RELIABILITY STOCHASTIC MODELING FOR REPAIRABLE PHYSICAL ASSETS. CASE STUDY APPLIED TO THE CHILEAN MINING

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1 Rlably Sochasc Modlg for Rparabl Physcal Asss. Cas sudy appld o h Chla Mg P. Vvros, A. Crspo, R. Tapa, F. Krsjapollr, V. Gozálz-Prda RELIABILITY STOCHASTIC MODELING FOR REPAIRABLE PHYSICAL ASSETS. CASE STUDY APPLIED TO THE CHILEAN MINING Pablo Vvros (* (** Adolfo Crspo (* Ré Tapa (*** Frdy Krsjapollr (* (** Vc Gozálz-Prda (* Idusral Egrg PhD. Idusral Egrg Idusral Egrg Idusral Egrg PhD. Idusral Egrg (* Uvrsy of Svll, Spa. School of Egrg. Idusral Maagm Dparm. Avda. Camo d los Dscubrmos, s/. Isla d la Caruja, Tfo: adolfo@us.s, vc.gozalzprda@gdls.com (** Uvrsdad Fdrco Saa María, Valparaíso, Chl. pablo.vvros@usm.cl, frdy.krsjapollr@usm.cl (*** RlPro SpA. Chl. r.apa@rlpro.pro Rcbdo: -- Acpado: DOI: ABSTRACT: Th rlably modllg, calculag ad projcg for dusral upm ad sysms ar oday a basc ad fudamal ask for rlably ad maac grs, rgardlss of h aur or gcs of hos dusral asss. I hs papr, h sochasc modls PRP, NHPP ad GRP ar xplad dal wh h corrspodg cocpual, mahmacal ad sochasc dvlopm. For ach modl, h rspcv cocpualzao ad paramrzao s aalysd dal. Th praccal applcao s dvlopd for a ral cas h mg dusry, whch shows sp by sp h appropra sochasc ad mahmacal dvlopm. Fally, hs rsarch bcoms a aalycal ad xplaaory procdur o h dfo, calculao, mhodology ad crra o b cosdrd for dusral asss paramrzao wh paral or ull pos maac dgradao. Ky Words: Rlably, Dgradao, Rparabl Asss, Smulao.. INTRODUCTION Th modl ad aalyss of rparabl upm ar of gra mporac, maly ordr o cras h prformac ord o rlably ad maac as par of h cos rduco hs las m. A rparabl sysm s dfd as: A sysm ha, afr falg o prform o or mor of s fucos sasfacorly, ca b rsord o fully sasfacory prformac by ay mhod ohr ha rplacm of h r sysm []. Dpdg o h yp of maac gv o upm, s possbl o fd 5 cass [2]: a Prfc maac or rparao: Maac oprao ha rsors h upm o h codo as good as w. b Mmum maac or rparao: Maac oprao ha rsors h upm o h codo as bad as old. c Imprfc maac or rparao: Maac oprao ha rsors h upm o h codo wors ha w bu br ha old. d Ovr-prfc maac or rparao: Maac oprao ha rsors h upm o h codo br ha w Dsrucv Maac or rparao: Maac oprao ha rsors h upm o h codo wors ha old. For a prfc maac, h mos commo dvlopd modl corrspods o h Prfc Rwal Procss (PRP. I, w assum ha rparg aco rsors h upm o a codo as good as w ad assums ha ms Publcacos DYNA SL -- c Mazarrdo º69-3º BILBAO (SPAIN Tl mal: dya@rvsadya.com Pag. / 6

