modified backtracking search optimization algorithm...

Research ArticleModified Backtracking Search Optimization AlgorithmInspired by Simulated Annealing for Constrained EngineeringOptimization Problems

HailongWang1 Zhongbo Hu 1 Yuqiu Sun1 Qinghua Su1 and Xuewen Xia2

1School of Information and Mathematics Yangtze University Jingzhou Hubei 434023 China2School of Software East China Jiaotong University Nanchang Jiangxi 330013 China

Correspondence should be addressed to Zhongbo Hu huzbdd126com

Received 10 October 2017 Accepted 20 December 2017 Published 13 February 2018

Academic Editor Silvia Conforto

Copyright copy 2018 Hailong Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The backtracking search optimization algorithm (BSA) is a population-based evolutionary algorithm for numerical optimizationproblems BSA has a powerful global exploration capacity while its local exploitation capability is relatively poor This affects theconvergence speed of the algorithm In this paper we propose a modified BSA inspired by simulated annealing (BSAISA) toovercome the deficiency of BSA In the BSAISA the amplitude control factor (119865) is modified based on the Metropolis criterion insimulated annealingThe redesigned119865 could be adaptively decreased as the number of iterations increases and it does not introduceextra parameters A self-adaptive 120576-constrainedmethod is used to handle the strict constraintsWe compared the performance of theproposed BSAISAwith BSA and other well-known algorithms when solving thirteen constrained benchmarks and five engineeringdesign problems The simulation results demonstrated that BSAISA is more effective than BSA and more competitive with otherwell-known algorithms in terms of convergence speed

1 Introduction

Optimization is an essential research objective in the fieldsof appliedmathematics and computer sciences Optimizationalgorithms mainly aim to obtain the global optimum foroptimization problems There are many different kinds ofoptimization problems in real world When an optimizationproblem has a simple and explicit gradient informationor requires relatively small budgets of allowed functionevaluations the implementation of classical optimizationtechniques such as mathematical programming often couldachieve efficient results [1] However many real-world engi-neering optimization problemsmay have complex nonlinearor nondifferentiable forms which make them difficult tobe tackled by using classical optimization techniques Theemergence of metaheuristic algorithms has overcome thedeficiencies of classical optimization techniques to someextent as they do not require gradient information andhave the ability to escape from local optima Metaheuristicalgorithms are mainly inspired from a variety of natural

phenomena andor biological social behavior Among thesemetaheuristic algorithms swarm intelligence algorithms andevolutionary algorithms perhaps are the most attractive [2]Swarm intelligence algorithms [3] generally simulate theintelligence behavior of swarms of creatures such as particleswarm optimization (PSO) [4] ant colony optimization(ACO) [5] cuckoo search (CS) [6] and the artificial beecolony (ABC) algorithm [7] These types of algorithmsgenerally are developed by inspirations from a series of com-plex behavior processes in swarms with mutual cooperationand self-organization in which ldquocooperationrdquo is their coreconceptThe evolutionary algorithms (EAs) [8 9] are inspiredby the mechanism of nature evolution in which ldquoevolutionrdquois the key idea Examples of EAs include genetic algorithm(GA) [10] differential evolution (DE) [11ndash14] covariancematrix adaptation evolution strategy (CMAES) [15] and thebacktracking search optimization algorithm (BSA) [16]

BSA is an iterative population-based EA which wasfirst proposed by Civicioglu in 2013 BSA has three basicgenetic operators selection mutation and crossover The

HindawiComputational Intelligence and NeuroscienceVolume 2018 Article ID 9167414 27 pageshttpsdoiorg10115520189167414

2 Computational Intelligence and Neuroscience

main difference between BSA and other similar algorithms isthat BSA possesses a memory for storing a population froma randomly chosen previous generation which is used togenerate the search-direction matrix for the next iterationIn addition BSA has a simple structure which makes itefficient fast and capable of solving multimodal problemsBSA has only one control parameter called the mix-ratewhich significantly reduces the sensitivity of the initial valuesto the algorithmrsquos parameters Due to these characteristicsin less than 4 years BSA has been employed successfullyto solve various engineering optimization problems such aspower systems [17ndash19] induction motor [20 21] antennaarrays [22 23] digital image processing [24 25] artificialneural networks [26ndash29] and energy and environmentalmanagement [30ndash32]

However BSA has a weak local exploitation capacity andits convergence speed is relatively slow Thus many studieshave attempted to improve the performance of BSA andsome modifications of BSA have been proposed to overcomethe deficiencies From the perspective of modified objectthe modifications of BSA can be divided into the followingfour categories It is noted that we consider classifying thepublication into the major modification category if it hasmore than one modification

(i) Modifications of the initial populations [33ndash38](ii) Modifications of the reproduction operators includ-

ing the mutation and crossover operators [39ndash47](iii) Modifications of the selection operators including

the local exploitation strategy [48ndash51](iv) Modifications of the control factor and parameter

[52ndash57]

The research on controlling parameters of EAs is oneof the most promising areas of research in evolutionarycomputation even a little modification of parameters inan algorithm can make a considerable difference [58] Inthe basic BSA the value of amplitude control factor (119865) isthe product of three and the standard normal distributionrandom number (ie 119865 = 3 sdot randn) which is often too largeor too small according to its formulationThismay give BSA apowerful global exploration capability at the early iterationshowever it also affects the later exploitation capability ofBSA Based on these considerations we focus mainly on theinfluence of the amplitude control factor (119865) on the BSAthat is the fourth of the categories defined above Duanand Luo [52] redesigned an adaptive 119865 based on the fitnessstatistics of population at each iteration Wang et al [53] andTian et al [54] proposed an adaptive 119865 based on Maxwell-Boltzmann distribution Askarzadeh and dos Santos Coelho[55] proposed an adaptive 119865 based on Burgerrsquos chaotic mapChen et al [56] redesigned an adaptive 119865 by introducingtwo extra parameters Nama et al [57] proposed a new 119865to adaptively change in the range of (045 199) and newmix-rate to randomly change in the range of (0 1) Thesemodifications of 119865 have achieved good effects in the BSA

Different from the modifications of 119865 in BSA describedabove a modified version of BSA (BSAISA) inspired bysimulated annealing (SA) is proposed in this paper In the

BSAISA 119865 based on iterations is redesigned by learning acharacteristic where SA can probabilistically accept a higherenergy state and the acceptance probability decreases withthe decrease in temperatureThe redesigned 119865 can adaptivelydecrease as the number of iterations increases without intro-ducing extra parameters This adaptive variation tendencyprovides an efficient tradeoff between early exploration andlater exploitation capability We verified the effectiveness andcompetitiveness of BSAISA in simulation experiments usingthirteen constrained benchmarks and five engineering designproblems in terms of convergence speed

The remainder of this paper is organized as followsSection 2 introduces the basic BSA As the main contributionof this paper a detailed explanation of BSAISA is presentedin Section 3 In Section 4 we present two sets of simulationexperiments in which we implemented BSAISA and BSA tosolve thirteen constrained optimization and five engineer-ing design problems The results are compared with thoseobtained by other well-known algorithms in terms of thesolution quality and function evaluations Finally we give ourconcluding remarks in Section 5

2 Backtracking Search OptimizationAlgorithm (BSA)

BSA is a population-based iterative EA BSA generates trialpopulations to take control of the amplitude of the search-direction matrix which provides a strong global explorationcapability BSA equiprobably uses two random crossoverstrategies to exchange the corresponding elements of individ-uals in populations and trial populations during the processof crossover Moreover BSA has two selection processesOne is used to select population from the current andhistorical populations the other is used to select the optimalpopulation In general BSA canbe divided into five processesinitialization selection Imutation crossover and selection II[16]

21 Initialization BSA generates initial population 119875 andinitial old population oldP using

119875119894119895 sim 119880 (low119895 up119895) oldP119894119895 sim 119880 (low119895 up119895) (1)

where 119875119894119895 and oldP119894119895 are the 119895th individual elements inthe problem dimension (119863) that falls in 119894th individualposition in the population size (119873) respectively low119895 andup119895 mean the lower boundary and the upper boundary ofthe 119895th dimension respectively and 119880 is a random uniformdistribution

22 Selection I BSArsquos selection I process is the beginning ofeach iteration It aims to reselect a new oldP for calculatingthe search direction based on population 119875 and historicalpopulation oldP The new oldP is reselected through the ldquoif-thenrdquo rule in

if 119886 lt 119887 then oldP fl 119875 | 119886 119887 sim 119880 (0 1) (2)

Computational Intelligence and Neuroscience 3

where fl is the update operation 119886 and 119887 represent randomnumbers between 0 and 1 The update operation (see (2))ensures BSA has amemory After oldP is reselected the orderof the individuals in oldP is randomly permuted by

oldP fl permuting (oldP) (3)

23 Mutation The mutation operator is used for generatingthe initial form of trial population119872 with

119872 = 119875 + 119865 sdot (oldP minus 119875) (4)

where F is the amplitude control factor of mutation operatorused to control the amplitude of the search direction Thevalue 119865 = 3 sdot randn where randn sim 119873(0 1) and 119873 isstandard normal distribution

24 Crossover In this process BSA generates the final formof trial population 119879 BSA equiprobably uses two crossoverstrategies to manipulate the selected elements of the individ-uals at each iteration Both the strategies generate differentbinary integer-valued matrices (map) of size 119873 sdot 119863 to selectthe elements of individuals that have to be manipulated

Strategy I uses the mix-rate parameter (mix-rate) tocontrol the numbers of elements of individuals that aremanipulated by using lceilmix-rate sdot rand sdot 119863rceil where mix-rate= 1 Strategy II manipulates the only one randomly selectedelement of individuals by using randi(119863) where randi sim119880(0119863)

The two strategies equiprobably are employed to manip-ulate the elements of individuals through the ldquoif-thenrdquo ruleif map119899119898 = 1 where 119899 isin 1 2 119873 and 119898 isin 1 2 119863then 119879 is updated with 119879119899119898 fl 119875119899119898

At the end of crossover process if some individuals in119879 have overflowed the allowed search space limits they willneed to be regenerated by using (1)

25 Selection II In BSArsquos selection II process the fitness val-ues in 119875119894 and 119879119894 are compared used to update the population119875 based on a greedy selection If 119879119894 have a better fitness valuethan 119875119894 then 119879119894 is updated to be 119875119894 The population 119875 isupdated by using

119875119894 = 119879119894 if fitness (119879119894) lt fitness (119875119894) 119875119894 else (5)

If the best individual of 119875 (119875best) has a better fitness valuethan the global minimum value obtained the global mini-mizer will be updated to be 119875best and the global minimumvalue will be updated to be the fitness value of 119875best3 Modified BSA Inspired by SA (BSAISA)

As mentioned in the introduction of this paper the researchwork on the control parameters of an algorithm is verymean-ingful and valuable In this paper in order to improve BSArsquosexploitation capability and convergence speed we propose amodified version of BSA (BSAISA) where the redesign of 119865 is

inspired by SAThemodified details of BSAISA are describedin this section First the structure of the modified 119865 isdescribed before we explain the detailed design principle forthe modified 119865 inspired by SA Subsequently two numericaltests are used to illustrate that the redesigned 119865 improvesthe convergence speed of the algorithm We introduce a self-adaptive 120576-constrained method for handling constraints atthe end of this section

31 Structure of the Adaptive Amplitude Control Factor119865 Themodified 119865 is a normally distributed random number whereits mean value is an exponential function and its variance isequal to 1 In BSAISA we redesign the adaptive 119865 to replacethe original version using

119865119894 sim 119873(exp(minus11003816100381610038161003816Δ11986811989410038161003816100381610038161119866 ) 1) (6)

where 119894 is the index of individuals119865119894 is the adaptive amplitudecontrol factor that corresponds to the 119894th individual |Δ119868119894|is the absolute value of the difference between the objectivefunction values of 119875119894 and 119879119894 (individual differences)119873 is thenormal distribution and 119866 is the current iteration

According to (6) the exponential function (mean value)decreases dynamically with the change in the number ofiterations (119866) and the individual differences (|Δ119868119894|) Basedon the probability density function curve of the normaldistribution the modified 119865 can be decreased adaptively asthe number of iterations increases Another characteristic ofthe modified F is that there are not any extra parameters

32 Design Principle of the Modified Amplitude Control Factor119865 The design principle of the modified 119865 is inspired by theMetropolis criterion in SA SA is a metaheuristic optimiza-tion technique based on physical behavior in nature SAbasedon theMonte Carlomethod was first proposed byMetropoliset al [59] and it was successfully introduced into the field ofcombinatorial optimization for solving complex optimizationproblems by Kirkpatrick et al [60]

The basic concept of SA derives from the process ofphysical annealing with solids An annealing process occurswhen a metal is heated to a molten state with a hightemperature then it is cooled slowly If the temperature isdecreased quickly the resulting crystal will havemany defectsand it is just metastable even the most stable crystallinestate will be achieved at all In other words this mayform a higher energy state than the most stable crystallinestate Therefore in order to reach the absolute minimumenergy state the temperature needs to be decreased at aslow rate SA simulates this process of annealing to searchthe global optimal solution in an optimization problemHowever accepting only the moves that lower the energyof system is like extremely rapid quenching thus SA uses aspecial and effective acceptance method that is Metropoliscriterion which can probabilistically accept the hill-climbingmoves (the higher energy moves) As a result the energy ofthe system evolves into a Boltzmann distribution during theprocess of the simulated annealing From this angle of view


it is no exaggeration to say that theMetropolis criterion is thecore of SA

TheMetropolis criterion can be expressed by the physicalsignificance of energy where the new energy state will beacceptedwhen the new energy state is lower than the previousenergy state and the new energy state will be probabilisticallyaccepted when the new energy state is higher than theprevious energy state This feature of SA can escape frombeing trapped in local minima especially in the early stagesof the search It can also be described as follows

(i) If Δ119864 = 119864119895 minus 119864119894 le 0 then the new state 119895 is acceptedand the energy with the displaced atom is used as the startingpoint for the next step where 119864 represents the energy of theatom Both 119894 and 119895 are the states of atoms and 119895 is the nextstate of 119894

(ii) If Δ119864 gt 0 then calculate the probability of 119875(Δ119864) =exp(minusΔ119864119896119879119888) and generate a random number 120579 which isa uniform distribution over (0 1) where 119896 is Boltzmannrsquosconstant (in general 119896 = 1) and119879119888 is the current temperatureIf 119875(Δ119864) gt 120579 then the new energy will be acceptedotherwise the previous energy is used to start the next step

Analysis 1 The Metropolis criterion states that SA has twocharacteristics (1) SA can probabilistically accept the higherenergy and (2) the acceptance probability of SA decreasesas the temperature decreases Therefore SA can reject andjump out of a local minimumwith a dynamic and decreasingprobability to continue exploiting the other solutions inthe state space This acceptance mechanism can enrich thediversity of energy states

Analysis 2 As shown in (4)119865 is used to control the amplitudeof population mutation in BSA thus 119865 is an important factorfor controlling population diversity If 119865 is excessively largethe diversity of the population will be too high and theconvergence speed of BSA will slow down If 119865 is excessivelysmall the diversity of the population will be reduced so it willbe difficult for BSA to obtain the global optimum and it maybe readily trapped by a local optimum Therefore adaptivelycontrolling the amplitude of 119865 is a key to accelerating theconvergence speed of the algorithm and maintaining its thepopulation diversity

Based on Analyses 1 and 2 it is clear that if 119865 candynamically decrease the convergence speed of BSA will beable to accelerate while maintaining the population diversityOn the other hand SA possesses this characteristic that itsacceptance probability can be dynamically reduced Basedon these two considerations we propose BSAISA with aredesigned 119865 which is inspired by SA More specifically thenew 119865 (see (6)) is redesigned by learning the formulation(119875(Δ119864)) of acceptance probability and its formulation hasbeen shown in the previous subsection

For the two formulas of the modified 119865 and 119875(Δ119864) theindividual difference (|Δ119868119894|) of a population or the energydifference (Δ119864) of a system will decrease as the number ofiterations increases in an algorithm and the temperature ofSA tends to decrease while the iteration of BSA tends toincrease As a result one can observe the correspondence

between modified 119865 and 119875(Δ119864) where the reciprocal ofindividual difference (1|Δ119868119894|) corresponds to the energydifference (Δ119864) of SA and the reciprocal of current iteration(1119866) corresponds to the current temperature (119879119888) of SA Inthis way the redesigned 119865 can be decreased adaptively as thenumber of iterations increases

33 Numerical Analysis of the Modified Amplitude ControlFactor 119865 In order to verify that the convergence speed ofthe basic BSA is improved with the modified 119865 two types(unimodal and multimodal) of unconstrained benchmarkfunctions are used to test the changing trends in 119865 and pop-ulation variances and best function values as the iterationsincreases respectively The two functions are Schwefel 12and Rastrigin and their detailed information is provided in[61]The two functions and the user parameters including thepopulations (119873) dimensions (119863) and maximum iterations(Max) are shown in Table 1 Three groups of test results arecompared in the tests including (1) the comparative curvesof the mean values of the modified 119865 and original 119865 forSchwefel 12 and Rastrigin (2) the comparative curves of themean values of BSA and BSAISA population variances fortwo functions and (3) the convergence curves of BSA andBSAISA for two functions They are depicted in Figures 1 2and 3 respectively Based on Figures 1ndash3 two results can beobserved as follows

(1) According to the trends of F in Figure 1 both theoriginal F and modified F are subject to changes from thenormal distribution The mean value of the original F doesnot tend to decrease as the number of iterations increases Bycontrast the mean value of the modified F exhibits a clearand fluctuating downward trend as the number of iterationsincreases

(2) According to Figure 2 the population variances ofBSAISA and BSA both exhibit a clear downward trend as thenumber of iterations increases The population variances ofBSAISA and BSA are almost same during the early iterationsThis illustrates that the modified F does not reduce thepopulation diversity in the early iterations In the middleand later iterations the population variances of BSAISAdecrease more quickly than that of BSA This illustrates thatthe modified F improves the convergence speed As can beseen from Figure 3 as the number of iterations increases thebest objective function value of BSAISA drops faster thanthat of BSA This shows that the modified F improves theconvergence speed of BSA Moreover BSAISA can find amore accurate solution at the same computational cost

Summary Based on the design principle and numericalanalysis of the modified 119865 the modified 119865 exhibits an overallfluctuating and downward trend which matches with theconcept of the acceptance probability in SA In particularduring the early iterations the modified 119865 is relativelylarge This allows BSAISA to search in a wide region whilemaintaining the population diversity of BSAISA As thenumber of iterations decreases the modified 119865 graduallyexhibits a decreasing trend This accelerates the convergencespeed of BSAISA In the later iterations the modified 119865 isrelatively small This enables BSAISA to fully search in the


Table 1 Two benchmark functions and the corresponding populations dimensions and max iterations

Name Objective function Range 119873 119863 Max

Schwefel 12 119891 (119909) = 119863sum119894=1

( 119894sum119895=1

119909119895)2

[minus100 100] 100 30 500

Rastrigin 119891 (119909) = 119863sum119894=1

(1199092119894 minus 10 cos (2120587119909119894) + 10) [minus512 512] 100 30 500

Note ldquo119873rdquo means populations ldquo119863rdquo means dimensions and ldquoMaxrdquo means maximum iterations

0 100 200 300 400 500minus05

0

05

1

15

Number of iterations

Mea

n va

lue o

f F

BSAISA(a) Modified 119865 on Schwefel 12

0 100 200 300 400 500minus10

minus5

0

5

10


Mea

n va

lue o

f F

BSA(b) Original 119865 on Schwefel 12

0 100 200 300 400 500minus05

0

05

1

15


Mea

n va

lue o

f F

BSAISA(c) Modified 119865 on Rastrigin

0 100 200 300 400 500minus10

minus5

0

5

10


Mea

n va

lue o

f F

BSA(d) Original 119865 on Rastrigin

Figure 1 Comparisons of modified 119865 and original 119865 for Schwefel 12 and Rastrigin


0 100 200 300 400 5000

500

1000

1500

2000

2500

3000

3500


Mea

n va

lue o

f pop

ulat

ion

varia

nces

BSABSAISA

(a) Schwefel 12

0 100 200 300 400 5000

2

4

6

8

10


Mea

n va

lue o

f pop

ulat

ion

varia

nces

BSABSAISA

(b) Rastrigin

Figure 2 The mean values of population variances versus number of iterations using BSA and BSAISA for two functions

0 100 200 300 400 5000

2

4

6

8

10


Best

func

tion

valu

e

BSABSAISA

times104

(a) Schwefel 12

BSABSAISA

0 100 200 300 400 5000

100

200

300

400

500


Best

func

tion

valu

e

(b) Rastrigin

Figure 3 Convergence curves of BSA and BSAISA for two functions

local region Therefore it can be concluded that BSAISA canadaptively control the amplitude of population mutation tochange its local exploitation capacity This may improve theconvergence speed of the algorithm Moreover the modified119865 does not introduce extra parameters so it does not increasethe sensitivity of BSAISA to the parameters

34 A Self-Adaptive 120576-Constrained Method for Handling Con-straints In general a constrained optimization problem can

be mathematically formulated as a minimization problem asfollows

min 119891 (119883)subject to 119892119894 (119883) le 0 119894 = 1 2 119898

ℎ119895 (119883) = 0 119895 = 1 2 119899(7)


where 119883 = (1199091 1199092 119909119863) isin 119877119863 is a 119863-dimensional vector119898 is the total number of inequality constraints and n is thetotal number of equality constraintsThe equality constraintsare transformed into inequality constraints by using |ℎ119895(119883)|minus120575 le 0 where 120575 is a very small degree of violation and120575 = 1119864 minus 4 in this paper The maximization problems aretransformed into minimization problems using minus119891(119883) Theconstraint violation 120593(119883) is given by

120593 (119883) = 119898sum119894=1

max (0 119892119894 (119883)) + 119899sum119895=1

max (0 10038161003816100381610038161003816ℎ119895 (119883)10038161003816100381610038161003816 minus 120575) (8)

Several constraint-handlingmethods have been proposedpreviously where the five most commonly used methodscomprise penalty functions feasibility and dominance rules

(FAD) stochastic ranking 120576-constrained methods and mul-tiobjectives concepts Among these five methods the 120576-constrained method is relatively effective and used widelyZhang et al [62] proposed a self-adaptive 120576-constrainedmethod (SA120576) to combine with the basic BSA for con-strained problems It has been verified that the SA120576 has astronger search efficiency and convergence than the fixed120576-constrained method and FAD In this paper the SA120576 isused to combine with BSAISA for constrained optimizationproblems which comprises the following two rules (1) ifthe constraint violations of two solutions are smaller thana given 120576 value or two solutions have the same constraintviolations the solution with a better objective function valueis preferred and (2) if not the solution with a smallerconstraint violation is preferred SA120576 could be expressed bythe following equations

the better one = 119891 (1198831)

119891(1198831) lt 119891 (1198832) if 120593 (1198831) 120593 (1198832) le 120576119891 (1198831) lt 119891 (1198832) if 120593 (1198831) = 120593 (1198832) 120593 (1198831) le 120593 (1198832) otherwise

(9)

where 120576 is a positive value that represents a tolerance relatedto constraint violationThe self-adaptive 120576 value is formulatedas the following equation

1205760 = 120593 (1198751205790 ) (10)

1205761 (0) = 1205760 (11)

1205762 (119905) = 120593(119879120579119905 ) if 1205760 gt Th11205760 else (12)

