An efficient algorithm for solving the machine chaining problem in cellular manufacturing

Computers ind. EngngVol. 22, No. 1, pp. 95-100, 1992 Printed in Great Britain

0360-8352/92 $5.00 + 0.00 Pergamon Press plc

AN EFFICIENT ALGORITHM FOR SOLVING THE MACHINE CHAINING PROBLEM IN CELLULAR MANUFACTURING WING S. CHOW l a n d OSTAP HAWALESHKA 2 tDepartment of Decision Sciences, School of Business, Hong Kong Baptist College, Hong Kong and 2Department of Mechanical and Industrial Engineering, Faculty of Engineering, University of Manitoba, Canada

(Received for publication 2 May 1991) Abstract--The machine chaining problem invariably arises when there is improper assignment of machines into machine cells in a cellular manufacturing environment, in which case a high intercellular movement of parts is observed. This paper examines this phenomenon and proposes an efficient algorithm for solving such a machine chaining problem.

I. INTRODUCTION

Cellular manufacturing (CM) is a production system that relies on the Group Technology concept as a means to facilitate better control of a shop floor. Therefore, the end solution to a CM problem is to segregate objects of a system into subsystems according to their similarity features. Since the size of the individual subsystems contains far fewer components than the original "undecomposed" problem, each subsystem therefore represents a simpler and more manageable entity. In essence, the basic information required to solve a CM problem is the machine-part incidence matrix--which consists of values of O's and l's, where 1 in an entry denotes the corresponding coordinate of a part that requires the service of that machine; or otherwise. All CM problems are resolved by manipulating the incidence matrix in a manner such that the grouping of all similar objects is possible. There are essentially two different approaches by which the manipulation of a machine-part incidence matrix is based: (a) the direct approach and, (b) the indirect approach. The direct approach to a CM problem includes those methods in which the grouping of similarity objects entails rearrangement of rows and columns of the original incidence matrix. In such an approach, machine cells (MC) and part families (PF) are formed simultaneously. The Rank Order Clustering (ROC) algorithm of King [1] and the Cluster Identification Algorithm (CIA) of Kusiak and Chow [2] are typical examples of such an approach. In contrast, the indirect approach involves the transformation of the original incidence matrix into a different form of information before data analysis is carried out. There are two ways in which data transformation can be done. The first is by transforming a machine-part incidence matrix into a part-based matrix in which the final result is in the form of part families. The corresponding machine cells of this approach have to be derived from the original incidence matrix. Conversely, the second way is by changing the original incidence matrix into a machine-based matrix; the result of which is based on machine cells. In view of the complexity of CM problems, the machine-based indirect approach is the favored method because the total number of machines involved in a CM problem is usually far less than the total number of parts, thus a lower computational effort is required [3]. Be this as it may, a more challenging problem that is yet to be resolved by researchers when using the machine-based indirect approach is the machine-chaining problem, which was first identified by King and Nakornchai [4]. In light of the intrinsic machine chaining problem in the machine-based approach, a lot of effort has been aimed at alleviating this problem by the introduction of resource constraints such as duplicated machines [5], material handling cost [6], capacity constraints [7], and alternative routing of parts [8]. Although these models provide a feasible solution for their respective CM environments, one is not sure how well these models actually perform. Considering all, it is indisputable therefore that an algorithm that would help solve the machine chaining problem is very much needed. In acknowledging the existence of such a problem and the inadequacies of the existing 95

96

WlrqG S. CHOW and OSTAPHAWALESHKA

algorithms in providing a solution, this paper aims at exploring an efficient algorithm just for this specific purpose. 2. MACHINE CHAINING PROBLEM

A machine chaining problem occurs as a result of an improper assignment of a machine M~ to a machine cell MC-j instead of a machine cell MC-k creating a high intercellular movement of parts in a CM environment. Clearly, such a phenomenon arises as a consequence of an inefficient algorithm. The common cause of the machine chaining problems in the literature is primarily due to those CM algorithms which consider each step of the grouping process in a disjointed manner. For example, the decision in grouping machine M~ into machine cell MC-j is strictly based on the similarity factor of machine Mi with the similarity of only one of the machines in machine cell MC-j and not with all machines in that machine cell. Seifoddini [9] has superficially touched on this issue by using the Single Linear Clustering Algorithm (SCLA) of McAuley [10]; and in a further attempt to solve such a problem, he presented the Average Cluster (AC) algorithm which he claimed, does not always eliminate the machine chaining problem. The major drawback of the SCLA and the AC algorithms lie in the definition of their similarity factor, which is computed as a percentage of the total number of similarity of parts required to be processed by both machines M~ and M s divided by the total number of parts to be processed by either machines M~ and M s. This similarity factor does not recognize the precise number of intercellular movement of parts between cells because the percentage in a cell with high intercellular movement could be the same as a cell with a low intercellular movement. One of the more efficient methods in recognizing the total number of intercellular movements of parts between two machines in a single stage algorithm is the commonality scores of Wei and Kern [11]. They defined the commonality scores of two machines as:

s~= S r(a,~,as~) k=i,p

where (p

F (aik, ask) =

-

1), if aik = ask = l

1,

if ai, = ajk = 0

0,

if a~ =A ask

and aik =

1, 0,

If machine i is used to processed part k Otherwise.

