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1 oyrh I. Rrd from " PRODING Aual RLIAILITY ad MAINTAINAILITY ymosum" UA Jauary -. Ths maral s osd hr wh rmsso of h I. uch rmsso of h I dos o ay way mly I dorsm of ay of Rlaof ororao's roducs or srvcs. Iral or rsoal us of hs maral s rmd. Howvr rmsso o rrrublsh hs maral for advrs or romooal uross or for cra w collcv works for rsal or rdsrbuo mus b obad from h I by wr o ubs-rmssos@.or. y choos o vw hs docum you ar o all rovsos of h coyrh laws roc.

2 Warray Prdco for Producs wh Radom rsss ad Usas Huaru Guo PhD Rlaof ororao Ara Mofor Rlaof ororao Adamaos Mas Rlaof ororao Dou Od Rlaof ororao Ky Words: warray rdco radom srsss radom usas srss-srh modl Mo arlo smulao UMMARY & ONLUION Mak accura warray rdcos s chall. I bcoms v mor chall wh roducs ar orad udr radom srsss ad radom usas. Tradoal mhod oly uss h avra valus of hs radom varabls for warray rdco. Th radomss of h varabls s ord whch may rsul accura rsuls. Ths ar rss mhods o solv hs ssu. oluos for wo dffr suaos ar dscussd. I h frs suao oly radom srsss ar h major cocr. I h scod suao boh radom srsss ad radom cusomr usas ar cosdrd. Two aalycal soluos h xac ad h aroxmad soluo ar rovdd for ach cas. Th comarso shows ha alhouh hy ar sml o us aroxmad soluos ca b vry dffr from h xac rsuls. I hs ar o oly h ma valu of h warray rdco bu also h varacs ad rvals of h rdco ar calculad. Ths s much br bcaus rval sma rovds mor formao ha a sml o sma. Th roosd mhods ca b ald o may dusrs such as lcroc auomobl ad hom alac comas. INTRODUTION Warray rdco s o of h mos mora ssus rlably r. I h rdco of warray rurs ora srsss ad cusomr usa daa mus b accoud for []. Usually omal or avra valus of h ora srsss ar usd. Howvr srsss ar of o cosa. Isad hy ar radom ad ca b dscrbd us dsrbuos. or xaml o vry usr accumulas mls r yar o a vhcl or dos vry usr r h sam umbr of as r wk o a rr. Thrfor us a sl cosa valu for a radom varabl h calculao s o arora. Th radomss of h srsss ad usa mus b cosdrd. y cosdr h radomss a cofdc rval rahr ha a sl valu of h warray rur ca b calculad. I hs ar mhods for oba h cofdc rval wll b rovdd. To accuraly rdc h warray rurs h lf-srss rlaosh ds o b sablshd frs maly hrouh acclrad ss. Oc h lf-srss rlaosh s obad h ffc of h radom srsss o h roduc lf ca b quaavly smad. I h follow scos h hory of lf-srss rlaoshs acclrad s s dscussd frs. Th mhods for rdc warray rurs for wo dffr suaos ar rovdd. I h frs suao h robably of falur dur h warray rod s affcd oly by radom srsss. I h scod suao h falur s a fuco of boh radom srsss ad radom cusomr usas. THORY ON ALRATD TTING AND LI- TR RLATIONHIP Th lf-srss rlaosh fuco xlas how srsss affc roduc lf. I hs fuco lf s rrsd by a rcl of h falur dsrbuo. I ral h fuco ca b wr a lo-lar form: s h srss or a rasformao of h srss. If w assum hr s oly o srss basd o dffr rasformaos h lf srss rlaosh quao ca b []: A Arrhus Ivrs Powr Law K Th rcl s slcd accord o h lf characrsc of dffr lf m dsrbuos. om ycal lf characrscs ar rsd Tabl. Dsrbuo Wbull xoal Loormal... df Probably Dsy Lf uco haracrsc cal Paramr Ma Lf l Mda Tabl Tycal Lf haracrscs Th lf-srss rlaosh ca b rad o a lfm $. 9 I

