1] provided that there is more than one machine-cell cluster, GE=I for a perfect block diagonal form and GE=0 for the diametric case. Freedom to select the appropriate weighting q allows the analyst to decide the relative importance between intercellular moves and voids in the cells. Ng [35] scrutinized GE as a performance measure. He noted that nonzero elements inside a diagonal block correspond to work processed in a machine cell with smaller material handling and setup costs. Alternatively, exceptional elements correspond to work that must be processed outside of the cells and thus has larger material handling and setup costs. For a typical example with q=.5 it was shown that the rate of change of GE with respect to a nonzero entry inside a diagonal block was three times that of an exceptional element which is contrary to their relative costs. By adjusting the weighting to q=.2 the same rate of change would decrease from three to 3/4, a more reasonable value but still maybe too large in practice to appropriately reflect the relative costs. It was also revealed for a large and sparse X and using q=.l that this rate of change can still be much larger than 1.0. Furthermore, it was shown that any matrix X with two completely decomposable diagonal blocks will have a grouping efficiency of at least (1-q)+q(MU). If q is to be small to overcome the above-mentioned problems then this minimum value for GE will always be quite large. Smaller values of q should probably be used if calculating GE but the value of its usage is questionable. Kumar and Chandrasekharan [32] also noted that the use of GE as a measure of performance has several shortcomings. They revealed that even quite poor solutions could yield a GE of .75 and thus the efficiency function GE has weak discriminating power. Also, the requirement that a weight be selected, possibly quite arbitrarily, to determine the relative importance of MU and ODV may be a difficulty. They proposed the Grouping eFficacy (GF) measure to overcome these shortcomings while retaining the desirable properties associated with the use of GE: GF = ( 1-PE)I(1 .DV)
(4)
where the Diagonal Voids (DV) is the number of zero elements in the block diagonals in the final configuration of the clustered machine-cell matrix to the total number of nonzero elements in X. GF will also take on values on [0,1].
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Ng [35] noticed with an example that the value of GF may be larger for a problem with five clusters than an apparently better four-cluster solution for the same problem. He proposed a modification to remedy this problem with GF called the Weighted Grouping eFficacy (WGF) which was derived by placing a weight of q on each entry inside the diagonal blocks and a weight of 1-q on exceptional elements given by: W G F = r(1 -PE)I[PE+r(1 + D V - P E ) ]
(s)
where r=ql(1-q) and q is a weighting just as for GE. If q=.5 then r=l, WGF=GF, and thus WGF is a generalization of GF. WGF will likewise have values on [0,1]. Miltenburg and Zhang [27] utilized one primary and two secondary performance measures. Their primary performance measure, called the Grouping Measure (GM), is also an attempt to overcome some similar supposed weaknesses with using the GE measure and is given by: G M = M U - DV
(6)
The value of GM is bounded by negative and positive unity, i.e., -I_
(7)
where dh(Xij)is the horizontal distance between a nonzero element and the diagonal and dv(X~j)is the vertical distance between a nonzero element and the diagonal. CLM is an overall indicator of the closeness of the nonzero elements to the diagonal, regardless of whether the nonzero elements are in a diagonal block or not. It is the average squared Euclidean distance of a nonzero element from the diagonal. CLM is not directly related to any of the CM design objectives but it may be a good gauge of cohesiveness for some circumstances just as TBE may be. Harhalakis, Nagi, and Proth [36] proposed the following three performance measures which have not yet been used in a study to compare cell formation procedures: 1) global efficiency - the ratio of the number of operations performed within the primary cells to the total number of operations, 2) group efficiencycalculated by subtracting from one the ratio of the actual number of external cells visited to the maximum number of cells that could be visited and 3) group technology efficiency- calculated by subtracting from one the ratio of actual number of intercellular moves required to the maximum number of intercellular moves possible.
