Optimal replacement policy for safety-related multi-component multi-state. systems

Transcription

1 Opmal replacemen polcy for safey-relaed mul-componen mul-sae sysems Mng Xu, Tao Chen 2,*, Xanhu Yang Deparmen of Auomaon, Tsnghua Unversy, Bejng 84, Chna 2 Dvson of Cvl, Chemcal and Envronmenal Engneerng, Unversy of Surrey, Guldford GU2 7XH, UK * Correspondng auhor. Tel.: ; Fax: Emal:.chen@surrey.ac.uk Absrac Ths paper nvesgaes replacemen schedulng for non-reparable safey-relaed sysems (SRS) wh mulple componens and saes. The am s o deermne he cos-mnmzng me for replacng SRS whle meeng he requred safey. Tradonally, such schedulng decsons are made whou consderng he neracon beween he SRS and he producon sysem under proecon, he neracon beng essenal o formulae he expeced cos o be mnmzed. In hs paper, he SRS s represened by a non-homogeneous connuous me Markov model, and s sae dsrbuon s evaluaed wh he ad of he unversal generang funcon. Moreover, a srucure funcon of SRS wh recursve propery s developed o evaluae he sae dsrbuon effcenly. These mehods form he bass o derve an explc expresson of he expeced sysem cos per un me, and o deermne he opmal me o replace he SRS. The proposed mehodology s demonsraed hrough an llusrave example. Keywords: Safey-relaed sysem; Schedule pon; Safey negry level; Unversal generang funcon; Srucure funcon. Inroducon Engneered sysems suffer from gradual performance deeroraon and unexpeced shock damage durng operaon. Therefore, manenance acves are essenal o preven sysem deeroraon and falure, or f complee prevenon s no possble, o mnmze he adverse mpac of her occurrences [, 2]. If he sysem s non-reparable, wll be replaced upon falure, whle mos reparable sysems can be resored o a funconng sae hrough manenance. Replacemen can be consdered as perfec manenance [2]. Generally, manenance acves can be classfed no wo caegores: correcve (un-planed), and prevenve (planned) [3, 4]. Correcve manenance refers o any acon ha resores he faled sysem o a workng sae. In conras, prevenve manenance refers o an acon carred ou when he sysem s sll funconng, wh he goal o resore he sysem o a specfed beer condon. In hs paper, we examne he replacemen schedule for non-reparable safey-relaed sysems (SRS) wh mulple componens and saes. SRS s wdely used n ndusry o reduce or preven rsk of he producon sysem (PS) beng proeced [5, 6]. Inernaonal sandards lke IEC 658 [7, 8] provde a means of ensurng ha safey s effecvely acheved by usng SRS. In order o comply wh hese sandards, SRS s usually quanfed hrough he safey negry level (SIL). For low demand mode of operaon, SILs are defned n erms of average probably of dangerous falure on demand (P avg ) as shown n Table. Ths paper seeks he opmal schedule pon when he SRS should be replaced, wh he objecve of mnmzng expeced sysem cos per un me under requred safey (or equvalenly, SIL).

2 Table Safey negry levels for low demand mode accordng o he IEC 658 sandard. SIL P avg 4-5 ~ ~ ~ -2-2 ~ - Numerous sudes have been repored n he leraure o mprove he manenance sraeges for PS; see [9-2] for examples. However, he manenance polcy for SRS, especally hose wh mulple componens and saes, has been under-explored. Mul-componen and mul-sae sysems are more complex han he radonal sysems wh a sngle componen or bnary saes, and hus hey requre specal echnques for modelng and analyss []. Curren sudes on he manenance of mul-componen and mul-sae sysems are prmarly focused on PS and may no be drecly exendable o SRS. For example, Hseh [3] examned a mul-sae deerorang PS wh sandby redundancy n whch componens may deerorae due o random falures. To acheve he requred sysem avalably wh mnmal lfecycle cos, Levn & Lsnansk [4] formulaed a jon redundancy and replacemen schedule opmzaon problem n whch he elemen verson, redundancy level, and replacemen nerval were opmzed smulaneously. The same auhors were he frs o consder elemen replacemen n mul-componen mul-sae sysems [5], and hey also addressed he correspondng mperfec repar schedulng problem [6]. Tan & Raghavan [9] developed a predcve manenance model for mul-componen and mul-sae sysems. Lu & Huang [] dscussed he opmal replacemen polcy for mul-sae sysems under mperfec manenance. However, hese aforemenoned model canno be drecly appled o SRS. When dealng wh SRS, one should consder no only he avalably, requred safey and manenance cos for SRS, bu also he neracon beween he SRS and PS beng proeced. Usually, some hdden modes of falure n SRS may be exposed only by he occurrence of a PS rsk. Conversely, SRS falure modes may affec he operaon of a PS. If hs neracon s no accouned for, subopmal soluons may maeralze. Neverheless, ncorporang he neracon beween SRS and PS sgnfcanly complcaes he desgn and opmzaon of replacemen polcy. For example, he SRS should be replaced mmedaely when some falure modes n he SRS lead o a shudown of he PS or he rsk of he PS exposes he hdden falure modes of he SRS. Oherwse, he SRS wll be replaced a a scheduled pon n order o acheve he requred safey n s lfecycle. Thus, f he neracon beween SRS and PS s no consdered, hs mporan nformaon s gnored when desgnng replacemen polcy. The major conrbuon of hs paper s o fnd he opmal replacemen polcy of mul-componen and mul-sae SRS wh consderaon of neracon beween SRS and PS. The SRS wll be modeled by a non-homogeneous connuous me Markov model (NHCTMM). We formulae a framework o faclae solvng hs model based on he mehod of unversal generang funcon (UGF). Especally, he srucure funcon of SRS wh recursve propery s developed, and sgnfcanly reduces he compuaonal burden. Furhermore, we derve an explc expresson of he expeced sysem cos per un me, and deermne he opmal schedule pon o replace he SRS usng mahemacal opmzaon echnque. The remander of he paper s organzed as follows. In secon 2, he SRS model s presened and he srucure funcon of SRS s developed o ad effcen compuaon. Secon 3 descrbes a replacemen model of SRS by jonly consderng SRS and PS n order o fnd he cos-mnmzng schedule pon. The proposed model s llusraed hrough he sudy of a hgh-negry pressure proecon sysem n Secon 4. Fnally, Secon 5 concludes he paper. 2

