Plant intelligence based metaheuristic optimization algorithms

46 Pages·2016·2.47 MB·English

by  Sinem Akyol

ArtifIntellRev(2017)47:417–462
DOI10.1007/s10462-016-9486-6
Plant intelligence based metaheuristic optimization
algorithms
SinemAkyol1 · BilalAlatas2
Publishedonline:30May2016
©SpringerScience+BusinessMediaDordrecht2016
Abstract Classical optimization algorithms are insufficient in large scale combinatorial
problems and in nonlinear problems. Hence, metaheuristic optimization algorithms have
been proposed. General purpose metaheuristic methods are evaluated in nine different
groups: biology-based, physics-based, social-based, music-based, chemical-based, sport-
based, mathematics-based, swarm-based, and hybrid methods which are combinations of
these.Studiesonplantsinrecentyearshaveshowedthatplantsexhibitintelligentbehaviors.
Accordingly,itisthoughtthatplantshavenervoussystem.Inthiswork,allofthealgorithms
andapplicationsaboutplantintelligencehavebeenfirstlycollectedandsearched.Information
isgivenaboutplantintelligencealgorithmssuchasFlowerPollinationAlgorithm,Invasive
WeedOptimization,PaddyFieldAlgorithm,RootMassOptimizationAlgorithm,Artificial
PlantOptimizationAlgorithm,SaplingGrowingupAlgorithm,PhotosyntheticAlgorithm,
PlantGrowthOptimization,RootGrowthAlgorithm,StrawberryAlgorithmasPlantProp-
agation Algorithm, Runner Root Algorithm, Path Planning Algorithm, and Rooted Tree
Optimization.
Keywords Plantintelligence·Globaloptimization·Metaheuristicmethods
1 Introduction
Mostof theoptimization algorithms needmathematical modelsfor systemmodelling and
objectivefunction.Establishmentofamathematicalmodelismostlyhardforcomplexsys-
tems.Eventhoughthemodelisestablished,itcannotbeusedduetohighcostofsolution
B
SinemAkyol
[email protected]
BilalAlatas
balatas@firat.edu.tr
1 DepartmentofComputerEngineering,TunceliUniversity,Tunceli,Turkey
2 DepartmentofSoftwareEngineering,FiratUniversity,Elazig,Turkey
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418 S.Akyol,B.Alatas
time.Classicaloptimizationalgorithmsareinsufficientinlarge-scalecombinationalandnon-
linearproblems.Suchalgorithmsarenoteffectiveinadaptationasolutionalgorithmfora
given problem. In most cases, this requires some assumptions whose approval of validity
canbedifficult.Generally,becauseofnaturalsolutionmechanismofclassicalalgorithms,
significantproblemismodeledthewaythatalgorithmcanhandleit.Solutionstrategyofclas-
sicaloptimizationalgorithmsgenerallydependsontypeofobjectiveandconstraints(linear,
non-linearetc.)anddependsontypeofvariables(integer,real)usedinmodellingofprob-
lem.Furthermore,effectivenessoftheclassicalalgorithmshighlydependsonsolutionspace
(convex,non-convexetc.),thenumberofdecisionvariable,andthenumberofconstraintsin
problemmodelling.
Another important deficiency is that; if there are different types of decision variables,
objectives,andconstraints,generalsolutionstrategyisnotpresentedforimplementedprob-
lemformulation.Inotherwords,mostofalgorithmssolvemodelswhichhavecertaintypeof
objectfunctionorconstraints.However,optimizationproblemsinmanydifferentareassuch
asmanagementscience,computer,andengineeringrequireconcurrentlydifferenttypesof
decision variables, object function, and constraints in their formulation. Therefore, meta-
heuristic optimization algorithms are proposed. These algorithms become quite popular
methods in recent years, due to their good computation power and easy conversion. In
other words, a metaheuristic program, written for a specific problem with a single objec-
tivefunction,canbeadaptedeasilytoamultiobjectiveversionofthisproblemoradifferent
problem.
Studiesinrecentyearsindicatethatplantsalsoexhibitintelligentbehaviors.According
tothis,itisthoughtthatplantshavenervoussystem.Forexample,inroots,importedlight
andpoisoninformationaretransmittedtogrowthcenterinroottipsandtherootsperform
orientationaccordingtothat.Additionally,itisthoughtthatplantsmakecontactwithexternal
worldusingelectriccurrent.Anexampleforthisisdefensemechanismwhichisshownagainst
toaphidorcaterpillarbyplants.Afterthefirstdevelopedattack,plantsproducesecretions
whichcanmakeworsetasteorwhichcanpoisontheirenemy.
