Reliability-Based Design Optimization Using Response ... · including the response surface method...

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Chwail KimSenior Researcher

Agency for Defense Development,P.O. Box 18 Jinhae,

Kyungnam 645-600, Koreae-mail: [email protected]

K. K. Choi1

Roy J. Carver ProfessorDepartment of Mechanical

and Industrial Engineering,College of Engineering,The University of Iowa,

Iowa City, IA 52242e-mail: [email protected]

Reliability-Based DesignOptimization Using ResponseSurface Method With PredictionInterval EstimationSince variances in the input variables of the engineering system cause subsequent vari-ances in the product output performance, reliability-based design optimization (RBDO) isgetting much attention recently. However, RBDO requires expensive computational time.Therefore, the response surface method is often used for computational efficiency insolving RBDO problems. A method to estimate the effect of the response surface error onthe RBDO result is developed in this paper. The effect of the error is expressed in termsof the prediction interval, which is utilized as the error metric for the response surfaceused for RBDO. The prediction interval provides upper and lower bounds for the confi-dence level that the design engineer specified. Using the prediction interval of the re-sponse surface, the upper and lower limits of the reliability are computed. The lower limitof reliability is compared with the target reliability to obtain a conservative optimumdesign and thus safeguard against the inaccuracy of the response surface. On the otherhand, in order to avoid obtaining a design that is too conservative, the developed methodalso constrains the upper limit of the reliability in the design optimization process. Theproposed procedure is combined with an adaptive sampling strategy to refine the re-sponse surface. Numerical examples show the usefulness and the efficiency of the pro-posed method. �DOI: 10.1115/1.2988476�

Keywords: reliability-based design optimization, response surface method, predictioninterval, moving least squares method

IntroductionThe existence of variances in the input variables of engineering

ystems due to manufacturing processes and operational condi-ions causes subsequent variances in the product output perfor-

ance. To obtain reliable designs, often a probabilistic designethod called reliability-based design optimization �RBDO� is

sed �1–7�. The RBDO formulation may involve the same costunction as deterministic optimization but contains probabilisticerformance constraints for considering the probability of theatisfaction/failure of the output performances.

In the practical industry design process, RBDO is necessary toake cost effective quality products, but it requires expensive

omputational time. Therefore, several metamodeling methods,ncluding the response surface method �RSM� �7–11� and the krig-ng method �12–14�, are used for RBDO problems for computa-ional efficiency.

In RSM-based RBDO, the accuracy of the response surfaceRS� is critical. Unfortunately, the conventional measures of theccuracy of RSM are based mostly on the error values at theampling points. Thus, in the conventional design process usingSM, when the global response surface is constructed using theseonventional error estimators, it is used as an accurate functionvaluator even though accuracy will depend on the location wherehe function needs to be estimated. Especially for RBDO, a smallesponse surface error could cause a significant error in the esti-ation of the reliability, and eventually, the optimized design may

1Corresponding author.Contributed by the Design Automation Committee of ASME for publication in the

OURNAL OF MECHANICAL DESIGN. Manuscript received July 24, 2007; final manuscripteceived July 30, 2008; published online October 7, 2008. Review conducted byissimos P. Mourelators. Paper presented at the Seventh World Congress on Struc-

ural and Multidisciplinary Optimization, 2007.

ournal of Mechanical Design Copyright © 20

ded 08 Oct 2008 to 128.255.53.136. Redistribution subject to ASM

not satisfy the target reliability. Thus, these error metrics do notprovide good information on the judgment of the accuracy of theresponse surface for use in RBDO, and a new error metric needsto be developed for a successful application of RSM to RBDO.

The inaccuracy of the metamodels can be interpreted as themetamodel uncertainty, where the true response is unknown ex-cept at the sampled points. Several researchers �15–17� studiedhow to quantify this kind of uncertainty. Hazelrigg �18� studiedthe estimation of the model error of mathematical models in en-gineering design. Vittal and Hajela �19� used confidence intervalsfor the estimation of reliability using RSM but did not extend it toapply to RBDO. Apley et al. �20� proposed the Bayesian predic-tion interval with the Gaussian random process �GRP� to quantifythe effect of metamodel uncertainty �they referred to this as “in-terpolation uncertainty”� for robust design optimization problems.They developed the closed-form prediction intervals and also dis-cussed additional simulations.

Martin and Simpson �21� studied how to quantify the modeluncertainty using the kriging model, and they �22� developed amethodology to quantify the model uncertainty impact with inputparameter uncertainty for a system-level decision making. Theyused the Monte Carlo simulation �MCS� and the coefficient ofdetermination for prediction, Rprediction

2 as a criterion to assess amodel quality. Mahadevan et al. �23� listed several kinds of un-certainties and errors such as the model form error �24,25�, thesolution approximation error, and the error in reliability analysis.They proposed an iterative method to include model errors inRBDO problems. A random variable that represented the modelerrors was added to the classical RBDO formulation, and MonteCarlo simulation was used to include the reliability analysis errorfrom a conventional first-order reliability method �FORM�.

Many recent papers have quantified the various kinds of uncer-

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ainties, including the metamodel uncertainty. However, very fewapers have applied the RBDO problem and also most of themave limited one-sided constraints.

