Reliability Optimization With Mixed Continuous-Discrete Random Variables and Parameters

Transcription

1 Subroto Gunawan Researh Fellow Panos Y. Papalambros Professor Department of Mehanial Engineering, University of Mihigan, Ann Arbor, MI 4809 Reliability Optimization With Mixe Continuous-Disrete Ranom Variables an Parameters Engineering esign problems frequently involve a mix of both ontinuous an isrete ranom variables an parameters. However, most methos in the literature eal with only the ontinuous or the isrete type, but not both. In partiular, no metho has yet aresse problems for whih the ranom omponents (variables an/or parameters) are ategorially isrete. This paper evelops an effiient optimization metho for problems involving mixe ontinuous-isrete ranom variables an parameters. The metho reues the number of funtion evaluations performe by systematially filtering the isrete ombinations use for estimating reliability base on their importane. This importane is assesse using the spatial istane from the feasible bounary an the probability of the isrete omponents. The metho is emonstrate in examples an is shown to be very effiient with only small errors. DOI: 0.5/ Keywors: filter, influene funtion, mixe ontinuous-isrete, reliability base esign optimization (RBDO) Introution Engineering esign problems frequently involve a mix of both ontinuous an isrete variables an parameters that are ranom. Due to this inherent ranomness, harateristis of a esign may iffer signifiantly from their nominal values. As suh, when optimizing the esign it is important to ensure that it is feasible regarless of the ranomness. An optimization approah that aounts for esign feasibility uner ranom onition is ommonly referre to as reliability base esign optimization RBDO. The term feasibility robust optimization is also use. Ranomness in the ontinuous omponents variables or parameters of a esign may be ue to variations in the esign s geometry, material properties, or operating environment. For instane, yiel strength of steel is typially ranom. The isrete omponents of a esign may also be ranom for numerous reasons: hanges in lifetime operation or manufaturing error. For example, some of the teeth in a gear may wear out an break within its esign lifetime. Similarly, uring the operation of a fuel ell system FCS some of the ells may fail for various reasons, an the number of faile ells is isrete an ranom. The gear an FCS examples involve the so-alle ategorially isrete ranom omponents beause values between the isrete jumps are not efine e.g., there is no gear with 8.3 teeth. A isrete ranom omponent an also result from limiting a ontinuous ranom omponent to a set of values. For example, thikness of a sheet metal is typially isrete ue to manufaturing praties. Researh in RBDO fouses almost entirely on problems with ontinuous ranom omponents. The two most popular approahes in ontinuous RBDO are the reliability inex approah RIA an its inverse, the performane measure approah PMA 6. These methos use the first-orer reliability metho Corresponing author. Contribute by the Design Theory an Methoology Committee of ASME for publiation in the JOURNAL OF MECHANICAL DESIGN. Manusript reeive May 9, 2005; final manusript revise January 25, Review onute by Wei Chen. Paper presente at the ASME 2005 Design Engineering Tehnial Conferenes an Computers an Information in Engineering Conferene DETC2005, Long Beah, California, USA, September 24 28, FORM an the seon-orer reliability metho SORM to alulate the reliability of a esign 7 0. RIA an PMA are both neste-loop algorithms an as suh not very effiient omputationally. Many methos have been evelope to improve their effiieny inluing the safety-fator approah SFA, the most probable point MPP importane sampling metho 2, the sequential optimization an reliability assessment SORA metho 3, the single-loop metho 4, an many others. A omparative stuy of SFA an SORA is given in Ref. 5. There is also onsierable researh in the moment mathing methos, also known as the first-orer seon moment FOSM approah. As the name iniates, these methos use first an seon orer erivatives to estimate the mean an variane of the propagate unertainty, respetively 6 8. There are also ontinuous RBDO methos that o not iretly fall into the RIA/PMA or FOSM ategories 9 2. Some work in RBDO for isrete ranom omponents is base on Taguhi s methoology in robust esign 22,23. However, as of the writing of this paper, methos for RBDO with mixe ontinuous-isrete ranom omponents are virtually nonexistent. There is only one paper in the literature ealing with this type of RBDO problem for optimization of strutures 24. In this work, the authors onvert the isrete part of the ranom omponents into ontinuous ones via aition of equivalent onstraints, an use ontinuous RBDO tehniques to solve the problem. This approah is not appliable to problems whose isrete ranom omponents are ategorial the number of gear teeth for instane. Also, reliability values alulate by this approah are somewhat ambiguous beause the alulation is performe for a ontinuous probability istribution that oes not neessarily reflet the isrete probabilities. There is a relate work in RBDO for a mix of ranom an interval omponents 25, but the metho annot be extene to a mix of ontinuous-isrete ranom omponents. The objetive of this artile is to evelop an effiient RBDO metho for problems involving a mix of ontinuous-isrete ranom variables an parameters. We use a probabilisti approah an assume that the relevant probabilities are known, an that the ranom omponents are inepenent. The work fouses stritly on esign reliability in terms of the onstraints of the optimization 58 / Vol. 29, FEBRUARY 2007 Copyright 2007 by ASME Transations of the ASME