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These measures do include the consideration of operation sequences but do not regard any data for part volumes. Compared to performing computer simulations, they are relatively easy to calculate. Numerous other authors have implemented performance measures implicitly through the objective functions of the mathematical models that they have developed for which there are many possibilities. A representative group of this work includes Stanfel [37, 38, and 39] who formed a linear integer programming problem with the objective of minimizing the total number of intercell transitions made by part groups. Kusiak [40] and Gunasingh and Lashkari [41] both proposed the maximization of total similarity for their linear integer programming formulations and this is a quite reasonable approach, especially if the similarities were developed with volume and/or routing data. Gunasingh and Lashkari [41] additionally used another objective function to minimize the total fixed cost of all machines in all cells less the savings in intercell movement costs that resulted form parts belonging to a particular machine cell. This latter savings appears to be quite difficult to estimate and the resulting performance measure is directly related to the CM objective of minimizing investment in new equipment but only slightly related to the objective of producing parts cell complete. Wei and Gaither [42] suggested several potential objective functions for their linear integer programming problems, including the minimization of intercell capacity imbalances, intercell shipments, and capacity underutilization. Shafer and Rogers [33] utilized goal programming to minimize deviations from several goals that included 1) minimize the amount setup times exceed zero, 2) maximize or minimize capacity utilization in each cell, 3) maximize or minimize the funds available to purchase new equipment, and 4) minimize the number of intercellular moves. In Table 2 is a categorization of the performance measures listed in Table 1 based upon the information the measures required for their calculation. Performance measures are categorized on the basis of whether or not they require part volume data and/or operation sequence data. Table 2 Categorization of p e r f o r m a n c e m e a s u r e s Part Volumes and Sequencing Not Considered
Part Volumes Considered
Simple Matching GeneralizedMatching
Weighted Intercelluar GlobalEfficiency Transfers Group Efficiency
Product Moment Correlation
Coefficient Total Bond Energy Proportion ExceptionalElements Machine Utilization Grouping Efficiency Grouping Efficacy Grouping Measure .Clustering Measure
Part Sequencing Considered
Both Part Volumes and Sequencing Considered
AverageWIP MaximumWIP Group Technology Average FlowTime MaximumFlowTime Efficiency Part Travel Distances Extra-Cellular Operations Longest Average
Queue
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The example machine-component matrices shown in Figure 1 were developed to help illustrate the limitations of some of the performance measures listed in the first column of Table 2. The simple matching, generalized matching, and the product moment correlation coefficient performance measures were not included in this comparison because only the final configuration of the machine-component matrix was developed and the value of calculating it is questionable unless the analyst has prior knowledge of an ideal configuration. In addition, the GE measure was not included because of its weak discriminating power and other evident problems associated with its use. Rather, the dominant measures of GF, WGF, and GM will be considered. Matrix B
Matrix A 1
2 3
4
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3
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5
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Matrix E
Matrix D 1
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4
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Performance Measure Machine Utilization (MU) Proportion Exceptional Elements (PE) Total Bond Energy (TBE) Grouping Efficacy (GF) Diagonal Voids (DV) Grouping Measure (GM)
Matrix A
Matrix B
Matrix C
Matrix D
Matrix E
1.00 0.64 12.00 0.36 0.00 1.00
0.56 0.00 12.00 0.56 0.79 -0.23
0.85 0.21 12.00 0.69 0.14 0.71
0.77 0.29 18.00 0.59 0.21 0.56
1.00 0.07 18.00 0.93 0.00 1.00
Figure 1. Example machine-component matrices and performance measures The machine-component Matrices A-C in Figure 1 all have the same configuration; however, the part families and machine cells are defined differently as is illustrated with the shaded regions. In Matrix A five small machine cells and part families have been defined, in Matrix B one large machine cell and part family has been defined, and in Matrix C two machine cells and part families have been defined. As is illustrated in Figure 1 for Matrices A-C, MU is often maximized by creating a larger number of small cells while PE may be minimized by creating a fewer number of larger cells. Also, note that the TBE measure is indifferent among the cell assignments shown in Matrices A-C. In fact, TBE is only sensitive to the configuration of the entire machine-component matrix and is indifferent to the actual
158
assignment of part types to families and machine types to cells. This would also be true for CLM. The GF measure is greatest for the more reasonable assignment shown in Matrix C and the GM is greatest and at its upper bound for Matrix A which is probably not a good configuration for CM. Note that GM will always be equal to 1.0 whenever the cells consist of all nonzero elements regardless of the number of exceptional elements. Matrices D and E were developed to illustrate another weakness associated with the TBE measure. Both machine-component Matrices D and E have the same number of rows, columns, and nonzero entries. However, while the machinecomponent Matrix E has more clearly defined clusters, TBE is the same for both matrices. It is interesting to note that the other five performance measures all appear to indicate that the layout in Matrix E is superior. The results for using different values of q for calculating WGF are listed in Table 3. Note that WGF is constant for Matrix B over all selected values of q. This is because WGF reduces to WGF=II(I+DV) whenever PE=0 and thus WGF is insensitive to q. A similar property is not found when DV=0 as seen for the results for Matrices A and E. For the case of DV=0, WGF=r(1-PE)I[PE+r(1-PE)] and is dependent upon r=ql(1-q) and thus remains sensitive to different values for q. Note that for the four matrices with PE>0, the ranking with respect to WGF is consistent over all values of q and thus consistent decisions regarding the relative performance of different cell formation procedures may be made.
