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Design of a cellular manufacturing system: A syntactic pattern recognition approach
Design of a Cellular Manufacturing System: A Syntactic Pattern Recognition Approach H.L. Wu, R. Venugopal, M.M. Barash, Purdue University, West Lafayette, Indiana
traditional functional system. One well-known method of designing cellular manufacturing groups is production flow analysis (PFA) developed by Burbidge. 1 This is a material flow simplification technique based on the assumption that the majority of components and machines in a factory belong to clearly defined families and groups, and that these groups and families can be identified by progressive analysis of the information contained in route cards. This technique provides an economic approach to cellular manufacture and in many circumstances this can be the only sensible approach.: The purpose of PFA can be summarized as: given the flow routines of the components, group components into families and machines into cells, in such a way that each component can be fully processed in a cell, using existing plant, tooling and processing methods. As Burbidge said in his book, 3 "It is comparatively simple to find the groups and families by eye with a small sample. The mental process used combines pattern recognition, the application of production know-how and intuition. However, it has proved to be surprisingly difficult to find a method suitable for the computer which will obtain the same results.'I n the past decade, a variety of approaches have been developed to implement PFA on computer. A review of these approaches is given in King and Nakornchai, 4 where they are classified as (l) similarity coefficient, (2) set theoretic, (3) evaluative, and (4) other analytical methods.
Abstract A syntactic pattern recognition approach is developed for formation of machining cells by classification of machining sequences. There are four steps in this approach: (1) primitive selection, (2) cluster analysis, (3) grammar inference, and (4) syntactic recognition. Tasks of grouping components to form families, identification of suitable manufacturing cells, and assignment of new products to cells are accomplished by means of syntactic pattern recognition techniques. Results obtained by testing this approach on two case problems are presented. Comparison with other techniques indicates that the pattern recognition approach has greater flexibility. Typical advantages are the capability to introduce cost measures which reflect the relative importance of machines as well as the ability to represent the type of material flow being modeled.
Keywords: Cellular Manufacturing System, Syntactic Pattern Recognition, Production Flow Analysis, Component Family, Grammar, Language.
Introduction In a batch type manufacturing environment, a cellular system which implements the group technology (GT) principle is considered better than the
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The purpose of this paper is to apply principles of syntactic pattern recognition for design of manufacturing cells. The method requires knowledge of formal language theory. The formalism of language theory provides the capability to model the following: (1) different types of material flow, (2) nonuniformity in importance of machines, and (3) classification of new components into an existing cellular set up. Two examples are presented to illustrate this approach.
A sentence in the language L(G) may be derived from the start symbol S by applying production rules from P. An example of generating a pattern of L(G) is as follows: i. From the start terminal S, applying the first rule of P, the subpattern 'aA" is obtained.
S~aA 2. Applying the third rule of P, a pattern 'ab" is obtained.
aA--ab
Syntactic Pattern Recognition
3. Since no further rules can be applied to 'ab', it is a final pattern. It is easily seen that the language L(G) is the set of all strings of the form a~b,n - 1.2 ..... In the context of manufacturing, the order in which the operations are performed can be written in the form of a grammar. For example, a part requiring the following operations: milling, drilling, boring, and grinding, in the given order, can be expressed in the form of a grammar as:
Syntactic pattern recognition borrows most of its analytical methods from formal language theory. In syntactic pattern recognition, complex patterns are represented in terms of simple subpatterns and relations among subpatterns. This is analogous to natural languages. For example, sentences are represented as a collection of words and grammar represents the relationship between words. Just as sentences can be decomposed into words and words can be decomposed into syllables, decomposition can be recursively applied to patterns and subpatterns until the simplest subpatterns are obtained. These simplest subpatterns are called primitives. Presumably, pattern primitives will have simple descriptions and can be easily recognized.
a
~.~
( Vt, Vn, P,S)
where: V t = {milling,drilling,boring,grinding} F n = {S,A,B,C} P = {S--milling A, A--drilling B, B~boring C, C~grinding}
The language that describes a set of patterns in terms of the structural relations and pattern primitives is called a pattern description language. The rules that define valid compositions of primitives into subpatterns and patterns are specified by the g r a m m a r of the pattern description language. Formally, a grammar G is defined as a 4-tuple (Vt, Vn, P,S), where Vt is a set of terminal symbols (the primitive symbols), Vn is a set of nonterminal symbols (the subpatterns), P is a set of production or rewrite rules (the structural component of a pattern), and S is the unique symbol called the start symbol. Consider the following g r a m m a r from Reference 6:
By applying the production rules, the sequence of operations to be performed on the part can be obtained. The concept of grammar can be used to abstract and simplify different complicated sequences. With the syntactic approach, large sets of complex patterns can be described using smaller sets of pattern primitives and grammatical rules. The recognition process is then accomplished by a syntactic analysis (parsing) of the string of primitives (terminal symbols) representing the pattern. More formally, a string (pattern) is said to belong to a certain language (class) L(G) if it can be derived from the productions of the grammar of L(G). Deciding if a string x is in L(G) is the parsing problem of formal language theory and the recognition problem of syntactic pattern recognition. The similarity between manufacturing ceils and grammars is immediately noticed by recognizing that each cell can speak a language (the family of components it can produce). Also the grammar of each cell is defined by the intrinsic nature of the cell.
