Journal of Computer and Communications, 2014, 2, 117-126
Published Online March 2014 in SciRes. http://www.scirp.org/journal/jcc
http://dx.doi.org/10.4236/jcc.2014.24016
How to cite this paper: Findik, O., Kiran, M.S. and Babaoğlu, I. (2014) Investigation Effects of Selection Mechanisms for
Gravitational Search Algorithm. Journal of Computer and Communications, 2, 117-126.
http://dx.doi.org/10.4236/jcc.2014.24016
Investigation Effects of Selection Mechanisms for
Gravitational Search Algorithm
Oğuz Findik1, Mustafa Servet Kiran2, Ismail Babaoğlu2
1Computer Engineering Department, Abant Izzet Baysal University, Bolu, Turkey
2Computer Engineering Department, Selçuk University, Konya, Turkey
Email: oguzf@ibu.edu.tr, mskiran@selcuk.edu.tr, ibabaoglu@selcu k.edu.tr
Received Novemb er 2013
Abstract
The gravitational search algorithm (GSA) is a population-based heuristic optimization technique
and has been proposed for solving continuous optimization problems. The GSA tries to obtain op-
timum or near optimum solution for the optimization problems by using interaction in all agents
or masses in the population. This paper proposes and analyzes fitness-based proportional (rou-
lette-wheel), tournament, rank-based and random selection mechanisms for choosing agents
which they act masses in the GSA. The proposed methods are applied to solve 23 numerical
benchmark functions, and obtained results are compared with the basic GSA algorithm. Experi-
mental results show that the proposed methods are better than the basic GSA in terms of solution
quality.
Keywords
Gravitational Search Algorithm; Roulette-Wheel Selection; Tournament Selection; Rank-Based
Selection; Random Selection; Continuous Optimization
1. Introduction
Optimization process in computer science is to find the best solution from all feasible solutions, in which the best
solution maximizes the profit function or minimizes the cost function. Especially if optimization problems have high
dimensions or non-linear characteristics, to find optimal solution is so hard because search space of optimization
problem increases exponentially with dimension increasing. To overcome this situation, many optimization algo-
rithms especially inspired from nature have been suggested in recent years such as particle swarm optimization de-
veloped by inspiring bird flocking or fish schooling [1], ant colony algorithm which simulates behavior of real ants
between nest and food source [2], bee colony algorithms which were inspired intelligent behavior of honey bees
[3,4]. In addition to these algorithms, some algorithms which were inspired by various natural events were devel-
oped. Such as harmony search algorithm inspired by natural musical performance process when a musician seek for
a better condition of harmony [5], genetic algorithm based on natural evolution [6] and the GSA simulated Newton’s
law of gravity [7].
The GSA is a heuristic optimization technique which is inspired by Newton’s law of gravity [7]. In this algorithm,
the main rule is that each agent has features attract masses to extend each other. When all agents attract each other,
O. Findik et al.
118
the effect which agents have big masses to solution is decrease, so to find optimum solution is difficult and the con-
vergence of the method to the optimum or near optimum has slowed. Which masses will be attraction effect and
which masses will be disregard is a problem in the GSA. To overcome this problem, selection mechanisms in the
GSA are analyzed in this study.
The rest of paper is organized as follows: Section 2 presents a literature review for the GSA. The basic GSA and
the selection mechanism are explained in Section 3. The experiments are presented in Section 4. Section 5 discusses
the experimental results and the study and finally, conclusion and future works are given in Section 6.
2. Literature Review on GSA
GSA is heuristic optimization technique which is inspired by Newton’s law of gravity. Each agent is named as
object and success in optimization algorithm are related as masses in GSA [7]. Using the mass of each agent and
distance of between agents are calculated new position of agent. Mass and distance are inversely. Effect of big
and close agent is more the solution.
While new position of each agent is calculated, to avoid fast convergence is used random number in basic
GSA. Main problem of fast convergence is to get stuck of local minimum the method. To overcome fast con-
vergence and local minimum, Han and Chang [8] suggested using chaotic variables instead of random numbers
in modified GSA.
Unconstrained acceleration has caused more diversification in the population. Khajehzadeh et al. [9] proposed
velocity clamping for velocities of the agents in order to prevent diversification in the population. Therefore,
velocity of the agent is constrained between maximum and minimum values of the velocities.
