2019 International Symposium on Nonlinear Theory and Its Applications
NOLTA2019, Kuala Lumpur, Malaysia, December 2-6, 2019.

Modification of Artificial Bee Colony Algorithm
under Time-Varying Condition
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 thebest solution with ABC, previous method and the proposedmethod. Object function which optimal solution moves circle of shape is used. We investigate characteristic of proposed method according to variance of normal distribution.
Best value and orbit of solutions for proposed method arebetter than those of other method.
1. Introduction
Optimization is to search optimal solution under condition. Benefit of optimization is high eﬃciency. Optimization is needed in various scenes. Combinatorial optimization problem is often solved by metaheuristic optimization
algorithms.
Many combinatorial optimization problems often resultsin local optima. Metaheuristic optimization algorithms areprovided good-quality solutions which are close to the optimal solution with less computational eﬀort 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, wechoose 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 datamining. 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 usedby our propose method.
ABC is idealized the social behavior of bees basedon their feeding characteristics. ABC was proposed byKaraboga Dervis in 2005 [2]. Artificial intelligence algorithms have been demonstrated to show eﬀectivenessand eﬃciency to solve diﬃcult optimization problems [3].
ABC performs higher ability in high dimensional optimization problem than other SI algorithms. However, demerit of
ABC is that it can not search time-varying function.
In previous study, we modified ABC for time-varying
function (ABCTV) [4]. However, we can not find accurate
solution. [5]In this study, we propose ABCTV in which the employed
bee is improved by predictive value. We compare the bestvalue 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, employedbee, onlooker bee and scout bee. It is suitable when theobject function is high dimension [6]. The procedure forsolving an optimization problem in the ABC algorithm isshown 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.
Step2.
(1) New locations vi is generated by Eq. (2) and calculated the fitness function value.

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(2) Based on the fitness, the best location and fitness
update.
step3.
(1) Based on the fitness, the probability pi is calculated
by Eq. (3).
(2) Select the number of employed bee i based on pi ,and Step1 is applied. When this procedure has been
repeated no times.
step4.
If function value of each i is better than function valueof all bees, the solution and the function best value areupdated.

In case of ABC, it can not be performed adapted to environmental changing such as shifts in the optimal solution orvariations of the cost function. The artificial bee colony algorithm for time-varying solution (ABCTV) is proposed.
ABCTV was modified 2 points. First, when employed
bee compares fitness function values between 2 locations,previous location is reevaluated for each simulation count.
Second, we calculate best fitness function value for eachsimulation count.
According to these modification, ABCTV can searchtime-varying solution. However, ABCTV can not search
accurate solution than the standard ABC. Figure 2 shows
flowchart of ABCTV.

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 following equations:1
i f g(xi ) ≥ 0

fi = (1)1 + g(xi ) 1 + |g(xi )| otherwise{xih + ϕ(xih − xmh )if j = hvi j =(2)xi jotherwisene∑
fn ,(3)pi = fi /i=1

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

Figure 2: Flowchart of ABCTV.
4. Proposed method
We propose ABCTV in which the employed bee improves by prediction. The employed bee searches a newlocation by Eq. (1) in the standard ABC. However, ABC
has problem point that the best solution follow for optima.
Therefore, we modify that the employed bee searches new
location by the prediction position. The prediction positionis calculated diﬀerence of the employed bee between current best location and the previous best location. According
to we assume the optimal solution is uniform motion, wepredict the location of the next optimal solution from thediﬀerence between the two locations. Furthermore, we use
equation of PSO to append the predictive value. In PSO,
position is calculated by Eq. (4).
vi jFigure 1: flowchart of ABC.

=xi j + ϕ(xi j − xk j ) + ψ(y j − xi j ),(4)

where y j is information of global best solution and xk j isinformation of local best solution. PSO use informationof global best solution to guide searching. This convert
the predictive value. From the beginning of the simulation
employed bee searches new location by using Eq. (5).

3. Previous method [4]The most of algorithms exclude time-varying solution.
However, when we search optimal solution, we need to apply the both solutions (time-varying and not time-varying).

