Abstract—There are many population based optimization

methods used for numeric functions and engineering problems.

The biggest problem of these methods is setting the balance

between exploration and exploitation. The artificial bee colony

algorithm proposed by Karaboga gives better results compared

to other known nature-inspired methods. Yet, while the ABC

algorithm is better in the exploration part, which is known as

exploring new places, it is not well enough in the exploitation

part, which is explained as exploiting the results found. To

overcome this problem, instead of random distribution of the

scout bees in the search space in ABC algorithm, this paper

proposed the Levy Flight ABC (LFABC) algorithm performing

the distribution using Levy Flight method. By this way, it was

ensured for the ABC algorithm to improve the exploitation. The

two methods were tested on 10 benchmark functions, and the

proposed method was seen to perform the results better.

Index Terms—Artificial bee colony, levy flight, levy

distribution, optimization.

I. INTRODUCTION

In the last 10-20 years, many population based algorithms

were proposed to solve numerical benchmark functions and

engineering problems, such as particle swarm optimization

[1], ant colony optimization [2], genetic algorithms [3],

artificial bee colony [4] and so on. These algorithms which

are also referred to as biological-inspired were generally

proposed through inspiration from social and foraging

behaviors of particular animals. Being inspired from social

and foraging behaviors of bees, ABC was first proposed in

2005 by Karaboga [4].

Showing a better performance compared to other

optimization techniques, the ABC algorithm [5], considering

its simplicity and efficiency, was used in the fields such as

function optimization [6], [7], vehicle routing problem [8],

data clustering [9], image processing and segmentation [10],

[11] , electric load forecasting [12], engineering design [13]

and so on.

There are two important points for biological-inspired

algorithms: exploration and exploitation. The exploration

part is concerned the ability of autonomously seeking for the

global optimum, whereas the exploitation part is related to

the ability of applying the existing knowledge to look for

better solutions [14]. Although the ABC algorithm is more

effective in many numerical benchmark functions compared

to other algorithms, the ABC algorithm has some deficiencies.

While ABC is good in exploration, it is not good enough in

Manuscript received February 9, 2013; revised April 14, 2013. This work

was supported by Scientific Research Project of Selcuk University.

Hüseyin Hakli and Harun Uğuz are with the Computer Engineering

Department, Selcuk University, Konya, Turkey (e-mail: hhakli@

selcuk.edu.tr, harun_uguz@ selcuk.edu.tr).

exploitation, which refers to ability of applying the existing

knowledge to look for better solutions. In other words, while

the ABC algorithm can perform global search better, it is

weak in local search. Many ABC variants were proposed to

overcome this problem. The common goal of the new

algorithms proposed was to strengthen exploitation by

ensuring the ABC algorithms perform local search better.

Inspired from the PSO algorithm, Zhu ve Kwong [15]

proposed gbest-guided ABC (GABC) algorithm by

incorporating the information of global best (gbest) solution

into the solution search equation to improve the exploitation.

And inspired from the DE algorithm, Gao et al. [16]

improved exploitation by ensuring the ABC algorithm

perform search only around the best solution. Xiang ve An [6]

proposed the ERABC algorithm, a combinatorial solution

search equation that is introduced to accelerate the search

process instead of ABC solution search equation, and a

chaotic search technique that is employed on scout bee phase.

Li et al. [14] used 3 different solution search equations in

parallel, and chose the one with better results, and developed

the ABC algorithm by performing this phase for both

employed and onlooker bees. Alatas [17] used chaotic maps

for parameter adaption, initial the artificial colony and scout

of employed bee to prevent the ABC to get stuck on local

solutions. In this paper, if improvement as much as the

predetermined limit value number was not attained in the

ABC algorithm, that food source was abandoned, and instead

of selecting a random food source within the search space, it

was ensured a new food source search be performed

according to Levy Flight distribution. GlobalMin (the food

source giving the best result at that time) information were

also used while searching for a new food source with this

distribution. Thus, by performing the search around

GlobalMin, it was that the ABC algorithm performs local

search more effectively, and that the scout bees that were not

useful enough become more useful. According to the

experimental results, the proposed method was seen to give

better results compared to the ABC algorithm and to prevent

being stuck on local minimum in several functions.

