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Arcle caon nfo: Guo, Wang W, Guo B, Peng R. Manenance Opmzaon for SysemsWh Dependen ompeng Rsks Usng a opula Funcon. Eksploaacja Nezawodnosc Manenance and Relably 03; 5 (): 9 7. hmng Guo Wenbn Wang Bo

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Arcle caon nfo: Guo, Wang W, Guo B, Peng R. Manenance Opmzaon for SysemsWh Dependen ompeng Rsks Usng a opula Funcon. Eksploaacja Nezawodnosc Manenance and Relably 03; 5 (): 9 7. hmng Guo Wenbn Wang Bo Guo Ru Peng Manenance Opmzaon for Sysems Wh Dependen ompeng Rsks Usng a opula Funcon Opymalzacja eksploaacj dla sysemów z zależnym zagrożenam konkurującym przy wykorzysanu funkcj kopuły Ths paper develops a jon copula relably model for sysems subjeced o dependen compeng rsks caused by wo degradaon processes and random shocks. The wo degradaon processes follow gamma processes and he random shocks follow a non-homogeneous Posson process (NHPP). Ther nerdependence relaonshp s modeled by a copula funcon, whch s deermned by a wo-sage mehod based on smulaed daa. s shown ha he proposed model can provde more precse resuls han he model whou consderng he dependen relaonshp. Through he proposed relably model, wo manenance models are suded and compared. s found ha he nspecon cos has sgnfcan effecs on he choosng of manenance polcy. Keywords: dependen compeng rsks, copula funcon, smulaed daa, degradaon, random shocks, manenance opmzaon. W nnejszej pracy opracowano wspólny model nezawodnośc z użycem kopuły dla sysemów poddawanych zależnym zagrożenom konkurującym powodowanym przez dwa procesy degradacj zaburzena losowe. Owe dwa procesy degradacj reprezenują yp procesu gamma, podczas gdy zaburzena losowe są ypem nejednorodnego procesu Possona (non-homogeneous Posson process - NHPP). ch zwązek wzajemnej zależnośc modelowany jes przy użycu funkcj kopuły, kóra jes wyznaczana na podsawe dwueapowej meody oparej o dane symulowane. Wykazano, ż proponowany model może zapewnć bardzej precyzyjne wynk nż model, w kórym ne ujęo zwązku zależnośc. W oparcu o proponowany model nezawodnośc, badane porównywane są dwa modele eksploaacj. Swerdzono, ż kosz przeglądu ma duży wpływ na wybór polyk eksploaacyjnej. Słowa kluczowe: zależne ryzyka konkurujące, funkcja kopuły, dane symulowane, degradacja, zaburzena losowe, opymalzacja eksploaacj.. nroducon ompeng rsks are que common suaons n ndusry for sysems or componens whch can be subjeced o more han one causes of falure a he same me and fal due o one of hem [7,9]. Therefore, s benefcal o consder he compeng rsks for he manenance schedulng. Many sudes rea he compeng rsks as ndependen falure processes. Lehman [7] nvesgaed a class of degradaon-hresholdshock models n whch he falure s caused by he compeng rsks of degradaon and rauma. Bocche e al. [] proposed a model o descrbe he compeng rsks caused by wear degradaon and hermal crackng for he cylnder lners n marne desel engne. Due o he complex feaures of lfeme daa, Jang [3] developed a compeng rsk model nvolvng a geomerc dsrbuon and an exponenal Posson dsrbuon o model bus-moor falure daa. L and Pham [8] presened an nspecon-manenance model for sysems subjeced o wo degradaon processes and random shocks. Zhu e al. [34] presened a manenance model ha maxmzes he un avalably by deermnng he degradaon hreshold level and he me o perform prevenve manenance (PM). Kharoufeh e al. [4] derved he sysem lfeme dsrbuon and he lmng average avalably for a perodcally nspeced sysem, whch s subjeced o degradaon and random shocks modulaed by a homogeneous Posson process. Wang e al. [30] suded he mpac of shocks on he produc and found ha he shocks had a sgnfcan mpac on he produc relably. The assumpon of s-ndependence beween compeng rsks may cause underesmaon or overesmaon of he sysem relably and has subsanal mpacs on manenance opmzaon [3]. Therefore, s essenal o ake accoun of he dependen relaonshp n order o model he relably more accuraely and make more approprae manenance sraegy. Some recen papers have ncorporaed he dependen relaonshp no he