Research ArticleModification of Fish Swarm Algorithm Based on LevyFlight and Firefly Behavior

Zhenrui Peng Kangli Dong Hong Yin and Yu Bai

School of Mechatronic Engineering Lanzhou Jiaotong University Lanzhou China

Correspondence should be addressed to Zhenrui Peng pzrui163com

Received 11 June 2018 Revised 18 July 2018 Accepted 9 August 2018 Published 13 September 2018

Academic Editor Rasit Koker

Copyright copy 2018 Zhenrui Peng et alis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Artificial fish swarm algorithm easily converges to local optimum especially in solving the global optimization problem ofmultidimensional and multiextreme value functions To overcome this drawback a novel fish swarm algorithm (LFFSA) based onLevy flight and firefly behavior is proposed LFFSA incorporates the moving strategy of firefly algorithm into two behaviorpatterns of fish swarm ie chasing behavior and preying behavior Furthermore Levy flight is introduced into the searchingstrategy To limit the search band nonlinear view and step size based on dynamic parameter are considered Finally the proposedalgorithm LFFSA is validated with several benchmark problems Numerical results demonstrate that LFFSA has a better per-formance in convergence speed and optimization accuracy than the other test algorithms

1 Introduction

Optimization plays an important role in modern industry aswell as in scientific world Due to the computational costs ofthe existing numerical methods researchers have to rely onmetaheuristic algorithms to solve complex optimizationproblems

Artificial fish swarm algorithm (AFSA) was proposed asa metaheuristic algorithm based on fish swarm behav-iorsmdashpreying behavior swarming behavior and chasingbehavior [1] With its simple principle and such good fea-tures as robustness and tolerance of parameter setting AFSAhas become an increasingly important tool in swarm in-telligence optimization Some algorithms derived fromAFSA have been presented Wang et al eliminated the steprestriction and added new leaping behavior to improve thestability of the algorithm [2] Another modified form ofAFSA used particle swarm optimization to reformulateAFSA integrated AFSA with communication behavior andcreated formulas for major AFSA parameters [3] Luan et aladopted normal distribution function Cauchy distributionfunction multiparent crossover operator mutation opera-tor and modified minimal generation gap model to over-come the drawback of slow convergence speed in later

iterations [4] Another improved AFSA with crossover andculture algorithm was proposed to enhance its optimizationefficiency [5] Hu et al integrated the merits of the self-adaptation strategy mutation strategy and hybrid strategyinto the social behaviors of AFSA [6]

Over the last decades AFSA and its improved algorithmshave been successfully applied to various engineering op-timization problems Kumar et al adopted AFSA to opti-mize renewable energy sources in a microgrid [7] Bycategorizing the social behaviors of fish swarm into foragingreproductive and random behaviors a novel artificial fishswarm algorithm was advocated for solving large-scalereliability-redundancy application problem [8] A de-rivative of AFSA was used to solve the multiobjective dis-assembly line balancing problem with fuzzy disassemblytimes [9] A binary fish swarm algorithm was presented tosolve profit-based unit commitment problem in generationcompanies [10]

Firefly algorithm (FA) is another metaheuristic algorithmbased on the idealized behavior of the flashing characteristicsof fireflies [11] FA can adaptively adjust the radius of theinduction and parallel search the optimum in multiple peaksFA has natural advantages in solving multimodal optimiza-tion problem Some drawbacks of searching strategy and

HindawiComputational Intelligence and NeuroscienceVolume 2018 Article ID 9827372 13 pageshttpsdoiorg10115520189827372

parameters in FA were targeted and improved by researchersA derivative of FA with directed movement of fireflies wasproposed by Farahani et al [12] Chaotic sequence was in-troduced into the basic FA to adjust parameters c and α byCoelho et al [13] Eagle Strategy which combines Levy flightsearch with FA was introduced by Yang and Deb [14] ElitistFA was presented [15] which tried to enhance the best so-lution position by generatingm uniform random vectors andmoving in the direction of best solution

Firefly algorithm and its modified forms were alsosuccessfully applied to numerous practical problems Jaga-theesan et al used firefly algorithm to design a controller foran automatic generation control of multiarea power thermalsystems [16] An FA-inspired band selection and optimizedextreme learning machine were proposed for hyperspectralimage classification [17] A self-adaptive firefly algorithmwas developed for placement of FACTS devices [18]Teshome et al modified FA to counteract some inherentproblems that may hinder the performance of the maximumpower point tracking [19] Alb et al used FA to solvea shieldingshunting electromagnetic problem [20] Mishraet al proposed a method for optimal placement of interlinepower flow controller by using FA [21] Other applicationsof improved FA include image compression [22] financialforecasting [23] image segmentation [24] structural opti-mization [25] classification problem [26] unconstrainedoptimization [27] economic dispatch problems [28] clus-tering [29] image retrieval [30] and mechanical optimaldesign [31] Some researchers presented comprehensivereviews of existing FA and its modified forms to encouragenew researchers to employ FA for solving their ownproblems [32 33]

At the same time numerous studies have shown thatLevy flight is similar to flight characteristics of given an-imals and insects [34ndash36] which has been widely used inswarm intelligence algorithms Subsequently Levy flighthas been applied to optimization and preliminary resultsshowed its potential capabilities Jensi and Jiji proposed anenhanced particle swarm optimization with Levy flight[37] Tang et al proposed a new framework of shuffled frog-leaping algorithm based on the exploration and exploita-tion mechanism by using Levy flight [38] Yahya and Sakaproposed a multiobjective artificial bee colony algorithmwith Levy flight and applied it to construction site layoutplanning [39]

Hybridization is recognized to be an important aspect ofhigh performing algorithms in recent years [40] Owing tosome drawbacks of traditional AFSA and FA they are notsuitable for solving highly nonlinear and multimodalproblems By integrating the merits of AFSA with Levy flightand FA this paper proposes a novel hybrid algorithmnamed LFFSA (fish swarm algorithm based on Levy flightand firefly behavior) for global optimization e highlightsof the new algorithm are as follows

(i) Attraction degree is involved in the definition ofartificial fish

(ii) Levy flight is used to adjust the search route ofartificial preying fishes

(iii) By analyzing the relationship between swarmingbehavior and chasing behavior unnecessary behavior(swarming behavior) is excluded instead of im-proving AFSA through adding new behaviors [3ndash6]

(iv) Time complexity of the improved algorithm is alsofurther analyzed to demonstrate the effectiveness ofthe improvement

e remainder of this paper is organized as followsSection 2 describes the basic AFSA FA and Levy flightrespectively Section 3 proposes and explains LFFSA algo-rithm in details in Section 4 the superiority of proposedalgorithm LFFSA is validated by several benchmark prob-lems Section 5 outlines the conclusion

2 Background

21 Artificial Fish Swarm Algorithm AFSA is a swarm in-telligence algorithm which can be employed to solve theoptimization problem by imitating swarming chasing andpreying behaviors of artificial fishes [1] As shown in Figure 1let Xi be the current position of one artificial fish Xv be theviewpoint of artificial fish at one moment Visual be the visualscope of each individual Xa and Xb be fishes within theVisual of Xi Step be the biggest step of artificial fish and δ bethe congestion factor of fish swarme food concentration isproportional to the fitness function f(X) e behaviorpatterns of fish swarms can be described as follows

Swarming behavior if f(Xc)gtf(Xi) where Xc is thecentral point inside the Visual of the point Xi swarmingbehavior is to be executed Take Xc as Xv e fish at Xi willtake a step toward the point Xc

Chasing behavior if the point (denoted by Xmax) havingthe best objective function value inside the Visual satisfiesf(Xmax)gtf(Xi) and if the Visual of Xi is not crowdedchasing behavior is to be executed Take Xmax as Xv e fishat Xi will take a step toward the point Xmax

Preying behavior preying behavior is tried in the fol-lowing situations

(1) f(Xc)ltf(Xi) f(Xmax)ltf(Xi) and the Visual isnot crowded

(2) e Visual is crowded

Here a point Xj inside the Visual of Xi is randomly se-lected Iff(Xj)gtf(Xi) the preying behavior is to be executedTake Xj as Xv e fish at Xi will take a step toward the pointXj Otherwise it will move a step randomly within its Visual

e best solution obtained in each iteration is marked asldquoboardrdquo After the specified iterations search process isterminated and the result on the ldquoboardrdquo is regarded as thefinal solution

For artificial preying fishes the position-updating can beformulated as

Xnext Xi + rand middotstep times Xj minusXi1113872 1113873

norm Xj minusXi1113872 1113873 (1)

where Xnext is the next position of artificial fish Xi is thecurrent position of artificial fish Xj is the position which has

2 Computational Intelligence and Neuroscience

a better objective function value rand is a random numberin [minus1 1] and norm(Xj minusXi) is the distance between twoposition vectors

For artificial swarming fishes the position-updating canbe formulated as

Xnext Xi + rand middotstep times Xc minusXi( 1113857

norm Xc minusXi( 1113857 (2)

For artificial chasing fishes the position-updating can beformulated as

Xnext Xi + rand middotstep times Xmax minusXi( 1113857

norm Xmax minusXi( 1113857 (3)

e flowchart of AFSA is shown in Figure 2

22 Firefly Algorithm Firefly algorithm (FA) [11] is anotherswarm intelligence algorithm It achieves swarming phe-nomenon by using the fluorescent signal between two fireflyindividuals

e attraction between fireflies depends on their lightintensities and attraction degree e light intensity is inproportion to the objective function value of fireflyrsquosposition Attraction degree is in proportion to the lightintensity e brighter the light intensity is the higherattraction degree will be Besides the farther the distanceis the lower the light intensity and attraction degree willbe

In the simplest form the light intensity I(r) varies withthe distance r monotonically and exponentially as

I(r) I0 eminuscr (4)

where I0 is the original light intensity and c is the lightabsorption coefficient As firefly attraction degree isproportional to the light intensity seen by adjacentfireflies the attraction degree beta of a firefly can bedefined as

beta beta0 middot eminuscr2 (5)

where beta0 is the attraction degree at r 0

e distance rij between any two fireflies i and j at xi andxj respectively is the Cartesian distance which is calculated as

rij xi minusxj

1113944

n

d1xid minus xjd1113872 1113873

11139741113972

(6)

where n is the dimensionality of the given problem

23 Levy Flight Levy flight is one kind of random searchingstrategy [35] Flying step satisfies a heavy-tailed Levy dis-tribution which can be represented by a clear power-lawequation as

AFSAbegins

Does Xi meet thecondition of swarming

behavior

N N Does Xi meet thecondition of chasing

behavior

Y Y

Y Ynext1 gt Ynext2N

Ni ge n

Y

Whether the stopcriterion is met

N

Y

End

Obtain the optimum solution

Xi = Xnext= Xnext2

Xi = Xnext= Xnext1

Xnext2Ynext2Xnext1Ynext1

ObtainXnext2Ynext2

Executepreying

behavior

ObtainXnext1Ynext1

i = i + 1

i = 1

Initialize theswarm X1X2hellipXn

Artificialfish Xi

Artificialfish Xi

Figure 2 Flowchart of AFSA

Xa

Xb

XiVisual

Xnext Step

Xv

Figure 1 Vision concept of the artificial fish

Computational Intelligence and Neuroscience 3

L(s) sim |s|1minusβ

(7)

where s is random Levy step For searching problems insidea wide range of unknown space the variance of Levymovement increases faster than the dimensional Brownianmovement

To some extent the foraging behavior of nature ani-mals is a kind of random movement behavior Becausenext movement usually depends on the current positionand the probability of moving to next position the ef-fectiveness of each random movement becomes greatlyimportant Recent studies show that Levy flight is one ofthe best searching strategies in random movement model[35 41ndash43]

3 Fish Swarm Algorithm Based on Levy Flightand Firefly Behavior

AFSA has several disadvantages in solving nonlinear andmultimodal problems Firstly AFSA uses swarming be-havior and chasing behavior to execute parallel search ina simple and fast way However after determining the di-rection each artificial fish moving with random step will beunable to approach the target point effectively Secondlyartificial fish will execute preying behavior when it does notmeet the conditions of swarming behavior and chasingbehavior is kind of searching strategy is inefficient andcan easily miss the optimum point irdly too manymoving patterns can increase the algorithm complexitywhich may cause slow convergence speed

e above-listed drawbacks of AFSA are improved in theproposed LFFSA FA has the unique moving strategy usingattraction between fireflies which can be used to fix therandom moving after determining the direction in AFSAAnd the preying behavior can be improved using Levy flightto specify the behavior of artificial fishes In LFFSA at-traction degree is involved in the definition of artificialfishes which allows each individual move according to at-traction degree Levy flight is also considered in the defi-nition of the artificial preying fish to avoid falling into thelocal optimum chasing behavior is excluded to decrease thealgorithm complexity

e flowchart of LFFSA is shown in Figure 3 Pseudocode of LFFSA is described in Algorithm 1 To show thedifference between AFSA and LFFSA vividly mechanisms ofboth algorithms are provided in Figure 4

e main improvements of LFFSA are summarized asfollows

(a) Improvement 1 FA-based moving strategy Attrac-tion degree is involved in the definition of artificialfishes which can be formulated as

betaij beta0 middot eminuscrij (8)

where rij is the Cartesian distance between artificialfishes i and j given by Equation (6) c is the lightintensity coefficient which can be set as a constantand beta0 the largest attraction degree is attractiondegree of an artificial fish at rij 0

e position-updating with preying behavior establishedby attraction degree can be formulated as

Xnext Xi + betaij Xj minusXi1113872 1113873 + α(randminus 05) (9)

where Xi is the current position of artificial fish ibetaij(Xj minusXi) is the attraction degree α is the step factorwhich is a constant between 0 and 1 rand is a numberchosen randomly in [minus1 1] and α(randminus 05) is to avoidfalling into the local optimum

(b) Improvement 2 inertia weight A linear inertiaweight is added into Equation (9) as

Xnext ωtXi + betaij Xj minusXi1113872 1113873 + α(randminus 05) (10)

ωt ωmax minus ωmax minusωmin( 1113857 middot genMaxgen (11)

where ωt is the weight size inherited from the lastposition of an artificial fish ωmax denotes the biggestweight ωmin is the minimum weight gen representsthe current iteration and Maxgen is the ultimateiteration

LFFSAbegins

Initialize theswarm X1X2hellipXn

i = 1

Does Xi meet thecondition of chasing

behavior

N

Y

Prey withLeacutevy fight

Xi = Xnext

i = i + 1 N i ge n

Y

Whether the stopcriterion is met

N

YObtain the

optimum solution

End

Moveaccording to

attractiondegree

Artificial fish Xi

Figure 3 Flowchart of LFFSA

4 Computational Intelligence and Neuroscience

Similarly the position-updating with chasing behaviorcan be formulated as

Xnext ωtXi + betaij Xmax minusXi( 1113857 + α(randminus 05) (12)

where Xmax is the position with the highest food concen-tration in the view of artificial fish Xi and other parametersare defined in Equations (10) and (11)

(c) Improvement 3 Levy flight-based search strategyLevy flight is involved in the definition of artificialfishes are executing preying behavior e move-ment can be formulated as

Xnext Xi + αoplus L(λ) (13)

L(λ) ϕ times μ|v|1β

Xi minusXbest( 1113857 (14)

where Xi is the current position of artificial fish i oplus isthe point to point multiplication L(λ) denotes arandom vector generated by Levy flight Xbest rep-resents the best fish on the ldquoboardrdquo μ tminusλ 1lt λlt 3and μ and v have the standard normal distributionμ sim N(0 ϕ2) v sim N(0 1) respectively where

ϕ Γ(1 + β)sin(πβ2)

Γ[(1 + β)2]β middot 2(βminus 1)21113896 1113897

1β

(15)

where Γ is the standard Gamma function(d) Improvement 4 nonlinear visual and step Visual and

step change nonlinearly and dynamically in LFFSAe updating equations are as follows

Visual ρ middot Visual + Visualmin (16)

Visual Preying unaccomplished

Chasing

Behavior unaccomplished

Chasingaccomplished

Preyingaccomplished

Movingrandomly

AFSA

Artificial fishes

Swarming Swarmingaccomplished

LFFSA Chasing accomplished

Artificial fishes with attraction degree

Chasingunaccomplished

Preying withLeacutevy flight

Chasing

Figure 4 Mechanisms of AFSA and LFFSA

(1) Initialization of X1 X2 Xn1113864 1113865(2) Evaluate fitness(3) while the stop criterion is not met do(4) while ilt n do(5) if Xi meets the condition of chasing behavior(6) Move with attraction degree and get a new Xi based on Equation (13)(7) else(8) Prey with Levy flight and get a new Xi based on Equations (11) and (14)(9) end if(10) end while(11) Update the ldquoboardrdquo(12) end while(13) Obtain the optimum solution

ALGORITHM 1 LFFSA

Computational Intelligence and Neuroscience 5

Step ρ middot Step + Stepmin (17)

ρ exp minus30 timesgen

Maxgen( )

s

( ) (18)

where Visualmin the minimum visual takes 0001Stepmin the minimum step takes 00002 ρ isa nonlinear weight and s represents an integer sgt 1Here s 3 Other parameters are the same as thosein Equation (11) e relationship between s and ρ isas shown in Figure 5

Besides dene the biggest distance between two articialshes as

maxD xmax minusxmin( )2 timesD

radic (19)

where xmax and xmin represent the upper bound and thelower bound of searching range respectively D denotes theD-dimension searching space e initial Visual is equal tomaxD and initial Step is equal to maxD8 en Visual andStep change dynamically according to Equations (16)ndash(18)

(e) Improvement 5 decrease of time complexitySwarming behavior in AFSA is excluded to decreasethe algorithm complexity

4 Numerical Simulation

41 Comparison of Convergence Accuracy LFFSA is vali-dated by numerical simulations Fish swarm algorithm withrey behavior (FFSA) dierential evolution (DE) algo-rithm self-adaptive dierential evolution (jDE) algorithmand the two basic algorithms (AFSA and FA) are comparedAll the algorithms are coded in Matlab 2014b e operatingsystem is windows 7 Simulation hardware is a PC with250GHz Inter Core i5 and 200GB Memory

Parameters shown in Table 1 are determined by trial anderror

e following benchmarks are chosen carefullyaccording to their features Functions Sphere Quartic andRosenbrock etc are simple unimodal problems FunctionsAckley Rastrigin and Schwefel etc are highly complexmultimodal problems with many local minima Schwefelfunction has a maximum value and other functions haveminimum values ese benchmarks are listed in Table 2[44 45] e solutions of 17 test functions obtained bydierent algorithms are compared To compare the con-vergence speed and accuracy of the algorithms clearly andcorrectly all functions are run 50 times for each algorithmrespectivelye results are averaged and plotted in Figure 6

From Figure 6 LFFSA can avoid local optimum and havebetter convergence accuracy compared with the other al-gorithms For AFSA and FA the solutions of most functionsare unsatisfactory the DE cannot nd ideal solutions of f3f4 f5 f6 f10 f11 f12 f14 and f15 the jDE has good accuracywhile solving some of those functions eg f1 f2 f3 and f8however solutions of f4 f10 f11 and f15 obtained by jDEare not so precise the LFFSA can obtain the ideal accuracy

for almost all functions although it cannot achieve a highprecision level like solutions of f2 obtained by jDE the FFSAis slightly worse than LFFSAe LFFSA outperforms jDE in10 benchmark functions while 2 functions are comparativeand 5 functions are worse

LFFSA is better than AFSA because Levy ight is able torestrict the movement step of AFSA to a very small areaaround the current position Furthermore the attractiondegree guides the sh moving Besides LFFSA can quicklylead the sh individual to the close-by optimal pointConsidering all the advantages discussed above the opti-mum solution can be found successfully by using LFFSAwhich outperforms the basic algorithms for all test functionsand outperforms jDE for several functions To observe thesearching capabilities of dierent algorithms directly theaverage median best and worst values obtained by dierentalgorithms are listed in Table 3 Results indicate that LFFSAcan nd ideal solutions and have a better robustness

42 Computational ComplexityAnalysis Time complexity isalso an important indicator in the analysis of algorithms Ifan algorithm is composed of several parts then its com-plexity is the sum of the complexities of these parts ealgorithm may consist of a loop executed many times andeach time is with a dierent complexity Time complexity of

Iteration0 500 1000 1500 2000

ρ

0

02

04

06

08

1

s = 3s = 12

s = 21s = 30

Figure 5 Value of ρ

Table 1 Parameter settings

Algorithms Parameters ValuesFA FFSA LFFSA β0 10FA FFSA LFFSA c 10AFSA FFSA LFFSA δ 0618AFSA FFSA LFFSA Trynumber 5DE jDE Scaling constant 05DE jDE Crossover constant 09All 6 algorithms Population 50

All 6 algorithms Maximum functionevaluations (FEs) 2 times 105

6 Computational Intelligence and Neuroscience

the algorithm is used to estimate the efficiency of the al-gorithm It is defined that the time complexity of the al-gorithm or the running time is O(f(n)) [46] Define N asthe population

In the definition of time complexity O(N2) and O(N)

are at different levels If the time complexity of one algorithmis O(N2) the time complexity of the other one is O(N) thenthe former algorithm is more complex In the other case ifthe time complexity of one algorithm is O(N2) while thetime complexity of the other one is O(N2 + N) theircomplexities are both O(N2)

e time complexity analysis of AFSA is provided inTable 4

From Table 4 the time complexity of AFSA is

O Maxgenlowast 3lowastN2

+ trynumberlowastN + 6lowastN1113872 11138731113872 1113873 (20)

Swarming behavior has N times of calculating conges-tion factor 1 time of judging and 1 time of movingerefore time complexity of swarming behavior isO(N2 + 2lowastN) Chasing behavior has N times of calculating

congestion factor N times of searching 1 time of judgingand 1 time of moving erefore time complexity of chasingbehavior is O(2lowastN2 + 2lowastN)

Time complexity analysis of LFFSA is listed in Table 5Due to the lack of swarming behavior time complexity

of LFFSA can be calculated as

O Maxgenlowast 2lowastN2

+ TrynumberlowastN + 4lowastN1113872 11138731113872 1113873 (21)

We can also obtain time complexity of FA

O Maxgenlowast N2

+ N1113872 11138731113872 1113873 (22)

A conclusion can be obtained that time complexities ofthe three algorithms are at the same level eir computa-tional complexities in the worst case are only the square ofthe training sample size

43 Experimental Complexity Analysis Time complexity isa rough estimate of time costemore accurate time cost ofan algorithm can only be validated by running it on

Table 2 Test functions

No Test functions Expression Optimum value Domain Df1 Sphere f(x) 1113936

Di1x

2i 0 (minus100 100)D 30

f2 Quartic f(x) 1113936Di1ix

4i 0 (minus128 128)D 30

f3 Ackley f(x) minus20 exp minus02

(1D)1113936Di1x

2i

1113969

minus exp[(1D)1113936Di1cos(2πxi)]1113882 1113883

+ 20 + e0 (minus32768 32768)D 30

f4 Rosenbrock f(x) 1113936Dminus1i1 100(xi+1 minusxi

2)2 + (1minusxi)2 0 (minus2048 2048)D 30

f5 Rastrigin1 f(x) 1113936Di1 xi

2 minus 10 cos(2πxi) + 101113864 1113865 0 (minus512 512)D 30

f6 Rastrigin2f(x) 1113936

Di1 yi

2 minus 10 cos(2πyi) + 101113864 11138650 (minus512 512)D 30

yi xi |xi|lt (12)

round(2xi)2 |xi|gt (12)1113896

f7 Schwefel f(x) 1113936Di1 xi middot sin

|xi|

11139681113864 1113865 4189829D (minus500 500)D 30

f8 Griewank f(x) (14000)1113936Di1x

2i minus1113937

Di1cos(xi

i

radic) + 1 0 (minus600 600)D 30

f9 Quadric f(x) 1113936Di1(1113936

ij1xj)

2 0 (minus100 100)D 30

f10 Schaffer1 f(x) 1113936Dminus1i1 ((sin2

xi+1

2 + xi2

1113968minus 05)1113864

(0001(xi+12 + xi

2) + 1)2) + 050 (minus100 100)D 30

f11 Schaffer2 f(x) 1113936Dminus1i1 (sin2

1113936Di1x

2i

1113969

minus 05)(0001(1113936Di1x

2i ) + 1)2 + 051113882 1113883 0 (minus100 100)D 30

f12 Maxmod f(x) max(|xi|) 0 (minus10 10)D 30

f13 Dixon and price (x1 minus 1)2 + 1113936Di1i(2xi

2 minusximinus1) 0 (minus10 10)D 30

f14 Powell f(x) 1113936D4i1 [(x4iminus 3 + 10x4iminus 2)

2 + 5(x4iminus 1 minusx4i)2 +

(x4iminus 2 minus 2x4iminus 1)2 + 10(x4iminus 3 minusx4i)

4]0 (minus4 5)D 28

f15 Zakharov f(x) 1113936Di1x

2i + (1113936

Di105ixi)

2 + (1113936Di105ixi)

4 0 (minus5 10)D 30

f16 Sin1 1113936Di1|xi sin(xi) + 01xi| 0 (minus10 10)D 30

f17 Sin2 f(x) minus1113936Di1sin(xi) sin20(ix2

i π) minus992784 (0 π)D 100

Computational Intelligence and Neuroscience 7

FEs104

log 1

0(f(x

))

ndash20

ndash15

ndash10

ndash5

0

5

10

FAAFSAFFSA

LFFSADEjDE

0 5 10 15 20

(a)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash40

ndash30

ndash20

ndash10

0

10

FEs1040 5 10 15 20

(b)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

FEs1040 5 10 15 20

(c)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash2

0

2

4

6

8

FEs1040 5 10 15 20

(d)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

2

3

FEs1040 5 10 15 20

(e)

log 1

0(f(x

))FAAFSAFFSA

LFFSADEjDE

ndash6

ndash4

ndash2

0

2

4

FEs1040 5 10 15 20

(f )

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

32

34

36

38

4

42

FEs1040 5 10 15 20

(g)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash10

ndash5

0

5

FEs1040 5 10 15 20

(h)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash15

ndash10

ndash5

0

5

FEs1040 5 10 15 20

(i)

Figure 6 Continued

8 Computational Intelligence and Neuroscience

computer Since dierent algorithms cannot reach the sameconvergence accuracy the test with xed convergence ac-curacy is not available erefore the test with max functionevaluations is conducted Running time of each algorithm iscounted by the explorer of MATLAB Parameter settings ofalgorithms are the same in Section 41 Average running timeof dierent algorithms is listed in Table 6 When function

evaluations are the same running speed of LFFSA is fasterthan that of AFSA while DE has the fastest running speedResults are quite in accord with those obtained by com-putational complexity analysis LFFSA and jDE are com-parative in experimental complexity Running time of FFSAis almost twice as much as that of LFFSA e improvementof LFFSA decreases time complexity to some extent

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash8

ndash6

ndash4

ndash2

0

2

FEs1040 5 10 15 20

(j)lo

g 10(

f(x))

FAAFSAFFSA

LFFSADEjDE

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

FEs1040 5 10 15 20

(k)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

FEs1040 5 10 15 20

(l)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash2

0

2

4

6

8

10

FEs1040 5 10 15 20

(m)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash5

0

5

FEs1040 5 10 15 20

(n)

log 1

0(f(x

))FAAFSAFFSA

LFFSADEjDE

ndash10

ndash5

0

5

10

FEs1040 5 10 15 20

(o)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

2

FEs1040 5 10 15 20

(p)

f(x)

FAAFSAFFSA

LFFSADEjDE

FEs1040 5 10 15 20

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

(q)

Figure 6 Iterative curves of test functions (a) f1 (b) f2 (c) f3 (d) f4 (e) f5 (f ) f6 (g) f7 (h) f8 (i) f9 (j) f10 (k) f11 (l) f12 (m) f13 (n) f14(o) f15 (p) f16 (q) f7

Computational Intelligence and Neuroscience 9

Table 3 Comparison of optimization results

No Items AFSA Std FA Std FFSA Std LFFSA Std DE Std jDE Std

f1

Worst 3014

0181

minus2798

0120

minus5132

0071

minus6743

0071

minus8544

0132

minus15021

0121Best 2402 minus3257 minus5233 minus8500 minus10371 minus17242Average 2656 minus3048 minus5145 minus7278 minus8924 minus16326Median 2651 minus3102 minus5193 minus6325 minus8586 minus16706

f2

Worst minus2322

0171

minus19245

0135

minus11178

0074

minus14207

0134

minus20506

0400

minus31644

0535Best minus4126 minus2813 minus11313 minus15585 minus21611 minus33250Average minus2812 minus2278 minus11273 minus14812 minus21054 minus32687Median minus2562 minus2197 minus11271 minus14834 minus20970 minus32844

f3

Worst 1386

0001

minus1776

0124

minus3908

0197

minus6121

0138

minus0027

0438

minus8934

0167Best 1307 minus2415 minus4647 minus6938 minus1281 minus9471Average 1587 minus2168 minus4147 minus6546 minus0049 minus9163Median 1586 minus2177 minus4225 minus6325 minus0035 minus9164

f4

Worst 2143

0087

22733

0097

minus20348

0125

minus1076

0376

1520

0003

1454

0025Best 1565 1043 minus2416 minus3405 1412 1363Average 1946 1476 minus2158 minus1946 1385 1312Median 1854 1385 minus2235 minus2325 1363 1287

f5

Worst 2310

0077

1864

0856

minus2846

0044

minus4385

0054

minus0579

0323

0898

6009Best 2096 1243 minus2982 minus5145 minus1591 minus12831Average 2236 1454 minus2946 minus4643 minus0999 minus3303Median 2136 1285 minus2435 minus4325 minus0963 0148

f6

Worst 2445

0038

1716

0133

minus2382

0076

minus3414

0048

1131

0042

0698

5049Best 2318 1255 minus2618 minus3606 1012 minus11404Average 2408 1571 minus2486 minus3489 1077 minus5466Median 2419 1601 minus2462 minus3487 1094 minus7632

f7

Worst 3815

0018

37846

0031

4186

736eminus 6

4156

434eminus 6

4099

296eminus 9

4087

403eminus 3Best 3945 3978 4099 4099 4099 4099Average 3813 3848 4099 4099 4099 4097Median 3736 3785 minus4099 minus4099 4099 4097

f8

Worst 0956

0133

minus0960

0223

minus5301

0051

minus6271

0055

minus989

0114

minusInf

0Best minus0644 minus0500 minus5444 minus6455 minus10255 minusInfAverage minus0735 minus0697 minus5357 minus6372 minus10071 minusInfMedian 0736 0685 minus5435 minus6325 minus10074 minusInf

f9

Worst minus11665

1057

minus976

1324

minus9347

0843

minus8695

1323

minus6848

1124

minus6131

2697Best minus14433 minus10574 minus12194 minus11937 minus10680 minus14831Average minus12786 minus10456 minus10764 minus10137 minus8178 minus9938Median 1136 1085 minus7435 minus6325 minus7745 minus9663

f10

Worst 0957

0131

0974

0223

minus5375

0049

minus6274

0056

0673

0032

0280

0110Best 0644 0497 minus5448 minus6486 0483 minus0022Average 0747 0649 minus5376 minus6348 0526 0230Median 1136 1085 minus7435 minus6325 0547 0211

f11

Worst minus0301

272eminus 05

minus0303

0009

minus4751

0071

minus5647

0085

minus0896

0103

minus1106

0135Best minus0301 minus0331 minus5011 minus5965 minus1107 minus1429Average minus0301 minus0313 minus4838 minus5804 minus0975 minus1364Median minus0301 minus0311 minus4835 minus5804 minus0896 minus1402

f12

Worst 0779

0037

minus1177

0163

minus2756

0035

minus3158

0037

minus0136

0052

minus1904

0221Best 0658 minus1638 minus2892 minus3287 minus0339 minus2526Average 0711 minus1388 minus2811 minus3221 minus0252 minus2291Median 0716 minus1376 minus2801 minus3216 0248 minus2374

f13

Worst 4286

0173

0574

0297

minus0602

0007

minus0602

0005

0039

0068

minus0176

538eminus 6Best 3706 minus0175 minus0605 minus0605 minus0162 minus0176Average 4147 minus0013 minus0603 minus0603 minus0088 minus0176Median 4213 minus0135 minus0603 minus6603 minus0086 minus0176

f14

Worst 2404

0084

0405

0318

minus3598

0117

minus4514

0101

0276

0211

minus2192

0401Best 2173 minus0576 minus4001 minus4869 minus0369 minus3409Average 2275 0069 minus3773 minus4656 minus0117 minus2867Median 2246 0133 minus3719 minus4651 minus0134 minus2924

10 Computational Intelligence and Neuroscience

44ParameterAnalysis ofLFFSA eeect of parameters onoptimization is analyzed in this section Taking Ackleyfunction as example Figure 7 shows the change of the ob-jective function value in the case of varying parametersTrynumber and β0 are proportional to the optimization resultTrynumber can impact the time complexity of the algorithmso the value should be appropriate not to aect the runningspeed e best value of c and δ is 25 and 12 respectively

5 Conclusion

LFFSA is proposed to improve the capability of AFSAwhich integrates the merits of both AFSA and FA Firstlythe searching characteristic of AFSA is studied by cal-culating the time complexity Secondly 17 benchmark testfunctions are used to verify LFFSA en time complexityof LFFSA is estimated Numerical results demonstrate thatLFFSA has a better performance in accuracy and speed ofoptimization to solve nonlinear optimization problemsthan the other test algorithms However the solutionobtained by LFFSA can be more precise and the way ofmodication could provide reference for those esectcientalgorithms eg DE and GWO

Table 3 Continued

No Items AFSA Std FA Std FFSA Std LFFSA Std DE Std jDE Std

f15

Worst 2588

00705

1505

0198

minus4281

0117

minus4975

0241

2241

0077

0635

0372Best 2366 minus4679 minus5792 minus72474 1972 minus0370Average 2490 minus4561 minus5408 minus63259 2140 0148Median 2506 minus4601 minus5456 minus63259 2136 0218

f16

Worst 1438

0058

minus1463

0253

minus2762

0026

minus3239

0032

minus2354

0042

minus3392

0433Best 1268 minus2296 minus2838 minus3334 minus2482 minus4646Average 1366 minus1987 minus2793 minus3282 minus2408 minus3868Median 1375 minus2016 minus2793 minus3284 minus2402 minus3751

f17

Worst minus22954

0643

minus25673

1721

minus70748

1908

minus79645

0131

minus47025

1403

minus63031

1942Best minus24885 minus32075 minus77054 minus80098 minus51440 minus68738Average minus23772 minus28641 minus74598 minus79996 minus48982 minus66539Median minus23731 minus28759 minus74907 minus80001 minus49035 minus66559

Table 4 Time complexity analysis of AFSA

Procedure of AFSA Time complexity(1) Initialization of N articial shes O(N)(2) Initialization of lsquoboardrsquo O(N)(3) Swarming behavior O(N2 + 2lowastN)(4) Chasing behavior O(2lowastN2 + 2lowastN)(5) Preying behavior O(TrynumberlowastN)(6) Judging of terminal condition O(1)(7) Information output of lsquoboardrsquo O(1)

Table 5 Time complexity analysis of LFFSA

Procedure of LFFSA Time complexity(1) Initialization of N articial shes O(N)(2) Initialization of lsquoboardrsquo O(N)(3) Chasing behavior O(2lowastN2 + 2lowastN)(4) Preying behavior O(trynumberlowastN)(5) Judging of terminal condition O(1)(6) Information output of lsquoboardrsquo O(1)

Table 6 Average running time of algorithms

NoRunning time (s)

AFSA FA FFSA LFFSA DE jDEf1 1023 747 185 753 464 608f2 1237 773 2111 710 432 653f3 1087 780 2073 896 412 825f4 1140 767 2049 1027 563 698f5 1053 766 1947 803 328 596f6 1972 610 2577 1637 771 1065f7 1053 743 1877 856 327 656f8 1233 803 2272 923 379 663f9 2083 720 2136 771 401 684f10 1627 846 2062 756 465 715f11 926 895 1374 635 363 667f12 965 534 1375 651 304 628f13 912 526 1436 1041 413 715f14 1395 574 1902 1031 648 986f15 945 539 1307 736 462 774f16 873 521 1388 1146 852 1887f17 1453 728 1841 596 332 644

β0

Valu

e

ndash8

ndash75

ndash7

ndash65

ndash6

ndash55

215050 1

(a)

γ

Valu

e

ndash63

ndash62

ndash61

ndash6

ndash59

ndash58

0 2 4 6 8 10

(b)

δ

Valu

e

ndash615ndash61

ndash605ndash6

ndash595ndash59

ndash585ndash58

215050 1

(c)

Trynumber

Valu

e

ndash64

ndash62

ndash6

ndash58

ndash56

ndash54

0 2 4 6 8 10

(d)

Figure 7 Test curves of parameters (a) Test curve of β0 (b) Testcurve of c (c) Test curve of δ (d) Test curve of Trynumber

Computational Intelligence and Neuroscience 11

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work is supported by National Natural ScienceFoundation of China (61463028) e authors wish to thankDr Yanliang Cui for his fruitful comments and suggestions

