J. Service Science & Management, 2010, 3, 494-500
doi:10.4236/jssm.2010.34056 Published Online December 2010 (http://www.SciRP.org/journal/jssm)
Manufacturing Cells Formation Based on Graph
Coloring
José Francisco Ferreira Ribeiro
School of Business, University of São Paulo, Ribeirão Preto, Brazil.
Received September 2nd, 2010; revised October 16th, 2010; accepted November 18th, 2010.
ABSTRACT
A method for cellular manufacturing design in Group Technology is presented in this paper. The proposed method
computes the dissimilarities between pa rts and or ganizes the produ ction system in pa rt-families a nd group-ma chin es. A
graph corresponding to the production system is generated and a coloring algorithm is activated in order to obtain a
number of cells equal to the desired number of cells. The corresp onding program was written in Matlab language and
runs on a microcomputer. The results obtained on several examples found in the literature are consistently equivalent
to or even better than those hitherto proposed, in terms of inter-cell moves and dimensions of the cells.
Keywords: Manufacturing Cells, Group Technology, Optimization, Heuristics, Graph Theory
1. Introduction
The design of manufacturing cells consists in partitioning
the set of parts to be manufactured in an industry into
families and the available machines into groups or cells,
so that each family is associated with one machine group,
and vice-versa. Each family-group pair constitutes a ma-
nufacturing cell [1].
This concept lies on grouping similar parts in families,
proposing to produce them in cells that have specially
selected machines to accomplish this. This procedure le-
ads to greater automation, set up time reduction, stan-
dardization of the tools used and a reduction of manu-
facturing cycles [2].
In a production system organized in manufacturing
cells, management, in turn, becomes simpler and more
efficient, a direct outcome of the dissolution of the global
production system in smaller subsystems. There is a
time-reduction of work post transferences, set up time of
machines, quantity of tools, order sizes and total manu-
facturing time [3,4].
However, the manufacturing cells project requires to
know the optimal solution of a very complex mathe-
matical problem [5]: Given the matrix incidence in the
form [parts × machines] or [parts × machine types],
where the types of machines would be the lathes, drills,
etc., the machines would be a specific lathe or a drill in
the available machine sector, the lines and columns of
this matrix must be rearranged, in order to provide it with
a block-diagonal structure.
The elements concentrated inside the diagonal blocks
constitute the manufacturing cells and the ones that are
outside the diagonal blocks are called inter-cells moves
and, in practice, considered undesirable. That is the rea-
son why, regarding the manufacturing cells, there is an
attempt to minimize the number of inter cell moves, at
the same time that a balance of workloads between the
different cells projected is sought.
A large number of techniques have been used in recent
years [6] to hit the block-diagonalization of the matrix,
designing manufacturing cells and implementing Group
Technology in the industries. Among these, we can cite
mathematical programming: [5,7,8-14]; branch and bo-
und: [15,16]; fuzzy logic: [17,18]; genetic algorithms:
[19-23]; neural networks: [24-26]; metaheuristcs like
tabu search and simulated annealing: [27-29]; data ana-
lysis: [2,30-33].
2. Graph of Manufacturing
A coloring of a graph G(V, A), with V a set of nodes and
A a set of edges, is an assignment of any color belonging
to set of colors C = {ci} for each node of V, where two
nodes connected by an edge of A cannot have the same
color; i.e., a coloring of G is a function f: V → C | if (i,
j)A  f(i) ≠ f(j). A k-coloring of G(V, A) is a color-
ing with k colors, i.e., a partition of V in k independent
sets of nodes. In this case, G is a k-coloring graph. The

