Cellular manufacturing: A taxonomic review framework

Journal of Manufacturing Systems Volume 13/No. 3

Cellular Manufacturing: A Taxonomic Review Framework O. Felix Offodile, Abraham Mehrez, and John Grznar, Kent State University, Kent, Ohio

Abstract

technology, their operations are referred to as cellular manufacturing (CM), a term which is often used interchangeably with group technology. The purpose of this paper is to provide a comprehensive taxonomic review of the cellular manufacturing literature. Traditional group technology literature (see Flynn and Jacobs ~ and their references) identifies two main steps concerned with GT and CM. First there is the classification and coding method, which uses codes to identify similarities between parts and resulting part families. The second approach employs the concept of part routing to guide the physical rearrangement of the production facility. We concentrate on this approach in our review. We give a general background of group technology and cellular manufacturing, and we present the taxonomic framework used to review the machinepart group formation models in cellular manufacturing. The taxonomic framework presented here is based on Mehrez, Rabinowitz, and Reisman's 2 five-level conceptual scheme for knowledge representation. The first level of the scheme is the reality framework and deals with problem definition and the relevant assumptions employed. We use this framework in the section on the group technology assumption domain. Level 3 of the scheme is the model framework and deals with the characteristics of the models. We use this framework to present cellular manufacturing model characteristics. The last level in the scheme is the statements framework, which we use to present the main properties and principal results of each CM model. Levels 2 and 4 of the authors' scheme deal with the logical frameworks of transforming reality to models and models to statements. These levels are beyond the scope of this paper and will not be presented, primarily because they are designed for model

The purpose of this paper is to employ a taxonomic framework for a comprehensive review of cellular manufacturing systems. Three classes of machine-part grouping techniques have been identified in cellular manufacturing: visual inspection, part coding and classification, and analysis of the production process. For this review, we concentrate on the latter approach and classify it by assumptions, characteristics, and main properties and results. A comprehensive review and discussions of various models are provided. Model assumptions and characteristics are summarized using a tabular framework. Finally, the paper provides directions for future research.

Keyvvords: Cellular Manufacturing, Group Technology

Introduction The group technology (GT) problem is that of classifying parts and machines into part families and machine cells for efficient production based on certain measurable commonalities. For example, by grouping similar parts one can take advantage of their similarities in design and manufacture. A part family may share the same setup, processing, routing, and so on, and may approach economies of scale of mass production. Similarly, by grouping machines together, intercellular travels can be reduced, thereby minimizing and in some cases eliminating material handling costs. Furthermore, reductions in setup time, manufacturing lead time, design variety, and work-in-process inventory can be achieved. The GT problem is therefore that of problem solving through the determination of appropriate family membership for the entities to be grouped. When two or more machines or machine centers are grouped together to form machine cells used in the manufacture of part families in group

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families using some classification scheme. Perhaps due to the design orientation and proprietary nature of most coding systems, PCA-based systems are not popularly known in the research literature for GT or CM. Kaparthi and Suresh 6 suggest, though, that the unpopularity of PCA-based systems might be due to their labor intensiveness, and thus they suggest automating the coding part of the method. Kusiak 7 echoes this belief by suggesting that the problem could be due to the expense and difficulty involved in implementing coding and classification systems; however, Hyer and Wemmerlov 8 and Levuhis 9 tend to suggest otherwise. For example, Hyer and Wemmerlov report that 62% of companies surveyed indicated that they used coding and classification systems; therefore the problem of finding groups with the weighted codes of the PCA could be one of the possible reasons for the slow development of

builders. As a survey piece, we are not privy to the logical frameworks the authors of the reviewed papers used in their models, and extracting them from the models is at best arduous. Also, we have found the three levels (levels 1, 3, and 5 of the referenced framework) used in our review to be adequate in satisfying this paper's objective. Furthermore, the complete five-level framework is particularly superior to other approaches because it presents a systematic procedure for knowledge representation and embodies five of the major components of problem solving: assumptions, problem transformation, model characteristics, model transformation, and model results. General discussion, conclusions, and directions for future research are included at the end of the paper.

Background Three methods for identifying machine-part families (Figure 1) have been identified in the literature 3 and are the basis of our taxonomic review framework. The first of these taxonomic groups is the ocular or visual method, and the other two methods are part characteristic based and production process based methods. Hybrid

Visual Inspection Based Method The visual inspection based method of group technology forms part families by studying part geometries and arranging them into families. The analyst simply reviews the parts and, based on experience, determines appropriate groups. This approach, although relatively inexpensive, is prone to error, relies very heavily on the expertise and experience of the user, and is rarely used in practice except with a fairly small number of parts. Burbidge a reports that this method has been used for solving large (2000 components) machine-part grouping problems; however, this assertion may be grossly over-optimistic. 5 Part Characteristic Based Systems The second method of group formation uses part characteristics to identify part families. Known as part coding and classification analysis (PCA), the approach uses a coding system to assign numerical weights to parts characteristics and identifies part

Figure 1 Taxonomic Review Framework for Group Technology

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grouping algorithms using PCA-based systems. Offodile ~° suggests a framework for converting the weighted codes of the PCA into a similarity measure that is amenable to many of the grouping algorithms in the literature as a means of alleviating this problem. PCA-based systems are traditionally design oriented or shape based and are therefore ideal for component variety reduction. Some PCA-based systems, for example in Opitz, 1~ incorporate production-based codes as supplemental codes and can therefore be used for production planning. Part characteristic based systems were the primary tool of GT in the 1960s and 1970s. Part characteristic based systems form part families by first coding the parts with respect to their design-based features, such as shapes, sizes, and tolerances. Several coding systems were then developed at the time, for example, the Opitz system. This concept of using design features for the purpose of describing and grouping similar parts was introduced by Mitrofanov. 12 Opitz, Eversheim, and Wiendhal ~3 and Opitz and Wiendhal ~4 later extended the idea to production cells, and Opitz developed a comprehensive coding and classification system for workpieces. Since these pioneering works, several other part coding and classification systems have been developed to facilitate part grouping: SAGT ls'16 and M I C L A S S . 17-19 PCA-based systems fall into one of three categories: monocode, polycode, or hybrid. For an extensive discussion on these categories and a comprehensive review of some coding systems, see Hyer and Wemmerlov 2° and Gallagher, Southern, and Knight. z~ Our objective here is not to provide a detailed review of the PCA literature, but to recognize its place in and contributions to the cellular manufacturing literature.

developer, Burbidge, 4 and after the initial data collection phase involves a series of three sequential procedures, as follows:

1. Creation of Packs. All parts in the analysis are sorted to identify those that require identical processing, and all parts are given a "pack number." In more recent applications of PFA, this step has been dropped and individual parts are used to create the PFA chart. 2. Production Flow Analysis Chart. A machinepart incidence matrix is determined by studying the manufacturing sequence and workloads for component parts and machines. The relationship C~/-- 1 if machine i processes component j, and 0 otherwise, is often used to record these sequences and workloads, resulting in a machine-part incidence matrix of 1's and O's. 3. Analysis. The PFA chart is rearranged to identify the groups of machines and parts. The rearrangement process is quite subjective and difficult, especially for fairly large machine-part matrices. Furthermore, it provides no basis for evaluating the goodness of the groups; however, it is the most crucial part of the grouping process. Some systematic methods for analyzing the machinepart incidence matrix have been developed to alleviate this problem. The most notable of these are the similarity coefficient based, array-based, and mathematical-based methods, which will be discussed in detail later in the paper. As stated elsewhere in this paper, our objective is to provide a review of the machine-part grouping literature in cellular manufacturing via the PFA branch of the proposed taxonomy in Figure 1. We would be remiss if we failed to mention previous efforts along these lines. Four review papers on group technology and cellular manufacturing (King and Nakornchai, 22 Mosier and Taube, z3 Wemmerlov and Hyer, 24 and Chu 2s) are reported in the literature. King and Nakornchai provide a comprehensive review of the existing machine-component group formation literature at the time. Further, they extend the rank order clustering (ROC) algorithm 26'27 to deal with the problem of storage and computational complexity. Mosier and Taube partition the literature into four research thrusts: part coding and classification, machine-part grouping, scheduling in a group technology environment, and group technol-

Production Process Based Systems Finally, the third method of group technology-and a method which has enjoyed the greatest amount of research attention--is production process based analysis. The objective in the productionbased approach to GT is to identify and group parts that share common processing requirements. Most production-oriented systems or production flow analysis (PFA) systems use route sheets to record the relationship between parts and the machines that process them. The method was popularized by its

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is known and available. We list the following assumptions to facilitate our discussion. Whenever appropriate, assumptions are further subdivided into a more detailed classification. Although we did not develop a tabular summary of the assumptions domain because they are not explicitly employed in some models, we list some assumptions here for reference because they are very important to any problem formulation and solution procedure. They are listed and suggested here for incorporation into future machine-part formation models for cellular manufacturing.

ogy implementation. Wemmerlov and Hyer present an impressive bibliography of the machine-part family formation problem in cellular manufacturing. In all, more than 70 group technology related works are cited and partitioned according to how the machine and/or part families were formed. Chu also provides a comprehensive bibliography of cellular manufacturing and partitions the literature into design and production-oriented systems. The latter classification is further partitioned into arraybased, hierarchical, nonhierarchical, mathematical, graph-theoretic, and heuristic approaches. The author also provides a classification based on the clustering criteria and the measure of performance. Neither King and Nakornchai nor Mosier and Taube provide an explicit or extensive taxonomic review framework. On the other hand, Wemmerlov and Hyer and Chu provide a taxonomic framework in one form or another; however, neither they nor the previous authors provide our three-level approach to knowledge representation. Finally, the accelerated growth of the literature in the past decade suggests the need for a more comprehensive review. Although all the reviews have some elements of taxonomy, they are deficient in either their timeliness or the extensiveness of their review of the literature. Chu's review is timely, but does not provide an extensive review of each of the papers listed in its taxonomic framework. King and Nakornchai extensively review each of the articles cited, but their work is no longer timely in view of the amount of research reported on the subject subsequently.

