A MODIFIED VORTEX SEARCH ALGORITHM FOR NUMERICAL FUNCTION OPTIMIZATION

International Journal of Artificial Intelligence and Applications (IJAIA), Vol. 7, No. 3, May 2016
DOI: 10.5121/ijaia.2016.7304 37
A MODIFIED VORTEX SEARCH
ALGORITHM FOR NUMERICAL FUNCTION
OPTIMIZATION
Berat Doğan
Department of Biomedical Engineering, Inonu University, Malatya, Turkey
ABSTRACT
The Vortex Search (VS) algorithm is one of the recently proposed metaheuristic algorithms which was
inspired from the vortical flow of the stirred fluids. Although the VS algorithm is shown to be a good
candidate for the solution of certain optimization problems, it also has some drawbacks. In the VS
algorithm, candidate solutions are generated around the current best solution by using a Gaussian
distribution at each iteration pass. This provides simplicity to the algorithm but it also leads to some
problems along. Especially, for the functions those have a number of local minimum points, to select a
single point to generate candidate solutions leads the algorithm to being trapped into a local minimum
point. Due to the adaptive step-size adjustment scheme used in the VS algorithm, the locality of the created
candidate solutions is increased at each iteration pass. Therefore, if the algorithm cannot escape a local
point as quickly as possible, it becomes much more difficult for the algorithm to escape from that point in
the latter iterations. In this study, a modified Vortex Search algorithm (MVS) is proposed to overcome
above mentioned drawback of the existing VS algorithm. In the MVS algorithm, the candidate solutions
are generated around a number of points at each iteration pass. Computational results showed that with
the help of this modification the global search ability of the existing VS algorithm is improved and the
MVS algorithm outperformed the existing VS algorithm, PSO2011 and ABC algorithms for the benchmark
numerical function set.
KEYWORDS
Metaheuristics, Numerical Function Optimization, Vortex Search Algorithm, Modified Vortex Search
Algorithm.
1. INTRODUCTION
In the past two decades, a number of metaheuristic algorithms have been proposed to solve
complex real-world optimization problems. Most of these algorithms are nature inspired methods
and therefore mimic natural metaphors such as, evolution of species (GA [1] and DE [2-3]),
annealing process (SA [4-5]), ant behaviour (ACO [6]), swarm behaviour (PSO [7] and ABC [8-
9]) etc. These algorithms make few or no assumptions for the problem at hand and provide fast
and robust solutions. Although, the solutions provided by metaheuristics may not be optimal
solutions, they are highly preferred because of their simplicity and flexibility.
Despite the high number of available metaheuristics, developing new metaheuristic algorithms is
still an active research area. In [10-15], a number of recently proposed metaheuristics can be
found. All of these metaheuristics have certain characteristics and thus each one may be more
successful on a certain optimization problem when compared to the others. The Vortex Search
(VS) algorithm [16] is one of these recently proposed metaheuristic algorithms which was
inspired from the vortical flow of the stirred fluids. The search behaviour of the VS algorithm is
modelled as a vortex pattern by using an adaptive step-size adjustment scheme. By this way, it is
aimed to have a good balance between the explorative and exploitative behaviour of the search.

38
The proposed VS algorithm was tested over 50 benchmark mathematical functions and the
obtained results compared to the single-solution based (Simulated Annealing, SA and Pattern
Search, PS) and population-based (Particle Swarm Optimization, PSO2011 and Artificial Bee
Colony, ABC) algorithms. A Wilcoxon-Signed Rank Test was performed to measure the pair-
wise statistical performances of the algorithms, the results of which indicated that the proposed
VS algorithm outperforms the SA, PS and ABC algorithms while being competitive with the
PSO2011 algorithm. Because of the simplicity of the proposed VS algorithm, a significant
decrease in the computational time of the 50 benchmark numerical functions was also achieved
when compared to the population-based algorithms. In some other studies [17-20], the VS
algorithm has also been successfully used for the solution of some real-world optimization
problems.
Although the proposed VS algorithm is a good candidate for the solution of optimization
problems, it also has some drawbacks. In the VS algorithm, candidate solutions are generated
around the current best solution by using a Gaussian distribution at each iteration pass. This
provides simplicity to the algorithm but it also leads to some problems along. Especially, for the
functions those have a number of local minimum points, to select a single point to generate
candidate solutions leads the algorithm to being trapped into a local minimum point. Due to the
adaptive step-size adjustment scheme used in the VS algorithm, the locality of the created
candidate solutions is increased at each iteration pass. Therefore, if the algorithm cannot escape a
local point as quickly as possible, it becomes much more difficult for the algorithm to escape
from that point in the latter iterations.
In this study, a modified Vortex Search algorithm (MVS) is proposed to overcome above
mentioned drawback of the existing VS algorithm. In the MVS algorithm, the candidate solutions
are generated around different points at each iteration pass. These points are iteratively updated
during the search process, details of which are given in the following section. The MVS
algorithm is again tested by using the 50 benchmark mathematical functions that was used earlier
in [16]. Because the SA and PS algorithms showed poor performances in [16], in this study these
two algorithms are excluded and the results are compared to the results those obtained by the VS
algorithm, PSO2011 and ABC algorithms. It is shown that, the MVS algorithm outperforms all
of these algorithms and can successfully find the known global minimum points of the functions
that the VS algorithm being trapped into the local minimum points earlier.
The remaining part of this paper is organized as follows. In the following section, first a brief
description of the VS algorithm is given. Then, the modification performed on the VS algorithm
is detailed and the MVS algorithm is introduced. Section 3 covers the experimental results and
discussion. Finally, Section 4 concludes the work.
2. METHODOLOGY
2.1. A Brief Description of the Vortex Search Algorithm
Let us consider a two-dimensional optimization problem. In a two dimensional space a vortex
pattern can be modelled by a number of nested circles. Here, the outer (largest) circle of the
vortex is first centered on the search space, where the initial center can be calculated using Eq. 1.
2
lowerlimitupperlimit
µ0
+
=
(1)
In Eq.1, upperlimit and lowerlimit are 1×d vectors that define the bound constraints of the
problem in d dimensional space. Then, a number of neighbor solutions )(sCt , (t represents