Inspiringbytheflowpollinationprocessoffloweryplants,FlowerPollinationAlgorithm
(FPA), has been developed in 2012 by Yang (2012). Invasive Weed Optimization (IWO),
inspiredfromthephenomenonofcolonizationofinvasiveweedsinnature,hasbeenproposed
by Mehrabian and Lucas (2006). It is based on ecology and weed biology. Paddy Field
Algorithm (PFA), which simulates growth process of paddy fields, is another plant based
metaheuristic algorithm and has been proposed by Premaratne et al. (2009). Root Mass
Optimization (RMO) is based on the process of root growth and has been proposed by
Qi et al. 2013. Artificial Plant Optimization Algorithm (APOA) simulates growth model
of plants which includes photosynthesis and phototropism mechanism (Zhao et al. 2011).
Sapling Growing up Algorithm (SGuA) is a computational method based on cultivating,
growingup,andmatingofsaplingsanditisdevelopedforefficientsolutionstosearchand
optimizationproblems(Karci2007a).
Photosynthetic Algorithm (PA) is based on the processes of Calvin–Benson cycle and
photorespiration cycle for the plants (Murase 2000). Plant Growth Optimization (PGO) is
proposed to simulate plant growth with a realistic way considering the spatial occupancy,
accountbranching,phototropism,andleafgrowth.Themainpurposeofthemodelistoselect
activepointbycomparingthemorphogenconcentrationforincreasingtheL-system(Caietal.
2008).RootGrowthAlgorithm(RGA)isanotherplantbasedmetaheuristicalgorithmand
simulatesrootgrowthofplantsbasedonL-system(Zhangetal.2014).
StrawberryalgorithmhasbeenproposedasanexemplarofPlantPropagationAlgorithm
(PPA).Itmapsanoptimizationproblemontosurvivaloptimizationproblemofstrawberry
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Plantintelligencebasedmetaheuristicoptimizationalgorithms 419
plantandadoptsstrategyofsurvivalintheenvironmentforsearchingpointsinthesearch
spacewhichgivethebestvaluesandultimatelythebestvalue(SalhiandFraga2011).Runner
RootAlgorithm(RRA)isinspiredbyplantssuchasstrawberryandspiderplantswhichare
spread through their runners and also which develop roots and root hairs for local search
formineralsandwaterresources(Merrikh-Bayat2015).PathPlanningAlgorithmisbased
onplantgrowthmechanism(PGPP)whichusesphototropism,negativegeotropism,apical
dominance,andbranchingasbasicrules(Zhouetal.2016).Oneofthemostrecentplantbased
algorithmisRootedTreeOptimization(RTO)anditisbasedontheintelligentbehaviorsof
rootswhichselecttheirorientationaccordingtothewetness(Labbietal.2016).Researches
aboutnewversionsandapplicationsofthesealgorithmsincrementallycontinue.
Inthiswork,inthesecondpart,theinformationisgivenaboutmetaheuristicoptimization
and it is mentioned that why metaheuristic algorithms are needed. In the third part, plant
intelligenceismentioned;optimizationalgorithmsdevelopedinspiredbyplantintelligence
areexaminedandrelatedworkswiththesealgorithmsaredescribed.FPA,IWO,PFA,RMO
Algorithm, APOA, SGuA, PA, PGO, RGA, Strawberry Algorithm as PPA, RRA, PGPP,
and RTO are explained. In the fourth part, discussions about general evaluations of these
algorithmsarepresentedandinthefifthpart,theworkisconcludedalongwithfutureworks.
2 Metaheuristicoptimization
Inmostofthereallifeproblems,solutionspaceoftheproblemisinfiniteoritistoolarge
forassessmentofallthesolutions.Therefore,withevaluatingsolutions,agoodsolutionis
neededtobefoundinacceptabletime.Actually,forsuchproblems,evaluatingsolutionsin
acceptabletimehasthesamemeaningwithevaluating“somesolutions”inthewholesolution
space. Selection of some solutions depending on what and how they are selected changes
accordingtometaheuristicmethod.Itcannotbeguaranteedthatoptimalsolutionisincluded
inthesolutionswhichgetinvolvedintheevaluation.Therefore,thesolution,proposedby
metaheuristicmethodsforanoptimizationproblem,mustbeperceivedasagoodsolution,
notasanoptimalsolution(Cura2008).