2 problem, an objetive robustness is not onsiere. The rest of the paper is organize as follows. Setion 2 provies the mathematial formulation of the mixe ontinuous-isrete optimization problem of interest. Setion 3 presents three methos to alulate mixe ontinuous-isrete reliability of a esign. Setion 4 emonstrates the appliation of the metho evelope with two examples. The paper onlues with a short summary in Se Problem Formulation Consier the eterministi mixe ontinuous-isrete optimization MCDO problem in Eq. minimize f x,p x= x,x Fig. Graphial representation of isrete reliability subjet to: g j x,p 0, where: x L x x U x k Z k R, j =,...,J k =,...,n p = p,p In this formulation, the esign variables x vary uring optimization, while the esign parameters p are fixe. The variable x ontains n ontinuous omponents x = x,...,x n an n isrete omponents x = x,...,x n. The ontinuous variables are boune by x L x x U, while eah isrete variable x k, k =,...,n, takes values from a set Z k whose number of elements is A k. Similarly, the parameter p ontains m ontinuous omponents p = p,...,p m an m isrete omponents p = p,...,p m. We assume there is no equality onstraint beause reliability in this ontext is efine only for inequality onstraints. In mixe ontinuous-isrete reliability optimization MCDRO, unertainty in the problem is moele as ranomness in variables an parameters, Eq. 2. minimize f X, P X = X, X subjet to: Pr g j X,P 0 R j, j =,...,J where: X = X,X an P = P,P 2 For this problem, the probabilisti onstraints are alulate base on the ranom variables X= X,X an ranom parameters P = P,P, an R j is the esire reliability of the jth onstraint speifie by the esigner. The probabilisti onstraint an be written also as F gj 0 R j, where F gj is the umulative istribution funtion CDF of g j. Sine we are not onsiering objetive robustness, minimization of the objetive is performe with respet to the mean values of the ranom variables X, given fixe mean values of the ranom parameters, P. The solution to Eq. 2 is a esign that is both optimal an satisfies the reliability onstraints, simply alle the reliability optimum. For simpliity of isussion, we assume there is no eterministi variable or parameter. Ranomness in X an P is moele as follows. Eah ontinuous ranom variable X i, i=,...,n, is assume ranomly istribute with a known probability ensity funtion PDF f Xi. Likewise, eah ontinuous ranom parameter P i, i=,...,m, is assume ranomly istribute with a PDF f Pi. The kth isrete ranom variable X k, k=,...,n, is assume isretely istribute aroun its mean value within a set W k Z k with a known probability mass funtion PMF f Xk. The number of elements in W k is B k A k. Similarly, the kth isrete ranom parameter P k, k =,...,m, is assume isretely istribute within a set S k R with a known PMF f Pk. The number of elements in S k is C k. All ranom variables an parameters are assume inepenent. For onveniene we efine two vetors Y = X,P an Y = X,P to be the ontinuous an isrete ranom omponents of the MCDRO, respetively. The number of isrete omponents in the problem is then n +m, while the number of ontinuous omponents is n +m. There are n +m isrete omponents, an eah omponent X k or P k an take a value from B k or C k number of hoies. So the total number of ombinations of all the n isrete hoies is D= m k= B k k= C k. Sine we assume inepenene, the joint PMF of the k isrete ombination is f X P,k n = m i= f Xi i= f Pi, for k=,...,d. Similarly, the joint PDF of the ontinuous omponent is f X P = n m i=f Xi i= f Pi. For simpliity of notation, we efine f XP,k f X P,k an f XP f X P to be the joint PMF an PDF, respetively. The most important an iffiult aspet of MCDRO is alulating the probabilisti onstraints effiiently. One these onstraints are alulate, we an then use a onventional optimization algorithm to fin the reliability optimum. The next setion provies a isussion on how to alulate these onstraints effiiently. 3 Reliability Analysis If there is no ontinuous omponent in the problem i.e., Y =, reliability of the jth onstraint an be alulate by summing the joint probability of all feasible isrete ombinations as shown in Eq. 3 Pr g j X,P 0 = f XP,k I j,k k= I j,k =, g j X k,p k 0 0, g j X k,p k 0, j =,...,J 3 Here I j,k is the feasibility iniator funtion for the jth onstraint at the kth isrete ombination, an X k an P k are the kth realization of X an P, respetively. Graphially, f XP,k an I j,k in Eq. 3 an be shown as stem plots on the salar g j axis with probability values for the orinate as shown in Fig.. The reliability value is then the sum of probabilities of all stems to the left of an inluing the onstraint bounary g j =0. When there are both ontinuous an isrete omponents in the problem, the iniator funtion I j,k is replae by the onitional probability that g j X,P is feasible given X k,p k. Theoretially, this probability is alulate by solving the following integral Pr g j X,P 0 X k,p k = F gj,k D 0 f XP X = g P j 0 X k,pk The reliability of g j is then as shown in Eq. 5 4 Journal of Mehanial Design FEBRUARY 2007, Vol. 29 / 59

