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A new fuzzy-c-means and assignment-technique-based cell formation algorithm to perform part-type-clusters and machine-type-clusters separately

Sani Susanto, Robert D. Kennedy and John W.H. Price Keywords: part-type-cluster, machine-type-cluster, cell fo rmation, fuzzy-c-partition, fuzzy-c-means Abstracts The incorporation of fuzziness in the cell formation problem by Chu and Hayya (1991) is a notable contribution, in which nonbinary classification logic is used. However, despite this development numerical illustrations performed in this research demonstrate that the Chu and Hayya approach can result in solutions with empty part-type-cluster(s) and/or empty machine-type-cluster(s). Further, it is noted that solutions based on the Chu and Hayya approach can contain non-empty part-type-cluster(s) being assigned to empty machine-type-cluster(s) and vice versa. Three strategies are offered in this research to overcome these inadequacies: the separate formation of part-type and machine-type-clusters, modification of the stopping criterion, and the adoption of the assignment technique in the formation of the final manufacturing-cell solutions. A new algorithm has resulted from these modifications and have been rigorously compared to the performance of the Chu and Hayya approach. In general, the new algorithm demonstrate superiorities in global efficiency and in generating feasible part-type-clusters, machine-type-clusters and manufacturing-cells.

1. Introduction

During the past 15 years or so, there has been a radical shift in the design of manufacturing planning and control systems using innovative concepts such as Just-in-Time, Optimisation Production Technology, Lean Production, the Integrated Factory, World Class Manufacturing, Total Quality Management, Computer Integrated Manufacturing, Flexible Manufacturing Systems and Group Technology or Cellular Manufacturing (CM).

Group Technology is a simple and commonly used technology in everyday life. Items are grouped by their similarities and separated by their differences often to create efficiencies in the control of items. Since the word technology is defined as “the application of knowledge”, Group Technology is “the application of knowledge about groups” (Wemmerlöv and Hyer, 1992).

Group Technology has received great attention in recent literature. One of the reasons is that Group Technology has proven to be a useful technology in solving some of the more complicated design and control problems associated with manufacturing systems. Successful implementation of Group Technology can have a profound effect on the profitability and overall operational efficiency of manufacturing. Group Technology can also simplify the design and process planning of new products by taking advantage of similarities in part-type design and manufacturing processes (Ng, 1996).

The most popular application of Group Technology in manufacturing systems is Cellular Manufacturing (CM). Although CM is just one realisation of Group Technology, CM is often used synonymously with Group Technology. Briefly, CM is a mode of operation where a portion of a factory’s production is manufactured in cells. The most distinctive characteristic of a manufacturing cell is that it contains dissimilar machine-types and/or processes closely located in one area, and that the cell is designed to manufacture a more or less well-defined set of similar part-types called a part-type-cluster (Wemmerlöv, 1988).

Cellular Manufacturing (CM) allows small and batch-type production to gain an economic advantage similar to that of mass production yet maintaining the high degree of flexibility associated with job-shop production.

One of the first, fundamental and most important problems faced in CM involves the decisions to decompose manufacturing systems into cells, also called the cell formation (CF) problem. With the CF problem, part-type-clusters and machine-type-clusters are identified such that (Chu, 1995):

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1. part-types with similar processing requirements (or design features, functions, materials) are produced in a manufacturing -cell sharing common resources such as machines, tools, and labour;

2. each part-type can be processed fully within a manufacturing-cell without the need for the movement across cells; and

3. capital investment in resources is maintained at a level co mpatible with corporate strategy. Much effort has been devoted to solve the CF problem (Moodie et.al., 1995). All CF

approaches up to 1989 (Xu and Wang, 1989), however, contained the implicit assumption that either a part-type (or machine-type) belongs to a part-type-cluster (or machine-type-cluster) or not. That is, the implicit assumption was based on the use of binary classification logic.

However, Xu and Wang (1989) and Chu and Hayya (1991) state that the relationships between part-type and part-type-cluster, as well as machine-type and machine-type-cluster, are often such as to not satisfy the binary classification logic.

Xu and Wang (1989) introduced fuzziness in the relationships between part-type and part-type-cluster and machine-type and machine-type-cluster in forming part-type-clusters and machine-type-clusters, they did this in the formation of manufacturing cells using the fuzzy c-means algorithm (Bezdek, 1981). Xu and Wang, however, used part-type design features (such as its overall length, maximum diameter, length to maximum diameter ratio etc.) as the basis of forming part-type-clusters. This approach is time consuming and inefficient when dealing with large numbers of part-types, each requiring a detailed analysis in their design features.

Chu and Hayya (1991) simplified Xu and Wang’s idea, in that manufacturing routing data, or the machine-type/part-type incidence matrices, were used as the only basis for the formation of part-type-clusters and machine-type-clusters. Since then this approach has been followed by Masnata and Settineri (1997) and Gindy et.al. (1995). In addition, Chu and Hayya’s work was slightly modified by Ponnambalam and Aravindan (1994) in the way in which the initial solution is generated.

However, it is noted by the authors that CF approaches based on Chu and Hayya’s method and subsequent modifications by others can result in a solution with empty part-type-cluster(s) and/or empty machine-type-cluster(s). This is unsatisfactory as effective Cellular Manufacturing requires non-empty part-type-clusters and non-empty machine-type-clusters solutions. Another problem noted by the authors is that solutions based on Chu and Hayya approach can contain a non-empty part-type-cluster assigned to an empty machine-type-cluster and vice versa. That is, part-types are to be processed in a manufacturing cell that contains no machine-types! Alternatively, machine-type-cluster(s) can exist that have no work assigned to them. Again, this results in the non-effective utilisation of cellular manufacturing resources.

In this article a modified Chu and Hayya approach will be developed to overcome these three difficulties. In section 2, the Chu and Hayya approach will be discussed. Its numerical example is illustrated as well as problems that may arise in implementing this approach. In section 3 some problems that may arise with Chu and Hayya’s approach are outlined. In section 4, strategies to handle problems outlined in section 3 are described. A numerical illustration of the new approach is included. In section 5 the summaries and conclusions are given.

2. The Chu and Hayya Fuzzy-c-means-based cell formation The Chu and Hayya Fuzzy c-means-based cell formation algorithm is simply

formulated as a constrained optimisation problem. The objective function and its related constraints are discussed in the following sub-section.

