PS-FW: A Hybrid Algorithm Based on Particle Swarm and …downloads.hindawi.com/journals/cin/2018/6094685.pdf · 2019. 7. 30. · ResearchArticle PS-FW: A Hybrid Algorithm Based on

Research ArticlePS-FW A Hybrid Algorithm Based on Particle Swarm andFireworks for Global Optimization

Shuangqing Chen Yang Liu LixinWei and Bing Guan

School of Petroleum Engineering Northeast Petroleum University Daqing 163318 China

Correspondence should be addressed to Yang Liu ly001nepueducn

Received 21 September 2017 Accepted 10 January 2018 Published 20 February 2018

Academic Editor Rasit Koker

Copyright copy 2018 Shuangqing Chen et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Particle swarm optimization (PSO) and fireworks algorithm (FWA) are two recently developed optimization methods which havebeen applied in various areas due to their simplicity and efficiency However when being applied to high-dimensional optimizationproblems PSO algorithm may be trapped in the local optima owing to the lack of powerful global exploration capability andfireworks algorithm is difficult to converge in some cases because of its relatively low local exploitation efficiency for noncorefireworks In this paper a hybrid algorithm called PS-FW is presented in which the modified operators of FWA are embeddedinto the solving process of PSO In the iteration process the abandonment and supplement mechanism is adopted to balance theexploration and exploitation ability of PS-FW and the modified explosion operator and the novel mutation operator are proposedto speed up the global convergence and to avoid prematurity To verify the performance of the proposed PS-FW algorithm22 high-dimensional benchmark functions have been employed and it is compared with PSO FWA stdPSO CPSO CLPSOFIPS Frankenstein and ALWPSO algorithms Results show that the PS-FW algorithm is an efficient robust and fast convergingoptimization method for solving global optimization problems

1 Introduction

Global optimization problems are common in engineeringand other related fields [1ndash3] and it is usually difficult to solvethe global optimization problems due to many local optimaand complex search space especially in high dimensionsFor solving optimization problems many methods havebeen reported in the past few years Recently the stochasticoptimization algorithms have attracted increasing attentionbecause they can get better solutions without any propertiesof the objective functions Therefore many effective meta-heuristic algorithms have been presented such as simulatedannealing (SA) [4] differential evolution (DE) [5] geneticalgorithm (GA) [6] particle swarm optimization (PSO) [7]ant colony optimization (ACO) [8] artificial bee colony(ABC) [9] and fireworks algorithm (FWA) [10]

Among these intelligent algorithms the PSO and FWAhave shown pretty outstanding performance in solving globaloptimization problems in the last several years PSO algo-rithm is a population-based algorithm originally proposed

by Kennedy and Eberhart [7] which is inspired by theforaging behavior of birds Fireworks algorithm is a newswarm intelligence algorithm that is motivated by observingfireworks explosion Owing to the less decision parame-ters simple implementation and good scalability PSO andFWA have been widely applied since they were proposedincluding shunting schedule optimization of electric multipleunits depot [11] optimal operation of trunk natural gaspipelines [12] location optimization of logistics distributioncenter [13] artificial neural networks design [14] warehouse-scheduling [15] fertilization optimization [16] power systemreconfiguration [17] and multimodal function optimization[18]

Although PSO and FWA are highly successful in solvingsome classes of global optimization problems there arecertain problems that need to be addressed when they areextended to handling complex high-dimensional optimiza-tion problemsThe PSO algorithm has a significant efficiencyin unimodal problems but it can easily be trapped in localoptima for multimodal problems Moreover the FWA is

HindawiComputational Intelligence and NeuroscienceVolume 2018 Article ID 6094685 27 pageshttpsdoiorg10115520186094685

2 Computational Intelligence and Neuroscience

difficult to converge for the optimization problems whichdo not have their optimal solutions at the origin This isbecause the two algorithms cannot keep the balance betweenthe exploration and exploitation properly Due to the optimalparticle dominating the solving process the PSO algorithmhas inferior swarm diversity in the later stage of iterations andrelatively poor exploration ability [19] while the fireworksand sparks in FWA are not well-informed by the wholeswarm [20] and the FWA framework lacks the local searchefficiency for noncore fireworks [21] In order to improvethe performance of PSO and FWA a considerable numberof modified algorithms have been proposed For exampleNickabadi et al presented AIWPSO algorithm in which anew adaptive inertia weight approach was adopted [22] Byembedding a reverse predictor and adding a repulsive forceinto the basic algorithm the RPPSO was developed [23]Wang and Liu used three strategies to ameliorate the standardalgorithm including best neighbor replacement abandonedmechanism and chaotic searching [24] Souravlias andParsopoulos introduced a PSO-based variant which coulddynamically assign different computational budget for eachparticle based on the quality of its neighbor [25] Based onself-adaption principle and bimodal Gaussian function theadvanced fireworks algorithm (AFWA) was proposed [26]Liu et al presented several methods for computing the explo-sion amplitude and number of sparks [27] Pei et al proposedto use the elite point of approximation landscape in thefireworks swarm and discussed the effectiveness of surrogate-assisted FWA [28] Zheng et al improved the new explosionoperator mutation operator selection strategy and mappingrules of FWA which led to the formation of enhancedfireworks algorithm (EFWA) [29 30] and dynamic searchin fireworks algorithm (dynFWA) [31] Zheng et al pro-posed the new cooperative FWA framework (CoFFWA) inwhich the independent selection method and crowdedness-avoiding cooperative strategy were contained [21] Li et alinvestigated the operators of FWA and introduced a novelguiding spark in FWA [32] and proposed the adaptivefireworks algorithm (AFWA) [33] and bare bones fireworksalgorithm (BBFWA) [34]

Hybrid algorithms can utilize various exploration andexploitation strategies for high-dimensional multimodaloptimization problems which have gradually become thenew research areas For example Valdez et al combined theadvantages of PSO with GA and proposed a modified hybridmethod [35] In the new PS-ABC algorithm introduced by Liet al the global optimum could be obtained by combiningthe local search phase in PSO with two global search phasesin ABC [19] Pandit et al presented the SPSO-DE in whichthe domain information of PSO and DE was shared withone another to overcome their respective weaknesses [36]Through changing the generation and selection strategy ofexplosive spark Gao and Diao proposed the CA-FWA [37]Zhang et al proposed BBO-FW algorithm which improvedthe interaction ability between fireworks [38] By combiningthe FWA with the operators of DE a novel hybrid optimiza-tion algorithm was proposed [20]

In this paper by utilizing the exploitation ability ofPSO and the exploration ability of FWA a novel hybrid

optimization algorithm called PS-FW is proposed Basedon the solving process of PSO algorithm the operators ofFWA are embedded into the update operation of the particleswarm In the iteration process in order to promote thebalance of exploitation and exploration ability of PS-FW wepresented three major techniques Firstly the abandonmentand supplement strategy is used to abandon a certainnumber of particles with poor quality and to supplementthe particle swarm with new individuals generated by FWAMeanwhile considering the information exchanges betweenthe optimal firework and its neighbor in each dimension themethod for obtaining the explosion amplitude is designed asadaptive and the mode of generating the explosion sparksis modified by combing the greedy algorithm Furthermorethe conventional Gaussian mutation operator is abandonedand the novel mutation operator based on the thought of thesocial cognition and learning is proposed The performanceof PS-FW is compared with several existing optimizationalgorithms The experimental results show that the proposedPS-FW is more efficacious in solving the global optimizationproblems

The rest of the paper is organized as follows Section 2describes the standard PSO and FWA Section 3 presentsthe PS-FW algorithm in which the algorithm details areproposed Section 4 introduces the simulation results over 22high-dimensional benchmark functions and the correspond-ing comparisons between PS-FW and other algorithms areexecuted Finally the conclusion is drawn in Section 5

2 Related Work

21 PSO Algorithm In PSO algorithm the particles scatter insearch space of the optimization problems and each particledenotes a feasible solution Each particle contains threeaspects of information the current position 119909119894 the velocityV119894 and the previous best position 119901119887119890119904119905119894 Assume that theoptimization problem is 119863-dimensional and 119872 representsthe size of the swarm population then the position andvelocity of 119894th (119894 = 1 2 119872) particle can be denoted as 119909119894 =(1199091198941 1199091198942 119909119894119863) and V119894 = (V1198941 V1198942 V119894119863) respectivelywhile the previous best position is represented as 119901119887119890119904119905119894 =(1199011198871198901199041199051198941 1199011198871198901199041199051198942 119901119887119890119904119905119894119863) Besides the best positionencountered by the entire particles so far is known as currentglobal best position 119892119887119890119904119905119894 = (1198921198871198901199041199051 1198921198871198901199041199052 119892119887119890119904119905119863)In each generation V119894 and 119909119894 are updated by the followingequations

V119894119896 (119905 + 1) = 119908 sdot V119894119896 (119905) + 1198881 sdot 1199031 sdot [119901119887119890119904119905119894119896 (119905) minus 119909119894119896 (119905)]+ 1198882 sdot 1199032 sdot [119892119887119890119904119905 (119905) minus 119909119894119896 (119905)] (1)

119909119894119896 (119905 + 1) = 119909119894119896 (119905) + V119894119896 (119905 + 1) (2)

where 1198881 and 1198882 are two learning factors that indicate theinfluence of the cognitive and social components 1199031 and 1199032are the random real numbers in interval [0 1] respectivelyand 119908 is the inertia weight which controls the convergencespeed of the algorithm

22 Fireworks Algorithm In FWA a firework or a sparkdenotes a potential solution of optimization problems while

Computational Intelligence and Neuroscience 3

the process of producing sparks from fireworks representsa search in the feasible space As in other optimizationalgorithms the optimal solutions are obtained by successiveiterations In each iteration the sparks can be produced bytwo ways the explosion and the Gaussian mutation Theexplosion of fireworks is dominated by the explosion ampli-tude and the number of explosion sparks Compared to thefireworks with lower fitness the fireworks with better fitnesswill have smaller explosion amplitude and more explosionsparks Suppose that 119873 denotes the number of fireworksthen the 119894th (119894 = 1 2 119873) firework can be denotedas 119909 = (1199091198941 1199091198942 119909119894119863) for 119863-dimensional optimizationproblems Besides the explosion amplitude can be obtainedby (3) and the sparks number can be calculated by (4)

119860 119894 = 119860 sdot 119891 (119909119894) minus 119910min + 120576sum119873119894=1 (119891 (119909119894) minus 119910min) + 120576 (3)

119904119894 = 119872119890 sdot 119910max minus 119891 (119909119894) + 120576sum119873119894=1 (119910max minus 119891 (119909119894)) + 120576 (4)

where 119891(119909) denotes the objective function value of the 119894thfirework 119894 = 1 2 119873 119860 119894 and 119904119894 are the explosion ampli-tude and the number of explosion sparks of the 119894th fireworkrespectively 119910max = max(119891(119909119894)) 119910min = min(119891(119909119894)) 119860 and119872119890 are two constants that dominate the explosion amplitudeand the number of explosion sparks respectively and 120576 is themachine epsilon

Moreover the bounds of 119904119894 are defined as follows

119904119894 =round (119886 sdot 119872119890) 119904119894 lt 119886 sdot 119872119890round (119887 sdot 119872119890) 119904119894 gt 119887 sdot 119872119890round (119904119894) otherwise

(5)

where 119886 119887 are two constants that control the minimum andmaximum of population size respectively

In order to generate each explosion spark of 119894th fireworkan offset is added to 119909119894 according to the following equation

119909119895119894 = 119909119894 + Δℎ (6)

where 119909119895119894 is the 119895th explosion spark of 119894th firework and Δℎ =119860 119894 sdot rand(minus1 1) sdot 119861 where 119861 is a 119863-dimensional vectorwhich has 119895119894 values of 1 and 119863 minus 119895119894 values of 0 where 119895119894denotes the number of randomly selected dimensions of 119909and 119895119894 = 119863 sdot rand() 119895 = 1 2 119904119894 where rand(minus1 1) andrand() are random numbers in the intervals [minus1 1] and [0 1]respectively

Another type of sparks known as the Gaussian sparksis generated based on the Gaussian mutation operator Ineach generation a certain number of Gaussian sparks aregenerated and each Gaussian spark is transformed from afireworkwhich is selected randomly For the selected firework119909119894 its Gaussian spark is generated based on

119909119895 = (119874 minus 119861119894) sdot 119909119894 + Gaussian (1 1) sdot 119909119894 sdot 119861119894 (7)

where 119909119895 is the 119895th Gaussian spark 119874 is a 119863-dimensionalvector whose values are 1 in each dimension 119861 is a 119863-dimensional vectorwhich has 119894 values of 1 and119863minus119894 values of0 119894 represents the number of randomly selected dimensionsof 119909119894 and 119894 = 119863 sdot rand() and Gaussian(1 minus1) represents arandom number subordinated to the Gaussian distributionwith the mean of 1 and the standard deviation of 1

For the purpose of passing information to the nextgeneration newfireworks populations are chosen to continuethe iteration All the fireworks the explosion sparks andGaussian sparks have the chance to be selected for the nextiteration The location with best fitness is kept for the nextgeneration while the other 119873 minus 1 locations are selectedbased on the selection operator and the selection operator isdenoted as follows

119877 (119883119894) = sum119895isin119870

119889 (119883119894 119883119895) = sum119895isin119870

10038171003817100381710038171003817119883119894 minus 11988311989510038171003817100381710038171003817 119901 (119883119894) = 119877 (119883119894)sum119896isin119870 119877 (119883119896)

(8)

where 119870 denotes the set comprised of all the originalfireworks and both types of sparks 119883119894 119883119895 and 119883119896 are 119894th119895th and 119896th location of 119870 respectively 119877(119883119894) is the distancebetween 119894th location and the rest of all the locations and119901(119883119894) denotes the probability of being selected for the 119894thlocation

3 Hybrid Optimization AlgorithmBased on PSO and FWA

The exploitation process focuses on utilizing the existinginformation to look for better solutions whereas the explo-ration process attaches importance to seek the optimalsolutions in the entire space For PSO under the guidanceof their historical best solutions and the current global bestsolution the particles can quickly find better solutions andthe excellent exploitation efficiency of algorithm is shown InFWA the fireworks can find the global optimal solution in thewhole search space by performing explosion and mutationoperations while the outstanding exploration capability ofFWA is demonstrated To utilize the advantages of the twoalgorithms a hybrid optimizationmethod (PS-FW) based onPSO and FWA is proposed

31 Feasibility Analysis The formation of a hybrid algorithmis mainly due to the effective combination of the operatorsof its composition algorithms in a certain way To clarifythe performance enhancement caused by combining the PSOalgorithm with fireworks algorithm we draw Figures 1 and2 to illustrate the optimization mechanism As shown inFigure 1 for standard PSO algorithm the 119894th particle movesfrom point 1 to point 4 under the common influence ofvelocity inertia self-cognition and social informationWhenthe operators of FWA are added the particle is transformedinto firework and performs explosion and mutation oper-ations and eventually reaches the position of firework orsparks such as point 5 shown in Figure 1 By performing the


Explosion

1

2

3

4

5

Mutation

xti

xt+1i

ti

gbestt

pbesttiw middot ti

c1 middot r1 middot (pbestti minus xti )

c2 middot r2 middot (gbestt minus xti )

Local optima region

Figure 1 Optimization mechanism of adding operators of FWA toPSO algorithm

Explosion

1

2

3

Mutation

Global optima region

xti xt+1

i

ti

gbestt

pbestti

5

4

Figure 2 Optimization mechanism of adding operators of PSO toFWA

operators of FWA the particle can explore better solutions inmultiple directions and jump out of the local optima region asdepicted in Figure 1 Thus we can argue that the operators ofFWA improve the global search ability of PSO algorithm Aswe know the searching region is determined by the explosionamplitude and fireworks with poor quality have biggeramplitude which may lead to an uncomprehensive searchwithout considering the cooperation with other fireworksWhen the firework with poor quality generates the explosionsparks and mutation sparks the new selected location mayskip over the global optima region without the attractionfrom the rest of fireworks and arrive at point 2 By adding theoperators of PSO after the 119894th firework updates its locationthe information of its own historical best location and currentglobal best location is taken into account then the newsolution is found in point 5 which is shown in Figure 2Therefore the operators of PSO could strengthen the localsearch efficiency of FWA Based on the above analysis itis concluded that the combination of PSO and FWA is aneffective way to form a superior optimization algorithm

32 The Abandonment and Supplement Mechanism Theparticles with their memory ability can be quickly converged

to the current optimal solution However the aggregationeffect of the particle swarm reduces the diversity of thepopulation which makes the search in the whole feasiblespace inefficient In this paper in order to enhance thebalance between exploitation ability and exploration ability ofPS-FW we adopt the abandonment and supplement strategywhich includes three main steps (i) All the particles in theparticle swarm 1199091 1199092 119909119872 are sorted in ascending orderThen the 119875num particles with better fitness are retained for thenext iteration and the FWnum (satisfying 119875num + FWnum =119872) particles with lower fitness are abandoned (ii) The 119875numexcellent individuals denoted as 1199091198651 1199091198652 119909119865119875num are usedto implement the explosion operator the mutation operatorand the selection operator (iii)The new individuals obtainedby the operators of FWA are added to the original populationto balance the number of particles and to generate the newparticle swarm for the next iteration The abandonmentand supplement strategy not only retains the informationof the excellent individuals so that they can participate inthe subsequent calculation but also avoids the individualswith poor quality wasting computing resources However theproblem arises how to determine 119875num For this throughanalyzing the process of solving the optimization problemswe should enhance the exploration ability of the algorithmand search the optimal solution in the global scope at earlystage of iterations which means the number of particlesexecuting the operators of FWA should be the majority Inthe later stage of iteration we should focus on searchingaround the current global optimal solution so the numberof excellent individuals retained in the algorithm shouldbe more Based on the discussion above the calculationof FWnum in this paper is shown in (9) in which FWnumdecreases with iteration process

FWnum = round [(FWmax minus FWmin) sdot (119868max minus 119905119868max)119903

+ FWmin] (9)

where FWmax and FWmin are the upper and lower bounds ofnumber of abandoned particles respectively 119868max is the max-imum number of iterations 119905 denotes the current number ofiterations round[] indicates that the values in brackets arerounded and 119903 represents a positive integer33 Modified Explosion Operator

331 Adaptive Explosion Amplitude Based on the analysisabove the definition of the explosion amplitude in standardFWA limits the diversity of the explosion sparks generated bythe excellent fireworks thus decreasing the local search abilityof algorithm In the enhanced fireworks algorithm (EFWA)[29] in order to avoid the weakness of the explosion ampli-tude generation in FWA a minimal explosion amplitudecheck mechanism is proposed which defines the explosionamplitude less than a certain threshold to obtain the samevalue as the threshold while the threshold is reducing withthe iteration process Suppose that 120575 denotes the threshold of


explosion amplitude then the explosion amplitude less thanthe threshold is defined as (10) in EFWA

119860 = 119860 init minus 119860 init minus 119860final119868maxsdot radic(2119868max minus 119905) 119905 (10)

where 119860 init and 119860final are the upper and lower bounds of theexplosion amplitude respectively

In this paper based on the minimal explosion amplitudedetection mechanism the basic explosion amplitude of eachfirework is calculated according to (3) and the explosionamplitude is adjusted by the following two methods(1)For the fireworkswhose explosion amplitude is greaterthan the threshold 120575 a control factor 120582 of the explosionamplitude is added The control factor makes the explosionsparks generated by the algorithm have larger search scopein the early stage of iterations which can effectively enhancethe exploration ability of the algorithm In the later stage ofiterations the explosion amplitude is reduced to improve thesearch efficiency around the current global optimal solutionThe adjustment of the explosion amplitude is shown in (11)and the control factor is calculated as shown in (12)

119860 119894 = 119860 119894 sdot 120582 forall119860 119894 gt 120575 (11)

120582 = 120582min sdot (120582max120582min)1(1+119905119868max) (12)

where 120582max and 120582min are the lower and upper bounds of thecontrol factor respectively(2) When the explosion amplitude of firework 119909119894 is lessthan the threshold the optimal firework and its neighborinformation are used to determine the explosion amplitudein the hybrid algorithm Since the PS-FW algorithm is basedon the framework of PSO the position of all individuals willapproach the current best position which leads to the fitnessof current optimal individual close to its neighbor individ-uals That is to say if the explosion amplitude of a fireworkis too small indicating that the firework may be locatednear the current best location therefore by consideringthe deviation information of all corresponding dimensionsbetween the current best firework and its neighbor fireworka new explosion amplitude of the firework 119909119894 is generatedThe explosion amplitude generation method can adaptivelyoptimize the solving process which can be interpreted fromtwo aspects When the algorithm is in the early iterationstage the position of fireworks is scattered and the deviationin dimensions between the optimal firework and its neigh-bor firework is larger which leads to the larger explosionamplitude and the improved probability of finding the globaloptimal solution As the algorithm enters the later iterationsthe fireworks gather around the current best location and theoffset of each dimension between the current best fireworkand its neighbor firework is reduced which results in thedecrement of explosion amplitude and the improvement ofthe local search ability for PS-FW There are two main stepsto obtain the explosion amplitude (i) Randomly select afirework 119909119895 around the current optimal firework according

to the fitness (ii) Update the explosion amplitude of the 119894thfirework according to the following equation

119860 119894 = sum119863119896=1 (10038161003816100381610038161003816119909best119896 minus 11990911989511989610038161003816100381610038161003816)119863 (13)

where119909best119896 denotes the value of the119896th dimension of currentoptimal firework

332 Modified Explosion Sparks Generation In FWA whengenerating an explosion spark the offsetΔℎ is only calculatedonce which results in the same changes for all the selecteddimensions and an ineffective search for different directionsIn the PS-FW algorithm proposed in this paper a newexplosion sparks generation method is introduced Firstlywhen generating the explosion sparks the location offset isperformed in all the dimensions of the fireworks insteadof randomly selecting part of dimensions Furthermore foreach dimension of the fireworks the different offsets arecalculated according to (14) thereby increasing the diversityof the explosion sparks and the global search capability ofthe hybrid algorithmMeanwhile suppose that 119909temp denotesthe 119894th firework without a location offset and 119909+ indicatesthe 119894th firework whose 119896th dimension adds a offset then 119909minusdenotes the 119894th firework whose 119896th dimension subtracts anoffset As shown in (15) inspired by greedy algorithm whenthe fireworks generate their explosion sparks the hybridalgorithm determines which offset to be selected based onthe value of objective function which can effectively improvethe local search capability of the algorithm and accelerate theconvergence

Δℎ119896 = 119860 sdot Gaussian (0 1) (14)

119909119895119894119896

= 119909119894119896 + Δℎ119896 119891 (119909+) le min (119891 (119909temp) 119891 (119909minus))119909119894119896 minus Δℎ119896 119891 (119909minus) le min (119891 (119909temp) 119891 (119909+))119909119894119896 119891 (119909temp) le min (119891 (119909+) 119891 (119909minus))

(15)

where 119909119895119894119896

and Δℎ119896 are the value and offset of the 119896thdimension of the 119895th explosion spark for the 119894th fireworkrespectively Gaussian(0 1) represents a random number thatfollows the standard normal distribution 119894 and 119895 are integersin the intervals [1 119875num] and [1 119904119894] respectively and min()indicates the minimum values in parentheses

Assume that num119864 denotes the total number of explosionsparks generated by all fireworks 119878min and 119878max represent thelower and upper bounds for the search scope and 119878min119896 and119878max119896 are corresponding to the bounds of 119896th dimensionrespectively Based on the explosion operator introducedin Sections 331 and 332 the detailed codes of explosionoperator are represented in Algorithm 1

34 Novel Mutation Operator As the Gaussian mutationoperator effectively increases the diversity of feasible solu-tions the performance of traditional FWA has been sig-nificantly improved However the numerical experiments


