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A linear assignment algorithm for formation of machine cells and part families in cellular manufacturing
Pergamon
Computers ind. Engng Vol. 35, Nos 1-2, pp. 81-84, 1998 © 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain PII: S0360-8352(98)00025-4 0360-8352/98 $19.00 + 0.00
A Linear Assignment Algorithm for Formation of Machine Cells and Part Families in Cellular Manufacturing Jun Wang Department of Mechanical & Automation Engineering The Chinese University of Hong Kong Shatin, New Territories, Hong Kong
Abstract-This paper presents a linear assignment algorithm for machine-cell and part-family formation to design cellular manufacturing systems. The present approach begins with the determination of part-family or machine-cell representatives by means of comparing similarity coefficients between parts or machines and finding a set of the least similar parts or machines. Using the group representatives and associated similarity coefficients, a linear assignment model is formulated for solving the formation problem by allocating the remaining parts or machines and maximizing a similarity index. Based on the formulated linear assignment model, a group formation algorithm is developed. © 1998 E l s e v i e r S c i e n c e Ltd. All rights reserved. Keywords: Clustering algorithm, machine-cell/part-family formation, cellular manufacturing 1. Introduction The design of a cellular manufacturing system usually begins with two group formation tasks: part-family formation and machine-cell formation. Part-family formation is to group parts with similar geometric characteristics or processing requirements to take advantage of their similarities for the design or manufacturing purpose. Machine-cell formation is to bring dissimilar machines together and dedicate them to the manufacture of one or more part families. The problem of clustering multivariate data into mutually exclusive groups is unfortunately known to be NPcomplete. The development of efficient group formation algorithms is thus always desirable. This paper presents a group formation algorithm for part-family/machine-cell formation. The present algorithm begins with the determination of part-family or machine-cell representatives by means of comparing similarity coefficients to find a set of the least similar parts or machines. Using the group representatives and associated similarity coefficients, a linear assignment model is formulated. Based on the linear assignment model, the group formation algorithm call the linear assignment algorithm is developed for assigning the remaining parts/machines to groups and deducing corresponding machine cells/part families from resulting part families/machine cells. 2. Existing Formation Models The p-median model is a zero-one integer linear program as follows (Kusiak, 1987): p
Maximize
p
~ ~ s,jz,j
(1)
i=l j = l P
subject to
~xij=
1, i = 1 , 2 . . . . . p;
(2)
j----1 P
xjj
=
;-- 1
81
(a)
23rd International Conference on Computers and Industrial Engineering
82
where sij denotes a similarity coefficient between parts i and j, xij is the binary decision variable defined as zij = 1 if part i is assigned to the family in which part j is the median or xij = 0 otherwise, p is the number of parts, and n is the desired number of families. In the above p-median model, the value of the objective function depends on which parts are selected as group medians as well as the value of similarity coefficients. As an alternative, the objective function in the following quadratic assignment model is defined as the sum of similarity coefficients between every pair of parts within every family or every pair of machines within every cell (Song and Hitomi, 1992): n
p
~_, ~ ~ s,jzikzjk
Maximize
(6)
k = l i=1 j < i n
~zjk
subject to
= 1, j = 1 , 2 , . . . , p ;
(7)
k=l P
xjk < up, k = 1 , 2 , . . . , n;
(8)
j=l
x j k e {0,1}, j = l , 2 , . . . , p ; k = l , 2 . . . , n ; where
Xjk
is the binary decision variable defined as
Xjk
-----
(9)
1 if part j is assigned to family k or
Xjk = 0 otherwise.