2 Rlably Sochasc Modlg for Rparabl Physcal Asss. Cas sudy appld o h Chla Mg P. Vvros, A. Crspo, R. Tapa, F. Krsjapollr, V. Gozálz-Prda bw falurs h upm ar dsrbud by a dcal ad dpd way. Th mos usd ad commo modl PRP s h Homogous Procsss of Poso (HPP, whch cosdrs ha h sysm o ags hr spols, dpdly of h prvous par of falurs. Tha s o say, s a procss whou mmory. Rgardg cas b, as bad as old s h oppos cas o wha happs cas a as good as w, sc s assumd ha h upm wll say afr h maac rvo h sam sa ha bfor ach falur. Ths cosdrao s basd ha h upm s complx, composd by hudrds of compos, wh may falur mods ad h fac ha rplacg or rparg a drmd compo wll o affc sgfcaly h global sa ad ag of h upm. I ohr words, h sysm s subjc o mmum rpars, whch dos o caus ay chag or cosdrabl mprovm. Th mos commo modl o rprs hs cas s hrough No Homogous Procsss of Poso (NHPP, hs cas h mos usd modl o rprs NHPP s calld Powr Law. I hs modl, s assumd a Wbull dsrbuo for h frs falur, ha lar s modfd ovr m. Alhough h modls HPP ad NHPP ar h mos usd, hy hav a praccal rsrco rgardg s applcao, sc a mor ralsc codo afr a rparg aco s wha w fd bw boh: wors ha w bu br ha old. I ordr o fd a gralzao o hs suao ad o dsgush bw HPP ad NHPP was cssary o cra h Gralzd Rwal Procss (GRP [3], whch sablshs a mprovm rao. Uforualy, h corporao of hs varabl ca complca h aalyc calculao of paramrs ad adjusms of probably. Thrfor, s applcably mahmac rms s complx. For hs raso, has b cosdrd soluos hrough h Mo Carlo smulao (MC bg o of h mos valdad mhods accordg h proposal dvlopd by auhor Krvsov [4] whr m srs of good fucog ar grad hrough h us of h vrs fuco of h probably dsrbuo (pdf ha has as a bas a radom varabl. Udrsadg h mporac ad applcably of mhods PRP, NHPP ad GRP hs papr roducs h cocpual, mahmacal ad sochasc dvlopm for ach o, as xplad ad prsd brfly h prvous paragraphs. Each modl s xplad ad dvlopd h followg way: cocpualzao ad paramrzg. I addo, ach modl wll b complmd wh a umrcal applcao, spcfcally corrspod o 2 pulp pumps (war, coppr cocra ad r maral usd h mg dusry of Chl, whch suffr dffr lvls of roso du o us sy, gographc hgh ad of cours accordg o h maac yp dvlopd s lf, plad or o plad (prvv or corrcv maac. Wh h abov, hs arcl bgs by roducg sochasc modls (PRP, NHPP ad GRP, h s prsd a brf of paramrzao procsss ad a umrcal aalyss applcao. Fally, h papr cocluds by summarzg h ma ls provdd by h arcl ad s applcao o h dusral scor. No h ovav corbuo of hs arcl volvg h applcably of IT ools rsolvg xsg modls. Such applcably s prsd hr as a sampl, ad wh a spcfc ral cas of Chla mg. Noaos PRP: Prfc Rwal Procss NHPP: No-Homogous Posso Procss GRP: Gralzd Rwal Procss λ(: Falur Ra of a lm a gv m : Fuco m bw falur - ad -h f(: Probably dsy fuco of falur (pdf of a lm wh oprao m F(: Probably dsy fuco of accumulad falur of a lm wh oprao m. Γ( : Gamma fuco. f(;θ: Probably dsy fuco of falur of a lm wh oprao m, wh forma ad scal paramrs gv by vcor 0. L(θ Lklhood fuco for paramrs vcor 0 a pdf gv. MTTF: Ma m o falur. MTTR: Ma m o rpar. â: Esmad valu of h paramr a (applcabl for all paramrs. A : Vrual ag of sysm a h mmda mom of h rpar of -h. T : Vrual ag (of oprao a h mmda mom of h rpar of -h. : Paramr whch sablshs h dfc of h rpar. TOL: Numrc grad o cosdr accpabl a dsrbuo adjusm hrough h lklhood maxmum. Publcacos DYNA SL -- c Mazarrdo º69-3º BILBAO (SPAIN Tl mal: dya@rvsadya.com Pag. 2 / 6