References

[1] X L Li Z J Shao J X Qian et al ldquoAn optimizing methodbased on autonomous animats fish-swarm algorithmrdquo Sys-tems Engineering-eory and Practice vol 22 pp 188ndash2002002 in Chinese

[2] C R Wang C L Zhou and J W Ma ldquoAn improved artificialfish-swarm algorithm and its application in feed-forwardneural networksrdquo in Proceedings of the Fourth InternationalConference on Machine Learning amp Cybernetics GuangzhouChina August 2005

[3] H C Tsai and Y H Lin ldquoModification of the fish swarmalgorithm with particle swarm optimization formulation andcommunication behaviorrdquo Applied Soft Computing vol 11no 8 pp 5367ndash5374 2011

[4] X Y Luan Z P Li and T Z Liu ldquoA novel attribute reductionalgorithm based on rough set and improved artificial fishswarm algorithmrdquo Neurocomputing vol 174 pp 522ndash5292016

[5] Y Wu X Z Gao Z Kai et al ldquoKnowledge-based artificialfish-swarm algorithmrdquo IFAC Proceedings Volumes vol 44no 1 pp 188ndash200 2011

[6] X T Hu H Q Zhang Z C Li et al ldquoA novel self-adaptationhybrid artificial fish-swarm algorithmrdquo IFAC ProceedingsVolumes vol 46 no 5 pp 583ndash588 2013

[7] K P Kumar B Saravanan and K S Swarup ldquoOptimizationof renewable energy sources in a microgrid using artificial fishswarm algorithmrdquo Energy Procedia vol 90 pp 107ndash113 2016

[8] Q He X T Hu H Ren et al ldquoA novel artificial fish swarmalgorithm for solving large-scale reliability-redundancy ap-plication problemrdquo ISA Transactions vol 59 pp 105ndash1132015

[9] Z Q Zhang K P Wang L X Zhu et al ldquoA Pareto improvedartificial fish swarm algorithm for solving a multi-objectivefuzzy disassembly line balancing problemrdquo Expert Systemswith Applications vol 86 pp 165ndash176 2017

[10] P K Singhal R Naresh and V Sharma ldquoBinary fish swarmalgorithm for profit-based unit commitment problem incompetitive electricity market with ramp rate constraintsrdquoGeneration Transmission and Distribution IET vol 9 no 13pp 1697ndash1707 2015

[11] X S Yang Nature-Inspired Metaheuristic Algorithmspp 83ndash96 Luniver Press London 2008

[12] S M Farahani B Nasiri A A Abshouri et al ldquoAn improvedfirefly algorithm with directed movementrdquo in Proceedings ofIEEE International Conference on Computer Science amp In-formation Technology Sichuan China June 2011

[13] L D S Coelho D L D A Bernert and V C Mariani ldquoAchaotic firefly algorithm applied to reliability-redundancy

optimizationrdquo Evolutionary Computation vol 30 pp 517ndash521 2011

[14] X S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo in Studies inComputational Intelligence vol 284 pp 101-111 SpringerBerlin Germany 2010

[15] S L Tilahun and C O Hong ldquoModified firefly algorithmrdquoJournal of Applied Mathematics vol 12 pp 2428ndash24392012

[16] K Jagatheesan B Anand S Samanta et al ldquoDesign ofa proportional-integral-derivative controller for an automaticgeneration control of multi-area power thermal systems usingfirefly algorithmrdquo IEEECAA Journal of Automatica Sinicapp 1ndash14 2016

[17] H Su Y Cai and Q Du ldquoFirefly-algorithm-inspiredframework with band selection and extreme learning ma-chine for hyperspectral image classificationrdquo IEEE Journal ofSelected Topics in Applied Earth Observations and RemoteSensing vol 10 no 1 pp 309ndash320 2016

[18] S Ranganathan M S Kalavathi and A R C ChristoberldquoSelf-adaptive firefly algorithm based multi-objectives formulti-type FACTS placementrdquo IET Generation Transmissionand Distribution vol 10 no 11 pp 188ndash200 2016

[19] D F Teshome C H Le Y W Lin et al ldquoA modified fireflyalgorithm for photovoltaic maximum power point trackingcontrol under partial shadingrdquo IEEE Journal of Emerging andSelected Topics in Power Electronics vol 5 no 2 pp 661ndash6712017

[20] M Alb P Alotto C Magele et al ldquoFirefly algorithm forfinding optimal shapes of electromagnetic devicesrdquo IEEETransactions on Magnetics vol 52 no 3 pp 1ndash4 2016

[21] A Mishra and V N K Gundavarapu ldquoLine utilisationfactor-based optimal allocation of IPFC and sizing usingfirefly algorithm for congestion managementrdquo GenerationTransmission and Distribution IET vol 10 no 1 pp 115ndash122 2016

[22] M H Horng ldquoVector quantization using the firefly algorithmfor image compressionrdquo Expert Systems with Applicationsvol 39 no 1 pp 078ndash1091 2012

[23] A Kazem E Sharifi F K Hussain et al ldquoSupport vectorregression with chaos-based firefly algorithm for stock marketprice forecastingrdquo Applied Soft Computing vol 13 no 2pp 947ndash958 2013

[24] L F He and S W Huang ldquoModified firefly algorithm basedmultilevel thresholding for color image segmentationrdquoNeurocomputing vol 240 pp 152ndash174 2017

[25] A H Gandomi X S Yang and A H Alavi ldquoMixed variablestructural optimization using Firefly Algorithmrdquo Computersand Structures vol 89 no 23-24 pp 2325ndash2336 2011

[26] M Alweshah and S Abdullah ldquoHybridizing firefly algorithmswith a probabilistic neural network for solving classificationproblemsrdquo Applied Soft Computing vol 35 pp 513ndash5242015

[27] R M Rizk-Allah E M Zaki A A El-Sawy et al ldquoHy-bridizing ant colony optimization with firefly algorithm forunconstrained optimization problemsrdquo Applied Mathematicsand Computation vol 224 pp 473ndash483 2013

[28] X S Yang S S S Hosseini and A H Gandomi ldquoFireflyalgorithm for solving non-convex economic dispatch prob-lems with valve loading effectrdquo Applied Soft Computingvol 12 no 3 pp 1180ndash1186 2012

[29] J Senthilnath S N Omkar and V Mani ldquoClustering usingfirefly algorithm performance studyrdquo Swarm and Evolu-tionary Computation vol 1 no 3 pp 164ndash171 2011

12 Computational Intelligence and Neuroscience

[30] T Kanimozhi and K Latha ldquoAn integrated approach to re-gion based image retrieval using firefly algorithm and supportvector machinerdquo Neurocomputing vol 151 pp 1099ndash11112015

[31] A Baykasoglu and F B Ozsoydan ldquoAdaptive firefly algorithmwith chaos for mechanical design optimization problemsrdquoApplied Soft Computing vol 36 pp 152ndash164 2015

[32] I Fister M Perc S M Kamal et al ldquoA review of chaos-basedfirefly algorithms perspectives and research challengesrdquoApplied Mathematics and Computation vol 252 pp 155ndash1652015

[33] I Fister I Fister X S Yang et al ldquoA comprehensive review offirefly algorithmsrdquo Swarm and Evolutionary Computationvol 13 pp 34ndash46 2013

[34] C T Brown L S Liebovitch and R Glendon ldquoLevy flights inDobe Jursquohoansi foraging patternsrdquo Human Ecology vol 35no 1 pp 129ndash138 2007

[35] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquoMathematics vol 226 no 2 pp 1830ndash1844 2012

[36] I Pavlyukevich ldquoCooling down Levy flightsrdquo Journal ofPhysics A Mathematical and eoretical vol 40 no 41pp 12299ndash12313 2007

[37] R Jensi and G W Jiji ldquoAn enhanced particle swarm opti-mization with Levy flight for global optimizationrdquo AppliedSoft Computing vol 43 pp 248ndash261 2016

[38] D Tang J Yang S Dong et al ldquoA Levy flight-based shuffledfrog-leaping algorithm and its applications for continuousoptimization problemsrdquo Applied Soft Computing vol 49pp 641ndash662 2016

[39] M Yahya and M P Saka ldquoConstruction site layout planningusing multi-objective artificial bee colony algorithmwith Levyflightsrdquo Automation in Construction vol 38 pp 14ndash29 2014

[40] C Blum and A Roli Hybrid Metaheuristics An IntroductionSpringer Berlin Germany 2008

[41] A M Reynolds and M A Frye ldquoFree-flight odor tracking indrosophila is consistent with an optimal intermittent scale-free searchrdquo PLos One vol 2 no 4 p e354 2007

[42] M F Shlesinger G M Zaslavsky and U Frisch Levy Flightsand Related Topics in Physics Springer Berlin HeidelbergGermany 1995

[43] M F Shlesinger ldquoMathematical physics search researchrdquoNature vol 443 no 7109 pp 281-282 2006

[44] S T Hsieh T Y Sun C C Liu et al ldquoEfficient populationutilization strategy for particle swarm optimizerrdquo IEEETransactions on Systems Man amp Cybernetics Part B vol 39no 2 pp 444ndash456 2009

[45] D Bratton and J Kennedy ldquoDefining a standard for particleswarm optimizationrdquo in Proceedings of IEEE Swarm In-telligence Symposium vol 107 pp 120ndash127 HonoluluHawaii April 2007

[46] U Manber Introduction to Algorithms A Creative ApproachAddison-Wesley Longman Publishing Co Inc Boston MAUSA 1989

Computational Intelligence and Neuroscience 13

Computer Games Technology

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Submit your manuscripts atwwwhindawicom

a better objective function value rand is a random numberin [minus1 1] and norm(Xj minusXi) is the distance between twoposition vectors

For artificial swarming fishes the position-updating canbe formulated as

Xnext Xi + rand middotstep times Xc minusXi( 1113857

norm Xc minusXi( 1113857 (2)

For artificial chasing fishes the position-updating can beformulated as

Xnext Xi + rand middotstep times Xmax minusXi( 1113857

norm Xmax minusXi( 1113857 (3)

e flowchart of AFSA is shown in Figure 2

22 Firefly Algorithm Firefly algorithm (FA) [11] is anotherswarm intelligence algorithm It achieves swarming phe-nomenon by using the fluorescent signal between two fireflyindividuals

e attraction between fireflies depends on their lightintensities and attraction degree e light intensity is inproportion to the objective function value of fireflyrsquosposition Attraction degree is in proportion to the lightintensity e brighter the light intensity is the higherattraction degree will be Besides the farther the distanceis the lower the light intensity and attraction degree willbe

In the simplest form the light intensity I(r) varies withthe distance r monotonically and exponentially as

I(r) I0 eminuscr (4)

where I0 is the original light intensity and c is the lightabsorption coefficient As firefly attraction degree isproportional to the light intensity seen by adjacentfireflies the attraction degree beta of a firefly can bedefined as

beta beta0 middot eminuscr2 (5)

where beta0 is the attraction degree at r 0

e distance rij between any two fireflies i and j at xi andxj respectively is the Cartesian distance which is calculated as

rij xi minusxj

1113944

n

d1xid minus xjd1113872 1113873

11139741113972

(6)

where n is the dimensionality of the given problem

23 Levy Flight Levy flight is one kind of random searchingstrategy [35] Flying step satisfies a heavy-tailed Levy dis-tribution which can be represented by a clear power-lawequation as

AFSAbegins

Does Xi meet thecondition of swarming

behavior

N N Does Xi meet thecondition of chasing

behavior

Y Y

Y Ynext1 gt Ynext2N

Ni ge n

Y

Whether the stopcriterion is met

N

Y

End

Obtain the optimum solution

Xi = Xnext= Xnext2

Xi = Xnext= Xnext1

Xnext2Ynext2Xnext1Ynext1

ObtainXnext2Ynext2

Executepreying

behavior

ObtainXnext1Ynext1

i = i + 1

i = 1

Initialize theswarm X1X2hellipXn

Artificialfish Xi

Artificialfish Xi

Figure 2 Flowchart of AFSA

Xa

Xb

XiVisual

Xnext Step

Xv

Figure 1 Vision concept of the artificial fish

Computational Intelligence and Neuroscience 3

L(s) sim |s|1minusβ

(7)

where s is random Levy step For searching problems insidea wide range of unknown space the variance of Levymovement increases faster than the dimensional Brownianmovement

To some extent the foraging behavior of nature ani-mals is a kind of random movement behavior Becausenext movement usually depends on the current positionand the probability of moving to next position the ef-fectiveness of each random movement becomes greatlyimportant Recent studies show that Levy flight is one ofthe best searching strategies in random movement model[35 41ndash43]

3 Fish Swarm Algorithm Based on Levy Flightand Firefly Behavior

AFSA has several disadvantages in solving nonlinear andmultimodal problems Firstly AFSA uses swarming be-havior and chasing behavior to execute parallel search ina simple and fast way However after determining the di-rection each artificial fish moving with random step will beunable to approach the target point effectively Secondlyartificial fish will execute preying behavior when it does notmeet the conditions of swarming behavior and chasingbehavior is kind of searching strategy is inefficient andcan easily miss the optimum point irdly too manymoving patterns can increase the algorithm complexitywhich may cause slow convergence speed

e above-listed drawbacks of AFSA are improved in theproposed LFFSA FA has the unique moving strategy usingattraction between fireflies which can be used to fix therandom moving after determining the direction in AFSAAnd the preying behavior can be improved using Levy flightto specify the behavior of artificial fishes In LFFSA at-traction degree is involved in the definition of artificialfishes which allows each individual move according to at-traction degree Levy flight is also considered in the defi-nition of the artificial preying fish to avoid falling into thelocal optimum chasing behavior is excluded to decrease thealgorithm complexity

e flowchart of LFFSA is shown in Figure 3 Pseudocode of LFFSA is described in Algorithm 1 To show thedifference between AFSA and LFFSA vividly mechanisms ofboth algorithms are provided in Figure 4

e main improvements of LFFSA are summarized asfollows

(a) Improvement 1 FA-based moving strategy Attrac-tion degree is involved in the definition of artificialfishes which can be formulated as

betaij beta0 middot eminuscrij (8)

where rij is the Cartesian distance between artificialfishes i and j given by Equation (6) c is the lightintensity coefficient which can be set as a constantand beta0 the largest attraction degree is attractiondegree of an artificial fish at rij 0

e position-updating with preying behavior establishedby attraction degree can be formulated as

Xnext Xi + betaij Xj minusXi1113872 1113873 + α(randminus 05) (9)

where Xi is the current position of artificial fish ibetaij(Xj minusXi) is the attraction degree α is the step factorwhich is a constant between 0 and 1 rand is a numberchosen randomly in [minus1 1] and α(randminus 05) is to avoidfalling into the local optimum

(b) Improvement 2 inertia weight A linear inertiaweight is added into Equation (9) as

Xnext ωtXi + betaij Xj minusXi1113872 1113873 + α(randminus 05) (10)

ωt ωmax minus ωmax minusωmin( 1113857 middot genMaxgen (11)

where ωt is the weight size inherited from the lastposition of an artificial fish ωmax denotes the biggestweight ωmin is the minimum weight gen representsthe current iteration and Maxgen is the ultimateiteration

LFFSAbegins

Initialize theswarm X1X2hellipXn

i = 1

Does Xi meet thecondition of chasing

behavior

N

Y

Prey withLeacutevy fight

Xi = Xnext

i = i + 1 N i ge n

Y

Whether the stopcriterion is met

N

YObtain the

optimum solution

End

Moveaccording to

attractiondegree

Artificial fish Xi

Figure 3 Flowchart of LFFSA

4 Computational Intelligence and Neuroscience

Similarly the position-updating with chasing behaviorcan be formulated as

Xnext ωtXi + betaij Xmax minusXi( 1113857 + α(randminus 05) (12)

where Xmax is the position with the highest food concen-tration in the view of artificial fish Xi and other parametersare defined in Equations (10) and (11)

(c) Improvement 3 Levy flight-based search strategyLevy flight is involved in the definition of artificialfishes are executing preying behavior e move-ment can be formulated as

Xnext Xi + αoplus L(λ) (13)

L(λ) ϕ times μ|v|1β

Xi minusXbest( 1113857 (14)

where Xi is the current position of artificial fish i oplus isthe point to point multiplication L(λ) denotes arandom vector generated by Levy flight Xbest rep-resents the best fish on the ldquoboardrdquo μ tminusλ 1lt λlt 3and μ and v have the standard normal distributionμ sim N(0 ϕ2) v sim N(0 1) respectively where

ϕ Γ(1 + β)sin(πβ2)

Γ[(1 + β)2]β middot 2(βminus 1)21113896 1113897

1β

(15)

where Γ is the standard Gamma function(d) Improvement 4 nonlinear visual and step Visual and

step change nonlinearly and dynamically in LFFSAe updating equations are as follows

Visual ρ middot Visual + Visualmin (16)

Visual Preying unaccomplished

Chasing

Behavior unaccomplished

Chasingaccomplished

Preyingaccomplished

Movingrandomly

AFSA

Artificial fishes

Swarming Swarmingaccomplished

LFFSA Chasing accomplished

Artificial fishes with attraction degree

Chasingunaccomplished

Preying withLeacutevy flight

Chasing

Figure 4 Mechanisms of AFSA and LFFSA

(1) Initialization of X1 X2 Xn1113864 1113865(2) Evaluate fitness(3) while the stop criterion is not met do(4) while ilt n do(5) if Xi meets the condition of chasing behavior(6) Move with attraction degree and get a new Xi based on Equation (13)(7) else(8) Prey with Levy flight and get a new Xi based on Equations (11) and (14)(9) end if(10) end while(11) Update the ldquoboardrdquo(12) end while(13) Obtain the optimum solution

ALGORITHM 1 LFFSA

Computational Intelligence and Neuroscience 5

Step ρ middot Step + Stepmin (17)

ρ exp minus30 timesgen

Maxgen( )

s

( ) (18)

where Visualmin the minimum visual takes 0001Stepmin the minimum step takes 00002 ρ isa nonlinear weight and s represents an integer sgt 1Here s 3 Other parameters are the same as thosein Equation (11) e relationship between s and ρ isas shown in Figure 5

Besides dene the biggest distance between two articialshes as

maxD xmax minusxmin( )2 timesD

radic (19)

where xmax and xmin represent the upper bound and thelower bound of searching range respectively D denotes theD-dimension searching space e initial Visual is equal tomaxD and initial Step is equal to maxD8 en Visual andStep change dynamically according to Equations (16)ndash(18)

(e) Improvement 5 decrease of time complexitySwarming behavior in AFSA is excluded to decreasethe algorithm complexity

4 Numerical Simulation

41 Comparison of Convergence Accuracy LFFSA is vali-dated by numerical simulations Fish swarm algorithm withrey behavior (FFSA) dierential evolution (DE) algo-rithm self-adaptive dierential evolution (jDE) algorithmand the two basic algorithms (AFSA and FA) are comparedAll the algorithms are coded in Matlab 2014b e operatingsystem is windows 7 Simulation hardware is a PC with250GHz Inter Core i5 and 200GB Memory

Parameters shown in Table 1 are determined by trial anderror

e following benchmarks are chosen carefullyaccording to their features Functions Sphere Quartic andRosenbrock etc are simple unimodal problems FunctionsAckley Rastrigin and Schwefel etc are highly complexmultimodal problems with many local minima Schwefelfunction has a maximum value and other functions haveminimum values ese benchmarks are listed in Table 2[44 45] e solutions of 17 test functions obtained bydierent algorithms are compared To compare the con-vergence speed and accuracy of the algorithms clearly andcorrectly all functions are run 50 times for each algorithmrespectivelye results are averaged and plotted in Figure 6

From Figure 6 LFFSA can avoid local optimum and havebetter convergence accuracy compared with the other al-gorithms For AFSA and FA the solutions of most functionsare unsatisfactory the DE cannot nd ideal solutions of f3f4 f5 f6 f10 f11 f12 f14 and f15 the jDE has good accuracywhile solving some of those functions eg f1 f2 f3 and f8however solutions of f4 f10 f11 and f15 obtained by jDEare not so precise the LFFSA can obtain the ideal accuracy

for almost all functions although it cannot achieve a highprecision level like solutions of f2 obtained by jDE the FFSAis slightly worse than LFFSAe LFFSA outperforms jDE in10 benchmark functions while 2 functions are comparativeand 5 functions are worse

LFFSA is better than AFSA because Levy ight is able torestrict the movement step of AFSA to a very small areaaround the current position Furthermore the attractiondegree guides the sh moving Besides LFFSA can quicklylead the sh individual to the close-by optimal pointConsidering all the advantages discussed above the opti-mum solution can be found successfully by using LFFSAwhich outperforms the basic algorithms for all test functionsand outperforms jDE for several functions To observe thesearching capabilities of dierent algorithms directly theaverage median best and worst values obtained by dierentalgorithms are listed in Table 3 Results indicate that LFFSAcan nd ideal solutions and have a better robustness

42 Computational ComplexityAnalysis Time complexity isalso an important indicator in the analysis of algorithms Ifan algorithm is composed of several parts then its com-plexity is the sum of the complexities of these parts ealgorithm may consist of a loop executed many times andeach time is with a dierent complexity Time complexity of

Iteration0 500 1000 1500 2000

ρ

0

02

04

06

08

1

s = 3s = 12

s = 21s = 30

Figure 5 Value of ρ

Table 1 Parameter settings

Algorithms Parameters ValuesFA FFSA LFFSA β0 10FA FFSA LFFSA c 10AFSA FFSA LFFSA δ 0618AFSA FFSA LFFSA Trynumber 5DE jDE Scaling constant 05DE jDE Crossover constant 09All 6 algorithms Population 50

All 6 algorithms Maximum functionevaluations (FEs) 2 times 105

6 Computational Intelligence and Neuroscience

the algorithm is used to estimate the efficiency of the al-gorithm It is defined that the time complexity of the al-gorithm or the running time is O(f(n)) [46] Define N asthe population

In the definition of time complexity O(N2) and O(N)

are at different levels If the time complexity of one algorithmis O(N2) the time complexity of the other one is O(N) thenthe former algorithm is more complex In the other case ifthe time complexity of one algorithm is O(N2) while thetime complexity of the other one is O(N2 + N) theircomplexities are both O(N2)

e time complexity analysis of AFSA is provided inTable 4

From Table 4 the time complexity of AFSA is

O Maxgenlowast 3lowastN2

+ trynumberlowastN + 6lowastN1113872 11138731113872 1113873 (20)

Swarming behavior has N times of calculating conges-tion factor 1 time of judging and 1 time of movingerefore time complexity of swarming behavior isO(N2 + 2lowastN) Chasing behavior has N times of calculating

congestion factor N times of searching 1 time of judgingand 1 time of moving erefore time complexity of chasingbehavior is O(2lowastN2 + 2lowastN)

Time complexity analysis of LFFSA is listed in Table 5Due to the lack of swarming behavior time complexity

of LFFSA can be calculated as

O Maxgenlowast 2lowastN2

+ TrynumberlowastN + 4lowastN1113872 11138731113872 1113873 (21)

We can also obtain time complexity of FA

O Maxgenlowast N2

+ N1113872 11138731113872 1113873 (22)

A conclusion can be obtained that time complexities ofthe three algorithms are at the same level eir computa-tional complexities in the worst case are only the square ofthe training sample size

43 Experimental Complexity Analysis Time complexity isa rough estimate of time costemore accurate time cost ofan algorithm can only be validated by running it on

Table 2 Test functions

No Test functions Expression Optimum value Domain Df1 Sphere f(x) 1113936

Di1x

2i 0 (minus100 100)D 30

f2 Quartic f(x) 1113936Di1ix

4i 0 (minus128 128)D 30

f3 Ackley f(x) minus20 exp minus02

(1D)1113936Di1x

2i

1113969

minus exp[(1D)1113936Di1cos(2πxi)]1113882 1113883

+ 20 + e0 (minus32768 32768)D 30

f4 Rosenbrock f(x) 1113936Dminus1i1 100(xi+1 minusxi

2)2 + (1minusxi)2 0 (minus2048 2048)D 30

f5 Rastrigin1 f(x) 1113936Di1 xi

2 minus 10 cos(2πxi) + 101113864 1113865 0 (minus512 512)D 30

f6 Rastrigin2f(x) 1113936

Di1 yi

2 minus 10 cos(2πyi) + 101113864 11138650 (minus512 512)D 30

yi xi |xi|lt (12)

round(2xi)2 |xi|gt (12)1113896

f7 Schwefel f(x) 1113936Di1 xi middot sin

|xi|

11139681113864 1113865 4189829D (minus500 500)D 30

f8 Griewank f(x) (14000)1113936Di1x

2i minus1113937

Di1cos(xi

i

radic) + 1 0 (minus600 600)D 30

f9 Quadric f(x) 1113936Di1(1113936

ij1xj)

2 0 (minus100 100)D 30

f10 Schaffer1 f(x) 1113936Dminus1i1 ((sin2

xi+1

2 + xi2

1113968minus 05)1113864

(0001(xi+12 + xi

2) + 1)2) + 050 (minus100 100)D 30

f11 Schaffer2 f(x) 1113936Dminus1i1 (sin2

1113936Di1x

2i

1113969

minus 05)(0001(1113936Di1x

2i ) + 1)2 + 051113882 1113883 0 (minus100 100)D 30

f12 Maxmod f(x) max(|xi|) 0 (minus10 10)D 30

f13 Dixon and price (x1 minus 1)2 + 1113936Di1i(2xi

2 minusximinus1) 0 (minus10 10)D 30

f14 Powell f(x) 1113936D4i1 [(x4iminus 3 + 10x4iminus 2)

2 + 5(x4iminus 1 minusx4i)2 +

(x4iminus 2 minus 2x4iminus 1)2 + 10(x4iminus 3 minusx4i)

4]0 (minus4 5)D 28

f15 Zakharov f(x) 1113936Di1x

2i + (1113936

Di105ixi)

2 + (1113936Di105ixi)

4 0 (minus5 10)D 30

f16 Sin1 1113936Di1|xi sin(xi) + 01xi| 0 (minus10 10)D 30

f17 Sin2 f(x) minus1113936Di1sin(xi) sin20(ix2

i π) minus992784 (0 π)D 100

Computational Intelligence and Neuroscience 7

FEs104

log 1

0(f(x

))

ndash20

ndash15

ndash10

ndash5

0

5

10

FAAFSAFFSA

LFFSADEjDE

0 5 10 15 20

(a)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash40

ndash30

ndash20

ndash10

0

10

FEs1040 5 10 15 20

(b)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

FEs1040 5 10 15 20

(c)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash2

0

2

4

6

8

FEs1040 5 10 15 20

(d)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

2

3

FEs1040 5 10 15 20

(e)

log 1

0(f(x

))FAAFSAFFSA

LFFSADEjDE

ndash6

ndash4

ndash2

0

2

4

FEs1040 5 10 15 20

(f )

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

32

34

36

38

4

42

FEs1040 5 10 15 20

(g)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash10

ndash5

0

5

FEs1040 5 10 15 20

(h)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash15

ndash10

ndash5

0

5

FEs1040 5 10 15 20

(i)

Figure 6 Continued

8 Computational Intelligence and Neuroscience

computer Since dierent algorithms cannot reach the sameconvergence accuracy the test with xed convergence ac-curacy is not available erefore the test with max functionevaluations is conducted Running time of each algorithm iscounted by the explorer of MATLAB Parameter settings ofalgorithms are the same in Section 41 Average running timeof dierent algorithms is listed in Table 6 When function

evaluations are the same running speed of LFFSA is fasterthan that of AFSA while DE has the fastest running speedResults are quite in accord with those obtained by com-putational complexity analysis LFFSA and jDE are com-parative in experimental complexity Running time of FFSAis almost twice as much as that of LFFSA e improvementof LFFSA decreases time complexity to some extent

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash8

ndash6

ndash4

ndash2

0

2

FEs1040 5 10 15 20

(j)lo

g 10(

f(x))

FAAFSAFFSA

LFFSADEjDE

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

FEs1040 5 10 15 20

(k)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

FEs1040 5 10 15 20

(l)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash2

0

2

4

6

8

10

FEs1040 5 10 15 20

(m)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash5

0

5

FEs1040 5 10 15 20

(n)

log 1

0(f(x

))FAAFSAFFSA

LFFSADEjDE

ndash10

ndash5

0

5

10

FEs1040 5 10 15 20

(o)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

2

FEs1040 5 10 15 20

(p)

f(x)

FAAFSAFFSA

LFFSADEjDE

FEs1040 5 10 15 20

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

(q)

Figure 6 Iterative curves of test functions (a) f1 (b) f2 (c) f3 (d) f4 (e) f5 (f ) f6 (g) f7 (h) f8 (i) f9 (j) f10 (k) f11 (l) f12 (m) f13 (n) f14(o) f15 (p) f16 (q) f7

Computational Intelligence and Neuroscience 9

Table 3 Comparison of optimization results

No Items AFSA Std FA Std FFSA Std LFFSA Std DE Std jDE Std

f1

Worst 3014

0181

minus2798

0120

minus5132

0071

minus6743

0071

minus8544

0132

minus15021

0121Best 2402 minus3257 minus5233 minus8500 minus10371 minus17242Average 2656 minus3048 minus5145 minus7278 minus8924 minus16326Median 2651 minus3102 minus5193 minus6325 minus8586 minus16706

f2

Worst minus2322

0171

minus19245

0135

minus11178

0074

minus14207

0134

minus20506

0400

minus31644

0535Best minus4126 minus2813 minus11313 minus15585 minus21611 minus33250Average minus2812 minus2278 minus11273 minus14812 minus21054 minus32687Median minus2562 minus2197 minus11271 minus14834 minus20970 minus32844

f3

Worst 1386

0001

minus1776

0124

minus3908

0197

minus6121

0138

minus0027

0438

minus8934

0167Best 1307 minus2415 minus4647 minus6938 minus1281 minus9471Average 1587 minus2168 minus4147 minus6546 minus0049 minus9163Median 1586 minus2177 minus4225 minus6325 minus0035 minus9164

f4

Worst 2143

0087

22733

0097

minus20348

0125

minus1076

0376

1520

0003

1454

0025Best 1565 1043 minus2416 minus3405 1412 1363Average 1946 1476 minus2158 minus1946 1385 1312Median 1854 1385 minus2235 minus2325 1363 1287

f5

Worst 2310

0077

1864

0856

minus2846

0044

minus4385

0054

minus0579

0323

0898

6009Best 2096 1243 minus2982 minus5145 minus1591 minus12831Average 2236 1454 minus2946 minus4643 minus0999 minus3303Median 2136 1285 minus2435 minus4325 minus0963 0148

f6

Worst 2445

0038

1716

0133

minus2382

0076

minus3414

0048

1131

0042

0698

5049Best 2318 1255 minus2618 minus3606 1012 minus11404Average 2408 1571 minus2486 minus3489 1077 minus5466Median 2419 1601 minus2462 minus3487 1094 minus7632

f7

Worst 3815

0018

37846

0031

4186

736eminus 6

4156

434eminus 6

4099

296eminus 9

4087

403eminus 3Best 3945 3978 4099 4099 4099 4099Average 3813 3848 4099 4099 4099 4097Median 3736 3785 minus4099 minus4099 4099 4097

f8

Worst 0956

0133

minus0960

0223

minus5301

0051

minus6271

0055

minus989

0114

minusInf

0Best minus0644 minus0500 minus5444 minus6455 minus10255 minusInfAverage minus0735 minus0697 minus5357 minus6372 minus10071 minusInfMedian 0736 0685 minus5435 minus6325 minus10074 minusInf

f9

Worst minus11665

1057

minus976

1324

minus9347

0843

minus8695

1323

minus6848

1124

minus6131

2697Best minus14433 minus10574 minus12194 minus11937 minus10680 minus14831Average minus12786 minus10456 minus10764 minus10137 minus8178 minus9938Median 1136 1085 minus7435 minus6325 minus7745 minus9663

f10

Worst 0957

0131

0974

0223

minus5375

0049

minus6274

0056

0673

0032

0280

0110Best 0644 0497 minus5448 minus6486 0483 minus0022Average 0747 0649 minus5376 minus6348 0526 0230Median 1136 1085 minus7435 minus6325 0547 0211

f11

Worst minus0301

272eminus 05

minus0303

0009

minus4751

0071

minus5647

0085

minus0896

0103

minus1106

0135Best minus0301 minus0331 minus5011 minus5965 minus1107 minus1429Average minus0301 minus0313 minus4838 minus5804 minus0975 minus1364Median minus0301 minus0311 minus4835 minus5804 minus0896 minus1402

f12

Worst 0779

0037

minus1177

0163

minus2756

0035

minus3158

0037

minus0136

0052

minus1904

0221Best 0658 minus1638 minus2892 minus3287 minus0339 minus2526Average 0711 minus1388 minus2811 minus3221 minus0252 minus2291Median 0716 minus1376 minus2801 minus3216 0248 minus2374

f13

Worst 4286

0173

0574

0297

minus0602

0007

minus0602

0005

0039

0068

minus0176

538eminus 6Best 3706 minus0175 minus0605 minus0605 minus0162 minus0176Average 4147 minus0013 minus0603 minus0603 minus0088 minus0176Median 4213 minus0135 minus0603 minus6603 minus0086 minus0176

f14

Worst 2404

0084

0405

0318

minus3598

0117

minus4514

0101

0276

0211

minus2192

0401Best 2173 minus0576 minus4001 minus4869 minus0369 minus3409Average 2275 0069 minus3773 minus4656 minus0117 minus2867Median 2246 0133 minus3719 minus4651 minus0134 minus2924

10 Computational Intelligence and Neuroscience

44ParameterAnalysis ofLFFSA eeect of parameters onoptimization is analyzed in this section Taking Ackleyfunction as example Figure 7 shows the change of the ob-jective function value in the case of varying parametersTrynumber and β0 are proportional to the optimization resultTrynumber can impact the time complexity of the algorithmso the value should be appropriate not to aect the runningspeed e best value of c and δ is 25 and 12 respectively

5 Conclusion

LFFSA is proposed to improve the capability of AFSAwhich integrates the merits of both AFSA and FA Firstlythe searching characteristic of AFSA is studied by cal-culating the time complexity Secondly 17 benchmark testfunctions are used to verify LFFSA en time complexityof LFFSA is estimated Numerical results demonstrate thatLFFSA has a better performance in accuracy and speed ofoptimization to solve nonlinear optimization problemsthan the other test algorithms However the solutionobtained by LFFSA can be more precise and the way ofmodication could provide reference for those esectcientalgorithms eg DE and GWO

Table 3 Continued

No Items AFSA Std FA Std FFSA Std LFFSA Std DE Std jDE Std

f15

Worst 2588

00705

1505

0198

minus4281

0117

minus4975

0241

2241

0077

0635

0372Best 2366 minus4679 minus5792 minus72474 1972 minus0370Average 2490 minus4561 minus5408 minus63259 2140 0148Median 2506 minus4601 minus5456 minus63259 2136 0218

f16

Worst 1438

0058

minus1463

0253

minus2762

0026

minus3239

0032

minus2354

0042

minus3392

0433Best 1268 minus2296 minus2838 minus3334 minus2482 minus4646Average 1366 minus1987 minus2793 minus3282 minus2408 minus3868Median 1375 minus2016 minus2793 minus3284 minus2402 minus3751

f17

Worst minus22954

0643

minus25673

1721

minus70748

1908

minus79645

0131

minus47025

1403

minus63031

1942Best minus24885 minus32075 minus77054 minus80098 minus51440 minus68738Average minus23772 minus28641 minus74598 minus79996 minus48982 minus66539Median minus23731 minus28759 minus74907 minus80001 minus49035 minus66559

Table 4 Time complexity analysis of AFSA

Procedure of AFSA Time complexity(1) Initialization of N articial shes O(N)(2) Initialization of lsquoboardrsquo O(N)(3) Swarming behavior O(N2 + 2lowastN)(4) Chasing behavior O(2lowastN2 + 2lowastN)(5) Preying behavior O(TrynumberlowastN)(6) Judging of terminal condition O(1)(7) Information output of lsquoboardrsquo O(1)

Table 5 Time complexity analysis of LFFSA

Procedure of LFFSA Time complexity(1) Initialization of N articial shes O(N)(2) Initialization of lsquoboardrsquo O(N)(3) Chasing behavior O(2lowastN2 + 2lowastN)(4) Preying behavior O(trynumberlowastN)(5) Judging of terminal condition O(1)(6) Information output of lsquoboardrsquo O(1)

Table 6 Average running time of algorithms

NoRunning time (s)

AFSA FA FFSA LFFSA DE jDEf1 1023 747 185 753 464 608f2 1237 773 2111 710 432 653f3 1087 780 2073 896 412 825f4 1140 767 2049 1027 563 698f5 1053 766 1947 803 328 596f6 1972 610 2577 1637 771 1065f7 1053 743 1877 856 327 656f8 1233 803 2272 923 379 663f9 2083 720 2136 771 401 684f10 1627 846 2062 756 465 715f11 926 895 1374 635 363 667f12 965 534 1375 651 304 628f13 912 526 1436 1041 413 715f14 1395 574 1902 1031 648 986f15 945 539 1307 736 462 774f16 873 521 1388 1146 852 1887f17 1453 728 1841 596 332 644