3 Fig. 2 Graphial representation of mixe ontinuous-isrete reliability Fig. 3 Different importane of ensity istributions Pr g j X,P 0 = F gj 0 = k= D f XP,k F gj,k Figure 2 shows the graphial representation of Eq. 5 on the g j axis. Unlike the isrete stem plot, the orinate in this figure is a weighte probability ensity. It shoul be emphasize that the urves shown are not PDF urves in that the area uner eah urve is not. For instane, the area uner the k=3 urve is learly larger than that of the k=5 urve. Rather, they are weighte ensity urves where eah urve represents one multipliation element in Eq. 5. So eah urve alreay inlues the weighting fator f XP,k. Following this efinition, the reliability of g j is the sum of the areas uner the urves to the left-han sie of an inluing g j =0. Analytial solution to the integral in Eq. 4 is generally not possible while numerial integration is prohibitively ineffiient. In the next subsetions we isuss three alternative methos to alulate mixe ontinuous-isrete reliability: Monte Carlo analysis MCA ; 2 full fatorial reliability analysis FFRA ; an 3 partial fatorial reliability analysis PFRA. 3. Monte Carlo Analysis. The simplest alternative to numerial integration is to use a Monte Carlo simulation MCS to approximate it. There are two approahes to use MCS to alulate the reliability F gj 0. In the first approah we use it to approximate F gj,k 0 only an then repeat it D times for all ombinations of the isrete ranom omponents. In this approah, the ontinuous ranom omponents are first isretize followe by a large number of experiments. Samples for the experiments are taken proportional to the probabilities of eah isretize bins; F gj,k 0 is then approximately equal to the ratio of feasible experiments where g j X,P 0 X k,p k to the total number of experiments. Sine f XP,k is known for all k, F gj 0 an be alulate using Eq. 5. In the seon approah we use MCS to iretly approximate F gj 0. Here, samples for the experiments are taken from both the ontinuous an the isrete ranom omponents. Samples for the isrete omponents are taken proportional to the probability mass of the isrete values. The imension of the first an seon approah is n +m an n +m + n +m, respetively. In general the seon approah requires more experiments for the same auray. However, the first approah nees to be repeate D times. Sine D is muh larger than n +m, overall the seon approah is generally more effiient than the first. For the rest of the artile we will use the term MCA to refer to the seon approah. MCA provies an exellent approximation to F gj 0 provie the number of experiments is large enough. In the absene of analytial solution, MCA results are often regare as the exat solution. However, the large number of funtion evaluations neee by the metho prevents its wiesprea use in pratial appliations. 3.2 Full Fatorial Reliability Analysis. An alternative 0 5 metho to alulate mixe ontinuous-isrete reliability is to use the FORM 8 0 to approximate the integral in Eq. 4. In this metho we alulate F gj,k 0 using FORM at all isrete ombinations, an then use them in Eq. 5. FORM has been reporte to be effiient in terms of funtion evaluations 8,9. However, here we have to perform FORM D times, one for eah isrete ombination. Sine D is generally large, the total number of funtion evaluations neee is still very high. If we an somehow limit the use of FORM only to those isrete ombinations that really nee it, we an rastially reue the number of funtion evaluations require. A metho to filter the isrete ombinations will be isusse next. 3.3 Partial Fatorial Reliability Analysis. In FFRA, the integral in Eq. 4 is alulate at all isrete ombinations. Mathematially, however, we really only nee to alulate F gj,k 0 s at those isrete ombinations that are lose to the onstraint bounary an whose f XP,k /s are large aoring to a metri that will be esribe later. Ifg j,k at a kth isrete ombination with the means X, P for the ontinuous omponents is far from the bounary, then F gj,k 0 an be approximate with I j,k with a very small error. Similarly, if f XP,k is very low, F gj,k 0 an also be approximate with I j,k with a maximum error of only 0.5 f XP,k. For illustration, onsier the following hypothetial weighte ensity istributions with D=4 as shown in Fig. 3. For the k= istribution, f XP, is very small an g j, is far from the bounary. So for this istribution, F gj, 0 I j, =.0 with very little error. Likewise, f XP,3 is also very small an its ontribution to the overall reliability is small. So although it is lose to the bounary, the error inue by approximating F gj,3 0 I j,3 =0 is small. In