research article bare-bones teaching-learning-based...

Research ArticleBare-Bones Teaching-Learning-Based Optimization

Feng Zou12 Lei Wang1 Xinhong Hei1 Debao Chen2 Qiaoyong Jiang1 and Hongye Li1

1 School of Computer Science and Engineering Xirsquoan University of Technology Xirsquoan 710048 China2 School of Physics and Electronic Information Huaibei Normal University Huaibei 235000 China

Correspondence should be addressed to Lei Wang wangleeei163com

Received 20 February 2014 Accepted 7 April 2014 Published 10 June 2014

Academic Editors S Balochian and Y Zhang

Copyright copy 2014 Feng Zou et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Teaching-learning-based optimization (TLBO) algorithm which simulates the teaching-learning process of the class room is oneof the recently proposed swarm intelligent (SI) algorithms In this paper a new TLBO variant called bare-bones teaching-learning-based optimization (BBTLBO) is presented to solve the global optimization problems In this method each learner of teacher phaseemploys an interactive learning strategy which is the hybridization of the learning strategy of teacher phase in the standard TLBOand Gaussian sampling learning based on neighborhood search and each learner of learner phase employs the learning strategyof learner phase in the standard TLBO or the new neighborhood search strategy To verify the performance of our approaches20 benchmark functions and two real-world problems are utilized Conducted experiments can been observed that the BBTLBOperforms significantly better than or at least comparable to TLBO and some existing bare-bones algorithms The results indicatethat the proposed algorithm is competitive to some other optimization algorithms

1 Introduction

Many real-life optimization problems are becomingmore andmore complex and difficult with the development of scientifictechnology So how to resolve these complex problems in anexactmanner within a reasonable time cost is very importantThe traditional optimization algorithms are difficult to solvethese complex nonlinear problems In recent years nature-inspired optimization algorithms which simulate naturalphenomena and have different design philosophies andcharacteristics such as evolutionary algorithms [1ndash3] andswarm intelligence algorithms [4ndash7] are a research fieldwhich simulates different natural phenomena to solve a widerange of problems In these algorithms the convergence rateof the algorithm is given prime importance for solving real-world optimization problems The ability of the algorithmsto obtain the global optima value is one aspect and the fasterconvergence is the other aspect

As a stochastic search scheme TLBO [8 9] is a newlypopulation-based algorithm based on swarm intelligence andhas characters of simple computation and rapid convergenceit has been extended to the function optimization engineer-ing optimization multiobjective optimization clustering

and so forth [9ndash17] TLBO is a parameter-free evolutionarytechnique and is also gaining popularity due to its abilityto achieve better results in comparatively faster convergencetime to genetic algorithms (GA) [1] particle swarm optimizer(PSO) [5] and artificial bee colony algorithm (ABC) [6]However in evolutionary computation research there havebeen always attempts to improve any given findings furtherand further This work is an attempt to improve the con-vergence characteristics of TLBO further without sacrificingthe accuracies obtained in TLBO and in some occasionstrying to even better the accuracies The aims of this paperare of threefold First authors propose an improved versionof TLBO namely BBTLBO Next the proposed techniqueis validated on unimodal and multimodal functions basedon different performance indicators The result of BBTLBOis compared with other algorithms Results of both thealgorithms are also compared using statistical paired 119905-testThirdly it is applied to solve the real-world optimizationproblem

The remainder of this paper is organized as followsThe TLBO algorithm is introduced in Section 2 Section 3presents a brief overview of some recently proposed

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 136920 17 pageshttpdxdoiorg1011552014136920

2 The Scientific World Journal

(1) Begin(2) Initialize119873 (number of learners) and119863 (number of dimensions)(3) Initialize learners119883 and evaluate all learners119883(4) Donate the best learner as Teacher and the mean of all learners119883 asMean(5) while (stopping condition not met)(6) for each learner119883

119894of the class Teaching phase

(7) TF = round(1 + rand(0 1))(8) for 119895 = 1 119863(9) 119899119890119908119883

119894119895= 119883119894119895+ rand(0 1) lowast (119879119890119886119888ℎ119890119903(119895) minus TF lowast119872119890119886119899(119895))

(10) endfor(11) Accept 119899119890119908119883

119894if 119891(119899119890119908119883

119894) is better than 119891(119883

119894)

(12) endfor(13) for each learner119883

119894of the class Learning phase

(14) Randomly select one learner119883119896 such that 119894 = 119896

(15) if 119891(119883119894) better 119891(119883

119896)

(16) for 119895 = 1 119863(17) 119899119890119908119883

119894119895= 119883119894119895+ rand(0 1) lowast (119883

119894119895minus 119883119896119895)

(18) endfor(19) else(20) for 119895 = 1 119863(21) 119899119890119908119883

119894119895= 119883119894119895+ rand(0 1) lowast (119883

119896119895minus 119883119894119895)

(22) endfor(23) endif(24) Accept 119899119890119908119883

119894if 119891(119899119890119908119883

119894) is better than 119891(119883

119894)

(25) endfor(26) Update the Teacher and theMean(27) endwhile(28) end

Algorithm 1 TLBO( )

bare-bones algorithms Section 4 describes the improvedteaching-learning-based optimization algorithm usingneighborhood search (BBTLBO) Section 5 presents the testson several benchmark functions and the experiments areconducted along with statistical tests The applications fortraining artificial neural network are shown in Section 6Conclusions are given in Section 7

2 Teaching-Learning-Based Optimization

Rao et al [8 9] first proposed a novel teaching-learning-based optimization (TLBO) inspired from the philosophy ofteaching and learning The TLBO algorithm is based on theeffect of the influence of a teacher on the output of learners ina class which is considered in terms of results or grades Theprocess ofworking of TLBO is divided into twopartsThefirstpart consists of ldquoteacher phaserdquo and the second part consistsof ldquolearner phaserdquo The ldquoteacher phaserdquo means learning fromthe teacher and the ldquolearner phaserdquo means learning throughthe interaction between learners

A good teacher is one who brings his or her learners up tohis or her level in terms of knowledge But in practice this isnot possible and a teacher can only move the mean of a classup to some extent depending on the capability of the classThis follows a random process depending on many factorsLet119872 be the mean and let 119879 be the teacher at any iteration 119879will try tomovemean119872 toward its own level so now the new

mean will be 119879 designated as119872new The solution is updatedaccording to the difference between the existing and the newmean according to the following expression

119899119890119908119883 = 119883 + 119903 times (119872new minus TF times119872) (1)

where TF is a teaching factor that decides the value of meanto be changed and 119903 is a random vector in which each elementis a random number in the range [0 1] The value of TF canbe either 1 or 2 which is again a heuristic step and decidedrandomly with equal probability as

TF = round [1 + rand (0 1)] (2)

Learners increase their knowledge by two differentmeans one through input from the teacher and the otherthrough interaction between themselves A learner interactsrandomly with other learners with the help of group discus-sions presentations formal communications and so forthA learner learns something new if the other learner hasmore knowledge than him or her Learner modification isexpressed as

119899119890119908119883119894=

119883119894+ 119903 lowast (119883

119894minus 119883119895) if119891 (119883

119894) lt 119891 (119883

119895)

119883119894+ 119903 lowast (119883

119895minus 119883119894) otherwise

(3)

As explained above the pseudocode for the implementa-tion of TLBO is summarized in Algorithm 1

The Scientific World Journal 3

3 Bare-Bones Algorithm

In this section we only presented a brief overview of somerecently proposed bare-bones algorithms

31 BBPSO and BBExp PSO is a swarm intelligence-basedalgorithm which is inspired by the behavior of birds flocking[5] In PSO each particle is attracted by its personal bestposition (119901best) and the global best position (119892best) foundso far Theoretical studies [18 19] proved that each particleconverges to the weighted average of 119901best and 119892best

lim119905rarrinfin

119883119894 (119905) =

1198881sdot 119892best + 1198882 sdot 119901best

1198881+ 1198882

(4)

where 1198881and 1198882are two leaning factors in PSO

Based on the convergence characteristic of PSO Kennedy[20] proposed a new PSO variant called bare-bones PSO(BBPSO) Bare-bones PSO retains the standard PSO socialcommunication but replaces dynamical particle update withsampling from a probability distribution based on 119892best and119901best119894 as follows

119909119894119895 (119905 + 1) = 119873(

119892best + 119901best119894119895 (119905)

2

100381610038161003816100381610038161003816119892best minus 119901best119894119895 (119905)

100381610038161003816100381610038161003816) (5)

where 119909119894119895(119905 + 1) is the 119895th dimension of the 119894th particle in

the population and119873 represents a Gaussian distributionwithmean (119892best + 119901best119894119895(119905))2 and standard deviation |119892best minus119901best119894119895(119905)|

Kennedy [20] proposed also an alternative version of theBBPSO denoted by BBExp where (5) is replaced by119909119894119895 (119905 + 1)

=

119873(

119892best + 119901best119894119895 (119905)

2

100381610038161003816100381610038161003816119892best minus 119901best119894119895 (119905)

100381610038161003816100381610038161003816) rand (0 1) gt 05

119901best119894119895 (119905) otherwise

(6)where rand (01) is a random value within [0 1] for the 119895thdimension For the alternative mechanism there is a 50chance that the search process is focusing on the previous bestpositions

32 BBDE GBDE and MGBDE Inspired by the BBPSOand DE Omran et al [21] proposed a new and efficientDE variant called bare-bones differential evolution (BBDE)The BBDE is a new almost parameter-free optimizationalgorithm that is a hybrid of the bare-bones particle swarmoptimizer and differential evolution Differential evolution isused tomutate for each particle the attractor associated withthat particle defined as a weighted average of its personal andneighborhood best positions For the BBDE the individual isupdated as follows119909119894119895 (119905 + 1)

= 1199011198943 119895(119905) + 1199032 sdot (1199091198941 119895

(119905) minus 1199091198942 119895(119905)) rand (0 1) gt CR

119901best1198943119895 (119905) otherwise(7)

where 1198941 1198942 and 119894

3are three indices chosen from the set

1 2 NP with 1198941= 1198942= 119894 rand (0 1) is a random value

within [0 1] for the 119895th dimension and 119901119894119895(119905) is defined by

119901119894119895 (119905 + 1) = 1199031119895 sdot 119901best119894119895 (119905) + (1 minus 1199032119895) 119892best119894 (119905) (8)

where 119901best and 119892best are personal best position and the globalbest position 119903

1119895 is a random value within [0 1] for the 119895th

dimensionBased on the idea that the Gaussian sampling is a fine

tuning procedure which starts during exploration and iscontinued to exploitation Wang et al [22] proposed a newparameter-freeDE algorithm calledGBDE In theGBDE themutation strategy uses a Gaussian sampling method which isdefined by

V119894119895 (119905 + 1)

=

119873(

119883best119895 (119905) + 119909119894119895 (119905)

2 rand (0 1) le CR or 119895 = 119895rand

10038161003816100381610038161003816119883best119895 (119905) minus 119909119894119895 (119905)

10038161003816100381610038161003816)

119909119894119895 (119905) otherwise(9)

where 119873 represents a Gaussian distribution with mean(119883best119895(119905)+119909119894119895(119905))2 and standard deviation |119883best119895(119905)minus119909119894119895(119905)|and CR is the probability of crossover

To balance the global search ability and convergence rateWang et al [22] proposed amodifiedGBDE (calledMGBDE)The mutation strategy uses a hybridization of GBDE andDEbest1 as follows

V119894119895 (119905 + 1)

=

119883best119895 (119905)+119865 sdot (1199091198941119895 (119905)minus1199091198942119895 (119905)) rand (0 1) le 05

119873(

119883best119895 (119905)+119909119894119895 (119905)

210038161003816100381610038161003816119883best119895 (119905)minus119909119894119895 (119905)

10038161003816100381610038161003816) otherwise

(10)

4 Proposed Algorithm BBTLBO

The bare-bones PSO utilizes this information by samplingcandidate solutions normally distributed around the for-mally derived attractor point That is the new position isgenerated by a Gaussian distribution for sampling the searchspace based on the 119892best and the 119901best at the current iterationAs a result the new position will be centered around theweighted average of 119901best and 119892best Generally speaking atthe initial evolutionary stages the search process focuses onexploration due to the large deviation With an increasingnumber of generations the deviation becomes smaller andthe search process will focus on exploitation From the searchbehavior of BBPSO the Gaussian sampling is a fine tuningprocedure which starts during exploration and is continuedto exploitation This can be beneficial for the search of manyevolutionary optimization algorithms Additionally the bare-bones PSO has no parameters to be tuned

Based on a previous explanation a new bare-bones TLBO(BBTLBO) with neighborhood search is proposed in this


Begin

Initialize learners size (NP) dimension (D) and hybridization factor (u)

Calculate the NTeacher and NMean of each learner

Modify each learner Xi in the class= + r lowast (NTeacher minus TF lowast NMean)= N((NTeacher + Xi)2 (NTeacher minus Xi))

newXi = u lowast newX1 + (1 minus u) lowast newX2Teacher phase

newXi better Xi

Xi = newXi Xi = Xi

Yes

Yes

Yes

Yes

No

No

No

No

Denote the NTeacheri and randomly select a Xk for each Xi

Learner phase

rand(0 1) lt 05

The original TLBO learning Neighborhood search strategy

newXi better Xi

Xi = newXi Xi = Xi

Termination criteria satisfied

End

gen = gen + 1

XiV1V2

Figure 1 Flow chart showing the working of BBTLBO algorithm

paper In fact for TLBO if the new learner has a betterfunction value than that of the old learner it is replacedwith the old one in the memory Otherwise the old one isretained in the memory In other words a greedy selectionmechanism is employed as the selection operation betweenthe old and the candidate one Hence the new teacher and thenew learner are the global best (119892best) and learnerrsquos personalbest (119901best) found so far respectively The complete flowchartof the BBTLBO algorithm is shown in Figure 1

41 Neighborhood Search It is known that birds of a featherflock together and people of a mind fall into the samegroup Just like evolutionary algorithms themselves thenotion of neighborhood is inspired by nature Neighborhoodtechnique is an efficient method to maintain diversity of

the solutions It plays an important role in evolutionaryalgorithms and is often introduced by researchers in orderto allow maintenance of a population of diverse individualsand improve the exploration capability of population-basedheuristic algorithms [23ndash26] In fact learners with similarinterests form different learning groups Because of his orher favor characteristic the learner maybe learns from theexcellent individual in the learning group

For the implementation of grouping various types ofconnected distances may be used Here we have used aring topology [27] based on the indexes of learners for thesake of simplicity In a ring topology the first individualis the neighbor of the last individual and vice versa Basedon the ring topology a 119896-neighborhood radius is definedwhere 119896 is a predefined integer number For each individual


NeighborhoodiXi

Ximinus1

Xi+1

Figure 2 Ring neighborhood topology with three members

its 119896-neighborhood radius consists of 2119896 + 1 individuals(including oneself) which are 119883

119894minus119896 119883

119894 119883

119894+119896 That is

the neighborhood size is 2119896 + 1 for a 119896-neighborhood Forsimplicity 119896 is set to 1 (Figure 2) in our algorithmThismeansthat there are 3 individuals in each learning group Oncegroups are constructed we can utilize them for updating thelearners of the corresponding group

42 Teacher Phase To balance the global and local searchability a modified interactive learning strategy is proposed inteacher phase In this learning phase each learner employs aninteractive learning strategy (the hybridization of the learningstrategy of teacher phase in the standard TLBO and Gaussiansampling learning) based on neighborhood search

In BBTLBO the updating formula of the learning for alearner 119883

119894in teacher phase is proposed by the hybridization

of the learning strategy of teacher phase and the Gaussiansampling learning as follows

1198811119895 (119905 + 1) = 119883119894119895 (119905) + rand (0 1)

sdot (119873119879119890119886119888ℎ119890119903119894119895 (119905) minus TF sdot 119873119872119890119886119899

119894119895 (119905))

1198812119895 (119905 + 1) = 119873(

119873119879119890119886119888ℎ119890119903119894119895 (119905) + 119873119872119890119886119899119894119895 (119905)

2

100381610038161003816100381610038161003816100381610038161003816

119873119879119890119886119888ℎ119890119903119894119895 (119905) minus 119873119872119890119886119899119894119895 (119905)

100381610038161003816100381610038161003816100381610038161003816

)

119899119890119908119883119894119895 (119905 + 1) = 119906 sdot 1198811119895 (119905 + 1) + (1 minus 119906) sdot 1198812119895 (119905 + 1)

(11)

where 119906 called the hybridization factor is a random numberin the range [0 1] for the 119895th dimension 119873119879119890119886119888ℎ119890119903 and119873119872119890119886119899 are the existing neighborhood best solution and theneighborhood mean solution of each learner and TF is ateaching factor which can be either 1 or 2 randomly

In the BBTLBO there is a (119906 lowast 100) chance that the119895th dimension of the 119894th learner in the population follows thebehavior of the learning strategy of teacher phase while theremaining (100 minus 119906lowast 100) follow the search behavior of theGaussian sampling in teacher phase This will be helpful tobalance the advantages of fast convergence rate (the attraction

of the learning strategy of teacher phase) and exploration (theGaussian sampling) in BBTLBO

43 Learner Phase At the same time in the learner phase alearner interacts randomly with other learners for enhancinghis or her knowledge in the class This learning method canbe treated as the global search strategy (shown in (3))

In this paper we introduce a new learning strategy inwhich each learner learns from the neighborhood teacher andthe other learner selected randomly of his or her correspond-ing neighborhood in learner phaseThis learningmethod canbe treated as the neighborhood search strategy Let 119899119890119908119883

119894

represent the interactive learning result of the learner119883119894This

neighborhood search strategy can be expressed as follows

119899119890119908119883119894119895= 119883119894119895+ 1199031lowast (119873119879119890119886119888ℎ119890119903

119894119895minus 119883119894119895)

+ 1199032lowast (119883119894119895minus 119883119896119895)

(12)

where 1199031and 1199032are random vectors in which each element

is a random number in the range [0 1] 119873119879119890119886119888ℎ119890119903 is theteacher of the learner 119883

119894rsquos corresponding neighborhood

and the learner 119883119896is selected randomly from the learnerrsquos

corresponding neighborhoodIn BBTLBO each learner is probabilistically learning by

means of the global search strategy or the neighborhoodsearch strategy in learner phaseThat is about 50of learnersin the population execute the learning strategy of learnerphase in the standard TLBO (shown in (3)) while theremaining 50execute neighborhood search strategy (shownin (12)) This will be helpful to balance the global search andlocal search in learner phase

Moreover compared to the original TLBO BBTLBOonlymodifies the learning strategies Therefore both the originalTLBO and BBTLBO have the same time complexity 119874 (NP sdot119863 sdot Genmax) where NP is the number of the population 119863is the number of dimensions and Genmax is the maximumnumber of generations

As explained above the pseudocode for the implementa-tion of BBTLBO is summarized in Algorithm 2

5 Functions Optimization

In this section to illustrate the effectiveness of the proposedmethod 20 benchmark functions are used to test the effi-ciency of BBTLBO To compare the search performance ofBBTLBO with some other methods other different algo-rithms are also simulated in the paper

51 Benchmark Functions Thedetails of 20 benchmark func-tions are shown in Table 1 Among 20 benchmark functions1198651to 1198659are unimodal functions and 119865

10to 11986520

are multi-modal functions The searching range and theory optima forall functions are also shown in Table 1

52 Parameter Settings All the experiments are carried outon the same machine with a Celoron 226GHz CPU 2GBmemory andWindows XP operating system withMatlab 79


(1) Begin(2) Initialize119873 (number of learners)119863 (number of dimensions) and hybridization factor 119906(3) Initialize learners 119883 and evaluate all learners119883(4) while (stopping condition not met)(5) for each learner119883


(6) TF = round(1 + rand(0 1))(7) Donate the119873 119879119890119886119888ℎ119890119903 and the119873 119872119890119886119899 in its neighborhood for each learner(8) Updating each learner according (11)(9) Accept 119899119890119908119883

119894if 119891(119899119890119908119883

119894) is better than 119891(119883

119894)




(13) if rand(0 1) lt 05(14) Updating each learner according (3)(15) else(16) Donate the119873119879119890119886119888ℎ119890119903 in its neighborhood for each learner(17) Updating each learner according (12)(18) endif(19) Accept 119899119890119908119883

119894if 119891(119899119890119908119883119894) is better than 119891(119883

119894)

(20) endfor(21) endwhile(22) end

Algorithm 2 BBTLBO( )

For the purpose of reducing statistical errors each algorithmis independently simulated 50 times For all algorithms thepopulation size was set to 20 Population-based stochasticalgorithms use the same stopping criterion that is reachinga certain number of function evaluations (FEs)

53 Effect of Variation in Parameter 119906 The hybridizationfactor u is set to 00 01 03 05 07 09 10 Comparativetests have been performed using different 119906 In our exper-iment the maximal FEs are used as ended condition ofalgorithm namely 40000 for all test functions Table 2 showsthe mean optimum solutions and the standard deviation ofthe solutions obtained using different hybridization factor119906 in the 50 independent runs The best results amongthe algorithms are shown in bold Figure 3 presents therepresentative convergence graphs of different benchmarkfunctions in terms of the mean fitness values achieved byusing different hybridization factor 119906 on all test functionsDue to the tight space limitation some sample graphs areillustrated

The comparisons in Table 2 and Figure 3 show that whenthe hybridization factor 119906 is set to 09 BBTLBOoffers the bestperformance on 20 test functions Hence the hybridizationfactor 119906 is set to 09 in the following experiments

54 Comparison of BBTLBO with Some Similar Bare-BonesAlgorithms In this section we compare BBTLBO with fiveother recently proposed three bare-bones DE variants andtwo bare-bones PSO algorithms Our experiment includestwo series of comparisons in terms of the solution accuracyand the solution convergence (convergence speed and successrate) We compared the performance of BBTLBO with other

similar bare-bones algorithms including BBPSO [20] BBExp[20] BBDE [21] GBDE [22] and MGBDE [22]

541 Comparisons on the Solution Accuracy In our exper-iment the maximal FEs are used as ended condition ofalgorithm namely 40000 for all test functions The resultsare shown in Table 3 in terms of the mean optimum solutionand the standard deviation of the solutions obtained in the 50independent runs by each algorithm on 20 test functionsThebest results among the algorithms are shown in bold Figure 4presents the convergence graphs of different benchmarkfunctions in terms of the mean fitness values achieved by 7algorithms for 50 independent runs Due to the tight spacelimitation some sample graphs are illustrated

From Table 3 it can be observed that the mean optimumsolution and the standard deviation of all algorithms performwell for the functions 119865

15and 11986517 Although BBExp performs

better than BBTLBO on function 1198659and MGBDE performs

better than BBTLBO on function 11986520 our approach BBTLBO

achieves better results than other algorithms on the rest of testfunctions Table 3 and Figure 4 conclude that the BBTLBOhas a good performance of the solution accuracy for testfunctions in this paper

542 Comparison of the Convergence Speed and SR In orderto compare the convergence speed and successful rate (SR)of different algorithms we select a threshold value of theobjective function for each test function For other functionsthe threshold values are listed in Table 4 In our experimentthe stopping criterion is that each algorithm is terminatedwhen the best fitness value so far is below the predefinedthreshold value (119879 Value) or the number of FEs reaches to


Table 1 Details of numerical benchmarks used

Function Formula 119863 Range Optima

Sphere 1198651(119909) =

119863

sum

119894=1

1199092

11989430 [minus100 100] 0

Sum square 1198652(119909) =

119863

sum

119894=1

1198941199092

11989430 [minus100 100] 0

Quadric 1198653(119909) =

119863

sum

119894=1

1198941199094

119894+ random(0 1) 30 [minus128 128] 0

Step 1198654(119909) =

119863

sum

119894=1

(lfloor119909119894+ 05rfloor)

2 30 [minus100 100] 0

Schwefel 12 1198655(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119895)

2

30 [minus100 100] 0

Schwefel 221 1198656(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863 30 [minus100 100] 0

Schwefel 222 1198657(119909) =

119863

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816+

119863

prod

119894=1

1003816100381610038161003816119909119894

100381610038161003816100381630 [minus10 10] 0

Zakharov 1198658(119909) =

119863

sum

119894=1

1199092

119894+ (

119863

sum

119894=1

05119894119909119894)

2

+ (

119863

sum

119894=1

05119894119909119894)

4

30 [minus100 100] 0

Rosenbrock 1198659(119909) =

119863minus1

sum

119894=1

lfloor100(1199092

119894minus 119909119894+1)2

+ (119909119894minus 1)2

rfloor 30 [minus2048 2048] 0

Ackley 11986510(119909) = 20 minus 20 exp((minus1

5)radic(

1

119863)

119863

sum

119894=1

1199092

119894) minus exp(( 1

119863)

119863

sum

119894=1

cos (2120587119909119894)) + 119890 30 [minus32 32] 0

Rastrigin 11986511(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10) 30 [minus512 512] 0

Weierstrass11986512(119909) =

119863

sum

119894=1

(

119896maxsum

119896=0

[119886119896 cos (2120587119887119896 (119909

119894+ 05))]) minus 119863

119896maxsum

119896=0

[119886119896 cos (2120587119887119896 times 05)]