3 Abbrevaons and nomenclaure A schedule pon o replace SRS a sysem dle me n case ( =, 2, 3, 4 ) C I () cos of sysem dleness for me C p penaly cos of C p penaly cos of C R cos of repar he SRS DG degraded sae E[T(A)] expeced sysem operang cycle me E[SC(A)] expeced sysem cos per operang cycle falure-dangerous sae FP full performance sae Falure-safe sae f x () probably densy of sae x (x G S ) of SRS a nsan G C se of saes of componen: G C ={OP,, } G S se of saes of SRS: G S ={FP, DG,, } g S () sae of subsysem combned by componens {, 2,, }: g s () G S HCTMM homogeneous connuous me Markov model I () he collecon of ses whch are ncluded n subs () and belong o MCS I () he collecon of ses whch are ncluded n subs () and belong o MCS NHCTMM non-homogeneous connuous me Markov model MCS collecon of he mnmal cu ses of SRS for MCS collecon of he mnmal cu ses of SRS for OP operaon sae P avg average probably of dangerous falure on demand PS producon sysem P () sae dsrbuon of componen a me P x () probably of sae x (x G S ) of SRS a nsan p SILx upper bound under SIL x p () probably of s () = x, (x G C ) a me nsan x p k () probably of SRS s ndvdual sae k (k=,2,, M) a nsan UGF unversal generang funcon V a se equals o {,, 2, 3} R() R()= gs ( ), S ( x ), and R()VV SC (A) cos n case SILx safey negry level x SRS safey-relaed sysem s () sae of componen a me nsan subs () he combnaon of componens {, 2,, } whch are n sae subs () he combnaon of componens {, 2,, } whch are n sae S (x ) he sysem sae when {s () = OP,, s - () = OP, s () = x, s + () = OP,, s m () = OP} T F occurrng me of falure (SRS and PS) x realzaon of s (): x G C sysem ranson nensy marx cos per un me dues o sysem dleness SRS sysem srucure funcon of SRS srucure funcon for componens/subsysems n SRS S( x) mappng beween S (x ) and he workng sae of SRS (A) expeced sysem cos per un me 3