1205761 (119905)=

1205762 (119905) if 1205762 (119905) gt Th2 1205762 (119905) lt 1205761 (119905 minus 1) 1205761 (119905 minus 1) else

(13)

120576 (119905) = 1205761 (119905) (1 minus 119905119879119888)

cp if 119905 le 1198791198880 else (14)

where 119905 is the number of the current iterations 120593(1198751205790 ) is theconstraint violation of the top 120579th individual in the initialpopulation 120593(119879120579119905 ) is the constraint violation of the top 120579thindividual in the trial population at the current iteration cpand 119879119888 are control parameters Th1 and Th2 are thresholdvalues 120576119905 is related to the iteration 119905 and functions 1205761(119905) and1205762(119905)

Firstly 1205760 is set as 120593(1198751205790 ) If the initial value 1205760 is biggerthanTh1 120593(119879120579119905 ) will be assigned to 1205762(119905) otherwise 1205760 will beassigned to 1205762(119905) Then if 1205762(119905) lt 1205761(119905 minus 1) and 1205762(119905) is biggerthanTh2 1205762(119905)will be assigned to 1205761(119905) otherwise 1205761(119905minus1)willbe assigned to 1205761(119905) Finally 120576119905 is updated as (14)The detailed

information of SA120576 can be acquired from [62] and the relatedparameter settings of SA120576 (the same as [62]) are presented inTable 2

To illustrate the changing trend of the self-adaptive 120576value vividly BSAISA with SA120576 is used to solve a well-knownbenchmark constrained function G10 in [61] The relatedparameters are set as 119873 = 30 120579 = 03119873 cp = 5 119879119888 = 2333Th1 = 10 and Th2 = 2 The changing trend of 120576 value isshown in Figure 4 Three sampling points that is 120576(500) =06033 120576(1000) = 01227 and 120576(2000) = 1194119864 minus 4 aremarked in Figure 4 As shown in Figure 4 it can be observedthat 120576 value declines very fast at first After it is smaller thanabout 2 it declines as an exponential wayThis changing trendof 120576 value could help algorithm to sufficiently search infeasibledomains near feasible domains

The pseudocode for BSAISA is showed in Pseudocode 1In Pseudocode 1 the modified adaptive 119865 is shown in lines(14)ndash(16)WhenBSAISAdeals with constrained optimizationproblems the code in line (8) and line (40) in Pseudocode 1should consider objective function value and constraintviolation simultaneously and SA120576 is applied to choose a bettersolution or best solution in line (42) and lines (47)-(48)

4 Experimental Studies

In this section two sets of simulation experiments wereexecuted to evaluate the effectiveness of the proposedBSAISA The first experiment set performed on 13 well-known benchmark constrained functions taken from [63](see Appendix A) These thirteen benchmarks contain dif-ferent properties as shown in Table 3 including the numberof variables (119863) objective function types the feasibility ratio(120588) constraint types and number and the number of activeconstraints in the optimum solution The second experiment


Table 2 The parameter setting of SA120576119879119888 = 02 lowast 119879max 120579 = 03 lowast 119873 cp = 5 Th1 = 10 Th2 = 2Note ldquo119879maxrdquo means the maximum iterations ldquo119873rdquo means populations

Table 3 Characters of the 13 benchmark functions

Fun 119863 Type 120588 () LI NI LE NE Activeg01 13 Quadratic 00003 9 0 0 0 6g02 20 Nonlinear 999973 1 1 0 0 1g03 10 Nonlinear 00000 0 0 0 1 1g04 5 Quadratic 270079 0 6 0 0 2g05 4 Nonlinear 00000 2 0 0 3 3g06 2 Nonlinear 00057 0 2 0 0 2g07 10 Quadratic 00003 3 5 0 0 6g08 2 Nonlinear 08581 0 2 0 0 0g09 7 Nonlinear 05199 0 4 0 0 2g10 8 Linear 00020 3 3 0 0 3g11 2 Quadratic 00973 0 0 0 1 1g12 3 Quadratic 47679 0 93 0 0 0g13 5 Nonlinear 00000 0 0 1 2 3Note ldquo119863rdquo is the number of variables ldquo120588rdquo represents feasibility ratio ldquoLIrdquo ldquoNIrdquo ldquoLErdquo and ldquoNErdquo represent linear inequality nonlinear inequality linear equalityand nonlinear equality respectively ldquoActiverdquo represents the number of active constraints at the global optimum

0 500 1000 1500 2000 2500

X 500Y 06033

Iteration

X 1000Y 01227

X 2000Y 00001194

1010

100

10minus10

10minus20

va

lue

Figure 4 Plot of 120576 value with iteration

is conducted on 5 engineering constrained optimizationproblems chosen from [64] (see Appendix B) These fiveproblems are the three-bar truss design problem (TTP)pressure vessel design problem (PVP) tensioncompressionspring design problem (TCSP) welded beam design problem(WBP) and speed reducer design problem (SRP) respec-tively These engineering problems include objective func-tions and constraints of various types and natures (quadraticcubic polynomial and nonlinear) with various number ofdesign variables (continuous integer mixed and discrete)

The recorded experimental results include the best func-tion value (Best) the worst function value (Worst) the meanfunction value (Mean) the standard deviation (Std) the best

solution (variables of best function value) the correspondingconstraint value and the number of function evaluations(FEs)The number of function evaluations can be consideredas a convergence rate or a computational cost

In order to evaluate the performance of BSAISA in termsof convergence speed the FEs are considered as the best FEscorresponding to the obtained best solution in this paperThe calculation of FEs are the product of population sizes(119873) and the number of iterations (Ibest) at which the bestfunction value is first obtained (ie FEs = 119873 lowast Ibest) Forexample if 2500 is themaximumnumber of iterations for oneminimization problem 119891(1999) = 00126653 119891(2000) =00126652 119891(2500) = 119891(2000) = 00126652 the Ibestvalue should be 2000However BSAISAneeds to evaluate theinitial historical population (oldP) so its actual FEs should beplus119873 (ie FEs = 119873 lowast Ibest + 119873)

41 Parameter Settings For the first experiment the mainparameters for 13 benchmark constrained functions are thesame as the following population size (119873) is set as 30 themaximum number of iterations (119879max) is set as 11665 There-fore BSAISArsquos maximum number of function evaluations(MFEs) should equal to 349980 (nearly 350000) The 13benchmarks were executed by using 30 independent runs

For the 5 real-world engineering design problems we useslightly different parameter settings since each problem hasdifferent natures that is TTP (119873 = 20 119879max = 1000) PVP(119873 = 20 119879max = 3000) TCSP (119873 = 20 119879max = 3000) WBP(119873 = 20 119879max = 3000) SRP (119873 = 20 119879max = 2000) The 6engineering problems were performed using 50 independentruns

The user parameters of all experiments are presentedin Table 4 Termination condition may be the maximum


Input ObjFun119873119863 max-iteration mix-rate low1119863 up1119863Output globalminimum globalminimizerrand sim 119880(0 1) randn sim 119873(0 1) 119908 = randint(sdot) randint sim 119880(sdot) 119908 isin 1 2 sdot

(1) function BSAISA(ObjFun 119863119873 max-iteration low up) Initialization(2) globalminimum = inf(3) for 119894 from 1 to 119873 do(4) for 119895 from 1 to 119863 do(5) 119875119894119895 = rand sdot (up119895 minus low119895) + low119895 Initialization of population 119875(6) oldP119894119895 = rand sdot (up119895 minus low119895) + low119895 Initialization of oldP(7) endlowast(8) fitness119875119894 = ObjFun(119875119894) fitnessoldP119894 = ObjFun(oldP119894) Initial-fitness values of 119875 and oldP(9) end(10) for iteration from 1 to max-iteration do Selection-I(11) if (119886 lt 119887 | 119886 119887 sim 119880(0 1)) then oldP fl 119875 end(12) oldP fl permuting(oldP)(13) Generation of Trail-Population∙(14) Δ119868119894 = abs(ObjFun(119875119894) minusObjFun(oldP119894)) Modified F based on SA∙(15) 119866 = iteration Modified F based on SA

∙(16) 119865119894 sim 119873(exp(minus1Δ1198681198941119866 ) 1) Modified F based on SA(17) 119872 = 119875 + 119865 sdot (oldP minus 119875) Mutation(18) map11198731119863 = 1 Initial-map is an119873-by-119863matrix of ones(19) if (119888 lt 119889 | 119888 119889 sim 119880(0 1)) then Crossover(20) for 119894 from 1 to 119873 do(21) map119894119906(1lceilmix-ratesdotrandsdot119863rceil) = 0 | 119906 = permuting(1 2 3 119863)(22) end(23) else(24) for 119894 from 1 to 119873 do map119894randi(119863) = 0 end(25) end(26) 119879 fl119872 Generation of Trial Population 119879(27) for 119894 from 1 to 119873 do(28) for 119895 from 1 to 119863 do(29) if map119894119895 = 1 then 119879119894119895 fl 119875119894119895(30) end(31) end(32) for 119894 from 1 to 119873 do(33) for 119895 from 1 to 119863 do(34) if (119879119894119895 lt low119894119895) or (119879119894119895 gt up119894119895) then(35) 119879119894119895 = rand sdot (up119895 minus low119895) + low119895(36) end(37) end(38) end(39) endlowast(40) fitness 119879119894 = ObjFun(119879119895) Seletion-II(41) for 119894 from 1 to 119873 dolowast(42) if fitness 119879119894 lt fitness 119875119894 then(43) fitness 119875119894 = fitness 119879119894(44) 119875119894 = 119879119894(45) end(46) endlowast(47) fitness 119875best = min(fitness 119875) | best isin 1 2 3 119873lowast(48) if fitness 119875best lt globalminimum then Export globalminimum and globalminimizer(49) globalminimum fl fitness 119875best(50) globalminimizer fl 119875best(51) end(52) end

Pseudocode 1 Pseudocode of BSAISA


Table 4 User parameters used for all experiments

Problem G01ndashG13 TTP PVP TCSP WBP SRP119873 30 20 20 20 20 20119879max 11665 1000 3000 3000 3000 2000Runs 30 50 50 50 50 50Note The expression ldquorunsrdquo denotes the number of independent runs

number of iterations CPU time or an allowable tolerancevalue between the last result and the best known functionvalue for most metaheuristics In this paper the maximumnumber of iterations is considered as termination conditionAll experiments were conducted on a computer with aIntel(R) Core(TM) i5-4590 CPU 330GHz and 4GB RAM

42 Simulation on Constrained Benchmark Problems In thissection BSAISA and BSA are performed on the 13 bench-marks simultaneously Their statistical results obtained from30 independent runs are listed in Tables 5 and 6 including thebest known function value (Best Known) and the obtainedBestMeanWorstStd values as well as the FEs The bestknown values for all the test problems are derived from [62]The best known values found by algorithms are highlighted inbold From Tables 5 and 6 it can be seen that BSAISA is ableto find the known optimal function values for G01 G04 G05G06 G08 G09 G011 G012 and G013 however BSA fails tofind the best known function value on G09 and G13 For therest of the functions BSAISA obtains the results very close tothe best known function values Moreover BSAISA requiresfewer FEs than BSA on G01 G03 G04 G05 G06 G07 G08G09 G011 andG012 Although the FEs of BSAISA are slightlyworse than that of BSA for G02 G10 and G13 BSAISA findsmore accurate best function values than BSA for G02 G03G07 G09 G10 and G13

To further compare BSAISA and BSA the function valueconvergence curves of 13 functions that have significantdifferences have been plotted as shown in Figures 5 and 6The horizontal axis represents the number of iterations whilethe vertical axis represents the difference of the objectivefunction value and the best known value For G02 G04G06 G08 G11 and G12 it is obvious that the convergencespeed of BSAISA is faster than that of BSA and its bestfunction value is also better than that of BSA For G01G03 G05 G07 G09 G10 and G13 it can be observed thatthe objective function values of BSAISA and BSA fluctuatein the early iterations and they decrease as the number ofiterations increases during the middle and late stages Thisillustrates that both the algorithms are able to escape the localoptimum under different degrees and the convergence curveof BSAISA drops still faster than that of BSA Therefore theexperiment results demonstrate that the convergence speedof the basic BSA is improved with our modified 119865

In order to further verify the competitiveness of BSAISAin aspect of convergence speed we compared BSAISA withsome classic and state-of-the-art approaches in terms of bestfunction value and function evaluations The best functionvalue and the corresponding FEs of each algorithm on 13

benchmarks are presented in Table 7 where the optimalresults are in bold on each function These compared algo-rithms are listed below

(1) Stochastic ranking (SR) [63]

(2) Filter simulated annealing (FSA) [65]

(3) Cultured differential evolution (CDE) [66]

(4) Agent based memetic algorithm (AMA) [64]

(5) Modified artificial bee colony (MABC) algorithm [67]

(6) Rough penalty genetic algorithm (RPGA) [68]

(7) BSA combined self-adaptive 120576 constrained method(BSA-SA120576) [62]

To compare these algorithms synthetically a simpleevaluation mechanism is used It can be explained as thebest function value (Best) is preferred and the functionevaluations (FEs) are secondary More specifically (1) if onealgorithm has a better Best than those of others on a functionthere is no need to consider FEs and the algorithm is superiorto other algorithms on this function (2) If two or morealgorithms have found the optimal Best on a function thealgorithm with the lowest FEs is considered as the winneron the function (3) Record the number of winners and thenumber of the optimal function values for each algorithmon the set of benchmarks and then give the sort for allalgorithms

From Table 7 it can be observed that the rank of these 8algorithms is as follows BSAISA CDE BSA-SA120576MABC SRRPGA FSA and AMA Among the 13 benchmarks BSAISAwins on 6 functions and it is able to find the optimal valuesof 10 functions This is better than all other algorithms thusBSAISA ranks the first The second algorithm CDE performsbetter on G02 G07 G09 and G10 than BSAISA but worseon G01 G03 G08 G11 G12 and G13 BSA-SA120576 obtains theoptimal function values of 10 functions but requires morefunction evaluations thanBSAISA andCDE so it should rankthe third MABC ranks the fourth It obtains the optimalfunction values of 7 functions which are fewer in numberthan those of the former three algorithms Both SR andRPGAhave found the same number of the optimal function valueswhile the former is the winner on G04 so SR is slightly betterthan RPGA As for the last two algorithms FSA and AMAjust perform well on three functions while FSA is the winneron G06 so FSA is slightly better than AMA

Based on the above comparison it can be concluded thatBSAISA is effective and competitive in terms of convergencespeed

43 Simulation on Engineering Design Problems In order toassess the optimization performance of BSAISA in real-worldengineering constrained optimization problems 5 well-known engineering constrained design problems includ-ing three-bar truss design pressure vessel design ten-sioncompression spring design welded beam design andspeed reducer design are considered in the second experi-ment


Table 5 The statistical results of BSAISA for 13 constrained benchmarks

Fun Known optimal Best Mean Worst Std FEsG01 minus15 minus15 minus15 minus15000000 808119864 minus 16 84630G02 minus0803619 minus0803599 minus0787688 minus0758565 114119864 minus 02 349500G03 minus1000500 minus1000498 minus1000481 minus1000441 135119864 minus 05 58560G04 minus30665538672 minus30665538672 minus30665538672 minus30665538672 109119864 minus 11 121650G05 5126496714 5126496714 5126496714 5126496714 585119864 minus 13 238410G06 minus6961813876 minus6961813876 minus6961813876 minus6961813876 185119864 minus 12 89550G07 24306209 24307381 24400881 24758205 102119864 minus 01 15060G08 minus00958250 minus00958250 minus0086683 minus0027263 237119864 minus 02 30930G09 680630057 680630057 680633025 680680043 911119864 minus 03 347760G10 7049248021 7049249056 7081241789 7326853581 624119864 + 01 346980G11 0749900 0749900 0749900 0749900 113119864 minus 16 87870G12 minus1 minus1 minus1 minus1 0 5430G13 00539415 00539415 01030000 04594033 980119864 minus 02 349800Note ldquoKnown optimalrdquo denotes the best known function values in the literatures ldquoBoldrdquo means the algorithm has found the best known function values Thesame as Table 6

Table 6 The statistical results of the basic BSA on 13 constrained benchmarks

Fun Known optimal Best Mean Worst Std FEsG01 minus15 minus15 minus15 minus15000000 660119864 minus 16 99300G02 minus0803619 minus0803255 minus0792760 minus0749326 910119864 minus 03 344580G03 minus1000500 minus1000488 minus0998905 minus0990600 266119864 minus 03 348960G04 minus30665538672 minus30665538672 minus30665538672 minus30665538672 111119864 minus 11 272040G05 5126496714 5126496714 5144041363 5275384724 399119864 + 01 299220G06 minus6961813876 minus6961813876 minus6961813876 minus6961813876 185119864 minus 12 111450G07 24306209 24307607 24344626 24399896 193119864 minus 02 347250G08 minus00958250 minus00958250 minus00958250 minus00958250 282119864 minus 17 73440G09 680630057 680630058 680630352 680632400 563119864 minus 04 348900G10 7049248021 7049278543 7053573853 7080192700 728119864 + 00 340020G11 0749900 0749900 0749900 0749900 113119864 minus 16 113250G12 minus1 minus1 minus1 minus1 0 13590G13 00539415 00539420 01816986 05477657 150119864 minus 01 347400

431 Three-Bar Truss Design Problem (TTP) The three-bar truss problem is one of the engineering minimizationtest problems for constrained algorithms The best feasiblesolution is obtained by BSAISA at 119909 = (0788675 0408248)with the objective function value 119891(119909) = 263895843 using8940 FEs The comparison of the best solutions obtainedfrom BSAISA BSA differential evolution with dynamicstochastic selection (DEDS) [69] hybrid evolutionary algo-rithm (HEAA) [70] hybrid particle swarm optimizationwithdifferential evolution (POS-DE) [71] differential evolutionwith level comparison (DELC) [72] andmine blast algorithm(MBA) [73] is presented in Table 8Their statistical results arelisted in Table 9

From Tables 8 and 9 BSAISA BSA DEDS HEAA PSO-DE and DELC all reach the best solution with the corre-sponding function value equal to 263895843 except MBAwith 263895852 However BSAISA requires the lowest FEs(only 8940) among all algorithms Its Std value is better thanBSA DEDS HEAA PSO-DE andMBA except DELCThese

comparative results indicate that BSAISA outperforms otheralgorithms in terms of computational cost and robustness forthis problem

Figure 7 depicts the convergence curves of BSAISA andBSA for the three-bar truss design problem where the valueof 119865(119909lowast) on the vertical axis equals 263895843 As shownin Figure 7 BSA achieves the global optimum at about 700iterations while BSAISA only reaches the global optimum atabout 400 iterations It can be concluded that the convergencespeed of BSAISA is faster than that of BSA for this problem

432 Pressure Vessel Design Problem (PVP) The pressurevessel design problemhas a nonlinear objective functionwiththree linear and one nonlinear inequality constraints andtwo discrete and two continuous design variables The valuesof the two discrete variables (1199091 1199092) should be the integermultiples of 00625 The best feasible solution is obtained byBSAISA at 119909 = (08750 04375 420984 1766366) with theobjective function value 119891(119909) = 60597143 using 31960 FEs


Table 7 Comparison of the best values and FEs obtained by BSAISA and other algorithms

Alg BSAISA SR FSA CDE AMA MABC RPGA BSA-SA120576Fun Best (FEs) Best (FEs) Best (FEs) Best (FEs) Best (FEs) Best (FEs) Best (FEs) Best (FEs)

G01 minus15 minus15000 minus14993316 minus15000000 minus15000 minus15000 minus15000 minus15000000(84630) (148200) (205748) (100100) (350000) (350000) (350000) (350000)




G05 5126497 5126497 51264981 5126571 5126512 5126487 5126544 5126497(238410) (51600) (47661) (100100) (350000) (350000) (350000) (350000)


G07 24307 24307 24311 24306 24315 24324 24333 24306(15060) (143000) (404501) (100100) (350000) (350000) (350000) (350000)


G09 680630057 680630 68063008 680630057 680645 680631 680631 6806301(347760) (111400) (324569) (100100) (350000) (350000) (350000) (350000)

G10 7049249 7054316 7059864 7049248 7281957 7058823 7049861 7049278(346980) (128400) (243520) (100100) (350000) (350000) (350000) (350000)

G11 0749900 0750 0749999 0749900 0750 0750 0749 0749900(87870) (11400) (23722) (100100) (350000) (350000) (350000) (350000)

G12 minus1 minus1000000 minus1000000 minus1000000 minus1000 minus1000 NA minus1000000(5430) (16400) (59355) (100100) (350000) (350000) (350000)

G13 00539415 0053957 00539498 0056180 0053947 0757 NA 00539415(349800) (69800) (120268) (100100) (350000) (350000) (350000)

Nu 6 + 10 1 + 5 1 + 3 4 + 10 0 + 3 1 + 7 0 + 5 0 + 10RK 1 5 7 2 8 4 6 3Note ldquoNArdquo means not available The same as Tables 8 10 11 12 13 14 15 and 17 ldquoRKrdquo represents the comprehensive ranking of each algorithm on the set ofbenchmarks ldquoNurdquo represents the sum of the number of winners and the number of the optimal function values for each algorithm on the set of benchmarks

Table 8 Comparison of best solutions for the three-bar truss design problem

Method DEDS HEAA PSO-DE DELC MBA BSA BSAISA1198831 0788675 0788680 0788675 0788675 0788675 0788675 07886751198832 0408248 0408234 0408248 0408248 0408560 0408248 04082481198921(119883) 177119864 minus 08 NA minus529119864 minus 11 NA minus529119864 minus 11 minus323119864 minus 12 01198922(119883) minus1464102 NA minus1463748 NA minus1463748 minus1464102 minus14641021198923(119883) minus0535898 NA minus0536252 NA minus0536252 minus0535898 minus0535898119891(119883) 263895843 263895843 263895843 263895843 263895852 263895843 263895843

Table 9 Comparison of statistical results for the three-bar truss design problem

Method Worst Mean Best Std FEsDEDS 263895849 263895843 263895843 97119864 minus 07 15000HEAA 263896099 263895865 263895843 49119864 minus 05 15000PSO-DE 263895843 263895843 263895843 45119864 minus 10 17600DELC 263895843 263895843 263895843 43119864 minus 14 10000MBA 263915983 263897996 263895852 393119864 minus 03 13280BSA 263895845 263895843 263895843 264119864 minus 07 13720BSAISA 263895843 263895843 263895843 575119864 minus 13 8940


0 2000 4000 6000Number of iterations

f(x

)minusf(x

lowast)(ＦＩＡ)

10minus2

10minus6

10minus10

10minus14

BSABSAISA

(a) Convergence curve on G01

0 3000 6000 9000 12000Number of iterations

f(x

)minusf(x

lowast)(ＦＩＡ)

100

10minus1

10minus2

10minus3

10minus4

10minus5

BSABSAISA

(b) Convergence curve on G02


f(x

)minusf(x

lowast)(ＦＩＡ)

102

100

10minus2

10minus4

10minus6

BSABSAISA

(c) Convergence curve on G03


f(x

)minusf(x

lowast)(ＦＩＡ)

103

100

10minus3

10minus6

10minus9

10minus12

BSABSAISA

(d) Convergence curve on G04


f(x

)minusf(x

lowast)(ＦＩＡ)

103

100

10minus3

10minus6

10minus9

10minus12

BSABSAISA

(e) Convergence curve on G05


f(x

)minusf(x

lowast)(ＦＩＡ)