G : (Vt, Vn, P,S) where:
Vt = {a,b} Vn = {S,A} P = {S--aA, A--aA, A~b} and , 4 , represents 'deduces'
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Cluster Analysis. Conventional clustering methods such as minimum spanning tree (MST), nearest (or k-nearest) neighbor classification rule, and the method of clustering centers can be extended to syntactic pattern recognition. We use the MST approach to group components having similar machining sequences. The distance between two strings x,y is defined as the smallest number of transformations required to derive y from x. Such a measure is called Levenshtein distance. 6 Three types of transformation are defined. They are deletion, insertion, and substitution. Consider the strings "abcd', 'ab', "ade', and 'ef', the distance between string 'abcd" and the other three strings are as follows:
For example, a cell of transfer line type will have a more restrictive g r a m m a r than a cell in which components can randomly move between machines.
The Method The data required for design of cellular manufacturing systems is the flow routines of all components. Flow routines specify machine sequences and times needed on various machines for each component. Only information on machine sequences is used in this approach. However, to decide on the n u m b e r of machines in each cell, information about machining times is needed. In this section the cell design method is described in detail using the following four steps: 1. 2. 3. 4.
Strings
abcd--ab abcd--ade abcd--ef
Primitive Selection. Cluster Analysis. Grammatic Inference. Syntactic Recognition.
Distance 2 deletions 2 deletions + 1 insertion 2 substitutions + 2 deletions
Weights can be given to each of these transformations. Giving each transformation a weight of unity, distances between string "abcd' and the other three strings will be 2, 3, and 4, respectively. Nonuniform weights can be used for different transformations, or for transformations involved with different strings. Measures obtained by the use of weighted t r a n s f o r m a t i o n s are called weighted Levenshtein distances. In the method used, distances between all pairs of strings are found. A string which is a subset of other strings is termed a dominated string. Dominated strings are found and the distances between the dominated strings and their dominating strings are given zero value. Only strings that are not dominated are used to form the minimum spanning tree (MST). The type of manufacturing cells under consideration is an important factor when determining dominance. In a random access system, strings that are substrings of others with machine codes occurring in any order are considered to be dominated. For example 'abcd' dominates "dcb'. However, in a unidirectional system, "abcd' does not dominate "dcb'. Grammaticlnference. After grouping components by cluster analysis, the c o m p o s i t i o n of machines required to form manufacturing cells is inferred. This is the process o f g r a m m a t i c inference. Two fundamental parameters affecting the configuration of a manufacturing cell are work load
Machine sequences are treated as strings which are used to form clusters. The so formed clusters are used to infer grammars which implicitly characterize their structural identities. The language generated by the g r a m m a r so inferred may be larger than the set consisting of the members of the cluster and may include some possible similar patterns due to the recursive nature of the grammar. Classification of new components is carried out by the use of distance measures 5 between strings (a syntactic pattern) and a language (a set of syntactic patterns generated by a grammar). The process is carried out by using an error correcting parser (ECP) algorithm. 6 PrimitiveSelection. Generally, this is the most complex part of any pattern recognition scheme. Invariably there is a trade-off between the complexity of the primitive chosen and the complexity of analysis (grammatic inference, recognition, etc.). However, for the problem on hand, primitives used are simple. Each machine is given a unique code such as 'a', or 'b', or 'c', etc. Each code indicates a different machine type. Using a coding system like the one proposed above would produce strings like 'abc', 'adef, "adefz'. .... and so on. Each string essentially indicates the sequence of machines on which different components are processed. Sequences are obtained for all component types under consideration and this data is used for further analysis.
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and material flow. Work load is concerned with the a m o u n t of load for each type of machine. The problem of work load is usually formulated and solved by the integer programming approach. The emphasis of this paper is put on material flow. Material flow is concerned with the machining sequences of components in a cell. Depending on the type of production system under study (unidirectional, bidirectional, multidirectional or random), material flow requirements are reflected by the g r a m m a r inferred. For example, in an unidirectional line system, the production rules of the inferred g r a m m a r should impose restriction on backtracking of material flow. If the system is a random-access one, then the inferred grammar should not impose any restrictions on material flow. Recognition o f N e w Components. One characteristic of the application of the syntactic pattern recognition approach to P F A is a systematic analysis of the assignment of new components to existing cells. A new component is classified as belonging to
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a particular cell if its route string is recognized by the inferred g r a m m a r of the cell. The recognition of a string by a grammar is easily done by a parsing algorithm. One simple yet powerful parsing algorithm is the Earley parsing algorithm as described in Reference 6. If none of the grammars of the existing cells accepts the new string, then the string should be assigned to such a cell where the distance between the language of this assigned cell and the new string is minimum. This is carried out by an algorithm called modified Earleys error correcting parsing (MEECP). The algorithm is explained in detail in Reference 6.
