Mathematical Computation September 2015, Volume 4, Issue 3, PP.60-64
Fuzzy Reliability Analysis of One Paralleling System Chenhui Liu, Jinhui Li, Yunfei Guo† Mathematics Department, Yanbian University, Yanji 133002, China †
Email: guoyunfei0413@sina.com
Abstract In this paper, we analyze the fuzzy reliability of paralleling system, focusing on the situation of the early failure mode. The early failure mode of fuzzy reliability mainly obeys the Weibull distribution. Based on this, we obtain the distribution function, density function, failure function, reliability function and the formula of the mean time to failure. For the paralleling system composed of multiple components,
C , C , , C 1
2
n
, especially when η =1 ,we also obtain the corresponding formulas. Finally, we cite an example to
verify the formula and deepen the understanding. Keywords: Paralleling System; Early Failure Mode; Fuzzy Reliability; Weibull Distribution
1 INTRODUCTION In 1965,Zadeh based on the precise mathematical set theory, gradually established operation, transformation rule and carried out the relevant theoretical research on the "fuzzy sets"[1]. In 1989, Li Tingjie and Gao He analyzed and deduced fuzzy reliability of paralleling system in the paper[2]. Zhang Minyue and Chen Yanli obtained the expression about reliability index by building and solving the model[3].
2 PRE- KNOWLEDGE Paralleling system is a system, which is failure when all of the components are failure. Graphic is the following.
FIG. 1 THE PARALLELING SYSTEM
Definition 2.1[4] Li Tingjie and Gao He defined the following symbols: C : System works normally C j :The jth works Ai
normally
:Fuzzy state set
They also obtained the following formulas: n n = = Cj P(C) P( C j ) C =j 1 =j 1 n C = 1 − C j j =1 If the failure of the parts were independent of each other, then
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(2.1)
n The system of common failure rate P (C) = ∏ C j j =1 P (C ∆Ai ) = P( Ai | C ) P(C) The system of fuzzy failure rate P (C j ∆Ai ) = P( Ai | C j ) P(C j ) The part of fuzzy failure rate
(2.2)
Definition 2.2[4] According to the general reliability theory and literature: = P(C) R= s , P (C j ) R j (2.3) P(C ∆ = Ai ) Rs , P(C j ∆ = Ai ) R j A | C ) µ (R ) = P(Ai |C) µ= (R ), P( A s i j A j Lemma 2.1[5] The failure distribution of paralleling system obeys weibull distribution Wei ( m,η ) . Weibull distribution’s two parameters are η , m .Among them η are referred to the characteristics of life (The distribution of the 0.632 digits), m are referred to the shape parameter and η > 0, m > 0 . Wei ( m,η ) ’s information is the following. i
Distribution function:
i
{ }
t m F (t ) = 1 − exp −( )
η
t>0
(2.4)
Density function: f(t) = [ F（t]' =
t m t m −1 ( ) ⋅ exp −( ) m η η η
t>0
(2.5)
Failure function: = λ (t)
f (t) m t m −1 = ( ) 1 − F (t) η η
t>0
(2.6)
Lemma 2.2 According to 2.4,2.5,2.6 above ,we can obtain that: The reliability function: t R(t) = 1 − F (t) = exp −( ) m η
t>0
(2.7)
The mean time to failure:
η 1 1 = = m t t λ (t) ( ) m −1 m( ) m −1
= MTTF
η η
t>0
(2.8)
η
3 FUZZY RELIABILITY ANALYSIS IN EARLY FAILURE MODE According to the Lemma 2.1,when m < 1 ,it is early failure mode. For a component paralleling system composed of n parts, C1 , C2 ,, Cn , The analysis of fuzzy reliability is the following. Assume that each part’s failure were independent of each other, and η = 1 , = m j m= , j 1, , n , and it obeys Wei ( m, η )
, then
Theorem 3.1 The fuzzy reliability function of system: n
m R= s (t) µ A (R s ){1 − ∏ [1 − exp( − t )]} j =1 i
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t >0
