- Email: [email protected]
Routes selection for the cell formation problem with alternative part process plans
~
Computers ind. Engng Vol.30, No. 3, pp. 423--431,1996
Pergamon
ROUTES PROBLEM
803604352(96)00011-3
SELECTION WITH
FOR
THE
ALTERNATIVE
CELL PART
Ltd 0.00
Copyright© 1996ElsevierScience Printedin GreatBritain.All rightsreserved 0360-8352/96 $15.00+
FORMATION PROCESS
PLANS
H A R K H W A N G l and PAEK REE 2 ~Department of Industrial Engineering, Korea Advanced Institute of Science and Technology, 373-1 Kusong-dong Yusong-gu, Taejon 305-701, Korea 2Department of Industrial Engineering, College of Engineering, Keimyung University, 1000 Shindang-dong Dalseo-gu, Taegu 704-701, Korea Abstract--This article proposes a two stage procedure for the cell formation problem with alternative process plans. At the first stage of the procedure,the route selectionproblem is solvedwith the objective of maximizing the sum of compatibility coefficientsamong selected process plans. At the second stage, part familiesare formed based on the result of the first stage using the p-median problem. The grouping solutions based on the proposed procedure are compared with those of the generalizedp-median model in terms of the number of exceptionalelements, grouping efficiencyand grouping efficacy.Copyright © 1996 ElsevierScience Ltd
1. INTRODUCTION In the manufacturing environment, the concept of group technology (GT) has been applied to enhance the productivity of batch type manufacturing systems. In GT, the parts to be produced are grouped into part families based on the similarity of the production processes or the design features. The machines which are needed to process parts in a part family are put together to form the manufacturing cell. These cells are building blocks of the cellular manufacturing (CM) system in which each cell is operated independently with minimal intercellular flows. Within each cell, the production flow can be streamlined as in a flow shop. Among the various benefits from CM reduced number of set-ups, reduced work-in-process inventories and reduced material handling costs are the critical ones. To design a CM system, we identify parts to be produced, part process plans and machines to carry out all the required operations. And then the cell formation process to group the parts into part families and assign machines to the families to form manufacturing cells if followed. In this study, our attention is focused on the cell formation. Numerous cell formation methods have appeared in the literature. Extensive reviews are available in the reports of Wemmerlov and Hyper [1], Chu [2] and Offodile et al. [3]. In most cell formation methods, parts are assumed to have a unique part process plan. In reality, there are usually alternative process plans for parts. And they use different sets of machines. Explicit consideration of alternative process plans invoke changes in the composition of all manufacturing cells so that lower capital investment in machines, more independent manufacturing cells and higher machine utilization can be achieved. Here, some of the cell formation methods that consider alternative process plans are reviewed briefly. Kusiak [4] proposed an integer programming model (p-median model) for part family formation. For the case of alternative process plans, he extended the model so that route selection and part family formation can be done simultaneously in the generalized p-median model. Kim and Won [5] formulated an assignment model where the sum of similarity coefficients between process plans in the same family can be maximized. From the solution of the model, part families and machine cells are partially identified and then they are removed from the incidence matrix. With the reduced problem, the procedure is repeated until all parts are covered. In the previous two models, part families are formed based on the 0-1 incidence matrix which shows machines required to process a part by a process plan. Other cell formation methods [6-13] require additional data such as cost and time of part processing operations, production volume of parts and machine capacity. In most of the previous works, the route selection and cell formation were simultaneously formulated in a single model. In Kusiak's generalized p-median model, route selection and part CAIE3o/3-E
423
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Hark Hwang and Pack Ree
family formation were done simultaneously with the objective of maximizing the total sum of similarity among process plans in the same cell. We suggest that better grouping solutions could be obtained by formulating and solving them separately in two stages. At the first stage, the route selection problem is formulated with newly defined compatibility coefficients. Two types of the compatibility coefficients were suggested. It is expected that the resulting set of process plans facilitates a grouping solution with highly independent manufacturing cells. At the second stage, part families are formed using the p-median model and machines are assigned to the families to complete cell formation. This article is organized as follows: In Section 2, two different sets of compatibility coefficients are defined and Section 3 describes the proposed two stage procedure. Through numerical experiments, computational results of the proposed procedure are compared with those of Kusiak's in Section 4. Concluding remarks appear in Section 5. 2. COMPATIBILITY COEFFICIENTS We are going to use the following notations in this study. M P R A
= = = =
number of different machines, number of different parts to be produced, total number of process plans, machine-process plan incidence matrix where A = {am,}, m = 1. . . . . M and r = 1. . . . . R. am~= 1 if machine m is required by route r, a,~ = 0 otherwise. p(r) = part number produced by process plan r, srs = similarity coefficient between process plan r and s, cr, = compatibility coefficient between process plan r and s, tpq = similarity coefficient between part p and q. At the first stage, process plans are selected for the parts from the list of alternative process plans under the condition of the one part-one process plan principle. We will call this set of process plans a covering set. A covering set which can result in a completely (or almost completely) blockdiagonalizable grouping solution is what we want to find at this stage. In the block-diagonal form of grouping solutions, parts in the same family should have many similarities in terms of machines required and parts from different part families should have very few or no similarities at all. In this regard, it is desirable for each route in a covering set to have highly similar process plans and highly dissimilar process plans simultaneously in the set so that clearly identifiable friends and foes of the process plan coexist in the covering set. If two process plans in a covering set have a mediocre similarity, then some machines are required by both process plans while other machines are required by only one of them. If any part associated with these two process plans is not in the same family, the commonly required machines result in intercellular movement of the part. For this reason, it would be advisable for two process plans having a mediocre similarity not to be together in a covering set. Since conventional similarity measures are mainly designed to cluster only similar process plans, they are not suitable for the purpose of route selection. In this regard, two different sets of coefficients called compatibility coefficients are proposed. According to these proposed measures, pairs of highly similar and highly dissimilar process plans get high scores. We expect that a covering set with high overall compatibility scores tends to have a block diagonalized grouping solution with minimal number of exceptional elements. The first set of compatibility coefficients (Type 1) are obtained by converting similarity coefficients and the second one is newly defined from the incidence matrix (Type 2).
