International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.4, August 2013

EVALUATION THE EFFICIENCY OF ARTIFICIAL BEE
COLONY AND THE FIREFLY ALGORITHM IN SOLVING
THE CONTINUOUS OPTIMIZATION PROBLEM
Seyyed Reza Khaze1, Isa maleki2, Sohrab Hojjatkhah3 and Ali Bagherinia4
1

Department of Computer Engineering, Dehdasht Branch, Islamic Azad University, Iran,

2

Department of Computer Engineering, Dehdasht Branch, Islamic Azad University, Iran,

3

Department of Computer Engineering, Dehdasht Branch, Islamic Azad University, Iran,

4

Department of Computer Engineering, Dehdasht Branch, Islamic Azad University, Iran,

khaze@iaudehdasht.ac.ir, khaze.reza@gmail.com
maleki@iaudehdasht.ac.ir, maleki.misa@gmail.com
hojjatkhah@gmail.com
ali.bagherinia@gmail.com

ABSTRACT
Now the Meta-Heuristic algorithms have been used vastly in solving the problem of continuous
optimization. In this paper the Artificial Bee Colony (ABC) algorithm and the Firefly Algorithm (FA) are
valuated. And for presenting the efficiency of the algorithms and also for more analysis of them, the
continuous optimization problems which are of the type of the problems of vast limit of answer and the
close optimized points are tested. So, in this paper the efficiency of the ABC algorithm and FA are
presented for solving the continuous optimization problems and also the said algorithms are studied from
the accuracy in reaching the optimized solution and the resulting time and the reliability of the optimized
answer points of view.

KEYWORDS
Meta-Heuristic Algorithm, Artificial Bee Colony (ABC), Firefly Algorithm (FA), Continuous Optimization

1. INTRODUCTION
Now the use of the Meta-Heuristic algorithms in accessing the optimized solution in the
continuous optimization problems has progressed a lot. According to the increase of the
complexity of the continuous optimization problems and the inability of the mathematical
methods for the optimized solution, the Meta Heuristic algorithms are the suitable solution for the
continuous optimization problems. The mathematical methods are used in many scientific and
engineering problems and cover a vast area of the different problems but despite the accurate
efficiency, the mathematical methods still face many problems solving the optimization problems.
The late researches and the struggles of the researchers have led to innovation of the algorithms
which have been inspired by the natural phenomenon, the ones which study the completion and
the behavior of the creatures of the nature and finally they have led to the Met-Heuristic
algorithms. The Meta-Heuristic algorithms have been efficient in solving the combined
optimization problems in finding the optimized solution [1, 2, 3 and 4].
Many Meta-Heuristic algorithms have been innovated inspiring the nature of which the Particle
Swarm Optimization (PSO) [5], Artificial Bee Colony (ABC) [6], Firefly Algorithm (FA) [7],
Bee Colony Optimization (BCO) [8] and Ants Colony Optimization (ACO) [9] could be pointed
out. The ABC algorithm [6, 10] is a Meta-Heuristic algorithm which is inspired by the mining
DOI:10.5121/ijcsa.2013.3403

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International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.4, August 2013