2.1 The Chu and Hayya Fuzzy-c-means-based cell formation problem formulation

Given a set of m machine-types and a set of n part-types which are to be grouped into c machine-type-clusters and c part-type-clusters respectively, (1< c < minm,n), let i. X x jk mxn= ( ) be the mxn machine-type/part-type incidence matrix, where x jk =1 if part-

type k needs machine-type j for an operation and x jk = 0 otherwise,

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ii. x k k k mkTx x x= ( , , ... , )1 2 be the attribute vector of part-type k, (Note: x j j j jn

Tx x x* ( , ,. .. , )= 1 2

are on the other hand called as the attribute vector of machine-type j)

iii. v Rim∈ is the centre of part-type cluster i, 1≤ ≤i c ,

iv. d ki k i= −x v is the Euclidean norm metric which measures “the distance” between

x k and cluster centre v i ,

v. u ki is the degree of membership of the attribute vector of part-type k in part-type cluster i,

vi. U u ki nxc= ( ) is an nxc matrix and,

vii. v v v v= ( , , ... , )1 2 c is an mxc matrix. Solve the following minimisation problem:

min ( , ) ( ) ( )imize J U u dki kii

c

k

n

22 2

11v =

==∑∑ (2.1.1)

subject to u k iki ∈ ∀[ , ] ,0 1 (2.1.2)

u kkii

c

=∑ = ∀

11 (2.1.3)

01

< < ∀=

∑u n ikik

n

(2.1.4)

Note: The set M fc of matrices U satisfying conditions (2.1.2)-(2.1.4) is called the fuzzy-c-partition for X, and is defined as

M U M u k i u r u n ifc nxc ki ki kik

n

i

c

= ∈ ∈ ∀ = ∀ < < ∀

==∑∑[ , ] , ; ;0 1 1 0

11

(2.1.5)

where M nxc is set of all real nxc matrices.

The solution to this constrained optimisation problems is expressed in the form of an algorithm, called the “Bezdek’s Fuzzy-c-means Clustering Algorithm”. Before outlining this algorithm, the theorem which serves as its basis is described.

Theorem 2.1 The Fuzzy-c-means Clustering Basic Theorem Let * be any inner product norm metric, and X have at least c < n distinct points, define ∀ =k , ,... , n1 2 the sets

I i i c;dk ki k i= ≤ ≤ = − =1 0x v (2.1.6)

I c Ik k

~

, ,.. .,= −1 2 (2.1.7)

then ( , )U M xMfc mxcv ∈ may be globally minimal for J2 in (2.1.1) only if

I udd

k kiki

khh

c=∅ ⇒ =

=

∑

12

1

(2.1.8)

or

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I u 0 for each i I and u 1k ki

~

k kii Ik

≠ ∅ ⇒ = ∈ =∈∑ (2.1.9)

and

v

x

i

ki kk

n

kik

n

u

ui c= ∀ =

=

=

∑

∑

2

1

2

1

12, ,... , (2.1.10a)

that is

vu x

ui c and j mji

ki jkk

n

kik

n= ∀ = ==

=

∑

∑

2

1

2

1

1 2 1 2, ,... , , , ... , (2.1.10b)

Proof: see Bezdek (1981)

2.2 Bezdek’s Fuzzy c-means Clustering Algorithm Bezdek(1981) proposed the fuzzy c-means algorithm to minimise J 2 in (2.1.1). The

detail steps of the algorithm are as follows:

Algorithm 2.2 Bezdek’s Fuzzy c-means Clustering Algorithm

Step 1: Choose a scalar ξ > 0 ; fix integer c, 2 ≤ <c n ; choose any inner product norm metric for

R m ; and initialise U Mfc( )0 ∈ . At step l, l = 0,1,2, ...,:

Step 2:

Calculate the c fuzzy cluster centres v k(l) with (2.1.10) and U (l) .

Step 3:

Update U(l) using (2.1.8) or (2.1.9) and v k(l) .

Step 4: (Stopping Criteria)

Compare U (l) to U (l )+1 in a convenient matrix norm, that is, if U U(l ) (l)+ − <1 ξ then stop;

otherwise, l l← + 1 and return to Step 2.

Note:

The fuzzy c-means algorithm has a number of algorithmic parameters, they are: c, U (0 ), ∗ and ξ .

A deeper analysis to the descent and convergence of fuzzy c-means by Bezdek(1981) showed that sequences (U )(l) (l), l , , , . . .v = 0 1 2 generated always conv erge (in theory) to strict local minima

(not just stationary point) of J 2 . These strict local minima are defined as optimal fuzzy clusterings.

Once U and v are obtained the part-type-clusters and machine-type-clusters are simultaneously formed based on the following rule:

Chu and Hayya’s Part-type and Machine-type-clusters and Manufacturing-Cells Formation Rule:

Rule 2.2.1: Part-type-Clusters Assignment Rule:

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Part-type k is assigned to part-type-cluster i if and only if: u u u uki k k kc= max , ,..., )1 2 (2.2.1)

Rule 2.2.2: Machine-type Clusters Assignment Rule: Machine-type j is assigned to machine-type cluster i if and only if: v v v vji j j2 jc= max , , . .. , 1 (2.2.2)

Rule 2.2.3: Manufacturing Cell Formation Rule: Manufacturing cell i, MC-i, is composed of PC -i, the part-type-cluster i and MC-i, the machine-type cluster i, that is MC-i = (MC-i,PC-i), where

PC i P u u u u P part type kk ki k k kc kk

n

− = = = −=

max , ,..., ;1 21

Υ (2.2.3)

and

MC i M v v v v M machine type jj ji j j2 jc jj

m

− = = = −=

max , ,... , ;11

Υ (2.2.4)

Since the method introduced by Chu and Hayya (1991) employs the resulting matrices U and v, their approach is labelled as the fuzzy c-means clustering approach of type (U -v). 2.4 Numerical Illustration of Chu and Hayya’s Approach The Chu and Hayya approach to CF problem was programmed in Turbo C++ , and run on an IBM PC/AT compatible. The approach was applied to the 12x10 machine-type/part-type incidence matrix described in Figure 1 using c = 3, Euclidean norm as the inner product

norm, ξ = 001. and the matrix U(0) = (u )ki( )0

10 3x as described in Table 1 (as suggested by Chu

and Hayya) was chosen as an initial solution. The final value of (u )ki and (v )ji are displayed in

Tables 2 and 3 respectively. After implementing Chu and Hayya’s Part-type and Machine-type-clusters and Manufacturing-Cells Formation Rule, three machine-types-clusters, three part-types-clusters and three manufacturing-cells are obtained, which are displayed in Figure 2. They are as follows: Machine-types-clusters: MC-1 = 1,2,3,4; MC-2 = 5,6,7,11,12 and MC-3 = 8,9,10 Part-types-clusters: PC-1 = 1,2,3,4; PC-2 = 5,6,7 and PC-3 = 8,9,10 Manufacturing -cells: MC-1 = MC-1,PC-1; MC-2 = MC-2,PC-2 and MC-3 = MC-3,PC-3

3. Problems Associated with Chu and Hayya’s Approach:

Chu and Hayya’s approach forms part-type-clusters and machine-type-clusters simultaneously. Based on computational experience it is observed that three problems may arise with this approach (Susanto et.al, 1997). These are described as follows.