This paper proposes to use the prediction interval �26–28� tostimate the effect of the response surface error on the RBDOesults. A number of different methods can be used to estimate thepproximation uncertainty. For the response surface, the movingeast squares method �MLSM� �29–32�, which is a locallyeighted approximation method, is used in this paper. The inter-al prediction can be used as an effective measure of accuracy forBDO. The proposed interval concept computes the probabilistic

nterval of the response based on the user-specified confidenceevel. Using the prediction interval of the response surface, thepper and lower limits of the reliability can be computed. Theower limit of reliability is compared with the target reliability inBDO, which will provide a conservative optimum design to

afeguard against the inaccuracy of the response surface. On thether hand, in order to avoid obtaining a too conservative designhat may not be close to the true optimum design, the proposed

ethod constrains the upper limit of the reliability in the RBDOrocess. The proposed procedure is combined with an adaptiveampling strategy that decides where to sample and how manyoints to sample. Therefore, this method can give a guideline forhe sampling location and the convergence criterion. Numericalxamples show the effectiveness and the efficiency of the pro-osed method.

Measure of Accuracy of the Response Surface

2.1 Conventional Error Metrics of the Response Surface.he conventional error metrics of the response surfaces are basedsually on the error values at the sampling points, such as theean square error. Even if the mean square error is very small, the

esponse surface results at the nonsampling points may not beood. That is, if the adjusted coefficient of determination, Radj

2 ,hich is a representative of the conventional accuracy measure, isery close to 1, then the approximation function passes throughery close to all the sampling points. However, that does not guar-ntee that the response surface is accurate at other nonsamplingoints, which could be very well candidate points for the optimumesign. Therefore, the conventional error metrics may not be ef-ective for design optimization, especially for RBDO since a smallesponse surface error could cause a significant reliability error,nd the optimum design result may not satisfy target reliabilities.

It is proposed to use the prediction interval of the responseurface using the MLSM �29–32�. This interval concept computeshe probabilistic interval of the response surface based on theonfidence level that the design engineer specified. This interval issed as a new measure of accuracy for RBDO.

2.2 New Measure of Accuracy for RBDO. This paper con-iders the effect of the response surface error on the optimumesign of RBDO. The error is expressed in terms of the prediction

Fig. 2 Projection of the estimated inter
reliability and „b… upper and lower limits of r
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interval, which is then used for RBDO. The prediction intervalprovides the upper and lower bounds for the given confidencelevel that the design engineer specified. Using the upper and lowerlimits and the response surface of the performance function, theupper and lower limits of the reliability can be computed. Thelower limit of the reliability is compared with the lower targetreliability index to obtain a conservative optimum design. How-ever, in order to avoid obtaining a too conservative design thatmay not be close to the true optimum design, the upper limit ofreliability is bounded above by the upper target reliability index.

Figure 1 shows the error in the estimation of the reliability inRBDO using RSM. Even though the reliability index �RSM usingthe response surface satisfies the target reliability index, the truereliability index �true may not satisfy the reliability constraint, andthus the design is not reliable. Note that in this figure and in thefollowing figures, the reliability index � �distance from the designpoint to the most probable failure point �MPP�� is meaningful onlyfor the U-space but not for the X-space. That is, the reliabilityindex shown in the X-space in these figures is used for the purposeof explanation.

Figure 2�a� shows the estimated interval of the response withthe upper and lower limit surfaces of the constraint function, GUand GL, respectively on the standard normal distribution U1-U2domain. Therefore, the reliability index, which is the distance be-tween the origin and MPP, can be evaluated as an interval��L

RSM,�URSM�. Figure 2�b� shows the upper and lower limits of the

reliability index on the original random variable X1-X2 domain.The conventional RSM-based RBDO method uses �RSM,

whereas the proposed method uses �LRSM and �U

RSM to find anoptimal design. �L

RSM is used to satisfy the lower target reliabilityindex, and �U

RSM is used to bind the upper target reliability indexin the RBDO process. The proposed design optimization formu-lation will be described in Sec. 4.

Fig. 1 Error in reliability estimation due to response surfaceerror

„a… interval of response and interval of
val. eliability.
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Confidence and Prediction Interval for the Movingeast Squares Method

3.1 Confidence Interval of the Response Surface of theoving Least Squares Method. Suppose that there are

-response values, yi, with respect to the changes of xij, whichenote the ith observation of variable xj. The relation can be ex-ressed as

yi = a0 + a1xi1 + a2xi2 + ¯ + akxik + �i

= a0 + �j=1

k

ajxij + �i, i = 1,2, . . . ,n �1�

here aj is the coefficient of the jth variable and �i is an approxi-ation error of the ith observation. The equation can be defined asmatrix form,

y = Xa + � �2�

here � is an �n�1� vector of approximation errors. It is assumed

The sum of squares of the residuals is

ournal of Mechanical Design


that the approximation error � has E��=0 and Var��=�2.For MLSM, the least squares function L�x� could be defined as

in the following equation, which is the weighted sum of squarederrors:

L�x� = �i=1

n

wiei2 = eTW�x�e = �y − Xa�TW�x��y − Xa� �3�

where x is the location where the approximation is sought. Thislocally weighted approximation can be performed from the con-sideration of effective data near the approximation location x, andthe data can be weighted according to the distance from the ap-proximation location x.

Note that the diagonal weighting matrix, W�x�, is not a con-stant matrix in MLSM. In other words, W�x� is a function of theapproximation location x, and it can be obtained by utilizingweighting functions. There are various types of weighting func-tions among which two functions are defined by the followingequation:

w�x − xi� = w�d� = �if d/R � 1 �polynomial 1 − 6�d/R�2 + 8�d/R�3 − 3�d/R�4

exponential exp�− d/R� �if d/R � 1, 0

�4�

here xi is a vector of the ith sampling point, d is the distanceetween x= �x1 ,x2 , . . . ,xNV�T and xi= �xi1 ,xi2 , . . . ,xiNV�T �i.e., d� j=1

NV�xj −xij�2 /NV, where NV is the number of variables�, w�d�s a weighting function at distance d, and R is the size of thepproximation region, which is a predefined value in this paper.