104

101

10minus2

10minus5

10minus8

10minus11

BSABSAISA

(f) Convergence curve on G06


f(x

)minusf(x

lowast)(ＦＩＡ)

102

101

100

10minus1

10minus2

10minus3

BSABSAISA

(g) Convergence curve on G07


f(x

)minusf(x

lowast)(ＦＩＡ)

102

10minus2

10minus6

10minus10

10minus14

10minus18

BSABSAISA

(h) Convergence curve on G08


f(x

)minusf(x

lowast)(ＦＩＡ)102

100

10minus2

10minus4

10minus6

BSABSAISA

(i) Convergence curve on G09

Figure 5 The convergence curves of the first 9 functions by BSAISA and BSA

For this problem BSAISA is compared with nine algo-rithms BSA BSA-SA120576 [62] DELC POS-DE genetic algo-rithms based on dominance tournament selection (GA-DT)[73] modified differential evolution (MDE) [74] coevolu-tionary particle swarm optimization (CPSO) [75] hybridparticle swarm optimization (HPSO) [76] and artificial beecolony algorithm (ABC) [77] The comparison of the bestsolutions obtained by BSAISA and other reported algorithmsis presented in Table 10 The statistical results of variousalgorithms are listed in Table 11

As shown Table 10 the obtained solution sets of allalgorithms satisfy the constraints for this problem BSAISABSA-SA120576 ABCDELC andHPSOfind the same considerablegood objective function value 60597143 which is slightly

worse than MDErsquos function value 60597143 It is worthmentioning that MBArsquos best solution was obtained at 119909 =(07802 03856 404292 1984964)with119891(119909)= 58893216 andthe corresponding constraint values equal to 119892119894(119909) = (0 0minus863645 minus415035) in [78] Though MBA finds a far betterfunction value than that of MDE its obtained variables (ie07802 and 03856) are not integermultiples of 00625 So theyare not listed in Table 10 to ensure a fair comparison FromTable 11 except forMDEwith the function value of 60597016BSAISA offers better function value results compared toGA-DT CPSO ABC and BSA Besides that BSAISA is farsuperior to other algorithms in terms of FEs Unfortunatelythe obtained Std value of BSAISA is relatively poor comparedwith others for this problem



f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA

103

101

10minus1

10minus3



f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA

10minus12

10minus16

10minus8

100

10minus4


f(x

)minusf(x

lowast)(ＦＩＡ)

0 100 200 300 400 500Number of iterations

BSABSAISA

10minus12

10minus16

10minus8

10minus1

10minus4


f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA


10minus8

10minus4

10minus2

10minus6


Figure 6 The convergence curves of the latter 4 functions by BSAISA and BSA

Table 10 Comparison of best solutions for the pressure vessel design problem

Method GA-DT MDE CPSO HPSO DELC ABC BSA-SA120576 BSA BSAISA1198831 08125 08125 08125 08125 08125 08125 08125 08125 081251198832 04375 04375 04375 04375 04375 04375 04375 04375 043751198833 420974 420984 420913 420984 420984 420984 420984 420984 4209841198834 1766540 1766360 1767465 1766366 1766366 1766366 1766366 1766366 17663661198921(119883) minus201119864 minus 03 0 minus137119864 minus 06 NA NA 0 minus95119864 minus 10 minus369119864 minus 08 01198922(119883) minus358119864 minus 02 minus0035881 minus359119864 minus 04 NA NA minus0035881 minus359119864 minus 2 minus0035881 minus00358811198923(119883) minus247593 minus00000 minus1187687 NA NA minus0000226 minus12119864 minus 4 minus0095446 01198924(119883) minus633460 minus633639 minus632535 NA NA minus63363 minus63363 minus632842 minus633634119891(119883) 60599463 60597017 60610777 60597143 60597143 60597143 60597143 60597150 60597143


Table 11 Comparison of statistical results for the pressure vessel design problem

Method Worst Mean Best Std FEsGA-DT 64693220 61772533 60599463 1309297 80000MDE 60597017 60597017 60597017 10119864 minus 12 24000CPSO 63638041 61471332 60610777 8645 30000HPSO 62886770 60999323 60597143 8620 81000DELC 60597143 60597143 60597143 21119864 minus 11 30000PSO-DE 60597143 60597143 60597143 10119864 minus 10 42100ABC NA 62453081 60597147 205119864 + 02 30000BSA-SA120576 61167804 60743682 60597143 171119864 + 01 80000BSA 67715969 62212861 60597150 203119864 + 02 60000BSAISA 71980054 64181935 60597143 304119864 + 02 16320


BSABSAISA

10minus1

101

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)

Figure 7 Convergence curves of BSAISA and BSA for the three-bartruss design problem

Figure 8 describes the convergence curves of BSAISAand BSA for the pressure vessel design problem where thevalue of119865(119909lowast) on the vertical axis equals 60597143 As shownin Figure 8 BSAISA is able to find the global optimum atabout 800 iterations and obtains a far more accurate functionvalue than that of BSA Moreover the convergence speed ofBSAISA is much faster than that of BSA

433 Tension Compression Spring Design Problem (TCSP)This design optimization problem has three continuousvariables and four nonlinear inequality constraints The bestfeasible solution is obtained by BSAISA at 119909 = (00516870356669 11291824) with 119891(119909) = 0012665 using 9440 FEsThis problem has been solved by other methods as followsGA-DT MDE CPSO HPSO DEDS HEAA DELC POS-DE ABC MBA BSA-SA120576 and Social Spider Optimization(SSOC) [79] The comparison of the best solutions obtainedfrom various algorithms is presented in Table 12 Theirstatistical results are listed in Table 13


BSABSAISA

106

104

102

100

10minus2

10minus4

F(x

)minusF(x

lowast)(ＦＩＡ)

Figure 8 Convergence curves of BSAISA and BSA for the pressurevessel design problem

From Tables 12 and 13 the vast majority of algorithmscan find the best function value 0012665 for this problemwhile GA-DT and CPSO fail to find it With regard to thecomputational cost (FEs) BSAISA only requires 9440 FEswhen it reaches the global optimum which is superior toall other algorithms except MBA with 7650 FEs Howeverthe Worst and Mean and Std values of BSAISA are betterthan those of MBA Consequently for this problem it can beconcluded that BSA has the obvious superiority in terms ofFEs over all other algorithms exceptMBAMoreover BSAISAhas a stronger robustness when compared with MBA alone

Figure 9 depicts the convergence curves of BSAISA andBSA for the tension compression spring design problemwhere the value of 119865(119909lowast) on the vertical axis equals 0012665From Figure 9 it can be observed that both BSAISA and BSAfall into a local optimum in the early iterations but they areable to successfully escape from the local optimum Howeverthe convergence speed of BSAISA is obviously faster than thatof BSA


Table 12 Comparison of best solutions for the tension compression spring design problem

Method 1198831 1198832 1198833 1198921(119883) 1198922(119883) 1198923(119883) 1198924(119883) 119891(119883)GA-DT 0051989 0363965 10890522 minus13119864 minus 05 minus21119864 minus 05 minus4061338 minus0722698 0012681MDE 0051688 0356692 11290483 minus0000000 minus0000000 minus4053734 minus0727747 0012665CPSO 0051728 0357644 11244543 minus845119864 minus 04 minus126119864 minus 05 minus4051300 minus0727090 0012675HPSO 0051706 0357126 11265083 NA NA NA NA 0012665DEDS 0051689 0356718 11288965 NA NA NA NA 0012665HEAA 0051690 0356729 11288294 NA NA NA NA 0012665DELC 0051689 0356718 11288966 NA NA NA NA 0012665ABC 0051749 0358179 11203763 minus0000000 minus0000000 minus4056663 minus0726713 0012665MBA 0051656 035594 11344665 0 0 minus4052248 minus0728268 0012665SSOC 0051689 0356718 11288965 NA NA NA NA 0012665BSA-SA120576 0051989 0356727 11288425 minus770119864 minus 09 minus330119864 minus 09 minus4054 minus0728 0012665BSA 0051694 0356845 11281488 minus105119864 minus 07 minus177 minus 08 minus4054037 minus0727640 0012665BSAISA 0051687 0356669 11291824 minus638119864 minus 10 minus153119864 minus 09 minus4053689 minus0727763 0012665

Table 13 Comparison of statistical results for the tension compression spring design problem

Method Worst Mean Best Std FEsGA-DT 0012973 0012742 0012681 590119864 minus 05 80000MDE 0012674 0012666 0012665 20119864 minus 6 24000CPSO 0012924 0012730 0012675 520119864 minus 05 23000HPSO 0012719 0012707 0012665 158119864 minus 05 81000DEDS 0012738 0012669 0012665 125119864 minus 05 24000HEAA 0012665 0012665 0012665 14119864 minus 09 24000DELC 0012666 0012665 0012665 13119864 minus 07 20000PSO-DE 0012665 0012665 0012665 12119864 minus 08 24950ABC NA 0012709 0012665 0012813 30000MBA 0012900 0012713 0012665 630119864 minus 05 7650SSOC 0012868 0012765 0012665 929119864 minus 05 25000BSA-SA120576 0012666 0012665 0012665 162119864 minus 07 80000BSA 0012669 0012666 0012665 724119864 minus 07 43220BSAISA 0012668 0012666 0012665 490119864 minus 07 9440


BSABSAISA

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)

Figure 9 Convergence curves of BSAISA and BSA for the tensioncompression spring design problem

434 Welded Beam Design Problem (WBP) The weldedbeam problem is a minimum cost problem with four con-tinuous design variables and subject to two linear and fivenonlinear inequality constraints The best feasible solution isobtained by BSAISA at 119909 = (0205730 3470489 90366240205730) with the objective function value 119891(119909) = 1724852using 29000 FEs

For this problem BSAISA is compared with many well-known algorithms as follows GA-DT MDE CPSO HPSODELC POS-DE ABC MBA BSA BSA-SA120576 and SSOCThebest solutions obtained from BSAISA and other well-knownalgorithms are listed in Table 14 The comparison of theirstatistical results is presented in Table 15

From Tables 14 and 15 except that the constraint valueof PSO is not available the obtained solution sets of allalgorithms satisfy the constraints for the problem Most ofalgorithms including BSAISA BSA BSA-SA120576 MDE HPSODELC POS-DE ABC and SSOC are able to find the bestfunction value 1724852 while GA-DT and CPSO and MBAfail to find it It should be admitted that DELC is superior


Table 14 Comparison of best solutions for the welded beam design problem

Method GA-DT MDE CPSO HPSO ABC MBA BSA-SA120576 BSA BSAISA1198831(ℎ) 0205986 0205730 0202369 0205730 0205730 0205729 0205730 0205730 02057301198832(119897) 3471328 3470489 3544214 3470489 3470489 3470493 3470489 3470489 34704891198833(119905) 9020224 9036624 904821 9036624 9036624 9036626 9036624 9036624 90366241198834(119887) 0206480 0205730 0205723 0205730 0205730 0205729 0205730 0205730 02057301198921(119883) minus0074092 minus0000335 minus12839796 NA 0000000 minus0001614 minus155119864 minus 10 minus532119864 minus 07 01198922(119883) minus0266227 minus0000753 minus1247467 NA minus0000002 minus0016911 minus430119864 minus 09 minus902119864 minus 06 01198923(119883) minus495119864 minus 04 minus0000000 minus149119864 minus 03 NA 0000000 minus240119864 minus 07 minus155119864 minus 15 minus786119864 minus 12 minus555119864 minus 171198924(119883) minus3430044 minus3432984 minus3429347 NA minus3432984 minus3432982 minus34330 minus3432984 minus34329841198925(119883) minus0080986 minus0080730 minus0079381 NA minus0080730 minus0080729 minus807119864 minus 02 minus0080730 minus00807301198926(119883) minus0235514 minus0235540 minus0235536 NA minus0235540 minus0235540 minus02355 minus0235540 minus02355401198927(119883) minus58666440 minus0000882 minus11681355 NA 0000000 minus0001464 minus185119864 minus 10 minus113119864 minus 07 minus546119864 minus 12119891(119883) 1728226 1724852 1728024 1724852 1724852 1724853 1724852 1724852 1724852

Table 15 Comparison of statistical results for the welded beam design problem

Method Worst Mean Best Std FEsGA-DT 1993408 1792654 1728226 747119864 minus 02 80000MDE 1724854 1724853 1724852 NA 24000CPSO 1782143 1748831 1728024 129119864 minus 02 24000HPSO 1814295 1749040 1724852 400119864 minus 02 81000DELC 1724852 1724852 1724852 41119864 minus 13 20000PSO-DE 1724852 1724852 1724852 67119864 minus 16 66600ABC NA 1741913 1724852 31119864 minus 02 30000MBA 1724853 1724853 1724853 694119864 minus 19 47340SSOC 1799332 1746462 1724852 257119864 minus 2 25000BSA-SA120576 1724852 1724852 1724852 811119864 minus 10 80000BSA 1724854 1724852 1724852 235119864 minus 07 45480BSAISA 1724854 1724852 1724852 296119864 minus 07 29000

to all other algorithms in terms of FEs and robustness forthis problem On the other hand except for DELC MDEand SSOC with FEs of 2000 2400 and 2500 respectivelyBSAISA requires fewer FEs than the remaining algorithms(excluding algorithms that do not reach the best solution)When considering the comparison of the Std values for thisproblem MBA exhibits its powerful robustness and BSAISAperforms better than most algorithms except MBA DELCPSO-DE BSA and BSA-SA120576

It is worth mentioning that from [74] the Worst MeanBest and Std value of MDE are given as 1724854 17248531724852 and 10119864 minus 15 respectively However the corre-sponding values of DELC equal 1724852 1724852 1724852and 41119864 minus 13 respectively where its Worst and Mean valuesare smaller than those of MDE while its Std is bigger thanthose ofMDE Sowe consider that the Std ofMDE is probablyan error data for this problem and we replace it with NA inTable 15

Figure 10 depicts the convergence curves of BSAISA andBSA for the welded beam design problem where the valueof 119865(119909lowast) on the vertical axis equals 1724852 Figure 10 showsthe convergence speed of BSAISA is faster than that of BSAremarkably

435 Speed Reducer Design Problem (SRP) This speedreducer design problemhas eleven constraints and six contin-uous design variables (1199091 1199092 1199094 1199095 1199096 1199097) and one integervariable (1199093) The best solution obtained from BSAISA is119909 = (3500000 07000000 17 7300000 7715320 33502155286654) with 119891(119909) = 2994471066 using 15860 FEsThe comparison of the best solutions obtained by BSAISAand other well-known algorithms is given in Table 16 Thestatistical results of BSAISA BSA MDE DEDS HEAADELC POS-DE ABCMBA and SSOC are listed in Table 17

As shown in Tables 16 and 17 the obtained solutionsets of all algorithms satisfy the constraints for this prob-lem BSAISA BSA DEDS and DELC are able to find thebest function value 2994471066 while the others do notAmong the four algorithms DEDS DELC and BSA require30000 30000 and 25640 FEs respectively However BSAISArequires only 15860 FEs when it reaches the same bestfunction value MBA fails to find the best known functionvalue thus BSAISA is better than MBA in this problemeven though MBA has lower FEs As for the comparisonof the Std among the four algorithms that achieve the bestknown function value BSAISA is worse than the othersHowever one thing that should bementioned is that themain


Table 16 Comparison of best solutions for the speed reducer design problem

Method MDE DEDS DELC HEAA POS-DE MBA BSA BSAISA1198831 3500010 3500000 3500000 3500023 3500000 3500000 3500000 35000001198832 0700000 0700000 0700000 0700000 0700000 0700000 0700000 07000001198833 17 17 17 17000013 17000000 17000000 17 171198834 7300156 7300000 7300000 7300428 7300000 7300033 7300000 73000001198835 7800027 7715320 7715320 7715377 7800000 7715772 7715320 77153201198836 3350221 3350215 3350215 3350231 3350215 3350218 3350215 33502151198837 5286685 5286654 5286654 5286664 5286683 5286654 5286654 5286654119891(119883) 2996356689 2994471066 2994471066 2994499107 2996348167 2994482453 2994471066 2994471066

Table 17 Comparison of statistical results for the speed reducer design problem

Method Worst Mean Best Std FEsMDE 2996390137 2996367220 2996356689 82119864 minus 03 24000DEDS 2994471066 2994471066 2994471066 358119864 minus 12 30000HEAA 2994752311 2994613368 2994499107 70119864 minus 02 40000DELC 2994471066 2994471066 2994471066 19119864 minus 12 30000PSO-DE 2996348204 2996348174 2996348167 64119864 minus 06 54350ABC NA 2997058412 2997058412 0 30000MBA 2999652444 2996769019 2994482453 156 6300SSOC 2996113298 2996113298 2996113298 134119864 minus 12 25000BSA 2994471066 2994471066 2994471066 987119864 minus 11 25640BSAISA 2994471095 2994471067 2994471066 540119864 minus 06 15860


BSABSAISA

101

10minus1

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)

Figure 10 Convergence curves of BSAISA and BSA for the weldedbeam design problem

purpose of the experiment is to compare the convergencespeed between BSAISA and other algorithms From thispoint of view it can be concluded that BSAISA has a betterperformance than other algorithms in terms of convergencespeed

Figure 11 depicts the convergence curves of BSAISA andBSA for the speed reducer design problem where the value of


BSABSAISA

104

101

10minus2

10minus5

10minus8

F(x

)minusF(x

lowast)(ＦＩＡ)

Figure 11 Convergence curves of BSAISA and BSA for the speedreducer design problem

119865(119909lowast) on the vertical axis equals 2994471066 Figure 11 showsthat the convergence speed of BSAISA is faster than that ofBSA

44 Comparisons Using Sign Test Sign Test [80] is one of themost popular statistical methods used to determine whethertwo algorithms are significantly different Recently Miao et


Table 18 Comparisons between BSAISA and other algorithms in Sign Tests

BSAISA-methods + asymp minus Total 119901 valueSR 11 1 1 13 0006FSA 12 0 1 13 0003CDE 8 0 5 13 0581AMA 13 0 0 13 0000MABC 10 2 1 13 0012RPGA 11 0 0 11 0001BSA-SA120576 8 4 1 13 0039Note The columns of ldquo+rdquo ldquoasymprdquo and ldquominusrdquo indicate the number of functions where BSAISA performs significantly better than almost the same as or significantlyworse than the compared algorithm respectively ldquo119901 valuerdquo denotes the probability value supporting the null hypothesis

al [81] utilized Sign Test method to analyze the performancesbetween their proposed modified algorithm and the originalone In this paper the two-tailed Sign Test with a significancelevel 005 is adopted to test the significant differences betweenthe results obtained by different algorithms and the testresults are given inTable 18The values of Best andFEs are twomost important criterions for the evaluations of algorithms inour paper they therefore should be chosen as the objectivesof the Sign Test The signs ldquo+rdquo ldquoasymprdquo and ldquominusrdquo representrespectively the fact that our BSAISA performs significantlybetter than almost the same as or significantly worse thanthe algorithm it is compared to The null hypothesis herein isthat the performances between BSAISA and one of the othersare not significantly differential

As shown in Table 18 the 119901 values of supporting the nullhypothesis of Sign Test for six pairs of algorithms (BSAISA-SR BSAISA-FSA BSAISA-AMA BSAISA-MABC BSAISA-RPGA and BSAISA-BSA-SA120576) are 0006 0003 0000 00120001 and 0039 respectively and thereby we can rejectthe null hypothesis This illustrates that the optimizationperformance of the proposed BSAISA is significantly betterthan those of the six algorithms The 119901 value of BSAISA-CDE is equal to 0581 which shows that we cannot reject thenull hypothesis However according to the related sign values(ldquo+rdquo ldquoasymprdquo and ldquominusrdquo) from Table 18 BSAISA is slightly worsethan CDE on 5 problems but wins on another 8 problemswhich illustrates that the proposed BSAISA has a relativelyexcellent competitiveness comparedwith theCDEGenerallythe statistical 119901 values and sign values validate that BSAISAhas the superiority compared to the other well-known algo-rithms on the constrained optimization problems

On the one hand all experimental results suggest thatthe proposed method improves the convergence speed ofBSA On the other hand the overall comparative results ofBSAISA and other well-known algorithms demonstrate thatBSAISA ismore effective and competitive for constrained andengineering optimization problems in terms of convergencespeed

5 Conclusions and Future Work

In this paper we proposed a modified version of BSAinspired by the Metropolis criterion in SA (BSAISA) TheMetropolis criterion may probabilistically accept a higher

energy state and the acceptance probability can decrease asthe temperature decreases which motivated us to redesignthe amplitude control factor 119865 so it can adaptively decreaseas the number of iterations increases The design principleand numerical analysis of the redesigned 119865 indicate thatthe change in 119865 could accelerate the convergence speed ofthe algorithm by improving the local exploitation capabilityFurthermore the redesigned 119865 does not introduce extraparameters We successfully implemented BSAISA to solvesome constrained optimization and engineering design prob-lems The experimental results demonstrated that BSAISAhas a faster convergence speed than BSA and it can effi-ciently balance the capacity for global exploration and localexploitation The comparisons of the results obtained byBSAISA and other well-known algorithms demonstrated thatBSAISA ismore effective and competitive for constrained andengineering optimization problems in terms of convergencespeed

This paper suggests that the proposed BSAISA has asuperiority in terms of convergence speed or computationalcost The downside of the proposed algorithm is of coursethat its robustness does not show enough superiority Soour future work is to further research into the robustness ofBSAISA on the basis of current research Niche techniqueis able to effectively maintain population diversity of evolu-tionary algorithms [82 83] How to combine BSAISA withniche technology to improve robustness of the algorithmmaydeserve to be studied in the future

Appendix

A Constrained Benchmark Problems

A1 Constrained Problem 01

min 119891 (119909) = 5 4sum119894=1

119909119894 minus 5 4sum119894=1

1199092119894 minus 13sum119894=5

119909119894subject to 1198921 (119909) = 21199091 + 21199092 + 11990910 + 11990911 minus 10

le 01198922 (119909) = 21199091 + 21199093 + 11990910 + 11990912 minus 10le 0


1198923 (119909) = 21199092 + 21199093 + 11990911 + 11990912 minus 10le 01198924 (119909) = minus81199091 + 11990910 le 01198925 (119909) = minus81199092 + 11990911 le 01198926 (119909) = minus81199093 + 11990913 le 01198927 (119909) = minus21199094 minus 1199095 + 11990910 le 01198928 (119909) = minus21199096 minus 1199097 + 11990911 le 01198929 (119909) = minus21199098 minus 1199099 + 11990912 le 00 le 119909119894 le 1 (119894 = 1 9)0 le 119909119894 le 100 (119894 = 10 11 12)0 le 11990913 le 1

(A1)


max 119891 (119909)

=1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816sum119899119894=1 cos4 (119909119894) minus 2prod119899119894=1cos2 (119909119894)

radicsum119899119894=1 11989411990921198941003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

subject to 1198921 (119909) = 075 minus 119899prod119894=1

119909119894 le 0

1198922 (119909) = 119899sum119894=1

119909119894 minus 75119899 le 0119899 = 20 0 le 119909119894 le 10 119894 = 1 119899

(A2)


max 119891 (119909) = (radic119899)119899 sdot 119899prod119894=1

119909119894subject to ℎ (119909) = 119899sum

119894=1

1199092119894 = 0119899 = 10 0 le 119909119894 le 1 119894 = 1 119899

(A3)


min 119891 (119909)= 5357854711990923 + 0835689111990911199095+ 372932391199091 minus 40792141

subject to 1198921 (119909)= 85334407 + 0005685811990921199095+ 0000626211990911199094 minus 0002205311990931199095minus 92 le 0