GSA starts to search with random solutions on the search space. An opposition based learning method is pro-
posed for the initialization and also continuation working of the GSA by Shaw et al. [10]. In this way, the con-
vergence rate of GSA has improved and the robustness of GSA is increased.
In order to improve global search ability of GSA, “Disruption” operator, inspired from astrophysics, has been
added to basic GSA [11]. The new method is used for optimizing 23 numeric functions and obtained results are
compared with basic GSA, PSO and GA. The experimental results show that proposed method is superior to ba-
sic GSA, PSO and GA.
Li and Zhou [12] developed a new version of GSA by combining the search strategy of PSO and the im-
proved GSA (IGSA) is applied to parameters identification of hydraulic turbine governing systems. The experi-
mental results show that (IGSA) is capable with respect to PSO, GA and basic GSA in terms of solution quality.
Niknama et al. [13] suggested self-adaptive GSA to improve convergence characteristic in basic GSA. Two
methods developed to improve solution in this technique. New solution in GSA moves independently the pre-
vious solution because GSA has not memory. Finding best solution is used to find new solution in the first tech-
nique. Second technique was developed for which solution has not local minimum. New solution was produced
using three different agent are selected in second technique. Produced solution was used new solution with
probability.
In addition, GSA algorithm has been used in many research fields such as clustering [14,15], image enhance-
ment [16], classification [17], secure communication [18] and filter modeling [19].
3. Gravitational Search Algorithm
One of the newest heuristic optimizers is gravitational search algorithm (GSA) which is based on law of gravity,
law of motion and interaction of masses [7]. In GSA, potential solutions correspond to the position of masses
and masses corresponds the fitness value of the solution produced for the optimization problem. By using law of
gravity and law of motion, Rashedi et al. [7] have defined interaction between the masses. The GSA is an itera-
tive algorithm and the algorithm is explained step by step as follows:
Step 1. Initialization
The masses are randomly produced on the solution space using Equation (1).
( )
minmax min
,
1,2, 1,2,,
ij jjj
XXrXXiN and jD
=+×−= …= …
(1)
where
,ij
X
is the ith mass position value on the jth dimension,
max
j
X
and
min
j
X
is the upper and lower
bound for the jth dimension, respectively, N is the number of masses and D is the dimensionality of the optimi-
O. Findik et al.
119
zation problem.
After the random solution is produced, the sizes of the masses are calculated as follows:
()
()
worst
best worst
()
() ()
i
i
fit tfitt
mt fitt fitt
=
(2)
1
()
() ()
i
iN
j
j
mt
Mt mt
=
=
(3)
where
()
i
Mt
is the size of inertial mass on the ith position on iteration time t,
is the size of the mass
which is on the ith position on iteration time t,
( )
i
fit t
is the fitness value of the mass on the ith position on
iteration time t,
( )
best
fit t
and
( )
worst
fit t
are the best and worst fitness values in the mass population on itera-
tion time t.
Step 2. Law of Gravity
In GSA, interaction between the masses is based on action-reaction. Force acting on a mass is calculated as
follows:
For each mass i,
For each dimension d,
( )( )( )( )
( )( )( )
( )
, ,,
,
1, 2,,
ij
d
i jidjd
ij
Mt Mt
F tGtXtXt
Rt
jN
ε
×
=× ×−
+
= …
(4)
where
()
,
d
ij
Ft
is the force acting on dth dimension of masses on ith position on iteration time t, () is the
gravitational constant number on iteration time t,
( )
i
Mt
and
( )
j
Mt
is the active and passive gravitational
masses on iteration time t, respectively,
( )
,id
Xt
is the dth dimension of masses on ith position,
( )
,jd
Xt
is the
dth dimension of masses on jth position at iteration time t,
( )
,ij
Rt
is the Euclidian distance between the mass
on ith position and the mass on jth position and is the small constant number.
After the force for each dimension is calculated using Equation (4), the total force on the mass is obtained as
follows:
,
1
() ()
N
dd
ij ij
j
Ftr Ft
=
= ×
(5 )
where () is the total force acting on dth dimension of the mass on ith position on iteration time t and
is a
random number produced in range of [0,1], which is used for providing stochastic characteristic to GSA.
Step 3. Law of Motion
The motion depends on acceleration of the mass and the acceleration of the mass is calculated as follows:
()
() ()
d
di
ii
Ft
at Mt
=
(6)
where
()
d
i
at
is the acceleration on the dth dimension of the mass ith position. The velocity and new position of
the mass are calculated as follows:
()( )
,
1 ()
d dd
iid ii
vtrvt at
+= ×+
(7)
()( )
1( 1)
d dd
i ii
XtXt vt+=+ +
(8)
where
( )
1
d
i
vt+
is the velocity for dth dimension of the mass on the ith position on iteration time
1t+
,
( )
1
d
i
Xt+
