Conservative Surrogate Model using Weighted Kriging Variance for Sampling-based RBDO

Transcription

1 9 th World Congress on Structural and Multdsclnary Otmzaton June 13-17, 011, Shzuoka, Jaan Conservatve Surrogate Model usng Weghted Krgng Varance for Samlng-based RBDO Lang Zhao 1, K.K. Cho, Ikn Lee 3, Davd Lamb 4 and Davd Gorsch 5 1,,3 Deartment of Mechancal & Industral Engneerng College of Engneerng The Unversty of Iowa Iowa Cty, IA 54, U.S.A. Emal: lazhao@engneerng.uowa.edu kkcho@engneerng.uowa.edu lee@engneerng.uowa.edu 4,5 US Army RDECOM/TARDEC Warren, MI , U.S.A. davd.lamb@us.army.ml gorschd@tacom.army.ml 1. Abstract In samlng-based relablty-based desgn otmzaton of ractcal comlex engneerng alcatons, the Monte Carlo smulaton (MCS) for stochastc senstvty analyss and robablty of falure calculaton s based on the redcton from the surrogate model for the erformance functons. When the number of samles used to construct the surrogate model s small, the redcton from the surrogate model becomes naccurate and thus MCS becomes naccurate as well. Therefore, to count n the redcton error from the surrogate model and assure the obtaned otmum desgn from samlng-based RBDO does satsfy the robablstc constrants, a conservatve surrogate model s needed. In ths aer, a conservatve surrogate model s constructed usng the weghted Krgng varance where the weght s determned by the relatve change n the corrected Akake nformaton crteron (AICc) of the dynamc Krgng model. The roosed conservatve surrogate model erforms better than the tradtonal Krgng redcton nterval aroach because t does not generate unnecessary local otmum; and t erforms better than the constant safety margn aroach because t adatvely counts n the uncertanty of the surrogate model n the lace that the samles are sarse. Numercal examles show that usng the roosed conservatve surrogate model for samlng-based RBDO s necessary to assure the otmum desgn satsfes the robablstc constrants when the number of samles s lmted.. Keywords: conservatve surrogate model, dynamc Krgng method, weghted Krgng varance, corrected Akake nformaton crteron (AICc). 3. Introducton In samlng-based RBDO, Monte Carlo smulaton (MCS) s used to carry out both the robablty of falure and the stochastc senstvty analyss [1, ]. To carry out MCS effcently n comlex engneerng alcatons, a surrogate model s used to redct the erformance functon at the MCS samles. Zhao et al. [3] develoed a dynamc Krgng method to accurately construct the redcton of the erformance functon for the MCS by usng the genetc algorthm for the bass functon selecton and the generalzed attern search for the correlaton arameter estmaton. When the surrogate model s not accurate enough, new samles are sequentally nserted to mrove the redcton accuracy [3]. However, nsertng a large number of new samles sequentally may not be alcable when the samles are from the hyscal exerments or when the comutatonal resource for smulaton s lmted. Therefore, the mrovement of the surrogate model cannot be from the ncrease of the samle numbers. When alyng the surrogate model for relablty-based desgn otmzaton when the number of samles s small, to assure the obtaned otmum desgn can satsfy the robablstc constrants, a conservatve surrogate model s needed. Pcheny [4] used both safety margn and safety factor aroaches to construct the conservatve surrogate model and concluded that when the Krgng method s used, both methods rovde smlar erformance n terms of the conservatveness and the accuracy. Hertog et al. [5] and Luna and Young [6] used the bootstrang method to estmate the Krgng redcton nterval to construct the conservatve surrogate model. The bootstrang varance s larger than the tradtonal Krgng redcton varance by consderng the uncertanty from the correlaton arameter n the Krgng method. However, the bootstrang rocedure s tme-consumng and not alcable for hgh-dmensonal roblems. Vana et al. [7] used cross-valdaton to estmate the safety margn for the

