An Improved Squirrel Search Algorithm for...

Research ArticleAn Improved Squirrel Search Algorithm for Optimization

Tongyi Zheng and Weili Luo

School of Civil Engineering Guangzhou University Guangzhou China

Correspondence should be addressed to Weili Luo wlluogzhueducn

Received 15 February 2019 Revised 5 May 2019 Accepted 28 May 2019 Published 1 July 2019

Academic Editor Alex Alexandridis

Copyright copy 2019 Tongyi Zheng andWeili LuoThis 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

Squirrel search algorithm (SSA) is a new biological-inspired optimization algorithm which has been proved to be more effectivefor solving unimodal multimodal and multidimensional optimization problems However similar to other swarm intelligence-based algorithms SSA also has its own disadvantages In order to get better global convergence ability an improved version ofSSA called ISSA is proposed in this paper Firstly an adaptive strategy of predator presence probability is proposed to balance theexploration and exploitation capabilities of the algorithm Secondly a normal cloudmodel is introduced to describe the randomnessand fuzziness of the foraging behavior of flying squirrels Thirdly a selection strategy between successive positions is incorporatedto preserve the best position of flying squirrel individuals Finally in order to enhance the local search ability of the algorithma dimensional search enhancement strategy is utilized 32 benchmark functions including unimodal multimodal and CEC 2014functions are used to test the global search ability of the proposed ISSA Experimental test results indicate that ISSA providescompetitive performance compared with the basic SSA and other four well-known state-of-the-art optimization algorithms

1 Introduction

Optimization is the process of determining decision variablesof a function in a way that the function is in its maximal orminimal value Many real-life engineering problems belongto optimization problems [1ndash3] in which decision variablesare determined in a way that the systems operate in their bestoptimal point Usually these problems are discontinuousnondifferentiable multimodal and nonconevx and thus theclassical gradient-based deterministic algorithms [4ndash6] arenot applicable

To overcome the drawbacks of the classical algorithmsa considerable number of stochastic optimization algorithmsknown as metaheuristic algorithms [7ndash9] have been devel-oped in recent decades These algorithms are mainly inspiredby biological behaviors or physical phenomena and can beroughly classified into three categories evolutionary algo-rithms swarm intelligence and physical-based algorithmsThe evolutionary algorithms simulate the evolution of thenature such as reproduction mutation recombination andselection in which a population tries to survive based on

the evaluation of fitness value in a given environmentGenetic algorithm (GA) [10] and evolution strategy (ES)[11] are the most popular evolutionary algorithms Swarmintelligence algorithms are the second category which arebased on movement of a swarm of creatures and imitatethe interaction of swarm and their environment in order toenhance their knowledge of a goal (eg food source) Themost well-known swarm intelligence algorithms are particleswarm optimization (PSO) [12] artificial bee colony (ABC)[13] algorithm ant colony optimization (ACO) [14] andgrey wolf optimization algorithm (GWO) [15] to name afew The physical-based algorithms are inspired by the basicphysical laws in universe such as gravitational force elec-tromagnetic force and inertia force Several representativesof this category are simulated annealing (SA) [16] big-bangbig-crunch (BB-BC) [17] and gravitational search algorithm(GSA) algorithm [18] Some of the recent nature-inspiredalgorithms are lightning attachment procedure optimization(LAPO) [19] spotted hyena optimizer (SHO) [20] weightedsuperposition attraction (WSA) [21] andmanymore [22 23]

HindawiComplexityVolume 2019 Article ID 6291968 31 pageshttpsdoiorg10115520196291968

2 Complexity

Unfortunately most of the abovementioned basic meta-heuristic algorithms fail to balance exploration and exploita-tion thereby yielding unsatisfactory performance for real-life complicated optimization problems Exploration standsfor global search ability and ensures the algorithm toreach all over the search space and then to find promisingregions whereas exploitation represents local search abilityand ensures the searching of optimum within the identifiedpromising regions Emphasizing the exploration capabilityonly results in a waste of computational resources on search-ing all over the interior regions of search space and thusreduces the convergence rate emphasizing the exploitationcapability only by contrast causes loss of population diversityearly and thus probably leads to premature convergence or tobe stuck in local optimalThis fact motivates the introductionof various strategies for improving the convergence rateand precision of the basic metaheuristic algorithms As anillustration premature convergence of PSO was preventedin CLPSO by proposing a comprehensive learning strategyto maintain the population diversity [24] a social learningcomponent called fitness-distance-ratio was employed toenhance local search capability of PSO [25] a self-organizinghierarchical PSO with time-varying acceleration coefficients(HPSO-TVAC) was introduced to efficiently control the localsearch and convergence to the global optimal solution [26]a distance-based locally informed PSO (LIPS) enabled thealgorithm to quickly converge to global optimal solutionwith high accuracy [27] Likewise many modified versionshave been proposed to enhance the global search ability ofthe basic ABC an improved-global-best-guide term witha nonlinear adjusting factor was employed to balance theexploration and exploitation [28] a multiobjective covari-ance guided artificial bee colony algorithm (M-CABC) wasproposed to obtain higher precision with quick convergencespeed when solving portfolio problems [29] the slow conver-gence and unsatisfactory solution accuracy were improved inthe variant IABC [30] As for fruit fly optimization algorithm(FOA) [31] an escape parameter was introduced in MFOAto escape from the local solution [32] and modified versionsfor balancing between exploration and exploitation abilitiesinclude for example IFOA [33]MSFOA [34] IFFO [35] andCMFOA [36]

Squirrel search algorithm (SSA) proposed byMohit et alin 2018 is a new and powerful global optimization algorithminspired by the natural dynamic foraging behavior of flyingsquirrels [37] In comparison with other swarm intelligenceoptimization algorithms SSA has the advantages of betterand efficient search space exploration because a seasonalmonitoring condition is incorporated Moreover three typesof trees (normal tree oak tree and hickory tree) are availablein the forest region preserving the population diversity andthus enhancing the exploration of the algorithm Test resultsof 33 benchmark functions and a real-time controller designproblem confirm the superiority of SSA in comparison withother well-known algorithms such as GA [10] PSO [25] BA(bat algorithm) [38] and FF (firefly algorithm) [39]

However SSA still suffers from premature convergenceand easily gets trapped in a local optimal solution especiallywhen solving highly complex problemsThe convergence rate

of SSA like other swarm intelligence algorithms depends onthe balance between exploration and exploitation capabilitiesIn other words an excellent performance in dealing withoptimization problems requires fine-tuning of the explo-ration and exploitation problem According to ldquono freelunchrdquo (NFL) theorem [40] no single optimization algorithmis able to achieve the best performance for all problems andSSA is not an exception Therefore there still exists room forimproving the accuracy and convergence rates of SSA

Based on the discussion above this study proposes animproved variant of SSA (ISSA) which employs four strate-gies to enhance the global search ability of SSA In brief themain contributions of this research can be summarized asfollows

(i) An adaptive strategy of predator presence probabilityis proposed which dynamically adjusts with the iterationprocess This strategy discourages premature convergenceand improves the intensive search ability of the algorithmespecially at the latter stages of search In this way a balancebetween the exploration and exploitation capabilities can beproperly managed

(ii)The proposed ISSA employs a normal cloud generator[41] to generate new locations for flying squirrels during thecourse of gliding which improves the exploration capabilityof SSAThis ismotivated by the fact that the gliding behaviorsof flying squirrels have characteristics of randomness andfuzziness which can be simultaneously described by thenormal cloud model [42]

(iii) A selection strategy between successive positions isproposed to maintain the best position of a flying squir-rel individual throughout the optimization process whichenhances the exploitation ability of the algorithm

(iv) A dimensional search enhancement strategy is orig-inally put forward and results in a better quality of the bestsolution in each iteration thereby strengthening the localsearch ability of the algorithm

The general properties of ISSA are evaluated against 32benchmark function including unimodal multimodal andCEC 2014 functions [43] Meanwhile its performance iscompared with the basic SSA and other four well-knownstate-of-the-art optimization algorithms

The rest of this paper is organized as follows Section 2briefly recapitulates the basic SSA Next the proposed ISSAis presented in detail in Section 3 Experimental comparisonsare illustrated in Section 4 Finally Section 5 gives theconcluding remarks

2 The Basic Squirrel SearchOptimization Algorithm

SSAmimics the dynamic foraging behavior of southern flyingsquirrels via gliding an effective mechanism used by smallmammals for travelling long distance in deciduous forest ofEurope and Asia [37] During warm weather the squirrelschange their locations by gliding from one tree to anotherin the forest and explore for food resources They can easilyfind acorn nuts for meeting daily energy needs After thatthey begin searching hickory nuts (the optimal food source)that are stored for winter During cold weather they become

Complexity 3

less active and maintain their energy requirements withthe storage of hickory nuts When the weather gets warmflying squirrels become active again The abovementionedprocess is repeated and continues throughout the life spaceof the squirrels which serves as a foundation of the SSAAccording to the food foraging strategy of flying squirrelsthe optimization SSA can bemodeled by the following phasesmathematically

21 Initialize the Algorithm Parameters Themain parametersof the SSA are the maximum number of iteration 119868119905119890119903119898119886119909the population size 119873119875 the number of decision variables nthe predator presence probability 119875119889119901 the scaling factor 119904119891the gliding constant 119866119888 and the upper and lower bounds fordecision variable 119865119878119880 and 119865119878119871 These parameters are set inthe beginning of the SSA procedure

22 Initialize Flying Squirrelsrsquo Locations and Their SortingThe flying squirrelsrsquo locations are randomly initialized in thesearch apace as follows

119865119878119894119895 = 119865119878119871 + rand ( ) lowast (119865119878119880 minus 119865119878119871) 119894 = 1 2 119873119875 119895 = 1 2 119899 (1)

where rand( ) is a uniformly distributed random number inthe range [0 1]

The fitness value 119891 = (11989111198912 119891119873119875) of an individualflying squirrelrsquos location is calculated by substituting the valueof decision variables into a fitness function

119891119894 = 119891119894 (1198651198781198941 1198651198781198942 119865119878119894119899) 119894 = 1 2 119873119875 (2)

Then the quality of food sources defined by the fitness value ofthe flying squirrelsrsquo locations is sorted in an ascending order

[119904119900119903119905119890119889 119891 119904119900119903119905119890 119894119899119889119890119909] = 119904119900119903119905 (119891) (3)

After sorting the food sources of each flying squirrelrsquoslocation three types of trees are categorized hickory tree(hickory nuts food source) oak tree (acorn nuts food source)and normal tree The location of the best food source (ieminimal fitness value) is regarded as the hickory nut tree(119865119878ℎ119905) the locations of the following three food sources aresupposed to be the acorn nuts trees (119865119878119886119905) and the rest areconsidered as normal trees (119865119878119899119905)

119865119878ℎ119905 = 119865119878 (119904119900119903119905119890 119894119899119889119890119909 (1)) (4)

119865119878119886119905 (1 3) = 119865119878 (119904119900119903119905119890 119894119899119889119890119909 (2 4)) (5)

119865119878119899119905 (1119873119875 minus 4) = 119865119878 (119904119900119903119905119890 119894119899119889119890119909 (5119873119875)) (6)

23 Generate NewLocations through Gliding Three scenariosmay appear after the dynamic gliding process of flyingsquirrels

Scenario 1 Flying squirrels on acorn nut trees tend to movetowards hickory nut treeThe new locations can be generatedas follows

119865119878119899119890119908119886119905 = 119865119878119900119897119889119886119905 + 119889119892119866119888 (119865119878119900119897119889ℎ119905 minus 119865119878119900119897119889119886119905 ) if 1198771 ge 119875119889119901119903119886119899119889119900119898 119897119900119888119886119905119894119900119899 119900119905ℎ119890119903119908119894119904119890 (7)

where 119889119892 is random gliding distance 1198771 is a function whichreturns a value from the uniform distribution on the interval[0 1] and 119866119888 is a gliding constantScenario 2 Some squirrels which are on normal trees maymove towards acornnut tree to fulfill their daily energy needsThe new locations can be generated as follows

119865119878119899119890119908119899119905 = 119865119878119900119897119889119899119905 + 119889119892119866119888 (119865119878119900119897119889119886119905 minus 119865119878119900119897119889119899119905 ) if 1198772 ge 119875119889119901119903119886119899119889119900119898 119897119900119888119886119905119894119900119899 119900119905ℎ119890119903119908119894119904119890 (8)

where1198772 is a functionwhich returns a value from the uniformdistribution on the interval [0 1]Scenario 3 Some flying squirrels on normal trees may movetowards hickory nut tree if they have already fulfilled theirdaily energy requirements In this scenario the new locationof squirrels can be generated as follows

119865119878119899119890119908119899119905 = 119865119878119900119897119889119899119905 + 119889119892119866119888 (119865119878119900119897119889ℎ119905 minus 119865119878119900119897119889119899119905 ) if 1198773 ge 119875119889119901119903119886119899119889119900119898 119897119900119888119886119905119894119900119899 119900119905ℎ119890119903119908119894119904119890 (9)

where1198773 is a functionwhich returns a value from the uniformdistribution on the interval [0 1]

In all scenarios gliding distance 119889119892 is considered to bein the interval between 9 and 20m [37] However this valueis quite large and may introduce large perturbations in (7)-(9) and hence may cause unsatisfactory performance of thealgorithm In order to achieve acceptable performance of thealgorithm a scaling factor (119904119891) is introduced as a divisor of119889119892 and its value is chosen to be 18 [37]

24 Check Seasonal Monitoring Condition The foragingbehavior of flying squirrels is significantly affected by seasonvariations [43]Therefore a seasonal monitoring condition isintroduced in the algorithm to prevent the algorithm frombeing trapped in local optimal solutions

A seasonal constant 119878119888 and its minimum value arecalculated firstly

119878119905119888 = radic 119899sum119896=1

(119865119878119905119886119905119896 minus 119865119878ℎ119905119896)2 119905 = 1 2 3 (10)

119878119888119898119894119899 = 10119864 minus 6365119868119905119890119903(119868119905119890119903max)25(11)

Then the seasonal monitoring condition is checked Underthe condition of 119878119905119888 lt 119878119888119898119894119899 the winter is over and the flyingsquirrels which lose their abilities to explore the forest will

4 Complexity

randomly relocate their searching positions for food sourceagain

119865119878119899119890119908119899119905 = 119865119878119871 + Levy (n) times (119865119878119880 minus 119865119878119871) (12)

where Levy distribution is a powerful mathematical tool toenhance the global exploration capability of most optimiza-tion algorithms [44]

Levy (119909) = 001 times 119903119886 times 120590100381610038161003816100381611990311988710038161003816100381610038161120573 (13)

where 119903119886 and 119903119887 are two functions which return a value fromthe uniform distribution on the interval [0 1] 120573 is a constant(120573 = 15 in this paper) and 120590 is calculated as follows

120590 = ( Γ (1 + 120573) times sin (1205871205732)Γ ((1 + 120573) 2) times 120573 times 2((120573minus1)2))1120573

(14)

where Γ(119909) = (x minus 1)25 Stopping Criterion The algorithm terminates if themaximum number of iterations is satisfied Otherwise thebehaviors of generating new locations and checking seasonalmonitoring condition are repeated

26 Procedure of the Basic SSA The pseudocode of SSA isprovided in Algorithm 1

3 The Improved Squirrel SearchOptimization Algorithm

This section presents an improved squirrel search optimiza-tion algorithm by introducing four strategies to enhance thesearching capability of the algorithm In the following thefour strategies will be presented in detail

31 An Adaptive Strategy of Predator Presence ProbabilityWhen flying squirrels generate new locations their naturalbehaviors are affected by the presence of predators and thischaracter is controlled by predator presence probability 119875119889119901In the early search stage flying squirrelsrsquo population is oftenfar away from the food source and its distribution range islarge thus it faces a great threat from predators With theevolution going on flying squirrelsrsquo locations are close to thefood source (an optimal solution) In this case the distri-bution range of flying squirrelsrsquo population is increasinglysmaller and less threats from predators are expectedThus toenhance the exploitation capacity of the SSA an adaptive 119875119889119901which dynamically varies as a function of iteration numberis adopted as follows

119875119889119901 = (119875119889119901119898119886119909 minus 119875119889119901119898119894119899) times (1 minus 119868119905119890119903119868119905119890119903119898119886119909)10+ 119875119889119901119898119894119899 (15)

where 119875119889119901119898119886119909 and 119875119889119901119898119894119899 are the maximum and minimumpredator presence probability respectively

32 Flying Squirrelsrsquo Random Position Generation Based onCloud Generator Under the condition of 1198771 1198772 1198773 lt 119875119889119901the flying squirrels randomly proceed gliding to the nextpotential food locations different individuals generally havedifferent judgments and their gliding directions and routinesvary In other words the foraging behavior of flying squirrelshas the characteristics of randomness and fuzziness Thesecharacteristics can be synthetically described and integratedby a normal cloudmodel In themodel a normal cloudmodelgenerator instead of uniformly distributed random functionsis used to reproduce new location for each flying squirrelThus (7)-(9) are replaced by the following equations

119865119878119899119890119908119886119905=

119865119878119900119897119889119886119905 + 119889119892119866119888 (119865119878119900119897119889ℎ119905 minus 119865119878119900119897119889119886119905 ) if 1198771 ge 119875119889119901119862119909 (119865119878119900119897119889119886119905 119864119899119867119890) 119900119905ℎ119890119903119908119894119904119890(16)

119865119878119899119890119908119899119905=

119865119878119900119897119889119899119905 + 119889119892119866119888 (119865119878119900119897119889119886119905 minus 119865119878119900119897119889119899119905 ) if 1198772 ge 119875119889119901119862119909 (119865119878119900119897119889119899119905 119864119899119867119890) 119900119905ℎ119890119903119908119894119904119890(17)

119865119878119899119890119908119899119905=

119865119878119900119897119889119899119905 + 119889119892119866119888 (119865119878119900119897119889ℎ119905 minus 119865119878119900119897119889119899119905 ) if 1198773 ge 119875119889119901119862119909 (119865119878119900119897119889119899119905 119864119899119867119890) 119900119905ℎ119890119903119908119894119904119890(18)

where 119864119899 (Entropy) represents the uncertainty measurementof a qualitative concept and 119867119890 (Hyper Entropy) is theuncertain degree of entropy 119864119899 [42] Specifically in (16)-(18) 119864119899 stands for the search radius and 119867119890 = 01119864119899 isused to represent the stability of the search In the earlyiterations a large 119864119899 is requested because the flying squirrelsrsquolocation is often far away froman optimal solution Under thecondition of final generations where the population locationis close to an optimal solution a smaller 119864119899 is appropriate forthe fine-tuning of solutions Therefore the search radius 119864119899dynamically changes with iteration number

119864119899 = 119864119899119898119886119909 times (1 minus 119868119905119890119903119868119905119890119903119898119886119909)10 (19)

where 119864119899119898119886119909 = (119865119878119880minus119865119878119871)4 is the maximum search radius

33 A Selection Strategy between Successive Positions Whennew positions of flying squirrels are generated it is possiblethat the new position is worse than the old oneThis suggeststhat the fitness value of each individual needs to be checkedafter the generation of new positions by comparing withthe old one in each iteration If the fitness value of thenew position is better than the old one the position of thecorresponding flying squirrel is updated by the new positionOtherwise the old position is reserved This strategy can bemathematically described by

119865119878119894 = 119865119878119899119890119908119894 if 119891119899119890119908119894 lt 119891119900119897119889119894119865119878119900119897119889119894 119900119905ℎ119890119903119908119894119904119890 (20)

Complexity 5

Set 119868119905119890119903119898119886119909119873119875 n 119875119889119901 119904119891 119866119888 119865119878119880 and 119865119878119871Randomly initialize the flying squirrels locations119865119878119894119895 = 119865119878119871 + rand() lowast (119865119878119880 minus 119865119878119871) 119894 = 1 2 119873119875 119895 = 1 2 119899Calculate fitness value119891119894 = 119891119894(1198651198781198941 1198651198781198942 119865119878119894119899) 119894 = 1 2 119873119875while 119868119905119890119903 lt 119868119905119890119903119898119886119909[119904119900119903119905119890119889 119891 119904119900119903119905119890 119894119899119889119890119909] = 119904119900119903119905(119891)119865119878ℎ119905 = 119865119878(119904119900119903119905119890 119894119899119889119890119909(1))119865119878119886119905(1 3) = 119865119878(119904119900119903119905119890 119894119899119889119890119909(2 4))119865119878119899119905(1119873119875 minus 4) = 119865119878(119904119900119903119905119890 119894119899119889119890119909(5119873119875))

Generate new locationsfor t = 1 n1 (n1 = total number of squirrels on acorn trees)

if 1198771 ge 119875119889119901 119865119878119899119890119908119886119905 = 119865119878119900119897119889119886119905 + 119889119892119866119888(119865119878119900119897119889ℎ119905 minus 119865119878119900119897119889119886119905 )else 119865119878119899119890119908119886119905 = 119903119886119899119889119900119898 119897119900119888119886119905119894119900119899end

endfor t =1 n2 (n2 = total number of squirrels on normal trees moving towards acorn trees)

if 1198772 ge 119875119889119901 119865119878119899119890119908119899119905 = 119865119878119900119897119889119899119905 + 119889119892119866119888(119865119878119900119897119889119886119905 minus 119865119878119900119897119889119899119905 )else 119865119878119899119890119908119899119905 = 119903119886119899119889119900119898 119897119900119888119886119905119894119900119899end

endfor t = 1 n3 (n3 = total number of squirrels on normal trees moving towards hickory trees)


end

119878119905119888 = radic 119899sum119896=1

(119865119878119905119886119905119896 minus 119865119878ℎ119905119896)2 119878119888119898119894119899 = 10119864 minus 6365119868119905119890119903(119868119905119890119903max)25

if 119878119905119888 lt 119878119888119898119894119899 119865119878119899119890119908119899119905 = 119865119878119871 + L evy(n) times (119865119878119880 minus 119865119878119871)end

Calculate fitness value of new locations119891119894 = 119891119894(1198651198781198991198901199081198941 1198651198781198991198901199081198942 119865119878119899119890119908119894119899 ) 119894 = 1 2 119873119875119868119905119890119903 = 119868119905119890119903 + 1end

Algorithm 1 Pseudocode of basic SSA

34 Enhance the Intensive Dimensional Search In the basicSSA all dimensions of one individual flying squirrel areupdated simultaneously The main drawback of this pro-cedure is that different dimensions are dependent and thechange of one dimension may have negative effects on otherspreventing them from finding the optimal variables in theirown dimensions To further enhance the intensive searchof each dimension the following steps are taken for eachiteration (i) find the best flying squirrel location (ii) generateone more solution based on the best flying squirrel locationby changing the value of one dimension while maintainingthe rest dimensions (iii) compare fitness values of the new-generated solution with the original one and reserve the

better one (iv) repeat steps (ii) and (iii) in other dimensionsindividually The new-generated solution is produced by

119865119878119899119890119908119887119890119904119905119895 = 119862119909 (119865119878119900119897119889119887119890119904119905119895 119864119899119867119890) 119895 = 1 2 119899 (21)

35 Procedure of ISSA Thepseudocode of SSA is provided inAlgorithm 2

4 Experimental Results and Analysis

The performance of proposed ISSA is verified and comparedwith five nature-inspired optimization algorithms includingthe basic SSA PSO [12] fruit fly optimization algorithm

6 Complexity

Set 119868119905119890119903119898119886119909119873119875 n 119875119889119901119898119886119909 119875119889119901119898119894119899 119904119891 119866119888 119865119878119880 and 119865119878119871Randomly initialize the flying squirrels locations119865119878119894119895 = 119865119878119871 + rand () lowast (119865119878119880 minus 119865119878119871) 119894 = 1 2 119873119875 119895 = 1 2 119899Calculate fitness value119891119894 = 119891119894(1198651198781198941 1198651198781198942 119865119878119894119899) 119894 = 1 2 119873119875while 119868119905119890119903 lt 119868119905119890119903119898119886119909 [119904119900119903119905119890119889 119891 119904119900119903119905119890 119894119899119889119890119909] = 119904119900119903119905(119891)119865119878ℎ119905 = 119865119878(119904119900119903119905119890 119894119899119889119890119909(1))119865119878119886119905(1 3) = 119865119878(119904119900119903119905119890 119894119899119889119890119909(2 4))119865119878119899119905(1119873119875 minus 4) = 119865119878(119904119900119903119905119890 119894119899119889119890119909(5119873119875))Generate new locations119875119889119901 = (119875119889119901119898119886119909 minus 119875119889119901119898119894119899) times (1 minus 119868119905119890119903119868119905119890119903119898119886119909 )10 + 119875119889119901119898119894119899for t = 1 n1 (n1 = total number of squirrels on acorn trees)

if 1198771 ge 119875119889119901 119865119878119899119890119908119886119905 = 119865119878119900119897119889119886119905 + 119889119892119866119888(119865119878119900119897119889ℎ119905 minus 119865119878119900119897119889119886119905 )else 119865119878119899119890119908119886119905 = 119862119909(119865119878119900119897119889119886119905 119864119899119867119890)end

endfor t = 1 n2 (n2 = total number of squirrels on normal trees moving towards acorn trees)

if 1198772 ge 119875119889119901 119865119878119899119890119908119899119905 = 119865119878119900119897119889119899119905 + 119889119892119866119888(119865119878119900119897119889119886119905 minus 119865119878119900119897119889119899119905 )else 119865119878119899119890119908119899119905 = 119862119909(119865119878119900119897119889119899119905 119864119899119867119890)end



end

119878119905119888 = radicsum119899119896=1 (119865119878119905119886119905119896 minus 119865119878ℎ119905119896)2 119878119888119898119894119899 = 10119864 minus 6365119868119905119890119903(119868119905119890119903max)25

if 119878119905119888 lt 119878119888119898119894119899 119865119878119899119890119908119899119905 = 119865119878119871 + L evy(n) times (119865119878119880 minus 119865119878119871)endCalculate fitness value of new locations119891119899119890119908119894 = 119891119894 (1198651198781198991198901199081198941 1198651198781198991198901199081198942 119865119878119899119890119908119894119899 ) 119894 = 1 2 119873119875if 119891119899119890119908119894 lt 119891119894 119865119878119894 = 119865119878119899119890119908119894119891119894 = 119891119899119890119908119894endEnhance intensive dimensional searchFind 119865119878119887119890119904119905 119891119887119890119904119905for j = 1n 119865119878119899119890119908119887119890119904119905119895 = 119862119909(119865119878119887119890119904119905119895 119864119899119867119890)Calculate fitness value of the new solution119891119899119890119908119887119890119904119905 = 119891(1198651198781198871198901199041199051 1198651198781198871198901199041199052 119865119878119899119890119908119887119890119904119905119895 119865119878119887119890119904119905119899)

if 119891119899119890119908119887119890119904119905 lt 119891119887119890119904119905 119865119878119887119890119904119905119895 = 119865119878119899119890119908119887119890119904119905119895119891119887119890119904119905 = 119891119899119890119908119887119890119904119905end

end 119868119905119890119903 = 119868119905119890119903 + 1end

Algorithm 2 Pseudocode of basic ISSA

Complexity 7

Table 1 Parametric settings of algorithms

Parameter ISSA SSA PSO CMFOA IFFO FOA119868119905119890119903119898119886119909 10000 10000 10000 10000 10000 10000119873119875 50 50 50 50 50 50119866119888 19 19 - - - -119904119891 18 18 - - - -119875119889119901119898119886119909 01 - - - - -119875119889119901119898119894119899 0001 - - - - -119875119889119901 - 01 - - - -1198621 and 1198622 - - 2 - - -119908 - - 09 - - -119864119899 119898119886119909 - - - (119880119861 minus 119880119871)4 - -120582119898119886119909 - - - - (119880119861 minus 119880119871)2 -120582119898119894119899 - - - - 000001 -119903119886119899119889119881119886119897119906119890 - - - - - 1

Table 2 Unimodal benchmark functions

Function Range Fmin

F1(119909) = 119899sum119894=1

1198941199092119894 [minus10 10] 0

F2(119909) = 119899sum119894=2

119894 (21199092119894 minus 119909119894minus1)2 + (1199091 minus 1)2 [minus10 10] 0

F3(119909) = minusexp(minus05 119899sum119894=1

1199092119894) [minus1 1] -1

F4(119909) = 119899sum119894=1

(106)(119894minus1)(119899minus1) 1199092119894 [minus100 100] 0

F5(119909) = 119899sum119894=1

1198941199094119894 + rand () [minus128 128] 0

F6(119909) = 119899minus1sum119894=1

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

F7(119909) = 119899sum119894=1

( 119894sum119895=1

1199092119895) [minus100 100] 0

F8(119909) = max 10038161003816100381610038161199091198941003816100381610038161003816 1 le 119894 le 119899 [minus100 100] 0

F9(119909) = 119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 119899prod119894=1

10038161003816100381610038161199091198941003816100381610038161003816 [minus10 10] 0

F10(119909) = 119899sum119894=1

1199092119894 [minus100 100] 0

F11(119909) = 119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816119894+1 [minus1 1] 0

(FOA) [31] and its two variations improved fruit fly opti-mization algorithm (IFFO) [35] and cloud model basedfly optimization algorithm (CMFOA) [36] 32 benchmarkfunctions are tested with a dimension being equal to 30 50or 100 These functions are frequently adopted for validatingglobal optimization algorithms among which F1-F11 areunimodal F13-F25 belong to multimodal and F26-F32 arecomposite functions in the IEEE CEC 2014 special section[43] Each function is calculated for ten independent runs inorder to better compare the results of different algorithms

Common parameters are set the same for all algorithmssuch as population size NP = 50 maximal iteration number119868119905119890119903119898119886119909 = 10000 Meanwhile the same set of initial randompopulations is used The algorithm-specific parameters arechosen the same as those used in the literature that introducesthe algorithm at the first time The parameters of PSO FOAIFFO CMFOA and SSA are chosen according to [12] [31][35] [36] and [37] respectively Table 1 summarizes bothcommon and algorithm-specific parameters for ISSA andother five algorithms The error value defined as (f (x) ndash

8 Complexity

Fmin) is recorded for the solution x where f (x) is the optimalfitness value of the function calculated by the algorithmsand Fmin is the true minimal value of the function Theaverage and standard deviation of the error values over allindependent runs are calculated

41 Test 1 Unimodal Functions Unimodal benchmark func-tions (Table 2) have one global optimum only and theyare commonly used for evaluating the exploitation capacityof optimization algorithms Tables 3ndash5 list the mean errorand standard deviation of the results obtained from eachalgorithm after ten runs at dimension n = 30 50 and 100respectively The best values are highlighted and markedin italic It is noted that difficulty in optimization ariseswith the increase in the dimension of a function becauseits search space increases exponentially [45] It is clear fromthe results that on most of unimodal functions ISSA hasbetter accuracy and convergence precision than other fivecounterpart algorithms which confirms that the proposedISSA has good exploitation ability As for F2 and F5 ISSA canobtain the same level of accurate mean error as IFFO whilethe former outperforms the latter under the condition of n =100 It is also found that both ISSA and CMFOA can achievethe true minimal value of F3 at n = 30 and 50 while ISSA issuperior at n = 100

Figures 1ndash3 show several representative convergencegraphs of ISSA and its competitors at n = 30 50 and 100respectively It can be observed that ISSA is able to convergeto the true value for most unimodal functions with thefastest convergence speed and highest accuracy while theconvergence results of PSO and FOA are far from satisfactoryThe IFFO and CMFOA with the improvements of searchradius though yield better convergence rates and accuracyin comparison with FOA but still cannot outperform theproposed ISSA It is also found that ISSA greatly improvesthe global convergence ability of SSA mainly because ofthe introduction of an adaptive strategy of 119875119889119901 a selectionstrategy between successive positions and enhancementin dimensional search In addition the accuracy of allalgorithms tends to decrease as the dimension increasesparticularly on F6 and F11

42 Test 2 Multimodal Functions Different from the uni-modal functions multimodal functions have one globaloptimal solution and multiple local optimal solutions andthe number of local optimal solutions exponentially increaseswith the increase of dimension This feature makes themsuitable for testing the exploration ability of an algorithmDetails of these multimodal functions are listed in Table 6The recorded results of statistical analysis over 10 inde-pendent runs are presented in Tables 7ndash9 for n = 3050 and 100 respectively It is revealed from these tablesthat ISSA is superior on F12 F13 F14 F16 F19 and F24regardless of dimension number On other functions ISSAtends to have comparable level of accuracy with some ofits competitors For example both ISSA and CMFOA areable to obtain the exact optimal solution of F21 and F22both ISSA and SSA have the same level of accuracy onF15 F18 and F23 It is noticeable that ISSA tends to get

better performance in accuracy on more functions as thedimension number increases This is mainly contributed bynormal cloud model based flying squirrelsrsquo random positiongeneration and dimensionally enhanced search These twostrategies can help the flying squirrels to escape from localoptimal

Figures 4ndash6 show the recorded convergence charac-teristics of algorithms for several multimodal benchmarkfunctions at n = 30 50 and 100 respectively It is evidentthat ISSA offers better global convergence rate and precisionin comparison with other five algorithms among which bothPSO and FOA are easy to be trapped to the local optimal andthe rest three algorithms (IFFO CMFOA and SSA) producefair convergence rates It is interesting to note that SSAbecomes much poorer as the dimension number increaseswhile ISSA still has excellent exploration ability and itsconvergence curve ranks No 1 at all iterations in the case of n= 100This is due to the incorporation of attributes regardingnormal cloud model generators and search enhancement oneach dimension

43 Test 3 CEC 2014 Benchmark Functions Next the bench-mark functions used in IEEE CEC 2014 are considered forinvestigating the balance between exploration and exploita-tion of optimization algorithms These functions includeseveral novel basic problems (eg with shifting and rotation)and hybrid and composite test problems In the presenttest seven CEC 2014 functions are selected with at leastone function in each group and the details are providedin Table 10 Statistical results obtained by different algo-rithms through 10 independent runs are recorded in Tables11ndash13 It is worth mentioning that CEC 2014 functions arespecifically designed to have complicated features and thusit is difficult to reach the global optimal for all algorithmsunder consideration Nevertheless in contrast to other fivealgorithms ISSA is able to get highly competitive results formost CEC 2014 functions in Table 10 especially at higherdimension number As a matter of fact ISSA always hasthe best solution at n = 100 although the solution is stillfar away from optimal The results of convergence studies(Figures 7ndash9) show that ISSAhas promising convergence per-formance with the comparison of other five algorithms Thesuperior performance of the proposed ISSA is mainly ben-efited from an equilibrium between global and local searchabilities because of the use of the four strategies describedin Section 3

44 Statistical Analysis In order to analyze the performanceof any two algorithms the most frequently used nonpara-metric statistical test Wilcoxonrsquos test [46] is considered forthe present work and results are summarized in Tables 14ndash16for n = 30 50 and 100 respectively The test is carriedout by considering the best solution of each algorithm oneach benchmark function with 10 independent runs and asignificance level of120572 =005 InTables 14ndash16 lsquo+rsquo sign indicatesthat the reference algorithm outperforms the compared one

Complexity 9

Table3Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

22612E-46

14374E

-13

18031E+0

338546

E-22

53419E-12

58887E+

01Std

39697E-46

10187E

-13

16038E

+02

1146

4E-22

26506

E-12

10717E

+01

F2Mean

53333E-01

600

00E-01

70475E

+04

666

67E-01

666

67E-01

12221E+0

2Std

28109E-01

21082E-01

18027E

+04

11102E

-16

364

14E-11

35452E+

01F3

Mean

000

00E+

00000

00E+

0047300

E-01

000

00E+

00860

42E-14

18586E

-02

Std

18504E

-1626168E-16

42225E-02

906

49E-17

27669E-14

19010E

-03

F4Mean

21268E-39

39943E-10

92617E

+07

37704

E-18

20345E-08

11112

E+07

Std

564

86E-39

32855E-10

18784E

+07

24165E-18

14277E

-08

30272E+

06F5

Mean

43164

E-03

93054E

-02

55376E+

0032231E-03

22754E-03

17965E

-02

Std

19931E-03

23058E-02

10210E

+00

13321E-03

1104

6E-03

31904

E-03

F6Mean

55447E-14

29695E+

0110

669E

+07

76347E

+00

89052E+

0012

862E

+04

Std

54894E-14

340

74E+

0118

313E

+06

590

03E+

0071150E

+00

44338E+

03F7

Mean

11996E

-44

65616E-12

16688E

+05

71055E

-22

60744

E-12

54708E+

03Std

304

49E-44

540

85E-12

17265E

+04

16506E

-22

29515E-12

49070E+

02F8

Mean

35080E-13

17850E

-03

45639E+

0127020E-11

264

40E-06

78561E+0

0Std

70894E

-1343281E-04

20578E+

00604

13E-12

24822E-07

17334E

+00

F9Mean

55772E-24

22148E-07

71792E

+01

23880E-11

204

15E-06

304

53E+

01Std

79227E

-24

67302E-08

30539E+

0141360

E-12

36636E-07

46515E+

01F10

Mean

26748E-44

44745E-13

13098E

+04

42105E-23

440

42E-13

36501E+

02Std

78461E-44

37055E-13

13116

E+03

10825E

-23

15887E

-13

43608E+

01F11

Mean

61803E-188

17833E

-60

18391E-03

78265E

-25

5117

6E-15

67271E-07

Std

000

00E+

0037202E-60

11147E

-03

65934E-25

76076E

-15

46836E-07

10 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

19373E

-45

244

08E-07

89004

E+03

36571E-21

32632E-11

28288E+

02Std

40611E

-45

72161E-08

10186E

+03

90310E

-22

82802E-12

18277E

+01

F2Mean

666

67E-01

21716E+

0080535E+

0512

514E

+00

666

67E-01

82661E+

02Std

82003E-16

22142E+

0011670E

+05

12485E

+00

25423E-10

77814E

+01

F3Mean

000

00E+

0076

318E

-11

866

02E-01

000

00E+

004110

0E-13

51246

E-02

Std

22204E-16

22120E-11

18360E

-02

23984E-16

14800E

-13

40430E-03

F4Mean

306

71E-38

97033E

-04

46756E+

0819

432E

-17

10621E-07

38893E+

07Std

69186E-38

344

64E-04

60135E+

0711627E

-17

76083E

-08

73301E+0

6F5

Mean

71557E

-03

29566

E-01

53164

E+01

10084E

-02

73580E

-03

10458E

-01

Std

23021E-03

33200

E-02

64915E+

0026523E-03

18100E

-03

26586E-02

F6Mean

43706

E-11

95471E+0

170

118E+

0777

331E+0

147194E+

0165331E+

04Std

95151E-11

35358E+

0153302E+

06344

63E+

0140976E+

0113

721E+0

4F7

Mean

12947E

-41

23079E-05

91659E

+05

69771E-21

64025E-11

28862E+

04Std

37876E-41

58814E-06

93287E

+04

31808E-21

26901E-11

28375E+

03F8

Mean

60872E-11

12706E

-01

67093E+

0184930E-11

71576E

-06

11919E

+01

Std

25158E-11

33391E-02

25011E

+00

11107E

-11

69032E-07

1040

4E+0

0F9

Mean

18289E

-23

38822E-04

20565E+

1060338E-11

63442E-06

11434E

+05

Std

28884E-23

72525E

-05

63864

E+10

64165E-12

90797E

-07

24588E+

05F10

Mean

12924E

-44

14594E

-06

41629E+

0422846

E-22

20175E-12

10536E

+03

Std

25807E-44

606

62E-07

37125E+

0341424E-23

52761E-13

59785E+

01F11

Mean

44745E-163

19208E

-58

92852E

-03

11169E

-24

51458E-15

16917E

-06

Std

000

00E+

0022612E-58

35776E-03

14674E

-24

82130E-15

94517E

-07

Complexity 11


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

52002E-44

1360

1E-01

64559E+

0467630E-20

540

42E-10

23688E+

03Std

89565E-44

28398E-02

27675E+

0318

636E

-20

10843E

-10

11782E

+02

F2Mean

25837E+

0028507E+

0189058E+

0610

018E

+01

11850E

+01

13921E+0

4Std

40923E+

0012

482E

+01

64125E+

0598

260E

+00

67378E+

0021479E+

03F3

Mean

19984E

-1517

506E

-05

99937E

-01

19984E

-1534570E-12

1946

0E-01

Std

27940E-16

33607E-06

19874E

-04

39686E-16

57323E-13

11259E

-02

F4Mean

14073E

-38

30139E+

0226151E+

0914

261E-16

10212E

-06

20499E+

08Std

15382E

-38

90551E+0

126783E+

0875

737E

-17

33853E-07

35388E+

07F5

Mean

17615E

-02

14345E

+00

59603E+

0237550E-02

29349E-02

12443E

+00

Std

42239E-03

13985E

-01

58412E+

0112

602E

-02

45825E-03

18471E-01

F6Mean

11417E

+01

58422E+

0242299E+

0817

988E

+02

16578E

+02

560

78E+

05Std

30258E+

0197

884E

+02

48581E+

0739022E+

01466

85E+

0165477E+

04F7

Mean

16881E-41

12984E

+01

64707E+

0611852E

-19

74831E-10

22865E+

05Std

34134E-41

22729E+

0033435E+

0531718E-20

95033E

-11

20650E+

04F8

Mean

45259E-08

39819E+

0085137E+

0117

042E

-04

29244

E-05

33956E+

01Std

29104E-08

41522E-01

12566E

+00

78665E

-05

27018E-06

47713E+

00F9

Mean

12222E

-22

36814E-01

72469E

+32

25070E-10

23795E-05

14112

E+27

Std

84369E-23

41963E-02

20633E+

3323874E-11

13879E

-06

44627E+

27F10

Mean

34254E-42

37261E-01

14940E

+05

20714E-21

18486E

-11

43041E+

03Std

966

08E-42

92561E-02

446

43E+

03464

07E-22

23956E-12

24348E+

02F11

Mean

1640

0E-12

315

040E

-52

37278E-02

65780E-24

3117

4E-14

10720E

-05

Std

51861E-123

16670E

-52

11165E

-02

72960E

-24

36245E-14

59475E-06

12 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

(b) F4

0 2000 4000 6000 8000 10000

0

5

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus15

minus5

(c) F6

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

minus50

(d) F7

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus2

minus4

minus6

minus8

minus10

minus12

minus14

(e) F8

0 2000 4000 6000 8000 10000

05

1015

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus5

minus10

minus20

minus25

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

Mea

n Er

rors

(log)

(h) F11

Figure 1 Convergence rate comparison for representative unimodal functions (n = 30)

Complexity 13

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus10

minus5

0

5

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

0

10

20

30

Mea

n Er

rors

(log)

2000 4000 6000 8000 100000Iteration

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

50

Mea

n Er

rors

(log)

(h) F11


14 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

20

Mea

n Er

rors

(log)

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

0

2

4

6

8

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

01020304050

Mea

n Er

rors

(log)

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus140

minus120

minus100

minus80

minus60

minus40

minus20

020

Mea

n Er

rors

(log)

(h) F11


Complexity 15

Table6Multim

odalbenchm

arkfunctio

ns

Functio

nRa

nge

Fmin

F12(119909)=

minus20exp(minus0

2radic1 119899119899 sum 119894=11199092 119894)minus

exp(1 119899119899 sum 119894=1co

s(2120587119909 119894))

+20+exp

(1 )[minus32

32]0

F13(119909)=

119899 sum 119894=11003816 1003816 1003816 1003816119909 119894sin(119909 119894)

+01119909 1198941003816 1003816 1003816 1003816

[minus1010]

0

F14(119909)=

119891 119904(1199091119909 2)

+sdotsdotsdot+119891 119904

(119909 119899119909 1)

[minus10010

0]0

119891 119904(119909119910)=

(1199092 +1199102 )025[sin2

(50(1199092 +

1199102 )01)+1

]F15(

119909)=119891 119904(1199091119909 2)

+sdotsdotsdot+119891 119904(

119909 1198991199091)

[minus10010

0]0

119891 119904(119909119910)=

05(sin2(radic 1199092+1199102

)minus05)

(1+0001

(1199092 +1199102 ))2

F16(119909)=

120587 11989910sin2

(120587119910 119894)+119899minus1 sum 119894=1

(119910 119894minus1 )2 [

1+10sin2

(120587119910 119894+1)]+

(119910 119899minus1 )2

+119899 sum 119894=1119906(119909 119894

10100

4)[minus50

50]0

119910 119894=1+1 4(119909

119894+1)

119906(119909 119894119886

119896119898)= 119896(119909 119894

minus119886)119898

119909 119894gt119886

0minus119886le

119909 119894le119886

119896(minus119909119894minus119886)119898

119909119894gt119886

F17(119909)=

1 4000119899 sum 119894=11199092 119894minus119899 prod 119894=1

cos(119909119894 radic 119894)+1

[minus10010

0]0

F18(119909)=

minus119899minus1 sum 119894=1(exp

(minus(1199092 119894+

1199092 119894+1+05

119909 119894119909 119894+1)

8)lowastc

os(4radic

1199092 119894+1199092 119894+1

+05119909 119894119909 119894+1))

[minus55]

1-n

F19(119909)=

119899 sum 119894=1(119909119894minus1)2

minus119899 sum 119894=2119909 119894119909 119894minus1

[minusn2n2 ]

119899(119899+4)(119899

minus1)minus6

F20 (119909 )=

sum119899minus1 119894=2(05

+(sin2(radic 1

001199092 119894+1199092 119894+1)minus0

5))(1+

0001(1199092 119894minus

2119909 119894119909119894minus1+1199092 119894minus1))2

[minus10010

0]0

F21(119909)=

119899 sum 119894=1[1199092 119894minus10

cos(2120587

119909 119894)+10]

[minus51251

2]0

F22(119909)=

119899 sum 119894=1[1199102 119894minus10

cos(2120587

119910 119894)+10]

119910 119894= 119909 119894

1003816 1003816 1003816 1003816119909 1198941003816 1003816 1003816 1003816lt05

119903119900119906119899119889(2119909

119894)2

1003816 1003816 1003816 10038161199091198941003816 1003816 1003816 1003816lt0

5[minus51

2512]

0

F23(119909)=

1minuscos(2120587

radic119899 sum 119894=11199092 119894)

+01radic119899 sum 119894=1

1199092 119894[minus10

0100]

0

F24(119909)=

119899 sum 119894=1119896119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 (119909119894+05

))]minus119899119896

119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 05

)][minus05

05]0

F25(119909)=

119899 sum 119896=1119899 sum 119895=1((1199102 119895119896 4000)minus

cos(119910 119895119896)+1

)119910119895119896=10

0(119909 119896minus1199092 119895

)2 +(1minus

1199092 119895)2[minus10

0100]

0

16 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmultim

odalbenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

51692E-14

21708E-07

16343E

+01

42641E-12

43970E-07

70983E

+00

Std

94813E-15

11785E

-07

45830E-01

51275E-13

53024E-08

45755E+

00F13

Mean

19651E-15

17670E

-07

30865E+

0138781E-12

12507E

-06

29660

E+01

Std

17016E

-1510

899E

-07

28749E+

00200

14E-12

19125E

-06

57790E+

00F14

Mean

28586E-11

47414E-02

21576E+

0235954E-05

17290E

-02

18705E

+02

Std

17874E

-1118

105E

-02

50836E+

0019

343E

-06

10857E

-03

50868E+

01F15

Mean

99552E

-01

46150E-01

12596E

+01

94983E

-01

10032E

+00

12147E

+01

Std

38926E-01

33522E-01

21495E-01

42966

E-01

35690E-01

17388E

-01

F16

Mean

15705E

-32

13069E

-15

56725E+

0650290E-25

99726E

-15

31482E+

00Std

28850E-48

57169E-16

17168E

+06

47027E-25

85374E-15

58054E-01

F17

Mean

13781E-02

10332E

-02

43352E+

0044332E-03

12793E

-02

10971E+0

0Std

14865E

-02

12632E

-02

42518E-01

79408E

-03

10155E

-02

10766E

-02

F18

Mean

50849E+

0038253E+

0020946

E+01

49225E+

00490

48E+

0021497E+

01Std

16014E

+00

14627E

+00

76856E

-01

21737E+

00204

11E+0

013

669E

+00

F19

Mean

268

41E-07

19292E

+02

49808E+

0519

677E

+02

240

98E+

0230226E+

04Std

32619E-08

15971E+0

214

706E

+05

16572E

+02

23149E+

0260289E+

03F2

0Mean

25989E-07

47006

E-06

33592E-02

44469E-08

18865E

-07

1540

6E-01

Std

59383E-07

73387E

-06

22456E-02

10350E

-07

31612E-07

56719E-02

F21

Mean

000

00E+

0070

841E-13

25769E+

02000

00E+

0045409E-11

30881E+

02Std

000

00E+

0045361E-13

90973E

+00

000

00E+

0019

882E

-11

27305E+

01F2

2Mean

000

00E+

007746

7E-13

23335E+

02000

00E+

00644

03E-11

25509E+

02Std

000

00E+

0036979E-13

15942E

+01

000

00E+

0033820E-11

26992E+

01F2

3Mean

93987E

-01

52987E-01

12199E

+01

13599E

+00

14399E

+00

21878E+

00Std

21705E-01

12517E

-01

49304

E-01

36576E-01

21705E-01

62731E-02

F24

Mean

14921E-14

37233E-04

32412E+

0147458E-09

42553E-03

26924E+

01Std

17226E

-1498

846E

-05

11649E

+00

28242E-09

42975E-04

35559E+

00F2

5Mean

29494E+

0110

724E

+02

11372E

+07

404

62E+

0193530E+

0092

421E+0

3Std

29743E+

0151800

E+01

31606

E+06

39685E+

0190392E+

0018

838E

+03

Complexity 17


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

85798E-14

24174E-04

18459E

+01

7404

4E-12

73673E

-07

82226E+

00Std

17360E

-1455274E-05

1944

7E-01

88139E-13

80222E-08

42517E+

00F13

Mean

22538E-15

29492E-04

71594E

+01

21041E-11

32004

E-06

60959E+

01Std

18688E

-1510

372E

-04

45394E+

0015

865E

-11

15334E

-06

44766

E+00

F14

Mean

71759E

-1120261E+

0043430E+

0277682E

-05

33324E-02

42669E+

02Std

24650E-11

50770E-01

14055E

+01

54975E-06

10537E

-03

80127E+

01F15

Mean

16716E

+00

12749E

+00

22241E+

0116

927E

+00

14937E

+00

21617E+

01Std

76572E

-01

43985E-01

33014E-01

47677E-01

63574E-01

54534E-01

F16

Mean

94233E-33

13057E

-09

76995E

+07

17755E

-24

846

48E-14

69921E+

00Std

14425E

-48

37533E-10

21712E+

0719

092E

-24

17429E

-13

89129E-01

F17

Mean

76377E

-03

14219E

-02

1160

6E+0

164039E-03

10080E

-02

1264

1E+0

0Std

57418E-03

21089E-02

46282E-01

70807E

-03

13952E

-02

16555E

-02

F18

Mean

83103E+

0079

047E

+00

39689E+

0189467E+

0096

041E+0

038726E+

01Std

260

72E+

0025432E+

0077616E

-01

78506E

-01

21029E+

0013

015E

+00

F19

Mean

45562E+

0126833E+

04806

68E+

0616

118E+

0413

155E

+04

70015E

+05

Std

38094E+

0121743E+

0421709E+

0612

498E

+04

1300

9E+0

497

174E

+04

F20

Mean

43064E-08

25702E-04

11519E

-01

52365E-08

16998E

-06

500

47E-01

Std

44294E-08

27576E-04

39417E-02

95247E

-08

49881E-06

26305E-01

F21

Mean

000

00E+

0011310E

-06

53146

E+02

000

00E+

0023711E

-10

58748E+

02Std

000

00E+

0033614E-07

32117E+

01000

00E+

0045437E-11

29507E+

01F2

2Mean

000

00E+

0016

167E

-06

48729E+

02000

00E+

00244

07E-10

52060

E+02

Std

000

00E+

0063216E-07

24382E+

01000

00E+

0075

889E

-11

42230E+

01F2

3Mean

13699E

+00

89987E-01

21237E+

0122699E+

0025899E+

0035955E+

00Std

23594E-01

666

67E-02

58033E-01

41913E-01

62973E-01

12247E

-01

F24

Mean

71054E

-1426826E-02

63090E+

0119

033E

-08

96037E

-03

47263E+

01Std

27621E-14

47780E-03

22392E+

0061075E-09

97071E-04

52689E+

00F2

5Mean

66563E+

0184722E+

0211275E

+08

65780E+

0139992E+

0188242E+

04Std

10992E

+02

2113

8E+0

221091E+

0794

954E

+01

43819E+

0116

832E

+04

18 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

18989E

-1316

584E

-01

19996E

+01

17809E

-11

14744E

-06

13554E

+01

Std

20566

E-14

53720E-02

90319E

-02

19159E

-12

18930E

-07

60821E+

00F13

Mean

22871E-15

17736E

-01

1944

7E+0

213

452E

-10

13291E-05

16379E

+02

Std

26741E-15

53611E-02

62653E+

0038592E-11

55001E-06

13313E

+01

F14

Mean

18736E

-1074

259E

+01

10132E

+03

22866

E-04

83534E-02

95534E

+02

Std

37223E-11

19144E

+01

18986E

+01

14283E

-05

10592E

-02

53523E+

01F15

Mean

26814E+

0010

178E

+01

47083E+

0128083E+

0034325E+

0045859E+

01Std

73851E-01

16238E

+00

22513E-01

46148E-01

60283E-01

69914E-01

F16

Mean

47116E-33

244

54E-04

90382E

+08

81890E-24

62347E-14

27647E+

03Std

72124E

-49

59650E-05

64985E+

0767958E-24

55604

E-14

44231E+

03F17

Mean

34494E-03

11896E

-02

37816E+

0134509E-03

41885E-03

21280E+

00Std

60565E-03

65363E-03

15922E

+00

46765E-03

86153E-03

54359E-02

F18

Mean

18033E

+01

17806E

+01

86826E+

0118

319E

+01

18828E

+01

82458E+

01Std

19652E

+00

38319E+

0093

222E

-01

29296E+

0025377E+

0015

159E

+00

F19

Mean

82462E+

0427944

E+06

48046

E+08

28415E+

0560265E+

0549201E+

07Std

55732E+

0489703E+

0596

715E

+07

24572E+

0527137E+

0572

772E

+06

F20

Mean

57130E-07

81688E-03

96848E

-01

13631E-06

27143E-05

21656E+

00Std

61122E-07

53195E-03

44542E-01

25155E-06

58766

E-05

80368E-01

F21

Mean

000

00E+

0051414E+

0013

305E

+03

000

00E+

0020026E-09

13623E

+03

Std

000

00E+

0017

825E

+00

22890E+

01000

00E+

0032815E-10

609

96E+

01F2

2Mean

000

00E+

0077

848E

+00

1260

9E+0

3000

00E+

0020383E-09

12745E

+03

Std

000

00E+

0023732E+

0029100

E+01

000

00E+

0029753E-10

59708E+

01F2

3Mean

25599E+

0020499E+

0039804

E+01

47099E+

0043699E+

0073

691E+0

0Std

36878E-01

15092E

-01

69296E-01

59151E-01

56184E-01

17989E

-01

F24

Mean

40927E-13

26229E+

0015

145E

+02

18874E

-07

29476E-02

10478E

+02

Std

88061E-14

63367E-01

42830E+

0037074E-08

17697E

-03

11873E

+01

F25

Mean

42987E+

028117

8E+0

313

524E

+09

56790E+

0244982E+

0218

038E

+06

Std

43423E+

0233128E+

0278

399E

+07

54327E+

0246926E+

0221315E+

05

Complexity 19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus8

minus6

minus4

minus2

02468

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(f) F24

Figure 4 Convergence rate comparison for representative multimodal functions (n = 30)

lsquo-rsquo sign indicates that the reference algorithm is inferior tothe compared one and lsquo=rsquo sign indicates that both algorithmshave comparable performances The results of last row of thetables show that the proposed ISSA has a larger number of lsquo+rsquocounts in comparison to other algorithms confirming thatISSA is better than the other five compared algorithms under95 level of significance

