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
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Distributed Sensor Networks
International Journal of
Advances in
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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
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Distributed Sensor Networks
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Volume 2014
International Journal of
ReconfigurableComputing
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Applied Computational Intelligence and Soft Computing
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Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
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Electrical and Computer Engineering
Journal of
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Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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ArtificialNeural Systems
Advances in
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RoboticsJournal of
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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
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
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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
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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Distributed Sensor Networks
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International Journal of
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Applied Computational Intelligence and Soft Computing
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Artificial Intelligence
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Electrical and Computer Engineering
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
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Distributed Sensor Networks
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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
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Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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httpwwwhindawicom Volume 2014
Advances in
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International Journal of
Biomedical Imaging
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ArtificialNeural Systems
Advances in
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RoboticsJournal of
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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
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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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
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
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Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Applied Computational Intelligence and Soft Computing
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Artificial Intelligence
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Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Advances in
Multimedia
International Journal of
Biomedical Imaging
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ArtificialNeural Systems
Advances in
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RoboticsJournal of
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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
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Human-ComputerInteraction
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014