Research Article Lévy-Flight Moth-Flame Algorithm for...

Research ArticleLeacutevy-Flight Moth-Flame Algorithm for Function Optimizationand Engineering Design Problems

Zhiming Li1 Yongquan Zhou12 Sen Zhang1 and Junmin Song1

1College of Information Science and Engineering Guangxi University for Nationalities Nanning 530006 China2Key Laboratory of Guangxi High Schools Complex System and Computational Intelligence Nanning 530006 China

Correspondence should be addressed to Yongquan Zhou yongquanzhou126com

Received 18 April 2016 Accepted 12 July 2016

Academic Editor Jose J Munoz

Copyright copy 2016 Zhiming Li 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 moth-flame optimization (MFO) algorithm is a novel nature-inspired heuristic paradigm The main inspiration of thisalgorithm is the navigation method of moths in nature called transverse orientation Moths fly in night by maintaining a fixedangle with respect to the moon a very effective mechanism for travelling in a straight line for long distances However these fancyinsects are trapped in a spiral path around artificial lights Aiming at the phenomenon that MFO algorithm has slow convergenceand low precision an improved version of MFO algorithm based on Levy-flight strategy which is named as LMFO is proposedLevy-flight can increase the diversity of the population against premature convergence and make the algorithm jump out of localoptimummore effectivelyThis approach is helpful to obtain a better trade-off between exploration and exploitation ability ofMFOthus which canmake LMFO faster andmore robust thanMFO And a comparison with ABC BA GGSA DA PSOGSA andMFOon 19 unconstrained benchmark functions and 2 constrained engineering design problems is tested These results demonstrate thesuperior performance of LMFO

1 Introduction

Optimization is a process of finding the best possible solu-tion(s) for a given problem In real world many problemscan be viewed as optimization problems Since the complex-ity of problems increases the need for new optimizationtechniques becomes more evident than before Over thepast several decades some kinds of methods have beenproposed to solve optimization problems and have madegreat progress For examplemathematical optimization tech-niques used to be the only tool for optimizing problemsbefore the proposal of heuristic optimization techniquesHowever these methods need to know the property ofoptimization problem such as continuity or differentiabilityIn recent years metaheuristic optimization algorithms havebecome more and more popular in optimization techniquesSome popular algorithms in this field are Genetic Algo-rithms (GA) [1 2] Particle Swarm Optimization (PSO) [3]Ant Colony Optimization (ACO) [4] Evolutionary Strategy(ES) [5] Differential Evolution (DE) [6] and EvolutionaryProgramming (EP) [7] The application of these algorithms

can be found in different branches of science and industryas well Despite the merits of these optimizers there is afundamental question here whether there is any optimizerfor solving all optimization problems According to the No-Free-Lunch (NFL) theorem [8] for optimization researchersare allowed to develop new algorithms solving optimizationproblems more effectively Some of the latest algorithmsare Artificial Bee Colony (ABC) algorithm [9] Bat Algo-rithm (BA) [10] Cuckoo Search (CS) algorithm [11] CuckooOptimization Algorithm (COA) [12] Gravitational SearchAlgorithm (GSA) [13] Charged System Search (CSS) [14]Firefly Algorithm (FA) [15] and Ray Optimization (RO) [16]and Dragonfly Algorithm (DA) [17]

Moth-flame optimization (MFO) [18] algorithm is a newmetaheuristic optimization method through imitating thenavigation method of moths in nature called transverseorientation In this algorithm moths and flames are bothsolutions The inventor of this algorithm Seyedali Mirjaliliproved that this algorithm is able to show very competitiveresults compared with other state-of-the-art metaheuristicoptimization algorithms However the MFO algorithm has

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 1423930 22 pageshttpdxdoiorg10115520161423930

2 Mathematical Problems in Engineering

been in research stage so far and convergence speed and cal-culation accuracy of this algorithm can be further advancedTo improve the performance of MFO a Levy-flight moth-flame optimization (LMFO) algorithm is proposed

We know that Levy-flight [11 19] has a strong abilityof strengthening global search and overcoming the problemof being trapped in local minima In order to make use ofthe good performance of Levy-flight we propose a Levy-flight moth-flame optimization MFO and Levy-flight havecomplementary advantages so the proposed algorithm canlead to a faster and more robust method The proposedalgorithm is verified on nineteen benchmark functions andtwo engineering problems

The rest of the paper is organized as follows Section 2presents a brief introduction to MFO and Levy-flight Animproved version of MFO algorithm LMFO is proposedin Section 3 The experimental results of test functions andengineering design problem are showed in Sections 4 and 5respectively Results and discussion are provided in Section 6Finally Section 7 concludes the work

2 Related Works

In this section a background about themoth-flame optimiza-tion algorithm and Levy-flight will be provided briefly

21 MFO Algorithm Moth-flame optimization [18] algo-rithm is a new metaheuristic optimization method which isproposed by Seyedali Mirjalili and based on the simulation ofthe behavior of moths for their special navigationmethods innightThey utilize a mechanism called transverse orientationfor navigation In this method a moth flies by maintaininga fixed angle with respect to the moon which is a veryeffective mechanism for travelling long distance in a straightpath because the moon is far away from the moth Thismechanism guarantees that moths fly along straight line innight However we usually observe that moths fly spirallyaround the lights In fact moths are tricked by artificial lightsand show such behaviors Since such light is extremely closeto the moon hence maintaining a similar angle to the lightsource causes a spiral fly path of moths

In theMFO algorithm the set ofmoths is represented in amatrix119872 For all the moths there is an array119874119872 for storingthe corresponding fitness valuesThe second key componentsin the algorithm are flames A matrix 119865 similar to the mothmatrix is considered For the flames it is also assumed thatthere is an array 119874119865 for storing the corresponding fitnessvalues

TheMFOalgorithm is a three-tuple that approximates theglobal optimal of the optimization problems and defined asfollows

MFO = (119868 119875 119879) (1)

119868 is a function that creates a random population of mothsand corresponding fitness values The methodical model ofthis function is as follows

119868 120601 997888rarr 119872119874119872 (2)

The 119875 function which is the main function moves themoths around the search space This function received thematrix of119872 and returns its updated one eventually

119875 119872 997888rarr 119872 (3)

The 119879 function returns true if the termination criterion issatisfied and false if the termination criterion is not satisfied

119879 119872 997888rarr true false (4)

With 119868 119875 and 119879 the general framework of the MFOalgorithm is defined as follows

119872 = 119868( )while 119879(119872) is equal to false

119872 = 119875(119872)

end

After the initialization the 119875 function is iteratively rununtil the 119879 function returns true For the sake of simulatingthe behavior of moths mathematically the position of eachmoth is updated with respect to a flame using the followingequation

119872119894= 119878 (119872

119894 119865119895) (5)

where 119872119894indicate the 119894th moth 119865

119895indicates the 119895th flame

and 119878 is the spiral functionAny types of spiral can be utilized here subject to the

following conditions

(1) Spiralrsquos initial point should start from the moth(2) Spiralrsquos final point should be the position of the flame(3) Fluctuation of the range of spiral should not exceed

the search space

Considering these points a logarithmic spiral is definedfor the MFO algorithm as follows

119878 (119872119894 119865119895) = 119863

119894sdot 119890119887119905sdot cos (2120587119905) + 119865

119895 (6)

where 119863119894indicates the distance of the 119894th moth for the 119895th

flame 119887 is a constant for defining the shape of the logarithmicspiral and 119905 is a random number in [minus1 1]

119863 is calculated as follows

119863119894=10038161003816100381610038161003816119865119895minus119872119894

10038161003816100381610038161003816 (7)



and119863119894indicates the distance of the 119894thmoth for the 119895th flame

Equation (6) describes the spiral flying path of mothsFrom this equation the next position of a moth is definedwith respect to a flameThe 119905 parameter in the spiral equationdefines how much the next position of the moth should beclose to the flame (119905 = minus1 is the closest position to the flamewhile 119905 = 1 shows the farthest)

A question thatmay rise here is that the position updatingin (6) only requires the moths to move towards a flame yet

Mathematical Problems in Engineering 3

(1) Initialize the position of moths(2) While (Iteration lt= Max iteration)(3) Update flame no using (8)(4) 119874119872 = FitnessFunction(119872)(5) if iteration = = 1(6) 119865 = sort(119872)(7) 119874119865 = sort(119874119872)(8) else(9) 119865 = sort(119872

119905minus 1119872

119905)

(10) 119874119865 = sort(119872119905minus1

119872119905)

(11) end(12) for 119894 = 1 119899

(13) for 119895 = 1 119889

(14) Update 119903 and 119905(15) Calculate119863 using (7) with respect to the corresponding moth(16) Update119872(119894 119895) using (5) and (6) with respect to the corresponding moth(17) end(18) end

Algorithm 1 MFO algorithm

it causes the MFO algorithm to be trapped in local optimaquickly In order to prevent this each moth is obliged toupdate its position using only one of the flames in (6)Another concern here is that the position updating of mothswith respect to 119899 different locations in the search space maydegrade the exploitation of the best promising solutions Toresolve this concern an adaptive mechanism provided thenumber of flames The following formula is utilized in thisregard

flame no = round(119873 minus 119897 lowast119873 minus 1

119879) (8)

where 119897 is the current number of iteration119873 is the maximumnumber of flames and 119879 indicates the maximum number ofiterations

The gradual decrement in number of flames balancesexploration and exploitation of the search space After allthe general steps of the 119875 function can be described inAlgorithm 1

As described in Algorithm 1 the 119875 function is executeduntil the 119879 function returns true After termination of the119875 function the best moth is returned as the best obtainedapproximation of the optimum

Note that the Quicksort method is utilized in MFO andthe sortrsquos computational complexity is119874(119899 log 119899) and119874(1198992) inthe best and worst case respectively (where 119899 is the numberof moths)

22 Levy-Flight Levy-flight was originally introduced by theFrench mathematician in 1937 named Paul Levy Levy-flightis a statistical description of motion that extends beyondthe more traditional Brownian motion discovered over onehundred years earlier A diverse range of both natural andartificial phenomena are now being described in terms ofLevy statistics [19]

Generally speaking animals looking for food is ran-dom from one place to another place A large number of

studies have shown that flight behavior of many animalsand insects has demonstrated the typical characteristics ofrandomness However the choice of the direction reliesonly on a mathematical model [20] which is called Levy-flight For instance many studies have shown that flightbehavior of many animals and insects has revealed the typicalcharacteristics of Levy-flight [21ndash24] According to [24] wecan know that fruit flies or Drosophila melanogaster exploretheir landscape utilizing a series of straight flight pathspunctuated by a sudden 90

∘ turn resulting in a Levy-flight-style fitful scale-free pattern Studies on human behaviorsuch as the Jursquohoansi hunter-gatherer foraging patterns [21]also show the typical feature of Levy-flight Pavlyukevich hasused Levy-flight in his research to present and theoreticallyjustify a new stochastic algorithm for global optimizationEven the light can be related to Levy-flight [20] SubsequentlyLevy-flight have been applied to optimization and optimalsearch and preliminary results show its promising capability[22 25]

3 The Proposed LMFO Approach

In order to increase the diversity of population against pre-mature convergence and accelerate the convergence speedthis paper proposes an improved Levy-flight moth-flameoptimization (LMFO) algorithm Levy-flight has the promi-nent properties to increase the diversity of populationsequentially which can make the algorithm effectively jumpout of the local optimum In other words this approach isbeneficial to obtain a better trade-off between the explorationand exploitation ability ofMFO So we let eachmoth performonce Levy-flight using (9) after the position updating whichis formulated as follows [11 26]

119883119905+1

119894= 119883119905

119894+ 119906 sign [rand minus 05] oplus Levy (120573) (9)


(1) Initialize the position of moths(2) While (Iteration lt= Max iteration)(3) Update flame no using (8)(4) 119874119872 = FitnessFunction(119872)(5) if iteration == 1(6) 119865 = sort(119872)(7) 119874119865 = sort(119874119872)(8) else(9) 119865 = sort(119872

119905minus 1119872

119905)

(10) 119874119865 = sort(119872119905minus1

119872119905)

(11) end(12) for 119894 = 1 119899

(13) for 119895 = 1 119889

(14) Update 119903 and 119905(15) Calculate119863 using (7) with respect to the corresponding moth(16) Update119872(119894 119895) using (5) and (6) with respect to the corresponding moth(17) end(18) for each search agent(19) Update the position of the current search agent using Levy-flight(20) end(21) Iteration = Iteration + 1(22) end

Algorithm 2 LMFO algorithm

where 119883119905119894is the 119894th moth or solution vector 119883

119894at iteration

119905 119906 is a random parameter which conforms to a uniformdistribution oplus is the dot product (entrywisemultiplications)and rand is a random number in [0 1] It should be notedhere that sign[rand minus 05] takes only three values 1 0 andminus1 And in (9) the combination of 119906 sign[rand minus 05] andLevy-flight can make moth walk more random That is tosay to get rid of local minima and improve global searchcapability are ensured via this combination in the basicMFOLevy-flight are a kind of random walk in which the steps aredetermined by the step lengths and the jumps conform to aLevy distribution as follows [11 27]

Levy (120573) sim 120583 = 119905minus1minus120573

(0 le 120573 le 2) (10)

Formula (11) is calculated as Levy random numbers

Levy (120573) sim120601 times 120583

|]|1120573 (11)

where 120583 and ] are both standard normal distributions Γ isa standard Gamma function 120573 = 15 and 120601 is defined asfollows

120601 = [Γ (1 + 120573) times sin (120587 times 1205732)

Γ (((1 + 120573) 2) times 120573 times 2(120573minus1)2)]

1120573

(12)

To sumup global search ability of the proposed algorithmis strengthened using random walk with Levy-flight to elim-inate the weakness of MFO its being trapped in local mini-mum is prevented and it is observed to give more successfulresults particularly for unimodal andmultimodal benchmarkfunctions Because of these features the proposed algorithm

has potential to provide superior performance compared toMFO In following section all kinds of benchmark functionsare hired to verify the effectiveness of the proposed algorithmThe main steps of Levy-flight moth-flame optimization canbe simply presented in Algorithm 2

4 Simulation Experiments

41 Simulation Platform All the algorithms are tested inMATLAB R2012a (714) and numerical experiment is set upon Intel Core (TM) i5-4590 Processor 330GHz 4GB RAMrunning on Windows 7

42 Benchmark Functions It is common in this field tobenchmark the performance of algorithm on a set of math-ematical functions with known global optimal The sameprocess is followed in which nineteen standard benchmarkfunctions are employed from the literature [27 28] as testbeds for comparison Three groups of benchmark functionswith different characteristics are selected to benchmark theperformance of the LMFO algorithm from different perspec-tives As shown in Tables 1ndash3 these benchmark functions aredivided into three groups unimodal functions multimodalfunctions and fixed-dimension multimodal functions Astheir names imply unimodal functions are suitable for bench-marking the exploitation and convergence of an algorithmsince they have one global optimum and no local optimaIn contrary multimodal functions have more than one opti-mum which makes them more challenging than unimodalfunctions One of the optima is called global optimum andthe rest are called local optima An algorithm should avoidall the local optima to approach and approximate the global


Table 1 Unimodal benchmark functions

Name Function Range Dim 119891min

Sphere 1198911(119909) =

119899

sum

119894=1

1199092

119894[minus100 100] 200 0

Schwefelrsquos 222 1198912(119909) =

119899

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119899

prod

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 [minus10 10] 200 0

Schwefelrsquos 12 1198913(119909) =

119899

sum

119894=1

(

119894

sum

119895=1

119909119895)

2

[minus100 100] 200 0

Schwefelrsquos 221 1198914(119909) = max

119894

10038161003816100381610038161199091198941003816100381610038161003816 1 le 119894 le 119899 [minus100 100] 200 0

Rosenbrock 1198915(119909) =

119899minus1

sum

119894=1

[100 (119909119894+1

minus 1199092

119894)2

+ (119909119894minus 1)2

] [minus30 30] 200 0

Step 1198916(119909) =

119899

sum

119894=1

(lfloor119909119894+ 05rfloor)

2

[minus100 100] 200 0

Quartic 1198917(119909) =

119899

sum

119894=1

1198941199094

119894+ random(0 1) [minus128 128] 200 0

XSYang-7 1198918(119909) =

119899

sum

119894=1

120576119895

1003816100381610038161003816100381610038161003816119909119894minus1

119894

1003816100381610038161003816100381610038161003816 120576119895isin ⋃[0 1] [minus5 5] 200 0

optimumTherefore exploration and local optima avoidanceof algorithms can be benchmarked by multimodal functionsThe mathematical formulation of the employed benchmarkfunctions is presented in Tables 1 2 and 3 respectivelyIn these three tables Range represent the boundary ofthe functionrsquos search space Dim denotes the dimension ofthe function and 119891min is the theoretical minimum of thefunction

Heuristic algorithms are stochastic optimization tech-niques and therefore they have to be run more than 10times for generating meaningful statistical results The bestobtained solution in the last iteration is calculated as themetrics of performance The same method is selected togenerate and report the results over 30 independent runsHowever average and standard deviation only compare theoverall performance of algorithms

To explore the performance of the proposed LMFOalgorithm some of the recent and well-known algorithmsin the literature are chosen ABC [9] BA [10] GGSA [29]DA [17] PSOGSA [30] and MFO [18] Note that 30 numbersearch agents and 1000 iterations are utilized for each of thealgorithms It should be noted that selection of the numberof moths (or other candidate solutions in other algorithms)should be done experimentally

In this paper Best Mean Worst and Std represent theoptimal fitness value mean fitness value worst fitness valueand standard deviation respectively Experimental results arelisted in Tables 4 5 and 6The best results are denoted in boldtype

Due to the stochastic nature of the algorithms statisticaltests should be conducted to confirm the significance ofthe results [31] The averages and standard deviation onlycompare the overall performance of the algorithms whilea statistical test considers each runrsquos results and proves thatthe results are statistically significant In order to determinewhether the results of LMFO differ from the best results of

ABC BA GGSA DA PSOGSA and MFO in a statisticalmethod a nonparametric test which is known as Wilcoxonrsquosrank-sum test [32 33] is performed at 5 significance levelTables 7 8 and 9 report the 119901 values produced byWilcoxonrsquostest for the pairwise comparison of the best value of sixgroups Such groups are formed by ABC versus LMFOBA versus LMFO GGSA versus LMFO DA versus MFOPSOGSA versus LMFO and MFO versus LMFO In general119901 values lt 005 can be considered as sufficient evidenceagainst the null hypothesis With the statistical test we canmake sure that the results are not generated by chance Thenonparametric Wilcoxon statistical test is conducted and thecalculated 119901 values are reported as metrics of significance aswell Experimental results of 119901 values rank-sum test are listedin Tables 7 8 and 9

43 Unimodal Benchmark Functions The unimodal bench-marks functions have only one globalminimumand there areno local minima for themTherefore these kinds of functionsare very suitable for benchmarking the convergence capabil-ity of algorithms According to the results of Table 4 LMFO isable to provide very competitive results As can be seen fromthis table LMFO outperforms all other algorithms in 119891

1sim1198918

Therefore the proposed algorithm has high performance tofind the global minimum of unimodal benchmark functionsAccording to the119901 values of119891

1sim1198918in Table 7 LMFO achieves

significant improvement in all the unimodal benchmarkfunctions compared to other algorithms Hence this provesthat LMFO has better performance than other algorithms inforging for global optimum solution of unimodal benchmarkfunctions

Figures 1ndash8 illustrate the averaged convergence curvesof all algorithms disposing unimodal benchmark functionsover 30 independent runs It can be noted here that allthe convergence curves in the following subsections are


Table2Multim

odalbenchm

arkfunctio

ns

Nam

eFu

nctio

nRa

nge

Dim

119891min

Rastr

igin

1198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

[minus512512]

200

0

Ackley

11989110(119909)=minus20exp(minus02radic1 119899

119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos(2120587119909119894))+20+119890

[minus3232]

200

0

Grie

wank

11989111=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos(

119909119894

radic119894

)+1

[minus60

060

0]200

0

Penalized

1

11989112(119909)=120587 11989910sin

(1205871199101)+

119899minus1

sum 119894=1

(1199101minus1)2

[1+10sin2(120587119910119894+1)]+(119910119899minus1)2

+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+119909119894+1

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)119898

119909119894gt119886

0minus119886lt119909119894lt119886

119896(minus119909119894minus119886)119898

119909119894ltminus119886

[minus5050]

200

0

Penalized

211989113(119909)=01sin2(31205871199091)+

119899

sum 119894=1

(119909119894minus1)2

[1+sin2(3120587119909119894+1)]+(119909119899minus1)2

[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

[minus5050]

200

0

Alpine

11989114(119909)=

119899

sum 119894=1

119909119894sin

(119909119894)+01119909119894

[minus1010]

200

0

Zakh

arov

11989115(119909)=

119899

sum 119894=1

1199092 119894+(

119899

sum 119894=1

05119894119909119894)

2

+(

119899

sum 119894=1

05119894119909119894)

4

[minus510]

200

0


Table3Fixed-dimensio

nmultim

odalbenchm

arkfunctio

ns

Nam

eFu

nctio

nRa

nge

Dim

119891min

Goldstein-Pric

e11989116(119909)=[1+(1199091+1199092+1)2

(19minus141199091+31199092 1minus141199092+611990911199092+31199092 2)]times[30+(21199091minus31199092)2

times(18minus321199091+121199092 1+481199092minus3611990911199092+271199092 2)]

[minus22]

23

DropWave

11989117(119909)=minus

1+cos(

12radic1199092 1+1199092 2)

(12)(1199092 1+1199092 2)+2

[minus512512]

2minus1

Schafferrsquos

F611989118(119909)=

sin2radic(1199092 1+1199092 2)minus05

[1+0001(1199092 1+1199092 2)]2

[minus100100]

2minus1

Easom

11989119=minuscos(1199091)timescos(1199092)times119890minus(1199091minus120587)2minus(1199092minus120587)2

[minus2lowast

pi2lowastpi]

2minus1


Table 4 Results of unimodal benchmark functions

Benchmark function Result AlgorithmABC BA GGSA DA PSOGSA MFO LMFO

1198911(119863 = 200)

Best 1989678 3657173 1627878 1513435 3483545 1402194 12921E minus 234Worst 3604923 456284 2426284 5251162 1422274 2400007 33309E minus 183Mean 2667704 4148078 1922404 2471682 9473512 1855884 11103E minus 184Std 4640383 2051524 1852355 1104655 2505192 2417767 0

1198912(119863 = 200)

Best 1080735 183E + 84 1497263 2627986 63756 4205512 85E minus 129Worst 1864672 39E + 99 1898577 25579923 806E + 40 7224955 73E minus 99Mean 1402811 145E + 98 1658829 13788594 269E + 39 5601173 24E minus 100Std 145252 706E + 98 9599018 56674292 147E + 40 6166749 13E minus 99

