J. Software Engineering & Applications, 2010, 3, 674-682
doi:10.4236/jsea.2010.37077 Published Online July 2010 (http://www.SciRP.org/journal/jsea)
Heuristic Approaches for Cell Formation in
Cellular Manufacturing
Shahram Saeedi1, Maghsud Solimanpur2, Iraj Mahdavi1, Nikbakhsh Javadian1
1Department of Industrial Eng., Mazandaran University of Science and Technology, Babol, Iran; 2Faculty of Engineering, Urmia
University, Urmia, Iran.
Received April 22nd, 2010; revised May 16th, 2010; accepted May 22nd, 2010.
ABSTRACT
Cellular Manufacturing System (CMS) is an application of Group Technology (GT) that allows decomposing a manu-
facturing system into subsystems. Grouping the machines and parts in a cellular manufacturing system, based on simi-
larities is known as cell formation problem (CFP) which is an NP-hard problem. In this paper, a mathematical model is
proposed for CFP and is solved using the Ant Colony Optimization (ACO), Genetic Algorithm (GA) and Simulated An-
nealing (SA) meta-heuristic methods and the results are compared. The computational results show that the GA method
is more effective in solving the model.
Keywords: Cell Formation Problem, Ant Colony Optimization, Genetic Algorithm, Simulated Annealing, Sequence
Data, Production Volume
1. Introduction
Cellular Manufacturing System (CMS) is an application
of the Group Technology (GT) philosophy that allows
decomposing a manufacturing system into subsystems
which makes its management easier than the entire man-
ufacturing system. It has been shown that CMS is an ac-
cepted solution to the problem of productivity in batch
production which includes a large portion of world man-
ufacturing [1]. The main idea in CMS is the principle of
“Similar things should be done similarly” which means
the similar manufacturing processes should be identified
and grouped in dedicated manufacturing cells.
Manufacturing systems employing CMS can improve
the productivity to a large extent. It has been found that
CMS can increase the productivity of manufacturing
system by three major factors [2]:
 improvement in quality of the work-force,
 increase in the availability of capital,
 improvement in the production technology.
Based on the simulation results performed by Morris
and Tersine [3], the superiority of CM over batch pro-
duction is significant especially when the setup/operation
ratio is high, demand is stable, one-way intercellular
flows and considerable materials handling are concerned.
Due to different solution approaches, different group-
ing solutions may be proposed for a certain problem.
Therefore there should be some criteria to compare these
solutions and choose the best one. There are several ob-
jectives to measure the effectiveness of CMS such as:
 Minimum number of intercellular/intracellular
moves,
 Greatest proportion of part operations performed
within a single cell,
 Maximum machine utilization,
 Minimal total costs by reducing set-up times, and
WIP (Work-in-Process),
 Minimal capital investment,
 Minimum number of voids in the cells.
With respect to the benefits mentioned above, CMS
has attracted the attention of researchers for the last dec-
ades. Some researches related to the work presented in
this paper are reviewed in the following.
Burbidge [4] defines group technology as: “an ap-
proach to the organization of work in which the organ-
izational units are relatively independent groups, each
responsible for the production of a given family of prod-
ucts”. In this approach, the main goal is to form manu-
facturing groups in which, some machines are located in
dedicated cells associated with some similar parts based
on a machine-part incidence matrix. In each cell, some
operations are done on the parts by machines, so that the
main objective is to maximize the intra-cell operations,
and to minimize the number of inter-cell movements

