IEEE Workshop on Nonlinear Circuit Networks
December 9-10, 2016

Firefly Algorithm combined with chaotic map
Masaki Moriyama, Masaki Takeuchi, Yoko Uwate and Yoshifumi Nishio
Departmrnt of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
E-mail: {moriyama, masaki, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract—Recently, nature-inspired metaheuristic optimization algorithms such as Firefly Algorithm(FA) has been developed. FA is idealized from the social behavior of fireflies basedon their flashing characteristics. Furthermore, FA combined withchaotic map is shown to be of benefit. In this study, we propose
an algorithm combined FA with Bernoulli shift map and Tent
map. Compared to the previous study, we investigate a difference
approach to insert chaotic map. We compare improved FA tothe conventional FA using 7 benchmark functions of Congress
on Evolutionary Computation 2013. In our study, improved FA
performs better than the conventional FA.

I. I NTRODUCTION
Swarm Intelligence is one of research territory of artificial
intelligence. Examples existing in nature are ant, bee, bird andfish. The swarm intelligence technique is important because
simple control regulation is more beneficial than complex
control regulation. The applications of swarm intelligence
technique are unmanned aircraft and a self-driving car. The
good points of this technique are not being high altitude andbeing able to be downsize of robots.
The non-linearity of many optimization problems often
results in local optima. To overcome this issue, metaheuristic
optimization algorithms are used. These optimization algorithms attempt to idealize social behavior or natural phenomena. Several metaheuristic optimization algorithms aredeveloped for global search. Such optimization algorithms develop more efficiency and solve larger problems. Metaheuristicalgorithms have Genetic Algorithm(GA), Ant Colony Optimization(ACO), Particle Swarm Optimization(PSO), Firefly
Algorithm(FA), etc. In our study, we use the FA.
FA idealizes the social behavior of fireflies based on their
flashing characteristics. In social insect colonies, each individual insect seems to have its own agenda and the group in total
appears to be highly organized. Nature-inspired Algorithms
have been demonstrated to show effectiveness and efficiency
to solve difficult optimization problems. A swarm is a group ofmulti-agent systems such as fireflies. Simple agents coordinate
their activities to solve the complex problem to multiple foragesites in dynamic environments. Therefore, FA is applicable formixed variable and engineering optimization.
Recent applications of nonlinear dynamics, especially ofchaos, have drawn attention in many fields. Chaos is seemingly
a random movement of deterministic system. Chaos system
has the properties of sensitivity to initial conditions. Therefore,
using chaotic system in image encryption can meet security
requirements. Moreover, the chaotic encryption algorithms

utilize one-dimensional chaos map, multi-dimensional chaos
map and ultra-dimensional chaos map. We drew attention to
one-dimensional chaotic map. Furthermore, FA combined withchaotic map is shown to be of benefit. In this study, we propose
an algorithm combined FA with Bernoulli shift map andTent map. Compared to the previous study, we investigate a
difference approach to insert chaotic map into the conventional
FA.
This paper illustrates FA combined with chaotic map.
Section 2 describes the conventional FA. Section 3 explains
our proposed method. Numerical simulation and simulation
results are discussed in Section 4. Section 5 is discussed theconclusion and outlines directions for further research.
II. F IREFLY A LGORITHM
First, we will introduce the behavior of fireflies. Firefly is
the one of insects. The flashing light of fireflies is an amazingsight in the summer sky in the tropical and temperate regions.
There are about 2000 firefly species in the world, and Japan
has about 40 firefly species. Moreover, most fireflies produce
short and rhythmic flashes. The pattern of flashes is often
unique for a particular species. Some species of fireflies caneven synchronize their flashes.
The light intensity at a particular distance r from thelight source obeys the inverse square law. Furthemore, theair absorbs light which becomes weaker and weaker as thedistance increases.
We can idealize some of the flashing characteristics offireflies so as to develop firefly-inspired algorithms. The conventional FA was developed by Xin-She Yang in 2007. We
use the following 3 idealized rules:- 50 -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 both decrease as
their distance increases. If there is no brighter one thana particular firefly, it will move randomly.
• The brightness of a firefly is affected or determined bythe landscape of the objective function.
•The 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 located at xi and xj respectively. The firefly i is
attracted to another more attractive firefly j and the movementof firefly i is determined by
xi = xi + β(xj − xi ) + αϵi(2)where α is the randomization parameter and ϵi is a random
vector which are drawn from a Gaussian distribution.
The parameter α(t) is defined by( −4 )t/tmax
10α(t) = α(0)(3)
0.9where t is the number of iteration. tmax is the maximum
number of t.

IV. N UMERICAL SIMULATION
We compare BFA and TFA to the conventional FA using
7 benchmark functions of Congress on Evolutionary Computation(CEC) 2013. Table I shows the functions we used. We
chose 2 unimodal functions(f1 and f2 ), 3 basic multimodalfunctions(f3 , f4 and f5 ) and 2 composition functions(f6 andf7 ).
In this simulation, the optimal solutions x∗ of these benchmark functions are shifted from 0 and the global optima f (x∗ )are not equal to 0. In addition, we assign the search range ofthese function is [−100, 100]D (D:Dimension), the number offirefly N is 30. Each numerical experiment is run 50 times.
Furthermore, we use β0 = 1.0, γ = 1.0 and tmax = 1500.
Table II shows the average, minimum and maximum error
value.