1198922 (119909)= minus85334407 minus 0005685811990921199095minus 0000626211990911199094 + 0002205311990931199095

le 01198923 (119909)= 8051249 + 0007131711990921199095+ 0002995511990911199092 + 0002181311990923minus 110 le 0

1198924 (119909)= minus8051249 minus 0007131711990921199095minus 0002995511990911199092 minus 0002181311990923+ 90 le 0

1198925 (119909)= 9300961 + 0004702611990931199095+ 0001254711990911199093 + 0001908511990931199094minus 25 le 0


78 le 1199091 le 10233 le 1199092 le 4527 le 119909119894 le 45 119894 = 3 4 5

(A4)


min 119891 (119909)= 31199091 + 000000111990931 + 21199092+ (00000023 ) 11990932


subject to 1198921 (119909) = minus1199094 + 1199093 minus 055 le 01198922 (119909) = minus1199093 + 1199094 minus 055 le 0ℎ3 (119909)= 1000 sin (minus1199093 minus 025)+ 1000 sin (minus1199094 minus 025) + 8948minus 1199091 = 0

ℎ4 (119909)= 1000 sin (1199093 minus 025)+ 1000 sin (1199093 minus 1199094 minus 025) + 8948minus 1199092 = 0

ℎ5 (119909)= 1000 sin (1199094 minus 025)+ 1000 sin (1199094 minus 1199093 minus 025)+ 12948 = 0

0 le 1199091 1199092 le 1200minus 055 le 1199093 1199094 le 055

(A5)


min 119891 (119909) = (1199091 minus 10)3 + (1199092 minus 20)3subject to 1198921 (119909) = minus (1199091 minus 5)2 minus (1199092 minus 5)2 + 100

le 01198922 (119909) = (1199091 minus 6)2 + (1199092 minus 5)2 minus 8281le 013 le 1199091 le 1000 le 1199092 le 100

(A6)


min 119891 (119909)= 11990921 + 11990922 + 11990911199092 minus 141199091 minus 161199092+ (1199093 minus 10)2 + 4 (1199094 minus 5)2+ (1199095 minus 3)2 + 2 (1199096 minus 1)2 + 511990927+ 7 (1199098 minus 11)2 + 2 (1199099 minus 10)2+ (11990910 minus 7)2 + 45

subject to 1198921 (119909)= minus105 + 41199091 + 51199092 minus 31199097 + 91199098 le 01198922 (119909) = 101199091 minus 81199092 minus 171199097 + 21199098 le 01198923 (119909) = minus81199091 + 21199092 + 51199099 minus 211990910 minus 12le 01198924 (119909)= 3 (1199091 minus 2)2 + 4 (1199092 minus 3)2 + 211990923minus 71199094 minus 120 le 0

1198925 (119909)= 511990921 + 81199092 + (1199093 minus 6)2 minus 21199094 minus 40le 01198926 (119909)= 11990921 + 2 (1199092 minus 2)2 minus 211990911199092 + 141199095minus 61199096 le 0

1198927 (119909)= 05 (1199091 minus 8)2 + 2 (1199092 minus 4)2 + 311990925minus 1199096 minus 30 le 0

1198928 (119909)= minus31199091 + 61199092 + 12 (1199099 minus 8)2 minus 711990910le 0minus 10 le 119909119894 le 10 119894 = 1 10

(A7)


min 119891 (119909) = minus sin3 (21205871199091) sin (21205871199092)11990931 (1199091 + 1199092)subject to 1198921 (119909) = 11990921 minus 1199092 + 1 le 0

1198922 (119909) = 1 minus 1199091 + (1199092 minus 4)2 le 00 le 119909119894 le 10 119894 = 1 2

(A8)


min 119891 (119909)= (1199091 minus 10)2 + 5 (1199092 minus 12)2 + 11990943+ 3 (1199094 minus 11)2 + 1011990965 + 711990926 + 11990947minus 411990961199097 minus 101199096 minus 81199097


subject to 1198921 (119909)= 211990921 + 311990942 + 1199093 + 411990924 + 51199095 minus 127le 01198922 (119909)= 71199091 + 31199092 + 1011990923 + 1199094 minus 1199095 minus 282le 01198923 (119909) = 231199091 + 11990922 + 611990926 minus 81199097 minus 196le 01198924 (119909)= 411990921 + 11990922 minus 311990911199092 + 211990923 + 51199096minus 111199097 le 0

minus 10 le 119909119894 le 10 119894 = 1 2 3 4 5 6 7(A9)


min 119891 (119909) = 1199091 + 1199092 + 1199093subject to 1198921 (119909) = minus1 + 00025 (1199094 + 1199096) le 0

1198922 (119909) = minus1 + 00025 (1199095 + 1199097 minus 1199094)le 01198923 (119909) = minus1 + 001 (1199098 minus 1199095) le 01198924 (119909)= minus11990911199096 + 833332521199094 + 1001199091minus 83333333 le 0

1198925 (119909)= minus11990921199097 + 12501199095 + 11990921199094 minus 12501199094le 01198926 (119909)= minus11990931199098 + 1250000 + 11990931199095 minus 25001199095le 0100 le 1199091 le 100001000 le 119909119894 le 10000 119894 = 2 310 le 119909119894 le 1000 119894 = 4 5 6 7 8

(A10)


min 119891 (119909) = 11990921 + (1199092 minus 1)2subject to ℎ (119909) = 1199092 minus 11990921 = 0

minus 1 le 119909119894 le 1 119894 = 1 2(A11)


max 119891 (119909)= (100 minus (1199091 minus 5)

2 minus (1199092 minus 5)2 minus (1199093 minus 5)2)2100subject to 119892 (119909)

= (1199091 minus 119901)2 + (1199092 minus 119902)2 + (1199093 minus 119903)2 minus 00625le 0119901 119902 119903 = 1 90 le 119909119894 le 10 119894 = 1 2 3

(A12)


min 119891 (119909) = 11989011990911199092119909311990941199095subject to ℎ1 (119909) = 11990921 + 11990922 + 11990923 + 11990924 + 11990925 minus 10

= 0ℎ2 (119909) = 11990921199093 minus 511990941199095 = 0ℎ3 (119909) = 11990931 + 11990932 + 1 = 0minus 23 le 119909119894 le 23 119894 = 1 2minus 32 le 119909119894 le 32 119894 = 3 4 5

(A13)

B Engineering Design Problems

B1 Three-Bar Truss Design Problem

min 119891 (119909) = (2radic21199091 + 1199092) times 119897subject to 1198921 (119909) = radic21199091 + 1199092radic211990921 + 211990911199092 P minus 120590 le 0

1198922 (119909) = 1199092radic211990921 + 211990911199092 P minus 120590 le 01198923 (119909) = 1radic21199092 + 1199091 P minus 120590 le 00 le 119909119894 le 1 119894 = 1 2119897 = 100 cmP = 2 kNcm2120590 = 2 kNcm2

(B1)


B2 Pressure Vessel Design Problem

min 119891 (119909)= 06224119909111990931199094 + 17781119909211990923+ 31661119909211199094 + 1984119909211199093

subject to 1198921 (119909) = minus1199091 + 001931199093 le 01198922 (119909) = minus1199092 + 0009541199093 le 01198923 (119909)= minus120587119909231199094 minus (43)12058711990933 + 1296000 le 01198924 (119909) = 1199094 minus 240 le 00 le 119909119894 le 100 119894 = 1 210 le 119909119894 le 200 119894 = 3 4

(B2)

B3 TensionCompression Spring Design Problem

min 119891 (119909) = (1199093 + 2) 119909211990921subject to 1198921 (119909) = minus119909321199093(7178511990941) + 1 le 0

1198922 (119909)= (411990922 minus 11990911199092)(12566 (119909211990931 minus 11990941)) +

1(510811990921)minus 1 le 0

1198923 (119909) = minus140451199091(119909221199093) + 1 le 01198924 (119909) = (1199091 + 1199092)15 minus 1 le 0005 le 1199091 le 200025 le 1199092 le 130200 le 1199093 le 1500

(B3)

B4 Welded Beam Design Problem

min 119891 (119909)= 110471119909211199092+ 00481111990931199094 (14 + 1199092)

subject to 1198921 (119909) = 120591 (119909) minus 120591max le 01198922 (119909) = 120590 (119909) minus 120590max le 0

1198923 (119909) = 1199091 minus 1199094 le 01198924 (119909)= 01047111990921 + 00481111990931199094 (14 + 1199092)minus 5 le 0

1198925 (119909) = 0125 minus 1199091 le 01198926 (119909) = 120575 (119909) minus 120575max le 01198927 (119909) = 119875 minus 119875119888 (119909) le 001 le 119909119894 le 2 119894 = 1 401 le 119909119894 le 10 119894 = 2 3

(B4)

where

120591 (119909) = radic(1205911015840)2 + 2120591101584012059110158401015840 11990922119877 + (12059110158401015840)21205911015840 = 119875radic211990911199092 120591

10158401015840 = 119872119877119869119872 = 119875(119871 + 11990922 )

119877 = radic119909224 + (1199091 + 11990932 )2

119869 = 2radic211990911199092 [1199092212 + (1199091 + 11990932 )2]120590 (119909) = 6119875119871119909411990923

120575 (119909) = 41198751198713119864119909331199094

119875119888 (119909) = 4013119864radic(119909231199096436)1198712 times (1 minus 11990932119871radic 1198644119866)

119875 = 6000 lb119871 = 14 in119864 = 30 times 106 psi119866 = 12 times 106 psi

120591max = 13600 psi120590max = 30000 psi120575max = 025 in

(B5)


B5 Speed Reducer Design Problem

min 119891 (119909)= 07854119909111990922 (3333311990923 + 1493341199093 minus 430934)minus 15081199091 (11990926 + 11990927) + 74777 (11990936 + 11990937)+ 07854 (119909411990926 + 119909511990927)

subject to 1198921 (119909) = 271199091119909221199093 minus 1 le 01198922 (119909) = 397511990911199092211990923 minus 1 le 0

1198923 (119909) = 193119909341199092119909461199093 minus 1 le 0

1198924 (119909) = 193119909351199092119909471199093 minus 1 le 0

1198925 (119909) = [(7451199094 (11990921199093))2 + 169 times 106]12

11011990936 minus 1le 0

1198926 (119909) = [(7451199095 (11990921199093))2 + 1575 times 106]128511990937 minus 1

le 01198927 (119909) = 1199092119909340 minus 1 le 01198928 (119909) = 511990921199091 minus 1 le 01198929 (119909) = 1199091121199092 minus 1 le 011989210 (119909) = 151199096 + 191199094 minus 1 le 011989211 (119909) = 111199097 + 191199095 minus 1 le 0

(B6)

where

26 le 1199091 le 3607 le 1199092 le 0817 le 1199093 le 2873 le 1199094 1199095 le 8329 le 1199096 le 3950 le 1199097 le 55

(B7)

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported in part by the National Natural Sci-ence Foundation of China (no 61663009) and the State KeyLaboratory of Silicate Materials for Architectures (WuhanUniversity of Technology SYSJJ2018-21)

References

[1] P Posik W Huyer and L Pal ldquoA comparison of global searchalgorithms for continuous black box optimizationrdquo Evolution-ary Computation vol 20 no 4 pp 509ndash541 2012

[2] A P Piotrowski M J Napiorkowski J J Napiorkowski andP M Rowinski ldquoSwarm Intelligence and Evolutionary Algo-rithms Performance versus speedrdquo Information Sciences vol384 pp 34ndash85 2017

[3] S Das andA Konar ldquoA swarm intelligence approach to the syn-thesis of two-dimensional IIR filtersrdquo Engineering Applicationsof Artificial Intelligence vol 20 no 8 pp 1086ndash1096 2007

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

[5] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[6] X S Yang and S Deb ldquoCuckoo search via LΘvy flightsrdquo inProceedings of the In World Congress on Nature BiologicallyInspired Computing pp 210ndash214 NaBIC 2009

[7] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep Erciyes University EngineeringFaculty Computer Engineering Department 2005

[8] J Q Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009

[9] S M Elsayed R A Sarker andD L Essam ldquoAdaptive Configu-ration of evolutionary algorithms for constrained optimizationrdquoApplied Mathematics and Computation vol 222 pp 680ndash7112013

[10] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence University ofMichigan Press Oxford UK1975

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

[12] 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 479ndash487 2016

[13] Q Su and Z Hu ldquoColor image quantization algorithm based onself-adaptive differential Evolutionrdquo Computational Intelligenceand Neuroscience vol 2013 Article ID 231916 8 pages 2013

[14] Z Hu Q Su and X Xia ldquoMultiobjective image color quan-tization algorithm based on self-adaptive hybrid differentialevolutionrdquo Computational Intelligence and Neuroscience vol2016 Article ID 2450431 12 pages 2016

[15] C Igel N Hansen and S Roth ldquoCovariance matrix adaptationfor multi-objective optimizationrdquo Evolutionary Computationvol 15 no 1 pp 1ndash28 2007


[16] P Civicioglu ldquoBacktracking search optimization algorithm fornumerical optimization problemsrdquo Applied Mathematics andComputation vol 219 no 15 pp 8121ndash8144 2013

[17] A El-Fergany ldquoOptimal allocation of multi-type distributedgenerators using backtracking search optimization algorithmrdquoInternational Journal of Electrical Power amp Energy Systems vol64 pp 1197ndash1205 2015

[18] M Modiri-Delshad and N A Rahim ldquoMulti-objective back-tracking search algorithm for economic emission dispatchproblemrdquo Applied Soft Computing vol 40 pp 479ndash494 2016

[19] S D Madasu M L S S Kumar and A K Singh ldquoComparableinvestigation of backtracking search algorithm in automaticgeneration control for two area reheat interconnected thermalpower systemrdquo Applied Soft Computing vol 55 pp 197ndash2102017

[20] J A Ali M A Hannan A Mohamed and M G M Abdol-rasol ldquoFuzzy logic speed controller optimization approach forinduction motor drive using backtracking search algorithmrdquoMeasurement vol 78 pp 49ndash62 2016

[21] M A Hannan J A Ali A Mohamed and M N Uddin ldquoARandom Forest Regression Based Space Vector PWM InverterController for the Induction Motor Driverdquo IEEE Transactionson Industrial Electronics vol 64 no 4 pp 2689ndash2699 2017

[22] K Guney A Durmus and S Basbug ldquoBacktracking searchoptimization algorithm for synthesis of concentric circularantenna arraysrdquo International Journal of Antennas and Propa-gation vol 2014 Article ID 250841 11 pages 2014

[23] R Muralidharan V Athinarayanan G K Mahanti and AMahanti ldquoQPSO versus BSA for failure correction of lineararray of mutually coupled parallel dipole antennas with fixedside lobe level and VSWRrdquo Advances in Electrical Engineeringvol 2014 Article ID 858290 7 pages 2014

[24] M Eskandari and O Toygar ldquoSelection of optimized featuresand weights on face-iris fusion using distance imagesrdquo Com-puter Vision and Image Understanding vol 137 article no 2225pp 63ndash75 2015

[25] U H Atasevar P Civicioglu E Besdok and C Ozkan ldquoAnew unsupervised change detection approach based on DWTimage fusion and backtracking search optimization algorithmfor optical remote sensing datardquo in Proceedings of the ISPRSTechnical Commission VII Mid-Term Symposium 2014 pp 15ndash18 October 2014

[26] S K Agarwal S Shah and R Kumar ldquoClassification of mentaltasks from EEG data using backtracking search optimizationbased neural classifierrdquo Neurocomputing vol 166 pp 397ndash4032015

[27] L Zhang and D Zhang ldquoEvolutionary cost-sensitive extremelearning machinerdquo IEEE Transactions on Neural Networks andLearning Systems vol 28 no 12 pp 3045ndash3060 2016

[28] F Zou D Chen S Li R Lu andM Lin ldquoCommunity detectionin complex networks Multi-objective discrete backtrackingsearch optimization algorithm with decompositionrdquo AppliedSoft Computing vol 53 pp 285ndash295 2017

[29] C Zhang J Zhou C Li W Fu and T Peng ldquoA compoundstructure of ELM based on feature selection and parameteroptimization using hybrid backtracking search algorithm forwind speed forecastingrdquo Energy Conversion and Managementvol 143 pp 360ndash376 2017

[30] C Lu L Gao X Li and P Chen ldquoEnergy-efficient multi-passturning operation using multi-objective backtracking searchalgorithmrdquo Journal of Cleaner Production vol 137 pp 1516ndash15312016

[31] M Akhtar M A Hannan R A Begum H Basri and EScavino ldquoBacktracking search algorithm in CVRP models forefficient solid waste collection and route optimizationrdquo WasteManagement vol 61 pp 117ndash128 2017

[32] M S Ahmed AMohamed T Khatib H Shareef R Z Homodand J A Ali ldquoReal time optimal schedule controller for homeenergy management system using new binary backtrackingsearch algorithmrdquo Energy and Buildings vol 138 pp 215ndash2272017

[33] S O Kolawole and H Duan ldquoBacktracking search algorithmfor non-aligned thrust optimization for satellite formationrdquo inProceedings of the 11th IEEE International Conference on Controland Automation (IEEE ICCA rsquo14) pp 738ndash743 June 2014

[34] Q Lin L Gao X Li and C Zhang ldquoA hybrid backtrackingsearch algorithm for permutation flow-shop scheduling prob-lemrdquo Computers amp Industrial Engineering vol 85 pp 437ndash4462015

[35] J Lin ldquoOppositional backtracking search optimization algo-rithm for parameter identification of hyperchaotic systemsrdquoNonlinear Dynamics vol 80 no 1-2 pp 209ndash219 2015

[36] Q L Xu N Guo and L Xu ldquoOpposition-based backtrackingsearch algorithm for numerical optimization problemsrdquo inProceedings of the In International Conference on IntelligentScience and Big Data Engineering pp 223ndash234 2015

[37] X Yuan B Ji Y Yuan R M Ikram X Zhang and Y HuangldquoAn efficient chaos embedded hybrid approach for hydro-thermal unit commitment problemrdquo Energy Conversion andManagement vol 91 pp 225ndash237 2015

[38] X Yuan XWu H Tian Y Yuan and R M Adnan ldquoParameteridentification of nonlinear muskingum model with backtrack-ing search algorithmrdquoWater ResourcesManagement vol 30 no8 pp 2767ndash2783 2016

[39] S Vitayasak and P Pongcharoen ldquoBacktracking search algo-rithm for designing a robustmachine layoutrdquoWITTransactionson Engineering Sciences vol 95 pp 411ndash420 2014

[40] S Vitayasak P Pongcharoen and C Hicks ldquoA tool for solvingstochastic dynamic facility layout problems with stochasticdemand using either a Genetic Algorithm or modified Back-tracking Search Algorithmrdquo International Journal of ProductionEconomics vol 190 pp 146ndash157 2017

[41] M Li H Zhao and X Weng ldquoBacktracking search optimiza-tion algorithmwith comprehensive learning strategyrdquo Journal ofSystems Engineering and Electronics vol 37 no 4 pp 958ndash9632015 (Chinese)

[42] W Zhao L Wang Y Yin B Wang and Y Wei ldquoAn improvedbacktracking search algorithm for constrained optimizationproblemsrdquo in Proceedings of the International Conference onKnowledge Science Engineering and Management pp 222ndash233Springer International Publishing 2014

[43] L Wang Y Zhong Y Yin W Zhao B Wang and Y XuldquoA hybrid backtracking search optimization algorithm withdifferential evolutionrdquo Mathematical Problems in Engineeringvol 2015 Article ID 769245 p 16 2015

[44] S Das D Mandal R Kar and S P Ghoshal ldquoInterference sup-pression of linear antenna arrays with combined BacktrackingSearch Algorithm and Differential Evolutionrdquo in Proceedings ofthe 3rd International Conference on Communication and SignalProcessing (ICCSP rsquo14) pp 162ndash166 April 2014

[45] S Das D Mandal R Kar and S P Ghoshal ldquoA new hybridizedbackscattering search optimization algorithm with differentialevolution for sidelobe suppression of uniformly excited con-centric circular antenna arraysrdquo International Journal of RF and


Microwave Computer-Aided Engineering vol 25 no 3 pp 262ndash268 2015

[46] S Mallick R Kar D Mandal and S P Ghoshal ldquoCMOSanalogue amplifier circuits optimisation using hybrid back-tracking search algorithm with differential evolutionrdquo Journalof Experimental and Theoretical Artificial Intelligence pp 1ndash312015

[47] D Chen F Zou R Lu and P Wang ldquoLearning backtrackingsearch optimisation algorithm and its applicationrdquo InformationSciences vol 376 pp 71ndash94 2017

[48] A F Ali ldquoA memetic backtracking search optimization algo-rithm for economic dispatch problemrdquo Egyptian ComputerScience Journal vol 39 no 2 pp 56ndash71 2015

[49] Y Wu Q Tang L Zhang and X He ldquoSolving stochastic two-sided assembly line balancing problem via hybrid backtrackingsearch optimization algorithmrdquo Journal of Wuhan University ofScience and Technology (Natural Science Edition) vol 39 no 2pp 121ndash127 2016 (Chinese)

[50] Z Su H Wang and P Yao ldquoA hybrid backtracking searchoptimization algorithm for nonlinear optimal control problemswith complex dynamic constraintsrdquo Neurocomputing vol 186pp 182ndash194 2016

[51] S Wang X Da M Li and T Han ldquoAdaptive backtrack-ing search optimization algorithm with pattern search fornumerical optimizationrdquo Journal of Systems Engineering andElectronics vol 27 no 2 Article ID 7514428 pp 395ndash406 2016

[52] H Duan and Q Luo ldquoAdaptive backtracking search algorithmfor induction magnetometer optimizationrdquo IEEE Transactionson Magnetics vol 50 no 12 pp 1ndash6 2014

[53] X J Wang S Y Liu and W K Tian ldquoImproved backtrackingsearch optimization algorithm with new effective mutationscale factor and greedy crossover strategyrdquo Journal of ComputerApplications vol 34 no 9 pp 2543ndash2546 2014 (Chinese)

[54] W K Tian S Y Liu and X J Wang ldquoStudy and improvementof backtracking search optimization algorithm based on differ-ential evolutionrdquoApplication Research of Computers vol 32 no6 pp 1653ndash1662 2015

[55] A Askarzadeh and L dos Santos Coelho ldquoA backtrackingsearch algorithm combined with Burgerrsquos chaotic map forparameter estimation of PEMFC electrochemical modelrdquo Inter-national Journal of Hydrogen Energy vol 39 pp 11165ndash111742014

[56] X Chen S Y Liu and YWang ldquoEmergency resources schedul-ing based on improved backtracking search optimization algo-rithmrdquo Computer Applications and Software vol 32 no 12 pp235ndash238 2015 (Chinese)

[57] S Nama A K Saha and S Ghosh ldquoImproved backtrackingsearch algorithm for pseudo dynamic active earth pressure onretaining wall supporting c-Φ backfillrdquoApplied Soft Computingvol 52 pp 885ndash897 2017

[58] G Karafotias M Hoogendoorn and A E Eiben ldquoParameterControl in Evolutionary Algorithms Trends and ChallengesrdquoIEEE Transactions on Evolutionary Computation vol 19 no 2pp 167ndash187 2015

