Hybrid Algorithm of Particle Swarm Optimization and Grey ... · In recent years, several numbers of...

Research ArticleHybrid Algorithm of Particle Swarm Optimization and GreyWolf Optimizer for Improving Convergence Performance

Narinder Singh and S B Singh

Department of Mathematics Punjabi University Patiala Punjab 147002 India

Correspondence should be addressed to Narinder Singh narindersinghgoriaymailcom

Received 9 June 2017 Revised 29 August 2017 Accepted 30 August 2017 Published 16 November 2017

Academic Editor N Shahzad

Copyright copy 2017 Narinder Singh and S B Singh This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A newly hybrid nature inspired algorithm called HPSOGWO is presented with the combination of Particle Swarm Optimization(PSO) and Grey Wolf Optimizer (GWO) The main idea is to improve the ability of exploitation in Particle Swarm Optimizationwith the ability of exploration in Grey Wolf Optimizer to produce both variantsrsquo strength Some unimodal multimodal and fixed-dimensionmultimodal test functions are used to check the solution quality and performance of HPSOGWOvariantThe numericaland statistical solutions show that the hybrid variant outperforms significantly the PSO and GWO variants in terms of solutionquality solution stability convergence speed and ability to find the global optimum

1 Introduction

In recent years several numbers of nature inspired opti-mization techniques have been developed These includeParticle Swarm Optimization (PSO) Gravitational Searchalgorithm (GSA) Genetic Algorithm (GA) EvolutionaryAlgorithm (EA) Deferential Evolution (DE) Ant ColonyOptimization (ACO) Biogeographically Based Optimization(BBO) Firefly algorithm (FA) and Bat algorithm (BA) Thecommon goal of these algorithms is to find the best qualityof solutions and better convergence performance In order todo this a nature inspired variant should be equipped withexploration and exploitation to ensure finding global opti-mum

Exploitation is the convergence capability to the mostexcellent result of the function near a good result and explo-ration is the capability of a variant to find whole parts offunction area Finally the goal of all nature inspired variantsis to balance the capability of exploration and exploitationcapably in order to search best global optimal solution in thesearch space As per Eiben and Schippers [1] exploitationand exploration in nature inspired computing are not under-standable due to lack of a usually accepted opinion and onthe other side with increase in one capability the other willweaken and vice versa

As per the above the existing nature inspired variants arecapable of solving several numbers of test and real life prob-lems It has been proved that there is no population-basedvariant which can perform generally enough to find thesolution of all types of optimization problems [2]

The Particle Swarm Optimization is one of the most usu-ally used evolutionary variants in hybrid techniques due to itscapability of searching global optimum convergence speedand simplicity

There are several studies in the text which have beenprepared to combine Particle Swarm Optimization variantwith other variants of metaheuristics such as hybrid ParticleSwarm Optimization with Genetic Algorithm (PSOGA) [34] Particle Swarm Optimization with Differential Evolution(PSODE) [5] and Particle Swarm Optimization with AntColony Optimization (PSOACO) [6] These hybrid algo-rithms are aimed at reducing the probability of trapping inlocal optimum Recently a newly nature inspired optimiza-tion technique is originated namely GSA [7] The varioustypes of hybrid variant of Particle Swarm Optimization hadbeen discussed below

Ahmed et al [8] presented hybrid variant of PSO calledHPSOM The main idea of the HPSOM was to integrate theParticle Swarm Optimization (PSO) with Genetic Algorithm(GA) mutation technique The performance of the hybrid

HindawiJournal of Applied MathematicsVolume 2017 Article ID 2030489 15 pageshttpsdoiorg10115520172030489

2 Journal of Applied Mathematics

variant is tested on several numbers of classical functionsand on the basis of results obtained authors have shown thathybrid variant outperforms significantly the Particle SwarmOptimization variant in terms of solution quality solutionstability convergence speed and ability to find the globaloptimum

Mirjalili andHashimrsquos [9] newly hybrid population-basedalgorithm (PSOGSA) was proposed with the combinationof PSO and Gravitational Search Algorithm (GSA) Themain idea is to integrate the capability of exploitation inParticle Swarm Optimization with the capability of explo-ration in Gravitation Search Algorithm to synthesize bothvariantsrsquo strength The performance of hybrid variant wastested on several numbers of benchmark functions Onthe basis of results obtained authors have proven that thehybrid variant possesses a better capability to escape fromlocal optimums with faster convergence than the PSO andGSA

Zhang et al [10] presented a hybrid variant combiningPSOwith back-propagation (BP) variant called PSO-BP algo-rithm This variant can make use of not only strong globalsearching ability of the PSOA but also strong local searchingability of the BP algorithm The convergence speed andconvergent accuracy performance of newly hybrid variant ofPSO were tested on several numbers of classical functionsOn the basis of experimental results authors have shownthat the hybrid variant is better than the BP and AdaptiveParticle Swarm Optimization Algorithm (APSOA) and BPalgorithm in terms of solution quality and convergencespeed

Ouyang et al [11] presented a hybrid PSO variant whichcombines the advantages of PSO and Nelder-Mead SimplexMethod (SM) variant is put forward to solve systems of non-linear equations and can be used to overcome the difficultyin selecting good initial guess for SM and inaccuracy of PSOdue to being easily trapped into local optimum

Experimental results show that the hybrid variant hasprecision high convergence rate and great robustness and itcan give suitable results of nonlinear equations

Yu et al [12] proposed a newly hybrid Particle SwarmOptimization variant to solve several problems by combiningmodified velocity model and space transformation searchExperimental studies on eight classical test problems revealthat the hybrid PSO holds good performance in solving bothmultimodal and unimodal problems

Yu et al [13] proposed a novel algorithm HPSO-DE bydeveloping a balanced parameter between PSO and DEThis quality of this hybrid variant has been tested on sev-eral numbers of benchmark functions In comparison withthe Particle SwarmOptimization Differential Evolution andHPSO-DE variants the newly hybrid variant finds betterquality solutions more frequently is more effective in obtain-ing better quality solutions and works in a more effectiveway

Abd-Elazim andAli [14] presented a newly hybrid variantcombined with bacterial foraging optimization algorithm(BFOA) and PSO namely bacterial swarm optimization(BSO) In this hybrid variant the search directions of tumblebehavior for each bacterium are oriented by the global best

location and the individualrsquos best location of Particle SwarmOptimization The performance of new variant has beencompared with the PSO variant and BFOA variant On thebasis of the obtained results they have shown the validityof the hybrid variant in tuning SVC compared with othermetaheuristics

Grey Wolf Optimizer is recently developed metaheuris-tics inspired from the hunting mechanism and leadershiphierarchy of grey wolves in nature and has been successfullyapplied for solving optimizing key values in the cryptographyalgorithms [15] feature subset selection [16] time forecasting[17] optimal power flow problem [18] economic dispatchproblems [19] flow shop scheduling problem [20] and opti-mal design of double layer grids [21] Several algorithms havealso been developed to improve the convergence perfor-mance of Grey Wolf Optimizer that includes parallelizedGWO [22 23] binary GWO [24] integration of DE withGWO [25] hybrid GWOwith Genetic Algorithm (GA) [26]hybrid DE with GWO [27] and hybrid Grey Wolf Optimizerusing Elite Opposition Based Learning Strategy and SimplexMethod [28]

Mittal et al [29] developed a modified variant of theGWO called modified Grey Wolf Optimizer (mGWO) Anexponential decay function is used to improve the exploita-tion and exploration in the search space over the course ofgenerations On the basis of the obtained results authorsproved that the modified variant benefits from high explo-ration in comparison to the standard Grey Wolf Optimizerand the performance of the variant is verified on severalnumbers of standard benchmark and real life NP hard prob-lems

S Singh and S B Singh [30] present a newly modi-fied approach of GWO called Mean Grey Wolf Optimizer(MGWO) This approach has been originated by modifyingthe position update (encircling behavior) equations of GWOMGWOapproach has been tested on various standard bench-mark functions and the accuracy of existing approach hasalso been verified with PSO and GWO In addition authorshave also considered five datasets classification that have beenutilized to check the accuracy of the modified variant Theobtained results are compared with the results using manydifferent metaheuristic approaches that is Grey Wolf Opti-mization Particle Swarm Optimization Population-BasedIncremental Learning (PBIL) Ant Colony Optimization(ACO) and so forth On the basis of statistical results it hasbeen observed that the modified variant is able to find bestsolutions in terms of high level of accuracy in classificationand improved local optima avoidance

N Singh and S B Singh [31] present a new hybrid swarmintelligence heuristics called HGWOSCA that is exercised ontwenty-two benchmark test problems five biomedical datasetproblems and one sine dataset problem Hybrid GWOSCAis combination of Grey Wolf Optimizer (GWO) used forexploitation phase and Sine Cosine Algorithm (SCA) forexploration phase in uncertain environment The movementdirections and speed of the grey wolf (alpha) are improvedusing position update equations of SCA The numericaland statistical solutions obtained with hybrid GWOSCAapproach are comparedwith othermetaheuristics approaches

Journal of Applied Mathematics 3

InitializationInitialize 119897 119886 119908 and 119888

119908 = 05 + rand()2Evaluate the fitness of agents by using (5)while (119905 lt max no of iter)for each search agentUpdate the velocity and position by using (6)end forUpdate 119897 119886 119908 and 119888Evaluate the fitness of all search agentsUpdate positon first three agents119905 = 119905 + 1end whilereturn first best search agent position

Pseudocode 1 Pseudocode of the proposed variant (HPSOGWO)

such as Particle Swarm Optimization (PSO) Ant LionOptimizer (ALO) Whale Optimization Algorithm (WOA)Hybrid Approach GWO (HAGWO) Mean GWO (MGWO)Grey Wolf Optimizer (GWO) and Sine Cosine Algorithm(SCA) Results demonstrate that newly hybrid approach canbe highly effective in solving benchmark and real life appli-cations with or without constrained and unknown searchareas

In this study we present a newly hybrid variant com-bining PSO and GWO variants named HPSOGWO We usetwenty-three unimodal multimodal and fixed-dimensionmultimodal functions to compare the performance ofhybrid variant with both standard PSO and standardGWO

The rest of the paper is structured as follows The Par-ticle Swarm Optimization (PSO) and Grey Wolf Optimizer(GWO) algorithm are discussed in Sections 2 and 3 TheHPSOGWOmathematical model and pseudocode (shown inPseudocode 1) are also discussed in Section 4Thebenchmarktested functions are presented in Section 5 and results anddiscussion are represented in Section 6 respectively Finallythe conclusion of the work is offered in Section 7

2 Particle Swarm Optimization Variant

The PSO algorithm was firstly introduced by Kennedy andEberhart in [32] and its fundamental judgment was primarilyinspired by the simulation of the social behavior of animalssuch as bird flocking and fish schooling While searching forfood the birds are either scattered or go together before theysettle on the position where they can find the food Whilethe birds are searching for food from one position to anotherthere is always a bird that can smell the food very well that isthe bird is aware of the position where the food can be foundhaving the correct food resource message Because they aretransmitting the message particularly the useful message atany period while searching for the food from one position toanother the birds will finally flock to the position where foodcan be found

This approach is learned from animalrsquos behavior tocalculate global optimization functionsproblems and everypartner of the swarmcrowd is called a particle In PSOtechnique the position of each partner of the crowd inthe global search space is updated by two mathematicalequations These mathematical equations are

V119896+1119894 = V119896119894 + 11988811199031 (119901119896119894 minus 119909119896119894 ) + 11988821199032 (119892best minus 119909119896119894 )119909119896+1119894 = 119909119896119894 + V119896+1119894 (1)

3 Grey Wolf Optimizer (GWO)

The above literature shows that there are many swarmintelligence approaches originated so far many of theminspired by search behaviors and hunting But there is noswarm intelligence approach in the literature mimicking theleadership hierarchy of grey wolves well known for theirpack hunting Motivated by various algorithms Mirjaliliet al [33] presented a new swarm intelligence approachknown as Grey Wolf Optimizer (GWO) inspired by the greywolves and investigate its abilities in solving standard andreal life applications The GWO variant mimics the huntingmechanism and leadership hierarchy of greywolves in natureIn GWO the crowd is split into four different groups suchas alpha beta delta and omega which are employed forsimulating the leadership hierarchy

Grey wolf belongs to Canidae family Grey wolves aremeasured as apex predators meaning that they are at thetop of the food chain Grey wolves mostly prefer to live ina pack The leaders are a female and a male known as alphasThe alpha (120572) is generally liable for making decisions aboutsleeping time to wake hunting and so on

