Improvement of the Reliability of a Series-Parallel System Subject to Modified Weibull Distribution with Fuzzy Parameters

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1 Joural of Mahmacs ad Sascs Rsarch Arcls Improvm of h Rlably of a Srs-Paralll Sysm Subjc o Modfd Wbull Dsrbuo wh Fuzzy Paramrs Nama Salah Youssf Tmraz Mahmacs Dparm, Faculy of Scc, Taa Uvrsy, Taa, Egyp Arcl hsory Rcvd: Rvsd: Accpd: Emal: sc_ama55@yahoo.com Absrac: Ths papr roduc a srs-paralll sysm cossd of dpd ad o-dcal compos wh lfms follow h modfd Wbull dsrbuo. Rduco mhod s roducd o mprov sysm rlably. Ohr mhods of ho ad cold sadby duplcao ar sablshd o mprov sysm rlably. A procdur for compug h rlably of h orgal sysm ad h mprovd sysms s prsd wh h paramrs of h modfd Wbull dsrbuo bcom fuzzy. Numrcal sudy s roducd o show h rsuls ad compar bw dffr mprovm mhods. Kywords: Rlably, Srs-Paralll Sysm, Fuzzy Numbrs, Mmbrshp Fuco, Modfd Wbull Dsrbuo, Maxmum llhood Mhod, Cofdc Irval, Cold Sadby, Ho Sadby, Rduco Mhod Iroduco Th rlably of a u of a sysm s h probably ha h u wll prform s spcfd fuco durg a rval of m udr gv codos. I gral, rlably of a sysm dpds o may facors such as h rlabls of h sysm us, h cofgurao of h sysm, ad sysm falur crra. I rlably suds, h goal s o prdc suabl rlably dcs for h sysm basd o compo falur daa ad sysm dsg (Hoylad ad Rausad, 004. I may cass, rlabls of praccal sysms should b mprovd o rach a cra lvl. Improvg h rlably of a sysm ca b achvd by dcrasg h falur ra of s compos or crasg h rpar ra of s rparabl compos. Improvg sysm rlably ca b achvd by ohr mhods such as ho duplcao mhod ad cold sadby duplcao mhod. By usg rduco mhod, sysm rlably s mprovd by rducg h falur ras of som of s us by a facor ρ whr (0<ρ<. I ho duplcao mhod, som of h sysm us ar duplcad paralll. I cold duplcao mhod, som of h sysm us ar duplcad by a cold rduda sadby compo va a swch whch ca b prfc or mprfc. Wbull dsrbuo s a flxbl dsrbuo ha usd sysm rlably aalyss. Wbull dsrbuo ca b usd o modl a vary of lf bhavors dpdg o dffr valus of h paramrs. La al. (003 roducd h modfd Wbull dsrbuo as a w lfm dsrbuo ha capabl of modlg a bahub shapd hazard ra fuco. Th proposd modl s drvd as a lmg cas of h Ba Igrad Modl ad has boh h Wbull dsrbuo ad Typ I xrm valu dsrbuo as spcal cass. Th modl ca b cosdrd as aohr usful gralzao of h Wbull dsrbuo. Fuzzy s hory was roducd by Zadh (965 ordr o gralz h hory of classcal ss. I a fuzzy s, s o cssary ha h lm s a full mmbr of h s or o a mmbr. I ca b a paral mmbr of h s. A fuzzy s s dfd by a fuco ha maps lms a doma of cocr o hr mmbrshp valu a s. Ths fuco s calld h mmbrshp fuco. Th mos ma characr of fuzzy s s ha mmbrshp fuco gvs vry lm a valu of [0, ] as s grad of mmbrshp. Th grad of mmbrshp has h characrsc ha a sgl valu combs h vdc for ad h vdc agas h lm (Nová al., Nama Salah Youssf Tmraz. Ths op accss arcl s dsrbud udr a Crav Commos Arbuo (CC- BY 3.0 lcs.