(1) Input 119875num particles sorted in ascending order according to their fitness(2) Initialize the location of fireworks 119909119894 = 119909119865119894 119894 = 1 2 119875num(3) for 119894 = 1 to 119875num do(4) Calculate the explosion amplitude 119860 119894 of 119894th firework by using (3)(5) Calculate the number of explosion sparks 119904119894 of 119894th firework by using (4)(6) Update the number of explosion sparks of 119894th firework by using (5)(7) if 119860 119894 gt 120575 do(8) Update the explosion amplitude of 119894th firework by using (11) and (12)(9) else do(10) Randomly select a firework 119909119895 around the current optimal firework(11) Update the explosion amplitude of 119894th firework by using (13)(12) end if(13) end for(14) Initialize the total number of explosion sparks num119864 = 0(15) for 119894 = 1 to 119875num do(16) for 119895 = 1 to 119904119894 do(17) Initialize the location of the 119895th explosion spark 119909119895119894 = 119909119894(18) for 119896 = 1 to119863 do(19) Calculate the offset by using (14)(20) Update the value of 119896th dimension of 119895th explosion spark by using (15)(21) if 119909119895

119894119896gt 119878max119896 or 119909119895119894119896 lt 119878min119896 do

(22) Update the 119909119895119894119896by using (17)

(23) end if(24) end for(25) num119864 = num119864 + 1(26) end for(27) end for(28) Output num119864 explosion sparks

Algorithm 1 Generating explosion sparks by the explosion operator of PS-FW

show that the combined application of Gaussian operatorand mapping operator makes the Gaussian sparks mostlyconcentrated around the zero point which is the reason whyFWA has the fast convergence speed for the problems withtheir optimal solutions at zero [31] In order to improve theadaptability of the algorithm for the nonzero optimizationproblems and maintain the contribution of the mutationoperator to the population diversity a newmutation operatoris proposed in the PS-FW Comparedwith the standard FWAthere are two main differences in this paper (i) In PS-FWwe randomly select a certain number of explosion sparks togenerate the mutation sparks instead of using the fireworksBecause the explosion sparks have better quality comparedto the fireworks based on (15) the mutation sparks generatedby the explosion sparks can effectively enrich the diversity ofthe population and have better global search ability (ii) Inthis paper the Gaussian random number is no longer used inmutation operator and the interactionmechanismof particlesin PSO is used for reference to design the mutation operatorThemutation sparks generated by our mutation operator cannot only maintain the better information of the explosionsparks but also have a proper movement towards the currentbest location which leads to promoting the convergence ofhybrid algorithm The proposed mutation operator is shownas follows

119909119894119896 = 1205831 sdot (119909best119896 minus 119909119895119896) + 1205832 sdot 119909119895119896 (16)

where 119909119894119896 and 119909119895119896 indicate the value of 119896th dimension of 119894thmutation spark and 119895th explosion spark respectively 119909best119896is the current optimal explosion spark 1205831 and 1205832 are therandom number in [0 1] and 119895 denotes the random integerof the interval [1 num119864] 119894 = 1 2 num119872 where num119872indicates the total number of mutation sparks

The detailed codes of mutation operator are representedin Algorithm 2

35Main Process of PS-FW In PS-FW the algorithm consistsof two main stages which are initialization stage and itera-tions stage In the initialization phase we need to initializethe position and velocity of the particle swarm as well as toinitialize the control parameters In the iterative phase thePS-FW algorithm inherits all the parameters and operatorsof the PSO algorithm and all particles are used as the maincarrier for storing feasible solutions Firstly in each iterationthe particles update their speed and position according tothe operators of the PSO algorithm and then perform theabandonment and supplement operation Besides in theprocess of generating the supplement particles by using theoperators of FWA we first generate num119864 explosion sparksaccording to the excellent 119875num particles and the modifiedexplosion operator then the fitness of the explosion sparksis given Secondly the num119872 mutation sparks are generatedby the explosion sparks and the novel mutation operatorFinally the FWnum supplement individuals are selected by the


(1) Input num119864 explosion sparks and best explosion spark119909best(2) for 119894 = 1 to num119872 do(3) Generate a random integer 119895 in the interval [1 num119864](4) Initialize the location of the 119894th mutation spark119909119894 = 119909119895(5) Calculate the number of dimensions to perform

the mutation 119894 = 119863 sdot rand()(6) Randomly select 119894 dimensions of 119909119894(7) for each dimension 119909119894119896 isin pre-selected 119894 dimensions

of 119909119894 do(8) Calculate the value of 119909119894119896 by using (16)(9) if 119909119894119896 gt 119878max119896 or 119909119894119896 lt 119878min119896 do(10) Update the value of 119909119894119896 by using (17)(11) end if(12) end for(13) end for(14) Output num119872 mutation sparks

Algorithm 2 Generating mutation sparks by the mutation opera-tor of PS-FW

combination of elite strategy and roulette strategyWhen eachiteration is completed it is judged whether the terminationcondition is satisfied If the stopping criterion is matched theiteration will be stopped and the best solutions are outputOtherwise the iteration phase will be repeated

In the procedures above there are two points to be noted(i) In the implementation process of the hybrid algorithmit is necessary to detect whether the position of individualsis within the feasible scope while the individuals consist ofparticles fireworks explosion sparks and mutation sparksAs shown in (17) if the position of individuals exceeds thefeasible scope it is adjusted by using the mapping criteria inthe EFWA algorithm [29]

119884119894119896 = 119878min119896 + 119890 sdot (119878max119896 minus 119878min119896)forall119884119894119896 gt 119878max119896 or 119884119894119896 lt 119878min119896 (17)

where 119884119894119896 indicates the value of the 119896th dimension of theindividual and 119890 is a random number in [0 1]

(ii) The selection strategy of FWA based on the densityof feasible solutions is abandoned in the PS-FW algorithmAlthough it is possible to maintain the diversity of thepopulation by selecting the location which has fewer indi-viduals around with a larger probability relatively more timeis wasted by calculating the spatial distance between theindividuals and the efficiency of the algorithm is reducedTherefore a selection strategy based on fitness is appliedin PS-FW which means the elite strategy is used to retainthe best individual directly into the next iteration and theremaining FWnum minus 1 locations are selected by the roulettecriterion according to the fitness

According to the description above themain codes of thePS-FW algorithm are given in Algorithm 3

4 Problems Experiments and Discussion

41 Test Problems In order to evaluate the efficacy and accu-racy of the proposed algorithm the performance of PS-FW istested by the 22 high-dimensional benchmark functionsThetest problemswhich consist ofmultimodal functions and uni-modal functions are listed in Table 1 and the correspondingoptimal solutions and search scope are presented in Table 1Compared with solving unimodal problems it is difficult tofind the global optimumofmultimodal problems because thelocal optimawill induce the optimization algorithmsrsquo fall intotheir surroundingsTherefore if the algorithm can efficientlyfind the optimal solutions of multimodal functions it canbe proved that the algorithm is an excellent optimizationalgorithm

42 Comparison of PS-FW with PSO and FWA In thissection we compare the performance of the PS-FW withthe PSO and FWA based on the 22 benchmark functions Inorder to explore global optimization capability of the threealgorithms on solving the high-dimensional optimizationproblem three experiments with different dimensions arecarried outThe dimensions of experiments are set to119863 = 30119863 = 60 and119863 = 100 respectively and each algorithm is usedto solve all the benchmark functions 20 times independentlyIn order to make a fair comparison the general controlparameters of algorithms such as the maximum number ofiterations (119868max) and the population size (119872) are set to beof the same value 119868max is set to 1000 and 119872 is set to 50 foreach function Besides the algorithms used in the experimentare coded by MATLAB 140 and the experiment platformis a personal computer with Core i5 202GHz CPU 4Gmemory and Windows 7 For the purpose of eliminating theimpact on performance caused by the difference in parametersettings themain control parameters of PS-FWalgorithm areconsistent with those of PSO and FWA and the other detailedcontrol parameters are shown in Table 2

For all the benchmark functions the mean and standarddeviation of best solutions obtained by PS-FW and otheralgorithms in 20 independent runs are recorded and theoptimization results are shown in Tables 3ndash5 Meanwhile theranks are also presented in tables and the three algorithmsare ranked mainly based on the mean of best solutions Inaddition the average convergence speed of the proposed PS-FW is compared with other algorithms for functions 1198911211989113 and 11989120 therefore the convergence curves are shown inFigure 3

According to the ranks shown in Tables 3ndash5 the averagevalues of best solutions for the proposed PS-FW outperformthose of the other algorithms Besides the performance ofPS-FW over standard deviation of best solutions is alsobetter than the rest of the algorithms For 22 problems with119863 = 30 the PS-FW can obtain the global optimum of1198912 1198913 1198914 1198915 1198916 1198918 11989112 11989115 11989117 11989118 11989120 and 11989121 whichshows excellent ability for solving optimization problems Asthe dimensions of problems increase the hybrid algorithmmaintains outstanding performance and obtains the optimalsolutions of the 10 functions except for functions 1198913 and 1198916compared with results in Table 3 When the dimensions of


(1) Input Objective function 119891(119909) and constraints(2) Initialization(3) Parameters initialization assign values to119872 119908max 119908min 1198881 1198882 119860119872119890 120576 120575 119886 119887 119903 num119872 119868max FWmax FWmin 120582min 120582max(4) Population initialization generate the random values for 119909119894 and V119894 of each particle in the feasible domain

calculate the 119892119887119890119904119905 of initial population(5) Set 119901119887119890119904119905119894 = 119909119894 (119894 = 1 2 119872) and 119905 = 0(6) Iterations(7) while 119905 le 119868max(8) 119905 = 119905 + 1(9) for 119894 = 1 to119872(10) for 119895 = 1 to119863(11) Update the velocity of particle 119909119894 by using (1)(12) Update the position of particle 119909119894 by using (2)(13) if 119909119894119896 gt 119878max119896 or 119909119894119896 lt 119878min119896(14) Update the value of 119909119894119896 by using (17)(15) end if(16) end for(17) end for(18) Calculate FWnum by using the (9)(19) Sort the particle population in ascending order and select the 119875num particles with better fitness(20) Generate num119864 explosion sparks by using Algorithm 1(21) Calculate the fitness of explosion sparks and storage the best explosion spark 119909best(22) Generate num119872 mutation sparks by using Algorithm 2(23) Select the FWnum individuals from the explosion sparks and mutation sparks by using the selection strategy(24) Combine the 119875num particles with FWnum individuals to generate the new population(25) Calculate 119892119887119890119904119905 and 119901119887119890119904119905119894 of new population(26) end while(27) Output 119892119887119890119904119905 = (1198921198871198901199041199051 1198921198871198901199041199052 119892119887119890119904119905119863)

Algorithm 3 The main codes of PS-FW algorithm

problems are 60 and 100 PS-FW can get the global optimumof functions 1198913 and 1198916 but not each run can succeed This isbecause functions1198913 and1198916 aremultimodal problems and thenumber of local optima increases rapidly as the dimensions ofthe problems increase which adds the difficulty of avoidingtrapping in the local optima In addition according to theranks and values shown in Tables 3ndash5 the PS-FW can get thehighest rank for all the functions It is also needed to point outthat the PS-FW obtains more stable solutions than PSO andFWA for all problems with the increasing of dimensionalityThe convergence speed of the three algorithms can be seenin Figure 3 and the descend rate of average best solutionsof PS-FW is obviously higher than the other two algorithmsThis is because the advantages of PSO and FWAare combinedinto the PS-FW so that the hybrid algorithm enhances itsglobal and local search ability Therefore PS-FW is efficientand robust in dealing with the high-dimensional benchmarkfunctions

From the above analysis it is possible to show that thePS-FW algorithm performs well in solving the functions inTable 1 However because the optimums of these functionsare mostly at the origin we need to further explore theperformance of PS-FW algorithm on the nonzero problemsThen the experiment of nonzero problems is carried outto prove the comprehensive performance of PS-FW In thisexperiment the optimums of test functions derived fromTable 1 are shifted and the specific values are displayed in

Table 6 In addition in order to achieve a fair comparisonbetween the experiments the parameters settings of threealgorithms are consistent with Table 2 and the dimension isset to 119863 = 30 The optimization results of three algorithmsare shown in Table 7 and the convergence curves of threealgorithms over functions 11989112 11989113 and 11989120 are displayed inFigure 4

From Table 7 we can know that the PS-FW algorithmkeeps high performance and can obtain the optimal solutionsof 11 functions in Table 6 Besides the PS-FW achieves thebest rank of three algorithms for all the functions withshift optimums which present the powerful solving abilityover optimization problems with nonzero optimums Bycomparing Table 7 with Table 3 it is known that fireworksalgorithm is relatively weak in searching for nonzero opti-mums However the PS-FW algorithm that derives fromthe fireworks algorithm and covers operators of PSO showsbetter performance which demonstrates the correctness ofthe combination of the two algorithms In addition theresult of PS-FW over function 16 is worse than the previousexperiment This is because 11989116 is a multimodal functionand the slight deviations from the optimums can cause thesignificant increase in the value of the objective function Byobserving the convergence curves in Figure 4 we can statethat the convergence speed of the PS-FW also remains fastIn order to determine whether the convergence performanceof PS-FW algorithm is superior to the other two algorithms


Table 1 The 22 high-dimensional benchmark functions

Name Function Search space Optimum

Sphere 1198911 (119909) = 119863sum119894=1

1199092119894 [minus100 100]119863 0Griewank 1198912 (119909) = 14000

119863sum119894=1

1199092119894 minus 119863prod119894=1

cos( 119909119894radic119894) + 1 [minus600 600]119863 0Rosenbrock 1198913 (119909) = 119863minus1sum

119894=1

[100 (119909119894+1 minus 1199092119894 )2 + (119909119894 minus 1)2] [minus5 10]119863 0Rastrigin 1198914 (119909) = 10119863 + 119863sum

119894=1

[1199092119894 minus 10 cos (2120587119909119894)] [minus512 512]119863 0

Noncontinuous Rastrigin

1198915(119909) = 119863sum119894=1

1199102119894 minus 10 cos(2120587119910119894) + 10119910119894 =

119909119894 10038161003816100381610038161199091198941003816100381610038161003816 lt 05round (2119909119894)2 10038161003816100381610038161199091198941003816100381610038161003816 ge 05

[minus5 10]119863 0

Ackley 1198916 (119909) = minus20 exp(minus02radic 1119863119863sum119894=1

1199092119894)minus exp( 1119863119863sum119894=1

cos (2120587119909119894)) + 20 + 119890 [minus30 30]119863 0Rotated Hyper-Ellipsoid 1198917 (119909) = 119863sum

119894=1

119894sum119895=1

1199092119895 [minus65536 65536]119863 0Noisy Quadric 1198918 (119909) = 119863sum

119894=1

1198941199094 + rand [minus128 128]119863 0Schwefelrsquos problem 221 1198919 (119909) = max

1le119894le119863

10038161003816100381610038161199091198941003816100381610038161003816 [minus100 100]119863 0Schwefelrsquos problem 222 11989110 (119909) = 119863sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 119863prod119894=1


119894=1

minus 119909119894 sin(radic10038161003816100381610038161199091198941003816100381610038161003816) [minus500 500]119889 minus4189829119863Step 11989112 (119909) = 119863sum

119894=1

([119909119894 + 05])2 [minus10 10]119863 0

Levy

11989113 (119909) = sin2 (1205871199101) + 119863minus1sum119894=1

(119910119894 minus 1)2 [1 + 10 sin2 (120587119910119894 + 1)]+ (119910119863 minus 1)2 [1 + sin2 (2120587119910119863)]119910119894 = 1 + 119909119894 minus 14

[minus10 10]119863 0

Powell Sum 11989114 (119909) = 119863sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816119894+1 [minus1 1]119863 0Sum squares 11989115 (119909) = 119863sum

119894=1

1198941199092119894 [minus10 10]119863 0Zakharov 11989116 (119909) = 119863sum

119894=1

1199092119894 + ( 119863sum119894=1

05119894119909119894)2 + ( 119863sum119894=1

05119894119909119894)4 [minus5 10]119863 0Mishra 7 11989117 (119909) = ( 119863prod

119894=1

119909119894 minus 119863)2 [minus119863119863]119863 0Weierstrass 11989118 (119909) = 119863sum

119894=1

[119896maxsum119896=0

(119886119896 cos (2120587119887119896 (119909119894 + 05))) minus 119863119896maxsum119896=0

119886119896 cos (120587119887119896)] [minus05 05]119863 0119886 = 05 119887 = 3 119896max = 20

Bent-Cigar 11989119 (119909) = 11990921 + 106 119863sum119894=1

1199092119894 [minus100 100]119863 0


Table 1 Continued


Trigonometric 2 11989120 (119909) = 1+ 119863sum119894=1

8 sin2 [7 (119909119894 minus 09)2]+6 sin2 [14 (119909119894 minus 09)2]+(119909 minus 09)2 [minus500 500]119863 1Quintic 11989121 (119909) = 119863sum

119894=1

100381610038161003816100381610038161199095119894 minus 31199094119894 + 41199093119894 + 21199092119894 minus 10119909119894 minus 410038161003816100381610038161003816 [minus10 10]119863 0Mishra 11 11989122 (119909) = [[

1119863119863sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + ( 119863prod119894=1

10038161003816100381610038161199091198941003816100381610038161003816)1119863]]2 [minus10 10]119863 0

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106

Aver

age b

est fi

tnes

s

(a) 11989112 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106Av

erag

e bes

t fitn

ess

(b) 11989112 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s(c) 11989112 with119863 = 100

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

(d) 11989113 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

(e) 11989113 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106Av

erag

e bes

t fitn

ess

(f) 11989113 with119863 = 100

PSOFWAPS-FW

200 400 600 800 10000Iteration

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

(g) 11989120 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

(h) 11989120 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

100101102103104105106107

Aver

age b

est fi

tnes

s

(i) 11989120 with119863 = 100

Figure 3 Convergence curves of PSO FWA and PS-FW for functions 11989112 11989113 and 11989120


10minus33

10minus23

10minus13

10minus3

107Av

erag

e bes

t fitn

ess

200 400 600 800 10000Iteration

PSOFWAPS-FW

(a) 11989112 with119863 = 30

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(b) 11989113 with119863 = 30

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(c) 11989120 with119863 = 30


Table 2 The parameter setting of the algorithms

Algorithm Parameter settings

PSO 119908(119905) = 119908max minus 119905119908max minus 119908min119868max 119908max = 095

119908min = 04 1198881 = 1198882 = 145FWA 119860 = 40119872119890 = 50 119886 = 004 119887 = 08

num119872 = 30 120576 = 1119864 minus 100PS-FW

119908(119905) = 119908max minus 119905119908max minus 119908min119868max 119908max = 095

119908min = 04 1198881 = 1198882 = 145 119860 = 40119872119890 = 50 119886 = 004 119887 = 08 num119872 = 30120576 = 1119864 minus 100 120575 = 1119864 minus 6 120582min = 1119864 minus 25120582max = 1 FWmax = 30 FWmin = 20 119903 = 2more clearly we compute the number of successful runs(success rate) and the average number of iterations in success-ful runs for each function in Table 6 The optimal solutionsobtained by different algorithms are various so we definethe convergence criterion for each functionThe convergencecriterion can be introduced as that if the best solutions 119891findfound by each of algorithms are satisfying (18) in a run [39]the run is considered to be successful and the minimumnumber of iterations satisfying the convergence criterion iscounted to calculate the average number of iterations10038161003816100381610038161003816119891find minus 119891opti10038161003816100381610038161003816 lt 120591 (18)

where119891opti is the optimumof function and 120591denotes the errorof algorithm

Suppose that ST denotes the number of successful runsAI indicates the average number of iterations in successfulruns and119880 denotes the iterations number when there are nosuccessful runs after 20 runs and its value is set to greater than119868max then Table 8 is shown as follows

According to the statistical results and ranks presented inTable 8 the success rate and the average iterations numberof PS-FW in 20 runs are both superior to other algorithmsFor all the benchmark functions in Table 6 the proposedPS-FW can satisfy the convergence criterion for all the 20

runs whereas the other algorithms can only converge tothe criterion for several functions In addition the PS-FWobtains the highest ranks for the average number of iterationsin successful runs and can converge to the criterion by arelatively small number of iterations In summary the PS-FW outperforms the other algorithms in terms of stabilityand convergence speed and is an efficacious algorithm foroptimization problems whose optimums are at origin or areshifted

43 Comparison of PS-FWwith PSOVariants In this sectionwe compare the performance of the proposed PS-FW withseveral existing variants of PSO which are introduced ina published paper The comparison is based on the 12benchmark functions introduced in the paper of Nickabadi etal [22] and the orders of functions are consistent with that inthis paper In order to make a fair comparison the run timesand maximum iterations of PS-FW are set to 30 and 200000respectively and the other parameters are set to be the sameas those in Section 42 The dimension of test problems isset to 119863 = 30 and the mean and standard deviation ofbest solutions obtained by algorithms are calculated Thecontrast results are presented in Table 9 and the rank of eachalgorithm is counted and shown

According to the results of Table 9 the PS-FW out-performs the other six PSO variants on both the averagevalues and standard deviation of best solutions after 200000iterations Among the 12 benchmark functions the PS-FWcan obtain the optimum of 10 functions which manifests thehighly powerful ability to find the global optimal solution Inaddition the PS-FW acquires the highest rank over almost allthe test problems except the function11989111 which indicates thePS-FW has significant improvement than other algorithmsBesides the analysis of numerical results obtained by PS-FWand other algorithms we applied the nonparametric statisti-cal tests to prove the superiority of the PS-FWThe Friedmantest and Bonferroni-Dunn test are adopted to compare theperformance of PS-FW with the other algorithms

The Friedman test is a multiple comparison test to detectthe significant differences among algorithms based on the


Table 3 Comparison of the optimization results obtained by PS-FW PSO and FWA with119863 = 30 for functions 1198911 to 11989122 (the best ranks aremarked in bold)

119891 119863 PSO FWA PS-FW

1198911 30 Mean 88371119864 + 01 13360119864 minus 151 58928119864 minus 264Std 43475119864 + 01 58057119864 minus 151 0Rank 3 2 1

1198912 30 Mean 71542119864 minus 02 0 0Std 12385119864 minus 01 0 0Rank 2 1 1

1198913 30 Mean 55766119864 + 02 26882119864 + 01 0Std 74828119864 + 02 83997119864 minus 01 0Rank 3 2 1

1198914 30 Mean 66547119864 + 01 0 0Std 36430119864 + 01 0 0Rank 2 1 1

1198915 30 Mean 65810119864 + 01 0 0Std 40117119864 + 01 0 0Rank 2 1 1

1198916 30 Mean 0 0 0Std 0 0 0Rank 1 1 1

1198917 30 Mean 14156119864 + 04 76585119864 minus 83 45128119864 minus 122Std 10006119864 + 04 33383119864 minus 82 18821119864 minus 121Rank 3 2 1

1198918 30 Mean 10419119864 minus 03 96596119864 minus 304 0Std 10584119864 minus 03 0 0Rank 3 2 1

1198919 30 Mean 63165119864 minus 01 74698119864 minus 54 31588119864 minus 97Std 60679119864 minus 01 23638119864 minus 53 12719119864 minus 96Rank 3 2 1


11989111 30 Mean minus72662119864 + 03 minus10511119864 + 04 minus12483119864 + 04Std 67867119864 + 02 19893119864 + 02 12661119864 + 02Rank 3 2 1

11989112 30 Mean 69734119864 minus 01 66542119864 minus 01 0Std 28586119864 minus 01 50080119864 minus 01 0Rank 3 2 1

11989113 30 Mean 17831119864 + 01 65460119864 + 00 14998119864 minus 32Std 86204119864 + 00 86700119864 minus 01 0Rank 3 2 1

11989114 30 Mean 66576119864 minus 08 45613119864 minus 191 21563119864 minus 291Std 54575119864 minus 08 0 0Rank 3 2 1

