July 24-25, Overview. Why the Reliability Issue is Important? Some Well-known Reliability Measures. Weibull and lognormal Probability Plots

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1 Par I: July 24-25, 204 Overview Why he Reliabiliy Issue is Impora? Reliabiliy Daa Paer Some Well-kow Reliabiliy Measures Weibull ad logormal Probabiliy Plos Maximum Likelihood Esimaor 2

2 Wha is Reliabiliy? Qualiy: Fiess for use The produc s overall abiliy o mee he cusomer s eeds. Reliabiliy: Time-orieed Qualiy (Log-erm Qualiy) (Dyamic qualiy) 3 Reliabiliy Issue a. () b. HQ/LC/QD (high qualiy, low cos, quick delivery) highly-reliable produc c. d. 4

3 To compee wih ohers, oday s maufacurers usually faced he followig decisio problems: How o Assess he Produc s MTTF? How o Deermie Opimal Bur-i ime? How o Selec he Bes Vedor? How o Improve he Produc s MTTF? 5 Reliabiliy Modelig ad Aalysis Sysem/Produc level Probabilisic Model s A B C D E R ( ) ( R ( ),, R ( )); * S MTTF ( MTTF,, MTTF ); S m m Compoe/iem level - Saisical Model Give a se of lifeime daa, say {,, }, how o make iferece he produc s lifeime iformaio, such as MTTF, Media Life, ec? 6

4 2 3 4 Case (a) Complee daa (before 960) Case (b) Cesored daa ( ) Case (c) No failure daa (980 -) 7 8

5 0* 0* 0* Cesored ime=0 9 Le S 0 : ormal use codiio, ad S0 S S2 S m are higher sresses l( T S) (ALT) Goal : To exrapolae he lifeime T a S Trasformer sress 0 0

6 Moivaig Example Twey specimes of a ew elecric isulaio are life-esed, half a 90 F ad he oher half a 240 F. The failure imes are as follows (i hours). A 90 0 F : 7228, 7228, 7228, 8448, 967, 967, 967, 967, 05*, 05*. A F : 75, 75, 52, 567, 67, 665, 665, 73, 76, 953. The mai goal of his lifeime es is o predic he produc s lifeime iformaio (such as MTTF) a a ormal use codiio, say 40. Degradaio Modelig QC d 0 f ime T if{ L( ) d} 2

7 Applicaios of Degradaio Models LED (ligh emiig diode) OLED (orgaic LED) LCD (Liquid crysal display) DLP (digial ligh processig) Food ad Drugs Meal Faigue Tesig Oher depedable sysems 3 ALT ADT Sep - Sress ALT Sep - Sress ADT Progressive - Sress ALT Progressive - Sress ADT 4

8 parameer (MTTF) Radom Sample Saisical Iferece ( X,, X ) Daa Reducio T ( X,, X ) 5 Seps of Reliabiliy Daa Aalysis Reliabiliy Measures (Iformaio) Some Well-kow Lifeime Disribuios Usig Probabiliy Plos o Assess he Produc s Lifeime Iformaio Usig MLE o Assess he Produc s Lifeime Iformaio 6

9 Ui I: Basic Reliabiliy Measures Reliabiliy fucio: R () Failure disribuio fucio: F () R () Failure desiy fucio: f() R() ' Hazard fucio: 00 p h perceile: h ( ) (l R ( )) p F p R p ' ( ) ( ) Mea-Time-To-Failure: MTTF f() d Rd () (a) (reliabiliy fucio) / ( sysem, equipme, compoe ) ( ormal use codiio ) ( a specified period of ime ) ( is ieded fucio, o failure ) ( probabiliy measure, ) 0 P( A) 8

10 (Reliabiliy Fucio) T/ ( lifeime ) T ormal use codiio R ( S) Pr T S 0 0 S 0 : ormal use codiio. 9 (b) (cdf) Ureliabiliy Fucio (Failure Disribuio Fucio) F () Pr{ T} Pr{ T } R ( ) 20

11 Cocep of Probabiliy Disribuio A mahemaical model ha relaes he value of he lifeime (T) wih he probabiliy of occurrece of ha value i he populaio. 2 (c) (pdf) Rage of Produc s Lifeime (T): Failure Desiy Fucio: f() Failure Disribuio Fucio [0, ) F () Pr{ T} f( xdx ) Reliabiliy Fucio R () Pr{ T} f( xdx ). 0 22