{vi j- 626 -=xih + ϕ(xih − xmh ) + ψ(x p − xih )if j = h(5)xi jotherwiseTable 1: Simulation results (time-varying)

αABC
ABCTV
proposed method(C=0.5)
proposed method(C=1.0)
proposed method(C=2.0)
proposed method(C=3.0)
proposed method(C=4.0)

0.01
1.08e-08
4.12e-07
2.26e-07
9.21e-08
1.55e-08
4.28e-09
4.31e-08

0.05
1.11e-06
3.34e-06
6.47e-06
3.84e-07
4.35e-07
4.28e-09
1.18e-06

0.10
4.11e-06
9.96e-05
6.75e-06
3.90e-07
8.59e-06
3.52e-06
1.99e-06

ψ is a uniform random parameter in [0,C], where C is anonnegative constant. In this study, we set 5 kinds C (0.5,1.0, 2.0, 3.0, 4.0).
5. Simulation results0.15
4.51e-06
7.33e-05
3.33e-05
6.39e-06
7.12e-06
1.05e-05
1.06e-05

0.20
3.12e-06
2.96e-04
2.62e-06
5.94e-06
2.92e-05
9.34e-07
3.16e-05

0.25
4.55e-06
1.43e-04
6.53e-06
1.07e-05
1.33e-05
4.01e-07
3.88e-06

0.30
6.08e-06
2.77e-05
1.56e-05
5.96e-05
6.71e-06
7.16e-06
1.56e-05

However, many of the proposed methods become less effective as α increases. This is reason that the movement of
the solution becomes discrete as α increases.
Secondly, we confirm that the proposed method has theability to search for not time-varying solutions. We use thethree benchmark functions in Table 2.

Firstly, in order to evaluate the performance of the propose method, we compare the result of Eq. (6).
(x1 − 250 − 125 sin αk)2g(x1 , x2 , k) = 1 − exp[−2 · 402(x2 − 250 + 125 sin αk)2−].
2 · 402

Table 2: Benchmark functions(6)function∑D 2
n=1 xn∑D2(x−√n=1 n 10cos(2πxn ) + 10)∑D∑D 220 − 20exp(−0.2 1/n n=1xn + e − exp(1/n n=1cos(2πxn ))NameSphere Function
Rastrigin Function
Ackley Function

Figure 3 shows the shape of this function.

Figure 4, 5 and 6 show the shape each function. Table 3
shows that the best value of propose method (C=3.0) for
not time-varying environment.

Figure 3: shape of time-varying function.
Equation (6) simulates a visual tracking problem. The
distribution of the minimum value represented by a Gaussian function moves continuously in the figure, and theproblem of tracking this movement is assumed. 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 approximated by simulation count. We express the constantvalue to use for simulation in Eq. (7).
ne = no = 50, tmax = 100000

Figure 4: Sphere function.

(7)

We compare the best value for 7 kinds movement speed of
optimal solution α (α = 0.01, 0.05, 0.10, 0.15, 0.20, 0.25,0.30). The best solution is the smallest value in results of
15 times simulations. Table 1 shows that the best value ofeach method. From Table 1, the best value of the propose
method is better than the best value of ABCTV. Especially,
the propose method when C = 3.0 has best performance.

- 627 -Figure 5: Rastrigin fuction.

the Institute of Systems, Control and Information Engineers, vol. 24, pp. 97-99, 2011.
[4] T. Nishida, “Modification of ABC Algorithm forAdaptation to Time-Varying Functions,” Electronics
and Communications in Japan, vol. 96, 2013.
[5] K. Kamiyotsumoto, T. Ott, Y. Uwate and Y. Nishio,“Artificial Bee Colony Algorithm for Time-Varying
Optimization,” Proceedings of RISP International
Workshop on Nonlinear Circuits, Communications andSignal Processing (NCSP’19), pp. 382-385, 2019.

Figure 6: Ackey fuction.

[6] 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.

Table 3: Simulation result (not time-varying)dimention
SphereRastrigin
Ackley

100
17.39
12.11

30
6.44e-243
92.80
19.23

50
6.11e-48
189.34
19.94

From Table 3, propose the method can search not timevarying solution.
6. Conclusion
This paper shows the ABC modified that employed bee
is changed by predictive value because this algorithm do
not applied for accurate time-varying solution. We tried
improvement of ABC, where employed bee searches newposition by predictive value. The predictive value is calculated by diﬀerence between best solutions by assuming that
optimal solution is constant motion. Furthermore, the predictive value is append searching equation of employed bee
by inspired from PSO. We set propose method has 5 kindsof range of random value. We compared the best values of
proposed method, previous method and ABC. As a result,even if optimal solution is discrete, 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, wetry to search high dimensional problems.
References[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 HighDimensional Function Optimization,” Transactions of- 628 -