Decomposition of manufacturing systems - Robotics and Automation ...

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IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4, NO. 5 , OCTOBER 1988

1

Decomposition of Manufacturing Systems ANDREW KUSIAK

AND

WING S . CHOW

A-

Abstract-An approach to decomposition of manufacturing systems known as the group technology (GT) is surveyed. GT allows to cluster machines into machine cells and parts into part families. There are two basic methods used for solving the GT problem: classification and modeling. Two variations of the classificationmethod, visual and coding, are briefly discussed. These formulations of the GT problem; namely, the matrix formulation, mathematical programming formulation, and algorithms for solving them are presented. Use of expert systems in GT is suggested. All concepts presented are illustrated with numerical examples.

I. INTRODUCTION HE BASIC IDEA behind Group Technology (GT) is to decompose a manufacturing system into subsystems which are easier to manage than the entire system. Introduction of GT in manufacturing has the following advantages:

AA

PARTS

T

1) 2) 3) 4) 5) 6) 7) 8) 9)

reduced production lead time (20-88 percent) reduced work in process (up to 88 percent) reduced labor (15-25 percent) reduced tooling (20-30 percent) reduced rework and scrap materials (15-75 percent) reduced setup time (20-60 percent) reduced order time delivery (13-136 percent) improved human relations reduced paper work.

For detail justification of the above advantages see [l], [2], PI, 1141, [17l, [19l, and [231-[271. To date, many GT survey papers have been published. They can be divided into two basic categories:

1) application papers, for example: Edwards [171, Edwards and Koenigsberger [ 161, Fazakerlay [191, and Schaffer

[W; 2) methological papers, for example: King and Nakornchai [31] and Mosier and Taube [49]. This paper focusses on models and algorithms developed for GT . There are two basic methods used for solving the GT problem: 1) classification 2) cluster analysis. Manuscript received June 10, 1987; revised January 28, 1988. This research was partially supported under grants from the Natural Sciences and Engineering Research Council of Canada and the University of Manitoba. A. Kusiak was with the Department of Mechanical and Industrial Engineering, The University of Manitoba, Winnipeg, Man., R3T 2N2, Canada. He is now with the Department of Industrial and Management Engineering, The University of Iowa, Iowa City, IA 52242. W. S. Chow is with the Department of Mechanical and Industrial Engineering, The University of Manitoba, Winnipeg, Man. R3T 2N2, Canada. IEEE Log Number 8821740.

BEFORE GROUPING

GROWED

PARTS

Fig. 1. Parts grouped into families using a visual method

The classification method is used to group parts into part families based on their design features. There are two variations of the classification method: visual method coding method. A. Visual Method

The visual method is a semi-systematic procedure where parts are grouped according to their similarity of the geometric shape as shown in Fig. 1, where ten parts have been grouped into four part families. Grouping parts using the visual method is dependent on personal preference. Therefore, this method is applicable in case where the number of parts is rather limited.

B. Coding Method In the coding method, parts can be classified on the basis of the following features:

1) 2) 3) 4) 5)

geometric shape and complexity dimensions type of material shape of raw material required accuracy of the finished part.

Using a coding system, each part is assigned a numerical and/or alphabetical code. Each digit of this code represents a feature of a part. The currently available coding systems differ with respect of the depth of coverage of the above five features. For example, a coding system may provide more information on the shape and dimension of a part, whereas another may emphasize more the accuracy of a part. There are three basic types of coding systems: monocode polycode hybrid.

0882-4967/88/1000-0457$01.00 O 1988 IEEE

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IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4, NO. 5 , OCTOBER 1988

MONOCODE

DIGIT 1

SYSTEM

SAMPLE MONOCODE

TT

3. BRONZ

3

2

DIGIT 2

1'' THICK AND STRAIGHT

STRAIGHT

1

2

HOLE

STRAIGHT DIGIT 3

a w

2: 5 2 w w d wWa

...

BRONZ W

... Fig. 2.

A monocode system and a monocode.

An example of a monocode system is illustrated in Fig. 2 . Features of each part are matched with a list of features corresponding to each node of the tree and a part code is generated. Since the monocode system has a tree structure, a digit selected at a particular node depends on the digit selected at the preceding node. To fully understand the representation of a part monocode, all its digits are required. For a given part, the length of its monocode is rather short compared to other coding systems [ 2 8 ] . For further discussion of coding systems, see for example: Ingram [28],Eckert [ 151, Schaffer [53],Dunlap and Hirlinger [131, Opitz and Wiendahl [52],and Gallagher and Knight [21]. In Section II, three formulations of the GT problem originating from cluster analysis are presented. Future developments in GT are discussed in Section III. Conclusions are drawn in Section IV.

PARTS

MACHINES

PF-1

MC-1

PF-2 GROUPED MACHINES AND PARTS

UNGROWED MACHINES AND PARTS

Fig. 3.

The physical machine layout.