The second top level in the hierarchy of greywolves is beta(120573) The beta (120573) are subordinate wolves that help the firstlevel wolf (alpha) in decision making or other pack actionsThe second level wolf (beta) should respect the first level wolf(alpha) but orders the other lower level cipliner for the packThe second level wolf (beta) reinforces the first level wolf(alpharsquos) orders throughout the pack and gives feedback to thealpha

The third level ranking grey wolf is omega (120574) This wolfplays the role of scapegoat Third level grey wolves alwayshave to submit to all the other dominant wolves They arethe last wolves that are permitted to eat It may seem thatthe third level wolves do not have a significant personalityin the pack but it can be observed that the entire pack facesinternal struggle and troubles in case of losing the omegaThis is due to the venting of violence and frustration of allwolves by the omega (120574)This assists fulfilling the whole packand maintaining the dominance structure

If a wolf is not an alpha (120572) beta (120573) or omega (120574) sheheis known as a subordinate (or delta (120575)) Delta (120575) wolves haveto submit to alpha (120572) and beta (120573) but they dominate theomega (120574)

In addition three main steps of hunting searching forprey encircling prey and attacking prey are implemented toperform optimization


The encircling behavior of each agent of the crowd iscalculated by the following mathematical equations

119889 = 10038161003816100381610038161003816119888 sdot 119909119901(119905) minus 119909 (119905)10038161003816100381610038161003816119909 (119905 + 1) = 119909119901(119905) minus 119886 sdot 119889 (2)

The vectors 119886 and 119888 are mathematically formulated as follows

119886 = 2119897 sdot 1199031119888 = 2 sdot 1199032 (3)

31Hunting In order tomathematically simulate the huntingbehavior we suppose that the alpha (120572) beta (120573) and delta (120575)have better knowledge about the potential location of preyThe following mathematical equations are developed in thisregard

997888rarr119889 120572 = 10038161003816100381610038161003816997888rarr119888 1 sdot 997888rarr119909 120572 minus 997888rarr11990910038161003816100381610038161003816 997888rarr119889 120573 = 10038161003816100381610038161003816997888rarr119888 2 sdot 997888rarr119909 120573 minus 997888rarr11990910038161003816100381610038161003816 997888rarr119889 120575 = 10038161003816100381610038161003816997888rarr119888 3 sdot 997888rarr119909 120575 minus 997888rarr11990910038161003816100381610038161003816997888rarr119909 1 = 997888rarr119909 120572 minus 997888rarr119886 1 sdot (997888rarr119889 120572) 997888rarr119909 2 = 997888rarr119909 120573 minus 997888rarr119886 2 sdot (997888rarr119889 120573) 997888rarr119909 3 = 997888rarr119909 120575 minus 997888rarr119886 3 sdot (997888rarr119889 120575)

997888rarr119909 1 + 997888rarr119909 2 + 997888rarr119909 33997888rarr119886 (sdot) = 2997888rarr119897 sdot 997888rarr119903 1 minus 997888rarr119897997888rarr119888 (sdot) = 2 sdot 997888rarr119903 2

(4)

32 Searching for Prey and Attacking Prey 119860 is random valuein gap [minus2119886 2119886] When random value |119860| lt 1 the wolves areforced to attack the prey Searching for prey is the explorationability and attacking the prey is the exploitation ability Thearbitrary values of 119860 are utilized to force the search to moveaway from the prey

When |119860| gt 1 the members of the population are en-forced to diverge from the prey

4 A Newly Hybrid Algorithm

Many researchers have presented several hybridization vari-ants for heuristic variants According to Talbi [34] twovariants can be hybridized in low level or high level with relayor coevolutionary techniques as heterogeneous or homoge-neous

In this text we hybridize Particle Swarm Optimizationwith Grey Wolf Optimizer algorithm using low-level coevo-lutionary mixed hybrid The hybrid is low level because we

merge the functionalities of both variants It is coevolutionarybecause we do not use both variants one after the other Inother ways they run in parallel It is mixed because thereare two distinct variants that are involved in generating finalsolutions of the problems On the basis of this modificationwe improve the ability of exploitation in Particle SwarmOptimization with the ability of exploration in Grey WolfOptimizer to produce both variantsrsquo strength

In HPSOGWO first three agentsrsquo position is updated inthe search space by the proposedmathematical equations (5)Instead of using usualmathematical equations we control theexploration and exploitation of the grey wolf in the searchspace by inertia constant The modified set of governingequations are

997888rarr119889 120572 = 10038161003816100381610038161003816997888rarr119888 1 sdot 997888rarr119909 120572 minus 119908 lowast 997888rarr11990910038161003816100381610038161003816997888rarr119889 120573 = 10038161003816100381610038161003816997888rarr119888 2 sdot 997888rarr119909 120573 minus 119908 lowast 997888rarr11990910038161003816100381610038161003816997888rarr119889 120575 = 10038161003816100381610038161003816997888rarr119888 3 sdot 997888rarr119909 120575 minus 119908 lowast 997888rarr11990910038161003816100381610038161003816 (5)

In order to combine PSO andGWOvariants the velocity andupdated equation are proposed as follows

V119896+1119894 = 119908 lowast (V119896119894 + 11988811199031 (1199091 minus 119909119896119894 ) + 11988821199032 (1199092 minus 119909119896119894 )+ 11988831199033 (1199093 minus 119909119896119894 ))

119909119896+1119894 = 119909119896119894 + V119896+1119894 (6)

5 Testing Functions

In this section twenty-three benchmark problems are used totest the ability of HPSOGWOThese problems can be dividedinto three different groups unimodal multimodal and fixed-dimension multimodal functions The exact details of thesetest problems are shown in Tables 1ndash3

6 Analysis and Discussion on the Results

The PSO GWO and HPSOGWO pseudocodes are coded inMATLAB R2013a and implemented on Intel HD Graphics15610158401015840 3GB Memory i5 Processor 430M 169 HD LCDPentium-Intel Core and 320GB HDD Number of searchagents is 30 maximum number of iterations is 500 1198881 = 1198882 =05 and 1198883 = 05 119908 = 05 + rand()2 and 119897 isin [2 0] all theseparameter settings are applied to test the quality of hybrid andother metaheuristics

In this paper our objective is to present the best suitableoptimal solution as compared to other metaheuristics Thebest optimal solutions and best statistical values achieved byHPSOGWO variant for unimodal functions are shown inTables 4 and 5 respectively

Firstly we tested the ability of HPSOGWO PSO andGWO variant that were run 30 times on each unimodalfunction The HPSOGWO GWO and PSO algorithms haveto be run at least more than ten times to search for the best


Table 1 Unimodal benchmark functions

Function Dim Range 119891min

1198651(119909) = 119899sum119894=1

1199092119894 30 [minus100 100] 0

Parameter space

218161412

108060402

0

minus100

minus50 minus50

5050

00

100

100

minus100

times104

x2

x 4

F1(x

1x

2x

3x

4)

1198652(119909) = 119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 119899prod119894=1

10038161003816100381610038161199091198941003816100381610038161003816 30 [minus10 10] 0

minus100

minus50 minus50

50500

0

100

100

minus100

x2

x 4

Parameter space

12000

10000

8000

6000

4000

2000

0

F2(x

1x

2x

3x

4)

1198653(119909) = 119899sum119894=1

( 119894sum119895minus1

119909119895)2 30 [minus100 100] 04

455

353

252

15

051

0

F3(x

1x

2x

3x

4)

minus100

minus50 minus50

5050

00

100

100

minus100

x2

x 4

Parameter space

times104

1198654(119909) = max11989410038161003816100381610038161199091198941003816100381610038161003816 1 le 119894 le 119899 30 [minus100 100] 0

Parameter space

F4(x

1x

2x

3x

4)

minus100

minus50 minus50

50

50403020

90807060

10

500

0

0

100

100

100

minus100

x2

x 4

1198655(119909) = 119899minus1sum119894=1

[100 (119909119894+1 minus 1199092119894 )2 + (119909119894 minus 1)2] 30 [minus30 30] 0

minus200

minus100 minus100

100

18

16

14

12

10

8

6

4

2

0

10000

200

200

minus200

x2

x 4

Parameter space

times1010

F5(x

1x

2x

3x

4)

1198656(119909) = 119899sum119894=1

([119909119894 + 05])2 30 [minus100 100] 0 F6(x

1x

2x

3x

4)

minus100

minus50minus50

50500

0

100

25

2

15

1

05

0

100

minus100

x2

x 4

times104

Parameter space

1198657(119909) = 119899sum119894=1

1198941199094119894 + rand [0 1) 30 [minus128 128] 0 F7(x

1x

2x

3x

4)

minus1minus05minus05

05050

0

1

25

35

2

3

4

15

1

05

0

1

minus1

x2

x 4

Parameter space

numerical or statistical solutions It is again a general methodthat an algorithm is run on a test problemmany times and thebest optimal solutions mean and standard deviation of thesuperior obtained results in the last generation are evaluatedas metrics of performance The performance of proposedhybrid variant is compared to PSO and GWO variant interms of best optimal and statistical results Similarly theconvergence performances of HPSOGWO PSO and GWOvariant have been compared on the basis of graph see Figures1(a)ndash1(g) On the basis of obtained results and convergenceperformance of the variants we concluded that HPSOGWOis more reliable in giving superior quality results with rea-sonable iterations and avoids premature convergence of thesearch process to local optimal point and provides superiorexploration of the search course

Further we noted that the unimodal problems are suitablefor standard exploitation Therefore these results prove thesuperior performance of HPSOGWO in terms of exploitingthe optimum

Secondly the performance of the proposed hybrid varianthas been tested on six multimodal benchmark functions Incontrast to the multimodal problems unimodal benchmark

problems have many local optima with the number risingexponentially with dimension This makes them appro-priate for benchmarking the exploration capability of anapproachThe numerical and statistical results obtained fromHPSOGWO PSO and GWO algorithms are shown in Tables6 and 7

Experimental results show that the proposed variantfinds a superior quality of solution without trapping in localmaximum and to attain faster convergence performance seeFigures 2(a)ndash2(f) respectively This approach outperformsGWO and PSO variants on the majority of the multimodalbenchmark functions The obtained solutions also prove thatthe HPSOGWO variant has merit in terms of exploration

Thirdly the suitable solutions of fixed-dimension multi-modal benchmark functions are illustrated in Tables 8 and9 The fixed dimensional benchmark functions have manylocal optima with the number growing exponentially withdimension This makes them fitting for benchmarking theexploration capacity of a variant Experimental numericaland statistical solutions have shown that the proposed variantis able to find superior quality of results onmaximumnumberof fixed dimensional multimodal benchmark functions as


Table2Multim

odalbenchm

arkfunctio

ns

Functio

nDim

Range

119891 min119865 8(119909

)=119899 sum 119894=1minus

119909 119894sin(radic1003816 1003816 1003816 1003816119909

1198941003816 1003816 1003816 1003816)30

[minus50050

0]minus418

9829times5

Para

met

er sp

ace

0200

400

600

800

1000

minus200

minus400

minus600

minus800

minus1000

F8(x1x2x3x4)

minus500

00

500

500

minus500

x 2x4

119865 9(119909)=119899 sum 119894=1[1199092 119894minus10

cos(2120587

119909 119894)+10]

30[minus51

2512]

0

minus5

minus5

0

20

40

60

80

100

5

5

00

x 2x4

F9(x1x2x3x4)

Para

met

er sp

ace

119865 10(119909)=minus

20exp(minus0

2radic1 119899119899 sum 119894=11199092 119894)minus

exp(1 119899119899 sum 119894=1co

s(2120587119909 119894))

+20+119890

30[minus32

32]0

F10(x1x2x3x4)

minus20

minus20

20

20

25

20

15

10 5 0

00

x 2x4

Para

met

er sp

ace

119865 11(119909)=

1 4000119899 sum 119894=11199092 119894

minus119899 prod 119894=1cos(119909119894 radic 119894)+1

30[minus60

0600]

0

F11(x1x2x3x4)

minus500

minus500

500

500

140

120

100

80 60

40

20 0

00

x 2x4

Para

met

er sp

ace

119865 12(119909)=120587 1198991

0sin(120587119910 119894)

+119899minus1 sum 119894=1(119910 119894minus

1)2 [1+1

0sin2(120587119910 119894+1)+(

119910 119899minus1)2]+

119899 sum 119894=1119906(119909 1198941

01004)

30[minus50

50]0

Para

met

er sp

ace

minus5

minus5

minus10

minus10

10

10

100

5

50

150

50

0

0

x 2x4

F12(x1x2x3x4)