They are criteria or computer methods identified in order to decide effective ones of
variousalternativemovementsforachievinganypurposeorthegoal.Suchalgorithmshave
convergenceproperty,buttheycannotguaranteeexactsolution,theycanonlyguaranteea
solutionwhichclosestoexactsolution.
Thereasonsofwhymetaheuristicalgorithmsareneededareasfollows:
(a) Optimizationproblemcanhaveastructurethattheprocessoffindingtheexactsolution
cannotbedefined
(b) Metaheuristicalgorithmscanbemuchsimplerfromthepointofdecisionmaker,interms
ofcomprehensibility.
(c) Metaheuristicalgorithmscanbeusedasapartofprocessoffindingtheexactsolution,
andlearningpurpose.
(d) Generally,themostdifficultpartsofrealworldproblems(whichpurposesandwhich
restrictionsmustbeused,whichalternativesmustbetested,howproblemsdatamustbe
collected)areneglectedinthedefinitionsmadewithmathematicalformulas.Faultiness
of the data used in process of determining model parameters can cause much larger
errorsthansub-optimalsolutionproducedbymetaheuristicapproach(Karabog˘a2011).
General purposed metaheuristic methods are evaluated in eight different groups which
arebiologybased,physicbased,swarmbased,socialbased,musicbased,chemistrybased,
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420 S.Akyol,B.Alatas
sport based, and math based. Furthermore, there are hybrid methods which are combina-
tion of these. These mentioned methods are presented in Fig. 1. Genetic Algorithm (GA)
(Holland1975),differentialevolutionalgorithm(StornandPrice1995),andbiogeography-
basedoptimization(Simon2008)arebiologybased;parliamentaryoptimizationalgorithm
(Borji2007),teaching-learning-basedoptimization(Ghasemietal.2015;Raoetal.2011),
andimperialistcompetitivealgorithm(Atashpaz-GargariandLucas2007a,b;Ghasemietal.
2014) are social based; artificial chemical reaction optimization algorithm (Alatas 2011a)
andchemicalreactionoptimization(LamandLi2010)arechemistrybased;harmonysearch
algorithm(Geemetal.2001)ismusicbased;gravitationalsearchalgorithm(Rashedietal.
2009),intelligentwaterdropsalgorithm(Shah-Hosseini2009),andchargedsystemsearch
(KavehandTalatahari2010)arephysicsbased;ParticleSwarmOptimization(PSO)algo-
rithm(KennedyandEberhart1995),catswarmoptimization(Chuetal.2006),antcolony
algorithm (Dorigo et al. 1991), monarch butterfly optimization (Wang et al. 2015), group
searchoptimizer(Heetal.2006, 2009),andcuckoosearchviaLevyflights(YangandSuash
2009)areswarmbased;leaguechampionshipalgorithm(Kashan2009)issportbased;and
base optimization algorithm (Salem 2012), and Matheuristics (Maniezzo et al. 2009) are
mathbasedalgorithmsandmethods.Culturalalgorithm(JinandReynolds1999)andcolo-
nialcompetitivedifferentialevolution(Ghasemietal.2016)canbeclassifiedasbothbiology
basedandsocialbasedalgorithm.
Althoughtherearemanysuccessfulsearchandoptimizationalgorithmsandtechniques
intheliterature;design,development,andimplementationofnewtechniquesisanimportant
task under the philosophy of improvement in the scientific field and always searching to
designbetter.Thebestalgorithmthatgivesthebestresultsforalltheproblemshasnotyet
beendesigned,thatiswhyconstantlynewartificialintelligenceoptimizationalgorithmsare
proposedorsomeefficientadditionsormodificationshavebeenperformedtotheexisting
algorithms.
3 Plantintelligenceoptimizationalgorithms
Asaresultofpreviousstudies,itisobservedthatplantshavegenderidentityandimmune
system.Furthermore,recentstudiesshowthatplantsexhibitintelligentbehavior.According
tothis,itisthoughtthatplantshavenervoussystem.Forexample,inroots,importedlight
andpoisondataaretransmittedtogrowthcentersinroottips,androotsperformorientation
accordingtothis.Asanotherconsideration,plantsareconsideredtobecontactingwiththe
externalworldbyelectriccurrents.Defensemechanism,whichisshownagainsttoaphidor
caterpillarbyplants,canbeshownasanexampleforthis.Afterthefirstattacktakesplace,
plantswillworsentheirtasteorproducesecretionswhichcanpoisontheirenemy.
