FUZZY RELIABILITY ANALYSIS OF COMPOUND SYSTEM BASED ON WEIBULL DISTRIBUTION

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1 IJAMML 3:1 (2015) Septeber 2015 ISSN: Available at DOI: FUZZY RELIABILITY ANALYSIS OF COMPOUND SYSTEM BASED ON WEIBULL DISTRIBUTION Yufei Guo Departet of Matheatics, Yabia Uiversity, Yaji, , P. R. Chia Abstract Data i the real world ca ot be recorded or collected precisely due to hua errors, achie errors, or soe other uexpected situatios. Ad reliability aalysis is a iportat research topic i egieerig. The data, however, ca ot be recorded precisely soeties as described above. The soe researchers pay attetio to the fuzzy reliability. I this paper, we are goig to aalyze the fuzzy reliability of copoud syste, which obeys the Weibull distributio. For the copoud syste coposed of ultiple copoets, C 1, C2,, C, especially whe η = 1, we obtai the correspodig fuzzy reliability fuctio. Besides, values of reliability of copoud syste with various ad have bee calculated. Fially, ecessary reliability aalysis is also provided i the paper. Keywords: copoud syste, fuzzy reliability, Weibull distributio. * Correspodig author. E-ail address: guoyufei0413@sia.co (Yufei Guo). Copyright 2015 Scietific Advaces Publishers 2010 Matheatics Subject Classificatio: 62N05. Subitted by Yewag Che. Received August 17, 2015; Revised August 29, 2015

2 32 Yufei Guo / IJAMML 3:1 (2015) Itroductio Zadeh [1] gradually established operatio, trasforatio rule ad carried out the relevat theoretical research o the fuzzy sets based o the precise atheatical set theory. Fuzzy set theory was first itroduced as a useful tool to copleet covetioal reliability theories i [2]. Sice the, there have bee a uber of approaches where the techology of fuzzy sets was used to evaluate syste reliability. For exaple, i Big et al. [3], a fuzzy liear regressio ethod is cobied with a fiite eleet ethod to evaluate the reliability of echaical structures. I this approach, a ebership fuctio of a triagular fuzzy uber is used to express the structure stress. I order to overcoe the liitatio of the traditioal failure ode, effects ad criticality aalysis, a fuzzy rule-based approach has bee ipleeted i [4, 5, 6]. Furtherore, Zio et al. [7] developed a fuzzy expert syste for hua reliability aalysis to elicitate factors ifluecig coditioal hua error for two depedece successive operator actios i a uclear power plat accidet. I Karii ad Hüllereier [8], fuzzy set theory has bee used to copleet probability theory to assess the risk of atural disaster whe statistical data ad/or physical kowledge are isufficiet for probabilistic aalysis. Meawhile, Dig ad Lisiaski [9] developed a fuzzy uiversal geeratig fuctio i which fuzzy ubers used to represet the state probability ad fuzzy copositio operators were itroduced to assess the reliability of a ulti-state syste. Moreover, Padey ad Tyagi [10] proposed a profust reliability to evaluate degradable systes ad a fuzzy ubers-based ethod to assess syste failure rate paraeters. Li ad Gao [11] aalyzed ad deduced fuzzy reliability of parallelig syste i the paper. Zhag ad Che [12] obtaied the expressio about reliability idex by buildig ad solvig the odel.

3 FUZZY RELIABILITY ANALYSIS OF COMPOUND / IJAMML 3:1 (2015) Pre- Kowledge Copoud syste i this paper is a syste whose graphic is i the followig: Figure 1. The copoud syste. Defiitio 2.1 ([11]). Li ad Gao defied the followig sybols: C : state that syste works orally. C j A : state that the j-th copoet works orally. : fuzzy state set. They also obtaied the followig forulas: C C = C j, j = 1 = 1 C j. j = 1 P( C ) = P( C ), j = 1 j (2.1) If the failure of the parts were idepedet each other, the The syste of coo failure rate The syste of fuzzy failure rate The part of fuzzy failure rate j P( C ) = C = 1 j, P( C A ) = P( A C ) P( C), P( C A ) = ( A j P C j ) P( C j ). (2.2)

4 34 Yufei Guo / IJAMML 3:1 (2015) Defiitio 2.2 ([11]). Accordig to the geeral reliability theory ad literature, we ca kow that P ( C) = R P( C ) = R, s, j j P ( C Ai ) = R ~ s, P( C A ) R j = ~, ~ j P ( A ) ( ), ( A i C = µ A s j ) A ( j ). R P C = µ R (2.3) Lea 2.1 ([13]). The failure distributio of parallelig syste obeys Weibull distributio Wei (, η). Two paraeters of Weibull distributio are η,. Aog the, η is referred to the characteristics of life, ad is, referred to the shape paraeter, where η > 0, > 0. Wei(, η)s iforatio is the followig: Distributio fuctio: () 1 exp t F t = ( ), t > 0. η (2.4) Desity fuctio: () [ ()] t 1 ( ) exp t f t = F t = ( ), t > 0. η η η (2.5) Failure fuctio: λ() t = f () t 1 F () t = t 1 ( ), t > 0. η η (2.6) Lea 2.2. Accordig to Equatios (2.4), (2.5), (2.6) above, we ca obtai that The reliability fuctio: () 1 () exp t R t = F t = ( ), t > 0. η (2.7)

