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Intelligent system
Intelligent Systems NORalt- I-IOIXAND
Fuzzy Reliability Analysis Based on Closed Fuzzy Numbers HSIEN-CHUNG WU
Operations Research/Industrial Engineering Group, Department of Mechanical Engineering, University of Texas at Austin, Austin, Texas 78712, USA
ABSTRACT The approach by proposing the fuzzy-valued probability measures to the conventional system reliability is discussed successfully in this paper. We consider the failed or functioning probability of each component in the system as nonnegative fuzzy numbers under the definition of fuzzy-valued probability measure, which is a more reasonable consideration in the real world. Under this setting, we can have the pivotal decomposition, and we still can improve the fuzzy reliability of the system by improving the fuzzy probability of each component. Furthermore, the membership function of the fuzzy reliability can be transformed into the nonlinear programming problem and can be solved by any current optimizer. ©Elsevier Science Inc. 1997
1.
INTRODUCTION
The theory of fuzzy reliability was proposed and developed by several authors. The recent collection of papers by Onisawa and Kacprzyk [10] gave many different approaches for fuzzy reliability. The conventional reliability is considered under the probability assumption and binary-state assumption. Cai et al. [2, 3, 4] gave a different insight by introducing the possibility assumption and the fuzzy-state assumption to replace the probability and binary-state assumptions. In the conventional system, we always give an exactly failed or functioning probability for each component. However, in practice, we cannot be sure of the exact value for each component. All we can know is that each component is failed or functioning with probability which is approximately equal to p. Cheng and Mon [6, 7] and Chen [5] gave this new approach. There is another point of view discussed in this p a p e r which considers the fuzzy-valued probability measures. Under this consideration, we can define
INFORMATION SCIENCES 103, 135-159 (1997) © Elsevier Science Inc. 1997 655 Avenue of the Americas, New York, NY 10010
0020-0255/97/$17.00 PII S0020-0255(97)00059-5
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the fuzzy conditional probability measure and in the meantime, we can discuss the pivotal decomposition. As a matter of fact, it is possible to extend most of the theorems for reliability theory under the sense of fuzzy-valued probability measures. In Section 2, we give some properties of fuzzy numbers which will be used in the rest of this paper. In Section 3, we discuss the fuzzy-valued probability measures and fuzzy-valued conditional probability measures. We need to introduce fuzzy conditional probability in order to derive the pivotal decomposition based on the fuzzy reliability, and we still can improve the fuzzy reliability of the system by improving the fuzzy probability of each component under the consideration of fuzziness, which will be discussed in Section 4. We also provide computational methods and an example to illustrate the membership of the fuzzy reliability. 2.
T H E A R I T H M E T I C OPERATIONS OF F U Z Z Y NUMBERS
We start this section by giving some properties of fuzzy numbers, which will be used in the rest of this paper. DEFINITION 2.1. Let X be a universal set. Then a fuzzy subset A of X is defined by its membership function ~A-: X ~ [0, 1]. We can also write the fuzzy set A as [(x,/ZA-(X)): X ~X}. We denote A , = {x: ~A-(X) >/ a} as the a-level set of A. DEFINITION 2.2. (i) A is called a normal fuzzy set if there exists x such that /~,/(x) = 1. (ii) A is called convex fuzzy set if tz~(Ax+(1-A)y)>~min{ix,i(x), gA-(Y)}, for A E [0, 1]. (That is, /zA- is a quasiconcave function). PROPOSITION 2.1 ([14]). A is a convex fuzzy set if and only/f{x: /ZA-(X)>~a} is a conuex set for all a. PROPOSITION 2.2 ([15]) (Resolution Identity). Let A be a fuzzy set with mernbershipfunction tx4 and A , = {x: tZA-(X)>/a}. Then,
~-(x)= sup alA~(x). c ~ [0, 1]
DEFINITION 2.3 ([12]). Let f ( x ) be a real-valued function on a topological space. If {x: f ( x ) >/a} is closed for every a, f ( x ) is said to be upper semicontinuous. DEFINITION 2.4. (i) rh is called a fuzzy number if rh is a normal convex fuzzy set and the a-level set, r h , is bounded Va :~ 0. The set of all fuzzy numbers is denoted as ~.
F U Z Z Y RELIABILITY ANALYSIS
137
(ii) rh is called a closed fuzzy number if rh is a fuzzy number and its membership function /x,~ is upper semicontinuous. The set of all closed fuzzy numbers is denoted as ~c~-tPROPOSITION 2.3. I f rh is a closed f u z z y n u m b e r then the ~-lecel set o f rh is" a closed interval, which is d e n o t e d by rh = [m- ,L , m- U ~ ]. ( T h a t is why we call rh as a closed f u z z y n u m b e r . ) Proof. Ix,~ is an upper semicontinuous function, so the c~-level set rh = {x: /x,~(x) >/cQ is a closed set (by Definition 2.3). Now rh is a convex fuzzy set, thus r h is a closed interval by Proposition 2.1. []
Next we consider the operations of fuzzy numbers. DEFINITION 2.5. Let rh and fi be fuzzy numbers. (i) The membership function of sum r h ¢ ~ is defined by /x~,e,~(z)= sup min{/x,~(x),/x~(y)}. x+v=z
(ii) The membership function of the opposite of rh is defined by /Xe,~(z ) = sup min/x~(x) = / x , ~ ( - z ) . z-
x
(iii) The difference of rh and fi is defined by rh e f t =rh ¢(@~). (iv) The membership function of product rh ® h is defined by
/x,~®~(z) = sup min{/x,~(x),/xe( y)}. ,y=~ (v) The membership function of the inverse of rh is defined by
z = l / x . x 4 = (I
(vi) The quotient of rh and h is defined by rh,~fi =rh ®(1/fi). REMARK 2.1. We
have
/x,~ee(z) = supx r_=min{/x,~(x),/xe(y)} and /x,~~,~~(z) = sup .... / v . ; ~ 0rain{/xr~(X), #~(Y)}" DEFINITION 2.6. Let ~ be a fuzzy number. (i) (ii) (iii) (iv)
ti is called ~ is called ~i is called ~ is called
a a a a
nonnegative fuzzy number if ix4(x)=0, V x < 0 . nonpositive fuzzy number if t x 4 ( x ) = 0, V x > 0. positive fuzzy number if /XA~(X)=0, Vx ~ 0. negative fuzzy number if /.t~(x)=0, Vx>~0.
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PROPOSITION 2.4 ([9])• Let rh and h be two fuzzy numbers. (i) rh ¢ fi, rh e h, and rh ® fi are also fuzzy numbers. (ii) If h is a positive or negative fuzzy number then rh • h is also a fuzzy number. Let "Qint" be any binary operation el.t, e i n t , ®int, o r (])in t between two closed intervals rh~,= [ m"/~ ~ , m- ,u] and h,~ = [ n-L , , n -U , ] . Then rh~Q) inth . is defined by /~aQ) intHa =" {Z ~ RIz = x o y , V x ~ rh, , V y ~ h e ,
where " o " is an usual binary operation + , - , × , o r / } . PROPOSITION 2.5 ([13], Theorem 2). Let Q be any binary operation ¢ , o , ®, or (D between two fuzzy numbers. If rh and h are two closed fuzzy numbers then,
PROPOSITION 2.6. Let rh and h be two closed fuzzy numbers. (i) rh ¢ h, rh 0 h, and Fn ® fi are also closed fuzzy numbers. (ii) If fi is positive or negative then rh ¢ h is also a closed fuzzy number. Proof. From Proposition 2.4, we just need to prove that /x,ho~ is upper semicontinuous. That is, we need to prove that (rhQh)~ is a closed set for all a, which is the immediate consequence of Proposition 2.5. []
PROPOSITION 2.7 ([8]). (i) The addition and multiplication for closed intervals are both associative and commutative. (ii) The addition and multiplication for closed fuzzy numbers are both associative and commutative. Proof. (ii) By (i) and Proposition 2.5.
[]
PROPOSITION 2.8. (i) If {t and [~ are two closed fuzzy numbers then (4eb)~=[a~~L +DL~,a~U+DV~land(~eD) =[~-b~,a,-b~].'u -u "L (ii) If ~ and D are two closed fuzzy numbers then (~i c D)~= • ~L -L [mm{a~b~, -L-U a-U-L" L -LDU a~b~ , b ~L, a ,-U'U b , } , max{a~b~,a~ , , a -U'L , b ~ , a ~-U'U b,}]. (iii) If ~ and b are two nonnegative closed fuzzy numbers then (gl ® D), = - L - L ~U'U [a~b~,a~b~]. (iv) I f fz is a nonnegative closed fuzzy number and [~ is a positive closed fuzzy number then ( 4 ¢ [~)~ - - [ a-L~ / b"~U, a-U ~ / b -~ L ]. Proof. If /~ is positive then b~ ~: 0 and /~ff* 0 for all a. The result follows from Propositions 2.5 and 2.6. []
F U Z Z Y RELIABILITY ANALYSIS
139
DEFINITION 2.7. (i) Let [a, b] and [c, d] be close intervals. We say that [a, b] ~int [C, d] if and only if a >~c and b >~d. (ii) Let 5 and/9 be two closed fuzzy numbers. We say that 5 ~/9 if and only if 5, ~i,t/9, for all a. (iii) Let 5 and /9 be two closed fuzzy numbers. We say that 5 _c~l/9 if and only if 5, c / 9 for all a. In this case, we say that 5 has Jess fuzziness than that of/9. (The spread of the membership function of 5 is less than or equal to the spread of the membership function of/9.) PROPOSITION 2.9. " ~ " and " ___c~" are partial orderings on Jccl (the set o f all closed fuzzy numbers). DEFINITION 2.8. We say that 5 is a crisp number with value m if its membership function is (~ /x~ (r) =
if r = m , otherwise.