2.1. Compatibility coefficients: Type 1 Let Srs be the similarity coefficient between process plans r and s. Also, let s , ~ , Sminand Smed bc the maximum, minimum and median of s,s's, respectively. The Ttype 1 compatibility coefficient crs is defined as follows:
Cell formation problem: route selection
425
1.0
e~
o
0.0 S rain
S reed
S max
Similarity Fig. 1. Similaritycoefficientsversus compatibilitycoefficients. forp(r) = p ( s )
Crs = 0
for p(r) # p ( s ) and s~, >I Smed, \Sma x -- Smed/
for p ( r ) # p ( s ) and S,~ 0 .
(1)
\Smed -- Smin/
The values of compatibility coefficients are in [0, 1]. They are obtained by converting similarity coefficients so that the most similar and the most dissimilar pairs of routes have their compatibility value of 1 and route pairs with mediocre similarity have their compatibility coefficient close to 0. The index n in Eqn (1) is introduced to generate different sets of compatibility coefficients. Depending on the value of n, values of compatibility coefficients and their distribution in [0, 1] can be varied. Figure 1 illustrates the relationship between similarity and type 1 compatibility coefficient. In this study, we used the well-known Jaccard's measure for the similarity coefficients.
2.2. Compatibility coefficients: Type 2 From the incidence matrix, we define the Type 2 compatibility coefficients as follows. crs = 0
forp(r) =p(s),
=[ar, - ~ "br,[/(ar, + ~ "br,) for p(r) # p ( s )
(2)
Where at, is the number of machines required by both process plans r and s, b,, be the number of machines required by either of the two process plans but not by both and ~ is a weighting factor. The values of compatibility coefficients are again in [0, 1] and if the two process plans are identical or totally different, c~s will be 1. ~ can be used to reflect the relative weight that the similarity and dissimilarity of a pair of process plans will have in determining compatibility coefficients. In this study, we tested ~ for the value of 1 and
r,s:p(r)~p(s)
/r,s:p(r)#p(s)
which is the ratio of the number of 1-1 matches and 0-1(1-0) matches in all pairs of process plans. For the latter value of ~, a~s- ~.b,s distributes around 0. 3. TWO STAGE PROCEDURE Utilizing compatibility coefficients defined in the previous section, a mathematical model for the route selection problem is formulated as follows;
426
Hark Hwang and Paek Ree R
R
~ ~ Cr,Yr Y~
Maximize
r=|
~
subject to:
(3)
s=l
r :p(r)
Y,=I =
forp=l,2
..... P
(4)
for r = 1, 2 . . . . . R
(5)
p
Yr E {0, 1}
Yr is 1 if process plan r is included in the covering set. Otherwise, it is 0. The objective function is to maximize the sum of compatibility coefficients of all pairs of process plans in a covering set. The constraints in (4) state that the part is allowed to have only one process plan in a covering set. At the second stage, part families are to be formed based on the generated covering set and machines are to be assigned. The machine-route incidence matrix A is reduced to MxP machinepart incidence matrix and similarity coefficients between parts are calculated. Part families are formed using the p-median problem whose objective function is to maximize the sum of similarity coefficients of part pairs in the same family. The p-median model and its generalized form are reproduced as follows. The p-median model: P
Maximize
P
tpqXpq
~
(6)
p=lq=l P
subject to:
~ Xpq = 1
for q = 1 , . . . , P
(7)
q=l P
Ex.
q=|
(8)
=K
for p = 1. . . . . P and q = 1. . . . , P
(9)
Xpq e {0, 1} f o r p : 1 . . . . . P and q = l . . . . . P
(10)
Xpq ~ Xqq
Xpq is 1, if part p belongs to part family q. Otherwise, it is zero. K is the number of part families to be formed. The generalized p-median model: R
Maximize
R
Z Z s,',W,'s
(11)
r=ls=l
R
~
subject to:
~ W,s=l
forp=l ..... P
(12)
r :p(r) = p s = 1 R
Z Wrr = K
(13)
r=l
W.~W~
forr=l
W,~s{0,1} f o r r = l
..... Rands=l ..... R
(14)
..... Rands=l
(15)
..... R
Wrs is 1, if process plan r is chosen and belongs to family s. Otherwise, it is 0. Once the part families are generated, then machines are assigned. For each machine, the number of exceptional elements and in-block blanks generated are counted when it is assigned to a family. We assign the machine to the family with the minimum number of exceptional elements. If there are ties, a family with the minimum number of in-block blanks is chosen for the machine. 4, NUMERICAL EXAMPLES Four example problems from the literature are solved by both Kusiak's method and our procedure for the purpose of comparison. The machine-route incidence matrices for the chosen
Cell formation problem: route selection
(a) Kusiak's[ 1] problem
(b) Sankaran's[ 10] problem Route Number 11111111112 12345678901234567890 Part Number I1 11223445555667889900 00001001100001010000 11110110000100111100 01000001111001000010 00111111011110111011 10100100101000100101 11110010000010111100
Route Number 11 12345678901 Part Number 11122334455 00101101101 01110010000 10011000110 llO00lllOlO
(c) Logendran's[ 16] problem Route Number" 11111111112222222222333 12345678901234567890123456789012 Part Number lllllllllll 11222334455566778899900111222344 OIlO0000110001001010001001001010 10010011010101100000100011011000 00010000001010010101010100100101 01001100100100100111001100100000 00100000001001011000100010010011 10010010100010001000101001001000 10001001010100010110010010010000
(d) Nagi's[ 11 ] problem Route Number 111111111122222222223333333333344444444455 123456789012345678901234567890123456789012345678901 Part Number 111111111111111111111112 112233445556677777788999999001112223334445556667890 001111111110000000000000000000000000000000000000000 000000000001100000011111111110000000000000000000000 000000000000000000000000000001111111110000000000000 000000000000000000000000000000000000000000000000111 000000000001111111100000000110000000000000000000000 101010100001000011101010101100000000000000000000000 010101010000111100010101010010000000000000000000000 000000000000000000000000000001111111110000000000000 110011001110000000000000000000000000000000000000000 000000000000000000000000000000000000001111111111000 000000000000000000000000000001110001110000000000000 111111111110000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000110 O000000000000000000000000000000000000011100011llO00 000000000000000000000000000000000000000000000000111 000000000000011111111111111110000000000000000000000 000000000000000000000000000000000000001111110001000 000000000100001001000001100001001001001001001000000 000000001000010010000110000000100100100100100100000 000000000010000100100000011000010010010010010010000 Fig. 2. Incidence matrices ofthe ~st problems:(a);Kusiak's [1] problem,(b); Sankaran's [1~ problem, (c); Logendran's [1~ problem, (d);Nagi's [11] problem.