11989115 30 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 3 Continued

119891 119863 PSO FWA PS-FW

11989117 30 Mean 0 98737119864 + 44 0Std 0 43038119864 + 45 0Rank 1 2 1

11989118 30 Mean 15069119864 + 01 0 0Std 40495119864 + 00 0 0Rank 2 1 1


11989120 30 Mean 38005119864 + 02 42079119864 + 01 1Std 85739119864 + 01 46125119864 + 00 0Rank 3 2 1

11989121 30 Mean 45577119864 + 01 171130119864 + 01 0Std 23091119864 + 01 21499119864 + 00 0Rank 3 2 1

11989122 30 Mean 70166119864 minus 01 11989119864 minus 149 35102119864 minus 292Std 59846119864 minus 01 52258119864 minus 149 0Rank 3 2 1

Average rank 25455 17273 1Overall rank 3 2 1

sets of data [40] The algorithms are ranked in Friedmantest which means the algorithm with the best performanceis ranked minimum the worst gets the maximum rank andso on In this section the mean and standard deviationof best solutions based on Table 9 are conducted with theFriedman test therefore the results are given in Table 10Through observing the results of Friedman test in Table 10 allthe 119901 value are lower than the level of significance considered120572 = 001 which indicates that the significant differencesamong the seven algorithms do exist According to the ranksobtained by the Friedman test in Table 10 the PS-FW has thebest performance on themean and standard deviation of bestsolutions followed by ALWPSO CLPSO and the other fouralgorithms Therefore we can conclude that the accuracy ofsolutions obtained by PS-FW is better than other algorithmsHowever the Friedman test can only detect whether there aresignificant differences among all the algorithms but is unableto conduct the proper comparisons between PS-FW and eachof the other algorithms Hence the Bonferroni-Dunn test isexecuted to check the superiority of PS-FW

The Bonferroni-Dunn test can be very intuitive to detectthe significant difference between the two or more algo-rithms For Bonferroni-Dunn test the judgment conditionfor the existence of significant difference between the twoalgorithms is that their mean ranks differ by at least thecritical difference (CD) and the equation of calculating thecritical difference is as follows [41]

CD120572 = 119902120572radic119873119894 (119873119894 + 1)6119873119891 (19)

where 119873119894 and 119873119891 are the number of algorithms and bench-mark functions and the critical values 119902120572 at the probabilitylevel 119886 are presented as follows

119902005 = 27711990201 = 254 (20)

By utilizing (19) and (20) the critical difference is shownas follows

CD005 = 244CD01 = 224 (21)

Here we carry out the Bonferroni-Dunn test for themean of best solutions success rate and average numberof iterations of successful runs on the basis of the ranksobtained by the Friedman test In order to provide a moreintuitive display of the results obtained by Bonferroni-Dunntest we illustrate the critical differences among the sevenalgorithms in Figure 5 For the purpose of comparing thealgorithms clearly a horizontal line which indicates thethreshold for the best performing algorithm (the one withpink color) is drawn in the graphs In addition another twolines which represent each level of significance consideredin the paper are also drawn and their heights are equalto the sum of minimum rank and the corresponding CDThen if the bars exceed the lines of significant level thecorresponding algorithms are proved to have worse per-formance than the best performing algorithm By observ-ing the results of Bonferroni-Dunn test in Figure 5(a) thebar of the PS-FW has the lowest height among all thealgorithms and the heights of bars corresponding to the



119891 119863 PSO FWA PS-FW


1198912 60 Mean 32482119864 + 00 0 0Std 96094119864 minus 01 0 0Rank 2 1 1

1198913 60 Mean 71638119864 + 04 45073119864 + 01 92568119864 minus 30Std 55811119864 + 04 18390119864 + 01 19330119864 minus 29Rank 3 2 1

1198914 60 Mean 32219119864 + 02 0 0Std 41863119864 + 01 0 0Rank 2 1 1

1198915 60 Mean 37498119864 + 02 0 0Std 53191119864 + 01 0 0Rank 2 1 1

1198916 60 Mean 13162119864 + 01 0 71054119864 minus 16Std 11773119864 + 00 0 14211119864 minus 15Rank 3 1 2


1198918 60 Mean 11343119864 + 00 12096119864 minus 288 0Std 32234119864 + 00 0 0Rank 3 2 1





11989113 60 Mean 62329119864 + 01 15355119864 + 01 14998119864 minus 32Std 20956119864 + 01 54415119864 + 00 0Rank 3 2 1


11989115 60 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 4 Continued

119891 119863 PSO FWA PS-FW

11989117 60 Mean 0 24945119864 + 145 0Std 0 57208119864 + 145 0Rank 1 2 1

11989118 60 Mean 39564119864 + 01 0 0Std 53138119864 + 00 0 0Rank 2 1 1


11989120 60 Mean 53645119864 + 03 14665119864 + 02 1Std 62256119864 + 03 28947119864 + 01 0Rank 3 2 1

11989121 60 Mean 19709119864 + 02 48085119864 + 01 0Std 28605119864 + 01 77355119864 + 00 0Rank 3 2 1

11989122 60 Mean 15314119864 + 00 15711119864 minus 142 13216119864 minus 280Std 59245119864 minus 01 47133119864 minus 142 0Rank 3 2 1


FIPS

CPSO

stdPs

o

PS-F

W

CLPS

O

AIW

PSO

Fran

kens

tein

Algorithms

Rank95 sig level

90 sig levelBest rank

0

2

4

6

8

Rank

s

(a) Mean

FIPS

CPSO

stdPs

o

PS-F

W

CLPS

O

AIW

PSO

Fran

kens

tein

Algorithms

Rank95 sig level


0

2

4

6

8

Rank

s

(b) Standard deviation

Figure 5The bar chart of Bonferroni-Dunn test for PS-FW and other PSO variants over mean and standard deviation of best solutions basedon Table 10

stdPSO CPSO FIPS and Frankenstein exceed the lines ofsignificant level which indicates that the PS-FW performssignificantly better than these four algorithms over thesolutions accuracy In addition the PS-FW acquires the bestrank over the standard deviation according to Figure 5(b)and the PS-FW has the obvious advantage compared to the

stdPSO CPSO FIPS and Frankenstein Therefore we canconclude that the PS-FW is the best performing algorithmfollowed by ALWPSO CLPSO and other four algorithmsand the advantages of PS-FW on the efficiency and solutionsaccuracy compared with other algorithms are definitelyproved



119891 119863 PSO FWA PS-FW


1198912 100 Mean 11830119864 + 02 0 0Std 51822119864 + 01 0 0Rank 2 1 1


1198914 100 Mean 47288119864 + 02 0 0Std 10713119864 + 02 0 0Rank 2 1 1

1198915 100 Mean 51626119864 + 02 0 0Std 14819119864 + 02 0 0Rank 2 1 1







11989112 100 Mean 12670119864 + 02 42335119864 + 00 0Std 48966119864 + 01 140825853 0Rank 3 2 1



11989115 100 Mean 0 0 0Std 0 0 0Rank 1 1 1


11989117 100 Mean 0 20047119864 + 232 0Std 0 67205119864 + 232 0Rank 1 2 1


Table 5 Continued

119891 119863 PSO FWA PS-FW

11989118 100 Mean 68687119864 + 01 0 0Std 13221119864 + 01 0 0Rank 2 1 1


11989120 100 Mean 90245119864 + 03 26557119864 + 02 1Std 38036119864 + 03 47674119864 + 01 0Rank 3 2 1

11989121 100 Mean 40256119864 + 03 91975119864 + 01 0Std 16131119864 + 04 17966119864 + 01 0Rank 3 2 1



Besides the above analysis we count the number ofsuccessful runs and the average number of iterations insuccessful runs for the PS-FW over 12 benchmark functionsand the statistical results are presented in Table 11 In thissection a successful run means the algorithm can obtain theoptimumwithin the 200000 iterations As shown in Table 11the PS-FW can converge to the optimal solution in each ofruns over the vast majority functions which manifests therobustness of PS-FW in solving the optimization problemsIn order to compare the convergence speed of PS-FW withother algorithms fairly the average numbers of iterations insuccessful runs are compared over the six functions 1198911 11989141198916 1198917 11989110 and 11989111 introduced in Nickabadi et alrsquos paperAccording to the numerical results in Table 11 the PS-FWcan converge to the optimal solution for all the six functionswithin 12000 iterations whereas the other algorithms havedifficulty in obtaining the optimum for functions 1198911 11989161198917 and 11989110 after 200000 iterations or can converge to theoptimum for functions119891411989111 with a lotmore iterations basedon the convergence curves in the paper by Nickabadi et alTherefore we can argue that the robustness and convergencespeed of PS-FW are superior to the other algorithms

44 Experiments to Analyze the PS-FW Control ParametersIn this section we investigate the impact of the controlparameters on the performance of PS-FW From the previousintroduction the PS-FW has several control parametersincluding the parameters adopted from PSO and FWA Herewe only analyze the three main control parameters which arethe control factors of explosion amplitudes 120582min 120582max and thenumber ofmutation sparks num119872 In order to test the impactof changes in control parameters on performance exhaus-tively six different combinations of parameters were selectedand experimented on Each set of parameters correspondsto 20 runs based on 22 functions introduced in Table 1 and

Table 6 The benchmark functions with shift optima

Name Original optima Shift optimaSphere [0 0 0] [70 70 70]Griewank [0 0 0] [70 70 70]Rastrigin [0 0 0] [3 3 3]NoncontinuousRastrigin [0 0 0] [5 5 5]Ackley [0 0 0] [20 20 20]RotatedHyper-Ellipsoid [0 0 0] [70 70 70]Schwefelrsquos problem221 [0 0 0] [70 70 70]Schwefelrsquos problem222 [0 0 0] [70 70 70]Step [minus05 minus05 minus05] [5 5 5]Levy [1 1 1] [5 5 5]Sum squares [0 0 0] [5 5 5]Zakharov [0 0 0] [5 5 5]Bent-Cigar [0 0 0] [70 70 70]Trigonometric 2 [09 09 09] [70 70 70]Mishra 11 [0 0 0] [5 5 5]

the dimensions of problems are set to 100 Moreover theother parameters settings of PS-FW except 120582min 120582max andnum119872 are the same as those in Section 42 In additionthe six combinations of control parameters are representedas six optimization strategies and their detailed parameterssettings are shown in Table 12 and the control parametersof Section 42 are marked as Strategy-1 and are presented Asshown in Table 12 we take a contrastingmethod that changesa parameter and keeps the other parameters unchanged


Table 7 Comparison of the optimization results obtained by PS-FW PSO and FWA for functions in Table 6 (the best ranks are marked inbold)

119891 119863 PSO FWA PS-FW


1198912 30 Mean 47829119864 + 00 62867119864 minus 01 0Std 15089119864 + 00 53523119864 minus 02 0Rank 3 2 1

1198914 30 Mean 12559119864 + 02 98052119864 + 00 0Std 47596119864 + 01 16323119864 + 00 0Rank 3 2 1

1198915 30 Mean 16140119864 + 02 22289119864 + 01 0Std 37649119864 + 01 27981119864 + 00 0Rank 3 2 1




11989110 30 Mean 16846119864 + 03 22370119864 + 02 0Std 26627119864 + 02 74690119864 + 01 0Rank 3 2 1

11989112 30 Mean 11359119864 + 02 21375119864 + 01 0Std 41907119864 + 01 29107119864 + 00 0Rank 3 2 1


11989115 30 Mean 0 29887119864 minus 04 0Std 0 13027119864 minus 03 0Rank 1 2 1


11989119 30 Mean 24875119864 + 09 22700119864 + 08 0Std 13163119864 + 09 27319119864 + 07 0Rank 3 2 1

11989120 30 Mean 20564119864 + 03 92562119864 + 02 1Std 79311119864 + 02 76748119864 + 01 0Rank 3 2 1




Table 8 Comparison of successful rates and average number ofiterations for PS-FW PSO and FWA with 120591 = 10minus4 for function 11989115and 120591 = 101 for other functions (the best ranks are marked in bold)

119891 PSO FWA PS-FW1198911ST 0 20 20Rank 2 1 1AI 119880 2017 284Rank 3 2 11198912ST 19 20 20Rank 2 1 1AI 96 46 28Rank 3 2 11198914ST 0 11 20Rank 3 2 1AI 119880 5848 2288Rank 3 2 11198915ST 0 0 20Rank 2 2 1AI 119880 119880 1049Rank 2 2 11198916ST 0 20 20Rank 2 1 1AI 119880 343 98Rank 3 2 11198917ST 0 0 20Rank 2 2 1AI 119880 119880 938Rank 2 2 11198919ST 0 0 20Rank 2 2 1AI 119880 119880 267Rank 2 2 111989110ST 0 0 20Rank 2 2 1AI 119880 119880 411Rank 2 2 111989112ST 0 0 20Rank 2 2 1AI 119880 119880 118Rank 2 2 111989113ST 0 0 20Rank 2 2 1AI 119880 119880 35Rank 2 2 111989115ST 20 19 20Rank 1 2 1AI 5053 6796 131Rank 2 3 1

Table 8 Continued119891 PSO FWA PS-FW11989116ST 16 0 20Rank 2 3 1AI 224 119880 2087Rank 2 3 111989119ST 0 0 20Rank 2 2 1AI 119880 119880 2089Rank 2 2 111989120ST 0 0 20Rank 2 2 1AI 119880 119880 1608Rank 2 2 111989122ST 20 20 20Rank 1 1 1AI 942 1232 93Rank 2 3 1

Average rank of ST 19 18 1Overall rank of AI 23 22 1

Then the optimization results and the corresponding ranksof different strategies are shown in Tables 13 and 14 andthe results focus on mean and standard deviation of bestsolutions obtained by different strategies From the results ofTables 13 and 14 the PS-FW with Strategy-6 and Strategy-7 has the best performance for almost all the benchmarkfunctions and can obtain the highest ranks over both themean and standard deviation of best solutions By adoptingStrategy-6 and Strategy-7 the PS-FW can get the optimumof 16 functions for the whole 20 runs especially includingthe functions 1198911 1198913 1198916 11989114 11989119 and 11989122 which cannot findthe global best solutions by other optimization strategies ofPS-FW Therefore the excellent performance of PS-FW withStrategy-6 and Strategy-7 proves the correctness of proposedmutation operator and indicates that increasing the numberof mutation sparks can enhance the global search capabilityof the algorithm However according to the ldquono free lunchtheoremrdquo [42] there is no algorithm that can perform betterthan others on all the problems hence the PS-FW withStrategy-6 and Strategy-7 has poor performance for function1198917 It is because function 1198917 has a wide search scope so thatthe solutions have little changes in the later iterations if 120582minis small which results in a relatively slow convergence speedfor PS-FW despite the increase in the number of mutationsparks For other strategies of PS-FW the different strategieshave their own advantages for various test functions the PS-FW with Strategy-1 performs well for functions 1198911 1198913 11989161198919 and 11989119 and the good solutions can be obtained by PS-FW over functions 1198917 11989116 under Strategy-2 and Strategy-3 Meanwhile the PS-FW with Strategy-4 and Strategy-5works well in solving the functions 11989110 and 11989122 In additionthe PS-FW can obtain the optimum of functions 1198912 11989141198915 1198918 11989112 11989115 11989117 11989118 11989120 and 11989121 and keep outstanding


Table 9 Comparison of the optimization results obtained by PS-FW and six PSO variants (the best ranks are marked in bold)

119891(119909) PS-FW stdPSO CPSO CLPSO FIPS Frankenstein AIWPSO1198911Mean 0 5198119864 minus 40 5146119864 minus 13 4894119864 minus 39 4588119864 minus 27 2409119864 minus 16 3370119864 minus 134Rank 1 3 7 4 5 6 2Std 0 11301119864 minus 78 77588119864 minus 25 67814119864 minus 78 19577119864 minus 53 20047119864 minus 31 51722119864 minus 267Rank 1 3 7 4 5 6 21198912Mean 0 21625119864 minus 02 21245119864 minus 02 0 24776119864 minus 04 14736119864 minus 03 28524119864 minus 02Rank 1 5 4 1 2 3 6Std 0 45019119864 minus 04 63144119864 minus 04 0 18266119864 minus 06 12846119864 minus 05 76640119864 minus 04Rank 1 4 5 1 2 3 61198913Mean 0 25404119864 + 01 82648119864 minus 01 13217119864 + 01 26714119864 + 01 28156119864 + 01 25003119864 + 00Rank 1 5 2 4 6 7 3Std 0 59031119864 + 02 23449119864 + 00 21480119864 + 02 20025119864 + 02 23132119864 + 02 15978119864 + 01Rank 1 7 2 5 4 6 31198914Mean 0 34757119864 + 01 36007119864 minus 13 0 58502119864 + 01 73836119864 + 01 16583119864 minus 01Rank 1 4 2 1 5 6 3Std 0 10636119864 + 02 15035119864 minus 24 0 19185119864 + 02 37055119864 + 02 21051119864 minus 01Rank 1 4 2 1 5 6 31198915Mean 0 20956119864 + 01 53717119864 minus 13 13333119864 minus 01 61883119864 + 01 70347119864 + 01 11842119864 minus 16Rank 1 5 3 4 6 7 2Std 0 18327119864 + 02 59437119864 minus 24 11954119864 minus 01 14013119864 + 02 29600119864 + 02 42073119864 minus 31Rank 1 6 3 4 5 7 21198916Mean 0 14921119864 minus 14 16091119864 minus 07 92371119864 minus 15 13856119864 minus 14 21792119864 minus 09 69870119864 minus 15Rank 1 5 7 3 4 6 2Std 0 18628119864 minus 29 78608119864 minus 14 66156119864 minus 30 23227119864 minus 29 17187119864 minus 18 42073119864 minus 31Rank 1 4 7 3 5 6 21198917Mean 0 14582119864 + 00 18889119864 + 03 19217119864 + 02 94634119864 + 00 17315119864 + 02 19570119864 minus 10Rank 1 3 7 6 4 5 2Std 0 11783119864 + 00 99106119864 + 06 38433119864 + 03 25976119864 + 01 91577119864 + 03 12012119864 minus 19Rank 1 3 7 5 4 6 21198918Mean 0 12375119864 minus 02 10764119864 minus 02 40642119864 minus 03 33047119864 minus 03 41690119864 minus 03 55241119864 minus 03Rank 1 7 6 3 2 4 5Std 0 23107119864 minus 05 27698119864 minus 05 96184119864 minus 07 86680119864 minus 07 24012119864 minus 06 15358119864 minus 05Rank 1 6 7 3 2 4 511989110Mean 0 34621119864 minus 26 54282119864 minus 14 99748119864 minus 39 26033119864 + 02 51953119864 + 04 18317119864 minus 137Rank 1 4 5 3 6 7 2Std 0 40873119864 minus 51 82868119864 minus 27 37661119864 minus 84 21785119864 + 04 11136119864 + 09 34534119864 minus 273Rank 1 4 5 3 6 7 211989111Mean minus12542119864 + 04 minus10995119864 + 04 minus12127119864 + 04 minus12546119864 + 04 minus11052119864 + 04 minus11221119864 + 04 minus12569119864 + 04Rank 3 7 5 2 6 4 1Std 14900119864 + 02 13753119864 + 05 33795119864 + 04 42567119864 + 03 94421119864 + 05 27708119864 + 05 11409119864 minus 25Rank 2 5 4 3 7 6 1


Table 9 Continued

119891(119909) PS-FW stdPSO CPSO CLPSO FIPS Frankenstein AIWPSO11989112Mean 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 1Std 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 111989113Mean 14998119864 minus 32 11422119864 minus 29 20913119864 minus 15 14998119864 minus 32 10273119864 minus 28 55136119864 minus 18 14998119864 minus 32Rank 1 2 5 1 3 4 1Std 0 32335119864 minus 57 12954119864 minus 29 12398119864 minus 94 10052119864 minus 56 14501119864 minus 34 12398119864 minus 94Rank 1 3 6 2 4 5 2

Table 10 The results of Friedman test for the PS-FW and otherPSO variants over themean and standard deviation of best solutionsbased on Table 9 (the best ranks are marked in bold)

Mean StdResults119873 12 12

Chi-square 3533 3718119901 value 372119864 minus 06 162119864 minus 06Friedman ranks of Algorithms

PS-FW 158 15stdPso 483 467CPSO 508 517CLPSO 317 325FIPS 475 467Frankenstein 558 575AIWPSO 3 3

performance in other functions under the whole seven strate-gies Therefore the robustness of the proposed algorithmis strongly proved To compare the convergence speeds fordifferent strategies of PS-FW the convergence curves overseveral functions are shown in Figure 6 By observing thecurves in Figure 6 the superiority of Strategy-6 and Strategy-7 in terms of convergence speed has been demonstratedand the PS-FW with all strategies can converge to solutionsthat are very close to the optimums Then we conduct theFriedman test and the Bonferroni-Dunn test for the meanand standard deviation of best solutions obtained by differentoptimization strategies so as to determine the impact degreeof each control parameter on the performance of PS-FWTheresults of Friedman test for different strategies of PS-FW areshown in Table 15 and the results of Bonferroni-Dunn test interms of mean and standard deviation based on Table 15 arepresented in Figures 7 and 8

According to the results of Friedman test in Table 15 the119901 value is lower than the level of significance considered120572 = 005 for both the mean and standard deviationof bets solutions which indicates that the performance ofseven strategies of PS-FW has the significant difference Byobserving the ranks obtained by the Friedman test in Table 15the PS-FWwith Strategy-7 has the best performance followed

Table 11The statistical results of PS-FW in terms of success rate andaverage number of iterations in successful runs for 12 benchmarkfunctions

Functions ST AT1198911 30 382801198912 30 88261198913 30 1126651198914 30 185381198915 30 213471198916 30 75511198917 30 591041198918 30 2281111989110 30 6304711989111 29 1100511989112 30 7516011989113 0 119880Table 12 The detailed parameters settings of the different opti-mization strategies for PS-FW (the square brackets represent therounding operations)

Strategies 120582max 120582min num119872Strategy-1 1 1119864 minus 25 30Strategy-2 1 1119864 minus 10 30Strategy-3 1 01 30Strategy-4 08 1119864 minus 25 30Strategy-5 06 1119864 minus 25 30Strategy-6 1 1119864 minus 25 [05 sdot num119864]Strategy-7 1 1119864 minus 25 [07 sdot num119864]

by Strategy-6 Strategy-1 and so on and the PS-FW withStrategy-2 performs the worst relative to other strategies overthe average values of best solutions In Bonferroni-Dunntest the values of critical difference are the same as those inSection 42 and the lines of best rank and significant level arealso drawn in Figures 7 and 8Through checking the bars cor-responding to the different strategies of PS-FW in Figure 7(a)the heights of bars for Strategy-1 to Strategy-5 exceed the linesof significant level Hence Strategy-7 represents the best com-bination of control parameters among all the seven strategies


Table 13 The mean standard deviation and corresponding ranks of best solutions obtained by different optimization strategies of PS-FWfor functions 1198911 to 11989113 (the best ranks are marked in bold)