3.1 The First Problem Associated with Chu and Hayya’s Approach:

The first problem associated with Chu and Hayya’s approach is that there is a distinct possibility that the approach will result in some empty part-type-cluster(s) and/or machine-type-cluster(s).

To illustrate this situation, the machine-type/part-type incidence matrix for 30 machine-types and 41 part-types as described in Figure 3 serves as the input data to the approach. In addition, the following parameters are chosen: c = 3, Euclidean norm as the inner product

norm, ξ = 001. and the matrix U (0) = (u )ki( )0

41 3x as described in Table 4 as an initial solution.

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As a result one empty part-type-cluster and one empty machine-type-cluster are obtained as follows:

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Machine-types-clusters: M C-1 = 4,5,6,7,8,9,13,14,15,16,17,18,19,20,24,25,26,27,28,29,30; MC-2 =1,2,3,10,11,12,21,22,23 and MC - 3 = Part-types-clusters: PC - 1 = 1,3,4,5,6,7,8,9,13,14,15,16,17,18,21,22,24,25,26,27,28,29,30,34,35,36,37,38, PC - 2 = 2,10,11,12,19,20,23,31,32,33,39,40,41 and PC - 3 = Since the number of expected non-empty part-type-clusters and non-empty machine-type-clusters are 3, this result is an infeasible solution. 3.2 The Second Problem Associated with Chu and Hayya’s Approach:

The second problem associated with Chu and Hayya’s approach is that it may end up with different numbers of non-empty part-type-clusters compared to the resulting number of non-empty machine-type clusters. To illustrate this situation, again machine-type/part-type incidence matrix in Figure 3 serves as the input data to the approach. In addition, the following

parameters are chosen: c = 4, Euclidean norm as the inner product norm, ξ = 0 01. and the

matrix U (0) = (u )ki( )0

41 4x as described in Table 5 as an initial solution.

As a result the number of non-empty part-type-cluster and non-empty machine-type clusters obtained are different, ie. 2 and 3 respectively. The final solution, which is infeasible, is as follows: Machine-types-clusters: M C-1 = 4,5,6,7,8,9,13,14,15,16,17,18,19,20,24,25,26,27,28,29,30; MC-2 = ; MC-3 = 1,2,3,10,11,12,21, 22,23 and MC - 4 = Part-types-clusters: PC - 1 = 1,3,4,5,6,7,8,9,13,14,15,16,17,21,22,25,26,27,28,29,30,34,35,36,37,38; PC - 2 = 18,24; PC - 3 = 2,10,11,12,19,20,23,31,32,33,39,40,41 and PC - 4 =

3.3 The Third Problem Associated with Chu and Hayya’s Approach: In case of a tie in Rule 2.2.1, that is max , , ... , u u uk k kc1 2 is achieved by more than

one part-type-cluster, then part-type k is directly assigned to the first (or the last) part-type-cluster achieving the maximum value. This is too restrictive, since it eliminates the opportunity for the other part-type-cluster(s) that achieved the maximum value to contain that particular part-type. A similar tie problem can arise in Machine-type Clusters Assignment Rule (ie. Rule 2.2.2). 4. Strategies to handle problems related to Chu and Hayya’s Approach

In Chu and Hayya’s approach, part-type-attribute vectors are exploited as the only basis for clustering. That is given part-type-attribute vectors x k (k=1,2,…,n), we have to find matrices

U and v that minimises functional J 2 as in (2.1.1). We employ Rule 2.2.1 to matrix U and Rule

2.2.2 to matrix v to obtain part-type-clusters and machine-type-clusters respectively. Due to the nature of stopping criteria in Step 4 of Algorithm 2.2, only one matrix U and one matrix v the final ones, are used. Thus, Chu and Hayya’s approach results in at most one set of c non-empty part-type-clusters and one set of c non-empty machine-type-clusters. Since there is no guarantee that these two sets can be obtained, the three problems mentioned in Sub-section 3.1 to 3.3 are often present. Modifications to Chu and Hayya’s approach to overco me these problems are really needed and are done through the following ways: i. forming the part-type and machine-type clusters separately ii. changing the stopping criteria in Chu and Hayya’s algorithm iii. applying the assignment technique to form manufacturing cells based on part-type and

machine-type clusters obtained from (i) as a result of the new stopping criteria in (ii).

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4.1 The First Modification: Forming The Part-type-clusters and Machine-type-clusters

Separately This modification is achieved through two separate clustering formations, called “The

First Half Manufacturing-cells Formation” and “The Second Half Clusters Formation”. The “The First Half Manufacturing-cells Formation”, in which machine-type-clusters are

formed, is exactly the same as that of Chu and Hayya’s approach. In this approach the given part-type-attribute vectors x k (k=1,2,…,n) are used to find matrices U and v to minimise J 2 as

in (2.1.1), but only matrices v will be used to determine the machine-type-clusters. In the “The Second Half Clusters Formation”, in which part-type-clusters are formed, the given machine-

type-attribute vectors x j j j jnTx x x* ( , ,. .. , )= 1 2 (j = 1,2, …, m) are used to find matrices U* and

v * such that the new functional J2* is minimised (note: the notations U* , v * and J 2

* are to be

defined in this Sub-section). Similar to “The First Half Clusters Formation”, only matrices v * will be used to determine the part-type-clusters. The First Half Manufacturing -cells Formation to form machine-type-clusters can be briefly reformulated as:

The Problem Formulation for “The First Half Manufacturing-cel ls Formation” Given

i. x k k k mkTx x x= ( , , ... , )1 2 , the attribute vector of part-type k (k=1,2, .., n) and,

ii. c, the number of desired part-type clusters, where 1 < c < minm,n. Find

i. the degree of membership matrix U u Mki nxc fc= ∈( ) and,

ii. the cluster centre matrix v v v v= ( , , ... , ) ,1 2 c where v i i i miTv v v= ( , ,. .. , )1 2 is the

centre of the i-th part-type cluster (i = 1, 2, ..,c), such that the following functional J 2 in (2.1.1)is minimised.

As mentioned before, in “The Second Half Manufacturing-cells Formation” the machine-type-attribute vectors of each of the machine-types are exploited to serve as the basis of the clustering process. This approach can be defined and formulated a follows:

The Problem Formulation for “The Second Half Clusters Formation”

Given

i. x j j j jnTx x x* ( , , ... , )= 1 2 , the attribute vector of machine-type j (j = 1,2, .., m),

ii. c, the number of desired machine-type clusters, where 1 < c < minm,n. Find

i. the degree of membership matrix U u Mjh mxc fc* * *( )= ∈ , where

M U M u j h u j u m hfc mxc jh jh jhj

m

h

c* * [ , ] , ; ;= ∈ ∈ ∀ = ∀ < < ∀

==∑∑0 1 1 0

11

(4.1.1)

ii. the cluster centre matrix v v v v* * * *( , , ... , ),= 1 2 c where vh h nhTv v v* * * *( , , ... , )= 1h 2 is the

centre of the i-th part-type cluster (h = 1, 2, ..,c) such that the following functional is minimised:

min ( , ) ( ) ( )* * * * *imize J U u djh jhh

c

j

m

22 2

11

v ===

∑∑ (4.1.2)

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4.2 The Second Modification: Change in the Stopping Criterion

It is expected that modifications will avoid the new approach from i. ending with some empty part-type-clusters (and yet empty machine-type-clusters) and ii. ending with some ties in assigning part-type into part-type-clusters (and yet machine-type into

machine-type-clusters). This can be done by changing the stopping criterion in Step 4 of the Algorithm 2.2.