ontrast, the k=2 istribution is very lose to the bounary an its f XP,2 is large. For this istribution I j,2 =.0 is not a goo approximation to F gj,2 0, an we nee to use FORM to alulate it more aurately. Similarly for the k=4 istribution, although it is far from the bounary, f XP,4 is so large that we annot neglet its area uner the urve to the left of g j =0. In this ase too, we nee to alulate F gj,4 0 more aurately. Notie in Fig. 3 that the areas to the left of g j =0 for ensity urves 3 an 4 seem omparable graphially. However, f XP,4 is larger than f XP,3 so numerially they are ifferent the area for ensity urve 4 is larger. From the previous isussion, it is apparent that we nee two piees of information to etermine if it is neessary to use FORM at the kth isrete ombination: f XP,k value; an 2 relative istane of g j,k from the onstraint bounary. The first piee of information is alreay available as part of our problem formulation an assumption; the seon piee of information is not as reaily available. To etermine how far g j,k is from the onstraint bounary, we efine an influene funtion h j,k g j,k :R R to be a monotonially ereasing mapping of the istane from the onstraint bounary. The funtion is maximum when g j,k =0, an graually 60 / Vol. 29, FEBRUARY 2007 Transations of the ASME

4 Fig. 5 Illustration of negligible importane Fig. 4 ereases as g j,k moves away from the bounary. Many types of influene funtion an be use, e.g., Gaussian or paraboli. In this PFRA metho we use the following influene funtion 6 h j,k g j,k = g j,k / ˆ gj, if g j,k 0 g j,k / ˆ gj, if g j,k 0 where is the CDF of a stanar normal istribution with zero mean an stanar eviation of ; an ˆ gj is a user-speifie parameter. A nie feature of this influene funtion is that it reaily 0. If we assume the weighte on- provies an estimate to F gj,k itional ensities are normally istribute with a stanar eviation of ˆ gj, then F gj,k Influene funtions for ifferent ˆ gj 0 Fˆ g j,k = h j,k, if g j,k 0 h j,k, if g j,k 0 7 In using Eqs. 6 an 7, we have invoke an assumption that the onitional ensities are normally istribute. However, this is only one way to efine an influene funtion. If the esigner has reasons to believe that the onitional ensities follow nonnormal istributions, this information may be use to onstrut a new influene funtion using the CDF of that non-normal istribution an the appropriate statistis as inputs. The h j,k funtion in Eq. 6 has a maximum value of 0.5 at g j,k =0, an graually eays with a istane from the bounary aoring to the error funtion erf. The eay rate of the funtion epens on the speifie ˆ gj, the larger it is the slower the eay. Figure 4 shows the graphs of h j,k for three values of ˆ gj. A large ˆ gj plaes more importane to ensity urves far away from the bounary an inreases the number of isrete ombinations hosen for FORM analysis. This in turn will inrease the total number of funtion evaluations. In general, however, the auray of the alulate reliability will also improve. In ontrast, a small ˆ gj will erease the number of funtion evaluations, but generally at the expense of auray. In our implementation, we use the value ˆ gj =0.. One very important remark regaring this hoie of value: sine ˆ gj is our approximation to the ensity urves stanar eviations, it is ritial that g j is numerially sale to be within the same orer of magnitue. In engineering problems, a typial saling proeure is to use the upper/lower boun of the onstraints as the normalizing fator. Using f XP,k an h j,k, we an now etermine if the kth isrete ombination is important enough to warrant a FORM analysis. We efine a quantity H j,k g j,k = f XP,k h j,k to be the importane funtion of the kth ombination. If H j,k is large, then it is neessary to perform FORM at this partiular ombination. If H j,k is small, then we an use Fˆ g j,k as an approximation. The step-by-step algorithm of the PFRA metho to alulate mixe ontinuous-isrete reliability of the jth onstraint is as follows: Step. Set ˆ gj =0. an a lower boun on the sum of normalize importane value SH min=0.95. Step 2. Calulate g j,k, h j,k, an H j,k for all k=,...,d. Calulate the approximation Fˆ g j,k. Step 3. Calulate the sum of all importane values SH= D k= H j,k. Step 4. If SH 0.00, approximate the reliability F gj 0 D k= f XP,k Fˆ g j,k, then stop. Else ontinue. Step 5. Calulate the normalize importane value H j,k=h j,k /SH for all k=,...,d. Step 6. Sort the H j,ks from largest to smallest. Start from the largest H j,k, selet NL D isrete ombinations suh that NL i= H j,i SH min. Step 7. From the selete NL ombinations, isar the ND NL ombinations whose H j,k Step 8. For the selete NP=NL ND ombinations, use FORM to alulate Fˆ g j,k. For the rest of the ombinations, keep the Fˆ g j,k ompute in Step 2. Step 9. The approximate reliability of the onstraint is F gj 0 D k= f XP,k Fˆ g j,k. Stop. There are two user-speifie parameters in the algorithm: ˆ gj an SH min. The effet of ˆ gj on the total number of funtion evaluations an the auray of the result has been isusse previously. The parameter SH min ontrols the minimum perentage of total H j,k s that are selete. The larger