3 dsrbuo. or xaml assum hr ar wo dd srsss. Th lf-srss rlaosh for o srss s Arrhus ad for h ohr srss s h vrs owr law. Th combd lf-srss rlaosh s: K A Th abov quao ca b rad o a lf dsrbuo such as h Wbull dsrbuo. or a Wbull dsrbuo. Th Wbull modl s: f 4 I h abov quaos -4 srss ca b hr a cosa or a radom varabl. I ordr o sma h aramrs quao 4 falur daa ar dd so acclrad lf ss ar coducd o oba falur daa frs. Th a smao mhod such as h maxmum lklhood smao ML mhod s ulzd for sma h modl aramrs []. I h follow scos w assum ha h modl aramrs hav b corrcly smad. W wll us h modl o rdc h robably of falurs for a warray rod wh h srsss ar radom. WARRANTY PRDITION AD ON RANDOM OPRATING TR. Thory o ucos of Radom abls osdr a roduc wh wo radom srsss: mraur ad vola. Is falur m dsrbuo s v quao 4. I h radoal mhod for warray rdco h avra srss valus ar usd h calculao. Th h rdcd robably of falur by h d of h warray rod of s: x 5 Isad of us h ma valu of ach srss w ca us hr dsrbuos o oba h xcd robably of falur. Assum h dsrbuo for s s ad all h srsss ar dd. Th xcd robably of falur s: x d d 6 Th varac of h robably of falur s: x d d 7 quao 5 s a aroxmao of quao 6. Ths s from h Taylor srs xaso. Accord o h Taylor srs xaso h valu of robably of falur ca b aroxmad by:...! 8 If w assum h rms wh ordr of or hhr ca b ord h xcd valu of h robably of falur a s: 9 If all h srsss ar dd h varac of h robably of falur a ca b aroxmad by: whr s h varac of h h srss. If w assum oly h rms wh ordr of or hhr ca b ord h xcd valu of h robably of falur a s:! ad h varac s:! All h drvavs h abov quaos ar calculad a h ma valus of h srsss. I hs ar aroxmaos quao 9 ad ar usd. Th frs ordr drvavs for ad ar: ;. xaml osdr a como udr wo srsss: mraur ad vola. Th mraur-ohrmal lf-srss rlaosh of quao s usd. Acclrad ss ar coducd ad a oro of h daa s v Tabl. Th lf dsrbuo s h Wbull dsrbuo. Th smad aramrs us ML ar: =.44; =986.8; =5.96; =.876 Th Wbull robably lo a h s srsss s v ur. rom h ds scfcaos s kow ha h ora mraur ad vola ar radom varabls follow ormal dsrbuos. Thy ar assumd o b dd

4 from ach aohr. Th dsrbuo aramrs ar v Tabl. Tm ald hrs Tmraur K Vola V Tabl alur Daa for h Two rsss xaml Rlaof ALTA Urlably Probably - Wbull Tm a=.44; =986.8; =5.96; = Probably Daa Tmraur-NoThrmal Wbull 48 = = rss Lvl Pos rss Lvl L 48 5 = = rss Lvl Pos rss Lvl L 78 = = rss Lvl Pos rss Lvl L 78 5 = = rss Lvl Pos rss Lvl L :5:6 AM ur Wbull Probably Plo of h Ts rsss rss Dsrbuo Paramrs Tmraur Normal N Vola Normal N 8 Tabl Dsrbuo for h Two Radom rsss Assum h warray m s 4 hours. Th xcd robably of falur ad s varac ca b obad us quao 9 ad. Th xcd valu s:. ad h varac s: Oc h ma ad varac of h robably of falur ar obad s cofdc bouds ca b asly calculad by assum ha h loarhm of s ormally dsrbud [5]: [ ] w w whr w x{ z [ ]} ad. Th ur ad lowr 9% wo-sdd bouds for hs xaml ar [.. 