Table 3 Weighted grouping efficacy for the matrices from Figure 1. for various values of q q
Matrix A
Matrix B
Matrix C
Matrix D
Matrix E
.5 .4 .3 .2 .1
0.36 0.27 0.19 0.12 0.06
0.56 0.56 0.56 0.56 0.56
0.69 0.63 0.56 0.45 0.28
0.59 0.52 0.45 0.33 0.20
0.93 0.90 0.85 0.78 0.60
In summary, of the performance measures that are not for considering operation sequences and part volumes, the family of WGF measures often appear to be one of the better choices for capturing both features of PE and DV. The value of q can be set to reflect managerial preferences for the relative costs of DV and PE but the relative ranking with respect to WGF for different configurations will usually not be altered. The simple matching, generalized matching, and product moment correlation coefficient are not related to any of the design objectives associated with CM and are only for considering how closely the clustered machine-component matrix matches the original randomly generated machine-component matrix. TBE and CLM are only for considering the final configuration of the machine-component matrix and do not consider how part families and machine cells are defined. In
159
addition, TBE may be considerably insensitive to the degree of cluster definition in the rearranged machine-component matrix. The PE performance measure may have a bias toward a small number of large cells while the MU measure may have a bias toward a large number of small cells. 4. A Framework for Comparing Cell Formation Solutions
In this section a framework for comparing alternate cell formation procedures is suggested. A basic prerequisite for a performance measure employed to compare alternate cell formation procedures should be that it is directly related to specific CM design objectives. The design objectives included in this framework are the following: 1) minimize intercellular transfers, 2) minimize machine setup time, 3) minimize the investment in new equipment, 4) maintain acceptable machine utilization levels, 5) improve quality, 6) reduce work-in-process levels, 7) reduce production lead times, 8) maintain on-time delivery performance, and 9) increase job satisfaction. Also, static and dynamic performance measures will be distinguished in the framework. Static performance measures are analytical or formula based. Dynamic performance measures require computer simulation models and analysis. Finally, performance measures used to specifically assess the achievement of the design goals (i.e., direct performance measures) and measures that approximate or indirectly assess the achievement of the design goals (i.e., surrogate performance measures) will also be distinguished in the framework. In the remainder of this section static, dynamic, direct, and surrogate performance measures for each of the CM design objectives will be discussed. The CM design objective most frequently assessed is the number of intercellular transfers required. Usually it is measured indirectly as the proportion of elements in the final rearranged machine-component matrix that are exceptional elements as defined with PE. Mosier [12] modified PE by including part volume data and derived the Volume Weighted Proportion of Exceptional Elements (VWPE): N M N M VWPE= T. T~V~e~kl3 3V~x~k i=1 k=l i=1 k=l
(8)
where V~ is the annual demand for part i. Likewise, a surrogate measure for assessing the proportion of operations that require intercellular transfers can be developed based upon operation sequence information. The Operations Sequences Proportion of Exceptional Elements (OSPE) is: N 0i-1 OSPE = 3 3 r i=lj=l
N
(9) i=1
where (z~j is set to one if operation j and operation j+l on part i are performed in different cells, zero otherwise, and O~ is the total number of operations required by
160
part type i. Finally, a measure for incorporating both operation sequence and part volume data to assess the actual proportion of operations requiring intercellular transfers may be developed. The Actual Proportion of Intercellular Operations (APIO) is:
N Ocl N APIO = Z T_,Via~j I T_,V,O~ i=1 j=l i=1
(10)
The example data shown in Figure 2 was developed to illustrate the four intercellular transfer measures of PE, VWPE, OSPE, and APIO. In Figure 2, machine routings, annual part type volume data, and three alternative machinecomponent assignments are provided. The results for the PE measure, which does not incorporate operation sequence or part type volume data, yields an equal ranking for all three machine-component assignments shown in Matrices A-C. When employing VWPE, the machine-component assignments shown in Matrices A and C are ranked as best. Employing the OSPE measure reveals that the machinepart assignment shown in Matrix B is best. The APIO measure ranks the machinecomponent assignment given in Matrix A as best. We thus note that when employing the three surrogate performance measures for the actual number of intercellular transfers, i.e., PE, VWPE, and OSPE, none of them consistently indicated the machine-component assignment shown in Matrix A as did APIO. Part T y p e