4 2. Model developmen Ths sudy reas SRS as a mul-componen and mul-sae sysem. In addon, SRS and PS wll be jonly consdered n he modelng and opmzaon framework so as o accoun for her neracon. Ths secon presens he assumpons for he sysem, he componen model, he model for he SRS by combnng mulple componens, and fnally he compuaonal procedure by explong he recursve propery of he sysem and he mehod of UGF. The abbrevaons and symbols are lsed as follows. 2. Sysem assumpons The model of sysem s based on he followng assumpons: ) The enre sysem s composed of PS (for producon) and SRS (for reducng or prevenng PS rsks) [5, 7]. 2) SRS s composed of mulple componens and each componen can experence wo caegores of falures: falure-safe () and falure-dangerous () [8]. In he absence of falure, he componen s n operaon (OP) sae. SRS s non-reparable when falure occurs, and a replacemen schedule wll ake place f needed. 3) Tme o falure ( or ) s dsrbued exponenally, whch mples ha he SRS s a Markov process [9]. In addon, sysem componens age over me, and hus he sae ranson nensy vares wh me. 4) The sae of any composon of componens s unambguously defned by he saes of hese componens and he naure of he neracon of he componens n SRS [8]. The occurrence of and evens n one or more componens may resul n he occurrence of and evens n SRS. 5) If a rsk of PS arses, he specfed safey funcon of SRS s requred o ac. A hs pon, f no even has occurred n SRS, he requred funcon of SRS can be mplemened and he PS can be aken o he safe sae. Oherwse f even occurred before he occurrence of PS rsk, SRS canno funcon as requred and a serous accden wll occur. Furhermore, we assume ha s unrecognzable by he operaor unl a serous accden occurs. 6) The PS wll be aken o a safe sae, f occurred n SRS. In addon, s always recognzable because resuls n shu-down of he plan. 2.2 Model of SRS componens Accordng o assumpon 2, each SRS componen has hree saes: OP, and. The sae of componen can be represened by he se GC 4 OP,, The sae s () of componen a any me nsan s a random varable akng a value from G C : s ()G C. The sae dsrbuon of componen a me can be represened by he se P ( ) p ( ), p ( ), p ( ) OP where p () represens he probably ha s () = x, (x G C ). x When descrbng he degradaon process of ndvdual componens, Lsnansk [2] employed he homogeneous connuous me Markov model (HCTMM). Bascally, assumes he me of ranson beween any wo saes follows a negave exponenal dsrbuon. The hypohess ha he ranson nensy o he nex sae only depends on he curren sae s applcable o componens havng no age effec. In fac, s more realsc o consder he case ha a componen s deeroraon process s no only relaed o he curren sae, bu also o he age of he componen [, 2]. Takng hs concep no accoun, he NHCTMM s ulzed n hs paper

5 o derve he sochasc behavor of ndvdual agng componens hrough consderng he age-relaed ncreasng sae ranson nensy [2, 22]. For a non-reparable agng componen, he sae-space dagram of an agng componen s llusraed n Fg., where () s he nensy of ranson from sae OP o sae, and () s he nensy of ranson from sae OP o sae. () () OP p () p () p () OP Fg. Sae-space dagram of componen The probably of componen n each sae can be expressed as: pop ( ) exp ( ) ( ) d () (2) p ( ) ( )exp ( ) ( ) s s ds d p ( ) p ( ) p ( ) (3) OP 2.3 Sae dsrbuon of SRS accordng o a Markov process () M,2 () M, () M M, K, K ( ) M, K() () M, K M, K() () M, () K,2 () K, () M, M M M- K () KK, K- () 2 () K,2 2, p () M p () DG M p () K () 2 pk p () p () FP Fg. 2 Sae-space dagram of SRS The saes of all m componens composng SRS a me are represened by he sae vecor 2 s ( ), s ( ),, s ( ). Suppose ha he SRS has M saes; s sae-space dagram s shown n Fg. 2 and can be m descrbed as a NHCTMM. Accordng o he srucure of SRS, he M saes can be classfed no four sub-saes: ) full performance sae (FP); 2) degraded sae (DG); 3) ; and 4). Thus, he se of saes of SRS can be represened by GS { FP, DG,, } Refer o he llusraon n Fg. 2, sae M ( M 3 ) s consdered as FP; he combnaon of saes M- o K 5

6 ( M K, K 2 ) s consdered as DG; he combnaon of saes K- o 2 s consdered as and sae s consdered as. As s always recognzed and he replacemen of SRS wll ake place subsequenly, no ranson exss from. The probably of he saes of a SRS a me can be represened by he probably of ndvdual saes pk () (k=, 2,, M): P ( ) p ( ) (4) FP M M P ( ) p ( ) (5) DG kk K k 2 k P ( ) p ( ) (6) k P ( ) p ( ) (7) Alhough NHCTMM s concepually sraghforward, s soluon nvolves complex operaons [23]. Solvng a NHCTMM n an effcen manner s ousde he scope of hs paper, and he formulaons gven n [24] are drecly used here o oban he dynamc sae probably of SRS. From Fg. 2, all he saes of ranson nensy can be summarzed n he marx M, M ( ) M, M 2 ( ) M,2( ) M,( ) M, M 2( ) M,2 ( ) M, ( ) 2,( ) (8) Furhermore, defne an auxlary funcon as follows 2 j (, 2, ) j, ( ) Then, he probably ha he SRS n sae M a me s: The probably ha he SRS n sae M- a me s: G j d (9) M p ( ) exp G(,, M) () M M M, M p ( ) p ( ) ( )exp G(,, M ) d () The probably ha SRS n sae M-2 a me can be consdered as he sum of probables of exsence of wo roues: ) he roue from M o M-2 drecly; 2) he roue from M o M- and hen from M- o M-2: p ( ) p ( ) p ( ) (2) 2 M 2 M 2 M 2 M 2( ) M ( ) M, M 2( )exp (,, 2) p p G M d (3) 2 2 M 2( ) M ( ) M, M ( )exp (, 2, ) M, M 2( 2)exp ( 2,, 2) 2 p p G M G M d d (4) In addon, he probably ha he sysem n sae (=, 2,,M-3) s he sum of all roues from M o, whch can be obaned n a smlar way [23]. The maxmum number of he roues of SRS s no more han 2 M+ -2-M, whch grows exponenally wh he ncrease of number of saes. When SRS becomes more and more complex, hs mehod s compuaonally neffcen or even nfeasble. Therefore, a more effcen mehod, based on UGF, s nroduced o deal wh hs problem. 2.4 The dsrbuon of saes of SRS accordng o UGF UGF s an essenal ool o oban he sae dsrbuon of he enre mul-sae sysem. Ths mehod s based 6