Manufacturing Cells Formation Based on Graph Coloring495
chromatic number λ(G) of G is the smaller number of
colors for which a k-coloring of G exists.
The graph coloring problem is NP-complete [34,35]. It
is a high combinatorial problem: for example, a complete
and exhaustive enumeration of all colorings consists in
an O(|A|.k|N|) algorithm.
But, some immediate results can be obtained of the
definition, such as: 1) A graph G is bi-chromatic if and
only if is bipartite; 2) A complete sub-graph of G with t
vertices Kt is t-chromatic; 3) A graph G with two or more
edges is at least 2-chromatic.
The concepts of graph coloring, clique (a complete
sub-graph of G) and independent set of nodes are close: 1)
k colors are necessary to coloring k nodes of a clique
with cardinality k, thus λ(G) is greater than or equal to
the cardinality of the greatest clique of G; 2) Let us con-
sider S1, .., Sk disjunctive subsets of S generated by the
k-coloring; then =  Si = S, i = 1, .., k, and each Si is an
independent set in which any couple of edges is not con-
nected.
To create the G(V, A) graph associated to the produc-
tion system, the following is done: 1) V = {collection of
parts to manufacture}; 2) A distance dij is calculated be-
tween any two pairs of parts to manufacture. There will
be an edge to aij connecting two vertices i and j if the
distance dij is greater or equal to the value of a critical
distance fixed a priori, which can be altered at any mo-
ment intending to increase or diminish the number of
cells of the graph and therefore, the number of projected
cells.
Given that 2 adjacent nodes cannot receive the same
color, the G coloring will not place in the same manu-
facturing cell parts whose distances are greater or equal
to an established value of critical distance. The parts re-
ceiving the same color will be grouped in the same fam-
ily of parts and will be manufactured by the same ma-
chine cell, hence distributed to the product families, ac-
cording to the number of operations executed in each
family of parts.
3. The Problem
The data of the problem are included in the matrix com-
prised of elements 0/1 [parts to produce × available ma-
chines], representative of the company’s production
process. From this matrix, the matrix of dissimilarities
[parts × parts] is obtained.
Afterwards, the manufacturing graph is developed and
a colored algorithm is applied to obtain the parts families
in a number identical to the allocated number of cells.
Once the families are established, the machine groups
or cells are obtained, attributing each of the machines to
the family that most needs its work.
3.1. Algorithm
The algorithm corresponding to the proposed method is
made up of four basic steps:
a) Determining the matrix for workloads;
b) Computing dissimilarities among parts;
c) While the number of cells is not obtained:
• Perform coloring of graph G(V, A);
• Obtain the families of parts;
• End While
d) Obtaining machine groups.
3.2. Load Matrix
From the initial data a matrix [parts × machines] is ob-
tained, which provides the total time spent by each part
transiting in the program of each machine. This matrix is
called work load matrix and its coefficients are calcu-
lated as follows:







k programi,j=j
loadi,j = unitidurationi,k
where:
1) unit[i] = unit number of parts[i] to manufacture;
2) duration[i, k] = duration of operation k on a part[i];
3) program[i, j] = type of machine j used to perform
operation k on a part[i].
3.3. Dissimilarities between Parts
The method of computing dissimilarities between parts
[36] considers the differences seen in the manufacturing
program of the parts. Given the parts Pi and Pj defined
by the vectors 0/1 (below), the dissimilarities are calcu-
lated.
1) Pi = [ai1, ai2, ... , aik, ... , aim]
2) Pj = [aj1, aj2, ... , ajk, ... , ajm]
where:
1) m = number of available machines
2)
0ifpart i uses machine k
aik= 1otherwise



The dissimilarity between parts Pi and Pj is:
 
m
k=1
dpi,pj = uaik,ajk


where:

aif aikajk
uaik,ajk = bifaik = ajk = 0
cifaik = ajk = 1





In the computational tests, a = 1, b = 0, c = -1.
3.4. Machine Groups
Once the graph coloring is performed and the family