REMARK 2.2. Suppose that /9 is a crisp number with value m. If d---c~/9 then it is easy to see that fi is also a crisp number with value m. That is, there is no fuzzy number which has less fuzziness than that of crisp number. PROPOSITION 2.10 ([8]). Let I, J, and K be closed intervals. (i) I ® (J ¢ K ) c_ ( I ® J ) • ( I ® K ) (subdistrikut&ity). (ii) I f J ® K is nonnegative then I ® ( J C K ) = ( I ® J ) ¢ ( I ® K ). (iii) If t is a real number then t( l c J ) = tl c tJ. PROPOSITION 2.11. Let d, /9, and ? be closed fuzzy numbers.
(i) ~ ® (/9 • ~)___c~(~ ®/9) • (~ ® ~). (ii) If/9 ® ~ is nonnegatiue then 5 ® (/9 ¢ ~ ) = (5 ®/9) ¢ ( fi ® ~). (iii) I f fi is a crisp number then f i ® ( / 9 ¢ ~ ) = ( 5 ® / 9 ) ¢ ( 5 ® 5 ) . Proof. By Propositions 2.10 and 2.5.
[]
PROPOSITION 2.12. Let fi and /~ be two closed fuzzy numbers. Then /)Cc~ 5 e ( / ) e S ) . If 5 is a crisp number then / 9 = S e ( / 9 e S ) . Proof. By Proposition 2.8, we have ( d e ( / 9 e S ) ) ~ c--c--La+. b, -a,-U ~/b~. then a~ -L = a-u, = m for all a. This completes the proof. []
REMARK 2.3. This means that /9 has less fuzziness than that of 5 ¢ (/9 e d). The more operations for fuzzy numbers will increase the fuzziness. The same situation occurs in Proposition 2.11 (i).
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PROPOSITION 2.13. Let 5, [~, and 6 be nonnegatiue and closed fuzzy numbers.
(i) If b is positiue and 5 • b = ~ then 5 C cl [~ ® ~. If [~ is a crisp number then 5 = [~ ® ~. (iN) I f [~c_d 5 then [~ ® ~ _Cclt~ ® C. Proof. (i) By Proposition 2.8, we have that ( g L /~-.U __ " L ua/o4 Ca -
-L__
implies
-
-L~U>~L~L
a~ - c~ b~ -j ~ ~,~
a~U~ / b~. L __ - c- Ua
- u _ - _ u [ / . _
as-%
. "-~ a~a
(iN) By Proposition 2.8, we have ( 5 ® ~ ) ~ ---a~c~ -L-L ~ b-L-L , c ~ =(/~®g)~" and ~ c , >~b~ c~ = (/9 ® ~) U. [] ( 5 ® -c). u = a-u-u "u-v PROPOSITION 2.14. Let {5i}7=1 and {bi}i= - ~ 1 be two sequences of closed fuzzy numbers. If SiC_ct [~i, v i = a . . . . . n then ~7 , 5iC_c, ~i~1 b i. ~L Proof. By Proposition 2.8, we have ( ~"i = 1 ai) ~=
(~.-
-
1 bi)a
L
v _ E n and ( ~/~ 1ai)~ -
-
,= l(ai)~
u
~<
En
-
,= l(bi)a
u
En.
"L
~_n i=
t= l(ai), >~ -
~L l(bi). =
u
= ( ~i n i bi)~.
[]
In order to define the fuzzy-valued probability measure in the next section, we need to introduce the limit of a sequence of fuzzy numbers, thus we first need to consider the metric between two fuzzy numbers. DEFINITION 2.9. Let A c_R" and B ___R". The Hausdorff metric is defined by /
dH(A,B)
=max~ sup inf [[a -bl[, sup inf ~acA
b~B
b~B
aeA
Ila-bl[ t.
DEFINITION 2.10 ([11]). We define the metric d% in .~1 as, d~,(5,/~)=
sup dH(5,,,[~,), 0
for any two closed fuzzy numbers.
REMARK 2.4 ([11]). (~ccl,d ~ ) is a complete metric space. LEMMA 2.1. Let 5 and f9 be two closed fuzzy numbers. Then we have dH(5,,,/~,, ) = max{I 5 L _/~L I, ]5~_/gu I}"
F U Z Z Y RELIABILITY ANALYSIS
141
Proof. We have
=max{
sup
inf ~l
_<:
Ix-yl,
sup
inf
Ix-yl}.
_< ~ U
Then we have the following cases. (i) If 4~ ~int/) . o r / ~ ~i~t 6~ then d , ( 4 , , / ~ , ) = max{l~ild -b~'c I,[4 U~-b~-C'I}. (ii) If 4._c/~ or/), c_4, then, d/t (4.,/~,~ ) = max{0, max{14 L - tS,~l, 1 4 ~ - / ~ I}} max{14,c-b~l,-L -v
/~,~'1}.
This completes the proof.
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PROPOSITION 2.15. Let 4 and [~ be two closed fuzzy numbers. Then d%( 4, [~) < • implies 14L - [~L[ < • and 1~,~- [~ [< • for all a ~ (0, 1]. Proof. It is obvious from Lemma 2.1.
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DEFINITION 2.11. Let {~in} be a sequence of closed fuzzy numbers. {4 n} is said to converge if there is a closed fuzzy number 4 with the following property: V• > 0, 3 N > 0 such that for n > N, we have d%(4,, ~)< e. We also say that the sequence {4,} converges to 4 and it is denoted as lim . . . . 4, = 4. If there is no such 6, the sequence {ti,} is said to diverge. PROPOSmON 2.16. Let {4,} be a sequence of closed fuzzy numbers. If lim~ ~ ~ 4~ exists then it is unique and (1i~4~)
=[Jim(4n)~,Jim(~7,),~']
for all ce. (Furthermore, ((tin) f } and {(a,,)~ - ~"} converge uniformly with respect to a.) Proof. Let l i m n ~ 4 , , = 4 . By Proposition 2.15, d j c ( 4 n , 4 ) < e implies -a~]
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DEFINITION 2.12. Let {4,} be a sequence of closed fuzzy numbers. If lim,_ ~ ~i~1 ai exists then we write oo
n
4,:
li'-m ~
n=l
n--*~
4 i,
i=1
otherwise the infinite (fuzzy) sum of sequence {4,} is said to diverge. PROPOSITION 2.17. If {4,} is a sequence of closed fuzzy numbers and lim. . . . ~'i~ 1 ai exists then we have ~
4. =
=
L
~
U
(a.)o,
ot
=
=1
[]
Proof. By Propositions 2.16 and 2.5. 3.
F U Z Z Y - V A L U E D PROBABILITY MEASURES
In this section, we shall discuss the fuzzy-valued probability measures and fuzzy-valued conditional probability measures. DEFINITION 3.1. Let 9- be a set of all fuzzy numbers and ~c~ be a set of all closed fuzzy numbers. We say that (i) f ( x ) is a fuzzy-valued function if j~ X ~ J . (ii) f(x) is a closed-fuzzy-valued function if fi X--*~I. DEFINITION 3.2. A fuzzy number/5 is called a fuzzy-probabilistic number if its membership function /xp(r)= 0 for r e [0, 1]. We denote ~e as a set of all fuzzy-probabilistic numbers. REMARK 3.1. (i) The fuzzy-probabilistic numbers are nonnegative fuzzy numbers. (ii) A fuzzy number 4 can induce a fuzzy-probabilistic number /5 by truncation. That is,
ixp(r)=(txa(r), 0,
if r ~ [0,1], otherwise.
We denote 0 and 1 as crisp numbers with values 0 and 1, respectively. Then 13 and 1 ~ J e . DEFINITION 3.3. By a fuzzy-valued probability measure t3 on a o--field (1), .~g), we mean a nonnegative closed-fuzzy-valued set function defined
F U Z Z Y RELIABILITY ANALYSIS
143
for all sets of ~" and satisfying the following conditions:
(i) P(O) = 6 (ii) 15(f~) ___cai (iii) f(U~= i Ai) = ~=1 15(Ai) exists for any sequence {.4i}~= 1 of disjoint events (.4i ('l.4j = Q~ for i -~j). We always regard 15(A) as a fuzzy-probabilistic number by truncation. Then 15(U~=1Ai)= @~=l fi(Ai) after truncating those two nonnegative fuzzy numbers as fuzzy-probabilistic numbers. The condition imposed on P by (iii) is called countable additivity. REMARK 3.2. If 1) is finite, ~" is taken to be the discrete o--field (all subsets of 12). In such a case, there are only finitely many events and hence, in particular, finitely many pairwise disjoint events. Then (iii) is reduced to (iii)': P is finitely additive; that is, for every collection of disjoint events {Ai}'/_ t, we have
f
A, = ~ f ( & ) . i=
i=1
PROPOSITION 3.1. Let -15 be a fuzzy-valued probability measure on the or-field (1~, ~'). (i) Let .4 and B be two events in the ~r-field (fl, ~{). If A GB then
f(B)~P(.4). (ii) Let A be an event in the cr-field ( fL J ) . Then 15(fD ~15(A).
