Joural of Mechacal Scece ad Techoloy 7 () (0) 78~789 www.sprerl.com/coe/78-9 DOI 0.007/s0-0-09-8 A effce approach o relaly-ased des opmzao wh he ehaced sequeal opmzao ad relaly assessme framewor Ho-Zho Hua *, udo Zha, De-Bao Me, Zhola Wa ad Yu Lu School of Mechacal, Elecroc, ad Idusral Eeer, versy of Elecroc Scece ad Techoloy of Cha No. 00, yua Aveue, Wes H-Tech Zoe, Chedu, Schua, 7, Cha (Mauscrp Receved Auus 7, 0; Revsed Jauary, 0; Acceped Jauary, 0) ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Asrac Relaly ased des opmzao (RBDO) has ee wdely mplemeed eeer pracces for hh safey ad relaly. I s a mpora challee o mprove compuaoal effcecy. Sequeal opmzao ad relaly assessme (SORA) has made rea effors o mprove compuaoal effcecy y decoupl a RBDO prolem o sequeal deermsc opmzao ad relaly aalyss as a sle-loop mehod. I hs paper, order o furher mprove compuaoal effcecy ad eed he applcao of he curre SORA mehod, a ehaced SORA (ESORA) s proposed y cosder cosa ad vary varaces of radom des varales whle eep he sequeal framewor. Some mahemacal eamples ad a eeer case are ve o llusrae he proposed mehod ad valdae he effcecy. Keywords: Relaly ased des opmzao; Sequeal opmzao ad relaly assessme; Compuaoal effcecy; Relaly aalyss; Sleloop mehod ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------. Iroduco Relaly ased des opmzao (RBDO) s a approach o achev relale decso whe cosder he radomess of des varales ad parameers whch maye come from maufacure, evrome ad so o [-]. The ypcal mahemacal formulao of RBDO s as follows: m f ( d,, ) d, s.. r( G ( d,, ) 0) Φ ( ), = h () d d d, L L d s a vecor of deermsc des varales; dcaes a vecor of mea values of radom varales = {,,, } whle represes a vecor of mea values of radom parameers = {,,, m }. f ( ) s he oecve fuco. G ( ), = h are performace fucos, ad r( G ( ) 0) s he proaly of success. Φ ( ) s he are relaly ad deoes he relaly de. Φ ( ) s he cumulave dsruo fuco (CDF) of he sadard ormal radom varale. The superscrps L ad deoe he lower ad upper oudares, respecvely. I * Correspod auhor. Tel.: +8 8 8 08, Fa.: +8 8 8 07 E-mal address: hzhua@uesc.edu.c Recommeded y Assocae Edor Tae Hee Lee KSME & Sprer 0 hs formulao, d, are des varales. Solv he RBDO prolem drecly wll volve doule loops: he ouer loop ad he er loop. The ouer loop s o mmze he oecve fuco whle relaly aalyss s performed he er loop. To effcely deal wh RBDO prolem, may mehods are developed o mprove he effcecy of relaly aalyss, such as relaly de approach (RIA), performace measure approach (MA) [7-9] ad ehac he effcecy of alorhm fd he mos proale po (M) [0-]. Alhouh he MA ca effcely decrease compuaoal epese relaly aalyss, he compuaoal epese for he lare-scale RBDO prolem wh doule loop approach s sll prohed. The wo ew classes for deal wh RBDO are proposed [-8]. I he frs class, he RBDO prolem s decoupled o sequeal deermsc opmzao ad relaly aalyss [-8]. Whe cosruc he deermsc cosras he deermsc opmzao, he sraey of cosra shf s adoped Ref. []; he sequeal opmzao ad relaly assessme (SORA) proposed Ref. [] ad ehaced oe Ref. [5] adops he sraey ha ulz s M of prevous cycle o oa he shf vecor of each radom des varale o each proalsc cosra. I he secod class, he RBDO prolem s covered o a deermsc opmzao y elma he relaly aalyss (perform he er loop) hrouh he KKT codo [7, 8].
