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1 Ken Academc Reposory Full ex documen (pdf) Caon for publshed verson Du, Hongyan and Chen, Lwen and Wu, Shaomn (27) Generalzed negraed mporance measure for sysem performance evaluaon: applcaon o a propeller plane sysem Manenance and Relably, 9 (2) pp ISS DOI hps://doorg/753/en2726 Lnk o record n KAR hp://karkenacuk/6549/ Documen Verson Publsher pdf Copyrgh & reuse Conen n he Ken Academc Reposory s made avalable for research purposes Unless oherwse saed all conen s proeced by copyrgh and n he absence of an open lcence (eg Creave Commons), permssons for furher reuse of conen should be sough from he publsher, auhor or oher copyrgh holder Versons of research The verson n he Ken Academc Reposory may dffer from he fnal publshed verson Users are advsed o check hp://karkenacuk for he saus of he paper Users should always ce he publshed verson of record Enqures For any furher enqures regardng he lcence saus of hs documen, please conac: researchsuppor@kenacuk If you beleve hs documen nfrnges copyrgh hen please conac he KAR admn eam wh he ake-down nformaon provded a hp://karkenacuk/conachml

2 Arcle caon nfo: DUI H, CHE L, WU S Generalzed negraed mporance measure for sysem performance evaluaon: applcaon o a propeller plane sysem Eksploaacja ezawodnosc Manenance and Relably 27; 9 (2): , hp://dxdoorg/753/en2726 Hongyan DUI Lwe CHE Shaomn WU Generalzed negraed mporance measure for sysem performance evaluaon: applcaon o a propeller plane sysem uogólnona mara znegrowanej ważnośc komponenów jako narzędze oceny wydajnośc sysemu: zasosowane w odnesenu do układu śmgłowca The negraed mporance measure (IIM) evaluaes he rae of sysem performance change due o a componen changng from one sae o anoher The IIM smply consders he scenaros where he ranson rae of a componen from one sae o anoher s consan Ths may conradc he assumpon of he degradaon, based on whch sysem performance s degradng and herefore he ranson rae may be ncreasng over me The Webull dsrbuon descrbes he lfe of a componen, whch has been used n many dfferen engneerng applcaons o model complex daa ses Ths paper exends he IIM o a new mporance measure ha consders he scenaros where he ranson rae of a componen degradng from one sae o anoher s a me-dependen funcon under he Webull dsrbuon I consders he condonal probably dsrbuon of a componen sojournng a a sae s he Webull dsrbuon, gven he nex sae ha componen wll jump o The research on he new mporance measure can denfy he mos mporan componen durng hree dfferen me perods of he sysem lfeme, whch s correspondng o he characerscs of Webull dsrbuons For llusraon, he paper hen derves some probablsc properes and apples he exended mporance measure o a real-world example (e, a propeller plane sysem) Keywords: sysem performance, mporance measure, Webull dsrbuon, ranson rae Mara znegrowanej ważnośc (IIM) pozwala ocenać szybkość zman wydajnośc sysemu powsałych w wynku przejśca elemenu sysemu z jednego sanu do drugego IIM pozwala rozważać scenarusze, w kórych szybkość przejśca elemenu z jednego sanu do drugego jes sała Jes o jednak sprzeczne z założenem degradacj, zgodne z kórym wydajność sysemu obnża sę, w zwązku z czym, szybkość przejśca może z upływem czasu ulegać zwększenu Rozkład Webulla opsuje żywoność danego elemenu, co wykorzysuje sę w welu różnych zasosowanach echncznych do modelowana złożonych zborów danych W przedsawonej pracy, rozszerzono IIM uzyskując nową marę ważnośc, kóra pozwala rozważać scenarusze, w kórych szybkość przejśca elemenu z jednego sanu do drugego w wynku degradacj jes zależną