A Manufacturing Cell Formation Algorithm with Minimum Inter-Cell ...

International Journal of Applied Engineering Research ISSN 0973-4562 Vol. 1 No.1 (2006) pp. 9-22 (c) Research India Publications http://www.ripublication.com/ijaer.htm

A Manufacturing Cell Formation Algorithm with Minimum Inter-Cell Movements Adnan M. Mukattash Dept. Industrial Engineering, Hashemite University Zarka, Jordan P.O. Box :150459 Zarka 13115, Jordan E-mail : [email protected] Mazin. B. Adel Dept. Industrial Engineering, University of Technology, Baghdad, Iraq Khaldoun K. Tahboub Dept. Industrial Engineering, University of Jordan, Amman, Jordan

Abstract The main objective of this paper is to construct an algorithm for the formation of manufacturing cells with unbounded cell sizes, such that intercell movements are minimized. The assignment of parts to cells is then performed such that the number of exceptional parts is minimized. The solution of the algorithm starts from some initial conditions. A closed interval for the solution is then specified with a lower bound of the minimum inter-cell movements for the initial conditions and an upper bound on the inter-cell movements of the last cell. A combinatorial proof is provided to show that there exists at least one solution starting from the initial conditions. Keywords: Flexible Manufacturing Cells, Group Technology, Cell Formation, Minimum Inter-Cell Movements

Introduction The basic problem in the design of Cellular Manufacturing (CM) systems lies in the identification and grouping of parts that share similar processes into families, and their associated machines into cells. According to Adil et al (1996), the ideal CM solution is achieved by having perfect diagonal blocks, i.e., when all parts in a family are fully processed within a single cell. Unfortunately, such a solution is rarely accomplished in practice. Parts that are manufactured by more than one machine cell are known as exceptional elements (or parts). A CM solution that yields a minimum

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A.M. Mukattash et al

number of exceptional parts is highly desirable since it minimizes intercellular movements. Hence, developing such CM has become an objective of many practitioners in the field of flexible manufacturing. The first step in cell design is the cell formation. The goal of cell formation is to create separable machine clusters and part families. The cell formation process often includes the identification of Exceptional Elements (EE) or parts. The existence of EEs in cell formation solutions is not a trivial problem (Kern and Wei, 1991). Exceptional elements create interactions between two independent manufacturing cells. They can be viewed as parts that require processing on machines in two or more cells. On the otherhand, a void in the cell formation solution indicates that a machine assigned to a cell is not required for the processing of a part in the cell. The exceptional element is created when a part requires processing on a machine that is not available in the allocated cell of the part. Exceptional elements and voids are the first two problems which will face the designer. Adil et al (1996), summarized the implications of voids and exceptional elements. From the above, it can be realized that, the first challange in cell formation is to minimize the number of exceptional elements. The second challenge in cell formation is specifying the number of cells solution. Many of the literature presented algorithms based on the assumption that the number of cells is specified in advance (e.g., Viswanathan (1995), Akturk and Balkose (1996), Dahel and Smith (1993), Boctor (1996)). Crama and Oosten (1996) pointed out that “it is sometimes necessary to extend this basic model with additional constraints in order to express limitations of a physical, technological or organizational nature. These constraints may express special relations between machines and/or parts. They may also be cardinality constraints on the size or the number of cells. Chen and Guerrero (1994) developed a general algorithm for cell formation in order to find the optimal solution with minimum inter-cell movements. Their algorithm is set to be terminated at any point to accommodate the manager’s desired number of cells. Unfortunately, their algorithm does not provide for finding possible alternative solutions. Rajamani and Aneja (1996) also developed an algorithm to solve the problem of cell formation by specifying the number of cells in advance. However, they pointed out that “sometimes it may not be economical or practical to achieve cell independence”. Finally, Mukattash et al (1997) developed a special algorithm for 2cell formation problems based on Stirling number. The main objective of this paper is to construct an algorithm, with unbounded or bounded cell sizes, such that inter-cell movements are minimized in order to handle the above two challanges. The algorithm starts with a specification of the required number of cells for the solution. This specification is translated into the initial conditions for starting the algorithm. Then, a closed interval for the solution is specified with a lower bound giving the minimum inter-cell movements for the initial conditions and an upper bound specifying the maximum allowable inter-cell movements of the last cell added to the solution. The output of the algorithm is cells with machines assigned to each cell. Finally, the assignment of parts to cells is then performed such that the number of exceptional parts (elements) is minimized. The

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Manufacturing Cell Formation Algorithm

merits of the proposed algorithm will be shown by comparing its performance to other solution methods applied to cited problems in the literature.