89]. To valda whhr h aroxmao rsuls ar accura ouh h xac aalycal soluos of h xcd robably of falur ad s varac ar also calculad us h rals quao 6 ad 7. Thy ar:. 5;. 6 As xcd hy ar slhly larr ha h aroxmad soluos bcaus h hh ordr rms ar ord quao 9 ad. Th xac aalycal soluo for h cofdc bouds of h robably of falur s [.6.]. or hs xaml h aroxmad ad xac aalycal soluos ar vry clos. I h abov aalyss h ucray radomss of h ora srsss s cosdrd h calculao. Th ucraly of h modl aramrs s o cosdrd. Th modl aramrs ar smad from h avalabl falur daa. Thr ucray ca b raly rducd f hr s a lar ouh saml sz or ouh hsory formao. Howvr h ucray causd by h radom srsss cao b rducd. I s mbddd h roduc orao. If o was o ra h ucray of modl aramrs h calculao quao 8 ca b xadd by ra modl aramrs as radom varabls. or dals las rfr o [4 5].. mulao Rsuls I sco. h xac ad aroxmad aalycal soluos ar rovdd for h xaml. oh soluos rqur sv comuao. I hs sco smulao soluos ar v. Us Mo arlo smulao o solv roblms wh radom srsss s asy ad srah forward. Th smulao rocdur s: Gra s of radom umbr for. alcula for ach s of. G h ma ad varac for h. orualy s o cssary o wr w smulao cod. mulao sofwar ackas such as Rlaof s RNO ca b usd. or hs xaml smulao rus wr coducd RNO ak abou mu o coml h smulao. Th ma ad varac for h robably of falur a warray m of 4 ar.4 ad.6. Ths valus ar vry clos o h xac aalycal soluos ad ar mor accura ha h aroxmad soluos. 4 WARRANTY PRDITION AD ON RANDOM TR AND UAG PROIL I sco w dscussd how o mak accura warray rdco by cosdr h radomss of radom srsss h calculao. I som alcaos such as wash machs h warray rur s o oly rlad o a radom srss such as load bu s also affcd by radom cusomr usa. I hs sco a srss-srh basd mhod wll b roosd o solv hs comlcad roblm.

5 4. Thory o rss-rh Modl Th srss-srh modl s wdly usd srucural rlably calculao [6-8]. Assum h srh dsrbuo s ad srss dsrbuo s. Th xcd robably of falur s dfd as: P x f x P x f x 4 whr x s h D cumulav dsrbuo fuco ad df robably dsy fuco. ur comars a srss dsrbuo ad a srh dsrbuo. Rlaof Wbull f Probably Dsy uco Rlaof ororao :46:47 PM Tm rh\daa : rss\daa : Pdf rh\daa Wbull-P ML RM MD M == Pdf L rss\daa Wbull-P ML RM MD M == Pdf L ur omarso of rss ad rh Dsrbuos Th larr h ovrla ara ur s h rar h robably of falur. Th srss-srh mhod s also radoally usd h auomobl dusry for warray rdco [9]. or xaml h usa dsrbuo a yar warray rod ca b houh of as srss ad h srh s h falur dsrbuo rms of mls. Th rdcd robably of falur h warray rod s calculad us quao 4. or h auomobl dusry a ycal warray olcy s yars ad 6 mls. o quao 4 ca b modfd o cosdr oly vhcls wh mla lss ha 6. Th modfd quao s: P 6 6 x f x 6 P x f x whr x s h robably of falur a mla of x ad f x s h df of h usa mla dsrbuo a x. or h auomobls oly h radom usa s cosdrd h calculao. or wash machs h warray rurs ar affcd o oly by h radom usa bu also by h radom load. I sco 4. a mhod of solv roblms lk h wash mach xaml wll b rovdd. 