od czasu funkcją rozkładu Webulla Przyjęo, że warunkowy rozkład prawdopodobeńswa elemenu przebywającego w pewnym sane jes rozkładem Webulla, gdze dany jes kolejny san do kórego ma przejść dany elemen Badana nad nową marą ważnośc umożlwają denykację najważnejszych elemenów podczas rzech różnych okresów czasu życa sysemu, co odpowada charakerysyce rozkładów Webulla Dla lusracj, wyprowadzono pewne właścwośc probablsyczne zasosowano rozszerzoną marę ważnośc do analzy przykładu rzeczywsego układu śmgłowca Słowa kluczowe: wydajność sysemu, mara ważnośc, rozkład Webulla, szybkość przejśca oaon x () sae of componen a me, x( ) =,,2,, M X () ( x( ), x2( ),, xn( )) : sae vecor of he componens a me Φ ( X( )) sysem srucure funcon a me and range {,,, M}, Φ ( X ( )) =Φ( x( ), x2( ),, xn( )) (, X( )) ( x( ),, x ( ),, x+ ( ),, xn( )) P, ml () Pr( x( ) = l x() = m) Pm, () Pr( x ( ) = m) b, ml ranson rae of componen from sae m o sae l c j manenance cos of mprovng he sysem from sae j o sae M { } ρ m, () Pr x ( ) < m ρ ml, () Pr { x ( ) < l x () = m} EKS P L O A T A C J A I I E Z A W O D O S C MA I T E A C E A D RELIABILITY VO L9, O 2,

3 Inroducon In relably engneerng, a bulk of research has been devoed o evaluae he conrbuon of componens o sysem relably and performance [4 For example, Yang e al [26 provded a mehod of smulang he relably of degradaon usng Mone Carlo prncple and cloud heory Cheng e al [6 proposed an approach o analyze he relably evaluaon based on fas Markov chan smulaons Leurondo e al [2 presened a model o evaluae he sysem performance wh localzed damage Werbńska-Wojcechowska and Zając [2 used a delay-me concep o analyze he sysem manenance performance The semnal work of relably mporance measure s creded o Brnbaum, who nroduce he well-known Brnbaum mporance n 969 [ Snce hen, a wde range of dfferen mporance measures have been nroduced For example, Wu and Chan [22 proposed he uly mporance of componen saes n mul-sae sysems Levn e al [3 generalzed he mporance measures for mul-sae elemens based on performance level resrcons Wu and Coolen [24 exended he Brnbaum mporance o a cos-based mporance measure Borgonovo e al [2, 3 proposed dfferenal mporance measure, and me-ndependen relably mporance for he rsk evaluaon Zha e al [27 presened a momen-ndependen mporance o evaluae he safey probably Tyrvänen [2 presened rsk mporance measures o analyze he dynamc relably Wu e al [23 proposed a componen manenance prory mporance o mprove he sysem performance Duu and Rauzy [9 exended he mporance measures o complex componens Kuo and Zhu [, 28 summarzed he conceps of mporance measures and her applcaon n relably and mahemacal programmng S, Du e al [7, 8 proposed he negraed mporance measure (IIM) of componen saes, whch evaluaes how he ranson of componen saes affecs he sysem performance n mul-sae sysems S, Du e al [9 suded he IIM from componen saes o he componen, whch can denfy he mos mporan componen for mprovng he sysem performance Du e al [7, 8 suded he IIM n sysem lfeme and sem-markov process o evaluae he change of he sysem performance, respecvely The IIM evaluaes he rae of sysem performance change due o a componen changng from one sae o anoher The IIM smply consders he scenaros when he ranson rae of a componen from one sae o anoher s consan Ths may conradc he assumpon of he degradaon, based on whch sysem performance s degradng and herefore he ranson rae may be ncreasng over me On he oher hand, he Webull dsrbuon s one of he mos commonly used lfeme dsrbuons n relably modelng and lfeme esng [25 I has been used n many dfferen engneerng applcaons o model