However, the use of commonality scores in their Linear Cell Clustering Algorithm (LCCA) does not solve the machine chaining problem because in their algorithm, each step of machines grouping was dealt with independently of each other. 3. SOLUTION

In solving the machine chaining problem, we need to regard all steps of the algorithm as being continuous--i.e, every preceding step is an input value to the next step of the solution. For instance, if machine Mi is grouped with machine Mj to form a machine cell MC - k, then in the next step of the grouping process we need to first transform machine cell M C - k into a single unit of machine and include it in the next computational step. This re-evaluation concept is important because without it the preceding solution has no meaning to the next step of the grouping process. To illustrate the transformation of M C - k as a single unit of machine, let us assume that machine Mi and machine Mj each consistes of 10 parts as shown in matrices (1) and (2). M i = (1, 0, 1, 0, 1, 0, 1, 0, 0, 0)

(1)

Ms = (1, 1, O, O, l, O, l, O, O, O)

(2)

Machine chaining problem

97

The transformation of MC - k into a new machine unit, M(t,j), is achieved by grouping matrices (1) and (2) into a new matrix according to the following rule: ~i

ifai(,)=l°raj(o =1

M(i,j), r =

Otherwise.

The new matrix for M C - k is shown in matrix (3). M(~j)= (1, 1, 1, 0, 1, 0, 1, 0, 0, 0)

(3)

To develop an algorithm that involve such a step, let us first consider a machine-part incidence matrix that consists of (m x n) elements. This incidence matrix is transformed into a (m x m) matrix by means of the commonality scores method of Wei and Kern. We then proceed to group the first two machines M~ and Mj that contribute to the highest commonality score. These machines will be treated as a new machine unit as shown in matrix (3) which is replaced in machines Mi and Mj in the incidence matrix. The above step will be repeated until all machines are grouped. This procedure is outlined in the following algorithm.

Algorithm Step 1. Step 2. Step 3.

Compute the commonality scores of the incidence matrix Group machines M i and M i which together contribute to the highest commonality score. Transform machines in Step 2 into a new machine unit Mk as follows:

M,,j),r =

Step 4. Step 5. Step 6.

1,

if ai(r)= 1 or aj(r)= 1

0,

Otherwise.

Formulate a dendrogram accordingly. Replace machines M~ and Mj with machine M k in the incidence matrix. If the number of machines in the new incidence matrix has only one unit of machine, then stop; otherwise proceed to Step 1.

The operation of such an algorithm is illustrated in Example 1.

Example 1 Solve the cellular manufacturing problem of a machine-part incidence matrix is depicted in matrix (4) Pi

P2

MI

1

1

M2

1

M5

P4 1

Ps

P6

P7

P8

1

1

1

1

1

1

P9

Plo PI! 1

1

1

M3 M4

P3

1

1

I

1

1

1

1

(4)

1 1

1

1 1

Solution Step 1.

The commonality scores of matrix (4) is as follows:

M1 M2 M3 M4 M5

M~ --

M2 21

M3 33

M4 21

M5 33

--

2

34

2

--

2

47

--

2

(5)

98

Step 2. Step 3.

WING S. CHOW and OSTAPHAWALESHKA

Machines M3 and M5 have the highest commonality score of 47 in matrix (5), therefore they are to be grouped. Let the new machine unit of grouping M3 and M5 as M(3.5) and its machine-part incidence matrix is as follows: M(3,5) = (0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0)

Step 4.

(6)

The following is the dendrogram derived from Step 2. ~3 . ~t15

Step 5.

Matrix (7) is the new incidence matrix:

MI M2 M4

Pl

P2

1

1

1

P4 1

1

P5

P6

P7

P8

P9 Plo P ,

1

1

1

1

1

1

1

1

1

M(3, 5)

Step 6. Step 1.

P3

1

1

1

1

1

1

(7)

1

Since the total number of machines in matrix (7)> 1, we proceed to Step 1. Since M 3 and M5 is the identical matrix, the commonality scores are the same as those in matrix (4) except that M 3 and M 5 are replaced by M(3,4) as follows:

MI

/

M2 M4

l

Mt --

M2 21

M4 21

M(3,5) 33

--

34 --

2 2

(8)

M(3, 5)

Step 2. Step 3.

Because machines M2 and M4 have the highest commonality score which is 34 as shown in matrix (8), they are to be grouped. Let the new machine unit of grouping M2 and M4 as M~2,4) and its machine-part incidence matrix is as follows (9)

M(2,4) = (1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1)

Step 4.

