Research Article Surrogate-Assisted Multiobjective...
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Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 695172, 13 pages http://dx.doi.org/10.1155/2013/695172 Research Article Surrogate-Assisted Multiobjective Evolutionary Algorithms for Structural Shape and Sizing Optimisation Tawatchai Kunakote and Sujin Bureerat Department of Mechanical Engineering, Faculty of Engineering, Khon Kaen University, Khon Kaen 40002, ailand Correspondence should be addressed to Sujin Bureerat; [email protected] Received 14 February 2013; Revised 4 June 2013; Accepted 5 June 2013 Academic Editor: Guilin Yang Copyright © 2013 T. Kunakote and S. Bureerat. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e work in this paper proposes the hybridisation of the well-established strength Pareto evolutionary algorithm (SPEA2) and some commonly used surrogate models. e surrogate models are introduced to an evolutionary optimisation process to enhance the performance of the optimiser when solving design problems with expensive function evaluation. Several surrogate models including quadratic function, radial basis function, neural network, and Kriging models are employed in combination with SPEA2 using real codes. e various hybrid optimisation strategies are implemented on eight simultaneous shape and sizing design problems of structures taking into account of structural weight, lateral bucking, natural frequency, and stress. Structural analysis is carried out by using a finite element procedure. e optimum results obtained are compared and discussed. e performance assessment is based on the hypervolume indicator. e performance of the surrogate models for estimating design constraints is investigated. It has been found that, by using a quadratic function surrogate model, the optimiser searching performance is greatly improved. 1. Introduction Since they have been invented for several decades, the implementation of evolutionary optimisers on a wide variety of real-world design problems has successfully been made in numerous fields. e algorithms are attractive and popular as they can responsd to unavoidable disadvantages of classical mathematical programming. e evolutionary algorithms (EAs) can deal with almost all kinds of design problem as their search mechanisms, to some extent, rely on randomi- sation and need no function derivatives, for example, [1– 4]. Recently, single-objective evolutionary methods that have outstanding performance in several applications are real- code ant colony optimisation (ACOR) [5], covariance matrix adaptation evolution strategy (CMA-ES) [6], and differential evolution (DE) [7]. e methods are robust and capable of reaching global optima. Most importantly, for multiple- objective design cases, they can be used to explore a Pareto optimal front of the problem within one simulation run. However, EAs have some unacceptable drawbacks that are a complete lack of consistency and low convergence rate. As a result, the obtained design results are considered near local optima for single objective cases and approximate Pareto fronts for multiobjective design [8]. With those aforementioned shortcomings, it is difficult or even impossible to employ the methods for solving the design problems with expensive function evaluation or limited number of function values available. Such obstructions can however be alleviated by introducing a surrogate model (SM) to an evolutionary optimisation procedure. With the use of such a model, an approximate function can be constructed, and inexpensive function prediction can be achieved. e hybridisation of EAs and SM for design optimisation has been studied since the last decade, and it has been found to greatly enhance the performance of EAs. Many researchers and engineers have focused their efforts on this research issue. e design problems can have single or multiple design objec- tives, for example, in [9–14] while mostly the demonstration problems are unconstrained or box-constrained. e hybrid EAs with SMs have been implemented on a wide variety of engineering application, for example, aerodynamics [15– 17], heat and mass transfer [18–20], and structures [13, 14, 21–23]. Design problems requiring a surrogate-assisted optimisation may be roughly categorised to have 9 classes
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 695172 13 pageshttpdxdoiorg1011552013695172
Research ArticleSurrogate-Assisted Multiobjective Evolutionary Algorithms forStructural Shape and Sizing Optimisation
Tawatchai Kunakote and Sujin Bureerat
Department of Mechanical Engineering Faculty of Engineering Khon Kaen University Khon Kaen 40002 Thailand
Correspondence should be addressed to Sujin Bureerat sujburkkuacth
Received 14 February 2013 Revised 4 June 2013 Accepted 5 June 2013
Academic Editor Guilin Yang
Copyright copy 2013 T Kunakote and S Bureerat This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Thework in this paper proposes the hybridisation of the well-established strength Pareto evolutionary algorithm (SPEA2) and somecommonly used surrogate models The surrogate models are introduced to an evolutionary optimisation process to enhance theperformance of the optimiserwhen solving design problemswith expensive function evaluation Several surrogatemodels includingquadratic function radial basis function neural network and Kriging models are employed in combination with SPEA2 using realcodes The various hybrid optimisation strategies are implemented on eight simultaneous shape and sizing design problems ofstructures taking into account of structural weight lateral bucking natural frequency and stress Structural analysis is carried outby using a finite element procedure The optimum results obtained are compared and discussed The performance assessment isbased on the hypervolume indicator The performance of the surrogate models for estimating design constraints is investigated Ithas been found that by using a quadratic function surrogate model the optimiser searching performance is greatly improved
1 Introduction
Since they have been invented for several decades theimplementation of evolutionary optimisers on a wide varietyof real-world design problems has successfully been made innumerous fieldsThe algorithms are attractive and popular asthey can responsd to unavoidable disadvantages of classicalmathematical programming The evolutionary algorithms(EAs) can deal with almost all kinds of design problem astheir search mechanisms to some extent rely on randomi-sation and need no function derivatives for example [1ndash4] Recently single-objective evolutionary methods that haveoutstanding performance in several applications are real-code ant colony optimisation (ACOR) [5] covariance matrixadaptation evolution strategy (CMA-ES) [6] and differentialevolution (DE) [7] The methods are robust and capableof reaching global optima Most importantly for multiple-objective design cases they can be used to explore a Paretooptimal front of the problem within one simulation runHowever EAs have some unacceptable drawbacks that area complete lack of consistency and low convergence rateAs a result the obtained design results are considered near
local optima for single objective cases and approximate Paretofronts for multiobjective design [8]
With those aforementioned shortcomings it is difficult oreven impossible to employ themethods for solving the designproblems with expensive function evaluation or limitednumber of function values available Such obstructions canhowever be alleviated by introducing a surrogate model (SM)to an evolutionary optimisation procedure With the use ofsuch a model an approximate function can be constructedand inexpensive function prediction can be achieved Thehybridisation of EAs and SM for design optimisation hasbeen studied since the last decade and it has been found togreatly enhance the performance of EAs Many researchersand engineers have focused their efforts on this research issueThedesign problems can have single ormultiple design objec-tives for example in [9ndash14] while mostly the demonstrationproblems are unconstrained or box-constrained The hybridEAs with SMs have been implemented on a wide varietyof engineering application for example aerodynamics [15ndash17] heat and mass transfer [18ndash20] and structures [1314 21ndash23] Design problems requiring a surrogate-assistedoptimisation may be roughly categorised to have 9 classes
2 Mathematical Problems in Engineering
as given in [10] Surrogate-assisted EAs not only employfunction approximation but also use some mathematicalprogramming to enhance the searching performance [24 25]A comprehensive survey on the surrogate models used forenhancing EAs performance can be found in [26ndash29]
The surrogate models commonly employed include poly-nomial regression techniques [11 20 22 26 27 30] radialbasis function interpolation (RBF) [9 11 12 16 18 27]artificial neural network (ANN) [9 15 23 26] supportvector machines [31] moving least square method [32] andGaussian process model (Kriging model) [10 16 25ndash27]Recent alternative surrogate-based approaches are concernedwith the approximation of field vectors (such as velocity andpressure fields for computational fluid dynamics and dis-placement and stress fields for structural analysis) rather thanapproximating objective and constraint functions directlyThe proper orthogonal decomposition (POD) technique isemployed for this task [33ndash35] With this concept a designercan deal with more complicated design problems such asdesign optimisation when aeroelasticity is taken into consid-eration Apart from that there exist optimisers with the use ofgrid-based computing integrating nongeneration evolution-ary algorithms surrogate models local search and Lamarck-ian learning process together This approach is termed asyn-chronousmetamodel-based evolutionary algorithms [36 37]A technique used in the design of computer experiment phaseis usually a Latin hypercube sampling technique Recentlythere have been several papers focusing on the developmentof optimum Latin hypercube sampling [38 39] Anotherinteresting approach is an infill sampling technique [40]
The most challenging task for surrogate-based optimi-sation is to improve the convergence rate To the authorsrsquobest knowledge the work in the literature mostly demon-strated applying the design approaches to unconstrained orbox-constrained optimisation which is simple to handle byusing EAs Nevertheless there are usually many nonlinearconstraints in the real-world design problems particularlyin structural optimisation The approximation of objectivefunction although being inaccurate can lead to real optima(referred to as ldquoBlessing of Uncertaintyrdquo in [11]) On the otherhand in estimating constraint functions the approximatefunction needs to be as precise as possible so that theevolutionary search can land in the real feasible regionInaccurate constraint function approximation can lead theoptimiser to an undesirable design space
In this paper the hybridisation of the well-establishedmultiobjective evolutionary algorithm (MOEA) namelystrength Pareto evolutionary algorithm [41] and some pop-ular surrogate models is developed The work is concernedwith the studies of using several SMs for performanceenhancement of multiobjective evolutionary algorithms insolving structural constrained optimisation A number ofsurrogate models including quadratic function radial basisfunction neural network and Kriging models are employedin combination with SPEA2 using real codes The varioushybrid optimisation strategies are implemented on eightsimultaneous shape and sizing design problems of a structurehaving design functions as structural weight lateral buckingnatural frequency and stress Structural analysis is carried out
by using finite element analysis (FEA) The optimum resultsobtained are compared and discussed The performance ofSMs for estimating design constraints is investigated It hasbeen found that by using a quadratic surrogate model thesearching performance of SPEA2 is greatly improved
2 Hybridisation of SPEA2 anda Surrogate Model
There have been several ways to properly integrate SMs intoEAs The hybrid algorithm proposed in this paper is similarto what is presented in [18] The SPEA2 with real codesis employed as the main MOEA procedure The surrogate-assisted SPEA is illustrated in Figure 1 The computationalsteps in the figure can be separated into two parts as themain SPEA procedure and the SPEA subproblem that usesan approximationmodelThemain SPEA is slightly modifiedfrom the original version presented in [41] The real coderecombination and mutation are similar to the work pre-sented in [18]The procedure starts with an initial populationan (empty) Pareto archive and some other predefined param-eters for example an external archive size and recombinationand mutation rates The fitness of the individuals in thepopulation is then assigned by performing actual functionevaluation The current population and their correspondingfitness values are brought to the surrogate environment assome of them are picked to build a surrogate model Havingsuch an approximation model the SPEA is employed tosolve the optimisation problem with approximate functionevaluation
The nondominated solutions from this stage are putback to the main SPEA procedure where their actual fit-ness values are determined The current population theexternal archive and the nondominated solutions fromperforming a surrogate-based optimisation are put togetherwhile their nondominated solutions are determined Theexternal archive is then updated by replacing it with thesenondominated solutions In case that the number of thenondominated solutions exceeds the predefined archive sizesome of them will be removed from the archive by meansof the nearest neighbourhoodmethod [41] Subsequently theso-called binary tournament selection is performed to selectsome members in the newly updated archive to reproduce aset of offspringThe algorithm goes back to the step of fitnessfunction evaluation and is repeated until the terminationcriterion is fulfilled
Design solutions are selected froma current population tobuild a surrogate model based upon the idea that they shouldbe evenly distributed throughout the design space Theproposed approach is operated on the domain of objectivefunctions which is somewhat similar to the adaptive gridalgorithm a Pareto archiving technique used in the Paretoarchive evolution strategy [8] The procedure to select thesolutions for surrogate modelling is illustrated in Figure 2On the objective domain all of the design points in thecurrent population are plotted and a proper grid is generatedcovering all the points A solution that is the closest to thecentre of each mesh is selected for function estimation
Mathematical Problems in Engineering 3
Stop
Postprocessing
No
Yes
Evaluate fitness values of
Environmental selection
Apply NNT if necessary
Binary tournament selection
Stop No
Yes
to build the SM model
Main procedure SM sub-problem
Preprocessing
Initialization k = 0
Pk f(Pk) and Ak =
Evaluate fitness values of
PkUAkUBk+1
Environmental selection
Find FindAk+1 from PkUAkUBk+1
Apply NNT if necessary
Binary tournament selectionPk1 = select(Ak+1)
Pk2 = recombine(Pk
1)
Pk+1 = mutate(Pk2)
Find f(Pk+1)
k = k + 1
Find f(Bt+1n ) the actual
function values of Bt+1n
t = t + 1
Qt2n = recombine(Qt
1n)
Qt+1n = mutate(Qt
2n)
Find f(Qt+1)
Qt1n = select(Bt+1
n )
Bt+1n from Bt
n UQtn
GtnUB
tn
Initialization t = 0
Qtn g(Qt
n) and Btn =
Pick the member from Pk and f(Pk)
Figure 1 Flowchart for surrogate-assisted SPEA
3 Surrogate Models
Let 119910 = 119891(x) are a function of a design vector x sized119899 times 1 Given a set of design solutions (or sampling points)X = [x1 x119873] and their corresponding function valuesY =
[1199101 119910
119873] selected during an evolutionary optimisation
process a surrogate or approximation model is constructedby means of curve fitting or interpolation Approximationmodels used in this study are as follows
31 Polynomial Regression The most commonly used poly-nomial surrogate model or a response surface model (RSM)is of the second-order polynomial or quadratic model whichcan be expressed as
119910 = 1205730+sum120573
119894119909119894+sum120573
119894119909119894119909119895 (1)
where 120573119894for 119894 = 0 (119899 + 1)(119899 + 2)2 are the polynomial
coefficients to be determined The coefficients can be foundby using a regression or least square technique [26]
32 Kriging Model A Kriging model (also known as aGaussian process model) used in this paper is the famousMATLAB toolbox named design and analysis of computerexperiments (DACE) [42] The estimation of function can bethought of as the combination of global and local approxima-tion models that is
119910 (x) = 119891 (x) + 119885 (x) (2)
where 119891(x) is a global regression model while 119885(x) is astochastic Gaussian process with zero mean and nonzerocovariance representing a localised deviation In this worka linear function is used for a global model which can beexpressed as
119891 = 1205730+
119899
sum
119894=1
120573119894119909119894= 120573119879f (3)
where 120573 = [1205730 120573
119899]119879 f = f(x) = [1 119909
1 1199092 119909
119899]119879 The
covariance of 119885(x) is expressed as
Cov (119885 (x119901) 119885 (x119902)) = 1205902R [119877 (x119901 x119902)] (4)
4 Mathematical Problems in Engineering
The selected points forbuilding a surrogate model
Grid centerf2
f1
Figure 2 Selection of sampling points for constructing a surrogatemodel
for 119901 119902 = 1 119873 where 119877 is the correlation functionbetween any two of the 119873 design points and R is thesymmetric correlation matrix size 119873 times 119873 with the unitydiagonal [26] The correlation function used herein is
119877 (x119901 x119902) = exp (minus(x119901 minus xq)119879120579 (x119901 minus xq)) (5)
where 120579119894are the unknown correlation parameters to be
determined by means of the maximum likelihood methodHaving found 120573 and 120579 the Kriging predictor can be achievedas
119910 = f (x)119879120573 + r119879 (x)Rminus1 (y minus F120573) (6)
where F = [f(x1) f(x2) f(x119899)]119879 and r119879(x) = [119877(x x1)119877(x x2) 119877(x x119873)] For more details see [42]
33 Radial Basis Function Interpolation The radial basisfunction interpolation has been used in a wide range of appli-cations such as integration between aerodynamic and finiteelement grids in aeroelastic analysis [43] The use of sucha model for surrogate-assisted evolutionary optimisation issaid to be commonplace [9 18 27]The approximate functioncan be written as
119910 (x) =119873
sum
119894=1
120572119894119870(
10038171003817100381710038171003817xminusx11989410038171003817100381710038171003817) (7)
where 120572119894are the coefficients to be determined and 119870 is a
radial basis kernel (Here it is set to be linear splines) Thecoefficients can be found from the 119873 sampling points asdetailed in Srisomporn and Bureerat 2008 [18]
34 Neural Network Artificial neural network (ANN) isdeveloped in response to the need to use complicated func-tion expression rather than a simple quadratic or radial basisfunction The model consists of interconnecting neurons ornodes that somewhat mimic the constructs of biologicalneurons as shown in Figure 3 ANN has been implemented
Table 1 Design variables and bounds for F1ndashF6
Design variables Bounds (mm)1199091= 119897119889
0 le 119897119889le 60
1199092= 119897119901
100 le 119897119901le 200
1199093= 1199031199011
15 le 1199031199011le 35
1199094= 1199031199012
15 le 1199031199012le 30
1199095= th 3 le th le 45
Input layer
Hidden layer
Output layer
Neuron or node
Figure 3 Neural network
on a wide variety of real world applications The use ofANN for surrogate-assisted optimisation can be found inthe literature [9] The neural network model used in thispaper is the feedforward multilayer perceptrons The detailsof the network in this work are the learning algorithm =backpropagation and network training function = Levenberg-Marquardt backpropagation
In this work a simple heuristic algorithm is used toconstruct a network topology The procedure starts with anetwork with two hidden layers while the number of nodes119873119877on each layer is randomly chosen with the bound 119873
119877isin
[15 50] The transfer function for the hidden layer can beeither hyperbolic tangent sigmoid or logarithmic sigmoid basedon randomisation If the termination criterion (training goal)is notmet randomly increase a number of nodes in the layersIf the number of nodes reaches the upper bound 119873
119877= 50
add the third hidden layer to the network where a transferfunction and node number are randomly chosen In case thatthe stopping criterion is not fulfilled increase the number ofnodes The algorithm repeats until the stopping criterion ismet
4 Testing Problems
Six testing multiobjective design problems are posed todesign a torque arm [44] structure shown in Figure 4 Sinceit is a plate-like structure under inplane loading designcriteria to be taken into consideration are weight stressbuckling and natural frequency The function calculationcan be carried out by using FEA where shell elements areemployed The design variables and their bound constraintsare given in Table 1 where other dimensions are set as 119897
119889=
30mm 119897119901= 190mm 119903
1199011= 26mm and 119903
1199012= 27mm
Mathematical Problems in Engineering 5
5066N
2789N
rp2rp1
R2 = 42mmr2 = 25mm
Fixedboundaryconditionsr1 = 40mm
th = platethickness
l0 = 100mm
R1 = 542
L = 420mm
lpld
Figure 4 Torque arm
The six biobjective optimisation problems are termed F1F2 F3 F4 F5 and F6 respectively and can be expressed as
objective functionmin 119891
1(x) max119891
2(x)
F1 1198911(x) = 119908 119891
2(x) = 120596
F2 1198911(x) = 119908 119891
2(x) = 120582
F3 1198911(x) = 119908 119891
1(x) = 120582 + 120596
1205960
subject to
120590von minus120590119910
119878119865
le 0
1 minus 120582 le 0
objective functionmin119891
1(x) max119891
2(x)
F4 1198911(x) = 119908 119891
2(x) = 120596
F5 1198911(x) = 119908 119891
2(x) = 120582
F6 1198911(x) = 119908 119891
2(x) = 120582 + 120596
1205960
subject to
120590von minus120590119910
119878119865
le 0
1 minus 120582 le 0
120596 minus 1205960le 0
(8)
where 119908 = structural weight 120596 = first mode natural fre-quency 120582 = lateral bucking factor (it is the ratio of criticalload to applied load herein) 120590von = maximum von Misesstress on the structure and 120596
0= 72688 rads Furthermore
material properties are Youngrsquos modulus 119864 = 60 times 106Ncm2Poissonrsquos ratio ] = 03 yield stress 120590
119910= 80000Ncm2 density
120588 = 000718 kgcm3 and safety factor 119878119865= 15
In fact the F1 F2 and F3 problems are similar to the F4F5 and F6 problems respectively except for the introductionof natural frequency constraints to the last three problemsThis means that the performance of several surrogate modelsin estimating a natural frequency during an optimisation
procedure can be examinedThe bound constraints in Table 1are not presented in the design problems as they can be dealtwith in the SPEA procedure It should be noted that thefinite element analysis for these design studies is not timeconsuming as the work focuses on a comparative study of theseveral multiobjective optimisers
Since the F1ndashF6 problems are small-scale other two testproblems are posed so as to investigate the capability of thosesurrogate models when dealing with large-scale problemsTwo design problems of a simple wing-box displayed inFigure 5 are expressed as
objective functionmin119891
1(x) max119891
2(x)
F7 1198911(x) = 119908 119891
2(x) = minus120582
F8 1198911(x) = 119908 119891
2(x) = minus (120596
1+ sdot sdot sdot + 120596
5)
subject to
120590von minus 120590119910 le 0
1 minus 120582 le 0
(9)
where 120596119894is the 119894th mode natural frequency of the wing-box
Thewing-box ismade up of Aluminiumwith Youngmodulus119864 = 70GPa Poissonrsquos ratio ] = 034 density 120588 = 2700 kgm3and yield strength 120590
119910119905= 100MPa The structure consists of 2
spars 4 ribs and 4 skins (front rear top and bottom skins)resulting in 52wing segments as shownThe thickness of eachsegment is assigned to be a design variable therefore thereare 52 design variableswith the lower andupper bounds being0002m and 0010m respectively This can be classified as alarge-scale sizing optimisation problem The wing is subjectto static loads on the front spar as shown in Figure 5 Shellelements are used for finite element analysis similarly to F1ndashF6
The multiobjective optimisers used in this paper arenamed as follows
(i) SPEA-O the original SPEA2without using a surrogatemodel
(ii) SPEA-Q SPEA using a quadratic response surfacemodel
6 Mathematical Problems in Engineering
Fixed BC
2467kN 1645kN1645kN 0822 kN
2m
01m
0125m
025m0125m
Figure 5 Simple wing-box
(iii) SPEA-R SPEAusing the radial basis function interpo-lation
(iv) SPEA-N SPEA using a neural network model(v) SPEA-K SPEA using a Kriging model
Each evolutionary algorithm is employed to solve eachproblem ten runs with 30 generations (50 for F7-F8) and apopulation size of 20 (50 for F7-F8) for SPEA-O For thealgorithms using an approximation model the number ofgenerations and population size used in the main procedureis 15 (25 for F7-F8) and 20 (50 for F7-F8) respectively whileon each generation additional 20 (50 solutions for F7-F8)solutions are taken from optimising a surrogate subproblemIn the surrogate subproblem the number of generationsand population size is set to be 100 and 100 respectivelyThe external Pareto archive size is set to be 100 Duringthe optimisation process constraints are handled by usingthe approach presented in [45] It should be noted thatthere have been a number of techniques used to handledesign constraints in evolutionary optimisation [46] Wechoose the aforementioned technique because it is simple andsufficiently effective
5 Results and Discussion
The performance comparison of the five MOEAs for solvingthe eight design problems is based on the hypervolume indi-cator Two types of performance assessment are conducted asdetailed in Tables 2ndash4 In Table 2 each cell is the normalisedmean value of 10 nondominated front hypervolumes obtainedfrom using the optimiser on the corresponding column forsolving the problem on the corresponding row For eachdesign problem the best and worst mean values are nor-malised to be ldquo1rdquo and ldquo0rdquo respectively The total normalisedmean values given in the table are used for evaluating theoverall performance of the surrogate-assisted SPEAs Fromthe results the best algorithm for F1 is SPEA-R while thesecond best is SPEA-Q SPEA-Q is the best for F2 while thesecond best is SPEA-NThe best optimisers for F3 F4 F5 andF6 are SPEA-K SPEA-R SPEA-K and SPEA-K respectivelywhereas the second best for those 4 design problems is SPEA-Q SPEA-O without using a surrogate model is the worstperformer for F2 F5 and F6 while SPEA-N is the worstperformer for F1 F3 and F4 The overall best performer
Table 2 Comparison of normalised average hypervolume
SPEA-O SPEA-Q SPEA-R SPEA-N SPEA-KF1 0429 0704 1000 0000 0523F2 0000 1000 0052 0917 0692F3 0577 0995 0745 0000 1000F4 0457 0916 1000 0000 0497F5 0000 0976 0640 0161 1000F6 0000 0673 0243 0007 1000sum F1ndashF6 1463 5264 3679 1084 4713F7 0811 1000 0724 NA 0000F8 0337 1000 0453 NA 0000sum F7-F8 1148 2000 1177 NA 0000
Table 3 Performance matrix of F1
SPEA-O SPEA-Q SPEA-R SPEA-N SPEA-KSPEA-O 0 0 1 0 0SPEA-Q 0 0 1 0 0SPEA-R 0 0 0 0 0SPEA-N 1 1 1 0 1SPEA-K 0 0 1 0 0Total 1 1 4 0 1Ranking 2 2 1 5 2
Table 4 Comparison of ranking scores by t-test
SPEA-O SPEA-Q SPEA-R SPEA-N SPEA-KF1 2 2 1 5 2F2 4 1 4 1 1F3 1 1 1 5 1F4 3 1 1 5 3F5 4 1 3 4 1F6 3 2 3 3 1sum F1ndashF6 17 8 13 23 9F7 1 1 1 NA 4F8 2 1 2 NA 4sum F7-F8 3 1 3 NA 8
based on the evaluation is SPEA-Q whereas the second bestis SPEA-K For the large-scale problems only SPEA-Q cansurpass SPEA-O SPEA-R is said to be equal to SPEA-OwhileSPEA-K gives that worst performance Note that SPEA-Ncannot be applied to these problems as it takes excessivelylong running time for network training It can be said thatonly the quadratic response surface model is useful for thelarge-scale wing-box design
The second performance assessment is based on thestatistical t-test For each design problem to compare the 5MOEAs a performance matrix T sized 5 times 5 whose elementsare full of ldquo0rdquo is generated For the element 119879
119894119895of the matrix
if the mean value of the 10 hypervolumes obtained frommethod 119895 is significantly larger than that obtained frommethod 119894 based on the statistical t-test at 95 confidencelevel the value of 119879
119894119895is set to be one Table 3 shows the
performance comparison of the problem F1 Having had theperformance matrix the 6th row on the table displays thetotal scores of each optimiser The ranking is made in such
Mathematical Problems in Engineering 7
0 500 1000 1500 2000 25000
2
4
6
8
10
12
14 F7
No of function evaluations
Hyp
ervo
lum
e
SPEA-OSPEA-Q
Figure 6 SPEA-O versus SPEA-Q for F7
a way that the best method is ranked as 1 while the worst isranked as 5The overall performance assessment based on thet-test is given in Table 4 where the last row on the table showsthe total score of all theMOEAsThe lower the score the betterthe performer Similar to the first assessment the overall bestmethod for F1ndashF6 based on this assessment is SPEA-Q whilethe close secondbest is SPEA-KTheworstmethod is SPEA-Owithout using a surrogate model For the large-scale testproblems the only surrogateassisted method that is superiorto the non-surrogate SPEA is SPEA-Q
Figures 6 and 7 show the search history as plots ofhypervolume against the number of function evaluations ofSPEA-O and SPEA-Q for solving F7 and F8 respectivelyThevalues of front hypervolume can fluctuate due to the use of aPareto archiving technique to remove some nondominatedsolutions from the Pareto archive From the figures it canbe seen that SPEA-Q requires approximately half of the totalnumber of function evaluations employed by SPEA-O to haveequally good nondominated fronts
The best approximate Pareto front of F1 obtained afternumerous optimisation runs of the surrogate-assisted SPEAsis chosen and plotted in Figure 8 whereas the structures ofsome selected design points are shown in Figure 9 Figure 10displays the best front of the F2 problemwhere some selecteddesign solutions are illustrated in Figure 11The best front andsome selected optimal structures of the design problemF3 aredisplayed in Figures 12 and 13 respectively Similarly the bestapproximate fronts and some selected design solutions of F4F5 and F6 are illustrated in Figures 14 15 16 17 18 and 19Figures 20 and 21 show the best fronts obtained from usingthe various SPEAs for solving F7 and F8 respectively It canbe seen that the best fronts are from SPEA-Q and SPEA-OThe SPEA-Q front does not totally dominate that of SPEA-O but they overlap each other while the SPEA-Q front hasobviously greater front extension
According to the aforementioned performance assess-ments it can be said that SPEA-R is the best algorithm in
0 500 1000 1500 2000 25000
2
4
6
8
10
12F8
No of function evaluations
Hyp
ervo
lum
e
SPEA-OSPEA-Q
Figure 7 SPEA-O versus SPEA-Q for F8
009 01 011 012 013 014 015 01608
085
09
095
1
105
11
115
12
1
2
3
4
5
6
Stru
ctur
al m
ass (
kg)
1120596
Figure 8 Approximate Pareto front of F1
the cases of the F1 design problem while the close secondbest is SPEA-Q When adding a natural frequency constraintto F1 becoming the F4 problem the overall best and secondbest methods are still SPEA-R and SPEA-Q respectivelyFor the F2 case and the F5 design problem which is F2with additional natural frequency constraint the top twoperformers are SPEA-Q and SPEA-K The same can be saidfor the case of F3 while SPEA-K outperforms the others forthe F6 design problem The surrogate models that is thequadratic response surface model the radial basis functioninterpolation and the Kriging model are said to enhance thesearching performance of SPEA for the small-scale problemsThe addition of constraints to the design problems can causethe searching performance of the hybrid algorithms TheSPEA using ANN is inferior to the other surrogate-assistedmethods or even the original SPEA2 without using a sur-rogate model because it requires efficient network topologyoptimisation before being used to estimate functions whilein this paper the simple heuristic approach is employed
8 Mathematical Problems in Engineering
= 37526 and 120596 = 83234= 45 and 120596 = 10354
= 33758 and 120596 = 73085 = 33779 and 120596 = 73032
= 33613 and 120596 = 72929 t = 30008 and 120596 = 63286
(1) t (2) t
(3) t (4) t
(5) t (6)
Figure 9 Selected design points from Figure 8
02 025 03 035 04 045 05 055 06 06508
09
1
11
12
13
14
15
1
2
3
4 5
6
Stru
ctur
al m
ass (
kg)
1120582
Figure 10 Approximate Pareto front of F2
= 45 and 120582 = 63261 = 39843 and 120582 = 3823
= 35599 and 120582 = 25781 = 35983 and 120582 = 25739
= 35555 and 120582 = 25201 = 30381 and 120582 = 16535
(1) t
(3) t
(5) t (6) t
(4) t
(2) t
Figure 11 Selected design solutions from Figure 10
Mathematical Problems in Engineering 9
01 015 02 025 03 035 04 045 0508
09
1
11
12
13
14
15
1
2
3
45
6
Stru
ctur
al m
ass (
kg)
1(120582 + 1205961205960)
Figure 12 Approximate Pareto front of F3
= 43963 and 120582 + 1205961205960 = 64596= 45 and 120582 + 1205961205960 = 75573
= 41201 and 120582 + 1205961205960 = 49251 = 35884 and 120582 + 1205961205960 = 35891
= 35532 and 120582 + 1205961205960 = 34666 = 3 and 120582 + 1205961205960 = 24581
(1) t (2) t
(4) t(3) t
(5) t (6) t
Figure 13 Selected design solutions from Figure 12
0095 01 0105 011 0115 012 0125 013 0135 014 0145085
09
095
1
105
11
115
12
1
2
3
4
56
Stru
ctur
al m
ass (
kg)
1120596
Figure 14 Approximate Pareto front of F4
10 Mathematical Problems in Engineering
= 45 and 120596 = 10349 = 42534 and 120596 = 96302
= 39481 and 120596 = 88068 = 36521 and 120596 = 80402
= 36501 and 120596 = 80223 = 33791 and 120596 = 73022
(1) t (2) t
(4) t
(6) t(5) t
(3) t
Figure 15 Selected design solutions from Figure 14
02 025 03 035 04 045 0509
095
1
105
11
115
12
125
13
135
14
1
2
3
45
6
Stru
ctur
al m
ass (
kg)
1120582
Figure 16 Approximate Pareto front of F5
= 45 and 120582 = 62885 = 45 and 120582 = 53614
= 40881 and 120582 = 42008 = 37402 and 120582 = 31575
= 3652 and 120582 = 30888 = 34047 and 120582 = 22103
(1) t (2) t
(4) t
(6) t
(3) t
(5) t
Figure 17 Selected design solutions from Figure 16
Mathematical Problems in Engineering 11
01 015 02 025 03 03509
095
1
105
11
115
12
125
13
135
14
1
2
3
4
56
Stru
ctur
al m
ass (
kg)
1(120582 + 1205961205960)
Figure 18 Approximate Pareto front of F6
= 45 and 120582 + 1205961205960 = 75573 = 44813 and 120582 + 1205961205960 = 65748
= 41914 and 120582 + 1205961205960 = 53343 = 38581 and 120582 + 1205961205960 = 4254
= 38571 and 120582 + 1205961205960 = 41494 = 34147 and 120582 + 1205961205960 = 30684
(1) t (2) t
(4) t
(6) t
(3) t
(5) t
