IEEE Workshop on Nonlinear Circuit Networks
December 7-8, 2018

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

Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, 770-8506, Japan† , Zurich University of Applied Sciences‡E-mail: {kamiyotsumoto, uwate, nishio}@ee.tokushima-u.ac.jp† ,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 shape is used.
We investigate characteristic of proposed method according tovariance of normal distribution. Best value and orbit of solutions
for proposed method are better than those of other method.

I. I NTRODUCTION
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 results inlocal optima. Metaheuristic optimization algorithms are solved
high performance in these problems. These algorithms aredeveloped 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 was bornfrom swarm of insect. Examples existing in nature are ant, bee
and firefly, etc. In insect colonies, each insect has its the groupin 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 propose method.
ABC is idealized the social behavior of bees based on
their feeding characteristics. ABC was proposed by Karaboga
Dervis in 2005 [2]. Artificial intelligence algorithms have been
demonstrated to show effectiveness and efficiency to solve
difficult optimization problems [3]. ABC performs higher
ability in high dimensional optimization problem than otherSI algorithms. However, demerit of ABC is that it can notsearch 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 isimproved by normal distribution. We compare the best value
and the orbit of the solution with ABC, ABCTV and theproposed method.
II. A RTIFICIAL B EE C OLONY A LGORITHM
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 andthe total 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
by Eq. (1). The best fi and initial arrangement arestored.
Step2. (1) New locations vi is generated and calculated thefitness function value.
(2) Based on the fitness, the best location and fitness
update.
step3. (1) Based on the fitness, the probability pi is calculated by Eq. (2).
(2) Select the number of employed bee i based onpi , and Step1 is applied. When this procedure has
been repeated no times.
step4. If function value of each i is better than function
value of all bees, the solution and the function bestvalue are updated.
step5. The employed bee which has not been generated thenew location.
Step6. Repeat steps 1 to 4 and output the solution.
Fitness function fi and probability pi are given by followingequations:1
if g(xi ) ≥ 0
fi =(1)
1 + g(x ) 1 + |g(xi )| otherwise- 15 -ipine∑= fi /
fn ,(2)i=1

where, g(xi ) expresses the objective function. Figure 1 shows
flowchart of ABC.

point that the employed bee which was able to search along
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).
f (x) = √Fig. 1.

flowchart of ABC.

III. PREVIOUS METHOD
The most of algorithms exclude time-varying solution. However, when we search optimal solution, we need to apply theboth solutions (time-varying and not time-varying). In caseof ABC, it can not be adapt fitness function to changing
object function. The artificial bee colony algorithm for timevarying solution (ABCTV) is proposed. ABCTV was modified
2 points. First, when employed bee compares fitness function
values between 2 locations, previous location is reevaluatedfor each simulation count. Second, we calculate best fitness function value for each simulation count. According tothese modification, ABCTV can search time-varying solution.
However, ABCTV can not search accurate solution than thestandard ABC. Figure 2 shows flowchart of ABCTV.

()(x − µ)2
exp −2σ 2
2πσ 2
1(3)

where σ 2 corresponds to the variance of normal distribution
and µ denotes the mean of normal distribution. Figure 1 showssome examples of probability density functions of normaldistribution. From Fig. 1, variance of normal distribution σ 2means range of probability and mean of normal distribution
expresses the position with the highest probability. In thissimulation, we set µ as position of employed bee which has thebest fitness value. According to this modification, we expectthat proposed method can search more accurate best solution
and orbit. We compare 6 kinds of variance by changing
the parameter 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 ofnormal distribution.

φ(x)xFig. 3.

Normal distribution.

V. S IMULATION RESULTIn order to evaluate the performance of the propose method,
we compare the result of Eq. (4).
(x1 − 250 − 125 sin αk)22 · 402(4)(x2 − 250 + 125 sin αk)2−].
2 · 402
Equation (4) simulates a visual tracking problem. The optimalfunction 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 similaition count. We express the constant value to use forsimulation in Eq. (5).
g(x1 , x2 , k) = 1 − exp[−Fig. 2.

Flowchart of ABCTV.

ne = no = 50, tmax = 100000, α = 0.01

IV. PROPOSED METHOD
We propose ABCTV in which the scout bee improves bynormal distribution. The scout bee searches a new location
randomly in the standard ABC. However, ABC has problem(5)First, we investigate the orbit of each algorithm. The resultsof orbit are summarized in Fig. 4 to Fig. 10.

- 16 -Fig. 6.

Fig. 4.

Orbit of ABC.

Fig. 8.

Orbit of propose method (σ 2 = 30.0).

Fig. 5.

Orbit of ABCTV.

Fig. 9.

Orbit of propose method (σ 2 = 50.0).

Orbit of propose method (σ 2 = 1.0).

Fig. 10.

Orbit of propose method (σ 2 = 100.0).

Figure 4 shows that ABC can not search optimal orbit.
However, ABCTV and proposed method can search optimalorbit. From Fig. 5, ABCTV can search to rough out optimal
orbit. From Fig. 6 to Fig. 10, the orbit of proposed method cansearch accurate optimal orbit as variance σ 2 decrease. Whenσ 2 is 100, dispersion of orbit is biased compared with that ofABCTV. The reason is that there is no diversity in determining
the mean of the normal distribution.

Fig. 7.

Orbit of propose method (σ 2 = 10.0).

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Next, we compare best value for each method. We show
best value of solution for each method by Table I. This
simulation shows the best solution out of 10 times. The bestsolution is the smallest value in results of 10 times simulations.

[5] W. Y. Szeto and Y. Wu and S. C. Ho, “An artificial bee colonyalgorithm for the capacitated vehicle routing problem,” European Journalof Operational Research, vol. 215, pp. 126-135, 2011.

TABLE IB EST SOLUTION OF EACH METHOD (α = 0.01).

methodABCABCTV
propose method (σ 2 = 1.0)
propose method (σ 2 = 10.0)
propose method (σ 2 = 30.0)
propose method (σ 2 = 50.0)
propose method (σ 2 = 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

From Table I, 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 2increase. 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 fornormal distribution when employed bee is initialized.
VI. C ONCLUSION
This paper shows the ABC modified that scout bee ischanged probability because this algorithm don’t applied for
accurate time-varying solution. We tried improvement of ABC,
where scout bee searches new position by probability basedon 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 searchbetter 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 themechanism of the proposed method in detail. Furthermore,
we try to search high dimensional problem.

R EFERENCES[1] X. S. Yang, “Nature-Inspired Metaheuristic Algorithms Second Edition,”
Luniver Press, 2010.
[2] D. Karaboga, “An idea based on honeybee swarm for numerical optimization,” Technical Report TR06, 2005.
[3] I. Iimura and S. Nakayama, “Search Performance Evaluation of Artificial
Bee Colony Algorithm on High-Dimensional Function Optimization,”Transactions of the Institute of Systems, Control and Information Engineers, vol. 24, pp. 97-99, 2011.
[4] T. Nishida, “Modification of ABC Algorithm for Adaptation to Time‐
Varying Functions,” Electronics and Communications in Japan, vol. 96,2013.

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