2017 RISP International Workshop on Nonlinear Circuits,
Communications and Signal Processing (NCSP'17)
Guam, USA, February 28th to March 3rd, 2017

Analysis of Fireﬂy Algorithm Combined with Chaotic Map
Masaki Moriyama, Masaki Takeuchi, Yoko Uwate and Yoshifumi Nishio
Department 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, optimization problems have drawn attention. To
solve the problems, nature-inspired metaheuristic optimization algorithms such as Fireﬂy Algorithm (FA) has been developed. FA is idealized from the social behavior of ﬁreﬂies
based on their ﬂashing characteristics. Furthermore, FA combined with chaotic map is shown to be of beneﬁt in the previous study. In our study, we investigate a different approach toinsert chaotic map. We compare improved FA to the conventional FA using benchmark functions of Congress on Evolutionary Computation 2013. In our study, improved FA performs better than the conventional FA.
1. Introduction
Swarm Intelligence is one of research territory of artiﬁcial
intelligence. Examples existing in nature are ant, bee, bird
and ﬁsh. Nevertheless individual thing has simple and limited
information, it shows advanced behavior as a whole when it
becomes a group. The swarm intelligence technique is important because simple control regulation is more beneﬁcial thancomplex control regulation. The applications of swarm intelligence technique are unmanned aircraft and a self-drivingcar. The good points of this technique are not being high altitude and being able to be downsize of robots.
Recently, optimization problems have drawn attention.
The examples of optimization problems are traveling salesman problem, knapsack problem, shortest path problem, etc.
These problems are very difﬁcult to solve, because the nonlinearity of many these problems often results in local optima. To overcome this issue, metaheuristic optimization algorithms are engaged in research. These optimization algorithms attempt to idealize social behavior or natural phenomena. In social insect colonies, each individual insect seems
to have its own agenda and the group in total appears tobe highly organized. Nature-inspired Algorithms have been
demonstrated to show effectiveness and efﬁciency to solve
difﬁcult optimization problems. A swarm is a group of multiagent systems such as ﬁreﬂies. Simple agents coordinate
their activities to solve the complex problem to multiple forage sites in dynamic environments [1]. Several metaheuris-

tic optimization algorithms are developed for global search.
Such optimization algorithms develop more efﬁciency andsolve larger problems. Metaheuristic algorithms have Genetic
Algorithm (GA), Ant Colony Optimization (ACO), Particle
Swarm Optimization (PSO), Fireﬂy Algorithm (FA), etc. In
our study, we use the FA. FA has also been applied to train
neural networks [2].
FA idealizes the social behavior of ﬁreﬂies based on theirﬂashing characteristics. Therefore, FA is applicable formixed variable and engineering optimization. For discrete
problems and combinatorial optimization, discrete versionsof FA have been developed with superior performance, which
can be used for traveling salesman problems, graph coloringand other applications [3].
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 [4]. Moreover, Chaos has a characteristic of an unpredictable movement. In the previous study,
FA combined with chaotic map is shown to be of beneﬁt [4].
We especially drew attention to one-dimensional chaotic map
because one-dimensional chaotic map is generated from elementary equation and easy to deal with. In our study, weinvestigated a different approach to insert one-dimensional
chaotic map into the conventional FA [5]. Moreover, we try
to analysis of FA combined with chaotic maps.
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.
2. The Conventional Fireﬂy Algorithm (FA) [6]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- 229 -

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 from the lightsource obeys the inverse square law. Furthermore, the air absorbs light which becomes weaker and weaker as the distanceincreases.
We can idealize some of the ﬂashing characteristics of ﬁreﬂies so as to develop ﬁreﬂy-inspired algorithms. The FA isswarm intelligence-based algorithm. Therefore, it has thesimilar advantages that other swarm intelligence-based algorithms have. The conventional FA was developed by Xin-She
Yang in 2007. We use the following 3 idealized rules:

simplest systems with the capability of generating chaotic
motion. One-dimensional maps are introduced as follows.
It generates chaotic sequences in (0, 1) assuming Eq. (5),Eq. (6) and Eq. (7). In the previous study, the author inserted
12 chaotic maps into the attractiveness of ﬁreﬂy β and thelight absorption coefﬁcient γ [4]. Each method is simulatedby Sphere Function and Griewank’s Function. In this study,
we insert Bernoulli shift map, Tent map and Logistic map into
the vector of random variable.
xi = xi + β(xj − xi ) + α(i + zi )(4)

The Bernoulli shift map(see Fig. 1) belongs to the class ofpiecewise linear maps similar to the Logistic map or the Kent
• All ﬁreﬂies are unisex so that one ﬁreﬂy will be attracted map. It is formulated as follows:
to other ﬁreﬂies regardless of their sex.