The rest of the paper is divided as follows. In Section II,

original ABC algorithm and Levy flight method are

presented. The proposed approach is detailed in Section III.

In Section IV, the experimental results and comparison of the

methods are presented. As a final, the paper is concluded with

the future works.

II. ABC AND LEVY FLIGHTS

A. Original Artificial Bee Colony

Artificial bee colony optimization algorithm was

developed in 2005 by being motivated from bee colonies by

Levy Flight Distribution for Scout Bee in Artificial Bee

Colony Algorithm

Hüseyin Hakli and Harun Uğuz

Lecture Notes on Software Engineering, Vol. 1, No. 3, August 2013

254DOI: 10.7763/LNSE.2013.V1.55

Karaboga [4]. ABC algorithm has three various bees:

employed, onlooker and scout. Colony is equal to both of

employed and onlooker bee, so half of the colony is

employed bees and another half is onlooker bees. In ABC

algorithm each food source stand for possible solution and

the amount of nectar of a food source represent the quality of

the possible solution. The number of food sources equals the

half of the colony size and equal to employed bees.

First, each employed bee randomly selects a food source

within the determined search space. This distribution is

performed according to the (1) below.

min max min

, (0, 1) ( )i j j j jx x rand x x (1)

where iX is D-dimensional vector, i=1, 2, 3…. SN (number

of food source), j=1, 2, 3, …, D, min

jx and max

jx are the

determined minimum and maximum limits.

After positions of employed bees are determined as such,

each employed bee selects different neighbor itself and

updates a randomly selected dimension according to the

following equation.

, , , , , ( )i j i j i j i j k jv x x x

(2)

where , i jv represents new value of jth dimension of ith particle,

, i jx represents jth dimension of ith particle, ,i j represents

the random value determined in the interval [-1,1], and

,k jx represents jth dimension of kth particle. If fitness value of

the new food source determined is better, the old food source

is forgotten by applying greedy search mechanism, and the

employed bee is updated with the new one. In the opposite

case, the old food source remains in the memory. After the

employed bees phase is completed, the probabilities are

determined for sharing quality of the solutions found with the

onlooker bees.

1

ii SN

jj

fitp

fit

(3)

where ifit denotes the fitness value of solution iX . The

fitness value ifit is defined as follows:

1

, ( ) 01 ( )

1 ( ) ( ) 0

i

ii

i i

if f Xf Xfit

f X if f X

(4)

Onlooker bees receive the information on quality of the

solutions (nectar amount of food source) from the employed

bees, and select the solution with better result. In Equation (3),

a pi probability is determined for each solution, the better the

quality of a solution, the higher its probability of being

selected. Then, the onlooker bees search for a new food

source using (2) around the food source they have selected,

and if the food source they find is better, they replace the old

food source with the newly found one.

Finally, if any improvement cannot be made as much as

the predetermined limit value number for a selected food

source, this food source is forgone, and the employed bee on

this source becomes scout bee. Scout bee randomly searches

for a food source within the determined search space.

B. Levy Flights

Levy flight [18] is a class of non-Gaussian random

processes whose random walks are drawn from Levy stable

distribution. This distribution is a simple power-law formula

L(s) ~ |s|-1-β where 0 < ß < 2 is an index. Mathematically

speaking, a simple version of Levy distribution can be

defined as [19], [20]:

3 2

1exp 0 ,

( , , ) 2 2( ) ( )

0 0

if sL s s s

if s

(5)

where γ>0 parameter is scale (controls the scale of

distribution) parameter, µ parameter is location or shift

parameter.

