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 [067 COPY 2 
 
 
 STX 
 
 FACULTY WORKING 
 PAPER NO. 1067 
 
 Efficient Algorithms for Grouping 
 Component — Processor Families 
 
 K. Ravi Kumar 
 Anthony Vannelli 
 
 jHE LIBRARY. OF. JHE 
 
 SEP 1 8 1984 
 
 College of Commerce and Business Administration 
 3ureau of Economic and Business Research 
 university of Illinois, Urbana-ChamDaign 
 
BEBR 
 
 FACULTY WORKING PAPER NO. 1067 
 College of Commerce and Business Administration 
 University of Illinois at Urbana-Champaign 
 August 1984 
 
 Efficient Algorithms for Grouping Component-Processor Families 
 
 K. Ravi Kumar, Assistant Professor 
 Department of Business Administration 
 
 Anthony Vannelli 
 University of Toronto 
 
Digitized by the Internet Archive 
 
 in 2011 with funding from 
 
 University of Illinois Urbana-Champaign 
 
 http://www.archive.org/details/efficientalgorit1067kuma 
 
Efficient Algorithms for Grouping Component - Processor Families 
 
 K. Ravi Kumar 
 
 Department of Business Administration 
 
 University of Illinois 
 
 at Champaign-Urbana 
 
 Champaign, Illinois 61820 
 
 Anthony Vannelli 
 
 Mathematical Sciences Department 
 
 IBM Thomas J. Watson Research Center 
 
 Yorktown Heights, NY 10511 
 
 and 
 
 Department of Industrial Engineering 
 
 University of Toronto 
 
 Toronto, Ontario M25 1A4 
 
 Abstract 
 
 The problem of grouping pact families is very important in the implementation of 
 Group Technology and Flexible Manufacturing System concepts. In this paper, we opt 
 to use existing routing sheet information to derive the component-processor groups. 
 The actual grouping is done by modelling the problem as an optimal k-decomposition 
 of weighted networks. Algorithms which are suitable for computer implementation 
 and large problems are developed to find an initial solution and for refining this 
 solution. Bounds on algorithm performance are constructed to give an estimate of the 
 quality of the generated solution. A numerical example illustrates these new techni- 
 ques. 
 
1. Introduction 
 
 The trend in product preferences is towards more customization and in terms of 
 manufacturing priorities, this implies smaller batch sizes (Reich, 1983). The conven- 
 tional approach in meeting such requirements is to utilize functional or process layouts. 
 However, statistical analysis indicate that such job shops have low machine utilization 
 rates and very high waiting times. A more efficient means of satisfying small batch 
 priorities is needed to alleviate the productivity lag in this sector of the manufacturing 
 industry, which comprises approximately 75% of all manufacturing. 
 
 Two technologies, which seek to solve this problem, are Group Technology (GT) 
 and Flexible Manufacturing Systems (FMS). They are similar in that they seek to 
 manufacture small lot sizes (in fact, a lot size of 1 in FMS) of "parts of similar 
 process, of somewhat dissimilar materials, geometry and size" (Mitrafanov, 1959). 
 They differ in that GT is conceptually a dedicated cell of machines: grouped, tooled 
 and scheduled as a unit while FMS seems to strive for flexibility in currently manufac- 
 tured products and those that may arise in the future. Also FMS is aimed at total 
 computer control while GT is satisfactorily implemented in a worker-machine environ- 
 ment. 
 
 Despite their differences, both GT and FMS share a common design problem - 
 identification of those "parts of similar process" implying both components and 
 processor types that will make up a GT cell or an FMS (Kusiak, 1984). The original 
 approaches to this problem were based on two different philosophies: 
 
 a) analyze a classified and coded data base of parts which reflect design shape, 
 engineering features as well as methods of manufacture (Hyde, 1981) 
 
 1 
 
b) analyze a route sheet data base which reflects the existing methods of manufac- 
 ture of parts currently being produced (Burbidge, 1975). 
 
 Clearly a) is a more global approach but it suffers due to the fact that classification 
 and coding can take thousands of man-hours without producing any productivity gains 
 in the short run. On the other hand, b) could be implemented quite easily but has 
 been criticized because it "simply perpetuates existing, often, poor practices" (Hyde, 
 1981). 
 
