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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3789-3809
© Research India Publications. http://www.ripublication.com
3789
K-means cluster algorithm-based evolutionary approach for constrained
multi-objective optimization
A. A. Mousa1,2, M. Higazy1,3, Yousria Abo-Elnaga4
1 Department of Mathematics and Statistics, Faculty of Sciences, Taif University, Saudi Arabia. 2Department of Basic Engineering Science, Faculty of Engineering, Shebin El-Kom, Menoufia University, Egypt.
3Department of physics and engineering mathematics, faculty of electronic engineering, Menofia University, Egypt. 4Department of Basic science, Higher Technological Institute, Tenth of Ramadan City, Egypt.
Abstract
One of the most popular approaches for generating non-dominated
solutions of a multi-objective optimization problem is the
evolutionary algorithms. The need of an approximation to the non-
dominated set for the decision maker for selecting a final preferred
solution is required in several real-life applications . Unfortunately,
the cardinality of the Pareto optimal solutions' set may be very large
or even infinite. On the other hand, due to the overflow of
information, the decision maker (DM) may not be concerned in
having an excessively large number of Pareto optimal solutions to
deal with. In this paper, a new enhanced evolutionary algorithm is
presented, our proposed algorithm has been enriched with modified
k-means cluster scheme, On the first hand, in phase I, K-means
clustering method is implemented to partition the population into a
determined number of subpopulation with-dynamic-size, where
distinct genetic algorithms (GAs) operators can be applied to each
sub-population, instead of one GAs operator applied to the whole
population. On the other hand, phase II uses K-means algorithm in
order to make the algorithms practical by allowing a decision maker
to control the resolution of the Pareto-set approximation by choosing
an appropriate k value ( no of required clusters). To prove the
excellence of the proposed approach compared to state-of-the-art
evolutionary algorithms, diverse numerical studies will be done using
a suite of multimodal test functions taken from the literature.
Keywords: K-means cluster algorithm, evolutionary approaches,
constrained optimization, multiobjective optimization
INTRODUCTION
Multiobjective Optimization Problems (MOPs) are those
having more than one objective. There are a set of possible
solutions and there is no unique optimum solution. these
solutions are optimal in the wider sense that no other solutions
in the search space are dominate them when all objectives are
considered. As in [1,2], these solutions are called Pareto
optimal solutions. Surely, MOPs appear in several areas of
knowledge for example economics [3-6], machine learning [6-
8] and electrical power system [9-12]. A few developed
evolutionary algorithms are constructed for solving the
constrained multi-objective optimization problems. In spite of
the several studies for solving the constrained optimization
problems; there are no enough studies concerning the
procedure for handling constraints. As an instance, in [13],
treating constraints as high-priority objectives was suggested
by Fonseca. In [14] by Harada, a few effective constraint-
handling guidelines were suggested and a Pareto descent
repair method was constructed. For MOPs, because the
objective vector cannot be directly assigned as the fitness
function value, it is always needed a correctly constructed
algorithm for fitness assignment. Using the Pareto dominance
relationship, the values of the fitness functions are assigned by
most of the existing MOEAs [15-16].
AL Malki et al.[17] have presented Hybrid Genetic K-Means
algorithm for Identifying the Most Significant Solutions from
Pareto Front. They have implemented clustering techniques to
organize and classify the solutions for various problems. In
[18], the Hybrid Genetic Algorithm with K-Means has been
presented by Al Malki et al. to solve the Clustering Problems.
They have succeeded to eliminate the empty cluster problem
by using a hybrid form of the k-means algorithm and GAs.
Their proposition was be proved by the results of simulation
tests via various data collections. The use of K-means
clustering technique in [19] enable the algorithm to implement
various operators of GA to each subpopulation rather than
utilizing a single GA operator for all population.
In this article, a new optimization algorithm is proposed
which runs in two stages: in the first stage, the algorithm
combines all principal characteristics of K-means with GA
clustering method and uses it as a search engine to produce
the correct Pareto optimal front. The use of K-means
clustering technique enable the algorithm to implement
various operators of GA to each subpopulation rather than
utilizing a single GA operator for all population.
Then in the second phase, k -means cluster algorithm is
adopted as reduction algorithm in order to improve the spread
of the solutions found so far. Our proposed algorithm has been
enriched with modified k-means cluster scheme, the use of k
-means also makes the algorithms practical by allowing a
decision maker to control the resolution of the Pareto-set
approximation by choosing an appropriate k value. k -means
clustering aims to partition n observations into k clusters in
which each observation belongs to the cluster with the
nearest mean. Finally, various kinds of multi-objective
benchmark problems have been reported to stress the
importance of hybridization algorithms in generating Pareto
optimal sets. Simulation results with the proposed approach
will be compared to those reported in the literature. The
comparison will demonstrate the superiority of the proposed
approach and confirms its potential to solve the multi-
objective optimization problems.
The organization of this paper is as follows: Multi-objective
optimization problem is presented in Section 2. Overview of
GAs are presented in Section 3. Clustering Algorithms are
introduced in Section 4. K-Means for clustering problems is
introduced in section 5. The proposed algorithm is presented
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© Research India Publications. http://www.ripublication.com
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in section 6. The Simulation results are discussed in Section 7.
Finally, we conclude the paper in Section8.
MULTIOBJECTIVE OPTIMIZATION (MO)
Mathematically, a general minimization problem of
M objectives can be presented as [20-21]:
Minimize: , 1, 2, ..., 1
subject to the constraints: 0, 1, 2, ..., .
i
j
f x f x i M
g x j J
and 1 2, ,..., ,nx x x x where the dimension of the
decision variable space is equal n , the i-th objective function
is
if x and the j-th inequality constraint is jg x . Then
the main task of the MO problem is to find x that optimize
if x . The evaluation of the solutions uses the concept of
the Pareto dominance because the notion of an optimum
solution in MO is different compared to the single objective
optimization (SO).
Definition 1: (Pareto dominance). A vector
1 2, ,..., Mu u u u is said to dominate a vector
1 2, ,..., Mv v v v (u dominate v denoted by u v ),
for a MO minimization problem, if and only if
,..., , ,..., : 2i i i ii i M u v i i M u v
where M is the dimension of the objective space.
Definition 2: (Pareto optimality). A solution ,u U is
called a Pareto optimal iff there is no other solution (v U ),
such that u is dominated by v . These solutions are called
non dominated solutions. The set of all such non dominated
solutions constitutes the Pareto-Optimal Set.
OVERVIEW OF THE GA
The techniques of natural selection are the primary concept of
GAs. Any optimization variable, (xn), is converted into a gene
as a real number or a string of bits. All variables' genes,
1,....., nx x, form a chromosome, that depicts all individuals.
Depending on the specific problem, an array of real numbers,
binary string, a list of components in a database could be
considered as a chromosome. Each possible solution is
represented by an individual, and the set of individuals form a
population. From a population the fittest is chosen for matting.