behavior of the bee colony for solving the continuous optimization problems of large space. ABC
algorithm starts working by creating the primary population of the random vectors. It works in a
way that in any repetition of the algorithm, the artificial bees randomly search the answers which
have been resulted in the previous repetition to find new answers. It is clear that the new answers
would not be necessarily better than the answers found in previous repetition. When any of the
artificial bees find a new answer, they will go back to the hive and will make decision for the next
move path in the next repetition. So, the optimization rate of the answers is calculated by the bees
and then the answer which is more befitting, will be selected as the search path in next repetition.
So, the area around the more optimized answers will be searched by more bees in the next
repetition. The search process continues until the needed conditions for ending the execution of
the program would be met.
The FA [7] is one of the newest Meta-Heuristic algorithms based on the swarm intelligence which
is used in solving the continuous optimization problems. First some artificial firefly are randomly
distributed in the problem space in FA, and then any firefly emit light, the intensity of which is in
conformity to the optimization rate of the point the firefly stands on. Then the light intensity of
any firefly is compared to the light intensity of the fireflies and the low light firefly goes toward
the intense lighted firefly. Also the most intense firefly moves around the problem for finding the
global optimized answer randomly. So, in FA the fireflies get in relationship with each other via
the light. The combination of these operations leads to the movement of the all fireflies toward
the more optimized points. In this paper, we study the ABC algorithm and FA and to show the
efficiency of these algorithms, will solve some continuous optimization functions.
The structure of the paper is as follows: in the section 2, we have introduce the related works; in
the section 3, the Meta-Heuristic algorithm is introduced; in the section 4, the analysis of ABC
and FA are for solving the continuous optimization problems has been studied; in the section 5,
we have evaluated the results of ABC and FA are; in the section 6, ABC and FA are discussed
and at finally in section 7, conclusion and future works is presented.

2. R ELATED WORKS
X. Lia [11] has used the Particle Swarm Optimization (PSO) Algorithm and Genetic Algorithm
(GA) to solve the continuous optimization problems. He has studied the PSO and GA algorithms
to test and evaluate the efficiency factor on 36 functions. To clearly show the efficiency of the
PSO and FA algorithms, he tested the functions in 30 dimensional spaces. The results of the tests
show that these algorithms have worked well on the 30 functions and found the optimized
answers. The researchers [12] have used the Dynamic PSO and Simple PSO to solve the
continuous mathematical functions. They have studied the functions in the 2 and 10 dimensional
spaces to show the efficiency of both algorithms. They have studied the parameters of the
Dynamic PSO and Simple PSO to analyze the mathematical functions and they have resulted that
Dynamic PSO is more efficient for solving the continuous problems and creates the answers close
to the optimized one. Reference [13] has used the ABC algorithm and the Differential Evolution
(DE) for solving the optimization problems in large scale. In this reference, a combined method
named DEM-ABC has been suggested. In the combined method for global convergence of the
ABC Algorithm, the DE Mutation Strategy is used. The combined algorithm is tested on some
functions in large scales. The results show that the combined algorithms more efficient than the
ABC Algorithm.
Researchers [14] have used ABC algorithm and Hybrid Artificial Bee Colony (HABC)
Algorithms for solving the continuous optimization problems. The DE algorithm is used for
optimizing the answers of the continuous optimization functions in HABC. They have tested the
ABC and HABC algorithms for evaluation and efficiency factors on 6 functions in the paper. To
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International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.4, August 2013

show the efficiency of the ABC and HABC algorithms, they have studied the functions in 30 and
60 dimensional spaces. The results show that HABC is more efficient than ABC algorithm.
Researchers [15] have used FA to solve the non-linear continuous functions. The results of the
tests show that FA is fast enough in convergence toward the optimized solution and finds the
optimized answer in a very short time. Reference [16] has used the ABC algorithm for solving the
continuous optimization functions. Improving the ABC algorithm, it is tried to go through
Exploitation and Local Search operations for reaching the optimized the answer in this reference.
The results show that the improve ABC algorithm has reached more optimized answer than the
ABC algorithm.
Researchers [17] have used the Bacterial Foraging Optimization Algorithm (BFOA) to study and
analyze the continuous functions. They have showed the efficiency of the BFOA by solving some
continuous functions. The goal of these functions is to find the optimized answers in multidimensional spaces. They have also studied the effects of the number of the bacteria on solving
the functions and have cited that accurate identification of the parameters makes BFOA be very
effective in optimizing the problems. Reference [18] has used the ABC algorithm to solve the
continuous optimization functions. To make better global search in this reference, some changes
has took place on exploration and exploitation operations using DE algorithm. The goal was to
make better global search and make ABC algorithm reach the answer as fast as possible. The
results of the tests show that the algorithm is able to find the optimized solution. The researchers
[19] have used FA and PSO algorithms to solve the continuous optimization problems. They have
cited that the PSO algorithm is very efficient in exploration and exploitation operations of the
continuous optimizations problems and also the FA is not very complex and is able to find the
optimized answer in a very short time.