In general, Levy distribution should be defined in terms of

Fourier transform

,20],exp[)(

kkF (6)

where α is a parameter within [-1,1] interval and known as

skewness or scale factor. An index of o stability β є (0, 2] is

also referred to as Levy index.

In particular, for β = 1, the integral can be carried out

analytically and is known as the Cauchy probability

distribution. Another special case when β= 2, the distribution

correspond to Gaussian distribution.

β and α parameters take a major part in determination of

the distribution. The parameter β controls the shape of the

probability distribution in such a way that one can obtain

different shapes of probability distribution, especially in the

tail region depending on the parameter β. Thus, the smaller β

parameter causes the distribution to make longer jumps since

there will be longer tail [21]. It makes longer jumps for

smaller values whereas it makes shorter jumps for bigger

values.

III. THE PROPOSED ALGORITHM LFABC

The ABC algorithm gives successful results for most

benchmark functions. Although it is better in setting the

balance between exploration and exploitation compared to

other algorithms, it has several deficiencies. While it can

perform better the global search by performing random

searches around each food source and cover the search space

optimally, it encounters several problems in the exploitation

part, and gets stuck on local minimums particularly in

complex multimodal functions. If the original ABC

algorithm fails to make improvement for a particular food

source as much as a predetermined limit value number, then

the employed bee having that source become scout bee. Then

this scout bee is randomly distributed within the search space,

boundaries of which are determined. In this case, scout bees

Lecture Notes on Software Engineering, Vol. 1, No. 3, August 2013

255

select a zone within the search space completely randomly

without using the results found until that moment as a result

of the iterations. It is a very low probability for this newly

selected food source to be better than the GlobalMin (best

value in current iteration) found until that moment. In this

paper, in order to utilize the scout bees better, they are made

find a new food source using Levy Flight distribution with

also the effect of GlobalMin instead of selecting a random

food source. By this way, it is sought to attain a better

solution using the best food source until that moment instead

of the food source that could not be improved. By performing

enough random searches, exploration of the ABC algorithm,

which is already good in exploration, is improved.

Let us review in some more detail how distribution is

performed using Levy flight. By Levy flight, new state of the

particle is calculated as [19]:

)(1 LevyXX tt (7)

α is the step size which should be related to the scales of

the problem of interest. In the proposed method α is random

number for all dimensions of particles.

)())((1 LevyDsizerandomXX tt (8)

The product means entry-wise multiplications.

A non-trivial scheme of generating step size s samples are

discussed in detail in[19], [20], it can be summarized as

)(01.0~)())((1

tt

j gbestxv

uLevyDsizerandoms

(9)

where u and v are drawn from normal distributions. That is

2 2~ (0, ) ~ (0, )u vu N v N (10)

with

1,2]2/)1[(

)2/sin()1(1

2)1(

vu

(11)

Here is standard Gamma

function.

One of the important points to be considered while

performing distribution by Levy flights is the value taken by

the β parameter. As stated before, Yang and Deb mentioned

that the β parameter gave different results at different values

in the trials conducted in the study named Multiobjective

cuckoo search for design optimization [19]. Accordingly, it

can be concluded that a different β

parameter for each

benchmark function gave a more effective result. Moreover,

in the Evolutionary Algorithms with Adaptive Levy

Mutations study of Chang-Yong Lee, different constant

values were set of the β parameter, procedures were

conducted by calculating the distribution for each of these

values and selecting the best of the offspring produced [21].

As seen in these two examples, β parameter substantially

affects distribution.

IV. EXPERIMENTS AND RESULTS

A. Benchmark Functions and Parameter Settings

Experiments are implemented with the population size of

50(namely, SN=25) and the control parameter limit equals

100 for all the test functions. The algorithms are examined on

benchmark functions with dimension (D) of 50. For all

benchmark function, number of iteration is determined as

10000.The bees are initialized in the search range [Xmin,

Xmax], where Xmax and Xmin are the maximum and

minimum value in search range. The experimental results in

Table II are the average of minimum values of the benchmark

functions for 30 runs. The stopping criterion is set number of

maximum iteration. The performance of LFABC is compared

with the original ABC. For LFABC, β parameter is set 1.5 in

Table II.