PF-1

11. CLUSTER ANALYSIS METHOD Cluster analysis is concerned with grouping of objects into homogenous clusters (groups) based on the object features. It has been applied in many areas such as: biology [18], data recognition [48], medicine [32], pattern recognition [57], production flow analysis [ 8 ] , [30],task selection [5 13, control engineering [%I, automated systems [33], [34], and expert systems [ 1 11. Waghodekar and Sahu [60] listed more than 400 references related to cluster analysis and group technology. The application of cluster analysis in GT is to group parts into part families (PF) and machines into machine cells (MC). The result of this grouping leads to 1 ) physical machine layout or 2 ) logical machine layout. The physical machine layout requires rearrangement of machines so that the shop floor is altered, as shown in Fig. 3 . On the other hand, using the logical machine layout machines are grouped into logical machine cells and the position of machines is not altered (see Fig. 4 ) . The logical grouping can be applied in case when the production content is changing frequently so that the physical machine layout is not justified. To model the GT problem, the following three formulations

A

LINGROWED MACHINES AND PARTS

PF-2

GROWED MACHINES AND

PARTS

Fig. 4. The logical machine layout.

are used: matrix formulation mathematical programming formulation graph formulation.

A . Matrix Formulation In the matrix formulation, a machine-part incidence matrix [au] is construced. The machine-part incidence matrix [au] consists of 0, 1 entries, where an entry 1 (0) indicates that machine i is used (not used) to process part j. Typically when an initial machine-part incidence matrix [au] is constructed, clusters of machines and parts are not visible. Clustering

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KUSIAK AND CHOW: DECOMPOSITION OF MANUFACTURING SYSTEMS

algorithms allow to transform an initial incidence matrix into a more structured (possibly block-diagonal) form. To illustrate the clustering concept consider the machine-part incidence matrix (1). Part Number 1 2 3 4 5

following approaches have been developed: production flow analysis similarity coefficient methods sorting-based algorithms bond energy algorithm cost-based method cluster identification algorithm.

Production Flow Analysis: To implement GT in manufacturing systems, Burbidge [8] introduced the concept of Production Flow Analysis (PFA). The PFA is applied at the following three levels: Rearranging rows and columns in matrix (1) results in matrix (2) Part Number PF-1 PF-2 1 3 2 4 5 Machine

(2)

MC-2 Two machine cells (clusters) MC-1 = {2,4}, MC-2 = { 1, 3) and two corresponding part families PF-1 = { 1, 3}, PF-2 = (2, 4, 5 } are visible in matrix (2). Clustering of a binary incidence matrix may result in the following two categories of clusters: 1) mutually separable clusters 2) partially separable clusters. The two categories of clusters are presented in matrices (2) and (3).

level 1: factory flow analysis level 2: group analysis level 3: line analysis. In level 1, through the analysis of the present state of part flows, the machine-part incidence matrix is generated. Based on the outcome of level 1, an attempt to identify machine cells is made in level 2. The identification can be achieved by rearranging rows and column of the machine-part matrix until clusters are visible. The generated clusters are used in the third level to analyze the flow pattern on the shop floor, determine layout of machines, and identify bottleneck machines. There are two major weaknesses for the PFA. First, the method is not systematic and second it is difficult for computerization. McAuley [47] has remedied the first weakness introducing the Single Linkage Cluster Analysis (SLCA). The latter method belongs to the class of similarity coefficient methods. Similarity Coefficient Methods: The Single Linkage Cluster Analysis (SLCA) is based on the similarity coefficient sii measure between two machines i and j and is computed as follows:

Part Number 1 2 3 4 5

d'(aik, ajk)= Matrix (3) cannot be separated into two disjoint clusters because of part 5 which is to be machined in two cells MC-1 and MC-2. Removing part 5 from matrix (3) results in the decomposition of matrix (3) into two separable machine cells MC-1 = {1,2},MC-2 = {3,4)andtwopartfamiliesPF-l = {1,2} and PF-2 = (3, 4). The two clusters are called partially separable clusters. To deal with the bottleneck part 5, one of the following three actions can be taken: 1) it can be machined in one machine cell and transferred to the other machine cell by a material-handing carrier; 2) it can be machined in a functional facility; 3) it can be subcontracted. To solve the matrix formulation of the GT problem the

d2(aik,ajk) =

t

1, 0,

if u,=a,k= I otherwise

0, 1,

if aik=a,k=O otherwise.

To solve the GT problem using SLCA, similarity coefficients for all possible pairs of machines are computed. Machine cells are generated based on a threshold value of the similarity coefficient. To illustrate the SLCA consider matrix (3). The similarity coefficients so are computed below and depicted in Fig. 5 .

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IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4, NO. 5 , OCTOBER 1988

Similarity

[ % I

100

-1

'w Machine Number

75-

25n-

Fig. 5.

I I

A tree of similarity coefficients.

n

The only part which requires processing in both machine cells MC-1 and MC-2 is part 5 with corresponding d'(ul5, uz5) = 1 . Another interesting similarity coefficient method was studied by De Witte [ 1 2 ] . He designed a clustering algorithm advocating the concept of some machines to be included in more than one machine cell. He divided all the available machines into 1) primary machines, 2) secondary machines, and 3) tertiary machines. To analyze the relationship between these machines three different similarity coefficients were used: 1) absolute similarity coefficient saij 2 ) mutual similarity coefficient srnV 3) single similarity coefficient ss,.

0

~ 2= 3 - = 0.