3. New Formation Algorithm The p-mediao model has a linear objective function, but its linear constraint coefficient matrices are not totally unimodular. The constraint matrix of the quadratic assignment model is totally unimodular, but its objective function is nonlinear. It is natural to explore an approach to combining the two types of models so that the resulting one is a linear assignment model. This is the starting point of the present approach. The key idea of the linear assignment algorithm lies in the selection of n most dissimilar parts or machines as group representatives. Many existing methods differ in the ways for determining group representatives. Since the objective of part-family/machine-cell formation is to group parts/machines with dissimilar design/manufacturing features into mutually exclusive groups, it is reasonable to conclude that the n least similar parts/machines are in n different groups. Instead of using group means or medians, the present algorithm uses the n most dissimilar parts or machines as group representatives and let each of them represent a single group. Without loss of generality, it is assumed hereafter that the similarity-coefficient matrix is symmetric; i.e., Vi,j, sij = sji. The group representatives can be determined by using the following recursive method: (rl, r2)
----
argmin(i,j)sij;
=
argminie{12,...,k_l}
(10) k-I
rk
~_s%,
k = 3,4,...,n;
(11)
j=l
where rk denotes the representative for group k, sij denotes the similarity coefficient between either parts or machines i and j. Eqn. (10) determines the pair of parts or machines associated with the least similarity coefficient. Eqn. (11) determines the remaining n - 2 most dissimilar group representatives one by one recursively based on the ones determined previously. Using n predetermined part-family representatives, the linear assignment model for part-family formation can be formulated as follows: Maximize
~-~-~SPrkXjk
(12}
j = l k--1
subject to
fiXjk=l, k=l
j=
1,2 . . . . . p;
(13)
23rd International Conference on Computers and Industrial Engineering
83
P
~],Xjk <_ up, k = 1 , 2 , . . . , n ;
(14)
j=l
x~k _> 0;
(15)
where xjk is the binary decision variable defined as the same as that in the quadratic assignment model for part-family formation. Although the linear assignment model for group formation can be solved by using any conventional linear programming methods such as the simplex method, a dedicated linear assignment algorithm specifically for solving the formation problem is usually more efficient. Step 1: Load the number of families n and the maximal number of parts per family up. Step 2: Load or compute the similarity coefficients between every pair of parts [sij]. Step 3: Determine group representatives using eqns. (10) and (11). Step 4: For j = 1 , 2 , . . . , p , if j /=rq, then assign part j to family q (i.e., set xjq = 1) where q = arg maxje{1.2,...,n } sir:. If the number of parts in family q is equal to up, remove family q from further consideration. Step 5: Deduce the corresponding machine cells based on the the resulting part families: for i = 1 , 2 , . . . , m ; if ~jej(i) xjq = maxt~]jej(i)xjl, then assign machine i to cell q (i.e., set yiq = 1) where J(i) = {jlaij = 1}. If the number of machines in cell q is equal to urn, remove cell q from further consideration. 4. Illustrative Examples
Example 1: Consider a part-family/machine-cell formation problem with four machines, five parts, two groups, and ten processes. The left and right matrices in Table 1 show respectively the original machine-part incidence matrixand the incidence matrix after block diagonalization using the linear assignment algorithm. The formation results in one exceptional element. The optimal solution to the part-family formation problem is: X l l = X31 ~--- X22 = X42 = X52 = 1. This is interpreted as follows: part-family 1 ={1,3} and part-family 2={2,4,5}. Two machine cells can then be deduced based on the resulting two part families and the machine-part incidence matrix: machine-cell 1={2,4}, machine-cell 2={1,3}. Example 2: Consider the example of 40 machines, 100 parts, and 10 groups (Chandrasekharan and Rajagopalan, 1987). Table 2 shows the inital and block-diagonalized incidence matrices through permuting rows and columns according to the optimal solution using the linear assignment algorithm (which is equivalent to the solution to the p-median model). 5. Concluding Remarks In this paper, a new approach is presented for part-family and machine-cell formation. Using a set of recursively determined group representatives, a linear assignment model is formulated. Based on the linear assignment model, a simple, but very effective and efficient group formation algorithm is developed for machine-cell and part-family formation. References Kusiak, A. (1987). The generalized group technology concept. International Journal of Production Research, 25, 561-569. Song, S. and Hitomi, K. (1992). Group technology cell formation for minimizing the intercell parts flow. International Journal of Production Research, 30, 2737-2753. Chandrasekharan, M. P. and Rajagopalan, R. (1987). ZODIAC - an algorithm for concurrent formation of part-families and machine-cells. International Journal of Production Research, 25, 835-850.