4. xaml A wash mach maufacurr coducd a survy o h usa rofl of s cusomrs. Th avra loads ad avra hours of us h mach wr rcordd for ach usr. c hs usa formao was avalabl h comay wad o us o mak mor ralsc smas of h falurs of h moors usd h wash mach for a 5 yar warray rod. rs acclrad s was coducd o sablsh h lf-srss load rlaosh. alurs wr rcordd hours. A Wbull-IPL vrs owr law modl was usd ad h aramrs wr smad from h falur daa. Thy ar: =.5; K=.69-5; =.5 Th df of hs falur m dsrbuo s dod as f. rom h survy daa h aalys foud h avra load s 7.6 lbs ad h avra usa durao r wk s.9 hours. Gv a 5 yar warray h oal avra ora m s 754 hours. Aly hs wo avra umbrs o h warray rdco h rdcd robably of falur s: x x K x Howvr bcaus mor formao s avalabl h cusomr usa daa sad of smly us h avra valus h dsrbuo of h load ad h ora hour ca b usd. Th aalys calculad h load dsrbuo from h survy daa us a Wbull dsrbuo. Th aramrs ar: =5; = 7.8 Th df of h load dsrbuo s dod as ls. I was foud ha h avra hours ad avra loads ar corrlad. or h cusomrs wh larr avra loads lor ora hours ar xcd. I ordr o ulz hs formao h aalys ald a Wbull dsrbuo o h ora hours ad rad h scal aramr as a fuco of load. A ral lo-lar fuco was usd for h -load rlao whch s: whr s h load. Th aramrs h modl for h 5 yar ora hour ar: =; =6.94; =.896 Ths df ora hour dsrbuo s dod as x. o far hr dsrbuos h lf dsrbuo of h moor h load dsrbuo ad h usa hour dsrbuo of cusomrs ar avalabl for h warray rur calculao. Th xcd robably of falur by h d of h 5 h yar ca b calculad us h srss-srh modl wh h hr dsrbuos. rs for a v load s h xcd robably of falur s: s T s T x s x s x s x s 6 Th ra load as a radom varabl h ovrall robably of falur s:

6 s T s l s ds x s x s l s ds Th varac of h robably of falur s: whr: x s x s l s ds x x s ; x x x s ; s s l s ; ; s s 7 8 quao 7 ad 8 ca b solvd umrcally. ommrcal sofwar ackas such as Mahcad ca b usd. or hs xaml h rsul for h ma valu quao 7 ad h varac from quao 8 ar:. 76 ;. 4 Us hs wo valus h 9% cofdc rval for h robably of falur s [.5.568]. quao 7 ca b show o b smlar o quao 6. Thrfor h aroxmao mhods for h ma ad varac sco ca b xdd o us hr. Howvr bcaus of h comlxy of h roblm h aroxmao rsuls ar o clos o h xac aalycal soluos aymor. To us h ma valu of h load ad usa o calcula h robably of falur w frs d o oba hs wo ma valus. Thy ar: ad x x x s l s ds 754 s sl s 7.6 Us hs wo valus h rdcd robably of falur s.9 as v quao 5. Ths valu s vry dffr from h xac aalycal soluo of.76 obad by quao 7. Thrfor h aroxmad aalycal soluo from h Taylor srs xaso s o accura for hs xaml. rom h abov calculao rocdur ca b s ha oba a aalycal soluo for h robably of falur bcoms vry chall wh mull radom varabls ar volvd. I bcoms v mor comlcad wh hs varabls ar corrlad. Thrfor smulao bcoms a aracv oo. I sco 4. h smulao soluos for h abov xaml ar rovdd. 4. mulao Rsuls Th smulao rocdur s: Gra a radom umbr for srss Us hs o ra a radom umbr of usa Us ad o calcula. Ra abov ss for ms ad G h ma ad varac for h. or hs xaml smulao rus wr coducd RNO ak abou.5 mus o coml h smulao. Th ma ad varac for h robably of falur a a warray m of 5 yars ar.75 ad.4. Ths rsuls ar vry clos o h xac aalycal soluos ad much br ha h aroxmao rsuls. 