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Research ArticleA Hybrid Algorithm Framework with Learning andComplementary Fusion Features for WhaleOptimization Algorithm

Wangyu Tong 12

1School of Computer Science and Technology Wuhan University of Technology Wuhan 430070 Hubei China2School of Computer Science Hubei University of Technology Wuhan 430068 Hubei China

Correspondence should be addressed to Wangyu Tong ruyu319hbuteducn

Received 29 October 2019 Accepted 16 January 2020 Published 18 February 2020

Academic Editor Cristian Mateos

Copyright copy 2020 Wangyu Tong (is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

It has been observed that the structure of whale optimization algorithm (WOA) is good at exploiting capability but it easily suffersfrom premature convergence Hybrid metaheuristics are of the most interesting recent trends for improving the performance ofWOA In this paper a hybrid algorithm framework with learning and complementary fusion features forWOA is designed calledhWOAlf First WOA is integrated with complementary feature operators to enhance exploration capability Second the proposedalgorithm framework adopts a learning parameter lp according to adaptive adjustment operator to replace the random parameterp To further verify the efficiency of the hWOAlf the DErand1 operator of differential evolution (DE) and the mutate operator ofbacktracking search optimization algorithm (BSA) are embedded into WOA respectively to form two new algorithms calledWOA-DE and WOA-BSA under the proposed framework Twenty-three benchmark functions and six engineering designproblems are employed to test the performance of WOA-DE and WOA-BSA Experimental results show that WOA-DE andWOA-BSA are competitive compared with some state-of-the-art algorithms

1 Introduction

Nowadays most function optimization problems and en-gineering design problems are turning out to be compli-cated which prevents traditional methods from providingaccurate or even satisfactory solutions However theemergence of the metaheuristic optimization algorithmsbreaks this situation and attracts the attention of manyresearchers because of their flexibility derivation-freemechanisms and local optima avoidance Generally met-aheuristic algorithms can be divided into two main cate-gories (e first category is evolutionary algorithms (EAs)that are a new family of optimization algorithms inspiredfrom the evolutionary procedure or intelligent behaviors oforganisms such as genetic algorithm (GA) [1] differentialevolution (DE) algorithm [2] biogeography-based algo-rithm (BBO) [3] backtracking search optimization algo-rithm (BSA) [4] grey prediction evolution algorithm

(GPEA) [5] and so on (e second category is swarm in-telligence algorithms (SIs) which mimic the social behaviorsof groups of animals (e most popular SI is particle swarmoptimization (PSO) [6] Other popular algorithms are ar-tificial bee colony (ABC) [7] cuckoo search (CS) [8] greywolf optimizer (GWO) [9] and so on

Whale optimization algorithm proposed byMirjaliliet alin 2016 [10] is one of the most recent SIs Al-though it has strong local search capability due to its bubble-net attacking method it is easy to get into local optimizationand is weak in exploration Hybrid metaheuristic algorithmsas one of the most efficient methods for enhancing the per-formance of algorithm have been testified to be more efficientthan the method that uses solely an algorithm Since WOAwas proposed it has been widely promoted in various fields bycombining with other algorithms One of themost widely usedstrategies to construct hybrid WOA is by simply hybridizingtwo algorithms according to a certain way For example

HindawiScientific ProgrammingVolume 2020 Article ID 5684939 25 pageshttpsdoiorg10115520205684939

Bentouati et al [11] designed a hybrid whale optimizationalgorithm and pattern search technique called WOA-PS andtested it in optimal power flow problem Xiong et al [12] putforward a hybrid algorithm called DEWOA which combinedthe exploration of DE with the exploitation of WOA forextracting the accurate parameters of PV models Luo et al[13] proposed a MDE-WOA algorithm which combinesmodified DEGL algorithm and WOA to solve global opti-mization problems Mafarja et al [14] introduced a hybridalgorithm with whale optimization algorithm and simulatedannealing in which two hybridization models are consideredfor solving the feature selection problem

Another strategy to construct hybridWOA is by combiningoperators with complementary fusion features into WOA toform a superior algorithm Trivedi et al [15] hybridized particleswarm optimization with whale optimization algorithm calledPSO-WOA to handle global numerical function optimization(e main idea is to combine best characteristic of both algo-rithms to form a novel mutation operator Korashy et al [16]hybridized whale optimization algorithm and grey wolf opti-mizer algorithm called WOA-GWO for optimal coordinationof direction overcurrent relays Xu et al [17] proposed amemetic WOA (MWOA) through incorporating a chaoticlocal search into WOA to enhance its exploitation abilityKaveh et al [18] attempted to enhance the original formulationof the WOA by hybridizing it with some concepts of thecolliding bodies optimization (CBO) (e new method knownas WOA-CBO algorithm is applied to building site layoutplanning problem Khalilpourazari et al [19] introduced anovel hybrid algorithm termed as sine-cosineWOA (SCWOA)for parameter optimization problem of multipass millingprocess to decrease total production time Singh and Hachimi[20] proposed a new hybrid whale optimization algorithm withmean grey wolf optimizer (WOA-MGWO) for global opti-mization Abdel-Basset et al [21] proposed a new algorithmwhich combines theWOAwith a local search strategy to defeatthe permutation flow shop scheduling problem Laskar et al[22] proposed a new population-based hybrid metaheuristicalgorithm called whale-PSO algorithm (HWPSO) to managecomplex optimization problems

In this paper a hybrid algorithm framework namedhWOAlf is proposed by combining learning and comple-mentary fusion features into WOA Specifically the pro-posed algorithm framework adopts a learning parameter lp

according to adaptive adjustment operator instead of therandom parameter p namely the selection of searchequation depends on the process of lp learning from thesubpopulation Additionally to improve the exploration ofWOA operators with complementary fusion features areemployed to balance exploration and exploitation capacity(is paper adopts the DErand1 operator of differentialevolution (DE) and the mutate operator of backtrackingsearch optimization algorithm (BSA) to form two new al-gorithms respectively called WOA-DE and WOA-BSA(en twenty-three benchmark functions and six engi-neering design problems are employed to test the perfor-mance of WOA-DE and WOA-BSA Experimental resultsshow that WOA-DE and WOA-BSA are competitivecompared with some state-of-the-art algorithms

(e main contributions of this paper can be summarizedas follows

(i) Proposing a hybrid algorithm framework for WOAnamed hWOAlf First designing a learning pa-rameter lp according to adaptive adjustment oper-ator Second combining complementary fusionfeature operators into WOA

(ii) Forming two new algorithms WOA-DE and WOA-BSA by complementary fusion features in which theDErand1 operator of DE and the mutation oper-ator of BSA are incorporated to WOA respectively

(e rest of the paper is organized as follows Section 2gives a brief review of WOA DE and BSA (e proposedhybrid algorithm framework is presented in Section 3 InSection 4 the experimental results and analysis are givenFinally the conclusions are shown in Section 5

2 Preliminary

21 Whale Optimization Algorithm (WOA) Whale opti-mization algorithm is a new swarm-based metaheuristicoptimization algorithm proposed by Mirjalili and Lewis in2016 [10] which mimics the intelligent foraging behavior ofhumpback whales (e algorithm is inspired by bubble-nethunting strategy WOA includes three operators to simulatethe search for prey encircling prey and bubble-net foragingbehavior of humpback whales Among them the encirclingprey and spiral bubble-net attacking method is the exploi-tation phase while searching for prey is the explorationphase(emathematical model of the two phase is presentedbelow

211 Stage 1 Exploitation Phase (Encircling PreyBubble-NetAttacking Method) During the exploitation phase hump-back whales update their position according to two mech-anisms shrinking encircling mechanism (encircling prey)and spiral updating position (spiral bubble-net attackingmethod) Shrinking encircling mechanism is represented bythe following equations

Xrarr

(t + 1) Xlowastrarr(t) minus A

rarrmiddot Disrarr

Disrarr

Crarr

middot Xlowastrarr(t) minus X

rarr(t)

111386811138681113868111386811138681113868

111386811138681113868111386811138681113868

(1)

where t represents the current iteration Xlowastrarr

(t) is the positionvector of the best solution obtained so far and X

rarr(t) is the

position vector Ararr

and Crarr

are coefficient vectors and thecalculation methods are as follows

Ararr

2 ararr

middot rrarr

minus ararr

Crarr

2 middot rrarr

(2)

where ararr is linearly decreased from 2 to 0 over the course of

iterations (in both exploration and exploitation phases) andrrarr is a random vector in [0 1]

(en the spiral updating position is mathematicallyexpressed by the following equation

2 Scientific Programming

Xrarr

(t + 1) Disprimerarr

middot ebl

middot cos(2πl) + Xlowastrarr(t)

Disprimerarr

Xlowastrarr(t) minus X

rarr(t)

111386811138681113868111386811138681113868

111386811138681113868111386811138681113868

(3)

where Disprimerarr

represents the distance of the ith humpbackwhale to best solution obtained so far b is a constant de-fining the logarithmic spiral shape l is a random number in[minus 1 1] and middot is the element-by-element multiplication

When whales capture their prey it is worth noting thatthe two mechanisms mentioned above that is shrinkingencircling mechanism and spiral updating position areexecuted simultaneously In order to imitate this behavior aprobability of 50 is assumed to choose between them (emathematical model is as follows

Xrarr


rarrmiddot Disrarr

if plt 05

Disprimerarr

middot ebl middot cos(2πl) + Xlowastrarr(t) if pge 05

⎧⎪⎨

⎪⎩

(4)

where p is a random number in [0 1]

212 Stage 2 Exploration Phase (Searching for Prey) In thisstage in order to enhance the exploration capability inWOA we update the position of whale according to arandomly chosen whale instead of the best whale found sofar (erefore a coefficient vector A with the random valuesgreater than 1 or less than -1 is used to force the whale tomove far away from the best known whale(emodel can bemathematically expressed as follows

Xrarr

(t + 1) Xrarr

rand(t) minus Ararr

middot Disrarr

Disrarr

Crarr

middot Xrarr

rand(t) minus Xrarr

(t)11138681113868111386811138681113868

11138681113868111386811138681113868(5)

where Xrarr

rand is a random position vector selected from thecurrent population

22 Differential Evolution (DE) Differential evolution (DE)algorithm is a population-based approach firstly proposed byStorn and Price in 1997 [2] and proven to be one of the mostpromising global search methods After generating initialpopulation the population is updated by looping mutationcrossover and selection operations (is paper adopts aclassic mutation operation called DErand1 In brief DEprocedures can be summarized in the following four steps

221 Step 1 Initialization At the beginning an initialpopulation is generated by means of random number dis-tribution We can initialize the jth dimension of the ithindividual according to

xij Lj + rand(0 1)lowast Uj minus Lj1113872 1113873 i 1 2 N

j 1 2 D(6)

where N is the population size D is the dimensional ofindividual rand(0 1) denotes an uniformly distributed

random number in [0 1] range and Uj and Lj denote theupper and lower bounds of the jth dimension respectively

222 Step 2 Mutation After the initialization DE enters itsmain loop Each target individual xi in the population is usedto generate a mutant individual vi via mutation operators(DErand)1 where the generated vi can be represented as

vi xr1 + Flowast xr2 minus xr3( 1113857 r1ne r2ne r3ne i (7)

where r1 r2 and r3 are randomly selected from the currentpopulation and F is the mutation control parameter to scalethe difference vector Similarly we give another five fre-quently used mutation operators as follows

(1) ldquoDErand2rdquo

vi xr1 + Flowast xr2 minus xr3( 1113857 + Flowast xr4 minus xr5( 1113857

r1ne r2ne r3ne r4ne r5ne i(8)

(2) ldquoDEbest1rdquo

vi xbest + Flowast xr1 minus xr2( 1113857 r1ne r2ne i (9)


vi xbest + Flowast xr1 minus xr2( 1113857 + Flowast xr3 minus xr4( 1113857

r1ne r2ne r3ne r4ne i(10)

(4) ldquoDErand-to-best1rdquo

vi xi + Flowast xbest minus xi( 1113857 + Flowast xr1 minus xr2( 1113857 r1ne r2ne i

(11)

(5) ldquoDEcurrent-to-best1rdquo

vi xi + Flowast xr1 minus xi( 1113857 + Flowast xr2 minus xr3( 1113857 r1ne r2ne r3ne i

(12)

where xbest is the individual that has the best fitness functionvalue

223 Step 3 Crossover (en a binomial crossover oper-ation is applied to the target individual xi and mutant in-dividual vi to generate a trial individual ui as follows

uij vij if rand(0 1)leCR or j jrand

xij otherwise⎧⎨

⎩ (13)

where jrand is a randomly chosen integer in [1 D] andCR isin [0 1] is called the crossover control parameter

Scientific Programming 3

224 Step 4 Selection Finally the greedy selection oper-ation is used to select the better one from the target indi-vidual xi and crossover individual ui into the nextgeneration (is operation is mainly based on the com-parison of the fitness value and is performed as shown below

xt+1i

ui if fitness ui( 1113857lt fitness xi( 1113857

xi otherwise1113896 (14)

where fitness is the fitness function

23 Backtracking Search Optimization Algorithm (BSA)Backtracking search optimization algorithm (BSA) is apopulation-based metaheuristic algorithm proposed byPinar Civicioglu in 2013 [4] Similar to most metaheuristicalgorithms the algorithm achieves the purpose of optimi-zation through the mutation crossover and selection ofpopulation What makes the BSA unique is its ability toremember historical populations which enables it to benefitfrom previous generation populations by mining historicalinformation (ere are five steps contained in original BSAcalled initialization selection-I mutation crossover andselection-II (e five steps of BSA are simply introduced asfollows

231 Step 1 Initialization BSA initializes the population Pand the historical population oldP with the following for-mula respectively

Pij sim U lowj upj1113872 1113873 (15)

oldPij sim U lowj upj1113872 1113873 (16)

where i 1 2 N j 1 2 D N and D representthe population size and the population dimension respec-tively U is the uniform distribution lowj and upj are thelower and upper boundaries of variables

232 Step 2 Selection-I Firstly the historical populationoldP is updated according to equation (17) (en the lo-cations of individuals in oldP are randomly changed asshown in equation (18)

if alt b(a b sim U(0 1)) then oldP P (17)

oldP ≔ permuting(oldP) (18)

where permuting(middot) performs a random permutation of theintegers from 1 to N

233 Step 3 Mutation (e mutation operator of BSA cangenerate the initial trial population which takes advantage ofthe historical information and the current information (emutation operation can be expressed as follows

Mij Pij + Flowast oldPij minus Pij1113872 1113873 (19)

where F is a control parameter and F 3 middot randn(rand n sim U(0 1)) (is operation gives the algorithmpowerful global search ability

234 Step 4 Crossover BSArsquos crossover process has twosteps (e first step generates a binary integer-valued matrixmap of size NlowastD (e second step is to determine thepositions of crossover individual elements in population Paccording to the generated matrix map and then exchangesuch individual elements in P with the corresponding po-sition elements in population M to obtain the final trialpopulation T (e crossover operation can be expressed asfollows

Vij Pij if mapij 1

Mij otherwise⎧⎨

⎩ (20)

After crossover operation some individuals of the trialpopulation T obtained at the end of BSArsquos crossover processmay overflow the allowed search space limits At this pointthe individuals beyond the boundary control will beregenerated according to equation (15)

235 Step 5 Selection-II (e greedy selection mechanism isemployed to maintain the most promising trial individualsinto the next generation Compare the fitness values of thetarget individuals and trial individuals if the fitness value oftrial individual is less than that of the target individual thenthe trial individual is accepted for the next generationotherwise the target individual is retained in the population(e selection operator is defined as

Pi Ti if fitness Ti( 1113857lt fitness Pi( 1113857

Pi otherwise1113896 (21)

where f(middot) is the objective function value of an individual

3 Proposed Hybrid Frameworkfor WOA hWOAlf

WOA which imitates the hunting behavior of humpbackwhales is one of the most recent SIs It includes three stagessearching for prey encircling prey and hunting prey (ebasic WOA has the disadvantage of weak exploration ca-pability (is paper proposes a hybrid algorithm framework(hWOAlf) to improve the global search capability of WOAOperators with complementary fusion features DErand1operator of DE and mutation operator BSA are embeddedinto WOA to form two new algorithms which includeWOA-DE and WOA-BSA respectively In addition alearning parameter lp adjusted by adaptive adjustmentoperator will replace the random parameter p of the originalWOA which can continuously balance exploration andexploitation through learning mechanism (e proposedframework consists of two main parts hybrid frameworkand learning parameter mechanism explained below


31 Proposed Hybrid Framework for WOA (e mutationoperators of equations (1) and (3) in WOA have strongexploitation capability but only the mutation operator ofequation (5) has a strong exploration capability (is makesWOA more capable of exploitation and less capable ofexploration In order to balance the exploration and ex-ploitation capabilities of WOA a generic framework flow-chart was developed and is shown in Figure 1 With thisframework BSA mutation operator (equation (7)) and DErand1 mutation operator (equation (19)) are used to im-prove the exploration capability of WOA

In order to have a clearer understanding of our proposedhWOAlf we summarize the four mutation operators used inthis framework as follows

(i) Operator 1 if plt r and|A|lt 1

Trarr

i Xlowastrarr

i minus Ararr

middot Crarr

middot Xlowastrarr

i minus Xrarr

i

111386811138681113868111386811138681113868

111386811138681113868111386811138681113868 (22)

(ii) Operator 2 if plt r and|A|ge 1

Trarr

i Xrarr

other minus Ararr

middot Crarr

middot Xrarr

other minus Xrarr

i

11138681113868111386811138681113868

11138681113868111386811138681113868 (23)

(iii) Operator 3 if pge r and|A|lt 1

Trarr

i Xlowastrarr

i minus Xrarr

i

111386811138681113868111386811138681113868

111386811138681113868111386811138681113868 middot ebl

middot cos(2πl) + Xlowastrarr

i (24)

(iv) Operator 4 if pge r and|A|ge 1

Trarr

i Xrarr

i + F1 middot oldXrarr

i minus Xrarr

i1113874 1113875or Trarr

i Xrarr

r1 + F2 middot Xrarr

r2 minus Xrarr

r31113874 1113875

(25)

where Ti is a test individual

32 Proposed Learning Parameter Mechanism In basicWOA p is a random number In this paper we design lp as afunction with learning strategy When the population is fullyevolved in the current iteration the following four numbersare recorded the number of individuals mutated by operator1 and operator 2 is n1 and the number of individualsmutated by these two operators that can successfully enterthe next generation is s1 similarly the number of indi-viduals mutated by operator 3 and operator 4 is n2 (enumber of individuals better than their parents after themutation of these two operators is s2 (e learning pa-rameter lp is designed as follows

lp 1 +(s1n1)

2 +(s1n1) +(s2n2) (26)

(e initial lp value is 05 and needs to be updated aftereach iteration In this paper lp is designed to control theexecution frequency of the search operators in equations

(22) and (23) and those in equations (24) and (25) In theprocess of algorithm evolution when lp is less than arandom number operator 1 and operator 2 are used tomutate otherwise operator 3 and operator 4 are selected tomutate (is continuous adaptive process will enable thealgorithm to choose themost suitable learning strategy in theevolution process

33 gte Pseudocode of hWOAlf For clarity the pseudocodeof hWOAlf is given in Algorithm 1

4 Experimental Results and Discussion

In this section to verify the validity of the proposed hybridalgorithm framework WOA-DE and WOA-BSA are dem-onstrated on twenty-three benchmark functions and sixengineering design problems Experimental results andcomparisons with other representative algorithms show thatthe proposed hWOAlf is effective and efficient All experi-ments are implemented in the software of MATLAB 2012a(e computer system running the program is Intel(R)Core(TM) i5-3230M CPU 260GHz and 4GB RAM

41Twenty-gtreeBenchmarkFunctions In order to evaluatethe performance of the proposed algorithm frameworkWOA-DE andWOA-BSA are compared with some EAs andSIs ie DE BBO BSA PSO HS GSA and WOA ontwenty-three benchmark functions listed in Tables 1ndash3(ese functions can be divided into three groups unimodalmultimodal and fixed-dimension multimodal (e uni-modal functions F1 minus F7 have only one global optimum sothey are used to investigate the exploitation capability whilethe remaining functions F8 minus F23 with more than two localoptima are utilized to evaluate the exploration ability Inaddition in the tables D denotes the number of functiondimensions Range is the boundary of the function searchspace and fmin is the theoretical optimum In order to havea fair comparison each experiment is independently run 30times for all the compared algorithms and the results ofevery run are recorded (e maximal iteration number(Maxiter) is used as the termination condition for all al-gorithms which is set to 1000 and population size is set to30

42 Comparison Results of Twenty-gtree BenchmarkFunctions Tables 4 and 5 tabulate the Best Mean and Stdresults of twenty-three benchmark functions for WOA-DEWOA-BSA and other seven compared algorithms (eresults of WOA-DE and WOA-BSA are shown in bold

421 Exploitation Capability Functions F1ndashF7 can be usedto evaluate the exploitation capability of the proposed al-gorithms since each of them has only one global optimalvalue (e results from Table 4 show that with respect to thebest WOA-DE and WOA-BSA rank first and second re-spectively for functions F1ndashF7 among all compared algo-rithms which means that the results obtained by WOA-DE


and WOA-BSA are closer to the theoretical optimal solu-tions on these functions andWOA-DE andWOA-BSA havevery strong exploitation capacity compared with other al-gorithms In terms of Mean and Std index WOA-BSAoutperforms the other algorithms while WOA-DE is not thebest but it is better than the other six algorithms

422 Exploration Capability Functions F8ndashF23 are multi-modal functions (ey include many local optima and thenumber of local optima increases exponentially with theincrease of dimension D So these functions are consideredto be very useful in evaluating the exploration capabilities ofalgorithms (e results of F8ndashF23 test functions for WOA-DE and WOA-BSA are shown in Table 5 where F8ndashF13 aremultimodal functions and F14ndashF23 are fixed-dimensionmultimodal functions (e results from Table 5 show thatWOA-DE and WOA-BSA have no advantage in terms ofBest Mean and Std compared to the other six algorithms(such as DE BBO BSA PSO HS and GSA) However bothof them are better than the original algorithm WOA whichindicates that our proposed hybrid algorithm framework caneffectively improve the performance of the original algo-rithm WOA

Figure 2 represents six convergence graphs obtained bythe nine algorithms for F1 F5 and F7 (unimodal functions)

F10 and F11 (multimodal functions) and F15 (hybrid com-position function) As can be seen from Figure 2 WOA-BSAhas faster convergence speed for F1 F5 F7 F10 F11 and F15meanwhile WOA-DE also has faster speed Based on theabove results we can see that WOA-DE and WOA-BSAhave competitive performance although they are not betterthan the compared algorithms in all aspects

43 Engineering Design Problems In order to further ex-amine the proposed hybrid framework it is applied to sixwell-known engineering design problems mixed the vari-ables taken from the optimization literatures (e descrip-tion of these problems including three-bar truss pressurevessel speed reducer gear train cantilever beam andI-beam is given in Appendix A Table 6 shows the parametersettings of six engineering problems according reference[23 24] whereN T andD represent the population size themaximum number of iterations and the dimension ofoptimization problems respectively However there are anumber of complex constraints in each problem whichmake it challenging to find the optimal solution satisfying allthe constraints So a constraint handling method such aspenalty functions feasibility and dominance rules stochasticranking multiobjective concepts ε-constrained methodsand Debrsquos heuristic constraints processing method [25 26]

Start

Is the stoppingcriterion satisfied

Generate a random initial population

Evaluate the fitness for each individual

Determine the best individual

Initialize the iteration counter t = 1 and thevariation probability lp = 05

For each individual

Update the individual if its trial vector is better

Update the best position found so far ifthere is a better position

Update a A C l and lp

Select a random individual Xrand from population

Set parameters of BSADE and WOA

lp lt rand

Generate Ti using equation (22)




Evaluate the fitness for the trial individual Ti

Update lp using equation (26)

|A| lt 1

Yes

Yes

No

No

No

Output the results

t = t + 1

Yes WOA

BSA or DE

|A| lt 1

Yes

No

Figure 1 Flowchart of the proposed hWOAlf


is necessary (is paper adopts Debrsquos heuristic constraintsprocessing method Each experiment is independently run30 times and the results obtained include the worst value(Worst) the mean value (Mean) the best function value(Best) the standard deviation (Std) and the functionevaluations (FEs)

431 gtree-Bar Truss Design Problem (ree-bar truss de-sign (as shown in Figure 3) is to minimize the weight of atruss with three bars subject to stress constraints (eproblem has two design variables and three nonlinear in-equality constraints (e formulation for this problem isshown in Appendix A1

Input ObjFun N D Tmax lb1 D ub1 D

Output Leaderminimum Xlowast

(1) rand sim U (0 1) randperm (N) sim random permutation of the integers 1 N normrnd (u d) sim random number chosen from anormal distribution with mean u and standard deviation d

(2) Initialization for i 1 to N do(3) for j 1 to D do(4) xij lbj + randlowast ubj Generate an initial whales population P X1 XN1113864 1113865

(5) end(6) end(7) finess ObjFun(P) Calculate the fitness of each search agent(8) Xlowast the best search agent(9) Leaderminimum the fitness of the best search agent(10) t 1 and lp 05(11) while tleTmax do(12) Mutation(13) Update the Position of search agents(14) for i 1 to N do(15) Update a A C and l(16) if lplt rand then(17) if |A|lt 1 then(18) the individual is mutated by equation (22)(19) else(20) Select a random individual Xrand the individual is mutated by equation (23)(21) end(22) else(23) if |A|lt 1 then(24) the individual is mutated by equation (24)(25) else(26) the individual is mutated by equation (25)(27) end(28) end(29) end(30) Checking allowable range(31) for i 1 to N do(32) for j 1 to D do(33) if xij lt lxj or xij gt uxj then(34) xij lxj + randlowast uxj

(35) end(36) end(37) end(38) Update the leader(39) for i 1 to N do(40) fitness(i) ObjFun(Xi) Calculate objective function values of T(41) if fitness(i)lt Leaderminimum then(42) Leaderminimum fitness(i)

(43) Xlowast Xi

(44) end(45) end(46) Update the learning parameter lp

(47) the lp is updated by equation (26)(48) t t + 1(49) end

ALGORITHM 1 (e pseudocode of hWOAlf


(e results of WOA-DE and WOA-BSA are comparedwith WOA dynamic stochastic selection (DEDS) [27]hybrid evolutionary algorithm (HEAA) [28] hybrid particleswarm optimization with differential evolution (PSO-DE)[29] differential evolution with level comparison (DELC)[30] mine blast algorithm (MBA) [31] and crow searchalgorithm (CSA) [32] (e best solutions obtained fromthese algorithms are presented in Table 7 From Table 7within the allowable error range of the constraint functionvalue the function value of the best solutions of WOA-DEand WOA-BSA is the smallest among all the comparisonalgorithms equaling to 263895711 As can be seen fromTable 8 although the FEs (equaling to 10000) obtained byDELC are the smallest WOA-DE and WOA-BSA find outthe current best solution of this problem In additionFigure 4 shows the convergence figure of WOA-DE WOA-BSA and WOA on three-bar truss design problem whichindicates the convergence speed of WOA-DE and WOA-BSA is faster than that of WOA (at is to say the proposedhybrid framework is effective

432 Pressure Vessel Design Problem (is problem aims tominimize the total cost of a cylindrical vessel subject toconstraints on the material forming and welding as shown inFigure 5 From this picture we can see that when the head ishemispherical and both ends of the cylindrical container aresealed by hemispherical heads Four optimization variablesare the thickness of the shell Ts (x1) thickness of the head Th

(x2) inner radius R (x3) and cylindrical section length L(x4) where Ts and Th need to be integer multiple of 00625while R (x3) and L (x4) are continuous variables In additionthe formulation of this problem is shown in Appendix A2

(e approaches applied to this problem include GAbased on coevolution model (CGA) [33] GA based on adominance-based tour tournament selection (DGA) [34]coevolutionary PSO (CPSO) [35] hybrid PSO (HPSO) [36]coevolutionary differential evolution (CDE) [37] CSA andWOA WOA-DE and WOA-BSA are compared with theseven algorithms listed above and the best solutions ob-tained by these methods are shown in Table 9 From Table 9we can see that the objective function values of the bestsolutions ofWOA-DE (equaling to 6059708057) andWOA-BSA (equaling to 6059708025) are superior to those ofcomparison algorithms Moreover as can be seen fromTable 10 the FEs obtained by the proposed WOA-DE andWOA-BSA are minimal equaling to 30240 and 30140 re-spectively (is suggests that WOA-DE and WOA-BSA can

find out the current best solution of this problem with thesmallest computational overhead Figure 6 shows thatWOA-DE and WOA-BSA have a faster convergence ratethan the original WOA on pressure vessel design issues

433 Speed Reducer Design Problem (e speed reducerdesign problem shown in Figure 7 is a practical designproblem that has been often used as a benchmark problemfor testing the performance of different optimization algo-rithms(e objective is to find the minimumweight of speedreducer under the constraints of bending stress of the gearteeth surface stress transverse deflections of the shafts andstresses in the shafts (ere are seven design variablesnamely the face width b (x1) the tooth module m (x2) thenumber of teeth in pinion z (x3) the length of the first shaftbetween bearings l1(x4) the length of the second shaftbetween bearings l2(x5) and the diameter of the first shaftd1(x6) and the second shaft d2(x7) (e mathematicalformulation of the problem is shown in Appendix A3

Tables 11 and 12 show the best solutions and statisticalresults of MDE DEDS DELC HEAA PSO-DE MBA andWOA From Table 11 it can be seen that the function valuesof the current best feasible solutions ofWOA-DE andWOA-BSA are equal to 2994469448 and 2994469656 respectively(e statistical results are given in Table 12 From the resultsof this table we can see that the FEs ofWOA-DE andWOA-BSA equaling to 15000 are better than those of the otheralgorithms except MBA equaling to 6300 (is indicates thatWOA-DE and WOA-BSA in the framework compared toother algorithms can obtain a relatively better solution bysacrificing a little computational overhead on this problemMoreover Figure 8 presents the convergence figure ofWOA-DE WOA-BSA and WOA on this problem FromFigure 8 the convergence speeds of WOA-DE and WOA-BSA are faster than WOA and their solutions are closer tothe optimal solution

434 Gear Train Design Problem Gear train design aims tominimize the cost of the gear ratio of the gear train It hasonly boundary constraints in parameters Variables to beoptimized are in discrete form since each gear has to have anintegral number of teeth In the problem the design vari-ables are nA(x1) nB(x2) nD(x3) and nF(x4) which arediscrete and must be integers (e gear ratio is defined asnBnDnFnA (e formulation of this problem is shown inAppendix A4 Figure 9 shows the speed reducer and itsparameters

Table 1 Description of unimodal benchmark functions

Function D Range fmin

F1(x) 1113936Di1x

2i 30 [minus 100 100] 0

F2(x) 1113936Di1|xi| + 1113937

Di1 |xi| 30 [minus 10 10] 0

F3(x) 1113936Di1(1113936

ij1 xj)

2 30 [minus 100 100] 0F4(x) max1leileD|xi| 30 [minus 100 100] 0F5(x) 1113936

Dminus 1i1 [100(xi+1 minus x2

i )2 + (xi minus 1)2] 30 [minus 30 30] 0F6(x) 1113936

Di1(xi + 05)2 30 [minus 100 100] 0

F7(x) 1113936Di1 ix4

i + random[0 1) 30 [minus 128 128] 0


Tabl

e2

Descriptio

nof

multim

odal

benchm

arkfunctio

ns

Functio

nD

Rang

efmin

F8(

x)

1113936

D i1

minusx

isin

(

|x

i|1113968

)30

[minus500

500]

minus4189829

times5

F9(

x)

1113936

D i1(

x2 i

minus10

cos(2π

xi)

+10

)30

[minus5125

12]

0

F10

(x

)

minus20

exp(

minus02

(1

D)1113936

D i1x

2 i

1113969

)minusexp(

(1

D)1113936

D i1cos

(2π

xi)

)+20

+e

30[minus323

2]0

F11

(x

)

(14000

)1113936D i1x

2 iminus

1113937D i1cos(

xi

iradic

)+1

30[minus600

600]

0F12

(x

)π

D10

sin(πy

1)+

1113936D

minus1

i1

(y

iminus1)

2 [1+10

sin2 (πy

i+1)

]+

(y

Dminus1)

21113966

1113967+

1113936D i1u

(x

i10

100

4)

30[minus505

0]0

yi

1

+x

i+14

u(

xi

ak

m)

k(

ximinus

a)m

x

igt

a

0minus

alt

xilt

a

k(

minusx

iminus

a)m

x

ilt

a

⎧⎪ ⎨ ⎪ ⎩

F13

(x

)01sin

2 (3π

x1)

+1113936

D i1

(x

iminus1)

2 [1+sin

2 (3π

xi+1

+1)

]+

(x

Dminus1)

2 [1+sin

2 (2π

xD

)]1113966

1113967+

1113936D i1u

(x

i51004)

30[minus505

0]0


Tabl

e3

Descriptio

nof

fixed-dim

ensio

nmultim

odal

benchm

arkfunctio

ns

Functio

nD

Rang

efmin

F14

(x

)

((1500)

+1113936

25 j1(1

j+

11139362 i1(

ximinus

aij

)6))

minus1

2[minus656

5]1

F15

(x

)

111393611 i1[

aiminus

(x1(

b2 i

+b

ix2)b

2 i+

bix

3+

x4)

]24

[minus5

5]000030

F16

(x

)4x

2 1minus21x

4 1+

(13)

x6 1

+x1x

2minus4x

2 2+4x

4 22

[minus5

5]minus10316

F17

(x

)

(x2

minus(514π

2 )x2 1

+(5π)

x1

minus6)

2+10

(1

minus(18π

))cos

x1

+10

2[minus5

5]0398

F18

(x

)

[1+

(x1

+x2

+1)

2 (19

minus14

x1

+3x

2 1minus14

x2

+6x

1x2

+3x

2 2)]

times[30

+(2x

1minus3x

2)2 (18

minus32

x1

+12

x2 1

+48

x2

minus36

x1x

2+27

x2 2)

]2

[minus2

2]3

F19

(x

)

minus1113936

4 i1c

iexp(

minus1113936

3 j1

aij

(x

jminus

pij

)2)

3[13

]minus386

F20

(x

)

minus1113936

4 i1c

iexp(

minus1113936

6 j1

aij

(x

jminus

pij

)2)

6[01

]minus332

F21

(x

)

minus1113936

5 i1[

(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus101232

F22

(x

)

minus1113936

7 i1[

(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus104028

F23

(x

)

minus1113936

1 i10

[(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus105363


Tabl

e4

Com

parisonof

optim

izationresults

obtained

fortheun

imod

albenchm

arkfunctio

ns

AL

Runs

30N

30a

ndT

1000

frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 1Be

st290E

minus12

(5)

0454643

(9)

0000908

(7)

104E

minus11

(6)

0042385

(8)

464E

minus17

(4)

276E

minus167(

3)692E

minus190(

1)162E

minus187(

2)

Mean

133E

minus11

(5)

1013223

(9)

0006257

(7)

973E

minus09

(6)

0063236

(8)

101E

minus16

(4)

146E

minus153(

3)107E

minus168(

2)145E

minus172(

1)

Std

998E

minus12

(4)

0311571

(8)

0010917

(7)

423E

minus08

(5)