 Our own contention is that for productivity purposes, b) should be implemented 
 for short-term gains and should be thought of as an intermediate step in the implemen- 
 tation of a). The criticism that existing practices are bad may be unfounded and the 
 redesign, once the classification and coding is complete, may be quite simple 
 (especially if the original grouping was a good approximation). Also the productivity 
 gains, with some form of grouping, through reduced set-up times, material handling 
 and waiting times, may be large enough to underwrite the classification and coding 
 projects as well as final redesigns. 
 
 In this paper, we will discuss some aspects of efficiently implementing the 
 grouping analysis using existing route sheet information. Specifically we will model 
 the methodology of grouping components and processors as a network decomposition 
 problem, and devise algorithms which aim at producing "good" solutions to the 
 problem. We also give methods to evaluate bow good the solutions are by construct- 
 ing bounds on the optimal solution. These algorithms can be efficiently implemented 
 for very large data sets, as is usually the case in such analysis, typically 2000 compo- 
 nents on 100 processors. In Section 2, we formulate the network model for the 
 component-processor grouping. An algorithm to obtain a good initial solution to the 
 problem is described in Section 3. In Section 4, we devise an efficient algorithm which 
 
 -2- 
 
seeks to improve the initial solution. A numerical example illustrates this new ap- 
 proach in Section 5 and Section 6 contains concluding remarks. 
 
 2. Component-Processor Grouping and the Network Decomposition Model 
 
 Burbidge (1975) proposes Production Flow Analysis as a technique to implement 
 GT. Within this analysis, there is a phase, called Group Analysis, which takes the 
 matrix of components (or packs of components) and processors needed by the compo- 
 nents and tries to rearrange this matrix such that block diagonalization (or approxi- 
 mately so) is achieved. For example consider Figure 1 which records the routing 
 requirements of 4 packs consisting of 7 parts. Pack 1, which is just one part, requires 
 machines A and C which pack 2, consisting of three parts, requires machines 8 and D. 
 This processor/pack information is depicted, in Figure 2, in matrix form with packs as 
 rows and machines as columns. Looking at this matrix, we can see packs 1 and 3 
 require only machines A and C while packs 2 and 4 need only machines B and D. 
 Thus, a family grouping of packs and machines is possible in which each pack in the 
 family can be processed by the group of machines in that family. This can be seen in 
 Figure 3, where we have exchanged rows 2 and 4 and also columns B and C, to create 
 a partition of the original matrix into diagonal blocks. Each block represents a family 
 consisting of packs and machines, with each pack of parts being processed using only 
 the machines in this family. Any off-diagonal entry, after this block diagonalization 
 process, represents interdependencies of the processing of a pack in one family with 
 the machines in another family. And, of course, the idea is to minimize the interde- 
 pendencies. 
 
f^cks Part Numbers Machines Required 
 
 1 5 A, C 
 
 2 10, 15, 20 B, D 
 
 3 25 D 
 
 4 30 C 
 
 Figure 1 
 
 Machines 
 
 Packs 
 
 1 
 2 
 3 
 4 
 
 B 
 
 C 
 
 m 
 
 Figure 2 
 
 Machines 
 
 Packs 
 
 1 
 4 
 3 
 2 
 
 5 
 
 Figure 3 
 
 4 - 
 
A variety of researchers have found that the process of block diagonalization is 
 not easy to implement on a computer. Burhidge (1975) points out that it is compara- 
 tively simple to find the families for a small sample using "pattern recognition, 
 application of production know-how and intuition. It has proved surprisingly difficult 
 to find a method suitable for the computer,...". Groover (1980) recognizes that "this 
 is the most subjective and most difficult step in production flow analysis, yet it is the 
 most crucial step in the procedure". El-Essawy and Torrance (1972) indicate that this 
 grouping process requires "an unjustifiably sophisticated procedure." Also, in real 
 world situations, the problem size can be very large, a typical value being 1400 parts 
 on 150 machines. The use of computers become increasingly necessary and efficient 
 algorithms even more so. 
 