While the stopping criterion in Step 4 is more based on “similarity” between two consecutive

matrices U(l) to U (l )+1 resulting from a series of iterations, the new stopping criterion reflects the effort to achieve the expectation mentioned above. The new stopping criteria is based on a given number called max_iterations. Thus, up to max_iterations, it is possible to get more than one successful part-type-clusters formation solution and/or one successful machine-type-clusters formation solution.

A successful machine-type-clusters formation solution is one that:

i. converges to c non-empty machine-type-clusters, and

ii. results in singleton set i v v v vji j j jc(l) (l) (l) (l), ,.. .,= max ; j = 1,2,. .. ,m1 2 , l is an integer, such

that , 1 ≤ ≤l max_iterations.

In addition, for each i (i=1,2,…,c), the set

MC i jv v v vji j j jc− = =(l) (l) (l) (l) (l), , ... ,max 1 2 (4.2.1)

is defined as the successful machine-type-cluster i obtained from the l-th iteration of “The First Half Manufacturing -cells Formation”. There will be at most max_iterations sets of c successful machine-type-clusters, when “The First Half Manufacturing-cells Formation” is terminated. A successful part-type-clusters formation solution is one that: i. converges to c non-empty part-type-clusters, and

ii. results in singleton set h v v v vkhs

ks

ks

kcs*( ) *( ) *( ) *( ), ,... ,= max ;k = 1,2,.. ., n1 2 , s is an integer,

such that , 1 ≤ ≤s max_ iterations. In addition, for each h (h=1,2,…,c), the set

PC k v v v vkhs

ks

ks

kcs− = =h , ,... ,(s) *( ) *( ) *( ) *( )max 1 2 (4.2.2)

is defined as the successful part-type-cluster h obtained from the s-th iteration of “The Second Half Manufacturing -cells Formation”. There will be at most max_iterations sets of c successful part-type-clusters, when “The Second Half Manufacturing-cells Formation” is terminated. 4.3 The Third Modification: The Application of The Assignment Technique The goal of the developed algorithm is to maximise the total non-zero entries in the diagonal-block of the final machine-type/part-type incidence matrix. When this is achieved, then the global efficiency of cellular manufacturing is maximised. The global efficiency of a cellular manufacturing is defined as the ratio of the total number of ones in the diagonal block of the final arranged machine-type/part-type incidence matrix to the total number of ones in that matrix. This goal can be achieved by implementing the assignment technique to the sets of c successful machine-type-clusters obtained from “The First Half Manufacturing -cells Formation” and the sets of c successful part-type-clusters obtained from “The Second Half Manufacturing-cells Formation”. Consider the situation of assigning c part-clusters to c machine-clusters and define

c x jkk PC

i(l), h(s)hi (s)(l)

=∈ −∈∑∑

j MC- (4.3.1)

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be the degree of conformance between PC − h (s) and MC − i (l) . Numerically, c i(l),h(s) represents the

total of ones in the block matrix as a result of the intersection between column(s) in

PC − h (s) and row(s) in MC − i (l) . Physically, this value means the utilisation of machine-types

belong to MC − i (l) by part-types belong to PC − h (s) . The assignment problem can be completely formulated as follows: Given ci(l),h(s) (i,h=1,2,…,c; l,s=1,2,…, max_iterations), determine Ωi(l),h(s) such that the quantity

ch=1

c

i=1

c

i(l),h(s) i(l),h(s)

l,s max_iterations

Ω∑∑=1,2,...,

(4.3.2)

is maximised, subject to

Ωi(l),h(s)h

==

∑ 11

c

for each i, l and s (4.3.3)

Ωi(l),h(s)i

==∑ 1

1

c

for each h, l and s (4.3.4)

Ωi(l), h(s)

, h(s)

i(l)

h(s)

i(l)

=

− − − − − −

− − − − − −

1

0

if the part type cluster PC is assigned to the machine type cluster MC

if the part type cluster PC isnot assigned to the machine type cluster MC(4.3.5)

This assignment problem can be solved by the Hungarian Method (Taha, 1992) or the Mack’s Method (Bunday, 1984). 4.4 The Detail Steps of The New Algorithm The steps involved in “The Fir st Half Clusters Formation” algorithm are as follows:

“The First Half Cluster Formation” Algorithm Step 1. Step 1a. Fix c, 2 ≤ <c m nmin , ;

Step 1b. Choose any inner product norm * on R m

Step 1c. Initialise U M fc( )0 ∈ .

Step 1d. Let success = 0; l = 0 and the number of maximum iterations be max_iterations. Step 2

Calculate the c fuzzy cluster centres v i(l) using (2.1.10a) or (2.1.10.b).

Step 3

If ∀ ∈ = =q the set A i v v v vq qi 1, 2, ... , m max q1 q2 qc, , , ... ,(l) (l) (l) (l) (l) is singleton

then

if Aqq

m(l)

==

1Υ 1, 2, .. ., c

then if l < max_iterations

then 1) success ← success + 1 2) at the l-th iteration and at the success-th successful machine-type-clusters

formation, assign machine-type q to machine -type-cluster i else 1) success ← success + 1

2) at the l-th iteration and at the success-th successful machine-type-clusters formation, assign machine-type q to machine-type-cluster i

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3) goto Step 6 else

if l = max_iterations then goto Step 6 else

if l = max_iterations then goto Step 6 Step 4

Determine I k and I k

~

as defined in (2.1.6) and (2.1.7) respectively, where k = 1, 2, ..., n

Step 5 Step 5a. l ← l + 1;

Step 5b. Calcula te u ki(l) as defined in (2.1.8) or (2.1.9), where k = 1,2, ..., n; i= 1,2, ..., c;

Step 5c. Goto Step 2. Step 6 Stop

“The Second Half Cluster Formation” to form part-type-clusters is based on the following theorem.