78 H.-Z. Hua e al. / Joural of Mechacal Scece ad Techoloy 7 () (0) 78~789 The oecve of hs paper s o furher mprove he compuaoal effcecy of he curre SORA y cosder oh cases of cosa ad vary varaces of radom des varales whle eep he sle-loop framewor. The effcecy of he proposed mehod s compared wh he es approaches Refs. [, ] wh several llusrave eamples. Ths paper s orazed as follows. I seco, he SORA mehod s refly revewed as well as he MA mehod. The ehaced SORA (ESORA) s proposed seco. Several eamples are used o llusrae he effcecy of he proposed mehod seco, followed y he coclusos seco 5.. Revew of SORA. Approach of MA Dffere from he deermsc opmzao, he feasly of proalsc cosras eeds o e checed whe mmz he oecve fuco RBDO. The h proalsc cosra s formulaed as follows: r( G ( d,, ) 0) = F (0) Φ( ) () G he cumulave dsruo fuco FG ( ) s descred as F (0) f (, ) d d () G = p p G ( ) 0, f, (, p ) s he o roaly Desy Fuco (DF) of, [9]. The frs order relaly mehod (FORM) s a M-ased mehod ad has ee wdely used for relaly aalyss of he RBDO. Based o he FORM, relaly de approach (RIA) ad performace measure approach (MA) are proposed [7, 0]. I he oh approaches, he radom varales ad parameers, -space should e rasformed o, he sadard ormal space us he Rosela rasformao. I has ee poed ou ha he MA s more sale ha he RIA [7]. Based o he MA, he proalsc cosra Eq. () ca e epressed as F G ( Φ( )) 0. So he RBDO formulao s rewre as [9]: m f ( d,, ) d,.. p = ( ( )) 0, G Φ = s G F h () G p d d d, L L G p ca e oaed y = ma G(, ), s.., =. The soluos of Eq. (5) are he M (, ) (5) ad he Deermsc opmzao performace measure Gp= G(, ), he -space ca e oaed us he verse Rosela rasformao. Whe he radom varales follow he ormal dsruo, he M -space ca e oaed y = + = +.. The M ( ) Afer rasformao, Gp G(, = ) = G(, ). Eq. () ca e rewre as: m f ( d,, ) d, s G h (7),( ),( ).. ( d,, ) 0, = L L d d d,.. Mehod of SORA Cycle Cycle RA L RA h The SORA s oe of he mos effce sle-loop mehods. The SORA decouples a RBDO prolem o sequeal deermsc opmzao ad relaly aalyss as show F. []. A each cycle, he deermsc opmzao s frs performed o oa he opmum of each des varale, ad he he relaly aalyss of each proalsc cosra s carred ou a he opmal po. If all he proaly cosras are o sasfed ad he value of oecve fuco s o sale, he Ms formao wll e used he e cycle. The flowchar of SORA s ve F.. Whe he h proalsc cosra s volaed Cycle (- ), also meas ha he performace measure a he M,( ),( ) does o sasfy G ( d,, ) 0 wh MA. The SORA uses a sraey of shf vecor o mae sure he M of Cycle fall o he deermsc feasle reo,( ),( ) G ( d,, ) 0 []. The shf vecor of he h proalsc cosra for Cycle s: s = (8) ( ),,( ),( ) Deermsc opmzao F.. Illusrao of SORA (RA: relaly aalyss). RA RA h s he vecor of mea values of radom varales,,( ),( ) ad s he correspod M of he h proalsc cosra oaed cycle (-). I should e oed ha each proalsc cosra has s ow shf vecor ecause each oe has s ow M. Wh he SORA mehod, he equvale deermsc opmzao of he oral RBDO prolem cycle s formulaed as: L ()