1. Layout Type: Straight Line, Rectangular, Square, Circular, Random 2. Setup Time: 2.1 Nonexistent 2.2 Exists: Sequence Dependent or Independent 3. Machine Clustering Requirements: 3.1 Mutually Separable 3.2 Partially Separable: Machine Duplication, Part Subcontraction, Both 4. Nature of Demand: Known (Time Independent) Unknown (Time Dependent) 5. Planning Horizon: Finite, Infinite 6. Batch Size: Constant & Equal per Machine Cell, Variable 7. Production Flow Policy: Backtracking Allowed, No Backtracking 8. Number of Machines per Machine Type: One, More than One 9. Machining (Operation) Times: Known (Deterministic), Probabilistic Arrangement of machines in a group technology cell is usually thought to be in the circular (arc) layout; however, several other layout configurations, such as line, rectangular, square, and even random, have been identified. 2s'29 Several advantages have been attributed to the circular topology, including the ability of the human operator to supervise more machines because walking distances between machines are minimal. Fazakerley 3° lists other social benefits of group technology and cellular manufacturing, including worker flexibility, importance of social group, reduced frustration, and improved status and job security. Most of the models reviewed in this paper deal with the problem of finding appropriate machinepart groups for a production facility and do not

Assumption Domain Level 1 of the five-level framework for knowledge representation deals with the reality framework or the analyst's perception of the problem to be solved. This perception leads to assumptions that the analyst uses to formulate and ultimately solve the problem. Some assumptions are general to cellular manufacturing, while others are specific to the particular model. In general, assumptions are divided along the structural lines of the models. For example, mathematical models will often assume that a given number of groups are to be formed and that the number of parts and machines to be grouped

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consider actual manufacturing process activities, such as scheduling and lot sizing. Consequently, setup times are in only a few of the n~dels. 31 Setup times can be sequence dependent or independent. In the former, processing order of the part is important, while in the latter it is not. When machine-part clusters are mutually separable, a machine (part) can belong to one and only one machine (part) group (family), unlike in partial separability where they could belong to more than one group (family). Partial separability (exceptional elements) results in inefficient operation of the cellular manufacturing system, and to improve cell efficiency machines are either duplicated, parts subcontracted, or both. Exceptional elements arise when large numbers of parts require processing on a machine or when parts require processing in more than one cell. In the former situation, capacity is limited, and to release the congestion more copies of the bottleneck machines are provided (duplication). Some models attain homogeneity in clusters by either subcontracting the exceptional parts or by forcing them to belong to one of the existing groups; 22'32'33 however, forcing a part to belong to a family with which it has little in common results in an inefficient operation of the machine cells. Many of the machine-part grouping problems deal with situations where demand for the components is assumed known and constant over the planning horizon. One exception to this assumption is the probabilistic model presented by Seifoddini. 34 When schedules are considered in determining production requirements of the machine groups, a planning horizon is often assumed and can be discrete or infinite depending on whether or not the schedule is repeatable. Production batch size for components can be either constant across all machine Cells or variable with respect to the technological requirements of the components. The former assumption is more often used than the latter and is a policy that is espoused in a just-in-time production environment. Often when a part leaves a machine cell it does so in a flow process manner with little or no chance of coming back to the cell. Within the various cells, however, the processing sequence of the parts determines their schedule, especially when a model assumes that intracellular material handling costs are negligible. 3s

It is generally assumed that the grouping problem involves m (-- 2) machines with one or more copies per machine type. The machining time of each machine for each operation on each part is also assumed known or varies probabilistically.

Model Characteristics The model framework (level 3) deals with the relevant characteristics of the models, such as their structures, properties, objectives, and constraints. We use this framework to present the formulation methodologies, problem data structures, clustering problems, solution approaches, decision variables, objective functions, and constraints for machinepart grouping in cellular manufacturing. We benchmark this domain as follows: 1. Model Structure: Matrix Formulation: Similarity Coefficient-Based, Array(Sorting) Based Math Programming: Integer Programming, Linear Programming, Dynamic Programming Graph Theory: Bipartite, Other Others: SiMulation, Expert Systems, Neural Networks, Fuzzy Sets 2. Problem Data Structure: Binary, Weighted, Either, Fractional 3. Clustering Problem: Part, Machine, Concurrent Formation of Groups 4. Solution Approach: 4.1 Heuristics 4.2 Hierarchical: Single Linkage, Average Linkage, Complete Linkage, Density Seeking, More than two methods 4.3 Nonhierarchical 4.4 Array-Based: Rank Ordering, Direct Clustering, Bond Energy, Cluster Identification, Occupancy Value 4.5 Assignment Model 4.6 Others: Linear Programming, Goal Programming, Graph Partitioning, Simulated Annealing, Fuzzy Mathematics (c-mean), Expert Systems, Neural Networks 5. Decision Variables: 5.1 Number of Machines of a given type to be assigned to a given cell 5.2 Number of Parts or Machines assigned to any given cell

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where

5.3 Number of Operations or Tool copies per part per group 5.4 Batch Size 6. Objectives: 6.1 Minimize Intercellular Travels 6.2 Minimize Intracellular Travels 6.3 Minimize Setup Time or Maximize Machine Scheduling Flexibility 6.4 Maximize Similarity (Minimize Dissimilarity) or Compatibility Measure 6.5 Minimize total production cost 6.6 Minimize exceptional elements costs (Subcontracting, Duplication, or Both) 6.7 Minimize Machine Idle Time 6.8 Maximize Machine Utilization 7. Constraints: 7.1 Number of Groups (cells or part families) 7.2 Number of parts per group 7.3 Number of machines per group 7.4 Machine capacity 7.5 Each Part, Machine, or Both belongs to one part family or machine group 7.6 Annual Operating Budget 7.7 Tool or Processing requirement of Parts Model

--- measure of similarity between machine m 1 and machine m z = number of parts processed by BM~M~ machines m~ and mz B g l m 2 = number of parts processed by only machine m 1 = number of parts processed by only B m iMz machine m 2 S mlm

In Eq. (7) uppercase subscripts indicate a 1 in the machine-part incidence matrix, while lowercase subscripts indicate a 0. The definition of similarity is such that O<--S,,,,m, < -- 1 inclusive, for minimum and maximum similarities, respectively. The result is an upper or lower triangular matrix of similarity measures between any pair of machines (parts). Appropriate machine-part groups are then formed using a grouping algorithm such as the single linkage, average linkage, and direct linkage algorithms. For I parts or machines, the resulting groups are said to be mutually separable if for any group G i C_ I and Gj _C I, i :k j, and elements of G i q~ Gj, partially separable otherwise. Many real-life problems are such that mutual separability might not be possible, resulting in exceptional elements of parts or bottleneck machines. Array-based methods group machines and parts without the benefit of a similarity measure. Rows and columns of the incidence matrix are rearranged until a diagonal pattern of mutually or partially separable clusters emerges. All array-based methods use the 0-1 machine-part incidence matrix as input. They have the common characteristic of forming machine-part groups simultaneously. The first known array-based algorithm specifically designed for group technology is the rank order clustering (ROC) algorithm developed by King. 26,27 Later, King and Nakornchai 22 extended the model to be able to handle large problems. Other models in this g r o u p i n c l u d e C h a n d r a s e k h a r a n and Rajagopalan, 39 Chan and Milner, 40 Kusiak, 41 and Kusiak and Chow, 32 who have provided some modifications to the basic model. Mathematical programming formulations are usually integer, linear, and dynamic in structure; however, 0-1 integer programming formulations tend to be used the most. Several authors have formulated the machine-component grouping

Structure

Structures of the grouping problem have been identified as either matrix, mathematical programming, or graph-theoretic formulations. 7'36 In a matrix formulation of the similarity type, the machine-part incidence matrix is converted to a table of machine (part) similarities, and machine (part) groups (families) are determined with the aid of a cluster-analytic algorithm. Alternatively, an algorithm that operates directly on the original machine-part incidence matrix is used to determine machine-part clusters concurrently in an arraybased mode. The objective is usually to maximize the similarity between the machines (parts) in the same group. As noted earlier, many models adopt the similarity coefficient approach for which the Jaccard similarity measure a7 has been used most often and has been proven to be most effective, as This measure is of the form given in Eq. (7) as defined by McAuley 2a for the case of two machines.