β0

Valu

e

ndash8

ndash75

ndash7

ndash65

ndash6

ndash55

215050 1

(a)

γ

Valu

e

ndash63

ndash62

ndash61

ndash6

ndash59

ndash58

0 2 4 6 8 10

(b)

δ

Valu

e

ndash615ndash61

ndash605ndash6

ndash595ndash59

ndash585ndash58

215050 1

(c)

Trynumber

Valu

e

ndash64

ndash62

ndash6

ndash58

ndash56

ndash54

0 2 4 6 8 10

(d)

Figure 7 Test curves of parameters (a) Test curve of β0 (b) Testcurve of c (c) Test curve of δ (d) Test curve of Trynumber

Computational Intelligence and Neuroscience 11

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work is supported by National Natural ScienceFoundation of China (61463028) e authors wish to thankDr Yanliang Cui for his fruitful comments and suggestions

References

[1] X L Li Z J Shao J X Qian et al ldquoAn optimizing methodbased on autonomous animats fish-swarm algorithmrdquo Sys-tems Engineering-eory and Practice vol 22 pp 188ndash2002002 in Chinese

[2] C R Wang C L Zhou and J W Ma ldquoAn improved artificialfish-swarm algorithm and its application in feed-forwardneural networksrdquo in Proceedings of the Fourth InternationalConference on Machine Learning amp Cybernetics GuangzhouChina August 2005

[3] H C Tsai and Y H Lin ldquoModification of the fish swarmalgorithm with particle swarm optimization formulation andcommunication behaviorrdquo Applied Soft Computing vol 11no 8 pp 5367ndash5374 2011

[4] X Y Luan Z P Li and T Z Liu ldquoA novel attribute reductionalgorithm based on rough set and improved artificial fishswarm algorithmrdquo Neurocomputing vol 174 pp 522ndash5292016

[5] Y Wu X Z Gao Z Kai et al ldquoKnowledge-based artificialfish-swarm algorithmrdquo IFAC Proceedings Volumes vol 44no 1 pp 188ndash200 2011

[6] X T Hu H Q Zhang Z C Li et al ldquoA novel self-adaptationhybrid artificial fish-swarm algorithmrdquo IFAC ProceedingsVolumes vol 46 no 5 pp 583ndash588 2013

[7] K P Kumar B Saravanan and K S Swarup ldquoOptimizationof renewable energy sources in a microgrid using artificial fishswarm algorithmrdquo Energy Procedia vol 90 pp 107ndash113 2016

[8] Q He X T Hu H Ren et al ldquoA novel artificial fish swarmalgorithm for solving large-scale reliability-redundancy ap-plication problemrdquo ISA Transactions vol 59 pp 105ndash1132015

[9] Z Q Zhang K P Wang L X Zhu et al ldquoA Pareto improvedartificial fish swarm algorithm for solving a multi-objectivefuzzy disassembly line balancing problemrdquo Expert Systemswith Applications vol 86 pp 165ndash176 2017

[10] P K Singhal R Naresh and V Sharma ldquoBinary fish swarmalgorithm for profit-based unit commitment problem incompetitive electricity market with ramp rate constraintsrdquoGeneration Transmission and Distribution IET vol 9 no 13pp 1697ndash1707 2015

[11] X S Yang Nature-Inspired Metaheuristic Algorithmspp 83ndash96 Luniver Press London 2008

[12] S M Farahani B Nasiri A A Abshouri et al ldquoAn improvedfirefly algorithm with directed movementrdquo in Proceedings ofIEEE International Conference on Computer Science amp In-formation Technology Sichuan China June 2011

[13] L D S Coelho D L D A Bernert and V C Mariani ldquoAchaotic firefly algorithm applied to reliability-redundancy

optimizationrdquo Evolutionary Computation vol 30 pp 517ndash521 2011

[14] X S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo in Studies inComputational Intelligence vol 284 pp 101-111 SpringerBerlin Germany 2010

[15] S L Tilahun and C O Hong ldquoModified firefly algorithmrdquoJournal of Applied Mathematics vol 12 pp 2428ndash24392012

[16] K Jagatheesan B Anand S Samanta et al ldquoDesign ofa proportional-integral-derivative controller for an automaticgeneration control of multi-area power thermal systems usingfirefly algorithmrdquo IEEECAA Journal of Automatica Sinicapp 1ndash14 2016

[17] H Su Y Cai and Q Du ldquoFirefly-algorithm-inspiredframework with band selection and extreme learning ma-chine for hyperspectral image classificationrdquo IEEE Journal ofSelected Topics in Applied Earth Observations and RemoteSensing vol 10 no 1 pp 309ndash320 2016

[18] S Ranganathan M S Kalavathi and A R C ChristoberldquoSelf-adaptive firefly algorithm based multi-objectives formulti-type FACTS placementrdquo IET Generation Transmissionand Distribution vol 10 no 11 pp 188ndash200 2016

[19] D F Teshome C H Le Y W Lin et al ldquoA modified fireflyalgorithm for photovoltaic maximum power point trackingcontrol under partial shadingrdquo IEEE Journal of Emerging andSelected Topics in Power Electronics vol 5 no 2 pp 661ndash6712017

[20] M Alb P Alotto C Magele et al ldquoFirefly algorithm forfinding optimal shapes of electromagnetic devicesrdquo IEEETransactions on Magnetics vol 52 no 3 pp 1ndash4 2016

[21] A Mishra and V N K Gundavarapu ldquoLine utilisationfactor-based optimal allocation of IPFC and sizing usingfirefly algorithm for congestion managementrdquo GenerationTransmission and Distribution IET vol 10 no 1 pp 115ndash122 2016

[22] M H Horng ldquoVector quantization using the firefly algorithmfor image compressionrdquo Expert Systems with Applicationsvol 39 no 1 pp 078ndash1091 2012

[23] A Kazem E Sharifi F K Hussain et al ldquoSupport vectorregression with chaos-based firefly algorithm for stock marketprice forecastingrdquo Applied Soft Computing vol 13 no 2pp 947ndash958 2013

[24] L F He and S W Huang ldquoModified firefly algorithm basedmultilevel thresholding for color image segmentationrdquoNeurocomputing vol 240 pp 152ndash174 2017

[25] A H Gandomi X S Yang and A H Alavi ldquoMixed variablestructural optimization using Firefly Algorithmrdquo Computersand Structures vol 89 no 23-24 pp 2325ndash2336 2011

[26] M Alweshah and S Abdullah ldquoHybridizing firefly algorithmswith a probabilistic neural network for solving classificationproblemsrdquo Applied Soft Computing vol 35 pp 513ndash5242015

[27] R M Rizk-Allah E M Zaki A A El-Sawy et al ldquoHy-bridizing ant colony optimization with firefly algorithm forunconstrained optimization problemsrdquo Applied Mathematicsand Computation vol 224 pp 473ndash483 2013

[28] X S Yang S S S Hosseini and A H Gandomi ldquoFireflyalgorithm for solving non-convex economic dispatch prob-lems with valve loading effectrdquo Applied Soft Computingvol 12 no 3 pp 1180ndash1186 2012

[29] J Senthilnath S N Omkar and V Mani ldquoClustering usingfirefly algorithm performance studyrdquo Swarm and Evolu-tionary Computation vol 1 no 3 pp 164ndash171 2011

12 Computational Intelligence and Neuroscience

[30] T Kanimozhi and K Latha ldquoAn integrated approach to re-gion based image retrieval using firefly algorithm and supportvector machinerdquo Neurocomputing vol 151 pp 1099ndash11112015

[31] A Baykasoglu and F B Ozsoydan ldquoAdaptive firefly algorithmwith chaos for mechanical design optimization problemsrdquoApplied Soft Computing vol 36 pp 152ndash164 2015

[32] I Fister M Perc S M Kamal et al ldquoA review of chaos-basedfirefly algorithms perspectives and research challengesrdquoApplied Mathematics and Computation vol 252 pp 155ndash1652015

[33] I Fister I Fister X S Yang et al ldquoA comprehensive review offirefly algorithmsrdquo Swarm and Evolutionary Computationvol 13 pp 34ndash46 2013

[34] C T Brown L S Liebovitch and R Glendon ldquoLevy flights inDobe Jursquohoansi foraging patternsrdquo Human Ecology vol 35no 1 pp 129ndash138 2007

[35] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquoMathematics vol 226 no 2 pp 1830ndash1844 2012

[36] I Pavlyukevich ldquoCooling down Levy flightsrdquo Journal ofPhysics A Mathematical and eoretical vol 40 no 41pp 12299ndash12313 2007

[37] R Jensi and G W Jiji ldquoAn enhanced particle swarm opti-mization with Levy flight for global optimizationrdquo AppliedSoft Computing vol 43 pp 248ndash261 2016

[38] D Tang J Yang S Dong et al ldquoA Levy flight-based shuffledfrog-leaping algorithm and its applications for continuousoptimization problemsrdquo Applied Soft Computing vol 49pp 641ndash662 2016

[39] M Yahya and M P Saka ldquoConstruction site layout planningusing multi-objective artificial bee colony algorithmwith Levyflightsrdquo Automation in Construction vol 38 pp 14ndash29 2014

[40] C Blum and A Roli Hybrid Metaheuristics An IntroductionSpringer Berlin Germany 2008

[41] A M Reynolds and M A Frye ldquoFree-flight odor tracking indrosophila is consistent with an optimal intermittent scale-free searchrdquo PLos One vol 2 no 4 p e354 2007

[42] M F Shlesinger G M Zaslavsky and U Frisch Levy Flightsand Related Topics in Physics Springer Berlin HeidelbergGermany 1995

[43] M F Shlesinger ldquoMathematical physics search researchrdquoNature vol 443 no 7109 pp 281-282 2006

[44] S T Hsieh T Y Sun C C Liu et al ldquoEfficient populationutilization strategy for particle swarm optimizerrdquo IEEETransactions on Systems Man amp Cybernetics Part B vol 39no 2 pp 444ndash456 2009

[45] D Bratton and J Kennedy ldquoDefining a standard for particleswarm optimizationrdquo in Proceedings of IEEE Swarm In-telligence Symposium vol 107 pp 120ndash127 HonoluluHawaii April 2007

[46] U Manber Introduction to Algorithms A Creative ApproachAddison-Wesley Longman Publishing Co Inc Boston MAUSA 1989

Computational Intelligence and Neuroscience 13

Computer Games Technology

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Submit your manuscripts atwwwhindawicom

L(s) sim |s|1minusβ

(7)

where s is random Levy step For searching problems insidea wide range of unknown space the variance of Levymovement increases faster than the dimensional Brownianmovement

To some extent the foraging behavior of nature ani-mals is a kind of random movement behavior Becausenext movement usually depends on the current positionand the probability of moving to next position the ef-fectiveness of each random movement becomes greatlyimportant Recent studies show that Levy flight is one ofthe best searching strategies in random movement model[35 41ndash43]

3 Fish Swarm Algorithm Based on Levy Flightand Firefly Behavior

AFSA has several disadvantages in solving nonlinear andmultimodal problems Firstly AFSA uses swarming be-havior and chasing behavior to execute parallel search ina simple and fast way However after determining the di-rection each artificial fish moving with random step will beunable to approach the target point effectively Secondlyartificial fish will execute preying behavior when it does notmeet the conditions of swarming behavior and chasingbehavior is kind of searching strategy is inefficient andcan easily miss the optimum point irdly too manymoving patterns can increase the algorithm complexitywhich may cause slow convergence speed

e above-listed drawbacks of AFSA are improved in theproposed LFFSA FA has the unique moving strategy usingattraction between fireflies which can be used to fix therandom moving after determining the direction in AFSAAnd the preying behavior can be improved using Levy flightto specify the behavior of artificial fishes In LFFSA at-traction degree is involved in the definition of artificialfishes which allows each individual move according to at-traction degree Levy flight is also considered in the defi-nition of the artificial preying fish to avoid falling into thelocal optimum chasing behavior is excluded to decrease thealgorithm complexity

e flowchart of LFFSA is shown in Figure 3 Pseudocode of LFFSA is described in Algorithm 1 To show thedifference between AFSA and LFFSA vividly mechanisms ofboth algorithms are provided in Figure 4

e main improvements of LFFSA are summarized asfollows

(a) Improvement 1 FA-based moving strategy Attrac-tion degree is involved in the definition of artificialfishes which can be formulated as

betaij beta0 middot eminuscrij (8)

where rij is the Cartesian distance between artificialfishes i and j given by Equation (6) c is the lightintensity coefficient which can be set as a constantand beta0 the largest attraction degree is attractiondegree of an artificial fish at rij 0

e position-updating with preying behavior establishedby attraction degree can be formulated as

Xnext Xi + betaij Xj minusXi1113872 1113873 + α(randminus 05) (9)

where Xi is the current position of artificial fish ibetaij(Xj minusXi) is the attraction degree α is the step factorwhich is a constant between 0 and 1 rand is a numberchosen randomly in [minus1 1] and α(randminus 05) is to avoidfalling into the local optimum

(b) Improvement 2 inertia weight A linear inertiaweight is added into Equation (9) as

Xnext ωtXi + betaij Xj minusXi1113872 1113873 + α(randminus 05) (10)

ωt ωmax minus ωmax minusωmin( 1113857 middot genMaxgen (11)

where ωt is the weight size inherited from the lastposition of an artificial fish ωmax denotes the biggestweight ωmin is the minimum weight gen representsthe current iteration and Maxgen is the ultimateiteration

LFFSAbegins

Initialize theswarm X1X2hellipXn

i = 1

Does Xi meet thecondition of chasing

behavior

N

Y

Prey withLeacutevy fight

Xi = Xnext

i = i + 1 N i ge n

Y

Whether the stopcriterion is met

N

YObtain the

optimum solution

End

Moveaccording to

attractiondegree

Artificial fish Xi

Figure 3 Flowchart of LFFSA

4 Computational Intelligence and Neuroscience

Similarly the position-updating with chasing behaviorcan be formulated as

Xnext ωtXi + betaij Xmax minusXi( 1113857 + α(randminus 05) (12)

where Xmax is the position with the highest food concen-tration in the view of artificial fish Xi and other parametersare defined in Equations (10) and (11)

(c) Improvement 3 Levy flight-based search strategyLevy flight is involved in the definition of artificialfishes are executing preying behavior e move-ment can be formulated as

Xnext Xi + αoplus L(λ) (13)

L(λ) ϕ times μ|v|1β

Xi minusXbest( 1113857 (14)

where Xi is the current position of artificial fish i oplus isthe point to point multiplication L(λ) denotes arandom vector generated by Levy flight Xbest rep-resents the best fish on the ldquoboardrdquo μ tminusλ 1lt λlt 3and μ and v have the standard normal distributionμ sim N(0 ϕ2) v sim N(0 1) respectively where

ϕ Γ(1 + β)sin(πβ2)

Γ[(1 + β)2]β middot 2(βminus 1)21113896 1113897

1β

(15)

where Γ is the standard Gamma function(d) Improvement 4 nonlinear visual and step Visual and

step change nonlinearly and dynamically in LFFSAe updating equations are as follows

Visual ρ middot Visual + Visualmin (16)

Visual Preying unaccomplished

Chasing

Behavior unaccomplished

Chasingaccomplished

Preyingaccomplished

Movingrandomly

AFSA

Artificial fishes

Swarming Swarmingaccomplished

LFFSA Chasing accomplished

Artificial fishes with attraction degree

Chasingunaccomplished

Preying withLeacutevy flight

Chasing

Figure 4 Mechanisms of AFSA and LFFSA

(1) Initialization of X1 X2 Xn1113864 1113865(2) Evaluate fitness(3) while the stop criterion is not met do(4) while ilt n do(5) if Xi meets the condition of chasing behavior(6) Move with attraction degree and get a new Xi based on Equation (13)(7) else(8) Prey with Levy flight and get a new Xi based on Equations (11) and (14)(9) end if(10) end while(11) Update the ldquoboardrdquo(12) end while(13) Obtain the optimum solution

ALGORITHM 1 LFFSA

Computational Intelligence and Neuroscience 5

Step ρ middot Step + Stepmin (17)

ρ exp minus30 timesgen

Maxgen( )

s

( ) (18)

where Visualmin the minimum visual takes 0001Stepmin the minimum step takes 00002 ρ isa nonlinear weight and s represents an integer sgt 1Here s 3 Other parameters are the same as thosein Equation (11) e relationship between s and ρ isas shown in Figure 5

Besides dene the biggest distance between two articialshes as

maxD xmax minusxmin( )2 timesD

radic (19)

where xmax and xmin represent the upper bound and thelower bound of searching range respectively D denotes theD-dimension searching space e initial Visual is equal tomaxD and initial Step is equal to maxD8 en Visual andStep change dynamically according to Equations (16)ndash(18)

(e) Improvement 5 decrease of time complexitySwarming behavior in AFSA is excluded to decreasethe algorithm complexity

4 Numerical Simulation

41 Comparison of Convergence Accuracy LFFSA is vali-dated by numerical simulations Fish swarm algorithm withrey behavior (FFSA) dierential evolution (DE) algo-rithm self-adaptive dierential evolution (jDE) algorithmand the two basic algorithms (AFSA and FA) are comparedAll the algorithms are coded in Matlab 2014b e operatingsystem is windows 7 Simulation hardware is a PC with250GHz Inter Core i5 and 200GB Memory

Parameters shown in Table 1 are determined by trial anderror

e following benchmarks are chosen carefullyaccording to their features Functions Sphere Quartic andRosenbrock etc are simple unimodal problems FunctionsAckley Rastrigin and Schwefel etc are highly complexmultimodal problems with many local minima Schwefelfunction has a maximum value and other functions haveminimum values ese benchmarks are listed in Table 2[44 45] e solutions of 17 test functions obtained bydierent algorithms are compared To compare the con-vergence speed and accuracy of the algorithms clearly andcorrectly all functions are run 50 times for each algorithmrespectivelye results are averaged and plotted in Figure 6

From Figure 6 LFFSA can avoid local optimum and havebetter convergence accuracy compared with the other al-gorithms For AFSA and FA the solutions of most functionsare unsatisfactory the DE cannot nd ideal solutions of f3f4 f5 f6 f10 f11 f12 f14 and f15 the jDE has good accuracywhile solving some of those functions eg f1 f2 f3 and f8however solutions of f4 f10 f11 and f15 obtained by jDEare not so precise the LFFSA can obtain the ideal accuracy

for almost all functions although it cannot achieve a highprecision level like solutions of f2 obtained by jDE the FFSAis slightly worse than LFFSAe LFFSA outperforms jDE in10 benchmark functions while 2 functions are comparativeand 5 functions are worse

LFFSA is better than AFSA because Levy ight is able torestrict the movement step of AFSA to a very small areaaround the current position Furthermore the attractiondegree guides the sh moving Besides LFFSA can quicklylead the sh individual to the close-by optimal pointConsidering all the advantages discussed above the opti-mum solution can be found successfully by using LFFSAwhich outperforms the basic algorithms for all test functionsand outperforms jDE for several functions To observe thesearching capabilities of dierent algorithms directly theaverage median best and worst values obtained by dierentalgorithms are listed in Table 3 Results indicate that LFFSAcan nd ideal solutions and have a better robustness

42 Computational ComplexityAnalysis Time complexity isalso an important indicator in the analysis of algorithms Ifan algorithm is composed of several parts then its com-plexity is the sum of the complexities of these parts ealgorithm may consist of a loop executed many times andeach time is with a dierent complexity Time complexity of

Iteration0 500 1000 1500 2000

ρ

0

02

04

06

08

1

s = 3s = 12

s = 21s = 30

Figure 5 Value of ρ

Table 1 Parameter settings

Algorithms Parameters ValuesFA FFSA LFFSA β0 10FA FFSA LFFSA c 10AFSA FFSA LFFSA δ 0618AFSA FFSA LFFSA Trynumber 5DE jDE Scaling constant 05DE jDE Crossover constant 09All 6 algorithms Population 50

All 6 algorithms Maximum functionevaluations (FEs) 2 times 105

6 Computational Intelligence and Neuroscience

the algorithm is used to estimate the efficiency of the al-gorithm It is defined that the time complexity of the al-gorithm or the running time is O(f(n)) [46] Define N asthe population

In the definition of time complexity O(N2) and O(N)

are at different levels If the time complexity of one algorithmis O(N2) the time complexity of the other one is O(N) thenthe former algorithm is more complex In the other case ifthe time complexity of one algorithm is O(N2) while thetime complexity of the other one is O(N2 + N) theircomplexities are both O(N2)

e time complexity analysis of AFSA is provided inTable 4

From Table 4 the time complexity of AFSA is

O Maxgenlowast 3lowastN2

+ trynumberlowastN + 6lowastN1113872 11138731113872 1113873 (20)

Swarming behavior has N times of calculating conges-tion factor 1 time of judging and 1 time of movingerefore time complexity of swarming behavior isO(N2 + 2lowastN) Chasing behavior has N times of calculating

congestion factor N times of searching 1 time of judgingand 1 time of moving erefore time complexity of chasingbehavior is O(2lowastN2 + 2lowastN)

Time complexity analysis of LFFSA is listed in Table 5Due to the lack of swarming behavior time complexity

of LFFSA can be calculated as

O Maxgenlowast 2lowastN2

+ TrynumberlowastN + 4lowastN1113872 11138731113872 1113873 (21)

We can also obtain time complexity of FA

O Maxgenlowast N2

+ N1113872 11138731113872 1113873 (22)

A conclusion can be obtained that time complexities ofthe three algorithms are at the same level eir computa-tional complexities in the worst case are only the square ofthe training sample size

43 Experimental Complexity Analysis Time complexity isa rough estimate of time costemore accurate time cost ofan algorithm can only be validated by running it on

Table 2 Test functions

No Test functions Expression Optimum value Domain Df1 Sphere f(x) 1113936

Di1x

2i 0 (minus100 100)D 30

f2 Quartic f(x) 1113936Di1ix

4i 0 (minus128 128)D 30

f3 Ackley f(x) minus20 exp minus02

(1D)1113936Di1x

2i

1113969

minus exp[(1D)1113936Di1cos(2πxi)]1113882 1113883

+ 20 + e0 (minus32768 32768)D 30

f4 Rosenbrock f(x) 1113936Dminus1i1 100(xi+1 minusxi

2)2 + (1minusxi)2 0 (minus2048 2048)D 30

f5 Rastrigin1 f(x) 1113936Di1 xi

2 minus 10 cos(2πxi) + 101113864 1113865 0 (minus512 512)D 30

f6 Rastrigin2f(x) 1113936

Di1 yi

2 minus 10 cos(2πyi) + 101113864 11138650 (minus512 512)D 30

yi xi |xi|lt (12)

round(2xi)2 |xi|gt (12)1113896

f7 Schwefel f(x) 1113936Di1 xi middot sin

|xi|

11139681113864 1113865 4189829D (minus500 500)D 30

f8 Griewank f(x) (14000)1113936Di1x

2i minus1113937

Di1cos(xi

i

radic) + 1 0 (minus600 600)D 30

f9 Quadric f(x) 1113936Di1(1113936

ij1xj)

2 0 (minus100 100)D 30

f10 Schaffer1 f(x) 1113936Dminus1i1 ((sin2

xi+1

2 + xi2

1113968minus 05)1113864

(0001(xi+12 + xi

2) + 1)2) + 050 (minus100 100)D 30

f11 Schaffer2 f(x) 1113936Dminus1i1 (sin2

1113936Di1x

2i

1113969

minus 05)(0001(1113936Di1x

2i ) + 1)2 + 051113882 1113883 0 (minus100 100)D 30

f12 Maxmod f(x) max(|xi|) 0 (minus10 10)D 30

f13 Dixon and price (x1 minus 1)2 + 1113936Di1i(2xi

2 minusximinus1) 0 (minus10 10)D 30

f14 Powell f(x) 1113936D4i1 [(x4iminus 3 + 10x4iminus 2)

2 + 5(x4iminus 1 minusx4i)2 +

(x4iminus 2 minus 2x4iminus 1)2 + 10(x4iminus 3 minusx4i)

4]0 (minus4 5)D 28

f15 Zakharov f(x) 1113936Di1x

2i + (1113936

Di105ixi)

2 + (1113936Di105ixi)

4 0 (minus5 10)D 30

f16 Sin1 1113936Di1|xi sin(xi) + 01xi| 0 (minus10 10)D 30

f17 Sin2 f(x) minus1113936Di1sin(xi) sin20(ix2

i π) minus992784 (0 π)D 100

Computational Intelligence and Neuroscience 7

FEs104

log 1

0(f(x

))

ndash20

ndash15

ndash10

ndash5

0

5

10

FAAFSAFFSA

LFFSADEjDE

0 5 10 15 20

(a)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash40

ndash30

ndash20

ndash10

0

10

FEs1040 5 10 15 20

(b)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

FEs1040 5 10 15 20

(c)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash2

0

2

4

6

8

FEs1040 5 10 15 20

(d)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

2

3

FEs1040 5 10 15 20

(e)

log 1

0(f(x

))FAAFSAFFSA

LFFSADEjDE

ndash6

ndash4

ndash2

0

2

4

FEs1040 5 10 15 20

(f )

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

32

34

36

38

4

42

FEs1040 5 10 15 20

(g)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash10

ndash5

0

5

FEs1040 5 10 15 20

(h)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash15

ndash10

ndash5

0

5

FEs1040 5 10 15 20

(i)

Figure 6 Continued

8 Computational Intelligence and Neuroscience

computer Since dierent algorithms cannot reach the sameconvergence accuracy the test with xed convergence ac-curacy is not available erefore the test with max functionevaluations is conducted Running time of each algorithm iscounted by the explorer of MATLAB Parameter settings ofalgorithms are the same in Section 41 Average running timeof dierent algorithms is listed in Table 6 When function

evaluations are the same running speed of LFFSA is fasterthan that of AFSA while DE has the fastest running speedResults are quite in accord with those obtained by com-putational complexity analysis LFFSA and jDE are com-parative in experimental complexity Running time of FFSAis almost twice as much as that of LFFSA e improvementof LFFSA decreases time complexity to some extent

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash8

ndash6

ndash4

ndash2

0

2

FEs1040 5 10 15 20

(j)lo

g 10(

f(x))

FAAFSAFFSA

LFFSADEjDE

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

FEs1040 5 10 15 20

(k)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

FEs1040 5 10 15 20

(l)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash2

0

2

4

6

8

10

FEs1040 5 10 15 20

(m)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash5

0

5

FEs1040 5 10 15 20

(n)

log 1

0(f(x

))FAAFSAFFSA

LFFSADEjDE

ndash10

ndash5

0

5

10

FEs1040 5 10 15 20

(o)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

2

FEs1040 5 10 15 20

(p)

f(x)

FAAFSAFFSA

LFFSADEjDE

FEs1040 5 10 15 20

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

(q)

Figure 6 Iterative curves of test functions (a) f1 (b) f2 (c) f3 (d) f4 (e) f5 (f ) f6 (g) f7 (h) f8 (i) f9 (j) f10 (k) f11 (l) f12 (m) f13 (n) f14(o) f15 (p) f16 (q) f7

Computational Intelligence and Neuroscience 9

Table 3 Comparison of optimization results

No Items AFSA Std FA Std FFSA Std LFFSA Std DE Std jDE Std

f1

Worst 3014

0181

minus2798

0120

minus5132

0071

minus6743

0071

minus8544

0132

minus15021

0121Best 2402 minus3257 minus5233 minus8500 minus10371 minus17242Average 2656 minus3048 minus5145 minus7278 minus8924 minus16326Median 2651 minus3102 minus5193 minus6325 minus8586 minus16706

f2

Worst minus2322

0171

minus19245

0135

minus11178

0074

minus14207

0134

minus20506

0400

minus31644

0535Best minus4126 minus2813 minus11313 minus15585 minus21611 minus33250Average minus2812 minus2278 minus11273 minus14812 minus21054 minus32687Median minus2562 minus2197 minus11271 minus14834 minus20970 minus32844

f3

Worst 1386

0001

minus1776

0124

minus3908

0197

minus6121

0138

minus0027

0438

minus8934

0167Best 1307 minus2415 minus4647 minus6938 minus1281 minus9471Average 1587 minus2168 minus4147 minus6546 minus0049 minus9163Median 1586 minus2177 minus4225 minus6325 minus0035 minus9164

f4

Worst 2143

0087

22733

0097

minus20348

0125

minus1076

0376

1520

0003

1454

0025Best 1565 1043 minus2416 minus3405 1412 1363Average 1946 1476 minus2158 minus1946 1385 1312Median 1854 1385 minus2235 minus2325 1363 1287

f5

Worst 2310

0077

1864

0856

minus2846

0044

minus4385

0054

minus0579

0323

0898

6009Best 2096 1243 minus2982 minus5145 minus1591 minus12831Average 2236 1454 minus2946 minus4643 minus0999 minus3303Median 2136 1285 minus2435 minus4325 minus0963 0148

f6

Worst 2445

0038

1716

0133

minus2382

0076

minus3414

0048

1131

0042

0698

5049Best 2318 1255 minus2618 minus3606 1012 minus11404Average 2408 1571 minus2486 minus3489 1077 minus5466Median 2419 1601 minus2462 minus3487 1094 minus7632

f7

Worst 3815

0018

37846

0031

4186

736eminus 6

4156

434eminus 6

4099

296eminus 9

4087

403eminus 3Best 3945 3978 4099 4099 4099 4099Average 3813 3848 4099 4099 4099 4097Median 3736 3785 minus4099 minus4099 4099 4097

f8

Worst 0956

0133

minus0960

0223

minus5301

0051

minus6271

0055

minus989

0114

minusInf

0Best minus0644 minus0500 minus5444 minus6455 minus10255 minusInfAverage minus0735 minus0697 minus5357 minus6372 minus10071 minusInfMedian 0736 0685 minus5435 minus6325 minus10074 minusInf

f9

Worst minus11665

1057

minus976

1324

minus9347

0843

minus8695

1323

minus6848

1124

minus6131

2697Best minus14433 minus10574 minus12194 minus11937 minus10680 minus14831Average minus12786 minus10456 minus10764 minus10137 minus8178 minus9938Median 1136 1085 minus7435 minus6325 minus7745 minus9663

f10

Worst 0957

0131

0974

0223

minus5375

0049

minus6274

0056

0673

0032

0280

0110Best 0644 0497 minus5448 minus6486 0483 minus0022Average 0747 0649 minus5376 minus6348 0526 0230Median 1136 1085 minus7435 minus6325 0547 0211

f11

Worst minus0301

272eminus 05

minus0303

0009

minus4751

0071

minus5647

0085

minus0896

0103

minus1106

0135Best minus0301 minus0331 minus5011 minus5965 minus1107 minus1429Average minus0301 minus0313 minus4838 minus5804 minus0975 minus1364Median minus0301 minus0311 minus4835 minus5804 minus0896 minus1402

f12

Worst 0779

0037

minus1177

0163

minus2756

0035

minus3158

0037

minus0136

0052

minus1904

0221Best 0658 minus1638 minus2892 minus3287 minus0339 minus2526Average 0711 minus1388 minus2811 minus3221 minus0252 minus2291Median 0716 minus1376 minus2801 minus3216 0248 minus2374

f13

Worst 4286

0173

0574

0297

minus0602

0007

minus0602

0005

0039

0068

minus0176

538eminus 6Best 3706 minus0175 minus0605 minus0605 minus0162 minus0176Average 4147 minus0013 minus0603 minus0603 minus0088 minus0176Median 4213 minus0135 minus0603 minus6603 minus0086 minus0176

f14

Worst 2404

0084

0405

0318

minus3598

0117

minus4514

0101

0276

0211

minus2192

0401Best 2173 minus0576 minus4001 minus4869 minus0369 minus3409Average 2275 0069 minus3773 minus4656 minus0117 minus2867Median 2246 0133 minus3719 minus4651 minus0134 minus2924

10 Computational Intelligence and Neuroscience

44ParameterAnalysis ofLFFSA eeect of parameters onoptimization is analyzed in this section Taking Ackleyfunction as example Figure 7 shows the change of the ob-jective function value in the case of varying parametersTrynumber and β0 are proportional to the optimization resultTrynumber can impact the time complexity of the algorithmso the value should be appropriate not to aect the runningspeed e best value of c and δ is 25 and 12 respectively

5 Conclusion

LFFSA is proposed to improve the capability of AFSAwhich integrates the merits of both AFSA and FA Firstlythe searching characteristic of AFSA is studied by cal-culating the time complexity Secondly 17 benchmark testfunctions are used to verify LFFSA en time complexityof LFFSA is estimated Numerical results demonstrate thatLFFSA has a better performance in accuracy and speed ofoptimization to solve nonlinear optimization problemsthan the other test algorithms However the solutionobtained by LFFSA can be more precise and the way ofmodication could provide reference for those esectcientalgorithms eg DE and GWO

Table 3 Continued

No Items AFSA Std FA Std FFSA Std LFFSA Std DE Std jDE Std

f15

Worst 2588

00705

1505

0198

minus4281

0117

minus4975

0241

2241

0077

0635

0372Best 2366 minus4679 minus5792 minus72474 1972 minus0370Average 2490 minus4561 minus5408 minus63259 2140 0148Median 2506 minus4601 minus5456 minus63259 2136 0218

f16

Worst 1438

0058

minus1463

0253

minus2762

0026

minus3239

0032

minus2354

0042

minus3392

0433Best 1268 minus2296 minus2838 minus3334 minus2482 minus4646Average 1366 minus1987 minus2793 minus3282 minus2408 minus3868Median 1375 minus2016 minus2793 minus3284 minus2402 minus3751

f17

Worst minus22954

0643

minus25673

1721

minus70748

1908

minus79645

0131

minus47025

1403

minus63031

1942Best minus24885 minus32075 minus77054 minus80098 minus51440 minus68738Average minus23772 minus28641 minus74598 minus79996 minus48982 minus66539Median minus23731 minus28759 minus74907 minus80001 minus49035 minus66559

Table 4 Time complexity analysis of AFSA

Procedure of AFSA Time complexity(1) Initialization of N articial shes O(N)(2) Initialization of lsquoboardrsquo O(N)(3) Swarming behavior O(N2 + 2lowastN)(4) Chasing behavior O(2lowastN2 + 2lowastN)(5) Preying behavior O(TrynumberlowastN)(6) Judging of terminal condition O(1)(7) Information output of lsquoboardrsquo O(1)

Table 5 Time complexity analysis of LFFSA

Procedure of LFFSA Time complexity(1) Initialization of N articial shes O(N)(2) Initialization of lsquoboardrsquo O(N)(3) Chasing behavior O(2lowastN2 + 2lowastN)(4) Preying behavior O(trynumberlowastN)(5) Judging of terminal condition O(1)(6) Information output of lsquoboardrsquo O(1)

Table 6 Average running time of algorithms

NoRunning time (s)

AFSA FA FFSA LFFSA DE jDEf1 1023 747 185 753 464 608f2 1237 773 2111 710 432 653f3 1087 780 2073 896 412 825f4 1140 767 2049 1027 563 698f5 1053 766 1947 803 328 596f6 1972 610 2577 1637 771 1065f7 1053 743 1877 856 327 656f8 1233 803 2272 923 379 663f9 2083 720 2136 771 401 684f10 1627 846 2062 756 465 715f11 926 895 1374 635 363 667f12 965 534 1375 651 304 628f13 912 526 1436 1041 413 715f14 1395 574 1902 1031 648 986f15 945 539 1307 736 462 774f16 873 521 1388 1146 852 1887f17 1453 728 1841 596 332 644

β0

Valu

e

ndash8

ndash75

ndash7

ndash65

ndash6

ndash55

215050 1

(a)

γ

Valu

e

ndash63

ndash62

ndash61

ndash6

ndash59

ndash58

0 2 4 6 8 10

(b)

δ

Valu

e

ndash615ndash61

ndash605ndash6

ndash595ndash59

ndash585ndash58

215050 1

(c)

Trynumber

Valu

e

ndash64

ndash62

ndash6

ndash58

ndash56

ndash54

0 2 4 6 8 10

(d)

Figure 7 Test curves of parameters (a) Test curve of β0 (b) Testcurve of c (c) Test curve of δ (d) Test curve of Trynumber

Computational Intelligence and Neuroscience 11

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work is supported by National Natural ScienceFoundation of China (61463028) e authors wish to thankDr Yanliang Cui for his fruitful comments and suggestions

References

[1] X L Li Z J Shao J X Qian et al ldquoAn optimizing methodbased on autonomous animats fish-swarm algorithmrdquo Sys-tems Engineering-eory and Practice vol 22 pp 188ndash2002002 in Chinese

[2] C R Wang C L Zhou and J W Ma ldquoAn improved artificialfish-swarm algorithm and its application in feed-forwardneural networksrdquo in Proceedings of the Fourth InternationalConference on Machine Learning amp Cybernetics GuangzhouChina August 2005

[3] H C Tsai and Y H Lin ldquoModification of the fish swarmalgorithm with particle swarm optimization formulation andcommunication behaviorrdquo Applied Soft Computing vol 11no 8 pp 5367ndash5374 2011

[4] X Y Luan Z P Li and T Z Liu ldquoA novel attribute reductionalgorithm based on rough set and improved artificial fishswarm algorithmrdquo Neurocomputing vol 174 pp 522ndash5292016