7 on he z-ransform and was frs proposed by Ushakov [24, 25]. UGF approach s recursve so as o reduce he problem complexy and compuaonal nensy by modularzng a sysem no s subsysems. Hence, hs mehod s adoped o oban he sae dsrbuon of SRS n hs paper. Noe ha he saes of all m componens whn a SRS a me are s ( ), s ( ),, s ( ) and hese saes 7 2 unambguously deermne he sae of SRS (assumpon 4). Therefore, he relaonshp beween he componen sae vecor and he SRS sae varable S can be expressed by he followng deermnsc funcon (called he sysem srucure funcon): 2 S s ( ), s ( ),, s ( ) (5) SRS In order o oban he sae dsrbuon of he SRS one can represen he sae dsrbuon of componen as: x OP x OP x OP,, m u ( z, ) p ( ) z p ( ) z p ( ) z p ( ) z (6) where Eq.(6) s called he z-ransform of dscree random varable of componen [26]. To formulae he unversal generang funcon of SRS, one needs o apply he composon operaor, : U ( z, ) u ( z, ),, u ( z, ) RSR RSR m x m x ( ),, x ( ) m x OP,, xm OP,, p z p z m j SRS px () z j x OP,, xm OP,, j P ( ) z P ( ) z P ( ) z P ( ) z FP DG FP DG Clearly, he oal number of combnaons of saes of a SRS wh m componens s 3 m [26]. For SRS wh a large number of componens, hs mehod requres an enormous number of evaluaons of he srucure funcon value. Forunaely, he srucure funcon of SRS can be defned recursvely and he saes of some subsysems correspondng o SRS can be obaned. Ths allows o acheve consderable reducon of compuaon. Accordng o Fg. 2, he saes of a SRS nclude FP, DG, and. Le S ( x ) xm m : (7) m G C G S be he mappng beween S (x ) and he workng sae of SRS. S (x ) = {s ()= OP,, s - () = OP, s () = x, s + () = OP,, s m () = OP} denoes all he componens are workng n OP sae excep componen. Based on assumpons 5 and 6, S ( x ) can be formulaed as FP, for x OP DG, for x or x and MCS, MCS S( x), for x, MCS, for x, MCS where MCS and MCS are he collecon of he mnmal cu ses (MCSs) of SRS for and, respecvely. MCS has been exensvely suded n he leraure [27-29], and a sysem s MCSs can be obaned by many ools such as RAM commander, CAFTA and BlockSm [3]. For he same SRS, and saes may have dfferen MCSs. We now dscuss he srucure funcon of SRS recursvely. When consderng he seres-parallel sysem srucure funcon, Levn e al [3] used a conservave approach o oban a smplfed srucure funcon wh he followng assumpons: ) Any elemen n a subsysem becomng wll always lead he relaed subsysem (8)

8 no an sae; 2) If he subsysem s composed of wo componens and s conneced n a parallel way, he subsysem wll be n an operaonal sae f eher componen s n he operaonal sae; 3) If he subsysem s composed of wo componens and s conneced n seres, he subsysem wll be n an operaonal sae only f boh of he wo componens are n operaonal sae; 4) In he remanng cases, he subsysem s n a sae. However, hs s an approxmae mehod as he probably of sae obaned by hs mehod may be hgher han he acual value when redundancy exss n he sysem. For hs reason, he above assumpons are no used n hs paper. Table 2 Srucure funcon Componen S ( x ) Subsysem g S (-) FP DG FP FP b DG b b a,b DG DG b DG/ / d c a b / e c a a,b a a a We encode all he m componens of SRS by negers from o m sequenally. Le g S (), m be he workng sae of subsysem {, 2,, } and g S () G S. Based on he assumpons 5 and 6, he srucure funcon s shown n Table 2. The sae of subsysem {, 2,, } can be obaned by combnng he subsysem {, 2,, -} and componen. Ths can be consdered n fve suaons. Suaon : If he subsysem {, 2,, -} s n sae (g S (-) = ) or componen makes he SRS n sae ( S x ( ) ), hen he subsysem {, 2,, } s n sae (g S () = ). Ths suaon s based on he assumpon ha sae s always recognzable (assumpon 6). The saes wh a superscrp a n Table 2 belong o hs suaon. Suaon 2: If any of he subsysem {, 2,, -} (or componen ) makes he SRS n FP sae ( S x ( ) FP ), hen he subsysem {, 2,, } wll be n he same sae as componen (or he subsysem {, 2,, -}). Ths s based on he fac ha he subsysem or componen n perfec condon before combnaon wll no affec he workng sae of he combned subsysem. The saes wh a superscrp b n Table 2 belong o hs suaon. Suaon 3: If he subsysem {, 2,, -} s n DG or sae (g S (-) = DG or ) and componen makes he SRS n sae ( S x ( ) ), hen he subsysem {, 2,, } s n sae (g S () = ). Clearly, snce componen has already made he SRS, s combnaon wh he subsysem {, 2,, -} can only worsen he suaon no maer wheher he subsysem s n DG or sae. Thus, FP and DG saes can be excluded for he new subsysem {, 2,, }. In addon, Eq. (8) ndcaes ha when componen s, he combned subsysem {, 2,, } canno be n sae, leavng he only possbly of. The saes wh a superscrp c n Table 2 belong o hs suaon. Suaon 4: If he subsysem {, 2,, -} s n DG sae (g S (-) = DG) and componen makes he SRS n DG sae ( S x ( ) DG ), hen he combned subsysem {, 2,, } s n DG, or sae (g S () = DG, or ), and he exac sae depends on he srucure of SRS. The saes wh a superscrp d n Table 2 belong o 8