39
the iteration index and initially 0=t ) are randomly generated around the initial center 0µ in the
d -dimensional space by using a Gaussian distribution. Here, { } nkssssC k ,...,2,1,...,,)( 210 ==
represents the solutions, and n represents the total number of candidate solutions. In Eq. 2, the
general form of the multivariate Gaussian distribution is given.






−Σ−−
Σ
=Σ −
)()(
2
1
exp
)2(
1
),|( 1
µµ
π
µ xxxp T
d
(2)
In Eq.2, d represents the dimension, x is the 1×d vector of a random variable, µ is the 1×d
vector of sample mean (center) and Σ is the covariance matrix. If the diagonal elements
(variances) of the values of Σ are equal and if the off-diagonal elements (covariance) are zero
(uncorrelated), then the resulting shape of the distribution will be spherical (which can be
considered circular for a two-dimensional problem, as in our case). Thus, the value of Σ can be
computed by using equal variances with zero covariance by using Eq. 3.
[ ] ddI ×⋅=Σ 2
σ (3)
In Eq. 3,
2
σ represents the variance of the distribution and I represents the dd × identity
matrix. The initial standard deviation ( 0σ ) of the distribution can be calculated by using Eq. 4.
2
)min()max(
0
lowerlimitupperlimit −
=σ
(4)
Here, 0σ can also be considered as the initial radius ( 0r ) of the outer circle for a two
dimensional optimization problem. Because a weak locality is required in the initial phases, 0r is
chosen to be a large value. Thus, a full coverage of the search space by the outer circle is
provided in the initial step. This process provides a bird's-eye view for the problem at hand.
In the selection phase, a solution (which is the best one) )(0
'
sCs ∈ is selected and memorized
from )(0 sC to replace the current circle center 0µ . Prior to the selection phase, the candidate
solutions must be ensured to be inside the search boundaries. For this purpose, the solutions that
exceed the boundaries are shifted into the boundaries, as in Eq. 5.






>+−⋅
≤≤
<+−⋅
=
ii
k
iii
ii
k
ii
k
ii
k
iii
i
k
upperlimits,lowerlimit)lowerlimitt(upperlimirand
upperlimitslowerlimit,s
lowerlimits,lowerlimit)lowerlimitt(upperlimirand
s
(5)
In Eq.5, nk ,...2,1= and di ,...,2,1= and rand is a uniformly distributed random number. Next,
the memorized best solution
'
s is assigned to be the center of the second circle (the inner one).
In the generation phase of the second step, the effective radius ( 1r ) of this new circle is reduced,
and then, a new set of solutions )(1 sC is generated around the new center. Note that in the
second step, the locality of the generated neighbors increased with the decreased radius. In the
selection phase of the second step, the new set of solutions )(1 sC is evaluated to select a solution
)(1
'
sCs ∈ . If the selected solution is better than the best solution found so far, then this solution
is assigned to be the new best solution and it is memorized. Next, the center of the third circle is

40
assigned to be the memorized best solution found so far. This process iterates until the
termination condition is met. An illustrative sketch of the process is given in Figure 1. In this
manner, once the algorithm is terminated, the resulting pattern appears as a vortex-like structure,
where the center of the smallest circle is the optimum point found by the algorithm. A
representative pattern is sketched in Figure 2 for a two-dimensional optimization problem for
which the upper and lower limits are between the [-10,10] interval. A description of the VS
algorithm is also provided in Figure 3.
The radius decrement process given in Figure 3 can be considered as a type of adaptive step-size
adjustment process which has critical importance on the performance of the VS algorithm. This
process should be performed in such a way that allows the algorithm to behave in an explorative
manner in the initial steps and in an exploitative manner in the latter steps. To achieve this type
of process, the value of the radius must be tuned properly during the search process. In the VS
algorithm, the inverse incomplete gamma function is used to decrease the value of the radius
during each iteration pass.
Figure 1. An illustrative sketch of the search process
-10 -5 0 5 10
-10
-5
0
5
10
Figure 2. A representative pattern showing the search boundaries (circles) of the VS algorithm after a
search process, which has a vortex-like structure.
The incomplete gamma function given in Eq. 6 most commonly arises in probability theory,
particularly in those applications involving the chi-square distribution [21].
∫ >= −−
x
at
adtteax
0
1
0),(γ (6)