3 Rlably Sochasc Modlg for Rparabl Physcal Asss. Cas sudy appld o h Chla Mg P. Vvros, A. Crspo, R. Tapa, F. Krsjapollr, V. Gozálz-Prda Noaos TQ: Tolrac ha corrspods o hghr or lowr prcag of h possbls of valu. Tabl: Noao. Wh h o of hghlghg h scfc ad chcal corbuo of hs arcl s cssary o mphasz hs roduco h followg cosdraos: Th wd rag ad varably of hr bhavour rurs h applcao of chus of varyg complxy ad dph, ha ca adap o h bs way o ach of h rals. Th varabl ha dfs ad codos h us of chus s h sa asss rmag afr rpar. I hs rgard, hr ar fv classfcaos rpar: Prfc, mmal, mprfc, ovr- prfc ad dsrucv. For prfc maac, s usd ad rcommdd Prfc Rwal Procss modl (PRP hrough homogous Posso procsss (HPP. For mmal rpar, grally rprsd by Nohomogous Posso Procss (NHPP, h mos wdly usd s "Powr Law" modl. Howvr h gra applcao of h aformod modls, hr ar varous suaos ha ar o covrd, sc h mos of cass rpar ar bw h prfc ad mmum codo (mprfc rpar. For hs suaos, ucas ad dvlops h "Gralzd Rwal Modl" (GRP. Morovr, sc h probably of fdg h valus ha gv h maxmum ovrall fuco of maxmum lklhood s vrually zro by h radom sarch, s cssary o df a olrac valu for h paral drvavs ( L/ B ad L/ ha hy ar machd o zro, sg hs olrac valu "TOL" as a accpabl rag o cosdr adjusg dsrbuo. For hs, h smulao chus ad spcally Mo Carlo mrg as a powrful alrav for rsoluo. I s rcommdd o chck rfrcs [5 ad 9] o sudy dals som advaags ad dfcs abou h modls. 2. PRESENTATION OF THE STOCHASTIC MODELS 2.. Prfc Rwal Procss (PRP PRP Cocpualzao Th Prfc Rwal Procss modl dscrbs h suao whch a rparabl sysm s rsord o a sa as good as w ad h ms bw falurs ar cosdrd dpd ad dcally dsrbud. Ths procss assums ha h upm rsors o a dcal codo o h orgal, as f s rplacd. Th graph of h falur ra dpdg o h m lapsd for upm wh growg falur ra, cosdrg h gral cas of a Wbull dsrbuo, would b h followg fgur. Fg. : Falur ra λ( PRP (Sourc: ow laborao As provd h formr graphc, h voluo of h falur ra s rs afr ach falur, vdly du o fac ha h upm rmas prfc codos. Th PRP modl posssss ma applcao ovr hos upm ha hav a compl maac ovr all s compos or f a 00% rplacm of h upm. I mahmacal rms: B h fucog m bw falurs -l ad h -h. Th, udr a PRP modl, ay m wll oby o h sam probably dsrbuo wh alrabl paramrs m, for xampl for h woparamr Wbull cas: Publcacos DYNA SL -- c Mazarrdo º69-3º BILBAO (SPAIN Tl mal: dya@rvsadya.com Pag. 3 / 6

4 Rlably Sochasc Modlg for Rparabl Physcal Asss. Cas sudy appld o h Chla Mg P. Vvros, A. Crspo, R. Tapa, F. Krsjapollr, V. Gozálz-Prda 0 f ( 0 < 0 Bg ad couous ad alrabl m. Th org of hs yp of dsrbuo s ordr o cosdr crasg or dcrasg falur ras alog h m from h las rpar. Bg ad h scal ad form paramrs rspcvly. PRP Paramrzao Modl Two- paramrs Wbull From a praccal po of vw, s characrzd by havg 2 paramrs, whr corrspods o h form paramr lkd o h wll-kow bah curv ad o h rspcv phas of lf cycl of h ass, ad h paramr kow as h scal paramr, whch s lkd drcly wh h varably ad dsprso of h lf daa ha h ass aalysd has. Th probably dsy fuco of falur (pdf corrspods o (: f ( ( I hs cas h falur ra s dfd as (2: λ( (2 Th ma m o falur (MTTF ad rlably R( ar (3: MTTF Γ +, R( (3 No ha Γ( corrspods gamma fuco (4: x x dx Γ ( 0 (4 Th Wbull fuco slf s a gralzao for h Expoal fuco, kowg ha, /λ ad γ0. Wh Wbull dsrbuo s possbl o rprs h sa of h ass for ay of 3 phass h bahub curv (lf cycl prspcv: fa moraly, usful lf or lf war ou. For paramrzao s rurd a maxmum lklhood fuco rsoluo, h a aural logarhm s appld parally wh rspc o ach paramr o fally drvd s ad uals o zro. I s prsd (5 ad (6: / ( (5 Publcacos DYNA SL -- c Mazarrdo º69-3º BILBAO (SPAIN Tl mal: dya@rvsadya.com Pag. 4 / 6

5 Rlably Sochasc Modlg for Rparabl Physcal Asss. Cas sudy appld o h Chla Mg P. Vvros, A. Crspo, R. Tapa, F. Krsjapollr, V. Gozálz-Prda + l ( l [ ] (2 (6 For h rsoluo of hs yp of adjusm, hr ar spcalzd sofwar applcaos. O of hs s RlPro [4] whch dsposs of advacd ad ffc algorhm for h rsoluo of hs kd of problms. Also, cas of d o clarfy ad xpad h cocps hadld hr, h followg rfrcs ar rcommdd [5, 6] No-Homogous Posso Procss NHPP Cocpualzao NHPP s a Posso procss wh a paramrc modl usd o rprs vs wh a occurrc of voluoal falur m ad always wh h sam dcy. Fg. 2: Falur ra λ( NHPP (Sourc: ow laborao Ths cas appls spcally for hos upm ha ar composd by may compos whr h rplacm of o of hm dos o affc h global rlably: cosdr a upm composd by hudrds of compo ha work srs, f o of hm fals, hs compo s rplacd ad h upm cous workg bu wh a lvl of was almos dcal o prvous o. For hs raso h NHPP modl appls for h calld mmum maac. Nx, fgur 2 prss h graph for h bhavour of h falur ra ovr m, bg hs complly accumulav bw o ad ohr falur. As w apprca h formr graphc, for h cas of NHPP, h falur ra rmas dpd o oal m lapsd. I h cas of NHPP, h fucos of h rlably ad falur probably ar xprssd as follows (fg. 3. Fg. 3: R( ad f( NHPP (Sourc: ow laborao Publcacos DYNA SL -- c Mazarrdo º69-3º BILBAO (SPAIN Tl mal: dya@rvsadya.com Pag. 5 / 6