relably modelng process. Su and Zhang [6] suded he relably assessmen for GaAs lasers based on compeng rsk model. The resuls show ha he dependence beween he raumac falure and degradaon has a grea nfluence on he accuracy of relably assessmen. onsderng he dependency beween wear falure and shock falure, Jang and o [] developed relably models wh wo classes of shock processes and a lnear degradaon process. The arrval of each shock mpacs boh he sof falure process and he hard falure process. Pan and Balakrshnan [] proposed o use a bvarae Brnbaun- Saunders dsrbuon o descrbe he dependen relaonshp beween he wo gamma degradaon processes and developed an nferenal mehod for he correspondng model parameers. Sngpurwalla [5] proposed a general framework for an apprecaon of compeng rsks and degradaon nvolvng nerdependen sochasc processes under he noon of a hazard poenal. Pan and Zhao [] reaed he Eks p l o a a c j a N e z a w o d n o s c Ma n e n a n c e a n d Relably Vo l.5, No., 03 9 Scence and Technology problem of acceleraed falure wh compeng causes of a degradaon falure mode and mulple raumac falure modes. Abbrng and van den Berg [] suded he dependen compeng rsks models wh a mxed proporonal hazard for each rsk. Wang and o [7] proposed a general modelng and analyss approach for relably predcon based on mulple degradaon measures and llusraed he approach wh mulvarae Normal dsrbuons. There has also been a growng neres n consderng he manenance opmzaon wh dependen compeng rsks n recen years. Kluke and Yang [5] suded he average avalably of mananed sysems subjec o shocks and graceful degradaon wh hdden falures. Huynh e al. [0] developed a dependen compeng rsk model by assumng he arrval rae of shocks as a funcon of he degradaon level, and proved he value of condon monorng o he manenance decson-makng. Laer Huynh e al. [] developed age-based manenance sraeges wh mnmal repars for sysems based on he same compeng rsk model. Wang and Pham [8] suded a mulobjecve opmzaon problem of mperfec prevenve manenance polcy for a sngle-un sysem subjeced o he dependen compeng rsks, by smulaneously maxmzng he sysem asympoc avalably and mnmzng he sysem cos rae. s assumed ha faal shocks wll cause he sysem o fal mmedaely, whereas nonfaal shocks wll ncrease he sysem degradaon level by a ceran cumulave shock amoun. n order o gve a more explc dependen relaonshp, hen [7] used he degradaon level as a varable of he arrval rae funcon of he faal shock, and an nspecon/replacemen polcy s dscussed based on he proposed model. asro [4] developed a dependen relaonshp for wo compeng falure modes n whch he non-mananable falure number affecs he mananable falure rae. The opmal number of PMs and he nerval beween successve PMs are deermned wh he objecve of mnmzng he expeced cos rae. Zequera and Bérenguer [3] suded he mperfec manenance polces wh he consderaon of wo compeng falure modes, where he hazard rae of he mananable falure mode depends on he hazard rae of he non-mananable falure mode. Deloux e al. [9] consdered a sysem wh wo falure mechansms due o an excessve deeroraon level and a shock. The opmal manenance sraegy s suded n an approach whch combnes sascal process conrol and condon-based manenance. Peng e al. [3] presened a prevenve manenance polcy for sysems subjeced o mulple compeng falures where he exernal random shocks conrbue o he nernal degradaon. Prevous researches have manly nvesgaed he dependence relaonshps among degradaon processes by mulvarae normal dsrbuon, and modeled he falure rae wh covaraes ec. Though he sysem relably funcons can be deduced drecly, hese approaches are nsuffcen o cope wh he complexy of he modern sysem n realy [9, 33]. opula s a powerful ool o model he dependence of random varables, and he copula based models allow for flexble specfcaon of he dependence srucure beween compeng random varables [3, 4]. Zhou [33] proposed a bvarae degradaon modelng framework based on gamma processes and copula funcon s used o descrbe he dependence beween performance characerscs. Wang and Pham [9] developed a flexble