1198913(119863 = 200)


1198914(119863 = 200)

Best 94265 8748997 2044273 270486 7625401 9526014 61E minus 117Worst 9740102 9212282 3130115 5969617 9874303 9815999 618E minus 82Mean 958655 9009865 2712328 4197914 9569831 9703653 206E minus 83Std 0778229 1330154 2551274 780909 590858 0687621 113E minus 82

1198915(119863 = 200)

Best 16111660 88328339 2399953 4717141 2565682 361E + 08 1969121Worst 94041761 159E + 08 5103632 47976603 482E + 08 892E + 08 1986685Mean 41344878 133E + 08 3740872 18257995 135E + 08 614E + 08 1982609Std 17567952 18028179 6431921 10933433 124E + 08 141E + 08 0459128

1198916(119863 = 200)

Best 1716543 3841553 14899 4422437 3366558 1366785 3965709Worst 3074379 4539055 24873 634734 1091236 2285115 4292694Mean 2392299 417903 1922047 2416716 7656132 1806698 4149763Std 3074406 1700178 2314721 124411 1901855 2467119 0632867

1198917(119863 = 200)

Best 5188092 0400252 6492429 923801 1189278 1181991 23E minus 06Worst 2992207 0744656 1477883 1817351 7166514 2686255 0000427Mean 1571676 0551746 9090199 5206488 1983666 1908788 856E minus 05Std 5873575 0072809 1742479 3934587 1047516 3607708 0000108

1198918(119863 = 200)


also averaged curves As may be seen from these curvesLMFO has the fastest convergence speed in all algorithmsFrom Table 4 and Figures 20ndash27 the LMFOrsquos Std is muchsmaller than other algorithms These show that LMFO hasa strong sense of stability and robust comparing from otheralgorithms

44Multimodal Benchmark Functions In contrast to the uni-modal benchmark functions multimodal benchmark func-tions have many local minima with the number increasing

exponentially with dimension This makes them suitable forbenchmarking the exploration ability of an algorithm So thefinal results are more important because these benchmarkfunctions can reflect the ability of the algorithm to escapefrom poor local optima and obtain the global optimum Thestatistical results of the algorithms onmultimodal benchmarkfunctions are presented in Table 5 As the results of BestWorst Mean and Std values show LMFO is also able toprovide very competitive results on the multimodal bench-mark functionsThese results show that the LMFO algorithm


Table 5 Results of multimodal benchmark functions


1198919(119863 = 200)


11989110(119863 = 200)


11989111(119863 = 200)


11989112(119863 = 200)

Best 7393398 386E + 08 2817774 1662384 7043398 703E + 08 888E minus 16Worst 167E + 08 685E + 08 7768577 23172033 205E + 09 192E + 09 888E minus 16Mean 58731093 559E + 08 1104376 2516012 728E + 08 13E + 09 888E minus 16Std 37177591 81914079 2035503 4641204 515E + 08 301E + 08 0

11989113(119863 = 200)

Best 19464968 125E + 09 6467005 2620391 26666517 168E + 09 194713Worst 34E + 08 209E + 09 3319003 183E + 08 247E + 09 404E + 09 1979025Mean 153E + 08 172E + 09 1684844 27183053 973E + 08 262E + 09 1962771Std 82296565 189E + 08 7247329 34759974 707E + 08 546E + 08 0072549

11989114(119863 = 200)


11989115(119863 = 200)


Table 6 Results of fixed-dimension multimodal benchmark functions


11989116(119863 = 2)

Best 3000547 3 3 3 3 3 3Worst 3052848 8400001 30 3179715 84 3 3000357Mean 3015989 156 7144215 3018584 84 3 3000061Std 0015909 2530177 9250619 0049376 2055036 183E minus 15 719E minus 05

11989117(119863 = 2)

Best minus1 minus1 minus1 minus1 minus1 minus1 minus1Worst minus098844 minus036913 minus093625 minus078575 minus093625 minus093625 minus1Mean minus099731 minus071344 minus0949 minus094932 minus09915 minus096175 minus1Std 0003196 0175563 0025938 0053641 0022043 0031767 0

11989118(119863 = 2)


11989119(119863 = 2)

Best minus1 minus1 minus1 minus1 minus1 minus1 minus1Worst minus099991 minus81E minus 05 minus81E minus 05 minus095227 minus1 minus1 minus099918Mean minus099999 minus080002 minus092221 minus099656 minus1 minus1 minus099973Std 203E minus 05 0406805 0257944 0010494 0 0 0000229


Table 7 Results of 119901-values rank-sum test on unimodal benchmark functions

1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918

ABC versus LMFO 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11BA versus LMFO 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11GGSA versus LMFO 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11DA versus LMFO 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11PSOGSA versus LMFO 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11MFO versus LMFO 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11 302E minus 11

Table 8 Results of 119901-values rank-sum test on multimodal benchmark functions

1198919

11989110

11989111

11989112

11989113

11989114

11989115

ABC versus LMFO 121E minus 12 121E minus 12 121E minus 12 302E minus 11 302E minus 11 302E minus 11 302E minus 11BA versus LMFO 121E minus 12 121E minus 12 121E minus 12 302E minus 11 302E minus 11 302E minus 11 302E minus 11GGSA versus LMFO 121E minus 12 121E minus 12 121E minus 12 302E minus 11 302E minus 11 302E minus 11 302E minus 11DA versus LMFO 121E minus 12 121E minus 12 121E minus 12 302E minus 11 302E minus 11 302E minus 11 302E minus 11PSOGSA versus LMFO 121E minus 12 121E minus 12 121E minus 12 302E minus 11 302E minus 11 302E minus 11 302E minus 11MFO versus LMFO 121E minus 12 121E minus 12 121E minus 12 302E minus 11 302E minus 11 302E minus 11 302E minus 11

Table 9 Results of 119901-values rank-sum test on fixed-dimensionmultimodal benchmark functions

11989116

11989117

11989118

11989119

ABC versus LMFO 302E minus 11 121E minus 12 121E minus 12 507E minus 10

BA versus LMFO 729E minus 03 121E minus 12 121E minus 12 677E minus 05

GGSA versus LMFO 770E minus 02 881E minus 10 513E minus 11 248E minus 08

DA versus LMFO 519E minus 02 456E minus 10 145E minus 11 296E minus 06PSOGSA versusLMFO

791E minus 09 418E minus 02 715E minus 13 121E minus 12

MFO versus LMFO 256E minus 11 689E minus 07 117E minus 13 121E minus 12

has merit in terms of exploration According to the 119901 valuesof 1198919sim11989115

reported in Table 8 LMFO achieves significantimprovement on 200-D compared to other algorithmsWhencomparing LMFO and other algorithms we can concludethat LMFO is significantly performing better with six groupsof comparison algorithms The 119901 values of 119891

9sim11989115

reportedin Table 8 are less than 005 which is strong evidence againstnull hypothesis Hence this evidence demonstrates that theresults of LMFO are statistically significant not occurring bycoincidence

Seen from Table 5 and Figures 9ndash15 the convergencerate of LMFO on the multimodal benchmark functions inmajority cases is better than other algorithms On the basisof Table 5 and Figures 9ndash15 we can draw a conclusion thatthe LMFO is able to avoid local minima in multimodalbenchmark functions with a good convergence speed FromTable 5 and Figures 28ndash37 the LMFOrsquos Std is much smaller

0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

times105

ABCBAGGSADA

PSOGSAMFOLMFO

f1

Figure 1 The convergence curves for 1198911

than other algorithms These show that LMFO has a strongsense of stability and robust comparing from other algo-rithms


0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

ABCBAGGSADA

PSOGSAMFOLMFO

10150

10100

1050

100

10minus50

10minus100

f2


0 200 400 600 800 10000

1

2

3

4

5

6

7

8

9

Iteration

Aver

age fi

tnes

s val

ue

times106

ABCBAGGSADA

PSOGSAMFOLMFO

f3


45 Fixed-Dimension Multimodal Benchmark Functions Forfixed-dimensionmultimodal benchmark functions with onlya few localminima the dimensions of themultimodal bench-mark functions are also small Under such circumstances itis difficult to judge the performance of individual algorithmThe major difference compared with multimodal functionsis that fixed-dimension multimodal functions appear to besimpler because of their low dimensions and a smallernumber of local minima In this experiment the resultsof Best Worst Mean and Std values of fixed-dimension

0 200 400 600 800 10000

102030405060708090

100

Iteration

Aver

age fi

tnes

s val

ue

f4

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

05

1

15

2

25

3

Iteration

Aver

age fi

tnes

s val

ue

f5times109

ABCBAGGSADA

PSOGSAMFOLMFO


multimodal benchmark functions are summarized inTable 6For all fixed-dimension multimodal functions LMFO cangive the best solution in terms of Best As Table 6 shows theLMFO algorithm provides the best results on two of fixed-dimension multimodal benchmark functionsThe results arefollowed by the MFO PSOGSA and ABC algorithms Inaddition the 119901 values of Wilcoxonrsquos rank-sum in Table 9show that the result of LMFO in 119891

16is not significantly

better than DA andGGSA algorithms (5 significance level)but it is significantly different compared with ABC MFOPSOGSA and BA In the remaining functions (119891

17 11989118 and


0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

f6times105

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Iteration

Aver

age fi

tnes

s val

ue

f7

ABCBAGGSADA

PSOGSAMFOLMFO


11989119) however the results of LMFO are significantly better

than other algorithms So it can be concluded that the resultsof LMFO in these benchmark functions are better than ABCBA GGSA DA PSOGSA and MFO

In addition the convergence rate of LMFO on the fixed-dimension benchmark functions with 2-dim can be shownin Figures 16ndash19 As can be seen from these figures it canbe claimed that LMFO has the faster convergence rate onfunctions 119891

17and 119891

18 From Figures 35ndash38 we can find that

0 200 400 600 800 10000

50

100

150

200

250

Iteration

Aver

age fi

tnes

s val

ue

f8

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

Iteration

Aver

age fi

tnes

s val

ue

f9

ABCBAGGSADA

PSOGSAMFOLMFO


all of the algorithms have a strong sense of stability except BAon fixed-dimension functions

Overall the results from Tables 4ndash6 Tables 7ndash9 Figures1ndash19 and Figures 20ndash38 show that the proposed method iseffective in not only optimizing unimodal and multimodalfunctions but also optimizing fixed-dimension multimodalfunctions

Since constraints are one of the major challenges insolving real problems and themain objective of designing theLMFO algorithm is to solve real problems two constrainedreal engineering problems are employed in the next section


0 200 400 600 800 10000

5

10

15

20

25

Iteration

Aver

age fi

tnes

s val

ue

f10

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

Iteration

Aver

age fi

tnes

s val

ue

f11

ABCBAGGSADA

PSOGSAMFOLMFO


to further investigate the performance of the MFO algorithmand provide a comprehensive study

5 LMFO for EngineeringOptimization Problems

In this section a set of two engineering problems (weldedbeam design and speed reducer design) is solved so as tofurther testify the performance of the proposed algorithmThere are some inequality constraints in real problems sothe LMFO algorithm should be capable of dealing with them

0 200 400 600 800 10000

1

2

3

4

5

6

7

Iteration

Aver

age fi

tnes

s val

ue

f12times109

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

2

4

6

8

10

12

14

Iteration

Aver

age fi

tnes

s val

ue

f13times109

ABCBAGGSADA

PSOGSAMFOLMFO


during optimization Several methods have been applied tohandle constraints in the literature penalty function specialoperators repaired algorithms separation of objectives andconstraints and hybrid methods [34] In this paper penaltymethod is employed to handle the constraints of welded beamand speed reducer

51 Welded Beam Design The objective is to evaluate theoptimal fabrication cost of a welded beam as shown inFigure 39 [35] The constraints of the beam are shear stress


0 200 400 600 800 10000

100

200

300

400

500

600

Iteration

Aver

age fi

tnes

s val

ue

f14

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

f151020

100

10minus20

10minus40

10minus60

10minus80

10minus100

10minus120

ABCBAGGSADA

PSOGSAMFOLMFO


(120591) bending stress in the beam (120579) buckling load on the bar(119875119888) end deflection of the beam (120575) and side constraintsThis problem has four variables that are thickness of weld

(ℎ) length of attached part of bar (119897) the height of the bar(119905) and thickness of the bar (119887) respectively This problem isformulated as follows

Consider = [1199091 1199092 1199093 1199094] = [ℎ 119897 119905 119887]

Minimize 119891 ()

= 1104711199092

11199092

0 200 400 600 800 10000

10

20

30

40

50

60

70

80

90

100

Iteration

Aver

age fi

tnes

s val

ue

f16

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

minus055

Iteration

Aver

age fi

tnes

s val

ue

f17

ABCBAGGSADA

PSOGSAMFOLMFO


+ 00481111990931199094(140 + 119909

2)

Subject to 1198921 () = 120591 () minus 120591max le 0

1198922 () = 120590 () minus 120590max le 0

1198923 () = 120575 () minus 120575max le 0

1198924 () = 119909

1minus 1199094le 0

1198925 () = 119875 minus 119875

119888 () le 0

1198926 () = 0125 minus 119909

1le 0


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

Iteration

Aver

age fi

tnes

s val

ue

f18

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

Iteration

Aver

age fi

tnes

s val

ue

f19

ABCBAGGSADA

PSOGSAMFOLMFO


1198927 ()

= 1104711199092

1

+ 00481111990931199094(140 + 119909

2) minus 50

le 0

Variable range 01 le 1199091le 2

01 le 1199092le 10

01 le 1199093le 10

01 le 1199094le 2

ABC BA GGSA DA PSOGSA MFO LMFO

0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f1times105

Figure 20 Standard deviation for 1198911


0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f2times1099



0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f3times106




0102030405060708090

100

Algorithms

Fitn

ess v

alue

f4



0

1

2

3

4

5

6

7

8

9

Algorithms

Fitn

ess v

alue

f5times108



005

115

225

335

445

Algorithms

Fitn

ess v

alue

f6times105



0

500

1000

1500

2000

2500

Algorithms

Fitn

ess v

alue

f7


ABC BA GGSA DA PSOGSA MFO LMFO0

20406080

100120140160180200

Algorithms

Fitn

ess v

alue

f8



0

500

1000

1500

2000

Algorithms

Fitn

ess v

alue

f9




0

2

4

6

8

10

12

14

16

18

20

Algorithms

Fitn

ess v

alue

f10



0

1000

2000

3000

4000

5000

Algorithms

Fitn

ess v

alue

f11



002040608

112141618

2

Algorithms

Fitn

ess v

alue

f12times109



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f13times109



0

50

100

150

200

Algorithms

Fitn

ess v

alue

f14



0

2000

4000

6000

8000

10000

Algorithms

Fitn

ess v

alue

f15




0

10

20

30

40

50

60

70

80

Algorithms

Fitn

ess v

alue

f16



minus1

minus09

minus08

minus07

minus06

minus05

minus04

Algorithms

Fitn

ess v

alue

f17


Where 120591 () = radic(1205911015840)2+ 2120591101584012059110158401015840

1199092

2119877+ (12059110158401015840)

2

1205911015840=

119875

radic211990911199092

12059110158401015840=119872119877

119869

119872 = 119875(119871 +1199092

2)

119877 = radic1199092

2

4+ (

1199091+ 1199093

2)

2

119869 = 2radic211990911199092[1199092

2

4+ (

1199091+ 1199093

2)

2

]

120590 () =6119875119871

11990941199092

3

120575 () =61198751198713

1198641199092

31199094

ABC BA GGSA DA PSOGSA MFO LMFOminus1

minus095minus09minus085minus08minus075minus07minus065minus06minus055

Algorithms

Fitn

ess v

alue

f18



minus1minus09minus08minus07minus06minus05minus04minus03minus02minus01

0

Algorithms

Fitn

ess v

alue

f19


119875119888 ()

=

4013119864radic1199092

31199096

436

1198712(1 minus

1199093

2119871

radic119864

4119866)

119875 = 6000119897119887

119871 = 14 in

120575max = 025 in

119864 = 30 times 16 psi

119866 = 12 times 106 psi

120591max = 13600 psi

120590max = 30000 psi(13)

Mirjalili tried to solve this problem using MFO [18]and GGSA [29 36] Coello Coello [37] and Deb [38 39]employed GA whereas Lee and Geem [40] used HS to solvethis problem Richardsonrsquos randommethod simplexmethod


Table 10 Comparison results of the welded beam design problem

Algorithm Optimum variables Optimal cost119867 119897 119905 119861

LMFO 02020 33575 90938 02061 17165MFO [18] 02057 34703 90364 02057 172452GGSA [29 36] 0215917 3314955 8896195 0215917 1770829GA (Coello Coello) [37] NA NA NA NA 18245GA (Deb) [38] NA NA NA NA 23800GA (Deb) [39] 02489 61730 81789 02533 24331HS (Lee and Geem) [40] 02442 62231 82915 02443 23807Random [41] 04575 47313 50853 06600 41185Simplex [41] 02792 56256 77512 02796 25307David [41] 02434 62552 82915 02444 23841APPROX [41] 02444 62189 82915 02444 23815

ht

l

P

L

t

Figure 39 Structure of welded beam design

l1

l2

d1

d2z

Figure 40 Structure of speed reducer design

Davidon-Fletcher-Powell and Griffith and Stewartrsquos succes-sive linear approximation are the mathematical approachesthat have been adopted by Ragsdell and Philips [41] for thisproblem The comparison results of the welded beam designproblem are shown in Table 10

The results of Table 10 show that the LMFO algorithmis able to find the best optimal design compared to otheralgorithms The results of LMFO are closely followed by theMFO and GGSA algorithms

52 Speed Reducer Design The objective function of thisproblem is to minimize the total weight of the speed reduceras illustrated in Figure 40 [42] The variables 119909

1sim1199097denote

the face width (119887) module of teeth (119898) number of teeth inthe pinion (119911) length of the first shaft between bearings (119897

1)

length of the second shaft between bearings (1198972) the diameter

of first (1198891) and second shafts (119889

2) respectively The mathe-

matical formulation of this problem can be summarized asfollows

Minimize 119891 ()

= 0785411990911199092

2(33333119909

2

3+ 149334119909

3minus 430934)

minus 15081199091(1199092

6+ 1199092

7) + 74777 (119909

3

6+ 1199093

7)

+ 07854 (11990941199092

6+ 11990951199092

7)

Subject to 1198921 () =

27

11990911199092

21199093

minus 1 le 0

1198922 () =

3975

11990911199092

21199092

3

minus 1 le 0

1198923 () =

1931199093

4

11990921199094

61199093

minus 1 le 0

1198924 () =

1931199093

5

11990921199095

71199093

minus 1 le 0

1198925 () =

[(745119909411990921199093)2+ 169 times 10

6]12

1101199093

6

minus 1 le 0

1198926 () =

[(745119909511990921199093)2+ 1575 times 10

6]12

851199093

7

minus 1

le 0

1198927 () =

11990921199093

40minus 1 le 0

1198928 () =

51199092

1199091

minus 1 le 0

1198929 () =

1199091

121199092

minus 1 le 0

11989210 () =

151199096+ 19

1199094

minus 1 le 0

11989211 () =

111199096+ 17

1199095

minus 1 le 0

where 26 le 1199091le 36


Table 11 Comparison results of the speed reducer design problem

Algorithm Optimal values for variables Optimum weight119887 (1199091) 119898 (119909

2) 119911 (119909

3) 119897

1(1199094) 119897

2(1199095) 119889

1(1199096) 119889

2(1199097)

LMFO 350411 07 17 73 733342 337164 528916 2994656Akhtar et al [43] 3506122 0700006 17 7549126 785933 3365576 5289773 300808Mezura-Montes et al [44] 3506163 0700831 17 7460181 7962143 33629 53090 3025005CS [11 45] 35015 07 17 76050 78181 33520 52875 3000981HCPS [46] 35 07 17 73 771532 3350215 5286654 299447107SCA [47] 35 07 17 7327602 7715321 3350267 5286655 2994744241(120583 + 120582) ES [5 48] 3499999 0699999 17 73 78 3350215 5286683 2996348094ABC [9 49] 3499999 07 17 73 78 3350215 5287800 2997058412

07 le 1199092le 08

17 le 1199093le 28

73 le 1199094le 83

73 le 1199095le 83

29 le 1199096le 39

50 le 1199097le 55

(14)

This problem has also been popular among researchersand optimized in many studiesThe heuristic algorithms thathave been employed to optimize this problem are Akhtar etal [43] Mezura-Montes et al [44] CS [11 45] HCPS [46]SCA [47] (120583 + 120582) ES [5 48] and ABC [9 49] The results ofthis problem are provided in Table 11 According to this tablethe LMFO and HCPS algorithms can find a design with theminimum weight for this problem

6 Results and Discussion

In this paper an improved version of MFO algorithm basedon Levy-flight strategy which is named as LMFO is pro-posed In order to benchmark the performance of LMFOnineteen unconstrained benchmark functions and two con-strained engineering design problems were conducted

According to the values of BestWorstMean and Std and119901 values in Section 4 the LMFO algorithm significantly out-performs others in terms of numerical optimization Thereare several reasons of why LMFO algorithm did perform wellon most of the test cases First Levy-flight strategy Levy-flight can increase the diversity of the population and makethe algorithm jump out of local optimum more effectivelyThis approach is helpful to make LMFO faster and morerobust than MFO Second update mechanism of moths inthis mechanism moths are required to update their positionswith respect to the best recent feasible flames Thereforethis approach promotes exploration of promising feasibleregions and is the main reason of the superiority of theLMFO algorithm Third Quicksort method is utilized inLMFO algorithmThese are the reasons why LMFOperformsbetter than other algorithms at the end of the results sectionAnother finding in the results is the poor performance ofABC BA and DA These three algorithms belong to the

class of swarm-based algorithms In contrary to evolutionaryalgorithms there is no mechanism for significant abruptmovements in the search space and this is likely to be thereason for the poor performance of ABC BA and DA

As we can see in Section 4 the LMFO has been demon-strated to perform better than or highly competitive withthe other algorithms The advantages of LMFO involveperforming simply and have few parameters to regulate Thework here proves the LMFO to be robust powerful andeffective over all types of benchmark functions Benchmarkevaluating is a good way for testing the performance of themetaheuristic algorithms but it also has some limitationsFor example different tuning parameter values in the opti-mization methods might lead to significant differences intheir performance Also benchmark test may arrive at fullydifferent conclusions if the termination criterion changes Ifwe change the population size or the number of iterations wemight draw a different conclusion

