a memetic algorithm based on multiple learning procedures for global optimal design of composite...
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Memetic Comp. (2014) 6:113–131DOI 10.1007/s12293-014-0132-z
REGULAR RESEARCH PAPER
A memetic algorithm based on multiple learning proceduresfor global optimal design of composite structures
Carlos Conceição António
Received: 7 August 2013 / Accepted: 24 March 2014 / Published online: 11 April 2014© Springer-Verlag Berlin Heidelberg 2014
Abstract The mechanical properties of composite struc-tures are sensitive to the choice of the material systems,the distribution of laminates on the structure, and the siz-ing at the laminate level. This sensitivity depends on manyvariables. To overcome the difficulties in attaining a globalsolution for the optimisation problem of composite struc-tures, a new strategy is proposed based on the hybridisa-tion of the memetic algorithm (MA) and the selfish gene(SG) algorithm. Instead of a local search, as performed inMAs, the selfish gene (SG) theory is applied, which fol-lows a different learning scheme in which the conventionalpopulation of the individuals is replaced by a virtual pop-ulation of alleles. The proposed approach, which is calledthe memetic-based selfish gene algorithm (MA + SG), is amixed model that applies multiple learning procedures toexplore the synergy of different cultural transmission rulesin the evolutionary process. The principal aspects of theapproach are as follows: co-evolution of multiple popula-tions, species conservation, migration rules, self-adaptivemultiple crossovers, local search in hybrid crossover withlocal genetic improvements, controlled mutation, individualage control, and feature-based allele statistical analysis. Todiscuss the capabilities of the proposed approach, numericalexamples are represented to compare the results of MA + SGwith those obtained using genetic algorithms (GA) and MA.The numerical results of the comparison tests showed thatthe GA and the MA maintained long periods without theevolution of the best-fitted individual/solutions. This behav-iour during evolution is associated with the slow matura-tion of the elite group of populations in the GA and MA
C. C. António (B)Faculdade de Engenharia, IDMEC, Universidade do Porto,Porto, Portugale-mail: [email protected]
approaches. This behaviour is avoided when the MA + SGproposed approach is used with computational cost benefits.
Keywords Memetic algorithm · Age structured · Selfishgene theory · Hybrid composite structures
1 Introduction
1.1 Memetic interpretation of evolution
The No-Free-Lunch Theorem that was formulated by Wolpertand Macready [1] changed the focus of computational intelli-gence from designing individual algorithms to designing thearchitectures of optimisation algorithms. This feature over-comes major drawbacks of the evolutionary algorithms (EAs)and the computational intelligence, which occur without theprior knowledge gained from experience [2]. Memetic algo-rithms (MAs) have been recently developed as extensionof the EAs by introducing individual learning as an inde-pendent procedure of the local refinement to accelerate thesearch. However, this perspective of MAs as an acceleratingthe search procedure is a notably simplistic view of the newmethod. Indeed, the work of Dawkins [3] introduced a dif-ferent interpretation of the classical Darwinism using genesinstead of individuals as the fundamental unit of evolution.This new interpretation of evolutionary rules was formalisedin the selfish gene Theory, which was presented by RichardDawkins [3].
Dawkins introduced the term “selfish gene” to express agene-centred vision of evolution. The selfish gene theory sug-gests that evolution is best viewed at the gene level insteadof the individual chromosome level. Furthermore, this theoryconsiders the selection mechanism of organisms or popula-tions as commonly being a selection based on genes. The
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term “meme” was also introduced by Dawkins as a unit ofthe human cultural evolution that is analogous to a gene inthe biological transmission, which suggests that selfish repli-cation may also model human cultural transmission [3]. Thememe as a cultural unit of transmission can be replicatedaccording to its perceived utility or popularity, copied andpropagated via inter-agent communication.
Following Darwinian principles of natural evolution andDawkins’ concept of meme, Moscato [4] proposed the MAsin 1989. The original approach to MAs was presented as afusion of population-based global search and heuristic learn-ing (local search). Indeed, an important aspect concerningMAs is the trade-off between the exploration abilities ofgenetic algorithms (GAs) and the exploitation abilities ofthe local-search strategy [5]. From this aspect, the globalcharacter of the search is given by the evolutionary nature ofthe approach, whereas the local search aspect is usually per-formed using intelligent local-search heuristics or other opti-misation methods. Today, the MAs cover several approachesthat are referred to in the literature [2,6–8] as the hybridevolutionary algorithm, the genetic local search, the culturalalgorithm, the lamarckian evolutionary algorithm, the Bald-winian evolutionary algorithm and the diffusion memeticalgorithm [9].
1.2 The importance of learning procedures
In computational intelligence, Dawkin’s idea of “meme”is often used to motivate the hybridisation of optimisationstrategies/evolutionary algorithms. In the context of memetictheory, an evolutionary algorithm maintains a populationcomposed using genotypes and memes. From this viewpoint,the genotypes represent solutions to a particular problem,whereas the memes represent strategies that can be usedto improve those solutions. The MAs were first introducedas a joint application of population-based global search andheuristic learning, where the latter is often viewed as a memeassociated with the local search. Sometimes, they are referredto as population-based meta-heuristic search methods, whichare inspired by Darwin’s theory of natural evolution andDawkins’ notion of meme as a unit of cultural evolution. Thisgenetic and cultural evolutionary symbiosis of MAs has beenmaterialised into the form of a hybrid global-local approachthat improves both the exploration and the exploitation prop-erties of the search [8].
Two basic forms of individual learning schemes in MAshave been investigated: Lamarckian learning and Baldwinianlearning [5–10]. In Lamarckian learning, the improvementobtained from the local search is reflected in the genotypeby including the locally improved individual into the popu-lation to compete for reproductive opportunities. This learn-ing process is based on the inheritance mechanism of theacquired traits, which were proposed by Lamarck in the nine-
teenth century. However, the Baldwin effect [11] suggests amechanism where evolution can be driven toward favourableadaptation without the referred inheritance. Thus, in Bald-winian learning [5–7,12], there are only changes in fitness ofthe individuals, and the improved genotype is not encodedback into the population. Baldwinian learning is associatedwith phenotypic plasticity, which is defined as the ability ofan organism to adapt to its environment [13,14]. The pureLamarckian approach or a probabilistic combination of thetwo approaches has been used in recent investigations [12].In the latter proposals, the improved fitness is always used,and the improved individual replaces the original one with agiven probability.
The Lamarckian learning accelerates the convergence ofthe evolutionary search, but a premature finish can occur[12]. However, a loss of population diversity is more unlikelywhen the Baldwinian learning is used, although the process isslower than Lamarkianism. The diversity can be lost becauseof the notably aggressive local search. This situation canoccur if, for example, every individual in the populationassumes a local optimum using a complete local search, orin the case of a partial local search, if the design space underconsideration has notably wide basins of attraction fromwhich crossover and mutation cannot easily escape [5,6,12].
The connection between learning and evolution is impor-tant in the proposed framework. Learning has a multileveleffect on the individual and its environment. The learningprocedure adapts the individual to the current environmen-tal changes faster than what evolution can do. As previ-ously mentioned, learning can indirectly influence the geneticmaterial of a species through the Baldwin effect or directlybe inherited in the Lamarckian viewpoint. An individual thatlearns can find a local optimum even if the genotype is notclose to the optimum. This effect increases the individual’schance of survival. However, learning requires a lifetime ofthe individual with associated costs. This criterion generatesa selection pressure related to the preference using the indi-viduals that have the correct phenotype from birth withoutany requirement for learning [15].
1.3 Selfish gene algorithm (SG)
The memetic algorithms that are based on genetic and cul-tural evolutions result from the new paradigm introduced bythe selfish gene theory of Dawkins [3], which was applied toalgorithmic optimisation. However, starting from Dawkins’theory, other algorithmic approaches have been proposed.In 1998, Corno et al. [16] proposed a SG, where the sur-vival of the fittest is considered a battle that is fought by thegenes instead of the individuals. Here, the population, whichis called the virtual population (VP), is similar to a storeroomof genes. The algorithm evolves and reproduces its effect on
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the statistical parameters of a virtual population, which isrelated to the allele’s frequencies in the same gene group.