6 Rlably Sochasc Modlg for Rparabl Physcal Asss. Cas sudy appld o h Chla Mg P. Vvros, A. Crspo, R. Tapa, F. Krsjapollr, V. Gozálz-Prda Havg as a bas h formr graphc, l s cosdr ha o upm has a falur a m. Afr bg rpard, h fucog s rsard ad bgs o work ha sam po. Th, h rlably fuco from, for a m ha rprss h lapsd m byod, wll b gvs as (7. R( R( > F ( > R( (7 Ths s calld by varous auhors [5] as Msso Tm, whr corrspods o lapsd caldar m. For a Wbull dsrbuo, from h prvous uao ad sc - corrspods o h oal m lapsd ul h las falur, ad h oal m (caldar lapsd afr gra h falur -h, wll b possbl o coclud h followg probably dsy fuco (8: f ( > xp NHPP Paramrzao Modl Wbull 2 paramrs I ordr o oba h paramrs ad, h lal rgrsso s o a choc. I s dally mad a adjusm by maxmum lklhood. Th lklhood fuco s xprssd as (9: (8 { } P( x [x, x +dx],.., f ( x ; θ L( θ f ( x ; θ (9 Whr ϴ corrspod o h vcor of h paramrs of dsrbuo o whch oby h f(. Morovr x corrspods o h lm -h of h sampl. As s wshd o oba maxmum lklhood bw h daa ad o pdf: f (;ϴ, h valus of h vcor ϴ ar adjusd wh h am o rach ha maxmum. Cocpually, paramrs ar sarchd ordr o br f o a sampl X,.,X such way ha h probably of h srs of valus ha b prsd a radom sampl b maxmal: Thus, h prs cas wh h smplfd lklhood fuco ad afr applyg ad paral drvavs ual o zro, h rsuls of h smaors for NHPP ar (0 ad (: / l (0 ( Whr corrspods o h lapsd m ul h falur -h ad h lapsd m ul h las falur. As h prvous cas, f s dd o clarfy ad xpad h cocps hadld hr, x rfrcs ar rcommdd [5] y [6] Gralzd Rwal Procss (GRP Publcacos DYNA SL -- c Mazarrdo º69-3º BILBAO (SPAIN Tl mal: dya@rvsadya.com Pag. 6 / 6

7 Rlably Sochasc Modlg for Rparabl Physcal Asss. Cas sudy appld o h Chla Mg P. Vvros, A. Crspo, R. Tapa, F. Krsjapollr, V. Gozálz-Prda GRP Cocpualzao Th radoal modls alrady show ar oly abl o modl 2 yps of maac: h complly prfc ad h complly mprfc. GRP modl s h gralzao for ay lvl of prfco ha has h maac, cludg h boh mod. GRP adds a w paramr, calld vrual ag. Th paramr A rprss h ag of h sysm a h mmda sa wh h -h rpar s carrd ou. I hs way, f A y, h lm has a m of fucog assocad o a probably dsrbuo codod for hs ag y. Tha s o say, all h falur ms hav dffr probably dsrbuos as h m passs by. Graphcally, h falur ra volvs as s show fgur 4. Fg. 4: Falur ra λ( GRP (Sourc: ow laborao Th way o corpora hs varabl s cosdrg ha upm bgs o opra wh cra was, whch s rflcd h rlably fuco. I hs mar, h accumulad rlably ad probably dsrbuo for + s (2: F( A y R( A y F( + y F( y F( y R( + y R( (2 By hs way, s clar ha hs vrual ag s h ag of was whch h upm bgs o work aga. Th rlably fuco rmas smlar o h Msso Tm oly ha hs dos o corrspod o a ral m lapsd, bu o a uval. X s h -h m of good fucog ad T h oal accumulad m lapsd ul falur -h, as follow (3: T x ( (3 Morovr, h paramr A s gv by (4: A A + x (2 (4 Usg (3, h would b (5: A T x (3 (5 Publcacos DYNA SL -- c Mazarrdo º69-3º BILBAO (SPAIN Tl mal: dya@rvsadya.com Pag. 7 / 6