is the dth dimension of the mass on ith position at iteration time
1t+
and
,id
r
is the random num-
ber produced for dth dimension of the mass ith position in range of [0,1].
Step 4. Termination
After new positions for the masses are obtained by using Equations (6)-(8), the fitness of the solutions are
O. Findik et al.
120
calculated using Equations (2) and (3). The obtained solution with best fitness value is stored and if a termina-
tion condition is met, the algorithm is terminated and the best solution is reported. Otherwise, running of the al-
gorithm is continued from the Step 2. Basic GSA is presented as a flowchart in Figure 1.
In the basic GSA, all agents in the population are used for calculating force acted on a mass. In order to in-
crease convergence and local search capabilities of the method, which agents will be used are decided for de-
termining acting force on a mass by using four selection mechanisms—roulette wheel, tournament, random and
rank-based selection mechanisms and these mechanisms are given below.
3.1. Random Selection
In order to provide enough diversification in the population, a certain number of particles are selected from the
population. Instead of whole population, the selected particles are used for calculating force acting on the mass.
In this selection mechanism, each agent has same selection probability and whole solution space is searched by
using this mechanism but the local search on the solution space is reduced.
3.2. Roulette Wheel Selection
This selection mechanism is based on the fitness of the solutions. For calculating force acting on the mass, a
certain number of particles are selected from the population using roulette wheel selection. Being selected
probability of a particle is given as follows:
1
i
iN
j
j
fit
pfit
=
=
(9 )
where
i
p
is the being selected probability of the ith particle,
i
fit
is the fitness value of the solution of ith par-
ticle and N is the number of particles. Being used roulette-wheel selection, it is aimed that the convergence rate
of the method is increased because the agent has been affected mostly good solutions obtained previous itera-
tion.
3.3. Tournament Selection
Tournament selection covers running several tourneys between the particles randomly selected from the popula-
tion. The winner of each tourney (the agent with the better fitness) is used for calculating force acting on the
Figure 1. The flow chart of the GSA.
O. Findik et al.
121
mass. The tournament size is important for the tournament selection mechanism because if the tournament is
larger, weak particles have a smaller chance to be selected. In the tournament selection, when the number of
tournament is increased, the agent has been affected mostly the best solution, and the solutions of the population
are quickly improved especially for unimodal functions. When the number of tournament is decreased, the solu-
tion with low fitness can be selected and the diversity in the population can increase and global search ability of
the method is improved.
3.4. Rank-Based Selection
Rank-based selection is an alternative selection mechanism used for obtaining chromosomes which will be sub-
jected to crossover operation in genetic algorithm [20]. In the rank-based selection, the agents are sorted in as-
cending order by using their fitness values. For each rank, a selection probability is calculated as follows:
( 1)
22 (1)( 1)
pos
Pos
pSP SPN
=− +×−×
(10)
where,
pos
p
is agent’s probability of being selected in position pos, N is the number of agents and SP is the se-
lective pressure. In the positions, position of least fit agent is first order 1 and the position of fittest agent is Nth
order.
4. Experimental Results
In order to investigate effects of the selection mechanisms to the performance of the GSA, 23 benchmark func-
tions taken from [7] are used and obtained results are compared the results of basic GSA.
4.1. Benchmark Functions
23 test functions given in Tables 1-3 and taken from [7] are divided to different groups. F1-F7 functions have
only one local minimum and this local minimum is global minimum. These functions are unimodal functions,
and used for investigating local search ability of the method. If a function has more than one local minimum,
this function is called as multimodal function and the global search capability of the method is tested on these
functions (F8-F13). Another difficulty for a method is dimensionality of the optimization problem [21,22].
While F14-F23 test functions are small-sized functions, the dimensionality for F1-F13 functions is taken as 30.
4.2. Control Parameters
The population size of the methods is taken as 50 in all experiments. The stopping criterion for the algorithms is
Table 1. The unimodal benchmark functions.
F.No D Range Function
1 30 [100,100]
2
11
()
D
i
i
Fx x
=
=
2 30 [10,10]
211
() DD
ii
ii
Fx xx
= =
= +
∑∏
3 30 [100,100]
2
311
()
Di
j
ij
Fx x
= =

