Hindawi Publishing CorporationComputational Intelligence and NeuroscienceVolume 2013 Article ID 453812 13 pageshttpdxdoiorg1011552013453812

Research ArticleA Novel Bat Algorithm Based on Differential Operator andLeacutevy Flights Trajectory

Jian Xie1 Yongquan Zhou12 and Huan Chen1

1 College of Information Science and Engineering Guangxi University for Nationalities Nanning Guangxi 530006 China2 Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis Guangxi University for Nationalities NanningGuangxi 530006 China

Correspondence should be addressed to Yongquan Zhou yongquanzhou126com

Received 16 August 2012 Accepted 1 February 2013

Academic Editor Christian W Dawson

Copyright copy 2013 Jian Xie et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Aiming at the phenomenon of slow convergence rate and low accuracy of bat algorithm a novel bat algorithm based on differentialoperator and Levy flights trajectory is proposed In this paper a differential operator is introduced to accelerate the convergencespeed of proposed algorithm which is similar to mutation strategy ldquoDEbest2rdquo in differential algorithm Levy flights trajectorycan ensure the diversity of the population against premature convergence and make the algorithm effectively jump out of localminima 14 typical benchmark functions and an instance of nonlinear equations are tested the simulation results not only showthat the proposed algorithm is feasible and effective but also demonstrate that this proposed algorithm has superior approximationcapabilities in high-dimensional space

1 Introduction

Nowadays since the evolutionary algorithm can solve someproblem that the traditional optimization algorithm cannotdo easy the evolutionary algorithms are widely applied indifferent fields such as the management science engineeringoptimization scientific computing More and more mod-ern metaheuristic algorithms inspired by nature or socialphenomenon are emerging and they become increasinglypopular for example particles swarms optimization (PSO)[1] firefly algorithm (FA) [2 3] artificial chemical reactionoptimization algorithm (ACROA) [4] glowworm swarmsoptimization (GSO) [5] invasive weed optimization (IWO)[6] differential evolution (DE) [7ndash9] bat algorithm (BA)[2 10] and so on [11ndash15] Some researchers have proposedtheir hybrid versions by combining two or more algorithms

Bat Algorithm (BA) is a novel metaheuristic optimizationalgorithm based on the echolocation behaviour of microbatswhich was proposed by Yang in 2010 [2 10] This algorithmgradually aroused peoplersquos close attention and which isincreasingly applied to different areas Tsai et al (2011)

proposed an improved EBA to solve numerical optimizationproblems [16] A multiobjective bat algorithm (MOBA) isproposed by Yang (2011) [17] which is first validated againsta subset of test functions and then applied to solve multi-objective design problems such as welded beam design In2012 Bora et al applied BA to solve the Brushless DCWheelMotor Problem [18] Although the basic BA has remarkableproperty compared against several traditional optimizationmethods the phenomenon of slow convergence rate andlow accuracy still exists Therefore in this paper we putforward an improved bat algorithm based on differentialoperator and Levy flights trajectory (DLBA) the purpose is toimprove the convergence rate and precision of bat algorithmAt the end of this paper we tested 14 typical benchmarkfunctions and applied them to solve nonlinear equationsthe simulation results not only showed that the proposedalgorithm is feasible and effective which is more robust butalso demonstrated the superior approximation capabilities inhigh-dimensional space

The rest of this study is organized as follows In Section 2the basic bat algorithm and Levy flights were described

2 Computational Intelligence and Neuroscience

In Section 3 we gave the design framework of DLBA Theimplementation and comparison of improved algorithm arepresented in Section 4 Finally we concluded this paper inSection 5

2 Bat Algorithm Leacutevy Flights andNonlinear Equations

21 Behaviour of Microbats Most of microbats have ad-vanced capability of echolocation These bats can emit a veryloud and short sound pulse the echo that reflects back fromthe surrounding objects is received by their extraordinary bigauricle Then this feedback information of echo is analyzedin their subtle brainThey not only can discriminate directionfor their own flight pathway according to the echo but alsocan distinguish different insects and obstacles to hunt preyand avoid a collision effectively in the day or night

22 Bat Algorithm (see [10]) First of all let us briefly reviewthe basics of the BA for single-objective optimization In thebasic BA developed by Yang in 2010 in order to propose thebat algorithm inspired by the echolocation characteristics ofmicrobats the following approximate or idealised rules wereused

IR1 All bats use echolocation to sense distance and they alsoldquoknowrdquo the difference between foodprey and backgroundbarriers in some magical way

IR2 Bats fly randomly with velocity V119894at position 119909

119894with

a fixed frequency 119891min varying wavelength 120582 and loudness1198600to search for prey They can automatically adjust the

wavelength (or frequency) of their emitted pulses and adjustthe rate of pulse emission 119903 isin [0 1] depending on theproximity of their target

IR3 Although the loudness can vary in many ways weassume that the loudness varies from a large (positive) 119860

0to

a minimum constant value 119860minIn addition for simplicity they also use the follow-

ing approximations in general the frequency 119891 in arange [119891min 119891max] corresponds to a range of wavelengths[120582min 120582max] In fact they just vary in the frequency whilefixed in the wavelength 120582 and assume 119891 isin [0 119891max] in theirimplementationThis is because 120582 and119891 are related due to thefact that 120582119891 = V is constant

In simulations they use virtual bats naturally to define theupdated rules of their positions 119909

119894and velocities V

119894in a D-

dimensional search spaceThe new solutions 119909119905119894and velocities

V119905119894at time step 119905 are given by

119891119894= 119891min + (119891max minus 119891min) 120573

V119905

119894= V119905minus1

119894+ (119909119905

119894minus 119909lowast) 119891119894

119909119905

119894= 119909119905minus1

119894+ V119905

119894

(1)

where 120573 isin [0 1] is a random vector drawn from a uniformdistribution Here 119909

lowastis the current global best location

(solution) which is located after comparing all the solutionsamong all the 119899 bats

For the local search part once a solution is selectedamong the current best solutions a new solution for each batis generated locally using random walk

119909new = 119909old + 120576119860 119905 (2)

where 120576 isin [minus1 1] is a random number while119860119905= ⟨119860119905

119894⟩ is the

average loudness of all the bats at this time stepFurthermore the loudness 119860

119894and the rate 119903

119894of pulse

emission have to be updated accordingly as the iterationsproceed These formulas are

119860119905+1

119894= 120572119860119905

119894 (3)

119903119905+1

119894= 1199030

119894[1 minus exp (minus120574119905)] (4)

where 120572 and 120574 are constantsBased on these approximations and idealization the

basic steps of the bat algorithm can be summarized in thePseudocode 1

23 Levy Flights Levy flights are Markov processes whichdiffer from regular Brownian motion whose individualjumps have lengths that are distributed with the probabilitydensity function (PDF) 120582(119909) decaying at large 119909 as 120582(119909) =|119909|minus1minus120572 with 0 lt 120572 lt 2 Due to the divergence of their var-

iance 120582(119909) ≃ 119909|minus1minus120572 extremely long jumps may occur andtypical trajectories are self-similar on all scales showingclusters of shorter jumps interspersed by long excursions [19]Levy flight has the following properties [20]

(1) Stability distribution of the sum of independentidentically distributed stable random variables equalto distribution of each variable

(2) Power law asymptotics (ldquoheavy tailsrdquo)(3) Generalized Central Limit Theorem The central limit

theorem states that the sum of a number of inde-pendent and identically distributed (iid) randomvariables with finite variances will tend to a normaldistribution as the number of variables grows

(4) Which has an infinite variance with an infinite meanvalue

Due to the these remarkable properties of stable distri-butions it is now believed that the Levy statistics provide aframework for the description ofmany natural phenomena inphysical chemical biological and economical systems froma general common point of view

Furthermore various studies have shown that the flightbehaviour of many animals and insects has demonstratedthe typical characteristics of Levy lights A recent studyby Reynolds and Frye shows that fruit flies or Drosophilamelanogaster explore their landscape using a series of straightflight paths punctuated by a sudden 90∘ turn leading to aLevy flight-style intermittent scale-free search pattern [21]Studies on human behaviour such as the Jursquohoansi hunter-gatherer foraging patterns also show the typical feature ofLevy flights [22] The conclusion that light is related to Levyflights is proposed by Barthelemy et al (2008) [23] The

Computational Intelligence and Neuroscience 3

Objective function 119891(119909) 119909 = [1199091 1199092 119909

119889]119879

Initialize the bat population 119909119894(119894 = 1 2 119899) and V

119894

Define pulse frequency 119891119894at 119909119894initialize pulse rates 119903

119894and the loudness 119860

119894

While (119905 ltMax number of iterations)Generate new solutions by adjusting frequencyand updating velocities and locationssolutions (1)if (rand gt 119903

119894)

Select a solution among the best solutionsGenerate a local solution around the selected best solution

end ifGenerate a new solution by flying randomlyif (rand lt 119860

119894amp 119891(119909

119894) lt 119891(119909

lowast))

Accept the new solutionsIncrease 119903

119894and reduce 119860

119894

end ifRank the bats and find the current best 119909

lowast

end whilePostprocess results and visualization

Pseudocode 1

study by Mercadier et al shows that the Levy flights ofphotons in hot atomic vapours (2009) [24] Subsequentlysuch behaviour has been applied to optimization and optimalsearch and preliminary results show its promising capability

24 Description of the Nonlinear Equations (see [25]) Thegeneral form of nonlinear equations with real variables isdescribed as follows

119892119894 (119883) = 0 119894 = 1 2 119899 (5)

where 119883 = (1199091 1199092 119909

119899) isin 119863 sub 119877

119899 119863 = (1199091 1199092 119909

119899) |

119886119894le 119909119894le 119887119894 119894 = 1 2 119899

In solving nonlinear equations process for DLBA thefitness function can be constructed by

119866 (119883) =

119899

sum

119894=1

1003816100381610038161003816119892119894 (119909)1003816100381610038161003816 (6)

So the solving of nonlinear equations can be translated intoan optimization problem in domain119863

min 119866 (119883)

st (1199091 1199092 119909

119899) isin 119863

(7)

Consequently the optimal value of (7) is exactly the solutionof (5)

3 DLBA

Inspired by Yangrsquos method we propose an improved batalgorithm based on differential operator and Levy-flightstrajectory (DLBA) based on the basic structure of BA andre-estimate the characters used in the original BA In DLBAnot only the movement of the bat is quite different from theoriginal BA but also the local search process is different

In DLBA the frequency fluctuates up and down whichcan change self-adaptively and the differential operator isintroduced which is similar to themutation operation of DEthe frequency 119891 is similar to the scale factor 119865 of DEbest2So the frequency updated formulae of a bat are defined asfollows

119891119905

1119894= ((119891

1min minus 1198911max)119905

119899119905

+ 1198911max)1205731 (8)

119891119905

2119894= ((119891

2max minus 1198912min)119905

119899119905

+ 1198912min)1205732 (9)

where 1205731 1205732isin [0 1] is a random vector drawn from a uni-

form distribution 1198911max = 119891

2max 1198911min = 1198912min 119899119905 is a

fixed parameter In DLBA the position 119909119905119894of each bat indi-

vidual are updated with (10) which is different from originalBAThis can preferably incorporate the echolocation charac-teristics of microbats

119909119905+1

119894= 119909119905

best + 119891119905

1119894(119909119905

1199031minus 119909119905

1199032) + 119891119905

2119894(119909119905

1199033minus 119909119905

1199034) (10)

where 119909119905best is the current global best location (solution)which is located after comparing all the solutions among allthe 119899 bats in 119905 generation 119909119905

119903119894is the bat individual in the bat

swarm and this can be achieved by randomizationIn addition Levy flight haves the prominent properties

increase the diversity of population sequentially which canmake the algorithmeffectively jumpout of the local optimumSo we let these bats perform the Levy flights with (11) beforethe position updating

119909119905

119894= 119905minus1

119894+ 120583 sign [rand minus 05] oplus Levy (11)

where 120583 is a random parameter drawn from a uniformdistribution sign oplusmeans entrywise multiplications rand isin[0 1] and random step length Levy obeys Levy distribution

Levy sim 119906 = 119905minus120582 (1 lt 120582 le 3) (12)

4 Computational Intelligence and Neuroscience

InputObjective function 119891(119909) 119909 = [1199091 1199092 119909

119889]119879 algorithmrsquos parameters 119899 119891min 119891max 120572 119899119905

BEGINFor 119894 = 1 To 119899 do

Initialize 119909119905119894 1199030119894 1198600119894

Calculating the initial fitnessEnd ForWhile (119905 lt itermax)

Update position use formula (8)sim(12)Evaluate fitness 119891(119909

119894) to find 119909best

For 119894 = 1 To 119899 doIf (rand gt 119903

119894)

Generate a 119909new around the selected best solution use formula (2)End If

End ForEvaluate fitness 119891(119909new)For 119894 = 1 To 119899 do

If (rand lt 119860119894)

Generate a 119909new around the selected best solution use formula (13)End If

End ForEvaluate fitness 119891(119909new)If (119891(119909119905+1best) lt 119891(119909

119905

best))Update 119903119905

119894 119860119905119894use formula (3) (14)

End If119905 = 119905 + 1

EndWhileENDOutput 119909best fitnessmin

Pseudocode 2

On the other hand each bat should have different valuesof loudness and pulse emission rate while the rate of pulseemission is relatively low and the loudness is relatively highDuring the search process the loudness usually decreaseswhile the rate of pulse emission increases Batsrsquo positionvariation is influenced by the pulse emission rate and loud-ness as well Firstly pulse emission rate 119903

119894causes fluctuation

of position using (2) sequentially more and more newposition can be explored Secondly loudness119860

119894is designed to

strengthen local search and to guide bats find better solutions

119909new = 119909best + 120578119903119905 (13)

where 120578 isin [minus1 1] is a random parameter and 119909best is thecurrent global best location (solution) in whole bats swarmWhile 119903

119905= ⟨119903119905

119894⟩ is the average pulse emission rate of all the

bats at this generationThe new rate of pulse emission 119903119905

119894and loudness119860119905

119894at time

step 119905 are given by

119903119905+1

119894= 119903119905

119894(

119905

119899119905

)

3

(14)

where 119903119905119894is time varying their loudness and emission rates

will be updated only if the best solution of the currentgeneration is better than the best solution of last generationwhich means that these bats are moving towards the optimalsolution The pseudocode of DLBA can be depicted as inPseudocode 2

4 The Simulation and Analysis

41 Parametric Studies The proposed DLBA is imple-mented in MATLAB simulation platform CPU Intel XeonE5405200GHz OS Microsoft Windows Server 2003Enterprise Edition SP2 RAM 1GHz Matlab VersionR2009a The parameter setting for BA and DLBA are listedin Table 1 the parameters of BA are recommended in theoriginal article

The stopping criterion can be defined in many ways Weadopt two terminated criteria we can use a given tolerance(Tol = 10119890 minus 5) for a test function that have certain mini-mum value in theory on the contrary each simulation runterminates when a certain number of function evaluations(FEs) have been reached In this paper FEs could be obtainedby population size multiplied by the number of iteration Inour experiment FEs = 24000 BA (= 300lowast40lowast2) DLBA (=200lowast40lowast3)

42 BenchmarkTest Function In order to validate the validityof DLBA we selected the 14 benchmark functions to experi-mentize The benchmark set include unimodal multimodalhigh-dimensional and low-dimensional unconstrained opti-mization benchmark functions where 119891

1ndash1198913are unimodal

functions 1198914ndash11989114

are multimodal functions 1198911ndash1198919have

certain theoretical minimum and 11989110ndash11989114

have uncertaintheoretical minimum

Computational Intelligence and Neuroscience 5

Table 1 The parameter set of BA and DLBA

Algorithms 119899 119891min 119891max 1198600

1198941199030

119894120572 120574 119899

119905

BA 40 0 100 (1 2) (0 01) 09 09 mdashDLBA 40 0 1 (1 2) (0 01) 09 mdash 5000

(1) 1198911 sphere function (the first function of De Jongrsquos test

set)

119891 (119909) =

119899

sum

119894=1

1199092

119894 minus10 le 119909

119894le 10 (15)

Here 119899 is dimensionality this function has a globalminimum119891min = 0 at 119909lowast = (0 0 0)

(2) 1198912 Schwegelrsquos problem 222

119891 (119909) =

119899

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816+

119899

prod

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816 minus10 le 119909

119894le 10 (16)

whose global minimum is obviously 119891min = 0 at 119909lowast= (0

0 0)(3) 1198913 Rosenbrock function

119891 (119909) =

119899minus1

sum

119894=1

[(119909119894minus 1)2+ 100(119909

119894+1minus 119909119894

2)

2

]

minus2408 le 119909119894le 2408

(17)

which has a global minimum 119891min = 0 at 119909lowast = (1 1 1)(4) 1198914 Eggcrate function

119891 (119909 119910) = 1199092+ 1199102+ 25 (sin2119909 + sin2119910)

(119909 119910) isin [minus2120587 2120587]

(18)

This 2-dimensional test function obviously gets the globalminimum 119891min = 0 at (0 0)

(5) 1198915 Ackleyrsquos function

119891 (119909) = minus20 exp[

[

minus

1

5

radic

1

119899

119899

sum

119894=1

1199092

119894]

]

minus exp[1119899

119899

sum

119894=1

cos (2120587119909119894)] + 20 + 119890

minus 30 le 119909 le 30

(19)

This function has a global minimum 119891min = 0 at 119909lowast=

(0 0 0) which is a multimodal function(6) 1198916 Griewangkrsquos function

119891 (119909) =

1

4000

119899

sum

119894=1

1199092

119894minus

119899

prod

119894=1

cos(119909119894

radic119894

) + 1 minus600 le 119909119894le 600

(20)

Its global minimum equal 119891min = 0 is obtainable for 119909lowast=

(0 0 0) the number of local minima for arbitrary 119899 is

unknown but in the two-dimensional case there are some500 local minima

(7) 1198917 Salomonrsquos function

119891 (119909) = minus cos(2120587radic119899

sum

119894=1

1199092

119894) + 01radic

119899

sum

119894=1

1199092

119894+ 1 minus5 le 119909

119894le 5

(21)

which has a global minimum 119891min = 0 at 119909lowast = (0 0 0)and is a multimodal function

(8) 1198918 Rastriginrsquos function

119891 (119909) = 10119899 +

119899

sum

119894=1

[1199092

119894minus 10 cos (2120587119909

119894)] minus512 le 119909

119894le 512

(22)

whose global minimum is 119891min = 0 at 119909lowast= (0 0 0)

for 119899 = 2 there are about 50 local minimizers arranged ina lattice-like configuration

(9) 1198919 Zakharovrsquos function

119891 (119909) =

119899

sum

119894=1

1199092

119894+ (

1

2

119899

sum

119894=1

119894119909119894)

2

+ (

1

2

119899

sum

119894=1

119894119909119894)

4

minus10 le 119909119894le 10

(23)

whose global minimum is 119891min = 0 at 119909lowast = (0 0 0) it isa multimodal function as well

(10) 11989110 Easomrsquos function

119891 (119909 119910)=minuscos (119909) cos (119910) exp [minus(119909minus120587)2+(119910minus120587)2]

minus10 le 119909 119910 le 10

(24)

whose global minimum is 119891min = minus1 at (120587 120587) it has manylocal minima

(11) 11989111 Schwegelrsquos function

119891 (119909) = minus

119899

sum

119894=1

119909119894sin(radic100381610038161003816

1003816119909119894

1003816100381610038161003816) minus500 le 119909

119894le 500 (25)

whose global minimum is 119891min asymp minus4189829119899 occuring at119909lowastasymp (4209687 4209687)(12) 119891

12 Shubertrsquos function

119891 (119909 119910)=[

5

sum

119894=1

119894 cos (119894+(119894+1) 119909)] sdot [5

sum

119894=1

119894 cos (119894+(119894+1) 119910)]

minus10 le 119909 119910 le 10

(26)

The number of local minima for this problem is not knownbut for 119899 = 2 the function has 760 local minima 18 of whichare global with 119891min asymp minus1867309

6 Computational Intelligence and Neuroscience

(13) 11989113 Xin-She Yangrsquos function

119891 (119909) = minus(

119899

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816) exp(minus

119899

sum

119894=1

1199092

119894) minus10 le 119909

119894le 10 (27)

which has multiple global minima for example for 119899 = 2 ithas 4 equal minima 119891min = minus1radic119890 asymp minus06065 at (05 05)(05-05) (minus05 05) and (minus05 minus05)

(14) 11989114 ldquoDrop Waverdquo function

119891 (119909) = minus

1 + cos(12radicsum119899119894=11199092

119894)

(12) (sum119899

119894=11199092

119894) + 2

minus512 le 119909119894le 512

(28)

This two-variable function is a multimodal test functionwhose global minimum is 119891min = minus1 occurs at 119909

lowast=

(0 0 0)

43 Comparison of Experimental Results The 2D landscapeof Schwegelrsquos function is shown in Figure 1 and this globalminimum can be found after about 720 FEs for 40 bats after6 iterations as shown in Figures 2 3 and 4

We adopt different terminated criteria aiming at differentbenchmark function we perform 100 times independentlyfor each test function and the record is given in Tables 2 and3

From Table 2 we can see that the DLBA performs muchbetter than the basic bat algorithm which converges muchfaster than BA under the fixed accuracy Tol = 10119890 minus 5 Inour experiment for the function 119891

4 we proposed that an

algorithm only costs 6 generations under the best situationand the average generations is 10 Furthermore for the firstgroup of test functions 119891

1ndash1198919 the DLBA averagely expend

439 generations when each function attains its terminatedcriteria however the BA needs 200 generations invariablyand the convergence speed of DLBA advances 29857 timesOn other hand the DLBAwhich obtained the accuracy of thesolution is much superior to BA which is more approximateto the theoretical value In aword it demonstrated thatDLBAhas fast convergence rate and high precision of the solutions

In Table 3 the five functions (11989110ndash11989114) independently

runs 100 times under the 24000 Fes we can clearly seethe precision of DLBA is obviously superior to the batalgorithm Some benchmark functions can easily attain thetheoretical optimal value In addition the standard deviationof DLBA is relatively low It shows that DLBA has superiorapproximation ability

Observe Tables 2 and 3 no matter high-dimensionalor low-dimensional we can find that DLBA can quicklyconverge to the global minima Furthermore Figures 5 67 8 and 9 show the convergence curves for some of thefunctions from a particular run of DLBA and BA which endat 200 generations The adaptive scheme generally convergesfaster than the basic scheme Here we select 119891

2(D = 20) 119891

4

(D = 2) 1198917(D = 5) 119891

11(D = 2) and 119891

13(D = 2)

DLBAnot only has superior approximation ability in low-dimensional space but also has excellent global search abilityin high-dimensional situation Table 4 is the experimental

result that DLBA performs 50 times independently under thehigh-dimensional situation As shown in Table 4 we can beconscious that the DLBA is effective under the multidimen-sional condition and acquired solution has higher accuracyeven approximate the theoretical value

Figures 10 11 12 13 and 14 are the distribution map ofoptimal fitness that is the selected functions independentlyperform 50 times under the multidimensional situationFigure 13 show that two ldquostraight linerdquo we can see in the figurethat DLBA reaches the global optimum(minus1) however BAfluctuates around 0 In order to display the fact of fluctuationwe magnify the two ldquostraight linerdquo and the amplifyingeffect are depicted in Figures 15 and 16 According to theexperimental results which are obtained from selected testfunctions DLBA presents higher precision than the originalBA on minimizing the outcome as the optimization goal

44 An Application for Solving Nonlinear Equations

Interval Arithmetic Benchmark (IAB) We consider onebenchmark problem proposed from interval arithmetic thebenchmark consists of the following system of [25]

0 = 1199091minus 025428722 minus 018324757119909

411990931199099

0 = 1199092minus 037842197 minus 016275449119909

1119909101199096

0 = 1199093minus 027162577 minus 016955071119909

1119909211990910

0 = 1199094minus 019807914 minus 015585316119909

711990911199096

0 = 1199095minus 044166728 minus 019950920119909

711990961199093

0 = 1199096minus 014654113 minus 018922793119909

8119909511990910

0 = 1199097minus 042937161 minus 021180486119909

211990951199098

0 = 1199098minus 007056438 minus 017081208119909

111990971199096

0 = 1199099minus 034504906 minus 019612740119909

1011990961199098

0 = 11990910minus 042651102 minus 021466544119909

411990981199091

(29)

Some of the solutions obtained as well as the functionvalues (which represent the values of the systemrsquos equationsobtained by replacing the variable values) are presented inTable 5 From Table 5 we can obviously observe that thefunction value of each equation solved byDLBA is superior towhich solved by EA [25] depending on the statistics of the 40function values the precision of function values is enhanced5199820E + 06 times by DLBA The convergence curve ofDLBA is depicted in Figure 17 Figure 17 show that DLBAhas fast convergence rate for solving nonlinear equationsThe achieved optimal fitness (the sum of function valuewith absolute value) in 50 times independent run is depictedin Figure 18 In order to reflect the precision of fitness wemagnify Figure 18 these optimal fitness that less than or equalto 0001 are plotted in Figure 19 We can found that there are32 times optimal fitness is less than 0001

Computational Intelligence and Neuroscience 7

Table 2 Comparison of BA and DLBA for several test functions under the first terminated criteria

TC BF Method Fitness IterationMin Mean Max Std Min Mean Max

Tol = 10119890 minus 5itermax = 200

1198911(119863 = 30) DLBA 430568008119864minus08 480356915119864minus06 996163402119864minus06 302060716119864minus06 10 176 26

BA 5681732462 13061959271 22382694528 3722960351 200 200 200

1198912(119863 = 20) DLBA 566761254119864minus07 633091639119864minus06 988929757119864minus06 248867687119864minus06 21 294 37

BA 2277136624 9335062167 273346090499 26941410301 200 200 200

1198913(119863 = 10) DLBA 566716240 739029403 828857171 037001430 200 200 200

BA 8224908050 29495948479 64076052284 10967161818 200 200 200

1198914(119863 = 2) DLBA 623475433119864minus09 393352701119864minus06 997022879119864minus06 277208752119864minus06 6 10 20

BA 105265280119864minus04 020705750 100771980 023116047 200 200 200

1198915(119863 = 5) DLBA 471737180119864minus07 640969812119864minus06 996088247119864minus06 271187009119864minus06 16 256 39

BA 115883977 302092450 586431360 069052427 200 200 200

1198916(119863 = 5) DLBA 138176777119864minus07 501027883119864minus06 992998256119864minus06 277338474119864minus06 11 184 56

BA 006234957 842851961 2990707979 612812390 200 200 200

1198917(119863 = 5) DLBA 778490674119864minus07 657288799119864minus06 997486019119864minus06 236637154119864minus06 18 46 114

BA 009987344 015318115 030215065 004923496 200 200 200

1198918(119863 = 5) DLBA 599983281119864minus08 447052788119864minus06 985695441119864minus06 281442645119864minus06 15 33 87

BA 724547772 1711077150 2590638177 437190073 200 200 200

1198919(119863 = 5) DLBA 665070016119864minus09 388692603119864minus06 969841027119864minus06 275444981119864minus06 10 151 23

BA 011106843 175623424 848211975 141343722 200 200 200

Table 3 Comparison of BA and DLBA for several test functions under the second terminated condition

TC BF Method FitnessMin Mean Max Std

FEs

11989110(119863 = 2) DLBA minus1 minus1 minus1 906493304119864 minus 17

BA minus099964845 minus098245546 minus090221232 001759602

11989111(119863 = 2) DLBA minus41898288727 minus41583523725 minus30054455266 1553948263

BA minus41898287874 minus40698044144 minus32810751671 2254615132

11989112(119863 = 5) DLBA minus18673090883 minus18673090883 minus18673090883 431613544119864 minus 13

BA minus18667541533 minus18401677767 minus17495599301 226333511

11989113(119863 = 2) DLBA minus060653066 minus060653066 minus060653066 863947249119864 minus 16

BA minus060652631 minus060411559 minus059656608 218888608119864 minus 03

11989114(119863 = 5) DLBA minus1 minus1 minus1 0

BA minus093624514 minus080454167 minus056977584 010270830

Table 4 Comparison of DLBA and BA under the high-dimensional situation

BF Method FitnessMin Mean Max Std

1198916(119863 = 128) DLBA 0 0 0 0

BA 258738759264 281140649000 296656723398 9396861133

1198919(119863 = 256) DLBA 266885939119864 minus 94 512093542119864 minus 69 256023720119864 minus 67 362071553119864 minus 68

BA 750339306546 825661302714 1150679192998 61337556609

1198918(119863 = 320) DLBA 0 0 0 0

BA 446859039467 472344624200 500857109760 13118818777

11989114(119863 = 512) DLBA minus1 minus1 minus1 0

BA minus148358700119864 minus 03 minus131084173119864 minus 03 minus120435493119864 minus 03 599288716119864 minus 05

1198911(119863 = 1024) DLBA 263790469119864 minus 109 473742970119864 minus 87 203861481119864 minus 85 289031313119864 minus 86

BA 1846348950157 1975226686813 2123923111285 56727878361

8 Computational Intelligence and Neuroscience

Table5Outcomeo

fDLB

AforIntervalA

rithm

eticBe

nchm

arkAnd

inCom

paris

onwith

EA

Metho

dEA

DLB

ASolutio

nsVa

riables

values

Functio

nsvalues

Varia

bles

values

Functio

nsvalues

Sol1

00464905115

02077959240

02578335208

722997083119864minus08

01013568357

02769798846

03810968901

minus284879905119864minus07

00840577820

01876863212

02787447331

minus263974619119864minus07

minus01388460309

03367887114

02006720536

306212480119864minus06

04943905739

00530391321

04452522319

774692656119864minus07

minus00760685163

02223730535

01491853777

133597657119864minus06

02475819110

01816084752

04320098178

minus787244534119864minus09

minus00170748156

00878963860

00734062202

341255029119864minus06

00003667535

03447200366

03459672005

324064555119864minus07

01481119311

02784227489

04273251429

minus118429644119864minus06

Sol2

01224819761

01318552790

02578332642

minus145675448119864minus07

01826200685

01964428361

03810968252

minus341069930119864minus07

02356779803

00364987069

02787462357

119723259119864minus06

minus00371150470

02354890155

02006687659

minus176074071119864minus08

03748181856

00675753064

04452514819

289773434119864minus07

02213311341

00739986588

01491839999

916571896119864minus08

00697813043

03607038292

04319795315

minus301421570119864minus05

00768058043

00059182979

00734021258

minus453845535119864minus07

minus00312153867

03767487763

03459672388

415572858119864minus07

01452667120

02811693568

04273281275

185998012119864minus06

Sol3

00633944399

01908436653

02578328072

minus595667763119864minus07

01017426933

02767897367

03810969084

minus239480772119864minus07

minus01051842285

03769063436

02787447816

minus204170456119864minus07

minus00477059943

02460900702

02006696507

671241513119864minus07

04149858326

00260337751

04452514467

minus438810984119864minus09

01215195321

00256054760

01491841089

189336147119864minus07

02539777159

01761486401

04320126793

297845790119864minus06

00843972823

01349869851

00734028724

779163472119864minus08

minus00534132992

03986395691

03459668311

330962522119864minus09

00880998746

03383563536

04273256312

minus646813249119864minus07

Sol4

01939820199

00603335280

02578382574

502240252119864minus06

00152114400

03633514726

03810978088

803072038119864minus07

01618654345

01097465792

02787430938

minus198215448119864minus06

00056985809

09114653768

02006688671

321352180119864minus08

01904538879

02502358229

04452573447

619105762119864minus06

minus01623604033

03089460561

01491746444

minus901451453119864minus06

01864448178

02428992222

04320068557

minus261979130119864minus06

minus00449302706

01144916285

00733954587

minus717750570119864minus06

01675935311

01774161896

03459538936

minus127733934119864minus05

minus00274959004

04539962587

04273210007

minus520897654119864minus06

Computational Intelligence and Neuroscience 9

3D surf

minus500

minus400

minus300

minus200

minus100

0100200300400500

z-la

bel

minus500minus400minus300minus200minus100

0100200300400500

minus500minus400minus300minus200minus100

0 100 200 300 400 500

Figure 1 The landscape of Schwegelrsquos function

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500

Figure 2 The locations of 40 bats in the initial phase

5 Conclusions

Aiming at the phenomenon of slow convergence rate andlow accuracy of bat algorithm we put forward animprovedbat algorithm with differential operator and Levy flightstrajectory (DLBA) based on the basic framework of batalgorithm (BA) the purpose is to improve the convergencerate and precision of bat algorithm In this paper we definethe frequency fluctuations up and down when the battracking prey which influence the batsrsquo location problemit is more graphic to simulate the batrsquos behavior Moreoverit can make the algorithm effectively jump out of the localoptimum to add Levy flights and differential operator themain reason is because Levy flight has prominent propertiesin the previously mentioned and the differential operatorcan guide bats find better solutions sequentially increase theconvergent speed

