Available online at ScienceDirect. Procedia CIRP 63 (2017 ) The 50th CIRP Conference on Manufacturing Systems

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1 Avaabe oe a SceceDrec Proceda CIRP 63 (27 ) The 5h CIRP Coferece o aufacur Sysems Reaby easureme for usae aufacur Sysems wh Faure Ieraco Lyu Xu a, *, Yahao Che a, Fore Brad a, Fexu Zhou a, oro Gva a,b a Isue of Advaced aufacur Techooy, Toj Uversy, Shaha 292, Cha b Dparme of echaca Eer, Poecco d ao, Va La asa, a 256, Iay * Correspod auhor. Te.: ; fax: E-ma address: Lyxu@oj.edu.c Absrac Reaby s oe of he mpora facors for maufacur sysem. os researches assume ha he faure s depede ad he compoes oy have wo saes, whch w ead o accurae resus. I hs paper, a reaby mode s proposed cosder boh faure eraco ad mu-sae propery of he maufacur sysem. Sar wh a wo-compoe sysem, a fuco of sae probaby uder he mpac of faure eraco s esabshed afer he aayss of faure eraco. The he mu-compoe sysem s decomposed o severa subsysems ad he faure eraco coeffce s esmaed each subsysem wh a Copua fuco ad he Grey mode mehod. Fay, he reaby mode s reazed wh he performace eera fuco whch s derved wh he UGF echque ad faure eraco coeffces. A exampe of a cyder ee maufacur sysem s suded, ad he resu s coser o he pracca daa. 27 The Auhors. Pubshed by Esever by Esever B.V. Ths B.V. s a ope access arce uder he CC BY-NC-ND cese (hp://creavecommos.or/ceses/by-c-d/4./). Peer-revew uder resposby of he scefc commee of The 5h CIRP Coferece o aufacur Sysems. Peer-revew uder resposby of he scefc commee of The 5h CIRP Coferece o aufacur Sysems Keywords: musae maufacur sysem; reaby measure; faure eraco; uversa eera fuco; rey mode. Iroduco Reaby descrbes he probaby of a sysem compe s expeced fuco dur a erva of me, whch ves a assessme of he overa performace of a sysem. A accurae reaby measure for a sysem uaraees s fucoay, effcecy ad safey. Wh he deveopme of s scae ad compexy, maufacur sysem pus forward hher requremes for reaby aayss. os of he reaby aayses are based o wo smpfed assumpos: Frs, cosder he subsysems ad he whoe sysem as a bary sysem, whch meas he sysem has a fuy operaoa sae ad a compeey faed sae. Secod, o faure eraco exss bewee he compoes. Faure eraco here refers o a preva pheomeo ha oe subsysem`s faure or deradao w affec he faure process of oher reaed subsysems. The sudy of musae sysems (SS) sared he md- 97s[2], sce he os of research have bee carred ou hs area. ehods adoped o reaby aayss of SS may cude: oe Caro Smuao[3,4], Sochasc Process Aayss[5], Uversa Geera Fuco(UGF)[6] ad so o. Oher mehods were aso deveoped. D e a.[7] deveoped he fuzzy uversa eera fuco cosder a musae sysem where performace raes ad correspod sae probabes are preseed as fuzzy vaues. Lsask[8] exeded he cassca reaby bock daram mehod o a reparabe musae sysem based o he combed radom processes ad he uversa eera fuco echque. Taboada e a.[9] deveoped a cusom eec aorhm o sove mupe objecve musae reaby opmzao des probems. Qa e a.[] proposed a ew dscrezed mode process o Bayesa beef eworks bass for he reaby of mu-sae mechaca sysems. The sudy meoed above focus o he musae propery of SS o exed he sudy o reaby The Auhors. Pubshed by Esever B.V. Ths s a ope access arce uder he CC BY-NC-ND cese (hp://creavecommos.or/ceses/by-c-d/4./). Peer-revew uder resposby of he scefc commee of The 5h CIRP Coferece o aufacur Sysems do:.6/j.procr