427
428
Hark Hwang and Pack Ree
(a) Similarity coefficients between process plans
1
2 3 4 5 6 7 8 9 10 11
process plans 1 2 3 4 5 6 7 8 9 10 11 .00 - - .33 .33 .33 .33 .33 .33 1.00 .00 .00 .33 .00 .33 1.00 .33 .00 .33 .00 .00 .33 .33 .33 .33 .33 .33 .00 .50 .00 .00 .33 .00 .33 .33 .00 .00 .33 .00 .33 1.00 .33 .50 .00 - 1.00 .33 .33 .50 .00 .33 .00 .33 .00 .00 .33 .50 .00 .33 .50 .00 .00
(b) Type I compatibility coefficients between process plans process plans 1 1
0.0
2 3 4 5 6 7
2
3
4
5
6
8 9 10 11 0.0 0.0 1.0 1.0 0.0 1.0 0.0 1.0 0.0 0.0 1.0 0.25 1.0 0.0 0.0 1.0 0.0 ..1.0 0.0 0.25 1.0 0.0 0.0 0.25 0.0 1.0 0.0 1.0 0.0 -oo 0.0 0.25 0.0 0.0 0.25 0.0 - ~ 0.0
7
- ~ - o o 0.0 0.0 0.0 0.0 -or 0.0 1.0 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 0.0 1.0 -oo 0.0
8
9 10 11
Fig. 3. Similarity and compatibility coefficients: (a); Similarity coefficients between process plans, (b); Type 1 compatibility coefficients between process plans. problems are shown in Fig. 2. We are going to illustrate the proposed procedure with Kusiak's problem. Figure 3(a) shows the similarity coefficients between process plans, where Smax, Smi, and Smed are 1.0, 0.0 and 0.333, respectively. According to Eqn. (1) with n = 1, the Type 1 compatibility process plans 1
2 3 4 5
6 7 8
9 10 11
1 .00
2
3
4
5
6
7
8
.24 1.00 .24 .24
.24 .24 .24 1.00
.24 1.00
.24
-ao -oo .24 .24 .24 .00 -oo .24 1.00 .24 .00 .24 .24 .24 .00 -0o 1.00 .00
.00
9
10 11 1.00 1.00
.24 1.00 .24 1.00 .24 1.00 .10 .24 .24 1.00 1.00
-oo 1.00 .24 .00 .24 1.00 .00 -oo .00
.24
.10
.24 .10 .24 1.00 .24 .10 .24 .10 .00 -oo .00
Fig. 4. Type 2 compatibility coefficients between process plans.
Cell formation problem: route selection
429
(a) Machine-part incidence
Machines
Parts 12345 01011 10100 01010 10100
1 2 3 4
(b) Similarity coefficients
Parts
1 2 3 4 5
Parts 1 2 3 4 5 0 0 1.0 0 0 0 0 1.0 0.5 0 0 0 0 0.5 0
(c) Cell formation
Cells
1 2
Parts I, 3 2,4, 5
Machines 2,4 1,3
Fig. 5. Part family formation: (a); Machine-part incidence, (b); Similarity coefficients, (c); Cell formation.
coefficients are c,~, = 1 for Sr, = 1.0 or 0.0, Crs = 0.25 for s,s = 0.5, and c,s = 0 for s,~ = 0.333 and they are shown in Fig. 3(b). Using Eqn (2) with g = 0.82, Type 2 compatibility coefficients are calculated and shown in Fig. 4. For example, consider route plans 1 and 4 of which a14 = 1 and bl4 = 2. So, the compatibility coefficient c14 = I1 - (0.82)(2)I/(1 + (0.82)(2)) = 0.242. Solving the route selection problem in Eqns (3)-(5) with two sets of compatibility coefficients in Figs 3(b) and 4, we obtained the identical covering set {2, 5, 7, 9, 11 }. Based on the covering set, the reduced machine-part incidence matrix is obtained as shown in Fig. 5(a). Then similarity coefficients between parts are computed and listed in Fig. 5(b). Part families are formed by solving the p-median problem in Eqns (6)-(10) where K is set 2 following the Kusiak's solution. The resulting cell formation is shown in Fig. 5(c).
Two stage procedure Type I Compatibility coefficients
Kuslak's generalized p-median method
Problems
ct= 1 257911
a = 082 257911
95,0 90.0 2 0 95.0 90.0 2 0 95.0 90.0 2 0 95.0 90.0 2 0
95.0 90.0 2 0
95.0 90.0 2 0
135610 12 14 15 18 20
13579 12 14 15 18 19 83.7 71.4 2 3 137812 14 15 17 20 22 24 27 30 31 86.0 70.3 3 5 246810 12 18 21 25 28 32 35 38 40 43 46 48 49 50 51 90.1 79.$ 5 1
13578 12 14 15 18 19 83.7 71.4 2 3 136810 14 15 17 20 22 24 27 30 31 85.7 67.6 3 5 135710 13 15 20 24 29 31 34 37 41 44 47 48 49 50 51 90.1 79.5 5 1
n=~c
Kusiak's[ 11 M = 4 P=-5 R = l l
Sankaran's[10l M = 6 P=10 R=20 Logcndran's[16] M= 7 I>=-14 R=32
Nagi's[ 1 l l M=20 P=20 R=51
Covering set
257911
CI CA K E' 95.0 ~ . 0 2 0 Covering set 1 4 5 7 8 13 14 16 17 19 Cl CA K E 78.8 ~ . 9 2 4 Covering set 2 3 6 9 1 1 14 15 18 20 23 24 27 30 31 C1 CA K 75.1 51.2 3 8 Covering set 1 3 5 7 1 1 12 18 21 23 28 31 34 37 39 42 45 48 49 50 51 CI CA K El 87.2 70.9 5 6
Type II Compatibility' coefficients
257911
n=2
:257911 14579 12 14 15 18 20
n=l 257911 13578 12 14 15 18 19
n=0.5
257911 14568 12 14 15 18 20
82.9 69.4 2 2 82.9 69.4 2 2 83.7 71.4 2 3 82.9 69.4 2 2 1 3 6 8 10 14 15 17 20 22 24 27 30 31 85.7 67.6 3 5 246810 12 18 21 25 28 31 34 37 41 44 47 48 49 50 51 90.1 79.5 5 I
1 3 6 8 10 14 15 17 211 22 ~ 24 27 30 31 85.7 67.6 3 5 246810 12 18 21 25 28 32 35 38 40 43 46 48 49 50 51 90.1 79.5 5 1
1 3 6 8 10 1 3 6 8 10 14 15 17 20 22 14 15 17 20 22 24 27 30 31 24 27 30 31 85.7 67.6 3 5 85.7 67.6 3 5 24689 135710 12 17 21 23 28 13 15 20 24 29 30 33 36 41 44 31 34 37 41 44 47 48 49 50 51 47 48 49 50 51 90.1 79.5 5 1 9 0 . 1 79.5 5 I
Fig. 6. Comparison of computational results by the Kusiak's method and the proposed procedure.