0010899

(6)

376E

minus17

(3)

791E

minus153(

2)0(

1)0(

1)

F 2Be

st399E

minus08

(5)

0271733

(8)

0006726

(7)

100E

minus05

(6)

0575724

(9)

350E

minus08

(4)

331E

minus114(

3)246E

minus124(

2)201E

minus124(

1)

Mean

680E

minus08

(5)

0324236

(8)

0020418

(7)

0001313

(6)

0727765

(9)

520E

minus08

(4)

988E

minus105(

3)847E

minus113(

1)127E

minus112(

2)

Std

195E

minus08

(5)

0041092

(8)

0014974

(7)

0004947

(6)

005865(

9)117E

minus08

(4)

494E

minus104(

3)421E

minus112(

1)575E

minus112(

2)

F 3Be

st1080803597(

9)83765444(

2)394952176

(5)

3208562

(1)

308450285

(8)

195538197

(4)

167730974

(7)

17510162(

3)532871248

(6)

Mean

157231259

(9)

135934273

(2)

87822206(

4)14904551(

1)5228201445(

5)409456876

(3)

9735512836(

7)1004518144(

8)7621728202(

6)

Std

2683155994(

6)34026326(

2)315589252

(4)

7677907

(1)

1591552755(

5)12983509(

3)3898890192(

7)598648178

(9)

4199058749(

8)

F 4Be

st1080885

(7)

0742244

(6)

3262876

(8)

0311111

(5)

4216935

(9)

135E

minus08

(3)

0019063

(4)

568E

minus10

(2)

430E

minus19

(1)

Mean

1575045

(5)

0972736

(2)

5969996

(6)

0563087

(1)

6113542

(7)

1264362

(3)

17704825(

9)14259819(

8)1562263

(4)

Std

038751(

3)0123737

(1)

157968(

6)0153909

(2)

1246667

(4)

1284382

(5)

1458112

(8)

17566092(

9)6519626

(7)

F 5Be

st25398273(

2)39474199(

7)4367835

(8)

11716823(

1)91050677(

9)25861836(

3)26607341(

5)25886812(

4)26835697(

6)

Mean

3831135

(5)

149884822

(8)

114032052

(7)

60388778(

6)342973593

(9)

37244874(

4)271471(

1)27512334(

3)27496506(

2)

Std

21417578(

4)181590232

(8)

35216402(

6)43241123(

7)492144758

(9)

24882941(

5)0485147

(1)

0547347

(3)

0539586

(2)

F 6Be

st288E

minus12

(2)

06227

(9)

000076(

4)251E

minus11

(3)

0048128

(8)

450E

minus17

(1)

0007148

(5)

0012986

(6)

0015406

(7)

Mean

890E

minus12

(2)

1071114

(9)

000788(

4)968E

minus09

(3)

0070002

(6)

103E

minus16

(1)

0057661

(5)

0263633

(8)

020125(

7)

Std

567E

minus12

(2)

0279422

(9)

0008319

(4)

300E

minus08

(3)

0011199

(5)

446E

minus17

(1)

0078559

(6)

0224346

(8)

0192754

(7)

F 7Be

st0016252

(6)

0002011

(4)

0012074

(5)

0037104

(9)

0035229

(8)

0016733

(7)

0000115

(2)

000016(

3)00001

(1)

Mean

0027322

(5)

0006396

(4)

0030674

(6)

0078177

(9)

0055369

(8)

0049762

(7)

0001935

(3)

00015

(1)

0001579

(2)

Std

0005743

(5)

0002154

(4)

0012799

(7)

0023769

(9)

0011801

(6)

0021587

(8)

0001959

(3)

0001692

(1)

0001761

(2)

Rank

Best

68

75

93

41

2Mean

57

64

82

33

1Std

15

64

71

23

1


Tabl

e5

Com

parisonof

optim

izationresults

obtained

forthemultim

odal

benchm

arkfunctio

ns

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 8Be

stminus1256948662(

1)minus9288859115(

5)minus123198552

(3)

minus8562082036(

8)minus1256947154(

2)minus3257715637(

9)minus8674343572(

7)minus8985661303(

6)minus1057783345(

4)

Mean

minus1256553865(

2)minus8297653051(

4)minus1198240966(

3)minus6118655936(

8)minus1256880513(

1)minus243432431

(9)

minus6658410197(

7)minus7600141184(

6)minus7657917372(

5)

Std

21623781(

2)497301094

(5)

22238506(

3)1574830603(

9)0909111

(1)

355937095

(4)

863971262

(7)

861181057

(6)

1183224502(

8)

F 9Be

st44307253(

7)29244538(

6)16598597(

4)14992992(

3)17949826(

5)14924381(

2)0(

1)0(

1)0(

1)

Mean

56783475(

8)53437985(

7)25788633(

4)46538306(

6)25466452(

3)25802587(

5)0(

1)3672248

(2)

0(1)

Std

6266486

(4)

14333435(

7)403673(

2)14270927(

6)4772648

(3)

6630417

(5)

0(1)

20113732(

8)0(

1)

F 10

Best

423E

minus07

(3)

0226971

(6)

0007536

(5)

300E

minus06

(4)

1080969

(7)

545E

minus09

(2)

888E

minus16

(1)

888E

minus16

(1)

888E

minus16

(1)

Mean

702E

minus07

(5)

0349244

(8)

0020786

(6)

008323(

7)2063648

(9)

770E

minus09

(4)

444E

minus15

(3)

420E

minus15

(2)

397E

minus15

(1)

Std

152E

minus07

(5)

0070199

(7)

0011333

(6)

0317495

(8)

0425232

(9)

150E

minus09

(4)

229E

minus15

(3)

130E

minus15

(1)

203E

minus15

(2)

F 11

Best

144E

minus11

(3)

0503713

(6)

0001224

(4)

742E

minus12

(2)

0120691

(5)

3009729

(7)

0(1)

0(1)

0(1)

Mean

980E

minus10

(1)

0766039

(8)

0026851

(6)

0011493

(5)

0618727

(7)

8568461

(9)

0003104

(4)

0001092

(2)

0001282

(3)

Std

307E

minus09

(1)

011031(

7)0029255

(6)

0009701

(4)

0275077

(8)

4065895

(9)

0011816

(5)

0005981

(2)

0007021

(3)

F 12

Best

239E

minus13

(3)

0001193

(6)

300E

minus06

(4)

338E

minus14

(2)

0000383

(5)

375E

minus19

(1)

0001352

(7)

0003179

(9)

0002984

(8)

Mean

100E

minus12

(1)

0002505

(4)

630E

minus05

(2)

0003456

(5)

0000581

(3)

0151627

(8)

0006188

(6)

0013432

(7)

0409258

(9)

Std

105E

minus12

(1)

0000844

(4)

800E

minus05

(2)

0018927

(7)

0000121

(3)

0210923

(8)

0006962

(6)

0004444

(5)

160174(

9)

F 13

Best

170E

minus12

(2)

0023921

(6)

800E

minus05

(4)

523E

minus12

(3)

0006145

(5)

594E

minus18

(1)

0025211

(7)

0122758

(9)

0080548

(8)

Mean

431E

minus12

(1)

0044601

(6)

0000741

(2)

0001831

(3)

0013234

(5)

0007331

(4)

0233207

(7)

0468482

(8)

047475(

9)

Std

269E

minus12

(1)

0012881

(5)

0000969

(2)

0004165

(3)

0005965

(4)

0025745

(6)

0181659

(7)

0238598

(9)

0224147

(8)

F 14

Best

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998118

(2)

0998004

(1)

0998004

(1)

0998004

(1)

Mean

0998004

(1)

6813974

(7)

0998004

(1)

3459539

(5)

0998004

(1)

4082277

(6)

247579(

3)3454612

(4)

2108894

(2)

Std

0(1)

5363613

(9)

412E

minus17

(2)

2863628

(7)

187E

minus14

(3)

2690671

(5)

2445473

(4)

3273113

(8)

2790461

(6)

F 15

Best

0000352

(3)

0000419

(5)

0000307

(1)

0000308

(2)

0000401

(4)

0001284

(6)

0000308

(2)

0000307

(1)

0000307

(1)

Mean

0000577

(4)

0006093

(9)

0000329

(1)

0000794

(6)

0004004

(8)

0002584

(7)

0000601

(5)

0000513

(3)

0000468

(2)

Std

0000109

(2)

0008761

(9)

410E

minus05

(1)

0000233

(3)

0007444

(8)

0002019

(7)

0000315

(4)

0000377

(5)

0000382

(6)

F 16

Best

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Mean

minus1031628

(1)

minus0977217

(2)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Std

671E

minus16

(4)

0207068

(8)

612E

minus16

(3)

678E

minus16

(5)

353E

minus08

(7)

513E

minus16

(1)

386E

minus11

(6)

671E

minus16

(4)

598E

minus16

(2)

F 17

Best

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

Mean

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397888

(2)

0397887

(1)

0397887

(1)

Std

0(1)

933E

minus11

(3)

0(1)

0(1)

437E

minus08

(4)

0(1)

930E

minus07

(5)

0(1)

125E

minus13

(2)

F 18

Best

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

Mean

3(1)

39(

3)3(

1)3(

1)3900137

(4)

3(1)

3000014

(2)

3(1)

3(1)

Std

127E

minus15

(2)

4929503

(9)

202E

minus15

(5)

152E

minus15

(3)

4929483

(8)

267E

minus15

(6)

380E

minus05

(7)

782E

minus16

(1)

166E

minus15

(4)

F 19

Best

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

Mean

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862628

(2)

minus3862782

(1)

minus3862782

(1)

Std

271E

minus15

(6)

229E

minus15

(1)

271E

minus15

(6)

270E

minus15

(5)

100E

minus06

(7)

239E

minus15

(2)

0000366

(8)

266E

minus15

(4)

251E

minus15

(3)

F 20

Best

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321994

(2)

minus3321995

(1)

minus3321989

(3)

minus3321995

(1)

minus3321995

(1)

Mean

minus3321989

(2)

minus3270475

(6)

minus3321995

(1)

minus3282364

(5)

minus330217(

3)minus3321995

(1)

minus3264959

(7)

minus3293815

(4)

minus3254599

(8)

Std

320E

minus05

(3)

0059923

(7)

173E

minus14

(2)

0057005

(6)

0045065

(4)

147E

minus15

(1)

0071826

(9)

0051995

(5)

0059944

(8)

F 21

Best

minus101532(

1)minus101532(

1)minus101532(

1)minus101532(

1)minus10153199(

2)minus101532(

1)minus10153181(

3)minus101532(

1)minus101532(

1)

Mean

minus10049112(

2)minus5725156

(8)

minus101532(

1)minus8062579

(6)

minus5406265

(9)

minus6043604

(7)

minus8882256

(4)

minus8882841

(3)

minus8462151

(5)

Std

0531474

(2)

3327602

(7)

115E

minus09

(1)

3091228

(6)

3673677

(9)

3495936

(8)

2378758

(3)

2378893

(4)

2686626

(5)


Tabl

e5

Con

tinued

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 22

Best

minus10402941(

1)minus10402941(

1)minus10402941(

1)minus10402941(

1)minus1040294

(2)

minus10402941(

1)minus1040292

(3)

minus10402941(

1)minus10402941(

1)

Mean

minus10211001(

2)minus5529595

(8)

minus10402941(

1)minus9521178

(3)

minus6545124

(7)

minus10402941(

1)minus8448955

(4)

minus7763089

(6)

minus796332(

5)

Std

1045029

(3)

3343034

(8)

169E

minus14

(2)

2005406

(4)

3686937

(9)

132E

minus15

(1)

2836906

(5)

3128103

(7)

286345(

6)

F 23

Best

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus10536335(

2)minus1053641

(1)

minus1053641

(1)

Mean

minus10536383(

2)minus5855753

(8)

minus1053641

(1)

minus916093(

4)minus5180275

(9)

minus10175835(

3)minus9003007

(5)

minus7990287

(7)

minus8647563

(6)

Std

0000108

(2)

3655594

(9)

111E

minus09

(1)

2570428

(4)

3329307

(8)

1485436

(3)

2591415

(5)

3225805

(7)

2733822

(6)

Rank

Best

18

32

75

64

2Mean

29

16

87

53

4Std

19

26

83

74

5


Table 13 shows the comparison of best solutions forartificial bee colony algorithm (ABC) [39] MBA CSAWOA WOA-DE and WOA-BSA From the table we can

see that the best solutions obtained by the six algorithms arethe same (equaling to 27e minus 12) In addition the statisticalresults of all six algorithms are shown in Table 14 WOA-DE

200 400 600 800 1000

10minus150

10minus100

10minus50

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(a)

200 400 600 800 1000

102

104

106

108

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(b)

Ave

rage

ln (f

lowast minus

f min

)

200 400 600 800 1000

10minus2

100

102

IterationDEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(c)

200 400 600 800 1000

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(d)

200 400 600 800 1000

10minus15

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(e)

200 400 600 800 1000

10minus4

10minus3

10minus2

10minus1

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(f )

Figure 2 Convergence graphs of six functions (a) F1 (b) F5 (c) F7 (d) F10 (e) F11 (f ) F15


with FEs of 940 and WOA-BSA with FEs of 1900 ranksecond and fourth respectively among all the six comparedalgorithms (e FEs of the top one are 60 belonging to ABCIn addition WOA-DE and WOA-BSA have faster con-vergence speeds than WOA from the function convergencegraphs of WOA-DE WOA-BSA and WOA (see Figure 10)

435 Cantilever Beam Design Problem (e objective ofcantilever beam design problem is to minimize the weight ofa cantilever beam consisting of five hollow sections with

fixed diameters as shown in Figure 11 (e design variablesare the heights of all beam elements and there is a constraintthat should not be violated by the final optimization design(e formulation of this problem is shown in Appendix A5

Table 15 provides the comparison results betweenWOA-DE WOA-BSA WOA MMA [40] GCAI [40] GCAII [40]cuckoo search algorithm (CS) [41] symbiotic organismssearch (SOS) [42] and moth-flame optimization algorithm(MOF) [43](e results in Table 15 show that WOA-DE andWOA-BSA can also effectively solve such problems Con-vergence plot using WOA-DE WOA-BSA and WOA for

Table 6 Parameter setting for engineering problems

Problem N T D(ree-bar truss 40 500 2Pressure vessel 20 5000 4Speed reducer 30 500 7Gear train 20 500 4Cantilever beam 30 1000 5I-beam 25 500 4

l l

l x1 x1x2

P

Figure 3 (ree-bar truss design problem

Table 7 Comparison of best solutions of the three-bar truss design problem

Method DEDS [27] HEAA [28] PSO-DE [29] DELC [30] MBA [31] CSA [32] WOA WOA-DE WOA-BSAX1 0788675 0788680 0788675 0788675 0788565 0788675 0786042 0788686 0788674X2 0408248 0408234 0408248 0408248 0408560 0408248 0415745 0408215 0408250g1(X) 177E minus 08 NA minus 529E minus 11 NA minus 529E minus 11 minus 169E minus 14 988E minus 07 100E minus 06 100E minus 06g2(X) minus 1464102 NA minus 1463748 NA minus 1463748 minus 1464102 minus 1455608 minus 1464139 minus 1464099g3(X) minus 0535898 NA minus 0536252 NA minus 0536252 minus 0535898 minus 0544390 minus 0535860 minus 0535900f (X) 263895843 263895843 263895843 263895843 263895852 263895843 263900852 263895711 263895711

Table 8 Comparison of statistical results of the three-bar truss design problem

Method Worst Mean Best Std FEsDEDS [27] 263895849 263895843 263895843 97e minus 07 15000HEAA [28] 263896099 263895865 263895843 49e minus 05 15000PSO-DE [29] 263895843 263895843 263895843 45e minus 10 17600DELC [30] 263895843 263895843 263895843 43e minus 14 10000MBA [31] 263915983 263897996 263895852 393e minus 03 13280CSA [32] 263895843 263895843 263895843 101e minus 10 25000WOA 266116794 264248699 263900852 0604342 20000WOA-DE 263936409 263897097 263895711 0007425 19920WOA-BSA 264011056 263912284 263895711 0028964 15200


0 100 200 300 400 500242

2422

2424

2426

2428

243

2432

2434

Number of iterations

Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA

Figure 4 Convergence plots using WOA WOA-DE and WOA-BSA for the three-bar truss design problem

Th

R

Ts

R

L

Figure 5 Pressure vessel design problem

Table 9 Comparison of best solutions of the pressure vessel design problem

Method CGA [33] DGA [34] CPSO [35] HPSO [36] CDE [37] CSA [32] WOA WOA-DE WOA-BSAX1 0812500 0812500 0812500 0812500 0812500 0812500 0812500 0812500 0812500X2 0437500 0437500 0437500 0437500 0437500 0437500 0437500 0437500 0437500X3 40323900 42097400 42091300 42098400 42098411 42098445 41825682 42098497 42098497X4 200000000 176654000 176746500 176636600 176637690 176636599 180046141 176635957 176635954g1(X) minus 342e minus 02 minus 201e minus 03 minus 137e minus 06 minus 880e minus 07 minus 667e minus 07 minus 402e minus 09 minus 0005264 995E minus 07 100E minus 06g2(X) minus 528e minus 02 minus 358e minus 02 minus 359e minus 04 minus 358e minus 02 minus 358e minus 02 minus 0035880 minus 0038483 minus 0035880 minus 0035880g3(X) minus 304402000 minus 24759300 minus 118768700 3122600 minus 3705123 minus 712e minus 04 minus 0000719 100E minus 06 100E minus 06g4(X) minus 400000000 minus 63346000 minus 63253500 minus 63363400 minus 63362310 minus 63363401 minus 59953859 minus 63364043 minus 63364046f (X) 6288744500 6059946300 6061077700 6059714300 6059734000 6059714363 6093211997 6059708057 6059708025

Table 10 Comparison of statistical results of the pressure vessel design problem

Method Worst Mean Best Std FEsCGA [33] 6308497000 6293843200 6288744500 7413300 900000DGA [34] 6469322000 6177253300 6059946300 130929700 80000CPSO [35] 6363804100 6147133200 6061077700 86450000 240000HPSO [36] 6288677000 6099932300 6059714300 86200000 81000CDE [37] 6371045500 6085230300 6059734000 43013000 204800CSA [32] 7332841621 6342499106 6059714363 384945416 250000WOA 8564987629 7124412382 6093211997 715900580 99940WOA-DE 44828494260 8946493319 6059708057 7565793120 30240WOA-BSA 12956012602 7114983582 6059708025 1679555214 30140


0 1000 2000 3000 4000 500035

4

45

5

55

6


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA

Figure 6 Convergence plots using WOA WOA-DE and WOA-BSA for pressure vessel design problem

l1

d1

d2l2

z1 z2

Figure 7 Speed reducer design problem

Table 11 Comparison of best solutions for the speed reducer design problem

Method MDE [38] DEDS [27] DELC [30] HEAA [28] PSO-DE [29] MBA [31] WOA WOA-DE WOA-BSAX1 3500010 3500000 3500000 3500023 3500000 3500000 3500000 3500000 3500000X2 0700000 0700000 0700000 0700000 0700000 0700000 0700000 0700000 0700000X3 17000000 17000000 17000000 17000013 17000000 17000000 10000007 17000000 17000000X4 7300156 7300000 7300000 7300428 7300000 7300033 7942261 7300000 7300000X5 7800027 7715320 7715320 7715377 7800000 7715772 7919344 7715310 7715310X6 3350221 3350215 3350215 3350231 3350215 3350218 3352296 3350213 3350213X7 5286685 5286654 5286654 5286664 5286683 5286654 5286722 5286652 5286653f(X) 2996356689 2994471066 299447106 2994499107 2996348167 2994482453 3005192607 2994469448 2994469656

Table 12 Comparison of statistical results for the speed reducer design problem

Method Worst Mean Best Std FEsMDE [38] 2996390137 2996367220 2996356689 82e minus 03 24000DEDS [27] 2994471066 2994471066 2994471066 358e minus 12 30000DELC [30] 2994471066 2994471066 2994471066 19e minus 12 30000HEAA [28] 2994752311 2994613368 2994499107 70e minus 02 40000PSO-DE [29] 2996348204 2996348174 2996348167 64e minus 06 54350MBA [31] 2999652444 2996769019 2994482453 1560000 6300WOA 3561958300 3065730259 3005192607 108307891 15000WOA-DE 3013442478 2998271376 2994469448 4750316 15000WOA-BSA 3007155564 2997610011 2994469656 3292733 15000


0 100 200 300 400 500345

35

355

36

365

37


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA

Figure 8 Convergence plots using WOA WOA-DE and WOA-BSA for speed reducer design problem

A D

B

C

Figure 9 Gear train design problem

Table 13 Comparison of best solutions of the gear train design problem

Method ABC [39] MBA [31] CSA [32] WOA WOA-DE WOA-BSAX1 49000000 43000000 49000000 43000000 49000000 43000000X2 16000000 16000000 19000000 16000000 19000000 19000000X3 19000000 19000000 16000000 19000000 16000000 16000000X4 43000000 49000000 43000000 49000000 43000000 49000000f(X) 27e minus 12 27e minus 12 27e minus 12 27e minus 12 27e minus 12 27e minus 12

Table 14 Comparison of statistical results of the gear train design problem

Method Worst Mean Best Std FEsABC [39] NA 36e minus 10 27e minus 12 552E minus 10 60MBA [31] 21e minus 08 25e minus 09 27e minus 12 394e minus 09 1120CSA [32] 32e minus 08 206e minus 09 27e minus 12 51e minus 09 100000WOA 273E minus 08 326E minus 09 27E minus 12 597E minus 09 6800WOA-DE 447E minus 08 480E minus 09 27E minus 12 951E minus 09 940WOA-BSA 398E minus 08 403E minus 09 27E minus 12 740E minus 09 1900


cantilever beam design problem (as shown in Figure 12)indicates that WOA-DE and WOA-BSA have faster rates ofconvergence which are close to the optimal solution

436 I-Beam Design Problem In this problem the goal is tominimize the vertical deflection of the beam shown inFigure 13 (ere are four design variables for this problemlength (x1) height (x2) and two thicknesses (x3 x4) (eproblem includes only one optimization constraint and isformulated in Appendix A6

Table 16 compares the best solutions of seven state-of-the-art methods including adaptive response surface method(ARSM) [44] improvedARSM (IARSM) [44] CS SOSWOAWOA-DE and WOA-BSA for I-beam design problem Fromthe results of this table we can see that the optimal valuesobtained by SOSWOA-DE andWOA-BSA are the same(econvergence plots of the three algorithms (WOA WOA-DEand WOA-BSA) are shown in Figure 14 We can see thatWOA-DE and WOA-BSA have faster convergence speed

44 Analysis and Discussion of Experimental ResultsTaking the experimental results on twenty-three benchmarkfunctions and six engineering design problems into con-sideration simultaneously the comprehensive performanceof the proposed hybrid framework is discussed based onWOA-DE and WOA-BSA from three aspects including thesolution accuracy convergence rate and algorithmicrobustness

441 Discussion of the Solution Accuracy (e Best values inthe above tables can reflect the solution accuracy of an al-gorithm in solving twenty-three benchmark functions and sixengineering design problems (e smaller the Best values thebetter the solution accuracy On the one hand compared witheight algorithms on twenty-three benchmark functionsWOA-DE ranks first among seven unimodal functionshowever it ranks fourth among multimodal functions whichmeans that WOA-DE is not competitive enough for morecomplex functions WOA-BSA ranks second for both uni-modal functions (except WOA-DE) and multimodal func-tions (except DE) (is shows that WOA-BSA is competitivein solving function optimization problems On the otherhand compared with some state-of-the-art algorithms onengineering design problems WOA-DE can find out thecurrent best optimal solutions in four problems includingthree-bar truss design problem speed reducer design prob-lem gear train design problem and I-beam design problemWOA-BSA can obtain the current best feasible solutions infour engineering problems All the above experimental resultsindicate that the proposed hybrid framework has excellentperformance with respect to solution accuracy

442 Discussion of the Convergence Rate (e FE valuesreflect the computational overhead of an algorithm whenfinding out the best solution of an optimization problem (esmaller the computational overhead of an algorithm the fasterthe convergence rate On the one hand when solving the six

0 100 200 300 400 500minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA

Figure 10 Convergence plots using WOA WOA-DE and WOA-BSA for gear train design problem

1 2 3 4 5

xi

xiConstantthickness

Figure 11 Cantilever beam design problem


engineering design problems although the FEs by WOA-DEare not the smallest its ranking among all compared algo-rithms is competitive (e convergence speed of WOA-BSAcompared withWOA-DE and originalWOA is far faster thanthe original WOA On the other hand when solving thetwenty-three benchmark functions all the compared algo-rithms search within the same FE limit At this time theconvergence rate of the compared algorithms can be reflectedby the trend of the convergence curves As illustrated inFigure 2 WOA-BSA and WOA-DE have better performancein convergence rate when compared with other algorithms

443 Discussion of the Algorithmic Robustness (e Meanand Std values in the above tables can reflect the robustness

of an algorithm (e smaller the Mean and Std values thehigher the robustness of an algorithm On the one hand ontwenty-three benchmark functions the Mean and Std valuesof WOA-DE rank third in unimodal functions and rankthird and fourth in multimodal functions respectively whilethe ranking of the Mean and Std values obtained by WOA-BSA is fourth and fifth for unimodal functions respectivelyFor multimodal functions the ranking of WOA-BSA is thefirst for the Mean and Std values which indicates that thealgorithm has strong robustness On the other hand on sixengineering problems unfortunately the Mean and Stdvalues obtained by WOA-DE are mediocre while the per-formance of WOA-BSA on Mean and Std values of 30 runtimes is also not competitive enough All in all theWOA-DEand WOA-BSA have better overall performance in terms ofrobustness but there is still much room for improvement

From the analysis of results based on the above two setsof experiments on WOA-DE and WOA-BSA we canprovide insight into the following two characteristics aboutthe proposed algorithm framework Firstly the proposedframework is an effective and generic technique forfunction optimization and engineering constrained prob-lems Secondly the overall performance of the frameworkis outstanding but there is still space to improve itsrobustness

0 200 400 600 800 100001

02

03

04

05

06

07

08

09


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA

Figure 12 Convergence plots using WOA WOA-DE and WOA-BSA for cantilever beam design problem

Table 15 Comparison results for cantilever design problem

Method x1 x2 x3 x4 x5 f(X)

MMA [40] 6010000 5300000 4490000 3490000 2150000 1340000GCAI [40] 6010000 5300000 4490000 3490000 2150000 1340000GCAII [40] 6010000 5300000 4490000 3490000 2150000 1340000CS [41] 6008900 5304900 4502300 3507700 2150400 1339990SOS [42] 6018780 5303440 4495870 3498960 2155640 1339960MFO [43] 5984872 5316727 4497333 3513616 2161620 1339988WOA 5649936 5132520 4730823 3669160 2462567 1350648WOA-DE 5776867 5139076 4972515 3507170 2197960 1347440WOA-BSA 5848097 5461858 4480823 3482342 2223440 1341385

P

h

b

tftwL

Figure 13 I-beam design problem


5 Conclusion

(e original WOA is easy to fall into local optimum when itcarries out the operation of prey encircling To address thisissue a hybrid algorithm framework with learning andcomplementary fusion features for WOA called hWOAlf isproposed in this paper First a learning parameter lp

according to adaptive adjustment operator is proposed tobalance exploration and exploitation capability Further-more the proposed hybrid algorithm framework calledhWOAlf takes advantage of the complementary fusionfeature operator to enhance the exploration capability ofWOA Two new algorithms WOA-DE and WOA-BSA areformed by embedding the DErand1 operator of DE and themutate operator intoWOA respectively(e effectiveness ofthe proposed hWOAlf framework is exhaustively demon-strated through comparing WOA-DE and WOA-BSA withsome state-of-the-art optimization algorithms on twenty-three continuous benchmark functions and six engineeringdesign problems

In the future there are two further studies to be doneon the one hand better parameter tuning strategies can bedeveloped to balance the exploration and exploitationcapabilities of WOA on the other hand more meta-heuristics are expected to be added to the proposed hybridframework

Appendix

A Engineering Design Problems

A1 gtree-Bar Truss Design Problem

minf(x) 22

radicx1 + x2( 1113857 times l

subject to

g1(x)

2

radicx1 + x2

2radic

x21 + 2x1x2

P minus σ le 0

g2(x) x2

2radic

x21 + 2x1x2

P minus σ le 0

g3(x) 1

2

radicx2 + x1

P minus σ le 0

0lexi le 1 i 1 2

l 100 cm

P 2 kNcm2

σ 2 kNcm2

(A1)

Table 16 Comparison results for I-beam design problem

Method x1 x2 x3 x4 f(X)

ARSM [44] 37050000 80000000 1710000 2310000 0015700IARSM [44] 48420000 79990000 0900000 2400000 0131000CS [41] 50000000 80000000 0900000 2321675 0013075SOS [42] 50000000 80000000 0900000 232179 0013074WOA 44000000 80000000 0900815 2644304 00131734WOA-DE 50000000 80000000 0900000 2321792 0013074WOA-BSA 50000000 80000000 0900000 2321792 0013074

0 100 200 300 400 500minus188

minus186

minus184

minus182

minus18

minus178

minus176


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA

Figure 14 Convergence plots using WOA WOA-DE and WOA-BSA for I-beam design problem


A2 Pressure Vessel Design Problem

minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0

g3(x) minus πx23x4 minus (43)πx

33 + 1296000le 0

g4(x) x4 minus 240le 0

0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)

A3 Speed Reducer Design Problem

minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)

A4 Gear Train Design Problem

minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)

A5 Cantilever Beam Design Problem

minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)

A6 I-Beam Design Problem

minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6

10le x1 le 50 10lex2 le 8009lex3 le 5 09le x4 le 5

(A7)

Data Availability

(e data used to support the findings of this study are in-cluded within the article In addition in order to better sharethe research results the LaTeX codes related to the data areavailable from the corresponding author upon request

Conflicts of Interest

(e author declares that there are no conflicts of interest

References

[1] J H Holland Adaptation in Natural and Artificial SystemsAn Introductory Analysis with Applications to Biology Con-trol and Artificial Intelligence MIT Press Cambridge MAUSA 1992

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

[3] D Simon ldquoBiogeography-based optimizationrdquo IEEE Trans-actions on Evolutionary Computation vol 12 no 6pp 702ndash713 2008

[4] P Civicioglu ldquoBacktracking search optimization algorithmfor numerical optimization problemsrdquo Applied Mathematicsand Computation vol 219 no 15 pp 8121ndash8144 2013

[5] Z Hu X Xu Q Su H Zhu and J Guo ldquoGrey predictionevolution algorithm for global optimizationrdquo AppliedMathematical Modelling vol 79 no 79 pp 145ndash160 2020

[6] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the ICNNrsquo95 - International Conference onNeural Networks IEEE Perth Australia November 1995

[7] D Karaboga An Idea Based on Honey Bee Swarm for Nu-merical Optimization Technical Report-tr06 Erciyes Uni-versity Engineering Faculty Computer EngineeringDepartment Kayseri Turkey 2005

[8] X S Yang and S Deb ldquoCuckoo search via Lvy flightsrdquo inProceedings of the World Congress on Nature amp BiologicallyInspired Computing (NaBIC) pp 210ndash214 IEEE KitakyushuJapan December 2009

[9] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf opti-mizerrdquo Advances in Engineering Software vol 69 pp 46ndash612014

[10] S Mirjalili and A Lewis ldquo(e whale optimization algorithmrdquoAdvances in Engineering Software vol 95 pp 51ndash67 2016

[11] B Bentouati L Chaib and S Chettih ldquoA hybrid whale al-gorithm and pattern search technique for optimal power flowproblemrdquo in Proceedings of the International Conference on


Modelling Identification and Control (ICMIC) pp 1048ndash1053 IEEE Dubai UAE 2016

[12] G Xiong J Zhang X Yuan D Shi Y He and G YaoldquoParameter extraction of solar photovoltaic models by meansof a hybrid differential evolution with whale optimizationalgorithmrdquo Solar Energy vol 176 pp 742ndash761 2018

[13] J Luo and B Shi ldquoA hybrid whale optimization algorithmbased on modified differential evolution for global optimi-zation problemsrdquo Applied Intelligence vol 49 no 5pp 1982ndash2000 2019

[14] M M Mafarja and S Mirjalili ldquoHybrid whale optimizationalgorithm with simulated annealing for feature selectionrdquoNeurocomputing vol 260 pp 302ndash312 2017

[15] I N Trivedi P Jangir A Kumar N Jangir and R Totlani ldquoAnovel hybrid PSO-WOA algorithm for global numericalfunctions optimizationrdquo Advances in Computer and Com-putational Sciences pp 53ndash60 2018

[16] A N Jadhav and N Gomathi ldquoWGC hybridization of ex-ponential grey wolf optimizer with whale optimization fordata clusteringrdquo Alexandria Engineering Journal vol 57no 3 pp 1569ndash1584 2018

[17] Z Xu Y Yu H Yachi J Ji Y Todo and S Gao ldquoA novelmemetic whale optimization algorithm for optimizationrdquoLecture Notes in Computer Science pp 384ndash396 2018

[18] A Kaveh and M Rastegar Moghaddam ldquoA hybrid WOA-CBO algorithm for construction site layout planning prob-lemrdquo Scientia Iranica vol 25 no 3 pp 1094ndash1104 2018

[19] S Khalilpourazari and S Khalilpourazary ldquoSCWOA an ef-ficient hybrid algorithm for parameter optimization of multi-pass milling processrdquo Journal of Industrial and ProductionEngineering vol 35 no 3 pp 135ndash147 2018

[20] N Singh and H Hachimi ldquoA new hybrid whale optimizeralgorithmwith mean strategy of grey wolf optimizer for globaloptimizationrdquo Mathematical and Computational Applica-tions vol 23 no 1 p 14 2018

[21] M Abdel-Basset G Manogaran D El-Shahat andS Mirjalili ldquoA hybrid whale optimization algorithm based onlocal search strategy for the permutation flow shop schedulingproblemrdquo Future Generation Computer Systems vol 85pp 129ndash145 2018

[22] N M Laskar K Guha I Chatterjee S ChandaK L Baishnab and P K Paul ldquoHWPSO a new hybrid whale-particle swarm optimization algorithm and its application inelectronic design optimization problemsrdquo Applied Intelli-gence vol 49 no 1 pp 265ndash291 2019

[23] H Wang Z Hu Y Sun et al ldquoModified backtracking searchoptimization algorithm inspired by simulated annealing forconstrained engineering optimization problemsrdquo Computa-tional Intelligence and Neuroscience vol 2018 Article ID9167414 27 pages 2018

[24] Z Li Z Hu Y Miao et al ldquoDeep-mining backtracking searchoptimization algorithm guided by collective wisdomrdquoMathematical Problems in Engineering vol 2019 Article ID2540102 2019

[25] H Wang Z Hu Y Sun Q Su and X Xia ldquoA novel modifiedBSA inspired by species evolution rule and simulatedannealing principle for constrained engineering optimizationproblemsrdquo Neural Computing and Applications vol 31 no 8pp 4157ndash4184 2019

[26] Z Hu Q Su X Yang and Z Xiong ldquoNot guaranteeingconvergence of differential evolution on a class of multimodalfunctionsrdquo Applied Soft Computing vol 41 pp 479ndash4872016