2 Lyu Xu e a. / Proceda CIRP 63 ( 27 ) aayses. However, reaby ca be fueced by varous facors. P e a.[] cosdered jo buffer sao a mu-sae maufacur ework. Ife ad fe buffer voume was dscussed ad he sudy dcaed ha he assumpo of fe buffer voume w overesmae he sysem reaby. Faure eraco commoy exss musae sysems, whch may aso fuece he performace of he sysem. Dur he operao of a maufacur sysem, faures of a u caused by corroso, ae, wear or shock damaes ke mproper maeace or overwork may crease he oad of oher us, he affecs he faure characerscs of he oher us ad eveuay eads o he faure of he us. Aeo has bee pad o he faure mechasm ad faure eraco he sysems` reaby as we. Nakaawa ad urhy[2] dvded faure eraco o hree ypes, whch s wdey used hs area. Accord o hs caeory, researchers expaded faure eraco o may aspecs. La ad Che[3] deveoped a opma perodca repaceme pocy for a mu-u sysem subjec o faure rae eraco bewee us. Su e a.[4,5] roduced he coceps of eracve faure, deveoped a aayca mode ad preses fve approaches o esmae he eracve faures. Gao e a. [6] esabshed a reaby mode of sysem ad a quas-perodc dyamc Preveve Repaceme o research he coexss of ype I ad II faure eraco. Q e a.[7] preseed wo perodca maeace cos modes for a wo-sae seres sysem ad a hree-sae seres sysem respecvey based o he hree ypes of faure eracos. Cosder he faure eraco ves rse o he aayss of he sysem, however he resu aways yeds he pracca daa. The sudes meoed above focus o he faure eraco suded ssue o reaby ad maeace eher a musae wo-compoe sysem or a bary mucompoe sysem. Sudy focus o musae mucompoe sysem wh faure eraco s s facy. Ths paper for he frs me cosders faure eraco a mu-compoe musae maufacur sysem(ss) o derve he reaby. oreover, based o a prevous sudy o he fuece of he deeerao[8]. The reaby s ve wh wo faure mechasms, amey faure eraco ad he deeerao. To defe he faure eraco, he copua fuco ad Grey mode s used o fd he parameers as a ew approach. The res of he paper s orazed as foows. The ex seco ves a aayss of a wo compoes musae maufacur sysem wh faure eraco. arkov cha s used o represe a he saes ad her rasos. Seco 3 proposes a decomposo mehod of he sysem based o fau correao of he compoes ad fures ou he faure eraco coeffce us Copua fuco ad rey mode mehod. The he reaby mode s cosruced afer v he performace eera fuco of oe compoe ad he whoe sysem by us he UGF echque ad mapp reaoshp. Seco 4 dedcaes o a case sudy o a hree processes ee cyder maufacur sysem. The paper cocudes ha cosder faure eraco musae sysem w ve a ower reaby. 2. Faure Traso A Two-compoe SS 2.. Defo ad Assumpo Cosder a sysem wh wo maches. Ay mache ca have dffere saes s fecyce, represeed by he se L {,,,}, where deoes he ew sae ad deoes he compeey faure sae. Very md assumpos are requred for he sequea sudy. These are foows: A eera faus are maaed mmedaey afer occurs ad o maeace me s cosdered. Wheever a sysem faure occurs, s deeced mmedaey ad oy oe u fas auray. A faed us are correcvey maaed bu cao be repared as ood as ew. A performace aayses are doe oy whe he sysem s a a seady sae. The sae raso caused by faure eraco s saaeous Faure raso represeao Accord o he faure eraco caeory [2], he faure raso for wo maches s usraed Fure.. Whe a mache