Research Article Lévy-Flight Krill Herd...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 682073 14 pageshttpdxdoiorg1011552013682073

Research ArticleLeacutevy-Flight Krill Herd Algorithm

Gaige Wang12 Lihong Guo1 Amir Hossein Gandomi3 Lihua Cao1 Amir Hossein Alavi4

Hong Duan5 and Jiang Li1

1 Changchun Institute of Optics Fine Mechanics and Physics Chinese Academy of Sciences Changchun Jilin 130033 China2University of Chinese Academy of Sciences Beijing 100039 China3Department of Civil Engineering University of Akron Akron OH 443253905 USA4Department of Civil and Environmental Engineering Engineering Building Michigan State University East Lansing MI 48824 USA5 School of Computer Science and Information Technology Northeast Normal University Changchun 130117 China

Correspondence should be addressed to Lihong Guo guolhciompaccn

Received 3 November 2012 Accepted 20 December 2012

Academic Editor Siamak Talatahari

Copyright copy 2013 Gaige Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

To improve the performance of the krill herd (KH) algorithm in this paper a Levy-flight krill herd (LKH) algorithm is proposedfor solving optimization tasks within limited computing time The improvement includes the addition of a new local Levy-flight(LLF) operator during the process when updating krill in order to improve its efficiency and reliability coping with global numericaloptimization problems The LLF operator encourages the exploitation and makes the krill individuals search the space carefully atthe end of the search The elitism scheme is also applied to keep the best krill during the process when updating the krill Fourteenstandard benchmark functions are used to verify the effects of these improvements and it is illustrated that in most cases theperformance of this novel metaheuristic LKHmethod is superior to or at least highly competitive with the standard KH and otherpopulation-based optimizationmethods Especially this newmethod can accelerate the global convergence speed to the true globaloptimum while preserving the main feature of the basic KH

1 Introduction

In current competitory world human beings make anattempt at extracting the maximum output or profit from arestricted amount of usable resources In the case of engi-neering optimization such as design optimization of tallsteel buildings [1] optimum design of gravity retaining walls[2] water geotechnical and transport engineering [3] andstructural optimization and design [4 5] engineers wouldattempt to design structures that satisfy all design require-ments with the minimum possible cost Most real-worldengineering optimization problems could be converted intogeneral global optimization problemsTherefore the study ofglobal optimization is of vital importance for the engineer-ing optimization In this issue many biological intelligenttechniques [6] as optimization tools have been developedand applied to solve engineering optimization problems forengineers A general classification way for these techniques isconsidering the nature of the techniques and optimization

techniques can be classified as two main groups classicalmethods and modern intelligent algorithms Classical meth-ods such as hill climbing have a rigorous move and willgenerate the same set of solutions if the iterations start withthe same initial starting point On the other hand modernintelligent algorithms often generate different solutions evenwith the same initial value However in general the finalsolutions though slightly different will converge to the sameoptimal values within a given accuracy The emergence ofmetaheuristic optimization algorithm as a blessing from theartificial intelligence and mathematical theorem has openedup a new facet to carry out the optimization of a functionRecently nature-inspired metaheuristic algorithms performpowerfully and efficiently in solving modern nonlinearnumerical global optimization problems To some extent allmetaheuristic algorithms make an attempt at relieving theconflict between diversificationexplorationrandomization(global search) and intensificationexploitation (local search)[7 8]

2 Mathematical Problems in Engineering

Inspired by nature these strongmetaheuristic algorithmshave been proposed to solve NP-hard tasks such as UCAVpath planning [9 10] test-sheet composition [11] and param-eter estimation [12] These kinds of metaheuristic methodsperform on a population of solutions and always find optimalor suboptimal solutions During the 1960s and 1970s com-puter scientists studied the possibility of formulating evolu-tion as an optimizationmethod and eventually this generateda subset of gradient free methods namely genetic algorithms(GAs) [13 14] In the last two decades a huge number oftechniques were developed on function optimization suchas bat algorithm (BA) [15 16] differential evolution (DE)[17 18] genetic programming (GP) [19] harmony search(HS) [20 21] particle swarm optimization (PSO) [22ndash24]cuckoo search (CS) [25 26] and more recently the krillherd (KH) algorithm [27] that is based on imitating the krillherding behavior in nature

Firstly proposed by Gandomi and Alavi in 2012 inspiredby the herding behavior of krill individuals KH algorithmis a novel swarm intelligence method for optimizing pos-sibly nondifferentiable and nonlinear complex functions incontinuous space [27] In KH the time-dependent positionof the krill individuals involves three main components (i)movement led by other individuals (ii) foraging motionand (iii) random physical diffusion One of the notableadvantages of theKHalgorithm is that derivative informationis unnecessary because it uses a random search instead ofa gradient search use in classical methods Moreover com-paring with other population-based metaheuristic methodsthis new method needs few control variables in principleonly a separate parameter Δt (time interval) to tune whichmakesKHeasy to implementmore robust and fits for parallelcomputation

KH is an effective and powerful algorithm in explorationbut at times it may trap into some local optima so that itcannot implement global search well For KH the searchdepends completely on random search so there is no guar-antee for a fast convergence In order to improve KH inoptimization problems a method has been proposed [28]which introduces a more focused mutation strategy into KHto add the diversity of population

On the other hand many researchers have centralized ontheories and applications of statistical techniques especiallyof Levy distributionAnd recently huge advances are acquiredin many fields One of these fields is the applications of Levyflight in optimization methods Previously Levy flights havebeen used together with some metaheuristic optimizationmethods such as firefly algorithm [29] cuckoo search [30]krill herd algorithm [31] and particle swarm optimization[32]

Firstly presented here an effective Levy-flight KH (LKH)method is originally proposed in this paper in order toaccelerate convergence speed thus making the approachmore feasible for a wider range of real-world engineeringapplications while keeping the desirable characteristics of theoriginal KH In LKH first of all a standard KH algorithmis implemented to shrink the search apace and select a goodcandidate solution set And then for more precise modelingof the krill behavior a local Levy-flight (LLF) operator is

added to the algorithmThis operator is applied to exploit thelimited promising area intensively to get better solutions soas to improve its efficiency and reliability for solving globalnumerical optimization problems The proposed method isevaluated on fourteen standard benchmark functions thathave ever been applied to verify optimization methodsin continuous optimization problems Experimental resultsshow that the LKH performs more efficiently and effectivelythan basic KH ABC ACO BA CS DE ES GA HS PBILand PSO

The structure of this paper is organized as followsSection 2 gives a description of basic KH algorithm and Levyflight in brief Our proposed LKH method is described indetail in Section 3 Subsequently our method is evaluatedthrough fourteen benchmark functions in Section 4 Inaddition the LKH is also compared with ABC ACO BA CSDE ES GA HS KH PBIL and PSO in that section FinallySection 5 involves the conclusion and proposals for futurework

2 Preliminary

At first in this section a background on the krill herd algo-rithm and Levy flight will be provided in brief

21 Krill Herd Algorithm Krill herd (KH) [27] is a newmeta-heuristic optimization method [4] for solving optimizationtasks which is based on the simulation of the herding of thekrill swarms in response to particular biological and envi-ronmental processes The time-dependent position of anindividual krill in 2D space is decided by three main actionspresented as follows

(i) movement affected by other krill individuals(ii) foraging action(iii) random diffusion

KH algorithm adopted the following Lagrangian modelin a d-dimensional decision space as in the following (1)

119889119883119894

119889119905= 119873119894+ 119865119894+ 119863119894 (1)

where 119873119894 119865119894 and 119863

119894are the motion led by other krill

individuals the foraging motion and the physical diffusionof the 119894th krill individual respectively

In movement affected by other krill individuals thedirection of motion 120572

119894 is approximately computed by the

target effect (target swarmdensity) local effect (a local swarmdensity) and a repulsive effect (repulsive swarm density) Fora krill individual this movement can be defined as

119873new119894

= 119873max

120572119894+ 120596119899119873

old119894

(2)

and 119873max is the maximum induced speed 120596

119899is the inertia

weight of the motion induced in [0 1] and 119873old119894

is the lastmotion induced

The foraging motion is estimated by the two maincomponents One is the food location and the other one is

Mathematical Problems in Engineering 3

Table1Be

nchm

arkfunctio

ns

Num

ber

Nam

eDefinitio

nF0

1Ac

kley

119891(119909)=

20+

119890minus

20sdot119890minus02sdotradic(1119899)sum119899 119894=11199092 119894minus

119890(1119899)sum119899 119894=1cos(2120587119909119894)

F02

Fletcher-Pow

ell119891(119909)=

119899 sum 119894=1

(119860119894minus

119861119894)2119860119894=

119899 sum 119895=1

(119886119894119895sin

120572119895+

119887119894119895cos120572119895)

119861119894=

119899 sum 119895=1

(119886119894119895sin

119909119895+

119887119894119895cos119909119895)

F03

Grie

wank

119891(119909)=

119899 sum 119894=1

1199092 119894

4000

minus

119899 prod 119894=1

cos(

119909119894

radic119894)

+1

F04

Penalty

1119891

(119909)=

120587 30

10sin2(1205871199101)+

119899minus1

sum 119894=1

(119910119894minus

1)2

sdot[1+

10sin2(120587119910119894+1)]+

(119910119899minus

1)2

+

119899 sum 119894=1

119906(119909119894101004)119910119894=

1+

025(119909119894+

1)

F05

Penalty

2119891(119909)=

01

sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus

1)2

sdot[1+sin2(3120587119909119894+1)]+

(119909119899minus

1)2

[1+sin2(2120587119909119899)]

+

119899 sum 119894=1

119906(11990911989451004)

F06

Quarticwith

noise

119891(119909)=

119899 sum 119894=1

(119894sdot1199094 119894+

119880(01))

F07

Rastrig

in119891(119909)=

10sdot119899+

119899 sum 119894=1

(1199092 119894minus

10sdotcos(2120587119909119894))

F08

Rosenb

rock

119891(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus

1199092 119894)2

+(119909119894minus

1)2

]

F09

Schw

efel226

119891(119909)=

4189829times

119863minus

119863 sum 119894=1

119909119894sin

(1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 100381612

)

F10

Schw

efel12

119891(119909)=

119899 sum 119894=1

(

119894 sum 119895=1

119909119895)