Instead of a local search as performed in MAs, the selfishgene algorithm follows a different learning scheme usingfeature-based allele’s statistical analysis. The conventionalpopulation of individuals considered in GAs is replaced by aVP. In the SG proposed by Corno et al. [16], the VP is updatedusing rewards and punishments for the winner and the loser ofa tournament, respectively. It is assumed that this trial occursmany times in the population, and after each competition,the winner has the opportunity to reproduce itself, whereasthe loser cannot generate any offspring. The result is thatall winning alleles slightly increase their frequency in theirrespective loci in the VP at the expense of the losing alleles[16]. Some authors proposed a new class of evolutionaryalgorithms that were denoted the estimation of distributionalgorithms (EDAs) with the selfish gene theory [17]. In theseapproaches, the new candidate solutions are generated bysampling from the distributions of alleles in the VP withoutusing mutation and crossover operators. Only the selectionoperator is directly executed on the entire population [17].The learning procedure is based on mutual information that isobtained from entropy and correlation measurements amongthe variables [17].
In recent developments of SG, the VP contains vectors ofthe accumulated frequencies of alleles following a probabil-ity distribution for each gene [18]. In this approach, whichwas proposed by Villagra et al. [18], a reward is only accu-mulated by the winner of the mating process between anindividual from the VP and random immigrants. The algo-rithm uses a multi-recombination procedure based on one ortwo virtual populations [18].
The previously mentioned different approaches of SGuse the statistical allele’s features or the mutual informationobtained from entropy and correlation measurements amongthe variables in a VP of alleles. Although the learning pro-cedures are different from the local search proposed in theMAs, in the SG approaches, the learning improvements arereflected in the genotype, which is similar to a Lamarckianprocedure. The optimisation of hybrid composite structuresthat consider the sizing, the topology and the material distrib-ution benefits from guaranteeing the population diversity anddepicting the local optima. Synergies are expected from thecombination of different types of Lamarckian learning proce-dures in terms of the exploitation and exploration propertiesof the design space.
1.4 State-of-the-art in hybrid composite structure design
The optimal stacking sequence design of laminated com-posite structure using GAs has been a notably importantresearch topic with increasing industrial applications. Theoptimisation techniques to design composite structures were
reviewed by Awad et al. [19], who recommended some newreal scenario proposals for civil engineering applications.However, few approaches address the design of hybrid com-posites using the material distribution as a design variableof the structure. In particular, Adali and Verijenko [20] stud-ied the effect of hybridisation on the dynamic response ofcomposite structures and concluded on the benefits of usingmulti-materials. Rahul et al. [21] minimised the cost/weightof hybrid laminates while maximising the strength underimpact loads using a GA. António [22–24] proposed a Hier-archical Genetic Algorithm based on co-evolution to opti-mise the hybrid composite shell structures that were rein-forced with beams. Baier et al. [25] combined the strengthand manufacturing requirements for structures that were builtwith metal parts and carbon composites, and they used aGA-based multi-criteria engineering. Liu et al. [26] devel-oped a bi-level strategy to design aircraft wing cover pan-els under compression and lateral pressure. The objectivewas to find the minimum mass design of stiffened panelsthat satisfies the buckling and strength constraints. A recentwork by Keller [27] illustrated the difficulties encounteredby the designer who focused on a global optimisation ofgeometrically partitioned composite shell structures. Kellerproposed an evolutionary algorithm scheme to specificallyoptimise the composite structures; this scheme encodes thenumber of global plies and their thickness, material and ori-entation.
The simultaneous use of different types of variablesin hybrid composite structures requires the developmentof robust design algorithms to achieve the global optimaldesign. In reality, the future applications in car, space andaeronautical industries will be multi-material constructions,where composites and advanced metals will be combined toachieve the best performance/cost balance. Recent publica-tion shows that few models have been developed to optimisethe multi-material construction with the sizing, the topologyand the material distribution in consideration [22–24,27].The optimal design of hybrid composite structures is an openissue, and the challenge is to find robust algorithms thatcover different types of variables (continuous and discretein nature), objectives and constraints for the global optimi-sation. The proposed MA + SG approach based on multiplelearning procedures is appropriate to answer the problemcomplexity such as to search the global optimal design ofhybrid composite structures.
1.5 Proposed MA + SG approach
The mechanical properties of laminated composite structuresare sensitive to the selections of materials, the spatial distri-bution of different materials in the structure, and the localorientation of fibre reinforcements in the laminate deals. Thissensitivity depends on many design variables. To overcome
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the difficulties of attaining the global solution for the opti-misation problem of composite structures, which involvesthe sizing, the topology and the material choice, a new strat-egy is proposed based on the hybridisation of the memeticalgorithm and the selfish gene algorithm (MA + SG).
The MAs and the SG are based on the same paradigm ofevolution centred on the gene level, and their difference isthe learning procedure. In this aspect, the hybridisation ofthe MA and the SG is a learning reinforcement to globallyoptimise the design of composite structures, which involvesthe sizing, the topology and the material choice.
In this work, first, the memetic properties of the previ-ously published hierarchical genetic algorithm (HGA) withthe age structure and the co-evolution of multi-populationsare addressed [22–24]. The memetic learning procedures ofthe HGA are studied to understand their exploitation andexploration algorithmic capacities.
Second, the application of the selfish gene theory isanalysed using a fusion of concepts that were proposed inmemetic algorithms and selfish gene algorithms. The agestructure of the memetic algorithm HGA [22–24] is appliedwith the feature-based allele’s statistical analysis, which isused in the learning procedure of the SG. Thus, the selfishgene theory in the version of Corno et al. [16] is appliedto the age-structured VP. Although the SG learning mecha-nism is Lamarckian, it is different from other mechanisms inthe previous HGA memetic algorithm [22–24]. Synergies areexpected from the combination of different types of Lamar-ckian learning procedures. In this paper, this challenge isperformed using a new approach called MA + SG based onmultiple learning procedures with a co-evolution of multi-ple populations. In MA + SG, all new individuals inherentlybelong to an enlarged VP population, whose age structure isupdated at each generation. According to the SG theory, newindividuals are generated using an innovative age parame-terised selfish gene (ApSG) crossover operator with mating-selection and offspring-generation mechanisms, which areinfluenced by the best alleles stored in the age-structuredVP.
The paper is organised as follows: Sect. 2 presents theformulation to optimise hybrid composites. The main fea-tures of the proposed approach of MA + SG are describedin Sect. 3. The Lamarckian and Baldwinian learning proce-dures of the MA + SG approach are detailed and analysedin Sects. 4 and 5, respectively. The optimisation results areprovided in Sect. 6. The effects of the new ApSG on theevolutionary search are discussed in Sect. 6. Conclusions onthe performance of the proposed approach are presented inSect. 7.
The proposed MA + SG approach aims to overcome somedifficulties and shortcomings of the existing approaches, inparticular, those associated with the optimisation of hybridcomposite structures:
• The design variables for composite structures are bothcontinuous and discrete by nature. This discontinuity ofthe design space is exacerbated for hybrid composites.The MA + SG approach is a good mean to overcome theexisting natural discontinuities in this type of structures.
• In a genetic/memetic search for the optimal design ofhybrid composite structures, it is easy to reach localoptima if the genetic operators are inefficient. The prema-ture anchorage at the local optima depends on the popula-tion diversity and the discontinuities of the design space.Different Lamarckian and Baldwinian learning proce-dures in the MA + SG approach are important to guar-antee the population diversity and to exchange informa-tion, which originates from different regions of the designspace.
• The optimal design of hybrid composite structures is mul-timodal optimisation problems because of the possibil-ity of finding different project solutions. The MA + SGapproach can intensify the search for alternative solutionsbecause of the learning procedures.
2 Optimisation problem formulations: designof composite structures
2.1 Design definition of hybrid composite laminates
A composite laminate is a material with bonded plies toobtain a homogenised macro-mechanic behaviour when thematerial is subjected to external applied loads. Each plyis a composite system made of a weakest material, whichis denoted by a matrix, and reinforced by long or shortfibres of the strongest material. The fibres are randomly dis-posed (short fibres) or disposed with a predefined orientation(long fibres). This work only considers composite systemswith long fibres, and the ply orientation is determined bythe angle of the longitudinal fibre axis relative to the lam-inate referential axis. The composite laminate structures inthis study are plates or shells that are reinforced with beamstiffeners.