SH min the more isrete ombinations will be selete. This will inrease the auray of the approximation, but with an inrease in the number of funtion evaluations. A erease in SH min will result in the opposite effet. In our implementation, we set SH min=0.95, i.e., 95% of the total H j,k s are selete. The ˆ gj an SH min values suggeste in this paper are base on our experiene in applying the algorithm. They are foun to result in less than % approximation error. The algorithm above ontains two filtering steps: negligible importane filtering Step 4 an onentrate importane filtering Step 7. Step 4 in the algorithm aounts for the ase where all the ensity urves are far from the onstraint bounary, as iniate by the low SH value see Fig. 5. In this ase, the Fˆ g j,k approximation is suffiient for all isrete ombinations, an no FORM analysis is neee at all. Step 7 in the algorithm aounts for the ase where only a few of the NL isrete ombinations selete in Step 6 are really important see Fig. 6. This is iniate by the very small H j,k s of the other ND ombinations. Note that in the above algorithm we have use a normal istri- Journal of Mehanial Design FEBRUARY 2007, Vol. 29 / 6

5 Table H values for X 2,P ombinations X 2 = P= P= P= P= Fig. 6 Illustration of onentrate importane bution to approximate F gj,k 0. If the atual weighte onitional ensities are nonsymmetri, then other forms of istributions might give better approximations. 3.4 Error an Effiieny. Of the three methos presente to alulate reliability, MCA is the most aurate followe by FFRA an PFRA. As mentione before, when analytial solution is not available, MCA solution is often regare as the true solution. If the urvature of the onstraint funtion is not too large, the FFRA solution will also be very lose to that of MCAs. Unfortunately, the FFRA error ue to FORM linearization is problem epenent, an there is no analytial estimation to it. Nevertheless, the error inue by FORM has been reporte to be small 8,9. Sine PFRA uses FORM, it is also affete by the linearization error. In aition to this error, PFRA also inues an error by approximating the onitional probabilities at some of the isrete ombinations. Like the FORM error, the atual amount of this error is problem epenent an is not possible to formulate analytially. In the best ase, all the ensity urves are far from the bounary negligible importane ase, an the PFRA error is zero. In the worst ase, however, the normal approximation an the ˆ gj estimate might be very ifferent from the atual ensity urves. In this situation, the maximum error of the onitional probabilities not alulate by FORM is 0.5 f XP,k. Sine we selet NP ombinations for FORM analysis, the total maximum error is D NP e max = k= 0.5 f XP,k 8 This error oes not inlue the FORM linearization error. Also, Eq. 8 involves the quantity NP that is problem epenent, an so the error alulation is also problem epenent. Sine we impose the SH min=0.95 seletion riterion in the PFRA algorithm Step 6, NP is generally large an the f XP,k s at the D NP ombinations are small. So overall the maximum error is small. Although MCA is the most aurate, it is also the most ineffiient of the three methos in terms of number of funtion evaluations FEs. The atual magnitue of MCA s FE epens on the problem s imension, but FE MCA 0 6 is typial. For the FFRA metho, FE FFRA = D FE FORM, where FE FORM is the number of funtion evaluations performe by FORM. There is no analytial formula for FE FORM, but it is reporte to be on the orer of O 0 0 O Due to the effiieny of FORM, in general FFRA is muh more effiient than MCA, even though FE FFRA involves the fator D. PFRA uses FORM at some, but not all, of the D isrete ombinations. The FE of PFRA is FE PFRA =D+ D FE FORM, where 0. The D in the first fator of the sum aounts for the evaluations of g j,k at all k=,...,d Step 2. In the best ase negligible importane ase, =0 an FE PFRA =D, i.e., no FORM analysis is neee at all. In the worst ase, PFRA performs FORM at all isrete ombinations. For this ase, = an FE PFRA =D +FE FORM FE FFRA for FE FORM, i.e., PFRA is as effiient as FFRA. The quantity ereases as the esign point moves away from the onstraint bounary, so PFRA is more effiient for esign points far from the bounary as is usually the ase in the early iterations of an optimization run. Overall, PFRA is at least as effiient as FFRA, while MCA is the least effiient of all. A stuy of the error an effiieny properties of MCA, FFRA, an espeially PFRA is given in the next setion through examples. 4 Demonstration Examples We apply the PFRA algorithm to two examples. In the first, we show a step-by-step implementation of the algorithm to alulate a mixe ontinuous-isrete reliability of a esign point with respet to a single inequality onstraint. In the seon, we show the appliation of PFRA in an optimization algorithm to solve a MCDRO problem. 