Design Optimization with Imprecise Random Variables

Transcription

1 The Insttute for Systems Research ISR Techncal Report Desgn Optmzaton wth Imprecse Random Varables RJeffrey W Herrmann ISR develops, apples and teaches advanced methodologes of desgn and analyss to solve complex, herarchcal, heterogeneous and dynamc problems of engneerng technology and systems for ndustry and government. ISR s a permanent nsttute of the Unversty of Maryland, wthn the A. James Clark School of Engneerng. It s a graduated Natonal Scence Foundaton Engneerng Research Center.

2 Desgn Optmzaton wth Imprecse Random Varables Jeffrey W. Herrmann Department of Mechancal Engneerng and Insttute for Systems Research Unversty of Maryland College Park, MD 074 ABSTRACT Desgn optmzaton s an mportant engneerng desgn actvty. Performng desgn optmzaton n the presence of uncertanty has been an actve area of research. The approaches used requre modelng the random varables usng precse probablty dstrbutons or representng uncertan quanttes as fuzzy sets. Ths work, however, consders problems n whch the random varables are descrbed wth mprecse probablty dstrbutons, whch are hghly relevant when there s lmted nformaton about the dstrbuton of a random varable. In partcular, ths paper formulates the mprecse probablty desgn optmzaton problem and presents an approach for solvng t. We present examples for llustratng the approach. 1. INTRODUCTION Desgn optmzaton s an mportant engneerng desgn actvty n automotve, aerospace, and other development processes. In general, desgn optmzaton determnes values for desgn varables such that an objectve functon s optmzed whle performance and other constrants are satsfed (Papalambros and Wlde, 000; Ravndran et al., 006; Arora, 004). The use of desgn optmzaton n engneerng desgn contnues to ncrease, drven by more powerful software packages and the formulaton of new desgn optmzaton problems motvated by the decson-based desgn (DBD) framework (Hazelrgg, 1998; Renaud and Gu, 006). Because many engneerng problems must be solved n the presence of uncertanty, developng approaches for solvng desgn optmzaton problems that have uncertan varables has been an actve area of research. The approaches used requre modelng the random varables usng precse probablty dstrbutons or representng uncertan quanttes as fuzzy sets. Haldar and Mahadevan (000) gve a 1

3 general ntroducton to relablty-based desgn optmzaton, and many dfferent soluton technques have been developed. For examples, see Lang et al. (007), Tu et al. (1999), Youn (007), and the references theren. Other approaches nclude evdence-based desgn optmzaton (Zhou and Mourelatos, 008b), possblty-based desgn optmzaton (Zhou and Mourelatos, 008a), and approaches that combne possbltes and probabltes (Nkolads, 007). Zhou and Mourelatos (008b) dscussed an evdence theory-based desgn optmzaton (EBDO) problem. They used a hybrd approach that frst solves a RBDO to get close to the optmal soluton and then generates response surfaces for the actve constrants and uses a dervatve-free optmzer to fnd a soluton. Dfferent models of uncertanty are more or less approprate gven The amount of nformaton avalable and the outlook of the decson-maker (desgn engneer) determnes the approprateness of dfferent models of uncertanty. No sngle model should be consdered unversally vald. In ths paper, we consder stuatons n whch there s nsuffcent nformaton about the random varables to model them wth precse probablty dstrbutons. Instead, mprecse probablty dstrbutons (descrbed n more detal below) are used to capture the lmted nformaton or knowledge. In the extreme case, the mprecse probablty dstrbuton may be a smple nterval. Ths paper presents an approach for solvng desgn optmzaton problems n whch the random varables are descrbed wth mprecse probablty dstrbutons because there exsts lmted nformaton about the uncertantes.. IMPRECISE PROBABILITIES In tradtonal probablty theory, the probablty of an event s defned by a sngle number between n the range [0, 1]. However, because ths may be napproprate n cases of ncomplete or conflctng nformaton, researchers have proposed theores of mprecse probabltes. For these stuatons, probabltes can be ntervals or sets, rather than precse numbers (Dempster, 1967; Walley, 1991; Wechselberger, 000). The theory of mprecse probabltes, formalzed by Walley (1991), uses the same fundamental noton of ratonalty as the work of de Fnett (1974, 1980). However, the theory allows a range of ndetermnacy prces at whch a decson-maker wll not enter a gamble as ether a buyer or a seller. These n turn correspond to ranges of probabltes. Imprecse probabltes have prevously been consdered n relablty analyss (Coolen, 004; Utkn, 004a, b) and engneerng desgn (Aughenbaugh and Pareds, 006; Lng et al., 006; Rekuc et al., 006).