Manufacturing Cells Formation Based on Graph Coloring 496
parts are obtained, the attribution of the machines to the
projected families is accomplished.
Such attribution is done according the number of op-
erations accomplished by the machines in each of the
families. One machine will always be attributed to the
family that executes the highest number of operations.
The pair [part family × machine group] comprises the
manufacturing cell.
4. Coloring Procedure
Provided that the problem of partitioning an X collection
of n objects with a dissimilarity of classes, in a fixed k
number, so that the diameter d(P) among the classes is
minimum. This corresponds to constructing a threshold
Gs graph, a partial sub-graph of G(V, A), that should be
k-colorable.
To color the graphs of the vertices, a technique based
on the algorithm by [37] was used. This algorithm enu-
merates all of the partitions in a fixed number of mini-
mum diameter classes. It is based on a threshold graph in
p colors, with each color defining a class. Many heuris-
tics to approximate the diameter and the partitions of
minimum diameter are enumerated only at the last stage.
Let d(P) the diameter of the partition of V into p
classes. A superior limit, s, for d(P) is heuristically de-
termined. Then, the k-colorations for Gs are enumerated.
As long as there is at least one possible k-coloration, the
value s is decreased and a new iteration is processed.
When the algorithm stops, the highest value obtained
is equal to the remaining partition diameters. To enumer-
ate all of the possible partitions of X, in minimum di-
ameter K classes, this algorithm is applied, which is
comprised of 3 stages and has an O(k|V|-k) complexity.
In the first stage a heuristic method is used to deter-
mine the maximum s, highest approximation to d(P),
such that Gs will be k-colorable. The maximum s is de-
termined by a method of dichotomic subdivisions of the
variation interval of the dissimilarities in a sequential
coloring method, in this case the saturation algorithm or
“Dsatur”.
In this algorithm, at each iteration, the rate of satura-
tion DSi(x) is defined as the number of colors already
employed by the neighbors of x. The procedure consists
of: coloring the most sizable vertex with the color 1; b) in
the following stages, take the vertex free of maximum
DS and color it with the least possible index color.
In the second stage, once the maximum s has been set,
all of the Gs colorings are enumerated in k colors. With
the aid of “Dsatur”, maximum if possible, a clique of Gs
is obtained, where each node represents a class.
Then, according the order of saturation, the other ver-
tices are colored in all possible manners. If a vertex is
near the colored vertices, these will not be eligible to be
used to color them. Thus, all of the partitions in k classes
of less than s diameter are obtained.
In the third stage, a decreasing order of the edges of
the dissimilarity values, starting from s, are considered.
An edge can be inserted in the graph as long as there is a
compatible partition, that is, if the edge connects differ-
ent types of nodes.
Thus, each inserted edge eliminates some previously
obtained partitions. The first edge that cannot be inserted,
since there would be no more compatible partitions left,
has equal value to the highest interclass dissimilarity
value. The remaining partitions for a fixed number of
classes are of minimum diameter.
5. Illustrative Example
Let the matrix PM[parts × machines], that informs the
use (PMij = 1) or not (PMij = 0) of part i by the machine j
(Table 1), and the matrix PP[parts × parts], that gives the
dissimilarity index between parts to be manufactured
(Table 2). Two manufacturing cells will be designed for
this workshop.
Figure 1 shows the 3-coloring graph solution for the
example. It is not acceptable, because the workshop must
be partitioned in 2 cells. The solution for the problem is
the 2-coloring graph shown in Figure 2.
After defining the families of parts, machines are as-
signed to the most demanding families. The distribution
of machines is carried out considering the number of
operations carried out in each family: a machine is as-
signed to the family in which it will be most used.
The matrix MC[parts × machines] in Table 3 presents
the workshop partitioned in 2 manufacturing cells and
this solution is very good: there are no inter-cell moves
and the dimensions of cells are equal to [2 × 2] for cell 1
and [3 × 2] for cell 2.
Table 1. Matrix PM [parts × machines].
M1 M2 M3 M4
P1 1 1
P2 1 1
P3 1 1
P4 1 1
P5 1
Table 2. Matrix PP [parts × parts].
P1 P2 P3 P4 P5
P1 —
P2 4 —
P3 0 4 —
P4 4 0 4 —
P5 3 1 3 1 —