[59] N Metropolis A W Rosenbluth M N Rosenbluth A HTeller and E Teller ldquoEquation of state calculations by fastcomputing machinesrdquo The Journal of Chemical Physics vol 21no 6 pp 1087ndash1092 1953

[60] S Kirkpatrick C D Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983

[61] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009

[62] C Zhang Q Lin L Gao and X Li ldquoBacktracking Search Algo-rithm with three constraint handling methods for constrainedoptimization problemsrdquo Expert Systems with Applications vol42 no 21 pp 7831ndash7845 2015

[63] T P Runarsson and X Yao ldquoStochastic ranking for constrainedevolutionary optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 4 no 3 pp 284ndash294 2000

[64] A S B Ullah R Sarker D Cornforth and C Lokan ldquoAMA Anew approach for solving constrained real-valued optimizationproblemsrdquo Soft Computing vol 13 no 8-9 pp 741ndash762 2009

[65] A RHedar andM Fukushima ldquoDerivative-free filter simulatedannealing method for constrained continuous global optimiza-tionrdquo Journal of Global Optimization vol 35 no 4 pp 521ndash5492006

[66] R L Becerra and C A Coello ldquoCultured differential evolu-tion for constrained optimizationrdquo Computer Methods AppliedMechanics and Engineering vol 195 no 33ndash36 pp 4303ndash43222006

[67] D Karaboga and B Akay ldquoA modified Artificial Bee Colony(ABC) algorithm for constrained optimization problemsrdquoApplied Soft Computing vol 11 no 3 pp 3021ndash3031 2011

[68] C-H Lin ldquoA rough penalty genetic algorithm for constrainedoptimizationrdquo Information Sciences vol 241 pp 119ndash137 2013

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

[70] Y Wang Z X Cai Y R Zhou and Z Fan ldquoConstrainedoptimization based on hybrid evolutionary algorithm andadaptive constraint-handling techniquerdquo Structural and Multi-disciplinary Optimization vol 37 no 4 pp 395ndash413 2009

[71] H Liu Z Cai and Y Wang ldquoHybridizing particle swarm opti-mization with differential evolution for constrained numericaland engineering optimizationrdquo Applied Soft Computing vol 10no 2 pp 629ndash640 2010

[72] L Wang and L-P Li ldquoAn effective differential evolution withlevel comparison for constrained engineering designrdquo Struc-tural andMultidisciplinary Optimization vol 41 no 6 pp 947ndash963 2010

[73] C A C Coello and E M Montes ldquoConstraint-handlingin genetic algorithms through the use of dominance-basedtournament selectionrdquo Advanced Engineering Informatics vol16 no 3 pp 193ndash203 2002

[74] E Mezura-Montes C A C Coello and J Velarsquozquez-ReyesldquoIncreasing successful offspring and diversity in differentialevolution for engineering designrdquo in Proceedings of the SeventhInternational Conference on Adaptive Computing in Design andManufacture (ACDM rsquo06) pp 131ndash139 2006

[75] Q He and L Wang ldquoAn effective co-evolutionary particleswarm optimization for constrained engineering design prob-lemsrdquo Engineering Applications of Artificial Intelligence vol 20no 1 pp 89ndash99 2007

[76] Q He and LWang ldquoA hybrid particle swarm optimization witha feasibility-based rule for constrained optimizationrdquo AppliedMathematics and Computation vol 186 no 2 pp 1407ndash14222007

[77] B Akay and D Karaboga ldquoArtificial bee colony algorithmfor large-scale problems and engineering design optimizationrdquoJournal of IntelligentManufacturing vol 23 no 4 pp 1001ndash10142012


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

[79] E Cuevas and M Cienfuegos ldquoA new algorithm inspired inthe behavior of the social-spider for constrained optimizationrdquoExpert Systems with Applications vol 41 no 2 pp 412ndash4252014

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

[81] Y Miao Q Su Z Hu and X Xia ldquoModified differentialevolution algorithm with onlooker bee operator for mixeddiscrete-continuous optimizationrdquo SpringerPlus vol 5 no 1article no 1914 2016

[82] E L Yu and P N Suganthan ldquoEnsemble of niching algorithmsrdquoInformation Sciences vol 180 no 15 pp 2815ndash2833 2010

[83] M Li D Lin and J Kou ldquoA hybrid niching PSO enhanced withrecombination-replacement crowding strategy for multimodalfunction optimizationrdquo Applied Soft Computing vol 12 no 3pp 975ndash987 2012

Computer Games Technology

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main difference between BSA and other similar algorithms isthat BSA possesses a memory for storing a population froma randomly chosen previous generation which is used togenerate the search-direction matrix for the next iterationIn addition BSA has a simple structure which makes itefficient fast and capable of solving multimodal problemsBSA has only one control parameter called the mix-ratewhich significantly reduces the sensitivity of the initial valuesto the algorithmrsquos parameters Due to these characteristicsin less than 4 years BSA has been employed successfullyto solve various engineering optimization problems such aspower systems [17ndash19] induction motor [20 21] antennaarrays [22 23] digital image processing [24 25] artificialneural networks [26ndash29] and energy and environmentalmanagement [30ndash32]

However BSA has a weak local exploitation capacity andits convergence speed is relatively slow Thus many studieshave attempted to improve the performance of BSA andsome modifications of BSA have been proposed to overcomethe deficiencies From the perspective of modified objectthe modifications of BSA can be divided into the followingfour categories It is noted that we consider classifying thepublication into the major modification category if it hasmore than one modification

(i) Modifications of the initial populations [33ndash38](ii) Modifications of the reproduction operators includ-

ing the mutation and crossover operators [39ndash47](iii) Modifications of the selection operators including

the local exploitation strategy [48ndash51](iv) Modifications of the control factor and parameter

[52ndash57]

The research on controlling parameters of EAs is oneof the most promising areas of research in evolutionarycomputation even a little modification of parameters inan algorithm can make a considerable difference [58] Inthe basic BSA the value of amplitude control factor (119865) isthe product of three and the standard normal distributionrandom number (ie 119865 = 3 sdot randn) which is often too largeor too small according to its formulationThismay give BSA apowerful global exploration capability at the early iterationshowever it also affects the later exploitation capability ofBSA Based on these considerations we focus mainly on theinfluence of the amplitude control factor (119865) on the BSAthat is the fourth of the categories defined above Duanand Luo [52] redesigned an adaptive 119865 based on the fitnessstatistics of population at each iteration Wang et al [53] andTian et al [54] proposed an adaptive 119865 based on Maxwell-Boltzmann distribution Askarzadeh and dos Santos Coelho[55] proposed an adaptive 119865 based on Burgerrsquos chaotic mapChen et al [56] redesigned an adaptive 119865 by introducingtwo extra parameters Nama et al [57] proposed a new 119865to adaptively change in the range of (045 199) and newmix-rate to randomly change in the range of (0 1) Thesemodifications of 119865 have achieved good effects in the BSA

Different from the modifications of 119865 in BSA describedabove a modified version of BSA (BSAISA) inspired bysimulated annealing (SA) is proposed in this paper In the

BSAISA 119865 based on iterations is redesigned by learning acharacteristic where SA can probabilistically accept a higherenergy state and the acceptance probability decreases withthe decrease in temperatureThe redesigned 119865 can adaptivelydecrease as the number of iterations increases without intro-ducing extra parameters This adaptive variation tendencyprovides an efficient tradeoff between early exploration andlater exploitation capability We verified the effectiveness andcompetitiveness of BSAISA in simulation experiments usingthirteen constrained benchmarks and five engineering designproblems in terms of convergence speed

The remainder of this paper is organized as followsSection 2 introduces the basic BSA As the main contributionof this paper a detailed explanation of BSAISA is presentedin Section 3 In Section 4 we present two sets of simulationexperiments in which we implemented BSAISA and BSA tosolve thirteen constrained optimization and five engineer-ing design problems The results are compared with thoseobtained by other well-known algorithms in terms of thesolution quality and function evaluations Finally we give ourconcluding remarks in Section 5

2 Backtracking Search OptimizationAlgorithm (BSA)

BSA is a population-based iterative EA BSA generates trialpopulations to take control of the amplitude of the search-direction matrix which provides a strong global explorationcapability BSA equiprobably uses two random crossoverstrategies to exchange the corresponding elements of individ-uals in populations and trial populations during the processof crossover Moreover BSA has two selection processesOne is used to select population from the current andhistorical populations the other is used to select the optimalpopulation In general BSA canbe divided into five processesinitialization selection Imutation crossover and selection II[16]

21 Initialization BSA generates initial population 119875 andinitial old population oldP using

119875119894119895 sim 119880 (low119895 up119895) oldP119894119895 sim 119880 (low119895 up119895) (1)

where 119875119894119895 and oldP119894119895 are the 119895th individual elements inthe problem dimension (119863) that falls in 119894th individualposition in the population size (119873) respectively low119895 andup119895 mean the lower boundary and the upper boundary ofthe 119895th dimension respectively and 119880 is a random uniformdistribution

22 Selection I BSArsquos selection I process is the beginning ofeach iteration It aims to reselect a new oldP for calculatingthe search direction based on population 119875 and historicalpopulation oldP The new oldP is reselected through the ldquoif-thenrdquo rule in

if 119886 lt 119887 then oldP fl 119875 | 119886 119887 sim 119880 (0 1) (2)





119872 = 119875 + 119865 sdot (oldP minus 119875) (4)












119865119894 sim 119873(exp(minus11003816100381610038161003816Δ11986811989410038161003816100381610038161119866 ) 1) (6)






















Schwefel 12 119891 (119909) = 119863sum119894=1

( 119894sum119895=1

119909119895)2

[minus100 100] 100 30 500

Rastrigin 119891 (119909) = 119863sum119894=1

(1199092119894 minus 10 cos (2120587119909119894) + 10) [minus512 512] 100 30 500


0 100 200 300 400 500minus05

0

05

1

15


Mea

n va

lue o

f F


0 100 200 300 400 500minus10

minus5

0

5

10


Mea

n va

lue o

f F


0 100 200 300 400 500minus05

0

05

1

15


Mea

n va

lue o

f F


0 100 200 300 400 500minus10

minus5

0

5

10


Mea

n va

lue o

f F




0 100 200 300 400 5000

500

1000

1500

2000

2500

3000

3500


Mea

n va

lue o

f pop

ulat

ion

varia

nces

BSABSAISA

(a) Schwefel 12

0 100 200 300 400 5000

2

4

6

8

10


Mea

n va

lue o

f pop

ulat

ion

varia

nces

BSABSAISA

(b) Rastrigin


0 100 200 300 400 5000

2

4

6

8

10


Best

func

tion

valu

e

BSABSAISA

times104

(a) Schwefel 12

BSABSAISA

0 100 200 300 400 5000

100

200

300

400

500


Best

func

tion

valu

e

(b) Rastrigin





min 119891 (119883)subject to 119892119894 (119883) le 0 119894 = 1 2 119898

ℎ119895 (119883) = 0 119895 = 1 2 119899(7)



120593 (119883) = 119898sum119894=1

max (0 119892119894 (119883)) + 119899sum119895=1

max (0 10038161003816100381610038161003816ℎ119895 (119883)10038161003816100381610038161003816 minus 120575) (8)



the better one = 119891 (1198831)


(9)


1205760 = 120593 (1198751205790 ) (10)

1205761 (0) = 1205760 (11)

1205762 (119905) = 120593(119879120579119905 ) if 1205760 gt Th11205760 else (12)

1205761 (119905)=


(13)

120576 (119905) = 1205761 (119905) (1 minus 119905119879119888)

cp if 119905 le 1198791198880 else (14)












0 500 1000 1500 2000 2500

X 500Y 06033

Iteration

X 1000Y 01227

X 2000Y 00001194

1010

100

10minus10

10minus20

va

lue


















































G05 5126497 5126497 51264981 5126571 5126512 5126487 5126544 5126497(238410) (51600) (47661) (100100) (350000) (350000) (350000) (350000)


G07 24307 24307 24311 24306 24315 24324 24333 24306(15060) (143000) (404501) (100100) (350000) (350000) (350000) (350000)


G09 680630057 680630 68063008 680630057 680645 680631 680631 6806301(347760) (111400) (324569) (100100) (350000) (350000) (350000) (350000)

G10 7049249 7054316 7059864 7049248 7281957 7058823 7049861 7049278(346980) (128400) (243520) (100100) (350000) (350000) (350000) (350000)

G11 0749900 0750 0749999 0749900 0750 0750 0749 0749900(87870) (11400) (23722) (100100) (350000) (350000) (350000) (350000)


G13 00539415 0053957 00539498 0056180 0053947 0757 NA 00539415(349800) (69800) (120268) (100100) (350000) (350000) (350000)








f(x

)minusf(x

lowast)(ＦＩＡ)

10minus2

10minus6

10minus10

10minus14

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

100

10minus1

10minus2

10minus3

10minus4

10minus5

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

100

10minus2

10minus4

10minus6

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

103

100

10minus3

10minus6

10minus9

10minus12

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

103

100

10minus3

10minus6

10minus9

10minus12

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

104

101

10minus2

10minus5

10minus8

10minus11

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

101

100

10minus1

10minus2

10minus3

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

10minus2

10minus6

10minus10

10minus14

10minus18

BSABSAISA



f(x

)minusf(x


100

10minus2

10minus4

10minus6

BSABSAISA








f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA

103

101

10minus1

10minus3



f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA

10minus12

10minus16

10minus8

100

10minus4


f(x

)minusf(x

lowast)(ＦＩＡ)


BSABSAISA

10minus12

10minus16

10minus8

10minus1

10minus4


f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA


10minus8

10minus4

10minus2

10minus6









BSABSAISA

10minus1

101

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

106

104

102

100

10minus2

10minus4

F(x

)minusF(x

lowast)(ＦＩＡ)










BSABSAISA

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





















BSABSAISA

101

10minus1

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

104

101

10minus2

10minus5

10minus8

F(x

)minusF(x

lowast)(ＦＩＡ)














Appendix



min 119891 (119909) = 5 4sum119894=1

119909119894 minus 5 4sum119894=1

1199092119894 minus 13sum119894=5

119909119894subject to 1198921 (119909) = 21199091 + 21199092 + 11990910 + 11990911 minus 10

le 01198922 (119909) = 21199091 + 21199093 + 11990910 + 11990912 minus 10le 0



(A1)


max 119891 (119909)


radicsum119899119894=1 11989411990921198941003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


119909119894 le 0

1198922 (119909) = 119899sum119894=1

119909119894 minus 75119899 le 0119899 = 20 0 le 119909119894 le 10 119894 = 1 119899

(A2)



119909119894subject to ℎ (119909) = 119899sum

119894=1

1199092119894 = 0119899 = 10 0 le 119909119894 le 1 119894 = 1 119899

(A3)


min 119891 (119909)= 5357854711990923 + 0835689111990911199095+ 372932391199091 minus 40792141



le 01198923 (119909)= 8051249 + 0007131711990921199095+ 0002995511990911199092 + 0002181311990923minus 110 le 0


1198925 (119909)= 9300961 + 0004702611990931199095+ 0001254711990911199093 + 0001908511990931199094minus 25 le 0


78 le 1199091 le 10233 le 1199092 le 4527 le 119909119894 le 45 119894 = 3 4 5

(A4)


min 119891 (119909)= 31199091 + 000000111990931 + 21199092+ (00000023 ) 11990932





0 le 1199091 1199092 le 1200minus 055 le 1199093 1199094 le 055

(A5)




(A6)







(A7)




(A8)





minus 10 le 119909119894 le 10 119894 = 1 2 3 4 5 6 7(A9)





(A10)



minus 1 le 119909119894 le 1 119894 = 1 2(A11)


max 119891 (119909)= (100 minus (1199091 minus 5)



(A12)




(A13)





(B1)



min 119891 (119909)= 06224119909111990931199094 + 17781119909211990923+ 31661119909211199094 + 1984119909211199093


(B2)



1198922 (119909)= (411990922 minus 11990911199092)(12566 (119909211990931 minus 11990941)) +

1(510811990921)minus 1 le 0


(B3)


min 119891 (119909)= 110471119909211199092+ 00481111990931199094 (14 + 1199092)


1198923 (119909) = 1199091 minus 1199094 le 01198924 (119909)= 01047111990921 + 00481111990931199094 (14 + 1199092)minus 5 le 0


(B4)

where

120591 (119909) = radic(1205911015840)2 + 2120591101584012059110158401015840 11990922119877 + (12059110158401015840)21205911015840 = 119875radic211990911199092 120591

10158401015840 = 119872119877119869119872 = 119875(119871 + 11990922 )

119877 = radic119909224 + (1199091 + 11990932 )2

119869 = 2radic211990911199092 [1199092212 + (1199091 + 11990932 )2]120590 (119909) = 6119875119871119909411990923

120575 (119909) = 41198751198713119864119909331199094




(B5)



min 119891 (119909)= 07854119909111990922 (3333311990923 + 1493341199093 minus 430934)minus 15081199091 (11990926 + 11990927) + 74777 (11990936 + 11990937)+ 07854 (119909411990926 + 119909511990927)


1198923 (119909) = 193119909341199092119909461199093 minus 1 le 0

1198924 (119909) = 193119909351199092119909471199093 minus 1 le 0

1198925 (119909) = [(7451199094 (11990921199093))2 + 169 times 106]12

11011990936 minus 1le 0

1198926 (119909) = [(7451199095 (11990921199093))2 + 1575 times 106]128511990937 minus 1


(B6)

where


(B7)



Acknowledgments


References





























































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in








119872 = 119875 + 119865 sdot (oldP minus 119875) (4)












119865119894 sim 119873(exp(minus11003816100381610038161003816Δ11986811989410038161003816100381610038161119866 ) 1) (6)






















Schwefel 12 119891 (119909) = 119863sum119894=1

( 119894sum119895=1

119909119895)2

[minus100 100] 100 30 500

Rastrigin 119891 (119909) = 119863sum119894=1

(1199092119894 minus 10 cos (2120587119909119894) + 10) [minus512 512] 100 30 500


0 100 200 300 400 500minus05

0

05

1

15


Mea

n va

lue o

f F


0 100 200 300 400 500minus10

minus5

0

5

10


Mea

n va

lue o

f F


0 100 200 300 400 500minus05

0

05

1

15


Mea

n va

lue o

f F


0 100 200 300 400 500minus10

minus5

0

5

10


Mea

n va

lue o

f F




0 100 200 300 400 5000

500

1000

1500

2000

2500

3000

3500


Mea

n va

lue o

f pop

ulat

ion

varia

nces

BSABSAISA

(a) Schwefel 12

0 100 200 300 400 5000

2

4

6

8

10


Mea

n va

lue o

f pop

ulat

ion

varia

nces

BSABSAISA

(b) Rastrigin


0 100 200 300 400 5000

2

4

6

8

10


Best

func

tion

valu

e

BSABSAISA

times104

(a) Schwefel 12

BSABSAISA

0 100 200 300 400 5000

100

200

300

400

500


Best

func

tion

valu

e

(b) Rastrigin





min 119891 (119883)subject to 119892119894 (119883) le 0 119894 = 1 2 119898

ℎ119895 (119883) = 0 119895 = 1 2 119899(7)



120593 (119883) = 119898sum119894=1

max (0 119892119894 (119883)) + 119899sum119895=1

max (0 10038161003816100381610038161003816ℎ119895 (119883)10038161003816100381610038161003816 minus 120575) (8)



the better one = 119891 (1198831)


(9)


1205760 = 120593 (1198751205790 ) (10)

1205761 (0) = 1205760 (11)

1205762 (119905) = 120593(119879120579119905 ) if 1205760 gt Th11205760 else (12)

1205761 (119905)=


(13)

120576 (119905) = 1205761 (119905) (1 minus 119905119879119888)

cp if 119905 le 1198791198880 else (14)












0 500 1000 1500 2000 2500

X 500Y 06033

Iteration

X 1000Y 01227

X 2000Y 00001194

1010

100

10minus10

10minus20

va

lue


















































G05 5126497 5126497 51264981 5126571 5126512 5126487 5126544 5126497(238410) (51600) (47661) (100100) (350000) (350000) (350000) (350000)


G07 24307 24307 24311 24306 24315 24324 24333 24306(15060) (143000) (404501) (100100) (350000) (350000) (350000) (350000)


G09 680630057 680630 68063008 680630057 680645 680631 680631 6806301(347760) (111400) (324569) (100100) (350000) (350000) (350000) (350000)

G10 7049249 7054316 7059864 7049248 7281957 7058823 7049861 7049278(346980) (128400) (243520) (100100) (350000) (350000) (350000) (350000)

G11 0749900 0750 0749999 0749900 0750 0750 0749 0749900(87870) (11400) (23722) (100100) (350000) (350000) (350000) (350000)


G13 00539415 0053957 00539498 0056180 0053947 0757 NA 00539415(349800) (69800) (120268) (100100) (350000) (350000) (350000)








f(x

)minusf(x

lowast)(ＦＩＡ)

10minus2

10minus6

10minus10

10minus14

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

100

10minus1

10minus2

10minus3

10minus4

10minus5

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

100

10minus2

10minus4

10minus6

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

103

100

10minus3

10minus6

10minus9

10minus12

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

103

100

10minus3

10minus6

10minus9

10minus12

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

104

101

10minus2

10minus5

10minus8

10minus11

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

101

100

10minus1

10minus2

10minus3

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

10minus2

10minus6

10minus10

10minus14

10minus18

BSABSAISA



f(x

)minusf(x


100

10minus2

10minus4

10minus6

BSABSAISA








f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA

103

101

10minus1

10minus3



f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA

10minus12

10minus16

10minus8

100

10minus4


f(x

)minusf(x

lowast)(ＦＩＡ)


BSABSAISA

10minus12

10minus16

10minus8

10minus1

10minus4


f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA


10minus8

10minus4

10minus2

10minus6









BSABSAISA

10minus1

101

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

106

104

102

100

10minus2

10minus4

F(x

)minusF(x

lowast)(ＦＩＡ)










BSABSAISA

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





















BSABSAISA

101

10minus1

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

104

101

10minus2

10minus5

10minus8

F(x

)minusF(x

lowast)(ＦＩＡ)














Appendix



min 119891 (119909) = 5 4sum119894=1

119909119894 minus 5 4sum119894=1

1199092119894 minus 13sum119894=5

119909119894subject to 1198921 (119909) = 21199091 + 21199092 + 11990910 + 11990911 minus 10

le 01198922 (119909) = 21199091 + 21199093 + 11990910 + 11990912 minus 10le 0



(A1)


max 119891 (119909)


radicsum119899119894=1 11989411990921198941003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


119909119894 le 0

1198922 (119909) = 119899sum119894=1

119909119894 minus 75119899 le 0119899 = 20 0 le 119909119894 le 10 119894 = 1 119899

(A2)



119909119894subject to ℎ (119909) = 119899sum

119894=1

1199092119894 = 0119899 = 10 0 le 119909119894 le 1 119894 = 1 119899

(A3)


min 119891 (119909)= 5357854711990923 + 0835689111990911199095+ 372932391199091 minus 40792141



le 01198923 (119909)= 8051249 + 0007131711990921199095+ 0002995511990911199092 + 0002181311990923minus 110 le 0


1198925 (119909)= 9300961 + 0004702611990931199095+ 0001254711990911199093 + 0001908511990931199094minus 25 le 0


