An Improved Firefly Algorithm Based on Newton's Law of ... · Algorithm based on Newton's law of...

An Improved Firefly Algorithm Based on

Newton's Law of Universal Gravitation

Yi lingzhi1, Xiao Weihong1, Yu Wenxin2, and Wang Genpin3

1 Hunan Province Cooperative Innovation Center for Wind Power Equipment and Energy Conversion, College of

Information Engineering, Xiangtan University. Xiangtan, Hunan province 411105, P. R. China 2

Hunan University of Science and Technology, Xiangtan, Hunan province 411201, P. R. China 3

Shenzhen Polytechnic, Shenzhen, Guangdong province 518000, P. R. China

E-mail: [email protected]

Abstract—Firefly Algorithm (FA) is a novel swarm

intelligence optimization algorithm. Due to the FA have low

precision defects, easily falling into local optimum value

when solving the global optimal value, an improved Firefly

Algorithm based on Newton's law of universal gravitation

was proposed in the paper. The proposed algorithm cites the

law of gravity, which builds a new evolutionary computation

model by using gravity as attractiveness between fireflies.

When the population falls into the local optimal region, the

proposed algorithm can improve firefly's diversity through

Gaussian mutation. Besides, the algorithm is convergent

with probability 1. With the experimental results on 4

standard test functions, the results show that the proposed

method is superior to FA in computational precision and

convergence rate.

Index Terms—optimal solution, Firefly Algorithm (FA),

gravity, Gaussian mutation

I. INTRODUCTION

Inspired by various natural phenomena in nature,

people put forward a series of intelligent optimization

algorithms to solve the complex optimization problem.

The swarm intelligence optimization algorithm includes:

ant colony optimization algorithm, particle swarm

optimization algorithm, artificial bee colony algorithm

and Firefly Algorithm [1]-[3]. In 2008, a Cambridge

scholar Yang proposed a swarm intelligence optimization

algorithm named Firefly Algorithm (FA) by studying the

mutual attraction and movement process of the fireflies

[4]. In this algorithm, attractiveness is proportional to

their brightness, thus for any two fireflies, the less

brighter one will move towards the brighter one. After

many groups of movements, all fireflies will be gathered

near the brightest fireflies, thus completing the final

search. Due to its simple structure, fewer parameters and

stronger searching ability, the algorithm has been

gradually applied in many fields, such as multi-modal

optimization problem, automatic control, price prediction

and image compression [5]-[8]. The application examples

show that FA can deal with optimization tasks more

naturally and efficiently, but some defects still exist in the

algorithm itself [9], [10]. Because the optimization ability

Manuscript received August 10, 2017; revised November 27, 2017.

of Firefly Algorithm depends on the interaction and

influence of fireflies, and the individual itself lacks the

mutation mechanism, so it is difficult for firefly particles

to jump out of the local optimum region. Especially in the

early stage of evolution, the super individuals in the

population tend to attract other individuals to gather

around them rapidly, which make the population diversity

decline greatly. At the late period of evolution, most of

the fireflies gathered near the optimal value, and the

population was prone to stagnation because of the loss of

further evolutionary power. In recent years, scholars have

made a lot of improvements on it.

Feng et al. proposed a kind of FA based on chaos

theory of population dynamics, which uses cubic

mapping chaos to initialize the firefly initial position and

achieves good results with better accuracy and faster

calculation speed [11]. Wang Mingbo et al. proposed a

new kind of swarm optimization based on simulated

annealing algorithm (MFA_SA) to overcome the

shortcoming of being easy to fall into local optimal

solution of the basic FA [12]. Gandomi et al. proposed a

new FA called FA with chaos (FAC), which introduced

different chaos maps into FA to adjust the control

parameters [13]. Fu Qiang et al. proposed a new FA

called FA based on improved evolutionism (IEFA) to

overcome the shortcoming of FA by increase the

attraction effect of the population optimal value on the

individual in the group [14].

In this paper, an improved Firefly Algorithm based on

Newton's law of universal gravitation was proposed to

solve the problems of FA. With the experimental results

on 4 standard test functions, the results show that this

method is superior to FA in computational precision and

convergence rate, and effectively solves the local

convergence of Firefly Algorithm.

II. FIREFLY ALGORITHM

In the Firefly Algorithm, there are two important issues:

brightness and attraction. The firefly brightness reflects

the location superiority and decides the moving direction.

The degree of attraction determines the firefly‟s moving

distance. The brightness and the degree of attraction are

constantly updated, so as to achieve the goal of

219© 2017 J. Adv. Inf. Technol.