Figure 19 Selected design solutions from Figure 18
More investigation on this issue would greatly improve theperformance of SPEA-N In cases of large-scale problemsonly SPEA using a quadratic function is superior to thenonsurrogate SPEA
6 Conclusions
The use of multiobjective evolutionary algorithm for thedesign of a torque arm leads to multiple optimum structuresfor decision making The hybridisation of SPEA and severalsurrogate models is developedThe various surrogate modelsincluding the quadratic regression technique the radialbasis function interpolation and the Kriging model canhelp improving the searching performance of the strengthPareto evolutionary algorithm For a design problem havingmass and natural frequency as objective functions and stress
and buckling as constraints SPEA using RBF is the bestmethod The SPEA using a quadratic regression is the bestfor the design case of F2 For the other multiobjective designproblems the best and the second best performers are SPEAusing the Krigingmodel and the quadratic RSM respectivelyNevertheless when considering all of the design problems theoverall top performer is SPEA2 using the quadratic regressionwhile the close second best uses theKrigingmodelMoreoverSPEA using a quadratic response surface model is the onlystrategy that is useful for solving large-scale problems Theuse of ANN for approximating constrained optimisationproblems is not effective as a proper network topologyoptimisation needs to be constructed before entering thesurrogate subproblem It is however possible to enhancethe performance of the ANN-based optimiser if an efficientnetwork topology optimisation is performed during theoptimisation process
12 Mathematical Problems in Engineering
20 25 30 35 40 45 50 55 60 65
F7
SPEA-OSPEA-Q
SPEA-RSPEA-K
f1
f2
minus60
minus50
minus40
minus30
minus20
minus10
0
Figure 20 Approximate Pareto front of F7
20 25 30 35 40 45 50 55 60
F8
SPEA-OSPEA-Q
SPEA-RSPEA-K
minus1000
minus950
minus900
minus850
minus800
minus750
minus700
minus650
minus600
minus550
f2
f1
Figure 21 Approximate Pareto front of F8
Acknowledgment
The authors are graceful for the support from the ThailandResearch Fund (RSA5280006)The first author also wishes tothank the office of higher education commission Thailandfor the CHE-PHD scholarship
References
[1] F Zong H Lin B Yu and X Pan ldquoDaily commute timeprediction based on genetic algorithmrdquoMathematical Problemsin Engineering vol 2012 Article ID 321574 20 pages 2012
[2] I Yao and D Han ldquoImproved barebones particle swarmoptimization with neighborhood search and its application on
ship designrdquo Mathematical Problems in Engineering vol 2013Article ID 175848 12 pages 2013
[3] Y Xu P Fan and L Yuan ldquoA simple and efficient artificial beecolony algorithmrdquo Mathematical Problems in Engineering vol2013 Article ID 526315 9 pages 2013
[4] S Talatahari E Khalili and S M Alavizadeh ldquoAcceleratedparticle swarm for optimum design of frame structuresrdquoMath-ematical Problems in Engineering vol 2013 Article ID 6498576 pages 2013
[5] K Socha and M Dorigo ldquoAnt colony optimization for contin-uous domainsrdquo European Journal of Operational Research vol185 no 3 pp 1155ndash1173 2008
[6] N Hansen S D Muller and P Koumoutsakos ldquoReducingthe time complexity of the derandomized evolution strategywith covariance matrix adaptation (CMA-ES)rdquo EvolutionaryComputation vol 11 no 1 pp 1ndash18 2003
[7] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[8] J D Knowles and D W Corne ldquoApproximating the nondom-inated front using the Pareto Archived Evolution StrategyrdquoEvolutionary Computation vol 8 no 2 pp 149ndash172 2000
[9] M Farina and P Amato ldquoLinked interpolation-optimizationstrategies for multicriteria optimization problemsrdquo Soft Com-puting vol 9 pp 54ndash65 2005
[10] J Knowles ldquoParEGO a hybrid algorithm with on-line land-scape approximation for expensive multiobjective optimizationproblemsrdquo IEEE Transactions on Evolutionary Computationvol 10 no 1 pp 50ndash66 2006
[11] Z Zhou Y S Ong M H Lim and B S Lee ldquoMemetic algo-rithm using multi-surrogates for computationally expensiveoptimization problemsrdquo Soft Computing vol 11 no 10 pp 957ndash971 2007
[12] R G Regis and C A Shoemaker ldquoLocal function approxima-tion in evolutionary algorithms for the optimization of costlyfunctionsrdquo IEEE Transactions on Evolutionary Computationvol 8 no 5 pp 490ndash505 2004
[13] N Pholdee and S Bureerat ldquoSurrogate-assisted evolutionaryoptimizers for multiobjective design of a torque arm structurerdquoAppliedMechanics andMaterials vol 101-102 pp 324ndash328 2011
[14] N Pholdee and S Bureerat ldquoPassive vibration control ofan automotive component using evolutionary optimisationrdquoJournal of Research and Applications in Mechanical Engineeringvol 1 pp 19ndash23 2011
[15] K C Giannakoglou D I Papadimitriou and I C KampolisldquoAerodynamic shape design using evolutionary algorithmsand new gradient-assisted metamodelsrdquo Computer Methods inApplied Mechanics and Engineering vol 195 no 44ndash47 pp6312ndash6329 2006
[16] Y S Ong K Y Lum P B Nair D M Shi and Z K ZhangldquoGlobal convergence of unconstrained and bound constrainedsurrogate-assisted evolutionary search in aerodynamic shapedesignrdquo in Proceedings of the Congress on Evolutionary Compu-tation (CEC rsquo03) pp 1856ndash1863 Canberra Australia December2003
[17] Y Tenne and S W Armfield ldquoA versatile surrogate-assistedmemetic algorithm for optimization of computationally expen-sive functions and its engineering applicationsrdquo Studies inComputational Intelligence vol 92 pp 43ndash72 2008
Mathematical Problems in Engineering 13
[18] S Srisomporn and S Bureerat ldquoGeometrical design of plate-fin heat sinks using hybridization of MOEA and RSMrdquo IEEETransactions onComponents and Packaging Technologies vol 31no 2 pp 351ndash360 2008
[19] S Kanyakam and S Bureerat ldquoComparative performance ofsurrogate-assisted MOEAs for geometrical design of pin-finheat sinksrdquo Applied Mathematics vol 2012 Article ID 53478314 pages 2012
[20] A Husain and K Y Kin ldquoOptimization of a microchannel heatsink with temperature dependent fluid propertiesrdquo Journal ofMaterials Processing Technology vol 208 no 1ndash3 pp 499ndash5062008
[21] J Yoo and C-Y Lee ldquoTopology optimization of a swing armtype actuator using the response surface methodrdquoMicrosystemTechnologies vol 13 no 1 pp 21ndash31 2007
[22] W Hu L G Yao and Z Z Hua ldquoOptimization of sheetmetal forming processes by adaptive response surface basedon intelligent sampling methodrdquo Journal of Materials ProcessingTechnology vol 197 no 1ndash3 pp 77ndash88 2008
[23] N Nariman-Zadeh A Darvizeh A Jamali and A MoeinildquoEvolutionary design of generalized polynomial neural net-works for modelling and prediction of explosive formingprocessrdquo Journal of Materials Processing Technology vol 164-165 pp 1561ndash1571 2005
[24] T Goel R Vaidyanathan R T Haftka W Shyy N V Queipoand K Tucker ldquoResponse surface approximation of Pareto opti-mal front in multi-objective optimizationrdquo Computer MethodsinAppliedMechanics and Engineering vol 196 no 4ndash6 pp 879ndash893 2007
[25] Z Zhou Y S Ong P B Nair A J Keane and K Y LumldquoCombining global and local surrogate models to accelerateevolutionary optimizationrdquo IEEE Transactions on Systems Manand Cybernetics Part C vol 37 no 1 pp 66ndash76 2007
[26] Y Jin ldquoA comprehensive survey of fitness approximation inevolutionary computationrdquo Soft Computing vol 9 pp 3ndash122005
[27] Y S Ong P B Nair A J Keane and K W Wong ldquoSurrogate-assisted evolutionary optimization frameworks for high-fidelityengineering design problemsrdquo in Knowledge Incorporation inEvolutionary Computation Studies in Fuzziness and Soft Com-puting Y Jin Ed pp 307ndash332 Springer Berlin Germany 2004
[28] Y Jin ldquoSurrogate-assisted evolutionary computation recentadvances and future challengesrdquo Swarm and Evolutionary Com-putation vol 1 pp 61ndash70 2011
[29] A I J Forrester and A J Keane ldquoRecent advances in surrogate-based optimizationrdquo Progress in Aerospace Sciences vol 45 no1ndash3 pp 50ndash79 2009
[30] K-T Chiang and F-P Chang ldquoApplication of response surfacemethodology in the parametric optimization of a pin-fin typeheat sinkrdquo International Communications in Heat and MassTransfer vol 33 no 7 pp 836ndash845 2006
[31] Y Tenne and S W Armfield ldquoA framework for memeticoptimization using variable global and local surrogate modelsrdquoSoft Computing vol 13 no 8-9 pp 781ndash793 2009
[32] P Breitkopf H Naceur A Rassineux and P Villon ldquoMovingleast squares response surface approximation formulation andmetal forming applicationsrdquo Computers and Structures vol 83no 17-18 pp 1411ndash1428 2005
[33] M Xiao B Raghavan P Breitkopf R F Coelho C Knopf-Lenoir and P Villon ldquoSurrogate models based on constrainedPOD for optimizationrdquo in Proceedings of the 4th European
Conference on Computational Mechanics (ECCM rsquo10) ParisFrance May 2010
[34] M Xiao P Breitkopf R F Coelho C Knopf-Lenoir MSidorkiewicz and P Villon ldquoModel reduction by CPOD andKriging application to the shape optimization of an intakeportrdquo Structural andMultidisciplinary Optimization vol 41 no4 pp 555ndash574 2010
[35] B RaghavanMHamdaouiM Xiao P Breitkopf and P VillonldquoA bi-level meta-modeling approach for structural optimizationusing modified POD bases and diffuse approximationrdquo Com-puters and Structures 2013
[36] B Raghavan and P Breitkopf ldquoAsynchronous evolutionaryshape optimization based on high-quality surrogates applica-tion to an air-conditioning ductrdquo Engineering with Computers2012
[37] E A Kontoleontos V G Asouti and K C GiannakoglouldquoAn asynchronous metamodel-assisted memetic algorithm forCFD-based shape optimizationrdquo Engineering Optimization vol44 no 2 pp 157ndash173 2012
[38] H Zhu L Liu T Long and L Peng ldquoA novel algorithmof maximin Latin hypercube design using successive localenumerationrdquo Engineering Optimization vol 44 no 5 pp 551ndash564 2012
[39] S D Georgiou and S Stylianou ldquoBlock-circulant matricesfor constructing optimal Latin hypercube designsrdquo Journal ofStatistical Planning and Inference vol 141 no 5 pp 1933ndash19432011
[40] J M Parr A J Keane A I J Forrester and C M E HoldenldquoInfill sampling criteria for surrogate-based optimization withconstraint handlingrdquo Engineering Optimization vol 44 no 10pp 1147ndash1166 2012
[41] E Zitzler M Laumanns and L Thiele ldquoSPEA2 improvingthe strength pareto evolutionary algorithm for multiobjectiveoptimizationrdquo inEvolutionaryMethods forDesign Optimizationand Control pp 95ndash100 2002
[42] S N Lophaven H B Neilson and J Sondergaard ldquoDACEa MATLAB kriging toolboxrdquo Tech Rep IMM-TR-2002-12Technical University of Denmark 2002
[43] R L Harder and R N Desmarais ldquoInterpolation using surfacesplinesrdquo Journal of Aircraft vol 9 no 2 pp 189ndash191 1972
[44] M E Botkin ldquoShape optimization with buckling and stressconstraintsrdquo AIAA Journal vol 34 no 2 pp 423ndash425 1996
[45] K Deb A Pratap and T Meyarivan ldquoConstrained test prob-lems for multi-objective evolutionary optimizationrdquo in Evolu-tionary Multi-Criterion Optimization (Zurich 2001) vol 1993of Lecture Notes in Computer Science pp 284ndash298 SpringerBerlin Germany 2001
[46] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
![Page 2: Research Article Surrogate-Assisted Multiobjective ...downloads.hindawi.com/journals/mpe/2013/695172.pdf · of engineering application, for example, aerodynamics [ ], heat and mass](https://reader034.fdocuments.net/reader034/viewer/2022052003/60164c5514f5a3172c3c2904/html5/thumbnails/2.jpg)
2 Mathematical Problems in Engineering
as given in [10] Surrogate-assisted EAs not only employfunction approximation but also use some mathematicalprogramming to enhance the searching performance [24 25]A comprehensive survey on the surrogate models used forenhancing EAs performance can be found in [26ndash29]
The surrogate models commonly employed include poly-nomial regression techniques [11 20 22 26 27 30] radialbasis function interpolation (RBF) [9 11 12 16 18 27]artificial neural network (ANN) [9 15 23 26] supportvector machines [31] moving least square method [32] andGaussian process model (Kriging model) [10 16 25ndash27]Recent alternative surrogate-based approaches are concernedwith the approximation of field vectors (such as velocity andpressure fields for computational fluid dynamics and dis-placement and stress fields for structural analysis) rather thanapproximating objective and constraint functions directlyThe proper orthogonal decomposition (POD) technique isemployed for this task [33ndash35] With this concept a designercan deal with more complicated design problems such asdesign optimisation when aeroelasticity is taken into consid-eration Apart from that there exist optimisers with the use ofgrid-based computing integrating nongeneration evolution-ary algorithms surrogate models local search and Lamarck-ian learning process together This approach is termed asyn-chronousmetamodel-based evolutionary algorithms [36 37]A technique used in the design of computer experiment phaseis usually a Latin hypercube sampling technique Recentlythere have been several papers focusing on the developmentof optimum Latin hypercube sampling [38 39] Anotherinteresting approach is an infill sampling technique [40]
The most challenging task for surrogate-based optimi-sation is to improve the convergence rate To the authorsrsquobest knowledge the work in the literature mostly demon-strated applying the design approaches to unconstrained orbox-constrained optimisation which is simple to handle byusing EAs Nevertheless there are usually many nonlinearconstraints in the real-world design problems particularlyin structural optimisation The approximation of objectivefunction although being inaccurate can lead to real optima(referred to as ldquoBlessing of Uncertaintyrdquo in [11]) On the otherhand in estimating constraint functions the approximatefunction needs to be as precise as possible so that theevolutionary search can land in the real feasible regionInaccurate constraint function approximation can lead theoptimiser to an undesirable design space
In this paper the hybridisation of the well-establishedmultiobjective evolutionary algorithm (MOEA) namelystrength Pareto evolutionary algorithm [41] and some pop-ular surrogate models is developed The work is concernedwith the studies of using several SMs for performanceenhancement of multiobjective evolutionary algorithms insolving structural constrained optimisation A number ofsurrogate models including quadratic function radial basisfunction neural network and Kriging models are employedin combination with SPEA2 using real codes The varioushybrid optimisation strategies are implemented on eightsimultaneous shape and sizing design problems of a structurehaving design functions as structural weight lateral buckingnatural frequency and stress Structural analysis is carried out
by using finite element analysis (FEA) The optimum resultsobtained are compared and discussed The performance ofSMs for estimating design constraints is investigated It hasbeen found that by using a quadratic surrogate model thesearching performance of SPEA2 is greatly improved
2 Hybridisation of SPEA2 anda Surrogate Model
There have been several ways to properly integrate SMs intoEAs The hybrid algorithm proposed in this paper is similarto what is presented in [18] The SPEA2 with real codesis employed as the main MOEA procedure The surrogate-assisted SPEA is illustrated in Figure 1 The computationalsteps in the figure can be separated into two parts as themain SPEA procedure and the SPEA subproblem that usesan approximationmodelThemain SPEA is slightly modifiedfrom the original version presented in [41] The real coderecombination and mutation are similar to the work pre-sented in [18]The procedure starts with an initial populationan (empty) Pareto archive and some other predefined param-eters for example an external archive size and recombinationand mutation rates The fitness of the individuals in thepopulation is then assigned by performing actual functionevaluation The current population and their correspondingfitness values are brought to the surrogate environment assome of them are picked to build a surrogate model Havingsuch an approximation model the SPEA is employed tosolve the optimisation problem with approximate functionevaluation
The nondominated solutions from this stage are putback to the main SPEA procedure where their actual fit-ness values are determined The current population theexternal archive and the nondominated solutions fromperforming a surrogate-based optimisation are put togetherwhile their nondominated solutions are determined Theexternal archive is then updated by replacing it with thesenondominated solutions In case that the number of thenondominated solutions exceeds the predefined archive sizesome of them will be removed from the archive by meansof the nearest neighbourhoodmethod [41] Subsequently theso-called binary tournament selection is performed to selectsome members in the newly updated archive to reproduce aset of offspringThe algorithm goes back to the step of fitnessfunction evaluation and is repeated until the terminationcriterion is fulfilled
Design solutions are selected froma current population tobuild a surrogate model based upon the idea that they shouldbe evenly distributed throughout the design space Theproposed approach is operated on the domain of objectivefunctions which is somewhat similar to the adaptive gridalgorithm a Pareto archiving technique used in the Paretoarchive evolution strategy [8] The procedure to select thesolutions for surrogate modelling is illustrated in Figure 2On the objective domain all of the design points in thecurrent population are plotted and a proper grid is generatedcovering all the points A solution that is the closest to thecentre of each mesh is selected for function estimation
Mathematical Problems in Engineering 3
Stop
Postprocessing
No
Yes
Evaluate fitness values of
Environmental selection
Apply NNT if necessary
Binary tournament selection
Stop No
Yes
to build the SM model
Main procedure SM sub-problem
Preprocessing
Initialization k = 0
Pk f(Pk) and Ak =
Evaluate fitness values of
PkUAkUBk+1
Environmental selection
Find FindAk+1 from PkUAkUBk+1
Apply NNT if necessary
Binary tournament selectionPk1 = select(Ak+1)
Pk2 = recombine(Pk
1)
Pk+1 = mutate(Pk2)
Find f(Pk+1)
k = k + 1
Find f(Bt+1n ) the actual
function values of Bt+1n
t = t + 1
Qt2n = recombine(Qt
1n)
Qt+1n = mutate(Qt
2n)
Find f(Qt+1)
Qt1n = select(Bt+1
n )
Bt+1n from Bt
n UQtn
GtnUB
tn
Initialization t = 0
Qtn g(Qt
n) and Btn =
Pick the member from Pk and f(Pk)
Figure 1 Flowchart for surrogate-assisted SPEA
3 Surrogate Models
Let 119910 = 119891(x) are a function of a design vector x sized119899 times 1 Given a set of design solutions (or sampling points)X = [x1 x119873] and their corresponding function valuesY =
[1199101 119910
119873] selected during an evolutionary optimisation
process a surrogate or approximation model is constructedby means of curve fitting or interpolation Approximationmodels used in this study are as follows
31 Polynomial Regression The most commonly used poly-nomial surrogate model or a response surface model (RSM)is of the second-order polynomial or quadratic model whichcan be expressed as
119910 = 1205730+sum120573
119894119909119894+sum120573
119894119909119894119909119895 (1)
where 120573119894for 119894 = 0 (119899 + 1)(119899 + 2)2 are the polynomial
coefficients to be determined The coefficients can be foundby using a regression or least square technique [26]
32 Kriging Model A Kriging model (also known as aGaussian process model) used in this paper is the famousMATLAB toolbox named design and analysis of computerexperiments (DACE) [42] The estimation of function can bethought of as the combination of global and local approxima-tion models that is
119910 (x) = 119891 (x) + 119885 (x) (2)
where 119891(x) is a global regression model while 119885(x) is astochastic Gaussian process with zero mean and nonzerocovariance representing a localised deviation In this worka linear function is used for a global model which can beexpressed as
119891 = 1205730+
119899
sum
119894=1
120573119894119909119894= 120573119879f (3)
where 120573 = [1205730 120573
119899]119879 f = f(x) = [1 119909
1 1199092 119909
119899]119879 The
covariance of 119885(x) is expressed as
Cov (119885 (x119901) 119885 (x119902)) = 1205902R [119877 (x119901 x119902)] (4)
4 Mathematical Problems in Engineering
The selected points forbuilding a surrogate model
Grid centerf2
f1
Figure 2 Selection of sampling points for constructing a surrogatemodel
for 119901 119902 = 1 119873 where 119877 is the correlation functionbetween any two of the 119873 design points and R is thesymmetric correlation matrix size 119873 times 119873 with the unitydiagonal [26] The correlation function used herein is
119877 (x119901 x119902) = exp (minus(x119901 minus xq)119879120579 (x119901 minus xq)) (5)
where 120579119894are the unknown correlation parameters to be
determined by means of the maximum likelihood methodHaving found 120573 and 120579 the Kriging predictor can be achievedas
119910 = f (x)119879120573 + r119879 (x)Rminus1 (y minus F120573) (6)
where F = [f(x1) f(x2) f(x119899)]119879 and r119879(x) = [119877(x x1)119877(x x2) 119877(x x119873)] For more details see [42]
33 Radial Basis Function Interpolation The radial basisfunction interpolation has been used in a wide range of appli-cations such as integration between aerodynamic and finiteelement grids in aeroelastic analysis [43] The use of sucha model for surrogate-assisted evolutionary optimisation issaid to be commonplace [9 18 27]The approximate functioncan be written as
119910 (x) =119873
sum
119894=1
120572119894119870(
10038171003817100381710038171003817xminusx11989410038171003817100381710038171003817) (7)
where 120572119894are the coefficients to be determined and 119870 is a
radial basis kernel (Here it is set to be linear splines) Thecoefficients can be found from the 119873 sampling points asdetailed in Srisomporn and Bureerat 2008 [18]
34 Neural Network Artificial neural network (ANN) isdeveloped in response to the need to use complicated func-tion expression rather than a simple quadratic or radial basisfunction The model consists of interconnecting neurons ornodes that somewhat mimic the constructs of biologicalneurons as shown in Figure 3 ANN has been implemented
Table 1 Design variables and bounds for F1ndashF6
Design variables Bounds (mm)1199091= 119897119889
0 le 119897119889le 60
1199092= 119897119901
100 le 119897119901le 200
1199093= 1199031199011
15 le 1199031199011le 35
1199094= 1199031199012
15 le 1199031199012le 30
1199095= th 3 le th le 45
Input layer
Hidden layer
Output layer
Neuron or node
Figure 3 Neural network
on a wide variety of real world applications The use ofANN for surrogate-assisted optimisation can be found inthe literature [9] The neural network model used in thispaper is the feedforward multilayer perceptrons The detailsof the network in this work are the learning algorithm =backpropagation and network training function = Levenberg-Marquardt backpropagation
In this work a simple heuristic algorithm is used toconstruct a network topology The procedure starts with anetwork with two hidden layers while the number of nodes119873119877on each layer is randomly chosen with the bound 119873
119877isin
[15 50] The transfer function for the hidden layer can beeither hyperbolic tangent sigmoid or logarithmic sigmoid basedon randomisation If the termination criterion (training goal)is notmet randomly increase a number of nodes in the layersIf the number of nodes reaches the upper bound 119873
119877= 50
add the third hidden layer to the network where a transferfunction and node number are randomly chosen In case thatthe stopping criterion is not fulfilled increase the number ofnodes The algorithm repeats until the stopping criterion ismet
4 Testing Problems
Six testing multiobjective design problems are posed todesign a torque arm [44] structure shown in Figure 4 Sinceit is a plate-like structure under inplane loading designcriteria to be taken into consideration are weight stressbuckling and natural frequency The function calculationcan be carried out by using FEA where shell elements areemployed The design variables and their bound constraintsare given in Table 1 where other dimensions are set as 119897
119889=
30mm 119897119901= 190mm 119903
1199011= 26mm and 119903
1199012= 27mm
Mathematical Problems in Engineering 5
5066N
2789N
rp2rp1
R2 = 42mmr2 = 25mm
Fixedboundaryconditionsr1 = 40mm
th = platethickness
l0 = 100mm
R1 = 542
L = 420mm
lpld
Figure 4 Torque arm
The six biobjective optimisation problems are termed F1F2 F3 F4 F5 and F6 respectively and can be expressed as
objective functionmin 119891
1(x) max119891
2(x)
F1 1198911(x) = 119908 119891
2(x) = 120596
F2 1198911(x) = 119908 119891
2(x) = 120582
F3 1198911(x) = 119908 119891
1(x) = 120582 + 120596
1205960
subject to
120590von minus120590119910
119878119865
le 0
1 minus 120582 le 0
objective functionmin119891
1(x) max119891
2(x)
F4 1198911(x) = 119908 119891
2(x) = 120596
F5 1198911(x) = 119908 119891
2(x) = 120582
F6 1198911(x) = 119908 119891
2(x) = 120582 + 120596
1205960
subject to
120590von minus120590119910
119878119865
le 0
1 minus 120582 le 0
120596 minus 1205960le 0
(8)
where 119908 = structural weight 120596 = first mode natural fre-quency 120582 = lateral bucking factor (it is the ratio of criticalload to applied load herein) 120590von = maximum von Misesstress on the structure and 120596
0= 72688 rads Furthermore
material properties are Youngrsquos modulus 119864 = 60 times 106Ncm2Poissonrsquos ratio ] = 03 yield stress 120590
119910= 80000Ncm2 density
120588 = 000718 kgcm3 and safety factor 119878119865= 15
In fact the F1 F2 and F3 problems are similar to the F4F5 and F6 problems respectively except for the introductionof natural frequency constraints to the last three problemsThis means that the performance of several surrogate modelsin estimating a natural frequency during an optimisation
procedure can be examinedThe bound constraints in Table 1are not presented in the design problems as they can be dealtwith in the SPEA procedure It should be noted that thefinite element analysis for these design studies is not timeconsuming as the work focuses on a comparative study of theseveral multiobjective optimisers
Since the F1ndashF6 problems are small-scale other two testproblems are posed so as to investigate the capability of thosesurrogate models when dealing with large-scale problemsTwo design problems of a simple wing-box displayed inFigure 5 are expressed as
objective functionmin119891
1(x) max119891
2(x)
F7 1198911(x) = 119908 119891
2(x) = minus120582
F8 1198911(x) = 119908 119891
2(x) = minus (120596
1+ sdot sdot sdot + 120596
5)
subject to
120590von minus 120590119910 le 0
1 minus 120582 le 0
(9)
where 120596119894is the 119894th mode natural frequency of the wing-box
Thewing-box ismade up of Aluminiumwith Youngmodulus119864 = 70GPa Poissonrsquos ratio ] = 034 density 120588 = 2700 kgm3and yield strength 120590
119910119905= 100MPa The structure consists of 2
spars 4 ribs and 4 skins (front rear top and bottom skins)resulting in 52wing segments as shownThe thickness of eachsegment is assigned to be a design variable therefore thereare 52 design variableswith the lower andupper bounds being0002m and 0010m respectively This can be classified as alarge-scale sizing optimisation problem The wing is subjectto static loads on the front spar as shown in Figure 5 Shellelements are used for finite element analysis similarly to F1ndashF6
The multiobjective optimisers used in this paper arenamed as follows
(i) SPEA-O the original SPEA2without using a surrogatemodel
(ii) SPEA-Q SPEA using a quadratic response surfacemodel
6 Mathematical Problems in Engineering
Fixed BC
2467kN 1645kN1645kN 0822 kN
2m
01m
0125m
025m0125m
Figure 5 Simple wing-box
(iii) SPEA-R SPEAusing the radial basis function interpo-lation
(iv) SPEA-N SPEA using a neural network model(v) SPEA-K SPEA using a Kriging model
Each evolutionary algorithm is employed to solve eachproblem ten runs with 30 generations (50 for F7-F8) and apopulation size of 20 (50 for F7-F8) for SPEA-O For thealgorithms using an approximation model the number ofgenerations and population size used in the main procedureis 15 (25 for F7-F8) and 20 (50 for F7-F8) respectively whileon each generation additional 20 (50 solutions for F7-F8)solutions are taken from optimising a surrogate subproblemIn the surrogate subproblem the number of generationsand population size is set to be 100 and 100 respectivelyThe external Pareto archive size is set to be 100 Duringthe optimisation process constraints are handled by usingthe approach presented in [45] It should be noted thatthere have been a number of techniques used to handledesign constraints in evolutionary optimisation [46] Wechoose the aforementioned technique because it is simple andsufficiently effective
5 Results and Discussion
The performance comparison of the five MOEAs for solvingthe eight design problems is based on the hypervolume indi-cator Two types of performance assessment are conducted asdetailed in Tables 2ndash4 In Table 2 each cell is the normalisedmean value of 10 nondominated front hypervolumes obtainedfrom using the optimiser on the corresponding column forsolving the problem on the corresponding row For eachdesign problem the best and worst mean values are nor-malised to be ldquo1rdquo and ldquo0rdquo respectively The total normalisedmean values given in the table are used for evaluating theoverall performance of the surrogate-assisted SPEAs Fromthe results the best algorithm for F1 is SPEA-R while thesecond best is SPEA-Q SPEA-Q is the best for F2 while thesecond best is SPEA-NThe best optimisers for F3 F4 F5 andF6 are SPEA-K SPEA-R SPEA-K and SPEA-K respectivelywhereas the second best for those 4 design problems is SPEA-Q SPEA-O without using a surrogate model is the worstperformer for F2 F5 and F6 while SPEA-N is the worstperformer for F1 F3 and F4 The overall best performer
Table 2 Comparison of normalised average hypervolume
SPEA-O SPEA-Q SPEA-R SPEA-N SPEA-KF1 0429 0704 1000 0000 0523F2 0000 1000 0052 0917 0692F3 0577 0995 0745 0000 1000F4 0457 0916 1000 0000 0497F5 0000 0976 0640 0161 1000F6 0000 0673 0243 0007 1000sum F1ndashF6 1463 5264 3679 1084 4713F7 0811 1000 0724 NA 0000F8 0337 1000 0453 NA 0000sum F7-F8 1148 2000 1177 NA 0000
Table 3 Performance matrix of F1
SPEA-O SPEA-Q SPEA-R SPEA-N SPEA-KSPEA-O 0 0 1 0 0SPEA-Q 0 0 1 0 0SPEA-R 0 0 0 0 0SPEA-N 1 1 1 0 1SPEA-K 0 0 1 0 0Total 1 1 4 0 1Ranking 2 2 1 5 2
Table 4 Comparison of ranking scores by t-test
SPEA-O SPEA-Q SPEA-R SPEA-N SPEA-KF1 2 2 1 5 2F2 4 1 4 1 1F3 1 1 1 5 1F4 3 1 1 5 3F5 4 1 3 4 1F6 3 2 3 3 1sum F1ndashF6 17 8 13 23 9F7 1 1 1 NA 4F8 2 1 2 NA 4sum F7-F8 3 1 3 NA 8
based on the evaluation is SPEA-Q whereas the second bestis SPEA-K For the large-scale problems only SPEA-Q cansurpass SPEA-O SPEA-R is said to be equal to SPEA-OwhileSPEA-K gives that worst performance Note that SPEA-Ncannot be applied to these problems as it takes excessivelylong running time for network training It can be said thatonly the quadratic response surface model is useful for thelarge-scale wing-box design
The second performance assessment is based on thestatistical t-test For each design problem to compare the 5MOEAs a performance matrix T sized 5 times 5 whose elementsare full of ldquo0rdquo is generated For the element 119879
119894119895of the matrix
if the mean value of the 10 hypervolumes obtained frommethod 119895 is significantly larger than that obtained frommethod 119894 based on the statistical t-test at 95 confidencelevel the value of 119879
119894119895is set to be one Table 3 shows the
performance comparison of the problem F1 Having had theperformance matrix the 6th row on the table displays thetotal scores of each optimiser The ranking is made in such
Mathematical Problems in Engineering 7
0 500 1000 1500 2000 25000
2
4
6
8
10
12
14 F7
No of function evaluations
Hyp
ervo
lum
e
SPEA-OSPEA-Q
Figure 6 SPEA-O versus SPEA-Q for F7
a way that the best method is ranked as 1 while the worst isranked as 5The overall performance assessment based on thet-test is given in Table 4 where the last row on the table showsthe total score of all theMOEAsThe lower the score the betterthe performer Similar to the first assessment the overall bestmethod for F1ndashF6 based on this assessment is SPEA-Q whilethe close secondbest is SPEA-KTheworstmethod is SPEA-Owithout using a surrogate model For the large-scale testproblems the only surrogateassisted method that is superiorto the non-surrogate SPEA is SPEA-Q
Figures 6 and 7 show the search history as plots ofhypervolume against the number of function evaluations ofSPEA-O and SPEA-Q for solving F7 and F8 respectivelyThevalues of front hypervolume can fluctuate due to the use of aPareto archiving technique to remove some nondominatedsolutions from the Pareto archive From the figures it canbe seen that SPEA-Q requires approximately half of the totalnumber of function evaluations employed by SPEA-O to haveequally good nondominated fronts
The best approximate Pareto front of F1 obtained afternumerous optimisation runs of the surrogate-assisted SPEAsis chosen and plotted in Figure 8 whereas the structures ofsome selected design points are shown in Figure 9 Figure 10displays the best front of the F2 problemwhere some selecteddesign solutions are illustrated in Figure 11The best front andsome selected optimal structures of the design problemF3 aredisplayed in Figures 12 and 13 respectively Similarly the bestapproximate fronts and some selected design solutions of F4F5 and F6 are illustrated in Figures 14 15 16 17 18 and 19Figures 20 and 21 show the best fronts obtained from usingthe various SPEAs for solving F7 and F8 respectively It canbe seen that the best fronts are from SPEA-Q and SPEA-OThe SPEA-Q front does not totally dominate that of SPEA-O but they overlap each other while the SPEA-Q front hasobviously greater front extension
According to the aforementioned performance assess-ments it can be said that SPEA-R is the best algorithm in
0 500 1000 1500 2000 25000
2
4
6
8
10
12F8
No of function evaluations
Hyp
ervo
lum
e
SPEA-OSPEA-Q
Figure 7 SPEA-O versus SPEA-Q for F8
009 01 011 012 013 014 015 01608
085
09
095
1
105
11
115
12
1
2
3
4
5
6
Stru
ctur
al m
ass (
kg)
1120596
Figure 8 Approximate Pareto front of F1