119886 = 05 119887 = 3 119896max = 20

30 [minus05 05] 0

Griewank 11986513(119909) =

119863

sum

119894=1

(1199092

119894

4000) minus

119899

prod

119894=1

cos(119909119894

radic119894

) + 1 30 [minus600 600] 0

Schwefel 11986514(119909) = 4189829119863 +

119863

sum

119894=1

(minus119909119894sinradicabs(119909

119894)) 30 [minus500 500] 0

Bohachevsky1 11986515(119909) = 119909

2

1+ 21199092

2minus 03 cos (3120587119909

1) minus 04 cos (4120587119909

2) + 07 2 [minus100 100] 0

Bohachevsky2 11986516(119909) = 119909

2

1+ 21199092

2minus 03cos (3120587119909

1) lowast cos (4120587119909

2) + 03 2 [minus100 100] 0

Bohachevsky3 11986517(119909) = 119909

2

1+ 21199092

2minus 03cos((3120587119909

1) + (4120587119909

2)) + 03 2 [minus100 100] 0

Shekel5 11986518(119909) = minus

5

sum

119894=1

[(119909 minus 119886119894)(119909 minus 119886

119894)119879+ 119888119894]

minus1

4 [0 10] minus101532

Shekel7 11986519(119909) = minus

7

sum

119894=1

[(119909 minus 119886119894)(119909 minus 119886

119894)119879+ 119888119894]

minus1

4 [0 10] minus104029

Shekel10 11986520(119909) = minus

10

sum

119894=1

[(119909 minus 119886119894)(119909 minus 119886

119894)119879+ 119888119894]

minus1

4 [0 10] minus105364

the maximal FEs 40000 The results are shown in Table 4in terms of the mean number of FEs (MFEs) required toconverge to the threshold and successful rate (SR) in the50 independent runs ldquoNaNrdquo represents that no runs of thecorresponding algorithm converged below the predefinedthreshold before meeting the maximum number of FEs Thebest results among the six algorithms are shown in boldface

FromTable 5 it can be observed that all algorithms hardlyconverge to the threshold for unimodal functions 119865

3 1198655 1198656

and 1198658and multimodal functions 119865

11 11986512 and 119865

14 BBTLBO

converges to the threshold except for functions 1198653 1198659 and

11986514 From the results of total average FEs BBTLBO converges

faster than other algorithms on all unimodal functions andthe majority of multimodal functions except for functions119865151198651611986519 and119865

20The acceleration rates between BBTLBO

and other algorithms are mostly 10 for functions 1198651 1198652 1198654

1198657 1198659 11986510 and 119865

13 From the results of total average SR

BBTLBO achieves the highest SR for those test functions ofwhich BBTLBO successfully converges to the threshold valueIt can be concluded that the BBTLBOhas a good performanceof convergence speed and successful rate (SR) of the solutionsfor test functions in this paper


Table2Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent119906

Fun

BBTL

BO(119906=00)

BBTL

BO(119906=01)

BBTL

BO(119906=03)

BBTL

BO(119906=05)

BBTL

BO(119906=07)

BBTL

BO(119906=09)

BBTL

BO(119906=10)

1198651

175119890minus001plusmn121119890+000

689119890minus071plusmn101119890minus070

123119890minus163plusmn00


00plusmn00

00plusmn00

00plusmn00

1198652





00plusmn00

00plusmn00

00plusmn00

1198653







199

eminus00

4plusmn113

eminus00

41198654

765119890+002plusmn583119890+002


00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

1198655

558119890+002plusmn653119890+002





216

eminus11

5plusmn110

eminus11


1198656

251119890+001plusmn534119890+000





363

eminus15

4plusmn134

eminus15


1198657






116

eminus18

8plusmn00


1198658

241119890+000plusmn307119890+000





107

eminus05

6plusmn439

eminus05


1198659

266

e+00

1plusmn179

e+00

0272119890+001plusmn317119890minus001

277119890+001plusmn318119890minus001

283119890+001plusmn278119890minus001

284119890+001plusmn267119890minus001

283119890+001plusmn341119890minus001

280119890+001plusmn387119890minus001

11986510830119890+000plusmn176119890+000



355

eminus01

5plusmn00

355

eminus01

5plusmn00

355

eminus01

5plusmn00

355

eminus01

5plusmn00

11986511374119890+001plusmn905119890+000

333119890+001plusmn118119890+001

271119890+001plusmn800119890+000

189119890+001plusmn114119890+001

573119890+000plusmn106119890+001

00plusmn00

00plusmn00

11986512815119890+000plusmn193119890+000

338119890minus001plusmn116119890+000

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00




00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986514

433

e+00

3plusmn679

e+00

2467119890+003plusmn610119890+002

517119890+003plusmn668119890+002

559119890+003plusmn685119890+002

553119890+003plusmn710119890+002

558119890+003plusmn780119890+002

540119890+003plusmn653119890+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518minus771119890+000plusmn347119890+000minus806119890+000plusmn339119890+000minus964119890+000plusmn181119890+000minus965119890+000plusmn176119890+000minus102

e+00

1plusmn677

eminus00

3minus985119890+000plusmn122119890+000minus993119890+000plusmn112119890+000

11986519minus769119890+000plusmn352119890+000minus813119890+000plusmn336119890+000minus987119890+000plusmn183119890+000minus103

e+00

1plusmn945

eminus00

1minus976119890+000plusmn195119890+000minus982119890+000plusmn178119890+000minus961119890+000plusmn199119890+000


e+00

1plusmn161

e+00



Table3Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

BBPS

OBB

Exp

BBDE

GBD

EMGBD

EBB

TLBO

1198651






00plusmn00

1198652

13800plusmn211119890+004

1000plusmn463119890+003


1400plusmn452119890+003


00plusmn00

1198653

132119890+000plusmn318119890+000





227eminus

004plusmn126

eminus004

1198654

560119890+000plusmn928119890+000

960119890minus001plusmn427119890+000

789119890+001plusmn305119890+002


108119890+000plusmn128119890+000

00plusmn00

1198655

124119890+004plusmn666119890+003

441119890+003plusmn337119890+003

209119890+000plusmn400119890+000

536119890+003plusmn326119890+003

757119890+002plusmn116119890+003

216eminus

115plusmn110

eminus114

1198656

167119890+001plusmn919119890+000

120119890+000plusmn522119890minus001

139119890+001plusmn447119890+000


110119890+000plusmn294119890+000

363eminus

154plusmn134

eminus153

1198657

234119890+001plusmn132119890+001

100119890+000plusmn303119890+000


600119890minus001plusmn240119890+000

200119890minus001plusmn141119890+000

116eminus

188plusmn00

1198658

187119890+002plusmn134119890+002

158119890+002plusmn700119890+001


172119890+002plusmn667119890+001

249119890+001plusmn199119890+001

107eminus

056plusmn439

eminus056

1198659

707119890+001plusmn148119890+002

357e+

001plusmn250e+

001

276119890+001plusmn106119890+001

317119890+001plusmn207119890+001

276119890+001plusmn146119890+001

283119890+001plusmn341119890minus001

11986510

106119890+001plusmn929119890+000

152119890+000plusmn511119890+000

134119890+000plusmn115119890+000

259119890+000plusmn645119890+000

554119890minus001plusmn279119890+000

355eminus

015plusmn00

11986511

116119890+002plusmn353119890+001

181119890+001plusmn728119890+000

676119890+001plusmn389119890+001

155119890+001plusmn596119890+000

203119890+001plusmn923119890+000

00plusmn00

11986512

273119890+000plusmn211119890+000


173119890+000plusmn132119890+000



00plusmn00

11986513






00plusmn00

11986514

364119890+003plusmn628119890+002

258119890+003plusmn551119890+002

230119890+003plusmn409119890+002

249119890+003plusmn541119890+002

260119890+003plusmn505119890+002

558e+

003plusmn780

e+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516


00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus560119890+000plusmn341119890+000

minus790119890+000plusmn274119890+000

minus709119890+000plusmn333119890+000

minus763119890+000plusmn286119890+000

minus801119890+000plusmn300119890+000

minus985e+

000plusmn122e+

000

11986519

minus597119890+000plusmn331119890+000

minus787119890+000plusmn303119890+000

minus621119890+000plusmn366119890+000

minus860119890+000plusmn268119890+000

minus837119890+000plusmn290119890+000

minus982e+

000plusmn178e+

000

11986520

minus581119890+000plusmn365119890+000

minus940119890+000plusmn242119890+000

minus602119890+000plusmn377119890+000

minus946e+

000plusmn224e+

000

minus938119890+000plusmn251119890+000

minus941119890+000plusmn243119890+000


0 05 1 15 2 25 3 35 4

0

50

FEs

minus200

minus150

minus100

minus50

log10

(mea

n fit

ness

)

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

(a) 1198657 Schwefel 222

0 05 1 15 2 25 3 35 4

0

10

FEs times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

minus60

minus50

minus40

minus30

minus20

minus10

log10

(mea

n fit

ness

)(b) 1198658 Zakharov

0 05 1 15 2 25 3 35 4

0

1

2

3

FEs

minus6

minus5

minus4

minus3

minus2

minus1

log10

(mea

n fit

ness

)

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

(c) 11986518 Shekel5

0 05 1 15 2 25 3 35 4

0

FEs

Mea

n fit

ness

minus12

minus10

minus8

minus6

minus4

minus2

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

(d) 11986511 Rastrigin

Figure 3 Comparison of the performance curves using different 119906

55 Comparison of BBTLBO with DE Variants PSO Variantsand Some TLBO Variants In this section we comparedthe performance of BBTLBO with other optimization algo-rithms including jDE [28] SaDE [29] PSOcfLocal [27]PSOwFIPS [30] and TLBO [8 9] In our experiment themaximal FEs are used as the stopping criterion of all algo-rithms namely 40000 for all test functions The results areshown in Table 5 in terms of the mean optimum solutionand the standard deviation of the solutions obtained in the50 independent runs by each algorithm on 20 test functions

where ldquo119908119905119897rdquo summarizes the competition results amongBBTLBO and other algorithms The best results among thealgorithms are shown in boldface

The comparisons in Table 5 show that that all algorithmsperform well for 119865

15 11986516 and 119865

17 Although SaDE outper-

forms BBTLBOon11986514 PSOcfLocal outperforms BBTLBOon

1198659and PSOwFIPS outperforms BBTLBO on 119865

19and 11986520 and

BBTLBO offers the highest accuracy on functions 1198653 1198654 1198655

1198657 1198658 11986510 11986511 and 119865

18 ldquo119908119905119897rdquo shows that BBTLBO offers

well accuracy for the majority of test functions in this paper


0 05 1 15 2 25 3 35 4

0

1

2

3

FEs

minus4

minus3

minus2

minus1

times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(a) 1198653 Quadric

0 05 1 15 2 25 3 35 41

15

2

25

3

35

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)(b) 1198659 Rosenbrock

0 05 1 15 2 25 3 35 433

34

35

36

37

38

39

4

41

42

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(c) 11986518 Shekel5

0

0

05 1 15 2 25 3 35 4FEs

Mea

n fit

ness

times104

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

(d) 11986514 Schwefel

Figure 4 Comparison of the performance curves using different algorithms

Table 5 concludes that BBTLBO has a good performance ofthe solution accuracy for all unimodal optimization problemsand most complex multimodal optimization problems

6 Two Real-World Optimization Problems

In this section to show the effectiveness of the proposedmethod the proposed BBTLBO algorithm is applied toestimate parameters of two real-world problems

61 Nonlinear Function Approximation The artificial neuralnetwork trained by our BBTLBO algorithm is a three-layer

Input x Output y

Desired output dBBTLBO algorithm

ANN

minus

Figure 5 BBTLBO-based ANN

feed-forward network and the basic structure of the proposedscheme is depicted in Figure 5 The inputs are connectedto all the hidden units which in turn all connected to all


Table 4 The mean number of FEs and SR with acceptable solutions using different algorithms

Fun 119905 value BBPSO BBExp BBDE GBDE MGBDE BBTLBOMFEs SR MFEs SR MFEs SR MFEs SR MFEs SR MFEs SR

1198651

1119864 minus 8 15922 100 17727 100 11042 100 19214 100 11440 100 1390 1001198652

1119864 minus 8 17515 54 19179 94 12243 100 20592 90 12634 100 1500 1001198653

1119864 minus 8 NaN 0 NaN 0 NaN 0 NaN 0 NaN 0 NaN 01198654

1119864 minus 8 11710 24 8120 84 3634 6 7343 40 4704 34 525 1001198655

1119864 minus 8 NaN 0 NaN 0 NaN 0 NaN 0 NaN 0 4100 1001198656


1119864 minus 8 17540 6 21191 90 17314 100 22684 94 15322 98 2144 1001198658


1119864 minus 2 17073 62 18404 42 14029 24 18182 52 17200 80 NaN 011986510

1119864 minus 8 24647 26 27598 90 18273 26 29172 82 18320 84 2110 10011986511


1119864 minus 8 NaN 0 25465 50 NaN 0 27317 64 19704 24 2471 10011986513

1119864 minus 8 16318 32 21523 58 11048 16 22951 64 14786 58 1470 10011986514


1119864 minus 8 658 100 1176 100 1274 100 1251 100 1206 100 799 10011986516

1119864 minus 8 657 98 1251 100 1294 100 1343 100 1308 100 813 10011986517

1119864 minus 8 995 100 2626 100 1487 100 2759 100 1921 100 973 10011986518

minus1015 1752 34 6720 44 2007 52 4377 32 8113 64 1684 9411986519

minus1040 2839 34 8585 48 1333 42 6724 50 3056 66 2215 9011986520

minus1053 1190 36 8928 74 1115 40 6548 76 5441 80 2822 82

the outputs The variables consist of neural network weightsand biases Suppose a three-layer forward neural networkarchitecture with 119872 input units 119873 hidden units and 119870

output units and the number of the variables is shown asfollows

119871 = (119872 + 1) lowast 119873 + (119873 + 1) lowast 119870 (13)

For neural network training the aim is to find a set ofweights with the smallest error measure Here the objectivefunction is the mean sum of squared errors (MSE) over alltraining patterns which is shown as follows

MSE = 1

119876 lowast 119870

119876

sum

119894=1

119870

sum

119895

1

2(119889119894119895minus 119910119894119895)2

(14)

where 119876 is the number of training data set 119870 is the numberof output units 119889

119894119895is desired output and 119910

119894119895is output inferred

from neural networkIn this example a three-layer feed-forward ANN with

one input unit five hidden units and one output unit isconstructed tomodel the curve of a nonlinear functionwhichis described by the following equation [31]

119910 = sin (2119909) exp (minus2119909) (15)

In this case activation function used in the output layer isthe sigma function and activation function used in the outputlayer is linear The number (dimension) of the variables is16 for BBTLBO-based ANN In order to train the ANN

200 pairs of data are chosen from the real model For eachalgorithm 50 runs are performed The other parametersare the same as those of the previous investigations Theresults are shown in Table 6 in terms of the mean MSEand the standard deviation obtained in the 50 independentruns for three methods Figure 6 shows the predicted timeseries for training and test using different algorithms It canconclude that the approximation achieved by BBTLBO hasgood performance

62 Tuning of PID Controller The continuous form of adiscrete-type PID controller with a small sampling period Δ119905is described as follows [32]

119906 [119896] = 119870119875 sdot 119890 (119896) + 119870119868 sdot

119896

sum

119894=1

119890 [119894] sdot Δ119905 + 119870119863 sdot119890 [119896] minus 119890 [119896 minus 1]

Δ119905

(16)

where 119906[119896] is the controlled output respectively 119890[119896] = 119903[119896]minus119910[119896] is the error signal 119903[119896] and 119910[119896] are the reference signaland the system output and 119870

119875 119870119868 and 119870

119863represent the

proportional integral and derivate gains respectivelyFor an unknown plant the goal of this problem is to

minimize the integral absolute error (IAE) which is given asfollow [32 33]

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903 (17)


Table5Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

jDE

SaDE

PSOcfLo

cal

PSOwFIPS

TLBO

BBTL

BO1198651






00plusmn00

1198652






00plusmn00

1198653






227eminus

004plusmn126

eminus004

1198654

211119890+001plusmn674119890+001

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

1198655

122119890+002plusmn137119890+002

428119890+001plusmn259119890+001

117119890+001plusmn930119890+000

260119890+003plusmn679119890+002


216eminus

115plusmn110

eminus114

1198656

306119890+001plusmn850119890+000

245119890+000plusmn260119890+000


266119890+000plusmn558119890minus001


363eminus

154plusmn134

eminus153

1198657






116eminus

188plusmn00

1198658

216119890+000plusmn416119890+000



586119890+001plusmn170119890+001


107eminus

056plusmn439

eminus056

1198659

249119890+001plusmn105119890+001

261119890+001plusmn107119890+000

240e+

001plusmn152

e+000

265119890+001plusmn354119890minus001

255119890+001plusmn501119890minus001

283119890+001plusmn341119890minus001

11986510






355eminus

015plusmn00

11986511

203119890+000plusmn194119890+000

386119890+000plusmn197119890+000

426119890+001plusmn106119890+001

115119890+002plusmn154119890+001

155119890+001plusmn809119890+000

00plusmn00

11986512



789119890minus001plusmn103119890+000

136119890+000plusmn741119890minus001

00plusmn00

00plusmn00

11986513





00plusmn00

00plusmn00

11986514

193119890+002plusmn142119890+002

135

e+002plusmn126e+

002

449119890+003plusmn825119890+002

396119890+003plusmn840119890+002

482119890+003plusmn686119890+002

558119890+003plusmn780119890+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus940119890+000plusmn210119890+000

minus925119890+000plusmn230119890+000

minus776119890+000plusmn342119890+000

minus979119890+000plusmn144119890+000

minus972119890+000plusmn142119890+000

minus985e+

000plusmn122e+

000

11986519

minus985119890+000plusmn190119890+000

minus987119890+000plusmn183119890+000

minus924119890+000plusmn270119890+000

minus104e+

001plusmn423eminus

009

minus922119890+000plusmn241119890+000

minus982119890+000plusmn178119890+000

11986520

minus965119890+000plusmn223119890+000

minus101119890+001plusmn159119890+000

minus963119890+000plusmn250119890+000

minus105e+

001plusmn101eminus

004

minus965119890+000plusmn223119890+000

minus941119890+000plusmn243119890+000

119908119905119897

1334

1244

1343

1244

1163

mdash


0 50 100 150 200 250 300 350 400 450 500

0

1

2

3 Convergence curves using different methods

Generation

Best

valu

e

minus5

minus4

minus3

minus2

minus1

TLBOBBTLBO

(a) Convergence curves

0 10 20 30 40 50 60 70

0

005

01

015

02

025

03

035 Approximation curves using different methods

Sample number

y(t)

minus005

ActualTLBO

BBTLBO

(b) Approximation curves

0 10 20 30 40 50 60 700

0002

0004

0006

0008

001

0012

0014

0016Error curves using different methods

Sample number

y(t)

TLBOBBTLBO

(c) Error curves


Table 6 Comparisons between BBTLBO and other algorithms onMSE

Algorithm Training error Testing errorMean Std Mean Std

TLBO 985119890 minus 004 926119890 minus 004 943119890 minus 004 918119890 minus 004

BBTLBO 345119890 minus 004 202119890 minus 004 276119890 minus 004 182119890 minus 004

where 119890(119905) and 119906(119905) are used to represent the system error andthe control output at time 119905 119905

119903is the rising time and 120596

119894(119894 = 1

2 3) are weight coefficientsTo avoid overshooting a penalty value is adopted in the

cost function That is once overshooting occurs the value

of overshooting is added to the cost function and the costfunction is given as follows [32 33]

if 119889119910 (119905) lt 0

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)

+1205964

1003816100381610038161003816119889119910 (119905)

1003816100381610038161003816) 119889119905 + 120596

3119905119903

else

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903

end

(18)


Table 7 Comparisons of parameters of PID controllers using different algorithms

Algorithm 119870119875

119870119868

119870119863

Overshoot () Peak time (s) Rise time (s) Cost function CPU time (s)GA 011257 002710 028792 290585 165000 105000 1634555 705900PSO 011772 001756 027737 104808 165000 065000 1160773 691000BBTLBO 011605 001661 025803 034261 180000 070000 1134300 704500

0 5 10 15 20 25 30 35 40 45 501

15

2

25

3

35

Generation

Best

valu

e

Performance curves using different methods

GAPSOBBTLBO

Figure 7 Performance curves using different methods

where 1205964is a coefficient and 120596

4≫ 1205961 119889119910(119905) = 119910(119905) minus119910(119905minus 1)

and 119910(119905) is the output of the controlled objectiveIn our simulation the formulas for the plant examined

are given as follows [34]

119866 (119904) =1958

1199043+ 1789119904

2+ 1033119904 + 1908

(19)

The system sampling time is Δ119905 = 005 second and thecontrol value 119906 is limited in the range of [minus10 10] Other rel-evant system variables are 119870

119875isin [0 1] 119870

119868isin [0 1] and 119870

119863isin

[0 1] The weight coefficients of the cost function are set as1205961= 0999 120596

2= 0001 120596

3= 2 and 120596 = 100 in this example

In the simulations the step response of PID controlsystem tuned by the proposed BBTLBO is compared withthat tuned by the standard genetic algorithm (GA) and thestandard PSO (PSO) The population sizes of GA PSO andBBTLBO are 50 and the corresponding maximum numbersof iterations are 50 50 and 50 respectively In addition thecrossover rate is set as 090 and the mutation rate is 010 forGA

The optimal parameters and the corresponding perfor-mance values of the PID controllers are listed in Table 7 andthe corresponding performance curves and step responsescurves are given in Figures 7 and 8 It can be seen fromFigure 7 and Table 7 that the PID controller tuned byBBTLBO has the minimum cost function and CPU time

0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

Time (s)

Step response curves using different methods

GAPSOBBTLBO

r in

you

t

Figure 8 Step response curves using different methods

Although PID controllers tuned by PSO have a smaller peaktime and rise time their maximum overshoots are muchlarger than the overshoot tuned by BBTLBO It concludes thatthe PID controller tuned by the BBTLBO could perform thebest control performance in the simulations

7 Conclusion

In this paper TLBO has been extended to BBTLBO whichuses the hybridization of the learning strategy in the stan-dard TLBO and Gaussian sampling learning to balance theexploration and the exploitation in teacher phase and uses amodified mutation operation so as to eliminate the duplicatelearners in learner phase The proposed BBTLBO algorithmis utilized to optimize 20 benchmark functions and tworeal-world optimization problems From the analysis andexperiments the BBTLBO algorithm significantly improvesthe performance of the original TLBO although it needs tospend more CPU time than the standard TLBO algorithmin each generation From the results compared with otheralgorithms on the 20 chosen test problems it can be observedthat the BBTLBO algorithm has good performance by usingneighborhood search more effectively to generate betterquality solutions although the BBTLBO algorithm does notalways have the best performance in all experiments cases ofthis paper It can be also observed that the BBTLBOalgorithm


gives the best performance on two real-world optimizationproblems compared with other algorithms in the paper

Further work includes research into neighborhood searchbased on different topological structures Moreover thealgorithm may be further applied to constrained dynamicand noisy single-objective and multiobjective optimizationdomain It is expected that BBTLBOwill be used tomore real-world optimization problems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was partially supported by the National NaturalScience Foundation of China (61100173 61100009 61272283and 61304082) This work is partially supported by theNatural Science Foundation of Anhui Province China (Grantno 1308085MF82) and the Doctoral Innovation Foundationof Xirsquoan University of Technology (207-002J1305)

References

[1] D E Goldberg Genetic Algorithms in Search Optimization andMachine Learning Addison-Wesley Reading Mass USA 1989

[2] L C Jiao and L Wang ldquoA novel genetic algorithm based onimmunityrdquo IEEE Transactions on SystemsMan and CyberneticsA Systems and Humans vol 30 no 5 pp 552ndash561 2000

[3] R Storn and K Price ldquoDifferential evolution a simple andefficient Heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[4] M Dorigo and T Stutzle Ant Colony Optimization MIT Press2004

[5] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[6] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

[7] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[8] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011

[9] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization an optimization method for continuousnon-linear large scale problemsrdquo Information Sciences vol 183no 1 pp 1ndash15 2012

[10] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization algorithm for unconstrained and con-strained real-parameter optimization problemsrdquo EngineeringOptimization vol 44 no 12 pp 1447ndash1462 2011

[11] V Togan ldquoDesign of planar steel frames using teaching-learningbased optimizationrdquo Engineering Structures vol 34 pp 225ndash232 2012

[12] R V Rao and V Patel ldquoAn elitist teaching-learning-based opti-mization algorithm for solving complex constrained optimiza-tion problemsrdquo International Journal of Industrial EngineeringComputations vol 3 pp 535ndash560 2012

[13] S O Degertekin and M S Hayalioglu ldquoSizing truss structuresusing teaching-learning-based optimizationrdquo Computers andStructures vol 119 pp 177ndash188 2013

[14] R V Rao and V Patel ldquoAn improved teaching-learning-basedoptimization algorithm for solving unconstrained optimizationproblemsrdquo Scientia Iranica vol 20 no 3 pp 710ndash720 2013

[15] R V Rao and V Patel ldquoMulti-objective optimization of com-bined Brayton and inverse Brayton cycles using advancedoptimization algorithmsrdquo Engineering Optimization vol 44 no8 pp 965ndash983 2011

[16] T Niknam F Golestaneh and M S Sadeghi ldquoTheta-multi-objective teaching-learning-based optimization for dynamiceconomic emission dispatchrdquo IEEE Systems Journal vol 6 no2 pp 341ndash352 2012

[17] R V Rao and V Patel ldquoMulti-objective optimization of heatexchangers using a modified teaching-learning-based opti-mization algorithmrdquo Applied Mathematical Modelling vol 37no 3 pp 1147ndash1162 2013

[18] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[19] F van den Bergh and A P Engelbrecht ldquoA study of particleswarm optimization particle trajectoriesrdquo Information Sciencesvol 176 no 8 pp 937ndash971 2006

[20] J Kennedy ldquoBare bones particle swarmsrdquo in Proceedings of theSwarm Intelligence Symposium (SIS rsquo03) pp 80ndash87 2003

[21] M G H Omran A P Engelbrecht and A Salman ldquoBarebones differential evolutionrdquo European Journal of OperationalResearch vol 196 no 1 pp 128ndash139 2009