78 le 1199091 le 10233 le 1199092 le 4527 le 119909119894 le 45 119894 = 3 4 5

(A4)


min 119891 (119909)= 31199091 + 000000111990931 + 21199092+ (00000023 ) 11990932





0 le 1199091 1199092 le 1200minus 055 le 1199093 1199094 le 055

(A5)




(A6)







(A7)




(A8)





minus 10 le 119909119894 le 10 119894 = 1 2 3 4 5 6 7(A9)





(A10)



minus 1 le 119909119894 le 1 119894 = 1 2(A11)


max 119891 (119909)= (100 minus (1199091 minus 5)



(A12)




(A13)





(B1)



min 119891 (119909)= 06224119909111990931199094 + 17781119909211990923+ 31661119909211199094 + 1984119909211199093


(B2)



1198922 (119909)= (411990922 minus 11990911199092)(12566 (119909211990931 minus 11990941)) +

1(510811990921)minus 1 le 0


(B3)


min 119891 (119909)= 110471119909211199092+ 00481111990931199094 (14 + 1199092)


1198923 (119909) = 1199091 minus 1199094 le 01198924 (119909)= 01047111990921 + 00481111990931199094 (14 + 1199092)minus 5 le 0


(B4)

where

120591 (119909) = radic(1205911015840)2 + 2120591101584012059110158401015840 11990922119877 + (12059110158401015840)21205911015840 = 119875radic211990911199092 120591

10158401015840 = 119872119877119869119872 = 119875(119871 + 11990922 )

119877 = radic119909224 + (1199091 + 11990932 )2

119869 = 2radic211990911199092 [1199092212 + (1199091 + 11990932 )2]120590 (119909) = 6119875119871119909411990923

120575 (119909) = 41198751198713119864119909331199094




(B5)



min 119891 (119909)= 07854119909111990922 (3333311990923 + 1493341199093 minus 430934)minus 15081199091 (11990926 + 11990927) + 74777 (11990936 + 11990937)+ 07854 (119909411990926 + 119909511990927)


1198923 (119909) = 193119909341199092119909461199093 minus 1 le 0

1198924 (119909) = 193119909351199092119909471199093 minus 1 le 0

1198925 (119909) = [(7451199094 (11990921199093))2 + 169 times 106]12

11011990936 minus 1le 0

1198926 (119909) = [(7451199095 (11990921199093))2 + 1575 times 106]128511990937 minus 1


(B6)

where


(B7)



Acknowledgments


References





























































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in





















Schwefel 12 119891 (119909) = 119863sum119894=1

( 119894sum119895=1

119909119895)2

[minus100 100] 100 30 500

Rastrigin 119891 (119909) = 119863sum119894=1

(1199092119894 minus 10 cos (2120587119909119894) + 10) [minus512 512] 100 30 500


0 100 200 300 400 500minus05

0

05

1

15


Mea

n va

lue o

f F


0 100 200 300 400 500minus10

minus5

0

5

10


Mea

n va

lue o

f F


0 100 200 300 400 500minus05

0

05

1

15


Mea

n va

lue o

f F


0 100 200 300 400 500minus10

minus5

0

5

10


Mea

n va

lue o

f F




0 100 200 300 400 5000

500

1000

1500

2000

2500

3000

3500


Mea

n va

lue o

f pop

ulat

ion

varia

nces

BSABSAISA

(a) Schwefel 12

0 100 200 300 400 5000

2

4

6

8

10


Mea

n va

lue o

f pop

ulat

ion

varia

nces

BSABSAISA

(b) Rastrigin


0 100 200 300 400 5000

2

4

6

8

10


Best

func

tion

valu

e

BSABSAISA

times104

(a) Schwefel 12

BSABSAISA

0 100 200 300 400 5000

100

200

300

400

500


Best

func

tion

valu

e

(b) Rastrigin





min 119891 (119883)subject to 119892119894 (119883) le 0 119894 = 1 2 119898

ℎ119895 (119883) = 0 119895 = 1 2 119899(7)



120593 (119883) = 119898sum119894=1

max (0 119892119894 (119883)) + 119899sum119895=1

max (0 10038161003816100381610038161003816ℎ119895 (119883)10038161003816100381610038161003816 minus 120575) (8)



the better one = 119891 (1198831)


(9)


1205760 = 120593 (1198751205790 ) (10)

1205761 (0) = 1205760 (11)

1205762 (119905) = 120593(119879120579119905 ) if 1205760 gt Th11205760 else (12)

1205761 (119905)=


(13)

120576 (119905) = 1205761 (119905) (1 minus 119905119879119888)

cp if 119905 le 1198791198880 else (14)












0 500 1000 1500 2000 2500

X 500Y 06033

Iteration

X 1000Y 01227

X 2000Y 00001194

1010

100

10minus10

10minus20

va

lue


















































G05 5126497 5126497 51264981 5126571 5126512 5126487 5126544 5126497(238410) (51600) (47661) (100100) (350000) (350000) (350000) (350000)


G07 24307 24307 24311 24306 24315 24324 24333 24306(15060) (143000) (404501) (100100) (350000) (350000) (350000) (350000)


G09 680630057 680630 68063008 680630057 680645 680631 680631 6806301(347760) (111400) (324569) (100100) (350000) (350000) (350000) (350000)

G10 7049249 7054316 7059864 7049248 7281957 7058823 7049861 7049278(346980) (128400) (243520) (100100) (350000) (350000) (350000) (350000)

G11 0749900 0750 0749999 0749900 0750 0750 0749 0749900(87870) (11400) (23722) (100100) (350000) (350000) (350000) (350000)


G13 00539415 0053957 00539498 0056180 0053947 0757 NA 00539415(349800) (69800) (120268) (100100) (350000) (350000) (350000)








f(x

)minusf(x

lowast)(ＦＩＡ)

10minus2

10minus6

10minus10

10minus14

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

100

10minus1

10minus2

10minus3

10minus4

10minus5

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

100

10minus2

10minus4

10minus6

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

103

100

10minus3

10minus6

10minus9

10minus12

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

103

100

10minus3

10minus6

10minus9

10minus12

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

104

101

10minus2

10minus5

10minus8

10minus11

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

101

100

10minus1

10minus2

10minus3

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

10minus2

10minus6

10minus10

10minus14

10minus18

BSABSAISA



f(x

)minusf(x


100

10minus2

10minus4

10minus6

BSABSAISA








f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA

103

101

10minus1

10minus3



f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA

10minus12

10minus16

10minus8

100

10minus4


f(x

)minusf(x

lowast)(ＦＩＡ)


BSABSAISA

10minus12

10minus16

10minus8

10minus1

10minus4


f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA


10minus8

10minus4

10minus2

10minus6









BSABSAISA

10minus1

101

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

106

104

102

100

10minus2

10minus4

F(x

)minusF(x

lowast)(ＦＩＡ)










BSABSAISA

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





















BSABSAISA

101

10minus1

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

104

101

10minus2

10minus5

10minus8

F(x

)minusF(x

lowast)(ＦＩＡ)














Appendix



min 119891 (119909) = 5 4sum119894=1

119909119894 minus 5 4sum119894=1

1199092119894 minus 13sum119894=5

119909119894subject to 1198921 (119909) = 21199091 + 21199092 + 11990910 + 11990911 minus 10

le 01198922 (119909) = 21199091 + 21199093 + 11990910 + 11990912 minus 10le 0



(A1)


max 119891 (119909)


radicsum119899119894=1 11989411990921198941003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


119909119894 le 0

1198922 (119909) = 119899sum119894=1

119909119894 minus 75119899 le 0119899 = 20 0 le 119909119894 le 10 119894 = 1 119899

(A2)



119909119894subject to ℎ (119909) = 119899sum

119894=1

1199092119894 = 0119899 = 10 0 le 119909119894 le 1 119894 = 1 119899

(A3)


min 119891 (119909)= 5357854711990923 + 0835689111990911199095+ 372932391199091 minus 40792141



le 01198923 (119909)= 8051249 + 0007131711990921199095+ 0002995511990911199092 + 0002181311990923minus 110 le 0


1198925 (119909)= 9300961 + 0004702611990931199095+ 0001254711990911199093 + 0001908511990931199094minus 25 le 0


78 le 1199091 le 10233 le 1199092 le 4527 le 119909119894 le 45 119894 = 3 4 5

(A4)


min 119891 (119909)= 31199091 + 000000111990931 + 21199092+ (00000023 ) 11990932





0 le 1199091 1199092 le 1200minus 055 le 1199093 1199094 le 055

(A5)




(A6)







(A7)




(A8)





minus 10 le 119909119894 le 10 119894 = 1 2 3 4 5 6 7(A9)





(A10)



minus 1 le 119909119894 le 1 119894 = 1 2(A11)


max 119891 (119909)= (100 minus (1199091 minus 5)



(A12)




(A13)





(B1)



min 119891 (119909)= 06224119909111990931199094 + 17781119909211990923+ 31661119909211199094 + 1984119909211199093


(B2)



1198922 (119909)= (411990922 minus 11990911199092)(12566 (119909211990931 minus 11990941)) +

1(510811990921)minus 1 le 0


(B3)


min 119891 (119909)= 110471119909211199092+ 00481111990931199094 (14 + 1199092)


1198923 (119909) = 1199091 minus 1199094 le 01198924 (119909)= 01047111990921 + 00481111990931199094 (14 + 1199092)minus 5 le 0


(B4)

where

120591 (119909) = radic(1205911015840)2 + 2120591101584012059110158401015840 11990922119877 + (12059110158401015840)21205911015840 = 119875radic211990911199092 120591

10158401015840 = 119872119877119869119872 = 119875(119871 + 11990922 )

119877 = radic119909224 + (1199091 + 11990932 )2

119869 = 2radic211990911199092 [1199092212 + (1199091 + 11990932 )2]120590 (119909) = 6119875119871119909411990923

120575 (119909) = 41198751198713119864119909331199094




(B5)



min 119891 (119909)= 07854119909111990922 (3333311990923 + 1493341199093 minus 430934)minus 15081199091 (11990926 + 11990927) + 74777 (11990936 + 11990937)+ 07854 (119909411990926 + 119909511990927)


1198923 (119909) = 193119909341199092119909461199093 minus 1 le 0

1198924 (119909) = 193119909351199092119909471199093 minus 1 le 0

1198925 (119909) = [(7451199094 (11990921199093))2 + 169 times 106]12

11011990936 minus 1le 0

1198926 (119909) = [(7451199095 (11990921199093))2 + 1575 times 106]128511990937 minus 1


(B6)

where


(B7)



Acknowledgments


References





























































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in





0 100 200 300 400 5000

500

1000

1500

2000

2500

3000

3500


Mea

n va

lue o

f pop

ulat

ion

varia

nces

BSABSAISA

(a) Schwefel 12

0 100 200 300 400 5000

2

4

6

8

10


Mea

n va

lue o

f pop

ulat

ion

varia

nces

BSABSAISA

(b) Rastrigin


0 100 200 300 400 5000

2

4

6

8

10


Best

func

tion

valu

e

BSABSAISA

times104

(a) Schwefel 12

BSABSAISA

0 100 200 300 400 5000

100

200

300

400

500


Best

func

tion

valu

e

(b) Rastrigin





min 119891 (119883)subject to 119892119894 (119883) le 0 119894 = 1 2 119898

ℎ119895 (119883) = 0 119895 = 1 2 119899(7)



120593 (119883) = 119898sum119894=1

max (0 119892119894 (119883)) + 119899sum119895=1

max (0 10038161003816100381610038161003816ℎ119895 (119883)10038161003816100381610038161003816 minus 120575) (8)



the better one = 119891 (1198831)


(9)


1205760 = 120593 (1198751205790 ) (10)

1205761 (0) = 1205760 (11)

1205762 (119905) = 120593(119879120579119905 ) if 1205760 gt Th11205760 else (12)

1205761 (119905)=


(13)

120576 (119905) = 1205761 (119905) (1 minus 119905119879119888)

cp if 119905 le 1198791198880 else (14)












0 500 1000 1500 2000 2500

X 500Y 06033

Iteration

X 1000Y 01227

X 2000Y 00001194

1010

100

10minus10

10minus20

va

lue


















































G05 5126497 5126497 51264981 5126571 5126512 5126487 5126544 5126497(238410) (51600) (47661) (100100) (350000) (350000) (350000) (350000)


G07 24307 24307 24311 24306 24315 24324 24333 24306(15060) (143000) (404501) (100100) (350000) (350000) (350000) (350000)


G09 680630057 680630 68063008 680630057 680645 680631 680631 6806301(347760) (111400) (324569) (100100) (350000) (350000) (350000) (350000)

G10 7049249 7054316 7059864 7049248 7281957 7058823 7049861 7049278(346980) (128400) (243520) (100100) (350000) (350000) (350000) (350000)

G11 0749900 0750 0749999 0749900 0750 0750 0749 0749900(87870) (11400) (23722) (100100) (350000) (350000) (350000) (350000)


G13 00539415 0053957 00539498 0056180 0053947 0757 NA 00539415(349800) (69800) (120268) (100100) (350000) (350000) (350000)








f(x

)minusf(x

lowast)(ＦＩＡ)

10minus2

10minus6

10minus10

10minus14

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

100

10minus1

10minus2

10minus3

10minus4

10minus5

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

100

10minus2

10minus4

10minus6

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

103

100

10minus3

10minus6

10minus9

10minus12

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

103

100

10minus3

10minus6

10minus9

10minus12

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

104

101

10minus2

10minus5

10minus8

10minus11

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

101

100

10minus1

10minus2

10minus3

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

10minus2

10minus6

10minus10

10minus14

10minus18

BSABSAISA



f(x

)minusf(x


100

10minus2

10minus4

10minus6

BSABSAISA








f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA

103

101

10minus1

10minus3



f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA

10minus12

10minus16

10minus8

100

10minus4


f(x

)minusf(x

lowast)(ＦＩＡ)


BSABSAISA

10minus12

10minus16

10minus8

10minus1

10minus4


f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA


10minus8

10minus4

10minus2

10minus6









BSABSAISA

10minus1

101

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

106

104

102

100

10minus2

10minus4

F(x

)minusF(x

lowast)(ＦＩＡ)










BSABSAISA

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





















BSABSAISA

101

10minus1

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

104

101

10minus2

10minus5

10minus8

F(x

)minusF(x

lowast)(ＦＩＡ)














Appendix



min 119891 (119909) = 5 4sum119894=1

119909119894 minus 5 4sum119894=1

1199092119894 minus 13sum119894=5

119909119894subject to 1198921 (119909) = 21199091 + 21199092 + 11990910 + 11990911 minus 10

le 01198922 (119909) = 21199091 + 21199093 + 11990910 + 11990912 minus 10le 0



(A1)


max 119891 (119909)


radicsum119899119894=1 11989411990921198941003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


119909119894 le 0

1198922 (119909) = 119899sum119894=1

119909119894 minus 75119899 le 0119899 = 20 0 le 119909119894 le 10 119894 = 1 119899

(A2)



119909119894subject to ℎ (119909) = 119899sum

119894=1

1199092119894 = 0119899 = 10 0 le 119909119894 le 1 119894 = 1 119899

(A3)


min 119891 (119909)= 5357854711990923 + 0835689111990911199095+ 372932391199091 minus 40792141



le 01198923 (119909)= 8051249 + 0007131711990921199095+ 0002995511990911199092 + 0002181311990923minus 110 le 0


1198925 (119909)= 9300961 + 0004702611990931199095+ 0001254711990911199093 + 0001908511990931199094minus 25 le 0


78 le 1199091 le 10233 le 1199092 le 4527 le 119909119894 le 45 119894 = 3 4 5

(A4)


min 119891 (119909)= 31199091 + 000000111990931 + 21199092+ (00000023 ) 11990932





0 le 1199091 1199092 le 1200minus 055 le 1199093 1199094 le 055

(A5)




(A6)







(A7)




(A8)





minus 10 le 119909119894 le 10 119894 = 1 2 3 4 5 6 7(A9)





(A10)



minus 1 le 119909119894 le 1 119894 = 1 2(A11)


max 119891 (119909)= (100 minus (1199091 minus 5)



(A12)




(A13)





(B1)



min 119891 (119909)= 06224119909111990931199094 + 17781119909211990923+ 31661119909211199094 + 1984119909211199093


(B2)



1198922 (119909)= (411990922 minus 11990911199092)(12566 (119909211990931 minus 11990941)) +

1(510811990921)minus 1 le 0


(B3)


min 119891 (119909)= 110471119909211199092+ 00481111990931199094 (14 + 1199092)


1198923 (119909) = 1199091 minus 1199094 le 01198924 (119909)= 01047111990921 + 00481111990931199094 (14 + 1199092)minus 5 le 0


(B4)

where

120591 (119909) = radic(1205911015840)2 + 2120591101584012059110158401015840 11990922119877 + (12059110158401015840)21205911015840 = 119875radic211990911199092 120591

10158401015840 = 119872119877119869119872 = 119875(119871 + 11990922 )

119877 = radic119909224 + (1199091 + 11990932 )2

119869 = 2radic211990911199092 [1199092212 + (1199091 + 11990932 )2]120590 (119909) = 6119875119871119909411990923

120575 (119909) = 41198751198713119864119909331199094




(B5)



min 119891 (119909)= 07854119909111990922 (3333311990923 + 1493341199093 minus 430934)minus 15081199091 (11990926 + 11990927) + 74777 (11990936 + 11990937)+ 07854 (119909411990926 + 119909511990927)


1198923 (119909) = 193119909341199092119909461199093 minus 1 le 0

1198924 (119909) = 193119909351199092119909471199093 minus 1 le 0

1198925 (119909) = [(7451199094 (11990921199093))2 + 169 times 106]12

11011990936 minus 1le 0

1198926 (119909) = [(7451199095 (11990921199093))2 + 1575 times 106]128511990937 minus 1


(B6)

where


(B7)



Acknowledgments


References





























































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in






120593 (119883) = 119898sum119894=1

max (0 119892119894 (119883)) + 119899sum119895=1

max (0 10038161003816100381610038161003816ℎ119895 (119883)10038161003816100381610038161003816 minus 120575) (8)



the better one = 119891 (1198831)


(9)


1205760 = 120593 (1198751205790 ) (10)

1205761 (0) = 1205760 (11)

1205762 (119905) = 120593(119879120579119905 ) if 1205760 gt Th11205760 else (12)

1205761 (119905)=


(13)

120576 (119905) = 1205761 (119905) (1 minus 119905119879119888)

cp if 119905 le 1198791198880 else (14)












0 500 1000 1500 2000 2500

X 500Y 06033

Iteration

X 1000Y 01227

X 2000Y 00001194

1010

100

10minus10

10minus20

va

lue


















































G05 5126497 5126497 51264981 5126571 5126512 5126487 5126544 5126497(238410) (51600) (47661) (100100) (350000) (350000) (350000) (350000)


G07 24307 24307 24311 24306 24315 24324 24333 24306(15060) (143000) (404501) (100100) (350000) (350000) (350000) (350000)


G09 680630057 680630 68063008 680630057 680645 680631 680631 6806301(347760) (111400) (324569) (100100) (350000) (350000) (350000) (350000)

G10 7049249 7054316 7059864 7049248 7281957 7058823 7049861 7049278(346980) (128400) (243520) (100100) (350000) (350000) (350000) (350000)

G11 0749900 0750 0749999 0749900 0750 0750 0749 0749900(87870) (11400) (23722) (100100) (350000) (350000) (350000) (350000)


G13 00539415 0053957 00539498 0056180 0053947 0757 NA 00539415(349800) (69800) (120268) (100100) (350000) (350000) (350000)








f(x

)minusf(x

lowast)(ＦＩＡ)

10minus2

10minus6

10minus10

10minus14

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

100

10minus1

10minus2

10minus3

10minus4

10minus5

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

100

10minus2

10minus4

10minus6

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

103

100

10minus3

10minus6

10minus9

10minus12

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

103

100

10minus3

10minus6

10minus9

10minus12

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

104

101

10minus2

10minus5

10minus8

10minus11

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

101

100

10minus1

10minus2

10minus3

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

10minus2

10minus6

10minus10

10minus14

10minus18

BSABSAISA



f(x

)minusf(x


100

10minus2

10minus4

10minus6

BSABSAISA








f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA

103

101

10minus1

10minus3



f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA

10minus12

10minus16

10minus8

100

10minus4


f(x

)minusf(x

lowast)(ＦＩＡ)


BSABSAISA

10minus12

10minus16

10minus8

10minus1

10minus4


f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA


10minus8

10minus4

10minus2

10minus6









BSABSAISA

10minus1

101

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

106

104

102

100

10minus2

10minus4

F(x

)minusF(x

lowast)(ＦＩＡ)










BSABSAISA

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





















BSABSAISA

101

10minus1

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

104

101

10minus2

10minus5

10minus8

F(x

)minusF(x

lowast)(ＦＩＡ)














Appendix



min 119891 (119909) = 5 4sum119894=1

119909119894 minus 5 4sum119894=1

1199092119894 minus 13sum119894=5

119909119894subject to 1198921 (119909) = 21199091 + 21199092 + 11990910 + 11990911 minus 10

le 01198922 (119909) = 21199091 + 21199093 + 11990910 + 11990912 minus 10le 0



(A1)


max 119891 (119909)


radicsum119899119894=1 11989411990921198941003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


119909119894 le 0

1198922 (119909) = 119899sum119894=1

119909119894 minus 75119899 le 0119899 = 20 0 le 119909119894 le 10 119894 = 1 119899

(A2)



119909119894subject to ℎ (119909) = 119899sum

119894=1

1199092119894 = 0119899 = 10 0 le 119909119894 le 1 119894 = 1 119899

(A3)


min 119891 (119909)= 5357854711990923 + 0835689111990911199095+ 372932391199091 minus 40792141



le 01198923 (119909)= 8051249 + 0007131711990921199095+ 0002995511990911199092 + 0002181311990923minus 110 le 0


1198925 (119909)= 9300961 + 0004702611990931199095+ 0001254711990911199093 + 0001908511990931199094minus 25 le 0


78 le 1199091 le 10233 le 1199092 le 4527 le 119909119894 le 45 119894 = 3 4 5

(A4)


min 119891 (119909)= 31199091 + 000000111990931 + 21199092+ (00000023 ) 11990932





0 le 1199091 1199092 le 1200minus 055 le 1199093 1199094 le 055

(A5)




(A6)







(A7)




(A8)





minus 10 le 119909119894 le 10 119894 = 1 2 3 4 5 6 7(A9)





(A10)



minus 1 le 119909119894 le 1 119894 = 1 2(A11)


max 119891 (119909)= (100 minus (1199091 minus 5)



(A12)




(A13)





(B1)



min 119891 (119909)= 06224119909111990931199094 + 17781119909211990923+ 31661119909211199094 + 1984119909211199093


(B2)



1198922 (119909)= (411990922 minus 11990911199092)(12566 (119909211990931 minus 11990941)) +

1(510811990921)minus 1 le 0


(B3)


min 119891 (119909)= 110471119909211199092+ 00481111990931199094 (14 + 1199092)


1198923 (119909) = 1199091 minus 1199094 le 01198924 (119909)= 01047111990921 + 00481111990931199094 (14 + 1199092)minus 5 le 0


(B4)

where

120591 (119909) = radic(1205911015840)2 + 2120591101584012059110158401015840 11990922119877 + (12059110158401015840)21205911015840 = 119875radic211990911199092 120591

10158401015840 = 119872119877119869119872 = 119875(119871 + 11990922 )