The polynomial weighting function, which is used in this paper,s expressed by a bell-shaped function. The weighting functionas the maximum value of 1 at 0, the normalized supporting size,nd the minimum value of 0 outside of the normalized support,.e., w�0�=1 and w�d /R�1�=0. Also the function decreasesmoothly from 1 to 0. The weighting matrix, W�x�, can be con-tructed using the weighting function in diagonal terms �29�,

W�x� = �w�x − x1� 0 ¯ 0

0 w�x − x2� ¯ 0

] ] ¯ ]

0 0 ¯ w�x − xn�� 5�

For MLSM, minimizing Eq. �3� gives the least squares estima-or b�x� of a as

b�x� = �XTW�x�X�−1XTW�x�y �6�

he variance property of b�x� is expressed by the covariance ma-rix

Cov�b�x�� = Cov��XTW�x�X�−1XTW�x�y�

= �XTW�x�X�−1XTW�x�Cov�y�

��XTW�x�X�−1XTW�x��T

= �2�XTW�x�X�−1XTW�x�W�x�X�XTW�x�X�−1

�7�

SSE = �i=1

n

�yi − yi�2 �8�

which has an n− p degree of freedom, where n is the number ofobservations and p is the number of terms. It can be shown�27,28� that

E�SSE� = �2�n − p� �9�

Thus, an estimator of �2 is given by

�2 = SSE/�n − p� �10�

The mean response at a particular point x is

�y x = x0Ta where x0

T = �1,xT� �11�

The estimated mean response at this point is

y�x� = x0Tb�x� �12�

The expected value and variance of the estimated mean responseare, respectively,

E�y�x�� = E�x0Tb�x�� = x0

Ta = �y x �13�

and

Var�y�x�� = �2x0T�XTW�x�X�−1XTW�x�W�x�X�XTW�x�X�−1x0

�14�

Using �2 instead of �2, the quantity

T =y�x� − �y x

Var�y�x��15�

where

Var�y�x�� = �2x0T�XTW�x�X�−1XTW�x�W�x�X�XTW�x�X�−1x0

has a t distribution with n− p degree of freedom. Thus,

P�− t�/2,n−p � T � t�/2,n−p� = 1 − � �16�

and

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P�y�x� − t�/2,n−pVar�y�x�� y x � y�x� + t�/2,n−p

Var�y�x��

= 1 − � �17�

Therefore, a 100�1−��% two-sided confidence interval on theean response at point x is

y�x� − t�/2,n−pVar�y�x�� y x � y�x� + t�/2,n−p

Var�y�x��

�18�

quation �18� implies that the mean response at point x will beithin the confidence interval with a probability of 100�1−��ased on the sampled information. In Eq. �18�, t�/2,n−p is a de-reasing function of � and �n− p�. Generally, as the probabilityncreases �i.e., � decreases�, the interval length increases. As theumber of sample points to generate the response surface in-reases, the interval length decreases if the estimated variance isot increased.

3.2 Prediction Interval of the Response Surface of Movingeast Squares Method. The confidence interval is for estimating

he interval of the mean response. However, the prediction inter-al is for predicting the interval of “the value of a single futurebservation” at a point. Therefore, the prediction interval of theesponse surface is used for the design optimization in this paper.

A point estimate for the future observation y0 at point x is

y�x� = x0Tb�x� �19�

here x0T= �1 xT�. The expected value of the prediction error is

E�y0 − y�x�� = �y x − �y x = 0 �20�

nd the variance of the prediction error is

ar�y0 − y�x��

= �2�1 + x0T�XTW�x�X�−1XTW�x�W�x�X�XTW�x�X�−1x0�

�21�

Using �2 instead of �2, the quantity

T =y0 − y�x�

Var�y0 − y�x��22�

here

ˆ ar�y0 − y�x��

= �2�1 + x0T�XTW�x�X�−1XTW�x�W�x�X�XTW�x�X�−1x0�

as a t distribution with an n− p degree of freedom. Thus,

P�− t�/2,n−p � T � t�/2,n−p� = 1 − � �23�nd

�y�x� − t�/2,n−pVar�y0 − y�x��

� y0 � y�x� + t�/2,n−pVar�y0 − y�x�� = 1 − � �24�

herefore, a 100�1−��% prediction interval for the future obser-ation at point x is

y�x� − t�/2,n−pVar�y0 − y�x�� y0 � y�x�

+ t�/2,n−pVar�y0 − y�x��

�25�

quation �25� implies that one future response at point x will beithin the prediction interval with a probability of 100�1−��%ased on the sampled information. In Eq. �25�, t�/2,n−p is a de-reasing function of � and �n− p�. As before, as the probabilityncreases, the interval length increases. As the number of sample
oints to generate the response surface increases, the interval
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length decreases. In addition, it can be seen from Eqs. �18� and

�25� that Var�y�x�� Var�y0− y�x�� since Var�y0− y�x��= �2

+Var�y�x��. Therefore, the prediction interval is always widerthan the confidence interval except when �2=0. This characteris-tic can also be observed in the example in Sec. 3.4.

3.3 On the Assumption of Randomness of Response Sur-face Approximation Error. This paper assumes that the responsesurface approximation error � can be treated as a random variablewith a normal distribution, which has E��=0 and Var��=�2.Therefore, it is investigated whether the assumption is reasonableby using test functions. The first test function is

X12X2/20 − 1 �26�

where �X1 ,X2�= �0,10�. From the explicit function, 36 points aresampled by a Latin-hypercube design, and the response surface isconstructed using MLSM with a full quadratic basis model.