8 Rlably Sochasc Modlg for Rparabl Physcal Asss. Cas sudy appld o h Chla Mg P. Vvros, A. Crspo, R. Tapa, F. Krsjapollr, V. Gozálz-Prda Whr s h paramr ha dcds h ffcvss of h rpar, hs way 0 mpls ha A 0, ha s o say vrual ag ual o 0. Thrfor 0 corrspod o a prfc rpar cas, ha s o say s complly ffcv. I h cas was h bgs o opra h sam par of h rlably fuco whr h upm fald. Ths would b (6: 0 < < : GRP 0 : PRP (HPP : NHPP (6 Plog h xsg rlao bw ral lf ad vrual ag ha volvs, s possbl o gra (s fgur 5 h followg comparav graphc for PRP, NHPP ad GRP. Fg. 5: Vrual ag V/S Ral ag PRP, NHPP y GRP (Sourc: ow laborao As NHPP, s drmd h codod rlably ad h rspcv pdf. Th, rlably wll b modld accordg o (7, (8 ad (9: R( > (7 F( > (8 f ( > (9 Paramrzao Modl GRP Th adjusm dvlopd s o h bass of a pdf wh wo-paramr Wbull (,, ad addg h paramr, so h w hav 3 paramrs o drm. Th mos commo approach for paramrs drmao, by maxmum lklhood, corrspods o h Lklhood Fuco. I ordr o solv hs, a paral drvav ach varabl s appld, h wll b obad a s of 3 uaos wh 3 ukow uas, hs ar:,,. Th paramrs,, ar h 3 valus o dfy. Sarchg hs paramrs s a vry xhausv procdur, as rurs mor prcso h procdur, grally s a log procss, so s suggsd o us h Mo Carlo smulao. Th sarchg of,, sars wh h smulao of ad, rpad by uform dsrbuos (20: Publcacos DYNA SL -- c Mazarrdo º69-3º BILBAO (SPAIN Tl mal: dya@rvsadya.com Pag. 8 / 6

9 Rlably Sochasc Modlg for Rparabl Physcal Asss. Cas sudy appld o h Chla Mg P. Vvros, A. Crspo, R. Tapa, F. Krsjapollr, V. Gozálz-Prda ~ U[0,] ~ U[0,0] (20 Smlarly, h smad paramr for h GRP modl ylds (2: 2 ( ( ( + + (2 Th, h procdur for h GRP adjusm s plod hrough h followg dagram of procss. S Fgur 6. [Sí] Fg. 6: Procss dagram for GRP modllg (Sourc: ow laborao As far as h probably o fd h valus ha gra h global maxmum of h maxmum lklhood fuco s vrually vald hrough radom sarch, s cssary o df a valu of olrac for h paral drvavs ( L/ B ad L/ ual o 0, bg cssary o fx hs valu of olrac TOL as a accpabl rak o cosdr ha s foud a global maxmum ad hs way o accp h rspcv dsrbuo adjusm. Cosdrg h addg of a w paramr, hs cas, always h adjusm GRP wll gv a hghr lklhood ha a PRP or NHPP adjusm. Nvrhlss, ordr o cosdr h xsc ad applcably of hs cass, s cssary o cou o slco crra. Ths s appld afr h adjusm hrough GRP oc obad h paramr. Publcacos DYNA SL -- c Mazarrdo º69-3º BILBAO (SPAIN Tl mal: dya@rvsadya.com Pag. 9 / 6

10 Rlably Sochasc Modlg for Rparabl Physcal Asss. Cas sudy appld o h Chla Mg P. Vvros, A. Crspo, R. Tapa, F. Krsjapollr, V. Gozálz-Prda As valu s always a couous valu, h probably o b xacly or 0 s praccally ull, hrfor s cosdrd a w olrac lvl, whch has b calld TQ. Ths olrac lvl corrspods o hghr ad lowr prcag of h possbls ha valu has. Th raso of hs valu ( paramr s o dfy wh would b mor appropra o cosdr a PRP or NHPP modl. Thrfor h praccal xprsso corrspods o (22: 0 + TQ < < TQ : GRP TQ : PRP TQ : NHPP (22 As prvous cas, o clarfy ad xpad h cocps hadld hr, rfrcs [5 ad 6] ar rcommdd. 3. NUMERICAL APPLICATION Accordg o prlmary cocpual ad aalycal dvlopm, procds o dvlop a praccal applcao, whch corrspods o h aalyss of a slurry pump (r maral, ID cod: P0, blogs o a procss volvd coppr mg. Th modl o b appld corrspods o GRP, gv h flxbly ad ably o gralz ad dscrma ay of h 3 modls xposd PRP, NHPP ad GRP. Tabl 2 shows h m rcords of good prformac whch hav b collcd by h y prformg dusral acvy, ad wll b cosdrd as h pu daa for h smulao. I ordr o udrsad ad aalys h applcably of h mhodology prsd ad s ffcs, h slcd m s ough. Th, procds o dvlop h probably dsrbuo adjusm. N of Falur Oprag m [h] N of Falur Oprag m [h] 860, , , , , , , , , , , , , , , , , , , ,33 680, , , ,30 Tabl. 2: Tm bw falurs for pump P0 [hours]. (Sourc: ow laborao Sp : Tolrac lvl Mus b dfd h olrac lvl for h paral drvavs as TOL 0.0 ad olrac for h valu s TQ 5 %. Sp 2: Dsrbuo paramrzao Oc olrac lvl s dfd, procds o apply GRP modl. For hs, s dcdd o us a compur ool: RlPro. I has b dvlopd from h prspcv of rsarch ad dusral applcao. Solvg h uaos wh RlPro, h followg paramrs ar obad. S Fgurs 7 ad 8 rspcvly. Publcacos DYNA SL -- c Mazarrdo º69-3º BILBAO (SPAIN Tl mal: dya@rvsadya.com Pag. 0 / 6