2 I Lraur, hr ar may paprs ha dal wh mprovg h rlably of mahmacal modls. Kha ad Ja (05 roducd rlably valuao of a sysm usg modfd Wbull dsrbuo. Ezza ad Rasoul (05 mprovd sysm rlably usg larxpoal dsrbuo fuco. El-Damcs (009 roducd rlably quvalc facors of a srsparalll sysm wh h sysm compos ar dpd ad dcal wh a lf dsrbuo of Wbull dsrbuo. Rlably quvalc facors for som sysms wh mxur Wbull falur ras wr roducd by Musafa (009. Poǵay al. (03 mprovd h rlably of compos sysm by usg rduco mhod ad ho duplcao mhod cosdrg h sysms survvor fuco. Rlad survvor quvalc fucos wr drvd all cass wh h compos lfm dsrbuo follows h gamma Wbull dsrbuo. Sc, h xac daa abou h paramrs of a probably dsrbuo cao b always avalabl so ha hs paramrs ca b xprssd as fuzzy umbrs. Ths fuzzy umbrs ca b drv from collcd daa or h opos of h xprs. El-Damcs ad Tmraz (05 proposd aalyss of avalably ad rlably of -ou-of- modl assumg h ras of h modl ar fuzzy. Chg ad Mo (993 usd h cofdc rval for aalyzg h fuzzy sysm rlably. Ch (994 prsd a w mhod for aalyzg h fuzzy sysm rlably usg arhmc opraos of fuzzy umbrs. Ch (996 prsd a w mhod for fuzzy sysm rlably aalyss basd o fuzzy m srs ad h α-cus arhmc opraos of fuzzy umbrs. I hs sudy, h rlably of a srs-paralll sysm cossg of dpd ad o-dcal compos s mprovd assumg ha h lfm of ach u follows modfd Wbull dsrbuo. W suppos ha h paramrs of h modfd Wbull dsrbuo ar fuzzy umbrs wh ragular mmbrshp fucos. Rlably fuco of h orgal sysm ad mprovd sysms s drvd accordg o rduco, ho duplcao, ad cold duplcao mhods. Fally, w sudy h modl umrcally. Ths papr s orgazd as follows. I Sco, rvw of modfd Wbull dsrbuo s dscussd. Maxmum llhood mhod s roducd o fd po smaors for h modfd Wbull dsrbuo. Dfos ad bascs of fuzzy umbrs ar dscussd. I Sco 3, h rlably fuco of h orgal sysm s dducd. I Sco 4, rduco mhod s appld o mprov h sysm rlably. I Sco 5, cold sadby duplcao mhod s prsd o cras sysm rlably. I Sco 6, ho sadby duplcao mhod s roducd o mprov h rlably of h sysm. I Sco 7, a procdur s roducd o show how h rvals for h sysm rlably ar obad. I Sco 8, a umrcal sudy s prsd o llusra h rsuls ad compar bw dffr mhods. Modfd Wbull Dsrbuo Th probably dsy fuco of h modfd Wbull dsrbuo s dfd as: ( = β( γ + λ f γ λ γ λ β β > 0, γ 0, λ 0, > 0 Th dsrbuo fuco s gv by:, ( γ λ β ( = ( F Th hazard ra fuco s gv by: ( ( γ λ = β γ + λ (3 h For λ=0, w oba h orgal Wbull dsrbuo. Dffr shaps of h probably dsy ad hazard ra fucos ar llusrad Fg. ad. Fg.. Dffr shaps of PDF vrsus m Fg.. Dffr shaps of HRF vrsus m 69