119877 = radic119909224 + (1199091 + 11990932 )2

119869 = 2radic211990911199092 [1199092212 + (1199091 + 11990932 )2]120590 (119909) = 6119875119871119909411990923

120575 (119909) = 41198751198713119864119909331199094




(B5)



min 119891 (119909)= 07854119909111990922 (3333311990923 + 1493341199093 minus 430934)minus 15081199091 (11990926 + 11990927) + 74777 (11990936 + 11990937)+ 07854 (119909411990926 + 119909511990927)


1198923 (119909) = 193119909341199092119909461199093 minus 1 le 0

1198924 (119909) = 193119909351199092119909471199093 minus 1 le 0

1198925 (119909) = [(7451199094 (11990921199093))2 + 169 times 106]12

11011990936 minus 1le 0

1198926 (119909) = [(7451199095 (11990921199093))2 + 1575 times 106]128511990937 minus 1


(B6)

where


(B7)



Acknowledgments


References





























































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in








0 500 1000 1500 2000 2500

X 500Y 06033

Iteration

X 1000Y 01227

X 2000Y 00001194

1010

100

10minus10

10minus20

va

lue


















































G05 5126497 5126497 51264981 5126571 5126512 5126487 5126544 5126497(238410) (51600) (47661) (100100) (350000) (350000) (350000) (350000)


G07 24307 24307 24311 24306 24315 24324 24333 24306(15060) (143000) (404501) (100100) (350000) (350000) (350000) (350000)


G09 680630057 680630 68063008 680630057 680645 680631 680631 6806301(347760) (111400) (324569) (100100) (350000) (350000) (350000) (350000)

G10 7049249 7054316 7059864 7049248 7281957 7058823 7049861 7049278(346980) (128400) (243520) (100100) (350000) (350000) (350000) (350000)

G11 0749900 0750 0749999 0749900 0750 0750 0749 0749900(87870) (11400) (23722) (100100) (350000) (350000) (350000) (350000)


G13 00539415 0053957 00539498 0056180 0053947 0757 NA 00539415(349800) (69800) (120268) (100100) (350000) (350000) (350000)








f(x

)minusf(x

lowast)(ＦＩＡ)

10minus2

10minus6

10minus10

10minus14

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

100

10minus1

10minus2

10minus3

10minus4

10minus5

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

100

10minus2

10minus4

10minus6

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

103

100

10minus3

10minus6

10minus9

10minus12

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

103

100

10minus3

10minus6

10minus9

10minus12

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

104

101

10minus2

10minus5

10minus8

10minus11

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

101

100

10minus1

10minus2

10minus3

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

10minus2

10minus6

10minus10

10minus14

10minus18

BSABSAISA



f(x

)minusf(x


100

10minus2

10minus4

10minus6

BSABSAISA








f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA

103

101

10minus1

10minus3



f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA

10minus12

10minus16

10minus8

100

10minus4


f(x

)minusf(x

lowast)(ＦＩＡ)


BSABSAISA

10minus12

10minus16

10minus8

10minus1

10minus4


f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA


10minus8

10minus4

10minus2

10minus6









BSABSAISA

10minus1

101

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

106

104

102

100

10minus2

10minus4

F(x

)minusF(x

lowast)(ＦＩＡ)










BSABSAISA

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





















BSABSAISA

101

10minus1

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

104

101

10minus2

10minus5

10minus8

F(x

)minusF(x

lowast)(ＦＩＡ)














Appendix



min 119891 (119909) = 5 4sum119894=1

119909119894 minus 5 4sum119894=1

1199092119894 minus 13sum119894=5

119909119894subject to 1198921 (119909) = 21199091 + 21199092 + 11990910 + 11990911 minus 10

le 01198922 (119909) = 21199091 + 21199093 + 11990910 + 11990912 minus 10le 0



(A1)


max 119891 (119909)


radicsum119899119894=1 11989411990921198941003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


119909119894 le 0

1198922 (119909) = 119899sum119894=1

119909119894 minus 75119899 le 0119899 = 20 0 le 119909119894 le 10 119894 = 1 119899

(A2)



119909119894subject to ℎ (119909) = 119899sum

119894=1

1199092119894 = 0119899 = 10 0 le 119909119894 le 1 119894 = 1 119899

(A3)


min 119891 (119909)= 5357854711990923 + 0835689111990911199095+ 372932391199091 minus 40792141



le 01198923 (119909)= 8051249 + 0007131711990921199095+ 0002995511990911199092 + 0002181311990923minus 110 le 0


1198925 (119909)= 9300961 + 0004702611990931199095+ 0001254711990911199093 + 0001908511990931199094minus 25 le 0


78 le 1199091 le 10233 le 1199092 le 4527 le 119909119894 le 45 119894 = 3 4 5

(A4)


min 119891 (119909)= 31199091 + 000000111990931 + 21199092+ (00000023 ) 11990932





0 le 1199091 1199092 le 1200minus 055 le 1199093 1199094 le 055

(A5)




(A6)







(A7)




(A8)





minus 10 le 119909119894 le 10 119894 = 1 2 3 4 5 6 7(A9)





(A10)



minus 1 le 119909119894 le 1 119894 = 1 2(A11)


max 119891 (119909)= (100 minus (1199091 minus 5)



(A12)




(A13)





(B1)



min 119891 (119909)= 06224119909111990931199094 + 17781119909211990923+ 31661119909211199094 + 1984119909211199093


(B2)



1198922 (119909)= (411990922 minus 11990911199092)(12566 (119909211990931 minus 11990941)) +

1(510811990921)minus 1 le 0


(B3)


min 119891 (119909)= 110471119909211199092+ 00481111990931199094 (14 + 1199092)


1198923 (119909) = 1199091 minus 1199094 le 01198924 (119909)= 01047111990921 + 00481111990931199094 (14 + 1199092)minus 5 le 0


(B4)

where

120591 (119909) = radic(1205911015840)2 + 2120591101584012059110158401015840 11990922119877 + (12059110158401015840)21205911015840 = 119875radic211990911199092 120591

10158401015840 = 119872119877119869119872 = 119875(119871 + 11990922 )

119877 = radic119909224 + (1199091 + 11990932 )2

119869 = 2radic211990911199092 [1199092212 + (1199091 + 11990932 )2]120590 (119909) = 6119875119871119909411990923

120575 (119909) = 41198751198713119864119909331199094




(B5)



min 119891 (119909)= 07854119909111990922 (3333311990923 + 1493341199093 minus 430934)minus 15081199091 (11990926 + 11990927) + 74777 (11990936 + 11990937)+ 07854 (119909411990926 + 119909511990927)


1198923 (119909) = 193119909341199092119909461199093 minus 1 le 0

1198924 (119909) = 193119909351199092119909471199093 minus 1 le 0

1198925 (119909) = [(7451199094 (11990921199093))2 + 169 times 106]12

11011990936 minus 1le 0

1198926 (119909) = [(7451199095 (11990921199093))2 + 1575 times 106]128511990937 minus 1


(B6)

where


(B7)



Acknowledgments


References





























































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in













































G05 5126497 5126497 51264981 5126571 5126512 5126487 5126544 5126497(238410) (51600) (47661) (100100) (350000) (350000) (350000) (350000)


G07 24307 24307 24311 24306 24315 24324 24333 24306(15060) (143000) (404501) (100100) (350000) (350000) (350000) (350000)


G09 680630057 680630 68063008 680630057 680645 680631 680631 6806301(347760) (111400) (324569) (100100) (350000) (350000) (350000) (350000)

G10 7049249 7054316 7059864 7049248 7281957 7058823 7049861 7049278(346980) (128400) (243520) (100100) (350000) (350000) (350000) (350000)

G11 0749900 0750 0749999 0749900 0750 0750 0749 0749900(87870) (11400) (23722) (100100) (350000) (350000) (350000) (350000)


G13 00539415 0053957 00539498 0056180 0053947 0757 NA 00539415(349800) (69800) (120268) (100100) (350000) (350000) (350000)








f(x

)minusf(x

lowast)(ＦＩＡ)

10minus2

10minus6

10minus10

10minus14

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

100

10minus1

10minus2

10minus3

10minus4

10minus5

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

100

10minus2

10minus4

10minus6

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

103

100

10minus3

10minus6

10minus9

10minus12

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

103

100

10minus3

10minus6

10minus9

10minus12

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

104

101

10minus2

10minus5

10minus8

10minus11

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

101

100

10minus1

10minus2

10minus3

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

10minus2

10minus6

10minus10

10minus14

10minus18

BSABSAISA



f(x

)minusf(x


100

10minus2

10minus4

10minus6

BSABSAISA








f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA

103

101

10minus1

10minus3



f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA

10minus12

10minus16

10minus8

100

10minus4


f(x

)minusf(x

lowast)(ＦＩＡ)


BSABSAISA

10minus12

10minus16

10minus8

10minus1

10minus4


f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA


10minus8

10minus4

10minus2

10minus6









BSABSAISA

10minus1

101

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

106

104

102

100

10minus2

10minus4

F(x

)minusF(x

lowast)(ＦＩＡ)










BSABSAISA

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





















BSABSAISA

101

10minus1

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

104

101

10minus2

10minus5

10minus8

F(x

)minusF(x

lowast)(ＦＩＡ)














Appendix



min 119891 (119909) = 5 4sum119894=1

119909119894 minus 5 4sum119894=1

1199092119894 minus 13sum119894=5

119909119894subject to 1198921 (119909) = 21199091 + 21199092 + 11990910 + 11990911 minus 10

le 01198922 (119909) = 21199091 + 21199093 + 11990910 + 11990912 minus 10le 0



(A1)


max 119891 (119909)


radicsum119899119894=1 11989411990921198941003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


119909119894 le 0

1198922 (119909) = 119899sum119894=1

119909119894 minus 75119899 le 0119899 = 20 0 le 119909119894 le 10 119894 = 1 119899

(A2)



119909119894subject to ℎ (119909) = 119899sum

119894=1

1199092119894 = 0119899 = 10 0 le 119909119894 le 1 119894 = 1 119899

(A3)


min 119891 (119909)= 5357854711990923 + 0835689111990911199095+ 372932391199091 minus 40792141



le 01198923 (119909)= 8051249 + 0007131711990921199095+ 0002995511990911199092 + 0002181311990923minus 110 le 0


1198925 (119909)= 9300961 + 0004702611990931199095+ 0001254711990911199093 + 0001908511990931199094minus 25 le 0


78 le 1199091 le 10233 le 1199092 le 4527 le 119909119894 le 45 119894 = 3 4 5

(A4)


min 119891 (119909)= 31199091 + 000000111990931 + 21199092+ (00000023 ) 11990932





0 le 1199091 1199092 le 1200minus 055 le 1199093 1199094 le 055

(A5)




(A6)







(A7)




(A8)





minus 10 le 119909119894 le 10 119894 = 1 2 3 4 5 6 7(A9)





(A10)



minus 1 le 119909119894 le 1 119894 = 1 2(A11)


max 119891 (119909)= (100 minus (1199091 minus 5)



(A12)




(A13)





(B1)



min 119891 (119909)= 06224119909111990931199094 + 17781119909211990923+ 31661119909211199094 + 1984119909211199093


(B2)



1198922 (119909)= (411990922 minus 11990911199092)(12566 (119909211990931 minus 11990941)) +

1(510811990921)minus 1 le 0


(B3)


min 119891 (119909)= 110471119909211199092+ 00481111990931199094 (14 + 1199092)


1198923 (119909) = 1199091 minus 1199094 le 01198924 (119909)= 01047111990921 + 00481111990931199094 (14 + 1199092)minus 5 le 0


(B4)

where

120591 (119909) = radic(1205911015840)2 + 2120591101584012059110158401015840 11990922119877 + (12059110158401015840)21205911015840 = 119875radic211990911199092 120591

10158401015840 = 119872119877119869119872 = 119875(119871 + 11990922 )

119877 = radic119909224 + (1199091 + 11990932 )2

119869 = 2radic211990911199092 [1199092212 + (1199091 + 11990932 )2]120590 (119909) = 6119875119871119909411990923

120575 (119909) = 41198751198713119864119909331199094




(B5)



min 119891 (119909)= 07854119909111990922 (3333311990923 + 1493341199093 minus 430934)minus 15081199091 (11990926 + 11990927) + 74777 (11990936 + 11990937)+ 07854 (119909411990926 + 119909511990927)


1198923 (119909) = 193119909341199092119909461199093 minus 1 le 0

1198924 (119909) = 193119909351199092119909471199093 minus 1 le 0

1198925 (119909) = [(7451199094 (11990921199093))2 + 169 times 106]12

11011990936 minus 1le 0

1198926 (119909) = [(7451199095 (11990921199093))2 + 1575 times 106]128511990937 minus 1


(B6)

where


(B7)



Acknowledgments


References





























































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in






f(x

)minusf(x

lowast)(ＦＩＡ)

10minus2

10minus6

10minus10

10minus14

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

100

10minus1

10minus2

10minus3

10minus4

10minus5

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

100

10minus2

10minus4

10minus6

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

103

100

10minus3

10minus6

10minus9

10minus12

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

103

100

10minus3

10minus6

10minus9

10minus12

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

104

101

10minus2

10minus5

10minus8

10minus11

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

101

100

10minus1

10minus2

10minus3

BSABSAISA



f(x

)minusf(x

lowast)(ＦＩＡ)

102

10minus2

10minus6

10minus10

10minus14

10minus18

BSABSAISA



f(x

)minusf(x


100

10minus2

10minus4

10minus6

BSABSAISA








f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA

103

101

10minus1

10minus3



f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA

10minus12

10minus16

10minus8

100

10minus4


f(x

)minusf(x

lowast)(ＦＩＡ)


BSABSAISA

10minus12

10minus16

10minus8

10minus1

10minus4


f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA


10minus8

10minus4

10minus2

10minus6









BSABSAISA

10minus1

101

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

106

104

102

100

10minus2

10minus4

F(x

)minusF(x

lowast)(ＦＩＡ)










BSABSAISA

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





















BSABSAISA

101

10minus1

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

104

101

10minus2

10minus5

10minus8

F(x

)minusF(x

lowast)(ＦＩＡ)














Appendix



min 119891 (119909) = 5 4sum119894=1

119909119894 minus 5 4sum119894=1

1199092119894 minus 13sum119894=5

119909119894subject to 1198921 (119909) = 21199091 + 21199092 + 11990910 + 11990911 minus 10

le 01198922 (119909) = 21199091 + 21199093 + 11990910 + 11990912 minus 10le 0



(A1)


max 119891 (119909)


radicsum119899119894=1 11989411990921198941003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


119909119894 le 0

1198922 (119909) = 119899sum119894=1

119909119894 minus 75119899 le 0119899 = 20 0 le 119909119894 le 10 119894 = 1 119899

(A2)



119909119894subject to ℎ (119909) = 119899sum

119894=1

1199092119894 = 0119899 = 10 0 le 119909119894 le 1 119894 = 1 119899

(A3)


min 119891 (119909)= 5357854711990923 + 0835689111990911199095+ 372932391199091 minus 40792141



le 01198923 (119909)= 8051249 + 0007131711990921199095+ 0002995511990911199092 + 0002181311990923minus 110 le 0


1198925 (119909)= 9300961 + 0004702611990931199095+ 0001254711990911199093 + 0001908511990931199094minus 25 le 0


78 le 1199091 le 10233 le 1199092 le 4527 le 119909119894 le 45 119894 = 3 4 5

(A4)


min 119891 (119909)= 31199091 + 000000111990931 + 21199092+ (00000023 ) 11990932





0 le 1199091 1199092 le 1200minus 055 le 1199093 1199094 le 055

(A5)




(A6)







(A7)




(A8)





minus 10 le 119909119894 le 10 119894 = 1 2 3 4 5 6 7(A9)





(A10)



minus 1 le 119909119894 le 1 119894 = 1 2(A11)


max 119891 (119909)= (100 minus (1199091 minus 5)



(A12)




(A13)





(B1)



min 119891 (119909)= 06224119909111990931199094 + 17781119909211990923+ 31661119909211199094 + 1984119909211199093


(B2)



1198922 (119909)= (411990922 minus 11990911199092)(12566 (119909211990931 minus 11990941)) +

1(510811990921)minus 1 le 0


(B3)


min 119891 (119909)= 110471119909211199092+ 00481111990931199094 (14 + 1199092)


1198923 (119909) = 1199091 minus 1199094 le 01198924 (119909)= 01047111990921 + 00481111990931199094 (14 + 1199092)minus 5 le 0


(B4)

where

120591 (119909) = radic(1205911015840)2 + 2120591101584012059110158401015840 11990922119877 + (12059110158401015840)21205911015840 = 119875radic211990911199092 120591

10158401015840 = 119872119877119869119872 = 119875(119871 + 11990922 )

119877 = radic119909224 + (1199091 + 11990932 )2

119869 = 2radic211990911199092 [1199092212 + (1199091 + 11990932 )2]120590 (119909) = 6119875119871119909411990923

120575 (119909) = 41198751198713119864119909331199094




(B5)



min 119891 (119909)= 07854119909111990922 (3333311990923 + 1493341199093 minus 430934)minus 15081199091 (11990926 + 11990927) + 74777 (11990936 + 11990937)+ 07854 (119909411990926 + 119909511990927)


1198923 (119909) = 193119909341199092119909461199093 minus 1 le 0

1198924 (119909) = 193119909351199092119909471199093 minus 1 le 0

1198925 (119909) = [(7451199094 (11990921199093))2 + 169 times 106]12

11011990936 minus 1le 0

1198926 (119909) = [(7451199095 (11990921199093))2 + 1575 times 106]128511990937 minus 1


(B6)

where


(B7)



Acknowledgments


References





























































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in






f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA

103

101

10minus1

10minus3



f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA

10minus12

10minus16

10minus8

100

10minus4


f(x

)minusf(x

lowast)(ＦＩＡ)


BSABSAISA

10minus12

10minus16

10minus8

10minus1

10minus4


f(x

)minusf(x

lowast)(ＦＩＡ)

BSABSAISA


10minus8

10minus4

10minus2

10minus6









BSABSAISA

10minus1

101

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

106

104

102

100

10minus2

10minus4

F(x

)minusF(x

lowast)(ＦＩＡ)










BSABSAISA

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





















BSABSAISA

101

10minus1

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

104

101

10minus2

10minus5

10minus8

F(x

)minusF(x

lowast)(ＦＩＡ)














Appendix



min 119891 (119909) = 5 4sum119894=1

119909119894 minus 5 4sum119894=1

1199092119894 minus 13sum119894=5

119909119894subject to 1198921 (119909) = 21199091 + 21199092 + 11990910 + 11990911 minus 10

le 01198922 (119909) = 21199091 + 21199093 + 11990910 + 11990912 minus 10le 0



(A1)


max 119891 (119909)


radicsum119899119894=1 11989411990921198941003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


119909119894 le 0

1198922 (119909) = 119899sum119894=1

119909119894 minus 75119899 le 0119899 = 20 0 le 119909119894 le 10 119894 = 1 119899

(A2)



119909119894subject to ℎ (119909) = 119899sum

119894=1

1199092119894 = 0119899 = 10 0 le 119909119894 le 1 119894 = 1 119899

(A3)


min 119891 (119909)= 5357854711990923 + 0835689111990911199095+ 372932391199091 minus 40792141



le 01198923 (119909)= 8051249 + 0007131711990921199095+ 0002995511990911199092 + 0002181311990923minus 110 le 0


1198925 (119909)= 9300961 + 0004702611990931199095+ 0001254711990911199093 + 0001908511990931199094minus 25 le 0


78 le 1199091 le 10233 le 1199092 le 4527 le 119909119894 le 45 119894 = 3 4 5

(A4)


min 119891 (119909)= 31199091 + 000000111990931 + 21199092+ (00000023 ) 11990932





0 le 1199091 1199092 le 1200minus 055 le 1199093 1199094 le 055

(A5)




(A6)







(A7)




(A8)





minus 10 le 119909119894 le 10 119894 = 1 2 3 4 5 6 7(A9)





(A10)



minus 1 le 119909119894 le 1 119894 = 1 2(A11)


max 119891 (119909)= (100 minus (1199091 minus 5)



(A12)




(A13)





(B1)



min 119891 (119909)= 06224119909111990931199094 + 17781119909211990923+ 31661119909211199094 + 1984119909211199093


(B2)



1198922 (119909)= (411990922 minus 11990911199092)(12566 (119909211990931 minus 11990941)) +

1(510811990921)minus 1 le 0


(B3)


min 119891 (119909)= 110471119909211199092+ 00481111990931199094 (14 + 1199092)


1198923 (119909) = 1199091 minus 1199094 le 01198924 (119909)= 01047111990921 + 00481111990931199094 (14 + 1199092)minus 5 le 0


(B4)

where

120591 (119909) = radic(1205911015840)2 + 2120591101584012059110158401015840 11990922119877 + (12059110158401015840)21205911015840 = 119875radic211990911199092 120591

10158401015840 = 119872119877119869119872 = 119875(119871 + 11990922 )

119877 = radic119909224 + (1199091 + 11990932 )2

119869 = 2radic211990911199092 [1199092212 + (1199091 + 11990932 )2]120590 (119909) = 6119875119871119909411990923

120575 (119909) = 41198751198713119864119909331199094




(B5)



min 119891 (119909)= 07854119909111990922 (3333311990923 + 1493341199093 minus 430934)minus 15081199091 (11990926 + 11990927) + 74777 (11990936 + 11990937)+ 07854 (119909411990926 + 119909511990927)


1198923 (119909) = 193119909341199092119909461199093 minus 1 le 0

1198924 (119909) = 193119909351199092119909471199093 minus 1 le 0

1198925 (119909) = [(7451199094 (11990921199093))2 + 169 times 106]12

11011990936 minus 1le 0

1198926 (119909) = [(7451199095 (11990921199093))2 + 1575 times 106]128511990937 minus 1


(B6)

where


(B7)



Acknowledgments


References





























































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in








BSABSAISA

10minus1

101

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

106

104

102

100

10minus2

10minus4

F(x

)minusF(x

lowast)(ＦＩＡ)










BSABSAISA

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





















BSABSAISA

101

10minus1

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

104

101

10minus2

10minus5

10minus8

F(x

)minusF(x

lowast)(ＦＩＡ)














Appendix



min 119891 (119909) = 5 4sum119894=1

119909119894 minus 5 4sum119894=1

1199092119894 minus 13sum119894=5

119909119894subject to 1198921 (119909) = 21199091 + 21199092 + 11990910 + 11990911 minus 10

le 01198922 (119909) = 21199091 + 21199093 + 11990910 + 11990912 minus 10le 0



(A1)


max 119891 (119909)


radicsum119899119894=1 11989411990921198941003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


119909119894 le 0

1198922 (119909) = 119899sum119894=1

119909119894 minus 75119899 le 0119899 = 20 0 le 119909119894 le 10 119894 = 1 119899

(A2)



119909119894subject to ℎ (119909) = 119899sum

119894=1

1199092119894 = 0119899 = 10 0 le 119909119894 le 1 119894 = 1 119899

(A3)


min 119891 (119909)= 5357854711990923 + 0835689111990911199095+ 372932391199091 minus 40792141



le 01198923 (119909)= 8051249 + 0007131711990921199095+ 0002995511990911199092 + 0002181311990923minus 110 le 0


1198925 (119909)= 9300961 + 0004702611990931199095+ 0001254711990911199093 + 0001908511990931199094minus 25 le 0