[22] H Wang S Rahnamayan H Sun and M G H OmranldquoGaussian bare-bones differential evolutionrdquo IEEE Transactionson Cybernetics vol 43 no 2 pp 634ndash647 2013

[23] X H Hu and R Eberhart ldquoMultiobjective optimization usingdynamic neighborhood particle swarm optimizationrdquo in Pro-ceedings of the Congress on Evolutionary Computation pp 677ndash1681 2002

[24] M G Omran A P Engelbrecht and A Salman ldquoUsingthe ring neighborhood topology with self-adaptive differentialevolutionrdquo in Advances in Natural Computation pp 976ndash979Springer Berlin Germany 2006

[25] X Li ldquoNiching without niching parameters particle swarmoptimization using a ring topologyrdquo IEEE Transactions onEvolutionary Computation vol 14 no 1 pp 150ndash169 2010

[26] I Maruta T H Kim D Song and T Sugie ldquoSynthesis of fixed-structure robust controllers using a constrained particle swarmoptimizer with cyclic neighborhood topologyrdquo Expert Systemswith Applications vol 40 no 9 pp 3595ndash3605 2013

[27] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the International Con-ference on Evolutionary Computation pp 1671ndash1676 HonoluluHawaii USA 2002

[28] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006


[29] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009

[30] R Mendes J Kennedy and J Neves ldquoThe fully informedparticle swarm simpler maybe betterrdquo IEEE Transactions onEvolutionary Computation vol 8 no 3 pp 204ndash210 2004

[31] F Herrera and M Lozano ldquoGradual distributed real-codedgenetic algorithmsrdquo IEEE Transactions on Evolutionary Compu-tation vol 4 no 1 pp 43ndash62 2000

[32] J Liu Advanced PID Control and MATLAB Simulation Elec-tronic Industry Press 2003

[33] J Zhang J Zhuang H Du and S Wang ldquoSelf-organizinggenetic algorithm based tuning of PID controllersrdquo InformationSciences vol 179 no 7 pp 1007ndash1017 2009

[34] R Haber-Haber R Haber M Schmittdiel and R M delToro ldquoA classic solution for the control of a high-performancedrilling processrdquo International Journal of Machine Tools andManufacture vol 47 no 15 pp 2290ndash2297 2007

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Distributed Sensor Networks


Advances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications


Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

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Multimedia


Biomedical Imaging


ArtificialNeural Systems

Advances in


RoboticsJournal of



Computational Intelligence and Neuroscience

Industrial EngineeringJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



(1) Begin(2) Initialize119873 (number of learners) and119863 (number of dimensions)(3) Initialize learners119883 and evaluate all learners119883(4) Donate the best learner as Teacher and the mean of all learners119883 asMean(5) while (stopping condition not met)(6) for each learner119883


(7) TF = round(1 + rand(0 1))(8) for 119895 = 1 119863(9) 119899119890119908119883

119894119895= 119883119894119895+ rand(0 1) lowast (119879119890119886119888ℎ119890119903(119895) minus TF lowast119872119890119886119899(119895))

(10) endfor(11) Accept 119899119890119908119883

119894if 119891(119899119890119908119883

119894) is better than 119891(119883

119894)




(15) if 119891(119883119894) better 119891(119883

119896)

(16) for 119895 = 1 119863(17) 119899119890119908119883

119894119895= 119883119894119895+ rand(0 1) lowast (119883

119894119895minus 119883119896119895)

(18) endfor(19) else(20) for 119895 = 1 119863(21) 119899119890119908119883

119894119895= 119883119894119895+ rand(0 1) lowast (119883

119896119895minus 119883119894119895)

(22) endfor(23) endif(24) Accept 119899119890119908119883

119894if 119891(119899119890119908119883

119894) is better than 119891(119883

119894)

(25) endfor(26) Update the Teacher and theMean(27) endwhile(28) end

Algorithm 1 TLBO( )

bare-bones algorithms Section 4 describes the improvedteaching-learning-based optimization algorithm usingneighborhood search (BBTLBO) Section 5 presents the testson several benchmark functions and the experiments areconducted along with statistical tests The applications fortraining artificial neural network are shown in Section 6Conclusions are given in Section 7

2 Teaching-Learning-Based Optimization

Rao et al [8 9] first proposed a novel teaching-learning-based optimization (TLBO) inspired from the philosophy ofteaching and learning The TLBO algorithm is based on theeffect of the influence of a teacher on the output of learners ina class which is considered in terms of results or grades Theprocess ofworking of TLBO is divided into twopartsThefirstpart consists of ldquoteacher phaserdquo and the second part consistsof ldquolearner phaserdquo The ldquoteacher phaserdquo means learning fromthe teacher and the ldquolearner phaserdquo means learning throughthe interaction between learners

A good teacher is one who brings his or her learners up tohis or her level in terms of knowledge But in practice this isnot possible and a teacher can only move the mean of a classup to some extent depending on the capability of the classThis follows a random process depending on many factorsLet119872 be the mean and let 119879 be the teacher at any iteration 119879will try tomovemean119872 toward its own level so now the new

mean will be 119879 designated as119872new The solution is updatedaccording to the difference between the existing and the newmean according to the following expression

119899119890119908119883 = 119883 + 119903 times (119872new minus TF times119872) (1)

where TF is a teaching factor that decides the value of meanto be changed and 119903 is a random vector in which each elementis a random number in the range [0 1] The value of TF canbe either 1 or 2 which is again a heuristic step and decidedrandomly with equal probability as

TF = round [1 + rand (0 1)] (2)

Learners increase their knowledge by two differentmeans one through input from the teacher and the otherthrough interaction between themselves A learner interactsrandomly with other learners with the help of group discus-sions presentations formal communications and so forthA learner learns something new if the other learner hasmore knowledge than him or her Learner modification isexpressed as

119899119890119908119883119894=

119883119894+ 119903 lowast (119883

119894minus 119883119895) if119891 (119883

119894) lt 119891 (119883

119895)

119883119894+ 119903 lowast (119883

119895minus 119883119894) otherwise

(3)

As explained above the pseudocode for the implementa-tion of TLBO is summarized in Algorithm 1





lim119905rarrinfin

119883119894 (119905) =


1198881+ 1198882

(4)



119909119894119895 (119905 + 1) = 119873(

119892best + 119901best119894119895 (119905)

2

100381610038161003816100381610038161003816119892best minus 119901best119894119895 (119905)

100381610038161003816100381610038161003816) (5)




=

119873(

119892best + 119901best119894119895 (119905)

2

100381610038161003816100381610038161003816119892best minus 119901best119894119895 (119905)

100381610038161003816100381610038161003816) rand (0 1) gt 05

119901best119894119895 (119905) otherwise



= 1199011198943 119895(119905) + 1199032 sdot (1199091198941 119895

(119905) minus 1199091198942 119895(119905)) rand (0 1) gt CR

119901best1198943119895 (119905) otherwise(7)

where 1198941 1198942 and 119894









V119894119895 (119905 + 1)

=

119873(

119883best119895 (119905) + 119909119894119895 (119905)

2 rand (0 1) le CR or 119895 = 119895rand

10038161003816100381610038161003816119883best119895 (119905) minus 119909119894119895 (119905)

10038161003816100381610038161003816)

119909119894119895 (119905) otherwise(9)



V119894119895 (119905 + 1)

=


119873(

119883best119895 (119905)+119909119894119895 (119905)

210038161003816100381610038161003816119883best119895 (119905)minus119909119894119895 (119905)

10038161003816100381610038161003816) otherwise

(10)





Begin





newXi better Xi

Xi = newXi Xi = Xi

Yes

Yes

Yes

Yes

No

No

No

No


Learner phase

rand(0 1) lt 05


newXi better Xi

Xi = newXi Xi = Xi


End

gen = gen + 1

XiV1V2







NeighborhoodiXi

Ximinus1

Xi+1



119894minus119896 119883

119894 119883

119894+119896 That is






1198811119895 (119905 + 1) = 119883119894119895 (119905) + rand (0 1)


119894119895 (119905))

1198812119895 (119905 + 1) = 119873(

119873119879119890119886119888ℎ119890119903119894119895 (119905) + 119873119872119890119886119899119894119895 (119905)

2

100381610038161003816100381610038161003816100381610038161003816

119873119879119890119886119888ℎ119890119903119894119895 (119905) minus 119873119872119890119886119899119894119895 (119905)

100381610038161003816100381610038161003816100381610038161003816

)

119899119890119908119883119894119895 (119905 + 1) = 119906 sdot 1198811119895 (119905 + 1) + (1 minus 119906) sdot 1198812119895 (119905 + 1)

(11)






119894



119899119890119908119883119894119895= 119883119894119895+ 1199031lowast (119873119879119890119886119888ℎ119890119903

119894119895minus 119883119894119895)

+ 1199032lowast (119883119894119895minus 119883119896119895)

(12)












10to 11986520







119894if 119891(119899119890119908119883

119894) is better than 119891(119883

119894)





119894if 119891(119899119890119908119883119894) is better than 119891(119883

119894)


















Sphere 1198651(119909) =

119863

sum

119894=1

1199092

11989430 [minus100 100] 0

Sum square 1198652(119909) =

119863

sum

119894=1

1198941199092

11989430 [minus100 100] 0

Quadric 1198653(119909) =

119863

sum

119894=1

1198941199094

119894+ random(0 1) 30 [minus128 128] 0

Step 1198654(119909) =

119863

sum

119894=1


2 30 [minus100 100] 0

Schwefel 12 1198655(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119895)

2

30 [minus100 100] 0

Schwefel 221 1198656(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863 30 [minus100 100] 0

Schwefel 222 1198657(119909) =

119863

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816+

119863

prod

119894=1

1003816100381610038161003816119909119894

100381610038161003816100381630 [minus10 10] 0

Zakharov 1198658(119909) =

119863

sum

119894=1

1199092

119894+ (

119863

sum

119894=1

05119894119909119894)

2

+ (

119863

sum

119894=1

05119894119909119894)

4

30 [minus100 100] 0

Rosenbrock 1198659(119909) =

119863minus1

sum

119894=1

lfloor100(1199092

119894minus 119909119894+1)2

+ (119909119894minus 1)2

rfloor 30 [minus2048 2048] 0


5)radic(

1

119863)

119863

sum

119894=1

1199092

119894) minus exp(( 1

119863)

119863

sum

119894=1

cos (2120587119909119894)) + 119890 30 [minus32 32] 0

Rastrigin 11986511(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10) 30 [minus512 512] 0

Weierstrass11986512(119909) =

119863

sum

119894=1

(

119896maxsum

119896=0

[119886119896 cos (2120587119887119896 (119909

119894+ 05))]) minus 119863

119896maxsum

119896=0

[119886119896 cos (2120587119887119896 times 05)]

119886 = 05 119887 = 3 119896max = 20

30 [minus05 05] 0

Griewank 11986513(119909) =

119863

sum

119894=1

(1199092

119894

4000) minus

119899

prod

119894=1

cos(119909119894

radic119894

) + 1 30 [minus600 600] 0

Schwefel 11986514(119909) = 4189829119863 +

119863

sum

119894=1


119894)) 30 [minus500 500] 0

Bohachevsky1 11986515(119909) = 119909

2

1+ 21199092

2minus 03 cos (3120587119909

1) minus 04 cos (4120587119909

2) + 07 2 [minus100 100] 0

Bohachevsky2 11986516(119909) = 119909

2

1+ 21199092

2minus 03cos (3120587119909

1) lowast cos (4120587119909

2) + 03 2 [minus100 100] 0

Bohachevsky3 11986517(119909) = 119909

2

1+ 21199092

2minus 03cos((3120587119909

1) + (4120587119909

2)) + 03 2 [minus100 100] 0

Shekel5 11986518(119909) = minus

5

sum

119894=1

[(119909 minus 119886119894)(119909 minus 119886

119894)119879+ 119888119894]

minus1

4 [0 10] minus101532

Shekel7 11986519(119909) = minus

7

sum

119894=1

[(119909 minus 119886119894)(119909 minus 119886

119894)119879+ 119888119894]

minus1

4 [0 10] minus104029

Shekel10 11986520(119909) = minus

10

sum

119894=1

[(119909 minus 119886119894)(119909 minus 119886

119894)119879+ 119888119894]

minus1

4 [0 10] minus105364



3 1198655 1198656


11 11986512 and 119865

14 BBTLBO






1198657 1198659 11986510 and 119865




Table2Com

paris

onsm

eanplusmnstd

ofthes

olutions


Fun

BBTL

BO(119906=00)

BBTL

BO(119906=01)

BBTL

BO(119906=03)

BBTL

BO(119906=05)

BBTL

BO(119906=07)

BBTL

BO(119906=09)

BBTL

BO(119906=10)

1198651

175119890minus001plusmn121119890+000




00plusmn00

00plusmn00

00plusmn00

1198652





00plusmn00

00plusmn00

00plusmn00

1198653







199

eminus00

4plusmn113

eminus00

41198654

765119890+002plusmn583119890+002


00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

1198655

558119890+002plusmn653119890+002





216

eminus11

5plusmn110

eminus11


1198656

251119890+001plusmn534119890+000





363

eminus15

4plusmn134

eminus15


1198657






116

eminus18

8plusmn00


1198658

241119890+000plusmn307119890+000





107

eminus05

6plusmn439

eminus05


1198659

266

e+00

1plusmn179

e+00

0272119890+001plusmn317119890minus001

277119890+001plusmn318119890minus001

283119890+001plusmn278119890minus001

284119890+001plusmn267119890minus001

283119890+001plusmn341119890minus001

280119890+001plusmn387119890minus001

11986510830119890+000plusmn176119890+000



355

eminus01

5plusmn00

355

eminus01

5plusmn00

355

eminus01

5plusmn00

355

eminus01

5plusmn00

11986511374119890+001plusmn905119890+000

333119890+001plusmn118119890+001

271119890+001plusmn800119890+000

189119890+001plusmn114119890+001

573119890+000plusmn106119890+001

00plusmn00

00plusmn00

11986512815119890+000plusmn193119890+000

338119890minus001plusmn116119890+000

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00




00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986514

433

e+00

3plusmn679

e+00

2467119890+003plusmn610119890+002

517119890+003plusmn668119890+002

559119890+003plusmn685119890+002

553119890+003plusmn710119890+002

558119890+003plusmn780119890+002

540119890+003plusmn653119890+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00


e+00

1plusmn677

eminus00



e+00

1plusmn945

eminus00



e+00

1plusmn161

e+00



Table3Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

BBPS

OBB

Exp

BBDE

GBD

EMGBD

EBB

TLBO

1198651






00plusmn00

1198652

13800plusmn211119890+004

1000plusmn463119890+003


1400plusmn452119890+003


00plusmn00

1198653

132119890+000plusmn318119890+000





227eminus

004plusmn126

eminus004

1198654

560119890+000plusmn928119890+000

960119890minus001plusmn427119890+000

789119890+001plusmn305119890+002


108119890+000plusmn128119890+000

00plusmn00

1198655

124119890+004plusmn666119890+003

441119890+003plusmn337119890+003

209119890+000plusmn400119890+000

536119890+003plusmn326119890+003

757119890+002plusmn116119890+003

216eminus

115plusmn110

eminus114

1198656

167119890+001plusmn919119890+000

120119890+000plusmn522119890minus001

139119890+001plusmn447119890+000


110119890+000plusmn294119890+000

363eminus

154plusmn134

eminus153

1198657

234119890+001plusmn132119890+001

100119890+000plusmn303119890+000


600119890minus001plusmn240119890+000

200119890minus001plusmn141119890+000

116eminus

188plusmn00

1198658

187119890+002plusmn134119890+002

158119890+002plusmn700119890+001


172119890+002plusmn667119890+001

249119890+001plusmn199119890+001

107eminus

056plusmn439

eminus056

1198659

707119890+001plusmn148119890+002

357e+

001plusmn250e+

001

276119890+001plusmn106119890+001

317119890+001plusmn207119890+001

276119890+001plusmn146119890+001

283119890+001plusmn341119890minus001

11986510

106119890+001plusmn929119890+000

152119890+000plusmn511119890+000

134119890+000plusmn115119890+000

259119890+000plusmn645119890+000

554119890minus001plusmn279119890+000

355eminus

015plusmn00

11986511

116119890+002plusmn353119890+001

181119890+001plusmn728119890+000

676119890+001plusmn389119890+001

155119890+001plusmn596119890+000

203119890+001plusmn923119890+000

00plusmn00

11986512

273119890+000plusmn211119890+000


173119890+000plusmn132119890+000



00plusmn00

11986513






00plusmn00

11986514

364119890+003plusmn628119890+002

258119890+003plusmn551119890+002

230119890+003plusmn409119890+002

249119890+003plusmn541119890+002

260119890+003plusmn505119890+002

558e+

003plusmn780

e+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516


00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus560119890+000plusmn341119890+000

minus790119890+000plusmn274119890+000

minus709119890+000plusmn333119890+000

minus763119890+000plusmn286119890+000

minus801119890+000plusmn300119890+000

minus985e+

000plusmn122e+

000

11986519

minus597119890+000plusmn331119890+000

minus787119890+000plusmn303119890+000

minus621119890+000plusmn366119890+000

minus860119890+000plusmn268119890+000

minus837119890+000plusmn290119890+000

minus982e+

000plusmn178e+

000

11986520

minus581119890+000plusmn365119890+000

minus940119890+000plusmn242119890+000

minus602119890+000plusmn377119890+000

minus946e+

000plusmn224e+

000

minus938119890+000plusmn251119890+000

minus941119890+000plusmn243119890+000


0 05 1 15 2 25 3 35 4

0

50

FEs

minus200

minus150

minus100

minus50

log10

(mea

n fit

ness

)

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

(a) 1198657 Schwefel 222

0 05 1 15 2 25 3 35 4

0

10

FEs times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

minus60

minus50

minus40

minus30

minus20

minus10

log10

(mea

n fit

ness


0 05 1 15 2 25 3 35 4

0

1

2

3

FEs

minus6

minus5

minus4

minus3

minus2

minus1

log10

(mea

n fit

ness

)

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

(c) 11986518 Shekel5

0 05 1 15 2 25 3 35 4

0

FEs

Mea

n fit

ness

minus12

minus10

minus8

minus6

minus4

minus2

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10






15 11986516 and 119865




19and 11986520 and


1198657 1198658 11986510 11986511 and 119865




0 05 1 15 2 25 3 35 4

0

1

2

3

FEs

minus4

minus3

minus2

minus1

times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(a) 1198653 Quadric

0 05 1 15 2 25 3 35 41

15

2

25

3

35

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness


0 05 1 15 2 25 3 35 433

34

35

36

37

38

39

4

41

42

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(c) 11986518 Shekel5

0

0

05 1 15 2 25 3 35 4FEs

Mea

n fit

ness

times104

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO







Input x Output y


ANN

minus






1198651

1119864 minus 8 15922 100 17727 100 11042 100 19214 100 11440 100 1390 1001198652

1119864 minus 8 17515 54 19179 94 12243 100 20592 90 12634 100 1500 1001198653


1119864 minus 8 11710 24 8120 84 3634 6 7343 40 4704 34 525 1001198655



1119864 minus 8 17540 6 21191 90 17314 100 22684 94 15322 98 2144 1001198658


1119864 minus 2 17073 62 18404 42 14029 24 18182 52 17200 80 NaN 011986510

1119864 minus 8 24647 26 27598 90 18273 26 29172 82 18320 84 2110 10011986511


1119864 minus 8 NaN 0 25465 50 NaN 0 27317 64 19704 24 2471 10011986513

1119864 minus 8 16318 32 21523 58 11048 16 22951 64 14786 58 1470 10011986514


1119864 minus 8 658 100 1176 100 1274 100 1251 100 1206 100 799 10011986516

1119864 minus 8 657 98 1251 100 1294 100 1343 100 1308 100 813 10011986517

1119864 minus 8 995 100 2626 100 1487 100 2759 100 1921 100 973 10011986518

minus1015 1752 34 6720 44 2007 52 4377 32 8113 64 1684 9411986519

minus1040 2839 34 8585 48 1333 42 6724 50 3056 66 2215 9011986520

minus1053 1190 36 8928 74 1115 40 6548 76 5441 80 2822 82



119871 = (119872 + 1) lowast 119873 + (119873 + 1) lowast 119870 (13)


MSE = 1

119876 lowast 119870

119876

sum

119894=1

119870

sum

119895

1

2(119889119894119895minus 119910119894119895)2

(14)






119910 = sin (2119909) exp (minus2119909) (15)




119906 [119896] = 119870119875 sdot 119890 (119896) + 119870119868 sdot

119896

sum

119894=1


Δ119905

(16)


119875 119870119868 and 119870

119863represent the



119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903 (17)


Table5Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

jDE

SaDE

PSOcfLo

cal

PSOwFIPS

TLBO

BBTL

BO1198651






00plusmn00

1198652






00plusmn00

1198653






227eminus

004plusmn126

eminus004

1198654

211119890+001plusmn674119890+001

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

1198655

122119890+002plusmn137119890+002

428119890+001plusmn259119890+001

117119890+001plusmn930119890+000

260119890+003plusmn679119890+002


216eminus

115plusmn110

eminus114

1198656

306119890+001plusmn850119890+000

245119890+000plusmn260119890+000


266119890+000plusmn558119890minus001


363eminus

154plusmn134

eminus153

1198657






116eminus

188plusmn00

1198658

216119890+000plusmn416119890+000



586119890+001plusmn170119890+001


107eminus

056plusmn439

eminus056

1198659

249119890+001plusmn105119890+001

261119890+001plusmn107119890+000

240e+

001plusmn152

e+000

265119890+001plusmn354119890minus001

255119890+001plusmn501119890minus001

283119890+001plusmn341119890minus001

11986510






355eminus

015plusmn00

11986511

203119890+000plusmn194119890+000

386119890+000plusmn197119890+000

426119890+001plusmn106119890+001

115119890+002plusmn154119890+001

155119890+001plusmn809119890+000

00plusmn00

11986512



789119890minus001plusmn103119890+000

136119890+000plusmn741119890minus001

00plusmn00

00plusmn00

11986513





00plusmn00

00plusmn00

11986514

193119890+002plusmn142119890+002

135

e+002plusmn126e+

002

449119890+003plusmn825119890+002

396119890+003plusmn840119890+002

482119890+003plusmn686119890+002

558119890+003plusmn780119890+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus940119890+000plusmn210119890+000

minus925119890+000plusmn230119890+000

minus776119890+000plusmn342119890+000

minus979119890+000plusmn144119890+000

minus972119890+000plusmn142119890+000

minus985e+

000plusmn122e+

000

11986519

minus985119890+000plusmn190119890+000

minus987119890+000plusmn183119890+000

minus924119890+000plusmn270119890+000

minus104e+

001plusmn423eminus

009

minus922119890+000plusmn241119890+000

minus982119890+000plusmn178119890+000

11986520

minus965119890+000plusmn223119890+000

minus101119890+001plusmn159119890+000

minus963119890+000plusmn250119890+000

minus105e+

001plusmn101eminus

004

minus965119890+000plusmn223119890+000

minus941119890+000plusmn243119890+000

119908119905119897

1334

1244

1343

1244

1163

mdash


0 50 100 150 200 250 300 350 400 450 500

0

1

2


Generation

Best

valu

e

minus5

minus4

minus3

minus2

minus1

TLBOBBTLBO


0 10 20 30 40 50 60 70

0

005

01

015

02

025

03


Sample number

y(t)

minus005

ActualTLBO

BBTLBO


0 10 20 30 40 50 60 700

0002

0004

0006

0008

001

0012

0014


Sample number

y(t)

TLBOBBTLBO

(c) Error curves








119894(119894 = 1




if 119889119910 (119905) lt 0

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)

+1205964

1003816100381610038161003816119889119910 (119905)

1003816100381610038161003816) 119889119905 + 120596

3119905119903

else

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903

end

(18)




119870119868

119870119863


0 5 10 15 20 25 30 35 40 45 501

15

2

25

3

35

Generation

Best

valu

e


GAPSOBBTLBO



4≫ 1205961 119889119910(119905) = 119910(119905) minus119910(119905minus 1)



119866 (119904) =1958

1199043+ 1789119904

2+ 1033119904 + 1908

(19)


119875isin [0 1] 119870

119868isin [0 1] and 119870

119863isin


2= 0001 120596




0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

Time (s)


GAPSOBBTLBO

r in

you

t



7 Conclusion







Acknowledgments


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







lim119905rarrinfin

119883119894 (119905) =


1198881+ 1198882

(4)



119909119894119895 (119905 + 1) = 119873(

119892best + 119901best119894119895 (119905)

2

100381610038161003816100381610038161003816119892best minus 119901best119894119895 (119905)

100381610038161003816100381610038161003816) (5)




=

119873(

119892best + 119901best119894119895 (119905)

2

100381610038161003816100381610038161003816119892best minus 119901best119894119895 (119905)

100381610038161003816100381610038161003816) rand (0 1) gt 05

119901best119894119895 (119905) otherwise



= 1199011198943 119895(119905) + 1199032 sdot (1199091198941 119895

(119905) minus 1199091198942 119895(119905)) rand (0 1) gt CR

119901best1198943119895 (119905) otherwise(7)

where 1198941 1198942 and 119894









V119894119895 (119905 + 1)

=

119873(

119883best119895 (119905) + 119909119894119895 (119905)

2 rand (0 1) le CR or 119895 = 119895rand

10038161003816100381610038161003816119883best119895 (119905) minus 119909119894119895 (119905)

10038161003816100381610038161003816)

119909119894119895 (119905) otherwise(9)



V119894119895 (119905 + 1)

=


119873(

119883best119895 (119905)+119909119894119895 (119905)

210038161003816100381610038161003816119883best119895 (119905)minus119909119894119895 (119905)

10038161003816100381610038161003816) otherwise

(10)