3 Radom Numbr Grao Usg h mhod of vrso w ca gra radom umbrs from h modfd Wbull dsrbuo as follows: ( F = β γ λ = u whr, u~uform(0,. Afr smpl calculao hs ylds: I( u I β γ λ I( u LambrW γ I γ β = xp γ (4 W ca us h prvous rlao o gra radom umbrs wh h paramrs β, γ ad λ ar ow whr LambrW s a fuco ha sasfs h followg rlao: LambrW(x * xp(lambrw(x = x Paramrs Esmao Th maxmum llhood mhod s usd o fd po smaors for h paramrs of h modfd Wbull dsrbuo as follows: L T,T,,T b a sampl of sz from a modfd Wbull dsrbuo. Th h llhood fuco s gv by: γ λ γ λ β = ( = β( γ + λ (5 L f logl = = Ta h logarhm of h llhood fuco: log( β + log( γ + λ γ λ β (6 = + ( γ log( + λ = = Th frs paral drvavs wh rspc o β, γ ad λ ar drvd as follows: γ λ γ λ = = (7 β β β = = = = + γ + β ( γ λ = = γ = ( log λ log ( (8 ( β γ,, = + λ + γ + λ ( β (9 = γ λ = = I ordr o oba h po smaors ˆ θ ( ˆ β, ˆ γ, ˆ = for h paramrs θ = (β, γ,, h prvous sysm of olar quaos ca b solvd umrcally afr quag hm o zro. Scod drvavs of h logarhm of h llhood fuco ar drvd ad h rsuls ar: I β = β γ β γ = ( γ = γ + = β λ = ( γ = logl = = = log( + ( θ logl + γ β λ β β γ = = β λ λ γ = γ + β = = ( γ + λ = γ λ ( log λ = λ γ γ + λ = ( log Th obsrvd formao marx s dfd as: λ λ β γ β λ β = β γ γ λ γ β λ γ λ λ I ordr o compu h sadard rror ad asympoc cofdc rval h usual larg sampl approxmao s usd whch h maxmum llhood smaors of θ ca b rad as approxmaly ormal. Hc as, ˆ β, ˆ γ, ˆ λ wll b h asympoc dsrbuo of h MLE ( gv by: whr, V = V ad: ˆj ˆ β ˆ ˆ ˆ β V V V 3 ˆ γ ~ N γ, Vˆ ˆ ˆ V V3 ˆ λ λ Vˆ ˆ ˆ 3 V3 V33 j θ= ˆ θ 70

4 V V V3 V V V3 = I V3 V3 V 33 ( θ s h approxma varac covarac marx ad I (θ s h vrs of h obsrvd formao marx. A Approxma 00( α% wo sdd cofdc rvals for β, γ ad λ ar, rspcvly, gv by: Fuzzy Paramrs ˆ β ± z Vˆ, ˆ γ ± z Vˆ, ˆ λ ± z Vˆ α α α 33 Now l us suppos ha h paramrs of h modfd Wbull dsrbuo ar fuzzy. A fuzzy s s dfd as a fuco ha maps lms a doma of cocr o hr mmbrshp valu a s ad hs fuco s calld mmbrshp fuco. Th mmbrshp fuco of a fuzzy s A s dod as µ A ad mmbrshp valu of x A s dod as µ A (x. Th doma of mmbrshp fuco s calld h uvrs of dscours. Dfo Djma al. (983 dfd fuzzy umbr as a gralzao of a rgular, ral umbr h ss ha dos o rfr o o sgl valu bu rahr o a cocd s of possbl valus, whr ach possbl valu has s ow wgh bw 0 ad. Dfo A mmbrshp fuco µ A (x of a fuzzy s A s a fuco µ A :X [0,] such ha vry lm x X has mmbrshp dgr µ A (x [0,]. W suppos ha h yp of h mmbrshp fucos s h ragular o. I gral, h ragular mmbrshp fuco s dfd as follows. Dfo 3 A ragular mmbrshp fuco s drmd by hr umbrs {a, b, c} whr a s h lows valu, b s h omal valu ad c s h maxmum valu. Th ragular mmbrshp fuco s gv by: 0, x< a ( x a/( b a, a x b ragl( xabc ;,, = ( c x /( c b, b x c' 0, x > c whr, h paramrs {a, b, c}, wh (a<b<c, drm h x coordas of h hr corrs of h udrlyg ragular mmbrshp fuco. Fg. 3. A srs-paralll sysm Orgal Sysm W cosdr a srs-paralll sysm cossd of subsysms cocd paralll ad ach subsysm cosss of us cocd srs for =,,,. Th sysm wll opra succssfully wh a las o subsysm s up (s Fg. 3. L R j ( b h rlably fuco of h u j (j =,,, of a subsysm, =,,,. Hc, h rlably fuco of h sysm wll b gv by: R( = Rj( = j= R( = = j= γj λj βj R( = xp = j= β γj λj j (0 If all us of h sysm ar dcal, h h rlably fuco wll b gv by: ( = xp{ β γ λ } R ( = Rduco Mhod I s assumd h rduco mhod ha h sysm dsg ca b mprovd by rducg h falur ras of a s of s compos by a facor ρ such ha 0<ρ< (Sarha al., 004; Sarha, 009. W suppos ha h falur ras of a s A of h us of ach subsysm ar dcrasd by mulplyg by a facor ρ ad hc sysm rlably fuco s obad as follows: R( = Rj, ρ( Rj( = j A j A R( = xp + whr: γj λj γj λj ρβj βj = j A j A ( 7