The quantitative analysis is also carried out for six algo-rithms with an index of mean absolute error (MAE) which isan effective performance index for ranking the optimizationalgorithms and is defined by [47]

MAE = sum119873119895=1 10038161003816100381610038161003816119898119895 minus 11989611989510038161003816100381610038161003816119873 (22)

20 Complexity

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

ISSASSAPSO

CMFOAIFFOFOA

1

2

3

4

5

6

7

8

Mea

n Er

rors

(log)

0 4000 6000 8000 100002000Iteration

(e) F19

ISSASSAPSO

CMFOAIFFOFOA

20000 6000 8000 100004000Iteration

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24


Table 10 CEC 2014 benchmark functions

Function Range FminF26 (CEC1 Rotated High Conditioned Elliptic Function) [minus100 100] 100F27 (CEC2 Rotated Bent Cigar Function) [minus100 100] 200F28 (CEC4 Shifted and Rotated Rosenbrockrsquos Function) [minus100 100] 400F29 (CEC17 Hybrid Function 1) [minus100 100] 1700F30 (CEC23 Composition Function 1) [minus100 100] 2300F31 (CEC24 Composition Function 2) [minus100 100] 2400F32 (CEC25 Composition Function 3) [minus100 100] 2500

Complexity 21

Table11

Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

31365E+

0413

133E

+06

11295E

+08

10017E

+06

864

75E+

0513

449E

+07

Std

18602E

+04

52974E+

0531387E+

0748689E+

0548607E+

0528405E+

06F2

7Mean

304

00E-10

1844

6E+0

484500

E+09

10512E

+04

12359E

+04

58535E+

08Std

61535E-10

14049E

+04

10125E

+09

12485E

+04

11922E

+04

38771E+

07F2

8Mean

42105E-01

46710E+

0173

819E

+02

4114

7E+0

137814E+

0114

226E

+02

Std

12624E

+00

31490E+

0199

455E

+01

47336E+

0134110E+

0129201E+

01F2

9Mean

75177E

+03

14891E+0

529286E+

0647277E+

0531099E+

0539826E+

05Std

33119E+

0368316E+

049190

4E+0

521021E+

0522686E+

0515

511E+0

5F3

0Mean

31524E+

0231524E+

0238129E+

0231524E+

0231524E+

0232568E+

02Std

85708E-12

19710E

-07

14082E

+01

11524E

-1145680E-11

58955E+

00F31

Mean

23483E+

0223172E+

0230117E+

0223811E

+02

23858E+

0224179E+

02Std

41748E+

0072

461E+0

048903E+

00560

97E+

0050249E+

0090

228E

+00

F32

Mean

20790E+

02206

03E+

0221884E+

0221485E+

0220975E+

0220633E+

02Std

41618E+

0032456E+

0030353E+

0087909E+

0057719E+

0016

880E

+00

22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof

WilcoxonrsquostestforISSAagainsto

ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24


where119898119895 is the mean of optimal values 119896119895 is the actual globaloptimal value and119873 is the number of samples In the presentwork119873 is the number of benchmark functions TheMAE ofall algorithms and their ranking for all functions are given inTable 17 It is clear to find that ISSA ranks No 1 and providesthe minimum MAE in all cases ISSA reaches the optimumsolution 653 times out of 960 runs (10 runs for each testfunction for n = 30 50 and 100 respectively) and comes in

the first rank as shown in Figure 10 It is concluded that ISSAprovides the best performance in comparison to other fiveoptimization algorithms

5 Conclusions

The SSA is a new algorithm for searching global optimizationbased on the food foraging behavior of the flying squirrels

28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29

Figure 7 Convergence rate comparison for representative CEC 2014 functions (n = 30)

Table 17 Ranking of algorithms using MAE

Algorithm MAE (n=30) MAE (n=50) MAE (n=100) RankISSA 12399E+03 96479E+03 40880E+04 1CMFOA 37162E+04 87565E+04 43758E+05 2IFFO 46305E+04 10037E+05 58554E+05 3SSA 46440E+04 10550E+05 17913E+06 4FOA 19074E+07 55874E+07 44101E+25 5PSO 27147E+08 14513E+09 22646E+31 6

To balance the exploration and exploitation capacities of theSSA an improved SSA (ISSA) is proposed in this paperby introducing four strategies namely an adaptive predatorpresence probability a normal cloudmodel generator a selec-tion strategy between successive positions and enhancementin dimensional search The adaptive strategy of predatorpresence probability assists to reach the potential searchregion at the early stage and then to implement the localsearch at the later stage The normal cloud model is utilizedto describe the randomness and fuzziness of the glidingbehavior of the flying squirrel population The selectionstrategy enhances the local search ability of an individualflying squirrel In addition enhancing dimensional search

results in a better quality of the best solution in eachiteration Extensive comparative studies are conducted for 32benchmark functions (including 11 unimodal functions 14multimodal functions and 7 CEC 2014 functions) Resultsconfirm that the proposed ISSA is a powerful optimiza-tion algorithm and in general significantly outperforms fivestate-of-the-art algorithms (PSO FOA IFFO CMFOA andSSA)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)

(d) F29Figure 9 Convergence rate comparison for representative CEC 2014 functions (n = 100)

30 Complexity

ISSA SSA PSO CMFOA IFFO FOA0

100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287

Figure 10 Comparison of algorithms in finding the global optimalsolution out of 960 runs

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by a research grant from NationalKey Research and Development Program of China (projectno 2017YFC1500400) and the National Natural ScienceFoundation of China (51808147)

References

[1] X S Yang Engineering Optimization An Introduction withMetaheuristic Applications Wiley Publishing 2010

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

[3] A Lopez-Jaimes and C A Coello Coello ldquoIncluding prefer-ences into a multiobjective evolutionary algorithm to deal withmany-objective engineering optimization problemsrdquo Informa-tion Sciences vol 277 pp 1ndash20 2014

[4] R A Monzingo R L Haupt and T W Miller ldquoGradient-based algorithms introduction to adaptive arraysrdquo IET DigitalLibrary pp 153ndash237 2011

[5] R JWilliams andD ZipserGradient-based learning algorithmsfor recurrent networks and their computational complexity Back-propagation L Erlbaum Associates Inc 1995

[6] F Ding and T Chen ldquoGradient based iterative algorithmsfor solving a class of matrix equationsrdquo IEEE Transactions onAutomatic Control vol 50 no 8 pp 1216ndash1221 2005

[7] M Mohammadi and N Ghadimi ldquoOptimal location andoptimized parameters for robust power system stabilizer usinghoneybee mating optimizationrdquo Complexity vol 21 no 1 pp242ndash258 2015

[8] O Abedinia N Amjady andA Ghasemi ldquoA newmetaheuristicalgorithm based on shark smell optimizationrdquo Complexity vol21 no 5 pp 97ndash116 2016

[9] D E Goldberg K Sastry andX Llora ldquoToward routine billion-variable optimization using genetic algorithmsrdquo Complexityvol 12 no 3 pp 27ndash29 2007

[10] J H Holland ldquoGenetic algorithms and the optimal allocationof trialsrdquo SIAM Journal on Computing vol 2 no 2 pp 88ndash1051973

[11] H Beyer and H Schwefel Evolution Strategies ndashA Comprehen-sive Introduction Kluwer Academic Publishers 2002

[12] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 IEEE 2002

[13] D Karaboga and B BasturkA Powerful and Efficient Algorithmfor Numerical Function Optimization Artificial Bee Colony(ABC) Algorithm Kluwer Academic Publishers 2007

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

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

[16] D Bertsimas and J Tsitsiklis ldquoSimulated annealingrdquo StatisticalScience vol 8 no 1 pp 10ndash15 1993

[17] Z Yin J Liu W Luo and Z Lu ldquoAn improved Big Bang-BigCrunch algorithm for structural damage detectionrdquo StructuralEngineering and Mechanics vol 68 no 6 pp 735ndash745 2018

[18] E Rashedi H Nezamabadi-pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 213pp 267ndash289 2010

[19] A F Nematollahi A Rahiminejad and B Vahidi ldquoA novelphysical based meta-heuristic optimization method known asLightning Attachment Procedure Optimizationrdquo Applied SoftComputing vol 59 pp 596ndash621 2017

[20] G Dhiman and V Kumar ldquoSpotted hyena optimizer A novelbio-inspired based metaheuristic technique for engineeringapplicationsrdquoAdvances in Engineering Software vol 114 pp 48ndash70 2017

[21] A Baykasoglu and S Akpinar ldquoWeightedSuperposition Attrac-tion (WSA) A swarm intelligence algorithm for optimizationproblems ndash Part 1 Unconstrained optimizationrdquo Applied SoftComputing vol 56 pp 520ndash540 2017

[22] A Kaveh and A Dadras ldquoA novel meta-heuristic optimizationalgorithm Thermal exchange optimizationrdquo Advances in Engi-neering Software vol 110 pp 69ndash84 2017

[23] A Tabari and A Ahmad ldquoA new optimization method electro-search algorithmrdquo in Computers amp Chemical Engineering vol10 pp 1ndash11 Chem UK 2017

[24] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[25] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Pro-ceedings of the Swarm Intelligence Symposium 2003 Sis lsquo03Proceedings of the IEEE pp 174ndash181 2012

[26] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[27] B Y Qu P N Suganthan and S Das ldquoA distance-based locallyinformed particle swarm model for multimodal optimizationrdquoIEEE Transactions on Evolutionary Computation vol 17 no 3pp 387ndash402 2013

[28] F Zhong H Li and S Zhong ldquoA modified ABC algorithmbased on improved-global-best-guided approach and adaptive-limit strategy for global optimizationrdquo Applied Soft Computingvol 46 pp 469ndash486 2016

Complexity 31

[29] D Kumar and K Mishra ldquoPortfolio optimization using novelco-variance guided Artificial Bee Colony algorithmrdquo Swarmand Evolutionary Computation vol 33 pp 119ndash130 2017

[30] S Ghambari and A Rahati ldquoAn improved artificial bee colonyalgorithm and its application to reliability optimization prob-lemsrdquo Applied Soft Computing vol 62 pp 736ndash767 2018

[31] W T Pan ldquoA new evolutionary computation approach fruitfly optimization algorithmrdquo in Proceedings of the Conference ofDigital Technology and InnovationManagement Taipei Taiwan2011

[32] W-T Pan ldquoUsing modified fruit fly optimisation algorithm toperform the function test and case studiesrdquo Connection Sciencevol 25 no 2-3 pp 151ndash160 2013

[33] L Wang Y Shi and S Liu ldquoAn improved fruit fly optimizationalgorithm and its application to joint replenishment problemsrdquoExpert Systems with Applications vol 42 no 9 pp 4310ndash43232015

[34] X F Yuan X S Dai J Y Zhao and Q He ldquoOn a novel multi-swarm fruit fly optimization algorithm and its applicationrdquoApplied Mathematics and Computation vol 233 pp 260ndash2712014

[35] Q-K Pan H-Y Sang J-H Duan and L Gao ldquoAn improvedfruit fly optimization algorithm for continuous function opti-mization problemsrdquo Knowledge-Based Systems vol 62 pp 69ndash83 2014

[36] T Zheng J Liu W Luo and Z Lu ldquoStructural damageidentification using cloud model based fruit fly optimizationalgorithmrdquo Structural Engineering andMechanics vol 67 no 3pp 245ndash254 2018

[37] M Jain V Singh and A Rani ldquoA novel nature-inspiredalgorithm for optimization Squirrel search algorithmrdquo Swarmand Evolutionary Computation 2018

[38] X-S Yang ldquoA new metaheuristic bat-inspired algorithmrdquo inProceedings of the Nature Inspired Cooperative Strategies forOptimization (NICSO 2010) vol 284 pp 65ndash74 Springer BerlinHeidelberg 2010

[39] I Fister I Fister Jr X-S Yang and J Brest ldquoA comprehensivereview of firefly algorithmsrdquo Swarm and Evolutionary Compu-tation vol 13 no 1 pp 34ndash46 2013

[40] DHWolpert andWGMacready ldquoNo free lunch theorems foroptimizationrdquo IEEE Transactions on Evolutionary Computationvol 1 no 1 pp 67ndash82 1997

[41] D Li H Meng and X Shi ldquoMembership clouds and member-ship clouds generatorrdquo International Journal of ComputationalResearch and Development vol 32 pp 15ndash20 1995

[42] D Li C Liu andWGan ldquoAnew cognitivemodel cloudmodelrdquoInternational Journal of Intelligent Systems vol 24 no 3 pp357ndash375 2009

[43] J Liang B Qu and P Suganthan ldquoProblem definitions andevaluation criteria for the CEC 2014 special session and com-petition on single objective real-parameter numerical optimiza-tionrdquo Zhengzhou China and Technical Report ComputationalIntelligence Laboratory Zhengzhou University Nanyang Tech-nological University Singapore 2014

[44] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NABIC rsquo09) pp 210ndash214 2009

[45] M Jamil and X-S Yang ldquoA literature survey of benchmarkfunctions for global optimisation problemsrdquo International Jour-nal of Mathematical Modelling andNumerical Optimisation vol4 no 2 pp 150ndash194 2013

[46] J Derrac S Garcıa D Molina and F Herrera ldquoA practicaltutorial on the use of nonparametric statistical tests as amethodology for comparing evolutionary and swarm intelli-gence algorithmsrdquo Swarm and Evolutionary Computation vol1 no 1 pp 3ndash18 2011

[47] E Nabil ldquoA modified flower pollination algorithm for globaloptimizationrdquo Expert Systems with Applications vol 57 pp 192ndash203 2016

Hindawiwwwhindawicom Volume 2018

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Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Dierential EquationsInternational Journal of

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Submit your manuscripts atwwwhindawicom

2 Complexity

Unfortunately most of the abovementioned basic meta-heuristic algorithms fail to balance exploration and exploita-tion thereby yielding unsatisfactory performance for real-life complicated optimization problems Exploration standsfor global search ability and ensures the algorithm toreach all over the search space and then to find promisingregions whereas exploitation represents local search abilityand ensures the searching of optimum within the identifiedpromising regions Emphasizing the exploration capabilityonly results in a waste of computational resources on search-ing all over the interior regions of search space and thusreduces the convergence rate emphasizing the exploitationcapability only by contrast causes loss of population diversityearly and thus probably leads to premature convergence or tobe stuck in local optimalThis fact motivates the introductionof various strategies for improving the convergence rateand precision of the basic metaheuristic algorithms As anillustration premature convergence of PSO was preventedin CLPSO by proposing a comprehensive learning strategyto maintain the population diversity [24] a social learningcomponent called fitness-distance-ratio was employed toenhance local search capability of PSO [25] a self-organizinghierarchical PSO with time-varying acceleration coefficients(HPSO-TVAC) was introduced to efficiently control the localsearch and convergence to the global optimal solution [26]a distance-based locally informed PSO (LIPS) enabled thealgorithm to quickly converge to global optimal solutionwith high accuracy [27] Likewise many modified versionshave been proposed to enhance the global search ability ofthe basic ABC an improved-global-best-guide term witha nonlinear adjusting factor was employed to balance theexploration and exploitation [28] a multiobjective covari-ance guided artificial bee colony algorithm (M-CABC) wasproposed to obtain higher precision with quick convergencespeed when solving portfolio problems [29] the slow conver-gence and unsatisfactory solution accuracy were improved inthe variant IABC [30] As for fruit fly optimization algorithm(FOA) [31] an escape parameter was introduced in MFOAto escape from the local solution [32] and modified versionsfor balancing between exploration and exploitation abilitiesinclude for example IFOA [33]MSFOA [34] IFFO [35] andCMFOA [36]

Squirrel search algorithm (SSA) proposed byMohit et alin 2018 is a new and powerful global optimization algorithminspired by the natural dynamic foraging behavior of flyingsquirrels [37] In comparison with other swarm intelligenceoptimization algorithms SSA has the advantages of betterand efficient search space exploration because a seasonalmonitoring condition is incorporated Moreover three typesof trees (normal tree oak tree and hickory tree) are availablein the forest region preserving the population diversity andthus enhancing the exploration of the algorithm Test resultsof 33 benchmark functions and a real-time controller designproblem confirm the superiority of SSA in comparison withother well-known algorithms such as GA [10] PSO [25] BA(bat algorithm) [38] and FF (firefly algorithm) [39]

However SSA still suffers from premature convergenceand easily gets trapped in a local optimal solution especiallywhen solving highly complex problemsThe convergence rate

of SSA like other swarm intelligence algorithms depends onthe balance between exploration and exploitation capabilitiesIn other words an excellent performance in dealing withoptimization problems requires fine-tuning of the explo-ration and exploitation problem According to ldquono freelunchrdquo (NFL) theorem [40] no single optimization algorithmis able to achieve the best performance for all problems andSSA is not an exception Therefore there still exists room forimproving the accuracy and convergence rates of SSA

Based on the discussion above this study proposes animproved variant of SSA (ISSA) which employs four strate-gies to enhance the global search ability of SSA In brief themain contributions of this research can be summarized asfollows

(i) An adaptive strategy of predator presence probabilityis proposed which dynamically adjusts with the iterationprocess This strategy discourages premature convergenceand improves the intensive search ability of the algorithmespecially at the latter stages of search In this way a balancebetween the exploration and exploitation capabilities can beproperly managed

(ii)The proposed ISSA employs a normal cloud generator[41] to generate new locations for flying squirrels during thecourse of gliding which improves the exploration capabilityof SSAThis ismotivated by the fact that the gliding behaviorsof flying squirrels have characteristics of randomness andfuzziness which can be simultaneously described by thenormal cloud model [42]

(iii) A selection strategy between successive positions isproposed to maintain the best position of a flying squir-rel individual throughout the optimization process whichenhances the exploitation ability of the algorithm

(iv) A dimensional search enhancement strategy is orig-inally put forward and results in a better quality of the bestsolution in each iteration thereby strengthening the localsearch ability of the algorithm

The general properties of ISSA are evaluated against 32benchmark function including unimodal multimodal andCEC 2014 functions [43] Meanwhile its performance iscompared with the basic SSA and other four well-knownstate-of-the-art optimization algorithms

The rest of this paper is organized as follows Section 2briefly recapitulates the basic SSA Next the proposed ISSAis presented in detail in Section 3 Experimental comparisonsare illustrated in Section 4 Finally Section 5 gives theconcluding remarks

2 The Basic Squirrel SearchOptimization Algorithm

SSAmimics the dynamic foraging behavior of southern flyingsquirrels via gliding an effective mechanism used by smallmammals for travelling long distance in deciduous forest ofEurope and Asia [37] During warm weather the squirrelschange their locations by gliding from one tree to anotherin the forest and explore for food resources They can easilyfind acorn nuts for meeting daily energy needs After thatthey begin searching hickory nuts (the optimal food source)that are stored for winter During cold weather they become

Complexity 3







119891119894 = 119891119894 (1198651198781198941 1198651198781198942 119865119878119894119899) 119894 = 1 2 119873119875 (2)


[119904119900119903119905119890119889 119891 119904119900119903119905119890 119894119899119889119890119909] = 119904119900119903119905 (119891) (3)


119865119878ℎ119905 = 119865119878 (119904119900119903119905119890 119894119899119889119890119909 (1)) (4)

119865119878119886119905 (1 3) = 119865119878 (119904119900119903119905119890 119894119899119889119890119909 (2 4)) (5)

119865119878119899119905 (1119873119875 minus 4) = 119865119878 (119904119900119903119905119890 119894119899119889119890119909 (5119873119875)) (6)



119865119878119899119890119908119886119905 = 119865119878119900119897119889119886119905 + 119889119892119866119888 (119865119878119900119897119889ℎ119905 minus 119865119878119900119897119889119886119905 ) if 1198771 ge 119875119889119901119903119886119899119889119900119898 119897119900119888119886119905119894119900119899 119900119905ℎ119890119903119908119894119904119890 (7)


119865119878119899119890119908119899119905 = 119865119878119900119897119889119899119905 + 119889119892119866119888 (119865119878119900119897119889119886119905 minus 119865119878119900119897119889119899119905 ) if 1198772 ge 119875119889119901119903119886119899119889119900119898 119897119900119888119886119905119894119900119899 119900119905ℎ119890119903119908119894119904119890 (8)


119865119878119899119890119908119899119905 = 119865119878119900119897119889119899119905 + 119889119892119866119888 (119865119878119900119897119889ℎ119905 minus 119865119878119900119897119889119899119905 ) if 1198773 ge 119875119889119901119903119886119899119889119900119898 119897119900119888119886119905119894119900119899 119900119905ℎ119890119903119908119894119904119890 (9)





119878119905119888 = radic 119899sum119896=1

(119865119878119905119886119905119896 minus 119865119878ℎ119905119896)2 119905 = 1 2 3 (10)

119878119888119898119894119899 = 10119864 minus 6365119868119905119890119903(119868119905119890119903max)25(11)


4 Complexity




Levy (119909) = 001 times 119903119886 times 120590100381610038161003816100381611990311988710038161003816100381610038161120573 (13)



(14)









119865119878119899119890119908119886119905=

119865119878119900119897119889119886119905 + 119889119892119866119888 (119865119878119900119897119889ℎ119905 minus 119865119878119900119897119889119886119905 ) if 1198771 ge 119875119889119901119862119909 (119865119878119900119897119889119886119905 119864119899119867119890) 119900119905ℎ119890119903119908119894119904119890(16)

119865119878119899119890119908119899119905=

119865119878119900119897119889119899119905 + 119889119892119866119888 (119865119878119900119897119889119886119905 minus 119865119878119900119897119889119899119905 ) if 1198772 ge 119875119889119901119862119909 (119865119878119900119897119889119899119905 119864119899119867119890) 119900119905ℎ119890119903119908119894119904119890(17)

119865119878119899119890119908119899119905=

119865119878119900119897119889119899119905 + 119889119892119866119888 (119865119878119900119897119889ℎ119905 minus 119865119878119900119897119889119899119905 ) if 1198773 ge 119875119889119901119862119909 (119865119878119900119897119889119899119905 119864119899119867119890) 119900119905ℎ119890119903119908119894119904119890(18)


119864119899 = 119864119899119898119886119909 times (1 minus 119868119905119890119903119868119905119890119903119898119886119909)10 (19)



119865119878119894 = 119865119878119899119890119908119894 if 119891119899119890119908119894 lt 119891119900119897119889119894119865119878119900119897119889119894 119900119905ℎ119890119903119908119894119904119890 (20)

Complexity 5








end

119878119905119888 = radic 119899sum119896=1

(119865119878119905119886119905119896 minus 119865119878ℎ119905119896)2 119878119888119898119894119899 = 10119864 minus 6365119868119905119890119903(119868119905119890119903max)25






119865119878119899119890119908119887119890119904119905119895 = 119862119909 (119865119878119900119897119889119887119890119904119905119895 119864119899119867119890) 119895 = 1 2 119899 (21)




6 Complexity







end



if 119891119899119890119908119887119890119904119905 lt 119891119887119890119904119905 119865119878119887119890119904119905119895 = 119865119878119899119890119908119887119890119904119905119895119891119887119890119904119905 = 119891119899119890119908119887119890119904119905end

end 119868119905119890119903 = 119868119905119890119903 + 1end


Complexity 7




Function Range Fmin

F1(119909) = 119899sum119894=1

1198941199092119894 [minus10 10] 0

F2(119909) = 119899sum119894=2



1199092119894) [minus1 1] -1

F4(119909) = 119899sum119894=1


F5(119909) = 119899sum119894=1

1198941199094119894 + rand () [minus128 128] 0

F6(119909) = 119899minus1sum119894=1


F7(119909) = 119899sum119894=1

( 119894sum119895=1

1199092119895) [minus100 100] 0

F8(119909) = max 10038161003816100381610038161199091198941003816100381610038161003816 1 le 119894 le 119899 [minus100 100] 0

F9(119909) = 119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 119899prod119894=1

10038161003816100381610038161199091198941003816100381610038161003816 [minus10 10] 0

F10(119909) = 119899sum119894=1

1199092119894 [minus100 100] 0

F11(119909) = 119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816119894+1 [minus1 1] 0



8 Complexity









Complexity 9


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

22612E-46

14374E

-13

18031E+0

338546

E-22

53419E-12

58887E+

01Std

39697E-46

10187E

-13

16038E

+02

1146

4E-22

26506

E-12

10717E

+01

F2Mean

53333E-01

600

00E-01

70475E

+04

666

67E-01

666

67E-01

12221E+0

2Std

28109E-01

21082E-01

18027E

+04

11102E

-16

364

14E-11

35452E+

01F3

Mean

000

00E+

00000

00E+

0047300

E-01

000

00E+

00860

42E-14

18586E

-02

Std

18504E

-1626168E-16

42225E-02

906

49E-17

27669E-14

19010E

-03

F4Mean

21268E-39

39943E-10

92617E

+07

37704

E-18

20345E-08

11112

E+07

Std

564

86E-39

32855E-10

18784E

+07

24165E-18

14277E

-08

30272E+

06F5

Mean

43164

E-03

93054E

-02

55376E+

0032231E-03

22754E-03

17965E

-02

Std

19931E-03

23058E-02

10210E

+00

13321E-03

1104

6E-03

31904

E-03

F6Mean

55447E-14

29695E+

0110

669E

+07

76347E

+00

89052E+

0012

862E

+04

Std

54894E-14

340

74E+

0118

313E

+06

590

03E+

0071150E

+00

44338E+

03F7

Mean

11996E

-44

65616E-12

16688E

+05

71055E

-22

60744

E-12

54708E+

03Std

304

49E-44

540

85E-12

17265E

+04

16506E

-22

29515E-12

49070E+

02F8

Mean

35080E-13

17850E

-03

45639E+

0127020E-11

264

40E-06

78561E+0

0Std

70894E

-1343281E-04

20578E+

00604

13E-12

24822E-07

17334E

+00

F9Mean

55772E-24

22148E-07

71792E

+01

23880E-11

204

15E-06

304

53E+

01Std

79227E

-24

67302E-08

30539E+

0141360

E-12

36636E-07

46515E+

01F10

Mean

26748E-44

44745E-13

13098E

+04

42105E-23

440

42E-13

36501E+

02Std

78461E-44

37055E-13

13116

E+03

10825E

-23

15887E

-13

43608E+

01F11

Mean

61803E-188

17833E

-60

18391E-03

78265E

-25

5117

6E-15

67271E-07

Std

000

00E+

0037202E-60

11147E

-03

65934E-25

76076E

-15

46836E-07

10 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

19373E

-45

244

08E-07

89004

E+03

36571E-21

32632E-11

28288E+

02Std

40611E

-45

72161E-08

10186E

+03

90310E

-22

82802E-12

18277E

+01

F2Mean

666

67E-01

21716E+

0080535E+

0512

514E

+00

666

67E-01

82661E+

02Std

82003E-16

22142E+

0011670E

+05

12485E

+00

25423E-10

77814E

+01

F3Mean

000

00E+

0076

318E

-11

866

02E-01

000

00E+

004110

0E-13

51246

E-02

Std

22204E-16

22120E-11

18360E

-02

23984E-16

14800E

-13

40430E-03

F4Mean

306

71E-38

97033E

-04

46756E+

0819

432E

-17

10621E-07

38893E+

07Std

69186E-38

344

64E-04

60135E+

0711627E

-17

76083E

-08

73301E+0

6F5

Mean

71557E

-03

29566

E-01

53164

E+01

10084E

-02

73580E

-03

10458E

-01

Std

23021E-03

33200

E-02

64915E+

0026523E-03

18100E

-03

26586E-02

F6Mean

43706

E-11

95471E+0

170

118E+

0777

331E+0

147194E+

0165331E+

04Std

95151E-11

35358E+

0153302E+

06344

63E+

0140976E+

0113

721E+0

4F7

Mean

12947E

-41

23079E-05

91659E

+05

69771E-21

64025E-11

28862E+

04Std

37876E-41

58814E-06

93287E

+04

31808E-21

26901E-11

28375E+

03F8

Mean

60872E-11

12706E

-01

67093E+

0184930E-11

71576E

-06

11919E

+01

Std

25158E-11

33391E-02

25011E

+00

11107E

-11

69032E-07

1040

4E+0

0F9

Mean

18289E

-23

38822E-04

20565E+

1060338E-11

63442E-06

11434E

+05

Std

28884E-23

72525E

-05

63864

E+10

64165E-12

90797E

-07

24588E+

05F10

Mean

12924E

-44

14594E

-06

41629E+

0422846

E-22

20175E-12

10536E

+03

Std

25807E-44

606

62E-07

37125E+

0341424E-23

52761E-13

59785E+

01F11

Mean

44745E-163

19208E

-58

92852E

-03

11169E

-24

51458E-15

16917E

-06

Std

000

00E+

0022612E-58

35776E-03

14674E

-24

82130E-15

94517E

-07

Complexity 11


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

52002E-44

1360

1E-01

64559E+

0467630E-20

540

42E-10

23688E+

03Std

89565E-44

28398E-02

27675E+

0318

636E

-20

10843E

-10

11782E

+02

F2Mean

25837E+

0028507E+

0189058E+

0610

018E

+01

11850E

+01

13921E+0

4Std

40923E+

0012

482E

+01

64125E+

0598

260E

+00

67378E+

0021479E+

03F3

Mean

19984E

-1517

506E

-05

99937E

-01

19984E

-1534570E-12

1946

0E-01

Std

27940E-16

33607E-06

19874E

-04

39686E-16

57323E-13

11259E

-02

F4Mean

14073E

-38

30139E+

0226151E+

0914

261E-16

10212E

-06

20499E+

08Std

15382E

-38

90551E+0

126783E+

0875

737E

-17

33853E-07

35388E+

07F5

Mean

17615E

-02

14345E

+00

59603E+

0237550E-02

29349E-02

12443E

+00

Std

42239E-03

13985E

-01

58412E+

0112

602E

-02

45825E-03

18471E-01

F6Mean

11417E

+01

58422E+

0242299E+

0817

988E

+02

16578E

+02

560

78E+

05Std

30258E+

0197

884E

+02

48581E+

0739022E+

01466

85E+

0165477E+

04F7

Mean

16881E-41

12984E

+01

64707E+

0611852E

-19

74831E-10

22865E+

05Std

34134E-41

22729E+

0033435E+

0531718E-20

95033E

-11

20650E+

04F8

Mean

45259E-08

39819E+

0085137E+

0117

042E

-04

29244

E-05

33956E+

01Std

29104E-08

41522E-01

12566E

+00

78665E

-05

27018E-06

47713E+

00F9

Mean

12222E

-22

36814E-01

72469E

+32

25070E-10

23795E-05

14112

E+27

Std

84369E-23

41963E-02

20633E+

3323874E-11

13879E

-06

44627E+

27F10

Mean

34254E-42

37261E-01

14940E

+05

20714E-21

18486E

-11

43041E+

03Std

966

08E-42

92561E-02

446

43E+

03464

07E-22

23956E-12

24348E+

02F11

Mean

1640

0E-12

315

040E

-52

37278E-02

65780E-24

3117

4E-14

10720E

-05

Std

51861E-123

16670E

-52

11165E

-02

72960E

-24

36245E-14

59475E-06

12 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

(b) F4

0 2000 4000 6000 8000 10000

0

5

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus15

minus5

(c) F6

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

minus50

(d) F7

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus2

minus4

minus6

minus8

minus10

minus12

minus14

(e) F8

0 2000 4000 6000 8000 10000

05

1015

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus5

minus10

minus20

minus25

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

Mea

n Er

rors

(log)

(h) F11


Complexity 13

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus10

minus5

0

5

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

0

10

20

30

Mea

n Er

rors

(log)

2000 4000 6000 8000 100000Iteration

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

50

Mea

n Er

rors

(log)

(h) F11


14 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

20

Mea

n Er

rors

(log)

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

0

2

4

6

8

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

01020304050

Mea

n Er

rors

(log)

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus140

minus120

minus100

minus80

minus60

minus40

minus20

020

Mea

n Er

rors

(log)

(h) F11


Complexity 15

Table6Multim

odalbenchm

arkfunctio

ns

Functio

nRa

nge

Fmin

F12(119909)=

minus20exp(minus0

2radic1 119899119899 sum 119894=11199092 119894)minus

exp(1 119899119899 sum 119894=1co

s(2120587119909 119894))

+20+exp

(1 )[minus32

32]0

F13(119909)=

119899 sum 119894=11003816 1003816 1003816 1003816119909 119894sin(119909 119894)

+01119909 1198941003816 1003816 1003816 1003816

[minus1010]

0

F14(119909)=

119891 119904(1199091119909 2)


(119909 119899119909 1)

[minus10010

0]0

119891 119904(119909119910)=

(1199092 +1199102 )025[sin2

(50(1199092 +

1199102 )01)+1

]F15(

119909)=119891 119904(1199091119909 2)


119909 1198991199091)

[minus10010

0]0

119891 119904(119909119910)=

05(sin2(radic 1199092+1199102

)minus05)

(1+0001

(1199092 +1199102 ))2

F16(119909)=

120587 11989910sin2

(120587119910 119894)+119899minus1 sum 119894=1

(119910 119894minus1 )2 [

1+10sin2

(120587119910 119894+1)]+

(119910 119899minus1 )2

+119899 sum 119894=1119906(119909 119894

10100

4)[minus50

50]0

119910 119894=1+1 4(119909

119894+1)

119906(119909 119894119886

119896119898)= 119896(119909 119894

minus119886)119898

119909 119894gt119886

0minus119886le

119909 119894le119886

119896(minus119909119894minus119886)119898

119909119894gt119886

F17(119909)=

1 4000119899 sum 119894=11199092 119894minus119899 prod 119894=1

cos(119909119894 radic 119894)+1

[minus10010

0]0

F18(119909)=


(minus(1199092 119894+

1199092 119894+1+05

119909 119894119909 119894+1)

8)lowastc

os(4radic

1199092 119894+1199092 119894+1

+05119909 119894119909 119894+1))

[minus55]

1-n

F19(119909)=

119899 sum 119894=1(119909119894minus1)2

minus119899 sum 119894=2119909 119894119909 119894minus1

[minusn2n2 ]

119899(119899+4)(119899

minus1)minus6

F20 (119909 )=

sum119899minus1 119894=2(05

+(sin2(radic 1

001199092 119894+1199092 119894+1)minus0

5))(1+

0001(1199092 119894minus

2119909 119894119909119894minus1+1199092 119894minus1))2

[minus10010

0]0

F21(119909)=

119899 sum 119894=1[1199092 119894minus10

cos(2120587

119909 119894)+10]

[minus51251

2]0

F22(119909)=

119899 sum 119894=1[1199102 119894minus10

cos(2120587

119910 119894)+10]

119910 119894= 119909 119894

1003816 1003816 1003816 1003816119909 1198941003816 1003816 1003816 1003816lt05

119903119900119906119899119889(2119909

119894)2

1003816 1003816 1003816 10038161199091198941003816 1003816 1003816 1003816lt0

5[minus51

2512]

0

F23(119909)=

1minuscos(2120587

radic119899 sum 119894=11199092 119894)

+01radic119899 sum 119894=1

1199092 119894[minus10

0100]

0

F24(119909)=

119899 sum 119894=1119896119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 (119909119894+05

))]minus119899119896

119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 05

)][minus05

05]0

F25(119909)=

119899 sum 119896=1119899 sum 119895=1((1199102 119895119896 4000)minus

cos(119910 119895119896)+1

)119910119895119896=10

0(119909 119896minus1199092 119895

)2 +(1minus

1199092 119895)2[minus10

0100]

0

16 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmultim

odalbenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

51692E-14

21708E-07

16343E

+01

42641E-12

43970E-07

70983E

+00

Std

94813E-15

11785E

-07

45830E-01

51275E-13

53024E-08

45755E+

00F13

Mean

19651E-15

17670E

-07

30865E+

0138781E-12

12507E

-06

29660

E+01

Std

17016E

-1510

899E

-07

28749E+

00200

14E-12

19125E

-06

57790E+

00F14

Mean

28586E-11

47414E-02

21576E+

0235954E-05

17290E

-02

18705E

+02

Std

17874E

-1118

105E

-02

50836E+

0019

343E

-06

10857E

-03

50868E+

01F15

Mean

99552E

-01

46150E-01

12596E

+01

94983E

-01

10032E

+00

12147E

+01

Std

38926E-01

33522E-01

21495E-01

42966

E-01

35690E-01

17388E

-01

F16

Mean

15705E

-32

13069E

-15

56725E+

0650290E-25

99726E

-15

31482E+

00Std

28850E-48

57169E-16

17168E

+06

47027E-25

85374E-15

58054E-01

F17

Mean

13781E-02

10332E

-02

43352E+

0044332E-03

12793E

-02

10971E+0

0Std

14865E

-02

12632E

-02

42518E-01

79408E

-03

10155E

-02

10766E

-02

F18

Mean

50849E+

0038253E+

0020946

E+01

49225E+

00490

48E+

0021497E+

01Std

16014E

+00

14627E

+00

76856E

-01

21737E+

00204

11E+0

013

669E

+00

F19

Mean

268

41E-07

19292E

+02

49808E+

0519

677E

+02

240

98E+

0230226E+

04Std

32619E-08

15971E+0

214

706E

+05

16572E

+02

23149E+

0260289E+

03F2

0Mean

25989E-07

47006

E-06

33592E-02

44469E-08

18865E

-07

1540

6E-01

Std

59383E-07

73387E

-06

22456E-02

10350E

-07

31612E-07

56719E-02

F21

Mean

000

00E+

0070

841E-13

25769E+

02000

00E+

0045409E-11

30881E+

02Std

000

00E+

0045361E-13

90973E

+00

000

00E+

0019

882E

-11

27305E+

01F2

2Mean

000

00E+

007746

7E-13

23335E+

02000

00E+

00644

03E-11

25509E+

02Std

000

00E+

0036979E-13

15942E

+01

000

00E+

0033820E-11

26992E+

01F2

3Mean

93987E

-01

52987E-01

12199E

+01

13599E

+00

14399E

+00

21878E+

00Std

21705E-01

12517E

-01

49304

E-01

36576E-01

21705E-01

62731E-02

F24

Mean

14921E-14

37233E-04

32412E+

0147458E-09

42553E-03

26924E+

01Std

17226E

-1498

846E

-05

11649E

+00

28242E-09

42975E-04

35559E+

00F2

5Mean

29494E+

0110

724E

+02

11372E

+07

404

62E+

0193530E+

0092

421E+0

3Std

29743E+

0151800

E+01

31606

E+06

39685E+

0190392E+

0018

838E

+03

Complexity 17


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

85798E-14

24174E-04

18459E

+01

7404

4E-12

73673E

-07

82226E+

00Std

17360E

-1455274E-05

1944

7E-01

88139E-13

80222E-08

42517E+

00F13

Mean

22538E-15

29492E-04

71594E

+01

21041E-11

32004

E-06

60959E+

01Std

18688E

-1510

372E

-04

45394E+

0015

865E

-11

15334E

-06

44766

E+00

F14

Mean

71759E

-1120261E+

0043430E+

0277682E

-05

33324E-02

42669E+

02Std

24650E-11

50770E-01

14055E

+01

54975E-06

10537E

-03

80127E+

01F15

Mean

16716E

+00

12749E

+00

22241E+

0116

927E

+00

14937E

+00

21617E+

01Std

76572E

-01

43985E-01

33014E-01

47677E-01

63574E-01

54534E-01

F16

Mean

94233E-33

13057E

-09

76995E

+07

17755E

-24

846

48E-14

69921E+

00Std

14425E

-48

37533E-10

21712E+

0719

092E

-24

17429E

-13

89129E-01

F17

Mean

76377E

-03

14219E

-02

1160

6E+0

164039E-03

10080E

-02

1264

1E+0

0Std

57418E-03

21089E-02

46282E-01

70807E

-03

13952E

-02

16555E

-02

F18

Mean

83103E+

0079

047E

+00

39689E+

0189467E+

0096

041E+0

038726E+

01Std

260

72E+

0025432E+

0077616E

-01

78506E

-01

21029E+

0013

015E

+00

F19

Mean

45562E+

0126833E+

04806

68E+

0616

118E+

0413

155E

+04

70015E

+05

Std

38094E+

0121743E+

0421709E+

0612

498E

+04

1300

9E+0

497

174E

+04

F20

Mean

43064E-08

25702E-04

11519E

-01

52365E-08

16998E

-06

500

47E-01

Std

44294E-08

27576E-04

39417E-02

95247E

-08

49881E-06

26305E-01

F21

Mean

000

00E+

0011310E

-06

53146

E+02

000

00E+

0023711E

-10

58748E+

02Std

000

00E+

0033614E-07

32117E+

01000

00E+

0045437E-11

29507E+

01F2

2Mean

000

00E+

0016

167E

-06

48729E+

02000

00E+

00244

07E-10

52060

E+02

Std

000

00E+

0063216E-07

24382E+

01000

00E+

0075

889E

-11

42230E+

01F2

3Mean

13699E

+00

89987E-01

21237E+

0122699E+

0025899E+

0035955E+

00Std

23594E-01

666

67E-02

58033E-01

41913E-01

62973E-01

12247E

-01

F24

Mean

71054E

-1426826E-02

63090E+

0119

033E

-08

96037E

-03

47263E+

01Std

27621E-14

47780E-03

22392E+

0061075E-09

97071E-04

52689E+

00F2

5Mean

66563E+

0184722E+

0211275E

+08

65780E+

0139992E+

0188242E+

04Std

10992E

+02

2113

8E+0

221091E+

0794

954E

+01

43819E+

0116

832E

+04

18 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

18989E

-1316

584E

-01

19996E

+01

17809E

-11

14744E

-06

13554E

+01

Std

20566

E-14

53720E-02

90319E

-02

19159E

-12

18930E

-07

60821E+

00F13

Mean

22871E-15

17736E

-01

1944

7E+0

213

452E

-10

13291E-05

16379E

+02

Std

26741E-15

53611E-02

62653E+

0038592E-11

55001E-06

13313E

+01

F14

Mean

18736E

-1074

259E

+01

10132E

+03

22866

E-04

83534E-02

95534E

+02

Std

37223E-11

19144E

+01

18986E

+01

14283E

-05

10592E

-02

53523E+

01F15

Mean

26814E+

0010

178E

+01

47083E+

0128083E+

0034325E+

0045859E+

01Std

73851E-01

16238E

+00

22513E-01

46148E-01

60283E-01

69914E-01

F16

Mean

47116E-33

244

54E-04

90382E

+08

81890E-24

62347E-14

27647E+

03Std

72124E

-49

59650E-05

64985E+

0767958E-24

55604

E-14

44231E+

03F17

Mean

34494E-03

11896E

-02

37816E+

0134509E-03

41885E-03

21280E+

00Std

60565E-03

65363E-03

15922E

+00

46765E-03

86153E-03

54359E-02

F18

Mean

18033E

+01

17806E

+01

86826E+

0118

319E

+01

18828E

+01

82458E+

01Std

19652E

+00

38319E+

0093

222E

-01

29296E+

0025377E+

0015

159E

+00

F19

Mean

82462E+

0427944

E+06

48046

E+08

28415E+

0560265E+

0549201E+

07Std

55732E+

0489703E+

0596

715E

+07

24572E+

0527137E+

0572

772E

+06

F20

Mean

57130E-07

81688E-03

96848E

-01

13631E-06

27143E-05

21656E+

00Std

61122E-07

53195E-03

44542E-01

25155E-06

58766

E-05

80368E-01

F21

Mean

000

00E+

0051414E+

0013

305E

+03

000

00E+

0020026E-09

13623E

+03

Std

000

00E+

0017

825E

+00

22890E+

01000

00E+

0032815E-10

609

96E+

01F2

2Mean

000

00E+

0077

848E

+00

1260

9E+0

3000

00E+

0020383E-09

12745E

+03

Std

000

00E+

0023732E+

0029100

E+01

000

00E+

0029753E-10

59708E+

01F2

3Mean

25599E+

0020499E+

0039804

E+01

47099E+

0043699E+

0073

691E+0

0Std

36878E-01

15092E

-01

69296E-01

59151E-01

56184E-01

17989E

-01

F24

Mean

40927E-13

26229E+

0015

145E

+02

18874E

-07

29476E-02

10478E

+02

Std

88061E-14

63367E-01

42830E+

0037074E-08

17697E

-03

11873E

+01

F25

Mean

42987E+

028117

8E+0

313

524E

+09

56790E+

0244982E+

0218

038E

+06

Std

43423E+

0233128E+

0278

399E

+07

54327E+

0246926E+

0221315E+

05

Complexity 19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus8

minus6

minus4

minus2

02468

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(f) F24




MAE = sum119873119895=1 10038161003816100381610038161003816119898119895 minus 11989611989510038161003816100381610038161003816119873 (22)

20 Complexity

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

ISSASSAPSO

CMFOAIFFOFOA

1

2

3

4

5

6

7

8

Mea

n Er

rors

(log)

0 4000 6000 8000 100002000Iteration

(e) F19

ISSASSAPSO

CMFOAIFFOFOA

20000 6000 8000 100004000Iteration

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




Complexity 21

Table11

Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

31365E+

0413

133E

+06

11295E

+08

10017E

+06

864

75E+

0513

449E

+07

Std

18602E

+04

52974E+

0531387E+

0748689E+

0548607E+

0528405E+

06F2

7Mean

304

00E-10

1844

6E+0

484500

E+09

10512E

+04

12359E

+04

58535E+

08Std

61535E-10

14049E

+04

10125E

+09

12485E

+04

11922E

+04

38771E+

07F2

8Mean

42105E-01

46710E+

0173

819E

+02

4114

7E+0

137814E+

0114

226E

+02

Std

12624E

+00

31490E+

0199

455E

+01

47336E+

0134110E+

0129201E+

01F2

9Mean

75177E

+03

14891E+0

529286E+

0647277E+

0531099E+

0539826E+

05Std

33119E+

0368316E+

049190

4E+0

521021E+

0522686E+

0515

511E+0

5F3

0Mean

31524E+

0231524E+

0238129E+

0231524E+

0231524E+

0232568E+

02Std

85708E-12

19710E

-07

14082E

+01

11524E

-1145680E-11

58955E+

00F31

Mean

23483E+

0223172E+

0230117E+

0223811E

+02

23858E+

0224179E+

02Std

41748E+

0072

461E+0

048903E+

00560

97E+

0050249E+

0090

228E

+00

F32

Mean

20790E+

02206

03E+

0221884E+

0221485E+

0220975E+

0220633E+

02Std

41618E+

0032456E+

0030353E+

0087909E+

0057719E+

0016

880E

+00

22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 3







119891119894 = 119891119894 (1198651198781198941 1198651198781198942 119865119878119894119899) 119894 = 1 2 119873119875 (2)


[119904119900119903119905119890119889 119891 119904119900119903119905119890 119894119899119889119890119909] = 119904119900119903119905 (119891) (3)


119865119878ℎ119905 = 119865119878 (119904119900119903119905119890 119894119899119889119890119909 (1)) (4)

119865119878119886119905 (1 3) = 119865119878 (119904119900119903119905119890 119894119899119889119890119909 (2 4)) (5)

119865119878119899119905 (1119873119875 minus 4) = 119865119878 (119904119900119903119905119890 119894119899119889119890119909 (5119873119875)) (6)



119865119878119899119890119908119886119905 = 119865119878119900119897119889119886119905 + 119889119892119866119888 (119865119878119900119897119889ℎ119905 minus 119865119878119900119897119889119886119905 ) if 1198771 ge 119875119889119901119903119886119899119889119900119898 119897119900119888119886119905119894119900119899 119900119905ℎ119890119903119908119894119904119890 (7)


119865119878119899119890119908119899119905 = 119865119878119900119897119889119899119905 + 119889119892119866119888 (119865119878119900119897119889119886119905 minus 119865119878119900119897119889119899119905 ) if 1198772 ge 119875119889119901119903119886119899119889119900119898 119897119900119888119886119905119894119900119899 119900119905ℎ119890119903119908119894119904119890 (8)