Case Study The problem taken for case study is the one used by Burbidge, l and H.M. Chan, et al. 2 The objective is to group a set of 43 components into clusters based on the similarity of flow routines of
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The Machine
Figure 1 C o m p o n e n t Matrix Presented
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the components. There are 16 different types of machine tools. The flow routines of components are expressed in matrix form in Figure 1. The rows are labeled with machine identifiers and the columns are labeled with component numbers. In the following, letters a to p are used to represent machine codes for the purpose of distinction. The problem is solved using the syntactic pattern recognition approach. Two examples of this problem based on different assumptions will be illustrated and results obtained are presented. Case1. In this example, all machines are given equal importance, and so unit Levenshtein distance is used. Also, material flows in the manufacturing cells are assumed to be unidirectional. Grouping sequences consisting of the same set of machines but occurring in different orders (such as 'abed" and 'dcba'), will cause the problem of backtracking. Results of Cluster Analysis. Result shows that the nondominated strings are {'efp', 'cfn', 'fghj', 'dehk', 'hklm', 'defho', 'bfhinp', "abfhip'}. These strings are used to form the MST based on distance measures. Distances which form the M S T are shown in Table 1. The d e n d r o g r a m of the MST so formed is shown in Figure 2.
Level 5 Level 4
I c/p
cfp, cfn dehk, defho bfhinp, abfhip
3
fghj, dehk
4 4 4
cfp,fghj cfp, hklm defho, abfhip
dehk
defho
fghj
hk/rn bfhinp abfhip
Level 1
work load balancing problem and is out of the scope of this paper. Results o f Grammatie Inference. For each cell formed, a g r a m m a r can be inferred. In this case, cells of unidirectional material flow are investigated. The result of grammatic inference for the cell consisting of the dominating components (strings) {'bfhinp', "abfhip'} is: a
=
{ Vn, Vt, P,S}
in which:
Vt= {a,b,fh, i,n,p} Vn= IS, U l ..... u61 and the production rules P are: S-- {a UI ,b U2fU3,h U4,iU5,n U6,p}
Strings
1 2 2
cfn
Level 2
Figure 2 The Dendrogram Developed from MST (Case I)
Table 1 Distances to Form MST (Case 1)
Distance
]
I
Level 3
UI-- {b U2 fU3 ,h U4,i US,n U6,p} U2-- {fU3 ,h U4,i U5 ,n U6,p U3-- h U4 ,i U5 ,n U6,p} U4-- {i U5 ,n U6,p U5--{nU6,p} U6-- {p} It is easy to see from the example that the types of machines needed are exactly the elements included in Vt. The g r a m m a r inferred restricts the material flow in the cell to be unidirectional. Only subsequences in the order abfhinp (e.g., 'ab', 'ap', "abf', 'ahp', etc.) are allowed. Sequences like 'ba', 'bhf?., 'npi', etc., are not allowed. Results of Recognition. Using the M E E C P algorithm, it is possible to classify new components to their appropriate cells. Assuming a new components sequence is 'abed" (a sequence which has not occurred before), the distance of this string to grammars of the five cells is shown in Table 3. Thus it is obvious that the new component should be assigned to cell 4.
Grouping components at various levels on the dendrogram gives different cell configurations. Results are shown in Table2. Results show that at level 3, the clusters obtained are the same as that of H. M. Chan et al. 2 Results of level 3 are used to exemplify grammatical inference and components recognition. Notice that in the approach a string may be dominated by several strings and thus has the flexibility of being assigned to more than one cell. Such assignments are to be carried out as a part of the
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Table 2 Different Clusterings on Dendrograms (Case 1)
Level
# of cells
1
8
2
7
3 4 5
5 4 1
Cells consists of
(cfp),(cfn),(dehk),(defho),(hklm),(bfhinp),(abfhip), (fghj) (cfp. cfn),(dehk),(defho),(hklm),(bfhinp),(abfhip), Oeghj) (cfp.cfn),(dehk, defho),(hklm),(bfhinp,abfhip),(fghj) (cfp, cfn),(dehk,defho,fghj),(hklm),(bfhinp,abfhip) (cfp, cfn,dehk, defho, hklm,bfhinp, abfhip,fghj)
Case 2. In this example, we consider a more realistic situation. All the machines are not equally important. This is due to a difference in costs of machines, or the availability of machines. Modeling nonuniformity is carried out by assigning different weights to different machines. The material flow of each cell is also assumed to be of the random-access type. Here the capability of a cell is flexible and sequences such as 'abcd' and "dcba' do not make any difference. Results of Cluster Analysis. As in the previous example, only the set of dominating strings is used for cluster analysis. It should be noticed that the set of dominating strings is not necessarily the same as that of the previous case because different types of material flow are assumed. Different Levenshtein weights are given to different machines. For the example chosen, machines are divided into three classes, each class having a particular weight. Machines of codes 'a' to 'c' are given a weight of 10, 'd' to 'i' are given 7, and 'j' to 'p' are given 1. The result of cluster analysis is given in the dendrogram as shown in Figure3. For this particular case, the set of nondominated strings remains the same, but the relative position of these strings are different in the dendrogram, which results in different cell formula-