2 conservatve surrogate model. Whle the cross-valdaton error s wdely used to estmate the redcton error, ths constant safety margn aroach does not dstngush the redcton error at dfferent samle locatons and wll yeld an over-conservatve surrogate model where the samles are aggregated around; and an under-conservatve surrogate model where the samles are sarse. In ths aer, a weghted Krgng varance s consdered to construct a more accurate conservatve surrogate model based the samle locatons. To evaluate the mortance from each samle and determne the weght for t, an accuracy measure of the surrogate model s needed frst. Under the Krgng framework, the Akake nformaton crteron (AIC) [8] s a relable ndcator to assess the accuracy of the surrogate model. Hurvch and Tsa [9] roosed a corrected AIC (AICc) to correct the bas n AIC when the number of samles s small. Burnham and Anderson [10] recommended usng AICc rather than AIC when the samle sze s small and showed that the AICc converges to AIC as the number of samles ncreases. Martn and Smson [11] comared the erformance of the corrected AIC assessment wth other methods and concluded that AICc rovdes the best accuracy assessment for surrogate model accuracy. In ths aer, the AICc s used to assess the accuracy changes n the surrogate model durng the cross-valdaton rocess, and an mortance functon value s assgned to each samle accordng to the relatve changes n AICc of the surrogate model and the weght s also assgned to each samle accordng to the mortance functon values. Then a weghted Krgng varance s calculated based on the weght values for each samle. By alyng ths weghted Krgng varance, one can construct the conservatve surrogate model for dynamc Krgng and use t for samlng-based RBDO to assure the obtaned otmum desgn can satsfy the robablstc constrant. The remander of ths aer s organzed n four sectons. rst, the backgrounds of the dynamc Krgng method, samlng-based RBDO, and the AICc are brefly summarzed. Then, the weghted Krgng varance usng the weght from the relatve change n AICc s ntroduced to construct the conservatve surrogate model. Thrd, the conservatve surrogate model s aled to samlng-based RBDO and the otmum desgn s verfed for the robablstc constrant usng numercal examles and comared wth the results usng the constant safety margn aroach. 4. Background of the Dynamc Krgng Method, Samlng-based RBDO, and AICc 4.1 Dynamc Krgng Method rst consder a unversal Krgng method. The outcomes are consdered as a realzaton of a stochastc rocess, and the redcted values are derved by alyng the stochastc rocess theory. Consder n samle onts T nd X [ x1, x,..., xn ] wth x R, and n resonses T Y [ y( x1), y( x),..., y( x n )] wth y( x ) R. In the Krgng method, the resonses at samles are consdered as a summaton of two arts as Y β e (1) The frst art of the rght-hand sde of Eq. (1), β, s consdered as the mean structure of the resonse, where =[ f( x ), 1,..., n] s a ( n K) f( x) fk ( x ), k 1,..., K reresents the user-defned bass functons, whch are usually n a smle olynomal form, such as1, xx,,.... The second art of the rght hand sde T n Eq. (1), e [ e( x1), e( x),..., e( x n )], s a realzaton of the stochastc rocess e( x ) that s assumed to have zero mean and covarance structure E[( e x)( e x)] R(, θ x, x ), where s the rocess varance and θ s the rocess correlaton arameter. The otmal choce of θ s defned as the maxmum lkelhood estmator (MLE), whch s the maxmzer of the lkelhood functon L, exressed as n 1 1 T 1 L R ex ( Yβ) R ( Yβ ) () where R s the symmetrc correlaton matrx wth - th comonent R R( θ, x, x ),, 1,..., n, and desgn matrx, and 1 ( ) T 1 ( ) T 1 Yβ R Yβ and 1 T 1 β R R Y are obtaned from the generalzed least square n regresson. Under the general decomoston of Eq. (1), the obectve s to redct the nose-free unbased resonse at a new ont of nterest x, exressed as T T 1 ( x) f β rr ( Yβ ) (3) where T 1 y krg r R( θ,x,x),..., R( θ,x,x). The redcton varance can be also obtaned as T 1 where u R rf. n ( ) (1 T ( T 1 ) 1 T 1 ) x u R ur R r (4)