the cases of the F1 design problem while the close secondbest is SPEA-Q When adding a natural frequency constraintto F1 becoming the F4 problem the overall best and secondbest methods are still SPEA-R and SPEA-Q respectivelyFor the F2 case and the F5 design problem which is F2with additional natural frequency constraint the top twoperformers are SPEA-Q and SPEA-K The same can be saidfor the case of F3 while SPEA-K outperforms the others forthe F6 design problem The surrogate models that is thequadratic response surface model the radial basis functioninterpolation and the Kriging model are said to enhance thesearching performance of SPEA for the small-scale problemsThe addition of constraints to the design problems can causethe searching performance of the hybrid algorithms TheSPEA using ANN is inferior to the other surrogate-assistedmethods or even the original SPEA2 without using a sur-rogate model because it requires efficient network topologyoptimisation before being used to estimate functions whilein this paper the simple heuristic approach is employed
8 Mathematical Problems in Engineering
= 37526 and 120596 = 83234= 45 and 120596 = 10354
= 33758 and 120596 = 73085 = 33779 and 120596 = 73032
= 33613 and 120596 = 72929 t = 30008 and 120596 = 63286
(1) t (2) t
(3) t (4) t
(5) t (6)
Figure 9 Selected design points from Figure 8
02 025 03 035 04 045 05 055 06 06508
09
1
11
12
13
14
15
1
2
3
4 5
6
Stru
ctur
al m
ass (
kg)
1120582
Figure 10 Approximate Pareto front of F2
= 45 and 120582 = 63261 = 39843 and 120582 = 3823
= 35599 and 120582 = 25781 = 35983 and 120582 = 25739
= 35555 and 120582 = 25201 = 30381 and 120582 = 16535
(1) t
(3) t
(5) t (6) t
(4) t
(2) t
Figure 11 Selected design solutions from Figure 10
Mathematical Problems in Engineering 9
01 015 02 025 03 035 04 045 0508
09
1
11
12
13
14
15
1
2
3
45
6
Stru
ctur
al m
ass (
kg)
1(120582 + 1205961205960)
Figure 12 Approximate Pareto front of F3
= 43963 and 120582 + 1205961205960 = 64596= 45 and 120582 + 1205961205960 = 75573
= 41201 and 120582 + 1205961205960 = 49251 = 35884 and 120582 + 1205961205960 = 35891
= 35532 and 120582 + 1205961205960 = 34666 = 3 and 120582 + 1205961205960 = 24581
(1) t (2) t
(4) t(3) t
(5) t (6) t
Figure 13 Selected design solutions from Figure 12
0095 01 0105 011 0115 012 0125 013 0135 014 0145085
09
095
1
105
11
115
12
1
2
3
4
56
Stru
ctur
al m
ass (
kg)
1120596
Figure 14 Approximate Pareto front of F4
10 Mathematical Problems in Engineering
= 45 and 120596 = 10349 = 42534 and 120596 = 96302
= 39481 and 120596 = 88068 = 36521 and 120596 = 80402
= 36501 and 120596 = 80223 = 33791 and 120596 = 73022
(1) t (2) t
(4) t
(6) t(5) t
(3) t
Figure 15 Selected design solutions from Figure 14
02 025 03 035 04 045 0509
095
1
105
11
115
12
125
13
135
14
1
2
3
45
6
Stru
ctur
al m
ass (
kg)
1120582
Figure 16 Approximate Pareto front of F5
= 45 and 120582 = 62885 = 45 and 120582 = 53614
= 40881 and 120582 = 42008 = 37402 and 120582 = 31575
= 3652 and 120582 = 30888 = 34047 and 120582 = 22103
(1) t (2) t
(4) t
(6) t
(3) t
(5) t
Figure 17 Selected design solutions from Figure 16
Mathematical Problems in Engineering 11
01 015 02 025 03 03509
095
1
105
11
115
12
125
13
135
14
1
2
3
4
56
Stru
ctur
al m
ass (
kg)
1(120582 + 1205961205960)
Figure 18 Approximate Pareto front of F6
= 45 and 120582 + 1205961205960 = 75573 = 44813 and 120582 + 1205961205960 = 65748
= 41914 and 120582 + 1205961205960 = 53343 = 38581 and 120582 + 1205961205960 = 4254
= 38571 and 120582 + 1205961205960 = 41494 = 34147 and 120582 + 1205961205960 = 30684
(1) t (2) t
(4) t
(6) t
(3) t
(5) t
Figure 19 Selected design solutions from Figure 18
More investigation on this issue would greatly improve theperformance of SPEA-N In cases of large-scale problemsonly SPEA using a quadratic function is superior to thenonsurrogate SPEA
6 Conclusions
The use of multiobjective evolutionary algorithm for thedesign of a torque arm leads to multiple optimum structuresfor decision making The hybridisation of SPEA and severalsurrogate models is developedThe various surrogate modelsincluding the quadratic regression technique the radialbasis function interpolation and the Kriging model canhelp improving the searching performance of the strengthPareto evolutionary algorithm For a design problem havingmass and natural frequency as objective functions and stress
and buckling as constraints SPEA using RBF is the bestmethod The SPEA using a quadratic regression is the bestfor the design case of F2 For the other multiobjective designproblems the best and the second best performers are SPEAusing the Krigingmodel and the quadratic RSM respectivelyNevertheless when considering all of the design problems theoverall top performer is SPEA2 using the quadratic regressionwhile the close second best uses theKrigingmodelMoreoverSPEA using a quadratic response surface model is the onlystrategy that is useful for solving large-scale problems Theuse of ANN for approximating constrained optimisationproblems is not effective as a proper network topologyoptimisation needs to be constructed before entering thesurrogate subproblem It is however possible to enhancethe performance of the ANN-based optimiser if an efficientnetwork topology optimisation is performed during theoptimisation process
12 Mathematical Problems in Engineering
20 25 30 35 40 45 50 55 60 65
F7
SPEA-OSPEA-Q
SPEA-RSPEA-K
f1
f2
minus60
minus50
minus40
minus30
minus20
minus10
0
Figure 20 Approximate Pareto front of F7
20 25 30 35 40 45 50 55 60
F8
SPEA-OSPEA-Q
SPEA-RSPEA-K
minus1000
minus950
minus900
minus850
minus800
minus750
minus700
minus650
minus600
minus550
f2
f1
Figure 21 Approximate Pareto front of F8
Acknowledgment
The authors are graceful for the support from the ThailandResearch Fund (RSA5280006)The first author also wishes tothank the office of higher education commission Thailandfor the CHE-PHD scholarship
References
[1] F Zong H Lin B Yu and X Pan ldquoDaily commute timeprediction based on genetic algorithmrdquoMathematical Problemsin Engineering vol 2012 Article ID 321574 20 pages 2012
[2] I Yao and D Han ldquoImproved barebones particle swarmoptimization with neighborhood search and its application on
ship designrdquo Mathematical Problems in Engineering vol 2013Article ID 175848 12 pages 2013
[3] Y Xu P Fan and L Yuan ldquoA simple and efficient artificial beecolony algorithmrdquo Mathematical Problems in Engineering vol2013 Article ID 526315 9 pages 2013
[4] S Talatahari E Khalili and S M Alavizadeh ldquoAcceleratedparticle swarm for optimum design of frame structuresrdquoMath-ematical Problems in Engineering vol 2013 Article ID 6498576 pages 2013
[5] K Socha and M Dorigo ldquoAnt colony optimization for contin-uous domainsrdquo European Journal of Operational Research vol185 no 3 pp 1155ndash1173 2008
[6] N Hansen S D Muller and P Koumoutsakos ldquoReducingthe time complexity of the derandomized evolution strategywith covariance matrix adaptation (CMA-ES)rdquo EvolutionaryComputation vol 11 no 1 pp 1ndash18 2003
[7] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[8] J D Knowles and D W Corne ldquoApproximating the nondom-inated front using the Pareto Archived Evolution StrategyrdquoEvolutionary Computation vol 8 no 2 pp 149ndash172 2000
[9] M Farina and P Amato ldquoLinked interpolation-optimizationstrategies for multicriteria optimization problemsrdquo Soft Com-puting vol 9 pp 54ndash65 2005
[10] J Knowles ldquoParEGO a hybrid algorithm with on-line land-scape approximation for expensive multiobjective optimizationproblemsrdquo IEEE Transactions on Evolutionary Computationvol 10 no 1 pp 50ndash66 2006
[11] Z Zhou Y S Ong M H Lim and B S Lee ldquoMemetic algo-rithm using multi-surrogates for computationally expensiveoptimization problemsrdquo Soft Computing vol 11 no 10 pp 957ndash971 2007
[12] R G Regis and C A Shoemaker ldquoLocal function approxima-tion in evolutionary algorithms for the optimization of costlyfunctionsrdquo IEEE Transactions on Evolutionary Computationvol 8 no 5 pp 490ndash505 2004
[13] N Pholdee and S Bureerat ldquoSurrogate-assisted evolutionaryoptimizers for multiobjective design of a torque arm structurerdquoAppliedMechanics andMaterials vol 101-102 pp 324ndash328 2011
[14] N Pholdee and S Bureerat ldquoPassive vibration control ofan automotive component using evolutionary optimisationrdquoJournal of Research and Applications in Mechanical Engineeringvol 1 pp 19ndash23 2011
[15] K C Giannakoglou D I Papadimitriou and I C KampolisldquoAerodynamic shape design using evolutionary algorithmsand new gradient-assisted metamodelsrdquo Computer Methods inApplied Mechanics and Engineering vol 195 no 44ndash47 pp6312ndash6329 2006
[16] Y S Ong K Y Lum P B Nair D M Shi and Z K ZhangldquoGlobal convergence of unconstrained and bound constrainedsurrogate-assisted evolutionary search in aerodynamic shapedesignrdquo in Proceedings of the Congress on Evolutionary Compu-tation (CEC rsquo03) pp 1856ndash1863 Canberra Australia December2003
[17] Y Tenne and S W Armfield ldquoA versatile surrogate-assistedmemetic algorithm for optimization of computationally expen-sive functions and its engineering applicationsrdquo Studies inComputational Intelligence vol 92 pp 43ndash72 2008
Mathematical Problems in Engineering 13
[18] S Srisomporn and S Bureerat ldquoGeometrical design of plate-fin heat sinks using hybridization of MOEA and RSMrdquo IEEETransactions onComponents and Packaging Technologies vol 31no 2 pp 351ndash360 2008
[19] S Kanyakam and S Bureerat ldquoComparative performance ofsurrogate-assisted MOEAs for geometrical design of pin-finheat sinksrdquo Applied Mathematics vol 2012 Article ID 53478314 pages 2012
[20] A Husain and K Y Kin ldquoOptimization of a microchannel heatsink with temperature dependent fluid propertiesrdquo Journal ofMaterials Processing Technology vol 208 no 1ndash3 pp 499ndash5062008
[21] J Yoo and C-Y Lee ldquoTopology optimization of a swing armtype actuator using the response surface methodrdquoMicrosystemTechnologies vol 13 no 1 pp 21ndash31 2007
[22] W Hu L G Yao and Z Z Hua ldquoOptimization of sheetmetal forming processes by adaptive response surface basedon intelligent sampling methodrdquo Journal of Materials ProcessingTechnology vol 197 no 1ndash3 pp 77ndash88 2008
[23] N Nariman-Zadeh A Darvizeh A Jamali and A MoeinildquoEvolutionary design of generalized polynomial neural net-works for modelling and prediction of explosive formingprocessrdquo Journal of Materials Processing Technology vol 164-165 pp 1561ndash1571 2005
[24] T Goel R Vaidyanathan R T Haftka W Shyy N V Queipoand K Tucker ldquoResponse surface approximation of Pareto opti-mal front in multi-objective optimizationrdquo Computer MethodsinAppliedMechanics and Engineering vol 196 no 4ndash6 pp 879ndash893 2007
[25] Z Zhou Y S Ong P B Nair A J Keane and K Y LumldquoCombining global and local surrogate models to accelerateevolutionary optimizationrdquo IEEE Transactions on Systems Manand Cybernetics Part C vol 37 no 1 pp 66ndash76 2007
[26] Y Jin ldquoA comprehensive survey of fitness approximation inevolutionary computationrdquo Soft Computing vol 9 pp 3ndash122005
[27] Y S Ong P B Nair A J Keane and K W Wong ldquoSurrogate-assisted evolutionary optimization frameworks for high-fidelityengineering design problemsrdquo in Knowledge Incorporation inEvolutionary Computation Studies in Fuzziness and Soft Com-puting Y Jin Ed pp 307ndash332 Springer Berlin Germany 2004
[28] Y Jin ldquoSurrogate-assisted evolutionary computation recentadvances and future challengesrdquo Swarm and Evolutionary Com-putation vol 1 pp 61ndash70 2011
[29] A I J Forrester and A J Keane ldquoRecent advances in surrogate-based optimizationrdquo Progress in Aerospace Sciences vol 45 no1ndash3 pp 50ndash79 2009
[30] K-T Chiang and F-P Chang ldquoApplication of response surfacemethodology in the parametric optimization of a pin-fin typeheat sinkrdquo International Communications in Heat and MassTransfer vol 33 no 7 pp 836ndash845 2006
[31] Y Tenne and S W Armfield ldquoA framework for memeticoptimization using variable global and local surrogate modelsrdquoSoft Computing vol 13 no 8-9 pp 781ndash793 2009
[32] P Breitkopf H Naceur A Rassineux and P Villon ldquoMovingleast squares response surface approximation formulation andmetal forming applicationsrdquo Computers and Structures vol 83no 17-18 pp 1411ndash1428 2005
[33] M Xiao B Raghavan P Breitkopf R F Coelho C Knopf-Lenoir and P Villon ldquoSurrogate models based on constrainedPOD for optimizationrdquo in Proceedings of the 4th European
Conference on Computational Mechanics (ECCM rsquo10) ParisFrance May 2010
[34] M Xiao P Breitkopf R F Coelho C Knopf-Lenoir MSidorkiewicz and P Villon ldquoModel reduction by CPOD andKriging application to the shape optimization of an intakeportrdquo Structural andMultidisciplinary Optimization vol 41 no4 pp 555ndash574 2010
[35] B RaghavanMHamdaouiM Xiao P Breitkopf and P VillonldquoA bi-level meta-modeling approach for structural optimizationusing modified POD bases and diffuse approximationrdquo Com-puters and Structures 2013
[36] B Raghavan and P Breitkopf ldquoAsynchronous evolutionaryshape optimization based on high-quality surrogates applica-tion to an air-conditioning ductrdquo Engineering with Computers2012
[37] E A Kontoleontos V G Asouti and K C GiannakoglouldquoAn asynchronous metamodel-assisted memetic algorithm forCFD-based shape optimizationrdquo Engineering Optimization vol44 no 2 pp 157ndash173 2012
[38] H Zhu L Liu T Long and L Peng ldquoA novel algorithmof maximin Latin hypercube design using successive localenumerationrdquo Engineering Optimization vol 44 no 5 pp 551ndash564 2012
[39] S D Georgiou and S Stylianou ldquoBlock-circulant matricesfor constructing optimal Latin hypercube designsrdquo Journal ofStatistical Planning and Inference vol 141 no 5 pp 1933ndash19432011
[40] J M Parr A J Keane A I J Forrester and C M E HoldenldquoInfill sampling criteria for surrogate-based optimization withconstraint handlingrdquo Engineering Optimization vol 44 no 10pp 1147ndash1166 2012
[41] E Zitzler M Laumanns and L Thiele ldquoSPEA2 improvingthe strength pareto evolutionary algorithm for multiobjectiveoptimizationrdquo inEvolutionaryMethods forDesign Optimizationand Control pp 95ndash100 2002
[42] S N Lophaven H B Neilson and J Sondergaard ldquoDACEa MATLAB kriging toolboxrdquo Tech Rep IMM-TR-2002-12Technical University of Denmark 2002
[43] R L Harder and R N Desmarais ldquoInterpolation using surfacesplinesrdquo Journal of Aircraft vol 9 no 2 pp 189ndash191 1972
[44] M E Botkin ldquoShape optimization with buckling and stressconstraintsrdquo AIAA Journal vol 34 no 2 pp 423ndash425 1996
[45] K Deb A Pratap and T Meyarivan ldquoConstrained test prob-lems for multi-objective evolutionary optimizationrdquo in Evolu-tionary Multi-Criterion Optimization (Zurich 2001) vol 1993of Lecture Notes in Computer Science pp 284ndash298 SpringerBerlin Germany 2001
[46] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011
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Stochastic AnalysisInternational Journal of
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Mathematical Problems in Engineering 3
Stop
Postprocessing
No
Yes
Evaluate fitness values of
Environmental selection
Apply NNT if necessary
Binary tournament selection
Stop No
Yes
to build the SM model
Main procedure SM sub-problem
Preprocessing
Initialization k = 0
Pk f(Pk) and Ak =
Evaluate fitness values of
PkUAkUBk+1
Environmental selection
Find FindAk+1 from PkUAkUBk+1
Apply NNT if necessary
Binary tournament selectionPk1 = select(Ak+1)
Pk2 = recombine(Pk
1)
Pk+1 = mutate(Pk2)
Find f(Pk+1)
k = k + 1
Find f(Bt+1n ) the actual
function values of Bt+1n
t = t + 1
Qt2n = recombine(Qt
1n)
Qt+1n = mutate(Qt
2n)
Find f(Qt+1)
Qt1n = select(Bt+1
n )
Bt+1n from Bt
n UQtn
GtnUB
tn
Initialization t = 0
Qtn g(Qt
n) and Btn =
Pick the member from Pk and f(Pk)
Figure 1 Flowchart for surrogate-assisted SPEA
3 Surrogate Models
Let 119910 = 119891(x) are a function of a design vector x sized119899 times 1 Given a set of design solutions (or sampling points)X = [x1 x119873] and their corresponding function valuesY =
[1199101 119910
119873] selected during an evolutionary optimisation
process a surrogate or approximation model is constructedby means of curve fitting or interpolation Approximationmodels used in this study are as follows
31 Polynomial Regression The most commonly used poly-nomial surrogate model or a response surface model (RSM)is of the second-order polynomial or quadratic model whichcan be expressed as
119910 = 1205730+sum120573
119894119909119894+sum120573
119894119909119894119909119895 (1)
where 120573119894for 119894 = 0 (119899 + 1)(119899 + 2)2 are the polynomial
coefficients to be determined The coefficients can be foundby using a regression or least square technique [26]
32 Kriging Model A Kriging model (also known as aGaussian process model) used in this paper is the famousMATLAB toolbox named design and analysis of computerexperiments (DACE) [42] The estimation of function can bethought of as the combination of global and local approxima-tion models that is
119910 (x) = 119891 (x) + 119885 (x) (2)
where 119891(x) is a global regression model while 119885(x) is astochastic Gaussian process with zero mean and nonzerocovariance representing a localised deviation In this worka linear function is used for a global model which can beexpressed as
119891 = 1205730+
119899
sum
119894=1
120573119894119909119894= 120573119879f (3)
where 120573 = [1205730 120573
119899]119879 f = f(x) = [1 119909
1 1199092 119909
119899]119879 The
covariance of 119885(x) is expressed as
Cov (119885 (x119901) 119885 (x119902)) = 1205902R [119877 (x119901 x119902)] (4)
4 Mathematical Problems in Engineering
The selected points forbuilding a surrogate model
Grid centerf2
f1
Figure 2 Selection of sampling points for constructing a surrogatemodel
for 119901 119902 = 1 119873 where 119877 is the correlation functionbetween any two of the 119873 design points and R is thesymmetric correlation matrix size 119873 times 119873 with the unitydiagonal [26] The correlation function used herein is
119877 (x119901 x119902) = exp (minus(x119901 minus xq)119879120579 (x119901 minus xq)) (5)
where 120579119894are the unknown correlation parameters to be
determined by means of the maximum likelihood methodHaving found 120573 and 120579 the Kriging predictor can be achievedas
119910 = f (x)119879120573 + r119879 (x)Rminus1 (y minus F120573) (6)
where F = [f(x1) f(x2) f(x119899)]119879 and r119879(x) = [119877(x x1)119877(x x2) 119877(x x119873)] For more details see [42]
33 Radial Basis Function Interpolation The radial basisfunction interpolation has been used in a wide range of appli-cations such as integration between aerodynamic and finiteelement grids in aeroelastic analysis [43] The use of sucha model for surrogate-assisted evolutionary optimisation issaid to be commonplace [9 18 27]The approximate functioncan be written as
119910 (x) =119873
sum
119894=1
120572119894119870(
10038171003817100381710038171003817xminusx11989410038171003817100381710038171003817) (7)
where 120572119894are the coefficients to be determined and 119870 is a
radial basis kernel (Here it is set to be linear splines) Thecoefficients can be found from the 119873 sampling points asdetailed in Srisomporn and Bureerat 2008 [18]
34 Neural Network Artificial neural network (ANN) isdeveloped in response to the need to use complicated func-tion expression rather than a simple quadratic or radial basisfunction The model consists of interconnecting neurons ornodes that somewhat mimic the constructs of biologicalneurons as shown in Figure 3 ANN has been implemented
Table 1 Design variables and bounds for F1ndashF6
Design variables Bounds (mm)1199091= 119897119889
0 le 119897119889le 60
1199092= 119897119901
100 le 119897119901le 200
1199093= 1199031199011
15 le 1199031199011le 35
1199094= 1199031199012
15 le 1199031199012le 30
1199095= th 3 le th le 45
Input layer
Hidden layer
Output layer
Neuron or node
Figure 3 Neural network
on a wide variety of real world applications The use ofANN for surrogate-assisted optimisation can be found inthe literature [9] The neural network model used in thispaper is the feedforward multilayer perceptrons The detailsof the network in this work are the learning algorithm =backpropagation and network training function = Levenberg-Marquardt backpropagation
In this work a simple heuristic algorithm is used toconstruct a network topology The procedure starts with anetwork with two hidden layers while the number of nodes119873119877on each layer is randomly chosen with the bound 119873
119877isin
[15 50] The transfer function for the hidden layer can beeither hyperbolic tangent sigmoid or logarithmic sigmoid basedon randomisation If the termination criterion (training goal)is notmet randomly increase a number of nodes in the layersIf the number of nodes reaches the upper bound 119873
119877= 50
add the third hidden layer to the network where a transferfunction and node number are randomly chosen In case thatthe stopping criterion is not fulfilled increase the number ofnodes The algorithm repeats until the stopping criterion ismet
4 Testing Problems
Six testing multiobjective design problems are posed todesign a torque arm [44] structure shown in Figure 4 Sinceit is a plate-like structure under inplane loading designcriteria to be taken into consideration are weight stressbuckling and natural frequency The function calculationcan be carried out by using FEA where shell elements areemployed The design variables and their bound constraintsare given in Table 1 where other dimensions are set as 119897
119889=
30mm 119897119901= 190mm 119903
1199011= 26mm and 119903
1199012= 27mm
Mathematical Problems in Engineering 5
5066N
2789N
rp2rp1
R2 = 42mmr2 = 25mm
Fixedboundaryconditionsr1 = 40mm
th = platethickness
l0 = 100mm
R1 = 542
L = 420mm
lpld
Figure 4 Torque arm
The six biobjective optimisation problems are termed F1F2 F3 F4 F5 and F6 respectively and can be expressed as
objective functionmin 119891
1(x) max119891
2(x)
F1 1198911(x) = 119908 119891
2(x) = 120596
F2 1198911(x) = 119908 119891
2(x) = 120582
F3 1198911(x) = 119908 119891
1(x) = 120582 + 120596
1205960
subject to
120590von minus120590119910
119878119865
le 0
1 minus 120582 le 0
objective functionmin119891
1(x) max119891
2(x)
F4 1198911(x) = 119908 119891
2(x) = 120596
F5 1198911(x) = 119908 119891
2(x) = 120582
F6 1198911(x) = 119908 119891
2(x) = 120582 + 120596
1205960
subject to
120590von minus120590119910
119878119865
le 0
1 minus 120582 le 0
120596 minus 1205960le 0
(8)
where 119908 = structural weight 120596 = first mode natural fre-quency 120582 = lateral bucking factor (it is the ratio of criticalload to applied load herein) 120590von = maximum von Misesstress on the structure and 120596
0= 72688 rads Furthermore
material properties are Youngrsquos modulus 119864 = 60 times 106Ncm2Poissonrsquos ratio ] = 03 yield stress 120590
119910= 80000Ncm2 density
120588 = 000718 kgcm3 and safety factor 119878119865= 15
In fact the F1 F2 and F3 problems are similar to the F4F5 and F6 problems respectively except for the introductionof natural frequency constraints to the last three problemsThis means that the performance of several surrogate modelsin estimating a natural frequency during an optimisation
procedure can be examinedThe bound constraints in Table 1are not presented in the design problems as they can be dealtwith in the SPEA procedure It should be noted that thefinite element analysis for these design studies is not timeconsuming as the work focuses on a comparative study of theseveral multiobjective optimisers
Since the F1ndashF6 problems are small-scale other two testproblems are posed so as to investigate the capability of thosesurrogate models when dealing with large-scale problemsTwo design problems of a simple wing-box displayed inFigure 5 are expressed as
objective functionmin119891
1(x) max119891
2(x)
F7 1198911(x) = 119908 119891
2(x) = minus120582
F8 1198911(x) = 119908 119891
2(x) = minus (120596
1+ sdot sdot sdot + 120596
5)
subject to
120590von minus 120590119910 le 0
1 minus 120582 le 0
(9)
where 120596119894is the 119894th mode natural frequency of the wing-box
Thewing-box ismade up of Aluminiumwith Youngmodulus119864 = 70GPa Poissonrsquos ratio ] = 034 density 120588 = 2700 kgm3and yield strength 120590
119910119905= 100MPa The structure consists of 2
spars 4 ribs and 4 skins (front rear top and bottom skins)resulting in 52wing segments as shownThe thickness of eachsegment is assigned to be a design variable therefore thereare 52 design variableswith the lower andupper bounds being0002m and 0010m respectively This can be classified as alarge-scale sizing optimisation problem The wing is subjectto static loads on the front spar as shown in Figure 5 Shellelements are used for finite element analysis similarly to F1ndashF6
The multiobjective optimisers used in this paper arenamed as follows
(i) SPEA-O the original SPEA2without using a surrogatemodel
(ii) SPEA-Q SPEA using a quadratic response surfacemodel
6 Mathematical Problems in Engineering
Fixed BC
2467kN 1645kN1645kN 0822 kN
2m
01m
0125m
025m0125m
Figure 5 Simple wing-box
(iii) SPEA-R SPEAusing the radial basis function interpo-lation
(iv) SPEA-N SPEA using a neural network model(v) SPEA-K SPEA using a Kriging model
Each evolutionary algorithm is employed to solve eachproblem ten runs with 30 generations (50 for F7-F8) and apopulation size of 20 (50 for F7-F8) for SPEA-O For thealgorithms using an approximation model the number ofgenerations and population size used in the main procedureis 15 (25 for F7-F8) and 20 (50 for F7-F8) respectively whileon each generation additional 20 (50 solutions for F7-F8)solutions are taken from optimising a surrogate subproblemIn the surrogate subproblem the number of generationsand population size is set to be 100 and 100 respectivelyThe external Pareto archive size is set to be 100 Duringthe optimisation process constraints are handled by usingthe approach presented in [45] It should be noted thatthere have been a number of techniques used to handledesign constraints in evolutionary optimisation [46] Wechoose the aforementioned technique because it is simple andsufficiently effective
5 Results and Discussion
The performance comparison of the five MOEAs for solvingthe eight design problems is based on the hypervolume indi-cator Two types of performance assessment are conducted asdetailed in Tables 2ndash4 In Table 2 each cell is the normalisedmean value of 10 nondominated front hypervolumes obtainedfrom using the optimiser on the corresponding column forsolving the problem on the corresponding row For eachdesign problem the best and worst mean values are nor-malised to be ldquo1rdquo and ldquo0rdquo respectively The total normalisedmean values given in the table are used for evaluating theoverall performance of the surrogate-assisted SPEAs Fromthe results the best algorithm for F1 is SPEA-R while thesecond best is SPEA-Q SPEA-Q is the best for F2 while thesecond best is SPEA-NThe best optimisers for F3 F4 F5 andF6 are SPEA-K SPEA-R SPEA-K and SPEA-K respectivelywhereas the second best for those 4 design problems is SPEA-Q SPEA-O without using a surrogate model is the worstperformer for F2 F5 and F6 while SPEA-N is the worstperformer for F1 F3 and F4 The overall best performer
Table 2 Comparison of normalised average hypervolume
SPEA-O SPEA-Q SPEA-R SPEA-N SPEA-KF1 0429 0704 1000 0000 0523F2 0000 1000 0052 0917 0692F3 0577 0995 0745 0000 1000F4 0457 0916 1000 0000 0497F5 0000 0976 0640 0161 1000F6 0000 0673 0243 0007 1000sum F1ndashF6 1463 5264 3679 1084 4713F7 0811 1000 0724 NA 0000F8 0337 1000 0453 NA 0000sum F7-F8 1148 2000 1177 NA 0000
Table 3 Performance matrix of F1
SPEA-O SPEA-Q SPEA-R SPEA-N SPEA-KSPEA-O 0 0 1 0 0SPEA-Q 0 0 1 0 0SPEA-R 0 0 0 0 0SPEA-N 1 1 1 0 1SPEA-K 0 0 1 0 0Total 1 1 4 0 1Ranking 2 2 1 5 2
Table 4 Comparison of ranking scores by t-test
SPEA-O SPEA-Q SPEA-R SPEA-N SPEA-KF1 2 2 1 5 2F2 4 1 4 1 1F3 1 1 1 5 1F4 3 1 1 5 3F5 4 1 3 4 1F6 3 2 3 3 1sum F1ndashF6 17 8 13 23 9F7 1 1 1 NA 4F8 2 1 2 NA 4sum F7-F8 3 1 3 NA 8
based on the evaluation is SPEA-Q whereas the second bestis SPEA-K For the large-scale problems only SPEA-Q cansurpass SPEA-O SPEA-R is said to be equal to SPEA-OwhileSPEA-K gives that worst performance Note that SPEA-Ncannot be applied to these problems as it takes excessivelylong running time for network training It can be said thatonly the quadratic response surface model is useful for thelarge-scale wing-box design
The second performance assessment is based on thestatistical t-test For each design problem to compare the 5MOEAs a performance matrix T sized 5 times 5 whose elementsare full of ldquo0rdquo is generated For the element 119879
119894119895of the matrix
if the mean value of the 10 hypervolumes obtained frommethod 119895 is significantly larger than that obtained frommethod 119894 based on the statistical t-test at 95 confidencelevel the value of 119879
119894119895is set to be one Table 3 shows the
performance comparison of the problem F1 Having had theperformance matrix the 6th row on the table displays thetotal scores of each optimiser The ranking is made in such
Mathematical Problems in Engineering 7
0 500 1000 1500 2000 25000
2
4
6
8
10
12
14 F7
No of function evaluations
Hyp
ervo
lum
e
SPEA-OSPEA-Q
Figure 6 SPEA-O versus SPEA-Q for F7
a way that the best method is ranked as 1 while the worst isranked as 5The overall performance assessment based on thet-test is given in Table 4 where the last row on the table showsthe total score of all theMOEAsThe lower the score the betterthe performer Similar to the first assessment the overall bestmethod for F1ndashF6 based on this assessment is SPEA-Q whilethe close secondbest is SPEA-KTheworstmethod is SPEA-Owithout using a surrogate model For the large-scale testproblems the only surrogateassisted method that is superiorto the non-surrogate SPEA is SPEA-Q
Figures 6 and 7 show the search history as plots ofhypervolume against the number of function evaluations ofSPEA-O and SPEA-Q for solving F7 and F8 respectivelyThevalues of front hypervolume can fluctuate due to the use of aPareto archiving technique to remove some nondominatedsolutions from the Pareto archive From the figures it canbe seen that SPEA-Q requires approximately half of the totalnumber of function evaluations employed by SPEA-O to haveequally good nondominated fronts
The best approximate Pareto front of F1 obtained afternumerous optimisation runs of the surrogate-assisted SPEAsis chosen and plotted in Figure 8 whereas the structures ofsome selected design points are shown in Figure 9 Figure 10displays the best front of the F2 problemwhere some selecteddesign solutions are illustrated in Figure 11The best front andsome selected optimal structures of the design problemF3 aredisplayed in Figures 12 and 13 respectively Similarly the bestapproximate fronts and some selected design solutions of F4F5 and F6 are illustrated in Figures 14 15 16 17 18 and 19Figures 20 and 21 show the best fronts obtained from usingthe various SPEAs for solving F7 and F8 respectively It canbe seen that the best fronts are from SPEA-Q and SPEA-OThe SPEA-Q front does not totally dominate that of SPEA-O but they overlap each other while the SPEA-Q front hasobviously greater front extension
According to the aforementioned performance assess-ments it can be said that SPEA-R is the best algorithm in
0 500 1000 1500 2000 25000
2
4
6
8
10
12F8
No of function evaluations
Hyp
ervo
lum
e
SPEA-OSPEA-Q
Figure 7 SPEA-O versus SPEA-Q for F8
009 01 011 012 013 014 015 01608
085
09
095
1
105
11
115
12
1
2
3
4
5
6
Stru
ctur
al m
ass (
kg)
1120596
Figure 8 Approximate Pareto front of F1
the cases of the F1 design problem while the close secondbest is SPEA-Q When adding a natural frequency constraintto F1 becoming the F4 problem the overall best and secondbest methods are still SPEA-R and SPEA-Q respectivelyFor the F2 case and the F5 design problem which is F2with additional natural frequency constraint the top twoperformers are SPEA-Q and SPEA-K The same can be saidfor the case of F3 while SPEA-K outperforms the others forthe F6 design problem The surrogate models that is thequadratic response surface model the radial basis functioninterpolation and the Kriging model are said to enhance thesearching performance of SPEA for the small-scale problemsThe addition of constraints to the design problems can causethe searching performance of the hybrid algorithms TheSPEA using ANN is inferior to the other surrogate-assistedmethods or even the original SPEA2 without using a sur-rogate model because it requires efficient network topologyoptimisation before being used to estimate functions whilein this paper the simple heuristic approach is employed
8 Mathematical Problems in Engineering
= 37526 and 120596 = 83234= 45 and 120596 = 10354
= 33758 and 120596 = 73085 = 33779 and 120596 = 73032
= 33613 and 120596 = 72929 t = 30008 and 120596 = 63286
(1) t (2) t
(3) t (4) t
(5) t (6)
Figure 9 Selected design points from Figure 8
02 025 03 035 04 045 05 055 06 06508
09
1
11
12
13
14
15
1
2
3
4 5
6
Stru
ctur
al m
ass (
kg)
1120582
Figure 10 Approximate Pareto front of F2
= 45 and 120582 = 63261 = 39843 and 120582 = 3823
= 35599 and 120582 = 25781 = 35983 and 120582 = 25739
= 35555 and 120582 = 25201 = 30381 and 120582 = 16535
(1) t
(3) t
(5) t (6) t
(4) t
(2) t
Figure 11 Selected design solutions from Figure 10
Mathematical Problems in Engineering 9
01 015 02 025 03 035 04 045 0508
09
1
11
12
13
14
15
1
2
3
45
6
Stru
ctur
al m
ass (
kg)
1(120582 + 1205961205960)
Figure 12 Approximate Pareto front of F3
= 43963 and 120582 + 1205961205960 = 64596= 45 and 120582 + 1205961205960 = 75573
= 41201 and 120582 + 1205961205960 = 49251 = 35884 and 120582 + 1205961205960 = 35891
= 35532 and 120582 + 1205961205960 = 34666 = 3 and 120582 + 1205961205960 = 24581
(1) t (2) t
(4) t(3) t
(5) t (6) t
Figure 13 Selected design solutions from Figure 12
0095 01 0105 011 0115 012 0125 013 0135 014 0145085
09
095
1
105
11
115
12
1
2
3
4
56
Stru
ctur
al m
ass (
kg)
1120596
Figure 14 Approximate Pareto front of F4
10 Mathematical Problems in Engineering
= 45 and 120596 = 10349 = 42534 and 120596 = 96302
= 39481 and 120596 = 88068 = 36521 and 120596 = 80402
= 36501 and 120596 = 80223 = 33791 and 120596 = 73022
(1) t (2) t
(4) t
(6) t(5) t
(3) t
Figure 15 Selected design solutions from Figure 14
02 025 03 035 04 045 0509
095
1
105
11
115
12
125
13
135
14
1
2
3
45
6
Stru
ctur
al m
ass (
kg)
1120582
Figure 16 Approximate Pareto front of F5
= 45 and 120582 = 62885 = 45 and 120582 = 53614
= 40881 and 120582 = 42008 = 37402 and 120582 = 31575
= 3652 and 120582 = 30888 = 34047 and 120582 = 22103
(1) t (2) t
(4) t
(6) t
(3) t
(5) t
Figure 17 Selected design solutions from Figure 16
Mathematical Problems in Engineering 11
01 015 02 025 03 03509
095
1
105
11
115
12
125
13
135
14
1
2
3
4
56
Stru
ctur
al m
ass (
kg)
1(120582 + 1205961205960)
Figure 18 Approximate Pareto front of F6
= 45 and 120582 + 1205961205960 = 75573 = 44813 and 120582 + 1205961205960 = 65748
= 41914 and 120582 + 1205961205960 = 53343 = 38581 and 120582 + 1205961205960 = 4254
= 38571 and 120582 + 1205961205960 = 41494 = 34147 and 120582 + 1205961205960 = 30684
(1) t (2) t
(4) t
(6) t
(3) t
(5) t