In Section 5 the results show that MFO outperformsother algorithms in themajority of real case studies Since thesearch space of these problems is unknown these results arestrong evidences for the applicability of LMFO in solving realproblems Due to the constrained nature of the case studiesin addition it can be stated that the LMFO algorithm is ableto optimize search spaces with infeasible regions as well Thisis due to the update mechanism of moths in which they arerequired to update their positions with respect to the bestrecent feasible flames Therefore this approach is the mainreason of the superiority of the LMFO algorithm

In our study nineteen benchmark functions have beenapplied to evaluate the performance of LMFO We alsotest our proposed method on the real-world engineeringproblems Moreover we will compare LMFO with otheroptimization algorithms

7 Conclusion and Future Works

Due to the limited performance of MFO Levy-flight strategyhas been introduced into the standard MFO to developa novel Levy-flight moth-flame optimization algorithm foroptimization problems As shown in Section 4 LMFO is veryefficient with an almost exponential convergence rate andthe results were compared to a wide range of algorithms forverification This observation is based on the comparison of


LMFOwith other algorithms to solve optimization problemsThe proposed algorithm proved its superior performanceon nineteen benchmark functions in terms of enhancedconvergence speed and modified avoidance of local minimaThis paper also identified and discussed the reasons forpoor performances of other algorithms It was observed thatthe swarm-based algorithms suffer from low explorationwhereas the LMFO does not

Furthermore this paper also considers solving two classi-cal engineering problems by using the LMFO algorithmThehigh level of exploration and exploitation of this algorithmwere the motivations for this study The comparative resultsin Section 5 show that the LMFO algorithm has high perfor-mance on challenging constrained problems with unknownspaces In this work the LMFO makes an attempt at takingmerits of the MFO and Levy-flight in order to avoid localoptimalWith both techniques combined LMFO can balanceexploration and exploitation and effectively solve complexproblems and real-world engineering problems

For future works two research directions can be rec-ommended Firstly we are going to apply the LMFO tosolve more real-world engineering problems Secondly it isrecommended to develop binary and multiobjective versionsof the MFO algorithm

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Science Foundation ofChina under Grants no 61463007 and 6153008

References

[1] J H Holland ldquoGenetic algorithmsrdquo Scientific American vol267 no 1 pp 66ndash72 1992

[2] J H Holland and J S Reitman ldquoCognitive systems based onadaptive algorithmsrdquo ACM SIGART Bulletin no 63 p 49 1977

[3] R Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 October1995

[4] A ColorniMDorigo andVManiezzo ldquoDistributed optimiza-tion by ant coloniesrdquo in Proceedings of the European Conferenceon Artificial Life pp 134ndash142 Paris France December 1991

[5] I Rechenberg Evolution Strategy NaturersquosWay of OptimizationOptimization Methods and Applications Possibilities and Limi-tations Springer Berlin Germany 1989

[6] 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

[7] X Yao Y Liu and G Lin ldquoEvolutionary programming madefasterrdquo IEEE Transactions on Evolutionary Computation vol 3no 2 pp 82ndash102 1999

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

[9] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[10] X-S Yang ldquoA new metaheuristic bat-inspired AlgorithmrdquoStudies in Computational Intelligence vol 284 pp 65ndash74 2010

[11] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the IEEEWorld Congress on Nature amp BiologicallyInspired Computing (NaBIC rsquo09) pp 210ndash214 CoimbatoreIndia December 2009

[12] R Rajabioun ldquoCuckoo optimization algorithmrdquo Applied SoftComputing vol 11 no 8 pp 5508ndash5518 2011

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

[14] A Kaveh and S Talatahari ldquoA novel heuristic optimizationmethod charged system searchrdquo Acta Mechanica vol 213 no3-4 pp 267ndash289 2010

[15] X S Yang ldquoFirefly algorithmrdquo in Engineering Optimization pp221ndash230 John Wiley amp Sons 2010

[16] A Kaveh and M Khayatazad ldquoA new meta-heuristic methodray optimizationrdquo Computers and Structures vol 112-113 no 4pp 283ndash294 2012

[17] S Mirjalili ldquoDragonfly algorithm a new meta-heuristic opti-mization technique for solving single-objective discrete andmulti-objective problemsrdquo Neural Computing and Applicationsvol 27 no 4 pp 1053ndash1073 2016

[18] S Mirjalili ldquoMoth-flame optimization algorithm a novelnature-inspired heuristic paradigmrdquo Knowledge-Based Systemsvol 89 pp 228ndash249 2015

[19] A F Kamaruzaman A M Zain S M Yusuf and A UdinldquoLevy flight algorithm for optimization problemsmdasha literaturereviewrdquo Applied Mechanics and Materials vol 421 pp 496ndash5012013

[20] P Barthelemy J Bertolotti andD SWiersma ldquoA Levy flight forlightrdquo Nature vol 453 no 7194 pp 495ndash498 2008

[21] C T Brown L S Liebovitch and R Glendon ldquoLevy flights indobe Jursquohoansi foraging patternsrdquo Human Ecology vol 35 no1 pp 129ndash138 2007

[22] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquo Journal of Computational Physics vol 226 no 2 pp1830ndash1844 2007

[23] I Pavlyukevich ldquoCooling down Levy flightsrdquo Journal of PhysicsAMathematical andTheoretical vol 40 no 41 pp 12299ndash123132007

[24] A M Reynolds and M A Frye ldquoFree-flight odor tracking inDrosophila is consistent with an optimal intermittent scale-freesearchrdquo PLoS ONE vol 2 no 4 article e354 2007

[25] M F Shlesinger ldquoMathematical physics search researchrdquoNature vol 443 no 7109 pp 281ndash282 2006

[26] J Xie Y Zhou and H Chen ldquoA novel bat algorithm based ondifferential operator and levy flights trajectoryrdquo ComputationalIntelligence and Neuroscience vol 2013 Article ID 453812 13pages 2013

[27] X S Yang ldquoAppendix a test problems in optimizationrdquo Engi-neering Optimization pp 261ndash266 2010


[28] K Tang X Li P N Suganthan Z Yang and T WeiseBenchmark Functions for the cecrsquo2008 Special Session and Com-petition on Large Scale Global Optimization Nature InspiredComputation and Applications Laboratory 2009

[29] M B Dowlatshahi and H Nezamabadi-Pour ldquoGGSA a group-ing gravitational search algorithm for data clusteringrdquo Engi-neering Applications of Artificial Intelligence vol 36 pp 114ndash1212014

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

[31] 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

[32] J Gibbons and S Chakraborti Nonparametric Statistical Infer-ence Springer Berlin Germany 2011

[33] D A Wolfe and M Hollander Nonparametric Statistical Meth-ods John Wiley amp Sons New York NY USA 2013

[34] C A C Coello ldquoTheoretical and numerical constraint-handling techniques used with evolutionary algorithms asurvey of the state of the artrdquo Computer Methods in AppliedMechanics and Engineering vol 191 no 11-12 pp 1245ndash12872002

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

[36] S Mirjalili and A Lewis ldquoAdaptive gbest-guided gravitationalsearch algorithmrdquo Neural Computing and Applications vol 25no 7-8 pp 1569ndash1584 2014

[37] C A Coello Coello ldquoConstraint-handling using an evolution-ary multiobjective optimization techniquerdquo Civil Engineeringand Environmental Systems vol 17 no 4 pp 319ndash346 2000

[38] K Deb ldquoOptimal design of a welded beam via genetic algo-rithmsrdquo AIAA Journal vol 29 no 11 pp 2013ndash2015 1991

[39] K Deb ldquoAn efficient constraint handling method for geneticalgorithmsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 186 no 2ndash4 pp 311ndash338 2000

[40] K S Lee and Z W Geem ldquoA new meta-heuristic algorithm forcontinuous engineering optimization harmony search theoryand practicerdquo Computer Methods in Applied Mechanics andEngineering vol 194 no 36-38 pp 3902ndash3933 2005

[41] K M Ragsdell and D T Phillips ldquoOptimal design of a classof welded structures using geometric programmingrdquo Journal ofEngineering for Industry vol 98 no 3 pp 1021ndash1025 1976

[42] A H Gandomi and X S Yang Benchmark Problems inStructural Optimization Computational Optimization Methodsand Algorithms Springer Berlin Germany 2011

[43] S Akhtar K Tai and T Ray ldquoA socio-behavioral simulationmodel for engineering design optimizationrdquo Engineering Opti-mization vol 34 no 4 pp 341ndash354 2002

[44] E Mezura-Montes C A C Coello and R Landa-BecerraldquoEngineering optimization using simple evolutionary algo-rithmrdquo in Proceedings of the 15th IEEE International Conferenceon Tools with Artificial Intelligence pp 149ndash156 IEEEComputerSociety November 2003

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

[46] W Long W-Z Zhang Y-F Huang and Y-X Chen ldquoAhybrid cuckoo search algorithm with feasibility-based rule forconstrained structural optimizationrdquo Journal of Central SouthUniversity vol 21 no 8 pp 3197ndash3204 2014

[47] T Ray andKM Liew ldquoSociety and civilization an optimizationalgorithm based on the simulation of social behaviorrdquo IEEETransactions onEvolutionaryComputation vol 7 no 4 pp 386ndash396 2003

[48] E Mezura-Montes and C A C Coello ldquoUseful infeasiblesolutions in engineering optimization with evolutionary algo-rithmsrdquo in Proceedings of the 4th Mexican international confer-ence on Advances in Artificial Intelligence (MICAI rsquo05) vol 3789pp 652ndash662 Springer Monterrey Mexico 2005

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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


been in research stage so far and convergence speed and cal-culation accuracy of this algorithm can be further advancedTo improve the performance of MFO a Levy-flight moth-flame optimization (LMFO) algorithm is proposed

We know that Levy-flight [11 19] has a strong abilityof strengthening global search and overcoming the problemof being trapped in local minima In order to make use ofthe good performance of Levy-flight we propose a Levy-flight moth-flame optimization MFO and Levy-flight havecomplementary advantages so the proposed algorithm canlead to a faster and more robust method The proposedalgorithm is verified on nineteen benchmark functions andtwo engineering problems

The rest of the paper is organized as follows Section 2presents a brief introduction to MFO and Levy-flight Animproved version of MFO algorithm LMFO is proposedin Section 3 The experimental results of test functions andengineering design problem are showed in Sections 4 and 5respectively Results and discussion are provided in Section 6Finally Section 7 concludes the work

2 Related Works

In this section a background about themoth-flame optimiza-tion algorithm and Levy-flight will be provided briefly

21 MFO Algorithm Moth-flame optimization [18] algo-rithm is a new metaheuristic optimization method which isproposed by Seyedali Mirjalili and based on the simulation ofthe behavior of moths for their special navigationmethods innightThey utilize a mechanism called transverse orientationfor navigation In this method a moth flies by maintaininga fixed angle with respect to the moon which is a veryeffective mechanism for travelling long distance in a straightpath because the moon is far away from the moth Thismechanism guarantees that moths fly along straight line innight However we usually observe that moths fly spirallyaround the lights In fact moths are tricked by artificial lightsand show such behaviors Since such light is extremely closeto the moon hence maintaining a similar angle to the lightsource causes a spiral fly path of moths

In theMFO algorithm the set ofmoths is represented in amatrix119872 For all the moths there is an array119874119872 for storingthe corresponding fitness valuesThe second key componentsin the algorithm are flames A matrix 119865 similar to the mothmatrix is considered For the flames it is also assumed thatthere is an array 119874119865 for storing the corresponding fitnessvalues

TheMFOalgorithm is a three-tuple that approximates theglobal optimal of the optimization problems and defined asfollows

MFO = (119868 119875 119879) (1)

119868 is a function that creates a random population of mothsand corresponding fitness values The methodical model ofthis function is as follows

119868 120601 997888rarr 119872119874119872 (2)

The 119875 function which is the main function moves themoths around the search space This function received thematrix of119872 and returns its updated one eventually

119875 119872 997888rarr 119872 (3)

The 119879 function returns true if the termination criterion issatisfied and false if the termination criterion is not satisfied

119879 119872 997888rarr true false (4)

With 119868 119875 and 119879 the general framework of the MFOalgorithm is defined as follows

119872 = 119868( )while 119879(119872) is equal to false

119872 = 119875(119872)

end

After the initialization the 119875 function is iteratively rununtil the 119879 function returns true For the sake of simulatingthe behavior of moths mathematically the position of eachmoth is updated with respect to a flame using the followingequation

119872119894= 119878 (119872

119894 119865119895) (5)



and 119878 is the spiral functionAny types of spiral can be utilized here subject to the

following conditions

(1) Spiralrsquos initial point should start from the moth(2) Spiralrsquos final point should be the position of the flame(3) Fluctuation of the range of spiral should not exceed

the search space

Considering these points a logarithmic spiral is definedfor the MFO algorithm as follows

119878 (119872119894 119865119895) = 119863

119894sdot 119890119887119905sdot cos (2120587119905) + 119865

119895 (6)

where 119863119894indicates the distance of the 119894th moth for the 119895th

flame 119887 is a constant for defining the shape of the logarithmicspiral and 119905 is a random number in [minus1 1]

119863 is calculated as follows

119863119894=10038161003816100381610038161003816119865119895minus119872119894

10038161003816100381610038161003816 (7)



and119863119894indicates the distance of the 119894thmoth for the 119895th flame

Equation (6) describes the spiral flying path of mothsFrom this equation the next position of a moth is definedwith respect to a flameThe 119905 parameter in the spiral equationdefines how much the next position of the moth should beclose to the flame (119905 = minus1 is the closest position to the flamewhile 119905 = 1 shows the farthest)

A question thatmay rise here is that the position updatingin (6) only requires the moths to move towards a flame yet



119905minus 1119872

119905)

(10) 119874119865 = sort(119872119905minus1

119872119905)

(11) end(12) for 119894 = 1 119899

(13) for 119895 = 1 119889





119879) (8)











119883119905+1

119894= 119883119905




119905minus 1119872

119905)

(10) 119874119865 = sort(119872119905minus1

119872119905)

(11) end(12) for 119894 = 1 119899

(13) for 119895 = 1 119889




119894at iteration



(0 le 120573 le 2) (10)


Levy (120573) sim120601 times 120583

|]|1120573 (11)


120601 = [Γ (1 + 120573) times sin (120587 times 1205732)


1120573

(12)









Sphere 1198911(119909) =

119899

sum

119894=1

1199092

119894[minus100 100] 200 0

Schwefelrsquos 222 1198912(119909) =

119899

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119899

prod

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 [minus10 10] 200 0


119899

sum

119894=1

(

119894

sum

119895=1

119909119895)

2

[minus100 100] 200 0


119894

10038161003816100381610038161199091198941003816100381610038161003816 1 le 119894 le 119899 [minus100 100] 200 0

Rosenbrock 1198915(119909) =

119899minus1

sum

119894=1

[100 (119909119894+1

minus 1199092

119894)2

+ (119909119894minus 1)2

] [minus30 30] 200 0

Step 1198916(119909) =

119899

sum

119894=1


2

[minus100 100] 200 0

Quartic 1198917(119909) =

119899

sum

119894=1

1198941199094

119894+ random(0 1) [minus128 128] 200 0

XSYang-7 1198918(119909) =

119899

sum

119894=1

120576119895

1003816100381610038161003816100381610038161003816119909119894minus1

119894

1003816100381610038161003816100381610038161003816 120576119895isin ⋃[0 1] [minus5 5] 200 0








1sim1198918






Table2Multim

odalbenchm

arkfunctio

ns

Nam

eFu

nctio

nRa

nge

Dim

119891min

Rastr

igin

1198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

[minus512512]

200

0

Ackley


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos(2120587119909119894))+20+119890

[minus3232]

200

0

Grie

wank

11989111=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos(

119909119894

radic119894

)+1

[minus60

060

0]200

0

Penalized

1

11989112(119909)=120587 11989910sin

(1205871199101)+

119899minus1

sum 119894=1

(1199101minus1)2

[1+10sin2(120587119910119894+1)]+(119910119899minus1)2

+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+119909119894+1

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)119898

119909119894gt119886

0minus119886lt119909119894lt119886

119896(minus119909119894minus119886)119898

119909119894ltminus119886

[minus5050]

200

0

Penalized

211989113(119909)=01sin2(31205871199091)+

119899

sum 119894=1

(119909119894minus1)2

[1+sin2(3120587119909119894+1)]+(119909119899minus1)2

[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

[minus5050]

200

0

Alpine

11989114(119909)=

119899

sum 119894=1

119909119894sin

(119909119894)+01119909119894

[minus1010]

200

0

Zakh

arov

11989115(119909)=

119899

sum 119894=1

1199092 119894+(

119899

sum 119894=1

05119894119909119894)

2

+(

119899

sum 119894=1

05119894119909119894)

4

[minus510]

200

0



nmultim

odalbenchm

arkfunctio

ns

Nam

eFu

nctio

nRa

nge

Dim

119891min

Goldstein-Pric

e11989116(119909)=[1+(1199091+1199092+1)2



[minus22]

23

DropWave

11989117(119909)=minus

1+cos(

12radic1199092 1+1199092 2)

(12)(1199092 1+1199092 2)+2

[minus512512]

2minus1

Schafferrsquos

F611989118(119909)=

sin2radic(1199092 1+1199092 2)minus05

[1+0001(1199092 1+1199092 2)]2

[minus100100]

2minus1

Easom


[minus2lowast

pi2lowastpi]

2minus1




1198911(119863 = 200)


1198912(119863 = 200)


1198913(119863 = 200)


1198914(119863 = 200)


1198915(119863 = 200)


1198916(119863 = 200)


1198917(119863 = 200)


1198918(119863 = 200)








1198919(119863 = 200)


11989110(119863 = 200)


11989111(119863 = 200)


11989112(119863 = 200)


11989113(119863 = 200)


11989114(119863 = 200)


11989115(119863 = 200)




11989116(119863 = 2)


11989117(119863 = 2)


11989118(119863 = 2)


11989119(119863 = 2)




1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918



1198919

11989110

11989111

11989112

11989113

11989114

11989115



11989116

11989117

11989118

11989119









9sim11989115



0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

times105

ABCBAGGSADA

PSOGSAMFOLMFO

f1




0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

ABCBAGGSADA

PSOGSAMFOLMFO

10150

10100

1050

100

10minus50

10minus100

f2


0 200 400 600 800 10000

1

2

3

4

5

6

7

8

9

Iteration

Aver

age fi

tnes

s val

ue

times106

ABCBAGGSADA

PSOGSAMFOLMFO

f3



0 200 400 600 800 10000

102030405060708090

100

Iteration

Aver

age fi

tnes

s val

ue

f4

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

05

1

15

2

25

3

Iteration

Aver

age fi

tnes

s val

ue

f5times109

ABCBAGGSADA

PSOGSAMFOLMFO





17 11989118 and


0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

f6times105

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Iteration

Aver

age fi

tnes

s val

ue

f7

ABCBAGGSADA

PSOGSAMFOLMFO





17and 119891


0 200 400 600 800 10000

50

100

150

200

250

Iteration

Aver

age fi

tnes

s val

ue

f8

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

Iteration

Aver

age fi

tnes

s val

ue

f9

ABCBAGGSADA

PSOGSAMFOLMFO






0 200 400 600 800 10000

5

10

15

20

25

Iteration

Aver

age fi

tnes

s val

ue

f10

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

Iteration

Aver

age fi

tnes

s val

ue

f11

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

1

2

3

4

5

6

7

Iteration

Aver

age fi

tnes

s val

ue

f12times109

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

2

4

6

8

10

12

14

Iteration

Aver

age fi

tnes

s val

ue

f13times109

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

100

200

300

400

500

600

Iteration

Aver

age fi

tnes

s val

ue

f14

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

f151020

100

10minus20

10minus40

10minus60

10minus80

10minus100

10minus120

ABCBAGGSADA

PSOGSAMFOLMFO




Consider = [1199091 1199092 1199093 1199094] = [ℎ 119897 119905 119887]

Minimize 119891 ()

= 1104711199092

11199092

0 200 400 600 800 10000

10

20

30

40

50

60

70

80

90

100

Iteration

Aver

age fi

tnes

s val

ue

f16

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

minus055

Iteration

Aver

age fi

tnes

s val

ue

f17

ABCBAGGSADA

PSOGSAMFOLMFO


+ 00481111990931199094(140 + 119909

2)


1198922 () = 120590 () minus 120590max le 0

1198923 () = 120575 () minus 120575max le 0

1198924 () = 119909

1minus 1199094le 0

1198925 () = 119875 minus 119875

119888 () le 0

1198926 () = 0125 minus 119909

1le 0


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

Iteration

Aver

age fi

tnes

s val

ue

f18

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

Iteration

Aver

age fi

tnes

s val

ue

f19

ABCBAGGSADA

PSOGSAMFOLMFO


1198927 ()

= 1104711199092

1

+ 00481111990931199094(140 + 119909

2) minus 50

le 0


01 le 1199092le 10

01 le 1199093le 10

01 le 1199094le 2


0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f1times105



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f2times1099



0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f3times106




0102030405060708090

100

Algorithms

Fitn

ess v

alue

f4



0

1

2

3

4

5

6

7

8

9

Algorithms

Fitn

ess v

alue

f5times108



005

115

225

335

445

Algorithms

Fitn

ess v

alue

f6times105



0

500

1000

1500

2000

2500

Algorithms

Fitn

ess v

alue

f7



20406080

100120140160180200

Algorithms

Fitn

ess v

alue

f8



0

500

1000

1500

2000

Algorithms

Fitn

ess v

alue

f9




0

2

4

6

8

10

12

14

16

18

20

Algorithms

Fitn

ess v

alue

f10



0

1000

2000

3000

4000

5000

Algorithms

Fitn

ess v

alue

f11



002040608

112141618

2

Algorithms

Fitn

ess v

alue

f12times109



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f13times109



0

50

100

150

200

Algorithms

Fitn

ess v

alue

f14



0

2000

4000

6000

8000

10000

Algorithms

Fitn

ess v

alue

f15




0

10

20

30

40

50

60

70

80

Algorithms

Fitn

ess v

alue

f16



minus1

minus09

minus08

minus07

minus06

minus05

minus04

Algorithms

Fitn

ess v

alue

f17


Where 120591 () = radic(1205911015840)2+ 2120591101584012059110158401015840

1199092

2119877+ (12059110158401015840)

2

1205911015840=

119875

radic211990911199092

12059110158401015840=119872119877

119869

119872 = 119875(119871 +1199092

2)

119877 = radic1199092

2

4+ (

1199091+ 1199093

2)

2

119869 = 2radic211990911199092[1199092

2

4+ (

1199091+ 1199093

2)

2

]