2

F11

Schw

efel222

119891(119909)=

119899 sum 119894=1

|119909119894|+

119899 prod 119894=1

|119909119894|

F12

Schw

efel221

119891(119909)=max 119894

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le

119894le

119899

F13

Sphere

119891(119909)=

119899 sum 119894=1

1199092 119894

F14

Step

119891(119909)=

6sdot119899+

119899 sum 119894=1

lfloor119909119894rfloor

lowastIn

benchm

arkfunctio

nF0

2them

atrix

elem

entsa 119899times119899b 119899times119899isin(minus100100)and120572119899times1isin(minus120587120587)ared

rawfro

mun

iform

distr

ibution

lowastIn

benchm

arkfunctio

nsF0

4andF0

5thed

efinitio

nof

thefun

ction119906(119909119894119886119896119898)isas

follo

ws119906(119909119894119886119896119898)=119896(119909119894minus119886)119898119909119894gt1198860minus119886le119909119894le119886119896(minus119909119894minus119886)119898119909119894ltminus119886


the prior knowledge about the food location For the 119894th krillindividual this motion can be approximately formulated asfollows

119865119894= 119881119891120573119894+ 120596119891119865old119894

(3)

where

120573119894= 120573

food119894

+ 120573best119894

(4)

and 119881119891is the foraging speed 120596

119891is the inertia weight of the

foraging motion between 0 and 1 119865old119894

is the last foragingmotion

The random diffusion of the krill individuals can be con-sidered to be a random process in essence This motion canbe described in terms of a maximum diffusion speed and arandom directional vector It can be indicated as follows

119863119894= 119863

max120575 (5)

where 119863max is the maximum diffusion speed and 120575 is the

random directional vector and its arrays are random valuesin [minus1 1]

Based on the three above-mentioned movements usingdifferent parameters of the motion during the time theposition vector of a krill individual during the interval 119905 to119905 + Δ119905 is expressed by the following equation

119883119894 (119905 + Δ119905) = 119883

119894 (119905) + Δ119905119889119883119894

119889119905 (6)

It should be noted that Δ119905 is one of the most importantparameters and should be fine-tuned in terms of the specificreal-world engineering optimization problemThis is becausethis parameter can be treated as a scale factor of the speedvector More details about the three main motions and KHalgorithm can be found in [27]

22 Levy Flights Usually the hunt of food by animals takesplace in the form of random or quasi-random That is to sayall animals feed in a walk path from one location to anotherat random However the direction it selects relies only on amathematical model [33] One of the remarkable models iscalled Levy flights

Levy flights are a class of random walk in which the stepsare determined in terms of the step lengths and the jumps aredistributed according to a Levy distribution More recentlyLevy flights have subsequently been applied to improve andoptimize searching In the case of CS the random walkingsteps of a cuckoo are determined by a Levy flight [34]

119883119905+1

119894= 119883119905

119894+ 120573119871 (119904 120582)

119871 (119904 120582) =120582Γ (120582) sin (1205871205822)

120587

1

1199041+120582 (119904 119904

0gt 0)

(7)

Here 120573 gt 0 is the step size scaling factor which shouldbe related to the scales of the problem of interestThe randomwalk via Levy flight is more efficient in exploring the searchspace as its step length ismuch longer in the long run Some ofthe new solutions should be generated by Levy walk aroundthe best solution obtained so far this will speed up the localsearch

3 Our Approach LKH

In general the standard KH algorithm is adept at exploringthe search space and locating the promising region of globaloptimal value but it is not relatively good at exploitingsolution In order to improve the exploitation of KH a newdistribution Levy flight performing local search called localLevy-flight (LLF) operator is introduced to form a novelLevy-flight krill herd (LKH) algorithm In LKH to beginwith standard KH algorithm with high convergence speedis used to shrink the search region to a more promisingarea And then LLF operator with good exploitation abilityis applied to exploit the limited area intensively to get bettersolutions In this way the strong exploration abilities of theoriginal KH and the exploitation abilities of the LLF operatorcan be fully extracted The difference between LKH and KHis that the LLF operator is used to perform local search andfine-tune the original KH generating a new solution for eachkrill instead of random walks originally used in KH As amatter of fact according to the figuration of LKH the originalKH in LKH focuses on the explorationdiversification atthe beginning of the search to evade trapping into localoptima in a multimodal landscape while later LLF oper-ator encourages the exploitationintensification and makesthe krill individuals search the space carefully at the endof the search Therefore our proposed LKH method canfully exploit the merits of different search techniques andovercome the lack of the exploitation of the KH and solve theconflict between exploration and exploitation effectively Thedetailed explanation of our method is described as follows

To start with standard KH algorithm utilizes three mainactions to search the promising area in the solution space anduse these actions to guide the generation of the candidatesolutions for the next generation It has been demonstratedthat [27] KH performs well in both convergence speed andfinal accuracy on unimodal problems and many simple mul-timodal problems Therefore in LKH we employ the meritof the fast convergence of KH to implement global searchIn addition KH is able to shrink the search region towardsthe promising area within a few generations However some-times KHrsquos performance on complex multimodal problemsis unsatisfying accordingly another search technique withgood exploitation ability is crucial to exploit the limited areacarefully to get optimal solutions

To improve the exploitation ability of the KH algorithmgenetic reproduction mechanisms have been incorporatedinto the standard KH algorithm Gandomi and Alavi haveproved that the KH II (KH with crossover operator only)performs the best among serials of KH methods [27] In ourpresent work we use a more focused local search techniquelocal Levy-flight (LLF) operator in the local search partof the LKH algorithm which can increase diversity of thepopulation in an attempt to avoid premature convergence andexploit a small region in the later run phase to refine the finalsolutions The main step of LLF operator used in the LKHalgorithm is presented in Algorithm 1

Here 119905 isin [0 119905max] and 119905max is the maximum of gen-erations 119889 is the number of decision variables NP is thesize of the parent population A is max Levy-flight step size


Begin119886 = 119860(119905

and2) Smaller step for local walk

NoSteps = lceil119889 lowast exprnd (2 lowast 119905max)rceilDetermine the next step size 119889119909 by performing Levy flight as shown in Section 22119903 = lceilNP lowast randrceilfor 119895 = 1 to d do

if rand le 05 Random number to determine direction119881119894(119895) = 119886 times 119889119909(119895) + 119883NPminus119903+1(119895)

else119881119894(119895) = 119886 times 119889119909(119895) minus 119883NPminus119903+1(119895)

end ifend for jEvaluate the offspring 119881

119894

if 119881119894is better than 119883

119894then

119883119894= 119881119894

end ifEnd

Algorithm 1 Local Levy-flight (LLF) operator

119883119894(119895) is the jth variable of the solution 119883

119894 119881119894is the offspring

lceil119889lowast exprnd(2 lowast 119905max)rceil is a random integer number between 1and119889lowastexprnd(2lowast119905max) drawn from exponential distributionexprnd(2lowast119905max) returns an array of random numbers chosenfrom the exponential distribution with mean parameter 2 lowast

119905max Similarly lceil119889 lowast randrceil is a random integer numberbetween 1 and d drawn from uniform distribution Andrand is a random real number in interval (0 1) drawn fromuniform distribution

In addition another important improvement is the addi-tion of elitism strategy into the LKH Clearly KH has somefundamental elitism However it can be further improvedAs with other population-based optimization algorithms wecombine some sort of elitism so as to store the optimalsolutions in the population Here we use a more centralizedelitismon the best solutions which can stop the best solutionsfrom being ruined by three motions and LLF operator inLKH In the main cycle of the LKH to start with theKEEP best solutions are retained in a variable KEEPKRILLGenerally speaking theKEEP worst solutions are substitutedby the KEEP best solutions at the end of the every iterationThere is a guarantee that this elitism strategy can make thewhole population not decline to the population with worsefitness than the former Note that we use an elitism strategyto save the property of the krill that has the best fitness inthe LKH process so even if three motions and LLF operatorcorrupt their corresponding krill we have retained it and canrecuperate to its preceding good status if needed

Based on the above analyses themain steps of Levy-flightkrill herd method can be simply presented in Algorithm 2

4 Simulation Experiments

In this section the performance of our proposed methodLKH is tested to global numerical optimization through aseries of experiments implemented in benchmark functions

To allow an unprejudiced comparison of CPU time allthe experiments were carried out on a PC with a Pentium IV

processor running at 20GHz 512 MB of RAM and a harddrive of 160GBOur executionwas compiled usingMATLABR2012b (80) running under Windows XP3 No commercialKH or other optimization tools were used in our simulationexperiments

Well-defined problem sets benefit for testing the perfor-mance of optimization algorithms proposed in this paperBased on numerical functions benchmark functions can beconsidered as objective functions to fulfill such tests In ourpresent study fourteen different benchmark functions areapplied to test our proposed metaheuristic LKH methodThe formulation of these benchmark functions are given inTable 1 and the properties of these benchmark functionsare presented in Table 2 More details of all the bench-mark functions can be found in [35 36] We must pointout that in [35] Yao et al have used 23 benchmarks totest optimization algorithms However for the other low-dimensional benchmark functions (such as 119889 = 2 4 and 6)all the methods perform almost identically with each other[37] because these low-dimensional benchmarks are toosimple to clarify the performance difference among differentmethodsTherefore in our present work only fourteen high-dimensional complex benchmarks are applied to verify ourproposed LKH algorithm

41 General Performance of LKH In order to explore themerits of LKH in this section we compared its perfor-mance on global numeric optimization problems with elevenpopulation-based optimization methods which are ABCACO BA CS DE ES GA HS KH PBIL and PSO ABC(artificial bee colony) [38] is an intelligent optimizationalgorithm based on the smart behavior of honey bee swarmACO (ant colony optimization) [39] is a swarm intelligencealgorithm for solving optimization problems which is basedon the pheromone deposition of ants BA (bat algorithm)[16] is a new powerful metaheuristic optimization methodinspired by the echolocation behavior of bats with varying


BeginStep 1 Initialization Set the generation counter 119905 = 1 initialize the population 119875 of NP

krill individuals randomly and each krill corresponds to a potential solution tothe given problem set the foraging speed 119881

119891 the maximum diffusion speed

119863max and the maximum induced speed 119873

max set max Levy flight step size 119860

and elitism parameterKEEP how many of the best krill to keep from one generation to the next

Step 2 Fitness evaluation Evaluate each krill individual according to its positionStep 3 While the termination criteria is not satisfied or t lt MaxGeneration do