4 Aughenbaugh and Herrmann (007, 008a, b) have compared technques usng mprecse probabltes to other statstcal approaches for makng relablty-based desgn decsons. The work descrbed n ths paper bulds upon these prevous results. In many engneerng applcatons, the relevant random varables (e.g., parameters or manufacturng errors) are contnuous varables. One common way to represent the mprecson n the probablty dstrbuton of such a random varable s a probablty box ( p-box ) that s a set of cumulatve probablty dstrbutons bounded by an upper dstrbuton F and a lower dstrbuton F. These bounds model the epstemc uncertanty about the probablty dstrbuton for the random varable. Of course, a tradtonal precse probablty dstrbuton s a specal case of a p-box, n whch the upper and lower bounds are equal. There are multple ways to construct a p-box for a random varable (Ferson et al., 003). In some cases, the type of dstrbuton s known (or assumed) but ts parameters are mprecse (such as an nterval for a mean). In other cases, the dstrbuton s constructed from sample data. Addtonally, one can create a p- box from a Dempster-Shafer structure, n whch ntervals (not ponts) wthn the range of the random varable are assgned probabltes. For more about p-boxes and the lnk between p-box representaton and Dempster-Shafer structures, see Ferson et al. (004). Functons of random varables that have mprecse probablty dstrbutons also have mprecse probablty dstrbutons. Methods for calculatng these convolutons are gven by Yager (1986), Wllamson and Downs (1990), and Berleant (1993, 1996). Wang (008) proposes a new nterval arthmetc that could be used as well. Therefore, p-boxes are a very general way to represent uncertanty. For computatonal purposes, n the approach below, we wll convert a p-box nto a canoncal Dempster-Shafer structure (Ferson et al. 003), whch wll necessarly be bounded. 3. DESIGN OPTIMIZATION WITH IMPRECISE PROBABILITIES In the mprecse probablty desgn optmzaton (IPDO) problem, there s a set of determnstc desgn varables for whch the desgner chooses values and a set of random varables, whch may be manufacturng errors, uncertan engneerng parameters, or other sources of uncertanty. Unlke other 3

5 work, ths formulaton does not nclude n the model random desgn varables. Such varables are typcally those n whch the desgner chooses the mean, but the actual value s random. In the IPDO formulaton presented here, each such quantty s modeled wth two quanttes: a determnstc desgn varable and a random parameter that represents the error of that varable. Ths does not lmt the scope of the model. For nstance, suppose we have a random desgn varable X that s a dmenson of a part. The mean of X, denoted μ X, s chosen by the desgner, but the dmenson s a normally dstrbuted random varable wth a standard devaton of σ. Examples of ths type of varable have been consdered n Zhou and Mourelatos (008a) and elsewhere. In ths formulaton, we replace the varable X wth X = d + Z, where the frst term, whch corresponds to the mean, s a determnstc desgn varable, and X X the second term s a random varable that s normally dstrbuted wth a mean of 0 and a standard devaton of σ. The general IPDO s formulated as follows: { ( dz) } ( dz) mn V f, d s.t. P g, 0 p = 1,, n L d d d U (1) In ths formulaton, k d R s the vector of determnstc desgn varables, and r Z R s the vector of random varables that have mprecse probablty dstrbutons. The probablstc constrants are functons of the determnstc desgn varables and the random varables. We want g ( ) dz, 0 (whch s the safe regon ) but wll be satsfed f the upper bound on the falure probablty s less than the target choose the upper probablty n order to be conservatve. p. We The functon f s the system performance, whch may be random, n whch case the functon V s a moment of that random performance, such as the upper lmt for the mean; thus V s a determnstc functon of d. In many cases, the objectve s specfed as a functon of only the determnstc desgn varables, n whch case we get the followng formulaton: mn f { ( dz) } ( d ) s.t. P g, 0 p = 1,, n L d d d d U () 4