Heuristic Approaches for Cell Formation in Cellular Manufacturing 675
(exceptional elements). It is shown that the machine-part
cell formation (MPCF) is a NP-hard problem [5]. There-
fore, it takes a long time to obtain an optimal solution for
medium-sized problems while it is computationally in-
tractable for large-sized problems. Thus, development
and application of heuristic techniques has attained the
interest of researchers in this area.
Joines et al. [6] offered a classification of the tech-
niques available for manufacturing cell formation. Indi-
vidual techniques are aggregated into methodological
groups including array-based clustering, hierarchical
clustering, non-hierarchical clustering, graph theoretic
approach, artificial intelligence, mathematical program-
ming, and heuristic approaches.
Table 1 provides a review of the researches related to
the current work in terms of the solution approach or
problem perspective. The works pointed out in this table
suffer from at least one of the following drawbacks:
1) Intercellular movements have been calculated re-
gardless of production volume though it is directly af-
fected by this parameter.
2) Sequence of operations has only been taken into
account in the calculation of similarity between the parts.
However, this parameter directly affects the number of
movements of parts between the cells.
3) In a large number of researches, the total number of
“ones” fell out of diagonal blocks is considered as a
measure of the number of intercellular movements be-
tween the cells. However, this value is seriously dictated
by the sequence through which parts are processed.
Suppose a certain operation of a part is processed out of
the associated cell. If this is the first or the last operation
of the part, a single intercellular movement takes place
whereas it is counted twice in otherwise. The mathe-
matical model attempted in this paper provides a formula
to calculate the intercellular movements in this way.
In this paper, a mathematical model is proposed for
solving the cell formation problem, and the model is
solved using Genetic Algorithm (GA), Simulated An-
nealing (SA) and Ant Colony Optimization (ACO). Per-
formance of these methods is compared using two exam-
ples selected from the literature. The comparison shows
Table 1. Summary of literature review
Reference Applied
Methodology
Sequence of
operation
Production
Volume
Exceptional
Elements
(Voids)
Intercellular
Movements
Islier [7] Ant algorithm No No No No
Prabhaharan et al. [8] Ant algorithm Yes Yes No Yes
Mak et al. [9] Ant algorithm Yes No No No
Spiliopoulos and Sofianopoulou [10] Ant algorithm Yes No No Yes
Kesen et al. [11] Ant algorithm Yes No No No
Satolgu and Suresh [12] Goal Programming No No No No
Kao and Fu [13] Clustering Algorithm No No No No
Pandian and Mahapatra [14] Neural Networks Yes No Yes Yes
Mahdavi et al. [15] Genetic Algorithm Yes No Yes No
Mahdavi and Shirazi [16] Heuristic Algorithm Yes No Yes No
Arkat et al. [17] Simulated Annealing No Yes No No
Ahi et al. [18] TOPSIS Yes No Yes No
Wang et al. [19] Scatter Search Yes Yes No No
Murugunandam et al. [20] GA + Tabu Search Yes Yes No No

Heuristic Approaches for Cell Formation in Cellular Manufacturing 676
the effectiveness of GA method.
2. Problem Formulations
2.1 Notations
C, M, P: Total number of Cells, Machines, and Parts.
i,j,c: Index of machines, parts and cells respectively.
Dj: Demand for part j.
Lc: Minimum number of machines which should be
assigned to cell c.
yjc: Boolean decision variable, which is 1 if part j is
assigned cell c, and 0 otherwise.
xic: Boolean decision variable, which is 1 if machine i
is assigned to cell c, and 0 otherwise.

ij: Boolean parameter, which is 1 if part j needs ma-
chine i for completion, and 0 otherwise.
βij: Boolean parameter, which is 1if machine i is the
first or the last machine needed for part j, and 0 other-
wise.
2.2 Mathematical Formulation
The objective function of the proposed model is to mini-
mize the total number of intercellular movements (f1) and
total number of voids (f2) which can be formulated as
below:
f1 =
11 1
CM P
ci j 
 Dj.

ij. (2 – βij) .yjc .(1 – xic); (1)
f2 = 
 
C
c
M
i
P
j11 1
Dj.(xic.yjc – 
ij. xic.yjc); (2)
Minimize f = f1 + f2
Subject to:
yjc ≤ 

M
i1
xic ;
 j,c (3)


C
c
ic
x
1
= 1;  i (4)


C
c
jc
y
1
= 1;  j (5)