 Very few attempts to grapple with this problem of grouping components and 
 processors have taken an analytical approach. King (1979, 1980) uses a Rank Order 
 Clustering (ROC) algorithm, while McCormick et al. (1972) use a sub-optimization 
 procedure on a restricted quadratic assignment model. King and Nakomchai (1982) 
 briefly review the approaches to this problem and extend the ROC algorithm to 
 perform more efficiently on the computer, regarding storage and CPU time. A major 
 problem in their algorithm is identification of bottleneck machines - this step is quite 
 arbitrary but is very crucial to the development of subsequent grouping. Also, the 
 methods discussed above do not take into account the annual volume of production 
 that is required of each component on the processor Le. the material handling cost- In 
 their formulations, each component is equally important in terms of cost irrespective of 
 volume. 
 
 We formulate this problem as an optimal k-decomposition problem in graph 
 theoretic terms. Instead of looking at block diagonalizations of matrices, we will, 
 
 -*- 
 
equivalency, look at decompositions of networks. For example. Figure 4 represents 
 the same information as Figures 1 and 2 but in network form. Also, Figure S repre- 
 sents the block diagonalized matrix Figure 3, which implies that the optimization 
 problem can also be formulated as : find a decomposition of the packs/machines 
 network such that there are minimal dependencies between the sub-networks. 
 
 Pa 
 
 Figure 4 
 
 Pr 
 
 Figure 5 
 
In network theory terms, let G — (K,£) be an undirected graph, where V is the 
 set of vertices (or nodes) and E is the set of edges (or arcs). In this case, let 
 
 K— {Pa'Pr) wnere ?a k t * ie sel °* P ac ' 15 °* P 31 ^ e -8- ^^^ products, subassemblies, 
 spare parts and P R is the set of processes e.g. beat treatment, de burring, painting. 
 The arc set £ contains all the interconnections between node set P A and node set P R 
 where each arc (i,j) represents the requirement (at least, currently) of processor j for 
 pack L 
 
 A k -decomposition of the graph G is obtained by deleting edges of G to obtain k 
 disconnected subgraphs G i - (V it £,), i — 1,2,..., k, and each of the vertices of G is 
 contained in exactly one of the node sets K { . Then, an optimal k -decomposition of a 
 graph G is a k -decomposition that minimizes the weight on the interconnections (or 
 edges) between the k subgraphs (Vannelli and Vidyasagar, 1984). The factor k 
 remains under management control and could be a policy variable. For instance, 
 k - 1 would not partition the graph at all while a large k (in comparison to the total 
 number of nodes) will tend to partition the graph very finely. In the first case, the 
 number of interdependencies is zero while in the latter, it will be quite high. So, there 
 is a trade-off to be made between the number of groups in the decomposition and the 
 amount of interdependencies. 
 
 To mathematically model the problem given a fixed k, let 
 
 0, if node i is not in subgraph j 
 
 : 
 
 if node i is in subgraph j. 
 
 Then, 
 
 k 
 2 x fj - 1, V i - 1,2,..., n 
 
 -7>- 
 
implying that each node (and there are n of them in total Le. {# of elements in P A ] + 
 {# of elements in P R ] - n) can only be in one subgraph. 
 
 Also, one could add a congestion constraint which restricts the cumber of nodes 
 in each subgraph; Le., : 
 
 / < 2 x ij * •• V J " 1 ' 2 -' k - 
 
 We can represent each arc of subgraph p (j> - 1,2^.., A:) by the node product 
 (x^ • XjJ). Then, arc (Lj) is in subgraph p if and only if x^ — x^ — 1. Let a^ be the 
 volume of component i that has to be processed through processor j (or even the profit 
 or productivity potential associated with i and j). We can now represent the sum of all 
 the arcs that belong within the k subgraphs by 
 
 *-l n T k 
 
 i-i y-i+i L/>»i J 
 
 Note that maximizing this quantity is equivalent to minimizing the sum of the interde- 
 pendencies of the k weighted subgraphs. 
 
 The optimal k -decomposition problem can now be stated as: 
 
 l»l y-/'+l p»\ 
 
 k 
 S t £ X ij" *■ Vi " 1 ' 2 — » " 
 
 /-I 
 
 /<2 *yS«. vy- 1.2 * 
 
 jr«-0 or 1. 
 
 (1) 
 
 8- 
 
This is the 0-1 quadratic programming problem with linear constraints which is 
 referred to as the quadratic assignment problem by King and Nakomchai (1982) and 
 this formulation is considered difficult to solve, since it is an NP -complete problem. 
 
 In the next two sections we develop algorithms to solve approximations of (1). A 
 starting k-decomposition and improved k-decomposition are found by these techni- 
 ques. We also develop bounds that convey how good the solution generated is, which 
 is a factor that other analytical approaches have failed to consider. 
 