5

Assuming the threshold value of the similarity coefficients, = 60 percent, from Fig. 5 the following machine cells are obtained: MC-1={1, 2) MC-2={3, 4). One of the disadvantages of the SLCA is that it fails to recognize the chaining problem resulted from the duplication of bottleneck machines [311. The use of the Average Linkage Clustering (ALC) algorithm to overcome the chaining problem was studied by Seifoddini and Wolfe [54]. They defined the similarity coefficient between any two clusters as an average of the similarity coefficient between all members of the two clusters. To solve the GT problem Seifoddini and Wolfe represented the machine-part incidence matrix using a binary machine code. The total number of intercellular movements (ICM) between two machine cells is computed as follows: n

d 3 (Ujk, Ujk)

ICM,= k=l

De Witte concluded that the best way of clustering is to start with coefficient saij and srnV and then use ssV to allocate the remaining unassigned machines. Sorting-Based Algorithms: The method of clustering based on sorting rows and columns of the machine-part incidence has been studied by many authors. King [30] developed the Rank Order Clustering (ROC) algorithm. His algorithm incorporates the following steps: STEP 1

For each row of the machine-part incidence matrix assign binary weight and calculate a decimal equivalent (weight). STEP 2 Sort rows of the binary matrix in decreasing order of the corresponding decimal weights. STEP 3 Repeat the above two steps for each column. STEP 4 Repeat the above steps until the position of each element in each row and column does not change. A weight for each row i and column j are calculated as follows:

where 1, Uik =

row i:

aik+O

if

ajk2n-k k=l

rEMC-I

0,

otherwise column j :

2

ak,2n-k.

k= I

Note that d 3 ( u i k , ujk) = 1 indicates that part k requires processing in both machine cells MC-i and MC-j, and thus part k is a bottleneck part. This problem can be solved either by removing part k from the two machine cells or adding an identical machine, say q, to each machine cell. The ALC algorithm is illustrated below. Consider the example illustrated in Fig. 5 where two clusters were obtained. The intercellular movement is computed as follows:

U l k = ( l , 1, 0, 0, 1) UZk=(O, 0, 1, 1, 1)

In the final matrix generated by the ROC algorithm clusters are identified visually. The ROC algorithm for matrix ( 1 ) is illustrated in Example 1. Example I : STEP 1 Assign binary weights to each row and calculate decimal equivalents

1

Part Number 2 3 4 5

Binary +24 2 3 2 2 2' 2O Weight

Decimal Equivalent

d 3 ( u l k , U Z k ) = ( O , 0, 0, 0, 1) q

2

20 10 0

2 Machine 3 Number 4

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KUSIAK AND CHOW: DECOMPOSITION OF MANUFACTURING SYSTEMS

STEP 2 Sorting the decimal weights in decreasing order results in the following matrix:

-

Machine Number 3 4

Part Number 1 2 3 4 5 1 1

1 1

1 1

1 1 1

2 4 Machine 1 Number 3

STEP 3 Repeating the above steps for each column produces the following matrix: Part Number 1 3 2 4 5

In the above matrix, two separable clusters are visiable. The matrix obtained is identical to matrix (2). The ROC algorithm was further extended in King and Nakornchai [3 11, and Chandrasekharan and Rajagopalan [101. In the latter paper, Chandrasekharan and Rajagopalan improved the ROC algorithm incorporating i) “block and slice” method and ii) hierarchical clustering method. Another interesting sorting-based algorithm, the Direct Cluster Algorithm (DCA), was developed by Chan and Milner [9]. The DCA consists of the following steps: STEP 1 Determine total number of “1’s” in each row and column in the machine-part incidence matrix. STEP 2 Sort each row in increasing order corresponding to the total number of “1’s.” STEP 3 Sort each column in decreasing order corresponding to the total number of “1’s.” STEP 4 Repeat the above steps until the position of each element in each row and column does not change. Based on the final outcome of above steps, some rows and columns in the machine-part matrix are to be rearranged until satisfactory clusters are obtained. Bond-Energy Algorithm: McCormick et al. [48] developed an interchange clustering algorithm called the BondEnergy Algorithm (BEA). The BEA seeks to form a blockdiagonal form by maximizing the measure of effectiveness which is defined as follows: m

[

Machine Number 3

Part Number 1

ME.0

-

ME= 112

Part Number 2 3 4

[: :] 1

n

2 aij[ai,j-l+ai,j+l+ai-l,i+ai+I,,].

i=l j = l

The ME is illustrated in Fig. 6. The BEA (McCormick et al., [48]) involves the following steps: STEP 1 Set i = 1. Select one of the columns arbitrarily.

2

3

1

1

1]

1

1

ME.4

Fig. 6. Value of the measure of effectiveness for two matrices.

STEP 2 Place each of the remaining n - i columns, one at 1 positions, and compute a time, in each of i each column’s contribution to the ME. Place the column that gives the largest incremental contribution to the ME in its best location. Increase i by 1 and repeat above step until i = n. STEP 3 When all the columns have been placed, repeat the procedure for the rows.

+

The BEA for matrix (1) is illustrated in Example 2. Example 2: STEP 1 Set i = 1. Select column 2. STEP 2 Placing each of the remaining columns in each of the i 1 positions, its contribution to the ME value is computed below

+

i = 1

Position

i

2 2 2 2

Column number

+

1

1

3 4 5

MEValue

0 0 2 1

Column 4 is placed in i + 1 position. Repeating the same procedure for the remaining columns lead to the following column order: (2, 4, 5, 1, 3). STEP 3 Repeating the above steps for rows results in the following matrix: Part Number 2 4 5 1 3 Machine

2 4 The value of ME for the above matrix is 9. Slagle et al. [56] developed a clustering algorithm based on the BEA and the Shortest Spanning Path (SPP) algorithm. Their concept was then extended in Bhat and Haupt [6]. They developed an algorithm where the matching between any two rows (columns) of the incidence matrix is calculated as follows:

m i j = C )aik-ajkI. k= I

The Bhat and Haupt [6] algorithm is similar to the McCormick et al. [48] algorithm. The difference between the two algorithms is that the Bhat and Haupt algorithm permutes rows

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IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4, NO, 5 , OCTOBER 1988

and columns of matrix A * A as opposed to permuting rows and calculating matchings in matrix A in the McCormick et al. algorithm. One can note that an entry mu of matrix M represents a distance between rows i and j of matrix A . Cost-Based Method: Askin and Subramanian [4] developed a clustering algorithm which considers the following manufacturing costs:

1) 2) 3) 4) 5)

fixed and variable machining cost setup cost production cycle inventory cost work-in-process inventory cost material handling cost.