3 In the dynamc Krgng method, the bass functon set f ( x) s no longer fxed. Instead, the otmal bass functon set s decded by the gven samles. rst, a number of olynomal functons are consdered as the canddate P functon. The hghest order P of the olynomal s decded by nd P n 1. A genetc algorthm s used to fnd the otmal subset of the bass functon by mnmzng the rocess varance. Second, a generalzed attern search algorthm s used to solve the roblem that s maxmzng Eq. (). Wth these two stes carred out, the dynamc Krgng can generate a more accurate surrogate model than the tradtonal unversal Krgng [3]. 4. Samlng-based RBDO The mathematcal formulaton of a general RBDO roblem s exressed as mnmze Cost( d) subect to Tar PG [ ( X) 0] P, 1,, nc (5) L U nd nr d dd, dr and XR T rv where d { d } μ(x ), 1~ nd s the desgn vector, whch s the mean value of the nd-dmensonal random rv T T varable vector X ={ X1, X,, X nd } ; ={ rv r rv r X X, X } where X and X stand for the random desgn Tar varable and random arameter comonents of the random nut X, resectvely; P s the target robablty of falure for the th constrant; and nc, nd, and nr are the number of robablstc constrants, desgn varables, and random varables lus arameters, resectvely. A relablty analyss for both the comonent and system levels nvolves calculaton of the robablty of falure, denoted by P, whch s defned usng a mult-dmensonal ntegral as P( ψ) P[ X] I ( ) f ( ; ) d EI ( ) nr x X x ψ x R X (6) where ψ s a vector of dstrbuton arameters, whch usually ncludes the mean (µ) and/or standard devaton (σ) of the random nut X X,, T 1 Xnr ; P reresents a robablty measure; s the falure set; f X ( x; ψ ) s a ont robablty densty functon (PD) of X; and E reresents the exectaton oerator. The falure set s defned as x: G ( ) 0 x for comonent relablty analyss of the th constrant functon G (x), and nc x: G ( ) 0 1 x and : nc ( ) 0 x G 1 x for the seres system and arallel system relablty analyss of nc erformance functons, resectvely. I ( x ) n Eq. (6) s called an ndcator functon and defned as 1, x I ( x ) (7) 0, otherwse In ths aer, snce the mean of X, μ,, T 1 nd s used as a desgn vector, the vector of dstrbuton arameters ψ s smly relaced wth µ for the comutaton of the robablty of falure n Eq. (6). Takng the artal dervatve of robablty of falure n Eq. (6) wth resect to the th desgn varable yelds P ( μ) I ( ) f ( ; ) d nr x R X x μ x (8) and the dfferental and ntegral oerators can be nterchanged usng the Lebnz s rule, gvng P ( μ) fx( x; μ) ln fx( x; μ) I ( ) ( ) ( ; ) nr d I f d nr x x x X x μ x R R (9) ln f ( ; ) X x μ EI ( x) snce I ( x ) s not a functon of. The artal dervatve of the log functon of the ont PD n Eq. (8) wth resect to s known as the frst-order score functon for and s denoted as (1) ln fx( ; ) s (; x μ ) x μ. (10) To comute the robablty of falure n Eq. (5) and the senstvty of robablty of falure n Eq. (8), statstcal samlng such as the Monte Carlo smulaton (MCS) at a gven desgn needs to be aled to true resonses, whch s comutatonally very exensve and almost rohbted. Hence, nstead of usng true resonses, whch are usually

4 obtaned from comuter smulatons, surrogate models are used be mlemented for the calculaton of the robablty of falure. To generate accurate surrogate models, ths aer uses the dynamc Krgng method, whch s dscussed n the revous secton. Denote the surrogate model obtaned by the dynamc Krgng method for the constrant functon G (X) as G ˆ ( X ). Then, by carryng out the MCS usng the conservatve surrogate model Gˆ ( X ), the robablstc constrants n Eq. (5) can be aroxmated as M 1 ( m) Tar P P[ G ( ) 0] ˆ ( ) X I x P (11) M ( m) where M s the MCS samle sze, x s the m th realzaton of X, and the falure set ˆ for the surrogate model s ˆ x: Gˆ ( x ) 0. Senstvty of the robablstc constrant n Eq. (8) s obtaned as defned as (1) ( m) where s ( x ; μ ) s obtaned usng Eq. (9). m1 P 1 x x μ (1) M ( m) (1) ( m) I ˆ ( ) s ( ; ) M m1 4.3 Corrected Akake nformaton crteron (AICc) AIC s orgnally roosed to evaluate