An Algorithm for Cell Formation With Unbounded Cell Size One can use the general recursion formula of Stirling number to find out exactly the number of ways to distribute n-distinguishable machines into p-indistinguishable cells with no cell empty. The general recursion formula of Stirling number is given below, in terms of machines distributed into cells as follows (Polya et al, 1983): A pn = A pn −−11 + p A pn −1

(1)

where, n: number of machines, and p: number of cells For example, there are 42525 ways to distribute ten machines into five cells, i.e., A = 42525. On the other hand, equation (1) provides more useful information relating to cell formation. To illustrate, let n = 4 and p = 2, then A24 =A13 + 2A23 = 1 + 2(3) = 7. The formula states that there are 7 ways to distribute 4 machines among 2 cells. However, it also states that allocating for machines to two cells is equivalent to allocating three machines to two cells then allocating the forth machine to one the cells. This result provides the main theme of the proposed algorithm. The challenge is now to choose the right machine that will provide a solution with minimum inter-cell movements. To develop the algorithm, we start with a limited number of machines (initial condition) equal to the number of the required cells. This limited number of machines must be chosen in such a way that leads to the most adequate solution(s). Also, the addition of the new machine to the initial condition must ensure that the addition of machine i will give less inter-cell movements than the addition of machine j. The addition process is continued until all the machines are assigned and the most adequate solution(s) is obtained. The path which leads to the most adequate solution is called the active path. 10 5

Assumptions The following assumptions are made for the derivation of the proposed cell formation algorithm: 1) 2) 3) 4) 5)

Number of cells is specified. Cell size is unbounded. Every cell in the initial condition(s) contains one and only one machine. Initial condition(s) must have minimum inter-cell movements. There is only one machine from each type of machines and one process plan for each part. 6) The operation sequence could be orderly rearranged.

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A.M. Mukattash et al 7) Processing times on all machines are equal. 8) Batch size is the same for all products.

Lower bound on the initial condition(s) The first step in deriving the general algorithm is to choose the initial condition(s) that leads to the most adequate solution(s) by passing through the active path(s). Next, we need to specify the closed interval of the most adequate solution in such a way that all the elements of the closed interval belong to the elements of Stirling number. Hence, the lower bound of the closed interval of the most adequate solution(s) represents the inter-cell movements of the initial condition(s). Given that, the lower bound is derived as follows: Let, A: the incidence matrix relating the parts to the machines. The elements of this matrix are either 0 or 1. aij =1 if part i is processed on machine j and it is 0 otherwise for all parts j and machines j. R : the set of all possible initial conditions To: some initial condition, then R = ∪ To for all possible initial conditions and To= ∪ tr where tr is any machine in the incidence matrix A and ∪ is the union operation. iA(π): value of inter-cell movements for any solution π. β: the lower bound of the closed interval containing To. Hence, β = iA(To) Now let,

(2)

p

ε j = ∑ a t r j , j =1,2,...,m

(3)

r =1

whereεj represents the number of ones in column j corresponding to machines t1,t2,…,tp in matrix A. Also,

if ε j = 0 0,  εj =  ε − 1, otherwise  j

(4)

where εj represents the inter-cell movements value of column j corresponding to machines t1, t2,…, tp in matrix A. Accordingly, m

β = ∑εj

(5)

j=1

Hence, β is inter-cell of matrix A corresponding to the machines t1, t2,…, tp in matrix A, and it is the minimum inter-cell.