s-dependen compeng rsk model o descrbe he dependence beween random shocks and he degradaon process by employng me-varyng copulas. Lo and Wlke [0] exended he copula graphc esmaor o model mulple dependen compeng rsks and appled he model o he unemploymen duraon daa from Germany. However, copula funcon has seldom been appled o model he dependence srucure n manenance opmzaon. n pracce, sysems are usually subjeced o compeng rsks nvolvng boh degradaon and shocks, as nvesgaed by many researchers [6, 0,, 5 and 30]. n hs paper, a sysem sufferng dependen compeng rsks caused by wo degradaon processes and random shocks s suded. Wh he dependence srucure modeled by copula funcon, a jon relably funcon s developed based on he smulaed daa and he manenance opmzaon s nvesgaed. The remanng paper s organzed as follows. Secon nvesgaes he sysem falure process and deduces he margnal relably funcon for he sysem sufferng wo degradaon falure processes and random shocks. Secon 3 develops he sysem relably model based on a copula funcon and provdes a parameer esmaon procedure based on smulaed daa. Secon 4 presens wo manenance models based on he jon copula relably funcon. n Secon 5, a numercal example s presened o llusrae he procedure o deermne he jon relably funcon and nvesgae he manenance opmzaon for he wo manenance polces.. Dependen compeng rsks onsder a sysem subjeced o compeng rsks due o wo degradaon processes and random shocks. The wo degradaon processes have a dependen relaonshp wh each oher as each shock causes a sudden ncremen jump o boh degradaon processes smulaneously. The sysem fals f he cumulave deeroraon of any degradaon process exceeds a ceran crcal falure hreshold... Degradaon processes whou random shocks Gamma processes have been exensvely adoped o descrbe he gradual degradaon phenomena e.g. corroson [6], crack growh [5]. Le X ( ), ( =, ) denoe he accumulaed deeroraon for he h degradaon process a me, where he nal sae of he sysem s perfec wh (0) 0 X =. Assume ha { ( ), 0} X, ( =, ) s a saonary gamma process where X( ) X( s) s gamma dsrbued for all 0 s . Whou consderng he nfluences of he shock process, X( ) X( s), 0 s has a gamma probably densy funcon (pdf) wh shape parameer α ( s) 0 and scale parameer β 0 : where Γ( α) α( s) α x ( s) β e x β f x α ( s), β ( ) = { x α s 0}, () Γ( ( )) α = u e u 0 du s he gamma funcon. { x 0} = f x 0 and { x 0} = 0 oherwse. The average deeroraon rae s u = α / β, and s varance s σ = α / β. Though he consan deeroraon rae may be unsuable for he realsc degradaon process, a monoonc ransformaon of he me scale can make he real deeroraon rae consan [3]. Wh he choce of α and β, such a process can be very flexble o model varous deeroraon behavors of he sysem. The sochasc process { X ( ), 0 } s me connuous and monooncally ncreasng, and he sysem fals once X ( ) exceeds a predeermned falure hreshold L. Though he sysem may be sll funconng afer crossng he falure hreshold, canno perform s funcon as requred and s regarded as faled for economcal or secury reasons. The me o falure (TTF) of he h degradaon process can be expressed as TL = nf{ X( ) L}, and s cumulave dsrbuon funcon (cdf) can be obaned as: 0 Eks p l o a a c j a N e z a w o d n o s c Ma n e n a n c e a n d Relably Vo l.5, No., 03 Scence and Technology Γ( α L FTL PTL PX L f x dx, β () ( ) ( ( ) ) ( ) ) = = = α, β =, () Γ ( α) L a a x e d where Γ (, ) =. x The pdf for TTF of he h degradaon process s f F u TL () = α α TL u u () = (ln( ) ( )) e du ( α) ψα, (3) Γ L β Γ ( a) where ψ ( a) = = ln Γ ( a) s called he dgamma funcon. Γ( a) a The relably funcon correspondng o he h degradaon process s ( L RTL FTL, ) () = () = Γ α β. (4) Γ( α).. Shock process Shocks may be generaed nernally whn componens or nroduced exernally from he envronmen ousde. Mos shocks are harmful o he sysem operaon, and can reduce he sysem resdual useful lfe. n hs paper, a cumulave shock model s employed o descrbe he shock process. The probables for he shock damages o occur n dfferen me nervals are assumed o be ndependen. The log-lnear process (LLP) s