Matting is done by merging genes from distinct parents to
generate a child, called a crossover. Finally, the children are
added to the population and the process repeats over again,
thus symbolizing an artificial Darwinian environment as
illustrated in Figure 1. The optimization shall stop if and only
if the population has converged or the generation has been
reached the maximum number.
Figure 1. GAs outline for optimization problems.
CLUSTERING ALGORITHM
Clustering is an approach of partitioning of data into similar
objects sets. Each set, called cluster, consists of mutually
similar objects and different to objects of other clusters [22].
Clustering is a very important area of research, which has
various applications in several fields such as in psychiatry
[23], market research [24], archaeology [25], pattern
recognition [26], medicine [27] and engineering [28].
In the literature, clustering has various proposed algorithms.
Due to its simplicity and accuracy, the K-means clustering is
possibly the most commonly-used clustering algorithm [29].
K-Means Clustering Technique
In 1967, Macqueen produced the K-means clustering
algorithm [30]. Due to the easiness of K-means clustering
algorithm, it is used in many fields. The K-means clustering
algorithm separates data into k sets as it is a partitioning
clustering approach [31]. The K-means clustering algorithm is
more prominent because it is an intelligent algorithm that can
cluster massive data rapidly and efficiently.
The basic idea of K-means algorithm is to classify the data D
into k different clusters where D is the data set, k is the
number of desired clusters. More precisely, the following
are the main steps of the K-means clustering algorithm
(Figure 2):
Step 1: Define a number of desired clusters, k.
Step 2: Choose the initial cluster centroids randomly, which
represent temporary means of the clusters.
Step 3: Compute the Euclidean distance from each object to
each cluster and each object is assigned to the closest
cluster with the smallest square distance.
Step 4: For each cluster, the new centroid is computed, and
each centroid value is now replaced by the respective
cluster centroid
Step 5: Repeat steps 3 and 4 until no point changes its cluster.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3789-3809
© Research India Publications. http://www.ripublication.com
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Figure (2): K-means algorithm procedure(Taken from [32]).
THE PROPOSED APPROACH
In this section, the proposed algorithm that we are dealing
with is informally described. Our proposed algorithm consists
of two phases. In phase I, K-means clustering technique is
implemented to partition the population into a determined
number of subpopulation with-dynamic-sizes. On the other
hand, phase II applies K-means algorithm to make the
algorithms practical by allowing the decision maker to control
the precision of the Pareto-set approximation by choosing a
suitable number of the needed clusters.
Phase I
Step1: Population Initialization: Two disconnected sub-
populations are used by the algorithm, the individuals that
initialized randomly satisfying the search space (The lower
and upper bounds) form the first one, while the reference
points that satisfying all constraints (feasible points) form the
second one. However, we have concentrated on how elitism
could be introduced to guarantee convergence to the true
Pareto-optimal solutions. For that, an “archiving/selection”
plan is proposed that ensures instantaneously the advance to
reach the Pareto-optimal set and coverage of the entire range
of the non-dominated solutions. An externally limited size
archive ( )tA of non-dominated solutions is preserved and
iteratively updated in the presence of new solutions based on
the concept of -dominance by the proposed algorithm.
Step 2: Repair Algorithm: By repairing infeasible individuals,
the main task of this algorithm is to recognize all feasible
individuals from the infeasible ones. Via this algorithm, the
infeasible individuals in a certain population are developed
gradually till they become feasible ones. As explained in [10],
the repair operation is done as follows. Let S be the feasible
domain and S be a search point (individual). Since better
reference point has better chances to be selected, the
algorithm selects one of the reference points, let it be r S ,
and generates random points (1 ) , [0,1] Z r
from the segment defined between,r , but the segment may
be extended equally [33, 34] on both sides determined by a
user specified parameter [0,1] .
In such a case the algorithm selects one of the reference points
(Better reference point has better chances to be selected), say
r S and creates random points
(1 ) , [0,1] Z r from the segment defined
between,r , but the segment may be extended equally [33,
34] on both sides determined by a user specified
parameter [0,1] . So, an up-to-date feasible individual is
represented as:
1
2
. (1 )., (1 2 ) , [0,1]
(1 ). .
z r
z r
(3)
Step3: Classifying a population according to non-domination:
In this step, the objective functions of each solution are
evaluated and the non-dominated sets of solutions are selected
according to non-domination concept by using the following
algorithm [35-36]. Figure(3) illustrate the classifying a
population according to non-domination.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3789-3809
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Start with m = 1.
The solutions mx and nx are compared for
domination for all 1, 2, , POPn N and m n .
If mx is dominated by nx , for any n, mark mx as
‘dominated’, and it is Inefficient.
Increase m by one and return to Step 2. If all
solutions in the population are considered, go to Step
5
The solutions which are not marked ‘dominated’ are
non-dominated solutions
Save the non-dominated solutions in an external
archive.
Figure (3): Classifying a population according to non-
domination
Step 4: Selection Stage: Through a weighted combination, a
dynamic-weight approach [37] is used to aggregate, the multi-
objective functions into a single combined fitness function.
The main characteristic of dynamic-weight approach is that
the weights attached to the multiple objective functions are
not fixed but randomly specified. Consequently, the direction
of the search is not specific. The multiple objective functions
are combined into a scalar fitness solution as follows:
( ) ... ( ) ... ( );1 1
Z w f x w f x w f xq qi i (4)
where x is an individual, Z is the combined fitness function,
( )f xi
is the ith objective function and w is a weighting-
vector with 0 wi 1, . . . , ;i q
where
11
qw
i i
. In general, the value of each weight can be
randomly determined as follows:
,
1
randomiw i = 1,2,...,q;qi random
j j
(5)
where randomi
and randomj
are non-negative random
real numbers.
After formulating the combined fitness function (Z) for
each chromosome by equation (4), the selection probability
for each chromosome is then defined by following linear
scaling function:
min
min1( )
pop
ii N
jj
Z Zprob
Z Z
(6)
where iprob
is the selection probability of a chromosome i
whose combined fitness function is iZ and
minZ is the worst
combined fitness in the population.
Finally, a pair of individual is chosen randomly from the
current population and the best solution with better selection
probability is chosen and subsequently copied in mating pool
[30]. This step has to be repeated frequently, until the size of
the mating pool is equivalent to the original population size.
Step 5: K-Means clustering technique: In this paper, K-means
clustering technique[38] was used to split the population to K
separated subpopulations with dynamic size, as shown in
Figure (4). By using this method, various operators of GA can
be applied to subpopulations in stead of applying a single GA
operator to the whole population.
Figure 4: The splitting of the population into K separated
subpopulations
Step6: Crossover operator: The crossover operator generates a
new offspring by mating (combing) two chromosomes
(parents). The concept behind crossover is that the new
offspring may take the best characteristics from the parents,
and it will be better than both parents. The Crossover occurs
during evolution process according to user-definable fixed
probability. In this step, three different crossover operators are
used which are horizontal band crossover, uniform crossover
and real part crossover which are described as follows:.