Sort the populationkrill from best to worstStore the KEEP best krill as KEEPKRILLfor 119894 = 1 NP (all krill) do

Perform the following motion calculationMotion induced by the presence of other individualsForaging motionPhysical diffusion

Update the krill individual position in the search space by (6)Fine-tune 119883

119905+1by performing LLF operator as shown in Algorithm 1

Evaluate each krill individual according to its new position 119883119905+1

end for 119894

Replace the KEEP worst krill with the KEEP best krill stored in KEEPKRILLSort the populationkrill from best to worst and find the current best119905 = 119905 + 1

Step 4 end whileStep 5 Post-processing the results and visualization

End

Algorithm 2 Levy-flight krill herd algorithm

pulse rates of emission and loudness CS (cuckoo search)[40] is ametaheuristic optimization algorithm inspired by theobligate brood parasitism of some cuckoo species by layingtheir eggs in the nests of other host birds DE (differentialevolution) [17] is a simple but excellent optimization methodthat uses the difference between two solutions to probabilisti-cally adapt a third solution AnES (evolutionary strategy) [41]is an algorithm that generally distributes equal importanceto mutation and recombination and that allows two or moreparents to reproduce an offspring A GA (genetic algorithm)[13] is a search heuristic that mimics the process of naturalevolution HS (harmony search) [20] is a new metaheuristicapproach inspired by behavior of musicianrsquo improvisationprocess PBIL (probability-based incremental learning) [42]is a type of genetic algorithm where the genotype of anentire population (probability vector) is evolved rather thanindividual members PSO (particle swarm optimization) [22]is also a swarm intelligence algorithm which is based onthe swarm behavior of fish and bird schooling in natureIn addition it should be noted that in [27] Gandomi andAlavi have proved that comparing all the algorithms the KHII (KH with crossover operator) performed the best whichconfirms the robustness of the KH algorithm Therefore inour work we use KH II as a standard KH algorithm

In our experiments we will use the same parameters forKH and LKH that are the foraging speed 119881

119891= 002 the

maximum diffusion speed 119863max

= 0005 the maximuminduced speed 119873

max= 001 and max Levy-flight step size

119860 = 10 (only for LKH) For ACO DE ES GA PBIL and

PSO we set the same parameters as [36 43] For ABC thenumber of colony size (employed bees and onlooker bees)NP = 50 the number of food sources 119865119900119900119889119873119906119898119887119890119903 = NP2andmaximum search times 119897119894119898119894119905 = 100 (a food source whichcould not be improved through ldquolimitrdquo trials is abandoned byits employed bee) For BA we set loudness L = 095 pulserate 119903 = 05 and scaling factor 120576 = 01 for CS a discoveryrate 119901

119886= 025 For HS we set harmony memory accepting

rate = 075 and pitch adjusting rate = 07We set population size NP = 50 andmaximum generation

Maxgen = 50 for each method We ran 100 Monte Carlosimulations of each method on each benchmark functionto get representative performances Tables 3 and 4 illustratethe results of the simulations Table 3 shows the averageminima found by each method averaged over 100 MonteCarlo runs Table 4 shows the absolute best minima found byeachmethod over 100MonteCarlo runsThat is to say Table 3shows the average performance of eachmethod while Table 4shows the best performance of each method The best valueachieved for each test problem is marked in bold Note thatthe normalizations in the tables are based on different scalesso values cannot be compared between the two tables Each ofthe functions in this study has 20 independent variables (ie119889 = 20)

From Table 3 we see that on average LKH is the mosteffective at finding objective function minimum on twelve ofthe fourteen benchmarks (F01ndashF08 F10 and F12ndashF14) ABCand GA are the secondmost effective performing the best onthe benchmarks F11 and F09 when multiple runs are made


Table 2 Properties of benchmark functions lb denotes lower bound ub denotes upper bound and opt denotes optimum point

Number Function lb ub opt Continuity ModalityF01 Ackley minus32768 32768 0 Continuous MultimodalF02 Fletcher-Powell minus120587 120587 0 Continuous MultimodalF03 Griewangk minus600 600 0 Continuous MultimodalF04 Penalty 1 minus50 50 0 Continuous MultimodalF05 Penalty 2 minus50 50 0 Continuous MultimodalF06 Quartic with noise minus128 128 1 Continuous MultimodalF07 Rastrigin minus512 512 0 Continuous MultimodalF08 Rosenbrock minus2048 2048 0 Continuous UnimodalF09 Schwefel 226 minus512 512 0 Continuous MultimodalF10 Schwefel 12 minus100 100 0 Continuous UnimodalF11 Schwefel 222 minus10 10 0 Continuous UnimodalF12 Schwefel 221 minus100 100 0 Continuous UnimodalF13 Sphere minus512 512 0 Continuous UnimodalF14 Step minus512 512 0 Discontinuous Unimodal

Table 3 Mean normalized optimization results in fourteen benchmark functions The values shown are the minimum objective functionvalues found by each algorithm averaged over 100 Monte Carlo simulations

ABC ACO BA CS DE ES GA HS KH LKH PBIL PSOF01 526 620 781 684 485 754 670 774 185 100 784 653F02 263 1001 1442 682 380 977 425 942 367 100 902 827F03 3149 1022 18264 6109 1719 7885 3257 15554 459 100 17691 6457F04 851198645 331198647 451198647 261198646 111198645 191198647 271198645 291198647 921198643 100 431198647 251198646

F05 121198646 171198647 281198647 301198646 291198645 131198647 511198645 221198647 351198644 100 261198647 311198646

F06 341198645 331198645 551198646 761198645 121198645 411198646 331198645 381198646 381198644 100 461198646 881198645

F07 122 230 342 264 199 316 204 289 125 100 318 234F08 1583 8599 8929 2557 1340 11042 2300 7523 534 100 9029 2664F09 177 111 393 286 224 273 100 333 210 193 347 335F10 5156 4379 12364 2915 6830 7256 4830 6646 3317 100 7549 5120F11 100 284 457 279 123 432 219 352 150 422 351 258F12 1465 913 1573 1105 1198 1452 1231 1491 246 100 1561 1252F13 561198643 151198644 321198644 121198644 311198643 331198644 111198644 301198644 85162 100 321198644 121198644

F14 20578 9569 111198643 42788 10381 70063 22785 101198643 2920 100 121198643 41103Time 240 322 108 202 195 203 238 277 466 430 100 237Total 1 0 0 0 0 0 1 0 0 12 0 0lowastThe values are normalized so that the minimum in each row is 100 These are not the absolute minima found by each algorithm but the average minimafound by each algorithm

respectively Table 4 shows that LKH performs the best ontwelve of the fourteen benchmarks which are F01ndashF04 F06ndashF08 and F10ndashF14 ACOandGA are the secondmost effectiveperforming the best on the benchmarks F05 and F09 whenmultiple runs are made respectively

Moreover the computational times of the twelve opti-mization methods were alike We collected the averagecomputational time of the optimization methods as appliedto the 14 benchmarks considered in this section The resultsare given in Table 3 From Table 3 PBIL was the quickestoptimizationmethod and LKHwas the eleventh fastest of thetwelve algorithms This is because that the evaluation of stepsize by Levy flight is too time consuming However we mustpoint out that in the vast majority of real-world engineering

applications it is the fitness function evaluation that is by farthe most expensive part of a population-based optimizationalgorithm

In addition in order to further prove the superiority of theproposed LKHmethod convergence plots of ABC ACO BACS DE ES GA HS KH LKH PBIL and PSO are illustratedin Figures 1ndash14 which mean the process of optimization Thevalues shown in Figures 1ndash14 are the average objective func-tion optimum obtained from 100 Monte Carlo simulationswhich are the true objective function value not normalizedMost importantly note that the best global solutions of thebenchmarks (F04 F05 F11 and F14) are illustrated in theform of the semilogarithmic convergence plots KH is shortfor KH II in the legends of the figures


Table 4 Best normalized optimization results in fourteen benchmark functions The values shown are the minimum objective functionvalues found by each algorithm


F05 221198645 100 171198648 181198647 191198646 201198648 211198644 201198648 731198644 270 311198648 281198647

F06 121198646 491198646 611198647 761198646 391198646 181198648 311198646 121198648 131198646 100 181198648 241198647

F07 200 453 630 495 368 628 321 475 198 100 590 490F08 1040 7089 5416 1503 1274 10961 1746 5883 556 100 7098 1922F09 460 218 910 698 628 780 100 1010 483 453 1001 814F10 56093 23620 70361 26575 92967 89367 28077 55377 26562 100 111198643 49502F11 2001 3233 7500 4409 2925 8322 3179 8827 2370 100 8651 5441F12 3939 1851 4536 2877 3487 4827 2382 4837 408 100 4544 3644F13 101198644 381198644 671198644 211198644 791198643 901198644 101198644 101198645 151198643 100 781198644 351198644

F14 59500 47000 651198643 171198643 63350 451198643 59200 581198643 11300 100 701198643 181198643

Total 0 1 0 0 0 0 1 0 0 12 0 0lowastThe values are normalized so that the minimum in each row is 100 These are the absolute best minima found by each algorithm

0 10 20 30 40 502

4

6

8

10

12

14

16

18

20

Number of generations

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO

Figure 1 Comparison of the performance of the different methodsfor the F01 Ackley function

Figure 1 shows the results obtained for the twelvemethodswhen the F01 Ackley function is applied From Figure 1clearly we can draw the conclusion that LKH is significantlysuperior to all the other algorithms during the process ofoptimization For other algorithms although slower KH IIeventually finds the global minimum close to LKH whileABC ACO BA CS DE ES GA HS PBIL and PSO fail tosearch the global minimum within the limited generationsHere all the algorithms show the almost same starting

0 10 20 30 40 50

2

4

6

8

10

12

Ben

chm

ark

func

tion

val

ue

times105

ABCACOBACSDEES

GAHSKHLKHPBILPSO


Figure 2 Comparison of the performance of the different methodsfor the F02 Fletcher-Powell function

point however LKH outperforms them with fast and stableconvergence rate

Figure 2 illustrates the optimization results for F02Fletcher-Powell function In this multimodal benchmarkproblem it is clear that LKH outperforms all other methodsduring the whole progress of optimization Other algorithmsdo not manage to succeed in this benchmark function withinmaximum number of generations At last ABC and KH IIconverge to the value that is significantly inferior to LKHrsquos