120590 () =6119875119871

11990941199092

3

120575 () =61198751198713

1198641199092

31199094



Algorithms

Fitn

ess v

alue

f18




0

Algorithms

Fitn

ess v

alue

f19


119875119888 ()

=

4013119864radic1199092

31199096

436

1198712(1 minus

1199093

2119871

radic119864

4119866)

119875 = 6000119897119887

119871 = 14 in

120575max = 025 in

119864 = 30 times 16 psi

119866 = 12 times 106 psi

120591max = 13600 psi

120590max = 30000 psi(13)






ht

l

P

L

t


l1

l2

d1

d2z





1sim1199097denote


1)





Minimize 119891 ()

= 0785411990911199092

2(33333119909

2

3+ 149334119909

3minus 430934)

minus 15081199091(1199092

6+ 1199092

7) + 74777 (119909

3

6+ 1199093

7)

+ 07854 (11990941199092

6+ 11990951199092

7)

Subject to 1198921 () =

27

11990911199092

21199093

minus 1 le 0

1198922 () =

3975

11990911199092

21199092

3

minus 1 le 0

1198923 () =

1931199093

4

11990921199094

61199093

minus 1 le 0

1198924 () =

1931199093

5

11990921199095

71199093

minus 1 le 0

1198925 () =

[(745119909411990921199093)2+ 169 times 10

6]12

1101199093

6

minus 1 le 0

1198926 () =

[(745119909511990921199093)2+ 1575 times 10

6]12

851199093

7

minus 1

le 0

1198927 () =

11990921199093

40minus 1 le 0

1198928 () =

51199092

1199091

minus 1 le 0

1198929 () =

1199091

121199092

minus 1 le 0

11989210 () =

151199096+ 19

1199094

minus 1 le 0

11989211 () =

111199096+ 17

1199095

minus 1 le 0

where 26 le 1199091le 36




2) 119911 (119909

3) 119897

1(1199094) 119897

2(1199095) 119889

1(1199096) 119889

2(1199097)


07 le 1199092le 08

17 le 1199093le 28

73 le 1199094le 83

73 le 1199095le 83

29 le 1199096le 39

50 le 1199097le 55

(14)















Competing Interests


Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119905minus 1119872

119905)

(10) 119874119865 = sort(119872119905minus1

119872119905)

(11) end(12) for 119894 = 1 119899

(13) for 119895 = 1 119889





119879) (8)











119883119905+1

119894= 119883119905




119905minus 1119872

119905)

(10) 119874119865 = sort(119872119905minus1

119872119905)

(11) end(12) for 119894 = 1 119899

(13) for 119895 = 1 119889




119894at iteration



(0 le 120573 le 2) (10)


Levy (120573) sim120601 times 120583

|]|1120573 (11)


120601 = [Γ (1 + 120573) times sin (120587 times 1205732)


1120573

(12)









Sphere 1198911(119909) =

119899

sum

119894=1

1199092

119894[minus100 100] 200 0

Schwefelrsquos 222 1198912(119909) =

119899

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119899

prod

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 [minus10 10] 200 0


119899

sum

119894=1

(

119894

sum

119895=1

119909119895)

2

[minus100 100] 200 0


119894

10038161003816100381610038161199091198941003816100381610038161003816 1 le 119894 le 119899 [minus100 100] 200 0

Rosenbrock 1198915(119909) =

119899minus1

sum

119894=1

[100 (119909119894+1

minus 1199092

119894)2

+ (119909119894minus 1)2

] [minus30 30] 200 0

Step 1198916(119909) =

119899

sum

119894=1


2

[minus100 100] 200 0

Quartic 1198917(119909) =

119899

sum

119894=1

1198941199094

119894+ random(0 1) [minus128 128] 200 0

XSYang-7 1198918(119909) =

119899

sum

119894=1

120576119895

1003816100381610038161003816100381610038161003816119909119894minus1

119894

1003816100381610038161003816100381610038161003816 120576119895isin ⋃[0 1] [minus5 5] 200 0








1sim1198918






Table2Multim

odalbenchm

arkfunctio

ns

Nam

eFu

nctio

nRa

nge

Dim

119891min

Rastr

igin

1198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

[minus512512]

200

0

Ackley


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos(2120587119909119894))+20+119890

[minus3232]

200

0

Grie

wank

11989111=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos(

119909119894

radic119894

)+1

[minus60

060

0]200

0

Penalized

1

11989112(119909)=120587 11989910sin

(1205871199101)+

119899minus1

sum 119894=1

(1199101minus1)2

[1+10sin2(120587119910119894+1)]+(119910119899minus1)2

+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+119909119894+1

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)119898

119909119894gt119886

0minus119886lt119909119894lt119886

119896(minus119909119894minus119886)119898

119909119894ltminus119886

[minus5050]

200

0

Penalized

211989113(119909)=01sin2(31205871199091)+

119899

sum 119894=1

(119909119894minus1)2

[1+sin2(3120587119909119894+1)]+(119909119899minus1)2

[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

[minus5050]

200

0

Alpine

11989114(119909)=

119899

sum 119894=1

119909119894sin

(119909119894)+01119909119894

[minus1010]

200

0

Zakh

arov

11989115(119909)=

119899

sum 119894=1

1199092 119894+(

119899

sum 119894=1

05119894119909119894)

2

+(

119899

sum 119894=1

05119894119909119894)

4

[minus510]

200

0



nmultim

odalbenchm

arkfunctio

ns

Nam

eFu

nctio

nRa

nge

Dim

119891min

Goldstein-Pric

e11989116(119909)=[1+(1199091+1199092+1)2



[minus22]

23

DropWave

11989117(119909)=minus

1+cos(

12radic1199092 1+1199092 2)

(12)(1199092 1+1199092 2)+2

[minus512512]

2minus1

Schafferrsquos

F611989118(119909)=

sin2radic(1199092 1+1199092 2)minus05

[1+0001(1199092 1+1199092 2)]2

[minus100100]

2minus1

Easom


[minus2lowast

pi2lowastpi]

2minus1




1198911(119863 = 200)


1198912(119863 = 200)


1198913(119863 = 200)


1198914(119863 = 200)


1198915(119863 = 200)


1198916(119863 = 200)


1198917(119863 = 200)


1198918(119863 = 200)








1198919(119863 = 200)


11989110(119863 = 200)


11989111(119863 = 200)


11989112(119863 = 200)


11989113(119863 = 200)


11989114(119863 = 200)


11989115(119863 = 200)




11989116(119863 = 2)


11989117(119863 = 2)


11989118(119863 = 2)


11989119(119863 = 2)




1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918



1198919

11989110

11989111

11989112

11989113

11989114

11989115



11989116

11989117

11989118

11989119









9sim11989115



0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

times105

ABCBAGGSADA

PSOGSAMFOLMFO

f1




0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

ABCBAGGSADA

PSOGSAMFOLMFO

10150

10100

1050

100

10minus50

10minus100

f2


0 200 400 600 800 10000

1

2

3

4

5

6

7

8

9

Iteration

Aver

age fi

tnes

s val

ue

times106

ABCBAGGSADA

PSOGSAMFOLMFO

f3



0 200 400 600 800 10000

102030405060708090

100

Iteration

Aver

age fi

tnes

s val

ue

f4

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

05

1

15

2

25

3

Iteration

Aver

age fi

tnes

s val

ue

f5times109

ABCBAGGSADA

PSOGSAMFOLMFO





17 11989118 and


0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

f6times105

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Iteration

Aver

age fi

tnes

s val

ue

f7

ABCBAGGSADA

PSOGSAMFOLMFO





17and 119891


0 200 400 600 800 10000

50

100

150

200

250

Iteration

Aver

age fi

tnes

s val

ue

f8

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

Iteration

Aver

age fi

tnes

s val

ue

f9

ABCBAGGSADA

PSOGSAMFOLMFO






0 200 400 600 800 10000

5

10

15

20

25

Iteration

Aver

age fi

tnes

s val

ue

f10

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

Iteration

Aver

age fi

tnes

s val

ue

f11

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

1

2

3

4

5

6

7

Iteration

Aver

age fi

tnes

s val

ue

f12times109

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

2

4

6

8

10

12

14

Iteration

Aver

age fi

tnes

s val

ue

f13times109

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

100

200

300

400

500

600

Iteration

Aver

age fi

tnes

s val

ue

f14

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

f151020

100

10minus20

10minus40

10minus60

10minus80

10minus100

10minus120

ABCBAGGSADA

PSOGSAMFOLMFO




Consider = [1199091 1199092 1199093 1199094] = [ℎ 119897 119905 119887]

Minimize 119891 ()

= 1104711199092

11199092

0 200 400 600 800 10000

10

20

30

40

50

60

70

80

90

100

Iteration

Aver

age fi

tnes

s val

ue

f16

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

minus055

Iteration

Aver

age fi

tnes

s val

ue

f17

ABCBAGGSADA

PSOGSAMFOLMFO


+ 00481111990931199094(140 + 119909

2)


1198922 () = 120590 () minus 120590max le 0

1198923 () = 120575 () minus 120575max le 0

1198924 () = 119909

1minus 1199094le 0

1198925 () = 119875 minus 119875

119888 () le 0

1198926 () = 0125 minus 119909

1le 0


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

Iteration

Aver

age fi

tnes

s val

ue

f18

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

Iteration

Aver

age fi

tnes

s val

ue

f19

ABCBAGGSADA

PSOGSAMFOLMFO


1198927 ()

= 1104711199092

1

+ 00481111990931199094(140 + 119909

2) minus 50

le 0


01 le 1199092le 10

01 le 1199093le 10

01 le 1199094le 2


0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f1times105



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f2times1099



0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f3times106




0102030405060708090

100

Algorithms

Fitn

ess v

alue

f4



0

1

2

3

4

5

6

7

8

9

Algorithms

Fitn

ess v

alue

f5times108



005

115

225

335

445

Algorithms

Fitn

ess v

alue

f6times105



0

500

1000

1500

2000

2500

Algorithms

Fitn

ess v

alue

f7



20406080

100120140160180200

Algorithms

Fitn

ess v

alue

f8



0

500

1000

1500

2000

Algorithms

Fitn

ess v

alue

f9




0

2

4

6

8

10

12

14

16

18

20

Algorithms

Fitn

ess v

alue

f10



0

1000

2000

3000

4000

5000

Algorithms

Fitn

ess v

alue

f11



002040608

112141618

2

Algorithms

Fitn

ess v

alue

f12times109



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f13times109



0

50

100

150

200

Algorithms

Fitn

ess v

alue

f14



0

2000

4000

6000

8000

10000

Algorithms

Fitn

ess v

alue

f15




0

10

20

30

40

50

60

70

80

Algorithms

Fitn

ess v

alue

f16



minus1

minus09

minus08

minus07

minus06

minus05

minus04

Algorithms

Fitn

ess v

alue

f17


Where 120591 () = radic(1205911015840)2+ 2120591101584012059110158401015840

1199092

2119877+ (12059110158401015840)

2

1205911015840=

119875

radic211990911199092

12059110158401015840=119872119877

119869

119872 = 119875(119871 +1199092

2)

119877 = radic1199092

2

4+ (

1199091+ 1199093

2)

2

119869 = 2radic211990911199092[1199092

2

4+ (

1199091+ 1199093

2)

2

]

120590 () =6119875119871

11990941199092

3

120575 () =61198751198713

1198641199092

31199094



Algorithms

Fitn

ess v

alue

f18




0

Algorithms

Fitn

ess v

alue

f19


119875119888 ()

=

4013119864radic1199092

31199096

436

1198712(1 minus

1199093

2119871

radic119864

4119866)

119875 = 6000119897119887

119871 = 14 in

120575max = 025 in

119864 = 30 times 16 psi

119866 = 12 times 106 psi

120591max = 13600 psi

120590max = 30000 psi(13)






ht

l

P

L

t


l1

l2

d1

d2z





1sim1199097denote


1)





Minimize 119891 ()

= 0785411990911199092

2(33333119909

2

3+ 149334119909

3minus 430934)

minus 15081199091(1199092

6+ 1199092

7) + 74777 (119909

3

6+ 1199093

7)

+ 07854 (11990941199092

6+ 11990951199092

7)

Subject to 1198921 () =

27

11990911199092

21199093

minus 1 le 0

1198922 () =

3975

11990911199092

21199092

3

minus 1 le 0

1198923 () =

1931199093

4

11990921199094

61199093

minus 1 le 0

1198924 () =

1931199093

5

11990921199095

71199093

minus 1 le 0

1198925 () =

[(745119909411990921199093)2+ 169 times 10

6]12

1101199093

6

minus 1 le 0

1198926 () =

[(745119909511990921199093)2+ 1575 times 10

6]12

851199093

7

minus 1

le 0

1198927 () =

11990921199093

40minus 1 le 0

1198928 () =

51199092

1199091

minus 1 le 0

1198929 () =

1199091

121199092

minus 1 le 0

11989210 () =

151199096+ 19

1199094

minus 1 le 0

11989211 () =

111199096+ 17

1199095

minus 1 le 0

where 26 le 1199091le 36




2) 119911 (119909

3) 119897

1(1199094) 119897

2(1199095) 119889

1(1199096) 119889

2(1199097)


07 le 1199092le 08

17 le 1199093le 28

73 le 1199094le 83

73 le 1199095le 83

29 le 1199096le 39

50 le 1199097le 55

(14)















Competing Interests


Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119905minus 1119872

119905)

(10) 119874119865 = sort(119872119905minus1

119872119905)

(11) end(12) for 119894 = 1 119899

(13) for 119895 = 1 119889




119894at iteration



(0 le 120573 le 2) (10)


Levy (120573) sim120601 times 120583

|]|1120573 (11)


120601 = [Γ (1 + 120573) times sin (120587 times 1205732)


1120573

(12)









Sphere 1198911(119909) =

119899

sum

119894=1

1199092

119894[minus100 100] 200 0

Schwefelrsquos 222 1198912(119909) =

119899

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119899

prod

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 [minus10 10] 200 0


119899

sum

119894=1

(

119894

sum

119895=1

119909119895)

2

[minus100 100] 200 0


119894

10038161003816100381610038161199091198941003816100381610038161003816 1 le 119894 le 119899 [minus100 100] 200 0

Rosenbrock 1198915(119909) =

119899minus1

sum

119894=1

[100 (119909119894+1

minus 1199092

119894)2

+ (119909119894minus 1)2

] [minus30 30] 200 0

Step 1198916(119909) =

119899

sum

119894=1


2

[minus100 100] 200 0

Quartic 1198917(119909) =

119899

sum

119894=1

1198941199094

119894+ random(0 1) [minus128 128] 200 0

XSYang-7 1198918(119909) =

119899

sum

119894=1

120576119895

1003816100381610038161003816100381610038161003816119909119894minus1

119894

1003816100381610038161003816100381610038161003816 120576119895isin ⋃[0 1] [minus5 5] 200 0








1sim1198918






Table2Multim

odalbenchm

arkfunctio

ns

Nam

eFu

nctio

nRa

nge

Dim

119891min

Rastr

igin

1198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

[minus512512]

200

0

Ackley


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos(2120587119909119894))+20+119890

[minus3232]

200

0

Grie

wank

11989111=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos(

119909119894

radic119894

)+1

[minus60

060

0]200

0

Penalized

1

11989112(119909)=120587 11989910sin

(1205871199101)+

119899minus1

sum 119894=1

(1199101minus1)2

[1+10sin2(120587119910119894+1)]+(119910119899minus1)2

+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+119909119894+1

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)119898

119909119894gt119886

0minus119886lt119909119894lt119886

119896(minus119909119894minus119886)119898

119909119894ltminus119886

[minus5050]

200

0

Penalized

211989113(119909)=01sin2(31205871199091)+

119899

sum 119894=1

(119909119894minus1)2

[1+sin2(3120587119909119894+1)]+(119909119899minus1)2

[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

[minus5050]

200

0

Alpine

11989114(119909)=

119899

sum 119894=1

119909119894sin

(119909119894)+01119909119894

[minus1010]

200

0

Zakh

arov

11989115(119909)=

119899

sum 119894=1

1199092 119894+(

119899

sum 119894=1

05119894119909119894)

2

+(

119899

sum 119894=1

05119894119909119894)

4

[minus510]

200

0



nmultim

odalbenchm

arkfunctio

ns

Nam

eFu

nctio

nRa

nge

Dim

119891min

Goldstein-Pric

e11989116(119909)=[1+(1199091+1199092+1)2



[minus22]

23

DropWave

11989117(119909)=minus

1+cos(

12radic1199092 1+1199092 2)

(12)(1199092 1+1199092 2)+2

[minus512512]

2minus1

Schafferrsquos

F611989118(119909)=

sin2radic(1199092 1+1199092 2)minus05

[1+0001(1199092 1+1199092 2)]2

[minus100100]

2minus1

Easom


[minus2lowast

pi2lowastpi]

2minus1




1198911(119863 = 200)


1198912(119863 = 200)


1198913(119863 = 200)


1198914(119863 = 200)


1198915(119863 = 200)


1198916(119863 = 200)


1198917(119863 = 200)


1198918(119863 = 200)








1198919(119863 = 200)


11989110(119863 = 200)


11989111(119863 = 200)


11989112(119863 = 200)


11989113(119863 = 200)


11989114(119863 = 200)


11989115(119863 = 200)




11989116(119863 = 2)


11989117(119863 = 2)


11989118(119863 = 2)


11989119(119863 = 2)




1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918



1198919

11989110

11989111

11989112

11989113

11989114

11989115



11989116

11989117

11989118

11989119









9sim11989115



0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

times105

ABCBAGGSADA

PSOGSAMFOLMFO

f1




0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

ABCBAGGSADA

PSOGSAMFOLMFO

10150

10100

1050

100

10minus50

10minus100

f2


0 200 400 600 800 10000

1

2

3

4

5

6

7

8

9

Iteration

Aver

age fi

tnes

s val

ue

times106

ABCBAGGSADA

PSOGSAMFOLMFO

f3



0 200 400 600 800 10000

102030405060708090

100

Iteration

Aver

age fi

tnes

s val

ue

f4

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

05

1

15

2

25

3

Iteration

Aver

age fi

tnes

s val

ue

f5times109

ABCBAGGSADA

PSOGSAMFOLMFO





17 11989118 and


0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

f6times105

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Iteration

Aver

age fi

tnes

s val

ue

f7

ABCBAGGSADA

PSOGSAMFOLMFO





17and 119891


0 200 400 600 800 10000

50

100

150

200

250

Iteration

Aver

age fi

tnes

s val

ue

f8

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

Iteration

Aver

age fi

tnes

s val

ue

f9

ABCBAGGSADA

PSOGSAMFOLMFO






0 200 400 600 800 10000

5

10

15

20

25

Iteration

Aver

age fi

tnes

s val

ue

f10

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

Iteration

Aver

age fi

tnes

s val

ue

f11

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

1

2

3

4

5

6

7

Iteration

Aver

age fi

tnes

s val

ue

f12times109

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

2

4

6

8

10

12

14

Iteration

Aver

age fi

tnes

s val

ue

f13times109

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

100

200

300

400

500

600

Iteration

Aver

age fi

tnes

s val

ue

f14

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

f151020

100

10minus20

10minus40

10minus60

10minus80

10minus100

10minus120

ABCBAGGSADA

PSOGSAMFOLMFO




Consider = [1199091 1199092 1199093 1199094] = [ℎ 119897 119905 119887]

Minimize 119891 ()

= 1104711199092

11199092

0 200 400 600 800 10000

10

20

30

40

50

60

70

80

90

100

Iteration

Aver

age fi

tnes

s val

ue

f16

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

minus055

Iteration

Aver

age fi

tnes

s val

ue

f17

ABCBAGGSADA

PSOGSAMFOLMFO


+ 00481111990931199094(140 + 119909

2)


1198922 () = 120590 () minus 120590max le 0

1198923 () = 120575 () minus 120575max le 0

1198924 () = 119909

1minus 1199094le 0

1198925 () = 119875 minus 119875

119888 () le 0

1198926 () = 0125 minus 119909

1le 0


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

Iteration

Aver

age fi

tnes

s val

ue

f18

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

Iteration

Aver

age fi

tnes

s val

ue

f19

ABCBAGGSADA

PSOGSAMFOLMFO


1198927 ()

= 1104711199092

1

+ 00481111990931199094(140 + 119909

2) minus 50

le 0


01 le 1199092le 10

01 le 1199093le 10

01 le 1199094le 2


0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f1times105



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f2times1099



0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f3times106




0102030405060708090

100

Algorithms

Fitn

ess v

alue

f4



0

1

2

3

4

5

6

7

8

9

Algorithms

Fitn

ess v

alue

f5times108



005

115

225

335

445

Algorithms

Fitn

ess v

alue

f6times105



0

500

1000

1500

2000

2500

Algorithms

Fitn

ess v

alue

f7



20406080

100120140160180200

Algorithms

Fitn

ess v

alue

f8



0

500

1000

1500

2000

Algorithms

Fitn

ess v

alue

f9




0

2

4

6

8

10

12

14

16

18

20

Algorithms

Fitn

ess v

alue

f10



0

1000

2000

3000

4000

5000

Algorithms

Fitn

ess v

alue

f11



002040608

112141618

2

Algorithms

Fitn

ess v

alue

f12times109



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f13times109



0

50

100

150

200

Algorithms

Fitn

ess v

alue

f14



0

2000

4000

6000

8000

10000

Algorithms

Fitn

ess v

alue

f15




0

10

20

30

40

50

60

70

80

Algorithms

Fitn

ess v

alue

f16



minus1

minus09

minus08

minus07

minus06

minus05

minus04

Algorithms

Fitn

ess v

alue

f17


Where 120591 () = radic(1205911015840)2+ 2120591101584012059110158401015840

1199092

2119877+ (12059110158401015840)

2

1205911015840=

119875

radic211990911199092

12059110158401015840=119872119877

119869

119872 = 119875(119871 +1199092

2)

119877 = radic1199092

2

4+ (

1199091+ 1199093

2)

2

119869 = 2radic211990911199092[1199092

2

4+ (

1199091+ 1199093

2)

2

]

120590 () =6119875119871

11990941199092

3

120575 () =61198751198713

1198641199092

31199094



Algorithms

Fitn

ess v

alue

f18




0

Algorithms

Fitn

ess v

alue

f19


119875119888 ()

=

4013119864radic1199092

31199096

436

1198712(1 minus

1199093

2119871

radic119864

4119866)

119875 = 6000119897119887

119871 = 14 in

120575max = 025 in

119864 = 30 times 16 psi

119866 = 12 times 106 psi

120591max = 13600 psi

120590max = 30000 psi(13)






ht

l

P

L

t


l1

l2

d1

d2z





1sim1199097denote


1)





Minimize 119891 ()