H.-Z. Hua e al. / Joural of Mechacal Scece ad Techoloy 7 () (0) 78~789 78 Wh he MA mehod, he M of a proalsc cosra could e oaed: G = ma G(, ) p, s.., =. () Follow he same way used Ref. [7], ased o Eq. () he relaoshp ewee he M (, ) ad he rade G(, ) he -space s:, G (, ) =, = G (, ), G (, ) =, = m G (, ), () ad he relaoshp of rade he -space ad -space s: F.. Flowchar of SORA. m f ( d,, ) d, s G h (9) ( ),,( ),( ).. ( d, s, ) 0, = L L d d d,. From Eqs. (7)-(9), he M of Cycle s appromaed he deermsc opmzao he SORA as: s = ( ),( ) ( ),,( ),( ) = +, = ;,( ),( ),( ),( ),( ) = +, = m.,( ),( ). Ehaced SORA for RBDO prolems (0) I hs paper, s assumed ha each performace fuco s eplc so ha s epresso of rade ca e oaed. Based o Eq. (), he follow formulao holds cycle : = +, =,( ),,( ), = +, = m.,( ),,( ), () From Eqs. (0) ad (), he M s appromaed he - space he h cycle of he SORA as:,( ),,( ),( ),( ),,( ),( ). () G (, ) G (, ) G ( d,, ), = = G ( d,, ), = =,, m. From Eqs. (), () ad (5), a he M (, ) -space, he follow formulao holds: = +, = = +, = m G( d,, ) =, = [,,, ], G( d,, ) =, = [,,, ] m, = [, ]. (5) he () Eq. () holds for each cycle, ad he M Cycle ca e oaed y = +, =, = +, = m, (7)
78 H.-Z. Hua e al. / Joural of Mechacal Scece ad Techoloy 7 () (0) 78~789 G( d,, ) =, = [,,, ],,, G( d,, ) =, = [,,, ] (8) m,,, = [, ]. Whe cosruc he deermsc cosras Cycle, he rade a he acual M ca o e oaed ecause he relaly aalyss s o performed. The rade a he acual M s appromaed he follow way. For he radom des varales wh cosa varaces, from Eq. (),,( ) ( ) whch s equvale o + (9), ( ),( ) ad he M of radom parameers s:,,( ). (0) By susu Eqs. (9) ad (0) o Eq. (8), he appromao of he rade a he M s oaed. A he frs cycle ( = ), he rade a he acual M s appromaed as he rade a he mea values of radom varales ad parameers. For he radom des varales wh vary varaces, wh Eqs. ()-(8) he follow formulao ca e oaed: F.. Flowchar of he ESORA.,,( ) ( ) ( ) whch s equvale o = +, =, = +, = m, G( d,, ) =, = [,,, ],,, m,,, () G( d,, ) =, = [,,, ] () = [, ]. For he radom des varales wh vary varaces (e.. = r, r s he cosa coeffce of varao), from Eq. () ( ). ( ),,( ) ( ) + () By corpora Eqs. (0) ad () o Eq. (), he rade a he M s oaed. A he frs cycle, he rade a he acual M s appromaed as he rade a he mea values of radom varales ad parameers. The dscusso aovemeoed s ased o he ormal radom varales ad parameers. For o-ormal radom varales ad parameers, he Racwz-Fessler wo-parameer equvale ormal mehod ca e used o oa he mea value ad varace of equvale ormal dsruo a a po of eres [, 7]. Mehods of choos parameers for some o-ormal radom dsruos o acheve he lear relaoshp ewee he mea value ad sadard devao are proposed Ref. []. The flowchar of he ESORA proposed hs paper s provded F.. Whe he performace fucos are all lear
