平成 28 年度電気関係学会四国支部連合大会 講演論文集 (2016 徳島大学)
2016 SHIKOKU-SECTION JOINT CONVENTION RECORD OF THE INSTITUTES OF ELECTRICAL AND RELATED ENGINEERS (TOKUSHIMA)

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Firefly Algorithm with Bernoulli Shift Map
Masaki MORIYAMA ,Masaki TAKEUCHI , Yoko UWATE and Yoshifumi NISHIO( Tokushima University)

1. Introduction

The Bernoulli shift map belongs to the class of piecewise
linear maps similar to the logistic map or the Kent map. It
is formulated as follows:

Recently, nature-inspired metaheuristic optimization algorithms such as Firefly Algorithm(FA) has been developed.
FA idealizes the social behavior of fireflies based on their
flashing characteristics. Furthermore, FA combined withchaotic map is to be of benefit.
In this study, we propose an algorithm that is combined
FA with Bernoulli shift map. Compared to the previousstudy, we investigate a difference approach to insert chaotic
map into the conventional FA.

zi+1 =• All fireflies are unisex so that one firefly will be attracted to other fireflies regardless of their sex.
• Attractiveness is proportional to the their brightness,thus for any two flashing fireflies, the less brighter one
will move towards the brighter one. The attractiveness is proportional to the brightness and they bothdecrease as their distance increases. If there is nobrighter one than a particular firefly, it will move randomly.

The attractiveness of firefly β is defined byα(t) = α(0)t/tmax

Namef (x*)

12345

Sphere Function
Rotated Rosenbrock’s Function
Rotated Griewank’s Function
Composition Function 2 (n=3,Unrotated)
Composition Function 5 (n=3,Rotated)

-1400
-900
-500
800
1100

Table 2: Simulation resultsFAff1f2(2)where α is the randomization parameter, and ǫi is a random
vector which are drawn from a Gaussian distribution.
The parameter α(t) is defined by
10−40.9No.

(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 located at xi and xj respectively. The firefly
i is attracted to another more attractive firefly j, and themovement of firefly i is determined by(5)

Table 1: 2013 CEC Benchmark Functions• The brightness of a firefly is affected or determined by
the landscape of the objective function.

xi = xi + β(xj − xi ) + αǫi(0 ≤ zi ≤ 0.5)
(0.5 ≤ zi ≤ 1).

We compare IFA to the conventional FA by using 5
benchmark functions of Congress on Evolutionary Computation 2013. In this experiment, the optimal solutions x∗ of these benchmark functions is shifted from 0,
and the global optima f (x∗ ) are not equal to 0. In addition, we assign the search range of these function is[−100, 100]D (D:Dimention), the number of firefly N is 30.
Each numerical experiment is run 50 times. Furthermore,
we use β0 is 1.0 and γ is 1.0. In this sumilation, we compare
the average of sumilation results. In results, IFA performs
better on 3 functions.

The conventional FA was developed by Xin-She Yang in2007. We use the following three idealized rules:

22zi
2zi − 14. Numerical experiment

2. Conventional Firefly Algorithmβ = β0 e −γrij



f3

f4(3)

where t is the number of iteration.

f5

3. Proposed method
We propose the improved FA(IFA). IFA is combined theconventional FA with Bernoulli shift map. Bernoulli shift
map is one of the one-dimensional chaotic maps, whichis the simplest systems with the capability of generating
chaotic motion. It generates chaotic sequences in (0, 1) assuming Eq. (5). In the previous study, the attractivenessof firefly and the light absorption coefficient are tuned withchaotic map. In IFA, we insert Bernoulli shift map into thevector of random variable.
xi = xi + β(xj − xi ) + α(ǫi + zi )
(4)

avg
min
max
avg
minmaxavg
minmaxavg
minmaxavg
minmaxIFA

6.45 × 10
4.32 × 10−4
9.66 × 10−4
2.73 × 101
2.54 × 101
2.85 × 101
5.63 × 10−1
4.15 × 10−2
2.23 × 100
3.31 × 103
6.08 × 102
6.27 × 103
2.33 × 102
2.01 × 102
2.53 × 102−4

6.26 × 10−4
3.27 × 10−4
1.07 × 10−3
2.72 × 101
2.53 × 101
2.83 × 101
4.06 × 10−1
7.28 × 10−2
1.70 × 100
3.46 × 103
1.36 × 103
6.13 × 103
2.33 × 102
2.19 × 102
2.50 × 102

5. Conclusion
This paper introduced the improved Firefly Algorithm(IFA). We tried to improve the conventional FA using
Bernoulli shift map. IFA performed better than the conventional FA.
In the future work, we will investigate Firefly Algorithm
by using more functions and insert other chaotic maps.

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