Horizontal band crossover :In this operator, two
horizontal crossover sites are selected randomly. The
information in the horizontal region, which is determined
by horizontal crossover sites, is exchanged between the
two parents based on a fixed probability [39].
Uniform crossover: In uniform crossover [40],
random (0,1) mask is generated. In the random mask the
‘0’ denotes bit unchanged, while ‘1’ represents bits
swapping.
Real part crossover: This operator works on the real
part itp of a chromosome to cross the real variables
itp of the chromosomes by exchanging the information
in column vectors of itp matrix [41]. The new power
matrices of off-springs ( 1)(itp offspring and
2)(itp offspring ) are created from The power
matrices ( itp ) of parent chromosomes, as follows:
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3789-3809
© Research India Publications. http://www.ripublication.com
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1 1 2 1
2 2 1 2
1 1
1
) ,
2) , ;
( ...,(1 ) ,...
( ...,(1 ) ,...
it
it
parent parent parent parent
Tj j
parent parent parent parent
Tj j
p offspring column column column column
p offspring column column column column
(7)
where is chosen randomly between 0 and 1, and
jcoulmn is the column vector in power matrix and“j” is a
random positive integer in range of [1,T].
Step 7: Mutation operator: This operator is used to explore
some of the points in the search space by altering few genes
value in an individual randomly from its initial state. The
mutation is generally occurs according to user-definable
mutation probability which is usually low value. In this step,
we used three different mutation methods which are one point
mutation, intelligent mutation and swap window mutation
which are described as follows:
Single point mutation: In single point mutation, a
single bit is chosen randomly from binary part (itu ) of
offspring and then its value is changed from ‘0’ to ‘1’
and vice versa [40-41].
Intelligent mutation: This mutation [42] searches
for (10) or (01) combinations in commitment schedule,
then randomly changing them to 00 or 11.
Swap window mutation: Swap window mutation
works on the binary part ( itu ) of a chromosome by
selecting: 1) two units at random 2) a time window of
width w between 1 and T and 3) the location of the
window, then exchanging the entries of the two units
which are in the window [43].
At any time instant, if status of any unit changed due to
mutation operator from 0 to 1, the corresponding output
power of this unit will be changed from 0 to real value chosen
at random from the range of min max[ , ]i ip p
.
Step 8: Combination stage: At this step [44,45], all
subpopulations are grouped once more to form a new
population, as shown in Figure (5).
Figure 5: Commination stage
Step 9: Update the Archive of Non-dominated Solution: The
proposed approach has an external archive of non-dominated
solutions which gets updated iteratively when new solutions
are found based on the concept of non-domination. The
Archive is updated in each iteration by copying the solutions
of current population tP to archive
tV and applying the
dominance criteria to remove all dominated solutions (i.e.,
each solution of tP has three probabilities as in Algorithm 1
in Figure (6) [60].
Algorithm 2:Update the Archive
( , )
|
{ }/{ }
t t
t
t t
t
t t
Input V X P
If Y V Y X then
V =V
Else if Y V X Y then
V V X Y
Else if
|
{ }
:
t
t t
t
Y V Y X then
V V X
End
Output V
Figure 6: Update the Archive of non-dominated solutions
Phase II: K-means Algorithm
Step 10: Centres 1 2, ,..., Kz z z of K initial cluster are
randomly chosen from the n observations
1 2, ,..., nx x x .
Step 11: A point 1, 1,2,...,x i n is assigned to cluster
, {1,2,..., }jC j k if and only if:
, 1,2,..., &i j i px z x z p K j p (8)
Step 12: Centres of new cluster 1 2, ,..., Kz z z are
computed as follows:
* 1, 1,2,..., ;
j i
i j
x Ci
z x i Kn
(9)
where in is the number of elements which are belonging
to cluster jC .
Step 13: If * , 1,2,....,i iz z i K
then the algorithm
terminate, otherwise go to step 11.
After this phase, we get an initial center for all
predetermined clusters.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3789-3809
© Research India Publications. http://www.ripublication.com
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Identifying the centroids of Pareto front
In order to identify one single point belong to the Pareto front
that represents certain cluster, the following algorithm as
shown in Figure (7) is presented. Figure (7) presents an
algorithm for identifying the centroid for each cluster in such
a way that it locate in the Pareto front while, Figure (8) shows
geometrically how the identifying mechanism occurs.
1 2
*
1 INPUT _ , 1,..., & 1,..., ,
0
INPUT , ,...,
For {1,..., }
Find | ( - ) ( - ) 1,2,..., , 1
j K
Kj Kj
j K
K
K Ki Ki k Ki K Kj
if x Cpartition matrix M m K K j n u
if x C
Z z z z
all K K do
z x x z Min x z i n u
* * * *
1 2
End for
OUTPUT Pareo front centriod: ( , ,...., ) KZ z z z
Figure 7: Algorithm for identifying the Pareto front centroids, taken from [17]
Figure 8: Algorithm for identifying the Pareto front centroids, taken from [17]
Results
The proposed algorithm is applied by various Pareto front of
MOO Test Instances for the CEC09 [46]; which are
containing different Pareto front characteristics to demonstrate
its ability to select the most compromise set of solution. The
problems cover different characteristics of MOOPs, namely
convex Pareto front, nonconvex Pareto Front and discrete
Pareto front and non-uniformity of solution distribution.
Twenty problems from CEC2009 contain different shapes of
Pareto front is selected and described in Table 1.
Table 1: MOO Test Instances for the CEC09
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3789-3809
© Research India Publications. http://www.ripublication.com
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f1
f 2
Figure 9: Pareto Front of problem UF2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f1
f 2
Figure 10: Pareto Front of problem UF1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f1
f 2
Figure 11: Pareto Front of problem UF4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f1
f 2
Figure 12: Pareto Front of problem UF3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
F1
F2
Figure 13: Pareto Front of problem UF6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
F1
F2
Figure 14: Pareto Front of problem UF5
00.1
0.20.3
0.40.5
0.60.7
0.80.9
1
00.1
0.20.3
0.40.5
0.60.7
0.80.9
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f1
f2
f 3
Figure 15: Pareto Front of problem UF8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f1
f 2
Figure 16: Pareto Front of problem UF7
00.1
0.20.3
0.40.5
0.60.7
0.80.9
1
00.1
0.20.3
0.40.5
0.60.7
0.80.9
1
0
0.2
0.4
0.6
0.8
1
f1
f2
f 3
Figure 17: Pareto Front of problem UF10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
00.10.20.30.40.50.60.70.80.910
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f1
f2
f 3
Figure 18: Pareto Front of problem UF9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f1
f 2
Figure 19: Pareto Front of problem CF2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f1
f 2
Figure 20: Pareto Front of problem CF1
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3789-3809
© Research India Publications. http://www.ripublication.com
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f1
f 2
Figure 21: Pareto Front of problem CF4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f1
f 2
Figure 22: Pareto Front of problem CF3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f1
f 2
Figure 23: Pareto Front of problem CF6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f1
f 2
Figure 24: Pareto Front of problem CF5
00.1
0.20.3
0.40.5
0.60.7
0.80.9
1
00.1
0.20.3
0.40.5
0.60.7
0.80.9
1
0
0.2
0.4
0.6
0.8
1
f1
f2
f 3
Figure 25: Pareto Front of problem CF8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f1
f 2
Figure 26: Pareto Front of problem CF7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
00.10.20.3
0.40.50.60.7
0.80.91
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f2f
1
f 3
Figure 27: Pareto Front of problem CF10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
00.10.20.30.40.50.60.70.80.91
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f2f
1
f 3
Figure 28: Pareto Front of problem CF9.