=

∑∑
4 30 [100,100]
4( )max{,1
i
i
Fxxi D= ≤≤
5 30 [30,30]
( )
122
51
1
( )100(1)
D
ii i
i
Fxxxx
+
=

=− +−

6 30 [100,100]
2
61
( )([0.5])
D
i
i
Fx x
=
= +
7 30 [1.28,1.28]
4
71
( )[0,1)
D
i
i
F xixrandom
=
= +
O. Findik et al.
122
Table 2. The multimodal benchmark functions.
F.No
D Range Function
8 30
[500,500]
( )
81
( )sin
D
ii
i
Fxxx
=
= −
9 30
[5.12,5.12]
( )
2
91
()10cos 210
D
ii
i
Fx xx
π
=
=−+


10 30
[32,32]
()
2
10 11
11
()20exp0.2expcos 220
DD
ii
ii
Fxxx e
nn
π
= =


=− −−++





∑∑
11 30
[600,600]
2
11 11
1
( )cos1
4000
DD i
i
ii
x
Fx xi
= =

=−+


∑∏
12 30
[50,50]
()()()()
122
2
12 11
11
( )10sin1110sin1(,5,100,4)
DD
iiD i
ii
F xyyyyux
D
πππ
+
= =

=+−++− +




∑∑
( )
( )
( )
,,, 0
m
ii
ii
m
ii
kxa xa
u x akmaxa
kxax a
−>
=−< <
−−<−
13 30
[50,50]
()()()()()
22
22 2
131 1
1
()0.1sin311 sin3111 sin21
( ,5,100,4)
D
i iDD
i
D
i
i
Fxx xxxx
ux
ππ π
=
=

=+−+++−+++
 

 

+
Table 3. The multimodal benchmark functions with fix dimensions.
F.No D
Range Function
14 2 [65.53,65.53]
1
25
2
14 11
11
() 5000 (
ji ij
i
Fx j xa
==


= +

+−

15 4 [5,5]
( )
2
2
11 12
15 2
134
()
ii
i
iii
x bbx
Fxa b bx x
=

+
= −

++


16 2 [5,5]
24624
161111 222
1
( )42.144
3
F xxxxxxxx=−+ +−+
17 2 [5,10] × [0,15]
2
2
172111
2
5.1 51
()6101cos()10
48
Fxxxxx
ππ π
 
=−+ −+−+
 
 
18 2 [5,5]
( )
( )
( )
( )
222
18121121 22
222
1 2112122
( )11191431463
302318 3212483627
F xxxxxxxxx
x xxxxxxx