119891(119909) Strategy-1 Strategy-2 Strategy-3 Strategy-4 Strategy-5 Strategy-6 Strategy-71198911Mean 97833119864 minus 245 66617119864 minus 217 81065119864 minus 224 14930119864 minus 224 68133119864 minus 231 0 0Rank 2 6 5 4 3 1 1Std 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 11198912Mean 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 1Std 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 11198913Mean 10341119864 minus 26 71483119864 minus 16 25737119864 minus 13 13156119864 minus 09 22836119864 minus 09 0 0Rank 2 3 4 5 6 1 1Std 38500119864 minus 26 13157119864 minus 15 71641119864 minus 13 42629119864 minus 09 45987119864 minus 09 0 0Rank 2 3 4 5 6 1 11198914Mean 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 1Std 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 11198915Mean 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 1Std 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 11198916Mean 71054119864 minus 16 23093119864 minus 15 14211119864 minus 15 23093119864 minus 15 24869119864 minus 15 0 0Rank 2 4 3 4 5 1 1Std 14211119864 minus 15 16945119864 minus 15 17405119864 minus 15 16945119864 minus 15 16281119864 minus 15 0 0Rank 2 4 5 4 3 1 11198917Mean 21860119864 minus 71 70151119864 minus 123 35034119864 minus 126 27732119864 minus 62 20900119864 minus 65 57053119864 minus 83 23724119864 minus 87Rank 5 2 1 7 6 4 3Std 47535119864 minus 71 18052119864 minus 122 12502119864 minus 125 12084119864 minus 61 90599119864 minus 65 57716119864 minus 83 99762119864 minus 87Rank 5 2 1 7 6 4 31198918Mean 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 1Std 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 11198919Mean 11555119864 minus 90 25372119864 minus 78 16308119864 minus 76 26199119864 minus 86 14655119864 minus 89 13155119864 minus 117 61364119864 minus 130Rank 3 6 7 5 4 2 1Std 27315119864 minus 90 11059119864 minus 77 47755119864 minus 76 77290119864 minus 86 62719119864 minus 89 57340119864 minus 117 26737119864 minus 129Rank 3 6 7 5 4 2 111989110Mean 22792119864 minus 128 55926119864 minus 118 91955119864 minus 124 30530119864 minus 130 28788119864 minus 130 67603119864 minus 161 16779119864 minus 167Rank 5 7 6 4 3 2 1Std 97764119864 minus 128 24326119864 minus 117 34455119864 minus 123 92801119864 minus 130 11346119864 minus 129 29329119864 minus 160 0Rank 5 7 6 3 4 2 1


Table 13 Continued

119891(119909) Strategy-1 Strategy-2 Strategy-3 Strategy-4 Strategy-5 Strategy-6 Strategy-711989111Mean minus41743119864 + 04 minus41279119864 + 04 minus41366119864 + 04 minus41366119864 + 04 minus41345119864 + 04 minus41757119864 + 04 minus41790119864 + 04Rank 3 6 4 4 5 2 1Std 43502119864 + 02 41356119864 + 02 35331119864 + 02 41470119864 + 02 34657119864 + 02 26837119864 + 02 14566119864 + 02Rank 7 5 4 6 3 2 111989112Mean 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 1Std 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 111989113Mean 14998119864 minus 32 14998119864 minus 32 14998119864 minus 32 14998119864 minus 32 14998119864 minus 32 14998119864 minus 32 14998119864 minus 32Rank 1 1 1 1 1 1 1Std 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 1

Strategy-1Strategy-2Strategy-3


Strategy-7

200 400 600 800 10000Iterations

10minus28410minus27410minus26410minus25410minus24410minus23410minus22410minus21410minus20410minus19410minus18410minus17410minus16410minus15410minus14410minus13410minus12410minus11410minus10410minus9410minus8410minus7410minus6410minus5410minus4410minus3410minus2410minus1410minus4106

Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7

10minus13610minus12610minus11610minus10610minus9610minus8610minus7610minus6610minus5610minus4610minus3610minus2610minus1610minus6104

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations

10minus17610minus16610minus15610minus14610minus13610minus12610minus11610minus10610minus9610minus8610minus7610minus6610minus5610minus4610minus3610minus2610minus1610minus6104

Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(d) 11989122Figure 6 Convergence curves of PS-FW with different strategies for functions 1198911 1198919 11989110 and 11989122



119891(119909) Strategy-1 Strategy-2 Strategy-3 Strategy-4 Strategy-5 Strategy-6 Strategy-711989114Mean 64751119864 minus 275 46790119864 minus 268 50050119864 minus 272 12035119864 minus 283 97967119864 minus 265 0 0Rank 3 5 4 2 6 1 1Std 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 111989115Mean 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 1Std 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 111989116Mean 24731119864 minus 93 25574119864 minus 102 10668119864 minus 102 92122119864 minus 91 78026119864 minus 91 25290119864 minus 114 17103119864 minus 116Rank 5 4 3 7 6 2 1Std 84009119864 minus 93 10215119864 minus 101 32290119864 minus 102 37019119864 minus 90 30225119864 minus 90 46404119864 minus 114 62900119864 minus 116Rank 5 4 3 7 6 2 111989117Mean 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 1Std 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 111989118Mean 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 1Std 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 111989119Mean 90096119864 minus 250 23878119864 minus 201 15857119864 minus 189 59464119864 minus 249 15925119864 minus 244 0 0Rank 2 5 6 3 4 1 1Std 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 111989120Mean 1 1 1 1 1 1 1Rank 1 1 1 1 1 1 1Std 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 111989121Mean 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 1Std 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 111989122Mean 49253119864 minus 273 85544119864 minus 231 14963119864 minus 229 38782119864 minus 275 43846119864 minus 276 0 0Rank 4 5 6 3 2 1 1Std 0 0 0 0 0 0 0Rank 1 1 1 1 1 1 1


0

2

4

6Ra

nks

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7Strategies

Rank95 sig level


(a) Strategy-7 as the best rank

0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level


(b) Strategy-6 as the best rank

Figure 7 The bar chart of Bonferroni-Dunn test for different strategies over the mean of best solutions based on Table 15

0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-7

Stra

tegy

-1

Strategies

Rank95 sig level



Figure 8 The bar chart of Bonferroni-Dunn test for different strategies over the standard deviation of best solutions based on Table 15

and the PS-FW with Strategy-7 performs significantly betterthan the other strategies except Strategy-6 In addition thePS-FW with Strategy-6 has significant superiority comparedwith Strategy-2 to Strategy-5 over the average values of bestsolutions based on Figure 7(b) Besides as shown in Figure 8the hybrid algorithm with different strategies has relativelysmall gaps in standard deviation Strategy-7 emerges as thebest performer over the standard deviation of best solutions

followed by Strategy-6 Strategy-1 and other strategies andStrategy-4 has the worst performance

Therefore based on the analysis above the solutionsaccuracy and convergence speed of PS-FW are determinedby the control parameters 120582min 120582max and num119872 Comparedwith 120582min and 120582max the number of mutation sparks hasa greater impact on the performance of PS-FW Hence wecan appropriately increase the number of mutation sparks


Table 15 The results of Friedman test for the different strategies ofPS-FW over the mean and standard deviation of optimal solutionsbased on Tables 13 and 14 (the best ranks are marked in bold)


Chi-square 4023 2238119901 value 410119864 minus 07 103119864 minus 03Friedman ranks of algorithms

Strategy-1 391 414Strategy-2 475 425Strategy-3 452 423Strategy-4 45 452Strategy-5 464 427Strategy-6 295 341Strategy-7 273 318

when solving the difficult multimodal global optimizationproblems In addition the value of 120582min can be increasedproperly for solving the optimization problems with largerange such as function 1198917 Considering that the increase inthe number ofmutation sparks will make the computing timelonger to improve the computational efficiency Strategy-1which ranks third in seven strategies is used to conduct theexperiments in Sections 42 and 43 in this paper As expectedwe should choose the suitable control parameters for variousproblems by taking all the aspects into consideration

5 Conclusion

In this paper a hybrid algorithm named PS-FW is proposedto solve the global optimization problems In PS-FW theexploitation capability is applied to find the optimal solutionand make the hybrid algorithm converge quickly whereasthe exploration ability of FWA is used to search for thebetter solutions in the entire feasible space Moreover theabandonment and supplement mechanism the modifiedexplosion operator and the novel mutation operator areproposed to enhance both the global and local search abilityof algorithmThen the validity of PS-FW is confirmed by the22 well-known high-dimensional benchmark functions Theresults show that PS-FW is an efficacious fast convergingand robust optimization algorithm by comparing with thePSO FWA stdPSO CPSO CLPSO FIPS Frankenstein andALWPSO over solving global optimization problems

The future work is to refine the PS-FW by testing morecomplex high-dimensional optimization problems Further-more we will try to apply the algorithm to multiobjectiveoptimization problems and real-world problems such as spa-tial layout optimization route optimization and structuralparameter optimization

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This study was funded by National Natural Science Founda-tion of China (nos 51674086 and 51534004) and NortheastPetroleum University Innovation Foundation for Postgradu-ate (no YJSCX2015-012NEPU)

References

[1] Y Tan Firework Algorithm A Novel Swarm Intelligence Opti-mization Method Springer Berlin Heidelberg Germany 2015

[2] N Islam S Rana R Ahsan and S Ghani ldquoAn OptimizedDesign of Network Arch Bridge using Global OptimizationAlgorithmrdquoAdvances in Structural Engineering vol 17 no 2 pp197ndash210 2014

[3] E Vinot V Reinbold and R Trigui ldquoGlobal Optimized Designof an Electric Variable Transmission for HEVsrdquo IEEE Trans-actions on Vehicular Technology vol 65 no 8 pp 6794ndash67982016

[4] N Gabere Simulated Annealing Driven Pattern Search Algo-rithms for Global Optimization University of the Witwater-srand Johannesburg South Africa 2007

[5] R Storn and K Price ldquoDifferential Evolution - A Simple andEfficient Heuristic for Global Optimization over ContinuousSpacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[6] P Kaelo andMM Ali ldquoIntegrated crossover rules in real codedgenetic algorithmsrdquo European Journal of Operational Researchvol 176 no 1 pp 60ndash76 2007

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

[8] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006

[9] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep Erciyes University Kayseri Turkey2005

[10] Y Tan and Y Zhu ldquoFireworks algorithm for optimizationrdquoAdvances in Swarm Intelligence pp 355ndash364 2010

[11] J Wang B Lin and J Jin ldquoOptimizing the shunting scheduleof electric multiple units depot using an enhanced particleswarm optimization algorithmrdquo Computational Intelligence andNeuroscience vol 2016 Article ID 5804626 2016

[12] X Wu C Li W Jia and Y He ldquoOptimal operation of trunknatural gas pipelines via an inertia-adaptive particle swarmoptimization algorithmrdquo Journal of Natural Gas Science andEngineering vol 21 pp 10ndash18 2014

[13] XHua XHu andWYuan ldquoResearch optimization on logisticsdistribution center location based on adaptive particle swarmalgorithmrdquo Optik - International Journal for Light and ElectronOptics vol 127 no 20 pp 8443ndash8450 2016

[14] B A Garroa and R A Vazquez ldquoDesigning artificial neuralnetworks using particle swarm optimization algorithmsrdquo Com-putational Intelligence and Neuroscience vol 2015 Article ID369298 20 pages 2015

[15] S Ye H Ma S Xu W Yang and M Fei ldquoAn effective fireworksalgorithm for warehouse-scheduling problemrdquo Transactions ofthe Institute of Measurement and Control vol 39 no 1 pp 75ndash85 2017


[16] Y Zheng Q Song and S Chen ldquoMultiobjective fireworks opti-mization for variable-rate fertilization in oil crop productionrdquoApplied Soft Computing vol 13 no 11 pp 4253ndash4263 2013

[17] A Mohamed Imran M Kowsalya and D P Kothari ldquoA novelintegration technique for optimal network reconfigurationand distributed generation placement in power distributionnetworksrdquo International Journal of Electrical Power amp EnergySystems vol 63 pp 461ndash472 2014

[18] J Li and Y Tan ldquoLoser-out tournament based fireworks algo-rithm for multi-modal function optimizationrdquo IEEE Transac-tions on Evolutionary Computation 2017

[19] Z Li W Wang Y Yan and Z Li ldquoPS-ABC A hybrid algo-rithm based on particle swarm and artificial bee colony forhigh-dimensional optimization problemsrdquo Expert Systems withApplications vol 42 no 22 pp 8881ndash8895 2015

[20] Y-J Zheng X-L Xu H-F Ling and S-Y Chen ldquoA hybridfireworks optimizationmethodwith differential evolution oper-atorsrdquo Neurocomputing vol 148 pp 75ndash82 2015

[21] S Zheng J Li A Janecek andY Tan ldquoA cooperative frameworkfor fireworks algorithmrdquo IEEE Transactions on ComputationalBiology and Bioinformatics vol 14 no 1 pp 27ndash41 2017

[22] A Nickabadi M M Ebadzadeh and R Safabakhsh ldquoA novelparticle swarm optimization algorithm with adaptive inertiaweightrdquo Applied Soft Computing vol 11 no 4 pp 3658ndash36702011

[23] L Li F Liu G Long P Guo and X Bie ldquoModified particleswarm optimization for BMDS interceptor resource planningrdquoApplied Intelligence vol 44 no 3 pp 471ndash488 2016

[24] C-F Wang and K Liu ldquoA novel particle swarm optimizationalgorithm for global optimizationrdquo Computational Intelligenceand Neuroscience vol 2016 Article ID 9482073 pp 1ndash9 2016

[25] D Souravlias and K E Parsopoulos ldquoParticle swarm optimiza-tion with neighborhood-based budget allocationrdquo InternationalJournal of Machine Learning and Cybernetics vol 7 no 3 pp451ndash477 2016

[26] J-J Xue Y Wang H Li X-F Meng and J-Y Xiao ldquoAdvancedfireworks algorithm and its application research in PID param-eters tuningrdquo Mathematical Problems in Engineering vol 2016Article ID 2534632 pp 1ndash9 2016

[27] J Liu S Zheng and Y Tan ldquoThe improvement on controllingexploration and exploitation of firework algorithmrdquo in Proceed-ings of the International Conference in Swarm Intelligence pp11ndash23 Springer Berlin Heidelberg Germany 2013

[28] Y Pei S Zheng Y Tan andH Takagi ldquoEffectiveness of approx-imation strategy in surrogate-assisted fireworks algorithmrdquoInternational Journal of Machine Learning and Cybernetics vol6 no 5 pp 795ndash810 2015

[29] S Zheng A Janecek and Y Tan ldquoEnhanced fireworks algo-rithmrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation vol 62 pp 2069ndash2077 Cancun Mexico June2013

[30] S Zheng C Yu J Li and Y Tan ldquoExponentially decreaseddimension number strategy based dynamic search fireworksalgorithm for solving CEC2015 competition problemsrdquo inProceedings of the IEEE Congress on Evolutionary Computation(CEC rsquo15) pp 1ndash8 Sendai Japan 2015

[31] S Zheng A Janecek J Li and Y Tan ldquoDynamic search infireworks algorithmrdquo in Proceedings of the 2014 IEEE Congresson Evolutionary Computation (CEC rsquo14) pp 3222ndash3229 ChinaJuly 2014

[32] J Li S Zheng and Y Tan ldquoThe Effect of Information Uti-lization Introducing a Novel Guiding Spark in the FireworksAlgorithmrdquo IEEE Transactions on Evolutionary Computationvol 21 no 1 pp 153ndash166 2017

[33] J Li S Zheng and Y Tan ldquoAdaptive fireworks algorithmrdquo inProceedings of the 2014 IEEE Congress on Evolutionary Compu-tation (CEC rsquo14) pp 3214ndash3221 Springer Berlin HeidelbergChina July 2014

[34] J Li and Y Tan ldquoThe bare bones fireworks algorithm Aminimalist global optimizerrdquo Applied Soft Computing vol 62pp 454ndash462 2018

[35] F Valdez P Melin and O Castillo ldquoModular Neural Networksarchitecture optimization with a new nature inspired methodusing a fuzzy combination of Particle Swarm Optimization andGenetic Algorithmsrdquo Information Sciences vol 270 pp 143ndash1532014

[36] M Pandit V Chaudhary H M Dubey and B K PanigrahildquoMulti-period wind integrated optimal dispatch using seriesPSO-DE with time-varying Gaussian membership functionbased fuzzy selectionrdquo International Journal of Electrical Poweramp Energy Systems vol 73 pp 259ndash272 2015

[37] H Gao and M Diao ldquoCultural firework algorithm and itsapplication for digital filters designrdquo International Journal ofModelling Identification and Control vol 14 no 4 pp 324ndash3312011

[38] B Zhang M-X Zhang and Y-J Zheng ldquoA hybridbiogeography-based optimization and fireworks algorithmrdquoin Proceedings of the 2014 IEEE Congress on EvolutionaryComputation (CEC rsquo14) pp 3200ndash3206 Beijing China July2014

[39] M J Amoshahy M Shamsi and M H Sedaaghi ldquoA novelflexible inertia weight particle swarm optimization algorithmrdquoPLoS ONE vol 11 no 8 Article ID e0161558 pp 1ndash42 2016

[40] M Friedman ldquoA comparison of alternative tests of significancefor the problem of m rankingsrdquo The Annals of MathematicalStatistics vol 11 no 1 pp 86ndash92 1940

[41] O J Dunn ldquoMultiple comparisons among meansrdquo Journal ofthe American Statistical Association vol 56 pp 52ndash64 1961

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

Computer Games Technology

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difficult to converge for the optimization problems whichdo not have their optimal solutions at the origin This isbecause the two algorithms cannot keep the balance betweenthe exploration and exploitation properly Due to the optimalparticle dominating the solving process the PSO algorithmhas inferior swarm diversity in the later stage of iterations andrelatively poor exploration ability [19] while the fireworksand sparks in FWA are not well-informed by the wholeswarm [20] and the FWA framework lacks the local searchefficiency for noncore fireworks [21] In order to improvethe performance of PSO and FWA a considerable numberof modified algorithms have been proposed For exampleNickabadi et al presented AIWPSO algorithm in which anew adaptive inertia weight approach was adopted [22] Byembedding a reverse predictor and adding a repulsive forceinto the basic algorithm the RPPSO was developed [23]Wang and Liu used three strategies to ameliorate the standardalgorithm including best neighbor replacement abandonedmechanism and chaotic searching [24] Souravlias andParsopoulos introduced a PSO-based variant which coulddynamically assign different computational budget for eachparticle based on the quality of its neighbor [25] Based onself-adaption principle and bimodal Gaussian function theadvanced fireworks algorithm (AFWA) was proposed [26]Liu et al presented several methods for computing the explo-sion amplitude and number of sparks [27] Pei et al proposedto use the elite point of approximation landscape in thefireworks swarm and discussed the effectiveness of surrogate-assisted FWA [28] Zheng et al improved the new explosionoperator mutation operator selection strategy and mappingrules of FWA which led to the formation of enhancedfireworks algorithm (EFWA) [29 30] and dynamic searchin fireworks algorithm (dynFWA) [31] Zheng et al pro-posed the new cooperative FWA framework (CoFFWA) inwhich the independent selection method and crowdedness-avoiding cooperative strategy were contained [21] Li et alinvestigated the operators of FWA and introduced a novelguiding spark in FWA [32] and proposed the adaptivefireworks algorithm (AFWA) [33] and bare bones fireworksalgorithm (BBFWA) [34]

Hybrid algorithms can utilize various exploration andexploitation strategies for high-dimensional multimodaloptimization problems which have gradually become thenew research areas For example Valdez et al combined theadvantages of PSO with GA and proposed a modified hybridmethod [35] In the new PS-ABC algorithm introduced by Liet al the global optimum could be obtained by combiningthe local search phase in PSO with two global search phasesin ABC [19] Pandit et al presented the SPSO-DE in whichthe domain information of PSO and DE was shared withone another to overcome their respective weaknesses [36]Through changing the generation and selection strategy ofexplosive spark Gao and Diao proposed the CA-FWA [37]Zhang et al proposed BBO-FW algorithm which improvedthe interaction ability between fireworks [38] By combiningthe FWA with the operators of DE a novel hybrid optimiza-tion algorithm was proposed [20]

In this paper by utilizing the exploitation ability ofPSO and the exploration ability of FWA a novel hybrid

optimization algorithm called PS-FW is proposed Basedon the solving process of PSO algorithm the operators ofFWA are embedded into the update operation of the particleswarm In the iteration process in order to promote thebalance of exploitation and exploration ability of PS-FW wepresented three major techniques Firstly the abandonmentand supplement strategy is used to abandon a certainnumber of particles with poor quality and to supplementthe particle swarm with new individuals generated by FWAMeanwhile considering the information exchanges betweenthe optimal firework and its neighbor in each dimension themethod for obtaining the explosion amplitude is designed asadaptive and the mode of generating the explosion sparksis modified by combing the greedy algorithm Furthermorethe conventional Gaussian mutation operator is abandonedand the novel mutation operator based on the thought of thesocial cognition and learning is proposed The performanceof PS-FW is compared with several existing optimizationalgorithms The experimental results show that the proposedPS-FW is more efficacious in solving the global optimizationproblems

The rest of the paper is organized as follows Section 2describes the standard PSO and FWA Section 3 presentsthe PS-FW algorithm in which the algorithm details areproposed Section 4 introduces the simulation results over 22high-dimensional benchmark functions and the correspond-ing comparisons between PS-FW and other algorithms areexecuted Finally the conclusion is drawn in Section 5

2 Related Work

21 PSO Algorithm In PSO algorithm the particles scatter insearch space of the optimization problems and each particledenotes a feasible solution Each particle contains threeaspects of information the current position 119909119894 the velocityV119894 and the previous best position 119901119887119890119904119905119894 Assume that theoptimization problem is 119863-dimensional and 119872 representsthe size of the swarm population then the position andvelocity of 119894th (119894 = 1 2 119872) particle can be denoted as 119909119894 =(1199091198941 1199091198942 119909119894119863) and V119894 = (V1198941 V1198942 V119894119863) respectivelywhile the previous best position is represented as 119901119887119890119904119905119894 =(1199011198871198901199041199051198941 1199011198871198901199041199051198942 119901119887119890119904119905119894119863) Besides the best positionencountered by the entire particles so far is known as currentglobal best position 119892119887119890119904119905119894 = (1198921198871198901199041199051 1198921198871198901199041199052 119892119887119890119904119905119863)In each generation V119894 and 119909119894 are updated by the followingequations

V119894119896 (119905 + 1) = 119908 sdot V119894119896 (119905) + 1198881 sdot 1199031 sdot [119901119887119890119904119905119894119896 (119905) minus 119909119894119896 (119905)]+ 1198882 sdot 1199032 sdot [119892119887119890119904119905 (119905) minus 119909119894119896 (119905)] (1)

119909119894119896 (119905 + 1) = 119909119894119896 (119905) + V119894119896 (119905 + 1) (2)

where 1198881 and 1198882 are two learning factors that indicate theinfluence of the cognitive and social components 1199031 and 1199032are the random real numbers in interval [0 1] respectivelyand 119908 is the inertia weight which controls the convergencespeed of the algorithm

22 Fireworks Algorithm In FWA a firework or a sparkdenotes a potential solution of optimization problems while








(5)



119909119895119894 = 119909119894 + Δℎ (6)






119877 (119883119894) = sum119895isin119870

119889 (119883119894 119883119895) = sum119895isin119870

10038171003817100381710038171003817119883119894 minus 11988311989510038171003817100381710038171003817 119901 (119883119894) = 119877 (119883119894)sum119896isin119870 119877 (119883119896)

(8)






Explosion

1

2

3

4

5

Mutation

xti

xt+1i

ti

gbestt

pbesttiw middot ti



Local optima region


Explosion

1

2

3

Mutation


xti xt+1

i

ti

gbestt

pbestti

5

4






+ FWmin] (9)








119860 119894 = 119860 119894 sdot 120582 forall119860 119894 gt 120575 (11)




119860 119894 = sum119863119896=1 (10038161003816100381610038161003816119909best119896 minus 11990911989511989610038161003816100381610038161003816)119863 (13)



Δℎ119896 = 119860 sdot Gaussian (0 1) (14)