The algorithm (Phase I) was implemented for the 20 problems
which are taken from CEC2009 with different Pareto front
Charctertistics as in figures (9-28).
Optimization of the above-formulated objective functions
using multi-objective EAs yields a set of Pareto-optimal
solutions, not give a single optimal solution. But the DM in
practical application needs to select a set of solutions with
limited size. From the above results the selected solutions
should have a diversity characteristics also it must cover the
entire Pareto front domain and allows the DM to attain a
reasonable representation of the Pareto-optimal front. In
addition, the results guarantee that the diversity among
selected solutions is achieved. Finally, the proposed algorithm
has a diversity preserving mechanism to overcome the cruise
of huge number of Pareto front as in figures(29-48)
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3789-3809
© Research India Publications. http://www.ripublication.com
3797
0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.82 Cluster Plot
f1
f 2
(0.72873,0.15143)
(0.22823,0.54985)
0 0.5 10
0.2
0.4
0.6
0.85 Cluster Plot
f1
f 2
(0.88689,0.058899)
(0.24024,0.51334)
(0.44444,0.33495)
(0.071071,0.74914)
(0.66216,0.18723)
0 0.5 10
0.5
17 Cluster Plot
f1
f 2
(0.045045,0.80052)
(0.44144,0.33643)
(0.5971,0.22785)
(0.91892,0.041713)
(0.29229,0.46076)
(0.15516,0.60904)
(0.75676,0.13049)
0 0.5 10
0.5
110 Cluster Plot
f1
f 2
(0.60561,0.22207)
(0.4955,0.29644)
(0.82983,0.089226)
(0.096096,0.69251)
(0.027027,0.84582)
(0.71722,0.15333)
(0.38789,0.37768)
(0.28378,0.46801)
(0.18519,0.57087)
(0.94344,0.028838)
Figure 29: Pareto front centroids of problem UF1for k=2,5,7, and 10 clusters
0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.82 Cluster Plot
f1
f 2
(0.22823,0.54985)
(0.72873,0.15143)
0 0.5 10
0.2
0.4
0.6
0.85 Cluster Plot
f1
f 2
(0.24024,0.51334)
(0.44444,0.33495)
(0.88689,0.058899)
(0.071071,0.74914)
(0.66216,0.18723)
0 0.5 10
0.5
17 Cluster Plot
f1
f 2
(0.44144,0.33643)
(0.15516,0.60904)
(0.5971,0.22785)
(0.29229,0.46076)
(0.75676,0.13049)
(0.045045,0.80052)
(0.91892,0.041713)
0 0.5 10
0.5
110 Cluster Plot
f1
f 2
(0.51301,0.28408)
(0.83884,0.08428)(0.73023,0.14566)
(0.2988,0.45407)
(0.4049,0.36415)
(0.03003,0.83739)
(0.1041,0.67978)
(0.94695,0.027018)
(0.62162,0.21182)
(0.1972,0.55711)
Figure 30: Pareto front centroids of problem UF2for k=2,5,7, and 10 clusters
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3789-3809
© Research India Publications. http://www.ripublication.com
3798
0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.82 Cluster Plot
f1
f 2
(0.72873,0.15143)
(0.22823,0.54985)
0 0.5 10
0.2
0.4
0.6
0.85 Cluster Plot
f1
f 2
(0.66216,0.18723)
(0.88689,0.058899)
(0.44444,0.33495)
(0.24024,0.51334)
(0.071071,0.74914)
0 0.5 10
0.5
17 Cluster Plot
f1
f 2
(0.59359,0.23013)
(0.75425,0.13194)
(0.044044,0.80276)
(0.15265,0.61226)
(0.28879,0.46402)
(0.91792,0.042243)
(0.43744,0.33946)
0 0.5 10
0.5
110 Cluster Plot
f1
f 2
(0.5025,0.29147)
(0.18919,0.56625)
(0.83283,0.087573)
(0.28929,0.46286)
(0.39439,0.37248)
(0.61211,0.21789)(0.72222,0.15037)
(0.028028,0.84296)
(0.098599,0.68846)
(0.94444,0.028317)
Figure 31: Pareto front centroids of problem UF3for k=2,5,7, and 10 clusters
0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
2 Cluster Plot
f1
f 2
(0.79079,0.35999)
(0.29029,0.88755)
0 0.5 10
0.5
15 Cluster Plot
f1
f 2
(0.57307,0.66844)
(0.75676,0.42482)
(0.92192,0.148)
(0.36436,0.86309)
(0.12613,0.97875)
0 0.5 10
0.5
17 Cluster Plot
f1
f 2 (0.70621,0.4999)
(0.42643,0.81621)
(0.26527,0.92725)
(0.57257,0.67055)
(0.94494,0.10605)
(0.09009,0.98915)
(0.82983,0.3102)
0 0.5 10
0.5
110 Cluster Plot
f1
f 2
(0.6001,0.63911)
(0.17668,0.96766)
(0.3984,0.84036)
(0.29029,0.9147)
(0.5015,0.74764)
(0.6952,0.51597)
(0.78679,0.38031)
(0.059059,0.99533)
(0.87437,0.23485)
(0.95896,0.079823)
Figure 32: Pareto front centroids of problem UF4for k=2,5,7, and 10 clusters
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3789-3809
© Research India Publications. http://www.ripublication.com
3799
0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
2 Cluster Plot
f1
f 2
(0.225,0.775)
(0.75,0.25)
0 0.5 10
0.5
15 Cluster Plot
f1
f 2
(0.1,0.9)
(0.725,0.275)
(0.525,0.475)
(0.325,0.675)
(0.925,0.075)
0 0.5 10
0.5
17 Cluster Plot
f1
f 2 (0.525,0.475)
(0.925,0.075)
(0.325,0.675)
(1,0)
(0.7,0.3)
(0.825,0.175)
(0.1,0.9)
0 0.5 10
0.5
110 Cluster Plot
f1
f 2
(0.7,0.3)
(0.575,0.425)
(0.45,0.55)
(0.075,0.925)
(0.275,0.725)
(1,0)
(0.825,0.175)
(0.925,0.075)
(0.75,0.25)
(0.65,0.35)