fas due o he fau of sef, such as corroso, ae ad so o, ca duce he faure of he reaed compoe refer o ache II Fure. wh a probaby of p; or creases he faure rae of mache II wh a probaby -p ad ves rse o he faure of mache II wh he accumuao of he damae. Whe he faure of mache II w duce a saaeous faure o mache I wh a probaby of q. Whe mache I fas due o a exera shock damae, he foow wo suao shoud be cosdered: The shock damae s que sma ha oy a eera faure occurs o mache I. The hazard w crease he faure rae of mache II wh a probaby of p` ad ead o faure whe he oa damae exceeds a specfed eve. Whe he faure of mache II caused by he damae w duce saaeous damae o mache I wh a probaby of q` ad ead o he faure of mache I. Oherwse, f he shock damae ves rse o a dramac desroy o mache I w ead o saaeous faure. The hazard w duce a saaeous faure of mache II wh a probaby of -p` ad resu faure of he sysem eveuay. -p` p` -p p q q` F. Faure raso bewee wo compoes 2.3. arkov mode for sae raso I s fecyce [, ), a maches w derade from sae o s compeey faure sae. The mache ca rasfer o

3 244 Lyu Xu e a. / Proceda CIRP 63 ( 27 ) ay ower sae from he sar sae due o he fuece of sef or he faure eraco. Cosder he sae deeerao ad faure eraco, he sae raso of a wo mache sysem s usraed Fure.2, where,j ad j, refer o he raso rae bewee dffere saes., 2,,2,2, -,, 2, 2 2,, ache I -p` p` -p p q -q ache II,2, 2,,2, F.2 Sae raso cosder he faure eraco -,, 2 Accord o he above aayss, oe ca fd ha he process of sae raso wh faure eraco s a sochasc process couous me erva [, ) ad fe sae space L () {,,,}, whch ves use of ohomoeeous arkov cha. Based o Fure.2, he sae raso marx of compoe s derved: I II I I (, ) (. ) (, 2) (, 2) (,) (,) I (, 2) (, 2) (,) (,) (,) (,) = faure eraco coeffce of mache I respec o mache II; () (, j) = probaby of sae raso ad Pr L ( ) jl ( ) (, j) Lm (2) 2,,, 2 () probaby of p, mak mache II work a ower sae. k Suppose he faure rae of mache II s r ( ) r wh represes he faure eraco coeffce. The we ca derve he probaby of each sae for mache II: p(, ) Pr( ( ); N ( )) Pr{ X ( ) },,, ;Pr{ N ( ) k} m 2 (,) sds (, 2) 2 exp ( ) ( ) x r exp (,) ( sds ) (, ) ( r ) r m exp (,) ( sds ) (,) (, j) j j ( ) exp( () ) dd ; jk j! exp ( sds ) XII = fuco of faure rae respec o me; NI =mache I`s faure umber me erva [,); = sae umber ha mache I passed from sae m o sae ; k = he k h faure. 3. Reaby aayss of u-compoe SS To measure he reaby of a SS wh faure eraco, decomposo of he sysem s eeded so ha he faure eraco oe subsysem ca be dscussed. Based o he eraco aayses, a Copua fuco of faure eraco coeffce s cosruced ad rey mode mehod w be used o deerme he faure eraco coeffce. The, he reaby mode wh faure eraco s reazed. S S 2 Sub Sub 2 Sub 3 (3) 2.4. Sae probaby aayss cosder he faure eraco (-) k S 3 Sub 4 For ay musae maufacur sysem, he faure rae of compoes w crease due o he operao of he sysem. The probaby of each sae s a resu of boh era ad exera facors of he compoe. I auso o he sae raso process of a wo us sysem Fure., suppos mache I ad II have a servce year of I ad II. The faure of mache I w foow a ohomoeeous Posso process wh mea-vaue fuco uxdx ( ) ad a esy rao of u () r p(), () where r s faure rae of mache I ad () s he shock damae rao fuco. Ad he sae probaby for mache I s derved paper[8]. Whe he