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Manufacturing Cell Formation Algorithm

Derivation of the initial condition(s) Let X be a (n×1) vector containing zero-one elements such that: X=(xi) (6) Where, xi= 0 or 1, i = 1,2,...,n such that, xi=1 means that machine i exists in one of the cells of T (where T is a set containing p-cells such that each cell contains only one machine), otherwise xi=0. In other words, if i∈ T then xi =1 if i∉ T then xi =0 Let, Sj be the sum of ones in column j, such that this column corresponds to machine i, i.e., n

Sj =

∑ x i a ij , j = 1,2,...,m

(7)

i =1

and

Sj = 0 0,  dj =  S − 1, otherwise  j

(8)

where dj represents the inter-cell in column j such that this column corresponds to machine i. Then, the inter-cell of the matrix A corresponding to T is the sum of the inter-cell of all the columns, denoted by Z, such that: m

Z=

∑

dj

(9)

j=1

Thus minimizing inter-cell movements can be formulated as follows: Min Z s.t. n

∑

xi = p

i =1

where minimum Z represents the inter-cell of all the initial conditions with lower bound β. Upper bound on the most adequate solution Let α represent the upper bound of the most adequate solution, then β ≤ iA(x) ≤ α (10) The inter-cell of α must be chosen to be very close to iA(x). For that it is assumed that: 1. (p-1) cells contain only one machine. 2. The last cell contains (n-p+1) machines. The above two conditions can be written as follows: C=(t1)(t2)(t3)...(tp-1) (tp tp+1...tn) (11)

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where Ci is the ith cell. Then the problem is to find Min(iA(c)). Let the vector X contain n-elements of xi, where, xi= 0 or 1, such that the number of ones is equal to (p-1). The elements of the vector X which are ones, will be put in a cell(s) which contains only one machine, and the other elements of the vector X (zeros) will be put in a cell which contains (n-p+1) machines. In other words: If ti= j∈ C then xj=1 1≤ i ≤ p-1 and xj=0 p≤ i ≤ n Let xi =1, then the conjugate of xi (denoted by xi ) is xi = 0 and vice versa. In general, the above conjugate relation can be written as follows: xi =1 - xi (12) Let, n

hj =

n

∑

xi aij , j = 1,2,....,m =

i =1

∑

i =1

n

aij -

∑

xi aij

(13)

i =1

where hj is the number of ones in column j of the last cell, Since X has (p-1) zero-one elements, and if xi = 0 then machine i is in the last cell. Let,

0, if h j = 0 fj =  1, otherwise

(14)

where fj is equal one, if column j has at least one non-zero element corresponding to the machines in the last cell, otherwise fj=0. Let, n

Gj =

∑

xi aij +fj = Sj + fj

(15)

i =1

where Gj represents the sum of ones of column j in the (p-1)-cells plus the ones in the last cell, under the condition of fj. Let,

0, if G j = 0 = dj  G j - 1, otherwise

(16)

where dj represents the inter-cell of column j of the selected state. In order to find the inter-cell for the first state [it is the selected state such that (p-1)-cells contain one machine each and the last cell contains (n-p+1)-machines] of matrix A, the problem is formulated as follows: m

Minimize Z =

∑

dj

j= 1

s.t. n

∑

xi = p -1

i =1

The above constraint represents the (p-1)-cells which contain one machine.

(17)

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Manufacturing Cell Formation Algorithm

Steps of adding a machine to previously allocated machines The initial condition for each cell contains one machine and has minimum inter-cell for this state (all cells contain one machine). The initial condition will be denoted by T0 with p-cells. The next step is to add a new machine different from those existing in the initial condition. The new machine will be added to the cells of the initial condition, then the inter-cell between this new machine and the machines in the initial condition is computed. The first addition of the new machine to the first cell will be denoted by T1(1), the second addition of the new machine to the second cell will be denoted by T1(2), and the last addition to the last cell will be denoted by T1(p). After that, we choose the machine from the above additions which has minimum inter-cell. This state will be denoted by T1. In other words, Min iA(T(k)1) = iA(T1) , k = 1,2,…,p