430
Hark Hwang and Pack Ree
With the Type 1 compatibility coefficients, we tested four different values (0.5, 1.0, 2.0, ~ ) of n in Eqn. (I) and two distinct values of ~ were tested with the Type 2 compatibility coefficients as mentioned earlier. Computational results are summarized in Fig. 6. In this figure, the source of the problem, the problem size (number of machines, parts and process plans), the covering set generated, the number of cells (K) and exceptional elements in the final grouping solution (E) are shown along with grouping efficiency (CI, Chandrasekharan et al. [14]) and grouping efficacy (CA, Kumar et al. [15]). CI and CA are the measures most frequently adopted to examine qualities of grouping solutions. In the generalized p-median model, the number of cells to be formed, K, should be given exogenously. Several different values of K are tried and the best one is chosen for each problem and it is used at the second stage of the proposed procedure for comparison on equal basis. All mathematical programming problems are solved using LINDO integer programming package installed on a 486 PC. The quadratic objective function of the route selection problem is linearized before they are solved. Compared to the Kusiak's method, the proposed procedure returns one identical grouping solution and three better solutions with both the Type 1 and Type 2 compatibility coeffÉcients. Varying n in Type 1 compatibility coefficients does not necessarily lead to different covering sets. In problems 2 and 4, different covering sets are found for a different value of n while identical covering sets are found in problem 1 and 3. Different covering sets are found for a different value of n, while identical covering sets are found in problem 1 and 3. Different covering sets results in one identical grouping solution in problem 4. With the Type 2 compatibility coefficients, the grouping solutions are as good as those of the Type 1 compatibility coefficients. 5. CONCLUSION
A two stage procedure for the cell formation problem with alternative part process plans is proposed. At the first stage, process plans are selected for parts based on the compatibility coefficients while respecting the one part-one route principle. Two different ways of defining compatibility coefficients are suggested. At the second stage, part families are formed and machines are assigned. Through four example problems of different size, the performance of the proposed procedure is compared with that of the Kusiak's generalized p-median model. The results show that grouping solutions generated by the proposed procedure are equal to or better than those of Kusiak's in terms of the number of exceptional elements, grouping efficiency and grouping efficacy regardless of the types of compatibility coefficients. It should be noticed that because of the quadratic objective function of the first stage the proposed procedure needs more computational time than the generalized p-median model. Developing a less time-consuming, heuristic for the two stage procedure can be considered for further studies.
REFERENCES 1. U. Wemmerov and N. L. Hyer. Research issues in cellular manufacturing. Int. J. Prod. Res. 25, 413-431 (1987). 2. C. H. Chu. Cluster analysis in manufacturing cellular formation. Omega 17, 289-295 (1989). 3. O. F. Offodile, A. Mehrez and J. Grznar. Cellular manufacturing: a taxonomic review framework. J. Manufact. Syst. 13, 196-220 (1994). 4. A. Kusiak. The generalized group technology concept. Int. J. Prod. Res. 25, 561-569 (1987). 5. Y. Won and S. Kim. An assignment method for the part-machine cell formation problem in the presence of multiple processing routes. Engng. Optimization 22, 231-240 (1994). 6. F. Choobineh. A framework for the design of cellular manufacturing system. Int. J. Prod. Res. 26, 1161-1172 (1988). 7. R. M. Sundaram and K. Doshi. Formation of part families to design cells with alternative routing consideration. Comp. Ind. Engng. 23, 59-62 (1992). 8. R. M. Sundaram and K. Doshi. Cellular manufacturing system design with alternative routing consideration. Comp. Ind. Engng. 25, 477-480 (1993). 9. T. Gupta. Design of manufacturing cells for flexible environment considering alternative routing. Int. J. Prod. Res. 31, 1259-1273 (1993). 10. S. Sankaran and R. Kasilingam. An integrated approach to cell formation and part routing in group technology manufacturing system. Engng. Optimization 16, 235-245 (1990). 11. R. Nagi, G. Harhalakis and J. Proth. Multiple routings and capacity consideration in group technology applications. Int. J. Prod. Res. 28, 2243-2257 (1990). 12. D. Rajamani, N. Singh and P. Aneja. Integrated design of cellular manufacturing systems in the presence of alternative process plans. Int. J. Prod. Res. 28, 1541-1554 (1990).
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13. S. L. Kang and U. Wemmerlov. A work load-oriented heuristic methodology for manufacturing cell formation allowing reallocation of operations. Eur. J. Oper. Res. 69, 292-311 (1993). 14. M. P. Chandrasekharan and R. Rajagopalan. An ideal seed non-hierarchical clustering algorithm for cellular manufacturing. Int. J. Prod. Res. 24, 451-464 (1986). 15. C. S. Kumar and M. P. Chandrasekharan. Grouping efficacy: a quantitative criterion for goodness of block diagonal forms of binary matrices in group technology. Int. J. Prod. Res. 28, 233-243 (1990). 16. R. Logendran, P. Ramakrishna and C. Sriskandarajah. Tabu search-based heuristics for cellular manufacturing systems in the presence of alternative process plans. Int. J. Prod. Res. 32, 273-297 (1994).