5 { } A,,..,, =,,, Cold Sadby Duplcao Mhod I hs mhod, w suppos ha a s C of compos of ach subsysm ar duplcad wh a dcal cold sadby u. Th rlably fuco of h mprovd sysm s obad as follows: Rc( = Rj, c( Rj( = j C j C Rc( = ( + γj λj γj λj γj λj βj βj βj = j C j C Rc j j = j= j C γj λj γj λj ( = xp β ( + β whr: { } C,,..,, =,,, Ho Sadby Duplcao Mhod (3 I hs mhod, w suppos ha a s of D compos of ach subsysm ar duplcad wh a ho sadby u. Th rlably fuco of h mprovd sysm s obad as follows: Rh( = Rj, h( Rj( = j D j D ( = ( γj βj λj γj βj λj γj βj λj Rh = j D j D γ λ β ( = xp βj ( j j Rh = j= j D Whr: Procdur { } D,,..,, =,,, j γj λj (4 I hs sco, w roduc a procdur o llusra how h rlably of h sysm s compud: Sp : Gra a radom sampl from h modfd Wbull dsrbuo a fxd valus of h paramrs (β, γ, by usg rlao (4. Sp : Subsug a s of quaos (7-(9 ad h quag hm o zro ˆ β, ˆ γ, ˆ λ by solvg h Sp 3: Oba h MLE ( rsula sysm of quaos umrcally Sp 4: Cosrucg h obsrvd formao marx ad compu s vrs Sp 5: Calcula h cofdc rvals for h paramrs (β, γ, a a lvl of sgfcac a Sp 6: Calcula h rvals for h fuzzy paramrs by subsug h followg rlao: ( α (, ( α cu( U M L+ cu M L U ˆ θ, ˆ L θ U = for θ = + (5 whr, a cu ={0,0.,0.,,} ad M s h po smaor of θ ad [L,U] s h cofdc rval lms of θ. Sp 7: Subsug quaos (0, (, (3 ad (4 o oba rvals for h fuzzy rlably fuco of h orgal sysm ad h mprovd sysms, rspcvly. Numrcal Sudy L us cosdr a srs-paralll sysm cosss of wo subsysms cocd paralll. Th frs subsysm cosss of wo us cocd srs ad h scod o cosss of o u. Suppos ha all us ar dcal. W wll apply our procdur o oba h lms for h fuzzy rlably of h sysm. Th rlably fuco of h orgal sysm wll b gv by: γ λ ( ( xp( γ λ = β ( xp( β R Now, w wll gra a radom sampl wh sz = 30 a (β, γ, = (0.,0.7,0.3 ad h rsul s: Subsug a s of quaos (7-(9 ad h ˆ β, ˆ γ, ˆ λ quag hm o zro. O ca oba h MLE ( by solvg h rsula sysm of quaos umrcally ad h valus ar: ( ˆ ˆ ˆ ( β, γ, λ = 0.9,0.763,0.6 = ( LM,, U Fg. 4. Comparso of h rlabls of h orgal sysm ad dffr mprovd sysms 7

6 I Fg. (4, w ca obsrv ha h rlably of h orgal sysm s mprovd by dffr ways. Cold sadby mhod s br ha ho duplcao mhod ad rduco mhod has dffr valus whch mprovs h sysm rlably. W cosruc h obsrvd formao marx ad h rsul s: I( θ = Tag h vrs of hs marx ylds h followg marx: I ( θ = Calcula h 95% cofdc rvals for h paramrs (β, γ, ad h rsuls ar: Cofdc rval for β = [0.00,0.7] Cofdc rval for γ = [0.088,.437] Cofdc rval for λ = [0.07,0.494] Now, w cosdr ha h paramrs β, γ ad λ ar ragular fuzzy umbrs h calcula