M
i
ic
x
1
≥ Lc;  c (6)
xic, yjc  {0,1}; (7)
Constraint (3) implies that assignment of a part to a
cell is subject to the presence of at least one machine in
that cell. Constraints (4) and (5) ensure that any part or
machine is assigned to only one cell. Constraint (6)
maintains the size of the cells and guarantees that at least
a predefined minimum number of parts will be assigned
to each cell.
3. The ACO Algorithm
Ant colony optimization was first developed by Dorigo
et al. [21] based on the behavior of real ants. Real ants
which live in colonies leave the nest to find food and
come back again at every time. Based on observations,
these ants always choose the shortest path to reach the
food. As soon as this shortest path is found by some ants,
the subsequent ants follow the same path. In fact there is
a complicated communication system controlling the
movement of ants. The secret of this communication is
based on a substance, called pheromone. Real ants lay a
substance known as pheromone on the ground when they
pass through a path. This substance is smelled by other
ants which leads them to follow the path traveled by
prior ants. The more ants pass on a path, the more phe-
romone is put on that path. Since the shorter path is trav-
eled fast, the density of pheromone on this path increases
faster than other paths. Therefore, a majority of ants in-
tend to travel on the shorter path after a given time. This
is the underlying mechanism of ACO which is imple-
mented to solve CFP in the following subsections.
3.1 Solution Representation and Evaluation
Suppose there are M machines and P parts to be clus-
tered into C cells. The relation of machines and parts,
which shows machine requirement of parts is normally
represented by a matrix named. In this paper, the ma-
chine-part incidence matrix indicates the production
process data as well. Specifically, an entry aij = k in the
matrix, means that operation k of part j needs machine
i for completion. Table 2 shows an example including
fifteen machines and twenty-five parts.
In the proposed algorithm, a solution is represented
with a string of length M + P. The first M characters of
the string show the cell number of the machines, and the
rest are used for the parts. For example, for M = 5, P = 4,
and C = 2, a solution can be represented as “212112211”
which means assign of machine 1 to 5 to cells 2,1,2,1 and
1 respectively and assignment of parts 1 to 4 to cells
2,2,1 and 1 respectively.
The generated solution may be infeasible, that is, the
selected machine and part, are assigned to a wrong cell.
In this case, the solution will be deleted, otherwise the
goodness (or efficiency) of the solution will be com-
puted.
3.2 Goodness Measurement
One of the most important steps in heuristic techniques is
the evaluation of the obtained solutions. In this step, the
goodness (or fitness) of the solution is calculated, and
based on the result, the solution may be deleted, kept, or
marked as good. The GA technique always keeps a popu-
lation of feasible best fitted chromosomes (obtained solu-
tions) and tries to achieve better solutions by mating the
parents. Similarly, the ACO model keeps a list of best
solutions ever found called elite list. When a new solution

Heuristic Approaches for Cell Formation in Cellular Manufacturing 677
Table 2. 15 × 25 machine-part matrix and demand for parts (Dj) of example 1
Parts 1 2 3 4 5 6 7 8 9 1011121314151616181920 21 22 23 2425
Dj 59 95 30 46 13 34 12 63 57 74 98593 75 22 24 61100 26 56 19 67 97 24 47
M1 1 1 1 1331 1
M2 2 2 2 4 2 5 2 2 8 2
M3 6 6 6 1 65
M4 2 5 3 222 21 2 2 22
M5 3 3 3 3 3 3 7
M6 3 4 3 2 3 23 4
M7 1 1 1 1 1 1
M8 7 2 454 44 4 4 4
M9 4 4 4 4 4 4 5
M10 4 1 5 4 44 3
M11 5 645 55 5 6 54
M12 1 1 111 12 1 11
M13 5 5 5 5 4
M14 6 3 4 333 33 3 3 33
M15 2 2 6 3 2 12 2 7
is obtained, the goodness (fitness) function is applied,
and based on the result, we decide to add the solution
to the elite list, or omit the solution and generate an-
other one.
In this model, considering the objective function de-
fined in Sub-section 2.3, the number of intercell move-
ments and the number of voids in cells is to be mini-
mized. So, the goodness function is defined as below:
goodness = f =
12
1
1ff


(8)
During the ACO, SA and GA iterations, the goodness
of each solution is calculated using Equation (8). The
constant value “1” is added to prevent division by zero.
3.3 The Ant Algorithm
Descriptive procedure of the proposed algorithm for
solving the attempted mathematical model is as follows:
Begin
1. Initialize
2. Generate a feasible random solution, and add
it to the elite list.
3. Evaluate the efficiency (goodness) of the so-
lution
4. Repeat
a. Generate another random solution,
based on pheromone trails.
b. Evaluate goodness, if better than the
worst solution in the elite list, add it
to elite list and delete the worst solu-
tion from list, update pheromone
trails.
c. Evaporate pheromone
d. Alter solutions periodically
5. Until stopping condition is met
End
This model uses a P = [Pck](C) × (M + P) pheromone ma-
trix in which, C, M, and P, are the number of cells, ma-
chines and parts, respectively. The initial value of Pck is 1.
So, to generate a random feasible solution in step 3, there
is no need to use the pheromone matrix.
Machines