The algorithm consists of three stages. In the first stage, parts are classified using a coding system. In stage two, an attempt to develop a feasible grouping between parts based on the manufacturing cost is performed. In stage three, the actual layout among a group of machine cells is analyzed. Cluster Identification Algorithm: Kusiak and Chow [4 11 applied the concept presented in Iri [29] to develop the Cluster Identification (CI) algorithm. The CI algorithm allows to check the existance of mutually separable clusters in a binary machine-part incidence matrix provided that they exist. The algorithm has relatively low computational time complexity of O(2mn). To cluster a machine-parts incidence matrix that does not have the block-diagonal structure embedded the cost analysis algorithm was developed [42]. In the cost analysis algorithm, with each column of the machine-part incidence matrix [ajj] cost cj of part j is associated. These costs are useful in handling the bottleneck parts. One can assign cost cj one of the following meaning: 1) subcontracting cost 2) part flow rate.

A part with high value of the flow rate between machines increases utilization of a material handling system. If the latter creates a problem, the decision to subcontract a part which heavily utilizes the material handling system can be considered. Application of the cost analysis algorithm [42] for solving a GT problem is illustrated in Example 3. Example 3: Determine the mutually separable clusters and parts to be subcontracted with the lowest possible subcontracting costs for the following incidence matrix and the corresponding part subcontracting costs. Part Number 1 2 3 4 5 6 7 8 9 10 1 1

1 2 3 Machine 4 Number 5 1 1 1 6 1/ 61 15 51 10 1 71 21 30 1 ]4 ] 7 [2.5 [ l 8: 70

Using the cost analysis algorithm the following machine cells are obtained: MC-1= { 1, 4, 7 )

MC-2={2, 3, 5 , 6 ) . The corresponding part families are as follows:

PF-1={2, 3, 6, 7 ) PF-1={5, 8, 10, 11). Parts 1 , 4, and 9 with the total cost of c1 + c4 + c9 = 2.5 + 6.0 + 2.0 = 10.5 have been subcontracted. Most of grouping algorithms presented do not provide optimal solutions and have high computational time complexity. Algorithms of McCormick et al. [48], Slagle et al. [56], Bhat and Haupt [6], King [30], and Kusiak [34] have computational time complexity O(m2n + n2m),where m is the number of rows and n is the number of columns in the binary machine-part incidence matrix. Computational time complexity of the cluster identification algorithm [41] is O(2mn) and the cost analysis algorithm [42] is O(2mn + n log n). An extensive computational analysis of Bhat and Haupt [6] and Kusiak [34] is included in the last reference.

B. Mathematical Programming Formulation Most mathematical programming models developed in GT consider a distance measure djj between part i and j . The distance measure du is a real-valued symmetric function obeying the following axioms [20]: reflexivity djj = 0 symmetry djj = djj triangle inequality djq = dip + dpq. The most commonly used distance measures are the following. 1) Minkowski distance measure [3]

Lk=l

J

where r is a positive integer and n is the number of parts. Two special cases of the above measure are widely used: absolute metric measure (for r = 1) Euclidean metric measure (for r = 2).

2) Weighted Minkowski distance measure [3]

1

1

There are two special cases: weighted absolute metric measure (for r = 1) weighted Euclidean metric measure (for r = 2).

3) Hamming distance measure [46] n

d;j=

Subcontracting Cost

C Z(aik, ajk) k= 1

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KUSIAK AND CHOW: DECOMPOSITION OF MANUFACTURING SYSTEMS

matrix of Hamming distances dijis obtained:

where

z(aik, ajk)=

Part Number 1 2 3 4 5

if aik#ajk otherwise.

1, 0,

Distance measures are also known as dissimilarity measures. Some similarity measures were discussed in the section on similarity coefficient methods. In this section, the following mathematical programming models are discussed: p-median model generalized p-median model quadratic programming model fractional programming model.

n p

and

x;= 1

subject to (5) n

(6)

j=I

xij=O, 1,

for a l l j = l , for all

i=l, j=l,

a . . ,

e . . ,

n ,n n.

x&=1.

For the two part families, the corresponding two machine cells are determined from matrix (1):

(4)

JJ,

x;= 1

PF-2={2, 4, 5).

MC-l= (2, 4)

The objective function of the p-median model is to maximize the total sum of distances between any two parts i and j .

xjj=P

x&=1

PF-l= { 1, 3)

dij distance measure between parts i and j .

,J-

5

Based on the definition of xij, two part families are formed:

otherwise

x.. 1 where [XI is the minimum integer value not smaller than x. STEP 2 Set rn = rn’. Choose rn initial nodes, one for each subgraph. STEP 3 Determine the common node of each subgraph. STEP 4 Remove the common node from the graph. Add the uncommon node to the corresponding subgraph. STEP 5 Repeat Steps 3 and 4 until every node is assigned. The algorithm is illustrated in Example 6 . Example 6: Determine the bottleneck nodes for the following graph.