[27] M Zhang W Luo and X Wang ldquoDifferential evolution withdynamic stochastic selection for constrained optimizationrdquoInformation Sciences vol 178 no 15 pp 3043ndash3074 2008

[28] Y Wang Z Cai Y Zhou and Z Fan ldquoConstrained opti-mization based on hybrid evolutionary algorithm andadaptive constraint-handling techniquerdquo Structural andMultidisciplinary Optimization vol 37 no 4 pp 395ndash4132009

[29] H Liu Z Cai and Y Wang ldquoHybridizing particle swarmoptimization with differential evolution for constrained nu-merical and engineering optimizationrdquo Applied Soft Com-puting vol 10 no 2 pp 629ndash640 2010

[30] L Wang and L-p Li ldquoAn effective differential evolution withlevel comparison for constrained engineering designrdquoStructural and Multidisciplinary Optimization vol 41 no 6pp 947ndash963 2010

[31] A Sadollah A Bahreininejad H Eskandar and M HamdildquoMine blast algorithm a new population based algorithm forsolving constrained engineering optimization problemsrdquoApplied Soft Computing vol 13 no 5 pp 2592ndash2612 2013

[32] A Askarzadeh ldquoA novel metaheuristic method for solvingconstrained engineering optimization problems Crow searchalgorithmrdquo Computers amp Structures vol 169 pp 1ndash12 2016

[33] C A C Coello ldquoUse of a self-adaptive penalty approach forengineering optimization problemsrdquo Computers in Industryvol 41 no 2 pp 113ndash127 2000

[34] C A C Coello and E M Montes ldquoConstraint-handling ingenetic algorithms through the use of dominance-basedtournament selectionrdquo Advanced Engineering Informaticsvol 16 no 3 pp 193ndash203 2002

[35] Q He and L Wang ldquoAn effective co-evolutionary particleswarm optimization for constrained engineering designproblemsrdquo Engineering Applications of Artificial Intelligencevol 20 no 1 pp 89ndash99 2007

[36] Q He and L Wang ldquoA hybrid particle swarm optimizationwith a feasibility-based rule for constrained optimizationrdquoApplied Mathematics and Computation vol 186 no 2pp 1407ndash1422 2007

[37] F-z Huang L Wang and Q He ldquoAn effective co-evolu-tionary differential evolution for constrained optimizationrdquoApplied Mathematics and Computation vol 186 no 1pp 340ndash356 2007

[38] E Mezura-Montes C A C Coello and J Velzquez-ReyesldquoIncreasing successful offspring and diversity in differentialevolution for engineering designrdquo in Proceedings of theSeventh International Conference on Adaptive Computing inDesign andManufacture pp 131ndash139 Bristol UK April 2006

[39] B Akay and D Karaboga ldquoArtificial bee colony algorithm forlarge-scale problems and engineering design optimizationrdquoJournal of Intelligent Manufacturing vol 23 no 4pp 1001ndash1014 2012

[40] H Chickermane and H C Gea ldquoStructural optimizationusing a new local approximation methodrdquo InternationalJournal for Numerical Methods in Engineering vol 39 no 5pp 829ndash846 1996

[41] A H Gandomi X-S Yang and A H Alavi ldquoCuckoo searchalgorithm a metaheuristic approach to solve structural op-timization problemsrdquo Engineering with Computers vol 29no 1 pp 17ndash35 2013

[42] M-Y Cheng and D Prayogo ldquoSymbiotic organisms search anew metaheuristic optimization algorithmrdquo Computers ampStructures vol 139 pp 98ndash112 2014


[43] S Mirjalili ldquoMoth-flame optimization algorithm a novelnature-inspired heuristic paradigmrdquo Knowledge-Based Sys-tems vol 89 pp 228ndash249 2015

[44] G G Wang ldquoAdaptive response surface method usinginherited Latin hypercube design pointsrdquo Journal of Me-chanical Design vol 125 no 2 pp 210ndash220 2003


Bentouati et al [11] designed a hybrid whale optimizationalgorithm and pattern search technique called WOA-PS andtested it in optimal power flow problem Xiong et al [12] putforward a hybrid algorithm called DEWOA which combinedthe exploration of DE with the exploitation of WOA forextracting the accurate parameters of PV models Luo et al[13] proposed a MDE-WOA algorithm which combinesmodified DEGL algorithm and WOA to solve global opti-mization problems Mafarja et al [14] introduced a hybridalgorithm with whale optimization algorithm and simulatedannealing in which two hybridization models are consideredfor solving the feature selection problem

Another strategy to construct hybridWOA is by combiningoperators with complementary fusion features into WOA toform a superior algorithm Trivedi et al [15] hybridized particleswarm optimization with whale optimization algorithm calledPSO-WOA to handle global numerical function optimization(e main idea is to combine best characteristic of both algo-rithms to form a novel mutation operator Korashy et al [16]hybridized whale optimization algorithm and grey wolf opti-mizer algorithm called WOA-GWO for optimal coordinationof direction overcurrent relays Xu et al [17] proposed amemetic WOA (MWOA) through incorporating a chaoticlocal search into WOA to enhance its exploitation abilityKaveh et al [18] attempted to enhance the original formulationof the WOA by hybridizing it with some concepts of thecolliding bodies optimization (CBO) (e new method knownas WOA-CBO algorithm is applied to building site layoutplanning problem Khalilpourazari et al [19] introduced anovel hybrid algorithm termed as sine-cosineWOA (SCWOA)for parameter optimization problem of multipass millingprocess to decrease total production time Singh and Hachimi[20] proposed a new hybrid whale optimization algorithm withmean grey wolf optimizer (WOA-MGWO) for global opti-mization Abdel-Basset et al [21] proposed a new algorithmwhich combines theWOAwith a local search strategy to defeatthe permutation flow shop scheduling problem Laskar et al[22] proposed a new population-based hybrid metaheuristicalgorithm called whale-PSO algorithm (HWPSO) to managecomplex optimization problems

In this paper a hybrid algorithm framework namedhWOAlf is proposed by combining learning and comple-mentary fusion features into WOA Specifically the pro-posed algorithm framework adopts a learning parameter lp

according to adaptive adjustment operator instead of therandom parameter p namely the selection of searchequation depends on the process of lp learning from thesubpopulation Additionally to improve the exploration ofWOA operators with complementary fusion features areemployed to balance exploration and exploitation capacity(is paper adopts the DErand1 operator of differentialevolution (DE) and the mutate operator of backtrackingsearch optimization algorithm (BSA) to form two new al-gorithms respectively called WOA-DE and WOA-BSA(en twenty-three benchmark functions and six engi-neering design problems are employed to test the perfor-mance of WOA-DE and WOA-BSA Experimental resultsshow that WOA-DE and WOA-BSA are competitivecompared with some state-of-the-art algorithms

(e main contributions of this paper can be summarizedas follows

(i) Proposing a hybrid algorithm framework for WOAnamed hWOAlf First designing a learning pa-rameter lp according to adaptive adjustment oper-ator Second combining complementary fusionfeature operators into WOA

(ii) Forming two new algorithms WOA-DE and WOA-BSA by complementary fusion features in which theDErand1 operator of DE and the mutation oper-ator of BSA are incorporated to WOA respectively

(e rest of the paper is organized as follows Section 2gives a brief review of WOA DE and BSA (e proposedhybrid algorithm framework is presented in Section 3 InSection 4 the experimental results and analysis are givenFinally the conclusions are shown in Section 5

2 Preliminary

21 Whale Optimization Algorithm (WOA) Whale opti-mization algorithm is a new swarm-based metaheuristicoptimization algorithm proposed by Mirjalili and Lewis in2016 [10] which mimics the intelligent foraging behavior ofhumpback whales (e algorithm is inspired by bubble-nethunting strategy WOA includes three operators to simulatethe search for prey encircling prey and bubble-net foragingbehavior of humpback whales Among them the encirclingprey and spiral bubble-net attacking method is the exploi-tation phase while searching for prey is the explorationphase(emathematical model of the two phase is presentedbelow

211 Stage 1 Exploitation Phase (Encircling PreyBubble-NetAttacking Method) During the exploitation phase hump-back whales update their position according to two mech-anisms shrinking encircling mechanism (encircling prey)and spiral updating position (spiral bubble-net attackingmethod) Shrinking encircling mechanism is represented bythe following equations

Xrarr


rarrmiddot Disrarr

Disrarr

Crarr

middot Xlowastrarr(t) minus X

rarr(t)

111386811138681113868111386811138681113868

111386811138681113868111386811138681113868

(1)

where t represents the current iteration Xlowastrarr

(t) is the positionvector of the best solution obtained so far and X

rarr(t) is the

position vector Ararr

and Crarr

are coefficient vectors and thecalculation methods are as follows

Ararr

2 ararr

middot rrarr

minus ararr

Crarr

2 middot rrarr

(2)

where ararr is linearly decreased from 2 to 0 over the course of

iterations (in both exploration and exploitation phases) andrrarr is a random vector in [0 1]

(en the spiral updating position is mathematicallyexpressed by the following equation


Xrarr


middot ebl


Disprimerarr


rarr(t)

111386811138681113868111386811138681113868

111386811138681113868111386811138681113868

(3)

where Disprimerarr



Xrarr


rarrmiddot Disrarr

if plt 05

Disprimerarr


⎧⎪⎨

⎪⎩

(4)



Xrarr

(t + 1) Xrarr

rand(t) minus Ararr

middot Disrarr

Disrarr

Crarr

middot Xrarr

rand(t) minus Xrarr

(t)11138681113868111386811138681113868

11138681113868111386811138681113868(5)

where Xrarr





j 1 2 D(6)
















(11)



(12)




xij otherwise⎧⎨

⎩ (13)




xt+1i






Pij sim U lowj upj1113872 1113873 (15)











Vij Pij if mapij 1

Mij otherwise⎧⎨

⎩ (20)












Trarr

i Xlowastrarr

i minus Ararr

middot Crarr

middot Xlowastrarr

i minus Xrarr

i

111386811138681113868111386811138681113868

111386811138681113868111386811138681113868 (22)


Trarr

i Xrarr

other minus Ararr

middot Crarr

middot Xrarr

other minus Xrarr

i

11138681113868111386811138681113868

11138681113868111386811138681113868 (23)


Trarr

i Xlowastrarr

i minus Xrarr

i

111386811138681113868111386811138681113868

111386811138681113868111386811138681113868 middot ebl


i (24)


Trarr

i Xrarr


i minus Xrarr

i1113874 1113875or Trarr

i Xrarr


r2 minus Xrarr

r31113874 1113875

(25)



lp 1 +(s1n1)

2 +(s1n1) +(s2n2) (26)















Start






For each individual






lp lt rand







|A| lt 1

Yes

Yes

No

No

No

Output the results

t = t + 1

Yes WOA

BSA or DE

|A| lt 1

Yes

No











(43) Xlowast Xi















F1(x) 1113936Di1x

2i 30 [minus 100 100] 0

F2(x) 1113936Di1|xi| + 1113937

Di1 |xi| 30 [minus 10 10] 0

F3(x) 1113936Di1(1113936

ij1 xj)




Di1(xi + 05)2 30 [minus 100 100] 0

F7(x) 1113936Di1 ix4

i + random[0 1) 30 [minus 128 128] 0


Tabl

e2

Descriptio

nof

multim

odal

benchm

arkfunctio

ns

Functio

nD

Rang

efmin

F8(

x)

1113936

D i1

minusx

isin

(

|x

i|1113968

)30

[minus500

500]

minus4189829

times5

F9(

x)

1113936

D i1(

x2 i

minus10

cos(2π

xi)

+10

)30

[minus5125

12]

0

F10

(x

)

minus20

exp(

minus02

(1

D)1113936

D i1x

2 i

1113969

)minusexp(

(1

D)1113936

D i1cos

(2π

xi)

)+20

+e

30[minus323

2]0

F11

(x

)

(14000

)1113936D i1x

2 iminus

1113937D i1cos(

xi

iradic

)+1

30[minus600

600]

0F12

(x

)π

D10

sin(πy

1)+

1113936D

minus1

i1

(y

iminus1)

2 [1+10

sin2 (πy

i+1)

]+

(y

Dminus1)

21113966

1113967+

1113936D i1u

(x

i10

100

4)

30[minus505

0]0

yi

1

+x

i+14

u(

xi

ak

m)

k(

ximinus

a)m

x

igt

a

0minus

alt

xilt

a

k(

minusx

iminus

a)m

x

ilt

a

⎧⎪ ⎨ ⎪ ⎩

F13

(x

)01sin

2 (3π

x1)

+1113936

D i1

(x

iminus1)

2 [1+sin

2 (3π

xi+1

+1)

]+

(x

Dminus1)

2 [1+sin

2 (2π

xD

)]1113966

1113967+

1113936D i1u

(x

i51004)

30[minus505

0]0


Tabl

e3

Descriptio

nof

fixed-dim

ensio

nmultim

odal

benchm

arkfunctio

ns

Functio

nD

Rang

efmin

F14

(x

)

((1500)

+1113936

25 j1(1

j+

11139362 i1(

ximinus

aij

)6))

minus1

2[minus656

5]1

F15

(x

)

111393611 i1[

aiminus

(x1(

b2 i

+b

ix2)b

2 i+

bix

3+

x4)

]24

[minus5

5]000030

F16

(x

)4x

2 1minus21x

4 1+

(13)

x6 1

+x1x

2minus4x

2 2+4x

4 22

[minus5

5]minus10316

F17

(x

)

(x2

minus(514π

2 )x2 1

+(5π)

x1

minus6)

2+10

(1

minus(18π

))cos

x1

+10

2[minus5

5]0398

F18

(x

)

[1+

(x1

+x2

+1)

2 (19

minus14

x1

+3x

2 1minus14

x2

+6x

1x2

+3x

2 2)]

times[30

+(2x

1minus3x

2)2 (18

minus32

x1

+12

x2 1

+48

x2

minus36

x1x

2+27

x2 2)

]2

[minus2

2]3

F19

(x

)

minus1113936

4 i1c

iexp(

minus1113936

3 j1

aij

(x

jminus

pij

)2)

3[13

]minus386

F20

(x

)

minus1113936

4 i1c

iexp(

minus1113936

6 j1

aij

(x

jminus

pij

)2)

6[01

]minus332

F21

(x

)

minus1113936

5 i1[

(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus101232

F22

(x

)

minus1113936

7 i1[

(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus104028

F23

(x

)

minus1113936

1 i10

[(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus105363


Tabl

e4

Com

parisonof

optim

izationresults

obtained

fortheun

imod

albenchm

arkfunctio

ns

AL

Runs

30N

30a

ndT

1000

frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 1Be

st290E

minus12

(5)

0454643

(9)

0000908

(7)

104E

minus11

(6)

0042385

(8)

464E

minus17

(4)

276E

minus167(

3)692E

minus190(

1)162E

minus187(

2)

Mean

133E

minus11

(5)

1013223

(9)

0006257

(7)

973E

minus09

(6)

0063236

(8)

101E

minus16

(4)

146E

minus153(

3)107E

minus168(

2)145E

minus172(

1)

Std

998E

minus12

(4)

0311571

(8)

0010917

(7)

423E

minus08

(5)

0010899

(6)

376E

minus17

(3)

791E

minus153(

2)0(

1)0(

1)

F 2Be

st399E

minus08

(5)

0271733

(8)

0006726

(7)

100E

minus05

(6)

0575724

(9)

350E

minus08

(4)

331E

minus114(

3)246E

minus124(

2)201E

minus124(

1)

Mean

680E

minus08

(5)

0324236

(8)

0020418

(7)

0001313

(6)

0727765

(9)

520E

minus08

(4)

988E

minus105(

3)847E

minus113(

1)127E

minus112(

2)

Std

195E

minus08

(5)

0041092

(8)

0014974

(7)

0004947

(6)

005865(

9)117E

minus08

(4)

494E

minus104(

3)421E

minus112(

1)575E

minus112(

2)

F 3Be

st1080803597(

9)83765444(

2)394952176

(5)

3208562

(1)

308450285

(8)

195538197

(4)

167730974

(7)

17510162(

3)532871248

(6)

Mean

157231259

(9)

135934273

(2)

87822206(

4)14904551(

1)5228201445(

5)409456876

(3)

9735512836(

7)1004518144(

8)7621728202(

6)

Std

2683155994(

6)34026326(

2)315589252

(4)

7677907

(1)

1591552755(

5)12983509(

3)3898890192(

7)598648178

(9)

4199058749(

8)

F 4Be

st1080885

(7)

0742244

(6)

3262876

(8)

0311111

(5)

4216935

(9)

135E

minus08

(3)

0019063

(4)

568E

minus10

(2)

430E

minus19

(1)

Mean

1575045

(5)

0972736

(2)

5969996

(6)

0563087

(1)

6113542

(7)

1264362

(3)

17704825(

9)14259819(

8)1562263

(4)

Std

038751(

3)0123737

(1)

157968(

6)0153909

(2)

1246667

(4)

1284382

(5)

1458112

(8)

17566092(

9)6519626

(7)

F 5Be

st25398273(

2)39474199(

7)4367835

(8)

11716823(

1)91050677(

9)25861836(

3)26607341(

5)25886812(

4)26835697(

6)

Mean

3831135

(5)

149884822

(8)

114032052

(7)

60388778(

6)342973593

(9)

37244874(

4)271471(

1)27512334(

3)27496506(

2)

Std

21417578(

4)181590232

(8)

35216402(

6)43241123(

7)492144758

(9)

24882941(

5)0485147

(1)

0547347

(3)

0539586

(2)

F 6Be

st288E

minus12

(2)

06227

(9)

000076(

4)251E

minus11

(3)

0048128

(8)

450E

minus17

(1)

0007148

(5)

0012986

(6)

0015406

(7)

Mean

890E

minus12

(2)

1071114

(9)

000788(

4)968E

minus09

(3)

0070002

(6)

103E

minus16

(1)

0057661

(5)

0263633

(8)

020125(

7)

Std

567E

minus12

(2)

0279422

(9)

0008319

(4)

300E

minus08

(3)

0011199

(5)

446E

minus17

(1)

0078559

(6)

0224346

(8)

0192754

(7)

F 7Be

st0016252

(6)

0002011

(4)

0012074

(5)

0037104

(9)

0035229

(8)

0016733

(7)

0000115

(2)

000016(

3)00001

(1)

Mean

0027322

(5)

0006396

(4)

0030674

(6)

0078177

(9)

0055369

(8)

0049762

(7)

0001935

(3)

00015

(1)

0001579

(2)

Std

0005743

(5)

0002154

(4)

0012799

(7)

0023769

(9)

0011801

(6)

0021587

(8)

0001959

(3)

0001692

(1)

0001761

(2)

Rank

Best

68

75

93

41

2Mean

57

64

82

33

1Std

15

64

71

23

1


Tabl

e5

Com

parisonof

optim

izationresults

obtained

forthemultim

odal

benchm

arkfunctio

ns

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 8Be

stminus1256948662(

1)minus9288859115(

5)minus123198552

(3)

minus8562082036(

8)minus1256947154(

2)minus3257715637(

9)minus8674343572(

7)minus8985661303(

6)minus1057783345(

4)

Mean

minus1256553865(

2)minus8297653051(

4)minus1198240966(

3)minus6118655936(

8)minus1256880513(

1)minus243432431

(9)

minus6658410197(

7)minus7600141184(

6)minus7657917372(

5)

Std

21623781(

2)497301094

(5)

22238506(

3)1574830603(

9)0909111

(1)

355937095

(4)

863971262

(7)

861181057

(6)

1183224502(

8)

F 9Be

st44307253(

7)29244538(

6)16598597(

4)14992992(

3)17949826(

5)14924381(

2)0(

1)0(

1)0(

1)

Mean

56783475(

8)53437985(

7)25788633(

4)46538306(

6)25466452(

3)25802587(

5)0(

1)3672248

(2)

0(1)

Std

6266486

(4)

14333435(

7)403673(

2)14270927(

6)4772648

(3)

6630417

(5)

0(1)

20113732(

8)0(

1)

F 10

Best

423E

minus07

(3)

0226971

(6)

0007536

(5)

300E

minus06

(4)

1080969

(7)

545E

minus09

(2)

888E

minus16

(1)

888E

minus16

(1)

888E

minus16

(1)

Mean

702E

minus07

(5)

0349244

(8)

0020786

(6)

008323(

7)2063648

(9)

770E

minus09

(4)

444E

minus15

(3)

420E

minus15

(2)

397E

minus15

(1)

Std

152E

minus07

(5)

0070199

(7)

0011333

(6)

0317495

(8)

0425232

(9)

150E

minus09

(4)

229E

minus15

(3)

130E

minus15

(1)

203E

minus15

(2)

F 11

Best

144E

minus11

(3)

0503713

(6)

0001224

(4)

742E

minus12

(2)

0120691

(5)

3009729

(7)

0(1)

0(1)

0(1)

Mean

980E

minus10

(1)

0766039

(8)

0026851

(6)

0011493

(5)

0618727

(7)

8568461

(9)

0003104

(4)

0001092

(2)

0001282

(3)

Std

307E

minus09

(1)

011031(

7)0029255

(6)

0009701

(4)

0275077

(8)

4065895

(9)

0011816

(5)

0005981

(2)

0007021

(3)

F 12

Best

239E

minus13

(3)

0001193

(6)

300E

minus06

(4)

338E

minus14

(2)

0000383

(5)

375E

minus19

(1)

0001352

(7)

0003179

(9)

0002984

(8)

Mean

100E

minus12

(1)

0002505

(4)

630E

minus05

(2)

0003456

(5)

0000581

(3)

0151627

(8)

0006188

(6)

0013432

(7)

0409258

(9)

Std

105E

minus12

(1)

0000844

(4)

800E

minus05

(2)

0018927

(7)

0000121

(3)

0210923

(8)

0006962

(6)

0004444

(5)

160174(

9)

F 13

Best

170E

minus12

(2)

0023921

(6)

800E

minus05

(4)

523E

minus12

(3)

0006145

(5)

594E

minus18

(1)

0025211

(7)

0122758

(9)

0080548

(8)

Mean

431E

minus12

(1)

0044601

(6)

0000741

(2)

0001831

(3)

0013234

(5)

0007331

(4)

0233207

(7)

0468482

(8)

047475(

9)

Std

269E

minus12

(1)

0012881

(5)

0000969

(2)

0004165

(3)

0005965

(4)

0025745

(6)

0181659

(7)

0238598

(9)

0224147

(8)

F 14

Best

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998118

(2)

0998004

(1)

0998004

(1)

0998004

(1)

Mean

0998004

(1)

6813974

(7)

0998004

(1)

3459539

(5)

0998004

(1)

4082277

(6)

247579(

3)3454612

(4)

2108894

(2)

Std

0(1)

5363613

(9)

412E

minus17

(2)

2863628

(7)

187E

minus14

(3)

2690671

(5)

2445473

(4)

3273113

(8)

2790461

(6)

F 15

Best

0000352

(3)

0000419

(5)

0000307

(1)

0000308

(2)

0000401

(4)

0001284

(6)

0000308

(2)

0000307

(1)

0000307

(1)

Mean

0000577

(4)

0006093

(9)

0000329

(1)

0000794

(6)

0004004

(8)

0002584

(7)

0000601

(5)

0000513

(3)

0000468

(2)

Std

0000109

(2)

0008761

(9)

410E

minus05

(1)

0000233

(3)

0007444

(8)

0002019

(7)

0000315

(4)

0000377

(5)

0000382

(6)

F 16

Best

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Mean

minus1031628

(1)

minus0977217

(2)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Std

671E

minus16

(4)

0207068

(8)

612E

minus16

(3)

678E

minus16

(5)

353E

minus08

(7)

513E

minus16

(1)

386E

minus11

(6)

671E

minus16

(4)

598E

minus16

(2)

F 17

Best

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

Mean

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397888

(2)

0397887

(1)

0397887

(1)

Std

0(1)

933E

minus11

(3)

0(1)

0(1)

437E

minus08

(4)

0(1)

930E

minus07

(5)

0(1)

125E

minus13

(2)

F 18

Best

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

Mean

3(1)

39(

3)3(

1)3(

1)3900137

(4)

3(1)

3000014

(2)

3(1)

3(1)

Std

127E

minus15

(2)

4929503

(9)

202E

minus15

(5)

152E

minus15

(3)

4929483

(8)

267E

minus15

(6)

380E

minus05

(7)

782E

minus16

(1)

166E

minus15

(4)

F 19

Best

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

Mean

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862628

(2)

minus3862782

(1)

minus3862782

(1)

Std

271E

minus15

(6)

229E

minus15

(1)

271E

minus15

(6)

270E

minus15

(5)

100E

minus06

(7)

239E

minus15

(2)

0000366

(8)

266E

minus15

(4)

251E

minus15

(3)

F 20

Best

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321994

(2)

minus3321995

(1)

minus3321989

(3)

minus3321995

(1)

minus3321995

(1)

Mean

minus3321989

(2)

minus3270475

(6)

minus3321995

(1)

minus3282364

(5)

minus330217(

3)minus3321995

(1)

minus3264959

(7)

minus3293815

(4)

minus3254599

(8)

Std

320E

minus05

(3)

0059923

(7)

173E

minus14

(2)

0057005

(6)

0045065

(4)

147E

minus15

(1)

0071826

(9)

0051995

(5)

0059944

(8)

F 21

Best

minus101532(

1)minus101532(

1)minus101532(

1)minus101532(

1)minus10153199(

2)minus101532(

1)minus10153181(

3)minus101532(

1)minus101532(

1)

Mean

minus10049112(

2)minus5725156

(8)

minus101532(

1)minus8062579

(6)

minus5406265

(9)

minus6043604

(7)

minus8882256

(4)

minus8882841

(3)

minus8462151

(5)

Std

0531474

(2)

3327602

(7)

115E

minus09

(1)

3091228

(6)

3673677

(9)

3495936

(8)

2378758

(3)

2378893

(4)

2686626

(5)


Tabl

e5

Con

tinued

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 22

Best

minus10402941(

1)minus10402941(

1)minus10402941(

1)minus10402941(

1)minus1040294

(2)

minus10402941(

1)minus1040292

(3)

minus10402941(

1)minus10402941(

1)

Mean

minus10211001(

2)minus5529595

(8)

minus10402941(

1)minus9521178

(3)

minus6545124

(7)

minus10402941(

1)minus8448955

(4)

minus7763089

(6)

minus796332(

5)

Std

1045029

(3)

3343034

(8)

169E

minus14

(2)

2005406

(4)

3686937

(9)

132E

minus15

(1)

2836906

(5)

3128103

(7)

286345(

6)

F 23

Best

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus10536335(

2)minus1053641

(1)

minus1053641

(1)

Mean

minus10536383(

2)minus5855753

(8)

minus1053641

(1)

minus916093(

4)minus5180275

(9)

minus10175835(

3)minus9003007

(5)

minus7990287

(7)

minus8647563

(6)

Std

0000108

(2)

3655594

(9)

111E

minus09

(1)

2570428

(4)

3329307

(8)

1485436

(3)

2591415

(5)

3225805

(7)

2733822

(6)

Rank

Best

18

32

75

64

2Mean

29

16

87

53

4Std

19

26

83

74

5




200 400 600 800 1000

10minus150

10minus100

10minus50

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(a)

200 400 600 800 1000

102

104

106

108

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(b)

Ave

rage

ln (f

lowast minus

f min

)

200 400 600 800 1000

10minus2

100

102

IterationDEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(c)

200 400 600 800 1000

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(d)

200 400 600 800 1000

10minus15

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(e)

200 400 600 800 1000

10minus4

10minus3

10minus2

10minus1

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(f )









l l

l x1 x1x2

P







0 100 200 300 400 500242

2422

2424

2426

2428

243

2432

2434


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


Th

R

Ts

R

L







0 1000 2000 3000 4000 500035

4

45

5

55

6


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


l1

d1

d2l2

z1 z2







0 100 200 300 400 500345

35

355

36

365

37


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


A D

B

C













0 100 200 300 400 500minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


1 2 3 4 5

xi

xiConstantthickness







0 200 400 600 800 100001

02

03

04

05

06

07

08

09


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA





P

h

b

tftwL



5 Conclusion




Appendix



minf(x) 22


subject to

g1(x)

2

radicx1 + x2

2radic

x21 + 2x1x2

P minus σ le 0

g2(x) x2

2radic

x21 + 2x1x2

P minus σ le 0

g3(x) 1

2

radicx2 + x1

P minus σ le 0

0lexi le 1 i 1 2

l 100 cm

P 2 kNcm2

σ 2 kNcm2

(A1)




0 100 200 300 400 500minus188

minus186

minus184

minus182

minus18

minus178

minus176


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA




minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0


33 + 1296000le 0


0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)


minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References

















































Xrarr


middot ebl


Disprimerarr


rarr(t)

111386811138681113868111386811138681113868

111386811138681113868111386811138681113868

(3)

where Disprimerarr



Xrarr


rarrmiddot Disrarr

if plt 05

Disprimerarr


⎧⎪⎨

⎪⎩

(4)



Xrarr

(t + 1) Xrarr

rand(t) minus Ararr

middot Disrarr

Disrarr

Crarr

middot Xrarr

rand(t) minus Xrarr

(t)11138681113868111386811138681113868

11138681113868111386811138681113868(5)

where Xrarr





j 1 2 D(6)
















(11)



(12)




xij otherwise⎧⎨

⎩ (13)




xt+1i






Pij sim U lowj upj1113872 1113873 (15)











Vij Pij if mapij 1

Mij otherwise⎧⎨

⎩ (20)












Trarr

i Xlowastrarr

i minus Ararr

middot Crarr

middot Xlowastrarr

i minus Xrarr

i

111386811138681113868111386811138681113868

111386811138681113868111386811138681113868 (22)


Trarr

i Xrarr

other minus Ararr

middot Crarr

middot Xrarr

other minus Xrarr

i

11138681113868111386811138681113868

11138681113868111386811138681113868 (23)


Trarr

i Xlowastrarr

i minus Xrarr

i

111386811138681113868111386811138681113868

111386811138681113868111386811138681113868 middot ebl


i (24)


Trarr

i Xrarr


i minus Xrarr

i1113874 1113875or Trarr

i Xrarr


r2 minus Xrarr

r31113874 1113875

(25)



lp 1 +(s1n1)

2 +(s1n1) +(s2n2) (26)















Start






For each individual






lp lt rand







|A| lt 1

Yes

Yes

No

No

No

Output the results

t = t + 1

Yes WOA

BSA or DE

|A| lt 1

Yes

No











(43) Xlowast Xi















F1(x) 1113936Di1x

2i 30 [minus 100 100] 0

F2(x) 1113936Di1|xi| + 1113937

Di1 |xi| 30 [minus 10 10] 0

F3(x) 1113936Di1(1113936

ij1 xj)




Di1(xi + 05)2 30 [minus 100 100] 0

F7(x) 1113936Di1 ix4

i + random[0 1) 30 [minus 128 128] 0


Tabl

e2

Descriptio

nof

multim

odal

benchm

arkfunctio

ns

Functio

nD

Rang

efmin

F8(

x)

1113936

D i1

minusx

isin

(

|x

i|1113968

)30

[minus500

500]

minus4189829

times5

F9(

x)

1113936

D i1(

x2 i

minus10

cos(2π

xi)

+10

)30

[minus5125

12]

0

F10

(x

)

minus20

exp(

minus02

(1

D)1113936

D i1x

2 i

1113969

)minusexp(

(1

D)1113936

D i1cos

(2π

xi)

)+20

+e

30[minus323

2]0

F11

(x

)

(14000

)1113936D i1x

2 iminus

1113937D i1cos(

xi

iradic

)+1

30[minus600

600]

0F12

(x

)π

D10

sin(πy

1)+

1113936D

minus1

i1

(y

iminus1)

2 [1+10

sin2 (πy

i+1)

]+

(y

Dminus1)

21113966

1113967+

1113936D i1u

(x

i10

100

4)

30[minus505

0]0

yi

1

+x

i+14

u(

xi

ak

m)

k(

ximinus

a)m

x

igt

a

0minus

alt

xilt

a

k(

minusx

iminus

a)m

x

ilt

a

⎧⎪ ⎨ ⎪ ⎩

F13

(x

)01sin

2 (3π

x1)

+1113936

D i1

(x

iminus1)

2 [1+sin

2 (3π

xi+1

+1)

]+

(x

Dminus1)

2 [1+sin

2 (2π

xD

)]1113966

1113967+

1113936D i1u

(x

i51004)

30[minus505

0]0


Tabl

e3

Descriptio

nof

fixed-dim

ensio

nmultim

odal

benchm

arkfunctio

ns

Functio

nD

Rang

efmin

F14

(x

)

((1500)

+1113936

25 j1(1

j+

11139362 i1(

ximinus

aij

)6))

minus1

2[minus656

5]1

F15

(x

)

111393611 i1[

aiminus

(x1(

b2 i

+b

ix2)b

2 i+

bix

3+

x4)

]24

[minus5

5]000030

F16

(x

)4x

2 1minus21x

4 1+

(13)

x6 1

+x1x

2minus4x

2 2+4x

4 22

[minus5

5]minus10316

F17

(x

)

(x2

minus(514π

2 )x2 1

+(5π)

x1

minus6)

2+10

(1

minus(18π

))cos

x1

+10

2[minus5

5]0398

F18

(x

)

[1+

(x1

+x2

+1)

2 (19

minus14

x1

+3x

2 1minus14

x2

+6x

1x2

+3x

2 2)]

times[30

+(2x

1minus3x

2)2 (18

minus32

x1

+12

x2 1

+48

x2

minus36

x1x

2+27

x2 2)

]2

[minus2

2]3

F19

(x

)

minus1113936

4 i1c

iexp(

minus1113936

3 j1

aij

(x

jminus

pij

)2)

3[13

]minus386

F20

(x

)

minus1113936

4 i1c

iexp(

minus1113936

6 j1

aij

(x

jminus

pij

)2)

6[01

]minus332

F21

(x

)

minus1113936

5 i1[

(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus101232

F22

(x

)

minus1113936

7 i1[

(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus104028

F23

(x

)

minus1113936

1 i10

[(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus105363


Tabl

e4

Com

parisonof

optim

izationresults

obtained

fortheun

imod

albenchm

arkfunctio

ns

AL

Runs

30N

30a

ndT

1000

frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 1Be

st290E

minus12

(5)

0454643

(9)

0000908

(7)

104E

minus11

(6)

0042385

(8)

464E

minus17

(4)

276E

minus167(

3)692E

minus190(

1)162E

minus187(

2)

Mean

133E

minus11

(5)

1013223

(9)

0006257

(7)

973E

minus09

(6)

0063236

(8)

101E

minus16

(4)

146E

minus153(

3)107E

minus168(

2)145E

minus172(

1)

Std

998E

minus12

(4)

0311571

(8)

0010917

(7)

423E

minus08

(5)

0010899

(6)

376E

minus17

(3)

791E

minus153(

2)0(

1)0(

1)

F 2Be

st399E

minus08

(5)

0271733

(8)

0006726

(7)

100E

minus05

(6)

0575724

(9)

350E

minus08

(4)

331E

minus114(

3)246E

minus124(

2)201E

minus124(

1)

Mean

680E

minus08

(5)

0324236

(8)

0020418

(7)

0001313

(6)

0727765

(9)

520E

minus08

(4)

988E

minus105(

3)847E

minus113(

1)127E

minus112(

2)

Std

195E

minus08

(5)