Figure 19 Selected design solutions from Figure 18
More investigation on this issue would greatly improve theperformance of SPEA-N In cases of large-scale problemsonly SPEA using a quadratic function is superior to thenonsurrogate SPEA
6 Conclusions
The use of multiobjective evolutionary algorithm for thedesign of a torque arm leads to multiple optimum structuresfor decision making The hybridisation of SPEA and severalsurrogate models is developedThe various surrogate modelsincluding the quadratic regression technique the radialbasis function interpolation and the Kriging model canhelp improving the searching performance of the strengthPareto evolutionary algorithm For a design problem havingmass and natural frequency as objective functions and stress
and buckling as constraints SPEA using RBF is the bestmethod The SPEA using a quadratic regression is the bestfor the design case of F2 For the other multiobjective designproblems the best and the second best performers are SPEAusing the Krigingmodel and the quadratic RSM respectivelyNevertheless when considering all of the design problems theoverall top performer is SPEA2 using the quadratic regressionwhile the close second best uses theKrigingmodelMoreoverSPEA using a quadratic response surface model is the onlystrategy that is useful for solving large-scale problems Theuse of ANN for approximating constrained optimisationproblems is not effective as a proper network topologyoptimisation needs to be constructed before entering thesurrogate subproblem It is however possible to enhancethe performance of the ANN-based optimiser if an efficientnetwork topology optimisation is performed during theoptimisation process
12 Mathematical Problems in Engineering
20 25 30 35 40 45 50 55 60 65
F7
SPEA-OSPEA-Q
SPEA-RSPEA-K
f1
f2
minus60
minus50
minus40
minus30
minus20
minus10
0
Figure 20 Approximate Pareto front of F7
20 25 30 35 40 45 50 55 60
F8
SPEA-OSPEA-Q
SPEA-RSPEA-K
minus1000
minus950
minus900
minus850
minus800
minus750
minus700
minus650
minus600
minus550
f2
f1
Figure 21 Approximate Pareto front of F8
Acknowledgment
The authors are graceful for the support from the ThailandResearch Fund (RSA5280006)The first author also wishes tothank the office of higher education commission Thailandfor the CHE-PHD scholarship
References
[1] F Zong H Lin B Yu and X Pan ldquoDaily commute timeprediction based on genetic algorithmrdquoMathematical Problemsin Engineering vol 2012 Article ID 321574 20 pages 2012
[2] I Yao and D Han ldquoImproved barebones particle swarmoptimization with neighborhood search and its application on
ship designrdquo Mathematical Problems in Engineering vol 2013Article ID 175848 12 pages 2013
[3] Y Xu P Fan and L Yuan ldquoA simple and efficient artificial beecolony algorithmrdquo Mathematical Problems in Engineering vol2013 Article ID 526315 9 pages 2013
[4] S Talatahari E Khalili and S M Alavizadeh ldquoAcceleratedparticle swarm for optimum design of frame structuresrdquoMath-ematical Problems in Engineering vol 2013 Article ID 6498576 pages 2013
[5] K Socha and M Dorigo ldquoAnt colony optimization for contin-uous domainsrdquo European Journal of Operational Research vol185 no 3 pp 1155ndash1173 2008
[6] N Hansen S D Muller and P Koumoutsakos ldquoReducingthe time complexity of the derandomized evolution strategywith covariance matrix adaptation (CMA-ES)rdquo EvolutionaryComputation vol 11 no 1 pp 1ndash18 2003
[7] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[8] J D Knowles and D W Corne ldquoApproximating the nondom-inated front using the Pareto Archived Evolution StrategyrdquoEvolutionary Computation vol 8 no 2 pp 149ndash172 2000
[9] M Farina and P Amato ldquoLinked interpolation-optimizationstrategies for multicriteria optimization problemsrdquo Soft Com-puting vol 9 pp 54ndash65 2005
[10] J Knowles ldquoParEGO a hybrid algorithm with on-line land-scape approximation for expensive multiobjective optimizationproblemsrdquo IEEE Transactions on Evolutionary Computationvol 10 no 1 pp 50ndash66 2006
[11] Z Zhou Y S Ong M H Lim and B S Lee ldquoMemetic algo-rithm using multi-surrogates for computationally expensiveoptimization problemsrdquo Soft Computing vol 11 no 10 pp 957ndash971 2007
[12] R G Regis and C A Shoemaker ldquoLocal function approxima-tion in evolutionary algorithms for the optimization of costlyfunctionsrdquo IEEE Transactions on Evolutionary Computationvol 8 no 5 pp 490ndash505 2004
[13] N Pholdee and S Bureerat ldquoSurrogate-assisted evolutionaryoptimizers for multiobjective design of a torque arm structurerdquoAppliedMechanics andMaterials vol 101-102 pp 324ndash328 2011
[14] N Pholdee and S Bureerat ldquoPassive vibration control ofan automotive component using evolutionary optimisationrdquoJournal of Research and Applications in Mechanical Engineeringvol 1 pp 19ndash23 2011
[15] K C Giannakoglou D I Papadimitriou and I C KampolisldquoAerodynamic shape design using evolutionary algorithmsand new gradient-assisted metamodelsrdquo Computer Methods inApplied Mechanics and Engineering vol 195 no 44ndash47 pp6312ndash6329 2006
[16] Y S Ong K Y Lum P B Nair D M Shi and Z K ZhangldquoGlobal convergence of unconstrained and bound constrainedsurrogate-assisted evolutionary search in aerodynamic shapedesignrdquo in Proceedings of the Congress on Evolutionary Compu-tation (CEC rsquo03) pp 1856ndash1863 Canberra Australia December2003
[17] Y Tenne and S W Armfield ldquoA versatile surrogate-assistedmemetic algorithm for optimization of computationally expen-sive functions and its engineering applicationsrdquo Studies inComputational Intelligence vol 92 pp 43ndash72 2008
Mathematical Problems in Engineering 13
[18] S Srisomporn and S Bureerat ldquoGeometrical design of plate-fin heat sinks using hybridization of MOEA and RSMrdquo IEEETransactions onComponents and Packaging Technologies vol 31no 2 pp 351ndash360 2008
[19] S Kanyakam and S Bureerat ldquoComparative performance ofsurrogate-assisted MOEAs for geometrical design of pin-finheat sinksrdquo Applied Mathematics vol 2012 Article ID 53478314 pages 2012
[20] A Husain and K Y Kin ldquoOptimization of a microchannel heatsink with temperature dependent fluid propertiesrdquo Journal ofMaterials Processing Technology vol 208 no 1ndash3 pp 499ndash5062008
[21] J Yoo and C-Y Lee ldquoTopology optimization of a swing armtype actuator using the response surface methodrdquoMicrosystemTechnologies vol 13 no 1 pp 21ndash31 2007
[22] W Hu L G Yao and Z Z Hua ldquoOptimization of sheetmetal forming processes by adaptive response surface basedon intelligent sampling methodrdquo Journal of Materials ProcessingTechnology vol 197 no 1ndash3 pp 77ndash88 2008
[23] N Nariman-Zadeh A Darvizeh A Jamali and A MoeinildquoEvolutionary design of generalized polynomial neural net-works for modelling and prediction of explosive formingprocessrdquo Journal of Materials Processing Technology vol 164-165 pp 1561ndash1571 2005
[24] T Goel R Vaidyanathan R T Haftka W Shyy N V Queipoand K Tucker ldquoResponse surface approximation of Pareto opti-mal front in multi-objective optimizationrdquo Computer MethodsinAppliedMechanics and Engineering vol 196 no 4ndash6 pp 879ndash893 2007
[25] Z Zhou Y S Ong P B Nair A J Keane and K Y LumldquoCombining global and local surrogate models to accelerateevolutionary optimizationrdquo IEEE Transactions on Systems Manand Cybernetics Part C vol 37 no 1 pp 66ndash76 2007
[26] Y Jin ldquoA comprehensive survey of fitness approximation inevolutionary computationrdquo Soft Computing vol 9 pp 3ndash122005
[27] Y S Ong P B Nair A J Keane and K W Wong ldquoSurrogate-assisted evolutionary optimization frameworks for high-fidelityengineering design problemsrdquo in Knowledge Incorporation inEvolutionary Computation Studies in Fuzziness and Soft Com-puting Y Jin Ed pp 307ndash332 Springer Berlin Germany 2004
[28] Y Jin ldquoSurrogate-assisted evolutionary computation recentadvances and future challengesrdquo Swarm and Evolutionary Com-putation vol 1 pp 61ndash70 2011
[29] A I J Forrester and A J Keane ldquoRecent advances in surrogate-based optimizationrdquo Progress in Aerospace Sciences vol 45 no1ndash3 pp 50ndash79 2009
[30] K-T Chiang and F-P Chang ldquoApplication of response surfacemethodology in the parametric optimization of a pin-fin typeheat sinkrdquo International Communications in Heat and MassTransfer vol 33 no 7 pp 836ndash845 2006
[31] Y Tenne and S W Armfield ldquoA framework for memeticoptimization using variable global and local surrogate modelsrdquoSoft Computing vol 13 no 8-9 pp 781ndash793 2009
[32] P Breitkopf H Naceur A Rassineux and P Villon ldquoMovingleast squares response surface approximation formulation andmetal forming applicationsrdquo Computers and Structures vol 83no 17-18 pp 1411ndash1428 2005
[33] M Xiao B Raghavan P Breitkopf R F Coelho C Knopf-Lenoir and P Villon ldquoSurrogate models based on constrainedPOD for optimizationrdquo in Proceedings of the 4th European
Conference on Computational Mechanics (ECCM rsquo10) ParisFrance May 2010
[34] M Xiao P Breitkopf R F Coelho C Knopf-Lenoir MSidorkiewicz and P Villon ldquoModel reduction by CPOD andKriging application to the shape optimization of an intakeportrdquo Structural andMultidisciplinary Optimization vol 41 no4 pp 555ndash574 2010
[35] B RaghavanMHamdaouiM Xiao P Breitkopf and P VillonldquoA bi-level meta-modeling approach for structural optimizationusing modified POD bases and diffuse approximationrdquo Com-puters and Structures 2013
[36] B Raghavan and P Breitkopf ldquoAsynchronous evolutionaryshape optimization based on high-quality surrogates applica-tion to an air-conditioning ductrdquo Engineering with Computers2012
[37] E A Kontoleontos V G Asouti and K C GiannakoglouldquoAn asynchronous metamodel-assisted memetic algorithm forCFD-based shape optimizationrdquo Engineering Optimization vol44 no 2 pp 157ndash173 2012
[38] H Zhu L Liu T Long and L Peng ldquoA novel algorithmof maximin Latin hypercube design using successive localenumerationrdquo Engineering Optimization vol 44 no 5 pp 551ndash564 2012
[39] S D Georgiou and S Stylianou ldquoBlock-circulant matricesfor constructing optimal Latin hypercube designsrdquo Journal ofStatistical Planning and Inference vol 141 no 5 pp 1933ndash19432011
[40] J M Parr A J Keane A I J Forrester and C M E HoldenldquoInfill sampling criteria for surrogate-based optimization withconstraint handlingrdquo Engineering Optimization vol 44 no 10pp 1147ndash1166 2012
[41] E Zitzler M Laumanns and L Thiele ldquoSPEA2 improvingthe strength pareto evolutionary algorithm for multiobjectiveoptimizationrdquo inEvolutionaryMethods forDesign Optimizationand Control pp 95ndash100 2002
[42] S N Lophaven H B Neilson and J Sondergaard ldquoDACEa MATLAB kriging toolboxrdquo Tech Rep IMM-TR-2002-12Technical University of Denmark 2002
[43] R L Harder and R N Desmarais ldquoInterpolation using surfacesplinesrdquo Journal of Aircraft vol 9 no 2 pp 189ndash191 1972
[44] M E Botkin ldquoShape optimization with buckling and stressconstraintsrdquo AIAA Journal vol 34 no 2 pp 423ndash425 1996
[45] K Deb A Pratap and T Meyarivan ldquoConstrained test prob-lems for multi-objective evolutionary optimizationrdquo in Evolu-tionary Multi-Criterion Optimization (Zurich 2001) vol 1993of Lecture Notes in Computer Science pp 284ndash298 SpringerBerlin Germany 2001
[46] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011
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Mathematical Problems in Engineering
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![Page 4: Research Article Surrogate-Assisted Multiobjective ...downloads.hindawi.com/journals/mpe/2013/695172.pdf · of engineering application, for example, aerodynamics [ ], heat and mass](https://reader034.fdocuments.net/reader034/viewer/2022052003/60164c5514f5a3172c3c2904/html5/thumbnails/4.jpg)
4 Mathematical Problems in Engineering
The selected points forbuilding a surrogate model
Grid centerf2
f1
Figure 2 Selection of sampling points for constructing a surrogatemodel
for 119901 119902 = 1 119873 where 119877 is the correlation functionbetween any two of the 119873 design points and R is thesymmetric correlation matrix size 119873 times 119873 with the unitydiagonal [26] The correlation function used herein is
119877 (x119901 x119902) = exp (minus(x119901 minus xq)119879120579 (x119901 minus xq)) (5)
where 120579119894are the unknown correlation parameters to be
determined by means of the maximum likelihood methodHaving found 120573 and 120579 the Kriging predictor can be achievedas
119910 = f (x)119879120573 + r119879 (x)Rminus1 (y minus F120573) (6)
where F = [f(x1) f(x2) f(x119899)]119879 and r119879(x) = [119877(x x1)119877(x x2) 119877(x x119873)] For more details see [42]
33 Radial Basis Function Interpolation The radial basisfunction interpolation has been used in a wide range of appli-cations such as integration between aerodynamic and finiteelement grids in aeroelastic analysis [43] The use of sucha model for surrogate-assisted evolutionary optimisation issaid to be commonplace [9 18 27]The approximate functioncan be written as
119910 (x) =119873
sum
119894=1
120572119894119870(
10038171003817100381710038171003817xminusx11989410038171003817100381710038171003817) (7)
where 120572119894are the coefficients to be determined and 119870 is a
radial basis kernel (Here it is set to be linear splines) Thecoefficients can be found from the 119873 sampling points asdetailed in Srisomporn and Bureerat 2008 [18]
34 Neural Network Artificial neural network (ANN) isdeveloped in response to the need to use complicated func-tion expression rather than a simple quadratic or radial basisfunction The model consists of interconnecting neurons ornodes that somewhat mimic the constructs of biologicalneurons as shown in Figure 3 ANN has been implemented
Table 1 Design variables and bounds for F1ndashF6
Design variables Bounds (mm)1199091= 119897119889
0 le 119897119889le 60
1199092= 119897119901
100 le 119897119901le 200
1199093= 1199031199011
15 le 1199031199011le 35
1199094= 1199031199012
15 le 1199031199012le 30
1199095= th 3 le th le 45
Input layer
Hidden layer
Output layer
Neuron or node
Figure 3 Neural network
on a wide variety of real world applications The use ofANN for surrogate-assisted optimisation can be found inthe literature [9] The neural network model used in thispaper is the feedforward multilayer perceptrons The detailsof the network in this work are the learning algorithm =backpropagation and network training function = Levenberg-Marquardt backpropagation
In this work a simple heuristic algorithm is used toconstruct a network topology The procedure starts with anetwork with two hidden layers while the number of nodes119873119877on each layer is randomly chosen with the bound 119873
119877isin
[15 50] The transfer function for the hidden layer can beeither hyperbolic tangent sigmoid or logarithmic sigmoid basedon randomisation If the termination criterion (training goal)is notmet randomly increase a number of nodes in the layersIf the number of nodes reaches the upper bound 119873
119877= 50
add the third hidden layer to the network where a transferfunction and node number are randomly chosen In case thatthe stopping criterion is not fulfilled increase the number ofnodes The algorithm repeats until the stopping criterion ismet
4 Testing Problems
Six testing multiobjective design problems are posed todesign a torque arm [44] structure shown in Figure 4 Sinceit is a plate-like structure under inplane loading designcriteria to be taken into consideration are weight stressbuckling and natural frequency The function calculationcan be carried out by using FEA where shell elements areemployed The design variables and their bound constraintsare given in Table 1 where other dimensions are set as 119897
119889=
30mm 119897119901= 190mm 119903
1199011= 26mm and 119903
1199012= 27mm
Mathematical Problems in Engineering 5
5066N
2789N
rp2rp1
R2 = 42mmr2 = 25mm
Fixedboundaryconditionsr1 = 40mm
th = platethickness
l0 = 100mm
R1 = 542
L = 420mm
lpld
Figure 4 Torque arm
The six biobjective optimisation problems are termed F1F2 F3 F4 F5 and F6 respectively and can be expressed as
objective functionmin 119891
1(x) max119891
2(x)
F1 1198911(x) = 119908 119891
2(x) = 120596
F2 1198911(x) = 119908 119891
2(x) = 120582
F3 1198911(x) = 119908 119891
1(x) = 120582 + 120596
1205960
subject to
120590von minus120590119910
119878119865
le 0
1 minus 120582 le 0
objective functionmin119891
1(x) max119891
2(x)
F4 1198911(x) = 119908 119891
2(x) = 120596
F5 1198911(x) = 119908 119891
2(x) = 120582
F6 1198911(x) = 119908 119891
2(x) = 120582 + 120596
1205960
subject to
120590von minus120590119910
119878119865
le 0
1 minus 120582 le 0
120596 minus 1205960le 0
(8)
where 119908 = structural weight 120596 = first mode natural fre-quency 120582 = lateral bucking factor (it is the ratio of criticalload to applied load herein) 120590von = maximum von Misesstress on the structure and 120596
0= 72688 rads Furthermore
material properties are Youngrsquos modulus 119864 = 60 times 106Ncm2Poissonrsquos ratio ] = 03 yield stress 120590
119910= 80000Ncm2 density
120588 = 000718 kgcm3 and safety factor 119878119865= 15
In fact the F1 F2 and F3 problems are similar to the F4F5 and F6 problems respectively except for the introductionof natural frequency constraints to the last three problemsThis means that the performance of several surrogate modelsin estimating a natural frequency during an optimisation
procedure can be examinedThe bound constraints in Table 1are not presented in the design problems as they can be dealtwith in the SPEA procedure It should be noted that thefinite element analysis for these design studies is not timeconsuming as the work focuses on a comparative study of theseveral multiobjective optimisers
Since the F1ndashF6 problems are small-scale other two testproblems are posed so as to investigate the capability of thosesurrogate models when dealing with large-scale problemsTwo design problems of a simple wing-box displayed inFigure 5 are expressed as
objective functionmin119891
1(x) max119891
2(x)
F7 1198911(x) = 119908 119891
2(x) = minus120582
F8 1198911(x) = 119908 119891
2(x) = minus (120596
1+ sdot sdot sdot + 120596
5)
subject to
120590von minus 120590119910 le 0
1 minus 120582 le 0
(9)
where 120596119894is the 119894th mode natural frequency of the wing-box
Thewing-box ismade up of Aluminiumwith Youngmodulus119864 = 70GPa Poissonrsquos ratio ] = 034 density 120588 = 2700 kgm3and yield strength 120590
119910119905= 100MPa The structure consists of 2
spars 4 ribs and 4 skins (front rear top and bottom skins)resulting in 52wing segments as shownThe thickness of eachsegment is assigned to be a design variable therefore thereare 52 design variableswith the lower andupper bounds being0002m and 0010m respectively This can be classified as alarge-scale sizing optimisation problem The wing is subjectto static loads on the front spar as shown in Figure 5 Shellelements are used for finite element analysis similarly to F1ndashF6
The multiobjective optimisers used in this paper arenamed as follows
(i) SPEA-O the original SPEA2without using a surrogatemodel
(ii) SPEA-Q SPEA using a quadratic response surfacemodel
6 Mathematical Problems in Engineering
Fixed BC
2467kN 1645kN1645kN 0822 kN
2m
01m
0125m
025m0125m
Figure 5 Simple wing-box
(iii) SPEA-R SPEAusing the radial basis function interpo-lation
(iv) SPEA-N SPEA using a neural network model(v) SPEA-K SPEA using a Kriging model
Each evolutionary algorithm is employed to solve eachproblem ten runs with 30 generations (50 for F7-F8) and apopulation size of 20 (50 for F7-F8) for SPEA-O For thealgorithms using an approximation model the number ofgenerations and population size used in the main procedureis 15 (25 for F7-F8) and 20 (50 for F7-F8) respectively whileon each generation additional 20 (50 solutions for F7-F8)solutions are taken from optimising a surrogate subproblemIn the surrogate subproblem the number of generationsand population size is set to be 100 and 100 respectivelyThe external Pareto archive size is set to be 100 Duringthe optimisation process constraints are handled by usingthe approach presented in [45] It should be noted thatthere have been a number of techniques used to handledesign constraints in evolutionary optimisation [46] Wechoose the aforementioned technique because it is simple andsufficiently effective
5 Results and Discussion
The performance comparison of the five MOEAs for solvingthe eight design problems is based on the hypervolume indi-cator Two types of performance assessment are conducted asdetailed in Tables 2ndash4 In Table 2 each cell is the normalisedmean value of 10 nondominated front hypervolumes obtainedfrom using the optimiser on the corresponding column forsolving the problem on the corresponding row For eachdesign problem the best and worst mean values are nor-malised to be ldquo1rdquo and ldquo0rdquo respectively The total normalisedmean values given in the table are used for evaluating theoverall performance of the surrogate-assisted SPEAs Fromthe results the best algorithm for F1 is SPEA-R while thesecond best is SPEA-Q SPEA-Q is the best for F2 while thesecond best is SPEA-NThe best optimisers for F3 F4 F5 andF6 are SPEA-K SPEA-R SPEA-K and SPEA-K respectivelywhereas the second best for those 4 design problems is SPEA-Q SPEA-O without using a surrogate model is the worstperformer for F2 F5 and F6 while SPEA-N is the worstperformer for F1 F3 and F4 The overall best performer
Table 2 Comparison of normalised average hypervolume
SPEA-O SPEA-Q SPEA-R SPEA-N SPEA-KF1 0429 0704 1000 0000 0523F2 0000 1000 0052 0917 0692F3 0577 0995 0745 0000 1000F4 0457 0916 1000 0000 0497F5 0000 0976 0640 0161 1000F6 0000 0673 0243 0007 1000sum F1ndashF6 1463 5264 3679 1084 4713F7 0811 1000 0724 NA 0000F8 0337 1000 0453 NA 0000sum F7-F8 1148 2000 1177 NA 0000
Table 3 Performance matrix of F1
SPEA-O SPEA-Q SPEA-R SPEA-N SPEA-KSPEA-O 0 0 1 0 0SPEA-Q 0 0 1 0 0SPEA-R 0 0 0 0 0SPEA-N 1 1 1 0 1SPEA-K 0 0 1 0 0Total 1 1 4 0 1Ranking 2 2 1 5 2
Table 4 Comparison of ranking scores by t-test
SPEA-O SPEA-Q SPEA-R SPEA-N SPEA-KF1 2 2 1 5 2F2 4 1 4 1 1F3 1 1 1 5 1F4 3 1 1 5 3F5 4 1 3 4 1F6 3 2 3 3 1sum F1ndashF6 17 8 13 23 9F7 1 1 1 NA 4F8 2 1 2 NA 4sum F7-F8 3 1 3 NA 8
based on the evaluation is SPEA-Q whereas the second bestis SPEA-K For the large-scale problems only SPEA-Q cansurpass SPEA-O SPEA-R is said to be equal to SPEA-OwhileSPEA-K gives that worst performance Note that SPEA-Ncannot be applied to these problems as it takes excessivelylong running time for network training It can be said thatonly the quadratic response surface model is useful for thelarge-scale wing-box design
The second performance assessment is based on thestatistical t-test For each design problem to compare the 5MOEAs a performance matrix T sized 5 times 5 whose elementsare full of ldquo0rdquo is generated For the element 119879
119894119895of the matrix
if the mean value of the 10 hypervolumes obtained frommethod 119895 is significantly larger than that obtained frommethod 119894 based on the statistical t-test at 95 confidencelevel the value of 119879
119894119895is set to be one Table 3 shows the
performance comparison of the problem F1 Having had theperformance matrix the 6th row on the table displays thetotal scores of each optimiser The ranking is made in such
Mathematical Problems in Engineering 7
0 500 1000 1500 2000 25000
2
4
6
8
10
12
14 F7
No of function evaluations
Hyp
ervo
lum
e
SPEA-OSPEA-Q
Figure 6 SPEA-O versus SPEA-Q for F7
a way that the best method is ranked as 1 while the worst isranked as 5The overall performance assessment based on thet-test is given in Table 4 where the last row on the table showsthe total score of all theMOEAsThe lower the score the betterthe performer Similar to the first assessment the overall bestmethod for F1ndashF6 based on this assessment is SPEA-Q whilethe close secondbest is SPEA-KTheworstmethod is SPEA-Owithout using a surrogate model For the large-scale testproblems the only surrogateassisted method that is superiorto the non-surrogate SPEA is SPEA-Q
Figures 6 and 7 show the search history as plots ofhypervolume against the number of function evaluations ofSPEA-O and SPEA-Q for solving F7 and F8 respectivelyThevalues of front hypervolume can fluctuate due to the use of aPareto archiving technique to remove some nondominatedsolutions from the Pareto archive From the figures it canbe seen that SPEA-Q requires approximately half of the totalnumber of function evaluations employed by SPEA-O to haveequally good nondominated fronts
The best approximate Pareto front of F1 obtained afternumerous optimisation runs of the surrogate-assisted SPEAsis chosen and plotted in Figure 8 whereas the structures ofsome selected design points are shown in Figure 9 Figure 10displays the best front of the F2 problemwhere some selecteddesign solutions are illustrated in Figure 11The best front andsome selected optimal structures of the design problemF3 aredisplayed in Figures 12 and 13 respectively Similarly the bestapproximate fronts and some selected design solutions of F4F5 and F6 are illustrated in Figures 14 15 16 17 18 and 19Figures 20 and 21 show the best fronts obtained from usingthe various SPEAs for solving F7 and F8 respectively It canbe seen that the best fronts are from SPEA-Q and SPEA-OThe SPEA-Q front does not totally dominate that of SPEA-O but they overlap each other while the SPEA-Q front hasobviously greater front extension
According to the aforementioned performance assess-ments it can be said that SPEA-R is the best algorithm in
0 500 1000 1500 2000 25000
2
4
6
8
10
12F8
No of function evaluations
Hyp
ervo
lum
e
SPEA-OSPEA-Q
Figure 7 SPEA-O versus SPEA-Q for F8
009 01 011 012 013 014 015 01608
085
09
095
1
105
11
115
12
1
2
3
4
5
6
Stru
ctur
al m
ass (
kg)
1120596
Figure 8 Approximate Pareto front of F1
the cases of the F1 design problem while the close secondbest is SPEA-Q When adding a natural frequency constraintto F1 becoming the F4 problem the overall best and secondbest methods are still SPEA-R and SPEA-Q respectivelyFor the F2 case and the F5 design problem which is F2with additional natural frequency constraint the top twoperformers are SPEA-Q and SPEA-K The same can be saidfor the case of F3 while SPEA-K outperforms the others forthe F6 design problem The surrogate models that is thequadratic response surface model the radial basis functioninterpolation and the Kriging model are said to enhance thesearching performance of SPEA for the small-scale problemsThe addition of constraints to the design problems can causethe searching performance of the hybrid algorithms TheSPEA using ANN is inferior to the other surrogate-assistedmethods or even the original SPEA2 without using a sur-rogate model because it requires efficient network topologyoptimisation before being used to estimate functions whilein this paper the simple heuristic approach is employed
8 Mathematical Problems in Engineering
= 37526 and 120596 = 83234= 45 and 120596 = 10354
= 33758 and 120596 = 73085 = 33779 and 120596 = 73032
= 33613 and 120596 = 72929 t = 30008 and 120596 = 63286
(1) t (2) t
(3) t (4) t
(5) t (6)
Figure 9 Selected design points from Figure 8
02 025 03 035 04 045 05 055 06 06508
09
1
11
12
13
14
15
1
2
3
4 5
6
Stru
ctur
al m
ass (
kg)
1120582
Figure 10 Approximate Pareto front of F2
= 45 and 120582 = 63261 = 39843 and 120582 = 3823
= 35599 and 120582 = 25781 = 35983 and 120582 = 25739
= 35555 and 120582 = 25201 = 30381 and 120582 = 16535
(1) t
(3) t
(5) t (6) t
(4) t
(2) t
Figure 11 Selected design solutions from Figure 10
Mathematical Problems in Engineering 9
01 015 02 025 03 035 04 045 0508
09
1
11
12
13
14
15
1
2
3
45
6
Stru
ctur
al m
ass (
kg)
1(120582 + 1205961205960)
Figure 12 Approximate Pareto front of F3
= 43963 and 120582 + 1205961205960 = 64596= 45 and 120582 + 1205961205960 = 75573
= 41201 and 120582 + 1205961205960 = 49251 = 35884 and 120582 + 1205961205960 = 35891
= 35532 and 120582 + 1205961205960 = 34666 = 3 and 120582 + 1205961205960 = 24581
(1) t (2) t
(4) t(3) t
(5) t (6) t
Figure 13 Selected design solutions from Figure 12
0095 01 0105 011 0115 012 0125 013 0135 014 0145085
09
095
1
105
11
115
12
1
2
3
4
56
Stru
ctur
al m
ass (
kg)
1120596
Figure 14 Approximate Pareto front of F4
10 Mathematical Problems in Engineering
= 45 and 120596 = 10349 = 42534 and 120596 = 96302
= 39481 and 120596 = 88068 = 36521 and 120596 = 80402
= 36501 and 120596 = 80223 = 33791 and 120596 = 73022
(1) t (2) t
(4) t
(6) t(5) t
(3) t
Figure 15 Selected design solutions from Figure 14
02 025 03 035 04 045 0509
095
1
105
11
115
12
125
13
135
14
1
2
3
45
6
Stru
ctur
al m
ass (
kg)
1120582
Figure 16 Approximate Pareto front of F5
= 45 and 120582 = 62885 = 45 and 120582 = 53614
= 40881 and 120582 = 42008 = 37402 and 120582 = 31575
= 3652 and 120582 = 30888 = 34047 and 120582 = 22103
(1) t (2) t
(4) t
(6) t
(3) t
(5) t
Figure 17 Selected design solutions from Figure 16
Mathematical Problems in Engineering 11
01 015 02 025 03 03509
095
1
105
11
115
12
125
13
135
14
1
2
3
4
56
Stru
ctur
al m
ass (
kg)
1(120582 + 1205961205960)
Figure 18 Approximate Pareto front of F6
= 45 and 120582 + 1205961205960 = 75573 = 44813 and 120582 + 1205961205960 = 65748
= 41914 and 120582 + 1205961205960 = 53343 = 38581 and 120582 + 1205961205960 = 4254
= 38571 and 120582 + 1205961205960 = 41494 = 34147 and 120582 + 1205961205960 = 30684
(1) t (2) t
(4) t
(6) t
(3) t
(5) t
Figure 19 Selected design solutions from Figure 18
More investigation on this issue would greatly improve theperformance of SPEA-N In cases of large-scale problemsonly SPEA using a quadratic function is superior to thenonsurrogate SPEA
6 Conclusions
The use of multiobjective evolutionary algorithm for thedesign of a torque arm leads to multiple optimum structuresfor decision making The hybridisation of SPEA and severalsurrogate models is developedThe various surrogate modelsincluding the quadratic regression technique the radialbasis function interpolation and the Kriging model canhelp improving the searching performance of the strengthPareto evolutionary algorithm For a design problem havingmass and natural frequency as objective functions and stress
and buckling as constraints SPEA using RBF is the bestmethod The SPEA using a quadratic regression is the bestfor the design case of F2 For the other multiobjective designproblems the best and the second best performers are SPEAusing the Krigingmodel and the quadratic RSM respectivelyNevertheless when considering all of the design problems theoverall top performer is SPEA2 using the quadratic regressionwhile the close second best uses theKrigingmodelMoreoverSPEA using a quadratic response surface model is the onlystrategy that is useful for solving large-scale problems Theuse of ANN for approximating constrained optimisationproblems is not effective as a proper network topologyoptimisation needs to be constructed before entering thesurrogate subproblem It is however possible to enhancethe performance of the ANN-based optimiser if an efficientnetwork topology optimisation is performed during theoptimisation process
12 Mathematical Problems in Engineering
20 25 30 35 40 45 50 55 60 65
F7
SPEA-OSPEA-Q
SPEA-RSPEA-K
f1
f2
minus60
minus50
minus40
minus30
minus20
minus10
0
Figure 20 Approximate Pareto front of F7
20 25 30 35 40 45 50 55 60
F8
SPEA-OSPEA-Q
SPEA-RSPEA-K
minus1000
minus950
minus900
minus850
minus800
minus750
minus700
minus650
minus600
minus550
f2
f1
Figure 21 Approximate Pareto front of F8
Acknowledgment
The authors are graceful for the support from the ThailandResearch Fund (RSA5280006)The first author also wishes tothank the office of higher education commission Thailandfor the CHE-PHD scholarship
References
[1] F Zong H Lin B Yu and X Pan ldquoDaily commute timeprediction based on genetic algorithmrdquoMathematical Problemsin Engineering vol 2012 Article ID 321574 20 pages 2012
[2] I Yao and D Han ldquoImproved barebones particle swarmoptimization with neighborhood search and its application on
ship designrdquo Mathematical Problems in Engineering vol 2013Article ID 175848 12 pages 2013
[3] Y Xu P Fan and L Yuan ldquoA simple and efficient artificial beecolony algorithmrdquo Mathematical Problems in Engineering vol2013 Article ID 526315 9 pages 2013
[4] S Talatahari E Khalili and S M Alavizadeh ldquoAcceleratedparticle swarm for optimum design of frame structuresrdquoMath-ematical Problems in Engineering vol 2013 Article ID 6498576 pages 2013
[5] K Socha and M Dorigo ldquoAnt colony optimization for contin-uous domainsrdquo European Journal of Operational Research vol185 no 3 pp 1155ndash1173 2008
[6] N Hansen S D Muller and P Koumoutsakos ldquoReducingthe time complexity of the derandomized evolution strategywith covariance matrix adaptation (CMA-ES)rdquo EvolutionaryComputation vol 11 no 1 pp 1ndash18 2003
[7] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[8] J D Knowles and D W Corne ldquoApproximating the nondom-inated front using the Pareto Archived Evolution StrategyrdquoEvolutionary Computation vol 8 no 2 pp 149ndash172 2000
[9] M Farina and P Amato ldquoLinked interpolation-optimizationstrategies for multicriteria optimization problemsrdquo Soft Com-puting vol 9 pp 54ndash65 2005
[10] J Knowles ldquoParEGO a hybrid algorithm with on-line land-scape approximation for expensive multiobjective optimizationproblemsrdquo IEEE Transactions on Evolutionary Computationvol 10 no 1 pp 50ndash66 2006
[11] Z Zhou Y S Ong M H Lim and B S Lee ldquoMemetic algo-rithm using multi-surrogates for computationally expensiveoptimization problemsrdquo Soft Computing vol 11 no 10 pp 957ndash971 2007
[12] R G Regis and C A Shoemaker ldquoLocal function approxima-tion in evolutionary algorithms for the optimization of costlyfunctionsrdquo IEEE Transactions on Evolutionary Computationvol 8 no 5 pp 490ndash505 2004
[13] N Pholdee and S Bureerat ldquoSurrogate-assisted evolutionaryoptimizers for multiobjective design of a torque arm structurerdquoAppliedMechanics andMaterials vol 101-102 pp 324ndash328 2011
[14] N Pholdee and S Bureerat ldquoPassive vibration control ofan automotive component using evolutionary optimisationrdquoJournal of Research and Applications in Mechanical Engineeringvol 1 pp 19ndash23 2011
[15] K C Giannakoglou D I Papadimitriou and I C KampolisldquoAerodynamic shape design using evolutionary algorithmsand new gradient-assisted metamodelsrdquo Computer Methods inApplied Mechanics and Engineering vol 195 no 44ndash47 pp6312ndash6329 2006
[16] Y S Ong K Y Lum P B Nair D M Shi and Z K ZhangldquoGlobal convergence of unconstrained and bound constrainedsurrogate-assisted evolutionary search in aerodynamic shapedesignrdquo in Proceedings of the Congress on Evolutionary Compu-tation (CEC rsquo03) pp 1856ndash1863 Canberra Australia December2003
[17] Y Tenne and S W Armfield ldquoA versatile surrogate-assistedmemetic algorithm for optimization of computationally expen-sive functions and its engineering applicationsrdquo Studies inComputational Intelligence vol 92 pp 43ndash72 2008
