RELIABILITY ASSESSMENT

Transcription

1 CHAPTER Rsk Aalyss Egeerg ad Ecoomcs RELIABILITY ASSESSMENT A. J. Clark School of Egeerg Deparme of Cvl ad Evromeal Egeerg 4b CHAPMAN HALL/CRC Rsk Aalyss for Egeerg Deparme of Cvl ad Evromeal Egeerg Uversy of Marylad, College Park CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. Avalably If he me o falure s characerzed by s mea, called mea me o falure (MTTF), ad he me o repar s characerzed by s mea, called mea me o repar (MTTR), a defo of hs probably of fdg a gve produc a fucog sae ca be gve by he followg rao for avalably (A): MTTF A (35) MTTF + MTTR

2 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. Relably, Falure Raes, ad Hazard Fucos As a radom varable, he me o falure (TTF or T for shor) s compleely defed by s relably fuco, R(). The relably fuco s defed as he probably ha a u or a compoe does o fal he me erval (,] or, equvalely, he probably ha he u or he compoe survves he me erval (, ], uder a specfed evrome. CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 3 Relably, Falure Raes, ad Hazard Fucos (co d) The probably par of hs defo of he TTF ca be expressed usg he relably fuco R() as follows: Pr probably T me o falure ay me perod R() Pr (T > ) (35)

3 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 4 Relably, Falure Raes, ad Hazard Fucos (co d) The relably fuco s also called he survvor (or survvorshp) fuco. Aoher fuco, ha ca compleely defe ay radom varable (e.g., me o falure as well as me o repar) s he cumulave dsrbuo fuco. Ths fuco s gve as F() - R() Pr (T ) (36) CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 5 Relably, Falure Raes, ad Hazard Fucos (co d) The CDF s he probably ha he produc does o survve he me erval (, ]. Assumg he TTF as a radom varable o be a couous posvely defed, ad F() o be dffereable, he CDF ca be wre as F( ) f ( x) dx for > (37a)

4 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 6 Relably, Falure Raes, ad Hazard Fucos (co d) Expoeal Dsrbuo The expoeal dsrbuo has a relably fuco R() as gve by R() exp(-λ) (38) λ falure rae cosa CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 7 Relably, Falure Raes, ad Hazard Fucos (co d) Webull Dsrbuo The relably fuco of he wo-parameer Webull dsrbuo s R() exp[-(/α) β ] (39) α scale parameer β shape parameer

5 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 8 Relably, Falure Raes, ad Hazard Fucos (co d) Logormal Dsrbuo The relably fuco of he logormal dsrbuo s gve by l( ) µ l( ) µ R( ) Φ Φ (4) σ σ µ log mea σ log sadard devao Φ(.) sadard ormal cumulave dsrbuo fuco Φ ( y) y x exp dx π (4) CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 9 Hazard Fucos The codoal probably Pr( < T + T > ) s he falure probably of a produc u he me erval (, + ], wh he codo ha he u s fucog a me, for small. Ths codoal probably ca be used as a bass for defg he hazard fuco for he u by expressg he codoal probably as

6 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. Hazard Fucos (co d) Pr( < T + T > ) f ( ) R( ) h( ) (4) h () f ( ) R( ) (43) h() hazard (or falure) rae fuco CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. Hazard Fucos (co d) The CDF, F(), for he me o falure, F(), ad he relably fuco, R(), ca always be expressed erms of he so-called cumulave hazard rae fuco (CHRF), H(), as follows: F( ) exp( H ( )) R( ) exp[ H ( )] (44) (45)

7 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. Hazard Fucos (co d) Based o Eq. 45, he CHRF ca be expressed hrough he respecve relably fuco as H ( ) l[ R( )] (46) I ca be show ha he cumulave hazard rae fuco ad he hazard (falure) rae fuco are relaed o each oher as h( ) dh ( ) d (47) CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 3 Hazard Fucos (co d) The cumulave hazard rae fuco ad s esmaes mus sasfy he followg codos: H ( ) Lm H() o-decreasg fuco ha ca be expressed as dh ( ) (48c) d ( H ()) h( ) (48a) (48b)