very flexble and has been wdely used o descrbe he occurrence of random evens, such as he wear of cylnder lner []. Here he shock process s descrbed by he LLP, and he random shocks are assumed o occur n a non-homogeneous Posson process (NHPP) wh nensy funcon λ( ) = re c, r (0, ), c (, + ). (5) Le N( ) denoe he number of shocks unl me, hen he expeced number of shocks unl me, denoed by W( ), s gven by r W EN re ds c e c cs ( ), () = [ ( )] = = c 0. (6) 0 r, c = 0 Furher, he probably dsrbuon of N( ) s ( W( )) P( N( ) n) e n! n W( ) = =. (7) The amoun of damage caused by he k h shock o he h degradaon process s denoed by S and S N k k ( µ, σ ). Furhermore, he accumulaed shock damages o he h degradaon process unl N( ) me s expressed as Z ( ) = S. k = onsder G( l) = P( Sk l) as he cdf for all S. The cdf for he accumulave shock damage o he h degradaon process ncurred by k he shock process s k N() PZ ( ( ) z) = P( Sk z) k= j ( j) W() z jµ = P( N( ) = 0) + G ( z) P( N( ) = j) = e + W W() Φ( ) ( ( )) e, (8) j= j= jσ j! ( ) where G j ( z) s he j-fold convoluon wh self..3. Degradaon processes wh random shocks Secon. nvesgaed he relably of he sysem subjeced o he degradaon process, whou consderng he nfluences nduced by he shock process. n praccal applcaons, he random shocks may exs and have mpacs on he degradaon processes. [9] n hs paper, he random shocks wll nduce a sudden ncremen o he degradaon process. onsderng he dependen relaonshp of degradaon processes and random shocks, he h degradaon process sae Y ( ) ncludes wo pars: he wear caused by he sysem agng and he sudden ncremens nduced by he random shocks. The h degradaon a me can be expressed as Y() = X() + Z(). Denoe he TTF for he h degradaon by T. The relably funcon for he h degradaon process wh random shock damages s gven by R() = PT ( ) = PY ( () L) = PX ( () + Z() L) = P( X () + Z () L N() = k) P( N() = k) k = 0 L ( k ) P( N() 0) P( X() L) P( N() k) P( X() z L) dg () z k = 0 k L W( ) Γ( a, bl) ( W( )) W( ) Γ( a, b( L z ( k ) = = + = + = e ( ) + e ( Γ( a ) k! k = 0 Γ( a ) )) ) dg ( z ). The pdf of TTF for he h degradaon process wh random shocks can be expressed as 3. Sysem relably analyss dr ( ) f ( ) =. (0) d The sysem falure occurs f any of he degradaon processes Y ( ) reaches he falure hreshold L. Therefore, he sysem relably a me s R( ) = P( Y ( ) L, Y ( ) L ) = P( X ( ) + Z ( ) L, X ( ) + Z ( ) L ) () f he wo degradaon processes are ndependen, he sysem relably funcon can be wren as R( ) = R ( ) R ( ). () (9) However, Eq. () s unable o provde he accurae sysem relably esmaon for our case, as here s dependency beween he wo degradaon processes due o he random shocks. s dffcul o calculae R( ) by Eq. () drecly, so we need o fnd anoher way o predc he relably of he sysem subjec o dependen compeng falures. Eks p l o a a c j a N e z a w o d n o s c Ma n e n a n c e a n d Relably Vo l.5, No., 03 Scence and Technology 3.. A opula approach A opula funcon s a powerful ool o model he dependence srucure of he compeng falure processes. One advanage of he copula funcon s ha he jon relably funcon can be modeled drecly hrough he unvarae margnal relably funcons of he ndvdual falure processes, (.e. F( ), F ( ) ) and he copula has no consrans on he unvarae margnal dsrbuon. The cdf of TTF for he wo degradaon processes can be expressed as F( ) = R( ) ( =, ), and he jon cdf of T and T s denoed by H(, ). Accordng o Sklar s heorem, here exss a unque copula such ha PT (, T ) = H(, ) = F ( ( ), F( ), Θ ), (3) where Θ s he parameer vecor of he copula funcon. Meanwhle, he jon relably funcon of he sysem wh and can be expressed as H(, ) = P( T , T ). (4) Because R ( ) and R ( ) are decreasng funcons, he sysem relably a me ( = = ) can be expressed wh he survval copula funcon as [8, 4] R () = H (, ) = = = R( ) + R( ) + F ( ( ), F( ), Θ) = = = R() + R() + F ( ( ), F(), Θ). (5) There s anoher approach o consruc he sysem relably wh a copula funcon, as shown n [9]. The jon relably funcon can be drecly modeled by a copula funcon and can be wren as R () = H(, ) = = = ( R( ), R( ), Θ) = = = ( R (), R (), Θ ). (6) The resuls of Eq. (5) and (6) may be dfferen, and we wll compare he wo approaches n Secon Parameer esmaon Assume ha he parameers of he margnal relably funcons for he degradaon processes are already gven. n order o predc he sysem relably, we need o esmae he copula parameers based on he known margnal dsrbuons. The pdf of he jon dsrbuon H(, ) can be denoed as f( ) as = =. Furher, we can oban f( ) from Eq. (5) as f() = f(, ) = = = ( R( ) + R( ) + F ( ( ), F( ), Θ )) = = (7) = f( ) + f( ) cf ( ( ), F ( ), Θ ) f( ) f( ) = = = f() + f() cf ( ( ), F( ), Θ) f( ) f( ), where cf ( ( ), F( ), Θ) = F ( ( ), F( ), Θ) s he F( ) F( ) copula densy funcon. Smlarly, f( ) for Eq. (6) s gven as f() = f(, ) = = = c( R( ), R( ), Θ) f( ) f( ) = =, (8) where cr ( ( ), R( ), Θ) = R ( ( ), R( ), Θ). R( ) R( ) n hs paper, he smulaed daa are used o esmae he parameers of he copula funcon and valdae he effecveness of he copula mehod. The proposed mehod can be dvded no wo sages. n he frs sage, we need o smulae he compeng falure processes o oban he sysem margnal relably sample wh he underlyng dependen relaonshp beween he degradaon processes and he shock process a dscree mes. The procedures are descrbed as follows: ompue he degradaon ncremen X ( ) ( =, ) of each degradaon process a = m ( m =,, ), where s he me sep for he degradaon process smulaon. Generae he shock arrval mes followng NHPP {,,..., }, ( n ) and he correspondng shock damages o each degradaon process { s, s,..., s n}. ompue he accumulaed shock damage a me as n Z ( ) = s k k = for each degradaon process. ompue he sysem relably R ˆ( ) = / { X ( ) + Z ( ) L } Num,where s an ndcaor j = Num j j funcon. =, f X( ) + Z( ) L and = 0 oherwse. Num s he oal number of smulaons. n he second sage, he Maxmum lkelhood esmaor (MLE) s used o esmae he copula funcon parameers based on he smulaed margnal relably sample. Below are he procedures: onsder N smulaed resuls for he degradaon processes, F( ), F ( ). Wh Eq. (7) and (8), whch are denoed by { j j } j=, N he log-lkelhood funcon for he bvarae copula can be expressed respecvely as N ln L( Θ) = ln c( F( j), F( j), Θ), (9) j= N ln L( Θ) = ln c( R( j), R( j), Θ). (0) j= Usng MLE, he copula parameers can be esmaed as Θ= ArgMax{ln L( Θ)}. () 4. Manenance models Ths secon presens wo knds of manenance polces based on he jon copula relably funcon for a non-reparable sysem. The frs polcy s a perodc nspecon/replacemen polcy and he decson varable for manenance decson maker s he nspecon nerval. The second polcy s an age-based manenance polcy and he decson varable s he replacemen age o be specfed. For boh manenance polces, he objecve s o mnmze he average manenance cos rae n long run. n Eks p l o a a c j a N e z a w o d n o s c Ma n e n a n c e a n d Relably Vo l.5, No., 03 Scence and Technology 4.. Perodc nspecon/replacemen polcy Due o cos reasons and oher praccal ssues, he sysem s nspeced a a perodc nerval τ. The nspecon s perfec and nsananeous wh a cos ncurred. When any of he wo degradaon processes wh he underlyng shock damages exceeds he pre-se hreshold, he sysem s deemed as faled hough sll runs unl he falure s denfed a he nex nspecon. n case when a sysem falure s denfed a an nspecon, s replaced nsanly wh a new one and he replacemen me s neglgble. The replacemen can be seen as a renewal. Denoe he accumulave manenance cos unl me as ( ). Accordng o he renewal heory, we have ( ) E[ R] lm =, () E[ TR] where E[ R ] s he expeced oal manenance cos n a renewal cycle, E[ TR ] s he expeced lengh of a renewal cycle. The manenance coss n a renewal cycle are composed of nspecon cos, replacemen cos and he delay me cos durng sysem falure perod. The delay me cos s ncurred by he loss of sysem performance durng he sysem falure perod. The expeced oal cos n a renewal cycle can be expressed as where ER [ ] = EN [ ] + DE[ ξ ] + R, (3) s he cos assocaed wh each nspecon, D s he delay me cos rae for he sysem falure duraon, R s he replacemen c

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