cl c2 c3 c4 c5 c6 Matrix A
Volume
10 20 20 10 20 100
Machine Routing
ml-m2-m3-m4 ml-m2-m4 m3-ml-m3-m2-m3 m6-m5-m4-m3 m6-m5-m3 m4-m5-m6 Matrix C
Matrix B
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Performance Measure
ProportionExceptionalElements(PE) Volume Weighted ProportionExceptionalElements(VWPE) OperationsSequence Proportionof ExceptionalElements(OSPE) Actual Proportionof IntercellularOperations(APIO)
Matrix A
Matrix B
Matrix C
0.20 0.11 0.18 0.10
0.20 0.25 0.14 0.22
0.20 0.11 0.27 0.18
Figure 2. Example part type routings, volumes & machine-component matrices.
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In addition to directly measuring the number of intercellular transfers statically using APIO, it can also be measured dynamically using computer simulation. For example, in the computer simulation models developed by Shafer and Meredith [14], a variable was updated each time a part traveled between two cells. Since both direct static and dynamic measures are available, a question that naturally arises is whether one method in some sense outperforms the other. Since computer simulation would tend to require more effort, APIO might be preferred. However, when using APIO one must implicitly assume that the demand for the parts is known and is relatively stable. If these assumptions are met then APIO may perform well for assessing the actual proportion of operations requiring an intercellular transfer. Alternatively, if these assumptions are not realistic, then computer simulation may be a more attractive choice to more accurately assess intercellular transfers. Another design goal associated with CM is to reduce setup times. Assessing setup times statically via an equation is very difficult and to do this a great deal of information such as part volumes, part routings, part processing sequences, and sequence dependent setup times for all parts at all machines is needed (for an example, see Shafer and Rogers [33]). In addition, a number of assumptions must be made. For example, it must usually be assumed that all parts are produced on each production cycle and that the processing sequence of the parts does not vary. As a result, setup times are probably best measured dynamically via computer solution. Measuring setup times directly using computer simulation also requires a great deal of data, most importantly sequence dependent setup times for each machine for all part type pairs processed by a given machine. Because it is often not practical to collect this data, setup times in the near future will most likely be measured indirectly. For example, assumptions about setup time reductions can be built into computer simulation models (see Flynn and Jacobs [3]). Assessing the investment in new equipment is perhaps the easiest CM design objective to measure. A direct static measure is to simply sum over all machine types the number of machines of the type needed times the cost of the machine. Assessing machine utilization levels can be accomplished statically or dynamically. Statically, ratios of capacity needed to capacity available have long been used. Likewise, resource utilization measures are among the most common measures employed in simulation analysis. Caution should be exercised, however, to not overly emphasize the machine utilization objective. For example, assuming adequate machine capacity, a shop's performance should be considered improved if its lead times and work-in-process levels are reduced regardless of the level of machine utilization. Likewise, a shop's performance should be considered improved whenever quality improves and setup times are reduced regardless of the level of machine utilization. Another important consideration related to machine utilization is how the machines are weighted. Average machine utilization with all machines weighted equally may frequently not be very appropriate because in most plants only a few machines are critical. Quality is perhaps the most difficult CM design objective to measure. Presently, only potential improvements in quality can be indirectly measured by considering the
162