[5] Y Wu X Z Gao Z Kai et al ldquoKnowledge-based artificialfish-swarm algorithmrdquo IFAC Proceedings Volumes vol 44no 1 pp 188ndash200 2011

[6] X T Hu H Q Zhang Z C Li et al ldquoA novel self-adaptationhybrid artificial fish-swarm algorithmrdquo IFAC ProceedingsVolumes vol 46 no 5 pp 583ndash588 2013

[7] K P Kumar B Saravanan and K S Swarup ldquoOptimizationof renewable energy sources in a microgrid using artificial fishswarm algorithmrdquo Energy Procedia vol 90 pp 107ndash113 2016

[8] Q He X T Hu H Ren et al ldquoA novel artificial fish swarmalgorithm for solving large-scale reliability-redundancy ap-plication problemrdquo ISA Transactions vol 59 pp 105ndash1132015

[9] Z Q Zhang K P Wang L X Zhu et al ldquoA Pareto improvedartificial fish swarm algorithm for solving a multi-objectivefuzzy disassembly line balancing problemrdquo Expert Systemswith Applications vol 86 pp 165ndash176 2017

[10] P K Singhal R Naresh and V Sharma ldquoBinary fish swarmalgorithm for profit-based unit commitment problem incompetitive electricity market with ramp rate constraintsrdquoGeneration Transmission and Distribution IET vol 9 no 13pp 1697ndash1707 2015

[11] X S Yang Nature-Inspired Metaheuristic Algorithmspp 83ndash96 Luniver Press London 2008

[12] S M Farahani B Nasiri A A Abshouri et al ldquoAn improvedfirefly algorithm with directed movementrdquo in Proceedings ofIEEE International Conference on Computer Science amp In-formation Technology Sichuan China June 2011

[13] L D S Coelho D L D A Bernert and V C Mariani ldquoAchaotic firefly algorithm applied to reliability-redundancy

optimizationrdquo Evolutionary Computation vol 30 pp 517ndash521 2011

[14] X S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo in Studies inComputational Intelligence vol 284 pp 101-111 SpringerBerlin Germany 2010

[15] S L Tilahun and C O Hong ldquoModified firefly algorithmrdquoJournal of Applied Mathematics vol 12 pp 2428ndash24392012

[16] K Jagatheesan B Anand S Samanta et al ldquoDesign ofa proportional-integral-derivative controller for an automaticgeneration control of multi-area power thermal systems usingfirefly algorithmrdquo IEEECAA Journal of Automatica Sinicapp 1ndash14 2016

[17] H Su Y Cai and Q Du ldquoFirefly-algorithm-inspiredframework with band selection and extreme learning ma-chine for hyperspectral image classificationrdquo IEEE Journal ofSelected Topics in Applied Earth Observations and RemoteSensing vol 10 no 1 pp 309ndash320 2016

[18] S Ranganathan M S Kalavathi and A R C ChristoberldquoSelf-adaptive firefly algorithm based multi-objectives formulti-type FACTS placementrdquo IET Generation Transmissionand Distribution vol 10 no 11 pp 188ndash200 2016

[19] D F Teshome C H Le Y W Lin et al ldquoA modified fireflyalgorithm for photovoltaic maximum power point trackingcontrol under partial shadingrdquo IEEE Journal of Emerging andSelected Topics in Power Electronics vol 5 no 2 pp 661ndash6712017

[20] M Alb P Alotto C Magele et al ldquoFirefly algorithm forfinding optimal shapes of electromagnetic devicesrdquo IEEETransactions on Magnetics vol 52 no 3 pp 1ndash4 2016

[21] A Mishra and V N K Gundavarapu ldquoLine utilisationfactor-based optimal allocation of IPFC and sizing usingfirefly algorithm for congestion managementrdquo GenerationTransmission and Distribution IET vol 10 no 1 pp 115ndash122 2016

[22] M H Horng ldquoVector quantization using the firefly algorithmfor image compressionrdquo Expert Systems with Applicationsvol 39 no 1 pp 078ndash1091 2012

[23] A Kazem E Sharifi F K Hussain et al ldquoSupport vectorregression with chaos-based firefly algorithm for stock marketprice forecastingrdquo Applied Soft Computing vol 13 no 2pp 947ndash958 2013

[24] L F He and S W Huang ldquoModified firefly algorithm basedmultilevel thresholding for color image segmentationrdquoNeurocomputing vol 240 pp 152ndash174 2017

[25] A H Gandomi X S Yang and A H Alavi ldquoMixed variablestructural optimization using Firefly Algorithmrdquo Computersand Structures vol 89 no 23-24 pp 2325ndash2336 2011

[26] M Alweshah and S Abdullah ldquoHybridizing firefly algorithmswith a probabilistic neural network for solving classificationproblemsrdquo Applied Soft Computing vol 35 pp 513ndash5242015

[27] R M Rizk-Allah E M Zaki A A El-Sawy et al ldquoHy-bridizing ant colony optimization with firefly algorithm forunconstrained optimization problemsrdquo Applied Mathematicsand Computation vol 224 pp 473ndash483 2013

[28] X S Yang S S S Hosseini and A H Gandomi ldquoFireflyalgorithm for solving non-convex economic dispatch prob-lems with valve loading effectrdquo Applied Soft Computingvol 12 no 3 pp 1180ndash1186 2012

[29] J Senthilnath S N Omkar and V Mani ldquoClustering usingfirefly algorithm performance studyrdquo Swarm and Evolu-tionary Computation vol 1 no 3 pp 164ndash171 2011

12 Computational Intelligence and Neuroscience

[30] T Kanimozhi and K Latha ldquoAn integrated approach to re-gion based image retrieval using firefly algorithm and supportvector machinerdquo Neurocomputing vol 151 pp 1099ndash11112015

[31] A Baykasoglu and F B Ozsoydan ldquoAdaptive firefly algorithmwith chaos for mechanical design optimization problemsrdquoApplied Soft Computing vol 36 pp 152ndash164 2015

[32] I Fister M Perc S M Kamal et al ldquoA review of chaos-basedfirefly algorithms perspectives and research challengesrdquoApplied Mathematics and Computation vol 252 pp 155ndash1652015

[33] I Fister I Fister X S Yang et al ldquoA comprehensive review offirefly algorithmsrdquo Swarm and Evolutionary Computationvol 13 pp 34ndash46 2013

[34] C T Brown L S Liebovitch and R Glendon ldquoLevy flights inDobe Jursquohoansi foraging patternsrdquo Human Ecology vol 35no 1 pp 129ndash138 2007

[35] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquoMathematics vol 226 no 2 pp 1830ndash1844 2012

[36] I Pavlyukevich ldquoCooling down Levy flightsrdquo Journal ofPhysics A Mathematical and eoretical vol 40 no 41pp 12299ndash12313 2007

[37] R Jensi and G W Jiji ldquoAn enhanced particle swarm opti-mization with Levy flight for global optimizationrdquo AppliedSoft Computing vol 43 pp 248ndash261 2016

[38] D Tang J Yang S Dong et al ldquoA Levy flight-based shuffledfrog-leaping algorithm and its applications for continuousoptimization problemsrdquo Applied Soft Computing vol 49pp 641ndash662 2016

[39] M Yahya and M P Saka ldquoConstruction site layout planningusing multi-objective artificial bee colony algorithmwith Levyflightsrdquo Automation in Construction vol 38 pp 14ndash29 2014

[40] C Blum and A Roli Hybrid Metaheuristics An IntroductionSpringer Berlin Germany 2008

[41] A M Reynolds and M A Frye ldquoFree-flight odor tracking indrosophila is consistent with an optimal intermittent scale-free searchrdquo PLos One vol 2 no 4 p e354 2007

[42] M F Shlesinger G M Zaslavsky and U Frisch Levy Flightsand Related Topics in Physics Springer Berlin HeidelbergGermany 1995

[43] M F Shlesinger ldquoMathematical physics search researchrdquoNature vol 443 no 7109 pp 281-282 2006

[44] S T Hsieh T Y Sun C C Liu et al ldquoEfficient populationutilization strategy for particle swarm optimizerrdquo IEEETransactions on Systems Man amp Cybernetics Part B vol 39no 2 pp 444ndash456 2009

[45] D Bratton and J Kennedy ldquoDefining a standard for particleswarm optimizationrdquo in Proceedings of IEEE Swarm In-telligence Symposium vol 107 pp 120ndash127 HonoluluHawaii April 2007

[46] U Manber Introduction to Algorithms A Creative ApproachAddison-Wesley Longman Publishing Co Inc Boston MAUSA 1989

Computational Intelligence and Neuroscience 13

Computer Games Technology

International Journal of

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Applied Computational Intelligence and Soft Computing

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Submit your manuscripts atwwwhindawicom

Similarly the position-updating with chasing behaviorcan be formulated as

Xnext ωtXi + betaij Xmax minusXi( 1113857 + α(randminus 05) (12)

where Xmax is the position with the highest food concen-tration in the view of artificial fish Xi and other parametersare defined in Equations (10) and (11)

(c) Improvement 3 Levy flight-based search strategyLevy flight is involved in the definition of artificialfishes are executing preying behavior e move-ment can be formulated as

Xnext Xi + αoplus L(λ) (13)

L(λ) ϕ times μ|v|1β

Xi minusXbest( 1113857 (14)

where Xi is the current position of artificial fish i oplus isthe point to point multiplication L(λ) denotes arandom vector generated by Levy flight Xbest rep-resents the best fish on the ldquoboardrdquo μ tminusλ 1lt λlt 3and μ and v have the standard normal distributionμ sim N(0 ϕ2) v sim N(0 1) respectively where

ϕ Γ(1 + β)sin(πβ2)

Γ[(1 + β)2]β middot 2(βminus 1)21113896 1113897

1β

(15)

where Γ is the standard Gamma function(d) Improvement 4 nonlinear visual and step Visual and

step change nonlinearly and dynamically in LFFSAe updating equations are as follows

Visual ρ middot Visual + Visualmin (16)

Visual Preying unaccomplished

Chasing

Behavior unaccomplished

Chasingaccomplished

Preyingaccomplished

Movingrandomly

AFSA

Artificial fishes

Swarming Swarmingaccomplished

LFFSA Chasing accomplished

Artificial fishes with attraction degree

Chasingunaccomplished

Preying withLeacutevy flight

Chasing

Figure 4 Mechanisms of AFSA and LFFSA

(1) Initialization of X1 X2 Xn1113864 1113865(2) Evaluate fitness(3) while the stop criterion is not met do(4) while ilt n do(5) if Xi meets the condition of chasing behavior(6) Move with attraction degree and get a new Xi based on Equation (13)(7) else(8) Prey with Levy flight and get a new Xi based on Equations (11) and (14)(9) end if(10) end while(11) Update the ldquoboardrdquo(12) end while(13) Obtain the optimum solution

ALGORITHM 1 LFFSA

Computational Intelligence and Neuroscience 5

Step ρ middot Step + Stepmin (17)

ρ exp minus30 timesgen

Maxgen( )

s

( ) (18)

where Visualmin the minimum visual takes 0001Stepmin the minimum step takes 00002 ρ isa nonlinear weight and s represents an integer sgt 1Here s 3 Other parameters are the same as thosein Equation (11) e relationship between s and ρ isas shown in Figure 5

Besides dene the biggest distance between two articialshes as

maxD xmax minusxmin( )2 timesD

radic (19)

where xmax and xmin represent the upper bound and thelower bound of searching range respectively D denotes theD-dimension searching space e initial Visual is equal tomaxD and initial Step is equal to maxD8 en Visual andStep change dynamically according to Equations (16)ndash(18)

(e) Improvement 5 decrease of time complexitySwarming behavior in AFSA is excluded to decreasethe algorithm complexity

4 Numerical Simulation

41 Comparison of Convergence Accuracy LFFSA is vali-dated by numerical simulations Fish swarm algorithm withrey behavior (FFSA) dierential evolution (DE) algo-rithm self-adaptive dierential evolution (jDE) algorithmand the two basic algorithms (AFSA and FA) are comparedAll the algorithms are coded in Matlab 2014b e operatingsystem is windows 7 Simulation hardware is a PC with250GHz Inter Core i5 and 200GB Memory

Parameters shown in Table 1 are determined by trial anderror

e following benchmarks are chosen carefullyaccording to their features Functions Sphere Quartic andRosenbrock etc are simple unimodal problems FunctionsAckley Rastrigin and Schwefel etc are highly complexmultimodal problems with many local minima Schwefelfunction has a maximum value and other functions haveminimum values ese benchmarks are listed in Table 2[44 45] e solutions of 17 test functions obtained bydierent algorithms are compared To compare the con-vergence speed and accuracy of the algorithms clearly andcorrectly all functions are run 50 times for each algorithmrespectivelye results are averaged and plotted in Figure 6

From Figure 6 LFFSA can avoid local optimum and havebetter convergence accuracy compared with the other al-gorithms For AFSA and FA the solutions of most functionsare unsatisfactory the DE cannot nd ideal solutions of f3f4 f5 f6 f10 f11 f12 f14 and f15 the jDE has good accuracywhile solving some of those functions eg f1 f2 f3 and f8however solutions of f4 f10 f11 and f15 obtained by jDEare not so precise the LFFSA can obtain the ideal accuracy

for almost all functions although it cannot achieve a highprecision level like solutions of f2 obtained by jDE the FFSAis slightly worse than LFFSAe LFFSA outperforms jDE in10 benchmark functions while 2 functions are comparativeand 5 functions are worse

LFFSA is better than AFSA because Levy ight is able torestrict the movement step of AFSA to a very small areaaround the current position Furthermore the attractiondegree guides the sh moving Besides LFFSA can quicklylead the sh individual to the close-by optimal pointConsidering all the advantages discussed above the opti-mum solution can be found successfully by using LFFSAwhich outperforms the basic algorithms for all test functionsand outperforms jDE for several functions To observe thesearching capabilities of dierent algorithms directly theaverage median best and worst values obtained by dierentalgorithms are listed in Table 3 Results indicate that LFFSAcan nd ideal solutions and have a better robustness

42 Computational ComplexityAnalysis Time complexity isalso an important indicator in the analysis of algorithms Ifan algorithm is composed of several parts then its com-plexity is the sum of the complexities of these parts ealgorithm may consist of a loop executed many times andeach time is with a dierent complexity Time complexity of

Iteration0 500 1000 1500 2000

ρ

0

02

04

06

08

1

s = 3s = 12

s = 21s = 30

Figure 5 Value of ρ

Table 1 Parameter settings

Algorithms Parameters ValuesFA FFSA LFFSA β0 10FA FFSA LFFSA c 10AFSA FFSA LFFSA δ 0618AFSA FFSA LFFSA Trynumber 5DE jDE Scaling constant 05DE jDE Crossover constant 09All 6 algorithms Population 50

All 6 algorithms Maximum functionevaluations (FEs) 2 times 105

6 Computational Intelligence and Neuroscience

the algorithm is used to estimate the efficiency of the al-gorithm It is defined that the time complexity of the al-gorithm or the running time is O(f(n)) [46] Define N asthe population

In the definition of time complexity O(N2) and O(N)

are at different levels If the time complexity of one algorithmis O(N2) the time complexity of the other one is O(N) thenthe former algorithm is more complex In the other case ifthe time complexity of one algorithm is O(N2) while thetime complexity of the other one is O(N2 + N) theircomplexities are both O(N2)

e time complexity analysis of AFSA is provided inTable 4

From Table 4 the time complexity of AFSA is

O Maxgenlowast 3lowastN2

+ trynumberlowastN + 6lowastN1113872 11138731113872 1113873 (20)

Swarming behavior has N times of calculating conges-tion factor 1 time of judging and 1 time of movingerefore time complexity of swarming behavior isO(N2 + 2lowastN) Chasing behavior has N times of calculating

congestion factor N times of searching 1 time of judgingand 1 time of moving erefore time complexity of chasingbehavior is O(2lowastN2 + 2lowastN)

Time complexity analysis of LFFSA is listed in Table 5Due to the lack of swarming behavior time complexity

of LFFSA can be calculated as

O Maxgenlowast 2lowastN2

+ TrynumberlowastN + 4lowastN1113872 11138731113872 1113873 (21)

We can also obtain time complexity of FA

O Maxgenlowast N2

+ N1113872 11138731113872 1113873 (22)

A conclusion can be obtained that time complexities ofthe three algorithms are at the same level eir computa-tional complexities in the worst case are only the square ofthe training sample size

43 Experimental Complexity Analysis Time complexity isa rough estimate of time costemore accurate time cost ofan algorithm can only be validated by running it on

Table 2 Test functions

No Test functions Expression Optimum value Domain Df1 Sphere f(x) 1113936

Di1x

2i 0 (minus100 100)D 30

f2 Quartic f(x) 1113936Di1ix

4i 0 (minus128 128)D 30

f3 Ackley f(x) minus20 exp minus02

(1D)1113936Di1x

2i

1113969

minus exp[(1D)1113936Di1cos(2πxi)]1113882 1113883

+ 20 + e0 (minus32768 32768)D 30

f4 Rosenbrock f(x) 1113936Dminus1i1 100(xi+1 minusxi

2)2 + (1minusxi)2 0 (minus2048 2048)D 30

f5 Rastrigin1 f(x) 1113936Di1 xi

2 minus 10 cos(2πxi) + 101113864 1113865 0 (minus512 512)D 30

f6 Rastrigin2f(x) 1113936

Di1 yi

2 minus 10 cos(2πyi) + 101113864 11138650 (minus512 512)D 30

yi xi |xi|lt (12)

round(2xi)2 |xi|gt (12)1113896

f7 Schwefel f(x) 1113936Di1 xi middot sin

|xi|

11139681113864 1113865 4189829D (minus500 500)D 30

f8 Griewank f(x) (14000)1113936Di1x

2i minus1113937

Di1cos(xi

i

radic) + 1 0 (minus600 600)D 30

f9 Quadric f(x) 1113936Di1(1113936

ij1xj)

2 0 (minus100 100)D 30

f10 Schaffer1 f(x) 1113936Dminus1i1 ((sin2

xi+1

2 + xi2

1113968minus 05)1113864

(0001(xi+12 + xi

2) + 1)2) + 050 (minus100 100)D 30

f11 Schaffer2 f(x) 1113936Dminus1i1 (sin2

1113936Di1x

2i

1113969

minus 05)(0001(1113936Di1x

2i ) + 1)2 + 051113882 1113883 0 (minus100 100)D 30

f12 Maxmod f(x) max(|xi|) 0 (minus10 10)D 30

f13 Dixon and price (x1 minus 1)2 + 1113936Di1i(2xi

2 minusximinus1) 0 (minus10 10)D 30

f14 Powell f(x) 1113936D4i1 [(x4iminus 3 + 10x4iminus 2)

2 + 5(x4iminus 1 minusx4i)2 +

(x4iminus 2 minus 2x4iminus 1)2 + 10(x4iminus 3 minusx4i)

4]0 (minus4 5)D 28

f15 Zakharov f(x) 1113936Di1x

2i + (1113936

Di105ixi)

2 + (1113936Di105ixi)

4 0 (minus5 10)D 30

f16 Sin1 1113936Di1|xi sin(xi) + 01xi| 0 (minus10 10)D 30

f17 Sin2 f(x) minus1113936Di1sin(xi) sin20(ix2

i π) minus992784 (0 π)D 100

Computational Intelligence and Neuroscience 7

FEs104

log 1

0(f(x

))

ndash20

ndash15

ndash10

ndash5

0

5

10

FAAFSAFFSA

LFFSADEjDE

0 5 10 15 20

(a)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash40

ndash30

ndash20

ndash10

0

10

FEs1040 5 10 15 20

(b)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

FEs1040 5 10 15 20

(c)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash2

0

2

4

6

8

FEs1040 5 10 15 20

(d)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

2

3

FEs1040 5 10 15 20

(e)

log 1

0(f(x

))FAAFSAFFSA

LFFSADEjDE

ndash6

ndash4

ndash2

0

2

4

FEs1040 5 10 15 20

(f )

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

32

34

36

38

4

42

FEs1040 5 10 15 20

(g)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash10

ndash5

0

5

FEs1040 5 10 15 20

(h)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash15

ndash10

ndash5

0

5

FEs1040 5 10 15 20

(i)

Figure 6 Continued

8 Computational Intelligence and Neuroscience

computer Since dierent algorithms cannot reach the sameconvergence accuracy the test with xed convergence ac-curacy is not available erefore the test with max functionevaluations is conducted Running time of each algorithm iscounted by the explorer of MATLAB Parameter settings ofalgorithms are the same in Section 41 Average running timeof dierent algorithms is listed in Table 6 When function

evaluations are the same running speed of LFFSA is fasterthan that of AFSA while DE has the fastest running speedResults are quite in accord with those obtained by com-putational complexity analysis LFFSA and jDE are com-parative in experimental complexity Running time of FFSAis almost twice as much as that of LFFSA e improvementof LFFSA decreases time complexity to some extent

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash8

ndash6

ndash4

ndash2

0

2

FEs1040 5 10 15 20

(j)lo

g 10(

f(x))

FAAFSAFFSA

LFFSADEjDE

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

FEs1040 5 10 15 20

(k)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

FEs1040 5 10 15 20

(l)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash2

0

2

4

6

8

10

FEs1040 5 10 15 20

(m)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash5

0

5

FEs1040 5 10 15 20

(n)

log 1

0(f(x

))FAAFSAFFSA

LFFSADEjDE

ndash10

ndash5

0

5

10

FEs1040 5 10 15 20

(o)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

2

FEs1040 5 10 15 20

(p)

f(x)

FAAFSAFFSA

LFFSADEjDE

FEs1040 5 10 15 20

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

(q)

Figure 6 Iterative curves of test functions (a) f1 (b) f2 (c) f3 (d) f4 (e) f5 (f ) f6 (g) f7 (h) f8 (i) f9 (j) f10 (k) f11 (l) f12 (m) f13 (n) f14(o) f15 (p) f16 (q) f7

Computational Intelligence and Neuroscience 9

Table 3 Comparison of optimization results

No Items AFSA Std FA Std FFSA Std LFFSA Std DE Std jDE Std

f1

Worst 3014

0181

minus2798

0120

minus5132

0071

minus6743

0071

minus8544

0132

minus15021

0121Best 2402 minus3257 minus5233 minus8500 minus10371 minus17242Average 2656 minus3048 minus5145 minus7278 minus8924 minus16326Median 2651 minus3102 minus5193 minus6325 minus8586 minus16706

f2

Worst minus2322

0171

minus19245

0135

minus11178

0074

minus14207

0134

minus20506

0400

minus31644

0535Best minus4126 minus2813 minus11313 minus15585 minus21611 minus33250Average minus2812 minus2278 minus11273 minus14812 minus21054 minus32687Median minus2562 minus2197 minus11271 minus14834 minus20970 minus32844

f3

Worst 1386

0001

minus1776

0124

minus3908

0197

minus6121

0138

minus0027

0438

minus8934

0167Best 1307 minus2415 minus4647 minus6938 minus1281 minus9471Average 1587 minus2168 minus4147 minus6546 minus0049 minus9163Median 1586 minus2177 minus4225 minus6325 minus0035 minus9164

f4

Worst 2143

0087

22733

0097

minus20348

0125

minus1076

0376

1520

0003

1454

0025Best 1565 1043 minus2416 minus3405 1412 1363Average 1946 1476 minus2158 minus1946 1385 1312Median 1854 1385 minus2235 minus2325 1363 1287

f5

Worst 2310

0077

1864

0856

minus2846

0044

minus4385

0054

minus0579

0323

0898

6009Best 2096 1243 minus2982 minus5145 minus1591 minus12831Average 2236 1454 minus2946 minus4643 minus0999 minus3303Median 2136 1285 minus2435 minus4325 minus0963 0148

f6

Worst 2445

0038

1716

0133

minus2382

0076

minus3414

0048

1131

0042

0698

5049Best 2318 1255 minus2618 minus3606 1012 minus11404Average 2408 1571 minus2486 minus3489 1077 minus5466Median 2419 1601 minus2462 minus3487 1094 minus7632

f7

Worst 3815

0018

37846

0031

4186

736eminus 6

4156

434eminus 6

4099

296eminus 9

4087

403eminus 3Best 3945 3978 4099 4099 4099 4099Average 3813 3848 4099 4099 4099 4097Median 3736 3785 minus4099 minus4099 4099 4097

f8

Worst 0956

0133

minus0960

0223

minus5301

0051

minus6271

0055

minus989

0114

minusInf

0Best minus0644 minus0500 minus5444 minus6455 minus10255 minusInfAverage minus0735 minus0697 minus5357 minus6372 minus10071 minusInfMedian 0736 0685 minus5435 minus6325 minus10074 minusInf

f9

Worst minus11665

1057

minus976

1324

minus9347

0843

minus8695

1323

minus6848

1124

minus6131

2697Best minus14433 minus10574 minus12194 minus11937 minus10680 minus14831Average minus12786 minus10456 minus10764 minus10137 minus8178 minus9938Median 1136 1085 minus7435 minus6325 minus7745 minus9663

f10

Worst 0957

0131

0974

0223

minus5375

0049

minus6274

0056

0673

0032

0280

0110Best 0644 0497 minus5448 minus6486 0483 minus0022Average 0747 0649 minus5376 minus6348 0526 0230Median 1136 1085 minus7435 minus6325 0547 0211

f11

Worst minus0301

272eminus 05

minus0303

0009

minus4751

0071

minus5647

0085

minus0896

0103

minus1106

0135Best minus0301 minus0331 minus5011 minus5965 minus1107 minus1429Average minus0301 minus0313 minus4838 minus5804 minus0975 minus1364Median minus0301 minus0311 minus4835 minus5804 minus0896 minus1402

f12

Worst 0779

0037

minus1177

0163

minus2756

0035

minus3158

0037

minus0136

0052

minus1904

0221Best 0658 minus1638 minus2892 minus3287 minus0339 minus2526Average 0711 minus1388 minus2811 minus3221 minus0252 minus2291Median 0716 minus1376 minus2801 minus3216 0248 minus2374

f13

Worst 4286

0173

0574

0297

minus0602

0007

minus0602

0005

0039

0068

minus0176

538eminus 6Best 3706 minus0175 minus0605 minus0605 minus0162 minus0176Average 4147 minus0013 minus0603 minus0603 minus0088 minus0176Median 4213 minus0135 minus0603 minus6603 minus0086 minus0176

f14

Worst 2404

0084

0405

0318

minus3598

0117

minus4514

0101

0276

0211

minus2192

0401Best 2173 minus0576 minus4001 minus4869 minus0369 minus3409Average 2275 0069 minus3773 minus4656 minus0117 minus2867Median 2246 0133 minus3719 minus4651 minus0134 minus2924

10 Computational Intelligence and Neuroscience

44ParameterAnalysis ofLFFSA eeect of parameters onoptimization is analyzed in this section Taking Ackleyfunction as example Figure 7 shows the change of the ob-jective function value in the case of varying parametersTrynumber and β0 are proportional to the optimization resultTrynumber can impact the time complexity of the algorithmso the value should be appropriate not to aect the runningspeed e best value of c and δ is 25 and 12 respectively

5 Conclusion

LFFSA is proposed to improve the capability of AFSAwhich integrates the merits of both AFSA and FA Firstlythe searching characteristic of AFSA is studied by cal-culating the time complexity Secondly 17 benchmark testfunctions are used to verify LFFSA en time complexityof LFFSA is estimated Numerical results demonstrate thatLFFSA has a better performance in accuracy and speed ofoptimization to solve nonlinear optimization problemsthan the other test algorithms However the solutionobtained by LFFSA can be more precise and the way ofmodication could provide reference for those esectcientalgorithms eg DE and GWO

Table 3 Continued

No Items AFSA Std FA Std FFSA Std LFFSA Std DE Std jDE Std

f15

Worst 2588

00705

1505

0198

minus4281

0117

minus4975

0241

2241

0077

0635

0372Best 2366 minus4679 minus5792 minus72474 1972 minus0370Average 2490 minus4561 minus5408 minus63259 2140 0148Median 2506 minus4601 minus5456 minus63259 2136 0218

f16

Worst 1438

0058

minus1463

0253

minus2762

0026

minus3239

0032

minus2354

0042

minus3392

0433Best 1268 minus2296 minus2838 minus3334 minus2482 minus4646Average 1366 minus1987 minus2793 minus3282 minus2408 minus3868Median 1375 minus2016 minus2793 minus3284 minus2402 minus3751

f17

Worst minus22954

0643

minus25673

1721

minus70748

1908

minus79645

0131

minus47025

1403

minus63031

1942Best minus24885 minus32075 minus77054 minus80098 minus51440 minus68738Average minus23772 minus28641 minus74598 minus79996 minus48982 minus66539Median minus23731 minus28759 minus74907 minus80001 minus49035 minus66559

Table 4 Time complexity analysis of AFSA

Procedure of AFSA Time complexity(1) Initialization of N articial shes O(N)(2) Initialization of lsquoboardrsquo O(N)(3) Swarming behavior O(N2 + 2lowastN)(4) Chasing behavior O(2lowastN2 + 2lowastN)(5) Preying behavior O(TrynumberlowastN)(6) Judging of terminal condition O(1)(7) Information output of lsquoboardrsquo O(1)

Table 5 Time complexity analysis of LFFSA

Procedure of LFFSA Time complexity(1) Initialization of N articial shes O(N)(2) Initialization of lsquoboardrsquo O(N)(3) Chasing behavior O(2lowastN2 + 2lowastN)(4) Preying behavior O(trynumberlowastN)(5) Judging of terminal condition O(1)(6) Information output of lsquoboardrsquo O(1)

Table 6 Average running time of algorithms

NoRunning time (s)

AFSA FA FFSA LFFSA DE jDEf1 1023 747 185 753 464 608f2 1237 773 2111 710 432 653f3 1087 780 2073 896 412 825f4 1140 767 2049 1027 563 698f5 1053 766 1947 803 328 596f6 1972 610 2577 1637 771 1065f7 1053 743 1877 856 327 656f8 1233 803 2272 923 379 663f9 2083 720 2136 771 401 684f10 1627 846 2062 756 465 715f11 926 895 1374 635 363 667f12 965 534 1375 651 304 628f13 912 526 1436 1041 413 715f14 1395 574 1902 1031 648 986f15 945 539 1307 736 462 774f16 873 521 1388 1146 852 1887f17 1453 728 1841 596 332 644

β0

Valu

e

ndash8

ndash75

ndash7

ndash65

ndash6

ndash55

215050 1

(a)

γ

Valu

e

ndash63

ndash62

ndash61

ndash6

ndash59

ndash58

0 2 4 6 8 10

(b)

δ

Valu

e

ndash615ndash61

ndash605ndash6

ndash595ndash59

ndash585ndash58

215050 1

(c)

Trynumber

Valu

e

ndash64

ndash62

ndash6

ndash58

ndash56

ndash54

0 2 4 6 8 10

(d)

Figure 7 Test curves of parameters (a) Test curve of β0 (b) Testcurve of c (c) Test curve of δ (d) Test curve of Trynumber

Computational Intelligence and Neuroscience 11

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work is supported by National Natural ScienceFoundation of China (61463028) e authors wish to thankDr Yanliang Cui for his fruitful comments and suggestions

References

[1] X L Li Z J Shao J X Qian et al ldquoAn optimizing methodbased on autonomous animats fish-swarm algorithmrdquo Sys-tems Engineering-eory and Practice vol 22 pp 188ndash2002002 in Chinese

[2] C R Wang C L Zhou and J W Ma ldquoAn improved artificialfish-swarm algorithm and its application in feed-forwardneural networksrdquo in Proceedings of the Fourth InternationalConference on Machine Learning amp Cybernetics GuangzhouChina August 2005

[3] H C Tsai and Y H Lin ldquoModification of the fish swarmalgorithm with particle swarm optimization formulation andcommunication behaviorrdquo Applied Soft Computing vol 11no 8 pp 5367ndash5374 2011

[4] X Y Luan Z P Li and T Z Liu ldquoA novel attribute reductionalgorithm based on rough set and improved artificial fishswarm algorithmrdquo Neurocomputing vol 174 pp 522ndash5292016

[5] Y Wu X Z Gao Z Kai et al ldquoKnowledge-based artificialfish-swarm algorithmrdquo IFAC Proceedings Volumes vol 44no 1 pp 188ndash200 2011

[6] X T Hu H Q Zhang Z C Li et al ldquoA novel self-adaptationhybrid artificial fish-swarm algorithmrdquo IFAC ProceedingsVolumes vol 46 no 5 pp 583ndash588 2013

[7] K P Kumar B Saravanan and K S Swarup ldquoOptimizationof renewable energy sources in a microgrid using artificial fishswarm algorithmrdquo Energy Procedia vol 90 pp 107ndash113 2016

[8] Q He X T Hu H Ren et al ldquoA novel artificial fish swarmalgorithm for solving large-scale reliability-redundancy ap-plication problemrdquo ISA Transactions vol 59 pp 105ndash1132015

[9] Z Q Zhang K P Wang L X Zhu et al ldquoA Pareto improvedartificial fish swarm algorithm for solving a multi-objectivefuzzy disassembly line balancing problemrdquo Expert Systemswith Applications vol 86 pp 165ndash176 2017

[10] P K Singhal R Naresh and V Sharma ldquoBinary fish swarmalgorithm for profit-based unit commitment problem incompetitive electricity market with ramp rate constraintsrdquoGeneration Transmission and Distribution IET vol 9 no 13pp 1697ndash1707 2015

[11] X S Yang Nature-Inspired Metaheuristic Algorithmspp 83ndash96 Luniver Press London 2008

[12] S M Farahani B Nasiri A A Abshouri et al ldquoAn improvedfirefly algorithm with directed movementrdquo in Proceedings ofIEEE International Conference on Computer Science amp In-formation Technology Sichuan China June 2011

[13] L D S Coelho D L D A Bernert and V C Mariani ldquoAchaotic firefly algorithm applied to reliability-redundancy

optimizationrdquo Evolutionary Computation vol 30 pp 517ndash521 2011

[14] X S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo in Studies inComputational Intelligence vol 284 pp 101-111 SpringerBerlin Germany 2010

[15] S L Tilahun and C O Hong ldquoModified firefly algorithmrdquoJournal of Applied Mathematics vol 12 pp 2428ndash24392012

[16] K Jagatheesan B Anand S Samanta et al ldquoDesign ofa proportional-integral-derivative controller for an automaticgeneration control of multi-area power thermal systems usingfirefly algorithmrdquo IEEECAA Journal of Automatica Sinicapp 1ndash14 2016

[17] H Su Y Cai and Q Du ldquoFirefly-algorithm-inspiredframework with band selection and extreme learning ma-chine for hyperspectral image classificationrdquo IEEE Journal ofSelected Topics in Applied Earth Observations and RemoteSensing vol 10 no 1 pp 309ndash320 2016

[18] S Ranganathan M S Kalavathi and A R C ChristoberldquoSelf-adaptive firefly algorithm based multi-objectives formulti-type FACTS placementrdquo IET Generation Transmissionand Distribution vol 10 no 11 pp 188ndash200 2016

[19] D F Teshome C H Le Y W Lin et al ldquoA modified fireflyalgorithm for photovoltaic maximum power point trackingcontrol under partial shadingrdquo IEEE Journal of Emerging andSelected Topics in Power Electronics vol 5 no 2 pp 661ndash6712017

[20] M Alb P Alotto C Magele et al ldquoFirefly algorithm forfinding optimal shapes of electromagnetic devicesrdquo IEEETransactions on Magnetics vol 52 no 3 pp 1ndash4 2016

[21] A Mishra and V N K Gundavarapu ldquoLine utilisationfactor-based optimal allocation of IPFC and sizing usingfirefly algorithm for congestion managementrdquo GenerationTransmission and Distribution IET vol 10 no 1 pp 115ndash122 2016

[22] M H Horng ldquoVector quantization using the firefly algorithmfor image compressionrdquo Expert Systems with Applicationsvol 39 no 1 pp 078ndash1091 2012

[23] A Kazem E Sharifi F K Hussain et al ldquoSupport vectorregression with chaos-based firefly algorithm for stock marketprice forecastingrdquo Applied Soft Computing vol 13 no 2pp 947ndash958 2013

[24] L F He and S W Huang ldquoModified firefly algorithm basedmultilevel thresholding for color image segmentationrdquoNeurocomputing vol 240 pp 152ndash174 2017

[25] A H Gandomi X S Yang and A H Alavi ldquoMixed variablestructural optimization using Firefly Algorithmrdquo Computersand Structures vol 89 no 23-24 pp 2325ndash2336 2011

[26] M Alweshah and S Abdullah ldquoHybridizing firefly algorithmswith a probabilistic neural network for solving classificationproblemsrdquo Applied Soft Computing vol 35 pp 513ndash5242015

[27] R M Rizk-Allah E M Zaki A A El-Sawy et al ldquoHy-bridizing ant colony optimization with firefly algorithm forunconstrained optimization problemsrdquo Applied Mathematicsand Computation vol 224 pp 473ndash483 2013

[28] X S Yang S S S Hosseini and A H Gandomi ldquoFireflyalgorithm for solving non-convex economic dispatch prob-lems with valve loading effectrdquo Applied Soft Computingvol 12 no 3 pp 1180ndash1186 2012