0 10 20 30 40 500

50

100

150

200

250

300

350

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO


Figure 3 Comparison of the performance of the different methodsfor the F03 Griewank function

Figure 3 shows the optimization results for F03 Griewankfunction From Figure 3 we can see that the figure showsthat there is a little difference between the performanceof LKH and KH II However from Table 3 and Figure 3we can conclude that LKH performs better than KH IIin this multimodal function Through carefully looking atFigure 6 ACO has a fast convergence initially towards theknownminimum as the procedure proceeds LKH gets closerand closer to the minimum while ACO comes into beingpremature and traps into the local minimum

Figure 4 shows the results for F04 Penalty 1 functionFrom Figure 4 clearly LKH outperforms all other methodsduring the whole progress of optimization in thismultimodalfunction Eventually KH II performs the second best atfinding the global minimum Although slower later DEperforms the third best at finding the global minimum

Figure 5 shows the performance achieved for F05 Penalty2 function For this multimodal function similar to the F04Penalty 2 function as shown in Figure 4 LKH is significantlysuperior to all the other algorithms during the process ofoptimization Here KH II shows a stable convergence ratein the whole optimization process and eventually it performsthe second best at finding the global minimum that issignificantly superior to the other algorithms

Figure 6 shows the results achieved for the twelve meth-ods when using the F06 Quartic (with noise) function Forthis case the figure shows that there is a little differenceamong the performance of DE GA KH II and LKH FromTable 3 and Figure 6 we can conclude that LKH performs thebest in thismultimodal function KH II DE andGAperformas well and have ranks of 2 3 and 4 respectively Throughcarefully looking at Figure 6 PSO has a fast convergence

0 10 20 30 40 50

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO

108

1010

106

104

102

100


Figure 4 Comparison of the performance of the different methodsfor the F04 Penalty 1 function

0 10 20 30 40 50

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKH

LKHPBILPSO

108

1010

106

104

102

100



initially towards the known minimum as the procedureproceeds LKH gets closer and closer to the minimum whilePSO comes into being premature and traps into the localminimum

Figure 7 shows the optimization results for the F07Rastrigin function In this multimodal benchmark problem


0 10 20 30 40 500

5

10

15

20

25

30

35

40

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO


Figure 6 Comparison of the performance of the different methodsfor the F06 Quartic (with noise) function

0 10 20 30 40 50

60

80

100

120

140

160

180

200

220

240

260

280

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO


Figure 7 Comparison of the performance of the different methodsfor the F07 Rastrigin function

it is obvious that LKH outperforms all other methods duringthe whole progress of optimization For other algorithms thefigure shows that there is little difference between the per-formance of ABC and KH II From Table 3 and Figure 7 wecan conclude that KH II performs slightly better than ABCin this multimodal function In addition other algorithms do

0 10 20 30 40 50

500

1000

1500

2000

2500

3000

3500

4000

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO


Figure 8 Comparison of the performance of the different methodsfor the F08 Rosenbrock function

notmanage to succeed in this benchmark functionwithin themaximum number of generations

Figure 8 shows the results for F08 Rosenbrock functionFrom Figure 8 we can conclude that LKH performs thebest in this unimodal function In addition KH II DEand ACO perform very well and have ranks of 2 3 and 4respectively Through carefully looking at Figure 8 PSO hasa fast convergence initially towards the known minimumhowever it is outperformed by LKH after 10 generationsFor other algorithms they do not manage to succeed inthis benchmark function within the maximum number ofgenerations

Figure 9 shows the equivalent results for the F09 Schwefel226 function From Figure 9 clearly GA is significantlysuperior to other algorithms including LKH during theprocess of optimization while ACO and ABC perform thesecond and the third best in this multimodal benchmarkfunction respectively Unfortunately LKH only performs thefourth in this multimodal benchmark function

Figure 10 shows the results for F10 Schwefel 12 functionFor this case LKH CS KH II and ACOperform the best andhave ranks of 1 2 3 and 4 respectively Looking carefully atFigure 7 LKH has the fastest and stable convergence rate atfinding the globalminimumand significantly outperforms allother approaches

Figure 11 shows the results for F11 Schwefel 222 functionFrom Figure 11 similar to the F09 Schwefel 226 functionas shown in Figure 9 it is clear that ABC is significantlysuperior to other algorithms including LKH during theprocess of optimization For other algorithms DE and KHII perform very well and have ranks of 2 and 3 respectivelyUnfortunately LKH only performs the tenth best in thisunimodal benchmark function among the twelve methods



0 10 20 30 40 501500

2000

2500

3000

3500

4000

4500

5000

5500

6000

6500

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKH

LKHPBILPSO

Figure 9 Comparison of the performance of the different methodsfor the F09 Schwefel 226 function

0 10 20 30 40 50

1

2

Ben

chm

ark

func

tion

val

ue

15

05

25

times104

ABCACOBACSDEES

GAHSKH

LKHPBILPSO


Figure 10 Comparison of the performance of the differentmethodsfor the F10 Schwefel 12 function

Figure 12 shows the results for F12 Schwefel 221 functionVery clearly LKH has the fastest convergence rate at findingthe global minimum and significantly outperforms all othermethods For other algorithms KH II and ACO that are onlyinferior to LKH perform very well and have ranks of 2 and 3respectively

0 10 20 30 40 50

Ben

chm

ark

func

tion

val

ue

105

107

106

104

103

102

101

ABCACOBACSDEES

GAHSKHLKHPBILPSO



0 10 20 30 40 50

10

20

30

40

50

60

70

80

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO



Figure 13 shows the results for F13 Sphere function FromFigure 13 LKH shows the fastest convergence rate at findingthe global minimum and significantly outperforms all othermethods In addition KH II DE andACOperform very welland have ranks of 2 3 and 4 respectively

Figure 14 shows the results for F14 Step function ClearlyLKH shows the fastest convergence rate at finding the


0 10 20 30 40 500

10

20

30

40

50

60

70

80

90

100

110

Ben

chm

ark

func

tion

val

ue


ABCACOBACSDEES

GAHSKHLKHPBILPSO

Figure 13 Comparison of the performance of the different methodsfor the F13 Sphere function

0 10 20 30 40 50

Ben

chm

ark

func

tion

val

ue

105

104

103

102

101

ABCACOBACSDEES

GAHSKHLKHPBILPSO


Figure 14 Comparison of the performance of the differentmethodsfor the F14 Step function

global minimum and significantly outperforms all otherapproaches Though slow KH II performs the second best atfinding the global minimum that is only inferior to the LKH

From the above analyses about Figures 1ndash14 we cancome to a conclusion that our proposed hybridmetaheuristicLKH algorithm significantly outperforms the other eleven

algorithms In general KH II is only inferior to LKH andperforms the second best among twelvemethods ABCACODE and GA perform the third best only inferior to theLKH and KH II ABC and GA especially perform betterthan LKH on benchmark functions F11 and F09 respectivelyFurthermore the illustration of benchmarks F04 F05 F06F08 and F10 shows that PSO has a faster convergence rateinitially while later it converges slower and slower to the trueobjective function value

42 Discussion For all of the standard benchmark functionsconsidered in this section the LKHmethod has been demon-strated to perform better than or at least highly competitivewith the standard KH and other eleven acclaimed state-of-the-art population-based methods The advantages ofLKH involve performing simply and easily and have fewparameters to regulate The work here proves the LKH to berobust powerful and effective over all types of benchmarkfunctions

Benchmark evaluating is a good way for testing theperformance of the metaheuristic methods but it is alsonot flawless and has some limitations First we did not domuch work painstakingly to carefully regulate the optimiza-tion methods in this section In general different tuningparameter values in the optimization methods might leadto significant differences in their performance Second real-world optimization problems may have little of a relationshipto benchmark functions Third benchmark tests may arriveat fully different conclusions if the grading criteria or problemsetup changes In our present work we looked into themean and best values obtained with some population sizeand after some number of iterations However we mightreach different conclusions if for example we change thepopulation size or look at howmany population size it needsto reach a certain function value or if we change the iterationDespite these caveats the benchmark results represented hereare prospective for LKH and show that this novel methodmight be capable of finding a niche among the plethora ofpopulation-based optimization methods

Note that running time is a bottleneck to the implemen-tation of many population-based optimization algorithmsIf an algorithm converges too slowly it will be impracticaland infeasible since it would take too long to search anoptimal or suboptimal solution LKH seems not to requirean unreasonable amount of computational time of the twelvecomparative optimization methods used in this paper LKHwas the eleventh fastest How to speed up the LKHrsquos conver-gence is worthy of further study

In our study 14 benchmark functions have been appliedto evaluate the performance of our LKH method we willtest our proposed method on more optimization problemssuch as the high-dimensional (d ge 20) CEC 2010 test suit[44] and the real-world engineering problems Moreoverwe will compare LKH with other optimization algorithmsIn addition we only consider the unconstrained functionoptimization in this study Our future work consists of addingthe other techniques into LKH for constrained optimizationproblems such as constrained real-parameter optimizationCEC 2010 test suit [45]


5 Conclusion and Future WorkDue to the limited performance of KH on complex problemsLLF operator has been introduced into the standard KHto develop a novel Levy-flight krill herd (LKH) algorithmfor optimization problems In LKH at first original KHalgorithm is applied to shrink the search region to a morepromising area Thereafter LLF operator is implementedas a critical complement to perform the local search toexploit the limited area intensively to get better solutions Inprinciple KH takes full advantage of the three motions in thepopulation and has experimentally demonstrated very goodperformance on themultimodal problems In a rugged regionof the fitness landscape KH may fail to proceed to bettersolutions [27] Then LLF operator is adaptively launched toreboost the search The LKH makes an attempt at takingmerits of the KH and Levy flight in order to avoid all krillgetting trapped in inferior local optimal regions The LKHenables the krill to havemore diverse exemplars to learn fromas the krill are updated each generation and also form newkrill to search in a larger search space With both techniquescombined LKHcan balance exploration and exploitation andeffectively solve complex multimodal problems

Furthermore this new method can speed up the globalconvergence rate without losing the strong robustness of thebasic KH From the analysis of the experimental results wecan see that the Levy-flight KH clearly improves the reliabilityof the global optimality and they also enhance the quality ofthe solutions Based on the results of the twelve methods onthe test problems we can conclude that LKH significantlyimproves the performances of the KH on most multimodaland unimodal problems In addition LKH is simple andimplements easily