119910 119894=1+119909

119894+1 4

119906(119909 119894119886119896

119898)= 119896(119909 119894

minus119886)119898

119909 119894gt119886

0minus119886lt

119909 119894lt119886

119896(minus119909119894minus119886)119898

119909 119894ltminus119886

119865 13(119909)=0

1 sin2( 3120587119909119894) +119899 sum 119894=1( 119909119894minus1)2

[ 1+sin2( 3120587119909119894+1)]

+( 119909119899minus1)2

[ 1+sin2( 2120587119909119899)] +

119899 sum 119894=1119906( 119909 1198945

1004)

30[minus50

50]0

F13(x1x2x3x4)

minus5

minus5

55

15

5

0

0

10

0x 2

x4

Para

met

er sp

ace


Table3Fixed-dimensio

nmultim

odalbenchm

arkfunctio

ns

Functio

nDim

Range

119891 min119865 14(119909

)=(1 500+25 sum 119895=1

1119895+sum

2 119894=1(119909 119894minus

119886 119894119895)6)minus1

2[minus65

65]1

500

400

450

300

350

200

250

100

150 050

Para

met

er sp

ace

50

50

100

100

00

minus100minus100

minus50

minus50

x 2x4

F14(x1x2x3x4)

119865 15(119909)=11 sum 119894=1[119886 119894minus

119909 1(1198872 119894+119887 1198941199092)

1198872 119894+119887 119894119909 119894+

119909 4]24

[minus55]

000030

F15(x1x2x3x4)

6 5 4 3 2 1 0

Para

met

er sp

ace

5

5

00

minus5

minus5

x 2x4

times105

119865 16(119909)=4

1199092 1minus211199094 1+1 31199096 1

+119909 1119909 2minus

41199092 2+41199094 2

2[minus5

5]minus10

316

F17(x1x2x1)

600

500

400

300

200

100 0

Para

met

er sp

ace

5

5

00

minus5

minus5

x 2x1

119865 17(119909)=(

119909 2minus51 412058721199092 1+5 120587119909 1

minus6)2

+10(1minus

1 8120587)cos119909 1+

102

[minus55]

0398

F17(x1x2x3x4)

600

500

400

300

200

100 0

Para

met

er sp

ace

5

5

00

minus5

minus5

x 2x4

119865 18(119909 )=

[1+(119909 1+

119909 2+1)2 (

19minus14119909 1

+31199092 1minus14

119909 2+6119909 11199092+31199092 2)]

times[30+(2119909

1minus31199092)2 times

(18minus32119909 1

+121199092 1+

48119909 2minus36

119909 1119909 2+27

1199092 2)]2

[minus22]

3

F18(x1x2x3x4)

Para

met

er sp

ace

3

25 2

15 1

05 0 5

5

00

minus5

minus5

x 2x4

times108

119865 19(119909)=minus

4 sum 119894=1119888 119894exp(minus3 sum 119895=1

119886 119894119895(119909119895minus119901 119894119895

)2 )3

[13]

minus386

F19(x1x2x3x4)

0

Para

met

er sp

ace

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus016

minus018

minus02 5

5

00

minus5

minus5

x 2x4

119865 20(119909)=minus

4 sum 119894=1119888 119894exp(minus6 sum 119895=1

119886 119894119895(119909119895minus119901 119894119895

)2 )6

[01]

minus332

F20(x1x2x3x4)

0

Para

met

er sp

ace

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus016

minus018

minus02 5

5

00

minus5

minus5

x 2x4

119865 21(119909)=minus

5 sum 119894=1[(119883minus

119886 119894)( 119883minus119886 119894)119879+119888 119894]minus1

4[01

0]minus10

1532

F21(x1x2x3x4)

0

Para

met

er sp

ace

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08 5

5

00

minus5

minus5

x 2x4


Table3Con

tinued

Functio

nDim

Range

119891 min119865 22(119909

)=minus7 sum 119894=1[(

119883minus119886 119894)(119883

minus119886 119894)119879+119888 119894]minus1

4[01

0]minus10

4028

F22(x1x2x3x4)

0

Para

met

er sp

ace

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08 5

5

00

minus5

minus5

x 2x4

119865 23(119909)=minus

10 sum 119894=1[(

119883minus119886 119894)(119883

minus119886 119894)119879+119888 119894]minus1

4[01

0]minus10

5363

F23(x1x2x3x4)

0

Para

met

er sp

ace

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08 5

5

00

minus5

minus5

x 2x4


Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

10minus5

10minus10

10minus15

10minus20

10minus25

GWOPSOHPSOGWO

(a)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

60

50

40

30

20

10Best

scor

e obt

aine

d so

far

100

105

1010

10minus5

10minus10

10minus15

GWOPSOHPSOGWO

(b)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

104

102

100

10minus2

10minus4

10minus6

GWOPSOHPSOGWO

(c)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

10minus2

10minus4

10minus6

GWOPSOHPSOGWO

(d)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far 60

50

40

30

20

10

108

106

104

102

GWOPSOHPSOGWO

(e)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

GWOPSOHPSOGWO

(f)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

101

102

10minus1

10minus2

GWOPSOHPSOGWO

(g)

Figure 1 Convergence curve of PSO GWO and HPSOGWO variants on unimodal functions


Table 4 PSO GWO and HPSOGWO numerical results of unimodal benchmark functions

Problem number PSO GWO HPSOGWOMin Max Min Max Min Max(1) 20865119890 minus 04 55486119890 + 04 21467119890 minus 04 75295119890 + 04 13425119890 minus 26 76471119890 + 04(2) 00144 44920119890 + 13 11966119890 minus 16 66754119890 + 09 102228119890 minus 16 66918119890 + 11(3) 730238 14347119890 + 05 59193119890 minus 07 10861119890 + 05 22578119890 minus 06 10826119890 + 05(4) 16407 806950 27185119890 minus 07 828389 10483119890 minus 07 844532(5) 822944 24409119890 + 08 285703 25251119890 + 08 279722 26243119890 + 08(6) 41362119890 minus 05 50030119890 + 04 07515 68231119890 + 04 02250 71425119890 + 04(7) 01172 917832 00020 1212115 00016 1254297

Table 5 PSO GWO and HPSOGWO statistical results of unimodal benchmark functions

Problem number PSO GWO HPSOGWO120583 120590 120583 120590 120583 120590(1) 8712610 55907119890 + 03 7163515 54709119890 + 03 7113572 51807119890 + 03(2) 89840119890 + 10 20089119890 + 12 13358119890 + 07 29853119890 + 08 13201119890 + 09 29479119890 + 10(3) 36652119890 + 03 13536119890 + 04 30724119890 + 03 12253119890 + 04 29968119890 + 03 11503119890 + 04(4) 53716 127250 38205 147334 29241 114636(5) 87517119890 + 05 15804119890 + 07 18961119890 + 06 17700119890 + 07 15766119890 + 06 15510119890 + 07(6) 7699903 48108119890 + 03 7916688 54560119890 + 03 6744328 49782119890 + 03(7) 233307 260342 08775 78283 07084 63324Table 6 PSO GWO and HPSOGWO numerical results of multimodal benchmark functions

Problem number PSO GWO HPSOGWOMin Max Min Max Min Max(8) minus47156119890 + 03 minus16695119890 + 03 minus55578119890 + 03 minus19564119890 + 03 minus73662119890 + 03 minus19443119890 + 03(9) 408164 4154710 144438 4194465 11369119890 minus 13 4506618(10) 20287 204933 12168119890 minus 13 205736 10036119890 minus 13 207754(11) 00099 5237574 00098 5488569 0 5856470(12) 18758119890 minus 06 58263119890 + 08 00790 49310119890 + 08 00279 62705119890 + 08(13) 98225119890 minus 06 12859119890 + 09 08013 10729119890 + 09 03948 13101119890 + 09

Table 7 PSO GWO and HPSOGWO statistical results of multimodal benchmark functions

Problem number PSO GWO HPSOGWO120583 120590 120583 120590 120583 120590(8) minus40864119890 + 03 5909621 minus39454119890 + 03 9749720 minus48885119890 + 03 12577119890 + 03(9) 1715439 1104524 469310 524809 292298 516528(10) 38566 37053 07880 31024 07320 30649(11) 420351 1172690 53535 415628 52721 390734(12) 24226119890 + 06 31868119890 + 07 40785119890 + 06 37525119890 + 07 28970119890 + 06 35106119890 + 07(13) 57599119890 + 06 72267119890 + 07 60898119890 + 06 69809119890 + 07 10604119890 + 07 68155119890 + 07

compared to PSO and GWO variants Further the con-vergence performance of these variants has been plottedin Figures 3(a)ndash3(j) All numerical and statistical solutionsdemonstrate that the hybrid variant has merit in terms ofexploration

Finally the accuracy of the newly hybrid approach hasbeen verified using starting and ending time of the CPU(TIC and TOC) CPU time and clock These results areprovided in Tables 10ndash12 respectively It may be seen thatthe hybrid algorithm solved most of the standard benchmark


Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1033

minus1034

minus1035

minus1036

minus1037

minus1038

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(a)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far 10

0

10minus5

10minus10

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(b)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

10minus5

10minus10

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(c)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

10minus5

10minus10

10minus15

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(d)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

105

10minus5

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(e)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

105

10minus5

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(f)Figure 2 Convergence curve of PSO GWO and HPSOGWO variants on multimodal functions

problems in the least time as compared to other metaheuris-tics

To sum up all simulation results assert that theHPSOGWO algorithm is very helpful in improving the effi-ciency of the PSO and GWO in terms of result quality as wellas computational efforts

7 Conclusion

In this article a newly hybrid variant is proposed utilizingstrengths ofGWOandPSOThemain idea behinddevelopingis to improve the ability of exploitation in Particle SwarmOptimization with the ability of exploration in Grey Wolf


Table 8 PSO GWO and HPSOGWO numerical results of fixed-dimension multimodal benchmark functions

Problem number PSO GWO HPSOGWOMin Max Min Max Min Max(14) 69033 1641829 29821 843717 19920 2146473(15) 88478119890 minus 04 00336 46589119890 minus 04 00182 40039119890 minus 04 04235(16) minus10316 11130 minus10316 25179 minus10316 25179(17) 03979 03979 03979 04269 03979 07183(18) 30002 185091 30001 204796 30000 305918(19) minus38528 minus29348 minus38560 minus36581 minus38625 minus19195(20) minus33220 minus32050 minus31769 minus21483 minus33220 minus23517(21) minus51008 minus03904 minus101504 minus04449 minus101517 minus03823(22) minus104027 minus08673 minus104015 minus06342 minus104027 minus04999(23) minus51756 minus05856 minus105336 minus10135 minus105359 minus06038

Table 9 PSO GWO and HPSOGWO statistical results of fixed-dimension multimodal benchmark functions

Problem number PSO GWO HPSOGWO120583 120590 120583 120590 120583 120590(14) 73915 71059 35216 63370 24176 55102(15) 00013 00021 92219119890 minus 04 00013 00016 00189(16) minus10203 01116 minus10298 00203 minus10227 01101(17) 03979 73060119890 minus 05 04012 00191 04008 00147(18) 42260 96184 31308 13724 30609 12347(19) minus38521 00610 minus38502 00270 minus38548 00173(20) minus30887 03856 minus30989 01063 minus32722 00962(21) minus47758 06949 minus71098 23615 minus77395 20893(22) minus77834 34737 minus69558 26253 minus74304 24135(23) minus46073 10954 minus73778 25355 minus73814 27851Table 10 Time-consuming results of unimodal benchmark functions

Problem PSO GWO HPSOGWOTIC and TOC CPU time Clock TIC and TOC CPU time Clock TIC and TOC CPU time Clock(1) 102847 00176203 1052 101117 00526014 1021 100342 00105 1002(2) 10319 0011 1011 101083 00257021 1041 100562 0011 1011(3) 100905 0051 1018 101281 0002 1025 10031 00211 1017(4) 100417 000031 1018 101785 000011 1036 0789512 000015 1014(5) 10151 00523051 1019 101501 00213012 1025 100190 000010 1012(6) 100567 00276408 1115 101479 00136071 1024 100471 00132091 1010(7) 1018 00765021 1016 100507 00367013 1037 100140 00116701 1014

Table 11 Time-consuming results of multimodal benchmark functions

Problem PSO GWO HPSOGWOTIC and TOC CPU time Clock TIC and TOC CPU time Clock TIC and TOC CPU time Clock(8) 101232 00232001 1017 100181 00260007 1004 100211 00108021 1000(9) 101371 0200301 1019 100617 00417009 1010 100781 000321 1008(10) 101709 00717003 1011 101117 00837056 108 100125 00215302 1009(11) 101731 00011 1012 10160 00011 1009 100157 0006 1002(12) 102019 00211005 1014 100397 00417042 1010 100719 00272012 1007(13) 101214 0431507 1019 10157 00266007 1007 101237 00117009 1001