= 0785411990911199092

2(33333119909

2

3+ 149334119909

3minus 430934)

minus 15081199091(1199092

6+ 1199092

7) + 74777 (119909

3

6+ 1199093

7)

+ 07854 (11990941199092

6+ 11990951199092

7)

Subject to 1198921 () =

27

11990911199092

21199093

minus 1 le 0

1198922 () =

3975

11990911199092

21199092

3

minus 1 le 0

1198923 () =

1931199093

4

11990921199094

61199093

minus 1 le 0

1198924 () =

1931199093

5

11990921199095

71199093

minus 1 le 0

1198925 () =

[(745119909411990921199093)2+ 169 times 10

6]12

1101199093

6

minus 1 le 0

1198926 () =

[(745119909511990921199093)2+ 1575 times 10

6]12

851199093

7

minus 1

le 0

1198927 () =

11990921199093

40minus 1 le 0

1198928 () =

51199092

1199091

minus 1 le 0

1198929 () =

1199091

121199092

minus 1 le 0

11989210 () =

151199096+ 19

1199094

minus 1 le 0

11989211 () =

111199096+ 17

1199095

minus 1 le 0

where 26 le 1199091le 36




2) 119911 (119909

3) 119897

1(1199094) 119897

2(1199095) 119889

1(1199096) 119889

2(1199097)


07 le 1199092le 08

17 le 1199093le 28

73 le 1199094le 83

73 le 1199095le 83

29 le 1199096le 39

50 le 1199097le 55

(14)















Competing Interests


Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Sphere 1198911(119909) =

119899

sum

119894=1

1199092

119894[minus100 100] 200 0

Schwefelrsquos 222 1198912(119909) =

119899

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 +

119899

prod

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 [minus10 10] 200 0


119899

sum

119894=1

(

119894

sum

119895=1

119909119895)

2

[minus100 100] 200 0


119894

10038161003816100381610038161199091198941003816100381610038161003816 1 le 119894 le 119899 [minus100 100] 200 0

Rosenbrock 1198915(119909) =

119899minus1

sum

119894=1

[100 (119909119894+1

minus 1199092

119894)2

+ (119909119894minus 1)2

] [minus30 30] 200 0

Step 1198916(119909) =

119899

sum

119894=1


2

[minus100 100] 200 0

Quartic 1198917(119909) =

119899

sum

119894=1

1198941199094

119894+ random(0 1) [minus128 128] 200 0

XSYang-7 1198918(119909) =

119899

sum

119894=1

120576119895

1003816100381610038161003816100381610038161003816119909119894minus1

119894

1003816100381610038161003816100381610038161003816 120576119895isin ⋃[0 1] [minus5 5] 200 0








1sim1198918






Table2Multim

odalbenchm

arkfunctio

ns

Nam

eFu

nctio

nRa

nge

Dim

119891min

Rastr

igin

1198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

[minus512512]

200

0

Ackley


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos(2120587119909119894))+20+119890

[minus3232]

200

0

Grie

wank

11989111=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos(

119909119894

radic119894

)+1

[minus60

060

0]200

0

Penalized

1

11989112(119909)=120587 11989910sin

(1205871199101)+

119899minus1

sum 119894=1

(1199101minus1)2

[1+10sin2(120587119910119894+1)]+(119910119899minus1)2

+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+119909119894+1

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)119898

119909119894gt119886

0minus119886lt119909119894lt119886

119896(minus119909119894minus119886)119898

119909119894ltminus119886

[minus5050]

200

0

Penalized

211989113(119909)=01sin2(31205871199091)+

119899

sum 119894=1

(119909119894minus1)2

[1+sin2(3120587119909119894+1)]+(119909119899minus1)2

[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

[minus5050]

200

0

Alpine

11989114(119909)=

119899

sum 119894=1

119909119894sin

(119909119894)+01119909119894

[minus1010]

200

0

Zakh

arov

11989115(119909)=

119899

sum 119894=1

1199092 119894+(

119899

sum 119894=1

05119894119909119894)

2

+(

119899

sum 119894=1

05119894119909119894)

4

[minus510]

200

0



nmultim

odalbenchm

arkfunctio

ns

Nam

eFu

nctio

nRa

nge

Dim

119891min

Goldstein-Pric

e11989116(119909)=[1+(1199091+1199092+1)2



[minus22]

23

DropWave

11989117(119909)=minus

1+cos(

12radic1199092 1+1199092 2)

(12)(1199092 1+1199092 2)+2

[minus512512]

2minus1

Schafferrsquos

F611989118(119909)=

sin2radic(1199092 1+1199092 2)minus05

[1+0001(1199092 1+1199092 2)]2

[minus100100]

2minus1

Easom


[minus2lowast

pi2lowastpi]

2minus1




1198911(119863 = 200)


1198912(119863 = 200)


1198913(119863 = 200)


1198914(119863 = 200)


1198915(119863 = 200)


1198916(119863 = 200)


1198917(119863 = 200)


1198918(119863 = 200)








1198919(119863 = 200)


11989110(119863 = 200)


11989111(119863 = 200)


11989112(119863 = 200)


11989113(119863 = 200)


11989114(119863 = 200)


11989115(119863 = 200)




11989116(119863 = 2)


11989117(119863 = 2)


11989118(119863 = 2)


11989119(119863 = 2)




1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918



1198919

11989110

11989111

11989112

11989113

11989114

11989115



11989116

11989117

11989118

11989119









9sim11989115



0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

times105

ABCBAGGSADA

PSOGSAMFOLMFO

f1




0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

ABCBAGGSADA

PSOGSAMFOLMFO

10150

10100

1050

100

10minus50

10minus100

f2


0 200 400 600 800 10000

1

2

3

4

5

6

7

8

9

Iteration

Aver

age fi

tnes

s val

ue

times106

ABCBAGGSADA

PSOGSAMFOLMFO

f3



0 200 400 600 800 10000

102030405060708090

100

Iteration

Aver

age fi

tnes

s val

ue

f4

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

05

1

15

2

25

3

Iteration

Aver

age fi

tnes

s val

ue

f5times109

ABCBAGGSADA

PSOGSAMFOLMFO





17 11989118 and


0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

f6times105

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Iteration

Aver

age fi

tnes

s val

ue

f7

ABCBAGGSADA

PSOGSAMFOLMFO





17and 119891


0 200 400 600 800 10000

50

100

150

200

250

Iteration

Aver

age fi

tnes

s val

ue

f8

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

Iteration

Aver

age fi

tnes

s val

ue

f9

ABCBAGGSADA

PSOGSAMFOLMFO






0 200 400 600 800 10000

5

10

15

20

25

Iteration

Aver

age fi

tnes

s val

ue

f10

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

Iteration

Aver

age fi

tnes

s val

ue

f11

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

1

2

3

4

5

6

7

Iteration

Aver

age fi

tnes

s val

ue

f12times109

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

2

4

6

8

10

12

14

Iteration

Aver

age fi

tnes

s val

ue

f13times109

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

100

200

300

400

500

600

Iteration

Aver

age fi

tnes

s val

ue

f14

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

f151020

100

10minus20

10minus40

10minus60

10minus80

10minus100

10minus120

ABCBAGGSADA

PSOGSAMFOLMFO




Consider = [1199091 1199092 1199093 1199094] = [ℎ 119897 119905 119887]

Minimize 119891 ()

= 1104711199092

11199092

0 200 400 600 800 10000

10

20

30

40

50

60

70

80

90

100

Iteration

Aver

age fi

tnes

s val

ue

f16

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

minus055

Iteration

Aver

age fi

tnes

s val

ue

f17

ABCBAGGSADA

PSOGSAMFOLMFO


+ 00481111990931199094(140 + 119909

2)


1198922 () = 120590 () minus 120590max le 0

1198923 () = 120575 () minus 120575max le 0

1198924 () = 119909

1minus 1199094le 0

1198925 () = 119875 minus 119875

119888 () le 0

1198926 () = 0125 minus 119909

1le 0


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

Iteration

Aver

age fi

tnes

s val

ue

f18

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

Iteration

Aver

age fi

tnes

s val

ue

f19

ABCBAGGSADA

PSOGSAMFOLMFO


1198927 ()

= 1104711199092

1

+ 00481111990931199094(140 + 119909

2) minus 50

le 0


01 le 1199092le 10

01 le 1199093le 10

01 le 1199094le 2


0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f1times105



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f2times1099



0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f3times106




0102030405060708090

100

Algorithms

Fitn

ess v

alue

f4



0

1

2

3

4

5

6

7

8

9

Algorithms

Fitn

ess v

alue

f5times108



005

115

225

335

445

Algorithms

Fitn

ess v

alue

f6times105



0

500

1000

1500

2000

2500

Algorithms

Fitn

ess v

alue

f7



20406080

100120140160180200

Algorithms

Fitn

ess v

alue

f8



0

500

1000

1500

2000

Algorithms

Fitn

ess v

alue

f9




0

2

4

6

8

10

12

14

16

18

20

Algorithms

Fitn

ess v

alue

f10



0

1000

2000

3000

4000

5000

Algorithms

Fitn

ess v

alue

f11



002040608

112141618

2

Algorithms

Fitn

ess v

alue

f12times109



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f13times109



0

50

100

150

200

Algorithms

Fitn

ess v

alue

f14



0

2000

4000

6000

8000

10000

Algorithms

Fitn

ess v

alue

f15




0

10

20

30

40

50

60

70

80

Algorithms

Fitn

ess v

alue

f16



minus1

minus09

minus08

minus07

minus06

minus05

minus04

Algorithms

Fitn

ess v

alue

f17


Where 120591 () = radic(1205911015840)2+ 2120591101584012059110158401015840

1199092

2119877+ (12059110158401015840)

2

1205911015840=

119875

radic211990911199092

12059110158401015840=119872119877

119869

119872 = 119875(119871 +1199092

2)

119877 = radic1199092

2

4+ (

1199091+ 1199093

2)

2

119869 = 2radic211990911199092[1199092

2

4+ (

1199091+ 1199093

2)

2

]

120590 () =6119875119871

11990941199092

3

120575 () =61198751198713

1198641199092

31199094



Algorithms

Fitn

ess v

alue

f18




0

Algorithms

Fitn

ess v

alue

f19


119875119888 ()

=

4013119864radic1199092

31199096

436

1198712(1 minus

1199093

2119871

radic119864

4119866)

119875 = 6000119897119887

119871 = 14 in

120575max = 025 in

119864 = 30 times 16 psi

119866 = 12 times 106 psi

120591max = 13600 psi

120590max = 30000 psi(13)






ht

l

P

L

t


l1

l2

d1

d2z





1sim1199097denote


1)





Minimize 119891 ()

= 0785411990911199092

2(33333119909

2

3+ 149334119909

3minus 430934)

minus 15081199091(1199092

6+ 1199092

7) + 74777 (119909

3

6+ 1199093

7)

+ 07854 (11990941199092

6+ 11990951199092

7)

Subject to 1198921 () =

27

11990911199092

21199093

minus 1 le 0

1198922 () =

3975

11990911199092

21199092

3

minus 1 le 0

1198923 () =

1931199093

4

11990921199094

61199093

minus 1 le 0

1198924 () =

1931199093

5

11990921199095

71199093

minus 1 le 0

1198925 () =

[(745119909411990921199093)2+ 169 times 10

6]12

1101199093

6

minus 1 le 0

1198926 () =

[(745119909511990921199093)2+ 1575 times 10

6]12

851199093

7

minus 1

le 0

1198927 () =

11990921199093

40minus 1 le 0

1198928 () =

51199092

1199091

minus 1 le 0

1198929 () =

1199091

121199092

minus 1 le 0

11989210 () =

151199096+ 19

1199094

minus 1 le 0

11989211 () =

111199096+ 17

1199095

minus 1 le 0

where 26 le 1199091le 36




2) 119911 (119909

3) 119897

1(1199094) 119897

2(1199095) 119889

1(1199096) 119889

2(1199097)


07 le 1199092le 08

17 le 1199093le 28

73 le 1199094le 83

73 le 1199095le 83

29 le 1199096le 39

50 le 1199097le 55

(14)















Competing Interests


Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table2Multim

odalbenchm

arkfunctio

ns

Nam

eFu

nctio

nRa

nge

Dim

119891min

Rastr

igin

1198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

[minus512512]

200

0

Ackley


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos(2120587119909119894))+20+119890

[minus3232]

200

0

Grie

wank

11989111=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos(

119909119894

radic119894

)+1

[minus60

060

0]200

0

Penalized

1

11989112(119909)=120587 11989910sin

(1205871199101)+

119899minus1

sum 119894=1

(1199101minus1)2

[1+10sin2(120587119910119894+1)]+(119910119899minus1)2

+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+119909119894+1

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)119898

119909119894gt119886

0minus119886lt119909119894lt119886

119896(minus119909119894minus119886)119898

119909119894ltminus119886

[minus5050]

200

0

Penalized

211989113(119909)=01sin2(31205871199091)+

119899

sum 119894=1

(119909119894minus1)2

[1+sin2(3120587119909119894+1)]+(119909119899minus1)2

[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

[minus5050]

200

0

Alpine

11989114(119909)=

119899

sum 119894=1

119909119894sin

(119909119894)+01119909119894

[minus1010]

200

0

Zakh

arov

11989115(119909)=

119899

sum 119894=1

1199092 119894+(

119899

sum 119894=1

05119894119909119894)

2

+(

119899

sum 119894=1

05119894119909119894)

4

[minus510]

200

0



nmultim

odalbenchm

arkfunctio

ns

Nam

eFu

nctio

nRa

nge

Dim

119891min

Goldstein-Pric

e11989116(119909)=[1+(1199091+1199092+1)2



[minus22]

23

DropWave

11989117(119909)=minus

1+cos(

12radic1199092 1+1199092 2)

(12)(1199092 1+1199092 2)+2

[minus512512]

2minus1

Schafferrsquos

F611989118(119909)=

sin2radic(1199092 1+1199092 2)minus05

[1+0001(1199092 1+1199092 2)]2

[minus100100]

2minus1

Easom


[minus2lowast

pi2lowastpi]

2minus1




1198911(119863 = 200)


1198912(119863 = 200)


1198913(119863 = 200)


1198914(119863 = 200)


1198915(119863 = 200)


1198916(119863 = 200)


1198917(119863 = 200)


1198918(119863 = 200)








1198919(119863 = 200)


11989110(119863 = 200)


11989111(119863 = 200)


11989112(119863 = 200)


11989113(119863 = 200)


11989114(119863 = 200)


11989115(119863 = 200)




11989116(119863 = 2)


11989117(119863 = 2)


11989118(119863 = 2)


11989119(119863 = 2)




1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918



1198919

11989110

11989111

11989112

11989113

11989114

11989115



11989116

11989117

11989118

11989119









9sim11989115



0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

times105

ABCBAGGSADA

PSOGSAMFOLMFO

f1




0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

ABCBAGGSADA

PSOGSAMFOLMFO

10150

10100

1050

100

10minus50

10minus100

f2


0 200 400 600 800 10000

1

2

3

4

5

6

7

8

9

Iteration

Aver

age fi

tnes

s val

ue

times106

ABCBAGGSADA

PSOGSAMFOLMFO

f3



0 200 400 600 800 10000

102030405060708090

100

Iteration

Aver

age fi

tnes

s val

ue

f4

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

05

1

15

2

25

3

Iteration

Aver

age fi

tnes

s val

ue

f5times109

ABCBAGGSADA

PSOGSAMFOLMFO





17 11989118 and


0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

f6times105

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Iteration

Aver

age fi

tnes

s val

ue

f7

ABCBAGGSADA

PSOGSAMFOLMFO





17and 119891


0 200 400 600 800 10000

50

100

150

200

250

Iteration

Aver

age fi

tnes

s val

ue

f8

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

Iteration

Aver

age fi

tnes

s val

ue

f9

ABCBAGGSADA

PSOGSAMFOLMFO






0 200 400 600 800 10000

5

10

15

20

25

Iteration

Aver

age fi

tnes

s val

ue

f10

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

Iteration

Aver

age fi

tnes

s val

ue

f11

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

1

2

3

4

5

6

7

Iteration

Aver

age fi

tnes

s val

ue

f12times109

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

2

4

6

8

10

12

14

Iteration

Aver

age fi

tnes

s val

ue

f13times109

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

100

200

300

400

500

600

Iteration

Aver

age fi

tnes

s val

ue

f14

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

f151020

100

10minus20

10minus40

10minus60

10minus80

10minus100

10minus120

ABCBAGGSADA

PSOGSAMFOLMFO




Consider = [1199091 1199092 1199093 1199094] = [ℎ 119897 119905 119887]

Minimize 119891 ()

= 1104711199092

11199092

0 200 400 600 800 10000

10

20

30

40

50

60

70

80

90

100

Iteration

Aver

age fi

tnes

s val

ue

f16

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

minus055

Iteration

Aver

age fi

tnes

s val

ue

f17

ABCBAGGSADA

PSOGSAMFOLMFO


+ 00481111990931199094(140 + 119909

2)


1198922 () = 120590 () minus 120590max le 0

1198923 () = 120575 () minus 120575max le 0

1198924 () = 119909

1minus 1199094le 0

1198925 () = 119875 minus 119875

119888 () le 0

1198926 () = 0125 minus 119909

1le 0


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

Iteration

Aver

age fi

tnes

s val

ue

f18

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

Iteration

Aver

age fi

tnes

s val

ue

f19

ABCBAGGSADA

PSOGSAMFOLMFO


1198927 ()

= 1104711199092

1

+ 00481111990931199094(140 + 119909

2) minus 50

le 0


01 le 1199092le 10

01 le 1199093le 10

01 le 1199094le 2


0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f1times105



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f2times1099



0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f3times106




0102030405060708090

100

Algorithms

Fitn

ess v

alue

f4



0

1

2

3

4

5

6

7

8

9

Algorithms

Fitn

ess v

alue

f5times108



005

115

225

335

445

Algorithms

Fitn

ess v

alue

f6times105



0

500

1000

1500

2000

2500

Algorithms

Fitn

ess v

alue

f7



20406080

100120140160180200

Algorithms

Fitn

ess v

alue

f8



0

500

1000

1500

2000

Algorithms

Fitn

ess v

alue

f9




0

2

4

6

8

10

12

14

16

18

20

Algorithms

Fitn

ess v

alue

f10



0

1000

2000

3000

4000

5000

Algorithms

Fitn

ess v

alue

f11



002040608

112141618

2

Algorithms

Fitn

ess v

alue

f12times109



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f13times109



0

50

100

150

200

Algorithms

Fitn

ess v

alue

f14



0

2000

4000

6000

8000

10000

Algorithms

Fitn

ess v

alue

f15




0

10

20

30

40

50

60

70

80

Algorithms

Fitn

ess v

alue

f16



minus1

minus09

minus08

minus07

minus06

minus05

minus04

Algorithms

Fitn

ess v

alue

f17


Where 120591 () = radic(1205911015840)2+ 2120591101584012059110158401015840

1199092

2119877+ (12059110158401015840)

2

1205911015840=

119875

radic211990911199092

12059110158401015840=119872119877

119869

119872 = 119875(119871 +1199092

2)

119877 = radic1199092

2

4+ (

1199091+ 1199093

2)

2

119869 = 2radic211990911199092[1199092

2

4+ (

1199091+ 1199093

2)

2

]

120590 () =6119875119871

11990941199092

3

120575 () =61198751198713

1198641199092

31199094



Algorithms

Fitn

ess v

alue

f18




0

Algorithms

Fitn

ess v

alue

f19


119875119888 ()

=

4013119864radic1199092

31199096

436

1198712(1 minus

1199093

2119871

radic119864

4119866)

119875 = 6000119897119887

119871 = 14 in

120575max = 025 in

119864 = 30 times 16 psi

119866 = 12 times 106 psi

120591max = 13600 psi

120590max = 30000 psi(13)






ht

l

P

L

t


l1

l2

d1

d2z





1sim1199097denote


1)





Minimize 119891 ()

= 0785411990911199092

2(33333119909

2

3+ 149334119909

3minus 430934)

minus 15081199091(1199092

6+ 1199092

7) + 74777 (119909

3

6+ 1199093

7)

+ 07854 (11990941199092

6+ 11990951199092

7)

Subject to 1198921 () =

27

11990911199092

21199093

minus 1 le 0

1198922 () =

3975

11990911199092

21199092

3

minus 1 le 0

1198923 () =

1931199093

4

11990921199094

61199093

minus 1 le 0

1198924 () =

1931199093

5

11990921199095

71199093

minus 1 le 0

1198925 () =

[(745119909411990921199093)2+ 169 times 10

6]12

1101199093

6

minus 1 le 0

1198926 () =

[(745119909511990921199093)2+ 1575 times 10

6]12

851199093

7

minus 1

le 0

1198927 () =

11990921199093

40minus 1 le 0

1198928 () =

51199092

1199091

minus 1 le 0

1198929 () =

1199091

121199092

minus 1 le 0

11989210 () =

151199096+ 19

1199094

minus 1 le 0

11989211 () =

111199096+ 17

1199095

minus 1 le 0

where 26 le 1199091le 36




2) 119911 (119909

3) 119897

1(1199094) 119897

2(1199095) 119889

1(1199096) 119889

2(1199097)


07 le 1199092le 08

17 le 1199093le 28

73 le 1199094le 83

73 le 1199095le 83

29 le 1199096le 39

50 le 1199097le 55

(14)















Competing Interests


Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











nmultim

odalbenchm

arkfunctio

ns

Nam

eFu

nctio

nRa

nge

Dim

119891min

Goldstein-Pric

e11989116(119909)=[1+(1199091+1199092+1)2



[minus22]

23

DropWave

11989117(119909)=minus

1+cos(

12radic1199092 1+1199092 2)

(12)(1199092 1+1199092 2)+2

[minus512512]

2minus1

Schafferrsquos

F611989118(119909)=

sin2radic(1199092 1+1199092 2)minus05

[1+0001(1199092 1+1199092 2)]2

[minus100100]

2minus1

Easom


[minus2lowast

pi2lowastpi]

2minus1




1198911(119863 = 200)


1198912(119863 = 200)


1198913(119863 = 200)


1198914(119863 = 200)


1198915(119863 = 200)


1198916(119863 = 200)


1198917(119863 = 200)


1198918(119863 = 200)








1198919(119863 = 200)


11989110(119863 = 200)


11989111(119863 = 200)


11989112(119863 = 200)


11989113(119863 = 200)


11989114(119863 = 200)


11989115(119863 = 200)




11989116(119863 = 2)


11989117(119863 = 2)


11989118(119863 = 2)


11989119(119863 = 2)