Figure 3 shows a frequency histogram and a normal quantileplot �or quantile-quantile plot� of the residuals from 36 samplingpoints using JMP, which is a statistical software �35�. The residualis the difference between the approximated response and thesampled value at the sampled location. If the normal quantile plotis a straight line, the distribution is a normal distribution �27�, andthe normal quantile plot in Fig. 3 shows that it is close to astraight line. In addition, the Shapiro–Wilk test �33,34� is used totest the normality of the approximate error. Generally, if a p-value�27,35� of the test statistic is greater than 0.05, we can say that thetest variable has a normal distribution. For this example, the teststatistic W is 0.9865 and the p-value is 0.9510. Thus, we can saythat the approximation error of the given example behaves like anormal random variable.

After the construction of the response surface, the error is cal-culated at 10201 points �101�101 points� to check the globalbehavior. The sample variance �2 from the 36 sampling points is0.117589, and the variance �2 from the 10201 sampling points is0.112380. This means that the sample variance is very close to thetrue variance, and the sample variance can be used to estimate thesystem characteristic.

The second normality test function is �36�

f = 1 − �Y − 6�2 − �Y − 6�3 + 0.6�Y − 6�4 − Z

where Y = 0.9063X1 + 0.4226X2

�27�Z = 0.4226X1 − 0.9063X2

�X1,X2� = �0,10�The first test function is mildly nonlinear, but the second one is

highly nonlinear. Figure 4 shows the graphical result of the nor-

Fig. 3 Normality test of the first test function

mality test, whereas according to the Shapiro–Wilk test, W is



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.9691 and the p-value is 0.4866. Thus, we can say that the ap-roximation error of the given example behaves like a normalandom variable.

3.4 Test of the Interval Estimation. The confidence and pre-iction intervals are tested for a system defined by

f�X� = 0.5X5 − 1.5X4 − 2.5X3 + 0.53X2 + 1.3X + 2.0 �28�

here X= �−1,1�. Evenly distributed 11 points are sampled toonstruct the response surface using MLSM. The intervals areomputed for the 95%, 90%, 80%, and 70% confidence levels.igure 5 shows the true response, Ytrue; the response surfacealue, Yrsm; the lower and upper limits of the confidence interval,IL and CIU, respectively; the lower and upper limits of the pre-iction interval, PIL and PIU, respectively; and the sampledoints, Experi. These figures show that as the confidence levelncreases, the confidence/prediction intervals become wider,hich should be the case.After constructing the response surface using MLSM, 1001 test

oints are sampled from the true function shown in Fig. 5. Forhese 1001 test points, the number of test points that are inside the

Fig. 4 Normality test of the second test function

Fig. 5 Test of confidence and prediction intervals. „a… Con
level�80%; and „d… confidence level�70%.


confidence/prediction interval is counted. Table 1 shows the testresults for different confidence levels. These test results show thatthe confidence interval may not be suitable for RBDO since thepercentage of test points within the confidence interval is less thanthe required confidence level, and thus the prediction interval isrecommended.

The sample variance �2 from the 11 sampling points is0.060493, and the variance �2 from the 1001 test points is0.042172. The sample variance �2 is larger than the variance �2,which could be treated as the true variance. Since the samplingpoints have larger variance than the true variance, the predictioninterval will cover the true responses up to the required confidencelevel. If the sample variance is smaller than the true variance, theprediction interval may not contain enough of the true responses.

4 Design Improvement of RBDO Using Response Sur-face Method Considering Prediction Interval

4.1 Main Concept of the Proposed Method. The conven-tional RBDO problem is defined by

Minimize f�d�

Subject to P�Gj�X� 0� 1 − ��− �t,j�, j = 1, . . . ,ncon

ence level�95%; „b… confidence level�90%; „c… confidence

Table 1 Test of confidence and prediction intervals

Case �a� �b� �c� �d�

Confidence level 95% 90% 80% 70%�1� No. of test points within the

confidence interval793 604 488 425

�2� Rate for confidence interval�=1 / total no. of test points* 100�

79.2% 60.3% 48.8% 42.5%

�3� No. of test points within theprediction interval

1001 1001 935 825

�4� Rate for prediction interval�=3 / total no. of test points* 100�

100% 100% 93.4% 82.4%

fid

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diL � di � di

U, i = 1, . . . ,ndv �29�

here X is the vector of random variables, d is the vector of meanalues of X, Gj is the limit state function �performance function�Gj 0 is regarded as failure�, �t,j is the required target reliabilityndex of the jth constraint, ndv is the number of design variables,nd ncon is the number of constraints. The RBDO problem can beeformulated, using the reliability index approach �RIA� �1–7�, as

Minimize f�d�

Subject to �t,j � Reliablity�Gj�X� 0�, j = 1, . . . ,ncon

�30�

diL � di � di

U, i = 1, . . . ,ndv

If RSM is used, the following problem is solved, instead of theroblem in Eq. �30�:

Minimize f�d�

Subject to �t,j � � jRSM, j = 1, . . . ,ncon �31�

diL � di � di

U, i = 1, . . . ,ndv

here � jRSM is the reliability index that is computed using the

esponse surface. However, this problem may yield an unreliableptimum design because of the approximation error of the re-ponse surface.

Thus, it is proposed to use the prediction interval in the designrocess. That is, the new RBDO formulation, using the predictionnterval, is defined as

Minimize f�d�

Subject to �t,j � �L,jRSM for the confidence level of �1 − ��

j = 1, . . . ,ncon �32�

diL � di � di

U, i = 1, . . . ,ndv

here �L,jRSM is the lower limit of the reliability using the predic-

ion interval obtained for the confidence level of �1−��. This for-ulation will yield a reliable optimum design. However, this for-ulation could yield a too conservative design that may not be

lose to the true optimum. Thus, it is proposed to bind the upperimit of the reliability �U,j

RSM in the design optimization formula-ion. The upper limit of the reliability of the performance functions considered only when the lower limit of the reliability becomesn active constraint. The proposed formulation is integrated withn adaptive sampling strategy that is performed at MPP in thisaper.