6 Frst, ths formulaton has the usual dffculty of computng the falure probablty for each constrant. Analytcally evaluatng the falure probablty s possble only n specal cases. An addtonal complcaton s the mprecson of the random varables, whch makes applyng standard RBDO technques dffcult. Now, Z has an mprecse jont probablty dstrbuton, whch can be consdered as a set H of precse jont probablty dstrbutons. For any precse jont probablty dstrbuton j { (, ) 0} P g dz be the probablty of volatng constrant when Z has that precse jont probablty dstrbuton. Then, we could reformulate the IPDO as the followng RBDO: Fj H, let mn f { ( d Z) } ( d ) s.t. P g, 0 p = 1,, n, F H j j L d d d d Unfortunately, because of the large number of constrants, ths reformulaton s not helpful unless the set H s lmted to a reasonable number of extreme dstrbutons that can be used as surrogates for the entre set. Research on ths topc s ongong and may provde a way to ncrease the computatonal effcency of IPDO n the future. U (3) Due to these dffcultes, we wll pursue a numercal approach. To do ths, we wll frst partton the constrants g ( ) d, Z 0 nto two sets. Set S 1 ncludes any constrant that can be rearranged so that (, ) = ( ( ) ( )) g d Z a h Z b d, where h ( ) b ( ) a s a postve scalar. Note g ( ) d, Z 0 f and only f Z d. The constrants that cannot be rearranged n ths way are placed n set S. For each constrant n S 1, we wll perform the convoluton needed to get the mprecse dstrbuton of h ( ) Z by combnng the Dempster-Shafer structures for the relevant random varables. Because the upper cumulatve probablty dstrbuton wll be a dscontnuous functon, we wll approxmate t wth ( ) { ( ) } F x P h Z x. Therefore, we can replace each of the constrants n 1 S by F ( b ( )) d p. 5

7 4. SOLUTION APPROACH To solve the IPDO, we wll use a sequental approach smlar to that of Du and Chen (004) and Zhou and Mourelatos (008a). A key part of the approach s to solve the followng determnstc optmzaton problem P gven values for the random varables n each constrant n S. Let constrant S n teraton k. k ( ) Z be specfc values for the random varables n s.t. mn f ( ( d )) (k) ( d Z ) ( d ) g, 0 S F b p S L d d d d (k) In the space of the desgn varables, the g ( ) U 1 dz, 0 constrants move the boundares of the safe (4) regon (by makng t smaller) n order to reduce the probablty of falure. However, t s stll necessary to determne the probablty of falure and compare t to the target. If t s too large, then we have to move that constrant some more. Gven these prelmnares, the complete approach follows: 1. Let k = 0. For S, let each component of value. k ( ) Z equal a value wthn the range of ts expected. Solve P to get the soluton 3. For S ( k + 1) d. { } ( { 1), 0} (, evaluate ( k + P g 1) d, Z ) 0. If ( k + P g ) pont s feasble; stop. Otherwse, for all constrants S k ( + 1) k ( ) Z = Z. For the others, fnd d Z p for all S, then the desgn where ( k + P g ) ( { 1), 0} ( 1) k+ Z by solvng the followng problem: d Z p, set 6