78 le 1199091 le 10233 le 1199092 le 4527 le 119909119894 le 45 119894 = 3 4 5

(A4)


min 119891 (119909)= 31199091 + 000000111990931 + 21199092+ (00000023 ) 11990932





0 le 1199091 1199092 le 1200minus 055 le 1199093 1199094 le 055

(A5)




(A6)







(A7)




(A8)





minus 10 le 119909119894 le 10 119894 = 1 2 3 4 5 6 7(A9)





(A10)



minus 1 le 119909119894 le 1 119894 = 1 2(A11)


max 119891 (119909)= (100 minus (1199091 minus 5)



(A12)




(A13)





(B1)



min 119891 (119909)= 06224119909111990931199094 + 17781119909211990923+ 31661119909211199094 + 1984119909211199093


(B2)



1198922 (119909)= (411990922 minus 11990911199092)(12566 (119909211990931 minus 11990941)) +

1(510811990921)minus 1 le 0


(B3)


min 119891 (119909)= 110471119909211199092+ 00481111990931199094 (14 + 1199092)


1198923 (119909) = 1199091 minus 1199094 le 01198924 (119909)= 01047111990921 + 00481111990931199094 (14 + 1199092)minus 5 le 0


(B4)

where

120591 (119909) = radic(1205911015840)2 + 2120591101584012059110158401015840 11990922119877 + (12059110158401015840)21205911015840 = 119875radic211990911199092 120591

10158401015840 = 119872119877119869119872 = 119875(119871 + 11990922 )

119877 = radic119909224 + (1199091 + 11990932 )2

119869 = 2radic211990911199092 [1199092212 + (1199091 + 11990932 )2]120590 (119909) = 6119875119871119909411990923

120575 (119909) = 41198751198713119864119909331199094




(B5)



min 119891 (119909)= 07854119909111990922 (3333311990923 + 1493341199093 minus 430934)minus 15081199091 (11990926 + 11990927) + 74777 (11990936 + 11990937)+ 07854 (119909411990926 + 119909511990927)


1198923 (119909) = 193119909341199092119909461199093 minus 1 le 0

1198924 (119909) = 193119909351199092119909471199093 minus 1 le 0

1198925 (119909) = [(7451199094 (11990921199093))2 + 169 times 106]12

11011990936 minus 1le 0

1198926 (119909) = [(7451199095 (11990921199093))2 + 1575 times 106]128511990937 minus 1


(B6)

where


(B7)



Acknowledgments


References





























































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in










BSABSAISA

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





















BSABSAISA

101

10minus1

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

104

101

10minus2

10minus5

10minus8

F(x

)minusF(x

lowast)(ＦＩＡ)














Appendix



min 119891 (119909) = 5 4sum119894=1

119909119894 minus 5 4sum119894=1

1199092119894 minus 13sum119894=5

119909119894subject to 1198921 (119909) = 21199091 + 21199092 + 11990910 + 11990911 minus 10

le 01198922 (119909) = 21199091 + 21199093 + 11990910 + 11990912 minus 10le 0



(A1)


max 119891 (119909)


radicsum119899119894=1 11989411990921198941003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


119909119894 le 0

1198922 (119909) = 119899sum119894=1

119909119894 minus 75119899 le 0119899 = 20 0 le 119909119894 le 10 119894 = 1 119899

(A2)



119909119894subject to ℎ (119909) = 119899sum

119894=1

1199092119894 = 0119899 = 10 0 le 119909119894 le 1 119894 = 1 119899

(A3)


min 119891 (119909)= 5357854711990923 + 0835689111990911199095+ 372932391199091 minus 40792141



le 01198923 (119909)= 8051249 + 0007131711990921199095+ 0002995511990911199092 + 0002181311990923minus 110 le 0


1198925 (119909)= 9300961 + 0004702611990931199095+ 0001254711990911199093 + 0001908511990931199094minus 25 le 0


78 le 1199091 le 10233 le 1199092 le 4527 le 119909119894 le 45 119894 = 3 4 5

(A4)


min 119891 (119909)= 31199091 + 000000111990931 + 21199092+ (00000023 ) 11990932





0 le 1199091 1199092 le 1200minus 055 le 1199093 1199094 le 055

(A5)




(A6)







(A7)




(A8)





minus 10 le 119909119894 le 10 119894 = 1 2 3 4 5 6 7(A9)





(A10)



minus 1 le 119909119894 le 1 119894 = 1 2(A11)


max 119891 (119909)= (100 minus (1199091 minus 5)



(A12)




(A13)





(B1)



min 119891 (119909)= 06224119909111990931199094 + 17781119909211990923+ 31661119909211199094 + 1984119909211199093


(B2)



1198922 (119909)= (411990922 minus 11990911199092)(12566 (119909211990931 minus 11990941)) +

1(510811990921)minus 1 le 0


(B3)


min 119891 (119909)= 110471119909211199092+ 00481111990931199094 (14 + 1199092)


1198923 (119909) = 1199091 minus 1199094 le 01198924 (119909)= 01047111990921 + 00481111990931199094 (14 + 1199092)minus 5 le 0


(B4)

where

120591 (119909) = radic(1205911015840)2 + 2120591101584012059110158401015840 11990922119877 + (12059110158401015840)21205911015840 = 119875radic211990911199092 120591

10158401015840 = 119872119877119869119872 = 119875(119871 + 11990922 )

119877 = radic119909224 + (1199091 + 11990932 )2

119869 = 2radic211990911199092 [1199092212 + (1199091 + 11990932 )2]120590 (119909) = 6119875119871119909411990923

120575 (119909) = 41198751198713119864119909331199094




(B5)



min 119891 (119909)= 07854119909111990922 (3333311990923 + 1493341199093 minus 430934)minus 15081199091 (11990926 + 11990927) + 74777 (11990936 + 11990937)+ 07854 (119909411990926 + 119909511990927)


1198923 (119909) = 193119909341199092119909461199093 minus 1 le 0

1198924 (119909) = 193119909351199092119909471199093 minus 1 le 0

1198925 (119909) = [(7451199094 (11990921199093))2 + 169 times 106]12

11011990936 minus 1le 0

1198926 (119909) = [(7451199095 (11990921199093))2 + 1575 times 106]128511990937 minus 1


(B6)

where


(B7)



Acknowledgments


References





























































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in




















BSABSAISA

101

10minus1

10minus3

10minus5

10minus7

F(x

)minusF(x

lowast)(ＦＩＡ)





BSABSAISA

104

101

10minus2

10minus5

10minus8

F(x

)minusF(x

lowast)(ＦＩＡ)














Appendix



min 119891 (119909) = 5 4sum119894=1

119909119894 minus 5 4sum119894=1

1199092119894 minus 13sum119894=5

119909119894subject to 1198921 (119909) = 21199091 + 21199092 + 11990910 + 11990911 minus 10

le 01198922 (119909) = 21199091 + 21199093 + 11990910 + 11990912 minus 10le 0



(A1)


max 119891 (119909)


radicsum119899119894=1 11989411990921198941003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


119909119894 le 0

1198922 (119909) = 119899sum119894=1

119909119894 minus 75119899 le 0119899 = 20 0 le 119909119894 le 10 119894 = 1 119899

(A2)



119909119894subject to ℎ (119909) = 119899sum

119894=1

1199092119894 = 0119899 = 10 0 le 119909119894 le 1 119894 = 1 119899

(A3)


min 119891 (119909)= 5357854711990923 + 0835689111990911199095+ 372932391199091 minus 40792141



le 01198923 (119909)= 8051249 + 0007131711990921199095+ 0002995511990911199092 + 0002181311990923minus 110 le 0


1198925 (119909)= 9300961 + 0004702611990931199095+ 0001254711990911199093 + 0001908511990931199094minus 25 le 0


78 le 1199091 le 10233 le 1199092 le 4527 le 119909119894 le 45 119894 = 3 4 5

(A4)


min 119891 (119909)= 31199091 + 000000111990931 + 21199092+ (00000023 ) 11990932





0 le 1199091 1199092 le 1200minus 055 le 1199093 1199094 le 055

(A5)




(A6)







(A7)




(A8)





minus 10 le 119909119894 le 10 119894 = 1 2 3 4 5 6 7(A9)





(A10)



minus 1 le 119909119894 le 1 119894 = 1 2(A11)


max 119891 (119909)= (100 minus (1199091 minus 5)



(A12)




(A13)





(B1)



min 119891 (119909)= 06224119909111990931199094 + 17781119909211990923+ 31661119909211199094 + 1984119909211199093


(B2)



1198922 (119909)= (411990922 minus 11990911199092)(12566 (119909211990931 minus 11990941)) +

1(510811990921)minus 1 le 0


(B3)


min 119891 (119909)= 110471119909211199092+ 00481111990931199094 (14 + 1199092)


1198923 (119909) = 1199091 minus 1199094 le 01198924 (119909)= 01047111990921 + 00481111990931199094 (14 + 1199092)minus 5 le 0


(B4)

where

120591 (119909) = radic(1205911015840)2 + 2120591101584012059110158401015840 11990922119877 + (12059110158401015840)21205911015840 = 119875radic211990911199092 120591

10158401015840 = 119872119877119869119872 = 119875(119871 + 11990922 )

119877 = radic119909224 + (1199091 + 11990932 )2

119869 = 2radic211990911199092 [1199092212 + (1199091 + 11990932 )2]120590 (119909) = 6119875119871119909411990923

120575 (119909) = 41198751198713119864119909331199094




(B5)



min 119891 (119909)= 07854119909111990922 (3333311990923 + 1493341199093 minus 430934)minus 15081199091 (11990926 + 11990927) + 74777 (11990936 + 11990937)+ 07854 (119909411990926 + 119909511990927)


1198923 (119909) = 193119909341199092119909461199093 minus 1 le 0

1198924 (119909) = 193119909351199092119909471199093 minus 1 le 0

1198925 (119909) = [(7451199094 (11990921199093))2 + 169 times 106]12

11011990936 minus 1le 0

1198926 (119909) = [(7451199095 (11990921199093))2 + 1575 times 106]128511990937 minus 1


(B6)

where


(B7)



Acknowledgments


References





























































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in














Appendix



min 119891 (119909) = 5 4sum119894=1

119909119894 minus 5 4sum119894=1

1199092119894 minus 13sum119894=5

119909119894subject to 1198921 (119909) = 21199091 + 21199092 + 11990910 + 11990911 minus 10

le 01198922 (119909) = 21199091 + 21199093 + 11990910 + 11990912 minus 10le 0



(A1)


max 119891 (119909)


radicsum119899119894=1 11989411990921198941003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


119909119894 le 0

1198922 (119909) = 119899sum119894=1

119909119894 minus 75119899 le 0119899 = 20 0 le 119909119894 le 10 119894 = 1 119899

(A2)



119909119894subject to ℎ (119909) = 119899sum

119894=1

1199092119894 = 0119899 = 10 0 le 119909119894 le 1 119894 = 1 119899

(A3)


min 119891 (119909)= 5357854711990923 + 0835689111990911199095+ 372932391199091 minus 40792141



le 01198923 (119909)= 8051249 + 0007131711990921199095+ 0002995511990911199092 + 0002181311990923minus 110 le 0


1198925 (119909)= 9300961 + 0004702611990931199095+ 0001254711990911199093 + 0001908511990931199094minus 25 le 0


78 le 1199091 le 10233 le 1199092 le 4527 le 119909119894 le 45 119894 = 3 4 5

(A4)


min 119891 (119909)= 31199091 + 000000111990931 + 21199092+ (00000023 ) 11990932





0 le 1199091 1199092 le 1200minus 055 le 1199093 1199094 le 055

(A5)




(A6)







(A7)




(A8)





minus 10 le 119909119894 le 10 119894 = 1 2 3 4 5 6 7(A9)





(A10)



minus 1 le 119909119894 le 1 119894 = 1 2(A11)


max 119891 (119909)= (100 minus (1199091 minus 5)



(A12)




(A13)





(B1)



min 119891 (119909)= 06224119909111990931199094 + 17781119909211990923+ 31661119909211199094 + 1984119909211199093


(B2)



1198922 (119909)= (411990922 minus 11990911199092)(12566 (119909211990931 minus 11990941)) +

1(510811990921)minus 1 le 0


(B3)


min 119891 (119909)= 110471119909211199092+ 00481111990931199094 (14 + 1199092)


1198923 (119909) = 1199091 minus 1199094 le 01198924 (119909)= 01047111990921 + 00481111990931199094 (14 + 1199092)minus 5 le 0


(B4)

where

120591 (119909) = radic(1205911015840)2 + 2120591101584012059110158401015840 11990922119877 + (12059110158401015840)21205911015840 = 119875radic211990911199092 120591

10158401015840 = 119872119877119869119872 = 119875(119871 + 11990922 )

119877 = radic119909224 + (1199091 + 11990932 )2

119869 = 2radic211990911199092 [1199092212 + (1199091 + 11990932 )2]120590 (119909) = 6119875119871119909411990923

120575 (119909) = 41198751198713119864119909331199094




(B5)



min 119891 (119909)= 07854119909111990922 (3333311990923 + 1493341199093 minus 430934)minus 15081199091 (11990926 + 11990927) + 74777 (11990936 + 11990937)+ 07854 (119909411990926 + 119909511990927)


1198923 (119909) = 193119909341199092119909461199093 minus 1 le 0

1198924 (119909) = 193119909351199092119909471199093 minus 1 le 0

1198925 (119909) = [(7451199094 (11990921199093))2 + 169 times 106]12

11011990936 minus 1le 0

1198926 (119909) = [(7451199095 (11990921199093))2 + 1575 times 106]128511990937 minus 1


(B6)

where


(B7)



Acknowledgments


References





























































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in






(A1)


max 119891 (119909)


radicsum119899119894=1 11989411990921198941003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


119909119894 le 0

1198922 (119909) = 119899sum119894=1

119909119894 minus 75119899 le 0119899 = 20 0 le 119909119894 le 10 119894 = 1 119899

(A2)



119909119894subject to ℎ (119909) = 119899sum

119894=1

1199092119894 = 0119899 = 10 0 le 119909119894 le 1 119894 = 1 119899

(A3)


min 119891 (119909)= 5357854711990923 + 0835689111990911199095+ 372932391199091 minus 40792141



le 01198923 (119909)= 8051249 + 0007131711990921199095+ 0002995511990911199092 + 0002181311990923minus 110 le 0


1198925 (119909)= 9300961 + 0004702611990931199095+ 0001254711990911199093 + 0001908511990931199094minus 25 le 0


78 le 1199091 le 10233 le 1199092 le 4527 le 119909119894 le 45 119894 = 3 4 5

(A4)


min 119891 (119909)= 31199091 + 000000111990931 + 21199092+ (00000023 ) 11990932





0 le 1199091 1199092 le 1200minus 055 le 1199093 1199094 le 055

(A5)




(A6)







(A7)




(A8)





minus 10 le 119909119894 le 10 119894 = 1 2 3 4 5 6 7(A9)





(A10)



minus 1 le 119909119894 le 1 119894 = 1 2(A11)


max 119891 (119909)= (100 minus (1199091 minus 5)



(A12)




(A13)





(B1)



min 119891 (119909)= 06224119909111990931199094 + 17781119909211990923+ 31661119909211199094 + 1984119909211199093


(B2)



1198922 (119909)= (411990922 minus 11990911199092)(12566 (119909211990931 minus 11990941)) +

1(510811990921)minus 1 le 0


(B3)


min 119891 (119909)= 110471119909211199092+ 00481111990931199094 (14 + 1199092)


1198923 (119909) = 1199091 minus 1199094 le 01198924 (119909)= 01047111990921 + 00481111990931199094 (14 + 1199092)minus 5 le 0


(B4)

where

120591 (119909) = radic(1205911015840)2 + 2120591101584012059110158401015840 11990922119877 + (12059110158401015840)21205911015840 = 119875radic211990911199092 120591

10158401015840 = 119872119877119869119872 = 119875(119871 + 11990922 )

119877 = radic119909224 + (1199091 + 11990932 )2

119869 = 2radic211990911199092 [1199092212 + (1199091 + 11990932 )2]120590 (119909) = 6119875119871119909411990923

120575 (119909) = 41198751198713119864119909331199094




(B5)



min 119891 (119909)= 07854119909111990922 (3333311990923 + 1493341199093 minus 430934)minus 15081199091 (11990926 + 11990927) + 74777 (11990936 + 11990937)+ 07854 (119909411990926 + 119909511990927)


1198923 (119909) = 193119909341199092119909461199093 minus 1 le 0

1198924 (119909) = 193119909351199092119909471199093 minus 1 le 0

1198925 (119909) = [(7451199094 (11990921199093))2 + 169 times 106]12

11011990936 minus 1le 0

1198926 (119909) = [(7451199095 (11990921199093))2 + 1575 times 106]128511990937 minus 1


(B6)

where


(B7)



Acknowledgments


References





























































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in








0 le 1199091 1199092 le 1200minus 055 le 1199093 1199094 le 055

(A5)




(A6)







(A7)




(A8)





minus 10 le 119909119894 le 10 119894 = 1 2 3 4 5 6 7(A9)





(A10)



minus 1 le 119909119894 le 1 119894 = 1 2(A11)


max 119891 (119909)= (100 minus (1199091 minus 5)



(A12)




(A13)





(B1)



min 119891 (119909)= 06224119909111990931199094 + 17781119909211990923+ 31661119909211199094 + 1984119909211199093


(B2)



1198922 (119909)= (411990922 minus 11990911199092)(12566 (119909211990931 minus 11990941)) +

1(510811990921)minus 1 le 0


(B3)


min 119891 (119909)= 110471119909211199092+ 00481111990931199094 (14 + 1199092)


1198923 (119909) = 1199091 minus 1199094 le 01198924 (119909)= 01047111990921 + 00481111990931199094 (14 + 1199092)minus 5 le 0


(B4)

where

120591 (119909) = radic(1205911015840)2 + 2120591101584012059110158401015840 11990922119877 + (12059110158401015840)21205911015840 = 119875radic211990911199092 120591

10158401015840 = 119872119877119869119872 = 119875(119871 + 11990922 )

119877 = radic119909224 + (1199091 + 11990932 )2

119869 = 2radic211990911199092 [1199092212 + (1199091 + 11990932 )2]120590 (119909) = 6119875119871119909411990923

120575 (119909) = 41198751198713119864119909331199094




(B5)



min 119891 (119909)= 07854119909111990922 (3333311990923 + 1493341199093 minus 430934)minus 15081199091 (11990926 + 11990927) + 74777 (11990936 + 11990937)+ 07854 (119909411990926 + 119909511990927)


1198923 (119909) = 193119909341199092119909461199093 minus 1 le 0

1198924 (119909) = 193119909351199092119909471199093 minus 1 le 0

1198925 (119909) = [(7451199094 (11990921199093))2 + 169 times 106]12

11011990936 minus 1le 0

1198926 (119909) = [(7451199095 (11990921199093))2 + 1575 times 106]128511990937 minus 1


(B6)

where


(B7)



Acknowledgments


References





























































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in






minus 10 le 119909119894 le 10 119894 = 1 2 3 4 5 6 7(A9)





(A10)



minus 1 le 119909119894 le 1 119894 = 1 2(A11)


max 119891 (119909)= (100 minus (1199091 minus 5)



(A12)




(A13)





(B1)



min 119891 (119909)= 06224119909111990931199094 + 17781119909211990923+ 31661119909211199094 + 1984119909211199093


(B2)



1198922 (119909)= (411990922 minus 11990911199092)(12566 (119909211990931 minus 11990941)) +

1(510811990921)minus 1 le 0


(B3)


min 119891 (119909)= 110471119909211199092+ 00481111990931199094 (14 + 1199092)


1198923 (119909) = 1199091 minus 1199094 le 01198924 (119909)= 01047111990921 + 00481111990931199094 (14 + 1199092)minus 5 le 0


(B4)

where

120591 (119909) = radic(1205911015840)2 + 2120591101584012059110158401015840 11990922119877 + (12059110158401015840)21205911015840 = 119875radic211990911199092 120591

10158401015840 = 119872119877119869119872 = 119875(119871 + 11990922 )

119877 = radic119909224 + (1199091 + 11990932 )2

119869 = 2radic211990911199092 [1199092212 + (1199091 + 11990932 )2]120590 (119909) = 6119875119871119909411990923

120575 (119909) = 41198751198713119864119909331199094




(B5)



min 119891 (119909)= 07854119909111990922 (3333311990923 + 1493341199093 minus 430934)minus 15081199091 (11990926 + 11990927) + 74777 (11990936 + 11990937)+ 07854 (119909411990926 + 119909511990927)


1198923 (119909) = 193119909341199092119909461199093 minus 1 le 0

1198924 (119909) = 193119909351199092119909471199093 minus 1 le 0

1198925 (119909) = [(7451199094 (11990921199093))2 + 169 times 106]12

11011990936 minus 1le 0

1198926 (119909) = [(7451199095 (11990921199093))2 + 1575 times 106]128511990937 minus 1


(B6)

where


(B7)



Acknowledgments


References





























































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in






min 119891 (119909)= 06224119909111990931199094 + 17781119909211990923+ 31661119909211199094 + 1984119909211199093


(B2)



1198922 (119909)= (411990922 minus 11990911199092)(12566 (119909211990931 minus 11990941)) +

1(510811990921)minus 1 le 0


(B3)


min 119891 (119909)= 110471119909211199092+ 00481111990931199094 (14 + 1199092)


1198923 (119909) = 1199091 minus 1199094 le 01198924 (119909)= 01047111990921 + 00481111990931199094 (14 + 1199092)minus 5 le 0


(B4)

where

120591 (119909) = radic(1205911015840)2 + 2120591101584012059110158401015840 11990922119877 + (12059110158401015840)21205911015840 = 119875radic211990911199092 120591

10158401015840 = 119872119877119869119872 = 119875(119871 + 11990922 )

119877 = radic119909224 + (1199091 + 11990932 )2

119869 = 2radic211990911199092 [1199092212 + (1199091 + 11990932 )2]120590 (119909) = 6119875119871119909411990923

120575 (119909) = 41198751198713119864119909331199094




(B5)



min 119891 (119909)= 07854119909111990922 (3333311990923 + 1493341199093 minus 430934)minus 15081199091 (11990926 + 11990927) + 74777 (11990936 + 11990937)+ 07854 (119909411990926 + 119909511990927)


1198923 (119909) = 193119909341199092119909461199093 minus 1 le 0

1198924 (119909) = 193119909351199092119909471199093 minus 1 le 0

1198925 (119909) = [(7451199094 (11990921199093))2 + 169 times 106]12

11011990936 minus 1le 0

1198926 (119909) = [(7451199095 (11990921199093))2 + 1575 times 106]128511990937 minus 1


(B6)

where


(B7)



Acknowledgments


References





























































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in






min 119891 (119909)= 07854119909111990922 (3333311990923 + 1493341199093 minus 430934)minus 15081199091 (11990926 + 11990927) + 74777 (11990936 + 11990937)+ 07854 (119909411990926 + 119909511990927)


1198923 (119909) = 193119909341199092119909461199093 minus 1 le 0

1198924 (119909) = 193119909351199092119909471199093 minus 1 le 0

1198925 (119909) = [(7451199094 (11990921199093))2 + 169 times 106]12

11011990936 minus 1le 0

1198926 (119909) = [(7451199095 (11990921199093))2 + 1575 times 106]128511990937 minus 1


(B6)

where


(B7)



Acknowledgments


References





























































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in

















































































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in




modified backtracking search optimization algorithm...

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