8 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 4 Hazard Fucos (co d) For he expoeal dsrbuo, he hazard (falure) rae fuco s cosa, ad s gve by (49) h() λ ad he expoeal cumulave hazard rae fuco s H() λ (5) CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 5 Hazard Fucos (co d) The Webull hazard (falure) rae fuco s a power law fuco, whch ca be wre as β β h ( ) (5) α α ad he respecve Webull cumulave hazard rae fuco s H() (/α) β (5)

9 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 6 Hazard Fucos (co d) For he logormal dsrbuo, he cumulave hazard (falure) rae fuco ca be obaed, usg Eqs. 46 ad 4, as l( ) µ H ( ) l Φ (53) σ µ log mea σ log sadard devao Φ(.) sadard ormal cumulave dsrbuo fuco CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 7 Hazard Fucos (co d) The logormal hazard (falure) rae fuco ca be obaed as he dervave of he correspodg CHRF: h( ) dh ( ) d µ l( ) φ σ σ µ l( ) Φ σ (54)

10 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 8 Seleco ad Fg Relably Models The bes lfeme dsrbuo for a gve produc s he oe based o he probablsc physcal model of he produc. Uforuaely, such models mgh o be avalable. Neverheless, he choce of he approprae dsrbuo should o be absoluely arbrary, ad a leas some physcal requremes mus be sasfed. CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 9 Seleco ad Fg Relably Models Complee Daa, Whou Cesorg If he avalable daa are complee,.e., whou cesorg, he followg emprcal relably (survvor) fuco,.e., esmae of he relably fuco, ca be used: < S ( ) < + ad,,..., < (55)

11 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. Seleco ad Fg Relably Models he h falure me deoed accordg o her ordered values (order sascs) as < <... < k k he umber of falures, ad s he sample sze I he case of complee daa wh dsc falures, k. The esmae ca also be appled o he Type I ad II rgh-cesored daa. CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. Seleco ad Fg Relably Models I he case of Type I cesorg, he me erval of S () esmao s (, T], where T s he es (or observao) durao. I he case of Type II cesorg, he respecve me erval s (, r ], where r s he larges observed falure me. Ths commoly-used esmae, S (), s called he emprcal survvor fuco.

12 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. Seleco ad Fg Relably Models Complee Daa, Whou Cesorg (co d) Based o 55, a esmae of he CDF of TTF ca be obaed as F ( ) S ( ) (56) F () esmae of he CDF of me o falure CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 3 Example 5: Sgle Falure Mode, Small Sample Whou Cesorg The sgle falure mode, o-cesored daa preseed Example are used o llusrae he esmao of a emprcal relably fuco usg Eq. 55. The sample sze hs case s 9. The TTFs ad he resuls of calculaos of he emprcal survvor fuco S () are gve Table 4.

13 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 4 Tme Order Number TTF (Years) Emprcal Survvor Fuco 9/9 6 8/ / / / / / / / / / / / / / / / / / /9 Table 4 Emprcal Survvor Fuco, S (), Based o Daa of Example CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 5 Survvorshp Value Tme o Falure (Years) Fgure 6.. Survvorshp Fuco for Sgle Falure Mode Whou Cesorg of Example 5

14 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 6 Example 6: Sgle Falure Mode, Small Sample, Type I Rgh Cesored Daa Equao 55 ca be appled o Type I ad II rgh cesored daa as was prevously saed, whch s llusraed hs example. The daa for hs example are gve Table as based o sgle falure mode, Type I rghcesored daa. CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 7 Example 6 (co d) Table. Example of Type I Rgh Cesored Daa ( Years) for Equpme Tme Order Number Tme (Years) TTF or TTC 7 TTF 4 TTF 3 5 TTF 8 TTF 3 TTF 37 TTF 4 TTF TTF me o falure, ad TTC me o cesorg TTF 9 5 TTC 5 TTC 5 TTC 5 TTC The TTFs ad he calculao resuls of he emprcal survvor fuco based o Eq. 55 are gve Table 5. The sample sze case s.

15 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 8 Example 6 (co d) Tme Order Number Tme o Falure, TTF (Years) Tme o Cesorg, TTF (Years) Emprcal Survvor Fuco Table 5 Emprcal Survvor Fuco, S (), Based o Daa Gve Table CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 9 Example 6 (co d) Cesorg was performed a he ed,.e., whou ay cesorg bewee falures. The emprcal survvor fuco he case of rgh cesorg does o reach he zero value o he rgh,.e., a he loges TTF observed. The resuls are ploed Fgure 7 as dvdual pos.