sae probaby of mache II s decded by sef ad he faure of mache I. Whe mache I fas, w fuece he performace of mache II wh a (-) h (+) F.3 A hree-process maufacur sysem Cosder a musae maufacur sysem wh hree processes S, S 2 ad S 3, showed F.3. Ay compoe of a process has dffere saes correspod o dffere performace raes. The feme of a compoes are radom varabes wh dsrbuo fuco F(), F () 2,, F ad he compoes sar from he ew sae a. 3.. Decomposo of he u-compoe maufacur sysem A maeras fow from he frs process o he foow procedures. Ay compoes of a prevous process`s faure

4 Lyu Xu e a. / Proceda CIRP 63 ( 27 ) w mpac he performace of he compoes he foow process. A decomposo of he sysem s show F.3 based o he eraco caused by maera fow. The whoe sysem s hus dvded o four subsysems: Sub (composed of (-) ad ), Sub 2 (composed of (-), k ad (+) ), Sub 3 (composed of (-)d, k ad (+) ) ad Sub 4 (composed of (-)d, h ad (+) ) Esmao he faure eraco. Copua s a fuco ha jos or coupes muvarae dsrbuo fucos o her oe-dmesoa mara dsrbuo fucos ad descrbes correao bewee he radom varabes. The reevace of dsrbuoa copuas for faure eraco s may because of Skar`s heorem, whch saes ha for a ve copua C ad mara dsrbuo fucos, he jo dsrbuo ca be obaed va: F( x, x,, x ) C( F( x ), F( x ),, F( x )) (4) 2 2 If F( x), F( x2),, F( x ) are couous, he C s uque; oherwse, C s uquey deermed o radom varabes of F. I auso o a musae maufacur sysem f we have he faure rae fucos ad he jo dsrbuo fuco of wo eraco compoes, he faure eraco coeffce ca be cacuaed by choos a proper Copua fuco foowed by esma parameers. Cosder a SS wh h subsysems, each subsysem s comprsed of severa maches. Se he faure rae fuco of he compoe as Px ( ),,2, ad he jo dsrbuo fuco as Px (, x2,, x ). Accord o he Copua fuco ad Skar`s heorem, a Copua fuco CPx ( ( ),, Px ( ); ) ca be oba, ad sasfy he foow reaoshp: Px (, x,, x) CPx ( ( ), Px ( ),, Px ( ); ) (5) 2 2 For ay maufacur sysem, he faure eraco exss oy bewee some compoes whe he ohers rema depede. So he faure rae fuco shoud wre as: m Px ( ) C( x; ) P( x) j j (6) C ( x ) = eera faure rae of compoes wh faure eraco; Pj ( x ) = faure rae of depede compoe; = eraco coeffce of compoe. Tradoay, copua parameers are deermed by ome Esmao ehod or axmum Lkehood Esmao ehod[9]. The mehod of mome esmao s fary smpe bu he resu may o ecessary suffce, whe he mehod of maxmum kehood esmao s o he corary. Grey sysem heory s a ew mehodooy ha focuses o probems vov sma daa ad poor formao. I deas wh ucera sysems wh paray kow formao hrouh sequece operaors, excava, ad exrac usefu formao from wha s avaabe ad ves a resu wh hh accuracy afer smper cacuao. The faure eraco coeffce s derved from he faure rae ad he correao bewee compoes, whch exacy mee he requreme for us Grey ode mehod. The ypca procedure of esma parameers wh Grey mode s he G(,) mode[2] Reaby mode wh UGF echque As meoed prevousy, he are sysem hs paper ca be decomposed o four subsysems. Take subsysem 2 as a exampe. Whe (-) fas, w have a mpac o mache k, ead o he crease of s faure rae ad he chae of he performace ad sae probabes. Suppose he creased faure rae of k s p, he creased performace of each sae s. The he performace eera fuco of k cosder he faure eraco ca be derved us he UGF echque: k ' k ( ) k ( ) k k k k u z p z u z p pz pz pz