2012 International Symposium on Nonlinear Theory and its Applications
NOLTA2012, Palma, Majorca, Spain, October 22-26, 2012

Investigation of Artificial Bee Colony Algorithmwith Two Kinds of Colonies
Yudai Shirasaki† , Yoko Uwate† and Yoshifumi Nishio††Dept. of Electrical and Electronic Eng.,
Tokushima University,
Tokushima 770–8506, Japan
phone: +81–88–656–7470, fax: +81–88–656–7471
Email: { shirasaki, uwate, nishio } @ee.tokushima-u.ac.jp
Abstract—Artiﬁcial Bee Colony (ABC) is wellknown as optimization algorithm based on a particular
intelligent foraging behavior of honeybee swarms. In
this study, We propose Artiﬁcial Bee Colony having
Two Kinds of Colonies (ABCTKC). It has diﬀerent
characteristic each group. We investigate the characteristic and the performance of ABCTKC by fourbenchmark functions and conﬁrm that this new searching strategy has high speed convergence.

interaction of diﬀerent characteristics colonies, ABCTKC is able to explore more eﬀectively than ABC and
ABCTC. First, we research the eﬀect of the information sharing and present the utility of ABCTKC. By
computer simulations, we conﬁrm that ABCTKC canconverge with higher speed than the conventional ABC
by avoiding the local minimum problem.
2. ABC with Two Kinds of Colonies
2.1. Behavior of real bees

1. Introduction
Karaboga has proposed an Artiﬁcial Bee Colony(ABC) [1] [2] algorithm in 2005. The ABC is known
as one of the optimization techniques used for optimization problems such as a linear programming, andthis is an evolutionary algorithm to simulate the intelligent foraging behavior of honeybee swarms [3] [4].
The ABC algorithm is proposed that base on real bee
behavior and consist of three kinds of honeybee, employed bee, onlooker bee and scout bee. Employed bee
has the role of searching and searches wide area. Onlooker bee has the role of development and searches
small area depending on employed bee searching informations. Scout bee that fell into local minima canescape from local minima by scout bee searching. ABC
can obtain high eﬃciency that combine these bees
searching. However, it has a disadvantage that convergence of the entire colony is slow.
In the previous study, we proposed Artiﬁcial Bee
Colony having Twin Colonies (ABCTC) [5]. ABCTC
divides one colony to two. In order to search eﬀectively, we propose the exchange of employed bee between colonies as new strategy of ABCTC. It has highspeed convergence in optimal solution because it caneﬃciently search having the exchange. The strategy
has the eﬀect having decrease search for local minima
in the exchange and can easy to escape from local minima.
In this study, we propose Artiﬁcial Bee Colony having Two Kind of Colonies (ABCTKC). In ABCTKC,
it has diﬀerent characteristics each colony. By the

One of the behavior of honeybee has divide intocolony. When the number of bee of a colony became
many numbers, the production eﬃciency of the colony
decreases. In order to clear up this problem, honeybee
divide into two colonies. From these, honeybee cankeep constant eﬃciency for searching.
2.2. Process of ABCTKC algorithm
The ABCTKC has two diﬀerent characteristiccolonies.
Two colonies are set to Colony1 andColony2, respectively. Colony1 has the same characteristic as ABC and ABCTC, Colony2 has diﬀerent
characteristic from Colony1. In Colony2, employed
bee search new solution using the current best resultof employed bee in Colony1 and the average of whole
position information in Colony2. The algorithm requires several parameters to be set; the number of bees
is M , employed bees Mg is 50% and onlooker bees Ml
is 50%. Initialization count is tRi . We explain theprocess of ABCTKC algorithm in detail as follows.
Step 1 (Initialization):
Let a generation step t = 0, the converge steptRi = 0 and the exchange step te . After the employed bees divided two colonies, disposes the employed bee i = (1, 2, ..., Me ) at random.
Step 2 (Search by employed bee):Colony1 :Employed bees search new food source each- 223 -

group. In order to search better food source, theemployed bee uses the following expression (1):
Xid (t + 1) = Xid (t) + Φid (Xid (t) − Xkd (t))(1)