Objective space

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

102

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(a)

Objective space

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

10minus1

10minus2

10minus3

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(b)Objective space

60

50

40

30

20

10Best

scor

e obt

aine

d so

far 10

0

10minus5

10minus10

10minus15

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(c)

Objective space

Best

scor

e obt

aine

d so

far 10

2

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

60

50

40

30

20

10

(d)Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far 10

2

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(e)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1003

minus1004

minus1005

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(f)Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1001

minus1002

minus1003

minus1004

minus1005

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(g)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(h)Figure 3 Continued


Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(i)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(j)

Figure 3 Convergence curve of PSO GWO and HPSOGWO variants on fixed-dimension multimodal functions

Table 12 Time-consuming results of fixed-dimension multimodal benchmark functions

Problem PSO GWO HPSOGWOTIC and TOC CPU time Clock TIC and TOC CPU time Clock TIC and TOC CPU time Clock(14) 101375 00317 1087 100787 00415123 1012 100423 00003 1005(15) 102471 08783879 1069 100581 00529014 1017 101247 00126391 1009(16) 101691 000097 1015 0897854 00372012 1008 101219 000161 1006(17) 101719 08788782 1098 102139 00156071 1019 100194 00136321 1014(18) 103241 003715 1045 100596 00166701 1024 100279 000107 1011(19) 100607 05198069 1096 101685 00196041 1036 10203 00017981 1024(20) 101307 059741 1045 101652 00413012 1006 100734 000231 1001(21) 100009 000077 1011 101727 00146781 1001 102751 000001 1000(22) 0906710 061981 0856 100582 00712109 1013 10071 002041 1008(23) 103272 00335917 1066 100522 00192509 1010 101017 00087552 1002

Optimizer to produce both variantsrsquo strength Twenty-threeclassical problems are used to test the quality of the hybridvariant compared to GWO and PSO Experimental solutionsproved that hybrid variant is more reliable in giving superiorquality of solutions with reasonable computational iterationas compared to PSO and GWO

Conflicts of Interest

The authors declare no conflicts of interest

References

[1] A E Eiben and C A Schippers ldquoOn evolutionary explorationand exploitationrdquo Fundamenta Informaticae vol 35 no 1ndash4 pp35ndash50 1998

[2] D HWolpert andWGMacready ldquoNo free lunch theorems foroptimizationrdquo IEEE Transactions on Evolutionary Computationvol 1 no 1 pp 67ndash82 1997

[3] X Lai and M Zhang ldquoAn efficient ensemble of GA and PSOfor real function optimizationrdquo in Proceedings of the 2nd IEEEInternational Conference on Computer Science and InformationTechnology (ICCSIT rsquo09) pp 651ndash655 Beijing China August2009

[4] A A A Esmin G Lambert-Torres and G B Alvarenga ldquoHy-brid evolutionary algorithm based on PSO and GA mutationrdquoin Proceedings of the 6th International Conference on HybridIntelligent Systems and 4th Conference on Neuro-Computing andEvolving Intelligence (HIS-NCEI rsquo06) Rio de Janeiro BrazilDecember 2006

[5] L Li B Xue B Niu L Tan and J Wang ldquoA novel PSO-DE-based hybrid algorithm for global optimizationrdquo in AdvancedIntelligent Computing Theories and Applications With Aspectsof Artificial Intelligence vol 5227 of Lecture Notes in ComputerScience pp 785ndash793 Springer Berlin Germany 2008

[6] N Holden and A A Freitas ldquoA hybrid PSOACO algorithm fordiscovering classification rules in data miningrdquo Journal of Arti-ficial Evolution and Applications vol 2008 Article ID 316145 11pages 2008

[7] E Rashedi H Nezamabadi-pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[8] A Ahmed A Esmin and S Matwin ldquoHPSOM a hybrid parti-cle swarmoptimization algorithmwith geneticmutationrdquo Inter-national Journal of Innovative Computing Information andControl vol 9 no 5 pp 1919ndash1934 2013

[9] S Mirjalili and S Z M Hashim ldquoA new hybrid PSOGSAalgorithm for function optimizationrdquo in Proceedings of the Inter-national Conference on Computer and Information Application(ICCIA rsquo10) pp 374ndash377 Tianjin China November 2010


[10] J R Zhang T M Lok andM R Lyu ldquoA hybrid particle swarmoptimization-back-propagation algorithm for feedforwardneu-ral network trainingrdquo Applied Mathematics and Computationvol 185 no 2 pp 1026ndash1037 2007

[11] A Ouyang Y Zhou and Q Luo ldquoHybrid particle swarm opti-mization algorithm for solving systems of nonlinear equationsrdquoin Proceedings of the IEEE International Conference on GranularComputing (GRC rsquo09) pp 460ndash465 Nanchang China August2009

[12] S Yu Z Wu H Wang Z Chen and H Zhong ldquoA hybridparticle swarm optimization algorithm based on space trans-formation search and a modified velocity modelrdquo InternationalJournal of Numerical Analysis amp Modeling vol 9 no 2 pp 371ndash377 2012

[13] X Yu J Cao H Shan L Zhu and J Guo ldquoAn adaptive hybridalgorithm based on particle swarm optimization and differ-ential evolution for global optimizationrdquo The Scientific WorldJournal vol 2014 Article ID 215472 16 pages 2014

[14] S M Abd-Elazim and E S Ali ldquoA hybrid particles swarm opti-mization and bacterial foraging for power system stabilityenhancementrdquo Complexity vol 21 no 2 pp 245ndash255 2015

[15] K Shankar and P Eswaran ldquoA secure visual secret share (VSS)creation scheme in visual cryptography using elliptic curvecryptography with optimization techniquerdquo Australian Journalof Basic amp Applied Science vol 9 no 36 pp 150ndash163 2015

[16] E Emary H M Zawbaa C Grosan and A E HassenianldquoFeature subset selection approach by gray-wolf optimizationrdquoin Afro-European Conference for Industrial Advancement vol334 of Advances in Intelligent Systems and Computing pp 1ndash13Springer Berlin Germany 2015

[17] Y Yusof and Z Mustaffa ldquoTime series forecasting of energycommodity using grey wolf optimizerrdquo in Proceedings of theInternational Multi Conference of Engineers and ComputerScientists (IMECS rsquo15) Hong Kong March 2015

[18] A A El-Fergany and H M Hasanien ldquoSingle and multi-objective optimal power flow using grey wolf optimizer anddifferential evolution algorithmsrdquo Electric Power Componentsand Systems vol 43 no 13 pp 1548ndash1559 2015

[19] V K Kamboj S K Bath and J S Dhillon ldquoSolution of non-convex economic load dispatch problem using GreyWolf Opti-mizerrdquo Neural Computing and Applications vol 27 no 8 pp1301ndash1316 2016

[20] G M Komaki and V Kayvanfar ldquoGrey Wolf Optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015

[21] S Gholizadeh ldquoOptimal design of double layer grids consid-ering nonlinear ehavior by sequential grey wolf algorithmrdquoJournal of Optimization in Civil Engineering vol 5 no 4 pp511ndash523 2015

[22] T-S Pan T-K Dao T-T Nguyen and S-C Chu ldquoA communi-cation strategy for paralleling grey wolf optimizerrdquo Advances inIntelligent Systems and Computing vol 388 pp 253ndash262 2015

[23] J Jayapriya andM Arock ldquoA parallel GWO technique for align-ingmultiplemolecular sequencesrdquo inProceedings of the Interna-tional Conference on Advances in Computing Communicationsand Informatics (ICACCI rsquo15) pp 210ndash215 IEEE Kochi IndiaAugust 2015

[24] E Emary H M Zawbaa and A E Hassanien ldquoBinary greywolf optimization approaches for feature selectionrdquo Neurocom-puting vol 172 pp 371ndash381 2016

[25] A Zhu C Xu Z Li J Wu and Z Liu ldquoHybridizing grey Wolfoptimization with differential evolution for global optimizationand test scheduling for 3D stacked SoCrdquo Journal of SystemsEngineering and Electronics vol 26 no 2 pp 317ndash328 2015

[26] M A Tawhid and A F Ali ldquoA Hybrid grey wolf optimizer andgenetic algorithm for minimizing potential energy functionrdquo inMemetic Computing pp 1ndash13 2017

[27] D Jitkongchuen ldquoA hybrid differential evolution with greyWolfoptimizer for continuous global optimizationrdquo in Proceedingsof the 7th International Conference on Information Technologyand Electrical Engineering (ICITEE rsquo15) pp 51ndash54 Chiang MaiThailand October 2015

[28] S Zhang Q Luo and Y Zhou ldquoHybrid grey wolf optimizerusing elite opposition-based learning strategy and simplexmethodrdquo International Journal of Computational Intelligenceand Applications vol 16 no 2 Article ID 1750012 2017

[29] N Mittal U Singh and B S Sohi ldquoModified grey wolf opti-mizer for global engineering optimizationrdquo Applied Compu-tational Intelligence and Soft Computing vol 2016 Article ID7950348 16 pages 2016

[30] S Singh and S B Singh ldquoMean grey wolf optimizerrdquo Evolution-ary Bioinformatics vol 13 pp 1ndash28 2017

[31] N Singh and S B Singh ldquoA novel hyrid GWO-SCA approachfor standard and realrdquo Engineering Science and Technology

[32] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995

[33] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014

[34] E-G Talbi ldquoA taxonomy of hybrid metaheuristicsrdquo Journal ofHeuristics vol 8 no 5 pp 541ndash564 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

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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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


variant is tested on several numbers of classical functionsand on the basis of results obtained authors have shown thathybrid variant outperforms significantly the Particle SwarmOptimization variant in terms of solution quality solutionstability convergence speed and ability to find the globaloptimum

Mirjalili andHashimrsquos [9] newly hybrid population-basedalgorithm (PSOGSA) was proposed with the combinationof PSO and Gravitational Search Algorithm (GSA) Themain idea is to integrate the capability of exploitation inParticle Swarm Optimization with the capability of explo-ration in Gravitation Search Algorithm to synthesize bothvariantsrsquo strength The performance of hybrid variant wastested on several numbers of benchmark functions Onthe basis of results obtained authors have proven that thehybrid variant possesses a better capability to escape fromlocal optimums with faster convergence than the PSO andGSA

Zhang et al [10] presented a hybrid variant combiningPSOwith back-propagation (BP) variant called PSO-BP algo-rithm This variant can make use of not only strong globalsearching ability of the PSOA but also strong local searchingability of the BP algorithm The convergence speed andconvergent accuracy performance of newly hybrid variant ofPSO were tested on several numbers of classical functionsOn the basis of experimental results authors have shownthat the hybrid variant is better than the BP and AdaptiveParticle Swarm Optimization Algorithm (APSOA) and BPalgorithm in terms of solution quality and convergencespeed

Ouyang et al [11] presented a hybrid PSO variant whichcombines the advantages of PSO and Nelder-Mead SimplexMethod (SM) variant is put forward to solve systems of non-linear equations and can be used to overcome the difficultyin selecting good initial guess for SM and inaccuracy of PSOdue to being easily trapped into local optimum

Experimental results show that the hybrid variant hasprecision high convergence rate and great robustness and itcan give suitable results of nonlinear equations

Yu et al [12] proposed a newly hybrid Particle SwarmOptimization variant to solve several problems by combiningmodified velocity model and space transformation searchExperimental studies on eight classical test problems revealthat the hybrid PSO holds good performance in solving bothmultimodal and unimodal problems

Yu et al [13] proposed a novel algorithm HPSO-DE bydeveloping a balanced parameter between PSO and DEThis quality of this hybrid variant has been tested on sev-eral numbers of benchmark functions In comparison withthe Particle SwarmOptimization Differential Evolution andHPSO-DE variants the newly hybrid variant finds betterquality solutions more frequently is more effective in obtain-ing better quality solutions and works in a more effectiveway

Abd-Elazim andAli [14] presented a newly hybrid variantcombined with bacterial foraging optimization algorithm(BFOA) and PSO namely bacterial swarm optimization(BSO) In this hybrid variant the search directions of tumblebehavior for each bacterium are oriented by the global best

location and the individualrsquos best location of Particle SwarmOptimization The performance of new variant has beencompared with the PSO variant and BFOA variant On thebasis of the obtained results they have shown the validityof the hybrid variant in tuning SVC compared with othermetaheuristics