In this work, the design of the hybrid composite materialstructures is optimised as follows: (1) optimise the size ofthe plate, the shell or the beam laminates; (2) optimise thestacking sequence of each laminate; (3) optimise the laminatedistribution for the structure. This natural decomposition ofthe optimisation problem leads to a multimodal-search land-scape of hybrid composite structures that can find severaloptimal solutions.
Figure 1 shows the type of composite structures that iden-tify the design variables. The design variables, which arerepresented by vector x, are defined as follows: the angleθi, j and the thickness t̄i, j of the ith ply of the jth plate orshell laminate are grouped in vectors θ and t̄, respectively;
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Fig. 1 Definition of the designvariables associated with the jthplate or shell hybrid laminateand the kth beam laminate
[ 1 / 2 ]beamlaminate
x
yz
plate/shelllaminate
s
Stacking sequence of symmetric (s)
laminate jπ
Composite system number
i, jθ
i, jt
kw
kh
/ … / m
the height h j and width w j of the rectangular cross sectionof the jth beam laminate are grouped in vectors h and w,respectively.
The macro-mechanical properties of each laminate dependon the previously defined design variables, the ply materialand the laminate distribution on the structure. The materialdistribution at the laminate and structure levels is definedby the variable π j , which is associated with the jth plate orshell laminate, as shown in Fig. 1. These variables, whichare related to the hybridisation, are important in the opti-mal design of laminated structures with multiple materials[22,23]. In the approach that we consider in this work, theyare grouped together as vector π. Only plate or shell lami-nates are considered in the optimisation process for the mate-rial distribution at laminate and structure levels.
2.2 Mathematical formulation of the optimisation problem
The optimisation problem is formulated to minimise theweight/cost of the structure W (t̄, w, h,π), which is sub-jected to structural-integrity constraints and has imposed ser-vice conditions, as defined in [22,28]. The hybrid compositestructures were analysed using the Finite Element Methodbased on the previously defined formulation with the Mar-guerre shell element and a Timoshenko beam element [22].The structural system with a non-linear behaviour is in equi-librium if the internal forces are equal to the external appliedloads. Because it is impossible to reach an exact equilibriumin the numerical process of the solution, the goal is to obtaina close state near equilibrium within a small error. Usingthe Total Lagrangian formulation, a measure of the error atthis equilibrium state is the vector of non-balanced forces�(d, λ) and is defined as
�(dt , λt ) = R(dt ) − λt F, (1)
where R is the vector of the equivalent nodal forces associatedwith the actual stress field of the structure, F is the equiva-lent nodal forces because of the external applied forces, λt is
a scale factor related to the load level t and dt is the corre-sponding displacement vector. The equilibrium path is tracedusing the arc-length method [29]. An approach similar to theone proposed by Budiansky [30] identifies the load factorsλb, which is associated with buckling, and λF P F , which isrelated to the first ply failure. The first ply failure is checkedusing the Huber–Mises law [28]. This procedure enables oneto obtain the corresponding critical displacements.
The constraints are imposed on the critical load factorand the critical displacement, both of which are associatedwith buckling and the first ply failure. A unified approach[28] for buckling and the first ply failure is used to check theintegrity of the hybrid composite structures using the conceptof a critical load factor λcri t , which is defined as
λcri t (x,π) = M I N [λb(x,π), λF P F (x,π)], (2)
where λb and λF P F are the critical buckling factor and theload factor that correspond to the first ply failure factor,respectively. Considering the equilibrium path, the criticaldisplacement dcrit can be associated with the buckling db orthe first ply failure dF P F , as follows
dcrit (x,π) = M AX [db(x,π), dF P F (x,π)] (3)
Thus, the plates and shells of hybrid composite structuresthat are reinforced with beams under static loading can beoptimised as follows
Minimise W (t̄, w,h,π) over θ, t̄, w, h,π, (4)
which is subject to
ϕ1(x,π) = 1 − λcri t (x,π)
λa≤ 0 (5)
ϕ2(x,π) = dcrit (x,π)
da− 1 ≤ 0, (6)
the size constraints
xlj ≤ x j ≤ xu
j , j = 1, ...N̄x (7)
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satisfy the equilibrium equation set
�(dt , λt , x,π) = R(dt , x,π) − λt F = 0, (8)
and the additional arc-length method equation is
Φ(dt , λt , x,π) = 0 (9)
In Eqs. (5) and (6), λa and da are the permitted critical loadfactor and the critical displacement, respectively. The sizeconstraints in Eq. (7) are associated with the bounds imposedon components x j of vector x of the design variables. Vectorx aggregates the design variables θ, t̄, w, h, which are associ-ated with the ply orientation and the sizing of each laminate,as defined in Sect. 2.1. Vector π is associated with the mate-rial distribution at the laminate and structure levels. In theequilibrium Eq. (8), R(dt , x,π) denotes the internal forcesin the structural system, dt is the displacement vector,λt is theload factor, and F is the vector of the applied external forces.The superscript t in Eqs. (8) and (9) denotes the incrementalnature of equilibrium in a non-linear analysis of compositestructures. The equilibrium path represented by Eq. (9) istraced using the arc-length method equation [28–30].
3 Memetic algorithm based on multiple learningprocedures
To optimise the composite structures with the sizing, thetopology and the material choice, the considered design vari-ables are continuous and discrete by nature. The encoding ofthese continuous and discrete variables using a single stringwith an appropriate code format in EAs, particularly in GAs,increases the computational search cost. Thus, decompo-sition approaches have been proposed in some compositeoptimisation models to increase the search efficiency. Thisdecomposition strategy is independent of the optimisationmethod. For example, Conceição António [28] proposed amultilevel GA based on a sequentially performed bileveldecomposition with different populations and fitness func-tions to improve the convergence properties and the effi-ciency. Murugan et al. [31] proposed a decomposition ofthe optimisation problem of aircraft composite structures atboth global and local levels. In this approach, the box-beamcross-sectional dimensions of the composite structure areglobally optimised, whereas the optimal ply angles of thosebox-beams are locally determined.
In general, the previous proposals attempt to overcomethe complexity and the specificity of optimising compositestructures to maintain the population’s diversity in the GAsearch. The formulation from (4) to (9) suggests a “natural”decomposition of the optimisation problem: the weight of thecomposite structure is related to the design variables t̄, h andw, the anisotropy of the composite laminates is related to θ
and the material choice at the ply, the laminate and the struc-ture levels are associated with π . Furthermore, the materialdistribution, which is represented by the variable vector π ,is associated with a discrete domain that encodes the pos-sible choices for the composite system (matrix/fibre). Fromprevious considerations, because the optimisation problemhas mixed variables, the feasible domain includes holes ordiscontinued regions that represent the discrete nature of thecoding [32].
The proposed MA + SG approach is a mixed modelthat applies multiple learning procedures to explore thesynergy of different cultural transmission rules in the evo-lutionary process. The principal aspects of this algorithmare co-evolution of multiple populations, species conser-vation, migration rules, self-adaptive multiple crossovers,local search in a hybrid crossover with the local geneticimprovements, controlled mutation, individual age controland feature-based allele’s statistical analysis. Most of theseaspects are associated with some type of problem knowl-edge and learning from evolution and can be classified asLamarckian (with genetic inheritance) or Baldwinian learn-ing (without genetic inheritance). These two types of learningprocedures are explained in detail in Sects. 4 and 5. Becausethe “natural” decomposition is used to reduce the complex-ity of the optimisation problem, the learning procedures canalso use specific heuristics that represent “memes”. Then, anappropriate research field to apply the multiple learning pro-cedures is the optimal design of composite structures, whichconsiders the sizing, topological and material distributiondesign variables.
The design variables in the optimisation of hybrid com-posite structures are both continuous and discrete by nature.They are represented by a single string with an appropri-ate code format in the MA. The representation increases thecost of the design space searching. Therefore, decompositionapproaches have been proposed in some composite optimi-sation models to increase the search efficiency [22–24,28].