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500minus400minus300minus200minus100

0100200300400500

Figure 3 The locations of 40 bats after six iterations

In addition batsrsquo position variation is influenced bythe pulse emission rate and loudness as well Firstly pulseemission rate causes update of position and more and morenew position can be explored consequently increasing thediversity of population Secondly loudness is designed tostrengthen local search and to guide bats find better solutions

In this paper we tested 14 typical benchmark functionsand applied them to solve nonlinear equations the simulationresults not only showed that the proposed algorithm is feasi-ble and effective which ismore robust but also demonstratedthe superior approximation capabilities in high-dimensionalspaceThis is not surprising as the aim of developing the newalgorithm was to try to use the advantages of existing algo-rithms and other interesting feature inspired by the fantasticbehavior of echolocation ofmicrobatsNumerically speakingthese can be translated into two crucial characteristics of themodern metaheuristics intensification and diversificationIntensification intends to search around the current bestsolutions and select the best candidates or solutions whilediversification makes sure that the algorithm can explore thesearch space efficiently

10 Computational Intelligence and Neuroscience

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500

Figure 4 The locations of 40 bats after fifteen iterations

0 20 40 60 80 100 120 140 160 180 2000

100200300400500600700800900

1000

Iteration

Fitn

ess

BADLBA

Figure 5 Convergence curves for the function 1198912

BADLBA

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12

Iteration

Fitn

ess

Figure 6 Convergence curves for the function 1198914

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07

Iteration

Fitn

ess

BADLBA

Figure 7 Convergence curves for the function 1198917

BADLBA

0 20 40 60 80 100 120 140 160 180 200minus420

minus410

minus400

minus390

minus380

minus370

minus360

minus350

minus340

Iteration

Fitn

ess

Figure 8 Convergence curves for the function 11989111

BADLBA

0 20 40 60 80 100 120 140 160 180 200minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Iteration

Fitn

ess

Figure 9 Convergence curves for the function 11989113

Computational Intelligence and Neuroscience 11

0 5 10 15 20 25 30 35 40 45 500

500

1000

1500

2000

2500

3000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 10 Distribution of optimal fitness for 1198916(D = 128)

0 5 10 15 20 25 30 35 40 45 500

2000

4000

6000

8000

10000

12000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 11 Distribution of optimal fitness for 1198919(D = 256)

BADLBA

0 5 10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000

6000

Number of running

Opt

imal

fitn

ess

Figure 12 Distribution of optimal fitness for 1198918(D = 320)

BADLBA

0 5 10 15 20 25 30 35 40 45 50minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Number of running

Opt

imal

fitn

ess

Figure 13 Distribution of optimal fitness for 11989114(D = 512)

BADLBA

Number of running

Opt

imal

fitn

ess

0 5 10 15 20 25 30 35 40 45 500

05

1

15

2

25times104

Figure 14 Distribution of optimal fitness for 1198911(D = 1024)

This potentially powerful optimization strategy can easilybe extended to study multiobjective optimization applica-tions with various constraints even to NP-hard problemsFurther studies can focus on the sensitivity and parameterstudies and their possible relationships with the convergencerate of the algorithm

Acknowledgments

This work is supported by the National Science Foundationof China under Grant no 61165015 Key Project of GuangxiScience Foundation under Grant no 2012GXNSFDA053028Key Project of Guangxi High School Science Foundationunder Grant no 20121ZD008 and the Fund by OpenResearch Fund Program of Key Lab of Intelligent Perception

12 Computational Intelligence and Neuroscience

0 5 10 15 20 25 30 35 40 45 50minus15

minus145

minus14

minus135

minus13

minus125

minus12

Number of running

Opt

imal

fitn

ess

BA

times10minus3

Figure 15 Details of BA line in Figure 12

0 5 10 15 20 25 30 35 40 45 50minus2minus18minus16minus14minus12minus1minus08minus06minus04minus02

0

Number of running

Opt

imal

fitn

ess

DLBA

Figure 16 Details of DLBA line in Figure 12

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7

Iteration

Fitn

ess

DLBA

Figure 17 Convergence curves for DLAB

0 5 10 15 20 25 30 35 40 45 500

002

004

006

008

01

012

014

016

018

Number of running

Opt

imal

fitn

ess

DLBA

Figure 18 Distribution of optimal fitness for DLAB

0 5 10 15 20 25 30 35 40 45 500

010203040506070809

1

Number of running

Opt

imal

fitn

ess

times10minus3

Figure 19 Distribution of optimal fitness less than 0001 inFigure 18

and Image Understanding of Ministry of Education of Chinaunder Grant no IPIU01201100

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of IEEE International Conference on NeuralNetworks (ICNN rsquo95) vol 4 pp 1942ndash1948 December 1995

[2] X-S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress Bristol UK 2nd edition 2010

[3] X-S Yang ldquoFirefly algorithms for multimodal optimizationrdquoin Proceedings of the 5th International Conference on StochasticAlgorithms Foundations and Applications (SAGA rsquo09) pp 169ndash178 2009

[4] B Alatas ldquoACROA artificial chemical reaction optimizationalgorithm for global optimizationrdquo Expert Systems with Appli-cations vol 38 no 10 pp 13170ndash13180 2011

Computational Intelligence and Neuroscience 13

[5] K N Krishnanand and D Ghose ldquoGlowworm warm opti-mization for searching higher dimensional spacesrdquo Studies inComputational Intelligence vol 248 pp 61ndash75 2009

[6] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[7] K Price R Storn and J Lampinen Differential Evolution-APractical Approach to Global Optimization Springer BerlinGermany 2005

[8] httpwwwicsiberkeleyedusim storn [9] R Storn and K Price ldquoDifferential evolutionmdasha simple and

efficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[10] X-S Yang ldquoA new metaheuristic Bat-inspired AlgorithmrdquoStudies in Computational Intelligence vol 284 pp 65ndash74 2010

[11] A Mucherino and O Seref ldquoMonkey search a novel meta-heuristic search for global optimizationrdquo in Proceedings of theConference on Data Mining Systems Analysis and Optimizationin Biomedicine vol 953 pp 162ndash173 March 2007

[12] KM Passino ldquoBiomimicry of bacterial foraging for distributedoptimization and controlrdquo IEEE Control Systems Magazine vol22 no 3 pp 52ndash67 2002

[13] G R Reynolds ldquoCultural algorithms theory and applicationrdquoin New Iders in Optimization D Corne M Dorigo and FGlover Eds pp 367ndash378 McGraw-Hill New York NY USA1999

[14] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[15] R Oftadeh M J Mahjoob and M Shariatpanahi ldquoA novelmeta-heuristic optimization algorithm inspired by group hunt-ing of animals hunting searchrdquo Computers and Mathematicswith Applications vol 60 no 7 pp 2087ndash2098 2010

[16] P W Tsai J S Pan B Y Liao M J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2011

[17] X-S Yang ldquoBat algorithm for multi-objective optimizationrdquoInternational Journal of Bio-Inspired Computation Bio-InspiredComputation (IJBIC) vol 3 no 5 pp 267ndash274 2011

[18] T C Bora L S Coelho and L Lebensztajn ldquoBat-Inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012

[19] B B Mandelbrot The Fractal Geometry of Nature WH Free-man New York NY USA 1983

[20] V A Chechkin R Metzler J Klafter and V Y GoncharldquoIntroduction to the theory of Levy flightsrdquo in AnomalousTransport Foundations and Applications pp 129ndash162 Wiley-VCH Cambridge UK 2008

[21] A M Reynolds and M A Frye ldquoFree-flight odor tracking inDrosophila is consistent with an optimal intermittent scale-freesearchrdquo PLoS ONE vol 2 no 4 p e354 2007

[22] C Brown L S Liebovitch andRGlendon ldquoLevy fights inDobeJursquohoansi foraging patternsrdquo Human Ecology vol 35 no 1 pp129ndash138 2007

[23] P Barthelemy J Bertolotti andD SWiersma ldquoA Levy flight forlightrdquo Nature vol 453 no 7194 pp 495ndash498 2008

[24] N Mercadier W Guerin M Chevrollier and R Kaiser ldquoLevyflights of photons in hot atomic vapoursrdquoNature Physics vol 5no 8 pp 602ndash605 2009

[25] C Grosan and A Abraham ldquoA new approach for solvingnonlinear equations systemsrdquo IEEE Transactions on SystemsMan and Cybernetics A vol 38 no 3 pp 698ndash714 2008

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

2 Computational Intelligence and Neuroscience

In Section 3 we gave the design framework of DLBA Theimplementation and comparison of improved algorithm arepresented in Section 4 Finally we concluded this paper inSection 5

2 Bat Algorithm Leacutevy Flights andNonlinear Equations

21 Behaviour of Microbats Most of microbats have ad-vanced capability of echolocation These bats can emit a veryloud and short sound pulse the echo that reflects back fromthe surrounding objects is received by their extraordinary bigauricle Then this feedback information of echo is analyzedin their subtle brainThey not only can discriminate directionfor their own flight pathway according to the echo but alsocan distinguish different insects and obstacles to hunt preyand avoid a collision effectively in the day or night

22 Bat Algorithm (see [10]) First of all let us briefly reviewthe basics of the BA for single-objective optimization In thebasic BA developed by Yang in 2010 in order to propose thebat algorithm inspired by the echolocation characteristics ofmicrobats the following approximate or idealised rules wereused

IR1 All bats use echolocation to sense distance and they alsoldquoknowrdquo the difference between foodprey and backgroundbarriers in some magical way

IR2 Bats fly randomly with velocity V119894at position 119909

119894with

a fixed frequency 119891min varying wavelength 120582 and loudness1198600to search for prey They can automatically adjust the

wavelength (or frequency) of their emitted pulses and adjustthe rate of pulse emission 119903 isin [0 1] depending on theproximity of their target

IR3 Although the loudness can vary in many ways weassume that the loudness varies from a large (positive) 119860

0to

a minimum constant value 119860minIn addition for simplicity they also use the follow-

ing approximations in general the frequency 119891 in arange [119891min 119891max] corresponds to a range of wavelengths[120582min 120582max] In fact they just vary in the frequency whilefixed in the wavelength 120582 and assume 119891 isin [0 119891max] in theirimplementationThis is because 120582 and119891 are related due to thefact that 120582119891 = V is constant

In simulations they use virtual bats naturally to define theupdated rules of their positions 119909

119894and velocities V

119894in a D-

dimensional search spaceThe new solutions 119909119905119894and velocities

V119905119894at time step 119905 are given by

119891119894= 119891min + (119891max minus 119891min) 120573

V119905

119894= V119905minus1

119894+ (119909119905

119894minus 119909lowast) 119891119894

119909119905

119894= 119909119905minus1

119894+ V119905

119894

(1)

where 120573 isin [0 1] is a random vector drawn from a uniformdistribution Here 119909

lowastis the current global best location

(solution) which is located after comparing all the solutionsamong all the 119899 bats

For the local search part once a solution is selectedamong the current best solutions a new solution for each batis generated locally using random walk

119909new = 119909old + 120576119860 119905 (2)

where 120576 isin [minus1 1] is a random number while119860119905= ⟨119860119905

119894⟩ is the

average loudness of all the bats at this time stepFurthermore the loudness 119860

119894and the rate 119903

119894of pulse

emission have to be updated accordingly as the iterationsproceed These formulas are

119860119905+1

119894= 120572119860119905

119894 (3)

119903119905+1

119894= 1199030

119894[1 minus exp (minus120574119905)] (4)

where 120572 and 120574 are constantsBased on these approximations and idealization the

basic steps of the bat algorithm can be summarized in thePseudocode 1

23 Levy Flights Levy flights are Markov processes whichdiffer from regular Brownian motion whose individualjumps have lengths that are distributed with the probabilitydensity function (PDF) 120582(119909) decaying at large 119909 as 120582(119909) =|119909|minus1minus120572 with 0 lt 120572 lt 2 Due to the divergence of their var-

iance 120582(119909) ≃ 119909|minus1minus120572 extremely long jumps may occur andtypical trajectories are self-similar on all scales showingclusters of shorter jumps interspersed by long excursions [19]Levy flight has the following properties [20]

(1) Stability distribution of the sum of independentidentically distributed stable random variables equalto distribution of each variable

(2) Power law asymptotics (ldquoheavy tailsrdquo)(3) Generalized Central Limit Theorem The central limit

theorem states that the sum of a number of inde-pendent and identically distributed (iid) randomvariables with finite variances will tend to a normaldistribution as the number of variables grows

(4) Which has an infinite variance with an infinite meanvalue

Due to the these remarkable properties of stable distri-butions it is now believed that the Levy statistics provide aframework for the description ofmany natural phenomena inphysical chemical biological and economical systems froma general common point of view

Furthermore various studies have shown that the flightbehaviour of many animals and insects has demonstratedthe typical characteristics of Levy lights A recent studyby Reynolds and Frye shows that fruit flies or Drosophilamelanogaster explore their landscape using a series of straightflight paths punctuated by a sudden 90∘ turn leading to aLevy flight-style intermittent scale-free search pattern [21]Studies on human behaviour such as the Jursquohoansi hunter-gatherer foraging patterns also show the typical feature ofLevy flights [22] The conclusion that light is related to Levyflights is proposed by Barthelemy et al (2008) [23] The

Computational Intelligence and Neuroscience 3

Objective function 119891(119909) 119909 = [1199091 1199092 119909

119889]119879

Initialize the bat population 119909119894(119894 = 1 2 119899) and V

119894

Define pulse frequency 119891119894at 119909119894initialize pulse rates 119903

119894and the loudness 119860

119894

While (119905 ltMax number of iterations)Generate new solutions by adjusting frequencyand updating velocities and locationssolutions (1)if (rand gt 119903

119894)

Select a solution among the best solutionsGenerate a local solution around the selected best solution

end ifGenerate a new solution by flying randomlyif (rand lt 119860

119894amp 119891(119909

119894) lt 119891(119909

lowast))

Accept the new solutionsIncrease 119903

119894and reduce 119860

119894

end ifRank the bats and find the current best 119909

lowast

end whilePostprocess results and visualization

Pseudocode 1

study by Mercadier et al shows that the Levy flights ofphotons in hot atomic vapours (2009) [24] Subsequentlysuch behaviour has been applied to optimization and optimalsearch and preliminary results show its promising capability

24 Description of the Nonlinear Equations (see [25]) Thegeneral form of nonlinear equations with real variables isdescribed as follows

119892119894 (119883) = 0 119894 = 1 2 119899 (5)

where 119883 = (1199091 1199092 119909

119899) isin 119863 sub 119877

119899 119863 = (1199091 1199092 119909

119899) |

119886119894le 119909119894le 119887119894 119894 = 1 2 119899

In solving nonlinear equations process for DLBA thefitness function can be constructed by

119866 (119883) =

119899

sum

119894=1

1003816100381610038161003816119892119894 (119909)1003816100381610038161003816 (6)

So the solving of nonlinear equations can be translated intoan optimization problem in domain119863

min 119866 (119883)

st (1199091 1199092 119909

119899) isin 119863

(7)

Consequently the optimal value of (7) is exactly the solutionof (5)

3 DLBA

Inspired by Yangrsquos method we propose an improved batalgorithm based on differential operator and Levy-flightstrajectory (DLBA) based on the basic structure of BA andre-estimate the characters used in the original BA In DLBAnot only the movement of the bat is quite different from theoriginal BA but also the local search process is different

In DLBA the frequency fluctuates up and down whichcan change self-adaptively and the differential operator isintroduced which is similar to themutation operation of DEthe frequency 119891 is similar to the scale factor 119865 of DEbest2So the frequency updated formulae of a bat are defined asfollows

119891119905

1119894= ((119891

1min minus 1198911max)119905

119899119905

+ 1198911max)1205731 (8)

119891119905

2119894= ((119891

2max minus 1198912min)119905

119899119905

+ 1198912min)1205732 (9)

where 1205731 1205732isin [0 1] is a random vector drawn from a uni-

form distribution 1198911max = 119891

2max 1198911min = 1198912min 119899119905 is a

fixed parameter In DLBA the position 119909119905119894of each bat indi-

vidual are updated with (10) which is different from originalBAThis can preferably incorporate the echolocation charac-teristics of microbats

119909119905+1

119894= 119909119905

best + 119891119905

1119894(119909119905

1199031minus 119909119905

1199032) + 119891119905

2119894(119909119905

1199033minus 119909119905

1199034) (10)

where 119909119905best is the current global best location (solution)which is located after comparing all the solutions among allthe 119899 bats in 119905 generation 119909119905

119903119894is the bat individual in the bat

swarm and this can be achieved by randomizationIn addition Levy flight haves the prominent properties

increase the diversity of population sequentially which canmake the algorithmeffectively jumpout of the local optimumSo we let these bats perform the Levy flights with (11) beforethe position updating

119909119905

119894= 119905minus1

119894+ 120583 sign [rand minus 05] oplus Levy (11)

where 120583 is a random parameter drawn from a uniformdistribution sign oplusmeans entrywise multiplications rand isin[0 1] and random step length Levy obeys Levy distribution

Levy sim 119906 = 119905minus120582 (1 lt 120582 le 3) (12)

4 Computational Intelligence and Neuroscience

InputObjective function 119891(119909) 119909 = [1199091 1199092 119909

119889]119879 algorithmrsquos parameters 119899 119891min 119891max 120572 119899119905

BEGINFor 119894 = 1 To 119899 do

Initialize 119909119905119894 1199030119894 1198600119894

Calculating the initial fitnessEnd ForWhile (119905 lt itermax)

Update position use formula (8)sim(12)Evaluate fitness 119891(119909

119894) to find 119909best

For 119894 = 1 To 119899 doIf (rand gt 119903

119894)

Generate a 119909new around the selected best solution use formula (2)End If

End ForEvaluate fitness 119891(119909new)For 119894 = 1 To 119899 do

If (rand lt 119860119894)

Generate a 119909new around the selected best solution use formula (13)End If

End ForEvaluate fitness 119891(119909new)If (119891(119909119905+1best) lt 119891(119909

119905

best))Update 119903119905

119894 119860119905119894use formula (3) (14)

End If119905 = 119905 + 1

EndWhileENDOutput 119909best fitnessmin

Pseudocode 2

On the other hand each bat should have different valuesof loudness and pulse emission rate while the rate of pulseemission is relatively low and the loudness is relatively highDuring the search process the loudness usually decreaseswhile the rate of pulse emission increases Batsrsquo positionvariation is influenced by the pulse emission rate and loud-ness as well Firstly pulse emission rate 119903

119894causes fluctuation

of position using (2) sequentially more and more newposition can be explored Secondly loudness119860

119894is designed to

strengthen local search and to guide bats find better solutions

119909new = 119909best + 120578119903119905 (13)

where 120578 isin [minus1 1] is a random parameter and 119909best is thecurrent global best location (solution) in whole bats swarmWhile 119903

119905= ⟨119903119905

119894⟩ is the average pulse emission rate of all the

bats at this generationThe new rate of pulse emission 119903119905

119894and loudness119860119905

119894at time

step 119905 are given by

119903119905+1

119894= 119903119905

119894(

119905

119899119905

)

3

(14)

where 119903119905119894is time varying their loudness and emission rates

will be updated only if the best solution of the currentgeneration is better than the best solution of last generationwhich means that these bats are moving towards the optimalsolution The pseudocode of DLBA can be depicted as inPseudocode 2

4 The Simulation and Analysis

41 Parametric Studies The proposed DLBA is imple-mented in MATLAB simulation platform CPU Intel XeonE5405200GHz OS Microsoft Windows Server 2003Enterprise Edition SP2 RAM 1GHz Matlab VersionR2009a The parameter setting for BA and DLBA are listedin Table 1 the parameters of BA are recommended in theoriginal article

The stopping criterion can be defined in many ways Weadopt two terminated criteria we can use a given tolerance(Tol = 10119890 minus 5) for a test function that have certain mini-mum value in theory on the contrary each simulation runterminates when a certain number of function evaluations(FEs) have been reached In this paper FEs could be obtainedby population size multiplied by the number of iteration Inour experiment FEs = 24000 BA (= 300lowast40lowast2) DLBA (=200lowast40lowast3)

42 BenchmarkTest Function In order to validate the validityof DLBA we selected the 14 benchmark functions to experi-mentize The benchmark set include unimodal multimodalhigh-dimensional and low-dimensional unconstrained opti-mization benchmark functions where 119891

1ndash1198913are unimodal

functions 1198914ndash11989114

are multimodal functions 1198911ndash1198919have

certain theoretical minimum and 11989110ndash11989114

have uncertaintheoretical minimum

Computational Intelligence and Neuroscience 5

Table 1 The parameter set of BA and DLBA

Algorithms 119899 119891min 119891max 1198600

1198941199030

119894120572 120574 119899

119905

BA 40 0 100 (1 2) (0 01) 09 09 mdashDLBA 40 0 1 (1 2) (0 01) 09 mdash 5000

(1) 1198911 sphere function (the first function of De Jongrsquos test

set)

119891 (119909) =

119899

sum

119894=1

1199092

119894 minus10 le 119909

119894le 10 (15)

Here 119899 is dimensionality this function has a globalminimum119891min = 0 at 119909lowast = (0 0 0)

(2) 1198912 Schwegelrsquos problem 222

119891 (119909) =

119899

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816+

119899

prod

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816 minus10 le 119909

119894le 10 (16)

whose global minimum is obviously 119891min = 0 at 119909lowast= (0

0 0)(3) 1198913 Rosenbrock function

119891 (119909) =

119899minus1

sum

119894=1

[(119909119894minus 1)2+ 100(119909

119894+1minus 119909119894

2)

2

]

minus2408 le 119909119894le 2408

(17)

which has a global minimum 119891min = 0 at 119909lowast = (1 1 1)(4) 1198914 Eggcrate function

119891 (119909 119910) = 1199092+ 1199102+ 25 (sin2119909 + sin2119910)

(119909 119910) isin [minus2120587 2120587]

(18)

This 2-dimensional test function obviously gets the globalminimum 119891min = 0 at (0 0)

(5) 1198915 Ackleyrsquos function

119891 (119909) = minus20 exp[

[

minus

1

5

radic

1

119899

119899

sum

119894=1

1199092

119894]

]

minus exp[1119899

119899

sum

119894=1

cos (2120587119909119894)] + 20 + 119890

minus 30 le 119909 le 30

(19)

This function has a global minimum 119891min = 0 at 119909lowast=

(0 0 0) which is a multimodal function(6) 1198916 Griewangkrsquos function

119891 (119909) =

1

4000

119899

sum

119894=1

1199092

119894minus

119899

prod

119894=1

cos(119909119894

radic119894

) + 1 minus600 le 119909119894le 600

(20)

Its global minimum equal 119891min = 0 is obtainable for 119909lowast=

(0 0 0) the number of local minima for arbitrary 119899 is

unknown but in the two-dimensional case there are some500 local minima

(7) 1198917 Salomonrsquos function

119891 (119909) = minus cos(2120587radic119899

sum

119894=1

1199092

119894) + 01radic

119899

sum

119894=1

1199092

119894+ 1 minus5 le 119909

119894le 5

(21)

which has a global minimum 119891min = 0 at 119909lowast = (0 0 0)and is a multimodal function

(8) 1198918 Rastriginrsquos function

119891 (119909) = 10119899 +

119899

sum

119894=1

[1199092

119894minus 10 cos (2120587119909

119894)] minus512 le 119909

119894le 512

(22)

whose global minimum is 119891min = 0 at 119909lowast= (0 0 0)

for 119899 = 2 there are about 50 local minimizers arranged ina lattice-like configuration

(9) 1198919 Zakharovrsquos function

119891 (119909) =

119899

sum

119894=1

1199092

119894+ (

1

2

119899

sum

119894=1

119894119909119894)

2

+ (

1

2

119899

sum

119894=1

119894119909119894)

4

minus10 le 119909119894le 10

(23)

whose global minimum is 119891min = 0 at 119909lowast = (0 0 0) it isa multimodal function as well

(10) 11989110 Easomrsquos function

119891 (119909 119910)=minuscos (119909) cos (119910) exp [minus(119909minus120587)2+(119910minus120587)2]

minus10 le 119909 119910 le 10

(24)

whose global minimum is 119891min = minus1 at (120587 120587) it has manylocal minima

(11) 11989111 Schwegelrsquos function

119891 (119909) = minus

119899

sum

119894=1

119909119894sin(radic100381610038161003816

1003816119909119894

1003816100381610038161003816) minus500 le 119909

119894le 500 (25)

whose global minimum is 119891min asymp minus4189829119899 occuring at119909lowastasymp (4209687 4209687)(12) 119891

12 Shubertrsquos function

119891 (119909 119910)=[

5

sum

119894=1

119894 cos (119894+(119894+1) 119909)] sdot [5

sum

119894=1

119894 cos (119894+(119894+1) 119910)]

minus10 le 119909 119910 le 10

(26)

The number of local minima for this problem is not knownbut for 119899 = 2 the function has 760 local minima 18 of whichare global with 119891min asymp minus1867309

6 Computational Intelligence and Neuroscience

(13) 11989113 Xin-She Yangrsquos function

119891 (119909) = minus(

119899

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816) exp(minus

119899

sum

119894=1

1199092

119894) minus10 le 119909

119894le 10 (27)

which has multiple global minima for example for 119899 = 2 ithas 4 equal minima 119891min = minus1radic119890 asymp minus06065 at (05 05)(05-05) (minus05 05) and (minus05 minus05)

(14) 11989114 ldquoDrop Waverdquo function

119891 (119909) = minus

1 + cos(12radicsum119899119894=11199092

119894)

(12) (sum119899

119894=11199092

119894) + 2

minus512 le 119909119894le 512

(28)

This two-variable function is a multimodal test functionwhose global minimum is 119891min = minus1 occurs at 119909

lowast=

(0 0 0)

43 Comparison of Experimental Results The 2D landscapeof Schwegelrsquos function is shown in Figure 1 and this globalminimum can be found after about 720 FEs for 40 bats after6 iterations as shown in Figures 2 3 and 4

We adopt different terminated criteria aiming at differentbenchmark function we perform 100 times independentlyfor each test function and the record is given in Tables 2 and3

From Table 2 we can see that the DLBA performs muchbetter than the basic bat algorithm which converges muchfaster than BA under the fixed accuracy Tol = 10119890 minus 5 Inour experiment for the function 119891

4 we proposed that an

algorithm only costs 6 generations under the best situationand the average generations is 10 Furthermore for the firstgroup of test functions 119891

1ndash1198919 the DLBA averagely expend

439 generations when each function attains its terminatedcriteria however the BA needs 200 generations invariablyand the convergence speed of DLBA advances 29857 timesOn other hand the DLBAwhich obtained the accuracy of thesolution is much superior to BA which is more approximateto the theoretical value In aword it demonstrated thatDLBAhas fast convergence rate and high precision of the solutions

In Table 3 the five functions (11989110ndash11989114) independently

runs 100 times under the 24000 Fes we can clearly seethe precision of DLBA is obviously superior to the batalgorithm Some benchmark functions can easily attain thetheoretical optimal value In addition the standard deviationof DLBA is relatively low It shows that DLBA has superiorapproximation ability

Observe Tables 2 and 3 no matter high-dimensionalor low-dimensional we can find that DLBA can quicklyconverge to the global minima Furthermore Figures 5 67 8 and 9 show the convergence curves for some of thefunctions from a particular run of DLBA and BA which endat 200 generations The adaptive scheme generally convergesfaster than the basic scheme Here we select 119891

2(D = 20) 119891

4

(D = 2) 1198917(D = 5) 119891

11(D = 2) and 119891

13(D = 2)

DLBAnot only has superior approximation ability in low-dimensional space but also has excellent global search abilityin high-dimensional situation Table 4 is the experimental

result that DLBA performs 50 times independently under thehigh-dimensional situation As shown in Table 4 we can beconscious that the DLBA is effective under the multidimen-sional condition and acquired solution has higher accuracyeven approximate the theoretical value

Figures 10 11 12 13 and 14 are the distribution map ofoptimal fitness that is the selected functions independentlyperform 50 times under the multidimensional situationFigure 13 show that two ldquostraight linerdquo we can see in the figurethat DLBA reaches the global optimum(minus1) however BAfluctuates around 0 In order to display the fact of fluctuationwe magnify the two ldquostraight linerdquo and the amplifyingeffect are depicted in Figures 15 and 16 According to theexperimental results which are obtained from selected testfunctions DLBA presents higher precision than the originalBA on minimizing the outcome as the optimization goal

44 An Application for Solving Nonlinear Equations

Interval Arithmetic Benchmark (IAB) We consider onebenchmark problem proposed from interval arithmetic thebenchmark consists of the following system of [25]

0 = 1199091minus 025428722 minus 018324757119909

411990931199099

0 = 1199092minus 037842197 minus 016275449119909

1119909101199096

0 = 1199093minus 027162577 minus 016955071119909

1119909211990910

0 = 1199094minus 019807914 minus 015585316119909

711990911199096

0 = 1199095minus 044166728 minus 019950920119909

711990961199093

0 = 1199096minus 014654113 minus 018922793119909

8119909511990910

0 = 1199097minus 042937161 minus 021180486119909

211990951199098

0 = 1199098minus 007056438 minus 017081208119909

111990971199096

0 = 1199099minus 034504906 minus 019612740119909

1011990961199098

0 = 11990910minus 042651102 minus 021466544119909

411990981199091

(29)