0041092

(8)

0014974

(7)

0004947

(6)

005865(

9)117E

minus08

(4)

494E

minus104(

3)421E

minus112(

1)575E

minus112(

2)

F 3Be

st1080803597(

9)83765444(

2)394952176

(5)

3208562

(1)

308450285

(8)

195538197

(4)

167730974

(7)

17510162(

3)532871248

(6)

Mean

157231259

(9)

135934273

(2)

87822206(

4)14904551(

1)5228201445(

5)409456876

(3)

9735512836(

7)1004518144(

8)7621728202(

6)

Std

2683155994(

6)34026326(

2)315589252

(4)

7677907

(1)

1591552755(

5)12983509(

3)3898890192(

7)598648178

(9)

4199058749(

8)

F 4Be

st1080885

(7)

0742244

(6)

3262876

(8)

0311111

(5)

4216935

(9)

135E

minus08

(3)

0019063

(4)

568E

minus10

(2)

430E

minus19

(1)

Mean

1575045

(5)

0972736

(2)

5969996

(6)

0563087

(1)

6113542

(7)

1264362

(3)

17704825(

9)14259819(

8)1562263

(4)

Std

038751(

3)0123737

(1)

157968(

6)0153909

(2)

1246667

(4)

1284382

(5)

1458112

(8)

17566092(

9)6519626

(7)

F 5Be

st25398273(

2)39474199(

7)4367835

(8)

11716823(

1)91050677(

9)25861836(

3)26607341(

5)25886812(

4)26835697(

6)

Mean

3831135

(5)

149884822

(8)

114032052

(7)

60388778(

6)342973593

(9)

37244874(

4)271471(

1)27512334(

3)27496506(

2)

Std

21417578(

4)181590232

(8)

35216402(

6)43241123(

7)492144758

(9)

24882941(

5)0485147

(1)

0547347

(3)

0539586

(2)

F 6Be

st288E

minus12

(2)

06227

(9)

000076(

4)251E

minus11

(3)

0048128

(8)

450E

minus17

(1)

0007148

(5)

0012986

(6)

0015406

(7)

Mean

890E

minus12

(2)

1071114

(9)

000788(

4)968E

minus09

(3)

0070002

(6)

103E

minus16

(1)

0057661

(5)

0263633

(8)

020125(

7)

Std

567E

minus12

(2)

0279422

(9)

0008319

(4)

300E

minus08

(3)

0011199

(5)

446E

minus17

(1)

0078559

(6)

0224346

(8)

0192754

(7)

F 7Be

st0016252

(6)

0002011

(4)

0012074

(5)

0037104

(9)

0035229

(8)

0016733

(7)

0000115

(2)

000016(

3)00001

(1)

Mean

0027322

(5)

0006396

(4)

0030674

(6)

0078177

(9)

0055369

(8)

0049762

(7)

0001935

(3)

00015

(1)

0001579

(2)

Std

0005743

(5)

0002154

(4)

0012799

(7)

0023769

(9)

0011801

(6)

0021587

(8)

0001959

(3)

0001692

(1)

0001761

(2)

Rank

Best

68

75

93

41

2Mean

57

64

82

33

1Std

15

64

71

23

1


Tabl

e5

Com

parisonof

optim

izationresults

obtained

forthemultim

odal

benchm

arkfunctio

ns

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 8Be

stminus1256948662(

1)minus9288859115(

5)minus123198552

(3)

minus8562082036(

8)minus1256947154(

2)minus3257715637(

9)minus8674343572(

7)minus8985661303(

6)minus1057783345(

4)

Mean

minus1256553865(

2)minus8297653051(

4)minus1198240966(

3)minus6118655936(

8)minus1256880513(

1)minus243432431

(9)

minus6658410197(

7)minus7600141184(

6)minus7657917372(

5)

Std

21623781(

2)497301094

(5)

22238506(

3)1574830603(

9)0909111

(1)

355937095

(4)

863971262

(7)

861181057

(6)

1183224502(

8)

F 9Be

st44307253(

7)29244538(

6)16598597(

4)14992992(

3)17949826(

5)14924381(

2)0(

1)0(

1)0(

1)

Mean

56783475(

8)53437985(

7)25788633(

4)46538306(

6)25466452(

3)25802587(

5)0(

1)3672248

(2)

0(1)

Std

6266486

(4)

14333435(

7)403673(

2)14270927(

6)4772648

(3)

6630417

(5)

0(1)

20113732(

8)0(

1)

F 10

Best

423E

minus07

(3)

0226971

(6)

0007536

(5)

300E

minus06

(4)

1080969

(7)

545E

minus09

(2)

888E

minus16

(1)

888E

minus16

(1)

888E

minus16

(1)

Mean

702E

minus07

(5)

0349244

(8)

0020786

(6)

008323(

7)2063648

(9)

770E

minus09

(4)

444E

minus15

(3)

420E

minus15

(2)

397E

minus15

(1)

Std

152E

minus07

(5)

0070199

(7)

0011333

(6)

0317495

(8)

0425232

(9)

150E

minus09

(4)

229E

minus15

(3)

130E

minus15

(1)

203E

minus15

(2)

F 11

Best

144E

minus11

(3)

0503713

(6)

0001224

(4)

742E

minus12

(2)

0120691

(5)

3009729

(7)

0(1)

0(1)

0(1)

Mean

980E

minus10

(1)

0766039

(8)

0026851

(6)

0011493

(5)

0618727

(7)

8568461

(9)

0003104

(4)

0001092

(2)

0001282

(3)

Std

307E

minus09

(1)

011031(

7)0029255

(6)

0009701

(4)

0275077

(8)

4065895

(9)

0011816

(5)

0005981

(2)

0007021

(3)

F 12

Best

239E

minus13

(3)

0001193

(6)

300E

minus06

(4)

338E

minus14

(2)

0000383

(5)

375E

minus19

(1)

0001352

(7)

0003179

(9)

0002984

(8)

Mean

100E

minus12

(1)

0002505

(4)

630E

minus05

(2)

0003456

(5)

0000581

(3)

0151627

(8)

0006188

(6)

0013432

(7)

0409258

(9)

Std

105E

minus12

(1)

0000844

(4)

800E

minus05

(2)

0018927

(7)

0000121

(3)

0210923

(8)

0006962

(6)

0004444

(5)

160174(

9)

F 13

Best

170E

minus12

(2)

0023921

(6)

800E

minus05

(4)

523E

minus12

(3)

0006145

(5)

594E

minus18

(1)

0025211

(7)

0122758

(9)

0080548

(8)

Mean

431E

minus12

(1)

0044601

(6)

0000741

(2)

0001831

(3)

0013234

(5)

0007331

(4)

0233207

(7)

0468482

(8)

047475(

9)

Std

269E

minus12

(1)

0012881

(5)

0000969

(2)

0004165

(3)

0005965

(4)

0025745

(6)

0181659

(7)

0238598

(9)

0224147

(8)

F 14

Best

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998118

(2)

0998004

(1)

0998004

(1)

0998004

(1)

Mean

0998004

(1)

6813974

(7)

0998004

(1)

3459539

(5)

0998004

(1)

4082277

(6)

247579(

3)3454612

(4)

2108894

(2)

Std

0(1)

5363613

(9)

412E

minus17

(2)

2863628

(7)

187E

minus14

(3)

2690671

(5)

2445473

(4)

3273113

(8)

2790461

(6)

F 15

Best

0000352

(3)

0000419

(5)

0000307

(1)

0000308

(2)

0000401

(4)

0001284

(6)

0000308

(2)

0000307

(1)

0000307

(1)

Mean

0000577

(4)

0006093

(9)

0000329

(1)

0000794

(6)

0004004

(8)

0002584

(7)

0000601

(5)

0000513

(3)

0000468

(2)

Std

0000109

(2)

0008761

(9)

410E

minus05

(1)

0000233

(3)

0007444

(8)

0002019

(7)

0000315

(4)

0000377

(5)

0000382

(6)

F 16

Best

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Mean

minus1031628

(1)

minus0977217

(2)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Std

671E

minus16

(4)

0207068

(8)

612E

minus16

(3)

678E

minus16

(5)

353E

minus08

(7)

513E

minus16

(1)

386E

minus11

(6)

671E

minus16

(4)

598E

minus16

(2)

F 17

Best

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

Mean

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397888

(2)

0397887

(1)

0397887

(1)

Std

0(1)

933E

minus11

(3)

0(1)

0(1)

437E

minus08

(4)

0(1)

930E

minus07

(5)

0(1)

125E

minus13

(2)

F 18

Best

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

Mean

3(1)

39(

3)3(

1)3(

1)3900137

(4)

3(1)

3000014

(2)

3(1)

3(1)

Std

127E

minus15

(2)

4929503

(9)

202E

minus15

(5)

152E

minus15

(3)

4929483

(8)

267E

minus15

(6)

380E

minus05

(7)

782E

minus16

(1)

166E

minus15

(4)

F 19

Best

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

Mean

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862628

(2)

minus3862782

(1)

minus3862782

(1)

Std

271E

minus15

(6)

229E

minus15

(1)

271E

minus15

(6)

270E

minus15

(5)

100E

minus06

(7)

239E

minus15

(2)

0000366

(8)

266E

minus15

(4)

251E

minus15

(3)

F 20

Best

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321994

(2)

minus3321995

(1)

minus3321989

(3)

minus3321995

(1)

minus3321995

(1)

Mean

minus3321989

(2)

minus3270475

(6)

minus3321995

(1)

minus3282364

(5)

minus330217(

3)minus3321995

(1)

minus3264959

(7)

minus3293815

(4)

minus3254599

(8)

Std

320E

minus05

(3)

0059923

(7)

173E

minus14

(2)

0057005

(6)

0045065

(4)

147E

minus15

(1)

0071826

(9)

0051995

(5)

0059944

(8)

F 21

Best

minus101532(

1)minus101532(

1)minus101532(

1)minus101532(

1)minus10153199(

2)minus101532(

1)minus10153181(

3)minus101532(

1)minus101532(

1)

Mean

minus10049112(

2)minus5725156

(8)

minus101532(

1)minus8062579

(6)

minus5406265

(9)

minus6043604

(7)

minus8882256

(4)

minus8882841

(3)

minus8462151

(5)

Std

0531474

(2)

3327602

(7)

115E

minus09

(1)

3091228

(6)

3673677

(9)

3495936

(8)

2378758

(3)

2378893

(4)

2686626

(5)


Tabl

e5

Con

tinued

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 22

Best

minus10402941(

1)minus10402941(

1)minus10402941(

1)minus10402941(

1)minus1040294

(2)

minus10402941(

1)minus1040292

(3)

minus10402941(

1)minus10402941(

1)

Mean

minus10211001(

2)minus5529595

(8)

minus10402941(

1)minus9521178

(3)

minus6545124

(7)

minus10402941(

1)minus8448955

(4)

minus7763089

(6)

minus796332(

5)

Std

1045029

(3)

3343034

(8)

169E

minus14

(2)

2005406

(4)

3686937

(9)

132E

minus15

(1)

2836906

(5)

3128103

(7)

286345(

6)

F 23

Best

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus10536335(

2)minus1053641

(1)

minus1053641

(1)

Mean

minus10536383(

2)minus5855753

(8)

minus1053641

(1)

minus916093(

4)minus5180275

(9)

minus10175835(

3)minus9003007

(5)

minus7990287

(7)

minus8647563

(6)

Std

0000108

(2)

3655594

(9)

111E

minus09

(1)

2570428

(4)

3329307

(8)

1485436

(3)

2591415

(5)

3225805

(7)

2733822

(6)

Rank

Best

18

32

75

64

2Mean

29

16

87

53

4Std

19

26

83

74

5




200 400 600 800 1000

10minus150

10minus100

10minus50

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(a)

200 400 600 800 1000

102

104

106

108

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(b)

Ave

rage

ln (f

lowast minus

f min

)

200 400 600 800 1000

10minus2

100

102

IterationDEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(c)

200 400 600 800 1000

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(d)

200 400 600 800 1000

10minus15

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(e)

200 400 600 800 1000

10minus4

10minus3

10minus2

10minus1

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(f )









l l

l x1 x1x2

P







0 100 200 300 400 500242

2422

2424

2426

2428

243

2432

2434


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


Th

R

Ts

R

L







0 1000 2000 3000 4000 500035

4

45

5

55

6


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


l1

d1

d2l2

z1 z2







0 100 200 300 400 500345

35

355

36

365

37


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


A D

B

C













0 100 200 300 400 500minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


1 2 3 4 5

xi

xiConstantthickness







0 200 400 600 800 100001

02

03

04

05

06

07

08

09


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA





P

h

b

tftwL



5 Conclusion




Appendix



minf(x) 22


subject to

g1(x)

2

radicx1 + x2

2radic

x21 + 2x1x2

P minus σ le 0

g2(x) x2

2radic

x21 + 2x1x2

P minus σ le 0

g3(x) 1

2

radicx2 + x1

P minus σ le 0

0lexi le 1 i 1 2

l 100 cm

P 2 kNcm2

σ 2 kNcm2

(A1)




0 100 200 300 400 500minus188

minus186

minus184

minus182

minus18

minus178

minus176


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA




minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0


33 + 1296000le 0


0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)


minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References


















































xt+1i






Pij sim U lowj upj1113872 1113873 (15)











Vij Pij if mapij 1

Mij otherwise⎧⎨

⎩ (20)












Trarr

i Xlowastrarr

i minus Ararr

middot Crarr

middot Xlowastrarr

i minus Xrarr

i

111386811138681113868111386811138681113868

111386811138681113868111386811138681113868 (22)


Trarr

i Xrarr

other minus Ararr

middot Crarr

middot Xrarr

other minus Xrarr

i

11138681113868111386811138681113868

11138681113868111386811138681113868 (23)


Trarr

i Xlowastrarr

i minus Xrarr

i

111386811138681113868111386811138681113868

111386811138681113868111386811138681113868 middot ebl


i (24)


Trarr

i Xrarr


i minus Xrarr

i1113874 1113875or Trarr

i Xrarr


r2 minus Xrarr

r31113874 1113875

(25)



lp 1 +(s1n1)

2 +(s1n1) +(s2n2) (26)















Start






For each individual






lp lt rand







|A| lt 1

Yes

Yes

No

No

No

Output the results

t = t + 1

Yes WOA

BSA or DE

|A| lt 1

Yes

No











(43) Xlowast Xi















F1(x) 1113936Di1x

2i 30 [minus 100 100] 0

F2(x) 1113936Di1|xi| + 1113937

Di1 |xi| 30 [minus 10 10] 0

F3(x) 1113936Di1(1113936

ij1 xj)




Di1(xi + 05)2 30 [minus 100 100] 0

F7(x) 1113936Di1 ix4

i + random[0 1) 30 [minus 128 128] 0


Tabl

e2

Descriptio

nof

multim

odal

benchm

arkfunctio

ns

Functio

nD

Rang

efmin

F8(

x)

1113936

D i1

minusx

isin

(

|x

i|1113968

)30

[minus500

500]

minus4189829

times5

F9(

x)

1113936

D i1(

x2 i

minus10

cos(2π

xi)

+10

)30

[minus5125

12]

0

F10

(x

)

minus20

exp(

minus02

(1

D)1113936

D i1x

2 i

1113969

)minusexp(

(1

D)1113936

D i1cos

(2π

xi)

)+20

+e

30[minus323

2]0

F11

(x

)

(14000

)1113936D i1x

2 iminus

1113937D i1cos(

xi

iradic

)+1

30[minus600

600]

0F12

(x

)π

D10

sin(πy

1)+

1113936D

minus1

i1

(y

iminus1)

2 [1+10

sin2 (πy

i+1)

]+

(y

Dminus1)

21113966

1113967+

1113936D i1u

(x

i10

100

4)

30[minus505

0]0

yi

1

+x

i+14

u(

xi

ak

m)

k(

ximinus

a)m

x

igt

a

0minus

alt

xilt

a

k(

minusx

iminus

a)m

x

ilt

a

⎧⎪ ⎨ ⎪ ⎩

F13

(x

)01sin

2 (3π

x1)

+1113936

D i1

(x

iminus1)

2 [1+sin

2 (3π

xi+1

+1)

]+

(x

Dminus1)

2 [1+sin

2 (2π

xD

)]1113966

1113967+

1113936D i1u

(x

i51004)

30[minus505

0]0


Tabl

e3

Descriptio

nof

fixed-dim

ensio

nmultim

odal

benchm

arkfunctio

ns

Functio

nD

Rang

efmin

F14

(x

)

((1500)

+1113936

25 j1(1

j+

11139362 i1(

ximinus

aij

)6))

minus1

2[minus656

5]1

F15

(x

)

111393611 i1[

aiminus

(x1(

b2 i

+b

ix2)b

2 i+

bix

3+

x4)

]24

[minus5

5]000030

F16

(x

)4x

2 1minus21x

4 1+

(13)

x6 1

+x1x

2minus4x

2 2+4x

4 22

[minus5

5]minus10316

F17

(x

)

(x2

minus(514π

2 )x2 1

+(5π)

x1

minus6)

2+10

(1

minus(18π

))cos

x1

+10

2[minus5

5]0398

F18

(x

)

[1+

(x1

+x2

+1)

2 (19

minus14

x1

+3x

2 1minus14

x2

+6x

1x2

+3x

2 2)]

times[30

+(2x

1minus3x

2)2 (18

minus32

x1

+12

x2 1

+48

x2

minus36

x1x

2+27

x2 2)

]2

[minus2

2]3

F19

(x

)

minus1113936

4 i1c

iexp(

minus1113936

3 j1

aij

(x

jminus

pij

)2)

3[13

]minus386

F20

(x

)

minus1113936

4 i1c

iexp(

minus1113936

6 j1

aij

(x

jminus

pij

)2)

6[01

]minus332

F21

(x

)

minus1113936

5 i1[

(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus101232

F22

(x

)

minus1113936

7 i1[

(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus104028

F23

(x

)

minus1113936

1 i10

[(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus105363


Tabl

e4

Com

parisonof

optim

izationresults

obtained

fortheun

imod

albenchm

arkfunctio

ns

AL

Runs

30N

30a

ndT

1000

frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 1Be

st290E

minus12

(5)

0454643

(9)

0000908

(7)

104E

minus11

(6)

0042385

(8)

464E

minus17

(4)

276E

minus167(

3)692E

minus190(

1)162E

minus187(

2)

Mean

133E

minus11

(5)

1013223

(9)

0006257

(7)

973E

minus09

(6)

0063236

(8)

101E

minus16

(4)

146E

minus153(

3)107E

minus168(

2)145E

minus172(

1)

Std

998E

minus12

(4)

0311571

(8)

0010917

(7)

423E

minus08

(5)

0010899

(6)

376E

minus17

(3)

791E

minus153(

2)0(

1)0(

1)

F 2Be

st399E

minus08

(5)

0271733

(8)

0006726

(7)

100E

minus05

(6)

0575724

(9)

350E

minus08

(4)

331E

minus114(

3)246E

minus124(

2)201E

minus124(

1)

Mean

680E

minus08

(5)

0324236

(8)

0020418

(7)

0001313

(6)

0727765

(9)

520E

minus08

(4)

988E

minus105(

3)847E

minus113(

1)127E

minus112(

2)

Std

195E

minus08

(5)

0041092

(8)

0014974

(7)

0004947

(6)

005865(

9)117E

minus08

(4)

494E

minus104(

3)421E

minus112(

1)575E

minus112(

2)

F 3Be

st1080803597(

9)83765444(

2)394952176

(5)

3208562

(1)

308450285

(8)

195538197

(4)

167730974

(7)

17510162(

3)532871248

(6)

Mean

157231259

(9)

135934273

(2)

87822206(

4)14904551(

1)5228201445(

5)409456876

(3)

9735512836(

7)1004518144(

8)7621728202(

6)

Std

2683155994(

6)34026326(

2)315589252

(4)

7677907

(1)

1591552755(

5)12983509(

3)3898890192(

7)598648178

(9)

4199058749(

8)

F 4Be

st1080885

(7)

0742244

(6)

3262876

(8)

0311111

(5)

4216935

(9)

135E

minus08

(3)

0019063

(4)

568E

minus10

(2)

430E

minus19

(1)

Mean

1575045

(5)

0972736

(2)

5969996

(6)

0563087

(1)

6113542

(7)

1264362

(3)

17704825(

9)14259819(

8)1562263

(4)

Std

038751(

3)0123737

(1)

157968(

6)0153909

(2)

1246667

(4)

1284382

(5)

1458112

(8)

17566092(

9)6519626

(7)

F 5Be

st25398273(

2)39474199(

7)4367835

(8)

11716823(

1)91050677(

9)25861836(

3)26607341(

5)25886812(

4)26835697(

6)

Mean

3831135

(5)

149884822

(8)

114032052

(7)

60388778(

6)342973593

(9)

37244874(

4)271471(

1)27512334(

3)27496506(

2)

Std

21417578(

4)181590232

(8)

35216402(

6)43241123(

7)492144758

(9)

24882941(

5)0485147

(1)

0547347

(3)

0539586

(2)

F 6Be

st288E

minus12

(2)

06227

(9)

000076(

4)251E

minus11

(3)

0048128

(8)

450E

minus17

(1)

0007148

(5)

0012986

(6)

0015406

(7)

Mean

890E

minus12

(2)

1071114

(9)

000788(

4)968E

minus09

(3)

0070002

(6)

103E

minus16

(1)

0057661

(5)

0263633

(8)

020125(

7)

Std

567E

minus12

(2)

0279422

(9)

0008319

(4)

300E

minus08

(3)

0011199

(5)

446E

minus17

(1)

0078559

(6)

0224346

(8)

0192754

(7)

F 7Be

st0016252

(6)

0002011

(4)

0012074

(5)

0037104

(9)

0035229

(8)

0016733

(7)

0000115

(2)

000016(

3)00001

(1)

Mean

0027322

(5)

0006396

(4)

0030674

(6)

0078177

(9)

0055369

(8)

0049762

(7)

0001935

(3)

00015

(1)

0001579

(2)

Std

0005743

(5)

0002154

(4)

0012799

(7)

0023769

(9)

0011801

(6)

0021587

(8)

0001959

(3)

0001692

(1)

0001761

(2)

Rank

Best

68

75

93

41

2Mean

57

64

82

33

1Std

15

64

71

23

1


Tabl

e5

Com

parisonof

optim

izationresults

obtained

forthemultim

odal

benchm

arkfunctio

ns

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 8Be

stminus1256948662(

1)minus9288859115(

5)minus123198552

(3)

minus8562082036(

8)minus1256947154(

2)minus3257715637(

9)minus8674343572(

7)minus8985661303(

6)minus1057783345(

4)

Mean

minus1256553865(

2)minus8297653051(

4)minus1198240966(

3)minus6118655936(

8)minus1256880513(

1)minus243432431

(9)

minus6658410197(

7)minus7600141184(

6)minus7657917372(

5)

Std

21623781(

2)497301094

(5)

22238506(

3)1574830603(

9)0909111

(1)

355937095

(4)

863971262

(7)

861181057

(6)

1183224502(

8)

F 9Be

st44307253(

7)29244538(

6)16598597(

4)14992992(

3)17949826(

5)14924381(

2)0(

1)0(

1)0(

1)

Mean

56783475(

8)53437985(

7)25788633(

4)46538306(

6)25466452(

3)25802587(

5)0(

1)3672248

(2)

0(1)

Std

6266486

(4)

14333435(

7)403673(

2)14270927(

6)4772648

(3)

6630417

(5)

0(1)

20113732(

8)0(

1)

F 10

Best

423E

minus07

(3)

0226971

(6)

0007536

(5)

300E

minus06

(4)

1080969

(7)

545E

minus09

(2)

888E

minus16

(1)

888E

minus16

(1)

888E

minus16

(1)

Mean

702E

minus07

(5)

0349244

(8)

0020786

(6)

008323(

7)2063648

(9)

770E

minus09

(4)

444E

minus15

(3)

420E

minus15

(2)

397E

minus15

(1)

Std

152E

minus07

(5)

0070199

(7)

0011333

(6)

0317495

(8)

0425232

(9)

150E

minus09

(4)

229E

minus15

(3)

130E

minus15

(1)

203E

minus15

(2)

F 11

Best

144E

minus11

(3)

0503713

(6)

0001224

(4)

742E

minus12

(2)

0120691

(5)

3009729

(7)

0(1)

0(1)

0(1)

Mean

980E

minus10

(1)

0766039

(8)

0026851

(6)

0011493

(5)

0618727

(7)

8568461

(9)

0003104

(4)

0001092

(2)

0001282

(3)

Std

307E

minus09

(1)

011031(

7)0029255

(6)

0009701

(4)

0275077

(8)

4065895

(9)

0011816

(5)

0005981

(2)

0007021

(3)

F 12

Best

239E

minus13

(3)

0001193

(6)

300E

minus06

(4)

338E

minus14

(2)

0000383

(5)

375E

minus19

(1)

0001352

(7)

0003179

(9)

0002984

(8)

Mean

100E

minus12

(1)

0002505

(4)

630E

minus05

(2)

0003456

(5)

0000581

(3)

0151627

(8)

0006188

(6)

0013432

(7)

0409258

(9)

Std

105E

minus12

(1)

0000844

(4)

800E

minus05

(2)

0018927

(7)

0000121

(3)

0210923

(8)

0006962

(6)

0004444

(5)

160174(

9)

F 13

Best

170E

minus12

(2)

0023921

(6)

800E

minus05

(4)

523E

minus12

(3)

0006145

(5)

594E

minus18

(1)

0025211

(7)

0122758

(9)

0080548

(8)

Mean

431E

minus12

(1)

0044601

(6)

0000741

(2)

0001831

(3)

0013234

(5)

0007331

(4)

0233207

(7)

0468482

(8)

047475(

9)

Std

269E

minus12

(1)

0012881

(5)

0000969

(2)

0004165

(3)

0005965

(4)

0025745

(6)

0181659

(7)

0238598

(9)

0224147

(8)

F 14

Best

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998118

(2)

0998004

(1)

0998004

(1)

0998004

(1)

Mean

0998004

(1)

6813974

(7)

0998004

(1)

3459539

(5)

0998004

(1)

4082277

(6)

247579(

3)3454612

(4)

2108894

(2)

Std

0(1)

5363613

(9)

412E

minus17

(2)

2863628

(7)

187E

minus14

(3)

2690671

(5)

2445473

(4)

3273113

(8)

2790461

(6)

F 15

Best

0000352

(3)

0000419

(5)

0000307

(1)

0000308

(2)

0000401

(4)

0001284

(6)

0000308

(2)

0000307

(1)

0000307

(1)

Mean

0000577

(4)

0006093

(9)

0000329

(1)

0000794

(6)

0004004

(8)

0002584

(7)

0000601

(5)

0000513

(3)

0000468

(2)

Std

0000109

(2)

0008761

(9)

410E

minus05

(1)

0000233

(3)

0007444

(8)

0002019

(7)

0000315

(4)

0000377

(5)

0000382

(6)

F 16

Best

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Mean

minus1031628

(1)

minus0977217

(2)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Std

671E

minus16

(4)

0207068

(8)

612E

minus16

(3)

678E

minus16

(5)

353E

minus08

(7)

513E

minus16

(1)

386E

minus11

(6)

671E

minus16

(4)

598E

minus16

(2)

F 17

Best

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

Mean

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397888

(2)

0397887

(1)

0397887

(1)

Std

0(1)

933E

minus11

(3)

0(1)

0(1)

437E

minus08

(4)

0(1)

930E

minus07

(5)

0(1)

125E

minus13

(2)

F 18

Best

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

Mean

3(1)

39(

3)3(

1)3(

1)3900137

(4)

3(1)

3000014

(2)

3(1)

3(1)

Std

127E

minus15

(2)

4929503

(9)

202E

minus15

(5)

152E

minus15

(3)

4929483

(8)

267E

minus15

(6)

380E

minus05

(7)

782E

minus16

(1)

166E

minus15

(4)

F 19

Best

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

Mean

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862628

(2)

minus3862782

(1)

minus3862782

(1)

Std

271E

minus15

(6)

229E

minus15

(1)

271E

minus15

(6)

270E

minus15

(5)

100E

minus06

(7)

239E

minus15

(2)

0000366

(8)

266E

minus15

(4)

251E

minus15

(3)

F 20

Best

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321994

(2)

minus3321995

(1)

minus3321989

(3)

minus3321995

(1)

minus3321995

(1)

Mean

minus3321989

(2)

minus3270475

(6)

minus3321995

(1)

minus3282364

(5)

minus330217(

3)minus3321995

(1)

minus3264959

(7)

minus3293815

(4)

minus3254599

(8)

Std

320E

minus05

(3)

0059923

(7)

173E

minus14

(2)

0057005

(6)

0045065

(4)

147E

minus15

(1)

0071826

(9)

0051995

(5)

0059944

(8)

F 21

Best

minus101532(

1)minus101532(

1)minus101532(

1)minus101532(

1)minus10153199(

2)minus101532(

1)minus10153181(

3)minus101532(

1)minus101532(

1)

Mean

minus10049112(

2)minus5725156

(8)

minus101532(

1)minus8062579

(6)

minus5406265

(9)

minus6043604

(7)

minus8882256

(4)

minus8882841

(3)

minus8462151

(5)

Std

0531474

(2)

3327602

(7)

115E

minus09

(1)

3091228

(6)

3673677

(9)

3495936

(8)

2378758

(3)

2378893

(4)

2686626

(5)


Tabl

e5

Con

tinued

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 22

Best

minus10402941(

1)minus10402941(

1)minus10402941(

1)minus10402941(

1)minus1040294

(2)

minus10402941(

1)minus1040292

(3)

minus10402941(

1)minus10402941(

1)

Mean

minus10211001(

2)minus5529595

(8)

minus10402941(

1)minus9521178

(3)

minus6545124

(7)

minus10402941(

1)minus8448955

(4)

minus7763089

(6)

minus796332(

5)

Std

1045029

(3)

3343034

(8)

169E

minus14

(2)

2005406

(4)

3686937

(9)

132E

minus15

(1)

2836906

(5)

3128103

(7)

286345(

6)

F 23

Best

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus10536335(

2)minus1053641

(1)

minus1053641

(1)

Mean

minus10536383(

2)minus5855753

(8)

minus1053641

(1)

minus916093(

4)minus5180275

(9)

minus10175835(

3)minus9003007

(5)

minus7990287

(7)

minus8647563

(6)

Std

0000108

(2)

3655594

(9)

111E

minus09

(1)

2570428

(4)

3329307

(8)

1485436

(3)

2591415

(5)

3225805

(7)

2733822

(6)

Rank

Best

18

32

75

64

2Mean

29

16

87

53

4Std

19

26

83

74

5




200 400 600 800 1000

10minus150

10minus100

10minus50

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(a)

200 400 600 800 1000

102

104

106

108

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(b)

Ave

rage

ln (f

lowast minus

f min

)

200 400 600 800 1000

10minus2

100

102

IterationDEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(c)

200 400 600 800 1000

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(d)

200 400 600 800 1000

10minus15

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(e)

200 400 600 800 1000

10minus4

10minus3

10minus2

10minus1

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(f )









l l

l x1 x1x2

P







0 100 200 300 400 500242

2422

2424

2426

2428

243

2432

2434


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


Th

R

Ts

R

L







0 1000 2000 3000 4000 500035

4

45

5

55

6


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


l1

d1

d2l2

z1 z2







0 100 200 300 400 500345

35

355

36

365

37


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


A D

B

C













0 100 200 300 400 500minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


1 2 3 4 5

xi

xiConstantthickness







0 200 400 600 800 100001

02

03

04

05

06

07

08

09


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA





P

h

b

tftwL



5 Conclusion




Appendix



minf(x) 22


subject to

g1(x)

2

radicx1 + x2

2radic

x21 + 2x1x2

P minus σ le 0

g2(x) x2

2radic

x21 + 2x1x2

P minus σ le 0

g3(x) 1

2

radicx2 + x1

P minus σ le 0

0lexi le 1 i 1 2

l 100 cm

P 2 kNcm2

σ 2 kNcm2

(A1)




0 100 200 300 400 500minus188

minus186

minus184

minus182

minus18

minus178

minus176


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA




minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0


33 + 1296000le 0


0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)


minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References




















































Trarr

i Xlowastrarr

i minus Ararr

middot Crarr

middot Xlowastrarr

i minus Xrarr

i

111386811138681113868111386811138681113868

111386811138681113868111386811138681113868 (22)


Trarr

i Xrarr

other minus Ararr

middot Crarr

middot Xrarr

other minus Xrarr

i

11138681113868111386811138681113868

11138681113868111386811138681113868 (23)


Trarr

i Xlowastrarr

i minus Xrarr

i

111386811138681113868111386811138681113868

111386811138681113868111386811138681113868 middot ebl


i (24)


Trarr

i Xrarr


i minus Xrarr

i1113874 1113875or Trarr

i Xrarr


r2 minus Xrarr

r31113874 1113875

(25)



lp 1 +(s1n1)

2 +(s1n1) +(s2n2) (26)















Start






For each individual






lp lt rand







|A| lt 1

Yes

Yes

No

No

No

Output the results

t = t + 1

Yes WOA

BSA or DE

|A| lt 1

Yes

No











(43) Xlowast Xi















F1(x) 1113936Di1x

2i 30 [minus 100 100] 0

F2(x) 1113936Di1|xi| + 1113937

Di1 |xi| 30 [minus 10 10] 0

F3(x) 1113936Di1(1113936

ij1 xj)




Di1(xi + 05)2 30 [minus 100 100] 0

F7(x) 1113936Di1 ix4

i + random[0 1) 30 [minus 128 128] 0


Tabl

e2

Descriptio

nof

multim

odal

benchm

arkfunctio

ns

Functio

nD

Rang

efmin

F8(

x)

1113936

D i1

minusx

isin

(

|x

i|1113968

)30

[minus500

500]

minus4189829

times5

F9(

x)

1113936

D i1(

x2 i

minus10

cos(2π

xi)

+10

)30

[minus5125

12]

0

F10

(x

)

minus20

exp(

minus02

(1

D)1113936

D i1x

2 i

1113969

)minusexp(

(1

D)1113936

D i1cos

(2π

xi)

)+20

+e

30[minus323

2]0

F11

(x

)

(14000

)1113936D i1x

2 iminus

1113937D i1cos(

xi

iradic

)+1

30[minus600

600]

0F12

(x

)π

D10

sin(πy

1)+

1113936D

minus1

i1

(y

iminus1)

2 [1+10

sin2 (πy

i+1)

]+

(y

Dminus1)

21113966

1113967+

1113936D i1u

(x

i10

100

4)

30[minus505

0]0

yi

1

+x

i+14

u(

xi

ak

m)

k(

ximinus

a)m

x

igt

a

0minus

alt

xilt

a

k(

minusx

iminus

a)m

x

ilt

a

⎧⎪ ⎨ ⎪ ⎩

F13

(x

)01sin

2 (3π

x1)

+1113936

D i1

(x

iminus1)

2 [1+sin

2 (3π

xi+1

+1)

]+

(x

Dminus1)

2 [1+sin

2 (2π

xD

)]1113966

1113967+

1113936D i1u

(x

i51004)

30[minus505

0]0


Tabl

e3

Descriptio

nof

fixed-dim

ensio

nmultim

odal

benchm

arkfunctio

ns

Functio

nD

Rang

efmin

F14

(x

)

((1500)

