Comparison of Weibayes and Markov Chain Monte Carlo methods for the reliability analysis of turbine nozzle components with right censored data only

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1 Comparson of Webayes and Markov Chan Mone Carlo mehods for he relably analyss of urbne nozzle componens wh rgh censored daa only Francesco Cannarle,2, Mchele Compare,2, Sara Maafrr 3, Fauso Carlevaro 3, Enrco Zo,2,4 Energy Deparmen, Polecnco d Mlano, Va la Masa 34, 256 Mlano, Ialy 2 Arams Srl, Va pergoles 5, Mlano, Ialy 3 General Elecrc-Nuovo Pgnone, Va Maeucc, Frenze, Ialy 4 Char on Sysem Scence and he Energec Challenge, Fondaon EDF, Cenrale Pars and Supelec, Pars, France Absrac The Webull dsrbuon s wdely used n relably engneerng o represen he componen falure behavour. The parameers of hs dsrbuon canno be esmaed by applyng he wdely used Maxmum Lkelhood Esmaon (MLE) mehod when he colleced feld daa conan rgh-censored mes only. To overcome hs lmaon, he Webayes mehod s ofen used n ndusral pracce: consss n seng he value of he shape parameer based on pror knowledge and, hen, esmang a lower confdence bound on he scale parameer. An alernave approach o esmae he Webull parameers reles on he Markov Chan Mone Carlo echnque, whn he Bayesan sascs framework. Ths echnque allows accommodang poor nformaon on he parameer values, whch s modeled by vague pror dsrbuons. In hs paper, a comparson beween he Webayes and MCMC approaches s proposed by way of a real ndusral case sudy concernng daa on Gas Turbne (GT) forced ouages due o he mechancal falure of a GT componen. Key Words: Webayes; Bayesan Analyss; Markov Chan Mone Carlo (MCMC); INTRODUCTION The Webull dsrbuon s probably he mos wdely used dsrbuon n relably engneerng (Crowder e al., 99). The reasons for hs populary s meanly due o he varey of shapes, and hus falure behavors, can accommodae. In general, boh he scale and shape parameers of he Webull dsrbuon can be easly esmaed by means of he Maxmum Lkelhood Esmaon (MLE) mehod, as falure me daases usually

2 conan boh acual falure mes and rgh-censored observaons (.e., no falure occurred durng he es me). However, when all he colleced daa are rgh-censored mes, MLE can no longer be appled (D Mao e al., 25). To overcome hs lmaon, he Webayes approach has been recenly nroduced (Abernehy, 28) and adoped by some wdely used commercal sofware ools for relably engneerng. The Webayes mehod asks he expers o gve a precse (.e., wh no uncerany) value of he shape parameer based on pror experence or knowledge. Then, provdes a lower confdence bound on he scale parameer. Such precse esmaon of he shape parameer may yeld msleadng resuls, f s wrong. The Bayesan paradgm offers an alernave approach o esmae he parameers of he Webull dsrbuon, whch are framed as random varables wh her own probably dsrbuons, so called pror dsrbuons. Poor nformaon on he parameer values can be accommodaed by vague pror dsrbuons and he Bayesan nference procedure allows adjusng hem based on he evdence comng from feld daa, also reducng he uncerany of he nal pror dsrbuons. The poseror dsrbuon hereby obaned encodes boh he pror knowledge of he exper and he sascal evdence colleced, and he poseror mean, medan and credble nervals of he falure me can be exraced. However, analycal approaches o derve he poseror dsrbuons are no always feasble n praccal cases, such as ha of Webull componens wh no falure experenced durng he es me and wh only non-nformave prors avalable. In hese cases, one can resor o he Markov Chan Mone Carlo (MCMC) algorhm, whch s an advanced Mone Carlo samplng echnque ha requres specfc heorec knowledge and experence for s use (Rober & Casella, 24). In hs work, Webayes and Bayesan approaches are compared by way of a case sudy concernng daa on Gas Turbne (GT) forced ouage due o he mechancal falure of a GT componen. The case sudy consdered n hs work s derved from a real ndusral applcaon. However, he deals on boh he componen and he degradaon mechansm ha lead o falure are no gven o proec he nellecual propery of General Elecrc (GE). The paper s organzed as follows: n Secon 2, bascs of relably analyss based on Webull dsrbuon are recalled; n Secon 3, he Webayes mehod s dscussed. Secon 4 presens Webull Bayesan analyss for dealng wh rgh-censored daa only. The applcaon of he mehodology o he GE s case sudy s descrbed n Secon 5, whereas n Secon 6 conclusons are drawn. 2 WEIBULL RELIABILITY ANALYSIS

