Optimum Maintenance of a System under two Types of Failure

Transcription

1 ntnatonal Jounal o Matals & tutual lablty Vol.4, o., Mah 6, 7-37 ntnatonal Jounal o Matals & tutual lablty Otmum Mantnan o a ystm un two ys o alu.. Baía * an M.D. Ba Datmnt o tatsts, C... Unvsty o Zaagoza, Maía Luna 3, 55 Zaagoza, an Abstat A mantnan mol o a systm subt to bo unval atastoh alus o val mno alus s snt. h om a tt by mans o an nston oly at o tms,,,. Moov t s assum a lss an t tstng, at s, als alams as wll as untt alus at an nston. val alus a mov by a mnmal a whas a t a ollows unval alus. A nwal o systm at val alu omlts mantnan atons. h tms o nstons an as a also tan nto aount. h t ost-mnmzng oly along an nnt tm san s also analyz. Kywo: Mantnan, Mnmal a, Otmum oly, lablty, Unval alu. ntouton h sgn o mantnan ols tuns out to b a ual ssu n lablty oy. alus u ablty to ulll s untons an a otn sonsbl o hug onom losss. h osts v om tv outon o owntm nu whn a systm als, la os non-nglgbl mony gus untly nvst n mantnan ous n o to vnt systms to al. h lassal ag lamnt an blo lamnt ols [] a sally sgn o val alus, whh a tt as soon as y ou. Howv many ngnng systms a subt to so-all unval alus whn nstons o sal tsts a n to sov m. alus o s sot ou n systms at altnat bo oatng an l os suh as sas o unts n stan-by mo. uty vs n gas onutons, nula ow lants, o alams a tyal amls o systms whh may ungo unval alus [4,,, ]. hus, o nston mgs u to unaoabl ost o a ontnuous montong. h mol n [] onss ost otmzaton by slton o a unqu ntval o bo nston an mantnan. h mol n [] ovs an mt nston oly whh hav non-zo obablty o als ostvs along w vntv atons o a systm subt to omtng alu tys. * Cosonng auo. -mal : gbaa@unza.s 7

2 A mantnan oly at suts a systm sntng two tys o alu s snt n s a. h val mno alus ty a mov by a mnmal a at bngs systm ba to oatng onton ust vous to alu as-ba-as-ol. A t a at stos systm to an as-goo-as-nw onton s a out to gt systm o unval atastoh alus ty U. Comuts sv as an aml o a systm at may ungo alus o bo tys. hus, a amag l o a vus a only tt by mans o ant-vus ogams at oally san omut. h om an b sonsbl o atastoh vnts suh as lost o whol nomaton ontan n ha s manwhl man untt. Howv alus at han n ow suly o omut sn a tt no soon y ou an, n gnal ts onsquns a lss motant. h wos [6, 8] ons an mt a whh s ahv w obablty o a t on w obablty -. h mol n [4] analyzs avalablty unton o a oally nst systm at ungos a numb o mt as bo bng tly a. h n [3] ooss a mantnan mol o a systm at anomly altnats oatng an l os but only t as a ons. Dnt obablst stutus as nng on ty o mantnan atons. h systm ltm ommonly aas to b shot at an mt a an n as at a t stoaton s a out. vlss, u to ta-ost o latt, sval mt as a allow vously to t a o vntual lamnt o unt. uh mt as ty to olong systm ltm as muh as ossbl. n s a w vlo an nston oly along w a mantnan ou o a systm whos alus a o ty o ty U n a anom way. W am at obtanng an otmal nston ntval at mnmzs t ost at o an nnt tm san. hs wo s oganz as ollows: mantnan oly s sb n ston whas ost unton an man sult onnng stn o an otmum oly a snt n ston 3. h last ston s vot to stuy o som amls at llustat otal sults.. h Mantnan Mol Cons a systm subt to two alu tys: a ty alu hans w obablty o a ty U w obablty q-. o tsts o nstons a a out at tms,,,... to tt ossbl oun o an unval alu. Whnv on o s nstons vals a alu, a t a stos systm to an as-goo-as-nw onton. Moov, mno alus a tt mmatly an a mnmal a at bngs systm ba to oatng onton ust vous to alu s ungon. n aton a t a s a out at val mno alu n o to vnt systm om toaton. W also ta nto aount ossblty o mt nstons so-all ty an ty statstal os. Bo nston an a tms a ons not nglgbl. gus an outln mantnan ou. otaton: - tm san btwn onsutv nstons - alu at osonng to tm to st alu - H umulatv alu at, H u u - α als alam obablty o a ty U alu at an nston - β obablty o an untt ty U alu at an nston - tm san to st ty U alu - tm san to - ntg at unton ty alu 8