3. META-H EURISTIC ALGORITHMS
The Meta-Heuristic algorithms are the tools for finding the answers or the answers close to the
optimized ones [20]. These algorithms utilize the two concepts of searching and cooperation
search the searching space of an optimization problem. So, the more powerful is an algorithm in
controlling these two parameters, the more able is the algorithm in finding the answers close to
the optimized one for the problem.

3.1 ABC
The ABC algorithm is innovated in 2005 by Karaboga inspiring the social life of the bees to solve
the optimization problems [6]. This algorithm is a simulation of the food search of the group of
the bees. The bees can be distributed in far and different distances to utilize the food resources
[21]. In ABC algorithm, the bees are classified in three groups: 1. Employed bees, 2. Onlooker
bees, 3. Scout bees.
The food search process starts by the employed bees. Each employed bee dances in a specific way
when finds food resource and the onlooker bees look at the dance of the employed bees to
understand the food resource location and the scout bees randomly look for the food in the around
environment.
In ABC algorithm the primary value stages of the employed, onlooker and scout bees are as
follows:

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International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.4, August 2013

Employed bees: In this stage the artificial bees searching around the food resource at xi point will
search for the better food resource at new location of vi [22]. Identification of the new food
resource takes place by the equation (1) [23].

vij  xij  ij ( xij  xkj )

j  1,2,..., n; k  1,2,..., SN
(1)

In equation (1), vi= [vi1, vi2,… vin] is the new location vector of the bees, xi= [xi1, xi2 , …, xin] is the
location vector of the ith bee, k (k≠j) is a correct random number in [1, SN] and the SN is the
number of the artificial bees. Φij is a random number uniformly distributed in [-1, 1]. The random
xi number selection from the problem limit is done by the equation (2) [23].

xij  L j  rand (0,1)  (U j  L j )
(2)
In equation (2), Uj and L j are the top limit and the down limit of the xi variable respectively and
the rand() is the random numbers function in (0, 1). When the new location of the food resource is
identified, the optimization of it must be calculated. So, the befitting rate of the xi vector is
identified according to the equation (3) [23].

1


fit i   1  f i
1  abs( f i )

fi  0
fi  0
(3)

Onlooker bees: In this stage any of the onlooker bees decide to search around the found food
resource by the specified possibility [22]. The onlooker bees make their selection according to the
possible values of the employed bees. So, the possibility of selection of the food resource by the
onlooker bees is calculated using the equation (4) [23].

pi 

fit i
SN

 fit

i

j 1

(4)
Scout bees: In ABC algorithm, if the known number of the repetitions would not lead to the
optimized answer, some of the bees leave their solution and become scout bees to randomly
search the limits of the problem for increasing the search process efficiency [22]. Execution of the
scout bees’ stage can increase the possibility of finding the global optimized answer.

3.2 FA
FA is of the algorithms based on the population which is introduced in 2008 by Yang [7]. First a
number of artificial bees are randomly produced in problem space in FA. Then a light intense is
related to any of the fireflies using the value found for the goal function at that point. The light
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International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.4, August 2013

intense valuing of any of the artificial fireflies is in a way that increasing the optimization rate of
the point a firefly stands on leads to increase of the light intense of it. The low light fireflies are
attracted toward the high light fireflies and this continues till all fireflies are gathered in one point
which is probably the global optimized point. Updating the law of the movement of the low light
bees toward the high light ones takes place using equation (5) [24].

x i  xi   0 e

rij2

1
( x j  x i )   ( rand  )
2

(5)
In equation (5), the values of α, β0 and γ are considered constant. α, β0 are selected from [0, 1] and
γ is selected from [0, ∞). Also, rij is the Euclid distance of the two fireflies which is identified as
equation (6) [24].
n

rij  xi  x j 

 (x

i, k

 x j , k )2

k 1

(6)
The absorption coefficient between two fireflies takes place using equation (7).