Proof. Since B=.4 U ( B \ A ) is a disjoint union, we have /2(B)=/2(A) /2( B \ .4) ~ t2( .4 ) (by Proposition 2.5). [] DEFINITION 3.4. Let .4 and B be two events in 12. We say that .4 and B are independent with respect to a fuzzy-valued probability measure /5 if and only if 15(.4 AB)=P(.4)®15(B). Then 155(.4AB)=IS(.4)®15(B) after truncating those two nonnegative fuzzy numbers as fuzzy-probabilistic numbers. DEFINITION 3.5. Let /5 be a fuzzy-valued probability measure and .4 ~ ft be such that 15(.4) positive (i.e., (15(.4))~ 4:0 and (15(.4))v ~ 0 for all oD. Then the fuzzy conditionalprobability, given .4, is the closed-fuzzy-valued set function denoted by P(.IA) and defined on 1~ as follows, 15(B1.4) =/5(.4 N B ) • 15(A), B ~ I~ . 15(BlA) is called the fuzzy conditional probability of B given .4. PROPOSITION 3.2. Let 15 be a fuzzy-valued probability measure and 15(A) be positive closed fuzzy number. Suppose that ~i~1 [15(A i N .4 ) • 15(A)] exists
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H.-C. WU
for any countable collection of disjoint euents {Ai}~ l. Then f(.[A) is a fuzzy-ualued probabifity measure. Proof. (i) f(O[A)=fi(Q~NA)O f ( A ) = f ( o ) O f(A)=0. (ii) t 6 ( [ I I A ) = f ( ~ N A ) O f ( A ) = f ( A ) O f ( A ) . Then by Proposition 2.8(iii), we have (f(~lA))~ " = ( f ( A ) ) ~ / ( f ( A ) ) ~ < 1 and (f(I~]A))~v =
(IA(A))~/(P(A))~ >/1 for all a. Thus f(ll]A)2~, 1. (rid We have
A i=
iA) [
]
:
P(AiNA ) O f ( A ) ,
=P iUl ( AiOA) O f ( A )
i=
(( AiGA}I~_:, is a sequence of disjoint events). Then
i=l f
U IAiA
AiNA
=
U
~c
: ~ (f(AinA))~ i=1
=
(P(A))
(by Proposition 2.17)
U
~_,(f(AiNA)Of(A))Zl i=1 vc
/~
(by Proposition 2.17)
i=1
a"
F U Z Z Y RELIABILITY ANALYSIS
145
Similarly, A iA i=
= a
(~
AlIA
.
i= ]
~
Thus 15
AiA i=
= (~) 15(ALIA). i=1
This completes the proof.
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COROLLARY 3.1. Let 15 be a fuzzy-valued probability measure. If ~ is finite then 15(. ]A ) is a fuzzy-valued probability measure. Proof. The proof in (iii) of preceding Proposition 3.2 is always true for finite case. The result follows from Remark 3.2. [] PROPOSITION 3.3. Let 15 be a fuzzy-valued probability measure and 15(A) be a positive closed fuzzy number. I r A and B are independent with respect to 15 then 15(B)c_ d 15(B~1). If ~5(A) is a crisp number with l~alue m 4:0 then P ( B ) =P(BIA). Proof. We have P ( B I A ) =15(A N B ) e 1 5 ( A ) = ( 1 5 ( A ) ® 1 5 ( B ) ) e P(A), thus, (15(B~A))~=(15(A)®15(B))2/(15(A))ll '
(by Proposition 2.8)
= (15(A))~(15(.))~/(15(~))~ .< (~(.))':, and
(15(8~) )~=( 15¢A)),~: (15(B) )~/( 15(A) )~ >~(15¢B))~ This completes the proof.
[]
REMARK3.3. (i) Intuitively, although A and B are independent, f ( B I A ) has more fuzziness than that of 15(B) since the fuzzy conditional probability /5(B[A) is considered under the fuzzy environment for given event A, thus P(B)_Ccl/5(B~A). Also we can say something as follows, we can exclude the independence, but we cannot exclude the fuzziness.
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H.-C. WU
Furthermore, this will affect the pivotal decomposition based on the fuzzy reliability. (ii) Although P(A) is a crisp number, /3(B)=/5(B~I) is still a closed fuzzy number since /5(A n B) is a closed fuzzy number. PROeOSmON 3.4 (Sub-total probability theorem). Let {Ai: i = 1,...,n} be a finite partition of fl and P be a fuzzy-valued probability measure. Suppose that each of lS(Ai ), Vi = 1.... , n is positive. Then we have n
P ( B ) ---c, ~
/3(B~Ai) ® P ( A i ) .
i=l
Proof. We have that B = UT=1(B n A i) is a disjoint union, thus n
P(B)=
(~ 1 5 ( B A A , ) i=1
(by definition of the fuzzy-valued probability measure) /7
---c, ~
/5(BIA~) ® f ( A ~ )
(by Propositions 2.13 (i) and 2.14).
i=1
This completes the proof.
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COROLLARY 3.2 (Total probability theorem). If each of lS(Ai) is a crisp number with value m i 4=O, V i = 1,..., n then,
P(B)
=
(~
/5(BIA~) ® f ( A , ) .
i=l
Proof. By Proposition 2.13 (i).
[]
REMARK 3.4. From Remark 3.3 (ii), we can see why /5(B) is still a closed fuzzy number although each of P ( A i) is a crisp number. 4. FUZZY RELIABILITY ANALYSIS In this section, we shall discuss the fuzzy reliability of series system, parallel system, and k-out-of-n system with identical components, and then we shall discuss the pivotal decomposition and the notion for improving
F U Z Z Y RELIABILITY ANALYSIS
147
the fuzzy system reliability. Finally, we shall provide computational methods and an example to illustrate the usefulness of the results discussed so far. DEFINITION 4.1. Let X and Y be two discrete random variables. We say that X and Y are independent if P{X=x, Y=y} =/~{X=x} ®/;{Y=y}. Let 1~ = {S, F} and ~t' be a o-field on 1~. Suppose that /~({S})=t5 and /~({F}) = 1 0/5. Then /;(1~) =/;({S}) •/~({F}) =/5 • (1 @/5) 2el 1 (by Proposition 2.12). Thus it is easy to see that P is a fuzzy-valued probability measure on (1~, Me). Let X be a Bernoulli random variable on the fuzzy probability space (1~, ~ ' , / ; ) . Then /;{X = 1} =/;({S}) =/5 and /~{X = 0} = P({F}) = 1 Ot5. REMARK 4.1. Let 4 = 1G/5. Then 1 e ~ =/5 since (1 @4)~ = 1 _ ~ v = 1 ( ] e / 5 ) V = [ l _ ( 1 - p -L -L and (1 eq)~ - v =p~ -v for all ce. This says that if ~ )]=p~ # { X = 0} =c~ then # { X = 1} = i ec~. In other words, # { X = 0} = i O # { X = 1} and # { X = 1} =1 e # { X = 0}. DEFINITION 4.2. The structure function of the given system with n components is 4,(X) = &(X l, X 2..... Xn), where each of X i is a Bernoulli random variable. We define the fuzzy reliability ? of the system as the fuzzy probability of d,(X)= 1, that is, ~=P{&(X)= 1}. We assume that all components are mutually independent. n ~X i, where each of X i is a Bernoulli (i) Seriessystem: Here &(X) -_ Hi= random variable, we have
~=/~{ 6(X) = 1} =
#
. {I~i=lXi=|}
=/~{X 1 = l , X 2 = 1 ..... X~ = 1} n
-
( ~ /~{Xi = l } i=1
?/
= ®A. i=1
(since mutually independent)
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H.-C. WU
We regard ~ as a fuzzy-probabilistic number by truncation. (ii) Parallel sytem: Here 6(X) = 1 - Vii"=1(1 - X i ) , we have = f{ 6(x)
= 1}
= ] e/5{ 6 ( X ) = 0}
(by Remark 4.1)
= i e/5{n7 ,(l - x , ) = a} = 1 ( 3 , 5 { 1 - X , = 1 , 1 - X 2 = 1 ..... 1 - X , = 1}
= ] e/5{x, = o.x2 =o ..... x . =o} =ie
®/5{x,=o} i=l
=ie
.=
(i ePi
)1 •
We regard i as a fuzzy-probabilistic number by truncation. We write k times
k times
pe-"*p=k/3
and
p®-..®p=/3
k.
(iii) k-out-of-n-system with identical components: We have
f=/5{6(X)=l}=/5
~xi>~k i=l
= (~/s/gXi=m m=k
.
~i=1
Now /5
Xi=m
=
/5{X 1 =1 .....
Xm=l,Xm+
I =0 ..... Xn=0
}
i
(since mutually independent)
mI
F U Z Z Y RELIABILITY ANALYSIS
149
Thus n
We regard t: as a fuzzy-probabilistic number by truncation. 4.1.