Journal of Advances in Information Technology Vol. 8, No. 4, November 2017

doi: 10.12720/jait.8.4.219-224

optimization. Mathematical description of FA described

below [15], [16].

Relative brightness is defined by

0ij

rI I e

(1)

where: 0I is the original light intensity, it determined by

the objective function. is the light absorption

coefficient, and ijr stands for the spatial distance between

firefly i and j . The intensity will gradually weakened

with increasing distance ijr and media absorption, thus the

light absorption coefficient is set to reflect this

characteristic.

Attraction is defined by

0ij

re

(2)

where 0 is the original attraction value.

Location is updated by

( 1) ( ) ( ( ) ( )) ( 1/ 2)i i j ix t x t x t x t rand (3)

where: xi and yi is the position of firefly i and j in the

space, and t indicates generation index. The parameter is the step factor, which is a random number with the

range [0, 1]; rand is a random factor which is uniformly

distributed on [0,1]. (rand-1/2) is the disturbance term, which can enhance search range and prevent the situation

that particles fall into local best.

TABLE I. PSEUDO CODE OF THE BASIC STEPS OF THE LOGFA

Improved Firefly Algorithm based on Newton's law of

universal gravitation

Objective function )(xf , Tdxxx ),...,( 1

Generate initial population of fireflies ),...,2,1( nixi Calculate the quality im

at ix according to Eq.6;

while (t < MaxGeneration)

for i=1: N all N fireflies

for j=1: N all N fireflies

if ji mm

Move firefly i towards j ;

end if

Evaluate new solutions and update the quality im

end for i

end for j

Rank the fireflies and find the current global best solution

end while

Post-process the results

III. IMPROVED FIREFLY ALGORITHM BASED ON

NEWTON'S LAW OF UNIVERSAL GRAVITATION (LOGFA)

Although FA has been gradually applied in many

fields, but some defects still exist in the algorithm itself,

such as slow convergence speed, easiness to stagnation,

premature convergence and low precision. In order to

solve this problem, this paper has made two

improvements. Firstly, we build a new evolutionary

computation model by using gravity as attractiveness

between fireflies, which makes the optimization

algorithm not only keep the evolutionary advantage of the

original algorithm, but also improve the accuracy of the

algorithm and convergence ability. Especially, the

algorithm still has higher precision and convergence

ability when the search range and initial value are larger.

Secondly, when the population falls into the local optimal

region, we can improve firefly's diversity through

Gaussian mutation. The basic steps of the LOGFA are

summarized by the pseudo code listed in Table I. Specific

measures for improvement are as follows.

A. Gravitational Evolution Model of LOGFA

In the Gravitational evolutionary model, the attraction

between the firefly particles is gravitation, and the motion

of the particles follows a certain degree of dynamical law.

The object with the highest mass occupies the best

position, that is, target value is the best. Under the

influence of attraction, smaller objects move toward

greater objects. In order to balance the global searching

ability in the early stage of evolution and the fast

convergence of the later stage, perturbation term with

step size contraction is introduced in the process of

position updating. In the perturbation term, the step factor

is decreasing by generation. If the position function value

after the location update is worse than the original

position, replace the new position with the original

position to ensure the search time and the accuracy of the

solution. The specific definition is as follows:

The gravity between firefly i and j is defined by

2

(t) (t)i jij

ij

m mF G

r (4)

where: ijr stands for the spatial distance between firefly i

and j, and ijm indicates the quality of the firefly i. G is the

gravitational constant of the time t, which is defined by

2

0ijreGG

(5)

where 0G is the original gravitational constant.

The quality im is calculated by

)()(

)()(

iworstibest

iworstii

xfxf

xfxfm

(6)

where: )( ixf indicates the target fitness value of the

firefly i at ix . If it is to solve the minimum problem,

)(min)(},..,2,1{

ini

ibest xfxf

, )(max)(},..,2,1{

ini

iworst xfxf

.

Conversely, if it is to solve the maximum problem, then

)(max)(},..,2,1{

ini

ibest xfxf

， )(min)(},..,2,1{

ini

iworst xfxf

.

Location is updated by

*

)2

1(

2

1)()1(

)(

)(

randatxtx

tm

tFa

iii

i

iji

(7)



where: rand is a random factor which is uniformly

distributed on [0, 1]. The parameter is the step factor, which is a random number with the range [0, 1]. The

parameter is the Step size contraction factor, which is a constant with the range [0.95, 1].