where X i (t) = (xi1 , xi2 , ..., xid ) is position informations in employed bees. k ∈ {1, 2, ..., Me } is
randomly chosen indexes for each groups. Although k is determined randomly, it has to be
diﬀerent from i. Φid is a random number between[−1,1]. d is dimension parameters.
Colony2:
In Colony2, the search method of employed bee
is diﬀerent from Colony1. Employed bee uses thefollowing expression (2):Xid (t + 1) =id (t) + rand[0, 1](Xid (t) − Xbest,d (t))− rand[0, 1](Xid (t) − Xave,d (t))(2)
where Xbest,d (t) is the current best position ofColony1 and Xave,d (t) is the average position ofwhole Colony2.
Step 3 (Evaluation of employed bee): Evaluate X i (t)and comparison of the current employed bee ﬁtness value f (X i (t)).
If f (X i (t)) ≤ f (X i (t + 1)): Let tRi = tRi + 1.
If f (X i (t)) > f (X i (t + 1)): Update the position
information on each X i , and tRi = 0.
Step 4 (Search by onlooker bee): Onlooker bee
searches new food sources depending on evaluation values of employed bees. As the evaluation
value of food source increases, the probability chosen as onlooker bee increases. The chosen probability Pi is calculated by the following expression(3):
f (Xi )Pi = ∑Mg(3)

n=1 f (Xn )Step 6 (Search by scout bee):
If Tlimit , initialize X i and tRi = 0. New X i issearched by scout bee and is calculated by thefollowing expression (5):Xid (t + 1) = Xmin + rand[0, 1](Xmax − Xmin ) (5)where Tlimit shows generation until carry out initialize X i and is calculated by the multiplicationof generation number T and dimension parameterd.
Step 7 (Exchange of employed bee): This Step 7 is
new step in ABCTC and ABCTKC.
If Te ≥ te : Exchange employed bee between
colonies and te = 0. The exchanged bee is determined by the provability of inverse proportional
to tRi . In other words, the larger tRi , the higher
is the likelihood of that employed bee to be selected. The selection probability Ei is calculatedby the following expression (6):( 1 )3Ei = ∑MTg Ri
1 3
n=1 ( TRi )Step 8 Let t = t + 1, go back to [Step 2] and repeat
until T = t.
3. Simulation
3.1. Benchmark Functions
In order to evaluate the performance of ABCTKC,
we apply ABCTKC to some benchmark problems.
The problems have the optimum (minimum) valuesof f (x) in the algorithm. We use the following fourbenchmarks.
1. Sphere function :f1 (x) =(4)

where Y j (t) = (xi1 , xi2 , ..., xid ) is position informations in onlooker bees.
Step 5 (Evaluation of onlooker bee):If f (X i (t)) ≤ f (Y j (t)): Let tRi = tRi + 1.
If f (X i (t)) > f (Y j (t)): Update the position information on each employed bee X i to onlooker
bee Y j , and tRi = 0.

d
∑

x2d ,

(7)

d=1

where x ∈ [−5.12, 5.12]d and the optimum solution x∗
are all [0, 0, ..., 0].
2. Rosenbrock function :Onlooker bee searching methods is the same as
employed, it shown following expression (4):
Yjd (t) = Xid (t) + Φid (Xid (t) − Xkd (t))(6)

f2 (x) =d−1∑(100(x2d+1 − xd )2 + (xd − 1)2 ),

(8)

d=1

where x ∈ [−2.048, 2.048]d and the optimum solutionx∗ are all [1, 1, ..., 1].
3. Rastrigin function :f3 (x) = 10d +

d
∑
( 2
)
xd − 10 cos(2πxd ) ,

(9)

d=1

where x ∈ [−5.12, 5.12]d and the optimum solution x∗
are all [0, 0, ..., 0].
4. Griewank function :- 224 -

f4 (x) =

d
∑
x2d

4000
d=1

+

d
∏

xd
cos( √ ) + 1,
d
d=1

(10)