The determined benchmark functions are numbered f1 to f10

given in Table I. All the functions are to be minimized. Table

1 shows characteristic(C), search range (Range), and

formulation of the functions. The functions f1-f2-f3-f7-f8 are

unimodal, f4-f5-f6-f9 are multimodal and f10 is shifted.

TABLE I: BENCHMARK FUNCTIONS(C: CHARACTERISTIC, U: UNIMODAL, M:

MULTIMODAL, S: SHIFTED)

Range C Function Formulation

[-100,

100] U Sphere

n

i

ixf1

2

1

[-100,

100] U Elliptic

n

i

i

ni xf1

2)1()1(6

2 )10(

[-10,

10] U

Rosenbro

ck

1

1

222

13 1100n

i

iii xxxf

[-5.12,

5.12] M Rastrigin

n

i

ii xxf1

2

4 10)2cos(10

[-600,

600] M

Griewan

k 1cos

4000

1

11

2

5

n

i

in

i

ii

xxf

[-32,

32] M Ackley

2

6

1

1

120exp 0.2

1exp cos(2 ) 20

D

i

i

D

i

i

f xD

x eD

[-10,

10] U

SumSqua

re

2

7

1

n

i

i

f ix

[-10,

10] U

Schwefel

2.22

n

i

n

i

ii xxf1 1

8

[-10,

10] M Alpine

n

i

iii xxxf1

9 1.0)sin(

[-100,

100] S

Shifted

Sphere oxzzf

n

i

i 1

2

10

B. Experimental Results

Table II shows the mean result of 30 runs with 10,000

iterations of ten benchmark functions (f1 to f10) with

dimension 50. The information about the mean optimum

solution, standard deviation, minimum and maximum value

and Wilcoxon test result by ABC and LFABC in 30 runs over

benchmark functions are given in Table I. The best mean

result and the best standard deviations of the benchmark

Lecture Notes on Software Engineering, Vol. 1, No. 3, August 2013

256

functions obtained by the algorithms are shown in bold.

According to Table II, the mean result of LFABC algorithm

performs better compared to ABC algorithm for the all

benchmark functions except f3. While ABC algorithm is

being trapped in local minimum, the LFABC algorithm

attained the optimum value for function f4 and f5. We can

conduct robustness comparisons of the algorithms by

checking the standard deviations. The fact inconsistency

between its results is high shows that it is not robust. LFABC

algorithm is a robust algorithm compared to ABC algorithm

for all benchmark functions except f3.

TABLE II: RESULTS FOR BENCHMARK FUNCTIONS (OPT: OPTIMUM STD:

STANDARD DEVIATION, MIN: MINIMUM, MAX: MAXIMUM, SIGN:

STATISTICAL TEST SIGN)