THE PIVOTAL
DECOMPOSITION
In this subsection, we just consider the Bernoulli random variables, thus ~ is finite. The fuzzy conditional probability can be used in our discussions for the pivotal decomposition by Corollary 3.1. Let (~/i; 0) denote the vector ~ with its ith component replaced by ~. So (li;P):(/~I ..... j0 i 1,],F),+I ..... fin ) and (6~;~)=(/3~ .... ,]~i_l,0, a/)i+ I ..... fin). We also use the following notations, (li,x)
-- (X 1 .....
Xi
i,l,xi~ 1..... xn),
(O~;x) --- ( x , . . . . . x~ t,O,x~+l . . . . . x n ) , ('i;X)
~-(X 1 .....
Xi
l,',Xi,
i .....
Xn).
PROPOSITION 4.1 ([1])(Pivotal decomposition of the structure function).
The following identity holds for any structure function 4) of order n,
~(x) =x~(l~;x) + (I -x~) ~(0i;x) Vx. THEOREM 4.1 (Sub-pivotal decomposition). Let ~-~(~) be the fuzzy reliability of the system and all components be mutually independent. Then,
(i)
~= ~(~) ---c,[ ~, ®P{~(1,; x) = llx~ = q]
(ii)
-~c.[,~ ®,~{~(.,;x)= 1ix,= 1}]. [0 e~,)®,~{~(0i; x)= .l~ = 0}]
150
H.-C. WU
Proof. We have ~(fi) = 6 { ~ ( X ) = 1}
--(~cl= [`6{Xi -~ 1} ®`6{(h(X) = 1]Xi = 1}] ¢ [`6{X i =0} ®`6{6(X) = liXi =0}] (by Propositions 3.4 and 2.7 for commutativity) = [pi ®`6{4)(li;X)= IlX,= 1}]
_~c,[~, ®`6(~(1,;x)= 1)1 * [0 o~,)®`6{~(0,;x)= 1)] (by Propositions 3.3, 2.13 (ii), and 2.14)
This completes the proof.
[]
COROLLARY 4.1 (Pivotal decomposition). If `6{Xi = 1} and `6(X i = 0} are crisp numbers then we have the pivotal decomposition,
r~r(P) ~ [/~i ®F(Xi;P)] ~ [(1 ~/~i)~r([~i;P)] • Proof. By Corollary 3.2 and Proposition 3.3.
[]
LEMMA 4.1. Let ~= ?(~) be the fuzzy reliability of the system. Then we haue ?(~) =`6{X~ = 1, (#(1/;X) = 1 = &(O~;X)} ¢`6{X i = 1, d)(1,; X) = 1, 4)(0i; X) = O}
• `6{x~ =
0, ~ , ( l ~ ; x )
¢`6{x~= 0, 6 ( l i ;
= 1 = 4,(0~; x ) }
x ) = 0, 6 ( % ; x )
= 1}.
151
F U Z Z Y RELIABILITY ANALYSIS Proof. We have
~(~) =/5{ ~b(X) = 1} = / ~ ( X / ~ ( I i ;X) + ( 1 - X i ) 1~)(0i ;X) = 1}
(by Proposition 4.1)
=15{Xiqb(1i; X) = 1, (1 -X,)~b(O,; X) = O}
~/~{X~b(1,;X) = 0 , ( 1 - X , ) ~b(O,;X) = 1} =/5{)(/= 1,&(li;X ) = 1 = &(Oi;X)}
• P{x
=
+P{x
= O, Ob(1,;X) = 1 = ~b(Oi;X)} =
= 1,
=o}
O, ¢b(1,;X) = O, ¢b(Oi; X ) = 1}.
This completes the proof.
[]
DEFINITION 4.3. The ith component is irrelevant to the structure d~(x) if ~b(x) is constant in x i, that is, ~b(li;x) = ~b(0i;x) for all (.i;x). Otherwise the ith component is relevant to the structure. THEOREM 4.2 (Pivotal decomposition). If the ith component is irrelevant to the structure ~b(x) then we have the pivotal decomposition,
As a matter of fact, we have t:(0i; ~ ) = ~(1i; ~)col Y(O). If fi i & a crisp number then Y(0i; ~) = Y(li; ~) = Y(~). Proof. We have
P(O) =/5{)(/= 1, ~b(li;X) = 1 = ~b(Oi; X)} • lS{Xi = O, ~b(Oi; X) = 1 = 6 ( l i ; X ) }
(by Lemma 4.1 and the irrelevance)
152
H.-C. WU = [15{ X~= 1} ®/5{&(l~;X) = 1}] • [/5[ Xi=O } ®/5{~b(Oi;X ) = 1}] (by the independence of components)
[/~/®r('ii~P)] ~ [(] ~Pi)®r(6i~P)]
(since 7 ( 0 g ; 0 ) = r ( l i ; 0 ) f o r the irrelevance in t h e / t h component and/~i ® (1 O/5i) is nonnegative, then by Proposition 2.11 (ii)) ~-c, ~ ® ?(13g;~) =Y(()i;~)
(by Propositions 2.12 and 2.13 (ii))
(since 1 is a crisp number with value 1).
The crisp case follows from Proposition 2.12.
[]
REMARK 4.2. In the conventional system, if the ith component is irrelevant to the structure &(x) then r=r(p)=r(Oi;p)=r(li;p), i.e., the system reliability is equal to the reliability whenever the ith component is failed or functioning. (It seems that we ignore the ith component because of the irrelevance.) (i) If/5 i is a crisp number then we have 7(13/;~)=~(1i;~)=7(~). However, here 7(~) is still a closed fuzzy number (i.e., the fuzzy reliability) since the others/Sj may not be crisp numbers. (ii) If/5 i is a fuzzy number then the fuzzy reliability, whenever the ith component is failed or functioning, has less fuzziness than that of the fuzzy reliability of the whole system. It is well behaved, since we seem to ignore the fuzziness for the ith component owing to the irrelevance. In practice, we expect that the system reliability r has membership 1 in the fuzzy system reliability. Now if the reliability r has membership 1 in the fuzzy system reliability &((3i;X), then the reliability r also has membership 1 in the original fuzzy system reliability ~(X) since (r(0i;p))l c-(r(P))l (the l-level set). This explains the equality r=r(p)=r(Oi;p)=r(li;p) for the conventional system.
F U Z Z Y RELIABILITY ANALYSIS
153
4.2. IMPROVING THE FUZZY SYSTEM RELIABILITY DEFINITION 4.4. A system with structure function 6(x) is called increasing if it satisfies the following conditions: (i) 6 ( 0 . . . . . 0 ) = 0 ,
6(1 ..... 1)=1
(ii) If x < y then 6(x) ~<6(Y). This definition says that the state of the increasing system cannot get worse if the state of any of its components was improved. The system is up if all its components are up, and down if all its components are down. THEOREM 4.3. Let ~ = ~(~) be the fuzzy reliability of an increasing system 6(X) which consists of independent components. Then we have: (i) if~ic_cl ~ then r(D)--d r(P'), where ~ =(/51 ..... /5, ,,/si,Pi+, ..... /5.) and ~' = (/51 ..... fii-1, fi;, Pi + p "" , /5,). (ii) if" p," ' ~~p~ - and (/5,)u- (/5,)~ = (Pi), -, u --(Pi), -, L for all a (if the membership function of f~ is the translation of the membership function of /si then/5i and/5~ satisfy those conditions) then r(P') ~ r(P). (iii) if /5~ ~ /5i and ~ is a crisp number then 7(~') ~ ~(~)). (iv) if~51 ~ Pi and /51,/si are two crisp numbers then 7(~') ~ r(O).
Proof. By Lemma 4.1, we have
P(~) = [/~{X~ = 1} ~ f ( 6 ( 1 , ; X ) = 1 = 6(Oi;X)} ]
[f(x, = 1) ® f(6(l,;X)
= 1,6(o,;x)
=o)]
(~ [ P ( X i=O} ~/t~{6(Oi;X ) : ] = 6(1/;X)}]
(~/~(~)
(the fourth term has empty event since 6 is increasing function)
= [~, ®P(6(1,;x)
= 1= 6(o,;x)}]
[/5/®,5(6(1i;X) = 1,6(O,;X) =0}]
[(i
f(6(o,; x)=1= 6(1,;x))]
154
H.-C. W U
Then, -
~
L
~
L
( r ( p ) ) . = ( Pi)~(/5{(5(1,;X) = 1 = &(0,; X)})~
+(/5,) ~(f{4,(L; x ) .
= 1,4,(o,; x)=o})~
+ ( 1 - ( / 5 , ) i)(/5{(;b (1,;X) = 1 = &(0,;X)})~ L
= [ 1 - ((/5i) ~'-(/5,) .)] (P{~b(1,;X) = 1 = &(0,;X)})~ ~
L
= 1,
x)
and .
~
U
( r ( p ) ) . = [1 + ( ( / s i ) ~ - ( / s i ) L ) ] ( P { d p ( l i ; X )
= 1 = qS(O,;X)})~
U " 1 , '~X ) = l,qb(0,'X)=0}) -t-(/s,),~(P{qb(
U.
(i) We have (f(A))~ ~ >10 and (P(A)) U >10 for any event A. thus (Y(~))~ >t (r(p)),~ and (?(~))v ~<(?(~,))v (since (/5,)~ >/( p,),~ "' L and (/5)~-..<(/5;)~). ~
r
L
(ii), (iii), and (iv) are obvious.