The proposed RSM-based RBDO formulation is to adaptivelydd sampling points until the following optimization process isompleted:

inimize f�d�

ubject to �t,L,j � �L,jRSM

if �L,jRSM is active impose the constraint �U,j

RSM � �t,U,j

�33�

for the confidence level of �1 − ��, j = 1, . . . ,ncon

diL � di � di

U, i = 1, . . . ,ndv

here �t,L,j and �t,U,j are the lower and upper target reliabilityndices, respectively, and �L,j

RSM and �U,jRSM are the lower and upper

imits of the reliability using the prediction interval, respectively.or the specified confidence level, if the interval ��t,L ,�t,U� ismall, the uncertainty of the response surfaces is low �i.e., accu-
ate response surface�. However, the smaller interval may require
21401-6 / Vol. 130, DECEMBER 2008


a larger number of sampling points. On the other hand, if theinterval is large, the uncertainty of the response surfaces is highand the optimum design obtained will not be close to the trueoptimum design. Therefore, there is a tradeoff between the uncer-tainty of the response surface and the number of sampling points.

4.2 Computational Process of the Proposed RSM-BasedRBDO Method. Figure 6 shows the overall computational pro-cess to solve the proposed formulation, which is described below.

�a� Take the initial sampling based on a conventional or auser-defined design of experiment.

�b� Construct the response surfaces from the analysis resultsat the sampling points.

�c� Solve the RBDO problem using the constraints �t,L,j��L,j

RSM, j=1, . . . ,ncon.�d� If the RBDO of step �c� is converged, go to the next step;

otherwise, iterate the RBDO process.�e� Check step �f� to step �g� for all constraints.�f� If the jth constraint �t,L,j ��L,j

RSM is active, go to the nextstep; otherwise, go to step �e�.

�g� If �U,jRSM��t,U,j is satisfied, go to step �e�; otherwise, save

the value �U,jRSM and go to step �e�.

�h� If all constraints are satisfied, finish the optimization; oth-erwise, sample an additional point at the MPP of thedominant constraint. The constraint for which ��U,j

RSM

−�t,U,j� is the maximum is the dominant constraint. Notethat this step can be applied only for the constraint where�t,L,j ��L,j

RSM is active. After the additional sampling atthe MPP of the dominant constraint, go to step �c�.

During the sequential adaptive sampling procedure, one addi-tional �sequential� point is sampled at the MPP of the dominantconstraint of the current design. This additional �sequential� pointwill be different since each sequence yields a different design.However, a more efficient sampling strategy should be investi-

Fig. 6 Computational flow chart of proposed formulation

gated thoroughly in the future.



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Numerical Example

5.1 Mathematical Design Problem. The given RBDO prob-em is defined to

inimize cos t�d1,d2� = d12 + d2

2

ubject to P�Gj�X� 0� 1 − ��− �t,j�, j = 1,2,3

G1�X� = X12X2/20 − 1

�34�G2�X� = �X1 + X2 − 5�2/30 + �X1 − X2 − 12�2/120

− X12X2/300 − 1

G3�X� = 80/�X12 + 8X2 + 5� − 1

0 � di � 10, i = 1,2

here Xi�N�di ,0.3� , i=1,2, and �t,j =3.0, j=1,2 ,3.The problem has two normal random variables X1 and X2 with

Fig. 7 Contour plots of fou

he corresponding mean values d1 and d2 and a variance of 0.3 for



both variables. The number of the probabilistic constraints is 3,and the target reliability index is 3.0 for all constraints. Figure7�a� shows the design domain and the contour plot of the objec-tive and the constraints.

5.2 RBDO Results of Different Formulations for theMathematical Problem. Four different design formulations areused to solve the given problem: �a� RBDO formulation using thetrue function as defined in Eq. �30�, �b� RBDO formulation usingthe response surface as defined in Eq. �31�, �c� RBDO formulationusing only the lower limit of the prediction interval of reliabilityas defined in Eq. �32�, and �d� the proposed RBDO formulationusing both the upper and lower limits of the prediction interval ofreliability as defined in Eq. �33�, with the adaptive samplingstrategy.

Sixteen points are initially sampled by the Latin-hypercube de-sign to construct the response surface and DOT is used as an opti-mizer �37�. The predefined parameters are �=0.1 �90% confi-dence level�, �t,j =�t,L,j =3.0, and �t,U,j =4.0.

fferent design formulations
r di
Table 2 and Fig. 7 show the optimum design results for differ-

DECEMBER 2008, Vol. 130 / 121401-7


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nt formulations. In Table 2, “Estimated ��Gi�� by RS” means thathe predicted values or intervals of reliability index of Gi using thebtained RSs at the optimal design for each formulation. “True�Gi��” means the true reliability of Gi at the optimal design forach formulation. True ��Gi�� is computed using the true func-ions to get the true reliability of the design for the purpose ofomparison. Therefore, true ��Gi�� is computed to test the perfor-ance of each formulation. Formulation �a� provides the accurate

esult but requires a sensitivity analysis. Formulation �b� may pro-ide a solution that is not reliable enough to satisfy the targeteliability because this formulation provides a point estimationsing RS. The estimated reliability of G1 using RS is 3.00449, andhe true reliability of G1 turns out to be 3.58429. Thus, the designs reliable with respect to this performance measure because thearget reliability index is 3.0. However, this first constraint hap-ens to be reliable due to the positive error, which we have noontrol of. The estimated reliability of G2 using RS is 3.00211,nd the true reliability of G2 is 2.90484, which is not reliable dueo the negative error. Therefore, formulation �b� cannot provide aonfident reliability of the optimal design.