Some of the solutions obtained as well as the functionvalues (which represent the values of the systemrsquos equationsobtained by replacing the variable values) are presented inTable 5 From Table 5 we can obviously observe that thefunction value of each equation solved byDLBA is superior towhich solved by EA [25] depending on the statistics of the 40function values the precision of function values is enhanced5199820E + 06 times by DLBA The convergence curve ofDLBA is depicted in Figure 17 Figure 17 show that DLBAhas fast convergence rate for solving nonlinear equationsThe achieved optimal fitness (the sum of function valuewith absolute value) in 50 times independent run is depictedin Figure 18 In order to reflect the precision of fitness wemagnify Figure 18 these optimal fitness that less than or equalto 0001 are plotted in Figure 19 We can found that there are32 times optimal fitness is less than 0001

Computational Intelligence and Neuroscience 7

Table 2 Comparison of BA and DLBA for several test functions under the first terminated criteria

TC BF Method Fitness IterationMin Mean Max Std Min Mean Max

Tol = 10119890 minus 5itermax = 200

1198911(119863 = 30) DLBA 430568008119864minus08 480356915119864minus06 996163402119864minus06 302060716119864minus06 10 176 26

BA 5681732462 13061959271 22382694528 3722960351 200 200 200

1198912(119863 = 20) DLBA 566761254119864minus07 633091639119864minus06 988929757119864minus06 248867687119864minus06 21 294 37

BA 2277136624 9335062167 273346090499 26941410301 200 200 200

1198913(119863 = 10) DLBA 566716240 739029403 828857171 037001430 200 200 200

BA 8224908050 29495948479 64076052284 10967161818 200 200 200

1198914(119863 = 2) DLBA 623475433119864minus09 393352701119864minus06 997022879119864minus06 277208752119864minus06 6 10 20

BA 105265280119864minus04 020705750 100771980 023116047 200 200 200

1198915(119863 = 5) DLBA 471737180119864minus07 640969812119864minus06 996088247119864minus06 271187009119864minus06 16 256 39

BA 115883977 302092450 586431360 069052427 200 200 200

1198916(119863 = 5) DLBA 138176777119864minus07 501027883119864minus06 992998256119864minus06 277338474119864minus06 11 184 56

BA 006234957 842851961 2990707979 612812390 200 200 200

1198917(119863 = 5) DLBA 778490674119864minus07 657288799119864minus06 997486019119864minus06 236637154119864minus06 18 46 114

BA 009987344 015318115 030215065 004923496 200 200 200

1198918(119863 = 5) DLBA 599983281119864minus08 447052788119864minus06 985695441119864minus06 281442645119864minus06 15 33 87

BA 724547772 1711077150 2590638177 437190073 200 200 200

1198919(119863 = 5) DLBA 665070016119864minus09 388692603119864minus06 969841027119864minus06 275444981119864minus06 10 151 23

BA 011106843 175623424 848211975 141343722 200 200 200

Table 3 Comparison of BA and DLBA for several test functions under the second terminated condition

TC BF Method FitnessMin Mean Max Std

FEs

11989110(119863 = 2) DLBA minus1 minus1 minus1 906493304119864 minus 17

BA minus099964845 minus098245546 minus090221232 001759602

11989111(119863 = 2) DLBA minus41898288727 minus41583523725 minus30054455266 1553948263

BA minus41898287874 minus40698044144 minus32810751671 2254615132

11989112(119863 = 5) DLBA minus18673090883 minus18673090883 minus18673090883 431613544119864 minus 13

BA minus18667541533 minus18401677767 minus17495599301 226333511

11989113(119863 = 2) DLBA minus060653066 minus060653066 minus060653066 863947249119864 minus 16

BA minus060652631 minus060411559 minus059656608 218888608119864 minus 03

11989114(119863 = 5) DLBA minus1 minus1 minus1 0

BA minus093624514 minus080454167 minus056977584 010270830

Table 4 Comparison of DLBA and BA under the high-dimensional situation

BF Method FitnessMin Mean Max Std

1198916(119863 = 128) DLBA 0 0 0 0

BA 258738759264 281140649000 296656723398 9396861133

1198919(119863 = 256) DLBA 266885939119864 minus 94 512093542119864 minus 69 256023720119864 minus 67 362071553119864 minus 68

BA 750339306546 825661302714 1150679192998 61337556609

1198918(119863 = 320) DLBA 0 0 0 0

BA 446859039467 472344624200 500857109760 13118818777

11989114(119863 = 512) DLBA minus1 minus1 minus1 0

BA minus148358700119864 minus 03 minus131084173119864 minus 03 minus120435493119864 minus 03 599288716119864 minus 05

1198911(119863 = 1024) DLBA 263790469119864 minus 109 473742970119864 minus 87 203861481119864 minus 85 289031313119864 minus 86

BA 1846348950157 1975226686813 2123923111285 56727878361

8 Computational Intelligence and Neuroscience

Table5Outcomeo

fDLB

AforIntervalA

rithm

eticBe

nchm

arkAnd

inCom

paris

onwith

EA

Metho

dEA

DLB

ASolutio

nsVa

riables

values

Functio

nsvalues

Varia

bles

values

Functio

nsvalues

Sol1

00464905115

02077959240

02578335208

722997083119864minus08

01013568357

02769798846

03810968901

minus284879905119864minus07

00840577820

01876863212

02787447331

minus263974619119864minus07

minus01388460309

03367887114

02006720536

306212480119864minus06

04943905739

00530391321

04452522319

774692656119864minus07

minus00760685163

02223730535

01491853777

133597657119864minus06

02475819110

01816084752

04320098178

minus787244534119864minus09

minus00170748156

00878963860

00734062202

341255029119864minus06

00003667535

03447200366

03459672005

324064555119864minus07

01481119311

02784227489

04273251429

minus118429644119864minus06

Sol2

01224819761

01318552790

02578332642

minus145675448119864minus07

01826200685

01964428361

03810968252

minus341069930119864minus07

02356779803

00364987069

02787462357

119723259119864minus06

minus00371150470

02354890155

02006687659

minus176074071119864minus08

03748181856

00675753064

04452514819

289773434119864minus07

02213311341

00739986588

01491839999

916571896119864minus08

00697813043

03607038292

04319795315

minus301421570119864minus05

00768058043

00059182979

00734021258

minus453845535119864minus07

minus00312153867

03767487763

03459672388

415572858119864minus07

01452667120

02811693568

04273281275

185998012119864minus06

Sol3

00633944399

01908436653

02578328072

minus595667763119864minus07

01017426933

02767897367

03810969084

minus239480772119864minus07

minus01051842285

03769063436

02787447816

minus204170456119864minus07

minus00477059943

02460900702

02006696507

671241513119864minus07

04149858326

00260337751

04452514467

minus438810984119864minus09

01215195321

00256054760

01491841089

189336147119864minus07

02539777159

01761486401

04320126793

297845790119864minus06

00843972823

01349869851

00734028724

779163472119864minus08

minus00534132992

03986395691

03459668311

330962522119864minus09

00880998746

03383563536

04273256312

minus646813249119864minus07

Sol4

01939820199

00603335280

02578382574

502240252119864minus06

00152114400

03633514726

03810978088

803072038119864minus07

01618654345

01097465792

02787430938

minus198215448119864minus06

00056985809

09114653768

02006688671

321352180119864minus08

01904538879

02502358229

04452573447

619105762119864minus06

minus01623604033

03089460561

01491746444

minus901451453119864minus06

01864448178

02428992222

04320068557

minus261979130119864minus06

minus00449302706

01144916285

00733954587

minus717750570119864minus06

01675935311

01774161896

03459538936

minus127733934119864minus05

minus00274959004

04539962587

04273210007

minus520897654119864minus06

Computational Intelligence and Neuroscience 9

3D surf

minus500

minus400

minus300

minus200

minus100

0100200300400500

z-la

bel

minus500minus400minus300minus200minus100

0100200300400500

minus500minus400minus300minus200minus100

0 100 200 300 400 500

Figure 1 The landscape of Schwegelrsquos function

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500

Figure 2 The locations of 40 bats in the initial phase

5 Conclusions

Aiming at the phenomenon of slow convergence rate andlow accuracy of bat algorithm we put forward animprovedbat algorithm with differential operator and Levy flightstrajectory (DLBA) based on the basic framework of batalgorithm (BA) the purpose is to improve the convergencerate and precision of bat algorithm In this paper we definethe frequency fluctuations up and down when the battracking prey which influence the batsrsquo location problemit is more graphic to simulate the batrsquos behavior Moreoverit can make the algorithm effectively jump out of the localoptimum to add Levy flights and differential operator themain reason is because Levy flight has prominent propertiesin the previously mentioned and the differential operatorcan guide bats find better solutions sequentially increase theconvergent speed

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500minus400minus300minus200minus100

0100200300400500

Figure 3 The locations of 40 bats after six iterations

In addition batsrsquo position variation is influenced bythe pulse emission rate and loudness as well Firstly pulseemission rate causes update of position and more and morenew position can be explored consequently increasing thediversity of population Secondly loudness is designed tostrengthen local search and to guide bats find better solutions

In this paper we tested 14 typical benchmark functionsand applied them to solve nonlinear equations the simulationresults not only showed that the proposed algorithm is feasi-ble and effective which ismore robust but also demonstratedthe superior approximation capabilities in high-dimensionalspaceThis is not surprising as the aim of developing the newalgorithm was to try to use the advantages of existing algo-rithms and other interesting feature inspired by the fantasticbehavior of echolocation ofmicrobatsNumerically speakingthese can be translated into two crucial characteristics of themodern metaheuristics intensification and diversificationIntensification intends to search around the current bestsolutions and select the best candidates or solutions whilediversification makes sure that the algorithm can explore thesearch space efficiently

10 Computational Intelligence and Neuroscience

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500

Figure 4 The locations of 40 bats after fifteen iterations

0 20 40 60 80 100 120 140 160 180 2000

100200300400500600700800900

1000

Iteration

Fitn

ess

BADLBA

Figure 5 Convergence curves for the function 1198912

BADLBA

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12

Iteration

Fitn

ess

Figure 6 Convergence curves for the function 1198914

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07

Iteration

Fitn

ess

BADLBA

Figure 7 Convergence curves for the function 1198917

BADLBA

0 20 40 60 80 100 120 140 160 180 200minus420

minus410

minus400

minus390

minus380

minus370

minus360

minus350

minus340

Iteration

Fitn

ess

Figure 8 Convergence curves for the function 11989111

BADLBA

0 20 40 60 80 100 120 140 160 180 200minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Iteration

Fitn

ess

Figure 9 Convergence curves for the function 11989113

Computational Intelligence and Neuroscience 11

0 5 10 15 20 25 30 35 40 45 500

500

1000

1500

2000

2500

3000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 10 Distribution of optimal fitness for 1198916(D = 128)

0 5 10 15 20 25 30 35 40 45 500

2000

4000

6000

8000

10000

12000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 11 Distribution of optimal fitness for 1198919(D = 256)

BADLBA

0 5 10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000

6000

Number of running

Opt

imal

fitn

ess

Figure 12 Distribution of optimal fitness for 1198918(D = 320)

BADLBA

0 5 10 15 20 25 30 35 40 45 50minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Number of running

Opt

imal

fitn

ess

Figure 13 Distribution of optimal fitness for 11989114(D = 512)

BADLBA

Number of running

Opt

imal

fitn

ess

0 5 10 15 20 25 30 35 40 45 500

05

1

15

2

25times104

Figure 14 Distribution of optimal fitness for 1198911(D = 1024)

This potentially powerful optimization strategy can easilybe extended to study multiobjective optimization applica-tions with various constraints even to NP-hard problemsFurther studies can focus on the sensitivity and parameterstudies and their possible relationships with the convergencerate of the algorithm

Acknowledgments

This work is supported by the National Science Foundationof China under Grant no 61165015 Key Project of GuangxiScience Foundation under Grant no 2012GXNSFDA053028Key Project of Guangxi High School Science Foundationunder Grant no 20121ZD008 and the Fund by OpenResearch Fund Program of Key Lab of Intelligent Perception

12 Computational Intelligence and Neuroscience

0 5 10 15 20 25 30 35 40 45 50minus15

minus145

minus14

minus135

minus13

minus125

minus12

Number of running

Opt

imal

fitn

ess

BA

times10minus3

Figure 15 Details of BA line in Figure 12

0 5 10 15 20 25 30 35 40 45 50minus2minus18minus16minus14minus12minus1minus08minus06minus04minus02

0

Number of running

Opt

imal

fitn

ess

DLBA

Figure 16 Details of DLBA line in Figure 12

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7

Iteration

Fitn

ess

DLBA

Figure 17 Convergence curves for DLAB

0 5 10 15 20 25 30 35 40 45 500

002

004

006

008

01

012

014

016

018

Number of running

Opt

imal

fitn

ess

DLBA

Figure 18 Distribution of optimal fitness for DLAB

0 5 10 15 20 25 30 35 40 45 500

010203040506070809

1

Number of running

Opt

imal

fitn

ess

times10minus3

Figure 19 Distribution of optimal fitness less than 0001 inFigure 18

and Image Understanding of Ministry of Education of Chinaunder Grant no IPIU01201100

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of IEEE International Conference on NeuralNetworks (ICNN rsquo95) vol 4 pp 1942ndash1948 December 1995

[2] X-S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress Bristol UK 2nd edition 2010

[3] X-S Yang ldquoFirefly algorithms for multimodal optimizationrdquoin Proceedings of the 5th International Conference on StochasticAlgorithms Foundations and Applications (SAGA rsquo09) pp 169ndash178 2009

[4] B Alatas ldquoACROA artificial chemical reaction optimizationalgorithm for global optimizationrdquo Expert Systems with Appli-cations vol 38 no 10 pp 13170ndash13180 2011

Computational Intelligence and Neuroscience 13

[5] K N Krishnanand and D Ghose ldquoGlowworm warm opti-mization for searching higher dimensional spacesrdquo Studies inComputational Intelligence vol 248 pp 61ndash75 2009

[6] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[7] K Price R Storn and J Lampinen Differential Evolution-APractical Approach to Global Optimization Springer BerlinGermany 2005

[8] httpwwwicsiberkeleyedusim storn [9] R Storn and K Price ldquoDifferential evolutionmdasha simple and

efficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[10] X-S Yang ldquoA new metaheuristic Bat-inspired AlgorithmrdquoStudies in Computational Intelligence vol 284 pp 65ndash74 2010

[11] A Mucherino and O Seref ldquoMonkey search a novel meta-heuristic search for global optimizationrdquo in Proceedings of theConference on Data Mining Systems Analysis and Optimizationin Biomedicine vol 953 pp 162ndash173 March 2007

[12] KM Passino ldquoBiomimicry of bacterial foraging for distributedoptimization and controlrdquo IEEE Control Systems Magazine vol22 no 3 pp 52ndash67 2002

[13] G R Reynolds ldquoCultural algorithms theory and applicationrdquoin New Iders in Optimization D Corne M Dorigo and FGlover Eds pp 367ndash378 McGraw-Hill New York NY USA1999

[14] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[15] R Oftadeh M J Mahjoob and M Shariatpanahi ldquoA novelmeta-heuristic optimization algorithm inspired by group hunt-ing of animals hunting searchrdquo Computers and Mathematicswith Applications vol 60 no 7 pp 2087ndash2098 2010

[16] P W Tsai J S Pan B Y Liao M J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2011

[17] X-S Yang ldquoBat algorithm for multi-objective optimizationrdquoInternational Journal of Bio-Inspired Computation Bio-InspiredComputation (IJBIC) vol 3 no 5 pp 267ndash274 2011

[18] T C Bora L S Coelho and L Lebensztajn ldquoBat-Inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012

[19] B B Mandelbrot The Fractal Geometry of Nature WH Free-man New York NY USA 1983

[20] V A Chechkin R Metzler J Klafter and V Y GoncharldquoIntroduction to the theory of Levy flightsrdquo in AnomalousTransport Foundations and Applications pp 129ndash162 Wiley-VCH Cambridge UK 2008

[21] A M Reynolds and M A Frye ldquoFree-flight odor tracking inDrosophila is consistent with an optimal intermittent scale-freesearchrdquo PLoS ONE vol 2 no 4 p e354 2007

[22] C Brown L S Liebovitch andRGlendon ldquoLevy fights inDobeJursquohoansi foraging patternsrdquo Human Ecology vol 35 no 1 pp129ndash138 2007

[23] P Barthelemy J Bertolotti andD SWiersma ldquoA Levy flight forlightrdquo Nature vol 453 no 7194 pp 495ndash498 2008

[24] N Mercadier W Guerin M Chevrollier and R Kaiser ldquoLevyflights of photons in hot atomic vapoursrdquoNature Physics vol 5no 8 pp 602ndash605 2009

[25] C Grosan and A Abraham ldquoA new approach for solvingnonlinear equations systemsrdquo IEEE Transactions on SystemsMan and Cybernetics A vol 38 no 3 pp 698ndash714 2008

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

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Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience 3

Objective function 119891(119909) 119909 = [1199091 1199092 119909

119889]119879

Initialize the bat population 119909119894(119894 = 1 2 119899) and V

119894

Define pulse frequency 119891119894at 119909119894initialize pulse rates 119903

119894and the loudness 119860

119894

While (119905 ltMax number of iterations)Generate new solutions by adjusting frequencyand updating velocities and locationssolutions (1)if (rand gt 119903

119894)

Select a solution among the best solutionsGenerate a local solution around the selected best solution

end ifGenerate a new solution by flying randomlyif (rand lt 119860

119894amp 119891(119909

119894) lt 119891(119909

lowast))

Accept the new solutionsIncrease 119903

119894and reduce 119860

119894

end ifRank the bats and find the current best 119909

lowast

end whilePostprocess results and visualization

Pseudocode 1

study by Mercadier et al shows that the Levy flights ofphotons in hot atomic vapours (2009) [24] Subsequentlysuch behaviour has been applied to optimization and optimalsearch and preliminary results show its promising capability

24 Description of the Nonlinear Equations (see [25]) Thegeneral form of nonlinear equations with real variables isdescribed as follows

119892119894 (119883) = 0 119894 = 1 2 119899 (5)

where 119883 = (1199091 1199092 119909

119899) isin 119863 sub 119877

119899 119863 = (1199091 1199092 119909

119899) |

119886119894le 119909119894le 119887119894 119894 = 1 2 119899

In solving nonlinear equations process for DLBA thefitness function can be constructed by

119866 (119883) =

119899

sum

119894=1

1003816100381610038161003816119892119894 (119909)1003816100381610038161003816 (6)

So the solving of nonlinear equations can be translated intoan optimization problem in domain119863

min 119866 (119883)

st (1199091 1199092 119909

119899) isin 119863

(7)

Consequently the optimal value of (7) is exactly the solutionof (5)

3 DLBA

Inspired by Yangrsquos method we propose an improved batalgorithm based on differential operator and Levy-flightstrajectory (DLBA) based on the basic structure of BA andre-estimate the characters used in the original BA In DLBAnot only the movement of the bat is quite different from theoriginal BA but also the local search process is different

In DLBA the frequency fluctuates up and down whichcan change self-adaptively and the differential operator isintroduced which is similar to themutation operation of DEthe frequency 119891 is similar to the scale factor 119865 of DEbest2So the frequency updated formulae of a bat are defined asfollows

119891119905

1119894= ((119891

1min minus 1198911max)119905

119899119905

+ 1198911max)1205731 (8)

119891119905

2119894= ((119891

2max minus 1198912min)119905

119899119905

+ 1198912min)1205732 (9)

where 1205731 1205732isin [0 1] is a random vector drawn from a uni-

form distribution 1198911max = 119891

2max 1198911min = 1198912min 119899119905 is a

fixed parameter In DLBA the position 119909119905119894of each bat indi-

vidual are updated with (10) which is different from originalBAThis can preferably incorporate the echolocation charac-teristics of microbats

119909119905+1

119894= 119909119905

best + 119891119905

1119894(119909119905

1199031minus 119909119905

1199032) + 119891119905

2119894(119909119905

1199033minus 119909119905

1199034) (10)

where 119909119905best is the current global best location (solution)which is located after comparing all the solutions among allthe 119899 bats in 119905 generation 119909119905

119903119894is the bat individual in the bat

swarm and this can be achieved by randomizationIn addition Levy flight haves the prominent properties

increase the diversity of population sequentially which canmake the algorithmeffectively jumpout of the local optimumSo we let these bats perform the Levy flights with (11) beforethe position updating

119909119905

119894= 119905minus1

119894+ 120583 sign [rand minus 05] oplus Levy (11)

where 120583 is a random parameter drawn from a uniformdistribution sign oplusmeans entrywise multiplications rand isin[0 1] and random step length Levy obeys Levy distribution

Levy sim 119906 = 119905minus120582 (1 lt 120582 le 3) (12)

4 Computational Intelligence and Neuroscience

InputObjective function 119891(119909) 119909 = [1199091 1199092 119909

119889]119879 algorithmrsquos parameters 119899 119891min 119891max 120572 119899119905

BEGINFor 119894 = 1 To 119899 do

Initialize 119909119905119894 1199030119894 1198600119894

Calculating the initial fitnessEnd ForWhile (119905 lt itermax)

Update position use formula (8)sim(12)Evaluate fitness 119891(119909

119894) to find 119909best

For 119894 = 1 To 119899 doIf (rand gt 119903

119894)

Generate a 119909new around the selected best solution use formula (2)End If

End ForEvaluate fitness 119891(119909new)For 119894 = 1 To 119899 do

If (rand lt 119860119894)

Generate a 119909new around the selected best solution use formula (13)End If

End ForEvaluate fitness 119891(119909new)If (119891(119909119905+1best) lt 119891(119909

119905

best))Update 119903119905

119894 119860119905119894use formula (3) (14)

End If119905 = 119905 + 1

EndWhileENDOutput 119909best fitnessmin

Pseudocode 2

On the other hand each bat should have different valuesof loudness and pulse emission rate while the rate of pulseemission is relatively low and the loudness is relatively highDuring the search process the loudness usually decreaseswhile the rate of pulse emission increases Batsrsquo positionvariation is influenced by the pulse emission rate and loud-ness as well Firstly pulse emission rate 119903

119894causes fluctuation

of position using (2) sequentially more and more newposition can be explored Secondly loudness119860

119894is designed to

strengthen local search and to guide bats find better solutions

119909new = 119909best + 120578119903119905 (13)

where 120578 isin [minus1 1] is a random parameter and 119909best is thecurrent global best location (solution) in whole bats swarmWhile 119903

119905= ⟨119903119905

119894⟩ is the average pulse emission rate of all the

bats at this generationThe new rate of pulse emission 119903119905

119894and loudness119860119905

119894at time

step 119905 are given by

119903119905+1

119894= 119903119905

119894(

119905

119899119905

)

3

(14)

where 119903119905119894is time varying their loudness and emission rates

will be updated only if the best solution of the currentgeneration is better than the best solution of last generationwhich means that these bats are moving towards the optimalsolution The pseudocode of DLBA can be depicted as inPseudocode 2

4 The Simulation and Analysis

41 Parametric Studies The proposed DLBA is imple-mented in MATLAB simulation platform CPU Intel XeonE5405200GHz OS Microsoft Windows Server 2003Enterprise Edition SP2 RAM 1GHz Matlab VersionR2009a The parameter setting for BA and DLBA are listedin Table 1 the parameters of BA are recommended in theoriginal article

The stopping criterion can be defined in many ways Weadopt two terminated criteria we can use a given tolerance(Tol = 10119890 minus 5) for a test function that have certain mini-mum value in theory on the contrary each simulation runterminates when a certain number of function evaluations(FEs) have been reached In this paper FEs could be obtainedby population size multiplied by the number of iteration Inour experiment FEs = 24000 BA (= 300lowast40lowast2) DLBA (=200lowast40lowast3)

42 BenchmarkTest Function In order to validate the validityof DLBA we selected the 14 benchmark functions to experi-mentize The benchmark set include unimodal multimodalhigh-dimensional and low-dimensional unconstrained opti-mization benchmark functions where 119891

1ndash1198913are unimodal

functions 1198914ndash11989114

are multimodal functions 1198911ndash1198919have

certain theoretical minimum and 11989110ndash11989114

have uncertaintheoretical minimum

Computational Intelligence and Neuroscience 5

Table 1 The parameter set of BA and DLBA

Algorithms 119899 119891min 119891max 1198600

1198941199030

119894120572 120574 119899

119905

BA 40 0 100 (1 2) (0 01) 09 09 mdashDLBA 40 0 1 (1 2) (0 01) 09 mdash 5000

(1) 1198911 sphere function (the first function of De Jongrsquos test

set)

119891 (119909) =

119899

sum

119894=1

1199092

119894 minus10 le 119909

119894le 10 (15)

Here 119899 is dimensionality this function has a globalminimum119891min = 0 at 119909lowast = (0 0 0)

(2) 1198912 Schwegelrsquos problem 222

119891 (119909) =

119899

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816+

119899

prod

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816 minus10 le 119909

119894le 10 (16)

whose global minimum is obviously 119891min = 0 at 119909lowast= (0

0 0)(3) 1198913 Rosenbrock function

119891 (119909) =

119899minus1

sum

119894=1

[(119909119894minus 1)2+ 100(119909

119894+1minus 119909119894

2)

2

]

minus2408 le 119909119894le 2408

(17)

which has a global minimum 119891min = 0 at 119909lowast = (1 1 1)(4) 1198914 Eggcrate function

119891 (119909 119910) = 1199092+ 1199102+ 25 (sin2119909 + sin2119910)

(119909 119910) isin [minus2120587 2120587]

(18)

This 2-dimensional test function obviously gets the globalminimum 119891min = 0 at (0 0)

(5) 1198915 Ackleyrsquos function

119891 (119909) = minus20 exp[

[

minus

1

5

radic

1

119899

119899

sum

119894=1

1199092

119894]

]

minus exp[1119899

119899

sum

119894=1

cos (2120587119909119894)] + 20 + 119890

minus 30 le 119909 le 30

(19)

This function has a global minimum 119891min = 0 at 119909lowast=

(0 0 0) which is a multimodal function(6) 1198916 Griewangkrsquos function

119891 (119909) =

1

4000

119899

sum

119894=1

1199092

119894minus

119899

prod

119894=1

cos(119909119894

radic119894

) + 1 minus600 le 119909119894le 600

(20)

Its global minimum equal 119891min = 0 is obtainable for 119909lowast=

(0 0 0) the number of local minima for arbitrary 119899 is

unknown but in the two-dimensional case there are some500 local minima

(7) 1198917 Salomonrsquos function

119891 (119909) = minus cos(2120587radic119899

sum

119894=1

1199092

119894) + 01radic

119899

sum

119894=1

1199092

119894+ 1 minus5 le 119909

119894le 5

(21)

which has a global minimum 119891min = 0 at 119909lowast = (0 0 0)and is a multimodal function

(8) 1198918 Rastriginrsquos function

119891 (119909) = 10119899 +

119899

sum

119894=1

[1199092

119894minus 10 cos (2120587119909

119894)] minus512 le 119909

119894le 512

(22)

whose global minimum is 119891min = 0 at 119909lowast= (0 0 0)

for 119899 = 2 there are about 50 local minimizers arranged ina lattice-like configuration

(9) 1198919 Zakharovrsquos function

119891 (119909) =

119899

sum

119894=1

1199092

119894+ (

1

2

119899

sum

119894=1

119894119909119894)

2

+ (

1

2

119899

sum

119894=1

119894119909119894)

4

minus10 le 119909119894le 10

(23)

whose global minimum is 119891min = 0 at 119909lowast = (0 0 0) it isa multimodal function as well

(10) 11989110 Easomrsquos function

119891 (119909 119910)=minuscos (119909) cos (119910) exp [minus(119909minus120587)2+(119910minus120587)2]

minus10 le 119909 119910 le 10

(24)

whose global minimum is 119891min = minus1 at (120587 120587) it has manylocal minima

(11) 11989111 Schwegelrsquos function

119891 (119909) = minus

119899

sum

119894=1

119909119894sin(radic100381610038161003816

1003816119909119894

1003816100381610038161003816) minus500 le 119909

119894le 500 (25)

whose global minimum is 119891min asymp minus4189829119899 occuring at119909lowastasymp (4209687 4209687)(12) 119891

12 Shubertrsquos function

119891 (119909 119910)=[

5

sum

119894=1

119894 cos (119894+(119894+1) 119909)] sdot [5

sum

119894=1

119894 cos (119894+(119894+1) 119910)]

minus10 le 119909 119910 le 10

(26)

The number of local minima for this problem is not knownbut for 119899 = 2 the function has 760 local minima 18 of whichare global with 119891min asymp minus1867309

6 Computational Intelligence and Neuroscience

(13) 11989113 Xin-She Yangrsquos function

119891 (119909) = minus(

119899

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816) exp(minus

119899

sum

119894=1

1199092

119894) minus10 le 119909

119894le 10 (27)

which has multiple global minima for example for 119899 = 2 ithas 4 equal minima 119891min = minus1radic119890 asymp minus06065 at (05 05)(05-05) (minus05 05) and (minus05 minus05)

(14) 11989114 ldquoDrop Waverdquo function

119891 (119909) = minus

1 + cos(12radicsum119899119894=11199092

119894)

(12) (sum119899

119894=11199092

119894) + 2

minus512 le 119909119894le 512

(28)

This two-variable function is a multimodal test functionwhose global minimum is 119891min = minus1 occurs at 119909

lowast=

(0 0 0)

43 Comparison of Experimental Results The 2D landscapeof Schwegelrsquos function is shown in Figure 1 and this globalminimum can be found after about 720 FEs for 40 bats after6 iterations as shown in Figures 2 3 and 4

We adopt different terminated criteria aiming at differentbenchmark function we perform 100 times independentlyfor each test function and the record is given in Tables 2 and3

From Table 2 we can see that the DLBA performs muchbetter than the basic bat algorithm which converges muchfaster than BA under the fixed accuracy Tol = 10119890 minus 5 Inour experiment for the function 119891

4 we proposed that an

algorithm only costs 6 generations under the best situationand the average generations is 10 Furthermore for the firstgroup of test functions 119891

1ndash1198919 the DLBA averagely expend

439 generations when each function attains its terminatedcriteria however the BA needs 200 generations invariablyand the convergence speed of DLBA advances 29857 timesOn other hand the DLBAwhich obtained the accuracy of thesolution is much superior to BA which is more approximateto the theoretical value In aword it demonstrated thatDLBAhas fast convergence rate and high precision of the solutions

In Table 3 the five functions (11989110ndash11989114) independently

runs 100 times under the 24000 Fes we can clearly seethe precision of DLBA is obviously superior to the batalgorithm Some benchmark functions can easily attain thetheoretical optimal value In addition the standard deviationof DLBA is relatively low It shows that DLBA has superiorapproximation ability