23rd International Conference on Computers and Industrial Engineering
84
Table 1: Initial and Permuted Mazhine-Part Incidence Matrices in E x a m p l e 1
1
2 I
1 1 1
3
4 1
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2 4
1 1
1 1 1
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Computers ind. Engng Vol. 35, Nos 1-2, pp. 81-84, 1998 © 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain PII: S0360-8352(98)00025-4 0360-8352/98 $19.00 + 0.00
A Linear Assignment Algorithm for Formation of Machine Cells and Part Families in Cellular Manufacturing Jun Wang Department of Mechanical & Automation Engineering The Chinese University of Hong Kong Shatin, New Territories, Hong Kong
Abstract-This paper presents a linear assignment algorithm for machine-cell and part-family formation to design cellular manufacturing systems. The present approach begins with the determination of part-family or machine-cell representatives by means of comparing similarity coefficients between parts or machines and finding a set of the least similar parts or machines. Using the group representatives and associated similarity coefficients, a linear assignment model is formulated for solving the formation problem by allocating the remaining parts or machines and maximizing a similarity index. Based on the formulated linear assignment model, a group formation algorithm is developed. © 1998 E l s e v i e r S c i e n c e Ltd. All rights reserved. Keywords: Clustering algorithm, machine-cell/part-family formation, cellular manufacturing 1. Introduction The design of a cellular manufacturing system usually begins with two group formation tasks: part-family formation and machine-cell formation. Part-family formation is to group parts with similar geometric characteristics or processing requirements to take advantage of their similarities for the design or manufacturing purpose. Machine-cell formation is to bring dissimilar machines together and dedicate them to the manufacture of one or more part families. The problem of clustering multivariate data into mutually exclusive groups is unfortunately known to be NPcomplete. The development of efficient group formation algorithms is thus always desirable. This paper presents a group formation algorithm for part-family/machine-cell formation. The present algorithm begins with the determination of part-family or machine-cell representatives by means of comparing similarity coefficients to find a set of the least similar parts or machines. Using the group representatives and associated similarity coefficients, a linear assignment model is formulated. Based on the linear assignment model, the group formation algorithm call the linear assignment algorithm is developed for assigning the remaining parts/machines to groups and deducing corresponding machine cells/part families from resulting part families/machine cells. 2. Existing Formation Models The p-median model is a zero-one integer linear program as follows (Kusiak, 1987): p
Maximize
p
~ ~ s,jz,j
(1)
i=l j = l P
subject to
~xij=
1, i = 1 , 2 . . . . . p;
(2)
j----1 P
xjj
=
;-- 1
81
(a)
23rd International Conference on Computers and Industrial Engineering
82
where sij denotes a similarity coefficient between parts i and j, xij is the binary decision variable defined as zij = 1 if part i is assigned to the family in which part j is the median or xij = 0 otherwise, p is the number of parts, and n is the desired number of families. In the above p-median model, the value of the objective function depends on which parts are selected as group medians as well as the value of similarity coefficients. As an alternative, the objective function in the following quadratic assignment model is defined as the sum of similarity coefficients between every pair of parts within every family or every pair of machines within every cell (Song and Hitomi, 1992): n
p
~_, ~ ~ s,jzikzjk
Maximize
(6)
k = l i=1 j < i n
~zjk
subject to
= 1, j = 1 , 2 , . . . , p ;
(7)
k=l P
xjk < up, k = 1 , 2 , . . . , n;
(8)
j=l
x j k e {0,1}, j = l , 2 , . . . , p ; k = l , 2 . . . , n ; where
Xjk
is the binary decision variable defined as
Xjk
-----
(9)
1 if part j is assigned to family k or
Xjk = 0 otherwise.