Computers ind. Engng Vol.30, No. 3, pp. 423--431,1996
Pergamon
ROUTES PROBLEM
803604352(96)00011-3
SELECTION WITH
FOR
THE
ALTERNATIVE
CELL PART
Ltd 0.00
Copyright© 1996ElsevierScience Printedin GreatBritain.All rightsreserved 0360-8352/96 $15.00+
FORMATION PROCESS
PLANS
H A R K H W A N G l and PAEK REE 2 ~Department of Industrial Engineering, Korea Advanced Institute of Science and Technology, 373-1 Kusong-dong Yusong-gu, Taejon 305-701, Korea 2Department of Industrial Engineering, College of Engineering, Keimyung University, 1000 Shindang-dong Dalseo-gu, Taegu 704-701, Korea Abstract--This article proposes a two stage procedure for the cell formation problem with alternative process plans. At the first stage of the procedure,the route selectionproblem is solvedwith the objective of maximizing the sum of compatibility coefficientsamong selected process plans. At the second stage, part familiesare formed based on the result of the first stage using the p-median problem. The grouping solutions based on the proposed procedure are compared with those of the generalizedp-median model in terms of the number of exceptionalelements, grouping efficiencyand grouping efficacy.Copyright © 1996 ElsevierScience Ltd
1. INTRODUCTION In the manufacturing environment, the concept of group technology (GT) has been applied to enhance the productivity of batch type manufacturing systems. In GT, the parts to be produced are grouped into part families based on the similarity of the production processes or the design features. The machines which are needed to process parts in a part family are put together to form the manufacturing cell. These cells are building blocks of the cellular manufacturing (CM) system in which each cell is operated independently with minimal intercellular flows. Within each cell, the production flow can be streamlined as in a flow shop. Among the various benefits from CM reduced number of set-ups, reduced work-in-process inventories and reduced material handling costs are the critical ones. To design a CM system, we identify parts to be produced, part process plans and machines to carry out all the required operations. And then the cell formation process to group the parts into part families and assign machines to the families to form manufacturing cells if followed. In this study, our attention is focused on the cell formation. Numerous cell formation methods have appeared in the literature. Extensive reviews are available in the reports of Wemmerlov and Hyper [1], Chu [2] and Offodile et al. [3]. In most cell formation methods, parts are assumed to have a unique part process plan. In reality, there are usually alternative process plans for parts. And they use different sets of machines. Explicit consideration of alternative process plans invoke changes in the composition of all manufacturing cells so that lower capital investment in machines, more independent manufacturing cells and higher machine utilization can be achieved. Here, some of the cell formation methods that consider alternative process plans are reviewed briefly. Kusiak [4] proposed an integer programming model (p-median model) for part family formation. For the case of alternative process plans, he extended the model so that route selection and part family formation can be done simultaneously in the generalized p-median model. Kim and Won [5] formulated an assignment model where the sum of similarity coefficients between process plans in the same family can be maximized. From the solution of the model, part families and machine cells are partially identified and then they are removed from the incidence matrix. With the reduced problem, the procedure is repeated until all parts are covered. In the previous two models, part families are formed based on the 0-1 incidence matrix which shows machines required to process a part by a process plan. Other cell formation methods [6-13] require additional data such as cost and time of part processing operations, production volume of parts and machine capacity. In most of the previous works, the route selection and cell formation were simultaneously formulated in a single model. In Kusiak's generalized p-median model, route selection and part CAIE3o/3-E
423
424
Hark Hwang and Pack Ree
family formation were done simultaneously with the objective of maximizing the total sum of similarity among process plans in the same cell. We suggest that better grouping solutions could be obtained by formulating and solving them separately in two stages. At the first stage, the route selection problem is formulated with newly defined compatibility coefficients. Two types of the compatibility coefficients were suggested. It is expected that the resulting set of process plans facilitates a grouping solution with highly independent manufacturing cells. At the second stage, part families are formed using the p-median model and machines are assigned to the families to complete cell formation. This article is organized as follows: In Section 2, two different sets of compatibility coefficients are defined and Section 3 describes the proposed two stage procedure. Through numerical experiments, computational results of the proposed procedure are compared with those of Kusiak's in Section 4. Concluding remarks appear in Section 5. 2. COMPATIBILITY COEFFICIENTS We are going to use the following notations in this study. M P R A
= = = =
number of different machines, number of different parts to be produced, total number of process plans, machine-process plan incidence matrix where A = {am,}, m = 1. . . . . M and r = 1. . . . . R. am~= 1 if machine m is required by route r, a,~ = 0 otherwise. p(r) = part number produced by process plan r, srs = similarity coefficient between process plan r and s, cr, = compatibility coefficient between process plan r and s, tpq = similarity coefficient between part p and q. At the first stage, process plans are selected for the parts from the list of alternative process plans under the condition of the one part-one process plan principle. We will call this set of process plans a covering set. A covering set which can result in a completely (or almost completely) blockdiagonalizable grouping solution is what we want to find at this stage. In the block-diagonal form of grouping solutions, parts in the same family should have many similarities in terms of machines required and parts from different part families should have very few or no similarities at all. In this regard, it is desirable for each route in a covering set to have highly similar process plans and highly dissimilar process plans simultaneously in the set so that clearly identifiable friends and foes of the process plan coexist in the covering set. If two process plans in a covering set have a mediocre similarity, then some machines are required by both process plans while other machines are required by only one of them. If any part associated with these two process plans is not in the same family, the commonly required machines result in intercellular movement of the part. For this reason, it would be advisable for two process plans having a mediocre similarity not to be together in a covering set. Since conventional similarity measures are mainly designed to cluster only similar process plans, they are not suitable for the purpose of route selection. In this regard, two different sets of coefficients called compatibility coefficients are proposed. According to these proposed measures, pairs of highly similar and highly dissimilar process plans get high scores. We expect that a covering set with high overall compatibility scores tends to have a block diagonalized grouping solution with minimal number of exceptional elements. The first set of compatibility coefficients (Type 1) are obtained by converting similarity coefficients and the second one is newly defined from the incidence matrix (Type 2).