Func. Opt. ABC LFABC Sign

f1 0 Mean 1,44E-15 6,22E-16 +

STD 1,87E-16 7,4E-17

Min 8,91E-16 5,16E-16

Max 1,85E-15 7,18E-16

f2 0 Mean 1,44E-15 5,53E-16 +

STD 2,59E-16 5,83E-17

Min 9,75E-16 4,64E-16

Max 2,03E-15 6,93E-16

f3 0 Mean 0,023337 0,027578 -

STD 0,035322 0,076159

Min 0,000754 2,11E-05

Max 0,177254 0,420718

f4 0 Mean 5,24E-14 0 +

STD 1,09E-13 0

Min 0 0

Max 5,19E-13 0

f5 0 Mean 7,8E-15 0 +

STD 1,03E-14 0

Min 1,11E-16 0

Max 4,27E-14 0

f6 0 Mean 1,17E-13 3,74E-14 +

STD 1,66E-14 3,58E-15

Min 9,24E-14 3,2E-14

Max 1,63E-13 4,26E-14

f7 0 Mean 1,45E-15 6,04E-16 +

STD 1,47E-16 7,55E-17

Min 1,17E-15 4,93E-16

Max 1,65E-15 7,18E-16

f8 0 Mean 3,24E-15 1,58E-15 +

STD 3,5E-16 8,79E-17

Min 2,54E-15 1,41E-15

Max 3,79E-15 1,81E-15

f9 0 Mean 3,37E-08 7,3E-10 +

STD 1,01E-07 2,02E-09

Min 8,79E-12 1,65E-15

Max 4,7E-07 1,06E-08

f10 0 Mean 1,46E-15 5,97E-16 +

STD 2,21E-16 7,55E-17

Min 9,98E-16 4,95E-16

Max 1,88E-15 7,35E-16

When examined Table II in general, it cannot be concluded

that the proposed method makes much more improvement

compared to the ABC algorithm. But the proposed method

found the optimum values by not being stuck on local

minimums for f4 and f5 functions. It also improved benchmark

functions (except f3) compared to the ABC algorithm.

Non-parametric statistical test has been implemented for

statistically significant of between LFABC and the ABC. We

used Wilcoxon’s test with significance level 0.05, and results

are shown in Table II. In the charts, in case Wilcoxon test was

at 0.05 significance level, + mark was used, otherwise, –

mark was used; no mark was used in case all results showed

the sale value or optimum value. From the results in Table II,

it can be observed that the LFABC algorithm’s performance

is significantly better compared to ABC algorithm for all

benchmark functions (except f3) in Table I.

V. CONCLUSION AND FUTURE WORKS

In order to overcome insufficiency of the ABC algorithm

in exploitation, this paper proposed the LFABC method

ensuring the scout bees in the ABC algorithm find a new food

source within the search space with also the effect of

GlobalMin. The original ABC algorithm was tested in 10

benchmark functions by the proposed method. While the

proposed method gave the optimum value by ensuring the

ABC algorithm escape from local minimums for several

functions, it was seen to make improvement in the other

functions.

As future work, a better improvement can be attained by

using the LFABC method together with other proposed

algorithms, and it can be used on different optimization

problems together with these methods. Furthermore, effect of

the β value, an important parameter for Levy Flight

distribution, on other functions can be examined and detailed.

VI. EDITORIAL POLICY

The submitting author is responsible for obtaining

agreement of all coauthors and any consent required from

sponsors before submitting a paper. It is the obligation of the

authors to cite relevant prior work.

Authors of rejected papers may revise and resubmit them

to the journal again.

ACKNOWLEDGMENT

This study has been supported by Scientific Research

Project of Selcuk University.

REFERENCES

[1] J. Kennedy and R. Eberhart, “Particle Swarm Optimization,” in Proc.

Sixth International Symposium on Micro Machine and Human Science,

Nagoya, Japan, 1995, pp. 39-43.

[2] M. Dorigo and G. D. Caro, “Ant colony optimization: a new

meta-heuristic,” in Proc. the 1999 Congress on Evolutionary

Computation, Washington, DC, 1999, pp. 1470-1477.

[3] K. S. Tang, K. F. Man, S. Kwong, and Q. He, “Genetic algorithms and

their applications,” IEEE Signal Processing Magazine, vol. 13, pp.

22-37, 1996.

[4] D. Karaboga, “An Idea Based On Honey Bee Swarm For Numerical

Optimization,” Technical Report-TR06, Erciyes University,

Engineering Faculty, Computer Engineering Department, 2005.

[5] D. Karaboga and B. Akay, “A comparative study of Artificial Bee

Colony algorithm,” Applied Mathematics and Computation, vol. 214,

pp. 108-132, 2009.

Lecture Notes on Software Engineering, Vol. 1, No. 3, August 2013

257

[6] W. Xiang and M. An, “An efficient and robust artificial bee colony

algorithm for numerical optimization,” Computers & Operations

Research, In Press, 2012.