4. Single Constraint Reliability. Consier a quarati onstraint g that is a funtion of one ontinuous variable X, one integer variable X 2, an one integer parameter P. The variables an parameters in this problem are ranom. Following the notation in Se. 2, X = X an X = X 2, where n = an n =; for the parameters, P = an P = P, where m =0 an m =; an so Y = X an Y = X 2, P. The objetive of this stuy is to alulate the reliability of a esign point whose means are X = X, X2 = 5.5,7 with respet to g. g X,P = 350 7X 2 +6X P 2 6X P +4X 2 P 5.8X 93.2X 2 63P Ranomness in the variables an the parameter is as follows all are inepenent. The variable X is ranomly istribute aoring to the normal PDF f X =N X,0.2. The integer variable X 2 is istribute aroun its nominal value within the set W = 5,6,7,8,9 aoring to PMF f X2 = 0.05,0.5,0.6,0.5,0.05. The integer parameter P is istribute within the set S = 2,3,4,5 with PMF f P = 0.05,0.3,0.45,0.2. Following our notation, B =5, C =4, an D=B C =20. The joint PMF of X 2 an P is shown below f X2 as olumns, f P as rows f XP = The step-by-step implementation of the PFRA algorithm to alulate reliability is as follows: Step. Set ˆ g =0. an SH min=0.95. Step 2. The g values at all isrete ombinations are alulate by substituting X =5.5 an the X 2, P ombination to Eq. 9. Using g, the h values are alulate using Eq. 6. The importane values H are obtaine by multiplying h with the joint pmf f XP in Eq. 0. The approximation Fˆ g at eah ombination is alulate using Eq. 7. The alulate H an Fˆ g are shown in Tables an 2, respetively. Step 3. The sum of the H,k s in Table is SH= 20 k= H,k 62 / Vol. 29, FEBRUARY 2007 Transations of the ASME

6 Table 2 Fˆ g values for X 2,P ombinations Table 5 Comparisons of MCA, FFRA, an PFRA results X 2 = P= P= P= P= = Step 4. From Step 3, SH So this is not a negligible ase; ontinue to Step 5. Step 5. The normalize importane values H is shown in Table 3. Step 6. Starting from the largest H in Table 3, NL=4 ombinations are selete suh that 4 k= H,k SH min. The ombinations selete are highlighte in Table 3. Step 7. From the 4 ombinations selete, none has a H,k So ND=0. Step 8. For the final NP=NL ND=4 ombinations selete, we use FORM to alulate Fˆ g. For the other ombinations, we keep the Fˆ g values from Table 2. The revise Fˆ g values are shown in Table 4. Step 9. The approximate reliability is F g 0 20 k= f XP,k Fˆ g,k = Stop. As a benhmark, we alulate the reliability using the MCA metho in whih,000,000 samples are taken from the X 2, P isrete sets an the isretize X istribution. The MCA reliability value is Fg 0 MCA = This value is onsiere to be the atual value. The absolute error of PFRA result is e =0.0042, less than the maximum error preite in Eq. 8 : e max = The relative error of PFRA is = F g 0 PFRA F g 0 MCA = F g 0 MCA less than %. For omparison, we also alulate the reliability using the FFRA metho. The ompute Fg 0 FFRA = is very lose to Fg 0 MCA. The isrepany between the two an be attribute to FORM linearization error as well as rounoff error. Table 5 shows a omparison of the F g 0 values an the relative errors obtaine using MCA, FFRA, an PFRA for ifferent esign points. Table 6 shows the number of funtion evaluations Table 3 H values for X 2,P ombinations X 2 = P= P= P= P= X, X2 MCA FFRA PFRA.7, , , , , , , , performe by eah metho. The perent reution in funtion evaluations from using PFRA ompare to FFRA: re = FE FFRA FE PFRA /FE FFRA is also shown. As seen in Table 5, PFRA provies an exellent estimate to MCA results for all ranges of reliability values. The relative error of the approximation is less than % for all eight esign points. At the same time, using PFRA results in a 4 44% reution in the number of funtion evaluations ompare to using FFRA as shown in Table Design of a Belleville Spring. The objetive of this problem is to optimize the Belleville spring shown in Fig. 7 for maximum rate loa P loa. This example is originally formulate by Siall 26, an is moifie here to be a MCDRO problem. All probabilities are assume inepenent. The esign variables are: external iameter e, internal iameter i, free height h, an thikness t. The variables e, i,h are ontinuous in meters, but ue to manufaturing praties the sheet metal thikness t is only available in multiples of 0.25 mm. There is ranomness in the ontinuous variables e, i,h, an eah is moele as a normal istribution with a stanar eviation of mm, mm, an mm, respetively 27. The spring thikness is also ranom an is isretely istribute aoring to the following probability Pr t = = 0.5, = t 0.2, = t ± , = t ± 0.5 The spring is to be mae from high strength steel Type C, but ue to manufaturing imperfetion its properties may vary. Table 7 Table 6 Comparison of FE MCA,FE FFRA, an FE PFRA X, X2 FE MCA FE FFRA FE PFRA re.7, 6,000, , 7,000, , 9,000, , 3,000, , 3,000, , 0,000, , 2,000, , 2,000, Table 4 Revise Fˆ g values for X 2,P ombinations X 2 = P= P= P= P= Fig. 7 A Belleville spring Journal of Mehanial Design FEBRUARY 2007, Vol. 29 / 63