The developed model in the proposed MA + SG approachis based on the co-evolution of multiple populations thatcover different regions of the design space with differ-ent design variables at each generation of the evolutionarysearch. The search for each population in the hierarchicaloptimisation algorithm proceeds for the same objective func-tions and constraints. However, the number of design vari-ables used at each generation is lower than the number ofvariables of the original optimisation problem. This “nat-ural” decomposition of the original problem is associatedwith the nature of the variables in the design of the hybridcomposite structures: the ply thickness and the cross sectionbeam variables are associated with the structural weight, theply orientation is associated with the anisotropy, the stackingsequence at the laminate level and the material distributionon the structure are associated with discrete material possi-
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Lamarckian learning based on
gene statistics
Isolation stage o f POP1
Evolve over
, t , h, w,
Evolve over
, t , h, w
Epoch = Epoch+1
Epoch = 1
Isolation stage of POP2
Migration flow
Evolve over
, for even generation;
t , h, w, for odd generation
Isolation stage of POP3
Stopevolution
?
Species control
Species control
ENDYesNo
START
Lamarckian learning based on
gene statistics
Lamarckian learning based on
local search
Baldwinian learning based on self-adaptive rules
VP
θ π
θ
θ ππ
Fig. 2 Flowchart of the MA + SG approach based on hybrid learningprocedures
bilities. However, in the MA + SG implementation, there isan additional advantage because the proposed decomposi-tion enables the application of different learning procedures.Figure 2 presents the most important aspects of the MA + SGframework. In this decomposition, the proposed MA + SGis based on a sequential hierarchical relation among threesub-populations, i.e., POP1, POP2 and POP3, which evolvein separated isolation stages followed by a migration flow ina co-evolutionary process.
Sub-population POP1 plays the role of the master pop-ulation that includes all design variables in the evolution-ary process. This sub-population provides the opportunityto screen the complete design space using the explorationcapacity of the search. The best individuals from POP1 arethe seeds for POP2, where the material identifier is main-tained constant. At this stage, the evolution depends only onthe search of the best anisotropy to reach the weight min-imisation. POP3 starts with the seeds from previous hierar-chic sub-populations (POP1 and POP2). At POP3, the evo-lutionary process is driven to an alternative use of mixeddesign variables to obtain the maximum diversity in the sub-population based on the equilibrium between exploration andexploitation of the search.
Even for a comparatively small chromosome structure,the size of the design space can be notably large. The geneticoperators drive the evolutionary search through more promis-ing regions to progressively reduce the design space. The
design space can be reduced by adopting a hierarchical topol-ogy because there are fewer intervening variables in the opti-misation process. Depending on the evolving population, dif-ferent design variables are considered in the optimisationmodel, which correspond to active and non-active segmentsof each chromosome, as shown in Fig. 2. The use of differentactive segments of the chromosome corresponds to a decom-position of the design space. The objective is to improve theexploration and exploitation of different regions of the designspace. Chromosome activation addresses the segmentationof the population in niches and is driven using the speciesconcept [22–24].
The concept of species is associated with the material dis-tribution topology in composite structures. In the proposedapproach, the individuals with chromosome segments corre-sponding to identical material selections and distributions onthe composite structure belong to the same species. All indi-viduals from the same species have equal genetic traces in thechromosomal segment associated with the material selectionat the ply, the laminate and the composite structure levels,whose representation is encoded in vector π [22]. The con-cept of species evolution in the multi-island GA was proposedby Siarry et al. [33] using a geometric-based definition. How-ever, the definition of species in the MA + SG approach ismaterial-based [22]. Furthermore, the introduced control ofnumber of individuals in the same species is similar to the pre-viously developed HGA [22,23]. Rules based on the speciesconcept are imposed on either isolation or migration stagesto overcome the predominance of a species and guarantee thediversity [22–24]. More specifically, the implementation ofthe species conservation paradigm is considered at the iso-lation and migration stages based on the following rules: 1.isolation stage: limit the number of individuals of the samespecies; and 2. migration stage: every candidate for migrationbelongs to a different species. The dominant and dominatedspecies are not previously known, and the appearance of newspecies results from co-evolution and local search, which is aLamarckian learning procedure. This procedure is not a com-pletely free evolutionary process because it can be concludedfrom the above positive discrimination in favour of speciesconservation. In reality, each species corresponds to a localoptimum, which is reflected in the genotype of the individ-uals inserted into the population. The species conservationparadigm is applied with the concept of an age-structured vir-tual population (VP). Most evolution-generated individualsare conserved and have a life cycle emulated in VP throughthe individual-age concept. From another viewpoint, the VPis simply considered a pool of genes fighting to replicate itsproperties, which is called “cultural transmission” [16].
The balance between exploration and exploitation isattained using both the species conservation paradigm, whichis applied to the virtual population, and the elitist strategy,which is adopted at each sub-population isolation stage. The
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flow chart in Fig. 2 shows the interaction of hierarchical pop-ulations with the age-structured virtual population.
Different types of learning procedures are used for eachsub-population evolutionary process [5,6]. The assignedLamarckian learning mechanisms (with genetic inheritance)are (1) the elitist hybrid crossover with genetic improvementbased on the local search to evolve sub-population POP2, (2)the controlled mutation based on environmental data of thestructural behaviour to evolve all sub-populations, and (3) thefeature-based allele’s statistics analysis, which is inspired inthe selfish gene algorithm of Corno et al. [16] and applied tosub-populations POP1 and POP3.
Two Baldwinian learning mechanisms (without geneticinheritance) are analysed: (1) the Self-adaptive rules basedon the success rate, which is associated to the phenotypicplasticity of the design space, and (2) the Individual agecontrol and life-cycle emulation, which is associated withphenotypic plasticity of age-structured VP.
4 Lamarckian learning mechanisms
The most important Lamarckian learning procedures are theelitist hybrid crossover with genetic improvements, the con-trolled mutation and the use of feature-based allele’s statis-tical analysis.
4.1 Elitist hybrid crossover with genetic improvement
The offspring generation mechanism is based on the Ham-ming metric. The offspring candidates are generated in neigh-bourhoods associated with the parents. Individual proper-ties of each candidate, such as the weight, energy and con-straint violation, are defined from both parents using approx-imations and heuristic rules [22–24]. A local optimisationprocess, which is driven on the genetic path defined by theparents considering the Hamming space, is performed to findthe best offspring for each couple. Only the best offspringcandidate is inserted back into the population. This geneticimprovement based on the local search on a defined trajectoryin the Hamming space is a type of Lamarckian learning.
4.2 Controlled mutation
The controlled mutation primarily incorporates the datarelated with the behaviour of the state variables of the struc-tural system [22,24]. In the first step to apply the SG theory,a relationship is established between the stress field in thecomposite structure and the genes of the chromosome thatis directly associated with the structural behaviour. The sec-ond step is to change the gene composition using a controlledmutation to improve the fitness of the selected individual andinsert it back into the population. This step is a self-adaptive
procedure based on local optimisation. The proposed Lamar-ckian model to mutate an individual (solution) is based onthe following steps:
• Step 1: Random selection of an individual (solution) fromthe elite group.
• Step 2: Rank in decreasing order the stress constraintvalues g(σk) ≤ 0 at each kth point of the compositestructure that corresponds to the selected individual.
• Step 3: Physical in situ identification of msup and min f
variables that are associated with the stress constraint val-ues at the top and bottom of the ranked list, respectively.The genes associated with these variables may possiblymutate.
• Step 4: If the stress constraints of the top group are vio-lated, then Mutation is performed to change the selectedindividual (solution) to avoid violating the constraints.The mutation only affects the genes of the selected vari-ables msup that are physically associated with the topgroup of the ranked list.
• Else• The mutation in the msup variables is not implemented.• End if• Step 5: If the stress constraints of the bottom group are
violated, then• The mutation in the min f variables is not implemented.• Else• The mutation is performed to minimise the weight of the
composite structure and the strain energy of the selectedindividual (solution). The mutation only affects the genesof the selected min f variables that are physically associ-ated with the bottom group of the ranked list.