H.-Z. Hua e al. / Joural of Mechacal Scece ad Techoloy 7 () (0) 78~789 785 fucos, ecause he rade of each performace fuco s cosa, he oral RBDO prolem s compleely rasformed o a deermsc opmzao prolem wh Eqs. (7) ad (). I oher words, he opmum of hs deermsc opmzao prolem s he opmal soluo of he oral RBDO prolem.. Numercal eamples I hs seco, wo mahemacal prolems from Ref. [] ad a speed reducer des eample from Ref. [0] are used o llusrae he effcecy of he proposed mehod. The opmal resuls of he proposed mehod are compared wh hose of he oral SORA ad he approaches Ref. []. I hs paper, all opmzao prolems are solved y meas of he fmco fuco he sofware Mala.. rolem wh lear cosras I hs prolem, he oecve fuco s olear ad he performace fucos are lear. All pu varales are ormal dsrued ( ) wh he dsruos as N (, ), =. The are relaly s 0.9985 =Φ () for each cosra []. The mahemacal formulao of hs RBDO s: Tale. Resuls ad comparso for eample wh vary varaces. r= 0.0 r= 0.5 Approaches f Cycles NFE Or. SORA a 85 Approach a [.0000, 8.0000, 9.0000, Approach a -.7 9 8.0000,.0000, Approach a.] 9 Doule loop a [.0000, 8.0000, ESORA.0000, 8.0000,.0000,.] N/A 80 -.7 N/A 7 Or. SORA a 88 Approach a [.0000,.79, Approach a.0000, 8.0000, -0.0 9 Approach a.7, 0.0] Doule loop a ESORA [.0000,.88,.0000, 8.0000,.7, 0.0] a Resuls from Ref. [], Resuls from our compuao. N/A 9-0.0 N/A 9 Tale. Resuls ad comparso for eample wh cosa varaces. m f ( ) = 5 5 5 s.. ( ( ) 0) R, = ( ) = ( + 5) ( ) = ( + 0) ( ) = ( + 8) ( ) = ( 7 + ) 0, 8, 8 8,, 0.. () = 0.0 = 0.5 Cycles NFE Cycles NFE Or. SORA a Approach a 5 5 Approach a Approach a Doule loop a N/A 59 N/A 79 ESORA N/A N/A 7 a Resuls from Ref. [], Resuls from our compuao. Opmums of he oral SORA (Or. SORA), approaches Ref. [], ad he ESORA wh vary varaces are ve Tale. The umer of fuco evaluao (NFE) s also lsed. I oh cases of r= 0.0 ad r= 0.5, he proposed ESORA effcely solves he RBDO prolem wh he sar po as [5 5 5 5 ], ad he opmal soluos of ESORA s almos he same as hose of Or. SORA ad approaches Ref. []. Alhouh he sar pos ad opmzao mehod adoped Ref. [] are uow, from Tale he NFE he ESORA s ovously less ha hose of Or. SORA ad approaches Ref. []. The reaso s ha ESORA he oral RBDO prolem s compleely rasformed o a deermsc opmzao ecause performace fucos are all lear. The resuls of wo cases of cosa varaces are show Tale. Because each performace fuco s lear, he NFE he ESORA s ovously less ha hose eeded Or. SORA ad approaches Ref. [].. rolem wh olear cosras I hs prolem, here are wo ormal radom des varales ad wh he mea values of, ad he sadard devaos of,. Three performace fucos are all olear ad he are relaly s 0.9985 =Φ () []. The mahemacal formulao of hs RBDO prolem s: m f ( ) = + s.. ( ( ) 0) R ( ) = ( + ) 0 ( + 5) ( ) ( ) = + 0 0 80 ( ) = ( ) + 8 + 5 0.0 0, =,. (5)