relationship of quality to other CM design objectives. For example, reductions in the number of intercellular transfers permits holding workers more accountable for product quality. Similarly, ensuring that machines are not overutilized facilitates higher quality because there is time available for preventive maintenance. Finally, reductions in work-in-process levels and lead times facilitate uncovering hidden problems in the production process and permits the discovery of quality problems sooner so that corrective action can be initiated. Thus, assessing the extent of quality improvements can be approximated by assessing the achievement of other CM design objectives. Finally, assessing work-in-process levels and lead times is best accomplished dynamically via computer simulation. Work-in-process levels can be measured as a time persistent variable and lead times as a simple average of individual lead times over all parts. Analytical models that rely upon statistically-based queueing theory require assumptions about the frequency distribution of arrivals and service times and are too often not easily applicable or impossible to apply to complex situations.
5. Summary and Future Research Choosing a measure of performance is part of the modeling of any production process and is ultimately up to the management's stated objectives. If appropriate value-based objectives of profit, revenue, or cost are available to judge the quality of CM configurations then they should be utilized. However, accurate data for such objectives are difficult to obtain at any point in time and especially difficult to obtain prior to implementing CM. In order to compare competing CM configuration proposals we must often judge them by how aligned they are with the basic objectives of CM. Most of the performance measures in this article may be rationally utilized to gauge a CM configuration depending upon the situation encountered and the managerially-stated objectives and in this respect there is not a truly dominant measure. If, however, no guidance is provided regarding the desired outcomes of converting to CM, other than the basic tenets of CM, then the WGF measure appears to be one of the better choices to incorporate for making static decisions because a tradeoff is made between having exceptional elements outside of the cells and the voids in the cells. WGF is even a better measure if in the formula PE can be replaced by APIO. An analyst may also want to separately consider MU, DV, APIO (or OSPE, VWPE, or PE if the data for APIO is not available), TBE, CLM, GE, and GM but the properties and weaknesses of several of these measures should not be overlooked. A large amount of research has been devoted to the development of cell formation procedures. Recently, a few studies have focused on comparing these cell formation procedures. The purpose of this paper is to review solution quality performance measures available and to offer a framework for comparing alternate cell formation procedures. Before useful comparisons of cell formation procedures can be made, meaningful performance measures must be developed. For a performance to be meaningful it should be related to one or more of the design
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objectives associated with CM. The following nine design objectives associated with CM were identified: 1) minimize intercellular transfers, 2) minimize machine setup time, 3) minimize the investment in new equipment, 4) maintain acceptable machine utilization levels, 5) improve quality, 6) reduce work-in-process, 7) reduce production lead times, 8) maintain on-time delivery performance, and 9) increase job satisfaction. Additionally, performance measures were classified as direct or surrogate, and as static or dynamic. In terms of assessing intercellular transfers, static and dynamic direct measures are available. A static measure may be more appropriate when product volumes are known and the product mix is relatively stable. Otherwise, computer simulation should be used to directly measure intercellular transfers. For assessing setup times, machine utilization, work-in-process levels, and lead times dynamic measures may often perform best. Because of the large amount of data required, setup times probably can only be practically measured indirectly, while the other three measures can be easily assessed directly. Measuring the investment required for new equipment can be measured directly via static measures. Finally, at present, the recommended approach for assessing potential quality improvements is to combine several performance measures such as machine utilization, work-in-process levels, lead times, and intercellular transfers. Combining these measures may often provide an indication of how much the manufacturing environment will support and enhance quality improvement activities.
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