119909119895119894119896


(15)

where 119909119895119894119896






119894119896gt 119878max119896 or 119909119895119894119896 lt 119878min119896 do




































Sphere 1198911 (119909) = 119863sum119894=1

1199092119894 [minus100 100]119863 0Griewank 1198912 (119909) = 14000

119863sum119894=1

1199092119894 minus 119863prod119894=1


119894=1


119894=1

[1199092119894 minus 10 cos (2120587119909119894)] [minus512 512]119863 0


1198915(119909) = 119863sum119894=1

1199102119894 minus 10 cos(2120587119910119894) + 10119910119894 =

119909119894 10038161003816100381610038161199091198941003816100381610038161003816 lt 05round (2119909119894)2 10038161003816100381610038161199091198941003816100381610038161003816 ge 05

[minus5 10]119863 0


1199092119894)minus exp( 1119863119863sum119894=1


119894=1

119894sum119895=1


119894=1


1le119894le119863


119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 119863prod119894=1


119894=1


119894=1

([119909119894 + 05])2 [minus10 10]119863 0

Levy

11989113 (119909) = sin2 (1205871199101) + 119863minus1sum119894=1


[minus10 10]119863 0

Powell Sum 11989114 (119909) = 119863sum119894=1


119894=1


119894=1

1199092119894 + ( 119863sum119894=1

05119894119909119894)2 + ( 119863sum119894=1


119894=1


119894=1

[119896maxsum119896=0

(119886119896 cos (2120587119887119896 (119909119894 + 05))) minus 119863119896maxsum119896=0

119886119896 cos (120587119887119896)] [minus05 05]119863 0119886 = 05 119887 = 3 119896max = 20

Bent-Cigar 11989119 (119909) = 11990921 + 106 119863sum119894=1

1199092119894 [minus100 100]119863 0


Table 1 Continued




119894=1


1119863119863sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + ( 119863prod119894=1

10038161003816100381610038161199091198941003816100381610038161003816)1119863]]2 [minus10 10]119863 0

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106

Aver

age b

est fi

tnes

s

(a) 11989112 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106Av

erag

e bes

t fitn

ess

(b) 11989112 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s(c) 11989112 with119863 = 100

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

(d) 11989113 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

(e) 11989113 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106Av

erag

e bes

t fitn

ess

(f) 11989113 with119863 = 100

PSOFWAPS-FW

200 400 600 800 10000Iteration

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

(g) 11989120 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

(h) 11989120 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

100101102103104105106107

Aver

age b

est fi

tnes

s

(i) 11989120 with119863 = 100



10minus33

10minus23

10minus13

10minus3

107Av

erag

e bes

t fitn

ess

200 400 600 800 10000Iteration

PSOFWAPS-FW

(a) 11989112 with119863 = 30

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(b) 11989113 with119863 = 30

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(c) 11989120 with119863 = 30





119908min = 04 1198881 = 1198882 = 145FWA 119860 = 40119872119890 = 50 119886 = 004 119887 = 08

num119872 = 30 120576 = 1119864 minus 100PS-FW












119891 119863 PSO FWA PS-FW




1198914 30 Mean 66547119864 + 01 0 0Std 36430119864 + 01 0 0Rank 2 1 1

1198915 30 Mean 65810119864 + 01 0 0Std 40117119864 + 01 0 0Rank 2 1 1

1198916 30 Mean 0 0 0Std 0 0 0Rank 1 1 1









11989115 30 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 3 Continued

119891 119863 PSO FWA PS-FW

11989117 30 Mean 0 98737119864 + 44 0Std 0 43038119864 + 45 0Rank 1 2 1

11989118 30 Mean 15069119864 + 01 0 0Std 40495119864 + 00 0 0Rank 2 1 1


11989120 30 Mean 38005119864 + 02 42079119864 + 01 1Std 85739119864 + 01 46125119864 + 00 0Rank 3 2 1

11989121 30 Mean 45577119864 + 01 171130119864 + 01 0Std 23091119864 + 01 21499119864 + 00 0Rank 3 2 1





CD120572 = 119902120572radic119873119894 (119873119894 + 1)6119873119891 (19)


119902005 = 27711990201 = 254 (20)


CD005 = 244CD01 = 224 (21)




119891 119863 PSO FWA PS-FW




1198914 60 Mean 32219119864 + 02 0 0Std 41863119864 + 01 0 0Rank 2 1 1

1198915 60 Mean 37498119864 + 02 0 0Std 53191119864 + 01 0 0Rank 2 1 1










11989115 60 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 4 Continued

119891 119863 PSO FWA PS-FW

11989117 60 Mean 0 24945119864 + 145 0Std 0 57208119864 + 145 0Rank 1 2 1

11989118 60 Mean 39564119864 + 01 0 0Std 53138119864 + 00 0 0Rank 2 1 1


11989120 60 Mean 53645119864 + 03 14665119864 + 02 1Std 62256119864 + 03 28947119864 + 01 0Rank 3 2 1

11989121 60 Mean 19709119864 + 02 48085119864 + 01 0Std 28605119864 + 01 77355119864 + 00 0Rank 3 2 1



FIPS

CPSO

stdPs

o

PS-F

W

CLPS

O

AIW

PSO

Fran

kens

tein

Algorithms

Rank95 sig level


0

2

4

6

8

Rank

s

(a) Mean

FIPS

CPSO

stdPs

o

PS-F

W

CLPS

O

AIW

PSO

Fran

kens

tein

Algorithms

Rank95 sig level


0

2

4

6

8

Rank

s







119891 119863 PSO FWA PS-FW


1198912 100 Mean 11830119864 + 02 0 0Std 51822119864 + 01 0 0Rank 2 1 1


1198914 100 Mean 47288119864 + 02 0 0Std 10713119864 + 02 0 0Rank 2 1 1

1198915 100 Mean 51626119864 + 02 0 0Std 14819119864 + 02 0 0Rank 2 1 1







11989112 100 Mean 12670119864 + 02 42335119864 + 00 0Std 48966119864 + 01 140825853 0Rank 3 2 1



11989115 100 Mean 0 0 0Std 0 0 0Rank 1 1 1


11989117 100 Mean 0 20047119864 + 232 0Std 0 67205119864 + 232 0Rank 1 2 1


Table 5 Continued

119891 119863 PSO FWA PS-FW

11989118 100 Mean 68687119864 + 01 0 0Std 13221119864 + 01 0 0Rank 2 1 1


11989120 100 Mean 90245119864 + 03 26557119864 + 02 1Std 38036119864 + 03 47674119864 + 01 0Rank 3 2 1

11989121 100 Mean 40256119864 + 03 91975119864 + 01 0Std 16131119864 + 04 17966119864 + 01 0Rank 3 2 1










119891 119863 PSO FWA PS-FW



1198914 30 Mean 12559119864 + 02 98052119864 + 00 0Std 47596119864 + 01 16323119864 + 00 0Rank 3 2 1

1198915 30 Mean 16140119864 + 02 22289119864 + 01 0Std 37649119864 + 01 27981119864 + 00 0Rank 3 2 1




11989110 30 Mean 16846119864 + 03 22370119864 + 02 0Std 26627119864 + 02 74690119864 + 01 0Rank 3 2 1

11989112 30 Mean 11359119864 + 02 21375119864 + 01 0Std 41907119864 + 01 29107119864 + 00 0Rank 3 2 1




11989119 30 Mean 24875119864 + 09 22700119864 + 08 0Std 13163119864 + 09 27319119864 + 07 0Rank 3 2 1

11989120 30 Mean 20564119864 + 03 92562119864 + 02 1Std 79311119864 + 02 76748119864 + 01 0Rank 3 2 1













Table 9 Continued
















Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

2

4

6Ra

nks

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level




0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-7

Stra

tegy

-1

Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in











(5)



119909119895119894 = 119909119894 + Δℎ (6)






119877 (119883119894) = sum119895isin119870

119889 (119883119894 119883119895) = sum119895isin119870

10038171003817100381710038171003817119883119894 minus 11988311989510038171003817100381710038171003817 119901 (119883119894) = 119877 (119883119894)sum119896isin119870 119877 (119883119896)

(8)






Explosion

1

2

3

4

5

Mutation

xti

xt+1i

ti

gbestt

pbesttiw middot ti



Local optima region


Explosion

1

2

3

Mutation


xti xt+1

i

ti

gbestt

pbestti

5

4






+ FWmin] (9)








119860 119894 = 119860 119894 sdot 120582 forall119860 119894 gt 120575 (11)




119860 119894 = sum119863119896=1 (10038161003816100381610038161003816119909best119896 minus 11990911989511989610038161003816100381610038161003816)119863 (13)



Δℎ119896 = 119860 sdot Gaussian (0 1) (14)

119909119895119894119896


(15)

where 119909119895119894119896






119894119896gt 119878max119896 or 119909119895119894119896 lt 119878min119896 do




































Sphere 1198911 (119909) = 119863sum119894=1

1199092119894 [minus100 100]119863 0Griewank 1198912 (119909) = 14000

119863sum119894=1

1199092119894 minus 119863prod119894=1


119894=1


119894=1

[1199092119894 minus 10 cos (2120587119909119894)] [minus512 512]119863 0


1198915(119909) = 119863sum119894=1

1199102119894 minus 10 cos(2120587119910119894) + 10119910119894 =

119909119894 10038161003816100381610038161199091198941003816100381610038161003816 lt 05round (2119909119894)2 10038161003816100381610038161199091198941003816100381610038161003816 ge 05

[minus5 10]119863 0


1199092119894)minus exp( 1119863119863sum119894=1


119894=1

119894sum119895=1


119894=1


1le119894le119863


119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 119863prod119894=1


119894=1


119894=1

([119909119894 + 05])2 [minus10 10]119863 0

Levy

11989113 (119909) = sin2 (1205871199101) + 119863minus1sum119894=1


[minus10 10]119863 0

Powell Sum 11989114 (119909) = 119863sum119894=1


119894=1


119894=1

1199092119894 + ( 119863sum119894=1

05119894119909119894)2 + ( 119863sum119894=1


119894=1


119894=1

[119896maxsum119896=0

(119886119896 cos (2120587119887119896 (119909119894 + 05))) minus 119863119896maxsum119896=0

119886119896 cos (120587119887119896)] [minus05 05]119863 0119886 = 05 119887 = 3 119896max = 20

Bent-Cigar 11989119 (119909) = 11990921 + 106 119863sum119894=1

1199092119894 [minus100 100]119863 0


Table 1 Continued




119894=1


1119863119863sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + ( 119863prod119894=1

10038161003816100381610038161199091198941003816100381610038161003816)1119863]]2 [minus10 10]119863 0

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106

Aver

age b

est fi

tnes

s

(a) 11989112 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106Av

erag

e bes

t fitn

ess

(b) 11989112 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s(c) 11989112 with119863 = 100

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

(d) 11989113 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

(e) 11989113 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106Av

erag

e bes

t fitn

ess

(f) 11989113 with119863 = 100

PSOFWAPS-FW

200 400 600 800 10000Iteration

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

(g) 11989120 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

(h) 11989120 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

100101102103104105106107

Aver

age b

est fi

tnes

s

(i) 11989120 with119863 = 100



10minus33

10minus23

10minus13

10minus3

107Av

erag

e bes

t fitn

ess

200 400 600 800 10000Iteration

PSOFWAPS-FW

(a) 11989112 with119863 = 30

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(b) 11989113 with119863 = 30

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(c) 11989120 with119863 = 30





119908min = 04 1198881 = 1198882 = 145FWA 119860 = 40119872119890 = 50 119886 = 004 119887 = 08

num119872 = 30 120576 = 1119864 minus 100PS-FW












119891 119863 PSO FWA PS-FW




1198914 30 Mean 66547119864 + 01 0 0Std 36430119864 + 01 0 0Rank 2 1 1

1198915 30 Mean 65810119864 + 01 0 0Std 40117119864 + 01 0 0Rank 2 1 1

1198916 30 Mean 0 0 0Std 0 0 0Rank 1 1 1









11989115 30 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 3 Continued

119891 119863 PSO FWA PS-FW

11989117 30 Mean 0 98737119864 + 44 0Std 0 43038119864 + 45 0Rank 1 2 1

11989118 30 Mean 15069119864 + 01 0 0Std 40495119864 + 00 0 0Rank 2 1 1


11989120 30 Mean 38005119864 + 02 42079119864 + 01 1Std 85739119864 + 01 46125119864 + 00 0Rank 3 2 1

11989121 30 Mean 45577119864 + 01 171130119864 + 01 0Std 23091119864 + 01 21499119864 + 00 0Rank 3 2 1





CD120572 = 119902120572radic119873119894 (119873119894 + 1)6119873119891 (19)


119902005 = 27711990201 = 254 (20)


CD005 = 244CD01 = 224 (21)




119891 119863 PSO FWA PS-FW




1198914 60 Mean 32219119864 + 02 0 0Std 41863119864 + 01 0 0Rank 2 1 1

1198915 60 Mean 37498119864 + 02 0 0Std 53191119864 + 01 0 0Rank 2 1 1










11989115 60 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 4 Continued

119891 119863 PSO FWA PS-FW

11989117 60 Mean 0 24945119864 + 145 0Std 0 57208119864 + 145 0Rank 1 2 1

11989118 60 Mean 39564119864 + 01 0 0Std 53138119864 + 00 0 0Rank 2 1 1


11989120 60 Mean 53645119864 + 03 14665119864 + 02 1Std 62256119864 + 03 28947119864 + 01 0Rank 3 2 1

11989121 60 Mean 19709119864 + 02 48085119864 + 01 0Std 28605119864 + 01 77355119864 + 00 0Rank 3 2 1



FIPS

CPSO

stdPs

o

PS-F

W

CLPS

O

AIW

PSO

Fran

kens

tein

Algorithms

Rank95 sig level


0

2

4

6

8

Rank

s

(a) Mean

FIPS

CPSO

stdPs

o

PS-F

W

CLPS

O

AIW

PSO

Fran

kens

tein

Algorithms

Rank95 sig level


0

2

4

6

8

Rank

s







119891 119863 PSO FWA PS-FW


1198912 100 Mean 11830119864 + 02 0 0Std 51822119864 + 01 0 0Rank 2 1 1


1198914 100 Mean 47288119864 + 02 0 0Std 10713119864 + 02 0 0Rank 2 1 1

1198915 100 Mean 51626119864 + 02 0 0Std 14819119864 + 02 0 0Rank 2 1 1







11989112 100 Mean 12670119864 + 02 42335119864 + 00 0Std 48966119864 + 01 140825853 0Rank 3 2 1



11989115 100 Mean 0 0 0Std 0 0 0Rank 1 1 1


11989117 100 Mean 0 20047119864 + 232 0Std 0 67205119864 + 232 0Rank 1 2 1


Table 5 Continued

119891 119863 PSO FWA PS-FW

11989118 100 Mean 68687119864 + 01 0 0Std 13221119864 + 01 0 0Rank 2 1 1


11989120 100 Mean 90245119864 + 03 26557119864 + 02 1Std 38036119864 + 03 47674119864 + 01 0Rank 3 2 1

11989121 100 Mean 40256119864 + 03 91975119864 + 01 0Std 16131119864 + 04 17966119864 + 01 0Rank 3 2 1










119891 119863 PSO FWA PS-FW



1198914 30 Mean 12559119864 + 02 98052119864 + 00 0Std 47596119864 + 01 16323119864 + 00 0Rank 3 2 1

1198915 30 Mean 16140119864 + 02 22289119864 + 01 0Std 37649119864 + 01 27981119864 + 00 0Rank 3 2 1




11989110 30 Mean 16846119864 + 03 22370119864 + 02 0Std 26627119864 + 02 74690119864 + 01 0Rank 3 2 1

11989112 30 Mean 11359119864 + 02 21375119864 + 01 0Std 41907119864 + 01 29107119864 + 00 0Rank 3 2 1




11989119 30 Mean 24875119864 + 09 22700119864 + 08 0Std 13163119864 + 09 27319119864 + 07 0Rank 3 2 1

11989120 30 Mean 20564119864 + 03 92562119864 + 02 1Std 79311119864 + 02 76748119864 + 01 0Rank 3 2 1













Table 9 Continued
















Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

2

4

6Ra

nks

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

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Stra

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-6

Stra

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-1

Stra

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-7Strategies

Rank95 sig level



0

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Stra

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Stra

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Stra

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Stra

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Stra

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Stra

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-7

Strategies

Rank95 sig level




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Rank

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Stra

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-3

Stra

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Stra

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Stra

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Stra

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Stra

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-7

Strategies

Rank95 sig level



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Stra

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Stra

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Stra

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-7

Stra

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-1

Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in





Explosion

1

2

3

4

5

Mutation

xti

xt+1i

ti

gbestt

pbesttiw middot ti



Local optima region


Explosion

1

2

3

Mutation


xti xt+1

i

ti

gbestt

pbestti

5

4






+ FWmin] (9)








119860 119894 = 119860 119894 sdot 120582 forall119860 119894 gt 120575 (11)




119860 119894 = sum119863119896=1 (10038161003816100381610038161003816119909best119896 minus 11990911989511989610038161003816100381610038161003816)119863 (13)



Δℎ119896 = 119860 sdot Gaussian (0 1) (14)

119909119895119894119896


(15)

where 119909119895119894119896






119894119896gt 119878max119896 or 119909119895119894119896 lt 119878min119896 do




































Sphere 1198911 (119909) = 119863sum119894=1

1199092119894 [minus100 100]119863 0Griewank 1198912 (119909) = 14000

119863sum119894=1

1199092119894 minus 119863prod119894=1


119894=1


119894=1

[1199092119894 minus 10 cos (2120587119909119894)] [minus512 512]119863 0


1198915(119909) = 119863sum119894=1

1199102119894 minus 10 cos(2120587119910119894) + 10119910119894 =

119909119894 10038161003816100381610038161199091198941003816100381610038161003816 lt 05round (2119909119894)2 10038161003816100381610038161199091198941003816100381610038161003816 ge 05

[minus5 10]119863 0


1199092119894)minus exp( 1119863119863sum119894=1


119894=1

119894sum119895=1


119894=1


1le119894le119863


119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 119863prod119894=1


119894=1


119894=1

([119909119894 + 05])2 [minus10 10]119863 0

Levy

11989113 (119909) = sin2 (1205871199101) + 119863minus1sum119894=1


[minus10 10]119863 0

Powell Sum 11989114 (119909) = 119863sum119894=1


119894=1


119894=1

1199092119894 + ( 119863sum119894=1

05119894119909119894)2 + ( 119863sum119894=1


119894=1


119894=1

[119896maxsum119896=0

(119886119896 cos (2120587119887119896 (119909119894 + 05))) minus 119863119896maxsum119896=0

119886119896 cos (120587119887119896)] [minus05 05]119863 0119886 = 05 119887 = 3 119896max = 20

Bent-Cigar 11989119 (119909) = 11990921 + 106 119863sum119894=1

1199092119894 [minus100 100]119863 0


Table 1 Continued




119894=1


1119863119863sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + ( 119863prod119894=1

10038161003816100381610038161199091198941003816100381610038161003816)1119863]]2 [minus10 10]119863 0

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106

Aver

age b

est fi

tnes

s

(a) 11989112 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106Av

erag

e bes

t fitn

ess

(b) 11989112 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s(c) 11989112 with119863 = 100

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

(d) 11989113 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

(e) 11989113 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106Av

erag

e bes

t fitn

ess

(f) 11989113 with119863 = 100

PSOFWAPS-FW

200 400 600 800 10000Iteration

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

(g) 11989120 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

(h) 11989120 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

100101102103104105106107

Aver

age b

est fi

tnes

s

(i) 11989120 with119863 = 100



10minus33

10minus23

10minus13

10minus3

107Av

erag

e bes

t fitn

ess

200 400 600 800 10000Iteration

PSOFWAPS-FW

(a) 11989112 with119863 = 30

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(b) 11989113 with119863 = 30

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(c) 11989120 with119863 = 30





119908min = 04 1198881 = 1198882 = 145FWA 119860 = 40119872119890 = 50 119886 = 004 119887 = 08

num119872 = 30 120576 = 1119864 minus 100PS-FW












119891 119863 PSO FWA PS-FW




1198914 30 Mean 66547119864 + 01 0 0Std 36430119864 + 01 0 0Rank 2 1 1

1198915 30 Mean 65810119864 + 01 0 0Std 40117119864 + 01 0 0Rank 2 1 1

1198916 30 Mean 0 0 0Std 0 0 0Rank 1 1 1









11989115 30 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 3 Continued

119891 119863 PSO FWA PS-FW

11989117 30 Mean 0 98737119864 + 44 0Std 0 43038119864 + 45 0Rank 1 2 1

11989118 30 Mean 15069119864 + 01 0 0Std 40495119864 + 00 0 0Rank 2 1 1


11989120 30 Mean 38005119864 + 02 42079119864 + 01 1Std 85739119864 + 01 46125119864 + 00 0Rank 3 2 1

11989121 30 Mean 45577119864 + 01 171130119864 + 01 0Std 23091119864 + 01 21499119864 + 00 0Rank 3 2 1





CD120572 = 119902120572radic119873119894 (119873119894 + 1)6119873119891 (19)


119902005 = 27711990201 = 254 (20)


CD005 = 244CD01 = 224 (21)




119891 119863 PSO FWA PS-FW




1198914 60 Mean 32219119864 + 02 0 0Std 41863119864 + 01 0 0Rank 2 1 1

1198915 60 Mean 37498119864 + 02 0 0Std 53191119864 + 01 0 0Rank 2 1 1










11989115 60 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 4 Continued

119891 119863 PSO FWA PS-FW

11989117 60 Mean 0 24945119864 + 145 0Std 0 57208119864 + 145 0Rank 1 2 1

11989118 60 Mean 39564119864 + 01 0 0Std 53138119864 + 00 0 0Rank 2 1 1


11989120 60 Mean 53645119864 + 03 14665119864 + 02 1Std 62256119864 + 03 28947119864 + 01 0Rank 3 2 1

11989121 60 Mean 19709119864 + 02 48085119864 + 01 0Std 28605119864 + 01 77355119864 + 00 0Rank 3 2 1



FIPS

CPSO

stdPs

o

PS-F

W

CLPS

O

AIW

PSO

Fran

kens

tein

Algorithms

Rank95 sig level


0

2

4

6

8

Rank

s

(a) Mean

FIPS

CPSO

stdPs

o

PS-F

W

CLPS

O

AIW

PSO

Fran

kens

tein

Algorithms

Rank95 sig level


0

2

4

6

8

Rank

s







119891 119863 PSO FWA PS-FW


1198912 100 Mean 11830119864 + 02 0 0Std 51822119864 + 01 0 0Rank 2 1 1


1198914 100 Mean 47288119864 + 02 0 0Std 10713119864 + 02 0 0Rank 2 1 1

1198915 100 Mean 51626119864 + 02 0 0Std 14819119864 + 02 0 0Rank 2 1 1







11989112 100 Mean 12670119864 + 02 42335119864 + 00 0Std 48966119864 + 01 140825853 0Rank 3 2 1



11989115 100 Mean 0 0 0Std 0 0 0Rank 1 1 1


11989117 100 Mean 0 20047119864 + 232 0Std 0 67205119864 + 232 0Rank 1 2 1


Table 5 Continued

119891 119863 PSO FWA PS-FW

11989118 100 Mean 68687119864 + 01 0 0Std 13221119864 + 01 0 0Rank 2 1 1


11989120 100 Mean 90245119864 + 03 26557119864 + 02 1Std 38036119864 + 03 47674119864 + 01 0Rank 3 2 1

11989121 100 Mean 40256119864 + 03 91975119864 + 01 0Std 16131119864 + 04 17966119864 + 01 0Rank 3 2 1










119891 119863 PSO FWA PS-FW



1198914 30 Mean 12559119864 + 02 98052119864 + 00 0Std 47596119864 + 01 16323119864 + 00 0Rank 3 2 1

1198915 30 Mean 16140119864 + 02 22289119864 + 01 0Std 37649119864 + 01 27981119864 + 00 0Rank 3 2 1




11989110 30 Mean 16846119864 + 03 22370119864 + 02 0Std 26627119864 + 02 74690119864 + 01 0Rank 3 2 1