Begin





newXi better Xi

Xi = newXi Xi = Xi

Yes

Yes

Yes

Yes

No

No

No

No


Learner phase

rand(0 1) lt 05


newXi better Xi

Xi = newXi Xi = Xi


End

gen = gen + 1

XiV1V2







NeighborhoodiXi

Ximinus1

Xi+1



119894minus119896 119883

119894 119883

119894+119896 That is






1198811119895 (119905 + 1) = 119883119894119895 (119905) + rand (0 1)


119894119895 (119905))

1198812119895 (119905 + 1) = 119873(

119873119879119890119886119888ℎ119890119903119894119895 (119905) + 119873119872119890119886119899119894119895 (119905)

2

100381610038161003816100381610038161003816100381610038161003816

119873119879119890119886119888ℎ119890119903119894119895 (119905) minus 119873119872119890119886119899119894119895 (119905)

100381610038161003816100381610038161003816100381610038161003816

)

119899119890119908119883119894119895 (119905 + 1) = 119906 sdot 1198811119895 (119905 + 1) + (1 minus 119906) sdot 1198812119895 (119905 + 1)

(11)






119894



119899119890119908119883119894119895= 119883119894119895+ 1199031lowast (119873119879119890119886119888ℎ119890119903

119894119895minus 119883119894119895)

+ 1199032lowast (119883119894119895minus 119883119896119895)

(12)












10to 11986520







119894if 119891(119899119890119908119883

119894) is better than 119891(119883

119894)





119894if 119891(119899119890119908119883119894) is better than 119891(119883

119894)


















Sphere 1198651(119909) =

119863

sum

119894=1

1199092

11989430 [minus100 100] 0

Sum square 1198652(119909) =

119863

sum

119894=1

1198941199092

11989430 [minus100 100] 0

Quadric 1198653(119909) =

119863

sum

119894=1

1198941199094

119894+ random(0 1) 30 [minus128 128] 0

Step 1198654(119909) =

119863

sum

119894=1


2 30 [minus100 100] 0

Schwefel 12 1198655(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119895)

2

30 [minus100 100] 0

Schwefel 221 1198656(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863 30 [minus100 100] 0

Schwefel 222 1198657(119909) =

119863

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816+

119863

prod

119894=1

1003816100381610038161003816119909119894

100381610038161003816100381630 [minus10 10] 0

Zakharov 1198658(119909) =

119863

sum

119894=1

1199092

119894+ (

119863

sum

119894=1

05119894119909119894)

2

+ (

119863

sum

119894=1

05119894119909119894)

4

30 [minus100 100] 0

Rosenbrock 1198659(119909) =

119863minus1

sum

119894=1

lfloor100(1199092

119894minus 119909119894+1)2

+ (119909119894minus 1)2

rfloor 30 [minus2048 2048] 0


5)radic(

1

119863)

119863

sum

119894=1

1199092

119894) minus exp(( 1

119863)

119863

sum

119894=1

cos (2120587119909119894)) + 119890 30 [minus32 32] 0

Rastrigin 11986511(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10) 30 [minus512 512] 0

Weierstrass11986512(119909) =

119863

sum

119894=1

(

119896maxsum

119896=0

[119886119896 cos (2120587119887119896 (119909

119894+ 05))]) minus 119863

119896maxsum

119896=0

[119886119896 cos (2120587119887119896 times 05)]

119886 = 05 119887 = 3 119896max = 20

30 [minus05 05] 0

Griewank 11986513(119909) =

119863

sum

119894=1

(1199092

119894

4000) minus

119899

prod

119894=1

cos(119909119894

radic119894

) + 1 30 [minus600 600] 0

Schwefel 11986514(119909) = 4189829119863 +

119863

sum

119894=1


119894)) 30 [minus500 500] 0

Bohachevsky1 11986515(119909) = 119909

2

1+ 21199092

2minus 03 cos (3120587119909

1) minus 04 cos (4120587119909

2) + 07 2 [minus100 100] 0

Bohachevsky2 11986516(119909) = 119909

2

1+ 21199092

2minus 03cos (3120587119909

1) lowast cos (4120587119909

2) + 03 2 [minus100 100] 0

Bohachevsky3 11986517(119909) = 119909

2

1+ 21199092

2minus 03cos((3120587119909

1) + (4120587119909

2)) + 03 2 [minus100 100] 0

Shekel5 11986518(119909) = minus

5

sum

119894=1

[(119909 minus 119886119894)(119909 minus 119886

119894)119879+ 119888119894]

minus1

4 [0 10] minus101532

Shekel7 11986519(119909) = minus

7

sum

119894=1

[(119909 minus 119886119894)(119909 minus 119886

119894)119879+ 119888119894]

minus1

4 [0 10] minus104029

Shekel10 11986520(119909) = minus

10

sum

119894=1

[(119909 minus 119886119894)(119909 minus 119886

119894)119879+ 119888119894]

minus1

4 [0 10] minus105364



3 1198655 1198656


11 11986512 and 119865

14 BBTLBO






1198657 1198659 11986510 and 119865




Table2Com

paris

onsm

eanplusmnstd

ofthes

olutions


Fun

BBTL

BO(119906=00)

BBTL

BO(119906=01)

BBTL

BO(119906=03)

BBTL

BO(119906=05)

BBTL

BO(119906=07)

BBTL

BO(119906=09)

BBTL

BO(119906=10)

1198651

175119890minus001plusmn121119890+000




00plusmn00

00plusmn00

00plusmn00

1198652





00plusmn00

00plusmn00

00plusmn00

1198653







199

eminus00

4plusmn113

eminus00

41198654

765119890+002plusmn583119890+002


00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

1198655

558119890+002plusmn653119890+002





216

eminus11

5plusmn110

eminus11


1198656

251119890+001plusmn534119890+000





363

eminus15

4plusmn134

eminus15


1198657






116

eminus18

8plusmn00


1198658

241119890+000plusmn307119890+000





107

eminus05

6plusmn439

eminus05


1198659

266

e+00

1plusmn179

e+00

0272119890+001plusmn317119890minus001

277119890+001plusmn318119890minus001

283119890+001plusmn278119890minus001

284119890+001plusmn267119890minus001

283119890+001plusmn341119890minus001

280119890+001plusmn387119890minus001

11986510830119890+000plusmn176119890+000



355

eminus01

5plusmn00

355

eminus01

5plusmn00

355

eminus01

5plusmn00

355

eminus01

5plusmn00

11986511374119890+001plusmn905119890+000

333119890+001plusmn118119890+001

271119890+001plusmn800119890+000

189119890+001plusmn114119890+001

573119890+000plusmn106119890+001

00plusmn00

00plusmn00

11986512815119890+000plusmn193119890+000

338119890minus001plusmn116119890+000

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00




00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986514

433

e+00

3plusmn679

e+00

2467119890+003plusmn610119890+002

517119890+003plusmn668119890+002

559119890+003plusmn685119890+002

553119890+003plusmn710119890+002

558119890+003plusmn780119890+002

540119890+003plusmn653119890+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00


e+00

1plusmn677

eminus00



e+00

1plusmn945

eminus00



e+00

1plusmn161

e+00



Table3Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

BBPS

OBB

Exp

BBDE

GBD

EMGBD

EBB

TLBO

1198651






00plusmn00

1198652

13800plusmn211119890+004

1000plusmn463119890+003


1400plusmn452119890+003


00plusmn00

1198653

132119890+000plusmn318119890+000





227eminus

004plusmn126

eminus004

1198654

560119890+000plusmn928119890+000

960119890minus001plusmn427119890+000

789119890+001plusmn305119890+002


108119890+000plusmn128119890+000

00plusmn00

1198655

124119890+004plusmn666119890+003

441119890+003plusmn337119890+003

209119890+000plusmn400119890+000

536119890+003plusmn326119890+003

757119890+002plusmn116119890+003

216eminus

115plusmn110

eminus114

1198656

167119890+001plusmn919119890+000

120119890+000plusmn522119890minus001

139119890+001plusmn447119890+000


110119890+000plusmn294119890+000

363eminus

154plusmn134

eminus153

1198657

234119890+001plusmn132119890+001

100119890+000plusmn303119890+000


600119890minus001plusmn240119890+000

200119890minus001plusmn141119890+000

116eminus

188plusmn00

1198658

187119890+002plusmn134119890+002

158119890+002plusmn700119890+001


172119890+002plusmn667119890+001

249119890+001plusmn199119890+001

107eminus

056plusmn439

eminus056

1198659

707119890+001plusmn148119890+002

357e+

001plusmn250e+

001

276119890+001plusmn106119890+001

317119890+001plusmn207119890+001

276119890+001plusmn146119890+001

283119890+001plusmn341119890minus001

11986510

106119890+001plusmn929119890+000

152119890+000plusmn511119890+000

134119890+000plusmn115119890+000

259119890+000plusmn645119890+000

554119890minus001plusmn279119890+000

355eminus

015plusmn00

11986511

116119890+002plusmn353119890+001

181119890+001plusmn728119890+000

676119890+001plusmn389119890+001

155119890+001plusmn596119890+000

203119890+001plusmn923119890+000

00plusmn00

11986512

273119890+000plusmn211119890+000


173119890+000plusmn132119890+000



00plusmn00

11986513






00plusmn00

11986514

364119890+003plusmn628119890+002

258119890+003plusmn551119890+002

230119890+003plusmn409119890+002

249119890+003plusmn541119890+002

260119890+003plusmn505119890+002

558e+

003plusmn780

e+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516


00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus560119890+000plusmn341119890+000

minus790119890+000plusmn274119890+000

minus709119890+000plusmn333119890+000

minus763119890+000plusmn286119890+000

minus801119890+000plusmn300119890+000

minus985e+

000plusmn122e+

000

11986519

minus597119890+000plusmn331119890+000

minus787119890+000plusmn303119890+000

minus621119890+000plusmn366119890+000

minus860119890+000plusmn268119890+000

minus837119890+000plusmn290119890+000

minus982e+

000plusmn178e+

000

11986520

minus581119890+000plusmn365119890+000

minus940119890+000plusmn242119890+000

minus602119890+000plusmn377119890+000

minus946e+

000plusmn224e+

000

minus938119890+000plusmn251119890+000

minus941119890+000plusmn243119890+000


0 05 1 15 2 25 3 35 4

0

50

FEs

minus200

minus150

minus100

minus50

log10

(mea

n fit

ness

)

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

(a) 1198657 Schwefel 222

0 05 1 15 2 25 3 35 4

0

10

FEs times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

minus60

minus50

minus40

minus30

minus20

minus10

log10

(mea

n fit

ness


0 05 1 15 2 25 3 35 4

0

1

2

3

FEs

minus6

minus5

minus4

minus3

minus2

minus1

log10

(mea

n fit

ness

)

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

(c) 11986518 Shekel5

0 05 1 15 2 25 3 35 4

0

FEs

Mea

n fit

ness

minus12

minus10

minus8

minus6

minus4

minus2

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10






15 11986516 and 119865




19and 11986520 and


1198657 1198658 11986510 11986511 and 119865




0 05 1 15 2 25 3 35 4

0

1

2

3

FEs

minus4

minus3

minus2

minus1

times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(a) 1198653 Quadric

0 05 1 15 2 25 3 35 41

15

2

25

3

35

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness


0 05 1 15 2 25 3 35 433

34

35

36

37

38

39

4

41

42

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(c) 11986518 Shekel5

0

0

05 1 15 2 25 3 35 4FEs

Mea

n fit

ness

times104

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO







Input x Output y


ANN

minus






1198651

1119864 minus 8 15922 100 17727 100 11042 100 19214 100 11440 100 1390 1001198652

1119864 minus 8 17515 54 19179 94 12243 100 20592 90 12634 100 1500 1001198653


1119864 minus 8 11710 24 8120 84 3634 6 7343 40 4704 34 525 1001198655



1119864 minus 8 17540 6 21191 90 17314 100 22684 94 15322 98 2144 1001198658


1119864 minus 2 17073 62 18404 42 14029 24 18182 52 17200 80 NaN 011986510

1119864 minus 8 24647 26 27598 90 18273 26 29172 82 18320 84 2110 10011986511


1119864 minus 8 NaN 0 25465 50 NaN 0 27317 64 19704 24 2471 10011986513

1119864 minus 8 16318 32 21523 58 11048 16 22951 64 14786 58 1470 10011986514


1119864 minus 8 658 100 1176 100 1274 100 1251 100 1206 100 799 10011986516

1119864 minus 8 657 98 1251 100 1294 100 1343 100 1308 100 813 10011986517

1119864 minus 8 995 100 2626 100 1487 100 2759 100 1921 100 973 10011986518

minus1015 1752 34 6720 44 2007 52 4377 32 8113 64 1684 9411986519

minus1040 2839 34 8585 48 1333 42 6724 50 3056 66 2215 9011986520

minus1053 1190 36 8928 74 1115 40 6548 76 5441 80 2822 82



119871 = (119872 + 1) lowast 119873 + (119873 + 1) lowast 119870 (13)


MSE = 1

119876 lowast 119870

119876

sum

119894=1

119870

sum

119895

1

2(119889119894119895minus 119910119894119895)2

(14)






119910 = sin (2119909) exp (minus2119909) (15)




119906 [119896] = 119870119875 sdot 119890 (119896) + 119870119868 sdot

119896

sum

119894=1


Δ119905

(16)


119875 119870119868 and 119870

119863represent the



119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903 (17)


Table5Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

jDE

SaDE

PSOcfLo

cal

PSOwFIPS

TLBO

BBTL

BO1198651






00plusmn00

1198652






00plusmn00

1198653






227eminus

004plusmn126

eminus004

1198654

211119890+001plusmn674119890+001

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

1198655

122119890+002plusmn137119890+002

428119890+001plusmn259119890+001

117119890+001plusmn930119890+000

260119890+003plusmn679119890+002


216eminus

115plusmn110

eminus114

1198656

306119890+001plusmn850119890+000

245119890+000plusmn260119890+000


266119890+000plusmn558119890minus001


363eminus

154plusmn134

eminus153

1198657






116eminus

188plusmn00

1198658

216119890+000plusmn416119890+000



586119890+001plusmn170119890+001


107eminus

056plusmn439

eminus056

1198659

249119890+001plusmn105119890+001

261119890+001plusmn107119890+000

240e+

001plusmn152

e+000

265119890+001plusmn354119890minus001

255119890+001plusmn501119890minus001

283119890+001plusmn341119890minus001

11986510






355eminus

015plusmn00

11986511

203119890+000plusmn194119890+000

386119890+000plusmn197119890+000

426119890+001plusmn106119890+001

115119890+002plusmn154119890+001

155119890+001plusmn809119890+000

00plusmn00

11986512



789119890minus001plusmn103119890+000

136119890+000plusmn741119890minus001

00plusmn00

00plusmn00

11986513





00plusmn00

00plusmn00

11986514

193119890+002plusmn142119890+002

135

e+002plusmn126e+

002

449119890+003plusmn825119890+002

396119890+003plusmn840119890+002

482119890+003plusmn686119890+002

558119890+003plusmn780119890+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus940119890+000plusmn210119890+000

minus925119890+000plusmn230119890+000

minus776119890+000plusmn342119890+000

minus979119890+000plusmn144119890+000

minus972119890+000plusmn142119890+000

minus985e+

000plusmn122e+

000

11986519

minus985119890+000plusmn190119890+000

minus987119890+000plusmn183119890+000

minus924119890+000plusmn270119890+000

minus104e+

001plusmn423eminus

009

minus922119890+000plusmn241119890+000

minus982119890+000plusmn178119890+000

11986520

minus965119890+000plusmn223119890+000

minus101119890+001plusmn159119890+000

minus963119890+000plusmn250119890+000

minus105e+

001plusmn101eminus

004

minus965119890+000plusmn223119890+000

minus941119890+000plusmn243119890+000

119908119905119897

1334

1244

1343

1244

1163

mdash


0 50 100 150 200 250 300 350 400 450 500

0

1

2


Generation

Best

valu

e

minus5

minus4

minus3

minus2

minus1

TLBOBBTLBO


0 10 20 30 40 50 60 70

0

005

01

015

02

025

03


Sample number

y(t)

minus005

ActualTLBO

BBTLBO


0 10 20 30 40 50 60 700

0002

0004

0006

0008

001

0012

0014


Sample number

y(t)

TLBOBBTLBO

(c) Error curves








119894(119894 = 1




if 119889119910 (119905) lt 0

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)

+1205964

1003816100381610038161003816119889119910 (119905)

1003816100381610038161003816) 119889119905 + 120596

3119905119903

else

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903

end

(18)




119870119868

119870119863


0 5 10 15 20 25 30 35 40 45 501

15

2

25

3

35

Generation

Best

valu

e


GAPSOBBTLBO



4≫ 1205961 119889119910(119905) = 119910(119905) minus119910(119905minus 1)



119866 (119904) =1958

1199043+ 1789119904

2+ 1033119904 + 1908

(19)


119875isin [0 1] 119870

119868isin [0 1] and 119870

119863isin


2= 0001 120596




0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

Time (s)


GAPSOBBTLBO

r in

you

t



7 Conclusion







Acknowledgments


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Begin





newXi better Xi

Xi = newXi Xi = Xi

Yes

Yes

Yes

Yes

No

No

No

No


Learner phase

rand(0 1) lt 05


newXi better Xi

Xi = newXi Xi = Xi


End

gen = gen + 1

XiV1V2







NeighborhoodiXi

Ximinus1

Xi+1



119894minus119896 119883

119894 119883

119894+119896 That is






1198811119895 (119905 + 1) = 119883119894119895 (119905) + rand (0 1)


119894119895 (119905))

1198812119895 (119905 + 1) = 119873(

119873119879119890119886119888ℎ119890119903119894119895 (119905) + 119873119872119890119886119899119894119895 (119905)

2

100381610038161003816100381610038161003816100381610038161003816

119873119879119890119886119888ℎ119890119903119894119895 (119905) minus 119873119872119890119886119899119894119895 (119905)

100381610038161003816100381610038161003816100381610038161003816

)

119899119890119908119883119894119895 (119905 + 1) = 119906 sdot 1198811119895 (119905 + 1) + (1 minus 119906) sdot 1198812119895 (119905 + 1)

(11)






119894



119899119890119908119883119894119895= 119883119894119895+ 1199031lowast (119873119879119890119886119888ℎ119890119903

119894119895minus 119883119894119895)

+ 1199032lowast (119883119894119895minus 119883119896119895)

(12)












10to 11986520







119894if 119891(119899119890119908119883

119894) is better than 119891(119883

119894)





119894if 119891(119899119890119908119883119894) is better than 119891(119883

119894)


















Sphere 1198651(119909) =

119863

sum

119894=1

1199092

11989430 [minus100 100] 0

Sum square 1198652(119909) =

119863

sum

119894=1

1198941199092

11989430 [minus100 100] 0

Quadric 1198653(119909) =

119863

sum

119894=1

1198941199094

119894+ random(0 1) 30 [minus128 128] 0

Step 1198654(119909) =

119863

sum

119894=1


2 30 [minus100 100] 0

Schwefel 12 1198655(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119895)

2

30 [minus100 100] 0

Schwefel 221 1198656(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863 30 [minus100 100] 0

Schwefel 222 1198657(119909) =

119863

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816+

119863

prod

119894=1

1003816100381610038161003816119909119894

100381610038161003816100381630 [minus10 10] 0

Zakharov 1198658(119909) =

119863

sum

119894=1

1199092

119894+ (

119863

sum

119894=1

05119894119909119894)

2

+ (

119863

sum

119894=1

05119894119909119894)

4

30 [minus100 100] 0

Rosenbrock 1198659(119909) =

119863minus1

sum

119894=1

lfloor100(1199092

119894minus 119909119894+1)2

+ (119909119894minus 1)2

rfloor 30 [minus2048 2048] 0


5)radic(

1

119863)

119863

sum

119894=1

1199092

119894) minus exp(( 1

119863)

119863

sum

119894=1

cos (2120587119909119894)) + 119890 30 [minus32 32] 0

Rastrigin 11986511(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10) 30 [minus512 512] 0

Weierstrass11986512(119909) =

119863

sum

119894=1

(

119896maxsum

119896=0

[119886119896 cos (2120587119887119896 (119909

119894+ 05))]) minus 119863

119896maxsum

119896=0

[119886119896 cos (2120587119887119896 times 05)]

119886 = 05 119887 = 3 119896max = 20

30 [minus05 05] 0

Griewank 11986513(119909) =

119863

sum

119894=1

(1199092

119894

4000) minus

119899

prod

119894=1

cos(119909119894

radic119894

) + 1 30 [minus600 600] 0

Schwefel 11986514(119909) = 4189829119863 +

119863

sum

119894=1


119894)) 30 [minus500 500] 0

Bohachevsky1 11986515(119909) = 119909

2

1+ 21199092

2minus 03 cos (3120587119909

1) minus 04 cos (4120587119909

2) + 07 2 [minus100 100] 0

Bohachevsky2 11986516(119909) = 119909

2

1+ 21199092

2minus 03cos (3120587119909

1) lowast cos (4120587119909

2) + 03 2 [minus100 100] 0

Bohachevsky3 11986517(119909) = 119909

2

1+ 21199092

2minus 03cos((3120587119909

1) + (4120587119909

2)) + 03 2 [minus100 100] 0

Shekel5 11986518(119909) = minus

5

sum

119894=1

[(119909 minus 119886119894)(119909 minus 119886

119894)119879+ 119888119894]

minus1

4 [0 10] minus101532

Shekel7 11986519(119909) = minus

7

sum

119894=1

[(119909 minus 119886119894)(119909 minus 119886

119894)119879+ 119888119894]

minus1

4 [0 10] minus104029

Shekel10 11986520(119909) = minus

10

sum

119894=1

[(119909 minus 119886119894)(119909 minus 119886

119894)119879+ 119888119894]

minus1

4 [0 10] minus105364



3 1198655 1198656


11 11986512 and 119865

14 BBTLBO






1198657 1198659 11986510 and 119865




Table2Com

paris

onsm

eanplusmnstd

ofthes

olutions


Fun

BBTL

BO(119906=00)

BBTL

BO(119906=01)

BBTL

BO(119906=03)

BBTL

BO(119906=05)

BBTL

BO(119906=07)

BBTL

BO(119906=09)

BBTL

BO(119906=10)

1198651

175119890minus001plusmn121119890+000




00plusmn00

00plusmn00

00plusmn00

1198652





00plusmn00

00plusmn00

00plusmn00

1198653







199

eminus00

4plusmn113

eminus00

41198654

765119890+002plusmn583119890+002


00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

1198655

558119890+002plusmn653119890+002





216

eminus11

5plusmn110

eminus11


1198656

251119890+001plusmn534119890+000





363

eminus15

4plusmn134

eminus15


1198657






116

eminus18

8plusmn00


1198658

241119890+000plusmn307119890+000





107

eminus05

6plusmn439

eminus05


1198659

266

e+00

1plusmn179

e+00

0272119890+001plusmn317119890minus001

277119890+001plusmn318119890minus001

283119890+001plusmn278119890minus001

284119890+001plusmn267119890minus001

283119890+001plusmn341119890minus001

280119890+001plusmn387119890minus001

11986510830119890+000plusmn176119890+000



355

eminus01

5plusmn00

355

eminus01

5plusmn00

355

eminus01

5plusmn00

355

eminus01

5plusmn00

11986511374119890+001plusmn905119890+000

333119890+001plusmn118119890+001

271119890+001plusmn800119890+000

189119890+001plusmn114119890+001

573119890+000plusmn106119890+001

00plusmn00

00plusmn00

11986512815119890+000plusmn193119890+000

338119890minus001plusmn116119890+000

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00




00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986514

433

e+00

3plusmn679

e+00

2467119890+003plusmn610119890+002

517119890+003plusmn668119890+002

559119890+003plusmn685119890+002

553119890+003plusmn710119890+002

558119890+003plusmn780119890+002

540119890+003plusmn653119890+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00


e+00

1plusmn677

eminus00



e+00

1plusmn945

eminus00



e+00

1plusmn161

e+00



Table3Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

BBPS

OBB

Exp

BBDE

GBD

EMGBD

EBB

TLBO

1198651






00plusmn00

1198652

13800plusmn211119890+004

1000plusmn463119890+003


1400plusmn452119890+003


00plusmn00

1198653

132119890+000plusmn318119890+000





227eminus

004plusmn126

eminus004

1198654

560119890+000plusmn928119890+000

960119890minus001plusmn427119890+000

789119890+001plusmn305119890+002


108119890+000plusmn128119890+000

00plusmn00

1198655

124119890+004plusmn666119890+003

441119890+003plusmn337119890+003

209119890+000plusmn400119890+000

536119890+003plusmn326119890+003

757119890+002plusmn116119890+003

216eminus

115plusmn110

eminus114

1198656

167119890+001plusmn919119890+000

120119890+000plusmn522119890minus001

139119890+001plusmn447119890+000


110119890+000plusmn294119890+000

363eminus

154plusmn134

eminus153

1198657

234119890+001plusmn132119890+001

100119890+000plusmn303119890+000


600119890minus001plusmn240119890+000

200119890minus001plusmn141119890+000

116eminus

188plusmn00

1198658

187119890+002plusmn134119890+002

158119890+002plusmn700119890+001


172119890+002plusmn667119890+001

249119890+001plusmn199119890+001

107eminus

056plusmn439

eminus056

1198659

707119890+001plusmn148119890+002

357e+

001plusmn250e+

001

276119890+001plusmn106119890+001

317119890+001plusmn207119890+001

276119890+001plusmn146119890+001

283119890+001plusmn341119890minus001

11986510

106119890+001plusmn929119890+000

152119890+000plusmn511119890+000

134119890+000plusmn115119890+000

259119890+000plusmn645119890+000

554119890minus001plusmn279119890+000

355eminus

015plusmn00

11986511

116119890+002plusmn353119890+001

181119890+001plusmn728119890+000

676119890+001plusmn389119890+001

155119890+001plusmn596119890+000

203119890+001plusmn923119890+000

00plusmn00

11986512

273119890+000plusmn211119890+000


173119890+000plusmn132119890+000



00plusmn00

11986513






00plusmn00

11986514

364119890+003plusmn628119890+002

258119890+003plusmn551119890+002

230119890+003plusmn409119890+002

249119890+003plusmn541119890+002

260119890+003plusmn505119890+002

558e+

003plusmn780

e+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516


00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus560119890+000plusmn341119890+000

minus790119890+000plusmn274119890+000

minus709119890+000plusmn333119890+000

minus763119890+000plusmn286119890+000

minus801119890+000plusmn300119890+000

minus985e+

000plusmn122e+

000

11986519

minus597119890+000plusmn331119890+000

minus787119890+000plusmn303119890+000

minus621119890+000plusmn366119890+000

minus860119890+000plusmn268119890+000

minus837119890+000plusmn290119890+000

minus982e+

000plusmn178e+

000

11986520

minus581119890+000plusmn365119890+000

minus940119890+000plusmn242119890+000

minus602119890+000plusmn377119890+000

minus946e+

000plusmn224e+

000

minus938119890+000plusmn251119890+000

minus941119890+000plusmn243119890+000


0 05 1 15 2 25 3 35 4

0

50

FEs

minus200

minus150

minus100

minus50

log10

(mea

n fit

ness

)