(3.1)
Proof. First we should infer the common reliability function of system: n
n
R s (t) = 1 − ∏ [1 − R j (t)] = 1 − ∏ [1 − exp(− t m )]
t >0
(3.2)
=j 1 =j 1
Then plug this formula into the following formula, obtain n
m R= s (t) µ A (R s ){1 − ∏ [1 − exp( − t )]} j =1
t >0
i
(3.3)
Theorem 3.2 The fuzzy failure function of system: n d µ A ( Rs ) λs (t) = ∏ λ j (t) − µ A ( Rs ) dt j =1
t>0
i
(3.4)
Proof. First we should infer the common failure function of system: i
−dRS = λ1 (t) ⋅ λ2 (t) ⋅ λ3 (t) ⋅ ⋅ ⋅ λn (t) RS dt
λs= (t)
n
= ∏ λ j (t)
(3.5)
t>0
j =1
Then plug this formula into the following formula, obtain d µ A ( Rs )R s − R s d µ A ( Rs ) − µ A ( Rs ) d R s dR − s = − = λs (t) = µ A ( Rs )R s dt Rs dt R s µ A ( Rs ) dt d µ A ( Rs ) −dRs d µ A ( Rs ) = − = λs − µ A ( Rs ) dt Rs dt µ A ( Rs ) dt dµ (R ) n A s = − λ t>0 (t) ∏ j µ A ( Rs ) dt j =1 i
i
i
i
i
i
i
i
(3.6)
i
i
Theorem 3.3 The fuzzy mean time to failure of system: i
1 n ~ d µ A ( Rs ) s ∏ λ j (t) − µ A ( Rs ) dt j =1 Proof. First we should infer the common mean time to failure of system: MTTF ==
t>0
(3.7)
i
i
1 = λs (t)
MTTF = s
1
t>0
n
(3.8)
∏ λ j (t) j =1
Then plug this formula into the following formula, obtain MTTF = ~
s
1 =
1 d µ A ( Rs ) ∏ λ j (t) − µ A ( Rs ) dt j =1
λs (t)
n
t>0
(3.9)
i
i
4 EXAMPLE Put electrical appliances into parallel. Assumes that the system is at the early stage of use, solve the very reliable fuzzy reliability and fuzzy failure rate of the paralleling system in 6 minutes. A1 : highly reliable state. Given that: (4.1) µ A (0.378) = 0.7
1
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see graphic below:
FIG. 2 THE PARALLELING SYSTEM
Because the elements belong to the early use, the failure rate function is the early failure type. According to (2.5), if 1 η =1,m = , then 2 1 1 1 −2 (t) ⋅ exp −(t) 2 2
f(t)=
t >0
(4.2)
Because t=6min=0.1h, Ai = Highly reliable , R s =0.378 , µ A (0.378) = 0.7 , according to (3.2), then 1
n
m R= s (t) µ A (R s ){1 − ∏ [1 − exp( − t )]} j =1 =0.7 × 1 − [1 − exp(-0.10.5 )]3 = 0.68605 i
t >0
}
{
(4.3)
Because µ A ( Rs ) is constant, according to (3.4), then i
n d µ A ( Rs ) λs (t) ∏ λ j (t) − = µ A ( Rs ) dt j =1 1 1 = ( × 0.1 2 )3 − 0 2 = 3.9528 t>0 i
i
MTTF = ~
s
1 =
λs (t)
1 =0.2530 3.9528
(4.4)
t>0
(4.5)
So the highly reliable fuzzy reliability is 0.68605, the fuzzy mean time to failure of system is 0.2530h.
5 CONCLUSIONS In this paper, we conclude that a series of formulas about fuzzy reliability of early failure mode in the paralleling system. We know that early failure mode obeys Weibull distribution.
REFERENCES [1]
Zadeh L.A. “Fuzzy Sets”. Information and Control. 1965, 8(3): 338-353
[2]
Li Tingjie, Gao He. “The fuzzy reliability of the Serial system.” Fuzzy Systems and Mathematics. 1989, 3(1): 38-45
[3]
Zhang Minyue, Chen Yanli. “Reliability analysis of different types of three components in parallel systems.” Journal of lanzhou university of technology. 37 (2): 127-130
[4]
Mao Shisong, Tang Yincai, Wang Lingling.”Reliable Statistics”. Higher education Press. 2008
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AUTHORS Chenhui Liu was was born on July 11th,
Yunfei Guo was born on April 13th, 1983
1995 in HeNan province. She is studying
in Jilin province, and received his M.S.
in Yanbian University.
degrees in Yanbian University, China in 2010. He is a Lecture of Yanbian University. His research interests are reliability and statistical analysis.
Jinhui Li was was born on March 20th, 1993 in JiangXi province. She is studying in Yanbian University.
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