=+ ++−+−++


× +−×−++−+

19 3 [0,1]
( )
43 2
19 11
( )exp
iijjij
ij
Fxcaxp
= =

=− −−


∑∑
20 6 [0,1]
( )
46 2
20 11
( )exp
iijjij
ij
Fxcax p
= =

=− −−


∑∑
21 4 [0,10]
( )( )
51
21 1
1
()
T
ii
i
Fxxa xac
=

=−− −+

22 4 [0,10]
( )()()
71
22 1
T
i ii
i
Fxxaxac
=

=−−−+

23 4 [0,10]
( )( )
10 1
23 1
1
()
T
ii
i
Fxxa xac
=

=−− −+

O. Findik et al.
123
maximum iteration number (MIN), and MIN is 1000 for F1-F13 functions and 500 for F14-F23 functions. The
gravitation constant (G) used in GSA depends on iteration time and is calculated as follows [7]:
( )
0
t
T
GtG e
α
= ×
(11 )
where
( )
Gt
is the gravitation constant at time step t,
0
G
is the initial gravitation constant which is taken as
100 in the initialization of the algorithm, is the maximum iteration number and is scaling factor and it is
set to 20.
Number of masses which will be act force is taken as 10, 15, 20, 25, 30, 35 and 40 in GSAF (GSA with rou-
lette wheel selection), GSAT (GSA with tournament selection), GSAR (GSA with random selection), GSAL
(GSA with rank-based selection). Experimental studies show that the more successful results are obtained when
the number of masses which will be act is taken as 40. Therefore, the number of masses which will be act is
taken as 40 in the comparisons. In GSA with rank-based selection mechanism, the linear ranking is used and the
selective pressure is taken as 2.
4.3. Compar isons
The mean results obtained by GSAF, GSAT and GSAR, GSAL are compared the mean results of basic GSA and
these results are presented in Table 4 for unimodal test functions, Table 5 for multimodal functions and Table 6
for the multimodal test functions with fix dimension.
According to Table 4, the results obtained by GSAR and GSAF methods are relatively better than the results
of basic GSA because these functions are unimodal functions and the local search ability of the GSAF and
GSAR is better than GSA.
Based on Table 5, due to the fact that the masses in basic GSA have been affected by all the masses in the
population, the diversity in the population is kept during the iterations and the results obtained by basic GSA are
slightly better than the results of proposed methods.
Table 4. The comparison of the methods on the unimodal test functions.
Func. Methods
GSA GSAR GSAF GSAT GSAL
F1 7.3E11 3.4E16 3.45E16 3.48E16 4.25E16
F2 4.03E5 8.18E08 8.06E08 8.39E08 8.8E08
F3 0.16E+3 111.9831 109.9879 114.1624 17.6635
F4 3.7E6 7.93E09 7.93E09 0.029325 8.86E09
F5 25.16 26.07533 29.79962 30.11498 25.4512
F6 8.3E11 0 0 0 0
F7 0.018 0.035454 0.031632 0.030072 0.032517
Table 5. The comparison of the methods on multimodal test functions.
Func. Methods
GSA GSAR GSAF GSAT GSAL
F8 2.8E+3 2778.19 2854.1 2786.07 2949.51
F9 15.32 21.45794 22.38657 25.66993 26.66489
F10 6.9E6 1.42E08 1.42E08 1.43E08 1.56E08
F11 0.29 1.469172 1.67303 1.3961 0.649063
F12 0.01 0.030507 0.058855 0.044521 2.56E18
F13 3.2E32 3.98E17 3.57E17 3.64E17 4.12E17
O. Findik et al.
124
Table 6. The comparison of the methods on the multimodal test functions with fix dimension.
Func. Methods
GSA GSAR GSAF GSAT GSAL
F14 3.70 5.307424 1.996672 56.55266 3.3022
F15 8.0E3 0.004322 0.004202 0.00765 0.0020
F16 1.03163 1.03163 1.03163 1.02012 1.03163
F17 0.3979 0.39789 0.3979 0.416527 0.3979
F18 3 3 3 3.243807 3
F19 3.7357 3.8603 3.8628 3.82561 3.8628
F20 2.0569 3.322 3.31792 2.86047 3.3220
F21 6.0748 6.15111 5.39355 2.49662 6.4399
F22 9.3389 10.4029 8.92953 2.05354 9.9150
F23 9.45 10.5364 9.33603 2.11848 10.5364
According to Table 6, the dimensionality is the important factor for the methods and the proposed methods
are better than the basic GSA, except GSAT. In the GSAT, masses have been affected from the same mass and
the diversity in the population has been lost and this has caused the stagnation of the population. This situation is