41
In Eq.6, 0>a is known as the shape parameter and 0≥x is a random variable. In conjunction
with the incomplete gamma function, its complementary ),( axΓ is usually also introduced (Eq.
7).
∫
∞
−−
>=Γ
x
at
adtteax 0),( 1
(7)
Thus, it follows that,
)(),(),( aaxax Γ=Γ+γ (8)
where )(aΓ is known as the gamma function. There exist many studies in the literature on
different proposed methods for the numerical calculation of the incomplete gamma function [22-
24]. MATLAB® also provides some tools for the calculation of the inverse incomplete gamma
(gammaincinv) function. The inverse incomplete gamma function (gammaincinv), computes the
inverse of the incomplete gamma function with respect to the integration limit x and represented
as gammaincinv(x,a) in MATLAB®.
Figure 3. A description of the VS algorithm
In Figure 4, the inverse incomplete gamma function is plotted for 1.0=x and [ ]1,0∈a . Here,
for our case the parameter a of the inverse incomplete gamma function defines the resolution of
the search. By equally sampling a values within [ ]1,0 interval at a certain step size, the

42
resolution of the search can be adjusted. For this purpose, at each iteration, a value of a is
computed by using the Eq.9
MaxItr
t
aat −= 0
(9)
where 0a is selected as 10 =a to ensure a full coverage of the search space at the first iteration,
t is the iteration index, and MaxItr represents the maximum number of iterations.
Let us consider an optimization problem defined within the [-10,10] region. The initial
radius 0r can be calculated with Eq. 10. Because 10 =a , the resulting function value is
1),()1( 0 ≈⋅ axvgammaincinx , which means 00 σ≈r as indicated before.
),()1( 000 axvgammaincinxr ⋅⋅= σ (10)
By means of Eq.4, the initial radius value 0r can be calculated as 100 ≈r . In Eq.11, a
general formula is also given to obtain the value of the radius at each iteration pass.
),()1(0 tt axvgammaincinxr ⋅⋅= σ (11)
Here, t represents the iteration index.
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
a
(1/x)*gammaincinv(x,a)
Figure 4. ),()1( axvgammaincinx ⋅ where 1.0=x and [ ]1,0∈a
2.2. The Modified Vortex Search Algorithm
The VS algorithm creates candidate solutions around a single point at each iteration pass. At the
first iteration, this point is the initial center 0µ which is determined with the upper and lower
limits of the problem at hand while in the latter iterations the center is shifted to the current best
position found so far. As mentioned before, this mechanism leads the VS algorithm to being
trapped into local minimum points for a number of functions.
To overcome above mentioned drawback, in this study a modified VS algorithm (MVS) is
proposed. In the MVS algorithm, candidate solutions are generated around multiple centers at