   0e

 rij

(7)
In equation (7), β0 shows the maximum of absorption and is identified in [0, 1]. The γ parameter
is the absorption coefficient and is identified in [0, ∞). The r parameter identifies the distance
between the fireflies and the value of it is calculated by equation (6). If β0=0, any of the fireflies
searches the problem space without any contribution of the other fireflies and the search takes
place randomly. Also, if γ=∞, this leads to the random search in problem space.

4. EVALUATION THE EFFICIENCY OF ABC
C ONTINUOUS O PTIMIZATION F UNCTIONS

AND

FA

FOR

SOLVING

THE

Searching the continuous optimization functions using the Meta-Heuristic algorithms leads to the
most optimized solution among the possible solutions. The Meta-Heuristic algorithms in solving
the continuous optimization problems are very efficient in getting close to the optimized answer.
These algorithms study the continuous optimization problems answers using the testing and the
better searching methods according to the problem structure and the complexity type of it and
produce accessing the optimized answers.

4.1 ABC in Solving the Continuous Optimization Functions
The goal of the optimization is to find the optimized solutions according to the limits and the
needs of the problems. It is possible for an optimization problem to have different solutions and
for selecting the optimized answer the goal function must be used. The flowchart of evaluation of
the goal function of the ABC algorithm for continuous optimization problems is showed in figure
(1).

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International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.4, August 2013

Figure 1: The Flowchart of ABC Algorithm for Solving the Continuous Optimization Functions

Figure (2) shows the quasi code of ABC algorithm for solving the continuous optimization
Functions.

Figure 2: The Quasi Code of ABC Algorithm for Solving the Continuous Optimization Functions.

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International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.4, August 2013

4.2 FA in Solving the Continuous Optimization Functions
Evaluation flowchart of the FA for solving the continuous optimization functions is shown in
figure (3).

Figure 3: Flowchart of the FA for Solving the Continuous Optimization Functions

Figure (4) shows the FA quasi code for solving continuous optimization functions

1. Initialize Parameters
2. Do
3. Evaluation Function
4. Evaluation Fitness
5. The Algorithm of Firefly Algorithm
for i=1 to n do
for j= 1 to i do
Move firefly i towards j
Move firefly i randomly
Evaluate new solutions
End for j
End for i
Rank the fireflies and find the current best
6. While (a stop criteria maximum iteration)

Figure 4: FA Quasi Code for Solving Continuous Optimization Functions

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International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.4, August 2013

5. EVALUATION AND R ESULTS
For evaluation and efficiency of the ABC and FA are, the Rastrigin two and three dimensional
function is studied [25]. This function holds many maximum and minimum points which have
made it be used as a test function for evaluation of the Meta-Heuristic Algorithms. So, for
comparison of the evaluation and efficiency of these algorithms, the Rastrigin function based on
optimized solution factor is used.


Two dimensional Rastrigin function
2

f1 ( x)  20   [ xi2  10 cos(2xi )]
i 1



Three dimensional Rastrigin function
3

f 2 ( x)  30   [ xi2  10 cos(2xi )]
i 1

The Meta-Heuristic algorithms are very delicate for their parameters and the settlement of the
parameters can affect their operation. The parameters settlement leads to more flexibility and
reliability of the algorithm. So, settlement of the parameters is one of the important factors in
reaching the optimized solution in continuous optimization problems. The population selection is
very important in Meta-Heuristic algorithms. If the population number is low, the problem will
soon be convergent and we will not get the favored answer or close to the global optimized
answer, and if the population number is high a long time is needed for the algorithm to be
convergent. So, the number of the population must be suitable and in conformity to the
optimization problem to get the optimized solution. For ABC and FA to effectively search the
functions space, the number of primary population is set 50 and the number of repetitions is set
100 for both algorithms. The results of Table (1) show that using the ABC and FA algorithm
makes getting the optimized solution possible. So, ABC and FA are well able to find the
optimized points in continuous optimization problems.
Table 1: Finding the Optimized Solution
Function
f1
f2