Observe Tables 2 and 3 no matter high-dimensionalor low-dimensional we can find that DLBA can quicklyconverge to the global minima Furthermore Figures 5 67 8 and 9 show the convergence curves for some of thefunctions from a particular run of DLBA and BA which endat 200 generations The adaptive scheme generally convergesfaster than the basic scheme Here we select 119891

2(D = 20) 119891

4

(D = 2) 1198917(D = 5) 119891

11(D = 2) and 119891

13(D = 2)

DLBAnot only has superior approximation ability in low-dimensional space but also has excellent global search abilityin high-dimensional situation Table 4 is the experimental

result that DLBA performs 50 times independently under thehigh-dimensional situation As shown in Table 4 we can beconscious that the DLBA is effective under the multidimen-sional condition and acquired solution has higher accuracyeven approximate the theoretical value

Figures 10 11 12 13 and 14 are the distribution map ofoptimal fitness that is the selected functions independentlyperform 50 times under the multidimensional situationFigure 13 show that two ldquostraight linerdquo we can see in the figurethat DLBA reaches the global optimum(minus1) however BAfluctuates around 0 In order to display the fact of fluctuationwe magnify the two ldquostraight linerdquo and the amplifyingeffect are depicted in Figures 15 and 16 According to theexperimental results which are obtained from selected testfunctions DLBA presents higher precision than the originalBA on minimizing the outcome as the optimization goal

44 An Application for Solving Nonlinear Equations

Interval Arithmetic Benchmark (IAB) We consider onebenchmark problem proposed from interval arithmetic thebenchmark consists of the following system of [25]

0 = 1199091minus 025428722 minus 018324757119909

411990931199099

0 = 1199092minus 037842197 minus 016275449119909

1119909101199096

0 = 1199093minus 027162577 minus 016955071119909

1119909211990910

0 = 1199094minus 019807914 minus 015585316119909

711990911199096

0 = 1199095minus 044166728 minus 019950920119909

711990961199093

0 = 1199096minus 014654113 minus 018922793119909

8119909511990910

0 = 1199097minus 042937161 minus 021180486119909

211990951199098

0 = 1199098minus 007056438 minus 017081208119909

111990971199096

0 = 1199099minus 034504906 minus 019612740119909

1011990961199098

0 = 11990910minus 042651102 minus 021466544119909

411990981199091

(29)

Some of the solutions obtained as well as the functionvalues (which represent the values of the systemrsquos equationsobtained by replacing the variable values) are presented inTable 5 From Table 5 we can obviously observe that thefunction value of each equation solved byDLBA is superior towhich solved by EA [25] depending on the statistics of the 40function values the precision of function values is enhanced5199820E + 06 times by DLBA The convergence curve ofDLBA is depicted in Figure 17 Figure 17 show that DLBAhas fast convergence rate for solving nonlinear equationsThe achieved optimal fitness (the sum of function valuewith absolute value) in 50 times independent run is depictedin Figure 18 In order to reflect the precision of fitness wemagnify Figure 18 these optimal fitness that less than or equalto 0001 are plotted in Figure 19 We can found that there are32 times optimal fitness is less than 0001

Computational Intelligence and Neuroscience 7

Table 2 Comparison of BA and DLBA for several test functions under the first terminated criteria

TC BF Method Fitness IterationMin Mean Max Std Min Mean Max

Tol = 10119890 minus 5itermax = 200

1198911(119863 = 30) DLBA 430568008119864minus08 480356915119864minus06 996163402119864minus06 302060716119864minus06 10 176 26

BA 5681732462 13061959271 22382694528 3722960351 200 200 200

1198912(119863 = 20) DLBA 566761254119864minus07 633091639119864minus06 988929757119864minus06 248867687119864minus06 21 294 37

BA 2277136624 9335062167 273346090499 26941410301 200 200 200

1198913(119863 = 10) DLBA 566716240 739029403 828857171 037001430 200 200 200

BA 8224908050 29495948479 64076052284 10967161818 200 200 200

1198914(119863 = 2) DLBA 623475433119864minus09 393352701119864minus06 997022879119864minus06 277208752119864minus06 6 10 20

BA 105265280119864minus04 020705750 100771980 023116047 200 200 200

1198915(119863 = 5) DLBA 471737180119864minus07 640969812119864minus06 996088247119864minus06 271187009119864minus06 16 256 39

BA 115883977 302092450 586431360 069052427 200 200 200

1198916(119863 = 5) DLBA 138176777119864minus07 501027883119864minus06 992998256119864minus06 277338474119864minus06 11 184 56

BA 006234957 842851961 2990707979 612812390 200 200 200

1198917(119863 = 5) DLBA 778490674119864minus07 657288799119864minus06 997486019119864minus06 236637154119864minus06 18 46 114

BA 009987344 015318115 030215065 004923496 200 200 200

1198918(119863 = 5) DLBA 599983281119864minus08 447052788119864minus06 985695441119864minus06 281442645119864minus06 15 33 87

BA 724547772 1711077150 2590638177 437190073 200 200 200

1198919(119863 = 5) DLBA 665070016119864minus09 388692603119864minus06 969841027119864minus06 275444981119864minus06 10 151 23

BA 011106843 175623424 848211975 141343722 200 200 200

Table 3 Comparison of BA and DLBA for several test functions under the second terminated condition

TC BF Method FitnessMin Mean Max Std

FEs

11989110(119863 = 2) DLBA minus1 minus1 minus1 906493304119864 minus 17

BA minus099964845 minus098245546 minus090221232 001759602

11989111(119863 = 2) DLBA minus41898288727 minus41583523725 minus30054455266 1553948263

BA minus41898287874 minus40698044144 minus32810751671 2254615132

11989112(119863 = 5) DLBA minus18673090883 minus18673090883 minus18673090883 431613544119864 minus 13

BA minus18667541533 minus18401677767 minus17495599301 226333511

11989113(119863 = 2) DLBA minus060653066 minus060653066 minus060653066 863947249119864 minus 16

BA minus060652631 minus060411559 minus059656608 218888608119864 minus 03

11989114(119863 = 5) DLBA minus1 minus1 minus1 0

BA minus093624514 minus080454167 minus056977584 010270830

Table 4 Comparison of DLBA and BA under the high-dimensional situation

BF Method FitnessMin Mean Max Std

1198916(119863 = 128) DLBA 0 0 0 0

BA 258738759264 281140649000 296656723398 9396861133

1198919(119863 = 256) DLBA 266885939119864 minus 94 512093542119864 minus 69 256023720119864 minus 67 362071553119864 minus 68

BA 750339306546 825661302714 1150679192998 61337556609

1198918(119863 = 320) DLBA 0 0 0 0

BA 446859039467 472344624200 500857109760 13118818777

11989114(119863 = 512) DLBA minus1 minus1 minus1 0

BA minus148358700119864 minus 03 minus131084173119864 minus 03 minus120435493119864 minus 03 599288716119864 minus 05

1198911(119863 = 1024) DLBA 263790469119864 minus 109 473742970119864 minus 87 203861481119864 minus 85 289031313119864 minus 86

BA 1846348950157 1975226686813 2123923111285 56727878361

8 Computational Intelligence and Neuroscience

Table5Outcomeo

fDLB

AforIntervalA

rithm

eticBe

nchm

arkAnd

inCom

paris

onwith

EA

Metho

dEA

DLB

ASolutio

nsVa

riables

values

Functio

nsvalues

Varia

bles

values

Functio

nsvalues

Sol1

00464905115

02077959240

02578335208

722997083119864minus08

01013568357

02769798846

03810968901

minus284879905119864minus07

00840577820

01876863212

02787447331

minus263974619119864minus07

minus01388460309

03367887114

02006720536

306212480119864minus06

04943905739

00530391321

04452522319

774692656119864minus07

minus00760685163

02223730535

01491853777

133597657119864minus06

02475819110

01816084752

04320098178

minus787244534119864minus09

minus00170748156

00878963860

00734062202

341255029119864minus06

00003667535

03447200366

03459672005

324064555119864minus07

01481119311

02784227489

04273251429

minus118429644119864minus06

Sol2

01224819761

01318552790

02578332642

minus145675448119864minus07

01826200685

01964428361

03810968252

minus341069930119864minus07

02356779803

00364987069

02787462357

119723259119864minus06

minus00371150470

02354890155

02006687659

minus176074071119864minus08

03748181856

00675753064

04452514819

289773434119864minus07

02213311341

00739986588

01491839999

916571896119864minus08

00697813043

03607038292

04319795315

minus301421570119864minus05

00768058043

00059182979

00734021258

minus453845535119864minus07

minus00312153867

03767487763

03459672388

415572858119864minus07

01452667120

02811693568

04273281275

185998012119864minus06

Sol3

00633944399

01908436653

02578328072

minus595667763119864minus07

01017426933

02767897367

03810969084

minus239480772119864minus07

minus01051842285

03769063436

02787447816

minus204170456119864minus07

minus00477059943

02460900702

02006696507

671241513119864minus07

04149858326

00260337751

04452514467

minus438810984119864minus09

01215195321

00256054760

01491841089

189336147119864minus07

02539777159

01761486401

04320126793

297845790119864minus06

00843972823

01349869851

00734028724

779163472119864minus08

minus00534132992

03986395691

03459668311

330962522119864minus09

00880998746

03383563536

04273256312

minus646813249119864minus07

Sol4

01939820199

00603335280

02578382574

502240252119864minus06

00152114400

03633514726

03810978088

803072038119864minus07

01618654345

01097465792

02787430938

minus198215448119864minus06

00056985809

09114653768

02006688671

321352180119864minus08

01904538879

02502358229

04452573447

619105762119864minus06

minus01623604033

03089460561

01491746444

minus901451453119864minus06

01864448178

02428992222

04320068557

minus261979130119864minus06

minus00449302706

01144916285

00733954587

minus717750570119864minus06

01675935311

01774161896

03459538936

minus127733934119864minus05

minus00274959004

04539962587

04273210007

minus520897654119864minus06

Computational Intelligence and Neuroscience 9

3D surf

minus500

minus400

minus300

minus200

minus100

0100200300400500

z-la

bel

minus500minus400minus300minus200minus100

0100200300400500

minus500minus400minus300minus200minus100

0 100 200 300 400 500

Figure 1 The landscape of Schwegelrsquos function

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500

Figure 2 The locations of 40 bats in the initial phase

5 Conclusions

Aiming at the phenomenon of slow convergence rate andlow accuracy of bat algorithm we put forward animprovedbat algorithm with differential operator and Levy flightstrajectory (DLBA) based on the basic framework of batalgorithm (BA) the purpose is to improve the convergencerate and precision of bat algorithm In this paper we definethe frequency fluctuations up and down when the battracking prey which influence the batsrsquo location problemit is more graphic to simulate the batrsquos behavior Moreoverit can make the algorithm effectively jump out of the localoptimum to add Levy flights and differential operator themain reason is because Levy flight has prominent propertiesin the previously mentioned and the differential operatorcan guide bats find better solutions sequentially increase theconvergent speed

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500minus400minus300minus200minus100

0100200300400500

Figure 3 The locations of 40 bats after six iterations

In addition batsrsquo position variation is influenced bythe pulse emission rate and loudness as well Firstly pulseemission rate causes update of position and more and morenew position can be explored consequently increasing thediversity of population Secondly loudness is designed tostrengthen local search and to guide bats find better solutions

In this paper we tested 14 typical benchmark functionsand applied them to solve nonlinear equations the simulationresults not only showed that the proposed algorithm is feasi-ble and effective which ismore robust but also demonstratedthe superior approximation capabilities in high-dimensionalspaceThis is not surprising as the aim of developing the newalgorithm was to try to use the advantages of existing algo-rithms and other interesting feature inspired by the fantasticbehavior of echolocation ofmicrobatsNumerically speakingthese can be translated into two crucial characteristics of themodern metaheuristics intensification and diversificationIntensification intends to search around the current bestsolutions and select the best candidates or solutions whilediversification makes sure that the algorithm can explore thesearch space efficiently

10 Computational Intelligence and Neuroscience

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500

Figure 4 The locations of 40 bats after fifteen iterations

0 20 40 60 80 100 120 140 160 180 2000

100200300400500600700800900

1000

Iteration

Fitn

ess

BADLBA

Figure 5 Convergence curves for the function 1198912

BADLBA

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12

Iteration

Fitn

ess

Figure 6 Convergence curves for the function 1198914

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07

Iteration

Fitn

ess

BADLBA

Figure 7 Convergence curves for the function 1198917

BADLBA

0 20 40 60 80 100 120 140 160 180 200minus420

minus410

minus400

minus390

minus380

minus370

minus360

minus350

minus340

Iteration

Fitn

ess

Figure 8 Convergence curves for the function 11989111

BADLBA

0 20 40 60 80 100 120 140 160 180 200minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Iteration

Fitn

ess

Figure 9 Convergence curves for the function 11989113

Computational Intelligence and Neuroscience 11

0 5 10 15 20 25 30 35 40 45 500

500

1000

1500

2000

2500

3000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 10 Distribution of optimal fitness for 1198916(D = 128)

0 5 10 15 20 25 30 35 40 45 500

2000

4000

6000

8000

10000

12000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 11 Distribution of optimal fitness for 1198919(D = 256)

BADLBA

0 5 10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000

6000

Number of running

Opt

imal

fitn

ess

Figure 12 Distribution of optimal fitness for 1198918(D = 320)

BADLBA

0 5 10 15 20 25 30 35 40 45 50minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Number of running

Opt

imal

fitn

ess

Figure 13 Distribution of optimal fitness for 11989114(D = 512)

BADLBA

Number of running

Opt

imal

fitn

ess

0 5 10 15 20 25 30 35 40 45 500

05

1

15

2

25times104

Figure 14 Distribution of optimal fitness for 1198911(D = 1024)

This potentially powerful optimization strategy can easilybe extended to study multiobjective optimization applica-tions with various constraints even to NP-hard problemsFurther studies can focus on the sensitivity and parameterstudies and their possible relationships with the convergencerate of the algorithm

Acknowledgments

This work is supported by the National Science Foundationof China under Grant no 61165015 Key Project of GuangxiScience Foundation under Grant no 2012GXNSFDA053028Key Project of Guangxi High School Science Foundationunder Grant no 20121ZD008 and the Fund by OpenResearch Fund Program of Key Lab of Intelligent Perception

12 Computational Intelligence and Neuroscience

0 5 10 15 20 25 30 35 40 45 50minus15

minus145

minus14

minus135

minus13

minus125

minus12

Number of running

Opt

imal

fitn

ess

BA

times10minus3

Figure 15 Details of BA line in Figure 12

0 5 10 15 20 25 30 35 40 45 50minus2minus18minus16minus14minus12minus1minus08minus06minus04minus02

0

Number of running

Opt

imal

fitn

ess

DLBA

Figure 16 Details of DLBA line in Figure 12

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7

Iteration

Fitn

ess

DLBA

Figure 17 Convergence curves for DLAB

0 5 10 15 20 25 30 35 40 45 500

002

004

006

008

01

012

014

016

018

Number of running

Opt

imal

fitn

ess

DLBA

Figure 18 Distribution of optimal fitness for DLAB

0 5 10 15 20 25 30 35 40 45 500

010203040506070809

1

Number of running

Opt

imal

fitn

ess

times10minus3

Figure 19 Distribution of optimal fitness less than 0001 inFigure 18

and Image Understanding of Ministry of Education of Chinaunder Grant no IPIU01201100

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of IEEE International Conference on NeuralNetworks (ICNN rsquo95) vol 4 pp 1942ndash1948 December 1995

[2] X-S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress Bristol UK 2nd edition 2010

[3] X-S Yang ldquoFirefly algorithms for multimodal optimizationrdquoin Proceedings of the 5th International Conference on StochasticAlgorithms Foundations and Applications (SAGA rsquo09) pp 169ndash178 2009

[4] B Alatas ldquoACROA artificial chemical reaction optimizationalgorithm for global optimizationrdquo Expert Systems with Appli-cations vol 38 no 10 pp 13170ndash13180 2011

Computational Intelligence and Neuroscience 13

[5] K N Krishnanand and D Ghose ldquoGlowworm warm opti-mization for searching higher dimensional spacesrdquo Studies inComputational Intelligence vol 248 pp 61ndash75 2009

[6] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[7] K Price R Storn and J Lampinen Differential Evolution-APractical Approach to Global Optimization Springer BerlinGermany 2005

[8] httpwwwicsiberkeleyedusim storn [9] R Storn and K Price ldquoDifferential evolutionmdasha simple and

efficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[10] X-S Yang ldquoA new metaheuristic Bat-inspired AlgorithmrdquoStudies in Computational Intelligence vol 284 pp 65ndash74 2010

[11] A Mucherino and O Seref ldquoMonkey search a novel meta-heuristic search for global optimizationrdquo in Proceedings of theConference on Data Mining Systems Analysis and Optimizationin Biomedicine vol 953 pp 162ndash173 March 2007

[12] KM Passino ldquoBiomimicry of bacterial foraging for distributedoptimization and controlrdquo IEEE Control Systems Magazine vol22 no 3 pp 52ndash67 2002

[13] G R Reynolds ldquoCultural algorithms theory and applicationrdquoin New Iders in Optimization D Corne M Dorigo and FGlover Eds pp 367ndash378 McGraw-Hill New York NY USA1999

[14] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[15] R Oftadeh M J Mahjoob and M Shariatpanahi ldquoA novelmeta-heuristic optimization algorithm inspired by group hunt-ing of animals hunting searchrdquo Computers and Mathematicswith Applications vol 60 no 7 pp 2087ndash2098 2010

[16] P W Tsai J S Pan B Y Liao M J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2011

[17] X-S Yang ldquoBat algorithm for multi-objective optimizationrdquoInternational Journal of Bio-Inspired Computation Bio-InspiredComputation (IJBIC) vol 3 no 5 pp 267ndash274 2011

[18] T C Bora L S Coelho and L Lebensztajn ldquoBat-Inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012

[19] B B Mandelbrot The Fractal Geometry of Nature WH Free-man New York NY USA 1983

[20] V A Chechkin R Metzler J Klafter and V Y GoncharldquoIntroduction to the theory of Levy flightsrdquo in AnomalousTransport Foundations and Applications pp 129ndash162 Wiley-VCH Cambridge UK 2008

[21] A M Reynolds and M A Frye ldquoFree-flight odor tracking inDrosophila is consistent with an optimal intermittent scale-freesearchrdquo PLoS ONE vol 2 no 4 p e354 2007

[22] C Brown L S Liebovitch andRGlendon ldquoLevy fights inDobeJursquohoansi foraging patternsrdquo Human Ecology vol 35 no 1 pp129ndash138 2007

[23] P Barthelemy J Bertolotti andD SWiersma ldquoA Levy flight forlightrdquo Nature vol 453 no 7194 pp 495ndash498 2008

[24] N Mercadier W Guerin M Chevrollier and R Kaiser ldquoLevyflights of photons in hot atomic vapoursrdquoNature Physics vol 5no 8 pp 602ndash605 2009

[25] C Grosan and A Abraham ldquoA new approach for solvingnonlinear equations systemsrdquo IEEE Transactions on SystemsMan and Cybernetics A vol 38 no 3 pp 698ndash714 2008

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

4 Computational Intelligence and Neuroscience

InputObjective function 119891(119909) 119909 = [1199091 1199092 119909

119889]119879 algorithmrsquos parameters 119899 119891min 119891max 120572 119899119905

BEGINFor 119894 = 1 To 119899 do

Initialize 119909119905119894 1199030119894 1198600119894

Calculating the initial fitnessEnd ForWhile (119905 lt itermax)

Update position use formula (8)sim(12)Evaluate fitness 119891(119909

119894) to find 119909best

For 119894 = 1 To 119899 doIf (rand gt 119903

119894)

Generate a 119909new around the selected best solution use formula (2)End If

End ForEvaluate fitness 119891(119909new)For 119894 = 1 To 119899 do

If (rand lt 119860119894)

Generate a 119909new around the selected best solution use formula (13)End If

End ForEvaluate fitness 119891(119909new)If (119891(119909119905+1best) lt 119891(119909

119905

best))Update 119903119905

119894 119860119905119894use formula (3) (14)

End If119905 = 119905 + 1

EndWhileENDOutput 119909best fitnessmin

Pseudocode 2

On the other hand each bat should have different valuesof loudness and pulse emission rate while the rate of pulseemission is relatively low and the loudness is relatively highDuring the search process the loudness usually decreaseswhile the rate of pulse emission increases Batsrsquo positionvariation is influenced by the pulse emission rate and loud-ness as well Firstly pulse emission rate 119903

119894causes fluctuation

of position using (2) sequentially more and more newposition can be explored Secondly loudness119860

119894is designed to

strengthen local search and to guide bats find better solutions

119909new = 119909best + 120578119903119905 (13)

where 120578 isin [minus1 1] is a random parameter and 119909best is thecurrent global best location (solution) in whole bats swarmWhile 119903

119905= ⟨119903119905

119894⟩ is the average pulse emission rate of all the

bats at this generationThe new rate of pulse emission 119903119905

119894and loudness119860119905

119894at time

step 119905 are given by

119903119905+1

119894= 119903119905

119894(

119905

119899119905

)

3

(14)

where 119903119905119894is time varying their loudness and emission rates

will be updated only if the best solution of the currentgeneration is better than the best solution of last generationwhich means that these bats are moving towards the optimalsolution The pseudocode of DLBA can be depicted as inPseudocode 2

4 The Simulation and Analysis

41 Parametric Studies The proposed DLBA is imple-mented in MATLAB simulation platform CPU Intel XeonE5405200GHz OS Microsoft Windows Server 2003Enterprise Edition SP2 RAM 1GHz Matlab VersionR2009a The parameter setting for BA and DLBA are listedin Table 1 the parameters of BA are recommended in theoriginal article

The stopping criterion can be defined in many ways Weadopt two terminated criteria we can use a given tolerance(Tol = 10119890 minus 5) for a test function that have certain mini-mum value in theory on the contrary each simulation runterminates when a certain number of function evaluations(FEs) have been reached In this paper FEs could be obtainedby population size multiplied by the number of iteration Inour experiment FEs = 24000 BA (= 300lowast40lowast2) DLBA (=200lowast40lowast3)

42 BenchmarkTest Function In order to validate the validityof DLBA we selected the 14 benchmark functions to experi-mentize The benchmark set include unimodal multimodalhigh-dimensional and low-dimensional unconstrained opti-mization benchmark functions where 119891

1ndash1198913are unimodal

functions 1198914ndash11989114

are multimodal functions 1198911ndash1198919have

certain theoretical minimum and 11989110ndash11989114

have uncertaintheoretical minimum

Computational Intelligence and Neuroscience 5

Table 1 The parameter set of BA and DLBA

Algorithms 119899 119891min 119891max 1198600

1198941199030

119894120572 120574 119899

119905

BA 40 0 100 (1 2) (0 01) 09 09 mdashDLBA 40 0 1 (1 2) (0 01) 09 mdash 5000

(1) 1198911 sphere function (the first function of De Jongrsquos test

set)

119891 (119909) =

119899

sum

119894=1

1199092

119894 minus10 le 119909

119894le 10 (15)

Here 119899 is dimensionality this function has a globalminimum119891min = 0 at 119909lowast = (0 0 0)

(2) 1198912 Schwegelrsquos problem 222

119891 (119909) =

119899

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816+

119899

prod

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816 minus10 le 119909

119894le 10 (16)

whose global minimum is obviously 119891min = 0 at 119909lowast= (0

0 0)(3) 1198913 Rosenbrock function

119891 (119909) =

119899minus1

sum

119894=1

[(119909119894minus 1)2+ 100(119909

119894+1minus 119909119894

2)

2

]

minus2408 le 119909119894le 2408

(17)

which has a global minimum 119891min = 0 at 119909lowast = (1 1 1)(4) 1198914 Eggcrate function

119891 (119909 119910) = 1199092+ 1199102+ 25 (sin2119909 + sin2119910)

(119909 119910) isin [minus2120587 2120587]

(18)

This 2-dimensional test function obviously gets the globalminimum 119891min = 0 at (0 0)

(5) 1198915 Ackleyrsquos function

119891 (119909) = minus20 exp[

[

minus

1

5

radic

1

119899

119899

sum

119894=1

1199092

119894]

]

minus exp[1119899

119899

sum

119894=1

cos (2120587119909119894)] + 20 + 119890

minus 30 le 119909 le 30

(19)

This function has a global minimum 119891min = 0 at 119909lowast=

(0 0 0) which is a multimodal function(6) 1198916 Griewangkrsquos function

119891 (119909) =

1

4000

119899

sum

119894=1

1199092

119894minus

119899

prod

119894=1

cos(119909119894

radic119894

) + 1 minus600 le 119909119894le 600

(20)

Its global minimum equal 119891min = 0 is obtainable for 119909lowast=

(0 0 0) the number of local minima for arbitrary 119899 is

unknown but in the two-dimensional case there are some500 local minima

(7) 1198917 Salomonrsquos function

119891 (119909) = minus cos(2120587radic119899

sum

119894=1

1199092

119894) + 01radic

119899

sum

119894=1

1199092

119894+ 1 minus5 le 119909

119894le 5

(21)

which has a global minimum 119891min = 0 at 119909lowast = (0 0 0)and is a multimodal function

(8) 1198918 Rastriginrsquos function

119891 (119909) = 10119899 +

119899

sum

119894=1

[1199092

119894minus 10 cos (2120587119909

119894)] minus512 le 119909

119894le 512

(22)

whose global minimum is 119891min = 0 at 119909lowast= (0 0 0)

for 119899 = 2 there are about 50 local minimizers arranged ina lattice-like configuration

(9) 1198919 Zakharovrsquos function

119891 (119909) =

119899

sum

119894=1

1199092

119894+ (

1

2

119899

sum

119894=1

119894119909119894)

2

+ (

1

2

119899

sum

119894=1

119894119909119894)

4

minus10 le 119909119894le 10

(23)

whose global minimum is 119891min = 0 at 119909lowast = (0 0 0) it isa multimodal function as well

(10) 11989110 Easomrsquos function

119891 (119909 119910)=minuscos (119909) cos (119910) exp [minus(119909minus120587)2+(119910minus120587)2]

minus10 le 119909 119910 le 10

(24)

whose global minimum is 119891min = minus1 at (120587 120587) it has manylocal minima

(11) 11989111 Schwegelrsquos function

119891 (119909) = minus

119899

sum

119894=1

119909119894sin(radic100381610038161003816

1003816119909119894

1003816100381610038161003816) minus500 le 119909

119894le 500 (25)

whose global minimum is 119891min asymp minus4189829119899 occuring at119909lowastasymp (4209687 4209687)(12) 119891

12 Shubertrsquos function

119891 (119909 119910)=[

5

sum

119894=1

119894 cos (119894+(119894+1) 119909)] sdot [5

sum

119894=1

119894 cos (119894+(119894+1) 119910)]

minus10 le 119909 119910 le 10

(26)

The number of local minima for this problem is not knownbut for 119899 = 2 the function has 760 local minima 18 of whichare global with 119891min asymp minus1867309

6 Computational Intelligence and Neuroscience

(13) 11989113 Xin-She Yangrsquos function

119891 (119909) = minus(

119899

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816) exp(minus

119899

sum

119894=1

1199092

119894) minus10 le 119909

119894le 10 (27)

which has multiple global minima for example for 119899 = 2 ithas 4 equal minima 119891min = minus1radic119890 asymp minus06065 at (05 05)(05-05) (minus05 05) and (minus05 minus05)

(14) 11989114 ldquoDrop Waverdquo function

119891 (119909) = minus

1 + cos(12radicsum119899119894=11199092

119894)

(12) (sum119899

119894=11199092

119894) + 2

minus512 le 119909119894le 512

(28)

This two-variable function is a multimodal test functionwhose global minimum is 119891min = minus1 occurs at 119909

lowast=

(0 0 0)

43 Comparison of Experimental Results The 2D landscapeof Schwegelrsquos function is shown in Figure 1 and this globalminimum can be found after about 720 FEs for 40 bats after6 iterations as shown in Figures 2 3 and 4

We adopt different terminated criteria aiming at differentbenchmark function we perform 100 times independentlyfor each test function and the record is given in Tables 2 and3

From Table 2 we can see that the DLBA performs muchbetter than the basic bat algorithm which converges muchfaster than BA under the fixed accuracy Tol = 10119890 minus 5 Inour experiment for the function 119891

4 we proposed that an

algorithm only costs 6 generations under the best situationand the average generations is 10 Furthermore for the firstgroup of test functions 119891

1ndash1198919 the DLBA averagely expend

439 generations when each function attains its terminatedcriteria however the BA needs 200 generations invariablyand the convergence speed of DLBA advances 29857 timesOn other hand the DLBAwhich obtained the accuracy of thesolution is much superior to BA which is more approximateto the theoretical value In aword it demonstrated thatDLBAhas fast convergence rate and high precision of the solutions

In Table 3 the five functions (11989110ndash11989114) independently

runs 100 times under the 24000 Fes we can clearly seethe precision of DLBA is obviously superior to the batalgorithm Some benchmark functions can easily attain thetheoretical optimal value In addition the standard deviationof DLBA is relatively low It shows that DLBA has superiorapproximation ability

Observe Tables 2 and 3 no matter high-dimensionalor low-dimensional we can find that DLBA can quicklyconverge to the global minima Furthermore Figures 5 67 8 and 9 show the convergence curves for some of thefunctions from a particular run of DLBA and BA which endat 200 generations The adaptive scheme generally convergesfaster than the basic scheme Here we select 119891

2(D = 20) 119891

4

(D = 2) 1198917(D = 5) 119891

11(D = 2) and 119891

13(D = 2)

DLBAnot only has superior approximation ability in low-dimensional space but also has excellent global search abilityin high-dimensional situation Table 4 is the experimental

result that DLBA performs 50 times independently under thehigh-dimensional situation As shown in Table 4 we can beconscious that the DLBA is effective under the multidimen-sional condition and acquired solution has higher accuracyeven approximate the theoretical value

Figures 10 11 12 13 and 14 are the distribution map ofoptimal fitness that is the selected functions independentlyperform 50 times under the multidimensional situationFigure 13 show that two ldquostraight linerdquo we can see in the figurethat DLBA reaches the global optimum(minus1) however BAfluctuates around 0 In order to display the fact of fluctuationwe magnify the two ldquostraight linerdquo and the amplifyingeffect are depicted in Figures 15 and 16 According to theexperimental results which are obtained from selected testfunctions DLBA presents higher precision than the originalBA on minimizing the outcome as the optimization goal

44 An Application for Solving Nonlinear Equations

Interval Arithmetic Benchmark (IAB) We consider onebenchmark problem proposed from interval arithmetic thebenchmark consists of the following system of [25]