3 In hs work, we assume a Webull Probably Densy Funcon (pdf), f (), for he sochasc falure me T: f( ) e,, () where s he shape parameer and α s he scale parameer of he dsrbuon. The correspondng relably funcon s gven by: R( ) e,, (2) whereas he hazard rae s: h( ),, (3) The shape parameer can be nerpreed as follows: < corresponds o a decreasng hazard rae behavor. Ths models he behavor of componens for whch he falure frequency s larger when hey are pu no servce and decreases over me. = n hs case, he hazard rae s consan, and he Webull dsrbuon reduces o an exponenal dsrbuon. > corresponds o an ncreasng hazard rae behavor, whch s ypcal of agng componens. Le be a posve random varable denong he me a whch a rgh-censorng kcks n: we observe eher falure me T or censorng me, whchever comes frs. I follows ha he observed daase s a collecon of random varables (Chrsensen e al., 2): Y mn( T, ) (4) In addon, he nformaon on wheher Y s an acual falure me or a censored observaon s known, and s modeled by he ndcaor varable T T (5) Tha s, δ s se o f we observe an acual falure me, oherwse s se o. We assume ha T and are sascally ndependen random varable, and ha he censorng dsrbuon does no depend on parameers and β. These condons are usually referred o as non-nformave censorng (Chrsensen e al., 2).

4 In pracce, he parameers of f () are unknown and need o be esmaed from daa. To do hs, le ( y, ) be ndependen dencally dsrbued observaons on =,,n uns. For convenence, he frs k observaons are falure mes and he remanng n-k observaons are rgh-censored oucomes. Then, he lkelhood of daa D=(y,δ), where y=(y,,yn) and δ=(δ,,δn), reads (Chrsensen e al., 2): L(, ) [ f ( y, )] [ R( y, )] D (6) or, equvalenly: n L(, D ) [ h( y, )] R( y, ) (7) The MLE echnque esmaes parameers α and β by maxmzng Eqs. (6) or (7), or her he logarhm. In deals, he ML esmaes ˆ and ˆ of parameers α and β, respecvely, are obaned by solvng he followng nonlnear equaons (Gonzalez-Gonzalez e al., 24): n y ˆ log( y) k n y log( y ) k ˆ ˆ (8) ˆ n k y ˆ ˆ (9) When he observaons are all rgh-censored (.e., no falure occurred), Eq. (9) becomes: n n y L(, D ) R( y, ) e () and he MLEs of parameers α and β no longer exs. To overcome hs problem, he Webayes mehod has been nroduced (Abernehy, 28), whch assumes ha he shape parameer β s known from eher pror experence or engneerng knowledge on he physcs of he falure. Bascally, f (y,,yn) s a se of samples drawn from he Webull dsrbuon n Eq. (2), hen y,, yn can be regarded as samples from an exponenal dsrbuon of mean me o falure, where β s known. Now, recall ha when he observaons from an exponenal dsrbuons

5 are all rgh-censored, hen a one-sded, lower ( )% confdence bound on s gven by (Zo, 27): n y log( ) () Therefore, a one-sded, lower ( )% confdence bound on he scale parameer s gven by: n y log( ) (2) The value of α correspondng o ε=.37 (.e., 63% confdence level),.63 ˆWeb, s he esmae ha some commercal sofware ools provde n oupu o he relably engneers. Ths value eners he manenance decson process. 3 BAYESIAN ANALYSIS Whn he Bayesan paradgm, boh scale parameer α and shape parameer are posve random varables. The pror knowledge on her varably s specfed n a jon pror dsrbuon wh pdf,. Informaon brough by daase D s combned wh, by means of he Bayes formula:, D,, D, L (, D), L dd (3) where he condonal pdf (, D ) s usually called poseror pdf (Rober & Casella, 24). As usual, n hs work we assume ha random varables α and are sascally ndependen. Ths mples ha he pror pdf, s he produc of he margnal pror pdfs and of α and, respecvely. Tha s:,, (4) When poor nformaon s avalable, non-nformave pror dsrbuons can be elced for boh parameers as, for example, he mproper exended Jeffery s pror (Al-Kuub & Hbrahm, 29). The Bayes formula n Eq. (3) wh lkelhood gven by Eq. () and prors by Eq. (4) gves a poseror dsrbuon whch s proporonal o:

6 n y (, D ) e ( ) ( ) (5) Eq. (5) defnes he kernel of an unknown pdf. Thus, a Markov Chan Mone Carlo (MCMC) algorhm (Casella & Berger, 24) can be exploed n order o oban samples from he poseror dsrbuon, whch wll be used o make poseror nference on parameers α and. 3. Markov Chan Mone Carlo MCMC s a famly of algorhms ha allow drawng samples from a probably dsrbuon g( θ), θθ (usually referred o as arge dsrbuon), whch are produced by an ergodc Markov chan X (Andreu & Thoms, 28). The man buldng block of hs class of algorhms s he Meropols-Hasngs (MH) algorhm. I requres he defnon of a famly of proposal dsrbuon qθ,, θθ, whch generae possble ransons for he Markov chan, say from θ o θ'. The ransons are acceped or rejeced accordng o he probably θqθ, θ θqθ, θ g r θθ, mn, g (6) Here, we focus on he (symmerc) random walk MH algorhm, n whch q, q symmerc probably densy q on Θ. In hs case, Eq. (6) reads: θ θ θ θ for some θ θ r, mn, g θθ g (7) As proposal dsrbuon, we choose he mulvarae Gaussan dsrbuon Normθ,, Σ μ wh vecor mean μ = and covarance marx Σ. Ths laer s a symmerc 22 x marx, wh 3 parameers o be se. The choce of her values s crcal for he convergence of he MH algorhm: large values of sandard devaons mprove he effecveness of he chan n spannng hroughou Θ, bu wh small effcency (.e., large number of rejeced samples). For hs, algorhms ha adapvely une he parameers of Σ have been devsed. In hs work, we have used he Adapve Random Walk Meropols-Hasngs (ARWMH) algorhm (Andreu & Thomas, 28). The pseudo-code of hs algorhm s brefly repored n he parcular case under sudy, where we are neresed n drawng from he jon poseror dsrbuon of α and ; ha s, n our case, g ( ) D, and q s a bvarae Gaussan dsrbuon (Andreu & Thomas, 28): θ,

7 . Inalze,, ρ and Σ, where and are he nal values for parameer α and respecvely; ρ s an nal value for parameer ρ,, whch eners he updang sep of covarance marx Σ n Eq. (2). Σ s he assgned sarng value of covarance marx of he proposal bvarae normal densy q ; s a user-valued consan whose value mus be se whn he nerval (,], where s se equal o p, and p 2 s he number of parameers of he kernel (Andreu & Thoms, 28). Ths quany eners he defnon of he sep-szes { } n Eq. (8). A eraon, gven,, ρ and Σ : 2. Sample 3. Compue 4. Sample u ~ Unf, 5. Se 6. Updae (, ), ~ Norm,, Σ (, D) r, mn, (, D), f u r (, ) f u r (8) ( ) ρ ρ ρ (9) ' Σ Σ ρ ρ Σ (2) and Afer M eraons of he ARWMH algorhm, we oban wo Markov chans,.e.,,, M,, M, whch are drawn from he margnal poserors D ( ) and D ( ), of α and, respecvely. To ponwse summarze he uncerany n he poseror dsrbuons, we consder he poseror medan ˆmed :

8 ˆ med P ˆ med D ( D) d (2) 2 and he poseror mean ˆmean E D ( D) d (22) ˆmean Inerval esmaon of parameer α can also be gven n erms of he % Credble Inerval (CI), whch s he smalles subse, CI c,, nf csup, c sup,, cnf csup D D d, cnf of R such ha: P (, ) ( ),, (23), In parcular, f we se n Eq. (23) c,, hen he value c ha solves Eq. (23) s he s he lower bound, c Lower sup of he one-sded % lower credble nerval CI nf. Muas muands, he defnons of poseror mean and medan, and credbly nerval gven for he scale parameer α are he same for parameer. 4 CASE STUDY In hs Secon, we llusrae he applcaon of he proposed mehods o he GT forced ouage due o a mechancal falure of GT componen, whch s assumed obeyng a Webull dsrbuon. We rely on a daase D conanng n 2 rgh-censored observaons. For confdenaly, hese values are no repored. To clearly hghlgh he dfference beween he wo mehods, we consder wo suaons:. Realsc problem: n hs case, he value of he shape parameer currenly used by GE s consdered. 2. Based problem: n hs case, he shape parameer s se o a value vary far from ha used by GE. 4. Realsc Problem Accordng o he GE pracce, he value of he shape parameer s se o ˆWeb 6 (24)