 3. A Heuristic Technique for Decomposing Undirected Graphs 
 
 In this section, we develop an algorithm for approximating the global solution of 
 the optimal k-decomposition problem (1). A modification of an eigenvector approach 
 introduced by Barnes (1982b) is used to accomplish this. 
 
 Barnes formulates an algorithm for approximating the optimal k-decomposition 
 problem with fixed subgraph size. 
 
 it— 1 n . k 
 
 Max 
 
 2 2 ( 2 a u x * x jp) 
 
 i.l Jmi+l V=l 
 
 (2) 
 
 k 
 s.t. 2 x tf - 1, Vi- 1,2,..., n 
 
 2 x ij- m j> v ->- ia--* 
 
 k 
 
 x„-0 or 1, J «/■■*. 
 /■l 
 
 The algorithm is a two-step procedure which first finds the k largest eigenvalues 
 ^]£^2- — -^A an< ^ tneu * corresponding eigenvectors u, of the admittance matrix A 
 (Cullum and Donath, 1974). An approximation of problem (2) is then found by 
 
 -9- 
 
solving the transportation problem 
 
 Max 2 2 (7=f) X * 
 
 (3) 
 » 
 ^ x (j - m 7 . V y - 1,2,..., k 
 
 'v^O- 
 
 Lawler (1976) shows that problem (3) is solved in 0(n ) time in the worst case. 
 However, this situation rarely arises in practice. 
 
 We now proceed to develop an algorithm for approximating the solution of 
 problem (1). We begin by first constructing upper and lower bounds on the number 
 of edges cut, E c for problem (1). Define 
 
 E u — sum of weighted edges cut by a k -decomposition 
 
 then, E C <E U . 
 
 It is also desirable to estimate how far E u is from E c . One would like to construct 
 a lower bound on E c in this case. Donath and Hoffman (1973) construct a simple 
 lower bound on E c for problem (2). Consider the matrix A, where 
 
 n 
 
 10- 
 
and calculate the k largest eigenvalues of A; that is, A 1 >X 2 >...>A Jk . A lower bound on 
 
 1 * - 
 £, is — - 2 X: m, (Donath and Hoffman, 1973). Thus, E e can always be bounded 
 
 c 2 i.i ' 
 
 for problem (1) by 
 
 -1 J \ imi <E e <E m . (4) 
 
 * i-l 
 
 The upper and lower bounds (4) on E c allow us to investigate the optimal 
 k -decomposition problem (1). Note that one fixes the number of nodes that belong to 
 each subgraph in the number of nodes that belong to each subgraph in the optimal 
 k -decomposition problem (2). Since one does not know a priori the number of nodes 
 to be fixed in each subgraph, a local optimum of problem (1) may be found by solving 
 all the transportation problems (3) where 
 
 r>mj>m 2 >... Mk-t 
 
 k 
 X m,-/! (5) 
 
 iml 
 
 m^Q, m i integer. 
 
 Clearly, this approach is unsuitable for problems containing many m/s satisfying (5). 
 However, the lower bound on E c given in (4) allows us to develop a more tractable 
 procedure for approximating the solution to problem (1). Note that the optimal 
 
 11 
 
solution of problem (1) is bounded by 
 
 Min -1 £ Km* 
 z i-i 
 
 S.L «>'«i>'n2>...>m jk >/ 
 
 (7) 
 k 
 2 m,.-/t 
 
 m^O, m f integer. 
 
 The solution of problem (7) can be obtained in greedy fashion as the following result 
 shows (Vannelli, 1984). 
 
 Theorem 1: The optimal solution of problem (7) is 
 
 m i — Minim^. "» ; >'w (+ i, «>mj>m 2 >...m A >r* 
 
 (8) 
 i-i * 
 
 and V m j m * rt ~^ J m j^ * or l ™^»^"* !»•••» 1 
 
 where m k + l - /. 
 
 Proof: We prove this result by contradiction. Assume that we can find 
 {mf. I - 1,2,..., k] such that m>m 1 >m 2 >...>m k >t and 
 
 -1 2 m,A,.<-l J w'X/. 
 ««i z i-i 
 
 where the mfs are obtained using formula (8). Letting X, - -— \ f , we have 
 
 * 
 2 (m'-rrtj) A, > 
 
 12 - 
 
and 
 
 k 
 
 By (9), there exists t e {k, k - 1,..., 1} such that 
 
 m ( —m ( > 0. 
 