[]
REMARK 4.3. (i) From Theorem 4.3 (i), in the ith component, if we replace the fuzzy probability/51 by another fuzzy probability/5i which has less fuzziness than that of/51 then the fuzzy reliability of the new system has less fuzziness than that of the original system. This seems reasonable from intuition. (ii) From Theorem 4.3 (ii), (iii), and (iv), we can improve the fuzzy system reliability by improving the fuzzy probability in the ith component.
4.3. COMPUTATIONAL METHODS AND EXAMPLE Suppose that each of/5 i is a nonnegative closed fuzzy number. Then by Proposition 2.6, the fuzzy reliability frO) is also a nonnegative closed fuzzy number. The a-level set of F(O) is A s = {r: /~(r) >/a} = [(f(~))~,(f(O))~]. By Proposition 2.2, the membership function of ~(~) is ~(r)=
sup a l A o ( r ) = s u p { a : O < a . . < l , ( 7 ( [ O ) ~ < r . . < ( ? ( ~ ) ) ~ } .
F U Z Z Y R E L I A B I L I T Y ANALYSIS
155
ThUS, p.(r) = M a x
a
subject to a ~< 1
(~(~))~r u
a>~0. METHOD 1. The membership of the reliability r is equal to zero for r ~ [(~(~))0L, (~(~))0v ]. That is, the preceding nonlinear program is infeasible for r~[(~(~))L,(~(~))0U], since (~(~))L is increasing with respect a and (~(~))~ is decreasing with respect 4. As a matter of fact, if we regard the fuzzy reliability ~(~) as a fuzzy-probabilistic number then we are just interested in r E [(~(~))0L,min{1,(fi(~))0u}]. For the further analysis, we can discard one of the constraints in the following ways. (i) If (7(~)) L -%Jr is redundant, since (7(~)) L is increasing with respect to a and (~(~))v is decreasing with respect a, that is,
Thus we solve the relaxed nonlinear program, /x(r) = Max
o(
subject to a~
(F(~))~L<-~r a>~0. (iii) If r/> (f(~))~ then the constraint (7(~)) L -%
(~(~))~L -~ .< (~O))~ .< (~(0))~' ~ r V s Thus we solve the relaxed nonlinear program ~(r) =Max
,~
subject to a ~< 1
U
(~(0))~>r a>~0. We can use the commercial optimizer GAMS (General Algebraic Modeling System) to solve the above nonlinear programs.
156
H.-C. W U
METHOD 2. Since (~(~))~ is increasing with respect to a and (~(~))~ is decreasing with respect to a, we have: (this follows from Method 1) (i) if (7(~))1L 4 r 4 ( 7 ( ~ ) ) ~ then ~z(r)= 1. (ii) if r 4 (~(~))~ then, p~(r) = max{ a : 0 4 a 4 1, a is the root of ( r_( p_ ) ) L~ - r
=
0},
(iii) if r > (F(~))~ then,
U /x(r) = m a x { a : 0 4 a 4 1 , a is the root of ( ~ ( ~ ) ) ~ - r = 0 } . We can use the commercial software MATLAB to find the roots of the preceding equations. EXAMPLE. We consider the following system. All components are mutually independent.
O
?@
®
The structure function is ~(x) =Xm(1- (1 - x 2 ) ( ! - x 3)). We have ~(~) =P{ ,~(X) = 1} =/5{X1(1 - (1 - X 2 ) ( 1 - X 3 ) ) = 1}
=/~{X 1 = 1 , ( 1 - X 2 ) ( 1 = ~ { x , = 1, (1 - x ~ )
- X 3 ) =0} = o, (! - x ~ )
= o}
• /~{X~ = 1,(1 - ) ( 2 ) = O , ( 1 - X 3 ) = 1}
• ,~{x~= 1,(1 -x~) = 1,(1 -x3) =o} =/~{X 1 = 1 , X 2 = i , X 3 = 1} */5{X, = 1 , X 2 = i,)(.3=0} O ]5{Xl = l , X 2 = 0 , X 3 = 1} = [p1~/~2~/~3] ~ [/01~t02~( ~- ~i03)] ~ [i O. ~('1 0/~2)~/~3].
FUZZY
RELIABILITY
ANALYSIS
157
Let /5i have m e m b e r s h i p function (with spread ~ri),
/zq(r) =
\-~/]
+1,
ifxi-tri<~r<~xi+(ri, otherwise.
Let ¢ri=0.2, V i = 1,2,3 and x I = x 2 = x 3 =0.8. T h e n each of/5~ is a closed fuzzy-probabilistic n u m b e r and, (/6i)
={r:/.t~,(r)>~c~}
={r: xi-~ri~/1-cr <~r<~xi+cri l~L--d-~} = [0.8 - 0.2~/1 - or, 0.8 + 0.2 l~-Z~-~ ]. Thus by Proposition 2.8, we have ( i ( ~ ) ) ~ = (/6~')3 + 2((/5~ ) 2 ( 1 - / 5 ~ )) = (0.8 - 0.2 l¢~L~-a ) s + 2(0.8 - 0.2¢1 - c~ )2(0.2 - 0 . 2 f f - c~ ) = 1.008 - 0 . 7 9 2 ¢ ] - - ~ - 0.24~ + 0.024c~v/1 - ce, and ( r ( p ) ) ~ = (p~')3 + 2((/5ff)2(1 - / 5 ~ ) )
=
(0.8
+
0.2 1¢T7~-~ ) 3 + 2(0.8 + 0.2 lOlLS-~ )2(0.2 + 0.2¢1 - ~-)
= 1.008 + 0.792~/1 - ce - 0.24c~ - 0.024c~gl - cr. H e r e , we consider the fuzzy probability, thus we can say that the functioning probability of each c o m p o n e n t is approximately equal to 0.8. F o r the conventional system with Pl = P z = P 3 =0.8, the reliability is r = r ( p ) = (0.8) 3 + 2(0.8)2(0.2) = 0.768. Thus in fuzzy system reliability, we expect that 0.768 will have m e m b e r s h i p 1. As a m a t t e r of fact, we have /z(0.768)= sup0 ,<, ,< 1 a 1A(0.768) = 1, since A 1 = [(?(0))~, (?(0))~ ] = [0.768, 0.768]. Now
158
H.-C. W U TABLE 1 Reliability r
Membership
0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.768 0.80 0.85 0.90 0.95 1.00
0.1729 0.3787 0.5399 0.6667 0.7661 0.8432 0.9016 0.9444 0.9739 0.9917 1.0000 0.9983 0.9893 0.9733 0.9510 0.9231
(r(P))0 = [0.216,1.8], if we regard the fuzzy reliability f(O) as a fuzzy-probabilistic n u m b e r then we are just interested in the reliability r ~ [0.216,1]. We use Method 2 to find the membership. Then we have Table 1. 5.
CONCLUSION
The attempt for considering the fuzzy-valued probability measure is proposed in this paper. We take advantage of using the closed fuzzy numbers to develop our techniques by just considering the lower and upper bounds of the closed intervals. In this case, the fuzzy system reliability is reduced to the conventional system reliability, which also makes the theory discussed so far m o r e tractable mathematically. Most of the structural properties of the coherent systems can be similarly extended under the consideration of fuzzy-valued probability measures. The author would like to thank the anonymous referees for their ualuable comments and suggestions.
REFERENCES 1. R. E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing: Probability Models, Holt, Rinehart and Winston, New York, 1975. 2. K. Y. Cai, C. Y. Wen, and M. L. Zhang, Fuzzy variables as a basis for a theory of fuzzy reliability in the possibility context, Fuzzy Sets and Systems 42:145-172 (1991).
FUZZY
RELIABILITY
ANALYSIS
159
3. K. Y. Cai, C. Y. Wen, and M. L. Zhang, Posbist reliability behavior of typical systems with two types of failures, Fuzzy Sets and Systems 43:17-32 (1991). 4. K. Y. Cai, C. Y. Wen, and M. L. Zhang, Fuzzy states as a basis for a theory of fuzzy reliability, Microelectron. Reliab. 33:2253-2263 (1993). 5. S.-M. Chen, Fuzzy system reliability analysis using fuzzy number arithmetic operations, Fuzzy Sets and Systems 64:31-38 (1994). 6. C.-H. Cheng and D.-L. Mon, Fuzzy system reliability analysis by the interval of confidence, Fuzzy Sets and Systems 56:29-35 (1993). 7. D.-L Mon and C.-H. Cheng, Fuzzy system reliability analysis for components with different membership functions, Fuzzy Sets and Systems 64:145-157 (1994). 8. R. E. Moore, lnten~al Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1966. 9. M. Mizumoto and K. Tanaka, The four operations of arithmetic on fuzzy numbers, Syst. Comput. Contr. 7:703-710 (1976). 10. T. Onisawa and J. Kacprzyk (Eds.), Reliability and Safety Analyses under Fuzziness, Physica-Verlag, Heidelberg, 1995. 11. M. L. Puri and D. A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114:409-422 (1986). 12. W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1986. 13. M. Sakawa and H. Yano, Feasibility and Pareto optimality for multiobjective linear and linear fractional programming problems with fuzzy parameters, in: J. L. Verdegacy and M. Delgado (Eds.), The Interface between Artificial Intelligence and Operations Research in Fuzzy Enl,ironment, Verlag TUV Rheinland, 1990, pp. 213-232. 14. L. A. Zadeh, Fuzzy sets, Inf. and Control 8:338-353 (1965). 15. L. A. Zadeh, The concept of linguistic variable and its application to approximate reasoning I, II and III, Inform. Sci. 8:199-249 (1975); 8:301-57 (1975); 9:43-80 (1975).