5 ONLUION I hs ar mhods for warray rdco of roducs wh radom ora srsss ad radom cusomr usas ar dscussd. Aalycal soluos ar rovdd for h cas suds. caus of h comlxy of h rocdur for oba h aalycal soluos h us of smulao o oba h soluos s also llusrad. Ulk h radoal mhod whch ca oly calcula h ma valu of h warray falurs ad ors h radomss of h srsss ad usas h roosd mhods ca calcula boh ma ad varac by ra h radomss of srsss ad usa o h calculao. Ths s much br bcaus rval sma rovds mor formao ha a sml o sma. omarsos show ha smulao rsuls ar vry accura. osquly should b rfrrd by rs v h comlxy of oba h aalycal soluos. RRN. A. Mas Rlably Prdcos asd o usomr Usa rss Profls Proc. A. Rlably & Maaably ym. Ja. 5.. Rlaof Acclrad Lf Ts Rfrc Rlaof Publcao 7.. W. Nlso Acclrad Ts: ascal Modls Ts Plas ad Daa Aalyss Nw York Joh Wly & os Rlaof Lf Daa Aalyss Rfrc Rlaof Publcao W. Q. Mkr ad L. A. scobar ascal Mhods for Rlably Daa Nw York Joh Wly & os J. Ta ad J. Zhao A Praccal Aroach or Prdc au Rlably Udr Radom yclc Load Rlably r ad ysm afy vol O. Gaudo ad J. L. olr alur Ra havor of omos ubjcd o Radom rsss Rlably r ad ysm afy vol H. ha ad Y. h A Praccal Rlably Aalyss Mhod for rs Rlably r ad ysm afy vol M. W. Lu ad R. J. Rudy Rlably Ts Tar

7 Dvlom Proc. A. Rlably & Maaably ym. Ja.. IOGRAPHI Huaru Guo PhD Drcor Thorcal Dvlom Rlaof ororao 45. assd LP Tucso Arzoa 857 UA -mal: Dr. Huaru Guo s h Drcor of Thorcal Dvlom a Rlaof ororao. H rcvd hs PhD ysms ad Idusral r from h Uvrsy of Arzoa. Hs dssrao focuss o rocss modl daoss ad corol for comlx maufacur rocsss. H has ublshd ars h aras of qualy r clud P ANOVA ad DO ad rlably r. Hs curr rsarch rss clud rarabl sysm modl acclrad lfdradao s warray daa aalyss ad robus omzao. Dr. Guo s a mmbr of R II ad I. Ara Mofor Maar mulao Grou Rlaof ororao 45. assd Loo Tucso Arzoa UA -mal: AraMofor@Rlaof.com Ms. Ara Mofor s h Maar of h mulao Grou a Rlaof ororao. Ovr h yars sh has layd a ky rol h ds ad dvlom of Rlaof's sofwar clud xsv volvm h lockm ad RNO roduc famls. Ms. Mofor holds a M.. dr Rlably ad Qualy r a.. hmcal r ad a.. omur cc all from h Uvrsy of Arzoa. Hr aras of rsarch ad rs clud sochasc v smulao ad sysm rlably ad maaably aalyss. Adamaos Mas Vc Prsd Produc Dvlom Rlaof ororao 45. assd LP Tucso Arzoa 857 UA -mal: Adamaos.Mas@Rlaof.com Mr. Mas s h Vc Prsd of roduc dvlom a Rlaof ororao. H flls a crcal rol h advacm of Rlaof's horcal rsarch ffors ad formulaos h subjcs of lf daa aalyss acclrad lf s ad sysm rlably ad maaably. H has layd a ky rol h dvlom of Rlaof's sofwar clud Wbull++ ALTA ad lockm ad has ublshd umrous ars o varous rlably mhods. Mr. Mas holds a. dr Mchacal r ad a M.. dr Rlably r from h Uvrsy of Arzoa. Dou Od Vc Prsd Rlaof ororao 45. assd LP Tucso Arzoa 857 UA -mal: Dou.Od@Rlasof.com Mr Od jod Rlaof 997 ad has srvd a vary of xcuv maam rols. I hs caacs h has b srumal h rowh ad voluo of h sals orazao hrouh h crao of sals las l dvlom sals smao ad rowh sras. Mr Od add h Uvrsy of Msoa ad s a mmbr of h ocy of Rlably rs.