the qualty of a model n statstcs based on the log-lkelhood functon [10]. It s a measure of the relatve goodness of ft of a statstcal model, and n general exressed as AIC k ln( L) (13) where k s the number of arameters n the statstcal model and L s the maxmzed value of the lkelhood functon for the estmated model. In the Krgng framework dscussed n Secton 4.1, the k s the number of dmenson of X, whch s nd, and L s the lkelhood functon as shown n Eq. (). When the samle sze s small,.e., n/k < 40 (n s the number of samle), whch s often the case n engneerng roblems, the corrected AIC (AICc) s used nstead to rovde an unbased estmaton, whch s exressed as kk ( 1) AICc AIC (14) n k 1 5 Samlng-based RBDO Usng Conservatve Surrogate Model 5.1 Weghted Krgng Predcton Varance for Conservatve Surrogate Model In Eq. (4), the Krgng redcton varance s calculated, and the C% uer bound of the redcton nterval s 1 C yvar ykrg (15) where s the nverse CD of the standard normal random varable. Ths redcton uer bound s usually used as the C% level conservatve surrogate model for the Krgng method, and often t s consdered as a varable safety margn aroach for constructng the conservatve surrogate model. Snce the Krgng method s an nterolaton method, the redcton varance becomes zero at the samle ont. As a result, the conservatve surrogate model y var s the same as the y krg at the samle onts. Ths nterolaton roerty of the Krgng method causes trouble when the conservatve surrogate model s used for an otmzaton roblem. Consder a 1-D examle, mn y (6x) sn(1x4) x [0,1] (16) The functon lot and the Krgng redcton usng seven onts are shown n g. 1.

5 wrong fgure!! gure 1: 1-D roblem wth 7 samles When the uer bound s used as the conservatve surrogate model and for otmzaton, the otmum x could be easly converged to the samle onts, whch are the local mnma of y var. Therefore, ths uer bound of the Krgng redcton nterval s not alcable for otmzaton. Another constant safety margn aroach s also often used for the conservatve surrogate model [7], where the safety margn s decded based on the emrcal CD of the cross-valdaton error. In ths constant safety margn aroach, the conservatve surrogate model s obtaned by shftng the Krgng redcton to a certan amount and s often exressed as yxv ykrg e XV,% C (17) where e XV,% C s the C ercentle of the cross-valdaton error. Ths conservatve surrogate model has the same dscreancy from the Krgng redcton regardless of the samle oston. Ths rases the roblem that the conservatve surrogate model cannot count n the dfferent uncertanty of the dscreancy due to dfferent samle locatons. Therefore, ths constant safety margn aroach may become over-conservatve f the samles are dense and under-conservatve f the samles are sarse. Consder the same examle shown n g.1. If there s no samle at x = 0.76, the safety margn Eq. (17) s s = and the conservatve surrogate model s shown n g.. The conservatve surrogate model s over-conservatve n the regon of [0, 0.55] and under-conservatve n the regon of [0.55, 1]. gure : 1-D roblem wth 6 samles

6 Accordng to the two examles dscussed above, one can see that when a conservatve surrogate model from the Krgng method s used for an otmzaton roblem, a varable safety margn aroach that has zero margn at the samle onts s not desrable because t generates unnecessary local otma regons; whereas a constant safety margn may not be desrable because t does not count n the effect from the samle oston. What s needed s a conservatve surrogate model that uses a varable safety margn that does not generate local otma and counts n the effect from the samle oston. In ths aer, a new conservatve surrogate model that combnes the varable safety factor aroach from Eq. (15) and the constant safety margn aroach from Eq. (17) s roosed. rst, to count n the effect from the samle oston, an mortance measure for each samle s needed. In ths aer, the AICc s chosen to quantfy how mortant a samle s for the Krgng redcton. The mortance functon for samle x s defned as ( ) AICc AICc Im( x ), 1,..., n (18) AICc ( ) where AICc s calculated usng all n samles and AICc s calculated by omttng x. rom Eq. (18), t shows that the larger Im( x ) s, the more mortant x s. Consder the same examle dscussed above. gs. 3(a)-(g) show the Krgng redctons when one of the samles s omtted, and the assocated mortance functon values are shown n g. 3(h) as well. The samle ont 6 at x = 0.76 s ndeed around the hghly nonlnear regon, and ts mortance functon value s the largest among all 7 samles. Therefore, ths mortance functon by the relatve change n AICc values can characterze how mortant one samle s for the Krgng redcton accuracy. (a) Krgng redcton w/o samle # (b) Krgng redcton w/o samle #3 (c) Krgng redcton w/o samle #4 (f) Krgng redcton w/o samle #5

7 (g) Krgng redcton w/o samle #6 (h) Im(x) functon values at each samle gure 3: Leave-one-out Krgng redcton After the mortance for each samle s calculated, to construct a weghted Krgng varance, a leave-one-out Krgng varance ( ) s calculated where the th samle x s omtted usng Eq. (4), and then the weghted Krgng varance s exressed as where the weght functon s defned as ( ) n weghted, wx ( ) 1 (19) wx ( ) 1/Im( x ) n 1 1/Im( x ) The weght functon decdes how much the leave-one-out Krgng varance ( ) contrbutes to the total Krgng varance when x s mssng accordng to the mortance functon value. nally, the conservatve surrogate model based on the weghted Krgng varance s 1 C ycon ykrg, weghted 100 (1) Accordng to Eq. (19), one can see that when x s omtted, the leave-one-out Krgng varance ( ) would not become zero at x, and therefore the total Krgng varance would not be zero at x and ndeed smoothes the conservatve surrogate model eventually. On the other hand, snce ( ) wll be larger at the lace where no samle s nearby, t wll make the total weghted Krgng varance become varable among the entre doman. Consder the same examle shown n gs. 4 and 5. In g. 4, the lnes labeled as Y_true, Y_krg, Y_xv,, Y_var and Y_con are the true resonse, the Krgng resonse, the conservatve surrogate model usng constant safety margn, the uer bound of Krgng redcton nterval, and the conservatve surrogate model usng the weghted Krgng varance, resectvely. When all seven samles are used and the conservatveness s set to be 90%, the conservatve surrogate model usng the weghted Krgng varance s smoother than the uer bound of the Krgng redcton nterval and closer to the true resonse than the conservatve surrogate model usng the safety margn. Moreover, f one mortant samle (.e., x = 0.76 n ths case) s mssng, the conservatve surrogate model usng the weghted Krgng varance would have a larger dscreancy than the conservatve surrogate model usng the constant safety margn aroach n the regon of [0.55, 1] and a smaller dscreancy n the regon of [0, 0.55], as shown n g. 5. These two cases ndcate that the conservatve surrogate model usng the weghted Krgng varance can adatvely dentfy the conservatveness accordng to samle locatons and rovde a smooth surrogate model that does not change the otmum regon of the orgnal resonse functon. (0)

8 gure 4: Conservatve surrogate model usng weghted Krgng varance (7 samles) gure 5: Conservatve surrogate model usng weghted Krgng varance (6 samles) 5.. Samlng-based RBDO usng the conservatve surrogate model When a lmted number of samles s used to generate the surrogate model for MCS n samlng-based RBDO, to assure the otmal desgn can satsfed the robablstc constrants, the surrogate models n Eqs. (11) and (1) need to be relaced by the conservatve surrogate model. Therefore, the conservatve surrogate model usng the weghted Krgng varance from Eq. (1) s used to reresent the orgnal erformance functon. The formulaton of samlng-based RBDO becomes mnmze Cost( d) ˆ c Tar subect to PG [ ( X) 0] P, 1,, nc () L U nd nr d dd dr XR, and where ˆ c G ( X) s the conservatve surrogate model for reresentng erformance functon G( X ). 6. Numercal Examle 6.1 -D RBDO Problem wth Hghly Nonlnear Constrant unctons Consder a -D RBDO roblem wth three robablstc constrants, exressed as