Receit,ed 19 Nol~ember 1996; ret~ised 22 January 1997
Fuzzy Reliability Analysis Based on Closed Fuzzy Numbers HSIEN-CHUNG WU
Operations Research/Industrial Engineering Group, Department of Mechanical Engineering, University of Texas at Austin, Austin, Texas 78712, USA
ABSTRACT The approach by proposing the fuzzy-valued probability measures to the conventional system reliability is discussed successfully in this paper. We consider the failed or functioning probability of each component in the system as nonnegative fuzzy numbers under the definition of fuzzy-valued probability measure, which is a more reasonable consideration in the real world. Under this setting, we can have the pivotal decomposition, and we still can improve the fuzzy reliability of the system by improving the fuzzy probability of each component. Furthermore, the membership function of the fuzzy reliability can be transformed into the nonlinear programming problem and can be solved by any current optimizer. ©Elsevier Science Inc. 1997
1.
INTRODUCTION
The theory of fuzzy reliability was proposed and developed by several authors. The recent collection of papers by Onisawa and Kacprzyk [10] gave many different approaches for fuzzy reliability. The conventional reliability is considered under the probability assumption and binary-state assumption. Cai et al. [2, 3, 4] gave a different insight by introducing the possibility assumption and the fuzzy-state assumption to replace the probability and binary-state assumptions. In the conventional system, we always give an exactly failed or functioning probability for each component. However, in practice, we cannot be sure of the exact value for each component. All we can know is that each component is failed or functioning with probability which is approximately equal to p. Cheng and Mon [6, 7] and Chen [5] gave this new approach. There is another point of view discussed in this p a p e r which considers the fuzzy-valued probability measures. Under this consideration, we can define
INFORMATION SCIENCES 103, 135-159 (1997) © Elsevier Science Inc. 1997 655 Avenue of the Americas, New York, NY 10010
0020-0255/97/$17.00 PII S0020-0255(97)00059-5
136
H.-C. WU
the fuzzy conditional probability measure and in the meantime, we can discuss the pivotal decomposition. As a matter of fact, it is possible to extend most of the theorems for reliability theory under the sense of fuzzy-valued probability measures. In Section 2, we give some properties of fuzzy numbers which will be used in the rest of this paper. In Section 3, we discuss the fuzzy-valued probability measures and fuzzy-valued conditional probability measures. We need to introduce fuzzy conditional probability in order to derive the pivotal decomposition based on the fuzzy reliability, and we still can improve the fuzzy reliability of the system by improving the fuzzy probability of each component under the consideration of fuzziness, which will be discussed in Section 4. We also provide computational methods and an example to illustrate the membership of the fuzzy reliability. 2.
T H E A R I T H M E T I C OPERATIONS OF F U Z Z Y NUMBERS
We start this section by giving some properties of fuzzy numbers, which will be used in the rest of this paper. DEFINITION 2.1. Let X be a universal set. Then a fuzzy subset A of X is defined by its membership function ~A-: X ~ [0, 1]. We can also write the fuzzy set A as [(x,/ZA-(X)): X ~X}. We denote A , = {x: ~A-(X) >/ a} as the a-level set of A. DEFINITION 2.2. (i) A is called a normal fuzzy set if there exists x such that /~,/(x) = 1. (ii) A is called convex fuzzy set if tz~(Ax+(1-A)y)>~min{ix,i(x), gA-(Y)}, for A E [0, 1]. (That is, /zA- is a quasiconcave function). PROPOSITION 2.1 ([14]). A is a convex fuzzy set if and only/f{x: /ZA-(X)>~a} is a conuex set for all a. PROPOSITION 2.2 ([15]) (Resolution Identity). Let A be a fuzzy set with mernbershipfunction tx4 and A , = {x: tZA-(X)>/a}. Then,
~-(x)= sup alA~(x). c ~ [0, 1]
DEFINITION 2.3 ([12]). Let f ( x ) be a real-valued function on a topological space. If {x: f ( x ) >/a} is closed for every a, f ( x ) is said to be upper semicontinuous. DEFINITION 2.4. (i) rh is called a fuzzy number if rh is a normal convex fuzzy set and the a-level set, r h , is bounded Va :~ 0. The set of all fuzzy numbers is denoted as ~.
F U Z Z Y RELIABILITY ANALYSIS
137
(ii) rh is called a closed fuzzy number if rh is a fuzzy number and its membership function /x,~ is upper semicontinuous. The set of all closed fuzzy numbers is denoted as ~c~-tPROPOSITION 2.3. I f rh is a closed f u z z y n u m b e r then the ~-lecel set o f rh is" a closed interval, which is d e n o t e d by rh = [m- ,L , m- U ~ ]. ( T h a t is why we call rh as a closed f u z z y n u m b e r . ) Proof. Ix,~ is an upper semicontinuous function, so the c~-level set rh = {x: /x,~(x) >/cQ is a closed set (by Definition 2.3). Now rh is a convex fuzzy set, thus r h is a closed interval by Proposition 2.1. []
Next we consider the operations of fuzzy numbers. DEFINITION 2.5. Let rh and fi be fuzzy numbers. (i) The membership function of sum r h ¢ ~ is defined by /x~,e,~(z)= sup min{/x,~(x),/x~(y)}. x+v=z
(ii) The membership function of the opposite of rh is defined by /Xe,~(z ) = sup min/x~(x) = / x , ~ ( - z ) . z-
x
(iii) The difference of rh and fi is defined by rh e f t =rh ¢(@~). (iv) The membership function of product rh ® h is defined by
/x,~®~(z) = sup min{/x,~(x),/xe( y)}. ,y=~ (v) The membership function of the inverse of rh is defined by
z = l / x . x 4 = (I
(vi) The quotient of rh and h is defined by rh,~fi =rh ®(1/fi). REMARK 2.1. We
have
/x,~ee(z) = supx r_=min{/x,~(x),/xe(y)} and /x,~~,~~(z) = sup .... / v . ; ~ 0rain{/xr~(X), #~(Y)}" DEFINITION 2.6. Let ~ be a fuzzy number. (i) (ii) (iii) (iv)
ti is called ~ is called ~i is called ~ is called
a a a a
nonnegative fuzzy number if ix4(x)=0, V x < 0 . nonpositive fuzzy number if t x 4 ( x ) = 0, V x > 0. positive fuzzy number if /XA~(X)=0, Vx ~ 0. negative fuzzy number if /.t~(x)=0, Vx>~0.
138
H.-C. WU
PROPOSITION 2.4 ([9])• Let rh and h be two fuzzy numbers. (i) rh ¢ fi, rh e h, and rh ® fi are also fuzzy numbers. (ii) If h is a positive or negative fuzzy number then rh • h is also a fuzzy number. Let "Qint" be any binary operation el.t, e i n t , ®int, o r (])in t between two closed intervals rh~,= [ m"/~ ~ , m- ,u] and h,~ = [ n-L , , n -U , ] . Then rh~Q) inth . is defined by /~aQ) intHa =" {Z ~ RIz = x o y , V x ~ rh, , V y ~ h e ,
where " o " is an usual binary operation + , - , × , o r / } . PROPOSITION 2.5 ([13], Theorem 2). Let Q be any binary operation ¢ , o , ®, or (D between two fuzzy numbers. If rh and h are two closed fuzzy numbers then,
PROPOSITION 2.6. Let rh and h be two closed fuzzy numbers. (i) rh ¢ h, rh 0 h, and Fn ® fi are also closed fuzzy numbers. (ii) If fi is positive or negative then rh ¢ h is also a closed fuzzy number. Proof. From Proposition 2.4, we just need to prove that /x,ho~ is upper semicontinuous. That is, we need to prove that (rhQh)~ is a closed set for all a, which is the immediate consequence of Proposition 2.5. []
PROPOSITION 2.7 ([8]). (i) The addition and multiplication for closed intervals are both associative and commutative. (ii) The addition and multiplication for closed fuzzy numbers are both associative and commutative. Proof. (ii) By (i) and Proposition 2.5.
[]
PROPOSITION 2.8. (i) If {t and [~ are two closed fuzzy numbers then (4eb)~=[a~~L +DL~,a~U+DV~land(~eD) =[~-b~,a,-b~].'u -u "L (ii) If ~ and D are two closed fuzzy numbers then (~i c D)~= • ~L -L [mm{a~b~, -L-U a-U-L" L -LDU a~b~ , b ~L, a ,-U'U b , } , max{a~b~,a~ , , a -U'L , b ~ , a ~-U'U b,}]. (iii) If ~ and b are two nonnegative closed fuzzy numbers then (gl ® D), = - L - L ~U'U [a~b~,a~b~]. (iv) I f fz is a nonnegative closed fuzzy number and [~ is a positive closed fuzzy number then ( 4 ¢ [~)~ - - [ a-L~ / b"~U, a-U ~ / b -~ L ]. Proof. If /~ is positive then b~ ~: 0 and /~ff* 0 for all a. The result follows from Propositions 2.5 and 2.6. []
F U Z Z Y RELIABILITY ANALYSIS
139
DEFINITION 2.7. (i) Let [a, b] and [c, d] be close intervals. We say that [a, b] ~int [C, d] if and only if a >~c and b >~d. (ii) Let 5 and/9 be two closed fuzzy numbers. We say that 5 ~/9 if and only if 5, ~i,t/9, for all a. (iii) Let 5 and /9 be two closed fuzzy numbers. We say that 5 _c~l/9 if and only if 5, c / 9 for all a. In this case, we say that 5 has Jess fuzziness than that of/9. (The spread of the membership function of 5 is less than or equal to the spread of the membership function of/9.) PROPOSITION 2.9. " ~ " and " ___c~" are partial orderings on Jccl (the set o f all closed fuzzy numbers). DEFINITION 2.8. We say that 5 is a crisp number with value m if its membership function is (~ /x~ (r) =
if r = m , otherwise.