Themetaheuristicoptimizationalgorithmswhicharedevelopedbyinspirationfromplant
intelligencehavebeenshowninFig.2andexplainedinsubsections.
3.1 FlowerPollinationAlgorithm(FPA)
3.1.1 Characteristicofflowerpollination
Itisestimatedthat,innature,thereareoveraquarterofamilliontypesoffloweryplants.
Approximately80%ofallplantspeciesarefloweryplant.Floweryplantshavebeenevolv-
ingformorethan125millionyears.Themainobjectiveofaplantisreproducingthrough
pollination.About90%ofpollensaretransferredbybioticpollinatorssuchasanimalsand
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Plantintelligencebasedmetaheuristicoptimizationalgorithms 421
Metaheuristic
Methods
Physics Social Music Swarm Chemistry Biology Sports Math
Hybrid
Based Based Based Based Based Based Based Based
Single-Point Multi-Point
Fixed Variable
Objective Objective
Function Function
Single Variable
Neighborhood Neighborhood
Structured Structured
Memory Memoryless
Fig.1 Metaheuristicmethods
Plant
Intelligence
Optimization
Algorithms
InOvpatsimiveiz aWtieoend PAaldgdoyr itFhimeld ORApotlgiomot irMzitahatmsiosn AOrAtpifltigicmoiarizilt ahPtmliaonnt GArSolgawopirnliitnghg mu p PhAotlgoosyrinththmetic POlapntimt Gizraotwiotnh
RAoolgt oGrirtohwmth PArolgPpoalargintahtt mion RAunlgnoerrit hRmoot PAoFlglllooinwraitethirom n ROopotitmeidz aTtrioene PaAthlg Porlaitnhnming
Fig.2 Plantintelligencemetaheuristicoptimizationalgorithms
insectsandabout10%ofpollensaretransferredbyabioticpollinatorssuchaswind.Biotic
pollinatorsvisitonlysometypesofflowers.Therefore,pollentransferofsametypesofflow-
erswillbemaximum.Thisprovidesadvantagesforpollinatorsintermsofresearchcostand
limitedmemory.Focusingonsomeunpredictablebutpotentiallymorerewardingnewflower
speciescangivebetterresults(Yangetal.2013).
Pollinationcanbeachievedbycross-pollinationorself-pollination.Cross-pollination,or
hybridizationcanoccurfromaflowersplantofadifferentplant.Self-pollinationoccursfrom
thesameflowerspollenordifferentflowersofthesameplant.Bioticcross-pollinationcan
occurinlong-distance,becauseoflongdistanceflightofbirds,bats,bees,etc.Thus,thisis
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422 S.Akyol,B.Alatas
consideredasglobalpollination.Inadditiontothis,beesandbirdsareabletomovewiththe
Levyflightbehavior(Yangetal.2013).
3.1.2 FPA
FPA,inspiredbytheflowpollinationprocessoffloweryplants,wasdevelopedin2012by
Yang(2012).Thefollowing4rulesareusedasamatterofconvenience.
(1) Biotic cross-pollination can be thought as global pollination process. Pollen-carrying
pollinatorsmoveaccordingtoaLevyflightprocess(Rule1).
(2) Self-pollinationandlocalpollinationareusedforlocalpollination(Rule2).
(3) Pollinatorssuchasbirdsandbeescandevelopflowerpersistence(Rule3).
(4) Aswitchprobability p ∈[0,1],whichisslightlybiasedtowardslocalpollination,can
controltheswitchingorinteractionofglobalandlocalpollination(Rule4).
Theaboverulesmustbeconvertedtoappropriateupdatingequationsforformulationof
updatingformulas.Forexample,flowerpollengametesaremovedbypollinatorssuchasbirds
and bees in the global pollination process. Pollen can be transferred over a long distance,
becausepollinatorscanoftenflyandmoveoveramuchlongerrange.Therefore,Rule1and
Rule3(flowerpersistence)canberepresentedmathematicallyas(1).