+1113936

25 j1(1

j+

11139362 i1(

ximinus

aij

)6))

minus1

2[minus656

5]1

F15

(x

)

111393611 i1[

aiminus

(x1(

b2 i

+b

ix2)b

2 i+

bix

3+

x4)

]24

[minus5

5]000030

F16

(x

)4x

2 1minus21x

4 1+

(13)

x6 1

+x1x

2minus4x

2 2+4x

4 22

[minus5

5]minus10316

F17

(x

)

(x2

minus(514π

2 )x2 1

+(5π)

x1

minus6)

2+10

(1

minus(18π

))cos

x1

+10

2[minus5

5]0398

F18

(x

)

[1+

(x1

+x2

+1)

2 (19

minus14

x1

+3x

2 1minus14

x2

+6x

1x2

+3x

2 2)]

times[30

+(2x

1minus3x

2)2 (18

minus32

x1

+12

x2 1

+48

x2

minus36

x1x

2+27

x2 2)

]2

[minus2

2]3

F19

(x

)

minus1113936

4 i1c

iexp(

minus1113936

3 j1

aij

(x

jminus

pij

)2)

3[13

]minus386

F20

(x

)

minus1113936

4 i1c

iexp(

minus1113936

6 j1

aij

(x

jminus

pij

)2)

6[01

]minus332

F21

(x

)

minus1113936

5 i1[

(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus101232

F22

(x

)

minus1113936

7 i1[

(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus104028

F23

(x

)

minus1113936

1 i10

[(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus105363


Tabl

e4

Com

parisonof

optim

izationresults

obtained

fortheun

imod

albenchm

arkfunctio

ns

AL

Runs

30N

30a

ndT

1000

frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 1Be

st290E

minus12

(5)

0454643

(9)

0000908

(7)

104E

minus11

(6)

0042385

(8)

464E

minus17

(4)

276E

minus167(

3)692E

minus190(

1)162E

minus187(

2)

Mean

133E

minus11

(5)

1013223

(9)

0006257

(7)

973E

minus09

(6)

0063236

(8)

101E

minus16

(4)

146E

minus153(

3)107E

minus168(

2)145E

minus172(

1)

Std

998E

minus12

(4)

0311571

(8)

0010917

(7)

423E

minus08

(5)

0010899

(6)

376E

minus17

(3)

791E

minus153(

2)0(

1)0(

1)

F 2Be

st399E

minus08

(5)

0271733

(8)

0006726

(7)

100E

minus05

(6)

0575724

(9)

350E

minus08

(4)

331E

minus114(

3)246E

minus124(

2)201E

minus124(

1)

Mean

680E

minus08

(5)

0324236

(8)

0020418

(7)

0001313

(6)

0727765

(9)

520E

minus08

(4)

988E

minus105(

3)847E

minus113(

1)127E

minus112(

2)

Std

195E

minus08

(5)

0041092

(8)

0014974

(7)

0004947

(6)

005865(

9)117E

minus08

(4)

494E

minus104(

3)421E

minus112(

1)575E

minus112(

2)

F 3Be

st1080803597(

9)83765444(

2)394952176

(5)

3208562

(1)

308450285

(8)

195538197

(4)

167730974

(7)

17510162(

3)532871248

(6)

Mean

157231259

(9)

135934273

(2)

87822206(

4)14904551(

1)5228201445(

5)409456876

(3)

9735512836(

7)1004518144(

8)7621728202(

6)

Std

2683155994(

6)34026326(

2)315589252

(4)

7677907

(1)

1591552755(

5)12983509(

3)3898890192(

7)598648178

(9)

4199058749(

8)

F 4Be

st1080885

(7)

0742244

(6)

3262876

(8)

0311111

(5)

4216935

(9)

135E

minus08

(3)

0019063

(4)

568E

minus10

(2)

430E

minus19

(1)

Mean

1575045

(5)

0972736

(2)

5969996

(6)

0563087

(1)

6113542

(7)

1264362

(3)

17704825(

9)14259819(

8)1562263

(4)

Std

038751(

3)0123737

(1)

157968(

6)0153909

(2)

1246667

(4)

1284382

(5)

1458112

(8)

17566092(

9)6519626

(7)

F 5Be

st25398273(

2)39474199(

7)4367835

(8)

11716823(

1)91050677(

9)25861836(

3)26607341(

5)25886812(

4)26835697(

6)

Mean

3831135

(5)

149884822

(8)

114032052

(7)

60388778(

6)342973593

(9)

37244874(

4)271471(

1)27512334(

3)27496506(

2)

Std

21417578(

4)181590232

(8)

35216402(

6)43241123(

7)492144758

(9)

24882941(

5)0485147

(1)

0547347

(3)

0539586

(2)

F 6Be

st288E

minus12

(2)

06227

(9)

000076(

4)251E

minus11

(3)

0048128

(8)

450E

minus17

(1)

0007148

(5)

0012986

(6)

0015406

(7)

Mean

890E

minus12

(2)

1071114

(9)

000788(

4)968E

minus09

(3)

0070002

(6)

103E

minus16

(1)

0057661

(5)

0263633

(8)

020125(

7)

Std

567E

minus12

(2)

0279422

(9)

0008319

(4)

300E

minus08

(3)

0011199

(5)

446E

minus17

(1)

0078559

(6)

0224346

(8)

0192754

(7)

F 7Be

st0016252

(6)

0002011

(4)

0012074

(5)

0037104

(9)

0035229

(8)

0016733

(7)

0000115

(2)

000016(

3)00001

(1)

Mean

0027322

(5)

0006396

(4)

0030674

(6)

0078177

(9)

0055369

(8)

0049762

(7)

0001935

(3)

00015

(1)

0001579

(2)

Std

0005743

(5)

0002154

(4)

0012799

(7)

0023769

(9)

0011801

(6)

0021587

(8)

0001959

(3)

0001692

(1)

0001761

(2)

Rank

Best

68

75

93

41

2Mean

57

64

82

33

1Std

15

64

71

23

1


Tabl

e5

Com

parisonof

optim

izationresults

obtained

forthemultim

odal

benchm

arkfunctio

ns

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 8Be

stminus1256948662(

1)minus9288859115(

5)minus123198552

(3)

minus8562082036(

8)minus1256947154(

2)minus3257715637(

9)minus8674343572(

7)minus8985661303(

6)minus1057783345(

4)

Mean

minus1256553865(

2)minus8297653051(

4)minus1198240966(

3)minus6118655936(

8)minus1256880513(

1)minus243432431

(9)

minus6658410197(

7)minus7600141184(

6)minus7657917372(

5)

Std

21623781(

2)497301094

(5)

22238506(

3)1574830603(

9)0909111

(1)

355937095

(4)

863971262

(7)

861181057

(6)

1183224502(

8)

F 9Be

st44307253(

7)29244538(

6)16598597(

4)14992992(

3)17949826(

5)14924381(

2)0(

1)0(

1)0(

1)

Mean

56783475(

8)53437985(

7)25788633(

4)46538306(

6)25466452(

3)25802587(

5)0(

1)3672248

(2)

0(1)

Std

6266486

(4)

14333435(

7)403673(

2)14270927(

6)4772648

(3)

6630417

(5)

0(1)

20113732(

8)0(

1)

F 10

Best

423E

minus07

(3)

0226971

(6)

0007536

(5)

300E

minus06

(4)

1080969

(7)

545E

minus09

(2)

888E

minus16

(1)

888E

minus16

(1)

888E

minus16

(1)

Mean

702E

minus07

(5)

0349244

(8)

0020786

(6)

008323(

7)2063648

(9)

770E

minus09

(4)

444E

minus15

(3)

420E

minus15

(2)

397E

minus15

(1)

Std

152E

minus07

(5)

0070199

(7)

0011333

(6)

0317495

(8)

0425232

(9)

150E

minus09

(4)

229E

minus15

(3)

130E

minus15

(1)

203E

minus15

(2)

F 11

Best

144E

minus11

(3)

0503713

(6)

0001224

(4)

742E

minus12

(2)

0120691

(5)

3009729

(7)

0(1)

0(1)

0(1)

Mean

980E

minus10

(1)

0766039

(8)

0026851

(6)

0011493

(5)

0618727

(7)

8568461

(9)

0003104

(4)

0001092

(2)

0001282

(3)

Std

307E

minus09

(1)

011031(

7)0029255

(6)

0009701

(4)

0275077

(8)

4065895

(9)

0011816

(5)

0005981

(2)

0007021

(3)

F 12

Best

239E

minus13

(3)

0001193

(6)

300E

minus06

(4)

338E

minus14

(2)

0000383

(5)

375E

minus19

(1)

0001352

(7)

0003179

(9)

0002984

(8)

Mean

100E

minus12

(1)

0002505

(4)

630E

minus05

(2)

0003456

(5)

0000581

(3)

0151627

(8)

0006188

(6)

0013432

(7)

0409258

(9)

Std

105E

minus12

(1)

0000844

(4)

800E

minus05

(2)

0018927

(7)

0000121

(3)

0210923

(8)

0006962

(6)

0004444

(5)

160174(

9)

F 13

Best

170E

minus12

(2)

0023921

(6)

800E

minus05

(4)

523E

minus12

(3)

0006145

(5)

594E

minus18

(1)

0025211

(7)

0122758

(9)

0080548

(8)

Mean

431E

minus12

(1)

0044601

(6)

0000741

(2)

0001831

(3)

0013234

(5)

0007331

(4)

0233207

(7)

0468482

(8)

047475(

9)

Std

269E

minus12

(1)

0012881

(5)

0000969

(2)

0004165

(3)

0005965

(4)

0025745

(6)

0181659

(7)

0238598

(9)

0224147

(8)

F 14

Best

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998118

(2)

0998004

(1)

0998004

(1)

0998004

(1)

Mean

0998004

(1)

6813974

(7)

0998004

(1)

3459539

(5)

0998004

(1)

4082277

(6)

247579(

3)3454612

(4)

2108894

(2)

Std

0(1)

5363613

(9)

412E

minus17

(2)

2863628

(7)

187E

minus14

(3)

2690671

(5)

2445473

(4)

3273113

(8)

2790461

(6)

F 15

Best

0000352

(3)

0000419

(5)

0000307

(1)

0000308

(2)

0000401

(4)

0001284

(6)

0000308

(2)

0000307

(1)

0000307

(1)

Mean

0000577

(4)

0006093

(9)

0000329

(1)

0000794

(6)

0004004

(8)

0002584

(7)

0000601

(5)

0000513

(3)

0000468

(2)

Std

0000109

(2)

0008761

(9)

410E

minus05

(1)

0000233

(3)

0007444

(8)

0002019

(7)

0000315

(4)

0000377

(5)

0000382

(6)

F 16

Best

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Mean

minus1031628

(1)

minus0977217

(2)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Std

671E

minus16

(4)

0207068

(8)

612E

minus16

(3)

678E

minus16

(5)

353E

minus08

(7)

513E

minus16

(1)

386E

minus11

(6)

671E

minus16

(4)

598E

minus16

(2)

F 17

Best

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

Mean

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397888

(2)

0397887

(1)

0397887

(1)

Std

0(1)

933E

minus11

(3)

0(1)

0(1)

437E

minus08

(4)

0(1)

930E

minus07

(5)

0(1)

125E

minus13

(2)

F 18

Best

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

Mean

3(1)

39(

3)3(

1)3(

1)3900137

(4)

3(1)

3000014

(2)

3(1)

3(1)

Std

127E

minus15

(2)

4929503

(9)

202E

minus15

(5)

152E

minus15

(3)

4929483

(8)

267E

minus15

(6)

380E

minus05

(7)

782E

minus16

(1)

166E

minus15

(4)

F 19

Best

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

Mean

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862628

(2)

minus3862782

(1)

minus3862782

(1)

Std

271E

minus15

(6)

229E

minus15

(1)

271E

minus15

(6)

270E

minus15

(5)

100E

minus06

(7)

239E

minus15

(2)

0000366

(8)

266E

minus15

(4)

251E

minus15

(3)

F 20

Best

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321994

(2)

minus3321995

(1)

minus3321989

(3)

minus3321995

(1)

minus3321995

(1)

Mean

minus3321989

(2)

minus3270475

(6)

minus3321995

(1)

minus3282364

(5)

minus330217(

3)minus3321995

(1)

minus3264959

(7)

minus3293815

(4)

minus3254599

(8)

Std

320E

minus05

(3)

0059923

(7)

173E

minus14

(2)

0057005

(6)

0045065

(4)

147E

minus15

(1)

0071826

(9)

0051995

(5)

0059944

(8)

F 21

Best

minus101532(

1)minus101532(

1)minus101532(

1)minus101532(

1)minus10153199(

2)minus101532(

1)minus10153181(

3)minus101532(

1)minus101532(

1)

Mean

minus10049112(

2)minus5725156

(8)

minus101532(

1)minus8062579

(6)

minus5406265

(9)

minus6043604

(7)

minus8882256

(4)

minus8882841

(3)

minus8462151

(5)

Std

0531474

(2)

3327602

(7)

115E

minus09

(1)

3091228

(6)

3673677

(9)

3495936

(8)

2378758

(3)

2378893

(4)

2686626

(5)


Tabl

e5

Con

tinued

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 22

Best

minus10402941(

1)minus10402941(

1)minus10402941(

1)minus10402941(

1)minus1040294

(2)

minus10402941(

1)minus1040292

(3)

minus10402941(

1)minus10402941(

1)

Mean

minus10211001(

2)minus5529595

(8)

minus10402941(

1)minus9521178

(3)

minus6545124

(7)

minus10402941(

1)minus8448955

(4)

minus7763089

(6)

minus796332(

5)

Std

1045029

(3)

3343034

(8)

169E

minus14

(2)

2005406

(4)

3686937

(9)

132E

minus15

(1)

2836906

(5)

3128103

(7)

286345(

6)

F 23

Best

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus10536335(

2)minus1053641

(1)

minus1053641

(1)

Mean

minus10536383(

2)minus5855753

(8)

minus1053641

(1)

minus916093(

4)minus5180275

(9)

minus10175835(

3)minus9003007

(5)

minus7990287

(7)

minus8647563

(6)

Std

0000108

(2)

3655594

(9)

111E

minus09

(1)

2570428

(4)

3329307

(8)

1485436

(3)

2591415

(5)

3225805

(7)

2733822

(6)

Rank

Best

18

32

75

64

2Mean

29

16

87

53

4Std

19

26

83

74

5




200 400 600 800 1000

10minus150

10minus100

10minus50

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(a)

200 400 600 800 1000

102

104

106

108

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(b)

Ave

rage

ln (f

lowast minus

f min

)

200 400 600 800 1000

10minus2

100

102

IterationDEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(c)

200 400 600 800 1000

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(d)

200 400 600 800 1000

10minus15

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(e)

200 400 600 800 1000

10minus4

10minus3

10minus2

10minus1

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(f )









l l

l x1 x1x2

P







0 100 200 300 400 500242

2422

2424

2426

2428

243

2432

2434


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


Th

R

Ts

R

L







0 1000 2000 3000 4000 500035

4

45

5

55

6


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


l1

d1

d2l2

z1 z2







0 100 200 300 400 500345

35

355

36

365

37


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


A D

B

C













0 100 200 300 400 500minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


1 2 3 4 5

xi

xiConstantthickness







0 200 400 600 800 100001

02

03

04

05

06

07

08

09


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA





P

h

b

tftwL



5 Conclusion




Appendix



minf(x) 22


subject to

g1(x)

2

radicx1 + x2

2radic

x21 + 2x1x2

P minus σ le 0

g2(x) x2

2radic

x21 + 2x1x2

P minus σ le 0

g3(x) 1

2

radicx2 + x1

P minus σ le 0

0lexi le 1 i 1 2

l 100 cm

P 2 kNcm2

σ 2 kNcm2

(A1)




0 100 200 300 400 500minus188

minus186

minus184

minus182

minus18

minus178

minus176


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA




minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0


33 + 1296000le 0


0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)


minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References






















































Start






For each individual






lp lt rand







|A| lt 1

Yes

Yes

No

No

No

Output the results

t = t + 1

Yes WOA

BSA or DE

|A| lt 1

Yes

No











(43) Xlowast Xi















F1(x) 1113936Di1x

2i 30 [minus 100 100] 0

F2(x) 1113936Di1|xi| + 1113937

Di1 |xi| 30 [minus 10 10] 0

F3(x) 1113936Di1(1113936

ij1 xj)




Di1(xi + 05)2 30 [minus 100 100] 0

F7(x) 1113936Di1 ix4

i + random[0 1) 30 [minus 128 128] 0


Tabl

e2

Descriptio

nof

multim

odal

benchm

arkfunctio

ns

Functio

nD

Rang

efmin

F8(

x)

1113936

D i1

minusx

isin

(

|x

i|1113968

)30

[minus500

500]

minus4189829

times5

F9(

x)

1113936

D i1(

x2 i

minus10

cos(2π

xi)

+10

)30

[minus5125

12]

0

F10

(x

)

minus20

exp(

minus02

(1

D)1113936

D i1x

2 i

1113969

)minusexp(

(1

D)1113936

D i1cos

(2π

xi)

)+20

+e

30[minus323

2]0

F11

(x

)

(14000

)1113936D i1x

2 iminus

1113937D i1cos(

xi

iradic

)+1

30[minus600

600]

0F12

(x

)π

D10

sin(πy

1)+

1113936D

minus1

i1

(y

iminus1)

2 [1+10

sin2 (πy

i+1)

]+

(y

Dminus1)

21113966

1113967+

1113936D i1u

(x

i10

100

4)

30[minus505

0]0

yi

1

+x

i+14

u(

xi

ak

m)

k(

ximinus

a)m

x

igt

a

0minus

alt

xilt

a

k(

minusx

iminus

a)m

x

ilt

a

⎧⎪ ⎨ ⎪ ⎩

F13

(x

)01sin

2 (3π

x1)

+1113936

D i1

(x

iminus1)

2 [1+sin

2 (3π

xi+1

+1)

]+

(x

Dminus1)

2 [1+sin

2 (2π

xD

)]1113966

1113967+

1113936D i1u

(x

i51004)

30[minus505

0]0


Tabl

e3

Descriptio

nof

fixed-dim

ensio

nmultim

odal

benchm

arkfunctio

ns

Functio

nD

Rang

efmin

F14

(x

)

((1500)

+1113936

25 j1(1

j+

11139362 i1(

ximinus

aij

)6))

minus1

2[minus656

5]1

F15

(x

)

111393611 i1[

aiminus

(x1(

b2 i

+b

ix2)b

2 i+

bix

3+

x4)

]24

[minus5

5]000030

F16

(x

)4x

2 1minus21x

4 1+

(13)

x6 1

+x1x

2minus4x

2 2+4x

4 22

[minus5

5]minus10316

F17

(x

)

(x2

minus(514π

2 )x2 1

+(5π)

x1

minus6)

2+10

(1

minus(18π

))cos

x1

+10

2[minus5

5]0398

F18

(x

)

[1+

(x1

+x2

+1)

2 (19

minus14

x1

+3x

2 1minus14

x2

+6x

1x2

+3x

2 2)]

times[30

+(2x

1minus3x

2)2 (18

minus32

x1

+12

x2 1

+48

x2

minus36

x1x

2+27

x2 2)

]2

[minus2

2]3

F19

(x

)

minus1113936

4 i1c

iexp(

minus1113936

3 j1

aij

(x

jminus

pij

)2)

3[13

]minus386

F20

(x

)

minus1113936

4 i1c

iexp(

minus1113936

6 j1

aij

(x

jminus

pij

)2)

6[01

]minus332

F21

(x

)

minus1113936

5 i1[

(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus101232

F22

(x

)

minus1113936

7 i1[

(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus104028

F23

(x

)

minus1113936

1 i10

[(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus105363


Tabl

e4

Com

parisonof

optim

izationresults

obtained

fortheun

imod

albenchm

arkfunctio

ns

AL

Runs

30N

30a

ndT

1000

frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 1Be

st290E

minus12

(5)

0454643

(9)

0000908

(7)

104E

minus11

(6)

0042385

(8)

464E

minus17

(4)

276E

minus167(

3)692E

minus190(

1)162E

minus187(

2)

Mean

133E

minus11

(5)

1013223

(9)

0006257

(7)

973E

minus09

(6)

0063236

(8)

101E

minus16

(4)

146E

minus153(

3)107E

minus168(

2)145E

minus172(

1)

Std

998E

minus12

(4)

0311571

(8)

0010917

(7)

423E

minus08

(5)

0010899

(6)

376E

minus17

(3)

791E

minus153(

2)0(

1)0(

1)

F 2Be

st399E

minus08

(5)

0271733

(8)

0006726

(7)

100E

minus05

(6)

0575724

(9)

350E

minus08

(4)

331E

minus114(

3)246E

minus124(

2)201E

minus124(

1)

Mean

680E

minus08

(5)

0324236

(8)

0020418

(7)

0001313

(6)

0727765

(9)

520E

minus08

(4)

988E

minus105(

3)847E

minus113(

1)127E

minus112(

2)

Std

195E

minus08

(5)

0041092

(8)

0014974

(7)

0004947

(6)

005865(

9)117E

minus08

(4)

494E

minus104(

3)421E

minus112(

1)575E

minus112(

2)

F 3Be

st1080803597(

9)83765444(

2)394952176

(5)

3208562

(1)

308450285

(8)

195538197

(4)

167730974

(7)

17510162(

3)532871248

(6)

Mean

157231259

(9)

135934273

(2)

87822206(

4)14904551(

1)5228201445(

5)409456876

(3)

9735512836(

7)1004518144(

8)7621728202(

6)

Std

2683155994(

6)34026326(

2)315589252

(4)

7677907

(1)

1591552755(

5)12983509(

3)3898890192(

7)598648178

(9)

4199058749(

8)

F 4Be

st1080885

(7)

0742244

(6)

3262876

(8)

0311111

(5)

4216935

(9)

135E

minus08

(3)

0019063

(4)

568E

minus10

(2)

430E

minus19

(1)

Mean

1575045

(5)

0972736

(2)

5969996

(6)

0563087

(1)

6113542

(7)

1264362

(3)

17704825(

9)14259819(

8)1562263

(4)

Std

038751(

3)0123737

(1)

157968(

6)0153909

(2)

1246667

(4)

1284382

(5)

1458112

(8)

17566092(

9)6519626

(7)

F 5Be

st25398273(

2)39474199(

7)4367835

(8)

11716823(

1)91050677(

9)25861836(

3)26607341(

5)25886812(

4)26835697(

6)

Mean

3831135

(5)

149884822

(8)

114032052

(7)

60388778(

6)342973593

(9)

37244874(

4)271471(

1)27512334(

3)27496506(

2)

Std

21417578(

4)181590232

(8)

35216402(

6)43241123(

7)492144758

(9)

24882941(

5)0485147

(1)

0547347

(3)

0539586

(2)

F 6Be

st288E

minus12

(2)

06227

(9)

000076(

4)251E

minus11

(3)

0048128

(8)

450E

minus17

(1)

0007148

(5)

0012986

(6)

0015406

(7)

Mean

890E

minus12

(2)

1071114

(9)

000788(

4)968E

minus09

(3)

0070002

(6)

103E

minus16

(1)

0057661

(5)

0263633

(8)

020125(

7)

Std

567E

minus12

(2)

0279422

(9)

0008319

(4)

300E

minus08

(3)

0011199

(5)

446E

minus17

(1)

0078559

(6)

0224346

(8)

0192754

(7)

F 7Be

st0016252

(6)

0002011

(4)

0012074

(5)

0037104

(9)

0035229

(8)

0016733

(7)

0000115

(2)

000016(

3)00001

(1)

Mean

0027322

(5)

0006396

(4)

0030674

(6)

0078177

(9)

0055369

(8)

0049762

(7)

0001935

(3)

00015

(1)

0001579

(2)

Std

0005743

(5)

0002154

(4)

0012799

(7)

0023769

(9)

0011801

(6)

0021587

(8)

0001959

(3)

0001692

(1)

0001761

(2)

Rank

Best

68

75

93

41

2Mean

57

64

82

33

1Std

15

64

71

23

1


Tabl

e5

Com

parisonof

optim

izationresults

obtained

forthemultim

odal

benchm

arkfunctio

ns

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 8Be

stminus1256948662(

1)minus9288859115(

5)minus123198552

(3)

minus8562082036(

8)minus1256947154(

2)minus3257715637(

9)minus8674343572(

7)minus8985661303(

6)minus1057783345(

4)

Mean

minus1256553865(

2)minus8297653051(

4)minus1198240966(

3)minus6118655936(

8)minus1256880513(

1)minus243432431

(9)

minus6658410197(

7)minus7600141184(

6)minus7657917372(

5)

Std

21623781(

2)497301094

(5)

22238506(

3)1574830603(

9)0909111

(1)

355937095

(4)

863971262

(7)

861181057

(6)

1183224502(

8)

F 9Be

st44307253(

7)29244538(

6)16598597(

4)14992992(

3)17949826(

5)14924381(

2)0(

1)0(

1)0(

1)

Mean

56783475(

8)53437985(

7)25788633(

4)46538306(

6)25466452(

3)25802587(

5)0(

1)3672248

(2)

0(1)

Std

6266486

(4)

14333435(

7)403673(

2)14270927(

6)4772648

(3)

6630417

(5)

0(1)

20113732(

8)0(

1)

F 10

Best

423E

minus07

(3)

0226971

(6)

0007536

(5)

300E

minus06

(4)

1080969

(7)

545E

minus09

(2)

888E

minus16

(1)

888E

minus16

(1)

888E

minus16

(1)

Mean

702E

minus07

(5)

0349244

(8)

0020786

(6)

008323(

7)2063648

(9)

770E

minus09

(4)

444E

minus15

(3)

420E

minus15

(2)

397E

minus15

(1)

Std

152E

minus07

(5)

0070199

(7)

0011333

(6)

0317495

(8)

0425232

(9)

150E

minus09

(4)

229E

minus15

(3)

130E

minus15

(1)

203E

minus15

(2)

F 11

Best

144E

minus11

(3)

0503713

(6)

0001224

(4)

742E

minus12

(2)

0120691

(5)

3009729

(7)

0(1)

0(1)

0(1)

Mean

980E

minus10

(1)

0766039

(8)

0026851

(6)

0011493

(5)

0618727

(7)

8568461

(9)

0003104

(4)

0001092

(2)

0001282

(3)

Std

307E

minus09

(1)

011031(

7)0029255

(6)

0009701

(4)

0275077

(8)

4065895

(9)

0011816

(5)

0005981

(2)

0007021

(3)

F 12

Best

239E

minus13

(3)

0001193

(6)

300E

minus06

(4)

338E

minus14

(2)

0000383

(5)

375E

minus19

(1)

0001352

(7)

0003179

(9)

0002984

(8)

Mean

100E

minus12

(1)

0002505

(4)

630E

minus05

(2)

0003456

(5)

0000581

(3)

0151627

(8)

0006188

(6)

0013432

(7)

0409258

(9)

Std

105E

minus12

(1)

0000844

(4)

800E

minus05

(2)

0018927

(7)

0000121

(3)

0210923

(8)

0006962

(6)

0004444

(5)

160174(

9)

F 13

Best

170E

minus12

(2)

0023921

(6)

800E

minus05

(4)

523E

minus12

(3)

0006145

(5)

594E

minus18

(1)

0025211

(7)

0122758

(9)

0080548

(8)

Mean

431E

minus12

(1)

0044601

(6)

0000741

(2)

0001831

(3)

0013234

(5)

0007331

(4)

0233207

(7)

0468482

(8)

047475(

9)

Std

269E

minus12

(1)

0012881

(5)

0000969

(2)

0004165

(3)

0005965

(4)

0025745

(6)

0181659

(7)

0238598

(9)

0224147

(8)

F 14

Best

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998118

(2)

0998004

(1)

0998004

(1)

0998004

(1)

Mean

0998004

(1)

6813974

(7)

0998004

(1)

3459539

(5)

0998004

(1)

4082277

(6)

247579(

3)3454612

(4)

2108894

(2)

Std

0(1)

5363613

(9)

412E

minus17

(2)

2863628

(7)

187E

minus14

(3)

2690671

(5)

2445473

(4)

3273113

(8)

2790461

(6)

F 15

Best

0000352

(3)

0000419

(5)

0000307

(1)

0000308

(2)

0000401

(4)

0001284

(6)

0000308

(2)

0000307

(1)

0000307

(1)

Mean

0000577

(4)

0006093

(9)

0000329

(1)

0000794

(6)

0004004

(8)

0002584

(7)

0000601

(5)

0000513

(3)

0000468

(2)

Std

0000109

(2)

0008761

(9)

410E

minus05

(1)

0000233

(3)

0007444

(8)

0002019

(7)

0000315

(4)

0000377

(5)

0000382

(6)

F 16

Best

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Mean

minus1031628

(1)

minus0977217

(2)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Std

671E

minus16

(4)

0207068

(8)

612E

minus16

(3)

678E

minus16

(5)

353E

minus08

(7)

513E

minus16

(1)

386E

minus11

(6)

671E

minus16

(4)

598E

minus16

(2)

F 17

Best

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

Mean

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397888

(2)

0397887

(1)

0397887

(1)

Std

0(1)

933E

minus11

(3)

0(1)

0(1)

437E

minus08

(4)

0(1)

930E

minus07

(5)

0(1)

125E

minus13

(2)

F 18

Best

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

Mean

3(1)

39(

3)3(

1)3(

1)3900137

(4)

3(1)

3000014

(2)

3(1)

3(1)

Std

127E

minus15

(2)

4929503

(9)

202E

minus15

(5)

152E

minus15

(3)

4929483

(8)

267E

minus15

(6)

380E

minus05

(7)

782E

minus16

(1)

166E

minus15

(4)

F 19

Best

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

Mean

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862628

(2)

minus3862782

(1)

minus3862782

(1)

Std

271E

minus15

(6)

229E

minus15

(1)

271E

minus15

(6)

270E

minus15

(5)

100E

minus06

(7)

239E

minus15

(2)

0000366

(8)

266E

minus15

(4)

251E

minus15

(3)

F 20

Best

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321994

(2)

minus3321995

(1)

minus3321989

(3)

minus3321995

(1)

minus3321995

(1)

Mean

minus3321989

(2)

minus3270475

(6)

minus3321995

(1)

minus3282364

(5)

minus330217(

3)minus3321995

(1)

minus3264959

(7)

minus3293815

(4)

minus3254599

(8)

Std

320E

minus05

(3)

0059923

(7)

173E

minus14

(2)

0057005

(6)

0045065

(4)

147E

minus15

(1)

0071826

(9)

0051995

(5)

0059944

(8)

F 21

Best

minus101532(

1)minus101532(

1)minus101532(

1)minus101532(

1)minus10153199(

2)minus101532(

1)minus10153181(

3)minus101532(

1)minus101532(

1)

Mean

minus10049112(

2)minus5725156

(8)

minus101532(

1)minus8062579

(6)

minus5406265

(9)

minus6043604

(7)

minus8882256

(4)

minus8882841

(3)

minus8462151

(5)

Std

0531474

(2)

3327602

(7)

115E

minus09

(1)

3091228

(6)

3673677

(9)

3495936

(8)

2378758

(3)

2378893

(4)

2686626

(5)


Tabl

e5

Con

tinued

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 22

Best

minus10402941(

1)minus10402941(

1)minus10402941(

1)minus10402941(

1)minus1040294

(2)

minus10402941(

1)minus1040292

(3)

minus10402941(

1)minus10402941(

1)

Mean

minus10211001(

2)minus5529595

(8)

minus10402941(

1)minus9521178

(3)

minus6545124

(7)

minus10402941(

1)minus8448955

(4)

minus7763089

(6)

minus796332(

5)

Std

1045029

(3)

3343034

(8)

169E

minus14

(2)

2005406

(4)

3686937

(9)

132E

minus15

(1)

2836906

(5)

3128103

(7)

286345(

6)

F 23

Best

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus10536335(

2)minus1053641

(1)

minus1053641

(1)

Mean

minus10536383(

2)minus5855753

(8)

minus1053641

(1)

minus916093(

4)minus5180275

(9)

minus10175835(

3)minus9003007

(5)

minus7990287

(7)

minus8647563

(6)

Std

0000108

(2)

3655594

(9)

111E

minus09

(1)

2570428

(4)

3329307

(8)

1485436

(3)

2591415

(5)

3225805

(7)

2733822

(6)

Rank

Best

18

32

75

64

2Mean

29

16

87

53

4Std

19

26

83

74

5




200 400 600 800 1000

10minus150

10minus100

10minus50

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(a)

200 400 600 800 1000

102

104

106

108

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(b)

Ave

rage

ln (f

lowast minus

f min

)

200 400 600 800 1000

10minus2

100

102

IterationDEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(c)

200 400 600 800 1000

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(d)

200 400 600 800 1000

10minus15

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(e)

200 400 600 800 1000

10minus4

10minus3

10minus2

10minus1

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(f )









l l

l x1 x1x2

P







0 100 200 300 400 500242

2422

2424

2426

2428

243

2432

2434


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


Th

R

Ts

R

L







0 1000 2000 3000 4000 500035

4

45

5

55

6


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


l1

d1

d2l2

z1 z2







0 100 200 300 400 500345

35

355

36

365

37


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


A D

B

C













0 100 200 300 400 500minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


1 2 3 4 5

xi

xiConstantthickness







0 200 400 600 800 100001

02

03

04

05

06

07

08

09


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA





P

h

b

tftwL



5 Conclusion




Appendix



minf(x) 22


subject to

g1(x)

2

radicx1 + x2

2radic

x21 + 2x1x2

P minus σ le 0

g2(x) x2

2radic

x21 + 2x1x2

P minus σ le 0

g3(x) 1

2

radicx2 + x1

P minus σ le 0

0lexi le 1 i 1 2

l 100 cm

P 2 kNcm2

σ 2 kNcm2

(A1)




0 100 200 300 400 500minus188

minus186

minus184

minus182

minus18

minus178

minus176


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA




minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0


33 + 1296000le 0


0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)


minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References

























































(43) Xlowast Xi















F1(x) 1113936Di1x

2i 30 [minus 100 100] 0

F2(x) 1113936Di1|xi| + 1113937

Di1 |xi| 30 [minus 10 10] 0

F3(x) 1113936Di1(1113936

ij1 xj)




Di1(xi + 05)2 30 [minus 100 100] 0

F7(x) 1113936Di1 ix4

i + random[0 1) 30 [minus 128 128] 0


Tabl

e2

Descriptio

nof

multim

odal

benchm

arkfunctio

ns

Functio

nD

Rang

efmin

F8(

x)

1113936

D i1

minusx

isin

(

|x

i|1113968

)30

[minus500

500]

minus4189829

times5

F9(

x)

1113936

D i1(

x2 i

minus10

cos(2π

xi)

+10

)30

[minus5125

12]

0

F10

(x

)

minus20

exp(

minus02

(1

D)1113936

D i1x

2 i

1113969

)minusexp(

(1

D)1113936

D i1cos

(2π

xi)

)+20

+e

30[minus323

2]0

F11

(x

)

(14000

)1113936D i1x

2 iminus

1113937D i1cos(

xi

iradic

)+1

30[minus600

600]

0F12

(x

)π

D10

sin(πy

1)+

1113936D

minus1

i1

(y

iminus1)

2 [1+10

sin2 (πy

i+1)

]+

(y

Dminus1)