26
Components
Distance with abed
i 2 3 4 5
cfp, cfn dehk, defho hklm bfhinp,abfhip fghj
3 3 4 2 4
Level 7 17
,s
Level 6
[
Level 5 11
8
'1
cfp
I
cfn
J
dehk defho fghj
Level 4 Level 3 Level 2
hklm bfhinp abfhip
Level 1
Figure 3 The Dendrogram and Distances (Case 2)
tions. Cells configured at level 4 of the d e n d r o g r a m are used for the processes followed. Results at Grammatic Inference. The g r a m m a r inferred for each cell should reflect the random material flow requirement. The language generated by such a g r a m m a r is the set of all strings which are produced by all permutations and combinations of all machines present in a cell. The g r a m m a r for the cell of dominating components {'bfhinp','abfhip] is:
G : {Vn, vt, e,s} in which:
Table 3 Distances of abcd and Languages of Five Existing Cells (Case 1)
Cell
Level 8
f
19
Vt = {a,b,fh, i,n,p} Vn = { s , u l ..... UT} and the production rules P are:
S-- a UI ,b U 2 f U 3 ,h U4,i U5 ,n U6,p U7 , a,b,fh, i, n,p} U I-- {b U2,fU3,h U4,i U5,n U6,p U7 ,b,f,h, i,n,p} U2-- {a UI ocU3 ,h U4,i U5 ,n U6,p U7 ,a,f h, i,n,p} U3-- {a UI ,b U2,h U4,i US,n U6,p U7 ,a, b,h, i,n,p} U4-- {a U I ,b U2 JU3 ,i US ,n U6,p U7 ,a,b,f i,n,p}
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components into groups. Material flow in a cell is then represented by the grammar inferred. Then the syntactic pattern recognition technique is used to classify a new component to a particular cell. This method is flexible in that different weight schemes can be used for the distance measurements between strings. In this case, economic effects of different transformations can be investigated a priori. By making the transformation penalties cost realistic, the economic effects of backtracking, quantity of components, and transferring among different cells can be modeled systematically. Also, it is easy to model different types of material flow and obtain corresponding cell configurations. The dendrogram established after cluster analysis provides a valuable basis for evaluation. At each level, nondominated strings and dominated strings can be combined to provide work load balance. Work load balancing and determination of number of machines in each cell can be solved by integer programming.
U5-- {a UI ,b U2,fU3 ,h U4 ,n U6,p U7 ,a,b,f h, n,p} U6-- {a UI ,b U2~fU3,h U4,i U5,p U7,a,b,f h, i,p} U7-- {a U I ,b U2 ocU3,h U4,i U5 ,n U6,a,b,f h, i,n} Results o f Recognition. For the unrestricted grammars inferred, the distance of a new string to a grammar is easily calculated without using parsing algorithms. The distance of a string to such a grammar is the summation of weights of machines which exist in the string and do not exist in the cell. The distances of string "abcd" to cells configured from level 4 of dendrogram are shown in Table 4. It is clear that the new component 'abcd" should be assigned to cell 4. Table 4 Distances of abcd and Languages of Five Cells (Case 2)
Cell
Components
Distance with abed
I 2 3 4 5
cfp, cfn dehk, defho hklm bfhinp, abfhip fghj
27 30 37
17 37
References 1. J.L. Burbidge. "'Production Flow Analysis", The Production Engineer, April 1971, pp. 139-152. 2. H.M. Chan, D.A. Milner. "Direct Clustering Algorithm for Group Formation in Cellular Manufacture", Journal o f Manufacturing Systems, Volume 1, Number I, 1982, pp. 65-75. 3. J.L. Burbidge. The Introduction of Group Technology, Heinemann, London, 1975. 4. J.R. King, V. Nakornchai. "Machine-Component Group Formation in Group Technology: Review and Extension", International Journal o f Production Research, Volume 20, Number 2, 1982, pp. 117-133. 5. K.S. Fu. "Recent Developments in Pattern Recognition", IEEE Transactions on Computers, Volume C-29, No. I0, October 1980, pp. 845-854. 6. K.S. Fu. Syntactic Pattern Recognition and Applications, PrenticeHall, 1982.
Concluding Remarks The method described gives a systematic approach to the P F A problem. The techniques and procedures used by syntactic pattern recognition are implemented to solve PFA problem. The process routes of components are represented as strings of machine codes. Cluster analysis based on distance measurements of strings is carried out to cluster
Author(s) Biography Herng-Lang Wu received his Ph.D. from the School of Industrial Engineering at Purdue University, West Lafayette, Indiana. His thesis topic was entitled "Computer Aided Configuration of F M S with Reference to Multi-spindle Head". Dr. Wu received his B.S. degree from Chung Yuan University (Taiwan, Republic of China) and a Master's degree from Texas A&M, both in Industrial Engineering. His current research interests are FMS, C A D / C A P P / C A M and the application of AI and OR techniques in manufacturing. Dr. Wu is currently working at AT&T Bell Laboratories. R. Venugopal received his Ph.D. from the School of Industrial Engineering at Purdue University. His thesis topic was entitled "Thermal Effects on the Accuracy of Numerically Controlled Machine Tools". Dr.