Formulation �c� provides the 90% confidence level that the op-imal design satisfies the target reliability of 3.0. Indeed Table 2hows that the estimated reliability intervals of G1, G2, and G3 are2.999,4.333�, �3.031,3.317�, and �4.460,12.300�, respectively, andhe true reliabilities are 4.16928, 3.09381, and 8.50783, respec-ively. We can see that all the true reliabilities are within therediction intervals. Note that the upper limit of the predictionnterval can be computed, but formulation �c� does not control thepper limit. As a result, the reliability of G2 is reasonable, but theeliability of G1 is too high, and thus, the optimum design is tooonservative.

Formulation �d� provides 90% confidence level that the reliabil-ty indices of active constraints of the optimum design are be-ween 3.0 and 4.0. Table 2 shows that the estimated reliabilityntervals of G1, G2, and G3 are �3.000,3.981�, �2.999,3.197�, and4.362,11.942�, respectively, and the true reliabilities are 3.48106,.03678, and 9.21225, respectively. The result of formulation �d�hows reasonable reliabilities for G1 and G2, which are 3.48106nd 3.03678, respectively. Formulation �d� is carried out usingdditional 14 sample points adaptively at the MPPs of the opti-um design obtained using the previous response surface, and

herefore a total of 30 sampling points are used during the wholeptimization process. Table 3 shows the additional sample points,nd Table 4 shows the optimization history with the adaptiveampling.

Figure 8 shows the contour plots of the design result of formu-ation �d� at different sampling stages; �a� initial 16 sampling �to-al 16 points�, �b� initial 16+ additional 5 �total 21 points�, �c�nitial 16+ additional 10 �total 26 points�, and �d� initial 16+ ad-

Table 2 Comparis

Design Formulation �a� Form

Method RBDO usingtrue function

RBresp

Response Real functionOpt. d1 3.275454 3Opt. d2 3.703557 3

Cost 24.44493 2Estimated ��G1� by RS N/AEstimated ��G2� by RS N/AEstimated ��G3� by RS N/A

True ��G1� 3.00890True ��G2� 3.00554True ��G3� 9.69834

No. of sample points N/A

itional 14 �total 30 points�. The small circles on the figures are

21401-8 / Vol. 130, DECEMBER 2008


the sample points at each stage. These figures show that the pre-diction interval is getting narrower as additional sample points areused, and Table 4 shows that the optimum point is converged tothe final result. Figure 8�a� is identical with Fig. 7�c� and 8�d� isthe same with Fig. 7�d�. The final result shows that at least 30points are necessary to satisfy the target reliability interval underthe given confidence level. It is not easy to determine the numberof sampling points that is required to successfully carry out theRSM-based RBDO. However, the proposed method provides anerror metric to effectively estimate the confidence level of theoptimized design of the RBDO problem. This is the major advan-tage of the proposed method. Obviously, the initial sampling andthe additional sampling strategy can affect the result significantly.Since this paper used the additional sampling strategy at the domi-nant MPP, as explained in Sec. 4.2, all the added points are lo-cated at the previous-step MPP of the first constraint. This adap-tive sampling strategy may not be effective, and therefore furtherresearch is required.

5.3 Reliability-Based Design Optimization of a Double-Folded-Spring System. In the microelectromechanical systems�MEMSs�, many types of actuators are used for motions of thedevice and double-folded-spring �DFS� �38� structures are com-monly used for flexible structures. Figure 9 shows a structure ofthe DFS system, and Ref. �39� describes the details of the system.When the x-directional force is applied to the system, the devicemoves to the x-direction. When the external force is removed, thedevice moves back to the original position due to the spring effect.When the device works, the rotational stability needs to be maxi-mized in order to avoid any interference of the device. Thus, thesystem can be optimized so that the spring structure achieves themaximum rotational spring stiffness subject to the required actua-

of design results

ation �b� Formulation �c� Formulation �d�

usinge surface

Prediction internallower limit

Proposedmethod

SM MLSM MLSM9073 3.577130 3.3973621458 3.895420 3.7836826575 27.97016 25.85832449 �2.999, 4.333� �3.000, 3.981�211 �3.031, 3.317� �2.999, 3.197�215 �4.460, 12.300� �4.362,11.942�429 4.16928 3.48106484 3.09381 3.03678467 8.50783 9.212256 16 30 �=16+14�

Table 3 Additional sample points

Total sample point X1 X2 Cost at sample point

17 2.78562 3.46751 19.783318 2.73810 3.42071 19.198519 2.71037 3.39929 18.901220 2.69212 3.38548 18.709021 2.67865 3.37551 18.569222 2.66799 3.36779 18.460223 2.66058 3.35916 18.362724 2.65307 3.35390 18.287425 2.64655 3.34939 18.222626 2.64089 3.34531 18.165427 2.63723 3.33934 18.106228 2.63264 3.33610 18.060429 2.62835 3.33339 18.019730 2.62454 3.33074 17.9820

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ion spring stiffness �x-directional translational stiffness�. TheEMS device has special characteristics like the scaling effect

nd etching process, which means that the system is under variousncertainties and therefore needs to apply RBDO.

