2010 IEEE Workshop on
Nonlinear Circuit Networks, Tokushima, Dec 3-4, 2010

Optimization Technique ofPheromone-Based Bees Algorithm
Yudai Shirasaki, Sho Shimomura, Masaki Sugimoto, Yoko Uwate and Yoshifumi Nishio
Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, 770-8506, Japan
Email: {shirasaki, s-sho, sugimoto, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract— Bees algorithm (BA) is an optimization algorithm
based on a particular intelligent behavior of honeybee swarms. In
this study, we propose a new BA containing one of the behaviorof honeybee in the natural world. The behavior has the action ofa pheromone. This pheromone has the effect to attract other bees.
Because many bees are attracted to the best site. Proposed BA
can leave from local minima more easily than previous methods.
We confirm the performances of proposed BA using the function
approximation called a benchmark problem. The pheromone ismore effective than standard BA.

I. I NTRODUCTION
Bees Algorithm (BA) [1] is generally known as a popular
optimization algorithm for the solution of object function andis an evolutionary algorithm to simulate the intelligent honeyforaging behavior of honeybee [2]. A honeybee is a social
insect which performs group behavior. It is very rare to makethis in the world of insect. In the natural world, honey foraging
behavior of the bee consists of two types, recruitment behavior
and navigation behavior. Recruitment strategies are used tocommunicate previous search experiences to other membersin the beehive. Navigation strategies are used to navigate inunknown world. And, the honey foraging bee consists of twotypes. There are search-bee and recruit-bee. Search-bee has arole which searches sources of honey and tells it to recruitbee. Recruit-bee is recruited to sources of honey by searchbee. Next, we explain honey foraging behavior of honeybee.
First, search-bee searches sources of honey at random. In orderto recruit recruit-bee in the beehive for the source of honey,
search-bee tells recruit-bee in the beehive the distance anddirection of the source by waggle dance. This dance has arole which gives the source of honey information (e.g. thedirection and the distance).
BA consists of a recruitment strategy and a navigate strategy
like behavior of honeybee. BA performs a kind of neighborhood searching combined with random search and can be used
for both combinatorial optimization problems and functional
optimization problems. The strong point of BA is that leavingfrom local minima is easy. However, BA has a weakness which
is difficult to converge.
In nature world, there is a theory that honeybee use
pheromone in order to look for sources of honey. Honeybee
have one of the most complex pheromonal communication
systems found in nature [3]. For example, drones emit apheromone that attracts other flying drones to promote drone

aggregations at sites suitable for mating with virgin queens.
However, the existing BAs do not use pheromone action for
navigation of recruitment.
In this study, we propose pheromone-based BA (PBBA).
The important feature of PBBA is that all bees are influenced
by pheromone. A pheromone is emitted by the best evaluation
bee among all the bees. When other bees fall into local
minima, they can leave from the local minima by being
attracted to the pheromone. Therefore, PBBA can easy toleave from local minima and convergence at high speed. In
Section II, we explain the algorithm of PBBA in detail. In
Section III, we perform basic numerical experiments by using
three algorithm methods including PBBA, and confirm theperformance of PBBA for functional optimization problems.
II. P HEROMONE -BASED B EES A LGORITHM
The existing BA algorithm does not use pheromones. The
proposed PBBA use pheromones. Pheromone acts to attract
bees. Therefore, if pheromone is disposed at the best evaluation site, pheromone attracts all other bees to the best
evaluation site.
The algorithm requires several parameters to be set; thenumber of search-bee M , the number of sites selected outof N visited sites, and the number of best sites out of selectedsites b.

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Step. 1 PBBA lets a generation step t = 0 and the converge step tRi = 0. PBBA disposes the search-bee
i = (1, 2, ..., M ) position X i = (xi1 , xi2 , ..., xiD )randomly. D is dimension parameters.
Step. 2 Comparison of the current search-bee fitness value
X i (t). Ranking zi is made into a high order of evaluation. Best site and good sites are chosen according
to a rank.
Step. 3 PBBA disposes the recruit-bee j = (1, 2, ..., Ni )around X i . The value of N is changed by the resultof [Step. 2]. PBBA assigns more recruit-bee j tosearch near to the best site. The recruit-bee position
rij is decided by following equations
rij d = xi (t) + ∆ϕij (t),∆ϕij = (Si (t)Qij , θij ),

(1)
(2)

where θij is the polar coordinate. Q is the randomvariable distributed uniformly on [0, 1]. θij is theamplitude and is the random variable distributed
uniformly on [0, 2π]. Sij is the largest search range.
Step. 4 PBBA evaluates the all current recruit-bee costf (Rij ). PBBA updates the search-bee position X ifor each i.
Step. 5 If X i (t + 1) ≤ X i (t); PBBA performs [Step. 6].
If X i (t + 1) > X i (t); PBBA updates each X i , lets
tRi = 0 and performs [Step. 7].
Step. 6 If tRi < TR ; PBBA lets tRi = 0 and initializes
Si (t). If tRi > TR ; Si (t + 1) = 0.9Si (t).
Step. 7 The current cost X i is evaluated. The global best
position xgbest among all Xi is updated. X i (t+1) isupdated according to X Gbest . The important featureof PBBA is that the all bees X are influenced bythe pheromone. The pheromone is disposed the best
position X Gbest in the whole group. Furthermore,pheromone strength differs with distance. The sitenearer, X Gbest attracts more strongly. The X i isupdated depending on the best position X Gbest .
M − (zi − 1)
,M −1
X i (t + 1) = X i (t) + di Q(X gbest − X i ),di = p ×(3)(4)

where the parameter p is positive acceleration coefficient.
Step. 8 PBBA let t = t + 1 and tRi = tRi + 1. PBBA goesback to [Step. 3], and repeats until t = T .
III. SIMULATION RESULTS
A. Benchmark Functions
In order to evaluate the performance of PBBA, we apply
PBBA to some benchmark problems. The problems have theoptimum (minimum) values of f (x) in the algorithm. We use
the following four benchmarks.
1. Rosenblock function :f1 (x) =D∑(100(x2d − xd;1 )2 + (1 − xd )2 ),(5)d=1