(cid:2) (cid:3)
xit+1 =xit +γL(λ) g∗−xit (1)
xt,isthepolleni,orx solutionvectoriniterationt,γ isascalefactortocontrolthestep
i i
size. L(λ)istheparametercorrespondingtothepollinationpower,morespeciallyitisthe
Levy-flights based step size, g∗ is the current best solution found among all solutions at
thecurrentiteration/generation.Levyflightcanbeusedeffectivelytosimulatepollinators’
movements(Yangetal.2013).
λ(cid:4)(λ)sin(πλ/2) 1
L ∼ ,(s (cid:5)s >0) (2)
π s1+λ 0
Here(cid:4)(λ)isthestandardgammafunctionandthisdistributionisvalidforlargestepss >0.
Itmustbe|s (cid:5)0|intheory,butinpractice,s canbeassmallas0.1.However,itisnottrivial
0 0
togeneratepseudo-randomstepsizeswhichconformtotheLevydistribution.Thereareafew
methodsfordrawingthesetypepseudo-randomnumbers.Oneofthemostefficientmethods
isusingtwoGaussiandistributionsU andV,whichistheso-calledMantegnaalgorithmfor
drawingstepsizes.
U (cid:2) (cid:3)
s = ,U ∼ N 0,σ2 , V ∼ N(0,1) (3)
|V|1/λ
(cid:2) (cid:3)
In(3),U ∼ 0,σ2 meansthatthesamplesaredrawnfromaGaussiannormaldistribution
withavarianceofσ2 andazeromean.Thevariancecanbecalculatedas(4)(Yangetal.
2013).
(cid:4) (cid:5)
(cid:4)(1+λ) sin(πλ/2) 1/λ
σ2 = . (4)
λ(cid:4)[(1+λ)/2] 2(λ−1)/2
Thisformulaseemscomplicated,butforagivenλvalueitisonlyaconstant.Forexample,
ifλ=1,thanthegammafunctionsbecome(cid:4)(1+λ)=1,(cid:4)([1+λ/2])=1.
(cid:4) (cid:5)
1 sin(πx1/2) 1/1
σ2 = . =1 (5)
1x1 20
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Plantintelligencebasedmetaheuristicoptimizationalgorithms 423
Flower Pollination Algorithm
Objective function min or max f(x), x=(x, x, ..., x )
1 2 d
Initialize a population of n flower/pollen gametes with random solutions.
Find the best solution in the population
Define a switch probability
Define a stopping criterion
While ()
For i=1 : n (all n flowers in the population)
If rand<p,
Draw a (d-dimensional) step vector L which obeys Levy distribution
Do global pollination via Eq. (1)
Else
Draw from a uniform distribution in [0,1]
Do local pollination via Eq. (6)
End If
Evaluate new solutions
If new solutions are better, than update them in the population
Find current best solution
End While
Output is the best solution found
Fig.3 PseudocodeofFPA
It is proved mathematically that Mantegna algorithm can produce random samples which
obeytherequireddistribution(2)correctly(Yangetal.2013).
Bothofrules(2)and(3)canberepresentedas(6)forlocalpollination.
(cid:6) (cid:7)
xt+1 =xt+∈ xt −xt (6)
i i j k
Here xt and xt are pollens in different flowers of the same plant species. This essentially
j k
mimicsflowerconstancyinalimitedneighborhood.Mathematically,ifxt andxt areselected
j k
fromthesamepopulationorcomefromthesamespecies;thisequivalentlybecomesalocal
randomwalkif∈isdrawnfromauniformdistributionin[0,1].
Inprinciple,flowerpollinationactivitiescanoccurbothatlocalandgloballevels,atall
scales.However,substantially,theflowersinthenot-far-awayoradjacentflowerpatchesare
morelikelytobepollinatedbylocalflowerpollenthanthosefaraway.Therefore,valueof
theswitchprobabilitycanbetakenas p=0.8.ThepseudocodeofFPAisshowninFig.3.
3.1.3 StudieswithFPA
Yangetal.(2013, 2014)usedFPAforsolvingmulti-objectiveoptimizationproblems.Pro-
posed algorithm was tested in multi-objective test functions and it has been seemed that
this algorithm has a better convergence speed compared to other algorithms Lenin (2014)
proposedahybridalgorithm,whichisacombinationofchaoticharmonysearchalgorithm
andFPA,forsolvingreactivepowerdispatchproblem.StandardFPAwasintegratedwiththe
harmonysearchalgorithmtoimprovethesearchaccuracy.