In the field of numerical optimization there are consider-able issues that deserve further study and somemore efficientoptimizationmethods should be developed depending on theanalysis of specific engineering problemOur futureworkwillfocus on the two issues On the one hand we would apply ourproposed LKH method to solve real-world civil engineeringoptimization problems [46] and obviously LKH can bea promising method for these optimization problems Onthe other hand we would develop more new metaheuristicmethods to solve optimization problems more efficiently andeffectively

AcknowledgmentsThisworkwas supported by the State Key Laboratory of LaserInteraction with Material Research Fund under Grant noSKLLIM0902-01 and Key Research Technology of Electric-discharge Nonchain Pulsed DF Laser under Grant no LXJJ-11-Q80

References

[1] S Gholizadeh and F Fattahi ldquoDesign optimization of tallsteel buildings by a modified particle swarm algorithmrdquo TheStructural Design of Tall and Special Buildings In press

[2] S Talatahari R Sheikholeslami M Shadfaran and M Pour-baba ldquoOptimumdesign of gravity retaining walls using charged

system search algorithmrdquo Mathematical Problems in Engineer-ing vol 2012 Article ID 301628 10 pages 2012

[3] X S Yang A H Gandomi S Talatahari and A H AlaviMetaheuristics in Water Geotechnical and Transport Engineer-ing Elsevier Waltham Mass USA 2013

[4] A H Gandomi X S Yang S Talatahari and A H AlaviMetaheuristic Applications in Structures and InfrastructuresElsevier Waltham Mass USA 2013

[5] S Gholizadeh and A Barzegar ldquoShape optimization of struc-tures for frequency constraints by sequential harmony searchalgorithmrdquo Engineering Optimization In press

[6] S Chen Y Zheng C Cattani and W Wang ldquoModeling ofbiological intelligence for SCM systemoptimizationrdquoComputa-tional and Mathematical Methods in Medicine vol 2010 ArticleID 769702 10 pages 2012

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

[8] X S Yang Engineering Optimization An Introduction withMetaheuristic Applications Wiley amp Sons NJ USA 2010

[9] G Wang L Guo H Duan L Liu H Wang and M ShaoldquoPath planning for uninhabited combat aerial vehicle usinghybrid meta-heuristic DEBBO algorithmrdquo Advanced ScienceEngineering and Medicine vol 4 no 6 pp 550ndash564 2012

[10] G Wang L Guo H Duan L Liu and H Wang ldquoA bat algo-rithm with mutation for UCAV path planningrdquo The ScientificWorld Journal vol 2012 Article ID 418946 15 pages 2012

[11] H DuanW Zhao GWang and X Feng ldquoTest-sheet composi-tion using analytic hierarchy process and hybrid metaheuristicalgorithmTSBBOrdquoMathematical Problems in Engineering vol2012 Article ID 712752 22 pages 2012

[12] W-H Ho and A L-F Chan ldquoHybrid Taguchi-differentialevolution algorithm for parameter estimation of differentialequation models with application to HIV dynamicsrdquo Mathe-matical Problems in Engineering vol 2011 Article ID 514756 14pages 2011

[13] D E Goldberg Genetic Algorithms in Search Optimization andMachine Learning Addison-Wesley New York NY USA 1998

[14] M Shahsavar A A Najafi and S T A Niaki ldquoStatistical designof genetic algorithms for combinatorial optimization problemsrdquoMathematical Problems in Engineering vol 2011 Article ID872415 17 pages 2011

[15] G Wang and L Guo ldquoA novel hybrid bat algorithm withharmony search for global numerical optimizationrdquo Journal ofApplied Mathematics In press

[16] X S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[17] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[18] A H Gandomi X-S Yang S Talatahari and S Deb ldquoCoupledeagle strategy and differential evolution for unconstrained andconstrained global optimizationrdquo Computers amp Mathematicswith Applications vol 63 no 1 pp 191ndash200 2012

[19] A H Gandomi and A H Alavi ldquoMulti-stage genetic pro-gramming a new strategy to nonlinear system modelingrdquoInformation Sciences vol 181 no 23 pp 5227ndash5239 2011

[20] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001


[21] G Wang and L Guo ldquoHybridizing harmony search with bio-geography based optimization for global numerical optimiza-tionrdquo Journal of Computational and Theoretical Nanoscience Inpress

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

[23] S Talatahari M Kheirollahi C Farahmandpour and A HGandomi ldquoA multi-stage particle swarm for optimum designof truss structuresrdquo Neural Computing amp Applications In press

[24] AHGandomi G J Yun X -S Yang and S Talatahari ldquoChaos-enhanced accelerated particle swarm optimizationrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 18no 2 pp 327ndash340 2013

[25] A H Gandomi X-S Yang and A H Alavi ldquoCuckoo searchalgorithm a metaheuristic approach to solve structural opti-mization problemsrdquo Engineering with Computers vol 29 no 1pp 1ndash19 2013

[26] G Wang L Guo H Duan L Liu H Wang and W JianboldquoA hybrid meta-heuristic DECS algorithm for UCAV pathplanningrdquo Journal of Information and Computational Sciencevol 9 no 16 pp 1ndash8 2012

[27] A H Gandomi and A H Alavi ldquoKrill Herd a new bio-inspiredoptimization algorithmrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4831ndash4845 2012

[28] G Wang L Guo H Duan H Wang L Liu and J Li ldquoIncor-porating mutation scheme into krill herd algorithm for globalnumerical optimizationrdquo Neural Computing and ApplicationsIn press

[29] G Wang L Guo H Duan H Wang and L Liu ldquoA newimproved firefly algorithm for global numerical optimizationrdquoJournal of Computational andTheoretical Nanoscience In press

[30] G Wang L Guo H Duan H Wang L Liu and M ShaoldquoA hybrid meta-heuristic DECS algorithm for UCAV three-dimension path planningrdquo The Scientific World Journal vol2012 Article ID 583973 11 pages 2012

[31] G Wang L Guo A H Gandomi et al ldquoA new improved krillherd algorithm for global numerical optimizationrdquo Neurocom-puting In press

[32] S Yang and J Lee ldquoMulti-basin particle swarm intelligencemethod for optimal calibration of parametric Levy modelsrdquoExpert Systems with Applications vol 39 no 1 pp 482ndash4932012

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

[34] A Natarajan S Subramanian and K Premalatha ldquoA compar-ative study of cuckoo search and bat algorithm for Bloom filteroptimisation in spam filteringrdquo International Journal of Bio-Inspired Computation vol 4 no 2 pp 89ndash99 2012

[35] X Yao Y Liu and G Lin ldquoEvolutionary programming madefasterrdquo IEEE Transactions on Evolutionary Computation vol 3no 2 pp 82ndash102 1999

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

[37] X Li J Wang J Zhou and M Yin ldquoA perturb biogeographybased optimization with mutation for global numerical opti-mizationrdquo Applied Mathematics and Computation vol 218 no2 pp 598ndash609 2011

[38] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony

(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[39] M Dorigo and T Stutzle Ant Colony Optimization MIT PressCambridge Mass USA 2004

[40] X S Yang and S Deb ldquoEngineering optimisation by cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 1 no 4 pp 330ndash343 2010

[41] H-G BeyerTheTheory of Evolution Strategies Springer BerlinGermany 2001

[42] B Shumeet ldquoPopulation-Based Incremental Learning AMethod for Integrating Genetic Search Based FunctionOptimization and Competitive Learningrdquo Tech Rep CMU-CS-94-163 Carnegie Mellon University Pittsburgh Pa USA1994

[43] G Wang L Guo H Duan L Liu and H Wang ldquoDynamicdeployment of wireless sensor networks by biogeography basedoptimization algorithmrdquo Journal of Sensor and Actuator Net-works vol 1 no 2 pp 86ndash96 2012

[44] K Tang X Li P N Suganthan Z Yang and T WeiseldquoBenchmark functions for the CECrsquo2010 special session andcompetition on large scale global optimizationrdquo Inspired Com-putation and Applications Laboratory USTC Hefei China2010

[45] R Mallipeddi and P Suganthan ldquoProblem definitions and eval-uation criteria for the CEC 2010 Competition on ConstrainedReal-ParameterOptimizationrdquo Nanyang Technological Univer-sity Singapore 2010

[46] P Lu S Chen and Y Zheng ldquoArtificial intelligence in civilengineeringrdquo Mathematical Problems in Engineering vol 2012Article ID 145974 22 pages 2012

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Inspired by nature these strongmetaheuristic algorithmshave been proposed to solve NP-hard tasks such as UCAVpath planning [9 10] test-sheet composition [11] and param-eter estimation [12] These kinds of metaheuristic methodsperform on a population of solutions and always find optimalor suboptimal solutions During the 1960s and 1970s com-puter scientists studied the possibility of formulating evolu-tion as an optimizationmethod and eventually this generateda subset of gradient free methods namely genetic algorithms(GAs) [13 14] In the last two decades a huge number oftechniques were developed on function optimization suchas bat algorithm (BA) [15 16] differential evolution (DE)[17 18] genetic programming (GP) [19] harmony search(HS) [20 21] particle swarm optimization (PSO) [22ndash24]cuckoo search (CS) [25 26] and more recently the krillherd (KH) algorithm [27] that is based on imitating the krillherding behavior in nature

Firstly proposed by Gandomi and Alavi in 2012 inspiredby the herding behavior of krill individuals KH algorithmis a novel swarm intelligence method for optimizing pos-sibly nondifferentiable and nonlinear complex functions incontinuous space [27] In KH the time-dependent positionof the krill individuals involves three main components (i)movement led by other individuals (ii) foraging motionand (iii) random physical diffusion One of the notableadvantages of theKHalgorithm is that derivative informationis unnecessary because it uses a random search instead ofa gradient search use in classical methods Moreover com-paring with other population-based metaheuristic methodsthis new method needs few control variables in principleonly a separate parameter Δt (time interval) to tune whichmakesKHeasy to implementmore robust and fits for parallelcomputation

KH is an effective and powerful algorithm in explorationbut at times it may trap into some local optima so that itcannot implement global search well For KH the searchdepends completely on random search so there is no guar-antee for a fast convergence In order to improve KH inoptimization problems a method has been proposed [28]which introduces a more focused mutation strategy into KHto add the diversity of population