Sm,m, =

BM~M2

(7)

BM,M~ + BM,m, + Bm,M~

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Journal of Manufacturing Systems Volume 13/No. 3

problem as a graph decomposition problem using graph and network theory. Owing to the combinatorial nature of graph theory, some of the models are solved using heuristic procedures. Rajagopalan and Batra; 3s Vannelli and Kumar; 4z and Kumar, Kusiak, and Vannelli 43 present some of the better known heuristic graph-theoretic models for solving the m a c h i n e - p a r t g r o u p i n g p r o b l e m . Latest advancements in computer technology have bolstered the use of simulation, expert systems, neural networks, and fuzzy set theory for modeling the machine-part grouping problem. These tools are clustered into the group " o t h e r " in our taxonomy.

ming formulations), any of the hierarchical and nonhierarchical approaches, array-based methods, optimization techniques, or other methods. Hierarchical solution approaches often employ single, average, or complete-linkage technologies. Single linkage implies that machines or machine groups (component or component groups) are admitted into another group based on the maximum similarity between any pair of machines (components) in the two groups, as opposed to their averages. Unfortunately, this method may sometimes admit machines (components) to families even when some elements of the group are more dissimilar than they are similar. Furthermore, some models have been solved employing cluster-analytic algorithms, such as rank order clustering, or any of the well-known mathematical tools, such as assignment model and linear programming.

Problem Data Structure The GT problem involves determining appropriate machine cells and part families from a binary or weighted machine-part incidence matrix. Binary machine-part incidence matrices are usually qualitative and are used to represent the state of a part on a machine. The state is 1 if the part is processed on the machine and 0 otherwise. On the other hand, weighted matrices measure both the state of the part and the magnitude of that state. An example of that measure could be the volume of the parts to be produced, setup time requirements, and so on. It can therefore assume any positive value or 0. A model that employs the weighted matrix approach to form part families in a coding and classification environment is presented by Offodile. lO Some models are general, however, and can be used for either the binary or weighted machine-part matrices. A fractional machine-part incidence matrix based on fuzzy mathematics is suggested by Chu and Hayya. a4

Decision Variables Decision variables represent actions or policy decisions concerning the system. These include the number of machine types that can be assigned to any group, parts or machines assigned to any group, and the number of parts per family.

Objectives Several objectives of group technology are cited in the literature. Ballakur and Steude148 list nine such objectives; however, the most common one seems to be minimization of intercellular material handling costs. This follows from the fact that intercellular material handling is a result of inefficiencies in forming the machine cells. Parts therefore belong to more than one cell (exceptional parts). Some other objectives include setup time minimization; maximization (minimization) of a similarity (dissimilarity) measure; minimization of total production cost, cost of exceptional elements, and machine idle times; and maximization of machine utilization. Kusiak 41 and Seifoddini and W o l f e 49 formulate models that minimize intercellular travels, while a model to maximize the sum of the compatibility index between all machines and parts is presented by Gunasingh and Lashkari. 5°'51 A drawback of many GT models is that they do not provide a means for evaluating the goodness of the resulting solution, especially on an absolute basis. An earlier attempt at providing a measure of

Clustering Problem While some m o d e l s address part family formation 1°'aLas or machine cell formation 28'46'a7 problems only, others address concurrent formation of machine cells and part families. 22"26'27"4° When the part (machine) family (cell) problem is solved, it is a simple matter to identify the machines (parts) that process them and assign them to the appropriate part (machine) family (cell).

Solution Approach Solution of the problem can be accomplished using heuristics (especially for 0-1 integer program-

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The similarity coefficient procedure is used to determine the machine-part groups for a binary machine-part incidence matrix. The model assumes a within-cell random, straight, rectangular, or square machine layout without diagonal movements. The author defines similarity as in Eq. (7) and uses Sneath's (see Sokal and Sheath 37) hierarchical single-linkage cluster-analytic algorithm to identify machine (component) clusters. As a means for evaluating various clusters that may result from the procedure, the author uses the objective of minimizing the number of intercell journeys and overall machine occupancy under the assumption of a sequence-independent production environment and without regard to machine loadings. The author further suggests that intercell journeys can be minimized by duplicating bottleneck machines, and the grouping problem may be constrained to limited cell size by, for example, space or logistics. An application of this algorithm to a practical problem is presented by Carrie. s

group efficiency is provided by Chandrasekharan and Rajagopalan. s2 The problem with the grouping efficiency measure is that it has a low discriminating power. This and other shortcomings of the measure are analyzed by Kumar and Chandrasekharan, s3 who propose an improvement called grouping efficacy. Constraints The most commonly stated constraint for the GT problem is the number of cells desired, and this is often specified as a parameter of the problem. 7 Both technological constraints, such as machine capacity and logistics, might constrain the number of parts or machines that can physically belong to a group. Also, budgetary constraints and processing requirements for the parts might further constrain the grouping problem. In some problems, especially where intercellular movements are not allowed, additional constraints are required to force the parts to be grouped and processed in one cell only. A summary of these characteristics is presented in Table 1 for the models reviewed in the paper, with the symbols corresponding to the first letter of the characteristic (X used if there are no choices in the characteristic). The main properties of the various models are further given in the next section.

Table 1 Model Characteristics

Main Output and Principal Results Main outputs and principal results of the models reviewed are summarized using level 5 (statements framework) of the knowledge representation framework. Specifically for each model, we follow our taxonomic framework and report on the assumptions and characteristics of the models as described earlier. The presentation is made in a critical review and chronological order format (within each taxonomic group) to aid in tracing developments in the field. The objective in this framework is to provide a descriptive summary of the main outputs and results of the models; therefore, we did not find it appropriate to employ a tabular summary for this purpose.

Y~

72

72

84

84

8S

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Model Structu~

S

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O

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B

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B

E

E

W

E

E

B

B

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B

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W

Cluster Problem

M

C

C

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C

M

C

C

C

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M

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X

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74.~ 75

75

80 80A 82

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P

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X

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X

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X

X

X

X

X F

S

S

S

S

S

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X

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X

X

P

P

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Matrix Formulation McAuley 2s first suggested a systematic approach to machine-part grouping in cellular manufacturing.

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Journal of Manufacturing Systems Volume

13/No. 3

McCormick, Schweitzer, and White s4 present a quadratic assignment-based cluster-analytic model called the bond energy algorithm (BEA). The authors define the bond strength between any two adjacent elements in a machine-part incidence matrix as their product, and the bond energy as the sum of the bond strengths. The objective is therefore to maximize the bond energy (similarity) of the matrix, resulting in a block diagonal matrix, if one exists. The BEA is a partial enumeration algorithm and can be applied to the machine-part incidence matrix of the 0-1 or weighted type. The final machine-part groups are independent of the initial ordering of the matrix and do not require that the number of groups and their sizes be specified a priori. Machines (parts) with high bond energy are then clustered using a type of permutation algorithm. De Witte ss defines three similarity coefficients that are based on the interdependencies between machines. Three classes of machines--primary, secondary, and tertiary--were defined based on whether only one of the machines is available to be

assigned to one group, whether several of the machines are available to be assigned to several cells, or whether enough of the machines are available to be assigned to all ceils. Three types of machine cells are therefore possible and are clustered using the graph-theoretic approach. King 26'z7 develops the ROC algorithm, which is the better known of the array-based clustering algorithms. The algorithm first assigns binary weights to each row and column of the machine-part incidence matrix. The binary weights for each row and column of the matrix are then converted to their decimal equivalents and alternately rearranged in decreasing order of magnitude until a diagonal pattern of machine-part clusters emerges (if it exists). Unlike some other algorithms, the ROC algorithm does not require that the number of groups be specified a priori. The ROC algorithm can be used to solve cluster-analytic problems with positive elements of nearly any size and converges very quickly, requiring only two iterations. A problem inherent in the algorithm is that the pattern of

Table 1 continued

Table 1 continued

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Journal of Manufacturing Systems Volume 13/No. 3

machine-part clusters may not be mutually separable, resulting in exceptional elements and bottleneck machines. The author suggests temporarily ignoring exceptional elements and bottleneck machines and reinstating them after the algorithm converges. The ROC algorithm has a computational complexity of a cubic order that makes it limited to only problems of moderate sizes. Furthermore, for relatively large problems, the binary weights increase, creating data storage and computer memory problems. Waghodekar and Sahu s6 show that the solution is also sensitive to the initial structure of the machine-part incidence matrix; however, the solution is precise and quick-converging for wellstructured problems. King and Nakornchai 2z extend the basic ROC model to improve its computational efficiency. The new algorithm (ROC2) simultaneously sorts several rows and columns, thus enabling it to solve problems of much larger dimensions. It employs a much quicker sorting procedure than the ROC algorithm.