(18)

In a similar manner, the minimum inter-cell for the second machine not included in T 1 is added to the cells, and the result will be as follows: (19) Min iA(T(k)2) = iA(T2) , k = 1,2,…,p In general the addition of a new machine to the previous machines can be represented as follows: Min iA(T(k)i) = iA(Ti) , i =1,2,…, n-p ;k = 1,2,…,p (20) The final result of all the additions is the most adequate solutions denoted by Tn-p. To choose the machine to be added to the previous machines, we first multiply the matrix A by At and the result will be a matrix of size (n×n) denoted by B. Matrix B will represent the inter-cell movement between the machines in the matrix A. The second step is to take out the columns of the initial condition(s) from matrix B. In order to choose the machine for the first addition, we look for the highest value in matrix B, such that the diagonal elements of matrix B are not chosen. Then the selected machine will be the machine in the column of matrix B which corresponds to the highest value. The process of selecting a new machine for addition can be represented as follows: h = Max b t j , k=1,2,...,p+i, j=1,2,...,n k j∉Ti tk ≠ j

(21)

If one finds more than one highest value in matrix B, then we put these values in a set H. Where, H = {j ∈ H b t j = h} k and, tp+i+1 ∈ H

(22)

To determine the inter-cell (iA) of adding the new machine to the previous machines (Ti), one can write,

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A.M. Mukattash et al m

iA(Ti)=

∑

εj

(23)

j= 1

where εj is the inter-cell of column j of matrix A, which corresponds to the machine in Ti, such that:

0, if ε j = 0  εj =  ε -1, otherwise  j

(24)

p

εj=

∑

µkj

(25)

k =1

where εj is the number of ones in column j in each cell of Ti, and µkj = 1 or 0. If the number of ones in cell k greater than zero, then µkj= 1, otherwise µkj = 0. If more than one initial condition exists, then the above steps are repeated for each initial condition, then from all the solutions of the initial conditions the solution(s) which has minimum inter-cell which is equal to or less than α is chosen. This minimum value represents the most adequate solution. If the sequence of the following elements T0,T1,...,Tn-p leads to the most adequate solution then we will show that the following relation is correct β≤ iA(Ti) ≤ α. For the lower bound condition β, if iA(Ti) < β, then Min iA(T) < β and this contradicts the assumption that Min iA(T) = β, for that β is less than or equal to iA(Ti). For the upper bound condition α, since we are adding a new machine to the previous machine(s), then the inter-cell movements will increase or will not change . Therefore, if i < j, then iA(Ti) ≤ iA(Tj) but the most adequate solution, i.e., with minimum inter-cell movements, is Tn-p , then, iA(Ti) ≤ iA(Tn-p) also, iA(Tn-p) ≤ α, then iA(Ti) ≤ α and β≤ iA(Ti) ≤ α Steps of the G-Algorithm Given an incidence matrix A of size (n×m) with elements equal to zero or one, such that n represents the number of machines, m represents the number of parts. Also, let p represents the number of cells. Step1: Start the algorithm, set i=0 Step 2: Determine the inter-cell between two machines by multiplying the matrix A by At, denoted by B=AAt, and bkj=iA((k)(j))