Grey Wolf Optimizer is recently developed metaheuris-tics inspired from the hunting mechanism and leadershiphierarchy of grey wolves in nature and has been successfullyapplied for solving optimizing key values in the cryptographyalgorithms [15] feature subset selection [16] time forecasting[17] optimal power flow problem [18] economic dispatchproblems [19] flow shop scheduling problem [20] and opti-mal design of double layer grids [21] Several algorithms havealso been developed to improve the convergence perfor-mance of Grey Wolf Optimizer that includes parallelizedGWO [22 23] binary GWO [24] integration of DE withGWO [25] hybrid GWOwith Genetic Algorithm (GA) [26]hybrid DE with GWO [27] and hybrid Grey Wolf Optimizerusing Elite Opposition Based Learning Strategy and SimplexMethod [28]

Mittal et al [29] developed a modified variant of theGWO called modified Grey Wolf Optimizer (mGWO) Anexponential decay function is used to improve the exploita-tion and exploration in the search space over the course ofgenerations On the basis of the obtained results authorsproved that the modified variant benefits from high explo-ration in comparison to the standard Grey Wolf Optimizerand the performance of the variant is verified on severalnumbers of standard benchmark and real life NP hard prob-lems

S Singh and S B Singh [30] present a newly modi-fied approach of GWO called Mean Grey Wolf Optimizer(MGWO) This approach has been originated by modifyingthe position update (encircling behavior) equations of GWOMGWOapproach has been tested on various standard bench-mark functions and the accuracy of existing approach hasalso been verified with PSO and GWO In addition authorshave also considered five datasets classification that have beenutilized to check the accuracy of the modified variant Theobtained results are compared with the results using manydifferent metaheuristic approaches that is Grey Wolf Opti-mization Particle Swarm Optimization Population-BasedIncremental Learning (PBIL) Ant Colony Optimization(ACO) and so forth On the basis of statistical results it hasbeen observed that the modified variant is able to find bestsolutions in terms of high level of accuracy in classificationand improved local optima avoidance

N Singh and S B Singh [31] present a new hybrid swarmintelligence heuristics called HGWOSCA that is exercised ontwenty-two benchmark test problems five biomedical datasetproblems and one sine dataset problem Hybrid GWOSCAis combination of Grey Wolf Optimizer (GWO) used forexploitation phase and Sine Cosine Algorithm (SCA) forexploration phase in uncertain environment The movementdirections and speed of the grey wolf (alpha) are improvedusing position update equations of SCA The numericaland statistical solutions obtained with hybrid GWOSCAapproach are comparedwith othermetaheuristics approaches























119886 = 2119897 sdot 1199031119888 = 2 sdot 1199032 (3)




(4)











119909119896+1119894 = 119909119896119894 + V119896+1119894 (6)

5 Testing Functions









1198651(119909) = 119899sum119894=1

1199092119894 30 [minus100 100] 0

Parameter space

218161412

108060402

0

minus100

minus50 minus50

5050

00

100

100

minus100

times104

x2

x 4

F1(x

1x

2x

3x

4)

1198652(119909) = 119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 119899prod119894=1

10038161003816100381610038161199091198941003816100381610038161003816 30 [minus10 10] 0

minus100

minus50 minus50

50500

0

100

100

minus100

x2

x 4

Parameter space

12000

10000

8000

6000

4000

2000

0

F2(x

1x

2x

3x

4)

1198653(119909) = 119899sum119894=1

( 119894sum119895minus1

119909119895)2 30 [minus100 100] 04

455

353

252

15

051

0

F3(x

1x

2x

3x

4)

minus100

minus50 minus50

5050

00

100

100

minus100

x2

x 4

Parameter space

times104

1198654(119909) = max11989410038161003816100381610038161199091198941003816100381610038161003816 1 le 119894 le 119899 30 [minus100 100] 0

Parameter space

F4(x

1x

2x

3x

4)

minus100

minus50 minus50

50

50403020

90807060

10

500

0

0

100

100

100

minus100

x2

x 4

1198655(119909) = 119899minus1sum119894=1


minus200

minus100 minus100

100

18

16

14

12

10

8

6

4

2

0

10000

200

200

minus200

x2

x 4

Parameter space

times1010

F5(x

1x

2x

3x

4)

1198656(119909) = 119899sum119894=1

([119909119894 + 05])2 30 [minus100 100] 0 F6(x

1x

2x

3x

4)

minus100

minus50minus50

50500

0

100

25

2

15

1

05

0

100

minus100

x2

x 4

times104

Parameter space

1198657(119909) = 119899sum119894=1

1198941199094119894 + rand [0 1) 30 [minus128 128] 0 F7(x

1x

2x

3x

4)


05050

0

1

25

35

2

3

4

15

1

05

0

1

minus1

x2

x 4

Parameter space








Table2Multim

odalbenchm

arkfunctio

ns

Functio

nDim

Range

119891 min119865 8(119909

)=119899 sum 119894=1minus

119909 119894sin(radic1003816 1003816 1003816 1003816119909

1198941003816 1003816 1003816 1003816)30

[minus50050

0]minus418

9829times5

Para

met

er sp

ace

0200

400

600

800

1000

minus200

minus400

minus600

minus800

minus1000

F8(x1x2x3x4)

minus500

00

500

500

minus500

x 2x4

119865 9(119909)=119899 sum 119894=1[1199092 119894minus10

cos(2120587

119909 119894)+10]

30[minus51

2512]

0

minus5

minus5

0

20

40

60

80

100

5

5

00

x 2x4

F9(x1x2x3x4)

Para

met

er sp

ace

119865 10(119909)=minus

20exp(minus0

2radic1 119899119899 sum 119894=11199092 119894)minus

exp(1 119899119899 sum 119894=1co

s(2120587119909 119894))

+20+119890

30[minus32

32]0

F10(x1x2x3x4)

minus20

minus20

20

20

25

20

15

10 5 0

00

x 2x4

Para

met

er sp

ace

119865 11(119909)=

1 4000119899 sum 119894=11199092 119894


30[minus60

0600]

0

F11(x1x2x3x4)

minus500

minus500

500

500

140

120

100

80 60

40

20 0

00

x 2x4

Para

met

er sp

ace

119865 12(119909)=120587 1198991

0sin(120587119910 119894)

+119899minus1 sum 119894=1(119910 119894minus

1)2 [1+1

0sin2(120587119910 119894+1)+(

119910 119899minus1)2]+

119899 sum 119894=1119906(119909 1198941

01004)

30[minus50

50]0

Para

met

er sp

ace

minus5

minus5

minus10

minus10

10

10

100

5

50

150

50

0

0

x 2x4

F12(x1x2x3x4)

119910 119894=1+119909

119894+1 4

119906(119909 119894119886119896

119898)= 119896(119909 119894

minus119886)119898

119909 119894gt119886

0minus119886lt

119909 119894lt119886

119896(minus119909119894minus119886)119898

119909 119894ltminus119886

119865 13(119909)=0

1 sin2( 3120587119909119894) +119899 sum 119894=1( 119909119894minus1)2

[ 1+sin2( 3120587119909119894+1)]

+( 119909119899minus1)2

[ 1+sin2( 2120587119909119899)] +

119899 sum 119894=1119906( 119909 1198945

1004)

30[minus50

50]0

F13(x1x2x3x4)

minus5

minus5

55

15

5

0

0

10

0x 2

x4

Para

met

er sp

ace



nmultim

odalbenchm

arkfunctio

ns

Functio

nDim

Range

119891 min119865 14(119909

)=(1 500+25 sum 119895=1

1119895+sum

2 119894=1(119909 119894minus

119886 119894119895)6)minus1

2[minus65

65]1

500

400

450

300

350

200

250

100

150 050

Para

met

er sp

ace

50

50

100

100

00

minus100minus100

minus50

minus50

x 2x4

F14(x1x2x3x4)

119865 15(119909)=11 sum 119894=1[119886 119894minus

119909 1(1198872 119894+119887 1198941199092)

1198872 119894+119887 119894119909 119894+

119909 4]24

[minus55]

000030

F15(x1x2x3x4)

6 5 4 3 2 1 0

Para

met

er sp

ace

5

5

00

minus5

minus5

x 2x4

times105

119865 16(119909)=4

1199092 1minus211199094 1+1 31199096 1

+119909 1119909 2minus

41199092 2+41199094 2

2[minus5

5]minus10

316

F17(x1x2x1)

600

500

400

300

200

100 0

Para

met

er sp

ace

5

5

00

minus5

minus5

x 2x1

119865 17(119909)=(

119909 2minus51 412058721199092 1+5 120587119909 1

minus6)2

+10(1minus

1 8120587)cos119909 1+

102

[minus55]

0398

F17(x1x2x3x4)

600

500

400

300

200

100 0

Para

met

er sp

ace

5

5

00

minus5

minus5

x 2x4

119865 18(119909 )=

[1+(119909 1+

119909 2+1)2 (

19minus14119909 1

+31199092 1minus14

119909 2+6119909 11199092+31199092 2)]

times[30+(2119909


(18minus32119909 1

+121199092 1+

48119909 2minus36

119909 1119909 2+27

1199092 2)]2

[minus22]

3

F18(x1x2x3x4)

Para

met

er sp

ace

3

25 2

15 1

05 0 5

5

00

minus5

minus5

x 2x4

times108

119865 19(119909)=minus

4 sum 119894=1119888 119894exp(minus3 sum 119895=1

119886 119894119895(119909119895minus119901 119894119895

)2 )3

[13]

minus386

F19(x1x2x3x4)

0

Para

met

er sp

ace

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus016

minus018

minus02 5

5

00

minus5

minus5

x 2x4

119865 20(119909)=minus

4 sum 119894=1119888 119894exp(minus6 sum 119895=1

119886 119894119895(119909119895minus119901 119894119895

)2 )6

[01]

minus332

F20(x1x2x3x4)

0

Para

met

er sp

ace

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus016

minus018

minus02 5

5

00

minus5

minus5

x 2x4

119865 21(119909)=minus

5 sum 119894=1[(119883minus

119886 119894)( 119883minus119886 119894)119879+119888 119894]minus1

4[01

0]minus10

1532

F21(x1x2x3x4)

0

Para

met

er sp

ace

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08 5

5

00

minus5

minus5

x 2x4


Table3Con

tinued

Functio

nDim

Range

119891 min119865 22(119909

)=minus7 sum 119894=1[(

119883minus119886 119894)(119883

minus119886 119894)119879+119888 119894]minus1

4[01

0]minus10

4028

F22(x1x2x3x4)

0

Para

met

er sp

ace

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08 5

5

00

minus5

minus5

x 2x4

119865 23(119909)=minus

10 sum 119894=1[(

119883minus119886 119894)(119883

minus119886 119894)119879+119888 119894]minus1

4[01

0]minus10

5363

F23(x1x2x3x4)

0

Para

met

er sp

ace

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08 5

5

00

minus5

minus5

x 2x4


Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

10minus5

10minus10

10minus15

10minus20

10minus25

GWOPSOHPSOGWO

(a)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

60

50

40

30

20

10Best

scor

e obt

aine

d so

far

100

105

1010

10minus5

10minus10

10minus15

GWOPSOHPSOGWO

(b)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

104

102

100

10minus2

10minus4

10minus6

GWOPSOHPSOGWO

(c)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

10minus2

10minus4

10minus6

GWOPSOHPSOGWO

(d)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far 60

50

40

30

20

10

108

106

104

102

GWOPSOHPSOGWO

(e)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

GWOPSOHPSOGWO

(f)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

101

102

10minus1

10minus2

GWOPSOHPSOGWO

(g)













Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1033

minus1034

minus1035

minus1036

minus1037

minus1038

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(a)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far 10

0

10minus5

10minus10

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(b)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

10minus5

10minus10

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(c)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

10minus5

10minus10

10minus15

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(d)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

105

10minus5

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(e)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

105

10minus5

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO




7 Conclusion











Objective space

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

102

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(a)

Objective space

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

10minus1

10minus2

10minus3

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(b)Objective space

60

50

40

30

20

10Best

scor

e obt

aine

d so

far 10

0

10minus5

10minus10

10minus15

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(c)

Objective space

Best

scor

e obt

aine

d so

far 10

2

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

60

50

40

30

20

10

(d)Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far 10

2

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(e)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1003

minus1004

minus1005

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(f)Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1001

minus1002

minus1003

minus1004

minus1005

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(g)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO



Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(i)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(j)







References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of


























119886 = 2119897 sdot 1199031119888 = 2 sdot 1199032 (3)




(4)











119909119896+1119894 = 119909119896119894 + V119896+1119894 (6)