Figure 33: Pareto front centroids of problem UF5for k=2,5,7, and 10 clusters
0 0.5 10
0.5
12 Cluster Plot
f1
f 2
(0.1875,0.8125)
(0.875,0.125)
0 0.5 10
0.5
15 Cluster Plot
f1
f 2
(0,1)
(0.45858,0.54142)
(0.375,0.625)
(0.29142,0.70858)
(0.875,0.125)
0 0.5 10
0.5
17 Cluster Plot
f1
f 2
(0,1)
(0.95833,0.041667)
(0.375,0.625)
(0.87462,0.12538)
(0.29142,0.70858)
(0.79129,0.20871)
(0.45858,0.54142)
0 0.5 10
0.5
110 Cluster Plot
f1
f 2
(0.87688,0.12312)
(0,1)
(0.3445,0.6555)
(0.97598,0.024024)
(0.40738,0.59262)
(0.28125,0.71875)
(0.4695,0.5305)
(0.77515,0.22485)
(0.9268,0.073198)
(0.8262,0.1738)
Figure 34: Pareto front centroids of problem UF6for k=2,5,7, and 10 clusters
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3789-3809
© Research India Publications. http://www.ripublication.com
3800
0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
2 Cluster Plot
f1
f 2
(0.24975,0.75025)
(0.75025,0.24975)
0 0.5 10
0.5
15 Cluster Plot
f1
f 2
(0.3023,0.6977)
(0.1006,0.8994)
(0.503,0.497)
(0.9014,0.098599)
(0.7027,0.2973)
0 0.5 10
0.5
17 Cluster Plot
f1
f 2
(0.78729,0.21271)
(0.35886,0.64114)
(0.92943,0.070571)
(0.64515,0.35485)
(0.5025,0.4975)
(0.21522,0.78478)
(0.071572,0.92843)
0 0.5 10
0.5
110 Cluster Plot
f1
f 2
(0.14665,0.85335)
(0.54505,0.45495)
(0.048549,0.95145)
(0.64515,0.35485)
(0.74575,0.25425)
(0.24525,0.75475)
(0.84735,0.15265)
(0.94945,0.050551)
(0.34484,0.65516)
(0.44494,0.55506)
Figure 35: Pareto front centroids of problem UF7for k=2,5,7, and 10 clusters
0.20.3
0.40.5
0.60.7
0.2
0.3
0.4
0.5
0.6
0.70.2
0.4
0.6
0.8
1
f1
2 Cluster Plot
X: 0.5784
Y: 0.5784
Z: 0.3602
X: 0.2387
Y: 0.2387
Z: 0.8945
f2
0.10.2
0.30.4
0.50.6
0.70.8
0.9
0
0.2
0.4
0.6
0.8
10.2
0.4
0.6
0.8
1
X: 0.8576
Y: 0.3519
Z: 0.2636
X: 0.604
Y: 0.2433
Z: 0.726
f1
5 Cluster Plot
X: 0.1443
Y: 0.1446
Z: 0.9644
X: 0.2443
Y: 0.6066
Z: 0.7232
X: 0.3523
Y: 0.8585
Z: 0.2605
f2
00.2
0.40.6
0.81
0
0.2
0.4
0.6
0.8
10
0.2
0.4
0.6
0.8
1
X: 0.902
Y: 0.2243
Z: 0.294
f1
X: 0.5542
Y: 0.1878
Z: 0.7878
X: 0.5661
Y: 0.5613
Z: 0.5738
7 Cluster Plot
X: 0.6796
Y: 0.682
Z: 0.1839
X: 0.1276
Y: 0.1277
Z: 0.9724
X: 0.1891
Y: 0.5552
Z: 0.7866
f2
X: 0.2239
Y: 0.9019
Z: 0.2948
00.2
0.40.6
0.81
0
0.2
0.4
0.6
0.8
10
0.2
0.4
0.6
0.8
1
X: 0.7776
Y: 0.1924
Z: 0.5734
X: 0.9316
Y: 0.2423
Z: 0.1952
f1
X: 0.4647
Y: 0.1775
Z: 0.8533
X: 0.6811
Y: 0.6748
Z: 0.2084
X: 0.5651
Y: 0.5375
Z: 0.6022
10 Cluster Plot
X: 0.0974
Y: 0.07408
Z: 0.987
X: 0.1607
Y: 0.3176
Z: 0.9259
X: 0.1966
Y: 0.5774
Z: 0.7775
f2
X: 0.2461
Y: 0.9366
Z: 0.1705
Figure 36: Pareto front centroids of problem UF8for k=2,5,7, and 10 clusters
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3789-3809
© Research India Publications. http://www.ripublication.com
3801
0.10.15
0.20.25
0.30.35
0.40.45
0.1
0.2
0.3
0.4
0.50.2
0.4
0.6
0.8
1
f1
X: 0.3763
Y: 0.3763
Z: 0.2475
2 Cluster Plot
X: 0.1237
Y: 0.1237
Z: 0.7525
f2
00.1
0.20.3
0.40.5
0.60.7
0.8
0
0.2
0.4
0.6
0.80
0.2
0.4
0.6
0.8
1
X: 0.7268
Y: 0.1006
Z: 0.1725
X: 0.4115
Y: 0.06112
Z: 0.5274
f1
5 Cluster Plot
X: 0.07219
Y: 0.07219
Z: 0.8556X: 0.06112
Y: 0.4115
Z: 0.5274
f2
X: 0.1006
Y: 0.7268
Z: 0.1725
00.1
0.20.3
0.40.5
0.60.7
0.8
0
0.2
0.4
0.6
0.80
0.2
0.4
0.6
0.8
1
X: 0.7257
Y: 0.1005
Z: 0.1738
f1
X: 0.4085
Y: 0.06081
Z: 0.5307
7 Cluster Plot
X: 0.08094
Y: 0.03988
Z: 0.8792
X: 0.03604
Y: 0.2445
Z: 0.7195X: 0.0633
Y: 0.4258
Z: 0.5109
f2
X: 0.09162
Y: 0.6068
Z: 0.3016
X: 0.1044
Y: 0.7969
Z: 0.09867
00.2
0.40.6
0.81
0
0.2
0.4
0.6
0.80
0.2
0.4
0.6
0.8
1
X: 0.8359
Y: 0.09752
Z: 0.06656
X: 0.5687
Y: 0.08362
Z: 0.3477
f1
X: 0.6936
Y: 0.1103
Z: 0.1962
X: 0.307
Y: 0.04581
Z: 0.6472
X: 0.4379
Y: 0.06527
Z: 0.4969
10 Cluster Plot
X: 0.176
Y: 0.02583
Z: 0.7982
X: 0.02834
Y: 0.05923
Z: 0.9124
X: 0.04459
Y: 0.3004
Z: 0.655
f2
X: 0.07952
Y: 0.5314
Z: 0.3891X: 0.104
Y: 0.7689
Z: 0.1271
Figure 37: Pareto front centroids of problem UF9for k=2,5,7, and 10 clusters
0.20.3
0.40.5
0.60.7
0.2
0.3
0.4
0.5
0.6
0.70.2
0.4
0.6
0.8
1
f1
2 Cluster Plot
X: 0.5784
Y: 0.5784
Z: 0.3602
X: 0.2387
Y: 0.2387
Z: 0.8945
f2
0.10.2
0.30.4
0.50.6
0.70.8
0.9
0
0.2
0.4
0.6
0.8
10.2
0.4
0.6
0.8
1
f1
5 Cluster Plot
f2
00.2
0.40.6
0.81
0
0.2
0.4
0.6
0.8
10.2
0.4
0.6
0.8
1
X: 0.9233
Y: 0.2441
Z: 0.2166
f1
X: 0.713
Y: 0.2712
Z: 0.6144
X: 0.4184
Y: 0.2058
Z: 0.8686
7 Cluster Plot
X: 0.6441
Y: 0.68
Z: 0.2747
X: 0.1
Y: 0.132
Z: 0.9762
X: 0.2346
Y: 0.6015
Z: 0.7342
f2
X: 0.224
Y: 0.9098
Z: 0.2744
00.2
0.40.6
0.81
0
0.2
0.4
0.6
0.8
10.2
0.4
0.6
0.8
1
X: 0.9466
Y: 0.1775
Z: 0.21
f1
X: 0.7991
Y: 0.5278
Z: 0.2278
X: 0.438
Y: 0.1731
Z: 0.869
X: 0.5358
Y: 0.5358
Z: 0.6303
10 Cluster Plot
X: 0.09875
Y: 0.09875
Z: 0.9835
X: 0.5278