43
each iterations pass. The search behavior of the MVS algorithm can be thought as a number of
parallel vortices that have different centers at each iteration pass. Initially, the centers of these
multiple vortices are selected as in the VS algorithm. Let us consider, the total number of centers
(or vortices) to be represented by m . Let us say, )(µtM represents the matrix that stores the
values of these m centers at each iteration pass and t represents the iteration index. Thus,
initially { } mlM l
,...,2,1,,...,,)( 0
2
0
1
00 == µµµµ and initial positions of these centers are computed
as in Eq. 12.
ml
lowerlimitupperlimitl
,...,2,1,
2
... 0
2
0
1
0 =
+
==== µµµ
(12)
Next, a number of candidate solutions are generated with a Gaussian distribution around these
initial centers by using the initial radius value 0r . In this case the total number of candidate
solutions is again selected to be n. But note that, these n solutions are generated around m
centers. Thus, one should select mn solutions around each center.
Let us say, { } mnkssssCS k
l
t
,...,2,1,...,,)( 21 == represents the subset of solutions generated
around the center ml ,...,2,1= for the iteration t . Then, the total solution set generated for the
iteration 0=t can be represented by { } mlCSCSCSsC l
,...,2,1,,...,,)( 0
2
0
1
00 == . In the selection
phase, for each subset of solutions, a solution (which is the best one) )(0
'
sCSs l
l ∈ is selected.
Prior to the selection phase it must be ensured that the candidate subsets of solutions are inside
the search boundaries. For this purpose, the solutions that exceed the boundaries are shifted into
the boundaries, as in Eq. 5. Let us say, the best solution of each subset is stored in a matrix
)( '
sPBestt at each iteration pass. Thus, for 0=t , { } mlssssPBest l ,...,2,1,,...,,)( ''
2
'
1
'
0 == . Note
that, the best solution of this matrix ( )( '
0 sPBest ) is also the best solution of the total candidate
solution set )(0 sC for the current iteration, which is represented as bestItr .
In the VS algorithm, at each iterations pass, the center is always shifted to the best solution found
so far, bests . However, in the MVS algorithm, there exist m centers which positions need to be
updated for the next iteration. The most important difference between the VS and MVS
algorithm arises from here. In the MVS algorithm, one of these centers is again shifted to the best
solution found so far, bests . But, the remaining 1−m centers are shifted to a new position
determined by the best positions generated around the each center at the iteration t and the best
position found so far, bests as shown in Eq. 13.
)( ''
bestll
l
t ssrands +⋅+=µ (13)
In Eq. 13, rand is a uniformly distributed random number, 1,...,2,1 −= ml and
)( '
1
'
sPBests tl −∈ . Thus, for 1=t , { } 1,...,2,1,,...,,)( 1
2
1
1
11 −== mlM l
µµµµ is determined by
using the )( '
0
'
sPBestsl ∈ positions and the best position found so far, bests . In Figure-5, an
illustrative sketch of the center update process is given for a two-dimensional problem. In Figure
5, only one center is considered.
In the MVS algorithm, the radius decrement process is held totally in the same way as it is done
in the VS algorithm. At each iteration pass, the radius is decreased by utilizing the inverse
incomplete gamma function and thus, the locality of the generated solutions is increased. In
Figure 6, a description of the MVS algorithm is provided.

44
Figure 5. An illustrative sketch of the center updating process for the MVS algorithm (only one center is
considered)
Figure 6. A description of the MVS algorithm
3. RESULTS
The proposed MVS algorithm is tested on 50 benchmark functions which were also used in [16]
to measure the performance of the VS algorithm. By using the same functions, in this study, the

45
performance of the MVS algorithm is compared to the VS, PSO2011 and ABC algorithms.
PSO2011 [25-26] is an extension of the standard PSO algorithm and the ABC algorithm is a
well-known optimization algorithm which was inspired from the collective behaviours of honey
bees.
The functions used in the experiments are listed in Table 1. For the formulations of the functions
listed in Table 1, please refer to the reference [16].
3.1. Algorithm Settings
The ABC and PSO2011 algorithms are selected to have a population size of 50, which is also the
number of neighborhood solutions of the proposed VS algorithm. The acceleration coefficients
( 1c and 2c ) of the PSO2011 algorithm are both set to 1.8, and the inertia coefficient is set to 0.6,
as in [27]. The limit value for the ABC algorithm is determined as limit = SN * D, where SN
represents the number of food sources and D represents the dimension. VS algorithm does not
have any additional parameters. Different from the VS algorithm, the MVS algorithm has the
parameter m , which represents the total number of centers.
3.2. Results
The proposed MVS algorithm is compared to the VS, PSO2011 and ABC algorithms by using
the 50 benchmark functions given in Table 1. For each algorithm, 30 different runs are
performed, and the mean and the best values are recorded. The maximum number of iterations is
selected to be 500,000. For the MATLAB® codes of the PSO2011, ABC, VS and MVS
algorithms please refer to [25], [28], [29] and [30]. For each algorithm, all of the functions are
run in parallel using a 32 core Intel® CPU 32 GB RAM workstation. For the first set of
experiments, results are given in Table 2.
As shown in Table 2, for the MVS algorithm two different cases are considered. In the first case,
the total number of candidate solutions is selected to be 50, which means 10 candidate solutions
are generated around each center for 5=m . In this case, the MVS algorithm can avoid from the
local minimum points for the functions F13, F16, F17, F22, F23, F41, and F43 which is not the
case for the VS algorithm. However, poor sampling of the search space for this case (10 points
around each center) leads the MVS algorithm to show a correspondingly poor performance on
the improvement of the found near optimal solutions (exploitation) for some of the functions.
This can be clearly seen in the pair-wise statistical comparison of the algorithms given in Table
3. The pair-wise statistical comparison of the algorithms is obtained by a Wilcoxon Signed-Rank
Test. The null hypothesis H0 for this test is: "There is no difference between the median of the
solutions produced by algorithm A and the median of the solutions produced by algorithm B for
the same benchmark problem", i.e., median (A) = median (B). To determine whether algorithm
A reached a statistically better solution than algorithm B, or if not, whether the alternative
hypothesis is valid, the sizes of the ranks provided by the Wilcoxon Signed-Rank Test (i.e., T+
and T-, as defined in [10]) are examined. Because an arithmetic precision value that is higher
than necessary makes it difficult to compare the local search abilities of the algorithms, during
the statistical pair-wise comparison, resulting values below are considered as 0. In Table 3, ‘+’
indicates cases in which the null hypothesis is rejected and the MVS algorithm exhibited a
statistically superior performance in the pair-wise Wilcoxon Signed-Rank Test at the 95%
significance level ( 05.0=α ); ‘-’ indicates cases in which the null hypothesis is rejected and the
MVS algorithm exhibited an inferior performance; and ‘=’ indicates cases in which there is no
statistical different between two algorithms. The last row of the Table 3 shows the total count of
(+/=/-) for the three statistical significance cases in the pair-wise comparison.

