IEEE Workshop on Nonlinear Circuit Networks
December 11-12, 2015

Firefly Algorithm Distinguishing between Males
and Females for Minimum Optimization Problems
Masaki Takeuchi

Haruna Matsushita

Yoko Uwate and Yoshifumi Nishio

Dept. of Electrical
Dept. of Electronics
Dept. of Electricaland Electronic Engineering,and Information Engineering,and Electronic Engineering,
Tokushima University
Kagawa University
Tokushima University
2-1 Minami-Josanjima,
2217-20 Hayashi-cho,
2-1 Minami-Josanjima,
Tokushima 770–8506, Japan
Takamatsu, Kagawa, 761-0396, Japan
Tokushima 770–8506, Japan
Email: masaki@ee.tokushima-u.ac.jp Email: haruna@eng.kagawa-u.ac.jp Email: {uwate, nishio}@ee.tokushima-u.ac.jp

Abstract—We propose Firefly Algorithm Distinguishing between Males and Females. This algorithm exists together withmales and females. In this study, we investigate the feature of
our proposed Firefly Algorithm by changing parameters andthe rate of females. Numerical experiments indicate that ourproposed Firefly Algorithm is superior to the conventional Firefly
Algorithm under some conditions.

I. I NTRODUCTION
The optimization problems have been important more andmore, recently. Most optimization problems are nonlinearwith many constraints. Consequently, optimization algorithms
require efficiency in order to find optimal solution. Stochasticalgorithms, one category of optimization algorithms, are efficient optimization problems. They have a deterministic component and a random component. Algorithms having only thedeterministic component are almost all local search algorithms.
There is a risk to be trapped at local optima such algorithms.
However, stochastic algorithms are possible to jump out such
locality by having random component.
One of stochastic algorithms is the swarm intelligencealgorithms. The swarm intelligence algorithms are based on
the behavior of animals and insects. Representative examples
are Particle Swarm Optimization (PSO) [1], Ant Colony Optimization (ACO), and Firefly Algorithm (FA) [2]–[5].
On the conventional FA, all fireflies are unisex. However,there are males and females in the real world. Therefore, we
distinguish sex of fireflies. This proposed method is called
Firefly Algorithm Distinguishing between Males and Females(FA-DMF). On FA-DMF, the movements of males and females
are defined from the physical differences. Therefore, themovements of males and females are different from each other.
We investigate the feature of FA-DMF by using famous two
test functions. 1 Numerical experiments indicate that FA-DMF
tends to increase randomness as increasing the rate of females.
This study is organized as follows: first, we explain theconventional Firefly Algorithm in Section II. Next, we describe
in detail of FA-DMF in Section III. Followed by, we show
numerical experiments. Finally, we conclude this study.

II. T HE C ONVENTIONAL F IREFLY A LGORITHM (FA) [2]
Firefly Algorithm (FA) has been developed by Yang, andit was based on the idealized behavior of the flashing characteristics of fireflies. It is suitable for multi-peak optimization
problems. The conventional FA is idealized these flashing
characteristics as the following three rules• all fireflies are unisex so that one firefly is attracted toother fireflies regardless of their sex;
•Attractiveness is proportional to their brightness, thus
for any two flashing fireflies, the less brighter one will
move towards the brighter one. The attractiveness isproportional to the brightness and they both decreaseas their distance increases. If no one is brighter than aparticular firefly, it moves randomly;

The brightness or light intensity of a firefly is affected ordetermined by the landscape of the objective function to
be optimized.
Attractiveness of firefly β is defined by•

β = β0 e−γrij
2(1)where γ is the light absorption coefficient, β0 is the attractiveness at rij = 0, and rij is the distance between any two
fireflies i and j at xi and xj . The movement of the firefly i isattracted to another more attractive firefly j, and is determinedbyxi = xi + ∆x,(2)∆x = β(xj − xi ) + αϵi ,(3)where xi is the position vector of firefly i, ϵi is the vectorof random variable, and α(t) is the randomization parameter.
The parameter α(t) is defined by( −4 )t/tmax
10α(t) = α(0),(4)
0.9where t is the number of iteration.