9 ( d d 10) ( d d 10) mnmze Cost( d) Tar subect to PG ( ( Xd ( )) 0) P.75%, 1 ~ 3 L U d d d, dr and XR where three constrants functons are X1 X G1 ( X) G ( X) 1 ( Y 6) ( Y 6) 0.6 ( Y 6) Z (4) 80 G3 ( X) 1 X1 8X 5 Y X1 where Z and are drawn n g. 6. The dstrbuton and desgn doman for each X varable are shown n Table 1. The ntal desgn ont s d 0 = [5 5]. Random Varables Table 1: Proertes of Random Varables Dstrbuton d L d 0 d U Standard Devaton X 1 Normal X Normal (3) gure 6: Cost and constrant functons lot It s worth mentonng that there s no need to construct the conservatve surrogate model from the ntal desgn ont. In the roosed RBDO rocess, the determnstc desgn otmzaton s carred out frst. The RBDO rocess starts from the determnstc otmum thereafter. The conservatve surrogate model wll be constructed for the RBDO rocess only. In ths examle, the determnstc otmum s x = [5.19, 0.74] (whch s the magenta cross n g. 7), and the number of samles n the local wndow s fxed to 10 samles from the Latn hyercube samlng method. Snce 10 samles may not be enough to construct the accurate surrogate models, the conservatve surrogate models usng the roosed weghted Krgng varance from Eq. (1) are generated for two actve constrants G 1 and G. or comarson study, the conservatve surrogate models usng the constant safety margn from Eq. (17) are also generated. The samlng-based RBDO s carred out usng these two dfferent conservatve surrogate models, and the otmum desgns are comared n Table and lotted n g. 7. The C% level s set to be 90% n ths examle. In g. 7, t can be seen that the surrogate model from the dynamc Krgng model tself, whch s the blue lne,

10 underestmates the true resonse and results n danger for the obtaned otmum desgn d = [4.743, ] (blue cross n g. 7) as the robablty of falure for G s.499%, whch s larger than the target robablty of falure.75%, as shown n Table. Therefore the conservatve surrogate model s ndeed necessary for countng n the uncertanty from the surrogate model and assurng that the otmum desgn can satsfy the robablstc constrants. By usng the weghted Krgng varance for the conservatve surrogate model, whch s the red lne n g.7, the obtaned RBDO otmum desgn s d = [4.700, ], whch s the red cross n g. 7, and the cost functon value, the robablty of falure for G 1, and the robablty of falure for G are ,.0654%, and %, resectvely. As a comarson, the constant safety margn aroach gves a more conservatve otmum desgn d = [4.6510, 1.605] (the green cross n g. 7) where the robablty of falure for G 1 and G are % and %, resectvely; and the cost functon value s rom the results n Table, one can see that the conservatve surrogate model usng the weghted Krgng varance can assure the otmum desgn satsfyng the robablstc constrants and has a better otmum desgn n terms of cost functon comared to the conservatve surrogate model usng the constant safety margn aroach. gure 7: Dfferent conservatve surrogate models and otmum desgns Table : Otmum desgns from dfferent surrogate models and the robablty of falure at the otmum Methods Cost Otmum Desgn P 1, % MCS (5M) P, % Dynamc Krgng , Surrogate Constant Safety Margn , Model Weghted Krgng Varance , Analytcal , When usng the surrogate model for samlng-based RBDO, the samle rofle may affect the result as well. A good samle rofle for dynamc Krgng may end u havng a better surrogate model. Therefore, to nvestgate whether the roosed weghted Krgng varance for the conservatve surrogate model has a stable erformance, a statstcal study s carred out. In ths statstcal study, 50 sets of 10-LHS samles are generated. or each samle set, the samlng-based RBDO s carred out usng both the weghted Krgng varance aroach and the constant safety margn aroach. The comarson for cost functon value and the robablty of falure at the otmum are shown n Table. 3. Methods Table: 3 Cost functon and robablty of falure at otmum desgn (50 trals) Cost (Medan) P 1, % (Medan) P, % (Medan) # of Volaton G 1 # of Volaton G Dynamc Krgng Constant Safety Margn