46
As shown in Table 3, the VS algorithm outperforms the MVS algorithm for the functions F5,
F10, F11, F38, and F42. But when the results given in Table 2 are compared for these functions,
it can be clearly seen that this difference mainly arises because of the poor exploitation ability of
the MVS algorithm with 50 candidate solutions. Therefore, another case in which the total
number of candidate solutions is selected to be 250 is considered for the MVS algorithm. In this
case, 50 candidate solutions are generated around each center for 5=m . As can be shown in
Table 2, the MVS algorithm with 250 candidate solutions performs better than the MVS
algorithm with 50 candidate solutions. Statistical pair-wise comparison of the algorithms for the
second case is also given in Table 4.
Table 1. Benchmark function set that is used in the experiments.
No Function Characteristics Range Dim. Min.
F1 Stepint Unimodal Separable [-5.12, 5.12] 5 0
F2 Step Unimodal Separable [-100, 100] 30 0
F3 Sphere Unimodal Separable [-100, 100] 30 0
F4 SumSquares Unimodal Separable [-10, 10] 30 0
F5 Quartic Unimodal Separable [-1.28, 1.28] 30 0
F6 Beale Unimodal Non-Separable [-4.5, 4.5] 5 0
F7 Easom Unimodal Non-Separable [-100, 100] 2 -1
F8 Matyas Unimodal Non-Separable [-10, 10] 2 0
F9 Colville Unimodal Non-Separable [-10, 10] 4 0
F10 Trid6 Unimodal Non-Separable [-D2
,D2
] 6 -50
F11 Trid10 Unimodal Non-Separable [-D2
,D2
] 10 -210
F12 Zakharov Unimodal Non-Separable [-5,10] 10 0
F13 Powell Unimodal Non-Separable [-4,5] 24 0
F14 Schwefel 2.22 Unimodal Non-Separable [-10, 10] 30 0
F15 Schwefel 1.2 Unimodal Non-Separable [-10, 10] 30 0
F16 Rosenbrock Unimodal Non-Separable [-30, 30] 30 0
F17 Dixon-Price Unimodal Non-Separable [-10, 10] 30 0
F18 Foxholes Multimodal Separable [-65.536, 65.536] 2 0.998
F19 Branin Multimodal Separable [-5,10]x[0,15] 2 0.398
F20 Bohachevsky1 Multimodal Separable [-100, 100] 2 0
F21 Booth Multimodal Separable [-10, 10] 2 0
F22 Rastrigin Multimodal Separable [-5.12, 5.12] 30 0
F23 Schwefel Multimodal Separable [-500, 500] 30 -12569.5
F24 Michalewicz2 Multimodal Separable [0, π ] 2 -1.8013
F27 Schaffer Multimodal Non-Separable [-100, 100] 2 0
F28 Six Hump Camel Back Multimodal Non-Separable [-5, 5] 2 -1.03163
F29 Bohachevsky2 Multimodal Non-Separable [-100, 100] 2 0
F30 Bohachevsky3 Multimodal Non-Separable [-100, 100] 2 0
F31 Shubert Multimodal Non-Separable [-10, 10] 2 -186.73
F32 GoldStein-Price Multimodal Non-Separable [-2, 2] 2 3
F33 Kowalik Multimodal Non-Separable [-5, 5] 4 0.00031
F34 Shekel5 Multimodal Non-Separable [0, 10] 4 -10.15
F37 Perm Multimodal Non-Separable [D, D] 4 0
F38 PowerSum Multimodal Non-Separable [0, D] 4 0
F39 Hartman3 Multimodal Non-Separable [0, 1] 3 -3.86
F40 Hartman6 Multimodal Non-Separable [0, 1] 6 -3.32
F41 Griewank Multimodal Non-Separable [-600, 600] 30 0