• End if
4.3 Feature-based allele statistical analysis
According to the selfish gene theory, the population can sim-ply be considered a pool of genes, and the individual genesfight for the same spot of the chromosome in the genotype[16]. The frequency of appearance of an allele increases theoverall success rate. According to the selfish gene theory,the survival of the fittest depends on the genes instead ofthe individuals. Unlike the traditional GA, an individual inthe SG theory is not as important to evolution because it isappropriated to consider the species conservation paradigm.The concept of a virtual population (VP) is used, and theSG affects the statistical parameters of this virtual popula-tion throughout the evolutionary process. Each new solutionis implicitly generated by changing the frequencies or theprobabilities of the alleles or the possible genes.
Assuming that the virtual population is age-structuredcontinuous, the statistical parameters of the allele populationare updated at each generation. Furthermore, each individ-
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Memetic Comp. (2014) 6:113–131 121
ual is eliminated after a lethal age, which indicates that theallele population is refreshed according to the evolution andthe VP diversity is guaranteed. Thus, most promising alle-les are selected for genes [16–18]. Then, new individuals aregenerated following the SG theory as follows:
• Step 1: Choose two individuals from an age-structuredvirtual population for a tournament. The individual withthe higher fitness is the winner;
• Step 2: The frequencies of alleles in the age-structuredvirtual population are weighted using the correspondingfitness of the chromosomes as a measure of the perfor-mance;
• Step 3: A stud is built using the previously obtained fre-quencies for the best alleles. This stud is a pseudo-parentfor the mating procedure [18];
• Step 4: An offspring is created using the winning indi-vidual in first step and the stud. A uniform parameterisedcrossover [22] is performed between the parents;
• Step 5: Go to the first step until the offspring populationis fully created;
• Step 6: Insert the offspring population in the age-structured virtual and current populations according theLamarckian learning rules.
This procedure is a pseudo-crossover scheme with a changedmating selection mechanism and an offspring generationmechanism, which are influenced by the best alleles in theage-structured virtual population (VP). The new operator iscalled the age parameterised selfish gene (ApSG) crossover.It finds the best solution using the interaction among theallele frequencies in the same gene group and by weight-ing these frequencies according to the corresponding fitnessof the individuals. Because binary and integer code formatsare used for the genotype, two independent procedures areadopted to weight the allele frequencies in the VP.
For each gene of the binary-formatted part of the chromo-some, the alleles can be “0” or “1”. In this case, the frequencyof an allele at the ith locus is weighted by the fitness of everyindividual of the VP with the same allele at that locus asfollows:
S0(i) =NV P∑
j=1,
if L(i, j)=0
F I T ( j) (10)
S1(i) =NV P∑
j=1,
if L(i, j)=1
F I T ( j) (11)
where L(i, j) is the allele’s value at the ith locus of the jthindividual of the VP, FIT is the fitness function and NV P isthe number of individuals in the actual age-structured VP.
The allele’s value Lst (i) at the ith locus of the part of thestud’s chromosome in binary format is
Lst (i) ={
0, if S0(i) ≥ S1(i)1, if S0(i) < S1(i)
(12)
For each gene of the encoded chromosome part using theinteger format, the alleles can assume different key numbersk. Each k represents a possible combination of materials thatdefines a stacking sequence of the composite laminate, asdefined in Fig. 1 of Sect. 2. Here, the frequency of allele kat the ith locus is weighted by the fitness of every individualfrom the VP with the same allele at that locus, as follows:
Sk(i) =NV P∑
j=1,
if L(i, j)=k
F I T ( j) (13)
Then, for the part of stud’s chromosome that is encoded usingthe integer format, the allele’s value Lst (i) at the ith locus is
Lst (i)= M, with SM (i)= M AX [Sk(i), k = 1, . . . , Ncb](14)
where Ncb is the number of total possible combinations ofmaterials that define a stacking sequence of the compositelaminate.
5 Baldwinian learning process
The Baldwin effect [11] suggests that evolution can be driventoward favourable adaptation without encoding the changesback into the individual genes. Two Baldwinian proceduresare adopted: the self-adaptive rules based on the success rateof evolution and the life-cycle emulation based on age control.
5.1 Self-adaptive rules based on the success rate
The adaptive rules use additional information related to thebehaviour of the state and the design variables of the struc-tural problem and are associated with some type of learning.At each generation, the self-adaptation of the genetic para-meters to the evolutionary conditions attempts to improve theefficiency of the genetic search.
The following crossover operators are used in the MA:Elitist hybrid crossover with genetic improvement (EHCgi)[22–24], Elitist parameterised uniform crossover (EpUC)[22–24] and age parameterised selfish gene (ApSG). A self-adaptive probability to select a crossover operator schemecan be established depending on the cumulative number ofsuccess events until the current generation of the evolution-ary process [24]. An event is considered successful if at leastone solution from the applied crossover is better than theworst-fitted solution of the elite group of the population. This
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122 Memetic Comp. (2014) 6:113–131
condition implies that the system learns from the evolution-ary behaviour of the population. The search engine changesduring the evolutionary process that influences the exchangeprocedure of the genetic material by a learning process. How-ever, because there is no inherited genetic transmission butonly a “cultural” transmission, which is associated with thefitness improvement (cumulative success event), this processis the Baldwin effect. Indeed, this self-adaptive procedure isan example of a Baldwinian learning process.
5.2 Individual age control and life cycle emulation
In the proposed approach, the eliminated individuals by theelitist strategy have a second opportunity to intervene in theevolutionary process. Indeed, all individuals generated in theevolutionary search inherently belong to an enlarged popula-tion with an age structure, where an individual is eliminatedwhen it reaches the lethal age (natural death). A continuousupdated model of deletion and reproduction was adopted forthe age-structured virtual population, as shown in Fig. 3. Thesize of the age-structured population changes with the evolu-tionary process. It is appropriate to conclude that the survivaltime decreases with the increasing age. Thus, the allele’sproperties in the VP are continuously updated. The transferof genetic characteristics is conditioned by the candidate age.The sexual maturity and potentiality of each individual in theVP are assumed to follow a normal probability distributionfunction (PDF) from birth (age = 1) until death at the lethalage, as shown in Fig. 3. Then, the parent is selected accordingto the potentiality of different individuals in the VP. The prob-ability of an individual to be selected as a parent depends onits age and corresponding position on the normal PDF curve.This dynamic parent-selection procedure emulates the indi-vidual behaviour along the life cycle. Individuals with agesat the tails of the normal density function are the youngestand oldest of the scale and have a notably low probabilityto be selected for recombination in the evolutionary process
123
n-1n
Population, tP
123
n-1n
Age
Death
Crossover
Age
0of
fspr
ing
Parent selection
Population, 1+tP
Age
)z(f z
zz
z
age=1 age=nz
Fig. 3 Deletion and reproduction rates in an age-structured virtual pop-ulation (VP)
[22]. Because the individual age does not affect the geneticcode, this procedure is based on the Baldwin effect, wherethe “cultural” transmission is associated with the life cycle.
6 Numerical results and discussion
6.1 Test case definition
To discuss the capabilities of the proposed memetic-basedselfish gene algorithm (MA + SG) with hybrid learning pro-cedures to address the multimodal optimisation of compositestructures, a numerical example is presented. A cylindricalshell with rib stiffeners, both of which are made of lami-nated composite materials, is considered as shown in Fig. 4.The shell is hinged on straight sides and free at its curvedboundaries. A central point load F = 4 kN is applied, andonly a quarter of the structure is considered for the struc-tural optimisation problem. Ten laminates are considered:four laminates that group the shell elements (from 1 to 4)and six others (from 5 to 10) that group the beam elementsof the rib stiffeners. The rib stiffeners are connected belowthe shell elements.