8 ( k+ 1) ( k+ 1) mn g (, ) ( k 1) d Z + Z ( k+ 1) ( k+ 1) ( k+ 1) ( d Z) ( d Z ) { } s.t. P g, g, = p Ths yelds a very bad (but not worst-case) value of those random varables used n that constrant. (A technque for solvng ths problem s descrbed below.) (5) 4. k = k + 1. Repeat steps and 3 untl a feasble desgn pont s found. At ths pont we have no proof that the algorthm wll converge, and the approach may fal on problems wth rregular objectve functons and constrants. Further analyss and expermentaton s needed to study ths aspect of the method. ( { 1), 0} We use the followng relablty analyss technque to determne f ( k + P g ) k+ ( 1) Z d Z p and to fnd. Ths relablty analyss technque corresponds roughly to solvng the nverse most probable pont problem dscussed n Du and Chen (004) or fndng the shftng vector of Zhou and Mourelatos ( k + 1) (008a). Gven d, set 0 ( k + p f = Wthout loss of generalty, we assume that g ( 1) d, ) Z s a functon of m random varables Z1,, Zm. The Dempster-Shafer structure of Z s represented by n equally lkely ntervals. Let N = nn 1 nm. Let N * = pn be the number of values and ntervals to save. Consder each of the N combnatons of ntervals for the random varables. For each combnaton, let each random varable range over ts nterval and fnd mn ( k+ g, the mnmum of g ( 1) d, ) Z for that combnaton. If mn g 0, add 1 N (the probablty of that combnaton) to p f. As the N combnatons of ntervals are checked, keep the those mnma. The fnal value of, m the values of Z1, Z that yeld the largest of the * N smallest mnma found along wth the values of Z1, Z that yeld ( { 1), 0} p f s used to estmate ( k + P g ) d Z. We set * N smallest mnma found., m k+ ( 1) Z equal to 5. EXAMPLES Ths secton presents three examples to llustrate the IPDO soluton method. The frst example has two desgn varables and three random varables: 7

9 The bounds d L = ( 0.01, 0.01) and U = ( 10,10) second one s n set S. mn d d 1 4 s.t. P z d1 {( ) ( ) } 1 1 P d + z d + z L + d d d d U d. Note that the frst constrant s n set S 1, whereas the (6) All three random varables have mprecse probablty dstrbutons. The random errors z 1 and z have the same dstrbuton, each characterzed by the ntervals ( k ) ( k ) /99, / 99, for k = 1,,100 (each nterval has a probablty of 0.01). Therefore, they can range from -1 to 1. The dstrbuton of random parameter z 3 s characterzed by the ntervals ( k ) ( k ) / 99, / 99 random parameter ranges from 1 to., for k = 1,,100 (each nterval has a probablty of 0.01). Ths For z 3, we wll approxmate ts upper cumulatve probablty as follows: 0 f x < 1 P{ z3 x} = ( x-1 ) f 1 x f x > 1.5 (7) The IPDO soluton approach begns wth z 1 and z both set to zero; that s, (0) Z = (0,0) : *(1) Ths yelds d = ( ,1.751) mn d d s.t. P z d1 d d L + d 0 d d d U, but the upper probablty of volatng the second constrant s too { } hgh. Our relablty analyss technque estmates that P ( d *(1) ) ( *(1) 1 z ) 1 d z = (8) 8

10 Because there are ( (1) (1) 1, ) ( , ) 10,000 = 100 combnatons of ntervals, we keep the worst 00 nterval lower bounds. z z = gves the best of these worst. (The frst superscrpt refers to the second constrant, the second to the teraton number.) Now we solve by addng the shftng vector to the second constrant: *() Ths yelds d = ( ,.6696) { ( ) ( ) } *() *() 1 1 mn d d 1 4 s.t. P z d1 (9) ( d ) ( d ) L + d d d d U. Our relablty analyss technque estmates that P 0 d + z d + z 0 = Thus, the upper probablty of volatng the second constrant s lower but stll too hgh. We also determne that ( (), () ) ( 0.777, ) best of the worst for ths desgn pont. z z = gves the 1 Now we solve wth the new shftng vector: *(3) Ths yelds d = ( 4.197,.6645) mn d d 1 4 s.t. P z d1 ( d ) ( d ) L + d d d d U. The frst constrant s not actve, but the upper probablty of (10) volatng the second constrant s now acceptable. Our relablty analyss technque estmates that { ( ) ( ) } *(3) *(3) 1 1 P 0 d + z d + z 0 = So the soluton s feasble, and the algorthm stops. 9