(0 ≤ zi ≤ 0.5)2zi
zi+1 =(5)• Attractiveness is proportional to the their brightness,2zi − 1 (0.5 ≤ zi ≤ 1).
thus for any two ﬂashing ﬁreﬂies, the less brighter one
will move towards the brighter one. The attractiveness is
The Tent map(see Fig. 2) is similar to the Logistic map.
proportional to the brightness and they both decrease as It displays some speciﬁc chaotic effects. This map is formutheir distance increases. If there is no brighter one than lated as follows:a particular ﬁreﬂy, it will move randomly.
(0 ≤ zi ≤ 0.5)2zi
zi+1 =(6)• The brightness of a ﬁreﬂy is affected or determined by
2 − 2zi (0.5 ≤ zi ≤ 1).
the landscape of the objective function.
The attractiveness of ﬁreﬂy β is deﬁned by2

β = (β0 − βmin )e−γrij + βmin

(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. βmin is theminimum value of β. The ﬁreﬂy i is attracted to another more
attractive ﬁreﬂy j and the movement of ﬁreﬂy i is determinedbyxi = xi + β(xj − xi ) + αi(2)where α is the randomization parameter and i is a random
vector which are drawn from a Gaussian distribution.
The parameter α(t) is deﬁned byα(t) = α(0)

10−4
0.9

Figure 1: Bernoulli shift map.

Figure 2: Tent map.

Although Logistic map(see Fig. 3) is a simple equation, it
is an interesting map that shows a complicated chaotic behavior. This map is formulated as follows:zi+1 = azi (1 − zi ), a = 4.0

t/tmax(3)

where t is the number of iteration. tmax is the maximum
number of t.
3. Our proposed method
We analyze the improved FA(BFA, TFA and LFA). BFA,
TFA and LFA are combined the conventional FA with
Bernoulli shift map, Tent map and Logistic map. These maps
are one of the one-dimensional chaotic maps, which is the- 230 -

Figure 3: Logistic map.

(7)

4. Numerical simulation
We compare BFA, TFA and LFA to the conventional FA
using 7 benchmark functions of Congress on Evolutionary
Computation(CEC) 2013 [7]. Table 1 shows the functions
we used. We chose 2 unimodal functions(f1 and f2 ), 3 basic multimodal functions(f3 , f4 and f5 ) and 2 composition
functions(f6 and f7 ). The function’s graphs(from Fig. 4 toFig. 10) we used are showed in last page.
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 searchrange of these function is [−100, 100]D (D:Dimension), thenumber of ﬁreﬂy N is 30. Each numerical experiment is
run 50 times. The average length of design variables L is100. Furthermore, we use β0 = 1.0, βmin = 0.2, γ = √1L ,D = 30, α(0) = 0.5 and tmax = 1500.
Table 2 shows the average, minimum and maximum error
value.
We chose these functions because these functions is popular function for optimization problems. Sphere Function isno dependency between variables. Weierstrass Function isreal-valued function advanced in 1872 by Karl Weierstrass.
This function is continuous. However it is almost Nondifferentiable. Historically, this function is very important as
example of pathological functions. Griewank’s Function isalmost gentle globally. However it has many local minima.
Griewank’s Function is said to be suitable for Simulated Annealing(SA) search. Composition Function 2 is generated
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 average error value of Table 2, BFA, TFA and
LFA performs better than the conventional FA in several functions. In the unimodal functions(f1 and f2 ), TFA is the best
performance. In the basic multimodal functions(f3 ), however, the result changed relatively little, we assume BFA’s
average error value is effective. From examining the ﬁndings, Rotated Rosenbrock’s Function may be not affected to alarge degree by our proposed algorithms. In the basic multimodal functions(f4 ), LFA is the best performance. In the basic multimodal functions(f5 ), BFA is the best performance.
Rotated Weierstrass Function(f4 ) and Rotated Griewank’s
Function(f5 ) has some local optima. These functions may beeasy to be affected by Our proposed algorithms. In the composition functions(f6 ), TFA and LFA is better than FA. In thecomposition functions(f7 ), the result changed relatively little. Further, FA, BFA and LFA’s average error value is samevalue.
In f2 , f4 and f5 , our proposed 3 algorithms perform better than the conventional FA. Therefore, FA combined withchaotic map especially have good effects in Discus Func-tion, Weierstrass Function and Griewank’s Function. In thissimulation, we also compare with BFA, TFA and LFA. BFA
performs best in 2 functions(f3 and f5 ). TFA performs
best in 3 functions(f1 , f2 and f6 ). LFA performs best in 1
function(f4 ). Hence, TFA is the best algorithm in our proposed 3 algorithms.
Next, we mainly compare the average error values of simulation results. In results, BFA performs better than FA on 5
functions(f1 , f2 , f3 , f4 and f5 ). We assume BFA is strong
for the unimodal functions and the basic multimodal functions. TFA performs better than FA on 5 functions(f1 , f2 , f4 ,f5 and f6 ). LFA performs better than FA on 4 functions(f2 ,f4 , f5 and f6 ).
Focusing attention on the form of map, the form of
Bernoulli shift map and Tent map is rectilinear. However,the form of Logistic map is rounded. In f1 , rectilinear
map(BFA and TFA) performs better than FA. In f6 , similar
form’s map(TFA and LFA) performs better than FA. Depending on the function, the form of map may inﬂuence the algorithm. LFA and FA’s average of error value is same resultin 3 functions(f1 , f3 and f7 ). Hence, we assume the map ofrounded form is apt not to develop.
As a result, we assumed FA combined with chaotic
map(BFA, TFA and LFA) is effective, compared to the conventional FA. However, the map of rectilinear is better than
the rounded form. We are able to be simple to apply because
one-dimensional chaotic map is easy equation.
5. Conclusion
This paper introduced analysis of the improved Fireﬂy Algorithm(BFA, TFA and LFA). We tried to improve the conventional FA using Bernoulli shift map, Tent map and Logistic map. We compared average error values of simulation
results. BFA, TFA and LFA performed better than the conventional FA. In light of the evidence, the form of map may
inﬂuence the algorithm depending on the function.
In the future work, we will investigate BFA, TFA and
LFA using more functions and insert other one-dimensional
chaotic maps. Moreover, we will reseach the relationship to
the form of map.
Acknowledgment
This work was partly supported by JSPS Grant-in-Aid for
Challenging Exploratory Research 26540127.
References[1] J. Senthilnath, S.N. Omkar and V. Mani, “Clustering
Using Fireﬂy Algorithm: Performance Study”, Swarmand Evolutionary Computation, 1, pp.164-171 (2011).