Chan and Milner 4° present the direct clustering algorithm (DCA) in which columns and rows are first arranged in their decreasing and increasing orders of magnitude determined by the number of l's in the machine-part incidence matrix. Columns and rows of the transformed matrix are further rearranged with the rows having left-most entries of 1 moved to the top of the matrix and columns having top-most entries moved to the left. The algorithm converges quickly and uses a progressive procedure to deal with exceptional elements and bottleneck machines. Different alternatives, such as redesigning parts, duplicating machines, forming independent clusters, and allowing intercell transfers, were suggested for dealing with cases of exceptional elements. Waghodekar and Sahu s6 propose a three-phase heuristic clustering algorithm that uses a producttype similarity coefficient measure. The first or additive similarity measure is similar to Eq. (7). The second, which the authors termed the product sim-

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Journal of Manufacturing Systems Volume 13/No. 3

ilarity measure, defined similarity between any pair of components as the ratio of the square of the number of components using the two machines to the product of the number of components using either machine. The third similarity measure was defined as the ratio of the square of the number of components using the two machines to the product of the number of common components processed in either machine, with respect to the remaining machines. The objective of the algorithm is to minimize the number of bottleneck machines without regard to chaining, space, and intracellular layout limitations that might exist when too many machines are clustered together. The product similarity measure was used to test a number of large problems and was shown to yield fewer numbers of exceptional elements compared to ROC 26'27 and ROC2. zz The algorithm has efficient computational techniques that enable it to find the machine-part groups in one run. More runs may be necessary if bottleneck machines exist. A matrix of m × m (m = number of machines) is used for the computation, unlike the m × n technique of ROC. Because n (number of parts) is usually much larger than m, the algorithm does not suffer from storage and memory problems of ROC. Finally, the algorithm can place machines close to one another by criterion other than their similarity or dissimilarity. Mosier and Taube a6 define two similarity measures for group technology. The first is a weighted definition of McAuley's z8 similarity measure to "incorporate the relative importance of each part." Called the additive similarity coefficient (ASC), the machine-part incidence matrix is converted to one that includes the volume of each part processed by the corresponding machine, such that the 0-1 incidence matrix is increased by a multiplier W " ) lm -- volume W of part i processed on machine 1m . The second definition of similarity is multiplicative in nature and is termed the multiplicative similarity coefficient (MSC). In either definition, the similarity coefficient takes on a value between -1 and + 1 for minimum and maximum similarities, respectively. The authors solved several test problems with McAuley's similarity measure, King's ROC, and then ASC and MSC and found that the methods (ASC and MSC) fare well against the other methods. Stanfel s7 deals with a deterministic clustering situation in which a limit is set on the maximum and

minimum cell sizes. The algorithm is divisive as it initially assumes that all machines belong to the same cell. The machines are then removed from this group based on cell size constraints, with the machine having the most parts to be processed removed first. The objective of the algorithm is to minimize the sum of intercell and extraneous machine transactions. The former is a function of exceptional components needing machines in another cell, while the latter is a function of within-cell machine idleness. In either situation the cells are imperfect, and the smaller the imperfections the better. Duplication of bottleneck machines is therefore encouraged. Chadrasekharan and Rajagopalan 39 provide an extension to the ROC algorithm. Called modified rank order clustering (MODROC), it is a combination of array and similarity coefficient based methods. The algorithm employs the so-called block-and-slice approach in which the columns corresponding to the largest block of l's on the top left-hand corner of a two-iteration solution of an ROC algorithm are sliced away. The ROC algorithm is progressively applied to the modified matrix until all columns are grouped. A measure of association is then defined for the resulting cells of part families, which are then hierarchically clustered using single-linkage technology with the objective of maximizing their similarities. It is claimed that the algorithm eliminates the deficiencies of ROC, such as the dependency of the solution on the initial form of the machine-part matrix. Further, an objective procedure was used to determine the bottleneck machines. Dutta et al. s8 present an external criterion based heuristic model for forming part families for flexible manufacturing systems. To facilitate design and processing information retrieval, parts to be grouped are coded by tooling requirements and processing methods. A number of dissimilarity measures were defined and used to reallocate parts to families based on their ability to reduce dissimilarity within part families. Order of reallocation is such that the most similar (least dissimilar) parts or groups are grouped first. The objective is to group parts with identical processing and tooling requirements together, thereby minimizing material handling and processing costs. Seifoddini and Wolfe s9 redefine Eq. (7) using a bit-level data storage technique, which they claim

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crossing out the rows with the selected machines and decreasing the number of machines for the remaining components by 1. The process continues until a diagonal pattern of mutually separable clusters results, if one exists. Otherwise, the method has the ability to directly indicate the existence of exceptional elements and bottleneck machines. Kusiak and C h o w 32'33 present cluster identification and cost analysis algorithms to solve the machine-part grouping problem. In the cluster identification algorithm (CIA), the machine-part incidence matrix is transformed into machine-part clusters using a form of cutting algorithm. The algorithm does not provide mechanisms for dealing with exceptional parts, however. This deficiency is corrected by the cost analysis algorithm (CAA). The machine-part incidence matrix includes subcontracting costs, which help the decision maker make intelligent decisions about which parts should be subcontracted. The algorithm basically forms machine clusters starting with parts that have maximum subcontracting costs. The result is that exceptional parts will have low subcontracting costs and will be easier to be subcontracted. The objective of the algorithm is therefore to minimize the cost of subcontracting (exceptional elements), subject to a limited cell size. Seifoddini and W o l f e 49 rise the average linkage clustering algorithm to generate alternative solutions to a machine-part clustering problem. In an attempt to find the best among alternative solutions, a material handling cost model was developed and used to evaluate each machine cell. Large cells result in less intercellular journeys, but may lead to higher incidence of intracellular journeys. On the other hand, small cells result in less intracellular journeys, but higher intercellular journeys might result. The size of the cells is a function of the threshold value. A CRAFF algorithm was then used to evaluate the cost of intra and intercellular journeys for each cell. The CRAFT input were the from-to chart, move-cost chart, and an initial layout for each cell. Mosier 62 define similarity coefficients that consider the relative importance of the parts in the grouping problem. They can therefore deal with values other than the 0-1 machine-part incidence matrix. The similarity coefficients and several others from the literature were then evaluated using

reduces the computational effort and data storage requirement. The authors' model employs the average linkage clustering (ALC) algorithm that minimizes the chaining problem of the single linkage clustering (SLC) algorithm. The model also specifically deals with bottleneck machines. With the objective of minimizing intercellular travels, bottleneck machines are duplicated with those creating the most intercellular journeys first. The process is continued for all bottleneck machines until no more machines generate intercellular journeys as specified by a threshold value. Askin and Subramanian 6° propose a three-stage heuristic procedure that considers work-in-process and cycle inventories, setup, intracell material handling, variable processing, and fixed machine costs. A combination of production flow analysis and part coding and classification analysis, the model assumes an infinite planning horizon and a constant mean demand time. Furthermore, shortages and lot splitting are not allowed, and each part belongs to an initial coding family. Each group is evaluated based on six cost parameters: setup, variable production, production cycle inventory, workin-process, material handling, and fixed machine costs. In the first stage of the procedure, a binary clustering algorithm (for example, King's 26'27 ROC) is used to process the machine-part incidence matrix. The second stage then performs a pairwise economic comparison of adjacent groups. The groups are combined if the cost of the resulting group is less than the sum of the cost of the two individual groups. Finally at stage 3, the resulting groups from stage 2 are further combined subject to machine capacity. Khator and Irani's 61 occupancy value (OV) method is designed to eliminate some of the limitations of the ROC and direct clustering algorithms, such as dependency of the final solution on the initial input matrix. The method instead relies on the choice of a good seed machine. It basically rearranges the machine-part incidence matrix based on the amount of machine usage by the parts. The parts with the smallest machine usage value are selected first and moved to the northwest corner of the matrix. Next the machines used for processing the parts are made part of the new matrix. Ties are broken by assigning the part (and its machines) with maximum OV first. The matrix is then updated by

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four clustering algorithms: single linkage, complete linkage, centroid, and Wards methods. Several measures of performance, including the between-cell transportation measure, were used to evaluate the measures of similarity. Results show that McAuley's z8 similarity coefficient measure outperformed the others, while performance of the grouping algorithm depends on the measure of performance used. For example, the complete-linkage algorithm was found to be the best when the measure of performance is minimization of intercell travels. Seifoddini n3 presents a comparative study of two of the most common similarity coefficient based algorithms: single and average linkage clustering algorithms (SLC and ALC). The authors found that although SLC was relatively easy to apply, it has the drawback of forming cells in which some members may be widely removed from others because of single linkage. Referred to as a chaining problem in the literature, it is characterized by a large number of machines per cell, and in many instances some machines do not contribute to the efficiency of the machine cells. Furthermore, SLC generally has more exceptional elements and bottleneck machines resulting in more intercellular travels. ALC overcomes these drawbacks at a cost of more computation time. This is primarily due to its average similarity criterion for a d m i t t i n g candidate machines to cells, which is always at a lower threshold value than maximum criterion of the SLC. Seifoddini64 deals with the problem of duplicating bottleneck machines in cellular manufacturing. The author defines a cost-based performance measure that captures the cost of duplicating bottleneck machines and compares it against the cost savings due to elimination of intercellular travels. If the former cost is lower, the machine is duplicated. As in his earlier efforts, 49 a CRAFT algorithm is used to find a good cell arrangement and its material handling costs. Stanfel 6s presents an integer programming model with a nonlinear objective function that minimizes intercellular material movement under machine loading constraints. Also, a Lagrangian relaxation approach with successive approximations is suggested, as is a set-covering problem. Solution procedures are then provided and discussed for the three formulations. In particular, for a given set of manufacturing cells, a part-type cell assignment

problem is solved, followed by a successive pairwise merging of cells, to optimize the objective function subject to prevailing constraints. The model assumes that the available number of different machine types, maximum number of parts, and maximum number of machine cells are known. The author's computational experience found that the Lagrangian method is the simplest to implement and is most likely to work well in practice. Wei and Kern ~ use a modified version of the similarity measure defined by Kusiak. 7 Rather than compute similarity for the parts, which are usually larger in number than the machines, they compute it for the machines and claim that it improves the converging time for their algorithm. The similarity score measures the relative value of clustering the machines within a given cell, and the algorithm can apparently solve problems even in the presence of exceptional parts. The clustering algorithm is of linear computational complexity and fast, making it easy to model fairly large problems on microcomputers. The algorithm creates the maximum number of machine cells and can be adapted to accommodate different constraints in specific situations. A two-stage sequence-independent heuristic algorithm for machine cell formation that minimizes material handling cost was presented by Harhalakis, Nagi, and Proth. 67 The first stage of the algorithm formulates a machine assignment model based on the cost of material flow. At this stage each of the machines is a cell. Two machine cells are then grouped together with the objective of minimizing their normalized intercell traffic under the constraint that the specified number of machines per cell is not exceeded. Cell aggregations are effected for the two cells, whose normalized intercell traffic is a maximum. In the second stage of the algorithm, intercell material handling costs of the resulting machine cells are maximized. The final cells are then evaluated using various measures of performance, such as global, group, and group technology efficiencies. Seifoddini68 compares the machine-component group analysis to the similarity coefficient method based (SCM) algorithms in terms of the number of bottleneck machines they have in the final solution. The author found that SCM-based models provide more satisfactory solutions in the presence of bottleneck machines than do machine-part group analysis based models. Further, because the machine-