11989112 30 Mean 11359119864 + 02 21375119864 + 01 0Std 41907119864 + 01 29107119864 + 00 0Rank 3 2 1




11989119 30 Mean 24875119864 + 09 22700119864 + 08 0Std 13163119864 + 09 27319119864 + 07 0Rank 3 2 1

11989120 30 Mean 20564119864 + 03 92562119864 + 02 1Std 79311119864 + 02 76748119864 + 01 0Rank 3 2 1













Table 9 Continued
















Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

2

4

6Ra

nks

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

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-6

Stra

tegy

-1

Stra

tegy

-7Strategies

Rank95 sig level



0

2

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s

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-3

Stra

tegy

-4

Stra

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-6

Stra

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-1

Stra

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-7

Strategies

Rank95 sig level




0

2

4

6

Rank

s

Stra

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-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-7

Stra

tegy

-1

Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in









119860 119894 = 119860 119894 sdot 120582 forall119860 119894 gt 120575 (11)




119860 119894 = sum119863119896=1 (10038161003816100381610038161003816119909best119896 minus 11990911989511989610038161003816100381610038161003816)119863 (13)



Δℎ119896 = 119860 sdot Gaussian (0 1) (14)

119909119895119894119896


(15)

where 119909119895119894119896






119894119896gt 119878max119896 or 119909119895119894119896 lt 119878min119896 do




































Sphere 1198911 (119909) = 119863sum119894=1

1199092119894 [minus100 100]119863 0Griewank 1198912 (119909) = 14000

119863sum119894=1

1199092119894 minus 119863prod119894=1


119894=1


119894=1

[1199092119894 minus 10 cos (2120587119909119894)] [minus512 512]119863 0


1198915(119909) = 119863sum119894=1

1199102119894 minus 10 cos(2120587119910119894) + 10119910119894 =

119909119894 10038161003816100381610038161199091198941003816100381610038161003816 lt 05round (2119909119894)2 10038161003816100381610038161199091198941003816100381610038161003816 ge 05

[minus5 10]119863 0


1199092119894)minus exp( 1119863119863sum119894=1


119894=1

119894sum119895=1


119894=1


1le119894le119863


119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 119863prod119894=1


119894=1


119894=1

([119909119894 + 05])2 [minus10 10]119863 0

Levy

11989113 (119909) = sin2 (1205871199101) + 119863minus1sum119894=1


[minus10 10]119863 0

Powell Sum 11989114 (119909) = 119863sum119894=1


119894=1


119894=1

1199092119894 + ( 119863sum119894=1

05119894119909119894)2 + ( 119863sum119894=1


119894=1


119894=1

[119896maxsum119896=0

(119886119896 cos (2120587119887119896 (119909119894 + 05))) minus 119863119896maxsum119896=0

119886119896 cos (120587119887119896)] [minus05 05]119863 0119886 = 05 119887 = 3 119896max = 20

Bent-Cigar 11989119 (119909) = 11990921 + 106 119863sum119894=1

1199092119894 [minus100 100]119863 0


Table 1 Continued




119894=1


1119863119863sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + ( 119863prod119894=1

10038161003816100381610038161199091198941003816100381610038161003816)1119863]]2 [minus10 10]119863 0

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106

Aver

age b

est fi

tnes

s

(a) 11989112 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106Av

erag

e bes

t fitn

ess

(b) 11989112 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s(c) 11989112 with119863 = 100

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

(d) 11989113 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

(e) 11989113 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106Av

erag

e bes

t fitn

ess

(f) 11989113 with119863 = 100

PSOFWAPS-FW

200 400 600 800 10000Iteration

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

(g) 11989120 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

(h) 11989120 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

100101102103104105106107

Aver

age b

est fi

tnes

s

(i) 11989120 with119863 = 100



10minus33

10minus23

10minus13

10minus3

107Av

erag

e bes

t fitn

ess

200 400 600 800 10000Iteration

PSOFWAPS-FW

(a) 11989112 with119863 = 30

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(b) 11989113 with119863 = 30

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(c) 11989120 with119863 = 30





119908min = 04 1198881 = 1198882 = 145FWA 119860 = 40119872119890 = 50 119886 = 004 119887 = 08

num119872 = 30 120576 = 1119864 minus 100PS-FW












119891 119863 PSO FWA PS-FW




1198914 30 Mean 66547119864 + 01 0 0Std 36430119864 + 01 0 0Rank 2 1 1

1198915 30 Mean 65810119864 + 01 0 0Std 40117119864 + 01 0 0Rank 2 1 1

1198916 30 Mean 0 0 0Std 0 0 0Rank 1 1 1









11989115 30 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 3 Continued

119891 119863 PSO FWA PS-FW

11989117 30 Mean 0 98737119864 + 44 0Std 0 43038119864 + 45 0Rank 1 2 1

11989118 30 Mean 15069119864 + 01 0 0Std 40495119864 + 00 0 0Rank 2 1 1


11989120 30 Mean 38005119864 + 02 42079119864 + 01 1Std 85739119864 + 01 46125119864 + 00 0Rank 3 2 1

11989121 30 Mean 45577119864 + 01 171130119864 + 01 0Std 23091119864 + 01 21499119864 + 00 0Rank 3 2 1





CD120572 = 119902120572radic119873119894 (119873119894 + 1)6119873119891 (19)


119902005 = 27711990201 = 254 (20)


CD005 = 244CD01 = 224 (21)




119891 119863 PSO FWA PS-FW




1198914 60 Mean 32219119864 + 02 0 0Std 41863119864 + 01 0 0Rank 2 1 1

1198915 60 Mean 37498119864 + 02 0 0Std 53191119864 + 01 0 0Rank 2 1 1










11989115 60 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 4 Continued

119891 119863 PSO FWA PS-FW

11989117 60 Mean 0 24945119864 + 145 0Std 0 57208119864 + 145 0Rank 1 2 1

11989118 60 Mean 39564119864 + 01 0 0Std 53138119864 + 00 0 0Rank 2 1 1


11989120 60 Mean 53645119864 + 03 14665119864 + 02 1Std 62256119864 + 03 28947119864 + 01 0Rank 3 2 1

11989121 60 Mean 19709119864 + 02 48085119864 + 01 0Std 28605119864 + 01 77355119864 + 00 0Rank 3 2 1



FIPS

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Rank95 sig level


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Rank95 sig level


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119891 119863 PSO FWA PS-FW


1198912 100 Mean 11830119864 + 02 0 0Std 51822119864 + 01 0 0Rank 2 1 1


1198914 100 Mean 47288119864 + 02 0 0Std 10713119864 + 02 0 0Rank 2 1 1

1198915 100 Mean 51626119864 + 02 0 0Std 14819119864 + 02 0 0Rank 2 1 1







11989112 100 Mean 12670119864 + 02 42335119864 + 00 0Std 48966119864 + 01 140825853 0Rank 3 2 1



11989115 100 Mean 0 0 0Std 0 0 0Rank 1 1 1


11989117 100 Mean 0 20047119864 + 232 0Std 0 67205119864 + 232 0Rank 1 2 1


Table 5 Continued

119891 119863 PSO FWA PS-FW

11989118 100 Mean 68687119864 + 01 0 0Std 13221119864 + 01 0 0Rank 2 1 1


11989120 100 Mean 90245119864 + 03 26557119864 + 02 1Std 38036119864 + 03 47674119864 + 01 0Rank 3 2 1

11989121 100 Mean 40256119864 + 03 91975119864 + 01 0Std 16131119864 + 04 17966119864 + 01 0Rank 3 2 1










119891 119863 PSO FWA PS-FW



1198914 30 Mean 12559119864 + 02 98052119864 + 00 0Std 47596119864 + 01 16323119864 + 00 0Rank 3 2 1

1198915 30 Mean 16140119864 + 02 22289119864 + 01 0Std 37649119864 + 01 27981119864 + 00 0Rank 3 2 1




11989110 30 Mean 16846119864 + 03 22370119864 + 02 0Std 26627119864 + 02 74690119864 + 01 0Rank 3 2 1

11989112 30 Mean 11359119864 + 02 21375119864 + 01 0Std 41907119864 + 01 29107119864 + 00 0Rank 3 2 1




11989119 30 Mean 24875119864 + 09 22700119864 + 08 0Std 13163119864 + 09 27319119864 + 07 0Rank 3 2 1

11989120 30 Mean 20564119864 + 03 92562119864 + 02 1Std 79311119864 + 02 76748119864 + 01 0Rank 3 2 1













Table 9 Continued
















Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

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4

6Ra

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Stra

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Stra

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5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in






119894119896gt 119878max119896 or 119909119895119894119896 lt 119878min119896 do




































Sphere 1198911 (119909) = 119863sum119894=1

1199092119894 [minus100 100]119863 0Griewank 1198912 (119909) = 14000

119863sum119894=1

1199092119894 minus 119863prod119894=1


119894=1


119894=1

[1199092119894 minus 10 cos (2120587119909119894)] [minus512 512]119863 0


1198915(119909) = 119863sum119894=1

1199102119894 minus 10 cos(2120587119910119894) + 10119910119894 =

119909119894 10038161003816100381610038161199091198941003816100381610038161003816 lt 05round (2119909119894)2 10038161003816100381610038161199091198941003816100381610038161003816 ge 05

[minus5 10]119863 0


1199092119894)minus exp( 1119863119863sum119894=1


119894=1

119894sum119895=1


119894=1


1le119894le119863


119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 119863prod119894=1


119894=1


119894=1

([119909119894 + 05])2 [minus10 10]119863 0

Levy

11989113 (119909) = sin2 (1205871199101) + 119863minus1sum119894=1


[minus10 10]119863 0

Powell Sum 11989114 (119909) = 119863sum119894=1


119894=1


119894=1

1199092119894 + ( 119863sum119894=1

05119894119909119894)2 + ( 119863sum119894=1


119894=1


119894=1

[119896maxsum119896=0

(119886119896 cos (2120587119887119896 (119909119894 + 05))) minus 119863119896maxsum119896=0

119886119896 cos (120587119887119896)] [minus05 05]119863 0119886 = 05 119887 = 3 119896max = 20

Bent-Cigar 11989119 (119909) = 11990921 + 106 119863sum119894=1

1199092119894 [minus100 100]119863 0


Table 1 Continued




119894=1


1119863119863sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + ( 119863prod119894=1

10038161003816100381610038161199091198941003816100381610038161003816)1119863]]2 [minus10 10]119863 0

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106

Aver

age b

est fi

tnes

s

(a) 11989112 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106Av

erag

e bes

t fitn

ess

(b) 11989112 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s(c) 11989112 with119863 = 100

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

(d) 11989113 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

(e) 11989113 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106Av

erag

e bes

t fitn

ess

(f) 11989113 with119863 = 100

PSOFWAPS-FW

200 400 600 800 10000Iteration

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

(g) 11989120 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

(h) 11989120 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

100101102103104105106107

Aver

age b

est fi

tnes

s

(i) 11989120 with119863 = 100



10minus33

10minus23

10minus13

10minus3

107Av

erag

e bes

t fitn

ess

200 400 600 800 10000Iteration

PSOFWAPS-FW

(a) 11989112 with119863 = 30

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(b) 11989113 with119863 = 30

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(c) 11989120 with119863 = 30





119908min = 04 1198881 = 1198882 = 145FWA 119860 = 40119872119890 = 50 119886 = 004 119887 = 08

num119872 = 30 120576 = 1119864 minus 100PS-FW












119891 119863 PSO FWA PS-FW




1198914 30 Mean 66547119864 + 01 0 0Std 36430119864 + 01 0 0Rank 2 1 1

1198915 30 Mean 65810119864 + 01 0 0Std 40117119864 + 01 0 0Rank 2 1 1

1198916 30 Mean 0 0 0Std 0 0 0Rank 1 1 1









11989115 30 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 3 Continued

119891 119863 PSO FWA PS-FW

11989117 30 Mean 0 98737119864 + 44 0Std 0 43038119864 + 45 0Rank 1 2 1

11989118 30 Mean 15069119864 + 01 0 0Std 40495119864 + 00 0 0Rank 2 1 1


11989120 30 Mean 38005119864 + 02 42079119864 + 01 1Std 85739119864 + 01 46125119864 + 00 0Rank 3 2 1

11989121 30 Mean 45577119864 + 01 171130119864 + 01 0Std 23091119864 + 01 21499119864 + 00 0Rank 3 2 1





CD120572 = 119902120572radic119873119894 (119873119894 + 1)6119873119891 (19)


119902005 = 27711990201 = 254 (20)


CD005 = 244CD01 = 224 (21)




119891 119863 PSO FWA PS-FW




1198914 60 Mean 32219119864 + 02 0 0Std 41863119864 + 01 0 0Rank 2 1 1

1198915 60 Mean 37498119864 + 02 0 0Std 53191119864 + 01 0 0Rank 2 1 1










11989115 60 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 4 Continued

119891 119863 PSO FWA PS-FW

11989117 60 Mean 0 24945119864 + 145 0Std 0 57208119864 + 145 0Rank 1 2 1

11989118 60 Mean 39564119864 + 01 0 0Std 53138119864 + 00 0 0Rank 2 1 1


11989120 60 Mean 53645119864 + 03 14665119864 + 02 1Std 62256119864 + 03 28947119864 + 01 0Rank 3 2 1

11989121 60 Mean 19709119864 + 02 48085119864 + 01 0Std 28605119864 + 01 77355119864 + 00 0Rank 3 2 1



FIPS

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Rank95 sig level


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Rank95 sig level


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119891 119863 PSO FWA PS-FW


1198912 100 Mean 11830119864 + 02 0 0Std 51822119864 + 01 0 0Rank 2 1 1


1198914 100 Mean 47288119864 + 02 0 0Std 10713119864 + 02 0 0Rank 2 1 1

1198915 100 Mean 51626119864 + 02 0 0Std 14819119864 + 02 0 0Rank 2 1 1







11989112 100 Mean 12670119864 + 02 42335119864 + 00 0Std 48966119864 + 01 140825853 0Rank 3 2 1



11989115 100 Mean 0 0 0Std 0 0 0Rank 1 1 1


11989117 100 Mean 0 20047119864 + 232 0Std 0 67205119864 + 232 0Rank 1 2 1


Table 5 Continued

119891 119863 PSO FWA PS-FW

11989118 100 Mean 68687119864 + 01 0 0Std 13221119864 + 01 0 0Rank 2 1 1


11989120 100 Mean 90245119864 + 03 26557119864 + 02 1Std 38036119864 + 03 47674119864 + 01 0Rank 3 2 1

11989121 100 Mean 40256119864 + 03 91975119864 + 01 0Std 16131119864 + 04 17966119864 + 01 0Rank 3 2 1










119891 119863 PSO FWA PS-FW



1198914 30 Mean 12559119864 + 02 98052119864 + 00 0Std 47596119864 + 01 16323119864 + 00 0Rank 3 2 1

1198915 30 Mean 16140119864 + 02 22289119864 + 01 0Std 37649119864 + 01 27981119864 + 00 0Rank 3 2 1




11989110 30 Mean 16846119864 + 03 22370119864 + 02 0Std 26627119864 + 02 74690119864 + 01 0Rank 3 2 1

11989112 30 Mean 11359119864 + 02 21375119864 + 01 0Std 41907119864 + 01 29107119864 + 00 0Rank 3 2 1




11989119 30 Mean 24875119864 + 09 22700119864 + 08 0Std 13163119864 + 09 27319119864 + 07 0Rank 3 2 1

11989120 30 Mean 20564119864 + 03 92562119864 + 02 1Std 79311119864 + 02 76748119864 + 01 0Rank 3 2 1













Table 9 Continued
















Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

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-7Strategies

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Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in































Sphere 1198911 (119909) = 119863sum119894=1

1199092119894 [minus100 100]119863 0Griewank 1198912 (119909) = 14000

119863sum119894=1

1199092119894 minus 119863prod119894=1


119894=1


119894=1

[1199092119894 minus 10 cos (2120587119909119894)] [minus512 512]119863 0


1198915(119909) = 119863sum119894=1

1199102119894 minus 10 cos(2120587119910119894) + 10119910119894 =

119909119894 10038161003816100381610038161199091198941003816100381610038161003816 lt 05round (2119909119894)2 10038161003816100381610038161199091198941003816100381610038161003816 ge 05

[minus5 10]119863 0


1199092119894)minus exp( 1119863119863sum119894=1


119894=1

119894sum119895=1


119894=1


1le119894le119863


119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 119863prod119894=1


119894=1


119894=1

([119909119894 + 05])2 [minus10 10]119863 0

Levy

11989113 (119909) = sin2 (1205871199101) + 119863minus1sum119894=1


[minus10 10]119863 0

Powell Sum 11989114 (119909) = 119863sum119894=1


119894=1


119894=1

1199092119894 + ( 119863sum119894=1

05119894119909119894)2 + ( 119863sum119894=1


119894=1


119894=1

[119896maxsum119896=0

(119886119896 cos (2120587119887119896 (119909119894 + 05))) minus 119863119896maxsum119896=0

119886119896 cos (120587119887119896)] [minus05 05]119863 0119886 = 05 119887 = 3 119896max = 20

Bent-Cigar 11989119 (119909) = 11990921 + 106 119863sum119894=1

1199092119894 [minus100 100]119863 0


Table 1 Continued




119894=1


1119863119863sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + ( 119863prod119894=1

10038161003816100381610038161199091198941003816100381610038161003816)1119863]]2 [minus10 10]119863 0

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106

Aver

age b

est fi

tnes

s

(a) 11989112 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106Av

erag

e bes

t fitn

ess

(b) 11989112 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s(c) 11989112 with119863 = 100

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

(d) 11989113 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

(e) 11989113 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106Av

erag

e bes

t fitn

ess

(f) 11989113 with119863 = 100

PSOFWAPS-FW

200 400 600 800 10000Iteration

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

(g) 11989120 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

(h) 11989120 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

100101102103104105106107

Aver

age b

est fi

tnes

s

(i) 11989120 with119863 = 100



10minus33

10minus23

10minus13

10minus3

107Av

erag

e bes

t fitn

ess

200 400 600 800 10000Iteration

PSOFWAPS-FW

(a) 11989112 with119863 = 30

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(b) 11989113 with119863 = 30

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(c) 11989120 with119863 = 30





119908min = 04 1198881 = 1198882 = 145FWA 119860 = 40119872119890 = 50 119886 = 004 119887 = 08

num119872 = 30 120576 = 1119864 minus 100PS-FW












119891 119863 PSO FWA PS-FW




1198914 30 Mean 66547119864 + 01 0 0Std 36430119864 + 01 0 0Rank 2 1 1

1198915 30 Mean 65810119864 + 01 0 0Std 40117119864 + 01 0 0Rank 2 1 1

1198916 30 Mean 0 0 0Std 0 0 0Rank 1 1 1









11989115 30 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 3 Continued

119891 119863 PSO FWA PS-FW

11989117 30 Mean 0 98737119864 + 44 0Std 0 43038119864 + 45 0Rank 1 2 1

11989118 30 Mean 15069119864 + 01 0 0Std 40495119864 + 00 0 0Rank 2 1 1


11989120 30 Mean 38005119864 + 02 42079119864 + 01 1Std 85739119864 + 01 46125119864 + 00 0Rank 3 2 1

11989121 30 Mean 45577119864 + 01 171130119864 + 01 0Std 23091119864 + 01 21499119864 + 00 0Rank 3 2 1





CD120572 = 119902120572radic119873119894 (119873119894 + 1)6119873119891 (19)


119902005 = 27711990201 = 254 (20)


CD005 = 244CD01 = 224 (21)




119891 119863 PSO FWA PS-FW




1198914 60 Mean 32219119864 + 02 0 0Std 41863119864 + 01 0 0Rank 2 1 1

1198915 60 Mean 37498119864 + 02 0 0Std 53191119864 + 01 0 0Rank 2 1 1










11989115 60 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 4 Continued

119891 119863 PSO FWA PS-FW

11989117 60 Mean 0 24945119864 + 145 0Std 0 57208119864 + 145 0Rank 1 2 1

11989118 60 Mean 39564119864 + 01 0 0Std 53138119864 + 00 0 0Rank 2 1 1


11989120 60 Mean 53645119864 + 03 14665119864 + 02 1Std 62256119864 + 03 28947119864 + 01 0Rank 3 2 1

11989121 60 Mean 19709119864 + 02 48085119864 + 01 0Std 28605119864 + 01 77355119864 + 00 0Rank 3 2 1



FIPS

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Rank95 sig level


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Rank95 sig level


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s







119891 119863 PSO FWA PS-FW


1198912 100 Mean 11830119864 + 02 0 0Std 51822119864 + 01 0 0Rank 2 1 1


1198914 100 Mean 47288119864 + 02 0 0Std 10713119864 + 02 0 0Rank 2 1 1

1198915 100 Mean 51626119864 + 02 0 0Std 14819119864 + 02 0 0Rank 2 1 1







11989112 100 Mean 12670119864 + 02 42335119864 + 00 0Std 48966119864 + 01 140825853 0Rank 3 2 1



11989115 100 Mean 0 0 0Std 0 0 0Rank 1 1 1


11989117 100 Mean 0 20047119864 + 232 0Std 0 67205119864 + 232 0Rank 1 2 1


Table 5 Continued

119891 119863 PSO FWA PS-FW

11989118 100 Mean 68687119864 + 01 0 0Std 13221119864 + 01 0 0Rank 2 1 1


11989120 100 Mean 90245119864 + 03 26557119864 + 02 1Std 38036119864 + 03 47674119864 + 01 0Rank 3 2 1

11989121 100 Mean 40256119864 + 03 91975119864 + 01 0Std 16131119864 + 04 17966119864 + 01 0Rank 3 2 1










119891 119863 PSO FWA PS-FW



1198914 30 Mean 12559119864 + 02 98052119864 + 00 0Std 47596119864 + 01 16323119864 + 00 0Rank 3 2 1

1198915 30 Mean 16140119864 + 02 22289119864 + 01 0Std 37649119864 + 01 27981119864 + 00 0Rank 3 2 1




11989110 30 Mean 16846119864 + 03 22370119864 + 02 0Std 26627119864 + 02 74690119864 + 01 0Rank 3 2 1

11989112 30 Mean 11359119864 + 02 21375119864 + 01 0Std 41907119864 + 01 29107119864 + 00 0Rank 3 2 1




11989119 30 Mean 24875119864 + 09 22700119864 + 08 0Std 13163119864 + 09 27319119864 + 07 0Rank 3 2 1

11989120 30 Mean 20564119864 + 03 92562119864 + 02 1Std 79311119864 + 02 76748119864 + 01 0Rank 3 2 1













Table 9 Continued
















Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

2

4

6Ra

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Stra

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Stra

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Stra

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Stra

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Stra

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Stra

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-7Strategies

Rank95 sig level



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Rank95 sig level




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Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in















Sphere 1198911 (119909) = 119863sum119894=1

1199092119894 [minus100 100]119863 0Griewank 1198912 (119909) = 14000

119863sum119894=1

1199092119894 minus 119863prod119894=1


119894=1


119894=1

[1199092119894 minus 10 cos (2120587119909119894)] [minus512 512]119863 0


1198915(119909) = 119863sum119894=1

1199102119894 minus 10 cos(2120587119910119894) + 10119910119894 =

119909119894 10038161003816100381610038161199091198941003816100381610038161003816 lt 05round (2119909119894)2 10038161003816100381610038161199091198941003816100381610038161003816 ge 05

[minus5 10]119863 0


1199092119894)minus exp( 1119863119863sum119894=1


119894=1

119894sum119895=1


119894=1


1le119894le119863


119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 119863prod119894=1


119894=1


119894=1

([119909119894 + 05])2 [minus10 10]119863 0

Levy

11989113 (119909) = sin2 (1205871199101) + 119863minus1sum119894=1


[minus10 10]119863 0

Powell Sum 11989114 (119909) = 119863sum119894=1


119894=1


119894=1

1199092119894 + ( 119863sum119894=1

05119894119909119894)2 + ( 119863sum119894=1


119894=1


119894=1

[119896maxsum119896=0

(119886119896 cos (2120587119887119896 (119909119894 + 05))) minus 119863119896maxsum119896=0