1 The extended version of this study is being reviewed for ISCAS’16 [6].

- 50 -TABLE IIN UMERICAL E XPERIMENT R ESULTS OF FA-DMFnamefemale percentage
ave
Rastrigin min
max
ave
Griewank min
max

conventional FA
10
2.22 × 101
9.95 × 100
4.28 × 101
2.83 × 10−4
1.23 × 10−4
7.68 × 10−3

2.63 × 101
1.21 × 101
4.66 × 101
3.71 × 10−4
1.29 × 10−4
8.59 × 10−320
2.07 × 101
1.09 × 101
3.68 × 101
5.23 × 10−4
1.21 × 10−4
7.65 × 10−330
1.83 × 101
7.96 × 100
2.79 × 101
7.39 × 10−4
1.41 × 10−4
1.02 × 10−2

III. FA-DMF
One of the rules of the conventional FA is that all fireflies areunisex. However, males and females exist in the real world.
Therefore, we distinguish sex of fireflies, that is, there are
two swarms in our proposed method. We call our proposed
method Firefly Algorithm Distinguishing between Males andFemales (FA-DMF). The movement of female is modeled fromthe physical differences. In the real world, females are bigger
than males and female eyes are smaller than male. Thus, inour proposed method, females move slower than males, andfemales have difficulty finding the flashes of other distantfireflies. In addition, we change the randomization parameterof female.
The female parameters α(t) and β, and the female movement x are determined with parameters V and W by(α(t) = α(0)1040.9)t/2tmax,

β = β0 e,x = x + ∆x/V.

(6)
(7)

In the proposed method, males are attracted to all fireflies,
while females are attracted to only males. Males move thesame as fireflies of the conventional FA.
IV. N UMERICAL E XPERIMENTS OF FA-DMF
We compare FA-DMF to the conventional FA with two test
functions (see Table I). These optimal solutions are f (x) = 0at x = 0.
TABLE IT HE T EST F UNCTIONSRastrigin

Formulaf (x) =n∑[-5.12,5.12]xi1 ∑ 2 ∏cos ( √ ) + 1xi −
4000i[-600,600]

i=1Griewankf (x) =range(x2i − 10 cos (2πxi ) + 10)nni=1i=1

In case of solving minimum problem, Rastrigin and Greiwank functions are multi-peak optimization problem. There
are a lot of local minimum and local maximum in Rastrigin
and Greiwank functions. The graph of Greiwank function hasgentler gradient than Rastrigin function.

FA-DMF50
1.90 × 101
7.96 × 100
3.18 × 101
1.02 × 10−3
1.58 × 10−4
1.51 × 10−260
1.88 × 101
1.09 × 101
2.98 × 101
1.97 × 10−3
2.07 × 10−4
1.27 × 10−270
1.97 × 101
1.09 × 101
3.28 × 101
2.82 × 10−3
1.77 × 10−4
2.00 × 10−280
2.10 × 101
1.19 × 101
3.48 × 101
5.13 × 10−3
3.75 × 10−4
2.55 × 10−290
2.34 × 101
1.39 × 101
4.08 × 101
1.04 × 100
1.03 × 100
1.06 × 100

Numerical experiment results are shown in Table II. Eachnumerical experiment is run 100 times. In each test function,
we define the number of dimensions N = 30, tmax = 1000,
V = 3 and W = 4. In this study, we change female percentage
from 10 to 90 every 10 percentage.
As increasing female percentage, average of Greiwank function is continuously increasing. On the other hands, average ofRastirigin function decrease slowly from 10 to 30, and increaseslightly from 30.
In case of numerical experiment of Rastrigin function, FADMF is almost superior to the conventional FA. In caseof numerical experiment of Greiwank function, FA-DMF issuperior to the conventional FA when female percentage isten. In addition, FA-DMF is superior to the conventional FA
about minimum and maximum when females percentage istwenty. Therefore, when the graph has gentle gradient, FADMF is inferior to the conventional FA.

(5)2−γrij/W

name40
1.93 × 101
1.09 × 101
3.48 × 101
1.00 × 10−3
1.50 × 10−3
7.75 × 10−3

V. C ONCLUSION
In this study, we investigated the feature of Firefly Algorithm Distinguishing between Males and Females (FA-DMF).
We apply our proposed Firefly Algorithm to two test functions.
Numerical experiments indicate that FA-DMF is superior tothe conventional FA under some conditions.
In the future work, we improve our proposed Firefly Algorithm, compare to other improved Firefly Algorithms, andapply to actual optimization problems.
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International Symposium on Circuits and Systems(ISCAS), 2016. (beingsubmitted)- 51 -