0 = 1199091minus 025428722 minus 018324757119909

411990931199099

0 = 1199092minus 037842197 minus 016275449119909

1119909101199096

0 = 1199093minus 027162577 minus 016955071119909

1119909211990910

0 = 1199094minus 019807914 minus 015585316119909

711990911199096

0 = 1199095minus 044166728 minus 019950920119909

711990961199093

0 = 1199096minus 014654113 minus 018922793119909

8119909511990910

0 = 1199097minus 042937161 minus 021180486119909

211990951199098

0 = 1199098minus 007056438 minus 017081208119909

111990971199096

0 = 1199099minus 034504906 minus 019612740119909

1011990961199098

0 = 11990910minus 042651102 minus 021466544119909

411990981199091

(29)

Some of the solutions obtained as well as the functionvalues (which represent the values of the systemrsquos equationsobtained by replacing the variable values) are presented inTable 5 From Table 5 we can obviously observe that thefunction value of each equation solved byDLBA is superior towhich solved by EA [25] depending on the statistics of the 40function values the precision of function values is enhanced5199820E + 06 times by DLBA The convergence curve ofDLBA is depicted in Figure 17 Figure 17 show that DLBAhas fast convergence rate for solving nonlinear equationsThe achieved optimal fitness (the sum of function valuewith absolute value) in 50 times independent run is depictedin Figure 18 In order to reflect the precision of fitness wemagnify Figure 18 these optimal fitness that less than or equalto 0001 are plotted in Figure 19 We can found that there are32 times optimal fitness is less than 0001

Computational Intelligence and Neuroscience 7

Table 2 Comparison of BA and DLBA for several test functions under the first terminated criteria

TC BF Method Fitness IterationMin Mean Max Std Min Mean Max

Tol = 10119890 minus 5itermax = 200

1198911(119863 = 30) DLBA 430568008119864minus08 480356915119864minus06 996163402119864minus06 302060716119864minus06 10 176 26

BA 5681732462 13061959271 22382694528 3722960351 200 200 200

1198912(119863 = 20) DLBA 566761254119864minus07 633091639119864minus06 988929757119864minus06 248867687119864minus06 21 294 37

BA 2277136624 9335062167 273346090499 26941410301 200 200 200

1198913(119863 = 10) DLBA 566716240 739029403 828857171 037001430 200 200 200

BA 8224908050 29495948479 64076052284 10967161818 200 200 200

1198914(119863 = 2) DLBA 623475433119864minus09 393352701119864minus06 997022879119864minus06 277208752119864minus06 6 10 20

BA 105265280119864minus04 020705750 100771980 023116047 200 200 200

1198915(119863 = 5) DLBA 471737180119864minus07 640969812119864minus06 996088247119864minus06 271187009119864minus06 16 256 39

BA 115883977 302092450 586431360 069052427 200 200 200

1198916(119863 = 5) DLBA 138176777119864minus07 501027883119864minus06 992998256119864minus06 277338474119864minus06 11 184 56

BA 006234957 842851961 2990707979 612812390 200 200 200

1198917(119863 = 5) DLBA 778490674119864minus07 657288799119864minus06 997486019119864minus06 236637154119864minus06 18 46 114

BA 009987344 015318115 030215065 004923496 200 200 200

1198918(119863 = 5) DLBA 599983281119864minus08 447052788119864minus06 985695441119864minus06 281442645119864minus06 15 33 87

BA 724547772 1711077150 2590638177 437190073 200 200 200

1198919(119863 = 5) DLBA 665070016119864minus09 388692603119864minus06 969841027119864minus06 275444981119864minus06 10 151 23

BA 011106843 175623424 848211975 141343722 200 200 200

Table 3 Comparison of BA and DLBA for several test functions under the second terminated condition

TC BF Method FitnessMin Mean Max Std

FEs

11989110(119863 = 2) DLBA minus1 minus1 minus1 906493304119864 minus 17

BA minus099964845 minus098245546 minus090221232 001759602

11989111(119863 = 2) DLBA minus41898288727 minus41583523725 minus30054455266 1553948263

BA minus41898287874 minus40698044144 minus32810751671 2254615132

11989112(119863 = 5) DLBA minus18673090883 minus18673090883 minus18673090883 431613544119864 minus 13

BA minus18667541533 minus18401677767 minus17495599301 226333511

11989113(119863 = 2) DLBA minus060653066 minus060653066 minus060653066 863947249119864 minus 16

BA minus060652631 minus060411559 minus059656608 218888608119864 minus 03

11989114(119863 = 5) DLBA minus1 minus1 minus1 0

BA minus093624514 minus080454167 minus056977584 010270830

Table 4 Comparison of DLBA and BA under the high-dimensional situation

BF Method FitnessMin Mean Max Std

1198916(119863 = 128) DLBA 0 0 0 0

BA 258738759264 281140649000 296656723398 9396861133

1198919(119863 = 256) DLBA 266885939119864 minus 94 512093542119864 minus 69 256023720119864 minus 67 362071553119864 minus 68

BA 750339306546 825661302714 1150679192998 61337556609

1198918(119863 = 320) DLBA 0 0 0 0

BA 446859039467 472344624200 500857109760 13118818777

11989114(119863 = 512) DLBA minus1 minus1 minus1 0

BA minus148358700119864 minus 03 minus131084173119864 minus 03 minus120435493119864 minus 03 599288716119864 minus 05

1198911(119863 = 1024) DLBA 263790469119864 minus 109 473742970119864 minus 87 203861481119864 minus 85 289031313119864 minus 86

BA 1846348950157 1975226686813 2123923111285 56727878361

8 Computational Intelligence and Neuroscience

Table5Outcomeo

fDLB

AforIntervalA

rithm

eticBe

nchm

arkAnd

inCom

paris

onwith

EA

Metho

dEA

DLB

ASolutio

nsVa

riables

values

Functio

nsvalues

Varia

bles

values

Functio

nsvalues

Sol1

00464905115

02077959240

02578335208

722997083119864minus08

01013568357

02769798846

03810968901

minus284879905119864minus07

00840577820

01876863212

02787447331

minus263974619119864minus07

minus01388460309

03367887114

02006720536

306212480119864minus06

04943905739

00530391321

04452522319

774692656119864minus07

minus00760685163

02223730535

01491853777

133597657119864minus06

02475819110

01816084752

04320098178

minus787244534119864minus09

minus00170748156

00878963860

00734062202

341255029119864minus06

00003667535

03447200366

03459672005

324064555119864minus07

01481119311

02784227489

04273251429

minus118429644119864minus06

Sol2

01224819761

01318552790

02578332642

minus145675448119864minus07

01826200685

01964428361

03810968252

minus341069930119864minus07

02356779803

00364987069

02787462357

119723259119864minus06

minus00371150470

02354890155

02006687659

minus176074071119864minus08

03748181856

00675753064

04452514819

289773434119864minus07

02213311341

00739986588

01491839999

916571896119864minus08

00697813043

03607038292

04319795315

minus301421570119864minus05

00768058043

00059182979

00734021258

minus453845535119864minus07

minus00312153867

03767487763

03459672388

415572858119864minus07

01452667120

02811693568

04273281275

185998012119864minus06

Sol3

00633944399

01908436653

02578328072

minus595667763119864minus07

01017426933

02767897367

03810969084

minus239480772119864minus07

minus01051842285

03769063436

02787447816

minus204170456119864minus07

minus00477059943

02460900702

02006696507

671241513119864minus07

04149858326

00260337751

04452514467

minus438810984119864minus09

01215195321

00256054760

01491841089

189336147119864minus07

02539777159

01761486401

04320126793

297845790119864minus06

00843972823

01349869851

00734028724

779163472119864minus08

minus00534132992

03986395691

03459668311

330962522119864minus09

00880998746

03383563536

04273256312

minus646813249119864minus07

Sol4

01939820199

00603335280

02578382574

502240252119864minus06

00152114400

03633514726

03810978088

803072038119864minus07

01618654345

01097465792

02787430938

minus198215448119864minus06

00056985809

09114653768

02006688671

321352180119864minus08

01904538879

02502358229

04452573447

619105762119864minus06

minus01623604033

03089460561

01491746444

minus901451453119864minus06

01864448178

02428992222

04320068557

minus261979130119864minus06

minus00449302706

01144916285

00733954587

minus717750570119864minus06

01675935311

01774161896

03459538936

minus127733934119864minus05

minus00274959004

04539962587

04273210007

minus520897654119864minus06

Computational Intelligence and Neuroscience 9

3D surf

minus500

minus400

minus300

minus200

minus100

0100200300400500

z-la

bel

minus500minus400minus300minus200minus100

0100200300400500

minus500minus400minus300minus200minus100

0 100 200 300 400 500

Figure 1 The landscape of Schwegelrsquos function

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500

Figure 2 The locations of 40 bats in the initial phase

5 Conclusions

Aiming at the phenomenon of slow convergence rate andlow accuracy of bat algorithm we put forward animprovedbat algorithm with differential operator and Levy flightstrajectory (DLBA) based on the basic framework of batalgorithm (BA) the purpose is to improve the convergencerate and precision of bat algorithm In this paper we definethe frequency fluctuations up and down when the battracking prey which influence the batsrsquo location problemit is more graphic to simulate the batrsquos behavior Moreoverit can make the algorithm effectively jump out of the localoptimum to add Levy flights and differential operator themain reason is because Levy flight has prominent propertiesin the previously mentioned and the differential operatorcan guide bats find better solutions sequentially increase theconvergent speed

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500minus400minus300minus200minus100

0100200300400500

Figure 3 The locations of 40 bats after six iterations

In addition batsrsquo position variation is influenced bythe pulse emission rate and loudness as well Firstly pulseemission rate causes update of position and more and morenew position can be explored consequently increasing thediversity of population Secondly loudness is designed tostrengthen local search and to guide bats find better solutions

In this paper we tested 14 typical benchmark functionsand applied them to solve nonlinear equations the simulationresults not only showed that the proposed algorithm is feasi-ble and effective which ismore robust but also demonstratedthe superior approximation capabilities in high-dimensionalspaceThis is not surprising as the aim of developing the newalgorithm was to try to use the advantages of existing algo-rithms and other interesting feature inspired by the fantasticbehavior of echolocation ofmicrobatsNumerically speakingthese can be translated into two crucial characteristics of themodern metaheuristics intensification and diversificationIntensification intends to search around the current bestsolutions and select the best candidates or solutions whilediversification makes sure that the algorithm can explore thesearch space efficiently

10 Computational Intelligence and Neuroscience

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500

Figure 4 The locations of 40 bats after fifteen iterations

0 20 40 60 80 100 120 140 160 180 2000

100200300400500600700800900

1000

Iteration

Fitn

ess

BADLBA

Figure 5 Convergence curves for the function 1198912

BADLBA

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12

Iteration

Fitn

ess

Figure 6 Convergence curves for the function 1198914

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07

Iteration

Fitn

ess

BADLBA

Figure 7 Convergence curves for the function 1198917

BADLBA

0 20 40 60 80 100 120 140 160 180 200minus420

minus410

minus400

minus390

minus380

minus370

minus360

minus350

minus340

Iteration

Fitn

ess

Figure 8 Convergence curves for the function 11989111

BADLBA

0 20 40 60 80 100 120 140 160 180 200minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Iteration

Fitn

ess

Figure 9 Convergence curves for the function 11989113

Computational Intelligence and Neuroscience 11

0 5 10 15 20 25 30 35 40 45 500

500

1000

1500

2000

2500

3000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 10 Distribution of optimal fitness for 1198916(D = 128)

0 5 10 15 20 25 30 35 40 45 500

2000

4000

6000

8000

10000

12000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 11 Distribution of optimal fitness for 1198919(D = 256)

BADLBA

0 5 10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000

6000

Number of running

Opt

imal

fitn

ess

Figure 12 Distribution of optimal fitness for 1198918(D = 320)

BADLBA

0 5 10 15 20 25 30 35 40 45 50minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Number of running

Opt

imal

fitn

ess

Figure 13 Distribution of optimal fitness for 11989114(D = 512)

BADLBA

Number of running

Opt

imal

fitn

ess

0 5 10 15 20 25 30 35 40 45 500

05

1

15

2

25times104

Figure 14 Distribution of optimal fitness for 1198911(D = 1024)

This potentially powerful optimization strategy can easilybe extended to study multiobjective optimization applica-tions with various constraints even to NP-hard problemsFurther studies can focus on the sensitivity and parameterstudies and their possible relationships with the convergencerate of the algorithm

Acknowledgments

This work is supported by the National Science Foundationof China under Grant no 61165015 Key Project of GuangxiScience Foundation under Grant no 2012GXNSFDA053028Key Project of Guangxi High School Science Foundationunder Grant no 20121ZD008 and the Fund by OpenResearch Fund Program of Key Lab of Intelligent Perception

12 Computational Intelligence and Neuroscience

0 5 10 15 20 25 30 35 40 45 50minus15

minus145

minus14

minus135

minus13

minus125

minus12

Number of running

Opt

imal

fitn

ess

BA

times10minus3

Figure 15 Details of BA line in Figure 12

0 5 10 15 20 25 30 35 40 45 50minus2minus18minus16minus14minus12minus1minus08minus06minus04minus02

0

Number of running

Opt

imal

fitn

ess

DLBA

Figure 16 Details of DLBA line in Figure 12

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7

Iteration

Fitn

ess

DLBA

Figure 17 Convergence curves for DLAB

0 5 10 15 20 25 30 35 40 45 500

002

004

006

008

01

012

014

016

018

Number of running

Opt

imal

fitn

ess

DLBA

Figure 18 Distribution of optimal fitness for DLAB

0 5 10 15 20 25 30 35 40 45 500

010203040506070809

1

Number of running

Opt

imal

fitn

ess

times10minus3

Figure 19 Distribution of optimal fitness less than 0001 inFigure 18

and Image Understanding of Ministry of Education of Chinaunder Grant no IPIU01201100

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of IEEE International Conference on NeuralNetworks (ICNN rsquo95) vol 4 pp 1942ndash1948 December 1995

[2] X-S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress Bristol UK 2nd edition 2010

[3] X-S Yang ldquoFirefly algorithms for multimodal optimizationrdquoin Proceedings of the 5th International Conference on StochasticAlgorithms Foundations and Applications (SAGA rsquo09) pp 169ndash178 2009

[4] B Alatas ldquoACROA artificial chemical reaction optimizationalgorithm for global optimizationrdquo Expert Systems with Appli-cations vol 38 no 10 pp 13170ndash13180 2011

Computational Intelligence and Neuroscience 13

[5] K N Krishnanand and D Ghose ldquoGlowworm warm opti-mization for searching higher dimensional spacesrdquo Studies inComputational Intelligence vol 248 pp 61ndash75 2009

[6] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[7] K Price R Storn and J Lampinen Differential Evolution-APractical Approach to Global Optimization Springer BerlinGermany 2005

[8] httpwwwicsiberkeleyedusim storn [9] R Storn and K Price ldquoDifferential evolutionmdasha simple and

efficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[10] X-S Yang ldquoA new metaheuristic Bat-inspired AlgorithmrdquoStudies in Computational Intelligence vol 284 pp 65ndash74 2010

[11] A Mucherino and O Seref ldquoMonkey search a novel meta-heuristic search for global optimizationrdquo in Proceedings of theConference on Data Mining Systems Analysis and Optimizationin Biomedicine vol 953 pp 162ndash173 March 2007

[12] KM Passino ldquoBiomimicry of bacterial foraging for distributedoptimization and controlrdquo IEEE Control Systems Magazine vol22 no 3 pp 52ndash67 2002

[13] G R Reynolds ldquoCultural algorithms theory and applicationrdquoin New Iders in Optimization D Corne M Dorigo and FGlover Eds pp 367ndash378 McGraw-Hill New York NY USA1999

[14] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[15] R Oftadeh M J Mahjoob and M Shariatpanahi ldquoA novelmeta-heuristic optimization algorithm inspired by group hunt-ing of animals hunting searchrdquo Computers and Mathematicswith Applications vol 60 no 7 pp 2087ndash2098 2010

[16] P W Tsai J S Pan B Y Liao M J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2011

[17] X-S Yang ldquoBat algorithm for multi-objective optimizationrdquoInternational Journal of Bio-Inspired Computation Bio-InspiredComputation (IJBIC) vol 3 no 5 pp 267ndash274 2011

[18] T C Bora L S Coelho and L Lebensztajn ldquoBat-Inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012

[19] B B Mandelbrot The Fractal Geometry of Nature WH Free-man New York NY USA 1983

[20] V A Chechkin R Metzler J Klafter and V Y GoncharldquoIntroduction to the theory of Levy flightsrdquo in AnomalousTransport Foundations and Applications pp 129ndash162 Wiley-VCH Cambridge UK 2008

[21] A M Reynolds and M A Frye ldquoFree-flight odor tracking inDrosophila is consistent with an optimal intermittent scale-freesearchrdquo PLoS ONE vol 2 no 4 p e354 2007

[22] C Brown L S Liebovitch andRGlendon ldquoLevy fights inDobeJursquohoansi foraging patternsrdquo Human Ecology vol 35 no 1 pp129ndash138 2007

[23] P Barthelemy J Bertolotti andD SWiersma ldquoA Levy flight forlightrdquo Nature vol 453 no 7194 pp 495ndash498 2008

[24] N Mercadier W Guerin M Chevrollier and R Kaiser ldquoLevyflights of photons in hot atomic vapoursrdquoNature Physics vol 5no 8 pp 602ndash605 2009

[25] C Grosan and A Abraham ldquoA new approach for solvingnonlinear equations systemsrdquo IEEE Transactions on SystemsMan and Cybernetics A vol 38 no 3 pp 698ndash714 2008

Submit your manuscripts athttpwwwhindawicom

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Applied Computational Intelligence and Soft Computing

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Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Electrical and Computer Engineering

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International Journal of

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ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience 5

Table 1 The parameter set of BA and DLBA

Algorithms 119899 119891min 119891max 1198600

1198941199030

119894120572 120574 119899

119905

BA 40 0 100 (1 2) (0 01) 09 09 mdashDLBA 40 0 1 (1 2) (0 01) 09 mdash 5000

(1) 1198911 sphere function (the first function of De Jongrsquos test

set)

119891 (119909) =

119899

sum

119894=1

1199092

119894 minus10 le 119909

119894le 10 (15)

Here 119899 is dimensionality this function has a globalminimum119891min = 0 at 119909lowast = (0 0 0)

(2) 1198912 Schwegelrsquos problem 222

119891 (119909) =

119899

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816+

119899

prod

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816 minus10 le 119909

119894le 10 (16)

whose global minimum is obviously 119891min = 0 at 119909lowast= (0

0 0)(3) 1198913 Rosenbrock function

119891 (119909) =

119899minus1

sum

119894=1

[(119909119894minus 1)2+ 100(119909

119894+1minus 119909119894

2)

2

]

minus2408 le 119909119894le 2408

(17)

which has a global minimum 119891min = 0 at 119909lowast = (1 1 1)(4) 1198914 Eggcrate function

119891 (119909 119910) = 1199092+ 1199102+ 25 (sin2119909 + sin2119910)

(119909 119910) isin [minus2120587 2120587]

(18)

This 2-dimensional test function obviously gets the globalminimum 119891min = 0 at (0 0)

(5) 1198915 Ackleyrsquos function

119891 (119909) = minus20 exp[

[

minus

1

5

radic

1

119899

119899

sum

119894=1

1199092

119894]

]

minus exp[1119899

119899

sum

119894=1

cos (2120587119909119894)] + 20 + 119890

minus 30 le 119909 le 30

(19)

This function has a global minimum 119891min = 0 at 119909lowast=

(0 0 0) which is a multimodal function(6) 1198916 Griewangkrsquos function

119891 (119909) =

1

4000

119899

sum

119894=1

1199092

119894minus

119899

prod

119894=1

cos(119909119894

radic119894

) + 1 minus600 le 119909119894le 600

(20)

Its global minimum equal 119891min = 0 is obtainable for 119909lowast=

(0 0 0) the number of local minima for arbitrary 119899 is

unknown but in the two-dimensional case there are some500 local minima

(7) 1198917 Salomonrsquos function

119891 (119909) = minus cos(2120587radic119899

sum

119894=1

1199092

119894) + 01radic

119899

sum

119894=1

1199092

119894+ 1 minus5 le 119909

119894le 5

(21)

which has a global minimum 119891min = 0 at 119909lowast = (0 0 0)and is a multimodal function

(8) 1198918 Rastriginrsquos function

119891 (119909) = 10119899 +

119899

sum

119894=1

[1199092

119894minus 10 cos (2120587119909

119894)] minus512 le 119909

119894le 512

(22)

whose global minimum is 119891min = 0 at 119909lowast= (0 0 0)

for 119899 = 2 there are about 50 local minimizers arranged ina lattice-like configuration

(9) 1198919 Zakharovrsquos function

119891 (119909) =

119899

sum

119894=1

1199092

119894+ (

1

2

119899

sum

119894=1

119894119909119894)

2

+ (

1

2

119899

sum

119894=1

119894119909119894)

4

minus10 le 119909119894le 10

(23)

whose global minimum is 119891min = 0 at 119909lowast = (0 0 0) it isa multimodal function as well

(10) 11989110 Easomrsquos function

119891 (119909 119910)=minuscos (119909) cos (119910) exp [minus(119909minus120587)2+(119910minus120587)2]

minus10 le 119909 119910 le 10

(24)

whose global minimum is 119891min = minus1 at (120587 120587) it has manylocal minima

(11) 11989111 Schwegelrsquos function

119891 (119909) = minus

119899

sum

119894=1

119909119894sin(radic100381610038161003816

1003816119909119894

1003816100381610038161003816) minus500 le 119909

119894le 500 (25)

whose global minimum is 119891min asymp minus4189829119899 occuring at119909lowastasymp (4209687 4209687)(12) 119891

12 Shubertrsquos function

119891 (119909 119910)=[

5

sum

119894=1

119894 cos (119894+(119894+1) 119909)] sdot [5

sum

119894=1

119894 cos (119894+(119894+1) 119910)]

minus10 le 119909 119910 le 10

(26)

The number of local minima for this problem is not knownbut for 119899 = 2 the function has 760 local minima 18 of whichare global with 119891min asymp minus1867309

6 Computational Intelligence and Neuroscience

(13) 11989113 Xin-She Yangrsquos function

119891 (119909) = minus(

119899

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816) exp(minus

119899

sum

119894=1

1199092

119894) minus10 le 119909

119894le 10 (27)

which has multiple global minima for example for 119899 = 2 ithas 4 equal minima 119891min = minus1radic119890 asymp minus06065 at (05 05)(05-05) (minus05 05) and (minus05 minus05)

(14) 11989114 ldquoDrop Waverdquo function

119891 (119909) = minus

1 + cos(12radicsum119899119894=11199092

119894)

(12) (sum119899

119894=11199092

119894) + 2

minus512 le 119909119894le 512

(28)

This two-variable function is a multimodal test functionwhose global minimum is 119891min = minus1 occurs at 119909

lowast=

(0 0 0)

43 Comparison of Experimental Results The 2D landscapeof Schwegelrsquos function is shown in Figure 1 and this globalminimum can be found after about 720 FEs for 40 bats after6 iterations as shown in Figures 2 3 and 4

We adopt different terminated criteria aiming at differentbenchmark function we perform 100 times independentlyfor each test function and the record is given in Tables 2 and3

From Table 2 we can see that the DLBA performs muchbetter than the basic bat algorithm which converges muchfaster than BA under the fixed accuracy Tol = 10119890 minus 5 Inour experiment for the function 119891

4 we proposed that an

algorithm only costs 6 generations under the best situationand the average generations is 10 Furthermore for the firstgroup of test functions 119891

1ndash1198919 the DLBA averagely expend

439 generations when each function attains its terminatedcriteria however the BA needs 200 generations invariablyand the convergence speed of DLBA advances 29857 timesOn other hand the DLBAwhich obtained the accuracy of thesolution is much superior to BA which is more approximateto the theoretical value In aword it demonstrated thatDLBAhas fast convergence rate and high precision of the solutions

In Table 3 the five functions (11989110ndash11989114) independently

runs 100 times under the 24000 Fes we can clearly seethe precision of DLBA is obviously superior to the batalgorithm Some benchmark functions can easily attain thetheoretical optimal value In addition the standard deviationof DLBA is relatively low It shows that DLBA has superiorapproximation ability

Observe Tables 2 and 3 no matter high-dimensionalor low-dimensional we can find that DLBA can quicklyconverge to the global minima Furthermore Figures 5 67 8 and 9 show the convergence curves for some of thefunctions from a particular run of DLBA and BA which endat 200 generations The adaptive scheme generally convergesfaster than the basic scheme Here we select 119891

2(D = 20) 119891

4

(D = 2) 1198917(D = 5) 119891

11(D = 2) and 119891

13(D = 2)

DLBAnot only has superior approximation ability in low-dimensional space but also has excellent global search abilityin high-dimensional situation Table 4 is the experimental

result that DLBA performs 50 times independently under thehigh-dimensional situation As shown in Table 4 we can beconscious that the DLBA is effective under the multidimen-sional condition and acquired solution has higher accuracyeven approximate the theoretical value

Figures 10 11 12 13 and 14 are the distribution map ofoptimal fitness that is the selected functions independentlyperform 50 times under the multidimensional situationFigure 13 show that two ldquostraight linerdquo we can see in the figurethat DLBA reaches the global optimum(minus1) however BAfluctuates around 0 In order to display the fact of fluctuationwe magnify the two ldquostraight linerdquo and the amplifyingeffect are depicted in Figures 15 and 16 According to theexperimental results which are obtained from selected testfunctions DLBA presents higher precision than the originalBA on minimizing the outcome as the optimization goal

44 An Application for Solving Nonlinear Equations

Interval Arithmetic Benchmark (IAB) We consider onebenchmark problem proposed from interval arithmetic thebenchmark consists of the following system of [25]

0 = 1199091minus 025428722 minus 018324757119909

411990931199099

0 = 1199092minus 037842197 minus 016275449119909

1119909101199096

0 = 1199093minus 027162577 minus 016955071119909

1119909211990910

0 = 1199094minus 019807914 minus 015585316119909

711990911199096

0 = 1199095minus 044166728 minus 019950920119909

711990961199093

0 = 1199096minus 014654113 minus 018922793119909

8119909511990910

0 = 1199097minus 042937161 minus 021180486119909

211990951199098

0 = 1199098minus 007056438 minus 017081208119909

111990971199096

0 = 1199099minus 034504906 minus 019612740119909

1011990961199098

0 = 11990910minus 042651102 minus 021466544119909

411990981199091

(29)

Some of the solutions obtained as well as the functionvalues (which represent the values of the systemrsquos equationsobtained by replacing the variable values) are presented inTable 5 From Table 5 we can obviously observe that thefunction value of each equation solved byDLBA is superior towhich solved by EA [25] depending on the statistics of the 40function values the precision of function values is enhanced5199820E + 06 times by DLBA The convergence curve ofDLBA is depicted in Figure 17 Figure 17 show that DLBAhas fast convergence rate for solving nonlinear equationsThe achieved optimal fitness (the sum of function valuewith absolute value) in 50 times independent run is depictedin Figure 18 In order to reflect the precision of fitness wemagnify Figure 18 these optimal fitness that less than or equalto 0001 are plotted in Figure 19 We can found that there are32 times optimal fitness is less than 0001

Computational Intelligence and Neuroscience 7

Table 2 Comparison of BA and DLBA for several test functions under the first terminated criteria

TC BF Method Fitness IterationMin Mean Max Std Min Mean Max

Tol = 10119890 minus 5itermax = 200

1198911(119863 = 30) DLBA 430568008119864minus08 480356915119864minus06 996163402119864minus06 302060716119864minus06 10 176 26

BA 5681732462 13061959271 22382694528 3722960351 200 200 200

1198912(119863 = 20) DLBA 566761254119864minus07 633091639119864minus06 988929757119864minus06 248867687119864minus06 21 294 37

BA 2277136624 9335062167 273346090499 26941410301 200 200 200

1198913(119863 = 10) DLBA 566716240 739029403 828857171 037001430 200 200 200

BA 8224908050 29495948479 64076052284 10967161818 200 200 200

1198914(119863 = 2) DLBA 623475433119864minus09 393352701119864minus06 997022879119864minus06 277208752119864minus06 6 10 20

BA 105265280119864minus04 020705750 100771980 023116047 200 200 200

1198915(119863 = 5) DLBA 471737180119864minus07 640969812119864minus06 996088247119864minus06 271187009119864minus06 16 256 39

BA 115883977 302092450 586431360 069052427 200 200 200

1198916(119863 = 5) DLBA 138176777119864minus07 501027883119864minus06 992998256119864minus06 277338474119864minus06 11 184 56

BA 006234957 842851961 2990707979 612812390 200 200 200

1198917(119863 = 5) DLBA 778490674119864minus07 657288799119864minus06 997486019119864minus06 236637154119864minus06 18 46 114

BA 009987344 015318115 030215065 004923496 200 200 200

1198918(119863 = 5) DLBA 599983281119864minus08 447052788119864minus06 985695441119864minus06 281442645119864minus06 15 33 87

BA 724547772 1711077150 2590638177 437190073 200 200 200

1198919(119863 = 5) DLBA 665070016119864minus09 388692603119864minus06 969841027119864minus06 275444981119864minus06 10 151 23

BA 011106843 175623424 848211975 141343722 200 200 200

Table 3 Comparison of BA and DLBA for several test functions under the second terminated condition

TC BF Method FitnessMin Mean Max Std

FEs

11989110(119863 = 2) DLBA minus1 minus1 minus1 906493304119864 minus 17

BA minus099964845 minus098245546 minus090221232 001759602

11989111(119863 = 2) DLBA minus41898288727 minus41583523725 minus30054455266 1553948263

BA minus41898287874 minus40698044144 minus32810751671 2254615132

11989112(119863 = 5) DLBA minus18673090883 minus18673090883 minus18673090883 431613544119864 minus 13

BA minus18667541533 minus18401677767 minus17495599301 226333511

11989113(119863 = 2) DLBA minus060653066 minus060653066 minus060653066 863947249119864 minus 16

BA minus060652631 minus060411559 minus059656608 218888608119864 minus 03

11989114(119863 = 5) DLBA minus1 minus1 minus1 0

BA minus093624514 minus080454167 minus056977584 010270830

Table 4 Comparison of DLBA and BA under the high-dimensional situation

BF Method FitnessMin Mean Max Std

1198916(119863 = 128) DLBA 0 0 0 0

BA 258738759264 281140649000 296656723398 9396861133

1198919(119863 = 256) DLBA 266885939119864 minus 94 512093542119864 minus 69 256023720119864 minus 67 362071553119864 minus 68

BA 750339306546 825661302714 1150679192998 61337556609

1198918(119863 = 320) DLBA 0 0 0 0

BA 446859039467 472344624200 500857109760 13118818777

11989114(119863 = 512) DLBA minus1 minus1 minus1 0

BA minus148358700119864 minus 03 minus131084173119864 minus 03 minus120435493119864 minus 03 599288716119864 minus 05

1198911(119863 = 1024) DLBA 263790469119864 minus 109 473742970119864 minus 87 203861481119864 minus 85 289031313119864 minus 86