12 (d) (Hazard Fucio) Pr (Fail i he ex Survive up o ) R()-R( ) R() R() R( ) h( ) lim 0 P(fail i he ex survive up o ) ( Isaaeous failure rae ) 23 R () R ( ) h ( ) lim 0 R() R() f() R () R () ie d h ( ) - l( R ( )) d 24

13 Bah-ub Curve h() h icreasig fucio if decreasig fucio if ( ) ( / )( / ) cosa fucio if 25 (e) (MTTF) MTTF E( T ) 0 0 xf ( x) dx Rxdx ( ) 26

14 (f) p-h Produc s p-h perceile ( p ) F( ) Pr{ T } p; p R( ) Pr{ T } p. p p p p F ( p) R ( p) ( : Media Life) Reliabiliy measures whe h() is kow R () e 0 hudu ( ) F () R () e ' f() F () h() e 0 0 hudu ( ) hudu ( ) 28

15 Reliabiliy Measures of Expoeial Reliabiliy fucio: R () e Failure disribuio fucio: F () e Failure desiy fucio: f() e Hazard fucio: h () 00 p h perceile: p l( p ) Mea-Time-To-Failure: MTTF 29 Weibull Disribuio Hazard fucio : Reliabiliy fucio : h( ) R ( ) e Failure desiy fucio : MTTF : IFR ( icreasig failure rae ) CFR ( cosa failure rae ) DFR ( decreasig failure rae ) f ( ) if if if e 30

16 Logormal Disribuio 2 Noaio: l T ~ N ; l () pdf : f ( ) l (2) R ( ) f( udu ) (3) MTTF e 2 2 l (4) h ( ) l ( icreasig decreasig a he a he firs secod ierval, he ierval ) 3 Log-locaio-scale Disribuio Noaio: l T Z; Z ~ g( ), G( ) l () pdf : f ( ) g l (2) R( ) f ( u) du G (3) MTTF R( ) d l g (4) h ( ) l G 0 32

17 Well-kow Log-locaio-scale Disribuio zexp( z) () Weibull Disribuio: g( ), 2 z /2 (2) Logormal Disribuio: g( z) e, zr 2 (3) Geeralized Gamma Disribuio: z e zr ( k) kzexp( z) g( ), z e zr (4) Geeralized log-burr Disribuio g( z), zr z k ( e / k) (5) log Logisic Disribuio g( z), zr z 2 ( e ) 33 Geeralized F-Disribuio l T Z 34

18 Geeralized F disribuio 35 Ui II: Two impora ools for aalyzig reliabiliy daa (A) Graphical Mehods (Probabiliy Plos) (a) (Weibull probabiliy plo) (b) (Logormal probabiliy plo) (B) Aalyical (Saisical) Mehods (Maximum Likelihood Esimaor) 36

19 ( / ) F ( ) e l{l( )} (l l ) F ( ) Y l{l( )}, X l, C l F ( ) Y X C (liear relaio), 37 l(l( )) F( ) b ( ) a b a l 38

20 Esimaor for ^ F( ) Fˆ ( ( i) ) i i ( ) ( media rak) (mea rak) 39 ) ; r 2) F ˆ ( ) () (2) ( r) () i 3) X l () i, Y l(l( )) F ˆ ( ) 4) Plo ( XY, ) 5) Check, fi ˆ (0.632), F ˆ () i 40

21 proporio fallig Weibull disribuio probabiliy paper 4 Mea ad media rak values for a sample FAILURE TIME (HRS) MEAN RANK MEDIAN RANK ˆ i (A) Mea Rak F( ( i) ) ( ) ˆ i 0.3 ( B) Media Rak F( ( i) ) ( )

22 99.0 proporio fallig Weibull disribuio probabiliy paper 43 Gridig wheel Life daa Fˆ ( ( i) ( i) {, ˆ i F( () i )} i locaio parameer ) 44

23 ) ˆ ( ˆ ˆ ) ( ) ( ) ( () i i i x F x ˆ 4900, ˆ plo) (weibull )} ˆ (, { 8 ) ( i i i F 45 46

24 Weibull probabiliy plo (cocave fucio) () Weibull locaio, scale, shape Reliabiliy fucio ( ) R () e l{ l R ( )} {l( ) l } () 47 3 () (2) (3) ˆ 0.90 x ( i) ( i) () ˆ, i Weibull probabiliy Plo 48

25 proporio fallig x (Logormal) T ~ LN( P( T P ( r 2 ; ) l ( T l( 0 ( F( )) ) P (lt r )) l (l l) ) l 0 ) 50