complex daa ses, such as lfe ess [5, faul dagnoss [4, 5, oral rrgaors [6, e al The Webull dsrbuon and s varans can accommodae ncreasng, consan or decreasng falure raes [ Thus, one may exend he IIM o a new mporance measure ha consders he scenaros where he ranson rae of a componen changng from one sae o anoher as a me-dependen funcon Typcally, one may consder he condonal probably dsrbuon of a componen sojournng n a sae s he Webull dsrbuon, gven he nex sae ha he componen wll jump o On he bass of such consderaon, hs paper proposes a new mporance measure The research on he new mporance measure can denfy he mos mporan componen durng hree dfferen me perods of he sysem lfeme, whch s correspondng o he characerscs of Webull dsrbuons The paper hen derves some probablsc properes I also analyzes he properes of he proposed mporance measure of he parallel-seres sysems and he seres-parallel sysems, respecvely A real-world example s borrowed o llusrae he proposed mporance measure The res of he paper s as follows Secon 2 exends he IIM The correspondng properes of IIM n he Webull dsrbuon are analyzed for ypcal parallel-seres sysem and seres-parallel sysem srucures n Secon 3 An applcaon s presened o llusrae he proposed mehod n Secon 4 Secon 5 gves he concluson of hs paper 2 The exended IIM for sysem performance under Webull dsrbuons In hs paper, he sae space of componen s {,,, M } and he sae space of he sysem s{,,, M} Sae represens he complee falure sae and sae M (M ) s he perfec funconng sae The saes are ordered from he complee falure sae o he perfec funconng sae We assume ha he levels of manenance cos and he sysem saes ( c c c c = ) are nversely proporonal The M M expeced manenance cos s M M C = cjpr( Φ ( X) = j) = ( cj cj+ Pr( Φ( X) j) S, Du e al [7 gave he followng IIM, as n Equaon ( IIM ml, Pm, bml, c j c j m X j l X j ( ) = ( ) ( + ) Pr ( Φ( ( ), ( )) ) Pr ( Φ( ( ), ( )) ) ( ) Pr ( ( ( ), ( )) ) Pr ( ( ( ), ( )) ), = Pm, bml, cj Φ m X = j Φ l X = j m< l Then IIM,ml () descrbes he rae of manenance cos loss due o a componen mprovng from sae m o sae l a me In Equaon (, b, ml, whch s he ranson rae of componen from sae m o sae l, s a quany ndependen of me Inuvely, b, ml may be a me-dependen quany as he componen s a deerorang/ageng un Hence, he IIM defned n Equaon ( s oo resrcve In order o arac wder applcaons, one may exend he IIM by nroducng he followng mporance measure Defnon The exended negraed mporance measure (EIIM) s gven by: EIIM ml, (, y)= P, m( ) b, ml ( y) cj Pr ( Φ( m(), X( )) = j) Pr ( Φ ( l (), X( )) = j), m< l In Equaon (2), b, ml ( y ) s a funcon of he sojourn me y An neresng queson s wha form of funcon b, ml ( y ) should be In he followng, we consder he case when he Webull dsrbuon s adoped Le he wo-parameer Webull dsrbuon W( ; θ, ) be denoed by: ( (2) W( ; θ, )= exp ( / θ), θ, >, (3) where θ and are he scale and shape parameers, respecvely We can hen oban he dsrbuon funcon of he sojourn me n sae m and he ranson rae for componen Denoe θ ml, he scale param- 28 EKS P L O A T A C J A I I E Z A W O D O S C MA I T E A C E A D RELIABILITY VO L9, O 2, 27