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[2] S. Nandy, P.P. Sarker and A. Das, “Analysis of NatureInspired Fireﬂy Algorithm Based Back-Propagation
Neural Network Training”, International Journal ofComputer Applications, 43(22), pp.8-16 (2012).
[3] X.-S. Yang and X. He, “Fireﬂy Algorithm: Recent Advances and Applications”, Int. J. Swarm Intelligence, 1,
1, pp.36-50 (2012).
[4] 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).
[5] M. Moriyama, M. Takeuchi, Y. Uwate and Y. Nishio,“Fireﬂy Algorithm Combined with Chaotic Map”, IEEE
Workshop on Nonlinear Circuit Networks (NCN’16),
pp.50-52 (2016).
[6] X.-S. Yang, “Nature-Inspired Metaheuristic Algorithms
Second Edition”, Luniver Press (2010).
[7] J.J. Liang, B.Y. Qu, P.N. Suganthan and Alfredo G.
Hernndez-Daz, “Problem Deﬁnitions and Evaluation
Criteria for the CEC2013 Special Session on RealParameter Optimization”, Technical Report 201212,
Computational Intelligence Laboratory, Zhengzhou
University, Zhengzhou, China, and Technical Report,
Nanyang Technological University, Singapore (2013).

No.
f1
f2
f3
f4
f5
f6
f7

f2

f3

f4

f5

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 × 105
2.73 × 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 × 10−4
3.91 × 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

LFA
6.45 × 10−4
2.60 × 10−4
1.00 × 10−3
1.20 × 105
7.29 × 104
1.85 × 105
2.73 × 101
2.59 × 101
2.85 × 101
9.70 × 100
4.08 × 100
1.59 × 101
5.15 × 10−1
7.00 × 10−2
2.24 × 100
3.26 × 103
9.72 × 102
6.11 × 103
2.33 × 102
2.21 × 102
2.43 × 102

Figure 5: f2 .
Figure 8: f5 .

Figure 6: f3 .

f (x*)
-1400
-1100
-900
-600
-500
800
1100

Table 2: Simulation results
f
f1

f6

Figure 4: f1 .

Table 1: 2013 CEC Benchmark Functions
Name
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)

Figure 9: f6 .

Figure 10: f7 .

Figure 7: f4 .

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