On the other hand many researchers have centralized ontheories and applications of statistical techniques especiallyof Levy distributionAnd recently huge advances are acquiredin many fields One of these fields is the applications of Levyflight in optimization methods Previously Levy flights havebeen used together with some metaheuristic optimizationmethods such as firefly algorithm [29] cuckoo search [30]krill herd algorithm [31] and particle swarm optimization[32]

Firstly presented here an effective Levy-flight KH (LKH)method is originally proposed in this paper in order toaccelerate convergence speed thus making the approachmore feasible for a wider range of real-world engineeringapplications while keeping the desirable characteristics of theoriginal KH In LKH first of all a standard KH algorithmis implemented to shrink the search apace and select a goodcandidate solution set And then for more precise modelingof the krill behavior a local Levy-flight (LLF) operator is

added to the algorithmThis operator is applied to exploit thelimited promising area intensively to get better solutions soas to improve its efficiency and reliability for solving globalnumerical optimization problems The proposed method isevaluated on fourteen standard benchmark functions thathave ever been applied to verify optimization methodsin continuous optimization problems Experimental resultsshow that the LKH performs more efficiently and effectivelythan basic KH ABC ACO BA CS DE ES GA HS PBILand PSO

The structure of this paper is organized as followsSection 2 gives a description of basic KH algorithm and Levyflight in brief Our proposed LKH method is described indetail in Section 3 Subsequently our method is evaluatedthrough fourteen benchmark functions in Section 4 Inaddition the LKH is also compared with ABC ACO BA CSDE ES GA HS KH PBIL and PSO in that section FinallySection 5 involves the conclusion and proposals for futurework

2 Preliminary

At first in this section a background on the krill herd algo-rithm and Levy flight will be provided in brief

21 Krill Herd Algorithm Krill herd (KH) [27] is a newmeta-heuristic optimization method [4] for solving optimizationtasks which is based on the simulation of the herding of thekrill swarms in response to particular biological and envi-ronmental processes The time-dependent position of anindividual krill in 2D space is decided by three main actionspresented as follows

(i) movement affected by other krill individuals(ii) foraging action(iii) random diffusion

KH algorithm adopted the following Lagrangian modelin a d-dimensional decision space as in the following (1)

119889119883119894

119889119905= 119873119894+ 119865119894+ 119863119894 (1)

where 119873119894 119865119894 and 119863

119894are the motion led by other krill

individuals the foraging motion and the physical diffusionof the 119894th krill individual respectively

In movement affected by other krill individuals thedirection of motion 120572

119894 is approximately computed by the

target effect (target swarmdensity) local effect (a local swarmdensity) and a repulsive effect (repulsive swarm density) Fora krill individual this movement can be defined as

119873new119894

= 119873max

120572119894+ 120596119899119873

old119894

(2)

and 119873max is the maximum induced speed 120596

119899is the inertia

weight of the motion induced in [0 1] and 119873old119894

is the lastmotion induced

The foraging motion is estimated by the two maincomponents One is the food location and the other one is


Table1Be

nchm

arkfunctio

ns

Num

ber

Nam

eDefinitio

nF0

1Ac

kley

119891(119909)=

20+

119890minus


119890(1119899)sum119899 119894=1cos(2120587119909119894)

F02

Fletcher-Pow

ell119891(119909)=

119899 sum 119894=1

(119860119894minus

119861119894)2119860119894=

119899 sum 119895=1

(119886119894119895sin

120572119895+

119887119894119895cos120572119895)

119861119894=

119899 sum 119895=1

(119886119894119895sin

119909119895+

119887119894119895cos119909119895)

F03

Grie

wank

119891(119909)=

119899 sum 119894=1

1199092 119894

4000

minus

119899 prod 119894=1

cos(

119909119894

radic119894)

+1

F04

Penalty

1119891

(119909)=

120587 30

10sin2(1205871199101)+

119899minus1

sum 119894=1

(119910119894minus

1)2

sdot[1+

10sin2(120587119910119894+1)]+

(119910119899minus

1)2

+

119899 sum 119894=1

119906(119909119894101004)119910119894=

1+

025(119909119894+

1)

F05

Penalty

2119891(119909)=

01

sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus

1)2

sdot[1+sin2(3120587119909119894+1)]+

(119909119899minus

1)2

[1+sin2(2120587119909119899)]

+

119899 sum 119894=1

119906(11990911989451004)

F06

Quarticwith

noise

119891(119909)=

119899 sum 119894=1

(119894sdot1199094 119894+

119880(01))

F07

Rastrig

in119891(119909)=

10sdot119899+

119899 sum 119894=1

(1199092 119894minus

10sdotcos(2120587119909119894))

F08

Rosenb

rock

119891(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus

1199092 119894)2

+(119909119894minus

1)2

]

F09

Schw

efel226

119891(119909)=

4189829times

119863minus

119863 sum 119894=1

119909119894sin

(1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 100381612

)

F10

Schw

efel12

119891(119909)=

119899 sum 119894=1

(

119894 sum 119895=1

119909119895)

2

F11

Schw

efel222

119891(119909)=

119899 sum 119894=1

|119909119894|+

119899 prod 119894=1

|119909119894|

F12

Schw

efel221

119891(119909)=max 119894

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le

119894le

119899

F13

Sphere

119891(119909)=

119899 sum 119894=1

1199092 119894

F14

Step

119891(119909)=

6sdot119899+

119899 sum 119894=1


lowastIn

benchm

arkfunctio

nF0

2them

atrix

elem


rawfro

mun

iform

distr

ibution

lowastIn

benchm

arkfunctio

nsF0

4andF0

5thed

efinitio

nof

thefun

ction119906(119909119894119886119896119898)isas

follo




119865119894= 119881119891120573119894+ 120596119891119865old119894

(3)

where

120573119894= 120573

food119894

+ 120573best119894

(4)






119863119894= 119863

max120575 (5)




119883119894 (119905 + Δ119905) = 119883

119894 (119905) + Δ119905119889119883119894

119889119905 (6)




119883119905+1

119894= 119883119905

119894+ 120573119871 (119904 120582)

119871 (119904 120582) =120582Γ (120582) sin (1205871205822)

120587

1

1199041+120582 (119904 119904

0gt 0)

(7)


3 Our Approach LKH






Begin119886 = 119860(119905






119894

if 119881119894is better than 119883

119894then

119883119894= 119881119894

end ifEnd



























end for 119894



End




119891= 002 the















F05 121198646 171198647 281198647 301198646 291198645 131198647 511198645 221198647 351198644 100 261198647 311198646

F06 341198645 331198645 551198646 761198645 121198645 411198646 331198645 381198646 381198644 100 461198646 881198645

F07 122 230 342 264 199 316 204 289 125 100 318 234F08 1583 8599 8929 2557 1340 11042 2300 7523 534 100 9029 2664F09 177 111 393 286 224 273 100 333 210 193 347 335F10 5156 4379 12364 2915 6830 7256 4830 6646 3317 100 7549 5120F11 100 284 457 279 123 432 219 352 150 422 351 258F12 1465 913 1573 1105 1198 1452 1231 1491 246 100 1561 1252F13 561198643 151198644 321198644 121198644 311198643 331198644 111198644 301198644 85162 100 321198644 121198644









F05 221198645 100 171198648 181198647 191198646 201198648 211198644 201198648 731198644 270 311198648 281198647

F06 121198646 491198646 611198647 761198646 391198646 181198648 311198646 121198648 131198646 100 181198648 241198647

F07 200 453 630 495 368 628 321 475 198 100 590 490F08 1040 7089 5416 1503 1274 10961 1746 5883 556 100 7098 1922F09 460 218 910 698 628 780 100 1010 483 453 1001 814F10 56093 23620 70361 26575 92967 89367 28077 55377 26562 100 111198643 49502F11 2001 3233 7500 4409 2925 8322 3179 8827 2370 100 8651 5441F12 3939 1851 4536 2877 3487 4827 2382 4837 408 100 4544 3644F13 101198644 381198644 671198644 211198644 791198643 901198644 101198644 101198645 151198643 100 781198644 351198644

F14 59500 47000 651198643 171198643 63350 451198643 59200 581198643 11300 100 701198643 181198643


0 10 20 30 40 502

4

6

8

10

12

14

16

18

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chm

ark

func

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val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO



0 10 20 30 40 50

2

4

6

8

10

12

Ben

chm

ark

func

tion

val

ue

times105

ABCACOBACSDEES

GAHSKHLKHPBILPSO






0 10 20 30 40 500

50

100

150

200

250

300

350

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO







0 10 20 30 40 50

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO

108

1010

106

104

102

100



0 10 20 30 40 50

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKH

LKHPBILPSO

108

1010

106

104

102

100






0 10 20 30 40 500

5

10

15

20

25

30

35

40

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO



0 10 20 30 40 50

60

80

100

120

140

160

180

200

220

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260

280

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO




0 10 20 30 40 50

500

1000

1500

2000

2500

3000

3500

4000

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO










0 10 20 30 40 501500

2000

2500

3000

3500

4000

4500

5000

5500

6000

6500

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKH

LKHPBILPSO


0 10 20 30 40 50

1

2

Ben

chm

ark

func

tion

val

ue

15

05

25

times104

ABCACOBACSDEES

GAHSKH

LKHPBILPSO




0 10 20 30 40 50

Ben

chm

ark

func

tion

val

ue

105

107

106

104

103

102

101

ABCACOBACSDEES

GAHSKHLKHPBILPSO



0 10 20 30 40 50

10

20

30

40

50

60

70

80

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO






0 10 20 30 40 500

10

20

30

40

50

60

70

80

90

100

110

Ben

chm

ark

func

tion

val

ue


ABCACOBACSDEES

GAHSKHLKHPBILPSO


0 10 20 30 40 50

Ben

chm

ark

func

tion

val

ue

105

104

103

102

101

ABCACOBACSDEES

GAHSKHLKHPBILPSO















References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table1Be

nchm

arkfunctio

ns

Num

ber

Nam

eDefinitio

nF0

1Ac

kley

119891(119909)=

20+

119890minus


119890(1119899)sum119899 119894=1cos(2120587119909119894)

F02

Fletcher-Pow

ell119891(119909)=

119899 sum 119894=1

(119860119894minus

119861119894)2119860119894=

119899 sum 119895=1

(119886119894119895sin

120572119895+

119887119894119895cos120572119895)

119861119894=

119899 sum 119895=1

(119886119894119895sin

119909119895+

119887119894119895cos119909119895)