z uk ( z) = varao of he eera fuco; p = probaby of he mache a sae ; = performace of he mache a sae ; = faure eraco coeffce. For he whoe sysem a mapp from compoes performace o sysem ca be reazed as foow: U( z) u ( z) u ( z) u ( z) u ( z) s s2 s3 s4 h ~ h ~ h ~ h ~ [ pz u( z)] [ pz u ( z)] [ p z u ( z)] [ p z u ( z)] Accord o he defo of reaby, he reaby mode of he sysem s esabshed: R ( U( z), ) s (,) (,2) (,3) s ( p z p z p z (,) (,2) (,3) k j p( (,) (,)) j ( (,) (,)) j z j (, ) k f p k ( (,) (,) ) f m ( (,) (,)) f z, ) p( r j (, ) ) = requred performace eve; ( ) = characersc fuco, akes whe r ; r = performace vaue. (9) (7) (8)

5 246 Lyu Xu e a. / Proceda CIRP 63 ( 27 ) Numerca Exampe The maufacur sysem usraed F.4 s used o mache a ee cyder wh hree processes. Each process has severa maches of he same characersc. The reaby of each mache s he resu of operao me, work oad ad maeace. Due o he dffere process me of each process, he uzao of each mache s o he same, whch eads o a dffere deeerao each mache. The probaby of sae raso due o s deeerao s ve Tabe. Accord o he decomposo proposed seco 3., he ee cyder maufacur sysem ca be dvded o four subsysems: Sub (comprsed of, 2 ad 3 ), Sub 2 (comprsed of, 2 ad 32 ), Sub 3 (comprsed of, ad 3 ), Sub 4 (comprsed of, ad 32 ). The he fuco of faure eraco coeffce ca be obaed from formua (6): P( X) C ( x; ) C ( x; ) C ( x; ) Sov he fuco by us rey mode mehod, he faure eraco coeffces are obaed Tabe 3. Tabe 3. Faure eraco coeffce Vaue Uz he probaby of sae raso, faure eraco coeffce ad he probaby of faure raso, he performace ad probaby a each sae of he mache are obaed Tabe 4: F.4 Ee cyder maufacur sysem Tabe. Probaby of he sae raso respec o deeerao (3,2).2 (3,). (3,) (,) (,) (,) (,).2 Whe fas, w duce he faure of 2 ad wh a probaby of p, p2 respecvey, ad ead o he faure of he sysem eveuay; or creases he faure rae of 2 ad wh he probaby of p ad p2,ad eads o he faure of 2 ad oy f he damae accumuaes o a proper eve. Whe he faure of 2 ad w duce a faure of wh a probaby of q2 ad q. The same faure raso exss bewee 2 respec o 3 ad 32, respec o 3 ad 32. The probaby of he faure raso s ve Tabe 2. Tabe 2. Probaby of faure raso pj pj q j Tabe 4. Performace ad probaby a each sae ache Sae performace Probaby / / Based o he cacuao above, he eera fuco of each mache cosder faure eraco ca be obaed from equao (7): u ( z).486z 2.283z.53z.75z u ( z) 2.87z 2.736z 2.765z u ( z) 2.288z 2.54z 2.83z u ( z).967z 2.499z 2.4z u ( z).468z 2.z 2.35z The he eera fuco of he sysem s obaed from formua (8): U ( z) u ( z) u ( z) u ( z) u ( z) u ( z) s z.z.8z Accord o he defo of reaby ad formua (9), he reaby of he sysem s: Rs s( U( Z), ) 95 5 S (.6z.z.8 z,35).67 If o faure eraco s cosdered, he eera fuco shoud be:

6 Lyu Xu e a. / Proceda CIRP 63 ( 27 ) U ( z) (.293z.45z.6z.4z s (.43z.279z..38 z ) (.43z.279z.38 z ) (.98z.63z.272 z ) (.98z.63z.272 z ) z.2z 36z The he reaby of he sysem s: Rs s( U( Z), ) S (.4z.2z 36 z,35).73 Compar he resus, ca be foud ha cosder he faure eraco w ve a ower resu reaby aayss, whch aso proves ha maufacur sysem s a compex sysem wh varous faure mechasm, cosder oy deeerao by sef w fae he error of he reaby aayss ad fuece he predco of performace of he sysem eveuay. Tak o accou wo faure mechasm he esabshed mode