Mathematical Problems in Engineering 13
[18] S Srisomporn and S Bureerat ldquoGeometrical design of plate-fin heat sinks using hybridization of MOEA and RSMrdquo IEEETransactions onComponents and Packaging Technologies vol 31no 2 pp 351ndash360 2008
[19] S Kanyakam and S Bureerat ldquoComparative performance ofsurrogate-assisted MOEAs for geometrical design of pin-finheat sinksrdquo Applied Mathematics vol 2012 Article ID 53478314 pages 2012
[20] A Husain and K Y Kin ldquoOptimization of a microchannel heatsink with temperature dependent fluid propertiesrdquo Journal ofMaterials Processing Technology vol 208 no 1ndash3 pp 499ndash5062008
[21] J Yoo and C-Y Lee ldquoTopology optimization of a swing armtype actuator using the response surface methodrdquoMicrosystemTechnologies vol 13 no 1 pp 21ndash31 2007
[22] W Hu L G Yao and Z Z Hua ldquoOptimization of sheetmetal forming processes by adaptive response surface basedon intelligent sampling methodrdquo Journal of Materials ProcessingTechnology vol 197 no 1ndash3 pp 77ndash88 2008
[23] N Nariman-Zadeh A Darvizeh A Jamali and A MoeinildquoEvolutionary design of generalized polynomial neural net-works for modelling and prediction of explosive formingprocessrdquo Journal of Materials Processing Technology vol 164-165 pp 1561ndash1571 2005
[24] T Goel R Vaidyanathan R T Haftka W Shyy N V Queipoand K Tucker ldquoResponse surface approximation of Pareto opti-mal front in multi-objective optimizationrdquo Computer MethodsinAppliedMechanics and Engineering vol 196 no 4ndash6 pp 879ndash893 2007
[25] Z Zhou Y S Ong P B Nair A J Keane and K Y LumldquoCombining global and local surrogate models to accelerateevolutionary optimizationrdquo IEEE Transactions on Systems Manand Cybernetics Part C vol 37 no 1 pp 66ndash76 2007
[26] Y Jin ldquoA comprehensive survey of fitness approximation inevolutionary computationrdquo Soft Computing vol 9 pp 3ndash122005
[27] Y S Ong P B Nair A J Keane and K W Wong ldquoSurrogate-assisted evolutionary optimization frameworks for high-fidelityengineering design problemsrdquo in Knowledge Incorporation inEvolutionary Computation Studies in Fuzziness and Soft Com-puting Y Jin Ed pp 307ndash332 Springer Berlin Germany 2004
[28] Y Jin ldquoSurrogate-assisted evolutionary computation recentadvances and future challengesrdquo Swarm and Evolutionary Com-putation vol 1 pp 61ndash70 2011
[29] A I J Forrester and A J Keane ldquoRecent advances in surrogate-based optimizationrdquo Progress in Aerospace Sciences vol 45 no1ndash3 pp 50ndash79 2009
[30] K-T Chiang and F-P Chang ldquoApplication of response surfacemethodology in the parametric optimization of a pin-fin typeheat sinkrdquo International Communications in Heat and MassTransfer vol 33 no 7 pp 836ndash845 2006
[31] Y Tenne and S W Armfield ldquoA framework for memeticoptimization using variable global and local surrogate modelsrdquoSoft Computing vol 13 no 8-9 pp 781ndash793 2009
[32] P Breitkopf H Naceur A Rassineux and P Villon ldquoMovingleast squares response surface approximation formulation andmetal forming applicationsrdquo Computers and Structures vol 83no 17-18 pp 1411ndash1428 2005
[33] M Xiao B Raghavan P Breitkopf R F Coelho C Knopf-Lenoir and P Villon ldquoSurrogate models based on constrainedPOD for optimizationrdquo in Proceedings of the 4th European
Conference on Computational Mechanics (ECCM rsquo10) ParisFrance May 2010
[34] M Xiao P Breitkopf R F Coelho C Knopf-Lenoir MSidorkiewicz and P Villon ldquoModel reduction by CPOD andKriging application to the shape optimization of an intakeportrdquo Structural andMultidisciplinary Optimization vol 41 no4 pp 555ndash574 2010
[35] B RaghavanMHamdaouiM Xiao P Breitkopf and P VillonldquoA bi-level meta-modeling approach for structural optimizationusing modified POD bases and diffuse approximationrdquo Com-puters and Structures 2013
[36] B Raghavan and P Breitkopf ldquoAsynchronous evolutionaryshape optimization based on high-quality surrogates applica-tion to an air-conditioning ductrdquo Engineering with Computers2012
[37] E A Kontoleontos V G Asouti and K C GiannakoglouldquoAn asynchronous metamodel-assisted memetic algorithm forCFD-based shape optimizationrdquo Engineering Optimization vol44 no 2 pp 157ndash173 2012
[38] H Zhu L Liu T Long and L Peng ldquoA novel algorithmof maximin Latin hypercube design using successive localenumerationrdquo Engineering Optimization vol 44 no 5 pp 551ndash564 2012
[39] S D Georgiou and S Stylianou ldquoBlock-circulant matricesfor constructing optimal Latin hypercube designsrdquo Journal ofStatistical Planning and Inference vol 141 no 5 pp 1933ndash19432011
[40] J M Parr A J Keane A I J Forrester and C M E HoldenldquoInfill sampling criteria for surrogate-based optimization withconstraint handlingrdquo Engineering Optimization vol 44 no 10pp 1147ndash1166 2012
[41] E Zitzler M Laumanns and L Thiele ldquoSPEA2 improvingthe strength pareto evolutionary algorithm for multiobjectiveoptimizationrdquo inEvolutionaryMethods forDesign Optimizationand Control pp 95ndash100 2002
[42] S N Lophaven H B Neilson and J Sondergaard ldquoDACEa MATLAB kriging toolboxrdquo Tech Rep IMM-TR-2002-12Technical University of Denmark 2002
[43] R L Harder and R N Desmarais ldquoInterpolation using surfacesplinesrdquo Journal of Aircraft vol 9 no 2 pp 189ndash191 1972
[44] M E Botkin ldquoShape optimization with buckling and stressconstraintsrdquo AIAA Journal vol 34 no 2 pp 423ndash425 1996
[45] K Deb A Pratap and T Meyarivan ldquoConstrained test prob-lems for multi-objective evolutionary optimizationrdquo in Evolu-tionary Multi-Criterion Optimization (Zurich 2001) vol 1993of Lecture Notes in Computer Science pp 284ndash298 SpringerBerlin Germany 2001
[46] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Stochastic AnalysisInternational Journal of
![Page 5: Research Article Surrogate-Assisted Multiobjective ...downloads.hindawi.com/journals/mpe/2013/695172.pdf · of engineering application, for example, aerodynamics [ ], heat and mass](https://reader034.fdocuments.net/reader034/viewer/2022052003/60164c5514f5a3172c3c2904/html5/thumbnails/5.jpg)
Mathematical Problems in Engineering 5
5066N
2789N
rp2rp1
R2 = 42mmr2 = 25mm
Fixedboundaryconditionsr1 = 40mm
th = platethickness
l0 = 100mm
R1 = 542
L = 420mm
lpld
Figure 4 Torque arm
The six biobjective optimisation problems are termed F1F2 F3 F4 F5 and F6 respectively and can be expressed as
objective functionmin 119891
1(x) max119891
2(x)
F1 1198911(x) = 119908 119891
2(x) = 120596
F2 1198911(x) = 119908 119891
2(x) = 120582
F3 1198911(x) = 119908 119891
1(x) = 120582 + 120596
1205960
subject to
120590von minus120590119910
119878119865
le 0
1 minus 120582 le 0
objective functionmin119891
1(x) max119891
2(x)
F4 1198911(x) = 119908 119891
2(x) = 120596
F5 1198911(x) = 119908 119891
2(x) = 120582
F6 1198911(x) = 119908 119891
2(x) = 120582 + 120596
1205960
subject to
120590von minus120590119910
119878119865
le 0
1 minus 120582 le 0
120596 minus 1205960le 0
(8)
where 119908 = structural weight 120596 = first mode natural fre-quency 120582 = lateral bucking factor (it is the ratio of criticalload to applied load herein) 120590von = maximum von Misesstress on the structure and 120596
0= 72688 rads Furthermore
material properties are Youngrsquos modulus 119864 = 60 times 106Ncm2Poissonrsquos ratio ] = 03 yield stress 120590
119910= 80000Ncm2 density
120588 = 000718 kgcm3 and safety factor 119878119865= 15
In fact the F1 F2 and F3 problems are similar to the F4F5 and F6 problems respectively except for the introductionof natural frequency constraints to the last three problemsThis means that the performance of several surrogate modelsin estimating a natural frequency during an optimisation
procedure can be examinedThe bound constraints in Table 1are not presented in the design problems as they can be dealtwith in the SPEA procedure It should be noted that thefinite element analysis for these design studies is not timeconsuming as the work focuses on a comparative study of theseveral multiobjective optimisers
Since the F1ndashF6 problems are small-scale other two testproblems are posed so as to investigate the capability of thosesurrogate models when dealing with large-scale problemsTwo design problems of a simple wing-box displayed inFigure 5 are expressed as
objective functionmin119891
1(x) max119891
2(x)
F7 1198911(x) = 119908 119891
2(x) = minus120582
F8 1198911(x) = 119908 119891
2(x) = minus (120596
1+ sdot sdot sdot + 120596
5)
subject to
120590von minus 120590119910 le 0
1 minus 120582 le 0
(9)
where 120596119894is the 119894th mode natural frequency of the wing-box
Thewing-box ismade up of Aluminiumwith Youngmodulus119864 = 70GPa Poissonrsquos ratio ] = 034 density 120588 = 2700 kgm3and yield strength 120590
119910119905= 100MPa The structure consists of 2
spars 4 ribs and 4 skins (front rear top and bottom skins)resulting in 52wing segments as shownThe thickness of eachsegment is assigned to be a design variable therefore thereare 52 design variableswith the lower andupper bounds being0002m and 0010m respectively This can be classified as alarge-scale sizing optimisation problem The wing is subjectto static loads on the front spar as shown in Figure 5 Shellelements are used for finite element analysis similarly to F1ndashF6
The multiobjective optimisers used in this paper arenamed as follows
(i) SPEA-O the original SPEA2without using a surrogatemodel
(ii) SPEA-Q SPEA using a quadratic response surfacemodel
6 Mathematical Problems in Engineering
Fixed BC
2467kN 1645kN1645kN 0822 kN
2m
01m
0125m
025m0125m
Figure 5 Simple wing-box
(iii) SPEA-R SPEAusing the radial basis function interpo-lation
(iv) SPEA-N SPEA using a neural network model(v) SPEA-K SPEA using a Kriging model
Each evolutionary algorithm is employed to solve eachproblem ten runs with 30 generations (50 for F7-F8) and apopulation size of 20 (50 for F7-F8) for SPEA-O For thealgorithms using an approximation model the number ofgenerations and population size used in the main procedureis 15 (25 for F7-F8) and 20 (50 for F7-F8) respectively whileon each generation additional 20 (50 solutions for F7-F8)solutions are taken from optimising a surrogate subproblemIn the surrogate subproblem the number of generationsand population size is set to be 100 and 100 respectivelyThe external Pareto archive size is set to be 100 Duringthe optimisation process constraints are handled by usingthe approach presented in [45] It should be noted thatthere have been a number of techniques used to handledesign constraints in evolutionary optimisation [46] Wechoose the aforementioned technique because it is simple andsufficiently effective
5 Results and Discussion
The performance comparison of the five MOEAs for solvingthe eight design problems is based on the hypervolume indi-cator Two types of performance assessment are conducted asdetailed in Tables 2ndash4 In Table 2 each cell is the normalisedmean value of 10 nondominated front hypervolumes obtainedfrom using the optimiser on the corresponding column forsolving the problem on the corresponding row For eachdesign problem the best and worst mean values are nor-malised to be ldquo1rdquo and ldquo0rdquo respectively The total normalisedmean values given in the table are used for evaluating theoverall performance of the surrogate-assisted SPEAs Fromthe results the best algorithm for F1 is SPEA-R while thesecond best is SPEA-Q SPEA-Q is the best for F2 while thesecond best is SPEA-NThe best optimisers for F3 F4 F5 andF6 are SPEA-K SPEA-R SPEA-K and SPEA-K respectivelywhereas the second best for those 4 design problems is SPEA-Q SPEA-O without using a surrogate model is the worstperformer for F2 F5 and F6 while SPEA-N is the worstperformer for F1 F3 and F4 The overall best performer
Table 2 Comparison of normalised average hypervolume
SPEA-O SPEA-Q SPEA-R SPEA-N SPEA-KF1 0429 0704 1000 0000 0523F2 0000 1000 0052 0917 0692F3 0577 0995 0745 0000 1000F4 0457 0916 1000 0000 0497F5 0000 0976 0640 0161 1000F6 0000 0673 0243 0007 1000sum F1ndashF6 1463 5264 3679 1084 4713F7 0811 1000 0724 NA 0000F8 0337 1000 0453 NA 0000sum F7-F8 1148 2000 1177 NA 0000
Table 3 Performance matrix of F1
SPEA-O SPEA-Q SPEA-R SPEA-N SPEA-KSPEA-O 0 0 1 0 0SPEA-Q 0 0 1 0 0SPEA-R 0 0 0 0 0SPEA-N 1 1 1 0 1SPEA-K 0 0 1 0 0Total 1 1 4 0 1Ranking 2 2 1 5 2
Table 4 Comparison of ranking scores by t-test
SPEA-O SPEA-Q SPEA-R SPEA-N SPEA-KF1 2 2 1 5 2F2 4 1 4 1 1F3 1 1 1 5 1F4 3 1 1 5 3F5 4 1 3 4 1F6 3 2 3 3 1sum F1ndashF6 17 8 13 23 9F7 1 1 1 NA 4F8 2 1 2 NA 4sum F7-F8 3 1 3 NA 8
based on the evaluation is SPEA-Q whereas the second bestis SPEA-K For the large-scale problems only SPEA-Q cansurpass SPEA-O SPEA-R is said to be equal to SPEA-OwhileSPEA-K gives that worst performance Note that SPEA-Ncannot be applied to these problems as it takes excessivelylong running time for network training It can be said thatonly the quadratic response surface model is useful for thelarge-scale wing-box design
The second performance assessment is based on thestatistical t-test For each design problem to compare the 5MOEAs a performance matrix T sized 5 times 5 whose elementsare full of ldquo0rdquo is generated For the element 119879
119894119895of the matrix
if the mean value of the 10 hypervolumes obtained frommethod 119895 is significantly larger than that obtained frommethod 119894 based on the statistical t-test at 95 confidencelevel the value of 119879
119894119895is set to be one Table 3 shows the
performance comparison of the problem F1 Having had theperformance matrix the 6th row on the table displays thetotal scores of each optimiser The ranking is made in such
Mathematical Problems in Engineering 7
0 500 1000 1500 2000 25000
2
4
6
8
10
12
14 F7
No of function evaluations
Hyp
ervo
lum
e
SPEA-OSPEA-Q
Figure 6 SPEA-O versus SPEA-Q for F7
a way that the best method is ranked as 1 while the worst isranked as 5The overall performance assessment based on thet-test is given in Table 4 where the last row on the table showsthe total score of all theMOEAsThe lower the score the betterthe performer Similar to the first assessment the overall bestmethod for F1ndashF6 based on this assessment is SPEA-Q whilethe close secondbest is SPEA-KTheworstmethod is SPEA-Owithout using a surrogate model For the large-scale testproblems the only surrogateassisted method that is superiorto the non-surrogate SPEA is SPEA-Q
Figures 6 and 7 show the search history as plots ofhypervolume against the number of function evaluations ofSPEA-O and SPEA-Q for solving F7 and F8 respectivelyThevalues of front hypervolume can fluctuate due to the use of aPareto archiving technique to remove some nondominatedsolutions from the Pareto archive From the figures it canbe seen that SPEA-Q requires approximately half of the totalnumber of function evaluations employed by SPEA-O to haveequally good nondominated fronts
The best approximate Pareto front of F1 obtained afternumerous optimisation runs of the surrogate-assisted SPEAsis chosen and plotted in Figure 8 whereas the structures ofsome selected design points are shown in Figure 9 Figure 10displays the best front of the F2 problemwhere some selecteddesign solutions are illustrated in Figure 11The best front andsome selected optimal structures of the design problemF3 aredisplayed in Figures 12 and 13 respectively Similarly the bestapproximate fronts and some selected design solutions of F4F5 and F6 are illustrated in Figures 14 15 16 17 18 and 19Figures 20 and 21 show the best fronts obtained from usingthe various SPEAs for solving F7 and F8 respectively It canbe seen that the best fronts are from SPEA-Q and SPEA-OThe SPEA-Q front does not totally dominate that of SPEA-O but they overlap each other while the SPEA-Q front hasobviously greater front extension
According to the aforementioned performance assess-ments it can be said that SPEA-R is the best algorithm in
0 500 1000 1500 2000 25000
2
4
6
8
10
12F8
No of function evaluations
Hyp
ervo
lum
e
SPEA-OSPEA-Q
Figure 7 SPEA-O versus SPEA-Q for F8
009 01 011 012 013 014 015 01608
085
09
095
1
105
11
115
12
1
2
3
4
5
6
Stru
ctur
al m
ass (
kg)
1120596
Figure 8 Approximate Pareto front of F1
the cases of the F1 design problem while the close secondbest is SPEA-Q When adding a natural frequency constraintto F1 becoming the F4 problem the overall best and secondbest methods are still SPEA-R and SPEA-Q respectivelyFor the F2 case and the F5 design problem which is F2with additional natural frequency constraint the top twoperformers are SPEA-Q and SPEA-K The same can be saidfor the case of F3 while SPEA-K outperforms the others forthe F6 design problem The surrogate models that is thequadratic response surface model the radial basis functioninterpolation and the Kriging model are said to enhance thesearching performance of SPEA for the small-scale problemsThe addition of constraints to the design problems can causethe searching performance of the hybrid algorithms TheSPEA using ANN is inferior to the other surrogate-assistedmethods or even the original SPEA2 without using a sur-rogate model because it requires efficient network topologyoptimisation before being used to estimate functions whilein this paper the simple heuristic approach is employed
8 Mathematical Problems in Engineering
= 37526 and 120596 = 83234= 45 and 120596 = 10354
= 33758 and 120596 = 73085 = 33779 and 120596 = 73032
= 33613 and 120596 = 72929 t = 30008 and 120596 = 63286
(1) t (2) t
(3) t (4) t
(5) t (6)
Figure 9 Selected design points from Figure 8
02 025 03 035 04 045 05 055 06 06508
09
1
11
12
13
14
15
1
2
3
4 5
6
Stru
ctur
al m
ass (
kg)
1120582
Figure 10 Approximate Pareto front of F2
= 45 and 120582 = 63261 = 39843 and 120582 = 3823
= 35599 and 120582 = 25781 = 35983 and 120582 = 25739
= 35555 and 120582 = 25201 = 30381 and 120582 = 16535
(1) t
(3) t
(5) t (6) t
(4) t
(2) t
Figure 11 Selected design solutions from Figure 10
Mathematical Problems in Engineering 9
01 015 02 025 03 035 04 045 0508
09
1
11
12
13
14
15
1
2
3
45
6
Stru
ctur
al m
ass (
kg)
1(120582 + 1205961205960)
Figure 12 Approximate Pareto front of F3
= 43963 and 120582 + 1205961205960 = 64596= 45 and 120582 + 1205961205960 = 75573
= 41201 and 120582 + 1205961205960 = 49251 = 35884 and 120582 + 1205961205960 = 35891
= 35532 and 120582 + 1205961205960 = 34666 = 3 and 120582 + 1205961205960 = 24581
(1) t (2) t
(4) t(3) t
(5) t (6) t
Figure 13 Selected design solutions from Figure 12
0095 01 0105 011 0115 012 0125 013 0135 014 0145085
09
095
1
105
11
115
12
1
2
3
4
56
Stru
ctur
al m
ass (
kg)
1120596
Figure 14 Approximate Pareto front of F4
10 Mathematical Problems in Engineering
= 45 and 120596 = 10349 = 42534 and 120596 = 96302
= 39481 and 120596 = 88068 = 36521 and 120596 = 80402
= 36501 and 120596 = 80223 = 33791 and 120596 = 73022
(1) t (2) t
(4) t
(6) t(5) t
(3) t
Figure 15 Selected design solutions from Figure 14
02 025 03 035 04 045 0509
095
1
105
11
115
12
125
13
135
14
1
2
3
45
6
Stru
ctur
al m
ass (
kg)
1120582
Figure 16 Approximate Pareto front of F5
= 45 and 120582 = 62885 = 45 and 120582 = 53614
= 40881 and 120582 = 42008 = 37402 and 120582 = 31575
= 3652 and 120582 = 30888 = 34047 and 120582 = 22103
(1) t (2) t
(4) t
(6) t
(3) t
(5) t
Figure 17 Selected design solutions from Figure 16
Mathematical Problems in Engineering 11
01 015 02 025 03 03509
095
1
105
11
115
12
125
13
135
14
1
2
3
4
56
Stru
ctur
al m
ass (
kg)
1(120582 + 1205961205960)
Figure 18 Approximate Pareto front of F6
= 45 and 120582 + 1205961205960 = 75573 = 44813 and 120582 + 1205961205960 = 65748
= 41914 and 120582 + 1205961205960 = 53343 = 38581 and 120582 + 1205961205960 = 4254
= 38571 and 120582 + 1205961205960 = 41494 = 34147 and 120582 + 1205961205960 = 30684
(1) t (2) t
(4) t
(6) t
(3) t
(5) t
Figure 19 Selected design solutions from Figure 18
More investigation on this issue would greatly improve theperformance of SPEA-N In cases of large-scale problemsonly SPEA using a quadratic function is superior to thenonsurrogate SPEA
6 Conclusions
The use of multiobjective evolutionary algorithm for thedesign of a torque arm leads to multiple optimum structuresfor decision making The hybridisation of SPEA and severalsurrogate models is developedThe various surrogate modelsincluding the quadratic regression technique the radialbasis function interpolation and the Kriging model canhelp improving the searching performance of the strengthPareto evolutionary algorithm For a design problem havingmass and natural frequency as objective functions and stress
and buckling as constraints SPEA using RBF is the bestmethod The SPEA using a quadratic regression is the bestfor the design case of F2 For the other multiobjective designproblems the best and the second best performers are SPEAusing the Krigingmodel and the quadratic RSM respectivelyNevertheless when considering all of the design problems theoverall top performer is SPEA2 using the quadratic regressionwhile the close second best uses theKrigingmodelMoreoverSPEA using a quadratic response surface model is the onlystrategy that is useful for solving large-scale problems Theuse of ANN for approximating constrained optimisationproblems is not effective as a proper network topologyoptimisation needs to be constructed before entering thesurrogate subproblem It is however possible to enhancethe performance of the ANN-based optimiser if an efficientnetwork topology optimisation is performed during theoptimisation process
12 Mathematical Problems in Engineering
20 25 30 35 40 45 50 55 60 65
F7
SPEA-OSPEA-Q
SPEA-RSPEA-K
f1
f2
minus60
minus50
minus40
minus30
minus20
minus10
0
Figure 20 Approximate Pareto front of F7
20 25 30 35 40 45 50 55 60
F8
SPEA-OSPEA-Q
SPEA-RSPEA-K
minus1000
minus950
minus900
minus850
minus800
minus750
minus700
minus650
minus600
minus550
f2
f1
Figure 21 Approximate Pareto front of F8
Acknowledgment
The authors are graceful for the support from the ThailandResearch Fund (RSA5280006)The first author also wishes tothank the office of higher education commission Thailandfor the CHE-PHD scholarship
References
[1] F Zong H Lin B Yu and X Pan ldquoDaily commute timeprediction based on genetic algorithmrdquoMathematical Problemsin Engineering vol 2012 Article ID 321574 20 pages 2012
[2] I Yao and D Han ldquoImproved barebones particle swarmoptimization with neighborhood search and its application on
ship designrdquo Mathematical Problems in Engineering vol 2013Article ID 175848 12 pages 2013
[3] Y Xu P Fan and L Yuan ldquoA simple and efficient artificial beecolony algorithmrdquo Mathematical Problems in Engineering vol2013 Article ID 526315 9 pages 2013
[4] S Talatahari E Khalili and S M Alavizadeh ldquoAcceleratedparticle swarm for optimum design of frame structuresrdquoMath-ematical Problems in Engineering vol 2013 Article ID 6498576 pages 2013
[5] K Socha and M Dorigo ldquoAnt colony optimization for contin-uous domainsrdquo European Journal of Operational Research vol185 no 3 pp 1155ndash1173 2008
[6] N Hansen S D Muller and P Koumoutsakos ldquoReducingthe time complexity of the derandomized evolution strategywith covariance matrix adaptation (CMA-ES)rdquo EvolutionaryComputation vol 11 no 1 pp 1ndash18 2003
[7] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[8] J D Knowles and D W Corne ldquoApproximating the nondom-inated front using the Pareto Archived Evolution StrategyrdquoEvolutionary Computation vol 8 no 2 pp 149ndash172 2000
[9] M Farina and P Amato ldquoLinked interpolation-optimizationstrategies for multicriteria optimization problemsrdquo Soft Com-puting vol 9 pp 54ndash65 2005
[10] J Knowles ldquoParEGO a hybrid algorithm with on-line land-scape approximation for expensive multiobjective optimizationproblemsrdquo IEEE Transactions on Evolutionary Computationvol 10 no 1 pp 50ndash66 2006
[11] Z Zhou Y S Ong M H Lim and B S Lee ldquoMemetic algo-rithm using multi-surrogates for computationally expensiveoptimization problemsrdquo Soft Computing vol 11 no 10 pp 957ndash971 2007
[12] R G Regis and C A Shoemaker ldquoLocal function approxima-tion in evolutionary algorithms for the optimization of costlyfunctionsrdquo IEEE Transactions on Evolutionary Computationvol 8 no 5 pp 490ndash505 2004
[13] N Pholdee and S Bureerat ldquoSurrogate-assisted evolutionaryoptimizers for multiobjective design of a torque arm structurerdquoAppliedMechanics andMaterials vol 101-102 pp 324ndash328 2011
[14] N Pholdee and S Bureerat ldquoPassive vibration control ofan automotive component using evolutionary optimisationrdquoJournal of Research and Applications in Mechanical Engineeringvol 1 pp 19ndash23 2011
[15] K C Giannakoglou D I Papadimitriou and I C KampolisldquoAerodynamic shape design using evolutionary algorithmsand new gradient-assisted metamodelsrdquo Computer Methods inApplied Mechanics and Engineering vol 195 no 44ndash47 pp6312ndash6329 2006
[16] Y S Ong K Y Lum P B Nair D M Shi and Z K ZhangldquoGlobal convergence of unconstrained and bound constrainedsurrogate-assisted evolutionary search in aerodynamic shapedesignrdquo in Proceedings of the Congress on Evolutionary Compu-tation (CEC rsquo03) pp 1856ndash1863 Canberra Australia December2003
[17] Y Tenne and S W Armfield ldquoA versatile surrogate-assistedmemetic algorithm for optimization of computationally expen-sive functions and its engineering applicationsrdquo Studies inComputational Intelligence vol 92 pp 43ndash72 2008
Mathematical Problems in Engineering 13
[18] S Srisomporn and S Bureerat ldquoGeometrical design of plate-fin heat sinks using hybridization of MOEA and RSMrdquo IEEETransactions onComponents and Packaging Technologies vol 31no 2 pp 351ndash360 2008
[19] S Kanyakam and S Bureerat ldquoComparative performance ofsurrogate-assisted MOEAs for geometrical design of pin-finheat sinksrdquo Applied Mathematics vol 2012 Article ID 53478314 pages 2012
[20] A Husain and K Y Kin ldquoOptimization of a microchannel heatsink with temperature dependent fluid propertiesrdquo Journal ofMaterials Processing Technology vol 208 no 1ndash3 pp 499ndash5062008
[21] J Yoo and C-Y Lee ldquoTopology optimization of a swing armtype actuator using the response surface methodrdquoMicrosystemTechnologies vol 13 no 1 pp 21ndash31 2007
[22] W Hu L G Yao and Z Z Hua ldquoOptimization of sheetmetal forming processes by adaptive response surface basedon intelligent sampling methodrdquo Journal of Materials ProcessingTechnology vol 197 no 1ndash3 pp 77ndash88 2008
[23] N Nariman-Zadeh A Darvizeh A Jamali and A MoeinildquoEvolutionary design of generalized polynomial neural net-works for modelling and prediction of explosive formingprocessrdquo Journal of Materials Processing Technology vol 164-165 pp 1561ndash1571 2005
[24] T Goel R Vaidyanathan R T Haftka W Shyy N V Queipoand K Tucker ldquoResponse surface approximation of Pareto opti-mal front in multi-objective optimizationrdquo Computer MethodsinAppliedMechanics and Engineering vol 196 no 4ndash6 pp 879ndash893 2007
[25] Z Zhou Y S Ong P B Nair A J Keane and K Y LumldquoCombining global and local surrogate models to accelerateevolutionary optimizationrdquo IEEE Transactions on Systems Manand Cybernetics Part C vol 37 no 1 pp 66ndash76 2007
[26] Y Jin ldquoA comprehensive survey of fitness approximation inevolutionary computationrdquo Soft Computing vol 9 pp 3ndash122005
[27] Y S Ong P B Nair A J Keane and K W Wong ldquoSurrogate-assisted evolutionary optimization frameworks for high-fidelityengineering design problemsrdquo in Knowledge Incorporation inEvolutionary Computation Studies in Fuzziness and Soft Com-puting Y Jin Ed pp 307ndash332 Springer Berlin Germany 2004
[28] Y Jin ldquoSurrogate-assisted evolutionary computation recentadvances and future challengesrdquo Swarm and Evolutionary Com-putation vol 1 pp 61ndash70 2011
[29] A I J Forrester and A J Keane ldquoRecent advances in surrogate-based optimizationrdquo Progress in Aerospace Sciences vol 45 no1ndash3 pp 50ndash79 2009
[30] K-T Chiang and F-P Chang ldquoApplication of response surfacemethodology in the parametric optimization of a pin-fin typeheat sinkrdquo International Communications in Heat and MassTransfer vol 33 no 7 pp 836ndash845 2006
[31] Y Tenne and S W Armfield ldquoA framework for memeticoptimization using variable global and local surrogate modelsrdquoSoft Computing vol 13 no 8-9 pp 781ndash793 2009
[32] P Breitkopf H Naceur A Rassineux and P Villon ldquoMovingleast squares response surface approximation formulation andmetal forming applicationsrdquo Computers and Structures vol 83no 17-18 pp 1411ndash1428 2005
[33] M Xiao B Raghavan P Breitkopf R F Coelho C Knopf-Lenoir and P Villon ldquoSurrogate models based on constrainedPOD for optimizationrdquo in Proceedings of the 4th European
Conference on Computational Mechanics (ECCM rsquo10) ParisFrance May 2010
[34] M Xiao P Breitkopf R F Coelho C Knopf-Lenoir MSidorkiewicz and P Villon ldquoModel reduction by CPOD andKriging application to the shape optimization of an intakeportrdquo Structural andMultidisciplinary Optimization vol 41 no4 pp 555ndash574 2010
[35] B RaghavanMHamdaouiM Xiao P Breitkopf and P VillonldquoA bi-level meta-modeling approach for structural optimizationusing modified POD bases and diffuse approximationrdquo Com-puters and Structures 2013
[36] B Raghavan and P Breitkopf ldquoAsynchronous evolutionaryshape optimization based on high-quality surrogates applica-tion to an air-conditioning ductrdquo Engineering with Computers2012
[37] E A Kontoleontos V G Asouti and K C GiannakoglouldquoAn asynchronous metamodel-assisted memetic algorithm forCFD-based shape optimizationrdquo Engineering Optimization vol44 no 2 pp 157ndash173 2012
[38] H Zhu L Liu T Long and L Peng ldquoA novel algorithmof maximin Latin hypercube design using successive localenumerationrdquo Engineering Optimization vol 44 no 5 pp 551ndash564 2012
[39] S D Georgiou and S Stylianou ldquoBlock-circulant matricesfor constructing optimal Latin hypercube designsrdquo Journal ofStatistical Planning and Inference vol 141 no 5 pp 1933ndash19432011
[40] J M Parr A J Keane A I J Forrester and C M E HoldenldquoInfill sampling criteria for surrogate-based optimization withconstraint handlingrdquo Engineering Optimization vol 44 no 10pp 1147ndash1166 2012
[41] E Zitzler M Laumanns and L Thiele ldquoSPEA2 improvingthe strength pareto evolutionary algorithm for multiobjectiveoptimizationrdquo inEvolutionaryMethods forDesign Optimizationand Control pp 95ndash100 2002
[42] S N Lophaven H B Neilson and J Sondergaard ldquoDACEa MATLAB kriging toolboxrdquo Tech Rep IMM-TR-2002-12Technical University of Denmark 2002
[43] R L Harder and R N Desmarais ldquoInterpolation using surfacesplinesrdquo Journal of Aircraft vol 9 no 2 pp 189ndash191 1972
[44] M E Botkin ldquoShape optimization with buckling and stressconstraintsrdquo AIAA Journal vol 34 no 2 pp 423ndash425 1996
[45] K Deb A Pratap and T Meyarivan ldquoConstrained test prob-lems for multi-objective evolutionary optimizationrdquo in Evolu-tionary Multi-Criterion Optimization (Zurich 2001) vol 1993of Lecture Notes in Computer Science pp 284ndash298 SpringerBerlin Germany 2001
[46] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Stochastic AnalysisInternational Journal of
![Page 6: Research Article Surrogate-Assisted Multiobjective ...downloads.hindawi.com/journals/mpe/2013/695172.pdf · of engineering application, for example, aerodynamics [ ], heat and mass](https://reader034.fdocuments.net/reader034/viewer/2022052003/60164c5514f5a3172c3c2904/html5/thumbnails/6.jpg)
6 Mathematical Problems in Engineering
Fixed BC
2467kN 1645kN1645kN 0822 kN
2m
01m
0125m
025m0125m
Figure 5 Simple wing-box
(iii) SPEA-R SPEAusing the radial basis function interpo-lation
(iv) SPEA-N SPEA using a neural network model(v) SPEA-K SPEA using a Kriging model
Each evolutionary algorithm is employed to solve eachproblem ten runs with 30 generations (50 for F7-F8) and apopulation size of 20 (50 for F7-F8) for SPEA-O For thealgorithms using an approximation model the number ofgenerations and population size used in the main procedureis 15 (25 for F7-F8) and 20 (50 for F7-F8) respectively whileon each generation additional 20 (50 solutions for F7-F8)solutions are taken from optimising a surrogate subproblemIn the surrogate subproblem the number of generationsand population size is set to be 100 and 100 respectivelyThe external Pareto archive size is set to be 100 Duringthe optimisation process constraints are handled by usingthe approach presented in [45] It should be noted thatthere have been a number of techniques used to handledesign constraints in evolutionary optimisation [46] Wechoose the aforementioned technique because it is simple andsufficiently effective
5 Results and Discussion
The performance comparison of the five MOEAs for solvingthe eight design problems is based on the hypervolume indi-cator Two types of performance assessment are conducted asdetailed in Tables 2ndash4 In Table 2 each cell is the normalisedmean value of 10 nondominated front hypervolumes obtainedfrom using the optimiser on the corresponding column forsolving the problem on the corresponding row For eachdesign problem the best and worst mean values are nor-malised to be ldquo1rdquo and ldquo0rdquo respectively The total normalisedmean values given in the table are used for evaluating theoverall performance of the surrogate-assisted SPEAs Fromthe results the best algorithm for F1 is SPEA-R while thesecond best is SPEA-Q SPEA-Q is the best for F2 while thesecond best is SPEA-NThe best optimisers for F3 F4 F5 andF6 are SPEA-K SPEA-R SPEA-K and SPEA-K respectivelywhereas the second best for those 4 design problems is SPEA-Q SPEA-O without using a surrogate model is the worstperformer for F2 F5 and F6 while SPEA-N is the worstperformer for F1 F3 and F4 The overall best performer
Table 2 Comparison of normalised average hypervolume
SPEA-O SPEA-Q SPEA-R SPEA-N SPEA-KF1 0429 0704 1000 0000 0523F2 0000 1000 0052 0917 0692F3 0577 0995 0745 0000 1000F4 0457 0916 1000 0000 0497F5 0000 0976 0640 0161 1000F6 0000 0673 0243 0007 1000sum F1ndashF6 1463 5264 3679 1084 4713F7 0811 1000 0724 NA 0000F8 0337 1000 0453 NA 0000sum F7-F8 1148 2000 1177 NA 0000
Table 3 Performance matrix of F1
SPEA-O SPEA-Q SPEA-R SPEA-N SPEA-KSPEA-O 0 0 1 0 0SPEA-Q 0 0 1 0 0SPEA-R 0 0 0 0 0SPEA-N 1 1 1 0 1SPEA-K 0 0 1 0 0Total 1 1 4 0 1Ranking 2 2 1 5 2
Table 4 Comparison of ranking scores by t-test
SPEA-O SPEA-Q SPEA-R SPEA-N SPEA-KF1 2 2 1 5 2F2 4 1 4 1 1F3 1 1 1 5 1F4 3 1 1 5 3F5 4 1 3 4 1F6 3 2 3 3 1sum F1ndashF6 17 8 13 23 9F7 1 1 1 NA 4F8 2 1 2 NA 4sum F7-F8 3 1 3 NA 8
based on the evaluation is SPEA-Q whereas the second bestis SPEA-K For the large-scale problems only SPEA-Q cansurpass SPEA-O SPEA-R is said to be equal to SPEA-OwhileSPEA-K gives that worst performance Note that SPEA-Ncannot be applied to these problems as it takes excessivelylong running time for network training It can be said thatonly the quadratic response surface model is useful for thelarge-scale wing-box design