78 H.-Z. Hua e al. / Joural of Mechacal Scece ad Techoloy 7 () (0) 78~789 Tale. Opmal resuls of eample wh cosa varaces. Des = 0.0 = 0.0 = 0.0 varales ESORA ESORA ESORA.0598.5999.8 0.000 0.000.7788 Oecve.098.099.9070 Cosras -0.5 -.50 -.08-0.87-0.0-5.80 0 - -0.905-5.8-0.859 Cycles NFE 5 Tale. Opmal resuls of eample wh vary varaces. Des r= 0.0 r= 0.0 r= 0.0 varales ESORA ESORA ESORA.0.8. 0.000 0.000.89 Oecve.07.8.8 Cosras -0.75 -.78 -.8-0.8-0.8 -.00 0-8 -.59-8.85-0.09 Cycles NFE 9 9 50 Tale 5. Comparsos of dffere approaches wh cosa varaces for eample. Or. SORA a Approach a Approach a Approach a = 0.0 = 0.0 = 0.0 Cycles NFE Cycles NFE Cycles NFE 87 7 95 Doule loop a N/A 9 N/A N/A 00 ESORA 5 Tale. Comparso of dffere approaches wh vary varaces for eample. r= 0.0 r= 0.0 r= 0.0 Cycles NFE Cycles NFE Cycles NFE Or. SORA a 89 8 795 Approach a 8 90 9 Approach a 8 90 8 Approach a 8 90 5 9 Doule loop a N/A 9 N/A N/A 79 ESORA 9 9 50 a Resuls from Ref. [], Resuls from our compuao. Bear roup Shaf Bear roup Shaf a Resuls from Ref. [], Resuls from our compuao. Wh he umercal olerace of 0.0%, he opmal soluos of ESORA wh he cases of cosa ad vary varaces are provded Tales ad respecvely. From Tales ad, whe he cosa varaces ad he cosa coeffce of varao crease, he NFE eds o crease ad also he oecve value. The performace measure a M of each proalsc cosra s less ha zero, whch dcaes ha each des po s feasle. Tales 5 ad provde he cycles ad NFE eeded he Or. SORA, approaches Ref. [], ad he ESORA. The ESORA effcely solves he RBDO prolem wo cycles for dffere cases of cosa varaces. The ESORA s more effce ha he doule loop mehod sce he NFE of he ESORA s less ha ha of doule loop mehod. The cycles eeded he ESORA are all o more ha hose of Or. SORA, ad approaches Ref. [].. Speed reducer des eample Ths eample s derved from Ref [0], show F.. I hs paper, s modfed as a RBDO prolem clud s radom des varales ad oe deermsc des varale. The properes of des varales of he speed reducer are ve F. 7. All radom varales are ormally dsrued. The sysem oecve fuco f s he speed reducer volume o e mmzed. The RBDO formulao s: ( ) ( ) ( ) 7 7 ( ) m f= 0.785. +.9.09.508 + + 7.77 + + 0.785 + 5 7 s.. ( 0 ) R, = 5 = 7 /( ) 0 : pper oud o he ed sress of he ear ooh. = 97.5/( ) 0 : pper oud o he coac sress of he ear ooh. =.9 /( ) 0 : pper oud o he rasverse defleco of he shaf. =.9 /( ) 0 : pper oud o he ras- 5 7 7 Gear Gear F.. Des varales of he speed reducer des.