Y: 0.7991
Z: 0.2278
X: 0.1731
Y: 0.438
Z: 0.869
X: 0.192
Y: 0.7556
Z: 0.6031
f2
X: 0.1775
Y: 0.9466
Z: 0.21
Figure 38: Pareto front centroids of problem UF10for k=2,5,7, and 10 clusters
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3789-3809
© Research India Publications. http://www.ripublication.com
3802
0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
2 Cluster Plot
f1
f 2
(0.75,0.25)
(0.225,0.775)
0 0.5 10
0.5
15 Cluster Plot
f1
f 2
(0.65,0.35)
(0.425,0.575)
(0.225,0.775)
(0.9,0.1)
(0.05,0.95)
0 0.5 10
0.5
17 Cluster Plot
f1
f 2
(0.3,0.7)
(0.55,0.45)
(0.15,0.85)
(0.025,0.975)
(0.725,0.275)
(0.425,0.575)
(0.925,0.075)
0 0.5 10
0.5
110 Cluster Plot
f1
f 2
(1,0)
(0.75,0.25)
(0.325,0.675)
(0.2,0.8)
(0.575,0.425)
(0.05,0.95)
(0.45,0.55)
(0.825,0.175)
(0.675,0.325)
(0.925,0.075)
Figure 39: Pareto front centroids of problem CF1 for k=2,5,7, and 10 clusters
0 0.2 0.4 0.6 0.80
0.5
12 Cluster Plot
f1
f 2
(0.078125,0.80558)
(0.78125,0.11907)
0 0.5 10
0.5
15 Cluster Plot
f1
f 2
(0,1)
(0.77861,0.11793)
(0.15625,0.61115)
(0.9262,0.037864)
(0.63366,0.2044)
0 0.5 10
0.5
17 Cluster Plot
f1
f 2
(0,1)
(0.67056,0.18203)
(0.2243,0.52666)(0.17376,0.58351)
(0.88997,0.05723)
(0.12632,0.64508)(0.082831,0.71295)
0 0.5 10
0.5
110 Cluster Plot
f1
f 2
(0.15032,0.61301)
(0.21611,0.53561)
(0,1)
(0.73908,0.14039)
(0.96244,0.019022)
(0.66726,0.18324)
(0.81222,0.098846)
(0.090456,0.70049)
(0.59676,0.22761)
(0.88667,0.058438)
Figure 40: Pareto front centroids of problem CF2 for k=2,5,7, and 10 clusters
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3789-3809
© Research India Publications. http://www.ripublication.com
3803
0 0.2 0.4 0.6 0.80.2
0.4
0.6
0.8
1
2 Cluster Plot
f1
f 2
(0,1)
(0.76828,0.38005)
0 0.5 10
0.5
15 Cluster Plot
f1
f 2
(0.90012,0.18938)
(0.55334,0.69286)
(0,1)
(0.6572,0.56725)
(0.96731,0.063944)
0 0.5 10
0.5
17 Cluster Plot
f1
f 2
(0.91748,0.15814)(0.88318,0.2199)
(0,1)
(0.95097,0.095565)
(0.6572,0.56725)
(0.55334,0.69286)
(0.98386,0.031933)
0 0.5 10
0.5
110 Cluster Plot
f1
f 2(0.88862,0.21017)
(0.93382,0.12781)
(0.55521,0.69164)
(0.62539,0.60879)
(0.5184,0.73114)
(0.97821,0.042946)
(0,1)
(0.59077,0.65089)
(0.69151,0.52173)(0.65907,0.56553)
Figure 41: Pareto front centroids of problem CF3 for k=2,5,7, and 10 clusters
0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
2 Cluster Plot
f1
f 2
(0.73003,0.35677)
(0.22996,0.77004)
0 0.5 10.2
0.4
0.6
0.8
1
5 Cluster Plot
f1
f 2
(0.90562,0.21938)
(0.29158,0.70842)
(0.70353,0.39396)
(0.096693,0.90331)
(0.49257,0.51849)
0 0.5 10
0.5
17 Cluster Plot
f1
f 2
(0.3497,0.6503)
(0.79535,0.32872)
(0.64809,0.42595)
(0.49456,0.51335)
(0.93223,0.19277)
(0.069138,0.93086)
(0.20892,0.79108)
0 0.5 10
0.5
110 Cluster Plot
f1
f 2 (0.54229,0.47926)
(0.95331,0.17169)
(0.43988,0.56012)
(0.1503,0.8497)
(0.65412,0.42294)
(0.85843,0.26657)
(0.0501,0.9499)
(0.2485,0.7515)
(0.34519,0.65481)
(0.75994,0.36102)
Figure 42: Pareto front centroids of problem CF4 for k=2,5,7, and 10 clusters
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3789-3809
© Research India Publications. http://www.ripublication.com
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0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
2 Cluster Plot
f1
f 2
(0.73003,0.35677)
(0.22996,0.77004)
0 0.5 10.2
0.4
0.6
0.8
1
5 Cluster Plot
f1
f 2
(0.70353,0.39396)
(0.096693,0.90331)
(0.90562,0.21938)
(0.49257,0.51849)
(0.29158,0.70842)
0 0.5 10
0.5
17 Cluster Plot
f1
f 2
(0.3497,0.6503)
(0.49456,0.51335)
(0.64809,0.42595)
(0.93223,0.19277)
(0.20892,0.79108)
(0.069138,0.93086)
(0.79535,0.32872)
0 0.5 10
0.5
110 Cluster Plot
f1
f 2
(0.24148,0.75852)
(0.85392,0.27108)
(0.75249,0.36666)
(0.53334,0.48437)
(0.43136,0.56864)
(0.95181,0.17319)
(0.048096,0.9519)
(0.33667,0.66333)
(0.64458,0.42771)
(0.14529,0.85471)
Figure 43: Pareto front centroids of problem CF5 for k=2,5,7, and 10 clusters
0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.82 Cluster Plot
f1
f 2
(0.2009,0.65208)
(0.70107,0.16264)
0 0.5 10
0.5
15 Cluster Plot
f1
f 2
(0.073647,0.85996)
(0.86952,0.085109)
(0.22946,0.59596)
(0.61599,0.19205)
(0.4023,0.36001)
0 0.5 10
0.5
17 Cluster Plot
f1
f 2
(0.41082,0.34861)
(0.28257,0.51598)
(0.91365,0.069151)
(0.56339,0.21906)
(0.053106,0.89757)
(0.73852,0.13465)
(0.16383,0.70027)
0 0.5 10
0.5
110 Cluster Plot
f1
f 2
(0.039579,0.92294)
(0.60291,0.19854)(0.71716,0.14202)
(0.20741,0.62882)
(0.83233,0.10183)
(0.39028,0.37253)
(0.29659,0.49547)
(0.49208,0.26345)
(0.94578,0.054741)
(0.12174,0.77192)
Figure 44: Pareto front centroids of problem CF6 for k=2,5,7, and 10 clusters
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3789-3809
