2019 RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing (NCSP2019)
Honolulu, Hawaii, USA, March 4-7, 2019

Artificial Bee Colony Algorithm for Time-Varying Optimization
Ken Kamiyotsumoto† , Yoko Uwate† , Thomas Ott‡ and Yoshifumi Nishio††

Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {kamiyotsumoto, uwate, nishio}@ee.tokushima-u.ac.jp‡Zurich University of Applied Sciences
Einsiedlerstrasse 31a, 8820 Waedenswil, Switzerland
E-mail: thomas.ott@zhaw.ch

Abstract
Recently, nature-inspired metaheuristic optimization algorithms such as Artificial Bee Colony Algorithm (ABC) is developed. ABC is based on the feeding behavior of bee herds.
ABC can not solve for time-varying function.
In this study, we offer a new ABC for time-varying function. We propose ABC in which the scout bee is improved
probability by normal distribution. We compare the best solution with ABC, previous method and the proposed method.
Object function which optimal solution moves circle of shapeis used. We investigate characteristic of proposed method according to variance of normal distribution. Best value andorbit of solutions for proposed method are better than thoseof other method.
1. Introduction
Optimization is to search optimal solution under condition.
Benefit of optimization is high efficiency. Optimization isneeded in various scenes. Combinatorial optimization problem is often solved by metaheuristic optimization algorithms.
Many combinatorial optimization problems often resultsin local optima. Metaheuristic optimization algorithms aresolved high performance in these problems. These algorithms are developed to solve more efficiency and larger problems. Metaheuristic optimization algorithms have Evolutionary Algorithm (EA), Swarm Intelligence (SI) algorithm, local
search, etc. In our study, we choose the SI.
SI is one of the artificial intelligence techniques. SI wasborn from swarm of insect. Examples existing in nature areant, bee and firefly, etc. In insect colonies, each insect has
its the group in total appears to be highly organized. The applications of SI technique are a self-driving car and data mining. The good points of this technique are smaller control system and multi control by simple systems. SI algorithms have
Artificial Bee Colony Algorithm (ABC), Ant Colony Optimization (ACO), Particle Swarm Optimization (PSO), Firefly

Algorithm (FA), etc [1]. The ABC is used by our proposemethod.
ABC is idealized the social behavior of bees based on theirfeeding characteristics. ABC was proposed by Karaboga
Dervis in 2005 [2]. Artificial intelligence algorithms havebeen demonstrated to show effectiveness and efficiency tosolve difficult optimization problems [3]. ABC performs
higher ability in high dimensional optimization problem thanother SI algorithms. However, demerit of ABC is that it cannot search time-varying function.
In previous study, we modified ABC for time-varying function (ABCTV) [4]. However, we can not find accurate solution.
In this study, we propose ABCTV in which the scout bee is
improved by normal distribution. We compare the best value
and the orbit of the solution with ABC, ABCTV and the proposed method.
2. Artificial Bee Colony Algorithm
Karaboga has proposed an Artificial Bee Colony (ABC)algorithm in 2005. The ABC algorithm is based on real bee
behavior and consist of 3 kinds of honeybee, employed bee,onlooker bee and scout bee. It is suitable when the object
function is high dimension [5]. The procedure for solving anoptimization problem in the ABC algorithm is shown below.
Step0.
Initialize each total number of employed bees ne andonlooker bee no , the colony size N = ne + no and thetotal number of iterations tmax .
Step1.
(1) Set the locations of the xi . i is the number of theemployed bee.
(2) Calculate the fitness fi in the initial arrangement byEq. (1). The best fi and initial arrangement are stored.

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2019 RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing (NCSP2019)
Honolulu, Hawaii, USA, March 4-7, 2019In case of ABC, it can not be adapt fitness function to changing object function. The artificial bee colony algorithm fortime-varying solution (ABCTV) is proposed. ABCTV wasmodified 2 points. First, when employed bee compares fitness function values between 2 locations, previous location isreevaluated for each simulation count. Second, we calculate
step3.
best fitness function value for each simulation count.
(1) Based on the fitness, the probability pi is calculated
According to these modification, ABCTV can search timeby Eq. (2).
varying solution. However, ABCTV can not search accurate(2) Select the number of employed bee i based on pi , solution than the standard ABC. Figure 2 shows flowchart ofand Step1 is applied. When this procedure has been re- ABCTV.
peated no times.

Step2.
(1) New locations vi is generated and calculated the fitness function value.
(2) Based on the fitness, the best location and fitness update.

step4.
If function value of each i is better than function valueof all bees, the solution and the function best value areupdated.
step5.
The employed bee which has not been generated the newlocation.
Step6.
Repeat steps 1 to 4 and output the solution.
Fitness function fi and probability pi are given by following equations:1
if g(xi ) ≥ 0
fi =(1)
1 + g(x ) 1 + |g(xi )|otherwise
ipine∑= fi /
fn ,(2)

Figure 2: Flowchart of ABCTV.
4. Proposed method

i=1
We propose ABCTV in which the scout bee improves bywhere, g(xi ) expresses the objective function. Figure 1 shows normal distribution. The scout bee searches a new location
flowchart of ABC.
randomly in the standard ABC. However, ABC has problem point that the employed bee which was able to searchalong the orbit is repositioned outside the orbit. Therefore,
we modify that the scout bee searches new location by normal distribution. From the beginning of the simulation scout
bee searches new location by using Eq. (3).