16 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 3 Survvorshp value Tme o Falure (Years) Fgure 7. Survvorshp Fuco for Sgle Falure Mode Wh Cesorg of Example 6 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 3 Example 7: Sgle Falure Mode, Large Sample Daa The daa hs example are based o Moe Carlo smulao. The TTFs ad he esmao resuls of he emprcal survvor fuco based o Eq. 55 are gve Table 6. The able shows oly a poro of daa sce he smulao process was carred ou for, smulao cycles.

17 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 3 Example 7 (co d) Table 6. Example 7 Daa ad Emprcal Survvor Fuco, S () Year 937 TTF (Years) Number of Falures Survvor Fuco. M M M M CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 33 Example 7 (co d) The complee daa se covers years from 937 o 6. For example, he survvorshp value a he year 974 s compued as (, 5), The emprcal survvorshp values are show Fgure 8.

18 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 34 Example 7 (co d) Survvorshp value Fed Daa Tme To Falure (Years) Fgure 8. Emprcal Survvor Fuco for Example 7 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 35 Seleco ad Fg Relably Models Samples Wh Cesorg I he hs case, he Kapla-Meer (or produc-lm) esmao procedure ca be appled o oba he survvor fuco ha accous for boh TTFs ad TTCs. The Kapla-Meer esmao procedure s based o a sample of ems, amog whch oly k values are dsc falure mes wh r observed falures. Therefore, r mus k (.e., r-k) repeaed (odsc) falure mes exs.

19 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 36 Seleco ad Fg Relably Models Samples Wh Cesorg (co d) The falure mes are deoed smlar o Eqs. 33a ad 33b, accordg o her ordered values: < <... < k, ad s equal o zero,.e.,. The umber of ems uder observao (cesorg) jus before j s deoed by j. The umber of falures a j s deoed by d j. The, he followg relaoshp holds: j + j d j (57) CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 37 Seleco ad Fg Relably Models Samples Wh Cesorg (co d) Uder hese codos, he produc-lm esmae of he relably fuco, S (), s gve by < j d j (58) S ( ) < + for,,..., k j j k < me o falure of a equpme

20 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 38 Seleco ad Fg Relably Models Samples Wh Cesorg (co d) For cases where d j,.e., oe falure a me j, Eq. 58 becomes (59) < < < + k S k j j j,,..., for ) ( CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 39 Seleco ad Fg Relably Models Samples Wh Cesorg (co d) For ucesored (complee) samples wh d j, he produc-lm esmae cocdes wh he emprcal S () gve by Eq. 55 as follows: For : ) ( j j j S For : ) ( j j j S For 3: 3 ) ( j j j S 3 3 M M M Therefore for ay : j j j j S ) (

21 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 4 Example 8: A Small Sample wh Two Falure Modes I hs example, lfe daa coss of mes o falure relaed o mulple falure modes (FMs). The relably fuco correspodg o each FM eeds o be esmaed usg Eq. 58. As a example, wo FMs,.e., FM ad FM, are cosdered here. CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 4 Example 8 (co d) Such TTF sample ca be represeed, for example, as follows: (FM) < (FM) < 3 (FM) < 4 (FM) <... < k (FM) For cases volvg more ha wo FMs a sample, he relably fuco for a specfc FM ca be esmaed by reag he TTFs assocaed wh falure modes oher ha FM as mes o cesorg (TTC).

22 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 4 Example 8 (co d) I should be oed ha cesorg meas ha a em survved up o he me of cesorg ad he em was removed from esg or servce. A sample of TTFs assocaed wh wo falure modes, sregh (FM) ad fague (FM), are show Table 7a. The calculaos of he emprcal survvor fuco based o Eq. 58 are gve Table 7a. CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 43 Table 7a. Example 8 Small Sample Daa ad Respecve Emprcal Survvor Fuco for Falure Mode S () Tme Order Number Tme o Falure (Years) Number of Occurreces of Falure Mode (Sregh) Number of Occurreces of Falure Mode (Fague) Emprcal Survvor Fuco for Falure Mode (Sregh)