Figure 9 shows four selected design variables, and Table 5hows the design bounds. The shape RBDO problem is defined to

aximize k � �

Table 4 Optimization history with

ample X1 �opt� X2 �opt� Cost g�1� g�2�

�16� 3.5771 3.8954 27.9702 0.0002 −0.0028 −17 3.5242 3.8610 27.3271 −0.0010 −0.0001 −18 3.4930 3.8445 26.9809 −0.0004 −0.0001 −19 3.4728 3.8337 26.7571 −0.0002 −0.0001 −20 3.4581 3.8257 26.5940 −0.0001 −0.0003 −21 3.4466 3.8194 26.4664 −0.0001 −0.0005 −22 3.4383 3.8121 26.3544 0.0000 0.0000 −23 3.4303 3.8077 26.2656 0.0000 −0.0002 −24 3.4234 3.8039 26.1890 0.0000 −0.0004 −25 3.4234 3.8039 26.1890 −0.0066 −0.0006 −26 3.4133 3.7952 26.0541 0.0000 0.0000 −27 3.4133 3.7952 26.0541 −0.0053 −0.0002 −28 3.4040 3.7900 25.9509 0.0000 −0.0003 −29 3.4040 3.7900 25.9509 −0.0044 −0.0005 −30 3.3974 3.7837 25.8583 0.0000 0.0000 −

Fig. 8 Contour plots of result of the formu



Subject to P�Gj�X� 0� 1 − ��− �t,j�, j = 1,2

G1 = k � x − 200 �35�

G2 = 300 − k � x

where �t,j = 1.5, Xi � N�di,0.01 � di�, i = 1, . . . ,4

ptive sampling for formulation „d…

� Beta1L Beta1U Beta2L Beta2U Beta3L Beta3U

178 2.9993 4.3325 3.0309 3.3173 4.4604 12.2996925 3.0033 4.3825 3.0009 3.2856 4.1447 12.6567835 3.0012 4.3533 2.9998 3.2756 4.0329 12.7234805 3.0005 4.3136 3.0008 3.2672 3.9972 12.7002803 3.0003 4.2729 3.0027 3.2600 3.9978 12.6427815 3.0002 4.2338 3.0049 3.2539 4.0173 12.5714837 3.0001 4.1969 2.9993 3.2406 4.0490 12.4947860 3.0001 4.1627 3.0016 3.2358 4.0843 12.4180886 3.0001 4.1309 3.0038 3.2315 4.1226 12.3424892 3.0232 4.1245 3.0059 3.2276 4.1399 12.2460941 3.0002 4.0738 2.9993 3.2154 4.2044 12.1976950 3.0188 4.0668 3.0012 3.2121 4.2263 12.1112991 3.0001 4.0242 3.0032 3.2092 4.2840 12.0649003 3.0155 4.0172 3.0049 3.2064 4.3078 11.9872042 3.0001 3.9806 2.9993 3.1965 4.3624 11.9422

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DECEMBER 2008, Vol. 130 / 121401-9


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The rotational stiffness k �� N �m� and the translational stiff-ess k �x ��N /�m� are computed using an ANSYS �40� nonlinearnite element �FE� analysis model. When the design variables arehanged, the shape of the FE model is changed and the meshes areutomatically updated. For the computational efficiency, the re-ponse surface method using MLSM is applied, and the proposedethod is applied for the design confidence of the RBDO result.

5.4 RBDO Results of Different Formulations for the DFSystem. Three different design formulations described in Sec. 5.2re used to solve the given problem: �b� RBDO formulation usinghe response surface, �c� RBDO formulation using only the lowerimit of the prediction interval of reliability, and �d� the proposedBDO formulation using both the upper and lower limits of therediction interval of reliability with the adaptive sampling strat-gy. Initially 40 points are sampled by the Latin-hypercube designor the entire design domain. The predefined parameters are �0.1 �90% confidence level�, �t,j =�t,L,j =1.5, and �t,U,j =3.0.Table 6 shows the design results of three formulations. Formu-

ation �b� provides the reliability of the active constraint G2 to be

Fig. 9 Design variables of the DFS system

Table 5 Comparison of design results

esign variable ��m� Lower limit Upper limit

1 �inner length� 100 250

2 �outer length ratio� 0.5 2.0

3 �spring length� 100 150

4 �spring thickness� 2 4

Table 6 Comparis

Design Formulation �b�

Method RBDO usingresponse surface

Response MLSMOpt. d1 249.993Opt. d2 0.697926Opt. d3 124.912Opt. d4 2.80122

Cost 1.35566�1010

Estimated ��G1� by RS 6.76482Estimated ��G2� by RS 1.50021

True ��G1� 8.0928True ��G2� 1.10647

No. of sample points 40

21401-10 / Vol. 130, DECEMBER 2008


1.50021, but the true reliability turns out to be 1.10647. Thus, thisformulation provides a design that is not reliable. Formulation �c�gives an optimal design that satisfies the target reliability of 1.5with 90% confidence level, while formulation �d� gives a designwhose active reliabilities are between 1.5 and 3.0 under the sameconfidence level. Table 6 shows that true reliabilities are withinthe corresponding prediction interval for formulations �c� and �d�.Formulation �d� is performed using additional 23 sample pointsaccording to the proposed sampling method, and therefore a totalof 63 sample points are used during the optimization process.Comparing the results of �c� and �d� gives an interesting observa-tion. Since formulation �d� uses more sample points, the confi-dence interval length is smaller than formulation �c� as expected.For the second constraint, �c� gives the prediction interval as�1.500,4.626� and �d� gives �1.500,2.986�. However, the true reli-abilities are 1.71623 and 2.32588, respectively, and the true reli-ability of case �c� is close to its lower bound. As a result, thedesign of case �c� is better than case �d� because �c� is less con-servative than �d� in this example. This could be due to the sam-pling method, which needs to be improved.