9 Ths value s derved from horough engneerng consderaons, no repored for confdenaly. The applcaon of he Webayes approach when consderng he one-sded 63% lower confdence bound on he scale parameer yelds.63 Web 662 (25) The MCMC bayesan analyss has been performed wh he followng pror dsrbuons: The generalzed mproper Jeffery pror dsrbuon a,, a (26) wh hyper-parameer a 2. Ths corresponds o a dffuse pror dsrbuon (.e., wh a wde suppor on R + ), whose large varably s coheren wh our poor knowledge base on s acual value. For he shape parameer, we have chosen a dsrbuon cenered on β=6, wh probably mass of.8 unformly dsrbued beween [5.8,6.2] (.e., symmercally on β=6), a probably mass of. unformly dsrbued n [,5.8], whereas he remanng probably mass of. s unformly dsrbued n [6.2,8]. Ths choce s jusfed by he followng consderaons: o To explo he GE pror knowledge we have pu a large poron of probably on he nerval [5.8, 6.2] ha encompasses he value ˆWeb 6 provded by sound engneerng consderaons. In fac, we expec ha f hs value s correc, he MCMC for shape parameer wll resul n a Markov chan movng no oo far from hs value. o The pror dsrbuon leaves a relavely small poron of probably n he remanng pars of he nerval [,8]. Ths s conssen wh our pror knowledge: we assume ha values n [,8] are plausble values for parameer β. o Values of scale parameers larger han 8 correspond o very small uncerany n he falure mes (values of sandard devaon almos 3% of he scale parameer). Ths suaon s no realsc. To draw samples from he poseror dsrbuon (, D ), he ARWMH algorhm descrbed n Secon 4 has been run for M=5,, eraons wh varables,, ρ Σ, and nalzed as repored n Table. ρ Σ

10 Log Scale Parameer Table. Values of parameer,, Σ and Then, we have appled:. a burn-n of 2,, samples,.e., he frs 2,, samples have been dscarded o elmnae he bas nroduced by he poson of he nal pon. 2. a sub-samplng (commonly referred o as hnnng) every samples o reduce he correlaon beween he successve pons of he Markov chans generaed by he algorhm. By so dong, he cardnaly of he orgnal Markov chans has reduced from 3,, o M 3 sample pons. The race plos (.e., he plo of he sampled pon, ordnae, vs he sample sep, abscssa) relevan o he Markov chans,, M and,, M are shown n Fgures and 2, respecvely. From hese Fgures, emerges ha here s good mxng,.e., he domans of he wo poseror dsrbuons D ( ) and D ( ) are well explored around he dsrbuon modes. In parcular, he race plo of,, M of ˆWeb n Fgure 2 shows ha hs chan ends o sample n proxmy 6, alhough samples are rarely (.e., wh small probably) drawn also from he remanng par of he suppor. 9.5 Traceplo Scale Parameer Sep x 4 Fgure : race plo of Markov chan,, M α (las sample pons).

11 Shape Parameer 8 Traceplo Shape Parameer Sep x 4 Fgure 2: race plo of Markov chan,, M β (las sample pons). Fgures 3 and 4 show he auocorrelaon plos of chan,, M and,, M, respecvely. Tha s, for l,2,..., 4 we measure he exen o whch he values of he chan a me ( l) and me are lnearly relaed, for every =,, M. From hese Fgures, resuls ha samples from boh Markov chans can be consdered almos uncorrelaed. Fgure 3: Auocorrelaon plo of Markov chan,, M α