47
F42 Ackley Multimodal Non-Separable [-32, 32] 30 0
F43 Penalized Multimodal Non-Separable [-50, 50] 30 0
F44 Penalized2 Multimodal Non-Separable [-50, 50] 30 0
F45 Langerman2 Multimodal Non-Separable [0, 10] 2 -1.08
F46 Langerman5 Multimodal Non-Separable [0, 10] 5 -1.5
F47 Langerman10 Multimodal Non-Separable [0, 10] 10 NA
F48 Fletcher Powell2 Multimodal Non-Separable [ ππ,− ] 2 0
Table 2. Statistical results of 30 runs obtained by PSO2011, ABC, VS and MVS algorithms (values <
16
10−
are considered as 0).
No Min. MVS (m = 5,
n = 50)
MVS (m = 5,
n =
250)
VS PSO2011 ABC
F1 0 Mean
StdDev
Best
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
F2 0 Mean
StdDev
Best
0
0
0
0
0
0
0.2
0.406838102
0
0.066666667
0.253708132
0
0
0
0
F3 0 Mean
StdDev
Best
0
0
0
0
0
0
0
0
0
0
0
0
2.78624E-16
0
2.23487E-16
F4 0 Mean
StdDev
Best
0
0
0
0
0
0
0
0
0
0
0
0
2.75098E-16
0
1.85594E-16
F5 0 Mean
StdDev
Best
0.000334677
0.000316498
3.62661E-07
0.000127606
7.01148E-05
5.59873E-06
0.000145026
7.30549E-05
5.54996E-05
1.64098E-05
5.56581E-06
7.13993E-06
0.013732963
0.002379448
0.008413424
F6 0 Mean
StdDev
Best
0
0
0
0
0
0
0
0
0
0
0
0
6.37598E-16
3.58687E-16
0
F7 -1 Mean
StdDev
Best
-1
0
-1
-1
0
-1
-1
0
-1
-1
0
-1
-1
0
-1
F8 0 Mean
StdDev
Best
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
F9 0 Mean
StdDev
Best
0
0
0
0
0
0
0
0
0
0
0
0
0.00576453
0.003966867
0.000383073
F10 -50 Mean
StdDev
Best
-50
2.88473E-14
-50
-50
2.97973E-14
-50
-50
2.96215E-14
-50
-50
3.61345E-14
-50
-50
4.94748E-14
-50
F11 -210 Mean
StdDev
Best
-210
6.64246E-13
-210
-210
5.51673E-13
-210
-210
6.19774E-13
-210
-210
2.30778E-13
-210
-210
9.62204E-12
-210
F12 0 Mean
StdDev
Best
0
0
0
0
0
0
0
0
0
0
0
0
7.56674E-14
3.76382E-14
2.31887E-14
F13 0 Mean
StdDev
7.59934E-09
4.14437E-08
3.88377E-10
1.27749E-09
1.43967E-05
2.27742E-06
2.04664E-07
1.21051E-08
9.09913E-05
1.42475E-05

48
Best 1.1432E-16 0 5.71959E-06 1.72679E-07 5.23427E-05
F14 0 Mean
StdDev
Best
3.79651E-05
0.000156663
0
0
0
0
0
0
0
1.094284383
0.870781136
0.107097937
8.51365E-16
0
6.93597E-16
F15 0 Mean
StdDev
Best
0
0
0
0
0
0
0
0
0
0
0
0
0.000760232
0.000440926
0.00027179
F16 0 Mean
StdDev
Best
1.20813E-07
2.94163E-07
1.14463E-12
3.51659E-08
5.41004E-08
1.85577E-13
0.367860114
1.130879848
9.42587E-05
0.930212233
1.714978077
0
0.003535257
0.003314818
7.08757E-05
F17 0 Mean
StdDev
Best
0
0
0
0
0
0
0.666666667
7.68909E-16
0.666666667
0.666666667
4.38309E-16
0.666666667
1.91607E-15
2.55403E-16
1.1447E-15
Table 2. (continued).
No Min. MVS (m = 5,
n = 50)
MVS (m = 5,
n = 250)
VS PSO2011 ABC
F18 0.998 Mean
StdDev
Best
0.998003838
0
0.998003838
0.998003838
0
0.998003838
0.998003838
0
0.998003838
34.26621987
126.6004794
0.998003838
0.998003933
4.33771E-07
0.998003838
F19 0.398 Mean
StdDev
Best
0.397887358
0
0.397887358
0.397887358
0
0.397887358
0.397887358
0
0.397887358
0.397887358
0
0.397887358
0.397887358
0
0.397887358
F20 0 Mean
StdDev
Best
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
F21 0 Mean
StdDev
Best
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
F22 0 Mean
StdDev
Best
4.14483E-16
8.95296E-16
0
0
3.24317E-16
0
57.60799224
13.94980276
33.82857771
26.11016129
5.686650032
16.91429893
0
0
0
F23 -12569.5 Mean
StdDev
Best
-12569.48662
3.63798E-12
-12569.48662
-12569.48662
3.02118E-12
-12569.48662
-11283.05416
352.1869262
-11799.62928
-8316.185447
463.9606712
-9466.201047
-12569.48662
1.85009E-12
-12569.48662
F24 -1.8013 Mean
StdDev
Best
-1.80130341
9.03362E-16
-1.80130341
-1.80130341
9.03362E-16
-1.80130341
-1.80130341
9.03362E-16
-1.80130341
-1.80130341
9.03362E-16
-1.80130341
-1.80130341
9.03362E-16
-1.80130341
F25 -4.6877 Mean
StdDev
Best
-4.653710247
0.051587389
-4.687658179
-4.668168867
0.02119113
-4.687658179
-4.670953055
0.020809276
-4.687658179
-4.67700874
0.036487971
-4.687658179
-4.687658179
2.60778E-15
-4.687658179
F26 -9.6602 Mean
StdDev
Best
-8.966488952
0.412225375
-9.556414106
-9.07030728
0.267235114
-9.513891389
-8.793361668
0.382153549
-9.410563187
-9.204154798
0.298287637
-9.660151716
-9.660151716
0
-9.660151716
F27 0 Mean
StdDev
Best
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
F28 -1.03163 Mean
StdDev
Best
-1.031628453
6.77522E-16
-1.031628453
-1.031628453
6.77522E-16
-1.031628453
-1.031628453
6.77522E-16
-1.031628453
-1.031628453
6.71219E-16
-1.031628453
-1.031628453
6.77522E-16
-1.031628453
F29 0 Mean
StdDev
Best
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
F30 0 Mean
StdDev
Best
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