119865119878119899119890119908119899119905 = 119865119878119900119897119889119899119905 + 119889119892119866119888 (119865119878119900119897119889ℎ119905 minus 119865119878119900119897119889119899119905 ) if 1198773 ge 119875119889119901119903119886119899119889119900119898 119897119900119888119886119905119894119900119899 119900119905ℎ119890119903119908119894119904119890 (9)





119878119905119888 = radic 119899sum119896=1

(119865119878119905119886119905119896 minus 119865119878ℎ119905119896)2 119905 = 1 2 3 (10)

119878119888119898119894119899 = 10119864 minus 6365119868119905119890119903(119868119905119890119903max)25(11)


4 Complexity




Levy (119909) = 001 times 119903119886 times 120590100381610038161003816100381611990311988710038161003816100381610038161120573 (13)



(14)









119865119878119899119890119908119886119905=

119865119878119900119897119889119886119905 + 119889119892119866119888 (119865119878119900119897119889ℎ119905 minus 119865119878119900119897119889119886119905 ) if 1198771 ge 119875119889119901119862119909 (119865119878119900119897119889119886119905 119864119899119867119890) 119900119905ℎ119890119903119908119894119904119890(16)

119865119878119899119890119908119899119905=

119865119878119900119897119889119899119905 + 119889119892119866119888 (119865119878119900119897119889119886119905 minus 119865119878119900119897119889119899119905 ) if 1198772 ge 119875119889119901119862119909 (119865119878119900119897119889119899119905 119864119899119867119890) 119900119905ℎ119890119903119908119894119904119890(17)

119865119878119899119890119908119899119905=

119865119878119900119897119889119899119905 + 119889119892119866119888 (119865119878119900119897119889ℎ119905 minus 119865119878119900119897119889119899119905 ) if 1198773 ge 119875119889119901119862119909 (119865119878119900119897119889119899119905 119864119899119867119890) 119900119905ℎ119890119903119908119894119904119890(18)


119864119899 = 119864119899119898119886119909 times (1 minus 119868119905119890119903119868119905119890119903119898119886119909)10 (19)



119865119878119894 = 119865119878119899119890119908119894 if 119891119899119890119908119894 lt 119891119900119897119889119894119865119878119900119897119889119894 119900119905ℎ119890119903119908119894119904119890 (20)

Complexity 5








end

119878119905119888 = radic 119899sum119896=1

(119865119878119905119886119905119896 minus 119865119878ℎ119905119896)2 119878119888119898119894119899 = 10119864 minus 6365119868119905119890119903(119868119905119890119903max)25






119865119878119899119890119908119887119890119904119905119895 = 119862119909 (119865119878119900119897119889119887119890119904119905119895 119864119899119867119890) 119895 = 1 2 119899 (21)




6 Complexity







end



if 119891119899119890119908119887119890119904119905 lt 119891119887119890119904119905 119865119878119887119890119904119905119895 = 119865119878119899119890119908119887119890119904119905119895119891119887119890119904119905 = 119891119899119890119908119887119890119904119905end

end 119868119905119890119903 = 119868119905119890119903 + 1end


Complexity 7




Function Range Fmin

F1(119909) = 119899sum119894=1

1198941199092119894 [minus10 10] 0

F2(119909) = 119899sum119894=2



1199092119894) [minus1 1] -1

F4(119909) = 119899sum119894=1


F5(119909) = 119899sum119894=1

1198941199094119894 + rand () [minus128 128] 0

F6(119909) = 119899minus1sum119894=1


F7(119909) = 119899sum119894=1

( 119894sum119895=1

1199092119895) [minus100 100] 0

F8(119909) = max 10038161003816100381610038161199091198941003816100381610038161003816 1 le 119894 le 119899 [minus100 100] 0

F9(119909) = 119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 119899prod119894=1

10038161003816100381610038161199091198941003816100381610038161003816 [minus10 10] 0

F10(119909) = 119899sum119894=1

1199092119894 [minus100 100] 0

F11(119909) = 119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816119894+1 [minus1 1] 0



8 Complexity









Complexity 9


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

22612E-46

14374E

-13

18031E+0

338546

E-22

53419E-12

58887E+

01Std

39697E-46

10187E

-13

16038E

+02

1146

4E-22

26506

E-12

10717E

+01

F2Mean

53333E-01

600

00E-01

70475E

+04

666

67E-01

666

67E-01

12221E+0

2Std

28109E-01

21082E-01

18027E

+04

11102E

-16

364

14E-11

35452E+

01F3

Mean

000

00E+

00000

00E+

0047300

E-01

000

00E+

00860

42E-14

18586E

-02

Std

18504E

-1626168E-16

42225E-02

906

49E-17

27669E-14

19010E

-03

F4Mean

21268E-39

39943E-10

92617E

+07

37704

E-18

20345E-08

11112

E+07

Std

564

86E-39

32855E-10

18784E

+07

24165E-18

14277E

-08

30272E+

06F5

Mean

43164

E-03

93054E

-02

55376E+

0032231E-03

22754E-03

17965E

-02

Std

19931E-03

23058E-02

10210E

+00

13321E-03

1104

6E-03

31904

E-03

F6Mean

55447E-14

29695E+

0110

669E

+07

76347E

+00

89052E+

0012

862E

+04

Std

54894E-14

340

74E+

0118

313E

+06

590

03E+

0071150E

+00

44338E+

03F7

Mean

11996E

-44

65616E-12

16688E

+05

71055E

-22

60744

E-12

54708E+

03Std

304

49E-44

540

85E-12

17265E

+04

16506E

-22

29515E-12

49070E+

02F8

Mean

35080E-13

17850E

-03

45639E+

0127020E-11

264

40E-06

78561E+0

0Std

70894E

-1343281E-04

20578E+

00604

13E-12

24822E-07

17334E

+00

F9Mean

55772E-24

22148E-07

71792E

+01

23880E-11

204

15E-06

304

53E+

01Std

79227E

-24

67302E-08

30539E+

0141360

E-12

36636E-07

46515E+

01F10

Mean

26748E-44

44745E-13

13098E

+04

42105E-23

440

42E-13

36501E+

02Std

78461E-44

37055E-13

13116

E+03

10825E

-23

15887E

-13

43608E+

01F11

Mean

61803E-188

17833E

-60

18391E-03

78265E

-25

5117

6E-15

67271E-07

Std

000

00E+

0037202E-60

11147E

-03

65934E-25

76076E

-15

46836E-07

10 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

19373E

-45

244

08E-07

89004

E+03

36571E-21

32632E-11

28288E+

02Std

40611E

-45

72161E-08

10186E

+03

90310E

-22

82802E-12

18277E

+01

F2Mean

666

67E-01

21716E+

0080535E+

0512

514E

+00

666

67E-01

82661E+

02Std

82003E-16

22142E+

0011670E

+05

12485E

+00

25423E-10

77814E

+01

F3Mean

000

00E+

0076

318E

-11

866

02E-01

000

00E+

004110

0E-13

51246

E-02

Std

22204E-16

22120E-11

18360E

-02

23984E-16

14800E

-13

40430E-03

F4Mean

306

71E-38

97033E

-04

46756E+

0819

432E

-17

10621E-07

38893E+

07Std

69186E-38

344

64E-04

60135E+

0711627E

-17

76083E

-08

73301E+0

6F5

Mean

71557E

-03

29566

E-01

53164

E+01

10084E

-02

73580E

-03

10458E

-01

Std

23021E-03

33200

E-02

64915E+

0026523E-03

18100E

-03

26586E-02

F6Mean

43706

E-11

95471E+0

170

118E+

0777

331E+0

147194E+

0165331E+

04Std

95151E-11

35358E+

0153302E+

06344

63E+

0140976E+

0113

721E+0

4F7

Mean

12947E

-41

23079E-05

91659E

+05

69771E-21

64025E-11

28862E+

04Std

37876E-41

58814E-06

93287E

+04

31808E-21

26901E-11

28375E+

03F8

Mean

60872E-11

12706E

-01

67093E+

0184930E-11

71576E

-06

11919E

+01

Std

25158E-11

33391E-02

25011E

+00

11107E

-11

69032E-07

1040

4E+0

0F9

Mean

18289E

-23

38822E-04

20565E+

1060338E-11

63442E-06

11434E

+05

Std

28884E-23

72525E

-05

63864

E+10

64165E-12

90797E

-07

24588E+

05F10

Mean

12924E

-44

14594E

-06

41629E+

0422846

E-22

20175E-12

10536E

+03

Std

25807E-44

606

62E-07

37125E+

0341424E-23

52761E-13

59785E+

01F11

Mean

44745E-163

19208E

-58

92852E

-03

11169E

-24

51458E-15

16917E

-06

Std

000

00E+

0022612E-58

35776E-03

14674E

-24

82130E-15

94517E

-07

Complexity 11


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

52002E-44

1360

1E-01

64559E+

0467630E-20

540

42E-10

23688E+

03Std

89565E-44

28398E-02

27675E+

0318

636E

-20

10843E

-10

11782E

+02

F2Mean

25837E+

0028507E+

0189058E+

0610

018E

+01

11850E

+01

13921E+0

4Std

40923E+

0012

482E

+01

64125E+

0598

260E

+00

67378E+

0021479E+

03F3

Mean

19984E

-1517

506E

-05

99937E

-01

19984E

-1534570E-12

1946

0E-01

Std

27940E-16

33607E-06

19874E

-04

39686E-16

57323E-13

11259E

-02

F4Mean

14073E

-38

30139E+

0226151E+

0914

261E-16

10212E

-06

20499E+

08Std

15382E

-38

90551E+0

126783E+

0875

737E

-17

33853E-07

35388E+

07F5

Mean

17615E

-02

14345E

+00

59603E+

0237550E-02

29349E-02

12443E

+00

Std

42239E-03

13985E

-01

58412E+

0112

602E

-02

45825E-03

18471E-01

F6Mean

11417E

+01

58422E+

0242299E+

0817

988E

+02

16578E

+02

560

78E+

05Std

30258E+

0197

884E

+02

48581E+

0739022E+

01466

85E+

0165477E+

04F7

Mean

16881E-41

12984E

+01

64707E+

0611852E

-19

74831E-10

22865E+

05Std

34134E-41

22729E+

0033435E+

0531718E-20

95033E

-11

20650E+

04F8

Mean

45259E-08

39819E+

0085137E+

0117

042E

-04

29244

E-05

33956E+

01Std

29104E-08

41522E-01

12566E

+00

78665E

-05

27018E-06

47713E+

00F9

Mean

12222E

-22

36814E-01

72469E

+32

25070E-10

23795E-05

14112

E+27

Std

84369E-23

41963E-02

20633E+

3323874E-11

13879E

-06

44627E+

27F10

Mean

34254E-42

37261E-01

14940E

+05

20714E-21

18486E

-11

43041E+

03Std

966

08E-42

92561E-02

446

43E+

03464

07E-22

23956E-12

24348E+

02F11

Mean

1640

0E-12

315

040E

-52

37278E-02

65780E-24

3117

4E-14

10720E

-05

Std

51861E-123

16670E

-52

11165E

-02

72960E

-24

36245E-14

59475E-06

12 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

(b) F4

0 2000 4000 6000 8000 10000

0

5

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus15

minus5

(c) F6

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

minus50

(d) F7

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus2

minus4

minus6

minus8

minus10

minus12

minus14

(e) F8

0 2000 4000 6000 8000 10000

05

1015

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus5

minus10

minus20

minus25

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

Mea

n Er

rors

(log)

(h) F11


Complexity 13

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus10

minus5

0

5

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

0

10

20

30

Mea

n Er

rors

(log)

2000 4000 6000 8000 100000Iteration

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

50

Mea

n Er

rors

(log)

(h) F11


14 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

20

Mea

n Er

rors

(log)

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

0

2

4

6

8

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

01020304050

Mea

n Er

rors

(log)

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus140

minus120

minus100

minus80

minus60

minus40

minus20

020

Mea

n Er

rors

(log)

(h) F11


Complexity 15

Table6Multim

odalbenchm

arkfunctio

ns

Functio

nRa

nge

Fmin

F12(119909)=

minus20exp(minus0

2radic1 119899119899 sum 119894=11199092 119894)minus

exp(1 119899119899 sum 119894=1co

s(2120587119909 119894))

+20+exp

(1 )[minus32

32]0

F13(119909)=

119899 sum 119894=11003816 1003816 1003816 1003816119909 119894sin(119909 119894)

+01119909 1198941003816 1003816 1003816 1003816

[minus1010]

0

F14(119909)=

119891 119904(1199091119909 2)


(119909 119899119909 1)

[minus10010

0]0

119891 119904(119909119910)=

(1199092 +1199102 )025[sin2

(50(1199092 +

1199102 )01)+1

]F15(

119909)=119891 119904(1199091119909 2)


119909 1198991199091)

[minus10010

0]0

119891 119904(119909119910)=

05(sin2(radic 1199092+1199102

)minus05)

(1+0001

(1199092 +1199102 ))2

F16(119909)=

120587 11989910sin2

(120587119910 119894)+119899minus1 sum 119894=1

(119910 119894minus1 )2 [

1+10sin2

(120587119910 119894+1)]+

(119910 119899minus1 )2

+119899 sum 119894=1119906(119909 119894

10100

4)[minus50

50]0

119910 119894=1+1 4(119909

119894+1)

119906(119909 119894119886

119896119898)= 119896(119909 119894

minus119886)119898

119909 119894gt119886

0minus119886le

119909 119894le119886

119896(minus119909119894minus119886)119898

119909119894gt119886

F17(119909)=

1 4000119899 sum 119894=11199092 119894minus119899 prod 119894=1

cos(119909119894 radic 119894)+1

[minus10010

0]0

F18(119909)=


(minus(1199092 119894+

1199092 119894+1+05

119909 119894119909 119894+1)

8)lowastc

os(4radic

1199092 119894+1199092 119894+1

+05119909 119894119909 119894+1))

[minus55]

1-n

F19(119909)=

119899 sum 119894=1(119909119894minus1)2

minus119899 sum 119894=2119909 119894119909 119894minus1

[minusn2n2 ]

119899(119899+4)(119899

minus1)minus6

F20 (119909 )=

sum119899minus1 119894=2(05

+(sin2(radic 1

001199092 119894+1199092 119894+1)minus0

5))(1+

0001(1199092 119894minus

2119909 119894119909119894minus1+1199092 119894minus1))2

[minus10010

0]0

F21(119909)=

119899 sum 119894=1[1199092 119894minus10

cos(2120587

119909 119894)+10]

[minus51251

2]0

F22(119909)=

119899 sum 119894=1[1199102 119894minus10

cos(2120587

119910 119894)+10]

119910 119894= 119909 119894

1003816 1003816 1003816 1003816119909 1198941003816 1003816 1003816 1003816lt05

119903119900119906119899119889(2119909

119894)2

1003816 1003816 1003816 10038161199091198941003816 1003816 1003816 1003816lt0

5[minus51

2512]

0

F23(119909)=

1minuscos(2120587

radic119899 sum 119894=11199092 119894)

+01radic119899 sum 119894=1

1199092 119894[minus10

0100]

0

F24(119909)=

119899 sum 119894=1119896119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 (119909119894+05

))]minus119899119896

119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 05

)][minus05

05]0

F25(119909)=

119899 sum 119896=1119899 sum 119895=1((1199102 119895119896 4000)minus

cos(119910 119895119896)+1

)119910119895119896=10

0(119909 119896minus1199092 119895

)2 +(1minus

1199092 119895)2[minus10

0100]

0

16 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmultim

odalbenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

51692E-14

21708E-07

16343E

+01

42641E-12

43970E-07

70983E

+00

Std

94813E-15

11785E

-07

45830E-01

51275E-13

53024E-08

45755E+

00F13

Mean

19651E-15

17670E

-07

30865E+

0138781E-12

12507E

-06

29660

E+01

Std

17016E

-1510

899E

-07

28749E+

00200

14E-12

19125E

-06

57790E+

00F14

Mean

28586E-11

47414E-02

21576E+

0235954E-05

17290E

-02

18705E

+02

Std

17874E

-1118

105E

-02

50836E+

0019

343E

-06

10857E

-03

50868E+

01F15

Mean

99552E

-01

46150E-01

12596E

+01

94983E

-01

10032E

+00

12147E

+01

Std

38926E-01

33522E-01

21495E-01

42966

E-01

35690E-01

17388E

-01

F16

Mean

15705E

-32

13069E

-15

56725E+

0650290E-25

99726E

-15

31482E+

00Std

28850E-48

57169E-16

17168E

+06

47027E-25

85374E-15

58054E-01

F17

Mean

13781E-02

10332E

-02

43352E+

0044332E-03

12793E

-02

10971E+0

0Std

14865E

-02

12632E

-02

42518E-01

79408E

-03

10155E

-02

10766E

-02

F18

Mean

50849E+

0038253E+

0020946

E+01

49225E+

00490

48E+

0021497E+

01Std

16014E

+00

14627E

+00

76856E

-01

21737E+

00204

11E+0

013

669E

+00

F19

Mean

268

41E-07

19292E

+02

49808E+

0519

677E

+02

240

98E+

0230226E+

04Std

32619E-08

15971E+0

214

706E

+05

16572E

+02

23149E+

0260289E+

03F2

0Mean

25989E-07

47006

E-06

33592E-02

44469E-08

18865E

-07

1540

6E-01

Std

59383E-07

73387E

-06

22456E-02

10350E

-07

31612E-07

56719E-02

F21

Mean

000

00E+

0070

841E-13

25769E+

02000

00E+

0045409E-11

30881E+

02Std

000

00E+

0045361E-13

90973E

+00

000

00E+

0019

882E

-11

27305E+

01F2

2Mean

000

00E+

007746

7E-13

23335E+

02000

00E+

00644

03E-11

25509E+

02Std

000

00E+

0036979E-13

15942E

+01

000

00E+

0033820E-11

26992E+

01F2

3Mean

93987E

-01

52987E-01

12199E

+01

13599E

+00

14399E

+00

21878E+

00Std

21705E-01

12517E

-01

49304

E-01

36576E-01

21705E-01

62731E-02

F24

Mean

14921E-14

37233E-04

32412E+

0147458E-09

42553E-03

26924E+

01Std

17226E

-1498

846E

-05

11649E

+00

28242E-09

42975E-04

35559E+

00F2

5Mean

29494E+

0110

724E

+02

11372E

+07

404

62E+

0193530E+

0092

421E+0

3Std

29743E+

0151800

E+01

31606

E+06

39685E+

0190392E+

0018

838E

+03

Complexity 17


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

85798E-14

24174E-04

18459E

+01

7404

4E-12

73673E

-07

82226E+

00Std

17360E

-1455274E-05

1944

7E-01

88139E-13

80222E-08

42517E+

00F13

Mean

22538E-15

29492E-04

71594E

+01

21041E-11

32004

E-06

60959E+

01Std

18688E

-1510

372E

-04

45394E+

0015

865E

-11

15334E

-06

44766

E+00

F14

Mean

71759E

-1120261E+

0043430E+

0277682E

-05

33324E-02

42669E+

02Std

24650E-11

50770E-01

14055E

+01

54975E-06

10537E

-03

80127E+

01F15

Mean

16716E

+00

12749E

+00

22241E+

0116

927E

+00

14937E

+00

21617E+

01Std

76572E

-01

43985E-01

33014E-01

47677E-01

63574E-01

54534E-01

F16

Mean

94233E-33

13057E

-09

76995E

+07

17755E

-24

846

48E-14

69921E+

00Std

14425E

-48

37533E-10

21712E+

0719

092E

-24

17429E

-13

89129E-01

F17

Mean

76377E

-03

14219E

-02

1160

6E+0

164039E-03

10080E

-02

1264

1E+0

0Std

57418E-03

21089E-02

46282E-01

70807E

-03

13952E

-02

16555E

-02

F18

Mean

83103E+

0079

047E

+00

39689E+

0189467E+

0096

041E+0

038726E+

01Std

260

72E+

0025432E+

0077616E

-01

78506E

-01

21029E+

0013

015E

+00

F19

Mean

45562E+

0126833E+

04806

68E+

0616

118E+

0413

155E

+04

70015E

+05

Std

38094E+

0121743E+

0421709E+

0612

498E

+04

1300

9E+0

497

174E

+04

F20

Mean

43064E-08

25702E-04

11519E

-01

52365E-08

16998E

-06

500

47E-01

Std

44294E-08

27576E-04

39417E-02

95247E

-08

49881E-06

26305E-01

F21

Mean

000

00E+

0011310E

-06

53146

E+02

000

00E+

0023711E

-10

58748E+

02Std

000

00E+

0033614E-07

32117E+

01000

00E+

0045437E-11

29507E+

01F2

2Mean

000

00E+

0016

167E

-06

48729E+

02000

00E+

00244

07E-10

52060

E+02

Std

000

00E+

0063216E-07

24382E+

01000

00E+

0075

889E

-11

42230E+

01F2

3Mean

13699E

+00

89987E-01

21237E+

0122699E+

0025899E+

0035955E+

00Std

23594E-01

666

67E-02

58033E-01

41913E-01

62973E-01

12247E

-01

F24

Mean

71054E

-1426826E-02

63090E+

0119

033E

-08

96037E

-03

47263E+

01Std

27621E-14

47780E-03

22392E+

0061075E-09

97071E-04

52689E+

00F2

5Mean

66563E+

0184722E+

0211275E

+08

65780E+

0139992E+

0188242E+

04Std

10992E

+02

2113

8E+0

221091E+

0794

954E

+01

43819E+

0116

832E

+04

18 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

18989E

-1316

584E

-01

19996E

+01

17809E

-11

14744E

-06

13554E

+01

Std

20566

E-14

53720E-02

90319E

-02

19159E

-12

18930E

-07

60821E+

00F13

Mean

22871E-15

17736E

-01

1944

7E+0

213

452E

-10

13291E-05

16379E

+02

Std

26741E-15

53611E-02

62653E+

0038592E-11

55001E-06

13313E

+01

F14

Mean

18736E

-1074

259E

+01

10132E

+03

22866

E-04

83534E-02

95534E

+02

Std

37223E-11

19144E

+01

18986E

+01

14283E

-05

10592E

-02

53523E+

01F15

Mean

26814E+

0010

178E

+01

47083E+

0128083E+

0034325E+

0045859E+

01Std

73851E-01

16238E

+00

22513E-01

46148E-01

60283E-01

69914E-01

F16

Mean

47116E-33

244

54E-04

90382E

+08

81890E-24

62347E-14

27647E+

03Std

72124E

-49

59650E-05

64985E+

0767958E-24

55604

E-14

44231E+

03F17

Mean

34494E-03

11896E

-02

37816E+

0134509E-03

41885E-03

21280E+

00Std

60565E-03

65363E-03

15922E

+00

46765E-03

86153E-03

54359E-02

F18

Mean

18033E

+01

17806E

+01

86826E+

0118

319E

+01

18828E

+01

82458E+

01Std

19652E

+00

38319E+

0093

222E

-01

29296E+

0025377E+

0015

159E

+00

F19

Mean

82462E+

0427944

E+06

48046

E+08

28415E+

0560265E+

0549201E+

07Std

55732E+

0489703E+

0596

715E

+07

24572E+

0527137E+

0572

772E

+06

F20

Mean

57130E-07

81688E-03

96848E

-01

13631E-06

27143E-05

21656E+

00Std

61122E-07

53195E-03

44542E-01

25155E-06

58766

E-05

80368E-01

F21

Mean

000

00E+

0051414E+

0013

305E

+03

000

00E+

0020026E-09

13623E

+03

Std

000

00E+

0017

825E

+00

22890E+

01000

00E+

0032815E-10

609

96E+

01F2

2Mean

000

00E+

0077

848E

+00

1260

9E+0

3000

00E+

0020383E-09

12745E

+03

Std

000

00E+

0023732E+

0029100

E+01

000

00E+

0029753E-10

59708E+

01F2

3Mean

25599E+

0020499E+

0039804

E+01

47099E+

0043699E+

0073

691E+0

0Std

36878E-01

15092E

-01

69296E-01

59151E-01

56184E-01

17989E

-01

F24

Mean

40927E-13

26229E+

0015

145E

+02

18874E

-07

29476E-02

10478E

+02

Std

88061E-14

63367E-01

42830E+

0037074E-08

17697E

-03

11873E

+01

F25

Mean

42987E+

028117

8E+0

313

524E

+09

56790E+

0244982E+

0218

038E

+06

Std

43423E+

0233128E+

0278

399E

+07

54327E+

0246926E+

0221315E+

05

Complexity 19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus8

minus6

minus4

minus2

02468

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(f) F24




MAE = sum119873119895=1 10038161003816100381610038161003816119898119895 minus 11989611989510038161003816100381610038161003816119873 (22)

20 Complexity

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

ISSASSAPSO

CMFOAIFFOFOA

1

2

3

4

5

6

7

8

Mea

n Er

rors

(log)

0 4000 6000 8000 100002000Iteration

(e) F19

ISSASSAPSO

CMFOAIFFOFOA

20000 6000 8000 100004000Iteration

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




Complexity 21

Table11

Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

31365E+

0413

133E

+06

11295E

+08

10017E

+06

864

75E+

0513

449E

+07

Std

18602E

+04

52974E+

0531387E+

0748689E+

0548607E+

0528405E+

06F2

7Mean

304

00E-10

1844

6E+0

484500

E+09

10512E

+04

12359E

+04

58535E+

08Std

61535E-10

14049E

+04

10125E

+09

12485E

+04

11922E

+04

38771E+

07F2

8Mean

42105E-01

46710E+

0173

819E

+02

4114

7E+0

137814E+

0114

226E

+02

Std

12624E

+00

31490E+

0199

455E

+01

47336E+

0134110E+

0129201E+

01F2

9Mean

75177E

+03

14891E+0

529286E+

0647277E+

0531099E+

0539826E+

05Std

33119E+

0368316E+

049190

4E+0

521021E+

0522686E+

0515

511E+0

5F3

0Mean

31524E+

0231524E+

0238129E+

0231524E+

0231524E+

0232568E+

02Std

85708E-12

19710E

-07

14082E

+01

11524E

-1145680E-11

58955E+

00F31

Mean

23483E+

0223172E+

0230117E+

0223811E

+02

23858E+

0224179E+

02Std

41748E+

0072

461E+0

048903E+

00560

97E+

0050249E+

0090

228E

+00

F32

Mean

20790E+

02206

03E+

0221884E+

0221485E+

0220975E+

0220633E+

02Std

41618E+

0032456E+

0030353E+

0087909E+

0057719E+

0016

880E

+00

22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








4 Complexity




Levy (119909) = 001 times 119903119886 times 120590100381610038161003816100381611990311988710038161003816100381610038161120573 (13)



(14)









119865119878119899119890119908119886119905=

119865119878119900119897119889119886119905 + 119889119892119866119888 (119865119878119900119897119889ℎ119905 minus 119865119878119900119897119889119886119905 ) if 1198771 ge 119875119889119901119862119909 (119865119878119900119897119889119886119905 119864119899119867119890) 119900119905ℎ119890119903119908119894119904119890(16)

119865119878119899119890119908119899119905=

119865119878119900119897119889119899119905 + 119889119892119866119888 (119865119878119900119897119889119886119905 minus 119865119878119900119897119889119899119905 ) if 1198772 ge 119875119889119901119862119909 (119865119878119900119897119889119899119905 119864119899119867119890) 119900119905ℎ119890119903119908119894119904119890(17)

119865119878119899119890119908119899119905=

119865119878119900119897119889119899119905 + 119889119892119866119888 (119865119878119900119897119889ℎ119905 minus 119865119878119900119897119889119899119905 ) if 1198773 ge 119875119889119901119862119909 (119865119878119900119897119889119899119905 119864119899119867119890) 119900119905ℎ119890119903119908119894119904119890(18)


119864119899 = 119864119899119898119886119909 times (1 minus 119868119905119890119903119868119905119890119903119898119886119909)10 (19)



119865119878119894 = 119865119878119899119890119908119894 if 119891119899119890119908119894 lt 119891119900119897119889119894119865119878119900119897119889119894 119900119905ℎ119890119903119908119894119904119890 (20)

Complexity 5








end

119878119905119888 = radic 119899sum119896=1

(119865119878119905119886119905119896 minus 119865119878ℎ119905119896)2 119878119888119898119894119899 = 10119864 minus 6365119868119905119890119903(119868119905119890119903max)25






119865119878119899119890119908119887119890119904119905119895 = 119862119909 (119865119878119900119897119889119887119890119904119905119895 119864119899119867119890) 119895 = 1 2 119899 (21)




6 Complexity







end



if 119891119899119890119908119887119890119904119905 lt 119891119887119890119904119905 119865119878119887119890119904119905119895 = 119865119878119899119890119908119887119890119904119905119895119891119887119890119904119905 = 119891119899119890119908119887119890119904119905end

end 119868119905119890119903 = 119868119905119890119903 + 1end


Complexity 7




Function Range Fmin

F1(119909) = 119899sum119894=1

1198941199092119894 [minus10 10] 0

F2(119909) = 119899sum119894=2



1199092119894) [minus1 1] -1

F4(119909) = 119899sum119894=1


F5(119909) = 119899sum119894=1

1198941199094119894 + rand () [minus128 128] 0

F6(119909) = 119899minus1sum119894=1


F7(119909) = 119899sum119894=1

( 119894sum119895=1

1199092119895) [minus100 100] 0

F8(119909) = max 10038161003816100381610038161199091198941003816100381610038161003816 1 le 119894 le 119899 [minus100 100] 0

F9(119909) = 119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 119899prod119894=1

10038161003816100381610038161199091198941003816100381610038161003816 [minus10 10] 0

F10(119909) = 119899sum119894=1

1199092119894 [minus100 100] 0

F11(119909) = 119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816119894+1 [minus1 1] 0



8 Complexity









Complexity 9


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

22612E-46

14374E

-13

18031E+0

338546

E-22

53419E-12

58887E+

01Std

39697E-46

10187E

-13

16038E

+02

1146

4E-22

26506

E-12

10717E

+01

F2Mean

53333E-01

600

00E-01

70475E

+04

666

67E-01

666

67E-01

12221E+0

2Std

28109E-01

21082E-01

18027E

+04

11102E

-16

364

14E-11

35452E+

01F3

Mean

000

00E+

00000

00E+

0047300

E-01

000

00E+

00860

42E-14

18586E

-02

Std

18504E

-1626168E-16

42225E-02

906

49E-17

27669E-14

19010E

-03

F4Mean

21268E-39

39943E-10

92617E

+07

37704

E-18

20345E-08

11112

E+07

Std

564

86E-39

32855E-10

18784E

+07

24165E-18

14277E

-08

30272E+

06F5

Mean

43164

E-03

93054E

-02

55376E+

0032231E-03

22754E-03

17965E

-02

Std

19931E-03

23058E-02

10210E

+00

13321E-03

1104

6E-03

31904

E-03

F6Mean

55447E-14

29695E+

0110

669E

+07

76347E

+00

89052E+

0012

862E

+04

Std

54894E-14

340

74E+

0118

313E

+06

590

03E+

0071150E

+00

44338E+

03F7

Mean

11996E

-44

65616E-12

16688E

+05

71055E

-22

60744

E-12

54708E+

03Std

304

49E-44

540

85E-12

17265E

+04

16506E

-22

29515E-12

49070E+

02F8

Mean

35080E-13

17850E

-03

45639E+

0127020E-11

264

40E-06

78561E+0

0Std

70894E

-1343281E-04

20578E+

00604

13E-12

24822E-07

17334E

+00

F9Mean

55772E-24

22148E-07

71792E

+01

23880E-11

204

15E-06

304

53E+

01Std

79227E

-24

67302E-08

30539E+

0141360

E-12

36636E-07

46515E+

01F10

Mean

26748E-44

44745E-13

13098E

+04

42105E-23

440

42E-13

36501E+

02Std

78461E-44

37055E-13

13116

E+03

10825E

-23

15887E

-13

43608E+

01F11

Mean

61803E-188

17833E

-60

18391E-03

78265E

-25

5117

6E-15

67271E-07

Std

000

00E+

0037202E-60

11147E

-03

65934E-25

76076E

-15

46836E-07

10 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

19373E

-45

244

08E-07

89004

E+03

36571E-21

32632E-11

28288E+

02Std

40611E

-45

72161E-08

10186E

+03

90310E

-22

82802E-12

18277E

+01

F2Mean

666

67E-01

21716E+

0080535E+

0512

514E

+00

666

67E-01

82661E+

02Std

82003E-16

22142E+

0011670E

+05

12485E

+00

25423E-10

77814E

+01

F3Mean

000

00E+

0076

318E

-11

866

02E-01

000

00E+

004110

0E-13

51246

E-02

Std

22204E-16

22120E-11

18360E

-02

23984E-16

14800E

-13

40430E-03

F4Mean

306

71E-38

97033E

-04

46756E+

0819

432E

-17

10621E-07

38893E+

07Std

69186E-38

344

64E-04

60135E+

0711627E

-17

76083E

-08

73301E+0

6F5

Mean

71557E

-03

29566

E-01

53164

E+01

10084E

-02

73580E

-03

10458E

-01

Std

23021E-03

33200

E-02

64915E+

0026523E-03

18100E

-03

26586E-02

F6Mean

43706

E-11

95471E+0

170

118E+

0777

331E+0

147194E+

0165331E+

04Std

95151E-11

35358E+

0153302E+

06344

63E+

0140976E+

0113

721E+0

4F7

Mean

12947E

-41

23079E-05

91659E

+05

69771E-21

64025E-11

28862E+

04Std

37876E-41

58814E-06

93287E

+04

31808E-21

26901E-11

28375E+

03F8

Mean

60872E-11

12706E

-01

67093E+

0184930E-11

71576E

-06

11919E

+01

Std

25158E-11

33391E-02

25011E

+00

11107E

-11

69032E-07

1040

4E+0

0F9

Mean

18289E

-23

38822E-04

20565E+

1060338E-11

63442E-06

11434E

+05

Std

28884E-23

72525E

-05

63864

E+10

64165E-12

90797E

-07

24588E+

05F10

Mean

12924E

-44

14594E

-06

41629E+

0422846

E-22

20175E-12

10536E

+03

Std

25807E-44

606

62E-07

37125E+

0341424E-23

52761E-13

59785E+

01F11

Mean

44745E-163

19208E

-58

92852E

-03

11169E

-24

51458E-15

16917E

-06

Std

000

00E+

0022612E-58

35776E-03

14674E

-24

82130E-15

94517E

-07

Complexity 11


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

52002E-44

1360

1E-01

64559E+

0467630E-20

540

42E-10

23688E+

03Std

89565E-44

28398E-02

27675E+

0318

636E

-20

10843E

-10

11782E

+02

F2Mean

25837E+

0028507E+

0189058E+

0610

018E

+01

11850E

+01

13921E+0

4Std

40923E+

0012

482E

+01

64125E+

0598

260E

+00

67378E+

0021479E+

03F3

Mean

19984E

-1517

506E

-05

99937E

-01

19984E

-1534570E-12

1946

0E-01

Std

27940E-16

33607E-06

19874E

-04

39686E-16

57323E-13

11259E

-02

F4Mean

14073E

-38

30139E+

0226151E+

0914

261E-16

10212E

-06

20499E+

08Std

15382E

-38

90551E+0

126783E+

0875

737E

-17

33853E-07

35388E+

07F5

Mean

17615E

-02

14345E

+00

59603E+

0237550E-02

29349E-02

12443E

+00

Std

42239E-03

13985E

-01

58412E+

0112

602E

-02

45825E-03

18471E-01

F6Mean

11417E

+01

58422E+

0242299E+

0817

988E

+02

16578E

+02

560

78E+

05Std

30258E+

0197

884E

+02

48581E+

0739022E+

01466

85E+

0165477E+

04F7

Mean

16881E-41

12984E

+01

64707E+

0611852E

-19

74831E-10

22865E+

05Std

34134E-41

22729E+

0033435E+

0531718E-20

95033E

-11

20650E+

04F8

Mean

45259E-08

39819E+

0085137E+

0117

042E

-04

29244

E-05

33956E+

01Std

29104E-08

41522E-01

12566E

+00

78665E

-05

27018E-06

47713E+

00F9

Mean

12222E

-22

36814E-01

72469E

+32

25070E-10

23795E-05

14112

E+27

Std

84369E-23

41963E-02

20633E+

3323874E-11

13879E

-06

44627E+

27F10

Mean

34254E-42

37261E-01

14940E

+05

20714E-21

18486E

-11

43041E+

03Std

966

08E-42

92561E-02

446

43E+

03464

07E-22

23956E-12

24348E+

02F11

Mean

1640

0E-12

315

040E

-52

37278E-02

65780E-24

3117

4E-14

10720E

-05

Std

51861E-123

16670E

-52

11165E

-02

72960E

-24

36245E-14

59475E-06

12 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

(b) F4

0 2000 4000 6000 8000 10000

0

5

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus15

minus5

(c) F6

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

minus50

(d) F7

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus2

minus4

minus6

minus8

minus10

minus12

minus14

(e) F8

0 2000 4000 6000 8000 10000

05

1015

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus5

minus10

minus20

minus25

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

Mea

n Er

rors

(log)

(h) F11


Complexity 13

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus10

minus5

0

5

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

0

10

20

30

Mea

n Er

rors

(log)

2000 4000 6000 8000 100000Iteration

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

50

Mea

n Er

rors

(log)

(h) F11


14 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

20

Mea

n Er

rors

(log)

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

0

2

4

6

8

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

01020304050

Mea

n Er

rors

(log)

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus140

minus120

minus100

minus80

minus60

minus40

minus20

020

Mea

n Er

rors

(log)

(h) F11


Complexity 15

Table6Multim

odalbenchm

arkfunctio

ns

Functio

nRa

nge

Fmin

F12(119909)=

minus20exp(minus0

2radic1 119899119899 sum 119894=11199092 119894)minus

exp(1 119899119899 sum 119894=1co

s(2120587119909 119894))

+20+exp

(1 )[minus32

32]0

F13(119909)=

119899 sum 119894=11003816 1003816 1003816 1003816119909 119894sin(119909 119894)

+01119909 1198941003816 1003816 1003816 1003816

[minus1010]

0

F14(119909)=

119891 119904(1199091119909 2)


(119909 119899119909 1)

[minus10010

0]0

119891 119904(119909119910)=

(1199092 +1199102 )025[sin2

(50(1199092 +

1199102 )01)+1

]F15(

119909)=119891 119904(1199091119909 2)


119909 1198991199091)

[minus10010

0]0

119891 119904(119909119910)=

05(sin2(radic 1199092+1199102

)minus05)

(1+0001

(1199092 +1199102 ))2

F16(119909)=

120587 11989910sin2

(120587119910 119894)+119899minus1 sum 119894=1

(119910 119894minus1 )2 [

1+10sin2

(120587119910 119894+1)]+

(119910 119899minus1 )2

+119899 sum 119894=1119906(119909 119894

10100

4)[minus50

50]0

119910 119894=1+1 4(119909

119894+1)

119906(119909 119894119886

119896119898)= 119896(119909 119894

minus119886)119898

119909 119894gt119886

0minus119886le

119909 119894le119886

119896(minus119909119894minus119886)119898

119909119894gt119886

F17(119909)=

1 4000119899 sum 119894=11199092 119894minus119899 prod 119894=1

cos(119909119894 radic 119894)+1

[minus10010

0]0

F18(119909)=


(minus(1199092 119894+

1199092 119894+1+05

119909 119894119909 119894+1)

8)lowastc

os(4radic

1199092 119894+1199092 119894+1

+05119909 119894119909 119894+1))

[minus55]

1-n

F19(119909)=

119899 sum 119894=1(119909119894minus1)2

minus119899 sum 119894=2119909 119894119909 119894minus1

[minusn2n2 ]

119899(119899+4)(119899

minus1)minus6

F20 (119909 )=

sum119899minus1 119894=2(05

+(sin2(radic 1

001199092 119894+1199092 119894+1)minus0

5))(1+

0001(1199092 119894minus

2119909 119894119909119894minus1+1199092 119894minus1))2

[minus10010

0]0

F21(119909)=

119899 sum 119894=1[1199092 119894minus10

cos(2120587

119909 119894)+10]

[minus51251

2]0

F22(119909)=

119899 sum 119894=1[1199102 119894minus10

cos(2120587

119910 119894)+10]

119910 119894= 119909 119894

1003816 1003816 1003816 1003816119909 1198941003816 1003816 1003816 1003816lt05

119903119900119906119899119889(2119909

119894)2

1003816 1003816 1003816 10038161199091198941003816 1003816 1003816 1003816lt0

5[minus51

2512]

0

F23(119909)=

1minuscos(2120587

radic119899 sum 119894=11199092 119894)

+01radic119899 sum 119894=1

1199092 119894[minus10

0100]

0

F24(119909)=

119899 sum 119894=1119896119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 (119909119894+05

))]minus119899119896

119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 05

)][minus05

05]0

F25(119909)=

119899 sum 119896=1119899 sum 119895=1((1199102 119895119896 4000)minus

cos(119910 119895119896)+1

)119910119895119896=10

0(119909 119896minus1199092 119895

)2 +(1minus

1199092 119895)2[minus10

0100]

0

16 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmultim

odalbenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

51692E-14

21708E-07

16343E

+01

42641E-12

43970E-07

70983E

+00

Std

94813E-15

11785E

-07

45830E-01

51275E-13

53024E-08

45755E+

00F13

Mean

19651E-15

17670E

-07

30865E+

0138781E-12

12507E

-06

29660

E+01

Std

17016E

-1510

899E

-07

28749E+

00200

14E-12

19125E

-06

57790E+

00F14

Mean

28586E-11

47414E-02

21576E+

0235954E-05

17290E

-02

18705E

+02

Std

17874E

-1118

105E

-02

50836E+

0019

343E

-06

10857E

-03

50868E+

01F15

Mean

99552E

-01

46150E-01

12596E

+01

94983E

-01

10032E

+00

12147E

+01

Std

38926E-01

33522E-01

21495E-01

42966

E-01

35690E-01

17388E

-01

F16

Mean

15705E

-32

13069E

-15

56725E+

0650290E-25

99726E

-15

31482E+

00Std

28850E-48

57169E-16

17168E

+06

47027E-25

85374E-15

58054E-01

F17

Mean

13781E-02

10332E

-02

43352E+

0044332E-03

12793E

-02

10971E+0

0Std

14865E

-02

12632E

-02

42518E-01

79408E

-03

10155E

-02

10766E

-02

F18

Mean

50849E+

0038253E+

0020946

E+01

49225E+

00490

48E+

0021497E+

01Std

16014E

+00

14627E

+00

76856E

-01

21737E+

00204

11E+0

013

669E

+00

F19

Mean

268

41E-07

19292E

+02

49808E+

0519

677E

+02

240

98E+

0230226E+

04Std

32619E-08

15971E+0

214

706E

+05

16572E

+02

23149E+

0260289E+

03F2

0Mean

25989E-07

47006

E-06

33592E-02

44469E-08

18865E

-07

1540

6E-01

Std

59383E-07

73387E

-06

22456E-02

10350E

-07

31612E-07

56719E-02

F21

Mean

000

00E+

0070

841E-13

25769E+

02000

00E+

0045409E-11

30881E+

02Std

000

00E+

0045361E-13

90973E

+00

000

00E+

0019

882E

-11

27305E+

01F2

2Mean

000

00E+

007746

7E-13

23335E+

02000

00E+

00644

03E-11

25509E+

02Std

000

00E+

0036979E-13

15942E

+01

000

00E+

0033820E-11

26992E+

01F2

3Mean

93987E

-01

52987E-01

12199E

+01

13599E

+00

14399E

+00

21878E+

00Std

21705E-01

12517E

-01

49304

E-01

36576E-01

21705E-01

62731E-02

F24

Mean

14921E-14

37233E-04

32412E+

0147458E-09

42553E-03

26924E+

01Std

17226E

-1498

846E

-05

11649E

+00

28242E-09

42975E-04

35559E+

00F2

5Mean

29494E+

0110

724E

+02

11372E

+07

404

62E+

0193530E+

0092

421E+0

3Std

29743E+

0151800

E+01

31606

E+06

39685E+

0190392E+

0018

838E

+03

Complexity 17


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

85798E-14

24174E-04

18459E

+01

7404

4E-12

73673E

-07

82226E+

00Std

17360E

-1455274E-05

1944

7E-01

88139E-13

80222E-08

42517E+

00F13

Mean

22538E-15

29492E-04

71594E

+01

21041E-11

32004

E-06

60959E+

01Std

18688E

-1510

372E

-04

45394E+

0015

865E

-11

15334E

-06

44766

E+00

F14

Mean

71759E

-1120261E+

0043430E+

0277682E

-05

33324E-02

42669E+

02Std

24650E-11

50770E-01

14055E

+01

54975E-06

10537E

-03

80127E+

01F15

Mean

16716E

+00

12749E

+00

22241E+

0116

927E

+00

14937E

+00

21617E+

01Std

76572E

-01

43985E-01

33014E-01

47677E-01

63574E-01

54534E-01

F16

Mean

94233E-33

13057E

-09

76995E

+07

17755E

-24

846

48E-14

69921E+

00Std

14425E

-48

37533E-10

21712E+

0719

092E

-24

17429E

-13

89129E-01

F17

Mean

76377E

-03

14219E

-02

1160

6E+0

164039E-03

10080E

-02

1264

1E+0

0Std

57418E-03

21089E-02

46282E-01

70807E

-03

13952E

-02

16555E

-02

F18

Mean

83103E+

0079

047E

+00

39689E+

0189467E+

0096

041E+0

038726E+

01Std

260

72E+

0025432E+

0077616E

-01

78506E

-01

21029E+

0013

015E

+00

F19

Mean

45562E+

0126833E+

04806

68E+

0616

118E+

0413

155E

+04

70015E

+05

Std

38094E+

0121743E+

0421709E+

0612

498E

+04

1300

9E+0

497

174E

+04

F20

Mean

43064E-08

25702E-04

11519E

-01

52365E-08

16998E

-06

500

47E-01

Std

44294E-08

27576E-04

39417E-02

95247E

-08

49881E-06

26305E-01

F21

Mean

000

00E+

0011310E

-06

53146

E+02

000

00E+

0023711E

-10

58748E+

02Std

000

00E+

0033614E-07

32117E+

01000

00E+

0045437E-11

29507E+

01F2

2Mean

000

00E+

0016

167E

-06

48729E+

02000

00E+

00244

07E-10

52060

E+02

Std

000

00E+

0063216E-07

24382E+

01000

00E+

0075

889E

-11

42230E+

01F2

3Mean

13699E

+00

89987E-01

21237E+

0122699E+

0025899E+

0035955E+

00Std

23594E-01

666

67E-02

58033E-01

41913E-01

62973E-01

12247E

-01

F24

Mean

71054E

-1426826E-02

63090E+

0119

033E

-08

96037E

-03

47263E+

01Std

27621E-14

47780E-03

22392E+

0061075E-09

97071E-04

52689E+

00F2

5Mean

66563E+

0184722E+

0211275E

+08

65780E+

0139992E+

0188242E+

04Std

10992E

+02

2113

8E+0

221091E+

0794

954E

+01

43819E+

0116

832E

+04

18 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

18989E

-1316

584E

-01

19996E

+01

17809E

-11

14744E

-06

13554E

+01

Std

20566

E-14

53720E-02

90319E

-02

19159E

-12

18930E

-07

60821E+

00F13

Mean

22871E-15

17736E

-01

1944

7E+0

213

452E

-10

13291E-05

16379E

+02

Std

26741E-15

53611E-02

62653E+

0038592E-11

55001E-06

13313E

+01

F14

Mean

18736E

-1074

259E

+01

10132E

+03

22866

E-04

83534E-02

95534E

+02

Std

37223E-11

19144E

+01

18986E

+01

14283E

-05

10592E

-02

53523E+

01F15

Mean

26814E+

0010

178E

+01

47083E+

0128083E+

0034325E+

0045859E+

01Std

73851E-01

16238E

+00

22513E-01

46148E-01

60283E-01

69914E-01

F16

Mean

47116E-33

244

54E-04

90382E

+08

81890E-24

62347E-14

27647E+

03Std

72124E

-49

59650E-05

64985E+

0767958E-24

55604

E-14

44231E+

03F17

Mean

34494E-03

11896E

-02

37816E+

0134509E-03

41885E-03

21280E+

00Std

60565E-03

65363E-03

15922E

+00

46765E-03

86153E-03

54359E-02

F18

Mean

18033E

+01

17806E

+01

86826E+

0118

319E

+01

18828E

+01

82458E+

01Std

19652E

+00

38319E+

0093

222E

-01

29296E+

0025377E+

0015

159E

+00

F19

Mean

82462E+

0427944

E+06

48046

E+08

28415E+

0560265E+

0549201E+

07Std

55732E+

0489703E+

0596

715E

+07

24572E+

0527137E+

0572

772E

+06

F20

Mean

57130E-07

81688E-03

96848E

-01

13631E-06

27143E-05

21656E+

00Std

61122E-07

53195E-03

44542E-01

25155E-06

58766

E-05

80368E-01

F21

Mean

000

00E+

0051414E+

0013

305E

+03

000

00E+

0020026E-09

13623E

+03

Std

000

00E+

0017

825E

+00

22890E+

01000

00E+

0032815E-10

609

96E+

01F2

2Mean

000

00E+

0077

848E

+00

1260

9E+0

3000

00E+

0020383E-09

12745E

+03

Std

000

00E+

0023732E+

0029100

E+01

000

00E+

0029753E-10

59708E+

01F2

3Mean

25599E+

0020499E+

0039804

E+01

47099E+

0043699E+

0073

691E+0

0Std

36878E-01

15092E

-01

69296E-01

59151E-01

56184E-01

17989E

-01

F24

Mean

40927E-13

26229E+

0015

145E

+02

18874E

-07

29476E-02

10478E

+02

Std

88061E-14

63367E-01

42830E+

0037074E-08

17697E

-03

11873E

+01

F25

Mean

42987E+

028117

8E+0

313

524E

+09

56790E+

0244982E+

0218

038E

+06

Std

43423E+

0233128E+

0278

399E

+07

54327E+

0246926E+

0221315E+

05

Complexity 19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus8

minus6

minus4

minus2

02468

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(f) F24




MAE = sum119873119895=1 10038161003816100381610038161003816119898119895 minus 11989611989510038161003816100381610038161003816119873 (22)

20 Complexity

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

ISSASSAPSO

CMFOAIFFOFOA

1

2

3

4

5

6

7

8

Mea

n Er

rors

(log)

0 4000 6000 8000 100002000Iteration

(e) F19

ISSASSAPSO

CMFOAIFFOFOA

20000 6000 8000 100004000Iteration

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




Complexity 21

Table11

Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

31365E+

0413

133E

+06

11295E

+08

10017E

+06

864

75E+

0513

449E

+07

Std

18602E

+04

52974E+

0531387E+

0748689E+

0548607E+

0528405E+

06F2

7Mean

304

00E-10

1844

6E+0

484500

E+09

10512E

+04

12359E

+04

58535E+

08Std

61535E-10

14049E

+04

10125E

+09

12485E

+04

11922E

+04

38771E+

07F2

8Mean

42105E-01

46710E+

0173

819E

+02

4114

7E+0

137814E+

0114

226E

+02

Std

12624E

+00

31490E+

0199

455E

+01

47336E+

0134110E+

0129201E+

01F2

9Mean

75177E

+03

14891E+0

529286E+

0647277E+

0531099E+

0539826E+

05Std

33119E+

0368316E+

049190

4E+0

521021E+

0522686E+

0515

511E+0

5F3

0Mean

31524E+

0231524E+

0238129E+

0231524E+

0231524E+

0232568E+

02Std

85708E-12

19710E

-07

14082E

+01

11524E

-1145680E-11

58955E+

00F31

Mean

23483E+

0223172E+

0230117E+

0223811E

+02

23858E+

0224179E+

02Std

41748E+

0072

461E+0

048903E+

00560

97E+

0050249E+

0090

228E

+00

F32

Mean

20790E+

02206

03E+

0221884E+

0221485E+

0220975E+

0220633E+

02Std

41618E+

0032456E+

0030353E+

0087909E+

0057719E+

0016

880E

+00

22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 5








end

119878119905119888 = radic 119899sum119896=1

(119865119878119905119886119905119896 minus 119865119878ℎ119905119896)2 119878119888119898119894119899 = 10119864 minus 6365119868119905119890119903(119868119905119890119903max)25