B. Gaussian Variation Strategy

If the firefly population does not evolve in many

iterations, it is judged that it has fallen into the local

optimal region. At this time, the Gaussian variation factor

is used to disturb the firefly population and restore its

evolutionary ability. Operation is as follows:

Firstly, all the fireflies in accordance with the size of

fitness sort. Then the worst firefly population is replaced

with the optimal firefly, resulting in an intermediate

population. Finally, the middle population is subjected to

Gaussian mutation [17] according to (8).

)1,0(Nxxx iii (8)

where )1,0(N is random vector for the Gaussian

distribution with mathematical expectation of 0 and

variance of 1.

In order to ensure that the individuals of the firefly

population are always searching effectively in the search

space, the domain constraints are processed for all the

firefly individuals. This measure can effectively pull the

individual back to the specified space when the individual

is out of the specified range. The domain constraint

processing is defined by

maxmax

minmin

xxx

xxxx

i

ii (9)

where: minx is the lower limit of the search space, and

maxx is the upper limit of the search space.

C. Convergence Analysis of LOGFA

In order to analyze the convergence of LOGFA, this

paper gives the analysis and proof based on the criteria

proposed by Solis [18].

Condition 1: )()),(( xfxDf , if S ,then

)()),(( fxDf ,

where: f is the target function, is the solution of the

algorithm in the iterative process. S is feasible solution

space, and ),( xD is the iterative result of the algorithm

at the 1t times.

Condition 2: 0)(, DvSD and 0))(1(

0

k

k Du .

where: )(Dv is the Lebesgue measure of D , )(Duk is the

probability measure ofthe algorithm‟s solution

in iteration k .

Theorem 1. Assume that f is measurable and

measurable space S is the measurable subset of nR . If the algorithm satisfies condition 1 and condition 2, then

1)(lim *

SXp kk

, that is, the algorithm is convergent

with probability 1. Where, *S is the global best position.

Theorem 2. LOGFA converges to the global optimal

solution with probability 1.

Proof.

According to the description of LOGFA, firefly after

each update position will not be worse than the original

position. If S , then )()),(( fxDf . In other

words, condition 1 is satisfied.

Assume that tkL is solution support set of firefly k in

the t-th iterative process. Fireflies are gathered in a

certain area with the increase of iteration times, and the

population was prone to stagnation because of the loss of

further evolutionary power. Therefore )(tkLv will

gradually decrease, then )()(1

SvSLvN

k

tk

, that is,

condition 2 is not satisfied. But the algorithm will

produces a variant solution when evolution stagnates, let

PLtk , then PLN

k

tk

1

. So SD , let 1)(01

N

kk Du ,

0])(1[lim))(1(10

kN

kk

kkk DuDu , that is to say

condition 2 is satisfied. Thus LOGFA converges to the

global optimal solution with probability 1. Proof is

completed.

IV. SIMULATION EXPERIMENT AND ANALYSIS

A. Standard Benchmark Functions

In order to verify the convergence speed and

optimization ability of LOGFA, 4 benchmark functions

are used for tests. All these functions are minimization

problems. The expressions of the 4 functions are given in

(10) to (13). The dimensions, the number of fireflies and

the global optimums of these functions are listed in Table

II.

22222222

2)4()4()4()4()4(1yxyxyxyx eeeef (10)

n

i

ixf1

22

(11)

)10)2cos(10(1

23

i

n

i

i xxf (12)

n

i

in

i

ii

xxf

11

24 )cos(

4000

11 (13)

B. Simulation Results and Analysis

This section will compare the standard Firefly

Algorithm, improved Firefly Algorithm based on

Newton's law of universal gravitation (LOGFA) and



Firefly Algorithm based on improved evolutionism

(IEFA) using the 4 benchmark functions.

In following tests, the algorithm parameters are set as

follows:

MaxGeneration=200, 3.0 , 1,10 ，

97.0 ， 10 G .