The second performance assessment is based on thestatistical t-test For each design problem to compare the 5MOEAs a performance matrix T sized 5 times 5 whose elementsare full of ldquo0rdquo is generated For the element 119879
119894119895of the matrix
if the mean value of the 10 hypervolumes obtained frommethod 119895 is significantly larger than that obtained frommethod 119894 based on the statistical t-test at 95 confidencelevel the value of 119879
119894119895is set to be one Table 3 shows the
performance comparison of the problem F1 Having had theperformance matrix the 6th row on the table displays thetotal scores of each optimiser The ranking is made in such
Mathematical Problems in Engineering 7
0 500 1000 1500 2000 25000
2
4
6
8
10
12
14 F7
No of function evaluations
Hyp
ervo
lum
e
SPEA-OSPEA-Q
Figure 6 SPEA-O versus SPEA-Q for F7
a way that the best method is ranked as 1 while the worst isranked as 5The overall performance assessment based on thet-test is given in Table 4 where the last row on the table showsthe total score of all theMOEAsThe lower the score the betterthe performer Similar to the first assessment the overall bestmethod for F1ndashF6 based on this assessment is SPEA-Q whilethe close secondbest is SPEA-KTheworstmethod is SPEA-Owithout using a surrogate model For the large-scale testproblems the only surrogateassisted method that is superiorto the non-surrogate SPEA is SPEA-Q
Figures 6 and 7 show the search history as plots ofhypervolume against the number of function evaluations ofSPEA-O and SPEA-Q for solving F7 and F8 respectivelyThevalues of front hypervolume can fluctuate due to the use of aPareto archiving technique to remove some nondominatedsolutions from the Pareto archive From the figures it canbe seen that SPEA-Q requires approximately half of the totalnumber of function evaluations employed by SPEA-O to haveequally good nondominated fronts
The best approximate Pareto front of F1 obtained afternumerous optimisation runs of the surrogate-assisted SPEAsis chosen and plotted in Figure 8 whereas the structures ofsome selected design points are shown in Figure 9 Figure 10displays the best front of the F2 problemwhere some selecteddesign solutions are illustrated in Figure 11The best front andsome selected optimal structures of the design problemF3 aredisplayed in Figures 12 and 13 respectively Similarly the bestapproximate fronts and some selected design solutions of F4F5 and F6 are illustrated in Figures 14 15 16 17 18 and 19Figures 20 and 21 show the best fronts obtained from usingthe various SPEAs for solving F7 and F8 respectively It canbe seen that the best fronts are from SPEA-Q and SPEA-OThe SPEA-Q front does not totally dominate that of SPEA-O but they overlap each other while the SPEA-Q front hasobviously greater front extension
According to the aforementioned performance assess-ments it can be said that SPEA-R is the best algorithm in
0 500 1000 1500 2000 25000
2
4
6
8
10
12F8
No of function evaluations
Hyp
ervo
lum
e
SPEA-OSPEA-Q
Figure 7 SPEA-O versus SPEA-Q for F8
009 01 011 012 013 014 015 01608
085
09
095
1
105
11
115
12
1
2
3
4
5
6
Stru
ctur
al m
ass (
kg)
1120596
Figure 8 Approximate Pareto front of F1
the cases of the F1 design problem while the close secondbest is SPEA-Q When adding a natural frequency constraintto F1 becoming the F4 problem the overall best and secondbest methods are still SPEA-R and SPEA-Q respectivelyFor the F2 case and the F5 design problem which is F2with additional natural frequency constraint the top twoperformers are SPEA-Q and SPEA-K The same can be saidfor the case of F3 while SPEA-K outperforms the others forthe F6 design problem The surrogate models that is thequadratic response surface model the radial basis functioninterpolation and the Kriging model are said to enhance thesearching performance of SPEA for the small-scale problemsThe addition of constraints to the design problems can causethe searching performance of the hybrid algorithms TheSPEA using ANN is inferior to the other surrogate-assistedmethods or even the original SPEA2 without using a sur-rogate model because it requires efficient network topologyoptimisation before being used to estimate functions whilein this paper the simple heuristic approach is employed
8 Mathematical Problems in Engineering
= 37526 and 120596 = 83234= 45 and 120596 = 10354
= 33758 and 120596 = 73085 = 33779 and 120596 = 73032
= 33613 and 120596 = 72929 t = 30008 and 120596 = 63286
(1) t (2) t
(3) t (4) t
(5) t (6)
Figure 9 Selected design points from Figure 8
02 025 03 035 04 045 05 055 06 06508
09
1
11
12
13
14
15
1
2
3
4 5
6
Stru
ctur
al m
ass (
kg)
1120582
Figure 10 Approximate Pareto front of F2
= 45 and 120582 = 63261 = 39843 and 120582 = 3823
= 35599 and 120582 = 25781 = 35983 and 120582 = 25739
= 35555 and 120582 = 25201 = 30381 and 120582 = 16535
(1) t
(3) t
(5) t (6) t
(4) t
(2) t
Figure 11 Selected design solutions from Figure 10
Mathematical Problems in Engineering 9
01 015 02 025 03 035 04 045 0508
09
1
11
12
13
14
15
1
2
3
45
6
Stru
ctur
al m
ass (
kg)
1(120582 + 1205961205960)
Figure 12 Approximate Pareto front of F3
= 43963 and 120582 + 1205961205960 = 64596= 45 and 120582 + 1205961205960 = 75573
= 41201 and 120582 + 1205961205960 = 49251 = 35884 and 120582 + 1205961205960 = 35891
= 35532 and 120582 + 1205961205960 = 34666 = 3 and 120582 + 1205961205960 = 24581
(1) t (2) t
(4) t(3) t
(5) t (6) t
Figure 13 Selected design solutions from Figure 12
0095 01 0105 011 0115 012 0125 013 0135 014 0145085
09
095
1
105
11
115
12
1
2
3
4
56
Stru
ctur
al m
ass (
kg)
1120596
Figure 14 Approximate Pareto front of F4
10 Mathematical Problems in Engineering
= 45 and 120596 = 10349 = 42534 and 120596 = 96302
= 39481 and 120596 = 88068 = 36521 and 120596 = 80402
= 36501 and 120596 = 80223 = 33791 and 120596 = 73022
(1) t (2) t
(4) t
(6) t(5) t
(3) t
Figure 15 Selected design solutions from Figure 14
02 025 03 035 04 045 0509
095
1
105
11
115
12
125
13
135
14
1
2
3
45
6
Stru
ctur
al m
ass (
kg)
1120582
Figure 16 Approximate Pareto front of F5
= 45 and 120582 = 62885 = 45 and 120582 = 53614
= 40881 and 120582 = 42008 = 37402 and 120582 = 31575
= 3652 and 120582 = 30888 = 34047 and 120582 = 22103
(1) t (2) t
(4) t
(6) t
(3) t
(5) t
Figure 17 Selected design solutions from Figure 16
Mathematical Problems in Engineering 11
01 015 02 025 03 03509
095
1
105
11
115
12
125
13
135
14
1
2
3
4
56
Stru
ctur
al m
ass (
kg)
1(120582 + 1205961205960)
Figure 18 Approximate Pareto front of F6
= 45 and 120582 + 1205961205960 = 75573 = 44813 and 120582 + 1205961205960 = 65748
= 41914 and 120582 + 1205961205960 = 53343 = 38581 and 120582 + 1205961205960 = 4254
= 38571 and 120582 + 1205961205960 = 41494 = 34147 and 120582 + 1205961205960 = 30684
(1) t (2) t
(4) t
(6) t
(3) t
(5) t
Figure 19 Selected design solutions from Figure 18
More investigation on this issue would greatly improve theperformance of SPEA-N In cases of large-scale problemsonly SPEA using a quadratic function is superior to thenonsurrogate SPEA
6 Conclusions
The use of multiobjective evolutionary algorithm for thedesign of a torque arm leads to multiple optimum structuresfor decision making The hybridisation of SPEA and severalsurrogate models is developedThe various surrogate modelsincluding the quadratic regression technique the radialbasis function interpolation and the Kriging model canhelp improving the searching performance of the strengthPareto evolutionary algorithm For a design problem havingmass and natural frequency as objective functions and stress
and buckling as constraints SPEA using RBF is the bestmethod The SPEA using a quadratic regression is the bestfor the design case of F2 For the other multiobjective designproblems the best and the second best performers are SPEAusing the Krigingmodel and the quadratic RSM respectivelyNevertheless when considering all of the design problems theoverall top performer is SPEA2 using the quadratic regressionwhile the close second best uses theKrigingmodelMoreoverSPEA using a quadratic response surface model is the onlystrategy that is useful for solving large-scale problems Theuse of ANN for approximating constrained optimisationproblems is not effective as a proper network topologyoptimisation needs to be constructed before entering thesurrogate subproblem It is however possible to enhancethe performance of the ANN-based optimiser if an efficientnetwork topology optimisation is performed during theoptimisation process
12 Mathematical Problems in Engineering
20 25 30 35 40 45 50 55 60 65
F7
SPEA-OSPEA-Q
SPEA-RSPEA-K
f1
f2
minus60
minus50
minus40
minus30
minus20
minus10
0
Figure 20 Approximate Pareto front of F7
20 25 30 35 40 45 50 55 60
F8
SPEA-OSPEA-Q
SPEA-RSPEA-K
minus1000
minus950
minus900
minus850
minus800
minus750
minus700
minus650
minus600
minus550
f2
f1
Figure 21 Approximate Pareto front of F8
Acknowledgment
The authors are graceful for the support from the ThailandResearch Fund (RSA5280006)The first author also wishes tothank the office of higher education commission Thailandfor the CHE-PHD scholarship
References
[1] F Zong H Lin B Yu and X Pan ldquoDaily commute timeprediction based on genetic algorithmrdquoMathematical Problemsin Engineering vol 2012 Article ID 321574 20 pages 2012
[2] I Yao and D Han ldquoImproved barebones particle swarmoptimization with neighborhood search and its application on
ship designrdquo Mathematical Problems in Engineering vol 2013Article ID 175848 12 pages 2013
[3] Y Xu P Fan and L Yuan ldquoA simple and efficient artificial beecolony algorithmrdquo Mathematical Problems in Engineering vol2013 Article ID 526315 9 pages 2013
[4] S Talatahari E Khalili and S M Alavizadeh ldquoAcceleratedparticle swarm for optimum design of frame structuresrdquoMath-ematical Problems in Engineering vol 2013 Article ID 6498576 pages 2013
[5] K Socha and M Dorigo ldquoAnt colony optimization for contin-uous domainsrdquo European Journal of Operational Research vol185 no 3 pp 1155ndash1173 2008
[6] N Hansen S D Muller and P Koumoutsakos ldquoReducingthe time complexity of the derandomized evolution strategywith covariance matrix adaptation (CMA-ES)rdquo EvolutionaryComputation vol 11 no 1 pp 1ndash18 2003
[7] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[8] J D Knowles and D W Corne ldquoApproximating the nondom-inated front using the Pareto Archived Evolution StrategyrdquoEvolutionary Computation vol 8 no 2 pp 149ndash172 2000
[9] M Farina and P Amato ldquoLinked interpolation-optimizationstrategies for multicriteria optimization problemsrdquo Soft Com-puting vol 9 pp 54ndash65 2005
[10] J Knowles ldquoParEGO a hybrid algorithm with on-line land-scape approximation for expensive multiobjective optimizationproblemsrdquo IEEE Transactions on Evolutionary Computationvol 10 no 1 pp 50ndash66 2006
[11] Z Zhou Y S Ong M H Lim and B S Lee ldquoMemetic algo-rithm using multi-surrogates for computationally expensiveoptimization problemsrdquo Soft Computing vol 11 no 10 pp 957ndash971 2007
[12] R G Regis and C A Shoemaker ldquoLocal function approxima-tion in evolutionary algorithms for the optimization of costlyfunctionsrdquo IEEE Transactions on Evolutionary Computationvol 8 no 5 pp 490ndash505 2004
[13] N Pholdee and S Bureerat ldquoSurrogate-assisted evolutionaryoptimizers for multiobjective design of a torque arm structurerdquoAppliedMechanics andMaterials vol 101-102 pp 324ndash328 2011
[14] N Pholdee and S Bureerat ldquoPassive vibration control ofan automotive component using evolutionary optimisationrdquoJournal of Research and Applications in Mechanical Engineeringvol 1 pp 19ndash23 2011
[15] K C Giannakoglou D I Papadimitriou and I C KampolisldquoAerodynamic shape design using evolutionary algorithmsand new gradient-assisted metamodelsrdquo Computer Methods inApplied Mechanics and Engineering vol 195 no 44ndash47 pp6312ndash6329 2006
[16] Y S Ong K Y Lum P B Nair D M Shi and Z K ZhangldquoGlobal convergence of unconstrained and bound constrainedsurrogate-assisted evolutionary search in aerodynamic shapedesignrdquo in Proceedings of the Congress on Evolutionary Compu-tation (CEC rsquo03) pp 1856ndash1863 Canberra Australia December2003
[17] Y Tenne and S W Armfield ldquoA versatile surrogate-assistedmemetic algorithm for optimization of computationally expen-sive functions and its engineering applicationsrdquo Studies inComputational Intelligence vol 92 pp 43ndash72 2008
Mathematical Problems in Engineering 13
[18] S Srisomporn and S Bureerat ldquoGeometrical design of plate-fin heat sinks using hybridization of MOEA and RSMrdquo IEEETransactions onComponents and Packaging Technologies vol 31no 2 pp 351ndash360 2008
[19] S Kanyakam and S Bureerat ldquoComparative performance ofsurrogate-assisted MOEAs for geometrical design of pin-finheat sinksrdquo Applied Mathematics vol 2012 Article ID 53478314 pages 2012
[20] A Husain and K Y Kin ldquoOptimization of a microchannel heatsink with temperature dependent fluid propertiesrdquo Journal ofMaterials Processing Technology vol 208 no 1ndash3 pp 499ndash5062008
[21] J Yoo and C-Y Lee ldquoTopology optimization of a swing armtype actuator using the response surface methodrdquoMicrosystemTechnologies vol 13 no 1 pp 21ndash31 2007
[22] W Hu L G Yao and Z Z Hua ldquoOptimization of sheetmetal forming processes by adaptive response surface basedon intelligent sampling methodrdquo Journal of Materials ProcessingTechnology vol 197 no 1ndash3 pp 77ndash88 2008
[23] N Nariman-Zadeh A Darvizeh A Jamali and A MoeinildquoEvolutionary design of generalized polynomial neural net-works for modelling and prediction of explosive formingprocessrdquo Journal of Materials Processing Technology vol 164-165 pp 1561ndash1571 2005
[24] T Goel R Vaidyanathan R T Haftka W Shyy N V Queipoand K Tucker ldquoResponse surface approximation of Pareto opti-mal front in multi-objective optimizationrdquo Computer MethodsinAppliedMechanics and Engineering vol 196 no 4ndash6 pp 879ndash893 2007
[25] Z Zhou Y S Ong P B Nair A J Keane and K Y LumldquoCombining global and local surrogate models to accelerateevolutionary optimizationrdquo IEEE Transactions on Systems Manand Cybernetics Part C vol 37 no 1 pp 66ndash76 2007
[26] Y Jin ldquoA comprehensive survey of fitness approximation inevolutionary computationrdquo Soft Computing vol 9 pp 3ndash122005
[27] Y S Ong P B Nair A J Keane and K W Wong ldquoSurrogate-assisted evolutionary optimization frameworks for high-fidelityengineering design problemsrdquo in Knowledge Incorporation inEvolutionary Computation Studies in Fuzziness and Soft Com-puting Y Jin Ed pp 307ndash332 Springer Berlin Germany 2004
[28] Y Jin ldquoSurrogate-assisted evolutionary computation recentadvances and future challengesrdquo Swarm and Evolutionary Com-putation vol 1 pp 61ndash70 2011
[29] A I J Forrester and A J Keane ldquoRecent advances in surrogate-based optimizationrdquo Progress in Aerospace Sciences vol 45 no1ndash3 pp 50ndash79 2009
[30] K-T Chiang and F-P Chang ldquoApplication of response surfacemethodology in the parametric optimization of a pin-fin typeheat sinkrdquo International Communications in Heat and MassTransfer vol 33 no 7 pp 836ndash845 2006
[31] Y Tenne and S W Armfield ldquoA framework for memeticoptimization using variable global and local surrogate modelsrdquoSoft Computing vol 13 no 8-9 pp 781ndash793 2009
[32] P Breitkopf H Naceur A Rassineux and P Villon ldquoMovingleast squares response surface approximation formulation andmetal forming applicationsrdquo Computers and Structures vol 83no 17-18 pp 1411ndash1428 2005
[33] M Xiao B Raghavan P Breitkopf R F Coelho C Knopf-Lenoir and P Villon ldquoSurrogate models based on constrainedPOD for optimizationrdquo in Proceedings of the 4th European
Conference on Computational Mechanics (ECCM rsquo10) ParisFrance May 2010
[34] M Xiao P Breitkopf R F Coelho C Knopf-Lenoir MSidorkiewicz and P Villon ldquoModel reduction by CPOD andKriging application to the shape optimization of an intakeportrdquo Structural andMultidisciplinary Optimization vol 41 no4 pp 555ndash574 2010
[35] B RaghavanMHamdaouiM Xiao P Breitkopf and P VillonldquoA bi-level meta-modeling approach for structural optimizationusing modified POD bases and diffuse approximationrdquo Com-puters and Structures 2013
[36] B Raghavan and P Breitkopf ldquoAsynchronous evolutionaryshape optimization based on high-quality surrogates applica-tion to an air-conditioning ductrdquo Engineering with Computers2012
[37] E A Kontoleontos V G Asouti and K C GiannakoglouldquoAn asynchronous metamodel-assisted memetic algorithm forCFD-based shape optimizationrdquo Engineering Optimization vol44 no 2 pp 157ndash173 2012
[38] H Zhu L Liu T Long and L Peng ldquoA novel algorithmof maximin Latin hypercube design using successive localenumerationrdquo Engineering Optimization vol 44 no 5 pp 551ndash564 2012
[39] S D Georgiou and S Stylianou ldquoBlock-circulant matricesfor constructing optimal Latin hypercube designsrdquo Journal ofStatistical Planning and Inference vol 141 no 5 pp 1933ndash19432011
[40] J M Parr A J Keane A I J Forrester and C M E HoldenldquoInfill sampling criteria for surrogate-based optimization withconstraint handlingrdquo Engineering Optimization vol 44 no 10pp 1147ndash1166 2012
[41] E Zitzler M Laumanns and L Thiele ldquoSPEA2 improvingthe strength pareto evolutionary algorithm for multiobjectiveoptimizationrdquo inEvolutionaryMethods forDesign Optimizationand Control pp 95ndash100 2002
[42] S N Lophaven H B Neilson and J Sondergaard ldquoDACEa MATLAB kriging toolboxrdquo Tech Rep IMM-TR-2002-12Technical University of Denmark 2002
[43] R L Harder and R N Desmarais ldquoInterpolation using surfacesplinesrdquo Journal of Aircraft vol 9 no 2 pp 189ndash191 1972
[44] M E Botkin ldquoShape optimization with buckling and stressconstraintsrdquo AIAA Journal vol 34 no 2 pp 423ndash425 1996
[45] K Deb A Pratap and T Meyarivan ldquoConstrained test prob-lems for multi-objective evolutionary optimizationrdquo in Evolu-tionary Multi-Criterion Optimization (Zurich 2001) vol 1993of Lecture Notes in Computer Science pp 284ndash298 SpringerBerlin Germany 2001
[46] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 7: Research Article Surrogate-Assisted Multiobjective ...downloads.hindawi.com/journals/mpe/2013/695172.pdf · of engineering application, for example, aerodynamics [ ], heat and mass](https://reader034.fdocuments.net/reader034/viewer/2022052003/60164c5514f5a3172c3c2904/html5/thumbnails/7.jpg)
Mathematical Problems in Engineering 7
0 500 1000 1500 2000 25000
2
4
6
8
10
12
14 F7
No of function evaluations
Hyp
ervo
lum
e
SPEA-OSPEA-Q
Figure 6 SPEA-O versus SPEA-Q for F7
a way that the best method is ranked as 1 while the worst isranked as 5The overall performance assessment based on thet-test is given in Table 4 where the last row on the table showsthe total score of all theMOEAsThe lower the score the betterthe performer Similar to the first assessment the overall bestmethod for F1ndashF6 based on this assessment is SPEA-Q whilethe close secondbest is SPEA-KTheworstmethod is SPEA-Owithout using a surrogate model For the large-scale testproblems the only surrogateassisted method that is superiorto the non-surrogate SPEA is SPEA-Q
Figures 6 and 7 show the search history as plots ofhypervolume against the number of function evaluations ofSPEA-O and SPEA-Q for solving F7 and F8 respectivelyThevalues of front hypervolume can fluctuate due to the use of aPareto archiving technique to remove some nondominatedsolutions from the Pareto archive From the figures it canbe seen that SPEA-Q requires approximately half of the totalnumber of function evaluations employed by SPEA-O to haveequally good nondominated fronts
The best approximate Pareto front of F1 obtained afternumerous optimisation runs of the surrogate-assisted SPEAsis chosen and plotted in Figure 8 whereas the structures ofsome selected design points are shown in Figure 9 Figure 10displays the best front of the F2 problemwhere some selecteddesign solutions are illustrated in Figure 11The best front andsome selected optimal structures of the design problemF3 aredisplayed in Figures 12 and 13 respectively Similarly the bestapproximate fronts and some selected design solutions of F4F5 and F6 are illustrated in Figures 14 15 16 17 18 and 19Figures 20 and 21 show the best fronts obtained from usingthe various SPEAs for solving F7 and F8 respectively It canbe seen that the best fronts are from SPEA-Q and SPEA-OThe SPEA-Q front does not totally dominate that of SPEA-O but they overlap each other while the SPEA-Q front hasobviously greater front extension
According to the aforementioned performance assess-ments it can be said that SPEA-R is the best algorithm in
0 500 1000 1500 2000 25000
2
4
6
8
10
12F8
No of function evaluations
Hyp
ervo
lum
e
SPEA-OSPEA-Q
Figure 7 SPEA-O versus SPEA-Q for F8
009 01 011 012 013 014 015 01608
085
09
095
1
105
11
115
12
1
2
3
4
5
6
Stru
ctur
al m
ass (
kg)
1120596
Figure 8 Approximate Pareto front of F1
the cases of the F1 design problem while the close secondbest is SPEA-Q When adding a natural frequency constraintto F1 becoming the F4 problem the overall best and secondbest methods are still SPEA-R and SPEA-Q respectivelyFor the F2 case and the F5 design problem which is F2with additional natural frequency constraint the top twoperformers are SPEA-Q and SPEA-K The same can be saidfor the case of F3 while SPEA-K outperforms the others forthe F6 design problem The surrogate models that is thequadratic response surface model the radial basis functioninterpolation and the Kriging model are said to enhance thesearching performance of SPEA for the small-scale problemsThe addition of constraints to the design problems can causethe searching performance of the hybrid algorithms TheSPEA using ANN is inferior to the other surrogate-assistedmethods or even the original SPEA2 without using a sur-rogate model because it requires efficient network topologyoptimisation before being used to estimate functions whilein this paper the simple heuristic approach is employed
8 Mathematical Problems in Engineering
= 37526 and 120596 = 83234= 45 and 120596 = 10354
= 33758 and 120596 = 73085 = 33779 and 120596 = 73032
= 33613 and 120596 = 72929 t = 30008 and 120596 = 63286
(1) t (2) t
(3) t (4) t
(5) t (6)
Figure 9 Selected design points from Figure 8
02 025 03 035 04 045 05 055 06 06508
09
1
11
12
13
14
15
1
2
3
4 5
6
Stru
ctur
al m
ass (
kg)
1120582
Figure 10 Approximate Pareto front of F2
= 45 and 120582 = 63261 = 39843 and 120582 = 3823
= 35599 and 120582 = 25781 = 35983 and 120582 = 25739
= 35555 and 120582 = 25201 = 30381 and 120582 = 16535
(1) t
(3) t
(5) t (6) t
(4) t
(2) t
Figure 11 Selected design solutions from Figure 10
Mathematical Problems in Engineering 9
01 015 02 025 03 035 04 045 0508
09
1
11
12
13
14
15
1
2
3
45
6
Stru
ctur
al m
ass (
kg)
1(120582 + 1205961205960)
Figure 12 Approximate Pareto front of F3
= 43963 and 120582 + 1205961205960 = 64596= 45 and 120582 + 1205961205960 = 75573
= 41201 and 120582 + 1205961205960 = 49251 = 35884 and 120582 + 1205961205960 = 35891
= 35532 and 120582 + 1205961205960 = 34666 = 3 and 120582 + 1205961205960 = 24581
(1) t (2) t
(4) t(3) t
(5) t (6) t
Figure 13 Selected design solutions from Figure 12
0095 01 0105 011 0115 012 0125 013 0135 014 0145085
09
095
1
105
11
115
12
1
2
3
4
56
Stru
ctur
al m
ass (
kg)
1120596
Figure 14 Approximate Pareto front of F4
10 Mathematical Problems in Engineering
= 45 and 120596 = 10349 = 42534 and 120596 = 96302
= 39481 and 120596 = 88068 = 36521 and 120596 = 80402
= 36501 and 120596 = 80223 = 33791 and 120596 = 73022
(1) t (2) t
(4) t
(6) t(5) t
(3) t
Figure 15 Selected design solutions from Figure 14
02 025 03 035 04 045 0509
095
1
105
11
115
12
125
13
135
14
1
2
3
45
6
Stru
ctur
al m
ass (
kg)
1120582
Figure 16 Approximate Pareto front of F5
= 45 and 120582 = 62885 = 45 and 120582 = 53614
= 40881 and 120582 = 42008 = 37402 and 120582 = 31575
= 3652 and 120582 = 30888 = 34047 and 120582 = 22103
(1) t (2) t
(4) t
(6) t
(3) t
(5) t
Figure 17 Selected design solutions from Figure 16
Mathematical Problems in Engineering 11
01 015 02 025 03 03509
095
1
105
11
115
12
125
13
135
14
1
2
3
4
56
Stru
ctur
al m
ass (
kg)
1(120582 + 1205961205960)
Figure 18 Approximate Pareto front of F6
= 45 and 120582 + 1205961205960 = 75573 = 44813 and 120582 + 1205961205960 = 65748
= 41914 and 120582 + 1205961205960 = 53343 = 38581 and 120582 + 1205961205960 = 4254
= 38571 and 120582 + 1205961205960 = 41494 = 34147 and 120582 + 1205961205960 = 30684
(1) t (2) t
(4) t
(6) t
(3) t
(5) t
Figure 19 Selected design solutions from Figure 18
More investigation on this issue would greatly improve theperformance of SPEA-N In cases of large-scale problemsonly SPEA using a quadratic function is superior to thenonsurrogate SPEA
6 Conclusions
The use of multiobjective evolutionary algorithm for thedesign of a torque arm leads to multiple optimum structuresfor decision making The hybridisation of SPEA and severalsurrogate models is developedThe various surrogate modelsincluding the quadratic regression technique the radialbasis function interpolation and the Kriging model canhelp improving the searching performance of the strengthPareto evolutionary algorithm For a design problem havingmass and natural frequency as objective functions and stress
and buckling as constraints SPEA using RBF is the bestmethod The SPEA using a quadratic regression is the bestfor the design case of F2 For the other multiobjective designproblems the best and the second best performers are SPEAusing the Krigingmodel and the quadratic RSM respectivelyNevertheless when considering all of the design problems theoverall top performer is SPEA2 using the quadratic regressionwhile the close second best uses theKrigingmodelMoreoverSPEA using a quadratic response surface model is the onlystrategy that is useful for solving large-scale problems Theuse of ANN for approximating constrained optimisationproblems is not effective as a proper network topologyoptimisation needs to be constructed before entering thesurrogate subproblem It is however possible to enhancethe performance of the ANN-based optimiser if an efficientnetwork topology optimisation is performed during theoptimisation process
12 Mathematical Problems in Engineering
20 25 30 35 40 45 50 55 60 65
F7
SPEA-OSPEA-Q
SPEA-RSPEA-K
f1
f2
minus60
minus50
minus40
minus30
minus20
minus10
0
Figure 20 Approximate Pareto front of F7
20 25 30 35 40 45 50 55 60
F8
SPEA-OSPEA-Q
SPEA-RSPEA-K
minus1000
minus950
minus900
minus850
minus800
minus750
minus700
minus650
minus600
minus550
f2
f1
Figure 21 Approximate Pareto front of F8
Acknowledgment
The authors are graceful for the support from the ThailandResearch Fund (RSA5280006)The first author also wishes tothank the office of higher education commission Thailandfor the CHE-PHD scholarship
References
[1] F Zong H Lin B Yu and X Pan ldquoDaily commute timeprediction based on genetic algorithmrdquoMathematical Problemsin Engineering vol 2012 Article ID 321574 20 pages 2012
[2] I Yao and D Han ldquoImproved barebones particle swarmoptimization with neighborhood search and its application on
ship designrdquo Mathematical Problems in Engineering vol 2013Article ID 175848 12 pages 2013
[3] Y Xu P Fan and L Yuan ldquoA simple and efficient artificial beecolony algorithmrdquo Mathematical Problems in Engineering vol2013 Article ID 526315 9 pages 2013
[4] S Talatahari E Khalili and S M Alavizadeh ldquoAcceleratedparticle swarm for optimum design of frame structuresrdquoMath-ematical Problems in Engineering vol 2013 Article ID 6498576 pages 2013
[5] K Socha and M Dorigo ldquoAnt colony optimization for contin-uous domainsrdquo European Journal of Operational Research vol185 no 3 pp 1155ndash1173 2008
[6] N Hansen S D Muller and P Koumoutsakos ldquoReducingthe time complexity of the derandomized evolution strategywith covariance matrix adaptation (CMA-ES)rdquo EvolutionaryComputation vol 11 no 1 pp 1ndash18 2003
[7] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[8] J D Knowles and D W Corne ldquoApproximating the nondom-inated front using the Pareto Archived Evolution StrategyrdquoEvolutionary Computation vol 8 no 2 pp 149ndash172 2000
[9] M Farina and P Amato ldquoLinked interpolation-optimizationstrategies for multicriteria optimization problemsrdquo Soft Com-puting vol 9 pp 54ndash65 2005
[10] J Knowles ldquoParEGO a hybrid algorithm with on-line land-scape approximation for expensive multiobjective optimizationproblemsrdquo IEEE Transactions on Evolutionary Computationvol 10 no 1 pp 50ndash66 2006
[11] Z Zhou Y S Ong M H Lim and B S Lee ldquoMemetic algo-rithm using multi-surrogates for computationally expensiveoptimization problemsrdquo Soft Computing vol 11 no 10 pp 957ndash971 2007
[12] R G Regis and C A Shoemaker ldquoLocal function approxima-tion in evolutionary algorithms for the optimization of costlyfunctionsrdquo IEEE Transactions on Evolutionary Computationvol 8 no 5 pp 490ndash505 2004
[13] N Pholdee and S Bureerat ldquoSurrogate-assisted evolutionaryoptimizers for multiobjective design of a torque arm structurerdquoAppliedMechanics andMaterials vol 101-102 pp 324ndash328 2011
[14] N Pholdee and S Bureerat ldquoPassive vibration control ofan automotive component using evolutionary optimisationrdquoJournal of Research and Applications in Mechanical Engineeringvol 1 pp 19ndash23 2011
[15] K C Giannakoglou D I Papadimitriou and I C KampolisldquoAerodynamic shape design using evolutionary algorithmsand new gradient-assisted metamodelsrdquo Computer Methods inApplied Mechanics and Engineering vol 195 no 44ndash47 pp6312ndash6329 2006
[16] Y S Ong K Y Lum P B Nair D M Shi and Z K ZhangldquoGlobal convergence of unconstrained and bound constrainedsurrogate-assisted evolutionary search in aerodynamic shapedesignrdquo in Proceedings of the Congress on Evolutionary Compu-tation (CEC rsquo03) pp 1856ndash1863 Canberra Australia December2003
[17] Y Tenne and S W Armfield ldquoA versatile surrogate-assistedmemetic algorithm for optimization of computationally expen-sive functions and its engineering applicationsrdquo Studies inComputational Intelligence vol 92 pp 43ndash72 2008
Mathematical Problems in Engineering 13
[18] S Srisomporn and S Bureerat ldquoGeometrical design of plate-fin heat sinks using hybridization of MOEA and RSMrdquo IEEETransactions onComponents and Packaging Technologies vol 31no 2 pp 351ndash360 2008
[19] S Kanyakam and S Bureerat ldquoComparative performance ofsurrogate-assisted MOEAs for geometrical design of pin-finheat sinksrdquo Applied Mathematics vol 2012 Article ID 53478314 pages 2012
[20] A Husain and K Y Kin ldquoOptimization of a microchannel heatsink with temperature dependent fluid propertiesrdquo Journal ofMaterials Processing Technology vol 208 no 1ndash3 pp 499ndash5062008
[21] J Yoo and C-Y Lee ldquoTopology optimization of a swing armtype actuator using the response surface methodrdquoMicrosystemTechnologies vol 13 no 1 pp 21ndash31 2007
[22] W Hu L G Yao and Z Z Hua ldquoOptimization of sheetmetal forming processes by adaptive response surface basedon intelligent sampling methodrdquo Journal of Materials ProcessingTechnology vol 197 no 1ndash3 pp 77ndash88 2008
[23] N Nariman-Zadeh A Darvizeh A Jamali and A MoeinildquoEvolutionary design of generalized polynomial neural net-works for modelling and prediction of explosive formingprocessrdquo Journal of Materials Processing Technology vol 164-165 pp 1561ndash1571 2005
[24] T Goel R Vaidyanathan R T Haftka W Shyy N V Queipoand K Tucker ldquoResponse surface approximation of Pareto opti-mal front in multi-objective optimizationrdquo Computer MethodsinAppliedMechanics and Engineering vol 196 no 4ndash6 pp 879ndash893 2007
[25] Z Zhou Y S Ong P B Nair A J Keane and K Y LumldquoCombining global and local surrogate models to accelerateevolutionary optimizationrdquo IEEE Transactions on Systems Manand Cybernetics Part C vol 37 no 1 pp 66ndash76 2007
[26] Y Jin ldquoA comprehensive survey of fitness approximation inevolutionary computationrdquo Soft Computing vol 9 pp 3ndash122005
[27] Y S Ong P B Nair A J Keane and K W Wong ldquoSurrogate-assisted evolutionary optimization frameworks for high-fidelityengineering design problemsrdquo in Knowledge Incorporation inEvolutionary Computation Studies in Fuzziness and Soft Com-puting Y Jin Ed pp 307ndash332 Springer Berlin Germany 2004
[28] Y Jin ldquoSurrogate-assisted evolutionary computation recentadvances and future challengesrdquo Swarm and Evolutionary Com-putation vol 1 pp 61ndash70 2011
[29] A I J Forrester and A J Keane ldquoRecent advances in surrogate-based optimizationrdquo Progress in Aerospace Sciences vol 45 no1ndash3 pp 50ndash79 2009
[30] K-T Chiang and F-P Chang ldquoApplication of response surfacemethodology in the parametric optimization of a pin-fin typeheat sinkrdquo International Communications in Heat and MassTransfer vol 33 no 7 pp 836ndash845 2006
[31] Y Tenne and S W Armfield ldquoA framework for memeticoptimization using variable global and local surrogate modelsrdquoSoft Computing vol 13 no 8-9 pp 781ndash793 2009
[32] P Breitkopf H Naceur A Rassineux and P Villon ldquoMovingleast squares response surface approximation formulation andmetal forming applicationsrdquo Computers and Structures vol 83no 17-18 pp 1411ndash1428 2005
[33] M Xiao B Raghavan P Breitkopf R F Coelho C Knopf-Lenoir and P Villon ldquoSurrogate models based on constrainedPOD for optimizationrdquo in Proceedings of the 4th European
Conference on Computational Mechanics (ECCM rsquo10) ParisFrance May 2010
[34] M Xiao P Breitkopf R F Coelho C Knopf-Lenoir MSidorkiewicz and P Villon ldquoModel reduction by CPOD andKriging application to the shape optimization of an intakeportrdquo Structural andMultidisciplinary Optimization vol 41 no4 pp 555ndash574 2010
[35] B RaghavanMHamdaouiM Xiao P Breitkopf and P VillonldquoA bi-level meta-modeling approach for structural optimizationusing modified POD bases and diffuse approximationrdquo Com-puters and Structures 2013
[36] B Raghavan and P Breitkopf ldquoAsynchronous evolutionaryshape optimization based on high-quality surrogates applica-tion to an air-conditioning ductrdquo Engineering with Computers2012
[37] E A Kontoleontos V G Asouti and K C GiannakoglouldquoAn asynchronous metamodel-assisted memetic algorithm forCFD-based shape optimizationrdquo Engineering Optimization vol44 no 2 pp 157ndash173 2012
[38] H Zhu L Liu T Long and L Peng ldquoA novel algorithmof maximin Latin hypercube design using successive localenumerationrdquo Engineering Optimization vol 44 no 5 pp 551ndash564 2012
[39] S D Georgiou and S Stylianou ldquoBlock-circulant matricesfor constructing optimal Latin hypercube designsrdquo Journal ofStatistical Planning and Inference vol 141 no 5 pp 1933ndash19432011
[40] J M Parr A J Keane A I J Forrester and C M E HoldenldquoInfill sampling criteria for surrogate-based optimization withconstraint handlingrdquo Engineering Optimization vol 44 no 10pp 1147ndash1166 2012