119865119878119899119890119908119887119890119904119905119895 = 119862119909 (119865119878119900119897119889119887119890119904119905119895 119864119899119867119890) 119895 = 1 2 119899 (21)




6 Complexity







end



if 119891119899119890119908119887119890119904119905 lt 119891119887119890119904119905 119865119878119887119890119904119905119895 = 119865119878119899119890119908119887119890119904119905119895119891119887119890119904119905 = 119891119899119890119908119887119890119904119905end

end 119868119905119890119903 = 119868119905119890119903 + 1end


Complexity 7




Function Range Fmin

F1(119909) = 119899sum119894=1

1198941199092119894 [minus10 10] 0

F2(119909) = 119899sum119894=2



1199092119894) [minus1 1] -1

F4(119909) = 119899sum119894=1


F5(119909) = 119899sum119894=1

1198941199094119894 + rand () [minus128 128] 0

F6(119909) = 119899minus1sum119894=1


F7(119909) = 119899sum119894=1

( 119894sum119895=1

1199092119895) [minus100 100] 0

F8(119909) = max 10038161003816100381610038161199091198941003816100381610038161003816 1 le 119894 le 119899 [minus100 100] 0

F9(119909) = 119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 119899prod119894=1

10038161003816100381610038161199091198941003816100381610038161003816 [minus10 10] 0

F10(119909) = 119899sum119894=1

1199092119894 [minus100 100] 0

F11(119909) = 119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816119894+1 [minus1 1] 0



8 Complexity









Complexity 9


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

22612E-46

14374E

-13

18031E+0

338546

E-22

53419E-12

58887E+

01Std

39697E-46

10187E

-13

16038E

+02

1146

4E-22

26506

E-12

10717E

+01

F2Mean

53333E-01

600

00E-01

70475E

+04

666

67E-01

666

67E-01

12221E+0

2Std

28109E-01

21082E-01

18027E

+04

11102E

-16

364

14E-11

35452E+

01F3

Mean

000

00E+

00000

00E+

0047300

E-01

000

00E+

00860

42E-14

18586E

-02

Std

18504E

-1626168E-16

42225E-02

906

49E-17

27669E-14

19010E

-03

F4Mean

21268E-39

39943E-10

92617E

+07

37704

E-18

20345E-08

11112

E+07

Std

564

86E-39

32855E-10

18784E

+07

24165E-18

14277E

-08

30272E+

06F5

Mean

43164

E-03

93054E

-02

55376E+

0032231E-03

22754E-03

17965E

-02

Std

19931E-03

23058E-02

10210E

+00

13321E-03

1104

6E-03

31904

E-03

F6Mean

55447E-14

29695E+

0110

669E

+07

76347E

+00

89052E+

0012

862E

+04

Std

54894E-14

340

74E+

0118

313E

+06

590

03E+

0071150E

+00

44338E+

03F7

Mean

11996E

-44

65616E-12

16688E

+05

71055E

-22

60744

E-12

54708E+

03Std

304

49E-44

540

85E-12

17265E

+04

16506E

-22

29515E-12

49070E+

02F8

Mean

35080E-13

17850E

-03

45639E+

0127020E-11

264

40E-06

78561E+0

0Std

70894E

-1343281E-04

20578E+

00604

13E-12

24822E-07

17334E

+00

F9Mean

55772E-24

22148E-07

71792E

+01

23880E-11

204

15E-06

304

53E+

01Std

79227E

-24

67302E-08

30539E+

0141360

E-12

36636E-07

46515E+

01F10

Mean

26748E-44

44745E-13

13098E

+04

42105E-23

440

42E-13

36501E+

02Std

78461E-44

37055E-13

13116

E+03

10825E

-23

15887E

-13

43608E+

01F11

Mean

61803E-188

17833E

-60

18391E-03

78265E

-25

5117

6E-15

67271E-07

Std

000

00E+

0037202E-60

11147E

-03

65934E-25

76076E

-15

46836E-07

10 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

19373E

-45

244

08E-07

89004

E+03

36571E-21

32632E-11

28288E+

02Std

40611E

-45

72161E-08

10186E

+03

90310E

-22

82802E-12

18277E

+01

F2Mean

666

67E-01

21716E+

0080535E+

0512

514E

+00

666

67E-01

82661E+

02Std

82003E-16

22142E+

0011670E

+05

12485E

+00

25423E-10

77814E

+01

F3Mean

000

00E+

0076

318E

-11

866

02E-01

000

00E+

004110

0E-13

51246

E-02

Std

22204E-16

22120E-11

18360E

-02

23984E-16

14800E

-13

40430E-03

F4Mean

306

71E-38

97033E

-04

46756E+

0819

432E

-17

10621E-07

38893E+

07Std

69186E-38

344

64E-04

60135E+

0711627E

-17

76083E

-08

73301E+0

6F5

Mean

71557E

-03

29566

E-01

53164

E+01

10084E

-02

73580E

-03

10458E

-01

Std

23021E-03

33200

E-02

64915E+

0026523E-03

18100E

-03

26586E-02

F6Mean

43706

E-11

95471E+0

170

118E+

0777

331E+0

147194E+

0165331E+

04Std

95151E-11

35358E+

0153302E+

06344

63E+

0140976E+

0113

721E+0

4F7

Mean

12947E

-41

23079E-05

91659E

+05

69771E-21

64025E-11

28862E+

04Std

37876E-41

58814E-06

93287E

+04

31808E-21

26901E-11

28375E+

03F8

Mean

60872E-11

12706E

-01

67093E+

0184930E-11

71576E

-06

11919E

+01

Std

25158E-11

33391E-02

25011E

+00

11107E

-11

69032E-07

1040

4E+0

0F9

Mean

18289E

-23

38822E-04

20565E+

1060338E-11

63442E-06

11434E

+05

Std

28884E-23

72525E

-05

63864

E+10

64165E-12

90797E

-07

24588E+

05F10

Mean

12924E

-44

14594E

-06

41629E+

0422846

E-22

20175E-12

10536E

+03

Std

25807E-44

606

62E-07

37125E+

0341424E-23

52761E-13

59785E+

01F11

Mean

44745E-163

19208E

-58

92852E

-03

11169E

-24

51458E-15

16917E

-06

Std

000

00E+

0022612E-58

35776E-03

14674E

-24

82130E-15

94517E

-07

Complexity 11


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

52002E-44

1360

1E-01

64559E+

0467630E-20

540

42E-10

23688E+

03Std

89565E-44

28398E-02

27675E+

0318

636E

-20

10843E

-10

11782E

+02

F2Mean

25837E+

0028507E+

0189058E+

0610

018E

+01

11850E

+01

13921E+0

4Std

40923E+

0012

482E

+01

64125E+

0598

260E

+00

67378E+

0021479E+

03F3

Mean

19984E

-1517

506E

-05

99937E

-01

19984E

-1534570E-12

1946

0E-01

Std

27940E-16

33607E-06

19874E

-04

39686E-16

57323E-13

11259E

-02

F4Mean

14073E

-38

30139E+

0226151E+

0914

261E-16

10212E

-06

20499E+

08Std

15382E

-38

90551E+0

126783E+

0875

737E

-17

33853E-07

35388E+

07F5

Mean

17615E

-02

14345E

+00

59603E+

0237550E-02

29349E-02

12443E

+00

Std

42239E-03

13985E

-01

58412E+

0112

602E

-02

45825E-03

18471E-01

F6Mean

11417E

+01

58422E+

0242299E+

0817

988E

+02

16578E

+02

560

78E+

05Std

30258E+

0197

884E

+02

48581E+

0739022E+

01466

85E+

0165477E+

04F7

Mean

16881E-41

12984E

+01

64707E+

0611852E

-19

74831E-10

22865E+

05Std

34134E-41

22729E+

0033435E+

0531718E-20

95033E

-11

20650E+

04F8

Mean

45259E-08

39819E+

0085137E+

0117

042E

-04

29244

E-05

33956E+

01Std

29104E-08

41522E-01

12566E

+00

78665E

-05

27018E-06

47713E+

00F9

Mean

12222E

-22

36814E-01

72469E

+32

25070E-10

23795E-05

14112

E+27

Std

84369E-23

41963E-02

20633E+

3323874E-11

13879E

-06

44627E+

27F10

Mean

34254E-42

37261E-01

14940E

+05

20714E-21

18486E

-11

43041E+

03Std

966

08E-42

92561E-02

446

43E+

03464

07E-22

23956E-12

24348E+

02F11

Mean

1640

0E-12

315

040E

-52

37278E-02

65780E-24

3117

4E-14

10720E

-05

Std

51861E-123

16670E

-52

11165E

-02

72960E

-24

36245E-14

59475E-06

12 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

(b) F4

0 2000 4000 6000 8000 10000

0

5

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus15

minus5

(c) F6

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

minus50

(d) F7

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus2

minus4

minus6

minus8

minus10

minus12

minus14

(e) F8

0 2000 4000 6000 8000 10000

05

1015

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus5

minus10

minus20

minus25

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

Mea

n Er

rors

(log)

(h) F11


Complexity 13

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus10

minus5

0

5

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

0

10

20

30

Mea

n Er

rors

(log)

2000 4000 6000 8000 100000Iteration

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

50

Mea

n Er

rors

(log)

(h) F11


14 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

20

Mea

n Er

rors

(log)

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

0

2

4

6

8

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

01020304050

Mea

n Er

rors

(log)

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus140

minus120

minus100

minus80

minus60

minus40

minus20

020

Mea

n Er

rors

(log)

(h) F11


Complexity 15

Table6Multim

odalbenchm

arkfunctio

ns

Functio

nRa

nge

Fmin

F12(119909)=

minus20exp(minus0

2radic1 119899119899 sum 119894=11199092 119894)minus

exp(1 119899119899 sum 119894=1co

s(2120587119909 119894))

+20+exp

(1 )[minus32

32]0

F13(119909)=

119899 sum 119894=11003816 1003816 1003816 1003816119909 119894sin(119909 119894)

+01119909 1198941003816 1003816 1003816 1003816

[minus1010]

0

F14(119909)=

119891 119904(1199091119909 2)


(119909 119899119909 1)

[minus10010

0]0

119891 119904(119909119910)=

(1199092 +1199102 )025[sin2

(50(1199092 +

1199102 )01)+1

]F15(

119909)=119891 119904(1199091119909 2)


119909 1198991199091)

[minus10010

0]0

119891 119904(119909119910)=

05(sin2(radic 1199092+1199102

)minus05)

(1+0001

(1199092 +1199102 ))2

F16(119909)=

120587 11989910sin2

(120587119910 119894)+119899minus1 sum 119894=1

(119910 119894minus1 )2 [

1+10sin2

(120587119910 119894+1)]+

(119910 119899minus1 )2

+119899 sum 119894=1119906(119909 119894

10100

4)[minus50

50]0

119910 119894=1+1 4(119909

119894+1)

119906(119909 119894119886

119896119898)= 119896(119909 119894

minus119886)119898

119909 119894gt119886

0minus119886le

119909 119894le119886

119896(minus119909119894minus119886)119898

119909119894gt119886

F17(119909)=

1 4000119899 sum 119894=11199092 119894minus119899 prod 119894=1

cos(119909119894 radic 119894)+1

[minus10010

0]0

F18(119909)=


(minus(1199092 119894+

1199092 119894+1+05

119909 119894119909 119894+1)

8)lowastc

os(4radic

1199092 119894+1199092 119894+1

+05119909 119894119909 119894+1))

[minus55]

1-n

F19(119909)=

119899 sum 119894=1(119909119894minus1)2

minus119899 sum 119894=2119909 119894119909 119894minus1

[minusn2n2 ]

119899(119899+4)(119899

minus1)minus6

F20 (119909 )=

sum119899minus1 119894=2(05

+(sin2(radic 1

001199092 119894+1199092 119894+1)minus0

5))(1+

0001(1199092 119894minus

2119909 119894119909119894minus1+1199092 119894minus1))2

[minus10010

0]0

F21(119909)=

119899 sum 119894=1[1199092 119894minus10

cos(2120587

119909 119894)+10]

[minus51251

2]0

F22(119909)=

119899 sum 119894=1[1199102 119894minus10

cos(2120587

119910 119894)+10]

119910 119894= 119909 119894

1003816 1003816 1003816 1003816119909 1198941003816 1003816 1003816 1003816lt05

119903119900119906119899119889(2119909

119894)2

1003816 1003816 1003816 10038161199091198941003816 1003816 1003816 1003816lt0

5[minus51

2512]

0

F23(119909)=

1minuscos(2120587

radic119899 sum 119894=11199092 119894)

+01radic119899 sum 119894=1

1199092 119894[minus10

0100]

0

F24(119909)=

119899 sum 119894=1119896119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 (119909119894+05

))]minus119899119896

119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 05

)][minus05

05]0

F25(119909)=

119899 sum 119896=1119899 sum 119895=1((1199102 119895119896 4000)minus

cos(119910 119895119896)+1

)119910119895119896=10

0(119909 119896minus1199092 119895

)2 +(1minus

1199092 119895)2[minus10

0100]

0

16 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmultim

odalbenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

51692E-14

21708E-07

16343E

+01

42641E-12

43970E-07

70983E

+00

Std

94813E-15

11785E

-07

45830E-01

51275E-13

53024E-08

45755E+

00F13

Mean

19651E-15

17670E

-07

30865E+

0138781E-12

12507E

-06

29660

E+01

Std

17016E

-1510

899E

-07

28749E+

00200

14E-12

19125E

-06

57790E+

00F14

Mean

28586E-11

47414E-02

21576E+

0235954E-05

17290E

-02

18705E

+02

Std

17874E

-1118

105E

-02

50836E+

0019

343E

-06

10857E

-03

50868E+

01F15

Mean

99552E

-01

46150E-01

12596E

+01

94983E

-01

10032E

+00

12147E

+01

Std

38926E-01

33522E-01

21495E-01

42966

E-01

35690E-01

17388E

-01

F16

Mean

15705E

-32

13069E

-15

56725E+

0650290E-25

99726E

-15

31482E+

00Std

28850E-48

57169E-16

17168E

+06

47027E-25

85374E-15

58054E-01

F17

Mean

13781E-02

10332E

-02

43352E+

0044332E-03

12793E

-02

10971E+0

0Std

14865E

-02

12632E

-02

42518E-01

79408E

-03

10155E

-02

10766E

-02

F18

Mean

50849E+

0038253E+

0020946

E+01

49225E+

00490

48E+

0021497E+

01Std

16014E

+00

14627E

+00

76856E

-01

21737E+

00204

11E+0

013

669E

+00

F19

Mean

268

41E-07

19292E

+02

49808E+

0519

677E

+02

240

98E+

0230226E+

04Std

32619E-08

15971E+0

214

706E

+05

16572E

+02

23149E+

0260289E+

03F2

0Mean

25989E-07

47006

E-06

33592E-02

44469E-08

18865E

-07

1540

6E-01

Std

59383E-07

73387E

-06

22456E-02

10350E

-07

31612E-07

56719E-02

F21

Mean

000

00E+

0070

841E-13

25769E+

02000

00E+

0045409E-11

30881E+

02Std

000

00E+

0045361E-13

90973E

+00

000

00E+

0019

882E

-11

27305E+

01F2

2Mean

000

00E+

007746

7E-13

23335E+

02000

00E+

00644

03E-11

25509E+

02Std

000

00E+

0036979E-13

15942E

+01

000

00E+

0033820E-11

26992E+

01F2

3Mean

93987E

-01

52987E-01

12199E

+01

13599E

+00

14399E

+00

21878E+

00Std

21705E-01

12517E

-01

49304

E-01

36576E-01

21705E-01

62731E-02

F24

Mean

14921E-14

37233E-04

32412E+

0147458E-09

42553E-03

26924E+

01Std

17226E

-1498

846E

-05

11649E

+00

28242E-09

42975E-04

35559E+

00F2

5Mean

29494E+

0110

724E

+02

11372E

+07

404

62E+

0193530E+

0092

421E+0

3Std

29743E+

0151800

E+01

31606

E+06

39685E+

0190392E+

0018

838E

+03

Complexity 17


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

85798E-14

24174E-04

18459E

+01

7404

4E-12

73673E

-07

82226E+

00Std

17360E

-1455274E-05

1944

7E-01

88139E-13

80222E-08

42517E+

00F13

Mean

22538E-15

29492E-04

71594E

+01

21041E-11

32004

E-06

60959E+

01Std

18688E

-1510

372E

-04

45394E+

0015

865E

-11

15334E

-06

44766

E+00

F14

Mean

71759E

-1120261E+

0043430E+

0277682E

-05

33324E-02

42669E+

02Std

24650E-11

50770E-01

14055E

+01

54975E-06

10537E

-03

80127E+

01F15

Mean

16716E

+00

12749E

+00

22241E+

0116

927E

+00

14937E

+00

21617E+

01Std

76572E

-01

43985E-01

33014E-01

47677E-01

63574E-01

54534E-01

F16

Mean

94233E-33

13057E

-09

76995E

+07

17755E

-24

846

48E-14

69921E+

00Std

14425E

-48

37533E-10

21712E+

0719

092E

-24

17429E

-13

89129E-01

F17

Mean

76377E

-03

14219E

-02

1160

6E+0

164039E-03

10080E

-02

1264

1E+0

0Std

57418E-03

21089E-02

46282E-01

70807E

-03

13952E

-02

16555E

-02

F18

Mean

83103E+

0079

047E

+00

39689E+

0189467E+

0096

041E+0

038726E+

01Std

260

72E+

0025432E+

0077616E

-01

78506E

-01

21029E+

0013

015E

+00

F19

Mean

45562E+

0126833E+

04806

68E+

0616

118E+

0413

155E

+04

70015E

+05

Std

38094E+

0121743E+

0421709E+

0612

498E

+04

1300

9E+0

497

174E

+04

F20

Mean

43064E-08

25702E-04

11519E

-01

52365E-08

16998E

-06

500

47E-01

Std

44294E-08

27576E-04

39417E-02

95247E

-08

49881E-06

26305E-01

F21

Mean

000

00E+

0011310E

-06

53146

E+02

000

00E+

0023711E

-10

58748E+

02Std

000

00E+

0033614E-07

32117E+

01000

00E+

0045437E-11

29507E+

01F2

2Mean

000

00E+

0016

167E

-06

48729E+

02000

00E+

00244

07E-10

52060

E+02

Std

000

00E+

0063216E-07

24382E+

01000

00E+

0075

889E

-11

42230E+

01F2

3Mean

13699E

+00

89987E-01

21237E+

0122699E+

0025899E+

0035955E+

00Std

23594E-01

666

67E-02

58033E-01

41913E-01

62973E-01

12247E

-01

F24

Mean

71054E

-1426826E-02

63090E+

0119

033E

-08

96037E

-03

47263E+

01Std

27621E-14

47780E-03

22392E+

0061075E-09

97071E-04

52689E+

00F2

5Mean

66563E+

0184722E+

0211275E

+08

65780E+

0139992E+

0188242E+

04Std

10992E

+02

2113

8E+0

221091E+

0794

954E

+01

43819E+

0116

832E

+04

18 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

18989E

-1316

584E

-01

19996E

+01

17809E

-11

14744E

-06

13554E

+01

Std

20566

E-14

53720E-02

90319E

-02

19159E

-12

18930E

-07

60821E+

00F13

Mean

22871E-15

17736E

-01

1944

7E+0

213

452E

-10

13291E-05

16379E

+02

Std

26741E-15

53611E-02

62653E+

0038592E-11

55001E-06

13313E

+01

F14

Mean

18736E

-1074

259E

+01

10132E

+03

22866

E-04

83534E-02

95534E

+02

Std

37223E-11

19144E

+01

18986E

+01

14283E

-05

10592E

-02

53523E+

01F15

Mean

26814E+

0010

178E

+01

47083E+

0128083E+

0034325E+

0045859E+

01Std

73851E-01

16238E

+00

22513E-01

46148E-01

60283E-01

69914E-01

F16

Mean

47116E-33

244

54E-04

90382E

+08

81890E-24

62347E-14

27647E+

03Std

72124E

-49

59650E-05

64985E+

0767958E-24

55604

E-14

44231E+

03F17

Mean

34494E-03

11896E

-02

37816E+

0134509E-03

41885E-03

21280E+

00Std

60565E-03

65363E-03

15922E

+00

46765E-03

86153E-03

54359E-02

F18

Mean

18033E

+01

17806E

+01

86826E+

0118

319E

+01

18828E

+01

82458E+

01Std

19652E

+00

38319E+

0093

222E

-01

29296E+

0025377E+

0015

159E

+00

F19

Mean

82462E+

0427944

E+06

48046

E+08

28415E+

0560265E+

0549201E+

07Std

55732E+

0489703E+

0596

715E

+07

24572E+

0527137E+

0572

772E

+06

F20

Mean

57130E-07

81688E-03

96848E

-01

13631E-06

27143E-05

21656E+

00Std

61122E-07

53195E-03

44542E-01

25155E-06

58766

E-05

80368E-01

F21

Mean

000

00E+

0051414E+

0013

305E

+03

000

00E+

0020026E-09

13623E

+03

Std

000

00E+

0017

825E

+00

22890E+

01000

00E+

0032815E-10

609

96E+

01F2

2Mean

000

00E+

0077

848E

+00

1260

9E+0

3000

00E+

0020383E-09

12745E

+03

Std

000

00E+

0023732E+

0029100

E+01

000

00E+

0029753E-10

59708E+

01F2

3Mean

25599E+

0020499E+

0039804

E+01

47099E+

0043699E+

0073

691E+0

0Std

36878E-01

15092E

-01

69296E-01

59151E-01

56184E-01

17989E

-01

F24

Mean

40927E-13

26229E+

0015

145E

+02

18874E

-07

29476E-02

10478E

+02

Std

88061E-14

63367E-01

42830E+

0037074E-08

17697E

-03

11873E

+01

F25

Mean

42987E+

028117

8E+0

313

524E

+09

56790E+

0244982E+

0218

038E

+06

Std

43423E+

0233128E+

0278

399E

+07

54327E+

0246926E+

0221315E+

05

Complexity 19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus8

minus6

minus4

minus2

02468

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(f) F24




MAE = sum119873119895=1 10038161003816100381610038161003816119898119895 minus 11989611989510038161003816100381610038161003816119873 (22)

20 Complexity

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

ISSASSAPSO

CMFOAIFFOFOA

1

2

3

4

5

6

7

8

Mea

n Er

rors

(log)

0 4000 6000 8000 100002000Iteration

(e) F19

ISSASSAPSO

CMFOAIFFOFOA

20000 6000 8000 100004000Iteration

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




Complexity 21

Table11

Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

31365E+

0413

133E

+06

11295E

+08

10017E

+06

864

75E+

0513

449E

+07

Std

18602E

+04

52974E+

0531387E+

0748689E+

0548607E+

0528405E+

06F2

7Mean

304

00E-10

1844

6E+0

484500

E+09

10512E

+04

12359E

+04

58535E+

08Std

61535E-10

14049E

+04

10125E

+09

12485E

+04

11922E

+04

38771E+

07F2

8Mean

42105E-01

46710E+

0173

819E

+02

4114

7E+0

137814E+

0114

226E

+02

Std

12624E

+00

31490E+

0199

455E

+01

47336E+

0134110E+

0129201E+

01F2

9Mean

75177E

+03

14891E+0

529286E+

0647277E+

0531099E+

0539826E+

05Std

33119E+

0368316E+

049190

4E+0

521021E+

0522686E+

0515

511E+0

5F3

0Mean

31524E+

0231524E+

0238129E+

0231524E+

0231524E+

0232568E+

02Std

85708E-12

19710E

-07

14082E

+01

11524E

-1145680E-11

58955E+

00F31

Mean

23483E+

0223172E+

0230117E+

0223811E

+02

23858E+

0224179E+

02Std

41748E+

0072

461E+0

048903E+

00560

97E+

0050249E+

0090

228E

+00

F32

Mean

20790E+

02206

03E+

0221884E+

0221485E+

0220975E+

0220633E+

02Std

41618E+

0032456E+

0030353E+

0087909E+

0057719E+

0016

880E

+00

22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








6 Complexity







end



if 119891119899119890119908119887119890119904119905 lt 119891119887119890119904119905 119865119878119887119890119904119905119895 = 119865119878119899119890119908119887119890119904119905119895119891119887119890119904119905 = 119891119899119890119908119887119890119904119905end

end 119868119905119890119903 = 119868119905119890119903 + 1end


Complexity 7




Function Range Fmin

F1(119909) = 119899sum119894=1

1198941199092119894 [minus10 10] 0

F2(119909) = 119899sum119894=2



1199092119894) [minus1 1] -1

F4(119909) = 119899sum119894=1


F5(119909) = 119899sum119894=1

1198941199094119894 + rand () [minus128 128] 0

F6(119909) = 119899minus1sum119894=1


F7(119909) = 119899sum119894=1

( 119894sum119895=1

1199092119895) [minus100 100] 0

F8(119909) = max 10038161003816100381610038161199091198941003816100381610038161003816 1 le 119894 le 119899 [minus100 100] 0

F9(119909) = 119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 119899prod119894=1

10038161003816100381610038161199091198941003816100381610038161003816 [minus10 10] 0

F10(119909) = 119899sum119894=1

1199092119894 [minus100 100] 0

F11(119909) = 119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816119894+1 [minus1 1] 0



8 Complexity









Complexity 9


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

22612E-46

14374E

-13

18031E+0

338546

E-22

53419E-12

58887E+

01Std

39697E-46

10187E

-13

16038E

+02

1146

4E-22

26506

E-12

10717E

+01

F2Mean

53333E-01

600

00E-01

70475E

+04

666

67E-01

666

67E-01

12221E+0

2Std

28109E-01

21082E-01

18027E

+04

11102E

-16

364

14E-11

35452E+

01F3

Mean

000

00E+

00000

00E+

0047300

E-01

000

00E+

00860

42E-14

18586E

-02

Std

18504E

-1626168E-16

42225E-02

906

49E-17

27669E-14

19010E

-03

F4Mean

21268E-39

39943E-10

92617E

+07

37704

E-18

20345E-08

11112

E+07

Std

564

86E-39

32855E-10

18784E

+07

24165E-18

14277E

-08

30272E+

06F5

Mean

43164

E-03

93054E

-02

55376E+

0032231E-03

22754E-03

17965E

-02

Std

19931E-03

23058E-02

10210E

+00

13321E-03

1104

6E-03

31904

E-03

F6Mean

55447E-14

29695E+

0110

669E

+07

76347E

+00

89052E+

0012

862E

+04

Std

54894E-14

340

74E+

0118

313E

+06

590

03E+

0071150E

+00

44338E+

03F7

Mean

11996E

-44

65616E-12

16688E

+05

71055E

-22

60744

E-12

54708E+

03Std

304

49E-44

540

85E-12

17265E

+04

16506E

-22

29515E-12

49070E+

02F8

Mean

35080E-13

17850E

-03

45639E+

0127020E-11

264

40E-06

78561E+0

0Std

70894E

-1343281E-04

20578E+

00604

13E-12

24822E-07

17334E

+00

F9Mean

55772E-24

22148E-07

71792E

+01

23880E-11

204

15E-06

304

53E+

01Std

79227E

-24

67302E-08

30539E+

0141360

E-12

36636E-07

46515E+

01F10

Mean

26748E-44

44745E-13

13098E

+04

42105E-23

440

42E-13

36501E+

02Std

78461E-44

37055E-13

13116

E+03

10825E

-23

15887E

-13

43608E+

01F11

Mean

61803E-188

17833E

-60

18391E-03

78265E

-25

5117

6E-15

67271E-07

Std

000

00E+

0037202E-60

11147E

-03

65934E-25

76076E

-15

46836E-07

10 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

19373E

-45

244

08E-07

89004

E+03

36571E-21

32632E-11

28288E+

02Std

40611E

-45

72161E-08

10186E

+03

90310E

-22

82802E-12

18277E

+01

F2Mean

666

67E-01

21716E+

0080535E+

0512

514E

+00

666

67E-01

82661E+

02Std

82003E-16

22142E+

0011670E

+05

12485E

+00

25423E-10

77814E

+01

F3Mean

000

00E+

0076

318E

-11

866

02E-01

000

00E+

004110

0E-13

51246

E-02

Std

22204E-16

22120E-11

18360E

-02

23984E-16

14800E

-13

40430E-03

F4Mean

306

71E-38

97033E

-04

46756E+

0819

432E

-17

10621E-07

38893E+

07Std

69186E-38

344

64E-04

60135E+

0711627E

-17

76083E

-08

73301E+0

6F5

Mean

71557E

-03

29566

E-01

53164

E+01

10084E

-02

73580E

-03

10458E

-01

Std

23021E-03

33200

E-02

64915E+

0026523E-03

18100E

-03

26586E-02

F6Mean

43706

E-11

95471E+0

170

118E+

0777

331E+0

147194E+

0165331E+

04Std

95151E-11

35358E+

0153302E+

06344

63E+

0140976E+

0113

721E+0

4F7

Mean

12947E

-41

23079E-05

91659E

+05

69771E-21

64025E-11

28862E+

04Std

37876E-41

58814E-06

93287E

+04

31808E-21

26901E-11

28375E+

03F8

Mean

60872E-11

12706E

-01

67093E+

0184930E-11

71576E

-06

11919E

+01

Std

25158E-11

33391E-02

25011E

+00

11107E

-11

69032E-07

1040

4E+0

0F9

Mean

18289E

-23

38822E-04

20565E+

1060338E-11

63442E-06

11434E

+05

Std

28884E-23

72525E

-05

63864

E+10

64165E-12

90797E

-07

24588E+

05F10

Mean

12924E

-44

14594E

-06

41629E+

0422846

E-22

20175E-12

10536E

+03

Std

25807E-44

606

62E-07

37125E+

0341424E-23

52761E-13

59785E+

01F11

Mean

44745E-163

19208E

-58

92852E

-03

11169E

-24

51458E-15

16917E

-06

Std

000

00E+

0022612E-58

35776E-03

14674E

-24

82130E-15

94517E

-07

Complexity 11


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

52002E-44

1360

1E-01

64559E+

0467630E-20

540

42E-10

23688E+

03Std

89565E-44

28398E-02

27675E+

0318

636E

-20

10843E

-10

11782E

+02

F2Mean

25837E+

0028507E+

0189058E+

0610

018E

+01

11850E

+01

13921E+0

4Std

40923E+

0012

482E

+01

64125E+

0598

260E

+00

67378E+

0021479E+

03F3

Mean

19984E

-1517

506E

-05

99937E

-01

19984E

-1534570E-12

1946

0E-01

Std

27940E-16

33607E-06

19874E

-04

39686E-16

57323E-13

11259E

-02

F4Mean

14073E

-38

30139E+

0226151E+

0914

261E-16

10212E

-06

20499E+

08Std

15382E

-38

90551E+0

126783E+

0875

737E

-17

33853E-07

35388E+

07F5

Mean

17615E

-02

14345E

+00

59603E+

0237550E-02

29349E-02

12443E

+00

Std

42239E-03

13985E

-01

58412E+

0112

602E

-02

45825E-03

18471E-01

F6Mean

11417E

+01

58422E+

0242299E+

0817

988E

+02

16578E

+02

560

78E+

05Std

30258E+

0197

884E

+02

48581E+

0739022E+

01466

85E+

0165477E+

04F7

Mean

16881E-41

12984E

+01

64707E+

0611852E

-19

74831E-10

22865E+

05Std

34134E-41

22729E+

0033435E+

0531718E-20

95033E

-11

20650E+

04F8

Mean

45259E-08

39819E+

0085137E+

0117

042E

-04

29244

E-05

33956E+

01Std

29104E-08

41522E-01

12566E

+00

78665E

-05

27018E-06

47713E+

00F9

Mean

12222E

-22

36814E-01

72469E

+32

25070E-10

23795E-05

14112

E+27

Std

84369E-23

41963E-02

20633E+

3323874E-11

13879E

-06

44627E+

27F10

Mean

34254E-42

37261E-01

14940E

+05

20714E-21

18486E

-11

43041E+

03Std

966

08E-42

92561E-02

446

43E+

03464

07E-22

23956E-12

24348E+

02F11

Mean

1640

0E-12

315

040E

-52

37278E-02

65780E-24

3117

4E-14

10720E

-05

Std

51861E-123

16670E

-52

11165E

-02

72960E

-24

36245E-14

59475E-06

12 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

(b) F4

0 2000 4000 6000 8000 10000

0

5

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus15

minus5

(c) F6

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

minus50

(d) F7

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus2

minus4

minus6

minus8

minus10

minus12

minus14

(e) F8

0 2000 4000 6000 8000 10000

05

1015

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus5

minus10

minus20

minus25

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

Mea

n Er

rors

(log)

(h) F11


Complexity 13

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus10

minus5

0

5

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

0

10

20

30

Mea

n Er

rors

(log)

2000 4000 6000 8000 100000Iteration

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

50

Mea

n Er

rors

(log)

(h) F11


14 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

20

Mea

n Er

rors

(log)

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

0

2

4

6

8

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

01020304050

Mea

n Er

rors

(log)

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus140

minus120

minus100

minus80

minus60

minus40

minus20

020

Mea

n Er

rors

(log)

(h) F11


Complexity 15

Table6Multim

odalbenchm

arkfunctio

ns

Functio

nRa

nge

Fmin

F12(119909)=

minus20exp(minus0

2radic1 119899119899 sum 119894=11199092 119894)minus

exp(1 119899119899 sum 119894=1co

s(2120587119909 119894))

+20+exp

(1 )[minus32

32]0

F13(119909)=

119899 sum 119894=11003816 1003816 1003816 1003816119909 119894sin(119909 119894)

+01119909 1198941003816 1003816 1003816 1003816

[minus1010]

0

F14(119909)=

119891 119904(1199091119909 2)


(119909 119899119909 1)

[minus10010

0]0

119891 119904(119909119910)=

(1199092 +1199102 )025[sin2

(50(1199092 +

1199102 )01)+1

]F15(

119909)=119891 119904(1199091119909 2)


119909 1198991199091)

[minus10010

0]0

119891 119904(119909119910)=

05(sin2(radic 1199092+1199102

)minus05)

(1+0001

(1199092 +1199102 ))2

F16(119909)=

120587 11989910sin2

(120587119910 119894)+119899minus1 sum 119894=1

(119910 119894minus1 )2 [

1+10sin2

(120587119910 119894+1)]+

(119910 119899minus1 )2

+119899 sum 119894=1119906(119909 119894

10100

4)[minus50

50]0

119910 119894=1+1 4(119909

119894+1)

119906(119909 119894119886

119896119898)= 119896(119909 119894

minus119886)119898

119909 119894gt119886

0minus119886le

119909 119894le119886

119896(minus119909119894minus119886)119898

119909119894gt119886

F17(119909)=

1 4000119899 sum 119894=11199092 119894minus119899 prod 119894=1

cos(119909119894 radic 119894)+1

[minus10010

0]0

F18(119909)=


(minus(1199092 119894+

1199092 119894+1+05

119909 119894119909 119894+1)

8)lowastc

os(4radic

1199092 119894+1199092 119894+1

+05119909 119894119909 119894+1))

[minus55]

1-n

F19(119909)=

119899 sum 119894=1(119909119894minus1)2

minus119899 sum 119894=2119909 119894119909 119894minus1

[minusn2n2 ]

119899(119899+4)(119899

minus1)minus6

F20 (119909 )=

sum119899minus1 119894=2(05

+(sin2(radic 1

001199092 119894+1199092 119894+1)minus0

5))(1+

0001(1199092 119894minus

2119909 119894119909119894minus1+1199092 119894minus1))2

[minus10010

0]0

F21(119909)=

119899 sum 119894=1[1199092 119894minus10

cos(2120587

119909 119894)+10]

[minus51251

2]0

F22(119909)=

119899 sum 119894=1[1199102 119894minus10

cos(2120587

119910 119894)+10]

119910 119894= 119909 119894

1003816 1003816 1003816 1003816119909 1198941003816 1003816 1003816 1003816lt05

119903119900119906119899119889(2119909

119894)2

1003816 1003816 1003816 10038161199091198941003816 1003816 1003816 1003816lt0

5[minus51

2512]

0

F23(119909)=

1minuscos(2120587

radic119899 sum 119894=11199092 119894)

+01radic119899 sum 119894=1

1199092 119894[minus10

0100]

0

F24(119909)=

119899 sum 119894=1119896119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 (119909119894+05

))]minus119899119896

119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 05

)][minus05

05]0

F25(119909)=

119899 sum 119896=1119899 sum 119895=1((1199102 119895119896 4000)minus

cos(119910 119895119896)+1

)119910119895119896=10

0(119909 119896minus1199092 119895

)2 +(1minus

1199092 119895)2[minus10

0100]

0

16 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmultim

odalbenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

51692E-14

21708E-07

16343E

+01

42641E-12

43970E-07

70983E

+00

Std

94813E-15

11785E

-07

45830E-01

51275E-13

53024E-08

45755E+

00F13

Mean

19651E-15

17670E

-07

30865E+

0138781E-12

12507E

-06

29660

E+01

Std

17016E

-1510

899E

-07

28749E+

00200

14E-12

19125E

-06

57790E+

00F14

Mean

28586E-11

47414E-02

21576E+

0235954E-05

17290E

-02

18705E

+02

Std

17874E

-1118

105E

-02

50836E+

0019

343E

-06

10857E

-03

50868E+

01F15

Mean

99552E

-01

46150E-01

12596E

+01

94983E

-01

10032E

+00

12147E

+01

Std

38926E-01

33522E-01

21495E-01

42966

E-01

35690E-01

17388E

-01

F16

Mean

15705E

-32

13069E

-15

56725E+

0650290E-25

99726E

-15

31482E+

00Std

28850E-48

57169E-16

17168E

+06

47027E-25

85374E-15

58054E-01

F17

Mean

13781E-02

10332E

-02

43352E+

0044332E-03

12793E

-02

10971E+0

0Std

14865E

-02

12632E

-02

42518E-01

79408E

-03

10155E

-02

10766E

-02

F18

Mean

50849E+

0038253E+

0020946

E+01

49225E+

00490

48E+

0021497E+

01Std

16014E

+00

14627E

+00

76856E

-01

21737E+

00204

11E+0

013

669E

+00

F19

Mean

268

41E-07

19292E

+02

49808E+

0519

677E

+02

240

98E+

0230226E+

04Std

32619E-08

15971E+0

214

706E

+05

16572E

+02

23149E+

0260289E+

03F2

0Mean

25989E-07

47006

E-06

33592E-02

44469E-08

18865E

-07

1540

6E-01

Std

59383E-07

73387E

-06

22456E-02

10350E

-07

31612E-07

56719E-02

F21

Mean

000

00E+

0070

841E-13

25769E+

02000

00E+

0045409E-11

30881E+

02Std

000

00E+

0045361E-13

90973E

+00

000

00E+

0019

882E

-11

27305E+

01F2

2Mean

000

00E+

007746

7E-13

23335E+

02000

00E+

00644

03E-11

25509E+

02Std

000

00E+

0036979E-13

15942E

+01

000

00E+

0033820E-11

26992E+

01F2

3Mean

93987E

-01

52987E-01

12199E

+01

13599E

+00

14399E

+00

21878E+

00Std

21705E-01

12517E

-01

49304

E-01

36576E-01

21705E-01

62731E-02

F24

Mean

14921E-14

37233E-04

32412E+

0147458E-09

42553E-03

26924E+

01Std

17226E

-1498

846E

-05

11649E

+00

28242E-09

42975E-04

35559E+

00F2

5Mean

29494E+

0110

724E

+02

11372E

+07

404

62E+

0193530E+

0092

421E+0

3Std

29743E+

0151800

E+01

31606

E+06

39685E+

0190392E+

0018

838E

+03

Complexity 17


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

85798E-14

24174E-04

18459E

+01

7404

4E-12

73673E

-07

82226E+

00Std

17360E

-1455274E-05

1944

7E-01

88139E-13

80222E-08

42517E+

00F13

Mean

22538E-15

29492E-04

71594E

+01

21041E-11

32004

E-06

60959E+

01Std

18688E

-1510

372E

-04

45394E+

0015

865E

-11

15334E

-06

44766

E+00

F14

Mean

71759E

-1120261E+

0043430E+

0277682E

-05

33324E-02

42669E+

02Std

24650E-11

50770E-01

14055E

+01

54975E-06

10537E

-03

80127E+

01F15

Mean

16716E

+00

12749E

+00

22241E+

0116

927E

+00

14937E

+00

21617E+

01Std

76572E

-01

43985E-01

33014E-01

47677E-01

63574E-01

54534E-01

F16

Mean

94233E-33

13057E

-09

76995E

+07

17755E

-24

846

48E-14

69921E+

00Std

14425E

-48

37533E-10

21712E+

0719

092E

-24

17429E

-13

89129E-01

F17

Mean

76377E

-03

14219E

-02

1160

6E+0

164039E-03

10080E

-02

1264

1E+0

0Std

57418E-03

21089E-02

46282E-01

70807E

-03

13952E

-02

16555E

-02

F18

Mean

83103E+

0079

047E

+00

39689E+

0189467E+

0096

041E+0

038726E+

01Std

260

72E+

0025432E+

0077616E

-01

78506E

-01

21029E+

0013

015E

+00

F19

Mean

45562E+

0126833E+

04806

68E+

0616

118E+

0413

155E

+04

70015E

+05

Std

38094E+

0121743E+

0421709E+

0612

498E

+04

1300

9E+0

497

174E

+04

F20

Mean

43064E-08

25702E-04

11519E

-01

52365E-08

16998E

-06

500

47E-01

Std

44294E-08

27576E-04

39417E-02

95247E

-08

49881E-06

26305E-01

F21

Mean

000

00E+

0011310E

-06

53146

E+02

000

00E+

0023711E

-10

58748E+

02Std

000

00E+

0033614E-07

32117E+

01000

00E+

0045437E-11

29507E+

01F2

2Mean

000

00E+

0016

167E

-06

48729E+

02000

00E+

00244

07E-10

52060

E+02

Std

000

00E+

0063216E-07

24382E+

01000

00E+

0075

889E

-11

42230E+

01F2

3Mean

13699E

+00

89987E-01

21237E+

0122699E+

0025899E+

0035955E+

00Std

23594E-01

666

67E-02

58033E-01

41913E-01

62973E-01

12247E

-01

F24

Mean

71054E

-1426826E-02

63090E+

0119

033E

-08

96037E

-03

47263E+

01Std

27621E-14

47780E-03

22392E+

0061075E-09

97071E-04

52689E+

00F2

5Mean

66563E+

0184722E+

0211275E

+08

65780E+

0139992E+

0188242E+

04Std

10992E

+02

2113

8E+0

221091E+

0794

954E

+01

43819E+

0116

832E

+04

18 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

18989E

-1316

584E

-01

19996E

+01

17809E

-11

14744E

-06

13554E

+01

Std

20566

E-14

53720E-02

90319E

-02

19159E

-12

18930E

-07

60821E+

00F13

Mean

22871E-15

17736E

-01

1944

7E+0

213

452E

-10

13291E-05

16379E

+02

Std

26741E-15

53611E-02

62653E+

0038592E-11

55001E-06

13313E

+01

F14

Mean

18736E

-1074

259E

+01

10132E

+03

22866

E-04

83534E-02

95534E

+02

Std

37223E-11

19144E

+01

18986E

+01

14283E

-05

10592E

-02

53523E+

01F15

Mean

26814E+

0010

178E

+01

47083E+

0128083E+

0034325E+

0045859E+

01Std

73851E-01

16238E

+00

22513E-01

46148E-01

60283E-01

69914E-01

F16

Mean

47116E-33

244

54E-04

90382E

+08

81890E-24

62347E-14

27647E+

03Std

72124E

-49

59650E-05

64985E+

0767958E-24

55604

E-14

44231E+

03F17

Mean

34494E-03

11896E

-02

37816E+

0134509E-03

41885E-03

21280E+

00Std

60565E-03

65363E-03

15922E

+00

46765E-03

86153E-03

54359E-02

F18

Mean

18033E

+01

17806E

+01

86826E+

0118

319E

+01

18828E

+01

82458E+

01Std

19652E

+00

38319E+

0093

222E

-01

29296E+

0025377E+

0015

159E

+00

F19

Mean

82462E+

0427944

E+06

48046

E+08

28415E+

0560265E+

0549201E+

07Std

55732E+

0489703E+

0596

715E

+07

24572E+

0527137E+

0572

772E

+06

F20

Mean

57130E-07

81688E-03

96848E

-01

13631E-06

27143E-05

21656E+

00Std

61122E-07

53195E-03

44542E-01

25155E-06

58766

E-05

80368E-01

F21

Mean

000

00E+

0051414E+

0013

305E

+03

000

00E+

0020026E-09

13623E

+03

Std

000

00E+

0017

825E

+00

22890E+

01000

00E+

0032815E-10

609

96E+

01F2

2Mean

000

00E+

0077

848E

+00

1260

9E+0

3000

00E+

0020383E-09

12745E

+03

Std

000

00E+

0023732E+

0029100

E+01

000

00E+

0029753E-10

59708E+

01F2

3Mean

25599E+

0020499E+

0039804

E+01

47099E+

0043699E+

0073

691E+0

0Std

36878E-01

15092E

-01

69296E-01

59151E-01

56184E-01

17989E

-01

F24

Mean

40927E-13

26229E+

0015

145E

+02

18874E

-07

29476E-02

10478E

+02

Std

88061E-14

63367E-01

42830E+

0037074E-08

17697E

-03

11873E

+01

F25

Mean

42987E+

028117

8E+0

313

524E

+09

56790E+

0244982E+

0218

038E

+06

Std

43423E+

0233128E+

0278

399E

+07

54327E+

0246926E+

0221315E+

05

Complexity 19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus8

minus6

minus4

minus2

02468

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(f) F24




MAE = sum119873119895=1 10038161003816100381610038161003816119898119895 minus 11989611989510038161003816100381610038161003816119873 (22)

20 Complexity

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

ISSASSAPSO

CMFOAIFFOFOA

1

2

3

4

5

6

7

8

Mea

n Er

rors

(log)

0 4000 6000 8000 100002000Iteration

(e) F19

ISSASSAPSO

CMFOAIFFOFOA

20000 6000 8000 100004000Iteration

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




Complexity 21

Table11

Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

31365E+

0413

133E

+06

11295E

+08

10017E

+06

864

75E+

0513

449E

+07

Std

18602E

+04

52974E+

0531387E+

0748689E+

0548607E+

0528405E+

06F2

7Mean

304

00E-10

1844

6E+0

484500

E+09

10512E

+04

12359E

+04

58535E+

08Std

61535E-10

14049E

+04

10125E

+09

12485E

+04

11922E

+04

38771E+

07F2

8Mean

42105E-01

46710E+

0173

819E

+02

4114

7E+0

137814E+

0114

226E

+02

Std

12624E

+00

31490E+

0199

455E

+01

47336E+

0134110E+

0129201E+

01F2

9Mean

75177E

+03

14891E+0

529286E+

0647277E+

0531099E+

0539826E+

05Std

33119E+

0368316E+

049190

4E+0

521021E+

0522686E+

0515

511E+0

5F3

0Mean

31524E+

0231524E+

0238129E+

0231524E+

0231524E+

0232568E+

02Std

85708E-12

19710E

-07

14082E

+01

11524E

-1145680E-11

58955E+

00F31

Mean

23483E+

0223172E+

0230117E+

0223811E

+02

23858E+

0224179E+

02Std

41748E+

0072

461E+0

048903E+

00560

97E+

0050249E+

0090

228E

+00

F32

Mean

20790E+

02206

03E+

0221884E+

0221485E+

0220975E+

0220633E+

02Std

41618E+

0032456E+

0030353E+

0087909E+

0057719E+

0016

880E

+00

22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 7




Function Range Fmin

F1(119909) = 119899sum119894=1

1198941199092119894 [minus10 10] 0

F2(119909) = 119899sum119894=2



1199092119894) [minus1 1] -1

F4(119909) = 119899sum119894=1


F5(119909) = 119899sum119894=1

1198941199094119894 + rand () [minus128 128] 0

F6(119909) = 119899minus1sum119894=1


F7(119909) = 119899sum119894=1

( 119894sum119895=1

1199092119895) [minus100 100] 0

F8(119909) = max 10038161003816100381610038161199091198941003816100381610038161003816 1 le 119894 le 119899 [minus100 100] 0

F9(119909) = 119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 119899prod119894=1

10038161003816100381610038161199091198941003816100381610038161003816 [minus10 10] 0

F10(119909) = 119899sum119894=1

1199092119894 [minus100 100] 0

F11(119909) = 119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816119894+1 [minus1 1] 0