12 Fgure 4: Auocorrelaon plo of Markov chan,, M To assess he convergence o he poseror dsrbuon,.e., he saonary of he wo Markov chans,, M and,, M summarzed n Table 2: we have exploed wo sandard dagnosc mehods, whose resuls are he Effecve Sample Sze (ESS), whch gves an esmae of he equvalen number of ndependen eraons ha he chan represens (Rober & Casella, 2). For example, he 3 samples from chan,, M conan 682 ndependen samples (Table 2, second column, second row), beng he nformaon n he remanng 838 samples already conaned n hose 682. The Geweke Tes akes wo nonoverlappng pars (usually he frs. and las.5 proporons) of he Markov chan and compares he means of boh pars, usng a dfference of means es o see f he wo pars, of he chan are from he same dsrbuon (null hypohess) (Rober & Casella, 2). From Table 2, emerges ha he algorhm has converged o he desred arge dsrbuon. Dagnosc Mehod β Commens ESS Passed Geweke Tes p-value The null hypohess of saonary canno be refused and Table 2: Resuls of some dagnosc mehods o assess saonary of he wo Markov chans,, M,, M Fgures 5 and 6 show he Kernel Densy Esmaon (KDE) (connuous lne) and he Emprcal Hsogram (EH) of he esmaed poseror pdfs D ( ) and D ( ), respecvely. The nformaon n he poseror dsrbuon s summarzed by he values of poseror mean and medan (Eqs. (2-22), respecvely), 95% CI, and one-sded 63% lower credbly bound for scale parameers. These are repored n Table 3, whereas Table 4 repors hose of he shape parameer.

13 Densy Densy x Esmaed poseror densy Scale Parameer -3.2 EH KDE Fgure 5: KDE and EH of poseror pdf D ( ) 3 Esmaed poseror densy Shape Parameer 2.5 EH KDE Fgure 6: KDE and EH of poseror pdf D ( ) ˆmean ˆmed.95,.95,.95 CI cnf, csup,.63 c Lower (65.7, 37) 92.7 Table 3: Poseror mean ( ˆmean ), poseror medan ( ˆmed ), 95% CI, and one sded 63% lower credbly bound (,.63 c Lower ) for scale parameer α. ˆmean ˆmed.95,.9 5,.95 CI cnf, csup,.63 c Lower (5.7, 7.37) 5.99 Table 4: Poseror mean ( ˆmean ), poseror medan ( ˆmed ), 95% CI, and one-sded 63% lower credbly bound (,.63 c Lower ) for scale parameer. The Bayesan framework gves a shape parameer value of almos 6 (eher when referrng o poseror medan or poseror mean). Ths means ha he avalable evdence suppors he pror knowledge on.63 β. Wh respec o he scale parameer, he one-sded 63% lower confdence bound =662 of he Web Webayes approach and he-one sded 63% lower credbly bound,.63 c Lower =92.7 esmaed by he

14 MCMC are que far from each oher (almos 3% of dfference). In parcular, he resul of he Webayes mehod s more conservave. 4.2 Based Problem The value of he shape parameer consdered n hs problem s ˆWeb.7,.e., far from ha used by GE n ndusral pracce. The am, n fac, s o propose a comparson of he Webayes and MCMC approaches, when he a pror knowledge s no suppored by he avalable evdence. The applcaon of he Webayes approach when consderng a one-sded 63% lower confdence bound on he scale parameer yelds.63 Web 47 (27) The MCMC bayesan analyss has been performed wh he followng pror dsrbuons: The generalzed mproper Jeffery pror dsrbuon of Equaon 26, wh hyper-parameer a 2 For he shape parameer, we have chosen a dsrbuon cenered on β=.7, wh a probably mass of.8 unformly dsrbued beween [.5,.9] (.e., symmercally on β=.7), a probably mass of. unformly dsrbued n [,.5], whereas he remanng probably mass of. s unformly dsrbued n [.9, 6]. Ths choce s jusfed by he followng consderaons: o To explo he assumed based pror knowledge, we have pu a large poron of probably on he nerval [.5,.9] ha encompasses he value ˆWeb.7. o The pror dsrbuon leaves a relavely small poron of probably n he remanng pars of he nerval [,6]. Ths s conssen wh our pror knowledge: we assume ha values n [,6] are plausble values for parameer β. To draw samples from he poseror dsrbuon (, D ), he ARWMH algorhm descrbed n Secon 3 has been run for M 5 eraons wh varables,, ρ Σ, and nalzed as repored n Table. Then, we have appled: ) a burn-n of,, samples. 2) a hnnng of 5 samples. By so dong, he cardnaly of he orgnal Markov chans has reduced from 4,, o M 8 sample pons. We have also checked he shape of he race plos, and appled he wo dagnosc mehods ESS, and Geweke ess. For brevy, hese are no repored. Fgures 7 and 8 show he KDE (red dash lne) and he EH of D ( ) and D ( ), respecvely, whereas he nformaon n he poseror dsrbuon s summarzed by n Table 5 and Table 6.