Theorem 4.4 The Fuzzy-c-means Clustering Basic Theorem Let * be any inner product norm metric, and X have at least c < n distinct points, define ∀ =j m1 2, ,.. ., the sets

I i i c;dj ji j i* * * *= ≤ ≤ = − =1 0x v (4.3.6)

I c Ij j

~ **, ,.. .,= −1 2 (4.3.7)

then ( )U M xMfc nxc* * *,v ∈ may be globally minimal for J 2

* in (4.1.2) only if

I ud

d

j ji

ji

jhh

c

* *

*

*

=∅ ⇒ =

=

∑

12

1

(4.3.8)

or

I u 0 for each i I and u 1j*

ji*

~

ji*

i I j*

≠ ∅ ⇒ = ∈ =∈∑j

*

(4.3.9)

and

v

x

i

ji jj

m

jij

m

u

ui c*

* *

*

, ,... ,= ∀ ==

=

∑

∑

( )

( )

2

1

2

1

12 (4.3.10a)

that is

vu x

ui c and k nki

ji kjj

m

jij

m*

*

, ,... , , ,... ,= ∀ = ==

=

∑

∑

( )

( )

2

1

2

1

1 2 1 2 (4.3.10b)

Proof: see Bezdek (1981)

12

The detail algorithm of “The Second Half Cluster Formation” are as follows:

“The Second Half Cluster Formation” Algorithm Step 1. Step 1a. Fix c, 2 ≤ <c m nmin , ;

Step 1b. Choose any inner product norm * on R n Step 1c. Initialise U M fc* *(0) ∈ .

Step 1d. Let SUCCESS = 0; l = 0 and the number of maximum iterations be max_iterations. Step 2

Calculate the c fuzzy cluster centres v i*(l) using (4.3.10a) or (4.3.10b).

Step 3

If ∀ ∈ = =p the set B h v v v vp ph p p pc 1, 2, ..., n max, , ,.. ., ,(l) *(l) *(l) *(l) *(l)1 2 is singleton

then

if Bpp

n(l)

==

1Υ 1, 2, ... , c

then if l < max_iterations

then 1) SUCCESS ← SUCCESS + 1 2) at the l-th iteration and at the SUCCESS-th successful part-type-cluster

formation, assign part-type p to part-type-cluster h else 1) SUCCESS ← SUCCESS + 1

2) at the l-th iteration and at the SUCCESS-th successful part-type-cluster formation, assign part-type p to part-type -cluster h

3) goto Step 6 else

if l = max_iterations then goto Step 6 else

if l = max_iterations then goto Step 6 Step 4

Determine I j* and I j

~ *

as defined in (4.3.6) and (4.3.7) respectively, where j = 1, 2, ..., m

Step 5 Step 5a. l ← l + 1;

Step 5b. Calculate u ji*(l) as defined in (4.3.8) or (4.3.9), where j= 1,2, ..., m; i= 1,2, ..., c;

Step 5c. Goto Step 2. Step 6 Stop

4.5 Computational results of the new algorithm The developed algorithm was tried using the same machine-type/part-type incidence matrix illustrated in Figure 3. For the case when the number of desired manufacturing cells is three (ie. c =3), and the number of maximum iterations is 10 (ie. max_iterations = 3), “The First Half Cluster Formation” algorithm resulted in 5 successful machine-type-clusters formation solutions, while “The Second Half Cluster Formation” algorithm produced 4 successful part-type-clusters formation solutions.

13

The five successful machine-type-clusters formation solutions are: 1. MC-1= 4,5,7,13,16,17,18,25,26; MC - 2 = 6,8,10,12,14,15,23,24,27,28; and MC-3=1,2,3,9,11,19,20,21,22,29,30

2. MC-1 = 4,5,7,13,16,17,18,24,26; MC - 2 = 2,6,8,10,12,14,15,23,25,27,28 and MC-3 = 1,3,9,11,19,20,21,22,29,30

3. MC - 1 = 4,5,6,7,8,13,15,16,17,18,24,26,27; MC - 2 = 2,3,10,12,14,23,25,28 and MC - 3 = 1,9,11,19,20,21,22,29,30

4. MC - 1 = 4,5,6,7,8,13,14,15,16,17,18,24,25,26,27,28; MC - 2 = 1,2,3,10,11,12,21,22,23and MC - 3 = 9,19,20,29,30

5. MC - 1 = 4,5,6,7,8,13,14,15,16,17,18,24,25,26,27,28,29; MC - 2 = 1,2,3,10,11,12,21,22,23 and MC - 3 = 9,19,20,30

The four successful part-type-clusters formation solutions are:

1. PC- 1 = 2,6,7,8,10,15,19,24,25,28,29,33,35,38,39; PC- 2 = 1,3,4,5,9,13,14,16,17,18,21,22,26,27,30,34,36,37,41 and PC- 3 = 11,12,20,23,31,32,40

2. PC- 1 = 6,8,15,19,24,25,28,29,33,35,38,41; PC- 2 = 1,3,4,5,7,9,13,14,16,17,21,22,26,27,30,34,36,37 and PC- 3 = 2,10,11,12,18,20,23,31,32,39,40

3. PC- 1 = 6,15,19,24,25,28,35,38; PC- 2 = 1 ,3,4,5,7,8,9,13,14,16,17,21,22,26,27,29,30,34,36,37 and PC- 3 = 2,10,11,12,18,20,23,31,32,33,39,40,41

4. PC- 1 = 19,25; PC- 2 = 1,3,4,5,6,7,8,9,13,14,15,16,17,21,22,26,27,28,29,30,34,35,36,37,38 and PC- 3 = 2,10,11,12,18,20,23,24,31,32,33,39,40,41

The assignment technique was then applied to the above machine-type-clusters and part-type-clusters to obtain the optimum manufacturing -cells (ie. the one that results in maximum number of ones in the final diagonal block of the rearranged initial machine-type/part-type incidence matrix). This technique advises us to select the following machine-type-clusters and part-type-clusters: The selected machine-type-clusters: MC - 1 = 4,5,6,7,8,13,14,15,16,17,18,24,25,26,27,28;MC - 2 = 1,2,3,10,11,12,21,22,23 and MC - 3 = 9,19,20,29,30

The selected part-type-clusters: PC - 1 = 6,15,19,24,25,28,35,38; PC - 2 = 1,3,4,5,7,8,9,13,14,16,17,21,22,26,27,29,30,34,36,37 and PC - 3 = 2,10,11,12,18,20,23,31,32,33,39,40,41

and the optimum manufacturing-cell is

MC = (MC-1,PC-1),(MC-2,PC-3),(MC-3,PC-2).

This final assignment is illustrated in Figure 4. The developed algorithm was also applied for the case when c=4 and max_iterations =10 and resulted in 2 successful machine-type-clusters formation solutions and 2 successful part-type-clusters formation solutions.