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

(a) 1198657 Schwefel 222

0 05 1 15 2 25 3 35 4

0

10

FEs times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

minus60

minus50

minus40

minus30

minus20

minus10

log10

(mea

n fit

ness


0 05 1 15 2 25 3 35 4

0

1

2

3

FEs

minus6

minus5

minus4

minus3

minus2

minus1

log10

(mea

n fit

ness

)

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

(c) 11986518 Shekel5

0 05 1 15 2 25 3 35 4

0

FEs

Mea

n fit

ness

minus12

minus10

minus8

minus6

minus4

minus2

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10






15 11986516 and 119865




19and 11986520 and


1198657 1198658 11986510 11986511 and 119865




0 05 1 15 2 25 3 35 4

0

1

2

3

FEs

minus4

minus3

minus2

minus1

times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(a) 1198653 Quadric

0 05 1 15 2 25 3 35 41

15

2

25

3

35

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness


0 05 1 15 2 25 3 35 433

34

35

36

37

38

39

4

41

42

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(c) 11986518 Shekel5

0

0

05 1 15 2 25 3 35 4FEs

Mea

n fit

ness

times104

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO







Input x Output y


ANN

minus






1198651

1119864 minus 8 15922 100 17727 100 11042 100 19214 100 11440 100 1390 1001198652

1119864 minus 8 17515 54 19179 94 12243 100 20592 90 12634 100 1500 1001198653


1119864 minus 8 11710 24 8120 84 3634 6 7343 40 4704 34 525 1001198655



1119864 minus 8 17540 6 21191 90 17314 100 22684 94 15322 98 2144 1001198658


1119864 minus 2 17073 62 18404 42 14029 24 18182 52 17200 80 NaN 011986510

1119864 minus 8 24647 26 27598 90 18273 26 29172 82 18320 84 2110 10011986511


1119864 minus 8 NaN 0 25465 50 NaN 0 27317 64 19704 24 2471 10011986513

1119864 minus 8 16318 32 21523 58 11048 16 22951 64 14786 58 1470 10011986514


1119864 minus 8 658 100 1176 100 1274 100 1251 100 1206 100 799 10011986516

1119864 minus 8 657 98 1251 100 1294 100 1343 100 1308 100 813 10011986517

1119864 minus 8 995 100 2626 100 1487 100 2759 100 1921 100 973 10011986518

minus1015 1752 34 6720 44 2007 52 4377 32 8113 64 1684 9411986519

minus1040 2839 34 8585 48 1333 42 6724 50 3056 66 2215 9011986520

minus1053 1190 36 8928 74 1115 40 6548 76 5441 80 2822 82



119871 = (119872 + 1) lowast 119873 + (119873 + 1) lowast 119870 (13)


MSE = 1

119876 lowast 119870

119876

sum

119894=1

119870

sum

119895

1

2(119889119894119895minus 119910119894119895)2

(14)






119910 = sin (2119909) exp (minus2119909) (15)




119906 [119896] = 119870119875 sdot 119890 (119896) + 119870119868 sdot

119896

sum

119894=1


Δ119905

(16)


119875 119870119868 and 119870

119863represent the



119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903 (17)


Table5Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

jDE

SaDE

PSOcfLo

cal

PSOwFIPS

TLBO

BBTL

BO1198651






00plusmn00

1198652






00plusmn00

1198653






227eminus

004plusmn126

eminus004

1198654

211119890+001plusmn674119890+001

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

1198655

122119890+002plusmn137119890+002

428119890+001plusmn259119890+001

117119890+001plusmn930119890+000

260119890+003plusmn679119890+002


216eminus

115plusmn110

eminus114

1198656

306119890+001plusmn850119890+000

245119890+000plusmn260119890+000


266119890+000plusmn558119890minus001


363eminus

154plusmn134

eminus153

1198657






116eminus

188plusmn00

1198658

216119890+000plusmn416119890+000



586119890+001plusmn170119890+001


107eminus

056plusmn439

eminus056

1198659

249119890+001plusmn105119890+001

261119890+001plusmn107119890+000

240e+

001plusmn152

e+000

265119890+001plusmn354119890minus001

255119890+001plusmn501119890minus001

283119890+001plusmn341119890minus001

11986510






355eminus

015plusmn00

11986511

203119890+000plusmn194119890+000

386119890+000plusmn197119890+000

426119890+001plusmn106119890+001

115119890+002plusmn154119890+001

155119890+001plusmn809119890+000

00plusmn00

11986512



789119890minus001plusmn103119890+000

136119890+000plusmn741119890minus001

00plusmn00

00plusmn00

11986513





00plusmn00

00plusmn00

11986514

193119890+002plusmn142119890+002

135

e+002plusmn126e+

002

449119890+003plusmn825119890+002

396119890+003plusmn840119890+002

482119890+003plusmn686119890+002

558119890+003plusmn780119890+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus940119890+000plusmn210119890+000

minus925119890+000plusmn230119890+000

minus776119890+000plusmn342119890+000

minus979119890+000plusmn144119890+000

minus972119890+000plusmn142119890+000

minus985e+

000plusmn122e+

000

11986519

minus985119890+000plusmn190119890+000

minus987119890+000plusmn183119890+000

minus924119890+000plusmn270119890+000

minus104e+

001plusmn423eminus

009

minus922119890+000plusmn241119890+000

minus982119890+000plusmn178119890+000

11986520

minus965119890+000plusmn223119890+000

minus101119890+001plusmn159119890+000

minus963119890+000plusmn250119890+000

minus105e+

001plusmn101eminus

004

minus965119890+000plusmn223119890+000

minus941119890+000plusmn243119890+000

119908119905119897

1334

1244

1343

1244

1163

mdash


0 50 100 150 200 250 300 350 400 450 500

0

1

2


Generation

Best

valu

e

minus5

minus4

minus3

minus2

minus1

TLBOBBTLBO


0 10 20 30 40 50 60 70

0

005

01

015

02

025

03


Sample number

y(t)

minus005

ActualTLBO

BBTLBO


0 10 20 30 40 50 60 700

0002

0004

0006

0008

001

0012

0014


Sample number

y(t)

TLBOBBTLBO

(c) Error curves








119894(119894 = 1




if 119889119910 (119905) lt 0

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)

+1205964

1003816100381610038161003816119889119910 (119905)

1003816100381610038161003816) 119889119905 + 120596

3119905119903

else

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903

end

(18)




119870119868

119870119863


0 5 10 15 20 25 30 35 40 45 501

15

2

25

3

35

Generation

Best

valu

e


GAPSOBBTLBO



4≫ 1205961 119889119910(119905) = 119910(119905) minus119910(119905minus 1)



119866 (119904) =1958

1199043+ 1789119904

2+ 1033119904 + 1908

(19)


119875isin [0 1] 119870

119868isin [0 1] and 119870

119863isin


2= 0001 120596




0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

Time (s)


GAPSOBBTLBO

r in

you

t



7 Conclusion







Acknowledgments


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




NeighborhoodiXi

Ximinus1

Xi+1



119894minus119896 119883

119894 119883

119894+119896 That is






1198811119895 (119905 + 1) = 119883119894119895 (119905) + rand (0 1)


119894119895 (119905))

1198812119895 (119905 + 1) = 119873(

119873119879119890119886119888ℎ119890119903119894119895 (119905) + 119873119872119890119886119899119894119895 (119905)

2

100381610038161003816100381610038161003816100381610038161003816

119873119879119890119886119888ℎ119890119903119894119895 (119905) minus 119873119872119890119886119899119894119895 (119905)

100381610038161003816100381610038161003816100381610038161003816

)

119899119890119908119883119894119895 (119905 + 1) = 119906 sdot 1198811119895 (119905 + 1) + (1 minus 119906) sdot 1198812119895 (119905 + 1)

(11)






119894



119899119890119908119883119894119895= 119883119894119895+ 1199031lowast (119873119879119890119886119888ℎ119890119903

119894119895minus 119883119894119895)

+ 1199032lowast (119883119894119895minus 119883119896119895)

(12)












10to 11986520







119894if 119891(119899119890119908119883

119894) is better than 119891(119883

119894)





119894if 119891(119899119890119908119883119894) is better than 119891(119883

119894)


















Sphere 1198651(119909) =

119863

sum

119894=1

1199092

11989430 [minus100 100] 0

Sum square 1198652(119909) =

119863

sum

119894=1

1198941199092

11989430 [minus100 100] 0

Quadric 1198653(119909) =

119863

sum

119894=1

1198941199094

119894+ random(0 1) 30 [minus128 128] 0

Step 1198654(119909) =

119863

sum

119894=1


2 30 [minus100 100] 0

Schwefel 12 1198655(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119895)

2

30 [minus100 100] 0

Schwefel 221 1198656(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863 30 [minus100 100] 0

Schwefel 222 1198657(119909) =

119863

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816+

119863

prod

119894=1

1003816100381610038161003816119909119894

100381610038161003816100381630 [minus10 10] 0

Zakharov 1198658(119909) =

119863

sum

119894=1

1199092

119894+ (

119863

sum

119894=1

05119894119909119894)

2

+ (

119863

sum

119894=1

05119894119909119894)

4

30 [minus100 100] 0

Rosenbrock 1198659(119909) =

119863minus1

sum

119894=1

lfloor100(1199092

119894minus 119909119894+1)2

+ (119909119894minus 1)2

rfloor 30 [minus2048 2048] 0


5)radic(

1

119863)

119863

sum

119894=1

1199092

119894) minus exp(( 1

119863)

119863

sum

119894=1

cos (2120587119909119894)) + 119890 30 [minus32 32] 0

Rastrigin 11986511(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10) 30 [minus512 512] 0

Weierstrass11986512(119909) =

119863

sum

119894=1

(

119896maxsum

119896=0

[119886119896 cos (2120587119887119896 (119909

119894+ 05))]) minus 119863

119896maxsum

119896=0

[119886119896 cos (2120587119887119896 times 05)]

119886 = 05 119887 = 3 119896max = 20

30 [minus05 05] 0

Griewank 11986513(119909) =

119863

sum

119894=1

(1199092

119894

4000) minus

119899

prod

119894=1

cos(119909119894

radic119894

) + 1 30 [minus600 600] 0

Schwefel 11986514(119909) = 4189829119863 +

119863

sum

119894=1


119894)) 30 [minus500 500] 0

Bohachevsky1 11986515(119909) = 119909

2

1+ 21199092

2minus 03 cos (3120587119909

1) minus 04 cos (4120587119909

2) + 07 2 [minus100 100] 0

Bohachevsky2 11986516(119909) = 119909

2

1+ 21199092

2minus 03cos (3120587119909

1) lowast cos (4120587119909

2) + 03 2 [minus100 100] 0

Bohachevsky3 11986517(119909) = 119909

2

1+ 21199092

2minus 03cos((3120587119909

1) + (4120587119909

2)) + 03 2 [minus100 100] 0

Shekel5 11986518(119909) = minus

5

sum

119894=1

[(119909 minus 119886119894)(119909 minus 119886

119894)119879+ 119888119894]

minus1

4 [0 10] minus101532

Shekel7 11986519(119909) = minus

7

sum

119894=1

[(119909 minus 119886119894)(119909 minus 119886

119894)119879+ 119888119894]

minus1

4 [0 10] minus104029

Shekel10 11986520(119909) = minus

10

sum

119894=1

[(119909 minus 119886119894)(119909 minus 119886

119894)119879+ 119888119894]

minus1

4 [0 10] minus105364



3 1198655 1198656


11 11986512 and 119865

14 BBTLBO






1198657 1198659 11986510 and 119865




Table2Com

paris

onsm

eanplusmnstd

ofthes

olutions


Fun

BBTL

BO(119906=00)

BBTL

BO(119906=01)

BBTL

BO(119906=03)

BBTL

BO(119906=05)

BBTL

BO(119906=07)

BBTL

BO(119906=09)

BBTL

BO(119906=10)

1198651

175119890minus001plusmn121119890+000




00plusmn00

00plusmn00

00plusmn00

1198652





00plusmn00

00plusmn00

00plusmn00

1198653







199

eminus00

4plusmn113

eminus00

41198654

765119890+002plusmn583119890+002


00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

1198655

558119890+002plusmn653119890+002





216

eminus11

5plusmn110

eminus11


1198656

251119890+001plusmn534119890+000





363

eminus15

4plusmn134

eminus15


1198657






116

eminus18

8plusmn00


1198658

241119890+000plusmn307119890+000





107

eminus05

6plusmn439

eminus05


1198659

266

e+00

1plusmn179

e+00

0272119890+001plusmn317119890minus001

277119890+001plusmn318119890minus001

283119890+001plusmn278119890minus001

284119890+001plusmn267119890minus001

283119890+001plusmn341119890minus001

280119890+001plusmn387119890minus001

11986510830119890+000plusmn176119890+000



355

eminus01

5plusmn00

355

eminus01

5plusmn00

355

eminus01

5plusmn00

355

eminus01

5plusmn00

11986511374119890+001plusmn905119890+000

333119890+001plusmn118119890+001

271119890+001plusmn800119890+000

189119890+001plusmn114119890+001

573119890+000plusmn106119890+001

00plusmn00

00plusmn00

11986512815119890+000plusmn193119890+000

338119890minus001plusmn116119890+000

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00




00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986514

433

e+00

3plusmn679

e+00

2467119890+003plusmn610119890+002

517119890+003plusmn668119890+002

559119890+003plusmn685119890+002

553119890+003plusmn710119890+002

558119890+003plusmn780119890+002

540119890+003plusmn653119890+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00


e+00

1plusmn677

eminus00



e+00

1plusmn945

eminus00



e+00

1plusmn161

e+00



Table3Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

BBPS

OBB

Exp

BBDE

GBD

EMGBD

EBB

TLBO

1198651






00plusmn00

1198652

13800plusmn211119890+004

1000plusmn463119890+003


1400plusmn452119890+003


00plusmn00

1198653

132119890+000plusmn318119890+000





227eminus

004plusmn126

eminus004

1198654

560119890+000plusmn928119890+000

960119890minus001plusmn427119890+000

789119890+001plusmn305119890+002


108119890+000plusmn128119890+000

00plusmn00

1198655

124119890+004plusmn666119890+003

441119890+003plusmn337119890+003

209119890+000plusmn400119890+000

536119890+003plusmn326119890+003

757119890+002plusmn116119890+003

216eminus

115plusmn110

eminus114

1198656

167119890+001plusmn919119890+000

120119890+000plusmn522119890minus001

139119890+001plusmn447119890+000


110119890+000plusmn294119890+000

363eminus

154plusmn134

eminus153

1198657

234119890+001plusmn132119890+001

100119890+000plusmn303119890+000


600119890minus001plusmn240119890+000

200119890minus001plusmn141119890+000

116eminus

188plusmn00

1198658

187119890+002plusmn134119890+002

158119890+002plusmn700119890+001


172119890+002plusmn667119890+001

249119890+001plusmn199119890+001

107eminus

056plusmn439

eminus056

1198659

707119890+001plusmn148119890+002

357e+

001plusmn250e+

001

276119890+001plusmn106119890+001

317119890+001plusmn207119890+001

276119890+001plusmn146119890+001

283119890+001plusmn341119890minus001

11986510

106119890+001plusmn929119890+000

152119890+000plusmn511119890+000

134119890+000plusmn115119890+000

259119890+000plusmn645119890+000

554119890minus001plusmn279119890+000

355eminus

015plusmn00

11986511

116119890+002plusmn353119890+001

181119890+001plusmn728119890+000

676119890+001plusmn389119890+001

155119890+001plusmn596119890+000

203119890+001plusmn923119890+000

00plusmn00

11986512

273119890+000plusmn211119890+000


173119890+000plusmn132119890+000



00plusmn00

11986513






00plusmn00

11986514

364119890+003plusmn628119890+002

258119890+003plusmn551119890+002

230119890+003plusmn409119890+002

249119890+003plusmn541119890+002

260119890+003plusmn505119890+002

558e+

003plusmn780

e+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516


00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus560119890+000plusmn341119890+000

minus790119890+000plusmn274119890+000

minus709119890+000plusmn333119890+000

minus763119890+000plusmn286119890+000

minus801119890+000plusmn300119890+000

minus985e+

000plusmn122e+

000

11986519

minus597119890+000plusmn331119890+000

minus787119890+000plusmn303119890+000

minus621119890+000plusmn366119890+000

minus860119890+000plusmn268119890+000

minus837119890+000plusmn290119890+000

minus982e+

000plusmn178e+

000

11986520

minus581119890+000plusmn365119890+000

minus940119890+000plusmn242119890+000

minus602119890+000plusmn377119890+000

minus946e+

000plusmn224e+

000

minus938119890+000plusmn251119890+000

minus941119890+000plusmn243119890+000


0 05 1 15 2 25 3 35 4

0

50

FEs

minus200

minus150

minus100

minus50

log10

(mea

n fit

ness

)

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

(a) 1198657 Schwefel 222

0 05 1 15 2 25 3 35 4

0

10

FEs times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

minus60

minus50

minus40

minus30

minus20

minus10

log10

(mea

n fit

ness


0 05 1 15 2 25 3 35 4

0

1

2

3

FEs

minus6

minus5

minus4

minus3

minus2

minus1

log10

(mea

n fit

ness

)

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

(c) 11986518 Shekel5

0 05 1 15 2 25 3 35 4

0

FEs

Mea

n fit

ness

minus12

minus10

minus8

minus6

minus4

minus2

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10






15 11986516 and 119865




19and 11986520 and


1198657 1198658 11986510 11986511 and 119865




0 05 1 15 2 25 3 35 4

0

1

2

3

FEs

minus4

minus3

minus2

minus1

times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(a) 1198653 Quadric

0 05 1 15 2 25 3 35 41

15

2

25

3

35

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness


0 05 1 15 2 25 3 35 433

34

35

36

37

38

39

4

41

42

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(c) 11986518 Shekel5

0

0

05 1 15 2 25 3 35 4FEs

Mea

n fit

ness

times104

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO







Input x Output y


ANN

minus






1198651

1119864 minus 8 15922 100 17727 100 11042 100 19214 100 11440 100 1390 1001198652

1119864 minus 8 17515 54 19179 94 12243 100 20592 90 12634 100 1500 1001198653


1119864 minus 8 11710 24 8120 84 3634 6 7343 40 4704 34 525 1001198655



1119864 minus 8 17540 6 21191 90 17314 100 22684 94 15322 98 2144 1001198658


1119864 minus 2 17073 62 18404 42 14029 24 18182 52 17200 80 NaN 011986510

1119864 minus 8 24647 26 27598 90 18273 26 29172 82 18320 84 2110 10011986511


1119864 minus 8 NaN 0 25465 50 NaN 0 27317 64 19704 24 2471 10011986513

1119864 minus 8 16318 32 21523 58 11048 16 22951 64 14786 58 1470 10011986514


1119864 minus 8 658 100 1176 100 1274 100 1251 100 1206 100 799 10011986516

1119864 minus 8 657 98 1251 100 1294 100 1343 100 1308 100 813 10011986517

1119864 minus 8 995 100 2626 100 1487 100 2759 100 1921 100 973 10011986518

minus1015 1752 34 6720 44 2007 52 4377 32 8113 64 1684 9411986519

minus1040 2839 34 8585 48 1333 42 6724 50 3056 66 2215 9011986520

minus1053 1190 36 8928 74 1115 40 6548 76 5441 80 2822 82



119871 = (119872 + 1) lowast 119873 + (119873 + 1) lowast 119870 (13)


MSE = 1

119876 lowast 119870

119876

sum

119894=1

119870

sum

119895

1

2(119889119894119895minus 119910119894119895)2

(14)






119910 = sin (2119909) exp (minus2119909) (15)




119906 [119896] = 119870119875 sdot 119890 (119896) + 119870119868 sdot

119896

sum

119894=1


Δ119905

(16)


119875 119870119868 and 119870

119863represent the



119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903 (17)


Table5Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

jDE

SaDE

PSOcfLo

cal

PSOwFIPS

TLBO

BBTL

BO1198651






00plusmn00

1198652






00plusmn00

1198653






227eminus

004plusmn126

eminus004

1198654

211119890+001plusmn674119890+001

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

1198655

122119890+002plusmn137119890+002

428119890+001plusmn259119890+001

117119890+001plusmn930119890+000

260119890+003plusmn679119890+002


216eminus

115plusmn110

eminus114

1198656

306119890+001plusmn850119890+000

245119890+000plusmn260119890+000


266119890+000plusmn558119890minus001


363eminus

154plusmn134

eminus153

1198657






116eminus

188plusmn00

1198658

216119890+000plusmn416119890+000



586119890+001plusmn170119890+001


107eminus

056plusmn439

eminus056

1198659

249119890+001plusmn105119890+001

261119890+001plusmn107119890+000

240e+

001plusmn152

e+000

265119890+001plusmn354119890minus001

255119890+001plusmn501119890minus001

283119890+001plusmn341119890minus001

11986510






355eminus

015plusmn00

11986511

203119890+000plusmn194119890+000

386119890+000plusmn197119890+000

426119890+001plusmn106119890+001

115119890+002plusmn154119890+001

155119890+001plusmn809119890+000

00plusmn00

11986512



789119890minus001plusmn103119890+000

136119890+000plusmn741119890minus001

00plusmn00

00plusmn00

11986513





00plusmn00

00plusmn00

11986514

193119890+002plusmn142119890+002

135

e+002plusmn126e+

002

449119890+003plusmn825119890+002

396119890+003plusmn840119890+002

482119890+003plusmn686119890+002

558119890+003plusmn780119890+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus940119890+000plusmn210119890+000

minus925119890+000plusmn230119890+000

minus776119890+000plusmn342119890+000

minus979119890+000plusmn144119890+000

minus972119890+000plusmn142119890+000

minus985e+

000plusmn122e+

000

11986519

minus985119890+000plusmn190119890+000

minus987119890+000plusmn183119890+000

minus924119890+000plusmn270119890+000

minus104e+

001plusmn423eminus

009

minus922119890+000plusmn241119890+000

minus982119890+000plusmn178119890+000

11986520

minus965119890+000plusmn223119890+000

minus101119890+001plusmn159119890+000

minus963119890+000plusmn250119890+000

minus105e+

001plusmn101eminus

004

minus965119890+000plusmn223119890+000

minus941119890+000plusmn243119890+000

119908119905119897

1334

1244

1343

1244

1163

mdash


0 50 100 150 200 250 300 350 400 450 500

0

1

2


Generation

Best

valu

e

minus5

minus4

minus3

minus2

minus1

TLBOBBTLBO


0 10 20 30 40 50 60 70

0

005

01

015

02

025

03


Sample number

y(t)

minus005

ActualTLBO

BBTLBO


0 10 20 30 40 50 60 700

0002

0004

0006

0008

001

0012

0014


Sample number

y(t)

TLBOBBTLBO

(c) Error curves








119894(119894 = 1




if 119889119910 (119905) lt 0

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)

+1205964

1003816100381610038161003816119889119910 (119905)

1003816100381610038161003816) 119889119905 + 120596

3119905119903

else

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903

end

(18)




119870119868

119870119863


0 5 10 15 20 25 30 35 40 45 501

15

2

25

3

35

Generation

Best

valu

e


GAPSOBBTLBO



4≫ 1205961 119889119910(119905) = 119910(119905) minus119910(119905minus 1)



119866 (119904) =1958

1199043+ 1789119904

2+ 1033119904 + 1908

(19)


119875isin [0 1] 119870

119868isin [0 1] and 119870

119863isin


2= 0001 120596




0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

Time (s)


GAPSOBBTLBO

r in

you

t



7 Conclusion







Acknowledgments


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







119894if 119891(119899119890119908119883

119894) is better than 119891(119883

119894)





119894if 119891(119899119890119908119883119894) is better than 119891(119883

119894)


















Sphere 1198651(119909) =

119863

sum

119894=1

1199092

11989430 [minus100 100] 0

Sum square 1198652(119909) =

119863

sum

119894=1

1198941199092

11989430 [minus100 100] 0

Quadric 1198653(119909) =

119863

sum

119894=1

1198941199094

119894+ random(0 1) 30 [minus128 128] 0

Step 1198654(119909) =

119863

sum

119894=1


2 30 [minus100 100] 0

Schwefel 12 1198655(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119895)

2

30 [minus100 100] 0

Schwefel 221 1198656(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863 30 [minus100 100] 0

Schwefel 222 1198657(119909) =

119863

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816+

119863

prod

119894=1

1003816100381610038161003816119909119894

100381610038161003816100381630 [minus10 10] 0

Zakharov 1198658(119909) =

119863

sum

119894=1

1199092

119894+ (

119863

sum

119894=1

05119894119909119894)