Fig. 13. The boundary graph corresponding to matrix (3).

@ m

Fig. 14. Two disjoint boundary subgraph obtained from the graph in Fig. 13.

One can note that by removing part p 5 two transition subgraphs are obtained as shown in Fig. 11. As the final outcome, the following two part families and their corresponding machine cells are obtained: PF-l={ 1, 2) PF-2={3, 4) M C - l = ( l , 2) MC-2= (3, 4).

Boundary Graph: A boundary graph consists of a hierarchy of bipartite graphs. At each level of the boundary graph, nodes of the bipartite graph represent either machines or parts. The boundary graph corresponding to matrix (1) in a form of two disjoint boundary graphs is shown in Fig. 12. Each boundary graph in Fig. 12 represents a cluster. To apply the boundary graph for determining bottleneck parts consider matrix (3). The boundary graph for matrix (3) is presented in Fig. 13. Removing parts p 5 from the boundary graph in Fig. 13 results in two boundary subgraphs shown in Fig. 14.

STEP 1 Assume that K = 3 rn‘ = [(5- 1)/3] = 2. STEP 2 Nodes 2 and 5 have been selected as starting nodes. STEP 3 Denote C { ( A ) } nodes in graph A C((A))^ “ C ( ( B ) } common nodes of graphs A to B . Then C((1)) = (2, 3) C{(2)) = ( 3 9 5 1 and C{(l))^C{(2)) = 3, i.e., the common node of C((1)) and C((2)). STEP 4 Node 3 is removed from C((1)) and C ( ( 2 ) ) . Adding nodes 2 and 5 to the corresponding subgraph gives C((1)) = (1, 2) and C((2)) = (4, 5). STEP 5 Since there are not any common nodes of C((1)) and C{(2)), and all nodes have been assigned, STOP. Let each node in the above example represent a part, then the solution for the above problem reads as follows: P F - l = ( I , 2) PF-2 = (4, 5 ) . The algorithm of Lee et al. [45] was further extended in Vannelli and Kumar [ S I .

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KUSIAK AND CHOW: DECOMPOSITION OF MANUFACTURING SYSTEMS

TABLE I CHARACTERISTICS OF GT PROBLEM FORMULATIONS AND SAMPLE GROUPING ALGORITHMS

Constraints GT Problem Formulation

Limited Number of Part Families

Limited Number of Parts in a Part Family

cost Consideration

McAuley [47], McCormick et al. [48] Slagle et al. [56], Bhat and Haupt [61, King [30], Chan and Milner [9], Kusiak [34]

no

no

no

Yes

Askin and Subramanian [4]

yes

Yes

Kusiak and Chow [41], [42]

no

no

Kusiak [34]

Matrix

Yes

no

Kusiak et al. [43]

no

Yes

Kusiak [38]

no

no

Laskhari et al. [44]

no

no

Lee et al. [45], Vanneli and Kumar [58]

Mathematical Programing

Graph

Sample Grouping Algorithms

no

For each of the three different formulations of the GT problem presented in this paper, the following three characteristics have been considered in Table I: limited number of part families (or machine cells), limited number of parts in each part family (or machines in each machine cell), costs (e.g., production, subcontracting). Based on these characteristics sample grouping algorithms have been assigned. 111. FUTURE DEVELOPMENTS IN GROUP TECHNOLOGY

The GT problem has the following features:

1) it might be formulated in a number of different ways; 2) computational complexity of these formulations varies; 3) it involves quantitative and qualitative data. The above features make the GT problem suitable for application of an expert system approach. The expert system should not only handle the qualitative data but also take advantages of available models and algorithms. A suitable rule-based system architecture was suggested in Kusiak 1361, [37] and is shown in Fig. 15. The tandem architecture includes an expert system and a set of models and algorithms. The expert system analyzes a GT problem and depending upon its type and size an appropriate model and algorithm is selected. The expert system interacts closely with the model and algorithm. For example, it may change parameters of the model, evaluate partial solution generated, or change search directions of the algorithm. The

Fig. 15. A tandem system architecture.

tandem architecture illustrated in Fig. 15 allows to consider realistic formulations of the GT problem. Kusiak [39] applied the tandem system architecture for solving the GT problem. The proposed expert system permits one to solve a large-scale GT problem in an acceptable computational time. The formulations discussed were actually simplified models of the industrial GT problem. As an example consider the following formulation of the industrial GT problem: Determine machine cells; for each machine cell, select a part family consisting of parts with minimum sum of subcontracting costs and select a suitable material-handling equipment with minimum corresponding cost subject to the following constraints: Processing time available at each machine is not exceeded. Constraint C2: Upper limit on the frequency of trips of material-handling carriers for each machine cell is not exceeded. Constraint C1:

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IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4, NO. 5, OCTOBER 1988

To present an impression of the expert system four sample decision rules which based on above constraints are shown.

Rule 1 IF THEN

constraint C1 is violated for machine i attempt to satisfy this constraint by:

include machines i , p in machine MC-k consider alternative process plan for parts violating constraint C4 constraint C4 is still violated remove from MC-k parts which require machines i, . * .,p .