2.1. Compatibility coefficients: Type 1 Let Srs be the similarity coefficient between process plans r and s. Also, let s , ~ , Sminand Smed bc the maximum, minimum and median of s,s's, respectively. The Ttype 1 compatibility coefficient crs is defined as follows:
Cell formation problem: route selection
425
1.0
e~
o
0.0 S rain
S reed
S max
Similarity Fig. 1. Similaritycoefficientsversus compatibilitycoefficients. forp(r) = p ( s )
Crs = 0
for p(r) # p ( s ) and s~, >I Smed, \Sma x -- Smed/
for p ( r ) # p ( s ) and S,~
(1)
\Smed -- Smin/
The values of compatibility coefficients are in [0, 1]. They are obtained by converting similarity coefficients so that the most similar and the most dissimilar pairs of routes have their compatibility value of 1 and route pairs with mediocre similarity have their compatibility coefficient close to 0. The index n in Eqn (1) is introduced to generate different sets of compatibility coefficients. Depending on the value of n, values of compatibility coefficients and their distribution in [0, 1] can be varied. Figure 1 illustrates the relationship between similarity and type 1 compatibility coefficient. In this study, we used the well-known Jaccard's measure for the similarity coefficients.
2.2. Compatibility coefficients: Type 2 From the incidence matrix, we define the Type 2 compatibility coefficients as follows. crs = 0
forp(r) =p(s),
=[ar, - ~ "br,[/(ar, + ~ "br,) for p(r) # p ( s )
(2)
Where at, is the number of machines required by both process plans r and s, b,, be the number of machines required by either of the two process plans but not by both and ~ is a weighting factor. The values of compatibility coefficients are again in [0, 1] and if the two process plans are identical or totally different, c~s will be 1. ~ can be used to reflect the relative weight that the similarity and dissimilarity of a pair of process plans will have in determining compatibility coefficients. In this study, we tested ~ for the value of 1 and
r,s:p(r)~p(s)
/r,s:p(r)#p(s)
which is the ratio of the number of 1-1 matches and 0-1(1-0) matches in all pairs of process plans. For the latter value of ~, a~s- ~.b,s distributes around 0. 3. TWO STAGE PROCEDURE Utilizing compatibility coefficients defined in the previous section, a mathematical model for the route selection problem is formulated as follows;
426
Hark Hwang and Paek Ree R
R
~ ~ Cr,Yr Y~
Maximize
r=|
~
subject to:
(3)
s=l
r :p(r)
Y,=I =
forp=l,2
..... P
(4)
for r = 1, 2 . . . . . R
(5)
p
Yr E {0, 1}
Yr is 1 if process plan r is included in the covering set. Otherwise, it is 0. The objective function is to maximize the sum of compatibility coefficients of all pairs of process plans in a covering set. The constraints in (4) state that the part is allowed to have only one process plan in a covering set. At the second stage, part families are to be formed based on the generated covering set and machines are to be assigned. The machine-route incidence matrix A is reduced to MxP machinepart incidence matrix and similarity coefficients between parts are calculated. Part families are formed using the p-median problem whose objective function is to maximize the sum of similarity coefficients of part pairs in the same family. The p-median model and its generalized form are reproduced as follows. The p-median model: P
Maximize
P
tpqXpq
~
(6)
p=lq=l P
subject to:
~ Xpq = 1
for q = 1 , . . . , P
(7)
q=l P
Ex.
q=|
(8)
=K
for p = 1. . . . . P and q = 1. . . . , P
(9)
Xpq e {0, 1} f o r p : 1 . . . . . P and q = l . . . . . P
(10)
Xpq ~ Xqq
Xpq is 1, if part p belongs to part family q. Otherwise, it is zero. K is the number of part families to be formed. The generalized p-median model: R
Maximize
R
Z Z s,',W,'s
(11)
r=ls=l
R
~
subject to:
~ W,s=l
forp=l ..... P
(12)
r :p(r) = p s = 1 R
Z Wrr = K
(13)
r=l
W.~W~
forr=l
W,~s{0,1} f o r r = l
..... Rands=l ..... R
(14)
..... Rands=l
(15)
..... R
Wrs is 1, if process plan r is chosen and belongs to family s. Otherwise, it is 0. Once the part families are generated, then machines are assigned. For each machine, the number of exceptional elements and in-block blanks generated are counted when it is assigned to a family. We assign the machine to the family with the minimum number of exceptional elements. If there are ties, a family with the minimum number of in-block blanks is chosen for the machine. 4, NUMERICAL EXAMPLES Four example problems from the literature are solved by both Kusiak's method and our procedure for the purpose of comparison. The machine-route incidence matrices for the chosen
Cell formation problem: route selection
(a) Kusiak's[ 1] problem
(b) Sankaran's[ 10] problem Route Number 11111111112 12345678901234567890 Part Number I1 11223445555667889900 00001001100001010000 11110110000100111100 01000001111001000010 00111111011110111011 10100100101000100101 11110010000010111100
Route Number 11 12345678901 Part Number 11122334455 00101101101 01110010000 10011000110 llO00lllOlO
(c) Logendran's[ 16] problem Route Number" 11111111112222222222333 12345678901234567890123456789012 Part Number lllllllllll 11222334455566778899900111222344 OIlO0000110001001010001001001010 10010011010101100000100011011000 00010000001010010101010100100101 01001100100100100111001100100000 00100000001001011000100010010011 10010010100010001000101001001000 10001001010100010110010010010000
(d) Nagi's[ 11 ] problem Route Number 111111111122222222223333333333344444444455 123456789012345678901234567890123456789012345678901 Part Number 111111111111111111111112 112233445556677777788999999001112223334445556667890 001111111110000000000000000000000000000000000000000 000000000001100000011111111110000000000000000000000 000000000000000000000000000001111111110000000000000 000000000000000000000000000000000000000000000000111 000000000001111111100000000110000000000000000000000 101010100001000011101010101100000000000000000000000 010101010000111100010101010010000000000000000000000 000000000000000000000000000001111111110000000000000 110011001110000000000000000000000000000000000000000 000000000000000000000000000000000000001111111111000 000000000000000000000000000001110001110000000000000 111111111110000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000110 O000000000000000000000000000000000000011100011llO00 000000000000000000000000000000000000000000000000111 000000000000011111111111111110000000000000000000000 000000000000000000000000000000000000001111110001000 000000000100001001000001100001001001001001001000000 000000001000010010000110000000100100100100100100000 000000000010000100100000011000010010010010010010000 Fig. 2. Incidence matrices ofthe ~st problems:(a);Kusiak's [1] problem,(b); Sankaran's [1~ problem, (c); Logendran's [1~ problem, (d);Nagi's [11] problem.