21113966

1113967+

1113936D i1u

(x

i10

100

4)

30[minus505

0]0

yi

1

+x

i+14

u(

xi

ak

m)

k(

ximinus

a)m

x

igt

a

0minus

alt

xilt

a

k(

minusx

iminus

a)m

x

ilt

a

⎧⎪ ⎨ ⎪ ⎩

F13

(x

)01sin

2 (3π

x1)

+1113936

D i1

(x

iminus1)

2 [1+sin

2 (3π

xi+1

+1)

]+

(x

Dminus1)

2 [1+sin

2 (2π

xD

)]1113966

1113967+

1113936D i1u

(x

i51004)

30[minus505

0]0


Tabl

e3

Descriptio

nof

fixed-dim

ensio

nmultim

odal

benchm

arkfunctio

ns

Functio

nD

Rang

efmin

F14

(x

)

((1500)

+1113936

25 j1(1

j+

11139362 i1(

ximinus

aij

)6))

minus1

2[minus656

5]1

F15

(x

)

111393611 i1[

aiminus

(x1(

b2 i

+b

ix2)b

2 i+

bix

3+

x4)

]24

[minus5

5]000030

F16

(x

)4x

2 1minus21x

4 1+

(13)

x6 1

+x1x

2minus4x

2 2+4x

4 22

[minus5

5]minus10316

F17

(x

)

(x2

minus(514π

2 )x2 1

+(5π)

x1

minus6)

2+10

(1

minus(18π

))cos

x1

+10

2[minus5

5]0398

F18

(x

)

[1+

(x1

+x2

+1)

2 (19

minus14

x1

+3x

2 1minus14

x2

+6x

1x2

+3x

2 2)]

times[30

+(2x

1minus3x

2)2 (18

minus32

x1

+12

x2 1

+48

x2

minus36

x1x

2+27

x2 2)

]2

[minus2

2]3

F19

(x

)

minus1113936

4 i1c

iexp(

minus1113936

3 j1

aij

(x

jminus

pij

)2)

3[13

]minus386

F20

(x

)

minus1113936

4 i1c

iexp(

minus1113936

6 j1

aij

(x

jminus

pij

)2)

6[01

]minus332

F21

(x

)

minus1113936

5 i1[

(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus101232

F22

(x

)

minus1113936

7 i1[

(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus104028

F23

(x

)

minus1113936

1 i10

[(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus105363


Tabl

e4

Com

parisonof

optim

izationresults

obtained

fortheun

imod

albenchm

arkfunctio

ns

AL

Runs

30N

30a

ndT

1000

frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 1Be

st290E

minus12

(5)

0454643

(9)

0000908

(7)

104E

minus11

(6)

0042385

(8)

464E

minus17

(4)

276E

minus167(

3)692E

minus190(

1)162E

minus187(

2)

Mean

133E

minus11

(5)

1013223

(9)

0006257

(7)

973E

minus09

(6)

0063236

(8)

101E

minus16

(4)

146E

minus153(

3)107E

minus168(

2)145E

minus172(

1)

Std

998E

minus12

(4)

0311571

(8)

0010917

(7)

423E

minus08

(5)

0010899

(6)

376E

minus17

(3)

791E

minus153(

2)0(

1)0(

1)

F 2Be

st399E

minus08

(5)

0271733

(8)

0006726

(7)

100E

minus05

(6)

0575724

(9)

350E

minus08

(4)

331E

minus114(

3)246E

minus124(

2)201E

minus124(

1)

Mean

680E

minus08

(5)

0324236

(8)

0020418

(7)

0001313

(6)

0727765

(9)

520E

minus08

(4)

988E

minus105(

3)847E

minus113(

1)127E

minus112(

2)

Std

195E

minus08

(5)

0041092

(8)

0014974

(7)

0004947

(6)

005865(

9)117E

minus08

(4)

494E

minus104(

3)421E

minus112(

1)575E

minus112(

2)

F 3Be

st1080803597(

9)83765444(

2)394952176

(5)

3208562

(1)

308450285

(8)

195538197

(4)

167730974

(7)

17510162(

3)532871248

(6)

Mean

157231259

(9)

135934273

(2)

87822206(

4)14904551(

1)5228201445(

5)409456876

(3)

9735512836(

7)1004518144(

8)7621728202(

6)

Std

2683155994(

6)34026326(

2)315589252

(4)

7677907

(1)

1591552755(

5)12983509(

3)3898890192(

7)598648178

(9)

4199058749(

8)

F 4Be

st1080885

(7)

0742244

(6)

3262876

(8)

0311111

(5)

4216935

(9)

135E

minus08

(3)

0019063

(4)

568E

minus10

(2)

430E

minus19

(1)

Mean

1575045

(5)

0972736

(2)

5969996

(6)

0563087

(1)

6113542

(7)

1264362

(3)

17704825(

9)14259819(

8)1562263

(4)

Std

038751(

3)0123737

(1)

157968(

6)0153909

(2)

1246667

(4)

1284382

(5)

1458112

(8)

17566092(

9)6519626

(7)

F 5Be

st25398273(

2)39474199(

7)4367835

(8)

11716823(

1)91050677(

9)25861836(

3)26607341(

5)25886812(

4)26835697(

6)

Mean

3831135

(5)

149884822

(8)

114032052

(7)

60388778(

6)342973593

(9)

37244874(

4)271471(

1)27512334(

3)27496506(

2)

Std

21417578(

4)181590232

(8)

35216402(

6)43241123(

7)492144758

(9)

24882941(

5)0485147

(1)

0547347

(3)

0539586

(2)

F 6Be

st288E

minus12

(2)

06227

(9)

000076(

4)251E

minus11

(3)

0048128

(8)

450E

minus17

(1)

0007148

(5)

0012986

(6)

0015406

(7)

Mean

890E

minus12

(2)

1071114

(9)

000788(

4)968E

minus09

(3)

0070002

(6)

103E

minus16

(1)

0057661

(5)

0263633

(8)

020125(

7)

Std

567E

minus12

(2)

0279422

(9)

0008319

(4)

300E

minus08

(3)

0011199

(5)

446E

minus17

(1)

0078559

(6)

0224346

(8)

0192754

(7)

F 7Be

st0016252

(6)

0002011

(4)

0012074

(5)

0037104

(9)

0035229

(8)

0016733

(7)

0000115

(2)

000016(

3)00001

(1)

Mean

0027322

(5)

0006396

(4)

0030674

(6)

0078177

(9)

0055369

(8)

0049762

(7)

0001935

(3)

00015

(1)

0001579

(2)

Std

0005743

(5)

0002154

(4)

0012799

(7)

0023769

(9)

0011801

(6)

0021587

(8)

0001959

(3)

0001692

(1)

0001761

(2)

Rank

Best

68

75

93

41

2Mean

57

64

82

33

1Std

15

64

71

23

1


Tabl

e5

Com

parisonof

optim

izationresults

obtained

forthemultim

odal

benchm

arkfunctio

ns

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 8Be

stminus1256948662(

1)minus9288859115(

5)minus123198552

(3)

minus8562082036(

8)minus1256947154(

2)minus3257715637(

9)minus8674343572(

7)minus8985661303(

6)minus1057783345(

4)

Mean

minus1256553865(

2)minus8297653051(

4)minus1198240966(

3)minus6118655936(

8)minus1256880513(

1)minus243432431

(9)

minus6658410197(

7)minus7600141184(

6)minus7657917372(

5)

Std

21623781(

2)497301094

(5)

22238506(

3)1574830603(

9)0909111

(1)

355937095

(4)

863971262

(7)

861181057

(6)

1183224502(

8)

F 9Be

st44307253(

7)29244538(

6)16598597(

4)14992992(

3)17949826(

5)14924381(

2)0(

1)0(

1)0(

1)

Mean

56783475(

8)53437985(

7)25788633(

4)46538306(

6)25466452(

3)25802587(

5)0(

1)3672248

(2)

0(1)

Std

6266486

(4)

14333435(

7)403673(

2)14270927(

6)4772648

(3)

6630417

(5)

0(1)

20113732(

8)0(

1)

F 10

Best

423E

minus07

(3)

0226971

(6)

0007536

(5)

300E

minus06

(4)

1080969

(7)

545E

minus09

(2)

888E

minus16

(1)

888E

minus16

(1)

888E

minus16

(1)

Mean

702E

minus07

(5)

0349244

(8)

0020786

(6)

008323(

7)2063648

(9)

770E

minus09

(4)

444E

minus15

(3)

420E

minus15

(2)

397E

minus15

(1)

Std

152E

minus07

(5)

0070199

(7)

0011333

(6)

0317495

(8)

0425232

(9)

150E

minus09

(4)

229E

minus15

(3)

130E

minus15

(1)

203E

minus15

(2)

F 11

Best

144E

minus11

(3)

0503713

(6)

0001224

(4)

742E

minus12

(2)

0120691

(5)

3009729

(7)

0(1)

0(1)

0(1)

Mean

980E

minus10

(1)

0766039

(8)

0026851

(6)

0011493

(5)

0618727

(7)

8568461

(9)

0003104

(4)

0001092

(2)

0001282

(3)

Std

307E

minus09

(1)

011031(

7)0029255

(6)

0009701

(4)

0275077

(8)

4065895

(9)

0011816

(5)

0005981

(2)

0007021

(3)

F 12

Best

239E

minus13

(3)

0001193

(6)

300E

minus06

(4)

338E

minus14

(2)

0000383

(5)

375E

minus19

(1)

0001352

(7)

0003179

(9)

0002984

(8)

Mean

100E

minus12

(1)

0002505

(4)

630E

minus05

(2)

0003456

(5)

0000581

(3)

0151627

(8)

0006188

(6)

0013432

(7)

0409258

(9)

Std

105E

minus12

(1)

0000844

(4)

800E

minus05

(2)

0018927

(7)

0000121

(3)

0210923

(8)

0006962

(6)

0004444

(5)

160174(

9)

F 13

Best

170E

minus12

(2)

0023921

(6)

800E

minus05

(4)

523E

minus12

(3)

0006145

(5)

594E

minus18

(1)

0025211

(7)

0122758

(9)

0080548

(8)

Mean

431E

minus12

(1)

0044601

(6)

0000741

(2)

0001831

(3)

0013234

(5)

0007331

(4)

0233207

(7)

0468482

(8)

047475(

9)

Std

269E

minus12

(1)

0012881

(5)

0000969

(2)

0004165

(3)

0005965

(4)

0025745

(6)

0181659

(7)

0238598

(9)

0224147

(8)

F 14

Best

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998118

(2)

0998004

(1)

0998004

(1)

0998004

(1)

Mean

0998004

(1)

6813974

(7)

0998004

(1)

3459539

(5)

0998004

(1)

4082277

(6)

247579(

3)3454612

(4)

2108894

(2)

Std

0(1)

5363613

(9)

412E

minus17

(2)

2863628

(7)

187E

minus14

(3)

2690671

(5)

2445473

(4)

3273113

(8)

2790461

(6)

F 15

Best

0000352

(3)

0000419

(5)

0000307

(1)

0000308

(2)

0000401

(4)

0001284

(6)

0000308

(2)

0000307

(1)

0000307

(1)

Mean

0000577

(4)

0006093

(9)

0000329

(1)

0000794

(6)

0004004

(8)

0002584

(7)

0000601

(5)

0000513

(3)

0000468

(2)

Std

0000109

(2)

0008761

(9)

410E

minus05

(1)

0000233

(3)

0007444

(8)

0002019

(7)

0000315

(4)

0000377

(5)

0000382

(6)

F 16

Best

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Mean

minus1031628

(1)

minus0977217

(2)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Std

671E

minus16

(4)

0207068

(8)

612E

minus16

(3)

678E

minus16

(5)

353E

minus08

(7)

513E

minus16

(1)

386E

minus11

(6)

671E

minus16

(4)

598E

minus16

(2)

F 17

Best

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

Mean

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397888

(2)

0397887

(1)

0397887

(1)

Std

0(1)

933E

minus11

(3)

0(1)

0(1)

437E

minus08

(4)

0(1)

930E

minus07

(5)

0(1)

125E

minus13

(2)

F 18

Best

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

Mean

3(1)

39(

3)3(

1)3(

1)3900137

(4)

3(1)

3000014

(2)

3(1)

3(1)

Std

127E

minus15

(2)

4929503

(9)

202E

minus15

(5)

152E

minus15

(3)

4929483

(8)

267E

minus15

(6)

380E

minus05

(7)

782E

minus16

(1)

166E

minus15

(4)

F 19

Best

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

Mean

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862628

(2)

minus3862782

(1)

minus3862782

(1)

Std

271E

minus15

(6)

229E

minus15

(1)

271E

minus15

(6)

270E

minus15

(5)

100E

minus06

(7)

239E

minus15

(2)

0000366

(8)

266E

minus15

(4)

251E

minus15

(3)

F 20

Best

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321994

(2)

minus3321995

(1)

minus3321989

(3)

minus3321995

(1)

minus3321995

(1)

Mean

minus3321989

(2)

minus3270475

(6)

minus3321995

(1)

minus3282364

(5)

minus330217(

3)minus3321995

(1)

minus3264959

(7)

minus3293815

(4)

minus3254599

(8)

Std

320E

minus05

(3)

0059923

(7)

173E

minus14

(2)

0057005

(6)

0045065

(4)

147E

minus15

(1)

0071826

(9)

0051995

(5)

0059944

(8)

F 21

Best

minus101532(

1)minus101532(

1)minus101532(

1)minus101532(

1)minus10153199(

2)minus101532(

1)minus10153181(

3)minus101532(

1)minus101532(

1)

Mean

minus10049112(

2)minus5725156

(8)

minus101532(

1)minus8062579

(6)

minus5406265

(9)

minus6043604

(7)

minus8882256

(4)

minus8882841

(3)

minus8462151

(5)

Std

0531474

(2)

3327602

(7)

115E

minus09

(1)

3091228

(6)

3673677

(9)

3495936

(8)

2378758

(3)

2378893

(4)

2686626

(5)


Tabl

e5

Con

tinued

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 22

Best

minus10402941(

1)minus10402941(

1)minus10402941(

1)minus10402941(

1)minus1040294

(2)

minus10402941(

1)minus1040292

(3)

minus10402941(

1)minus10402941(

1)

Mean

minus10211001(

2)minus5529595

(8)

minus10402941(

1)minus9521178

(3)

minus6545124

(7)

minus10402941(

1)minus8448955

(4)

minus7763089

(6)

minus796332(

5)

Std

1045029

(3)

3343034

(8)

169E

minus14

(2)

2005406

(4)

3686937

(9)

132E

minus15

(1)

2836906

(5)

3128103

(7)

286345(

6)

F 23

Best

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus10536335(

2)minus1053641

(1)

minus1053641

(1)

Mean

minus10536383(

2)minus5855753

(8)

minus1053641

(1)

minus916093(

4)minus5180275

(9)

minus10175835(

3)minus9003007

(5)

minus7990287

(7)

minus8647563

(6)

Std

0000108

(2)

3655594

(9)

111E

minus09

(1)

2570428

(4)

3329307

(8)

1485436

(3)

2591415

(5)

3225805

(7)

2733822

(6)

Rank

Best

18

32

75

64

2Mean

29

16

87

53

4Std

19

26

83

74

5




200 400 600 800 1000

10minus150

10minus100

10minus50

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(a)

200 400 600 800 1000

102

104

106

108

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(b)

Ave

rage

ln (f

lowast minus

f min

)

200 400 600 800 1000

10minus2

100

102

IterationDEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(c)

200 400 600 800 1000

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(d)

200 400 600 800 1000

10minus15

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(e)

200 400 600 800 1000

10minus4

10minus3

10minus2

10minus1

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(f )









l l

l x1 x1x2

P







0 100 200 300 400 500242

2422

2424

2426

2428

243

2432

2434


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


Th

R

Ts

R

L







0 1000 2000 3000 4000 500035

4

45

5

55

6


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


l1

d1

d2l2

z1 z2







0 100 200 300 400 500345

35

355

36

365

37


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


A D

B

C













0 100 200 300 400 500minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


1 2 3 4 5

xi

xiConstantthickness







0 200 400 600 800 100001

02

03

04

05

06

07

08

09


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA





P

h

b

tftwL



5 Conclusion




Appendix



minf(x) 22


subject to

g1(x)

2

radicx1 + x2

2radic

x21 + 2x1x2

P minus σ le 0

g2(x) x2

2radic

x21 + 2x1x2

P minus σ le 0

g3(x) 1

2

radicx2 + x1

P minus σ le 0

0lexi le 1 i 1 2

l 100 cm

P 2 kNcm2

σ 2 kNcm2

(A1)




0 100 200 300 400 500minus188

minus186

minus184

minus182

minus18

minus178

minus176


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA




minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0


33 + 1296000le 0


0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)


minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References



























































F1(x) 1113936Di1x

2i 30 [minus 100 100] 0

F2(x) 1113936Di1|xi| + 1113937

Di1 |xi| 30 [minus 10 10] 0

F3(x) 1113936Di1(1113936

ij1 xj)




Di1(xi + 05)2 30 [minus 100 100] 0

F7(x) 1113936Di1 ix4

i + random[0 1) 30 [minus 128 128] 0


Tabl

e2

Descriptio

nof

multim

odal

benchm

arkfunctio

ns

Functio

nD

Rang

efmin

F8(

x)

1113936

D i1

minusx

isin

(

|x

i|1113968

)30

[minus500

500]

minus4189829

times5

F9(

x)

1113936

D i1(

x2 i

minus10

cos(2π

xi)

+10

)30

[minus5125

12]

0

F10

(x

)

minus20

exp(

minus02

(1

D)1113936

D i1x

2 i

1113969

)minusexp(

(1

D)1113936

D i1cos

(2π

xi)

)+20

+e

30[minus323

2]0

F11

(x

)

(14000

)1113936D i1x

2 iminus

1113937D i1cos(

xi

iradic

)+1

30[minus600

600]

0F12

(x

)π

D10

sin(πy

1)+

1113936D

minus1

i1

(y

iminus1)

2 [1+10

sin2 (πy

i+1)

]+

(y

Dminus1)

21113966

1113967+

1113936D i1u

(x

i10

100

4)

30[minus505

0]0

yi

1

+x

i+14

u(

xi

ak

m)

k(

ximinus

a)m

x

igt

a

0minus

alt

xilt

a

k(

minusx

iminus

a)m

x

ilt

a

⎧⎪ ⎨ ⎪ ⎩

F13

(x

)01sin

2 (3π

x1)

+1113936

D i1

(x

iminus1)

2 [1+sin

2 (3π

xi+1

+1)

]+

(x

Dminus1)

2 [1+sin

2 (2π

xD

)]1113966

1113967+

1113936D i1u

(x

i51004)

30[minus505

0]0


Tabl

e3

Descriptio

nof

fixed-dim

ensio

nmultim

odal

benchm

arkfunctio

ns

Functio

nD

Rang

efmin

F14

(x

)

((1500)

+1113936

25 j1(1

j+

11139362 i1(

ximinus

aij

)6))

minus1

2[minus656

5]1

F15

(x

)

111393611 i1[

aiminus

(x1(

b2 i

+b

ix2)b

2 i+

bix

3+

x4)

]24

[minus5

5]000030

F16

(x

)4x

2 1minus21x

4 1+

(13)

x6 1

+x1x

2minus4x

2 2+4x

4 22

[minus5

5]minus10316

F17

(x

)

(x2

minus(514π

2 )x2 1

+(5π)

x1

minus6)

2+10

(1

minus(18π

))cos

x1

+10

2[minus5

5]0398

F18

(x

)

[1+

(x1

+x2

+1)

2 (19

minus14

x1

+3x

2 1minus14

x2

+6x

1x2

+3x

2 2)]

times[30

+(2x

1minus3x

2)2 (18

minus32

x1

+12

x2 1

+48

x2

minus36

x1x

2+27

x2 2)

]2

[minus2

2]3

F19

(x

)

minus1113936

4 i1c

iexp(

minus1113936

3 j1

aij

(x

jminus

pij

)2)

3[13

]minus386

F20

(x

)

minus1113936

4 i1c

iexp(

minus1113936

6 j1

aij

(x

jminus

pij

)2)

6[01

]minus332

F21

(x

)

minus1113936

5 i1[

(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus101232

F22

(x

)

minus1113936

7 i1[

(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus104028

F23

(x

)

minus1113936

1 i10

[(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus105363


Tabl

e4

Com

parisonof

optim

izationresults

obtained

fortheun

imod

albenchm

arkfunctio

ns

AL

Runs

30N

30a

ndT

1000

frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 1Be

st290E

minus12

(5)

0454643

(9)

0000908

(7)

104E

minus11

(6)

0042385

(8)

464E

minus17

(4)

276E

minus167(

3)692E

minus190(

1)162E

minus187(

2)

Mean

133E

minus11

(5)

1013223

(9)

0006257

(7)

973E

minus09

(6)

0063236

(8)

101E

minus16

(4)

146E

minus153(

3)107E

minus168(

2)145E

minus172(

1)

Std

998E

minus12

(4)

0311571

(8)

0010917

(7)

423E

minus08

(5)

0010899

(6)

376E

minus17

(3)

791E

minus153(

2)0(

1)0(

1)

F 2Be

st399E

minus08

(5)

0271733

(8)

0006726

(7)

100E

minus05

(6)

0575724

(9)

350E

minus08

(4)

331E

minus114(

3)246E

minus124(

2)201E

minus124(

1)

Mean

680E

minus08

(5)

0324236

(8)

0020418

(7)

0001313

(6)

0727765

(9)

520E

minus08

(4)

988E

minus105(

3)847E

minus113(

1)127E

minus112(

2)

Std

195E

minus08

(5)

0041092

(8)

0014974

(7)

0004947

(6)

005865(

9)117E

minus08

(4)

494E

minus104(

3)421E

minus112(

1)575E

minus112(

2)

F 3Be

st1080803597(

9)83765444(

2)394952176

(5)

3208562

(1)

308450285

(8)

195538197

(4)

167730974

(7)

17510162(

3)532871248

(6)

Mean

157231259

(9)

135934273

(2)

87822206(

4)14904551(

1)5228201445(

5)409456876

(3)

9735512836(

7)1004518144(

8)7621728202(

6)

Std

2683155994(

6)34026326(

2)315589252

(4)

7677907

(1)

1591552755(

5)12983509(

3)3898890192(

7)598648178

(9)

4199058749(

8)

F 4Be

st1080885

(7)

0742244

(6)

3262876

(8)

0311111

(5)

4216935

(9)

135E

minus08

(3)

0019063

(4)

568E

minus10

(2)

430E

minus19

(1)

Mean

1575045

(5)

0972736

(2)

5969996

(6)

0563087

(1)

6113542

(7)

1264362

(3)

17704825(

9)14259819(

8)1562263

(4)

Std

038751(

3)0123737

(1)

157968(

6)0153909

(2)

1246667

(4)

1284382

(5)

1458112

(8)

17566092(

9)6519626

(7)

F 5Be

st25398273(

2)39474199(

7)4367835

(8)

11716823(

1)91050677(

9)25861836(

3)26607341(

5)25886812(

4)26835697(

6)

Mean

3831135

(5)

149884822

(8)

114032052

(7)

60388778(

6)342973593

(9)

37244874(

4)271471(

1)27512334(

3)27496506(

2)

Std

21417578(

4)181590232

(8)

35216402(

6)43241123(

7)492144758

(9)

24882941(

5)0485147

(1)

0547347

(3)

0539586

(2)

F 6Be

st288E

minus12

(2)

06227

(9)

000076(

4)251E

minus11

(3)

0048128

(8)

450E

minus17

(1)

0007148

(5)

0012986

(6)

0015406

(7)

Mean

890E

minus12

(2)

1071114

(9)

000788(

4)968E

minus09

(3)

0070002

(6)

103E

minus16

(1)

0057661

(5)

0263633

(8)

020125(

7)

Std

567E

minus12

(2)

0279422

(9)

0008319

(4)

300E

minus08

(3)

0011199

(5)

446E

minus17

(1)

0078559

(6)

0224346

(8)

0192754

(7)

F 7Be

st0016252

(6)

0002011

(4)

0012074

(5)

0037104

(9)

0035229

(8)

0016733

(7)

0000115

(2)

000016(

3)00001

(1)

Mean

0027322

(5)

0006396

(4)

0030674

(6)

0078177

(9)

0055369

(8)

0049762

(7)

0001935

(3)

00015

(1)

0001579

(2)

Std

0005743

(5)

0002154

(4)

0012799

(7)

0023769

(9)

0011801

(6)

0021587

(8)

0001959

(3)

0001692

(1)

0001761

(2)

Rank

Best

68

75

93

41

2Mean

57

64

82

33

1Std

15

64

71

23

1


Tabl

e5

Com

parisonof

optim

izationresults

obtained

forthemultim

odal

benchm

arkfunctio

ns

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 8Be

stminus1256948662(

1)minus9288859115(

5)minus123198552

(3)

minus8562082036(

8)minus1256947154(

2)minus3257715637(

9)minus8674343572(

7)minus8985661303(

6)minus1057783345(

4)

Mean

minus1256553865(

2)minus8297653051(

4)minus1198240966(

3)minus6118655936(

8)minus1256880513(

1)minus243432431

(9)

minus6658410197(

7)minus7600141184(

6)minus7657917372(

5)

Std

21623781(

2)497301094

(5)

22238506(

3)1574830603(

9)0909111

(1)

355937095

(4)

863971262

(7)

861181057

(6)

1183224502(

8)

F 9Be

st44307253(

7)29244538(

6)16598597(

4)14992992(

3)17949826(

5)14924381(

2)0(

1)0(

1)0(

1)

Mean

56783475(

8)53437985(

7)25788633(

4)46538306(

6)25466452(

3)25802587(

5)0(

1)3672248

(2)

0(1)

Std

6266486

(4)

14333435(

7)403673(

2)14270927(

6)4772648

(3)

6630417

(5)

0(1)

20113732(

8)0(

1)

F 10

Best

423E

minus07

(3)

0226971

(6)

0007536

(5)

300E

minus06

(4)

1080969

(7)

545E

minus09

(2)

888E

minus16

(1)

888E

minus16

(1)

888E

minus16

(1)

Mean

702E

minus07

(5)

0349244

(8)

0020786

(6)

008323(

7)2063648

(9)

770E

minus09

(4)

444E

minus15

(3)

420E

minus15

(2)

397E

minus15

(1)

Std

152E

minus07

(5)

0070199

(7)

0011333

(6)

0317495

(8)

0425232

(9)

150E

minus09

(4)

229E

minus15

(3)

130E

minus15

(1)

203E

minus15

(2)

F 11

Best

144E

minus11

(3)

0503713

(6)

0001224

(4)

742E

minus12

(2)

0120691

(5)

3009729

(7)

0(1)

0(1)

0(1)

Mean

980E

minus10

(1)

0766039

(8)

0026851

(6)

0011493

(5)

0618727

(7)

8568461

(9)

0003104

(4)

0001092

(2)

0001282

(3)

Std

307E

minus09

(1)

011031(

7)0029255

(6)

0009701

(4)

0275077

(8)

4065895

(9)

0011816

(5)

0005981

(2)

0007021

(3)

F 12

Best

239E

minus13

(3)

0001193

(6)

300E

minus06

(4)

338E

minus14

(2)

0000383

(5)

375E

minus19

(1)

0001352

(7)

0003179

(9)

0002984

(8)

Mean

100E

minus12

(1)

0002505

(4)

630E

minus05

(2)

0003456

(5)

0000581

(3)

0151627

(8)

0006188

(6)

0013432

(7)

0409258

(9)

Std

105E

minus12

(1)

0000844

(4)

800E

minus05

(2)

0018927

(7)

0000121

(3)

0210923

(8)

0006962

(6)

0004444

(5)

160174(

9)

F 13

Best

170E

minus12

(2)

0023921

(6)

800E

minus05

(4)

523E

minus12

(3)

0006145

(5)

594E

minus18

(1)

0025211

(7)

0122758

(9)

0080548

(8)

Mean

431E

minus12

(1)

0044601

(6)

0000741

(2)

0001831

(3)

0013234

(5)

0007331

(4)

0233207

(7)

0468482

(8)

047475(

9)

Std

269E

minus12

(1)

0012881

(5)

0000969

(2)

0004165

(3)

0005965

(4)

0025745

(6)

0181659

(7)

0238598

(9)

0224147

(8)

F 14

Best

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998118

(2)

0998004

(1)

0998004

(1)

0998004

(1)

Mean

0998004

(1)

6813974

(7)

0998004

(1)

3459539

(5)

0998004

(1)

4082277

(6)

247579(

3)3454612

(4)

2108894

(2)

Std

0(1)

5363613

(9)

412E

minus17

(2)

2863628

(7)

187E

minus14

(3)

2690671

(5)

2445473

(4)

3273113

(8)

2790461

(6)

F 15

Best

0000352

(3)

0000419

(5)

0000307

(1)

0000308

(2)

0000401

(4)

0001284

(6)

0000308

(2)

0000307

(1)

0000307

(1)

Mean

0000577

(4)

0006093

(9)

0000329

(1)

0000794

(6)

0004004

(8)

0002584

(7)

0000601

(5)

0000513

(3)

0000468

(2)

Std

0000109

(2)

0008761

(9)

410E

minus05

(1)

0000233

(3)

0007444

(8)

0002019

(7)

0000315

(4)

0000377

(5)

0000382

(6)

F 16

Best

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Mean

minus1031628

(1)

minus0977217

(2)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Std

671E

minus16

(4)

0207068

(8)

612E

minus16

(3)

678E

minus16

(5)

353E

minus08

(7)

513E

minus16

(1)

386E

minus11

(6)

671E

minus16

(4)

598E

minus16

(2)

F 17

Best

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

Mean

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397888

(2)

0397887

(1)

0397887

(1)

Std

0(1)

933E

minus11

(3)

0(1)

0(1)

437E

minus08

(4)

0(1)

930E

minus07

(5)

0(1)

125E

minus13

(2)

F 18

Best

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

Mean

3(1)

39(

3)3(

1)3(

1)3900137

(4)

3(1)

3000014

(2)

3(1)

3(1)

Std

127E

minus15

(2)

4929503

(9)

202E

minus15

(5)

152E

minus15

(3)

4929483

(8)

267E

minus15

(6)

380E

minus05

(7)

782E

minus16

(1)

166E

minus15

(4)

F 19

Best

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

Mean

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862628

(2)

minus3862782

(1)

minus3862782

(1)

Std

271E

minus15

(6)

229E

minus15

(1)

271E

minus15

(6)

270E

minus15

(5)

100E

minus06

(7)

239E

minus15

(2)

0000366

(8)

266E

minus15

(4)

251E

minus15

(3)

F 20

Best

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321994

(2)

minus3321995

(1)

minus3321989

(3)

minus3321995

(1)

minus3321995

(1)

Mean

minus3321989

(2)

minus3270475

(6)

minus3321995

(1)

minus3282364

(5)

minus330217(

3)minus3321995

(1)

minus3264959

(7)

minus3293815

(4)

minus3254599

(8)

Std

320E

minus05

(3)

0059923

(7)

173E

minus14

(2)

0057005

(6)

0045065

(4)

147E

minus15

(1)

0071826

(9)

0051995

(5)

0059944

(8)

F 21

Best

minus101532(

1)minus101532(

1)minus101532(

1)minus101532(

1)minus10153199(

2)minus101532(

1)minus10153181(

3)minus101532(

1)minus101532(

1)

Mean

minus10049112(

2)minus5725156

(8)

minus101532(

1)minus8062579

(6)

minus5406265

(9)

minus6043604

(7)

minus8882256

(4)

minus8882841

(3)

minus8462151

(5)

Std

0531474

(2)

3327602

(7)

115E

minus09

(1)

3091228

(6)

3673677

(9)

3495936

(8)

2378758

(3)

2378893

(4)

2686626

(5)


Tabl

e5

Con

tinued

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 22

Best

minus10402941(

1)minus10402941(

1)minus10402941(

1)minus10402941(

1)minus1040294

(2)

minus10402941(

1)minus1040292

(3)

minus10402941(

1)minus10402941(

1)

Mean

minus10211001(

2)minus5529595

(8)

minus10402941(

1)minus9521178

(3)

minus6545124

(7)

minus10402941(

1)minus8448955

(4)

minus7763089

(6)

minus796332(

5)

Std

1045029

(3)

3343034

(8)

169E

minus14

(2)

2005406

(4)

3686937

(9)

132E

minus15

(1)

2836906

(5)

3128103

(7)

286345(

6)

F 23

Best

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus10536335(

2)minus1053641

(1)

minus1053641

(1)

Mean

minus10536383(

2)minus5855753

(8)

minus1053641

(1)

minus916093(

4)minus5180275

(9)

minus10175835(

3)minus9003007

(5)

minus7990287

(7)

minus8647563

(6)

Std

0000108

(2)

3655594

(9)

111E

minus09

(1)

2570428

(4)

3329307

(8)

1485436

(3)

2591415

(5)

3225805

(7)

2733822

(6)

Rank

Best

18

32

75

64

2Mean

29

16

87

53

4Std

19

26

83

74

5




200 400 600 800 1000

10minus150

10minus100

10minus50

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(a)

200 400 600 800 1000

102

104

106

108

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(b)

Ave

rage

ln (f

lowast minus

f min

)

200 400 600 800 1000

10minus2

100

102

IterationDEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(c)

200 400 600 800 1000

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(d)

200 400 600 800 1000

10minus15

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(e)

200 400 600 800 1000

10minus4

10minus3

10minus2

10minus1

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(f )









l l

l x1 x1x2

P







0 100 200 300 400 500242

2422

2424

2426

2428

243

2432

2434


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


Th

R

Ts

R

L







0 1000 2000 3000 4000 500035

4

45

5

55

6


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


l1

d1

d2l2

z1 z2







0 100 200 300 400 500345

35

355

36

365

37


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


A D

B

C













0 100 200 300 400 500minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


1 2 3 4 5

xi

xiConstantthickness







0 200 400 600 800 100001

02

03

04

05

06

07

08

09


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA





P

h

b

tftwL



5 Conclusion




Appendix



minf(x) 22


subject to

g1(x)

2

radicx1 + x2

2radic

x21 + 2x1x2

P minus σ le 0

g2(x) x2

2radic

x21 + 2x1x2

P minus σ le 0

g3(x) 1

2

radicx2 + x1

P minus σ le 0

0lexi le 1 i 1 2

l 100 cm

P 2 kNcm2

σ 2 kNcm2

(A1)




0 100 200 300 400 500minus188

minus186

minus184

minus182

minus18

minus178

minus176


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA




minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0


33 + 1296000le 0


0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)


minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References

















































Tabl

e2

Descriptio

nof

multim

odal

benchm

arkfunctio

ns

Functio

nD

Rang

efmin

F8(

x)

1113936

D i1

minusx

isin

(

|x

i|1113968

)30

[minus500

500]

minus4189829

times5

F9(

x)

1113936

D i1(

x2 i

minus10

cos(2π

xi)

+10

)30

[minus5125

12]

0

F10

(x

)

minus20

exp(

minus02

(1

D)1113936

D i1x

2 i

1113969

)minusexp(

(1

D)1113936

D i1cos

(2π

xi)

)+20

+e

30[minus323

2]0

F11

(x

)

(14000

)1113936D i1x

2 iminus

1113937D i1cos(

xi

iradic

)+1

30[minus600

600]