balanced in other selection mechanisms by using fitness values or randomness.
5. Results and Discussion
In this study, we used four selection mechanisms- roulette wheel, tournament, random and rank-based selections
and obtained the better results than GSA. Experimental results show that the selection mechanisms directly af-
fect the performance of GSA because to obtain a new position for the agent in GSA is important for the perfor-
mance of GSA. For the population-based heuristic approaches, information sharing and interaction between the
agents describe behavior of the method and to be selected the agents using their fitness values provides to obtain
high quality solutions for the numerical benchmark functions. Experimental results show that fitness-based se-
lection mechanisms such as roulette-wheel and tournament is appropriate for the unimodal and multimodal
functions with fix dimension, but for multimodal functions with huge local minimums these mechanisms have
caused early saturation of the population, to lost diversification in the population and to get stuck of local mini-
mums. Therefore, the random, rank-based selection mechanisms are more appropriate for solving the multimod-
al functions instead of fitness-based selection mechanisms. But it should be mentioned that the random selection
mechanism can cause the slow convergence to the optimum or optimums and reduce the local search ability of
the method. The effect of the selection mechanism used for generating new chromosomes is known on the Ge-
netic algorithm in the literature but in this study, the effect of selection mechanism for the GSA has been inves-
tigated and obtained results are used for comparing and discussion. In GSA, the selection techniques do not have
the same effect with the GA because velocity updating equation of GSA does not work likewise crossover of
GA. The obtained results show that the proposed techniques for GSA have positive effect on solving the numer-
ical function. Consequently, it is shown that the solution quality is improved by using selection mechanism in
original GSA and, the suitable selection mechanism for GSA should be used in order to obtain more quality so-
lution depending on structure of the optimization problem.
6. Conclusion and Future Works
We analyzed four selection mechanisms for the GSA on the 23 benchmark functions based on solution quality.
The experimental results show that using the appropriate selection mechanism provides to obtain the quality so-
lutions. Because GSA with the selection mechanism has high performance on the continuous optimization prob-
lems, our future works include applications of proposed method in various optimization problems.
O. Findik et al.
125
Acknowledgements
This study has been supported by Scientific Research Project of Selçuk University.
References
[1] Kennedy, J. and Eberhart, R.C. (1995) Particle Swarm Optimization. Proceedings of International Conference on Neu-
ral Networks, 4, 1942-1948. http://dx.doi.org/10.1109/ICNN.1995.488968
[2] Dorigo, M., Maniezzo, V. and Colorni, A. (1996) The Ant System: Optimization by a Colony of Cooperating Agents.
IEEE Transactions on Systems, Man, and CyberneticsPart B, 26, 1-13.
[3] Pham, D.T., Ghanbarzadeh, A., Koc, E., Otri, S., Rahim, S. and Zaidi, M. (2006) The Bees Algorithm: A Novel Tool
for Complex Optimisation Problems. Proceedings of Intelligent Production Machines and Systems (IPROMS) Confer-
ence, 454-459.
[4] Karaboga, D. and Basturk, B. (2007) A Powerful and Efficient Algorithm for Numerical Function Optimization: Ar-
tificial Bee Colony (ABC) Algorithm. Journal of Global Optimization, 39, 459-171.
http://dx.doi.org/10.1007/s10898-007-9149-x
[5] Geem, Z.W., Kim, J.H. and Loganathan, G.V. (2001) A New Heuristic Optimization Algorithm: Harmony Search. Si-