[29] J Senthilnath S N Omkar and V Mani ldquoClustering usingfirefly algorithm performance studyrdquo Swarm and Evolu-tionary Computation vol 1 no 3 pp 164ndash171 2011

12 Computational Intelligence and Neuroscience

[30] T Kanimozhi and K Latha ldquoAn integrated approach to re-gion based image retrieval using firefly algorithm and supportvector machinerdquo Neurocomputing vol 151 pp 1099ndash11112015

[31] A Baykasoglu and F B Ozsoydan ldquoAdaptive firefly algorithmwith chaos for mechanical design optimization problemsrdquoApplied Soft Computing vol 36 pp 152ndash164 2015

[32] I Fister M Perc S M Kamal et al ldquoA review of chaos-basedfirefly algorithms perspectives and research challengesrdquoApplied Mathematics and Computation vol 252 pp 155ndash1652015

[33] I Fister I Fister X S Yang et al ldquoA comprehensive review offirefly algorithmsrdquo Swarm and Evolutionary Computationvol 13 pp 34ndash46 2013

[34] C T Brown L S Liebovitch and R Glendon ldquoLevy flights inDobe Jursquohoansi foraging patternsrdquo Human Ecology vol 35no 1 pp 129ndash138 2007

[35] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquoMathematics vol 226 no 2 pp 1830ndash1844 2012

[36] I Pavlyukevich ldquoCooling down Levy flightsrdquo Journal ofPhysics A Mathematical and eoretical vol 40 no 41pp 12299ndash12313 2007

[37] R Jensi and G W Jiji ldquoAn enhanced particle swarm opti-mization with Levy flight for global optimizationrdquo AppliedSoft Computing vol 43 pp 248ndash261 2016

[38] D Tang J Yang S Dong et al ldquoA Levy flight-based shuffledfrog-leaping algorithm and its applications for continuousoptimization problemsrdquo Applied Soft Computing vol 49pp 641ndash662 2016

[39] M Yahya and M P Saka ldquoConstruction site layout planningusing multi-objective artificial bee colony algorithmwith Levyflightsrdquo Automation in Construction vol 38 pp 14ndash29 2014

[40] C Blum and A Roli Hybrid Metaheuristics An IntroductionSpringer Berlin Germany 2008

[41] A M Reynolds and M A Frye ldquoFree-flight odor tracking indrosophila is consistent with an optimal intermittent scale-free searchrdquo PLos One vol 2 no 4 p e354 2007

[42] M F Shlesinger G M Zaslavsky and U Frisch Levy Flightsand Related Topics in Physics Springer Berlin HeidelbergGermany 1995

[43] M F Shlesinger ldquoMathematical physics search researchrdquoNature vol 443 no 7109 pp 281-282 2006

[44] S T Hsieh T Y Sun C C Liu et al ldquoEfficient populationutilization strategy for particle swarm optimizerrdquo IEEETransactions on Systems Man amp Cybernetics Part B vol 39no 2 pp 444ndash456 2009

[45] D Bratton and J Kennedy ldquoDefining a standard for particleswarm optimizationrdquo in Proceedings of IEEE Swarm In-telligence Symposium vol 107 pp 120ndash127 HonoluluHawaii April 2007

[46] U Manber Introduction to Algorithms A Creative ApproachAddison-Wesley Longman Publishing Co Inc Boston MAUSA 1989

Computational Intelligence and Neuroscience 13

Computer Games Technology

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Submit your manuscripts atwwwhindawicom

Step ρ middot Step + Stepmin (17)

ρ exp minus30 timesgen

Maxgen( )

s

( ) (18)

where Visualmin the minimum visual takes 0001Stepmin the minimum step takes 00002 ρ isa nonlinear weight and s represents an integer sgt 1Here s 3 Other parameters are the same as thosein Equation (11) e relationship between s and ρ isas shown in Figure 5

Besides dene the biggest distance between two articialshes as

maxD xmax minusxmin( )2 timesD

radic (19)

where xmax and xmin represent the upper bound and thelower bound of searching range respectively D denotes theD-dimension searching space e initial Visual is equal tomaxD and initial Step is equal to maxD8 en Visual andStep change dynamically according to Equations (16)ndash(18)

(e) Improvement 5 decrease of time complexitySwarming behavior in AFSA is excluded to decreasethe algorithm complexity

4 Numerical Simulation

41 Comparison of Convergence Accuracy LFFSA is vali-dated by numerical simulations Fish swarm algorithm withrey behavior (FFSA) dierential evolution (DE) algo-rithm self-adaptive dierential evolution (jDE) algorithmand the two basic algorithms (AFSA and FA) are comparedAll the algorithms are coded in Matlab 2014b e operatingsystem is windows 7 Simulation hardware is a PC with250GHz Inter Core i5 and 200GB Memory

Parameters shown in Table 1 are determined by trial anderror

e following benchmarks are chosen carefullyaccording to their features Functions Sphere Quartic andRosenbrock etc are simple unimodal problems FunctionsAckley Rastrigin and Schwefel etc are highly complexmultimodal problems with many local minima Schwefelfunction has a maximum value and other functions haveminimum values ese benchmarks are listed in Table 2[44 45] e solutions of 17 test functions obtained bydierent algorithms are compared To compare the con-vergence speed and accuracy of the algorithms clearly andcorrectly all functions are run 50 times for each algorithmrespectivelye results are averaged and plotted in Figure 6

From Figure 6 LFFSA can avoid local optimum and havebetter convergence accuracy compared with the other al-gorithms For AFSA and FA the solutions of most functionsare unsatisfactory the DE cannot nd ideal solutions of f3f4 f5 f6 f10 f11 f12 f14 and f15 the jDE has good accuracywhile solving some of those functions eg f1 f2 f3 and f8however solutions of f4 f10 f11 and f15 obtained by jDEare not so precise the LFFSA can obtain the ideal accuracy

for almost all functions although it cannot achieve a highprecision level like solutions of f2 obtained by jDE the FFSAis slightly worse than LFFSAe LFFSA outperforms jDE in10 benchmark functions while 2 functions are comparativeand 5 functions are worse

LFFSA is better than AFSA because Levy ight is able torestrict the movement step of AFSA to a very small areaaround the current position Furthermore the attractiondegree guides the sh moving Besides LFFSA can quicklylead the sh individual to the close-by optimal pointConsidering all the advantages discussed above the opti-mum solution can be found successfully by using LFFSAwhich outperforms the basic algorithms for all test functionsand outperforms jDE for several functions To observe thesearching capabilities of dierent algorithms directly theaverage median best and worst values obtained by dierentalgorithms are listed in Table 3 Results indicate that LFFSAcan nd ideal solutions and have a better robustness

42 Computational ComplexityAnalysis Time complexity isalso an important indicator in the analysis of algorithms Ifan algorithm is composed of several parts then its com-plexity is the sum of the complexities of these parts ealgorithm may consist of a loop executed many times andeach time is with a dierent complexity Time complexity of

Iteration0 500 1000 1500 2000

ρ

0

02

04

06

08

1

s = 3s = 12

s = 21s = 30

Figure 5 Value of ρ

Table 1 Parameter settings

Algorithms Parameters ValuesFA FFSA LFFSA β0 10FA FFSA LFFSA c 10AFSA FFSA LFFSA δ 0618AFSA FFSA LFFSA Trynumber 5DE jDE Scaling constant 05DE jDE Crossover constant 09All 6 algorithms Population 50

All 6 algorithms Maximum functionevaluations (FEs) 2 times 105

6 Computational Intelligence and Neuroscience

the algorithm is used to estimate the efficiency of the al-gorithm It is defined that the time complexity of the al-gorithm or the running time is O(f(n)) [46] Define N asthe population

In the definition of time complexity O(N2) and O(N)

are at different levels If the time complexity of one algorithmis O(N2) the time complexity of the other one is O(N) thenthe former algorithm is more complex In the other case ifthe time complexity of one algorithm is O(N2) while thetime complexity of the other one is O(N2 + N) theircomplexities are both O(N2)

e time complexity analysis of AFSA is provided inTable 4

From Table 4 the time complexity of AFSA is

O Maxgenlowast 3lowastN2

+ trynumberlowastN + 6lowastN1113872 11138731113872 1113873 (20)

Swarming behavior has N times of calculating conges-tion factor 1 time of judging and 1 time of movingerefore time complexity of swarming behavior isO(N2 + 2lowastN) Chasing behavior has N times of calculating

congestion factor N times of searching 1 time of judgingand 1 time of moving erefore time complexity of chasingbehavior is O(2lowastN2 + 2lowastN)

Time complexity analysis of LFFSA is listed in Table 5Due to the lack of swarming behavior time complexity

of LFFSA can be calculated as

O Maxgenlowast 2lowastN2

+ TrynumberlowastN + 4lowastN1113872 11138731113872 1113873 (21)

We can also obtain time complexity of FA

O Maxgenlowast N2

+ N1113872 11138731113872 1113873 (22)

A conclusion can be obtained that time complexities ofthe three algorithms are at the same level eir computa-tional complexities in the worst case are only the square ofthe training sample size

43 Experimental Complexity Analysis Time complexity isa rough estimate of time costemore accurate time cost ofan algorithm can only be validated by running it on

Table 2 Test functions

No Test functions Expression Optimum value Domain Df1 Sphere f(x) 1113936

Di1x

2i 0 (minus100 100)D 30

f2 Quartic f(x) 1113936Di1ix

4i 0 (minus128 128)D 30

f3 Ackley f(x) minus20 exp minus02

(1D)1113936Di1x

2i

1113969

minus exp[(1D)1113936Di1cos(2πxi)]1113882 1113883

+ 20 + e0 (minus32768 32768)D 30

f4 Rosenbrock f(x) 1113936Dminus1i1 100(xi+1 minusxi

2)2 + (1minusxi)2 0 (minus2048 2048)D 30

f5 Rastrigin1 f(x) 1113936Di1 xi

2 minus 10 cos(2πxi) + 101113864 1113865 0 (minus512 512)D 30

f6 Rastrigin2f(x) 1113936

Di1 yi

2 minus 10 cos(2πyi) + 101113864 11138650 (minus512 512)D 30

yi xi |xi|lt (12)

round(2xi)2 |xi|gt (12)1113896

f7 Schwefel f(x) 1113936Di1 xi middot sin

|xi|

11139681113864 1113865 4189829D (minus500 500)D 30

f8 Griewank f(x) (14000)1113936Di1x

2i minus1113937

Di1cos(xi

i

radic) + 1 0 (minus600 600)D 30

f9 Quadric f(x) 1113936Di1(1113936

ij1xj)

2 0 (minus100 100)D 30

f10 Schaffer1 f(x) 1113936Dminus1i1 ((sin2

xi+1

2 + xi2

1113968minus 05)1113864

(0001(xi+12 + xi

2) + 1)2) + 050 (minus100 100)D 30

f11 Schaffer2 f(x) 1113936Dminus1i1 (sin2

1113936Di1x

2i

1113969

minus 05)(0001(1113936Di1x

2i ) + 1)2 + 051113882 1113883 0 (minus100 100)D 30

f12 Maxmod f(x) max(|xi|) 0 (minus10 10)D 30

f13 Dixon and price (x1 minus 1)2 + 1113936Di1i(2xi

2 minusximinus1) 0 (minus10 10)D 30

f14 Powell f(x) 1113936D4i1 [(x4iminus 3 + 10x4iminus 2)

2 + 5(x4iminus 1 minusx4i)2 +

(x4iminus 2 minus 2x4iminus 1)2 + 10(x4iminus 3 minusx4i)

4]0 (minus4 5)D 28

f15 Zakharov f(x) 1113936Di1x

2i + (1113936

Di105ixi)

2 + (1113936Di105ixi)

4 0 (minus5 10)D 30

f16 Sin1 1113936Di1|xi sin(xi) + 01xi| 0 (minus10 10)D 30

f17 Sin2 f(x) minus1113936Di1sin(xi) sin20(ix2

i π) minus992784 (0 π)D 100

Computational Intelligence and Neuroscience 7

FEs104

log 1

0(f(x

))

ndash20

ndash15

ndash10

ndash5

0

5

10

FAAFSAFFSA

LFFSADEjDE

0 5 10 15 20

(a)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash40

ndash30

ndash20

ndash10

0

10

FEs1040 5 10 15 20

(b)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

FEs1040 5 10 15 20

(c)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash2

0

2

4

6

8

FEs1040 5 10 15 20

(d)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

2

3

FEs1040 5 10 15 20

(e)

log 1

0(f(x

))FAAFSAFFSA

LFFSADEjDE

ndash6

ndash4

ndash2

0

2

4

FEs1040 5 10 15 20

(f )

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

32

34

36

38

4

42

FEs1040 5 10 15 20

(g)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash10

ndash5

0

5

FEs1040 5 10 15 20

(h)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash15

ndash10

ndash5

0

5

FEs1040 5 10 15 20

(i)

Figure 6 Continued

8 Computational Intelligence and Neuroscience

computer Since dierent algorithms cannot reach the sameconvergence accuracy the test with xed convergence ac-curacy is not available erefore the test with max functionevaluations is conducted Running time of each algorithm iscounted by the explorer of MATLAB Parameter settings ofalgorithms are the same in Section 41 Average running timeof dierent algorithms is listed in Table 6 When function

evaluations are the same running speed of LFFSA is fasterthan that of AFSA while DE has the fastest running speedResults are quite in accord with those obtained by com-putational complexity analysis LFFSA and jDE are com-parative in experimental complexity Running time of FFSAis almost twice as much as that of LFFSA e improvementof LFFSA decreases time complexity to some extent

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash8

ndash6

ndash4

ndash2

0

2

FEs1040 5 10 15 20

(j)lo

g 10(

f(x))

FAAFSAFFSA

LFFSADEjDE

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

FEs1040 5 10 15 20

(k)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

FEs1040 5 10 15 20

(l)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash2

0

2

4

6

8

10

FEs1040 5 10 15 20

(m)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash5

0

5

FEs1040 5 10 15 20

(n)

log 1

0(f(x

))FAAFSAFFSA

LFFSADEjDE

ndash10

ndash5

0

5

10

FEs1040 5 10 15 20

(o)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

2

FEs1040 5 10 15 20

(p)

f(x)

FAAFSAFFSA

LFFSADEjDE

FEs1040 5 10 15 20

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

(q)

Figure 6 Iterative curves of test functions (a) f1 (b) f2 (c) f3 (d) f4 (e) f5 (f ) f6 (g) f7 (h) f8 (i) f9 (j) f10 (k) f11 (l) f12 (m) f13 (n) f14(o) f15 (p) f16 (q) f7

Computational Intelligence and Neuroscience 9

Table 3 Comparison of optimization results

No Items AFSA Std FA Std FFSA Std LFFSA Std DE Std jDE Std

f1

Worst 3014

0181

minus2798

0120

minus5132

0071

minus6743

0071

minus8544

0132

minus15021

0121Best 2402 minus3257 minus5233 minus8500 minus10371 minus17242Average 2656 minus3048 minus5145 minus7278 minus8924 minus16326Median 2651 minus3102 minus5193 minus6325 minus8586 minus16706

f2

Worst minus2322

0171

minus19245

0135

minus11178

0074

minus14207

0134

minus20506

0400

minus31644

0535Best minus4126 minus2813 minus11313 minus15585 minus21611 minus33250Average minus2812 minus2278 minus11273 minus14812 minus21054 minus32687Median minus2562 minus2197 minus11271 minus14834 minus20970 minus32844

f3

Worst 1386

0001

minus1776

0124

minus3908

0197

minus6121

0138

minus0027

0438

minus8934

0167Best 1307 minus2415 minus4647 minus6938 minus1281 minus9471Average 1587 minus2168 minus4147 minus6546 minus0049 minus9163Median 1586 minus2177 minus4225 minus6325 minus0035 minus9164

f4

Worst 2143

0087

22733

0097

minus20348

0125

minus1076

0376

1520

0003

1454

0025Best 1565 1043 minus2416 minus3405 1412 1363Average 1946 1476 minus2158 minus1946 1385 1312Median 1854 1385 minus2235 minus2325 1363 1287

f5

Worst 2310

0077

1864

0856

minus2846

0044

minus4385

0054

minus0579

0323

0898

6009Best 2096 1243 minus2982 minus5145 minus1591 minus12831Average 2236 1454 minus2946 minus4643 minus0999 minus3303Median 2136 1285 minus2435 minus4325 minus0963 0148

f6

Worst 2445

0038

1716

0133

minus2382

0076

minus3414

0048

1131

0042

0698

5049Best 2318 1255 minus2618 minus3606 1012 minus11404Average 2408 1571 minus2486 minus3489 1077 minus5466Median 2419 1601 minus2462 minus3487 1094 minus7632

f7

Worst 3815

0018

37846

0031

4186

736eminus 6

4156

434eminus 6

4099

296eminus 9

4087

403eminus 3Best 3945 3978 4099 4099 4099 4099Average 3813 3848 4099 4099 4099 4097Median 3736 3785 minus4099 minus4099 4099 4097

f8

Worst 0956

0133

minus0960

0223

minus5301

0051

minus6271

0055

minus989

0114

minusInf

0Best minus0644 minus0500 minus5444 minus6455 minus10255 minusInfAverage minus0735 minus0697 minus5357 minus6372 minus10071 minusInfMedian 0736 0685 minus5435 minus6325 minus10074 minusInf

f9

Worst minus11665

1057

minus976

1324

minus9347

0843

minus8695

1323

minus6848

1124

minus6131

2697Best minus14433 minus10574 minus12194 minus11937 minus10680 minus14831Average minus12786 minus10456 minus10764 minus10137 minus8178 minus9938Median 1136 1085 minus7435 minus6325 minus7745 minus9663

f10

Worst 0957

0131

0974

0223

minus5375

0049

minus6274

0056

0673

0032

0280

0110Best 0644 0497 minus5448 minus6486 0483 minus0022Average 0747 0649 minus5376 minus6348 0526 0230Median 1136 1085 minus7435 minus6325 0547 0211

f11

Worst minus0301

272eminus 05

minus0303

0009

minus4751

0071

minus5647

0085

minus0896

0103

minus1106

0135Best minus0301 minus0331 minus5011 minus5965 minus1107 minus1429Average minus0301 minus0313 minus4838 minus5804 minus0975 minus1364Median minus0301 minus0311 minus4835 minus5804 minus0896 minus1402

f12

Worst 0779

0037

minus1177

0163

minus2756

0035

minus3158

0037

minus0136

0052

minus1904

0221Best 0658 minus1638 minus2892 minus3287 minus0339 minus2526Average 0711 minus1388 minus2811 minus3221 minus0252 minus2291Median 0716 minus1376 minus2801 minus3216 0248 minus2374

f13

Worst 4286

0173

0574

0297

minus0602

0007

minus0602

0005

0039

0068

minus0176

538eminus 6Best 3706 minus0175 minus0605 minus0605 minus0162 minus0176Average 4147 minus0013 minus0603 minus0603 minus0088 minus0176Median 4213 minus0135 minus0603 minus6603 minus0086 minus0176

f14

Worst 2404

0084

0405

0318

minus3598

0117

minus4514

0101

0276

0211

minus2192

0401Best 2173 minus0576 minus4001 minus4869 minus0369 minus3409Average 2275 0069 minus3773 minus4656 minus0117 minus2867Median 2246 0133 minus3719 minus4651 minus0134 minus2924

10 Computational Intelligence and Neuroscience

44ParameterAnalysis ofLFFSA eeect of parameters onoptimization is analyzed in this section Taking Ackleyfunction as example Figure 7 shows the change of the ob-jective function value in the case of varying parametersTrynumber and β0 are proportional to the optimization resultTrynumber can impact the time complexity of the algorithmso the value should be appropriate not to aect the runningspeed e best value of c and δ is 25 and 12 respectively

5 Conclusion

LFFSA is proposed to improve the capability of AFSAwhich integrates the merits of both AFSA and FA Firstlythe searching characteristic of AFSA is studied by cal-culating the time complexity Secondly 17 benchmark testfunctions are used to verify LFFSA en time complexityof LFFSA is estimated Numerical results demonstrate thatLFFSA has a better performance in accuracy and speed ofoptimization to solve nonlinear optimization problemsthan the other test algorithms However the solutionobtained by LFFSA can be more precise and the way ofmodication could provide reference for those esectcientalgorithms eg DE and GWO

Table 3 Continued

No Items AFSA Std FA Std FFSA Std LFFSA Std DE Std jDE Std

f15

Worst 2588

00705

1505

0198

minus4281

0117

minus4975

0241

2241

0077

0635

0372Best 2366 minus4679 minus5792 minus72474 1972 minus0370Average 2490 minus4561 minus5408 minus63259 2140 0148Median 2506 minus4601 minus5456 minus63259 2136 0218

f16

Worst 1438

0058

minus1463

0253

minus2762

0026

minus3239

0032

minus2354

0042

minus3392

0433Best 1268 minus2296 minus2838 minus3334 minus2482 minus4646Average 1366 minus1987 minus2793 minus3282 minus2408 minus3868Median 1375 minus2016 minus2793 minus3284 minus2402 minus3751

f17

Worst minus22954

0643

minus25673

1721

minus70748

1908

minus79645

0131

minus47025

1403

minus63031

1942Best minus24885 minus32075 minus77054 minus80098 minus51440 minus68738Average minus23772 minus28641 minus74598 minus79996 minus48982 minus66539Median minus23731 minus28759 minus74907 minus80001 minus49035 minus66559

Table 4 Time complexity analysis of AFSA

Procedure of AFSA Time complexity(1) Initialization of N articial shes O(N)(2) Initialization of lsquoboardrsquo O(N)(3) Swarming behavior O(N2 + 2lowastN)(4) Chasing behavior O(2lowastN2 + 2lowastN)(5) Preying behavior O(TrynumberlowastN)(6) Judging of terminal condition O(1)(7) Information output of lsquoboardrsquo O(1)

Table 5 Time complexity analysis of LFFSA

Procedure of LFFSA Time complexity(1) Initialization of N articial shes O(N)(2) Initialization of lsquoboardrsquo O(N)(3) Chasing behavior O(2lowastN2 + 2lowastN)(4) Preying behavior O(trynumberlowastN)(5) Judging of terminal condition O(1)(6) Information output of lsquoboardrsquo O(1)

Table 6 Average running time of algorithms

NoRunning time (s)

AFSA FA FFSA LFFSA DE jDEf1 1023 747 185 753 464 608f2 1237 773 2111 710 432 653f3 1087 780 2073 896 412 825f4 1140 767 2049 1027 563 698f5 1053 766 1947 803 328 596f6 1972 610 2577 1637 771 1065f7 1053 743 1877 856 327 656f8 1233 803 2272 923 379 663f9 2083 720 2136 771 401 684f10 1627 846 2062 756 465 715f11 926 895 1374 635 363 667f12 965 534 1375 651 304 628f13 912 526 1436 1041 413 715f14 1395 574 1902 1031 648 986f15 945 539 1307 736 462 774f16 873 521 1388 1146 852 1887f17 1453 728 1841 596 332 644

β0

Valu

e

ndash8

ndash75

ndash7

ndash65

ndash6

ndash55

215050 1

(a)

γ

Valu

e

ndash63

ndash62

ndash61

ndash6

ndash59

ndash58

0 2 4 6 8 10

(b)

δ

Valu

e

ndash615ndash61

ndash605ndash6

ndash595ndash59

ndash585ndash58

215050 1

(c)

Trynumber

Valu

e

ndash64

ndash62

ndash6

ndash58

ndash56

ndash54

0 2 4 6 8 10

(d)

Figure 7 Test curves of parameters (a) Test curve of β0 (b) Testcurve of c (c) Test curve of δ (d) Test curve of Trynumber

Computational Intelligence and Neuroscience 11

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work is supported by National Natural ScienceFoundation of China (61463028) e authors wish to thankDr Yanliang Cui for his fruitful comments and suggestions

References

[1] X L Li Z J Shao J X Qian et al ldquoAn optimizing methodbased on autonomous animats fish-swarm algorithmrdquo Sys-tems Engineering-eory and Practice vol 22 pp 188ndash2002002 in Chinese

[2] C R Wang C L Zhou and J W Ma ldquoAn improved artificialfish-swarm algorithm and its application in feed-forwardneural networksrdquo in Proceedings of the Fourth InternationalConference on Machine Learning amp Cybernetics GuangzhouChina August 2005

[3] H C Tsai and Y H Lin ldquoModification of the fish swarmalgorithm with particle swarm optimization formulation andcommunication behaviorrdquo Applied Soft Computing vol 11no 8 pp 5367ndash5374 2011

[4] X Y Luan Z P Li and T Z Liu ldquoA novel attribute reductionalgorithm based on rough set and improved artificial fishswarm algorithmrdquo Neurocomputing vol 174 pp 522ndash5292016

[5] Y Wu X Z Gao Z Kai et al ldquoKnowledge-based artificialfish-swarm algorithmrdquo IFAC Proceedings Volumes vol 44no 1 pp 188ndash200 2011

[6] X T Hu H Q Zhang Z C Li et al ldquoA novel self-adaptationhybrid artificial fish-swarm algorithmrdquo IFAC ProceedingsVolumes vol 46 no 5 pp 583ndash588 2013

[7] K P Kumar B Saravanan and K S Swarup ldquoOptimizationof renewable energy sources in a microgrid using artificial fishswarm algorithmrdquo Energy Procedia vol 90 pp 107ndash113 2016

[8] Q He X T Hu H Ren et al ldquoA novel artificial fish swarmalgorithm for solving large-scale reliability-redundancy ap-plication problemrdquo ISA Transactions vol 59 pp 105ndash1132015

[9] Z Q Zhang K P Wang L X Zhu et al ldquoA Pareto improvedartificial fish swarm algorithm for solving a multi-objectivefuzzy disassembly line balancing problemrdquo Expert Systemswith Applications vol 86 pp 165ndash176 2017

[10] P K Singhal R Naresh and V Sharma ldquoBinary fish swarmalgorithm for profit-based unit commitment problem incompetitive electricity market with ramp rate constraintsrdquoGeneration Transmission and Distribution IET vol 9 no 13pp 1697ndash1707 2015

[11] X S Yang Nature-Inspired Metaheuristic Algorithmspp 83ndash96 Luniver Press London 2008

[12] S M Farahani B Nasiri A A Abshouri et al ldquoAn improvedfirefly algorithm with directed movementrdquo in Proceedings ofIEEE International Conference on Computer Science amp In-formation Technology Sichuan China June 2011

[13] L D S Coelho D L D A Bernert and V C Mariani ldquoAchaotic firefly algorithm applied to reliability-redundancy

optimizationrdquo Evolutionary Computation vol 30 pp 517ndash521 2011

[14] X S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo in Studies inComputational Intelligence vol 284 pp 101-111 SpringerBerlin Germany 2010

[15] S L Tilahun and C O Hong ldquoModified firefly algorithmrdquoJournal of Applied Mathematics vol 12 pp 2428ndash24392012

[16] K Jagatheesan B Anand S Samanta et al ldquoDesign ofa proportional-integral-derivative controller for an automaticgeneration control of multi-area power thermal systems usingfirefly algorithmrdquo IEEECAA Journal of Automatica Sinicapp 1ndash14 2016

[17] H Su Y Cai and Q Du ldquoFirefly-algorithm-inspiredframework with band selection and extreme learning ma-chine for hyperspectral image classificationrdquo IEEE Journal ofSelected Topics in Applied Earth Observations and RemoteSensing vol 10 no 1 pp 309ndash320 2016

[18] S Ranganathan M S Kalavathi and A R C ChristoberldquoSelf-adaptive firefly algorithm based multi-objectives formulti-type FACTS placementrdquo IET Generation Transmissionand Distribution vol 10 no 11 pp 188ndash200 2016

[19] D F Teshome C H Le Y W Lin et al ldquoA modified fireflyalgorithm for photovoltaic maximum power point trackingcontrol under partial shadingrdquo IEEE Journal of Emerging andSelected Topics in Power Electronics vol 5 no 2 pp 661ndash6712017

[20] M Alb P Alotto C Magele et al ldquoFirefly algorithm forfinding optimal shapes of electromagnetic devicesrdquo IEEETransactions on Magnetics vol 52 no 3 pp 1ndash4 2016

[21] A Mishra and V N K Gundavarapu ldquoLine utilisationfactor-based optimal allocation of IPFC and sizing usingfirefly algorithm for congestion managementrdquo GenerationTransmission and Distribution IET vol 10 no 1 pp 115ndash122 2016

[22] M H Horng ldquoVector quantization using the firefly algorithmfor image compressionrdquo Expert Systems with Applicationsvol 39 no 1 pp 078ndash1091 2012

[23] A Kazem E Sharifi F K Hussain et al ldquoSupport vectorregression with chaos-based firefly algorithm for stock marketprice forecastingrdquo Applied Soft Computing vol 13 no 2pp 947ndash958 2013

[24] L F He and S W Huang ldquoModified firefly algorithm basedmultilevel thresholding for color image segmentationrdquoNeurocomputing vol 240 pp 152ndash174 2017

[25] A H Gandomi X S Yang and A H Alavi ldquoMixed variablestructural optimization using Firefly Algorithmrdquo Computersand Structures vol 89 no 23-24 pp 2325ndash2336 2011

[26] M Alweshah and S Abdullah ldquoHybridizing firefly algorithmswith a probabilistic neural network for solving classificationproblemsrdquo Applied Soft Computing vol 35 pp 513ndash5242015

[27] R M Rizk-Allah E M Zaki A A El-Sawy et al ldquoHy-bridizing ant colony optimization with firefly algorithm forunconstrained optimization problemsrdquo Applied Mathematicsand Computation vol 224 pp 473ndash483 2013

[28] X S Yang S S S Hosseini and A H Gandomi ldquoFireflyalgorithm for solving non-convex economic dispatch prob-lems with valve loading effectrdquo Applied Soft Computingvol 12 no 3 pp 1180ndash1186 2012

[29] J Senthilnath S N Omkar and V Mani ldquoClustering usingfirefly algorithm performance studyrdquo Swarm and Evolu-tionary Computation vol 1 no 3 pp 164ndash171 2011

12 Computational Intelligence and Neuroscience

[30] T Kanimozhi and K Latha ldquoAn integrated approach to re-gion based image retrieval using firefly algorithm and supportvector machinerdquo Neurocomputing vol 151 pp 1099ndash11112015

[31] A Baykasoglu and F B Ozsoydan ldquoAdaptive firefly algorithmwith chaos for mechanical design optimization problemsrdquoApplied Soft Computing vol 36 pp 152ndash164 2015

[32] I Fister M Perc S M Kamal et al ldquoA review of chaos-basedfirefly algorithms perspectives and research challengesrdquoApplied Mathematics and Computation vol 252 pp 155ndash1652015

[33] I Fister I Fister X S Yang et al ldquoA comprehensive review offirefly algorithmsrdquo Swarm and Evolutionary Computationvol 13 pp 34ndash46 2013

[34] C T Brown L S Liebovitch and R Glendon ldquoLevy flights inDobe Jursquohoansi foraging patternsrdquo Human Ecology vol 35no 1 pp 129ndash138 2007

[35] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquoMathematics vol 226 no 2 pp 1830ndash1844 2012

[36] I Pavlyukevich ldquoCooling down Levy flightsrdquo Journal ofPhysics A Mathematical and eoretical vol 40 no 41pp 12299ndash12313 2007

[37] R Jensi and G W Jiji ldquoAn enhanced particle swarm opti-mization with Levy flight for global optimizationrdquo AppliedSoft Computing vol 43 pp 248ndash261 2016

[38] D Tang J Yang S Dong et al ldquoA Levy flight-based shuffledfrog-leaping algorithm and its applications for continuousoptimization problemsrdquo Applied Soft Computing vol 49pp 641ndash662 2016

[39] M Yahya and M P Saka ldquoConstruction site layout planningusing multi-objective artificial bee colony algorithmwith Levyflightsrdquo Automation in Construction vol 38 pp 14ndash29 2014

[40] C Blum and A Roli Hybrid Metaheuristics An IntroductionSpringer Berlin Germany 2008

[41] A M Reynolds and M A Frye ldquoFree-flight odor tracking indrosophila is consistent with an optimal intermittent scale-free searchrdquo PLos One vol 2 no 4 p e354 2007

[42] M F Shlesinger G M Zaslavsky and U Frisch Levy Flightsand Related Topics in Physics Springer Berlin HeidelbergGermany 1995

[43] M F Shlesinger ldquoMathematical physics search researchrdquoNature vol 443 no 7109 pp 281-282 2006

[44] S T Hsieh T Y Sun C C Liu et al ldquoEfficient populationutilization strategy for particle swarm optimizerrdquo IEEETransactions on Systems Man amp Cybernetics Part B vol 39no 2 pp 444ndash456 2009

[45] D Bratton and J Kennedy ldquoDefining a standard for particleswarm optimizationrdquo in Proceedings of IEEE Swarm In-telligence Symposium vol 107 pp 120ndash127 HonoluluHawaii April 2007

[46] U Manber Introduction to Algorithms A Creative ApproachAddison-Wesley Longman Publishing Co Inc Boston MAUSA 1989

Computational Intelligence and Neuroscience 13

Computer Games Technology

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Submit your manuscripts atwwwhindawicom

the algorithm is used to estimate the efficiency of the al-gorithm It is defined that the time complexity of the al-gorithm or the running time is O(f(n)) [46] Define N asthe population

In the definition of time complexity O(N2) and O(N)

are at different levels If the time complexity of one algorithmis O(N2) the time complexity of the other one is O(N) thenthe former algorithm is more complex In the other case ifthe time complexity of one algorithm is O(N2) while thetime complexity of the other one is O(N2 + N) theircomplexities are both O(N2)

e time complexity analysis of AFSA is provided inTable 4

From Table 4 the time complexity of AFSA is

O Maxgenlowast 3lowastN2

+ trynumberlowastN + 6lowastN1113872 11138731113872 1113873 (20)

Swarming behavior has N times of calculating conges-tion factor 1 time of judging and 1 time of movingerefore time complexity of swarming behavior isO(N2 + 2lowastN) Chasing behavior has N times of calculating

congestion factor N times of searching 1 time of judgingand 1 time of moving erefore time complexity of chasingbehavior is O(2lowastN2 + 2lowastN)

Time complexity analysis of LFFSA is listed in Table 5Due to the lack of swarming behavior time complexity

of LFFSA can be calculated as

O Maxgenlowast 2lowastN2

+ TrynumberlowastN + 4lowastN1113872 11138731113872 1113873 (21)

We can also obtain time complexity of FA

O Maxgenlowast N2

+ N1113872 11138731113872 1113873 (22)

A conclusion can be obtained that time complexities ofthe three algorithms are at the same level eir computa-tional complexities in the worst case are only the square ofthe training sample size

43 Experimental Complexity Analysis Time complexity isa rough estimate of time costemore accurate time cost ofan algorithm can only be validated by running it on

Table 2 Test functions

No Test functions Expression Optimum value Domain Df1 Sphere f(x) 1113936

Di1x

2i 0 (minus100 100)D 30

f2 Quartic f(x) 1113936Di1ix

4i 0 (minus128 128)D 30

f3 Ackley f(x) minus20 exp minus02

(1D)1113936Di1x

2i

1113969

minus exp[(1D)1113936Di1cos(2πxi)]1113882 1113883

+ 20 + e0 (minus32768 32768)D 30

f4 Rosenbrock f(x) 1113936Dminus1i1 100(xi+1 minusxi

2)2 + (1minusxi)2 0 (minus2048 2048)D 30

f5 Rastrigin1 f(x) 1113936Di1 xi

2 minus 10 cos(2πxi) + 101113864 1113865 0 (minus512 512)D 30

f6 Rastrigin2f(x) 1113936

Di1 yi

2 minus 10 cos(2πyi) + 101113864 11138650 (minus512 512)D 30

yi xi |xi|lt (12)

round(2xi)2 |xi|gt (12)1113896

f7 Schwefel f(x) 1113936Di1 xi middot sin

|xi|

11139681113864 1113865 4189829D (minus500 500)D 30

f8 Griewank f(x) (14000)1113936Di1x

2i minus1113937

Di1cos(xi

i

radic) + 1 0 (minus600 600)D 30

f9 Quadric f(x) 1113936Di1(1113936

ij1xj)

2 0 (minus100 100)D 30

f10 Schaffer1 f(x) 1113936Dminus1i1 ((sin2

xi+1

2 + xi2

1113968minus 05)1113864

(0001(xi+12 + xi

2) + 1)2) + 050 (minus100 100)D 30

f11 Schaffer2 f(x) 1113936Dminus1i1 (sin2

1113936Di1x

2i

1113969

minus 05)(0001(1113936Di1x

2i ) + 1)2 + 051113882 1113883 0 (minus100 100)D 30

f12 Maxmod f(x) max(|xi|) 0 (minus10 10)D 30

f13 Dixon and price (x1 minus 1)2 + 1113936Di1i(2xi

2 minusximinus1) 0 (minus10 10)D 30

f14 Powell f(x) 1113936D4i1 [(x4iminus 3 + 10x4iminus 2)

2 + 5(x4iminus 1 minusx4i)2 +

(x4iminus 2 minus 2x4iminus 1)2 + 10(x4iminus 3 minusx4i)