h rvals for hm by usg rlao (5 ad h rsuls ar show Tabl. Th rvals for h fuzzy rlably fuco of h orgal sysm wll b calculad from h followg rlao: γ ɶ L λl ( xp( ɶ ɶ βl ɶ ɶ γl λl ( xp( ɶ βl, γ ɶ U L ( xp( ɶ ɶ λ βu γ ɶ U λu ( xp( ɶ ɶ βu ɶ (, ɶ ( RL RU = Subsug h valus of h fuzzy paramrs from Tabl h abov rlao, w oba h rvals for h fuzzy rlably fuco of h orgal sysm a a fxd m ( = ad h rsuls ar show Tabl. Also, w oba h rvals for h fuzzy rlably fucos of h mprovd sysms usg rduco, cold sadby ad ho duplcao mhods a a fxd m ( = ad h rsuls ar show Tabls 3-5, rspcvly. Tabl. Calculaos of h rvals for fuzzy paramrs ˆ β, ˆ γ, ˆ λ ( α cu ɶ βl, ɶ β U ɶ γl, ɶ γ U ɶ λl, ɶ λ U 0.0 [0.00, 0.7] [0.088,.437] [0.07, 0.494] 0. [0.09, 0.66] [0.55,.504] [0.050, 0.57] 0. [0.039, 0.36] [0.3,.57] [0.073, 0.540] 0.3 [0.049, 0.46] [0.90,.639] [0.097, 0.563] 0.4 [0.059, 0.56] [0.358,.706] [0.0, 0.587] 0.5 [0.069, 0.66] [0.45,.774] [0.44, 0.60] 0.6 [0.079, 0.75] [0.493,.84] [0.67, 0.633] 0.7 [0.089, 0.85] [0.560,.908] [0.90, 0.657] 0.8 [0.099, 0.95] [0.68,.976] [0.4, 0.680] 0.9 [0.09, 0.305] [0.695,.043] [0.37, 0.703].0 [0.9, 0.35] [0.763,.] [0.6, 0.77] Tabl. Calculaos of h rvals for h fuzzy rlably fuco a m = a cu L, U 0.0 [0.40, 0.999] 0. [0.87, 0.997] 0. [0.40, 0.994] 0.3 [0.0, 0.990] 0.4 [0.083, 0.983] 0.5 [0.047, 0.974] 0.6 [0.03, 0.96] 0.7 [0.09, 0.944] 0.8 [0.00, 0.9] 0.9 [0.005, 0.893].0 [0.00, 0.857] Tabl 3. Calculaos of rvals of fuzzy rlably fuco usg rduco mhod a m = {,0} {,0} α cu p L' {0,} {0,} p U p L' {,0} {,0} p U p L' p U ' {,} {,} p L p U {,} {,} p L' p U 0.0 [0.80, 0.999] [0.477, 0.999] [0.370, 0.999] [0.505, 0.999] [0.566, 0.999] 0. [0.0, 0.998] [0.4, 0.998] [0.30, 0.998] [0.445, 0.999] [0.503, 0.999] 0. [0.66, 0.996] [0.366, 0.997] [0.37, 0.997] [0.385, 0.998] [0.437, 0.998] 0.3 [0.0, 0.99] [0.33, 0.995] [0.79, 0.995] [0.36, 0.996] [0.37, 0.997] 0.4 [0.083, 0.987] [0.6, 0.99] [0.9, 0.99] [0.7, 0.993] [0.308, 0.995] 0.5 [0.055, 0.980] [0.5, 0.986] [0.089, 0.986] [0., 0.989] [0.50, 0.993] 0.6 [0.035, 0.970] [0.75, 0.980] [0.059, 0.979] [0.78, 0.984] [0.99, 0.989] 0.7 [0.0, 0.956] [0.37, 0.970] [0.037, 0.969] [0.38, 0.977] [0.5, 0.984] 0.8 [0.0, 0.937] [0.04, 0.958] [0.0, 0.956] [0.05, 0.967] [0.4, 0.976] 0.9 [0.006, 0.94] [0.077, 0.94] [0.0, 0.939] [0.077, 0.954] [0.08, 0.967].0 [0.003, 0.884] [0.054, 0.9] [0.005, 0.96] [0.054, 0.937] [0.057, 0.954] 73

7 Tabl 4. Calculaos of rvals for fuzzy rlably fuco usg cold sadby mhod a m = {,0} {,0} a cu C L' {0,} {0,} C U C L' {,0} {,0} C U C L' {,} {,} C U C L' {,} {,} C U C L' C U 0.0 [0.93, 0.999] [0.55, 0.999] [0.43, 0.999] [0.583, 0.999] [0.664, 0.999] 0. [0.8, 0.998] [0.476, 0.999] [0.343, 0.999] [0.503, 0.999] [0.576, 0.999] 0. [0.69, 0.997] [0.398, 0.999] [0.58, 0.999] [0.48, 0.999] [0.48, 0.999] 0.3 [0., 0.994] [0.3, 0.999] [0.85, 0.999] [0.337, 0.999] [0.385, 0.999] 0.4 [0.08, 