3. Stretch function :f3 (x) =D∑((x2d + 1 + x2d )0.25 (1 + sin(50(x2d + 1 + x2d )0.1 )2 , (7)d=1where x ∈ [−10, 10]D and the optimum solution x∗ are all[0, 0, ..., 0].
4. Rastrigin function :f4 (x) = 10D +where x ∈ [−5.12, 5.12]D and the optimum solution x∗ areall [0, 0, ..., 0].
The functions f1 and f2 are unimodal function, f3 and f4
are multimodal functions which have numerous local minima.
Therefore, in the case of multimodal functions, the solution
lapsed into local minima easily. All the functions have thedimension parameter D. In this study, D is set 100 toinvestigate the performance for different dimensions. We carryout the simulation 10 times with 2000 generations.
B. Simulation Results
In order to evaluate the performance of PBBA and investigate behavior of PBBA, we compare the three algorithms;
BA, PBBA, and particle swarm optimization (PSO) [4]. BA
is the general BA method which does not use pheromone.
PBBA is the proposed algorithm explained in Section II. PSO
is popular optimization method for the solution of object
function, and has the character similar to the pheromone ofPBBA. Therefore, we used PSO for the comparison. The
parameters of BA and PBBA are set as follows;
M = 20, b = 2, e = 6, N1 = 9, N2 = 5, zi = 0.02
where b is the number of the best sites, e is the number of theelite sites, N1 is the number of recruit-bee in best sites, andN2 is the number of recruit-bee in elite sites.
The parameters of the PSO was set as follows;
c1 = c2 = 1.8, w = 0.5, K = 60
where c1 and c2 are the acceleration coefficients, w is theinertia weight, and K is the number of particles. K is thesame as the total number of bees in BA and PBBA.
The result for the 100 dimensional functions are summarized in Table I.
TABLE IC OMPARISON OF BA, PSO AND PBBA RESULTS WITH D = 100.

f

2. Sphere function :f2 (x) =f1

x2d ,(8)d=1where x ∈ [−2.048, 2.048]D and the optimum solution x∗ are
all [1, 1, ..., 1].

D∑

D∑
( 2)
xd − 10 cos(2πxd ) ,(6)f2where x ∈ [−5.12, 5.12]D and the optimum solution x∗ are
all [0, 0, ..., 0].

f3

d=1

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f4

Avg.
Min.
Avg.
Min.
Avg.
Min.
Avg.
Min.

BA
136.9796.20
9.03e-016
9.43e-017
259.34245.91770.07702.54

PSO
177.3692.43
0.23
1.81e-006
195.04165.58439.41358.25

PBBA
97.40
95.73
3.47e-022
3.71e-023
150.81123.87200.98163.17

5510PSOBAPBBA

Gbest ( f (Pg))

Gbest ( f (Pg))

104

103

102

10PSOBAPBBA

010−510−1010−1510−2010−25110100200

400

600

0800 1000 1200 1400 1600 1800 2000

200400600

800 1000 1200 1400 1600 1800 2000

Generation ( t )Generation ( t )

(a)

(b)
42.5

10

Gbest ( f (Pg))

Gbest ( f (Pg))

10PSO
BAPBBA

2.4

10

2.3

10

PSO
BAPBBA

3102.2102100200400

600

800 1000 1200 1400 1600 1800 2000

0200400600

800 1000 1200 1400 1600 1800 2000

Generation ( t )Generation ( t )(c)(d)Fig. 1. Average gbest value of every generation for 100-dimensional four functions. (a) Rosenblock function. (b) Sphere function. (c) Stretch function.
(d) Rastrigin function.

The performances of PBBA are the best evaluation among
the three algorithms. Because it is difficult for PSO to find theoptimum solution for high-dimensional functions. PSO lapse
into the local minima and prematurely converges. From Fig. 1,
this is because that the pheromone action of PBBA worked
effectively to multimodal functions. From these results, wecan refer that PBBA is the effective algorithm, and that it isworth to add the action of pheromone to BA.
IV. C ONCLUSIONSIn this study, we have proposed pheromone-based bees algorithm (PBBA) and have confirmed that the PBBA could obtain
the effective results especially for two functional optimization
problems in used function approximation. A pheromone has
an effect which attracts a bee. Therefore, PBBA was able

to converge at high speed on the optimal solution without
lapsing into local minima. Because of this, this pheromone
was effective. In future works, we want to add different effects
to the pheromone action.
R EFERENCES[1] D.T. Pham, A. Ghanbarzaden, E. Koc, S. Otri, S. Rahim and M.
Zaidi, “The Bees Algorithm - A Novel Tool for Complex Optimization
Problems,” Intelligent Production Machines and Systems, pp. 454-459,
Cardiff University, UK, 2006.
[2] K. Guney and M. Onay, “Bees algorithm for interference suppression oflinear antenna arrays by controlling the phase-only and both the amplitude
and phase,” Faculty of Engineering, Erciyes University, 2007.
[3] M.S. Blum, Honey Bee Pheromones in the Hive and the Honey Bee,pp. 385–389, Dadant and Sons, 1992.
[4] J. Kennedy and R. Eberhart, “Particle swarm optimization,” Proc. Int.
Conf. on Neural Networks, vol. 4, pp. 1942–1948, 1995.

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