III. O UR PROPOSED METHOD
We propose the improved FA(BFA and TFA). BFA and TFA
are combined the conventional FA with Bernoulli shift mapand Tent map. These maps are one of the one-dimensional
chaotic maps, which is the simplest systems with the capability of generating chaotic motion. One-dimensional maps areintroduced as follows. It generates chaotic sequences in (0, 1)
assuming Eq. (5) and Eq. (6). In the previous study, the author
inserted 12 chaotic maps into the attractiveness of firefly β.
Moreover the author inserted 12 chaotic maps into the lightabsorption coefficient γ. Each method is simulated by Sphere
Function and Griewank’s Function. In this study, we insertBernoulli shift map and Tent map into the vector of randomvariable.
xi = xi + β(xj − xi ) + α(ϵi + zi )(4)

The Bernoulli shift map(Fig. 1) belongs to the class ofpiecewise linear maps similar to the logistic map or the Kent
map. It is formulated as follows:{2zi(0 ≤ zi ≤ 0.5)
zi+1 =(5)2zi − 1 (0.5 ≤ zi ≤ 1).
The Tent map(Fig. 2) is similar to the logistic map. It
displays some specific chaotic effects. This map is formulatedas follows:{2zi(0 ≤ zi ≤ 0.5)
zi+1 =(6)2 − 2zi (0.5 ≤ zi ≤ 1).

Fig. 1.

Bernoulli shift map.

Fig. 2.

Tent map.

TABLE I
2013 CEC B ENCHMARK F UNCTIONS

No.
1234567Name
Sphere Function
Rotated Discus Function
Rotated Rosenbrock’s Function
Rotated Weierstrass Function
Rotated Griewank’s Function
Composition Function 2 (n=3, Unrotated)
Composition Function 5 (n=3, Rotated)f (x*)-1400-1100-900-600-500
800
1100

We chose these functions because these functions is popularfunction for optimization problems.
Sphere Function is no dependency between variables.
Weierstrass Function is real-valued function advanced in 1872
by Karl Weierstrass. This function is continuous. However
it is almost Nondifferentiable. Historically, this funtion isvery important as exaple of pathological functions. Griewank’s
Function is almost gentle globally. However it has many
local minima. Griewank’s Function is said to be suitable forSimulated Annealing(SA) search. Composition Function 2 isgenerated from Schwefel’s Function. Moreover Composition
Function 5 is generated from Rotated Schwefel’s Function,
Rotated Rastrigin’s Function and Rotated Weierstrass Function. Moreover we tried to use these functions for FA.
Showing in Table II, BFA and TFA performs better thanthe conventional FA. Both BFA and TFA performs better than
FA in f1 , f2 , f4 and f5 . In the unimodal functions(f1 andf2 ), TFA is the best perfomance. In the basic multimodalfunctions(f3 , f4 and f5 ), BFA is the best perfomance. In thecomposition functions(f6 ), TFA is better than FA and BFA.
In the composition functions(f7 ), the result changed relatively
little. Therefore, We assumed BFA and TFA is effective.
In this simulation, we compare the average error values ofsimulation results. In results, BFA performs better than FA on
5 functions(f1 , f2 , f3 , f4 and f5 ). TFA performs better thanFA on 5 functions(f1 , f2 , f4 , f5 and f6 ).

- 51 -TABLE IIS IMULATION RESULTS

ff1

f2

f3

f4

f5

f6

f7

avg
min
max
avg
min
max
avg
min
max
avg
min
max
avg
min
max
avg
min
max
avg
min
max

FA
6.45 × 10−4
4.32 × 10−4
9.66 × 10−4
1.22 × 105
7.55 × 104
2.14 × 1052.73 × 101
2.54 × 101
2.85 × 101
1.04 × 101
7.07 × 100
1.59 × 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

BFA
6.26 × 10−4
3.27 × 10−4
1.07 × 10−3
1.19 × 105
6.61 × 104
1.86 × 105
2.72 × 101
2.53 × 101
2.83 × 101
9.86 × 100
3.68 × 100
1.73 × 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

TFA
6.11 × 10−43.91 × 10−4
8.97 × 10−4
1.14 × 105
7.12 × 104
1.68 × 105
2.73 × 101
2.59 × 101
2.90 × 101
1.03 × 101
5.69 × 100
1.50 × 101
5.39 × 10−1
3.08 × 10−2
1.93 × 100
3.11 × 103
1.48 × 103
6.21 × 103
2.34 × 102
2.14 × 102
2.53 × 102

V. C ONCLUSION
This paper introduced the improved Firefly Algorithm(BFAand TFA). We tried to improve the conventional FA using
Bernoulli shift map and Tent map. We compared average error
values of simulation results. BFA and TFA performed better
than the conventional FA. Futhermore, we assumed BFA andTFA is effective.
In the future work, we will investigate BFA and TFA using
more functions and insert other chaotic maps. Futhermore, wewill compare BFA and TFA to FA inserted only chaotic maps.
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