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partitioned into two sets. The first set contains parts requiring two operations or less and are then removed from the problem. Set 2 of the problem, containing the rest of the parts, is then further partitioned into two mutually exclusive groups. One group is used to hold parts that have backtracks in their operational sequence, while the other is used to hold those that do not. Stage 3 of the algorithm identifies parts with identical operational sequences and groups them together. Finally in stage 4, the algorithm computes the similarity measure between the groups found at stage 3. The operation sequences with no backtracks are then merged in their descending order of similarity until an a priori established threshold value is reached. G u 71 presents a two-stage process-based approach for the machine-part grouping problem in group technology. In the first stage, a clusteranalytic algorithm based on processing similarities of the parts is used to form the part families that are used to form the machine cells in the second stage. The resulting machine cells are found using the processing requirements of the part families and the available machines that can provide them. Two 0-1 input requirement matrices are used in the model. One matrix defines the relationship between parts and processes, while the other defines the relationship between machines and processes. The model is different from the many machine-part grouping models in the literature in that it can be used for both classical 0-1 machine-part incidence matrices and those that include machines that can perform more than a single operation on the parts. Gupta 72 evaluates the performance of four of the better known hierarchical clustering algorithms. Specifically, the single, average, complete, and weighted-average linkage clustering algorithms are evaluated with respect to their chaining effects using a production data based measure of similarity to establish the degree of association between the machines. Clusters were analyzed at four different levels of their formation, and the criteria of sizes of the largest and smallest cells and their ratios to the number of machines in the clustering problem were used as measures of chaining severity. The study found that cell sizes are proportional to the clustering stage, increasing with higher levels of clustering. Furthermore, it concluded that the choice of clustering method influences the chaining problem,

component group analysis methods use only the 0-1 machine-component incidence matrix as input, they cannot incorporate such parameters as production volume, operation sequence, and operation processing time. These parameters, however, can be used to measure the desirability of a given bottleneck situation, that is, either to duplicate the machine or not. Machine-component based algorithms are, however, simpler and easier to apply than SCMbased ones. Finally, the SCM models find groups in two stages. They first find machine groups and then corresponding parts are added to the machine groups where they belong. Seifoddini 34 presents a probabilistic demand model for machine cell formation. The model first assigns a probability function to the product mix and machine-part incidence matrix. Several alternative solutions are then generated for all possible machine-component charts. Finally the cost of intercellular material handling is used to evaluate solution alternatives. The solution alternative with the minimum cost is then selected. Tam n9 defines a similarity coefficient that incorporates both machine resource requirements and operations sequences of the parts. The coefficients are actually a measure of dissimilarity (or distance measures) rather than similarity. A density linkage clustering technique, known as the kth-nearest neighbor (KNN) cluster method is then used to find the groups of parts. Actually the algorithm is a singlelinkage technology except that the similarities are determined using a distance measure. Furthermore, the threshold level at which two groups are joined together does not have to be specified a priori. The main advantage of the model is that part families resulting from it allow machines to "interleave between identical operations of different parts," thereby minimizing setup times and intracell travels. Vakharia and Wemmerlov 7° present a four-stage cell formation algorithm that is based on the operational or material flow sequence of the parts. It is basically a similarity coefficient based heuristic that considers processing sequences and machine requirements and demand of the parts. A similarity measure between any pair of parts is defined in terms of their operational sequence with the objective of generating mutually separable clusters with a pure flowline structure. Stage 1 of the algorithm is data collection, while in stage 2 the problem is

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with chaining getting progressively worse in the order of complete, weighted-average, average, and single-linkage technologies. Similar findings were also reported by Gupta and Seifoddini. 73 K u s i a k 74 develops three heuristic branching algorithms that use different branching schemes to solve the GT problem with bottleneck parts and machines. All three algorithms use the efficient cluster identification algorithm az at each node for branchand-bound decisions. The first algorithm is used to solve simple grouping problems with no restrictions on the number of machines or parts in the machine cells or part families. The second algorithm is designed to solve problems with restriction on the size of the machine cells, while the third heuristic screens machines and parts to identify bottlenecks. The algorithms are simple and can be modified to incorporate various constraints. They can detect bottleneck machines and produce very good solutions. Taboun, Sankaran, and Bhole 7s report on the results of a simulation model used to investigate the performance of groups formed using three similarity measures based on machine, tool, and processing requirements for the parts. The groups were effected using the single linkage clustering technology. A statistical analysis of the resulting groups indicated that the three strategies were significantly different. The study showed that part families formed in accordance with their required machine similarities performed better than those formed by either their tool or processing requirements. In particular, in a test of the impact of part mix and part arrival levels on the three grouping criteria, it was found that the number of parts produced by the cells is a function of the grouping criterion, with processing sequence similarity groups having the least number of parts produced. Furthermore, the variation in productivity is more in the order of part mix and part arrival, with grouping criteria having the least effect.

model assumes that operating times, machine capacities, and manufacturing requirements for each part are known. Kusiak 4 proposes a pattern recognition based part grouping for flexible manufacturing systems. The proposed model can easily be interfaced to a computer-aided design (CAD) database to access technological and geometric part attributes. The data extracted from the CAD database in the form of part geometry is first recognized and then coded using a coding and classification system. Next, similarity between parts is calculated and used to form part families, which can be a parameter of the model or determined through optimization. Two cluster-analytic models (p-median and matrix) were presented for the part grouping problem. In the p-median formulation, a distance minimization objective in an integer programming model is presented subject to each part belonging to only one part family, there being exactly p number of part families, and the jth part family being formed only when the corresponding element is a median. In the matrix formulation model, the number of part families is not a parameter, but is determined by solving the clustering problem. Furthermore, the author investigated the computational complexity of some clustering algorithms (BEA; ROC; Slagle, Chang, and Heller; 78 and Bhat and Haupt 79) and proposed the rank energy algorithm (REA) for identifying part families for flexible manufacturing systems. The algorithm is a modification of a matching algorithm by Bhat and Haupt and combines some features of King's 26'27 ROC algorithm. The proposed REA was shown to have a computational complexity of 0 (IM z + MI2), where M is the number of machines and I is the number of parts. Selvam and Balasubramanian 8° present an integer programming formulation for grouping components based on their operation sequences. The objective of the model is to minimize the sum of material handling and machine idle costs subject to limitations on the number of groups and subject to each component belonging to only one group, the one that minimizes the objective. A covering technique heuristic algorithm that uses a measure reflecting the similarity between processing sequence of the components and maximum number of groups as input is used to find groups of components. Some parameters used to determine

Mathematical Programming P u r c h e c k 76'77 u s e s linear programming to solve the machine-part grouping problem with the objective of minimizing the total capacity cost of the facilities. A coding and classification system is used to code the part and its manufacturing process. The codes are then converted into a similarity measure, which is used to form part families. Parts are not restricted to belonging to only one family. The

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then formulated and solved as an integer programming problem with the objective of maximizing the similarity measure, subject to each part belonging to one and only one part family. The model assumes that the number of part families is known a priori and uses that number as a constraint of the problem. By generating a set of different process plans for each part and selecting only one of them for a part family, an improved pattern of diagonal clusters was obtained. Steudel and Ballakur 8~ present a two-stage dynamic programming heuristic model for the machine grouping problem in flexible manufacturing systems. The authors introduce a new similarity coefficient measure called cell bond strengths (CBS), which is shown to be more robust at capturing the relationship between the machines, with respect to the parts incident on them. The measure is based on the processing times of the parts on the machines, and the operation setup times are ignored. The first stage of the model uses dynamic programming to determine an optimum " c h a i n " of machines for which the sum of their bond strengths are maximized. In the second stage, a heuristic procedure is used to partition the chain to form cells subject to a restriction on the maximum number of machines that can be included in a cell. The authors claim that the procedure is consistent in finding machine-part groups and that the model does not depend on the initial arrangement of the machinepart incidence load matrix. Choobineh sz adopts a modified Jaccard similarity measure [Eq. (7)] that uses operations sequences and proposes an integer programming formulation approach. The algorithm uses a two-stage procedure that first tries to identify natural part families using the similarity measure without any specification on the number of such part families. In the second stage, a linear integer programming formulation is proposed for forming and determining the configuration of the machine cells. The objective function of the model minimizes the sum of the production costs and the acquisition and maintenance costs of the machines. Furthermore, the model identifies the type and number of machines in each cell and the assignment of the part families to the machine cells. Constraints of the model include machine capacity, budget, and that the parts must belong to only one part family.