1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918



1198919

11989110

11989111

11989112

11989113

11989114

11989115



11989116

11989117

11989118

11989119









9sim11989115



0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

times105

ABCBAGGSADA

PSOGSAMFOLMFO

f1




0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

ABCBAGGSADA

PSOGSAMFOLMFO

10150

10100

1050

100

10minus50

10minus100

f2


0 200 400 600 800 10000

1

2

3

4

5

6

7

8

9

Iteration

Aver

age fi

tnes

s val

ue

times106

ABCBAGGSADA

PSOGSAMFOLMFO

f3



0 200 400 600 800 10000

102030405060708090

100

Iteration

Aver

age fi

tnes

s val

ue

f4

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

05

1

15

2

25

3

Iteration

Aver

age fi

tnes

s val

ue

f5times109

ABCBAGGSADA

PSOGSAMFOLMFO





17 11989118 and


0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

f6times105

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Iteration

Aver

age fi

tnes

s val

ue

f7

ABCBAGGSADA

PSOGSAMFOLMFO





17and 119891


0 200 400 600 800 10000

50

100

150

200

250

Iteration

Aver

age fi

tnes

s val

ue

f8

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

Iteration

Aver

age fi

tnes

s val

ue

f9

ABCBAGGSADA

PSOGSAMFOLMFO






0 200 400 600 800 10000

5

10

15

20

25

Iteration

Aver

age fi

tnes

s val

ue

f10

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

Iteration

Aver

age fi

tnes

s val

ue

f11

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

1

2

3

4

5

6

7

Iteration

Aver

age fi

tnes

s val

ue

f12times109

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

2

4

6

8

10

12

14

Iteration

Aver

age fi

tnes

s val

ue

f13times109

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

100

200

300

400

500

600

Iteration

Aver

age fi

tnes

s val

ue

f14

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

f151020

100

10minus20

10minus40

10minus60

10minus80

10minus100

10minus120

ABCBAGGSADA

PSOGSAMFOLMFO




Consider = [1199091 1199092 1199093 1199094] = [ℎ 119897 119905 119887]

Minimize 119891 ()

= 1104711199092

11199092

0 200 400 600 800 10000

10

20

30

40

50

60

70

80

90

100

Iteration

Aver

age fi

tnes

s val

ue

f16

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

minus055

Iteration

Aver

age fi

tnes

s val

ue

f17

ABCBAGGSADA

PSOGSAMFOLMFO


+ 00481111990931199094(140 + 119909

2)


1198922 () = 120590 () minus 120590max le 0

1198923 () = 120575 () minus 120575max le 0

1198924 () = 119909

1minus 1199094le 0

1198925 () = 119875 minus 119875

119888 () le 0

1198926 () = 0125 minus 119909

1le 0


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

Iteration

Aver

age fi

tnes

s val

ue

f18

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

Iteration

Aver

age fi

tnes

s val

ue

f19

ABCBAGGSADA

PSOGSAMFOLMFO


1198927 ()

= 1104711199092

1

+ 00481111990931199094(140 + 119909

2) minus 50

le 0


01 le 1199092le 10

01 le 1199093le 10

01 le 1199094le 2


0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f1times105



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f2times1099



0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f3times106




0102030405060708090

100

Algorithms

Fitn

ess v

alue

f4



0

1

2

3

4

5

6

7

8

9

Algorithms

Fitn

ess v

alue

f5times108



005

115

225

335

445

Algorithms

Fitn

ess v

alue

f6times105



0

500

1000

1500

2000

2500

Algorithms

Fitn

ess v

alue

f7



20406080

100120140160180200

Algorithms

Fitn

ess v

alue

f8



0

500

1000

1500

2000

Algorithms

Fitn

ess v

alue

f9




0

2

4

6

8

10

12

14

16

18

20

Algorithms

Fitn

ess v

alue

f10



0

1000

2000

3000

4000

5000

Algorithms

Fitn

ess v

alue

f11



002040608

112141618

2

Algorithms

Fitn

ess v

alue

f12times109



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f13times109



0

50

100

150

200

Algorithms

Fitn

ess v

alue

f14



0

2000

4000

6000

8000

10000

Algorithms

Fitn

ess v

alue

f15




0

10

20

30

40

50

60

70

80

Algorithms

Fitn

ess v

alue

f16



minus1

minus09

minus08

minus07

minus06

minus05

minus04

Algorithms

Fitn

ess v

alue

f17


Where 120591 () = radic(1205911015840)2+ 2120591101584012059110158401015840

1199092

2119877+ (12059110158401015840)

2

1205911015840=

119875

radic211990911199092

12059110158401015840=119872119877

119869

119872 = 119875(119871 +1199092

2)

119877 = radic1199092

2

4+ (

1199091+ 1199093

2)

2

119869 = 2radic211990911199092[1199092

2

4+ (

1199091+ 1199093

2)

2

]

120590 () =6119875119871

11990941199092

3

120575 () =61198751198713

1198641199092

31199094



Algorithms

Fitn

ess v

alue

f18




0

Algorithms

Fitn

ess v

alue

f19


119875119888 ()

=

4013119864radic1199092

31199096

436

1198712(1 minus

1199093

2119871

radic119864

4119866)

119875 = 6000119897119887

119871 = 14 in

120575max = 025 in

119864 = 30 times 16 psi

119866 = 12 times 106 psi

120591max = 13600 psi

120590max = 30000 psi(13)






ht

l

P

L

t


l1

l2

d1

d2z





1sim1199097denote


1)





Minimize 119891 ()

= 0785411990911199092

2(33333119909

2

3+ 149334119909

3minus 430934)

minus 15081199091(1199092

6+ 1199092

7) + 74777 (119909

3

6+ 1199093

7)

+ 07854 (11990941199092

6+ 11990951199092

7)

Subject to 1198921 () =

27

11990911199092

21199093

minus 1 le 0

1198922 () =

3975

11990911199092

21199092

3

minus 1 le 0

1198923 () =

1931199093

4

11990921199094

61199093

minus 1 le 0

1198924 () =

1931199093

5

11990921199095

71199093

minus 1 le 0

1198925 () =

[(745119909411990921199093)2+ 169 times 10

6]12

1101199093

6

minus 1 le 0

1198926 () =

[(745119909511990921199093)2+ 1575 times 10

6]12

851199093

7

minus 1

le 0

1198927 () =

11990921199093

40minus 1 le 0

1198928 () =

51199092

1199091

minus 1 le 0

1198929 () =

1199091

121199092

minus 1 le 0

11989210 () =

151199096+ 19

1199094

minus 1 le 0

11989211 () =

111199096+ 17

1199095

minus 1 le 0

where 26 le 1199091le 36




2) 119911 (119909

3) 119897

1(1199094) 119897

2(1199095) 119889

1(1199096) 119889

2(1199097)


07 le 1199092le 08

17 le 1199093le 28

73 le 1199094le 83

73 le 1199095le 83

29 le 1199096le 39

50 le 1199097le 55

(14)















Competing Interests


Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198911(119863 = 200)


1198912(119863 = 200)


1198913(119863 = 200)


1198914(119863 = 200)


1198915(119863 = 200)


1198916(119863 = 200)


1198917(119863 = 200)


1198918(119863 = 200)








1198919(119863 = 200)


11989110(119863 = 200)


11989111(119863 = 200)


11989112(119863 = 200)


11989113(119863 = 200)


11989114(119863 = 200)


11989115(119863 = 200)




11989116(119863 = 2)


11989117(119863 = 2)


11989118(119863 = 2)


11989119(119863 = 2)




1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918



1198919

11989110

11989111

11989112

11989113

11989114

11989115



11989116

11989117

11989118

11989119









9sim11989115



0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

times105

ABCBAGGSADA

PSOGSAMFOLMFO

f1




0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

ABCBAGGSADA

PSOGSAMFOLMFO

10150

10100

1050

100

10minus50

10minus100

f2


0 200 400 600 800 10000

1

2

3

4

5

6

7

8

9

Iteration

Aver

age fi

tnes

s val

ue

times106

ABCBAGGSADA

PSOGSAMFOLMFO

f3



0 200 400 600 800 10000

102030405060708090

100

Iteration

Aver

age fi

tnes

s val

ue

f4

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

05

1

15

2

25

3

Iteration

Aver

age fi

tnes

s val

ue

f5times109

ABCBAGGSADA

PSOGSAMFOLMFO





17 11989118 and


0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

f6times105

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Iteration

Aver

age fi

tnes

s val

ue

f7

ABCBAGGSADA

PSOGSAMFOLMFO





17and 119891


0 200 400 600 800 10000

50

100

150

200

250

Iteration

Aver

age fi

tnes

s val

ue

f8

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

Iteration

Aver

age fi

tnes

s val

ue

f9

ABCBAGGSADA

PSOGSAMFOLMFO






0 200 400 600 800 10000

5

10

15

20

25

Iteration

Aver

age fi

tnes

s val

ue

f10

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

Iteration

Aver

age fi

tnes

s val

ue

f11

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

1

2

3

4

5

6

7

Iteration

Aver

age fi

tnes

s val

ue

f12times109

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

2

4

6

8

10

12

14

Iteration

Aver

age fi

tnes

s val

ue

f13times109

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

100

200

300

400

500

600

Iteration

Aver

age fi

tnes

s val

ue

f14

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

f151020

100

10minus20

10minus40

10minus60

10minus80

10minus100

10minus120

ABCBAGGSADA

PSOGSAMFOLMFO




Consider = [1199091 1199092 1199093 1199094] = [ℎ 119897 119905 119887]

Minimize 119891 ()

= 1104711199092

11199092

0 200 400 600 800 10000

10

20

30

40

50

60

70

80

90

100

Iteration

Aver

age fi

tnes

s val

ue

f16

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

minus055

Iteration

Aver

age fi

tnes

s val

ue

f17

ABCBAGGSADA

PSOGSAMFOLMFO


+ 00481111990931199094(140 + 119909

2)


1198922 () = 120590 () minus 120590max le 0

1198923 () = 120575 () minus 120575max le 0

1198924 () = 119909

1minus 1199094le 0

1198925 () = 119875 minus 119875

119888 () le 0

1198926 () = 0125 minus 119909

1le 0


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

Iteration

Aver

age fi

tnes

s val

ue

f18

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

Iteration

Aver

age fi

tnes

s val

ue

f19

ABCBAGGSADA

PSOGSAMFOLMFO


1198927 ()

= 1104711199092

1

+ 00481111990931199094(140 + 119909

2) minus 50

le 0


01 le 1199092le 10

01 le 1199093le 10

01 le 1199094le 2


0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f1times105



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f2times1099



0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f3times106




0102030405060708090

100

Algorithms

Fitn

ess v

alue

f4



0

1

2

3

4

5

6

7

8

9

Algorithms

Fitn

ess v

alue

f5times108



005

115

225

335

445

Algorithms

Fitn

ess v

alue

f6times105



0

500

1000

1500

2000

2500

Algorithms

Fitn

ess v

alue

f7



20406080

100120140160180200

Algorithms

Fitn

ess v

alue

f8



0

500

1000

1500

2000

Algorithms

Fitn

ess v

alue

f9




0

2

4

6

8

10

12

14

16

18

20

Algorithms

Fitn

ess v

alue

f10



0

1000

2000

3000

4000

5000

Algorithms

Fitn

ess v

alue

f11



002040608

112141618

2

Algorithms

Fitn

ess v

alue

f12times109



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f13times109



0

50

100

150

200

Algorithms

Fitn

ess v

alue

f14



0

2000

4000

6000

8000

10000

Algorithms

Fitn

ess v

alue

f15




0

10

20

30

40

50

60

70

80

Algorithms

Fitn

ess v

alue

f16



minus1

minus09

minus08

minus07

minus06

minus05

minus04

Algorithms

Fitn

ess v

alue

f17


Where 120591 () = radic(1205911015840)2+ 2120591101584012059110158401015840

1199092

2119877+ (12059110158401015840)

2

1205911015840=

119875

radic211990911199092

12059110158401015840=119872119877

119869

119872 = 119875(119871 +1199092

2)

119877 = radic1199092

2

4+ (

1199091+ 1199093

2)

2

119869 = 2radic211990911199092[1199092

2

4+ (

1199091+ 1199093

2)

2

]

120590 () =6119875119871

11990941199092

3

120575 () =61198751198713

1198641199092

31199094



Algorithms

Fitn

ess v

alue

f18




0

Algorithms

Fitn

ess v

alue

f19


119875119888 ()

=

4013119864radic1199092

31199096

436

1198712(1 minus

1199093

2119871

radic119864

4119866)

119875 = 6000119897119887

119871 = 14 in

120575max = 025 in

119864 = 30 times 16 psi

119866 = 12 times 106 psi

120591max = 13600 psi

120590max = 30000 psi(13)






ht

l

P

L

t


l1

l2

d1

d2z





1sim1199097denote


1)





Minimize 119891 ()

= 0785411990911199092

2(33333119909

2

3+ 149334119909

3minus 430934)

minus 15081199091(1199092

6+ 1199092

7) + 74777 (119909

3

6+ 1199093

7)

+ 07854 (11990941199092

6+ 11990951199092

7)

Subject to 1198921 () =

27

11990911199092

21199093

minus 1 le 0

1198922 () =

3975

11990911199092

21199092

3

minus 1 le 0

1198923 () =

1931199093

4

11990921199094

61199093

minus 1 le 0

1198924 () =

1931199093

5

11990921199095

71199093

minus 1 le 0

1198925 () =

[(745119909411990921199093)2+ 169 times 10

6]12

1101199093

6

minus 1 le 0

1198926 () =

[(745119909511990921199093)2+ 1575 times 10

6]12

851199093

7

minus 1

le 0

1198927 () =

11990921199093

40minus 1 le 0

1198928 () =

51199092

1199091

minus 1 le 0

1198929 () =

1199091

121199092

minus 1 le 0

11989210 () =

151199096+ 19

1199094

minus 1 le 0

11989211 () =

111199096+ 17

1199095

minus 1 le 0

where 26 le 1199091le 36




2) 119911 (119909

3) 119897

1(1199094) 119897

2(1199095) 119889

1(1199096) 119889

2(1199097)


07 le 1199092le 08

17 le 1199093le 28

73 le 1199094le 83

73 le 1199095le 83

29 le 1199096le 39

50 le 1199097le 55

(14)















Competing Interests


Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198919(119863 = 200)


11989110(119863 = 200)


11989111(119863 = 200)


11989112(119863 = 200)


11989113(119863 = 200)


11989114(119863 = 200)


11989115(119863 = 200)




11989116(119863 = 2)


11989117(119863 = 2)


11989118(119863 = 2)


11989119(119863 = 2)




1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918



1198919

11989110

11989111

11989112

11989113

11989114

11989115



11989116

11989117

11989118

11989119









9sim11989115



0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

times105

ABCBAGGSADA

PSOGSAMFOLMFO

f1




0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

ABCBAGGSADA

PSOGSAMFOLMFO

10150

10100

1050

100

10minus50

10minus100

f2


0 200 400 600 800 10000

1

2

3

4

5

6

7

8

9

Iteration

Aver

age fi

tnes

s val

ue

times106

ABCBAGGSADA

PSOGSAMFOLMFO

f3



0 200 400 600 800 10000

102030405060708090

100

Iteration

Aver

age fi

tnes

s val

ue

f4

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

05

1

15

2

25

3

Iteration

Aver

age fi

tnes

s val

ue

f5times109

ABCBAGGSADA

PSOGSAMFOLMFO





17 11989118 and


0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

f6times105

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Iteration

Aver

age fi

tnes

s val

ue

f7

ABCBAGGSADA

PSOGSAMFOLMFO





17and 119891


0 200 400 600 800 10000

50

100

150

200

250

Iteration

Aver

age fi

tnes

s val

ue

f8

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

Iteration

Aver

age fi

tnes

s val

ue

f9

ABCBAGGSADA

PSOGSAMFOLMFO






0 200 400 600 800 10000

5

10

15

20

25

Iteration

Aver

age fi

tnes

s val

ue

f10

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

Iteration

Aver

age fi

tnes

s val

ue

f11

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

1

2

3

4

5

6

7

Iteration

Aver

age fi

tnes

s val

ue

f12times109

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

2

4

6

8

10

12

14

Iteration

Aver

age fi

tnes

s val

ue

f13times109

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

100

200

300

400

500

600

Iteration

Aver

age fi

tnes

s val

ue

f14

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

f151020

100

10minus20

10minus40

10minus60

10minus80

10minus100

10minus120

ABCBAGGSADA

PSOGSAMFOLMFO




Consider = [1199091 1199092 1199093 1199094] = [ℎ 119897 119905 119887]

Minimize 119891 ()

= 1104711199092

11199092

0 200 400 600 800 10000

10

20

30

40

50

60

70

80

90

100

Iteration

Aver

age fi

tnes

s val

ue

f16

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

minus055

Iteration

Aver

age fi

tnes

s val

ue

f17

ABCBAGGSADA

PSOGSAMFOLMFO


+ 00481111990931199094(140 + 119909

2)


1198922 () = 120590 () minus 120590max le 0

1198923 () = 120575 () minus 120575max le 0

1198924 () = 119909

1minus 1199094le 0

1198925 () = 119875 minus 119875

119888 () le 0

1198926 () = 0125 minus 119909

1le 0


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

Iteration

Aver

age fi

tnes

s val

ue

f18

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

Iteration

Aver

age fi

tnes

s val

ue

f19

ABCBAGGSADA

PSOGSAMFOLMFO


1198927 ()

= 1104711199092

1

+ 00481111990931199094(140 + 119909

2) minus 50

le 0


01 le 1199092le 10

01 le 1199093le 10

01 le 1199094le 2


0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f1times105



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f2times1099



0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f3times106




0102030405060708090

100

Algorithms

Fitn

ess v

alue

f4



0

1

2

3

4

5

6

7

8

9

Algorithms

Fitn

ess v

alue

f5times108



005

115

225

335

445

Algorithms

Fitn

ess v

alue

f6times105



0

500

1000

1500

2000

2500

Algorithms

Fitn

ess v

alue

f7



20406080

100120140160180200

Algorithms

Fitn

ess v

alue

f8



0

500

1000

1500

2000

Algorithms

Fitn

ess v

alue

f9




0

2

4

6

8

10

12

14

16

18

20

Algorithms

Fitn

ess v

alue

f10



0

1000

2000

3000

4000

5000

Algorithms

Fitn

ess v

alue

f11



002040608

112141618

2

Algorithms

Fitn

ess v

alue

f12times109



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f13times109



0

50

100

150

200

Algorithms

Fitn

ess v

alue

f14



0

2000

4000

6000

8000

10000

Algorithms

Fitn

ess v

alue

f15




0

10

20

30

40

50

60

70

80

Algorithms

Fitn

ess v

alue

f16



minus1

minus09

minus08

minus07

minus06

minus05

minus04

Algorithms

Fitn

ess v

alue

f17


Where 120591 () = radic(1205911015840)2+ 2120591101584012059110158401015840

1199092

2119877+ (12059110158401015840)

2

1205911015840=

119875

radic211990911199092

12059110158401015840=119872119877

119869

119872 = 119875(119871 +1199092

2)

119877 = radic1199092

2

4+ (

1199091+ 1199093

2)

2

119869 = 2radic211990911199092[1199092

2

4+ (

1199091+ 1199093

2)

2

]

120590 () =6119875119871

11990941199092

3

120575 () =61198751198713

1198641199092

31199094



Algorithms

Fitn

ess v

alue

f18




0

Algorithms

Fitn

ess v

alue

f19


119875119888 ()

=

4013119864radic1199092

31199096

436

1198712(1 minus

1199093

2119871

radic119864

4119866)

119875 = 6000119897119887

119871 = 14 in

120575max = 025 in

119864 = 30 times 16 psi

119866 = 12 times 106 psi

120591max = 13600 psi

120590max = 30000 psi(13)






ht

l

P

L

t


l1

l2

d1

d2z





1sim1199097denote


1)





Minimize 119891 ()

= 0785411990911199092

2(33333119909

2

3+ 149334119909

3minus 430934)

minus 15081199091(1199092

6+ 1199092

7) + 74777 (119909

3

6+ 1199093

7)

+ 07854 (11990941199092

6+ 11990951199092

7)

Subject to 1198921 () =

27

11990911199092

21199093

minus 1 le 0

1198922 () =

3975

11990911199092

21199092

3

minus 1 le 0

1198923 () =

1931199093

4

11990921199094

61199093

minus 1 le 0

1198924 () =

1931199093

5

11990921199095

71199093

minus 1 le 0

1198925 () =

[(745119909411990921199093)2+ 169 times 10

6]12

1101199093

6

minus 1 le 0

1198926 () =

[(745119909511990921199093)2+ 1575 times 10

6]12

851199093

7

minus 1

le 0

1198927 () =

11990921199093

40minus 1 le 0

1198928 () =

51199092

1199091

minus 1 le 0

1198929 () =

1199091

121199092

minus 1 le 0

11989210 () =

151199096+ 19

1199094

minus 1 le 0

11989211 () =

111199096+ 17

1199095

minus 1 le 0

where 26 le 1199091le 36




2) 119911 (119909

3) 119897

1(1199094) 119897

2(1199095) 119889

1(1199096) 119889

2(1199097)


07 le 1199092le 08

17 le 1199093le 28

73 le 1199094le 83

73 le 1199095le 83

29 le 1199096le 39

50 le 1199097le 55

(14)















Competing Interests


Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918



1198919

11989110

11989111

11989112

11989113

11989114

11989115



11989116

11989117

11989118

11989119









9sim11989115



0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

times105

ABCBAGGSADA

PSOGSAMFOLMFO

f1




0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

ABCBAGGSADA

PSOGSAMFOLMFO

10150

10100

1050

100

10minus50

10minus100

f2


0 200 400 600 800 10000

1

2

3

4

5

6

7

8

9

Iteration

Aver

age fi

tnes

s val

ue

times106

ABCBAGGSADA

PSOGSAMFOLMFO

f3



0 200 400 600 800 10000

102030405060708090

100

Iteration

Aver

age fi

tnes

s val

ue

f4

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

05

1

15

2

25

3

Iteration

Aver

age fi

tnes

s val

ue

f5times109

ABCBAGGSADA

PSOGSAMFOLMFO





17 11989118 and


0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

f6times105

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Iteration

Aver

age fi

tnes

s val

ue

f7

ABCBAGGSADA

PSOGSAMFOLMFO





17and 119891


0 200 400 600 800 10000

50

100

150

200

250

Iteration

Aver

age fi

tnes

s val

ue

f8

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

Iteration

Aver

age fi

tnes

s val

ue

f9

ABCBAGGSADA

PSOGSAMFOLMFO






0 200 400 600 800 10000

5

10

15

20

25

Iteration

Aver

age fi

tnes

s val

ue

f10

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

Iteration

Aver

age fi

tnes

s val

ue

f11

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

1

2

3

4

5

6

7

Iteration

Aver

age fi

tnes

s val

ue

f12times109

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

2

4

6

8

10

12

14

Iteration

Aver

age fi

tnes

s val

ue

f13times109

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

100

200

300

400

500

600

Iteration

Aver

age fi

tnes

s val

ue

f14

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

f151020

100

10minus20

10minus40

10minus60

10minus80

10minus100

10minus120

ABCBAGGSADA

PSOGSAMFOLMFO




Consider = [1199091 1199092 1199093 1199094] = [ℎ 119897 119905 119887]