0.99] [0.5, 0.999] [0.4, 0.999] [0.60, 0.999] [0.94, 0.999] 0.5 [0.054, 0.985] [0.89, 0.998] [0.079, 0.998] [0.94, 0.999] [0.5, 0.999] 0.6 [0.034, 0.977] [0.37, 0.997] [0.048, 0.996] [0.40, 0.998] [0.5, 0.999] 0.7 [0.00, 0.967] [0.093, 0.994] [0.07, 0.994] [0.094, 0.996] [0.00, 0.999] 0.8 [0.0, 0.95] [0.060, 0.99] [0.04, 0.990] [0.060, 0.994] [0.063, 0.998] 0.9 [0.006, 0.93] [0.036, 0.985] [0.007, 0.983] [0.036, 0.990] [0.037, 0.997].0 [0.003, 0.907] [0.00, 0.977] [0.003, 0.973] [0.00, 0.985] [0.00, 0.995] Tabl 5. Calculaos of rvals for fuzzy rlably usg ho duplcao mhod a m = {,0} {,0} α-cu H L' {0,} {0,} H U H L' {,0} {,0} H U H L' H U ' {,} {,} H L H U {,} {,} H L' H U 0.0 [0.67, 0.999] [0.396, 0.999] [0.35, 0.999] [0.48, 0.999] [0.456, 0.999] 0. [0.06, 0.998] [0.3, 0.999] [0.4, 0.999] [0.337, 0.999] [0.366, 0.999] 0. [0.53, 0.997] [0.49, 0.999] [0.76, 0.999] [0.60, 0.999] [0.80, 0.999] 0.3 [0.09, 0.994] [0.86, 0.999] [0.3, 0.999] [0.93, 0.999] [0.05, 0.999] 0.4 [0.075, 0.990] [0.33, 0.998] [0.08, 0.998] [0.37, 0.999] [0.44, 0.999] 0.5 [0.049, 0.985] [0.09, 0.997] [0.054, 0.996] [0.093, 0.998] [0.097, 0.999] 0.6 [0.03, 0.976] [0.060, 0.994] [0.033, 0.994] [0.06, 0.996] [0.063, 0.999] 0.7 [0.09, 0.965] [0.037,0.990] [0.00, 0.989] [0.037, 0.993] [0.038, 0.998] 0.8 [0.0, 0.948] [0.0, 0.983] [0.0, 0.98] [0.0, 0.989] [0.0, 0.996] 0.9 [0.005, 0.97] [0.0, 0.973] [0.006, 0.970] [0.0, 0.98] [0.0, 0.99].0 [0.00, 0.899] [0.005, 0.958] [0.00, 0.953] [0.005, 0.97] [0.005, 0.986] Cocluso I hs papr, w aalyzd a srs-paralll sysm cossd of dpd ad o-dcal compos wh h lfms of h compos follow modfd Wbull dsrbuo. I lraur, h paramrs of h lfm dsrbuo wr cosdrd o b ow. Howvr, xac valus of h paramrs of ay dsrbuo ar of uow. Hr, h paramrs of h modfd Wbull dsrbuo wr cosdrd o b fuzzy whch mas ha hy ar o logr rad as fxd umbrs. Rduco mhod was usd o mprov h rlably of h orgal sysm. Also, cold ad ho duplcao mhods wr roducd o cras sysm rlably. A procdur was addd o show how h rvals for fuzzy rlably fuco ar compud. Numrcal sudy was proposd o compar bw dffr mhods. Th valus obad Tabls 4 ad 5 show ha cold sadby mhod s br ha ho duplcao mhod for mprovg sysm rlably. As fuur wor, a sudy of a sysm assumg ha s compos ar dpd ca b rad. Also, crasg h umbr of sadby us ca b usd o cras sysms rlably. Nw dsrbuos ca b usd o modl h lfms of h compos of a sysm. Acowldgm I would ha h rvwrs for hr comms whch mprovd my papr. Fudg Iformao Thr s o fudg sourc for hs arcl. Ehcs Ths arcl s orgal. Rfrcs Ch, S.M., 994. Fuzzy sysm rlably aalyss usg fuzzy umbr arhmc opraos. Fuzzy Ss Sys., 64: DOI: 0.06/065-04( Ch, S.M., 996. Nw mhod for fuzzy sysm rlably aalyss, cybrcs ad sysms. I. J., 7: DOI: 0.080/ Chg, C.H. ad D.L. Mo, 993. Fuzzy sysm rlably aalyss by rval of cofdc. Fuzzy Ss Sys., 56: DOI: 0.06/065-04(93908-H 74

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