2

+ (

119863

sum

119894=1

05119894119909119894)

4

30 [minus100 100] 0

Rosenbrock 1198659(119909) =

119863minus1

sum

119894=1

lfloor100(1199092

119894minus 119909119894+1)2

+ (119909119894minus 1)2

rfloor 30 [minus2048 2048] 0


5)radic(

1

119863)

119863

sum

119894=1

1199092

119894) minus exp(( 1

119863)

119863

sum

119894=1

cos (2120587119909119894)) + 119890 30 [minus32 32] 0

Rastrigin 11986511(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10) 30 [minus512 512] 0

Weierstrass11986512(119909) =

119863

sum

119894=1

(

119896maxsum

119896=0

[119886119896 cos (2120587119887119896 (119909

119894+ 05))]) minus 119863

119896maxsum

119896=0

[119886119896 cos (2120587119887119896 times 05)]

119886 = 05 119887 = 3 119896max = 20

30 [minus05 05] 0

Griewank 11986513(119909) =

119863

sum

119894=1

(1199092

119894

4000) minus

119899

prod

119894=1

cos(119909119894

radic119894

) + 1 30 [minus600 600] 0

Schwefel 11986514(119909) = 4189829119863 +

119863

sum

119894=1


119894)) 30 [minus500 500] 0

Bohachevsky1 11986515(119909) = 119909

2

1+ 21199092

2minus 03 cos (3120587119909

1) minus 04 cos (4120587119909

2) + 07 2 [minus100 100] 0

Bohachevsky2 11986516(119909) = 119909

2

1+ 21199092

2minus 03cos (3120587119909

1) lowast cos (4120587119909

2) + 03 2 [minus100 100] 0

Bohachevsky3 11986517(119909) = 119909

2

1+ 21199092

2minus 03cos((3120587119909

1) + (4120587119909

2)) + 03 2 [minus100 100] 0

Shekel5 11986518(119909) = minus

5

sum

119894=1

[(119909 minus 119886119894)(119909 minus 119886

119894)119879+ 119888119894]

minus1

4 [0 10] minus101532

Shekel7 11986519(119909) = minus

7

sum

119894=1

[(119909 minus 119886119894)(119909 minus 119886

119894)119879+ 119888119894]

minus1

4 [0 10] minus104029

Shekel10 11986520(119909) = minus

10

sum

119894=1

[(119909 minus 119886119894)(119909 minus 119886

119894)119879+ 119888119894]

minus1

4 [0 10] minus105364



3 1198655 1198656


11 11986512 and 119865

14 BBTLBO






1198657 1198659 11986510 and 119865




Table2Com

paris

onsm

eanplusmnstd

ofthes

olutions


Fun

BBTL

BO(119906=00)

BBTL

BO(119906=01)

BBTL

BO(119906=03)

BBTL

BO(119906=05)

BBTL

BO(119906=07)

BBTL

BO(119906=09)

BBTL

BO(119906=10)

1198651

175119890minus001plusmn121119890+000




00plusmn00

00plusmn00

00plusmn00

1198652





00plusmn00

00plusmn00

00plusmn00

1198653







199

eminus00

4plusmn113

eminus00

41198654

765119890+002plusmn583119890+002


00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

1198655

558119890+002plusmn653119890+002





216

eminus11

5plusmn110

eminus11


1198656

251119890+001plusmn534119890+000





363

eminus15

4plusmn134

eminus15


1198657






116

eminus18

8plusmn00


1198658

241119890+000plusmn307119890+000





107

eminus05

6plusmn439

eminus05


1198659

266

e+00

1plusmn179

e+00

0272119890+001plusmn317119890minus001

277119890+001plusmn318119890minus001

283119890+001plusmn278119890minus001

284119890+001plusmn267119890minus001

283119890+001plusmn341119890minus001

280119890+001plusmn387119890minus001

11986510830119890+000plusmn176119890+000



355

eminus01

5plusmn00

355

eminus01

5plusmn00

355

eminus01

5plusmn00

355

eminus01

5plusmn00

11986511374119890+001plusmn905119890+000

333119890+001plusmn118119890+001

271119890+001plusmn800119890+000

189119890+001plusmn114119890+001

573119890+000plusmn106119890+001

00plusmn00

00plusmn00

11986512815119890+000plusmn193119890+000

338119890minus001plusmn116119890+000

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00




00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986514

433

e+00

3plusmn679

e+00

2467119890+003plusmn610119890+002

517119890+003plusmn668119890+002

559119890+003plusmn685119890+002

553119890+003plusmn710119890+002

558119890+003plusmn780119890+002

540119890+003plusmn653119890+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00


e+00

1plusmn677

eminus00



e+00

1plusmn945

eminus00



e+00

1plusmn161

e+00



Table3Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

BBPS

OBB

Exp

BBDE

GBD

EMGBD

EBB

TLBO

1198651






00plusmn00

1198652

13800plusmn211119890+004

1000plusmn463119890+003


1400plusmn452119890+003


00plusmn00

1198653

132119890+000plusmn318119890+000





227eminus

004plusmn126

eminus004

1198654

560119890+000plusmn928119890+000

960119890minus001plusmn427119890+000

789119890+001plusmn305119890+002


108119890+000plusmn128119890+000

00plusmn00

1198655

124119890+004plusmn666119890+003

441119890+003plusmn337119890+003

209119890+000plusmn400119890+000

536119890+003plusmn326119890+003

757119890+002plusmn116119890+003

216eminus

115plusmn110

eminus114

1198656

167119890+001plusmn919119890+000

120119890+000plusmn522119890minus001

139119890+001plusmn447119890+000


110119890+000plusmn294119890+000

363eminus

154plusmn134

eminus153

1198657

234119890+001plusmn132119890+001

100119890+000plusmn303119890+000


600119890minus001plusmn240119890+000

200119890minus001plusmn141119890+000

116eminus

188plusmn00

1198658

187119890+002plusmn134119890+002

158119890+002plusmn700119890+001


172119890+002plusmn667119890+001

249119890+001plusmn199119890+001

107eminus

056plusmn439

eminus056

1198659

707119890+001plusmn148119890+002

357e+

001plusmn250e+

001

276119890+001plusmn106119890+001

317119890+001plusmn207119890+001

276119890+001plusmn146119890+001

283119890+001plusmn341119890minus001

11986510

106119890+001plusmn929119890+000

152119890+000plusmn511119890+000

134119890+000plusmn115119890+000

259119890+000plusmn645119890+000

554119890minus001plusmn279119890+000

355eminus

015plusmn00

11986511

116119890+002plusmn353119890+001

181119890+001plusmn728119890+000

676119890+001plusmn389119890+001

155119890+001plusmn596119890+000

203119890+001plusmn923119890+000

00plusmn00

11986512

273119890+000plusmn211119890+000


173119890+000plusmn132119890+000



00plusmn00

11986513






00plusmn00

11986514

364119890+003plusmn628119890+002

258119890+003plusmn551119890+002

230119890+003plusmn409119890+002

249119890+003plusmn541119890+002

260119890+003plusmn505119890+002

558e+

003plusmn780

e+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516


00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus560119890+000plusmn341119890+000

minus790119890+000plusmn274119890+000

minus709119890+000plusmn333119890+000

minus763119890+000plusmn286119890+000

minus801119890+000plusmn300119890+000

minus985e+

000plusmn122e+

000

11986519

minus597119890+000plusmn331119890+000

minus787119890+000plusmn303119890+000

minus621119890+000plusmn366119890+000

minus860119890+000plusmn268119890+000

minus837119890+000plusmn290119890+000

minus982e+

000plusmn178e+

000

11986520

minus581119890+000plusmn365119890+000

minus940119890+000plusmn242119890+000

minus602119890+000plusmn377119890+000

minus946e+

000plusmn224e+

000

minus938119890+000plusmn251119890+000

minus941119890+000plusmn243119890+000


0 05 1 15 2 25 3 35 4

0

50

FEs

minus200

minus150

minus100

minus50

log10

(mea

n fit

ness

)

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

(a) 1198657 Schwefel 222

0 05 1 15 2 25 3 35 4

0

10

FEs times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

minus60

minus50

minus40

minus30

minus20

minus10

log10

(mea

n fit

ness


0 05 1 15 2 25 3 35 4

0

1

2

3

FEs

minus6

minus5

minus4

minus3

minus2

minus1

log10

(mea

n fit

ness

)

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

(c) 11986518 Shekel5

0 05 1 15 2 25 3 35 4

0

FEs

Mea

n fit

ness

minus12

minus10

minus8

minus6

minus4

minus2

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10






15 11986516 and 119865




19and 11986520 and


1198657 1198658 11986510 11986511 and 119865




0 05 1 15 2 25 3 35 4

0

1

2

3

FEs

minus4

minus3

minus2

minus1

times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(a) 1198653 Quadric

0 05 1 15 2 25 3 35 41

15

2

25

3

35

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness


0 05 1 15 2 25 3 35 433

34

35

36

37

38

39

4

41

42

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(c) 11986518 Shekel5

0

0

05 1 15 2 25 3 35 4FEs

Mea

n fit

ness

times104

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO







Input x Output y


ANN

minus






1198651

1119864 minus 8 15922 100 17727 100 11042 100 19214 100 11440 100 1390 1001198652

1119864 minus 8 17515 54 19179 94 12243 100 20592 90 12634 100 1500 1001198653


1119864 minus 8 11710 24 8120 84 3634 6 7343 40 4704 34 525 1001198655



1119864 minus 8 17540 6 21191 90 17314 100 22684 94 15322 98 2144 1001198658


1119864 minus 2 17073 62 18404 42 14029 24 18182 52 17200 80 NaN 011986510

1119864 minus 8 24647 26 27598 90 18273 26 29172 82 18320 84 2110 10011986511


1119864 minus 8 NaN 0 25465 50 NaN 0 27317 64 19704 24 2471 10011986513

1119864 minus 8 16318 32 21523 58 11048 16 22951 64 14786 58 1470 10011986514


1119864 minus 8 658 100 1176 100 1274 100 1251 100 1206 100 799 10011986516

1119864 minus 8 657 98 1251 100 1294 100 1343 100 1308 100 813 10011986517

1119864 minus 8 995 100 2626 100 1487 100 2759 100 1921 100 973 10011986518

minus1015 1752 34 6720 44 2007 52 4377 32 8113 64 1684 9411986519

minus1040 2839 34 8585 48 1333 42 6724 50 3056 66 2215 9011986520

minus1053 1190 36 8928 74 1115 40 6548 76 5441 80 2822 82



119871 = (119872 + 1) lowast 119873 + (119873 + 1) lowast 119870 (13)


MSE = 1

119876 lowast 119870

119876

sum

119894=1

119870

sum

119895

1

2(119889119894119895minus 119910119894119895)2

(14)






119910 = sin (2119909) exp (minus2119909) (15)




119906 [119896] = 119870119875 sdot 119890 (119896) + 119870119868 sdot

119896

sum

119894=1


Δ119905

(16)


119875 119870119868 and 119870

119863represent the



119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903 (17)


Table5Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

jDE

SaDE

PSOcfLo

cal

PSOwFIPS

TLBO

BBTL

BO1198651






00plusmn00

1198652






00plusmn00

1198653






227eminus

004plusmn126

eminus004

1198654

211119890+001plusmn674119890+001

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

1198655

122119890+002plusmn137119890+002

428119890+001plusmn259119890+001

117119890+001plusmn930119890+000

260119890+003plusmn679119890+002


216eminus

115plusmn110

eminus114

1198656

306119890+001plusmn850119890+000

245119890+000plusmn260119890+000


266119890+000plusmn558119890minus001


363eminus

154plusmn134

eminus153

1198657






116eminus

188plusmn00

1198658

216119890+000plusmn416119890+000



586119890+001plusmn170119890+001


107eminus

056plusmn439

eminus056

1198659

249119890+001plusmn105119890+001

261119890+001plusmn107119890+000

240e+

001plusmn152

e+000

265119890+001plusmn354119890minus001

255119890+001plusmn501119890minus001

283119890+001plusmn341119890minus001

11986510






355eminus

015plusmn00

11986511

203119890+000plusmn194119890+000

386119890+000plusmn197119890+000

426119890+001plusmn106119890+001

115119890+002plusmn154119890+001

155119890+001plusmn809119890+000

00plusmn00

11986512



789119890minus001plusmn103119890+000

136119890+000plusmn741119890minus001

00plusmn00

00plusmn00

11986513





00plusmn00

00plusmn00

11986514

193119890+002plusmn142119890+002

135

e+002plusmn126e+

002

449119890+003plusmn825119890+002

396119890+003plusmn840119890+002

482119890+003plusmn686119890+002

558119890+003plusmn780119890+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus940119890+000plusmn210119890+000

minus925119890+000plusmn230119890+000

minus776119890+000plusmn342119890+000

minus979119890+000plusmn144119890+000

minus972119890+000plusmn142119890+000

minus985e+

000plusmn122e+

000

11986519

minus985119890+000plusmn190119890+000

minus987119890+000plusmn183119890+000

minus924119890+000plusmn270119890+000

minus104e+

001plusmn423eminus

009

minus922119890+000plusmn241119890+000

minus982119890+000plusmn178119890+000

11986520

minus965119890+000plusmn223119890+000

minus101119890+001plusmn159119890+000

minus963119890+000plusmn250119890+000

minus105e+

001plusmn101eminus

004

minus965119890+000plusmn223119890+000

minus941119890+000plusmn243119890+000

119908119905119897

1334

1244

1343

1244

1163

mdash


0 50 100 150 200 250 300 350 400 450 500

0

1

2


Generation

Best

valu

e

minus5

minus4

minus3

minus2

minus1

TLBOBBTLBO


0 10 20 30 40 50 60 70

0

005

01

015

02

025

03


Sample number

y(t)

minus005

ActualTLBO

BBTLBO


0 10 20 30 40 50 60 700

0002

0004

0006

0008

001

0012

0014


Sample number

y(t)

TLBOBBTLBO

(c) Error curves








119894(119894 = 1




if 119889119910 (119905) lt 0

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)

+1205964

1003816100381610038161003816119889119910 (119905)

1003816100381610038161003816) 119889119905 + 120596

3119905119903

else

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903

end

(18)




119870119868

119870119863


0 5 10 15 20 25 30 35 40 45 501

15

2

25

3

35

Generation

Best

valu

e


GAPSOBBTLBO



4≫ 1205961 119889119910(119905) = 119910(119905) minus119910(119905minus 1)



119866 (119904) =1958

1199043+ 1789119904

2+ 1033119904 + 1908

(19)


119875isin [0 1] 119870

119868isin [0 1] and 119870

119863isin


2= 0001 120596




0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

Time (s)


GAPSOBBTLBO

r in

you

t



7 Conclusion







Acknowledgments


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






Sphere 1198651(119909) =

119863

sum

119894=1

1199092

11989430 [minus100 100] 0

Sum square 1198652(119909) =

119863

sum

119894=1

1198941199092

11989430 [minus100 100] 0

Quadric 1198653(119909) =

119863

sum

119894=1

1198941199094

119894+ random(0 1) 30 [minus128 128] 0

Step 1198654(119909) =

119863

sum

119894=1


2 30 [minus100 100] 0

Schwefel 12 1198655(119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119895)

2

30 [minus100 100] 0

Schwefel 221 1198656(119909) = max 1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863 30 [minus100 100] 0

Schwefel 222 1198657(119909) =

119863

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816+

119863

prod

119894=1

1003816100381610038161003816119909119894

100381610038161003816100381630 [minus10 10] 0

Zakharov 1198658(119909) =

119863

sum

119894=1

1199092

119894+ (

119863

sum

119894=1

05119894119909119894)

2

+ (

119863

sum

119894=1

05119894119909119894)

4

30 [minus100 100] 0

Rosenbrock 1198659(119909) =

119863minus1

sum

119894=1

lfloor100(1199092

119894minus 119909119894+1)2

+ (119909119894minus 1)2

rfloor 30 [minus2048 2048] 0


5)radic(

1

119863)

119863

sum

119894=1

1199092

119894) minus exp(( 1

119863)

119863

sum

119894=1

cos (2120587119909119894)) + 119890 30 [minus32 32] 0

Rastrigin 11986511(119909) =

119863

sum

119894=1

(1199092

119894minus 10 cos (2120587119909

119894) + 10) 30 [minus512 512] 0

Weierstrass11986512(119909) =

119863

sum

119894=1

(

119896maxsum

119896=0

[119886119896 cos (2120587119887119896 (119909

119894+ 05))]) minus 119863

119896maxsum

119896=0

[119886119896 cos (2120587119887119896 times 05)]

119886 = 05 119887 = 3 119896max = 20

30 [minus05 05] 0

Griewank 11986513(119909) =

119863

sum

119894=1

(1199092

119894

4000) minus

119899

prod

119894=1

cos(119909119894

radic119894

) + 1 30 [minus600 600] 0

Schwefel 11986514(119909) = 4189829119863 +

119863

sum

119894=1


119894)) 30 [minus500 500] 0

Bohachevsky1 11986515(119909) = 119909

2

1+ 21199092

2minus 03 cos (3120587119909

1) minus 04 cos (4120587119909

2) + 07 2 [minus100 100] 0

Bohachevsky2 11986516(119909) = 119909

2

1+ 21199092

2minus 03cos (3120587119909

1) lowast cos (4120587119909

2) + 03 2 [minus100 100] 0

Bohachevsky3 11986517(119909) = 119909

2

1+ 21199092

2minus 03cos((3120587119909

1) + (4120587119909

2)) + 03 2 [minus100 100] 0

Shekel5 11986518(119909) = minus

5

sum

119894=1

[(119909 minus 119886119894)(119909 minus 119886

119894)119879+ 119888119894]

minus1

4 [0 10] minus101532

Shekel7 11986519(119909) = minus

7

sum

119894=1

[(119909 minus 119886119894)(119909 minus 119886

119894)119879+ 119888119894]

minus1

4 [0 10] minus104029

Shekel10 11986520(119909) = minus

10

sum

119894=1

[(119909 minus 119886119894)(119909 minus 119886

119894)119879+ 119888119894]

minus1

4 [0 10] minus105364



3 1198655 1198656


11 11986512 and 119865

14 BBTLBO






1198657 1198659 11986510 and 119865




Table2Com

paris

onsm

eanplusmnstd

ofthes

olutions


Fun

BBTL

BO(119906=00)

BBTL

BO(119906=01)

BBTL

BO(119906=03)

BBTL

BO(119906=05)

BBTL

BO(119906=07)

BBTL

BO(119906=09)

BBTL

BO(119906=10)

1198651

175119890minus001plusmn121119890+000




00plusmn00

00plusmn00

00plusmn00

1198652





00plusmn00

00plusmn00

00plusmn00

1198653







199

eminus00

4plusmn113

eminus00

41198654

765119890+002plusmn583119890+002


00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

1198655

558119890+002plusmn653119890+002





216

eminus11

5plusmn110

eminus11


1198656

251119890+001plusmn534119890+000





363

eminus15

4plusmn134

eminus15


1198657






116

eminus18

8plusmn00


1198658

241119890+000plusmn307119890+000





107

eminus05

6plusmn439

eminus05


1198659

266

e+00

1plusmn179

e+00

0272119890+001plusmn317119890minus001

277119890+001plusmn318119890minus001

283119890+001plusmn278119890minus001

284119890+001plusmn267119890minus001

283119890+001plusmn341119890minus001

280119890+001plusmn387119890minus001

11986510830119890+000plusmn176119890+000



355

eminus01

5plusmn00

355

eminus01

5plusmn00

355

eminus01

5plusmn00

355

eminus01

5plusmn00

11986511374119890+001plusmn905119890+000

333119890+001plusmn118119890+001

271119890+001plusmn800119890+000

189119890+001plusmn114119890+001

573119890+000plusmn106119890+001

00plusmn00

00plusmn00

11986512815119890+000plusmn193119890+000

338119890minus001plusmn116119890+000

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00




00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986514

433

e+00

3plusmn679

e+00

2467119890+003plusmn610119890+002

517119890+003plusmn668119890+002

559119890+003plusmn685119890+002

553119890+003plusmn710119890+002

558119890+003plusmn780119890+002

540119890+003plusmn653119890+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00


e+00

1plusmn677

eminus00



e+00

1plusmn945

eminus00



e+00

1plusmn161

e+00



Table3Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

BBPS

OBB

Exp

BBDE

GBD

EMGBD

EBB

TLBO

1198651






00plusmn00

1198652

13800plusmn211119890+004

1000plusmn463119890+003


1400plusmn452119890+003


00plusmn00

1198653

132119890+000plusmn318119890+000





227eminus

004plusmn126

eminus004

1198654

560119890+000plusmn928119890+000

960119890minus001plusmn427119890+000

789119890+001plusmn305119890+002


108119890+000plusmn128119890+000

00plusmn00

1198655

124119890+004plusmn666119890+003

441119890+003plusmn337119890+003

209119890+000plusmn400119890+000

536119890+003plusmn326119890+003

757119890+002plusmn116119890+003

216eminus

115plusmn110

eminus114

1198656

167119890+001plusmn919119890+000

120119890+000plusmn522119890minus001

139119890+001plusmn447119890+000


110119890+000plusmn294119890+000

363eminus

154plusmn134

eminus153

1198657

234119890+001plusmn132119890+001

100119890+000plusmn303119890+000


600119890minus001plusmn240119890+000

200119890minus001plusmn141119890+000

116eminus

188plusmn00

1198658

187119890+002plusmn134119890+002

158119890+002plusmn700119890+001


172119890+002plusmn667119890+001

249119890+001plusmn199119890+001

107eminus

056plusmn439

eminus056

1198659

707119890+001plusmn148119890+002

357e+

001plusmn250e+

001

276119890+001plusmn106119890+001

317119890+001plusmn207119890+001

276119890+001plusmn146119890+001

283119890+001plusmn341119890minus001

11986510

106119890+001plusmn929119890+000

152119890+000plusmn511119890+000

134119890+000plusmn115119890+000

259119890+000plusmn645119890+000

554119890minus001plusmn279119890+000

355eminus

015plusmn00

11986511

116119890+002plusmn353119890+001

181119890+001plusmn728119890+000

676119890+001plusmn389119890+001

155119890+001plusmn596119890+000

203119890+001plusmn923119890+000

00plusmn00

11986512

273119890+000plusmn211119890+000


173119890+000plusmn132119890+000



00plusmn00

11986513






00plusmn00

11986514

364119890+003plusmn628119890+002

258119890+003plusmn551119890+002

230119890+003plusmn409119890+002

249119890+003plusmn541119890+002

260119890+003plusmn505119890+002

558e+

003plusmn780

e+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516


00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus560119890+000plusmn341119890+000

minus790119890+000plusmn274119890+000

minus709119890+000plusmn333119890+000

minus763119890+000plusmn286119890+000

minus801119890+000plusmn300119890+000

minus985e+

000plusmn122e+

000

11986519

minus597119890+000plusmn331119890+000

minus787119890+000plusmn303119890+000

minus621119890+000plusmn366119890+000

minus860119890+000plusmn268119890+000

minus837119890+000plusmn290119890+000

minus982e+

000plusmn178e+

000

11986520

minus581119890+000plusmn365119890+000

minus940119890+000plusmn242119890+000

minus602119890+000plusmn377119890+000

minus946e+

000plusmn224e+

000

minus938119890+000plusmn251119890+000

minus941119890+000plusmn243119890+000


0 05 1 15 2 25 3 35 4

0

50

FEs

minus200

minus150

minus100

minus50

log10

(mea

n fit

ness

)

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

(a) 1198657 Schwefel 222

0 05 1 15 2 25 3 35 4

0

10

FEs times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

minus60

minus50

minus40

minus30

minus20

minus10

log10

(mea

n fit

ness


0 05 1 15 2 25 3 35 4

0

1

2

3

FEs

minus6

minus5

minus4

minus3

minus2

minus1

log10

(mea

n fit

ness

)

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

(c) 11986518 Shekel5

0 05 1 15 2 25 3 35 4

0

FEs

Mea

n fit

ness

minus12

minus10

minus8

minus6

minus4

minus2

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10






15 11986516 and 119865




19and 11986520 and


1198657 1198658 11986510 11986511 and 119865




0 05 1 15 2 25 3 35 4

0

1

2

3

FEs

minus4

minus3

minus2

minus1

times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(a) 1198653 Quadric

0 05 1 15 2 25 3 35 41

15

2

25

3

35

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness


0 05 1 15 2 25 3 35 433

34

35

36

37

38

39

4

41

42

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(c) 11986518 Shekel5

0

0

05 1 15 2 25 3 35 4FEs

Mea

n fit

ness

times104

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO







Input x Output y


ANN

minus






1198651

1119864 minus 8 15922 100 17727 100 11042 100 19214 100 11440 100 1390 1001198652

1119864 minus 8 17515 54 19179 94 12243 100 20592 90 12634 100 1500 1001198653


1119864 minus 8 11710 24 8120 84 3634 6 7343 40 4704 34 525 1001198655



1119864 minus 8 17540 6 21191 90 17314 100 22684 94 15322 98 2144 1001198658


1119864 minus 2 17073 62 18404 42 14029 24 18182 52 17200 80 NaN 011986510

1119864 minus 8 24647 26 27598 90 18273 26 29172 82 18320 84 2110 10011986511


1119864 minus 8 NaN 0 25465 50 NaN 0 27317 64 19704 24 2471 10011986513

1119864 minus 8 16318 32 21523 58 11048 16 22951 64 14786 58 1470 10011986514


1119864 minus 8 658 100 1176 100 1274 100 1251 100 1206 100 799 10011986516

1119864 minus 8 657 98 1251 100 1294 100 1343 100 1308 100 813 10011986517

1119864 minus 8 995 100 2626 100 1487 100 2759 100 1921 100 973 10011986518

minus1015 1752 34 6720 44 2007 52 4377 32 8113 64 1684 9411986519

minus1040 2839 34 8585 48 1333 42 6724 50 3056 66 2215 9011986520

minus1053 1190 36 8928 74 1115 40 6548 76 5441 80 2822 82



119871 = (119872 + 1) lowast 119873 + (119873 + 1) lowast 119870 (13)