4]0 (minus4 5)D 28

f15 Zakharov f(x) 1113936Di1x

2i + (1113936

Di105ixi)

2 + (1113936Di105ixi)

4 0 (minus5 10)D 30

f16 Sin1 1113936Di1|xi sin(xi) + 01xi| 0 (minus10 10)D 30

f17 Sin2 f(x) minus1113936Di1sin(xi) sin20(ix2

i π) minus992784 (0 π)D 100

Computational Intelligence and Neuroscience 7

FEs104

log 1

0(f(x

))

ndash20

ndash15

ndash10

ndash5

0

5

10

FAAFSAFFSA

LFFSADEjDE

0 5 10 15 20

(a)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash40

ndash30

ndash20

ndash10

0

10

FEs1040 5 10 15 20

(b)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

FEs1040 5 10 15 20

(c)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash2

0

2

4

6

8

FEs1040 5 10 15 20

(d)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

2

3

FEs1040 5 10 15 20

(e)

log 1

0(f(x

))FAAFSAFFSA

LFFSADEjDE

ndash6

ndash4

ndash2

0

2

4

FEs1040 5 10 15 20

(f )

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

32

34

36

38

4

42

FEs1040 5 10 15 20

(g)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash10

ndash5

0

5

FEs1040 5 10 15 20

(h)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash15

ndash10

ndash5

0

5

FEs1040 5 10 15 20

(i)

Figure 6 Continued

8 Computational Intelligence and Neuroscience

computer Since dierent algorithms cannot reach the sameconvergence accuracy the test with xed convergence ac-curacy is not available erefore the test with max functionevaluations is conducted Running time of each algorithm iscounted by the explorer of MATLAB Parameter settings ofalgorithms are the same in Section 41 Average running timeof dierent algorithms is listed in Table 6 When function

evaluations are the same running speed of LFFSA is fasterthan that of AFSA while DE has the fastest running speedResults are quite in accord with those obtained by com-putational complexity analysis LFFSA and jDE are com-parative in experimental complexity Running time of FFSAis almost twice as much as that of LFFSA e improvementof LFFSA decreases time complexity to some extent

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash8

ndash6

ndash4

ndash2

0

2

FEs1040 5 10 15 20

(j)lo

g 10(

f(x))

FAAFSAFFSA

LFFSADEjDE

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

FEs1040 5 10 15 20

(k)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

FEs1040 5 10 15 20

(l)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash2

0

2

4

6

8

10

FEs1040 5 10 15 20

(m)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash5

0

5

FEs1040 5 10 15 20

(n)

log 1

0(f(x

))FAAFSAFFSA

LFFSADEjDE

ndash10

ndash5

0

5

10

FEs1040 5 10 15 20

(o)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

2

FEs1040 5 10 15 20

(p)

f(x)

FAAFSAFFSA

LFFSADEjDE

FEs1040 5 10 15 20

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

(q)

Figure 6 Iterative curves of test functions (a) f1 (b) f2 (c) f3 (d) f4 (e) f5 (f ) f6 (g) f7 (h) f8 (i) f9 (j) f10 (k) f11 (l) f12 (m) f13 (n) f14(o) f15 (p) f16 (q) f7

Computational Intelligence and Neuroscience 9

Table 3 Comparison of optimization results

No Items AFSA Std FA Std FFSA Std LFFSA Std DE Std jDE Std

f1

Worst 3014

0181

minus2798

0120

minus5132

0071

minus6743

0071

minus8544

0132

minus15021

0121Best 2402 minus3257 minus5233 minus8500 minus10371 minus17242Average 2656 minus3048 minus5145 minus7278 minus8924 minus16326Median 2651 minus3102 minus5193 minus6325 minus8586 minus16706

f2

Worst minus2322

0171

minus19245

0135

minus11178

0074

minus14207

0134

minus20506

0400

minus31644

0535Best minus4126 minus2813 minus11313 minus15585 minus21611 minus33250Average minus2812 minus2278 minus11273 minus14812 minus21054 minus32687Median minus2562 minus2197 minus11271 minus14834 minus20970 minus32844

f3

Worst 1386

0001

minus1776

0124

minus3908

0197

minus6121

0138

minus0027

0438

minus8934

0167Best 1307 minus2415 minus4647 minus6938 minus1281 minus9471Average 1587 minus2168 minus4147 minus6546 minus0049 minus9163Median 1586 minus2177 minus4225 minus6325 minus0035 minus9164

f4

Worst 2143

0087

22733

0097

minus20348

0125

minus1076

0376

1520

0003

1454

0025Best 1565 1043 minus2416 minus3405 1412 1363Average 1946 1476 minus2158 minus1946 1385 1312Median 1854 1385 minus2235 minus2325 1363 1287

f5

Worst 2310

0077

1864

0856

minus2846

0044

minus4385

0054

minus0579

0323

0898

6009Best 2096 1243 minus2982 minus5145 minus1591 minus12831Average 2236 1454 minus2946 minus4643 minus0999 minus3303Median 2136 1285 minus2435 minus4325 minus0963 0148

f6

Worst 2445

0038

1716

0133

minus2382

0076

minus3414

0048

1131

0042

0698

5049Best 2318 1255 minus2618 minus3606 1012 minus11404Average 2408 1571 minus2486 minus3489 1077 minus5466Median 2419 1601 minus2462 minus3487 1094 minus7632

f7

Worst 3815

0018

37846

0031

4186

736eminus 6

4156

434eminus 6

4099

296eminus 9

4087

403eminus 3Best 3945 3978 4099 4099 4099 4099Average 3813 3848 4099 4099 4099 4097Median 3736 3785 minus4099 minus4099 4099 4097

f8

Worst 0956

0133

minus0960

0223

minus5301

0051

minus6271

0055

minus989

0114

minusInf

0Best minus0644 minus0500 minus5444 minus6455 minus10255 minusInfAverage minus0735 minus0697 minus5357 minus6372 minus10071 minusInfMedian 0736 0685 minus5435 minus6325 minus10074 minusInf

f9

Worst minus11665

1057

minus976

1324

minus9347

0843

minus8695

1323

minus6848

1124

minus6131

2697Best minus14433 minus10574 minus12194 minus11937 minus10680 minus14831Average minus12786 minus10456 minus10764 minus10137 minus8178 minus9938Median 1136 1085 minus7435 minus6325 minus7745 minus9663

f10

Worst 0957

0131

0974

0223

minus5375

0049

minus6274

0056

0673

0032

0280

0110Best 0644 0497 minus5448 minus6486 0483 minus0022Average 0747 0649 minus5376 minus6348 0526 0230Median 1136 1085 minus7435 minus6325 0547 0211

f11

Worst minus0301

272eminus 05

minus0303

0009

minus4751

0071

minus5647

0085

minus0896

0103

minus1106

0135Best minus0301 minus0331 minus5011 minus5965 minus1107 minus1429Average minus0301 minus0313 minus4838 minus5804 minus0975 minus1364Median minus0301 minus0311 minus4835 minus5804 minus0896 minus1402

f12

Worst 0779

0037

minus1177

0163

minus2756

0035

minus3158

0037

minus0136

0052

minus1904

0221Best 0658 minus1638 minus2892 minus3287 minus0339 minus2526Average 0711 minus1388 minus2811 minus3221 minus0252 minus2291Median 0716 minus1376 minus2801 minus3216 0248 minus2374

f13

Worst 4286

0173

0574

0297

minus0602

0007

minus0602

0005

0039

0068

minus0176

538eminus 6Best 3706 minus0175 minus0605 minus0605 minus0162 minus0176Average 4147 minus0013 minus0603 minus0603 minus0088 minus0176Median 4213 minus0135 minus0603 minus6603 minus0086 minus0176

f14

Worst 2404

0084

0405

0318

minus3598

0117

minus4514

0101

0276

0211

minus2192

0401Best 2173 minus0576 minus4001 minus4869 minus0369 minus3409Average 2275 0069 minus3773 minus4656 minus0117 minus2867Median 2246 0133 minus3719 minus4651 minus0134 minus2924

10 Computational Intelligence and Neuroscience

44ParameterAnalysis ofLFFSA eeect of parameters onoptimization is analyzed in this section Taking Ackleyfunction as example Figure 7 shows the change of the ob-jective function value in the case of varying parametersTrynumber and β0 are proportional to the optimization resultTrynumber can impact the time complexity of the algorithmso the value should be appropriate not to aect the runningspeed e best value of c and δ is 25 and 12 respectively

5 Conclusion

LFFSA is proposed to improve the capability of AFSAwhich integrates the merits of both AFSA and FA Firstlythe searching characteristic of AFSA is studied by cal-culating the time complexity Secondly 17 benchmark testfunctions are used to verify LFFSA en time complexityof LFFSA is estimated Numerical results demonstrate thatLFFSA has a better performance in accuracy and speed ofoptimization to solve nonlinear optimization problemsthan the other test algorithms However the solutionobtained by LFFSA can be more precise and the way ofmodication could provide reference for those esectcientalgorithms eg DE and GWO

Table 3 Continued

No Items AFSA Std FA Std FFSA Std LFFSA Std DE Std jDE Std

f15

Worst 2588

00705

1505

0198

minus4281

0117

minus4975

0241

2241

0077

0635

0372Best 2366 minus4679 minus5792 minus72474 1972 minus0370Average 2490 minus4561 minus5408 minus63259 2140 0148Median 2506 minus4601 minus5456 minus63259 2136 0218

f16

Worst 1438

0058

minus1463

0253

minus2762

0026

minus3239

0032

minus2354

0042

minus3392

0433Best 1268 minus2296 minus2838 minus3334 minus2482 minus4646Average 1366 minus1987 minus2793 minus3282 minus2408 minus3868Median 1375 minus2016 minus2793 minus3284 minus2402 minus3751

f17

Worst minus22954

0643

minus25673

1721

minus70748

1908

minus79645

0131

minus47025

1403

minus63031

1942Best minus24885 minus32075 minus77054 minus80098 minus51440 minus68738Average minus23772 minus28641 minus74598 minus79996 minus48982 minus66539Median minus23731 minus28759 minus74907 minus80001 minus49035 minus66559

Table 4 Time complexity analysis of AFSA

Procedure of AFSA Time complexity(1) Initialization of N articial shes O(N)(2) Initialization of lsquoboardrsquo O(N)(3) Swarming behavior O(N2 + 2lowastN)(4) Chasing behavior O(2lowastN2 + 2lowastN)(5) Preying behavior O(TrynumberlowastN)(6) Judging of terminal condition O(1)(7) Information output of lsquoboardrsquo O(1)

Table 5 Time complexity analysis of LFFSA

Procedure of LFFSA Time complexity(1) Initialization of N articial shes O(N)(2) Initialization of lsquoboardrsquo O(N)(3) Chasing behavior O(2lowastN2 + 2lowastN)(4) Preying behavior O(trynumberlowastN)(5) Judging of terminal condition O(1)(6) Information output of lsquoboardrsquo O(1)

Table 6 Average running time of algorithms

NoRunning time (s)

AFSA FA FFSA LFFSA DE jDEf1 1023 747 185 753 464 608f2 1237 773 2111 710 432 653f3 1087 780 2073 896 412 825f4 1140 767 2049 1027 563 698f5 1053 766 1947 803 328 596f6 1972 610 2577 1637 771 1065f7 1053 743 1877 856 327 656f8 1233 803 2272 923 379 663f9 2083 720 2136 771 401 684f10 1627 846 2062 756 465 715f11 926 895 1374 635 363 667f12 965 534 1375 651 304 628f13 912 526 1436 1041 413 715f14 1395 574 1902 1031 648 986f15 945 539 1307 736 462 774f16 873 521 1388 1146 852 1887f17 1453 728 1841 596 332 644

β0

Valu

e

ndash8

ndash75

ndash7

ndash65

ndash6

ndash55

215050 1

(a)

γ

Valu

e

ndash63

ndash62

ndash61

ndash6

ndash59

ndash58

0 2 4 6 8 10

(b)

δ

Valu

e

ndash615ndash61

ndash605ndash6

ndash595ndash59

ndash585ndash58

215050 1

(c)

Trynumber

Valu

e

ndash64

ndash62

ndash6

ndash58

ndash56

ndash54

0 2 4 6 8 10

(d)

Figure 7 Test curves of parameters (a) Test curve of β0 (b) Testcurve of c (c) Test curve of δ (d) Test curve of Trynumber

Computational Intelligence and Neuroscience 11

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work is supported by National Natural ScienceFoundation of China (61463028) e authors wish to thankDr Yanliang Cui for his fruitful comments and suggestions

References

[1] X L Li Z J Shao J X Qian et al ldquoAn optimizing methodbased on autonomous animats fish-swarm algorithmrdquo Sys-tems Engineering-eory and Practice vol 22 pp 188ndash2002002 in Chinese

[2] C R Wang C L Zhou and J W Ma ldquoAn improved artificialfish-swarm algorithm and its application in feed-forwardneural networksrdquo in Proceedings of the Fourth InternationalConference on Machine Learning amp Cybernetics GuangzhouChina August 2005

[3] H C Tsai and Y H Lin ldquoModification of the fish swarmalgorithm with particle swarm optimization formulation andcommunication behaviorrdquo Applied Soft Computing vol 11no 8 pp 5367ndash5374 2011

[4] X Y Luan Z P Li and T Z Liu ldquoA novel attribute reductionalgorithm based on rough set and improved artificial fishswarm algorithmrdquo Neurocomputing vol 174 pp 522ndash5292016

[5] Y Wu X Z Gao Z Kai et al ldquoKnowledge-based artificialfish-swarm algorithmrdquo IFAC Proceedings Volumes vol 44no 1 pp 188ndash200 2011

[6] X T Hu H Q Zhang Z C Li et al ldquoA novel self-adaptationhybrid artificial fish-swarm algorithmrdquo IFAC ProceedingsVolumes vol 46 no 5 pp 583ndash588 2013

[7] K P Kumar B Saravanan and K S Swarup ldquoOptimizationof renewable energy sources in a microgrid using artificial fishswarm algorithmrdquo Energy Procedia vol 90 pp 107ndash113 2016

[8] Q He X T Hu H Ren et al ldquoA novel artificial fish swarmalgorithm for solving large-scale reliability-redundancy ap-plication problemrdquo ISA Transactions vol 59 pp 105ndash1132015

[9] Z Q Zhang K P Wang L X Zhu et al ldquoA Pareto improvedartificial fish swarm algorithm for solving a multi-objectivefuzzy disassembly line balancing problemrdquo Expert Systemswith Applications vol 86 pp 165ndash176 2017

[10] P K Singhal R Naresh and V Sharma ldquoBinary fish swarmalgorithm for profit-based unit commitment problem incompetitive electricity market with ramp rate constraintsrdquoGeneration Transmission and Distribution IET vol 9 no 13pp 1697ndash1707 2015

[11] X S Yang Nature-Inspired Metaheuristic Algorithmspp 83ndash96 Luniver Press London 2008

[12] S M Farahani B Nasiri A A Abshouri et al ldquoAn improvedfirefly algorithm with directed movementrdquo in Proceedings ofIEEE International Conference on Computer Science amp In-formation Technology Sichuan China June 2011

[13] L D S Coelho D L D A Bernert and V C Mariani ldquoAchaotic firefly algorithm applied to reliability-redundancy

optimizationrdquo Evolutionary Computation vol 30 pp 517ndash521 2011

[14] X S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo in Studies inComputational Intelligence vol 284 pp 101-111 SpringerBerlin Germany 2010

[15] S L Tilahun and C O Hong ldquoModified firefly algorithmrdquoJournal of Applied Mathematics vol 12 pp 2428ndash24392012

[16] K Jagatheesan B Anand S Samanta et al ldquoDesign ofa proportional-integral-derivative controller for an automaticgeneration control of multi-area power thermal systems usingfirefly algorithmrdquo IEEECAA Journal of Automatica Sinicapp 1ndash14 2016

[17] H Su Y Cai and Q Du ldquoFirefly-algorithm-inspiredframework with band selection and extreme learning ma-chine for hyperspectral image classificationrdquo IEEE Journal ofSelected Topics in Applied Earth Observations and RemoteSensing vol 10 no 1 pp 309ndash320 2016

[18] S Ranganathan M S Kalavathi and A R C ChristoberldquoSelf-adaptive firefly algorithm based multi-objectives formulti-type FACTS placementrdquo IET Generation Transmissionand Distribution vol 10 no 11 pp 188ndash200 2016

[19] D F Teshome C H Le Y W Lin et al ldquoA modified fireflyalgorithm for photovoltaic maximum power point trackingcontrol under partial shadingrdquo IEEE Journal of Emerging andSelected Topics in Power Electronics vol 5 no 2 pp 661ndash6712017

[20] M Alb P Alotto C Magele et al ldquoFirefly algorithm forfinding optimal shapes of electromagnetic devicesrdquo IEEETransactions on Magnetics vol 52 no 3 pp 1ndash4 2016

[21] A Mishra and V N K Gundavarapu ldquoLine utilisationfactor-based optimal allocation of IPFC and sizing usingfirefly algorithm for congestion managementrdquo GenerationTransmission and Distribution IET vol 10 no 1 pp 115ndash122 2016

[22] M H Horng ldquoVector quantization using the firefly algorithmfor image compressionrdquo Expert Systems with Applicationsvol 39 no 1 pp 078ndash1091 2012

[23] A Kazem E Sharifi F K Hussain et al ldquoSupport vectorregression with chaos-based firefly algorithm for stock marketprice forecastingrdquo Applied Soft Computing vol 13 no 2pp 947ndash958 2013

[24] L F He and S W Huang ldquoModified firefly algorithm basedmultilevel thresholding for color image segmentationrdquoNeurocomputing vol 240 pp 152ndash174 2017

[25] A H Gandomi X S Yang and A H Alavi ldquoMixed variablestructural optimization using Firefly Algorithmrdquo Computersand Structures vol 89 no 23-24 pp 2325ndash2336 2011

[26] M Alweshah and S Abdullah ldquoHybridizing firefly algorithmswith a probabilistic neural network for solving classificationproblemsrdquo Applied Soft Computing vol 35 pp 513ndash5242015

[27] R M Rizk-Allah E M Zaki A A El-Sawy et al ldquoHy-bridizing ant colony optimization with firefly algorithm forunconstrained optimization problemsrdquo Applied Mathematicsand Computation vol 224 pp 473ndash483 2013

[28] X S Yang S S S Hosseini and A H Gandomi ldquoFireflyalgorithm for solving non-convex economic dispatch prob-lems with valve loading effectrdquo Applied Soft Computingvol 12 no 3 pp 1180ndash1186 2012

[29] J Senthilnath S N Omkar and V Mani ldquoClustering usingfirefly algorithm performance studyrdquo Swarm and Evolu-tionary Computation vol 1 no 3 pp 164ndash171 2011

12 Computational Intelligence and Neuroscience

[30] T Kanimozhi and K Latha ldquoAn integrated approach to re-gion based image retrieval using firefly algorithm and supportvector machinerdquo Neurocomputing vol 151 pp 1099ndash11112015

[31] A Baykasoglu and F B Ozsoydan ldquoAdaptive firefly algorithmwith chaos for mechanical design optimization problemsrdquoApplied Soft Computing vol 36 pp 152ndash164 2015

[32] I Fister M Perc S M Kamal et al ldquoA review of chaos-basedfirefly algorithms perspectives and research challengesrdquoApplied Mathematics and Computation vol 252 pp 155ndash1652015

[33] I Fister I Fister X S Yang et al ldquoA comprehensive review offirefly algorithmsrdquo Swarm and Evolutionary Computationvol 13 pp 34ndash46 2013

[34] C T Brown L S Liebovitch and R Glendon ldquoLevy flights inDobe Jursquohoansi foraging patternsrdquo Human Ecology vol 35no 1 pp 129ndash138 2007

[35] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquoMathematics vol 226 no 2 pp 1830ndash1844 2012

[36] I Pavlyukevich ldquoCooling down Levy flightsrdquo Journal ofPhysics A Mathematical and eoretical vol 40 no 41pp 12299ndash12313 2007

[37] R Jensi and G W Jiji ldquoAn enhanced particle swarm opti-mization with Levy flight for global optimizationrdquo AppliedSoft Computing vol 43 pp 248ndash261 2016

[38] D Tang J Yang S Dong et al ldquoA Levy flight-based shuffledfrog-leaping algorithm and its applications for continuousoptimization problemsrdquo Applied Soft Computing vol 49pp 641ndash662 2016

[39] M Yahya and M P Saka ldquoConstruction site layout planningusing multi-objective artificial bee colony algorithmwith Levyflightsrdquo Automation in Construction vol 38 pp 14ndash29 2014

[40] C Blum and A Roli Hybrid Metaheuristics An IntroductionSpringer Berlin Germany 2008

[41] A M Reynolds and M A Frye ldquoFree-flight odor tracking indrosophila is consistent with an optimal intermittent scale-free searchrdquo PLos One vol 2 no 4 p e354 2007

[42] M F Shlesinger G M Zaslavsky and U Frisch Levy Flightsand Related Topics in Physics Springer Berlin HeidelbergGermany 1995

[43] M F Shlesinger ldquoMathematical physics search researchrdquoNature vol 443 no 7109 pp 281-282 2006

[44] S T Hsieh T Y Sun C C Liu et al ldquoEfficient populationutilization strategy for particle swarm optimizerrdquo IEEETransactions on Systems Man amp Cybernetics Part B vol 39no 2 pp 444ndash456 2009

[45] D Bratton and J Kennedy ldquoDefining a standard for particleswarm optimizationrdquo in Proceedings of IEEE Swarm In-telligence Symposium vol 107 pp 120ndash127 HonoluluHawaii April 2007

[46] U Manber Introduction to Algorithms A Creative ApproachAddison-Wesley Longman Publishing Co Inc Boston MAUSA 1989

Computational Intelligence and Neuroscience 13

Computer Games Technology

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Submit your manuscripts atwwwhindawicom

FEs104

log 1

0(f(x

))

ndash20

ndash15

ndash10

ndash5

0

5

10

FAAFSAFFSA

LFFSADEjDE

0 5 10 15 20

(a)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash40

ndash30

ndash20

ndash10

0

10

FEs1040 5 10 15 20

(b)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

FEs1040 5 10 15 20

(c)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash2

0

2

4

6

8

FEs1040 5 10 15 20

(d)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

2

3

FEs1040 5 10 15 20

(e)

log 1

0(f(x

))FAAFSAFFSA

LFFSADEjDE

ndash6

ndash4

ndash2

0

2

4

FEs1040 5 10 15 20

(f )

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

32

34

36

38

4

42

FEs1040 5 10 15 20

(g)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash10

ndash5

0

5

FEs1040 5 10 15 20

(h)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash15

ndash10

ndash5

0

5

FEs1040 5 10 15 20

(i)

Figure 6 Continued

8 Computational Intelligence and Neuroscience

computer Since dierent algorithms cannot reach the sameconvergence accuracy the test with xed convergence ac-curacy is not available erefore the test with max functionevaluations is conducted Running time of each algorithm iscounted by the explorer of MATLAB Parameter settings ofalgorithms are the same in Section 41 Average running timeof dierent algorithms is listed in Table 6 When function

evaluations are the same running speed of LFFSA is fasterthan that of AFSA while DE has the fastest running speedResults are quite in accord with those obtained by com-putational complexity analysis LFFSA and jDE are com-parative in experimental complexity Running time of FFSAis almost twice as much as that of LFFSA e improvementof LFFSA decreases time complexity to some extent

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash8

ndash6

ndash4

ndash2

0

2

FEs1040 5 10 15 20

(j)lo

g 10(

f(x))

FAAFSAFFSA

LFFSADEjDE

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

FEs1040 5 10 15 20

(k)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

FEs1040 5 10 15 20

(l)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash2

0

2

4

6

8

10

FEs1040 5 10 15 20

(m)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash5

0

5

FEs1040 5 10 15 20

(n)

log 1

0(f(x

))FAAFSAFFSA

LFFSADEjDE

ndash10

ndash5

0

5

10

FEs1040 5 10 15 20

(o)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

2

FEs1040 5 10 15 20

(p)

f(x)

FAAFSAFFSA

LFFSADEjDE

FEs1040 5 10 15 20

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

(q)

Figure 6 Iterative curves of test functions (a) f1 (b) f2 (c) f3 (d) f4 (e) f5 (f ) f6 (g) f7 (h) f8 (i) f9 (j) f10 (k) f11 (l) f12 (m) f13 (n) f14(o) f15 (p) f16 (q) f7

Computational Intelligence and Neuroscience 9

Table 3 Comparison of optimization results

No Items AFSA Std FA Std FFSA Std LFFSA Std DE Std jDE Std

f1

Worst 3014

0181

minus2798

0120

minus5132

0071

minus6743

0071

minus8544

0132

minus15021

0121Best 2402 minus3257 minus5233 minus8500 minus10371 minus17242Average 2656 minus3048 minus5145 minus7278 minus8924 minus16326Median 2651 minus3102 minus5193 minus6325 minus8586 minus16706

f2

Worst minus2322

0171

minus19245

0135

minus11178

0074

minus14207

0134

minus20506

0400

minus31644

0535Best minus4126 minus2813 minus11313 minus15585 minus21611 minus33250Average minus2812 minus2278 minus11273 minus14812 minus21054 minus32687Median minus2562 minus2197 minus11271 minus14834 minus20970 minus32844

f3

Worst 1386

0001

minus1776

0124

minus3908

0197

minus6121

0138

minus0027

0438

minus8934

0167Best 1307 minus2415 minus4647 minus6938 minus1281 minus9471Average 1587 minus2168 minus4147 minus6546 minus0049 minus9163Median 1586 minus2177 minus4225 minus6325 minus0035 minus9164

f4

Worst 2143

0087

22733

0097

minus20348

0125

minus1076

0376

1520

0003

1454

0025Best 1565 1043 minus2416 minus3405 1412 1363Average 1946 1476 minus2158 minus1946 1385 1312Median 1854 1385 minus2235 minus2325 1363 1287

f5

Worst 2310

0077

1864

0856

minus2846

0044

minus4385

0054

minus0579

0323

0898

6009Best 2096 1243 minus2982 minus5145 minus1591 minus12831Average 2236 1454 minus2946 minus4643 minus0999 minus3303Median 2136 1285 minus2435 minus4325 minus0963 0148

f6

Worst 2445

0038

1716

0133

minus2382

0076

minus3414

0048

1131

0042

0698

5049Best 2318 1255 minus2618 minus3606 1012 minus11404Average 2408 1571 minus2486 minus3489 1077 minus5466Median 2419 1601 minus2462 minus3487 1094 minus7632

f7

Worst 3815

0018

37846

0031

4186

736eminus 6

4156

434eminus 6

4099

296eminus 9

4087

403eminus 3Best 3945 3978 4099 4099 4099 4099Average 3813 3848 4099 4099 4099 4097Median 3736 3785 minus4099 minus4099 4099 4097

f8

Worst 0956

0133

minus0960

0223

minus5301

0051

minus6271

0055

minus989

0114

minusInf

0Best minus0644 minus0500 minus5444 minus6455 minus10255 minusInfAverage minus0735 minus0697 minus5357 minus6372 minus10071 minusInfMedian 0736 0685 minus5435 minus6325 minus10074 minusInf

f9

Worst minus11665

1057

minus976

1324

minus9347

0843

minus8695

1323

minus6848

1124

minus6131

2697Best minus14433 minus10574 minus12194 minus11937 minus10680 minus14831Average minus12786 minus10456 minus10764 minus10137 minus8178 minus9938Median 1136 1085 minus7435 minus6325 minus7745 minus9663

f10

Worst 0957

0131

0974

0223

minus5375

0049

minus6274

0056

0673

0032

0280

0110Best 0644 0497 minus5448 minus6486 0483 minus0022Average 0747 0649 minus5376 minus6348 0526 0230Median 1136 1085 minus7435 minus6325 0547 0211

f11

Worst minus0301

272eminus 05

minus0303

0009

minus4751

0071

minus5647

0085

minus0896

0103

minus1106

0135Best minus0301 minus0331 minus5011 minus5965 minus1107 minus1429Average minus0301 minus0313 minus4838 minus5804 minus0975 minus1364Median minus0301 minus0311 minus4835 minus5804 minus0896 minus1402

f12

Worst 0779

0037

minus1177

0163

minus2756

0035

minus3158

0037

minus0136

0052

minus1904

0221Best 0658 minus1638 minus2892 minus3287 minus0339 minus2526Average 0711 minus1388 minus2811 minus3221 minus0252 minus2291Median 0716 minus1376 minus2801 minus3216 0248 minus2374

f13

Worst 4286

0173

0574

0297

minus0602

0007

minus0602

0005

0039

0068

minus0176

538eminus 6Best 3706 minus0175 minus0605 minus0605 minus0162 minus0176Average 4147 minus0013 minus0603 minus0603 minus0088 minus0176Median 4213 minus0135 minus0603 minus6603 minus0086 minus0176

f14

Worst 2404

0084

0405

0318

minus3598

0117

minus4514

0101

0276

0211

minus2192

0401Best 2173 minus0576 minus4001 minus4869 minus0369 minus3409Average 2275 0069 minus3773 minus4656 minus0117 minus2867Median 2246 0133 minus3719 minus4651 minus0134 minus2924

10 Computational Intelligence and Neuroscience

44ParameterAnalysis ofLFFSA eeect of parameters onoptimization is analyzed in this section Taking Ackleyfunction as example Figure 7 shows the change of the ob-jective function value in the case of varying parametersTrynumber and β0 are proportional to the optimization resultTrynumber can impact the time complexity of the algorithmso the value should be appropriate not to aect the runningspeed e best value of c and δ is 25 and 12 respectively

5 Conclusion

LFFSA is proposed to improve the capability of AFSAwhich integrates the merits of both AFSA and FA Firstlythe searching characteristic of AFSA is studied by cal-culating the time complexity Secondly 17 benchmark testfunctions are used to verify LFFSA en time complexityof LFFSA is estimated Numerical results demonstrate thatLFFSA has a better performance in accuracy and speed ofoptimization to solve nonlinear optimization problemsthan the other test algorithms However the solutionobtained by LFFSA can be more precise and the way ofmodication could provide reference for those esectcientalgorithms eg DE and GWO

Table 3 Continued

No Items AFSA Std FA Std FFSA Std LFFSA Std DE Std jDE Std

f15

Worst 2588

00705

1505

0198

minus4281

0117

minus4975

0241

2241

0077

0635

0372Best 2366 minus4679 minus5792 minus72474 1972 minus0370Average 2490 minus4561 minus5408 minus63259 2140 0148Median 2506 minus4601 minus5456 minus63259 2136 0218

f16

Worst 1438

0058

minus1463

0253

minus2762

0026

minus3239

0032

minus2354

0042

minus3392

0433Best 1268 minus2296 minus2838 minus3334 minus2482 minus4646Average 1366 minus1987 minus2793 minus3282 minus2408 minus3868Median 1375 minus2016 minus2793 minus3284 minus2402 minus3751

f17

Worst minus22954

0643

minus25673

1721

minus70748

1908

minus79645

0131

minus47025

1403

minus63031

1942Best minus24885 minus32075 minus77054 minus80098 minus51440 minus68738Average minus23772 minus28641 minus74598 minus79996 minus48982 minus66539Median minus23731 minus28759 minus74907 minus80001 minus49035 minus66559

Table 4 Time complexity analysis of AFSA

Procedure of AFSA Time complexity(1) Initialization of N articial shes O(N)(2) Initialization of lsquoboardrsquo O(N)(3) Swarming behavior O(N2 + 2lowastN)(4) Chasing behavior O(2lowastN2 + 2lowastN)(5) Preying behavior O(TrynumberlowastN)(6) Judging of terminal condition O(1)(7) Information output of lsquoboardrsquo O(1)

Table 5 Time complexity analysis of LFFSA

Procedure of LFFSA Time complexity(1) Initialization of N articial shes O(N)(2) Initialization of lsquoboardrsquo O(N)(3) Chasing behavior O(2lowastN2 + 2lowastN)(4) Preying behavior O(trynumberlowastN)(5) Judging of terminal condition O(1)(6) Information output of lsquoboardrsquo O(1)

Table 6 Average running time of algorithms

NoRunning time (s)

AFSA FA FFSA LFFSA DE jDEf1 1023 747 185 753 464 608f2 1237 773 2111 710 432 653f3 1087 780 2073 896 412 825f4 1140 767 2049 1027 563 698f5 1053 766 1947 803 328 596f6 1972 610 2577 1637 771 1065f7 1053 743 1877 856 327 656f8 1233 803 2272 923 379 663f9 2083 720 2136 771 401 684f10 1627 846 2062 756 465 715f11 926 895 1374 635 363 667f12 965 534 1375 651 304 628f13 912 526 1436 1041 413 715f14 1395 574 1902 1031 648 986f15 945 539 1307 736 462 774f16 873 521 1388 1146 852 1887f17 1453 728 1841 596 332 644

β0

Valu

e

ndash8

ndash75

ndash7

ndash65

ndash6

ndash55

215050 1

(a)

γ

Valu

e

ndash63

ndash62

ndash61

ndash6

ndash59

ndash58

0 2 4 6 8 10

(b)

δ

Valu

e

ndash615ndash61

ndash605ndash6

ndash595ndash59

ndash585ndash58

215050 1

(c)

Trynumber

Valu

e

ndash64

ndash62

ndash6

ndash58

ndash56

ndash54

0 2 4 6 8 10

(d)

Figure 7 Test curves of parameters (a) Test curve of β0 (b) Testcurve of c (c) Test curve of δ (d) Test curve of Trynumber

Computational Intelligence and Neuroscience 11

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work is supported by National Natural ScienceFoundation of China (61463028) e authors wish to thankDr Yanliang Cui for his fruitful comments and suggestions

References

[1] X L Li Z J Shao J X Qian et al ldquoAn optimizing methodbased on autonomous animats fish-swarm algorithmrdquo Sys-tems Engineering-eory and Practice vol 22 pp 188ndash2002002 in Chinese

[2] C R Wang C L Zhou and J W Ma ldquoAn improved artificialfish-swarm algorithm and its application in feed-forwardneural networksrdquo in Proceedings of the Fourth InternationalConference on Machine Learning amp Cybernetics GuangzhouChina August 2005

[3] H C Tsai and Y H Lin ldquoModification of the fish swarmalgorithm with particle swarm optimization formulation andcommunication behaviorrdquo Applied Soft Computing vol 11no 8 pp 5367ndash5374 2011

[4] X Y Luan Z P Li and T Z Liu ldquoA novel attribute reductionalgorithm based on rough set and improved artificial fishswarm algorithmrdquo Neurocomputing vol 174 pp 522ndash5292016

[5] Y Wu X Z Gao Z Kai et al ldquoKnowledge-based artificialfish-swarm algorithmrdquo IFAC Proceedings Volumes vol 44no 1 pp 188ndash200 2011

[6] X T Hu H Q Zhang Z C Li et al ldquoA novel self-adaptationhybrid artificial fish-swarm algorithmrdquo IFAC ProceedingsVolumes vol 46 no 5 pp 583ndash588 2013

[7] K P Kumar B Saravanan and K S Swarup ldquoOptimizationof renewable energy sources in a microgrid using artificial fishswarm algorithmrdquo Energy Procedia vol 90 pp 107ndash113 2016

[8] Q He X T Hu H Ren et al ldquoA novel artificial fish swarmalgorithm for solving large-scale reliability-redundancy ap-plication problemrdquo ISA Transactions vol 59 pp 105ndash1132015

[9] Z Q Zhang K P Wang L X Zhu et al ldquoA Pareto improvedartificial fish swarm algorithm for solving a multi-objectivefuzzy disassembly line balancing problemrdquo Expert Systemswith Applications vol 86 pp 165ndash176 2017

[10] P K Singhal R Naresh and V Sharma ldquoBinary fish swarmalgorithm for profit-based unit commitment problem incompetitive electricity market with ramp rate constraintsrdquoGeneration Transmission and Distribution IET vol 9 no 13pp 1697ndash1707 2015

[11] X S Yang Nature-Inspired Metaheuristic Algorithmspp 83ndash96 Luniver Press London 2008

[12] S M Farahani B Nasiri A A Abshouri et al ldquoAn improvedfirefly algorithm with directed movementrdquo in Proceedings ofIEEE International Conference on Computer Science amp In-formation Technology Sichuan China June 2011

[13] L D S Coelho D L D A Bernert and V C Mariani ldquoAchaotic firefly algorithm applied to reliability-redundancy

optimizationrdquo Evolutionary Computation vol 30 pp 517ndash521 2011

[14] X S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo in Studies inComputational Intelligence vol 284 pp 101-111 SpringerBerlin Germany 2010

[15] S L Tilahun and C O Hong ldquoModified firefly algorithmrdquoJournal of Applied Mathematics vol 12 pp 2428ndash24392012

[16] K Jagatheesan B Anand S Samanta et al ldquoDesign ofa proportional-integral-derivative controller for an automaticgeneration control of multi-area power thermal systems usingfirefly algorithmrdquo IEEECAA Journal of Automatica Sinicapp 1ndash14 2016