WangandZhou(2014)proposeddimensionbydimensionimprovementbasedFPA,for
multi-objective optimization problem. They also applied local neighborhood search strat-
egyinthisimprovedalgorithmforenhancingthelocalsearchingability.Yangetal.(2013)
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424 S.Akyol,B.Alatas
indicated that, it is important to balance exploration and exploitation for any metaheuris-
tic algorithm. Because, interaction of these two components could significantly affect the
efficiency.Therefore,theystudiedFPAwithEagleStrategy.
Abdel-Raouf et al. (2014a) used FPA with chaos theory for solving definite integral.
Additionally, they proposed a hybrid method which was a combination of FPA and PSO
forimprovingsearchaccuracy.Theyuseditforsolvingconstrainedoptimizationproblems
(Abdel-Raoufetal.2014b).TheyalsoproposedanewhybridalgorithmcombinedwithFPA
andchaoticharmonysearchalgorithm,tosolveSudokupuzzles(Abdel-Raoufetal.2014c).
Sundareswaran et al. proposed a modification for steps of traditional GA, used for PVM
inverter,byimitatingflowerpollinationandsubsequentseedproductioninplants(Sulaiman
et al. 2014). Prathiba et al. set the real power generations using FPA, for minimizing the
fuelcost,ineconomicloaddispatch,whichisthemainoptimizationtaskinpowersystem
operation(Prathibaetal.2014).
ŁukasikandKowalski(2015)studiedFPAforcontinuousoptimizationandtheycompared
solutions with PSO. Platt used FPA in the calculation of dew point pressures of a system
exhibiting double retrograde vaporization. The main idea was to apply a new algorithmic
structureinahardnonlinearalgebraicsystemarisingfromreal-worldsituations(Platt2014).
KanagasabaiandRavindhranathReddy(2014)proposedacombinationofFPAandPSOfor
solvingoptimalreactivepowerdispatchproblem.Sakibetal.2014comparedFPAwithBat
Algorithm. They tested these two algorithms on both unimodal and multimodal, low and
highdimensionalcontinuousfunctionsandtheyobservedthatFPAgavebetterresults.
3.2 InvasiveWeedOptimization(IWO)
3.2.1 Theinspirationphenomenon
IWO,inspiredfromthephenomenonofcolonizationofinvasiveweedsinnature,isproposed
byMehrabianandLucas(2006).IWOisbasedonecologyandweedbiology.Itwasseemed
that mimicking invasive weeds properties, leads a powerful optimization algorithm. In a
croppingfield,weedcolonization’sbehaviorcanbeexplainedasfollows:
Weeds invade a cropping system by the way of disperse. They occupy suitable fields
betweenplants.Eachinvasiveweedtakestheunusedresourcesinthefield,andbecomesa
floweringplant,andproducesnewinvasiveweedsindependently.Numberofnewinvasive
weedproducedbyeachfloweringherbdependsonfitnessoffloweringplantsinthecolony.
Theseweedsprovidebetteradaptationtotheenvironmentandgrowfasterbytakingmore
unusedresourcesandproducemoreseeds.Thenewproducedweedsrandomlyspreadover
thefieldandgrowtofloweringplants.Thisprocesscontinuesuntilreachingthemaximum
numberofweedinthefieldbecauseoflimitedresources.Weedswithbetterfitnesscanonly
surviveandcanproducenewplants.Thecompetitionbetweenweedscausesthemtobecome
welladaptedandevolvedovertime(KarimkashiandKishk2010).
3.2.2 Algorithm
Thenewkeytermsusedforexplainingthisalgorithmshouldbeintroduced,beforeconsid-
eringthealgorithmprocess.SomeofthesetermsareshowninTable1.Eachindividualor
agentiscalledasaseed,orasetcontainingavalueofeachoptimizationvariable.Eachseed
grows to a flowering plant in colony. The meaning of a plant is an individual or an agent
afterevaluatingitsfitness.Therefore,growingaseedtoaplantcorrespondstoevaluatingthe
fitnessofanagent(KarimkashiandKishk2010).