[41] E Zitzler M Laumanns and L Thiele ldquoSPEA2 improvingthe strength pareto evolutionary algorithm for multiobjectiveoptimizationrdquo inEvolutionaryMethods forDesign Optimizationand Control pp 95ndash100 2002
[42] S N Lophaven H B Neilson and J Sondergaard ldquoDACEa MATLAB kriging toolboxrdquo Tech Rep IMM-TR-2002-12Technical University of Denmark 2002
[43] R L Harder and R N Desmarais ldquoInterpolation using surfacesplinesrdquo Journal of Aircraft vol 9 no 2 pp 189ndash191 1972
[44] M E Botkin ldquoShape optimization with buckling and stressconstraintsrdquo AIAA Journal vol 34 no 2 pp 423ndash425 1996
[45] K Deb A Pratap and T Meyarivan ldquoConstrained test prob-lems for multi-objective evolutionary optimizationrdquo in Evolu-tionary Multi-Criterion Optimization (Zurich 2001) vol 1993of Lecture Notes in Computer Science pp 284ndash298 SpringerBerlin Germany 2001
[46] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 8: Research Article Surrogate-Assisted Multiobjective ...downloads.hindawi.com/journals/mpe/2013/695172.pdf · of engineering application, for example, aerodynamics [ ], heat and mass](https://reader034.fdocuments.net/reader034/viewer/2022052003/60164c5514f5a3172c3c2904/html5/thumbnails/8.jpg)
8 Mathematical Problems in Engineering
= 37526 and 120596 = 83234= 45 and 120596 = 10354
= 33758 and 120596 = 73085 = 33779 and 120596 = 73032
= 33613 and 120596 = 72929 t = 30008 and 120596 = 63286
(1) t (2) t
(3) t (4) t
(5) t (6)
Figure 9 Selected design points from Figure 8
02 025 03 035 04 045 05 055 06 06508
09
1
11
12
13
14
15
1
2
3
4 5
6
Stru
ctur
al m
ass (
kg)
1120582
Figure 10 Approximate Pareto front of F2
= 45 and 120582 = 63261 = 39843 and 120582 = 3823
= 35599 and 120582 = 25781 = 35983 and 120582 = 25739
= 35555 and 120582 = 25201 = 30381 and 120582 = 16535
(1) t
(3) t
(5) t (6) t
(4) t
(2) t
Figure 11 Selected design solutions from Figure 10
Mathematical Problems in Engineering 9
01 015 02 025 03 035 04 045 0508
09
1
11
12
13
14
15
1
2
3
45
6
Stru
ctur
al m
ass (
kg)
1(120582 + 1205961205960)
Figure 12 Approximate Pareto front of F3
= 43963 and 120582 + 1205961205960 = 64596= 45 and 120582 + 1205961205960 = 75573
= 41201 and 120582 + 1205961205960 = 49251 = 35884 and 120582 + 1205961205960 = 35891
= 35532 and 120582 + 1205961205960 = 34666 = 3 and 120582 + 1205961205960 = 24581
(1) t (2) t
(4) t(3) t
(5) t (6) t
Figure 13 Selected design solutions from Figure 12
0095 01 0105 011 0115 012 0125 013 0135 014 0145085
09
095
1
105
11
115
12
1
2
3
4
56
Stru
ctur
al m
ass (
kg)
1120596
Figure 14 Approximate Pareto front of F4
10 Mathematical Problems in Engineering
= 45 and 120596 = 10349 = 42534 and 120596 = 96302
= 39481 and 120596 = 88068 = 36521 and 120596 = 80402
= 36501 and 120596 = 80223 = 33791 and 120596 = 73022
(1) t (2) t
(4) t
(6) t(5) t
(3) t
Figure 15 Selected design solutions from Figure 14
02 025 03 035 04 045 0509
095
1
105
11
115
12
125
13
135
14
1
2
3
45
6
Stru
ctur
al m
ass (
kg)
1120582
Figure 16 Approximate Pareto front of F5
= 45 and 120582 = 62885 = 45 and 120582 = 53614
= 40881 and 120582 = 42008 = 37402 and 120582 = 31575
= 3652 and 120582 = 30888 = 34047 and 120582 = 22103
(1) t (2) t
(4) t
(6) t
(3) t
(5) t
Figure 17 Selected design solutions from Figure 16
Mathematical Problems in Engineering 11
01 015 02 025 03 03509
095
1
105
11
115
12
125
13
135
14
1
2
3
4
56
Stru
ctur
al m
ass (
kg)
1(120582 + 1205961205960)
Figure 18 Approximate Pareto front of F6
= 45 and 120582 + 1205961205960 = 75573 = 44813 and 120582 + 1205961205960 = 65748
= 41914 and 120582 + 1205961205960 = 53343 = 38581 and 120582 + 1205961205960 = 4254
= 38571 and 120582 + 1205961205960 = 41494 = 34147 and 120582 + 1205961205960 = 30684
(1) t (2) t
(4) t
(6) t
(3) t
(5) t
Figure 19 Selected design solutions from Figure 18
More investigation on this issue would greatly improve theperformance of SPEA-N In cases of large-scale problemsonly SPEA using a quadratic function is superior to thenonsurrogate SPEA
6 Conclusions
The use of multiobjective evolutionary algorithm for thedesign of a torque arm leads to multiple optimum structuresfor decision making The hybridisation of SPEA and severalsurrogate models is developedThe various surrogate modelsincluding the quadratic regression technique the radialbasis function interpolation and the Kriging model canhelp improving the searching performance of the strengthPareto evolutionary algorithm For a design problem havingmass and natural frequency as objective functions and stress
and buckling as constraints SPEA using RBF is the bestmethod The SPEA using a quadratic regression is the bestfor the design case of F2 For the other multiobjective designproblems the best and the second best performers are SPEAusing the Krigingmodel and the quadratic RSM respectivelyNevertheless when considering all of the design problems theoverall top performer is SPEA2 using the quadratic regressionwhile the close second best uses theKrigingmodelMoreoverSPEA using a quadratic response surface model is the onlystrategy that is useful for solving large-scale problems Theuse of ANN for approximating constrained optimisationproblems is not effective as a proper network topologyoptimisation needs to be constructed before entering thesurrogate subproblem It is however possible to enhancethe performance of the ANN-based optimiser if an efficientnetwork topology optimisation is performed during theoptimisation process
12 Mathematical Problems in Engineering
20 25 30 35 40 45 50 55 60 65
F7
SPEA-OSPEA-Q
SPEA-RSPEA-K
f1
f2
minus60
minus50
minus40
minus30
minus20
minus10
0
Figure 20 Approximate Pareto front of F7
20 25 30 35 40 45 50 55 60
F8
SPEA-OSPEA-Q
SPEA-RSPEA-K
minus1000
minus950
minus900
minus850
minus800
minus750
minus700
minus650
minus600
minus550
f2
f1
Figure 21 Approximate Pareto front of F8
Acknowledgment
The authors are graceful for the support from the ThailandResearch Fund (RSA5280006)The first author also wishes tothank the office of higher education commission Thailandfor the CHE-PHD scholarship
References
[1] F Zong H Lin B Yu and X Pan ldquoDaily commute timeprediction based on genetic algorithmrdquoMathematical Problemsin Engineering vol 2012 Article ID 321574 20 pages 2012
[2] I Yao and D Han ldquoImproved barebones particle swarmoptimization with neighborhood search and its application on
ship designrdquo Mathematical Problems in Engineering vol 2013Article ID 175848 12 pages 2013
[3] Y Xu P Fan and L Yuan ldquoA simple and efficient artificial beecolony algorithmrdquo Mathematical Problems in Engineering vol2013 Article ID 526315 9 pages 2013
[4] S Talatahari E Khalili and S M Alavizadeh ldquoAcceleratedparticle swarm for optimum design of frame structuresrdquoMath-ematical Problems in Engineering vol 2013 Article ID 6498576 pages 2013
[5] K Socha and M Dorigo ldquoAnt colony optimization for contin-uous domainsrdquo European Journal of Operational Research vol185 no 3 pp 1155ndash1173 2008
[6] N Hansen S D Muller and P Koumoutsakos ldquoReducingthe time complexity of the derandomized evolution strategywith covariance matrix adaptation (CMA-ES)rdquo EvolutionaryComputation vol 11 no 1 pp 1ndash18 2003
[7] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[8] J D Knowles and D W Corne ldquoApproximating the nondom-inated front using the Pareto Archived Evolution StrategyrdquoEvolutionary Computation vol 8 no 2 pp 149ndash172 2000
[9] M Farina and P Amato ldquoLinked interpolation-optimizationstrategies for multicriteria optimization problemsrdquo Soft Com-puting vol 9 pp 54ndash65 2005
[10] J Knowles ldquoParEGO a hybrid algorithm with on-line land-scape approximation for expensive multiobjective optimizationproblemsrdquo IEEE Transactions on Evolutionary Computationvol 10 no 1 pp 50ndash66 2006
[11] Z Zhou Y S Ong M H Lim and B S Lee ldquoMemetic algo-rithm using multi-surrogates for computationally expensiveoptimization problemsrdquo Soft Computing vol 11 no 10 pp 957ndash971 2007
[12] R G Regis and C A Shoemaker ldquoLocal function approxima-tion in evolutionary algorithms for the optimization of costlyfunctionsrdquo IEEE Transactions on Evolutionary Computationvol 8 no 5 pp 490ndash505 2004
[13] N Pholdee and S Bureerat ldquoSurrogate-assisted evolutionaryoptimizers for multiobjective design of a torque arm structurerdquoAppliedMechanics andMaterials vol 101-102 pp 324ndash328 2011
[14] N Pholdee and S Bureerat ldquoPassive vibration control ofan automotive component using evolutionary optimisationrdquoJournal of Research and Applications in Mechanical Engineeringvol 1 pp 19ndash23 2011
[15] K C Giannakoglou D I Papadimitriou and I C KampolisldquoAerodynamic shape design using evolutionary algorithmsand new gradient-assisted metamodelsrdquo Computer Methods inApplied Mechanics and Engineering vol 195 no 44ndash47 pp6312ndash6329 2006
[16] Y S Ong K Y Lum P B Nair D M Shi and Z K ZhangldquoGlobal convergence of unconstrained and bound constrainedsurrogate-assisted evolutionary search in aerodynamic shapedesignrdquo in Proceedings of the Congress on Evolutionary Compu-tation (CEC rsquo03) pp 1856ndash1863 Canberra Australia December2003
[17] Y Tenne and S W Armfield ldquoA versatile surrogate-assistedmemetic algorithm for optimization of computationally expen-sive functions and its engineering applicationsrdquo Studies inComputational Intelligence vol 92 pp 43ndash72 2008
Mathematical Problems in Engineering 13
[18] S Srisomporn and S Bureerat ldquoGeometrical design of plate-fin heat sinks using hybridization of MOEA and RSMrdquo IEEETransactions onComponents and Packaging Technologies vol 31no 2 pp 351ndash360 2008
[19] S Kanyakam and S Bureerat ldquoComparative performance ofsurrogate-assisted MOEAs for geometrical design of pin-finheat sinksrdquo Applied Mathematics vol 2012 Article ID 53478314 pages 2012
[20] A Husain and K Y Kin ldquoOptimization of a microchannel heatsink with temperature dependent fluid propertiesrdquo Journal ofMaterials Processing Technology vol 208 no 1ndash3 pp 499ndash5062008
[21] J Yoo and C-Y Lee ldquoTopology optimization of a swing armtype actuator using the response surface methodrdquoMicrosystemTechnologies vol 13 no 1 pp 21ndash31 2007
[22] W Hu L G Yao and Z Z Hua ldquoOptimization of sheetmetal forming processes by adaptive response surface basedon intelligent sampling methodrdquo Journal of Materials ProcessingTechnology vol 197 no 1ndash3 pp 77ndash88 2008
[23] N Nariman-Zadeh A Darvizeh A Jamali and A MoeinildquoEvolutionary design of generalized polynomial neural net-works for modelling and prediction of explosive formingprocessrdquo Journal of Materials Processing Technology vol 164-165 pp 1561ndash1571 2005
[24] T Goel R Vaidyanathan R T Haftka W Shyy N V Queipoand K Tucker ldquoResponse surface approximation of Pareto opti-mal front in multi-objective optimizationrdquo Computer MethodsinAppliedMechanics and Engineering vol 196 no 4ndash6 pp 879ndash893 2007
[25] Z Zhou Y S Ong P B Nair A J Keane and K Y LumldquoCombining global and local surrogate models to accelerateevolutionary optimizationrdquo IEEE Transactions on Systems Manand Cybernetics Part C vol 37 no 1 pp 66ndash76 2007
[26] Y Jin ldquoA comprehensive survey of fitness approximation inevolutionary computationrdquo Soft Computing vol 9 pp 3ndash122005
[27] Y S Ong P B Nair A J Keane and K W Wong ldquoSurrogate-assisted evolutionary optimization frameworks for high-fidelityengineering design problemsrdquo in Knowledge Incorporation inEvolutionary Computation Studies in Fuzziness and Soft Com-puting Y Jin Ed pp 307ndash332 Springer Berlin Germany 2004
[28] Y Jin ldquoSurrogate-assisted evolutionary computation recentadvances and future challengesrdquo Swarm and Evolutionary Com-putation vol 1 pp 61ndash70 2011
[29] A I J Forrester and A J Keane ldquoRecent advances in surrogate-based optimizationrdquo Progress in Aerospace Sciences vol 45 no1ndash3 pp 50ndash79 2009
[30] K-T Chiang and F-P Chang ldquoApplication of response surfacemethodology in the parametric optimization of a pin-fin typeheat sinkrdquo International Communications in Heat and MassTransfer vol 33 no 7 pp 836ndash845 2006
[31] Y Tenne and S W Armfield ldquoA framework for memeticoptimization using variable global and local surrogate modelsrdquoSoft Computing vol 13 no 8-9 pp 781ndash793 2009
[32] P Breitkopf H Naceur A Rassineux and P Villon ldquoMovingleast squares response surface approximation formulation andmetal forming applicationsrdquo Computers and Structures vol 83no 17-18 pp 1411ndash1428 2005
[33] M Xiao B Raghavan P Breitkopf R F Coelho C Knopf-Lenoir and P Villon ldquoSurrogate models based on constrainedPOD for optimizationrdquo in Proceedings of the 4th European
Conference on Computational Mechanics (ECCM rsquo10) ParisFrance May 2010
[34] M Xiao P Breitkopf R F Coelho C Knopf-Lenoir MSidorkiewicz and P Villon ldquoModel reduction by CPOD andKriging application to the shape optimization of an intakeportrdquo Structural andMultidisciplinary Optimization vol 41 no4 pp 555ndash574 2010
[35] B RaghavanMHamdaouiM Xiao P Breitkopf and P VillonldquoA bi-level meta-modeling approach for structural optimizationusing modified POD bases and diffuse approximationrdquo Com-puters and Structures 2013
[36] B Raghavan and P Breitkopf ldquoAsynchronous evolutionaryshape optimization based on high-quality surrogates applica-tion to an air-conditioning ductrdquo Engineering with Computers2012
[37] E A Kontoleontos V G Asouti and K C GiannakoglouldquoAn asynchronous metamodel-assisted memetic algorithm forCFD-based shape optimizationrdquo Engineering Optimization vol44 no 2 pp 157ndash173 2012
[38] H Zhu L Liu T Long and L Peng ldquoA novel algorithmof maximin Latin hypercube design using successive localenumerationrdquo Engineering Optimization vol 44 no 5 pp 551ndash564 2012
[39] S D Georgiou and S Stylianou ldquoBlock-circulant matricesfor constructing optimal Latin hypercube designsrdquo Journal ofStatistical Planning and Inference vol 141 no 5 pp 1933ndash19432011
[40] J M Parr A J Keane A I J Forrester and C M E HoldenldquoInfill sampling criteria for surrogate-based optimization withconstraint handlingrdquo Engineering Optimization vol 44 no 10pp 1147ndash1166 2012
[41] E Zitzler M Laumanns and L Thiele ldquoSPEA2 improvingthe strength pareto evolutionary algorithm for multiobjectiveoptimizationrdquo inEvolutionaryMethods forDesign Optimizationand Control pp 95ndash100 2002
[42] S N Lophaven H B Neilson and J Sondergaard ldquoDACEa MATLAB kriging toolboxrdquo Tech Rep IMM-TR-2002-12Technical University of Denmark 2002
[43] R L Harder and R N Desmarais ldquoInterpolation using surfacesplinesrdquo Journal of Aircraft vol 9 no 2 pp 189ndash191 1972
[44] M E Botkin ldquoShape optimization with buckling and stressconstraintsrdquo AIAA Journal vol 34 no 2 pp 423ndash425 1996
[45] K Deb A Pratap and T Meyarivan ldquoConstrained test prob-lems for multi-objective evolutionary optimizationrdquo in Evolu-tionary Multi-Criterion Optimization (Zurich 2001) vol 1993of Lecture Notes in Computer Science pp 284ndash298 SpringerBerlin Germany 2001
[46] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 9: Research Article Surrogate-Assisted Multiobjective ...downloads.hindawi.com/journals/mpe/2013/695172.pdf · of engineering application, for example, aerodynamics [ ], heat and mass](https://reader034.fdocuments.net/reader034/viewer/2022052003/60164c5514f5a3172c3c2904/html5/thumbnails/9.jpg)
Mathematical Problems in Engineering 9
01 015 02 025 03 035 04 045 0508
09
1
11
12
13
14
15
1
2
3
45
6
Stru
ctur
al m
ass (
kg)
1(120582 + 1205961205960)
Figure 12 Approximate Pareto front of F3
= 43963 and 120582 + 1205961205960 = 64596= 45 and 120582 + 1205961205960 = 75573
= 41201 and 120582 + 1205961205960 = 49251 = 35884 and 120582 + 1205961205960 = 35891
= 35532 and 120582 + 1205961205960 = 34666 = 3 and 120582 + 1205961205960 = 24581
(1) t (2) t
(4) t(3) t
(5) t (6) t
Figure 13 Selected design solutions from Figure 12
0095 01 0105 011 0115 012 0125 013 0135 014 0145085
09
095
1
105
11
115
12
1
2
3
4
56
Stru
ctur
al m
ass (
kg)
1120596
Figure 14 Approximate Pareto front of F4
10 Mathematical Problems in Engineering
= 45 and 120596 = 10349 = 42534 and 120596 = 96302
= 39481 and 120596 = 88068 = 36521 and 120596 = 80402
= 36501 and 120596 = 80223 = 33791 and 120596 = 73022
(1) t (2) t
(4) t
(6) t(5) t
(3) t
Figure 15 Selected design solutions from Figure 14
02 025 03 035 04 045 0509
095
1
105
11
115
12
125
13
135
14
1
2
3
45
6
Stru
ctur
al m
ass (
kg)
1120582
Figure 16 Approximate Pareto front of F5
= 45 and 120582 = 62885 = 45 and 120582 = 53614
= 40881 and 120582 = 42008 = 37402 and 120582 = 31575
= 3652 and 120582 = 30888 = 34047 and 120582 = 22103
(1) t (2) t
(4) t
(6) t
(3) t
(5) t
Figure 17 Selected design solutions from Figure 16
Mathematical Problems in Engineering 11
01 015 02 025 03 03509
095
1
105
11
115
12
125
13
135
14
1
2
3
4
56
Stru
ctur
al m
ass (
kg)
1(120582 + 1205961205960)
Figure 18 Approximate Pareto front of F6
= 45 and 120582 + 1205961205960 = 75573 = 44813 and 120582 + 1205961205960 = 65748
= 41914 and 120582 + 1205961205960 = 53343 = 38581 and 120582 + 1205961205960 = 4254
= 38571 and 120582 + 1205961205960 = 41494 = 34147 and 120582 + 1205961205960 = 30684
(1) t (2) t
(4) t
(6) t
(3) t
(5) t
Figure 19 Selected design solutions from Figure 18
More investigation on this issue would greatly improve theperformance of SPEA-N In cases of large-scale problemsonly SPEA using a quadratic function is superior to thenonsurrogate SPEA
6 Conclusions
The use of multiobjective evolutionary algorithm for thedesign of a torque arm leads to multiple optimum structuresfor decision making The hybridisation of SPEA and severalsurrogate models is developedThe various surrogate modelsincluding the quadratic regression technique the radialbasis function interpolation and the Kriging model canhelp improving the searching performance of the strengthPareto evolutionary algorithm For a design problem havingmass and natural frequency as objective functions and stress
and buckling as constraints SPEA using RBF is the bestmethod The SPEA using a quadratic regression is the bestfor the design case of F2 For the other multiobjective designproblems the best and the second best performers are SPEAusing the Krigingmodel and the quadratic RSM respectivelyNevertheless when considering all of the design problems theoverall top performer is SPEA2 using the quadratic regressionwhile the close second best uses theKrigingmodelMoreoverSPEA using a quadratic response surface model is the onlystrategy that is useful for solving large-scale problems Theuse of ANN for approximating constrained optimisationproblems is not effective as a proper network topologyoptimisation needs to be constructed before entering thesurrogate subproblem It is however possible to enhancethe performance of the ANN-based optimiser if an efficientnetwork topology optimisation is performed during theoptimisation process
12 Mathematical Problems in Engineering
20 25 30 35 40 45 50 55 60 65
F7
SPEA-OSPEA-Q
SPEA-RSPEA-K
f1
f2
minus60
minus50
minus40
minus30
minus20
minus10
0
Figure 20 Approximate Pareto front of F7
20 25 30 35 40 45 50 55 60
F8
SPEA-OSPEA-Q
SPEA-RSPEA-K
minus1000
minus950
minus900
minus850
minus800
minus750
minus700
minus650
minus600
minus550
f2
f1
Figure 21 Approximate Pareto front of F8
Acknowledgment
The authors are graceful for the support from the ThailandResearch Fund (RSA5280006)The first author also wishes tothank the office of higher education commission Thailandfor the CHE-PHD scholarship
References
[1] F Zong H Lin B Yu and X Pan ldquoDaily commute timeprediction based on genetic algorithmrdquoMathematical Problemsin Engineering vol 2012 Article ID 321574 20 pages 2012
[2] I Yao and D Han ldquoImproved barebones particle swarmoptimization with neighborhood search and its application on
ship designrdquo Mathematical Problems in Engineering vol 2013Article ID 175848 12 pages 2013
[3] Y Xu P Fan and L Yuan ldquoA simple and efficient artificial beecolony algorithmrdquo Mathematical Problems in Engineering vol2013 Article ID 526315 9 pages 2013
[4] S Talatahari E Khalili and S M Alavizadeh ldquoAcceleratedparticle swarm for optimum design of frame structuresrdquoMath-ematical Problems in Engineering vol 2013 Article ID 6498576 pages 2013
[5] K Socha and M Dorigo ldquoAnt colony optimization for contin-uous domainsrdquo European Journal of Operational Research vol185 no 3 pp 1155ndash1173 2008
[6] N Hansen S D Muller and P Koumoutsakos ldquoReducingthe time complexity of the derandomized evolution strategywith covariance matrix adaptation (CMA-ES)rdquo EvolutionaryComputation vol 11 no 1 pp 1ndash18 2003
[7] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[8] J D Knowles and D W Corne ldquoApproximating the nondom-inated front using the Pareto Archived Evolution StrategyrdquoEvolutionary Computation vol 8 no 2 pp 149ndash172 2000
[9] M Farina and P Amato ldquoLinked interpolation-optimizationstrategies for multicriteria optimization problemsrdquo Soft Com-puting vol 9 pp 54ndash65 2005
[10] J Knowles ldquoParEGO a hybrid algorithm with on-line land-scape approximation for expensive multiobjective optimizationproblemsrdquo IEEE Transactions on Evolutionary Computationvol 10 no 1 pp 50ndash66 2006
[11] Z Zhou Y S Ong M H Lim and B S Lee ldquoMemetic algo-rithm using multi-surrogates for computationally expensiveoptimization problemsrdquo Soft Computing vol 11 no 10 pp 957ndash971 2007
[12] R G Regis and C A Shoemaker ldquoLocal function approxima-tion in evolutionary algorithms for the optimization of costlyfunctionsrdquo IEEE Transactions on Evolutionary Computationvol 8 no 5 pp 490ndash505 2004
[13] N Pholdee and S Bureerat ldquoSurrogate-assisted evolutionaryoptimizers for multiobjective design of a torque arm structurerdquoAppliedMechanics andMaterials vol 101-102 pp 324ndash328 2011
[14] N Pholdee and S Bureerat ldquoPassive vibration control ofan automotive component using evolutionary optimisationrdquoJournal of Research and Applications in Mechanical Engineeringvol 1 pp 19ndash23 2011
[15] K C Giannakoglou D I Papadimitriou and I C KampolisldquoAerodynamic shape design using evolutionary algorithmsand new gradient-assisted metamodelsrdquo Computer Methods inApplied Mechanics and Engineering vol 195 no 44ndash47 pp6312ndash6329 2006
[16] Y S Ong K Y Lum P B Nair D M Shi and Z K ZhangldquoGlobal convergence of unconstrained and bound constrainedsurrogate-assisted evolutionary search in aerodynamic shapedesignrdquo in Proceedings of the Congress on Evolutionary Compu-tation (CEC rsquo03) pp 1856ndash1863 Canberra Australia December2003
[17] Y Tenne and S W Armfield ldquoA versatile surrogate-assistedmemetic algorithm for optimization of computationally expen-sive functions and its engineering applicationsrdquo Studies inComputational Intelligence vol 92 pp 43ndash72 2008
Mathematical Problems in Engineering 13
[18] S Srisomporn and S Bureerat ldquoGeometrical design of plate-fin heat sinks using hybridization of MOEA and RSMrdquo IEEETransactions onComponents and Packaging Technologies vol 31no 2 pp 351ndash360 2008
[19] S Kanyakam and S Bureerat ldquoComparative performance ofsurrogate-assisted MOEAs for geometrical design of pin-finheat sinksrdquo Applied Mathematics vol 2012 Article ID 53478314 pages 2012
[20] A Husain and K Y Kin ldquoOptimization of a microchannel heatsink with temperature dependent fluid propertiesrdquo Journal ofMaterials Processing Technology vol 208 no 1ndash3 pp 499ndash5062008
[21] J Yoo and C-Y Lee ldquoTopology optimization of a swing armtype actuator using the response surface methodrdquoMicrosystemTechnologies vol 13 no 1 pp 21ndash31 2007
[22] W Hu L G Yao and Z Z Hua ldquoOptimization of sheetmetal forming processes by adaptive response surface basedon intelligent sampling methodrdquo Journal of Materials ProcessingTechnology vol 197 no 1ndash3 pp 77ndash88 2008
[23] N Nariman-Zadeh A Darvizeh A Jamali and A MoeinildquoEvolutionary design of generalized polynomial neural net-works for modelling and prediction of explosive formingprocessrdquo Journal of Materials Processing Technology vol 164-165 pp 1561ndash1571 2005
[24] T Goel R Vaidyanathan R T Haftka W Shyy N V Queipoand K Tucker ldquoResponse surface approximation of Pareto opti-mal front in multi-objective optimizationrdquo Computer MethodsinAppliedMechanics and Engineering vol 196 no 4ndash6 pp 879ndash893 2007
[25] Z Zhou Y S Ong P B Nair A J Keane and K Y LumldquoCombining global and local surrogate models to accelerateevolutionary optimizationrdquo IEEE Transactions on Systems Manand Cybernetics Part C vol 37 no 1 pp 66ndash76 2007
[26] Y Jin ldquoA comprehensive survey of fitness approximation inevolutionary computationrdquo Soft Computing vol 9 pp 3ndash122005
[27] Y S Ong P B Nair A J Keane and K W Wong ldquoSurrogate-assisted evolutionary optimization frameworks for high-fidelityengineering design problemsrdquo in Knowledge Incorporation inEvolutionary Computation Studies in Fuzziness and Soft Com-puting Y Jin Ed pp 307ndash332 Springer Berlin Germany 2004
[28] Y Jin ldquoSurrogate-assisted evolutionary computation recentadvances and future challengesrdquo Swarm and Evolutionary Com-putation vol 1 pp 61ndash70 2011
[29] A I J Forrester and A J Keane ldquoRecent advances in surrogate-based optimizationrdquo Progress in Aerospace Sciences vol 45 no1ndash3 pp 50ndash79 2009
[30] K-T Chiang and F-P Chang ldquoApplication of response surfacemethodology in the parametric optimization of a pin-fin typeheat sinkrdquo International Communications in Heat and MassTransfer vol 33 no 7 pp 836ndash845 2006
[31] Y Tenne and S W Armfield ldquoA framework for memeticoptimization using variable global and local surrogate modelsrdquoSoft Computing vol 13 no 8-9 pp 781ndash793 2009
[32] P Breitkopf H Naceur A Rassineux and P Villon ldquoMovingleast squares response surface approximation formulation andmetal forming applicationsrdquo Computers and Structures vol 83no 17-18 pp 1411ndash1428 2005
[33] M Xiao B Raghavan P Breitkopf R F Coelho C Knopf-Lenoir and P Villon ldquoSurrogate models based on constrainedPOD for optimizationrdquo in Proceedings of the 4th European
Conference on Computational Mechanics (ECCM rsquo10) ParisFrance May 2010
[34] M Xiao P Breitkopf R F Coelho C Knopf-Lenoir MSidorkiewicz and P Villon ldquoModel reduction by CPOD andKriging application to the shape optimization of an intakeportrdquo Structural andMultidisciplinary Optimization vol 41 no4 pp 555ndash574 2010
[35] B RaghavanMHamdaouiM Xiao P Breitkopf and P VillonldquoA bi-level meta-modeling approach for structural optimizationusing modified POD bases and diffuse approximationrdquo Com-puters and Structures 2013
[36] B Raghavan and P Breitkopf ldquoAsynchronous evolutionaryshape optimization based on high-quality surrogates applica-tion to an air-conditioning ductrdquo Engineering with Computers2012
[37] E A Kontoleontos V G Asouti and K C GiannakoglouldquoAn asynchronous metamodel-assisted memetic algorithm forCFD-based shape optimizationrdquo Engineering Optimization vol44 no 2 pp 157ndash173 2012
[38] H Zhu L Liu T Long and L Peng ldquoA novel algorithmof maximin Latin hypercube design using successive localenumerationrdquo Engineering Optimization vol 44 no 5 pp 551ndash564 2012
[39] S D Georgiou and S Stylianou ldquoBlock-circulant matricesfor constructing optimal Latin hypercube designsrdquo Journal ofStatistical Planning and Inference vol 141 no 5 pp 1933ndash19432011
[40] J M Parr A J Keane A I J Forrester and C M E HoldenldquoInfill sampling criteria for surrogate-based optimization withconstraint handlingrdquo Engineering Optimization vol 44 no 10pp 1147ndash1166 2012
[41] E Zitzler M Laumanns and L Thiele ldquoSPEA2 improvingthe strength pareto evolutionary algorithm for multiobjectiveoptimizationrdquo inEvolutionaryMethods forDesign Optimizationand Control pp 95ndash100 2002
[42] S N Lophaven H B Neilson and J Sondergaard ldquoDACEa MATLAB kriging toolboxrdquo Tech Rep IMM-TR-2002-12Technical University of Denmark 2002
[43] R L Harder and R N Desmarais ldquoInterpolation using surfacesplinesrdquo Journal of Aircraft vol 9 no 2 pp 189ndash191 1972
[44] M E Botkin ldquoShape optimization with buckling and stressconstraintsrdquo AIAA Journal vol 34 no 2 pp 423ndash425 1996
[45] K Deb A Pratap and T Meyarivan ldquoConstrained test prob-lems for multi-objective evolutionary optimizationrdquo in Evolu-tionary Multi-Criterion Optimization (Zurich 2001) vol 1993of Lecture Notes in Computer Science pp 284ndash298 SpringerBerlin Germany 2001
[46] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 10: Research Article Surrogate-Assisted Multiobjective ...downloads.hindawi.com/journals/mpe/2013/695172.pdf · of engineering application, for example, aerodynamics [ ], heat and mass](https://reader034.fdocuments.net/reader034/viewer/2022052003/60164c5514f5a3172c3c2904/html5/thumbnails/10.jpg)
10 Mathematical Problems in Engineering
= 45 and 120596 = 10349 = 42534 and 120596 = 96302
= 39481 and 120596 = 88068 = 36521 and 120596 = 80402
= 36501 and 120596 = 80223 = 33791 and 120596 = 73022
(1) t (2) t
(4) t
(6) t(5) t
(3) t
Figure 15 Selected design solutions from Figure 14
02 025 03 035 04 045 0509
095
1
105
11
115
12
125
13
135
14
1
2
3
45
6
Stru
ctur
al m
ass (
kg)
1120582
Figure 16 Approximate Pareto front of F5
= 45 and 120582 = 62885 = 45 and 120582 = 53614
= 40881 and 120582 = 42008 = 37402 and 120582 = 31575
= 3652 and 120582 = 30888 = 34047 and 120582 = 22103
(1) t (2) t
(4) t
(6) t
(3) t
(5) t
Figure 17 Selected design solutions from Figure 16
Mathematical Problems in Engineering 11
01 015 02 025 03 03509
095
1
105
11
115
12
125
13
135
14
1
2
3
4
56
Stru
ctur
al m
ass (
kg)
1(120582 + 1205961205960)
Figure 18 Approximate Pareto front of F6
= 45 and 120582 + 1205961205960 = 75573 = 44813 and 120582 + 1205961205960 = 65748
= 41914 and 120582 + 1205961205960 = 53343 = 38581 and 120582 + 1205961205960 = 4254
= 38571 and 120582 + 1205961205960 = 41494 = 34147 and 120582 + 1205961205960 = 30684
(1) t (2) t
(4) t
(6) t
(3) t
(5) t
Figure 19 Selected design solutions from Figure 18
More investigation on this issue would greatly improve theperformance of SPEA-N In cases of large-scale problemsonly SPEA using a quadratic function is superior to thenonsurrogate SPEA
6 Conclusions
The use of multiobjective evolutionary algorithm for thedesign of a torque arm leads to multiple optimum structuresfor decision making The hybridisation of SPEA and severalsurrogate models is developedThe various surrogate modelsincluding the quadratic regression technique the radialbasis function interpolation and the Kriging model canhelp improving the searching performance of the strengthPareto evolutionary algorithm For a design problem havingmass and natural frequency as objective functions and stress
and buckling as constraints SPEA using RBF is the bestmethod The SPEA using a quadratic regression is the bestfor the design case of F2 For the other multiobjective designproblems the best and the second best performers are SPEAusing the Krigingmodel and the quadratic RSM respectivelyNevertheless when considering all of the design problems theoverall top performer is SPEA2 using the quadratic regressionwhile the close second best uses theKrigingmodelMoreoverSPEA using a quadratic response surface model is the onlystrategy that is useful for solving large-scale problems Theuse of ANN for approximating constrained optimisationproblems is not effective as a proper network topologyoptimisation needs to be constructed before entering thesurrogate subproblem It is however possible to enhancethe performance of the ANN-based optimiser if an efficientnetwork topology optimisation is performed during theoptimisation process
12 Mathematical Problems in Engineering
20 25 30 35 40 45 50 55 60 65
F7
SPEA-OSPEA-Q
SPEA-RSPEA-K
f1
f2
minus60
minus50
minus40
minus30
minus20
minus10
0
Figure 20 Approximate Pareto front of F7
20 25 30 35 40 45 50 55 60
F8
SPEA-OSPEA-Q
SPEA-RSPEA-K
minus1000
minus950
minus900
minus850
minus800
minus750
minus700
minus650
minus600
minus550
f2
f1
Figure 21 Approximate Pareto front of F8
Acknowledgment
The authors are graceful for the support from the ThailandResearch Fund (RSA5280006)The first author also wishes tothank the office of higher education commission Thailandfor the CHE-PHD scholarship
References
[1] F Zong H Lin B Yu and X Pan ldquoDaily commute timeprediction based on genetic algorithmrdquoMathematical Problemsin Engineering vol 2012 Article ID 321574 20 pages 2012
[2] I Yao and D Han ldquoImproved barebones particle swarmoptimization with neighborhood search and its application on
ship designrdquo Mathematical Problems in Engineering vol 2013Article ID 175848 12 pages 2013
[3] Y Xu P Fan and L Yuan ldquoA simple and efficient artificial beecolony algorithmrdquo Mathematical Problems in Engineering vol2013 Article ID 526315 9 pages 2013
[4] S Talatahari E Khalili and S M Alavizadeh ldquoAcceleratedparticle swarm for optimum design of frame structuresrdquoMath-ematical Problems in Engineering vol 2013 Article ID 6498576 pages 2013
[5] K Socha and M Dorigo ldquoAnt colony optimization for contin-uous domainsrdquo European Journal of Operational Research vol185 no 3 pp 1155ndash1173 2008
[6] N Hansen S D Muller and P Koumoutsakos ldquoReducingthe time complexity of the derandomized evolution strategywith covariance matrix adaptation (CMA-ES)rdquo EvolutionaryComputation vol 11 no 1 pp 1ndash18 2003
[7] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[8] J D Knowles and D W Corne ldquoApproximating the nondom-inated front using the Pareto Archived Evolution StrategyrdquoEvolutionary Computation vol 8 no 2 pp 149ndash172 2000
[9] M Farina and P Amato ldquoLinked interpolation-optimizationstrategies for multicriteria optimization problemsrdquo Soft Com-puting vol 9 pp 54ndash65 2005
[10] J Knowles ldquoParEGO a hybrid algorithm with on-line land-scape approximation for expensive multiobjective optimizationproblemsrdquo IEEE Transactions on Evolutionary Computationvol 10 no 1 pp 50ndash66 2006
[11] Z Zhou Y S Ong M H Lim and B S Lee ldquoMemetic algo-rithm using multi-surrogates for computationally expensiveoptimization problemsrdquo Soft Computing vol 11 no 10 pp 957ndash971 2007
[12] R G Regis and C A Shoemaker ldquoLocal function approxima-tion in evolutionary algorithms for the optimization of costlyfunctionsrdquo IEEE Transactions on Evolutionary Computationvol 8 no 5 pp 490ndash505 2004
[13] N Pholdee and S Bureerat ldquoSurrogate-assisted evolutionaryoptimizers for multiobjective design of a torque arm structurerdquoAppliedMechanics andMaterials vol 101-102 pp 324ndash328 2011
[14] N Pholdee and S Bureerat ldquoPassive vibration control ofan automotive component using evolutionary optimisationrdquoJournal of Research and Applications in Mechanical Engineeringvol 1 pp 19ndash23 2011
[15] K C Giannakoglou D I Papadimitriou and I C KampolisldquoAerodynamic shape design using evolutionary algorithmsand new gradient-assisted metamodelsrdquo Computer Methods inApplied Mechanics and Engineering vol 195 no 44ndash47 pp6312ndash6329 2006