8 Complexity









Complexity 9


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

22612E-46

14374E

-13

18031E+0

338546

E-22

53419E-12

58887E+

01Std

39697E-46

10187E

-13

16038E

+02

1146

4E-22

26506

E-12

10717E

+01

F2Mean

53333E-01

600

00E-01

70475E

+04

666

67E-01

666

67E-01

12221E+0

2Std

28109E-01

21082E-01

18027E

+04

11102E

-16

364

14E-11

35452E+

01F3

Mean

000

00E+

00000

00E+

0047300

E-01

000

00E+

00860

42E-14

18586E

-02

Std

18504E

-1626168E-16

42225E-02

906

49E-17

27669E-14

19010E

-03

F4Mean

21268E-39

39943E-10

92617E

+07

37704

E-18

20345E-08

11112

E+07

Std

564

86E-39

32855E-10

18784E

+07

24165E-18

14277E

-08

30272E+

06F5

Mean

43164

E-03

93054E

-02

55376E+

0032231E-03

22754E-03

17965E

-02

Std

19931E-03

23058E-02

10210E

+00

13321E-03

1104

6E-03

31904

E-03

F6Mean

55447E-14

29695E+

0110

669E

+07

76347E

+00

89052E+

0012

862E

+04

Std

54894E-14

340

74E+

0118

313E

+06

590

03E+

0071150E

+00

44338E+

03F7

Mean

11996E

-44

65616E-12

16688E

+05

71055E

-22

60744

E-12

54708E+

03Std

304

49E-44

540

85E-12

17265E

+04

16506E

-22

29515E-12

49070E+

02F8

Mean

35080E-13

17850E

-03

45639E+

0127020E-11

264

40E-06

78561E+0

0Std

70894E

-1343281E-04

20578E+

00604

13E-12

24822E-07

17334E

+00

F9Mean

55772E-24

22148E-07

71792E

+01

23880E-11

204

15E-06

304

53E+

01Std

79227E

-24

67302E-08

30539E+

0141360

E-12

36636E-07

46515E+

01F10

Mean

26748E-44

44745E-13

13098E

+04

42105E-23

440

42E-13

36501E+

02Std

78461E-44

37055E-13

13116

E+03

10825E

-23

15887E

-13

43608E+

01F11

Mean

61803E-188

17833E

-60

18391E-03

78265E

-25

5117

6E-15

67271E-07

Std

000

00E+

0037202E-60

11147E

-03

65934E-25

76076E

-15

46836E-07

10 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

19373E

-45

244

08E-07

89004

E+03

36571E-21

32632E-11

28288E+

02Std

40611E

-45

72161E-08

10186E

+03

90310E

-22

82802E-12

18277E

+01

F2Mean

666

67E-01

21716E+

0080535E+

0512

514E

+00

666

67E-01

82661E+

02Std

82003E-16

22142E+

0011670E

+05

12485E

+00

25423E-10

77814E

+01

F3Mean

000

00E+

0076

318E

-11

866

02E-01

000

00E+

004110

0E-13

51246

E-02

Std

22204E-16

22120E-11

18360E

-02

23984E-16

14800E

-13

40430E-03

F4Mean

306

71E-38

97033E

-04

46756E+

0819

432E

-17

10621E-07

38893E+

07Std

69186E-38

344

64E-04

60135E+

0711627E

-17

76083E

-08

73301E+0

6F5

Mean

71557E

-03

29566

E-01

53164

E+01

10084E

-02

73580E

-03

10458E

-01

Std

23021E-03

33200

E-02

64915E+

0026523E-03

18100E

-03

26586E-02

F6Mean

43706

E-11

95471E+0

170

118E+

0777

331E+0

147194E+

0165331E+

04Std

95151E-11

35358E+

0153302E+

06344

63E+

0140976E+

0113

721E+0

4F7

Mean

12947E

-41

23079E-05

91659E

+05

69771E-21

64025E-11

28862E+

04Std

37876E-41

58814E-06

93287E

+04

31808E-21

26901E-11

28375E+

03F8

Mean

60872E-11

12706E

-01

67093E+

0184930E-11

71576E

-06

11919E

+01

Std

25158E-11

33391E-02

25011E

+00

11107E

-11

69032E-07

1040

4E+0

0F9

Mean

18289E

-23

38822E-04

20565E+

1060338E-11

63442E-06

11434E

+05

Std

28884E-23

72525E

-05

63864

E+10

64165E-12

90797E

-07

24588E+

05F10

Mean

12924E

-44

14594E

-06

41629E+

0422846

E-22

20175E-12

10536E

+03

Std

25807E-44

606

62E-07

37125E+

0341424E-23

52761E-13

59785E+

01F11

Mean

44745E-163

19208E

-58

92852E

-03

11169E

-24

51458E-15

16917E

-06

Std

000

00E+

0022612E-58

35776E-03

14674E

-24

82130E-15

94517E

-07

Complexity 11


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

52002E-44

1360

1E-01

64559E+

0467630E-20

540

42E-10

23688E+

03Std

89565E-44

28398E-02

27675E+

0318

636E

-20

10843E

-10

11782E

+02

F2Mean

25837E+

0028507E+

0189058E+

0610

018E

+01

11850E

+01

13921E+0

4Std

40923E+

0012

482E

+01

64125E+

0598

260E

+00

67378E+

0021479E+

03F3

Mean

19984E

-1517

506E

-05

99937E

-01

19984E

-1534570E-12

1946

0E-01

Std

27940E-16

33607E-06

19874E

-04

39686E-16

57323E-13

11259E

-02

F4Mean

14073E

-38

30139E+

0226151E+

0914

261E-16

10212E

-06

20499E+

08Std

15382E

-38

90551E+0

126783E+

0875

737E

-17

33853E-07

35388E+

07F5

Mean

17615E

-02

14345E

+00

59603E+

0237550E-02

29349E-02

12443E

+00

Std

42239E-03

13985E

-01

58412E+

0112

602E

-02

45825E-03

18471E-01

F6Mean

11417E

+01

58422E+

0242299E+

0817

988E

+02

16578E

+02

560

78E+

05Std

30258E+

0197

884E

+02

48581E+

0739022E+

01466

85E+

0165477E+

04F7

Mean

16881E-41

12984E

+01

64707E+

0611852E

-19

74831E-10

22865E+

05Std

34134E-41

22729E+

0033435E+

0531718E-20

95033E

-11

20650E+

04F8

Mean

45259E-08

39819E+

0085137E+

0117

042E

-04

29244

E-05

33956E+

01Std

29104E-08

41522E-01

12566E

+00

78665E

-05

27018E-06

47713E+

00F9

Mean

12222E

-22

36814E-01

72469E

+32

25070E-10

23795E-05

14112

E+27

Std

84369E-23

41963E-02

20633E+

3323874E-11

13879E

-06

44627E+

27F10

Mean

34254E-42

37261E-01

14940E

+05

20714E-21

18486E

-11

43041E+

03Std

966

08E-42

92561E-02

446

43E+

03464

07E-22

23956E-12

24348E+

02F11

Mean

1640

0E-12

315

040E

-52

37278E-02

65780E-24

3117

4E-14

10720E

-05

Std

51861E-123

16670E

-52

11165E

-02

72960E

-24

36245E-14

59475E-06

12 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

(b) F4

0 2000 4000 6000 8000 10000

0

5

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus15

minus5

(c) F6

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

minus50

(d) F7

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus2

minus4

minus6

minus8

minus10

minus12

minus14

(e) F8

0 2000 4000 6000 8000 10000

05

1015

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus5

minus10

minus20

minus25

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

Mea

n Er

rors

(log)

(h) F11


Complexity 13

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus10

minus5

0

5

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

0

10

20

30

Mea

n Er

rors

(log)

2000 4000 6000 8000 100000Iteration

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

50

Mea

n Er

rors

(log)

(h) F11


14 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

20

Mea

n Er

rors

(log)

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

0

2

4

6

8

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

01020304050

Mea

n Er

rors

(log)

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus140

minus120

minus100

minus80

minus60

minus40

minus20

020

Mea

n Er

rors

(log)

(h) F11


Complexity 15

Table6Multim

odalbenchm

arkfunctio

ns

Functio

nRa

nge

Fmin

F12(119909)=

minus20exp(minus0

2radic1 119899119899 sum 119894=11199092 119894)minus

exp(1 119899119899 sum 119894=1co

s(2120587119909 119894))

+20+exp

(1 )[minus32

32]0

F13(119909)=

119899 sum 119894=11003816 1003816 1003816 1003816119909 119894sin(119909 119894)

+01119909 1198941003816 1003816 1003816 1003816

[minus1010]

0

F14(119909)=

119891 119904(1199091119909 2)


(119909 119899119909 1)

[minus10010

0]0

119891 119904(119909119910)=

(1199092 +1199102 )025[sin2

(50(1199092 +

1199102 )01)+1

]F15(

119909)=119891 119904(1199091119909 2)


119909 1198991199091)

[minus10010

0]0

119891 119904(119909119910)=

05(sin2(radic 1199092+1199102

)minus05)

(1+0001

(1199092 +1199102 ))2

F16(119909)=

120587 11989910sin2

(120587119910 119894)+119899minus1 sum 119894=1

(119910 119894minus1 )2 [

1+10sin2

(120587119910 119894+1)]+

(119910 119899minus1 )2

+119899 sum 119894=1119906(119909 119894

10100

4)[minus50

50]0

119910 119894=1+1 4(119909

119894+1)

119906(119909 119894119886

119896119898)= 119896(119909 119894

minus119886)119898

119909 119894gt119886

0minus119886le

119909 119894le119886

119896(minus119909119894minus119886)119898

119909119894gt119886

F17(119909)=

1 4000119899 sum 119894=11199092 119894minus119899 prod 119894=1

cos(119909119894 radic 119894)+1

[minus10010

0]0

F18(119909)=


(minus(1199092 119894+

1199092 119894+1+05

119909 119894119909 119894+1)

8)lowastc

os(4radic

1199092 119894+1199092 119894+1

+05119909 119894119909 119894+1))

[minus55]

1-n

F19(119909)=

119899 sum 119894=1(119909119894minus1)2

minus119899 sum 119894=2119909 119894119909 119894minus1

[minusn2n2 ]

119899(119899+4)(119899

minus1)minus6

F20 (119909 )=

sum119899minus1 119894=2(05

+(sin2(radic 1

001199092 119894+1199092 119894+1)minus0

5))(1+

0001(1199092 119894minus

2119909 119894119909119894minus1+1199092 119894minus1))2

[minus10010

0]0

F21(119909)=

119899 sum 119894=1[1199092 119894minus10

cos(2120587

119909 119894)+10]

[minus51251

2]0

F22(119909)=

119899 sum 119894=1[1199102 119894minus10

cos(2120587

119910 119894)+10]

119910 119894= 119909 119894

1003816 1003816 1003816 1003816119909 1198941003816 1003816 1003816 1003816lt05

119903119900119906119899119889(2119909

119894)2

1003816 1003816 1003816 10038161199091198941003816 1003816 1003816 1003816lt0

5[minus51

2512]

0

F23(119909)=

1minuscos(2120587

radic119899 sum 119894=11199092 119894)

+01radic119899 sum 119894=1

1199092 119894[minus10

0100]

0

F24(119909)=

119899 sum 119894=1119896119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 (119909119894+05

))]minus119899119896

119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 05

)][minus05

05]0

F25(119909)=

119899 sum 119896=1119899 sum 119895=1((1199102 119895119896 4000)minus

cos(119910 119895119896)+1

)119910119895119896=10

0(119909 119896minus1199092 119895

)2 +(1minus

1199092 119895)2[minus10

0100]

0

16 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmultim

odalbenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

51692E-14

21708E-07

16343E

+01

42641E-12

43970E-07

70983E

+00

Std

94813E-15

11785E

-07

45830E-01

51275E-13

53024E-08

45755E+

00F13

Mean

19651E-15

17670E

-07

30865E+

0138781E-12

12507E

-06

29660

E+01

Std

17016E

-1510

899E

-07

28749E+

00200

14E-12

19125E

-06

57790E+

00F14

Mean

28586E-11

47414E-02

21576E+

0235954E-05

17290E

-02

18705E

+02

Std

17874E

-1118

105E

-02

50836E+

0019

343E

-06

10857E

-03

50868E+

01F15

Mean

99552E

-01

46150E-01

12596E

+01

94983E

-01

10032E

+00

12147E

+01

Std

38926E-01

33522E-01

21495E-01

42966

E-01

35690E-01

17388E

-01

F16

Mean

15705E

-32

13069E

-15

56725E+

0650290E-25

99726E

-15

31482E+

00Std

28850E-48

57169E-16

17168E

+06

47027E-25

85374E-15

58054E-01

F17

Mean

13781E-02

10332E

-02

43352E+

0044332E-03

12793E

-02

10971E+0

0Std

14865E

-02

12632E

-02

42518E-01

79408E

-03

10155E

-02

10766E

-02

F18

Mean

50849E+

0038253E+

0020946

E+01

49225E+

00490

48E+

0021497E+

01Std

16014E

+00

14627E

+00

76856E

-01

21737E+

00204

11E+0

013

669E

+00

F19

Mean

268

41E-07

19292E

+02

49808E+

0519

677E

+02

240

98E+

0230226E+

04Std

32619E-08

15971E+0

214

706E

+05

16572E

+02

23149E+

0260289E+

03F2

0Mean

25989E-07

47006

E-06

33592E-02

44469E-08

18865E

-07

1540

6E-01

Std

59383E-07

73387E

-06

22456E-02

10350E

-07

31612E-07

56719E-02

F21

Mean

000

00E+

0070

841E-13

25769E+

02000

00E+

0045409E-11

30881E+

02Std

000

00E+

0045361E-13

90973E

+00

000

00E+

0019

882E

-11

27305E+

01F2

2Mean

000

00E+

007746

7E-13

23335E+

02000

00E+

00644

03E-11

25509E+

02Std

000

00E+

0036979E-13

15942E

+01

000

00E+

0033820E-11

26992E+

01F2

3Mean

93987E

-01

52987E-01

12199E

+01

13599E

+00

14399E

+00

21878E+

00Std

21705E-01

12517E

-01

49304

E-01

36576E-01

21705E-01

62731E-02

F24

Mean

14921E-14

37233E-04

32412E+

0147458E-09

42553E-03

26924E+

01Std

17226E

-1498

846E

-05

11649E

+00

28242E-09

42975E-04

35559E+

00F2

5Mean

29494E+

0110

724E

+02

11372E

+07

404

62E+

0193530E+

0092

421E+0

3Std

29743E+

0151800

E+01

31606

E+06

39685E+

0190392E+

0018

838E

+03

Complexity 17


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

85798E-14

24174E-04

18459E

+01

7404

4E-12

73673E

-07

82226E+

00Std

17360E

-1455274E-05

1944

7E-01

88139E-13

80222E-08

42517E+

00F13

Mean

22538E-15

29492E-04

71594E

+01

21041E-11

32004

E-06

60959E+

01Std

18688E

-1510

372E

-04

45394E+

0015

865E

-11

15334E

-06

44766

E+00

F14

Mean

71759E

-1120261E+

0043430E+

0277682E

-05

33324E-02

42669E+

02Std

24650E-11

50770E-01

14055E

+01

54975E-06

10537E

-03

80127E+

01F15

Mean

16716E

+00

12749E

+00

22241E+

0116

927E

+00

14937E

+00

21617E+

01Std

76572E

-01

43985E-01

33014E-01

47677E-01

63574E-01

54534E-01

F16

Mean

94233E-33

13057E

-09

76995E

+07

17755E

-24

846

48E-14

69921E+

00Std

14425E

-48

37533E-10

21712E+

0719

092E

-24

17429E

-13

89129E-01

F17

Mean

76377E

-03

14219E

-02

1160

6E+0

164039E-03

10080E

-02

1264

1E+0

0Std

57418E-03

21089E-02

46282E-01

70807E

-03

13952E

-02

16555E

-02

F18

Mean

83103E+

0079

047E

+00

39689E+

0189467E+

0096

041E+0

038726E+

01Std

260

72E+

0025432E+

0077616E

-01

78506E

-01

21029E+

0013

015E

+00

F19

Mean

45562E+

0126833E+

04806

68E+

0616

118E+

0413

155E

+04

70015E

+05

Std

38094E+

0121743E+

0421709E+

0612

498E

+04

1300

9E+0

497

174E

+04

F20

Mean

43064E-08

25702E-04

11519E

-01

52365E-08

16998E

-06

500

47E-01

Std

44294E-08

27576E-04

39417E-02

95247E

-08

49881E-06

26305E-01

F21

Mean

000

00E+

0011310E

-06

53146

E+02

000

00E+

0023711E

-10

58748E+

02Std

000

00E+

0033614E-07

32117E+

01000

00E+

0045437E-11

29507E+

01F2

2Mean

000

00E+

0016

167E

-06

48729E+

02000

00E+

00244

07E-10

52060

E+02

Std

000

00E+

0063216E-07

24382E+

01000

00E+

0075

889E

-11

42230E+

01F2

3Mean

13699E

+00

89987E-01

21237E+

0122699E+

0025899E+

0035955E+

00Std

23594E-01

666

67E-02

58033E-01

41913E-01

62973E-01

12247E

-01

F24

Mean

71054E

-1426826E-02

63090E+

0119

033E

-08

96037E

-03

47263E+

01Std

27621E-14

47780E-03

22392E+

0061075E-09

97071E-04

52689E+

00F2

5Mean

66563E+

0184722E+

0211275E

+08

65780E+

0139992E+

0188242E+

04Std

10992E

+02

2113

8E+0

221091E+

0794

954E

+01

43819E+

0116

832E

+04

18 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

18989E

-1316

584E

-01

19996E

+01

17809E

-11

14744E

-06

13554E

+01

Std

20566

E-14

53720E-02

90319E

-02

19159E

-12

18930E

-07

60821E+

00F13

Mean

22871E-15

17736E

-01

1944

7E+0

213

452E

-10

13291E-05

16379E

+02

Std

26741E-15

53611E-02

62653E+

0038592E-11

55001E-06

13313E

+01

F14

Mean

18736E

-1074

259E

+01

10132E

+03

22866

E-04

83534E-02

95534E

+02

Std

37223E-11

19144E

+01

18986E

+01

14283E

-05

10592E

-02

53523E+

01F15

Mean

26814E+

0010

178E

+01

47083E+

0128083E+

0034325E+

0045859E+

01Std

73851E-01

16238E

+00

22513E-01

46148E-01

60283E-01

69914E-01

F16

Mean

47116E-33

244

54E-04

90382E

+08

81890E-24

62347E-14

27647E+

03Std

72124E

-49

59650E-05

64985E+

0767958E-24

55604

E-14

44231E+

03F17

Mean

34494E-03

11896E

-02

37816E+

0134509E-03

41885E-03

21280E+

00Std

60565E-03

65363E-03

15922E

+00

46765E-03

86153E-03

54359E-02

F18

Mean

18033E

+01

17806E

+01

86826E+

0118

319E

+01

18828E

+01

82458E+

01Std

19652E

+00

38319E+

0093

222E

-01

29296E+

0025377E+

0015

159E

+00

F19

Mean

82462E+

0427944

E+06

48046

E+08

28415E+

0560265E+

0549201E+

07Std

55732E+

0489703E+

0596

715E

+07

24572E+

0527137E+

0572

772E

+06

F20

Mean

57130E-07

81688E-03

96848E

-01

13631E-06

27143E-05

21656E+

00Std

61122E-07

53195E-03

44542E-01

25155E-06

58766

E-05

80368E-01

F21

Mean

000

00E+

0051414E+

0013

305E

+03

000

00E+

0020026E-09

13623E

+03

Std

000

00E+

0017

825E

+00

22890E+

01000

00E+

0032815E-10

609

96E+

01F2

2Mean

000

00E+

0077

848E

+00

1260

9E+0

3000

00E+

0020383E-09

12745E

+03

Std

000

00E+

0023732E+

0029100

E+01

000

00E+

0029753E-10

59708E+

01F2

3Mean

25599E+

0020499E+

0039804

E+01

47099E+

0043699E+

0073

691E+0

0Std

36878E-01

15092E

-01

69296E-01

59151E-01

56184E-01

17989E

-01

F24

Mean

40927E-13

26229E+

0015

145E

+02

18874E

-07

29476E-02

10478E

+02

Std

88061E-14

63367E-01

42830E+

0037074E-08

17697E

-03

11873E

+01

F25

Mean

42987E+

028117

8E+0

313

524E

+09

56790E+

0244982E+

0218

038E

+06

Std

43423E+

0233128E+

0278

399E

+07

54327E+

0246926E+

0221315E+

05

Complexity 19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus8

minus6

minus4

minus2

02468

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(f) F24




MAE = sum119873119895=1 10038161003816100381610038161003816119898119895 minus 11989611989510038161003816100381610038161003816119873 (22)

20 Complexity

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

ISSASSAPSO

CMFOAIFFOFOA

1

2

3

4

5

6

7

8

Mea

n Er

rors

(log)

0 4000 6000 8000 100002000Iteration

(e) F19

ISSASSAPSO

CMFOAIFFOFOA

20000 6000 8000 100004000Iteration

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




Complexity 21

Table11

Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

31365E+

0413

133E

+06

11295E

+08

10017E

+06

864

75E+

0513

449E

+07

Std

18602E

+04

52974E+

0531387E+

0748689E+

0548607E+

0528405E+

06F2

7Mean

304

00E-10

1844

6E+0

484500

E+09

10512E

+04

12359E

+04

58535E+

08Std

61535E-10

14049E

+04

10125E

+09

12485E

+04

11922E

+04

38771E+

07F2

8Mean

42105E-01

46710E+

0173

819E

+02

4114

7E+0

137814E+

0114

226E

+02

Std

12624E

+00

31490E+

0199

455E

+01

47336E+

0134110E+

0129201E+

01F2

9Mean

75177E

+03

14891E+0

529286E+

0647277E+

0531099E+

0539826E+

05Std

33119E+

0368316E+

049190

4E+0

521021E+

0522686E+

0515

511E+0

5F3

0Mean

31524E+

0231524E+

0238129E+

0231524E+

0231524E+

0232568E+

02Std

85708E-12

19710E

-07

14082E

+01

11524E

-1145680E-11

58955E+

00F31

Mean

23483E+

0223172E+

0230117E+

0223811E

+02

23858E+

0224179E+

02Std

41748E+

0072

461E+0

048903E+

00560

97E+

0050249E+

0090

228E

+00

F32

Mean

20790E+

02206

03E+

0221884E+

0221485E+

0220975E+

0220633E+

02Std

41618E+

0032456E+

0030353E+

0087909E+

0057719E+

0016

880E

+00

22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








8 Complexity









Complexity 9


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

22612E-46

14374E

-13

18031E+0

338546

E-22

53419E-12

58887E+

01Std

39697E-46

10187E

-13

16038E

+02

1146

4E-22

26506

E-12

10717E

+01

F2Mean

53333E-01

600

00E-01

70475E

+04

666

67E-01

666

67E-01

12221E+0

2Std

28109E-01

21082E-01

18027E

+04

11102E

-16

364

14E-11

35452E+

01F3

Mean

000

00E+

00000

00E+

0047300

E-01

000

00E+

00860

42E-14

18586E

-02

Std

18504E

-1626168E-16

42225E-02

906

49E-17

27669E-14

19010E

-03

F4Mean

21268E-39

39943E-10

92617E

+07

37704

E-18

20345E-08

11112

E+07

Std

564

86E-39

32855E-10

18784E

+07

24165E-18

14277E

-08

30272E+

06F5

Mean

43164

E-03

93054E

-02

55376E+

0032231E-03

22754E-03

17965E

-02

Std

19931E-03

23058E-02

10210E

+00

13321E-03

1104

6E-03

31904

E-03

F6Mean

55447E-14

29695E+

0110

669E

+07

76347E

+00

89052E+

0012

862E

+04

Std

54894E-14

340

74E+

0118

313E

+06

590

03E+

0071150E

+00

44338E+

03F7

Mean

11996E

-44

65616E-12

16688E

+05

71055E

-22

60744

E-12

54708E+

03Std

304

49E-44

540

85E-12

17265E

+04

16506E

-22

29515E-12

49070E+

02F8

Mean

35080E-13

17850E

-03

45639E+

0127020E-11

264

40E-06

78561E+0

0Std

70894E

-1343281E-04

20578E+

00604

13E-12

24822E-07

17334E

+00

F9Mean

55772E-24

22148E-07

71792E

+01

23880E-11

204

15E-06

304

53E+

01Std

79227E

-24

67302E-08

30539E+

0141360

E-12

36636E-07

46515E+

01F10

Mean

26748E-44

44745E-13

13098E

+04

42105E-23

440

42E-13

36501E+

02Std

78461E-44

37055E-13

13116

E+03

10825E

-23

15887E

-13

43608E+

01F11

Mean

61803E-188

17833E

-60

18391E-03

78265E

-25

5117

6E-15

67271E-07

Std

000

00E+

0037202E-60

11147E

-03

65934E-25

76076E

-15

46836E-07

10 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

19373E

-45

244

08E-07

89004

E+03

36571E-21

32632E-11

28288E+

02Std

40611E

-45

72161E-08

10186E

+03

90310E

-22

82802E-12

18277E

+01

F2Mean

666

67E-01

21716E+

0080535E+

0512

514E

+00

666

67E-01

82661E+

02Std

82003E-16

22142E+

0011670E

+05

12485E

+00

25423E-10

77814E

+01

F3Mean

000

00E+

0076

318E

-11

866

02E-01

000

00E+

004110

0E-13

51246

E-02

Std

22204E-16

22120E-11

18360E

-02

23984E-16

14800E

-13

40430E-03

F4Mean

306

71E-38

97033E

-04

46756E+

0819

432E

-17

10621E-07

38893E+

07Std

69186E-38

344

64E-04

60135E+

0711627E

-17

76083E

-08

73301E+0

6F5

Mean

71557E

-03

29566

E-01

53164

E+01

10084E

-02

73580E

-03

10458E

-01

Std

23021E-03

33200

E-02

64915E+

0026523E-03

18100E

-03

26586E-02

F6Mean

43706

E-11

95471E+0

170

118E+

0777

331E+0

147194E+

0165331E+

04Std

95151E-11

35358E+

0153302E+

06344

63E+

0140976E+

0113

721E+0

4F7

Mean

12947E

-41

23079E-05

91659E

+05

69771E-21

64025E-11

28862E+

04Std

37876E-41

58814E-06

93287E

+04

31808E-21

26901E-11

28375E+

03F8

Mean

60872E-11

12706E

-01

67093E+

0184930E-11

71576E

-06

11919E

+01

Std

25158E-11

33391E-02

25011E

+00

11107E

-11

69032E-07

1040

4E+0

0F9

Mean

18289E

-23

38822E-04

20565E+

1060338E-11

63442E-06

11434E

+05

Std

28884E-23

72525E

-05

63864

E+10

64165E-12

90797E

-07

24588E+

05F10

Mean

12924E

-44

14594E

-06

41629E+

0422846

E-22

20175E-12

10536E

+03

Std

25807E-44

606

62E-07

37125E+

0341424E-23

52761E-13

59785E+

01F11

Mean

44745E-163

19208E

-58

92852E

-03

11169E

-24

51458E-15

16917E

-06

Std

000

00E+

0022612E-58

35776E-03

14674E

-24

82130E-15

94517E

-07

Complexity 11


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

52002E-44

1360

1E-01

64559E+

0467630E-20

540

42E-10

23688E+

03Std

89565E-44

28398E-02

27675E+

0318

636E

-20

10843E

-10

11782E

+02

F2Mean

25837E+

0028507E+

0189058E+

0610

018E

+01

11850E

+01

13921E+0

4Std

40923E+

0012

482E

+01

64125E+

0598

260E

+00

67378E+

0021479E+

03F3

Mean

19984E

-1517

506E

-05

99937E

-01

19984E

-1534570E-12

1946

0E-01

Std

27940E-16

33607E-06

19874E

-04

39686E-16

57323E-13

11259E

-02

F4Mean

14073E

-38

30139E+

0226151E+

0914

261E-16

10212E

-06

20499E+

08Std

15382E

-38

90551E+0

126783E+

0875

737E

-17

33853E-07

35388E+

07F5

Mean

17615E

-02

14345E

+00

59603E+

0237550E-02

29349E-02

12443E

+00

Std

42239E-03

13985E

-01

58412E+

0112

602E

-02

45825E-03

18471E-01

F6Mean

11417E

+01

58422E+

0242299E+

0817

988E

+02

16578E

+02

560

78E+

05Std

30258E+

0197

884E

+02

48581E+

0739022E+

01466

85E+

0165477E+

04F7

Mean

16881E-41

12984E

+01

64707E+

0611852E

-19

74831E-10

22865E+

05Std

34134E-41

22729E+

0033435E+

0531718E-20

95033E

-11

20650E+

04F8

Mean

45259E-08

39819E+

0085137E+

0117

042E

-04

29244

E-05

33956E+

01Std

29104E-08

41522E-01

12566E

+00

78665E

-05

27018E-06

47713E+

00F9

Mean

12222E

-22

36814E-01

72469E

+32

25070E-10

23795E-05

14112

E+27

Std

84369E-23

41963E-02

20633E+

3323874E-11

13879E

-06

44627E+

27F10

Mean

34254E-42

37261E-01

14940E

+05

20714E-21

18486E

-11

43041E+

03Std

966

08E-42

92561E-02

446

43E+

03464

07E-22

23956E-12

24348E+

02F11

Mean

1640

0E-12

315

040E

-52

37278E-02

65780E-24

3117

4E-14

10720E

-05

Std

51861E-123

16670E

-52

11165E

-02

72960E

-24

36245E-14

59475E-06

12 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

(b) F4

0 2000 4000 6000 8000 10000

0

5

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus15

minus5

(c) F6

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

minus50

(d) F7

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus2

minus4

minus6

minus8

minus10

minus12

minus14

(e) F8

0 2000 4000 6000 8000 10000

05

1015

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus5

minus10

minus20

minus25

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

Mea

n Er

rors

(log)

(h) F11


Complexity 13

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus10

minus5

0

5

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

0

10

20

30

Mea

n Er

rors

(log)

2000 4000 6000 8000 100000Iteration

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

50

Mea

n Er

rors

(log)

(h) F11


14 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

20

Mea

n Er

rors

(log)

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

0

2

4

6

8

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

01020304050

Mea

n Er

rors

(log)

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus140

minus120

minus100

minus80

minus60

minus40

minus20

020

Mea

n Er

rors

(log)

(h) F11


Complexity 15

Table6Multim

odalbenchm

arkfunctio

ns

Functio

nRa

nge

Fmin

F12(119909)=

minus20exp(minus0

2radic1 119899119899 sum 119894=11199092 119894)minus

exp(1 119899119899 sum 119894=1co

s(2120587119909 119894))

+20+exp

(1 )[minus32

32]0

F13(119909)=

119899 sum 119894=11003816 1003816 1003816 1003816119909 119894sin(119909 119894)

+01119909 1198941003816 1003816 1003816 1003816

[minus1010]

0

F14(119909)=

119891 119904(1199091119909 2)


(119909 119899119909 1)

[minus10010

0]0

119891 119904(119909119910)=

(1199092 +1199102 )025[sin2

(50(1199092 +

1199102 )01)+1

]F15(

119909)=119891 119904(1199091119909 2)


119909 1198991199091)

[minus10010

0]0

119891 119904(119909119910)=

05(sin2(radic 1199092+1199102

)minus05)

(1+0001

(1199092 +1199102 ))2

F16(119909)=

120587 11989910sin2

(120587119910 119894)+119899minus1 sum 119894=1

(119910 119894minus1 )2 [

1+10sin2

(120587119910 119894+1)]+

(119910 119899minus1 )2

+119899 sum 119894=1119906(119909 119894

10100

4)[minus50

50]0

119910 119894=1+1 4(119909

119894+1)

119906(119909 119894119886

119896119898)= 119896(119909 119894

minus119886)119898

119909 119894gt119886

0minus119886le

119909 119894le119886

119896(minus119909119894minus119886)119898

119909119894gt119886

F17(119909)=

1 4000119899 sum 119894=11199092 119894minus119899 prod 119894=1

cos(119909119894 radic 119894)+1

[minus10010

0]0

F18(119909)=


(minus(1199092 119894+

1199092 119894+1+05

119909 119894119909 119894+1)

8)lowastc

os(4radic

1199092 119894+1199092 119894+1

+05119909 119894119909 119894+1))

[minus55]

1-n

F19(119909)=

119899 sum 119894=1(119909119894minus1)2

minus119899 sum 119894=2119909 119894119909 119894minus1

[minusn2n2 ]

119899(119899+4)(119899

minus1)minus6

F20 (119909 )=

sum119899minus1 119894=2(05

+(sin2(radic 1

001199092 119894+1199092 119894+1)minus0

5))(1+

0001(1199092 119894minus

2119909 119894119909119894minus1+1199092 119894minus1))2

[minus10010

0]0

F21(119909)=

119899 sum 119894=1[1199092 119894minus10

cos(2120587

119909 119894)+10]

[minus51251

2]0

F22(119909)=

119899 sum 119894=1[1199102 119894minus10

cos(2120587

119910 119894)+10]

119910 119894= 119909 119894

1003816 1003816 1003816 1003816119909 1198941003816 1003816 1003816 1003816lt05

119903119900119906119899119889(2119909

119894)2

1003816 1003816 1003816 10038161199091198941003816 1003816 1003816 1003816lt0

5[minus51

2512]

0

F23(119909)=

1minuscos(2120587

radic119899 sum 119894=11199092 119894)

+01radic119899 sum 119894=1

1199092 119894[minus10

0100]

0

F24(119909)=

119899 sum 119894=1119896119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 (119909119894+05

))]minus119899119896

119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 05

)][minus05

05]0

F25(119909)=

119899 sum 119896=1119899 sum 119895=1((1199102 119895119896 4000)minus

cos(119910 119895119896)+1

)119910119895119896=10

0(119909 119896minus1199092 119895

)2 +(1minus

1199092 119895)2[minus10

0100]

0

16 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmultim

odalbenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

51692E-14

21708E-07

16343E

+01

42641E-12

43970E-07

70983E

+00

Std

94813E-15

11785E

-07

45830E-01

51275E-13

53024E-08

45755E+

00F13

Mean

19651E-15

17670E

-07

30865E+

0138781E-12

12507E

-06

29660

E+01

Std

17016E

-1510

899E

-07

28749E+

00200

14E-12

19125E

-06

57790E+

00F14

Mean

28586E-11

47414E-02

21576E+

0235954E-05

17290E

-02

18705E

+02

Std

17874E

-1118

105E

-02

50836E+

0019

343E

-06

10857E

-03

50868E+

01F15

Mean

99552E

-01

46150E-01

12596E

+01

94983E

-01

10032E

+00

12147E

+01

Std

38926E-01

33522E-01

21495E-01

42966

E-01

35690E-01

17388E

-01

F16

Mean

15705E

-32

13069E

-15

56725E+

0650290E-25

99726E

-15

31482E+

00Std

28850E-48

57169E-16

17168E

+06

47027E-25

85374E-15

58054E-01

F17

Mean

13781E-02

10332E

-02

43352E+

0044332E-03

12793E

-02

10971E+0

0Std

14865E

-02

12632E

-02

42518E-01

79408E

-03

10155E

-02

10766E

-02

F18

Mean

50849E+

0038253E+

0020946

E+01

49225E+

00490

48E+

0021497E+

01Std

16014E

+00

14627E

+00

76856E

-01

21737E+

00204

11E+0

013

669E

+00

F19

Mean

268

41E-07

19292E

+02

49808E+

0519

677E

+02

240

98E+

0230226E+

04Std

32619E-08

15971E+0

214

706E

+05

16572E

+02

23149E+

0260289E+

03F2

0Mean

25989E-07

47006

E-06

33592E-02

44469E-08

18865E

-07

1540

6E-01

Std

59383E-07

73387E

-06

22456E-02

10350E

-07

31612E-07

56719E-02

F21

Mean

000

00E+

0070

841E-13

25769E+

02000

00E+

0045409E-11

30881E+

02Std

000

00E+

0045361E-13

90973E

+00

000

00E+

0019

882E

-11

27305E+

01F2

2Mean

000

00E+

007746

7E-13

23335E+

02000

00E+

00644

03E-11

25509E+

02Std

000

00E+

0036979E-13

15942E

+01

000

00E+

0033820E-11

26992E+

01F2

3Mean

93987E

-01

52987E-01

12199E

+01

13599E

+00

14399E

+00

21878E+

00Std

21705E-01

12517E

-01

49304

E-01

36576E-01

21705E-01

62731E-02

F24

Mean

14921E-14

37233E-04

32412E+

0147458E-09

42553E-03

26924E+

01Std

17226E

-1498

846E

-05

11649E

+00

28242E-09

42975E-04

35559E+

00F2

5Mean

29494E+

0110

724E

+02

11372E

+07

404

62E+

0193530E+

0092

421E+0

3Std

29743E+

0151800

E+01

31606

E+06

39685E+

0190392E+

0018

838E

+03

Complexity 17


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

85798E-14

24174E-04

18459E

+01

7404

4E-12

73673E

-07

82226E+

00Std

17360E

-1455274E-05

1944

7E-01

88139E-13

80222E-08

42517E+

00F13

Mean

22538E-15

29492E-04

71594E

+01

21041E-11

32004

E-06

60959E+

01Std

18688E

-1510

372E

-04

45394E+

0015

865E

-11

15334E

-06

44766

E+00

F14

Mean

71759E

-1120261E+

0043430E+

0277682E

-05

33324E-02

42669E+

02Std

24650E-11

50770E-01

14055E

+01

54975E-06

10537E

-03

80127E+

01F15

Mean

16716E

+00

12749E

+00

22241E+

0116

927E

+00

14937E

+00

21617E+

01Std

76572E

-01

43985E-01

33014E-01

47677E-01

63574E-01

54534E-01

F16

Mean

94233E-33

13057E

-09

76995E

+07

17755E

-24

846

48E-14

69921E+

00Std

14425E

-48

37533E-10

21712E+

0719

092E

-24

17429E

-13

89129E-01

F17

Mean

76377E

-03

14219E

-02

1160

6E+0

164039E-03

10080E

-02

1264

1E+0

0Std

57418E-03

21089E-02

46282E-01

70807E

-03

13952E

-02

16555E

-02

F18

Mean

83103E+

0079

047E

+00

39689E+

0189467E+

0096

041E+0

038726E+

01Std

260

72E+

0025432E+

0077616E

-01

78506E

-01

21029E+

0013

015E

+00

F19

Mean

45562E+

0126833E+

04806

68E+

0616

118E+

0413

155E

+04

70015E

+05

Std

38094E+

0121743E+

0421709E+

0612

498E

+04

1300

9E+0

497

174E

+04

F20

Mean

43064E-08

25702E-04

11519E

-01

52365E-08

16998E

-06

500

47E-01

Std

44294E-08

27576E-04

39417E-02

95247E

-08

49881E-06

26305E-01

F21

Mean

000

00E+

0011310E

-06

53146

E+02

000

00E+

0023711E

-10

58748E+

02Std

000

00E+

0033614E-07

32117E+

01000

00E+

0045437E-11

29507E+

01F2

2Mean

000

00E+

0016

167E

-06

48729E+

02000

00E+

00244

07E-10

52060

E+02

Std

000

00E+

0063216E-07

24382E+

01000

00E+

0075

889E

-11

42230E+

01F2

3Mean

13699E

+00

89987E-01

21237E+

0122699E+

0025899E+

0035955E+

00Std

23594E-01

666

67E-02

58033E-01

41913E-01

62973E-01

12247E

-01

F24

Mean

71054E

-1426826E-02

63090E+

0119

033E

-08

96037E

-03

47263E+

01Std

27621E-14

47780E-03

22392E+

0061075E-09

97071E-04

52689E+

00F2

5Mean

66563E+

0184722E+

0211275E

+08

65780E+

0139992E+

0188242E+

04Std

10992E

+02

2113

8E+0

221091E+

0794

954E

+01

43819E+

0116

832E

+04

18 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

18989E

-1316

584E

-01

19996E

+01

17809E

-11

14744E

-06

13554E

+01

Std

20566

E-14

53720E-02

90319E

-02

19159E

-12

18930E

-07

60821E+

00F13

Mean

22871E-15

17736E

-01

1944

7E+0

213

452E

-10

13291E-05

16379E

+02

Std

26741E-15

53611E-02

62653E+

0038592E-11

55001E-06

13313E

+01

F14

Mean

18736E

-1074

259E

+01

10132E

+03

22866

E-04

83534E-02

95534E

+02

Std

37223E-11

19144E

+01

18986E

+01

14283E

-05

10592E

-02

53523E+

01F15

Mean

26814E+

0010

178E

+01

47083E+

0128083E+

0034325E+

0045859E+

01Std

73851E-01

16238E

+00

22513E-01

46148E-01

60283E-01

69914E-01

F16

Mean

47116E-33

244

54E-04

90382E

+08

81890E-24

62347E-14

27647E+

03Std

72124E

-49

59650E-05

64985E+

0767958E-24

55604

E-14

44231E+

03F17

Mean

34494E-03

11896E

-02

37816E+

0134509E-03

41885E-03

21280E+

00Std

60565E-03

65363E-03

15922E

+00

46765E-03

86153E-03

54359E-02

F18

Mean

18033E

+01

17806E

+01

86826E+

0118

319E

+01

18828E

+01

82458E+

01Std

19652E

+00

38319E+

0093

222E

-01

29296E+

0025377E+

0015

159E

+00

F19

Mean

82462E+

0427944

E+06

48046

E+08

28415E+

0560265E+

0549201E+

07Std

55732E+

0489703E+

0596

715E

+07

24572E+

0527137E+

0572

772E

+06

F20

Mean

57130E-07

81688E-03

96848E

-01

13631E-06

27143E-05

21656E+

00Std

61122E-07

53195E-03

44542E-01

25155E-06

58766

E-05

80368E-01

F21

Mean

000

00E+

0051414E+

0013

305E

+03

000

00E+

0020026E-09

13623E

+03

Std

000

00E+

0017

825E

+00

22890E+

01000

00E+

0032815E-10

609

96E+

01F2

2Mean

000

00E+

0077

848E

+00

1260

9E+0

3000

00E+

0020383E-09

12745E

+03

Std

000

00E+

0023732E+

0029100

E+01

000

00E+

0029753E-10

59708E+

01F2

3Mean

25599E+

0020499E+

0039804

E+01

47099E+

0043699E+

0073

691E+0

0Std

36878E-01

15092E

-01

69296E-01

59151E-01

56184E-01

17989E

-01

F24

Mean

40927E-13

26229E+

0015

145E

+02

18874E

-07

29476E-02

10478E

+02

Std

88061E-14

63367E-01

42830E+

0037074E-08

17697E

-03

11873E

+01

F25

Mean

42987E+

028117

8E+0

313

524E

+09

56790E+

0244982E+

0218

038E

+06

Std

43423E+

0233128E+

0278

399E

+07

54327E+

0246926E+

0221315E+

05

Complexity 19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus8

minus6

minus4

minus2

02468

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(f) F24




MAE = sum119873119895=1 10038161003816100381610038161003816119898119895 minus 11989611989510038161003816100381610038161003816119873 (22)

20 Complexity

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

ISSASSAPSO

CMFOAIFFOFOA

1

2

3

4

5

6

7

8

Mea

n Er

rors

(log)

0 4000 6000 8000 100002000Iteration

(e) F19

ISSASSAPSO

CMFOAIFFOFOA

20000 6000 8000 100004000Iteration

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




Complexity 21

Table11

Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

31365E+

0413

133E

+06

11295E

+08

10017E

+06

864

75E+

0513

449E

+07

Std

18602E

+04

52974E+

0531387E+

0748689E+

0548607E+

0528405E+

06F2

7Mean

304

00E-10

1844

6E+0

484500

E+09

10512E

+04

12359E

+04

58535E+

08Std

61535E-10

14049E

+04

10125E

+09

12485E

+04

11922E

+04

38771E+

07F2

8Mean

42105E-01

46710E+

0173

819E

+02

4114

7E+0

137814E+

0114

226E

+02

Std

12624E

+00

31490E+

0199

455E

+01

47336E+

0134110E+

0129201E+

01F2

9Mean

75177E

+03

14891E+0

529286E+

0647277E+

0531099E+

0539826E+

05Std

33119E+

0368316E+

049190

4E+0

521021E+

0522686E+

0515

511E+0

5F3

0Mean

31524E+

0231524E+

0238129E+

0231524E+

0231524E+

0232568E+

02Std

85708E-12

19710E

-07

14082E

+01

11524E

-1145680E-11

58955E+

00F31

Mean

23483E+

0223172E+

0230117E+

0223811E

+02

23858E+

0224179E+

02Std

41748E+

0072

461E+0

048903E+

00560

97E+

0050249E+

0090

228E

+00

F32

Mean

20790E+

02206

03E+

0221884E+

0221485E+

0220975E+

0220633E+

02Std

41618E+

0032456E+

0030353E+

0087909E+

0057719E+

0016

880E

+00

22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 9


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

22612E-46

14374E

-13

18031E+0

338546

E-22

53419E-12

58887E+

01Std

39697E-46

10187E

-13

16038E

+02

1146

4E-22

26506

E-12

10717E

+01

F2Mean

53333E-01

600

00E-01

70475E

+04

666

67E-01

666

67E-01

12221E+0

2Std

28109E-01

21082E-01

18027E

+04

11102E

-16

364

14E-11

35452E+

01F3

Mean

000

00E+

00000

00E+

0047300

E-01

000

00E+

00860

42E-14

18586E

-02

Std

18504E

-1626168E-16

42225E-02

906

49E-17

27669E-14

19010E

-03

F4Mean

21268E-39

39943E-10

92617E

+07

37704

E-18

20345E-08

11112

E+07

Std

564

86E-39

32855E-10

18784E

+07

24165E-18

14277E

-08

30272E+

06F5

Mean

43164

E-03

93054E

-02

55376E+

0032231E-03

22754E-03

17965E

-02

Std

19931E-03

23058E-02

10210E

+00

13321E-03

1104

6E-03

31904

E-03

F6Mean

55447E-14

29695E+

0110

669E

+07

76347E

+00

89052E+

0012

862E

+04

Std

54894E-14

340

74E+

0118

313E

+06

590

03E+

0071150E

+00

44338E+

03F7

Mean

11996E

-44

65616E-12

16688E

+05

71055E

-22

60744

E-12

54708E+

03Std

304

49E-44

540

85E-12

17265E

+04

16506E

-22

29515E-12

49070E+

02F8

Mean

35080E-13

17850E

-03

45639E+

0127020E-11

264

40E-06

78561E+0

0Std

70894E

-1343281E-04

20578E+

00604

13E-12

24822E-07

17334E

+00

F9Mean

55772E-24

22148E-07

71792E

+01

23880E-11

204

15E-06

304

53E+

01Std

79227E

-24

67302E-08

30539E+

0141360

E-12

36636E-07

46515E+

01F10

Mean

26748E-44

44745E-13

13098E

+04

42105E-23

440

42E-13

36501E+

02Std

78461E-44

37055E-13

13116

E+03

10825E

-23

15887E

-13

43608E+

01F11

Mean

61803E-188

17833E

-60

18391E-03

78265E

-25

5117

6E-15

67271E-07

Std

000

00E+

0037202E-60

11147E

-03

65934E-25

76076E

-15

46836E-07

10 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

19373E

-45

244

08E-07

89004

E+03

36571E-21

32632E-11

28288E+

02Std

40611E

-45

72161E-08

10186E

+03

90310E

-22

82802E-12

18277E

+01

F2Mean

666

67E-01

21716E+

0080535E+

0512

514E

+00

666

67E-01

82661E+

02Std

82003E-16

22142E+

0011670E

+05

12485E

+00

25423E-10

77814E

+01

F3Mean

000

00E+

0076

318E

-11

866

02E-01

000

00E+

004110

0E-13

51246

E-02

Std

22204E-16

22120E-11

18360E

-02

23984E-16

14800E

-13

40430E-03

F4Mean

306

71E-38

97033E

-04

46756E+

0819

432E

-17

10621E-07

38893E+

07Std

69186E-38

344

64E-04

60135E+

0711627E

-17

76083E

-08

73301E+0

6F5

Mean

71557E

-03

29566

E-01

53164

E+01

10084E

-02

73580E

-03

10458E

-01

Std

23021E-03

33200

E-02

64915E+

0026523E-03

18100E

-03

26586E-02

F6Mean

43706

E-11

95471E+0

170

118E+

0777

331E+0

147194E+

0165331E+

04Std

95151E-11

35358E+

0153302E+

06344

63E+

0140976E+

0113

721E+0

4F7

Mean

12947E

-41

23079E-05

91659E

+05

69771E-21

64025E-11

28862E+

04Std

37876E-41

58814E-06

93287E

+04

31808E-21

26901E-11

28375E+

03F8

Mean

60872E-11

12706E

-01

67093E+

0184930E-11

71576E

-06

11919E

+01

Std

25158E-11

33391E-02

25011E

+00

11107E

-11

69032E-07

1040

4E+0

0F9

Mean

18289E

-23

38822E-04

20565E+

1060338E-11

63442E-06

11434E

+05

Std

28884E-23

72525E

-05

63864

E+10

64165E-12

90797E

-07

24588E+

05F10

Mean

12924E

-44

14594E

-06

41629E+

0422846

E-22

20175E-12

10536E

+03

Std

25807E-44

606

62E-07

37125E+

0341424E-23

52761E-13

59785E+

01F11

Mean

44745E-163

19208E

-58

92852E

-03

11169E

-24

51458E-15

16917E

-06

Std

000

00E+

0022612E-58

35776E-03

14674E

-24

82130E-15

94517E

-07

Complexity 11


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

52002E-44

1360

1E-01

64559E+

0467630E-20

540

42E-10

23688E+

03Std

89565E-44

28398E-02

27675E+

0318

636E

-20

10843E

-10

11782E

+02

F2Mean

25837E+

0028507E+

0189058E+

0610

018E

+01

11850E

+01

13921E+0

4Std

40923E+

0012

482E

+01

64125E+

0598

260E

+00

67378E+

0021479E+

03F3

Mean

19984E

-1517

506E

-05

99937E

-01

19984E

-1534570E-12

1946

0E-01

Std

27940E-16

33607E-06

19874E

-04

39686E-16

57323E-13

11259E

-02

F4Mean

14073E

-38

30139E+

0226151E+

0914

261E-16

10212E

-06

20499E+

08Std

15382E

-38

90551E+0

126783E+

0875

737E

-17

33853E-07

35388E+

07F5

Mean

17615E

-02

14345E

+00

59603E+

0237550E-02

29349E-02

12443E

+00

Std

42239E-03

13985E

-01

58412E+

0112

602E

-02

45825E-03

18471E-01

F6Mean

11417E

+01

58422E+

0242299E+

0817

988E

+02

16578E

+02

560

78E+

05Std

30258E+

0197

884E

+02

48581E+

0739022E+

01466

85E+

0165477E+

04F7

Mean

16881E-41

12984E

+01

64707E+

0611852E

-19

74831E-10

22865E+

05Std

34134E-41

22729E+

0033435E+

0531718E-20

95033E

-11

20650E+

04F8

Mean

45259E-08

39819E+

0085137E+

0117

042E

-04

29244

E-05

33956E+

01Std

29104E-08

41522E-01

12566E

+00

78665E

-05

27018E-06

47713E+

00F9

Mean

12222E

-22

36814E-01

72469E

+32

25070E-10

23795E-05

14112

E+27

Std

84369E-23

41963E-02

20633E+

3323874E-11

13879E

-06

44627E+

27F10

Mean

34254E-42

37261E-01

14940E

+05

20714E-21

18486E

-11

43041E+

03Std

966

08E-42

92561E-02

446

43E+

03464

07E-22

23956E-12

24348E+

02F11

Mean

1640

0E-12

315

040E

-52

37278E-02

65780E-24

3117

4E-14

10720E

-05

Std

51861E-123

16670E

-52

11165E

-02

72960E

-24

36245E-14

59475E-06

12 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

(b) F4

0 2000 4000 6000 8000 10000

0

5

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus15

minus5

(c) F6

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

minus50

(d) F7

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus2

minus4

minus6

minus8

minus10

minus12

minus14

(e) F8

0 2000 4000 6000 8000 10000

05

1015

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus5

minus10

minus20

minus25

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

Mea

n Er

rors

(log)

(h) F11


Complexity 13

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus10

minus5

0

5

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

0

10

20

30

Mea

n Er

rors

(log)

2000 4000 6000 8000 100000Iteration

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

50

Mea

n Er

rors

(log)

(h) F11


14 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

20

Mea

n Er

rors

(log)

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

0

2

4

6

8

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

01020304050

Mea

n Er

rors

(log)

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus140

minus120

minus100

minus80

minus60

minus40

minus20

020

Mea

n Er

rors

(log)

(h) F11


Complexity 15

Table6Multim

odalbenchm

arkfunctio

ns

Functio

nRa

nge

Fmin

F12(119909)=

minus20exp(minus0

2radic1 119899119899 sum 119894=11199092 119894)minus

exp(1 119899119899 sum 119894=1co

s(2120587119909 119894))

+20+exp

(1 )[minus32

32]0

F13(119909)=

119899 sum 119894=11003816 1003816 1003816 1003816119909 119894sin(119909 119894)

+01119909 1198941003816 1003816 1003816 1003816

[minus1010]

0

F14(119909)=

119891 119904(1199091119909 2)


(119909 119899119909 1)

[minus10010

0]0

119891 119904(119909119910)=

(1199092 +1199102 )025[sin2

(50(1199092 +

1199102 )01)+1

]F15(

119909)=119891 119904(1199091119909 2)


119909 1198991199091)

[minus10010

0]0

119891 119904(119909119910)=

05(sin2(radic 1199092+1199102

)minus05)

(1+0001

(1199092 +1199102 ))2

F16(119909)=

120587 11989910sin2

(120587119910 119894)+119899minus1 sum 119894=1

(119910 119894minus1 )2 [

1+10sin2

(120587119910 119894+1)]+

(119910 119899minus1 )2

+119899 sum 119894=1119906(119909 119894

10100

4)[minus50

50]0

119910 119894=1+1 4(119909

119894+1)

119906(119909 119894119886

119896119898)= 119896(119909 119894

minus119886)119898

119909 119894gt119886

0minus119886le

119909 119894le119886

119896(minus119909119894minus119886)119898

119909119894gt119886

F17(119909)=

1 4000119899 sum 119894=11199092 119894minus119899 prod 119894=1

cos(119909119894 radic 119894)+1

[minus10010

0]0

F18(119909)=


(minus(1199092 119894+

1199092 119894+1+05

119909 119894119909 119894+1)

8)lowastc

os(4radic

1199092 119894+1199092 119894+1

+05119909 119894119909 119894+1))

[minus55]

1-n

F19(119909)=

119899 sum 119894=1(119909119894minus1)2

minus119899 sum 119894=2119909 119894119909 119894minus1

[minusn2n2 ]

119899(119899+4)(119899

minus1)minus6

F20 (119909 )=

sum119899minus1 119894=2(05

+(sin2(radic 1

001199092 119894+1199092 119894+1)minus0

5))(1+

0001(1199092 119894minus

2119909 119894119909119894minus1+1199092 119894minus1))2

[minus10010

0]0

F21(119909)=

119899 sum 119894=1[1199092 119894minus10

cos(2120587

119909 119894)+10]