F03

Grie

wank

119891(119909)=

119899 sum 119894=1

1199092 119894

4000

minus

119899 prod 119894=1

cos(

119909119894

radic119894)

+1

F04

Penalty

1119891

(119909)=

120587 30

10sin2(1205871199101)+

119899minus1

sum 119894=1

(119910119894minus

1)2

sdot[1+

10sin2(120587119910119894+1)]+

(119910119899minus

1)2

+

119899 sum 119894=1

119906(119909119894101004)119910119894=

1+

025(119909119894+

1)

F05

Penalty

2119891(119909)=

01

sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus

1)2

sdot[1+sin2(3120587119909119894+1)]+

(119909119899minus

1)2

[1+sin2(2120587119909119899)]

+

119899 sum 119894=1

119906(11990911989451004)

F06

Quarticwith

noise

119891(119909)=

119899 sum 119894=1

(119894sdot1199094 119894+

119880(01))

F07

Rastrig

in119891(119909)=

10sdot119899+

119899 sum 119894=1

(1199092 119894minus

10sdotcos(2120587119909119894))

F08

Rosenb

rock

119891(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus

1199092 119894)2

+(119909119894minus

1)2

]

F09

Schw

efel226

119891(119909)=

4189829times

119863minus

119863 sum 119894=1

119909119894sin

(1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 100381612

)

F10

Schw

efel12

119891(119909)=

119899 sum 119894=1

(

119894 sum 119895=1

119909119895)

2

F11

Schw

efel222

119891(119909)=

119899 sum 119894=1

|119909119894|+

119899 prod 119894=1

|119909119894|

F12

Schw

efel221

119891(119909)=max 119894

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le

119894le

119899

F13

Sphere

119891(119909)=

119899 sum 119894=1

1199092 119894

F14

Step

119891(119909)=

6sdot119899+

119899 sum 119894=1


lowastIn

benchm

arkfunctio

nF0

2them

atrix

elem


rawfro

mun

iform

distr

ibution

lowastIn

benchm

arkfunctio

nsF0

4andF0

5thed

efinitio

nof

thefun

ction119906(119909119894119886119896119898)isas

follo




119865119894= 119881119891120573119894+ 120596119891119865old119894

(3)

where

120573119894= 120573

food119894

+ 120573best119894

(4)






119863119894= 119863

max120575 (5)




119883119894 (119905 + Δ119905) = 119883

119894 (119905) + Δ119905119889119883119894

119889119905 (6)




119883119905+1

119894= 119883119905

119894+ 120573119871 (119904 120582)

119871 (119904 120582) =120582Γ (120582) sin (1205871205822)

120587

1

1199041+120582 (119904 119904

0gt 0)

(7)


3 Our Approach LKH






Begin119886 = 119860(119905






119894

if 119881119894is better than 119883

119894then

119883119894= 119881119894

end ifEnd



























end for 119894



End




119891= 002 the















F05 121198646 171198647 281198647 301198646 291198645 131198647 511198645 221198647 351198644 100 261198647 311198646

F06 341198645 331198645 551198646 761198645 121198645 411198646 331198645 381198646 381198644 100 461198646 881198645

F07 122 230 342 264 199 316 204 289 125 100 318 234F08 1583 8599 8929 2557 1340 11042 2300 7523 534 100 9029 2664F09 177 111 393 286 224 273 100 333 210 193 347 335F10 5156 4379 12364 2915 6830 7256 4830 6646 3317 100 7549 5120F11 100 284 457 279 123 432 219 352 150 422 351 258F12 1465 913 1573 1105 1198 1452 1231 1491 246 100 1561 1252F13 561198643 151198644 321198644 121198644 311198643 331198644 111198644 301198644 85162 100 321198644 121198644









F05 221198645 100 171198648 181198647 191198646 201198648 211198644 201198648 731198644 270 311198648 281198647

F06 121198646 491198646 611198647 761198646 391198646 181198648 311198646 121198648 131198646 100 181198648 241198647

F07 200 453 630 495 368 628 321 475 198 100 590 490F08 1040 7089 5416 1503 1274 10961 1746 5883 556 100 7098 1922F09 460 218 910 698 628 780 100 1010 483 453 1001 814F10 56093 23620 70361 26575 92967 89367 28077 55377 26562 100 111198643 49502F11 2001 3233 7500 4409 2925 8322 3179 8827 2370 100 8651 5441F12 3939 1851 4536 2877 3487 4827 2382 4837 408 100 4544 3644F13 101198644 381198644 671198644 211198644 791198643 901198644 101198644 101198645 151198643 100 781198644 351198644

F14 59500 47000 651198643 171198643 63350 451198643 59200 581198643 11300 100 701198643 181198643


0 10 20 30 40 502

4

6

8

10

12

14

16

18

20


Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO



0 10 20 30 40 50

2

4

6

8

10

12

Ben

chm

ark

func

tion

val

ue

times105

ABCACOBACSDEES

GAHSKHLKHPBILPSO






0 10 20 30 40 500

50

100

150

200

250

300

350

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO







0 10 20 30 40 50

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO

108

1010

106

104

102

100



0 10 20 30 40 50

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKH

LKHPBILPSO

108

1010

106

104

102

100






0 10 20 30 40 500

5

10

15

20

25

30

35

40

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO



0 10 20 30 40 50

60

80

100

120

140

160

180

200

220

240

260

280

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO




0 10 20 30 40 50

500

1000

1500

2000

2500

3000

3500

4000

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO










0 10 20 30 40 501500

2000

2500

3000

3500

4000

4500

5000

5500

6000

6500

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKH

LKHPBILPSO


0 10 20 30 40 50

1

2

Ben

chm

ark

func

tion

val

ue

15

05

25

times104

ABCACOBACSDEES

GAHSKH

LKHPBILPSO




0 10 20 30 40 50

Ben

chm

ark

func

tion

val

ue

105

107

106

104

103

102

101

ABCACOBACSDEES

GAHSKHLKHPBILPSO



0 10 20 30 40 50

10

20

30

40

50

60

70

80

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO






0 10 20 30 40 500

10

20

30

40

50

60

70

80

90

100

110

Ben

chm

ark

func

tion

val

ue


ABCACOBACSDEES

GAHSKHLKHPBILPSO


0 10 20 30 40 50

Ben

chm

ark

func

tion

val

ue

105

104

103

102

101

ABCACOBACSDEES

GAHSKHLKHPBILPSO















References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119865119894= 119881119891120573119894+ 120596119891119865old119894

(3)

where

120573119894= 120573

food119894

+ 120573best119894

(4)






119863119894= 119863

max120575 (5)




119883119894 (119905 + Δ119905) = 119883

119894 (119905) + Δ119905119889119883119894

119889119905 (6)




119883119905+1

119894= 119883119905

119894+ 120573119871 (119904 120582)

119871 (119904 120582) =120582Γ (120582) sin (1205871205822)

120587

1

1199041+120582 (119904 119904

0gt 0)

(7)


3 Our Approach LKH






Begin119886 = 119860(119905






119894

if 119881119894is better than 119883

119894then

119883119894= 119881119894

end ifEnd



























end for 119894



End




119891= 002 the















F05 121198646 171198647 281198647 301198646 291198645 131198647 511198645 221198647 351198644 100 261198647 311198646

F06 341198645 331198645 551198646 761198645 121198645 411198646 331198645 381198646 381198644 100 461198646 881198645

F07 122 230 342 264 199 316 204 289 125 100 318 234F08 1583 8599 8929 2557 1340 11042 2300 7523 534 100 9029 2664F09 177 111 393 286 224 273 100 333 210 193 347 335F10 5156 4379 12364 2915 6830 7256 4830 6646 3317 100 7549 5120F11 100 284 457 279 123 432 219 352 150 422 351 258F12 1465 913 1573 1105 1198 1452 1231 1491 246 100 1561 1252F13 561198643 151198644 321198644 121198644 311198643 331198644 111198644 301198644 85162 100 321198644 121198644









F05 221198645 100 171198648 181198647 191198646 201198648 211198644 201198648 731198644 270 311198648 281198647

F06 121198646 491198646 611198647 761198646 391198646 181198648 311198646 121198648 131198646 100 181198648 241198647

F07 200 453 630 495 368 628 321 475 198 100 590 490F08 1040 7089 5416 1503 1274 10961 1746 5883 556 100 7098 1922F09 460 218 910 698 628 780 100 1010 483 453 1001 814F10 56093 23620 70361 26575 92967 89367 28077 55377 26562 100 111198643 49502F11 2001 3233 7500 4409 2925 8322 3179 8827 2370 100 8651 5441F12 3939 1851 4536 2877 3487 4827 2382 4837 408 100 4544 3644F13 101198644 381198644 671198644 211198644 791198643 901198644 101198644 101198645 151198643 100 781198644 351198644

F14 59500 47000 651198643 171198643 63350 451198643 59200 581198643 11300 100 701198643 181198643


0 10 20 30 40 502

4

6

8

10

12

14

16

18

20


Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO



0 10 20 30 40 50

2

4

6

8

10

12

Ben

chm

ark

func

tion

val

ue

times105

ABCACOBACSDEES

GAHSKHLKHPBILPSO






0 10 20 30 40 500

50

100

150

200

250

300

350

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO







0 10 20 30 40 50

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO

108

1010

106

104

102

100



0 10 20 30 40 50

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKH

LKHPBILPSO

108

1010

106

104

102

100






0 10 20 30 40 500

5

10

15

20

25

30

35

40

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO



0 10 20 30 40 50

60

80

100

120

140

160

180

200

220

240

260

280

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO




0 10 20 30 40 50

500

1000

1500

2000

2500

3000

3500

4000

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO










0 10 20 30 40 501500

2000

2500

3000

3500

4000

4500

5000

5500

6000

6500

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKH

LKHPBILPSO


0 10 20 30 40 50

1

2

Ben

chm

ark

func

tion

val

ue

15

05

25

times104

ABCACOBACSDEES

GAHSKH

LKHPBILPSO




0 10 20 30 40 50

Ben

chm

ark

func

tion

val

ue

105

107

106

104

103

102

101

ABCACOBACSDEES

GAHSKHLKHPBILPSO



0 10 20 30 40 50

10

20

30

40

50

60

70

80

Ben

chm

ark

func

tion

val

ue

ABCACOBACSDEES

GAHSKHLKHPBILPSO






0 10 20 30 40 500

10

20

30

40

50

60

70

80

90

100

110

Ben

chm

ark

func

tion

val

ue


ABCACOBACSDEES

GAHSKHLKHPBILPSO


0 10 20 30 40 50

Ben

chm

ark

func

tion

val

ue

105

104

103

102

101

ABCACOBACSDEES

GAHSKHLKHPBILPSO















References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Begin119886 = 119860(119905