THEN ELSE

The number of machines in each cell does not exceed its upper limit of alternatively the dimension (e.g., the length) of each machine cell is not exceeded. Constraint C4: Some machines have to be included in the same cell because of technological requirements.

Constraint C3:

e ,

AND IF THEN

The above formulation of the GT problem and the solution obtained using the tandem expert system are illustrated in Example 7. Example 7: Solve the GT problem for the following data: a) Matrix of processing times (tij is a processing time part j on machine i )

______

1

2

3

4

4

Part Number 6 7 8

5

21 20

10

22 2 6

10

AND IF THEN ELSE

i) introducing multiple machine i OR ii) considering alternative process plans for parts produced on machine i none of the two can be implemented solve a corresponding knapsack problem for parts produced on machine i the code of any of the removed parts begins with S place it on the list of parts to be subcontracted place it on the list of parts to be manufactured in the functional manufacturing facility.

Rule 2

AND IF THEN

constraint C2 is not violated for a robot and an AGV the robot utilization rate U 4 0.40 and the AGV utilization rate U 1 0.65 select the AGV.

Rule 3 IF THEN AND IF THEN

constraint C3 is violated for machine cell MC-k attempt to replace basic process plans with alternative process plans constraint C3 still Giolated remove from this machine cell parts violating this constraint.

18 7

7

Machine Number

6 7

b) A vector of production costs for p a r t j [ ~ j ] = [ 6 0 ,55, 15, 13, 10, 20, 8, 18, 40, 5, 31, 301. c) A vector of maximum processing time available on each machine

[7'i]=[40,

40, 40, 50, 50, 60, 2OlT.

d) The total number of machines in each cell must not exceed N = 3. e) Machines 2 and 5 have to be included in the same cell due to the technological requirement. f ) The vector of frequency of trips of an automated guided vehicle (AGV) for handling part j [T;]=[ll,

30, 2.5, -, 6, 10, -, 6, 7, 15, 18, 141.

g) The vector of frequency of trips of a robot for handling part j [ T f ] = [ l l , 30, 5, 3, 6, 15, 10, 12, 7, -, 36, 281. h) The maximum frequency of trips which can be handled by an AGV and a robot F A= 40

F R= 100.

Applying the tandem expert system resulted in the following machine cells: MC-1= (2, 5)

Rule 4 IF AND IF

5

25

Note that the symbol S has been used to denote parts which the management is willing to subcontract. IF

8

6

3 1

AND IF THEN

1 2

10

35

16

12

8

5 [tijl =

9 1 0 1 1

constraint C4 is violated for machine i, * . .,p including machines i, * * p in machine cell MC-k does not violate constraint C3

MC-2={3, 7)

e ,

MC-3 = { 1, 4, 6)

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KUSIAK AND CHOW: DECOMPOSITION OF MANUFACTURING SYSTEMS

n

I

I

I I

I I I

1

I

I

FUNCTIONAL MANUFACTURING FACILITY

I I I I

AUTOMATED STORACE/RETRIEVAL SYSTEM

I

MACHINE CELL MC-3

I

Fig. 16. A structure of the automated manufacturing system in Example 7

and the corresponding part families:

PF-1= { 1, 6, 9) PF-2= ( 5 , 10, 12) PF-3={2, 4, 7, 8, l l } . Part p 3 is to be subcontracted. A cellular structure of the automated manufacturing system is presented in Fig. 16. One can note in Fig. 16 that to transfer material two AGV’s and a robot have been selected. 1V. CONCLUSION In this paper, the following two methods of solving the Group Technology (GT) problem have been discussed: classification cluster analysis. The classification method is used to group parts into part families based on their design features. Two variations of the classification method; namely, visual and coding method, were outlined. The visual method is applicable in case when the number of parts is rather limited. Therefore, the coding methods which are more commonly used are discussed. The cluster analysis approach was used to group machines into machine cells and parts into part families. To model the GT problem, three clustering formulations were presented: the matrix formulation, mathematical programming formulation, and graph formulation. Since the GT problem is NP complete, heuristic algorithms are most likely to be used for solving large-scale industrial problems. The results of grouping leads to the physical machine layout or logical machine layout. The latter is used in case when the production content changes rather frequently. To model the GT problem, the generalized matrix formulation seems to be the most appropriate. The authors are in the process of developing an expert system for solving this

formulation. The expert system is linked with a heuristic algorithm. For introduction to this concept see Kusiak [39] and Kumar et al. [33]. There have been some doubts whether the cellular concept is applicable to automated manufacturing systems. Some studies have shown that grouping machines into machine cells may limit the system flexibility. However, industrial applications have proven that it is virtually impossible to implement a large-scale automated manufacturing system without using the cellular concept. There are at least four crucial factors which support this thesis: 1) Volume of information: Volume of information in a large-scale automated manufacturing system is typically large and it is too expensive to effectively process information without the system’s decomposition. 2) Material handling system: In a typical automated manufacturing system, automated material handling carriers are used, for example, Automated Guided Vehicles (AGV’s) or robots. Each of the two carriers (an AGV or robot) can tend only a limited number of machines. This limitation for fixed handling robots is imposed by: maximum number of trips an AGV can make per time unit AGV guidance path, for example, a wire, which should not be intersected by another guidance path within the AGV’s working area. An AGV while approaching an intersection slows down and this, in turn, reduces its utilization. 3 ) Technological requirements: Some machines have to be grouped together due to technological requirements, for example, a forging machine and a heat treatment station. 4) Management: Although in most of the currently operating automated manufacturing systems the degree of automation is higher than in the classical manufacturing systems, humans will be, for a long time, a part of these new manufacturing systems. Due to limited size of each machine cell, cellular manufacturing system is easier to manage than a

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IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4, NO. 5 , OCTOBER 1988

functionally organized system. It seems that this view has been given especially serious consideration in Europe and Japan.