H.-Z. Hua e al. / Joural of Mechacal Scece ad Techoloy 7 () (0) 78~789 787 Tale 7. Dsruo deals of radom des varales he speed reducer des. Deermsc & radom varales Descrpo Sadard devao Dsruo Lower oud Mea of ear face wdh 0.0 Normal.. Mea of eeh module 0.0 Normal 0..0 Numer of eeh of po - - 7 8 Mea of dsace ewee ears 0.0 Normal 7. 8. 5 Mea of dsace ewee ears 0.0 Normal 7. 8. Mea of dameer of shaf 0.005 Normal.9.9 7 Mea of dameer of shaf 0.005 Normal 5 5.5 pper oud verse defleco of he shaf : 5= A / B 00 0 : pper oud o he sresses of he shaf : = A / B 850 0 : pper oud o he sresses of he shaf : 7= 0 0, 8= / 0 ad 9= / + 5 0 : Dmesoal resrcos ased o space; 0 = (.5 +.9) / 0 : Des codo for he shaf ased o epereces: = (. +.9)/ 0 7 : Des codo for he 5 shaf ased o epereces. 75 A=.9 0 +, 75 5 A= 57.5 0 + 0.5 0.5, B 0. 7 =. B 0. = ad The are relaly s 0.9985 =Φ () for each proalsc cosra. The opmal soluos oaed y he Or. SORA ad he ESROA wh he cosa varaces as, = 0.0, 0.0, 0.0, = = 5, = 0.005 are ve 7 Tale 8. The sar pos are he same for oh mehods as [.5 0. 8.8..0 5.099]. The same sraey ad codo are adoped for oh mehods ha he covere crero s 0, = ad 0.0% for he value of he oecve fuco. From Tale 8, he cycles ad NFE eeded ESORA are oh less ha hose of he Or. SORA whch dcaes ha he ESORA s more effce ha he Or. SORA. The opmal soluos of Or. SORA ad ESORA wh vary varaces are provded Tale 9. The sar pos are he same for oh mehods as [.5 0. 8.8..0 5.099]. The covere crero s 0, = ad 0.0% for he value of oecve fuco. For oh cases of r= 0.008 ad r= 0.0, he cycles ad NFE eeded he ESORA are oh much less ha hose of Or. SORA especally whe he value of cosa coeffce of varao creases whch dcaes ha he ESORA s much more effce ha he Or. SORA. Tale 8. Opmal soluos of speed reducer des wh cosa varaces. Des varales Or. SORA ESORA.000.000 0.8 0.8 8 8 7.000 7.000 5 7.970 7.99.9.8 7 5.0 5.0 Oecve 087.57 087.57 5. Coclusos Cosras -0.09-0.095-0.0-0.0-0.55-0.55-0.899-0.899 5 -. -. -7.80-7.80 7-7.9-7.90 8 -.75 -.755 9-0.5-0.5 0-0.079-0.078-0.05-0.05 Cycles NFE 55 7 SORA s oe of he mos effce sle loop mehods for deal wh RBDO. I hs paper, a ehaced SORA s proposed wh he am of furher mprov he compuaoal effcecy cosder oh cases of cosa ad vary varaces whle eep he sle loop framewor. I he ESORA, whe he performace fucos are lear, he oral RBDO prolem s compleely rasformed o a deermsc opmzao prolem. Whe he performace fucos are o all lear, he deermsc opmzao, he rade a he acual M s appromaed us he acual M
788 H.-Z. Hua e al. / Joural of Mechacal Scece ad Techoloy 7 () (0) 78~789 Tale 9. Opmal soluo of speed reducer des wh vary varaces. Des varales ad he mea values of radom varales of prevous cycle, ad he mea values of radom varales of curre cycle whle he rade s appromaed a he mea value of he radom des varales ad parameers a he frs cycle. As demosraed y he eamples, he ESORA performs more effcely compared wh he curre approaches whe he performace fucos are all lear. The cycles eeded ESORA for RBDO wh olear cosras are o more ha hose eeded he compared approaches whle he NFE ca e reduced us he same sar pos ad opmzao mehods he compared approaches. I he speed