5 Testing Functions









1198651(119909) = 119899sum119894=1

1199092119894 30 [minus100 100] 0

Parameter space

218161412

108060402

0

minus100

minus50 minus50

5050

00

100

100

minus100

times104

x2

x 4

F1(x

1x

2x

3x

4)

1198652(119909) = 119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 119899prod119894=1

10038161003816100381610038161199091198941003816100381610038161003816 30 [minus10 10] 0

minus100

minus50 minus50

50500

0

100

100

minus100

x2

x 4

Parameter space

12000

10000

8000

6000

4000

2000

0

F2(x

1x

2x

3x

4)

1198653(119909) = 119899sum119894=1

( 119894sum119895minus1

119909119895)2 30 [minus100 100] 04

455

353

252

15

051

0

F3(x

1x

2x

3x

4)

minus100

minus50 minus50

5050

00

100

100

minus100

x2

x 4

Parameter space

times104

1198654(119909) = max11989410038161003816100381610038161199091198941003816100381610038161003816 1 le 119894 le 119899 30 [minus100 100] 0

Parameter space

F4(x

1x

2x

3x

4)

minus100

minus50 minus50

50

50403020

90807060

10

500

0

0

100

100

100

minus100

x2

x 4

1198655(119909) = 119899minus1sum119894=1


minus200

minus100 minus100

100

18

16

14

12

10

8

6

4

2

0

10000

200

200

minus200

x2

x 4

Parameter space

times1010

F5(x

1x

2x

3x

4)

1198656(119909) = 119899sum119894=1

([119909119894 + 05])2 30 [minus100 100] 0 F6(x

1x

2x

3x

4)

minus100

minus50minus50

50500

0

100

25

2

15

1

05

0

100

minus100

x2

x 4

times104

Parameter space

1198657(119909) = 119899sum119894=1

1198941199094119894 + rand [0 1) 30 [minus128 128] 0 F7(x

1x

2x

3x

4)


05050

0

1

25

35

2

3

4

15

1

05

0

1

minus1

x2

x 4

Parameter space








Table2Multim

odalbenchm

arkfunctio

ns

Functio

nDim

Range

119891 min119865 8(119909

)=119899 sum 119894=1minus

119909 119894sin(radic1003816 1003816 1003816 1003816119909

1198941003816 1003816 1003816 1003816)30

[minus50050

0]minus418

9829times5

Para

met

er sp

ace

0200

400

600

800

1000

minus200

minus400

minus600

minus800

minus1000

F8(x1x2x3x4)

minus500

00

500

500

minus500

x 2x4

119865 9(119909)=119899 sum 119894=1[1199092 119894minus10

cos(2120587

119909 119894)+10]

30[minus51

2512]

0

minus5

minus5

0

20

40

60

80

100

5

5

00

x 2x4

F9(x1x2x3x4)

Para

met

er sp

ace

119865 10(119909)=minus

20exp(minus0

2radic1 119899119899 sum 119894=11199092 119894)minus

exp(1 119899119899 sum 119894=1co

s(2120587119909 119894))

+20+119890

30[minus32

32]0

F10(x1x2x3x4)

minus20

minus20

20

20

25

20

15

10 5 0

00

x 2x4

Para

met

er sp

ace

119865 11(119909)=

1 4000119899 sum 119894=11199092 119894


30[minus60

0600]

0

F11(x1x2x3x4)

minus500

minus500

500

500

140

120

100

80 60

40

20 0

00

x 2x4

Para

met

er sp

ace

119865 12(119909)=120587 1198991

0sin(120587119910 119894)

+119899minus1 sum 119894=1(119910 119894minus

1)2 [1+1

0sin2(120587119910 119894+1)+(

119910 119899minus1)2]+

119899 sum 119894=1119906(119909 1198941

01004)

30[minus50

50]0

Para

met

er sp

ace

minus5

minus5

minus10

minus10

10

10

100

5

50

150

50

0

0

x 2x4

F12(x1x2x3x4)

119910 119894=1+119909

119894+1 4

119906(119909 119894119886119896

119898)= 119896(119909 119894

minus119886)119898

119909 119894gt119886

0minus119886lt

119909 119894lt119886

119896(minus119909119894minus119886)119898

119909 119894ltminus119886

119865 13(119909)=0

1 sin2( 3120587119909119894) +119899 sum 119894=1( 119909119894minus1)2

[ 1+sin2( 3120587119909119894+1)]

+( 119909119899minus1)2

[ 1+sin2( 2120587119909119899)] +

119899 sum 119894=1119906( 119909 1198945

1004)

30[minus50

50]0

F13(x1x2x3x4)

minus5

minus5

55

15

5

0

0

10

0x 2

x4

Para

met

er sp

ace



nmultim

odalbenchm

arkfunctio

ns

Functio

nDim

Range

119891 min119865 14(119909

)=(1 500+25 sum 119895=1

1119895+sum

2 119894=1(119909 119894minus

119886 119894119895)6)minus1

2[minus65

65]1

500

400

450

300

350

200

250

100

150 050

Para

met

er sp

ace

50

50

100

100

00

minus100minus100

minus50

minus50

x 2x4

F14(x1x2x3x4)

119865 15(119909)=11 sum 119894=1[119886 119894minus

119909 1(1198872 119894+119887 1198941199092)

1198872 119894+119887 119894119909 119894+

119909 4]24

[minus55]

000030

F15(x1x2x3x4)

6 5 4 3 2 1 0

Para

met

er sp

ace

5

5

00

minus5

minus5

x 2x4

times105

119865 16(119909)=4

1199092 1minus211199094 1+1 31199096 1

+119909 1119909 2minus

41199092 2+41199094 2

2[minus5

5]minus10

316

F17(x1x2x1)

600

500

400

300

200

100 0

Para

met

er sp

ace

5

5

00

minus5

minus5

x 2x1

119865 17(119909)=(

119909 2minus51 412058721199092 1+5 120587119909 1

minus6)2

+10(1minus

1 8120587)cos119909 1+

102

[minus55]

0398

F17(x1x2x3x4)

600

500

400

300

200

100 0

Para

met

er sp

ace

5

5

00

minus5

minus5

x 2x4

119865 18(119909 )=

[1+(119909 1+

119909 2+1)2 (

19minus14119909 1

+31199092 1minus14

119909 2+6119909 11199092+31199092 2)]

times[30+(2119909


(18minus32119909 1

+121199092 1+

48119909 2minus36

119909 1119909 2+27

1199092 2)]2

[minus22]

3

F18(x1x2x3x4)

Para

met

er sp

ace

3

25 2

15 1

05 0 5

5

00

minus5

minus5

x 2x4

times108

119865 19(119909)=minus

4 sum 119894=1119888 119894exp(minus3 sum 119895=1

119886 119894119895(119909119895minus119901 119894119895

)2 )3

[13]

minus386

F19(x1x2x3x4)

0

Para

met

er sp

ace

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus016

minus018

minus02 5

5

00

minus5

minus5

x 2x4

119865 20(119909)=minus

4 sum 119894=1119888 119894exp(minus6 sum 119895=1

119886 119894119895(119909119895minus119901 119894119895

)2 )6

[01]

minus332

F20(x1x2x3x4)

0

Para

met

er sp

ace

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus016

minus018

minus02 5

5

00

minus5

minus5

x 2x4

119865 21(119909)=minus

5 sum 119894=1[(119883minus

119886 119894)( 119883minus119886 119894)119879+119888 119894]minus1

4[01

0]minus10

1532

F21(x1x2x3x4)

0

Para

met

er sp

ace

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08 5

5

00

minus5

minus5

x 2x4


Table3Con

tinued

Functio

nDim

Range

119891 min119865 22(119909

)=minus7 sum 119894=1[(

119883minus119886 119894)(119883

minus119886 119894)119879+119888 119894]minus1

4[01

0]minus10

4028

F22(x1x2x3x4)

0

Para

met

er sp

ace

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08 5

5

00

minus5

minus5

x 2x4

119865 23(119909)=minus

10 sum 119894=1[(

119883minus119886 119894)(119883

minus119886 119894)119879+119888 119894]minus1

4[01

0]minus10

5363

F23(x1x2x3x4)

0

Para

met

er sp

ace

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08 5

5

00

minus5

minus5

x 2x4


Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

10minus5

10minus10

10minus15

10minus20

10minus25

GWOPSOHPSOGWO

(a)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

60

50

40

30

20

10Best

scor

e obt

aine

d so

far

100

105

1010

10minus5

10minus10

10minus15

GWOPSOHPSOGWO

(b)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

104

102

100

10minus2

10minus4

10minus6

GWOPSOHPSOGWO

(c)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

10minus2

10minus4

10minus6

GWOPSOHPSOGWO

(d)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far 60

50

40

30

20

10

108

106

104

102

GWOPSOHPSOGWO

(e)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

GWOPSOHPSOGWO

(f)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

101

102

10minus1

10minus2

GWOPSOHPSOGWO

(g)













Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1033

minus1034

minus1035

minus1036

minus1037

minus1038

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(a)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far 10

0

10minus5

10minus10

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(b)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

10minus5

10minus10

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(c)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

10minus5

10minus10

10minus15

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(d)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

105

10minus5

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(e)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

105

10minus5

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO




7 Conclusion











Objective space

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

102

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(a)

Objective space

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

10minus1

10minus2

10minus3

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(b)Objective space

60

50

40

30

20

10Best

scor

e obt

aine

d so

far 10

0

10minus5

10minus10

10minus15

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(c)

Objective space

Best

scor

e obt

aine

d so

far 10

2

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

60

50

40

30

20

10

(d)Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far 10

2

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(e)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1003

minus1004

minus1005

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(f)Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1001

minus1002

minus1003

minus1004

minus1005

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(g)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO



Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(i)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(j)







References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







1198651(119909) = 119899sum119894=1

1199092119894 30 [minus100 100] 0

Parameter space

218161412

108060402

0

minus100

minus50 minus50

5050

00

100

100

minus100

times104

x2

x 4

F1(x

1x

2x

3x

4)

1198652(119909) = 119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 119899prod119894=1

10038161003816100381610038161199091198941003816100381610038161003816 30 [minus10 10] 0

minus100

minus50 minus50

50500

0

100

100

minus100

x2

x 4

Parameter space

12000

10000

8000

6000

4000

2000

0

F2(x

1x

2x

3x

4)

1198653(119909) = 119899sum119894=1

( 119894sum119895minus1

119909119895)2 30 [minus100 100] 04

455

353

252

15

051

0

F3(x

1x

2x

3x

4)

minus100

minus50 minus50

5050

00

100

100

minus100

x2

x 4

Parameter space

times104

1198654(119909) = max11989410038161003816100381610038161199091198941003816100381610038161003816 1 le 119894 le 119899 30 [minus100 100] 0

Parameter space

F4(x

1x

2x

3x

4)

minus100

minus50 minus50

50

50403020

90807060

10

500

0

0

100

100

100

minus100

x2

x 4

1198655(119909) = 119899minus1sum119894=1


minus200

minus100 minus100

100

18

16

14

12

10

8

6

4

2

0

10000

200

200

minus200

x2

x 4

Parameter space

times1010

F5(x

1x

2x

3x

4)

1198656(119909) = 119899sum119894=1

([119909119894 + 05])2 30 [minus100 100] 0 F6(x

1x

2x

3x

4)

minus100

minus50minus50

50500

0

100

25

2

15

1

05

0

100

minus100

x2

x 4

times104

Parameter space

1198657(119909) = 119899sum119894=1

1198941199094119894 + rand [0 1) 30 [minus128 128] 0 F7(x

1x

2x

3x

4)


05050

0

1

25

35

2

3

4

15

1

05

0

1

minus1

x2

x 4

Parameter space








Table2Multim

odalbenchm

arkfunctio

ns

Functio

nDim

Range

119891 min119865 8(119909

)=119899 sum 119894=1minus

119909 119894sin(radic1003816 1003816 1003816 1003816119909

1198941003816 1003816 1003816 1003816)30

[minus50050

0]minus418

9829times5

Para

met

er sp

ace

0200

400

600

800

1000

minus200

minus400

minus600

minus800

minus1000

F8(x1x2x3x4)

minus500

00

500

500

minus500

x 2x4

119865 9(119909)=119899 sum 119894=1[1199092 119894minus10

cos(2120587

119909 119894)+10]

30[minus51

2512]

0

minus5

minus5

0

20

40

60

80

100

5

5

00

x 2x4

F9(x1x2x3x4)

Para

met

er sp

ace

119865 10(119909)=minus

20exp(minus0

2radic1 119899119899 sum 119894=11199092 119894)minus

exp(1 119899119899 sum 119894=1co

s(2120587119909 119894))