The mechanical properties of the ply laminate materialsare presented in Table 1 [34] and denoted by the longitudinalstrength X, the transversal strength Y, the shear strength S,the longitudinal Young modulus E1, the transversal elasticmodulus E2, the shear modulus G12, Poisson’s ratio ν andthe specific weight of the ply material ρ. One material fromTable 1 is selected for each ply. The pair material/stackingsequence, which is defined by variable π j , is a combinationof different materials for symmetric shell laminates. Fourcomposite systems are considered for ply laminates in thestudy: one carbon/epoxy composite, two glass/epoxy com-posites and one Kevlar/epoxy composite. The Kevlar/epoxyis considered a possible material choice only for the innerply of the symmetric laminates. The remaining materials arefree selections, and at least two materials must be consideredfor a hybrid composite laminate construction. Then, thereare 33 possible combinations of these four materials for thestacking sequence π j considering the defined rules and sixplies in the symmetric jth composite shell laminate construc-tion. The beam laminates have six plies that are made frommaterial number 2 (Table 1), which does not change duringthe optimisation process.
Considering the optimisation problem from Eqs. (4) to (9)and using an appropriate formulation for the genetic search,the global fitness function FIT is maximised as
Maximise F I T = C̄1 − β1W (t̄, w, h,π)
−2∑
i=1
Γi [ϕi (θ, t̄, w, h,π)] (15)
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Memetic Comp. (2014) 6:113–131 123
Fig. 4 Geometry and laminatedistribution of the compositecylindrical shell reinforced withstiffeners
θ θ
RL1
y
F
L
Lm
x
R=25.4 [m]
L = R/10
L1/m =19.983
z
stiffener
Table 1 Mechanical propertiesof the materials in the compositelaminate
Material E1 (GPa) E2 (GPa) G12 (GPa) ν
Type Code
CFRP: T300/N5208 1 181.00 10.3 7.17 0.28
GFRP: Scotchply 1002 2 38.60 8.27 4.14 0.26
GFRP: E-glass/epoxy 3 43.00 8.90 4.50 0.27
KFRP: Kev 49/epoxy 4 76.00 5.50 2.30 0.34
X (MPa) Y (MPa) S (MPa) ρ (kg/m3)
CFRP: T300/N5208 1 1,500 40 68 1,600
GFRP: Scotchply 1002 2 1,062 31 72 1,800
GFRP: E-glass/epoxy 3 1,280 49 69 2,000
KFRP: Kev 49/epoxy 4 1,400 12 34 1,460
with Γi [ϕi (θ, t̄, w, h,π)]=
{0 if ϕi (θ, t̄, w, h,π) ≤ 0K̄i [ϕi (x,π)]qi if ϕi (θ, t̄,w,h, π) > 0
i = 1, 2
(16)
where Γi [ϕi (θ, t̄, w, h,π)] represents each constraint termthat is associated with the constraint ϕi (θ, t̄, w, h,π) andthe constants in Eq. (15) are C̄1 = 10, 000 and β1 = 10.The lower bound of the critical load factor defined in Eq. (5)is λa = 0.5. The maximum allowed critical displacementin buckling or FPF failure modes in Eq. (6) is da = 1.3 ×10−1m. The size constraints on the design variables are
−90o ≤ θi, j ≤ 90o
1.2 × 10−3m ≤ t i, j ≤ 2.4 × 10−3m
2.0 × 10−2m ≤ h j ≤ 4.0 × 10−2m
5.0 × 10−3m ≤ w j ≤ 1.5 × 10−2m
(17)
A code binary format for the design variables θ, t̄, w, h isadopted with four digits for the ply angle and three digits forthe remaining variables. An integer format is adopted for the
material distribution design variable π. Each sub-populationof the MA + SG has 15 individuals. The elite group hasfive individuals, the mutation group has three individualseach in all populations, and three individuals participate ineach migration flow among the three MA + SG populations(POP1, POP2 and POP3). The Isolation periods of the threeMA + SG populations perform on a sequence of 8-8-8 gen-erations. Each referred sequence is denoted by an epoch.
In the age-structured VP, the lethal age is 15 genera-tions. In the self-adaptive procedure to select the crossoveroperator, which is referred to in Sect. 5.1, the bounds onthe probability of selection of the crossover operator EpUCin populations POP1, POP2 and POP3 are γ in f = 0.5and γ sup= 0.85. These limits are established to control theinfluence of the elitism of the Elitist parameterised uniformcrossover (EpUC).
6.2 Analysis of the evolutionary process
The MA + SG populations evolve during 15 epochs for a mul-timodal optimisation of the proposed composite structure.The self-adaptation of the evolutionary parameters to select
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124 Memetic Comp. (2014) 6:113–131
Table 2 Self-adaptiveprobability to select a crossoveroperator in MA + SG evolution
Epoch POP1 POP2 POP3
EpUC ApSG EHCgi EpUC EpUC ApSG
1 0.500 0.500
2 0.500 0.500
3 0.571 0.429
4 0.727 0.273
5 0.692 0.308
6 0.647 0.353
7 0.681 0.319
8 0.500 0.500 0.500 0.500 0.696 0.304
9 0.680 0.320
10 0.692 0.308
11 0.667 0.333
12 0.636 0.364
13 0.647 0.353
14 0.658 0.342
15 0.641 0.359
73
75
77
79
81
83
85
0 2 4 6 8 10 12 14 16
Epoch
Wei
gh
t [k
g]
1
2
3
4
5
Solutions:
73
83
93
103
113
123
0 1 2 3
Epoch
Wei
ght [
kg]
Fig. 5 Weight evolution of the individuals (solutions) of the elite group
the crossover is considered a Baldwinian learning processthat results from the success rate of replacement-based fit-ness and its cultural transmission to the next generationswithout genetic transmission. The probability of selectingan appropriate crossover scheme is updated along the evo-lutionary process, and it is associated with the cumulativesuccess events as defined in reference [24]. For the presentedexample, the probabilities of selecting the crossover oper-ators are shown in Table 2. Self-adaptive probabilities are
calculated for each MA + SG population (POP1, POP2, andPOP3) during the evolutionary process.
The influence of the lower limit on the selection probabil-ity of the crossover operator EpUC, γ in f = 0.5, is observedin POP1 and PO2, which implies that the cumulative suc-cess of the crossover operator EpUC in POP1 and POP2 islow. However, a minimum selection probability was assumedbecause of the elitist properties of EpUC. On the contrary, theselection probability of the crossover operator EpUC is highfor POP3, which is associated with the success of elitism insub-population POP3. The analysis of the self-adaptive prop-erties in Table 2 reveals the influence of different memeticlearning procedures on the sub-populations.
The mutation operators Implicit Mutation and ControlledMutation [22] have identical selection probabilities. Theimplicit mutation [22,28] represents the needs for populationdiversity, whereas the controlled mutation is a Lamarckianlearning process.
Because the example is a multimodal optimisation prob-lem of composite structures, it is possible to obtain differentand alternative optimal designs. The evolution of the elite-group solutions in 15 epochs is shown in Fig. 5, and thecorresponding results for the five best optimal-design solu-tions are presented in Tables 3 and 4. Only shell laminates areconsidered for the material distribution optimisation. Tables3 and 4 are complementary. Each line of Table 4 reports anoptimal distribution of the shell laminates for the structure(πk, k = 1, . . . , 4). Each shell laminate of Table 3 has anoptimal stacking sequence (πk) in Table 4. Although someoptimal solutions for the shell laminates have similar valuesfor the θ and t̄ components, they present different materialdistributions, as highlighted in the last column of Table 4.