3 - lng o a yl - K numb o nstons vous to a ty U alu, - K numb o nstons vous to K ty alu, K - K 3 numb o nstons om a ty U alu untl ts tton. als Alam t a st 3 4 n t g. hm o a yl nng at t a ollowng ty alu als Alam hs nston tts ty U st n J Downtm t g. hm o a yl nng at t a ollowng a ty U alu - numb o nstons n a yl vous to a ty U alu o - numb o als alams n a yl - numb o nstons n a yl - L numb o mnmal as n a yl - O systm utm - t nston tm - t U tm o t a at ollows a ty U alu - t tm o t a ollowng ty alu ty alu h nt ooston ontans som bas sults lat to ag nnt mnmal a [5] at wll b us all ov s atl. ooston. Un mol assumtons, ollowng sults hol: 9

4 3 a h nsty an lablty untons osonng to tm to st ty U alu, a qh q qh b h nsty an lablty untons o tm to ty alu, a! H H! H H an a nnnt anom vaabls. A yl not by s tm san btwn two onsutv nwals o systm an onsttuts y o unlyng obablst mol. n s mol a yl s omlt at t a at s a at ty U alu o ty alu. h nt om ovs vous sults at la to obtan an ts t ost, [ ]. n what ollows a ollton o aulay anom vaabls,,, wll b us. h osonng nsty an lablty untons a! H H! H H n aton, w shall ma us o mtu not by Z wh Z snts a anom vaabl w ollowng obablty unton: q Z,,,,- h untons blow snt a ual ntst n oomng sults:

5 3 hom. Un mol assumtons, sults blow a sats: a Lt an not stvly ollowng vnts: a yl s omlt at t a at ollows a ty U alu o w t a at ollows ty alu. ts osonng obablts a:, b h man numb o nstons n a yl s β h man numb o als alams n a yl s α h t utm n a yl s O h man lng o a yl w no nstons an as s gvn by β h man lng o nstons an as U t t t t β oo: a om ooston, t s v <! H H qh

6 an a omlmntay vnts an, hn, sult hols. b h numb o nstons n a yl s ss as K K K 3 wh A nots nato unton osonng to vnt A. By mans o ooston w obtan ollowng tatons K H! <, < H q qh n aton K <, < qh H! H K 3 s a gomt anom vaabl w aamt β, nnnt om an. hs at along w at a o hom la to 3 K 3 β K 3 an sult n at b o hom ollows by tang tatons n. h numb o als alams n a yl,, whn numb o nstons vous to a ty U alu o ty alu,, s nown s a bnomal anom vaabl w aamts n an α. ho, α h ollowng taton s obtan by usng sam alulatons o oo n at b o hom : K K an sult hols. 3

7 h utm o s ss as O 4 h nt tatons a v by mans o ooston : as wll as H H qh q! qh H H! h sult s obtan by tang tatons n 4. h lng o a yl wout nstons an as s ss as ollows 5 K K 3 an sult hols by tang tatons along w, 3, an 5. h lng o bo nstons an as n a yl s tu t t h sult s u by tang tatons n ogong sson along w ats a an b o hom. h lng o a yl,, s obtan by aton o an, o 3. Cost unton an Man sults n s ston w ous on obtanng ost o a yl as wll as h ost unton. Moov w am at stuyng ontons guaantng stn o an otmum oly,. h ollowng osts a assum: - untay ost o nston - untay ost o als alam - ost o t a o a ty U alu - ost o t a at - m ost o mnmal a at - ost at u to owntm ty alu ty alu 33