the similarity matrix of the grouping problem include processing sequence of each component, its production volume per period, and material handling costs between any two cells. Han and Ham 4s present an integer goal programming cluster-analytic model for forming part families in group technology. The input to the model is the part classification codes whose sameness was determined using the absolute Minkowski distance metric. The objective of the model is to lexicographically minimize these distances, subject to parts being in the same family codes or significant similar digits, a part being a member of only one family, and all variables being integers. The output of the model is basically an optimized part list of the dissimilarity between all the parts. Kumar, Kusiak, and Vannelli 43 formulate the grouping problem as an optimal k-decomposition problem in graph theory using 0-1 quadratic programming. The machine-part incidence matrix is converted to its network equivalent with one set of the vertices denoting the machines and the other the parts. The arcs of the network represent the relationship between the connected machines and parts. The grouping problem then becomes that of decomposing the graph into k subgraphs that minimize the weights on the edges, k may be specified as a parameter or determined through optimization. The model is approximated and solved by a linear transportation model using a two-stage polynomialbounded algorithm. In the first stage, the partitioning problem is approximated by a linear transportation problem. The output of this stage becomes an input to the second stage, which capitalizes on the bipartite nature of the grouping problem. Bounds on the optimal solution are used to measure efficiency of the algorithm, which minimizes the sum of the interdependencies of the weighted subgraphs subject to each node belonging to only one subgraph with a limited number of nodes per group. Kusiak 7 presents the p-median clustering problem using integer programming formulation. The measure of similarity between any two parts is given by the sum of the number of common elements, including O's, in the machine-part incidence matrix. The similarity coefficients S~ between any two entities i and j are therefore strictly > 0, unlike traditional similarity measures in which 0 -< S~ --< 1. The resulting n × n matrix of similarity measures is

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Co and Araar a3 present a three-stage mathematical model for solving the machine-part grouping problem. Operations are fast assigned to machines with the objective of maximizing machine utilization by minimizing the deviation between the workload assigned to a machine and its capacity. A 0-1 integer programming model is proposed for this stage. In the second stage of the model, an extended version of King's ~'27 rank order clustering algorithm is used to cluster machines based on the similarity of the operations. Finally in the third stage, a direct-search algorithm is used to find the number of machines in a cell and the cell composition. Chu and Lee 38 present a 0-1 integer programming formulation for the part-family formation problem in group technology. The model is a modified p-median formulation with less constraints and decision variables. It therefore takes less CPU time to converge. The objective of the model is to maximize the sum of the similarities among the machine cells, subject to each part belonging to only one cell, a part belonging to only one existing cell, and there being a limited number of part families. A modified Jaccard similarity coefficient measure [Eq. (7)] is proposed and shown to provide better clustering results, especially in the presence of exceptional elements. The model also has the capacity to optimally determine the desired number of groups. Gunasingh and Lashkari s°'s4 propose two 0-1 integer programming formulations based on tooling requirements of the components (parts) in each family, available tooling on the machines, and processing times. The models assume that part families are known and place limitations on the number of machines that can be admitted in each cell. The first formulation deals with maximization of the compatibility between machines and parts, and the second formulation seeks to minimize the cost of allocating the machines and the cost of intercell travels. The objective of the model is to maximize machine and part compatibilities subject to availability of number of each machine type and restrictions on cell size. Other variants (Lagrangian relaxation and sequential modeling approaches) of these basic models have also been suggested by the authors (for details, see Lashkari and Gunasingh ss and Gunasingh and Lashkari, sl respectively). Vakharia, Askin, and Sen s6 present a 0-1 integer programming formulation with the objective of

minimizing the total cost of machines required and intercell material handling costs, subject to each part being completely processed in each cell, machines required per cell, number of each machines per cell, and number of cells visited by each part. It assumes that the process plans and demand for the parts are known, with the parts understood to be composites that could be identified with a coding and classification system. Further, each part must visit each cell at least once. The model simultaneously considers a multiple-design objective and provides lower bounds on the optimal solution. It is solved by relaxing the integrality restriction on the number of machines required per cell. A heuristic procedure is developed to obtain near-optimal solutions to the grouping problem. Srinivasan, Narendran, and Mahadevan 87 propose an integer programming formulation that was solved with the assignment algorithm as an alternative to the p-median formulation. The authors argue that the latter model is deficient in that it requires the number of groups to be specified as a parameter and might not necessarily correspond to the optimal solution. Furthermore, they point out that the problem with the p-median formulation is that it often requires that the problem be solved more than once to determine the true value of the number of groups, with the first solution usually requiring the use of another algorithm. Indeed, groups occur naturally for well-structured problems, and specifying the number of groups a priori might mar the analyst's chances of identifying these natural clusters. The proposed assignment model uses the similarity coefficient of the machines and components as input for identifying initial machine cells and part families. The resulting part families are then merged with other part families with which they are a subset. The part families are then assigned to machine cells with the objective of minimizing bottleneck machines and exceptional parts. Boctor 8a presents a mixed-integer linear programming formulation of the machine grouping problem that allows the analyst more flexibility in controlling cell size. Some integrality conditions were relaxed to improve the model's computational efficiency and feasibility. The model is then solved using the simulated annealing algorithm, which proved to be helpful in solving large-scale problems. The objective of the model is to minimize the

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is then solved heuristically. The component-PCB matrix is first o r g a n i z e d using King and Narkonchai's 22 rank order clustering algorithm. A sequential search procedure is then used to add parts to a group subject to machine capacity availability. Srinivasan and Narendran 92 use the assignment method to obtain initial seeds for a nonhierarchical algorithm with the objective of minimizing intercell movements and machine idling. The algorithm begins by calculating the similarity for row (machine) vectors. Next an assignment problem is solved to maximize the similarity between machines. The output (solution) of the assignment model forms initial part families and machine cells, which are used as the input (seed) for a nonhierarchical clustering algorithm. These part families and machine cells are then alternately clustered by rows and columns of the incidence matrix until blockdiagonal groups emerge. The authors employ a "maximum density rule" that assigns a vector of parts to a seed part family for which it has the maximum number of l's.

number of exceptional elements, hence intercell material handling costs, subject to each machine (part) being assigned to only one cell (part family) and an allowable number of machine cells not exceeding a prespecified number. The number of cells is specified as a parameter of the problem. Jain, Kasilingam, and Bhole 89 present a 0-1 integer programming formulation for simultaneous formation of machine cells and tool provisioning problems in flexible manufacturing systems. The general objective of the model is to minimize the cost of processing the parts and cost of tools and machines, subject to the capacity available at the machines, tool lives, and processing requirements of the parts. The solution of the model specifies the number of machines and copies of the tools needed to minimize production cost of the system. The model assumes that information on tool requirements, processing times, processing costs, and production quantities for each part are available and that information about machine and tool compatibility, tool lives, machine cost, and available processing times on a machine are also available. Logendran 9° presents a quadratic 0-1 programming model for forming machine cells, with the objective of minimizing inter and intracellular material movement. Constraints of the model are that each part (machine) must be assigned to only one family (cell), all operations on any part must be completed in a machine belonging to its cell, and availability of machine capacity. Three dissimilarity measures are used to identify key machines on which machines for given cells are clustered. The model assumes that the number of operations, cells, and machines per cell; times per operation; and machine capacities are known. A four-phase algorithmic procedure of key workstation identification, clustering, improvement, and assignment of parts to the cells is then used to identify machine cells. Results of the analysis show that the Jaccard similarity measure 2a is the most effective for forming machine cells. Similar findings are also reported by Logendran and West. 91 Maimon and Shtub 31 present a nonlinear mixedinteger programming formulation for grouping printed circuit boards (PCB) for assembly, with the objective of minimizing setup times, subject to availability of setup times, consideration of all component parts, and machine capacity availability. The model

Graph Theory Purcheck 93 introduces a lattice-theoretic method that maximizes scheduling flexibility. For computational convenience, data are first coded, sorted by size, and then ordered by significance. A type of similarity measure is used to make between-object comparisons, and codes are then listed by their significant digits. Parts are then iteratively grouped, first by affiliation, and then groups are merged based on their affinity with one another. Rajagopalan and Batra 35 develop a graphtheoretic model that uses cliques of machine graphs for finding machine cells. The vertices of the graph correspond to the machines and its edges correspond to the relationships between the machines and the components that are incident on them. The relationships are established using a form of similarity measure and a predefined threshold value. The model assumes a stable demand for the product over a finite planning horizon, existence of unique code numbers for production machines, consistent operation sequences, and accurate setup and machining times. The objective of the model is to minimize intercellular material handling costs subject to machine capacity constraints. The model places a limit on the number of different machines that can be