9 hs suaon. Suaon 5: If he subsysem {, 2,, -} s n sae (g S (-) = ) and componen makes he SRS n DG sae ( S x ( ) DG ), hen he subsysem {, 2,, } s n or sae (g S () = or ), and he exac sae also depends on he srucure of SRS. The saes wh a superscrp e n Table 2 belong o hs suaon. From he analyss above, he workng sae of SRS canno be deermned unquely whou he srucure nformaon of SRS for a specfc applcaon (suaons 4 and 5). Thus, for deermnng he srucure funcon unquely, more nformaon abou he SRS s needed. In he numercal realzaon of he composon operaors, we encode he saes FP, DG, and by negers,, 2 and 3, respecvely. Table 2 can be rewren as Table 3. Table 3 Srucure funcon wh saes encoded by negers Componen S ( x ) Subsysem g S (-) (FP) (DG) 2 () 3 () (FP) 2 3 (DG) / 2/ () 2 2/ () Gven se V = {,, 2, 3}, hus S ( x ), g S (-) V. Le R() be he ordered par g ( ), S ( x ), S and hus R()VV. Then, based on Table 3 and srucure nformaon of SRS he srucure funcon can be obaned by g ( ), S ( x ) S S g () g ( ) g ( ), S ( x ), m S S I ( ) x x MCS, x subs ( ) I ( ) x x MCS, x subs ( ) max gs ( ), S ( x ), for R( ), and R( ) 2,, for R( ),, I ( ), I ( ) 2, for R( ), or R( ) 2,, I ( ), I ( ) 3, for R( ), or R( ) 2,, I ( ) where subs () and subs () denoe he combned componens {, 2,, } beng and, respecvely. I () denoes he collecon of ses whch are ncluded n subs () and belong o MCS, and I () s he collecon of ses whch are nclude n subs () and belong o MCS. For example, a 2oo3 sysem wh componens, 2 and 3 has, {,2},{2,3},{,3} MCS MCS. When we consder subsysem {, 2}, f componens and 2 are n sae, hen subs (2) = {, 2}, subs (2) =, I (2) ={,2} and I =. In Eq. (9), he frs row gs ( ), S ( x ) max gs ( ), S ( x ) (9) corresponds o he saes enclosed by he doed lne n Table 3. Also, he equaon gs ( ), S ( x ) =, 2 or 3 dsngushes he 9

10 saes n bold fon n Table 3 accordng o he srucure of SRS (he mnmal cu ses of SRS for and ). The recursve algorhm can be derved by Eqs. (8)(9). Furhermore, gven he sae dsrbuon of componen accordng o he z-ransform: S ( OP) S ( ) S ( ) (, ) OP ( ) ( ) ( ) u z p z p z p z, (2) he unversal generang funcon of SRS can be obaned recursvely as U ( z, ) u ( z, ),, u ( z, ) K 2 3 FP DG..., u ( z, ), u2( z, ), u3( z, ),..., uk ( z, ) P ( ) z P ( ) z P ( ) z P ( ) z (2) Fnally, he probably densy funcon of each sae of SRS s dpx () fx ( ), x FP, DG,, (22) d 3. Manenance model and polcy 3. Sysem cos In lgh of SRS degradaon and falure, a schedule pon A s proposed wh he objecve of mnmzng expeced sysem cos per un me. Sysem cos consdered n hs paper comprses cos due o sysem dleness, replacemen of he SRS, and penaly of and. Sysem dleness arses when he SRS needs o be replaced or PS o be resumed. Le CI () denoe he cos of sysem dleness ha akes a lnear form n dle me : CI ( ),,. CR denoes he cos due o replacemen of SRS. The penaly coss C and Cp are he p coss ncurred wh and, respecvely. Usually Cp Cp. 3.2 Manenance polcy The replacemen of SRS wll ake place a he schedule pon A f he me o ceran falures T F A or a he me T F f T F < A. The me beween wo replacemens of SRS s consdered as an operang cycle. Accordng o he falure effec and neracon beween SRS and PS, he falures ypes of he enre sysem can be classfed no four caegores as shown n Fg. 3. Case : occurs a ceran me T F ( A). Replacemen of SRS wll ake place a T F wh duraon a, and he cos of hs case s SC( A) CI ( a) Cp CR. Case 2: A dangerous even of PS occurs afer a T F ( A). Replacemen of SRS wll ake place a T F wh duraon a 2, and he cos of hs case s SC2( A) CI ( a2) Cp CR. Case 3: Case or case 2 does no happen beween ( A). Replacemen of SRS wll ake place a he schedule pon A wh duraon a 3, and he cos of hs case s SC3( A) CI( a3) CR. Case 4: A dangerous even of PS occurs a T ( A) and he SRS can reduce he rsk appropraely. As no F and occur, replacemen of SRS s no needed. Afer sysem resar, he SRS wll connue o work unl he occurrence of one of he four cases (, 2, 3 or 4). However, case 4 s no ndependen, and has been ncluded n case, case 2 and case 3. Neverheless, case 4 s needed n cos calculaon. Assume ha he duraon of