8 34 hom. Un mol assumtons, ollowng sults hol: a h man ost nu u to mnmal as n a yl s m CM b h man ost o a yl s gvn by C Ψ α wh m Ψ β h ost unton s ss as ollows α Q, Ψ 6 oo: a h numb o mnmal as n a yl, L, s a anom vaabl w ollowng obablty unton, q L,,..., L h ost nu n a yl u to mnmal as s gvn by L m CM Hn, t ollows m m m m m m m m q q L CM b h ost o a yl s ss as

9 C CM O h sult s v om taton o vous omula along w sults n a, b,, n hom an a n hom. om now on ost at o an nnt tm san wll b ons obtv unton. h unamntal nwal om [9] stats at as tm gos by suh a unton onvgs almost suly to ato btwn t ost o a yl an ts man lng, at s Q, C an sult n 6 hols. h ollowng om s onn w stn o an otmum oly at mnmzs ost unton. hom 3. ov at < an a natual numb a gvn, sts an otmum nston ntval,, < < mnmzng ost unton n 6 an only Ψ <. Moov s on o oots o quaton blow: A A α A Ψ t 7 wh A t β β t t oo: Bo untons an an lm lm U satsy ollowng ots as shown n []. a non nasng w an non ngatv. lm lm lm lm lm lm Lt us assum at Ψ <. h ogong ots nsu at sts < < suh at Ψ 35

10 n aton o > Ψ < ho, om 6 s v at Q, < o all <. Moov, om lmtng ots onnng an an n hom, an quaton 6 t s u at along w ats lm Q, an lm Q, an stn o a nt mnmum,, s onlu. Cons now at Ψ. om 6 nqualty blow s obtan: Q, lm Q, t ollows at otmum nston ntval mnmzng ost unton s. hus, otmum oly onssts o ayng out no nston. h onton n 7 s obtan by ntaton n 6 an sttng vatv qual to zo. h nssay an sunt onton o stn o a nt otmum oly, Ψ <, s quvalnt to ollowng nqualty O > β m o ma mantnan wll wo ong, systm utm shoul b gat an ombnaton o osts on ght han s o nqualty. s s as, osts nu u to mantnan oly a omnsat by ts tuns. W st to otmum numb o ty alus vous to t a,, w ons sam ou as n [7]. vs at Q, mn Q, 4. amls m to alu s assum to b an onntal anom vaabl w man. h λ obablts o als alam an untt alu at nston satsy, stvly, α. 5 an.5. β aml : λ.5,.5,.5,.3,.5,,,... t t., t. 5 U m 36

11 abl. Otmum oly an ost Q, aml : λ.,.5,,.5, 3,,,,... t t, t. 5 U abl. Otmum oly an ost Q, abls an show bo otmum nston ntval,, an otmum numb o ty alus,, vous to t a as wll as osonng otmum ost, Q,. h man atu n abl s at gat valu o owntm,, hgh nston quny an otmum ost. abl vals at gat obablty o a ty U alu,, lss qunt nston an low otmum ost. m Anowlgmnts hs wo has bn suot by Unvsty o Zaagoza-baa un ot B4-C-. ns. Balow.., oshan. Mamatal hoy o lablty. AM, Baía.., Ba M.D., Camos C.A. Otmzaton o nston ntvals bas on ost. Jounal o Al obablty 38, Baía.., Ba M.D., Camos C.A. Otmal nston an vntv mantnan o unts w val an unval alus. lablty ngnng an ystm aty 78, Bswas A., aa J., aa. Avalablty o a oally nst systm mantan un an mt-a oly. ansatons on lablty 3, 5, Blo H.W., Bogs W.., avts,.h. Ag nnt mnmal a. Jounal o Al obablty, Bown M., oshan. mt a. Jounal o Al obablty, aagawa. o an squntal vntv mantnan ols. Jounal o Al obablty 3, aagawa., asu K. Otmal ols o a systm w mt mantnan. ansatons on lablty -36, oss. ntouton to obablty Mols, 7 ton. Aam ss,.. Vauo J.K. On tm-nnt avalablty an mantnan otmzaton o stanby unts un vaous mantnan ols. lablty ngnng an ystm aty 56, Vauo J.K. Avalablty an ost untons o oally nst vntvly mantan unts. lablty ngnng an ystm aty 63, Zqua.., Bngu C. Otmal shulng o non-t nstons. MA Jounal o Managmnt Mamats, 5, oomng. 37