BA 1846348950157 1975226686813 2123923111285 56727878361

8 Computational Intelligence and Neuroscience

Table5Outcomeo

fDLB

AforIntervalA

rithm

eticBe

nchm

arkAnd

inCom

paris

onwith

EA

Metho

dEA

DLB

ASolutio

nsVa

riables

values

Functio

nsvalues

Varia

bles

values

Functio

nsvalues

Sol1

00464905115

02077959240

02578335208

722997083119864minus08

01013568357

02769798846

03810968901

minus284879905119864minus07

00840577820

01876863212

02787447331

minus263974619119864minus07

minus01388460309

03367887114

02006720536

306212480119864minus06

04943905739

00530391321

04452522319

774692656119864minus07

minus00760685163

02223730535

01491853777

133597657119864minus06

02475819110

01816084752

04320098178

minus787244534119864minus09

minus00170748156

00878963860

00734062202

341255029119864minus06

00003667535

03447200366

03459672005

324064555119864minus07

01481119311

02784227489

04273251429

minus118429644119864minus06

Sol2

01224819761

01318552790

02578332642

minus145675448119864minus07

01826200685

01964428361

03810968252

minus341069930119864minus07

02356779803

00364987069

02787462357

119723259119864minus06

minus00371150470

02354890155

02006687659

minus176074071119864minus08

03748181856

00675753064

04452514819

289773434119864minus07

02213311341

00739986588

01491839999

916571896119864minus08

00697813043

03607038292

04319795315

minus301421570119864minus05

00768058043

00059182979

00734021258

minus453845535119864minus07

minus00312153867

03767487763

03459672388

415572858119864minus07

01452667120

02811693568

04273281275

185998012119864minus06

Sol3

00633944399

01908436653

02578328072

minus595667763119864minus07

01017426933

02767897367

03810969084

minus239480772119864minus07

minus01051842285

03769063436

02787447816

minus204170456119864minus07

minus00477059943

02460900702

02006696507

671241513119864minus07

04149858326

00260337751

04452514467

minus438810984119864minus09

01215195321

00256054760

01491841089

189336147119864minus07

02539777159

01761486401

04320126793

297845790119864minus06

00843972823

01349869851

00734028724

779163472119864minus08

minus00534132992

03986395691

03459668311

330962522119864minus09

00880998746

03383563536

04273256312

minus646813249119864minus07

Sol4

01939820199

00603335280

02578382574

502240252119864minus06

00152114400

03633514726

03810978088

803072038119864minus07

01618654345

01097465792

02787430938

minus198215448119864minus06

00056985809

09114653768

02006688671

321352180119864minus08

01904538879

02502358229

04452573447

619105762119864minus06

minus01623604033

03089460561

01491746444

minus901451453119864minus06

01864448178

02428992222

04320068557

minus261979130119864minus06

minus00449302706

01144916285

00733954587

minus717750570119864minus06

01675935311

01774161896

03459538936

minus127733934119864minus05

minus00274959004

04539962587

04273210007

minus520897654119864minus06

Computational Intelligence and Neuroscience 9

3D surf

minus500

minus400

minus300

minus200

minus100

0100200300400500

z-la

bel

minus500minus400minus300minus200minus100

0100200300400500

minus500minus400minus300minus200minus100

0 100 200 300 400 500

Figure 1 The landscape of Schwegelrsquos function

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500

Figure 2 The locations of 40 bats in the initial phase

5 Conclusions

Aiming at the phenomenon of slow convergence rate andlow accuracy of bat algorithm we put forward animprovedbat algorithm with differential operator and Levy flightstrajectory (DLBA) based on the basic framework of batalgorithm (BA) the purpose is to improve the convergencerate and precision of bat algorithm In this paper we definethe frequency fluctuations up and down when the battracking prey which influence the batsrsquo location problemit is more graphic to simulate the batrsquos behavior Moreoverit can make the algorithm effectively jump out of the localoptimum to add Levy flights and differential operator themain reason is because Levy flight has prominent propertiesin the previously mentioned and the differential operatorcan guide bats find better solutions sequentially increase theconvergent speed

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500minus400minus300minus200minus100

0100200300400500

Figure 3 The locations of 40 bats after six iterations

In addition batsrsquo position variation is influenced bythe pulse emission rate and loudness as well Firstly pulseemission rate causes update of position and more and morenew position can be explored consequently increasing thediversity of population Secondly loudness is designed tostrengthen local search and to guide bats find better solutions

In this paper we tested 14 typical benchmark functionsand applied them to solve nonlinear equations the simulationresults not only showed that the proposed algorithm is feasi-ble and effective which ismore robust but also demonstratedthe superior approximation capabilities in high-dimensionalspaceThis is not surprising as the aim of developing the newalgorithm was to try to use the advantages of existing algo-rithms and other interesting feature inspired by the fantasticbehavior of echolocation ofmicrobatsNumerically speakingthese can be translated into two crucial characteristics of themodern metaheuristics intensification and diversificationIntensification intends to search around the current bestsolutions and select the best candidates or solutions whilediversification makes sure that the algorithm can explore thesearch space efficiently

10 Computational Intelligence and Neuroscience

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500

Figure 4 The locations of 40 bats after fifteen iterations

0 20 40 60 80 100 120 140 160 180 2000

100200300400500600700800900

1000

Iteration

Fitn

ess

BADLBA

Figure 5 Convergence curves for the function 1198912

BADLBA

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12

Iteration

Fitn

ess

Figure 6 Convergence curves for the function 1198914

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07

Iteration

Fitn

ess

BADLBA

Figure 7 Convergence curves for the function 1198917

BADLBA

0 20 40 60 80 100 120 140 160 180 200minus420

minus410

minus400

minus390

minus380

minus370

minus360

minus350

minus340

Iteration

Fitn

ess

Figure 8 Convergence curves for the function 11989111

BADLBA

0 20 40 60 80 100 120 140 160 180 200minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Iteration

Fitn

ess

Figure 9 Convergence curves for the function 11989113

Computational Intelligence and Neuroscience 11

0 5 10 15 20 25 30 35 40 45 500

500

1000

1500

2000

2500

3000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 10 Distribution of optimal fitness for 1198916(D = 128)

0 5 10 15 20 25 30 35 40 45 500

2000

4000

6000

8000

10000

12000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 11 Distribution of optimal fitness for 1198919(D = 256)

BADLBA

0 5 10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000

6000

Number of running

Opt

imal

fitn

ess

Figure 12 Distribution of optimal fitness for 1198918(D = 320)

BADLBA

0 5 10 15 20 25 30 35 40 45 50minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Number of running

Opt

imal

fitn

ess

Figure 13 Distribution of optimal fitness for 11989114(D = 512)

BADLBA

Number of running

Opt

imal

fitn

ess

0 5 10 15 20 25 30 35 40 45 500

05

1

15

2

25times104

Figure 14 Distribution of optimal fitness for 1198911(D = 1024)

This potentially powerful optimization strategy can easilybe extended to study multiobjective optimization applica-tions with various constraints even to NP-hard problemsFurther studies can focus on the sensitivity and parameterstudies and their possible relationships with the convergencerate of the algorithm

Acknowledgments

This work is supported by the National Science Foundationof China under Grant no 61165015 Key Project of GuangxiScience Foundation under Grant no 2012GXNSFDA053028Key Project of Guangxi High School Science Foundationunder Grant no 20121ZD008 and the Fund by OpenResearch Fund Program of Key Lab of Intelligent Perception

12 Computational Intelligence and Neuroscience

0 5 10 15 20 25 30 35 40 45 50minus15

minus145

minus14

minus135

minus13

minus125

minus12

Number of running

Opt

imal

fitn

ess

BA

times10minus3

Figure 15 Details of BA line in Figure 12

0 5 10 15 20 25 30 35 40 45 50minus2minus18minus16minus14minus12minus1minus08minus06minus04minus02

0

Number of running

Opt

imal

fitn

ess

DLBA

Figure 16 Details of DLBA line in Figure 12

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7

Iteration

Fitn

ess

DLBA

Figure 17 Convergence curves for DLAB

0 5 10 15 20 25 30 35 40 45 500

002

004

006

008

01

012

014

016

018

Number of running

Opt

imal

fitn

ess

DLBA

Figure 18 Distribution of optimal fitness for DLAB

0 5 10 15 20 25 30 35 40 45 500

010203040506070809

1

Number of running

Opt

imal

fitn

ess

times10minus3

Figure 19 Distribution of optimal fitness less than 0001 inFigure 18

and Image Understanding of Ministry of Education of Chinaunder Grant no IPIU01201100

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of IEEE International Conference on NeuralNetworks (ICNN rsquo95) vol 4 pp 1942ndash1948 December 1995

[2] X-S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress Bristol UK 2nd edition 2010

[3] X-S Yang ldquoFirefly algorithms for multimodal optimizationrdquoin Proceedings of the 5th International Conference on StochasticAlgorithms Foundations and Applications (SAGA rsquo09) pp 169ndash178 2009

[4] B Alatas ldquoACROA artificial chemical reaction optimizationalgorithm for global optimizationrdquo Expert Systems with Appli-cations vol 38 no 10 pp 13170ndash13180 2011

Computational Intelligence and Neuroscience 13

[5] K N Krishnanand and D Ghose ldquoGlowworm warm opti-mization for searching higher dimensional spacesrdquo Studies inComputational Intelligence vol 248 pp 61ndash75 2009

[6] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[7] K Price R Storn and J Lampinen Differential Evolution-APractical Approach to Global Optimization Springer BerlinGermany 2005

[8] httpwwwicsiberkeleyedusim storn [9] R Storn and K Price ldquoDifferential evolutionmdasha simple and

efficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[10] X-S Yang ldquoA new metaheuristic Bat-inspired AlgorithmrdquoStudies in Computational Intelligence vol 284 pp 65ndash74 2010

[11] A Mucherino and O Seref ldquoMonkey search a novel meta-heuristic search for global optimizationrdquo in Proceedings of theConference on Data Mining Systems Analysis and Optimizationin Biomedicine vol 953 pp 162ndash173 March 2007

[12] KM Passino ldquoBiomimicry of bacterial foraging for distributedoptimization and controlrdquo IEEE Control Systems Magazine vol22 no 3 pp 52ndash67 2002

[13] G R Reynolds ldquoCultural algorithms theory and applicationrdquoin New Iders in Optimization D Corne M Dorigo and FGlover Eds pp 367ndash378 McGraw-Hill New York NY USA1999

[14] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[15] R Oftadeh M J Mahjoob and M Shariatpanahi ldquoA novelmeta-heuristic optimization algorithm inspired by group hunt-ing of animals hunting searchrdquo Computers and Mathematicswith Applications vol 60 no 7 pp 2087ndash2098 2010

[16] P W Tsai J S Pan B Y Liao M J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2011

[17] X-S Yang ldquoBat algorithm for multi-objective optimizationrdquoInternational Journal of Bio-Inspired Computation Bio-InspiredComputation (IJBIC) vol 3 no 5 pp 267ndash274 2011

[18] T C Bora L S Coelho and L Lebensztajn ldquoBat-Inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012

[19] B B Mandelbrot The Fractal Geometry of Nature WH Free-man New York NY USA 1983

[20] V A Chechkin R Metzler J Klafter and V Y GoncharldquoIntroduction to the theory of Levy flightsrdquo in AnomalousTransport Foundations and Applications pp 129ndash162 Wiley-VCH Cambridge UK 2008

[21] A M Reynolds and M A Frye ldquoFree-flight odor tracking inDrosophila is consistent with an optimal intermittent scale-freesearchrdquo PLoS ONE vol 2 no 4 p e354 2007

[22] C Brown L S Liebovitch andRGlendon ldquoLevy fights inDobeJursquohoansi foraging patternsrdquo Human Ecology vol 35 no 1 pp129ndash138 2007

[23] P Barthelemy J Bertolotti andD SWiersma ldquoA Levy flight forlightrdquo Nature vol 453 no 7194 pp 495ndash498 2008

[24] N Mercadier W Guerin M Chevrollier and R Kaiser ldquoLevyflights of photons in hot atomic vapoursrdquoNature Physics vol 5no 8 pp 602ndash605 2009

[25] C Grosan and A Abraham ldquoA new approach for solvingnonlinear equations systemsrdquo IEEE Transactions on SystemsMan and Cybernetics A vol 38 no 3 pp 698ndash714 2008

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 Computational Intelligence and Neuroscience

(13) 11989113 Xin-She Yangrsquos function

119891 (119909) = minus(

119899

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816) exp(minus

119899

sum

119894=1

1199092

119894) minus10 le 119909

119894le 10 (27)

which has multiple global minima for example for 119899 = 2 ithas 4 equal minima 119891min = minus1radic119890 asymp minus06065 at (05 05)(05-05) (minus05 05) and (minus05 minus05)

(14) 11989114 ldquoDrop Waverdquo function

119891 (119909) = minus

1 + cos(12radicsum119899119894=11199092

119894)

(12) (sum119899

119894=11199092

119894) + 2

minus512 le 119909119894le 512

(28)

This two-variable function is a multimodal test functionwhose global minimum is 119891min = minus1 occurs at 119909

lowast=

(0 0 0)

43 Comparison of Experimental Results The 2D landscapeof Schwegelrsquos function is shown in Figure 1 and this globalminimum can be found after about 720 FEs for 40 bats after6 iterations as shown in Figures 2 3 and 4

We adopt different terminated criteria aiming at differentbenchmark function we perform 100 times independentlyfor each test function and the record is given in Tables 2 and3

From Table 2 we can see that the DLBA performs muchbetter than the basic bat algorithm which converges muchfaster than BA under the fixed accuracy Tol = 10119890 minus 5 Inour experiment for the function 119891

4 we proposed that an

algorithm only costs 6 generations under the best situationand the average generations is 10 Furthermore for the firstgroup of test functions 119891

1ndash1198919 the DLBA averagely expend

439 generations when each function attains its terminatedcriteria however the BA needs 200 generations invariablyand the convergence speed of DLBA advances 29857 timesOn other hand the DLBAwhich obtained the accuracy of thesolution is much superior to BA which is more approximateto the theoretical value In aword it demonstrated thatDLBAhas fast convergence rate and high precision of the solutions

In Table 3 the five functions (11989110ndash11989114) independently

runs 100 times under the 24000 Fes we can clearly seethe precision of DLBA is obviously superior to the batalgorithm Some benchmark functions can easily attain thetheoretical optimal value In addition the standard deviationof DLBA is relatively low It shows that DLBA has superiorapproximation ability

Observe Tables 2 and 3 no matter high-dimensionalor low-dimensional we can find that DLBA can quicklyconverge to the global minima Furthermore Figures 5 67 8 and 9 show the convergence curves for some of thefunctions from a particular run of DLBA and BA which endat 200 generations The adaptive scheme generally convergesfaster than the basic scheme Here we select 119891

2(D = 20) 119891

4

(D = 2) 1198917(D = 5) 119891

11(D = 2) and 119891

13(D = 2)

DLBAnot only has superior approximation ability in low-dimensional space but also has excellent global search abilityin high-dimensional situation Table 4 is the experimental

result that DLBA performs 50 times independently under thehigh-dimensional situation As shown in Table 4 we can beconscious that the DLBA is effective under the multidimen-sional condition and acquired solution has higher accuracyeven approximate the theoretical value

Figures 10 11 12 13 and 14 are the distribution map ofoptimal fitness that is the selected functions independentlyperform 50 times under the multidimensional situationFigure 13 show that two ldquostraight linerdquo we can see in the figurethat DLBA reaches the global optimum(minus1) however BAfluctuates around 0 In order to display the fact of fluctuationwe magnify the two ldquostraight linerdquo and the amplifyingeffect are depicted in Figures 15 and 16 According to theexperimental results which are obtained from selected testfunctions DLBA presents higher precision than the originalBA on minimizing the outcome as the optimization goal

44 An Application for Solving Nonlinear Equations

Interval Arithmetic Benchmark (IAB) We consider onebenchmark problem proposed from interval arithmetic thebenchmark consists of the following system of [25]

0 = 1199091minus 025428722 minus 018324757119909

411990931199099

0 = 1199092minus 037842197 minus 016275449119909

1119909101199096

0 = 1199093minus 027162577 minus 016955071119909

1119909211990910

0 = 1199094minus 019807914 minus 015585316119909

711990911199096

0 = 1199095minus 044166728 minus 019950920119909

711990961199093

0 = 1199096minus 014654113 minus 018922793119909

8119909511990910

0 = 1199097minus 042937161 minus 021180486119909

211990951199098

0 = 1199098minus 007056438 minus 017081208119909

111990971199096

0 = 1199099minus 034504906 minus 019612740119909

1011990961199098

0 = 11990910minus 042651102 minus 021466544119909

411990981199091

(29)

Some of the solutions obtained as well as the functionvalues (which represent the values of the systemrsquos equationsobtained by replacing the variable values) are presented inTable 5 From Table 5 we can obviously observe that thefunction value of each equation solved byDLBA is superior towhich solved by EA [25] depending on the statistics of the 40function values the precision of function values is enhanced5199820E + 06 times by DLBA The convergence curve ofDLBA is depicted in Figure 17 Figure 17 show that DLBAhas fast convergence rate for solving nonlinear equationsThe achieved optimal fitness (the sum of function valuewith absolute value) in 50 times independent run is depictedin Figure 18 In order to reflect the precision of fitness wemagnify Figure 18 these optimal fitness that less than or equalto 0001 are plotted in Figure 19 We can found that there are32 times optimal fitness is less than 0001

Computational Intelligence and Neuroscience 7

Table 2 Comparison of BA and DLBA for several test functions under the first terminated criteria

TC BF Method Fitness IterationMin Mean Max Std Min Mean Max

Tol = 10119890 minus 5itermax = 200

1198911(119863 = 30) DLBA 430568008119864minus08 480356915119864minus06 996163402119864minus06 302060716119864minus06 10 176 26

BA 5681732462 13061959271 22382694528 3722960351 200 200 200

1198912(119863 = 20) DLBA 566761254119864minus07 633091639119864minus06 988929757119864minus06 248867687119864minus06 21 294 37

BA 2277136624 9335062167 273346090499 26941410301 200 200 200

1198913(119863 = 10) DLBA 566716240 739029403 828857171 037001430 200 200 200

BA 8224908050 29495948479 64076052284 10967161818 200 200 200

1198914(119863 = 2) DLBA 623475433119864minus09 393352701119864minus06 997022879119864minus06 277208752119864minus06 6 10 20

BA 105265280119864minus04 020705750 100771980 023116047 200 200 200

1198915(119863 = 5) DLBA 471737180119864minus07 640969812119864minus06 996088247119864minus06 271187009119864minus06 16 256 39

BA 115883977 302092450 586431360 069052427 200 200 200

1198916(119863 = 5) DLBA 138176777119864minus07 501027883119864minus06 992998256119864minus06 277338474119864minus06 11 184 56

BA 006234957 842851961 2990707979 612812390 200 200 200

1198917(119863 = 5) DLBA 778490674119864minus07 657288799119864minus06 997486019119864minus06 236637154119864minus06 18 46 114

BA 009987344 015318115 030215065 004923496 200 200 200

1198918(119863 = 5) DLBA 599983281119864minus08 447052788119864minus06 985695441119864minus06 281442645119864minus06 15 33 87

BA 724547772 1711077150 2590638177 437190073 200 200 200

1198919(119863 = 5) DLBA 665070016119864minus09 388692603119864minus06 969841027119864minus06 275444981119864minus06 10 151 23

BA 011106843 175623424 848211975 141343722 200 200 200

Table 3 Comparison of BA and DLBA for several test functions under the second terminated condition

TC BF Method FitnessMin Mean Max Std

FEs

11989110(119863 = 2) DLBA minus1 minus1 minus1 906493304119864 minus 17

BA minus099964845 minus098245546 minus090221232 001759602

11989111(119863 = 2) DLBA minus41898288727 minus41583523725 minus30054455266 1553948263

BA minus41898287874 minus40698044144 minus32810751671 2254615132

11989112(119863 = 5) DLBA minus18673090883 minus18673090883 minus18673090883 431613544119864 minus 13

BA minus18667541533 minus18401677767 minus17495599301 226333511

11989113(119863 = 2) DLBA minus060653066 minus060653066 minus060653066 863947249119864 minus 16

BA minus060652631 minus060411559 minus059656608 218888608119864 minus 03

11989114(119863 = 5) DLBA minus1 minus1 minus1 0

BA minus093624514 minus080454167 minus056977584 010270830

Table 4 Comparison of DLBA and BA under the high-dimensional situation

BF Method FitnessMin Mean Max Std

1198916(119863 = 128) DLBA 0 0 0 0

BA 258738759264 281140649000 296656723398 9396861133

1198919(119863 = 256) DLBA 266885939119864 minus 94 512093542119864 minus 69 256023720119864 minus 67 362071553119864 minus 68

BA 750339306546 825661302714 1150679192998 61337556609

1198918(119863 = 320) DLBA 0 0 0 0

BA 446859039467 472344624200 500857109760 13118818777

11989114(119863 = 512) DLBA minus1 minus1 minus1 0

BA minus148358700119864 minus 03 minus131084173119864 minus 03 minus120435493119864 minus 03 599288716119864 minus 05

1198911(119863 = 1024) DLBA 263790469119864 minus 109 473742970119864 minus 87 203861481119864 minus 85 289031313119864 minus 86

BA 1846348950157 1975226686813 2123923111285 56727878361

8 Computational Intelligence and Neuroscience

Table5Outcomeo

fDLB

AforIntervalA

rithm

eticBe

nchm

arkAnd

inCom

paris

onwith

EA

Metho

dEA

DLB

ASolutio

nsVa

riables

values

Functio

nsvalues

Varia

bles

values

Functio

nsvalues

Sol1

00464905115

02077959240

02578335208

722997083119864minus08

01013568357

02769798846

03810968901

minus284879905119864minus07

00840577820

01876863212

02787447331

minus263974619119864minus07

minus01388460309

03367887114

02006720536

306212480119864minus06

04943905739

00530391321

04452522319

774692656119864minus07

minus00760685163

02223730535

01491853777

133597657119864minus06

02475819110

01816084752

04320098178

minus787244534119864minus09

minus00170748156

00878963860

00734062202

341255029119864minus06

00003667535

03447200366

03459672005

324064555119864minus07

01481119311

02784227489

04273251429

minus118429644119864minus06

Sol2

01224819761

01318552790

02578332642

minus145675448119864minus07

01826200685

01964428361

03810968252

minus341069930119864minus07

02356779803

00364987069

02787462357

119723259119864minus06

minus00371150470

02354890155

02006687659

minus176074071119864minus08

03748181856

00675753064

04452514819

289773434119864minus07

02213311341

00739986588

01491839999

916571896119864minus08

00697813043

03607038292

04319795315

minus301421570119864minus05

00768058043

00059182979

00734021258

minus453845535119864minus07

minus00312153867

03767487763

03459672388

415572858119864minus07

01452667120

02811693568

04273281275

185998012119864minus06

Sol3

00633944399

01908436653

02578328072

minus595667763119864minus07

01017426933

02767897367

03810969084

minus239480772119864minus07

minus01051842285

03769063436

02787447816

minus204170456119864minus07

minus00477059943

02460900702

02006696507

671241513119864minus07

04149858326

00260337751

04452514467

minus438810984119864minus09

01215195321

00256054760

01491841089

189336147119864minus07

02539777159

01761486401

04320126793

297845790119864minus06

00843972823

01349869851

00734028724

779163472119864minus08

minus00534132992

03986395691

03459668311

330962522119864minus09

00880998746

03383563536

04273256312

minus646813249119864minus07

Sol4

01939820199

00603335280

02578382574

502240252119864minus06

00152114400

03633514726

03810978088

803072038119864minus07

01618654345

01097465792

02787430938

minus198215448119864minus06

00056985809

09114653768

02006688671

321352180119864minus08

01904538879

02502358229

04452573447

619105762119864minus06

minus01623604033

03089460561

01491746444

minus901451453119864minus06

01864448178

02428992222

04320068557

minus261979130119864minus06

minus00449302706

01144916285

00733954587

minus717750570119864minus06

01675935311

01774161896

03459538936

minus127733934119864minus05

minus00274959004

04539962587

04273210007

minus520897654119864minus06

Computational Intelligence and Neuroscience 9

3D surf

minus500

minus400

minus300

minus200

minus100

0100200300400500

z-la

bel

minus500minus400minus300minus200minus100

0100200300400500

minus500minus400minus300minus200minus100

0 100 200 300 400 500

Figure 1 The landscape of Schwegelrsquos function

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500

Figure 2 The locations of 40 bats in the initial phase

5 Conclusions

Aiming at the phenomenon of slow convergence rate andlow accuracy of bat algorithm we put forward animprovedbat algorithm with differential operator and Levy flightstrajectory (DLBA) based on the basic framework of batalgorithm (BA) the purpose is to improve the convergencerate and precision of bat algorithm In this paper we definethe frequency fluctuations up and down when the battracking prey which influence the batsrsquo location problemit is more graphic to simulate the batrsquos behavior Moreoverit can make the algorithm effectively jump out of the localoptimum to add Levy flights and differential operator themain reason is because Levy flight has prominent propertiesin the previously mentioned and the differential operatorcan guide bats find better solutions sequentially increase theconvergent speed

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500minus400minus300minus200minus100

0100200300400500

Figure 3 The locations of 40 bats after six iterations

In addition batsrsquo position variation is influenced bythe pulse emission rate and loudness as well Firstly pulseemission rate causes update of position and more and morenew position can be explored consequently increasing thediversity of population Secondly loudness is designed tostrengthen local search and to guide bats find better solutions

In this paper we tested 14 typical benchmark functionsand applied them to solve nonlinear equations the simulationresults not only showed that the proposed algorithm is feasi-ble and effective which ismore robust but also demonstratedthe superior approximation capabilities in high-dimensionalspaceThis is not surprising as the aim of developing the newalgorithm was to try to use the advantages of existing algo-rithms and other interesting feature inspired by the fantasticbehavior of echolocation ofmicrobatsNumerically speakingthese can be translated into two crucial characteristics of themodern metaheuristics intensification and diversificationIntensification intends to search around the current bestsolutions and select the best candidates or solutions whilediversification makes sure that the algorithm can explore thesearch space efficiently

10 Computational Intelligence and Neuroscience

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500

Figure 4 The locations of 40 bats after fifteen iterations

0 20 40 60 80 100 120 140 160 180 2000

100200300400500600700800900

1000

Iteration

Fitn

ess

BADLBA

Figure 5 Convergence curves for the function 1198912

BADLBA

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12

Iteration

Fitn

ess

Figure 6 Convergence curves for the function 1198914

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07

Iteration

Fitn

ess

BADLBA

Figure 7 Convergence curves for the function 1198917

BADLBA

0 20 40 60 80 100 120 140 160 180 200minus420

minus410

minus400

minus390

minus380

minus370

minus360

minus350

minus340

Iteration

Fitn

ess

Figure 8 Convergence curves for the function 11989111

BADLBA

0 20 40 60 80 100 120 140 160 180 200minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Iteration

Fitn

ess

Figure 9 Convergence curves for the function 11989113

Computational Intelligence and Neuroscience 11

0 5 10 15 20 25 30 35 40 45 500

500

1000

1500

2000

2500

3000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 10 Distribution of optimal fitness for 1198916(D = 128)

0 5 10 15 20 25 30 35 40 45 500

2000

4000

6000

8000

10000

12000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 11 Distribution of optimal fitness for 1198919(D = 256)

BADLBA

0 5 10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000

6000

Number of running

Opt

imal

fitn

ess

Figure 12 Distribution of optimal fitness for 1198918(D = 320)

BADLBA

0 5 10 15 20 25 30 35 40 45 50minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Number of running

Opt

imal

fitn

ess

Figure 13 Distribution of optimal fitness for 11989114(D = 512)

BADLBA

Number of running

Opt

imal

fitn

ess

0 5 10 15 20 25 30 35 40 45 500

05

1

15

2

25times104

Figure 14 Distribution of optimal fitness for 1198911(D = 1024)

This potentially powerful optimization strategy can easilybe extended to study multiobjective optimization applica-tions with various constraints even to NP-hard problemsFurther studies can focus on the sensitivity and parameterstudies and their possible relationships with the convergencerate of the algorithm

Acknowledgments

This work is supported by the National Science Foundationof China under Grant no 61165015 Key Project of GuangxiScience Foundation under Grant no 2012GXNSFDA053028Key Project of Guangxi High School Science Foundationunder Grant no 20121ZD008 and the Fund by OpenResearch Fund Program of Key Lab of Intelligent Perception

12 Computational Intelligence and Neuroscience

0 5 10 15 20 25 30 35 40 45 50minus15

minus145

minus14

minus135

minus13

minus125

minus12

Number of running

Opt

imal

fitn

ess

BA

times10minus3

Figure 15 Details of BA line in Figure 12

0 5 10 15 20 25 30 35 40 45 50minus2minus18minus16minus14minus12minus1minus08minus06minus04minus02

0

Number of running

Opt

imal

fitn

ess

DLBA

Figure 16 Details of DLBA line in Figure 12

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7

Iteration

Fitn

ess

DLBA

Figure 17 Convergence curves for DLAB

0 5 10 15 20 25 30 35 40 45 500

002

004

006

008

01

012

014

016

018

Number of running

Opt

imal

fitn

ess

DLBA

Figure 18 Distribution of optimal fitness for DLAB

0 5 10 15 20 25 30 35 40 45 500

010203040506070809

1

Number of running

Opt

imal

fitn

ess

times10minus3

Figure 19 Distribution of optimal fitness less than 0001 inFigure 18

and Image Understanding of Ministry of Education of Chinaunder Grant no IPIU01201100

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of IEEE International Conference on NeuralNetworks (ICNN rsquo95) vol 4 pp 1942ndash1948 December 1995

[2] X-S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress Bristol UK 2nd edition 2010

[3] X-S Yang ldquoFirefly algorithms for multimodal optimizationrdquoin Proceedings of the 5th International Conference on StochasticAlgorithms Foundations and Applications (SAGA rsquo09) pp 169ndash178 2009

[4] B Alatas ldquoACROA artificial chemical reaction optimizationalgorithm for global optimizationrdquo Expert Systems with Appli-cations vol 38 no 10 pp 13170ndash13180 2011

Computational Intelligence and Neuroscience 13

[5] K N Krishnanand and D Ghose ldquoGlowworm warm opti-mization for searching higher dimensional spacesrdquo Studies inComputational Intelligence vol 248 pp 61ndash75 2009

[6] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[7] K Price R Storn and J Lampinen Differential Evolution-APractical Approach to Global Optimization Springer BerlinGermany 2005

[8] httpwwwicsiberkeleyedusim storn [9] R Storn and K Price ldquoDifferential evolutionmdasha simple and

efficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[10] X-S Yang ldquoA new metaheuristic Bat-inspired AlgorithmrdquoStudies in Computational Intelligence vol 284 pp 65ndash74 2010