17

Manufacturing Cell Formation Algorithm

Step 3: Find the initial condition T0 by finding the minimum inter-cell of T, where T = (t1)(t2)...(tp) and Min iA(T) = iA(Ti) = β. Step 4: Find the upper bound value of α, where α is minimum inter-cell of C , α = Min iA(C) ,C = (t1)(t2) …(tp-1)(tptp+1 …tn) Step 5: i = i+1 Step 6: Choose the machine from the set R = {tp+1, …,tn}, which has Max bkj where k represents the machine of Ti, and j belongs to the set R R = R/{j}. Step 7: Find Min iA(Ti(k)) k=1,2,…,p in order to find Ti. Step 8: If i < n-p go to step 5. Step 9: If α < iA(Tn-p) i =1 and Min iA(T1(k)) without the previous state T1 and go to step 5 Step 10: If α > iA(Tn-p), α=iA(Tn-p). Ιf there exists another initial conditions ,i = 0: go to step 5, else Stop. Step 11: Now, let the most adequate solution be Tn-p. To show that the algorithm leads to the most adequate solution, we will start working on the algorithm backward. This means that, instead of adding a new machine, we will take out a machine from the last cell. At the same time we take into consideration that every cell has at least one machine in it. Then we will get the following inverse sequence Tn-p, Tn-p-1, Tn-p-2,…, T0 (26) This inverse sequence is equivalent to the sequence of the algorithm. To show that the initial condition exists in the inverse sequence, we assume that the inverse sequence does not have the initial condition and does not pass through it. Since T0 ∉ R (where R represents the set which contains all the elements of T0, such that iA(T0) = β) then iA(T0 ) ≥ β+1, and iA(Ti)= ρi ≥ β+1, ρi ≥ ρi -1 , i = 1,2,…,p (27) (28) iA(Tn-p)=ρn-p ≥ β +1 Let, Tn-p = Cn-p1 Cn-p2…Cn-pp, Tn-p ∈ Sn-pp (29) where, Cn-pi is machine cell i of Tn-p, i=1,2,…,p then, iA[(t1)(t2) …(tp)] ≥ β+1 , ∀ ti∈Cn-pi.

(30)

Because, if the above equality is less than (β+1), then [(t1)(t2)…(tp)] must belong to R. Since there must be an T0 ∈ R such that iA(T0) = β where ; T0=(t1)(t2) …(tp), then there exists at least t1 and t2 in the same cell, of Tn-p cells. Such that the intersection of t1 with its elements is minimum compared with the other elements (t2,…,tp), then we put t1 in another cell, such that the intersection of t1 with this cell is

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A.M. Mukattash et al

maximum compared with other cells. From the above condition of t1 we have two situations: We may find T0 inside the last cell (Tn-p), which means that the most adequate solution exists and the initial condition also exists, and this is contradictory to the assumption, that the inverse sequence does not have the initial condition and does pass through it. Or we get a value smaller than ρn-p which means (Tn-p) is not most adequate because, if (Tn-p) is the most adequate solution then it is impossible to be decreased. Finally, to evaluate the performance of the proposed algorithm with unbounded cell n

size for a given A p , the number of ways to determine all the possible initial n

conditions is equal to ( p ), and the total number of all iterations through the active path is equal to (n-p)p. For the upper bound of the most adequate solution (α), the   total number of iterations is equal to  n  . Hence, the total number of iterations p −1

n

required is equal to ( p ) + (n-p)p +

   n      p −1

.

Part Assignment to the Generated Cells In this section, the developed G-algorithm is utilized in forming machine cells with minimum inter-cell movements and assigning parts to these machine cells so as to achieve minimum number of exceptions, voids, or sum of exceptions and voids. Part assignment with minimum exceptional elements The output from the general algorithm is groups of machines that have minimum inter-cell movements. The objective here is to achieve minimum number of ones outside the cells. Let, X be the most adequate solution written in the form: X = C1 C2 …Cp = ( t h +1 t 2 ... t h )( t h +1 ... t h )...( t h +1 ... t h ) o 1 1 2 p −1 p where h0 = 0 and hp= n.

(30)

Moreover, the result obtained from the above general algorithm may be written in matrix form [M], where the elements of matrix [M] are: mij = a t j , j =1,2,…,m. (31) i

Hence, the assignment of parts into these cells takes the following steps: Let, p1: be the first cell p2: be the second cell # pp: be the pth cell such that, pi≠ϕ.