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traditional functional system. One well-known method of designing cellular manufacturing groups is production flow analysis (PFA) developed by Burbidge. 1 This is a material flow simplification technique based on the assumption that the majority of components and machines in a factory belong to clearly defined families and groups, and that these groups and families can be identified by progressive analysis of the information contained in route cards. This technique provides an economic approach to cellular manufacture and in many circumstances this can be the only sensible approach.: The purpose of PFA can be summarized as: given the flow routines of the components, group components into families and machines into cells, in such a way that each component can be fully processed in a cell, using existing plant, tooling and processing methods. As Burbidge said in his book, 3 "It is comparatively simple to find the groups and families by eye with a small sample. The mental process used combines pattern recognition, the application of production know-how and intuition. However, it has proved to be surprisingly difficult to find a method suitable for the computer which will obtain the same results.'I n the past decade, a variety of approaches have been developed to implement PFA on computer. A review of these approaches is given in King and Nakornchai, 4 where they are classified as (l) similarity coefficient, (2) set theoretic, (3) evaluative, and (4) other analytical methods.
Abstract A syntactic pattern recognition approach is developed for formation of machining cells by classification of machining sequences. There are four steps in this approach: (1) primitive selection, (2) cluster analysis, (3) grammar inference, and (4) syntactic recognition. Tasks of grouping components to form families, identification of suitable manufacturing cells, and assignment of new products to cells are accomplished by means of syntactic pattern recognition techniques. Results obtained by testing this approach on two case problems are presented. Comparison with other techniques indicates that the pattern recognition approach has greater flexibility. Typical advantages are the capability to introduce cost measures which reflect the relative importance of machines as well as the ability to represent the type of material flow being modeled.
Keywords: Cellular Manufacturing System, Syntactic Pattern Recognition, Production Flow Analysis, Component Family, Grammar, Language.
Introduction In a batch type manufacturing environment, a cellular system which implements the group technology (GT) principle is considered better than the
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Journal of Manufacturing Systems Volume 5/No. 2
The purpose of this paper is to apply principles of syntactic pattern recognition for design of manufacturing cells. The method requires knowledge of formal language theory. The formalism of language theory provides the capability to model the following: (1) different types of material flow, (2) nonuniformity in importance of machines, and (3) classification of new components into an existing cellular set up. Two examples are presented to illustrate this approach.
A sentence in the language L(G) may be derived from the start symbol S by applying production rules from P. An example of generating a pattern of L(G) is as follows: i. From the start terminal S, applying the first rule of P, the subpattern 'aA" is obtained.
S~aA 2. Applying the third rule of P, a pattern 'ab" is obtained.
aA--ab
Syntactic Pattern Recognition
3. Since no further rules can be applied to 'ab', it is a final pattern. It is easily seen that the language L(G) is the set of all strings of the form a~b,n - 1.2 ..... In the context of manufacturing, the order in which the operations are performed can be written in the form of a grammar. For example, a part requiring the following operations: milling, drilling, boring, and grinding, in the given order, can be expressed in the form of a grammar as:
Syntactic pattern recognition borrows most of its analytical methods from formal language theory. In syntactic pattern recognition, complex patterns are represented in terms of simple subpatterns and relations among subpatterns. This is analogous to natural languages. For example, sentences are represented as a collection of words and grammar represents the relationship between words. Just as sentences can be decomposed into words and words can be decomposed into syllables, decomposition can be recursively applied to patterns and subpatterns until the simplest subpatterns are obtained. These simplest subpatterns are called primitives. Presumably, pattern primitives will have simple descriptions and can be easily recognized.
a
~.~
( Vt, Vn, P,S)
where: V t = {milling,drilling,boring,grinding} F n = {S,A,B,C} P = {S--milling A, A--drilling B, B~boring C, C~grinding}
The language that describes a set of patterns in terms of the structural relations and pattern primitives is called a pattern description language. The rules that define valid compositions of primitives into subpatterns and patterns are specified by the g r a m m a r of the pattern description language. Formally, a grammar G is defined as a 4-tuple (Vt, Vn, P,S), where Vt is a set of terminal symbols (the primitive symbols), Vn is a set of nonterminal symbols (the subpatterns), P is a set of production or rewrite rules (the structural component of a pattern), and S is the unique symbol called the start symbol. Consider the following g r a m m a r from Reference 6:
By applying the production rules, the sequence of operations to be performed on the part can be obtained. The concept of grammar can be used to abstract and simplify different complicated sequences. With the syntactic approach, large sets of complex patterns can be described using smaller sets of pattern primitives and grammatical rules. The recognition process is then accomplished by a syntactic analysis (parsing) of the string of primitives (terminal symbols) representing the pattern. More formally, a string (pattern) is said to belong to a certain language (class) L(G) if it can be derived from the productions of the grammar of L(G). Deciding if a string x is in L(G) is the parsing problem of formal language theory and the recognition problem of syntactic pattern recognition. The similarity between manufacturing ceils and grammars is immediately noticed by recognizing that each cell can speak a language (the family of components it can produce). Also the grammar of each cell is defined by the intrinsic nature of the cell.