Minimize 119891 ()

= 1104711199092

11199092

0 200 400 600 800 10000

10

20

30

40

50

60

70

80

90

100

Iteration

Aver

age fi

tnes

s val

ue

f16

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

minus055

Iteration

Aver

age fi

tnes

s val

ue

f17

ABCBAGGSADA

PSOGSAMFOLMFO


+ 00481111990931199094(140 + 119909

2)


1198922 () = 120590 () minus 120590max le 0

1198923 () = 120575 () minus 120575max le 0

1198924 () = 119909

1minus 1199094le 0

1198925 () = 119875 minus 119875

119888 () le 0

1198926 () = 0125 minus 119909

1le 0


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

Iteration

Aver

age fi

tnes

s val

ue

f18

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

Iteration

Aver

age fi

tnes

s val

ue

f19

ABCBAGGSADA

PSOGSAMFOLMFO


1198927 ()

= 1104711199092

1

+ 00481111990931199094(140 + 119909

2) minus 50

le 0


01 le 1199092le 10

01 le 1199093le 10

01 le 1199094le 2


0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f1times105



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f2times1099



0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f3times106




0102030405060708090

100

Algorithms

Fitn

ess v

alue

f4



0

1

2

3

4

5

6

7

8

9

Algorithms

Fitn

ess v

alue

f5times108



005

115

225

335

445

Algorithms

Fitn

ess v

alue

f6times105



0

500

1000

1500

2000

2500

Algorithms

Fitn

ess v

alue

f7



20406080

100120140160180200

Algorithms

Fitn

ess v

alue

f8



0

500

1000

1500

2000

Algorithms

Fitn

ess v

alue

f9




0

2

4

6

8

10

12

14

16

18

20

Algorithms

Fitn

ess v

alue

f10



0

1000

2000

3000

4000

5000

Algorithms

Fitn

ess v

alue

f11



002040608

112141618

2

Algorithms

Fitn

ess v

alue

f12times109



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f13times109



0

50

100

150

200

Algorithms

Fitn

ess v

alue

f14



0

2000

4000

6000

8000

10000

Algorithms

Fitn

ess v

alue

f15




0

10

20

30

40

50

60

70

80

Algorithms

Fitn

ess v

alue

f16



minus1

minus09

minus08

minus07

minus06

minus05

minus04

Algorithms

Fitn

ess v

alue

f17


Where 120591 () = radic(1205911015840)2+ 2120591101584012059110158401015840

1199092

2119877+ (12059110158401015840)

2

1205911015840=

119875

radic211990911199092

12059110158401015840=119872119877

119869

119872 = 119875(119871 +1199092

2)

119877 = radic1199092

2

4+ (

1199091+ 1199093

2)

2

119869 = 2radic211990911199092[1199092

2

4+ (

1199091+ 1199093

2)

2

]

120590 () =6119875119871

11990941199092

3

120575 () =61198751198713

1198641199092

31199094



Algorithms

Fitn

ess v

alue

f18




0

Algorithms

Fitn

ess v

alue

f19


119875119888 ()

=

4013119864radic1199092

31199096

436

1198712(1 minus

1199093

2119871

radic119864

4119866)

119875 = 6000119897119887

119871 = 14 in

120575max = 025 in

119864 = 30 times 16 psi

119866 = 12 times 106 psi

120591max = 13600 psi

120590max = 30000 psi(13)






ht

l

P

L

t


l1

l2

d1

d2z





1sim1199097denote


1)





Minimize 119891 ()

= 0785411990911199092

2(33333119909

2

3+ 149334119909

3minus 430934)

minus 15081199091(1199092

6+ 1199092

7) + 74777 (119909

3

6+ 1199093

7)

+ 07854 (11990941199092

6+ 11990951199092

7)

Subject to 1198921 () =

27

11990911199092

21199093

minus 1 le 0

1198922 () =

3975

11990911199092

21199092

3

minus 1 le 0

1198923 () =

1931199093

4

11990921199094

61199093

minus 1 le 0

1198924 () =

1931199093

5

11990921199095

71199093

minus 1 le 0

1198925 () =

[(745119909411990921199093)2+ 169 times 10

6]12

1101199093

6

minus 1 le 0

1198926 () =

[(745119909511990921199093)2+ 1575 times 10

6]12

851199093

7

minus 1

le 0

1198927 () =

11990921199093

40minus 1 le 0

1198928 () =

51199092

1199091

minus 1 le 0

1198929 () =

1199091

121199092

minus 1 le 0

11989210 () =

151199096+ 19

1199094

minus 1 le 0

11989211 () =

111199096+ 17

1199095

minus 1 le 0

where 26 le 1199091le 36




2) 119911 (119909

3) 119897

1(1199094) 119897

2(1199095) 119889

1(1199096) 119889

2(1199097)


07 le 1199092le 08

17 le 1199093le 28

73 le 1199094le 83

73 le 1199095le 83

29 le 1199096le 39

50 le 1199097le 55

(14)















Competing Interests


Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

ABCBAGGSADA

PSOGSAMFOLMFO

10150

10100

1050

100

10minus50

10minus100

f2


0 200 400 600 800 10000

1

2

3

4

5

6

7

8

9

Iteration

Aver

age fi

tnes

s val

ue

times106

ABCBAGGSADA

PSOGSAMFOLMFO

f3



0 200 400 600 800 10000

102030405060708090

100

Iteration

Aver

age fi

tnes

s val

ue

f4

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

05

1

15

2

25

3

Iteration

Aver

age fi

tnes

s val

ue

f5times109

ABCBAGGSADA

PSOGSAMFOLMFO





17 11989118 and


0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

f6times105

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Iteration

Aver

age fi

tnes

s val

ue

f7

ABCBAGGSADA

PSOGSAMFOLMFO





17and 119891


0 200 400 600 800 10000

50

100

150

200

250

Iteration

Aver

age fi

tnes

s val

ue

f8

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

Iteration

Aver

age fi

tnes

s val

ue

f9

ABCBAGGSADA

PSOGSAMFOLMFO






0 200 400 600 800 10000

5

10

15

20

25

Iteration

Aver

age fi

tnes

s val

ue

f10

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

Iteration

Aver

age fi

tnes

s val

ue

f11

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

1

2

3

4

5

6

7

Iteration

Aver

age fi

tnes

s val

ue

f12times109

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

2

4

6

8

10

12

14

Iteration

Aver

age fi

tnes

s val

ue

f13times109

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

100

200

300

400

500

600

Iteration

Aver

age fi

tnes

s val

ue

f14

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

f151020

100

10minus20

10minus40

10minus60

10minus80

10minus100

10minus120

ABCBAGGSADA

PSOGSAMFOLMFO




Consider = [1199091 1199092 1199093 1199094] = [ℎ 119897 119905 119887]

Minimize 119891 ()

= 1104711199092

11199092

0 200 400 600 800 10000

10

20

30

40

50

60

70

80

90

100

Iteration

Aver

age fi

tnes

s val

ue

f16

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

minus055

Iteration

Aver

age fi

tnes

s val

ue

f17

ABCBAGGSADA

PSOGSAMFOLMFO


+ 00481111990931199094(140 + 119909

2)


1198922 () = 120590 () minus 120590max le 0

1198923 () = 120575 () minus 120575max le 0

1198924 () = 119909

1minus 1199094le 0

1198925 () = 119875 minus 119875

119888 () le 0

1198926 () = 0125 minus 119909

1le 0


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

Iteration

Aver

age fi

tnes

s val

ue

f18

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

Iteration

Aver

age fi

tnes

s val

ue

f19

ABCBAGGSADA

PSOGSAMFOLMFO


1198927 ()

= 1104711199092

1

+ 00481111990931199094(140 + 119909

2) minus 50

le 0


01 le 1199092le 10

01 le 1199093le 10

01 le 1199094le 2


0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f1times105



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f2times1099



0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f3times106




0102030405060708090

100

Algorithms

Fitn

ess v

alue

f4



0

1

2

3

4

5

6

7

8

9

Algorithms

Fitn

ess v

alue

f5times108



005

115

225

335

445

Algorithms

Fitn

ess v

alue

f6times105



0

500

1000

1500

2000

2500

Algorithms

Fitn

ess v

alue

f7



20406080

100120140160180200

Algorithms

Fitn

ess v

alue

f8



0

500

1000

1500

2000

Algorithms

Fitn

ess v

alue

f9




0

2

4

6

8

10

12

14

16

18

20

Algorithms

Fitn

ess v

alue

f10



0

1000

2000

3000

4000

5000

Algorithms

Fitn

ess v

alue

f11



002040608

112141618

2

Algorithms

Fitn

ess v

alue

f12times109



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f13times109



0

50

100

150

200

Algorithms

Fitn

ess v

alue

f14



0

2000

4000

6000

8000

10000

Algorithms

Fitn

ess v

alue

f15




0

10

20

30

40

50

60

70

80

Algorithms

Fitn

ess v

alue

f16



minus1

minus09

minus08

minus07

minus06

minus05

minus04

Algorithms

Fitn

ess v

alue

f17


Where 120591 () = radic(1205911015840)2+ 2120591101584012059110158401015840

1199092

2119877+ (12059110158401015840)

2

1205911015840=

119875

radic211990911199092

12059110158401015840=119872119877

119869

119872 = 119875(119871 +1199092

2)

119877 = radic1199092

2

4+ (

1199091+ 1199093

2)

2

119869 = 2radic211990911199092[1199092

2

4+ (

1199091+ 1199093

2)

2

]

120590 () =6119875119871

11990941199092

3

120575 () =61198751198713

1198641199092

31199094



Algorithms

Fitn

ess v

alue

f18




0

Algorithms

Fitn

ess v

alue

f19


119875119888 ()

=

4013119864radic1199092

31199096

436

1198712(1 minus

1199093

2119871

radic119864

4119866)

119875 = 6000119897119887

119871 = 14 in

120575max = 025 in

119864 = 30 times 16 psi

119866 = 12 times 106 psi

120591max = 13600 psi

120590max = 30000 psi(13)






ht

l

P

L

t


l1

l2

d1

d2z





1sim1199097denote


1)





Minimize 119891 ()

= 0785411990911199092

2(33333119909

2

3+ 149334119909

3minus 430934)

minus 15081199091(1199092

6+ 1199092

7) + 74777 (119909

3

6+ 1199093

7)

+ 07854 (11990941199092

6+ 11990951199092

7)

Subject to 1198921 () =

27

11990911199092

21199093

minus 1 le 0

1198922 () =

3975

11990911199092

21199092

3

minus 1 le 0

1198923 () =

1931199093

4

11990921199094

61199093

minus 1 le 0

1198924 () =

1931199093

5

11990921199095

71199093

minus 1 le 0

1198925 () =

[(745119909411990921199093)2+ 169 times 10

6]12

1101199093

6

minus 1 le 0

1198926 () =

[(745119909511990921199093)2+ 1575 times 10

6]12

851199093

7

minus 1

le 0

1198927 () =

11990921199093

40minus 1 le 0

1198928 () =

51199092

1199091

minus 1 le 0

1198929 () =

1199091

121199092

minus 1 le 0

11989210 () =

151199096+ 19

1199094

minus 1 le 0

11989211 () =

111199096+ 17

1199095

minus 1 le 0

where 26 le 1199091le 36




2) 119911 (119909

3) 119897

1(1199094) 119897

2(1199095) 119889

1(1199096) 119889

2(1199097)


07 le 1199092le 08

17 le 1199093le 28

73 le 1199094le 83

73 le 1199095le 83

29 le 1199096le 39

50 le 1199097le 55

(14)















Competing Interests


Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 200 400 600 800 10000

1

2

3

4

5

6

Iteration

Aver

age fi

tnes

s val

ue

f6times105

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Iteration

Aver

age fi

tnes

s val

ue

f7

ABCBAGGSADA

PSOGSAMFOLMFO





17and 119891


0 200 400 600 800 10000

50

100

150

200

250

Iteration

Aver

age fi

tnes

s val

ue

f8

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

Iteration

Aver

age fi

tnes

s val

ue

f9

ABCBAGGSADA

PSOGSAMFOLMFO






0 200 400 600 800 10000

5

10

15

20

25

Iteration

Aver

age fi

tnes

s val

ue

f10

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

Iteration

Aver

age fi

tnes

s val

ue

f11

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

1

2

3

4

5

6

7

Iteration

Aver

age fi

tnes

s val

ue

f12times109

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

2

4

6

8

10

12

14

Iteration

Aver

age fi

tnes

s val

ue

f13times109

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

100

200

300

400

500

600

Iteration

Aver

age fi

tnes

s val

ue

f14

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

f151020

100

10minus20

10minus40

10minus60

10minus80

10minus100

10minus120

ABCBAGGSADA

PSOGSAMFOLMFO




Consider = [1199091 1199092 1199093 1199094] = [ℎ 119897 119905 119887]

Minimize 119891 ()

= 1104711199092

11199092

0 200 400 600 800 10000

10

20

30

40

50

60

70

80

90

100

Iteration

Aver

age fi

tnes

s val

ue

f16

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

minus055

Iteration

Aver

age fi

tnes

s val

ue

f17

ABCBAGGSADA

PSOGSAMFOLMFO


+ 00481111990931199094(140 + 119909

2)


1198922 () = 120590 () minus 120590max le 0

1198923 () = 120575 () minus 120575max le 0

1198924 () = 119909

1minus 1199094le 0

1198925 () = 119875 minus 119875

119888 () le 0

1198926 () = 0125 minus 119909

1le 0


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

Iteration

Aver

age fi

tnes

s val

ue

f18

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

Iteration

Aver

age fi

tnes

s val

ue

f19

ABCBAGGSADA

PSOGSAMFOLMFO


1198927 ()

= 1104711199092

1

+ 00481111990931199094(140 + 119909

2) minus 50

le 0


01 le 1199092le 10

01 le 1199093le 10

01 le 1199094le 2


0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f1times105



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f2times1099



0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f3times106




0102030405060708090

100

Algorithms

Fitn

ess v

alue

f4



0

1

2

3

4

5

6

7

8

9

Algorithms

Fitn

ess v

alue

f5times108



005

115

225

335

445

Algorithms

Fitn

ess v

alue

f6times105



0

500

1000

1500

2000

2500

Algorithms

Fitn

ess v

alue

f7



20406080

100120140160180200

Algorithms

Fitn

ess v

alue

f8



0

500

1000

1500

2000

Algorithms

Fitn

ess v

alue

f9




0

2

4

6

8

10

12

14

16

18

20

Algorithms

Fitn

ess v

alue

f10



0

1000

2000

3000

4000

5000

Algorithms

Fitn

ess v

alue

f11



002040608

112141618

2

Algorithms

Fitn

ess v

alue

f12times109



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f13times109



0

50

100

150

200

Algorithms

Fitn

ess v

alue

f14



0

2000

4000

6000

8000

10000

Algorithms

Fitn

ess v

alue

f15




0

10

20

30

40

50

60

70

80

Algorithms

Fitn

ess v

alue

f16



minus1

minus09

minus08

minus07

minus06

minus05

minus04

Algorithms

Fitn

ess v

alue

f17


Where 120591 () = radic(1205911015840)2+ 2120591101584012059110158401015840

1199092

2119877+ (12059110158401015840)

2

1205911015840=

119875

radic211990911199092

12059110158401015840=119872119877

119869

119872 = 119875(119871 +1199092

2)

119877 = radic1199092

2

4+ (

1199091+ 1199093

2)

2

119869 = 2radic211990911199092[1199092

2

4+ (

1199091+ 1199093

2)

2

]

120590 () =6119875119871

11990941199092

3

120575 () =61198751198713

1198641199092

31199094



Algorithms

Fitn

ess v

alue

f18




0

Algorithms

Fitn

ess v

alue

f19


119875119888 ()

=

4013119864radic1199092

31199096

436

1198712(1 minus

1199093

2119871

radic119864

4119866)

119875 = 6000119897119887

119871 = 14 in

120575max = 025 in

119864 = 30 times 16 psi

119866 = 12 times 106 psi

120591max = 13600 psi

120590max = 30000 psi(13)






ht

l

P

L

t


l1

l2

d1

d2z





1sim1199097denote


1)





Minimize 119891 ()

= 0785411990911199092

2(33333119909

2

3+ 149334119909

3minus 430934)

minus 15081199091(1199092

6+ 1199092

7) + 74777 (119909

3

6+ 1199093

7)

+ 07854 (11990941199092

6+ 11990951199092

7)

Subject to 1198921 () =

27

11990911199092

21199093

minus 1 le 0

1198922 () =

3975

11990911199092

21199092

3

minus 1 le 0

1198923 () =

1931199093

4

11990921199094

61199093

minus 1 le 0

1198924 () =

1931199093

5

11990921199095

71199093

minus 1 le 0

1198925 () =

[(745119909411990921199093)2+ 169 times 10

6]12

1101199093

6

minus 1 le 0

1198926 () =

[(745119909511990921199093)2+ 1575 times 10

6]12

851199093

7

minus 1

le 0

1198927 () =

11990921199093

40minus 1 le 0

1198928 () =

51199092

1199091

minus 1 le 0

1198929 () =

1199091

121199092

minus 1 le 0

11989210 () =

151199096+ 19

1199094

minus 1 le 0

11989211 () =

111199096+ 17

1199095

minus 1 le 0

where 26 le 1199091le 36




2) 119911 (119909

3) 119897

1(1199094) 119897

2(1199095) 119889

1(1199096) 119889

2(1199097)


07 le 1199092le 08

17 le 1199093le 28

73 le 1199094le 83

73 le 1199095le 83

29 le 1199096le 39

50 le 1199097le 55

(14)















Competing Interests


Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 200 400 600 800 10000

5

10

15

20

25

Iteration

Aver

age fi

tnes

s val

ue

f10

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

Iteration

Aver

age fi

tnes

s val

ue

f11

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

1

2

3

4

5

6

7

Iteration

Aver

age fi

tnes

s val

ue

f12times109

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 10000

2

4

6

8

10

12

14

Iteration

Aver

age fi

tnes

s val

ue

f13times109

ABCBAGGSADA

PSOGSAMFOLMFO





0 200 400 600 800 10000

100

200

300

400

500

600

Iteration

Aver

age fi

tnes

s val

ue

f14

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

f151020

100

10minus20

10minus40

10minus60

10minus80

10minus100

10minus120

ABCBAGGSADA

PSOGSAMFOLMFO




Consider = [1199091 1199092 1199093 1199094] = [ℎ 119897 119905 119887]

Minimize 119891 ()

= 1104711199092

11199092

0 200 400 600 800 10000

10

20

30

40

50

60

70

80

90

100

Iteration

Aver

age fi

tnes

s val

ue

f16

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

minus055

Iteration

Aver

age fi

tnes

s val

ue

f17

ABCBAGGSADA

PSOGSAMFOLMFO


+ 00481111990931199094(140 + 119909

2)


1198922 () = 120590 () minus 120590max le 0

1198923 () = 120575 () minus 120575max le 0

1198924 () = 119909

1minus 1199094le 0

1198925 () = 119875 minus 119875

119888 () le 0

1198926 () = 0125 minus 119909

1le 0


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

Iteration

Aver

age fi

tnes

s val

ue

f18

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

Iteration

Aver

age fi

tnes

s val

ue

f19

ABCBAGGSADA

PSOGSAMFOLMFO


1198927 ()

= 1104711199092

1

+ 00481111990931199094(140 + 119909

2) minus 50

le 0


01 le 1199092le 10

01 le 1199093le 10

01 le 1199094le 2


0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f1times105



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f2times1099



0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f3times106




0102030405060708090

100

Algorithms

Fitn

ess v

alue

f4



0

1

2

3

4

5

6

7

8

9

Algorithms

Fitn

ess v

alue

f5times108



005

115

225

335

445

Algorithms

Fitn

ess v

alue

f6times105



0

500

1000

1500

2000

2500

Algorithms

Fitn

ess v

alue

f7



20406080

100120140160180200

Algorithms

Fitn

ess v

alue

f8



0

500

1000

1500

2000

Algorithms

Fitn

ess v

alue

f9




0

2

4

6

8

10

12

14

16

18

20

Algorithms

Fitn

ess v

alue

f10



0

1000

2000

3000

4000

5000

Algorithms

Fitn

ess v

alue

f11



002040608

112141618

2

Algorithms

Fitn

ess v

alue

f12times109



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f13times109



0

50

100

150

200

Algorithms

Fitn

ess v

alue

f14



0

2000

4000

6000

8000

10000

Algorithms

Fitn

ess v

alue

f15




0

10

20

30

40

50

60

70

80

Algorithms

Fitn

ess v

alue

f16



minus1

minus09

minus08

minus07

minus06

minus05

minus04

Algorithms

Fitn

ess v

alue

f17


Where 120591 () = radic(1205911015840)2+ 2120591101584012059110158401015840

1199092

2119877+ (12059110158401015840)

2

1205911015840=

119875

radic211990911199092

12059110158401015840=119872119877

119869

119872 = 119875(119871 +1199092

2)

119877 = radic1199092

2

4+ (

1199091+ 1199093

2)

2

119869 = 2radic211990911199092[1199092

2

4+ (

1199091+ 1199093

2)

2

]

120590 () =6119875119871

11990941199092

3

120575 () =61198751198713

1198641199092

31199094



Algorithms

Fitn

ess v

alue

f18




0

Algorithms

Fitn

ess v

alue

f19


119875119888 ()

=

4013119864radic1199092

31199096

436

1198712(1 minus

1199093

2119871

radic119864

4119866)

119875 = 6000119897119887

119871 = 14 in

120575max = 025 in

119864 = 30 times 16 psi

119866 = 12 times 106 psi

120591max = 13600 psi

120590max = 30000 psi(13)






ht

l

P

L

t


l1

l2

d1

d2z





1sim1199097denote


1)





Minimize 119891 ()

= 0785411990911199092

2(33333119909

2

3+ 149334119909

3minus 430934)

minus 15081199091(1199092

6+ 1199092

7) + 74777 (119909

3

6+ 1199093

7)

+ 07854 (11990941199092

6+ 11990951199092

7)