+20+119890

30[minus32

32]0

F10(x1x2x3x4)

minus20

minus20

20

20

25

20

15

10 5 0

00

x 2x4

Para

met

er sp

ace

119865 11(119909)=

1 4000119899 sum 119894=11199092 119894


30[minus60

0600]

0

F11(x1x2x3x4)

minus500

minus500

500

500

140

120

100

80 60

40

20 0

00

x 2x4

Para

met

er sp

ace

119865 12(119909)=120587 1198991

0sin(120587119910 119894)

+119899minus1 sum 119894=1(119910 119894minus

1)2 [1+1

0sin2(120587119910 119894+1)+(

119910 119899minus1)2]+

119899 sum 119894=1119906(119909 1198941

01004)

30[minus50

50]0

Para

met

er sp

ace

minus5

minus5

minus10

minus10

10

10

100

5

50

150

50

0

0

x 2x4

F12(x1x2x3x4)

119910 119894=1+119909

119894+1 4

119906(119909 119894119886119896

119898)= 119896(119909 119894

minus119886)119898

119909 119894gt119886

0minus119886lt

119909 119894lt119886

119896(minus119909119894minus119886)119898

119909 119894ltminus119886

119865 13(119909)=0

1 sin2( 3120587119909119894) +119899 sum 119894=1( 119909119894minus1)2

[ 1+sin2( 3120587119909119894+1)]

+( 119909119899minus1)2

[ 1+sin2( 2120587119909119899)] +

119899 sum 119894=1119906( 119909 1198945

1004)

30[minus50

50]0

F13(x1x2x3x4)

minus5

minus5

55

15

5

0

0

10

0x 2

x4

Para

met

er sp

ace



nmultim

odalbenchm

arkfunctio

ns

Functio

nDim

Range

119891 min119865 14(119909

)=(1 500+25 sum 119895=1

1119895+sum

2 119894=1(119909 119894minus

119886 119894119895)6)minus1

2[minus65

65]1

500

400

450

300

350

200

250

100

150 050

Para

met

er sp

ace

50

50

100

100

00

minus100minus100

minus50

minus50

x 2x4

F14(x1x2x3x4)

119865 15(119909)=11 sum 119894=1[119886 119894minus

119909 1(1198872 119894+119887 1198941199092)

1198872 119894+119887 119894119909 119894+

119909 4]24

[minus55]

000030

F15(x1x2x3x4)

6 5 4 3 2 1 0

Para

met

er sp

ace

5

5

00

minus5

minus5

x 2x4

times105

119865 16(119909)=4

1199092 1minus211199094 1+1 31199096 1

+119909 1119909 2minus

41199092 2+41199094 2

2[minus5

5]minus10

316

F17(x1x2x1)

600

500

400

300

200

100 0

Para

met

er sp

ace

5

5

00

minus5

minus5

x 2x1

119865 17(119909)=(

119909 2minus51 412058721199092 1+5 120587119909 1

minus6)2

+10(1minus

1 8120587)cos119909 1+

102

[minus55]

0398

F17(x1x2x3x4)

600

500

400

300

200

100 0

Para

met

er sp

ace

5

5

00

minus5

minus5

x 2x4

119865 18(119909 )=

[1+(119909 1+

119909 2+1)2 (

19minus14119909 1

+31199092 1minus14

119909 2+6119909 11199092+31199092 2)]

times[30+(2119909


(18minus32119909 1

+121199092 1+

48119909 2minus36

119909 1119909 2+27

1199092 2)]2

[minus22]

3

F18(x1x2x3x4)

Para

met

er sp

ace

3

25 2

15 1

05 0 5

5

00

minus5

minus5

x 2x4

times108

119865 19(119909)=minus

4 sum 119894=1119888 119894exp(minus3 sum 119895=1

119886 119894119895(119909119895minus119901 119894119895

)2 )3

[13]

minus386

F19(x1x2x3x4)

0

Para

met

er sp

ace

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus016

minus018

minus02 5

5

00

minus5

minus5

x 2x4

119865 20(119909)=minus

4 sum 119894=1119888 119894exp(minus6 sum 119895=1

119886 119894119895(119909119895minus119901 119894119895

)2 )6

[01]

minus332

F20(x1x2x3x4)

0

Para

met

er sp

ace

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus016

minus018

minus02 5

5

00

minus5

minus5

x 2x4

119865 21(119909)=minus

5 sum 119894=1[(119883minus

119886 119894)( 119883minus119886 119894)119879+119888 119894]minus1

4[01

0]minus10

1532

F21(x1x2x3x4)

0

Para

met

er sp

ace

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08 5

5

00

minus5

minus5

x 2x4


Table3Con

tinued

Functio

nDim

Range

119891 min119865 22(119909

)=minus7 sum 119894=1[(

119883minus119886 119894)(119883

minus119886 119894)119879+119888 119894]minus1

4[01

0]minus10

4028

F22(x1x2x3x4)

0

Para

met

er sp

ace

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08 5

5

00

minus5

minus5

x 2x4

119865 23(119909)=minus

10 sum 119894=1[(

119883minus119886 119894)(119883

minus119886 119894)119879+119888 119894]minus1

4[01

0]minus10

5363

F23(x1x2x3x4)

0

Para

met

er sp

ace

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08 5

5

00

minus5

minus5

x 2x4


Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

10minus5

10minus10

10minus15

10minus20

10minus25

GWOPSOHPSOGWO

(a)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

60

50

40

30

20

10Best

scor

e obt

aine

d so

far

100

105

1010

10minus5

10minus10

10minus15

GWOPSOHPSOGWO

(b)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

104

102

100

10minus2

10minus4

10minus6

GWOPSOHPSOGWO

(c)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

10minus2

10minus4

10minus6

GWOPSOHPSOGWO

(d)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far 60

50

40

30

20

10

108

106

104

102

GWOPSOHPSOGWO

(e)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

GWOPSOHPSOGWO

(f)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

101

102

10minus1

10minus2

GWOPSOHPSOGWO

(g)













Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1033

minus1034

minus1035

minus1036

minus1037

minus1038

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(a)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far 10

0

10minus5

10minus10

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(b)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

10minus5

10minus10

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(c)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

10minus5

10minus10

10minus15

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(d)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

105

10minus5

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(e)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

105

10minus5

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO




7 Conclusion











Objective space

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

102

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(a)

Objective space

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

10minus1

10minus2

10minus3

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(b)Objective space

60

50

40

30

20

10Best

scor

e obt

aine

d so

far 10

0

10minus5

10minus10

10minus15

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(c)

Objective space

Best

scor

e obt

aine

d so

far 10

2

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

60

50

40

30

20

10

(d)Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far 10

2

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(e)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1003

minus1004

minus1005

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(f)Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1001

minus1002

minus1003

minus1004

minus1005

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(g)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO



Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(i)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(j)







References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Table2Multim

odalbenchm

arkfunctio

ns

Functio

nDim

Range

119891 min119865 8(119909

)=119899 sum 119894=1minus

119909 119894sin(radic1003816 1003816 1003816 1003816119909

1198941003816 1003816 1003816 1003816)30

[minus50050

0]minus418

9829times5

Para

met

er sp

ace

0200

400

600

800

1000

minus200

minus400

minus600

minus800

minus1000

F8(x1x2x3x4)

minus500

00

500

500

minus500

x 2x4

119865 9(119909)=119899 sum 119894=1[1199092 119894minus10

cos(2120587

119909 119894)+10]

30[minus51

2512]

0

minus5

minus5

0

20

40

60

80

100

5

5

00

x 2x4

F9(x1x2x3x4)

Para

met

er sp

ace

119865 10(119909)=minus

20exp(minus0

2radic1 119899119899 sum 119894=11199092 119894)minus

exp(1 119899119899 sum 119894=1co

s(2120587119909 119894))

+20+119890

30[minus32

32]0

F10(x1x2x3x4)

minus20

minus20

20

20

25

20

15

10 5 0

00

x 2x4

Para

met

er sp

ace

119865 11(119909)=

1 4000119899 sum 119894=11199092 119894


30[minus60

0600]

0

F11(x1x2x3x4)

minus500

minus500

500

500

140

120

100

80 60

40

20 0

00

x 2x4

Para

met

er sp

ace

119865 12(119909)=120587 1198991

0sin(120587119910 119894)

+119899minus1 sum 119894=1(119910 119894minus

1)2 [1+1

0sin2(120587119910 119894+1)+(

119910 119899minus1)2]+

119899 sum 119894=1119906(119909 1198941

01004)

30[minus50

50]0

Para

met

er sp

ace

minus5

minus5

minus10

minus10

10

10

100

5

50

150

50

0

0

x 2x4

F12(x1x2x3x4)

119910 119894=1+119909

119894+1 4

119906(119909 119894119886119896

119898)= 119896(119909 119894

minus119886)119898

119909 119894gt119886

0minus119886lt

119909 119894lt119886

119896(minus119909119894minus119886)119898

119909 119894ltminus119886

119865 13(119909)=0

1 sin2( 3120587119909119894) +119899 sum 119894=1( 119909119894minus1)2

[ 1+sin2( 3120587119909119894+1)]

+( 119909119899minus1)2

[ 1+sin2( 2120587119909119899)] +

119899 sum 119894=1119906( 119909 1198945

1004)

30[minus50

50]0

F13(x1x2x3x4)

minus5

minus5

55

15

5

0

0

10

0x 2

x4

Para

met

er sp

ace



nmultim

odalbenchm

arkfunctio

ns

Functio

nDim

Range

119891 min119865 14(119909

)=(1 500+25 sum 119895=1

1119895+sum

2 119894=1(119909 119894minus

119886 119894119895)6)minus1

2[minus65

65]1

500

400

450

300

350

200

250

100

150 050

Para

met

er sp

ace

50

50

100

100

00

minus100minus100

minus50

minus50

x 2x4

F14(x1x2x3x4)

119865 15(119909)=11 sum 119894=1[119886 119894minus

119909 1(1198872 119894+119887 1198941199092)

1198872 119894+119887 119894119909 119894+

119909 4]24

[minus55]

000030

F15(x1x2x3x4)

6 5 4 3 2 1 0

Para

met

er sp

ace

5

5

00

minus5

minus5

x 2x4

times105

119865 16(119909)=4

1199092 1minus211199094 1+1 31199096 1

+119909 1119909 2minus

41199092 2+41199094 2

2[minus5

5]minus10

316

F17(x1x2x1)

600

500

400

300

200

100 0

Para

met

er sp

ace

5

5

00

minus5

minus5

x 2x1

119865 17(119909)=(

119909 2minus51 412058721199092 1+5 120587119909 1

minus6)2

+10(1minus

1 8120587)cos119909 1+

102

[minus55]

0398

F17(x1x2x3x4)

600

500

400

300

200

100 0

Para

met

er sp

ace

5

5

00

minus5

minus5

x 2x4

119865 18(119909 )=

[1+(119909 1+

119909 2+1)2 (

19minus14119909 1

+31199092 1minus14

119909 2+6119909 11199092+31199092 2)]

times[30+(2119909


(18minus32119909 1

+121199092 1+

48119909 2minus36

119909 1119909 2+27

1199092 2)]2

[minus22]

3

F18(x1x2x3x4)

Para

met

er sp

ace

3

25 2

15 1

05 0 5

5

00

minus5

minus5

x 2x4

times108

119865 19(119909)=minus

4 sum 119894=1119888 119894exp(minus3 sum 119895=1

119886 119894119895(119909119895minus119901 119894119895

)2 )3

[13]

minus386

F19(x1x2x3x4)

0

Para

met

er sp

ace

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus016

minus018

minus02 5

5

00

minus5

minus5

x 2x4

119865 20(119909)=minus

4 sum 119894=1119888 119894exp(minus6 sum 119895=1

119886 119894119895(119909119895minus119901 119894119895

)2 )6

[01]

minus332

F20(x1x2x3x4)

0

Para

met

er sp

ace

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus016

minus018

minus02 5

5

00

minus5

minus5

x 2x4

119865 21(119909)=minus

5 sum 119894=1[(119883minus

119886 119894)( 119883minus119886 119894)119879+119888 119894]minus1

4[01

0]minus10

1532

F21(x1x2x3x4)

0

Para

met

er sp

ace

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08 5

5

00

minus5

minus5

x 2x4


Table3Con

tinued

Functio

nDim

Range

119891 min119865 22(119909

)=minus7 sum 119894=1[(

119883minus119886 119894)(119883

minus119886 119894)119879+119888 119894]minus1

4[01

0]minus10

4028

F22(x1x2x3x4)