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Plantintelligencebasedmetaheuristicoptimizationalgorithms 425
Table1 SomeofthekeytermsusedinIWO
Agent/seed Eachindividualcontainingavalueofeachoptimization
variableinthecolony
Fitness Avaluerepresentingthegoodnessofsolutionforeachseed
Plant Aseedafterevaluatingitsfitness
Colony allseedsorindividuals
Sizeofpopulation Numberofplantsinthecolony
Maximumnumberofplants Themaximumnumberofplantsallowedtoproducenewseed
inthecolony
Forsimulatingthecolonizingbehaviorofweedsfollowingstepsareacceptedandflow-
chartofthesimulationisshowninFig.4.
1. Primarily, N parameters(variable)thatneedtobeoptimizedmustbeselected.Then,in
N-dimensionalsolutionspace,amaximumandaminimumvaluemustbeassignedfor
eachofthesevariables(Definingthesolutionspace)(KarimkashiandKishk2010).
2. A finite number of seeds are distributed randomly on the defined solution space. In
another words, each seed randomly takes position in N-dimensional solution space.
Positionofeachseedisaninitialsolution,whichcontainsN valuesfortheN variables,
ofoptimizationproblem(Initializingapopulation).
3. Eachofinitialseedgrowstoafloweringplant.Thefitnessfunction,definedforrepre-
sentinggoodnessofthesolution,returnsafitnessvalueforeachseed.Seediscalledas
aplant,afterassigningthefitnessvaluetothecorrespondingseed(Evaluatefitnessof
eachindividual)(KarimkashiandKishk2010).
4. Flowering plants are ranked according to fitness value assigned to them, before they
producenewseeds.Then,eachfloweringplantproducesseedsaccordingtoitsranking
inthecolony.Inotherwords,thenumberofseedproductionofeachplantdependson
itsfitnessvalueorrankinganditincreasesfromtheminimumpossibleseeds(S )to
min
maximumpossibleseeds(S ).Theseseeds,whichsolvetheproblembettercorrespond
max
theplantswhicharemoreadaptedtothecolonyandtherefore,theyproducemoreseed.
Thisstep,byallowingalloftheplantstoparticipateinthereproductioncontest,adds
animportantpropertytothealgorithm(Rankingpopulationandproducingnewseeds)
(KarimkashiandKishk2010).
5. Inthisstep,producedseedsarespreadtolocationofproducedplantwithequal-average
bynormallydistributedrandomnumbersonthesearchspace.Atthepresenttimestep,
thestandarddeviation(SD)canbeexpressedby(7).
(iter −iter)n (cid:2) (cid:3)
σ = max σ −σ +σ (7)
iter (iter )n iinitial final final
max
Here,iter isnumberofmaximumiteration.σ andσ areinitialandfinal
max iinitial final
standarddeviation respectively, andn is nonlinear modulation index.Algorithm starts
withahighinitialSDwhichcanbeexploredbyoptimizerthroughthewholesolution
space.Byincreasingthenumberofiterations,forfindingtheglobaloptimalsolution,SD
valueisdecreasedgraduallytosearcharoundthelocalminimumormaximum(Disper-
sion)(KarimkashiandKishk2010).
6. New seeds grow to flowering plant, after all seeds have found their positions on the
searchspace,andthen,theyarerankedwiththeirparents.Theplantsinlowrankingsin
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426 S.Akyol,B.Alatas
Start Finish
Define the solution
space
Keep the best individual
Create the initial population Yes
No
Finished?
Evaluate the fitness of each individual and
rank the population
Evaluate the fitness of each individual
Eliminate the individuals with low and rank the population
fitness for reaching the maximum
number of plants
Reproduce each individual Disperse the new seeds over
according to its rank the solution space
Fig.4 FlowchartofIWOalgorithm
thecolonyareeliminatedforreachingnumberofmaximumplantinthecolony(P ).
max
Thenumberoffitnessevaluation,populationsize,ismorethanthemaximumnumberof
plantsinthecolony(Competitiveexclusion)(KarimkashiandKishk2010).
7. Survivedplantsproducenewseedsaccordingtotheirrankinginthecolony.Theprocess
isrepeatedatstep3,untilthemaximumnumberofiterationisreachedorfitnesscriterion
ismet(Repeat)(KarimkashiandKishk2010).
3.2.3 Selectionofcontrolparametervalues
Threeparametersamongallparametersaffecttheconvergenceofthealgorithm;initialSS,
σ , final SS, σ , and nonlinear modulation index, n, must be tuned carefully for
iinitial final
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