19

Manufacturing Cell Formation Algorithm

Let, n

∑

rjz =

mij , z = 1,2,…,p ; j=1,2,…,m

(32)

i =1 i ≠ h z −1 +1,.....hz

where rjz is the number of exceptions in column j and cell z. If Min {rjz}= rjk , then j∈ pk , z = 1,2,…,p

(33)

where k is the kth cell in column j which has minimum exceptions (αj). (rjk= αj).Then the new assignment of parts is given by: p1p2p…pp = ( t f ... t f )...( t f +1 ... t f ) (34) − o

1

p 1

p

and, f0 = 1, fp= m Such that the number of ones outside the diagonal must be minimum. Next the columns of matrix (M) are rearranged according to the above new arrangement of parts to obtain matrix (N). Finally, the total minimum number of ones outside the cells is equal to: m

∑

αj

(35)

j= 1

Part assignment with minimum voids The objective here is to achieve minimum number of voids inside the cells. The same steps of minimum exceptions are followed except that rjz is changed as follows: hz

rjz =

∑

mij

(36)

i = h z−1 +1

where rjz is the number of voids in column j and cell z and, mij=1-mij then continue the same steps as minimum exceptions.

(37)

Part assignment with minimum sum of exceptions and voids The objective here is to assign parts to cells so as to minimize the sum of the number of voids inside the diagonal blocks and number of ones outside the blocks. Then, follow the same steps as minimum exception except that (rjz) is changed as follows: hz

n

rjz =

∑

i =1 i ≠ h z −1 +1,.....hz

mij +

∑

mij

i = h z−1 + 1

(38) where rjz the sum of exceptions and voids in column j and cell z and, mij=1-mij then follow the same steps as minimum exceptions.

(39)

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A.M. Mukattash et al

Testing The G-Algorithm To illustrate the operational procedures for assigning parts to machine cells by applying the G-Algorithm with unbounded cell size, the researchers used the data of the problem used by Chen and Guerrero (1994). The results of applying the GAlgorithm with unbounded cell size (p=3) for forming cells of machines, followed by part assignment, are shown in Table 1 and Table 2, respectively.

Table 1: Resulting cell clusters for minimum exceptions.

4 1 2 3 5 6

11 1 0 0 0 0 0

1 1 1 0 1 0 0

2 1 1 1 0 0 0

3 4 5 6 8 9 10 12 13 14 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 0 1 0 0 0 1 0 1 1 0 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 0 0 0 1 Minimum number of exceptions =10

15 1 1 1 0 0 0

7 0 0 0 1 0 1

Table 2: Resulting cell clusters for minimum exceptions and voids.

4 1 2 3 5 6

1 1 1 0 1 0 0

2 1 1 1 0 0 0

8 11 15 3 9 10 12 13 4 5 1 1 1 0 0 0 0 1 0 0 1 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 1 0 0 0 1 0 1 1 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 Minimum number of voids and exceptions=24

6 0 0 1 0 1 1

7 0 0 0 1 0 1

14 0 0 0 1 1 1

Table 1 represents part assignment with minimum inter-cell movements while Table 2 represents part assignment with minimum sum of voids and exceptions. From the above tables, it is clear that part assignment with minimum sum of voids and exceptions will affect the number of parts inside the cell. Minimization of voids inside the cell will affect the efficiency of the solution. Comparison with other algorithms In order to evaluate the performance of the proposed approach, the solution of different problems taken from the literature are compared to the solutions obtained from the G-algorithm. Table 3 summarizes these results.

21

Manufacturing Cell Formation Algorithm Table 3: Computational results for problems taken from the literature. Method used DCA ROC2 ART1/KS G- algorithm Kusiak-Chow King G- algorithm King Kusiak-Chow G- algorithm ART1 G- algorithm Ideal seed G- algorithm Scaling G- algorithm

Problem size (n, p) 9, 9

8, 7

6, 10

8, 10 8, 20 7, 11

# cells obtained 3 3 3 3 2 2 2 2 2 2 2 2 3 3 2 2

# exceptions 1 1 1 1 0 0 0 3 0 0 0 0 9 9 2 2

Sum of voids and exceptions 5 5 5 5

15 15

The results show that the proposed approach obtained solutions with minimum number of exceptions or minimum sum of voids and exceptions that were comparable to the results of the selected problems.