reducer de eample, he same sar pos ad he opmzao mehod are ulzed, he cycles ad NFE of ESORA are much less ha hose of he oral SORA especally whe he value of cosa coeffce of varao creases, whch dcaes ha he ESORA s much more effce ha he oral SORA. Acowledme r= 0.008 r= 0.0 Or. SORA ESORA Or. SORA Ths research was parally suppored y he Naoal Naural Scece Foudao of Cha uder corac umer 50750, he Naoal Hh Techoloy Research ad Developme roram of Cha (8 roram) uder corac ESORA.000.000.000.000 0.85 0.8 0.87 0.87 7 7 7 7 7.000 7.000 7.7 7.70 5 8.0870 8.0870 8.000 8.000.0.9.5.5 7 5.9 5.8 5.50 5.50 Oecve 07.757 07.75 0.80 0.80 Cosras -0.057-0.058-0.05-0.05-0.797-0.79-0.908-0.908-0.50-0.5-0.5-0.5-0.8978-0.8978-0.898-0.897 5-77.8-77.8-9.0708-9.0708-59.78-59.78-7.5-7.5 7-8.98-8.98-8.8-8.8 8 -.79 -.79 -.7088 -.7089 9-0.75-0.750-0.9-0.9 0-0.0-0.0-0.099-0.099-0.08-0.08-0.07-0.07 Cycles 8 5 NFE 879 580 7 umer 007AA0Z0, ad The Fudameal Research Fuds for he Ceral verses of Cha uder corac umer ZYG00J09. Refereces [] S. Km, S. Ju, H. Ka, Y. ar ad D. H. Lee, Relaly ased opmal des of a helcoper cosder aual varao of amospherc emperaure, Joural of Mechacal Scece ad Techoloy, 5 (5) (0) 095-0. [] Y. J. ar, S. Ju, S. Km ad D. H. Lee, Des opmzao of a loop hea ppe o cool a lhum o aery ooard a mlary arcraf, Joural of Mechacal Scece ad Techoloy, () (00) 09-8. [] Z. Wa, G. L, H-Z Hua,. L. Zha ad Y. L, Relaly-ased des opmzao for o-ooms cosder sreh deradao ad radom oal wor me, Joural of Mechacal Scece ad Techoloy, (7) (0) 05-09. [] H. Z. Hua,. Zha, D. B. Me, Y. Lu ad Y. F. L, Muldscplary des opmzao wh dscree ad couous varales of varous uceraes, Ieraoal Joural of Compuaoal Iellece Sysems, 5 () (0) 9-0. [5] H. Z. Hua, H. Yu,. Zha, S. Ze ad Z. Wa, Collaorave opmzao wh verse relaly for muldscplary sysems uceray aalyss, Eeer Opmzao, (8) (00) 7-77. [] H. Z. Hua ad. Zha, Des opmzao wh dscree ad couous varales of aleaory ad epsemc uceraes, Joural of Mechacal Des, () (009) 000--000-8. [7] J. Tu ad K. K. Cho, A ew sudy o relaly ased des opmzao, Joural of Mechacal Des, () (999) 557-5. [8] B. D. You ad K. K. Cho, Selec proalsc approaches for relaly-ased des opmzao, AIAA Joural, () (00) -. [9] B. D. You, K. K. Cho ad L. Du, Erched performace measure approach for relaly ased des opmzao, AIAA Joural, () (005) 87-88. [0] B. D. You, K. K. Cho ad Y. H. ar, Hyrd aalyss mehod for relaly-ased des opmzao, Joural of Mechacal Des, 5 () (00) -. [] B. D. You, K. K. Cho ad L. Du, Adapve proaly aalyss us a ehaced hyrd mea value mehod, Srucural ad Muldscplary Opmzao, 9 () (005) -8. []. Du ad W. Che, A mos proale po ased mehod for uceray aalyss, roc. of ASME 000 DETC/CIE, Balmore, Marylad, SA (000) DETC000/DAC-. [] Y. T. Wu, Y. Sh, R. H. Sues ad M. A. Cesare, Safey facor ased approach for proaly ased des opmzao, roc. of d AIAA/ASME/ASC/AHS/ASC SDM Coferece ad Eho Seale, Washo, SA (00) AIAA-00-5. []. Du ad W. Che, Sequeal opmzao ad relaly