mulation, 76, 60-68. http://dx.doi.org/10.1177/003754970107600201
[6] Holland, J.H. (1975) Adaptation in Natural and Artificial Systems. The University of Michigan Press, Ann Arbor.
[7] Rashedi, E., Nezamabadi-Pour, H. and Saryazdi, S. (2009) GSA: A Gravitational Search Algorithm. Information Sci-
ences, 179, 2232-2248. http://dx.doi.org/10.1016/j.ins.2009.03.004
[8] Han, X. and Chang, X. (2012) A Chaotic Digital Secure Communication Based on a Modified Gravitational Search
Algorithm Filter. Information Sciences, 208, 14-27. http://dx.doi.org/10.1016/j.ins.2012.04.039
[9] Khajehzadeh, M. , Taha, M. R., El-Shafie, A. and Eslami, M. (2012) A Modified Gravitational Search Algorithm for
Slope Stability Analysis. Engineering Applications of Artificial Intelligence, 25, 1589-1597.
http://dx.doi.org/10.1016/j.engappai.2012.01.011
[10] Shaw, B., Mukherjee, V. and Ghoshal, S.P. (2012) A Novel Opposition-Based Gravitational Search Algorithm for
Combined Economic and Emission Dispatch Problems of Power Systems. Electrical Power and Energy Systems, 35,
21-33. http://dx.doi.org/10.1016/j.ijepes.2011.08.012
[11] Sarafrazi, S., Nezamabadi-Pour, H. and Saryazdi, S. (2011) Disruption: A New Operator in Gravitational Search Algo-
rithm. Scientia Iranica, 18, 539-548. http://dx.doi.org/10.1016/j.scient.2011.04.003
[12] Li, C. and Zhou, J. (2011) Parameters Identification of Hydraulic Turbine Governing System Using Improved Gravita-
tional Search Algorithm. Energy Conversion and Management, 52, 374-381.
http://dx.doi.org/10.1016/j.enconman.2010.07.012
[13] Niknama, T., Golestaneh, F. and Malekpour, A. (2012) Probabilistic Energy and Operation Management of a Micro-
grid Containing Wind/Photovoltaic/Fuel Cell Generation and Energy Storage Devices Based on Point Estimate Method
and Self-Adaptive Gravitational Search Algorithm. Energy, 43, 427-437.
http://dx.doi.org/10.1016/j.energy.2012.03.064
[14] Yin, M., Hu, Y., Yang, F., Li, X. and Gu, W. (2011) A Novel Hybrid K-Harmonic Means and Gravitational Search
Algorithm Approach for Clustering. Expert Systems with Applications, 38, 9319-9324.
http://dx.doi.org/10.1016/j.eswa.2011.01.018
[15] Hatamloua, A., Abdullah, S. and Nezamabadi-pour, H. (2012) A Combined Approach for Clustering Based on K-
Means and Gravitational Search Algorithms. Swarm and Evolutionary Computation, 6, 47-52.
http://dx.doi.org/10.1016/j.swevo.2012.02.003
[16] Zhao, W. (2011) Adaptive Image Enhancement Based on Gravitational Search Algorithm. Procedia Engineering, 15,
3288-3292. http://dx.doi.org/10.1016/j.proeng.2011.08.617
[17] Bahrololoum, A., Nezamabadi-pour, H., Bahrololoum, H. and Saeed, M. (2012) A Prototype Classifier Based on Gra-
vitational Search Algorithm. Applied Soft Computing, 12, 819-825. http://dx.doi.org/10.1016/j.asoc.2011.10.008
[18] Han, X. and Chang, X. (2012) Chaotic Secure Communication Based on a Gravitational Search Algorithm Filter. En-
gineering Applications of Artificial Intelligence, 25, 766-774. http://dx.doi.org/10.1016/j.engappai.2012.01.014
[19] Rashedi, E., Nezama badi-pour, H. and Saryazdi, S. (2011) Filter Modeling Using Gravitational Search Algorithm. En-
gineering Applications of Artificial Intelligence, 24, 117-122. http://dx.doi.org/10.1016/j.engappai.2010.05.007
[20] Pohlheimi, H. (2006) GEATbx: Genetic and Evolutionary Algorithm Toolbox for Use with MATLAB Documentation.
http://www.geatbx.com/docu/index.html
O. Findik et al.
126
[21] Boyer, D.O., Martinez, C.H. and Pedrajas, N.G. (2005) Crossover Operator for Evolutionary Algorithms Based on
Population Features. Journal of Artificial Intelligence Research, 24, 1-48.
[22] Karaboga, D. and Akay, B. (2009) A Comparative Study of Artificial Bee Colony Algorithm. Applied Mathematics
and Computation, 214, 108-132. http://dx.doi.org/10.1016/j.amc.2009.03.090