TABLE II. STANDARD BENCHMARK FUNCTIONS

Function name Function Dimensions Population number Global optimum

Fourpeaks 1f

2 20 -2

Sphere 2f 10 50 0

Rastrigin 3f 10 50 0

Griewank 4f

10 50 0

TABLE III. SIMULATION RESULTS OF FOUR FUNCTION

Function Algorithm Range Minimum Maximum Average Variance

Fourpeaks

FA (-5,5) -1.9999 -1.9998 -1.9999 2.4943E-09

(-50,50) -4.4069E-3

-8.08E-120 -2.9458E-4 9.3747E-07

IEFA

(-5,5) -2 -2 -2 0

(-50,50) -2 -9.73E-34 -1.2292 0.7723

LOGFA

(-5,5) -2 -2 -2 0

(-50,50) -2 -2 -2 0

Sphere

FA

(-10,10) 101.1423 170.0588 131.6443 300.9770

(-100,100) 5694.2374 19575.1563 12906.5488 14249138.45

IEFA

(-10,10) 4.1978 47.0799 26.5774 146.6322

(-100,100) 2504.7592 13066.0866 6169.1311 5231248.87

LOGFA

(-10,10) 2.34E-12 9.78E-05 5.0927E-06 4.5247E-10

(-100,100) 2.02E-11 4.23E-04 2.8840E-05 9.1340E-09

Rastrigin

FA

(-5,5) 2.5681 47.1232 26.0191 196.1422

(-10,10) 1.52E+02 2.09E+02 187.1096 251.0512

IEFA

(-5,5) 1.42E-11 5.49E-05 5.7586E-06 2.07995E-10

(-10,10) 5.70E-06 18.2706 3.4905 23.9473

LOGFA

(-5,5) 1.60E-11 1.35E-06 1.6683E-07 1.53986E-13

(-10,10) 9.04E-09 1.90E-04 2.0478E-05 2.0384E-09

Griewank

FA

(-5,5)

4.036E-03 4.3112E-01 1.8701E-01 3.1864E-02

(-50,50) 1.4688 2.0702 1.8217 2.7158E-02

IEFA

(-5,5) 1.55E-13 2.01E-06 2.0495E-07 2.8973E-13

(-50,50) 0.4811 1.4830 0.9613 7.8882E-02

LOGFA

(-5,5) 1.12E-14 1.63E-08 1.9630E-09 2.3288E-17

(-50,50) 9.49E-12 2.44E-07 2.8929E-08 3.8471E-15

Each algorithm will run 20 times independently for

each test function. The average value of each function,

the optimal solution, the worst solution and the variance

are obtained. The specific calculation results are shown in

Table III. The evolution curves of the four test functions

are shown in Fig. 1 to Fig. 4.

As we can see from Table III: By comparing the

desired minimum number of iterations, the maximum

number of iterations and the average number of iterations,

we can conclude that LOGFA shows greater evolutionary

optimization than FA, it has better robustness and

adaptability. LOGFA is only one or two high accuracy

than IEFA when the search range is small. However,

when the search range becomes larger, LOGFA can still

achieve high accuracy, showing a strong stability. At this

moment, IEFA accuracy is relatively low.

As we can see from Fig. 1 to Fig. 4: FA appeared the

phenomenon of premature convergence, and the precision

is not high. The convergence speed of IEFA and LOGFA

is very fast, and it is very good to avoid falling into the

local optimal region. However, the convergence accuracy

of LOGFA is higher.



Figure 1.

Evolution curves of fourpeaks.

Figure 2.

Evolution curves of sphere.

Figure.3

evolution curves of rastrigi.

Figure 4.

Evolution curves of griewank.

In general, LOGFA has better ability to search the

optimal value than the other two algorithms, and the

accuracy of the optimal value is higher, which shows

strong global optimization ability and higher search

precision. Additionally, LOGFA has better robustness,

adaptability and stability.

V.CONCLUSIONS

Due to the FA have low precision defects, easily

falling into local optimum value when solving the global

optimal value, an improved Firefly Algorithm based on

Newton's law of universal gravitation was proposed in

this paper. The experimental results show that LOGFA

has the best accuracy and stability, which can balance the

global and local search capability well, and overcome the

shortcomings of the basic Firefly Algorithm. It is can be

applied to complex industrial process optimization, such

as power plant optimal control, to solve actual

engineering problems with high profits.

ACKNOWLEDGEMENT

This work was supported by the National Natural

Science Foundation of China (61572416),Hunan

Provincial Natural science Zhuzhou United foundation

(2016JJ5033), JCYJ2016042911221382)”, Key discipline

of Hunan Province „Information and Communication

Engineering‟ in „12th Five Year Plan‟. Hunan University

of Science and Technology Doctor Startup Fund Grant

No.E51664. Hunan Provincial Department of Education

Science Research Project Grant No.16C0639.

REFERENCES

[1] I. Fister, X. S. Yang, et al., “Modified Firefly Algorithm using quaternion representation,” Expert Systems with Applications, vol.

40, no. 18, pp. 7220-7230, 2013.

[2] X. S. Yang, “Firefly Algorithm,” Lévy Flights and Global Optimization, vol. 20, pp. 209-218, 2010.

[3] A. H. Gandomi, X. S. Yang, and A. H. Alavi, “Mixed variable structural optimization using Firefly Algorithm,” Computers &

Structures, vol. 89, no. 23-24, pp. 2325-2336, 2011.

[4] X. S. Yang, “Stochastic test functions and design optimisation,” International Journal of Bio-Inspired Computation, vol. 2, no. 2,

pp. 78-84, 2010.