23 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 44 Example 8 (co d) Table 7b provdes he compuaoal deals of he emprcal survvorshp values for falure mode, wh he sample sze ad c j umber of ems cesored a me j. A me order 7 of Tables 7a ad 7b, S (6.) -/ Smlarly a he me order umber 9 of hese ables, S (49.6) (-/6)(-/4).65 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 45 Table 7b. Example 8 Compuaoal Deals for Emprcal Survvor Fuco for Falure Mode S () Tme Tme o Order Falure Number (Years) j Number of Falures for Mode Number of Cesorgs for Mode Emprcal Survvor Fuco for Mode j j d j c j d j- - c j- (-d j / j ) / / / /

24 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 46 Example 9: Large Sample wh Two Falure Modes Two falure modes, sregh (FM) ad fague (FM), are smulaed hs example. A poro of hese daa relaed o oe compoe s examed here. The full sample sze s,, The TTFs ad he resuls of calculaos of he emprcal survvor fuco based o Eq. 58 are gve Table 8. CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 47 Example 9 (co d) Table 8. Daa ad Emprcal Survvor Fuco for Falure Mode S () Year Tme o Falure (Years) Number of Occurreces of Falure Mode (Sregh) Number of Occurreces of Falure Mode (Fague) Survvor Fuco for Falure Mode (Sregh)

25 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 48 Example 9 (co d) The complee daa se covers years from 984 ll 6. The resuls are ploed Fgure 9 as a sep fuco. The fgure also shows he fed relably fuco usg loglear rasformao ad regresso as dscussed Example. CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 49 Example 9 (co d) Survvorshp value Daa Fed 5 5 Tme o Falure (Years) Fgure 9. Emprcal Survvor Fuco for Example 9

26 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 5 Seleco ad Fg Relably Models Paramerc Relably Fucos Besdes he radoal dsrbuo esmao mehods, such as he mehod of momes ad maxmum lkelhood descrbed Appedx A, he emprcal survvor fucos ca be used o f aalycal relably fucos. Afer evaluag a emprcal relably fuco, a aalycal paramerc hazard rae fucos, such as gve by Eqs. 45 ad 47, ca be fed usg he emprcal survvorshp fuco obaed from lfe daa. CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 5 Seleco ad Fg Relably Models Paramerc Relably Fucos (co d) The Webull relably fuco was used sudes performed for he U. S. Army Corps of Egeers as provded Eq. 39 cludg he expoeal relably fuco as s specfc case. Also, he relably fuco havg a polyomal cumulave hazard fuco was used as follows: R() exp(-h()) H() a + a + a (6a) (6b)

27 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 5 Seleco ad Fg Relably Models Paramerc Relably Fucos (co d) Therefore, he hazard fuco s gve by h() a + a (6c) For he specal case where he parameers a ad a are equal o zero, Eq. 6b reduces o he expoeal dsrbuo. For he specal case where he parameers a ad a are zeros, he Eq. 6b reduces o he specfc case of he Webull dsrbuo wh he shape parameer of (Raylegh dsrbuo) CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 53 Seleco ad Fg Relably Models Parameer Esmao Usg Loglear Trasformao Eqs. 6a o 6c provde expoeal models wh parameers a, a, ad a. The logarhmc rasformao of Eqs. 6a o 6c leads o -l(r()) a + a (6a) -l(r()) a + a + a (6b)

28 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 54 Seleco ad Fg Relably Models Parameer Esmao Usg Loglear Trasformao Usg y o deoe he lef sde of hese equao,.e., y - l(r()), he followg soluos ca be obaed for he parameers a s accordg o Eq. 6a: ( ) y y a x a y a CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 55 Seleco ad Fg Relably Models Parameer Esmao Usg Loglear Trasformao The parameers of Eq. 6b ca obaed by solvg he followg se of smulaeous equaos: y a a a y a a a y a a a 4 3 3

29 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 56 Example : Loglear Trasformao for Parameer Esmao for Example 7 Daa For Example 7 daa, he loglear leas square esmao gves he followg values of he parameer esmaes: a.638 a (/Year) a.85 (/Year) CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 57 Example : (co d) All he model parameers esmaes are of hgh sascal sgfcace. The mulple adjused correlao coeffce squared (R ) s.999 dcag a good f. The fed values of relably fuco ad he respecve emprcal survvor fuco are gve Table 9 ad Fgure 8. Example 4. of your Texbook shows aoher example of esmag he parameers.