 In addition, we claim 
 
 m*-m,.>0 V I - 1,2 /. (10) 
 
 To show that (10) is true, assume that m ; — m f <0 for some i e \\, 2,..., t—l}. Then, 
 there exists a partition \m t } such that 
 
 m 
 
 \. » rrtj , + 1 for some j e {I, 2,..., /- 1} 
 
 •• • - 
 
 •• 
 
 m, — m, otherwise 
 
 This implies m ( has not been chosen according to (8). Therefore 
 1 • » * 
 
 Since Xj<X 2 <...< X A , then 
 
 t k 
 
 2 (m'-m,) X, < - 2 (J"*-" 1 ;) X,. 
 i-l m/ + l 
 
 We have a contradiction. ■ 
 
 Problem (7) is easily solved in the optimal 2 -decomposition case. Since the 
 largest eigenvalue of A is Xj — 0, m 2 is chosen to be the smallest integer value greater 
 than or equal to / and satisfying m^ + m 2 ■■ n. This result shows that the best 
 
 - 13- 
 
weighted cut decomposes the graph into one large subgraph and one smaller one. In 
 general, we find that the solution of problem (7) by (8) yields a weighted cut which 
 decomposes the original graph into an equal number of large and small subgraphs 
 containing almost « and / nodes respectively. 
 
 The previous discussion leads to the following heuristic technique for solving 
 problem (1) and for obtaining bounds on E c . 
 
 Algorithm 1 
 
 Step 1: Choose the number of desired subgraphs k, an upper bound « and lower 
 bound / on subgraph size. 
 
 Step 2: Solve problem (7) by formula (8) to obtain «>mj>m 2 >... >m k >? 
 
 Step 3: Find the k largest eigenvalues Aj<X 2 ^ — ^^jk an( ^ corresponding eigenvec- 
 tors u. of the admittance matrix A. A local optimum for problem (1) is 
 found by solving the one transportation problem 
 
 k It y H. 
 
 Min 
 
 (ID 
 
 -2 S(-7^)*f 
 
 s.t. 2 x u - l v ' - | A—" 
 /-I 
 
 *,;><>• 
 
 Step 4: Upper and lower bounds on the sum of the edges cut E c for problem (1) 
 
 - 14 
 
are 
 
 
 2 ,-, 
 
 where £ M (mj^..,m A ) is the sum of the edges cut by the k -decomposition 
 solution obtained by solving problem (11) and \ ] >X 2 >..-X Jk are the k 
 largest eigenvalues of A. 
 Tighter bounds on E c for problem (1) can be obtained in (Barnes, 1982a). 
 
 4. Improving an Existing It-Decomposition 
 
 In general, a k-decomposition obtained by Algorithm 1 will not be optimal, even 
 locally. For such a partition, it may be possible to decrease E u (m J ,...,m k ) by inter- 
 changing nodes in the k subgraphs. Kernighan and Lin (1970) describe an 0{n 2 ) 
 routine for performing this. However, this technique assumes that the subgraph sizes 
 remains fixed and that only two subgraphs are considered at one time. This can be 
 very limiting when applied to the bounded subgraph constraints considered in problem 
 (1). 
 
 In general, one would like to know if it is possible to interchange subsets of the k 
 subgraphs to decrease the number of interconnections between subgraphs. In this 
 section, we describe a new technique for performing this sequence of interchanges. 
 We take advantage of the bipartite graph structure of GT to accomplish this. The new 
 k-decomposition improvement method is solved in polynomial time. A linear transpor- 
 tation problem is again solved at each step. 
 
 An improved k-decomposition of the existing k-decomposition is generated as 
 follows. For each row element i e P A , let s(ij) denote the sum of the elements in 
 row i that are in subgraph Si. If row i is in the pth subgraph, then let a(i) — s(i,p). 
 