4 eer of componen n sae m, gven ha he nex sae s l, and he shape parameer of componen Then he ranson rae b, ml ( y ) s bml, ( y) / θml, y / θ, ml = ( )( ) Accordng o Equaon (2), he expresson of he EIIM under he Webull dsrbuon s as followng: Defnon 2 The EIIM under he Webull dsrbuon s gven by: EIIMml, ( y, )= P, m()( / θml, )( y/ θ, ml ) ( cj cj+ Pr ( Φ( m (), X( )) j) Pr ( Φ( l ( ), X()) j) Pm, = () ( )( ) = / θ ml, y/ θ, ml c j Pr ( Φ( m (), X( )) j) Pr ( Φ( l ( ), X()) = j) (4) I s known ha wh dfferen shape parameers, he Webull dsrbuon becomes dfferen dsrbuons, such as he exponenal dsrbuon, he Raylegh dsrbuon, he normal dsrbuon Thus, wh dfferen shape parameers, Equaon (4) can be convered no dfferen expressons under dfferen dsrbuons When M =, he mul-sae sysem reduces o a bnary sysem Equaon (4) can be convered no: EIIM, y, P, / θ, y/ θ, c Pr ( Φ( (), X( )) = ) Pr ( Φ ( ( ), X()) = ) (5) ( )= ()( )( ) I s obvous ha he las erm n he rgh hand sde of Equaon (5) s he Brnbaum mporance measure (BM) of componen, as n Equaon (6): BM()= Pr ( Φ( ( ), X ()) = ) Pr ( Φ( (), X( )) = ) (6) From Equaons (5) and 6, he EIIM of componen s a generalzaon of BM based on he sysem performance We wll dscuss he dfference beween BM and EIIM for he change of dfferen parameers n secon 4 3 Characerscs of EIIM for ypcal sysem srucures We now dscuss he properes of he EIIM for parallel-seres sysem and seres-parallel sysem srucures Fg gves he srucure of a ypcal parallel-seres sysem, where [j represens he componen ha s locaed n row and column j The correspondng srucure funcon of he sysem s Φ( X( )) max mn X[ j () j = { } Fg A parallel-seres sysem The srucure of a ypcal seres-parallel sysem s as n Fg 2, where [j represens he componen whch s locaed n column and row j The correspondng srucure funcon of he sysem s Φ( X( )) mn max X () = { [ j } j Fg 2 A seres-parallel sysem Assume ha he lfeme dsrbuon funcon of componen [j follows Webull dsrbuon W ; θ j ml, j proposons: ( [, [ ) We can oban he followng EKS P L O A T A C J A I I E Z A W O D O S C MA I T E A C E A D RELIABILITY VO L9, O 2, 27 28

5 Proposon In a parallel-seres sysem, assume =, hen EIIM (, y) EIIM ( y, ) only when P[ j, m() / θ[ j, m( m+ αm( )( ρ[ j, m( )) [ j ( ) [ j [ 2j2 [ 2 j2 P[ j, m() / θ [ 2j2, m( m+ αm( 2)( ρ[ j, m()) [ j, mm ( + [ 2j2, mm ( + Proof Accordng o he srucure funcon n he parallel-seres sysems and Equaon (4), he rgh hand sde of Equaon (4) can be convered no: ( cj cj+ Pr ( Φ( m ( ), X ()) j) = ( cj cj+ Pr(max{ mn { X[ h ( )},, mn { X[( h ()}, h h mn{ X[ (),, X[ ( j (), m, X[ ( j+ (),, X[ ()}, mn { X [( + h ( )},, h + mn { X[ h ()}) j h = ( cj cj+ Pr( mn { X[ h ( )} j,, mn { X[( h ()} j, h h mn{ X[ ( ),, X[( j ( ), mx, [( j+ ( ),, X[ ( )} j, mn { X[( + h ( )} j,, h mn { X[ h ()} j) h + For k {, 2,, }, k, j {,,, }, we can oban: Pr( mn { X[ kh ( )} j) = Pr( mn { X[ kh ()} > j) h h k k = Pr( X[ k ( ) > j,, X[ k ( ) > j) = ( [ kh, j( )) k ρ k mn{ X[ ( ),, X[ ( j ( ), m, X[ ( j+ ( ),, X[ ( )} m, so for k =, j { m,, }, we have : Pr(mn{ X[ ( ),, X[ ( j ( ), m, X[ ( j+ ( ),, X[ ( )} j) =, and for k =, j {,, m }, we have: Pr(mn{ X[ ( ),, X[( j ( ), mx, [( j+ ( ),, X[ ( )} j) = Pr(mn{ X[ ( ),, X[( j ( ), mx, [( j+ ( ),, X[ ()} > j) = Pr( X[ ( ) > j,, X[( j ( ) > j, X[( j+ () > j,, X[ j () > ) k = ( ρ[ h, j ( )), j So we can ge: ( cj cj+ Pr Φ( m ( ), X ()) j ( ) m k k = ( cj cj+ [ ( ρ [ kh, j ()) [ ( ρ [ h, j ()) + ( cj cj+ [ ( ρ[ kh, j ()) k=, k, j m k=, k Then he rgh hand sde of Equaon (4) s: ( cj cj + Pr ( Φ( m( ), X()) j) Pr ( Φ( l(), X( )) j) l k l k = ( cj cj+ ) [ ( ρ[ kh, j ()) ( cj c j+ [ ( ρ[ kh, j( )) [ ( ρ[ h, j ( )) m k=, k m k=, k, j l k = ( cj cj+ [ ( ρ[ kh, j ()) ( ρ[ h, j ( )) m k=, k, j 282 EKS P L O A T A C J A I I E Z A W O D O S C MA I T E A C E A D RELIABILITY VO L9, O 2, 27