0F12

(x

)π

D10

sin(πy

1)+

1113936D

minus1

i1

(y

iminus1)

2 [1+10

sin2 (πy

i+1)

]+

(y

Dminus1)

21113966

1113967+

1113936D i1u

(x

i10

100

4)

30[minus505

0]0

yi

1

+x

i+14

u(

xi

ak

m)

k(

ximinus

a)m

x

igt

a

0minus

alt

xilt

a

k(

minusx

iminus

a)m

x

ilt

a

⎧⎪ ⎨ ⎪ ⎩

F13

(x

)01sin

2 (3π

x1)

+1113936

D i1

(x

iminus1)

2 [1+sin

2 (3π

xi+1

+1)

]+

(x

Dminus1)

2 [1+sin

2 (2π

xD

)]1113966

1113967+

1113936D i1u

(x

i51004)

30[minus505

0]0


Tabl

e3

Descriptio

nof

fixed-dim

ensio

nmultim

odal

benchm

arkfunctio

ns

Functio

nD

Rang

efmin

F14

(x

)

((1500)

+1113936

25 j1(1

j+

11139362 i1(

ximinus

aij

)6))

minus1

2[minus656

5]1

F15

(x

)

111393611 i1[

aiminus

(x1(

b2 i

+b

ix2)b

2 i+

bix

3+

x4)

]24

[minus5

5]000030

F16

(x

)4x

2 1minus21x

4 1+

(13)

x6 1

+x1x

2minus4x

2 2+4x

4 22

[minus5

5]minus10316

F17

(x

)

(x2

minus(514π

2 )x2 1

+(5π)

x1

minus6)

2+10

(1

minus(18π

))cos

x1

+10

2[minus5

5]0398

F18

(x

)

[1+

(x1

+x2

+1)

2 (19

minus14

x1

+3x

2 1minus14

x2

+6x

1x2

+3x

2 2)]

times[30

+(2x

1minus3x

2)2 (18

minus32

x1

+12

x2 1

+48

x2

minus36

x1x

2+27

x2 2)

]2

[minus2

2]3

F19

(x

)

minus1113936

4 i1c

iexp(

minus1113936

3 j1

aij

(x

jminus

pij

)2)

3[13

]minus386

F20

(x

)

minus1113936

4 i1c

iexp(

minus1113936

6 j1

aij

(x

jminus

pij

)2)

6[01

]minus332

F21

(x

)

minus1113936

5 i1[

(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus101232

F22

(x

)

minus1113936

7 i1[

(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus104028

F23

(x

)

minus1113936

1 i10

[(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus105363


Tabl

e4

Com

parisonof

optim

izationresults

obtained

fortheun

imod

albenchm

arkfunctio

ns

AL

Runs

30N

30a

ndT

1000

frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 1Be

st290E

minus12

(5)

0454643

(9)

0000908

(7)

104E

minus11

(6)

0042385

(8)

464E

minus17

(4)

276E

minus167(

3)692E

minus190(

1)162E

minus187(

2)

Mean

133E

minus11

(5)

1013223

(9)

0006257

(7)

973E

minus09

(6)

0063236

(8)

101E

minus16

(4)

146E

minus153(

3)107E

minus168(

2)145E

minus172(

1)

Std

998E

minus12

(4)

0311571

(8)

0010917

(7)

423E

minus08

(5)

0010899

(6)

376E

minus17

(3)

791E

minus153(

2)0(

1)0(

1)

F 2Be

st399E

minus08

(5)

0271733

(8)

0006726

(7)

100E

minus05

(6)

0575724

(9)

350E

minus08

(4)

331E

minus114(

3)246E

minus124(

2)201E

minus124(

1)

Mean

680E

minus08

(5)

0324236

(8)

0020418

(7)

0001313

(6)

0727765

(9)

520E

minus08

(4)

988E

minus105(

3)847E

minus113(

1)127E

minus112(

2)

Std

195E

minus08

(5)

0041092

(8)

0014974

(7)

0004947

(6)

005865(

9)117E

minus08

(4)

494E

minus104(

3)421E

minus112(

1)575E

minus112(

2)

F 3Be

st1080803597(

9)83765444(

2)394952176

(5)

3208562

(1)

308450285

(8)

195538197

(4)

167730974

(7)

17510162(

3)532871248

(6)

Mean

157231259

(9)

135934273

(2)

87822206(

4)14904551(

1)5228201445(

5)409456876

(3)

9735512836(

7)1004518144(

8)7621728202(

6)

Std

2683155994(

6)34026326(

2)315589252

(4)

7677907

(1)

1591552755(

5)12983509(

3)3898890192(

7)598648178

(9)

4199058749(

8)

F 4Be

st1080885

(7)

0742244

(6)

3262876

(8)

0311111

(5)

4216935

(9)

135E

minus08

(3)

0019063

(4)

568E

minus10

(2)

430E

minus19

(1)

Mean

1575045

(5)

0972736

(2)

5969996

(6)

0563087

(1)

6113542

(7)

1264362

(3)

17704825(

9)14259819(

8)1562263

(4)

Std

038751(

3)0123737

(1)

157968(

6)0153909

(2)

1246667

(4)

1284382

(5)

1458112

(8)

17566092(

9)6519626

(7)

F 5Be

st25398273(

2)39474199(

7)4367835

(8)

11716823(

1)91050677(

9)25861836(

3)26607341(

5)25886812(

4)26835697(

6)

Mean

3831135

(5)

149884822

(8)

114032052

(7)

60388778(

6)342973593

(9)

37244874(

4)271471(

1)27512334(

3)27496506(

2)

Std

21417578(

4)181590232

(8)

35216402(

6)43241123(

7)492144758

(9)

24882941(

5)0485147

(1)

0547347

(3)

0539586

(2)

F 6Be

st288E

minus12

(2)

06227

(9)

000076(

4)251E

minus11

(3)

0048128

(8)

450E

minus17

(1)

0007148

(5)

0012986

(6)

0015406

(7)

Mean

890E

minus12

(2)

1071114

(9)

000788(

4)968E

minus09

(3)

0070002

(6)

103E

minus16

(1)

0057661

(5)

0263633

(8)

020125(

7)

Std

567E

minus12

(2)

0279422

(9)

0008319

(4)

300E

minus08

(3)

0011199

(5)

446E

minus17

(1)

0078559

(6)

0224346

(8)

0192754

(7)

F 7Be

st0016252

(6)

0002011

(4)

0012074

(5)

0037104

(9)

0035229

(8)

0016733

(7)

0000115

(2)

000016(

3)00001

(1)

Mean

0027322

(5)

0006396

(4)

0030674

(6)

0078177

(9)

0055369

(8)

0049762

(7)

0001935

(3)

00015

(1)

0001579

(2)

Std

0005743

(5)

0002154

(4)

0012799

(7)

0023769

(9)

0011801

(6)

0021587

(8)

0001959

(3)

0001692

(1)

0001761

(2)

Rank

Best

68

75

93

41

2Mean

57

64

82

33

1Std

15

64

71

23

1


Tabl

e5

Com

parisonof

optim

izationresults

obtained

forthemultim

odal

benchm

arkfunctio

ns

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 8Be

stminus1256948662(

1)minus9288859115(

5)minus123198552

(3)

minus8562082036(

8)minus1256947154(

2)minus3257715637(

9)minus8674343572(

7)minus8985661303(

6)minus1057783345(

4)

Mean

minus1256553865(

2)minus8297653051(

4)minus1198240966(

3)minus6118655936(

8)minus1256880513(

1)minus243432431

(9)

minus6658410197(

7)minus7600141184(

6)minus7657917372(

5)

Std

21623781(

2)497301094

(5)

22238506(

3)1574830603(

9)0909111

(1)

355937095

(4)

863971262

(7)

861181057

(6)

1183224502(

8)

F 9Be

st44307253(

7)29244538(

6)16598597(

4)14992992(

3)17949826(

5)14924381(

2)0(

1)0(

1)0(

1)

Mean

56783475(

8)53437985(

7)25788633(

4)46538306(

6)25466452(

3)25802587(

5)0(

1)3672248

(2)

0(1)

Std

6266486

(4)

14333435(

7)403673(

2)14270927(

6)4772648

(3)

6630417

(5)

0(1)

20113732(

8)0(

1)

F 10

Best

423E

minus07

(3)

0226971

(6)

0007536

(5)

300E

minus06

(4)

1080969

(7)

545E

minus09

(2)

888E

minus16

(1)

888E

minus16

(1)

888E

minus16

(1)

Mean

702E

minus07

(5)

0349244

(8)

0020786

(6)

008323(

7)2063648

(9)

770E

minus09

(4)

444E

minus15

(3)

420E

minus15

(2)

397E

minus15

(1)

Std

152E

minus07

(5)

0070199

(7)

0011333

(6)

0317495

(8)

0425232

(9)

150E

minus09

(4)

229E

minus15

(3)

130E

minus15

(1)

203E

minus15

(2)

F 11

Best

144E

minus11

(3)

0503713

(6)

0001224

(4)

742E

minus12

(2)

0120691

(5)

3009729

(7)

0(1)

0(1)

0(1)

Mean

980E

minus10

(1)

0766039

(8)

0026851

(6)

0011493

(5)

0618727

(7)

8568461

(9)

0003104

(4)

0001092

(2)

0001282

(3)

Std

307E

minus09

(1)

011031(

7)0029255

(6)

0009701

(4)

0275077

(8)

4065895

(9)

0011816

(5)

0005981

(2)

0007021

(3)

F 12

Best

239E

minus13

(3)

0001193

(6)

300E

minus06

(4)

338E

minus14

(2)

0000383

(5)

375E

minus19

(1)

0001352

(7)

0003179

(9)

0002984

(8)

Mean

100E

minus12

(1)

0002505

(4)

630E

minus05

(2)

0003456

(5)

0000581

(3)

0151627

(8)

0006188

(6)

0013432

(7)

0409258

(9)

Std

105E

minus12

(1)

0000844

(4)

800E

minus05

(2)

0018927

(7)

0000121

(3)

0210923

(8)

0006962

(6)

0004444

(5)

160174(

9)

F 13

Best

170E

minus12

(2)

0023921

(6)

800E

minus05

(4)

523E

minus12

(3)

0006145

(5)

594E

minus18

(1)

0025211

(7)

0122758

(9)

0080548

(8)

Mean

431E

minus12

(1)

0044601

(6)

0000741

(2)

0001831

(3)

0013234

(5)

0007331

(4)

0233207

(7)

0468482

(8)

047475(

9)

Std

269E

minus12

(1)

0012881

(5)

0000969

(2)

0004165

(3)

0005965

(4)

0025745

(6)

0181659

(7)

0238598

(9)

0224147

(8)

F 14

Best

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998118

(2)

0998004

(1)

0998004

(1)

0998004

(1)

Mean

0998004

(1)

6813974

(7)

0998004

(1)

3459539

(5)

0998004

(1)

4082277

(6)

247579(

3)3454612

(4)

2108894

(2)

Std

0(1)

5363613

(9)

412E

minus17

(2)

2863628

(7)

187E

minus14

(3)

2690671

(5)

2445473

(4)

3273113

(8)

2790461

(6)

F 15

Best

0000352

(3)

0000419

(5)

0000307

(1)

0000308

(2)

0000401

(4)

0001284

(6)

0000308

(2)

0000307

(1)

0000307

(1)

Mean

0000577

(4)

0006093

(9)

0000329

(1)

0000794

(6)

0004004

(8)

0002584

(7)

0000601

(5)

0000513

(3)

0000468

(2)

Std

0000109

(2)

0008761

(9)

410E

minus05

(1)

0000233

(3)

0007444

(8)

0002019

(7)

0000315

(4)

0000377

(5)

0000382

(6)

F 16

Best

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Mean

minus1031628

(1)

minus0977217

(2)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Std

671E

minus16

(4)

0207068

(8)

612E

minus16

(3)

678E

minus16

(5)

353E

minus08

(7)

513E

minus16

(1)

386E

minus11

(6)

671E

minus16

(4)

598E

minus16

(2)

F 17

Best

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

Mean

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397888

(2)

0397887

(1)

0397887

(1)

Std

0(1)

933E

minus11

(3)

0(1)

0(1)

437E

minus08

(4)

0(1)

930E

minus07

(5)

0(1)

125E

minus13

(2)

F 18

Best

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

Mean

3(1)

39(

3)3(

1)3(

1)3900137

(4)

3(1)

3000014

(2)

3(1)

3(1)

Std

127E

minus15

(2)

4929503

(9)

202E

minus15

(5)

152E

minus15

(3)

4929483

(8)

267E

minus15

(6)

380E

minus05

(7)

782E

minus16

(1)

166E

minus15

(4)

F 19

Best

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

Mean

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862628

(2)

minus3862782

(1)

minus3862782

(1)

Std

271E

minus15

(6)

229E

minus15

(1)

271E

minus15

(6)

270E

minus15

(5)

100E

minus06

(7)

239E

minus15

(2)

0000366

(8)

266E

minus15

(4)

251E

minus15

(3)

F 20

Best

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321994

(2)

minus3321995

(1)

minus3321989

(3)

minus3321995

(1)

minus3321995

(1)

Mean

minus3321989

(2)

minus3270475

(6)

minus3321995

(1)

minus3282364

(5)

minus330217(

3)minus3321995

(1)

minus3264959

(7)

minus3293815

(4)

minus3254599

(8)

Std

320E

minus05

(3)

0059923

(7)

173E

minus14

(2)

0057005

(6)

0045065

(4)

147E

minus15

(1)

0071826

(9)

0051995

(5)

0059944

(8)

F 21

Best

minus101532(

1)minus101532(

1)minus101532(

1)minus101532(

1)minus10153199(

2)minus101532(

1)minus10153181(

3)minus101532(

1)minus101532(

1)

Mean

minus10049112(

2)minus5725156

(8)

minus101532(

1)minus8062579

(6)

minus5406265

(9)

minus6043604

(7)

minus8882256

(4)

minus8882841

(3)

minus8462151

(5)

Std

0531474

(2)

3327602

(7)

115E

minus09

(1)

3091228

(6)

3673677

(9)

3495936

(8)

2378758

(3)

2378893

(4)

2686626

(5)


Tabl

e5

Con

tinued

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 22

Best

minus10402941(

1)minus10402941(

1)minus10402941(

1)minus10402941(

1)minus1040294

(2)

minus10402941(

1)minus1040292

(3)

minus10402941(

1)minus10402941(

1)

Mean

minus10211001(

2)minus5529595

(8)

minus10402941(

1)minus9521178

(3)

minus6545124

(7)

minus10402941(

1)minus8448955

(4)

minus7763089

(6)

minus796332(

5)

Std

1045029

(3)

3343034

(8)

169E

minus14

(2)

2005406

(4)

3686937

(9)

132E

minus15

(1)

2836906

(5)

3128103

(7)

286345(

6)

F 23

Best

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus10536335(

2)minus1053641

(1)

minus1053641

(1)

Mean

minus10536383(

2)minus5855753

(8)

minus1053641

(1)

minus916093(

4)minus5180275

(9)

minus10175835(

3)minus9003007

(5)

minus7990287

(7)

minus8647563

(6)

Std

0000108

(2)

3655594

(9)

111E

minus09

(1)

2570428

(4)

3329307

(8)

1485436

(3)

2591415

(5)

3225805

(7)

2733822

(6)

Rank

Best

18

32

75

64

2Mean

29

16

87

53

4Std

19

26

83

74

5




200 400 600 800 1000

10minus150

10minus100

10minus50

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(a)

200 400 600 800 1000

102

104

106

108

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(b)

Ave

rage

ln (f

lowast minus

f min

)

200 400 600 800 1000

10minus2

100

102

IterationDEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(c)

200 400 600 800 1000

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(d)

200 400 600 800 1000

10minus15

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(e)

200 400 600 800 1000

10minus4

10minus3

10minus2

10minus1

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(f )









l l

l x1 x1x2

P







0 100 200 300 400 500242

2422

2424

2426

2428

243

2432

2434


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


Th

R

Ts

R

L







0 1000 2000 3000 4000 500035

4

45

5

55

6


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


l1

d1

d2l2

z1 z2







0 100 200 300 400 500345

35

355

36

365

37


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


A D

B

C













0 100 200 300 400 500minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


1 2 3 4 5

xi

xiConstantthickness







0 200 400 600 800 100001

02

03

04

05

06

07

08

09


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA





P

h

b

tftwL



5 Conclusion




Appendix



minf(x) 22


subject to

g1(x)

2

radicx1 + x2

2radic

x21 + 2x1x2

P minus σ le 0

g2(x) x2

2radic

x21 + 2x1x2

P minus σ le 0

g3(x) 1

2

radicx2 + x1

P minus σ le 0

0lexi le 1 i 1 2

l 100 cm

P 2 kNcm2

σ 2 kNcm2

(A1)




0 100 200 300 400 500minus188

minus186

minus184

minus182

minus18

minus178

minus176


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA




minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0


33 + 1296000le 0


0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)


minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References

















































Tabl

e3

Descriptio

nof

fixed-dim

ensio

nmultim

odal

benchm

arkfunctio

ns

Functio

nD

Rang

efmin

F14

(x

)

((1500)

+1113936

25 j1(1

j+

11139362 i1(

ximinus

aij

)6))

minus1

2[minus656

5]1

F15

(x

)

111393611 i1[

aiminus

(x1(

b2 i

+b

ix2)b

2 i+

bix

3+

x4)

]24

[minus5

5]000030

F16

(x

)4x

2 1minus21x

4 1+

(13)

x6 1

+x1x

2minus4x

2 2+4x

4 22

[minus5

5]minus10316

F17

(x

)

(x2

minus(514π

2 )x2 1

+(5π)

x1

minus6)

2+10

(1

minus(18π

))cos

x1

+10

2[minus5

5]0398

F18

(x

)

[1+

(x1

+x2

+1)

2 (19

minus14

x1

+3x

2 1minus14

x2

+6x

1x2

+3x

2 2)]

times[30

+(2x

1minus3x

2)2 (18

minus32

x1

+12

x2 1

+48

x2

minus36

x1x

2+27

x2 2)

]2

[minus2

2]3

F19

(x

)

minus1113936

4 i1c

iexp(

minus1113936

3 j1

aij

(x

jminus

pij

)2)

3[13

]minus386

F20

(x

)

minus1113936

4 i1c

iexp(

minus1113936

6 j1

aij

(x

jminus

pij

)2)

6[01

]minus332

F21

(x

)

minus1113936

5 i1[

(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus101232

F22

(x

)

minus1113936

7 i1[

(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus104028

F23

(x

)

minus1113936

1 i10

[(X

minusa

i)(

Xminus

ai)

T+

c i]minus

14

[01

0]minus105363


Tabl

e4

Com

parisonof

optim

izationresults

obtained

fortheun

imod

albenchm

arkfunctio

ns

AL

Runs

30N

30a

ndT

1000

frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 1Be

st290E

minus12

(5)

0454643

(9)

0000908

(7)

104E

minus11

(6)

0042385

(8)

464E

minus17

(4)

276E

minus167(

3)692E

minus190(

1)162E

minus187(

2)

Mean

133E

minus11

(5)

1013223

(9)

0006257

(7)

973E

minus09

(6)

0063236

(8)

101E

minus16

(4)

146E

minus153(

3)107E

minus168(

2)145E

minus172(

1)

Std

998E

minus12

(4)

0311571

(8)

0010917

(7)

423E

minus08

(5)

0010899

(6)

376E

minus17

(3)

791E

minus153(

2)0(

1)0(

1)

F 2Be

st399E

minus08

(5)

0271733

(8)

0006726

(7)

100E

minus05

(6)

0575724

(9)

350E

minus08

(4)

331E

minus114(

3)246E

minus124(

2)201E

minus124(

1)

Mean

680E

minus08

(5)

0324236

(8)

0020418

(7)

0001313

(6)

0727765

(9)

520E

minus08

(4)

988E

minus105(

3)847E

minus113(

1)127E

minus112(

2)

Std

195E

minus08

(5)

0041092

(8)

0014974

(7)

0004947

(6)

005865(

9)117E

minus08

(4)

494E

minus104(

3)421E

minus112(

1)575E

minus112(

2)

F 3Be

st1080803597(

9)83765444(

2)394952176

(5)

3208562

(1)

308450285

(8)

195538197

(4)

167730974

(7)

17510162(

3)532871248

(6)

Mean

157231259

(9)

135934273

(2)

87822206(

4)14904551(

1)5228201445(

5)409456876

(3)

9735512836(

7)1004518144(

8)7621728202(

6)

Std

2683155994(

6)34026326(

2)315589252

(4)

7677907

(1)

1591552755(

5)12983509(

3)3898890192(

7)598648178

(9)

4199058749(

8)

F 4Be

st1080885

(7)

0742244

(6)

3262876

(8)

0311111

(5)

4216935

(9)

135E

minus08

(3)

0019063

(4)

568E

minus10

(2)

430E

minus19

(1)

Mean

1575045

(5)

0972736

(2)

5969996

(6)

0563087

(1)

6113542

(7)

1264362

(3)

17704825(

9)14259819(

8)1562263

(4)

Std

038751(

3)0123737

(1)

157968(

6)0153909

(2)

1246667

(4)

1284382

(5)

1458112

(8)

17566092(

9)6519626

(7)

F 5Be

st25398273(

2)39474199(

7)4367835

(8)

11716823(

1)91050677(

9)25861836(

3)26607341(

5)25886812(

4)26835697(

6)

Mean

3831135

(5)

149884822

(8)

114032052

(7)

60388778(

6)342973593

(9)

37244874(

4)271471(

1)27512334(

3)27496506(

2)

Std

21417578(

4)181590232

(8)

35216402(

6)43241123(

7)492144758

(9)

24882941(

5)0485147

(1)

0547347

(3)

0539586

(2)

F 6Be

st288E

minus12

(2)

06227

(9)

000076(

4)251E

minus11

(3)

0048128

(8)

450E

minus17

(1)

0007148

(5)

0012986

(6)

0015406

(7)

Mean

890E

minus12

(2)

1071114

(9)

000788(

4)968E

minus09

(3)

0070002

(6)

103E

minus16

(1)

0057661

(5)

0263633

(8)

020125(

7)

Std

567E

minus12

(2)

0279422

(9)

0008319

(4)

300E

minus08

(3)

0011199

(5)

446E

minus17

(1)

0078559

(6)

0224346

(8)

0192754

(7)

F 7Be

st0016252

(6)

0002011

(4)

0012074

(5)

0037104

(9)

0035229

(8)

0016733

(7)

0000115

(2)

000016(

3)00001

(1)

Mean

0027322

(5)

0006396

(4)

0030674

(6)

0078177

(9)

0055369

(8)

0049762

(7)

0001935

(3)

00015

(1)

0001579

(2)

Std

0005743

(5)

0002154

(4)

0012799

(7)

0023769

(9)

0011801

(6)

0021587

(8)

0001959

(3)

0001692

(1)

0001761

(2)

Rank

Best

68

75

93

41

2Mean

57

64

82

33

1Std

15

64

71

23

1


Tabl

e5

Com

parisonof

optim

izationresults

obtained

forthemultim

odal

benchm

arkfunctio

ns

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 8Be

stminus1256948662(

1)minus9288859115(

5)minus123198552

(3)

minus8562082036(

8)minus1256947154(

2)minus3257715637(

9)minus8674343572(

7)minus8985661303(

6)minus1057783345(

4)

Mean

minus1256553865(

2)minus8297653051(

4)minus1198240966(

3)minus6118655936(

8)minus1256880513(

1)minus243432431

(9)

minus6658410197(

7)minus7600141184(

6)minus7657917372(

5)

Std

21623781(

2)497301094

(5)

22238506(

3)1574830603(

9)0909111

(1)

355937095

(4)

863971262

(7)

861181057

(6)

1183224502(

8)

F 9Be

st44307253(

7)29244538(

6)16598597(

4)14992992(

3)17949826(

5)14924381(

2)0(

1)0(

1)0(

1)

Mean

56783475(

8)53437985(

7)25788633(

4)46538306(

6)25466452(

3)25802587(

5)0(

1)3672248

(2)

0(1)

Std

6266486

(4)

14333435(

7)403673(

2)14270927(

6)4772648

(3)

6630417

(5)

0(1)

20113732(

8)0(

1)

F 10

Best

423E

minus07

(3)

0226971

(6)

0007536

(5)

300E

minus06

(4)

1080969

(7)

545E

minus09

(2)

888E

minus16

(1)

888E

minus16

(1)

888E

minus16

(1)

Mean

702E

minus07

(5)

0349244

(8)

0020786

(6)

008323(

7)2063648

(9)

770E

minus09

(4)

444E

minus15

(3)

420E

minus15

(2)

397E

minus15

(1)

Std

152E

minus07

(5)

0070199

(7)

0011333

(6)

0317495

(8)

0425232

(9)

150E

minus09

(4)

229E

minus15

(3)

130E

minus15

(1)

203E

minus15

(2)

F 11

Best

144E

minus11

(3)

0503713

(6)

0001224

(4)

742E

minus12

(2)

0120691

(5)

3009729

(7)

0(1)

0(1)

0(1)

Mean

980E

minus10

(1)

0766039

(8)

0026851

(6)

0011493

(5)

0618727

(7)

8568461

(9)

0003104

(4)

0001092

(2)

0001282

(3)

Std

307E

minus09

(1)

011031(

7)0029255

(6)

0009701

(4)

0275077

(8)

4065895

(9)

0011816

(5)

0005981

(2)

0007021

(3)

F 12

Best

239E

minus13

(3)

0001193

(6)

300E

minus06

(4)

338E

minus14

(2)

0000383

(5)

375E

minus19

(1)

0001352

(7)

0003179

(9)

0002984

(8)

Mean

100E

minus12

(1)

0002505

(4)

630E

minus05

(2)

0003456

(5)

0000581

(3)

0151627

(8)

0006188

(6)

0013432

(7)

0409258

(9)

Std

105E

minus12

(1)

0000844

(4)

800E

minus05

(2)

0018927

(7)

0000121

(3)

0210923

(8)

0006962

(6)

0004444

(5)

160174(

9)

F 13

Best

170E

minus12

(2)

0023921

(6)

800E

minus05

(4)

523E

minus12

(3)

0006145

(5)

594E

minus18

(1)

0025211

(7)

0122758

(9)

0080548

(8)

Mean

431E

minus12

(1)

0044601

(6)

0000741

(2)

0001831

(3)

0013234

(5)

0007331

(4)

0233207

(7)

0468482

(8)

047475(

9)

Std

269E

minus12

(1)

0012881

(5)

0000969

(2)

0004165

(3)

0005965

(4)

0025745

(6)

0181659

(7)

0238598

(9)

0224147

(8)

F 14

Best

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998118

(2)

0998004

(1)

0998004

(1)

0998004

(1)

Mean

0998004

(1)

6813974

(7)

0998004

(1)

3459539

(5)

0998004

(1)

4082277

(6)

247579(

3)3454612

(4)

2108894

(2)

Std

0(1)

5363613

(9)

412E

minus17

(2)

2863628

(7)

187E

minus14

(3)

2690671

(5)

2445473

(4)

3273113

(8)

2790461

(6)

F 15

Best

0000352

(3)

0000419

(5)

0000307

(1)

0000308

(2)

0000401

(4)

0001284

(6)

0000308

(2)

0000307

(1)

0000307

(1)

Mean

0000577

(4)

0006093

(9)

0000329

(1)

0000794

(6)

0004004

(8)

0002584

(7)

0000601

(5)

0000513

(3)

0000468

(2)

Std

0000109

(2)

0008761

(9)

410E

minus05

(1)

0000233

(3)

0007444

(8)

0002019

(7)

0000315

(4)

0000377

(5)

0000382

(6)

F 16

Best

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Mean

minus1031628

(1)

minus0977217

(2)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Std

671E

minus16

(4)

0207068

(8)

612E

minus16

(3)

678E

minus16

(5)

353E

minus08

(7)

513E

minus16

(1)

386E

minus11

(6)

671E

minus16

(4)

598E

minus16

(2)

F 17

Best

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

Mean

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397888

(2)

0397887

(1)

0397887

(1)

Std

0(1)

933E

minus11

(3)

0(1)

0(1)

437E

minus08

(4)

0(1)

930E

minus07

(5)

0(1)

125E

minus13

(2)

F 18

Best

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

Mean

3(1)

39(

3)3(

1)3(

1)3900137

(4)

3(1)

3000014

(2)

3(1)

3(1)

Std

127E

minus15

(2)

4929503

(9)

202E

minus15

(5)

152E

minus15

(3)

4929483

(8)

267E

minus15

(6)

380E

minus05

(7)

782E

minus16

(1)

166E

minus15

(4)

F 19

Best

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

Mean

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862628

(2)

minus3862782

(1)

minus3862782

(1)

Std

271E

minus15

(6)

229E

minus15

(1)

271E

minus15

(6)

270E

minus15

(5)

100E

minus06

(7)

239E

minus15

(2)

0000366

(8)

266E

minus15

(4)

251E

minus15

(3)

F 20

Best

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321994

(2)

minus3321995

(1)

minus3321989

(3)

minus3321995

(1)

minus3321995

(1)

Mean

minus3321989

(2)

minus3270475

(6)

minus3321995

(1)

minus3282364

(5)

minus330217(

3)minus3321995

(1)

minus3264959

(7)

minus3293815

(4)

minus3254599

(8)

Std

320E

minus05

(3)

0059923

(7)

173E

minus14

(2)

0057005

(6)

0045065

(4)

147E

minus15

(1)

0071826

(9)

0051995

(5)

0059944

(8)

F 21

Best

minus101532(

1)minus101532(

1)minus101532(

1)minus101532(

1)minus10153199(

2)minus101532(

1)minus10153181(

3)minus101532(

1)minus101532(

1)

Mean

minus10049112(

2)minus5725156

(8)

minus101532(

1)minus8062579

(6)

minus5406265

(9)

minus6043604

(7)

minus8882256

(4)

minus8882841

(3)

minus8462151

(5)

Std

0531474

(2)

3327602

(7)

115E

minus09

(1)

3091228

(6)

3673677

(9)

3495936

(8)

2378758

(3)

2378893

(4)

2686626

(5)


Tabl

e5

Con

tinued

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 22

Best

minus10402941(

1)minus10402941(

1)minus10402941(

1)minus10402941(

1)minus1040294

(2)

minus10402941(

1)minus1040292

(3)

minus10402941(

1)minus10402941(

1)

Mean

minus10211001(

2)minus5529595

(8)

minus10402941(

1)minus9521178

(3)

minus6545124

(7)

minus10402941(

1)minus8448955

(4)

minus7763089

(6)

minus796332(

5)

Std

1045029

(3)

3343034

(8)

169E

minus14

(2)

2005406

(4)

3686937

(9)

132E

minus15

(1)

2836906

(5)

3128103

(7)

286345(

6)

F 23

Best

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus10536335(

2)minus1053641

(1)

minus1053641

(1)

Mean

minus10536383(

2)minus5855753

(8)

minus1053641

(1)

minus916093(

4)minus5180275

(9)

minus10175835(

3)minus9003007

(5)

minus7990287

(7)

minus8647563

(6)

Std

0000108

(2)

3655594

(9)

111E

minus09

(1)

2570428

(4)

3329307

(8)

1485436

(3)

2591415

(5)

3225805

(7)

2733822

(6)

Rank

Best

18

32

75

64

2Mean

29

16

87

53

4Std

19

26

83

74

5




200 400 600 800 1000

10minus150

10minus100

10minus50

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(a)

200 400 600 800 1000

102

104

106

108

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(b)

Ave

rage

ln (f

lowast minus

f min

)

200 400 600 800 1000

10minus2

100

102

IterationDEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(c)

200 400 600 800 1000

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(d)

200 400 600 800 1000

10minus15

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(e)

200 400 600 800 1000

10minus4

10minus3

10minus2

10minus1

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(f )









l l

l x1 x1x2

P







0 100 200 300 400 500242

2422

2424

2426

2428

243

2432

2434


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


Th

R

Ts

R

L







0 1000 2000 3000 4000 500035

4

45

5

55

6


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


l1

d1

d2l2

z1 z2







0 100 200 300 400 500345

35

355

36

365

37


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


A D

B

C













0 100 200 300 400 500minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


1 2 3 4 5

xi

xiConstantthickness







0 200 400 600 800 100001

02

03

04

05

06

07

08

09


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA





P

h

b

tftwL



5 Conclusion




Appendix



minf(x) 22


subject to

g1(x)

2

radicx1 + x2

2radic

x21 + 2x1x2

P minus σ le 0

g2(x) x2

2radic

x21 + 2x1x2

P minus σ le 0

g3(x) 1

2

radicx2 + x1

P minus σ le 0

0lexi le 1 i 1 2

l 100 cm

P 2 kNcm2

σ 2 kNcm2

(A1)




0 100 200 300 400 500minus188

minus186

minus184

minus182

minus18

minus178

minus176


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA




minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0


33 + 1296000le 0


0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)


minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References

















































Tabl

e4

Com

parisonof

optim

izationresults

obtained

fortheun

imod

albenchm

arkfunctio

ns

AL

Runs

30N

30a

ndT

1000

frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 1Be

st290E

minus12

(5)

0454643

(9)

0000908

(7)

104E

minus11

(6)

0042385

(8)

464E

minus17

(4)

276E

minus167(

3)692E

minus190(

1)162E

minus187(

2)

Mean

133E

minus11

(5)

1013223

(9)

0006257

(7)

973E

minus09

(6)

0063236

(8)

101E

minus16

(4)

146E

minus153(

3)107E

minus168(

2)145E

minus172(

1)

Std

998E

minus12

(4)

0311571

(8)

0010917

(7)

423E

minus08

(5)

0010899

(6)

376E

minus17

(3)

791E

minus153(

2)0(

1)0(

1)

F 2Be

st399E

minus08

(5)

0271733

(8)

0006726

(7)

100E

minus05

(6)

0575724

(9)

350E

minus08

(4)

331E

minus114(

3)246E

minus124(

2)201E

minus124(

1)

Mean

680E

minus08

(5)

0324236

(8)

0020418

(7)

0001313

(6)

0727765

(9)

520E

minus08

(4)

988E

minus105(

3)847E

minus113(

1)127E

minus112(

2)

Std

195E

minus08

(5)

0041092

(8)

0014974

(7)

0004947

(6)

005865(

9)117E

minus08

(4)

494E

minus104(

3)421E

minus112(

1)575E

minus112(

2)

F 3Be

st1080803597(

9)83765444(

2)394952176

(5)

3208562

(1)

308450285

(8)

195538197

(4)

167730974

(7)

17510162(

3)532871248

(6)

Mean

157231259

(9)

135934273

(2)

87822206(

4)14904551(

1)5228201445(

5)409456876

(3)

9735512836(

7)1004518144(

8)7621728202(

6)

Std

2683155994(

6)34026326(

2)315589252

(4)

7677907

(1)

1591552755(

5)12983509(

3)3898890192(

7)598648178

(9)

4199058749(

8)

F 4Be

st1080885

(7)

0742244

(6)

3262876

(8)

0311111

(5)

4216935

(9)

135E

minus08

(3)

0019063

(4)

568E

minus10

(2)

430E

minus19

(1)

Mean

1575045

(5)

0972736

(2)

5969996

(6)

0563087

(1)

6113542

(7)

1264362

(3)

17704825(

9)14259819(

8)1562263

(4)

Std

038751(

3)0123737

(1)