G : (Vt, Vn, P,S) where:
Vt = {a,b} Vn = {S,A} P = {S--aA, A--aA, A~b} and , 4 , represents 'deduces'
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Cluster Analysis. Conventional clustering methods such as minimum spanning tree (MST), nearest (or k-nearest) neighbor classification rule, and the method of clustering centers can be extended to syntactic pattern recognition. We use the MST approach to group components having similar machining sequences. The distance between two strings x,y is defined as the smallest number of transformations required to derive y from x. Such a measure is called Levenshtein distance. 6 Three types of transformation are defined. They are deletion, insertion, and substitution. Consider the strings "abcd', 'ab', "ade', and 'ef', the distance between string 'abcd" and the other three strings are as follows:
For example, a cell of transfer line type will have a more restrictive g r a m m a r than a cell in which components can randomly move between machines.
The Method The data required for design of cellular manufacturing systems is the flow routines of all components. Flow routines specify machine sequences and times needed on various machines for each component. Only information on machine sequences is used in this approach. However, to decide on the n u m b e r of machines in each cell, information about machining times is needed. In this section the cell design method is described in detail using the following four steps: 1. 2. 3. 4.
Strings
abcd--ab abcd--ade abcd--ef
Primitive Selection. Cluster Analysis. Grammatic Inference. Syntactic Recognition.
Distance 2 deletions 2 deletions + 1 insertion 2 substitutions + 2 deletions
Weights can be given to each of these transformations. Giving each transformation a weight of unity, distances between string "abcd' and the other three strings will be 2, 3, and 4, respectively. Nonuniform weights can be used for different transformations, or for transformations involved with different strings. Measures obtained by the use of weighted t r a n s f o r m a t i o n s are called weighted Levenshtein distances. In the method used, distances between all pairs of strings are found. A string which is a subset of other strings is termed a dominated string. Dominated strings are found and the distances between the dominated strings and their dominating strings are given zero value. Only strings that are not dominated are used to form the minimum spanning tree (MST). The type of manufacturing cells under consideration is an important factor when determining dominance. In a random access system, strings that are substrings of others with machine codes occurring in any order are considered to be dominated. For example 'abcd' dominates "dcb'. However, in a unidirectional system, "abcd' does not dominate "dcb'. Grammaticlnference. After grouping components by cluster analysis, the c o m p o s i t i o n of machines required to form manufacturing cells is inferred. This is the process o f g r a m m a t i c inference. Two fundamental parameters affecting the configuration of a manufacturing cell are work load
Machine sequences are treated as strings which are used to form clusters. The so formed clusters are used to infer grammars which implicitly characterize their structural identities. The language generated by the g r a m m a r so inferred may be larger than the set consisting of the members of the cluster and may include some possible similar patterns due to the recursive nature of the grammar. Classification of new components is carried out by the use of distance measures 5 between strings (a syntactic pattern) and a language (a set of syntactic patterns generated by a grammar). The process is carried out by using an error correcting parser (ECP) algorithm. 6 PrimitiveSelection. Generally, this is the most complex part of any pattern recognition scheme. Invariably there is a trade-off between the complexity of the primitive chosen and the complexity of analysis (grammatic inference, recognition, etc.). However, for the problem on hand, primitives used are simple. Each machine is given a unique code such as 'a', or 'b', or 'c', etc. Each code indicates a different machine type. Using a coding system like the one proposed above would produce strings like 'abc', 'adef, "adefz'. .... and so on. Each string essentially indicates the sequence of machines on which different components are processed. Sequences are obtained for all component types under consideration and this data is used for further analysis.
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and material flow. Work load is concerned with the a m o u n t of load for each type of machine. The problem of work load is usually formulated and solved by the integer programming approach. The emphasis of this paper is put on material flow. Material flow is concerned with the machining sequences of components in a cell. Depending on the type of production system under study (unidirectional, bidirectional, multidirectional or random), material flow requirements are reflected by the g r a m m a r inferred. For example, in an unidirectional line system, the production rules of the inferred g r a m m a r should impose restriction on backtracking of material flow. If the system is a random-access one, then the inferred grammar should not impose any restrictions on material flow. Recognition o f N e w Components. One characteristic of the application of the syntactic pattern recognition approach to P F A is a systematic analysis of the assignment of new components to existing cells. A new component is classified as belonging to
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a particular cell if its route string is recognized by the inferred g r a m m a r of the cell. The recognition of a string by a grammar is easily done by a parsing algorithm. One simple yet powerful parsing algorithm is the Earley parsing algorithm as described in Reference 6. If none of the grammars of the existing cells accepts the new string, then the string should be assigned to such a cell where the distance between the language of this assigned cell and the new string is minimum. This is carried out by an algorithm called modified Earleys error correcting parsing (MEECP). The algorithm is explained in detail in Reference 6.