6 A las, we can oban: EIIM[ j, mm ( (, y) P[, j, m [ j / θ[ j, ml y / θ[ j, ml + = ()( ) [ j ( ) k ( cm cm+ ( ρ[ kh, m( )) ( ρ[ h ( )),, m k=, k, j EIIM[ j, mm ( (, y) P[, j, m [ j / θ[ j, ml y / θ[ j, ml + = ()( ) [ j ( ) k ( cm cm+ ( ρ[ kh, m( )) ( ρ[ h ( )), m k=, k, j Then we have: EIIM[ j, mm ( )(, y) EIIM[ j, mm ( )( y, ) + + [ j [ j 2 P[, j, m ()( [ j / θ[ j, ml )( y/ θ[ j, ml ) ( ρ[ h, m( )) P [, j, m()( [ j / θ[ j, ml)( y / θ [ 2j2, ml ) ( ρ[ h ( )) 2, m 2 ( ρ[ h, m( )) ( ρ[ j, m( )) ( ( )) ( ρ[ h, 2 m ρ[ j, m()) [ j [ j P[, j, m [ j / θ[ j, ml y / θ ()( )( [ j, ml ) P[ j m() j / 2, 2, ( [ θ [ j, ml y / θ )( [ 2j2, ml ) αm( )( ρ[ j, m( )) α ( )( ρ m 2 [ j, m( )) [ j = [ 2 j2 [ j [ 2j2 P[, j, m() ( / θ [ j, m( m+ ) P[, j, m() / ( θ[ 2j2, m( m+ ) αm( )( ρ[ j, m( )) m 2 α ( )( ρ [ j m( )), Snce he parallel sysem and he seres sysem are dual sysems, accordng o Proposon, we can oban Proposon 2 Proposon 2 In a seres-parallel sysem, assume =, hen P[ j, m() / θ[ j, m( m+ βm( ) ρ[ j, m() [ j P[ j, m() / θ [ 2j2, m( m+ βm( 2) ρ[ j, m() [ j [ 2j2 [ 2 j2 EIIM (, y) EIIM (, y) [ j, mm ( + [ 2j2, mm ( + only when Proof Smlarly o Proposon, he proof can be esablshed In Proposon, f we consder he suaon ha here s only one row, he parallel-seres sysem reduces a seres sysem In Proposon 2, f we consder he suaon ha here s only one column, he seres-parallel sysem reduces a parallel sysem Accordng o Proposons and 2, we have Corollares and 2 Corollary In a parallel-seres sysem, assume =, hen [ j [ j2 EIIM (, y) EIIM (, y) only when [ j, m( m+ [ j2, m( m+ P[ j, m() / θ[ j, m( m+ ρ[ j, m() [ j ( ) P[ j, m( ) / θ 2 [ j2, mm ( + ρ[ j, m() 2 [ j2 Proof Smlarly o Proposon, he proof can be esablshed EKS P L O A T A C J A I I E Z A W O D O S C MA I T E A C E A D RELIABILITY VO L9, O 2,

7 Corollary 2 In a seres-parallel sysem, assume [ j = [ j, hen [ j, m( m+ [ j2, m( m+ 2 EIIM (, y) EIIM (, y) only when P[ j, m() / θ[ j, m( m+ ρ[ j, m() [ j P[ j, m() 2 ρ ( / θ[ j2, mm ( + ) [ j2, m () [ j2 Proof Smlarly o Proposon 2, he proof can be esablshed From Proposons, 2 and Corollares, 2, n a parallel-seres sysem or seres-parallel sysem, f he ranson rae funcon of componen from sae m o an adjacen sae s larger han ha of componen j, hen he effec of componen on he sysem performance s larger han ha of componen j Ths also means ha we can denfy he mos mporan componen n he sysem when consderng he effecs of mprovemen beween adjacen saes of a componen on sysem performance, a me If wo componens are n he same degradng sae, hen he componen wh larger mporance value should be mananed frs n he operaon of a parallel-seres sysem or seres-parallel sysem o ge he larger mprovemen of sysem performance 4 Applcaon o a propeller plane sysem Fg 3 A propeller plane sysem In hs secon, we use a real-world example o llusrae he applcaon of he EIIM A propeller plane manly consss of engnes, 2, 3, 4 (componens, 2, 3, 4), conrol panel (componen 5), and wngs, 2 (componens 6, 7), as shown n Fg 3 The four componens (e, engnes) consues a 2-ou-of-4: G sysem Tha s, n order o ensure safey flgh, a leas wo engnes mus be workng We assume ha when he propeller plane sysem fals, he manenance cos s mllon CY In order o analyze he dfference beween BM and EIIM, we assume ha he lfeme of all he componens follow he Webull dsrbuon wh he same scale parameers and dfferen shape parameers =, =, = 3, and θ, = Fg 4 llusraes he dfference beween BM and EIIM of engnes and oher componens for dfferen shape parameers From Fg 4, we have he followng fndngs If he shape parameer of a componen s less han, he ranson rae decreases wh me, he BM of a componen s bgger han s EIIM If he shape parameer of a componen s equal o, he ranson rae s b, ( y) = / θ, = based on Proposon 2, and ( )= () () whch s ndependen of he so- IIM, y, P, BM journ me If he shape parameer of a componen s more han, he ranson rae