[17] H Su Y Cai and Q Du ldquoFirefly-algorithm-inspiredframework with band selection and extreme learning ma-chine for hyperspectral image classificationrdquo IEEE Journal ofSelected Topics in Applied Earth Observations and RemoteSensing vol 10 no 1 pp 309ndash320 2016

[18] S Ranganathan M S Kalavathi and A R C ChristoberldquoSelf-adaptive firefly algorithm based multi-objectives formulti-type FACTS placementrdquo IET Generation Transmissionand Distribution vol 10 no 11 pp 188ndash200 2016

[19] D F Teshome C H Le Y W Lin et al ldquoA modified fireflyalgorithm for photovoltaic maximum power point trackingcontrol under partial shadingrdquo IEEE Journal of Emerging andSelected Topics in Power Electronics vol 5 no 2 pp 661ndash6712017

[20] M Alb P Alotto C Magele et al ldquoFirefly algorithm forfinding optimal shapes of electromagnetic devicesrdquo IEEETransactions on Magnetics vol 52 no 3 pp 1ndash4 2016

[21] A Mishra and V N K Gundavarapu ldquoLine utilisationfactor-based optimal allocation of IPFC and sizing usingfirefly algorithm for congestion managementrdquo GenerationTransmission and Distribution IET vol 10 no 1 pp 115ndash122 2016

[22] M H Horng ldquoVector quantization using the firefly algorithmfor image compressionrdquo Expert Systems with Applicationsvol 39 no 1 pp 078ndash1091 2012

[23] A Kazem E Sharifi F K Hussain et al ldquoSupport vectorregression with chaos-based firefly algorithm for stock marketprice forecastingrdquo Applied Soft Computing vol 13 no 2pp 947ndash958 2013

[24] L F He and S W Huang ldquoModified firefly algorithm basedmultilevel thresholding for color image segmentationrdquoNeurocomputing vol 240 pp 152ndash174 2017

[25] A H Gandomi X S Yang and A H Alavi ldquoMixed variablestructural optimization using Firefly Algorithmrdquo Computersand Structures vol 89 no 23-24 pp 2325ndash2336 2011

[26] M Alweshah and S Abdullah ldquoHybridizing firefly algorithmswith a probabilistic neural network for solving classificationproblemsrdquo Applied Soft Computing vol 35 pp 513ndash5242015

[27] R M Rizk-Allah E M Zaki A A El-Sawy et al ldquoHy-bridizing ant colony optimization with firefly algorithm forunconstrained optimization problemsrdquo Applied Mathematicsand Computation vol 224 pp 473ndash483 2013

[28] X S Yang S S S Hosseini and A H Gandomi ldquoFireflyalgorithm for solving non-convex economic dispatch prob-lems with valve loading effectrdquo Applied Soft Computingvol 12 no 3 pp 1180ndash1186 2012

[29] J Senthilnath S N Omkar and V Mani ldquoClustering usingfirefly algorithm performance studyrdquo Swarm and Evolu-tionary Computation vol 1 no 3 pp 164ndash171 2011

12 Computational Intelligence and Neuroscience

[30] T Kanimozhi and K Latha ldquoAn integrated approach to re-gion based image retrieval using firefly algorithm and supportvector machinerdquo Neurocomputing vol 151 pp 1099ndash11112015

[31] A Baykasoglu and F B Ozsoydan ldquoAdaptive firefly algorithmwith chaos for mechanical design optimization problemsrdquoApplied Soft Computing vol 36 pp 152ndash164 2015

[32] I Fister M Perc S M Kamal et al ldquoA review of chaos-basedfirefly algorithms perspectives and research challengesrdquoApplied Mathematics and Computation vol 252 pp 155ndash1652015

[33] I Fister I Fister X S Yang et al ldquoA comprehensive review offirefly algorithmsrdquo Swarm and Evolutionary Computationvol 13 pp 34ndash46 2013

[34] C T Brown L S Liebovitch and R Glendon ldquoLevy flights inDobe Jursquohoansi foraging patternsrdquo Human Ecology vol 35no 1 pp 129ndash138 2007

[35] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquoMathematics vol 226 no 2 pp 1830ndash1844 2012

[36] I Pavlyukevich ldquoCooling down Levy flightsrdquo Journal ofPhysics A Mathematical and eoretical vol 40 no 41pp 12299ndash12313 2007

[37] R Jensi and G W Jiji ldquoAn enhanced particle swarm opti-mization with Levy flight for global optimizationrdquo AppliedSoft Computing vol 43 pp 248ndash261 2016

[38] D Tang J Yang S Dong et al ldquoA Levy flight-based shuffledfrog-leaping algorithm and its applications for continuousoptimization problemsrdquo Applied Soft Computing vol 49pp 641ndash662 2016

[39] M Yahya and M P Saka ldquoConstruction site layout planningusing multi-objective artificial bee colony algorithmwith Levyflightsrdquo Automation in Construction vol 38 pp 14ndash29 2014

[40] C Blum and A Roli Hybrid Metaheuristics An IntroductionSpringer Berlin Germany 2008

[41] A M Reynolds and M A Frye ldquoFree-flight odor tracking indrosophila is consistent with an optimal intermittent scale-free searchrdquo PLos One vol 2 no 4 p e354 2007

[42] M F Shlesinger G M Zaslavsky and U Frisch Levy Flightsand Related Topics in Physics Springer Berlin HeidelbergGermany 1995

[43] M F Shlesinger ldquoMathematical physics search researchrdquoNature vol 443 no 7109 pp 281-282 2006

[44] S T Hsieh T Y Sun C C Liu et al ldquoEfficient populationutilization strategy for particle swarm optimizerrdquo IEEETransactions on Systems Man amp Cybernetics Part B vol 39no 2 pp 444ndash456 2009

[45] D Bratton and J Kennedy ldquoDefining a standard for particleswarm optimizationrdquo in Proceedings of IEEE Swarm In-telligence Symposium vol 107 pp 120ndash127 HonoluluHawaii April 2007

[46] U Manber Introduction to Algorithms A Creative ApproachAddison-Wesley Longman Publishing Co Inc Boston MAUSA 1989

Computational Intelligence and Neuroscience 13

Computer Games Technology

International Journal of

Hindawiwwwhindawicom Volume 2018

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Journal ofEngineeringVolume 2018

Advances in

FuzzySystems

Hindawiwwwhindawicom

Volume 2018

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ReconfigurableComputing

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

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Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Human-ComputerInteraction

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Scientic Programming

Submit your manuscripts atwwwhindawicom

computer Since dierent algorithms cannot reach the sameconvergence accuracy the test with xed convergence ac-curacy is not available erefore the test with max functionevaluations is conducted Running time of each algorithm iscounted by the explorer of MATLAB Parameter settings ofalgorithms are the same in Section 41 Average running timeof dierent algorithms is listed in Table 6 When function

evaluations are the same running speed of LFFSA is fasterthan that of AFSA while DE has the fastest running speedResults are quite in accord with those obtained by com-putational complexity analysis LFFSA and jDE are com-parative in experimental complexity Running time of FFSAis almost twice as much as that of LFFSA e improvementof LFFSA decreases time complexity to some extent

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash8

ndash6

ndash4

ndash2

0

2

FEs1040 5 10 15 20

(j)lo

g 10(

f(x))

FAAFSAFFSA

LFFSADEjDE

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

FEs1040 5 10 15 20

(k)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

FEs1040 5 10 15 20

(l)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash2

0

2

4

6

8

10

FEs1040 5 10 15 20

(m)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash5

0

5

FEs1040 5 10 15 20

(n)

log 1

0(f(x

))FAAFSAFFSA

LFFSADEjDE

ndash10

ndash5

0

5

10

FEs1040 5 10 15 20

(o)

log 1

0(f(x

))

FAAFSAFFSA

LFFSADEjDE

ndash4

ndash3

ndash2

ndash1

0

1

2

FEs1040 5 10 15 20

(p)

f(x)

FAAFSAFFSA

LFFSADEjDE

FEs1040 5 10 15 20

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

(q)

Figure 6 Iterative curves of test functions (a) f1 (b) f2 (c) f3 (d) f4 (e) f5 (f ) f6 (g) f7 (h) f8 (i) f9 (j) f10 (k) f11 (l) f12 (m) f13 (n) f14(o) f15 (p) f16 (q) f7

Computational Intelligence and Neuroscience 9

Table 3 Comparison of optimization results

No Items AFSA Std FA Std FFSA Std LFFSA Std DE Std jDE Std

f1

Worst 3014

0181

minus2798

0120

minus5132

0071

minus6743

0071

minus8544

0132

minus15021

0121Best 2402 minus3257 minus5233 minus8500 minus10371 minus17242Average 2656 minus3048 minus5145 minus7278 minus8924 minus16326Median 2651 minus3102 minus5193 minus6325 minus8586 minus16706

f2

Worst minus2322

0171

minus19245

0135

minus11178

0074

minus14207

0134

minus20506

0400

minus31644

0535Best minus4126 minus2813 minus11313 minus15585 minus21611 minus33250Average minus2812 minus2278 minus11273 minus14812 minus21054 minus32687Median minus2562 minus2197 minus11271 minus14834 minus20970 minus32844

f3

Worst 1386

0001

minus1776

0124

minus3908

0197

minus6121

0138

minus0027

0438

minus8934

0167Best 1307 minus2415 minus4647 minus6938 minus1281 minus9471Average 1587 minus2168 minus4147 minus6546 minus0049 minus9163Median 1586 minus2177 minus4225 minus6325 minus0035 minus9164

f4

Worst 2143

0087

22733

0097

minus20348

0125

minus1076

0376

1520

0003

1454

0025Best 1565 1043 minus2416 minus3405 1412 1363Average 1946 1476 minus2158 minus1946 1385 1312Median 1854 1385 minus2235 minus2325 1363 1287

f5

Worst 2310

0077

1864

0856

minus2846

0044

minus4385

0054

minus0579

0323

0898

6009Best 2096 1243 minus2982 minus5145 minus1591 minus12831Average 2236 1454 minus2946 minus4643 minus0999 minus3303Median 2136 1285 minus2435 minus4325 minus0963 0148

f6

Worst 2445

0038

1716

0133

minus2382

0076

minus3414

0048

1131

0042

0698

5049Best 2318 1255 minus2618 minus3606 1012 minus11404Average 2408 1571 minus2486 minus3489 1077 minus5466Median 2419 1601 minus2462 minus3487 1094 minus7632

f7

Worst 3815

0018

37846

0031

4186

736eminus 6

4156

434eminus 6

4099

296eminus 9

4087

403eminus 3Best 3945 3978 4099 4099 4099 4099Average 3813 3848 4099 4099 4099 4097Median 3736 3785 minus4099 minus4099 4099 4097

f8

Worst 0956

0133

minus0960

0223

minus5301

0051

minus6271

0055

minus989

0114

minusInf

0Best minus0644 minus0500 minus5444 minus6455 minus10255 minusInfAverage minus0735 minus0697 minus5357 minus6372 minus10071 minusInfMedian 0736 0685 minus5435 minus6325 minus10074 minusInf

f9

Worst minus11665

1057

minus976

1324

minus9347

0843

minus8695

1323

minus6848

1124

minus6131

2697Best minus14433 minus10574 minus12194 minus11937 minus10680 minus14831Average minus12786 minus10456 minus10764 minus10137 minus8178 minus9938Median 1136 1085 minus7435 minus6325 minus7745 minus9663

f10

Worst 0957

0131

0974

0223

minus5375

0049

minus6274

0056

0673

0032

0280

0110Best 0644 0497 minus5448 minus6486 0483 minus0022Average 0747 0649 minus5376 minus6348 0526 0230Median 1136 1085 minus7435 minus6325 0547 0211

f11

Worst minus0301

272eminus 05

minus0303

0009

minus4751

0071

minus5647

0085

minus0896

0103

minus1106

0135Best minus0301 minus0331 minus5011 minus5965 minus1107 minus1429Average minus0301 minus0313 minus4838 minus5804 minus0975 minus1364Median minus0301 minus0311 minus4835 minus5804 minus0896 minus1402

f12

Worst 0779

0037

minus1177

0163

minus2756

0035

minus3158

0037

minus0136

0052

minus1904

0221Best 0658 minus1638 minus2892 minus3287 minus0339 minus2526Average 0711 minus1388 minus2811 minus3221 minus0252 minus2291Median 0716 minus1376 minus2801 minus3216 0248 minus2374

f13

Worst 4286

0173

0574

0297

minus0602

0007

minus0602

0005

0039

0068

minus0176

538eminus 6Best 3706 minus0175 minus0605 minus0605 minus0162 minus0176Average 4147 minus0013 minus0603 minus0603 minus0088 minus0176Median 4213 minus0135 minus0603 minus6603 minus0086 minus0176

f14

Worst 2404

0084

0405

0318

minus3598

0117

minus4514

0101

0276

0211

minus2192

0401Best 2173 minus0576 minus4001 minus4869 minus0369 minus3409Average 2275 0069 minus3773 minus4656 minus0117 minus2867Median 2246 0133 minus3719 minus4651 minus0134 minus2924

10 Computational Intelligence and Neuroscience

44ParameterAnalysis ofLFFSA eeect of parameters onoptimization is analyzed in this section Taking Ackleyfunction as example Figure 7 shows the change of the ob-jective function value in the case of varying parametersTrynumber and β0 are proportional to the optimization resultTrynumber can impact the time complexity of the algorithmso the value should be appropriate not to aect the runningspeed e best value of c and δ is 25 and 12 respectively

5 Conclusion

LFFSA is proposed to improve the capability of AFSAwhich integrates the merits of both AFSA and FA Firstlythe searching characteristic of AFSA is studied by cal-culating the time complexity Secondly 17 benchmark testfunctions are used to verify LFFSA en time complexityof LFFSA is estimated Numerical results demonstrate thatLFFSA has a better performance in accuracy and speed ofoptimization to solve nonlinear optimization problemsthan the other test algorithms However the solutionobtained by LFFSA can be more precise and the way ofmodication could provide reference for those esectcientalgorithms eg DE and GWO

Table 3 Continued

No Items AFSA Std FA Std FFSA Std LFFSA Std DE Std jDE Std

f15

Worst 2588

00705

1505

0198

minus4281

0117

minus4975

0241

2241

0077

0635

0372Best 2366 minus4679 minus5792 minus72474 1972 minus0370Average 2490 minus4561 minus5408 minus63259 2140 0148Median 2506 minus4601 minus5456 minus63259 2136 0218

f16

Worst 1438

0058

minus1463

0253

minus2762

0026

minus3239

0032

minus2354

0042

minus3392

0433Best 1268 minus2296 minus2838 minus3334 minus2482 minus4646Average 1366 minus1987 minus2793 minus3282 minus2408 minus3868Median 1375 minus2016 minus2793 minus3284 minus2402 minus3751

f17

Worst minus22954

0643

minus25673

1721

minus70748

1908

minus79645

0131

minus47025

1403

minus63031

1942Best minus24885 minus32075 minus77054 minus80098 minus51440 minus68738Average minus23772 minus28641 minus74598 minus79996 minus48982 minus66539Median minus23731 minus28759 minus74907 minus80001 minus49035 minus66559

Table 4 Time complexity analysis of AFSA

Procedure of AFSA Time complexity(1) Initialization of N articial shes O(N)(2) Initialization of lsquoboardrsquo O(N)(3) Swarming behavior O(N2 + 2lowastN)(4) Chasing behavior O(2lowastN2 + 2lowastN)(5) Preying behavior O(TrynumberlowastN)(6) Judging of terminal condition O(1)(7) Information output of lsquoboardrsquo O(1)

Table 5 Time complexity analysis of LFFSA

Procedure of LFFSA Time complexity(1) Initialization of N articial shes O(N)(2) Initialization of lsquoboardrsquo O(N)(3) Chasing behavior O(2lowastN2 + 2lowastN)(4) Preying behavior O(trynumberlowastN)(5) Judging of terminal condition O(1)(6) Information output of lsquoboardrsquo O(1)

Table 6 Average running time of algorithms

NoRunning time (s)

AFSA FA FFSA LFFSA DE jDEf1 1023 747 185 753 464 608f2 1237 773 2111 710 432 653f3 1087 780 2073 896 412 825f4 1140 767 2049 1027 563 698f5 1053 766 1947 803 328 596f6 1972 610 2577 1637 771 1065f7 1053 743 1877 856 327 656f8 1233 803 2272 923 379 663f9 2083 720 2136 771 401 684f10 1627 846 2062 756 465 715f11 926 895 1374 635 363 667f12 965 534 1375 651 304 628f13 912 526 1436 1041 413 715f14 1395 574 1902 1031 648 986f15 945 539 1307 736 462 774f16 873 521 1388 1146 852 1887f17 1453 728 1841 596 332 644

β0

Valu

e

ndash8

ndash75

ndash7

ndash65

ndash6

ndash55

215050 1

(a)

γ

Valu

e

ndash63

ndash62

ndash61

ndash6

ndash59

ndash58

0 2 4 6 8 10

(b)

δ

Valu

e

ndash615ndash61

ndash605ndash6

ndash595ndash59

ndash585ndash58

215050 1

(c)

Trynumber

Valu

e

ndash64

ndash62

ndash6

ndash58

ndash56

ndash54

0 2 4 6 8 10

(d)

Figure 7 Test curves of parameters (a) Test curve of β0 (b) Testcurve of c (c) Test curve of δ (d) Test curve of Trynumber

Computational Intelligence and Neuroscience 11

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work is supported by National Natural ScienceFoundation of China (61463028) e authors wish to thankDr Yanliang Cui for his fruitful comments and suggestions

References

[1] X L Li Z J Shao J X Qian et al ldquoAn optimizing methodbased on autonomous animats fish-swarm algorithmrdquo Sys-tems Engineering-eory and Practice vol 22 pp 188ndash2002002 in Chinese

[2] C R Wang C L Zhou and J W Ma ldquoAn improved artificialfish-swarm algorithm and its application in feed-forwardneural networksrdquo in Proceedings of the Fourth InternationalConference on Machine Learning amp Cybernetics GuangzhouChina August 2005

[3] H C Tsai and Y H Lin ldquoModification of the fish swarmalgorithm with particle swarm optimization formulation andcommunication behaviorrdquo Applied Soft Computing vol 11no 8 pp 5367ndash5374 2011

[4] X Y Luan Z P Li and T Z Liu ldquoA novel attribute reductionalgorithm based on rough set and improved artificial fishswarm algorithmrdquo Neurocomputing vol 174 pp 522ndash5292016

[5] Y Wu X Z Gao Z Kai et al ldquoKnowledge-based artificialfish-swarm algorithmrdquo IFAC Proceedings Volumes vol 44no 1 pp 188ndash200 2011

[6] X T Hu H Q Zhang Z C Li et al ldquoA novel self-adaptationhybrid artificial fish-swarm algorithmrdquo IFAC ProceedingsVolumes vol 46 no 5 pp 583ndash588 2013

[7] K P Kumar B Saravanan and K S Swarup ldquoOptimizationof renewable energy sources in a microgrid using artificial fishswarm algorithmrdquo Energy Procedia vol 90 pp 107ndash113 2016

[8] Q He X T Hu H Ren et al ldquoA novel artificial fish swarmalgorithm for solving large-scale reliability-redundancy ap-plication problemrdquo ISA Transactions vol 59 pp 105ndash1132015

[9] Z Q Zhang K P Wang L X Zhu et al ldquoA Pareto improvedartificial fish swarm algorithm for solving a multi-objectivefuzzy disassembly line balancing problemrdquo Expert Systemswith Applications vol 86 pp 165ndash176 2017

[10] P K Singhal R Naresh and V Sharma ldquoBinary fish swarmalgorithm for profit-based unit commitment problem incompetitive electricity market with ramp rate constraintsrdquoGeneration Transmission and Distribution IET vol 9 no 13pp 1697ndash1707 2015

[11] X S Yang Nature-Inspired Metaheuristic Algorithmspp 83ndash96 Luniver Press London 2008

[12] S M Farahani B Nasiri A A Abshouri et al ldquoAn improvedfirefly algorithm with directed movementrdquo in Proceedings ofIEEE International Conference on Computer Science amp In-formation Technology Sichuan China June 2011

[13] L D S Coelho D L D A Bernert and V C Mariani ldquoAchaotic firefly algorithm applied to reliability-redundancy

optimizationrdquo Evolutionary Computation vol 30 pp 517ndash521 2011

[14] X S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo in Studies inComputational Intelligence vol 284 pp 101-111 SpringerBerlin Germany 2010

[15] S L Tilahun and C O Hong ldquoModified firefly algorithmrdquoJournal of Applied Mathematics vol 12 pp 2428ndash24392012

[16] K Jagatheesan B Anand S Samanta et al ldquoDesign ofa proportional-integral-derivative controller for an automaticgeneration control of multi-area power thermal systems usingfirefly algorithmrdquo IEEECAA Journal of Automatica Sinicapp 1ndash14 2016

[17] H Su Y Cai and Q Du ldquoFirefly-algorithm-inspiredframework with band selection and extreme learning ma-chine for hyperspectral image classificationrdquo IEEE Journal ofSelected Topics in Applied Earth Observations and RemoteSensing vol 10 no 1 pp 309ndash320 2016

[18] S Ranganathan M S Kalavathi and A R C ChristoberldquoSelf-adaptive firefly algorithm based multi-objectives formulti-type FACTS placementrdquo IET Generation Transmissionand Distribution vol 10 no 11 pp 188ndash200 2016

[19] D F Teshome C H Le Y W Lin et al ldquoA modified fireflyalgorithm for photovoltaic maximum power point trackingcontrol under partial shadingrdquo IEEE Journal of Emerging andSelected Topics in Power Electronics vol 5 no 2 pp 661ndash6712017

[20] M Alb P Alotto C Magele et al ldquoFirefly algorithm forfinding optimal shapes of electromagnetic devicesrdquo IEEETransactions on Magnetics vol 52 no 3 pp 1ndash4 2016

[21] A Mishra and V N K Gundavarapu ldquoLine utilisationfactor-based optimal allocation of IPFC and sizing usingfirefly algorithm for congestion managementrdquo GenerationTransmission and Distribution IET vol 10 no 1 pp 115ndash122 2016

[22] M H Horng ldquoVector quantization using the firefly algorithmfor image compressionrdquo Expert Systems with Applicationsvol 39 no 1 pp 078ndash1091 2012

[23] A Kazem E Sharifi F K Hussain et al ldquoSupport vectorregression with chaos-based firefly algorithm for stock marketprice forecastingrdquo Applied Soft Computing vol 13 no 2pp 947ndash958 2013

[24] L F He and S W Huang ldquoModified firefly algorithm basedmultilevel thresholding for color image segmentationrdquoNeurocomputing vol 240 pp 152ndash174 2017

[25] A H Gandomi X S Yang and A H Alavi ldquoMixed variablestructural optimization using Firefly Algorithmrdquo Computersand Structures vol 89 no 23-24 pp 2325ndash2336 2011

[26] M Alweshah and S Abdullah ldquoHybridizing firefly algorithmswith a probabilistic neural network for solving classificationproblemsrdquo Applied Soft Computing vol 35 pp 513ndash5242015

[27] R M Rizk-Allah E M Zaki A A El-Sawy et al ldquoHy-bridizing ant colony optimization with firefly algorithm forunconstrained optimization problemsrdquo Applied Mathematicsand Computation vol 224 pp 473ndash483 2013

[28] X S Yang S S S Hosseini and A H Gandomi ldquoFireflyalgorithm for solving non-convex economic dispatch prob-lems with valve loading effectrdquo Applied Soft Computingvol 12 no 3 pp 1180ndash1186 2012

[29] J Senthilnath S N Omkar and V Mani ldquoClustering usingfirefly algorithm performance studyrdquo Swarm and Evolu-tionary Computation vol 1 no 3 pp 164ndash171 2011

12 Computational Intelligence and Neuroscience

[30] T Kanimozhi and K Latha ldquoAn integrated approach to re-gion based image retrieval using firefly algorithm and supportvector machinerdquo Neurocomputing vol 151 pp 1099ndash11112015

[31] A Baykasoglu and F B Ozsoydan ldquoAdaptive firefly algorithmwith chaos for mechanical design optimization problemsrdquoApplied Soft Computing vol 36 pp 152ndash164 2015

[32] I Fister M Perc S M Kamal et al ldquoA review of chaos-basedfirefly algorithms perspectives and research challengesrdquoApplied Mathematics and Computation vol 252 pp 155ndash1652015

[33] I Fister I Fister X S Yang et al ldquoA comprehensive review offirefly algorithmsrdquo Swarm and Evolutionary Computationvol 13 pp 34ndash46 2013

[34] C T Brown L S Liebovitch and R Glendon ldquoLevy flights inDobe Jursquohoansi foraging patternsrdquo Human Ecology vol 35no 1 pp 129ndash138 2007

[35] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquoMathematics vol 226 no 2 pp 1830ndash1844 2012

[36] I Pavlyukevich ldquoCooling down Levy flightsrdquo Journal ofPhysics A Mathematical and eoretical vol 40 no 41pp 12299ndash12313 2007

[37] R Jensi and G W Jiji ldquoAn enhanced particle swarm opti-mization with Levy flight for global optimizationrdquo AppliedSoft Computing vol 43 pp 248ndash261 2016

[38] D Tang J Yang S Dong et al ldquoA Levy flight-based shuffledfrog-leaping algorithm and its applications for continuousoptimization problemsrdquo Applied Soft Computing vol 49pp 641ndash662 2016

[39] M Yahya and M P Saka ldquoConstruction site layout planningusing multi-objective artificial bee colony algorithmwith Levyflightsrdquo Automation in Construction vol 38 pp 14ndash29 2014

[40] C Blum and A Roli Hybrid Metaheuristics An IntroductionSpringer Berlin Germany 2008

[41] A M Reynolds and M A Frye ldquoFree-flight odor tracking indrosophila is consistent with an optimal intermittent scale-free searchrdquo PLos One vol 2 no 4 p e354 2007

[42] M F Shlesinger G M Zaslavsky and U Frisch Levy Flightsand Related Topics in Physics Springer Berlin HeidelbergGermany 1995

[43] M F Shlesinger ldquoMathematical physics search researchrdquoNature vol 443 no 7109 pp 281-282 2006

[44] S T Hsieh T Y Sun C C Liu et al ldquoEfficient populationutilization strategy for particle swarm optimizerrdquo IEEETransactions on Systems Man amp Cybernetics Part B vol 39no 2 pp 444ndash456 2009

[45] D Bratton and J Kennedy ldquoDefining a standard for particleswarm optimizationrdquo in Proceedings of IEEE Swarm In-telligence Symposium vol 107 pp 120ndash127 HonoluluHawaii April 2007

[46] U Manber Introduction to Algorithms A Creative ApproachAddison-Wesley Longman Publishing Co Inc Boston MAUSA 1989

Computational Intelligence and Neuroscience 13

Computer Games Technology

International Journal of

Hindawiwwwhindawicom Volume 2018

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Journal ofEngineeringVolume 2018

Advances in

FuzzySystems

Hindawiwwwhindawicom

Volume 2018

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Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

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Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Human-ComputerInteraction

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Scientic Programming

Submit your manuscripts atwwwhindawicom

Table 3 Comparison of optimization results

No Items AFSA Std FA Std FFSA Std LFFSA Std DE Std jDE Std

f1

Worst 3014

0181

minus2798

0120

minus5132

0071

minus6743

0071

minus8544

0132

minus15021

0121Best 2402 minus3257 minus5233 minus8500 minus10371 minus17242Average 2656 minus3048 minus5145 minus7278 minus8924 minus16326Median 2651 minus3102 minus5193 minus6325 minus8586 minus16706

f2

Worst minus2322

0171

minus19245

0135

minus11178

0074

minus14207

0134

minus20506

0400

minus31644

0535Best minus4126 minus2813 minus11313 minus15585 minus21611 minus33250Average minus2812 minus2278 minus11273 minus14812 minus21054 minus32687Median minus2562 minus2197 minus11271 minus14834 minus20970 minus32844

f3

Worst 1386

0001

minus1776

0124

minus3908

0197

minus6121

0138

minus0027

0438

minus8934

0167Best 1307 minus2415 minus4647 minus6938 minus1281 minus9471Average 1587 minus2168 minus4147 minus6546 minus0049 minus9163Median 1586 minus2177 minus4225 minus6325 minus0035 minus9164

f4

Worst 2143

0087

22733

0097

minus20348

0125

minus1076

0376

1520

0003

1454

0025Best 1565 1043 minus2416 minus3405 1412 1363Average 1946 1476 minus2158 minus1946 1385 1312Median 1854 1385 minus2235 minus2325 1363 1287

f5

Worst 2310

0077

1864

0856

minus2846

0044

minus4385

0054

minus0579

0323

0898

6009Best 2096 1243 minus2982 minus5145 minus1591 minus12831Average 2236 1454 minus2946 minus4643 minus0999 minus3303Median 2136 1285 minus2435 minus4325 minus0963 0148

f6

Worst 2445

0038

1716

0133

minus2382

0076

minus3414

0048

1131

0042

0698

5049Best 2318 1255 minus2618 minus3606 1012 minus11404Average 2408 1571 minus2486 minus3489 1077 minus5466Median 2419 1601 minus2462 minus3487 1094 minus7632

f7

Worst 3815

0018

37846

0031

4186

736eminus 6

4156

434eminus 6

4099

296eminus 9

4087

403eminus 3Best 3945 3978 4099 4099 4099 4099Average 3813 3848 4099 4099 4099 4097Median 3736 3785 minus4099 minus4099 4099 4097

f8

Worst 0956

0133

minus0960

0223

minus5301

0051

minus6271

0055

minus989

0114

minusInf

0Best minus0644 minus0500 minus5444 minus6455 minus10255 minusInfAverage minus0735 minus0697 minus5357 minus6372 minus10071 minusInfMedian 0736 0685 minus5435 minus6325 minus10074 minusInf

f9

Worst minus11665

1057

minus976

1324

minus9347

0843

minus8695

1323

minus6848

1124

minus6131

2697Best minus14433 minus10574 minus12194 minus11937 minus10680 minus14831Average minus12786 minus10456 minus10764 minus10137 minus8178 minus9938Median 1136 1085 minus7435 minus6325 minus7745 minus9663

f10

Worst 0957

0131

0974

0223

minus5375

0049

minus6274

0056

0673

0032

0280

0110Best 0644 0497 minus5448 minus6486 0483 minus0022Average 0747 0649 minus5376 minus6348 0526 0230Median 1136 1085 minus7435 minus6325 0547 0211

f11

Worst minus0301

272eminus 05

minus0303

0009

minus4751

0071

minus5647

0085

minus0896

0103

minus1106

0135Best minus0301 minus0331 minus5011 minus5965 minus1107 minus1429Average minus0301 minus0313 minus4838 minus5804 minus0975 minus1364Median minus0301 minus0311 minus4835 minus5804 minus0896 minus1402

f12

Worst 0779

0037

minus1177

0163

minus2756

0035

minus3158

0037

minus0136

0052

minus1904

0221Best 0658 minus1638 minus2892 minus3287 minus0339 minus2526Average 0711 minus1388 minus2811 minus3221 minus0252 minus2291Median 0716 minus1376 minus2801 minus3216 0248 minus2374

f13

Worst 4286

0173

0574

0297

minus0602

0007

minus0602

0005

0039

0068

minus0176

538eminus 6Best 3706 minus0175 minus0605 minus0605 minus0162 minus0176Average 4147 minus0013 minus0603 minus0603 minus0088 minus0176Median 4213 minus0135 minus0603 minus6603 minus0086 minus0176

f14

Worst 2404

0084

0405

0318

minus3598

0117

minus4514

0101

0276

0211

minus2192

0401Best 2173 minus0576 minus4001 minus4869 minus0369 minus3409Average 2275 0069 minus3773 minus4656 minus0117 minus2867Median 2246 0133 minus3719 minus4651 minus0134 minus2924

10 Computational Intelligence and Neuroscience

44ParameterAnalysis ofLFFSA eeect of parameters onoptimization is analyzed in this section Taking Ackleyfunction as example Figure 7 shows the change of the ob-jective function value in the case of varying parametersTrynumber and β0 are proportional to the optimization resultTrynumber can impact the time complexity of the algorithmso the value should be appropriate not to aect the runningspeed e best value of c and δ is 25 and 12 respectively

5 Conclusion

LFFSA is proposed to improve the capability of AFSAwhich integrates the merits of both AFSA and FA Firstlythe searching characteristic of AFSA is studied by cal-culating the time complexity Secondly 17 benchmark testfunctions are used to verify LFFSA en time complexityof LFFSA is estimated Numerical results demonstrate thatLFFSA has a better performance in accuracy and speed ofoptimization to solve nonlinear optimization problemsthan the other test algorithms However the solutionobtained by LFFSA can be more precise and the way ofmodication could provide reference for those esectcientalgorithms eg DE and GWO

Table 3 Continued

No Items AFSA Std FA Std FFSA Std LFFSA Std DE Std jDE Std

f15

Worst 2588

00705

1505

0198

minus4281

0117

minus4975

0241

2241

0077

0635

0372Best 2366 minus4679 minus5792 minus72474 1972 minus0370Average 2490 minus4561 minus5408 minus63259 2140 0148Median 2506 minus4601 minus5456 minus63259 2136 0218

f16

Worst 1438

0058

minus1463

0253

minus2762

0026

minus3239

0032

minus2354

0042

minus3392

0433Best 1268 minus2296 minus2838 minus3334 minus2482 minus4646Average 1366 minus1987 minus2793 minus3282 minus2408 minus3868Median 1375 minus2016 minus2793 minus3284 minus2402 minus3751

f17

Worst minus22954

0643

minus25673

1721

minus70748

1908

minus79645

0131

minus47025

1403

minus63031

1942Best minus24885 minus32075 minus77054 minus80098 minus51440 minus68738Average minus23772 minus28641 minus74598 minus79996 minus48982 minus66539Median minus23731 minus28759 minus74907 minus80001 minus49035 minus66559

Table 4 Time complexity analysis of AFSA

Procedure of AFSA Time complexity(1) Initialization of N articial shes O(N)(2) Initialization of lsquoboardrsquo O(N)(3) Swarming behavior O(N2 + 2lowastN)(4) Chasing behavior O(2lowastN2 + 2lowastN)(5) Preying behavior O(TrynumberlowastN)(6) Judging of terminal condition O(1)(7) Information output of lsquoboardrsquo O(1)

Table 5 Time complexity analysis of LFFSA

Procedure of LFFSA Time complexity(1) Initialization of N articial shes O(N)(2) Initialization of lsquoboardrsquo O(N)(3) Chasing behavior O(2lowastN2 + 2lowastN)(4) Preying behavior O(trynumberlowastN)(5) Judging of terminal condition O(1)(6) Information output of lsquoboardrsquo O(1)

Table 6 Average running time of algorithms

NoRunning time (s)

AFSA FA FFSA LFFSA DE jDEf1 1023 747 185 753 464 608f2 1237 773 2111 710 432 653f3 1087 780 2073 896 412 825f4 1140 767 2049 1027 563 698f5 1053 766 1947 803 328 596f6 1972 610 2577 1637 771 1065f7 1053 743 1877 856 327 656f8 1233 803 2272 923 379 663f9 2083 720 2136 771 401 684f10 1627 846 2062 756 465 715f11 926 895 1374 635 363 667f12 965 534 1375 651 304 628f13 912 526 1436 1041 413 715f14 1395 574 1902 1031 648 986f15 945 539 1307 736 462 774f16 873 521 1388 1146 852 1887f17 1453 728 1841 596 332 644

β0

Valu

e

ndash8

ndash75

ndash7

ndash65

ndash6

ndash55

215050 1

(a)

γ

Valu

e

ndash63

ndash62

ndash61

ndash6

ndash59

ndash58

0 2 4 6 8 10

(b)

δ

Valu

e

ndash615ndash61

ndash605ndash6

ndash595ndash59

ndash585ndash58

215050 1

(c)

Trynumber

Valu

e

ndash64

ndash62

ndash6

ndash58

ndash56

ndash54

0 2 4 6 8 10

(d)

Figure 7 Test curves of parameters (a) Test curve of β0 (b) Testcurve of c (c) Test curve of δ (d) Test curve of Trynumber

Computational Intelligence and Neuroscience 11

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work is supported by National Natural ScienceFoundation of China (61463028) e authors wish to thankDr Yanliang Cui for his fruitful comments and suggestions

References

[1] X L Li Z J Shao J X Qian et al ldquoAn optimizing methodbased on autonomous animats fish-swarm algorithmrdquo Sys-tems Engineering-eory and Practice vol 22 pp 188ndash2002002 in Chinese

[2] C R Wang C L Zhou and J W Ma ldquoAn improved artificialfish-swarm algorithm and its application in feed-forwardneural networksrdquo in Proceedings of the Fourth InternationalConference on Machine Learning amp Cybernetics GuangzhouChina August 2005

[3] H C Tsai and Y H Lin ldquoModification of the fish swarmalgorithm with particle swarm optimization formulation andcommunication behaviorrdquo Applied Soft Computing vol 11no 8 pp 5367ndash5374 2011

[4] X Y Luan Z P Li and T Z Liu ldquoA novel attribute reductionalgorithm based on rough set and improved artificial fishswarm algorithmrdquo Neurocomputing vol 174 pp 522ndash5292016