[11] A Mucherino and O Seref ldquoMonkey search a novel meta-heuristic search for global optimizationrdquo in Proceedings of theConference on Data Mining Systems Analysis and Optimizationin Biomedicine vol 953 pp 162ndash173 March 2007

[12] KM Passino ldquoBiomimicry of bacterial foraging for distributedoptimization and controlrdquo IEEE Control Systems Magazine vol22 no 3 pp 52ndash67 2002

[13] G R Reynolds ldquoCultural algorithms theory and applicationrdquoin New Iders in Optimization D Corne M Dorigo and FGlover Eds pp 367ndash378 McGraw-Hill New York NY USA1999

[14] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[15] R Oftadeh M J Mahjoob and M Shariatpanahi ldquoA novelmeta-heuristic optimization algorithm inspired by group hunt-ing of animals hunting searchrdquo Computers and Mathematicswith Applications vol 60 no 7 pp 2087ndash2098 2010

[16] P W Tsai J S Pan B Y Liao M J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2011

[17] X-S Yang ldquoBat algorithm for multi-objective optimizationrdquoInternational Journal of Bio-Inspired Computation Bio-InspiredComputation (IJBIC) vol 3 no 5 pp 267ndash274 2011

[18] T C Bora L S Coelho and L Lebensztajn ldquoBat-Inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012

[19] B B Mandelbrot The Fractal Geometry of Nature WH Free-man New York NY USA 1983

[20] V A Chechkin R Metzler J Klafter and V Y GoncharldquoIntroduction to the theory of Levy flightsrdquo in AnomalousTransport Foundations and Applications pp 129ndash162 Wiley-VCH Cambridge UK 2008

[21] A M Reynolds and M A Frye ldquoFree-flight odor tracking inDrosophila is consistent with an optimal intermittent scale-freesearchrdquo PLoS ONE vol 2 no 4 p e354 2007

[22] C Brown L S Liebovitch andRGlendon ldquoLevy fights inDobeJursquohoansi foraging patternsrdquo Human Ecology vol 35 no 1 pp129ndash138 2007

[23] P Barthelemy J Bertolotti andD SWiersma ldquoA Levy flight forlightrdquo Nature vol 453 no 7194 pp 495ndash498 2008

[24] N Mercadier W Guerin M Chevrollier and R Kaiser ldquoLevyflights of photons in hot atomic vapoursrdquoNature Physics vol 5no 8 pp 602ndash605 2009

[25] C Grosan and A Abraham ldquoA new approach for solvingnonlinear equations systemsrdquo IEEE Transactions on SystemsMan and Cybernetics A vol 38 no 3 pp 698ndash714 2008

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience 7

Table 2 Comparison of BA and DLBA for several test functions under the first terminated criteria

TC BF Method Fitness IterationMin Mean Max Std Min Mean Max

Tol = 10119890 minus 5itermax = 200

1198911(119863 = 30) DLBA 430568008119864minus08 480356915119864minus06 996163402119864minus06 302060716119864minus06 10 176 26

BA 5681732462 13061959271 22382694528 3722960351 200 200 200

1198912(119863 = 20) DLBA 566761254119864minus07 633091639119864minus06 988929757119864minus06 248867687119864minus06 21 294 37

BA 2277136624 9335062167 273346090499 26941410301 200 200 200

1198913(119863 = 10) DLBA 566716240 739029403 828857171 037001430 200 200 200

BA 8224908050 29495948479 64076052284 10967161818 200 200 200

1198914(119863 = 2) DLBA 623475433119864minus09 393352701119864minus06 997022879119864minus06 277208752119864minus06 6 10 20

BA 105265280119864minus04 020705750 100771980 023116047 200 200 200

1198915(119863 = 5) DLBA 471737180119864minus07 640969812119864minus06 996088247119864minus06 271187009119864minus06 16 256 39

BA 115883977 302092450 586431360 069052427 200 200 200

1198916(119863 = 5) DLBA 138176777119864minus07 501027883119864minus06 992998256119864minus06 277338474119864minus06 11 184 56

BA 006234957 842851961 2990707979 612812390 200 200 200

1198917(119863 = 5) DLBA 778490674119864minus07 657288799119864minus06 997486019119864minus06 236637154119864minus06 18 46 114

BA 009987344 015318115 030215065 004923496 200 200 200

1198918(119863 = 5) DLBA 599983281119864minus08 447052788119864minus06 985695441119864minus06 281442645119864minus06 15 33 87

BA 724547772 1711077150 2590638177 437190073 200 200 200

1198919(119863 = 5) DLBA 665070016119864minus09 388692603119864minus06 969841027119864minus06 275444981119864minus06 10 151 23

BA 011106843 175623424 848211975 141343722 200 200 200

Table 3 Comparison of BA and DLBA for several test functions under the second terminated condition

TC BF Method FitnessMin Mean Max Std

FEs

11989110(119863 = 2) DLBA minus1 minus1 minus1 906493304119864 minus 17

BA minus099964845 minus098245546 minus090221232 001759602

11989111(119863 = 2) DLBA minus41898288727 minus41583523725 minus30054455266 1553948263

BA minus41898287874 minus40698044144 minus32810751671 2254615132

11989112(119863 = 5) DLBA minus18673090883 minus18673090883 minus18673090883 431613544119864 minus 13

BA minus18667541533 minus18401677767 minus17495599301 226333511

11989113(119863 = 2) DLBA minus060653066 minus060653066 minus060653066 863947249119864 minus 16

BA minus060652631 minus060411559 minus059656608 218888608119864 minus 03

11989114(119863 = 5) DLBA minus1 minus1 minus1 0

BA minus093624514 minus080454167 minus056977584 010270830

Table 4 Comparison of DLBA and BA under the high-dimensional situation

BF Method FitnessMin Mean Max Std

1198916(119863 = 128) DLBA 0 0 0 0

BA 258738759264 281140649000 296656723398 9396861133

1198919(119863 = 256) DLBA 266885939119864 minus 94 512093542119864 minus 69 256023720119864 minus 67 362071553119864 minus 68

BA 750339306546 825661302714 1150679192998 61337556609

1198918(119863 = 320) DLBA 0 0 0 0

BA 446859039467 472344624200 500857109760 13118818777

11989114(119863 = 512) DLBA minus1 minus1 minus1 0

BA minus148358700119864 minus 03 minus131084173119864 minus 03 minus120435493119864 minus 03 599288716119864 minus 05

1198911(119863 = 1024) DLBA 263790469119864 minus 109 473742970119864 minus 87 203861481119864 minus 85 289031313119864 minus 86

BA 1846348950157 1975226686813 2123923111285 56727878361

8 Computational Intelligence and Neuroscience

Table5Outcomeo

fDLB

AforIntervalA

rithm

eticBe

nchm

arkAnd

inCom

paris

onwith

EA

Metho

dEA

DLB

ASolutio

nsVa

riables

values

Functio

nsvalues

Varia

bles

values

Functio

nsvalues

Sol1

00464905115

02077959240

02578335208

722997083119864minus08

01013568357

02769798846

03810968901

minus284879905119864minus07

00840577820

01876863212

02787447331

minus263974619119864minus07

minus01388460309

03367887114

02006720536

306212480119864minus06

04943905739

00530391321

04452522319

774692656119864minus07

minus00760685163

02223730535

01491853777

133597657119864minus06

02475819110

01816084752

04320098178

minus787244534119864minus09

minus00170748156

00878963860

00734062202

341255029119864minus06

00003667535

03447200366

03459672005

324064555119864minus07

01481119311

02784227489

04273251429

minus118429644119864minus06

Sol2

01224819761

01318552790

02578332642

minus145675448119864minus07

01826200685

01964428361

03810968252

minus341069930119864minus07

02356779803

00364987069

02787462357

119723259119864minus06

minus00371150470

02354890155

02006687659

minus176074071119864minus08

03748181856

00675753064

04452514819

289773434119864minus07

02213311341

00739986588

01491839999

916571896119864minus08

00697813043

03607038292

04319795315

minus301421570119864minus05

00768058043

00059182979

00734021258

minus453845535119864minus07

minus00312153867

03767487763

03459672388

415572858119864minus07

01452667120

02811693568

04273281275

185998012119864minus06

Sol3

00633944399

01908436653

02578328072

minus595667763119864minus07

01017426933

02767897367

03810969084

minus239480772119864minus07

minus01051842285

03769063436

02787447816

minus204170456119864minus07

minus00477059943

02460900702

02006696507

671241513119864minus07

04149858326

00260337751

04452514467

minus438810984119864minus09

01215195321

00256054760

01491841089

189336147119864minus07

02539777159

01761486401

04320126793

297845790119864minus06

00843972823

01349869851

00734028724

779163472119864minus08

minus00534132992

03986395691

03459668311

330962522119864minus09

00880998746

03383563536

04273256312

minus646813249119864minus07

Sol4

01939820199

00603335280

02578382574

502240252119864minus06

00152114400

03633514726

03810978088

803072038119864minus07

01618654345

01097465792

02787430938

minus198215448119864minus06

00056985809

09114653768

02006688671

321352180119864minus08

01904538879

02502358229

04452573447

619105762119864minus06

minus01623604033

03089460561

01491746444

minus901451453119864minus06

01864448178

02428992222

04320068557

minus261979130119864minus06

minus00449302706

01144916285

00733954587

minus717750570119864minus06

01675935311

01774161896

03459538936

minus127733934119864minus05

minus00274959004

04539962587

04273210007

minus520897654119864minus06

Computational Intelligence and Neuroscience 9

3D surf

minus500

minus400

minus300

minus200

minus100

0100200300400500

z-la

bel

minus500minus400minus300minus200minus100

0100200300400500

minus500minus400minus300minus200minus100

0 100 200 300 400 500

Figure 1 The landscape of Schwegelrsquos function

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500

Figure 2 The locations of 40 bats in the initial phase

5 Conclusions

Aiming at the phenomenon of slow convergence rate andlow accuracy of bat algorithm we put forward animprovedbat algorithm with differential operator and Levy flightstrajectory (DLBA) based on the basic framework of batalgorithm (BA) the purpose is to improve the convergencerate and precision of bat algorithm In this paper we definethe frequency fluctuations up and down when the battracking prey which influence the batsrsquo location problemit is more graphic to simulate the batrsquos behavior Moreoverit can make the algorithm effectively jump out of the localoptimum to add Levy flights and differential operator themain reason is because Levy flight has prominent propertiesin the previously mentioned and the differential operatorcan guide bats find better solutions sequentially increase theconvergent speed

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500minus400minus300minus200minus100

0100200300400500

Figure 3 The locations of 40 bats after six iterations

In addition batsrsquo position variation is influenced bythe pulse emission rate and loudness as well Firstly pulseemission rate causes update of position and more and morenew position can be explored consequently increasing thediversity of population Secondly loudness is designed tostrengthen local search and to guide bats find better solutions

In this paper we tested 14 typical benchmark functionsand applied them to solve nonlinear equations the simulationresults not only showed that the proposed algorithm is feasi-ble and effective which ismore robust but also demonstratedthe superior approximation capabilities in high-dimensionalspaceThis is not surprising as the aim of developing the newalgorithm was to try to use the advantages of existing algo-rithms and other interesting feature inspired by the fantasticbehavior of echolocation ofmicrobatsNumerically speakingthese can be translated into two crucial characteristics of themodern metaheuristics intensification and diversificationIntensification intends to search around the current bestsolutions and select the best candidates or solutions whilediversification makes sure that the algorithm can explore thesearch space efficiently

10 Computational Intelligence and Neuroscience

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500

Figure 4 The locations of 40 bats after fifteen iterations

0 20 40 60 80 100 120 140 160 180 2000

100200300400500600700800900

1000

Iteration

Fitn

ess

BADLBA

Figure 5 Convergence curves for the function 1198912

BADLBA

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12

Iteration

Fitn

ess

Figure 6 Convergence curves for the function 1198914

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07

Iteration

Fitn

ess

BADLBA

Figure 7 Convergence curves for the function 1198917

BADLBA

0 20 40 60 80 100 120 140 160 180 200minus420

minus410

minus400

minus390

minus380

minus370

minus360

minus350

minus340

Iteration

Fitn

ess

Figure 8 Convergence curves for the function 11989111

BADLBA

0 20 40 60 80 100 120 140 160 180 200minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Iteration

Fitn

ess

Figure 9 Convergence curves for the function 11989113

Computational Intelligence and Neuroscience 11

0 5 10 15 20 25 30 35 40 45 500

500

1000

1500

2000

2500

3000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 10 Distribution of optimal fitness for 1198916(D = 128)

0 5 10 15 20 25 30 35 40 45 500

2000

4000

6000

8000

10000

12000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 11 Distribution of optimal fitness for 1198919(D = 256)

BADLBA

0 5 10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000

6000

Number of running

Opt

imal

fitn

ess

Figure 12 Distribution of optimal fitness for 1198918(D = 320)

BADLBA

0 5 10 15 20 25 30 35 40 45 50minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Number of running

Opt

imal

fitn

ess

Figure 13 Distribution of optimal fitness for 11989114(D = 512)

BADLBA

Number of running

Opt

imal

fitn

ess

0 5 10 15 20 25 30 35 40 45 500

05

1

15

2

25times104

Figure 14 Distribution of optimal fitness for 1198911(D = 1024)

This potentially powerful optimization strategy can easilybe extended to study multiobjective optimization applica-tions with various constraints even to NP-hard problemsFurther studies can focus on the sensitivity and parameterstudies and their possible relationships with the convergencerate of the algorithm

Acknowledgments

This work is supported by the National Science Foundationof China under Grant no 61165015 Key Project of GuangxiScience Foundation under Grant no 2012GXNSFDA053028Key Project of Guangxi High School Science Foundationunder Grant no 20121ZD008 and the Fund by OpenResearch Fund Program of Key Lab of Intelligent Perception

12 Computational Intelligence and Neuroscience

0 5 10 15 20 25 30 35 40 45 50minus15

minus145

minus14

minus135

minus13

minus125

minus12

Number of running

Opt

imal

fitn

ess

BA

times10minus3

Figure 15 Details of BA line in Figure 12

0 5 10 15 20 25 30 35 40 45 50minus2minus18minus16minus14minus12minus1minus08minus06minus04minus02

0

Number of running

Opt

imal

fitn

ess

DLBA

Figure 16 Details of DLBA line in Figure 12

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7

Iteration

Fitn

ess

DLBA

Figure 17 Convergence curves for DLAB

0 5 10 15 20 25 30 35 40 45 500

002

004

006

008

01

012

014

016

018

Number of running

Opt

imal

fitn

ess

DLBA

Figure 18 Distribution of optimal fitness for DLAB

0 5 10 15 20 25 30 35 40 45 500

010203040506070809

1

Number of running

Opt

imal

fitn

ess

times10minus3

Figure 19 Distribution of optimal fitness less than 0001 inFigure 18

and Image Understanding of Ministry of Education of Chinaunder Grant no IPIU01201100

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of IEEE International Conference on NeuralNetworks (ICNN rsquo95) vol 4 pp 1942ndash1948 December 1995

[2] X-S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress Bristol UK 2nd edition 2010

[3] X-S Yang ldquoFirefly algorithms for multimodal optimizationrdquoin Proceedings of the 5th International Conference on StochasticAlgorithms Foundations and Applications (SAGA rsquo09) pp 169ndash178 2009

[4] B Alatas ldquoACROA artificial chemical reaction optimizationalgorithm for global optimizationrdquo Expert Systems with Appli-cations vol 38 no 10 pp 13170ndash13180 2011

Computational Intelligence and Neuroscience 13

[5] K N Krishnanand and D Ghose ldquoGlowworm warm opti-mization for searching higher dimensional spacesrdquo Studies inComputational Intelligence vol 248 pp 61ndash75 2009

[6] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[7] K Price R Storn and J Lampinen Differential Evolution-APractical Approach to Global Optimization Springer BerlinGermany 2005

[8] httpwwwicsiberkeleyedusim storn [9] R Storn and K Price ldquoDifferential evolutionmdasha simple and

efficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[10] X-S Yang ldquoA new metaheuristic Bat-inspired AlgorithmrdquoStudies in Computational Intelligence vol 284 pp 65ndash74 2010

[11] A Mucherino and O Seref ldquoMonkey search a novel meta-heuristic search for global optimizationrdquo in Proceedings of theConference on Data Mining Systems Analysis and Optimizationin Biomedicine vol 953 pp 162ndash173 March 2007

[12] KM Passino ldquoBiomimicry of bacterial foraging for distributedoptimization and controlrdquo IEEE Control Systems Magazine vol22 no 3 pp 52ndash67 2002

[13] G R Reynolds ldquoCultural algorithms theory and applicationrdquoin New Iders in Optimization D Corne M Dorigo and FGlover Eds pp 367ndash378 McGraw-Hill New York NY USA1999

[14] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[15] R Oftadeh M J Mahjoob and M Shariatpanahi ldquoA novelmeta-heuristic optimization algorithm inspired by group hunt-ing of animals hunting searchrdquo Computers and Mathematicswith Applications vol 60 no 7 pp 2087ndash2098 2010

[16] P W Tsai J S Pan B Y Liao M J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2011

[17] X-S Yang ldquoBat algorithm for multi-objective optimizationrdquoInternational Journal of Bio-Inspired Computation Bio-InspiredComputation (IJBIC) vol 3 no 5 pp 267ndash274 2011

[18] T C Bora L S Coelho and L Lebensztajn ldquoBat-Inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012

[19] B B Mandelbrot The Fractal Geometry of Nature WH Free-man New York NY USA 1983

[20] V A Chechkin R Metzler J Klafter and V Y GoncharldquoIntroduction to the theory of Levy flightsrdquo in AnomalousTransport Foundations and Applications pp 129ndash162 Wiley-VCH Cambridge UK 2008

[21] A M Reynolds and M A Frye ldquoFree-flight odor tracking inDrosophila is consistent with an optimal intermittent scale-freesearchrdquo PLoS ONE vol 2 no 4 p e354 2007

[22] C Brown L S Liebovitch andRGlendon ldquoLevy fights inDobeJursquohoansi foraging patternsrdquo Human Ecology vol 35 no 1 pp129ndash138 2007

[23] P Barthelemy J Bertolotti andD SWiersma ldquoA Levy flight forlightrdquo Nature vol 453 no 7194 pp 495ndash498 2008

[24] N Mercadier W Guerin M Chevrollier and R Kaiser ldquoLevyflights of photons in hot atomic vapoursrdquoNature Physics vol 5no 8 pp 602ndash605 2009

[25] C Grosan and A Abraham ldquoA new approach for solvingnonlinear equations systemsrdquo IEEE Transactions on SystemsMan and Cybernetics A vol 38 no 3 pp 698ndash714 2008

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

8 Computational Intelligence and Neuroscience

Table5Outcomeo

fDLB

AforIntervalA

rithm

eticBe

nchm

arkAnd

inCom

paris

onwith

EA

Metho

dEA

DLB

ASolutio

nsVa

riables

values

Functio

nsvalues

Varia

bles

values

Functio

nsvalues

Sol1

00464905115

02077959240

02578335208

722997083119864minus08

01013568357

02769798846

03810968901

minus284879905119864minus07

00840577820

01876863212

02787447331

minus263974619119864minus07

minus01388460309

03367887114

02006720536

306212480119864minus06

04943905739

00530391321

04452522319

774692656119864minus07

minus00760685163

02223730535

01491853777

133597657119864minus06

02475819110

01816084752

04320098178

minus787244534119864minus09

minus00170748156

00878963860

00734062202

341255029119864minus06

00003667535

03447200366

03459672005

324064555119864minus07

01481119311

02784227489

04273251429

minus118429644119864minus06

Sol2

01224819761

01318552790

02578332642

minus145675448119864minus07

01826200685

01964428361

03810968252

minus341069930119864minus07

02356779803

00364987069

02787462357

119723259119864minus06

minus00371150470

02354890155

02006687659

minus176074071119864minus08

03748181856

00675753064

04452514819

289773434119864minus07

02213311341

00739986588

01491839999

916571896119864minus08

00697813043

03607038292

04319795315

minus301421570119864minus05

00768058043

00059182979

00734021258

minus453845535119864minus07

minus00312153867

03767487763

03459672388

415572858119864minus07

01452667120

02811693568

04273281275

185998012119864minus06

Sol3

00633944399

01908436653

02578328072

minus595667763119864minus07

01017426933

02767897367

03810969084

minus239480772119864minus07

minus01051842285

03769063436

02787447816

minus204170456119864minus07

minus00477059943

02460900702

02006696507

671241513119864minus07

04149858326

00260337751

04452514467

minus438810984119864minus09

01215195321

00256054760

01491841089

189336147119864minus07

02539777159

01761486401

04320126793

297845790119864minus06

00843972823

01349869851

00734028724

779163472119864minus08

minus00534132992

03986395691

03459668311

330962522119864minus09

00880998746

03383563536

04273256312

minus646813249119864minus07

Sol4

01939820199

00603335280

02578382574

502240252119864minus06

00152114400

03633514726

03810978088

803072038119864minus07

01618654345

01097465792

02787430938

minus198215448119864minus06

00056985809

09114653768

02006688671

321352180119864minus08

01904538879

02502358229

04452573447

619105762119864minus06

minus01623604033

03089460561

01491746444

minus901451453119864minus06

01864448178

02428992222

04320068557

minus261979130119864minus06

minus00449302706

01144916285

00733954587

minus717750570119864minus06

01675935311

01774161896

03459538936

minus127733934119864minus05

minus00274959004

04539962587

04273210007

minus520897654119864minus06

Computational Intelligence and Neuroscience 9

3D surf

minus500

minus400

minus300

minus200

minus100

0100200300400500

z-la

bel

minus500minus400minus300minus200minus100

0100200300400500

minus500minus400minus300minus200minus100

0 100 200 300 400 500

Figure 1 The landscape of Schwegelrsquos function

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500

Figure 2 The locations of 40 bats in the initial phase

5 Conclusions

Aiming at the phenomenon of slow convergence rate andlow accuracy of bat algorithm we put forward animprovedbat algorithm with differential operator and Levy flightstrajectory (DLBA) based on the basic framework of batalgorithm (BA) the purpose is to improve the convergencerate and precision of bat algorithm In this paper we definethe frequency fluctuations up and down when the battracking prey which influence the batsrsquo location problemit is more graphic to simulate the batrsquos behavior Moreoverit can make the algorithm effectively jump out of the localoptimum to add Levy flights and differential operator themain reason is because Levy flight has prominent propertiesin the previously mentioned and the differential operatorcan guide bats find better solutions sequentially increase theconvergent speed

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500minus400minus300minus200minus100

0100200300400500

Figure 3 The locations of 40 bats after six iterations

In addition batsrsquo position variation is influenced bythe pulse emission rate and loudness as well Firstly pulseemission rate causes update of position and more and morenew position can be explored consequently increasing thediversity of population Secondly loudness is designed tostrengthen local search and to guide bats find better solutions

In this paper we tested 14 typical benchmark functionsand applied them to solve nonlinear equations the simulationresults not only showed that the proposed algorithm is feasi-ble and effective which ismore robust but also demonstratedthe superior approximation capabilities in high-dimensionalspaceThis is not surprising as the aim of developing the newalgorithm was to try to use the advantages of existing algo-rithms and other interesting feature inspired by the fantasticbehavior of echolocation ofmicrobatsNumerically speakingthese can be translated into two crucial characteristics of themodern metaheuristics intensification and diversificationIntensification intends to search around the current bestsolutions and select the best candidates or solutions whilediversification makes sure that the algorithm can explore thesearch space efficiently

10 Computational Intelligence and Neuroscience

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500

Figure 4 The locations of 40 bats after fifteen iterations

0 20 40 60 80 100 120 140 160 180 2000

100200300400500600700800900

1000

Iteration

Fitn

ess

BADLBA

Figure 5 Convergence curves for the function 1198912

BADLBA

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12

Iteration

Fitn

ess

Figure 6 Convergence curves for the function 1198914

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07

Iteration

Fitn

ess

BADLBA

Figure 7 Convergence curves for the function 1198917

BADLBA

0 20 40 60 80 100 120 140 160 180 200minus420

minus410

minus400

minus390

minus380

minus370

minus360

minus350

minus340

Iteration

Fitn

ess

Figure 8 Convergence curves for the function 11989111

BADLBA

0 20 40 60 80 100 120 140 160 180 200minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Iteration

Fitn

ess

Figure 9 Convergence curves for the function 11989113

Computational Intelligence and Neuroscience 11

0 5 10 15 20 25 30 35 40 45 500

500

1000

1500

2000

2500

3000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 10 Distribution of optimal fitness for 1198916(D = 128)

0 5 10 15 20 25 30 35 40 45 500

2000

4000

6000

8000

10000

12000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 11 Distribution of optimal fitness for 1198919(D = 256)

BADLBA

0 5 10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000

6000

Number of running

Opt

imal

fitn

ess

Figure 12 Distribution of optimal fitness for 1198918(D = 320)

BADLBA

0 5 10 15 20 25 30 35 40 45 50minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Number of running

Opt

imal

fitn

ess

Figure 13 Distribution of optimal fitness for 11989114(D = 512)

BADLBA

Number of running

Opt

imal

fitn

ess

0 5 10 15 20 25 30 35 40 45 500

05

1

15

2

25times104

Figure 14 Distribution of optimal fitness for 1198911(D = 1024)

This potentially powerful optimization strategy can easilybe extended to study multiobjective optimization applica-tions with various constraints even to NP-hard problemsFurther studies can focus on the sensitivity and parameterstudies and their possible relationships with the convergencerate of the algorithm

Acknowledgments

This work is supported by the National Science Foundationof China under Grant no 61165015 Key Project of GuangxiScience Foundation under Grant no 2012GXNSFDA053028Key Project of Guangxi High School Science Foundationunder Grant no 20121ZD008 and the Fund by OpenResearch Fund Program of Key Lab of Intelligent Perception

12 Computational Intelligence and Neuroscience

0 5 10 15 20 25 30 35 40 45 50minus15

minus145

minus14

minus135

minus13

minus125

minus12

Number of running

Opt

imal

fitn

ess

BA

times10minus3

Figure 15 Details of BA line in Figure 12

0 5 10 15 20 25 30 35 40 45 50minus2minus18minus16minus14minus12minus1minus08minus06minus04minus02

0

Number of running

Opt

imal

fitn

ess

DLBA

Figure 16 Details of DLBA line in Figure 12

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7

Iteration

Fitn

ess

DLBA

Figure 17 Convergence curves for DLAB

0 5 10 15 20 25 30 35 40 45 500

002

004

006

008

01

012

014

016

018

Number of running

Opt

imal

fitn

ess

DLBA

Figure 18 Distribution of optimal fitness for DLAB

0 5 10 15 20 25 30 35 40 45 500

010203040506070809

1

Number of running

Opt

imal

fitn

ess

times10minus3

Figure 19 Distribution of optimal fitness less than 0001 inFigure 18

and Image Understanding of Ministry of Education of Chinaunder Grant no IPIU01201100

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of IEEE International Conference on NeuralNetworks (ICNN rsquo95) vol 4 pp 1942ndash1948 December 1995

[2] X-S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress Bristol UK 2nd edition 2010

[3] X-S Yang ldquoFirefly algorithms for multimodal optimizationrdquoin Proceedings of the 5th International Conference on StochasticAlgorithms Foundations and Applications (SAGA rsquo09) pp 169ndash178 2009

[4] B Alatas ldquoACROA artificial chemical reaction optimizationalgorithm for global optimizationrdquo Expert Systems with Appli-cations vol 38 no 10 pp 13170ndash13180 2011

Computational Intelligence and Neuroscience 13

[5] K N Krishnanand and D Ghose ldquoGlowworm warm opti-mization for searching higher dimensional spacesrdquo Studies inComputational Intelligence vol 248 pp 61ndash75 2009

[6] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[7] K Price R Storn and J Lampinen Differential Evolution-APractical Approach to Global Optimization Springer BerlinGermany 2005

[8] httpwwwicsiberkeleyedusim storn [9] R Storn and K Price ldquoDifferential evolutionmdasha simple and

efficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[10] X-S Yang ldquoA new metaheuristic Bat-inspired AlgorithmrdquoStudies in Computational Intelligence vol 284 pp 65ndash74 2010

[11] A Mucherino and O Seref ldquoMonkey search a novel meta-heuristic search for global optimizationrdquo in Proceedings of theConference on Data Mining Systems Analysis and Optimizationin Biomedicine vol 953 pp 162ndash173 March 2007

[12] KM Passino ldquoBiomimicry of bacterial foraging for distributedoptimization and controlrdquo IEEE Control Systems Magazine vol22 no 3 pp 52ndash67 2002

[13] G R Reynolds ldquoCultural algorithms theory and applicationrdquoin New Iders in Optimization D Corne M Dorigo and FGlover Eds pp 367ndash378 McGraw-Hill New York NY USA1999

[14] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[15] R Oftadeh M J Mahjoob and M Shariatpanahi ldquoA novelmeta-heuristic optimization algorithm inspired by group hunt-ing of animals hunting searchrdquo Computers and Mathematicswith Applications vol 60 no 7 pp 2087ndash2098 2010

[16] P W Tsai J S Pan B Y Liao M J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2011

[17] X-S Yang ldquoBat algorithm for multi-objective optimizationrdquoInternational Journal of Bio-Inspired Computation Bio-InspiredComputation (IJBIC) vol 3 no 5 pp 267ndash274 2011

[18] T C Bora L S Coelho and L Lebensztajn ldquoBat-Inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012

[19] B B Mandelbrot The Fractal Geometry of Nature WH Free-man New York NY USA 1983

[20] V A Chechkin R Metzler J Klafter and V Y GoncharldquoIntroduction to the theory of Levy flightsrdquo in AnomalousTransport Foundations and Applications pp 129ndash162 Wiley-VCH Cambridge UK 2008

[21] A M Reynolds and M A Frye ldquoFree-flight odor tracking inDrosophila is consistent with an optimal intermittent scale-freesearchrdquo PLoS ONE vol 2 no 4 p e354 2007

[22] C Brown L S Liebovitch andRGlendon ldquoLevy fights inDobeJursquohoansi foraging patternsrdquo Human Ecology vol 35 no 1 pp129ndash138 2007

[23] P Barthelemy J Bertolotti andD SWiersma ldquoA Levy flight forlightrdquo Nature vol 453 no 7194 pp 495ndash498 2008

[24] N Mercadier W Guerin M Chevrollier and R Kaiser ldquoLevyflights of photons in hot atomic vapoursrdquoNature Physics vol 5no 8 pp 602ndash605 2009

[25] C Grosan and A Abraham ldquoA new approach for solvingnonlinear equations systemsrdquo IEEE Transactions on SystemsMan and Cybernetics A vol 38 no 3 pp 698ndash714 2008