[16] Y S Ong K Y Lum P B Nair D M Shi and Z K ZhangldquoGlobal convergence of unconstrained and bound constrainedsurrogate-assisted evolutionary search in aerodynamic shapedesignrdquo in Proceedings of the Congress on Evolutionary Compu-tation (CEC rsquo03) pp 1856ndash1863 Canberra Australia December2003
[17] Y Tenne and S W Armfield ldquoA versatile surrogate-assistedmemetic algorithm for optimization of computationally expen-sive functions and its engineering applicationsrdquo Studies inComputational Intelligence vol 92 pp 43ndash72 2008
Mathematical Problems in Engineering 13
[18] S Srisomporn and S Bureerat ldquoGeometrical design of plate-fin heat sinks using hybridization of MOEA and RSMrdquo IEEETransactions onComponents and Packaging Technologies vol 31no 2 pp 351ndash360 2008
[19] S Kanyakam and S Bureerat ldquoComparative performance ofsurrogate-assisted MOEAs for geometrical design of pin-finheat sinksrdquo Applied Mathematics vol 2012 Article ID 53478314 pages 2012
[20] A Husain and K Y Kin ldquoOptimization of a microchannel heatsink with temperature dependent fluid propertiesrdquo Journal ofMaterials Processing Technology vol 208 no 1ndash3 pp 499ndash5062008
[21] J Yoo and C-Y Lee ldquoTopology optimization of a swing armtype actuator using the response surface methodrdquoMicrosystemTechnologies vol 13 no 1 pp 21ndash31 2007
[22] W Hu L G Yao and Z Z Hua ldquoOptimization of sheetmetal forming processes by adaptive response surface basedon intelligent sampling methodrdquo Journal of Materials ProcessingTechnology vol 197 no 1ndash3 pp 77ndash88 2008
[23] N Nariman-Zadeh A Darvizeh A Jamali and A MoeinildquoEvolutionary design of generalized polynomial neural net-works for modelling and prediction of explosive formingprocessrdquo Journal of Materials Processing Technology vol 164-165 pp 1561ndash1571 2005
[24] T Goel R Vaidyanathan R T Haftka W Shyy N V Queipoand K Tucker ldquoResponse surface approximation of Pareto opti-mal front in multi-objective optimizationrdquo Computer MethodsinAppliedMechanics and Engineering vol 196 no 4ndash6 pp 879ndash893 2007
[25] Z Zhou Y S Ong P B Nair A J Keane and K Y LumldquoCombining global and local surrogate models to accelerateevolutionary optimizationrdquo IEEE Transactions on Systems Manand Cybernetics Part C vol 37 no 1 pp 66ndash76 2007
[26] Y Jin ldquoA comprehensive survey of fitness approximation inevolutionary computationrdquo Soft Computing vol 9 pp 3ndash122005
[27] Y S Ong P B Nair A J Keane and K W Wong ldquoSurrogate-assisted evolutionary optimization frameworks for high-fidelityengineering design problemsrdquo in Knowledge Incorporation inEvolutionary Computation Studies in Fuzziness and Soft Com-puting Y Jin Ed pp 307ndash332 Springer Berlin Germany 2004
[28] Y Jin ldquoSurrogate-assisted evolutionary computation recentadvances and future challengesrdquo Swarm and Evolutionary Com-putation vol 1 pp 61ndash70 2011
[29] A I J Forrester and A J Keane ldquoRecent advances in surrogate-based optimizationrdquo Progress in Aerospace Sciences vol 45 no1ndash3 pp 50ndash79 2009
[30] K-T Chiang and F-P Chang ldquoApplication of response surfacemethodology in the parametric optimization of a pin-fin typeheat sinkrdquo International Communications in Heat and MassTransfer vol 33 no 7 pp 836ndash845 2006
[31] Y Tenne and S W Armfield ldquoA framework for memeticoptimization using variable global and local surrogate modelsrdquoSoft Computing vol 13 no 8-9 pp 781ndash793 2009
[32] P Breitkopf H Naceur A Rassineux and P Villon ldquoMovingleast squares response surface approximation formulation andmetal forming applicationsrdquo Computers and Structures vol 83no 17-18 pp 1411ndash1428 2005
[33] M Xiao B Raghavan P Breitkopf R F Coelho C Knopf-Lenoir and P Villon ldquoSurrogate models based on constrainedPOD for optimizationrdquo in Proceedings of the 4th European
Conference on Computational Mechanics (ECCM rsquo10) ParisFrance May 2010
[34] M Xiao P Breitkopf R F Coelho C Knopf-Lenoir MSidorkiewicz and P Villon ldquoModel reduction by CPOD andKriging application to the shape optimization of an intakeportrdquo Structural andMultidisciplinary Optimization vol 41 no4 pp 555ndash574 2010
[35] B RaghavanMHamdaouiM Xiao P Breitkopf and P VillonldquoA bi-level meta-modeling approach for structural optimizationusing modified POD bases and diffuse approximationrdquo Com-puters and Structures 2013
[36] B Raghavan and P Breitkopf ldquoAsynchronous evolutionaryshape optimization based on high-quality surrogates applica-tion to an air-conditioning ductrdquo Engineering with Computers2012
[37] E A Kontoleontos V G Asouti and K C GiannakoglouldquoAn asynchronous metamodel-assisted memetic algorithm forCFD-based shape optimizationrdquo Engineering Optimization vol44 no 2 pp 157ndash173 2012
[38] H Zhu L Liu T Long and L Peng ldquoA novel algorithmof maximin Latin hypercube design using successive localenumerationrdquo Engineering Optimization vol 44 no 5 pp 551ndash564 2012
[39] S D Georgiou and S Stylianou ldquoBlock-circulant matricesfor constructing optimal Latin hypercube designsrdquo Journal ofStatistical Planning and Inference vol 141 no 5 pp 1933ndash19432011
[40] J M Parr A J Keane A I J Forrester and C M E HoldenldquoInfill sampling criteria for surrogate-based optimization withconstraint handlingrdquo Engineering Optimization vol 44 no 10pp 1147ndash1166 2012
[41] E Zitzler M Laumanns and L Thiele ldquoSPEA2 improvingthe strength pareto evolutionary algorithm for multiobjectiveoptimizationrdquo inEvolutionaryMethods forDesign Optimizationand Control pp 95ndash100 2002
[42] S N Lophaven H B Neilson and J Sondergaard ldquoDACEa MATLAB kriging toolboxrdquo Tech Rep IMM-TR-2002-12Technical University of Denmark 2002
[43] R L Harder and R N Desmarais ldquoInterpolation using surfacesplinesrdquo Journal of Aircraft vol 9 no 2 pp 189ndash191 1972
[44] M E Botkin ldquoShape optimization with buckling and stressconstraintsrdquo AIAA Journal vol 34 no 2 pp 423ndash425 1996
[45] K Deb A Pratap and T Meyarivan ldquoConstrained test prob-lems for multi-objective evolutionary optimizationrdquo in Evolu-tionary Multi-Criterion Optimization (Zurich 2001) vol 1993of Lecture Notes in Computer Science pp 284ndash298 SpringerBerlin Germany 2001
[46] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 11: Research Article Surrogate-Assisted Multiobjective ...downloads.hindawi.com/journals/mpe/2013/695172.pdf · of engineering application, for example, aerodynamics [ ], heat and mass](https://reader034.fdocuments.net/reader034/viewer/2022052003/60164c5514f5a3172c3c2904/html5/thumbnails/11.jpg)
Mathematical Problems in Engineering 11
01 015 02 025 03 03509
095
1
105
11
115
12
125
13
135
14
1
2
3
4
56
Stru
ctur
al m
ass (
kg)
1(120582 + 1205961205960)
Figure 18 Approximate Pareto front of F6
= 45 and 120582 + 1205961205960 = 75573 = 44813 and 120582 + 1205961205960 = 65748
= 41914 and 120582 + 1205961205960 = 53343 = 38581 and 120582 + 1205961205960 = 4254
= 38571 and 120582 + 1205961205960 = 41494 = 34147 and 120582 + 1205961205960 = 30684
(1) t (2) t
(4) t
(6) t
(3) t
(5) t
Figure 19 Selected design solutions from Figure 18
More investigation on this issue would greatly improve theperformance of SPEA-N In cases of large-scale problemsonly SPEA using a quadratic function is superior to thenonsurrogate SPEA
6 Conclusions
The use of multiobjective evolutionary algorithm for thedesign of a torque arm leads to multiple optimum structuresfor decision making The hybridisation of SPEA and severalsurrogate models is developedThe various surrogate modelsincluding the quadratic regression technique the radialbasis function interpolation and the Kriging model canhelp improving the searching performance of the strengthPareto evolutionary algorithm For a design problem havingmass and natural frequency as objective functions and stress
and buckling as constraints SPEA using RBF is the bestmethod The SPEA using a quadratic regression is the bestfor the design case of F2 For the other multiobjective designproblems the best and the second best performers are SPEAusing the Krigingmodel and the quadratic RSM respectivelyNevertheless when considering all of the design problems theoverall top performer is SPEA2 using the quadratic regressionwhile the close second best uses theKrigingmodelMoreoverSPEA using a quadratic response surface model is the onlystrategy that is useful for solving large-scale problems Theuse of ANN for approximating constrained optimisationproblems is not effective as a proper network topologyoptimisation needs to be constructed before entering thesurrogate subproblem It is however possible to enhancethe performance of the ANN-based optimiser if an efficientnetwork topology optimisation is performed during theoptimisation process
12 Mathematical Problems in Engineering
20 25 30 35 40 45 50 55 60 65
F7
SPEA-OSPEA-Q
SPEA-RSPEA-K
f1
f2
minus60
minus50
minus40
minus30
minus20
minus10
0
Figure 20 Approximate Pareto front of F7
20 25 30 35 40 45 50 55 60
F8
SPEA-OSPEA-Q
SPEA-RSPEA-K
minus1000
minus950
minus900
minus850
minus800
minus750
minus700
minus650
minus600
minus550
f2
f1
Figure 21 Approximate Pareto front of F8
Acknowledgment
The authors are graceful for the support from the ThailandResearch Fund (RSA5280006)The first author also wishes tothank the office of higher education commission Thailandfor the CHE-PHD scholarship
References
[1] F Zong H Lin B Yu and X Pan ldquoDaily commute timeprediction based on genetic algorithmrdquoMathematical Problemsin Engineering vol 2012 Article ID 321574 20 pages 2012
[2] I Yao and D Han ldquoImproved barebones particle swarmoptimization with neighborhood search and its application on
ship designrdquo Mathematical Problems in Engineering vol 2013Article ID 175848 12 pages 2013
[3] Y Xu P Fan and L Yuan ldquoA simple and efficient artificial beecolony algorithmrdquo Mathematical Problems in Engineering vol2013 Article ID 526315 9 pages 2013
[4] S Talatahari E Khalili and S M Alavizadeh ldquoAcceleratedparticle swarm for optimum design of frame structuresrdquoMath-ematical Problems in Engineering vol 2013 Article ID 6498576 pages 2013
[5] K Socha and M Dorigo ldquoAnt colony optimization for contin-uous domainsrdquo European Journal of Operational Research vol185 no 3 pp 1155ndash1173 2008
[6] N Hansen S D Muller and P Koumoutsakos ldquoReducingthe time complexity of the derandomized evolution strategywith covariance matrix adaptation (CMA-ES)rdquo EvolutionaryComputation vol 11 no 1 pp 1ndash18 2003
[7] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[8] J D Knowles and D W Corne ldquoApproximating the nondom-inated front using the Pareto Archived Evolution StrategyrdquoEvolutionary Computation vol 8 no 2 pp 149ndash172 2000
[9] M Farina and P Amato ldquoLinked interpolation-optimizationstrategies for multicriteria optimization problemsrdquo Soft Com-puting vol 9 pp 54ndash65 2005
[10] J Knowles ldquoParEGO a hybrid algorithm with on-line land-scape approximation for expensive multiobjective optimizationproblemsrdquo IEEE Transactions on Evolutionary Computationvol 10 no 1 pp 50ndash66 2006
[11] Z Zhou Y S Ong M H Lim and B S Lee ldquoMemetic algo-rithm using multi-surrogates for computationally expensiveoptimization problemsrdquo Soft Computing vol 11 no 10 pp 957ndash971 2007
[12] R G Regis and C A Shoemaker ldquoLocal function approxima-tion in evolutionary algorithms for the optimization of costlyfunctionsrdquo IEEE Transactions on Evolutionary Computationvol 8 no 5 pp 490ndash505 2004
[13] N Pholdee and S Bureerat ldquoSurrogate-assisted evolutionaryoptimizers for multiobjective design of a torque arm structurerdquoAppliedMechanics andMaterials vol 101-102 pp 324ndash328 2011
[14] N Pholdee and S Bureerat ldquoPassive vibration control ofan automotive component using evolutionary optimisationrdquoJournal of Research and Applications in Mechanical Engineeringvol 1 pp 19ndash23 2011
[15] K C Giannakoglou D I Papadimitriou and I C KampolisldquoAerodynamic shape design using evolutionary algorithmsand new gradient-assisted metamodelsrdquo Computer Methods inApplied Mechanics and Engineering vol 195 no 44ndash47 pp6312ndash6329 2006
[16] Y S Ong K Y Lum P B Nair D M Shi and Z K ZhangldquoGlobal convergence of unconstrained and bound constrainedsurrogate-assisted evolutionary search in aerodynamic shapedesignrdquo in Proceedings of the Congress on Evolutionary Compu-tation (CEC rsquo03) pp 1856ndash1863 Canberra Australia December2003
[17] Y Tenne and S W Armfield ldquoA versatile surrogate-assistedmemetic algorithm for optimization of computationally expen-sive functions and its engineering applicationsrdquo Studies inComputational Intelligence vol 92 pp 43ndash72 2008
Mathematical Problems in Engineering 13
[18] S Srisomporn and S Bureerat ldquoGeometrical design of plate-fin heat sinks using hybridization of MOEA and RSMrdquo IEEETransactions onComponents and Packaging Technologies vol 31no 2 pp 351ndash360 2008
[19] S Kanyakam and S Bureerat ldquoComparative performance ofsurrogate-assisted MOEAs for geometrical design of pin-finheat sinksrdquo Applied Mathematics vol 2012 Article ID 53478314 pages 2012
[20] A Husain and K Y Kin ldquoOptimization of a microchannel heatsink with temperature dependent fluid propertiesrdquo Journal ofMaterials Processing Technology vol 208 no 1ndash3 pp 499ndash5062008
[21] J Yoo and C-Y Lee ldquoTopology optimization of a swing armtype actuator using the response surface methodrdquoMicrosystemTechnologies vol 13 no 1 pp 21ndash31 2007
[22] W Hu L G Yao and Z Z Hua ldquoOptimization of sheetmetal forming processes by adaptive response surface basedon intelligent sampling methodrdquo Journal of Materials ProcessingTechnology vol 197 no 1ndash3 pp 77ndash88 2008
[23] N Nariman-Zadeh A Darvizeh A Jamali and A MoeinildquoEvolutionary design of generalized polynomial neural net-works for modelling and prediction of explosive formingprocessrdquo Journal of Materials Processing Technology vol 164-165 pp 1561ndash1571 2005
[24] T Goel R Vaidyanathan R T Haftka W Shyy N V Queipoand K Tucker ldquoResponse surface approximation of Pareto opti-mal front in multi-objective optimizationrdquo Computer MethodsinAppliedMechanics and Engineering vol 196 no 4ndash6 pp 879ndash893 2007
[25] Z Zhou Y S Ong P B Nair A J Keane and K Y LumldquoCombining global and local surrogate models to accelerateevolutionary optimizationrdquo IEEE Transactions on Systems Manand Cybernetics Part C vol 37 no 1 pp 66ndash76 2007
[26] Y Jin ldquoA comprehensive survey of fitness approximation inevolutionary computationrdquo Soft Computing vol 9 pp 3ndash122005
[27] Y S Ong P B Nair A J Keane and K W Wong ldquoSurrogate-assisted evolutionary optimization frameworks for high-fidelityengineering design problemsrdquo in Knowledge Incorporation inEvolutionary Computation Studies in Fuzziness and Soft Com-puting Y Jin Ed pp 307ndash332 Springer Berlin Germany 2004
[28] Y Jin ldquoSurrogate-assisted evolutionary computation recentadvances and future challengesrdquo Swarm and Evolutionary Com-putation vol 1 pp 61ndash70 2011
[29] A I J Forrester and A J Keane ldquoRecent advances in surrogate-based optimizationrdquo Progress in Aerospace Sciences vol 45 no1ndash3 pp 50ndash79 2009
[30] K-T Chiang and F-P Chang ldquoApplication of response surfacemethodology in the parametric optimization of a pin-fin typeheat sinkrdquo International Communications in Heat and MassTransfer vol 33 no 7 pp 836ndash845 2006
[31] Y Tenne and S W Armfield ldquoA framework for memeticoptimization using variable global and local surrogate modelsrdquoSoft Computing vol 13 no 8-9 pp 781ndash793 2009
[32] P Breitkopf H Naceur A Rassineux and P Villon ldquoMovingleast squares response surface approximation formulation andmetal forming applicationsrdquo Computers and Structures vol 83no 17-18 pp 1411ndash1428 2005
[33] M Xiao B Raghavan P Breitkopf R F Coelho C Knopf-Lenoir and P Villon ldquoSurrogate models based on constrainedPOD for optimizationrdquo in Proceedings of the 4th European
Conference on Computational Mechanics (ECCM rsquo10) ParisFrance May 2010
[34] M Xiao P Breitkopf R F Coelho C Knopf-Lenoir MSidorkiewicz and P Villon ldquoModel reduction by CPOD andKriging application to the shape optimization of an intakeportrdquo Structural andMultidisciplinary Optimization vol 41 no4 pp 555ndash574 2010
[35] B RaghavanMHamdaouiM Xiao P Breitkopf and P VillonldquoA bi-level meta-modeling approach for structural optimizationusing modified POD bases and diffuse approximationrdquo Com-puters and Structures 2013
[36] B Raghavan and P Breitkopf ldquoAsynchronous evolutionaryshape optimization based on high-quality surrogates applica-tion to an air-conditioning ductrdquo Engineering with Computers2012
[37] E A Kontoleontos V G Asouti and K C GiannakoglouldquoAn asynchronous metamodel-assisted memetic algorithm forCFD-based shape optimizationrdquo Engineering Optimization vol44 no 2 pp 157ndash173 2012
[38] H Zhu L Liu T Long and L Peng ldquoA novel algorithmof maximin Latin hypercube design using successive localenumerationrdquo Engineering Optimization vol 44 no 5 pp 551ndash564 2012
[39] S D Georgiou and S Stylianou ldquoBlock-circulant matricesfor constructing optimal Latin hypercube designsrdquo Journal ofStatistical Planning and Inference vol 141 no 5 pp 1933ndash19432011
[40] J M Parr A J Keane A I J Forrester and C M E HoldenldquoInfill sampling criteria for surrogate-based optimization withconstraint handlingrdquo Engineering Optimization vol 44 no 10pp 1147ndash1166 2012
[41] E Zitzler M Laumanns and L Thiele ldquoSPEA2 improvingthe strength pareto evolutionary algorithm for multiobjectiveoptimizationrdquo inEvolutionaryMethods forDesign Optimizationand Control pp 95ndash100 2002
[42] S N Lophaven H B Neilson and J Sondergaard ldquoDACEa MATLAB kriging toolboxrdquo Tech Rep IMM-TR-2002-12Technical University of Denmark 2002
[43] R L Harder and R N Desmarais ldquoInterpolation using surfacesplinesrdquo Journal of Aircraft vol 9 no 2 pp 189ndash191 1972
[44] M E Botkin ldquoShape optimization with buckling and stressconstraintsrdquo AIAA Journal vol 34 no 2 pp 423ndash425 1996
[45] K Deb A Pratap and T Meyarivan ldquoConstrained test prob-lems for multi-objective evolutionary optimizationrdquo in Evolu-tionary Multi-Criterion Optimization (Zurich 2001) vol 1993of Lecture Notes in Computer Science pp 284ndash298 SpringerBerlin Germany 2001
[46] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 12: Research Article Surrogate-Assisted Multiobjective ...downloads.hindawi.com/journals/mpe/2013/695172.pdf · of engineering application, for example, aerodynamics [ ], heat and mass](https://reader034.fdocuments.net/reader034/viewer/2022052003/60164c5514f5a3172c3c2904/html5/thumbnails/12.jpg)
12 Mathematical Problems in Engineering
20 25 30 35 40 45 50 55 60 65
F7
SPEA-OSPEA-Q
SPEA-RSPEA-K
f1
f2
minus60
minus50
minus40
minus30
minus20
minus10
0
Figure 20 Approximate Pareto front of F7
20 25 30 35 40 45 50 55 60
F8
SPEA-OSPEA-Q
SPEA-RSPEA-K
minus1000
minus950
minus900
minus850
minus800
minus750
minus700
minus650
minus600
minus550
f2
f1
Figure 21 Approximate Pareto front of F8
Acknowledgment
The authors are graceful for the support from the ThailandResearch Fund (RSA5280006)The first author also wishes tothank the office of higher education commission Thailandfor the CHE-PHD scholarship
References
[1] F Zong H Lin B Yu and X Pan ldquoDaily commute timeprediction based on genetic algorithmrdquoMathematical Problemsin Engineering vol 2012 Article ID 321574 20 pages 2012
[2] I Yao and D Han ldquoImproved barebones particle swarmoptimization with neighborhood search and its application on
ship designrdquo Mathematical Problems in Engineering vol 2013Article ID 175848 12 pages 2013
[3] Y Xu P Fan and L Yuan ldquoA simple and efficient artificial beecolony algorithmrdquo Mathematical Problems in Engineering vol2013 Article ID 526315 9 pages 2013
[4] S Talatahari E Khalili and S M Alavizadeh ldquoAcceleratedparticle swarm for optimum design of frame structuresrdquoMath-ematical Problems in Engineering vol 2013 Article ID 6498576 pages 2013
[5] K Socha and M Dorigo ldquoAnt colony optimization for contin-uous domainsrdquo European Journal of Operational Research vol185 no 3 pp 1155ndash1173 2008
[6] N Hansen S D Muller and P Koumoutsakos ldquoReducingthe time complexity of the derandomized evolution strategywith covariance matrix adaptation (CMA-ES)rdquo EvolutionaryComputation vol 11 no 1 pp 1ndash18 2003
[7] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[8] J D Knowles and D W Corne ldquoApproximating the nondom-inated front using the Pareto Archived Evolution StrategyrdquoEvolutionary Computation vol 8 no 2 pp 149ndash172 2000
[9] M Farina and P Amato ldquoLinked interpolation-optimizationstrategies for multicriteria optimization problemsrdquo Soft Com-puting vol 9 pp 54ndash65 2005
[10] J Knowles ldquoParEGO a hybrid algorithm with on-line land-scape approximation for expensive multiobjective optimizationproblemsrdquo IEEE Transactions on Evolutionary Computationvol 10 no 1 pp 50ndash66 2006
[11] Z Zhou Y S Ong M H Lim and B S Lee ldquoMemetic algo-rithm using multi-surrogates for computationally expensiveoptimization problemsrdquo Soft Computing vol 11 no 10 pp 957ndash971 2007
[12] R G Regis and C A Shoemaker ldquoLocal function approxima-tion in evolutionary algorithms for the optimization of costlyfunctionsrdquo IEEE Transactions on Evolutionary Computationvol 8 no 5 pp 490ndash505 2004
[13] N Pholdee and S Bureerat ldquoSurrogate-assisted evolutionaryoptimizers for multiobjective design of a torque arm structurerdquoAppliedMechanics andMaterials vol 101-102 pp 324ndash328 2011
[14] N Pholdee and S Bureerat ldquoPassive vibration control ofan automotive component using evolutionary optimisationrdquoJournal of Research and Applications in Mechanical Engineeringvol 1 pp 19ndash23 2011
[15] K C Giannakoglou D I Papadimitriou and I C KampolisldquoAerodynamic shape design using evolutionary algorithmsand new gradient-assisted metamodelsrdquo Computer Methods inApplied Mechanics and Engineering vol 195 no 44ndash47 pp6312ndash6329 2006
[16] Y S Ong K Y Lum P B Nair D M Shi and Z K ZhangldquoGlobal convergence of unconstrained and bound constrainedsurrogate-assisted evolutionary search in aerodynamic shapedesignrdquo in Proceedings of the Congress on Evolutionary Compu-tation (CEC rsquo03) pp 1856ndash1863 Canberra Australia December2003
[17] Y Tenne and S W Armfield ldquoA versatile surrogate-assistedmemetic algorithm for optimization of computationally expen-sive functions and its engineering applicationsrdquo Studies inComputational Intelligence vol 92 pp 43ndash72 2008
Mathematical Problems in Engineering 13
[18] S Srisomporn and S Bureerat ldquoGeometrical design of plate-fin heat sinks using hybridization of MOEA and RSMrdquo IEEETransactions onComponents and Packaging Technologies vol 31no 2 pp 351ndash360 2008
[19] S Kanyakam and S Bureerat ldquoComparative performance ofsurrogate-assisted MOEAs for geometrical design of pin-finheat sinksrdquo Applied Mathematics vol 2012 Article ID 53478314 pages 2012
[20] A Husain and K Y Kin ldquoOptimization of a microchannel heatsink with temperature dependent fluid propertiesrdquo Journal ofMaterials Processing Technology vol 208 no 1ndash3 pp 499ndash5062008
[21] J Yoo and C-Y Lee ldquoTopology optimization of a swing armtype actuator using the response surface methodrdquoMicrosystemTechnologies vol 13 no 1 pp 21ndash31 2007
[22] W Hu L G Yao and Z Z Hua ldquoOptimization of sheetmetal forming processes by adaptive response surface basedon intelligent sampling methodrdquo Journal of Materials ProcessingTechnology vol 197 no 1ndash3 pp 77ndash88 2008
[23] N Nariman-Zadeh A Darvizeh A Jamali and A MoeinildquoEvolutionary design of generalized polynomial neural net-works for modelling and prediction of explosive formingprocessrdquo Journal of Materials Processing Technology vol 164-165 pp 1561ndash1571 2005
[24] T Goel R Vaidyanathan R T Haftka W Shyy N V Queipoand K Tucker ldquoResponse surface approximation of Pareto opti-mal front in multi-objective optimizationrdquo Computer MethodsinAppliedMechanics and Engineering vol 196 no 4ndash6 pp 879ndash893 2007
[25] Z Zhou Y S Ong P B Nair A J Keane and K Y LumldquoCombining global and local surrogate models to accelerateevolutionary optimizationrdquo IEEE Transactions on Systems Manand Cybernetics Part C vol 37 no 1 pp 66ndash76 2007
[26] Y Jin ldquoA comprehensive survey of fitness approximation inevolutionary computationrdquo Soft Computing vol 9 pp 3ndash122005
[27] Y S Ong P B Nair A J Keane and K W Wong ldquoSurrogate-assisted evolutionary optimization frameworks for high-fidelityengineering design problemsrdquo in Knowledge Incorporation inEvolutionary Computation Studies in Fuzziness and Soft Com-puting Y Jin Ed pp 307ndash332 Springer Berlin Germany 2004
[28] Y Jin ldquoSurrogate-assisted evolutionary computation recentadvances and future challengesrdquo Swarm and Evolutionary Com-putation vol 1 pp 61ndash70 2011
[29] A I J Forrester and A J Keane ldquoRecent advances in surrogate-based optimizationrdquo Progress in Aerospace Sciences vol 45 no1ndash3 pp 50ndash79 2009
[30] K-T Chiang and F-P Chang ldquoApplication of response surfacemethodology in the parametric optimization of a pin-fin typeheat sinkrdquo International Communications in Heat and MassTransfer vol 33 no 7 pp 836ndash845 2006
[31] Y Tenne and S W Armfield ldquoA framework for memeticoptimization using variable global and local surrogate modelsrdquoSoft Computing vol 13 no 8-9 pp 781ndash793 2009
[32] P Breitkopf H Naceur A Rassineux and P Villon ldquoMovingleast squares response surface approximation formulation andmetal forming applicationsrdquo Computers and Structures vol 83no 17-18 pp 1411ndash1428 2005
[33] M Xiao B Raghavan P Breitkopf R F Coelho C Knopf-Lenoir and P Villon ldquoSurrogate models based on constrainedPOD for optimizationrdquo in Proceedings of the 4th European
Conference on Computational Mechanics (ECCM rsquo10) ParisFrance May 2010
[34] M Xiao P Breitkopf R F Coelho C Knopf-Lenoir MSidorkiewicz and P Villon ldquoModel reduction by CPOD andKriging application to the shape optimization of an intakeportrdquo Structural andMultidisciplinary Optimization vol 41 no4 pp 555ndash574 2010
[35] B RaghavanMHamdaouiM Xiao P Breitkopf and P VillonldquoA bi-level meta-modeling approach for structural optimizationusing modified POD bases and diffuse approximationrdquo Com-puters and Structures 2013
[36] B Raghavan and P Breitkopf ldquoAsynchronous evolutionaryshape optimization based on high-quality surrogates applica-tion to an air-conditioning ductrdquo Engineering with Computers2012
[37] E A Kontoleontos V G Asouti and K C GiannakoglouldquoAn asynchronous metamodel-assisted memetic algorithm forCFD-based shape optimizationrdquo Engineering Optimization vol44 no 2 pp 157ndash173 2012
[38] H Zhu L Liu T Long and L Peng ldquoA novel algorithmof maximin Latin hypercube design using successive localenumerationrdquo Engineering Optimization vol 44 no 5 pp 551ndash564 2012
[39] S D Georgiou and S Stylianou ldquoBlock-circulant matricesfor constructing optimal Latin hypercube designsrdquo Journal ofStatistical Planning and Inference vol 141 no 5 pp 1933ndash19432011
[40] J M Parr A J Keane A I J Forrester and C M E HoldenldquoInfill sampling criteria for surrogate-based optimization withconstraint handlingrdquo Engineering Optimization vol 44 no 10pp 1147ndash1166 2012
[41] E Zitzler M Laumanns and L Thiele ldquoSPEA2 improvingthe strength pareto evolutionary algorithm for multiobjectiveoptimizationrdquo inEvolutionaryMethods forDesign Optimizationand Control pp 95ndash100 2002
[42] S N Lophaven H B Neilson and J Sondergaard ldquoDACEa MATLAB kriging toolboxrdquo Tech Rep IMM-TR-2002-12Technical University of Denmark 2002
[43] R L Harder and R N Desmarais ldquoInterpolation using surfacesplinesrdquo Journal of Aircraft vol 9 no 2 pp 189ndash191 1972
[44] M E Botkin ldquoShape optimization with buckling and stressconstraintsrdquo AIAA Journal vol 34 no 2 pp 423ndash425 1996
[45] K Deb A Pratap and T Meyarivan ldquoConstrained test prob-lems for multi-objective evolutionary optimizationrdquo in Evolu-tionary Multi-Criterion Optimization (Zurich 2001) vol 1993of Lecture Notes in Computer Science pp 284ndash298 SpringerBerlin Germany 2001
[46] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 13: Research Article Surrogate-Assisted Multiobjective ...downloads.hindawi.com/journals/mpe/2013/695172.pdf · of engineering application, for example, aerodynamics [ ], heat and mass](https://reader034.fdocuments.net/reader034/viewer/2022052003/60164c5514f5a3172c3c2904/html5/thumbnails/13.jpg)
Mathematical Problems in Engineering 13
[18] S Srisomporn and S Bureerat ldquoGeometrical design of plate-fin heat sinks using hybridization of MOEA and RSMrdquo IEEETransactions onComponents and Packaging Technologies vol 31no 2 pp 351ndash360 2008
[19] S Kanyakam and S Bureerat ldquoComparative performance ofsurrogate-assisted MOEAs for geometrical design of pin-finheat sinksrdquo Applied Mathematics vol 2012 Article ID 53478314 pages 2012
[20] A Husain and K Y Kin ldquoOptimization of a microchannel heatsink with temperature dependent fluid propertiesrdquo Journal ofMaterials Processing Technology vol 208 no 1ndash3 pp 499ndash5062008
[21] J Yoo and C-Y Lee ldquoTopology optimization of a swing armtype actuator using the response surface methodrdquoMicrosystemTechnologies vol 13 no 1 pp 21ndash31 2007
[22] W Hu L G Yao and Z Z Hua ldquoOptimization of sheetmetal forming processes by adaptive response surface basedon intelligent sampling methodrdquo Journal of Materials ProcessingTechnology vol 197 no 1ndash3 pp 77ndash88 2008
[23] N Nariman-Zadeh A Darvizeh A Jamali and A MoeinildquoEvolutionary design of generalized polynomial neural net-works for modelling and prediction of explosive formingprocessrdquo Journal of Materials Processing Technology vol 164-165 pp 1561ndash1571 2005
[24] T Goel R Vaidyanathan R T Haftka W Shyy N V Queipoand K Tucker ldquoResponse surface approximation of Pareto opti-mal front in multi-objective optimizationrdquo Computer MethodsinAppliedMechanics and Engineering vol 196 no 4ndash6 pp 879ndash893 2007
[25] Z Zhou Y S Ong P B Nair A J Keane and K Y LumldquoCombining global and local surrogate models to accelerateevolutionary optimizationrdquo IEEE Transactions on Systems Manand Cybernetics Part C vol 37 no 1 pp 66ndash76 2007
[26] Y Jin ldquoA comprehensive survey of fitness approximation inevolutionary computationrdquo Soft Computing vol 9 pp 3ndash122005
[27] Y S Ong P B Nair A J Keane and K W Wong ldquoSurrogate-assisted evolutionary optimization frameworks for high-fidelityengineering design problemsrdquo in Knowledge Incorporation inEvolutionary Computation Studies in Fuzziness and Soft Com-puting Y Jin Ed pp 307ndash332 Springer Berlin Germany 2004
[28] Y Jin ldquoSurrogate-assisted evolutionary computation recentadvances and future challengesrdquo Swarm and Evolutionary Com-putation vol 1 pp 61ndash70 2011
[29] A I J Forrester and A J Keane ldquoRecent advances in surrogate-based optimizationrdquo Progress in Aerospace Sciences vol 45 no1ndash3 pp 50ndash79 2009
[30] K-T Chiang and F-P Chang ldquoApplication of response surfacemethodology in the parametric optimization of a pin-fin typeheat sinkrdquo International Communications in Heat and MassTransfer vol 33 no 7 pp 836ndash845 2006
[31] Y Tenne and S W Armfield ldquoA framework for memeticoptimization using variable global and local surrogate modelsrdquoSoft Computing vol 13 no 8-9 pp 781ndash793 2009
[32] P Breitkopf H Naceur A Rassineux and P Villon ldquoMovingleast squares response surface approximation formulation andmetal forming applicationsrdquo Computers and Structures vol 83no 17-18 pp 1411ndash1428 2005
[33] M Xiao B Raghavan P Breitkopf R F Coelho C Knopf-Lenoir and P Villon ldquoSurrogate models based on constrainedPOD for optimizationrdquo in Proceedings of the 4th European
Conference on Computational Mechanics (ECCM rsquo10) ParisFrance May 2010
[34] M Xiao P Breitkopf R F Coelho C Knopf-Lenoir MSidorkiewicz and P Villon ldquoModel reduction by CPOD andKriging application to the shape optimization of an intakeportrdquo Structural andMultidisciplinary Optimization vol 41 no4 pp 555ndash574 2010
[35] B RaghavanMHamdaouiM Xiao P Breitkopf and P VillonldquoA bi-level meta-modeling approach for structural optimizationusing modified POD bases and diffuse approximationrdquo Com-puters and Structures 2013
[36] B Raghavan and P Breitkopf ldquoAsynchronous evolutionaryshape optimization based on high-quality surrogates applica-tion to an air-conditioning ductrdquo Engineering with Computers2012
[37] E A Kontoleontos V G Asouti and K C GiannakoglouldquoAn asynchronous metamodel-assisted memetic algorithm forCFD-based shape optimizationrdquo Engineering Optimization vol44 no 2 pp 157ndash173 2012
[38] H Zhu L Liu T Long and L Peng ldquoA novel algorithmof maximin Latin hypercube design using successive localenumerationrdquo Engineering Optimization vol 44 no 5 pp 551ndash564 2012
[39] S D Georgiou and S Stylianou ldquoBlock-circulant matricesfor constructing optimal Latin hypercube designsrdquo Journal ofStatistical Planning and Inference vol 141 no 5 pp 1933ndash19432011
[40] J M Parr A J Keane A I J Forrester and C M E HoldenldquoInfill sampling criteria for surrogate-based optimization withconstraint handlingrdquo Engineering Optimization vol 44 no 10pp 1147ndash1166 2012
[41] E Zitzler M Laumanns and L Thiele ldquoSPEA2 improvingthe strength pareto evolutionary algorithm for multiobjectiveoptimizationrdquo inEvolutionaryMethods forDesign Optimizationand Control pp 95ndash100 2002
[42] S N Lophaven H B Neilson and J Sondergaard ldquoDACEa MATLAB kriging toolboxrdquo Tech Rep IMM-TR-2002-12Technical University of Denmark 2002
[43] R L Harder and R N Desmarais ldquoInterpolation using surfacesplinesrdquo Journal of Aircraft vol 9 no 2 pp 189ndash191 1972
[44] M E Botkin ldquoShape optimization with buckling and stressconstraintsrdquo AIAA Journal vol 34 no 2 pp 423ndash425 1996
[45] K Deb A Pratap and T Meyarivan ldquoConstrained test prob-lems for multi-objective evolutionary optimizationrdquo in Evolu-tionary Multi-Criterion Optimization (Zurich 2001) vol 1993of Lecture Notes in Computer Science pp 284ndash298 SpringerBerlin Germany 2001
[46] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 14: Research Article Surrogate-Assisted Multiobjective ...downloads.hindawi.com/journals/mpe/2013/695172.pdf · of engineering application, for example, aerodynamics [ ], heat and mass](https://reader034.fdocuments.net/reader034/viewer/2022052003/60164c5514f5a3172c3c2904/html5/thumbnails/14.jpg)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