where x ∈ [−512, 512]d and the optimum solution x∗
are all [0, 0, ..., 0].
All the functions have the dimension parameter d.
In this study, d is set 20 or 50 to investigate the performance and the defaults of employed bee is same eachalgorithms. We carry out the simulation 100 times
with T = 10000 generations.
3.2. Simulation Results
We investigate the performance of ABCTKC thatcompare with the conventional ABC by tour benchmark functions. In this study, we calculated the percentage of found optimal solution and the average ofgeneration until found optimal solution. First, we investigate the inﬂuence of the exchange of employed
bees between colonies for benchmark functions (seeFig. 1). The best It is shown in x-axis. The It showsinterval timing that carry out exchange of employed
bee between colonies and used to calculate the Te . The
frequency of exchange is calculated from multiplicationof generation times and It . In the case of It = 0.0001,
the exchange is carried out every generation. In thecase of It = 1, the exchange is not done. The y-axis
shows average evaluation value of the simulation 100
times. Next, we consider the number of exchanges ofbees for one time. In this study, the maximum number of exchanges of bees is set from one point to ﬁvepoints. The exchanges are performed when the bee did
not update evaluation value. From Fig. 1, we foundthat its optimal parameters are very similar in d = 20
and d = 50 each function. If we apply ABCTKC to
new function in high dimension, we can use optimal
parameters that were determined by low dimension.
From this, we can ﬁnd optimal parameters that are obtained in little calculation. In comparison with ABC,
we determined optimal parameters of ABCTKC withreference to these investigations.
In Table 1, it is shown comparison results with ABC
and ABCTKC in d = 20 and d = 50. In d = 20, ABCTKC is better results than ABC in Sphere, Rosenbrockand Griewank function. In Rastrigin function, becauseof the diﬀerence between MAX and MIN by ABCTKC’s result, ABCTKC obtains better results withsmall number of trials. In d = 50, ABCTKC is better
results than ABC in all functions. Particularly, ABCTKC is more eﬀectively in Rastrigin and Griewank
functions. From results, we found that ABCTKC isobtained better results when the number of exchange
times in every generation was increased. So, it is necessary to investigate behaviors by increasing the maximum number of exchange. Further, we found thatABCTKC is eﬀective in high dimensions, particularly,
multimodal functions. This is considered to be because Colony2 having high neighborhood search ability acted commutatively well with Colony1. It wasalso good to that have an unbalanced ratio of information sharing between colonies.
4. Conclusion
In this study, we proposed Artiﬁcial Bee Colony having Two Kinds of Colonies (ABCTKC) to improve theconvergence speed of the conventional ABC. We investigated the performance of ABCTKC by comparison
with ABC. From results, we conﬁrmed ABCTKC has
high dependability of parameters. Its optimal parameters are very similar in d = 20 and d = 50 for each function. As results, we found that ABCTKC obtained
better results when the number of exchange times inevery generation was increased. In future works, wewould like to perform closer parametric examination.
We will analyze the behavior of ABCTKC by more
multi colonies.
Acknowledgment
This work was partly supported by JSPS Grant-inAid for Young Scientists 23700269.
References[1] D. Karaboga, “An idea based on honeybee swarm
for numerical optimization,” Technical Report
TR06, Erciyes University, Computer Engineering
Department, 2005.
[2] D. Karaboga and B. Akay, “A comparative studyof artiﬁcial bee colony algorithm,” Applied Mathematics and Computation 214, pp. 108-132, Erciyes University, Computer Engineering Department, 2009.
[3] D.T. Pham, A. Ghanbarzaden, E. Koc, S. Otri,
S. Rahim and M. Zaidi, “The bees algorithm,”
Technical Note, Manufacturing Engineering Centre, Cardiﬀ University, UK, 2005.
[4] K. Guney and M. Onay, “Bees algorithm for interference suppression of linear antenna arrays bycontrolling the phase-only and both the amplitude
and phase,” Faculty of Engineering, Erciyes University, 2007.
[5] Y. Shirasaki, Y. Uwate and Y. Nishio, “Artiﬁcial bee colony having twin colonies,” Proc. of
Asia Paciﬁc Conference on Circuits and Systems,
2012.(Being submitted.)- 225 -

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 1: Performance of ABCTKC as changing parameter. (a) Sphere in d = 20. (b) Rosenbrock in d = 20.
(c) Rastrigin in d = 20. (d) Griewank in d = 20. (e) Sphere in d = 100. (f) Rosenbrock in d = 100.
(g) Rastrigin in d = 100. (h) Griewank in d = 100.

Table 1: Comparison of the performance of ABC and ABCTKC for d = 20.
Function
SphereRosenbrock
Rastrigin
Griewank

Algorithm
ABC
ABCTKCABC
ABCTKCABC
ABCTKCABC
ABCTKC

Ave
8.71E−255
4.33E-289
1.29
0.94
0.25
0.8
0
0

Min
4.34E−261
4.82E−300
0.78
0.47
0
0
0
0

Max
4.65E−253
3.21E−251
2.01
1.42
18.7
3.19
0
0

Optimal solution found [％]
NULL
NULL
NULL
NULL
86
30
100
100

Generation ave
NULL
NULL
NULL
NULL
7710
8130
916
1053

Table 2: Comparison of the performance of ABC and ABCTKC for d = 50.
Function
SphereRosenbrock
Rastrigin
Griewank

Algorithm
ABC
ABCTKCABC
ABCTKCABC
ABCTKCABC
ABCTKC

Ave
9.77E−97
6.34E-125
12.83
11.4
254.5
12.1
1.09E−16
3.33E-18

Min
1.21E−98
5.34E−129
8.38
8.06
203.2
4.97
0
0

Max
1.19E−95
3.86E−123
19.75
15.8
295.3
20.5
1.11E−16
1.11E−16- 226 -

Optimal solution found [％]
NULL
NULL
NULL
NULL
NULL
NULL
2
97

Generation ave
NULL
NULL
NULL
NULL
NULL
NULL
8929
5890