[7] F. Kang, J. Li, and H. Li, “Artificial bee colony algorithm and pattern

search hybridized for global optimization,” Applied Soft Computing, In

Press, 2013.

[8] W. Y. Szeto, Y. Wu, and S. C. Ho, “An artificial bee colony algorithm

for the capacitated vehicle routing problem,” European Journal of

Operational Research, vol. 215, pp. 126-135, 2011.

[9] X. Yan, Y. Zhu, W. Zou, and L. Wang, “A new approach for data

clustering using hybrid artificial bee colony algorithm,”

Neurocomputing, vol. 97, pp. 241-250, 2012.

[10] J. Yu and H. Duan, “Artificial Bee Colony approach to information

granulation-based fuzzy radial basis function neural networks for

image fusion,” Optik, In Press, 2012.

[11] M.-H. Horng, “Multilevel thresholding selection based on the artificial

bee colony algorithm for image segmentation,” Expert Systems with

Applications, vol. 38, pp. 13785-13791, 2011.

[12] W.-C. Hong, “Electric load forecasting by seasonal recurrent SVR

(support vector regression) with chaotic artificial bee colony

algorithm,” Energy, vol. 36, pp. 5568-5578, 2011.

[13] W.-D. Chang, “Nonlinear CSTR control system design using an

artificial bee colony algorithm,” Simulation Modelling Practice and

Theory, vol. 31, pp. 1-9, 2013.

[14] G. Li, P. Niu, and X. Xiao, “Development and investigation of efficient

artificial bee colony algorithm for numerical function optimization,”

Applied Soft Computing, vol. 12, pp. 320-332, 2012.

[15] G. Zhu and S. Kwong, “Gbest-guided artificial bee colony algorithm

for numerical function optimization,” Applied Mathematics and

Computation, vol. 217, pp. 3166-3173, 2010.

[16] W. Gao, S. Liu, and L. Huang, “A global best artificial bee colony

algorithm for global optimization,” Journal of Computational and

Applied Mathematics, vol. 236, pp. 2741-2753, 2012.

[17] B. Alatas, “Chaotic bee colony algorithms for global numerical

optimization,” Expert Systems with Applications, vol. 37, pp.

5682-5687, 2010.

[18] A. V. Chechkin, R. Metzler, J. Klafter, and V. Y. Gonchar,

“Introduction To The Theory Of Lévy Flights,” in R. Klages, G.

Radons, and I. M. Sokolov (Eds.), Anomalous Transport: Foundations

and Applications, John Wiley & Sons, pp. 129-162, 2008.

[20] X.-S. Yang, Engineering Optimization An Introduction with

Metaheuristic Applications, first ed., New Jersey: John Wiley and Sons,

2010.

[21] C.-Y. Lee and X. Yao, “Evolutionary Algorithms with Adaptive Levy

Mutations,” in Proc. the 2001 Congress on Evolutionary Computation,

Seoul, South Korea, 2001, pp. 568-575.

H. Hakli was born in Turkey on September 3, 1989. He

graduated valedictorian from the Department of

Computer Engineering at Selcuk University in 2011. He

is receiving his MS degree (thesis phase) from the same

department and same university. He has been a research

assistant in Department of Computer Engineering since

2011. His interest areas include optimization,

nature-inspired algorithms, machine learning and

artificial intelligence systems.

H. Uğuz was born in Turkey in 1977. He graduated from

the Department of Computer Engineering at Selcuk

University in 1999. He received his MS degree from the

same department and same university in 2002. His PhD

degree was received from the Department of Electrical

and Electronics Engineering at Selcuk University in

2007. He has been an associate professor in the

Department of Computer Engineering since 2007. His

interest areas include hidden Markov models, machine learning and artificial

intelligence systems.

Lecture Notes on Software Engineering, Vol. 1, No. 3, August 2013

258

[19] X.-S.Yang and S. Deb, “Multiobjective cuckoo search for design

optimization,” Computers & Operations Research.