11 resarng he enre sysem s a 4 and he cos of hs case s SC4( A) CI( a4) CR. Usually, he sysem dle me sasfes a 2 >a >a 3 and a >a 4 accordng o he ypes of falure evens. A case case 2 E E a E CH a 2 case 3 case 4 E CH a 4 a 3 Fg. 3. Four cases of sysem falures ypes Le T(A) be he sysem operang cycle me. The expeced sysem operang cycle me E[T(A)] can be obaned as A A A E T( A) ( a ) f ( ) d ( a ) f ( ) d f ( ) d 2 CH 3 A A A ( A a ) f ( ) d f ( ) d f ( ) d CH (23) As shown n Eq. (23), f SRS s an deal sysem wh no falures ( and ) n s lfe cycle, hen replacemen wll ake place a he schedule pon A. The expeced sysem operang cycle me can be smplfed o A+a 3. In addon, he expeced sysem cos per operang cycle s A A A E SC( A) SC ( A) f ( ) d SC ( A) f ( ) df ( ) d 2 CH 3 4 A A A CH d SC ( A) f ( ) d f ( ) d f ( ) d SC ( A) f ( ) d f ( ) d f ( ) d A A CH A A A CH f ( ) d f ( ) d f ( ) d A f ( ) f ( ) f ( ) d CH (24) Smlarly, f no falure ( or ) occurs n SRS s lfe cycle and replacemen akes place a he schedule pon A A, hen he expeced sysem cos per operang cycle can be smplfed o SC3( A) SC4( A) fch ( x) dx, whch only depends on he me o replacemen schedule A and he rsk densy f CH (x) of PS. Therefore, he expeced sysem cos per un me s E SC( A) ( A) (25) E T( A) Furhermore, he SRS s requred o sasfy a ceran SIL x, and p SILx s he upper bound under SIL x (x=, 2, 3, 4)

12 and gven n Table. Hence he P avg should no be greaer han he upper bound p SILx. The objecve of he model s o mnmze he expeced sysem cos per un me under he consran of requred safey E T ( A) mn ( A) (26) x A f () ddx SILx (27) ET ( A) P ( A) p, x,2,3,4 avg Ths opmzaon problem can be solved by a sequenal quadrac programmng avalable n many compuaon sofware packages. In hs paper, he Malab Opmzaon Toolbox s used o produce he resuls n he followng example. The major compuaonal cos of he proposed model s o oban he enre SRS saes dsrbuon (FP, DG, and ) by usng UGF. However, he sysem srucure of a general SRS canno smply be derved from he combnaon of subsysems and componens n some cases. To solve hs problem, MCSs are used o dsngush he saes of SRS. Thus, he compuaonal cos manly depends on he complexy of solvng he MCSs of SRS, a problem ha has been exensvely suded n he leraure [27-29]. Even n 999, a sysem wh 82 componens, and as a resul 272 MCSs, can be solved whn hree mnues [32]. The sae-of-he-ar commercal sofware ools, such as RAM commander, CAFTA and BlockSm [3] are exremely powerful and capable of handlng much larger problems. Furhermore, f he sysem s so complex as o exceed he capacy of hese ools, he mehod of sysem modularzaon and approxmaon can sll be used [33]. 4 An llusrave example PS PS2 Pressure vessel Safey valve(sv) Fg. 4 Hgh negry pressure proecon sysem Consder a hgh negry pressure proecon sysem ha composes of one safey valve (SV) and wo pressure sensors PS and PS2 as shown n Fg. 4 [5]. The gas flows o a pressure vessel (PV) hrough SV. If one or boh pressure sensors ssue a sgnal ha he pressure s oo hgh, he SV wll swch off. Each componen (PS, PS2 or PV) can experence and, and f any of he componens suffer, he SV wll be closed and he enre sysem wll shudown. For llusraon purpose, he age-relaed ranson nenses as a funcon of me are gven n Table 4. The PS s me o rsk s assumed o be an exponenally dsrbued random varable wh pdf f () CH and me-varyng falure rae ( CH )