Case Study The problem taken for case study is the one used by Burbidge, l and H.M. Chan, et al. 2 The objective is to group a set of 43 components into clusters based on the similarity of flow routines of
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The Machine
Figure 1 C o m p o n e n t Matrix Presented
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Journal of Manufacturing Systems Volume 5/No. 2
the components. There are 16 different types of machine tools. The flow routines of components are expressed in matrix form in Figure 1. The rows are labeled with machine identifiers and the columns are labeled with component numbers. In the following, letters a to p are used to represent machine codes for the purpose of distinction. The problem is solved using the syntactic pattern recognition approach. Two examples of this problem based on different assumptions will be illustrated and results obtained are presented. Case1. In this example, all machines are given equal importance, and so unit Levenshtein distance is used. Also, material flows in the manufacturing cells are assumed to be unidirectional. Grouping sequences consisting of the same set of machines but occurring in different orders (such as 'abed" and 'dcba'), will cause the problem of backtracking. Results of Cluster Analysis. Result shows that the nondominated strings are {'efp', 'cfn', 'fghj', 'dehk', 'hklm', 'defho', 'bfhinp', "abfhip'}. These strings are used to form the MST based on distance measures. Distances which form the M S T are shown in Table 1. The d e n d r o g r a m of the MST so formed is shown in Figure 2.
Level 5 Level 4
I c/p
cfp, cfn dehk, defho bfhinp, abfhip
3
fghj, dehk
4 4 4
cfp,fghj cfp, hklm defho, abfhip
dehk
defho
fghj
hk/rn bfhinp abfhip
Level 1
work load balancing problem and is out of the scope of this paper. Results o f Grammatie Inference. For each cell formed, a g r a m m a r can be inferred. In this case, cells of unidirectional material flow are investigated. The result of grammatic inference for the cell consisting of the dominating components (strings) {'bfhinp', "abfhip'} is: a
=
{ Vn, Vt, P,S}
in which:
Vt= {a,b,fh, i,n,p} Vn= IS, U l ..... u61 and the production rules P are: S-- {a UI ,b U2fU3,h U4,iU5,n U6,p}
Strings
1 2 2
cfn
Level 2
Figure 2 The Dendrogram Developed from MST (Case I)
Table 1 Distances to Form MST (Case 1)
Distance
]
I
Level 3
UI-- {b U2 fU3 ,h U4,i US,n U6,p} U2-- {fU3 ,h U4,i U5 ,n U6,p U3-- h U4 ,i U5 ,n U6,p} U4-- {i U5 ,n U6,p U5--{nU6,p} U6-- {p} It is easy to see from the example that the types of machines needed are exactly the elements included in Vt. The g r a m m a r inferred restricts the material flow in the cell to be unidirectional. Only subsequences in the order abfhinp (e.g., 'ab', 'ap', "abf', 'ahp', etc.) are allowed. Sequences like 'ba', 'bhf?., 'npi', etc., are not allowed. Results of Recognition. Using the M E E C P algorithm, it is possible to classify new components to their appropriate cells. Assuming a new components sequence is 'abed" (a sequence which has not occurred before), the distance of this string to grammars of the five cells is shown in Table 3. Thus it is obvious that the new component should be assigned to cell 4.
Grouping components at various levels on the dendrogram gives different cell configurations. Results are shown in Table2. Results show that at level 3, the clusters obtained are the same as that of H. M. Chan et al. 2 Results of level 3 are used to exemplify grammatical inference and components recognition. Notice that in the approach a string may be dominated by several strings and thus has the flexibility of being assigned to more than one cell. Such assignments are to be carried out as a part of the
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Table 2 Different Clusterings on Dendrograms (Case 1)
Level
# of cells
1
8
2
7
3 4 5
5 4 1
Cells consists of
(cfp),(cfn),(dehk),(defho),(hklm),(bfhinp),(abfhip), (fghj) (cfp. cfn),(dehk),(defho),(hklm),(bfhinp),(abfhip), Oeghj) (cfp.cfn),(dehk, defho),(hklm),(bfhinp,abfhip),(fghj) (cfp, cfn),(dehk,defho,fghj),(hklm),(bfhinp,abfhip) (cfp, cfn,dehk, defho, hklm,bfhinp, abfhip,fghj)
Case 2. In this example, we consider a more realistic situation. All the machines are not equally important. This is due to a difference in costs of machines, or the availability of machines. Modeling nonuniformity is carried out by assigning different weights to different machines. The material flow of each cell is also assumed to be of the random-access type. Here the capability of a cell is flexible and sequences such as 'abcd' and "dcba' do not make any difference. Results of Cluster Analysis. As in the previous example, only the set of dominating strings is used for cluster analysis. It should be noticed that the set of dominating strings is not necessarily the same as that of the previous case because different types of material flow are assumed. Different Levenshtein weights are given to different machines. For the example chosen, machines are divided into three classes, each class having a particular weight. Machines of codes 'a' to 'c' are given a weight of 10, 'd' to 'i' are given 7, and 'j' to 'p' are given 1. The result of cluster analysis is given in the dendrogram as shown in Figure3. For this particular case, the set of nondominated strings remains the same, but the relative position of these strings are different in the dendrogram, which results in different cell formula-