H.-Z. Hua e al. / Joural of Mechacal Scece ad Techoloy 7 () (0) 78~789 789 assessme mehod for effce proalsc des, Joural of Mechacal Des, () (00) 5-. [5] T. M. Cho ad B. C. Lee, Relaly-ased des opmzao us cove appromaos ad sequeal opmzao ad relaly assessme mehod, Joural of Mechacal Scece ad Techoloy, () (00) 79-8. []. L. Y ad W. Che, Ehaced sequeal opmzao ad relaly assessme mehod for proalsc opmzao wh vary des varace, Srucures ad Ifrasrucure Eeer, (-) (00) -75. [7] J. La, Z.. Mourelaos ad J. Tu, A sle-loop mehod for relaly-ased des opmzao, roc. of ASME 00 IDETC/CIE, Sal Lae Cy, T, SA (00) 9-0. [8] S. Sha ad G. G. Wa, Relaly des space ad complee sle-loop relaly-ased des opmzao, Relaly Eeer & Sysem Safey, 9 (8) (008) 8-0. [9] R. Racwz ad B. Flessler, Srucural relaly uder comed radom load sequeces, Compuers & Srucures, 9 (5) (978) 89-9. [0] L. Du ad K. K. Cho, A verse aalyss mehod for des opmzao wh oh sascal ad fuzzy uceraes, Srucural ad Muldscplary Opmzao, 7 () (008) 07-9. Ho-Zho Hua s a professor ad he Dea of he School of Mechacal, Elecroc, ad Idusral Eeer, versy of Elecroc Scece ad Techoloy of Cha. He has held vs appomes a several uverses he SA, Caada, ad Asa. He receved a h.d. deree Relaly Eeer from Shaha Jaoo versy, Cha ad has pulshed 50 oural papers ad 5 oos felds of relaly eeer, opmzao des, fuzzy ses heory, ad produc developme. He s a Fellow of ISEAM (Ieraoal Socey of Eeer Asse Maaeme), ad a memer of ESRA (Europea Safey ad Relaly Assocao) Techcal Commee o Sysem Relaly, a Reoal Edor of Ieraoal Joural of Relaly ad Applcaos, a Edoral Board Memer of Ieraoal Joural of Relaly, Qualy ad Safey Eeer, Ieraoal Joural of Qualy, Sascs, ad Relaly, Ieraoal Joural of Relaly ad Qualy erformace, Ieraoal Joural of erformaly Eeer, Advaces Fuzzy Ses ad Sysems, ad The Ope Mechacal Eeer Joural. He receved he Wllam A. J. Goloms Award from he Isue of Idusral Eeers 00, ad he Bes aper Award of he 8h Ieraoal Coferece o Froers of Des ad Maufacur 008. Hs curre research eress clude sysem relaly aalyss, warray, maeace pla ad opmzao, compuaoal ellece produc des. opmzao. udo Zha receved he B.S. deree Mechacal Eeer from Jaa versy ad h.d. deree from versy of Elecroc Scece ad Techoloy of Cha. Hs ma research eress clude relaly ased des ad opmzao, relaly ased muldscplary des ad De-Bao Me s currely a h.d. sude sudy a he versy of Elecroc Scece ad Techoloy of Cha. He receved a B.S. deree Mechacal Eeer from Norhwes A&F versy. Hs ma research eress clude relaly ased des ad opmzao, relaly ased muldscplary des ad opmzao. Zhola Wa receved he h.d. deree Mecharocs Eeer from he versy of Elecroc Scece ad Techoloy of Cha. He s currely he assocae professor he School of Mechacal, Elecroc, ad Idusral Eeer a versy of Elecroc Scece ad Techoloy of Cha. He has ee a vs scholar he Deparme of Mechacal ad Aerospace Eeer, Mssour versy of Scece ad Techoloy from 007 o 008. Hs research eress clude relaly-ased des ad rous des. Yu Lu s a Assocae rofessor he School of Mechacal, Elecroc, ad Idusral Eeer, a he versy of Elecroc Scece ad Techoloy of Cha. He receved hs h.d. deree Mecharocs Eeer from he versy of Elecroc Scece ad Techoloy of Cha. He was a Vs re-docoral Fellow he Deparme of Mechacal Eeer a Norhweser versy, Evaso,.S.A. from 008 o 00. Hs research eress clude sysem relaly model ad aalyss, maeace decsos, prooscs ad healh maaeme, ad des uder uceray.