119886119896 cos (120587119887119896)] [minus05 05]119863 0119886 = 05 119887 = 3 119896max = 20

Bent-Cigar 11989119 (119909) = 11990921 + 106 119863sum119894=1

1199092119894 [minus100 100]119863 0


Table 1 Continued




119894=1


1119863119863sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + ( 119863prod119894=1

10038161003816100381610038161199091198941003816100381610038161003816)1119863]]2 [minus10 10]119863 0

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106

Aver

age b

est fi

tnes

s

(a) 11989112 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106Av

erag

e bes

t fitn

ess

(b) 11989112 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s(c) 11989112 with119863 = 100

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

(d) 11989113 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

(e) 11989113 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106Av

erag

e bes

t fitn

ess

(f) 11989113 with119863 = 100

PSOFWAPS-FW

200 400 600 800 10000Iteration

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

(g) 11989120 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

(h) 11989120 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

100101102103104105106107

Aver

age b

est fi

tnes

s

(i) 11989120 with119863 = 100



10minus33

10minus23

10minus13

10minus3

107Av

erag

e bes

t fitn

ess

200 400 600 800 10000Iteration

PSOFWAPS-FW

(a) 11989112 with119863 = 30

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(b) 11989113 with119863 = 30

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(c) 11989120 with119863 = 30





119908min = 04 1198881 = 1198882 = 145FWA 119860 = 40119872119890 = 50 119886 = 004 119887 = 08

num119872 = 30 120576 = 1119864 minus 100PS-FW












119891 119863 PSO FWA PS-FW




1198914 30 Mean 66547119864 + 01 0 0Std 36430119864 + 01 0 0Rank 2 1 1

1198915 30 Mean 65810119864 + 01 0 0Std 40117119864 + 01 0 0Rank 2 1 1

1198916 30 Mean 0 0 0Std 0 0 0Rank 1 1 1









11989115 30 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 3 Continued

119891 119863 PSO FWA PS-FW

11989117 30 Mean 0 98737119864 + 44 0Std 0 43038119864 + 45 0Rank 1 2 1

11989118 30 Mean 15069119864 + 01 0 0Std 40495119864 + 00 0 0Rank 2 1 1


11989120 30 Mean 38005119864 + 02 42079119864 + 01 1Std 85739119864 + 01 46125119864 + 00 0Rank 3 2 1

11989121 30 Mean 45577119864 + 01 171130119864 + 01 0Std 23091119864 + 01 21499119864 + 00 0Rank 3 2 1





CD120572 = 119902120572radic119873119894 (119873119894 + 1)6119873119891 (19)


119902005 = 27711990201 = 254 (20)


CD005 = 244CD01 = 224 (21)




119891 119863 PSO FWA PS-FW




1198914 60 Mean 32219119864 + 02 0 0Std 41863119864 + 01 0 0Rank 2 1 1

1198915 60 Mean 37498119864 + 02 0 0Std 53191119864 + 01 0 0Rank 2 1 1










11989115 60 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 4 Continued

119891 119863 PSO FWA PS-FW

11989117 60 Mean 0 24945119864 + 145 0Std 0 57208119864 + 145 0Rank 1 2 1

11989118 60 Mean 39564119864 + 01 0 0Std 53138119864 + 00 0 0Rank 2 1 1


11989120 60 Mean 53645119864 + 03 14665119864 + 02 1Std 62256119864 + 03 28947119864 + 01 0Rank 3 2 1

11989121 60 Mean 19709119864 + 02 48085119864 + 01 0Std 28605119864 + 01 77355119864 + 00 0Rank 3 2 1



FIPS

CPSO

stdPs

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PS-F

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CLPS

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Algorithms

Rank95 sig level


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Rank

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(a) Mean

FIPS

CPSO

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Algorithms

Rank95 sig level


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119891 119863 PSO FWA PS-FW


1198912 100 Mean 11830119864 + 02 0 0Std 51822119864 + 01 0 0Rank 2 1 1


1198914 100 Mean 47288119864 + 02 0 0Std 10713119864 + 02 0 0Rank 2 1 1

1198915 100 Mean 51626119864 + 02 0 0Std 14819119864 + 02 0 0Rank 2 1 1







11989112 100 Mean 12670119864 + 02 42335119864 + 00 0Std 48966119864 + 01 140825853 0Rank 3 2 1



11989115 100 Mean 0 0 0Std 0 0 0Rank 1 1 1


11989117 100 Mean 0 20047119864 + 232 0Std 0 67205119864 + 232 0Rank 1 2 1


Table 5 Continued

119891 119863 PSO FWA PS-FW

11989118 100 Mean 68687119864 + 01 0 0Std 13221119864 + 01 0 0Rank 2 1 1


11989120 100 Mean 90245119864 + 03 26557119864 + 02 1Std 38036119864 + 03 47674119864 + 01 0Rank 3 2 1

11989121 100 Mean 40256119864 + 03 91975119864 + 01 0Std 16131119864 + 04 17966119864 + 01 0Rank 3 2 1










119891 119863 PSO FWA PS-FW



1198914 30 Mean 12559119864 + 02 98052119864 + 00 0Std 47596119864 + 01 16323119864 + 00 0Rank 3 2 1

1198915 30 Mean 16140119864 + 02 22289119864 + 01 0Std 37649119864 + 01 27981119864 + 00 0Rank 3 2 1




11989110 30 Mean 16846119864 + 03 22370119864 + 02 0Std 26627119864 + 02 74690119864 + 01 0Rank 3 2 1

11989112 30 Mean 11359119864 + 02 21375119864 + 01 0Std 41907119864 + 01 29107119864 + 00 0Rank 3 2 1




11989119 30 Mean 24875119864 + 09 22700119864 + 08 0Std 13163119864 + 09 27319119864 + 07 0Rank 3 2 1

11989120 30 Mean 20564119864 + 03 92562119864 + 02 1Std 79311119864 + 02 76748119864 + 01 0Rank 3 2 1













Table 9 Continued
















Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

2

4

6Ra

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Stra

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Stra

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-7Strategies

Rank95 sig level



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Rank95 sig level




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Rank95 sig level



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Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in







Sphere 1198911 (119909) = 119863sum119894=1

1199092119894 [minus100 100]119863 0Griewank 1198912 (119909) = 14000

119863sum119894=1

1199092119894 minus 119863prod119894=1


119894=1


119894=1

[1199092119894 minus 10 cos (2120587119909119894)] [minus512 512]119863 0


1198915(119909) = 119863sum119894=1

1199102119894 minus 10 cos(2120587119910119894) + 10119910119894 =

119909119894 10038161003816100381610038161199091198941003816100381610038161003816 lt 05round (2119909119894)2 10038161003816100381610038161199091198941003816100381610038161003816 ge 05

[minus5 10]119863 0


1199092119894)minus exp( 1119863119863sum119894=1


119894=1

119894sum119895=1


119894=1


1le119894le119863


119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 119863prod119894=1


119894=1


119894=1

([119909119894 + 05])2 [minus10 10]119863 0

Levy

11989113 (119909) = sin2 (1205871199101) + 119863minus1sum119894=1


[minus10 10]119863 0

Powell Sum 11989114 (119909) = 119863sum119894=1


119894=1


119894=1

1199092119894 + ( 119863sum119894=1

05119894119909119894)2 + ( 119863sum119894=1


119894=1


119894=1

[119896maxsum119896=0

(119886119896 cos (2120587119887119896 (119909119894 + 05))) minus 119863119896maxsum119896=0

119886119896 cos (120587119887119896)] [minus05 05]119863 0119886 = 05 119887 = 3 119896max = 20

Bent-Cigar 11989119 (119909) = 11990921 + 106 119863sum119894=1

1199092119894 [minus100 100]119863 0


Table 1 Continued




119894=1


1119863119863sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + ( 119863prod119894=1

10038161003816100381610038161199091198941003816100381610038161003816)1119863]]2 [minus10 10]119863 0

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106

Aver

age b

est fi

tnes

s

(a) 11989112 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106Av

erag

e bes

t fitn

ess

(b) 11989112 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s(c) 11989112 with119863 = 100

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

(d) 11989113 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

(e) 11989113 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106Av

erag

e bes

t fitn

ess

(f) 11989113 with119863 = 100

PSOFWAPS-FW

200 400 600 800 10000Iteration

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

(g) 11989120 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

(h) 11989120 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

100101102103104105106107

Aver

age b

est fi

tnes

s

(i) 11989120 with119863 = 100



10minus33

10minus23

10minus13

10minus3

107Av

erag

e bes

t fitn

ess

200 400 600 800 10000Iteration

PSOFWAPS-FW

(a) 11989112 with119863 = 30

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(b) 11989113 with119863 = 30

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(c) 11989120 with119863 = 30





119908min = 04 1198881 = 1198882 = 145FWA 119860 = 40119872119890 = 50 119886 = 004 119887 = 08

num119872 = 30 120576 = 1119864 minus 100PS-FW












119891 119863 PSO FWA PS-FW




1198914 30 Mean 66547119864 + 01 0 0Std 36430119864 + 01 0 0Rank 2 1 1

1198915 30 Mean 65810119864 + 01 0 0Std 40117119864 + 01 0 0Rank 2 1 1

1198916 30 Mean 0 0 0Std 0 0 0Rank 1 1 1









11989115 30 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 3 Continued

119891 119863 PSO FWA PS-FW

11989117 30 Mean 0 98737119864 + 44 0Std 0 43038119864 + 45 0Rank 1 2 1

11989118 30 Mean 15069119864 + 01 0 0Std 40495119864 + 00 0 0Rank 2 1 1


11989120 30 Mean 38005119864 + 02 42079119864 + 01 1Std 85739119864 + 01 46125119864 + 00 0Rank 3 2 1

11989121 30 Mean 45577119864 + 01 171130119864 + 01 0Std 23091119864 + 01 21499119864 + 00 0Rank 3 2 1





CD120572 = 119902120572radic119873119894 (119873119894 + 1)6119873119891 (19)


119902005 = 27711990201 = 254 (20)


CD005 = 244CD01 = 224 (21)




119891 119863 PSO FWA PS-FW




1198914 60 Mean 32219119864 + 02 0 0Std 41863119864 + 01 0 0Rank 2 1 1

1198915 60 Mean 37498119864 + 02 0 0Std 53191119864 + 01 0 0Rank 2 1 1










11989115 60 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 4 Continued

119891 119863 PSO FWA PS-FW

11989117 60 Mean 0 24945119864 + 145 0Std 0 57208119864 + 145 0Rank 1 2 1

11989118 60 Mean 39564119864 + 01 0 0Std 53138119864 + 00 0 0Rank 2 1 1


11989120 60 Mean 53645119864 + 03 14665119864 + 02 1Std 62256119864 + 03 28947119864 + 01 0Rank 3 2 1

11989121 60 Mean 19709119864 + 02 48085119864 + 01 0Std 28605119864 + 01 77355119864 + 00 0Rank 3 2 1



FIPS

CPSO

stdPs

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PS-F

W

CLPS

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Algorithms

Rank95 sig level


0

2

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8

Rank

s

(a) Mean

FIPS

CPSO

stdPs

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PS-F

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CLPS

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Algorithms

Rank95 sig level


0

2

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Rank

s







119891 119863 PSO FWA PS-FW


1198912 100 Mean 11830119864 + 02 0 0Std 51822119864 + 01 0 0Rank 2 1 1


1198914 100 Mean 47288119864 + 02 0 0Std 10713119864 + 02 0 0Rank 2 1 1

1198915 100 Mean 51626119864 + 02 0 0Std 14819119864 + 02 0 0Rank 2 1 1







11989112 100 Mean 12670119864 + 02 42335119864 + 00 0Std 48966119864 + 01 140825853 0Rank 3 2 1



11989115 100 Mean 0 0 0Std 0 0 0Rank 1 1 1


11989117 100 Mean 0 20047119864 + 232 0Std 0 67205119864 + 232 0Rank 1 2 1


Table 5 Continued

119891 119863 PSO FWA PS-FW

11989118 100 Mean 68687119864 + 01 0 0Std 13221119864 + 01 0 0Rank 2 1 1


11989120 100 Mean 90245119864 + 03 26557119864 + 02 1Std 38036119864 + 03 47674119864 + 01 0Rank 3 2 1

11989121 100 Mean 40256119864 + 03 91975119864 + 01 0Std 16131119864 + 04 17966119864 + 01 0Rank 3 2 1










119891 119863 PSO FWA PS-FW



1198914 30 Mean 12559119864 + 02 98052119864 + 00 0Std 47596119864 + 01 16323119864 + 00 0Rank 3 2 1

1198915 30 Mean 16140119864 + 02 22289119864 + 01 0Std 37649119864 + 01 27981119864 + 00 0Rank 3 2 1




11989110 30 Mean 16846119864 + 03 22370119864 + 02 0Std 26627119864 + 02 74690119864 + 01 0Rank 3 2 1

11989112 30 Mean 11359119864 + 02 21375119864 + 01 0Std 41907119864 + 01 29107119864 + 00 0Rank 3 2 1




11989119 30 Mean 24875119864 + 09 22700119864 + 08 0Std 13163119864 + 09 27319119864 + 07 0Rank 3 2 1

11989120 30 Mean 20564119864 + 03 92562119864 + 02 1Std 79311119864 + 02 76748119864 + 01 0Rank 3 2 1













Table 9 Continued
















Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

2

4

6Ra

nks

Stra

tegy

-2

Stra

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-3

Stra

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-4

Stra

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Stra

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Stra

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-7Strategies

Rank95 sig level



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Rank95 sig level




0

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-7

Strategies

Rank95 sig level



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2

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s

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Stra

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Stra

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Stra

tegy

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Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in





Table 1 Continued




119894=1


1119863119863sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + ( 119863prod119894=1

10038161003816100381610038161199091198941003816100381610038161003816)1119863]]2 [minus10 10]119863 0

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106

Aver

age b

est fi

tnes

s

(a) 11989112 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106Av

erag

e bes

t fitn

ess

(b) 11989112 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s(c) 11989112 with119863 = 100

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

(d) 11989113 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

(e) 11989113 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

10minus34

10minus24

10minus14

10minus4

106Av

erag

e bes

t fitn

ess

(f) 11989113 with119863 = 100

PSOFWAPS-FW

200 400 600 800 10000Iteration

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

(g) 11989120 with119863 = 30

200 400 600 800 10000Iteration

PSOFWAPS-FW

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

(h) 11989120 with119863 = 60

200 400 600 800 10000Iteration

PSOFWAPS-FW

100101102103104105106107

Aver

age b

est fi

tnes

s

(i) 11989120 with119863 = 100



10minus33

10minus23

10minus13

10minus3

107Av

erag

e bes

t fitn

ess

200 400 600 800 10000Iteration

PSOFWAPS-FW

(a) 11989112 with119863 = 30

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(b) 11989113 with119863 = 30

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(c) 11989120 with119863 = 30





119908min = 04 1198881 = 1198882 = 145FWA 119860 = 40119872119890 = 50 119886 = 004 119887 = 08

num119872 = 30 120576 = 1119864 minus 100PS-FW












119891 119863 PSO FWA PS-FW




1198914 30 Mean 66547119864 + 01 0 0Std 36430119864 + 01 0 0Rank 2 1 1

1198915 30 Mean 65810119864 + 01 0 0Std 40117119864 + 01 0 0Rank 2 1 1

1198916 30 Mean 0 0 0Std 0 0 0Rank 1 1 1









11989115 30 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 3 Continued

119891 119863 PSO FWA PS-FW

11989117 30 Mean 0 98737119864 + 44 0Std 0 43038119864 + 45 0Rank 1 2 1

11989118 30 Mean 15069119864 + 01 0 0Std 40495119864 + 00 0 0Rank 2 1 1


11989120 30 Mean 38005119864 + 02 42079119864 + 01 1Std 85739119864 + 01 46125119864 + 00 0Rank 3 2 1

11989121 30 Mean 45577119864 + 01 171130119864 + 01 0Std 23091119864 + 01 21499119864 + 00 0Rank 3 2 1





CD120572 = 119902120572radic119873119894 (119873119894 + 1)6119873119891 (19)


119902005 = 27711990201 = 254 (20)


CD005 = 244CD01 = 224 (21)




119891 119863 PSO FWA PS-FW




1198914 60 Mean 32219119864 + 02 0 0Std 41863119864 + 01 0 0Rank 2 1 1

1198915 60 Mean 37498119864 + 02 0 0Std 53191119864 + 01 0 0Rank 2 1 1










11989115 60 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 4 Continued

119891 119863 PSO FWA PS-FW

11989117 60 Mean 0 24945119864 + 145 0Std 0 57208119864 + 145 0Rank 1 2 1

11989118 60 Mean 39564119864 + 01 0 0Std 53138119864 + 00 0 0Rank 2 1 1


11989120 60 Mean 53645119864 + 03 14665119864 + 02 1Std 62256119864 + 03 28947119864 + 01 0Rank 3 2 1

11989121 60 Mean 19709119864 + 02 48085119864 + 01 0Std 28605119864 + 01 77355119864 + 00 0Rank 3 2 1



FIPS

CPSO

stdPs

o

PS-F

W

CLPS

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Fran

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Algorithms

Rank95 sig level


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2

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Rank

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(a) Mean

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Algorithms

Rank95 sig level


0

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119891 119863 PSO FWA PS-FW


1198912 100 Mean 11830119864 + 02 0 0Std 51822119864 + 01 0 0Rank 2 1 1


1198914 100 Mean 47288119864 + 02 0 0Std 10713119864 + 02 0 0Rank 2 1 1

1198915 100 Mean 51626119864 + 02 0 0Std 14819119864 + 02 0 0Rank 2 1 1







11989112 100 Mean 12670119864 + 02 42335119864 + 00 0Std 48966119864 + 01 140825853 0Rank 3 2 1



11989115 100 Mean 0 0 0Std 0 0 0Rank 1 1 1


11989117 100 Mean 0 20047119864 + 232 0Std 0 67205119864 + 232 0Rank 1 2 1


Table 5 Continued

119891 119863 PSO FWA PS-FW

11989118 100 Mean 68687119864 + 01 0 0Std 13221119864 + 01 0 0Rank 2 1 1


11989120 100 Mean 90245119864 + 03 26557119864 + 02 1Std 38036119864 + 03 47674119864 + 01 0Rank 3 2 1

11989121 100 Mean 40256119864 + 03 91975119864 + 01 0Std 16131119864 + 04 17966119864 + 01 0Rank 3 2 1










119891 119863 PSO FWA PS-FW



1198914 30 Mean 12559119864 + 02 98052119864 + 00 0Std 47596119864 + 01 16323119864 + 00 0Rank 3 2 1

1198915 30 Mean 16140119864 + 02 22289119864 + 01 0Std 37649119864 + 01 27981119864 + 00 0Rank 3 2 1




11989110 30 Mean 16846119864 + 03 22370119864 + 02 0Std 26627119864 + 02 74690119864 + 01 0Rank 3 2 1

11989112 30 Mean 11359119864 + 02 21375119864 + 01 0Std 41907119864 + 01 29107119864 + 00 0Rank 3 2 1




11989119 30 Mean 24875119864 + 09 22700119864 + 08 0Std 13163119864 + 09 27319119864 + 07 0Rank 3 2 1

11989120 30 Mean 20564119864 + 03 92562119864 + 02 1Std 79311119864 + 02 76748119864 + 01 0Rank 3 2 1













Table 9 Continued
















Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

2

4

6Ra

nks

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7Strategies

Rank95 sig level



0

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s

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Stra

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Stra

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-4

Stra

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Stra

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-6

Stra

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-1

Stra

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-7

Strategies

Rank95 sig level




0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level



0

2

4

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Rank

s

Stra

tegy

-2

Stra

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-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-7

Stra

tegy

-1

Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in





10minus33

10minus23

10minus13

10minus3

107Av

erag

e bes

t fitn

ess

200 400 600 800 10000Iteration

PSOFWAPS-FW

(a) 11989112 with119863 = 30

10minus33

10minus23

10minus13

10minus3

107

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(b) 11989113 with119863 = 30

100

101

102

103

104

105

106

Aver

age b

est fi

tnes

s

200 400 600 800 10000Iteration

PSOFWAPS-FW

(c) 11989120 with119863 = 30





119908min = 04 1198881 = 1198882 = 145FWA 119860 = 40119872119890 = 50 119886 = 004 119887 = 08

num119872 = 30 120576 = 1119864 minus 100PS-FW












119891 119863 PSO FWA PS-FW




1198914 30 Mean 66547119864 + 01 0 0Std 36430119864 + 01 0 0Rank 2 1 1

1198915 30 Mean 65810119864 + 01 0 0Std 40117119864 + 01 0 0Rank 2 1 1

1198916 30 Mean 0 0 0Std 0 0 0Rank 1 1 1









11989115 30 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 3 Continued

119891 119863 PSO FWA PS-FW

11989117 30 Mean 0 98737119864 + 44 0Std 0 43038119864 + 45 0Rank 1 2 1

11989118 30 Mean 15069119864 + 01 0 0Std 40495119864 + 00 0 0Rank 2 1 1


11989120 30 Mean 38005119864 + 02 42079119864 + 01 1Std 85739119864 + 01 46125119864 + 00 0Rank 3 2 1

11989121 30 Mean 45577119864 + 01 171130119864 + 01 0Std 23091119864 + 01 21499119864 + 00 0Rank 3 2 1





CD120572 = 119902120572radic119873119894 (119873119894 + 1)6119873119891 (19)


119902005 = 27711990201 = 254 (20)


CD005 = 244CD01 = 224 (21)




119891 119863 PSO FWA PS-FW




1198914 60 Mean 32219119864 + 02 0 0Std 41863119864 + 01 0 0Rank 2 1 1

1198915 60 Mean 37498119864 + 02 0 0Std 53191119864 + 01 0 0Rank 2 1 1










11989115 60 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 4 Continued

119891 119863 PSO FWA PS-FW

11989117 60 Mean 0 24945119864 + 145 0Std 0 57208119864 + 145 0Rank 1 2 1

11989118 60 Mean 39564119864 + 01 0 0Std 53138119864 + 00 0 0Rank 2 1 1


11989120 60 Mean 53645119864 + 03 14665119864 + 02 1Std 62256119864 + 03 28947119864 + 01 0Rank 3 2 1

11989121 60 Mean 19709119864 + 02 48085119864 + 01 0Std 28605119864 + 01 77355119864 + 00 0Rank 3 2 1



FIPS

CPSO

stdPs

o

PS-F

W

CLPS

O

AIW

PSO

Fran

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Algorithms

Rank95 sig level


0

2

4

6

8

Rank

s

(a) Mean

FIPS

CPSO

stdPs

o

PS-F

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CLPS

O

AIW

PSO

Fran

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Algorithms

Rank95 sig level


0

2

4

6

8

Rank

s







119891 119863 PSO FWA PS-FW


1198912 100 Mean 11830119864 + 02 0 0Std 51822119864 + 01 0 0Rank 2 1 1


1198914 100 Mean 47288119864 + 02 0 0Std 10713119864 + 02 0 0Rank 2 1 1

1198915 100 Mean 51626119864 + 02 0 0Std 14819119864 + 02 0 0Rank 2 1 1







11989112 100 Mean 12670119864 + 02 42335119864 + 00 0Std 48966119864 + 01 140825853 0Rank 3 2 1



11989115 100 Mean 0 0 0Std 0 0 0Rank 1 1 1


11989117 100 Mean 0 20047119864 + 232 0Std 0 67205119864 + 232 0Rank 1 2 1


Table 5 Continued

119891 119863 PSO FWA PS-FW

11989118 100 Mean 68687119864 + 01 0 0Std 13221119864 + 01 0 0Rank 2 1 1


11989120 100 Mean 90245119864 + 03 26557119864 + 02 1Std 38036119864 + 03 47674119864 + 01 0Rank 3 2 1