15 Densy Densy In hs based case sudy, he esmaons of Webayes and MCMC mehods are very dfferen from each oher. On he conrary, he esmaons of he MCMC approach for he wo dfferen a pror sengs are no very far. Ths seems o sugges ha he MCMC mehod offers more robusness han he Webayes approach, as he Bayesan framework allows adjusng he nal esmaon of β, even f s no conssen wh he gahered evdence. On he conrary, he possbly of adjusng he pror esmaon based on he gahered evdence s no offered by he Webayes mehod. Ths may consue a lmaon for he Webayes mehod, as leads o provde wrong resuls f he value of he shape parameer s no correcly se. On he oher sde, hs dsadvanage s couner-balanced by he fac ha Webayes s smpler and faser han MCMC. However, addonal research work needs o be carred ou boh o nvesgae under whch condons and o whch exen Bayesan framework resuls o be more flexble han Webayes, and o explore he sensvy of he accuracy of he resuls he wo mehods o he daase cardnaly, nformaon conen of he pror knowledge, ec..9 x -3 Esmaed poseror densy Scale Parameer EH KDE Fgure 5: KDE and EH of poseror pdf D ( ).4.2 Esmaed poseror densy Shape Parameer EH KDE Fgure 6: KDE and EH of poseror pdf D ( )

16 ˆmean ˆmed.95,.95,.95 CI cnf, csup,.63 c Lower (65.47, 3883) 23 Table 5: Poseror mean ( ˆmean ), poseror medan ( ˆmed ), 95% CI ( CI.95 ) and he-one sded 63% lower credbly bound ( c,.63 Lower ) for scale parameer ˆmean ˆmed.95,.9 5,.95 CI cnf, csup,.63 c Lower (.69, 5.96) Table 6: Poseror mean ( ˆmean ), poseror medan ( ˆmed ), 95% CI ( CI credbly bound ( c,.63 Lower.95 )for scale parameer ) and he-one sded 63% lower 5 CONCLUSION In hs work, we have consdered he realsc problem n relably analyss of Webull parameer esmaons when he colleced feld daa conan rgh-censored mes only. The Webayes mehod and he Bayesan approach have been appled o a real ndusral case sudy concernng daa of GT forced ouages due o a mechancal falure of a componen. Ths applcaon has shown ha he wo mehods provde he same value for he shape parameer, bu dfferen values for he scale parameers. 6 REFERENCES Abernehy, R. 28. The new Webull handbook. Ffh Edon. Al-Kuub, H.S. & Ibrahm, N.A. 29. Bayes esmaor for exponenal dsrbuon wh exenson of Jeffery pror nformaon. Malaysan Journal of Mahemacal Scences 3(2): AAA. Andreu, C. & Thoms, J. 28. A uoral on adapve MCMC. Sascs and Compung 8 (4): Chrsensen, R., Johnson, W., Branscum, A., Hanson, T. 2. Bayesan Ideas and Daa Analyss: An Inroducon for Scenss and Sascans. CRC Press.

17 Crowder, M. J., Kmber, A. C., Smh, R. L., Sweeng, T. J. (99). Sascal Analyss of Relably Daa. CRC Press. D Mao, F., Compare, M., Maafrr, S., Zo, E. 25. A double-loop Mone Carlo approach for Par Lfe Daa Base reconsrucon and scheduled manenance mprovemen, Proceedngs of he European Safey and Relably Conference, ESREL 24, pp Gonzalez-Gonzalez, D., Canu-Sfuenes, M., Praga-Alejo, R., Flores-Hermosllo, B., Zuñga- Salazar, R. 24. Fuzzy relably analyss wh only censored daa. Engneerng Applcaons of Arfcal Inellgence 32: Rober, C.P. & Casella, G. 24. Mone Carlo sascal mehods. New York: Sprnger. Rober, C.P. & Casella, G. 2. Inroducon o Mone Carlo mehods wh R. New York: Sprnger. Zo, E., 27. An nroducon o he bascs of relably and rsk analyss. Sngapore: World Scenfc Publshng Company.