26 ( F( )) 0.5 b a b ( ) a x 0 l 5 Logormal disribuio probabiliy paper 52

27 Moivaig Example (Revisied) Twey specimes of a ew elecric isulaio are life-esed, half a 90 F ad he oher half a 240 F. The failure imes are as follows (i hours). A 90 0 F : 7228, 7228, 7228, 8448, 967, 967, 967, 967, 05*, 05*. A F : 75, 75, 52, 567, 67, 665, 665, 73, 76, 953. The mai goal of his lifeime es is o predic he produc s lifeime iformaio (such as MTTF) a a ormal use codiio, say Logormal Probabiliy Plo of Elecric Isulaio Daa Probabiliy Plo (Idividual Fi) for lifeime Logormal Cesorig Colum i failure or cesore - ML Esimaes Perce emperaure Table of Saisics Loc S cale A D * F C lifeime

28 Weibull Probabiliy Plo of Elecric Isulaio Daa Probabiliy Plo (Idividual Fi) for lifeime Weibull Cesorig Colum i failure or cesore - ML Esimaes Perce emperaure Table of Saisics Shape Scale AD* F C lifeime How o cosruc GG3 Probabiliy Plo? T ~ GG ( k,, ) 3 u k ( ) e u PT ( ) du 0 ( k) Gk (( ) ) Gk ( F( )) ( ) G F l k ( ( )) (l l ) 56

29 Maximum likelihood mehod (MLE) Mehods of momes Miimum-Chi-square mehod Miimum-disace mehod Leas square esimaes (LSE) Bayes Esimaes 57 Moivaio of MLE (I) pp p = 0.25 p = 0.75p : S 4 Pr{ S 3 p} C p ( p) Pr{ S4 3 p 0.75} Pr{ S4 3 p 0.25}

30 Joi pdf ad likelihood fucio probabiliy fucio Likelihood fucio f( x,, x ) f( xi ) Example i xi xi f( x,, x ) ( ) i L( x,, x) f( xi ) Example i xi xi i L( x,, x ) ( ) 59 Maximum Likelihood Esimae Le L be he likelihood of a sample, where L L( ; x,, x ) L; L i i f( xi ) if xi is observed F( xi ) if xi is cesored i The he maximum likelihood esimaor of values of ˆ ha maximizes L. ˆ arg max { (,, )} L x x is he 60

31 (Weibull disribuio) T ~ Weibull(, ) f (, ) ( ) exp ( ) ; 0, 0, 0 F (, ) exp ( ) 6 Joi Pdf of Weibull disribuio L(, ) f(, ) i i i i ( ) ( ) exp( ( ) ) i i l(, ) l L(, ) i [l l ( )l i ( ) ] i 62

32 MLE for Weibull disribuio i l(, ) i i i l(, ) [ l l i ( ) l( )] i i Seig hese parial derivaives equal o 0, ad solve by umerical mehod. (Lawless, 2003) 63 Miiab Oupu (Weibull) 64

33 (Logormal disribuio) Defiiio: X f x LN 2 ~ (, ) if 2 2 (, ) exp ( ) ; 0 Aoher expressio lx x 2x Y X N f y 2 l ~ (, ) y y (, ) exp ( ) ; 65 Joi pdf of Logormal disribuio L 2 2 (, ) Li (, ) i i i f 2 Y( yi, ) yi exp ( ) l 2 2 (, ) l L(, ) yi l(2 ) l ( ) 2 2 i 2 66

34 MLE for logormal disribuio 2 yi l(, ) 0 2 i 2 ( yi ) 2 i l(, ) ( ) ( ) 2 yi ˆ is MLE of i ( y ˆ ) ˆ is MLE of i 2 2 i 2 67 Miiab Oupu (Logormal) 68

35 Log-locaio-scale disribuio (cesored daa) li l i i L(, ) L (, ; daa ) ( g( )) G( ) i i i i i i i if i-h daa is a failure observaio 0 if i-h daa is a righ cesored observaio l(, ; DATA) l L(, ; DATA) l i l i il l i l g( g( ))} il G( g( )) i 69 : (cesored daa) i i i i i i i L(, ) L (, ; daa ) f ( ;, ) F( ;, ) i if i-h daa is a failure observaio 0 if i-h daa is a righ cesored observaio i l(, ; DATA) l L(, ; DATA) i F l f( ;, ) l ( ;, ) i i i i i [l l ( )l ] ( ) i i i i 70