Table 7 shows the optimization history with the adaptive sam-pling for formulation �d�. “Xi�opt�” is the optimal point of thedesign variables, and “Xi�add�” is the additional sample point ateach stage. Another observation is the history of the length of theprediction interval in Table 7. In some stage of the history, theinterval length becomes longer than the previous stage eventhough an additional sample point is used. When a sample point isadded, if the sample variance is increased, then the predictioninterval length could be increased. From this observation, the fu-ture study about a more effective sampling method is required. Itis noted that all the prediction intervals in the examples success-fully include the true reliability indices, and thus the developedprediction interval is applicable to RBDO. Figure 10 shows thefinite element model of the optimal design shape for the formula-tion �d� case.

6 ConclusionIn this paper, a method to estimate the effect of the response

surface error on the RBDO result is developed. The predictioninterval is utilized as an error metric for the response surface usedfor RBDO. The MLSM is used to generate the response surface.Using the assumption that the response surface approximation er-ror can be treated as a random variable with normal distribution,the prediction interval is used to establish a confidence level. Thelower limit of the reliability obtained from the prediction interval,which is generated based on the confidence level that the designengineer has specified, is used in the RBDO formulation to ensurea reliable optimum design. On the other hand, in order to avoidobtaining a design that is too conservative, the developed methodalso bounds the upper limit of the reliability obtained from theprediction interval in the design optimization process. The pro-

of design results

Formulation �c� Formulation �d�

Prediction intervallower limit

Proposedmethod

MLSM MLSM249.992 249.998

0.500046 0.679818130.491 110.3072.90288 2.41748

1.34131�1010 1.32654�1010

�3.461, 9.143� �4.081, 6.233��1.500, 4.626� �1.500, 2.986�

7.85957 5.704751.71623 2.32588

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osed procedure is combined with an adaptive sampling strategyo sample at MPP additionally to refine the response surface untiloth limits are satisfied. Therefore, this method can give a guide-ine for the sampling location and the convergence criterion. The

ost important advantage of the proposed method is that theethod gives how much the design is reliable quantitatively �i.e.,

�LRSM,�U

RSM�� under the required confidence level and how manydditional sampling points are necessary when using RSM. Theumerical example shows the usefulness and the computationalfficiency of the proposed method. To refine the proposed method,urther research on the efficient additional sampling strategy andhe accuracy of the prediction interval is required.

cknowledgmentThis research was supported by the Korea Research Foundation

rant funded by the Korean Government �MOEHRD� �KRF-005-214-D00231� and the Automotive Research Center that isponsored by the U.S. Army TARDEC.

eferences�1� Tu, J., and Choi, K. K., 1999, “A New Study on Reliability-Based Design

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ig. 10 The optimized finite element model for the formulationd…

Table 7 Optimization history with

ample X1 �opt� X2 �opt� X3 �opt� X4 �opt� X1 �add� X2

�40� 249.99 0.500 130.49 2.903 —41 250.00 0.530 129.42 2.844 249.85 042 249.99 0.538 131.93 2.901 250.01 043 249.99 0.997 150.00 3.292 250.02 044 249.99 1.075 150.00 3.296 250.15 045 250.00 1.120 107.65 2.319 250.15 146 250.00 0.989 100.02 2.152 250.05 147 240.63 0.501 100.05 2.170 249.98 048 250.00 0.500 100.01 2.166 240.68 049 245.78 1.764 100.01 2.138 250.01 050 250.00 2.000 100.00 2.157 245.72 151 250.00 2.000 100.00 2.175 250.06 252 250.00 2.000 100.00 2.179 250.04 253 250.00 2.000 100.00 2.182 250.04 254 237.32 1.480 130.36 2.884 250.04 255 233.21 1.381 130.27 2.878 237.36 156 223.13 0.628 135.79 3.006 233.25 157 229.76 0.500 141.73 3.116 223.17 058 227.43 0.500 145.63 3.195 229.83 059 236.11 0.555 130.06 2.867 227.50 060 250.00 0.577 116.07 2.547 236.13 061 250.00 0.655 109.41 2.396 249.98 062 249.98 0.630 108.64 2.379 249.97 063 250.00 0.680 110.31 2.417 249.95 0

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d� X3 �add� X4 �add� Beta1L Beta1U Beta2L Beta2U

— — 3.461 9.143 1.500 4.6260 129.21 2.936 4.167 8.893 1.500 4.1409 128.12 2.877 4.628 9.253 1.500 3.9847 130.63 2.933 5.203 13.666 1.500 4.9917 148.94 3.329 6.457 13.924 1.500 4.6905 148.91 3.332 2.919 6.232 1.500 3.9800 106.33 2.351 1.681 5.732 1.500 4.7159 98.84 2.187 1.500 5.664 1.500 4.7150 101.19 2.134 2.730 5.778 1.500 3.7390 98.74 2.199 1.500 6.866 1.500 5.4764 101.20 2.103 2.422 6.338 1.500 4.5260 98.80 2.192 3.339 6.236 1.500 3.6050 98.78 2.209 3.559 6.225 1.500 3.3760 98.78 2.214 3.668 6.218 1.500 3.2640 98.78 2.216 4.922 8.020 1.500 3.4150 129.04 2.916 4.953 7.961 1.501 3.3781 128.96 2.910 5.568 9.078 1.500 3.4668 134.50 3.039 6.285 11.244 1.500 3.8519 140.52 3.150 6.588 12.448 1.500 4.0359 144.45 3.230 5.318 8.394 1.500 3.3275 128.77 2.900 4.344 6.854 1.500 3.1627 114.78 2.580 3.981 6.196 1.500 3.0405 108.12 2.428 3.980 6.159 1.500 3.0110 107.35 2.412 4.081 6.233 1.500 2.986

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—.50.52.53.99.07.12.98.50.50.76.00.00.00.00.48.38.62.49.49.55.57.65.63

Model Uncertainty in Robust Design With Computer Experiments,” ASME J.

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