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience 9

3D surf

minus500

minus400

minus300

minus200

minus100

0100200300400500

z-la

bel

minus500minus400minus300minus200minus100

0100200300400500

minus500minus400minus300minus200minus100

0 100 200 300 400 500

Figure 1 The landscape of Schwegelrsquos function

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500

Figure 2 The locations of 40 bats in the initial phase

5 Conclusions

Aiming at the phenomenon of slow convergence rate andlow accuracy of bat algorithm we put forward animprovedbat algorithm with differential operator and Levy flightstrajectory (DLBA) based on the basic framework of batalgorithm (BA) the purpose is to improve the convergencerate and precision of bat algorithm In this paper we definethe frequency fluctuations up and down when the battracking prey which influence the batsrsquo location problemit is more graphic to simulate the batrsquos behavior Moreoverit can make the algorithm effectively jump out of the localoptimum to add Levy flights and differential operator themain reason is because Levy flight has prominent propertiesin the previously mentioned and the differential operatorcan guide bats find better solutions sequentially increase theconvergent speed

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500minus400minus300minus200minus100

0100200300400500

Figure 3 The locations of 40 bats after six iterations

In addition batsrsquo position variation is influenced bythe pulse emission rate and loudness as well Firstly pulseemission rate causes update of position and more and morenew position can be explored consequently increasing thediversity of population Secondly loudness is designed tostrengthen local search and to guide bats find better solutions

In this paper we tested 14 typical benchmark functionsand applied them to solve nonlinear equations the simulationresults not only showed that the proposed algorithm is feasi-ble and effective which ismore robust but also demonstratedthe superior approximation capabilities in high-dimensionalspaceThis is not surprising as the aim of developing the newalgorithm was to try to use the advantages of existing algo-rithms and other interesting feature inspired by the fantasticbehavior of echolocation ofmicrobatsNumerically speakingthese can be translated into two crucial characteristics of themodern metaheuristics intensification and diversificationIntensification intends to search around the current bestsolutions and select the best candidates or solutions whilediversification makes sure that the algorithm can explore thesearch space efficiently

10 Computational Intelligence and Neuroscience

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500

Figure 4 The locations of 40 bats after fifteen iterations

0 20 40 60 80 100 120 140 160 180 2000

100200300400500600700800900

1000

Iteration

Fitn

ess

BADLBA

Figure 5 Convergence curves for the function 1198912

BADLBA

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12

Iteration

Fitn

ess

Figure 6 Convergence curves for the function 1198914

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07

Iteration

Fitn

ess

BADLBA

Figure 7 Convergence curves for the function 1198917

BADLBA

0 20 40 60 80 100 120 140 160 180 200minus420

minus410

minus400

minus390

minus380

minus370

minus360

minus350

minus340

Iteration

Fitn

ess

Figure 8 Convergence curves for the function 11989111

BADLBA

0 20 40 60 80 100 120 140 160 180 200minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Iteration

Fitn

ess

Figure 9 Convergence curves for the function 11989113

Computational Intelligence and Neuroscience 11

0 5 10 15 20 25 30 35 40 45 500

500

1000

1500

2000

2500

3000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 10 Distribution of optimal fitness for 1198916(D = 128)

0 5 10 15 20 25 30 35 40 45 500

2000

4000

6000

8000

10000

12000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 11 Distribution of optimal fitness for 1198919(D = 256)

BADLBA

0 5 10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000

6000

Number of running

Opt

imal

fitn

ess

Figure 12 Distribution of optimal fitness for 1198918(D = 320)

BADLBA

0 5 10 15 20 25 30 35 40 45 50minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Number of running

Opt

imal

fitn

ess

Figure 13 Distribution of optimal fitness for 11989114(D = 512)

BADLBA

Number of running

Opt

imal

fitn

ess

0 5 10 15 20 25 30 35 40 45 500

05

1

15

2

25times104

Figure 14 Distribution of optimal fitness for 1198911(D = 1024)

This potentially powerful optimization strategy can easilybe extended to study multiobjective optimization applica-tions with various constraints even to NP-hard problemsFurther studies can focus on the sensitivity and parameterstudies and their possible relationships with the convergencerate of the algorithm

Acknowledgments

This work is supported by the National Science Foundationof China under Grant no 61165015 Key Project of GuangxiScience Foundation under Grant no 2012GXNSFDA053028Key Project of Guangxi High School Science Foundationunder Grant no 20121ZD008 and the Fund by OpenResearch Fund Program of Key Lab of Intelligent Perception

12 Computational Intelligence and Neuroscience

0 5 10 15 20 25 30 35 40 45 50minus15

minus145

minus14

minus135

minus13

minus125

minus12

Number of running

Opt

imal

fitn

ess

BA

times10minus3

Figure 15 Details of BA line in Figure 12

0 5 10 15 20 25 30 35 40 45 50minus2minus18minus16minus14minus12minus1minus08minus06minus04minus02

0

Number of running

Opt

imal

fitn

ess

DLBA

Figure 16 Details of DLBA line in Figure 12

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7

Iteration

Fitn

ess

DLBA

Figure 17 Convergence curves for DLAB

0 5 10 15 20 25 30 35 40 45 500

002

004

006

008

01

012

014

016

018

Number of running

Opt

imal

fitn

ess

DLBA

Figure 18 Distribution of optimal fitness for DLAB

0 5 10 15 20 25 30 35 40 45 500

010203040506070809

1

Number of running

Opt

imal

fitn

ess

times10minus3

Figure 19 Distribution of optimal fitness less than 0001 inFigure 18

and Image Understanding of Ministry of Education of Chinaunder Grant no IPIU01201100

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of IEEE International Conference on NeuralNetworks (ICNN rsquo95) vol 4 pp 1942ndash1948 December 1995

[2] X-S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress Bristol UK 2nd edition 2010

[3] X-S Yang ldquoFirefly algorithms for multimodal optimizationrdquoin Proceedings of the 5th International Conference on StochasticAlgorithms Foundations and Applications (SAGA rsquo09) pp 169ndash178 2009

[4] B Alatas ldquoACROA artificial chemical reaction optimizationalgorithm for global optimizationrdquo Expert Systems with Appli-cations vol 38 no 10 pp 13170ndash13180 2011

Computational Intelligence and Neuroscience 13

[5] K N Krishnanand and D Ghose ldquoGlowworm warm opti-mization for searching higher dimensional spacesrdquo Studies inComputational Intelligence vol 248 pp 61ndash75 2009

[6] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[7] K Price R Storn and J Lampinen Differential Evolution-APractical Approach to Global Optimization Springer BerlinGermany 2005

[8] httpwwwicsiberkeleyedusim storn [9] R Storn and K Price ldquoDifferential evolutionmdasha simple and

efficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[10] X-S Yang ldquoA new metaheuristic Bat-inspired AlgorithmrdquoStudies in Computational Intelligence vol 284 pp 65ndash74 2010

[11] A Mucherino and O Seref ldquoMonkey search a novel meta-heuristic search for global optimizationrdquo in Proceedings of theConference on Data Mining Systems Analysis and Optimizationin Biomedicine vol 953 pp 162ndash173 March 2007

[12] KM Passino ldquoBiomimicry of bacterial foraging for distributedoptimization and controlrdquo IEEE Control Systems Magazine vol22 no 3 pp 52ndash67 2002

[13] G R Reynolds ldquoCultural algorithms theory and applicationrdquoin New Iders in Optimization D Corne M Dorigo and FGlover Eds pp 367ndash378 McGraw-Hill New York NY USA1999

[14] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[15] R Oftadeh M J Mahjoob and M Shariatpanahi ldquoA novelmeta-heuristic optimization algorithm inspired by group hunt-ing of animals hunting searchrdquo Computers and Mathematicswith Applications vol 60 no 7 pp 2087ndash2098 2010

[16] P W Tsai J S Pan B Y Liao M J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2011

[17] X-S Yang ldquoBat algorithm for multi-objective optimizationrdquoInternational Journal of Bio-Inspired Computation Bio-InspiredComputation (IJBIC) vol 3 no 5 pp 267ndash274 2011

[18] T C Bora L S Coelho and L Lebensztajn ldquoBat-Inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012

[19] B B Mandelbrot The Fractal Geometry of Nature WH Free-man New York NY USA 1983

[20] V A Chechkin R Metzler J Klafter and V Y GoncharldquoIntroduction to the theory of Levy flightsrdquo in AnomalousTransport Foundations and Applications pp 129ndash162 Wiley-VCH Cambridge UK 2008

[21] A M Reynolds and M A Frye ldquoFree-flight odor tracking inDrosophila is consistent with an optimal intermittent scale-freesearchrdquo PLoS ONE vol 2 no 4 p e354 2007

[22] C Brown L S Liebovitch andRGlendon ldquoLevy fights inDobeJursquohoansi foraging patternsrdquo Human Ecology vol 35 no 1 pp129ndash138 2007

[23] P Barthelemy J Bertolotti andD SWiersma ldquoA Levy flight forlightrdquo Nature vol 453 no 7194 pp 495ndash498 2008

[24] N Mercadier W Guerin M Chevrollier and R Kaiser ldquoLevyflights of photons in hot atomic vapoursrdquoNature Physics vol 5no 8 pp 602ndash605 2009

[25] C Grosan and A Abraham ldquoA new approach for solvingnonlinear equations systemsrdquo IEEE Transactions on SystemsMan and Cybernetics A vol 38 no 3 pp 698ndash714 2008

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

10 Computational Intelligence and Neuroscience

minus500 minus400 minus300 minus200 minus100 0 100 200 300 400 500minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500

Figure 4 The locations of 40 bats after fifteen iterations

0 20 40 60 80 100 120 140 160 180 2000

100200300400500600700800900

1000

Iteration

Fitn

ess

BADLBA

Figure 5 Convergence curves for the function 1198912

BADLBA

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12

Iteration

Fitn

ess

Figure 6 Convergence curves for the function 1198914

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07

Iteration

Fitn

ess

BADLBA

Figure 7 Convergence curves for the function 1198917

BADLBA

0 20 40 60 80 100 120 140 160 180 200minus420

minus410

minus400

minus390

minus380

minus370

minus360

minus350

minus340

Iteration

Fitn

ess

Figure 8 Convergence curves for the function 11989111

BADLBA

0 20 40 60 80 100 120 140 160 180 200minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Iteration

Fitn

ess

Figure 9 Convergence curves for the function 11989113

Computational Intelligence and Neuroscience 11

0 5 10 15 20 25 30 35 40 45 500

500

1000

1500

2000

2500

3000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 10 Distribution of optimal fitness for 1198916(D = 128)

0 5 10 15 20 25 30 35 40 45 500

2000

4000

6000

8000

10000

12000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 11 Distribution of optimal fitness for 1198919(D = 256)

BADLBA

0 5 10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000

6000

Number of running

Opt

imal

fitn

ess

Figure 12 Distribution of optimal fitness for 1198918(D = 320)

BADLBA

0 5 10 15 20 25 30 35 40 45 50minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Number of running

Opt

imal

fitn

ess

Figure 13 Distribution of optimal fitness for 11989114(D = 512)

BADLBA

Number of running

Opt

imal

fitn

ess

0 5 10 15 20 25 30 35 40 45 500

05

1

15

2

25times104

Figure 14 Distribution of optimal fitness for 1198911(D = 1024)

This potentially powerful optimization strategy can easilybe extended to study multiobjective optimization applica-tions with various constraints even to NP-hard problemsFurther studies can focus on the sensitivity and parameterstudies and their possible relationships with the convergencerate of the algorithm

Acknowledgments

This work is supported by the National Science Foundationof China under Grant no 61165015 Key Project of GuangxiScience Foundation under Grant no 2012GXNSFDA053028Key Project of Guangxi High School Science Foundationunder Grant no 20121ZD008 and the Fund by OpenResearch Fund Program of Key Lab of Intelligent Perception

12 Computational Intelligence and Neuroscience

0 5 10 15 20 25 30 35 40 45 50minus15

minus145

minus14

minus135

minus13

minus125

minus12

Number of running

Opt

imal

fitn

ess

BA

times10minus3

Figure 15 Details of BA line in Figure 12

0 5 10 15 20 25 30 35 40 45 50minus2minus18minus16minus14minus12minus1minus08minus06minus04minus02

0

Number of running

Opt

imal

fitn

ess

DLBA

Figure 16 Details of DLBA line in Figure 12

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7

Iteration

Fitn

ess

DLBA

Figure 17 Convergence curves for DLAB

0 5 10 15 20 25 30 35 40 45 500

002

004

006

008

01

012

014

016

018

Number of running

Opt

imal

fitn

ess

DLBA

Figure 18 Distribution of optimal fitness for DLAB

0 5 10 15 20 25 30 35 40 45 500

010203040506070809

1

Number of running

Opt

imal

fitn

ess

times10minus3

Figure 19 Distribution of optimal fitness less than 0001 inFigure 18

and Image Understanding of Ministry of Education of Chinaunder Grant no IPIU01201100

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of IEEE International Conference on NeuralNetworks (ICNN rsquo95) vol 4 pp 1942ndash1948 December 1995

[2] X-S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress Bristol UK 2nd edition 2010

[3] X-S Yang ldquoFirefly algorithms for multimodal optimizationrdquoin Proceedings of the 5th International Conference on StochasticAlgorithms Foundations and Applications (SAGA rsquo09) pp 169ndash178 2009

[4] B Alatas ldquoACROA artificial chemical reaction optimizationalgorithm for global optimizationrdquo Expert Systems with Appli-cations vol 38 no 10 pp 13170ndash13180 2011

Computational Intelligence and Neuroscience 13

[5] K N Krishnanand and D Ghose ldquoGlowworm warm opti-mization for searching higher dimensional spacesrdquo Studies inComputational Intelligence vol 248 pp 61ndash75 2009

[6] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[7] K Price R Storn and J Lampinen Differential Evolution-APractical Approach to Global Optimization Springer BerlinGermany 2005

[8] httpwwwicsiberkeleyedusim storn [9] R Storn and K Price ldquoDifferential evolutionmdasha simple and

efficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[10] X-S Yang ldquoA new metaheuristic Bat-inspired AlgorithmrdquoStudies in Computational Intelligence vol 284 pp 65ndash74 2010

[11] A Mucherino and O Seref ldquoMonkey search a novel meta-heuristic search for global optimizationrdquo in Proceedings of theConference on Data Mining Systems Analysis and Optimizationin Biomedicine vol 953 pp 162ndash173 March 2007

[12] KM Passino ldquoBiomimicry of bacterial foraging for distributedoptimization and controlrdquo IEEE Control Systems Magazine vol22 no 3 pp 52ndash67 2002

[13] G R Reynolds ldquoCultural algorithms theory and applicationrdquoin New Iders in Optimization D Corne M Dorigo and FGlover Eds pp 367ndash378 McGraw-Hill New York NY USA1999

[14] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[15] R Oftadeh M J Mahjoob and M Shariatpanahi ldquoA novelmeta-heuristic optimization algorithm inspired by group hunt-ing of animals hunting searchrdquo Computers and Mathematicswith Applications vol 60 no 7 pp 2087ndash2098 2010

[16] P W Tsai J S Pan B Y Liao M J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2011

[17] X-S Yang ldquoBat algorithm for multi-objective optimizationrdquoInternational Journal of Bio-Inspired Computation Bio-InspiredComputation (IJBIC) vol 3 no 5 pp 267ndash274 2011

[18] T C Bora L S Coelho and L Lebensztajn ldquoBat-Inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012

[19] B B Mandelbrot The Fractal Geometry of Nature WH Free-man New York NY USA 1983

[20] V A Chechkin R Metzler J Klafter and V Y GoncharldquoIntroduction to the theory of Levy flightsrdquo in AnomalousTransport Foundations and Applications pp 129ndash162 Wiley-VCH Cambridge UK 2008

[21] A M Reynolds and M A Frye ldquoFree-flight odor tracking inDrosophila is consistent with an optimal intermittent scale-freesearchrdquo PLoS ONE vol 2 no 4 p e354 2007

[22] C Brown L S Liebovitch andRGlendon ldquoLevy fights inDobeJursquohoansi foraging patternsrdquo Human Ecology vol 35 no 1 pp129ndash138 2007

[23] P Barthelemy J Bertolotti andD SWiersma ldquoA Levy flight forlightrdquo Nature vol 453 no 7194 pp 495ndash498 2008

[24] N Mercadier W Guerin M Chevrollier and R Kaiser ldquoLevyflights of photons in hot atomic vapoursrdquoNature Physics vol 5no 8 pp 602ndash605 2009

[25] C Grosan and A Abraham ldquoA new approach for solvingnonlinear equations systemsrdquo IEEE Transactions on SystemsMan and Cybernetics A vol 38 no 3 pp 698ndash714 2008

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience 11

0 5 10 15 20 25 30 35 40 45 500

500

1000

1500

2000

2500

3000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 10 Distribution of optimal fitness for 1198916(D = 128)

0 5 10 15 20 25 30 35 40 45 500

2000

4000

6000

8000

10000

12000

Number of running

Opt

imal

fitn

ess

BADLBA

Figure 11 Distribution of optimal fitness for 1198919(D = 256)

BADLBA

0 5 10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000

6000

Number of running

Opt

imal

fitn

ess

Figure 12 Distribution of optimal fitness for 1198918(D = 320)

BADLBA

0 5 10 15 20 25 30 35 40 45 50minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Number of running

Opt

imal

fitn

ess

Figure 13 Distribution of optimal fitness for 11989114(D = 512)

BADLBA

Number of running

Opt

imal

fitn

ess

0 5 10 15 20 25 30 35 40 45 500

05

1

15

2

25times104

Figure 14 Distribution of optimal fitness for 1198911(D = 1024)

This potentially powerful optimization strategy can easilybe extended to study multiobjective optimization applica-tions with various constraints even to NP-hard problemsFurther studies can focus on the sensitivity and parameterstudies and their possible relationships with the convergencerate of the algorithm

Acknowledgments

This work is supported by the National Science Foundationof China under Grant no 61165015 Key Project of GuangxiScience Foundation under Grant no 2012GXNSFDA053028Key Project of Guangxi High School Science Foundationunder Grant no 20121ZD008 and the Fund by OpenResearch Fund Program of Key Lab of Intelligent Perception

12 Computational Intelligence and Neuroscience

0 5 10 15 20 25 30 35 40 45 50minus15

minus145

minus14

minus135

minus13

minus125

minus12

Number of running

Opt

imal

fitn

ess

BA

times10minus3

Figure 15 Details of BA line in Figure 12

0 5 10 15 20 25 30 35 40 45 50minus2minus18minus16minus14minus12minus1minus08minus06minus04minus02

0

Number of running

Opt

imal

fitn

ess

DLBA

Figure 16 Details of DLBA line in Figure 12

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7

Iteration

Fitn

ess

DLBA

Figure 17 Convergence curves for DLAB

0 5 10 15 20 25 30 35 40 45 500

002

004

006

008

01

012

014

016

018

Number of running

Opt

imal

fitn

ess

DLBA

Figure 18 Distribution of optimal fitness for DLAB

0 5 10 15 20 25 30 35 40 45 500

010203040506070809

1

Number of running

Opt

imal

fitn

ess

times10minus3

Figure 19 Distribution of optimal fitness less than 0001 inFigure 18

and Image Understanding of Ministry of Education of Chinaunder Grant no IPIU01201100

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of IEEE International Conference on NeuralNetworks (ICNN rsquo95) vol 4 pp 1942ndash1948 December 1995

[2] X-S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress Bristol UK 2nd edition 2010

[3] X-S Yang ldquoFirefly algorithms for multimodal optimizationrdquoin Proceedings of the 5th International Conference on StochasticAlgorithms Foundations and Applications (SAGA rsquo09) pp 169ndash178 2009

[4] B Alatas ldquoACROA artificial chemical reaction optimizationalgorithm for global optimizationrdquo Expert Systems with Appli-cations vol 38 no 10 pp 13170ndash13180 2011

Computational Intelligence and Neuroscience 13

[5] K N Krishnanand and D Ghose ldquoGlowworm warm opti-mization for searching higher dimensional spacesrdquo Studies inComputational Intelligence vol 248 pp 61ndash75 2009

[6] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[7] K Price R Storn and J Lampinen Differential Evolution-APractical Approach to Global Optimization Springer BerlinGermany 2005

[8] httpwwwicsiberkeleyedusim storn [9] R Storn and K Price ldquoDifferential evolutionmdasha simple and

efficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[10] X-S Yang ldquoA new metaheuristic Bat-inspired AlgorithmrdquoStudies in Computational Intelligence vol 284 pp 65ndash74 2010

[11] A Mucherino and O Seref ldquoMonkey search a novel meta-heuristic search for global optimizationrdquo in Proceedings of theConference on Data Mining Systems Analysis and Optimizationin Biomedicine vol 953 pp 162ndash173 March 2007

[12] KM Passino ldquoBiomimicry of bacterial foraging for distributedoptimization and controlrdquo IEEE Control Systems Magazine vol22 no 3 pp 52ndash67 2002

[13] G R Reynolds ldquoCultural algorithms theory and applicationrdquoin New Iders in Optimization D Corne M Dorigo and FGlover Eds pp 367ndash378 McGraw-Hill New York NY USA1999

[14] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[15] R Oftadeh M J Mahjoob and M Shariatpanahi ldquoA novelmeta-heuristic optimization algorithm inspired by group hunt-ing of animals hunting searchrdquo Computers and Mathematicswith Applications vol 60 no 7 pp 2087ndash2098 2010

[16] P W Tsai J S Pan B Y Liao M J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2011

[17] X-S Yang ldquoBat algorithm for multi-objective optimizationrdquoInternational Journal of Bio-Inspired Computation Bio-InspiredComputation (IJBIC) vol 3 no 5 pp 267ndash274 2011

[18] T C Bora L S Coelho and L Lebensztajn ldquoBat-Inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012

[19] B B Mandelbrot The Fractal Geometry of Nature WH Free-man New York NY USA 1983

[20] V A Chechkin R Metzler J Klafter and V Y GoncharldquoIntroduction to the theory of Levy flightsrdquo in AnomalousTransport Foundations and Applications pp 129ndash162 Wiley-VCH Cambridge UK 2008

[21] A M Reynolds and M A Frye ldquoFree-flight odor tracking inDrosophila is consistent with an optimal intermittent scale-freesearchrdquo PLoS ONE vol 2 no 4 p e354 2007

[22] C Brown L S Liebovitch andRGlendon ldquoLevy fights inDobeJursquohoansi foraging patternsrdquo Human Ecology vol 35 no 1 pp129ndash138 2007

[23] P Barthelemy J Bertolotti andD SWiersma ldquoA Levy flight forlightrdquo Nature vol 453 no 7194 pp 495ndash498 2008

[24] N Mercadier W Guerin M Chevrollier and R Kaiser ldquoLevyflights of photons in hot atomic vapoursrdquoNature Physics vol 5no 8 pp 602ndash605 2009

[25] C Grosan and A Abraham ldquoA new approach for solvingnonlinear equations systemsrdquo IEEE Transactions on SystemsMan and Cybernetics A vol 38 no 3 pp 698ndash714 2008

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

12 Computational Intelligence and Neuroscience

0 5 10 15 20 25 30 35 40 45 50minus15

minus145

minus14

minus135

minus13

minus125

minus12

Number of running

Opt

imal

fitn

ess

BA

times10minus3

Figure 15 Details of BA line in Figure 12

0 5 10 15 20 25 30 35 40 45 50minus2minus18minus16minus14minus12minus1minus08minus06minus04minus02

0

Number of running

Opt

imal

fitn

ess

DLBA

Figure 16 Details of DLBA line in Figure 12

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7

Iteration

Fitn

ess

DLBA

Figure 17 Convergence curves for DLAB

0 5 10 15 20 25 30 35 40 45 500

002

004

006

008

01

012

014

016

018

Number of running

Opt

imal

fitn

ess

DLBA

Figure 18 Distribution of optimal fitness for DLAB

0 5 10 15 20 25 30 35 40 45 500

010203040506070809

1

Number of running

Opt

imal

fitn

ess

times10minus3

Figure 19 Distribution of optimal fitness less than 0001 inFigure 18

and Image Understanding of Ministry of Education of Chinaunder Grant no IPIU01201100

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of IEEE International Conference on NeuralNetworks (ICNN rsquo95) vol 4 pp 1942ndash1948 December 1995

[2] X-S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress Bristol UK 2nd edition 2010

[3] X-S Yang ldquoFirefly algorithms for multimodal optimizationrdquoin Proceedings of the 5th International Conference on StochasticAlgorithms Foundations and Applications (SAGA rsquo09) pp 169ndash178 2009

[4] B Alatas ldquoACROA artificial chemical reaction optimizationalgorithm for global optimizationrdquo Expert Systems with Appli-cations vol 38 no 10 pp 13170ndash13180 2011

Computational Intelligence and Neuroscience 13

[5] K N Krishnanand and D Ghose ldquoGlowworm warm opti-mization for searching higher dimensional spacesrdquo Studies inComputational Intelligence vol 248 pp 61ndash75 2009

[6] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[7] K Price R Storn and J Lampinen Differential Evolution-APractical Approach to Global Optimization Springer BerlinGermany 2005

[8] httpwwwicsiberkeleyedusim storn [9] R Storn and K Price ldquoDifferential evolutionmdasha simple and

efficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[10] X-S Yang ldquoA new metaheuristic Bat-inspired AlgorithmrdquoStudies in Computational Intelligence vol 284 pp 65ndash74 2010

[11] A Mucherino and O Seref ldquoMonkey search a novel meta-heuristic search for global optimizationrdquo in Proceedings of theConference on Data Mining Systems Analysis and Optimizationin Biomedicine vol 953 pp 162ndash173 March 2007

[12] KM Passino ldquoBiomimicry of bacterial foraging for distributedoptimization and controlrdquo IEEE Control Systems Magazine vol22 no 3 pp 52ndash67 2002

[13] G R Reynolds ldquoCultural algorithms theory and applicationrdquoin New Iders in Optimization D Corne M Dorigo and FGlover Eds pp 367ndash378 McGraw-Hill New York NY USA1999

[14] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[15] R Oftadeh M J Mahjoob and M Shariatpanahi ldquoA novelmeta-heuristic optimization algorithm inspired by group hunt-ing of animals hunting searchrdquo Computers and Mathematicswith Applications vol 60 no 7 pp 2087ndash2098 2010

[16] P W Tsai J S Pan B Y Liao M J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2011

[17] X-S Yang ldquoBat algorithm for multi-objective optimizationrdquoInternational Journal of Bio-Inspired Computation Bio-InspiredComputation (IJBIC) vol 3 no 5 pp 267ndash274 2011

[18] T C Bora L S Coelho and L Lebensztajn ldquoBat-Inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012

[19] B B Mandelbrot The Fractal Geometry of Nature WH Free-man New York NY USA 1983

[20] V A Chechkin R Metzler J Klafter and V Y GoncharldquoIntroduction to the theory of Levy flightsrdquo in AnomalousTransport Foundations and Applications pp 129ndash162 Wiley-VCH Cambridge UK 2008

[21] A M Reynolds and M A Frye ldquoFree-flight odor tracking inDrosophila is consistent with an optimal intermittent scale-freesearchrdquo PLoS ONE vol 2 no 4 p e354 2007

[22] C Brown L S Liebovitch andRGlendon ldquoLevy fights inDobeJursquohoansi foraging patternsrdquo Human Ecology vol 35 no 1 pp129ndash138 2007

[23] P Barthelemy J Bertolotti andD SWiersma ldquoA Levy flight forlightrdquo Nature vol 453 no 7194 pp 495ndash498 2008

[24] N Mercadier W Guerin M Chevrollier and R Kaiser ldquoLevyflights of photons in hot atomic vapoursrdquoNature Physics vol 5no 8 pp 602ndash605 2009

[25] C Grosan and A Abraham ldquoA new approach for solvingnonlinear equations systemsrdquo IEEE Transactions on SystemsMan and Cybernetics A vol 38 no 3 pp 698ndash714 2008

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience 13

[5] K N Krishnanand and D Ghose ldquoGlowworm warm opti-mization for searching higher dimensional spacesrdquo Studies inComputational Intelligence vol 248 pp 61ndash75 2009

[6] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[7] K Price R Storn and J Lampinen Differential Evolution-APractical Approach to Global Optimization Springer BerlinGermany 2005

[8] httpwwwicsiberkeleyedusim storn [9] R Storn and K Price ldquoDifferential evolutionmdasha simple and

efficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[10] X-S Yang ldquoA new metaheuristic Bat-inspired AlgorithmrdquoStudies in Computational Intelligence vol 284 pp 65ndash74 2010

[11] A Mucherino and O Seref ldquoMonkey search a novel meta-heuristic search for global optimizationrdquo in Proceedings of theConference on Data Mining Systems Analysis and Optimizationin Biomedicine vol 953 pp 162ndash173 March 2007

[12] KM Passino ldquoBiomimicry of bacterial foraging for distributedoptimization and controlrdquo IEEE Control Systems Magazine vol22 no 3 pp 52ndash67 2002

[13] G R Reynolds ldquoCultural algorithms theory and applicationrdquoin New Iders in Optimization D Corne M Dorigo and FGlover Eds pp 367ndash378 McGraw-Hill New York NY USA1999

[14] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[15] R Oftadeh M J Mahjoob and M Shariatpanahi ldquoA novelmeta-heuristic optimization algorithm inspired by group hunt-ing of animals hunting searchrdquo Computers and Mathematicswith Applications vol 60 no 7 pp 2087ndash2098 2010

[16] P W Tsai J S Pan B Y Liao M J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2011

[17] X-S Yang ldquoBat algorithm for multi-objective optimizationrdquoInternational Journal of Bio-Inspired Computation Bio-InspiredComputation (IJBIC) vol 3 no 5 pp 267ndash274 2011

[18] T C Bora L S Coelho and L Lebensztajn ldquoBat-Inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012

[19] B B Mandelbrot The Fractal Geometry of Nature WH Free-man New York NY USA 1983

[20] V A Chechkin R Metzler J Klafter and V Y GoncharldquoIntroduction to the theory of Levy flightsrdquo in AnomalousTransport Foundations and Applications pp 129ndash162 Wiley-VCH Cambridge UK 2008

[21] A M Reynolds and M A Frye ldquoFree-flight odor tracking inDrosophila is consistent with an optimal intermittent scale-freesearchrdquo PLoS ONE vol 2 no 4 p e354 2007

[22] C Brown L S Liebovitch andRGlendon ldquoLevy fights inDobeJursquohoansi foraging patternsrdquo Human Ecology vol 35 no 1 pp129ndash138 2007

[23] P Barthelemy J Bertolotti andD SWiersma ldquoA Levy flight forlightrdquo Nature vol 453 no 7194 pp 495ndash498 2008

[24] N Mercadier W Guerin M Chevrollier and R Kaiser ldquoLevyflights of photons in hot atomic vapoursrdquoNature Physics vol 5no 8 pp 602ndash605 2009

[25] C Grosan and A Abraham ldquoA new approach for solvingnonlinear equations systemsrdquo IEEE Transactions on SystemsMan and Cybernetics A vol 38 no 3 pp 698ndash714 2008

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

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Distributed Sensor Networks

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Advances in

FuzzySystems

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International Journal of

ReconfigurableComputing

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Applied Computational Intelligence and Soft Computing

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Electrical and Computer Engineering

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Computer Networks and Communications

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Biomedical Imaging

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Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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