26
Components
Distance with abed
i 2 3 4 5
cfp, cfn dehk, defho hklm bfhinp,abfhip fghj
3 3 4 2 4
Level 7 17
,s
Level 6
[
Level 5 11
8
'1
cfp
I
cfn
J
dehk defho fghj
Level 4 Level 3 Level 2
hklm bfhinp abfhip
Level 1
Figure 3 The Dendrogram and Distances (Case 2)
tions. Cells configured at level 4 of the d e n d r o g r a m are used for the processes followed. Results at Grammatic Inference. The g r a m m a r inferred for each cell should reflect the random material flow requirement. The language generated by such a g r a m m a r is the set of all strings which are produced by all permutations and combinations of all machines present in a cell. The g r a m m a r for the cell of dominating components {'bfhinp','abfhip] is:
G : {Vn, vt, e,s} in which:
Table 3 Distances of abcd and Languages of Five Existing Cells (Case 1)
Cell
Level 8
f
19
Vt = {a,b,fh, i,n,p} Vn = { s , u l ..... UT} and the production rules P are:
S-- a UI ,b U 2 f U 3 ,h U4,i U5 ,n U6,p U7 , a,b,fh, i, n,p} U I-- {b U2,fU3,h U4,i U5,n U6,p U7 ,b,f,h, i,n,p} U2-- {a UI ocU3 ,h U4,i U5 ,n U6,p U7 ,a,f h, i,n,p} U3-- {a UI ,b U2,h U4,i US,n U6,p U7 ,a, b,h, i,n,p} U4-- {a U I ,b U2 JU3 ,i US ,n U6,p U7 ,a,b,f i,n,p}
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Journal o f Manufacturing Systems Volume 5/No. 2
components into groups. Material flow in a cell is then represented by the grammar inferred. Then the syntactic pattern recognition technique is used to classify a new component to a particular cell. This method is flexible in that different weight schemes can be used for the distance measurements between strings. In this case, economic effects of different transformations can be investigated a priori. By making the transformation penalties cost realistic, the economic effects of backtracking, quantity of components, and transferring among different cells can be modeled systematically. Also, it is easy to model different types of material flow and obtain corresponding cell configurations. The dendrogram established after cluster analysis provides a valuable basis for evaluation. At each level, nondominated strings and dominated strings can be combined to provide work load balance. Work load balancing and determination of number of machines in each cell can be solved by integer programming.
U5-- {a UI ,b U2,fU3 ,h U4 ,n U6,p U7 ,a,b,f h, n,p} U6-- {a UI ,b U2~fU3,h U4,i U5,p U7,a,b,f h, i,p} U7-- {a U I ,b U2 ocU3,h U4,i U5 ,n U6,a,b,f h, i,n} Results o f Recognition. For the unrestricted grammars inferred, the distance of a new string to a grammar is easily calculated without using parsing algorithms. The distance of a string to such a grammar is the summation of weights of machines which exist in the string and do not exist in the cell. The distances of string "abcd" to cells configured from level 4 of dendrogram are shown in Table 4. It is clear that the new component 'abcd" should be assigned to cell 4. Table 4 Distances of abcd and Languages of Five Cells (Case 2)
Cell
Components
Distance with abed
I 2 3 4 5
cfp, cfn dehk, defho hklm bfhinp, abfhip fghj
27 30 37
17 37
References 1. J.L. Burbidge. "'Production Flow Analysis", The Production Engineer, April 1971, pp. 139-152. 2. H.M. Chan, D.A. Milner. "Direct Clustering Algorithm for Group Formation in Cellular Manufacture", Journal o f Manufacturing Systems, Volume 1, Number I, 1982, pp. 65-75. 3. J.L. Burbidge. The Introduction of Group Technology, Heinemann, London, 1975. 4. J.R. King, V. Nakornchai. "Machine-Component Group Formation in Group Technology: Review and Extension", International Journal o f Production Research, Volume 20, Number 2, 1982, pp. 117-133. 5. K.S. Fu. "Recent Developments in Pattern Recognition", IEEE Transactions on Computers, Volume C-29, No. I0, October 1980, pp. 845-854. 6. K.S. Fu. Syntactic Pattern Recognition and Applications, PrenticeHall, 1982.
Concluding Remarks The method described gives a systematic approach to the P F A problem. The techniques and procedures used by syntactic pattern recognition are implemented to solve PFA problem. The process routes of components are represented as strings of machine codes. Cluster analysis based on distance measurements of strings is carried out to cluster
Author(s) Biography Herng-Lang Wu received his Ph.D. from the School of Industrial Engineering at Purdue University, West Lafayette, Indiana. His thesis topic was entitled "Computer Aided Configuration of F M S with Reference to Multi-spindle Head". Dr. Wu received his B.S. degree from Chung Yuan University (Taiwan, Republic of China) and a Master's degree from Texas A&M, both in Industrial Engineering. His current research interests are FMS, C A D / C A P P / C A M and the application of AI and OR techniques in manufacturing. Dr. Wu is currently working at AT&T Bell Laboratories. R. Venugopal received his Ph.D. from the School of Industrial Engineering at Purdue University. His thesis topic was entitled "Thermal Effects on the Accuracy of Numerically Controlled Machine Tools". Dr.
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