11 The second example s the mathematcal example from Zhou and Mourelatos (008a). In our verson, the problem has two desgn varables = ( d, d ) ( z, z ) 1 d and two random varables, the error for each one: 1 Z =. The bounds are 0 10 for both desgn varables. The objectve s to mnmze the sum d of the desgn varables. In the terms of our general IPDO, we have the followng relatonshps: f ( d ) = d1+ d ( d, Z) = ( + ) ( + ) 0 ( d Z) ( ) ( ) ( d, Z) = 75 ( + ) 8( + ) g d z d z g, = 4 d + z + d + z 5 + d + z d z 1 10 g d z d z Both random varables have the same mprecse probablty dstrbuton, whch s characterzed by the ntervals ( k ) ( k ) / 99, / ). Therefore, they can range from -1 to 1., for k = 1,,100 (each nterval has a probablty of (11) We cannot separate any of the constrants, so 1 S = and S = { } 1,,3. The two random varables have the same mprecse probablty dstrbuton, whch s approxmately an mprecse unform dstrbuton wth a lower bound n the range [-1, 0.5] and the upper bound n the range [0.5, 1]. Fgure 1 shows the actual p-box z1 (or z) Fgure 1. The p-box for z 1 (and z ). 10

12 Frst, we wll solve the determnstc optmzaton problem wth both random varables replaced by 0. We get d = ( ,.066) functon value s , the same optmal soluton as Zhou and Mourelatos (008a). The objectve (0) Next, we let = ( 0,0) Z for all S and solve problem P. Ths yelds the soluton { (1), 0} (1) d = ( ,.066). P g ( ) zero for the thrd constrant. d Z s greater than 0.0 for the frst two constrants but equals When evaluatng ths probablty, we have to compare 10,000 combnatons, n whch each combnaton has an nterval from the p-box for z 1 and an nterval from the p-box for z. For each combnaton, we must fnd the mnmal value of g over those values of z 1 and z. Based on the mathematcal analyss of the constrants, t s possble to develop smple rules to dentfy the values of z 1 and z that gve the mnmum for that combnaton. Let mn max z1, z1 and mn max z, z be the ntervals that form the combnaton. For the frst constrant, the mnmum s found at z mn z =, and z 1 s ether an endpont of the nterval or d1. For the second mn mn constrant, the mnmum s found at one of the followng fve ponts: ( 1, ) z z, mn mn max mn max max max max ( 6.4 d1 0.6d 0.6 z, z ), ( z1, z ), ( z1, d1 d 0.6z1 ), or ( 1, ) z z. For thrd constrant, the mnmum s found at z max z =, and z 1 s one of the endponts of ts nterval. From ths algorthm we set ( z 1(1), z 1(1) ) = ( 0.711, 1) and ( (1), (1) ) ( , 0.818) 1 z z =. We wll use 1 these two vectors n the frst two constrants as we try to fnd a feasble soluton n the next teraton. For the thrd constrant, whch was already feasble, we let ( 3(1), 3(1) ) ( 0,0) z z =. 1 () The second teraton of the problem yelds the soluton d = ( ,3.455) { (1), 0} objectve functon equals P g ( ). At ths pont, the d Z s agan greater than 0.0 for the frst two 11