13 Table 4 Parameers for each componen of SRS Componen Inal condon Transon nenses (year - ) p () p () p () () () OP Sae dsrbuons of SRS We encode he componens PS, PS2 and PV by negers, 2, 3. The MCSs of SRS for and are MCS {,2},{3} and MCS {},{2},{3}. Accordng o Eq. (8), S ( OP), X S ( ) 3 (=,2), S OP, S and S 3 ( ) 3 ( ) 2 ( ), ( ) 3 3. The componen sae dsrbuon can be obaned by Eq.(), (2) and (3), and he UGFs can be formulaed as follows: For componen (=, 2), u ( z, ) p ( ) z p ( ) z p ( ) z (28) 3 OP For componen 3, u (, ) ( ) ( ) ( ) 3 z pop z p z p z (29) The process of obanng he SRS dsrbuon usng he recursve approach s as follows: U ( z, ) u ( z, ) u ( z, ) 2 2 p ( ) p ( ) z p ( ) p ( ) p ( ) p ( ) z p ( ) p ( ) z OP OP OP OP p ( ) p ( ) p ( ) p ( ) p ( ) p ( ) p ( ) p ( ) p ( ) p ( ) z OP OP U ( z, ) U ( z, ) u ( z, ) 2 3 p ( ) p ( ) p ( ) z p ( ) p ( ) p ( ) p ( ) p ( ) p ( ) z OP OP OP OP OP OP OP p ( ) p ( ) p ( ) p ( ) p ( ) p ( ) p ( ) p ( ) p ( ) OP OP OP OP p ( ) p ( ) p ( ) p ( ) p ( ) p ( ) z z OP P ( ) z P ( ) z P ( ) z P ( ) z 2 3 FP DG 2 3 (3) (3) where P ( ) P ( ) P ( ). FP DG Thus, he sae dsrbuons of SRS are 2 3 PFP ( ) pop ( ) pop ( ) pop ( ) PDG ( ) pop ( ) p ( ) pop ( ) p ( ) pop ( ) pop ( ) P ( ) p ( ) p ( ) pop ( ) pop ( ) pop ( ) p ( ) pop ( ) p ( ) p ( ) p ( ) pop ( ) p ( ) 2 3 p ( ) p ( ) p ( ) P () Fg. 5 clearly llusraes he me-varyng characersc of he probably funcons for and saes 3 (32)

14 Expeced sysem cos per un me Expeced sysem cos per un me of he SRS. Accordng o he model presened n secon 2, he SRS wll evenually fall no eher or sae, ha s P P lm ( ) ( ). Ths phenomenon can be clearly verfed from Fg P() P() P()+P() (years) Fg. 5 The me-varyng probably funcons for and saes of he SRS. 4.2 Opmal schedule pon for replacemen Assume ha hs SRS requres SIL, hus p SILx = - [7, 8]. The assocaed parameers are abulaed n Table 5. Table 5 Safey and replacemen parameers p SILx C R C p C p a a 2 a 3 a Fg. 6 (a) plos he expeced sysem cos per un me as a funcon of schedule pon, whch shows ha he opmal schedule pon s A * =.372 (years) gvng he mnmum cos ( A ) = 675. The correspondng average probably of falure on demand, dsplayed n Fg. 6(b), s P avg (A * ) =.299. Fg. 6 also ndcae ha f we only consder he requred SIL, all schedule pons before 3.32 years s feasble. Should replacemen ake place a A=3.32 years whou he gudance of he proposed model, he expeced cos would become 7269 per un me, whch s 8% hgher han he opmal polcy Schedule pon(years) Average probably of falure on demand (a) (b) Fg. 6 (a) Expeced sysem cos per un me as a funcon of schedule pon A (b) Expeced sysem cos per un me as a funcon of P avg 4