157968(

6)0153909

(2)

1246667

(4)

1284382

(5)

1458112

(8)

17566092(

9)6519626

(7)

F 5Be

st25398273(

2)39474199(

7)4367835

(8)

11716823(

1)91050677(

9)25861836(

3)26607341(

5)25886812(

4)26835697(

6)

Mean

3831135

(5)

149884822

(8)

114032052

(7)

60388778(

6)342973593

(9)

37244874(

4)271471(

1)27512334(

3)27496506(

2)

Std

21417578(

4)181590232

(8)

35216402(

6)43241123(

7)492144758

(9)

24882941(

5)0485147

(1)

0547347

(3)

0539586

(2)

F 6Be

st288E

minus12

(2)

06227

(9)

000076(

4)251E

minus11

(3)

0048128

(8)

450E

minus17

(1)

0007148

(5)

0012986

(6)

0015406

(7)

Mean

890E

minus12

(2)

1071114

(9)

000788(

4)968E

minus09

(3)

0070002

(6)

103E

minus16

(1)

0057661

(5)

0263633

(8)

020125(

7)

Std

567E

minus12

(2)

0279422

(9)

0008319

(4)

300E

minus08

(3)

0011199

(5)

446E

minus17

(1)

0078559

(6)

0224346

(8)

0192754

(7)

F 7Be

st0016252

(6)

0002011

(4)

0012074

(5)

0037104

(9)

0035229

(8)

0016733

(7)

0000115

(2)

000016(

3)00001

(1)

Mean

0027322

(5)

0006396

(4)

0030674

(6)

0078177

(9)

0055369

(8)

0049762

(7)

0001935

(3)

00015

(1)

0001579

(2)

Std

0005743

(5)

0002154

(4)

0012799

(7)

0023769

(9)

0011801

(6)

0021587

(8)

0001959

(3)

0001692

(1)

0001761

(2)

Rank

Best

68

75

93

41

2Mean

57

64

82

33

1Std

15

64

71

23

1


Tabl

e5

Com

parisonof

optim

izationresults

obtained

forthemultim

odal

benchm

arkfunctio

ns

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 8Be

stminus1256948662(

1)minus9288859115(

5)minus123198552

(3)

minus8562082036(

8)minus1256947154(

2)minus3257715637(

9)minus8674343572(

7)minus8985661303(

6)minus1057783345(

4)

Mean

minus1256553865(

2)minus8297653051(

4)minus1198240966(

3)minus6118655936(

8)minus1256880513(

1)minus243432431

(9)

minus6658410197(

7)minus7600141184(

6)minus7657917372(

5)

Std

21623781(

2)497301094

(5)

22238506(

3)1574830603(

9)0909111

(1)

355937095

(4)

863971262

(7)

861181057

(6)

1183224502(

8)

F 9Be

st44307253(

7)29244538(

6)16598597(

4)14992992(

3)17949826(

5)14924381(

2)0(

1)0(

1)0(

1)

Mean

56783475(

8)53437985(

7)25788633(

4)46538306(

6)25466452(

3)25802587(

5)0(

1)3672248

(2)

0(1)

Std

6266486

(4)

14333435(

7)403673(

2)14270927(

6)4772648

(3)

6630417

(5)

0(1)

20113732(

8)0(

1)

F 10

Best

423E

minus07

(3)

0226971

(6)

0007536

(5)

300E

minus06

(4)

1080969

(7)

545E

minus09

(2)

888E

minus16

(1)

888E

minus16

(1)

888E

minus16

(1)

Mean

702E

minus07

(5)

0349244

(8)

0020786

(6)

008323(

7)2063648

(9)

770E

minus09

(4)

444E

minus15

(3)

420E

minus15

(2)

397E

minus15

(1)

Std

152E

minus07

(5)

0070199

(7)

0011333

(6)

0317495

(8)

0425232

(9)

150E

minus09

(4)

229E

minus15

(3)

130E

minus15

(1)

203E

minus15

(2)

F 11

Best

144E

minus11

(3)

0503713

(6)

0001224

(4)

742E

minus12

(2)

0120691

(5)

3009729

(7)

0(1)

0(1)

0(1)

Mean

980E

minus10

(1)

0766039

(8)

0026851

(6)

0011493

(5)

0618727

(7)

8568461

(9)

0003104

(4)

0001092

(2)

0001282

(3)

Std

307E

minus09

(1)

011031(

7)0029255

(6)

0009701

(4)

0275077

(8)

4065895

(9)

0011816

(5)

0005981

(2)

0007021

(3)

F 12

Best

239E

minus13

(3)

0001193

(6)

300E

minus06

(4)

338E

minus14

(2)

0000383

(5)

375E

minus19

(1)

0001352

(7)

0003179

(9)

0002984

(8)

Mean

100E

minus12

(1)

0002505

(4)

630E

minus05

(2)

0003456

(5)

0000581

(3)

0151627

(8)

0006188

(6)

0013432

(7)

0409258

(9)

Std

105E

minus12

(1)

0000844

(4)

800E

minus05

(2)

0018927

(7)

0000121

(3)

0210923

(8)

0006962

(6)

0004444

(5)

160174(

9)

F 13

Best

170E

minus12

(2)

0023921

(6)

800E

minus05

(4)

523E

minus12

(3)

0006145

(5)

594E

minus18

(1)

0025211

(7)

0122758

(9)

0080548

(8)

Mean

431E

minus12

(1)

0044601

(6)

0000741

(2)

0001831

(3)

0013234

(5)

0007331

(4)

0233207

(7)

0468482

(8)

047475(

9)

Std

269E

minus12

(1)

0012881

(5)

0000969

(2)

0004165

(3)

0005965

(4)

0025745

(6)

0181659

(7)

0238598

(9)

0224147

(8)

F 14

Best

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998118

(2)

0998004

(1)

0998004

(1)

0998004

(1)

Mean

0998004

(1)

6813974

(7)

0998004

(1)

3459539

(5)

0998004

(1)

4082277

(6)

247579(

3)3454612

(4)

2108894

(2)

Std

0(1)

5363613

(9)

412E

minus17

(2)

2863628

(7)

187E

minus14

(3)

2690671

(5)

2445473

(4)

3273113

(8)

2790461

(6)

F 15

Best

0000352

(3)

0000419

(5)

0000307

(1)

0000308

(2)

0000401

(4)

0001284

(6)

0000308

(2)

0000307

(1)

0000307

(1)

Mean

0000577

(4)

0006093

(9)

0000329

(1)

0000794

(6)

0004004

(8)

0002584

(7)

0000601

(5)

0000513

(3)

0000468

(2)

Std

0000109

(2)

0008761

(9)

410E

minus05

(1)

0000233

(3)

0007444

(8)

0002019

(7)

0000315

(4)

0000377

(5)

0000382

(6)

F 16

Best

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Mean

minus1031628

(1)

minus0977217

(2)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Std

671E

minus16

(4)

0207068

(8)

612E

minus16

(3)

678E

minus16

(5)

353E

minus08

(7)

513E

minus16

(1)

386E

minus11

(6)

671E

minus16

(4)

598E

minus16

(2)

F 17

Best

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

Mean

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397888

(2)

0397887

(1)

0397887

(1)

Std

0(1)

933E

minus11

(3)

0(1)

0(1)

437E

minus08

(4)

0(1)

930E

minus07

(5)

0(1)

125E

minus13

(2)

F 18

Best

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

Mean

3(1)

39(

3)3(

1)3(

1)3900137

(4)

3(1)

3000014

(2)

3(1)

3(1)

Std

127E

minus15

(2)

4929503

(9)

202E

minus15

(5)

152E

minus15

(3)

4929483

(8)

267E

minus15

(6)

380E

minus05

(7)

782E

minus16

(1)

166E

minus15

(4)

F 19

Best

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

Mean

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862628

(2)

minus3862782

(1)

minus3862782

(1)

Std

271E

minus15

(6)

229E

minus15

(1)

271E

minus15

(6)

270E

minus15

(5)

100E

minus06

(7)

239E

minus15

(2)

0000366

(8)

266E

minus15

(4)

251E

minus15

(3)

F 20

Best

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321994

(2)

minus3321995

(1)

minus3321989

(3)

minus3321995

(1)

minus3321995

(1)

Mean

minus3321989

(2)

minus3270475

(6)

minus3321995

(1)

minus3282364

(5)

minus330217(

3)minus3321995

(1)

minus3264959

(7)

minus3293815

(4)

minus3254599

(8)

Std

320E

minus05

(3)

0059923

(7)

173E

minus14

(2)

0057005

(6)

0045065

(4)

147E

minus15

(1)

0071826

(9)

0051995

(5)

0059944

(8)

F 21

Best

minus101532(

1)minus101532(

1)minus101532(

1)minus101532(

1)minus10153199(

2)minus101532(

1)minus10153181(

3)minus101532(

1)minus101532(

1)

Mean

minus10049112(

2)minus5725156

(8)

minus101532(

1)minus8062579

(6)

minus5406265

(9)

minus6043604

(7)

minus8882256

(4)

minus8882841

(3)

minus8462151

(5)

Std

0531474

(2)

3327602

(7)

115E

minus09

(1)

3091228

(6)

3673677

(9)

3495936

(8)

2378758

(3)

2378893

(4)

2686626

(5)


Tabl

e5

Con

tinued

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 22

Best

minus10402941(

1)minus10402941(

1)minus10402941(

1)minus10402941(

1)minus1040294

(2)

minus10402941(

1)minus1040292

(3)

minus10402941(

1)minus10402941(

1)

Mean

minus10211001(

2)minus5529595

(8)

minus10402941(

1)minus9521178

(3)

minus6545124

(7)

minus10402941(

1)minus8448955

(4)

minus7763089

(6)

minus796332(

5)

Std

1045029

(3)

3343034

(8)

169E

minus14

(2)

2005406

(4)

3686937

(9)

132E

minus15

(1)

2836906

(5)

3128103

(7)

286345(

6)

F 23

Best

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus10536335(

2)minus1053641

(1)

minus1053641

(1)

Mean

minus10536383(

2)minus5855753

(8)

minus1053641

(1)

minus916093(

4)minus5180275

(9)

minus10175835(

3)minus9003007

(5)

minus7990287

(7)

minus8647563

(6)

Std

0000108

(2)

3655594

(9)

111E

minus09

(1)

2570428

(4)

3329307

(8)

1485436

(3)

2591415

(5)

3225805

(7)

2733822

(6)

Rank

Best

18

32

75

64

2Mean

29

16

87

53

4Std

19

26

83

74

5




200 400 600 800 1000

10minus150

10minus100

10minus50

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(a)

200 400 600 800 1000

102

104

106

108

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(b)

Ave

rage

ln (f

lowast minus

f min

)

200 400 600 800 1000

10minus2

100

102

IterationDEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(c)

200 400 600 800 1000

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(d)

200 400 600 800 1000

10minus15

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(e)

200 400 600 800 1000

10minus4

10minus3

10minus2

10minus1

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(f )









l l

l x1 x1x2

P







0 100 200 300 400 500242

2422

2424

2426

2428

243

2432

2434


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


Th

R

Ts

R

L







0 1000 2000 3000 4000 500035

4

45

5

55

6


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


l1

d1

d2l2

z1 z2







0 100 200 300 400 500345

35

355

36

365

37


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


A D

B

C













0 100 200 300 400 500minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


1 2 3 4 5

xi

xiConstantthickness







0 200 400 600 800 100001

02

03

04

05

06

07

08

09


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA





P

h

b

tftwL



5 Conclusion




Appendix



minf(x) 22


subject to

g1(x)

2

radicx1 + x2

2radic

x21 + 2x1x2

P minus σ le 0

g2(x) x2

2radic

x21 + 2x1x2

P minus σ le 0

g3(x) 1

2

radicx2 + x1

P minus σ le 0

0lexi le 1 i 1 2

l 100 cm

P 2 kNcm2

σ 2 kNcm2

(A1)




0 100 200 300 400 500minus188

minus186

minus184

minus182

minus18

minus178

minus176


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA




minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0


33 + 1296000le 0


0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)


minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References

















































Tabl

e5

Com

parisonof

optim

izationresults

obtained

forthemultim

odal

benchm

arkfunctio

ns

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 8Be

stminus1256948662(

1)minus9288859115(

5)minus123198552

(3)

minus8562082036(

8)minus1256947154(

2)minus3257715637(

9)minus8674343572(

7)minus8985661303(

6)minus1057783345(

4)

Mean

minus1256553865(

2)minus8297653051(

4)minus1198240966(

3)minus6118655936(

8)minus1256880513(

1)minus243432431

(9)

minus6658410197(

7)minus7600141184(

6)minus7657917372(

5)

Std

21623781(

2)497301094

(5)

22238506(

3)1574830603(

9)0909111

(1)

355937095

(4)

863971262

(7)

861181057

(6)

1183224502(

8)

F 9Be

st44307253(

7)29244538(

6)16598597(

4)14992992(

3)17949826(

5)14924381(

2)0(

1)0(

1)0(

1)

Mean

56783475(

8)53437985(

7)25788633(

4)46538306(

6)25466452(

3)25802587(

5)0(

1)3672248

(2)

0(1)

Std

6266486

(4)

14333435(

7)403673(

2)14270927(

6)4772648

(3)

6630417

(5)

0(1)

20113732(

8)0(

1)

F 10

Best

423E

minus07

(3)

0226971

(6)

0007536

(5)

300E

minus06

(4)

1080969

(7)

545E

minus09

(2)

888E

minus16

(1)

888E

minus16

(1)

888E

minus16

(1)

Mean

702E

minus07

(5)

0349244

(8)

0020786

(6)

008323(

7)2063648

(9)

770E

minus09

(4)

444E

minus15

(3)

420E

minus15

(2)

397E

minus15

(1)

Std

152E

minus07

(5)

0070199

(7)

0011333

(6)

0317495

(8)

0425232

(9)

150E

minus09

(4)

229E

minus15

(3)

130E

minus15

(1)

203E

minus15

(2)

F 11

Best

144E

minus11

(3)

0503713

(6)

0001224

(4)

742E

minus12

(2)

0120691

(5)

3009729

(7)

0(1)

0(1)

0(1)

Mean

980E

minus10

(1)

0766039

(8)

0026851

(6)

0011493

(5)

0618727

(7)

8568461

(9)

0003104

(4)

0001092

(2)

0001282

(3)

Std

307E

minus09

(1)

011031(

7)0029255

(6)

0009701

(4)

0275077

(8)

4065895

(9)

0011816

(5)

0005981

(2)

0007021

(3)

F 12

Best

239E

minus13

(3)

0001193

(6)

300E

minus06

(4)

338E

minus14

(2)

0000383

(5)

375E

minus19

(1)

0001352

(7)

0003179

(9)

0002984

(8)

Mean

100E

minus12

(1)

0002505

(4)

630E

minus05

(2)

0003456

(5)

0000581

(3)

0151627

(8)

0006188

(6)

0013432

(7)

0409258

(9)

Std

105E

minus12

(1)

0000844

(4)

800E

minus05

(2)

0018927

(7)

0000121

(3)

0210923

(8)

0006962

(6)

0004444

(5)

160174(

9)

F 13

Best

170E

minus12

(2)

0023921

(6)

800E

minus05

(4)

523E

minus12

(3)

0006145

(5)

594E

minus18

(1)

0025211

(7)

0122758

(9)

0080548

(8)

Mean

431E

minus12

(1)

0044601

(6)

0000741

(2)

0001831

(3)

0013234

(5)

0007331

(4)

0233207

(7)

0468482

(8)

047475(

9)

Std

269E

minus12

(1)

0012881

(5)

0000969

(2)

0004165

(3)

0005965

(4)

0025745

(6)

0181659

(7)

0238598

(9)

0224147

(8)

F 14

Best

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998004

(1)

0998118

(2)

0998004

(1)

0998004

(1)

0998004

(1)

Mean

0998004

(1)

6813974

(7)

0998004

(1)

3459539

(5)

0998004

(1)

4082277

(6)

247579(

3)3454612

(4)

2108894

(2)

Std

0(1)

5363613

(9)

412E

minus17

(2)

2863628

(7)

187E

minus14

(3)

2690671

(5)

2445473

(4)

3273113

(8)

2790461

(6)

F 15

Best

0000352

(3)

0000419

(5)

0000307

(1)

0000308

(2)

0000401

(4)

0001284

(6)

0000308

(2)

0000307

(1)

0000307

(1)

Mean

0000577

(4)

0006093

(9)

0000329

(1)

0000794

(6)

0004004

(8)

0002584

(7)

0000601

(5)

0000513

(3)

0000468

(2)

Std

0000109

(2)

0008761

(9)

410E

minus05

(1)

0000233

(3)

0007444

(8)

0002019

(7)

0000315

(4)

0000377

(5)

0000382

(6)

F 16

Best

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Mean

minus1031628

(1)

minus0977217

(2)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

minus1031628

(1)

Std

671E

minus16

(4)

0207068

(8)

612E

minus16

(3)

678E

minus16

(5)

353E

minus08

(7)

513E

minus16

(1)

386E

minus11

(6)

671E

minus16

(4)

598E

minus16

(2)

F 17

Best

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

Mean

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397887

(1)

0397888

(2)

0397887

(1)

0397887

(1)

Std

0(1)

933E

minus11

(3)

0(1)

0(1)

437E

minus08

(4)

0(1)

930E

minus07

(5)

0(1)

125E

minus13

(2)

F 18

Best

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

3(1)

Mean

3(1)

39(

3)3(

1)3(

1)3900137

(4)

3(1)

3000014

(2)

3(1)

3(1)

Std

127E

minus15

(2)

4929503

(9)

202E

minus15

(5)

152E

minus15

(3)

4929483

(8)

267E

minus15

(6)

380E

minus05

(7)

782E

minus16

(1)

166E

minus15

(4)

F 19

Best

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

Mean

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862782

(1)

minus3862628

(2)

minus3862782

(1)

minus3862782

(1)

Std

271E

minus15

(6)

229E

minus15

(1)

271E

minus15

(6)

270E

minus15

(5)

100E

minus06

(7)

239E

minus15

(2)

0000366

(8)

266E

minus15

(4)

251E

minus15

(3)

F 20

Best

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321995

(1)

minus3321994

(2)

minus3321995

(1)

minus3321989

(3)

minus3321995

(1)

minus3321995

(1)

Mean

minus3321989

(2)

minus3270475

(6)

minus3321995

(1)

minus3282364

(5)

minus330217(

3)minus3321995

(1)

minus3264959

(7)

minus3293815

(4)

minus3254599

(8)

Std

320E

minus05

(3)

0059923

(7)

173E

minus14

(2)

0057005

(6)

0045065

(4)

147E

minus15

(1)

0071826

(9)

0051995

(5)

0059944

(8)

F 21

Best

minus101532(

1)minus101532(

1)minus101532(

1)minus101532(

1)minus10153199(

2)minus101532(

1)minus10153181(

3)minus101532(

1)minus101532(

1)

Mean

minus10049112(

2)minus5725156

(8)

minus101532(

1)minus8062579

(6)

minus5406265

(9)

minus6043604

(7)

minus8882256

(4)

minus8882841

(3)

minus8462151

(5)

Std

0531474

(2)

3327602

(7)

115E

minus09

(1)

3091228

(6)

3673677

(9)

3495936

(8)

2378758

(3)

2378893

(4)

2686626

(5)


Tabl

e5

Con

tinued

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 22

Best

minus10402941(

1)minus10402941(

1)minus10402941(

1)minus10402941(

1)minus1040294

(2)

minus10402941(

1)minus1040292

(3)

minus10402941(

1)minus10402941(

1)

Mean

minus10211001(

2)minus5529595

(8)

minus10402941(

1)minus9521178

(3)

minus6545124

(7)

minus10402941(

1)minus8448955

(4)

minus7763089

(6)

minus796332(

5)

Std

1045029

(3)

3343034

(8)

169E

minus14

(2)

2005406

(4)

3686937

(9)

132E

minus15

(1)

2836906

(5)

3128103

(7)

286345(

6)

F 23

Best

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus10536335(

2)minus1053641

(1)

minus1053641

(1)

Mean

minus10536383(

2)minus5855753

(8)

minus1053641

(1)

minus916093(

4)minus5180275

(9)

minus10175835(

3)minus9003007

(5)

minus7990287

(7)

minus8647563

(6)

Std

0000108

(2)

3655594

(9)

111E

minus09

(1)

2570428

(4)

3329307

(8)

1485436

(3)

2591415

(5)

3225805

(7)

2733822

(6)

Rank

Best

18

32

75

64

2Mean

29

16

87

53

4Std

19

26

83

74

5




200 400 600 800 1000

10minus150

10minus100

10minus50

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(a)

200 400 600 800 1000

102

104

106

108

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(b)

Ave

rage

ln (f

lowast minus

f min

)

200 400 600 800 1000

10minus2

100

102

IterationDEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(c)

200 400 600 800 1000

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(d)

200 400 600 800 1000

10minus15

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(e)

200 400 600 800 1000

10minus4

10minus3

10minus2

10minus1

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(f )









l l

l x1 x1x2

P







0 100 200 300 400 500242

2422

2424

2426

2428

243

2432

2434


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


Th

R

Ts

R

L







0 1000 2000 3000 4000 500035

4

45

5

55

6


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


l1

d1

d2l2

z1 z2







0 100 200 300 400 500345

35

355

36

365

37


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


A D

B

C













0 100 200 300 400 500minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


1 2 3 4 5

xi

xiConstantthickness







0 200 400 600 800 100001

02

03

04

05

06

07

08

09


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA





P

h

b

tftwL



5 Conclusion




Appendix



minf(x) 22


subject to

g1(x)

2

radicx1 + x2

2radic

x21 + 2x1x2

P minus σ le 0

g2(x) x2

2radic

x21 + 2x1x2

P minus σ le 0

g3(x) 1

2

radicx2 + x1

P minus σ le 0

0lexi le 1 i 1 2

l 100 cm

P 2 kNcm2

σ 2 kNcm2

(A1)




0 100 200 300 400 500minus188

minus186

minus184

minus182

minus18

minus178

minus176


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA




minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0


33 + 1296000le 0


0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)


minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References

















































Tabl

e5

Con

tinued

AL

Runs

30N

30and

T1000

Frank

DE

BBO

BSA

PSO

HS

GSA

WOA

WOA-D

EWOA-BSA

F 22

Best

minus10402941(

1)minus10402941(

1)minus10402941(

1)minus10402941(

1)minus1040294

(2)

minus10402941(

1)minus1040292

(3)

minus10402941(

1)minus10402941(

1)

Mean

minus10211001(

2)minus5529595

(8)

minus10402941(

1)minus9521178

(3)

minus6545124

(7)

minus10402941(

1)minus8448955

(4)

minus7763089

(6)

minus796332(

5)

Std

1045029

(3)

3343034

(8)

169E

minus14

(2)

2005406

(4)

3686937

(9)

132E

minus15

(1)

2836906

(5)

3128103

(7)

286345(

6)

F 23

Best

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus1053641

(1)

minus10536335(

2)minus1053641

(1)

minus1053641

(1)

Mean

minus10536383(

2)minus5855753

(8)

minus1053641

(1)

minus916093(

4)minus5180275

(9)

minus10175835(

3)minus9003007

(5)

minus7990287

(7)

minus8647563

(6)

Std

0000108

(2)

3655594

(9)

111E

minus09

(1)

2570428

(4)

3329307

(8)

1485436

(3)

2591415

(5)

3225805

(7)

2733822

(6)

Rank

Best

18

32

75

64

2Mean

29

16

87

53

4Std

19

26

83

74

5




200 400 600 800 1000

10minus150

10minus100

10minus50

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(a)

200 400 600 800 1000

102

104

106

108

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(b)

Ave

rage

ln (f

lowast minus

f min

)

200 400 600 800 1000

10minus2

100

102

IterationDEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(c)

200 400 600 800 1000

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(d)

200 400 600 800 1000

10minus15

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(e)

200 400 600 800 1000

10minus4

10minus3

10minus2

10minus1

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(f )









l l

l x1 x1x2

P







0 100 200 300 400 500242

2422

2424

2426

2428

243

2432

2434


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


Th

R

Ts

R

L







0 1000 2000 3000 4000 500035

4

45

5

55

6


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


l1

d1

d2l2

z1 z2







0 100 200 300 400 500345

35

355

36

365

37


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


A D

B

C













0 100 200 300 400 500minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


1 2 3 4 5

xi

xiConstantthickness







0 200 400 600 800 100001

02

03

04

05

06

07

08

09


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA





P

h

b

tftwL



5 Conclusion




Appendix



minf(x) 22


subject to

g1(x)

2

radicx1 + x2

2radic

x21 + 2x1x2

P minus σ le 0

g2(x) x2

2radic

x21 + 2x1x2

P minus σ le 0

g3(x) 1

2

radicx2 + x1

P minus σ le 0

0lexi le 1 i 1 2

l 100 cm

P 2 kNcm2

σ 2 kNcm2

(A1)




0 100 200 300 400 500minus188

minus186

minus184

minus182

minus18

minus178

minus176


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA




minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0


33 + 1296000le 0


0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)


minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References



















































200 400 600 800 1000

10minus150

10minus100

10minus50

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(a)

200 400 600 800 1000

102

104

106

108

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(b)

Ave

rage

ln (f

lowast minus

f min

)

200 400 600 800 1000

10minus2

100

102

IterationDEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(c)

200 400 600 800 1000

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(d)

200 400 600 800 1000

10minus15

10minus10

10minus5

100

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(e)

200 400 600 800 1000

10minus4

10minus3

10minus2

10minus1

Iteration

Ave

rage

ln (f

lowast minus

f min

)

DEBBOBSA

PSOHSGSA

WOAWOA-DEWOA-BSA

(f )









l l

l x1 x1x2

P







0 100 200 300 400 500242

2422

2424

2426

2428

243

2432

2434


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


Th

R

Ts

R

L







0 1000 2000 3000 4000 500035

4

45

5

55

6


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


l1

d1

d2l2

z1 z2







0 100 200 300 400 500345

35

355

36

365

37


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


A D

B

C













0 100 200 300 400 500minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


1 2 3 4 5

xi

xiConstantthickness







0 200 400 600 800 100001

02

03

04

05

06

07

08

09


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA





P

h

b

tftwL



5 Conclusion




Appendix



minf(x) 22


subject to

g1(x)

2

radicx1 + x2

2radic

x21 + 2x1x2

P minus σ le 0

g2(x) x2

2radic

x21 + 2x1x2

P minus σ le 0

g3(x) 1

2

radicx2 + x1

P minus σ le 0

0lexi le 1 i 1 2

l 100 cm

P 2 kNcm2

σ 2 kNcm2

(A1)




0 100 200 300 400 500minus188

minus186

minus184

minus182

minus18

minus178

minus176


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA




minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0


33 + 1296000le 0


0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)


minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References























































l l

l x1 x1x2

P







0 100 200 300 400 500242

2422

2424

2426

2428

243

2432

2434


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


Th

R

Ts

R

L







0 1000 2000 3000 4000 500035

4

45

5

55

6


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


l1

d1

d2l2

z1 z2







0 100 200 300 400 500345

35

355

36

365

37


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


A D

B

C













0 100 200 300 400 500minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


1 2 3 4 5

xi

xiConstantthickness







0 200 400 600 800 100001

02

03

04

05

06

07

08

09


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA





P

h

b

tftwL



5 Conclusion




Appendix



minf(x) 22


subject to

g1(x)

2

radicx1 + x2

2radic

x21 + 2x1x2

P minus σ le 0

g2(x) x2

2radic

x21 + 2x1x2

P minus σ le 0

g3(x) 1

2

radicx2 + x1

P minus σ le 0

0lexi le 1 i 1 2

l 100 cm

P 2 kNcm2

σ 2 kNcm2

(A1)




0 100 200 300 400 500minus188

minus186

minus184

minus182

minus18

minus178

minus176


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA




minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0


33 + 1296000le 0


0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)


minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References

















































0 100 200 300 400 500242

2422

2424

2426

2428

243

2432

2434


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


Th

R

Ts

R

L







0 1000 2000 3000 4000 500035

4

45

5

55

6


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


l1

d1

d2l2

z1 z2







0 100 200 300 400 500345

35

355

36

365

37


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


A D

B

C













0 100 200 300 400 500minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


1 2 3 4 5

xi

xiConstantthickness







0 200 400 600 800 100001

02

03

04

05

06

07

08

09


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA





P

h

b

tftwL



5 Conclusion




Appendix



minf(x) 22


subject to

g1(x)

2

radicx1 + x2

2radic

x21 + 2x1x2

P minus σ le 0

g2(x) x2

2radic

x21 + 2x1x2

P minus σ le 0

g3(x) 1

2

radicx2 + x1

P minus σ le 0

0lexi le 1 i 1 2

l 100 cm

P 2 kNcm2

σ 2 kNcm2

(A1)




0 100 200 300 400 500minus188

minus186

minus184

minus182

minus18

minus178

minus176


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA




minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0


33 + 1296000le 0


0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)


minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References

















































0 1000 2000 3000 4000 500035

4

45

5

55

6


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


l1

d1

d2l2

z1 z2







0 100 200 300 400 500345

35

355

36

365

37


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


A D

B

C













0 100 200 300 400 500minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


1 2 3 4 5

xi

xiConstantthickness







0 200 400 600 800 100001

02

03

04

05

06

07

08

09


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA





P

h

b

tftwL



5 Conclusion




Appendix



minf(x) 22


subject to

g1(x)

2

radicx1 + x2

2radic

x21 + 2x1x2

P minus σ le 0

g2(x) x2

2radic

x21 + 2x1x2

P minus σ le 0

g3(x) 1

2

radicx2 + x1

P minus σ le 0

0lexi le 1 i 1 2

l 100 cm

P 2 kNcm2

σ 2 kNcm2

(A1)




0 100 200 300 400 500minus188

minus186

minus184

minus182

minus18

minus178

minus176


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA




minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0


33 + 1296000le 0


0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)


minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References

















































0 100 200 300 400 500345

35

355

36

365

37


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


A D

B

C













0 100 200 300 400 500minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


1 2 3 4 5

xi

xiConstantthickness







0 200 400 600 800 100001

02

03

04

05

06

07

08

09


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA





P

h

b

tftwL



5 Conclusion




Appendix



minf(x) 22


subject to

g1(x)

2

radicx1 + x2

2radic

x21 + 2x1x2

P minus σ le 0

g2(x) x2

2radic

x21 + 2x1x2

P minus σ le 0

g3(x) 1

2

radicx2 + x1

P minus σ le 0

0lexi le 1 i 1 2

l 100 cm

P 2 kNcm2

σ 2 kNcm2

(A1)




0 100 200 300 400 500minus188

minus186

minus184

minus182

minus18

minus178

minus176


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA




minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0


33 + 1296000le 0


0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)


minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References























































0 100 200 300 400 500minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA


1 2 3 4 5

xi

xiConstantthickness







0 200 400 600 800 100001

02

03

04

05

06

07

08

09


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA





P

h

b

tftwL



5 Conclusion




Appendix



minf(x) 22


subject to

g1(x)

2

radicx1 + x2

2radic

x21 + 2x1x2

P minus σ le 0

g2(x) x2

2radic

x21 + 2x1x2

P minus σ le 0

g3(x) 1

2

radicx2 + x1

P minus σ le 0

0lexi le 1 i 1 2

l 100 cm

P 2 kNcm2

σ 2 kNcm2

(A1)




0 100 200 300 400 500minus188

minus186

minus184

minus182

minus18

minus178

minus176


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA




minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0


33 + 1296000le 0


0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)


minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References





















































0 200 400 600 800 100001

02

03

04

05

06

07

08

09


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA





P

h

b

tftwL



5 Conclusion




Appendix



minf(x) 22


subject to

g1(x)

2

radicx1 + x2

2radic

x21 + 2x1x2

P minus σ le 0

g2(x) x2

2radic

x21 + 2x1x2

P minus σ le 0

g3(x) 1

2

radicx2 + x1

P minus σ le 0

0lexi le 1 i 1 2

l 100 cm

P 2 kNcm2

σ 2 kNcm2

(A1)




0 100 200 300 400 500minus188

minus186

minus184

minus182

minus18

minus178

minus176


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA




minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0


33 + 1296000le 0


0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)


minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References

















































5 Conclusion




Appendix



minf(x) 22


subject to

g1(x)

2

radicx1 + x2

2radic

x21 + 2x1x2

P minus σ le 0

g2(x) x2

2radic

x21 + 2x1x2

P minus σ le 0

g3(x) 1

2

radicx2 + x1

P minus σ le 0

0lexi le 1 i 1 2

l 100 cm

P 2 kNcm2

σ 2 kNcm2

(A1)




0 100 200 300 400 500minus188

minus186

minus184

minus182

minus18

minus178

minus176


Mea

n lo

g 10F

(x)

WOAWOA-DEWOA-BSA




minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0


33 + 1296000le 0


0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)


minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References


















































minf(x) 06224x1x3x4 + 17781x2x23

+ 31661x21x4 + 1984x

21x3

subject to

g1(x) minus x1 + 00193x3 le 0

g2(x) minus x2 + 000954x3 le 0


33 + 1296000le 0


0lexi le 100 i 1 2

10le xi le 200 i 3 4

(A2)


minf(x) 07854x1x22 33333x

23 + 149334x3 minus 4309341113872 1113873 minus 1508x1 x

26 + x

271113872 1113873

+ 74777 x36 + x

371113872 1113873 + 07854 x4x

26 + x5x

271113872 1113873

subject to

g1(x) 27

x1x22x3

minus 1le 0

g2(x) 3975

x1x22x

23

minus 1le 0

g3(x) 193x3

4x2x

46x3

minus 1le 0

g4(x) 193x3

5x2x

47x3

minus 1le 0

g5(x) 745x4 x2x3( 1113857( 1113857

2+ 169 times 1061113960 1113961

12

110x36

minus 1le 0

g6(x) 745x5 x2x3( 1113857( 1113857

2+ 1575 times 1061113960 1113961

12

85x37

minus 1le 0

g7(x) x2x3

40minus 1le 0

g8(x) 5x2

x1minus 1le 0

g9(x) x1

12x2minus 1le 0

g10(x) 15x6 + 19

x4minus 1le 0

g11(x) 11x7 + 19

x5minus 1le 0

(A3)


where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References

















































where

26le x1 le 36

07le x2 le 08

17le x3 le 28

73le x4 x5 le 83

29le x6 le 39

50le x7 le 55

(A4)


minf(x) 1

6931minus

x3x2

x1x41113888 1113889

2

subject to

12le x1 x2 x3 x4 le 60

(A5)


minf(x) 06224 x1 + x2 + x3 + x4 + x5( 1113857

subject to

g1(x) 61x31

+27x32

+19x33

+7x34

+1x35

minus 1le 0

001le x1 x2 x3 x4 x5 le 100

(A6)


minf(x) 5000

x3 x2 minus 2x4( 11138573121113872 1113873 + x1x

34( 11138576( 1113857 + 2x1x4 x2 minus x4( 11138572( 1113857

2

subject to

g1(x) 2x1x3 + x3 x2 minus 2x4( 1113857le 300

g2(x) 180000x2

x3 x2 minus 2x4( 11138573

+ 2x1x4 4x24 + 3x2 x2 minus 2x4( 1113857( 1113857

+15000x1

x2 minus 2x4( 1113857x33 + 2x3x

31le 6


(A7)

Data Availability




References

















































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