13 constrants but equals zero for the thrd constrant. We set ( 1(), 1() ) ( 1, ) z z = and 1 ( z () () 1, z ) = ( , 1). For the thrd constrant, ( 3() 3() 1, ) ( 0,0) z z =. (3) The thrd teraton of the problem yelds the soluton d = ( ,3.540) { (1), 0} functon equals P g ( ) for the thrd constrant, so the soluton s feasble.. At ths pont, the objectve d Z s less than 0.0 for the frst two constrants and equals zero The thrd example that we consder s the optmzaton of a thn-walled pressure vessel. Our formulaton s based on the RBDO formulaton of Zhou and Mourelatos (008b). The problem was orgnally ntroduced by Lews and Mstree (1997). The problem has three desgn varables: the radus R, the mdsecton length L, and the wall thckness t. The objectve s to maxmze the volume of the pressure vessel. Fve constrants ensure that the desgn s strong enough to resst the nternal pressure (wth a safety factor of ) and meets geometrc requrements. In our formulaton there are three random varables: the manufacturng error of the radus, the nternal pressure P, and the materal yeldng strength Y. In the terms of our general IPDO, we have d= ( R, Lt, ) and = ( z P Y ) relatonshps: Z 1,, and the followng ( d ) f = πr + πr L 1 ( d, Z) = ( + + ) g ty P R z t 1 1 ( ) ( ) ( ) ( d Z) ( ) ( d, Z) = 60 ( + ( + ) + ) ( d, Z) = 1 ( + + ) ( d, Z) = + 5 ( ), = g R z t t Y P R z R z t t g L R z t 3 1 g4 R z1 t g R z t 5 1 We set p = 0.0 for = 1,,5. The bounds on the desgn varables are the followng ranges: 5 R 4, 10 L 48, and 0.5 t rearranged as follows:. The last three constrants, whch form set S = { } 1 3, 4,5 (1), can be 1

14 1 ( ) ( ( )) ( dz) ( d) g, = h ( z ) b = z L+ R+ t ( dz, ) = ( ) ( d) = ( + 1) ( dz, ) = ( ) ( d) = ( 5 ) g h z b z R t g h z b z t R Therefore, S = { 1, }. Each of the three random varables has an mprecse probablty dstrbuton. The mprecse probablty dstrbuton of the nternal pressure P s approxmately an mprecse normal dstrbuton. The mprecse mean has a range of [975, 105]. The standard devaton s precsely 50. The mprecse probablty dstrbuton of the materal yeldng strength Y s also approxmately an mprecse normal dstrbuton. The mprecse mean has a range of [53500, 66500]. The standard devaton s precsely (13) The actual p-boxes used for these two random varables are constructed as follows: for k = 1,,100, let fk ( k ) = 1 / 00, whch therefore ranges from to The k-th nterval n the p-box for the nternal pressure P s Φ ( f ), Φ ( f ) k k, and the k-th nterval n the p-box for the materal yeldng strength Y s Φ ( f ), Φ ( f ) k k. The mprecse probablty dstrbuton of z 1, the manufacturng error of the radus, s based on data gven by Zhou and Mourelatos (008b), who assume that we have 100 sample ponts for the error, but the data are gven only n bns as follows: 3 ponts are n the range [-4.5, -3], 45 ponts are n the range [-3, 0], 49 ponts are n the range [0, 3], ponts are n the range [3, 4.5], and 1 pont s n the range [4.5, 6]. Fgure shows the correspondng p-box for z 1 and the curve we use for approxmatng the upper bound of ths L p-box. An approxmaton s created for each part of the p-box. For x z, z U where the lower left corner of the upper bound s ( L L U U z, F ) and the upper rght corner of the upper bound s (, ) z F, the L x z = 1 z z U U L approxmaton F ( x) F ( F F ) U L. Fgure 3 shows the correspondng p-box for z 1 and the curve we use for approxmatng the upper bound of ths p-box. Because ( ) { ( Z ) } F x P h x, the approxmaton reduces the feasble regon. If a soluton s feasble wth respect to the approxmaton, then t s feasble wth respect to the orgnal p-box. 13

15 Pbox Approxmaton Fgure. The p-box for z 1 and the approxmaton for ts upper bound. z1 Pbox Approxmaton z1 Fgure 3. The p-box for z 1 and the approxmaton for ts upper bound. Frst, we wll solve the determnstc optmzaton problem wth all three random varables replaced by constants: Z = ( z P Y ) = ( ). = ( RLt,, ) = ( 11.75,36,0.5) 1,, 0,1000,60000 d s the optmal soluton that we found. The pressure vessel volume equals,410. Note that the probablty of falure for constrants 14