The two successful machine-type-clusters formation solutions are: 1. MC-1 = 8,9,15,19,20,28,29,30; MC-2 = 5,6,14,16,18,24,25; MC-3 = 1,2,3,4,10,11,12,13,23,27 and MC-4 = 7,17,21,22,26

2. MC-1 = 8,9,19,20,27,28,29,30; MC-2 = 5,6,7,14,15,16,18,24,25,26; MC-3 = 1,2,3,4,10,11,12,13,22,23 and MC - 4 = 17,21

The two successful part-type-clusters formation solutions are:

1. PC-1 = 3,13,17,21,30; PC-2 = 4,5,11,16,18,20,26,27,34,36,37; PC-3 = 2,10,12,23,31,32,33,39,40,41 and PC- 4 = 1,6,7,8,9,14,15,19,22,24,25,28,29,35,38

2. PC-1 = 3,13,17,21,22,30,37; PC-2 = 5,11,16,18,27,34,36; PC-3 = 2,10,12,20,23,31,32,33,39,40,41 and PC- 4 = 1,4,6,7,8,9,14,15,19,24,25,26,28,29,35,38

Again, the assignment technique was then applied to the above machine-type-clusters and part-type-clusters to obtain the optimum manufacturing -cells. This technique advises us to select the following machine-type-clusters and part-type-clusters: The selected machine-type-clusters: MC-1 = 8,9,19,20,27,28,29,30; MC-2 = 5,6,7,14,15,16,18,24,25,26; MC-3 = 1,2,3,4,10,11,12,13,22,23 and MC - 4 = 17,21

The selected part-type-clusters: PC-1 = 3,13,17,21,30; PC-2 = 4,5,11,16,18,20,26,27,34,36,37; PC-3 = 2,10,12,23,31,32,33,39,40,41 and PC- 4 = 1,6,7,8,9,14,15,19,22,24,25,28,29,35,38

14

and the optimum manufacturing-cell is

MC = (MC-1,PC-1),(MC-2,PC-2),(MC-3,PC-3),(MC-4,PC-4). This final assignment is illustrated in Figure 5. These results demonstrate the validity of the developed algorithm as when the Chu and Hayya approach was followed to solve the same input data an infeasible solution was obtained as discussed in sub-section 3.1 and 3.2. In addition this new approach will result in manufacturing -cells with higher global efficiency as supported by the result obtained by comparing the two approaches using 28 different 40x40 machine-type/part-type incidence matrices with different densities and c values (Refer to Table 6). 5. Conclusions In this article, an outline of the Chu and Hayya approach to cell formation problems was presented and it was noted that this technique has shortcomings for some of the cases investigated. Strategies to overcome these problems were studied, they included the separate formation of part-type-clusters and machine-clusters followed by the assignment techniques. In addition, this new method increased the global efficiency of the resulting manufacturing -cells. 6. Acknowledgements The research reported in this paper has been supported by a Monash University Postgraduate Publication Award received from the Monash University Research Training Support Branch. The authors also wish to acknowledge the helpful suggestions recived from the reviewers of this paper. References Bezdek, J.C., 1981. Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum

Press, New York. Bunday, B.D., Basic Linear Programming , Edward Arnold, London; Baltimore; Md, USA,

1984. Chu, C.H., 1995. “Recent Advances in Mathematical Programming for Cell Formation”,

Planning, Design and Analysis of Cellular Manufacturing Systems, 1995, ed. Kamrani, A., Parsaei, H.R., and Liles, D.H., p. 3-46, Elsevier Science B.B., Amsterdam, The Netherlands.

Chu, C.H., and Hayya, J.C., 1991. “A Fuzzy Clustering Approach to Manufacturing Cell Formation”, International Journal of Production Research, Vol. 29, No. 7, p. 1475-1487.

Gindy, N.N.Z., Ratchev, T.M., and Case, K., 1995. “Component Grouping for Group Technology Applications - a Fuzzy Clustering Approach with Validity Measure”, International Journal of Production Research, Vol. 33, No. 9, p. 2493-2509.

Kumar, K.R., and Vannelli, A., 1987. “Strategic subcontracting for efficient disaggregated manufacturing”, International Journal of Production Research, Vol. 25, No. 12, p. 1715-1728.

McAuley, J., 1972. “Machine grouping for efficient production”, The Production Engineer, Vol. 51, p. 53-57

Masnata, A., and Settineri, L., 1997. “An Application of Fuzzy Clustering to Cellular Manufacturing”, International Journal of Production Research, Vol. 35, No. 4, p. 1077-1094.

Moodie, C., Uzsoy, R., and Yih, Y., 1995. Manufacturing Cells: A Systems Engineering View . Taylor & Francis Ltd., Bristol, PA, USA.

Ng, S.M., 1996. “On the Characterization and Measure of Machine Cells in Group Technology”, Operations Research, Vol. 44, No. 5, p. 735-744.

15

Ponnambalam, S.G., and Aravindan, P., 1994. “Design of Cellular Manufacturing Systems Using Objective Functional Clustering Algorithms”, International Journal Advanced Manufacturing Technology, Vol. 9, p. 390-397.

Susanto, S., Kennedy, R.D., and Price, J.W.H., 1997. A preliminary study of a clustering and assignment problem-based cell formation problem. Proceedings of The International Conference on Manufacturing Automation (ICMA ‘97), Vol. 1, p. 95-104.

Taha, H., Operations Research: an introduction, 5th. ed., p. 214, Macmillan Publishing Company, Don Mills, Ontario, 1992.

Wemmerlöv, U., 1988. Production Planning and Control Procedures for Cellular Manufacturing Systems: Concepts and Practices. The Library of American Production, Falls Church, Virginia, USA.

Wemmerlöv, U., and Hyer, N.L., 1992. “Group Technology”, Handbook of Industrial Engineering, ed. Salvendy, G., John Wiley & Sons, Inc., New York, USA.

Xu, H., and Wang, H.-P (Ben), 1989. “Part Family Formation for Group Technology Applications Based on Fuzzy Mathematics”, International Journal of Production Research, Vol. 27, No. 9, p. 1637-1651.

16

Appendices

Figure 1 . The 12x10 initial machine -type/ part-

type incidence matrix (Source: McAuley,J., 1972)

Part-type (k)

Initial degree of membership of part-type k to part-type-cluster i

(u )ki

1 2 3 01 1.000 0.000 0.000 02 0.000 1.000 0.000 03 0.000 0.000 1.000 04 1.000 0.000 0.000 05 0.000 1.000 0.000 06 0.000 0.000 1.000 07 1.000 0.000 0.000 08 0.000 1.000 0.000 09 0.000 0.000 1.000 10 1.000 0.000 0.000

Table 1. Initial solution to the application of Chu and Hayya approach to input data as described in Figure 1.