15 Opmal schedule pon (years) Expeced sysem cos per un me 4.3 Impac of parameers Table 6 The paramerc values used n analyss Parameer Values 4. 5, 4.5 5, 5 5, 5.5 5, 6. 5 C R 4, 45, 5, 55, 6 C p 8, 9,,, 2 C p 4. 4, 4.5 4, 5. 4, 5.5 4, 6. 4 In hs secon, we nvesgae he effec of cos-relaed parameers on replacemen schedulng and he correspondng cos hrough sensvy analyss. The mpac of ndvdual parameers s quanfed by varyng ha parameer accordng o Table 6, whle keepng he remanng parameers fxed o he nomnal value gven n Table Scaled cos-relaed parameers (a) CR Cp Cp Scaled cos-relaed parameers (b) CR Cp Cp Fg. 7 (a) Opmal schedule pon, correspondng o scaled cos-relaed parameers (b) Expeced sysem cos per un me, correspondng o scaled cos-relaed parameers Fg. 7 llusraes he effecs of cos-relaed parameers on opmal schedule pon and expeced sysem cos per un me. Clearly, a large value of resuls n hgher sysem cos per un me due o dleness, as shown n Fg. 7 (b), and requres a longer opmal schedule pon A * o manan he cos (Fg. 7 (a)). The mpac of cos C R has he same rend as, bu smaller magnude han, ha of. In addon, he effec of cos, C p and C p, due o and, has smlar rend: hey boh have negave mpac on opmal schedule pon A * and posve mpac on he expeced sysem cos, hough he effec of C p appears o be more promnen. In summary, he analyss reveals ha n pracce, specal aenon should be pad o he parameers ha have large mpac o acheve mnmum expeced sysem cos per un me. 5. Concluson Ths paper presens a replacemen model for a non-reparable SRS. The sae dsrbuon of SRS are analyzed based on he srucure of SRS, s componens sae dsrbuon, he agng behavor of he componens, and he relaonshp beween SRS and PS. The sae dsrbuon s obaned by usng a NHCTMM. The soluon of such NHCTMM can be dramacally smplfed usng he UGF mehod and he recursve propery of he srucure funcon. The neracon beween SRS and PS s accouned for when devsng he objecve funcon,.e., 5

16 expeced sysem cos per un me, o be mnmzed. An llusrave example s suded o demonsrae ha an opmal schedule pon A * can be obaned o mnmze he sysem cos per un me whle meeng he requred safey level. Neverheless, he proposed model has ceran resrcons. Frs, he SRS s non-reparable, whle n realy, may be repared and re-used. Consderng a reparable SRS wll sgnfcanly complcae he objecve funcon. In addon, he sysem dle me a (=,2,3,4) due o replacng SRS or resorng PS may no be consan, bu random varables n pracce. Boh ssues are currenly beng nvesgaed. Acknowledgemens Ths work was parally suppored by Mnsry of Ralways of Chna (Scenfc Research and Developmen Program No. 28X3-C), and Naonal Naural Scence Foundaon of Chna (Gran No ). References [] Yu L, Hong-Zhong H. Opmal Replacemen Polcy for Mul-Sae Sysem Under Imperfec Manenance. Relably, IEEE Transacons on. 2;59: [2] Wang H. A survey of manenance polces of deerorang sysems. European Journal of Operaonal Research. 22;39: [3] Bevlacqua M, Bragla M. The analyc herarchy process appled o manenance sraegy selecon. Relably Engneerng & Sysem Safey. 2;7:7-83. [4] Zhao YX. On prevenve manenance polcy of a crcal relably level for sysem subjec o degradaon. Relably Engneerng & Sysem Safey. 23;79:3-8. [5] Rouvroye JL. Enhanced markov analyss as a mehod o assess safey n he process. Technsche Unverse Endhoven, Duch;2. [6] Rouvroye JL, van den Blek EG. Comparng safey analyss echnques. Relably Engneerng & Sysem Safey. 22;75: [7] Inernaonal Elecroechncal Commsson.Funconal safey of elecrcal/elecronc/programmable elecronc safey-relaed sysems. IEC 658,Pars-7,s Ed,Geneva, Swzerland, 998. [8] Inernaonal Elecroechncal Commsson.Funconal safey of elecrcal/elecronc/programmable elecronc safey-relaed sysems. IEC 658,Pars-7, 2nd Ed,Geneva, Swzerland, 2. [9] Tan CM, Raghavan N. A framework o praccal predcve manenance modelng for mul-sae sysems. Relably Engneerng & Sysem Safey. 28;93:38-5. [] Mohandas K, Chaudhur D, Rao BVA. Opmal perodc replacemen for a deerorang producon sysem wh nspecon and mnmal repar. Relably Engneerng & Sysem Safey. 992;37:73-7. [] Wang W, Banjevc D, Pech M. A mul-componen and mul-falure mode nspecon model based on he delay me concep. Relably Engneerng & Sysem Safey. 2;95:92-2. [2] Cho DI, Parlar M. A survey of manenance models for mul-un sysems. European Journal of Operaonal Research. 99;5:-23. [3] Hseh C-C. Replacemen and sandby redundancy polces n a deerorang sysem wh agng and random shocks. Compuers & Operaons Research. 25;32: [4] Levn G, Lsnansk A. Jon redundancy and manenance opmzaon for mulsae seres-parallel sysems. Relably Engneerng & Sysem Safey. 999;64: [5] Levn G, Lsnansk A. Opmal replacemen schedulng n mul-sae seres parallel sysems. Qualy & Relably Engneerng Inernaonal. 2;6: [6] Levn G, Lsnansk A. Opmzaon of mperfec prevenve manenance for mul-sae sysems. 6

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