The following is the dendrogram derived from Step 2.

Step 5.

Matrix (10) is the new incidence matrix:

MI

PI F 1

M(2.4) l M(3, 5)

Step 6.

1

P2 1

P3

1

P4

1 1

P5

P6

P7

P8

1

1

1

1

1

1 1

P9

Plo Pll 1

1 1

1

(10)

1

Since the total number of machines in matrix (10) > 1, we proceed to Step 1. Figure 1 is the final dendrogram formulation for the problem depicted in matrix (4):

Fig. 1. The final dendrogram solution for Example 1.

Machine chaining problem Models:

Linear cell clustering algorithm (LCCA)

Average clustering algorithm (AC algorithm)

Dendr°gram:~5

99

I~1 1~121~4

Proposed algorithm

M~!, ~1 r~12r~4

r~3 rv~s~1M~ 4

Total number of machine cells:

2

2

2

Total number of bottleneck parts:

4

4

3

Fig. 2. The clustering result of incidence matrix (4). 4. ANALYSIS

If one decides to have two machine cells from a CM problem as shown in matrix (4), then the two machine cells can be obtained by removing the last linkage line, i.e. the bottom line of the dendrogram in Fig. 1. In doing so, we would have two machine cells namely, machine cell MC - 1 which consists of (Ms, Ms) and machine cell M C - 2 which contains of (M~, M2, M4). The bottleneck parts for this grouping is shown in matrix (11).

P9 Ptl MC-2

M2

1

1

P4 1

M4

1

1

1

P5 1

l

1

P7 P2 1

1

l

1

P6 P8 Pm

P3

1

1

M3

1

1

1

1

M5

1

1

1

1

MI

MC-1

Pl 1

1

(ll)

In analyzing matrix (1 l), one can see that parts P6, Ps and Pl0 are the bottleneck parts for machine cells MC - 1 and MC - 2. These parts need to be processed first in machine cell MC - 2 before they are transferred to machine cell MC - 1. The total number of bottleneck parts for this problem is 3. In order to evaluate the efficiency of this algorithm, a comparison of the solution obtained by this method to that by other existing algorithms is needed. For this purpose, we have selected two algorithms for comparison: (a) the AC algorithm of Seifoddini [9] and; (b) the LCCA of Wei and Kern [ll]. The result is presented in Fig. (2). As can be seen in Fig. 2, the AC algorithm and the LCCA assigned machine M~ to the machine cell of MC - j which consists of machines M 3 and M 5. This is due to the aforementioned reason. It is not surprising to see that the AC algorithm and LCCA solution gives a higher intercellular movement in a CM environment when compared to our proposed algorithm. CONCLUSION

When solving a cellular manufacturing problem by means of the machines assignment method, one commonly encounters the so-called machine chaining problem which generates a high intercellular movement of parts in a shop floor. In this paper, we have discussed the potential sources of such a problem and proposed an efficient algorithm as the solution. This algorithm has been illustrated with an example and its solution compared with that by other existing algorithms. REFERENCES 1. J. R. King. Machine-component grouping in production flow analysis: An approach using a rank order clustering algorithm. Int. J. Prod. Res. lg, 213-232 (1980). 2. A. Kusiak and W. S. Chow. An algorithm for cluster identification. SMC-17, 696-699 (1987). 3. K. R. Gunasingh and R. S. Lashkari. The cell formation problem in cellular manufacturing systems--a sequential modeling approach. Computers ind. Engng, 16, 469-476 (1989). 4. J. R. King and V. Nakornachai. Machine-component group formation in group technology. Int. J. Prod. Res. 29, 117-133 (1982).

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5. H. Seifoddini. Duplication process in machine cells formation in group technology, liE Trans. 21, 382-391 (1989). 6. H. Seifoddini and P. H. Wolfe. Selection of a threshold value based on material handling cost in machine-component grouping, liE Trans. 19, 266-270 (1987). 7. A. Kusiak. Knowledge-based group technology. In Expert Systems: Strategies and Solutions in Manufacturing Design and Planning (Edited by A. Kusiak). Society of Manufacturing Engineers, pp. 259-299 (1988). 8. R. J. Kasilingam and S. Sankaran. Simultaneous cell formation and route selection in flexible manufacturing systems. In The Impact of Information Technology on Systems Management (Edited by C. Yau), Proc. Int. Conf. on Systems Management '90, 11-13 June 1990, Victoria Hotel, Hong Kong, pp. 349 354. 9. H. Seifoddini. A note on the similarity coefficient method and the problem of improper machine assignment in group technology problem. Int. J. Prod. Res. 27, 1161 1165 (1989). 10. J. McAuley. Machine grouping for efficient production. Prod. Engr, Feb., 53-57 (1972). 11. J. C. Wei and G. M. Kern. Commonality analysis: A linear cell clustering algorithm for group technology. Int. J. Prod. Res. 27 2053-2062 (1989).