Range of search Space
±30
±30

ABC
0.0059
0.0136

FA
0.1287
0.7516

To show the efficiency of ABC and FA are, the convergence diagram is used. The operation of
the algorithms in convergence toward the optimized answer or the suitable number of the
repetitions is showed in figure (5). Studying the diagrams show that first the starting answers of
the algorithms are randomly selected from the answer space and then by repeating the algorithms,
the value of the goal function will get close to the optimized answer.

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International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.4, August 2013

Figure 5: The Convergence Diagram of ABC and FA for Solving the Two and Three Dimensional Rastrigin
Function

Table (2) shows the number of the repetition of the execution of ABC and FA are for two
dimensional Rastrigin function. The reason of using the repetitive repetitions is to show the
existence of probability in the structure of these algorithms. The results of Table (2) show the fact
that ABC algorithm is more efficient in finding the global optimized points.
Table 2: Comparison of the Results of ABC and FA for f1 function
Algorithm
ABC
FA

20
1.0372
1.5273

40
1.1610
1.3604

Iteration
60
80
1.0056 0.0148
1.1312 1.0228

100
0.0059
0.1287

Figure (6) shows comparison diagram of the results of ABC and FA for f1 function.

Figure 6: The Comparison Diagram of the Results of ABC and FA for f1 Function

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International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.4, August 2013

Table (3) shows the repetition number of execution of ABC and FA for three dimensional
Rastrigin function. As it is seen, ABC algorithm is more efficient in solving the continuous
optimization functions of large dimensions and is more able in finding the global optimized
points. So, ABC algorithm is able to retain the balance of the local and global search of the
problem despite the increase in the dimensions of it in an optimized way.
Table 3: Comparison of the Results of ABC and FA for f2 Function
Algorithm
ABC
FA

20
1.6723
2.2512

40
1.1688
2.1298

Iteration
60
80
1.6165
0.0826
1.6228
1.1743

100
0.0136
0.7516

Figure (7) shows comparison diagram of the results of ABC and FA for f2 function.

Figure 7: The Comparison Diagram of the Results of ABC and FA for f2 Function

6. DISCUSSION
ABC and FA are efficient algorithms in different solving the continuous optimization problems.
These algorithms use the quality parameters number the value of which could be settled easily.
Also the speed of convergence of ABC and FA is very high in probability of finding the global
optimized answer. SO, it is possible to find the optimized solution in continuous optimization
problems using ABC and FA are. One of the merits of ABC algorithm is the abandonment
phenomenon. It means the time the employed bees would not find the optimized solutions after
some repetitions, and then they transform to the scout bees again and move in random paths to
start searching for optimized solutions. By this way the solutions which are not optimized will be
abandoned and again the global optimized points are searched. So, the behaviour of the bees for
finding the optimized points is a combination of the two methods of local and global searching.
In comparison to the other Meta-Heuristic algorithms, ABC algorithm is a sample one to some
extent because this algorithm just uses the three basic specifications of colony population, the
maximum number of the repetitions and the abandonment factor. So, the implementation of ABC
is very simple from calculation point of view and if the right values are used for its parameters,
there would be high probability of finding the global
optimized answer. For escaping the local optimization, ABC algorithm acts in a way that when it
faces such location, the bees will be transferred to other parts of the search space and then will
search the optimized answers there and will repeat this till reaching the global optimized answer.
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International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.4, August 2013