7 Table 7 Steel properties an their probabilities Table 8 Comparison of MCA, FFRA, an PFRA optima Steel type E GPa aw MPa shows the four possible isrete variations of the steel properties, an the probabilities of eah variation. For ease of referral, we assign the letter A, B, C, an D to eah of the four possible variations. In this table, the quantities E an aw are the elasti moulus an allowable stress of the steel, respetively. So in this problem X = e, i,h an X = t, an n =3 an n =. For the parameters, P = an P = steel type, an m =0 an m =. The ontinuous omponents are Y = e, i,h, while the isrete omponents are Y = t,steel type. The PDFs of the ontinuous variable are f e =N e,0.0866, f i =N i,0.0767, an f h =N h, For the isrete variable, W = t 0.5, t 0.25, t, t +0.25, t +0.5 with a PMF f t = 0.05,0.2,0.5,0.2,0.05 an B =5. For the isrete parameter: S = A,B,C,D withapmff steel = 0.,0.25,0.6,0.05 an C =4. The number of ombinations of the isrete omponents is D=B C =20. The spring optimization is onstraine by two esign onstraints, maximum allowable stress an maximum mass, an five geometri onstraints. The formulation of the eterministi MCDO is shown below notie how the onstraints are numerially sale maximize f x,p = P loa x= e, i,h,t subjet to: g x,p max / aw 0 g 2 x,p m/m max 0 g 3 x,p h min /h 0 g 4 x,p h + t /l 0 g 5 x,p e / max 0 g 6 x,p.25 i / e 0 Probability A B C D g 7 x,p 0.3 e i /h 0 2 Here P loa is the rate loa N, max is the maximum stress Pa, an m is the spring mass kg. Details of the eterministi problem an be foun in Ref. 28. The optimum of the eterministi MCDO is e, i,h,t * = 0.3,0.232,5.0,8.0 using the baseline steel Type C as the material. The t * an h * values shown are in mm. The maximum loa * of this eterministi optimum is P loa =67.82 kn. The onstraint values are g * = 0.009, 0.7,0, 0.35,0, , where g 3 an g 5 are ative. Base on the speifie PDFs an PMFs, the reliabilities of this optimal point w.r.t. the onstraints are F * gj 0 = 0.376,0.994,0.5,.0,0.976,0.982,.0. As an be seen, the eterministi optimum has low reliabilities in terms of g an g Reliability Optimization. For reliability optimization, all onstraints are replae with probabilisti ones. The lower reliability boun for all probabilisti onstraints is set to be R j =0.99 for j=,...,7. We solve the MCDRO using the MOST algorithm as implemente in the ommerial software isight 9.0. MOST is an MCA FFRA PFRA e m i m h mm t mm P loa kn F g F g FE 4,470,000 77,752 0,866 optimization algorithm that ombines SQP an branh-an-boun algorithms to solve mixe ontinuous-isrete problems 29. For omparison, we solve the problem three times, eah using MCA, FFRA, an PFRA for reliability alulation. For fairness, the same starting point e, i,h,t = 0.3,0.2,5.0,7.0 is use in all three runs. For the MCA metho,,000 samples are use this number provies a relatively aurate reliability while still pratially manageable. For the FFRA metho, the HL-RF algorithm 9 is hosen for FORM alulation. For the PFRA metho, the parameters are speifie to be ˆ gj =0. an SH min=0.95. The Fˆ g values j at the selete ombinations are also alulate with FORM using the HL-RF algorithm. Table 8 shows the reliability optima obtaine an the total number of funtion evaluations performe. In ounting FE, eah alulation of either the objetive or onstraint funtion is onsiere one evaluation. The table also shows the reliabilities of the optima in terms of g an g 3. Reliability values in terms of the other onstraints are the same for all three optima. We see in Table 8 that the MCA optimum has the highest P loa while PFRA optimum has the lowest. However, we also see that the MCA optimum oes not quite satisfy the thir reliability onstraint. A possible explanation is that the number of samples use is not large enough so that the reliability values alulate by MCA are noisy. Even with the relatively low number of samples, however, the optimization alreay require more than 4 million funtion evaluations. This observation further emonstrates the impratiality of MCA. The FFRA s P loa is slightly lower than that of MCAs, but it satisfies all the reliability onstraints. In terms of effiieny, the FE FFRA value is also signifiantly lower than FE MCA. PERA s P loa is lower than FFRA s. This may be ause by the inonsistent hanges in the reliability values an hene not-as-aurate graient values alulate using PFRA. Nevertheless, in return for this 7% erease in P loa, we gain a signifiantly larger 37% erease in total funtion evaluations. These results suggest a potential hybri optimization algorithm utilizing both PFRA an FFRA to ahieve onvergene an effiieny. The PFRA metho offers a way to reue the omputational requirement of solving a MCDRO problem signifiantly by sarifiing a little auray. Even with the reution, however, the total omputational ost may still be rather high. One way to further inrease effiieny is to use a more effiient tehnique other than FORM to alulate the onitional probability. Another way is to use parallelization to alulate the onitional probability at eah isrete ombination. These options merit further investigation. 5 Summary This paper presents a metho to reue the number of funtion evaluations neee to alulate a mixe ontinuous-isrete reliability while maintaining auray. Unlike the FFRA metho, PFRA uses FORM to alulate the ontinuous onitional probabilities only at some of the isrete ombinations. These isrete ombinations are systematially selete base on their importane, whih in turn epens on the relative istane from the 64 / Vol. 29, FEBRUARY 2007 Transations of the ASME