[minus51251

2]0

F22(119909)=

119899 sum 119894=1[1199102 119894minus10

cos(2120587

119910 119894)+10]

119910 119894= 119909 119894

1003816 1003816 1003816 1003816119909 1198941003816 1003816 1003816 1003816lt05

119903119900119906119899119889(2119909

119894)2

1003816 1003816 1003816 10038161199091198941003816 1003816 1003816 1003816lt0

5[minus51

2512]

0

F23(119909)=

1minuscos(2120587

radic119899 sum 119894=11199092 119894)

+01radic119899 sum 119894=1

1199092 119894[minus10

0100]

0

F24(119909)=

119899 sum 119894=1119896119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 (119909119894+05

))]minus119899119896

119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 05

)][minus05

05]0

F25(119909)=

119899 sum 119896=1119899 sum 119895=1((1199102 119895119896 4000)minus

cos(119910 119895119896)+1

)119910119895119896=10

0(119909 119896minus1199092 119895

)2 +(1minus

1199092 119895)2[minus10

0100]

0

16 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmultim

odalbenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

51692E-14

21708E-07

16343E

+01

42641E-12

43970E-07

70983E

+00

Std

94813E-15

11785E

-07

45830E-01

51275E-13

53024E-08

45755E+

00F13

Mean

19651E-15

17670E

-07

30865E+

0138781E-12

12507E

-06

29660

E+01

Std

17016E

-1510

899E

-07

28749E+

00200

14E-12

19125E

-06

57790E+

00F14

Mean

28586E-11

47414E-02

21576E+

0235954E-05

17290E

-02

18705E

+02

Std

17874E

-1118

105E

-02

50836E+

0019

343E

-06

10857E

-03

50868E+

01F15

Mean

99552E

-01

46150E-01

12596E

+01

94983E

-01

10032E

+00

12147E

+01

Std

38926E-01

33522E-01

21495E-01

42966

E-01

35690E-01

17388E

-01

F16

Mean

15705E

-32

13069E

-15

56725E+

0650290E-25

99726E

-15

31482E+

00Std

28850E-48

57169E-16

17168E

+06

47027E-25

85374E-15

58054E-01

F17

Mean

13781E-02

10332E

-02

43352E+

0044332E-03

12793E

-02

10971E+0

0Std

14865E

-02

12632E

-02

42518E-01

79408E

-03

10155E

-02

10766E

-02

F18

Mean

50849E+

0038253E+

0020946

E+01

49225E+

00490

48E+

0021497E+

01Std

16014E

+00

14627E

+00

76856E

-01

21737E+

00204

11E+0

013

669E

+00

F19

Mean

268

41E-07

19292E

+02

49808E+

0519

677E

+02

240

98E+

0230226E+

04Std

32619E-08

15971E+0

214

706E

+05

16572E

+02

23149E+

0260289E+

03F2

0Mean

25989E-07

47006

E-06

33592E-02

44469E-08

18865E

-07

1540

6E-01

Std

59383E-07

73387E

-06

22456E-02

10350E

-07

31612E-07

56719E-02

F21

Mean

000

00E+

0070

841E-13

25769E+

02000

00E+

0045409E-11

30881E+

02Std

000

00E+

0045361E-13

90973E

+00

000

00E+

0019

882E

-11

27305E+

01F2

2Mean

000

00E+

007746

7E-13

23335E+

02000

00E+

00644

03E-11

25509E+

02Std

000

00E+

0036979E-13

15942E

+01

000

00E+

0033820E-11

26992E+

01F2

3Mean

93987E

-01

52987E-01

12199E

+01

13599E

+00

14399E

+00

21878E+

00Std

21705E-01

12517E

-01

49304

E-01

36576E-01

21705E-01

62731E-02

F24

Mean

14921E-14

37233E-04

32412E+

0147458E-09

42553E-03

26924E+

01Std

17226E

-1498

846E

-05

11649E

+00

28242E-09

42975E-04

35559E+

00F2

5Mean

29494E+

0110

724E

+02

11372E

+07

404

62E+

0193530E+

0092

421E+0

3Std

29743E+

0151800

E+01

31606

E+06

39685E+

0190392E+

0018

838E

+03

Complexity 17


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

85798E-14

24174E-04

18459E

+01

7404

4E-12

73673E

-07

82226E+

00Std

17360E

-1455274E-05

1944

7E-01

88139E-13

80222E-08

42517E+

00F13

Mean

22538E-15

29492E-04

71594E

+01

21041E-11

32004

E-06

60959E+

01Std

18688E

-1510

372E

-04

45394E+

0015

865E

-11

15334E

-06

44766

E+00

F14

Mean

71759E

-1120261E+

0043430E+

0277682E

-05

33324E-02

42669E+

02Std

24650E-11

50770E-01

14055E

+01

54975E-06

10537E

-03

80127E+

01F15

Mean

16716E

+00

12749E

+00

22241E+

0116

927E

+00

14937E

+00

21617E+

01Std

76572E

-01

43985E-01

33014E-01

47677E-01

63574E-01

54534E-01

F16

Mean

94233E-33

13057E

-09

76995E

+07

17755E

-24

846

48E-14

69921E+

00Std

14425E

-48

37533E-10

21712E+

0719

092E

-24

17429E

-13

89129E-01

F17

Mean

76377E

-03

14219E

-02

1160

6E+0

164039E-03

10080E

-02

1264

1E+0

0Std

57418E-03

21089E-02

46282E-01

70807E

-03

13952E

-02

16555E

-02

F18

Mean

83103E+

0079

047E

+00

39689E+

0189467E+

0096

041E+0

038726E+

01Std

260

72E+

0025432E+

0077616E

-01

78506E

-01

21029E+

0013

015E

+00

F19

Mean

45562E+

0126833E+

04806

68E+

0616

118E+

0413

155E

+04

70015E

+05

Std

38094E+

0121743E+

0421709E+

0612

498E

+04

1300

9E+0

497

174E

+04

F20

Mean

43064E-08

25702E-04

11519E

-01

52365E-08

16998E

-06

500

47E-01

Std

44294E-08

27576E-04

39417E-02

95247E

-08

49881E-06

26305E-01

F21

Mean

000

00E+

0011310E

-06

53146

E+02

000

00E+

0023711E

-10

58748E+

02Std

000

00E+

0033614E-07

32117E+

01000

00E+

0045437E-11

29507E+

01F2

2Mean

000

00E+

0016

167E

-06

48729E+

02000

00E+

00244

07E-10

52060

E+02

Std

000

00E+

0063216E-07

24382E+

01000

00E+

0075

889E

-11

42230E+

01F2

3Mean

13699E

+00

89987E-01

21237E+

0122699E+

0025899E+

0035955E+

00Std

23594E-01

666

67E-02

58033E-01

41913E-01

62973E-01

12247E

-01

F24

Mean

71054E

-1426826E-02

63090E+

0119

033E

-08

96037E

-03

47263E+

01Std

27621E-14

47780E-03

22392E+

0061075E-09

97071E-04

52689E+

00F2

5Mean

66563E+

0184722E+

0211275E

+08

65780E+

0139992E+

0188242E+

04Std

10992E

+02

2113

8E+0

221091E+

0794

954E

+01

43819E+

0116

832E

+04

18 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

18989E

-1316

584E

-01

19996E

+01

17809E

-11

14744E

-06

13554E

+01

Std

20566

E-14

53720E-02

90319E

-02

19159E

-12

18930E

-07

60821E+

00F13

Mean

22871E-15

17736E

-01

1944

7E+0

213

452E

-10

13291E-05

16379E

+02

Std

26741E-15

53611E-02

62653E+

0038592E-11

55001E-06

13313E

+01

F14

Mean

18736E

-1074

259E

+01

10132E

+03

22866

E-04

83534E-02

95534E

+02

Std

37223E-11

19144E

+01

18986E

+01

14283E

-05

10592E

-02

53523E+

01F15

Mean

26814E+

0010

178E

+01

47083E+

0128083E+

0034325E+

0045859E+

01Std

73851E-01

16238E

+00

22513E-01

46148E-01

60283E-01

69914E-01

F16

Mean

47116E-33

244

54E-04

90382E

+08

81890E-24

62347E-14

27647E+

03Std

72124E

-49

59650E-05

64985E+

0767958E-24

55604

E-14

44231E+

03F17

Mean

34494E-03

11896E

-02

37816E+

0134509E-03

41885E-03

21280E+

00Std

60565E-03

65363E-03

15922E

+00

46765E-03

86153E-03

54359E-02

F18

Mean

18033E

+01

17806E

+01

86826E+

0118

319E

+01

18828E

+01

82458E+

01Std

19652E

+00

38319E+

0093

222E

-01

29296E+

0025377E+

0015

159E

+00

F19

Mean

82462E+

0427944

E+06

48046

E+08

28415E+

0560265E+

0549201E+

07Std

55732E+

0489703E+

0596

715E

+07

24572E+

0527137E+

0572

772E

+06

F20

Mean

57130E-07

81688E-03

96848E

-01

13631E-06

27143E-05

21656E+

00Std

61122E-07

53195E-03

44542E-01

25155E-06

58766

E-05

80368E-01

F21

Mean

000

00E+

0051414E+

0013

305E

+03

000

00E+

0020026E-09

13623E

+03

Std

000

00E+

0017

825E

+00

22890E+

01000

00E+

0032815E-10

609

96E+

01F2

2Mean

000

00E+

0077

848E

+00

1260

9E+0

3000

00E+

0020383E-09

12745E

+03

Std

000

00E+

0023732E+

0029100

E+01

000

00E+

0029753E-10

59708E+

01F2

3Mean

25599E+

0020499E+

0039804

E+01

47099E+

0043699E+

0073

691E+0

0Std

36878E-01

15092E

-01

69296E-01

59151E-01

56184E-01

17989E

-01

F24

Mean

40927E-13

26229E+

0015

145E

+02

18874E

-07

29476E-02

10478E

+02

Std

88061E-14

63367E-01

42830E+

0037074E-08

17697E

-03

11873E

+01

F25

Mean

42987E+

028117

8E+0

313

524E

+09

56790E+

0244982E+

0218

038E

+06

Std

43423E+

0233128E+

0278

399E

+07

54327E+

0246926E+

0221315E+

05

Complexity 19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus8

minus6

minus4

minus2

02468

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(f) F24




MAE = sum119873119895=1 10038161003816100381610038161003816119898119895 minus 11989611989510038161003816100381610038161003816119873 (22)

20 Complexity

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

ISSASSAPSO

CMFOAIFFOFOA

1

2

3

4

5

6

7

8

Mea

n Er

rors

(log)

0 4000 6000 8000 100002000Iteration

(e) F19

ISSASSAPSO

CMFOAIFFOFOA

20000 6000 8000 100004000Iteration

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




Complexity 21

Table11

Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

31365E+

0413

133E

+06

11295E

+08

10017E

+06

864

75E+

0513

449E

+07

Std

18602E

+04

52974E+

0531387E+

0748689E+

0548607E+

0528405E+

06F2

7Mean

304

00E-10

1844

6E+0

484500

E+09

10512E

+04

12359E

+04

58535E+

08Std

61535E-10

14049E

+04

10125E

+09

12485E

+04

11922E

+04

38771E+

07F2

8Mean

42105E-01

46710E+

0173

819E

+02

4114

7E+0

137814E+

0114

226E

+02

Std

12624E

+00

31490E+

0199

455E

+01

47336E+

0134110E+

0129201E+

01F2

9Mean

75177E

+03

14891E+0

529286E+

0647277E+

0531099E+

0539826E+

05Std

33119E+

0368316E+

049190

4E+0

521021E+

0522686E+

0515

511E+0

5F3

0Mean

31524E+

0231524E+

0238129E+

0231524E+

0231524E+

0232568E+

02Std

85708E-12

19710E

-07

14082E

+01

11524E

-1145680E-11

58955E+

00F31

Mean

23483E+

0223172E+

0230117E+

0223811E

+02

23858E+

0224179E+

02Std

41748E+

0072

461E+0

048903E+

00560

97E+

0050249E+

0090

228E

+00

F32

Mean

20790E+

02206

03E+

0221884E+

0221485E+

0220975E+

0220633E+

02Std

41618E+

0032456E+

0030353E+

0087909E+

0057719E+

0016

880E

+00

22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








10 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

19373E

-45

244

08E-07

89004

E+03

36571E-21

32632E-11

28288E+

02Std

40611E

-45

72161E-08

10186E

+03

90310E

-22

82802E-12

18277E

+01

F2Mean

666

67E-01

21716E+

0080535E+

0512

514E

+00

666

67E-01

82661E+

02Std

82003E-16

22142E+

0011670E

+05

12485E

+00

25423E-10

77814E

+01

F3Mean

000

00E+

0076

318E

-11

866

02E-01

000

00E+

004110

0E-13

51246

E-02

Std

22204E-16

22120E-11

18360E

-02

23984E-16

14800E

-13

40430E-03

F4Mean

306

71E-38

97033E

-04

46756E+

0819

432E

-17

10621E-07

38893E+

07Std

69186E-38

344

64E-04

60135E+

0711627E

-17

76083E

-08

73301E+0

6F5

Mean

71557E

-03

29566

E-01

53164

E+01

10084E

-02

73580E

-03

10458E

-01

Std

23021E-03

33200

E-02

64915E+

0026523E-03

18100E

-03

26586E-02

F6Mean

43706

E-11

95471E+0

170

118E+

0777

331E+0

147194E+

0165331E+

04Std

95151E-11

35358E+

0153302E+

06344

63E+

0140976E+

0113

721E+0

4F7

Mean

12947E

-41

23079E-05

91659E

+05

69771E-21

64025E-11

28862E+

04Std

37876E-41

58814E-06

93287E

+04

31808E-21

26901E-11

28375E+

03F8

Mean

60872E-11

12706E

-01

67093E+

0184930E-11

71576E

-06

11919E

+01

Std

25158E-11

33391E-02

25011E

+00

11107E

-11

69032E-07

1040

4E+0

0F9

Mean

18289E

-23

38822E-04

20565E+

1060338E-11

63442E-06

11434E

+05

Std

28884E-23

72525E

-05

63864

E+10

64165E-12

90797E

-07

24588E+

05F10

Mean

12924E

-44

14594E

-06

41629E+

0422846

E-22

20175E-12

10536E

+03

Std

25807E-44

606

62E-07

37125E+

0341424E-23

52761E-13

59785E+

01F11

Mean

44745E-163

19208E

-58

92852E

-03

11169E

-24

51458E-15

16917E

-06

Std

000

00E+

0022612E-58

35776E-03

14674E

-24

82130E-15

94517E

-07

Complexity 11


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

52002E-44

1360

1E-01

64559E+

0467630E-20

540

42E-10

23688E+

03Std

89565E-44

28398E-02

27675E+

0318

636E

-20

10843E

-10

11782E

+02

F2Mean

25837E+

0028507E+

0189058E+

0610

018E

+01

11850E

+01

13921E+0

4Std

40923E+

0012

482E

+01

64125E+

0598

260E

+00

67378E+

0021479E+

03F3

Mean

19984E

-1517

506E

-05

99937E

-01

19984E

-1534570E-12

1946

0E-01

Std

27940E-16

33607E-06

19874E

-04

39686E-16

57323E-13

11259E

-02

F4Mean

14073E

-38

30139E+

0226151E+

0914

261E-16

10212E

-06

20499E+

08Std

15382E

-38

90551E+0

126783E+

0875

737E

-17

33853E-07

35388E+

07F5

Mean

17615E

-02

14345E

+00

59603E+

0237550E-02

29349E-02

12443E

+00

Std

42239E-03

13985E

-01

58412E+

0112

602E

-02

45825E-03

18471E-01

F6Mean

11417E

+01

58422E+

0242299E+

0817

988E

+02

16578E

+02

560

78E+

05Std

30258E+

0197

884E

+02

48581E+

0739022E+

01466

85E+

0165477E+

04F7

Mean

16881E-41

12984E

+01

64707E+

0611852E

-19

74831E-10

22865E+

05Std

34134E-41

22729E+

0033435E+

0531718E-20

95033E

-11

20650E+

04F8

Mean

45259E-08

39819E+

0085137E+

0117

042E

-04

29244

E-05

33956E+

01Std

29104E-08

41522E-01

12566E

+00

78665E

-05

27018E-06

47713E+

00F9

Mean

12222E

-22

36814E-01

72469E

+32

25070E-10

23795E-05

14112

E+27

Std

84369E-23

41963E-02

20633E+

3323874E-11

13879E

-06

44627E+

27F10

Mean

34254E-42

37261E-01

14940E

+05

20714E-21

18486E

-11

43041E+

03Std

966

08E-42

92561E-02

446

43E+

03464

07E-22

23956E-12

24348E+

02F11

Mean

1640

0E-12

315

040E

-52

37278E-02

65780E-24

3117

4E-14

10720E

-05

Std

51861E-123

16670E

-52

11165E

-02

72960E

-24

36245E-14

59475E-06

12 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

(b) F4

0 2000 4000 6000 8000 10000

0

5

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus15

minus5

(c) F6

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

minus50

(d) F7

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus2

minus4

minus6

minus8

minus10

minus12

minus14

(e) F8

0 2000 4000 6000 8000 10000

05

1015

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus5

minus10

minus20

minus25

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

Mea

n Er

rors

(log)

(h) F11


Complexity 13

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus10

minus5

0

5

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

0

10

20

30

Mea

n Er

rors

(log)

2000 4000 6000 8000 100000Iteration

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

50

Mea

n Er

rors

(log)

(h) F11


14 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

20

Mea

n Er

rors

(log)

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

0

2

4

6

8

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

01020304050

Mea

n Er

rors

(log)

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus140

minus120

minus100

minus80

minus60

minus40

minus20

020

Mea

n Er

rors

(log)

(h) F11


Complexity 15

Table6Multim

odalbenchm

arkfunctio

ns

Functio

nRa

nge

Fmin

F12(119909)=

minus20exp(minus0

2radic1 119899119899 sum 119894=11199092 119894)minus

exp(1 119899119899 sum 119894=1co

s(2120587119909 119894))

+20+exp

(1 )[minus32

32]0

F13(119909)=

119899 sum 119894=11003816 1003816 1003816 1003816119909 119894sin(119909 119894)

+01119909 1198941003816 1003816 1003816 1003816

[minus1010]

0

F14(119909)=

119891 119904(1199091119909 2)


(119909 119899119909 1)

[minus10010

0]0

119891 119904(119909119910)=

(1199092 +1199102 )025[sin2

(50(1199092 +

1199102 )01)+1

]F15(

119909)=119891 119904(1199091119909 2)


119909 1198991199091)

[minus10010

0]0

119891 119904(119909119910)=

05(sin2(radic 1199092+1199102

)minus05)

(1+0001

(1199092 +1199102 ))2

F16(119909)=

120587 11989910sin2

(120587119910 119894)+119899minus1 sum 119894=1

(119910 119894minus1 )2 [

1+10sin2

(120587119910 119894+1)]+

(119910 119899minus1 )2

+119899 sum 119894=1119906(119909 119894

10100

4)[minus50

50]0

119910 119894=1+1 4(119909

119894+1)

119906(119909 119894119886

119896119898)= 119896(119909 119894

minus119886)119898

119909 119894gt119886

0minus119886le

119909 119894le119886

119896(minus119909119894minus119886)119898

119909119894gt119886

F17(119909)=

1 4000119899 sum 119894=11199092 119894minus119899 prod 119894=1

cos(119909119894 radic 119894)+1

[minus10010

0]0

F18(119909)=


(minus(1199092 119894+

1199092 119894+1+05

119909 119894119909 119894+1)

8)lowastc

os(4radic

1199092 119894+1199092 119894+1

+05119909 119894119909 119894+1))

[minus55]

1-n

F19(119909)=

119899 sum 119894=1(119909119894minus1)2

minus119899 sum 119894=2119909 119894119909 119894minus1

[minusn2n2 ]

119899(119899+4)(119899

minus1)minus6

F20 (119909 )=

sum119899minus1 119894=2(05

+(sin2(radic 1

001199092 119894+1199092 119894+1)minus0

5))(1+

0001(1199092 119894minus

2119909 119894119909119894minus1+1199092 119894minus1))2

[minus10010

0]0

F21(119909)=

119899 sum 119894=1[1199092 119894minus10

cos(2120587

119909 119894)+10]

[minus51251

2]0

F22(119909)=

119899 sum 119894=1[1199102 119894minus10

cos(2120587

119910 119894)+10]

119910 119894= 119909 119894

1003816 1003816 1003816 1003816119909 1198941003816 1003816 1003816 1003816lt05

119903119900119906119899119889(2119909

119894)2

1003816 1003816 1003816 10038161199091198941003816 1003816 1003816 1003816lt0

5[minus51

2512]

0

F23(119909)=

1minuscos(2120587

radic119899 sum 119894=11199092 119894)

+01radic119899 sum 119894=1

1199092 119894[minus10

0100]

0

F24(119909)=

119899 sum 119894=1119896119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 (119909119894+05

))]minus119899119896

119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 05

)][minus05

05]0

F25(119909)=

119899 sum 119896=1119899 sum 119895=1((1199102 119895119896 4000)minus

cos(119910 119895119896)+1

)119910119895119896=10

0(119909 119896minus1199092 119895

)2 +(1minus

1199092 119895)2[minus10

0100]

0

16 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmultim

odalbenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

51692E-14

21708E-07

16343E

+01

42641E-12

43970E-07

70983E

+00

Std

94813E-15

11785E

-07

45830E-01

51275E-13

53024E-08

45755E+

00F13

Mean

19651E-15

17670E

-07

30865E+

0138781E-12

12507E

-06

29660

E+01

Std

17016E

-1510

899E

-07

28749E+

00200

14E-12

19125E

-06

57790E+

00F14

Mean

28586E-11

47414E-02

21576E+

0235954E-05

17290E

-02

18705E

+02

Std

17874E

-1118

105E

-02

50836E+

0019

343E

-06

10857E

-03

50868E+

01F15

Mean

99552E

-01

46150E-01

12596E

+01

94983E

-01

10032E

+00

12147E

+01

Std

38926E-01

33522E-01

21495E-01

42966

E-01

35690E-01

17388E

-01

F16

Mean

15705E

-32

13069E

-15

56725E+

0650290E-25

99726E

-15

31482E+

00Std

28850E-48

57169E-16

17168E

+06

47027E-25

85374E-15

58054E-01

F17

Mean

13781E-02

10332E

-02

43352E+

0044332E-03

12793E

-02

10971E+0

0Std

14865E

-02

12632E

-02

42518E-01

79408E

-03

10155E

-02

10766E

-02

F18

Mean

50849E+

0038253E+

0020946

E+01

49225E+

00490

48E+

0021497E+

01Std

16014E

+00

14627E

+00

76856E

-01

21737E+

00204

11E+0

013

669E

+00

F19

Mean

268

41E-07

19292E

+02

49808E+

0519

677E

+02

240

98E+

0230226E+

04Std

32619E-08

15971E+0

214

706E

+05

16572E

+02

23149E+

0260289E+

03F2

0Mean

25989E-07

47006

E-06

33592E-02

44469E-08

18865E

-07

1540

6E-01

Std

59383E-07

73387E

-06

22456E-02

10350E

-07

31612E-07

56719E-02

F21

Mean

000

00E+

0070

841E-13

25769E+

02000

00E+

0045409E-11

30881E+

02Std

000

00E+

0045361E-13

90973E

+00

000

00E+

0019

882E

-11

27305E+

01F2

2Mean

000

00E+

007746

7E-13

23335E+

02000

00E+

00644

03E-11

25509E+

02Std

000

00E+

0036979E-13

15942E

+01

000

00E+

0033820E-11

26992E+

01F2

3Mean

93987E

-01

52987E-01

12199E

+01

13599E

+00

14399E

+00

21878E+

00Std

21705E-01

12517E

-01

49304

E-01

36576E-01

21705E-01

62731E-02

F24

Mean

14921E-14

37233E-04

32412E+

0147458E-09

42553E-03

26924E+

01Std

17226E

-1498

846E

-05

11649E

+00

28242E-09

42975E-04

35559E+

00F2

5Mean

29494E+

0110

724E

+02

11372E

+07

404

62E+

0193530E+

0092

421E+0

3Std

29743E+

0151800

E+01

31606

E+06

39685E+

0190392E+

0018

838E

+03

Complexity 17


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

85798E-14

24174E-04

18459E

+01

7404

4E-12

73673E

-07

82226E+

00Std

17360E

-1455274E-05

1944

7E-01

88139E-13

80222E-08

42517E+

00F13

Mean

22538E-15

29492E-04

71594E

+01

21041E-11

32004

E-06

60959E+

01Std

18688E

-1510

372E

-04

45394E+

0015

865E

-11

15334E

-06

44766

E+00

F14

Mean

71759E

-1120261E+

0043430E+

0277682E

-05

33324E-02

42669E+

02Std

24650E-11

50770E-01

14055E

+01

54975E-06

10537E

-03

80127E+

01F15

Mean

16716E

+00

12749E

+00

22241E+

0116

927E

+00

14937E

+00

21617E+

01Std

76572E

-01

43985E-01

33014E-01

47677E-01

63574E-01

54534E-01

F16

Mean

94233E-33

13057E

-09

76995E

+07

17755E

-24

846

48E-14

69921E+

00Std

14425E

-48

37533E-10

21712E+

0719

092E

-24

17429E

-13

89129E-01

F17

Mean

76377E

-03

14219E

-02

1160

6E+0

164039E-03

10080E

-02

1264

1E+0

0Std

57418E-03

21089E-02

46282E-01

70807E

-03

13952E

-02

16555E

-02

F18

Mean

83103E+

0079

047E

+00

39689E+

0189467E+

0096

041E+0

038726E+

01Std

260

72E+

0025432E+

0077616E

-01

78506E

-01

21029E+

0013

015E

+00

F19

Mean

45562E+

0126833E+

04806

68E+

0616

118E+

0413

155E

+04

70015E

+05

Std

38094E+

0121743E+

0421709E+

0612

498E

+04

1300

9E+0

497

174E

+04

F20

Mean

43064E-08

25702E-04

11519E

-01

52365E-08

16998E

-06

500

47E-01

Std

44294E-08

27576E-04

39417E-02

95247E

-08

49881E-06

26305E-01

F21

Mean

000

00E+

0011310E

-06

53146

E+02

000

00E+

0023711E

-10

58748E+

02Std

000

00E+

0033614E-07

32117E+

01000

00E+

0045437E-11

29507E+

01F2

2Mean

000

00E+

0016

167E

-06

48729E+

02000

00E+

00244

07E-10

52060

E+02

Std

000

00E+

0063216E-07

24382E+

01000

00E+

0075

889E

-11

42230E+

01F2

3Mean

13699E

+00

89987E-01

21237E+

0122699E+

0025899E+

0035955E+

00Std

23594E-01

666

67E-02

58033E-01

41913E-01

62973E-01

12247E

-01

F24

Mean

71054E

-1426826E-02

63090E+

0119

033E

-08

96037E

-03

47263E+

01Std

27621E-14

47780E-03

22392E+

0061075E-09

97071E-04

52689E+

00F2

5Mean

66563E+

0184722E+

0211275E

+08

65780E+

0139992E+

0188242E+

04Std

10992E

+02

2113

8E+0

221091E+

0794

954E

+01

43819E+

0116

832E

+04

18 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

18989E

-1316

584E

-01

19996E

+01

17809E

-11

14744E

-06

13554E

+01

Std

20566

E-14

53720E-02

90319E

-02

19159E

-12

18930E

-07

60821E+

00F13

Mean

22871E-15

17736E

-01

1944

7E+0

213

452E

-10

13291E-05

16379E

+02

Std

26741E-15

53611E-02

62653E+

0038592E-11

55001E-06

13313E

+01

F14

Mean

18736E

-1074

259E

+01

10132E

+03

22866

E-04

83534E-02

95534E

+02

Std

37223E-11

19144E

+01

18986E

+01

14283E

-05

10592E

-02

53523E+

01F15

Mean

26814E+

0010

178E

+01

47083E+

0128083E+

0034325E+

0045859E+

01Std

73851E-01

16238E

+00

22513E-01

46148E-01

60283E-01

69914E-01

F16

Mean

47116E-33

244

54E-04

90382E

+08

81890E-24

62347E-14

27647E+

03Std

72124E

-49

59650E-05

64985E+

0767958E-24

55604

E-14

44231E+

03F17

Mean

34494E-03

11896E

-02

37816E+

0134509E-03

41885E-03

21280E+

00Std

60565E-03

65363E-03

15922E

+00

46765E-03

86153E-03

54359E-02

F18

Mean

18033E

+01

17806E

+01

86826E+

0118

319E

+01

18828E

+01

82458E+

01Std

19652E

+00

38319E+

0093

222E

-01

29296E+

0025377E+

0015

159E

+00

F19

Mean

82462E+

0427944

E+06

48046

E+08

28415E+

0560265E+

0549201E+

07Std

55732E+

0489703E+

0596

715E

+07

24572E+

0527137E+

0572

772E

+06

F20

Mean

57130E-07

81688E-03

96848E

-01

13631E-06

27143E-05

21656E+

00Std

61122E-07

53195E-03

44542E-01

25155E-06

58766

E-05

80368E-01

F21

Mean

000

00E+

0051414E+

0013

305E

+03

000

00E+

0020026E-09

13623E

+03

Std

000

00E+

0017

825E

+00

22890E+

01000

00E+

0032815E-10

609

96E+

01F2

2Mean

000

00E+

0077

848E

+00

1260

9E+0

3000

00E+

0020383E-09

12745E

+03

Std

000

00E+

0023732E+

0029100

E+01

000

00E+

0029753E-10

59708E+

01F2

3Mean

25599E+

0020499E+

0039804

E+01

47099E+

0043699E+

0073

691E+0

0Std

36878E-01

15092E

-01

69296E-01

59151E-01

56184E-01

17989E

-01

F24

Mean

40927E-13

26229E+

0015

145E

+02

18874E

-07

29476E-02

10478E

+02

Std

88061E-14

63367E-01

42830E+

0037074E-08

17697E

-03

11873E

+01

F25

Mean

42987E+

028117

8E+0

313

524E

+09

56790E+

0244982E+

0218

038E

+06

Std

43423E+

0233128E+

0278

399E

+07

54327E+

0246926E+

0221315E+

05

Complexity 19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus8

minus6

minus4

minus2

02468

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(f) F24




MAE = sum119873119895=1 10038161003816100381610038161003816119898119895 minus 11989611989510038161003816100381610038161003816119873 (22)

20 Complexity

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

ISSASSAPSO

CMFOAIFFOFOA

1

2

3

4

5

6

7

8

Mea

n Er

rors

(log)

0 4000 6000 8000 100002000Iteration

(e) F19

ISSASSAPSO

CMFOAIFFOFOA

20000 6000 8000 100004000Iteration

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




Complexity 21

Table11

Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

31365E+

0413

133E

+06

11295E

+08

10017E

+06

864

75E+

0513

449E

+07

Std

18602E

+04

52974E+

0531387E+

0748689E+

0548607E+

0528405E+

06F2

7Mean

304

00E-10

1844

6E+0

484500

E+09

10512E

+04

12359E

+04

58535E+

08Std

61535E-10

14049E

+04

10125E

+09

12485E

+04

11922E

+04

38771E+

07F2

8Mean

42105E-01

46710E+

0173

819E

+02

4114

7E+0

137814E+

0114

226E

+02

Std

12624E

+00

31490E+

0199

455E

+01

47336E+

0134110E+

0129201E+

01F2

9Mean

75177E

+03

14891E+0

529286E+

0647277E+

0531099E+

0539826E+

05Std

33119E+

0368316E+

049190

4E+0

521021E+

0522686E+

0515

511E+0

5F3

0Mean

31524E+

0231524E+

0238129E+

0231524E+

0231524E+

0232568E+

02Std

85708E-12

19710E

-07

14082E

+01

11524E

-1145680E-11

58955E+

00F31

Mean

23483E+

0223172E+

0230117E+

0223811E

+02

23858E+

0224179E+

02Std

41748E+

0072

461E+0

048903E+

00560

97E+

0050249E+

0090

228E

+00

F32

Mean

20790E+

02206

03E+

0221884E+

0221485E+

0220975E+

0220633E+

02Std

41618E+

0032456E+

0030353E+

0087909E+

0057719E+

0016

880E

+00

22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 11


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonun

imod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F1Mean

52002E-44

1360

1E-01

64559E+

0467630E-20

540

42E-10

23688E+

03Std

89565E-44

28398E-02

27675E+

0318

636E

-20

10843E

-10

11782E

+02

F2Mean

25837E+

0028507E+

0189058E+

0610

018E

+01

11850E

+01

13921E+0

4Std

40923E+

0012

482E

+01

64125E+

0598

260E

+00

67378E+

0021479E+

03F3

Mean

19984E

-1517

506E

-05

99937E

-01

19984E

-1534570E-12

1946

0E-01

Std

27940E-16

33607E-06

19874E

-04

39686E-16

57323E-13

11259E

-02

F4Mean

14073E

-38

30139E+

0226151E+

0914

261E-16

10212E

-06

20499E+

08Std

15382E

-38

90551E+0

126783E+

0875

737E

-17

33853E-07

35388E+

07F5

Mean

17615E

-02

14345E

+00

59603E+

0237550E-02

29349E-02

12443E

+00

Std

42239E-03

13985E

-01

58412E+

0112

602E

-02

45825E-03

18471E-01

F6Mean

11417E

+01

58422E+

0242299E+

0817

988E

+02

16578E

+02

560

78E+

05Std

30258E+

0197

884E

+02

48581E+

0739022E+

01466

85E+

0165477E+

04F7

Mean

16881E-41

12984E

+01

64707E+

0611852E

-19

74831E-10

22865E+

05Std

34134E-41

22729E+

0033435E+

0531718E-20

95033E

-11

20650E+

04F8

Mean

45259E-08

39819E+

0085137E+

0117

042E

-04

29244

E-05

33956E+

01Std

29104E-08

41522E-01

12566E

+00

78665E

-05

27018E-06

47713E+

00F9

Mean

12222E

-22

36814E-01

72469E

+32

25070E-10

23795E-05

14112

E+27

Std

84369E-23

41963E-02

20633E+

3323874E-11

13879E

-06

44627E+

27F10

Mean

34254E-42

37261E-01

14940E

+05

20714E-21

18486E

-11

43041E+

03Std

966

08E-42

92561E-02

446

43E+

03464

07E-22

23956E-12

24348E+

02F11

Mean

1640

0E-12

315

040E

-52

37278E-02

65780E-24

3117

4E-14

10720E

-05

Std

51861E-123

16670E

-52

11165E

-02

72960E

-24

36245E-14

59475E-06

12 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

(b) F4

0 2000 4000 6000 8000 10000

0

5

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus15

minus5

(c) F6

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

minus50

(d) F7

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus2

minus4

minus6

minus8

minus10

minus12

minus14

(e) F8

0 2000 4000 6000 8000 10000

05

1015

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus5

minus10

minus20

minus25

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

Mea

n Er

rors

(log)

(h) F11


Complexity 13

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus10

minus5

0

5

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

0

10

20

30

Mea

n Er

rors

(log)

2000 4000 6000 8000 100000Iteration

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

50

Mea

n Er

rors

(log)

(h) F11


14 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

20

Mea

n Er

rors

(log)

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

0

2

4

6

8

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

01020304050

Mea

n Er

rors

(log)

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus140

minus120

minus100

minus80

minus60

minus40

minus20

020

Mea

n Er

rors

(log)

(h) F11


Complexity 15

Table6Multim

odalbenchm

arkfunctio

ns

Functio

nRa

nge

Fmin

F12(119909)=

minus20exp(minus0

2radic1 119899119899 sum 119894=11199092 119894)minus

exp(1 119899119899 sum 119894=1co

s(2120587119909 119894))

+20+exp

(1 )[minus32

32]0

F13(119909)=

119899 sum 119894=11003816 1003816 1003816 1003816119909 119894sin(119909 119894)

+01119909 1198941003816 1003816 1003816 1003816

[minus1010]

0

F14(119909)=

119891 119904(1199091119909 2)


(119909 119899119909 1)

[minus10010

0]0

119891 119904(119909119910)=

(1199092 +1199102 )025[sin2

(50(1199092 +

1199102 )01)+1

]F15(

119909)=119891 119904(1199091119909 2)


119909 1198991199091)

[minus10010

0]0

119891 119904(119909119910)=

05(sin2(radic 1199092+1199102

)minus05)

(1+0001

(1199092 +1199102 ))2

F16(119909)=

120587 11989910sin2

(120587119910 119894)+119899minus1 sum 119894=1

(119910 119894minus1 )2 [

1+10sin2

(120587119910 119894+1)]+

(119910 119899minus1 )2

+119899 sum 119894=1119906(119909 119894

10100

4)[minus50

50]0

119910 119894=1+1 4(119909

119894+1)

119906(119909 119894119886

119896119898)= 119896(119909 119894

minus119886)119898

119909 119894gt119886

0minus119886le

119909 119894le119886

119896(minus119909119894minus119886)119898

119909119894gt119886

F17(119909)=

1 4000119899 sum 119894=11199092 119894minus119899 prod 119894=1

cos(119909119894 radic 119894)+1

[minus10010

0]0

F18(119909)=


(minus(1199092 119894+

1199092 119894+1+05

119909 119894119909 119894+1)

8)lowastc

os(4radic

1199092 119894+1199092 119894+1

+05119909 119894119909 119894+1))

[minus55]

1-n

F19(119909)=

119899 sum 119894=1(119909119894minus1)2

minus119899 sum 119894=2119909 119894119909 119894minus1

[minusn2n2 ]

119899(119899+4)(119899

minus1)minus6

F20 (119909 )=

sum119899minus1 119894=2(05

+(sin2(radic 1

001199092 119894+1199092 119894+1)minus0

5))(1+

0001(1199092 119894minus

2119909 119894119909119894minus1+1199092 119894minus1))2

[minus10010

0]0

F21(119909)=

119899 sum 119894=1[1199092 119894minus10

cos(2120587

119909 119894)+10]

[minus51251

2]0

F22(119909)=

119899 sum 119894=1[1199102 119894minus10

cos(2120587

119910 119894)+10]

119910 119894= 119909 119894

1003816 1003816 1003816 1003816119909 1198941003816 1003816 1003816 1003816lt05

119903119900119906119899119889(2119909

119894)2

1003816 1003816 1003816 10038161199091198941003816 1003816 1003816 1003816lt0

5[minus51

2512]

0

F23(119909)=

1minuscos(2120587

radic119899 sum 119894=11199092 119894)

+01radic119899 sum 119894=1

1199092 119894[minus10

0100]

0

F24(119909)=

119899 sum 119894=1119896119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 (119909119894+05

))]minus119899119896

119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 05

)][minus05

05]0

F25(119909)=

119899 sum 119896=1119899 sum 119895=1((1199102 119895119896 4000)minus

cos(119910 119895119896)+1

)119910119895119896=10

0(119909 119896minus1199092 119895

)2 +(1minus

1199092 119895)2[minus10

0100]

0

16 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmultim

odalbenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

51692E-14

21708E-07

16343E

+01

42641E-12

43970E-07

70983E

+00

Std

94813E-15

11785E

-07

45830E-01

51275E-13

53024E-08

45755E+

00F13

Mean

19651E-15

17670E

-07

30865E+

0138781E-12

12507E

-06

29660

E+01

Std

17016E

-1510

899E

-07

28749E+

00200

14E-12

19125E

-06

57790E+

00F14

Mean

28586E-11

47414E-02

21576E+

0235954E-05

17290E

-02

18705E

+02

Std

17874E

-1118

105E

-02

50836E+

0019

343E

-06

10857E

-03

50868E+

01F15

Mean

99552E

-01

46150E-01

12596E

+01

94983E

-01

10032E

+00

12147E

+01

Std

38926E-01

33522E-01

21495E-01

42966

E-01

35690E-01

17388E

-01

F16

Mean

15705E

-32

13069E

-15

56725E+

0650290E-25

99726E

-15

31482E+

00Std

28850E-48

57169E-16

17168E

+06

47027E-25

85374E-15

58054E-01

F17

Mean

13781E-02

10332E

-02

43352E+

0044332E-03

12793E

-02

10971E+0

0Std

14865E

-02

12632E

-02

42518E-01

79408E

-03

10155E

-02

10766E

-02

F18

Mean

50849E+

0038253E+

0020946

E+01

49225E+

00490

48E+

0021497E+

01Std

16014E

+00

14627E

+00

76856E

-01

21737E+

00204

11E+0

013

669E

+00

F19

Mean

268

41E-07

19292E

+02

49808E+

0519

677E

+02

240

98E+

0230226E+

04Std

32619E-08

15971E+0

214

706E

+05

16572E

+02

23149E+

0260289E+

03F2

0Mean

25989E-07

47006

E-06

33592E-02

44469E-08

18865E

-07

1540

6E-01

Std

59383E-07

73387E

-06

22456E-02

10350E

-07

31612E-07

56719E-02

F21

Mean

000

00E+

0070

841E-13

25769E+

02000

00E+

0045409E-11

30881E+

02Std

000

00E+

0045361E-13

90973E

+00

000

00E+

0019

882E

-11

27305E+

01F2

2Mean

000

00E+

007746

7E-13

23335E+

02000

00E+

00644

03E-11

25509E+

02Std

000

00E+

0036979E-13

15942E

+01

000

00E+

0033820E-11

26992E+

01F2

3Mean

93987E

-01

52987E-01

12199E

+01

13599E

+00

14399E

+00

21878E+

00Std

21705E-01

12517E

-01

49304

E-01

36576E-01

21705E-01

62731E-02

F24

Mean

14921E-14

37233E-04

32412E+

0147458E-09

42553E-03

26924E+

01Std

17226E

-1498

846E

-05

11649E

+00

28242E-09

42975E-04

35559E+

00F2

5Mean

29494E+

0110

724E

+02

11372E

+07

404

62E+

0193530E+

0092

421E+0

3Std

29743E+

0151800

E+01

31606

E+06

39685E+

0190392E+

0018

838E

+03

Complexity 17


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

85798E-14

24174E-04

18459E

+01

7404

4E-12

73673E

-07

82226E+

00Std

17360E

-1455274E-05

1944

7E-01

88139E-13

80222E-08

42517E+

00F13

Mean

22538E-15

29492E-04

71594E

+01

21041E-11

32004

E-06

60959E+

01Std

18688E

-1510

372E

-04

45394E+

0015

865E

-11

15334E

-06

44766

E+00

F14

Mean

71759E

-1120261E+

0043430E+

0277682E

-05

33324E-02

42669E+

02Std

24650E-11

50770E-01

14055E

+01

54975E-06

10537E

-03

80127E+

01F15

Mean

16716E

+00

12749E

+00

22241E+

0116

927E

+00

14937E

+00

21617E+

01Std

76572E

-01

43985E-01

33014E-01

47677E-01

63574E-01

54534E-01

F16

Mean

94233E-33

13057E

-09

76995E

+07

17755E

-24

846

48E-14

69921E+

00Std

14425E

-48

37533E-10

21712E+

0719

092E

-24

17429E

-13

89129E-01

F17

Mean

76377E

-03

14219E

-02

1160

6E+0

164039E-03

10080E

-02

1264

1E+0

0Std

57418E-03

21089E-02

46282E-01

70807E

-03

13952E

-02

16555E

-02

F18

Mean

83103E+

0079

047E

+00

39689E+

0189467E+

0096

041E+0

038726E+

01Std

260

72E+

0025432E+

0077616E

-01

78506E

-01

21029E+

0013

015E

+00

F19

Mean

45562E+

0126833E+

04806

68E+

0616

118E+

0413

155E

+04

70015E

+05

Std

38094E+

0121743E+

0421709E+

0612

498E

+04

1300

9E+0

497

174E

+04

F20

Mean

43064E-08

25702E-04

11519E

-01

52365E-08

16998E

-06

500

47E-01

Std

44294E-08

27576E-04

39417E-02

95247E

-08

49881E-06

26305E-01

F21

Mean

000

00E+

0011310E

-06

53146

E+02

000

00E+

0023711E

-10

58748E+

02Std

000

00E+

0033614E-07

32117E+

01000

00E+

0045437E-11

29507E+

01F2

2Mean

000

00E+

0016

167E

-06

48729E+

02000

00E+

00244

07E-10

52060

E+02

Std

000

00E+

0063216E-07

24382E+

01000

00E+

0075

889E

-11

42230E+

01F2

3Mean

13699E

+00

89987E-01

21237E+

0122699E+

0025899E+

0035955E+

00Std

23594E-01

666

67E-02

58033E-01

41913E-01

62973E-01

12247E

-01

F24

Mean

71054E

-1426826E-02

63090E+

0119

033E

-08

96037E

-03

47263E+

01Std

27621E-14

47780E-03

22392E+

0061075E-09

97071E-04

52689E+

00F2

5Mean

66563E+

0184722E+

0211275E

+08

65780E+

0139992E+

0188242E+

04Std

10992E

+02

2113

8E+0

221091E+

0794

954E

+01

43819E+

0116

832E

+04

18 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

18989E

-1316

584E

-01

19996E

+01

17809E

-11

14744E

-06

13554E

+01

Std

20566

E-14

53720E-02

90319E

-02

19159E

-12

18930E

-07

60821E+

00F13

Mean

22871E-15

17736E

-01

1944

7E+0

213

452E

-10

13291E-05

16379E

+02

Std

26741E-15

53611E-02

62653E+

0038592E-11

55001E-06

13313E

+01

F14

Mean

18736E

-1074

259E

+01

10132E

+03

22866

E-04

83534E-02

95534E

+02

Std

37223E-11

19144E

+01

18986E

+01

14283E

-05

10592E

-02

53523E+

01F15

Mean

26814E+

0010

178E

+01

47083E+

0128083E+

0034325E+

0045859E+

01Std

73851E-01

16238E

+00

22513E-01

46148E-01

60283E-01

69914E-01

F16

Mean

47116E-33

244

54E-04

90382E

+08

81890E-24

62347E-14

27647E+

03Std

72124E

-49

59650E-05

64985E+

0767958E-24

55604

E-14

44231E+

03F17

Mean

34494E-03

11896E

-02

37816E+

0134509E-03

41885E-03

21280E+

00Std

60565E-03

65363E-03

15922E

+00

46765E-03

86153E-03

54359E-02

F18

Mean

18033E

+01

17806E

+01

86826E+

0118

319E

+01

18828E

+01

82458E+

01Std

19652E

+00

38319E+

0093

222E

-01

29296E+

0025377E+

0015

159E

+00

F19

Mean

82462E+

0427944

E+06

48046

E+08

28415E+

0560265E+

0549201E+

07Std

55732E+

0489703E+

0596

715E

+07

24572E+

0527137E+

0572

772E

+06

F20

Mean

57130E-07

81688E-03

96848E

-01

13631E-06

27143E-05

21656E+

00Std

61122E-07

53195E-03

44542E-01

25155E-06

58766

E-05

80368E-01

F21

Mean

000

00E+

0051414E+

0013

305E

+03

000

00E+

0020026E-09

13623E

+03

Std

000

00E+

0017

825E

+00

22890E+

01000

00E+

0032815E-10

609

96E+

01F2

2Mean

000

00E+

0077

848E

+00

1260

9E+0

3000

00E+

0020383E-09

12745E

+03

Std

000

00E+

0023732E+

0029100

E+01

000

00E+

0029753E-10

59708E+

01F2

3Mean

25599E+

0020499E+

0039804

E+01

47099E+

0043699E+

0073

691E+0

0Std

36878E-01

15092E

-01

69296E-01

59151E-01

56184E-01

17989E

-01

F24

Mean

40927E-13

26229E+

0015

145E

+02

18874E

-07

29476E-02

10478E

+02

Std

88061E-14

63367E-01

42830E+

0037074E-08

17697E

-03

11873E

+01

F25

Mean

42987E+

028117

8E+0

313

524E

+09

56790E+

0244982E+

0218

038E

+06

Std

43423E+

0233128E+

0278

399E

+07

54327E+

0246926E+

0221315E+

05

Complexity 19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus8

minus6

minus4

minus2

02468

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(f) F24




MAE = sum119873119895=1 10038161003816100381610038161003816119898119895 minus 11989611989510038161003816100381610038161003816119873 (22)

20 Complexity

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

ISSASSAPSO

CMFOAIFFOFOA

1

2

3

4

5

6

7

8

Mea

n Er

rors

(log)

0 4000 6000 8000 100002000Iteration

(e) F19

ISSASSAPSO

CMFOAIFFOFOA

20000 6000 8000 100004000Iteration

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




Complexity 21

Table11

Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

31365E+

0413

133E

+06

11295E

+08

10017E

+06

864

75E+

0513

449E

+07

Std

18602E

+04

52974E+

0531387E+

0748689E+

0548607E+

0528405E+

06F2

7Mean

304

00E-10

1844

6E+0

484500

E+09

10512E

+04

12359E

+04

58535E+

08Std

61535E-10

14049E

+04

10125E

+09

12485E

+04

11922E

+04

38771E+

07F2

8Mean

42105E-01

46710E+

0173

819E

+02

4114

7E+0

137814E+

0114

226E

+02

Std

12624E

+00

31490E+

0199

455E

+01

47336E+

0134110E+

0129201E+

01F2

9Mean

75177E

+03

14891E+0

529286E+

0647277E+

0531099E+

0539826E+

05Std

33119E+

0368316E+

049190

4E+0

521021E+

0522686E+

0515

511E+0

5F3

0Mean

31524E+

0231524E+

0238129E+

0231524E+

0231524E+

0232568E+

02Std

85708E-12

19710E

-07

14082E

+01

11524E

-1145680E-11

58955E+

00F31

Mean

23483E+

0223172E+

0230117E+

0223811E

+02

23858E+

0224179E+

02Std

41748E+

0072

461E+0

048903E+

00560

97E+

0050249E+

0090

228E

+00

F32

Mean

20790E+

02206

03E+

0221884E+

0221485E+

0220975E+

0220633E+

02Std

41618E+

0032456E+

0030353E+

0087909E+

0057719E+

0016

880E

+00

22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








12 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

(b) F4

0 2000 4000 6000 8000 10000

0

5

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus15

minus5

(c) F6

0 2000 4000 6000 8000 10000

0

10

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus10

minus20

minus30

minus40

minus50

(d) F7

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus2

minus4

minus6

minus8

minus10

minus12

minus14

(e) F8

0 2000 4000 6000 8000 10000

05

1015

Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus5

minus10

minus20

minus25

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

Mea

n Er

rors

(log)

(h) F11


Complexity 13

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus10

minus5

0

5

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

0

10

20

30

Mea

n Er

rors

(log)

2000 4000 6000 8000 100000Iteration

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

50

Mea

n Er

rors

(log)

(h) F11


14 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

20

Mea

n Er

rors

(log)

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

0

2

4

6

8

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

01020304050

Mea

n Er

rors

(log)

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus140

minus120

minus100

minus80

minus60

minus40

minus20

020

Mea

n Er

rors

(log)

(h) F11


Complexity 15

Table6Multim

odalbenchm

arkfunctio

ns

Functio

nRa

nge

Fmin

F12(119909)=

minus20exp(minus0

2radic1 119899119899 sum 119894=11199092 119894)minus

exp(1 119899119899 sum 119894=1co

s(2120587119909 119894))

+20+exp

(1 )[minus32

32]0

F13(119909)=

119899 sum 119894=11003816 1003816 1003816 1003816119909 119894sin(119909 119894)

+01119909 1198941003816 1003816 1003816 1003816

[minus1010]

0

F14(119909)=

119891 119904(1199091119909 2)


(119909 119899119909 1)

[minus10010

0]0

119891 119904(119909119910)=

(1199092 +1199102 )025[sin2

(50(1199092 +

1199102 )01)+1

]F15(

119909)=119891 119904(1199091119909 2)


119909 1198991199091)

[minus10010

0]0

119891 119904(119909119910)=

05(sin2(radic 1199092+1199102

)minus05)