49
F31 -186.73 Mean
StdDev
Best
-186.7309088
3.25344E-14
-186.7309088
-186.7309088
2.93854E-14
-186.7309088
-186.7309088
3.76909E-14
-186.7309088
-186.7309088
4.49449E-13
-186.7309088
-186.7309088
1.18015E-14
-186.7309088
F32 3 Mean
StdDev
Best
3
1.25607E-15
3
3
1.51835E-15
3
3
1.44961E-15
3
3
1.22871E-15
3
3
1.7916E-15
3
F33 0.00031 Mean
StdDev
Best
0.000307486
0
0.000307486
0.000307486
0
0.000307486
0.000307486
0
0.000307486
0.000307486
0
0.000307486
0.000319345
5.4385E-06
0.00030894
F34 -10.15 Mean
StdDev
Best
-10.15319968
6.8481E-15
-10.15319968
-10.15319968
7.01294E-15
-10.15319968
-10.15319968
7.2269E-15
-10.15319968
-9.363375596
2.081063878
-10.15319968
-10.15319968
7.2269E-15
-10.15319968
No Min. MVS (m = 5,
n = 50)
MVS (m = 5,
n = 250)
VS PSO2011 ABC
F35-10.4 Mean
StdDev
Best
-10.40294057
1.51161E-15
-10.40294057
-10.40294057
1.64931E-15
-10.40294057
-10.40294057
1.61598E-15
-10.40294057
-10.40294057
1.80672E-15
-10.40294057
-10.40294057
1.04311E-15
-10.40294057
F36-10.53 Mean
StdDev
Best
-10.53640982
2.05998E-15
-10.53640982
-10.53640982
1.61598E-15
-10.53640982
-10.53640982
1.47518E-15
-10.53640982
-10.53640982
0
-10.53640982
-10.53640982
2.13774E-15
-10.53640982
F370 Mean
StdDev
Best
0.003329829
0.002357345
1.59342E-09
0.003422267
0.002379677
4.15798E-11
0.002815467
0.002374325
0
0.002854996
0.007218334
1.30581E-08
0.003526435
0.001604834
0.00097117
F380 Mean
StdDev
Best
4.70363E-05
6.70904E-05
3.91829E-08
1.39476E-06
8.77149E-07
6.72858E-11
1.78046E-06
1.28089E-06
4.82E-09
3.14986E-05
6.43525E-05
1.50435E-11
0.000288005
0.00013892
5.82234E-05
F39-3.86 Mean
StdDev
Best
-3.862782148
2.65431E-15
-3.862782148
-3.862782148
2.68234E-15
-3.862782148
-3.862782148
2.69625E-15
-3.862782148
-3.862782148
2.71009E-15
-3.862782148
-3.862782148
2.71009E-15
-3.862782148
F40-3.32 Mean
StdDev
Best
-3.322368011
6.54548E-16
-3.322368011
-3.322368011
5.71336E-16
-3.322368011
-3.322368011
5.14996E-16
-3.322368011
-3.318394475
0.021763955
-3.322368011
-3.322368011
6.54548E-16
-3.322368011
F410 Mean
StdDev
Best
0
0
0
0
0
0
0.032798017
0.018570459
0.00739604
0.004761038
0.008047673
0
0
0
0
F420 Mean
StdDev
Best
1.49806E-14
3.29641E-15
7.99361E-15
1.29674E-14
3.31178E-15
7.99361E-15
1.15463E-14
3.61345E-15
7.99361E-15
0.660186991
0.711496752
7.99361E-15
2.44545E-14
3.02083E-15
2.22045E-14
F430 Mean
StdDev
Best
0
0
0
0
0
0
0.114662313
0.532276418
0
0.024187276
0.080213839
0
2.63417E-16
0
1.29727E-16
F440 Mean
StdDev
Best
0
0
0
0
0
0
0
0
0
0
0
0
2.7797E-16
0
2.22214E-16
F45-1.08 Mean
StdDev
Best
-1.080938442
4.70125E-16
-1.080938442
-1.080938442
4.51681E-16
-1.080938442
-1.080938442
4.51681E-16
-1.080938442
-1.080938442
4.51681E-16
-1.080938442
-1.080938442
4.96507E-16
-1.080938442
F46-1.5 Mean
StdDev
Best
-1.499999223
6.77522E-16
-1.499999223
-1.499999223
6.77522E-16
-1.499999223
-1.499999223
6.77522E-16
-1.499999223
-1.499999223
6.77522E-16
-1.499999223
-1.499999223
1.05365E-15
-1.499999223
F47NA Mean
StdDev
Best
-1.403866666
0.219139813
-1.5
-1.320499999
0.279253038
-1.5
-1.271399999
0.313658787
-1.5
-1.069011938
0.422205043
-1.5
-1.482016588
0.097662612
-1.499998488