MSE = 1

119876 lowast 119870

119876

sum

119894=1

119870

sum

119895

1

2(119889119894119895minus 119910119894119895)2

(14)






119910 = sin (2119909) exp (minus2119909) (15)




119906 [119896] = 119870119875 sdot 119890 (119896) + 119870119868 sdot

119896

sum

119894=1


Δ119905

(16)


119875 119870119868 and 119870

119863represent the



119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903 (17)


Table5Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

jDE

SaDE

PSOcfLo

cal

PSOwFIPS

TLBO

BBTL

BO1198651






00plusmn00

1198652






00plusmn00

1198653






227eminus

004plusmn126

eminus004

1198654

211119890+001plusmn674119890+001

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

1198655

122119890+002plusmn137119890+002

428119890+001plusmn259119890+001

117119890+001plusmn930119890+000

260119890+003plusmn679119890+002


216eminus

115plusmn110

eminus114

1198656

306119890+001plusmn850119890+000

245119890+000plusmn260119890+000


266119890+000plusmn558119890minus001


363eminus

154plusmn134

eminus153

1198657






116eminus

188plusmn00

1198658

216119890+000plusmn416119890+000



586119890+001plusmn170119890+001


107eminus

056plusmn439

eminus056

1198659

249119890+001plusmn105119890+001

261119890+001plusmn107119890+000

240e+

001plusmn152

e+000

265119890+001plusmn354119890minus001

255119890+001plusmn501119890minus001

283119890+001plusmn341119890minus001

11986510






355eminus

015plusmn00

11986511

203119890+000plusmn194119890+000

386119890+000plusmn197119890+000

426119890+001plusmn106119890+001

115119890+002plusmn154119890+001

155119890+001plusmn809119890+000

00plusmn00

11986512



789119890minus001plusmn103119890+000

136119890+000plusmn741119890minus001

00plusmn00

00plusmn00

11986513





00plusmn00

00plusmn00

11986514

193119890+002plusmn142119890+002

135

e+002plusmn126e+

002

449119890+003plusmn825119890+002

396119890+003plusmn840119890+002

482119890+003plusmn686119890+002

558119890+003plusmn780119890+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus940119890+000plusmn210119890+000

minus925119890+000plusmn230119890+000

minus776119890+000plusmn342119890+000

minus979119890+000plusmn144119890+000

minus972119890+000plusmn142119890+000

minus985e+

000plusmn122e+

000

11986519

minus985119890+000plusmn190119890+000

minus987119890+000plusmn183119890+000

minus924119890+000plusmn270119890+000

minus104e+

001plusmn423eminus

009

minus922119890+000plusmn241119890+000

minus982119890+000plusmn178119890+000

11986520

minus965119890+000plusmn223119890+000

minus101119890+001plusmn159119890+000

minus963119890+000plusmn250119890+000

minus105e+

001plusmn101eminus

004

minus965119890+000plusmn223119890+000

minus941119890+000plusmn243119890+000

119908119905119897

1334

1244

1343

1244

1163

mdash


0 50 100 150 200 250 300 350 400 450 500

0

1

2


Generation

Best

valu

e

minus5

minus4

minus3

minus2

minus1

TLBOBBTLBO


0 10 20 30 40 50 60 70

0

005

01

015

02

025

03


Sample number

y(t)

minus005

ActualTLBO

BBTLBO


0 10 20 30 40 50 60 700

0002

0004

0006

0008

001

0012

0014


Sample number

y(t)

TLBOBBTLBO

(c) Error curves








119894(119894 = 1




if 119889119910 (119905) lt 0

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)

+1205964

1003816100381610038161003816119889119910 (119905)

1003816100381610038161003816) 119889119905 + 120596

3119905119903

else

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903

end

(18)




119870119868

119870119863


0 5 10 15 20 25 30 35 40 45 501

15

2

25

3

35

Generation

Best

valu

e


GAPSOBBTLBO



4≫ 1205961 119889119910(119905) = 119910(119905) minus119910(119905minus 1)



119866 (119904) =1958

1199043+ 1789119904

2+ 1033119904 + 1908

(19)


119875isin [0 1] 119870

119868isin [0 1] and 119870

119863isin


2= 0001 120596




0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

Time (s)


GAPSOBBTLBO

r in

you

t



7 Conclusion







Acknowledgments


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Table2Com

paris

onsm

eanplusmnstd

ofthes

olutions


Fun

BBTL

BO(119906=00)

BBTL

BO(119906=01)

BBTL

BO(119906=03)

BBTL

BO(119906=05)

BBTL

BO(119906=07)

BBTL

BO(119906=09)

BBTL

BO(119906=10)

1198651

175119890minus001plusmn121119890+000




00plusmn00

00plusmn00

00plusmn00

1198652





00plusmn00

00plusmn00

00plusmn00

1198653







199

eminus00

4plusmn113

eminus00

41198654

765119890+002plusmn583119890+002


00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

1198655

558119890+002plusmn653119890+002





216

eminus11

5plusmn110

eminus11


1198656

251119890+001plusmn534119890+000





363

eminus15

4plusmn134

eminus15


1198657






116

eminus18

8plusmn00


1198658

241119890+000plusmn307119890+000





107

eminus05

6plusmn439

eminus05


1198659

266

e+00

1plusmn179

e+00

0272119890+001plusmn317119890minus001

277119890+001plusmn318119890minus001

283119890+001plusmn278119890minus001

284119890+001plusmn267119890minus001

283119890+001plusmn341119890minus001

280119890+001plusmn387119890minus001

11986510830119890+000plusmn176119890+000



355

eminus01

5plusmn00

355

eminus01

5plusmn00

355

eminus01

5plusmn00

355

eminus01

5plusmn00

11986511374119890+001plusmn905119890+000

333119890+001plusmn118119890+001

271119890+001plusmn800119890+000

189119890+001plusmn114119890+001

573119890+000plusmn106119890+001

00plusmn00

00plusmn00

11986512815119890+000plusmn193119890+000

338119890minus001plusmn116119890+000

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00




00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986514

433

e+00

3plusmn679

e+00

2467119890+003plusmn610119890+002

517119890+003plusmn668119890+002

559119890+003plusmn685119890+002

553119890+003plusmn710119890+002

558119890+003plusmn780119890+002

540119890+003plusmn653119890+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00


e+00

1plusmn677

eminus00



e+00

1plusmn945

eminus00



e+00

1plusmn161

e+00



Table3Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

BBPS

OBB

Exp

BBDE

GBD

EMGBD

EBB

TLBO

1198651






00plusmn00

1198652

13800plusmn211119890+004

1000plusmn463119890+003


1400plusmn452119890+003


00plusmn00

1198653

132119890+000plusmn318119890+000





227eminus

004plusmn126

eminus004

1198654

560119890+000plusmn928119890+000

960119890minus001plusmn427119890+000

789119890+001plusmn305119890+002


108119890+000plusmn128119890+000

00plusmn00

1198655

124119890+004plusmn666119890+003

441119890+003plusmn337119890+003

209119890+000plusmn400119890+000

536119890+003plusmn326119890+003

757119890+002plusmn116119890+003

216eminus

115plusmn110

eminus114

1198656

167119890+001plusmn919119890+000

120119890+000plusmn522119890minus001

139119890+001plusmn447119890+000


110119890+000plusmn294119890+000

363eminus

154plusmn134

eminus153

1198657

234119890+001plusmn132119890+001

100119890+000plusmn303119890+000


600119890minus001plusmn240119890+000

200119890minus001plusmn141119890+000

116eminus

188plusmn00

1198658

187119890+002plusmn134119890+002

158119890+002plusmn700119890+001


172119890+002plusmn667119890+001

249119890+001plusmn199119890+001

107eminus

056plusmn439

eminus056

1198659

707119890+001plusmn148119890+002

357e+

001plusmn250e+

001

276119890+001plusmn106119890+001

317119890+001plusmn207119890+001

276119890+001plusmn146119890+001

283119890+001plusmn341119890minus001

11986510

106119890+001plusmn929119890+000

152119890+000plusmn511119890+000

134119890+000plusmn115119890+000

259119890+000plusmn645119890+000

554119890minus001plusmn279119890+000

355eminus

015plusmn00

11986511

116119890+002plusmn353119890+001

181119890+001plusmn728119890+000

676119890+001plusmn389119890+001

155119890+001plusmn596119890+000

203119890+001plusmn923119890+000

00plusmn00

11986512

273119890+000plusmn211119890+000


173119890+000plusmn132119890+000



00plusmn00

11986513






00plusmn00

11986514

364119890+003plusmn628119890+002

258119890+003plusmn551119890+002

230119890+003plusmn409119890+002

249119890+003plusmn541119890+002

260119890+003plusmn505119890+002

558e+

003plusmn780

e+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516


00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus560119890+000plusmn341119890+000

minus790119890+000plusmn274119890+000

minus709119890+000plusmn333119890+000

minus763119890+000plusmn286119890+000

minus801119890+000plusmn300119890+000

minus985e+

000plusmn122e+

000

11986519

minus597119890+000plusmn331119890+000

minus787119890+000plusmn303119890+000

minus621119890+000plusmn366119890+000

minus860119890+000plusmn268119890+000

minus837119890+000plusmn290119890+000

minus982e+

000plusmn178e+

000

11986520

minus581119890+000plusmn365119890+000

minus940119890+000plusmn242119890+000

minus602119890+000plusmn377119890+000

minus946e+

000plusmn224e+

000

minus938119890+000plusmn251119890+000

minus941119890+000plusmn243119890+000


0 05 1 15 2 25 3 35 4

0

50

FEs

minus200

minus150

minus100

minus50

log10

(mea

n fit

ness

)

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

(a) 1198657 Schwefel 222

0 05 1 15 2 25 3 35 4

0

10

FEs times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

minus60

minus50

minus40

minus30

minus20

minus10

log10

(mea

n fit

ness


0 05 1 15 2 25 3 35 4

0

1

2

3

FEs

minus6

minus5

minus4

minus3

minus2

minus1

log10

(mea

n fit

ness

)

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

(c) 11986518 Shekel5

0 05 1 15 2 25 3 35 4

0

FEs

Mea

n fit

ness

minus12

minus10

minus8

minus6

minus4

minus2

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10






15 11986516 and 119865




19and 11986520 and


1198657 1198658 11986510 11986511 and 119865




0 05 1 15 2 25 3 35 4

0

1

2

3

FEs

minus4

minus3

minus2

minus1

times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(a) 1198653 Quadric

0 05 1 15 2 25 3 35 41

15

2

25

3

35

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness


0 05 1 15 2 25 3 35 433

34

35

36

37

38

39

4

41

42

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(c) 11986518 Shekel5

0

0

05 1 15 2 25 3 35 4FEs

Mea

n fit

ness

times104

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO







Input x Output y


ANN

minus






1198651

1119864 minus 8 15922 100 17727 100 11042 100 19214 100 11440 100 1390 1001198652

1119864 minus 8 17515 54 19179 94 12243 100 20592 90 12634 100 1500 1001198653


1119864 minus 8 11710 24 8120 84 3634 6 7343 40 4704 34 525 1001198655



1119864 minus 8 17540 6 21191 90 17314 100 22684 94 15322 98 2144 1001198658


1119864 minus 2 17073 62 18404 42 14029 24 18182 52 17200 80 NaN 011986510

1119864 minus 8 24647 26 27598 90 18273 26 29172 82 18320 84 2110 10011986511


1119864 minus 8 NaN 0 25465 50 NaN 0 27317 64 19704 24 2471 10011986513

1119864 minus 8 16318 32 21523 58 11048 16 22951 64 14786 58 1470 10011986514


1119864 minus 8 658 100 1176 100 1274 100 1251 100 1206 100 799 10011986516

1119864 minus 8 657 98 1251 100 1294 100 1343 100 1308 100 813 10011986517

1119864 minus 8 995 100 2626 100 1487 100 2759 100 1921 100 973 10011986518

minus1015 1752 34 6720 44 2007 52 4377 32 8113 64 1684 9411986519

minus1040 2839 34 8585 48 1333 42 6724 50 3056 66 2215 9011986520

minus1053 1190 36 8928 74 1115 40 6548 76 5441 80 2822 82



119871 = (119872 + 1) lowast 119873 + (119873 + 1) lowast 119870 (13)


MSE = 1

119876 lowast 119870

119876

sum

119894=1

119870

sum

119895

1

2(119889119894119895minus 119910119894119895)2

(14)






119910 = sin (2119909) exp (minus2119909) (15)




119906 [119896] = 119870119875 sdot 119890 (119896) + 119870119868 sdot

119896

sum

119894=1


Δ119905

(16)


119875 119870119868 and 119870

119863represent the



119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903 (17)


Table5Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

jDE

SaDE

PSOcfLo

cal

PSOwFIPS

TLBO

BBTL

BO1198651






00plusmn00

1198652






00plusmn00

1198653






227eminus

004plusmn126

eminus004

1198654

211119890+001plusmn674119890+001

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

1198655

122119890+002plusmn137119890+002

428119890+001plusmn259119890+001

117119890+001plusmn930119890+000

260119890+003plusmn679119890+002


216eminus

115plusmn110

eminus114

1198656

306119890+001plusmn850119890+000

245119890+000plusmn260119890+000


266119890+000plusmn558119890minus001


363eminus

154plusmn134

eminus153

1198657






116eminus

188plusmn00

1198658

216119890+000plusmn416119890+000



586119890+001plusmn170119890+001


107eminus

056plusmn439

eminus056

1198659

249119890+001plusmn105119890+001

261119890+001plusmn107119890+000

240e+

001plusmn152

e+000

265119890+001plusmn354119890minus001

255119890+001plusmn501119890minus001

283119890+001plusmn341119890minus001

11986510






355eminus

015plusmn00

11986511

203119890+000plusmn194119890+000

386119890+000plusmn197119890+000

426119890+001plusmn106119890+001

115119890+002plusmn154119890+001

155119890+001plusmn809119890+000

00plusmn00

11986512



789119890minus001plusmn103119890+000

136119890+000plusmn741119890minus001

00plusmn00

00plusmn00

11986513





00plusmn00

00plusmn00

11986514

193119890+002plusmn142119890+002

135

e+002plusmn126e+

002

449119890+003plusmn825119890+002

396119890+003plusmn840119890+002

482119890+003plusmn686119890+002

558119890+003plusmn780119890+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus940119890+000plusmn210119890+000

minus925119890+000plusmn230119890+000

minus776119890+000plusmn342119890+000

minus979119890+000plusmn144119890+000

minus972119890+000plusmn142119890+000

minus985e+

000plusmn122e+

000

11986519

minus985119890+000plusmn190119890+000

minus987119890+000plusmn183119890+000

minus924119890+000plusmn270119890+000

minus104e+

001plusmn423eminus

009

minus922119890+000plusmn241119890+000

minus982119890+000plusmn178119890+000

11986520

minus965119890+000plusmn223119890+000

minus101119890+001plusmn159119890+000

minus963119890+000plusmn250119890+000

minus105e+

001plusmn101eminus

004

minus965119890+000plusmn223119890+000

minus941119890+000plusmn243119890+000

119908119905119897

1334

1244

1343

1244

1163

mdash


0 50 100 150 200 250 300 350 400 450 500

0

1

2


Generation

Best

valu

e

minus5

minus4

minus3

minus2

minus1

TLBOBBTLBO


0 10 20 30 40 50 60 70

0

005

01

015

02

025

03


Sample number

y(t)

minus005

ActualTLBO

BBTLBO


0 10 20 30 40 50 60 700

0002

0004

0006

0008

001

0012

0014


Sample number

y(t)

TLBOBBTLBO

(c) Error curves








119894(119894 = 1




if 119889119910 (119905) lt 0

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)

+1205964

1003816100381610038161003816119889119910 (119905)

1003816100381610038161003816) 119889119905 + 120596

3119905119903

else

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903

end

(18)




119870119868

119870119863


0 5 10 15 20 25 30 35 40 45 501

15

2

25

3

35

Generation

Best

valu

e


GAPSOBBTLBO



4≫ 1205961 119889119910(119905) = 119910(119905) minus119910(119905minus 1)



119866 (119904) =1958

1199043+ 1789119904

2+ 1033119904 + 1908

(19)


119875isin [0 1] 119870

119868isin [0 1] and 119870

119863isin


2= 0001 120596




0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

Time (s)


GAPSOBBTLBO

r in

you

t



7 Conclusion







Acknowledgments


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Table3Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

BBPS

OBB

Exp

BBDE

GBD

EMGBD

EBB

TLBO

1198651






00plusmn00

1198652

13800plusmn211119890+004

1000plusmn463119890+003


1400plusmn452119890+003


00plusmn00

1198653

132119890+000plusmn318119890+000





227eminus

004plusmn126

eminus004

1198654

560119890+000plusmn928119890+000

960119890minus001plusmn427119890+000

789119890+001plusmn305119890+002


108119890+000plusmn128119890+000

00plusmn00

1198655

124119890+004plusmn666119890+003

441119890+003plusmn337119890+003

209119890+000plusmn400119890+000

536119890+003plusmn326119890+003

757119890+002plusmn116119890+003

216eminus

115plusmn110

eminus114

1198656

167119890+001plusmn919119890+000

120119890+000plusmn522119890minus001

139119890+001plusmn447119890+000


110119890+000plusmn294119890+000

363eminus

154plusmn134

eminus153

1198657

234119890+001plusmn132119890+001

100119890+000plusmn303119890+000


600119890minus001plusmn240119890+000

200119890minus001plusmn141119890+000

116eminus

188plusmn00

1198658

187119890+002plusmn134119890+002

158119890+002plusmn700119890+001


172119890+002plusmn667119890+001

249119890+001plusmn199119890+001

107eminus

056plusmn439

eminus056

1198659

707119890+001plusmn148119890+002

357e+

001plusmn250e+

001

276119890+001plusmn106119890+001

317119890+001plusmn207119890+001

276119890+001plusmn146119890+001

283119890+001plusmn341119890minus001

11986510

106119890+001plusmn929119890+000

152119890+000plusmn511119890+000

134119890+000plusmn115119890+000

259119890+000plusmn645119890+000

554119890minus001plusmn279119890+000

355eminus

015plusmn00

11986511

116119890+002plusmn353119890+001

181119890+001plusmn728119890+000

676119890+001plusmn389119890+001

155119890+001plusmn596119890+000

203119890+001plusmn923119890+000

00plusmn00

11986512

273119890+000plusmn211119890+000


173119890+000plusmn132119890+000



00plusmn00

11986513






00plusmn00

11986514

364119890+003plusmn628119890+002

258119890+003plusmn551119890+002

230119890+003plusmn409119890+002

249119890+003plusmn541119890+002

260119890+003plusmn505119890+002

558e+

003plusmn780

e+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516


00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus560119890+000plusmn341119890+000

minus790119890+000plusmn274119890+000

minus709119890+000plusmn333119890+000

minus763119890+000plusmn286119890+000

minus801119890+000plusmn300119890+000

minus985e+

000plusmn122e+

000

11986519

minus597119890+000plusmn331119890+000

minus787119890+000plusmn303119890+000

minus621119890+000plusmn366119890+000

minus860119890+000plusmn268119890+000

minus837119890+000plusmn290119890+000

minus982e+

000plusmn178e+

000

11986520

minus581119890+000plusmn365119890+000

minus940119890+000plusmn242119890+000

minus602119890+000plusmn377119890+000

minus946e+

000plusmn224e+

000

minus938119890+000plusmn251119890+000

minus941119890+000plusmn243119890+000


0 05 1 15 2 25 3 35 4

0

50

FEs

minus200

minus150

minus100

minus50

log10

(mea

n fit

ness

)

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

(a) 1198657 Schwefel 222

0 05 1 15 2 25 3 35 4

0

10

FEs times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

minus60

minus50

minus40

minus30

minus20

minus10

log10

(mea

n fit

ness


0 05 1 15 2 25 3 35 4

0

1

2

3

FEs

minus6

minus5

minus4

minus3

minus2

minus1

log10

(mea

n fit

ness

)

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

(c) 11986518 Shekel5

0 05 1 15 2 25 3 35 4

0

FEs

Mea

n fit

ness

minus12

minus10

minus8

minus6

minus4

minus2

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10






15 11986516 and 119865




19and 11986520 and


1198657 1198658 11986510 11986511 and 119865




0 05 1 15 2 25 3 35 4

0

1

2

3

FEs

minus4

minus3

minus2

minus1

times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(a) 1198653 Quadric

0 05 1 15 2 25 3 35 41

15

2

25

3

35

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness


0 05 1 15 2 25 3 35 433

34

35

36

37

38

39

4

41

42

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(c) 11986518 Shekel5

0

0

05 1 15 2 25 3 35 4FEs

Mea

n fit

ness

times104

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO







Input x Output y


ANN

minus






1198651

1119864 minus 8 15922 100 17727 100 11042 100 19214 100 11440 100 1390 1001198652

1119864 minus 8 17515 54 19179 94 12243 100 20592 90 12634 100 1500 1001198653


1119864 minus 8 11710 24 8120 84 3634 6 7343 40 4704 34 525 1001198655



1119864 minus 8 17540 6 21191 90 17314 100 22684 94 15322 98 2144 1001198658


1119864 minus 2 17073 62 18404 42 14029 24 18182 52 17200 80 NaN 011986510

1119864 minus 8 24647 26 27598 90 18273 26 29172 82 18320 84 2110 10011986511


1119864 minus 8 NaN 0 25465 50 NaN 0 27317 64 19704 24 2471 10011986513

1119864 minus 8 16318 32 21523 58 11048 16 22951 64 14786 58 1470 10011986514


1119864 minus 8 658 100 1176 100 1274 100 1251 100 1206 100 799 10011986516

1119864 minus 8 657 98 1251 100 1294 100 1343 100 1308 100 813 10011986517

1119864 minus 8 995 100 2626 100 1487 100 2759 100 1921 100 973 10011986518

minus1015 1752 34 6720 44 2007 52 4377 32 8113 64 1684 9411986519

minus1040 2839 34 8585 48 1333 42 6724 50 3056 66 2215 9011986520

minus1053 1190 36 8928 74 1115 40 6548 76 5441 80 2822 82



119871 = (119872 + 1) lowast 119873 + (119873 + 1) lowast 119870 (13)


MSE = 1

119876 lowast 119870

119876

sum

119894=1

119870

sum

119895

1

2(119889119894119895minus 119910119894119895)2

(14)






119910 = sin (2119909) exp (minus2119909) (15)




119906 [119896] = 119870119875 sdot 119890 (119896) + 119870119868 sdot

119896

sum

119894=1


Δ119905

(16)


119875 119870119868 and 119870

119863represent the



119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903 (17)


Table5Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

jDE

SaDE

PSOcfLo

cal

PSOwFIPS

TLBO

BBTL

BO1198651






00plusmn00

1198652






00plusmn00

1198653






227eminus

004plusmn126

eminus004

1198654

211119890+001plusmn674119890+001

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

1198655

122119890+002plusmn137119890+002

428119890+001plusmn259119890+001

117119890+001plusmn930119890+000

260119890+003plusmn679119890+002


216eminus

115plusmn110

eminus114

1198656

306119890+001plusmn850119890+000

245119890+000plusmn260119890+000


266119890+000plusmn558119890minus001


363eminus

154plusmn134

eminus153

1198657






116eminus

188plusmn00

1198658

216119890+000plusmn416119890+000



586119890+001plusmn170119890+001


107eminus

056plusmn439

eminus056

1198659

249119890+001plusmn105119890+001

261119890+001plusmn107119890+000

240e+

001plusmn152

e+000

265119890+001plusmn354119890minus001

255119890+001plusmn501119890minus001

283119890+001plusmn341119890minus001

11986510






355eminus

015plusmn00

11986511

203119890+000plusmn194119890+000

386119890+000plusmn197119890+000

426119890+001plusmn106119890+001

115119890+002plusmn154119890+001

155119890+001plusmn809119890+000

00plusmn00

11986512



789119890minus001plusmn103119890+000

136119890+000plusmn741119890minus001

00plusmn00

00plusmn00

11986513





00plusmn00

00plusmn00

11986514

193119890+002plusmn142119890+002

135

e+002plusmn126e+

002

449119890+003plusmn825119890+002

396119890+003plusmn840119890+002

482119890+003plusmn686119890+002

558119890+003plusmn780119890+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus940119890+000plusmn210119890+000

minus925119890+000plusmn230119890+000

minus776119890+000plusmn342119890+000

minus979119890+000plusmn144119890+000

minus972119890+000plusmn142119890+000

minus985e+

000plusmn122e+

000

11986519

minus985119890+000plusmn190119890+000

minus987119890+000plusmn183119890+000

minus924119890+000plusmn270119890+000

minus104e+

001plusmn423eminus

009

minus922119890+000plusmn241119890+000

minus982119890+000plusmn178119890+000

11986520

minus965119890+000plusmn223119890+000

minus101119890+001plusmn159119890+000

minus963119890+000plusmn250119890+000

minus105e+

001plusmn101eminus

004

minus965119890+000plusmn223119890+000

minus941119890+000plusmn243119890+000

119908119905119897

1334

1244

1343

1244

1163

mdash


0 50 100 150 200 250 300 350 400 450 500

0

1

2


Generation

Best

valu

e

minus5

minus4

minus3

minus2

minus1

TLBOBBTLBO


0 10 20 30 40 50 60 70

0

005

01

015

02

025

03


Sample number

y(t)

minus005

ActualTLBO

BBTLBO


0 10 20 30 40 50 60 700

0002

0004

0006

0008

001

0012

0014


Sample number

y(t)