(1+0001

(1199092 +1199102 ))2

F16(119909)=

120587 11989910sin2

(120587119910 119894)+119899minus1 sum 119894=1

(119910 119894minus1 )2 [

1+10sin2

(120587119910 119894+1)]+

(119910 119899minus1 )2

+119899 sum 119894=1119906(119909 119894

10100

4)[minus50

50]0

119910 119894=1+1 4(119909

119894+1)

119906(119909 119894119886

119896119898)= 119896(119909 119894

minus119886)119898

119909 119894gt119886

0minus119886le

119909 119894le119886

119896(minus119909119894minus119886)119898

119909119894gt119886

F17(119909)=

1 4000119899 sum 119894=11199092 119894minus119899 prod 119894=1

cos(119909119894 radic 119894)+1

[minus10010

0]0

F18(119909)=


(minus(1199092 119894+

1199092 119894+1+05

119909 119894119909 119894+1)

8)lowastc

os(4radic

1199092 119894+1199092 119894+1

+05119909 119894119909 119894+1))

[minus55]

1-n

F19(119909)=

119899 sum 119894=1(119909119894minus1)2

minus119899 sum 119894=2119909 119894119909 119894minus1

[minusn2n2 ]

119899(119899+4)(119899

minus1)minus6

F20 (119909 )=

sum119899minus1 119894=2(05

+(sin2(radic 1

001199092 119894+1199092 119894+1)minus0

5))(1+

0001(1199092 119894minus

2119909 119894119909119894minus1+1199092 119894minus1))2

[minus10010

0]0

F21(119909)=

119899 sum 119894=1[1199092 119894minus10

cos(2120587

119909 119894)+10]

[minus51251

2]0

F22(119909)=

119899 sum 119894=1[1199102 119894minus10

cos(2120587

119910 119894)+10]

119910 119894= 119909 119894

1003816 1003816 1003816 1003816119909 1198941003816 1003816 1003816 1003816lt05

119903119900119906119899119889(2119909

119894)2

1003816 1003816 1003816 10038161199091198941003816 1003816 1003816 1003816lt0

5[minus51

2512]

0

F23(119909)=

1minuscos(2120587

radic119899 sum 119894=11199092 119894)

+01radic119899 sum 119894=1

1199092 119894[minus10

0100]

0

F24(119909)=

119899 sum 119894=1119896119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 (119909119894+05

))]minus119899119896

119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 05

)][minus05

05]0

F25(119909)=

119899 sum 119896=1119899 sum 119895=1((1199102 119895119896 4000)minus

cos(119910 119895119896)+1

)119910119895119896=10

0(119909 119896minus1199092 119895

)2 +(1minus

1199092 119895)2[minus10

0100]

0

16 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmultim

odalbenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

51692E-14

21708E-07

16343E

+01

42641E-12

43970E-07

70983E

+00

Std

94813E-15

11785E

-07

45830E-01

51275E-13

53024E-08

45755E+

00F13

Mean

19651E-15

17670E

-07

30865E+

0138781E-12

12507E

-06

29660

E+01

Std

17016E

-1510

899E

-07

28749E+

00200

14E-12

19125E

-06

57790E+

00F14

Mean

28586E-11

47414E-02

21576E+

0235954E-05

17290E

-02

18705E

+02

Std

17874E

-1118

105E

-02

50836E+

0019

343E

-06

10857E

-03

50868E+

01F15

Mean

99552E

-01

46150E-01

12596E

+01

94983E

-01

10032E

+00

12147E

+01

Std

38926E-01

33522E-01

21495E-01

42966

E-01

35690E-01

17388E

-01

F16

Mean

15705E

-32

13069E

-15

56725E+

0650290E-25

99726E

-15

31482E+

00Std

28850E-48

57169E-16

17168E

+06

47027E-25

85374E-15

58054E-01

F17

Mean

13781E-02

10332E

-02

43352E+

0044332E-03

12793E

-02

10971E+0

0Std

14865E

-02

12632E

-02

42518E-01

79408E

-03

10155E

-02

10766E

-02

F18

Mean

50849E+

0038253E+

0020946

E+01

49225E+

00490

48E+

0021497E+

01Std

16014E

+00

14627E

+00

76856E

-01

21737E+

00204

11E+0

013

669E

+00

F19

Mean

268

41E-07

19292E

+02

49808E+

0519

677E

+02

240

98E+

0230226E+

04Std

32619E-08

15971E+0

214

706E

+05

16572E

+02

23149E+

0260289E+

03F2

0Mean

25989E-07

47006

E-06

33592E-02

44469E-08

18865E

-07

1540

6E-01

Std

59383E-07

73387E

-06

22456E-02

10350E

-07

31612E-07

56719E-02

F21

Mean

000

00E+

0070

841E-13

25769E+

02000

00E+

0045409E-11

30881E+

02Std

000

00E+

0045361E-13

90973E

+00

000

00E+

0019

882E

-11

27305E+

01F2

2Mean

000

00E+

007746

7E-13

23335E+

02000

00E+

00644

03E-11

25509E+

02Std

000

00E+

0036979E-13

15942E

+01

000

00E+

0033820E-11

26992E+

01F2

3Mean

93987E

-01

52987E-01

12199E

+01

13599E

+00

14399E

+00

21878E+

00Std

21705E-01

12517E

-01

49304

E-01

36576E-01

21705E-01

62731E-02

F24

Mean

14921E-14

37233E-04

32412E+

0147458E-09

42553E-03

26924E+

01Std

17226E

-1498

846E

-05

11649E

+00

28242E-09

42975E-04

35559E+

00F2

5Mean

29494E+

0110

724E

+02

11372E

+07

404

62E+

0193530E+

0092

421E+0

3Std

29743E+

0151800

E+01

31606

E+06

39685E+

0190392E+

0018

838E

+03

Complexity 17


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

85798E-14

24174E-04

18459E

+01

7404

4E-12

73673E

-07

82226E+

00Std

17360E

-1455274E-05

1944

7E-01

88139E-13

80222E-08

42517E+

00F13

Mean

22538E-15

29492E-04

71594E

+01

21041E-11

32004

E-06

60959E+

01Std

18688E

-1510

372E

-04

45394E+

0015

865E

-11

15334E

-06

44766

E+00

F14

Mean

71759E

-1120261E+

0043430E+

0277682E

-05

33324E-02

42669E+

02Std

24650E-11

50770E-01

14055E

+01

54975E-06

10537E

-03

80127E+

01F15

Mean

16716E

+00

12749E

+00

22241E+

0116

927E

+00

14937E

+00

21617E+

01Std

76572E

-01

43985E-01

33014E-01

47677E-01

63574E-01

54534E-01

F16

Mean

94233E-33

13057E

-09

76995E

+07

17755E

-24

846

48E-14

69921E+

00Std

14425E

-48

37533E-10

21712E+

0719

092E

-24

17429E

-13

89129E-01

F17

Mean

76377E

-03

14219E

-02

1160

6E+0

164039E-03

10080E

-02

1264

1E+0

0Std

57418E-03

21089E-02

46282E-01

70807E

-03

13952E

-02

16555E

-02

F18

Mean

83103E+

0079

047E

+00

39689E+

0189467E+

0096

041E+0

038726E+

01Std

260

72E+

0025432E+

0077616E

-01

78506E

-01

21029E+

0013

015E

+00

F19

Mean

45562E+

0126833E+

04806

68E+

0616

118E+

0413

155E

+04

70015E

+05

Std

38094E+

0121743E+

0421709E+

0612

498E

+04

1300

9E+0

497

174E

+04

F20

Mean

43064E-08

25702E-04

11519E

-01

52365E-08

16998E

-06

500

47E-01

Std

44294E-08

27576E-04

39417E-02

95247E

-08

49881E-06

26305E-01

F21

Mean

000

00E+

0011310E

-06

53146

E+02

000

00E+

0023711E

-10

58748E+

02Std

000

00E+

0033614E-07

32117E+

01000

00E+

0045437E-11

29507E+

01F2

2Mean

000

00E+

0016

167E

-06

48729E+

02000

00E+

00244

07E-10

52060

E+02

Std

000

00E+

0063216E-07

24382E+

01000

00E+

0075

889E

-11

42230E+

01F2

3Mean

13699E

+00

89987E-01

21237E+

0122699E+

0025899E+

0035955E+

00Std

23594E-01

666

67E-02

58033E-01

41913E-01

62973E-01

12247E

-01

F24

Mean

71054E

-1426826E-02

63090E+

0119

033E

-08

96037E

-03

47263E+

01Std

27621E-14

47780E-03

22392E+

0061075E-09

97071E-04

52689E+

00F2

5Mean

66563E+

0184722E+

0211275E

+08

65780E+

0139992E+

0188242E+

04Std

10992E

+02

2113

8E+0

221091E+

0794

954E

+01

43819E+

0116

832E

+04

18 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

18989E

-1316

584E

-01

19996E

+01

17809E

-11

14744E

-06

13554E

+01

Std

20566

E-14

53720E-02

90319E

-02

19159E

-12

18930E

-07

60821E+

00F13

Mean

22871E-15

17736E

-01

1944

7E+0

213

452E

-10

13291E-05

16379E

+02

Std

26741E-15

53611E-02

62653E+

0038592E-11

55001E-06

13313E

+01

F14

Mean

18736E

-1074

259E

+01

10132E

+03

22866

E-04

83534E-02

95534E

+02

Std

37223E-11

19144E

+01

18986E

+01

14283E

-05

10592E

-02

53523E+

01F15

Mean

26814E+

0010

178E

+01

47083E+

0128083E+

0034325E+

0045859E+

01Std

73851E-01

16238E

+00

22513E-01

46148E-01

60283E-01

69914E-01

F16

Mean

47116E-33

244

54E-04

90382E

+08

81890E-24

62347E-14

27647E+

03Std

72124E

-49

59650E-05

64985E+

0767958E-24

55604

E-14

44231E+

03F17

Mean

34494E-03

11896E

-02

37816E+

0134509E-03

41885E-03

21280E+

00Std

60565E-03

65363E-03

15922E

+00

46765E-03

86153E-03

54359E-02

F18

Mean

18033E

+01

17806E

+01

86826E+

0118

319E

+01

18828E

+01

82458E+

01Std

19652E

+00

38319E+

0093

222E

-01

29296E+

0025377E+

0015

159E

+00

F19

Mean

82462E+

0427944

E+06

48046

E+08

28415E+

0560265E+

0549201E+

07Std

55732E+

0489703E+

0596

715E

+07

24572E+

0527137E+

0572

772E

+06

F20

Mean

57130E-07

81688E-03

96848E

-01

13631E-06

27143E-05

21656E+

00Std

61122E-07

53195E-03

44542E-01

25155E-06

58766

E-05

80368E-01

F21

Mean

000

00E+

0051414E+

0013

305E

+03

000

00E+

0020026E-09

13623E

+03

Std

000

00E+

0017

825E

+00

22890E+

01000

00E+

0032815E-10

609

96E+

01F2

2Mean

000

00E+

0077

848E

+00

1260

9E+0

3000

00E+

0020383E-09

12745E

+03

Std

000

00E+

0023732E+

0029100

E+01

000

00E+

0029753E-10

59708E+

01F2

3Mean

25599E+

0020499E+

0039804

E+01

47099E+

0043699E+

0073

691E+0

0Std

36878E-01

15092E

-01

69296E-01

59151E-01

56184E-01

17989E

-01

F24

Mean

40927E-13

26229E+

0015

145E

+02

18874E

-07

29476E-02

10478E

+02

Std

88061E-14

63367E-01

42830E+

0037074E-08

17697E

-03

11873E

+01

F25

Mean

42987E+

028117

8E+0

313

524E

+09

56790E+

0244982E+

0218

038E

+06

Std

43423E+

0233128E+

0278

399E

+07

54327E+

0246926E+

0221315E+

05

Complexity 19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus8

minus6

minus4

minus2

02468

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(f) F24




MAE = sum119873119895=1 10038161003816100381610038161003816119898119895 minus 11989611989510038161003816100381610038161003816119873 (22)

20 Complexity

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

ISSASSAPSO

CMFOAIFFOFOA

1

2

3

4

5

6

7

8

Mea

n Er

rors

(log)

0 4000 6000 8000 100002000Iteration

(e) F19

ISSASSAPSO

CMFOAIFFOFOA

20000 6000 8000 100004000Iteration

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




Complexity 21

Table11

Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

31365E+

0413

133E

+06

11295E

+08

10017E

+06

864

75E+

0513

449E

+07

Std

18602E

+04

52974E+

0531387E+

0748689E+

0548607E+

0528405E+

06F2

7Mean

304

00E-10

1844

6E+0

484500

E+09

10512E

+04

12359E

+04

58535E+

08Std

61535E-10

14049E

+04

10125E

+09

12485E

+04

11922E

+04

38771E+

07F2

8Mean

42105E-01

46710E+

0173

819E

+02

4114

7E+0

137814E+

0114

226E

+02

Std

12624E

+00

31490E+

0199

455E

+01

47336E+

0134110E+

0129201E+

01F2

9Mean

75177E

+03

14891E+0

529286E+

0647277E+

0531099E+

0539826E+

05Std

33119E+

0368316E+

049190

4E+0

521021E+

0522686E+

0515

511E+0

5F3

0Mean

31524E+

0231524E+

0238129E+

0231524E+

0231524E+

0232568E+

02Std

85708E-12

19710E

-07

14082E

+01

11524E

-1145680E-11

58955E+

00F31

Mean

23483E+

0223172E+

0230117E+

0223811E

+02

23858E+

0224179E+

02Std

41748E+

0072

461E+0

048903E+

00560

97E+

0050249E+

0090

228E

+00

F32

Mean

20790E+

02206

03E+

0221884E+

0221485E+

0220975E+

0220633E+

02Std

41618E+

0032456E+

0030353E+

0087909E+

0057719E+

0016

880E

+00

22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 13

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

Mea

n Er

rors

(log)

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus15

minus10

minus5

0

5

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

0

10

20

30

Mea

n Er

rors

(log)

2000 4000 6000 8000 100000Iteration

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus200

minus150

minus100

minus50

0

50

Mea

n Er

rors

(log)

(h) F11


14 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

20

Mea

n Er

rors

(log)

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

0

2

4

6

8

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

01020304050

Mea

n Er

rors

(log)

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus140

minus120

minus100

minus80

minus60

minus40

minus20

020

Mea

n Er

rors

(log)

(h) F11


Complexity 15

Table6Multim

odalbenchm

arkfunctio

ns

Functio

nRa

nge

Fmin

F12(119909)=

minus20exp(minus0

2radic1 119899119899 sum 119894=11199092 119894)minus

exp(1 119899119899 sum 119894=1co

s(2120587119909 119894))

+20+exp

(1 )[minus32

32]0

F13(119909)=

119899 sum 119894=11003816 1003816 1003816 1003816119909 119894sin(119909 119894)

+01119909 1198941003816 1003816 1003816 1003816

[minus1010]

0

F14(119909)=

119891 119904(1199091119909 2)


(119909 119899119909 1)

[minus10010

0]0

119891 119904(119909119910)=

(1199092 +1199102 )025[sin2

(50(1199092 +

1199102 )01)+1

]F15(

119909)=119891 119904(1199091119909 2)


119909 1198991199091)

[minus10010

0]0

119891 119904(119909119910)=

05(sin2(radic 1199092+1199102

)minus05)

(1+0001

(1199092 +1199102 ))2

F16(119909)=

120587 11989910sin2

(120587119910 119894)+119899minus1 sum 119894=1

(119910 119894minus1 )2 [

1+10sin2

(120587119910 119894+1)]+

(119910 119899minus1 )2

+119899 sum 119894=1119906(119909 119894

10100

4)[minus50

50]0

119910 119894=1+1 4(119909

119894+1)

119906(119909 119894119886

119896119898)= 119896(119909 119894

minus119886)119898

119909 119894gt119886

0minus119886le

119909 119894le119886

119896(minus119909119894minus119886)119898

119909119894gt119886

F17(119909)=

1 4000119899 sum 119894=11199092 119894minus119899 prod 119894=1

cos(119909119894 radic 119894)+1

[minus10010

0]0

F18(119909)=


(minus(1199092 119894+

1199092 119894+1+05

119909 119894119909 119894+1)

8)lowastc

os(4radic

1199092 119894+1199092 119894+1

+05119909 119894119909 119894+1))

[minus55]

1-n

F19(119909)=

119899 sum 119894=1(119909119894minus1)2

minus119899 sum 119894=2119909 119894119909 119894minus1

[minusn2n2 ]

119899(119899+4)(119899

minus1)minus6

F20 (119909 )=

sum119899minus1 119894=2(05

+(sin2(radic 1

001199092 119894+1199092 119894+1)minus0

5))(1+

0001(1199092 119894minus

2119909 119894119909119894minus1+1199092 119894minus1))2

[minus10010

0]0

F21(119909)=

119899 sum 119894=1[1199092 119894minus10

cos(2120587

119909 119894)+10]

[minus51251

2]0

F22(119909)=

119899 sum 119894=1[1199102 119894minus10

cos(2120587

119910 119894)+10]

119910 119894= 119909 119894

1003816 1003816 1003816 1003816119909 1198941003816 1003816 1003816 1003816lt05

119903119900119906119899119889(2119909

119894)2

1003816 1003816 1003816 10038161199091198941003816 1003816 1003816 1003816lt0

5[minus51

2512]

0

F23(119909)=

1minuscos(2120587

radic119899 sum 119894=11199092 119894)

+01radic119899 sum 119894=1

1199092 119894[minus10

0100]

0

F24(119909)=

119899 sum 119894=1119896119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 (119909119894+05

))]minus119899119896

119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 05

)][minus05

05]0

F25(119909)=

119899 sum 119896=1119899 sum 119895=1((1199102 119895119896 4000)minus

cos(119910 119895119896)+1

)119910119895119896=10

0(119909 119896minus1199092 119895

)2 +(1minus

1199092 119895)2[minus10

0100]

0

16 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmultim

odalbenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

51692E-14

21708E-07

16343E

+01

42641E-12

43970E-07

70983E

+00

Std

94813E-15

11785E

-07

45830E-01

51275E-13

53024E-08

45755E+

00F13

Mean

19651E-15

17670E

-07

30865E+

0138781E-12

12507E

-06

29660

E+01

Std

17016E

-1510

899E

-07

28749E+

00200

14E-12

19125E

-06

57790E+

00F14

Mean

28586E-11

47414E-02

21576E+

0235954E-05

17290E

-02

18705E

+02

Std

17874E

-1118

105E

-02

50836E+

0019

343E

-06

10857E

-03

50868E+

01F15

Mean

99552E

-01

46150E-01

12596E

+01

94983E

-01

10032E

+00

12147E

+01

Std

38926E-01

33522E-01

21495E-01

42966

E-01

35690E-01

17388E

-01

F16

Mean

15705E

-32

13069E

-15

56725E+

0650290E-25

99726E

-15

31482E+

00Std

28850E-48

57169E-16

17168E

+06

47027E-25

85374E-15

58054E-01

F17

Mean

13781E-02

10332E

-02

43352E+

0044332E-03

12793E

-02

10971E+0

0Std

14865E

-02

12632E

-02

42518E-01

79408E

-03

10155E

-02

10766E

-02

F18

Mean

50849E+

0038253E+

0020946

E+01

49225E+

00490

48E+

0021497E+

01Std

16014E

+00

14627E

+00

76856E

-01

21737E+

00204

11E+0

013

669E

+00

F19

Mean

268

41E-07

19292E

+02

49808E+

0519

677E

+02

240

98E+

0230226E+

04Std

32619E-08

15971E+0

214

706E

+05

16572E

+02

23149E+

0260289E+

03F2

0Mean

25989E-07

47006

E-06

33592E-02

44469E-08

18865E

-07

1540

6E-01

Std

59383E-07

73387E

-06

22456E-02

10350E

-07

31612E-07

56719E-02

F21

Mean

000

00E+

0070

841E-13

25769E+

02000

00E+

0045409E-11

30881E+

02Std

000

00E+

0045361E-13

90973E

+00

000

00E+

0019

882E

-11

27305E+

01F2

2Mean

000

00E+

007746

7E-13

23335E+

02000

00E+

00644

03E-11

25509E+

02Std

000

00E+

0036979E-13

15942E

+01

000

00E+

0033820E-11

26992E+

01F2

3Mean

93987E

-01

52987E-01

12199E

+01

13599E

+00

14399E

+00

21878E+

00Std

21705E-01

12517E

-01

49304

E-01

36576E-01

21705E-01

62731E-02

F24

Mean

14921E-14

37233E-04

32412E+

0147458E-09

42553E-03

26924E+

01Std

17226E

-1498

846E

-05

11649E

+00

28242E-09

42975E-04

35559E+

00F2

5Mean

29494E+

0110

724E

+02

11372E

+07

404

62E+

0193530E+

0092

421E+0

3Std

29743E+

0151800

E+01

31606

E+06

39685E+

0190392E+

0018

838E

+03

Complexity 17


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

85798E-14

24174E-04

18459E

+01

7404

4E-12

73673E

-07

82226E+

00Std

17360E

-1455274E-05

1944

7E-01

88139E-13

80222E-08

42517E+

00F13

Mean

22538E-15

29492E-04

71594E

+01

21041E-11

32004

E-06

60959E+

01Std

18688E

-1510

372E

-04

45394E+

0015

865E

-11

15334E

-06

44766

E+00

F14

Mean

71759E

-1120261E+

0043430E+

0277682E

-05

33324E-02

42669E+

02Std

24650E-11

50770E-01

14055E

+01

54975E-06

10537E

-03

80127E+

01F15

Mean

16716E

+00

12749E

+00

22241E+

0116

927E

+00

14937E

+00

21617E+

01Std

76572E

-01

43985E-01

33014E-01

47677E-01

63574E-01

54534E-01

F16

Mean

94233E-33

13057E

-09

76995E

+07

17755E

-24

846

48E-14

69921E+

00Std

14425E

-48

37533E-10

21712E+

0719

092E

-24

17429E

-13

89129E-01

F17

Mean

76377E

-03

14219E

-02

1160

6E+0

164039E-03

10080E

-02

1264

1E+0

0Std

57418E-03

21089E-02

46282E-01

70807E

-03

13952E

-02

16555E

-02

F18

Mean

83103E+

0079

047E

+00

39689E+

0189467E+

0096

041E+0

038726E+

01Std

260

72E+

0025432E+

0077616E

-01

78506E

-01

21029E+

0013

015E

+00

F19

Mean

45562E+

0126833E+

04806

68E+

0616

118E+

0413

155E

+04

70015E

+05

Std

38094E+

0121743E+

0421709E+

0612

498E

+04

1300

9E+0

497

174E

+04

F20

Mean

43064E-08

25702E-04

11519E

-01

52365E-08

16998E

-06

500

47E-01

Std

44294E-08

27576E-04

39417E-02

95247E

-08

49881E-06

26305E-01

F21

Mean

000

00E+

0011310E

-06

53146

E+02

000

00E+

0023711E

-10

58748E+

02Std

000

00E+

0033614E-07

32117E+

01000

00E+

0045437E-11

29507E+

01F2

2Mean

000

00E+

0016

167E

-06

48729E+

02000

00E+

00244

07E-10

52060

E+02

Std

000

00E+

0063216E-07

24382E+

01000

00E+

0075

889E

-11

42230E+

01F2

3Mean

13699E

+00

89987E-01

21237E+

0122699E+

0025899E+

0035955E+

00Std

23594E-01

666

67E-02

58033E-01

41913E-01

62973E-01

12247E

-01

F24

Mean

71054E

-1426826E-02

63090E+

0119

033E

-08

96037E

-03

47263E+

01Std

27621E-14

47780E-03

22392E+

0061075E-09

97071E-04

52689E+

00F2

5Mean

66563E+

0184722E+

0211275E

+08

65780E+

0139992E+

0188242E+

04Std

10992E

+02

2113

8E+0

221091E+

0794

954E

+01

43819E+

0116

832E

+04

18 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

18989E

-1316

584E

-01

19996E

+01

17809E

-11

14744E

-06

13554E

+01

Std

20566

E-14

53720E-02

90319E

-02

19159E

-12

18930E

-07

60821E+

00F13

Mean

22871E-15

17736E

-01

1944

7E+0

213

452E

-10

13291E-05

16379E

+02

Std

26741E-15

53611E-02

62653E+

0038592E-11

55001E-06

13313E

+01

F14

Mean

18736E

-1074

259E

+01

10132E

+03

22866

E-04

83534E-02

95534E

+02

Std

37223E-11

19144E

+01

18986E

+01

14283E

-05

10592E

-02

53523E+

01F15

Mean

26814E+

0010

178E

+01

47083E+

0128083E+

0034325E+

0045859E+

01Std

73851E-01

16238E

+00

22513E-01

46148E-01

60283E-01

69914E-01

F16

Mean

47116E-33

244

54E-04

90382E

+08

81890E-24

62347E-14

27647E+

03Std

72124E

-49

59650E-05

64985E+

0767958E-24

55604

E-14

44231E+

03F17

Mean

34494E-03

11896E

-02

37816E+

0134509E-03

41885E-03

21280E+

00Std

60565E-03

65363E-03

15922E

+00

46765E-03

86153E-03

54359E-02

F18

Mean

18033E

+01

17806E

+01

86826E+

0118

319E

+01

18828E

+01

82458E+

01Std

19652E

+00

38319E+

0093

222E

-01

29296E+

0025377E+

0015

159E

+00

F19

Mean

82462E+

0427944

E+06

48046

E+08

28415E+

0560265E+

0549201E+

07Std

55732E+

0489703E+

0596

715E

+07

24572E+

0527137E+

0572

772E

+06

F20

Mean

57130E-07

81688E-03

96848E

-01

13631E-06

27143E-05

21656E+

00Std

61122E-07

53195E-03

44542E-01

25155E-06

58766

E-05

80368E-01

F21

Mean

000

00E+

0051414E+

0013

305E

+03

000

00E+

0020026E-09

13623E

+03

Std

000

00E+

0017

825E

+00

22890E+

01000

00E+

0032815E-10

609

96E+

01F2

2Mean

000

00E+

0077

848E

+00

1260

9E+0

3000

00E+

0020383E-09

12745E

+03

Std

000

00E+

0023732E+

0029100

E+01

000

00E+

0029753E-10

59708E+

01F2

3Mean

25599E+

0020499E+

0039804

E+01

47099E+

0043699E+

0073

691E+0

0Std

36878E-01

15092E

-01

69296E-01

59151E-01

56184E-01

17989E

-01

F24

Mean

40927E-13

26229E+

0015

145E

+02

18874E

-07

29476E-02

10478E

+02

Std

88061E-14

63367E-01

42830E+

0037074E-08

17697E

-03

11873E

+01

F25

Mean

42987E+

028117

8E+0

313

524E

+09

56790E+

0244982E+

0218

038E

+06

Std

43423E+

0233128E+

0278

399E

+07

54327E+

0246926E+

0221315E+

05

Complexity 19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus8

minus6

minus4

minus2

02468

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(f) F24




MAE = sum119873119895=1 10038161003816100381610038161003816119898119895 minus 11989611989510038161003816100381610038161003816119873 (22)

20 Complexity

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

ISSASSAPSO

CMFOAIFFOFOA

1

2

3

4

5

6

7

8

Mea

n Er

rors

(log)

0 4000 6000 8000 100002000Iteration

(e) F19

ISSASSAPSO

CMFOAIFFOFOA

20000 6000 8000 100004000Iteration

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




Complexity 21

Table11

Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

31365E+

0413

133E

+06

11295E

+08

10017E

+06

864

75E+

0513

449E

+07

Std

18602E

+04

52974E+

0531387E+

0748689E+

0548607E+

0528405E+

06F2

7Mean

304

00E-10

1844

6E+0

484500

E+09

10512E

+04

12359E

+04

58535E+

08Std

61535E-10

14049E

+04

10125E

+09

12485E

+04

11922E

+04

38771E+

07F2

8Mean

42105E-01

46710E+

0173

819E

+02

4114

7E+0

137814E+

0114

226E

+02

Std

12624E

+00

31490E+

0199

455E

+01

47336E+

0134110E+

0129201E+

01F2

9Mean

75177E

+03

14891E+0

529286E+

0647277E+

0531099E+

0539826E+

05Std

33119E+

0368316E+

049190

4E+0

521021E+

0522686E+

0515

511E+0

5F3

0Mean

31524E+

0231524E+

0238129E+

0231524E+

0231524E+

0232568E+

02Std

85708E-12

19710E

-07

14082E

+01

11524E

-1145680E-11

58955E+

00F31

Mean

23483E+

0223172E+

0230117E+

0223811E

+02

23858E+

0224179E+

02Std

41748E+

0072

461E+0

048903E+

00560

97E+

0050249E+

0090

228E

+00

F32

Mean

20790E+

02206

03E+

0221884E+

0221485E+

0220975E+

0220633E+

02Std

41618E+

0032456E+

0030353E+

0087909E+

0057719E+

0016

880E

+00

22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








14 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(a) F1

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus40

minus30

minus20

minus10

0

10

20

Mea

n Er

rors

(log)

(b) F4

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

0

2

4

6

8

10

Mea

n Er

rors

(log)

(c) F6

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F7

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus8

minus6

minus4

minus2

0

2

Mea

n Er

rors

(log)

(e) F8

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus30

minus20

minus10

01020304050

Mea

n Er

rors

(log)

(f) F9

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus50

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(g) F10

0 2000 4000 6000 8000 10000Iteration

ISSASSA

PSOCMFOA

IFFOFOA

minus140

minus120

minus100

minus80

minus60

minus40

minus20

020

Mea

n Er

rors

(log)

(h) F11


Complexity 15

Table6Multim

odalbenchm

arkfunctio

ns

Functio

nRa

nge

Fmin

F12(119909)=

minus20exp(minus0

2radic1 119899119899 sum 119894=11199092 119894)minus

exp(1 119899119899 sum 119894=1co

s(2120587119909 119894))

+20+exp

(1 )[minus32

32]0

F13(119909)=

119899 sum 119894=11003816 1003816 1003816 1003816119909 119894sin(119909 119894)

+01119909 1198941003816 1003816 1003816 1003816

[minus1010]

0

F14(119909)=

119891 119904(1199091119909 2)


(119909 119899119909 1)

[minus10010

0]0

119891 119904(119909119910)=

(1199092 +1199102 )025[sin2

(50(1199092 +

1199102 )01)+1

]F15(

119909)=119891 119904(1199091119909 2)


119909 1198991199091)

[minus10010

0]0

119891 119904(119909119910)=

05(sin2(radic 1199092+1199102

)minus05)

(1+0001

(1199092 +1199102 ))2

F16(119909)=

120587 11989910sin2

(120587119910 119894)+119899minus1 sum 119894=1

(119910 119894minus1 )2 [

1+10sin2

(120587119910 119894+1)]+

(119910 119899minus1 )2

+119899 sum 119894=1119906(119909 119894

10100

4)[minus50

50]0

119910 119894=1+1 4(119909

119894+1)

119906(119909 119894119886

119896119898)= 119896(119909 119894

minus119886)119898

119909 119894gt119886

0minus119886le

119909 119894le119886

119896(minus119909119894minus119886)119898

119909119894gt119886

F17(119909)=

1 4000119899 sum 119894=11199092 119894minus119899 prod 119894=1

cos(119909119894 radic 119894)+1

[minus10010

0]0

F18(119909)=


(minus(1199092 119894+

1199092 119894+1+05

119909 119894119909 119894+1)

8)lowastc

os(4radic

1199092 119894+1199092 119894+1

+05119909 119894119909 119894+1))

[minus55]

1-n

F19(119909)=

119899 sum 119894=1(119909119894minus1)2

minus119899 sum 119894=2119909 119894119909 119894minus1

[minusn2n2 ]

119899(119899+4)(119899

minus1)minus6

F20 (119909 )=

sum119899minus1 119894=2(05

+(sin2(radic 1

001199092 119894+1199092 119894+1)minus0

5))(1+

0001(1199092 119894minus

2119909 119894119909119894minus1+1199092 119894minus1))2

[minus10010

0]0

F21(119909)=

119899 sum 119894=1[1199092 119894minus10

cos(2120587

119909 119894)+10]

[minus51251

2]0

F22(119909)=

119899 sum 119894=1[1199102 119894minus10

cos(2120587

119910 119894)+10]

119910 119894= 119909 119894

1003816 1003816 1003816 1003816119909 1198941003816 1003816 1003816 1003816lt05

119903119900119906119899119889(2119909

119894)2

1003816 1003816 1003816 10038161199091198941003816 1003816 1003816 1003816lt0

5[minus51

2512]

0

F23(119909)=

1minuscos(2120587

radic119899 sum 119894=11199092 119894)

+01radic119899 sum 119894=1

1199092 119894[minus10

0100]

0

F24(119909)=

119899 sum 119894=1119896119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 (119909119894+05

))]minus119899119896

119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 05

)][minus05

05]0

F25(119909)=

119899 sum 119896=1119899 sum 119895=1((1199102 119895119896 4000)minus

cos(119910 119895119896)+1

)119910119895119896=10

0(119909 119896minus1199092 119895

)2 +(1minus

1199092 119895)2[minus10

0100]

0

16 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmultim

odalbenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

51692E-14

21708E-07

16343E

+01

42641E-12

43970E-07

70983E

+00

Std

94813E-15

11785E

-07

45830E-01

51275E-13

53024E-08

45755E+

00F13

Mean

19651E-15

17670E

-07

30865E+

0138781E-12

12507E

-06

29660

E+01

Std

17016E

-1510

899E

-07

28749E+

00200

14E-12

19125E

-06

57790E+

00F14

Mean

28586E-11

47414E-02

21576E+

0235954E-05

17290E

-02

18705E

+02

Std

17874E

-1118

105E

-02

50836E+

0019

343E

-06

10857E

-03

50868E+

01F15

Mean

99552E

-01

46150E-01

12596E

+01

94983E

-01

10032E

+00

12147E

+01

Std

38926E-01

33522E-01

21495E-01

42966

E-01

35690E-01

17388E

-01

F16

Mean

15705E

-32

13069E

-15

56725E+

0650290E-25

99726E

-15

31482E+

00Std

28850E-48

57169E-16

17168E

+06

47027E-25

85374E-15

58054E-01

F17

Mean

13781E-02

10332E

-02

43352E+

0044332E-03

12793E

-02

10971E+0

0Std

14865E

-02

12632E

-02

42518E-01

79408E

-03

10155E

-02

10766E

-02

F18

Mean

50849E+

0038253E+

0020946

E+01

49225E+

00490

48E+

0021497E+

01Std

16014E

+00

14627E

+00

76856E

-01

21737E+

00204

11E+0

013

669E

+00

F19

Mean

268

41E-07

19292E

+02

49808E+

0519

677E

+02

240

98E+

0230226E+

04Std

32619E-08

15971E+0

214

706E

+05

16572E

+02

23149E+

0260289E+

03F2

0Mean

25989E-07

47006

E-06

33592E-02

44469E-08

18865E

-07

1540

6E-01

Std

59383E-07

73387E

-06

22456E-02

10350E

-07

31612E-07

56719E-02

F21

Mean

000

00E+

0070

841E-13

25769E+

02000

00E+

0045409E-11

30881E+

02Std

000

00E+

0045361E-13

90973E

+00

000

00E+

0019

882E

-11

27305E+

01F2

2Mean

000

00E+

007746

7E-13

23335E+

02000

00E+

00644

03E-11

25509E+

02Std

000

00E+

0036979E-13

15942E

+01

000

00E+

0033820E-11

26992E+

01F2

3Mean

93987E

-01

52987E-01

12199E

+01

13599E

+00

14399E

+00

21878E+

00Std

21705E-01

12517E

-01

49304

E-01

36576E-01

21705E-01

62731E-02

F24

Mean

14921E-14

37233E-04

32412E+

0147458E-09

42553E-03

26924E+

01Std

17226E

-1498

846E

-05

11649E

+00

28242E-09

42975E-04

35559E+

00F2

5Mean

29494E+

0110

724E

+02

11372E

+07

404

62E+

0193530E+

0092

421E+0

3Std

29743E+

0151800

E+01

31606

E+06

39685E+

0190392E+

0018

838E

+03

Complexity 17


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

85798E-14

24174E-04

18459E

+01

7404

4E-12

73673E

-07

82226E+

00Std

17360E

-1455274E-05

1944

7E-01

88139E-13

80222E-08

42517E+

00F13

Mean

22538E-15

29492E-04

71594E

+01

21041E-11

32004

E-06

60959E+

01Std

18688E

-1510

372E

-04

45394E+

0015

865E

-11

15334E

-06

44766

E+00

F14

Mean

71759E

-1120261E+

0043430E+

0277682E

-05

33324E-02

42669E+

02Std

24650E-11

50770E-01

14055E

+01

54975E-06

10537E

-03

80127E+

01F15

Mean

16716E

+00

12749E

+00

22241E+

0116

927E

+00

14937E

+00

21617E+

01Std

76572E

-01

43985E-01

33014E-01

47677E-01

63574E-01

54534E-01

F16

Mean

94233E-33

13057E

-09

76995E

+07

17755E

-24

846

48E-14

69921E+

00Std

14425E

-48

37533E-10

21712E+

0719

092E

-24

17429E

-13

89129E-01

F17

Mean

76377E

-03

14219E

-02

1160

6E+0

164039E-03

10080E

-02

1264

1E+0

0Std

57418E-03

21089E-02

46282E-01

70807E

-03

13952E

-02

16555E

-02

F18

Mean

83103E+

0079

047E

+00

39689E+

0189467E+

0096

041E+0

038726E+

01Std

260

72E+

0025432E+

0077616E

-01

78506E

-01

21029E+

0013

015E

+00

F19

Mean

45562E+

0126833E+

04806

68E+

0616

118E+

0413

155E

+04

70015E

+05

Std

38094E+

0121743E+

0421709E+

0612

498E

+04

1300

9E+0

497

174E

+04

F20

Mean

43064E-08

25702E-04

11519E

-01

52365E-08

16998E

-06

500

47E-01

Std

44294E-08

27576E-04

39417E-02

95247E

-08

49881E-06

26305E-01

F21

Mean

000

00E+

0011310E

-06

53146

E+02

000

00E+

0023711E

-10

58748E+

02Std

000

00E+

0033614E-07

32117E+

01000

00E+

0045437E-11

29507E+

01F2

2Mean

000

00E+

0016

167E

-06

48729E+

02000

00E+

00244

07E-10

52060

E+02

Std

000

00E+

0063216E-07

24382E+

01000

00E+

0075

889E

-11

42230E+

01F2

3Mean

13699E

+00

89987E-01

21237E+

0122699E+

0025899E+

0035955E+

00Std

23594E-01

666

67E-02

58033E-01

41913E-01

62973E-01

12247E

-01

F24

Mean

71054E

-1426826E-02

63090E+

0119

033E

-08

96037E

-03

47263E+

01Std

27621E-14

47780E-03

22392E+

0061075E-09

97071E-04

52689E+

00F2

5Mean

66563E+

0184722E+

0211275E

+08

65780E+

0139992E+

0188242E+

04Std

10992E

+02

2113

8E+0

221091E+

0794

954E

+01

43819E+

0116

832E

+04

18 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

18989E

-1316

584E

-01

19996E

+01

17809E

-11

14744E

-06

13554E

+01

Std

20566

E-14

53720E-02

90319E

-02

19159E

-12

18930E

-07

60821E+

00F13

Mean

22871E-15

17736E

-01

1944

7E+0

213

452E

-10

13291E-05

16379E

+02

Std

26741E-15

53611E-02

62653E+

0038592E-11

55001E-06

13313E

+01

F14

Mean

18736E

-1074

259E

+01

10132E

+03

22866

E-04

83534E-02

95534E

+02

Std

37223E-11

19144E

+01

18986E

+01

14283E

-05

10592E

-02

53523E+

01F15

Mean

26814E+

0010

178E

+01

47083E+

0128083E+

0034325E+

0045859E+

01Std

73851E-01

16238E

+00

22513E-01

46148E-01

60283E-01

69914E-01

F16

Mean

47116E-33

244

54E-04

90382E

+08

81890E-24

62347E-14

27647E+

03Std

72124E

-49

59650E-05

64985E+

0767958E-24

55604

E-14

44231E+

03F17

Mean

34494E-03

11896E

-02

37816E+

0134509E-03

41885E-03

21280E+

00Std

60565E-03

65363E-03

15922E

+00

46765E-03

86153E-03

54359E-02

F18

Mean

18033E

+01

17806E

+01

86826E+

0118

319E

+01

18828E

+01

82458E+

01Std

19652E

+00

38319E+

0093

222E

-01

29296E+

0025377E+

0015

159E

+00

F19

Mean

82462E+

0427944

E+06

48046

E+08

28415E+

0560265E+

0549201E+

07Std

55732E+

0489703E+

0596

715E

+07

24572E+

0527137E+

0572

772E

+06

F20

Mean

57130E-07

81688E-03

96848E

-01

13631E-06

27143E-05

21656E+

00Std

61122E-07

53195E-03

44542E-01

25155E-06

58766

E-05

80368E-01

F21

Mean

000

00E+

0051414E+

0013

305E

+03

000

00E+

0020026E-09

13623E

+03

Std

000

00E+

0017

825E

+00

22890E+

01000

00E+

0032815E-10

609

96E+

01F2

2Mean

000

00E+

0077

848E

+00

1260

9E+0

3000

00E+

0020383E-09

12745E

+03

Std

000

00E+

0023732E+

0029100

E+01

000

00E+

0029753E-10

59708E+

01F2

3Mean

25599E+

0020499E+

0039804

E+01

47099E+

0043699E+

0073

691E+0

0Std

36878E-01

15092E

-01

69296E-01

59151E-01

56184E-01

17989E

-01

F24

Mean

40927E-13

26229E+

0015

145E

+02

18874E

-07

29476E-02

10478E

+02

Std

88061E-14

63367E-01

42830E+

0037074E-08

17697E

-03

11873E

+01

F25

Mean

42987E+

028117

8E+0

313

524E

+09

56790E+

0244982E+

0218

038E

+06

Std

43423E+

0233128E+

0278

399E

+07

54327E+

0246926E+

0221315E+

05

Complexity 19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus8

minus6

minus4

minus2

02468

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(f) F24




MAE = sum119873119895=1 10038161003816100381610038161003816119898119895 minus 11989611989510038161003816100381610038161003816119873 (22)

20 Complexity

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

ISSASSAPSO

CMFOAIFFOFOA

1

2

3

4

5

6

7

8

Mea

n Er

rors

(log)

0 4000 6000 8000 100002000Iteration

(e) F19

ISSASSAPSO

CMFOAIFFOFOA

20000 6000 8000 100004000Iteration

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




Complexity 21

Table11

Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

31365E+

0413

133E

+06

11295E

+08

10017E

+06

864

75E+

0513

449E

+07

Std

18602E

+04

52974E+

0531387E+

0748689E+

0548607E+

0528405E+

06F2

7Mean

304

00E-10

1844

6E+0

484500

E+09

10512E

+04

12359E

+04

58535E+

08Std

61535E-10

14049E

+04

10125E

+09

12485E

+04

11922E

+04

38771E+

07F2

8Mean

42105E-01

46710E+

0173

819E

+02

4114

7E+0

137814E+

0114

226E

+02

Std

12624E

+00

31490E+

0199

455E

+01

47336E+

0134110E+

0129201E+

01F2

9Mean

75177E

+03

14891E+0

529286E+

0647277E+

0531099E+

0539826E+

05Std

33119E+

0368316E+

049190

4E+0

521021E+

0522686E+

0515

511E+0

5F3

0Mean

31524E+

0231524E+

0238129E+

0231524E+

0231524E+

0232568E+

02Std

85708E-12

19710E

-07

14082E

+01

11524E

-1145680E-11

58955E+

00F31

Mean

23483E+

0223172E+

0230117E+

0223811E

+02

23858E+

0224179E+

02Std

41748E+

0072

461E+0

048903E+

00560

97E+

0050249E+

0090

228E

+00

F32

Mean

20790E+

02206

03E+

0221884E+

0221485E+

0220975E+

0220633E+

02Std

41618E+

0032456E+

0030353E+

0087909E+

0057719E+

0016

880E

+00

22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 15

Table6Multim

odalbenchm

arkfunctio

ns

Functio

nRa

nge

Fmin

F12(119909)=

minus20exp(minus0

2radic1 119899119899 sum 119894=11199092 119894)minus

exp(1 119899119899 sum 119894=1co

s(2120587119909 119894))

+20+exp

(1 )[minus32

32]0

F13(119909)=

119899 sum 119894=11003816 1003816 1003816 1003816119909 119894sin(119909 119894)

+01119909 1198941003816 1003816 1003816 1003816

[minus1010]

0

F14(119909)=

119891 119904(1199091119909 2)


(119909 119899119909 1)

[minus10010

0]0

119891 119904(119909119910)=

(1199092 +1199102 )025[sin2

(50(1199092 +

1199102 )01)+1

]F15(

119909)=119891 119904(1199091119909 2)


119909 1198991199091)

[minus10010

0]0

119891 119904(119909119910)=

05(sin2(radic 1199092+1199102

)minus05)

(1+0001

(1199092 +1199102 ))2

F16(119909)=

120587 11989910sin2

(120587119910 119894)+119899minus1 sum 119894=1

(119910 119894minus1 )2 [

1+10sin2

(120587119910 119894+1)]+

(119910 119899minus1 )2

+119899 sum 119894=1119906(119909 119894

10100

4)[minus50

50]0

119910 119894=1+1 4(119909

119894+1)

119906(119909 119894119886

119896119898)= 119896(119909 119894

minus119886)119898

119909 119894gt119886

0minus119886le

119909 119894le119886

119896(minus119909119894minus119886)119898

119909119894gt119886

F17(119909)=

1 4000119899 sum 119894=11199092 119894minus119899 prod 119894=1

cos(119909119894 radic 119894)+1

[minus10010

0]0

F18(119909)=


(minus(1199092 119894+

1199092 119894+1+05

119909 119894119909 119894+1)

8)lowastc

os(4radic

1199092 119894+1199092 119894+1

+05119909 119894119909 119894+1))

[minus55]

1-n

F19(119909)=

119899 sum 119894=1(119909119894minus1)2

minus119899 sum 119894=2119909 119894119909 119894minus1

[minusn2n2 ]

119899(119899+4)(119899

minus1)minus6

F20 (119909 )=

sum119899minus1 119894=2(05

+(sin2(radic 1

001199092 119894+1199092 119894+1)minus0

5))(1+

0001(1199092 119894minus

2119909 119894119909119894minus1+1199092 119894minus1))2

[minus10010

0]0

F21(119909)=

119899 sum 119894=1[1199092 119894minus10

cos(2120587

119909 119894)+10]

[minus51251

2]0

F22(119909)=

119899 sum 119894=1[1199102 119894minus10

cos(2120587

119910 119894)+10]

119910 119894= 119909 119894

1003816 1003816 1003816 1003816119909 1198941003816 1003816 1003816 1003816lt05

119903119900119906119899119889(2119909

119894)2

1003816 1003816 1003816 10038161199091198941003816 1003816 1003816 1003816lt0

5[minus51

2512]

0

F23(119909)=

1minuscos(2120587

radic119899 sum 119894=11199092 119894)

+01radic119899 sum 119894=1

1199092 119894[minus10

0100]

0

F24(119909)=

119899 sum 119894=1119896119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 (119909119894+05

))]minus119899119896

119898119886119909 sum 119896=0[119886119896 co

s(2120587119887119896 05

)][minus05

05]0

F25(119909)=

119899 sum 119896=1119899 sum 119895=1((1199102 119895119896 4000)minus

cos(119910 119895119896)+1

)119910119895119896=10

0(119909 119896minus1199092 119895

)2 +(1minus

1199092 119895)2[minus10

0100]