[5] Y Wu X Z Gao Z Kai et al ldquoKnowledge-based artificialfish-swarm algorithmrdquo IFAC Proceedings Volumes vol 44no 1 pp 188ndash200 2011

[6] X T Hu H Q Zhang Z C Li et al ldquoA novel self-adaptationhybrid artificial fish-swarm algorithmrdquo IFAC ProceedingsVolumes vol 46 no 5 pp 583ndash588 2013

[7] K P Kumar B Saravanan and K S Swarup ldquoOptimizationof renewable energy sources in a microgrid using artificial fishswarm algorithmrdquo Energy Procedia vol 90 pp 107ndash113 2016

[8] Q He X T Hu H Ren et al ldquoA novel artificial fish swarmalgorithm for solving large-scale reliability-redundancy ap-plication problemrdquo ISA Transactions vol 59 pp 105ndash1132015

[9] Z Q Zhang K P Wang L X Zhu et al ldquoA Pareto improvedartificial fish swarm algorithm for solving a multi-objectivefuzzy disassembly line balancing problemrdquo Expert Systemswith Applications vol 86 pp 165ndash176 2017

[10] P K Singhal R Naresh and V Sharma ldquoBinary fish swarmalgorithm for profit-based unit commitment problem incompetitive electricity market with ramp rate constraintsrdquoGeneration Transmission and Distribution IET vol 9 no 13pp 1697ndash1707 2015

[11] X S Yang Nature-Inspired Metaheuristic Algorithmspp 83ndash96 Luniver Press London 2008

[12] S M Farahani B Nasiri A A Abshouri et al ldquoAn improvedfirefly algorithm with directed movementrdquo in Proceedings ofIEEE International Conference on Computer Science amp In-formation Technology Sichuan China June 2011

[13] L D S Coelho D L D A Bernert and V C Mariani ldquoAchaotic firefly algorithm applied to reliability-redundancy

optimizationrdquo Evolutionary Computation vol 30 pp 517ndash521 2011

[14] X S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo in Studies inComputational Intelligence vol 284 pp 101-111 SpringerBerlin Germany 2010

[15] S L Tilahun and C O Hong ldquoModified firefly algorithmrdquoJournal of Applied Mathematics vol 12 pp 2428ndash24392012

[16] K Jagatheesan B Anand S Samanta et al ldquoDesign ofa proportional-integral-derivative controller for an automaticgeneration control of multi-area power thermal systems usingfirefly algorithmrdquo IEEECAA Journal of Automatica Sinicapp 1ndash14 2016

[17] H Su Y Cai and Q Du ldquoFirefly-algorithm-inspiredframework with band selection and extreme learning ma-chine for hyperspectral image classificationrdquo IEEE Journal ofSelected Topics in Applied Earth Observations and RemoteSensing vol 10 no 1 pp 309ndash320 2016

[18] S Ranganathan M S Kalavathi and A R C ChristoberldquoSelf-adaptive firefly algorithm based multi-objectives formulti-type FACTS placementrdquo IET Generation Transmissionand Distribution vol 10 no 11 pp 188ndash200 2016

[19] D F Teshome C H Le Y W Lin et al ldquoA modified fireflyalgorithm for photovoltaic maximum power point trackingcontrol under partial shadingrdquo IEEE Journal of Emerging andSelected Topics in Power Electronics vol 5 no 2 pp 661ndash6712017

[20] M Alb P Alotto C Magele et al ldquoFirefly algorithm forfinding optimal shapes of electromagnetic devicesrdquo IEEETransactions on Magnetics vol 52 no 3 pp 1ndash4 2016

[21] A Mishra and V N K Gundavarapu ldquoLine utilisationfactor-based optimal allocation of IPFC and sizing usingfirefly algorithm for congestion managementrdquo GenerationTransmission and Distribution IET vol 10 no 1 pp 115ndash122 2016

[22] M H Horng ldquoVector quantization using the firefly algorithmfor image compressionrdquo Expert Systems with Applicationsvol 39 no 1 pp 078ndash1091 2012

[23] A Kazem E Sharifi F K Hussain et al ldquoSupport vectorregression with chaos-based firefly algorithm for stock marketprice forecastingrdquo Applied Soft Computing vol 13 no 2pp 947ndash958 2013

[24] L F He and S W Huang ldquoModified firefly algorithm basedmultilevel thresholding for color image segmentationrdquoNeurocomputing vol 240 pp 152ndash174 2017

[25] A H Gandomi X S Yang and A H Alavi ldquoMixed variablestructural optimization using Firefly Algorithmrdquo Computersand Structures vol 89 no 23-24 pp 2325ndash2336 2011

[26] M Alweshah and S Abdullah ldquoHybridizing firefly algorithmswith a probabilistic neural network for solving classificationproblemsrdquo Applied Soft Computing vol 35 pp 513ndash5242015

[27] R M Rizk-Allah E M Zaki A A El-Sawy et al ldquoHy-bridizing ant colony optimization with firefly algorithm forunconstrained optimization problemsrdquo Applied Mathematicsand Computation vol 224 pp 473ndash483 2013

[28] X S Yang S S S Hosseini and A H Gandomi ldquoFireflyalgorithm for solving non-convex economic dispatch prob-lems with valve loading effectrdquo Applied Soft Computingvol 12 no 3 pp 1180ndash1186 2012

[29] J Senthilnath S N Omkar and V Mani ldquoClustering usingfirefly algorithm performance studyrdquo Swarm and Evolu-tionary Computation vol 1 no 3 pp 164ndash171 2011

12 Computational Intelligence and Neuroscience

[30] T Kanimozhi and K Latha ldquoAn integrated approach to re-gion based image retrieval using firefly algorithm and supportvector machinerdquo Neurocomputing vol 151 pp 1099ndash11112015

[31] A Baykasoglu and F B Ozsoydan ldquoAdaptive firefly algorithmwith chaos for mechanical design optimization problemsrdquoApplied Soft Computing vol 36 pp 152ndash164 2015

[32] I Fister M Perc S M Kamal et al ldquoA review of chaos-basedfirefly algorithms perspectives and research challengesrdquoApplied Mathematics and Computation vol 252 pp 155ndash1652015

[33] I Fister I Fister X S Yang et al ldquoA comprehensive review offirefly algorithmsrdquo Swarm and Evolutionary Computationvol 13 pp 34ndash46 2013

[34] C T Brown L S Liebovitch and R Glendon ldquoLevy flights inDobe Jursquohoansi foraging patternsrdquo Human Ecology vol 35no 1 pp 129ndash138 2007

[35] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquoMathematics vol 226 no 2 pp 1830ndash1844 2012

[36] I Pavlyukevich ldquoCooling down Levy flightsrdquo Journal ofPhysics A Mathematical and eoretical vol 40 no 41pp 12299ndash12313 2007

[37] R Jensi and G W Jiji ldquoAn enhanced particle swarm opti-mization with Levy flight for global optimizationrdquo AppliedSoft Computing vol 43 pp 248ndash261 2016

[38] D Tang J Yang S Dong et al ldquoA Levy flight-based shuffledfrog-leaping algorithm and its applications for continuousoptimization problemsrdquo Applied Soft Computing vol 49pp 641ndash662 2016

[39] M Yahya and M P Saka ldquoConstruction site layout planningusing multi-objective artificial bee colony algorithmwith Levyflightsrdquo Automation in Construction vol 38 pp 14ndash29 2014

[40] C Blum and A Roli Hybrid Metaheuristics An IntroductionSpringer Berlin Germany 2008

[41] A M Reynolds and M A Frye ldquoFree-flight odor tracking indrosophila is consistent with an optimal intermittent scale-free searchrdquo PLos One vol 2 no 4 p e354 2007

[42] M F Shlesinger G M Zaslavsky and U Frisch Levy Flightsand Related Topics in Physics Springer Berlin HeidelbergGermany 1995

[43] M F Shlesinger ldquoMathematical physics search researchrdquoNature vol 443 no 7109 pp 281-282 2006

[44] S T Hsieh T Y Sun C C Liu et al ldquoEfficient populationutilization strategy for particle swarm optimizerrdquo IEEETransactions on Systems Man amp Cybernetics Part B vol 39no 2 pp 444ndash456 2009

[45] D Bratton and J Kennedy ldquoDefining a standard for particleswarm optimizationrdquo in Proceedings of IEEE Swarm In-telligence Symposium vol 107 pp 120ndash127 HonoluluHawaii April 2007

[46] U Manber Introduction to Algorithms A Creative ApproachAddison-Wesley Longman Publishing Co Inc Boston MAUSA 1989

Computational Intelligence and Neuroscience 13

Computer Games Technology

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Advances in

FuzzySystems

Hindawiwwwhindawicom

Volume 2018

International Journal of

ReconfigurableComputing

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

thinspArtificial Intelligence

Hindawiwwwhindawicom Volumethinsp2018

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawiwwwhindawicom Volume 2018

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Computational Intelligence and Neuroscience

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018

Human-ComputerInteraction

Advances in

Hindawiwwwhindawicom Volume 2018

Scientic Programming

Submit your manuscripts atwwwhindawicom

44ParameterAnalysis ofLFFSA eeect of parameters onoptimization is analyzed in this section Taking Ackleyfunction as example Figure 7 shows the change of the ob-jective function value in the case of varying parametersTrynumber and β0 are proportional to the optimization resultTrynumber can impact the time complexity of the algorithmso the value should be appropriate not to aect the runningspeed e best value of c and δ is 25 and 12 respectively

5 Conclusion

LFFSA is proposed to improve the capability of AFSAwhich integrates the merits of both AFSA and FA Firstlythe searching characteristic of AFSA is studied by cal-culating the time complexity Secondly 17 benchmark testfunctions are used to verify LFFSA en time complexityof LFFSA is estimated Numerical results demonstrate thatLFFSA has a better performance in accuracy and speed ofoptimization to solve nonlinear optimization problemsthan the other test algorithms However the solutionobtained by LFFSA can be more precise and the way ofmodication could provide reference for those esectcientalgorithms eg DE and GWO

Table 3 Continued

No Items AFSA Std FA Std FFSA Std LFFSA Std DE Std jDE Std

f15

Worst 2588

00705

1505

0198

minus4281

0117

minus4975

0241

2241

0077

0635

0372Best 2366 minus4679 minus5792 minus72474 1972 minus0370Average 2490 minus4561 minus5408 minus63259 2140 0148Median 2506 minus4601 minus5456 minus63259 2136 0218

f16

Worst 1438

0058

minus1463

0253

minus2762

0026

minus3239

0032

minus2354

0042

minus3392

0433Best 1268 minus2296 minus2838 minus3334 minus2482 minus4646Average 1366 minus1987 minus2793 minus3282 minus2408 minus3868Median 1375 minus2016 minus2793 minus3284 minus2402 minus3751

f17

Worst minus22954

0643

minus25673

1721

minus70748

1908

minus79645

0131

minus47025

1403

minus63031

1942Best minus24885 minus32075 minus77054 minus80098 minus51440 minus68738Average minus23772 minus28641 minus74598 minus79996 minus48982 minus66539Median minus23731 minus28759 minus74907 minus80001 minus49035 minus66559

Table 4 Time complexity analysis of AFSA

Procedure of AFSA Time complexity(1) Initialization of N articial shes O(N)(2) Initialization of lsquoboardrsquo O(N)(3) Swarming behavior O(N2 + 2lowastN)(4) Chasing behavior O(2lowastN2 + 2lowastN)(5) Preying behavior O(TrynumberlowastN)(6) Judging of terminal condition O(1)(7) Information output of lsquoboardrsquo O(1)

Table 5 Time complexity analysis of LFFSA

Procedure of LFFSA Time complexity(1) Initialization of N articial shes O(N)(2) Initialization of lsquoboardrsquo O(N)(3) Chasing behavior O(2lowastN2 + 2lowastN)(4) Preying behavior O(trynumberlowastN)(5) Judging of terminal condition O(1)(6) Information output of lsquoboardrsquo O(1)

Table 6 Average running time of algorithms

NoRunning time (s)

AFSA FA FFSA LFFSA DE jDEf1 1023 747 185 753 464 608f2 1237 773 2111 710 432 653f3 1087 780 2073 896 412 825f4 1140 767 2049 1027 563 698f5 1053 766 1947 803 328 596f6 1972 610 2577 1637 771 1065f7 1053 743 1877 856 327 656f8 1233 803 2272 923 379 663f9 2083 720 2136 771 401 684f10 1627 846 2062 756 465 715f11 926 895 1374 635 363 667f12 965 534 1375 651 304 628f13 912 526 1436 1041 413 715f14 1395 574 1902 1031 648 986f15 945 539 1307 736 462 774f16 873 521 1388 1146 852 1887f17 1453 728 1841 596 332 644

β0

Valu

e

ndash8

ndash75

ndash7

ndash65

ndash6

ndash55

215050 1

(a)

γ

Valu

e

ndash63

ndash62

ndash61

ndash6

ndash59

ndash58

0 2 4 6 8 10

(b)

δ

Valu

e

ndash615ndash61

ndash605ndash6

ndash595ndash59

ndash585ndash58

215050 1

(c)

Trynumber

Valu

e

ndash64

ndash62

ndash6

ndash58

ndash56

ndash54

0 2 4 6 8 10

(d)

Figure 7 Test curves of parameters (a) Test curve of β0 (b) Testcurve of c (c) Test curve of δ (d) Test curve of Trynumber

Computational Intelligence and Neuroscience 11

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work is supported by National Natural ScienceFoundation of China (61463028) e authors wish to thankDr Yanliang Cui for his fruitful comments and suggestions

References

[1] X L Li Z J Shao J X Qian et al ldquoAn optimizing methodbased on autonomous animats fish-swarm algorithmrdquo Sys-tems Engineering-eory and Practice vol 22 pp 188ndash2002002 in Chinese

[2] C R Wang C L Zhou and J W Ma ldquoAn improved artificialfish-swarm algorithm and its application in feed-forwardneural networksrdquo in Proceedings of the Fourth InternationalConference on Machine Learning amp Cybernetics GuangzhouChina August 2005

[3] H C Tsai and Y H Lin ldquoModification of the fish swarmalgorithm with particle swarm optimization formulation andcommunication behaviorrdquo Applied Soft Computing vol 11no 8 pp 5367ndash5374 2011

[4] X Y Luan Z P Li and T Z Liu ldquoA novel attribute reductionalgorithm based on rough set and improved artificial fishswarm algorithmrdquo Neurocomputing vol 174 pp 522ndash5292016

[5] Y Wu X Z Gao Z Kai et al ldquoKnowledge-based artificialfish-swarm algorithmrdquo IFAC Proceedings Volumes vol 44no 1 pp 188ndash200 2011

[6] X T Hu H Q Zhang Z C Li et al ldquoA novel self-adaptationhybrid artificial fish-swarm algorithmrdquo IFAC ProceedingsVolumes vol 46 no 5 pp 583ndash588 2013

[7] K P Kumar B Saravanan and K S Swarup ldquoOptimizationof renewable energy sources in a microgrid using artificial fishswarm algorithmrdquo Energy Procedia vol 90 pp 107ndash113 2016

[8] Q He X T Hu H Ren et al ldquoA novel artificial fish swarmalgorithm for solving large-scale reliability-redundancy ap-plication problemrdquo ISA Transactions vol 59 pp 105ndash1132015

[9] Z Q Zhang K P Wang L X Zhu et al ldquoA Pareto improvedartificial fish swarm algorithm for solving a multi-objectivefuzzy disassembly line balancing problemrdquo Expert Systemswith Applications vol 86 pp 165ndash176 2017

[10] P K Singhal R Naresh and V Sharma ldquoBinary fish swarmalgorithm for profit-based unit commitment problem incompetitive electricity market with ramp rate constraintsrdquoGeneration Transmission and Distribution IET vol 9 no 13pp 1697ndash1707 2015

[11] X S Yang Nature-Inspired Metaheuristic Algorithmspp 83ndash96 Luniver Press London 2008

[12] S M Farahani B Nasiri A A Abshouri et al ldquoAn improvedfirefly algorithm with directed movementrdquo in Proceedings ofIEEE International Conference on Computer Science amp In-formation Technology Sichuan China June 2011

[13] L D S Coelho D L D A Bernert and V C Mariani ldquoAchaotic firefly algorithm applied to reliability-redundancy

optimizationrdquo Evolutionary Computation vol 30 pp 517ndash521 2011

[14] X S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo in Studies inComputational Intelligence vol 284 pp 101-111 SpringerBerlin Germany 2010

[15] S L Tilahun and C O Hong ldquoModified firefly algorithmrdquoJournal of Applied Mathematics vol 12 pp 2428ndash24392012

[16] K Jagatheesan B Anand S Samanta et al ldquoDesign ofa proportional-integral-derivative controller for an automaticgeneration control of multi-area power thermal systems usingfirefly algorithmrdquo IEEECAA Journal of Automatica Sinicapp 1ndash14 2016

[17] H Su Y Cai and Q Du ldquoFirefly-algorithm-inspiredframework with band selection and extreme learning ma-chine for hyperspectral image classificationrdquo IEEE Journal ofSelected Topics in Applied Earth Observations and RemoteSensing vol 10 no 1 pp 309ndash320 2016

[18] S Ranganathan M S Kalavathi and A R C ChristoberldquoSelf-adaptive firefly algorithm based multi-objectives formulti-type FACTS placementrdquo IET Generation Transmissionand Distribution vol 10 no 11 pp 188ndash200 2016

[19] D F Teshome C H Le Y W Lin et al ldquoA modified fireflyalgorithm for photovoltaic maximum power point trackingcontrol under partial shadingrdquo IEEE Journal of Emerging andSelected Topics in Power Electronics vol 5 no 2 pp 661ndash6712017

[20] M Alb P Alotto C Magele et al ldquoFirefly algorithm forfinding optimal shapes of electromagnetic devicesrdquo IEEETransactions on Magnetics vol 52 no 3 pp 1ndash4 2016

[21] A Mishra and V N K Gundavarapu ldquoLine utilisationfactor-based optimal allocation of IPFC and sizing usingfirefly algorithm for congestion managementrdquo GenerationTransmission and Distribution IET vol 10 no 1 pp 115ndash122 2016

[22] M H Horng ldquoVector quantization using the firefly algorithmfor image compressionrdquo Expert Systems with Applicationsvol 39 no 1 pp 078ndash1091 2012

[23] A Kazem E Sharifi F K Hussain et al ldquoSupport vectorregression with chaos-based firefly algorithm for stock marketprice forecastingrdquo Applied Soft Computing vol 13 no 2pp 947ndash958 2013

[24] L F He and S W Huang ldquoModified firefly algorithm basedmultilevel thresholding for color image segmentationrdquoNeurocomputing vol 240 pp 152ndash174 2017

[25] A H Gandomi X S Yang and A H Alavi ldquoMixed variablestructural optimization using Firefly Algorithmrdquo Computersand Structures vol 89 no 23-24 pp 2325ndash2336 2011

[26] M Alweshah and S Abdullah ldquoHybridizing firefly algorithmswith a probabilistic neural network for solving classificationproblemsrdquo Applied Soft Computing vol 35 pp 513ndash5242015

[27] R M Rizk-Allah E M Zaki A A El-Sawy et al ldquoHy-bridizing ant colony optimization with firefly algorithm forunconstrained optimization problemsrdquo Applied Mathematicsand Computation vol 224 pp 473ndash483 2013

[28] X S Yang S S S Hosseini and A H Gandomi ldquoFireflyalgorithm for solving non-convex economic dispatch prob-lems with valve loading effectrdquo Applied Soft Computingvol 12 no 3 pp 1180ndash1186 2012

[29] J Senthilnath S N Omkar and V Mani ldquoClustering usingfirefly algorithm performance studyrdquo Swarm and Evolu-tionary Computation vol 1 no 3 pp 164ndash171 2011

12 Computational Intelligence and Neuroscience

[30] T Kanimozhi and K Latha ldquoAn integrated approach to re-gion based image retrieval using firefly algorithm and supportvector machinerdquo Neurocomputing vol 151 pp 1099ndash11112015

[31] A Baykasoglu and F B Ozsoydan ldquoAdaptive firefly algorithmwith chaos for mechanical design optimization problemsrdquoApplied Soft Computing vol 36 pp 152ndash164 2015

[32] I Fister M Perc S M Kamal et al ldquoA review of chaos-basedfirefly algorithms perspectives and research challengesrdquoApplied Mathematics and Computation vol 252 pp 155ndash1652015

[33] I Fister I Fister X S Yang et al ldquoA comprehensive review offirefly algorithmsrdquo Swarm and Evolutionary Computationvol 13 pp 34ndash46 2013

[34] C T Brown L S Liebovitch and R Glendon ldquoLevy flights inDobe Jursquohoansi foraging patternsrdquo Human Ecology vol 35no 1 pp 129ndash138 2007

[35] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquoMathematics vol 226 no 2 pp 1830ndash1844 2012

[36] I Pavlyukevich ldquoCooling down Levy flightsrdquo Journal ofPhysics A Mathematical and eoretical vol 40 no 41pp 12299ndash12313 2007

[37] R Jensi and G W Jiji ldquoAn enhanced particle swarm opti-mization with Levy flight for global optimizationrdquo AppliedSoft Computing vol 43 pp 248ndash261 2016

[38] D Tang J Yang S Dong et al ldquoA Levy flight-based shuffledfrog-leaping algorithm and its applications for continuousoptimization problemsrdquo Applied Soft Computing vol 49pp 641ndash662 2016

[39] M Yahya and M P Saka ldquoConstruction site layout planningusing multi-objective artificial bee colony algorithmwith Levyflightsrdquo Automation in Construction vol 38 pp 14ndash29 2014

[40] C Blum and A Roli Hybrid Metaheuristics An IntroductionSpringer Berlin Germany 2008

[41] A M Reynolds and M A Frye ldquoFree-flight odor tracking indrosophila is consistent with an optimal intermittent scale-free searchrdquo PLos One vol 2 no 4 p e354 2007

[42] M F Shlesinger G M Zaslavsky and U Frisch Levy Flightsand Related Topics in Physics Springer Berlin HeidelbergGermany 1995

[43] M F Shlesinger ldquoMathematical physics search researchrdquoNature vol 443 no 7109 pp 281-282 2006

[44] S T Hsieh T Y Sun C C Liu et al ldquoEfficient populationutilization strategy for particle swarm optimizerrdquo IEEETransactions on Systems Man amp Cybernetics Part B vol 39no 2 pp 444ndash456 2009

[45] D Bratton and J Kennedy ldquoDefining a standard for particleswarm optimizationrdquo in Proceedings of IEEE Swarm In-telligence Symposium vol 107 pp 120ndash127 HonoluluHawaii April 2007

[46] U Manber Introduction to Algorithms A Creative ApproachAddison-Wesley Longman Publishing Co Inc Boston MAUSA 1989

Computational Intelligence and Neuroscience 13

Computer Games Technology

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Advances in

FuzzySystems

Hindawiwwwhindawicom

Volume 2018

International Journal of

ReconfigurableComputing

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

thinspArtificial Intelligence

Hindawiwwwhindawicom Volumethinsp2018

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawiwwwhindawicom Volume 2018

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Computational Intelligence and Neuroscience

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018

Human-ComputerInteraction

Advances in

Hindawiwwwhindawicom Volume 2018

Scientic Programming

Submit your manuscripts atwwwhindawicom

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work is supported by National Natural ScienceFoundation of China (61463028) e authors wish to thankDr Yanliang Cui for his fruitful comments and suggestions

References

[1] X L Li Z J Shao J X Qian et al ldquoAn optimizing methodbased on autonomous animats fish-swarm algorithmrdquo Sys-tems Engineering-eory and Practice vol 22 pp 188ndash2002002 in Chinese

[2] C R Wang C L Zhou and J W Ma ldquoAn improved artificialfish-swarm algorithm and its application in feed-forwardneural networksrdquo in Proceedings of the Fourth InternationalConference on Machine Learning amp Cybernetics GuangzhouChina August 2005

[3] H C Tsai and Y H Lin ldquoModification of the fish swarmalgorithm with particle swarm optimization formulation andcommunication behaviorrdquo Applied Soft Computing vol 11no 8 pp 5367ndash5374 2011

[4] X Y Luan Z P Li and T Z Liu ldquoA novel attribute reductionalgorithm based on rough set and improved artificial fishswarm algorithmrdquo Neurocomputing vol 174 pp 522ndash5292016

[5] Y Wu X Z Gao Z Kai et al ldquoKnowledge-based artificialfish-swarm algorithmrdquo IFAC Proceedings Volumes vol 44no 1 pp 188ndash200 2011

[6] X T Hu H Q Zhang Z C Li et al ldquoA novel self-adaptationhybrid artificial fish-swarm algorithmrdquo IFAC ProceedingsVolumes vol 46 no 5 pp 583ndash588 2013

[7] K P Kumar B Saravanan and K S Swarup ldquoOptimizationof renewable energy sources in a microgrid using artificial fishswarm algorithmrdquo Energy Procedia vol 90 pp 107ndash113 2016

[8] Q He X T Hu H Ren et al ldquoA novel artificial fish swarmalgorithm for solving large-scale reliability-redundancy ap-plication problemrdquo ISA Transactions vol 59 pp 105ndash1132015

[9] Z Q Zhang K P Wang L X Zhu et al ldquoA Pareto improvedartificial fish swarm algorithm for solving a multi-objectivefuzzy disassembly line balancing problemrdquo Expert Systemswith Applications vol 86 pp 165ndash176 2017

[10] P K Singhal R Naresh and V Sharma ldquoBinary fish swarmalgorithm for profit-based unit commitment problem incompetitive electricity market with ramp rate constraintsrdquoGeneration Transmission and Distribution IET vol 9 no 13pp 1697ndash1707 2015

[11] X S Yang Nature-Inspired Metaheuristic Algorithmspp 83ndash96 Luniver Press London 2008

[12] S M Farahani B Nasiri A A Abshouri et al ldquoAn improvedfirefly algorithm with directed movementrdquo in Proceedings ofIEEE International Conference on Computer Science amp In-formation Technology Sichuan China June 2011

[13] L D S Coelho D L D A Bernert and V C Mariani ldquoAchaotic firefly algorithm applied to reliability-redundancy

optimizationrdquo Evolutionary Computation vol 30 pp 517ndash521 2011

[14] X S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo in Studies inComputational Intelligence vol 284 pp 101-111 SpringerBerlin Germany 2010

[15] S L Tilahun and C O Hong ldquoModified firefly algorithmrdquoJournal of Applied Mathematics vol 12 pp 2428ndash24392012

[16] K Jagatheesan B Anand S Samanta et al ldquoDesign ofa proportional-integral-derivative controller for an automaticgeneration control of multi-area power thermal systems usingfirefly algorithmrdquo IEEECAA Journal of Automatica Sinicapp 1ndash14 2016

[17] H Su Y Cai and Q Du ldquoFirefly-algorithm-inspiredframework with band selection and extreme learning ma-chine for hyperspectral image classificationrdquo IEEE Journal ofSelected Topics in Applied Earth Observations and RemoteSensing vol 10 no 1 pp 309ndash320 2016

[18] S Ranganathan M S Kalavathi and A R C ChristoberldquoSelf-adaptive firefly algorithm based multi-objectives formulti-type FACTS placementrdquo IET Generation Transmissionand Distribution vol 10 no 11 pp 188ndash200 2016

[19] D F Teshome C H Le Y W Lin et al ldquoA modified fireflyalgorithm for photovoltaic maximum power point trackingcontrol under partial shadingrdquo IEEE Journal of Emerging andSelected Topics in Power Electronics vol 5 no 2 pp 661ndash6712017

[20] M Alb P Alotto C Magele et al ldquoFirefly algorithm forfinding optimal shapes of electromagnetic devicesrdquo IEEETransactions on Magnetics vol 52 no 3 pp 1ndash4 2016

[21] A Mishra and V N K Gundavarapu ldquoLine utilisationfactor-based optimal allocation of IPFC and sizing usingfirefly algorithm for congestion managementrdquo GenerationTransmission and Distribution IET vol 10 no 1 pp 115ndash122 2016

[22] M H Horng ldquoVector quantization using the firefly algorithmfor image compressionrdquo Expert Systems with Applicationsvol 39 no 1 pp 078ndash1091 2012

[23] A Kazem E Sharifi F K Hussain et al ldquoSupport vectorregression with chaos-based firefly algorithm for stock marketprice forecastingrdquo Applied Soft Computing vol 13 no 2pp 947ndash958 2013

[24] L F He and S W Huang ldquoModified firefly algorithm basedmultilevel thresholding for color image segmentationrdquoNeurocomputing vol 240 pp 152ndash174 2017

[25] A H Gandomi X S Yang and A H Alavi ldquoMixed variablestructural optimization using Firefly Algorithmrdquo Computersand Structures vol 89 no 23-24 pp 2325ndash2336 2011

[26] M Alweshah and S Abdullah ldquoHybridizing firefly algorithmswith a probabilistic neural network for solving classificationproblemsrdquo Applied Soft Computing vol 35 pp 513ndash5242015

[27] R M Rizk-Allah E M Zaki A A El-Sawy et al ldquoHy-bridizing ant colony optimization with firefly algorithm forunconstrained optimization problemsrdquo Applied Mathematicsand Computation vol 224 pp 473ndash483 2013

[28] X S Yang S S S Hosseini and A H Gandomi ldquoFireflyalgorithm for solving non-convex economic dispatch prob-lems with valve loading effectrdquo Applied Soft Computingvol 12 no 3 pp 1180ndash1186 2012

[29] J Senthilnath S N Omkar and V Mani ldquoClustering usingfirefly algorithm performance studyrdquo Swarm and Evolu-tionary Computation vol 1 no 3 pp 164ndash171 2011

12 Computational Intelligence and Neuroscience

[30] T Kanimozhi and K Latha ldquoAn integrated approach to re-gion based image retrieval using firefly algorithm and supportvector machinerdquo Neurocomputing vol 151 pp 1099ndash11112015

[31] A Baykasoglu and F B Ozsoydan ldquoAdaptive firefly algorithmwith chaos for mechanical design optimization problemsrdquoApplied Soft Computing vol 36 pp 152ndash164 2015

[32] I Fister M Perc S M Kamal et al ldquoA review of chaos-basedfirefly algorithms perspectives and research challengesrdquoApplied Mathematics and Computation vol 252 pp 155ndash1652015

[33] I Fister I Fister X S Yang et al ldquoA comprehensive review offirefly algorithmsrdquo Swarm and Evolutionary Computationvol 13 pp 34ndash46 2013

[34] C T Brown L S Liebovitch and R Glendon ldquoLevy flights inDobe Jursquohoansi foraging patternsrdquo Human Ecology vol 35no 1 pp 129ndash138 2007

[35] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquoMathematics vol 226 no 2 pp 1830ndash1844 2012

[36] I Pavlyukevich ldquoCooling down Levy flightsrdquo Journal ofPhysics A Mathematical and eoretical vol 40 no 41pp 12299ndash12313 2007

[37] R Jensi and G W Jiji ldquoAn enhanced particle swarm opti-mization with Levy flight for global optimizationrdquo AppliedSoft Computing vol 43 pp 248ndash261 2016

[38] D Tang J Yang S Dong et al ldquoA Levy flight-based shuffledfrog-leaping algorithm and its applications for continuousoptimization problemsrdquo Applied Soft Computing vol 49pp 641ndash662 2016

[39] M Yahya and M P Saka ldquoConstruction site layout planningusing multi-objective artificial bee colony algorithmwith Levyflightsrdquo Automation in Construction vol 38 pp 14ndash29 2014

[40] C Blum and A Roli Hybrid Metaheuristics An IntroductionSpringer Berlin Germany 2008

[41] A M Reynolds and M A Frye ldquoFree-flight odor tracking indrosophila is consistent with an optimal intermittent scale-free searchrdquo PLos One vol 2 no 4 p e354 2007

[42] M F Shlesinger G M Zaslavsky and U Frisch Levy Flightsand Related Topics in Physics Springer Berlin HeidelbergGermany 1995

[43] M F Shlesinger ldquoMathematical physics search researchrdquoNature vol 443 no 7109 pp 281-282 2006

[44] S T Hsieh T Y Sun C C Liu et al ldquoEfficient populationutilization strategy for particle swarm optimizerrdquo IEEETransactions on Systems Man amp Cybernetics Part B vol 39no 2 pp 444ndash456 2009

[45] D Bratton and J Kennedy ldquoDefining a standard for particleswarm optimizationrdquo in Proceedings of IEEE Swarm In-telligence Symposium vol 107 pp 120ndash127 HonoluluHawaii April 2007

[46] U Manber Introduction to Algorithms A Creative ApproachAddison-Wesley Longman Publishing Co Inc Boston MAUSA 1989

Computational Intelligence and Neuroscience 13

Computer Games Technology

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Advances in

FuzzySystems

Hindawiwwwhindawicom

Volume 2018

International Journal of

ReconfigurableComputing

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

thinspArtificial Intelligence

Hindawiwwwhindawicom Volumethinsp2018

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawiwwwhindawicom Volume 2018

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Computational Intelligence and Neuroscience

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018

Human-ComputerInteraction

Advances in

Hindawiwwwhindawicom Volume 2018

Scientic Programming

Submit your manuscripts atwwwhindawicom

[30] T Kanimozhi and K Latha ldquoAn integrated approach to re-gion based image retrieval using firefly algorithm and supportvector machinerdquo Neurocomputing vol 151 pp 1099ndash11112015

[31] A Baykasoglu and F B Ozsoydan ldquoAdaptive firefly algorithmwith chaos for mechanical design optimization problemsrdquoApplied Soft Computing vol 36 pp 152ndash164 2015

[32] I Fister M Perc S M Kamal et al ldquoA review of chaos-basedfirefly algorithms perspectives and research challengesrdquoApplied Mathematics and Computation vol 252 pp 155ndash1652015

[33] I Fister I Fister X S Yang et al ldquoA comprehensive review offirefly algorithmsrdquo Swarm and Evolutionary Computationvol 13 pp 34ndash46 2013

[34] C T Brown L S Liebovitch and R Glendon ldquoLevy flights inDobe Jursquohoansi foraging patternsrdquo Human Ecology vol 35no 1 pp 129ndash138 2007

[35] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquoMathematics vol 226 no 2 pp 1830ndash1844 2012

[36] I Pavlyukevich ldquoCooling down Levy flightsrdquo Journal ofPhysics A Mathematical and eoretical vol 40 no 41pp 12299ndash12313 2007

[37] R Jensi and G W Jiji ldquoAn enhanced particle swarm opti-mization with Levy flight for global optimizationrdquo AppliedSoft Computing vol 43 pp 248ndash261 2016

[38] D Tang J Yang S Dong et al ldquoA Levy flight-based shuffledfrog-leaping algorithm and its applications for continuousoptimization problemsrdquo Applied Soft Computing vol 49pp 641ndash662 2016

[39] M Yahya and M P Saka ldquoConstruction site layout planningusing multi-objective artificial bee colony algorithmwith Levyflightsrdquo Automation in Construction vol 38 pp 14ndash29 2014

[40] C Blum and A Roli Hybrid Metaheuristics An IntroductionSpringer Berlin Germany 2008

[41] A M Reynolds and M A Frye ldquoFree-flight odor tracking indrosophila is consistent with an optimal intermittent scale-free searchrdquo PLos One vol 2 no 4 p e354 2007

[42] M F Shlesinger G M Zaslavsky and U Frisch Levy Flightsand Related Topics in Physics Springer Berlin HeidelbergGermany 1995

[43] M F Shlesinger ldquoMathematical physics search researchrdquoNature vol 443 no 7109 pp 281-282 2006

[44] S T Hsieh T Y Sun C C Liu et al ldquoEfficient populationutilization strategy for particle swarm optimizerrdquo IEEETransactions on Systems Man amp Cybernetics Part B vol 39no 2 pp 444ndash456 2009

[45] D Bratton and J Kennedy ldquoDefining a standard for particleswarm optimizationrdquo in Proceedings of IEEE Swarm In-telligence Symposium vol 107 pp 120ndash127 HonoluluHawaii April 2007

[46] U Manber Introduction to Algorithms A Creative ApproachAddison-Wesley Longman Publishing Co Inc Boston MAUSA 1989

Computational Intelligence and Neuroscience 13

Computer Games Technology

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Advances in

FuzzySystems

Hindawiwwwhindawicom

Volume 2018

International Journal of

ReconfigurableComputing

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

thinspArtificial Intelligence

Hindawiwwwhindawicom Volumethinsp2018

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawiwwwhindawicom Volume 2018

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Computational Intelligence and Neuroscience

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018

Human-ComputerInteraction

Advances in

Hindawiwwwhindawicom Volume 2018

Scientic Programming

Submit your manuscripts atwwwhindawicom

Computer Games Technology

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Advances in

FuzzySystems

Hindawiwwwhindawicom

Volume 2018

International Journal of

ReconfigurableComputing

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

thinspArtificial Intelligence

Hindawiwwwhindawicom Volumethinsp2018

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawiwwwhindawicom Volume 2018

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Computational Intelligence and Neuroscience

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018

Human-ComputerInteraction

Advances in

Hindawiwwwhindawicom Volume 2018

Scientic Programming

Submit your manuscripts atwwwhindawicom