REMARK 2.2. Suppose that /9 is a crisp number with value m. If d---c~/9 then it is easy to see that fi is also a crisp number with value m. That is, there is no fuzzy number which has less fuzziness than that of crisp number. PROPOSITION 2.10 ([8]). Let I, J, and K be closed intervals. (i) I ® (J ¢ K ) c_ ( I ® J ) • ( I ® K ) (subdistrikut&ity). (ii) I f J ® K is nonnegative then I ® ( J C K ) = ( I ® J ) ¢ ( I ® K ). (iii) If t is a real number then t( l c J ) = tl c tJ. PROPOSITION 2.11. Let d, /9, and ? be closed fuzzy numbers.
(i) ~ ® (/9 • ~)___c~(~ ®/9) • (~ ® ~). (ii) If/9 ® ~ is nonnegatiue then 5 ® (/9 ¢ ~ ) = (5 ®/9) ¢ ( fi ® ~). (iii) I f fi is a crisp number then f i ® ( / 9 ¢ ~ ) = ( 5 ® / 9 ) ¢ ( 5 ® 5 ) . Proof. By Propositions 2.10 and 2.5.
[]
PROPOSITION 2.12. Let fi and /~ be two closed fuzzy numbers. Then /)Cc~ 5 e ( / ) e S ) . If 5 is a crisp number then / 9 = S e ( / 9 e S ) . Proof. By Proposition 2.8, we have ( d e ( / 9 e S ) ) ~ c--c--La+. b, -a,-U ~
REMARK 2.3. This means that /9 has less fuzziness than that of 5 ¢ (/9 e d). The more operations for fuzzy numbers will increase the fuzziness. The same situation occurs in Proposition 2.11 (i).
140
H.-C. W U
PROPOSITION 2.13. Let 5, [~, and 6 be nonnegatiue and closed fuzzy numbers.
(i) If b is positiue and 5 • b = ~ then 5 C cl [~ ® ~. If [~ is a crisp number then 5 = [~ ® ~. (iN) I f [~c_d 5 then [~ ® ~ _Cclt~ ® C. Proof. (i) By Proposition 2.8, we have that ( g L /~-.U __ " L ua/o4 Ca -
-L__
implies
-
-L~U>~L~L
a~ - c~ b~ -j ~ ~,~
a~U~ / b~. L __ - c- Ua
- u _ - _ u [ / . _
as-%
. "-~ a~a
(iN) By Proposition 2.8, we have ( 5 ® ~ ) ~ ---a~c~ -L-L ~ b-L-L , c ~ =(/~®g)~" and ~ c , >~b~ c~ = (/9 ® ~) U. [] ( 5 ® -c). u = a-u-u "u-v PROPOSITION 2.14. Let {5i}7=1 and {bi}i= - ~ 1 be two sequences of closed fuzzy numbers. If SiC_ct [~i, v i = a . . . . . n then ~7 , 5iC_c, ~i~1 b i. ~L Proof. By Proposition 2.8, we have ( ~"i = 1 ai) ~=
(~.-
-
1 bi)a
L
v _ E n and ( ~/~ 1ai)~ -
-
,= l(ai)~
u
~<
En
-
,= l(bi)a
u
En.
"L
~_n i=
t= l(ai), >~ -
~L l(bi). =
u
= ( ~i n i bi)~.
[]
In order to define the fuzzy-valued probability measure in the next section, we need to introduce the limit of a sequence of fuzzy numbers, thus we first need to consider the metric between two fuzzy numbers. DEFINITION 2.9. Let A c_R" and B ___R". The Hausdorff metric is defined by /
dH(A,B)
=max~ sup inf [[a -bl[, sup inf ~acA
b~B
b~B
aeA
Ila-bl[ t.
DEFINITION 2.10 ([11]). We define the metric d% in .~1 as, d~,(5,/~)=
sup dH(5,,,[~,), 0
for any two closed fuzzy numbers.
REMARK 2.4 ([11]). (~ccl,d ~ ) is a complete metric space. LEMMA 2.1. Let 5 and f9 be two closed fuzzy numbers. Then we have dH(5,,,/~,, ) = max{I 5 L _/~L I, ]5~_/gu I}"
F U Z Z Y RELIABILITY ANALYSIS
141
Proof. We have
=max{
sup
inf ~l
_<:
Ix-yl,
sup
inf
Ix-yl}.
_< ~ U
Then we have the following cases. (i) If 4~ ~int/) . o r / ~ ~i~t 6~ then d , ( 4 , , / ~ , ) = max{l~ild -b~'c I,[4 U~-b~-C'I}. (ii) If 4._c/~ or/), c_4, then, d/t (4.,/~,~ ) = max{0, max{14 L - tS,~l, 1 4 ~ - / ~ I}} max{14,c-b~l,-L -v
/~,~'1}.
This completes the proof.
[]
PROPOSITION 2.15. Let 4 and [~ be two closed fuzzy numbers. Then d%( 4, [~) < • implies 14L - [~L[ < • and 1~,~- [~ [< • for all a ~ (0, 1]. Proof. It is obvious from Lemma 2.1.
[]
DEFINITION 2.11. Let {~in} be a sequence of closed fuzzy numbers. {4 n} is said to converge if there is a closed fuzzy number 4 with the following property: V• > 0, 3 N > 0 such that for n > N, we have d%(4,, ~)< e. We also say that the sequence {4,} converges to 4 and it is denoted as lim . . . . 4, = 4. If there is no such 6, the sequence {ti,} is said to diverge. PROPOSmON 2.16. Let {4,} be a sequence of closed fuzzy numbers. If lim~ ~ ~ 4~ exists then it is unique and (1i~4~)
=[Jim(4n)~,Jim(~7,),~']
for all ce. (Furthermore, ((tin) f } and {(a,,)~ - ~"} converge uniformly with respect to a.) Proof. Let l i m n ~ 4 , , = 4 . By Proposition 2.15, d j c ( 4 n , 4 ) < e implies -a~]
142
H.-C. WU
DEFINITION 2.12. Let {4,} be a sequence of closed fuzzy numbers. If lim,_ ~ ~i~1 ai exists then we write oo
n
4,:
li'-m ~
n=l
n--*~
4 i,
i=1
otherwise the infinite (fuzzy) sum of sequence {4,} is said to diverge. PROPOSITION 2.17. If {4,} is a sequence of closed fuzzy numbers and lim. . . . ~'i~ 1 ai exists then we have ~
4. =
=
L
~
U
(a.)o,
ot
=
=1
[]
Proof. By Propositions 2.16 and 2.5. 3.
F U Z Z Y - V A L U E D PROBABILITY MEASURES
In this section, we shall discuss the fuzzy-valued probability measures and fuzzy-valued conditional probability measures. DEFINITION 3.1. Let 9- be a set of all fuzzy numbers and ~c~ be a set of all closed fuzzy numbers. We say that (i) f ( x ) is a fuzzy-valued function if j~ X ~ J . (ii) f(x) is a closed-fuzzy-valued function if fi X--*~I. DEFINITION 3.2. A fuzzy number/5 is called a fuzzy-probabilistic number if its membership function /xp(r)= 0 for r e [0, 1]. We denote ~e as a set of all fuzzy-probabilistic numbers. REMARK 3.1. (i) The fuzzy-probabilistic numbers are nonnegative fuzzy numbers. (ii) A fuzzy number 4 can induce a fuzzy-probabilistic number /5 by truncation. That is,
ixp(r)=(txa(r), 0,
if r ~ [0,1], otherwise.
We denote 0 and 1 as crisp numbers with values 0 and 1, respectively. Then 13 and 1 ~ J e . DEFINITION 3.3. By a fuzzy-valued probability measure t3 on a o--field (1), .~g), we mean a nonnegative closed-fuzzy-valued set function defined
F U Z Z Y RELIABILITY ANALYSIS
143
for all sets of ~" and satisfying the following conditions:
(i) P(O) = 6 (ii) 15(f~) ___cai (iii) f(U~= i Ai) = ~=1 15(Ai) exists for any sequence {.4i}~= 1 of disjoint events (.4i ('l.4j = Q~ for i -~j). We always regard 15(A) as a fuzzy-probabilistic number by truncation. Then 15(U~=1Ai)= @~=l fi(Ai) after truncating those two nonnegative fuzzy numbers as fuzzy-probabilistic numbers. The condition imposed on P by (iii) is called countable additivity. REMARK 3.2. If 1) is finite, ~" is taken to be the discrete o--field (all subsets of 12). In such a case, there are only finitely many events and hence, in particular, finitely many pairwise disjoint events. Then (iii) is reduced to (iii)': P is finitely additive; that is, for every collection of disjoint events {Ai}'/_ t, we have
f
A, = ~ f ( & ) . i=
i=1
PROPOSITION 3.1. Let -15 be a fuzzy-valued probability measure on the or-field (1~, ~'). (i) Let .4 and B be two events in the ~r-field (fl, ~{). If A GB then
f(B)~P(.4). (ii) Let A be an event in the cr-field ( fL J ) . Then 15(fD ~15(A).
