2017 IEEE Workshop on Nonlinear Signal Processing, Feb. 10, 2017

Research of 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
I. I NTRODUCTION
Recently, nature-inspired metaheuristic optimization algorithms such as Firefly Algorithm (FA) have been developed.
In a previous study, FA combined with chaotic map wasconfirmed to be of benefit.
In this study, we investigate an FA algorithms which is
combined FA with a one-dimensional chaotic map. Comparedto the previous study, we investigate a different approach toinsert chaotic map.

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.
The average length of design variables L is 100. Furthermore,
we use β0 = 1.0, βmin = 0.2, γ = √1L , D = 30 andtmax = 1500.
Table II shows the average error values.

II. F IREFLY A LGORITHM (FA)
FA is idealized from some of the flashing characteristics offireflies [1].
The attractiveness of firefly β is defined byTABLE I
2013 CEC BENCHMARK FUNCTIONS

β = (β0 − βmin )e−γrij + βmin2No.
f1
f2
f3
f4
f5
f6
f7

(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 . βmin is the minimum
value of β. The firefly i is attracted to another more attractive
firefly j and the movement of firefly i is determined by
xi = xi + β(xj − xi ) + αϵi

ff1
f2
f3
f4
f5
f6
f7

where α is the randomization parameter and ϵi is a random
vector which is drawn from a Gaussian distribution.

xi = xi + β(xj − xi ) + α(ϵi + z i )where z i is one-dimensional chaotic map.

(3)

IV. N UMERICAL SIMULATION
We compare BFA, TFA and LFA with the conventional FA
using 7 benchmark functions of Congress on EvolutionaryComputation (CEC) 2013. Table I shows the functions weused. We chose 2 unimodal functions, 3 basic multimodal
functions and 2 composition functions.

f (x*)
-1400
-1100
-900
-600
-500
8001100

TABLE IIS IMULATION RESULTS(2)

III. O UR PROPOSED METHOD
We introduce the improved FA (BFA, TFA and LFA). BFA,
TFA and LFA are combined FA with Bernoulli shift map,
Tent map and Logistic map, respectively. These maps are oneof the one-dimensional chaotic maps, which is the simplest
systems generating chaotic motion. In the previous study, A.H.
Gandomi et al inserted chaotic maps into the attractiveness offirefly β and the light absorption coefficient γ [2]. In this study,
we insert a chaotic map into the vector of random variable.

NameSphere 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)

FA6.45 × 10−41.22 × 105
2.73 × 101
1.04 × 101
5.63 × 10−1
3.31 × 103
2.33 × 102

BFA
6.26 × 10−4
1.19 × 105
2.72 × 101
9.86 × 100
4.06 × 10−1
3.46 × 103
2.33 × 102

TFA
6.11 × 10−4
1.14 × 105
2.73 × 101
1.03 × 101
5.39 × 10−1
3.11 × 103
2.34 × 102

LFA
6.45 × 10−4
1.20 × 105
2.73 × 101
9.70 × 100
5.15 × 10−1
3.26 × 103
2.33 × 102

In this simulation, we compare the average error values ofBFA, TFA and LFA. BFA performs best for 2 functions (f3 andf5 ). TFA performs best for 3 functions (f1 , f2 and f6 ). LFA
performs best for a function (f4 ). In f7 , improved FA does
not perform better than FA. Hence, TFA is the best algorithm
among our proposed three algorithms.
V. C ONCLUSION
This study introduced the improved Firefly Algorithm (BFA,TFA and LFA). We compared average error values of simulation results. TFA is the best algorithm among our proposed
three algorithms.
R EFERENCES[1] X.-S. Yang, “Nature-Inspired Metaheuristic Algorithms Second Edition”,
Luniver Press (2010).
[2] A.H. Gandomi, X.-S. Yang, S. Talatahari and A.H. Alavi, “Firefly
Algorithm with Chaos”, Commun Nonlinear Sci Numer Simulat, 18,
pp.89-98 (2013).

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