427
428
Hark Hwang and Pack Ree
(a) Similarity coefficients between process plans
1
2 3 4 5 6 7 8 9 10 11
process plans 1 2 3 4 5 6 7 8 9 10 11 .00 - - .33 .33 .33 .33 .33 .33 1.00 .00 .00 .33 .00 .33 1.00 .33 .00 .33 .00 .00 .33 .33 .33 .33 .33 .33 .00 .50 .00 .00 .33 .00 .33 .33 .00 .00 .33 .00 .33 1.00 .33 .50 .00 - 1.00 .33 .33 .50 .00 .33 .00 .33 .00 .00 .33 .50 .00 .33 .50 .00 .00
(b) Type I compatibility coefficients between process plans process plans 1 1
0.0
2 3 4 5 6 7
2
3
4
5
6
8 9 10 11 0.0 0.0 1.0 1.0 0.0 1.0 0.0 1.0 0.0 0.0 1.0 0.25 1.0 0.0 0.0 1.0 0.0 ..1.0 0.0 0.25 1.0 0.0 0.0 0.25 0.0 1.0 0.0 1.0 0.0 -oo 0.0 0.25 0.0 0.0 0.25 0.0 - ~ 0.0
7
- ~ - o o 0.0 0.0 0.0 0.0 -or 0.0 1.0 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 0.0 1.0 -oo 0.0
8
9 10 11
Fig. 3. Similarity and compatibility coefficients: (a); Similarity coefficients between process plans, (b); Type 1 compatibility coefficients between process plans. problems are shown in Fig. 2. We are going to illustrate the proposed procedure with Kusiak's problem. Figure 3(a) shows the similarity coefficients between process plans, where Smax, Smi, and Smed are 1.0, 0.0 and 0.333, respectively. According to Eqn. (1) with n = 1, the Type 1 compatibility process plans 1
2 3 4 5
6 7 8
9 10 11
1 .00
2
3
4
5
6
7
8
.24 1.00 .24 .24
.24 .24 .24 1.00
.24 1.00
.24
-ao -oo .24 .24 .24 .00 -oo .24 1.00 .24 .00 .24 .24 .24 .00 -0o 1.00 .00
.00
9
10 11 1.00 1.00
.24 1.00 .24 1.00 .24 1.00 .10 .24 .24 1.00 1.00
-oo 1.00 .24 .00 .24 1.00 .00 -oo .00
.24
.10
.24 .10 .24 1.00 .24 .10 .24 .10 .00 -oo .00
Fig. 4. Type 2 compatibility coefficients between process plans.
Cell formation problem: route selection
429
(a) Machine-part incidence
Machines
Parts 12345 01011 10100 01010 10100
1 2 3 4
(b) Similarity coefficients
Parts
1 2 3 4 5
Parts 1 2 3 4 5 0 0 1.0 0 0 0 0 1.0 0.5 0 0 0 0 0.5 0
(c) Cell formation
Cells
1 2
Parts I, 3 2,4, 5
Machines 2,4 1,3
Fig. 5. Part family formation: (a); Machine-part incidence, (b); Similarity coefficients, (c); Cell formation.
coefficients are c,~, = 1 for Sr, = 1.0 or 0.0, Crs = 0.25 for s,s = 0.5, and c,s = 0 for s,~ = 0.333 and they are shown in Fig. 3(b). Using Eqn (2) with g = 0.82, Type 2 compatibility coefficients are calculated and shown in Fig. 4. For example, consider route plans 1 and 4 of which a14 = 1 and bl4 = 2. So, the compatibility coefficient c14 = I1 - (0.82)(2)I/(1 + (0.82)(2)) = 0.242. Solving the route selection problem in Eqns (3)-(5) with two sets of compatibility coefficients in Figs 3(b) and 4, we obtained the identical covering set {2, 5, 7, 9, 11 }. Based on the covering set, the reduced machine-part incidence matrix is obtained as shown in Fig. 5(a). Then similarity coefficients between parts are computed and listed in Fig. 5(b). Part families are formed by solving the p-median problem in Eqns (6)-(10) where K is set 2 following the Kusiak's solution. The resulting cell formation is shown in Fig. 5(c).
Two stage procedure Type I Compatibility coefficients
Kuslak's generalized p-median method
Problems
ct= 1 257911
a = 082 257911
95,0 90.0 2 0 95.0 90.0 2 0 95.0 90.0 2 0 95.0 90.0 2 0
95.0 90.0 2 0
95.0 90.0 2 0
135610 12 14 15 18 20
13579 12 14 15 18 19 83.7 71.4 2 3 137812 14 15 17 20 22 24 27 30 31 86.0 70.3 3 5 246810 12 18 21 25 28 32 35 38 40 43 46 48 49 50 51 90.1 79.$ 5 1
13578 12 14 15 18 19 83.7 71.4 2 3 136810 14 15 17 20 22 24 27 30 31 85.7 67.6 3 5 135710 13 15 20 24 29 31 34 37 41 44 47 48 49 50 51 90.1 79.5 5 1
n=~c
Kusiak's[ 11 M = 4 P=-5 R = l l
Sankaran's[10l M = 6 P=10 R=20 Logcndran's[16] M= 7 I>=-14 R=32
Nagi's[ 1 l l M=20 P=20 R=51
Covering set
257911
CI CA K E' 95.0 ~ . 0 2 0 Covering set 1 4 5 7 8 13 14 16 17 19 Cl CA K E 78.8 ~ . 9 2 4 Covering set 2 3 6 9 1 1 14 15 18 20 23 24 27 30 31 C1 CA K 75.1 51.2 3 8 Covering set 1 3 5 7 1 1 12 18 21 23 28 31 34 37 39 42 45 48 49 50 51 CI CA K El 87.2 70.9 5 6
Type II Compatibility' coefficients
257911
n=2
:257911 14579 12 14 15 18 20
n=l 257911 13578 12 14 15 18 19
n=0.5
257911 14568 12 14 15 18 20
82.9 69.4 2 2 82.9 69.4 2 2 83.7 71.4 2 3 82.9 69.4 2 2 1 3 6 8 10 14 15 17 20 22 24 27 30 31 85.7 67.6 3 5 246810 12 18 21 25 28 31 34 37 41 44 47 48 49 50 51 90.1 79.5 5 I
1 3 6 8 10 14 15 17 211 22 ~ 24 27 30 31 85.7 67.6 3 5 246810 12 18 21 25 28 32 35 38 40 43 46 48 49 50 51 90.1 79.5 5 1
1 3 6 8 10 1 3 6 8 10 14 15 17 20 22 14 15 17 20 22 24 27 30 31 24 27 30 31 85.7 67.6 3 5 85.7 67.6 3 5 24689 135710 12 17 21 23 28 13 15 20 24 29 30 33 36 41 44 31 34 37 41 44 47 48 49 50 51 47 48 49 50 51 90.1 79.5 5 1 9 0 . 1 79.5 5 I
Fig. 6. Comparison of computational results by the Kusiak's method and the proposed procedure.