11 Rlably Sochasc Modlg for Rparabl Physcal Asss. Cas sudy appld o h Chla Mg P. Vvros, A. Crspo, R. Tapa, F. Krsjapollr, V. Gozálz-Prda Fg. 7: Paramrzao procss for GRP, RlPro Sofwar. (Sourc: ow laborao Fg. 8: Paramr smao adjusm GRP. RlPro Sofwar. (Sourc: ow laborao From h paramrs obad ( 986,067; 2,026 ad 0,92, h paral drvavs ar solvd ad h accpac of GRP dsrbuo adjusm ad lvl of olrac ar vrfd proprly (23. [l( L] [l( L] < 7,76 0 TOL 0,0 0, < TOL 0,0 (23 Rgardg h valu, has o (24: TQ 0,05 < 0,92 < TQ 0,95 (24 Thrfor, drmg a accpabl adjusm soluo by maxmum lklhood wh paral drvavs (ualy guara of h f, ad a paramr wh a umrc valu bw h rag 0.05 < <0.95, s possbl o affrm Publcacos DYNA SL -- c Mazarrdo º69-3º BILBAO (SPAIN Tl mal: dya@rvsadya.com Pag. / 6

12 Rlably Sochasc Modlg for Rparabl Physcal Asss. Cas sudy appld o h Chla Mg P. Vvros, A. Crspo, R. Tapa, F. Krsjapollr, V. Gozálz-Prda ha h us of GRP modl s suabl for h cas. For hs, h us of a radoal adjusm would b complly corrc. Sp 3: Aalyss Wh hs rsul, o possbl aalyss s o projc (corrcly h upm falurs, ad drcly h ra of cras of h frucy of falur ad h dcrasg of oprao ms. Fg. 9: P.D.F. of GRP modl, for lapsd m -. RlPro Sofwar. (Sourc: ow laborao Th xpcd m of corrc prformac, a h prvous sa ha h oprao s rcovry (afr falur, s drmd by h dffrc bw h xpcacy of probably dsy fuco (basd o h oal lapsd m ad h vrual ag of h ass. Graphcally, Fgurs 9 ad 0 rprs. Fg. 0: Expcd TBF accordg o h oprag m lapsd ul h las rvo. RlPro Sofwar. (Sourc: ow laborao To sma h umbr of accumulad falurs ovr m, s possbl o cosdr ha h frs falur occurs a h MTBF, xpcd valu a h bgg of h oprao of h upm. Thus s possbl o projc rcursvly wh h xpcd MTBF h vry occurrc of falur. Obvously, hs s a gralzao ad smplfcao of h problm. Fally, s possbl o oba h graph of cumulav umbr of vs for a oal m of oprao. S Fgur wh h faul bhavour accumulad v/s oal m of oprao. Clarly a rsg rd ad acclrad falur m lss oprao s dfd. Ths s udrsood as acv agg m. Aalycally (25 o (30: [ > ] E ( f ( > d (25 Publcacos DYNA SL -- c Mazarrdo º69-3º BILBAO (SPAIN Tl mal: dya@rvsadya.com Pag. 2 / 6

13 Rlably Sochasc Modlg for Rparabl Physcal Asss. Cas sudy appld o h Chla Mg P. Vvros, A. Crspo, R. Tapa, F. Krsjapollr, V. Gozálz-Prda Pag. 3 / 6 Publcacos DYNA SL -- c Mazarrdo º69-3º BILBAO (SPAIN Tl mal: dya@rvsadya.com ] [ ( > E MTBF (26 ( ( ( > d f MTBF (27 (28 (29 ( (, (, (, ( ( Γ + Γ + Γ + s s p R MTBF MTBF dp p MTBF s s (30 ( d MTBF ( d MTBF