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in a cell under the assumption that intracell movements are negligible, both in terms of handling costs and cost of machine idleness; however, machines may reside in several groups (duplication). The model employs a three-phase procedure in which the machine-part incidence matrix is first used to construct a graph with the properties specified earlier. Graph theory is then used to identify groups with strong relationships. Next a graph-partitioning algorithm is used to form machine cells with these seed groups. Finally in the third stage, components are allocated to cells. Cells are then evaluated by calculating their intercell movements and machine loads. The first stage of the method is based on Jaccard's similarity measure z8 given in Eq. (7). Chakravarty and Shtub 94 present two design procedures that consider scheduling decisions for generating efficient machine layouts in group technology. The model assumes an infinite planning horizon, constant demand with no shortages, and availability of machine capacity and production route cards. Technological dependencies between the machines required to process any part are represented by a graph whose nodes represent machines and whose arcs represent technological precedence relationships. A mixture of assignment and heuristic algorithms is used to find the optimal sequence. The objective of the model is to minimize the sum of setup and inventory carrying costs. Chandrasekharan and Rajagopalan sz present a three-step nonhierarchical algorithm for concurrent formation of part families and machine cells in group technology. The grouping problem is first formulated as a bipartite graph in which the set of vertices on one half of the graph comprises the machines and the other set comprise the components. An upper bound is then established for the number of groups and used to initiate a nonhierarchical grouping algorithm. Next, with the dual objective of maximizing machine utilization and intercell movements (efficiency), the rows and columns of the machine-part incidence matrix are rearranged with the cells having the maximum efficiency assigned first. Lastly, the output from the last stage is further improved by defining "ideal" seeds (centroids of the groups) for the groups. The algorithm is repeated to form a diagonalized matrix of clusters. A measure of effectiveness called grouping efficiency is then used to test the goodness

of the solution. The algorithm is basically a modified MacQueen's 95 method and uses the last k vectors, rather than the first, as the initial seeds. The authors argue that the use of the first k vectors often leads to singleton clusters. Chandrasekharan and Rajagopalan 96 present an extended and improved version of the method that uses a nonhierarchical clustering method for group formation. Part families and machine cells are formed first using natural clusters. Ideal seeds are then generated and used to form mutually separable clusters. Robinson and Duckstein 97 propose the use of polyhedral dynamics in set theory for the machinepart grouping problem. Machines are represented with the vertices of the polyhedra, while parts are represented with the edges. Part groupings are formed based on their processing requirements, and the strength of the relationship between parts in the same family is determined using a complexity measure. The model can identify potential bottleneck machines and exceptional parts. Vannelli and Kumar 42 present a graph-theoretic model that minimizes the number of bottleneck machines (parts) that can be duplicated (subcontracted) to form mutually separable clusters. The minimal bottleneck problem in group technology is shown to be equivalent to that of finding the minimal cut-nodes of a bipartite graph subject to disconnecting the graph into m subgroups subject to specified cell sizes. A dynamic programming heuristic is then used to find the minimal cut-nodes for a specified number of groups and number of machines per group. Kumar and Vannelli 98 extend the algorithm to consider the situation with weighted graphs to identify parts that can be subcontracted to minimize subcontracting cost. Others Ballakur and Steudel 4s present a sequenceindependent heuristic procedure that considers within-cell machine utilization, workload fractions, maximum cell size, and percentage of operation completion per part per cell for concurrently forming machine-part groups. The objective of the model is to minimize intercell material handling costs (intercell moves) subject to capacity availability and cell size restrictions. The model assumes that the number of machines of type m, its total capacity T m, number of operations Ti,,, for part i on

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highest degree of membership. The model also considers workload balance between machines while assigning parts to part families. A comparative study of the model and an 0-1 integer programming procedure shows that fuzzy clustering exhibits superior performance in terms of computation times. Furthermore, the authors claim that the fuzzy algorithm is insensitive to exceptional elements, especially for a relatively small number of groups. Kaparthi and Suresh n present a patternrecognition approach that uses neural network technology for forming part families. The input to the neural network system is part drawings. The system then codes parts using the Opitz system format. Ten neurons, each representing a value in the Opitz system, are used as the output. Each digit in the system is represented by a neuron that is trained to recognize each digit separately. The part drawing is represented as pixels whose bitmaps are used as input to the net. The bit assumes a value of 1 if dark and a 0 if light. The obvious advantage of the model is that it introduces neural network technology as a viable alternative for the grouping problem. In particular, the model proposes automatic generation of part codes and families that will alleviate the labor intensiveness and cost of part coding for group technology. The use of a neural network for the part family formation problem is also suggested by Kao and Moon. ~o~

machine m are known parameters of the model. A two-stage procedure is then employed to assign machines to each cell using a prespecified threshold value ("cell admission factor") subject to the maximum size of the cell ("cell size upper limit"). K u s i a k 99 presents a model that is based on expert systems and optimization technologies for solving the generalized GT problem. The input matrix is of the weighted variety and represents processing times of parts on corresponding machines. No mathematical formulation of the problem is presented; however, the objective of the model is stated as that of minimizing material handling costs for the part family having maximum production costs. The model is solved heuristically subject to limitations on machine capacity, cell size, material handling system capabilities, machine dimensions, and technological requirements. A two-stage solutionusing expert systems technology and a heuristic algorithm is employed. The heuristic clustering algorithm is used to generate partial solutions, which are then evaluated using the expert system that modifies the direction in which the solution could be searched. The influences of ratio of setup to process time, material transfer time between workcenters, demand stability, and within-cell workflow on the attractiveness of cellular manufacturing layouts are investigated by Morris and Tersine. ~00 A comparative evaluation of these factors on process and cellular layouts shows that the operating variables do not favor cellular layouts over process layouts. The authors hypothesize that the ideal cellular manufacturing layouts are distinguished by high ratio of setup to process time, stable demand, high material movement between process departments, and unidirectional workflow within the cell. Chu and Hayya 44 formulate the machine cell-part family formation problem using a fuzzy c-means clustering algorithm. The formulation recognizes the possibility of a part belonging to more than one family and offers some advantages over conventional clustering methods. The objective of the model is to minimize a sum of squares error function subject to a nonbinary matrix with a value between 0 and 1; each part may belong to more than one family with different degrees of membership, and each part family consists of at least one part. A part is assigned to a group with which it has the

Related Literature in Cellular Manufacturing Contributions to the cellular manufacturing literature abound in ways other than for machine cell or part family formation. For example, Kern and Wei m°z present a systematic procedure for identifying and reducing intercell movements caused by the presence of exceptional elements. Traditionally, exceptional elements are eliminated by either duplicating bottleneck machines or subcontracting exceptional parts. The authors suggest a cost tradeoff analysis comparing the cost of machine duplication with that of part subcontracting. The alternative that provides the lowest cost per intercell transfer is selected. Gupta and Tompkins ~°3 present a discrete-event simulation model that examines the dynamic behavior of part families in group technology. They find that the number of intercellular movements is lower for larger cell sizes than for smaller ones.

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Boucher and Muckstadt 1°4 provide a method for estimating and evaluating the cost of converting from functional to group technology layouts. They note that setup times, production lead times, and support functions are the major determining differences between functional and group technology layouts. For example, in group technology layouts, setup times are reduced due to the similarity of machine operations that make elimination of certain machine changeover times possible. There is therefore less need to produce large lot sizes to justify the setup times. Inventory costs are therefore reduced and labor productivity is increased. Wemmerlov and Hyer l°s report that most cellular manufacturing system designs do not specify objective functions, such that the cell assessment process is done independent of the cell formation process. The authors note that the cell formation problem is a multiobjective one, and they enumerate nine such possible objectives, including minimization of throughput times, setup times, inventories, and intercell material movement. Sule 1°6 presents a systematic procedure for analyzing production capacity problems in cellular manufacturing. The procedure starts with the output from any conventional grouping algorithm and a machine capacity requirement matrix. Then employing a four-phase procedure, the author provides a procedure that determines the number of machines with their groupings and the amount of intercell material movement needed to process all components to minimize total cost.

focus of this three-level review framework. Formation of machine cells and part families in cellular manufacturing has been accomplished using similarity coefficient based hierarchical algorithms such as the single, 28 average, 49 and complete-linkagenz algorithms. Similarity coefficient methods usually operate on the 0-1 machine-part incidence matrix with the objective of maximizing the similarity of members of the same group. Usually, similarity coefficient based models find groups in two steps. First machine cells (part families) are formed and then the corresponding part families (machine cells) are assigned to the groups. Machine-part groups can be formed concurrently, however, using arraybased algorithms such as rank order clustering 26'z7 and direct clustering. 4° Other techniques for machine-part grouping include mathematical programming models, graphtheoretical models, fuzzy mathematics, and neural network technology. That this many tools have been employed for solving this problem is a testimony to its complexity. Indeed, the general grouping problem is NP-complete. 1°7 Therefore, simplifying assumptions are made to make the problem manageable. For example, it is often assumed (implicitly) that demand for parts is known and constant over the planning horizon; however, this assumption may not be tenable in the real world. Seifoddini 34 presents the only known model with a probabilistic demand pattern. All models reviewed in this paper are of the static variety in the sense that all machines and parts are assumed to be available at time 0 and do not change over time. In future research, this somewhat restrictive assumption can be relaxed in favor of a more practical dynamic shop condition. So that the greatest amount of benefit can be derived from cellular manufacturing, other possible areas for future research must be investigated. A majority of the models, especially mathematical ones, specify the number of groups and number of machines and parts per group as parameters of the problem. This practice cannot be justified in many production environments for obvious reasons, not the least of which are limitations of space and other logistical and technological limitations. Future models should be able to specify the optimal number of groups, makeup of the cells, and the optimal production mix subject to these technological and