[5] S. Cao, J. Wang, X. Gu, A Wireless Sensor Network Location Algorithm Based on Firefly Algorithm, AsiaSim, 2012, pp. 18-26.

[6] L. D. S. Coelho and V. C. Mariani, “Firefly Algorithm approach based on chaotic Tinkerbell map applied to multivariable PID

controller tuning,” Computers & Mathematics with Applications,

vol. 64, no. 8, pp. 2371-2382, 2012.

[7] A. Kazem, E. Sharifi, et al., “Support vector regression with chaos-based Firefly Algorithm for stock market price forecasting,”

Applied Soft Computing, vol. 13, no. 2, pp. 947-958, 2013.

[8] M. H. Horng, “Vector quantization using the Firefly Algorithm for image compression,” Expert Systems with Applications An

International Journal, vol. 39, no. 1, pp. 1078-1091, 2012.

[9] M. Y. Cheng, Z. W. Ni, et al., “Overview on glowworm swarm optimization or Firefly Algorithm,” Computer Science, 2015.

[10] X. S. Yang, “Multiobjective Firefly Algorithm for continuous optimization,” Engineering with Computers, 2013.

[11] Y. Feng, J. Liu, and H. E. Yichao, “Chaos-based dynamic population Firefly Algorithm,” Journal of Computer Applications,

vol. 33, no. 3, pp. 796-792, 2013.

[12] Wang M, Qiang F U, Tong N, et al. “Multi-group Firefly Algorithm based on simulated annealing mechanism,” Journal of

Computer Applications, 2015.



[13] A. H. Gandomi, X. S. Yang, et al., “Firefly Algorithm with chaos,” Communications in Nonlinear Science & Numerical

Simulation, vol. 18, no. 1, pp. 89-98, 2013.

[14] F. U. Qiang, N. Tong, et al., “Firefly Algorithm based on improved evolutionism,” Computer Science, 2014.

[15] W. Li, “Study on leapfrog Firefly Algorithm and its application in the radio spectrum allocation,” Microcomputer & Its Applications,

2015.

[16] C. P. Liu and Y. E. Chun-Ming, “Firefly Algorithm with chaotic search strategy,” Journal of Systems & Management, 2013.

[17] W. Xia, M. X. Cheng, and M. D. Liu, “Improved ISPO algorithm based on Gaussian mutation,” Application Research of Computers,

vol. 30, no. 4, pp. 986-985, 2013.

[18] F. J. Solis and R. J. B. Wets “Minimization by random search techniques,” Mathematics of Operations Research, vol. 6, no. 1,

pp. 19-30, 1981.

Ling-Zhi Yi was born on March 27, 1966 in Hunan in China. She

graduated from the Xiangtan University in 1986. And after then she

worked in Xiangtan University, act as the leader of the electricity-

electronics committee of Xiangtan University. She is elected as a core

young teacher of Hunan province in 1999, and as the person for 121

project of Hunan province in 2007. Up to today, she has published 100

papers in some famous magazines with 4 patents for invention and 18

software copyrights, 51papers cited by SCI and EI index included. She

is now an expert in National Energy Conservation Center, senior

member of Chinese Association of Automation and executive director

in Institute of Automation of Hunan

Wei-Hong Xiao Received the bachelor's degree in electronic

information engineering from Hunan University of Science and

Technology, Xiangtan, China, in 2016.He presently attend

Xiangtan University, pursuing a Master's degree in electrical

engineering, and his research direction is intelligent algorithm.

WenXin Yu received the B.Sc. degree in applied mathematics from

Hebei Normal University, Shijiazhuang, China, in 2005, and the M.S.

degree in wavelet analysis from Changsha University of Science &

Technology, Changsha, in 2008. And Ph.D. degree in electrical

engineering from Hunan University. Changsha, in 2015.

He joined the Hunan University of Science and Technology, where he c

urrently is lecturer of School of Information and Electrical Engineering.

His interests include the research of Fault diagnosis, signal processing, a

nd wavelet analysis and its application.

Genping Wang was born in 1966 in Jiangxi in China. He received the

B.E. degree from Zhejiang University in 1988. He received his Ph.D.

degree in Industrial Automation from Zhejiang University, Hangzhou,

Zhejiang, in 1998. His major field of study including signal and

processing, control technology and application of computer.



An Improved Firefly Algorithm Based on Newton's Law of ... · Algorithm based on Newton's law of...

Documents

Transcript of An Improved Firefly Algorithm Based on Newton's Law of ... · Algorithm based on Newton's law of...