30 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 58 Table 9. Emprcal Survvor Fuco, S (), ad Fed Relably Fuco Usg Loglear Trasformao ad Regresso for Example Year Tme o Falure (Years) Number of Falures Survvor Fuco Fed Relably Fuco 937. _ M M M M M _ M M M M M CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 59 Example (co d) Survvorshp value Fed Daa Tme To Falure (Years) Fgure 8. Fed Relably Fucos usg Lolear Trasformao ad Regresso for Example

31 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 6 Seleco ad Fg Relably Models Nolear Model Esmao The model provded by Eqs. 6a ad 6b s olear wh respec o me wh hree parameers. The parameers ca be esmaed, ad errors ca be aalyzed usg olear regresso aalyss procedures. The esmao of olear model parameers ca be esseally based o usg umercal opmzao mehods. CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 6 Seleco ad Fg Relably Models Nolear Model Esmao (co d) The parameer esmaes ca be obaed usg he quas Newo mehod of opmzao. A umercal algorhm ca be advsed for hs purpose, or commercally avalable sofware, such as STATISTICA ad s Nolear Esmao procedure, ca be used. Example 4. ad 3 of your Texbook llusrae he olear esmao procedures.

32 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 6 Seleco ad Fg Relably Models Probably Plog Probably plos are vsual represeaos ha show relably daa ad prelmary esmao of assumed TTF dsrbuo parameers, by graphg rasformed values of a emprcal survvor fuco (or CDF) versus me (or rasformed me) o a specally cosruced probably paper. Relably daa ha follow he uderlyg dsrbuo of a probably paper ype wll fall o a sragh le. CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 63 Example 4: Probably Plog of Webull Dsrbuo for he Daa of Example 8 A rasformao of he relably Webull fuco ca be developed by akg he logarhm of he relably fuco of Eq. 39 wce as follows: l l β l β lα R( ) (6)

33 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 64 Example 4 (co d) Le y l l ad x l R( ) () y s herefore lear x wh slope β. Replacg R() by he respecve emprcal survvor fuco,.e., S (), a lear regresso procedure ca be used o f he followg le o he rasformed daa: CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 65 Example 4 (co d) y(x) bx + a The dsrbuo parameers ca be esmaed as follows: β b ad α exp(-a/β) The values of hese esmaes for he daa of Example 8 are β.5554 α a -7.94

34 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 66 Example 4 (co d) The fed relably fuco ad he respecve emprcal survvor fuco are gve Table 3. The respecve probably plo s gve Fgure. The sum of he squared resduals for he Webull dsrbuo fed usg he probably paper s.7, whch s worse compared o.8 based o he olear esmao Example 4-3 of he exbook. CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 67 Table 3. Emprcal Survvor Fuco, S (), ad Fed Webull Relably Fuco Usg Probably Paper for Example 4 Year Tme o Falure (Years) Number of Occurreces of Falure Mode (Sregh) Survvor Fuco for Falure Mode (Sregh). M M M M Probably Paper Fed Relably Fuco - M

35 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 68 Example 4 (co d) l l(/r()) Daa Fed l(tme) Fgure. Webull Probably Paper Plog for Example 4 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 69 Seleco ad Fg Relably Models Assessme of Hazard Fucos Oce he parameers of uderlyg lfe dsrbuos are kow,.e., esmaed, assessg he cumulave hazard fuco (CHRF) ad hazard (falure) rae fuco s reduced o applyg Eq. 46 ad 47, respecvely. Two examples of he hazard fucos calculaos are provded for demosrao purposes: Relably fuco wh a polyomal CHRF (Eq. 6) Based o he Webull relably fuco from Example 4

36 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 7 Example 5: Hazard Fuco Assessme from a Polyomal Cumulave Hazard Fuco Example 4- demosraed he developme of a polyomal cumulave hazard fuco from relably daa. The resulg relably fuco expressed accordg o Eq. 6 wh he esmaed parameers s as follows: R() exp( ) CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 7 Example 5 (co d) Usg Eq. 46, he CHRF s H() where s me years. The respecve hazard (falure) rae fuco s he dervave of H(), as provded by Eq. 47, herefore ca be wre as h()