Summary and Directions for Future Research In this paper, a taxonomic framework is proposed and used to review the literature on cellular manufacturing systems. The framework classifies group technology into three taxonomic groups: ocular, part coding and classification analysis, and production flow analysis. A three-level (assumptions, characteristics, and main properties domains) taxonomic framework for knowledge representation is then used to present a comprehensive review of the production flow analysis based cluster-analytic models. Specifically, the systematic procedure for machine-part grouping (analysis) of the PFA is the

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logistical constraints. For this to be possible, though, a robust formulation of an objective function of the problem subject to these constraints must be undertaken. In particular, layout requirements of the machines must be taken into consideration. Very few models consider processing and sequencing requirements of the parts. 89'97 Furthermore, only a handful of models consider machine loadings and p r o d u c t i o n v o l u m e s for components. 46'8° It has usually been sufficient to assume that there is an interaction between a part and a machine by entering a 1 at their intersection on the machine-part incidence matrix. Although this simplification (0-1) of the true state of the interaction between the machine and part is still NP-hard, more work needs to be done with matrices of the weighted variety. 4n'n8 It is well known that practical machine-part grouping problems do not lend themselves to partitioning into mutually separable clusters. Yet the majority of algorithms in the literature are developed for, and work best with, well-structured machine-part incidence matrices. Clusters occur naturally, and the test of any good algorithm should be to find the natural clusters and separate them from the exceptional elements. Researchers must acknowledge and accept this reality and develop algorithms that deal with more pressing problems once the best machine-part groups have been determined. Problems such as trade-off analysis between machine duplication, part subcontracting, and intercellular material handling costs should be addressed and presented as a decision model to the decision maker. Perhaps due to the just-in-time philosophy of minimizing and eliminating material movement and wasted motions, some researchers have tried to force exceptional elements to belong to a group to attain mutual separability in machine-part clusters; however, forcing exceptional parts into families with which they have little or nothing in common, or duplicating machines without any economic considerations, might be counterproductive to the essence of group technology and might even result in more cost than intercellular material movement or subcontracting. Indeed, with recent advancements in computer technology, where robots, automatic guided vehicle systems, and automatic transporters are readily available, material handling costs might even be less expensive over time. Obviously, more

research and the test of time are needed to support or refute this observation. Finally, of all the possible objectives used in the machine-part problem and listed earlier in the paper, minimization of intercellular material handling costs (exceptional elements) is the most widely used. As noted above, this objective may now be trivial or less important because of advancements in computer technology. Other objectives have been suggested'~ and listed elsewhere in this paper. Given the large number of possible objectives that could be specified, one is tempted to suggest a more robust objective function that can incorporate as many of the known (and yet to be defined) objectives as possible. Invariably, goal programming, simulation, expert systems, and neural network based models are possibilities. Each of these technologies have been suggested: goal programming by Han and Ham, 4s simulation by Gupta and Tompkins, 1°3 expert systems by Kusiak, 99 and neural networks by Kao and Moon 1°1 and Kaparthi and Suresh; 6 however, a more complete analysis is required.

Acknowledgment This work was supported in part by a research grant from the Kent State University Research Council and from the Department of Industrial Engineering and Management and the Center of Robotics at Ben Gurion University.

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57. L.E. Stanfel, "Machine Clustering for Economic Production," Engineering Costs and Production Economics (v9, 1985), pp73-81. 58. S.P. Dutta et al., "A Heuristic Procedure for Determining Manufacturing Families from Design-Based Grouping for Flexible Manufacturing Systems," Computers and Industrial Engineering (vl0, n3, 1986), pp193-201. 59. H. Seifoddini and P.M. Wolfe, "Application of the Similarity Coefficient Method in Group Technology," HE Transactions (v18, n3, 1986), pp271-277. 60. R.G. Askin and S.P. Subramanian, "A Cost-Based Heuristic for Group Technology Configuration," International Journal of Production Research (v25, nl, 1987), ppl01-113. 61. S.K. Khator and S.A. Irani, "Cell Formation in Group Technology: A New Approach," Computers and Industrial Engineering (v12, n2, 1987), pp131-142. 62. C.T. Mosier, "An Experiment Investigating the Application of Clustering Procedures and Similarity Coefficients to the GT Machine Cell Formation Problem," International Journal of Production Research (v27, nl0, 1989), p1811. 63. H. Seifoddini, "Single Linkage Versus Average Linkage Clustering in Machine Cells Formation Applications," Computers and Industrial Engineering (vl6, n3, 1989), pp419-426. 64. H. Seifoddini, "Duplication Process in Machine Cells Formation in Group Technology," liE Transactions (v21, n4, 1989), pp382-388. 65. L.E. Stanfel, "Successive Approximations Procedures for a Cellular Manufacturing Problem with Machine Loading Constraints," Engineering Costs and Production Economics (v 17, 1989), pp135-147. 66. J.C. Wei and G.M. Kern, "Commonality Analysis: A Linear Cell Clustering Algorithm for Group Technology," International Journal of Production Research (v27, n 12, 1989), pp2053-2062. 67. G. Harhalakis, R. Nagi, and J.M. Proth, "An Efficient Heuristic in Manufacturing Cell Formation for Group Technology Applications," International Journal of Production Research (v28, nl, 1990), pp185-198. 68. H. Seifoddini, "Machine-Component Group Analysis Versus the Similarity Coefficient Method in Cellular Manufacturing Applications," Computers and Industrial Engineering (v18, n3, 1990), pp333-339. 69. K.Y. Tam, "An Operation Sequence Based Similarity Coefficient for Part Families Formation," Journal of Manufacturing Systems (v9, nl, 1990), pp55-68. 70. A.J. Vakharia and U. Wemmerlov, "Designing a Cellular Manufacturing System: A Materials Flow Approach Based on Operation Sequence," liE Transactions (v22, nl, 1990), pp84-97. 71. P. Gu, "Process-Based Machine Grouping for Cellular Manufacturing Systems," Computers in Indust~ (v17, 1991), pp9-17. 72. T. Gupta, "Clustering Algorithms for the Design of a Cellular Manufacturing System--An Analysis of Their Performance," Computers and Industrial Engineering (v20, n4, 1991), pp461-468. 73. T. Gupta and H. Seifoddini, "Clustering Algorithms for the Design of a Cellular Manufacturing System--An Analysis of Their Performance," Computers and Industrial Engineering (v19, nl-4, 1990), pp432-436. 74. A. Kusiak, "Branching Algorithms for Solving the Group Technology Problem," Journal of Manufacturing Systems (vl0, n4, 1991 ), pp332-343. 75. S.M. Taboun, S. Sankaran, and S. Bhole, "Comparison and Evaluation of Similarity Measures in Group Technology," Computers and Industrial Engineering (v20, n3, 1991), pp343-353. 76. G.F.K. Purcheck, "A Mathematical Classification as a Basis for the Design of Group-Technology Production Cells," Production Engineer (v54, 1974), pp35-48. 77. G.F.K. Purcheck, "A Linear-Programming Method for the Combinatorial Grouping of an Incomplete Power Set," Journal of Cybernetics (v5, 1975), pp51-76. "78. J.R. Slagle, C.L. Chang, and S.R. Heller, "A Clustering--A Data-Reorganization Algorithm," IEEE Transactions on Systems, Man, and Cybernetics (v5, 1975), pp125-128. 79. M.V. Bhat and A. Haupt, "An Efficient Clustering Algorithm," IEEE Transactions on Systems, Man, and Cybernetics (v6, 1975), pp61-64. 80. R.P. Selvan and K.N. Balasubramanian, "Algorithmic Grouping of Operation Sequences," Engineering Costs and Production

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Journal of Production Research (v29, n8, 1991), pp1535-1547. 103. R.M. Gupta and J.A. Tompkins, " A n Examination of the Dynamic Behavior of Part-Families in Group Technology," International Journal of Production Research (v20, nl, 1982), pp73-86. 104. T.O. Boucher and J.A. Muckstadt, "Cost Estimating Methods for Evaluating the Conversion from a Functional Manufacturing Layout to Group Technology," liE Transactions (v17, n3, 1985), pp268-275. 105. U. Wemmerlov and N.L. Hyer, "Research Issues in Cellular Manufacturing," International Journal of Production Research (v25, n3, 1987), pp413-431. 106. D.R. Sule, "Machine Capacity Planning in Group Technology," International Journal of Production Research (v29, n9, 1991 ), pp 1909-1922. 10'7. J.K. Lenstra, "Clustering a Data Array and the TravelingSalesman Problem," Operations Research (v22, 1974), pp413-414.

Authors' Biographies O. Felix Offodile is an assistant professor of administrative sciences at Kent State University. He received his BS, MS, and PhD degrees in industrial engineering from Texas Tech University. His current research interests are in the areas of robotics, manufacturing automation management, scheduling, and inventory management. Dr. Offodile is a senior member of liE and SME and a member of APICS and TIMS/ORSA. Abraham Mehrez received his PhD in operations research from Johns Hopkins University. He has more than 100 publications in various management sciences journals. His research interests are in risk and decision analyses. John Grznar received his BBA in industrial management and MBA from Kent State University, where he is currently pursuing a PhD in operations management. He is a member of APICS and IIE.

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