ABC algorithm is well efficient in multi variable functions and also the functions which have
local minimums and maximums. The movement of the employed bees to go further than the
unsuitable places makes the algorithm work well facing the problems of very high dimensions
and also the problems in which the population is primarily unsuitably distributed. So, the success
rate of ABC algorithm in solving the optimization problems is high. The employed bees try to get
close to the optimized areas an any stage in ABC and finally they try to find the optimized
solutions by the help of scout bees. One of the merits of FA in solving the continuous
optimization of complex functions is that it is able to change status from one optimization point to
the other one. In FA, if the best solution is not found, the search is not stopped and it is done
around the neighbour of the previous points to find the optimized solution. ABC and FA use the
random variables in continuous optimization process. In accurate words, the answers of these
algorithms are probable in their nature. In fact in these algorithms the search process takes place
around the answers of the previous stage. So, these algorithms are able to escape the local
optimized points. So, these algorithms must be executed repetitively to reach the optimized
solution. In ABC algorithm the befitting function is used to get the optimized points and the
problem space is studied more accurately by the cooperation. But in FA as the fireflies gather the
more lighted firefly, it is not possible to get the optimized points well and this leads to increase of
the speed of convergence to the local optimization points.

7. CONCLUSION AND FUTURE WORKS
In this paper we have studied the ABC and FA for solving the continuous optimization problems
of vast area and the answers close to the optimized one. To evaluate the efficiency of the
algorithms, two types of comparison from answer accuracy and reliability in convergence points
have been studied. After studying the results, it became clear that the probability of convergence
and getting the optimized answer via ABC algorithm is a little more than FA in solving the
continuous optimization problems. ABC algorithm is more compatible in solving such problems
and is also more powerful from answering point and when it is convergent, it is faster than
FA.This is not generalizable to the other problems and such verdict is not right for all
optimization problems. So, ABC algorithm is more reliable in reaching the global optimized
points in comparison to FA algorithm in solving the problems like functions of continuous
optimization. We hope in future will use other Meta-Heuristic algorithms for studying the
continuous optimization problems of high dimensions and more optimized points presenting this
paper.

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Authors
Seyyed Reza Khaze is a Lecturer and Member of the Research Committee of
the Department of Computer Engineering, Dehdasht Branch, Islamic Azad
University, Iran. He is a Member of Editorial Board and Review Board in
Several International Journals and National Conferences. His interested research
areas are in the Software Cost Estimation, Machine learning, Data Mining,
Optimization and Artificial Intelligence.

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International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.4, August 2013

Isa Maleki is a Lecturer and Member of The Research Committee of The
Department of Computer Engineering, Dehdasht Branch, Islamic Azad
University, Iran. He Also Has Research Collaboration with Dehdasht
Universities Research Association Ngo. He is a Member of Review Board in
Several National Conferences. His Interested Research Areas Are in the
Software Cost Estimation, Machine learning, Data Mining, Optimization and
Artificial Intelligence.
Sohrab Hojjatkhah is Currently Head of the Department of Computer
Engineering, Dehdasht Branch, Islamic Azad University, Iran. He Has a
Bachelor's Degree In Software Engineering, Received From Amir Kabir
University, Iran, Then Received A Master's Degree In Artificial Intelligence
From The University Of Shiraz And Currently PhD Candidate In Department Of
Computer Engineering At Science And Research Branch, Islamic Azad
University, Iran. His Interested Research Areas Are In The Image Processing,
Speech Processing, Machine Learning, Data Mining And Artificial Intelligence.
Ali Bagherinia is a lecturer and member of the Research Committee of the
Department of Computer Engineering, Dehdasht Branch, Islamic Azad
University, Iran. He Has a Currently PhD Candidate In Department Of
Computer Engineering At Science And Research Branch, Islamic Azad
University, Iran. His Interested Research Areas Are in the Wireless Sensor
Networks, data mining, optimization and artificial intelligence.

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