11989121 100 Mean 40256119864 + 03 91975119864 + 01 0Std 16131119864 + 04 17966119864 + 01 0Rank 3 2 1










119891 119863 PSO FWA PS-FW



1198914 30 Mean 12559119864 + 02 98052119864 + 00 0Std 47596119864 + 01 16323119864 + 00 0Rank 3 2 1

1198915 30 Mean 16140119864 + 02 22289119864 + 01 0Std 37649119864 + 01 27981119864 + 00 0Rank 3 2 1




11989110 30 Mean 16846119864 + 03 22370119864 + 02 0Std 26627119864 + 02 74690119864 + 01 0Rank 3 2 1

11989112 30 Mean 11359119864 + 02 21375119864 + 01 0Std 41907119864 + 01 29107119864 + 00 0Rank 3 2 1




11989119 30 Mean 24875119864 + 09 22700119864 + 08 0Std 13163119864 + 09 27319119864 + 07 0Rank 3 2 1

11989120 30 Mean 20564119864 + 03 92562119864 + 02 1Std 79311119864 + 02 76748119864 + 01 0Rank 3 2 1













Table 9 Continued
















Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

2

4

6Ra

nks

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7Strategies

Rank95 sig level



0

2

4

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Rank

s

Stra

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Stra

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-3

Stra

tegy

-4

Stra

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Stra

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-6

Stra

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-1

Stra

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-7

Strategies

Rank95 sig level




0

2

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Rank

s

Stra

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-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-7

Stra

tegy

-1

Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in






119891 119863 PSO FWA PS-FW




1198914 30 Mean 66547119864 + 01 0 0Std 36430119864 + 01 0 0Rank 2 1 1

1198915 30 Mean 65810119864 + 01 0 0Std 40117119864 + 01 0 0Rank 2 1 1

1198916 30 Mean 0 0 0Std 0 0 0Rank 1 1 1









11989115 30 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 3 Continued

119891 119863 PSO FWA PS-FW

11989117 30 Mean 0 98737119864 + 44 0Std 0 43038119864 + 45 0Rank 1 2 1

11989118 30 Mean 15069119864 + 01 0 0Std 40495119864 + 00 0 0Rank 2 1 1


11989120 30 Mean 38005119864 + 02 42079119864 + 01 1Std 85739119864 + 01 46125119864 + 00 0Rank 3 2 1

11989121 30 Mean 45577119864 + 01 171130119864 + 01 0Std 23091119864 + 01 21499119864 + 00 0Rank 3 2 1





CD120572 = 119902120572radic119873119894 (119873119894 + 1)6119873119891 (19)


119902005 = 27711990201 = 254 (20)


CD005 = 244CD01 = 224 (21)




119891 119863 PSO FWA PS-FW




1198914 60 Mean 32219119864 + 02 0 0Std 41863119864 + 01 0 0Rank 2 1 1

1198915 60 Mean 37498119864 + 02 0 0Std 53191119864 + 01 0 0Rank 2 1 1










11989115 60 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 4 Continued

119891 119863 PSO FWA PS-FW

11989117 60 Mean 0 24945119864 + 145 0Std 0 57208119864 + 145 0Rank 1 2 1

11989118 60 Mean 39564119864 + 01 0 0Std 53138119864 + 00 0 0Rank 2 1 1


11989120 60 Mean 53645119864 + 03 14665119864 + 02 1Std 62256119864 + 03 28947119864 + 01 0Rank 3 2 1

11989121 60 Mean 19709119864 + 02 48085119864 + 01 0Std 28605119864 + 01 77355119864 + 00 0Rank 3 2 1



FIPS

CPSO

stdPs

o

PS-F

W

CLPS

O

AIW

PSO

Fran

kens

tein

Algorithms

Rank95 sig level


0

2

4

6

8

Rank

s

(a) Mean

FIPS

CPSO

stdPs

o

PS-F

W

CLPS

O

AIW

PSO

Fran

kens

tein

Algorithms

Rank95 sig level


0

2

4

6

8

Rank

s







119891 119863 PSO FWA PS-FW


1198912 100 Mean 11830119864 + 02 0 0Std 51822119864 + 01 0 0Rank 2 1 1


1198914 100 Mean 47288119864 + 02 0 0Std 10713119864 + 02 0 0Rank 2 1 1

1198915 100 Mean 51626119864 + 02 0 0Std 14819119864 + 02 0 0Rank 2 1 1







11989112 100 Mean 12670119864 + 02 42335119864 + 00 0Std 48966119864 + 01 140825853 0Rank 3 2 1



11989115 100 Mean 0 0 0Std 0 0 0Rank 1 1 1


11989117 100 Mean 0 20047119864 + 232 0Std 0 67205119864 + 232 0Rank 1 2 1


Table 5 Continued

119891 119863 PSO FWA PS-FW

11989118 100 Mean 68687119864 + 01 0 0Std 13221119864 + 01 0 0Rank 2 1 1


11989120 100 Mean 90245119864 + 03 26557119864 + 02 1Std 38036119864 + 03 47674119864 + 01 0Rank 3 2 1

11989121 100 Mean 40256119864 + 03 91975119864 + 01 0Std 16131119864 + 04 17966119864 + 01 0Rank 3 2 1










119891 119863 PSO FWA PS-FW



1198914 30 Mean 12559119864 + 02 98052119864 + 00 0Std 47596119864 + 01 16323119864 + 00 0Rank 3 2 1

1198915 30 Mean 16140119864 + 02 22289119864 + 01 0Std 37649119864 + 01 27981119864 + 00 0Rank 3 2 1




11989110 30 Mean 16846119864 + 03 22370119864 + 02 0Std 26627119864 + 02 74690119864 + 01 0Rank 3 2 1

11989112 30 Mean 11359119864 + 02 21375119864 + 01 0Std 41907119864 + 01 29107119864 + 00 0Rank 3 2 1




11989119 30 Mean 24875119864 + 09 22700119864 + 08 0Std 13163119864 + 09 27319119864 + 07 0Rank 3 2 1

11989120 30 Mean 20564119864 + 03 92562119864 + 02 1Std 79311119864 + 02 76748119864 + 01 0Rank 3 2 1













Table 9 Continued
















Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

2

4

6Ra

nks

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level




0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

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-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

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-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-7

Stra

tegy

-1

Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in





Table 3 Continued

119891 119863 PSO FWA PS-FW

11989117 30 Mean 0 98737119864 + 44 0Std 0 43038119864 + 45 0Rank 1 2 1

11989118 30 Mean 15069119864 + 01 0 0Std 40495119864 + 00 0 0Rank 2 1 1


11989120 30 Mean 38005119864 + 02 42079119864 + 01 1Std 85739119864 + 01 46125119864 + 00 0Rank 3 2 1

11989121 30 Mean 45577119864 + 01 171130119864 + 01 0Std 23091119864 + 01 21499119864 + 00 0Rank 3 2 1





CD120572 = 119902120572radic119873119894 (119873119894 + 1)6119873119891 (19)


119902005 = 27711990201 = 254 (20)


CD005 = 244CD01 = 224 (21)




119891 119863 PSO FWA PS-FW




1198914 60 Mean 32219119864 + 02 0 0Std 41863119864 + 01 0 0Rank 2 1 1

1198915 60 Mean 37498119864 + 02 0 0Std 53191119864 + 01 0 0Rank 2 1 1










11989115 60 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 4 Continued

119891 119863 PSO FWA PS-FW

11989117 60 Mean 0 24945119864 + 145 0Std 0 57208119864 + 145 0Rank 1 2 1

11989118 60 Mean 39564119864 + 01 0 0Std 53138119864 + 00 0 0Rank 2 1 1


11989120 60 Mean 53645119864 + 03 14665119864 + 02 1Std 62256119864 + 03 28947119864 + 01 0Rank 3 2 1

11989121 60 Mean 19709119864 + 02 48085119864 + 01 0Std 28605119864 + 01 77355119864 + 00 0Rank 3 2 1



FIPS

CPSO

stdPs

o

PS-F

W

CLPS

O

AIW

PSO

Fran

kens

tein

Algorithms

Rank95 sig level


0

2

4

6

8

Rank

s

(a) Mean

FIPS

CPSO

stdPs

o

PS-F

W

CLPS

O

AIW

PSO

Fran

kens

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Algorithms

Rank95 sig level


0

2

4

6

8

Rank

s







119891 119863 PSO FWA PS-FW


1198912 100 Mean 11830119864 + 02 0 0Std 51822119864 + 01 0 0Rank 2 1 1


1198914 100 Mean 47288119864 + 02 0 0Std 10713119864 + 02 0 0Rank 2 1 1

1198915 100 Mean 51626119864 + 02 0 0Std 14819119864 + 02 0 0Rank 2 1 1







11989112 100 Mean 12670119864 + 02 42335119864 + 00 0Std 48966119864 + 01 140825853 0Rank 3 2 1



11989115 100 Mean 0 0 0Std 0 0 0Rank 1 1 1


11989117 100 Mean 0 20047119864 + 232 0Std 0 67205119864 + 232 0Rank 1 2 1


Table 5 Continued

119891 119863 PSO FWA PS-FW

11989118 100 Mean 68687119864 + 01 0 0Std 13221119864 + 01 0 0Rank 2 1 1


11989120 100 Mean 90245119864 + 03 26557119864 + 02 1Std 38036119864 + 03 47674119864 + 01 0Rank 3 2 1

11989121 100 Mean 40256119864 + 03 91975119864 + 01 0Std 16131119864 + 04 17966119864 + 01 0Rank 3 2 1










119891 119863 PSO FWA PS-FW



1198914 30 Mean 12559119864 + 02 98052119864 + 00 0Std 47596119864 + 01 16323119864 + 00 0Rank 3 2 1

1198915 30 Mean 16140119864 + 02 22289119864 + 01 0Std 37649119864 + 01 27981119864 + 00 0Rank 3 2 1




11989110 30 Mean 16846119864 + 03 22370119864 + 02 0Std 26627119864 + 02 74690119864 + 01 0Rank 3 2 1

11989112 30 Mean 11359119864 + 02 21375119864 + 01 0Std 41907119864 + 01 29107119864 + 00 0Rank 3 2 1




11989119 30 Mean 24875119864 + 09 22700119864 + 08 0Std 13163119864 + 09 27319119864 + 07 0Rank 3 2 1

11989120 30 Mean 20564119864 + 03 92562119864 + 02 1Std 79311119864 + 02 76748119864 + 01 0Rank 3 2 1













Table 9 Continued
















Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

2

4

6Ra

nks

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level




0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-7

Stra

tegy

-1

Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in






119891 119863 PSO FWA PS-FW




1198914 60 Mean 32219119864 + 02 0 0Std 41863119864 + 01 0 0Rank 2 1 1

1198915 60 Mean 37498119864 + 02 0 0Std 53191119864 + 01 0 0Rank 2 1 1










11989115 60 Mean 0 0 0Std 0 0 0Rank 1 1 1



Table 4 Continued

119891 119863 PSO FWA PS-FW

11989117 60 Mean 0 24945119864 + 145 0Std 0 57208119864 + 145 0Rank 1 2 1

11989118 60 Mean 39564119864 + 01 0 0Std 53138119864 + 00 0 0Rank 2 1 1


11989120 60 Mean 53645119864 + 03 14665119864 + 02 1Std 62256119864 + 03 28947119864 + 01 0Rank 3 2 1

11989121 60 Mean 19709119864 + 02 48085119864 + 01 0Std 28605119864 + 01 77355119864 + 00 0Rank 3 2 1



FIPS

CPSO

stdPs

o

PS-F

W

CLPS

O

AIW

PSO

Fran

kens

tein

Algorithms

Rank95 sig level


0

2

4

6

8

Rank

s

(a) Mean

FIPS

CPSO

stdPs

o

PS-F

W

CLPS

O

AIW

PSO

Fran

kens

tein

Algorithms

Rank95 sig level


0

2

4

6

8

Rank

s







119891 119863 PSO FWA PS-FW


1198912 100 Mean 11830119864 + 02 0 0Std 51822119864 + 01 0 0Rank 2 1 1


1198914 100 Mean 47288119864 + 02 0 0Std 10713119864 + 02 0 0Rank 2 1 1

1198915 100 Mean 51626119864 + 02 0 0Std 14819119864 + 02 0 0Rank 2 1 1







11989112 100 Mean 12670119864 + 02 42335119864 + 00 0Std 48966119864 + 01 140825853 0Rank 3 2 1



11989115 100 Mean 0 0 0Std 0 0 0Rank 1 1 1


11989117 100 Mean 0 20047119864 + 232 0Std 0 67205119864 + 232 0Rank 1 2 1


Table 5 Continued

119891 119863 PSO FWA PS-FW

11989118 100 Mean 68687119864 + 01 0 0Std 13221119864 + 01 0 0Rank 2 1 1


11989120 100 Mean 90245119864 + 03 26557119864 + 02 1Std 38036119864 + 03 47674119864 + 01 0Rank 3 2 1

11989121 100 Mean 40256119864 + 03 91975119864 + 01 0Std 16131119864 + 04 17966119864 + 01 0Rank 3 2 1










119891 119863 PSO FWA PS-FW



1198914 30 Mean 12559119864 + 02 98052119864 + 00 0Std 47596119864 + 01 16323119864 + 00 0Rank 3 2 1

1198915 30 Mean 16140119864 + 02 22289119864 + 01 0Std 37649119864 + 01 27981119864 + 00 0Rank 3 2 1




11989110 30 Mean 16846119864 + 03 22370119864 + 02 0Std 26627119864 + 02 74690119864 + 01 0Rank 3 2 1

11989112 30 Mean 11359119864 + 02 21375119864 + 01 0Std 41907119864 + 01 29107119864 + 00 0Rank 3 2 1




11989119 30 Mean 24875119864 + 09 22700119864 + 08 0Std 13163119864 + 09 27319119864 + 07 0Rank 3 2 1

11989120 30 Mean 20564119864 + 03 92562119864 + 02 1Std 79311119864 + 02 76748119864 + 01 0Rank 3 2 1













Table 9 Continued
















Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

2

4

6Ra

nks

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level




0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-7

Stra

tegy

-1

Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in





Table 4 Continued

119891 119863 PSO FWA PS-FW

11989117 60 Mean 0 24945119864 + 145 0Std 0 57208119864 + 145 0Rank 1 2 1

11989118 60 Mean 39564119864 + 01 0 0Std 53138119864 + 00 0 0Rank 2 1 1


11989120 60 Mean 53645119864 + 03 14665119864 + 02 1Std 62256119864 + 03 28947119864 + 01 0Rank 3 2 1

11989121 60 Mean 19709119864 + 02 48085119864 + 01 0Std 28605119864 + 01 77355119864 + 00 0Rank 3 2 1



FIPS

CPSO

stdPs

o

PS-F

W

CLPS

O

AIW

PSO

Fran

kens

tein

Algorithms

Rank95 sig level


0

2

4

6

8

Rank

s

(a) Mean

FIPS

CPSO

stdPs

o

PS-F

W

CLPS

O

AIW

PSO

Fran

kens

tein

Algorithms

Rank95 sig level


0

2

4

6

8

Rank

s







119891 119863 PSO FWA PS-FW


1198912 100 Mean 11830119864 + 02 0 0Std 51822119864 + 01 0 0Rank 2 1 1


1198914 100 Mean 47288119864 + 02 0 0Std 10713119864 + 02 0 0Rank 2 1 1

1198915 100 Mean 51626119864 + 02 0 0Std 14819119864 + 02 0 0Rank 2 1 1







11989112 100 Mean 12670119864 + 02 42335119864 + 00 0Std 48966119864 + 01 140825853 0Rank 3 2 1



11989115 100 Mean 0 0 0Std 0 0 0Rank 1 1 1


11989117 100 Mean 0 20047119864 + 232 0Std 0 67205119864 + 232 0Rank 1 2 1


Table 5 Continued

119891 119863 PSO FWA PS-FW

11989118 100 Mean 68687119864 + 01 0 0Std 13221119864 + 01 0 0Rank 2 1 1


11989120 100 Mean 90245119864 + 03 26557119864 + 02 1Std 38036119864 + 03 47674119864 + 01 0Rank 3 2 1

11989121 100 Mean 40256119864 + 03 91975119864 + 01 0Std 16131119864 + 04 17966119864 + 01 0Rank 3 2 1










119891 119863 PSO FWA PS-FW



1198914 30 Mean 12559119864 + 02 98052119864 + 00 0Std 47596119864 + 01 16323119864 + 00 0Rank 3 2 1

1198915 30 Mean 16140119864 + 02 22289119864 + 01 0Std 37649119864 + 01 27981119864 + 00 0Rank 3 2 1




11989110 30 Mean 16846119864 + 03 22370119864 + 02 0Std 26627119864 + 02 74690119864 + 01 0Rank 3 2 1

11989112 30 Mean 11359119864 + 02 21375119864 + 01 0Std 41907119864 + 01 29107119864 + 00 0Rank 3 2 1




11989119 30 Mean 24875119864 + 09 22700119864 + 08 0Std 13163119864 + 09 27319119864 + 07 0Rank 3 2 1

11989120 30 Mean 20564119864 + 03 92562119864 + 02 1Std 79311119864 + 02 76748119864 + 01 0Rank 3 2 1













Table 9 Continued
















Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

2

4

6Ra

nks

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level




0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-7

Stra

tegy

-1

Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in






119891 119863 PSO FWA PS-FW


1198912 100 Mean 11830119864 + 02 0 0Std 51822119864 + 01 0 0Rank 2 1 1


1198914 100 Mean 47288119864 + 02 0 0Std 10713119864 + 02 0 0Rank 2 1 1

1198915 100 Mean 51626119864 + 02 0 0Std 14819119864 + 02 0 0Rank 2 1 1







11989112 100 Mean 12670119864 + 02 42335119864 + 00 0Std 48966119864 + 01 140825853 0Rank 3 2 1



11989115 100 Mean 0 0 0Std 0 0 0Rank 1 1 1


11989117 100 Mean 0 20047119864 + 232 0Std 0 67205119864 + 232 0Rank 1 2 1


Table 5 Continued

119891 119863 PSO FWA PS-FW

11989118 100 Mean 68687119864 + 01 0 0Std 13221119864 + 01 0 0Rank 2 1 1


11989120 100 Mean 90245119864 + 03 26557119864 + 02 1Std 38036119864 + 03 47674119864 + 01 0Rank 3 2 1

11989121 100 Mean 40256119864 + 03 91975119864 + 01 0Std 16131119864 + 04 17966119864 + 01 0Rank 3 2 1










119891 119863 PSO FWA PS-FW



1198914 30 Mean 12559119864 + 02 98052119864 + 00 0Std 47596119864 + 01 16323119864 + 00 0Rank 3 2 1

1198915 30 Mean 16140119864 + 02 22289119864 + 01 0Std 37649119864 + 01 27981119864 + 00 0Rank 3 2 1




11989110 30 Mean 16846119864 + 03 22370119864 + 02 0Std 26627119864 + 02 74690119864 + 01 0Rank 3 2 1

11989112 30 Mean 11359119864 + 02 21375119864 + 01 0Std 41907119864 + 01 29107119864 + 00 0Rank 3 2 1




11989119 30 Mean 24875119864 + 09 22700119864 + 08 0Std 13163119864 + 09 27319119864 + 07 0Rank 3 2 1

11989120 30 Mean 20564119864 + 03 92562119864 + 02 1Std 79311119864 + 02 76748119864 + 01 0Rank 3 2 1













Table 9 Continued
















Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

2

4

6Ra

nks

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level




0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-7

Stra

tegy

-1

Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in





Table 5 Continued

119891 119863 PSO FWA PS-FW

11989118 100 Mean 68687119864 + 01 0 0Std 13221119864 + 01 0 0Rank 2 1 1


11989120 100 Mean 90245119864 + 03 26557119864 + 02 1Std 38036119864 + 03 47674119864 + 01 0Rank 3 2 1

11989121 100 Mean 40256119864 + 03 91975119864 + 01 0Std 16131119864 + 04 17966119864 + 01 0Rank 3 2 1










119891 119863 PSO FWA PS-FW



1198914 30 Mean 12559119864 + 02 98052119864 + 00 0Std 47596119864 + 01 16323119864 + 00 0Rank 3 2 1

1198915 30 Mean 16140119864 + 02 22289119864 + 01 0Std 37649119864 + 01 27981119864 + 00 0Rank 3 2 1




11989110 30 Mean 16846119864 + 03 22370119864 + 02 0Std 26627119864 + 02 74690119864 + 01 0Rank 3 2 1

11989112 30 Mean 11359119864 + 02 21375119864 + 01 0Std 41907119864 + 01 29107119864 + 00 0Rank 3 2 1




11989119 30 Mean 24875119864 + 09 22700119864 + 08 0Std 13163119864 + 09 27319119864 + 07 0Rank 3 2 1

11989120 30 Mean 20564119864 + 03 92562119864 + 02 1Std 79311119864 + 02 76748119864 + 01 0Rank 3 2 1













Table 9 Continued
















Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

2

4

6Ra

nks

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level




0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-7

Stra

tegy

-1

Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in






119891 119863 PSO FWA PS-FW



1198914 30 Mean 12559119864 + 02 98052119864 + 00 0Std 47596119864 + 01 16323119864 + 00 0Rank 3 2 1

1198915 30 Mean 16140119864 + 02 22289119864 + 01 0Std 37649119864 + 01 27981119864 + 00 0Rank 3 2 1




11989110 30 Mean 16846119864 + 03 22370119864 + 02 0Std 26627119864 + 02 74690119864 + 01 0Rank 3 2 1

11989112 30 Mean 11359119864 + 02 21375119864 + 01 0Std 41907119864 + 01 29107119864 + 00 0Rank 3 2 1




11989119 30 Mean 24875119864 + 09 22700119864 + 08 0Std 13163119864 + 09 27319119864 + 07 0Rank 3 2 1

11989120 30 Mean 20564119864 + 03 92562119864 + 02 1Std 79311119864 + 02 76748119864 + 01 0Rank 3 2 1













Table 9 Continued
















Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

2

4

6Ra

nks

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level




0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-7

Stra

tegy

-1

Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in














Table 9 Continued
















Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

2

4

6Ra

nks

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level




0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-7

Stra

tegy

-1

Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in








Table 9 Continued
















Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

2

4

6Ra

nks

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level




0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-7

Stra

tegy

-1

Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in





Table 9 Continued
















Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

2

4

6Ra

nks

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level




0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-7

Stra

tegy

-1

Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in








Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

2

4

6Ra

nks

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level




0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-7

Stra

tegy

-1

Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in





Table 13 Continued




Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(a) 1198911



Strategy-7


Aver

age b

est fi

tnes

s

200 400 600 800 10000Iterations

(b) 1198919



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s

(c) 11989110



Strategy-7

200 400 600 800 10000Iterations


Aver

age b

est fi

tnes

s






0

2

4

6Ra

nks

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level




0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-7

Stra

tegy

-1

Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in








0

2

4

6Ra

nks

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level




0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-7

Stra

tegy

-1

Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in





0

2

4

6Ra

nks

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level




0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-1

Stra

tegy

-7

Strategies

Rank95 sig level



0

2

4

6

Rank

s

Stra

tegy

-2

Stra

tegy

-3

Stra

tegy

-4

Stra

tegy

-5

Stra

tegy

-6

Stra

tegy

-7

Stra

tegy

-1

Strategies

Rank95 sig level













5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in










5 Conclusion





Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in





































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in









Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in




Documents

PS-FW: A Hybrid Algorithm Based on Particle Swarm and …downloads.hindawi.com/journals/cin/2018/6094685.pdf · 2019. 7. 30. · ResearchArticle PS-FW: A Hybrid Algorithm Based on