© Research India Publications. http://www.ripublication.com
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0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.82 Cluster Plot
f1
f 2
(0.2009,0.65208)
(0.70107,0.16264)
0 0.5 10
0.5
15 Cluster Plot
f1
f 2
(0.4023,0.36001)
(0.073647,0.85996)
(0.22946,0.59596)
(0.61599,0.19205)
(0.86952,0.085109)
0 0.5 10
0.5
17 Cluster Plot
f1
f 2
(0.73952,0.13425)
(0.16533,0.69778)
(0.053607,0.89664)
(0.28457,0.51311)
(0.56488,0.21819)
(0.41283,0.34625)
(0.91416,0.068949)
0 0.5 10
0.5
110 Cluster Plot
f1
f 2 (0.28958,0.50539)
(0.48463,0.26968)
(0.71218,0.14432)
(0.11723,0.77982)
(0.038076,0.92579)
(0.82932,0.10274)
(0.38327,0.38113)
(0.59588,0.20206)
(0.2009,0.63918)
(0.94478,0.055249)
Figure 45: Pareto front centroids of problem CF7 for k=2,5,7, and 10 clusters
0.40.45
0.50.55
0.60.65
0.70.75
0.8
0
0.2
0.4
0.6
0.80.494
0.496
0.498
0.5
0.502
0.504
X: 0.7923
Y: 3.019e-09
Z: 0.495
f1
2 Cluster Plot
f2
X: 0.4062
Y: 0.6018
Z: 0.5012
00.2
0.40.6
0.81
0
0.2
0.4
0.6
0.8
10.2
0.4
0.6
0.8
1
X: 0.8899
Y: 3.638e-09
Z: 0.3847
X: 0.7963
Y: 0.4598
Z: 0.3342
f1
5 Cluster Plot
X: 0.315
Y: 0.3175
Z: 0.8465
X: 0.5565
Y: 0.7268
Z: 0.3268
f2
X: 0
Y: 0.8905
Z: 0.3837
00.2
0.40.6
0.81
0
0.2
0.4
0.6
0.8
10.2
0.4
0.6
0.8
1
X: 0.9527
Y: 3.824e-09
Z: 0.2604
X: 0.6643
Y: 2.437e-09
Z: 0.7256
f1
X: 0.7587
Y: 0.5766
Z: 0.2368
7 Cluster Plot
X: 0.5243
Y: 0.5044
Z: 0.6559
X: 0.1807
Y: 0.308
Z: 0.9032X: 0.4757
Y: 0.824
Z: 0.2636
f2
X: 0
Y: 0.8986
Z: 0.3709
00.2
0.40.6
0.81
0
0.2
0.4
0.6
0.8
10
0.2
0.4
0.6
0.8
1
X: 0.9729
Y: 3.91e-09
Z: 0.1991
X: 0.8086
Y: 3.446e-09
Z: 0.5748
f1
X: 0.4982
Y: 0.1402
Z: 0.8362
X: 0.7685
Y: 0.5923
Z: 0.1774
X: 0.6782
Y: 0.4989
Z: 0.5157
10 Cluster Plot
X: 0.1614
Y: 0.2446
Z: 0.9388
X: 0.4775
Y: 0.827
Z: 0.2544
f2
X: 0
Y: 0.722
Z: 0.6736
X: 0
Y: 0.9608
Z: 0.2381
Figure 46: Pareto front centroids of problem CF8 for k=2,5,7, and 10 clusters
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3789-3809
© Research India Publications. http://www.ripublication.com
3806
0.10.2
0.30.4
0.50.6
0.70.8
0.4
0.42
0.44
0.46
0.48
0.50.2
0.4
0.6
0.8
1
f1
2 Cluster Plot
X: 0.7026
Y: 0.4964
Z: 0.3558
X: 0.1266
Y: 0.4236
Z: 0.8154
f2
00.2
0.40.6
0.81
0.2
0.4
0.6
0.8
10.2
0.4
0.6
0.8
1
X: 0.8818
Y: 0.3061
Z: 0.2855
f1
X: 0.5128
Y: 0.3597
Z: 0.7411
5 Cluster Plot
X: 0.5712
Y: 0.7429
Z: 0.2903
X: 0.05313
Y: 0.2518
Z: 0.947
f2
X: 0
Y: 0.7788
Z: 0.5836
00.2
0.40.6
0.81
0
0.2
0.4
0.6
0.8
10.2
0.4
0.6
0.8
1
X: 0.8946
Y: 0.3161
Z: 0.2466
f1
X: 0.6493
Y: 0.2278
Z: 0.6995
7 Cluster Plot
X: 0.585
Y: 0.7577
Z: 0.2387
X: 0.4373
Y: 0.5657
Z: 0.6786
X: 0.08263
Y: 0.1856
Z: 0.9643
f2
X: 0.001069
Y: 0.5657
Z: 0.8139
X: 0
Y: 0.8761
Z: 0.4514
00.2
0.40.6
0.81
0
0.2
0.4
0.6
0.8
10
0.2
0.4
0.6
0.8
1
X: 0.9136
Y: 0.3232
Z: 0.1796
X: 0.7852
Y: 0.2767
Z: 0.5302
f1
X: 0.517
Y: 0.1808
Z: 0.8226
10 Cluster Plot
X: 0.4675
Y: 0.6056
Z: 0.6351
X: 0.5595
Y: 0.7246
Z: 0.3874X: 0.6026
Y: 0.7805
Z: 0.1261
X: 0.3187
Y: 0.4123
Z: 0.8461
X: 0.0361
Y: 0.1733
Z: 0.975
f2
X: 0
Y: 0.5668
Z: 0.8133
X: 0
Y: 0.8764
Z: 0.4508
Figure 47: Pareto front centroids of problem CF9 for k=2,5,7, and 10 clusters
0.10.2
0.30.4
0.50.6
0.70.8
0.4
0.42
0.44
0.46
0.48
0.50.2
0.4
0.6
0.8
1
f1
2 Cluster Plot
X: 0.1266
Y: 0.4236
Z: 0.8154
f2
00.2
0.40.6
0.81
0.2
0.4
0.6
0.8
10.2
0.4
0.6
0.8
1
f1
X: 0.5128
Y: 0.3597
Z: 0.7411
5 Cluster Plot
X: 0.5712
Y: 0.7429
Z: 0.2903
X: 0.05313
Y: 0.2518
Z: 0.947
f2
X: 0
Y: 0.7788
Z: 0.5836
00.2
0.40.6
0.81
0
0.2
0.4
0.6
0.8
10.2
0.4
0.6
0.8
1
X: 0.8948
Y: 0.3164
Z: 0.2462
f1
X: 0.6502
Y: 0.2277
Z: 0.6987
7 Cluster Plot
X: 0.08528
Y: 0.1879
Z: 0.9632
X: 0.5859
Y: 0.7589
Z: 0.2342
X: 0.4418
Y: 0.572
Z: 0.6705
f2
X: 0.001298
Y: 0.5663
Z: 0.8135
X: 0
Y: 0.8762
Z: 0.4511
00.2
0.40.6
0.81
0
0.2
0.4
0.6
0.8
10
0.2
0.4
0.6
0.8
1
X: 0.9194
Y: 0.3256
Z: 0.1524
X: 0.8284
Y: 0.2918
Z: 0.4535
f1
X: 0.6407
Y: 0.2217
Z: 0.7216
X: 0.321
Y: 0.2171
Z: 0.91
10 Cluster Plot
X: 0.535
Y: 0.6929
Z: 0.4685
X: 0.4052
Y: 0.5271
Z: 0.7378
X: 0.5989
Y: 0.7757
Z: 0.1577
X: 0.01529
Y: 0.1777
Z: 0.9763
f2
X: 0
Y: 0.5702
Z: 0.811X: 0
Y: 0.8774
Z: 0.4492
Figure 48: Pareto front centroids of problem CF10 for k=2,5,7, and 10 clusters
Solving MOOPs using multi-objective EAs not gives a single
optimal solution (i.e., only one point), but it gives a set of
Pareto optimal solutions (nondominated solutions), in which
one objective value cannot be improved without sacrificing
other objectives values. But, for all practical applications we
need to select one solution or a set of limited number of