36 (cesored) 2 2 (, ; ) i(, ; i) i L DATA L daa i f y 2 i 2 i ( i;, ) [ F( yi;, )] l DATA L DATA 2 2 (, ; ) l (, ; ) i 2 2 il f( yi;, ) i[ F( yi;, )] 7 Likelihood Cofidece Ierval 95% joi cofidece ierval ca be obaied by: L(, daa) 2 {(, ) 2l{ } 0.05(2)}; L( ˆ, ˆ daa) ( ˆ, ˆ):MLE for (, ) 72

37 Miiab Oupu (Weibull) 73 Miiab Oupu (Logormal) 74

38 Life-Sress Model Logormal (Weibull) Arrheius Model l( T S) ( S) ; ( S) /(273 S); ~ N if T LN S SEV if T Weib S 2 (0,), ~ ( ( ), ) (0,), ~ (exp( ( )), ) 75 Saisical Life-Sress ALT Model disribuio Sress relaio Arrheius Model Iverse Power Model Weibull disribuio R( ( s) s) e e ( ) ( s) r0 r (273s) R( ) ( s) ( ) ( s) e r 0 ( s r ) Logormal disribuio l ( s) R( s) ( ) r ( ) 0 r s (273 s) l ( s) R( ) ( ) ( s) (l r0 ) r (l s) 76

39 Quesios: () How may sresses should be used? (2) How may observaios should be ake? (3) Which(Samplig) cesorig pla is beer? (4) Wha is he opimal cesorig ime? (5) How o esimae ( S 0 )? MTTF S )? (6) How o cosruc a cofidece ierval for ( S 0 )? ( 0 77 The Need for Acceleraed Tess Need imely iformaio o high reliabiliy producs. Moder producs desiged o las for years or decades. Acceleraed Tess (ATs) used for imely assessme of reliabiliy of produc compoes ad maerials. Tess a high levels of use rae, emperaure, volage, pressure, humidiy, ec. Esimae life a use codiios. Noe : Esimaio / predicio from ALTs ivolves exrapolaio. 78

40 Mehods of Acceleraio Mehod : Icrease he use rae of he produc (High use rae reduces es ime) Mehod 2 : Use higher sress o icrease rae of failure (Acceleraed life es ; ALT) Mehod 3 : Use higher sress o icrease he degradaio rae of he produc, ad he degradaio measuremes are used o esimae/assess he produc s lifeime. (ADT) Nelso (990), Meeker & Escobar (993)) 79 The Arrheius Model (for emperaure) Le be MTTF (or T0 ) of he produc where : Ae H (T K H :acivaio eergy ) K : Bolzma's cosa (8.67x0 T ev/k) absolue emperaure (i degree kelvi) -5 80

41 AF (acceleraed facor) ( ) e H K ( ) T T 2 Example: H 0.5, T 25 C, AF 6.4 Example 2 : o T o C H 0., T 45 o C, T 2 AF o C 8 Arrheius AF uder ( H 0.5,( H.0)) Lower\Higher (00) (268) (679) (637) (3767) (8305) (7597) (35938) (70933) (28) (76) (92) (463) (065) (2348) (4976) (063) (20059) (8.6) (23.2) (58.8) (42) (326) (79) (524) (3) (64) (2.8) (7.6) (9.3) (46.6) (07) (237) (50) (02) (2020) (2.7) (6.8) (6.4) (37.7) (83.) (76) (360) (70) (2.5) (6.) (4.) (3.0) (65.7) (34) (265) (2.4) (5.5) (2.2) (25.9) (52.9) (04) (2.3) (5.) (0.8) (22) (43.3) (2.2) (4.7) (9.5) (8.8) (2.) (4.3) (8.5) (2.0) (4.0) 35.4 (2.0) 82

42 Likelihood fucio of Logormal ALT daa l 2 2 (,, ) l{ L(,, )} m m ( r) log (l /(273 S ) m i i 2 ij i i 2 i jd jc l ij /(273 Si ) log( ( )) 2 83 MLE oupu for ALT daa (Weibull) 84

43 MLE oupu for ALT daa (Logormal) C 25 C

44 ˆ ˆ ˆ 0.46 ˆ 0 ˆ ˆ C 85 C proporio fallig

45 C 85 C proporio fallig Exercises Daa ses Lifeime disribuios Geeralized Gamma Disribuio: z e zr ( k) kzexp( z) g( ), Geeralized log-burr Disribuio g( z), zr z k ( e / k) Log-Logisic Disribuio g( z), zr z 2 ( e ) 90