14 Rlably Sochasc Modlg for Rparabl Physcal Asss. Cas sudy appld o h Chla Mg P. Vvros, A. Crspo, R. Tapa, F. Krsjapollr, V. Gozálz-Prda Fg. : Forcas for Numbr of Falurs. RlPro Sofwar. (Sourc: ow laborao I addo, h sofwar ool usd (RlPro allows dagrammg h m voluo curvs: probably dsy fuco f(, h cumulav probably fuco F(, rlably R( ad h falur ra λ(. S Fgur 2. Fg. 2: Curvs f(, F(, R( ad λ(, RlPro Sofwar. (Sourc: ow laborao By dfaul, RlPro graphs h curvs whch wll com udr h lms afr h x vs (ral sadard. Wh hs s possbl o o how uckly dgrads lm. Aalysg graphc curvs Fgur 2, h drorao of rlably curv afr h occurrc of a falur s dfd. A h sam m s possbl o o ha h curv of probably dsy fuco crass s dsy, approachg vry m o h org valus. For h falur ra, afr ach falur s crmd uval rvals. 4. FUTURE LINES OF APPLICATION Ths mhodology s focusd o praccal applcao, so furhr rsarch may focus o dffr applcao xampls for dffr yps of machs (usd mg or aohr scor. I such furhr applcaos, may b rsg o spcfy h codos of us of, whch maac s carrd ou, h oal oprag m, ulzao ms, vromal or procss codos, c. Wh hs wll b possbl o compar h rsuls obad for boh yps of machry accordg o dffr boudary codos. 5. CONCLUSIONS Publcacos DYNA SL -- c Mazarrdo º69-3º BILBAO (SPAIN Tl mal: dya@rvsadya.com Pag. 4 / 6

15 Rlably Sochasc Modlg for Rparabl Physcal Asss. Cas sudy appld o h Chla Mg P. Vvros, A. Crspo, R. Tapa, F. Krsjapollr, V. Gozálz-Prda Th rlably modl s a ssal aspc for h maagm ad opmzao of physcal dusral asss. I ordr o lar dal h sp by sp of ach modl s a fudamal ask o apply ffcv ad corrcly ach modl. Dvrs rsarchs om h procss of rsoluo ad oly prs fal rsuls dcag h us of a modl ad h us of som compud ool wh grad algorhm. I was dfd dffr rsarchs, whch movad h rsarch am o dvlop a spcfc cocpual par ad rsoluo praccal for ach sochasc paramrc modl formr mod. Ths s fudamal o rcogz h valu of hs work ad s corbuo for fuur rsarchrs who wsh o lar ad apply hs kowldg. For hs raso, hs rsarch bcoms a aalyc ad xplcav procdur abou h dfo, calculao mhodology ad crra ha mus b cosdrd o paramrz dusral asss udr cra dgradao lvl afr maac, complmg addo s aalyss wh a umrc applcao ha allows dmosrag sp by sp h mahmac dvlopm as appropra. Th praccal cas was dvlopd mg dusry of Chl. I s worh mog hr h fru lack of fdback bw usrs ad maufacurrs of upm, rsulg gorac by maufacurrs abou h ral waksss of h machs. Th prformac of asss udr dal codos (lab s s xrmly dffrcs comparg wh ral procss codos. As a scod phas of hs rsarch for poal publcao, h rsarch am s aalyzg a prsorg o h rspcv paramr ha prss h curr arcl, whch corrspods o h dfcao whhr h modl s oparamrc. Th mhod s paramrc (MP f h modlg fs a probably dsrbuo fuco kow; o h ohr had, f you cao mak hs assumpo, h mhod s oparamrc (MNP. Thr ar also modls ha coa a poro of h paramrc fuco ad o o, hs ar h sm-paramrc mhods (MSP. Th lar classfcao (MSP s of gra rs for rsarch ad applcao, sc pracc grally h asss ar subjc o may varabls ha classcal modls o cludd h modlg ad aalyss of falur ra, for xampl: oprag mpraur, workload, dagoss of lubrcas (pars pr mllo, c. Ths varabls ar o cosa ad ca caus chags h rlably of a compo, whch s cssary o aalyz ad uafy for dsgg a ffcv ad ffc maac polcy. I addo, s xpcd o corpora o h aalyss h TRP modl (Trd-Rwal- Procss [8], dscrbd ad sudd dal by Ldvs, Elvbakk ad Hgglad [9], hs bg a dffr modl of mprfc rpar, wh smlar characrscs o NHPP. Modr mhods allow modlg basd o hs vromal facors ad srss, bu hy ar boudd o assumpos or rsrcos o h umbr of facors o aalyz, whch maks mor complx applcao ad obvously hr sysmac us h rlably aalyss. 6. APPRECIATION Th auhors would lk o ackowldg h suppor of h Scfc Char of MM BLad for Oprao ad Maac Tchology a Tabah Uvrsy, Mada, Saud Araba. 7. BIBLIOGRAPHY [] Aschr, H., & Fgold, H. (984. Rparabl sysms rlably: modlg, frc, mscocpos ad hr causs. Nw York, NY: Marcl Dkkr. [2] Vbr, B., Nagod, M., & Fajdga, M. (2008. Gralzd rwal procss for rparabl sysms basd o f Wbull mxur. Rlably Egrg ad Sysm Safy, 93, pp do:0.06/j.rss [3] Kjma, M., & Suma, N. (986. A usful gralzao of rwal hory: coug procss govrd by o-gav markova crms. Joural Appld Probably, 23, pp do:0.2307/3247. [4] Krvsov, V. (2000. Mo Carlo approach o modlg ad smao of h gralzd rwal procss rparabl sysm rlably aalyss, Ph.D dssrao, Uvrsy of Marylad. [5] Muhammad, M., Abd Majd, M. A., & Ibrahm, N. A. (2009. A cas sudy of rlably assssm for crfugal pumps a prochmcal pla. I: 4h World Cogrss o Egrg Ass Maagm, Ahs. do:0.007/ _44. [6] Wag, P., & Co, D. W. (2005. Rparabl sysms rlably rd ss ad valuao. Aual Rlably ad Maaably Symposum, 20, pp [7] Kumar, U., & Klfsjö, B. (992. Rlably aalyss of hydraulc sysms of LHD machs usg h powr law procss modl. Rlably ad Egrg ad Sysm Safy, 35, pp do:0.06/ ( Publcacos DYNA SL -- c Mazarrdo º69-3º BILBAO (SPAIN Tl mal: dya@rvsadya.com Pag. 5 / 6