Heuristic Approaches for Cell Formation in Cellular Manufacturing 678
4. Genetic Algorithm
In this method, an initial population of solutions (chro-
mosomes) is generated randomly while subsequent pop-
ulations are generated by choosing good parents and
mating them. The mating may cause to worse, better, or
even infeasible solutions. By keeping better solutions in
population and omitting bad ones, the algorithm con-
verges stage by stage and after a number of iterations, the
local or global optimum will be found.
4.1 The GA Algorithm
Begin
1. Generate initial population containing N chromo-
somes.
2. Compute the fitness of chromosomes in current
population.
3. Generate the next population
a. Choose two best parents randomly from the
current population.
b. Mate the parents and generate two children
(Crossover operator).
c. Apply the mutation operator.
d. Compute the fitness of children.
4. Repeat step 3 until termination condition is met.
End
In the proposed algorithm, the size of initial population is
1000 and the mating candidate parents are chosen by rou-
lette wheel method. Chromosomes are represented as de-
scribed in Subsection 3.1. The probability of crossing over
and mutation is considered as 0.8 and 0.2 respectively.
5. Simulated Annealing
The Simulated Annealing (SA) algorithm is derived from
metallurgy and thermodynamics which incorporated a
temperature parameter into the minimization parameter. A
high temperature expands the search space, and a lower
temperature restricts the exploration. The procedure starts
from a high temperature and ends at a low temperature.
At each temperature, a number of iterations are done.
Some heuristic algorithms like Hill Climbing tech-
nique, may found the Local Optimum instead of the
Global optimum because the movements leading to a
new point worse than the current point are not allowed.
SA algorithm allows non-improving movements to be
taken in the hope of escaping the local optimum with a
probability depending on the procedure temperature and
the amount of the badness of the solution.
5.1 The SA Algorithm
Using the same representation described for solutions in
Subsection 3.1, the SA algorithm can be written as fol-
lows:
Begin
1. Initialize Temp.
2. Find a feasible solution, called x.
3. Compute its goodness f(x).
4. Repeat until frozen
a. Do 1000 times
i. y: = FindNeighbour(x).
ii. Delta: = f(x) - f(y).
iii. if Delta > 0 then x: = y ; Ac-
cept y).
iv. else if U(0,1) < e-(Delta)/Temp
then x: = y (Accept y).
v. else reject y.
b. End Do.
5. Temp: = Temp * 0.95.
6. The Solution will be best so far.
End
The Algorithm starts at the temperature of 5000 with a
feasible solution. The neighbor of a solution is obtained
by making some changes in the solution (some parts or
machines are randomly moved from one cell to other
one).
6. Examples
The proposed algorithms are applied to solve two
benchmark problems (15 × 25 and 20 × 35 sizes) avail-
able in the literature. The algorithms are implemented in
C Language and are executed on a Pentium IV PC.
Example 1
The first example consists of fifteen machines and
twenty-five parts which are grouped in three cells. The
machine-part incidence matrix of the problem with is
given in Table 2. The table also indicates demand of
each part generated by a discrete uniform distribution in
[1..100]. The solution obtained by the proposed ACO,
SA and GA algorithms for this problem is shown in Ta-
ble 3. The obtained values for f1, f2 and f are 863, 803 and
1666 respectively in all three methods.
Example 2
The second example is adopted from Boe and Cheng
[22] with 35 parts, 20 machines and four cells. Table 4
and Table 5 show the machine-cell incidence matrix and
the solution obtained by ACO, GA and SA algorithms.
Since the demand for parts are not included in [22], these
values are randomly generated using a discrete uniform
distribution in [1..100]. The final obtained value of the
objective function is 4562 in all three methods.
Figures 1 and 2 show the number of iterations and
convergence speed of different algorithms. In these fig-
ures, the horizontal axis shows the number of iterations
(number of generations in GA, number of ants in ACO,
and iterations of SA) and the vertical axis shows the val-
ue of the objective function.