8 bounary an the probability of the isrete omponents. In the numerial example, the PFRA result is foun to be in exellent agreement with the MCA value along with a signifiant improvement in omputational effiieny. The relative error of the approximation is less than %, an the number of funtion evaluations ereases by as muh as 44%. When use in reliability optimization, PFRA is also foun to perform well. The PFRA optimum is slightly inferior to the FFRA optimum 7% lower, but the total number of funtion evaluations is erease by 37%. Aknowlegment This researh was partially supporte by the Automotive Researh Center ARC, a U.S. Army Center of Exellene in Moeling an Simulation of Groun Vehiles at the University of Mihigan, an by NSF Grant No. DMI This support is gratefully aknowlege. The opinions expresse in the paper are those of the authors an o not neessarily reflet those of the sponsors. Referenes Pu, Y., Das, P. K., an Faulkner, D., 997, A Strategy for Reliability-Base Optimization, Eng. Strut., 9 3, pp Li, H., an Foshi, R. O., 998, An Inverse Reliability Metho an Its Appliation, Strut. Safety, 20 3, pp Yu, X., Chang, K., an Choi, K. K., 998, Probabilisti Strutural Durability Preition, AIAA J., 36 4, Tu, J., Choi, K. K., an Park, Y. H., 999, A New Stuy on Reliability-Base Design Optimization, ASME J. Meh. Des., 2, pp Du, X., an Chen, W., 200, A Most Probable Point Base Metho for Unertainty Analysis, J. Design Manuf. Autom., 4, pp Lee, J. O., Yang, Y. S., an Ruy, W. S., 2002, A Comparative Stuy on Reliability-Inex an Target-Performane-Base Probabilisti Strutural Design Optimization, Comput. Strut., 80, pp Cizelj, L., Mavko, B., an Riesh-Oppermann, H., 994, Appliation of First an Seon Orer Reliability Methos in the Safety Assessment of Crake Steam Generator Tubes, Nul. Eng. Des., 47, pp Zhao, Y. G., an Ono, T., 999, A General Proeure for First/Seon-Orer Reliability Metho FORM/SORM, Strut. Safety, 2, pp Youn, B. D., an Choi, K. K., 2004, Seleting Probabilisti Approahes for Reliability-Base Design Optimization, AIAA J., 42, pp Chiralaksanakul, A., an Mahaevan, S., 2004, Reliability-Base Design Optimization Methos, DETC04/DAC-57456, Proeeings of DETC 04, Salt Lake City, UT, Sept. 28 Ot. 2. Wu, Y. T., an Wang, W., 998, Effiient Probabilisti Design by Converting Reliability Constraints to Approximately Equivalent Deterministi Constraints, J. Integr. Des. Proess Si., 2 4, pp Du, X., an Chen, W., 2000, Towars a Better Unerstaning of Moeling Feasibility Robustness in Engineering Design, ASME J. Meh. Des., 22, pp Du, X., an Chen, W., 2004, Sequential Optimization an Reliability Assessment Metho for Effiient Probabilisti Design, ASME J. Meh. Des., 26, pp Liang, J., Mourelatos, Z. P., an Tu, J., 2004, A Single-Loop Metho for Reliability-Base Design Optimization, DETC04/DAC-57255, Proeeings of DETC 04, Salt Lake City, UT, Sept. 28 Ot Yang, R. J., an Gu, L., 2004, Experiene with Approximate Reliability- Base Optimization Methos, Strut. Multiisip. Optim., 26, pp Wu, Y. T., Millwater, H. R., an Cruse, T. A., 990, Avane Probabilisti Strutural Analysis Metho for Impliit Performane Funtions, AIAA J., 28 9, pp Parkinson, A., Sorensen, C., an Pourhassan, N., 993, A General Approah for Robust Optimal Design, ASME J. Meh. Des., 5, pp Jung, D. H., an Lee, B. C., 2002, Development of a Simple an Effiient Metho for Robust Optimization, Int. J. Numer. Methos Eng., 53, pp Yu, J. C., an Ishii, K., 998, Design for Robustness Base on Manufaturing Variation Patterns, ASME J. Meh. Des., 20, pp Yu, J. C., an Ho, W. C., 2000, Moifie Sequential Programming for Feasibility Robustness of Constraine Design Optimization, DETC00/DAC- 453, Proeeings of DETC 00, Baltimore, MD, Sept Gunawan, S., an Azarm, S., 2005, A Feasibility Robust Optimization Metho Using Sensitivity Region Conept, ASME J. Meh. Des., 27, pp Roy, R., Parmee, I. C., an Purhase, G., 996, Sensitivity Analysis of Engineering Designs Using Taguhi s Methoology, DETC96/DAC-455, Proeeings of DETC 96, Irvine, CA, Aug Lee, K. H., an Park, G. J., 2002, Robust Optimization in Disrete Design Spae for Constraine Problems, AIAA J., 40 4, pp Stoki, R., Kolanek, K., Jeno, S., an Kleiber, M., 200, Stuy on Disrete Optimization Tehniques in Reliability-Base Optimization of Truss Strutures, Comput. Strut., 79, pp Du, X., an Sujianto, A., 2003, Reliability-Base Design with the Mixture of Ranom an Interval Variables, DETC03/DAC-48709, Proeeings of DETC 03, Chiago, IL, Sept Siall, J. N., 982, Optimal Engineering Design Priniples an Appliations, Marel Dekker, New York. 27 Hirokawa, N., an Fujita, K., 2002, Mini-Max Type Formulation of Strit Robust Design Optimization Uner Correlative Variation, DETC02/DAC- 3404, Proeeings of DETC 02, Montreal, Canaa, Sept. 29 Ot Gunawan, S., 2004, Parameter Sensitivity Measures for Single Objetive, Multi-Objetive, an Feasibility Robust Design Optimization, Ph.D. issertation, University of Marylan, College Park, MD. 29 Tseng, C. H., Wang, L. W., an Ling, S. F., 995, Enhaning Branh-an- Boun Metho for Strutural Optimization, J. Strut. Eng., 2 5, pp Journal of Mehanial Design FEBRUARY 2007, Vol. 29 / 65