37 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 7 Example 5 (co d) The resuls of hese calculaos are gve Table 4 ad Fgure. Takg o accou ha he hazard rae fucos are used for projecos, he able covers years from 99 ll. I ca be observed from he fgure ha he hazard (falure) rae fuco s creasg me, whch shows agg of he gve equpme. CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 73 Example 5 (co d) Table 4. Hazard (Falure) Rae ad Cumulave Hazard Rae Fucos for Relably Fuco wh a Polyomal CHRF for Example 4- Daa ad Example 5 Compuaos Year 98 Tme o Falure (Years) 43 Hazard Rae Fuco.995 Cumulave Hazard Rae Fuco.6369 M M M M

38 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 74 Example 5 (co d) CHRF HRF Tme o Falure (Years) Fgure. Cumulave Hazard Rae Fuco (CHRF) ad Hazard Rae Fuco (HRF) for Example 5 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 75 Example 6: Assessg he Hazard Fuco for he Webull Dsrbuo Ths example s based o he Webull relably fuco obaed usg probably plog Example 4. The Webull CHRF H() s gve by Eq. 5 ad he respecve hazard (falure) rae fuco h() by Eq. 5.

39 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 76 Example 6 (co d): Assessg he Hazard Fuco for he Webull Dsrbuo Usg hese equaos ad he esmaes of he dsrbuo parameers from Example 4, he followg expressos for H() ad h() ca be obaed: H() (/54346.).5554 h() (.5554/54346.)( /54346.) x -7 (/54346.) CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 77 Example 6 (co d) The resulg hazard fucos are gve Table 5. Corary o he prevous example, he hazard (falure) rae fuco hs case s decreasg me, whch shows ha he gve u s mprovg wh respec o falure mode whch mgh o be realsc. If s o realsc, a dffere probably dsrbuo should be cosdered.

40 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 78 Example 6 (co d) Table 5. Hazard (Falure) Rae ad Cumulave Hazard Rae Fucos for Webull Relably Fuco for Example 4 Daa ad Example 6 Compuaos Year 985 Tme o Falure (Years) Hazard Rae Fuco.35 M M M M Cumulave Hazard Rae Fuco E E E-5.33 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 79 Bayesa Mehods The procedures dscussed he prevous secos are relaed o he so-called sascal ferece. Applyg ay of such procedures s usually assocaed wh some assumpos, e.g., a sample s composed of ucorrelaed decally dsrbued radom varables. The decally dsrbued propery ca be saed accordg o a specfc dsrbuo, e.g., he expoeal or Webull dsrbuo.

41 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 8 Bayesa Mehods Such a assumpo somemes s checked usg approprae hypohess esg procedures. Neverheless, eve f he correspodg hypohess s o rejeced, hese characerscs cao be ake wh absolue ceray. I he framework of sascs, daa resul from observaos, ess, measuremes, polls, ec. These daa ca be vewed as objecve formao. CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 8 Bayesa Mehods Types of Iformao The ypes of formao avalable o egeers ca be classfed as:. Objecve formao based o expermeal resuls, or observaos; ad. Subjecve formao based o experece, uo, oher prevous problems ha are smlar o he oe uder cosderao, or he physcs of he problem.

42 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 8 Bayesa Mehods Bayesa Probables Problems wh boh objecve ad subjecve ypes of formao. The subjecve probables are assumed o cosue a pror kowledge abou a parameer, wh gaed objecve formao (or probables). Combg he wo ype produces poseror kowledge. The combao s performed based o Bayes heorem. CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 83 Bayesa Mehods Bayes Theorem Sample Space S A A A3 A4 A5 E P(A )P(E A ) P(A E) P(A )P(E A ) + P(A )P(E A ) P(A )P(E A )

43 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 84 Bayesa Mehods Example (co d): Defecve Producs Cosder Le 3 of he hree maufacurg les. The hree les maufacure %, 3%, ad 5% of he compoes, respecvely. The qualy assurace deparme of he producg facory deermed ha he probably of havg defecve producs from les,, ad 3 are.,., ad., respecvely. CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 85 Bayesa Mehods Example (co d): Defecve Producs The followg eves were defed: L Compoe produced by le L Compoe produced by le L 3 Compoe produced by le 3 D Defecve compoe Therefore, he followg probables are gve: P(D L ). P(D L ). P(D L 3 ).