Heuristic Approaches for Cell Formation in Cellular Manufacturing 679
Table 3. The solution obtained by the proposed ACO, SA, and GA algorithms for example 1
1 4 5 7 14 1722 2 6 8 131516183 910111219 20 21 232425
M7 1 1 1 1 1 1
M2 2 2 2 2 2 2 2 4 5 8
M5 3 3 3 3 3 3 7
M13 4 4 4 4 4 4 5
M9 5 5 5 5 4
M1 3 1 1 1 1 3 1 1
M15 6 2 2 3 2 1 2 2 7
M6 3 3 2 3 2 3 4 4
M10 1 4 5 4 4 4 3
M12 111 1 1 1 2 1 1 1
M4 5 232 2 2 2 1 2 2 2 2
M14 6 343 3 3 3 3 3 3 3 3
M8 7 24 5 4 4 4 4 4 4
M11 56 4 5 5 5 5 6 5 4
M3 6 6 6 1 6 5
Table 4. The machine-cell incidence matrix of example 2
1 2 3 4 5 6 7 8 910 11 12 13 14 15 16 17 1819 20 21 222324 25 26 27 28 29 30 31 32 33 34 35
1 5 4 5 6 514623 5 414
2 1 1 3 1 363 3
3 1 1 2 11 1
4 2 2 4 2 41
5 1 1 111
6 1 2 11 3
7 2 2 3 1 1 22513414 2 4 2 3
8 3 4 5 3325 2
9 2 3 22 21
10 3 4 1322
11 1 1 1 1 1 1
12 2 2 2 2 2 2
13 3 5 3 5
14 4 3 2 6 4 462 4
15 3 3 3 3 3 1 1
16 5 5 6 2 25 6 5
17 4 3 5 4432 3
18 5 4 7 5 57 5
19 4 4 4 4 2 2 1
20 4 5 433
Machines

Heuristic Approaches for Cell Formation in Cellular Manufacturing 680
Table 5. The solution obtained by the proposed ACO, SA, and GA algorithms for example 2
1 3 5 15 17 18 20 23 25 29 32 34 35 2710 12 13 24 27 31814 16 19 22 26 4 6 9 11 21 28 30 33
1 5 5 1 4 6 3 4 1 4 26 4 5 5
3 1 1 2 1 1 1
7 2 2 3 2 2 1 4 2 3 1112534 4
8 3 4 3 3 2 5 2 5
17 4 3 5 4 4 3 2 3
2 3 1131336
4 224241
13 3535
14 4 43264624
18 5 547575
5 1 1 1 11
6 1211 3
9 2 2 1 232
10 341322
20 3 4543
11 1 1 1 1 11
12 2 2 2 2 22
15 3 3 3 3 311
16 2 5 25 5 5 6 6
19 1 4 4 4 422
0
2000
4000
6000
8000
10000
12000
13569103137 171 205 239273 307 341375 409 443 477511 545 579
SA
ACO
GA
Figure 1. Convergence speed of GA, ACO and SA for example
Objective Function
Iterations

Heuristic Approaches for Cell Formation in Cellular Manufacturing 681
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
1336597129 161 193 225257 289 321 353385 417 449 481 513 545 577
SA
ACO
GA
Figure 2. Convergence speed of GA, ACO and SA for example 2
Table 6. Computational time of GA, ACO and SA
Meta-Heuristic Method Example 1 Example 2
GA 10 14
SA 37 48
ACO 251 263
The computational time (in seconds) of different algo-
rithms for examples 1 and 2 are shown in Table 6.
7. Conclusions
This paper discusses that the sequence of operations and
the production volume are two major factors to be con-
sidered in the design of CMS. Despite this fact, it has not
been taken into account in a majority of researches
available in the literature. To capture this fact, a new
model for solving cell formation problem in CMS is
proposed. Due to the NP-hardness of the formulated
problem, three solution approaches based on ACO, GA
and SA are used to solve the model. The objective func-
tion of the model is to minimize the total number of in-
tercellular movements and the number of voids. The total
number of cells is defined as a constant parameter in the
algorithm.
The computational results show that the proposed al-
gorithms are effective in minimizing the total number of
voids and intercellular movements.
As shown in Figures 1 and 2, the GA algorithm has
obtained the optimum value faster than other techniques.
The attempted mathematical model can be further ex-
tended by considering alternate process plans for each
part, machine redundancy, processing time of each op-
eration, etc.
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