solutions; which should preserve the diversity of Pareto front.
This paper presents hybrid genetic K-means approach which
intends to not only make Pareto front tractable but to select
the most compromise set of solution depending on the DM
choice. The proposed methodology is tested by various
problems of MOO test instances: the Special Session and
Competition 2009 (CEC09) to demonstrate its ability to select
the most compromise set of solution. Results of simulation
experiments using several problems form CEC09 prove our
claim and show the ability of the proposed algorithm to solve
the clustering problem for the Pareto front. From the above
results the selected solution has a diversity characteristics also
cover the entire Pareto front domain and allows the DM to
attain a reasonable representation of the Pareto-optimal front.
Finally, the proposed algorithm has a diversity preserving
mechanism to overcome the cruise of huge number of Pareto
front.
CONCLUSION
Solving MOOPs using multi-objective EAs not gives a single
optimal solution, but a set of Pareto-optimal solutions, in
which one objective cannot be improved without sacrificing
other objectives. But, for practical applications we need to
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 6 (2018) pp. 3789-3809
© Research India Publications. http://www.ripublication.com
3807
select one solution or a set of limited number of solutions;
which preserve the diversity of Pareto front. In this paper, a
new enhanced evolutionary algorithm is presented, our
proposed algorithm has been enriched with modified k-means
cluster scheme, On the first hand, in phase I, K-means
clustering technique is implemented to partition the
population into a determined number of subpopulation with-
dynamic-sizes. In this way, different genetic algorithms (GAs)
operators can apply to each sub-population, instead of one
GAs operator applied to the whole population. On the other
hand, phase II applies K-means algorithm to make the
algorithms practical by allowing the decision maker to control
the precision of the Pareto-set approximation by choosing a
suitable number of the needed clusters. The proposed
methodology is examined by various problems of MOO test
instances: the Special Session and Competition 2009 (CEC09)
to demonstrate its ability to select the most compromise set of
solution. Results of simulation experiments using several
problems form CEC09 prove our claim and show the ability
of the proposed algorithm to solve the clustering problem for
the Pareto front.
The main conclusions of the research work presented in this
roject can be summarized as follows:
1. In order to improve the performance of the GAs,
hybridization of the GAs with k-means is presented.
This algorithm is made of a classical genetic
algorithm based on the ideas of co-evolution coupled
with a k-means as cluster technique in order to
improve the spread of the solutions found so far.
2. The numerical analysis shows that our hybridization
algorithm is very efficient for benchmark problems.
3. The proposed approach was capable to eliminate the
drawbacks of GAs
4. Allowing a decision maker to control the precision of
Pareto front by defining desired number of cluster k
values.
5. Empirical results show that our approach is very
efficient in detecting the comprise set of solution
from the Pareto set.
6. The proposed algorithm has been effectively applied
to deal with the MOP, with no limitation in solving
higher-dimensional problems.
7. The proposed approach was capable to find well
distributed Pareto-front.
8. The success of our approach on most of the test
problems not only provides confidence but also stress
the importance of hybrid evolutionary algorithms in
solving multiobjective optimization problems.
9. The reality of using the proposed approach to handle
complex problems of realistic dimensions has been
approved due to procedure simplicity.
Recommendations for Future Researches
Several possible areas for future work have arisen from this
research concerning cluster analysis. Those are summarized as
follows:
1. An implementation of the proposed algorithm for
solving real problems.
2. Performance of the proposed algorithm can also be
improved by further investigation of the various
GAs parameters.
3. Evolutionary Algorithms, such as GAs, are a very
important area of investigation and have personally
provided a great deal of interesting work and
stimulation in the complexity and diversity of
applications in the last few years has been
enormous.
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