Subject to 1198921 () =

27

11990911199092

21199093

minus 1 le 0

1198922 () =

3975

11990911199092

21199092

3

minus 1 le 0

1198923 () =

1931199093

4

11990921199094

61199093

minus 1 le 0

1198924 () =

1931199093

5

11990921199095

71199093

minus 1 le 0

1198925 () =

[(745119909411990921199093)2+ 169 times 10

6]12

1101199093

6

minus 1 le 0

1198926 () =

[(745119909511990921199093)2+ 1575 times 10

6]12

851199093

7

minus 1

le 0

1198927 () =

11990921199093

40minus 1 le 0

1198928 () =

51199092

1199091

minus 1 le 0

1198929 () =

1199091

121199092

minus 1 le 0

11989210 () =

151199096+ 19

1199094

minus 1 le 0

11989211 () =

111199096+ 17

1199095

minus 1 le 0

where 26 le 1199091le 36




2) 119911 (119909

3) 119897

1(1199094) 119897

2(1199095) 119889

1(1199096) 119889

2(1199097)


07 le 1199092le 08

17 le 1199093le 28

73 le 1199094le 83

73 le 1199095le 83

29 le 1199096le 39

50 le 1199097le 55

(14)















Competing Interests


Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 200 400 600 800 10000

100

200

300

400

500

600

Iteration

Aver

age fi

tnes

s val

ue

f14

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000Iteration

Aver

age fi

tnes

s val

ue (l

og)

f151020

100

10minus20

10minus40

10minus60

10minus80

10minus100

10minus120

ABCBAGGSADA

PSOGSAMFOLMFO




Consider = [1199091 1199092 1199093 1199094] = [ℎ 119897 119905 119887]

Minimize 119891 ()

= 1104711199092

11199092

0 200 400 600 800 10000

10

20

30

40

50

60

70

80

90

100

Iteration

Aver

age fi

tnes

s val

ue

f16

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

minus055

Iteration

Aver

age fi

tnes

s val

ue

f17

ABCBAGGSADA

PSOGSAMFOLMFO


+ 00481111990931199094(140 + 119909

2)


1198922 () = 120590 () minus 120590max le 0

1198923 () = 120575 () minus 120575max le 0

1198924 () = 119909

1minus 1199094le 0

1198925 () = 119875 minus 119875

119888 () le 0

1198926 () = 0125 minus 119909

1le 0


0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

Iteration

Aver

age fi

tnes

s val

ue

f18

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

Iteration

Aver

age fi

tnes

s val

ue

f19

ABCBAGGSADA

PSOGSAMFOLMFO


1198927 ()

= 1104711199092

1

+ 00481111990931199094(140 + 119909

2) minus 50

le 0


01 le 1199092le 10

01 le 1199093le 10

01 le 1199094le 2


0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f1times105



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f2times1099



0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f3times106




0102030405060708090

100

Algorithms

Fitn

ess v

alue

f4



0

1

2

3

4

5

6

7

8

9

Algorithms

Fitn

ess v

alue

f5times108



005

115

225

335

445

Algorithms

Fitn

ess v

alue

f6times105



0

500

1000

1500

2000

2500

Algorithms

Fitn

ess v

alue

f7



20406080

100120140160180200

Algorithms

Fitn

ess v

alue

f8



0

500

1000

1500

2000

Algorithms

Fitn

ess v

alue

f9




0

2

4

6

8

10

12

14

16

18

20

Algorithms

Fitn

ess v

alue

f10



0

1000

2000

3000

4000

5000

Algorithms

Fitn

ess v

alue

f11



002040608

112141618

2

Algorithms

Fitn

ess v

alue

f12times109



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f13times109



0

50

100

150

200

Algorithms

Fitn

ess v

alue

f14



0

2000

4000

6000

8000

10000

Algorithms

Fitn

ess v

alue

f15




0

10

20

30

40

50

60

70

80

Algorithms

Fitn

ess v

alue

f16



minus1

minus09

minus08

minus07

minus06

minus05

minus04

Algorithms

Fitn

ess v

alue

f17


Where 120591 () = radic(1205911015840)2+ 2120591101584012059110158401015840

1199092

2119877+ (12059110158401015840)

2

1205911015840=

119875

radic211990911199092

12059110158401015840=119872119877

119869

119872 = 119875(119871 +1199092

2)

119877 = radic1199092

2

4+ (

1199091+ 1199093

2)

2

119869 = 2radic211990911199092[1199092

2

4+ (

1199091+ 1199093

2)

2

]

120590 () =6119875119871

11990941199092

3

120575 () =61198751198713

1198641199092

31199094



Algorithms

Fitn

ess v

alue

f18




0

Algorithms

Fitn

ess v

alue

f19


119875119888 ()

=

4013119864radic1199092

31199096

436

1198712(1 minus

1199093

2119871

radic119864

4119866)

119875 = 6000119897119887

119871 = 14 in

120575max = 025 in

119864 = 30 times 16 psi

119866 = 12 times 106 psi

120591max = 13600 psi

120590max = 30000 psi(13)






ht

l

P

L

t


l1

l2

d1

d2z





1sim1199097denote


1)





Minimize 119891 ()

= 0785411990911199092

2(33333119909

2

3+ 149334119909

3minus 430934)

minus 15081199091(1199092

6+ 1199092

7) + 74777 (119909

3

6+ 1199093

7)

+ 07854 (11990941199092

6+ 11990951199092

7)

Subject to 1198921 () =

27

11990911199092

21199093

minus 1 le 0

1198922 () =

3975

11990911199092

21199092

3

minus 1 le 0

1198923 () =

1931199093

4

11990921199094

61199093

minus 1 le 0

1198924 () =

1931199093

5

11990921199095

71199093

minus 1 le 0

1198925 () =

[(745119909411990921199093)2+ 169 times 10

6]12

1101199093

6

minus 1 le 0

1198926 () =

[(745119909511990921199093)2+ 1575 times 10

6]12

851199093

7

minus 1

le 0

1198927 () =

11990921199093

40minus 1 le 0

1198928 () =

51199092

1199091

minus 1 le 0

1198929 () =

1199091

121199092

minus 1 le 0

11989210 () =

151199096+ 19

1199094

minus 1 le 0

11989211 () =

111199096+ 17

1199095

minus 1 le 0

where 26 le 1199091le 36




2) 119911 (119909

3) 119897

1(1199094) 119897

2(1199095) 119889

1(1199096) 119889

2(1199097)


07 le 1199092le 08

17 le 1199093le 28

73 le 1199094le 83

73 le 1199095le 83

29 le 1199096le 39

50 le 1199097le 55

(14)















Competing Interests


Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 200 400 600 800 1000minus1

minus095

minus09

minus085

minus08

minus075

minus07

minus065

minus06

Iteration

Aver

age fi

tnes

s val

ue

f18

ABCBAGGSADA

PSOGSAMFOLMFO


0 200 400 600 800 1000minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

Iteration

Aver

age fi

tnes

s val

ue

f19

ABCBAGGSADA

PSOGSAMFOLMFO


1198927 ()

= 1104711199092

1

+ 00481111990931199094(140 + 119909

2) minus 50

le 0


01 le 1199092le 10

01 le 1199093le 10

01 le 1199094le 2


0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f1times105



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f2times1099



0

05

1

15

2

25

3

35

4

45

Algorithms

Fitn

ess v

alue

f3times106




0102030405060708090

100

Algorithms

Fitn

ess v

alue

f4



0

1

2

3

4

5

6

7

8

9

Algorithms

Fitn

ess v

alue

f5times108



005

115

225

335

445

Algorithms

Fitn

ess v

alue

f6times105



0

500

1000

1500

2000

2500

Algorithms

Fitn

ess v

alue

f7



20406080

100120140160180200

Algorithms

Fitn

ess v

alue

f8



0

500

1000

1500

2000

Algorithms

Fitn

ess v

alue

f9




0

2

4

6

8

10

12

14

16

18

20

Algorithms

Fitn

ess v

alue

f10



0

1000

2000

3000

4000

5000

Algorithms

Fitn

ess v

alue

f11



002040608

112141618

2

Algorithms

Fitn

ess v

alue

f12times109



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f13times109



0

50

100

150

200

Algorithms

Fitn

ess v

alue

f14



0

2000

4000

6000

8000

10000

Algorithms

Fitn

ess v

alue

f15




0

10

20

30

40

50

60

70

80

Algorithms

Fitn

ess v

alue

f16



minus1

minus09

minus08

minus07

minus06

minus05

minus04

Algorithms

Fitn

ess v

alue

f17


Where 120591 () = radic(1205911015840)2+ 2120591101584012059110158401015840

1199092

2119877+ (12059110158401015840)

2

1205911015840=

119875

radic211990911199092

12059110158401015840=119872119877

119869

119872 = 119875(119871 +1199092

2)

119877 = radic1199092

2

4+ (

1199091+ 1199093

2)

2

119869 = 2radic211990911199092[1199092

2

4+ (

1199091+ 1199093

2)

2

]

120590 () =6119875119871

11990941199092

3

120575 () =61198751198713

1198641199092

31199094



Algorithms

Fitn

ess v

alue

f18




0

Algorithms

Fitn

ess v

alue

f19


119875119888 ()

=

4013119864radic1199092

31199096

436

1198712(1 minus

1199093

2119871

radic119864

4119866)

119875 = 6000119897119887

119871 = 14 in

120575max = 025 in

119864 = 30 times 16 psi

119866 = 12 times 106 psi

120591max = 13600 psi

120590max = 30000 psi(13)






ht

l

P

L

t


l1

l2

d1

d2z





1sim1199097denote


1)





Minimize 119891 ()

= 0785411990911199092

2(33333119909

2

3+ 149334119909

3minus 430934)

minus 15081199091(1199092

6+ 1199092

7) + 74777 (119909

3

6+ 1199093

7)

+ 07854 (11990941199092

6+ 11990951199092

7)

Subject to 1198921 () =

27

11990911199092

21199093

minus 1 le 0

1198922 () =

3975

11990911199092

21199092

3

minus 1 le 0

1198923 () =

1931199093

4

11990921199094

61199093

minus 1 le 0

1198924 () =

1931199093

5

11990921199095

71199093

minus 1 le 0

1198925 () =

[(745119909411990921199093)2+ 169 times 10

6]12

1101199093

6

minus 1 le 0

1198926 () =

[(745119909511990921199093)2+ 1575 times 10

6]12

851199093

7

minus 1

le 0

1198927 () =

11990921199093

40minus 1 le 0

1198928 () =

51199092

1199091

minus 1 le 0

1198929 () =

1199091

121199092

minus 1 le 0

11989210 () =

151199096+ 19

1199094

minus 1 le 0

11989211 () =

111199096+ 17

1199095

minus 1 le 0

where 26 le 1199091le 36




2) 119911 (119909

3) 119897

1(1199094) 119897

2(1199095) 119889

1(1199096) 119889

2(1199097)


07 le 1199092le 08

17 le 1199093le 28

73 le 1199094le 83

73 le 1199095le 83

29 le 1199096le 39

50 le 1199097le 55

(14)















Competing Interests


Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0102030405060708090

100

Algorithms

Fitn

ess v

alue

f4



0

1

2

3

4

5

6

7

8

9

Algorithms

Fitn

ess v

alue

f5times108



005

115

225

335

445

Algorithms

Fitn

ess v

alue

f6times105



0

500

1000

1500

2000

2500

Algorithms

Fitn

ess v

alue

f7



20406080

100120140160180200

Algorithms

Fitn

ess v

alue

f8



0

500

1000

1500

2000

Algorithms

Fitn

ess v

alue

f9




0

2

4

6

8

10

12

14

16

18

20

Algorithms

Fitn

ess v

alue

f10



0

1000

2000

3000

4000

5000

Algorithms

Fitn

ess v

alue

f11



002040608

112141618

2

Algorithms

Fitn

ess v

alue

f12times109



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f13times109



0

50

100

150

200

Algorithms

Fitn

ess v

alue

f14



0

2000

4000

6000

8000

10000

Algorithms

Fitn

ess v

alue

f15




0

10

20

30

40

50

60

70

80

Algorithms

Fitn

ess v

alue

f16



minus1

minus09

minus08

minus07

minus06

minus05

minus04

Algorithms

Fitn

ess v

alue

f17


Where 120591 () = radic(1205911015840)2+ 2120591101584012059110158401015840

1199092

2119877+ (12059110158401015840)

2

1205911015840=

119875

radic211990911199092

12059110158401015840=119872119877

119869

119872 = 119875(119871 +1199092

2)

119877 = radic1199092

2

4+ (

1199091+ 1199093

2)

2

119869 = 2radic211990911199092[1199092

2

4+ (

1199091+ 1199093

2)

2

]

120590 () =6119875119871

11990941199092

3

120575 () =61198751198713

1198641199092

31199094



Algorithms

Fitn

ess v

alue

f18




0

Algorithms

Fitn

ess v

alue

f19


119875119888 ()

=

4013119864radic1199092

31199096

436

1198712(1 minus

1199093

2119871

radic119864

4119866)

119875 = 6000119897119887

119871 = 14 in

120575max = 025 in

119864 = 30 times 16 psi

119866 = 12 times 106 psi

120591max = 13600 psi

120590max = 30000 psi(13)






ht

l

P

L

t


l1

l2

d1

d2z





1sim1199097denote


1)





Minimize 119891 ()

= 0785411990911199092

2(33333119909

2

3+ 149334119909

3minus 430934)

minus 15081199091(1199092

6+ 1199092

7) + 74777 (119909

3

6+ 1199093

7)

+ 07854 (11990941199092

6+ 11990951199092

7)

Subject to 1198921 () =

27

11990911199092

21199093

minus 1 le 0

1198922 () =

3975

11990911199092

21199092

3

minus 1 le 0

1198923 () =

1931199093

4

11990921199094

61199093

minus 1 le 0

1198924 () =

1931199093

5

11990921199095

71199093

minus 1 le 0

1198925 () =

[(745119909411990921199093)2+ 169 times 10

6]12

1101199093

6

minus 1 le 0

1198926 () =

[(745119909511990921199093)2+ 1575 times 10

6]12

851199093

7

minus 1

le 0

1198927 () =

11990921199093

40minus 1 le 0

1198928 () =

51199092

1199091

minus 1 le 0

1198929 () =

1199091

121199092

minus 1 le 0

11989210 () =

151199096+ 19

1199094

minus 1 le 0

11989211 () =

111199096+ 17

1199095

minus 1 le 0

where 26 le 1199091le 36




2) 119911 (119909

3) 119897

1(1199094) 119897

2(1199095) 119889

1(1199096) 119889

2(1199097)


07 le 1199092le 08

17 le 1199093le 28

73 le 1199094le 83

73 le 1199095le 83

29 le 1199096le 39

50 le 1199097le 55

(14)















Competing Interests


Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0

2

4

6

8

10

12

14

16

18

20

Algorithms

Fitn

ess v

alue

f10



0

1000

2000

3000

4000

5000

Algorithms

Fitn

ess v

alue

f11



002040608

112141618

2

Algorithms

Fitn

ess v

alue

f12times109



0

05

1

15

2

25

3

35

4

Algorithms

Fitn

ess v

alue

f13times109



0

50

100

150

200

Algorithms

Fitn

ess v

alue

f14



0

2000

4000

6000

8000

10000

Algorithms

Fitn

ess v

alue

f15




0

10

20

30

40

50

60

70

80

Algorithms

Fitn

ess v

alue

f16



minus1

minus09

minus08

minus07

minus06

minus05

minus04

Algorithms

Fitn

ess v

alue

f17


Where 120591 () = radic(1205911015840)2+ 2120591101584012059110158401015840

1199092

2119877+ (12059110158401015840)

2

1205911015840=

119875

radic211990911199092

12059110158401015840=119872119877

119869

119872 = 119875(119871 +1199092

2)

119877 = radic1199092

2

4+ (

1199091+ 1199093

2)

2

119869 = 2radic211990911199092[1199092

2

4+ (

1199091+ 1199093

2)

2

]

120590 () =6119875119871

11990941199092

3

120575 () =61198751198713

1198641199092

31199094



Algorithms

Fitn

ess v

alue

f18




0

Algorithms

Fitn

ess v

alue

f19


119875119888 ()

=

4013119864radic1199092

31199096

436

1198712(1 minus

1199093

2119871

radic119864

4119866)

119875 = 6000119897119887

119871 = 14 in

120575max = 025 in

119864 = 30 times 16 psi

119866 = 12 times 106 psi

120591max = 13600 psi

120590max = 30000 psi(13)






ht

l

P

L

t


l1

l2

d1

d2z





1sim1199097denote


1)





Minimize 119891 ()

= 0785411990911199092

2(33333119909

2

3+ 149334119909

3minus 430934)

minus 15081199091(1199092

6+ 1199092

7) + 74777 (119909

3

6+ 1199093

7)

+ 07854 (11990941199092

6+ 11990951199092

7)

Subject to 1198921 () =

27

11990911199092

21199093

minus 1 le 0

1198922 () =

3975

11990911199092

21199092

3

minus 1 le 0

1198923 () =

1931199093

4

11990921199094

61199093

minus 1 le 0

1198924 () =

1931199093

5

11990921199095

71199093

minus 1 le 0

1198925 () =

[(745119909411990921199093)2+ 169 times 10

6]12

1101199093

6

minus 1 le 0

1198926 () =

[(745119909511990921199093)2+ 1575 times 10

6]12

851199093

7

minus 1

le 0

1198927 () =

11990921199093

40minus 1 le 0

1198928 () =

51199092

1199091

minus 1 le 0

1198929 () =

1199091

121199092

minus 1 le 0

11989210 () =

151199096+ 19

1199094

minus 1 le 0

11989211 () =

111199096+ 17

1199095

minus 1 le 0

where 26 le 1199091le 36




2) 119911 (119909

3) 119897

1(1199094) 119897

2(1199095) 119889

1(1199096) 119889

2(1199097)


07 le 1199092le 08

17 le 1199093le 28

73 le 1199094le 83

73 le 1199095le 83

29 le 1199096le 39

50 le 1199097le 55

(14)















Competing Interests


Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0

10

20

30

40

50

60

70

80

Algorithms

Fitn

ess v

alue

f16



minus1

minus09

minus08

minus07

minus06

minus05

minus04

Algorithms

Fitn

ess v

alue

f17


Where 120591 () = radic(1205911015840)2+ 2120591101584012059110158401015840

1199092

2119877+ (12059110158401015840)

2

1205911015840=

119875

radic211990911199092

12059110158401015840=119872119877

119869

119872 = 119875(119871 +1199092

2)

119877 = radic1199092

2

4+ (

1199091+ 1199093

2)

2

119869 = 2radic211990911199092[1199092

2

4+ (

1199091+ 1199093

2)

2

]

120590 () =6119875119871

11990941199092

3

120575 () =61198751198713

1198641199092

31199094



Algorithms

Fitn

ess v

alue

f18




0

Algorithms

Fitn

ess v

alue

f19


119875119888 ()

=

4013119864radic1199092

31199096

436

1198712(1 minus

1199093

2119871

radic119864

4119866)

119875 = 6000119897119887

119871 = 14 in

120575max = 025 in

119864 = 30 times 16 psi

119866 = 12 times 106 psi

120591max = 13600 psi

120590max = 30000 psi(13)






ht

l

P

L

t


l1

l2

d1

d2z





1sim1199097denote


1)





Minimize 119891 ()

= 0785411990911199092

2(33333119909

2

3+ 149334119909

3minus 430934)

minus 15081199091(1199092

6+ 1199092

7) + 74777 (119909

3

6+ 1199093

7)

+ 07854 (11990941199092

6+ 11990951199092

7)

Subject to 1198921 () =

27

11990911199092

21199093

minus 1 le 0

1198922 () =

3975

11990911199092

21199092

3

minus 1 le 0

1198923 () =

1931199093

4

11990921199094

61199093

minus 1 le 0

1198924 () =

1931199093

5

11990921199095

71199093

minus 1 le 0

1198925 () =

[(745119909411990921199093)2+ 169 times 10

6]12

1101199093

6

minus 1 le 0

1198926 () =

[(745119909511990921199093)2+ 1575 times 10

6]12

851199093

7

minus 1

le 0

1198927 () =

11990921199093

40minus 1 le 0

1198928 () =

51199092

1199091

minus 1 le 0

1198929 () =

1199091

121199092

minus 1 le 0

11989210 () =

151199096+ 19

1199094

minus 1 le 0

11989211 () =

111199096+ 17

1199095

minus 1 le 0

where 26 le 1199091le 36




2) 119911 (119909

3) 119897

1(1199094) 119897

2(1199095) 119889

1(1199096) 119889

2(1199097)


07 le 1199092le 08

17 le 1199093le 28

73 le 1199094le 83

73 le 1199095le 83

29 le 1199096le 39

50 le 1199097le 55

(14)















Competing Interests


Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













ht

l

P

L

t


l1

l2

d1

d2z





1sim1199097denote


1)





Minimize 119891 ()

= 0785411990911199092

2(33333119909

2

3+ 149334119909

3minus 430934)

minus 15081199091(1199092

6+ 1199092

7) + 74777 (119909

3

6+ 1199093

7)

+ 07854 (11990941199092

6+ 11990951199092

7)

Subject to 1198921 () =

27

11990911199092

21199093

minus 1 le 0

1198922 () =

3975

11990911199092

21199092

3

minus 1 le 0

1198923 () =

1931199093

4

11990921199094

61199093

minus 1 le 0

1198924 () =

1931199093

5

11990921199095

71199093

minus 1 le 0

1198925 () =

[(745119909411990921199093)2+ 169 times 10

6]12

1101199093

6

minus 1 le 0

1198926 () =

[(745119909511990921199093)2+ 1575 times 10

6]12

851199093

7

minus 1

le 0

1198927 () =

11990921199093

40minus 1 le 0

1198928 () =

51199092

1199091

minus 1 le 0

1198929 () =

1199091

121199092

minus 1 le 0

11989210 () =

151199096+ 19

1199094

minus 1 le 0

11989211 () =

111199096+ 17

1199095

minus 1 le 0

where 26 le 1199091le 36




2) 119911 (119909

3) 119897

1(1199094) 119897

2(1199095) 119889

1(1199096) 119889

2(1199097)


07 le 1199092le 08

17 le 1199093le 28

73 le 1199094le 83

73 le 1199095le 83

29 le 1199096le 39

50 le 1199097le 55

(14)















Competing Interests


Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2) 119911 (119909

3) 119897

1(1199094) 119897

2(1199095) 119889

1(1199096) 119889

2(1199097)


07 le 1199092le 08

17 le 1199093le 28

73 le 1199094le 83

73 le 1199095le 83

29 le 1199096le 39

50 le 1199097le 55

(14)















Competing Interests


Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Competing Interests


Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Lévy-Flight Moth-Flame Algorithm for...

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Transcript of Research Article Lévy-Flight Moth-Flame Algorithm for...