ncreases wh me, and componen EIIM s also hgher han s BM When fxng y= and θ, =, we can oban he BM and EIIM of engnes wh he change of shape parameer as n Fg 5, whch s correspondng o Fgs 4 (a), (c), and (e) for =, =, = 3 When he shape parameer becomes bgger, he value of EIIM also be- (a) Imporance of engnes when = (b) Imporance of oher componens when = and oher componens (c) Imporance of engnes when = (d) Imporance of oher componens when = and oher componens (e) Imporance of engnes when = 3 and oher componens (f) Imporance of oher componens when = 3 and oher componens Fg 4 Dfference beween BM and EIIM of engnes and oher componens 284 EKS P L O A T A C J A I I E Z A W O D O S C MA I T E A C E A D RELIABILITY VO L9, O 2, 27

8 relables of all engnes are he same So BM of all engnes are he same when =, as n Fg 6(a) When θ, = and y=3, he value of EIIM of engnes ncreases wh he ncremen of shape parameers, whch s correspondng o Fg 6(b) Fg 5 Imporance of engnes wh he change of shape parameer 5 Conclusons and fuure work Ths paper exends he negraed mporance measure o a new measure and sudes he proposons of he new measure The resuls are useful for manenance managers o evaluae whch componen sae generaes he mos mprovemen n provdng he sysem performance I ypcally consders he condonal probably dsrbuon of a componen sojournng a a sae s he Webull dsrbuon, gven he nex sae ha componen wll jump o Then he dfference beween he Brnbaum mporance and he negraed mporance measure of a componen wh dfferen shape parameers s dscussed If he shape parameer s smaller han, he ranson rae decreases wh me, and he Brnbaum mporance measure of a componen s bgger han s exended negraed mporance measure If he shape parameer of a componen equals o, he ranson rae s consan, (a) BM of engnes Fg 6 Dfference of mporance among engnes (b) EIIM of engnes comes bgger han BM more and more Ths s because ha / θ, becomes bgger, and he EIIM s relaed o shape parameers As dscussed above, he four engnes consue a 2-ou-of-4: G sysem In case all he four engnes follow dfferen Webull dsrbuons, how do he engnes affec he sysem performance? To fnd he answer, we wll analyze he dfference of mporance among engnes We assume θ, =, y = 3, 5 = 6 = 7 = 3, and for he shape parameer of engnes, = 3, 2 = 2, 3 =, 4 = The dfference beween he BM and he IIM among engnes s shown n Fg 6 From Fg 6, he order of BM and EIIM of four engnes s engne > engne 2 > engne 3 > engne 4 When θ, = and =, accordng o Equaon (3), W( ; θ,, )= exp ( / θ, e = Tha s o say ha he whch s ndependen of he sojourn me of a componen n a sae If he shape parameer of a componen s greaer han, he ranson rae ncreases wh me, and componen exended negraed mporance measure s hgher han s BM In he fuure work, we wll dscuss he oher dsrbuons when componen manenance cos changes wh he falure rae and me The cos can ake dfferen ypes no consderaon, such as he consran of oher resources (fnance, exernal facor), opporuny cos, and so on Acknowledgemens The auhors graefully acknowledge he fnancal suppors for hs research from he aonal aural Youh Scence Foundaon of Chna (os 7573, 6443, 75768) References Brnbaum Z W On he mporance of dfferen componens n a mul-componen sysem, n Mulvarae Analyss II ew York: Academc Press 969: Borgonovo E Dfferenal, crcaly and Brnbaum mporance measures: An applcaon o basc even, groups and SSCs n even rees and bnary decson dagrams Relably Engneerng & Sysem Safey 27; 92: , hps://doorg/6/jress26923 EKS P L O A T A C J A I I E Z A W O D O S C MA I T E A C E A D RELIABILITY VO L9, O 2,

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