 - 15- 
 
We now calculate the change in the sum of the interconnections that result in moving 
 node i in subgraph S to subgraph S., j+p. Equivalently, we calculate the net gain in 
 off -diagonal elements in moving row i in block p to block j. This is done by 
 
 Note that l R (iJ)<0 implies that by moving node i to subgraph S., the number of 
 interconnections E u is reduced by h R (ij). Since the graph representation of GT is a 
 bipartite graph, we can determine which nodes can be moved to other subgraphs so 
 that the existing k-decomposition is improved by keeping the columns fixed. This is 
 obtained by solving the following problem 
 
 k m 
 
 rowmin(0- Min J J M'V)*,y 
 
 Jml f-1 
 
 k 
 
 s.t. 2 x o r- *• v ' " u m(m " \ p A D 
 
 (12) 
 
 2 X ij < m-m JR (t), V;- 1,2,...,* 
 ««1 
 
 *i£° 
 
 where rrtj R (f) is the number of nodes in P R that are in subgraph 5 at iteration /. A 
 similar procedure can be found for the columns of B. Letting A c (i' v /) represent the net 
 gain in off-diagonal elements in moving column i in block p to block j, we can deter- 
 mine which nodes can be moved to other subgraphs so that the existing k- 
 decomposition is improved by keeping the rows fixed. In this case, we solve the linear 
 
 16 
 
transportation problem 
 
 k n 
 colmin(0- Min J 2 V'VXk,;/ 
 
 S.L J J. /y - 1, V I - U^,|»(« - | P* | ) . 
 
 (13) 
 
 n 
 i-1 
 
 ->^° 
 
 where m^(/) is the number of nodes in P A that are in subgraph 5 • at iteration /. 
 
 Note that problems (12) and (13) are easy to solve transportation problems. 
 Alternating row and column changes, which most decrease the number of interconnec- 
 tions are performed at each iteration. After a finite number of steps, no improvement 
 will be possible and a local optimum is obtained. The constraints imposed on prob- 
 lems (12) and (13) assume that the bounded subgraph constraints of problem (1) are 
 satisfied We tie these ideas together in the following algorithm. 
 
 Algorithm 2 
 
 Step 0: / - 1 
 
 Step 1: Given a It-decomposition {S[ f ^ ,..., SJf^}, determine its matrix B representa- 
 tion. Calculate 
 
 l R (iJ) A net gain in interconnections in moving row i to subgraph 5J ' 
 
 A c (; J) A net gain in interconnections in moving column i to subgraph 5, 
 Step 2: Solve problems (12) and (13). 
 Step 3: Calculate min {rowmin(0, colmin(/')} - c(/). If c(t) - 0, go to Step 6. 
 
 -17- 
 
Step 4: If c(f) - rowmin(/)<0, use solution of problem (12) to permute the 
 appropriate rows of B, otherwise permute the appropriate columns of B. 
 
 Step 5: / *- I + 1 
 
 Go to Step 1. 
 
 Step 6: The new cut is E u + Z c(t) The best local k-decomposition is 
 
 Algorithm 2 is illustrated on the following example. 
 Example 4.1 
 
 Consider the following 7-node example. 
 
 
 
 Figure 6 
 
 We seek the optimal 2 -decomposition where / - 2 and * - 4 in problem 1. Assume 
 that the initial 2 -decomposition is: 
 
 S\" - {1,4,7} and S^ - {2,3,5,6} 
 
 Step 1: 
 
 18 
 
Step 2: 
 
 Step 3: 
 
 Step 4: 
 
 Step 5: 
 Step 1: 
 
 B- 
 
 7 5 6 
 
 < 
 OM 
 
 ljo 
 ~i i V \ 
 
 1 J 
 
 a r o o i -n 
 
 A *" L-i i o oj 
 
 The solution of problem (12) is to interchange rows 1 and 4. The solution 
 
 of problem (13) is to move column 2 to S \ l) 
 
 rowmin (1) — -2 
 
 colmin (1) ■» 
 
 c(l) - -2. 
 
 Permute rows 1 and 4 of B 
 
 7 5 6 
 
 B- 
 
 — 
 
 — 
 
 1 
 
 
 
 1 
 
 
 
 T V" 
 
 
 
 1 
 
 _ 
 
 — 
 
 l <—2. Go to Step 1. 
 
 .(2) 
 
 ■ (2) 
 
 {3,4,7}, S^'- { 1,2,5,6} 
 
 A *" Li 1 o oj 
 
 Step 2: No rows or columns can be moved by solving (12) and (13) 
 Step 3: c(2) - 0. Go to Step 6. 
 