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Memetic Comp. (2014) 6:113–131 125
Table 3 Decoded results formultiple optimal designsolutions that correspond to thefirst two segments of thechromosome [t̄i, j , h j , w j (mm)and θi, j (◦)] for the MA + SGapproach
Laminate Design variable Optimal design solutions
Shell Beam 1st 2nd 3rd 4th 5th
1 t̄1,1/ θ1,1 1.20/−6 Equal to Equal to Equal to Equal to
1 t̄2,1/ θ2,1 1.20/−18 1st 1st 1st 1st
1 t̄3,1/ θ3,1 1.20/−30 Solution Solution Solution Solution
2 t̄1,2/ θ1,2 1.20/42 Equal to Equal to Equal to Equal to
2 t̄2,2/ θ2,2 1.20/−66 1st 1st 1st 1st
2 t̄3,2/ θ3,2 1.20/90 Solution Solution Solution Solution
3 t̄1,3/ θ1,3 1.20/−6 1.20 /−6 Equal to Equal to Equal to
3 t̄2,3/ θ2,3 1.20/42 1.20/18 2nd 1st 1st
3 t̄3,3/ θ3,3 1.20/54 1.20/78 Solution Solution Solution
4 t̄1,4/ θ1,4 1.20/78 Equal to Equal to Equal to Equal to
4 t̄2,4/ θ2,4 1.20/−66 1st 1st 1st 1st
4 t̄3,4/ θ3,4 1.20/−18 Solution Solution Solution Solution
5 h1/w1 34.3/5.0
6 h2/w2 34.3/5.0 Equal to Equal to Equal to Equal to
7 h3/w3 20.0/7.9 1st 1st 1st 1st
8 h4/w4 20.0/6.4 Solution Solution Solution Solution
9 h5/w5 34.3/5.0
10 h6/w6 20.0/5.0
Table 4 Decoded results formultiple optimal designs thatcorrespond to the third segmentof each chromosome (plymaterial properties as defined inTable 1) for the MA + SGapproach
Optimal design solutions Optimal stacking sequences of shell laminates on structure
π1 π2 π3 π4
1st [1/1/4]S [1/1/4]S [1/1/4]S [2/1/4]S
2nd [1/1/4]S [1/1/4]S [1/1/4]S [1/1/2]S
3rd [1/1/4]S [1/1/4]S [1/1/4]S [1/2/1]S
4th [1/1/4]S [1/1/4]S [1/1/4]S [3/1/4]S
5th [1/1/4]S [1/1/4]S [1/1/4]S [2/2/4]S
The fourth shell laminate is located at the top right corner ofthe composite structure in Fig. 4.
Table 3 presents the optimal solutions for the cross sectiondesign variables of beam laminates, which are grouped intovectors h and w. For a complete structure, the five designs inTables 3 and 4 correspond to alternative optimal designs asa consequence of the multimodal weight minimisation.
The lower bound of the ply thickness of the shell lam-inates, which is defined in Eq. (17), is attained. Thus, thepossible minimum weight of shell laminates that are allowedbased on the size and the loading constraints of the optimi-sation problem is obtained. Most optimal material distrib-ution use the optimal stacking sequence [1/1/4]S, which isbuilt using carbon/epoxy (T300/N5208) and Kevlar/epoxy(Kev49/epoxy) composite systems. Only for the last laminateπ4 are other material compositions selected. Those solutions
are rational because of a combination of high-strength mate-rial (carbon) and heavy material (Kevlar) in the core of theshell laminate.
The efficiency of the hybrid learning procedures inMA + SG is measured by the success rate during the iso-lation period (per epoch) of each MA + SG population. Asuccess event indicates that in each generation, at least onenewly generated individual (solution) from the crossover ormutation is better than the worst-fitted individual (solution)in the elite group of the previous population. This last indi-vidual is the most probable candidate to be eliminated fromthe elite group in the next generation [24]. Then, the successrate is defined in each population as:
Success rate (%)
= sum of success events per epoch
total number of events per epoch× 100 (18)
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126 Memetic Comp. (2014) 6:113–131
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Epoch
Su
cces
s ra
te (
%)
in P
OP
1Controlled Mutation
Implicit Mutation
Crossover, ApSG
Crossover, EpUC
Fig. 6 Success rates of crossover and mutation in POP1
for the crossover and mutation operators independently. Theresults obtained using the self-adaptive and memetic pro-cedures are summarised in Figs. 6, 7, and 8 for the threepopulations of MA + SG. In population POP1, during thefirst epoch, the controlled mutation and the new crossoveroperator, ApSG, have full success rate; after the first epoch,the evolution is mainly supported by the crossover opera-tors, as shown in Fig. 6. The controlled mutation is the mostimportant operator for the evolution of population POP2 witha success rate of approximately 100 % in most epochs, asshown in Fig. 7. There are contributions from the crossoveroperators, and Fig. 7 shows that the controlled mutation oper-ator notably effectively contributes to the success of the evo-lutionary process. This behaviour was previously referredto in reference [24] for the case study with three compos-ite systems; however, it is exacerbated here because of theselfish gene concepts, which are associated with the age-structured virtual population in an MA. Furthermore, thecomplete success of a controlled mutation is only observedfor population POP2, where the Elitist Hybrid Crossover withgenetic improvement (EHCgi) [22] is applied. This successis a consequence of the local search promoted by both oper-ators. Thus, there is a particular synergy between both oper-ators because of the Baldwin effect, as reported by otherresearchers [5].
The evolution of population POP3 is generally based onthe crossover operators as shown in Fig. 8. Comparing theresults in Figs. 6, 7 and 8 with the similar results in [24],the benefits of introducing the new operator ApSG can beconcluded. Furthermore, the constant success of operatorsin populations POP2 and POP3 shows that the hierarchical
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Epoch
Su
cces
s ra
te (
%)
in P
OP
2
Controlled Mutation
Implicit MutationCrossover, EHCgi
Crossover, EpUC
Fig. 7 Success rates of crossover and mutation in POP2
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Epoch
Su
cces
s ra
te (
%)
in P
OP
3Controlled Mutation
Implicit Mutation
Crossover, ApSG
Crossover, EpUC
Fig. 8 Success rates of crossover and mutation in POP3
decomposition of the optimisation problem is important inthe evolutionary process.
The above-referred behaviour of the genetic operatorsApSG is studied in more details. The merits of the high-est fitted individuals in the age-structured VP at the first andfifteenth epochs are compared in Fig. 9. It can be concludedthat the quality of the individual’s merit of the VP improvesthroughout the evolutionary process. This result is particu-larly important to extract the feature-based allele’s statisticalanalysis, which is necessary for the evolution progress. Thedistribution of population’s age during the evolution in the
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Memetic Comp. (2014) 6:113–131 127
9000
9050
9100
9150
9200
9250
9300
0 2 4 6 8 10 12 14 16
age
Fit
nes
s (V
P)
1st epoch 15th epoch
Fig. 9 Best individuals for the age-structured VP
0
2
4
6
8
10
12
14
16
18
20
0 2 4 6 8 10 12 14
age
nu
mb
er o
f in
div
idu
als
in V
P
1st epoch
15th epoch
Fig. 10 Distribution of population age in the age-structured VP at thefirst and fifteenth epochs
age-structured VP is described in Fig. 10 for the first and fif-teenth epochs. The age-structured VP size decreases becauseof the lethal age deletion process and the decreasing gener-ation rate for sufficiently different individuals (clones areavoided).
74
76
78
80
82
84
86
88
90
92
94
96
98
100
102
104
0 100 200 300 400 500 600 700 800
fitness evaluations (1st - 3rd epoch)
wei
gh
t [k
g]
Best solution, MA + SGBest solution, GABest solution, MA
Fig. 11 Comparison of the evolution paths of MA + SG, GA and MAfor the 1st–3rd epochs
6.3 Comparison with other algorithms
The obtained results of the proposed MA + SG approach arecompared with those of a simple GA and an MA. The compar-ison is done for identical numbers of fitness evaluations thatcorrespond to identical computational costs. The comparisonanalysis has two objectives: (i) to analyse the influence ofmultiple learning procedures and (ii) to analyse the influenceof the new ApSG operator based on the SG algorithm withallele statistics. The comparison of MA + SG with both GAand MA can identify the Baldwin effect through the impactof phenotypic plasticity on the evolution. The phenotypicplasticity is defined as the ability of an organism to adapt toits environment, which is passed here through self-adaptiverules at the isolation stage and the life cycle emulation foran age-structured VP. As mentioned in Sect. 1.2, this effectenables one to explore the neighbouring regions of a pheno-typic space [13,14] according to its local plasticity.