Part-type number (k)

Final degree of membership of part-type k to

part-type-cluster i (u )ki

Decision: part-type k is

assigned to part-type-cluster

1 2 3

01 0.943 0.032 0.025 1 02 0.943 0.032 0.025 1 03 0.630 0.157 0.213 1 04 0.427 0.285 0.290 1 05 0.014 0.974 0.012 2 06 0.014 0.974 0.012 2 07 0.179 0.539 0.282 2 08 0.018 0.019 0.963 3 09 0.018 0.019 0.963 3 10 0.061 0.066 0.873 3

Table 2. The degree of membership of part-types and its assignment to part-type-clusters

Machine-type

number (j)

(v )ji

Decision: machine-type j is

assigned to

1 2 3 machine-type-cluster

01 0.985 0.047 0.046 1 02 0.985 0.047 0.046 1 03 0.909 0.012 0.017 1 04 0.743 0.001 0.000 1 05 0.014 0.951 0.028 2 06 0.014 0.951 0.028 2 07 0.076 0.860 0.030 2 08 0.002 0.002 0.926 3 09 0.002 0.002 0.926 3 10 0.000 0.000 0.656 3 11 0.743 0.826 0.001 2 12 0.743 0.826 0.001 2

Table 3. The part-type-cluster centres for machine -types and machine-types assignment

Part -types

0 0 0 0 0 0 0 0 0 1 1 2 3 4 5 6 7 8 9 0

01 1 1 1 1 0 0 0 0 0 0 02 1 1 1 1 0 0 0 0 0 0 03 1 1 1 0 0 0 0 0 0 0 04 1 1 0 0 0 0 0 0 0 0 Machine- 05 0 0 0 0 1 1 1 0 0 0 types 06 0 0 0 0 1 1 1 0 0 0 07 0 0 0 1 1 1 0 0 0 0 08 0 0 0 0 0 0 0 1 1 1

09 0 0 0 0 0 0 0 1 1 1 10 0 0 0 0 0 0 0 1 1 0

17

Figure 2. The final solution for machine-type/ part-type incidence matrix in Figure 1 with c = 3.

Figure3: The 30x41 initial machine-type/part-type incidence

matrix (Source: Kumar and Vaneli, 1987)

Part-types 0 0 0 0 0 0 0 0 0 1 1 2 3 4 5 6 7 8 9 0 01 1 1 1 1 0 0 0 0 0 0 02 1 1 1 1 0 0 0 0 0 0 03 1 1 1 0 0 0 0 0 0 0 04 1 1 0 0 0 0 0 0 0 0 05 0 0 0 0 1 1 1 0 0 0 Machine- 06 0 0 0 0 1 1 1 0 0 0 types 07 0 0 0 1 1 1 0 0 0 0 11 1 1 0 0 1 1 0 0 0 0 12 1 1 0 0 1 1 0 0 0 0 08 0 0 0 0 0 0 0 1 1 1 09 0 0 0 0 0 0 0 1 1 1 10 0 0 0 0 0 0 0 1 1 0

Part-types 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 01 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 1 02 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 03 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 04 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 05 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 06 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 07 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 08 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 09 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 11 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 12 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 Machine- 14 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 types 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 19 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 20 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 21 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 0 22 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 23 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 25 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 27 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 29 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 30 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

18

Part-type (k) Initial degree of membership of part-type k to

part -type -cluster i (u )ki

1 2 3 01 1.000 0.000 0.000 02 0.000 1.000 0.000 03 0.000 0.000 1.000 04 1.000 0.000 0.000 05 0.000 1.000 0.000 06 0.000 0.000 1.000 07 1.000 0.000 0.000 08 0.000 1.000 0.000 09 0.000 0.000 1.000 10 1.000 0.000 0.000 11 0.000 1.000 0.000 12 0.000 0.000 1.000 13 1.000 0.000 0.000 14 0.000 1.000 0.000 15 0.000 0.000 1.000 16 1.000 0.000 0.000 17 0.000 1.000 0.000 18 0.000 0.000 1.000 19 1.000 0.000 0.000 20 0.000 1.000 0.000 21 0.000 0.000 1.000 22 1.000 0.000 0.000 23 0.000 1.000 0.000 24 0.000 0.000 1.000 25 1.000 0.000 0.000 26 0.000 1.000 0.000 27 0.000 0.000 1.000 28 1.000 0.000 0.000 29 0.000 1.000 0.000 30 0.000 0.000 1.000 31 1.000 0.000 0.000 32 0.000 1.000 0.000 33 0.000 0.000 1.000 34 1.000 0.000 0.000 35 0.000 1.000 0.000 36 0.000 0.000 1.000 37 1.000 0.000 0.000 38 0.000 1.000 0.000 39 0.000 0.000 1.000 40 1.000 0.000 0.000 41 0.000 1.000 0.000

Table 4. Initial solution to the application of Chu and Hayya approach to input data as described in Figure 3.

Part-type (k) Initial degree of membership of part-type k to part-type-cluster i (u )ki

1 2 3 4 01 1.000 0.000 0.000 0.000 02 0.000 1.000 0.000 0.000 03 0.000 0.000 1.000 1.000 04 0.000 0.000 0.000 1.000 05 1.000 0.000 0.000 0.000 06 0.000 1.000 0.000 0.000 07 0.000 0.000 1.000 1.000 08 0.000 0.000 0.000 1.000 09 1.000 0.000 0.000 0.000 10 0.000 1.000 0.000 0.000 11 0.000 0.000 1.000 1.000 12 0.000 0.000 0.000 1.000 13 1.000 0.000 0.000 0.000 14 0.000 1.000 0.000 0.000 15 0.000 0.000 1.000 1.000 16 0.000 0.000 0.000 1.000 17 1.000 0.000 0.000 0.000 18 0.000 1.000 0.000 0.000 19 0.000 0.000 1.000 1.000 20 0.000 0.000 0.000 1.000 21 1.000 0.000 0.000 0.000 22 0.000 1.000 0.000 0.000 23 0.000 0.000 1.000 1.000 24 0.000 0.000 0.000 1.000 25 1.000 0.000 0.000 0.000 26 0.000 1.000 0.000 0.000 27 0.000 0.000 1.000 1.000 28 0.000 0.000 0.000 1.000 29 1.000 0.000 0.000 0.000 30 0.000 1.000 0.000 0.000 31 0.000 0.000 1.000 1.000 32 0.000 0.000 0.000 1.000 33 1.000 0.000 0.000 0.000 34 0.000 1.000 0.000 0.000 35 0.000 0.000 1.000 1.000 36 0.000 0.000 0.000 1.000 37 1.000 0.000 0.000 0.000 38 0.000 1.000 0.000 0.000 39 0.000 0.000 1.000 1.000 40 0.000 0.000 0.000 1.000 41 1.000 0.000 0.000 0.000

Table 5. Initial solution to the application of Chu and Hayya approach to input data as described in Figure 3.

19

Figure 4. The final solution for machine-type/ part-type incidence matrix in Figure 3 with c=3.