compcaes he compuao bu ves a more accurae resu. 5. Cocuso Ths paper esabshed a ew reaby aayss mode cosder he faure eraco he musae maufacur sysem. The mode cosdered wo faure mechasm of a maufacur sysem, amey faure eraco ad deeerao of he compoes. To prese he fuece of faure eraco, faure eraco coeffce was roduced, ad derved wh a copua fuco ad Grey mode mehod. A case sudy was proposed o vadae he mode. The resu dcaes ha despe he faure eraco w overesmae sysem reaby, whch aso shows ha he proposed mehod s more accurae ad reasoabe. The proposed mode aso provdes a bass for maeace pocy sudy. To acheve a more pracca sh, preveve maeace pocy cosder he faure eraco w be furher sudy he fuure. Ackowedemes Ths research s paray suppored by Naoa Scece ad Techooy ajor Projec of he sry of Scece ad Techooy of Cha (2ZX45-) ad Shaha ucpa Scece ad Techooy Commsso (6652). Refereces [] Lsask A, Freke I, D Y. u-sae Sysem Reaby Aayss ad Opmzao for Eeers ad Idusra aaers[j]. 2. [2] urchad J D. Fudamea coceps ad reaos for reaby aayss of mu-sae sysems.[j]. Reaby & Fau Tree Aayss: [3] Ramrez-arquez J E, Co D W. A oe-caro smuao approach for approxma mu-sae wo-erma reaby[j]. Reaby Eeer & Sysem Safey, 25, 87(2): [4] Ya J J, Lu F, Fa L I. Reaby ode ad Smuao for Phased sso Sysem wh u-mode Faures[J]. Fre Coro & Commad Coro, 2, 36(2): [5] Ave T, Jese U. Sochasc modes reaby /[]. Sprer, 999. [6] Youssef A A, ohb A, Emarahy H A. Avaaby Assessme of u-sae aufacur Sysems Us Uversa Geera Fuco[J]. CIRP Aas - aufacur Techooy, 26, 55(): [7] D Y, Lsask A. Fuzzy uversa eera fucos for mu-sae sysem reaby assessme[j]. Fuzzy Ses & Sysems, 28, 59(3): [8] Lsask A. Exeded bock daram mehod for a mu-sae sysem reaby assessme[j]. Reaby Eeer & Sysem Safey, 27, 92(2):6-67. [9] Taboada H A, Espru J F, Co D W. OS-GA: A u-objecve u-sae Geec Aorhm for Sysem Reaby Opmzao Des Probems[J]. IEEE Trasacos o Reaby, 28, 57():82-9. [] Qa W X, Y X W, Xe L Y. Dscrezed ode Process of Reaby of u-sae echaca Sysems[J]. Dobe Daxue Xuebao/joura of Norheaser Uversy, 28, 29(): [] Cha P C, L Y K, Che J C. Sysem reaby for a mu-sae maufacur ework wh jo buffer saos[j]. Joura of aufacur Sysems, 27, 42:7-78. [2] Nakaawa T, urhy D N P. Opma repaceme poces for a wo-u sysem wh faure eracos[j]. RAIRO - Operaos Research, 993, 27(4). [3] La T, Che Y C. Opma perodc repaceme pocy for a wo-u sysem wh faure rae eraco[j]. The Ieraoa Joura of Advaced aufacur Techooy, 26, 29(3): [4] Su Y, a L, ahew J, e a. A Aayca ode for Ieracve Faures[J]. Reaby Eeer & Sysem Safey, 26, 9(5): [5] Su Y, a L. Esmae of eracve coeffces[c]// Reaby, aaaby ad Safey, 29. ICRS 29. 8h Ieraoa Coferece o. IEEE, 29:3-34. [6] Gao W K, Zha Z S, Lu Y, e a. Reaby mode ad dyamc repaceme pocy for wo-u parae sysem wh faure eracos[j]. Jsuaj Jche Zhzao Xo/compuer Ieraed aufacur Sysems Cms, 25, 2(2):5-58. [7] Q G, Ya G. aeace erva decso modes for a sysem wh faure eraco[j]. Joura of aufacur Sysems, 25, 36:9 4 [8] Zhou F X, A-P LI, Na X, e a. Reaby aayss of mu-sae maufacur sysems based o performace deradao[j]. Jsuaj Jche Zhzao Xo/compuer Ieraed aufacur Sysems Cms, 24, 2(6): [9] B Raves. A roduco o copuas[]// A roduco o copuas /. Sprer, 26:xx,3 [2] Hsu C C, Che C Y. Appcaos of mproved rey predco mode for power demad forecas[j]. Eery Coverso & aaeme, 23, 44(4):4-4