44 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 86 Bayesa Mehods Example (co d): Defecve Producs Sce hese eves are o depede, he jo probables ca be deermed as follows: P(DIL ) P(D L )P(L ).(.). P(DIL P(DIL 3 ) P(D L ) P(D L 3 )P(L ).(.3).3 )P(L ).(.5). 3 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 87 Bayesa Mehods Example (co d): Defecve Producs The heorem of oal probably ca be used o deerme he probably of a defecve compoe as follows: P(D) P(D L) P(L) + P(D L) + P(D L3) P(L3).(.) +.(.3) +.(.5) Therefore, o he average, 5% of he compoes produced by he facory are defecve.

45 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 88 Bayesa Mehods Example (co d): Defecve Producs Because of he hgh corbuo of Le 3 o he defecve probably, a qualy assurace egeer subjeced he le o furher aalyss. The defecve probably for Le 3 was assumed o be.. A examao of he source of hs probably revealed ha s subjecve, ad also s ucera. A beer descrpo of hs probably ca be as show a fgure he form of a pror dscree dsrbuo for he probably. The dsrbuo s called P P (p). CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 89 Bayesa Mehods.35 Example (co d): Defecve Producs Pror Probably, PP(p) Defecve Probably, p The mea probably p based o hs dsrbuo s: p.5(.5)+.(.5)+.5(.3)+.(.)+.5(.3)+.3(.).975 whch s approxmaely..

46 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 9 Bayesa Mehods Example (co d): Defecve Producs Now assume ha a compoe from Le 3 was esed ad foud o be defecve, he subjecve pror dsrbuo eeds o be revsed o reflec he ew (objecve) formao. The revsed dsrbuo s called he poseror dsrbuo (P P(p)), ad ca be compued as follows: P(A )P(E A ) P(A E) P(A )P(E A ) + P(A )P(E A ) P(A )P(E A ) CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 9 Bayesa Mehods Example (co d): Defecve Producs.5(.5) P P(.5) (.) P P(.) (.3) P P(.5) P(A )P(E A ) P(A E) P(A )P(E A ) + P(A )P(E A ) P(A )P(E A ).(.) P P(.) (.3) P P(.5) (.) P P(.3)

47 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 9 Bayesa Mehods Example (co d): Defecve Producs The resulg probables add up o. The mea probably p based o he poseror dsrbuo s: P.5(.658) +.(.536) +.5(.7848) +.(.53) +.5( ) +.3(.5899).8354 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 93 Bayesa Mehods Example (co d): Defecve Producs The poseror mea probably (.8354) s larger ha he pror mea probably (.975). The crease s due o he deeced falure from he es. Now assume ha a secod compoe from Le was esed ad foud o be defecve, he poseror dsrbuo eeds o be revsed o reflec he ew (objecve) formao. The revsed poseror dsrbuo bulds o he curre poseror dsrbuo, reag as a pror dsrbuo.

48 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 94 Bayesa Mehods Example (co d): Defecve Producs Probably Pror Pos. D Pos. D Pos. 3D Pos. 4D Pos. 5D Pos. 6D Pos. 7D Pos. 8D Pos. 9D Pos. D E E-6 6.E Average, p Normalzg Facor, ND The las row of he able s he ormalzg facor for cases where a o-defecve compoe resuls from a es. The facor hs case s deoed ND he able. For example, he ormalzg facor (ND) case of a odefecve es accordg o he pror dsrbuo s: ND.5(-.5) +.(-.5) +.5(-.3) + + (.3)(-.).855 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 95 Bayesa Mehods Example (co d): Defecve Producs. Poseror Probably Average Probably Normalzg Facor Tes Number

49 CHAPTER 4b. RELIABILITY ASSESSMENT Slde No. 96 Bayesa Mehods Example (co d): Defecve Producs If he ex 8 ess resul oe odefecve ad seve defecve compoes, he resulg poseror dsrbuos are show he able. I ca be observed from he fgure ha he average probably s approachg.3 as more ad more defecve ess are obaed. The average probably cao exceed.3 because he pror dsrbuo has zero probably values for p values larger ha.3. Also, he effec of a o-defecve compoe o he poseror probables ca be see hs fgure.