50
F480 Mean
StdDev
Best
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
F490 Mean
StdDev
Best
0
0
0
0
0
0
0
0
0
3.083487114
4.389694328
0
1.48707E-12
8.11041E-12
3.1715E-16
F500 Mean
StdDev
Best
0
0
0
0
0
0
0
0
0
580.0839029
1280.698395
0
1.111095363
0.598962098
0.182153237
Table 3. Pair-wise statistical comparison of the MVS (m = 5, n = 50) algorithm by Wilcoxon Signed-Rank
Test ( 05.0=α )

51
Table 4. Pair-wise statistical comparison of the MVS (m = 5, n = 250) algorithm by Wilcoxon Signed-
Rank Test ( 05.0=α ).
As shown in Table 4, MVS algorithm with 250 candidate solutions can successfully improve the
near optimal solutions and thus performs better than all of the other algorithms.
In [31], authors stated that after a sufficient value for colony size, any increment in the value
does not improve the performance of the ABC algorithm significantly. For the test problems
carried out in [31] colony sizes of 10, 50 and 100 are used for the ABC algorithm. It is shown

52
that although from 10 to 50 the performance of the ABC algorithm significantly increased, there
is not any significant difference between the performances achieved by 50 and 100 colony sizes.
Similarly, for the PSO algorithm it is reported that, PSO with different population sizes has
almost the similar performance which means the performance of PSO is not sensitive to the
population size [32]. Based on the above considerations, in this study a comparison of the MVS
algorithm to the ABC and PSO2011 algorithms with a different population size is not performed.
For the VS algorithm it is expected to achieve better exploitation ability with an increased
number of candidate solutions. But the problem with the VS algorithm is with its global search
ability rather than the local search ability for some of the functions listed above. Therefore, a
comparison of the MVS (m = 5, n = 50) to VS algorithm with 50 candidate solutions is thought
to be enough to show the improvement achieved by the modification performed on the VS
algorithm.
In Figure 7, the average computational time of 30 runs for 500,000 iterations is also provided for
the MVS (m = 5, n = 50), MVS (m = 5, n = 250), VS, PSO2011 and ABC algorithms. As shown
in this figure, the required computational time to perform 500,000 iterations with the MVS
algorithm is slightly increased when compared to the VS algorithm. However, even for the MVS
(m = 5, n = 250) algorithm the required computational time to perform 500,000 iterations is still
lower than the PSO2011 and ABC algorithms.
Figure 7. Average computational time of 30 runs for 50 benchmark functions (500,000 iterations).
4. CONCLUSIONS
This paper presents a modified VS algorithm in which the global search ability of the existing
VS algorithm is improved. This is achieved by using multiple centers during the candidate
solution generation phase of the algorithm at each iteration pass. In the VS algorithm, only one
center is used for this purpose and this usually leads the algorithm to being trapped into a local
minimum point for some of the benchmark functions. Although the complexity of the existing
VS algorithm is a bit increased, there is not any significant difference between the computational
time of the modified VS algorithm and the existing VS algorithm. Computational results showed
that the MVS algorithm outperforms the existing VS algorithm, PSO2011 and ABC algorithms
for the benchmark numerical function set.
In the future studies, the MVS algorithm will be used for the solution of some real world
optimization problems.
0 5 10 15 20 25 30 35 40 45 50
0
1000
2000
3000
4000
5000
6000
7000
Functions
Seconds
MVS (m = 5, n = 250)
MVS (m = 5, n = 50)
VS
ABC
PSO2011

53
REFERENCES
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Authors
Dr. Berat Doğan received his BSc. degree in Electronics Engineering from
Erciyes University, Turkey, 2006. He received his MSc. degree in Biomedical
Engineering from Istanbul Technical University, Turkey, 2009. He received his
PhD. in Electronics Engineering at Istanbul Technical University, Turkey, 2015.
Between 2008-2009 he worked as a software engineer at Nortel Networks Netas
Telecommunication Inc. Then, from 2009 to July 2015 he worked as a Research
Assistant at Istanbul Technical University. Now he is working as an Assistant
Professor at Inonu University, Malatya, Turkey. His research interests include
optimization algorithms, pattern recognition, biomedical signal and image
processing, and bioinformatics.

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