0

Para

met

er sp

ace

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08 5

5

00

minus5

minus5

x 2x4

119865 23(119909)=minus

10 sum 119894=1[(

119883minus119886 119894)(119883

minus119886 119894)119879+119888 119894]minus1

4[01

0]minus10

5363

F23(x1x2x3x4)

0

Para

met

er sp

ace

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08 5

5

00

minus5

minus5

x 2x4


Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

10minus5

10minus10

10minus15

10minus20

10minus25

GWOPSOHPSOGWO

(a)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

60

50

40

30

20

10Best

scor

e obt

aine

d so

far

100

105

1010

10minus5

10minus10

10minus15

GWOPSOHPSOGWO

(b)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

104

102

100

10minus2

10minus4

10minus6

GWOPSOHPSOGWO

(c)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

10minus2

10minus4

10minus6

GWOPSOHPSOGWO

(d)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far 60

50

40

30

20

10

108

106

104

102

GWOPSOHPSOGWO

(e)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

GWOPSOHPSOGWO

(f)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

101

102

10minus1

10minus2

GWOPSOHPSOGWO

(g)













Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1033

minus1034

minus1035

minus1036

minus1037

minus1038

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(a)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far 10

0

10minus5

10minus10

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(b)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

10minus5

10minus10

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(c)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

10minus5

10minus10

10minus15

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(d)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

105

10minus5

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(e)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

105

10minus5

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO




7 Conclusion











Objective space

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

102

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(a)

Objective space

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

10minus1

10minus2

10minus3

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(b)Objective space

60

50

40

30

20

10Best

scor

e obt

aine

d so

far 10

0

10minus5

10minus10

10minus15

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(c)

Objective space

Best

scor

e obt

aine

d so

far 10

2

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

60

50

40

30

20

10

(d)Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far 10

2

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(e)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1003

minus1004

minus1005

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(f)Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1001

minus1002

minus1003

minus1004

minus1005

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(g)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO



Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(i)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(j)







References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






nmultim

odalbenchm

arkfunctio

ns

Functio

nDim

Range

119891 min119865 14(119909

)=(1 500+25 sum 119895=1

1119895+sum

2 119894=1(119909 119894minus

119886 119894119895)6)minus1

2[minus65

65]1

500

400

450

300

350

200

250

100

150 050

Para

met

er sp

ace

50

50

100

100

00

minus100minus100

minus50

minus50

x 2x4

F14(x1x2x3x4)

119865 15(119909)=11 sum 119894=1[119886 119894minus

119909 1(1198872 119894+119887 1198941199092)

1198872 119894+119887 119894119909 119894+

119909 4]24

[minus55]

000030

F15(x1x2x3x4)

6 5 4 3 2 1 0

Para

met

er sp

ace

5

5

00

minus5

minus5

x 2x4

times105

119865 16(119909)=4

1199092 1minus211199094 1+1 31199096 1

+119909 1119909 2minus

41199092 2+41199094 2

2[minus5

5]minus10

316

F17(x1x2x1)

600

500

400

300

200

100 0

Para

met

er sp

ace

5

5

00

minus5

minus5

x 2x1

119865 17(119909)=(

119909 2minus51 412058721199092 1+5 120587119909 1

minus6)2

+10(1minus

1 8120587)cos119909 1+

102

[minus55]

0398

F17(x1x2x3x4)

600

500

400

300

200

100 0

Para

met

er sp

ace

5

5

00

minus5

minus5

x 2x4

119865 18(119909 )=

[1+(119909 1+

119909 2+1)2 (

19minus14119909 1

+31199092 1minus14

119909 2+6119909 11199092+31199092 2)]

times[30+(2119909


(18minus32119909 1

+121199092 1+

48119909 2minus36

119909 1119909 2+27

1199092 2)]2

[minus22]

3

F18(x1x2x3x4)

Para

met

er sp

ace

3

25 2

15 1

05 0 5

5

00

minus5

minus5

x 2x4

times108

119865 19(119909)=minus

4 sum 119894=1119888 119894exp(minus3 sum 119895=1

119886 119894119895(119909119895minus119901 119894119895

)2 )3

[13]

minus386

F19(x1x2x3x4)

0

Para

met

er sp

ace

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus016

minus018

minus02 5

5

00

minus5

minus5

x 2x4

119865 20(119909)=minus

4 sum 119894=1119888 119894exp(minus6 sum 119895=1

119886 119894119895(119909119895minus119901 119894119895

)2 )6

[01]

minus332

F20(x1x2x3x4)

0

Para

met

er sp

ace

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus016

minus018

minus02 5

5

00

minus5

minus5

x 2x4

119865 21(119909)=minus

5 sum 119894=1[(119883minus

119886 119894)( 119883minus119886 119894)119879+119888 119894]minus1

4[01

0]minus10

1532

F21(x1x2x3x4)

0

Para

met

er sp

ace

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08 5

5

00

minus5

minus5

x 2x4


Table3Con

tinued

Functio

nDim

Range

119891 min119865 22(119909

)=minus7 sum 119894=1[(

119883minus119886 119894)(119883

minus119886 119894)119879+119888 119894]minus1

4[01

0]minus10

4028

F22(x1x2x3x4)

0

Para

met

er sp

ace

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08 5

5

00

minus5

minus5

x 2x4

119865 23(119909)=minus

10 sum 119894=1[(

119883minus119886 119894)(119883

minus119886 119894)119879+119888 119894]minus1

4[01

0]minus10

5363

F23(x1x2x3x4)

0

Para

met

er sp

ace

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08 5

5

00

minus5

minus5

x 2x4


Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

10minus5

10minus10

10minus15

10minus20

10minus25

GWOPSOHPSOGWO

(a)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

60

50

40

30

20

10Best

scor

e obt

aine

d so

far

100

105

1010

10minus5

10minus10

10minus15

GWOPSOHPSOGWO

(b)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

104

102

100

10minus2

10minus4

10minus6

GWOPSOHPSOGWO

(c)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

10minus2

10minus4

10minus6

GWOPSOHPSOGWO

(d)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far 60

50

40

30

20

10

108

106

104

102

GWOPSOHPSOGWO

(e)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

GWOPSOHPSOGWO

(f)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

101

102

10minus1

10minus2

GWOPSOHPSOGWO

(g)













Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1033

minus1034

minus1035

minus1036

minus1037

minus1038

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(a)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far 10

0

10minus5

10minus10

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(b)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

10minus5

10minus10

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(c)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

10minus5

10minus10

10minus15

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(d)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

105

10minus5

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(e)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

105

10minus5

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO




7 Conclusion











Objective space

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

102

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(a)

Objective space

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

10minus1

10minus2

10minus3

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(b)Objective space

60

50

40

30

20

10Best

scor

e obt

aine

d so

far 10

0

10minus5

10minus10

10minus15

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(c)

Objective space

Best

scor

e obt

aine

d so

far 10

2

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

60

50

40

30

20

10

(d)Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far 10

2

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(e)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1003

minus1004

minus1005

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(f)Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1001

minus1002

minus1003

minus1004

minus1005

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(g)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO



Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(i)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(j)







References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Table3Con

tinued

Functio

nDim

Range

119891 min119865 22(119909

)=minus7 sum 119894=1[(

119883minus119886 119894)(119883

minus119886 119894)119879+119888 119894]minus1

4[01

0]minus10

4028

F22(x1x2x3x4)

0

Para

met

er sp

ace

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08 5

5

00

minus5

minus5

x 2x4

119865 23(119909)=minus

10 sum 119894=1[(

119883minus119886 119894)(119883

minus119886 119894)119879+119888 119894]minus1

4[01

0]minus10

5363

F23(x1x2x3x4)

0

Para

met

er sp

ace

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08 5

5

00

minus5

minus5

x 2x4


Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

10minus5

10minus10

10minus15

10minus20

10minus25

GWOPSOHPSOGWO

(a)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

60

50

40

30

20

10Best

scor

e obt

aine

d so

far

100

105

1010

10minus5

10minus10

10minus15

GWOPSOHPSOGWO

(b)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

104

102

100

10minus2

10minus4

10minus6

GWOPSOHPSOGWO

(c)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

10minus2

10minus4

10minus6

GWOPSOHPSOGWO

(d)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far 60

50

40

30

20

10

108

106

104

102

GWOPSOHPSOGWO

(e)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

GWOPSOHPSOGWO

(f)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

101

102

10minus1

10minus2

GWOPSOHPSOGWO

(g)













Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1033

minus1034

minus1035

minus1036

minus1037

minus1038

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(a)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far 10

0

10minus5

10minus10

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(b)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

10minus5

10minus10

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(c)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

10minus5

10minus10

10minus15

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(d)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

105

10minus5

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(e)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

105

10minus5

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO




7 Conclusion











Objective space

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

102

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(a)

Objective space

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

10minus1

10minus2

10minus3

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(b)Objective space

60

50

40

30

20

10Best

scor

e obt

aine

d so

far 10

0

10minus5

10minus10

10minus15

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(c)

Objective space

Best

scor

e obt

aine

d so

far 10

2

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

60

50

40

30

20

10

(d)Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far 10

2

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(e)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1003

minus1004

minus1005

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(f)Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1001

minus1002

minus1003

minus1004

minus1005

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(g)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO



Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(i)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(j)







References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

10minus5

10minus10

10minus15

10minus20

10minus25

GWOPSOHPSOGWO

(a)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

60

50

40

30

20

10Best

scor

e obt

aine

d so

far

100

105

1010

10minus5

10minus10

10minus15

GWOPSOHPSOGWO

(b)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

104

102

100

10minus2

10minus4

10minus6

GWOPSOHPSOGWO

(c)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

10minus2

10minus4

10minus6

GWOPSOHPSOGWO

(d)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far 60

50

40

30

20

10

108

106

104

102

GWOPSOHPSOGWO

(e)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

GWOPSOHPSOGWO

(f)

Objective space

50 100 150 200 250 300 350 400 450 500

Iteration

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

100

101

102

10minus1

10minus2

GWOPSOHPSOGWO

(g)













Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1033

minus1034

minus1035

minus1036

minus1037

minus1038

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(a)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far 10

0

10minus5

10minus10

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(b)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

10minus5

10minus10

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(c)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

10minus5

10minus10

10minus15

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(d)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

105

10minus5

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(e)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

105

10minus5

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO




7 Conclusion











Objective space

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

102

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(a)

Objective space

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

10minus1

10minus2

10minus3

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(b)Objective space

60

50

40

30

20

10Best

scor

e obt

aine

d so

far 10

0

10minus5

10minus10

10minus15

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(c)

Objective space

Best

scor

e obt

aine

d so

far 10

2

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

60

50

40

30

20

10

(d)Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far 10

2

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(e)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1003

minus1004

minus1005

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(f)Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1001

minus1002

minus1003

minus1004

minus1005

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(g)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO



Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(i)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(j)







References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of















Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1033

minus1034

minus1035

minus1036

minus1037

minus1038

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(a)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far 10

0

10minus5

10minus10

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(b)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

10minus5

10minus10

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(c)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

10minus5

10minus10

10minus15

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(d)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

105

10minus5

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(e)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

100

105

10minus5

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO




7 Conclusion











Objective space

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

102

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(a)

Objective space

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

10minus1

10minus2

10minus3

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(b)Objective space

60

50

40

30

20

10Best

scor

e obt

aine

d so

far 10

0

10minus5

10minus10

10minus15

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(c)

Objective space

Best

scor

e obt

aine

d so

far 10

2

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

60

50

40

30

20

10

(d)Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far 10

2

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(e)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1003

minus1004

minus1005

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(f)Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1001

minus1002

minus1003

minus1004

minus1005

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(g)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO



Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(i)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(j)







References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of













Objective space

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

102

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(a)

Objective space

Best

scor

e obt

aine

d so

far

60

50

40

30

20

10

10minus1

10minus2

10minus3

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(b)Objective space

60

50

40

30

20

10Best

scor

e obt

aine

d so

far 10

0

10minus5

10minus10

10minus15

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(c)

Objective space

Best

scor

e obt

aine

d so

far 10

2

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

60

50

40

30

20

10

(d)Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far 10

2

101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(e)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1003

minus1004

minus1005

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(f)Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus1001

minus1002

minus1003

minus1004

minus1005

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(g)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO



Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(i)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(j)







References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far

minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(i)

Objective space

60

50

40

30

20

10

Best

scor

e obt

aine

d so

far minus100

minus101

50 100 150 200 250 300 350 400 450 500

Iteration

GWOPSOHPSOGWO

(j)







References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Hybrid Algorithm of Particle Swarm Optimization and Grey ... · In recent years, several numbers of...

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Transcript of Hybrid Algorithm of Particle Swarm Optimization and Grey ... · In recent years, several numbers of...