11 Weghted Krgng Varance In Table 3, when usng the dynamc Krgng method tself, the otmum desgns volate the robablstc constrant G 8 tmes out of 50 trals. It shows that the dynamc Krgng redcton ether underestmates the true resonse or overestmates the true resonse at a rough 50% chance for each way. By usng the weghted Krgng varance for the conservatve surrogate model, the obtaned otmum desgns volate the robablstc constrant G three tmes. As a comarson, f the constant safety margn s used for the conservatve surrogate model, the number of volatons for G s 1 tme out of 50 trals. Constrant G 1 s not volated n both conservatve surrogate modelng cases due to ts lnearty. The medan cost functon value at the otmum (whch s ) usng the constant safety margn s larger than the one by usng the weghted Krgng varance, whch s ( ). It s worth mentonng that snce the conservatveness level n ths examle s set to be 90%, both the constant safety margn aroach and the weghted Krgng varance aroach satsfy the conservatveness. However, the weghted Krgng varance aroach rovdes a better otmum n terms of the cost functon value. 7. Concluson When alyng the surrogate model for otmzaton roblems, the conservatve surrogate model usng the uer bound of the Krgng redcton nterval s not desrable because t generates multle local otmums at the samle onts. The conservatve surrogate model usng the constant safety margn does not dstngush the uncertanty of the surrogate model at dfferent samle locatons, and often the obtaned otmum becomes over-conservatve where samles are dense and under-conservatve where samles are sarse. A weghted Krgng varance usng the changes n AICc of the Krgng model to quantfy the mortance from each samle ont s roosed to construct a conservatve surrogate model that does not generate unnecessary local otmum and dstngushes the uncertanty of the surrogate model at dfferent samle locatons. By alyng the weghted Krgng varance for the conservatve surrogate model n the samlng-based RBDO roblem, the obtaned otmum satsfes the robablstc constrants at the conservatveness level and acheves a better otmum n terms of cost functon value comared wth the otmum usng the constant safety margn. 8. Acknowledgement Research s suorted by the Automotve Research Center, whch s sonsored by the U.S. Army Tank Automotve Research, Develoment and Engneerng Center (TARDEC) and Army Research Offce (ARO). 9. References [1] I. Lee., K.K. Cho., Y. Noh., L. Zhao., and D. Gorsch., Samlng-Based Stochastc Senstvty Analyss Usng Score unctons for RBDO Problems Wth Correlated Random Varables, Journal of Mechancal Desgn, , Vol. 133, 011. [] I. Lee., K.K. Cho., and L. Zhao., Samlng-Based RBDO Usng the Dynamc Krgng (D-Krgng) Method and Stochastc Senstvty Analyss, Submtted to Structural and Multdsclnary Otmzaton, 011. [3] L. Zhao., K.K., Cho., and I. Lee., A Metamodelng Method Usng Dynamc Krgng for Desgn Otmzaton, acceted by AIAA Journal, 011. [4] V. Pcheny., Imrovng Accuracy and Comensatng for Uncertanty n Surrogate Modelng, Ph.D Dssertaton, Unv. of lorda, Ganesvlle, L, 009. [5] D. Hertog., J. Klenen., and A. Sem., The correct Krgng varance estmated by bootstrang, Journal of the Oeratonal Research Socety, Vol. 57, , 006. [6] S.D. Luna and A. Young., The bootstra and Krgng Predcton Intervals, Scandnavan J Stat, Vol. 30, , 003. [7].A.C. Vana, V. Pcheny, and R.T. Haftka, "Usng cross valdaton to desgn conservatve surrogates," AIAA Journal, Vol. 48, No.10, , 010. [8] H. Akake., A new look at the statstcal model dentfcaton, IEEE Transacton on Automatc Control, Vol. 19, No. 6, , [9] C.M. Hurvch., and C.L. Tsa, Regresson and Tme Seres Model Selecton n Small Samles, Bometrka, Vol. 76, , [10] K.P. Burnham and D.R. Anderson., Model Selecton and Multmodel Inference: A Practcal Informaton-Theoretc Aroach, Srnger, 00. [11] J. Martn., and T.W. Smson., Use of Krgng Model to Aroxmate Determnstc Comuter Models, AIAA Journal, Vol. 43, No. 4, , 005.