 Step 6: The new cut is £ 2 -2 - 3-2 - 1 and s[ 2) - {3,4,7} s£ 2) - { 1,2,5,6} is a 
 local optimal 2-decomposition satisfying problem ( 1 ) with « — 4, I — 2. 
 
 19- 
 
5. A Numerical Example 
 
 The most difficult aspect of implementing Algorithms 1 and 2 is to determine the 
 linear coefficients of the transportation problems. This is accomplished in Algorithm 1 
 by finding the k largest eigenvalues and their corresponding eigenvectors. In the case 
 of very large sparse symmetric matrices (dimension 2000 or greater), an available 
 FORTRAN code (Cullum and Willoughby, 1984b) allows one to handle such prob- 
 lems. An APL code called PUSH is currently being used to find the linear coefficients 
 $ R (iJ) and b c (i,j) for any partition {s[ f) ,..., S[ e) ) in Algorithm 2. 
 
 We illustrate the use of Algorithms 1 and 2 on the following 23 machine by 20 
 pack GT problem given in Figure 7 (Groover, 1980). We attempt to find an optimal 
 2-decomposition of this problem where « — 29 and I — 14. We solve problem (3) to 
 decompose the graph into two subgraphs containing 19 and 24 nodes. This yields the 
 matrix representation B 1 (20 edges cut). This is shown in Figure 8. Using Algorithm 
 2, machines 4, 16, 19 and 20 are moved to subgraph S 2 yielding matrix representation 
 B (14 edges cut). Finally, note that pack 42 can be moved to subgraph S 2 yielding 13 
 edges cut; see Figure 9. We can bound the number of edges cut, E c by Algorithm 1 
 
 6 < E c < 13. 
 
 20 
 
Components 
 
 1 
 
 i 
 
 2 
 
 3 
 
 4 
 
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 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
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 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 25 26 27 28 29 3Q 31 32 33 34 35 36 37 38 39 40 HI 42 43 
 
 Figure 7 
 
 
 
 
 
 
 
 
 
 
 
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 20 edges cut 
 Figure 8 
 
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 13 edges cut 
 Figure 9 
 
 6. Conclusions 
 
 One of the fundamental design problems faced, when a manufacturing plant is 
 deciding to implement the new technologies of flexible manufacturing systems or group 
 technology, is the question of what parts to produce on which machines. In this paper, 
 using the philosophy of production flow analysis (Burbidge, 1975), we have modelled 
 the grouping of parts/packs and machines as a weighted bipartite graph decomposition 
 problem. 
 
 While this formulation is an NP -complete problem, we have developed a two 
 phase polynomially bounded algorithm for approximating the optimal solution. Phase 
 one approximates the original graph partitioning problem by an easily solved linear 
 transportation problem. The output of this algorithm is to be viewed as a good 
 starting solution to the grouping problem, which can be improved by the phase two 
 
 -22 
 
algorithm. The phase two algorithm takes advantage of the bipartite graph structure of 
 the GT problem to accomplish this- Both these algorithms have been designed with 
 the large-scale nature of these problems in mind, and are easily implemented on 
 computers. 
 
 Given that these algorithms may not be optimal, we have derived bounds on the 
 optimal solution which indicates how good these algorithms perform. This allows the 
 users to decide whether to seek a better solution. Also imbedded in our algorithms is 
 the flexibility for users to perform sensitivity analysis on the number of groups that 
 may be decided. 
 
 Further issues that have to be dealt with in this line of research are the conse- 
 quences of the bottleneck machines (those that are needed by more than one group) 
 and efficient methods to analyze the costs of perfect decomposition or decoupling of 
 these groups. These problems are currently being investigated. 
 
 23 - 
 
References 
 
 Barnes, E.R., "Partitioning, Spectra and Linear Programming", Proceedings of 
 the 25th Anniversary Waterloo Conference in Combinatorics, (June 1982). 
 
 Barnes, E.R., "An Algorithm for Partitioning the Nodes of a Graph", SIAM J. 
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 Burbidge, J.L., The Introduction of Group Technology, Halsted Press, John 
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 Cullum, J. and Do oath, W.E., "A Block Lancos Algorithm for Computing the 
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 24 
 
McCormick, W.T., Schweitzer, P.J. and White, T.E., "Problem Decomposition 
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 25- 
 
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