REFERENCES Anonymous, “How to predict the benefit of group technology?,” Prod. Engineer, pp. 51-54, Feb. 1980. Anonymous, “Product-based group technology pay off,” Mach. Prod. Eng., vol. 3, pp. 37-39, 1980. T. S . Arthanari and Y. Dodge, Mathematical Programming in Statistics. New York, NY: Wiley, 1981. R. Askin and S . Subramanian, “A cost-based heuristic for group technology configuration.” Int. J . Prod. Res., vol. 25, no. 1, pp. 101-114, 1987. A. Ballakur, “Streamlining material flows for just-in-time applications,” in Proc. 1986Fall IIE Conf., (Boston, MA), pp. 51-56, 1986. M. V. Bhat and A. Haupt, “An efficient clustering algorithm,” IEEE Trans. Syst., Man, Cybern., vol. SMC-6, no. 1, pp. 61-64, 1976. J. L. Burbidge, “The simplification of material flow systems,” Int. J. Prod. Res., vol. 20, no. 3., pp. 339-347, 1982. -, “Production flow analysis,” Prod. Engineer, pp. 139-152, AprJMay 1971. H. M. Chan and D. A. Milner, “Direct clustering algorithm for group formation in cellular manufacturing,” J. Manuf. Syst., vol. 1, no. 1, pp. 65-74, 1982. M. P. Chandrasekharan and R. Rajagopalan, “MODROC: An extension of rank order clustering for group technology,” Int. J . Prod. Res., vol. 24, no. 5, pp. 1221-1233, 1986. Y. Cheng and K. S . Fu, “Conceptual clustering in knowledge organization,” IEEE Trans. Putt. Anal. Mach. Intell., vol. PAMI-7, no. 5, pp. 592-598, 1985. J . De Witte, “The use of similarity coefficients in production flow analysis,” Int. J. Prod. Res., vol. 18, no. 4, pp. 503-514, 1980. G. C. Dunlap and C. R. Hirlinger, “Well planned coding, classification system offers company-wide synergistic benefits,” Indust. Eng., vol. 15, no. 1, pp. 78-83. 1983. F. R. E. Durie, “A survey of group technology and its potential for user application in the UK,” Prod. Engineer, pp. 51-61, Feb. 1970. R. L. Eckert, “Codes and classification systems,” Amer. Much., pp. 88-92, Dec. 1975. G. A. B. Edwards and F. Koenigsberger, “Group technology, the cell system and machine tools,’’ Prod. Engineer, pp. 249-256, July/Aug. 1973. G. A. B. Edwards, “The family grouping philosophy,” Int. J. Prod. Res., vol. 9, no. 3, pp. 337-353, 1971. B. Everitt, Cluster Analysis. New York, NY: Halsted, 1980. G. M. Fazakerlay, “Group technology: Social benefits and social problems,” Prod. Engineer, pp. 383-386, Oct. 1974. K. S. Fu, “Recent developments in pattern recognition,” IEEE Trans. Comput., vol. C-29, no. 10, pp. 845-854, 1980. C. C. Gallagher and W. A. Knight, Group Technology. London, UK: Butterworths, 1973. F. Glover and E. Woolsey, “Converting the 0-1 polynomial problem to a 0-1 linear programming,” Oper. Res., vol. 22, pp. 180-182, 1974. T. J . Greene and R. P. Sadowski, “Cellular manufacturing control,” J. Manuf. Syst., vol. 2, no. 2, pp. 137-145, 1983. R. D. Holtz, “GT and CAPP cut work-in-process time 80%, part 1,” Assembly Eng., vol. 21, no. 6, pp. 24-27, 1978. -, “GT and CAPP cut work-in-process time 80%: Part 2,” Assembly Eng., vol. 21, no. 7, pp. 16-19, 1978. A. Houtzeel and C. S. Brown, “A management overview of group technology,” in N. L. Hyer and R. E. King, Eds., Group Technology at Work, Society of Manufacturing Engineers, Michigan, pp. 3-16, 1984. N. L. Hyer and R. E. King, Eds., Group Technology at Work, Society of Manufacturing Engineers, Michigan, 1978. F. B. Ingram. “Group technology,” Prod. Inventory Manage. J., Forth Quarter, 1982. M. Iri, “On the synthesis of the loop cut set matrices and the related problem,” in K. Kondo, Ed., RAAG Memoirs, vol. 4, pp. 376-410, 1968. J . R. King, ‘ ‘Machine-component group formulation in production flow analysis: An approach using a rank order clustering algorithm,” Int. J . Prod. Res., vol. 18, no. 2, pp. 213-232, 1980. J. R. King and V. Nakornchai, “Machine-component group formation