5 FUZZY RELIABILITY ANALYSIS OF COMPOUND / IJAMML 3:1 (2015) Fuzzy Reliability Aalysis i Early Failure Mode Accordig to the Lea 2.1, whe < 1, it is i early failure ode. For a copoet parallelig syste coposed of parts, C, C2,, C. The aalysis of fuzzy reliability is i the followig. 1 Assue that each part s failure is idepedet each other, ad η = 1, j =, j = 1,,, ad it obeys Wei (, η), the fuzzy reliability fuctio of copoud syste ca be obtaied. Theore 3.1. The fuzzy reliability fuctio of copoud syste: 2 () t = µ ( R )( 2 e ) exp( t ), t > 0, R ~ s A s (3.1) whe η = 1 ad j =, j = 1,,. Proof. First of all, whe η = 1 R j j () t = 1 F () t = exp ( t ), j = 1,,, t > 0. Fro referece [14], we ca obtai that R () = 1 1 = 2 s t Rj Rj R j= 1 j= 1 j= 1 2 ( ) ( ) 2 j t exp t j = 2 exp j= 1 j= 1 2 j 2 exp j exp = 2. t t j (3.2) j= 1 j= 1 Whe =, j = 1,,, we ca obtai that j s () t = 2 exp ( t ) exp ( t ) R 2 = 2 ( 2 e ) exp ( t ).

6 36 Yufei Guo / IJAMML 3:1 (2015) So R () t = µ ( R ) R () t = µ ( R )( 2 e ) exp( t ). ~ s A s s A s 2 Fro Theore 3.1, we ca see that fuzzy reliability fuctio of copoud syste R ~ s() t is the fuctio about tie t, which icludes two paraeters ad. Differet values of R ~ s () t with differet ad are provided i Table 1. Table 1. Reliability of copoud syste with various ad Fro Table 1, we ca see the effects of two paraeters ad. First of all, the reliability would icrease whe icreases for each give, this is because that the syste is i the early failure ode, ad the failure rate would decrease with the icreasig. O the other had, the reliability would decrease whe icreases for each give, reasos for this is that ore copoets i the series syste, lower reliability would syste becoes.

7 FUZZY RELIABILITY ANALYSIS OF COMPOUND / IJAMML 3:1 (2015) Figure 2 ad Figure 3 provide ore ituitive explaatio. Figure 2. Reliability whe settig = 3. Figure 3. Reliability whe settig = 1/2.

8 38 Yufei Guo / IJAMML 3:1 (2015) Reliability aalysis about copoud syste have bee give above, as for fuzzy reliability, we would regard µ A ( Rs ) as the costat which is the ea µ A ( Rs ), ad because R () ( ) (), ~ s t = µ A Rs Rs t the reliability aalysis are the sae as fuzzy reliability aalysis i this eaig. 4. Coclusio I this paper, we aalyze the fuzzy reliability of copoud syste, which obeys the Weibull distritutio. For the copoud syste coposed of ultiple copoets, C, C2,, C, especially whe η = 1, we obtai 1 the correspodig fuzzy reliability fuctio. Fially, ecessary reliability aalysis is also provided i the paper. Refereces [1] L. A. Zadeh, Fuzzy sets, Iforatio ad Cotrol 8(3) (1965), [2] T. Oisawa, Fuzzy theory i reliability aalysis, Fuzzy Sets Syst. 30(3) (1989), [3] L. Big, Z. Meili ad X. Kai, A practical egieerig ethod for fuzzy reliability aalysis of echaical structures, Reliab. Eg. Syst. Saf. 67(3) (2000), [4] J. B. Bowles ad C. E. Pelaez, Fuzzy logic prioritizatio of failures i a syste failure ode, effects ad criticality aalysis, Reliab. Eg. Syst. Saf. 50(2) (2005), [5] H. Gargaa ad S. K. Chaturvedi, Criticality assesset odels for failure ode effects ad criticality aalysis usig fuzzy logic, IEEE Tras. Reliab. 60(1) (2011), [6] K. Xu, L. C. Tag, M. Xie, S. L. Ho ad M. L. Zhu, Fuzzy assesset of FMEA for egie systes, Reliab. Eg. Syst. Saf. 75(1) (2002), [7] E. Zio, P. Baraldi, M. Librizzi, L. Podofillii ad V. N. Dag, A fuzzy set-based approach for odelig depedece aog hua errors, Fuzzy Sets Syst. 160(13) (2009), [8] I. Karii ad E. Hüllereier, Risk assesset syste of atural hazards: A ew approach based o fuzzy probability, Fuzzy Sets Syst. 158(9) (2007), [9] Y. Dig ad A. Lisiaski, Fuzzy uiversal geeratig fuctios for ulti-state syste reliability assesset, Fuzzy Sets Syst. 159(3) (2008),

9 FUZZY RELIABILITY ANALYSIS OF COMPOUND / IJAMML 3:1 (2015) [10] D. Padey ad S. K. Tyagi, Profust reliability of a gracefully degradable syste, Fuzzy Sets Syst. 158(7) (2007), [11] Li Tigjie ad Gao He, The fuzzy reliability of the serial syste, Fuzzy Systes ad Matheatics 3(1) (1989), [12] Zhag Miyue ad Che Yali, Reliability aalysis of differet types of three copoets i parallel systes, Joural of Lazhou Uiversity of Techology 37(2) (2011), [13] Mao Shisog, Tag Yicai ad Wag Liglig, Reliable Statistics, Higher Educatio Press, [14] Che Dexi, Fuzzy Reliability of Copoud Systes (I), Syste Egieerig-Theory & Practice, g