Manufacturing Cells Formation Based on Graph Coloring497
Figure 1. Three-coloring graph.
Figure 2. Two-coloring graph.
Table 3. Matrix MC [parts × machines].
M2 M4 M1 M3
P1 1 1
P3 1 1
P2 1 1
P4 1 1
P5 1
6. Computational Results
The corresponding program to the method presented was
written in Matlab language and is incorporated in a mi-
crocomputer: program MCF (Manufacturing Cells For-
mation).
Up to now, the results obtained about literature exam-
ples (E) are equivalent or better than those proposed,
with regards to the number of inter-cells movements (I)
and the dimensions of the cells (D).
Table 4 summarizes 15 tests conducted on classical
examples extracted from literature. The results obtained
(S) induced a diminishing of the number of inter-cell
moves for the first 3 examples of Table 4, at the same
time that the cell dimensions continued the same or prac-
tically the same up to what is currently known.
The computational time (T) required to solve all the
problems dealt always under 10 seconds. The time mea-
surements were done in a Pentium 2.20 GHz, 1.99 GB.
In addition to testing the 15 examples in order to
compare the results, we generated 1000 random samples
with 10 to 100 machines and 10 to 1000 parts, for per-
formance tests. Figures 3 and Figure 4 show the per-
formance of the method in terms of inter-cell movements
(maximum = 243) and computational time (maximum =
12 minutes).
Table 4. Tests.
E S I D
1 [32] [32]
MCF
4
2
4 × 6, 5 × 6
4 × 6, 5 × 6
2 [31] [31]
MCF
24
16
5 × 4, 10 × 6, 15 × 6
5 × 3, 10 × 6, 15 × 7
3 [31] [31]
MCF
15
14
6 × 5, 4 × 4, 4 × 5, 3 × 3, 3 × 3
5 × 4, 4 × 3, 4 × 5, 3 × 4, 4 × 4
4 [31] [31]
MCF
1
1
7 × 7, 6 × 5, 4 × 5, 3 × 3
7 × 7, 6 × 5, 4 × 5, 3 × 3
5 [38] [38]
MCF
0
0
2 × 5, 3 × 3, 2 × 6, 3 × 6
2 × 5, 3 × 3, 2 × 6, 3 × 6
6 [1] [32]
MCF
0
0
16 × 8, 7 × 4, 20 × 8
16 × 8, 7 × 4, 20 × 8
7 [1] [32]
MCF
1
1
—
13 × 5, 7 × 4, 23 × 10
8 [39] [39]
MCF
6
6
—
12 × 9, 10 × 8, 19 × 13
9 [39] [40]
MCF
3
3
—
25 × 19, 16 × 11
10 [31] [31]
MCF
0
0
6 × 4, 9 × 5, 5 × 3
6 × 4, 9 × 5, 5 × 3
11 [41] [41]
MCF
0
0
4 × 5, 3 × 5, 3 × 5
4 × 5, 3 × 5, 3 × 5
12 [32] [32]
MCF
1
1
4 × 3, 3 × 4
3 × 3, 4 × 4
13 [5] [5]
MCF
0
0
2 × 2, 3 × 2
2 × 2, 3 × 2
14 [42] [42]
MCF
2
2
3 × 2, 4 × 3
3 × 2, 4 × 3
15 [43] [43]
MCF
2
2
3 × 2, 4 × 3
3 × 2, 4 × 3
Figure 3. Inter-cell moves.

Manufacturing Cells Formation Based on Graph Coloring 498
7. Conclusions
A method of coloring graphs to assist the manufacturing
cells project was presented and discussed in the present
article. The method is grounded on a procedure that un-
derstands computing dissimilarities among parts and the
coloring of a threshold graph, which aims at grouping the
parts in families and the machines in subsets, so that a
single subset of machines corresponds to each family of
parts and vice versa.
The obtained solution tries to minimize the number of
inter-cell moves, to the extent that it tries to group the
parts with closely corresponding manufacturing pro-
grams and separate the parts with different programs.
Figure 4. Computational time.
Table 5. Manufacturing cells (I = 2).
M 1 M 2 M 4 M 6 M 8 M 11 M 3 M 5 M 7 M 9 M 10 M 12
P1 1 1 1 1 1
P2 1 1 1 1 1
P3 1 1 1 1
P5 1 1 1 1 1
P6 1 1 1 1 1
P4 1 1 1 1
P7 1 1 1
P8 1 1 1
P9 1 1 1 1 1
Table 6. Manufacturing cells (I = 0).
M 3 M 5 M 7 M 9 M 10 M 11 M 1 M 2 M 4 M 6 M 8 M 12
P4 1 1 1 1
P5 1 1 1 1 1
P7 1 1 1
P8 1 1 1
P1 1 1 1 1 1
P2 1 1 1 1 1
P3 1 1 1 1
P6 1 1 1 1 1
P9 1 1 1 1 1
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