16 Rlably Sochasc Modlg for Rparabl Physcal Asss. Cas sudy appld o h Chla Mg P. Vvros, A. Crspo, R. Tapa, F. Krsjapollr, V. Gozálz-Prda [8] Lopra, C. M., & Maoas, E. C. (20. Aplcacó dl aálss d daos rcurrs sobr rrupors FL245 Ircoxó Elécrca. S.A. Rvsa Colombaa d Esadísca, 34(2, pp [9] Wckma, G. R., Shll, R. L., & Marvl, J. H. (200. Modlg h rlably of rparabl sysms avao dusry. Compurs ad Idusral Egrg, 40, pp do:0.06/s ( [0] Mas, A., & Zhao, W. (2005. Modlg ad aalyss of rparabl sysm wh gral rpar. IEEE Procdgs aual Rlably ad Maaably Symposum. do:0.09/rams [] Damaso, V. C., & Garca, P. A. (2009. Tsg ad prvv maac schdulg opmzao for agg sysms modld by gralzd rwal procss. Psu. Opr. Vol.29 (3, pp hp://dx.do.org/0.590/s [2] Malaya, Y. K., L, M. N., Bma, J. M., & Karcch, R. (2002. Sofwar rlably growh wh s covrag. IEEE Tras. Rl., 5, pp do:0.09/tr [3] Yaz, M., Joglar, F., & Modarrs, M. (2002. Gralzd rwal procss for aalyss of rparabl sysm wh lmd falur xprc. Rlably Egrg & Sysm Safy, 77, pp do:0.06/s ( [4] RlPro, Rlably ad Produco, Aalyss ad Smulao. (204. RlPro SpA. Rrvd Aprl 0, 204. hp:// [5] Slvaov, A. I., & Yudkvch, E. (972. Fudamos d la oría d vjcmo d la mauara. [6] Soskov, B. (972. Fudamos d la oría y dl cálculo d Fabldad. Mr. [7] Bbbgo, M., Ch-Dw, L., & Zks, R. (2009. Balacg bur- ad msso ms vroms wh caasrophc ad rparabl falurs. Rlably Egrg & Sysm Safy, 94(8, pp do:0.06/j.rss [8] Mara Luz Gámz, Bo Hry Ldvs. (205. Noparamrc smao rd-rwal procsss. Rlably Egrg & Sysm Safy. do:0.06/j.rss [9] B.H. Ldvs, G. Elvbakk, K. Hgglad. Th rd-rwal procss for sascal aalyss of rparabl sysms. Tchomrcs2003; 45:3 44. Publcacos DYNA SL -- c Mazarrdo º69-3º BILBAO (SPAIN Tl mal: dya@rvsadya.com Pag. 6 / 6