in group technology: Review and extension,” J . Prod. Res., vol. 20, no. 2, pp. 117-133, 1982. T. D. Klastorin “The p-median problem for cluster analysis: A comparison test using the mixture model approach,” Manag. Sci., vol. 31, no. 1. pp. 1134-1146, 1982. S . R. T. Kumar, S . Joshi, R. L. Kashyap, C. L. Moodie, and T. C. Chang, “Expert system in industrial engineering,” Int. J . Prod. Res., vol. 24, pp. 1107-1125, 1986. A. Kusiak, “The part families problem in flexible manufacturing systems,” Annals Oper. Res., vol. 3, pp. 279-300, 1985. -, “The generalized p-median problem,” Working Paper 20187, Dep. Mech. Indust. Eng., Univ. of Manitoba, Winnipeg, Canada, 1987. -, “Artificial intelligence and operations research in flexible manufacturing system,” Information Syst. Operat. Res. (INFOR), vol. 25, no. 1, pp. 2-12, 1987. -, “Design of expert systems for scheduling automated manufacturing systems,” Indust. Eng., vol. 19, no. 7, pp. 42-46, 1987. -, “The generalized group technology concept,” Int. J. Prod. Res., vol. 25, no. 4, pp. 561-569, 1987. -, “An expert system for group technology,” Indust. Eng., vol. 19, no. 10, pp. 56-61, 1987. K. R. Kumar, A. Kusiak, and A. Vannelli, “Grouping parts and components in flexible manufacturing systems,” Eur. 1.Operat. Res., vol. 24, no. 3, pp. 387-397, 1986. A. Kusiak and W. S. Chow, “An algorithm for cluster identification,” IEEE Trans. Syst., Man, Cybern., vol. SMC-17, no. 4, pp. 696-699, 1987. -, “Efficient solving of the group technology problem,” J. Manuf. Syst., vol. 6, no. 2, pp. 117-124, 1987. A. Kusiak, A. Vannelli, and K. R. Kumar, “Clustering analysis: Models and algorithms,” Contr. Cybern., vol. 15, no. 2, pp. 139154, 1986. R. S. Lashkari, S. P. Dutta, and G . Nadoli, “Part family formation in flexible manufacturing systems-An integer programming approach,” in A. Kusiak, Ed., Modern Production Management Systems. Amsterdam, The Netherlands: North Holland, 1987, pp. 627-635. J. L. Lee, W. G. Vogt, and M. H. Mickle, “Calculation of shortest paths by optimal decomposition,” IEEE Trans. Syst., Man. Cybern., vol. SMC-12, pp. 410415, 1982. R. C. T. Lee, “Clustering analysis and its applications,” in J . T. Tou, Ed., Advances in Information Systems Science. New York, NY: Plenum, 1981. J. McAuley, “Machine grouping for efficient production,” Prod. Engineer, pp. 53-57, Feb. 1972. W. T. McCormick, P. J . Schweitzer, and T. W . White,” Problem decomposition and data reorganization by cluster technique, ” Oper. Res., vol. 20, no. 5, pp. 993-1009, 1972. C. Mosier and L. Taube, “The facets of group technology and their implementation-A state of art survey ,” OMEGA, vol. 13, no. 5, pp. 381-391, 1985. J. M. Mulvey and H. P. Crowder, “Clustering analysis: An application of Lagangian relaxation,” Manag. Sci., vol. 25, no. 4, pp. 329-340, 1979. Y. Nagai, S. Tenda, and T . Shingu, “Determination of similar task types by the use of the multidimensional classification method: Towards improving quality of work life and job satisfaction,” I n f . J . Prod. Res., vol. 18, no. 3, pp. 307-332, 1980. H. Opitz and H. P. Wiendahl, “Group technology and manufacturing system for small and medium quantity production,” Int. J . Prod. Res., vol. 9, no. 1, pp. 181-203, 1971. G. Schaffer, “Implementing CIM,” Amer. Machinist, pp. 151-174, Aug. 1981. H. Seifoddini and P. M. Wolfe, “Application of the similarity coefficient method in group technology,” IIE Trans., vol. 18, no. 3, pp. 271-277, 1986. D. D. Siljak and M. F. Sezer, “Nested decomposition into weakly coupled components,” in IFAC 9th World Congr., (Budapest, Hungary, July 2-6, 1984). J . L. Slagle, C. L. Chang, and S . R. Heller, “A clustering and data recognition algorithm,” IEEE Trans. Syst., Man, Cybern., vol. SMC-5, pp. 125-128, 1975. J. T. Tou and R. C. Gonzalez. Pattern Recognition Principles. Reading, MA: Addison-Wesley, 1974. A. Vannelli and K. R. Kumar, “A method for finding minimal bottleneck cells for grouping part-machine families,” Inr. J . Prod. Res., vol. 24, no. 2, pp. 387-400, 1986.

KUSIAK AND CHOW: DECOMPOSITION OF MANUFACTURING SYSTEMS

[59] W. 1. Vos, “Group technology, what can it do for you?,” in Vth fnt. Con$ on Production Reseurch (Amsterdam, The Netherlands, Aug. 12-16, 1979). [60] P. H. Waghodekar and S. Sahu, “Group technology: A research bibliography,” OPSEARCH, vol. 20, no. 4, pp. 225-249, 1983.

Andrew Kusisk for a photograph and biography please see page 402 of the August 1988 issue of this JOURNAL.

47 1 Wing S. Chow received the B.Sc. degree in applied mathematics from the University of Winnipeg, Winnipeg, Man., Canada, the B A . degree in statistics and the M.Sc. degree in operations research from the University of Manitoba, Winnipeg, and is a Ph.D. student in the Department of Mechanical and Industrial Engineering at that university. His interests are in group technology and scheduling flexible manufacturing system.