0

16 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmultim

odalbenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

51692E-14

21708E-07

16343E

+01

42641E-12

43970E-07

70983E

+00

Std

94813E-15

11785E

-07

45830E-01

51275E-13

53024E-08

45755E+

00F13

Mean

19651E-15

17670E

-07

30865E+

0138781E-12

12507E

-06

29660

E+01

Std

17016E

-1510

899E

-07

28749E+

00200

14E-12

19125E

-06

57790E+

00F14

Mean

28586E-11

47414E-02

21576E+

0235954E-05

17290E

-02

18705E

+02

Std

17874E

-1118

105E

-02

50836E+

0019

343E

-06

10857E

-03

50868E+

01F15

Mean

99552E

-01

46150E-01

12596E

+01

94983E

-01

10032E

+00

12147E

+01

Std

38926E-01

33522E-01

21495E-01

42966

E-01

35690E-01

17388E

-01

F16

Mean

15705E

-32

13069E

-15

56725E+

0650290E-25

99726E

-15

31482E+

00Std

28850E-48

57169E-16

17168E

+06

47027E-25

85374E-15

58054E-01

F17

Mean

13781E-02

10332E

-02

43352E+

0044332E-03

12793E

-02

10971E+0

0Std

14865E

-02

12632E

-02

42518E-01

79408E

-03

10155E

-02

10766E

-02

F18

Mean

50849E+

0038253E+

0020946

E+01

49225E+

00490

48E+

0021497E+

01Std

16014E

+00

14627E

+00

76856E

-01

21737E+

00204

11E+0

013

669E

+00

F19

Mean

268

41E-07

19292E

+02

49808E+

0519

677E

+02

240

98E+

0230226E+

04Std

32619E-08

15971E+0

214

706E

+05

16572E

+02

23149E+

0260289E+

03F2

0Mean

25989E-07

47006

E-06

33592E-02

44469E-08

18865E

-07

1540

6E-01

Std

59383E-07

73387E

-06

22456E-02

10350E

-07

31612E-07

56719E-02

F21

Mean

000

00E+

0070

841E-13

25769E+

02000

00E+

0045409E-11

30881E+

02Std

000

00E+

0045361E-13

90973E

+00

000

00E+

0019

882E

-11

27305E+

01F2

2Mean

000

00E+

007746

7E-13

23335E+

02000

00E+

00644

03E-11

25509E+

02Std

000

00E+

0036979E-13

15942E

+01

000

00E+

0033820E-11

26992E+

01F2

3Mean

93987E

-01

52987E-01

12199E

+01

13599E

+00

14399E

+00

21878E+

00Std

21705E-01

12517E

-01

49304

E-01

36576E-01

21705E-01

62731E-02

F24

Mean

14921E-14

37233E-04

32412E+

0147458E-09

42553E-03

26924E+

01Std

17226E

-1498

846E

-05

11649E

+00

28242E-09

42975E-04

35559E+

00F2

5Mean

29494E+

0110

724E

+02

11372E

+07

404

62E+

0193530E+

0092

421E+0

3Std

29743E+

0151800

E+01

31606

E+06

39685E+

0190392E+

0018

838E

+03

Complexity 17


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

85798E-14

24174E-04

18459E

+01

7404

4E-12

73673E

-07

82226E+

00Std

17360E

-1455274E-05

1944

7E-01

88139E-13

80222E-08

42517E+

00F13

Mean

22538E-15

29492E-04

71594E

+01

21041E-11

32004

E-06

60959E+

01Std

18688E

-1510

372E

-04

45394E+

0015

865E

-11

15334E

-06

44766

E+00

F14

Mean

71759E

-1120261E+

0043430E+

0277682E

-05

33324E-02

42669E+

02Std

24650E-11

50770E-01

14055E

+01

54975E-06

10537E

-03

80127E+

01F15

Mean

16716E

+00

12749E

+00

22241E+

0116

927E

+00

14937E

+00

21617E+

01Std

76572E

-01

43985E-01

33014E-01

47677E-01

63574E-01

54534E-01

F16

Mean

94233E-33

13057E

-09

76995E

+07

17755E

-24

846

48E-14

69921E+

00Std

14425E

-48

37533E-10

21712E+

0719

092E

-24

17429E

-13

89129E-01

F17

Mean

76377E

-03

14219E

-02

1160

6E+0

164039E-03

10080E

-02

1264

1E+0

0Std

57418E-03

21089E-02

46282E-01

70807E

-03

13952E

-02

16555E

-02

F18

Mean

83103E+

0079

047E

+00

39689E+

0189467E+

0096

041E+0

038726E+

01Std

260

72E+

0025432E+

0077616E

-01

78506E

-01

21029E+

0013

015E

+00

F19

Mean

45562E+

0126833E+

04806

68E+

0616

118E+

0413

155E

+04

70015E

+05

Std

38094E+

0121743E+

0421709E+

0612

498E

+04

1300

9E+0

497

174E

+04

F20

Mean

43064E-08

25702E-04

11519E

-01

52365E-08

16998E

-06

500

47E-01

Std

44294E-08

27576E-04

39417E-02

95247E

-08

49881E-06

26305E-01

F21

Mean

000

00E+

0011310E

-06

53146

E+02

000

00E+

0023711E

-10

58748E+

02Std

000

00E+

0033614E-07

32117E+

01000

00E+

0045437E-11

29507E+

01F2

2Mean

000

00E+

0016

167E

-06

48729E+

02000

00E+

00244

07E-10

52060

E+02

Std

000

00E+

0063216E-07

24382E+

01000

00E+

0075

889E

-11

42230E+

01F2

3Mean

13699E

+00

89987E-01

21237E+

0122699E+

0025899E+

0035955E+

00Std

23594E-01

666

67E-02

58033E-01

41913E-01

62973E-01

12247E

-01

F24

Mean

71054E

-1426826E-02

63090E+

0119

033E

-08

96037E

-03

47263E+

01Std

27621E-14

47780E-03

22392E+

0061075E-09

97071E-04

52689E+

00F2

5Mean

66563E+

0184722E+

0211275E

+08

65780E+

0139992E+

0188242E+

04Std

10992E

+02

2113

8E+0

221091E+

0794

954E

+01

43819E+

0116

832E

+04

18 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

18989E

-1316

584E

-01

19996E

+01

17809E

-11

14744E

-06

13554E

+01

Std

20566

E-14

53720E-02

90319E

-02

19159E

-12

18930E

-07

60821E+

00F13

Mean

22871E-15

17736E

-01

1944

7E+0

213

452E

-10

13291E-05

16379E

+02

Std

26741E-15

53611E-02

62653E+

0038592E-11

55001E-06

13313E

+01

F14

Mean

18736E

-1074

259E

+01

10132E

+03

22866

E-04

83534E-02

95534E

+02

Std

37223E-11

19144E

+01

18986E

+01

14283E

-05

10592E

-02

53523E+

01F15

Mean

26814E+

0010

178E

+01

47083E+

0128083E+

0034325E+

0045859E+

01Std

73851E-01

16238E

+00

22513E-01

46148E-01

60283E-01

69914E-01

F16

Mean

47116E-33

244

54E-04

90382E

+08

81890E-24

62347E-14

27647E+

03Std

72124E

-49

59650E-05

64985E+

0767958E-24

55604

E-14

44231E+

03F17

Mean

34494E-03

11896E

-02

37816E+

0134509E-03

41885E-03

21280E+

00Std

60565E-03

65363E-03

15922E

+00

46765E-03

86153E-03

54359E-02

F18

Mean

18033E

+01

17806E

+01

86826E+

0118

319E

+01

18828E

+01

82458E+

01Std

19652E

+00

38319E+

0093

222E

-01

29296E+

0025377E+

0015

159E

+00

F19

Mean

82462E+

0427944

E+06

48046

E+08

28415E+

0560265E+

0549201E+

07Std

55732E+

0489703E+

0596

715E

+07

24572E+

0527137E+

0572

772E

+06

F20

Mean

57130E-07

81688E-03

96848E

-01

13631E-06

27143E-05

21656E+

00Std

61122E-07

53195E-03

44542E-01

25155E-06

58766

E-05

80368E-01

F21

Mean

000

00E+

0051414E+

0013

305E

+03

000

00E+

0020026E-09

13623E

+03

Std

000

00E+

0017

825E

+00

22890E+

01000

00E+

0032815E-10

609

96E+

01F2

2Mean

000

00E+

0077

848E

+00

1260

9E+0

3000

00E+

0020383E-09

12745E

+03

Std

000

00E+

0023732E+

0029100

E+01

000

00E+

0029753E-10

59708E+

01F2

3Mean

25599E+

0020499E+

0039804

E+01

47099E+

0043699E+

0073

691E+0

0Std

36878E-01

15092E

-01

69296E-01

59151E-01

56184E-01

17989E

-01

F24

Mean

40927E-13

26229E+

0015

145E

+02

18874E

-07

29476E-02

10478E

+02

Std

88061E-14

63367E-01

42830E+

0037074E-08

17697E

-03

11873E

+01

F25

Mean

42987E+

028117

8E+0

313

524E

+09

56790E+

0244982E+

0218

038E

+06

Std

43423E+

0233128E+

0278

399E

+07

54327E+

0246926E+

0221315E+

05

Complexity 19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus8

minus6

minus4

minus2

02468

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(f) F24




MAE = sum119873119895=1 10038161003816100381610038161003816119898119895 minus 11989611989510038161003816100381610038161003816119873 (22)

20 Complexity

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

ISSASSAPSO

CMFOAIFFOFOA

1

2

3

4

5

6

7

8

Mea

n Er

rors

(log)

0 4000 6000 8000 100002000Iteration

(e) F19

ISSASSAPSO

CMFOAIFFOFOA

20000 6000 8000 100004000Iteration

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




Complexity 21

Table11

Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

31365E+

0413

133E

+06

11295E

+08

10017E

+06

864

75E+

0513

449E

+07

Std

18602E

+04

52974E+

0531387E+

0748689E+

0548607E+

0528405E+

06F2

7Mean

304

00E-10

1844

6E+0

484500

E+09

10512E

+04

12359E

+04

58535E+

08Std

61535E-10

14049E

+04

10125E

+09

12485E

+04

11922E

+04

38771E+

07F2

8Mean

42105E-01

46710E+

0173

819E

+02

4114

7E+0

137814E+

0114

226E

+02

Std

12624E

+00

31490E+

0199

455E

+01

47336E+

0134110E+

0129201E+

01F2

9Mean

75177E

+03

14891E+0

529286E+

0647277E+

0531099E+

0539826E+

05Std

33119E+

0368316E+

049190

4E+0

521021E+

0522686E+

0515

511E+0

5F3

0Mean

31524E+

0231524E+

0238129E+

0231524E+

0231524E+

0232568E+

02Std

85708E-12

19710E

-07

14082E

+01

11524E

-1145680E-11

58955E+

00F31

Mean

23483E+

0223172E+

0230117E+

0223811E

+02

23858E+

0224179E+

02Std

41748E+

0072

461E+0

048903E+

00560

97E+

0050249E+

0090

228E

+00

F32

Mean

20790E+

02206

03E+

0221884E+

0221485E+

0220975E+

0220633E+

02Std

41618E+

0032456E+

0030353E+

0087909E+

0057719E+

0016

880E

+00

22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








16 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmultim

odalbenchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

51692E-14

21708E-07

16343E

+01

42641E-12

43970E-07

70983E

+00

Std

94813E-15

11785E

-07

45830E-01

51275E-13

53024E-08

45755E+

00F13

Mean

19651E-15

17670E

-07

30865E+

0138781E-12

12507E

-06

29660

E+01

Std

17016E

-1510

899E

-07

28749E+

00200

14E-12

19125E

-06

57790E+

00F14

Mean

28586E-11

47414E-02

21576E+

0235954E-05

17290E

-02

18705E

+02

Std

17874E

-1118

105E

-02

50836E+

0019

343E

-06

10857E

-03

50868E+

01F15

Mean

99552E

-01

46150E-01

12596E

+01

94983E

-01

10032E

+00

12147E

+01

Std

38926E-01

33522E-01

21495E-01

42966

E-01

35690E-01

17388E

-01

F16

Mean

15705E

-32

13069E

-15

56725E+

0650290E-25

99726E

-15

31482E+

00Std

28850E-48

57169E-16

17168E

+06

47027E-25

85374E-15

58054E-01

F17

Mean

13781E-02

10332E

-02

43352E+

0044332E-03

12793E

-02

10971E+0

0Std

14865E

-02

12632E

-02

42518E-01

79408E

-03

10155E

-02

10766E

-02

F18

Mean

50849E+

0038253E+

0020946

E+01

49225E+

00490

48E+

0021497E+

01Std

16014E

+00

14627E

+00

76856E

-01

21737E+

00204

11E+0

013

669E

+00

F19

Mean

268

41E-07

19292E

+02

49808E+

0519

677E

+02

240

98E+

0230226E+

04Std

32619E-08

15971E+0

214

706E

+05

16572E

+02

23149E+

0260289E+

03F2

0Mean

25989E-07

47006

E-06

33592E-02

44469E-08

18865E

-07

1540

6E-01

Std

59383E-07

73387E

-06

22456E-02

10350E

-07

31612E-07

56719E-02

F21

Mean

000

00E+

0070

841E-13

25769E+

02000

00E+

0045409E-11

30881E+

02Std

000

00E+

0045361E-13

90973E

+00

000

00E+

0019

882E

-11

27305E+

01F2

2Mean

000

00E+

007746

7E-13

23335E+

02000

00E+

00644

03E-11

25509E+

02Std

000

00E+

0036979E-13

15942E

+01

000

00E+

0033820E-11

26992E+

01F2

3Mean

93987E

-01

52987E-01

12199E

+01

13599E

+00

14399E

+00

21878E+

00Std

21705E-01

12517E

-01

49304

E-01

36576E-01

21705E-01

62731E-02

F24

Mean

14921E-14

37233E-04

32412E+

0147458E-09

42553E-03

26924E+

01Std

17226E

-1498

846E

-05

11649E

+00

28242E-09

42975E-04

35559E+

00F2

5Mean

29494E+

0110

724E

+02

11372E

+07

404

62E+

0193530E+

0092

421E+0

3Std

29743E+

0151800

E+01

31606

E+06

39685E+

0190392E+

0018

838E

+03

Complexity 17


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

85798E-14

24174E-04

18459E

+01

7404

4E-12

73673E

-07

82226E+

00Std

17360E

-1455274E-05

1944

7E-01

88139E-13

80222E-08

42517E+

00F13

Mean

22538E-15

29492E-04

71594E

+01

21041E-11

32004

E-06

60959E+

01Std

18688E

-1510

372E

-04

45394E+

0015

865E

-11

15334E

-06

44766

E+00

F14

Mean

71759E

-1120261E+

0043430E+

0277682E

-05

33324E-02

42669E+

02Std

24650E-11

50770E-01

14055E

+01

54975E-06

10537E

-03

80127E+

01F15

Mean

16716E

+00

12749E

+00

22241E+

0116

927E

+00

14937E

+00

21617E+

01Std

76572E

-01

43985E-01

33014E-01

47677E-01

63574E-01

54534E-01

F16

Mean

94233E-33

13057E

-09

76995E

+07

17755E

-24

846

48E-14

69921E+

00Std

14425E

-48

37533E-10

21712E+

0719

092E

-24

17429E

-13

89129E-01

F17

Mean

76377E

-03

14219E

-02

1160

6E+0

164039E-03

10080E

-02

1264

1E+0

0Std

57418E-03

21089E-02

46282E-01

70807E

-03

13952E

-02

16555E

-02

F18

Mean

83103E+

0079

047E

+00

39689E+

0189467E+

0096

041E+0

038726E+

01Std

260

72E+

0025432E+

0077616E

-01

78506E

-01

21029E+

0013

015E

+00

F19

Mean

45562E+

0126833E+

04806

68E+

0616

118E+

0413

155E

+04

70015E

+05

Std

38094E+

0121743E+

0421709E+

0612

498E

+04

1300

9E+0

497

174E

+04

F20

Mean

43064E-08

25702E-04

11519E

-01

52365E-08

16998E

-06

500

47E-01

Std

44294E-08

27576E-04

39417E-02

95247E

-08

49881E-06

26305E-01

F21

Mean

000

00E+

0011310E

-06

53146

E+02

000

00E+

0023711E

-10

58748E+

02Std

000

00E+

0033614E-07

32117E+

01000

00E+

0045437E-11

29507E+

01F2

2Mean

000

00E+

0016

167E

-06

48729E+

02000

00E+

00244

07E-10

52060

E+02

Std

000

00E+

0063216E-07

24382E+

01000

00E+

0075

889E

-11

42230E+

01F2

3Mean

13699E

+00

89987E-01

21237E+

0122699E+

0025899E+

0035955E+

00Std

23594E-01

666

67E-02

58033E-01

41913E-01

62973E-01

12247E

-01

F24

Mean

71054E

-1426826E-02

63090E+

0119

033E

-08

96037E

-03

47263E+

01Std

27621E-14

47780E-03

22392E+

0061075E-09

97071E-04

52689E+

00F2

5Mean

66563E+

0184722E+

0211275E

+08

65780E+

0139992E+

0188242E+

04Std

10992E

+02

2113

8E+0

221091E+

0794

954E

+01

43819E+

0116

832E

+04

18 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

18989E

-1316

584E

-01

19996E

+01

17809E

-11

14744E

-06

13554E

+01

Std

20566

E-14

53720E-02

90319E

-02

19159E

-12

18930E

-07

60821E+

00F13

Mean

22871E-15

17736E

-01

1944

7E+0

213

452E

-10

13291E-05

16379E

+02

Std

26741E-15

53611E-02

62653E+

0038592E-11

55001E-06

13313E

+01

F14

Mean

18736E

-1074

259E

+01

10132E

+03

22866

E-04

83534E-02

95534E

+02

Std

37223E-11

19144E

+01

18986E

+01

14283E

-05

10592E

-02

53523E+

01F15

Mean

26814E+

0010

178E

+01

47083E+

0128083E+

0034325E+

0045859E+

01Std

73851E-01

16238E

+00

22513E-01

46148E-01

60283E-01

69914E-01

F16

Mean

47116E-33

244

54E-04

90382E

+08

81890E-24

62347E-14

27647E+

03Std

72124E

-49

59650E-05

64985E+

0767958E-24

55604

E-14

44231E+

03F17

Mean

34494E-03

11896E

-02

37816E+

0134509E-03

41885E-03

21280E+

00Std

60565E-03

65363E-03

15922E

+00

46765E-03

86153E-03

54359E-02

F18

Mean

18033E

+01

17806E

+01

86826E+

0118

319E

+01

18828E

+01

82458E+

01Std

19652E

+00

38319E+

0093

222E

-01

29296E+

0025377E+

0015

159E

+00

F19

Mean

82462E+

0427944

E+06

48046

E+08

28415E+

0560265E+

0549201E+

07Std

55732E+

0489703E+

0596

715E

+07

24572E+

0527137E+

0572

772E

+06

F20

Mean

57130E-07

81688E-03

96848E

-01

13631E-06

27143E-05

21656E+

00Std

61122E-07

53195E-03

44542E-01

25155E-06

58766

E-05

80368E-01

F21

Mean

000

00E+

0051414E+

0013

305E

+03

000

00E+

0020026E-09

13623E

+03

Std

000

00E+

0017

825E

+00

22890E+

01000

00E+

0032815E-10

609

96E+

01F2

2Mean

000

00E+

0077

848E

+00

1260

9E+0

3000

00E+

0020383E-09

12745E

+03

Std

000

00E+

0023732E+

0029100

E+01

000

00E+

0029753E-10

59708E+

01F2

3Mean

25599E+

0020499E+

0039804

E+01

47099E+

0043699E+

0073

691E+0

0Std

36878E-01

15092E

-01

69296E-01

59151E-01

56184E-01

17989E

-01

F24

Mean

40927E-13

26229E+

0015

145E

+02

18874E

-07

29476E-02

10478E

+02

Std

88061E-14

63367E-01

42830E+

0037074E-08

17697E

-03

11873E

+01

F25

Mean

42987E+

028117

8E+0

313

524E

+09

56790E+

0244982E+

0218

038E

+06

Std

43423E+

0233128E+

0278

399E

+07

54327E+

0246926E+

0221315E+

05

Complexity 19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus8

minus6

minus4

minus2

02468

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(f) F24




MAE = sum119873119895=1 10038161003816100381610038161003816119898119895 minus 11989611989510038161003816100381610038161003816119873 (22)

20 Complexity

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

ISSASSAPSO

CMFOAIFFOFOA

1

2

3

4

5

6

7

8

Mea

n Er

rors

(log)

0 4000 6000 8000 100002000Iteration

(e) F19

ISSASSAPSO

CMFOAIFFOFOA

20000 6000 8000 100004000Iteration

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




Complexity 21

Table11

Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

31365E+

0413

133E

+06

11295E

+08

10017E

+06

864

75E+

0513

449E

+07

Std

18602E

+04

52974E+

0531387E+

0748689E+

0548607E+

0528405E+

06F2

7Mean

304

00E-10

1844

6E+0

484500

E+09

10512E

+04

12359E

+04

58535E+

08Std

61535E-10

14049E

+04

10125E

+09

12485E

+04

11922E

+04

38771E+

07F2

8Mean

42105E-01

46710E+

0173

819E

+02

4114

7E+0

137814E+

0114

226E

+02

Std

12624E

+00

31490E+

0199

455E

+01

47336E+

0134110E+

0129201E+

01F2

9Mean

75177E

+03

14891E+0

529286E+

0647277E+

0531099E+

0539826E+

05Std

33119E+

0368316E+

049190

4E+0

521021E+

0522686E+

0515

511E+0

5F3

0Mean

31524E+

0231524E+

0238129E+

0231524E+

0231524E+

0232568E+

02Std

85708E-12

19710E

-07

14082E

+01

11524E

-1145680E-11

58955E+

00F31

Mean

23483E+

0223172E+

0230117E+

0223811E

+02

23858E+

0224179E+

02Std

41748E+

0072

461E+0

048903E+

00560

97E+

0050249E+

0090

228E

+00

F32

Mean

20790E+

02206

03E+

0221884E+

0221485E+

0220975E+

0220633E+

02Std

41618E+

0032456E+

0030353E+

0087909E+

0057719E+

0016

880E

+00

22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 17


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

85798E-14

24174E-04

18459E

+01

7404

4E-12

73673E

-07

82226E+

00Std

17360E

-1455274E-05

1944

7E-01

88139E-13

80222E-08

42517E+

00F13

Mean

22538E-15

29492E-04

71594E

+01

21041E-11

32004

E-06

60959E+

01Std

18688E

-1510

372E

-04

45394E+

0015

865E

-11

15334E

-06

44766

E+00

F14

Mean

71759E

-1120261E+

0043430E+

0277682E

-05

33324E-02

42669E+

02Std

24650E-11

50770E-01

14055E

+01

54975E-06

10537E

-03

80127E+

01F15

Mean

16716E

+00

12749E

+00

22241E+

0116

927E

+00

14937E

+00

21617E+

01Std

76572E

-01

43985E-01

33014E-01

47677E-01

63574E-01

54534E-01

F16

Mean

94233E-33

13057E

-09

76995E

+07

17755E

-24

846

48E-14

69921E+

00Std

14425E

-48

37533E-10

21712E+

0719

092E

-24

17429E

-13

89129E-01

F17

Mean

76377E

-03

14219E

-02

1160

6E+0

164039E-03

10080E

-02

1264

1E+0

0Std

57418E-03

21089E-02

46282E-01

70807E

-03

13952E

-02

16555E

-02

F18

Mean

83103E+

0079

047E

+00

39689E+

0189467E+

0096

041E+0

038726E+

01Std

260

72E+

0025432E+

0077616E

-01

78506E

-01

21029E+

0013

015E

+00

F19

Mean

45562E+

0126833E+

04806

68E+

0616

118E+

0413

155E

+04

70015E

+05

Std

38094E+

0121743E+

0421709E+

0612

498E

+04

1300

9E+0

497

174E

+04

F20

Mean

43064E-08

25702E-04

11519E

-01

52365E-08

16998E

-06

500

47E-01

Std

44294E-08

27576E-04

39417E-02

95247E

-08

49881E-06

26305E-01

F21

Mean

000

00E+

0011310E

-06

53146

E+02

000

00E+

0023711E

-10

58748E+

02Std

000

00E+

0033614E-07

32117E+

01000

00E+

0045437E-11

29507E+

01F2

2Mean

000

00E+

0016

167E

-06

48729E+

02000

00E+

00244

07E-10

52060

E+02

Std

000

00E+

0063216E-07

24382E+

01000

00E+

0075

889E

-11

42230E+

01F2

3Mean

13699E

+00

89987E-01

21237E+

0122699E+

0025899E+

0035955E+

00Std

23594E-01

666

67E-02

58033E-01

41913E-01

62973E-01

12247E

-01

F24

Mean

71054E

-1426826E-02

63090E+

0119

033E

-08

96037E

-03

47263E+

01Std

27621E-14

47780E-03

22392E+

0061075E-09

97071E-04

52689E+

00F2

5Mean

66563E+

0184722E+

0211275E

+08

65780E+

0139992E+

0188242E+

04Std

10992E

+02

2113

8E+0

221091E+

0794

954E

+01

43819E+

0116

832E

+04

18 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

18989E

-1316

584E

-01

19996E

+01

17809E

-11

14744E

-06

13554E

+01

Std

20566

E-14

53720E-02

90319E

-02

19159E

-12

18930E

-07

60821E+

00F13

Mean

22871E-15

17736E

-01

1944

7E+0

213

452E

-10

13291E-05

16379E

+02

Std

26741E-15

53611E-02

62653E+

0038592E-11

55001E-06

13313E

+01

F14

Mean

18736E

-1074

259E

+01

10132E

+03

22866

E-04

83534E-02

95534E

+02

Std

37223E-11

19144E

+01

18986E

+01

14283E

-05

10592E

-02

53523E+

01F15

Mean

26814E+

0010

178E

+01

47083E+

0128083E+

0034325E+

0045859E+

01Std

73851E-01

16238E

+00

22513E-01

46148E-01

60283E-01

69914E-01

F16

Mean

47116E-33

244

54E-04

90382E

+08

81890E-24

62347E-14

27647E+

03Std

72124E

-49

59650E-05

64985E+

0767958E-24

55604

E-14

44231E+

03F17

Mean

34494E-03

11896E

-02

37816E+

0134509E-03

41885E-03

21280E+

00Std

60565E-03

65363E-03

15922E

+00

46765E-03

86153E-03

54359E-02

F18

Mean

18033E

+01

17806E

+01

86826E+

0118

319E

+01

18828E

+01

82458E+

01Std

19652E

+00

38319E+

0093

222E

-01

29296E+

0025377E+

0015

159E

+00

F19

Mean

82462E+

0427944

E+06

48046

E+08

28415E+

0560265E+

0549201E+

07Std

55732E+

0489703E+

0596

715E

+07

24572E+

0527137E+

0572

772E

+06

F20

Mean

57130E-07

81688E-03

96848E

-01

13631E-06

27143E-05

21656E+

00Std

61122E-07

53195E-03

44542E-01

25155E-06

58766

E-05

80368E-01

F21

Mean

000

00E+

0051414E+

0013

305E

+03

000

00E+

0020026E-09

13623E

+03

Std

000

00E+

0017

825E

+00

22890E+

01000

00E+

0032815E-10

609

96E+

01F2

2Mean

000

00E+

0077

848E

+00

1260

9E+0

3000

00E+

0020383E-09

12745E

+03

Std

000

00E+

0023732E+

0029100

E+01

000

00E+

0029753E-10

59708E+

01F2

3Mean

25599E+

0020499E+

0039804

E+01

47099E+

0043699E+

0073

691E+0

0Std

36878E-01

15092E

-01

69296E-01

59151E-01

56184E-01

17989E

-01

F24

Mean

40927E-13

26229E+

0015

145E

+02

18874E

-07

29476E-02

10478E

+02

Std

88061E-14

63367E-01

42830E+

0037074E-08

17697E

-03

11873E

+01

F25

Mean

42987E+

028117

8E+0

313

524E

+09

56790E+

0244982E+

0218

038E

+06

Std

43423E+

0233128E+

0278

399E

+07

54327E+

0246926E+

0221315E+

05

Complexity 19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus8

minus6

minus4

minus2

02468

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(f) F24




MAE = sum119873119895=1 10038161003816100381610038161003816119898119895 minus 11989611989510038161003816100381610038161003816119873 (22)

20 Complexity

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

ISSASSAPSO

CMFOAIFFOFOA

1

2

3

4

5

6

7

8

Mea

n Er

rors

(log)

0 4000 6000 8000 100002000Iteration

(e) F19

ISSASSAPSO

CMFOAIFFOFOA

20000 6000 8000 100004000Iteration

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




Complexity 21

Table11

Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

31365E+

0413

133E

+06

11295E

+08

10017E

+06

864

75E+

0513

449E

+07

Std

18602E

+04

52974E+

0531387E+

0748689E+

0548607E+

0528405E+

06F2

7Mean

304

00E-10

1844

6E+0

484500

E+09

10512E

+04

12359E

+04

58535E+

08Std

61535E-10

14049E

+04

10125E

+09

12485E

+04

11922E

+04

38771E+

07F2

8Mean

42105E-01

46710E+

0173

819E

+02

4114

7E+0

137814E+

0114

226E

+02

Std

12624E

+00

31490E+

0199

455E

+01

47336E+

0134110E+

0129201E+

01F2

9Mean

75177E

+03

14891E+0

529286E+

0647277E+

0531099E+

0539826E+

05Std

33119E+

0368316E+

049190

4E+0

521021E+

0522686E+

0515

511E+0

5F3

0Mean

31524E+

0231524E+

0238129E+

0231524E+

0231524E+

0232568E+

02Std

85708E-12

19710E

-07

14082E

+01

11524E

-1145680E-11

58955E+

00F31

Mean

23483E+

0223172E+

0230117E+

0223811E

+02

23858E+

0224179E+

02Std

41748E+

0072

461E+0

048903E+

00560

97E+

0050249E+

0090

228E

+00

F32

Mean

20790E+

02206

03E+

0221884E+

0221485E+

0220975E+

0220633E+

02Std

41618E+

0032456E+

0030353E+

0087909E+

0057719E+

0016

880E

+00

22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








18 Complexity


obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonmulti-mod

albenchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F12

Mean

18989E

-1316

584E

-01

19996E

+01

17809E

-11

14744E

-06

13554E

+01

Std

20566

E-14

53720E-02

90319E

-02

19159E

-12

18930E

-07

60821E+

00F13

Mean

22871E-15

17736E

-01

1944

7E+0

213

452E

-10

13291E-05

16379E

+02

Std

26741E-15

53611E-02

62653E+

0038592E-11

55001E-06

13313E

+01

F14

Mean

18736E

-1074

259E

+01

10132E

+03

22866

E-04

83534E-02

95534E

+02

Std

37223E-11

19144E

+01

18986E

+01

14283E

-05

10592E

-02

53523E+

01F15

Mean

26814E+

0010

178E

+01

47083E+

0128083E+

0034325E+

0045859E+

01Std

73851E-01

16238E

+00

22513E-01

46148E-01

60283E-01

69914E-01

F16

Mean

47116E-33

244

54E-04

90382E

+08

81890E-24

62347E-14

27647E+

03Std

72124E

-49

59650E-05

64985E+

0767958E-24

55604

E-14

44231E+

03F17

Mean

34494E-03

11896E

-02

37816E+

0134509E-03

41885E-03

21280E+

00Std

60565E-03

65363E-03

15922E

+00

46765E-03

86153E-03

54359E-02

F18

Mean

18033E

+01

17806E

+01

86826E+

0118

319E

+01

18828E

+01

82458E+

01Std

19652E

+00

38319E+

0093

222E

-01

29296E+

0025377E+

0015

159E

+00

F19

Mean

82462E+

0427944

E+06

48046

E+08

28415E+

0560265E+

0549201E+

07Std

55732E+

0489703E+

0596

715E

+07

24572E+

0527137E+

0572

772E

+06

F20

Mean

57130E-07

81688E-03

96848E

-01

13631E-06

27143E-05

21656E+

00Std

61122E-07

53195E-03

44542E-01

25155E-06

58766

E-05

80368E-01

F21

Mean

000

00E+

0051414E+

0013

305E

+03

000

00E+

0020026E-09

13623E

+03

Std

000

00E+

0017

825E

+00

22890E+

01000

00E+

0032815E-10

609

96E+

01F2

2Mean

000

00E+

0077

848E

+00

1260

9E+0

3000

00E+

0020383E-09

12745E

+03

Std

000

00E+

0023732E+

0029100

E+01

000

00E+

0029753E-10

59708E+

01F2

3Mean

25599E+

0020499E+

0039804

E+01

47099E+

0043699E+

0073

691E+0

0Std

36878E-01

15092E

-01

69296E-01

59151E-01

56184E-01

17989E

-01

F24

Mean

40927E-13

26229E+

0015

145E

+02

18874E

-07

29476E-02

10478E

+02

Std

88061E-14

63367E-01

42830E+

0037074E-08

17697E

-03

11873E

+01

F25

Mean

42987E+

028117

8E+0

313

524E

+09

56790E+

0244982E+

0218

038E

+06

Std

43423E+

0233128E+

0278

399E

+07

54327E+

0246926E+

0221315E+

05

Complexity 19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus8

minus6

minus4

minus2

02468

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(f) F24




MAE = sum119873119895=1 10038161003816100381610038161003816119898119895 minus 11989611989510038161003816100381610038161003816119873 (22)

20 Complexity

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

ISSASSAPSO

CMFOAIFFOFOA

1

2

3

4

5

6

7

8

Mea

n Er

rors

(log)

0 4000 6000 8000 100002000Iteration

(e) F19

ISSASSAPSO

CMFOAIFFOFOA

20000 6000 8000 100004000Iteration

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




Complexity 21

Table11

Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

31365E+

0413

133E

+06

11295E

+08

10017E

+06

864

75E+

0513

449E

+07

Std

18602E

+04

52974E+

0531387E+

0748689E+

0548607E+

0528405E+

06F2

7Mean

304

00E-10

1844

6E+0

484500

E+09

10512E

+04

12359E

+04

58535E+

08Std

61535E-10

14049E

+04

10125E

+09

12485E

+04

11922E

+04

38771E+

07F2

8Mean

42105E-01

46710E+

0173

819E

+02

4114

7E+0

137814E+

0114

226E

+02

Std

12624E

+00

31490E+

0199

455E

+01

47336E+

0134110E+

0129201E+

01F2

9Mean

75177E

+03

14891E+0

529286E+

0647277E+

0531099E+

0539826E+

05Std

33119E+

0368316E+

049190

4E+0

521021E+

0522686E+

0515

511E+0

5F3

0Mean

31524E+

0231524E+

0238129E+

0231524E+

0231524E+

0232568E+

02Std

85708E-12

19710E

-07

14082E

+01

11524E

-1145680E-11

58955E+

00F31

Mean

23483E+

0223172E+

0230117E+

0223811E

+02

23858E+

0224179E+

02Std

41748E+

0072

461E+0

048903E+

00560

97E+

0050249E+

0090

228E

+00

F32

Mean

20790E+

02206

03E+

0221884E+

0221485E+

0220975E+

0220633E+

02Std

41618E+

0032456E+

0030353E+

0087909E+

0057719E+

0016

880E

+00

22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus8

minus6

minus4

minus2

02468

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(f) F24




MAE = sum119873119895=1 10038161003816100381610038161003816119898119895 minus 11989611989510038161003816100381610038161003816119873 (22)

20 Complexity

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

ISSASSAPSO

CMFOAIFFOFOA

1

2

3

4

5

6

7

8

Mea

n Er

rors

(log)

0 4000 6000 8000 100002000Iteration

(e) F19

ISSASSAPSO

CMFOAIFFOFOA

20000 6000 8000 100004000Iteration

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




Complexity 21

Table11

Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

31365E+

0413

133E

+06

11295E

+08

10017E

+06

864

75E+

0513

449E

+07

Std

18602E

+04

52974E+

0531387E+

0748689E+

0548607E+

0528405E+

06F2

7Mean

304

00E-10

1844

6E+0

484500

E+09

10512E

+04

12359E

+04

58535E+

08Std

61535E-10

14049E

+04

10125E

+09

12485E

+04

11922E

+04

38771E+

07F2

8Mean

42105E-01

46710E+

0173

819E

+02

4114

7E+0

137814E+

0114

226E

+02

Std

12624E

+00

31490E+

0199

455E

+01

47336E+

0134110E+

0129201E+

01F2

9Mean

75177E

+03

14891E+0

529286E+

0647277E+

0531099E+

0539826E+

05Std

33119E+

0368316E+

049190

4E+0

521021E+

0522686E+

0515

511E+0

5F3

0Mean

31524E+

0231524E+

0238129E+

0231524E+

0231524E+

0232568E+

02Std

85708E-12

19710E

-07

14082E

+01

11524E

-1145680E-11

58955E+

00F31

Mean

23483E+

0223172E+

0230117E+

0223811E

+02

23858E+

0224179E+

02Std

41748E+

0072

461E+0

048903E+

00560

97E+

0050249E+

0090

228E

+00

F32

Mean

20790E+

02206

03E+

0221884E+

0221485E+

0220975E+

0220633E+

02Std

41618E+

0032456E+

0030353E+

0087909E+

0057719E+

0016

880E

+00

22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








20 Complexity

0 2000 4000 6000 8000 10000

02

Iteration

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus12

minus10

minus8

minus6

minus4

minus2

024

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

ISSASSAPSO

CMFOAIFFOFOA

1

2

3

4

5

6

7

8

Mea

n Er

rors

(log)

0 4000 6000 8000 100002000Iteration

(e) F19

ISSASSAPSO

CMFOAIFFOFOA

20000 6000 8000 100004000Iteration

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




Complexity 21

Table11

Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

31365E+

0413

133E

+06

11295E

+08

10017E

+06

864

75E+

0513

449E

+07

Std

18602E

+04

52974E+

0531387E+

0748689E+

0548607E+

0528405E+

06F2

7Mean

304

00E-10

1844

6E+0

484500

E+09

10512E

+04

12359E

+04

58535E+

08Std

61535E-10

14049E

+04

10125E

+09

12485E

+04

11922E

+04

38771E+

07F2

8Mean

42105E-01

46710E+

0173

819E

+02

4114

7E+0

137814E+

0114

226E

+02

Std

12624E

+00

31490E+

0199

455E

+01

47336E+

0134110E+

0129201E+

01F2

9Mean

75177E

+03

14891E+0

529286E+

0647277E+

0531099E+

0539826E+

05Std

33119E+

0368316E+

049190

4E+0

521021E+

0522686E+

0515

511E+0

5F3

0Mean

31524E+

0231524E+

0238129E+

0231524E+

0231524E+

0232568E+

02Std

85708E-12

19710E

-07

14082E

+01

11524E

-1145680E-11

58955E+

00F31

Mean

23483E+

0223172E+

0230117E+

0223811E

+02

23858E+

0224179E+

02Std

41748E+

0072

461E+0

048903E+

00560

97E+

0050249E+

0090

228E

+00

F32

Mean

20790E+

02206

03E+

0221884E+

0221485E+

0220975E+

0220633E+

02Std

41618E+

0032456E+

0030353E+

0087909E+

0057719E+

0016

880E

+00

22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 21

Table11

Statisticalresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=30

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

31365E+

0413

133E

+06

11295E

+08

10017E

+06

864

75E+

0513

449E

+07

Std

18602E

+04

52974E+

0531387E+

0748689E+

0548607E+

0528405E+

06F2

7Mean

304

00E-10

1844

6E+0

484500

E+09

10512E

+04

12359E

+04

58535E+

08Std

61535E-10

14049E

+04

10125E

+09

12485E

+04

11922E

+04

38771E+

07F2

8Mean

42105E-01

46710E+

0173

819E

+02

4114

7E+0

137814E+

0114

226E

+02

Std

12624E

+00

31490E+

0199

455E

+01

47336E+

0134110E+

0129201E+

01F2

9Mean

75177E

+03

14891E+0

529286E+

0647277E+

0531099E+

0539826E+

05Std

33119E+

0368316E+

049190

4E+0

521021E+

0522686E+

0515

511E+0

5F3

0Mean

31524E+

0231524E+

0238129E+

0231524E+

0231524E+

0232568E+

02Std

85708E-12

19710E

-07

14082E

+01

11524E

-1145680E-11

58955E+

00F31

Mean

23483E+

0223172E+

0230117E+

0223811E

+02

23858E+

0224179E+

02Std

41748E+

0072

461E+0

048903E+

00560

97E+

0050249E+

0090

228E

+00

F32

Mean

20790E+

02206

03E+

0221884E+

0221485E+

0220975E+

0220633E+

02Std

41618E+

0032456E+

0030353E+

0087909E+

0057719E+

0016

880E

+00

22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








22 Complexity

Table12Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=50

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

26771E+

05264

58E+

064113

9E+0

828678E+

0621673E+

0645383E+

07Std

10247E

+05

11716E

+06

85387E+

07804

25E+

0545535E+

0511975E

+07

F27

Mean

63168E+

0310

319E

+04

24705E+

1011223E

+04

11413E

+04

17003E

+09

Std

10293E

+04

11213E

+04

15153E

+09

97927E

+03

10930E

+04

21837E+

08F2

8Mean

64225E+

0189987E+

0122396E+

0310

089E

+02

85303E+

0122261E+

02Std

50934E+

0111705E

+01

300

13E+

0240299E+

0141667E+

0157160

E+01

F29

Mean

33693E+

0452699E+

052115

8E+0

747974E+

0560921E+

05240

66E+

06Std

18553E

+04

31305E+

0535783E+

0623522E+

0543922E+

0587454E+

05F3

0Mean

34400

E+02

34400

E+02

53872E+

0234400

E+02

34400

E+02

38544

E+02

Std

26860

E-12

65963E-07

38691E+

0126516E-12

33520E-12

10309E

+01

F31

Mean

26752E+

0226538E+

02460

79E+

0226825E+

0226586E+

0231213E+

02Std

50026E+

0070

454E

+00

68300

E+00

444

49E+

0039383E+

0036751E+

00F32

Mean

21061E+

0221388E+

0227124E+

0221691E+

0221542E+

0222054E+

02Std

55300

E+00

59914E+

0011291E+0

162484E+

0052166

E+00

52494E+

00

Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 23

Table13Statistic

alresults

obtained

byISSA

SSA

PSO

CMFO

AIFF

Oand

FOAthroug

h10

independ

entrun

sonCE

C2014

benchm

arkfunctio

nswith

n=100

Fun

ISSA

SSA

PSO

CMFO

AIFFO

FOA

F26

Mean

10395E

+06

49662E+

0719

596E

+09

10516E

+07

15208E

+07

28282E+

08Std

36972E+

0556939E+

0621605E+

0835784E+

0650169E+

0644860

E+07

F27

Mean

14837E

+04

58871E+

0510

093E

+11

264

10E+

0437388E+

0471189E

+09

Std

15318E

+04

10255E

+05

1009

9E+10

28473E+

0441209E+

0432998E+

08F2

8Mean

13263E

+02

24979E+

0211962E

+04

22607E+

0223713E+

0284991E+

02Std

43021E+

0170

814E

+01

14132E

+03

45595E+

01246

42E+

0110

057E

+02

F29

Mean

16986E

+05

42648E+

0617618E

+08

31738E+

0628874E+

0618

248E

+07

Std

62432E+

0411220E

+06

27101E+

0742353E+

0513

296E

+06

62005E+

06F3

0Mean

34823E+

0234875E+

0214

344E

+03

34910E+

0234901E+

0257172E+

02Std

62960

E-11

43294E-01

15590E

+02

91883E

-01

9300

0E-01

28371E+

01F31

Mean

34722E+

0235878E+

0292

092E

+02

35108E+

0234814E+

0250149E+

02Std

10958E

+01

37623E+

0024898E+

0110

734E

+01

10706E

+01

10838E

+01

F32

Mean

24544E+

0225216E+

0252841E+

0226036E+

0226337E+

0229287E+

02Std

15945E

+01

13749E

+01

24285E+

0112

685E

+01

15913E

+01

11210E

+01

24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








24 Complexity

Table14R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=30

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F115

723E

-03

+54503E-11

+21431E-06

+12

930E

-04

+31274E-08

+F2

59105E-01

-59726E-07

+16

785E

-01

-16

785E

-01

-17438E

-06

+F3

18034E

-01

-56302E-11

+66374E-01

-44113E-06

+18

978E

-10

+F4

39391E-03

+80559E-08

+80897E-04

+14

754E

-03

+10

215E

-06

+F5

75194E

-07

+35327E-08

+22706

E-01

-42611E

-02

+15

497E

-06

+F6

22263E-02

+18

702E

-08

+27096E-03

+33147E-03

+73

030E

-06

+F7

39878E-03

+21023E-10

+26126E-07

+11038E

-04

+58740

E-11

+F8

37778E-07

+12

311E-13

+22556E-07

+88317E-11

+16

744E

-07

+F9

25658E-06

+39583E-05

+20251E-08

+27652E-08

+68325E-02

-F10

40986E-03

+15

715E

-10

+62372E-07

+10

581E-05

+75

777E

-10

+F11

16385E

-01

-55101E -0 4

+45288E-03

+62300

E-02

-14

019E

-03

+F12

25148E-04

+17

221E-15

+88689E-10

+82337E-10

+840

91E-04

+F13

62223E-04

+82292E-11

+17434E

-04

+68585E-02

-56801E-08

+F14

16770E

-05

+35961E-16

+60168E-13

+240

86E-12

+10

063E

-06

+F15

91211E-03

+42859E-14

+79

924E

-01

-96

191E-01

-12

100E

-14

+F16

49253E-05

+24808E-06

+81048E-03

+49672E-03

+35094E-08

+F17

52276E-01

-11956E

-10

+16

338E

-01

-87704

E-01

-12

329E

-18

+F18

59605E-02

-73103E

-10

+75245E

-01

-83423E-01

-14

080E

-08

+F19

40911E

-03

+20151E-06

+45217E-03

+93

504E

-03

+69674E-08

+F2

089857E-02

-10

735E

-03

+29254E-01

-76

513E

-01

-12

493E

-05

+F2

180383E-04

+13

653E

-14

+=

49618E-05

+51686E-11

+F2

296

507E

-05

+51321E-12

+=

19712E

-04

+25703E-10

+F2

310

362E

-03

+37568E-14

+16

044E

-02

+19

660E

-04

+74

376E

-08

+F24

82001E-07

+16

038E

-14

+48491E-04

+16

951E-10

+18

472E

-09

+F2

514

795E

-03

+12

097E

-06

+19

763E

-01

-43929E-02

-82364

E-08

+F2

629892E-05

+12

127E

-06

+13

438E

-04

+38826E-04

+11510E

-07

+F2

724771E-03

+77

797E

-10

+25931E-02

+95

563E

-03

-38874E-12

+F2

811525E

-03

+21817E-09

+23075E-02

+76

652E

-03

+10

245E

-07

+F2

999

588E

-05

+340

16E-06

+61373E-05

+21918E-03

+23509E-05

+F3

090

190E

-02

-12

454E

-07

+71059E

-05

+16

503E

-06

+33480E-04

+F31

25587E-01

-98

592E

-11

+22578E-01

-13

543E

-01

-79

203E

-02

-F32

31415E-01

-55580E-06

+71757E

-02

-20510E-01

-34 882E-01

-+-

293

320

2010

239

293

Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 25

Table15R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=50

(120572=005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F120377E-06

+51683E-10

+44186E-07

+55764

E-07

+3111

3E-12

+F2

60105E-02

-42014E-09

+17

277E

-01

-244

22E-02

+91

132E

-11

+F3

17250E

-06

+13

907E

-16

+98

022E

-02

-10

738E

-05

+18

638E

-11

+F4

93262E

-06

+14

595E

-09

+50379E-04

+16

848E

-03

+42472E-08

+F5

57607E-10

+92

006E

-10

+23798E-02

+81251E-01

-10

642E

-06

+F6

13107E

-05

+13

362E

-11

+56932E-05

+53828E-03

+10

919E

-07

+F7

57850E-07

+18

163E

-10

+67859E-05

+35922E-05

+13

335E

-10

+F8

75219E

-07

+22270E-14

+33394E-02

+11235E

-10

+460

85E-11

+F9

39321E-08

+33513E-01

-26869E-10

+37640

E-09

+17

549E

-01

-F10

32994E-05

+55796E-11

+30272E-08

+72

141E-07

+97

090E

-13

+F11

24950E-02

+18

0 32 E

-05

+39453E-02

+78

893E

-02

-30964

E-04

+F12

22790E-07

+25730E-19

+82015E-10

+33180E-10

+17

587E

-04

+F13

860

55E-06

+26273E-12

+23293E-03

+99

266E

-05

+98

054E

-12

+F14

500

86E-07

+62475E-15

+70

383E

-12

+506

88E-15

+4114

6E-08

+F15

17136E

-01

-13

728E

-13

+94

200E

-01

-59423E-01

-33136E-15

+F16

16083E

-06

+13

679E

-06

+16

464E

-02

+15

895E

-01

-13

483E

-09

+F17

290

46E-01

-39668E-14

+68720E-01

-62215E-01

-29446

E-18

+F18

66743E-01

-11386E

-10

+43569E-01

-20341E-01

-45540

E-11

+F19

36286E-03

+92

080E

-07

+27891E-03

+10

982E

-02

+28723E-09

+F2

016

305E

-02

+68713E-06

+80834E-01

-31893E-01

-19

845E

-04

+F2

121300

E-06

+17

078E

-12

+=

49113E-08

+32451E-13

+F2

220294E-05

+31368E-13

+=

31089E-06

+23903E-11

+F2

312

107E

-04

+60776E-15

+77

875E

-06

+70

901E-05

+17

113E-09

+F24

25888E-08

+14

322E

-14

+404

14E-06

+17

080E

-10

+40917E-10

+F2

531276E-06

+39758E-08

+98

360E

-01

-49413E-01

-45773E-08

+F2

613

214E

-04

+99

102E

-08

+41042E-06

+17402E

-07

+79

545E

-07

+F2

716

043E

-01

-19

505E

-12

+34341E-01

-39881E-01

-14

412E

-09

+F2

812

130E

-01

-58692E-09

+13

887E

-01

-42578E-01

-264

64E-04

+F2

984658E-04

+16

521E-08

+200

73E-04

+27477E-03

+11585E

-05

+F3

094

213E

-04

+67411E

-08

+53101E-04

+546

40E-04

+47099E-07

+F31

46697E-01

-42833E-14

+79

775E

-01

-40133E-01

-11364E

-10

+F32

27813E-01

-24129E-07

+61643E-02

-83535E-02

-6355 2E-03

++-

248

311

1911

2012

311

26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








26 Complexity

Table16R

esultsof


ther

sixalgorithm

sfor

each

benchm

arkfunctio

nwith

10independ

entrun

sand

n=100(120572=

005)

Fun

SSAvs

ISSA

PSOvs

ISSA

CMFO

Avs

ISSA

IFFO

vsISSA

FOAvs

ISSA

p-value

win

p-value

win

p-value

win

p-value

win

p-value

win

F110

378E

-07

+78

176E

-14

+11254E

-06

+73355E

-08

+29716E-13

+F2

42836E-05

+82177E-12

+49949E-02

+26382E-03

+72

835E

-09

+F3

49896E-08

+78

338E

-35

+35536E-02

+13

895E

-08

+11550E

-12

+F4

23331E-06

+19

205E

-10

+21416E-04

+52932E-06

+19

678E

-08

+F5

1260

0E-10

+12

963E

-10

+17

828E

-03

+10

868E

-05

+50309E-09

+F6

98970E

-02

-53354E-10

+47015E-06

+16

844E

-05

+61888E-10

+F7

22243E-08

+41865E-13

+87771E-07

+13

044E

-09

+62464

E-11

+F8

22556E-10

+53495E-18

+74

894E

-05

+79

906E

-11

+31999E-09

+F9

49870E-10

+29549E-01

-10

030E

-10

+12

423E

-12

+34344

E-01

-F10

46494E-07

+304

86E-15

+19

111E-07

+15

614E

-09

+94

423E

-13

+F11

18990E

-02

+22724E-06

+19

056E

-02

+23614E-02

+29444

E-04

+F12

43699E-06

+12

600E

-22

+32460

E-10

+14

367E

-09

+600

50E-05

+F13

24541E-06

+59980E-15

+15

823E

-06

+31849E-05

+24334E-11

+F14

63858E-07

+45807E-17

+22981E-12

+12

864E

-09

+86555E-13

+F15

17146E

-07

+22593E-17

+70

366E

-01

-99

469E

-02

-51238E-16

+F16

39761E-07

+8113

5E-12

+41494E-03

+62574E-03

+79

491E-02

+F17

10397E

-02

+67363E-14

+99

961E-01

-83209E-01

-79

210E

-16

+F18

86191E-01

-17

179E

-15

+79

452E

-01

-43052E-01

-17

688E

-13

+F19

590

40E-06

+75

177E

-08

+33686E-03

+46936E-05

+47998E-09

+F2

090

127E

-04

+72

610E

-05

+37345E-01

-18

813E

-01

-13

324E

-05

+F2

176

534E

-06

+21239E-17

+=

12438E

-08

+11562E

-13

+F2

226358E-06

+29856E-16

+=

44818E-09

+17

365E

-13

+F2

334130E-03

+466

44E-17

+28070E-06

+78

756E

-06

+590

44E-11

+F24

36618E-07

+18

577E

-15

+60981E-08

+16

105E

-12

+47301E-10

+F2

564937E-12

+11756E

-12

+51565E-01

-92

513E

-01

-69216E-10

+F2

656291E-10

+36946

E-10

+13740E

-05

+12

241E-05

+94

839E

-09

+F2

752495E-08

+15

615E

-10

+18

874E

-01

-12

714E

-01

-15

781E-13

+F2

8260

66E-03

+86946

E-10

+75

687E

-04

+43007E-05

+36968E-09

+F2

984514E-07

+71725E

-09

+266

46E-09

+87814E-05

+71732E

-06

+F3

044636E-03

+38618E-09

+15

805E

-02

+27858E-02

+12

999E

-09

+F31

13273E

-02

+18

782E

-13

+52897E-01

-78

331E-01

-604

88E-11

+F32

37345E-01

-86751E-10

+93

177E

-02

-61812E-03

+20169E-06

++-

293

311

228

257

311

Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 27

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 80002000 10000Iteration

minus14

minus12

minus10

minus8

minus6

minus4

minus2

02

Mea

n Er

rors

(log)

(a) F12

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(b) F13

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus10

minus8

minus6

minus4

minus2

0

2

4

Mea

n Er

rors

(log)

(c) F14

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus40

minus30

minus20

minus10

0

10

Mea

n Er

rors

(log)

(d) F16

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(e) F19

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

minus15

minus10

minus5

0

5

Mea

n Er

rors

(log)

(f) F24




5 Conclusions


28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








28 Complexity

0 2000 4000 6000 8000 10000Iteration

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

(a) F26

0

5

10

15

Mea

n Er

rors

(log)

ISSASSAPSO

CMFOAIFFOFOA

minus5

minus10

2000 4000 6000 8000 100000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

minus1

0

1

2

3

4

5

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

100004000 6000 800020000Iteration

3

4

5

6

7

8

9

Mea

n Er

rors

(log)

(d) F29






Data Availability


Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 29

ISSASSAPSO

CMFOAIFFOFOA

0 4000 6000 8000 100002000Iteration

5

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2

4

6

8

10

12

Mea

n Er

rors

(log)

20000 6000 8000 100004000Iteration

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

152

253

354

455

55

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

4

5

6

7

8

9

10

Mea

n Er

rors

(log)

2000 4000 60000 100008000Iteration

(d) F29


ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

6

7

8

9

10

11

Mea

n Er

rors

(log)

(a) F26

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

456789

101112

Mea

n Er

rors

(log)

(b) F27

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

225

335

445

555

6

Mea

n Er

rors

(log)

(c) F28

ISSASSAPSO

CMFOAIFFOFOA

2000 4000 6000 8000 100000Iteration

5

6

7

8

9

10

Mea

n Er

rors

(log)


30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








30 Complexity


100

200

300

400

500

600

700

Algorithm

Num

ber o

f fou

nd o

ptim

ums

653

502

586 580

10287




Acknowledgments


References





























Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 31



























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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