TLBOBBTLBO

(c) Error curves








119894(119894 = 1




if 119889119910 (119905) lt 0

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)

+1205964

1003816100381610038161003816119889119910 (119905)

1003816100381610038161003816) 119889119905 + 120596

3119905119903

else

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903

end

(18)




119870119868

119870119863


0 5 10 15 20 25 30 35 40 45 501

15

2

25

3

35

Generation

Best

valu

e


GAPSOBBTLBO



4≫ 1205961 119889119910(119905) = 119910(119905) minus119910(119905minus 1)



119866 (119904) =1958

1199043+ 1789119904

2+ 1033119904 + 1908

(19)


119875isin [0 1] 119870

119868isin [0 1] and 119870

119863isin


2= 0001 120596




0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

Time (s)


GAPSOBBTLBO

r in

you

t



7 Conclusion







Acknowledgments


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0 05 1 15 2 25 3 35 4

0

50

FEs

minus200

minus150

minus100

minus50

log10

(mea

n fit

ness

)

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

(a) 1198657 Schwefel 222

0 05 1 15 2 25 3 35 4

0

10

FEs times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

minus60

minus50

minus40

minus30

minus20

minus10

log10

(mea

n fit

ness


0 05 1 15 2 25 3 35 4

0

1

2

3

FEs

minus6

minus5

minus4

minus3

minus2

minus1

log10

(mea

n fit

ness

)

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10

(c) 11986518 Shekel5

0 05 1 15 2 25 3 35 4

0

FEs

Mea

n fit

ness

minus12

minus10

minus8

minus6

minus4

minus2

times104

u = 00

u = 01

u = 03

u = 05

u = 07

u = 09

u = 10






15 11986516 and 119865




19and 11986520 and


1198657 1198658 11986510 11986511 and 119865




0 05 1 15 2 25 3 35 4

0

1

2

3

FEs

minus4

minus3

minus2

minus1

times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(a) 1198653 Quadric

0 05 1 15 2 25 3 35 41

15

2

25

3

35

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness


0 05 1 15 2 25 3 35 433

34

35

36

37

38

39

4

41

42

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(c) 11986518 Shekel5

0

0

05 1 15 2 25 3 35 4FEs

Mea

n fit

ness

times104

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO







Input x Output y


ANN

minus






1198651

1119864 minus 8 15922 100 17727 100 11042 100 19214 100 11440 100 1390 1001198652

1119864 minus 8 17515 54 19179 94 12243 100 20592 90 12634 100 1500 1001198653


1119864 minus 8 11710 24 8120 84 3634 6 7343 40 4704 34 525 1001198655



1119864 minus 8 17540 6 21191 90 17314 100 22684 94 15322 98 2144 1001198658


1119864 minus 2 17073 62 18404 42 14029 24 18182 52 17200 80 NaN 011986510

1119864 minus 8 24647 26 27598 90 18273 26 29172 82 18320 84 2110 10011986511


1119864 minus 8 NaN 0 25465 50 NaN 0 27317 64 19704 24 2471 10011986513

1119864 minus 8 16318 32 21523 58 11048 16 22951 64 14786 58 1470 10011986514


1119864 minus 8 658 100 1176 100 1274 100 1251 100 1206 100 799 10011986516

1119864 minus 8 657 98 1251 100 1294 100 1343 100 1308 100 813 10011986517

1119864 minus 8 995 100 2626 100 1487 100 2759 100 1921 100 973 10011986518

minus1015 1752 34 6720 44 2007 52 4377 32 8113 64 1684 9411986519

minus1040 2839 34 8585 48 1333 42 6724 50 3056 66 2215 9011986520

minus1053 1190 36 8928 74 1115 40 6548 76 5441 80 2822 82



119871 = (119872 + 1) lowast 119873 + (119873 + 1) lowast 119870 (13)


MSE = 1

119876 lowast 119870

119876

sum

119894=1

119870

sum

119895

1

2(119889119894119895minus 119910119894119895)2

(14)






119910 = sin (2119909) exp (minus2119909) (15)




119906 [119896] = 119870119875 sdot 119890 (119896) + 119870119868 sdot

119896

sum

119894=1


Δ119905

(16)


119875 119870119868 and 119870

119863represent the



119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903 (17)


Table5Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

jDE

SaDE

PSOcfLo

cal

PSOwFIPS

TLBO

BBTL

BO1198651






00plusmn00

1198652






00plusmn00

1198653






227eminus

004plusmn126

eminus004

1198654

211119890+001plusmn674119890+001

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

1198655

122119890+002plusmn137119890+002

428119890+001plusmn259119890+001

117119890+001plusmn930119890+000

260119890+003plusmn679119890+002


216eminus

115plusmn110

eminus114

1198656

306119890+001plusmn850119890+000

245119890+000plusmn260119890+000


266119890+000plusmn558119890minus001


363eminus

154plusmn134

eminus153

1198657






116eminus

188plusmn00

1198658

216119890+000plusmn416119890+000



586119890+001plusmn170119890+001


107eminus

056plusmn439

eminus056

1198659

249119890+001plusmn105119890+001

261119890+001plusmn107119890+000

240e+

001plusmn152

e+000

265119890+001plusmn354119890minus001

255119890+001plusmn501119890minus001

283119890+001plusmn341119890minus001

11986510






355eminus

015plusmn00

11986511

203119890+000plusmn194119890+000

386119890+000plusmn197119890+000

426119890+001plusmn106119890+001

115119890+002plusmn154119890+001

155119890+001plusmn809119890+000

00plusmn00

11986512



789119890minus001plusmn103119890+000

136119890+000plusmn741119890minus001

00plusmn00

00plusmn00

11986513





00plusmn00

00plusmn00

11986514

193119890+002plusmn142119890+002

135

e+002plusmn126e+

002

449119890+003plusmn825119890+002

396119890+003plusmn840119890+002

482119890+003plusmn686119890+002

558119890+003plusmn780119890+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus940119890+000plusmn210119890+000

minus925119890+000plusmn230119890+000

minus776119890+000plusmn342119890+000

minus979119890+000plusmn144119890+000

minus972119890+000plusmn142119890+000

minus985e+

000plusmn122e+

000

11986519

minus985119890+000plusmn190119890+000

minus987119890+000plusmn183119890+000

minus924119890+000plusmn270119890+000

minus104e+

001plusmn423eminus

009

minus922119890+000plusmn241119890+000

minus982119890+000plusmn178119890+000

11986520

minus965119890+000plusmn223119890+000

minus101119890+001plusmn159119890+000

minus963119890+000plusmn250119890+000

minus105e+

001plusmn101eminus

004

minus965119890+000plusmn223119890+000

minus941119890+000plusmn243119890+000

119908119905119897

1334

1244

1343

1244

1163

mdash


0 50 100 150 200 250 300 350 400 450 500

0

1

2


Generation

Best

valu

e

minus5

minus4

minus3

minus2

minus1

TLBOBBTLBO


0 10 20 30 40 50 60 70

0

005

01

015

02

025

03


Sample number

y(t)

minus005

ActualTLBO

BBTLBO


0 10 20 30 40 50 60 700

0002

0004

0006

0008

001

0012

0014


Sample number

y(t)

TLBOBBTLBO

(c) Error curves








119894(119894 = 1




if 119889119910 (119905) lt 0

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)

+1205964

1003816100381610038161003816119889119910 (119905)

1003816100381610038161003816) 119889119905 + 120596

3119905119903

else

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903

end

(18)




119870119868

119870119863


0 5 10 15 20 25 30 35 40 45 501

15

2

25

3

35

Generation

Best

valu

e


GAPSOBBTLBO



4≫ 1205961 119889119910(119905) = 119910(119905) minus119910(119905minus 1)



119866 (119904) =1958

1199043+ 1789119904

2+ 1033119904 + 1908

(19)


119875isin [0 1] 119870

119868isin [0 1] and 119870

119863isin


2= 0001 120596




0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

Time (s)


GAPSOBBTLBO

r in

you

t



7 Conclusion







Acknowledgments


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0 05 1 15 2 25 3 35 4

0

1

2

3

FEs

minus4

minus3

minus2

minus1

times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(a) 1198653 Quadric

0 05 1 15 2 25 3 35 41

15

2

25

3

35

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness


0 05 1 15 2 25 3 35 433

34

35

36

37

38

39

4

41

42

FEs times104

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO

log10

(mea

n fit

ness

)

(c) 11986518 Shekel5

0

0

05 1 15 2 25 3 35 4FEs

Mea

n fit

ness

times104

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

BBPSOBBExpBBDE

GBDEMGBDEBBTLBO







Input x Output y


ANN

minus






1198651

1119864 minus 8 15922 100 17727 100 11042 100 19214 100 11440 100 1390 1001198652

1119864 minus 8 17515 54 19179 94 12243 100 20592 90 12634 100 1500 1001198653


1119864 minus 8 11710 24 8120 84 3634 6 7343 40 4704 34 525 1001198655



1119864 minus 8 17540 6 21191 90 17314 100 22684 94 15322 98 2144 1001198658


1119864 minus 2 17073 62 18404 42 14029 24 18182 52 17200 80 NaN 011986510

1119864 minus 8 24647 26 27598 90 18273 26 29172 82 18320 84 2110 10011986511


1119864 minus 8 NaN 0 25465 50 NaN 0 27317 64 19704 24 2471 10011986513

1119864 minus 8 16318 32 21523 58 11048 16 22951 64 14786 58 1470 10011986514


1119864 minus 8 658 100 1176 100 1274 100 1251 100 1206 100 799 10011986516

1119864 minus 8 657 98 1251 100 1294 100 1343 100 1308 100 813 10011986517

1119864 minus 8 995 100 2626 100 1487 100 2759 100 1921 100 973 10011986518

minus1015 1752 34 6720 44 2007 52 4377 32 8113 64 1684 9411986519

minus1040 2839 34 8585 48 1333 42 6724 50 3056 66 2215 9011986520

minus1053 1190 36 8928 74 1115 40 6548 76 5441 80 2822 82



119871 = (119872 + 1) lowast 119873 + (119873 + 1) lowast 119870 (13)


MSE = 1

119876 lowast 119870

119876

sum

119894=1

119870

sum

119895

1

2(119889119894119895minus 119910119894119895)2

(14)






119910 = sin (2119909) exp (minus2119909) (15)




119906 [119896] = 119870119875 sdot 119890 (119896) + 119870119868 sdot

119896

sum

119894=1


Δ119905

(16)


119875 119870119868 and 119870

119863represent the



119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903 (17)


Table5Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

jDE

SaDE

PSOcfLo

cal

PSOwFIPS

TLBO

BBTL

BO1198651






00plusmn00

1198652






00plusmn00

1198653






227eminus

004plusmn126

eminus004

1198654

211119890+001plusmn674119890+001

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

1198655

122119890+002plusmn137119890+002

428119890+001plusmn259119890+001

117119890+001plusmn930119890+000

260119890+003plusmn679119890+002


216eminus

115plusmn110

eminus114

1198656

306119890+001plusmn850119890+000

245119890+000plusmn260119890+000


266119890+000plusmn558119890minus001


363eminus

154plusmn134

eminus153

1198657






116eminus

188plusmn00

1198658

216119890+000plusmn416119890+000



586119890+001plusmn170119890+001


107eminus

056plusmn439

eminus056

1198659

249119890+001plusmn105119890+001

261119890+001plusmn107119890+000

240e+

001plusmn152

e+000

265119890+001plusmn354119890minus001

255119890+001plusmn501119890minus001

283119890+001plusmn341119890minus001

11986510






355eminus

015plusmn00

11986511

203119890+000plusmn194119890+000

386119890+000plusmn197119890+000

426119890+001plusmn106119890+001

115119890+002plusmn154119890+001

155119890+001plusmn809119890+000

00plusmn00

11986512



789119890minus001plusmn103119890+000

136119890+000plusmn741119890minus001

00plusmn00

00plusmn00

11986513





00plusmn00

00plusmn00

11986514

193119890+002plusmn142119890+002

135

e+002plusmn126e+

002

449119890+003plusmn825119890+002

396119890+003plusmn840119890+002

482119890+003plusmn686119890+002

558119890+003plusmn780119890+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus940119890+000plusmn210119890+000

minus925119890+000plusmn230119890+000

minus776119890+000plusmn342119890+000

minus979119890+000plusmn144119890+000

minus972119890+000plusmn142119890+000

minus985e+

000plusmn122e+

000

11986519

minus985119890+000plusmn190119890+000

minus987119890+000plusmn183119890+000

minus924119890+000plusmn270119890+000

minus104e+

001plusmn423eminus

009

minus922119890+000plusmn241119890+000

minus982119890+000plusmn178119890+000

11986520

minus965119890+000plusmn223119890+000

minus101119890+001plusmn159119890+000

minus963119890+000plusmn250119890+000

minus105e+

001plusmn101eminus

004

minus965119890+000plusmn223119890+000

minus941119890+000plusmn243119890+000

119908119905119897

1334

1244

1343

1244

1163

mdash


0 50 100 150 200 250 300 350 400 450 500

0

1

2


Generation

Best

valu

e

minus5

minus4

minus3

minus2

minus1

TLBOBBTLBO


0 10 20 30 40 50 60 70

0

005

01

015

02

025

03


Sample number

y(t)

minus005

ActualTLBO

BBTLBO


0 10 20 30 40 50 60 700

0002

0004

0006

0008

001

0012

0014


Sample number

y(t)

TLBOBBTLBO

(c) Error curves








119894(119894 = 1




if 119889119910 (119905) lt 0

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)

+1205964

1003816100381610038161003816119889119910 (119905)

1003816100381610038161003816) 119889119905 + 120596

3119905119903

else

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903

end

(18)




119870119868

119870119863


0 5 10 15 20 25 30 35 40 45 501

15

2

25

3

35

Generation

Best

valu

e


GAPSOBBTLBO



4≫ 1205961 119889119910(119905) = 119910(119905) minus119910(119905minus 1)



119866 (119904) =1958

1199043+ 1789119904

2+ 1033119904 + 1908

(19)


119875isin [0 1] 119870

119868isin [0 1] and 119870

119863isin


2= 0001 120596




0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

Time (s)


GAPSOBBTLBO

r in

you

t



7 Conclusion







Acknowledgments


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






1198651

1119864 minus 8 15922 100 17727 100 11042 100 19214 100 11440 100 1390 1001198652

1119864 minus 8 17515 54 19179 94 12243 100 20592 90 12634 100 1500 1001198653


1119864 minus 8 11710 24 8120 84 3634 6 7343 40 4704 34 525 1001198655



1119864 minus 8 17540 6 21191 90 17314 100 22684 94 15322 98 2144 1001198658


1119864 minus 2 17073 62 18404 42 14029 24 18182 52 17200 80 NaN 011986510

1119864 minus 8 24647 26 27598 90 18273 26 29172 82 18320 84 2110 10011986511


1119864 minus 8 NaN 0 25465 50 NaN 0 27317 64 19704 24 2471 10011986513

1119864 minus 8 16318 32 21523 58 11048 16 22951 64 14786 58 1470 10011986514


1119864 minus 8 658 100 1176 100 1274 100 1251 100 1206 100 799 10011986516

1119864 minus 8 657 98 1251 100 1294 100 1343 100 1308 100 813 10011986517

1119864 minus 8 995 100 2626 100 1487 100 2759 100 1921 100 973 10011986518

minus1015 1752 34 6720 44 2007 52 4377 32 8113 64 1684 9411986519

minus1040 2839 34 8585 48 1333 42 6724 50 3056 66 2215 9011986520

minus1053 1190 36 8928 74 1115 40 6548 76 5441 80 2822 82



119871 = (119872 + 1) lowast 119873 + (119873 + 1) lowast 119870 (13)


MSE = 1

119876 lowast 119870

119876

sum

119894=1

119870

sum

119895

1

2(119889119894119895minus 119910119894119895)2

(14)






119910 = sin (2119909) exp (minus2119909) (15)




119906 [119896] = 119870119875 sdot 119890 (119896) + 119870119868 sdot

119896

sum

119894=1


Δ119905

(16)


119875 119870119868 and 119870

119863represent the



119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903 (17)


Table5Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

jDE

SaDE

PSOcfLo

cal

PSOwFIPS

TLBO

BBTL

BO1198651






00plusmn00

1198652






00plusmn00

1198653






227eminus

004plusmn126

eminus004

1198654

211119890+001plusmn674119890+001

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

1198655

122119890+002plusmn137119890+002

428119890+001plusmn259119890+001

117119890+001plusmn930119890+000

260119890+003plusmn679119890+002


216eminus

115plusmn110

eminus114

1198656

306119890+001plusmn850119890+000

245119890+000plusmn260119890+000


266119890+000plusmn558119890minus001


363eminus

154plusmn134

eminus153

1198657






116eminus

188plusmn00

1198658

216119890+000plusmn416119890+000



586119890+001plusmn170119890+001


107eminus

056plusmn439

eminus056

1198659

249119890+001plusmn105119890+001

261119890+001plusmn107119890+000

240e+

001plusmn152

e+000

265119890+001plusmn354119890minus001

255119890+001plusmn501119890minus001

283119890+001plusmn341119890minus001

11986510






355eminus

015plusmn00

11986511

203119890+000plusmn194119890+000

386119890+000plusmn197119890+000

426119890+001plusmn106119890+001

115119890+002plusmn154119890+001

155119890+001plusmn809119890+000

00plusmn00

11986512



789119890minus001plusmn103119890+000

136119890+000plusmn741119890minus001

00plusmn00

00plusmn00

11986513





00plusmn00

00plusmn00

11986514

193119890+002plusmn142119890+002

135

e+002plusmn126e+

002

449119890+003plusmn825119890+002

396119890+003plusmn840119890+002

482119890+003plusmn686119890+002

558119890+003plusmn780119890+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus940119890+000plusmn210119890+000

minus925119890+000plusmn230119890+000

minus776119890+000plusmn342119890+000

minus979119890+000plusmn144119890+000

minus972119890+000plusmn142119890+000

minus985e+

000plusmn122e+

000

11986519

minus985119890+000plusmn190119890+000

minus987119890+000plusmn183119890+000

minus924119890+000plusmn270119890+000

minus104e+

001plusmn423eminus

009

minus922119890+000plusmn241119890+000

minus982119890+000plusmn178119890+000

11986520

minus965119890+000plusmn223119890+000

minus101119890+001plusmn159119890+000

minus963119890+000plusmn250119890+000

minus105e+

001plusmn101eminus

004

minus965119890+000plusmn223119890+000

minus941119890+000plusmn243119890+000

119908119905119897

1334

1244

1343

1244

1163

mdash


0 50 100 150 200 250 300 350 400 450 500

0

1

2


Generation

Best

valu

e

minus5

minus4

minus3

minus2

minus1

TLBOBBTLBO


0 10 20 30 40 50 60 70

0

005

01

015

02

025

03


Sample number

y(t)

minus005

ActualTLBO

BBTLBO


0 10 20 30 40 50 60 700

0002

0004

0006

0008

001

0012

0014


Sample number

y(t)

TLBOBBTLBO

(c) Error curves








119894(119894 = 1




if 119889119910 (119905) lt 0

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)

+1205964

1003816100381610038161003816119889119910 (119905)

1003816100381610038161003816) 119889119905 + 120596

3119905119903

else

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903

end

(18)




119870119868

119870119863


0 5 10 15 20 25 30 35 40 45 501

15

2

25

3

35

Generation

Best

valu

e


GAPSOBBTLBO



4≫ 1205961 119889119910(119905) = 119910(119905) minus119910(119905minus 1)



119866 (119904) =1958

1199043+ 1789119904

2+ 1033119904 + 1908

(19)


119875isin [0 1] 119870

119868isin [0 1] and 119870

119863isin


2= 0001 120596




0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

Time (s)


GAPSOBBTLBO

r in

you

t



7 Conclusion







Acknowledgments


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Table5Com

paris

onsm

eanplusmnstd

ofthes

olutions

usingdifferent

algorithm

s

Fun

jDE

SaDE

PSOcfLo

cal

PSOwFIPS

TLBO

BBTL

BO1198651






00plusmn00

1198652






00plusmn00

1198653






227eminus

004plusmn126

eminus004

1198654

211119890+001plusmn674119890+001

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

1198655

122119890+002plusmn137119890+002

428119890+001plusmn259119890+001

117119890+001plusmn930119890+000

260119890+003plusmn679119890+002


216eminus

115plusmn110

eminus114

1198656

306119890+001plusmn850119890+000

245119890+000plusmn260119890+000


266119890+000plusmn558119890minus001


363eminus

154plusmn134

eminus153

1198657






116eminus

188plusmn00

1198658

216119890+000plusmn416119890+000



586119890+001plusmn170119890+001


107eminus

056plusmn439

eminus056

1198659

249119890+001plusmn105119890+001

261119890+001plusmn107119890+000

240e+

001plusmn152

e+000

265119890+001plusmn354119890minus001

255119890+001plusmn501119890minus001

283119890+001plusmn341119890minus001

11986510






355eminus

015plusmn00

11986511

203119890+000plusmn194119890+000

386119890+000plusmn197119890+000

426119890+001plusmn106119890+001

115119890+002plusmn154119890+001

155119890+001plusmn809119890+000

00plusmn00

11986512



789119890minus001plusmn103119890+000

136119890+000plusmn741119890minus001

00plusmn00

00plusmn00

11986513





00plusmn00

00plusmn00

11986514

193119890+002plusmn142119890+002

135

e+002plusmn126e+

002

449119890+003plusmn825119890+002

396119890+003plusmn840119890+002

482119890+003plusmn686119890+002

558119890+003plusmn780119890+002

11986515

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986516

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986517

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

00plusmn00

11986518

minus940119890+000plusmn210119890+000

minus925119890+000plusmn230119890+000

minus776119890+000plusmn342119890+000

minus979119890+000plusmn144119890+000

minus972119890+000plusmn142119890+000

minus985e+

000plusmn122e+

000

11986519

minus985119890+000plusmn190119890+000

minus987119890+000plusmn183119890+000

minus924119890+000plusmn270119890+000

minus104e+

001plusmn423eminus

009

minus922119890+000plusmn241119890+000

minus982119890+000plusmn178119890+000

11986520

minus965119890+000plusmn223119890+000

minus101119890+001plusmn159119890+000

minus963119890+000plusmn250119890+000

minus105e+

001plusmn101eminus

004

minus965119890+000plusmn223119890+000

minus941119890+000plusmn243119890+000

119908119905119897

1334

1244

1343

1244

1163

mdash


0 50 100 150 200 250 300 350 400 450 500

0

1

2


Generation

Best

valu

e

minus5

minus4

minus3

minus2

minus1

TLBOBBTLBO


0 10 20 30 40 50 60 70

0

005

01

015

02

025

03


Sample number

y(t)

minus005

ActualTLBO

BBTLBO


0 10 20 30 40 50 60 700

0002

0004

0006

0008

001

0012

0014


Sample number

y(t)

TLBOBBTLBO

(c) Error curves








119894(119894 = 1




if 119889119910 (119905) lt 0

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)

+1205964

1003816100381610038161003816119889119910 (119905)

1003816100381610038161003816) 119889119905 + 120596

3119905119903

else

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903

end

(18)




119870119868

119870119863


0 5 10 15 20 25 30 35 40 45 501

15

2

25

3

35

Generation

Best

valu

e


GAPSOBBTLBO



4≫ 1205961 119889119910(119905) = 119910(119905) minus119910(119905minus 1)



119866 (119904) =1958

1199043+ 1789119904

2+ 1033119904 + 1908

(19)


119875isin [0 1] 119870

119868isin [0 1] and 119870

119863isin


2= 0001 120596




0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

Time (s)


GAPSOBBTLBO

r in

you

t



7 Conclusion







Acknowledgments


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0 50 100 150 200 250 300 350 400 450 500

0

1

2


Generation

Best

valu

e

minus5

minus4

minus3

minus2

minus1

TLBOBBTLBO


0 10 20 30 40 50 60 70

0

005

01

015

02

025

03


Sample number

y(t)

minus005

ActualTLBO

BBTLBO


0 10 20 30 40 50 60 700

0002

0004

0006

0008

001

0012

0014


Sample number

y(t)

TLBOBBTLBO

(c) Error curves








119894(119894 = 1




if 119889119910 (119905) lt 0

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)

+1205964

1003816100381610038161003816119889119910 (119905)

1003816100381610038161003816) 119889119905 + 120596

3119905119903

else

119891 (119905) = int

infin

0

(1205961 |119890 (119905)| + 1205962119906

2(119905)) 119889119905 + 1205963119905119903

end

(18)




119870119868

119870119863


0 5 10 15 20 25 30 35 40 45 501

15

2

25

3

35

Generation

Best

valu

e


GAPSOBBTLBO



4≫ 1205961 119889119910(119905) = 119910(119905) minus119910(119905minus 1)



119866 (119904) =1958

1199043+ 1789119904

2+ 1033119904 + 1908

(19)


119875isin [0 1] 119870

119868isin [0 1] and 119870

119863isin


2= 0001 120596




0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

Time (s)


GAPSOBBTLBO

r in

you

t



7 Conclusion







Acknowledgments


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






119870119868

119870119863


0 5 10 15 20 25 30 35 40 45 501

15

2

25

3

35

Generation

Best

valu

e


GAPSOBBTLBO



4≫ 1205961 119889119910(119905) = 119910(119905) minus119910(119905minus 1)



119866 (119904) =1958

1199043+ 1789119904

2+ 1033119904 + 1908

(19)


119875isin [0 1] 119870

119868isin [0 1] and 119870

119863isin


2= 0001 120596




0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

Time (s)


GAPSOBBTLBO

r in

you

t



7 Conclusion







Acknowledgments


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in








Acknowledgments


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in

















Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



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