Summary and Concluding Remarks An Algorithm for the formation of machine cells with unbounded cell size has been derived. To reduce the number of ways of forming p-cells from n-machines, the generation of cells starts from an initial condition. The initial condition is based on dissimilarity between the machines. Machines having maximum dissimilarity (having minimum inter-cell movements) were chosen for the initial condition(s).The number of machines to start with the initial condition is specified in advance by the designer. This means that the designer has the flexibility to choose the required number of cells. Moreover, the designer can choose the minimum number of machines in each cell. After choosing the initial condition, the next step is to add a new machine to the initial condition so as the inter-cell movements are minimum. A criteria to find an upper bound of the most adequate solution has been derived. The inter-cell movements of the added machine should not exceed the upper bound. The output of the general algorithm with unbounded cell size, is groups (cells) of machines having minimum number of inter-cell movements. To assign parts to these cells, the designer has the flexibility to choose between three cases, namely, minimize the number of exceptions, minimize the number of voids, or minimize the sum of the “number of voids and the number of exceptions”. The benefits of this algorithm is clear when the designer wants to change the traditional practices of manufacturing from functional

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A.M. Mukattash et al

production into small to medium-sized batch-oriented production. The benefits are achieved by having the flexibility of choosing the required number of cells. Moreover, the designer has the flexibility of choosing the most adequate number of cells by solving the problem for different values of p(required number of cells). Moreover, the above algorithm, has been tested on examples available in the literature and the results were comparable to those present in the literature.

References [1] [2]

[3] [4]

[5]

[6]

[7] [8]

[9]

[10]

[11] [12] [13]

Adil, G.K., Rajamani, D., and Strong, D., “Cell formation considering alternate routings”, Int. J. Prod. Res., Vol. 34, No. 5, pp. 1361-1380, 1996. Akturk, M. S., and Balkose, H.O., “Part-machine grouping using a multiobjective cluster analysis”, Int. J. Prod. Res., Vol. 34, No. 8, pp. 2299-2315, 1996. Boctor, F.F., “The minimum-cost, machine-part cell formation problem”, Int. J. Prod. Res., Vol. 34, No. 4, pp. 1045-1063, 1996. Boe, W.J., and Cheng, C.H., “A close neighbor algorithm for designing cellular manufacturing systems”, Int. J. Prod. Res., Vol. 29, No. 10, pp. 20972116, 1991. Chandrasekharan, M.O., and Rajagopalan, R., “An ideal seed non-hierarchical clustering algorithm for cellular manufacturing”, Int. J. Prod. Res., Vol. 24, No. 2, pp. 451- 464, 1986. Chen, H.G, and Guerrero, H.H., “A general search algorithm for cell formation in group technology”, Int. J. Prod. Res., Vol. 32, No. 11, pp. 27112724, 1994. Crama, Y. and Oosten, M., “Models for machine-part grouping in cellular manufacturing”, Int. J. Prod. Res., Vol. 34, No. 6, pp. 1693-1713, 1996. Dahel, N. E., and Smith, S. B., “Designing flexibility into cellular manufacturing systems”, Int. J. Prod. Res., Vol. 31, No. 4, pp. 933-945, 1993. Kern, G. M., and Wei, J.C., “The cost of eliminating exceptional elements in group technology cell formation”, Int. J. Prod. Res., Vol. 29, No. 8, pp. 15351547, 1991. Mukkatash, A., Adil, M., and Tahboub, K., “A two cell formation algorithm with minimum inter-cell movements”, Al-Azhar fifth international engineering conference, Al-Azhar University, Cairo, Dec.,1997. Polya, G., Tarjan, R.E., and Eoods, D.R , Notes on introductory combinatorics, Birkhauser, Boston, 1983. Rajamani, D., Singh, N. and Aneja, Y. P., “Design of cellular manufacturing systems”, Int. J. Prod. Res., Vol. 34, No. 7, pp. 1917-1928, 1996. Viswanathan, S., “Configuring cellular manufacturing systems: a quadratic integer programming formulation and a simple interchange heuristic”, Int. J. Prod. Res., Vol. 33, No. 2, pp. 361-376, 1995.