3. New Formation Algorithm The p-mediao model has a linear objective function, but its linear constraint coefficient matrices are not totally unimodular. The constraint matrix of the quadratic assignment model is totally unimodular, but its objective function is nonlinear. It is natural to explore an approach to combining the two types of models so that the resulting one is a linear assignment model. This is the starting point of the present approach. The key idea of the linear assignment algorithm lies in the selection of n most dissimilar parts or machines as group representatives. Many existing methods differ in the ways for determining group representatives. Since the objective of part-family/machine-cell formation is to group parts/machines with dissimilar design/manufacturing features into mutually exclusive groups, it is reasonable to conclude that the n least similar parts/machines are in n different groups. Instead of using group means or medians, the present algorithm uses the n most dissimilar parts or machines as group representatives and let each of them represent a single group. Without loss of generality, it is assumed hereafter that the similarity-coefficient matrix is symmetric; i.e., Vi,j, sij = sji. The group representatives can be determined by using the following recursive method: (rl, r2)
----
argmin(i,j)sij;
=
argminie{12,...,k_l}
(10) k-I
rk
~_s%,
k = 3,4,...,n;
(11)
j=l
where rk denotes the representative for group k, sij denotes the similarity coefficient between either parts or machines i and j. Eqn. (10) determines the pair of parts or machines associated with the least similarity coefficient. Eqn. (11) determines the remaining n - 2 most dissimilar group representatives one by one recursively based on the ones determined previously. Using n predetermined part-family representatives, the linear assignment model for part-family formation can be formulated as follows: Maximize
~-~-~SPrkXjk
(12}
j = l k--1
subject to
fiXjk=l, k=l
j=
1,2 . . . . . p;
(13)
23rd International Conference on Computers and Industrial Engineering
83
P
~],Xjk <_ up, k = 1 , 2 , . . . , n ;
(14)
j=l
x~k _> 0;
(15)
where xjk is the binary decision variable defined as the same as that in the quadratic assignment model for part-family formation. Although the linear assignment model for group formation can be solved by using any conventional linear programming methods such as the simplex method, a dedicated linear assignment algorithm specifically for solving the formation problem is usually more efficient. Step 1: Load the number of families n and the maximal number of parts per family up. Step 2: Load or compute the similarity coefficients between every pair of parts [sij]. Step 3: Determine group representatives using eqns. (10) and (11). Step 4: For j = 1 , 2 , . . . , p , if j /=rq, then assign part j to family q (i.e., set xjq = 1) where q = arg maxje{1.2,...,n } sir:. If the number of parts in family q is equal to up, remove family q from further consideration. Step 5: Deduce the corresponding machine cells based on the the resulting part families: for i = 1 , 2 , . . . , m ; if ~jej(i) xjq = maxt~]jej(i)xjl, then assign machine i to cell q (i.e., set yiq = 1) where J(i) = {jlaij = 1}. If the number of machines in cell q is equal to urn, remove cell q from further consideration. 4. Illustrative Examples
Example 1: Consider a part-family/machine-cell formation problem with four machines, five parts, two groups, and ten processes. The left and right matrices in Table 1 show respectively the original machine-part incidence matrixand the incidence matrix after block diagonalization using the linear assignment algorithm. The formation results in one exceptional element. The optimal solution to the part-family formation problem is: X l l = X31 ~--- X22 = X42 = X52 = 1. This is interpreted as follows: part-family 1 ={1,3} and part-family 2={2,4,5}. Two machine cells can then be deduced based on the resulting two part families and the machine-part incidence matrix: machine-cell 1={2,4}, machine-cell 2={1,3}. Example 2: Consider the example of 40 machines, 100 parts, and 10 groups (Chandrasekharan and Rajagopalan, 1987). Table 2 shows the inital and block-diagonalized incidence matrices through permuting rows and columns according to the optimal solution using the linear assignment algorithm (which is equivalent to the solution to the p-median model). 5. Concluding Remarks In this paper, a new approach is presented for part-family and machine-cell formation. Using a set of recursively determined group representatives, a linear assignment model is formulated. Based on the linear assignment model, a simple, but very effective and efficient group formation algorithm is developed for machine-cell and part-family formation. References Kusiak, A. (1987). The generalized group technology concept. International Journal of Production Research, 25, 561-569. Song, S. and Hitomi, K. (1992). Group technology cell formation for minimizing the intercell parts flow. International Journal of Production Research, 30, 2737-2753. Chandrasekharan, M. P. and Rajagopalan, R. (1987). ZODIAC - an algorithm for concurrent formation of part-families and machine-cells. International Journal of Production Research, 25, 835-850.
23rd International Conference on Computers and Industrial Engineering
84
Table 1: Initial and Permuted Mazhine-Part Incidence Matrices in E x a m p l e 1
1
2 I
1 1 1
3
4 1
5 1
2 4
1 1
1 1 1
3 1 1
2
1
1
5
1 1 1 '
3
1
4
1
1
(a)
1
(b)
Table 2: Initial and Permuted Ma~:hine-Part Incidence Matrices in E x a m p l e 2 I
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