The simple GA is based on the same operators (mutationand crossover) that are applied in population POP1, exceptthe age parameterised selfish gene (ApSG) operator. Thehierarchical structure is not used in the simple GA, and thegenetic parameter values are identical to those used in POP1.In the MA approach, the co-evolution of multiple populationsis implemented so that the new ApSG crossover operator isdiscarded, and the remaining operators and genetic parame-ters of the proposed MA + SG approach are maintained. Thedirect effect of using the new ApSG operator is perceived
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128 Memetic Comp. (2014) 6:113–131
73,0
73,5
74,0
74,5
75,0
75,5
76,0
76,5
77,0
77,5
78,0
78,5
79,0
79,5
80,0
825 1075 1325 1575 1825 2075 2325
fitness evaluations (4th - 9th epoch)
wei
gh
t [k
g]
Best solution, MA + SGBest solution, GABest solution, MA
Fig. 12 Comparison of the evolution paths of MA + SG, MA and GAfor the 4th–9th epochs
73,0
73,5
74,0
74,5
75,0
75,5
76,0
76,5
77,0
77,5
78,0
2445 2695 2945 3195 3445 3695 3945
fitness evaluations (10th - 15th epoch)
wei
gh
t [k
g]
Best solution, MA + SGBest solution, GABest solution, MA
Fig. 13 Comparison of the evolution path of MA + SG, MA and GAfor the 10th–15th epochs
from the results in Figs. 11, 12, 13 in different scales. Theevolution history is divided by epoch intervals: Fig. 11 for the1st–3rd epochs, Fig. 12 for the 4th–9th epochs and Fig. 13 for
the 10th–15th epochs. Each optimisation algorithm searchesthe design space using a different path according to the FreeLunch Theorem for genetic search [1]. The optimal solutionsfor the GA and MA approaches are reported in Tables 5 and 6.
The three algorithms exhibit a close behaviour during thefirst 125 fitness evaluations, as shown in Fig. 11. However,the proposed MA + SG approach performs better than theGA and MA approaches during most of the evolution timefrom the 1st epoch to the 3rd epoch.
In Fig. 12, GA and MA evolve for a long period to main-tain the weight of the best solutions through the 4th epoch tothe 9th epoch. This similar behaviour of GA and MA duringthis intermediate evolution period is associated with the mat-uration of the elite group. In this context, the ApSG operatorin the MA + SG approach is visibly beneficial because thebehaviour of the age-structured VP based on the SG theorywith allele’s statistical analysis is more explorative.
In the last period (10th–15th epochs), the MA approachdoes not recover the weight difference when compared withthe MA + SG results, as shown in Fig. 13. In addition, thesimple GA has difficulties at the final path of the evolution-ary process and does not reach the optimal weight, whichMA + SG obtains during the considered evolution period.
For the three plotted periods, the proposed MA + SGapproach shows the synergy of multiple learning proceduresbased on Lamarckian or Baldwinian concepts. Furthermore,it is remarkable that the positive influence of the SG the-ory materialised the age parameterised selfish gene (ApSG)operator. Finally, although the weight differences are morequalitative than quantitative, they might become significantlyimportant in real scenarios, such as aircraft or automobileindustrial applications, where the scale effects are effective.
7 Conclusions
In this paper, the application of the selfish gene the-ory was analysed using a fusion of concepts proposed inmemetic algorithms and selfish gene algorithms. The pro-posed approach, named the memetic-based selfish gene algo-rithm (MA + SG), was studied and applied to the optimi-sation of hybrid composite shell structures with stiffeners.First, the memetic features, mainly the learning procedures,were analysed and classified according to the Lamarckianand Baldwinian learning procedures. The co-evolution ofmultiple populations, species conservation, migration rules,self-adaptive multiple crossovers, local search in the hybridcrossover with the local genetic improvements, controlledmutation, individual age control and feature-based allele sta-tistics analysis are the most important details of the proposedapproach. Second, the selfish gene theory was applied toan age-structured virtual population to improve the searchperformance. A new operator, which is denoted by the age
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Table 5 Decoded results for thebest optimal design solutionsthat correspond to the first twosegments of the chromosome[t̄i, j , h j , w j (mm) and θi, j (◦)]for the GA and MA approaches
Laminate Design variable Optimal design solutions
Shell Beam GA MA
1 t̄1,1/ θ1,1 1.20/−6 1.20/90
1 t̄2,1/ θ2,1 1.20/−18 1.20/−6
1 t̄3,1/ θ3,1 1.20/90 1.20/42
2 t̄1,2/ θ1,2 1.20/30 1.20/42
2 t̄2,2/ θ2,2 1.20/30 1.20/90
2 t̄3,2/ θ3,2 1.20/78 1.20/78
3 t̄1,3/ θ1,3 1.20/−6 1.20/−6
3 t̄2,3/ θ2,3 1.20/6 1.20/54
3 t̄3,3/ θ3,3 1.20/−30 1.20/−6
4 t̄1,4/ θ1,4 1.20/−18 1.20/6
4 t̄2,4/ θ2,4 1.20/90 1.20/−78
4 t̄3,4/ θ3,4 1.20/−30 1.20/66
5 h1/w1 20.0/15.0 20.0/6.4
6 h2/w2 22.9/13.6 34.3/5.0
7 h3/w3 31.4/13.6 31.4/13.6
8 h4/w4 20.0/6.4 20.0/6.4
9 h5/w5 22.9/13.6 20.0/12.1
10 h6/w6 28.6/15.0 31.4/7.9
Table 6 Decoded results for the best optimal design solutions that correspond to the third segment of each chromosome (the ply material propertiesare defined in Table 1) for the GA and MA approaches
Optimal design solutions Optimal stacking sequences of shell laminates on structure
π1 π2 π3 π4
GA [1/1/4]S [1/1/4]S [1/1/4]S [1/2/1]S
MA [1/1/4]S [1/1/4]S [1/1/4]S [1/1/4]S
parameterised selfish gene (ApSG) crossover, was proposed.The operator is based on a crossover scheme that resultsfrom changes in the mating selection mechanism and the off-spring generation mechanism, which are applied to an age-structured virtual population.
The results show that the MA + SG approach is promis-ing when we optimise hybrid composite structures. Someimportant conclusions can be drawn:
• With the co-evolution of populations and multiple learn-ing procedures, it is possible to obtain different and alter-native designs for the multimodal optimisation problemof composite structures.
• The optimal solutions of the MA + SG approach are wiseby considering a combination of high-strength material(carbon) and heavier material (Kevlar) in the core of theshell laminate. This result proves the efficiency of theproposed approach.
• The efficiency of the hybrid learning procedures in theMA + SG approach was measured and verified by com-paring with the results from a simple GA (without learn-ing). The differences are particularly relevant at the end ofthe evolutionary search history. For the same number offitness evaluations (same computational cost), the searchusing GA does not reach the same minimum weightvalue.
• The benefits of the selfish gene theory through the newcrossover operator ApSG in the MA + SG approach wererevealed by comparing the results with the MA approach:in general, the results obtained from the MA + SGapproach are better than the MA results throughout theevolutionary search history.
• The incorporation data, which are related to the behaviourof the state variables of the hybrid composite structures,are obtained using a controlled mutation operator, whichinduces a positive behaviour in the MA + SG approach as
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measured by the success rate. This result is particularlyrelevant because of the synergetic effects with the EHCgicrossover operator.
• The search in the isolation stage of the sub-populationsexhibited exploitative properties, whereas the behaviourof the age-structured VP based on the SG theory with theallele’s statistical analysis was more explorative.
• The quality of the individual’s merit of the VP improvedduring the evolutionary process. This result is particularlyimportant to extract the feature-based allele’s statisticalanalysis, which is necessary for the evolution progresssupported by this Lamarckian learning.
• The numerical results of the comparison tests show thatthe GA and the MA maintain long periods without evo-lution of the best-fitted individual/solutions. This behav-iour during the evolution is associated to the slow mat-uration of the elite group of populations in the GA andMA approaches. The benefits of the ApSG operator inthe MA + SG approach were confirmed in this context.
The optimal design of hybrid composite structures is an openissue, and the challenge is to find robust algorithms thatcover different types of variables (continuous and discretein nature), objectives and constraints for the global optimi-sation. The proposed MA + SG approach based on multiplelearning procedures is appropriate to overcome the difficul-ties that are associated with the search of a global optimaldesign of hybrid composite structures.
Acknowledgments The author acknowledges “FCT-Fundação paraa Ciência e Tecnologia” of Portugal for the financial support throughstrategic project PEst-OE/EME/UI225/2011.
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