16 4 and 5 equals the probablty that z 1 s less than equal to zero, whch s mprecse but can be qute large, so t s not a feasble soluton to the IPDO problem. 1(0) (0) Next, we let = ( 0,1000, 60000) d (1) = Z = Z and solve problem P. Ths yelds the soluton ( , ,0.3186). (The optmzaton requred 348 functon evaluatons.) The pressure vessel volume equals 6,57. The relablty analyss technque shows that, for all S, { ( ) } P g d (1), Z 0 = 0, so the soluton s feasble, and the algorthm stops. 6. COMPARISON TO RBDO The IPDO addresses stuatons n whch probablty dstrbutons are not precse. An alternatve approach s to use a tradtonal RBDO approach whle varyng the probablty dstrbutons of the random varables. The basc dea s to loop over dfferent combnatons of the dstrbutons for the random varables. For each combnaton, we solve a tradtonal RBDO problem to get a soluton. Ths procedure wll yeld a set of solutons and gves the desgner some dea of where good solutons le. But t s not clear how a desgner should select a soluton from ths set. Another alternatve s to remove the mprecson. For nstance, one can replace each mprecse probablty dstrbuton by the maxmum entropy probablty dstrbuton that fts wthn the p-box. Ths yelds an RBDO problem. For the frst example n Secton 5, we can model z 1, z, and z 3 wth unform dstrbutons. The range for z 1 and z s [-1, 1], and the range for z 3 s [1, ]. Solvng the RBDO problem yelds the soluton d = ( 4.7,.513). The objectve functon value s , whch s better than that of the more conservatve IPDO soluton, but the probablty of falure of ths new soluton s greater than the desred target for some of the probablty dstrbutons n the p-boxes of the random varables. 7. SUMMARY Ths paper ntroduced the mprecse probablty desgn optmzaton (IPDO) problem, n whch there s a set of determnstc desgn varables for whch the desgner chooses values and a set of mprecse random varables, whch may be manufacturng errors, uncertan engneerng parameters, or other sources of 15

17 uncertanty. Ths paper presented a sequental approach for solvng ths problem. To avod unnecessary calculatons, the approach parttons the constrants nto two sets. By explotng ther specal structure, the cumulatve probablty dstrbutons for constrants n the frst set are calculated only once and then replaced wth an approxmaton. After ths, the approach solves a seres of determnstc optmzaton problems and shfts selected constrants n each teraton n order to reduce the probablty of falure. We have used examples to llustrate the usefulness of the approach. The results show that the proposed IPDO approach fnds hgh-qualty feasble solutons, though the computatonal effort s ncreased because of the computatonal effort of the relablty analyss technque and the teratons needed to converge to a soluton. Although ths work was motvated by problems n whch only mprecse probablty dstrbutons are avalable, the approach s use of Dempster-Shafer structures makes t compatble wth other approaches wthn the doman of evdence theory as well (Shafer, 1976). Future work wll focus on mprovng the computatonal effcency and stablty of the approach by consderng adaptve loop-methods smlar to those proposed by Youn (007). ACKNOWLEDGEMENTS Ths paper reuses background materal and references from prevous papers wrtten wth co-author Jason Aughenbaugh, whose assstance the author greatly apprecates. REFERENCES Arora, Jasbr S., Introducton to Optmum Desgn, nd edton, Elsever Academc Press, Amsterdam, 004. Aughenbaugh, Jason M., and Jeffrey W. Herrmann, Updatng uncertanty assessments: a comparson of statstcal approaches, Proceedngs of IDETC/CIE 007, ASME Internatonal Desgn Engneerng Techncal Conferences & Computers and Informaton n Engneerng Conference, September 4-7, 007, Las Vegas, Nevada, DETC Aughenbaugh, Jason M., and Jeffrey W. Herrmann, A Comparson of Informaton Management usng Imprecse Probabltes and Precse Bayesan Updatng of Relablty Estmates, Proceedngs of 16

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