430
Hark Hwang and Pack Ree
With the Type 1 compatibility coefficients, we tested four different values (0.5, 1.0, 2.0, ~ ) of n in Eqn. (I) and two distinct values of ~ were tested with the Type 2 compatibility coefficients as mentioned earlier. Computational results are summarized in Fig. 6. In this figure, the source of the problem, the problem size (number of machines, parts and process plans), the covering set generated, the number of cells (K) and exceptional elements in the final grouping solution (E) are shown along with grouping efficiency (CI, Chandrasekharan et al. [14]) and grouping efficacy (CA, Kumar et al. [15]). CI and CA are the measures most frequently adopted to examine qualities of grouping solutions. In the generalized p-median model, the number of cells to be formed, K, should be given exogenously. Several different values of K are tried and the best one is chosen for each problem and it is used at the second stage of the proposed procedure for comparison on equal basis. All mathematical programming problems are solved using LINDO integer programming package installed on a 486 PC. The quadratic objective function of the route selection problem is linearized before they are solved. Compared to the Kusiak's method, the proposed procedure returns one identical grouping solution and three better solutions with both the Type 1 and Type 2 compatibility coeffÉcients. Varying n in Type 1 compatibility coefficients does not necessarily lead to different covering sets. In problems 2 and 4, different covering sets are found for a different value of n while identical covering sets are found in problem 1 and 3. Different covering sets are found for a different value of n, while identical covering sets are found in problem 1 and 3. Different covering sets results in one identical grouping solution in problem 4. With the Type 2 compatibility coefficients, the grouping solutions are as good as those of the Type 1 compatibility coefficients. 5. CONCLUSION
A two stage procedure for the cell formation problem with alternative part process plans is proposed. At the first stage, process plans are selected for parts based on the compatibility coefficients while respecting the one part-one route principle. Two different ways of defining compatibility coefficients are suggested. At the second stage, part families are formed and machines are assigned. Through four example problems of different size, the performance of the proposed procedure is compared with that of the Kusiak's generalized p-median model. The results show that grouping solutions generated by the proposed procedure are equal to or better than those of Kusiak's in terms of the number of exceptional elements, grouping efficiency and grouping efficacy regardless of the types of compatibility coefficients. It should be noticed that because of the quadratic objective function of the first stage the proposed procedure needs more computational time than the generalized p-median model. Developing a less time-consuming, heuristic for the two stage procedure can be considered for further studies.
REFERENCES 1. U. Wemmerov and N. L. Hyer. Research issues in cellular manufacturing. Int. J. Prod. Res. 25, 413-431 (1987). 2. C. H. Chu. Cluster analysis in manufacturing cellular formation. Omega 17, 289-295 (1989). 3. O. F. Offodile, A. Mehrez and J. Grznar. Cellular manufacturing: a taxonomic review framework. J. Manufact. Syst. 13, 196-220 (1994). 4. A. Kusiak. The generalized group technology concept. Int. J. Prod. Res. 25, 561-569 (1987). 5. Y. Won and S. Kim. An assignment method for the part-machine cell formation problem in the presence of multiple processing routes. Engng. Optimization 22, 231-240 (1994). 6. F. Choobineh. A framework for the design of cellular manufacturing system. Int. J. Prod. Res. 26, 1161-1172 (1988). 7. R. M. Sundaram and K. Doshi. Formation of part families to design cells with alternative routing consideration. Comp. Ind. Engng. 23, 59-62 (1992). 8. R. M. Sundaram and K. Doshi. Cellular manufacturing system design with alternative routing consideration. Comp. Ind. Engng. 25, 477-480 (1993). 9. T. Gupta. Design of manufacturing cells for flexible environment considering alternative routing. Int. J. Prod. Res. 31, 1259-1273 (1993). 10. S. Sankaran and R. Kasilingam. An integrated approach to cell formation and part routing in group technology manufacturing system. Engng. Optimization 16, 235-245 (1990). 11. R. Nagi, G. Harhalakis and J. Proth. Multiple routings and capacity consideration in group technology applications. Int. J. Prod. Res. 28, 2243-2257 (1990). 12. D. Rajamani, N. Singh and P. Aneja. Integrated design of cellular manufacturing systems in the presence of alternative process plans. Int. J. Prod. Res. 28, 1541-1554 (1990).
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13. S. L. Kang and U. Wemmerlov. A work load-oriented heuristic methodology for manufacturing cell formation allowing reallocation of operations. Eur. J. Oper. Res. 69, 292-311 (1993). 14. M. P. Chandrasekharan and R. Rajagopalan. An ideal seed non-hierarchical clustering algorithm for cellular manufacturing. Int. J. Prod. Res. 24, 451-464 (1986). 15. C. S. Kumar and M. P. Chandrasekharan. Grouping efficacy: a quantitative criterion for goodness of block diagonal forms of binary matrices in group technology. Int. J. Prod. Res. 28, 233-243 (1990). 16. R. Logendran, P. Ramakrishna and C. Sriskandarajah. Tabu search-based heuristics for cellular manufacturing systems in the presence of alternative process plans. Int. J. Prod. Res. 32, 273-297 (1994).