Part-types 0 1 1 2 2 2 3 3 0 1 1 1 1 2 2 3 3 3 3 4 4 0 0 0 0 0 0 0 1 1 1 1 2 2 2 2 2 3 3 3 3 6 5 9 4 5 8 5 8 2 0 1 2 8 0 3 1 2 3 9 0 1 1 3 4 5 7 8 9 3 4 6 7 1 2 6 7 9 0 4 6 7 04 0 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 06 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 07 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 08 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 13 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 16 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 24 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 27 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

20

Figure 5. The final solution for machine-type/ part-type incidence matrix in Figure 3 with c = 4.

Part-types 0 1 1 2 3 0 0 1 1 1 2 2 2 3 3 3 0 1 1 2 3 3 3 3 4 4 0 0 0 0 0 1 1 1 2 2 2 2 2 3

3 3 3 7 1 0 4 5 1 68 0 6 7 4 6 7 2 0 2 3 1 2 3 9 0 1 1 6 7 8 9 4 5 9 2 4 5 8 9 5 8 08 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 1 0 0 09 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 20 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 29 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 30 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 05 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 06 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 07 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 0 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Machine- 15 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 types 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 18 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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Global for

Efficiency c=3 of Global

for Efficiency c=4 of Global

for Efficiency c=4 of

Data source file

Density The Chu and Hayya algorithm

The developed algorithm

The Chu and Hayya algorithm

The developed algorithm

The Chu and Hayya algorithm

The developed algorithm

40x40a.dat 0.153 53.69 65.98 60.66 65.57 55.33 36.48

40x40b.dat 0.188 67.77 63.46 52.49 57.48 43.85 41.86

40x40c.dat 0.209 58.51 58.21 48.66 49.25 49.85 52.84

40x40d.dat 0.228 55.22 65.66 fails 41.21 fails 46.70

40x40e.dat 0.253 59.26 64.94 46.17 49.63 41.73 41.98

40x40f.dat 0.282 54.77 55.43 47.67 46.56 fails 35.48

40x40g.dat 0.317 54.83 64.10 fails 47.93 42.01 35.90

40x40h.dat 0.341 51.10 56.23 46.70 45.42 37.91 37.18

40x40i.dat 0.361 49.57 61.01 42.29 46.27 fails 38.13

40x40j.dat 0.387 48.30 61.55 45.07 42.65 48.79 41.68

40x40k.dat 0.411 48.48 50.00 41.19 45.74 39.67 39.82

40x40l.dat 0.438 55.21 52.92 40.80 43.65 39.23 32.24

40x40m.dat 0.467 46.72 55.02 40.29 50.87 34.54 37.48

40x40n.dat 0.483 45.28 56.40 42.17 42.82 32.47 38.03

40x40o.dat 0.516 48.73 57.70 43.27 52.48 fails 34.55

40x40p.dat 0.543 55.93 55.35 fails 41.20 46.95 43.96

40x40q.dat 0.559 47.09 55.03 36.47 45.41 36.58 39.82

40x40r.dat 0.584 43.53 42.46 36.26 37.11 30.05 31.44

40x40s.dat 0.621 48.39 51.51 46.18 42.66 fails 35.61

40x40t.dat 0.643 42.76 50.44 44.12 50.44 fails 34.01

40x40u.dat 0.667 45.74 55.39 36.27 53.14 fails 33.27

40x40v.dat 0.680 45.40 48.35 38.05 39.34 30.88 34.65

40x40w.dat 0.710 41.11 51.94 29.67 45.42 fails 30.46

40x40x.dat 0.741 39.04 42.75 39.38 37.52 28.58 32.88

40x40y.dat 0.780 38.70 41.67 34.54 39.42 25.24 35.10

40x40z.dat 0.783 40.70 48.60 29.85 35.75 29.77 38.31

40x401.dat 0.814 43.70 52.30 36.41 39.32 fails 29.65

40x402.dat 0.856 38.98 42.19 29.64 41.24 28.47 30.58

Table 6. The global efficiency resulting from the Chu and Hayya and the new developed algorithm for number of manufacturing-cells (c) equals to 3,4 and 5 for different data source files (files are available from the authors)

Authors:

Sani Susanto, Department of Industrial Engineering, Faculty of Industrial Technology, Catholic University of Parahyangan, Jl. Ciumbuleuit 94, Bandung -40141, West Java, Indonesia, phone/fax: 62-022-5202552 E-mail: ssusanto@home.unpar.ac.id. Sani Susanto is a lecturer in the Department of Industrial Engineering, Faculty of Industrial Technology at Catholic University of Parahyangan. He holds a Bachelor Degree in Mathematics from Bandung Institute of Technology (Indonesia, 1987), a Bachelor Degree in Agrosocioeconomics from the University of Padjadjaran

(Indonesia, 1991), a Masters Degree in Industrial and Engineering Management from Bandung Institute of Technology (Indonesia, 1992) and a PhD in Industrial and Engineering Management from Monash University (Australia, 1998). His research interests lie in the areas of Production Systems,

22

Operations Management, the application of Quantitative Techniques in Industrial Engineering, and Engineering Education. He is a member of Institute of Industrial Engineers Australia, Unesco International Centre for Engineering Education and Special Interest Group Working Group 5.7.

Dr. Robert Damian Kennedy, Mechanical Engineering Department, Monash University-Caulfield Campus, PO Box 197, Caulfield East, Victoria 3145 Australia, Ph/Fax: 61-3-9903 2175 Damian.Kennedy@eng.monash.edu.au Dr. Robert Damian Kennedy is currently Executive Director of the Monash Centre for Manufacturing and Industrial Engineering (MonMIEC), Monash University, Australia. He holds the degrees of BE, Grad.Dip.Mgmt (RMIT, Australia), MEngSci (Northwestern,

Ill) and PhD (West Virginia). Prior to joining the Engineering Faculty at Monash in 1980, Dr. Kennedy was a practicing design engineer in industry specialising in microelectronics design and optical communications research and development. At Monash he teaches subjects in Production Planning a nd Control, Computer Simulation, Design of Productive Systems and Engineering Management. His research interests lie in the development of industrial engineering theory as it relates to the design of productive systems. He is an active consultant to industry in the development of productivity and performance in the manufacturing and service industries.

Associate Professor John W.H. Price, Mechanical Engineering Department, Monash University-Caulfield Campus, PO Box 197, Caulfield East, Victoria 3145 Australia, Ph: (61)(03) 9903 2868, Fax: 61-3-9903 2766, E-mail: John.Price@eng.monash.edu.au Associate Professor John W.H. Price is at the Mechanical Engineering Department, Monash University, Australia. He holds the degrees of BE, MEngSci (Melbourne), PhD and DIC (London, Imperial College).

He was employed until 1992 by the State Electricity Commission of Victoria as the Senior Plant Integrity Engineer. From 1978 to 1984 he worked in the UK Nuclear industry for the National Nuclear Corporation (NNC) on the integrity of the pressure vessels in several nuclear reactor systems. At Monash he teaches Manufacturing, Design and Engineering Management subjects and researches in the area of structural failure. He is an active consultant, widely used by industry to investigate accidents and failures.