119894

if 119881119894is better than 119883

119894then

119883119894= 119881119894

end ifEnd



























end for 119894



End




119891= 002 the















F05 121198646 171198647 281198647 301198646 291198645 131198647 511198645 221198647 351198644 100 261198647 311198646

F06 341198645 331198645 551198646 761198645 121198645 411198646 331198645 381198646 381198644 100 461198646 881198645

F07 122 230 342 264 199 316 204 289 125 100 318 234F08 1583 8599 8929 2557 1340 11042 2300 7523 534 100 9029 2664F09 177 111 393 286 224 273 100 333 210 193 347 335F10 5156 4379 12364 2915 6830 7256 4830 6646 3317 100 7549 5120F11 100 284 457 279 123 432 219 352 150 422 351 258F12 1465 913 1573 1105 1198 1452 1231 1491 246 100 1561 1252F13 561198643 151198644 321198644 121198644 311198643 331198644 111198644 301198644 85162 100 321198644 121198644









F05 221198645 100 171198648 181198647 191198646 201198648 211198644 201198648 731198644 270 311198648 281198647

F06 121198646 491198646 611198647 761198646 391198646 181198648 311198646 121198648 131198646 100 181198648 241198647

F07 200 453 630 495 368 628 321 475 198 100 590 490F08 1040 7089 5416 1503 1274 10961 1746 5883 556 100 7098 1922F09 460 218 910 698 628 780 100 1010 483 453 1001 814F10 56093 23620 70361 26575 92967 89367 28077 55377 26562 100 111198643 49502F11 2001 3233 7500 4409 2925 8322 3179 8827 2370 100 8651 5441F12 3939 1851 4536 2877 3487 4827 2382 4837 408 100 4544 3644F13 101198644 381198644 671198644 211198644 791198643 901198644 101198644 101198645 151198643 100 781198644 351198644

F14 59500 47000 651198643 171198643 63350 451198643 59200 581198643 11300 100 701198643 181198643


0 10 20 30 40 502

4

6

8

10

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14

16

18

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Ben

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func

tion

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ABCACOBACSDEES

GAHSKHLKHPBILPSO



0 10 20 30 40 50

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8

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Ben

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func

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ABCACOBACSDEES

GAHSKHLKHPBILPSO






0 10 20 30 40 500

50

100

150

200

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Ben

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func

tion

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ABCACOBACSDEES

GAHSKHLKHPBILPSO







0 10 20 30 40 50

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GAHSKHLKHPBILPSO

108

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Ben

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ABCACOBACSDEES

GAHSKHLKHPBILPSO



0 10 20 30 40 50

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Ben

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ABCACOBACSDEES

GAHSKHLKHPBILPSO




0 10 20 30 40 50

500

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Ben

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ABCACOBACSDEES

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0 10 20 30 40 501500

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ABCACOBACSDEES

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ABCACOBACSDEES

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GAHSKHLKHPBILPSO



0 10 20 30 40 50

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Ben

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func

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ABCACOBACSDEES

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0 10 20 30 40 500

10

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0 10 20 30 40 50

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GAHSKHLKHPBILPSO















References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















end for 119894



End




119891= 002 the















F05 121198646 171198647 281198647 301198646 291198645 131198647 511198645 221198647 351198644 100 261198647 311198646

F06 341198645 331198645 551198646 761198645 121198645 411198646 331198645 381198646 381198644 100 461198646 881198645

F07 122 230 342 264 199 316 204 289 125 100 318 234F08 1583 8599 8929 2557 1340 11042 2300 7523 534 100 9029 2664F09 177 111 393 286 224 273 100 333 210 193 347 335F10 5156 4379 12364 2915 6830 7256 4830 6646 3317 100 7549 5120F11 100 284 457 279 123 432 219 352 150 422 351 258F12 1465 913 1573 1105 1198 1452 1231 1491 246 100 1561 1252F13 561198643 151198644 321198644 121198644 311198643 331198644 111198644 301198644 85162 100 321198644 121198644









F05 221198645 100 171198648 181198647 191198646 201198648 211198644 201198648 731198644 270 311198648 281198647

F06 121198646 491198646 611198647 761198646 391198646 181198648 311198646 121198648 131198646 100 181198648 241198647

F07 200 453 630 495 368 628 321 475 198 100 590 490F08 1040 7089 5416 1503 1274 10961 1746 5883 556 100 7098 1922F09 460 218 910 698 628 780 100 1010 483 453 1001 814F10 56093 23620 70361 26575 92967 89367 28077 55377 26562 100 111198643 49502F11 2001 3233 7500 4409 2925 8322 3179 8827 2370 100 8651 5441F12 3939 1851 4536 2877 3487 4827 2382 4837 408 100 4544 3644F13 101198644 381198644 671198644 211198644 791198643 901198644 101198644 101198645 151198643 100 781198644 351198644

F14 59500 47000 651198643 171198643 63350 451198643 59200 581198643 11300 100 701198643 181198643


0 10 20 30 40 502

4

6

8

10

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14

16

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Ben

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func

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ABCACOBACSDEES

GAHSKHLKHPBILPSO



0 10 20 30 40 50

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Ben

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func

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ABCACOBACSDEES

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0 10 20 30 40 500

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Ben

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0 10 20 30 40 500

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Ben

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ABCACOBACSDEES

GAHSKHLKHPBILPSO



0 10 20 30 40 50

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Ben

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ABCACOBACSDEES

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0 10 20 30 40 50

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Ben

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0 10 20 30 40 501500

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0 10 20 30 40 50

Ben

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ABCACOBACSDEES

GAHSKHLKHPBILPSO



0 10 20 30 40 50

10

20

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Ben

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func

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ABCACOBACSDEES

GAHSKHLKHPBILPSO






0 10 20 30 40 500

10

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0 10 20 30 40 50

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104

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102

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GAHSKHLKHPBILPSO















References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














F05 121198646 171198647 281198647 301198646 291198645 131198647 511198645 221198647 351198644 100 261198647 311198646

F06 341198645 331198645 551198646 761198645 121198645 411198646 331198645 381198646 381198644 100 461198646 881198645

F07 122 230 342 264 199 316 204 289 125 100 318 234F08 1583 8599 8929 2557 1340 11042 2300 7523 534 100 9029 2664F09 177 111 393 286 224 273 100 333 210 193 347 335F10 5156 4379 12364 2915 6830 7256 4830 6646 3317 100 7549 5120F11 100 284 457 279 123 432 219 352 150 422 351 258F12 1465 913 1573 1105 1198 1452 1231 1491 246 100 1561 1252F13 561198643 151198644 321198644 121198644 311198643 331198644 111198644 301198644 85162 100 321198644 121198644









F05 221198645 100 171198648 181198647 191198646 201198648 211198644 201198648 731198644 270 311198648 281198647

F06 121198646 491198646 611198647 761198646 391198646 181198648 311198646 121198648 131198646 100 181198648 241198647

F07 200 453 630 495 368 628 321 475 198 100 590 490F08 1040 7089 5416 1503 1274 10961 1746 5883 556 100 7098 1922F09 460 218 910 698 628 780 100 1010 483 453 1001 814F10 56093 23620 70361 26575 92967 89367 28077 55377 26562 100 111198643 49502F11 2001 3233 7500 4409 2925 8322 3179 8827 2370 100 8651 5441F12 3939 1851 4536 2877 3487 4827 2382 4837 408 100 4544 3644F13 101198644 381198644 671198644 211198644 791198643 901198644 101198644 101198645 151198643 100 781198644 351198644

F14 59500 47000 651198643 171198643 63350 451198643 59200 581198643 11300 100 701198643 181198643


0 10 20 30 40 502

4

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Ben

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ABCACOBACSDEES

GAHSKHLKHPBILPSO



0 10 20 30 40 50

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ABCACOBACSDEES

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0 10 20 30 40 500

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Ben

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0 10 20 30 40 500

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0 10 20 30 40 50

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0 10 20 30 40 50

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0 10 20 30 40 501500

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0 10 20 30 40 50

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0 10 20 30 40 500

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GAHSKHLKHPBILPSO















References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












F05 221198645 100 171198648 181198647 191198646 201198648 211198644 201198648 731198644 270 311198648 281198647

F06 121198646 491198646 611198647 761198646 391198646 181198648 311198646 121198648 131198646 100 181198648 241198647

F07 200 453 630 495 368 628 321 475 198 100 590 490F08 1040 7089 5416 1503 1274 10961 1746 5883 556 100 7098 1922F09 460 218 910 698 628 780 100 1010 483 453 1001 814F10 56093 23620 70361 26575 92967 89367 28077 55377 26562 100 111198643 49502F11 2001 3233 7500 4409 2925 8322 3179 8827 2370 100 8651 5441F12 3939 1851 4536 2877 3487 4827 2382 4837 408 100 4544 3644F13 101198644 381198644 671198644 211198644 791198643 901198644 101198644 101198645 151198643 100 781198644 351198644

F14 59500 47000 651198643 171198643 63350 451198643 59200 581198643 11300 100 701198643 181198643


0 10 20 30 40 502

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References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 500

50

100

150

200

250

300

350

Ben

chm

ark

func

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References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 500

5

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15

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25

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chm

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References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0 10 20 30 40 501500

2000

2500

3000

3500

4000

4500

5000

5500

6000

6500

Ben

chm

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func

tion

val

ue


ABCACOBACSDEES

GAHSKHLKHPBILPSO


0 10 20 30 40 50

Ben

chm

ark

func

tion

val

ue

105

104

103

102

101

ABCACOBACSDEES

GAHSKHLKHPBILPSO















References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 500

10

20

30

40

50

60

70

80

90

100

110

Ben

chm

ark

func

tion

val

ue


ABCACOBACSDEES

GAHSKHLKHPBILPSO


0 10 20 30 40 50

Ben

chm

ark

func

tion

val

ue

105

104

103

102

101

ABCACOBACSDEES

GAHSKHLKHPBILPSO















References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Lévy-Flight Krill Herd...

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