()
1(x − µ)2f (x) = √exp −2σ 22πσ 2Figure 1: flowchart of ABC.

(3)

where σ 2 corresponds to the variance of normal distribution
and µ denotes the mean of normal distribution. Figure 1
shows some examples of probability density functions of normal distribution.

From Fig. 1, variance of normal distribution σ 2 means
range of probability and mean of normal distribution ex3. Previous method
presses the position with the highest probability. In this simThe most of algorithms exclude time-varying solution. ulation, we set µ as position of employed bee which has theHowever, when we search optimal solution, we need to ap- best fitness value. According to this modification, we expect
ply the both solutions (time-varying and not time-varying). that proposed method can search more accurate best solution383

2019 RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing (NCSP2019)
Honolulu, Hawaii, USA, March 4-7, 2019

φ(x)

x

Figure 3: Normal distribution.
Figure 5: Orbit of ABCTV.

and orbit. We compare 6 kinds of variance by changing theparameter σ 2 (σ 2 =1.0, 10.0, 30.0, 50.0 and 100.0). We investigate best value and orbit of solutions by variance of normal
distribution.
5. Simulation result
In order to evaluate the performance of the propose
method, we compare the result of Eq. (4).
(x1 − 250 − 125 sin αk)2
g(x1 , x2 , k) = 1 − exp[−2 · 402(x2 − 250 + 125 sin αk)2−].
2 · 402

Figure 6: Orbit of propose method (σ 2 = 1.0).
(4)Equation (4) simulates a visual tracking problem. The optimal function value is 0, and position along a circle of radius
125 centered at (x1 , x2 ) = (250, 250), then return its original
position in (2π/α) steps. k means time that is approximatedby simulation count. We express the constant value to use forsimulation in Eq. (5).
ne = no = 50, tmax = 100000, α = 0.01

(5)

Figure 7: Orbit of propose method (σ 2 = 10.0).

First, we investigate the orbit of each algorithm. The results of orbit are summarized in Figs. 4 to 10.

Figure 8: Orbit of propose method (σ 2 = 30.0).

Figure 4: Orbit of ABC.

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2019 RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing (NCSP2019)
Honolulu, Hawaii, USA, March 4-7, 2019

From Table 1, it is clear that the best value of proposed
method better than best value of ABCTV. Furthermore, proposed method has good effect for best value as variance σ 2 increase. However, proposed method when σ 2 = 10.0 is good
performance without depending on that trend. The reason is
we estimate that there is a proper range of probability for normal distribution when employed bee is initialized.
6. Conclusion
Figure 9: Orbit of propose method (σ 2 = 50.0).

Figure 10: Orbit of propose method (σ 2 = 100.0).

This paper shows the ABC modified that scout bee ischanged probability because this algorithm do not applied
for accurate time-varying solution. We tried improvementof ABC, where scout bee searches new position by probability based on normal distribution. We set 5 kinds of variance of normal distribution and mean by position of best employed bee. We compared the best values and orbit of proposed method, previous method and ABC. As a result, when
variance of normal distribution is small, proposed method
could search better orbit than other algorithms. When variance of normal distribution is large, proposed method could
search better value than other algorithms. Therefore, propose
method is better performance than other methods.
In the future work, we would like to investigate the mechanism of the proposed method in detail. Furthermore, we try
to search high dimensional problems.

Figure 4 shows that ABC can not search optimal orbit.
References
However, ABCTV and proposed method can search optimalorbit. From Fig. 5, ABCTV can search to rough out optimal [1] X. S. Yang, “Nature-Inspired Metaheuristic Algorithms
Second Edition,” Luniver Press, 2010.
orbit. From Figs. 6 to 10, the orbit of proposed method can2
search accurate optimal orbit as variance σ decrease. When[2] D. Karaboga, “An idea based on honeybee swarm for nuσ 2 is 100, dispersion of orbit is biased compared with that ofmerical optimization,” Technical Report TR06, 2005.
ABCTV. The reason is that there is no diversity in determining the mean of the normal distribution.
[3] I. Iimura and S. Nakayama, “Search Performance EvalNext, we compare best value for each method. We show bestuation of Artificial Bee Colony Algorithm on Highvalue of solution for each method by Table 1. This simulation
Dimensional Function Optimization,” Transactions of theshows the best solution out of 10 times. The best solution is
Institute of Systems, Control and Information Engineers,
the smallest value in results of 10 times simulations.
vol. 24, pp. 97-99, 2011.
Table 1: Best solution of each method (α = 0.01).

methodABCABCTV
propose method (σ2 = 1.0)
propose method (σ2 = 10.0)
propose method (σ2 = 30.0)
propose method (σ2 = 50.0)
propose method (σ2 = 100.0)

best value
1.93010e-09
2.29477e-07
3.73631e-07
9.92083e-08
3.06453e-07
2.30944e-07
6.78707e-08[4] T. Nishida, “Modification of ABC Algorithm for Adaptation to Time-Varying Functions,” Electronics and Communications in Japan, vol. 96, 2013.
[5] W. Y. Szeto and 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.

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