Proof. Since B=.4 U ( B \ A ) is a disjoint union, we have /2(B)=/2(A) /2( B \ .4) ~ t2( .4 ) (by Proposition 2.5). [] DEFINITION 3.4. Let .4 and B be two events in 12. We say that .4 and B are independent with respect to a fuzzy-valued probability measure /5 if and only if 15(.4 AB)=P(.4)®15(B). Then 155(.4AB)=IS(.4)®15(B) after truncating those two nonnegative fuzzy numbers as fuzzy-probabilistic numbers. DEFINITION 3.5. Let /5 be a fuzzy-valued probability measure and .4 ~ ft be such that 15(.4) positive (i.e., (15(.4))~ 4:0 and (15(.4))v ~ 0 for all oD. Then the fuzzy conditionalprobability, given .4, is the closed-fuzzy-valued set function denoted by P(.IA) and defined on 1~ as follows, 15(B1.4) =/5(.4 N B ) • 15(A), B ~ I~ . 15(BlA) is called the fuzzy conditional probability of B given .4. PROPOSITION 3.2. Let 15 be a fuzzy-valued probability measure and 15(A) be positive closed fuzzy number. Suppose that ~i~1 [15(A i N .4 ) • 15(A)] exists
144
H.-C. WU
for any countable collection of disjoint euents {Ai}~ l. Then f(.[A) is a fuzzy-ualued probabifity measure. Proof. (i) f(O[A)=fi(Q~NA)O f ( A ) = f ( o ) O f(A)=0. (ii) t 6 ( [ I I A ) = f ( ~ N A ) O f ( A ) = f ( A ) O f ( A ) . Then by Proposition 2.8(iii), we have (f(~lA))~ " = ( f ( A ) ) ~ / ( f ( A ) ) ~ < 1 and (f(I~]A))~v =
(IA(A))~/(P(A))~ >/1 for all a. Thus f(ll]A)2~, 1. (rid We have
A i=
iA) [
]
:
P(AiNA ) O f ( A ) ,
=P iUl ( AiOA) O f ( A )
i=
(( AiGA}I~_:, is a sequence of disjoint events). Then
i=l f
U IAiA
AiNA
=
U
~c
: ~ (f(AinA))~ i=1
=
(P(A))
(by Proposition 2.17)
U
~_,(f(AiNA)Of(A))Zl i=1 vc
/~
(by Proposition 2.17)
i=1
a"
F U Z Z Y RELIABILITY ANALYSIS
145
Similarly, A iA i=
= a
(~
AlIA
.
i= ]
~
Thus 15
AiA i=
= (~) 15(ALIA). i=1
This completes the proof.
[]
COROLLARY 3.1. Let 15 be a fuzzy-valued probability measure. If ~ is finite then 15(. ]A ) is a fuzzy-valued probability measure. Proof. The proof in (iii) of preceding Proposition 3.2 is always true for finite case. The result follows from Remark 3.2. [] PROPOSITION 3.3. Let 15 be a fuzzy-valued probability measure and 15(A) be a positive closed fuzzy number. I r A and B are independent with respect to 15 then 15(B)c_ d 15(B~1). If ~5(A) is a crisp number with l~alue m 4:0 then P ( B ) =P(BIA). Proof. We have P ( B I A ) =15(A N B ) e 1 5 ( A ) = ( 1 5 ( A ) ® 1 5 ( B ) ) e P(A), thus, (15(B~A))~=(15(A)®15(B))2/(15(A))ll '
(by Proposition 2.8)
= (15(A))~(15(.))~/(15(~))~ .< (~(.))':, and
(15(8~) )~=( 15¢A)),~: (15(B) )~/( 15(A) )~ >~(15¢B))~ This completes the proof.
[]
REMARK3.3. (i) Intuitively, although A and B are independent, f ( B I A ) has more fuzziness than that of 15(B) since the fuzzy conditional probability /5(B[A) is considered under the fuzzy environment for given event A, thus P(B)_Ccl/5(B~A). Also we can say something as follows, we can exclude the independence, but we cannot exclude the fuzziness.
146
H.-C. WU
Furthermore, this will affect the pivotal decomposition based on the fuzzy reliability. (ii) Although P(A) is a crisp number, /3(B)=/5(B~I) is still a closed fuzzy number since /5(A n B) is a closed fuzzy number. PROeOSmON 3.4 (Sub-total probability theorem). Let {Ai: i = 1,...,n} be a finite partition of fl and P be a fuzzy-valued probability measure. Suppose that each of lS(Ai ), Vi = 1.... , n is positive. Then we have n
P ( B ) ---c, ~
/3(B~Ai) ® P ( A i ) .
i=l
Proof. We have that B = UT=1(B n A i) is a disjoint union, thus n
P(B)=
(~ 1 5 ( B A A , ) i=1
(by definition of the fuzzy-valued probability measure) /7
---c, ~
/5(BIA~) ® f ( A ~ )
(by Propositions 2.13 (i) and 2.14).
i=1
This completes the proof.
[]
COROLLARY 3.2 (Total probability theorem). If each of lS(Ai) is a crisp number with value m i 4=O, V i = 1,..., n then,
P(B)
=
(~
/5(BIA~) ® f ( A , ) .
i=l
Proof. By Proposition 2.13 (i).
[]
REMARK 3.4. From Remark 3.3 (ii), we can see why /5(B) is still a closed fuzzy number although each of P ( A i) is a crisp number. 4. FUZZY RELIABILITY ANALYSIS In this section, we shall discuss the fuzzy reliability of series system, parallel system, and k-out-of-n system with identical components, and then we shall discuss the pivotal decomposition and the notion for improving
F U Z Z Y RELIABILITY ANALYSIS
147
the fuzzy system reliability. Finally, we shall provide computational methods and an example to illustrate the usefulness of the results discussed so far. DEFINITION 4.1. Let X and Y be two discrete random variables. We say that X and Y are independent if P{X=x, Y=y} =/~{X=x} ®/;{Y=y}. Let 1~ = {S, F} and ~t' be a o-field on 1~. Suppose that /~({S})=t5 and /~({F}) = 1 0/5. Then /;(1~) =/;({S}) •/~({F}) =/5 • (1 @/5) 2el 1 (by Proposition 2.12). Thus it is easy to see that P is a fuzzy-valued probability measure on (1~, Me). Let X be a Bernoulli random variable on the fuzzy probability space (1~, ~ ' , / ; ) . Then /;{X = 1} =/;({S}) =/5 and /~{X = 0} = P({F}) = 1 Ot5. REMARK 4.1. Let 4 = 1G/5. Then 1 e ~ =/5 since (1 @4)~ = 1 _ ~ v = 1 ( ] e / 5 ) V = [ l _ ( 1 - p -L -L and (1 eq)~ - v =p~ -v for all ce. This says that if ~ )]=p~ # { X = 0} =c~ then # { X = 1} = i ec~. In other words, # { X = 0} = i O # { X = 1} and # { X = 1} =1 e # { X = 0}. DEFINITION 4.2. The structure function of the given system with n components is 4,(X) = &(X l, X 2..... Xn), where each of X i is a Bernoulli random variable. We define the fuzzy reliability ? of the system as the fuzzy probability of d,(X)= 1, that is, ~=P{&(X)= 1}. We assume that all components are mutually independent. n ~X i, where each of X i is a Bernoulli (i) Seriessystem: Here &(X) -_ Hi= random variable, we have
~=/~{ 6(X) = 1} =
#
. {I~i=lXi=|}
=/~{X 1 = l , X 2 = 1 ..... X~ = 1} n
-
( ~ /~{Xi = l } i=1
?/
= ®A. i=1
(since mutually independent)
148
H.-C. WU
We regard ~ as a fuzzy-probabilistic number by truncation. (ii) Parallel sytem: Here 6(X) = 1 - Vii"=1(1 - X i ) , we have = f{ 6(x)
= 1}
= ] e/5{ 6 ( X ) = 0}
(by Remark 4.1)
= i e/5{n7 ,(l - x , ) = a} = 1 ( 3 , 5 { 1 - X , = 1 , 1 - X 2 = 1 ..... 1 - X , = 1}
= ] e/5{x, = o.x2 =o ..... x . =o} =ie
®/5{x,=o} i=l
=ie
.=
(i ePi
)1 •
We regard i as a fuzzy-probabilistic number by truncation. We write k times
k times
pe-"*p=k/3
and
p®-..®p=/3
k.
(iii) k-out-of-n-system with identical components: We have
f=/5{6(X)=l}=/5
~xi>~k i=l
= (~/s/gXi=m m=k
.
~i=1
Now /5
Xi=m
=
/5{X 1 =1 .....
Xm=l,Xm+
I =0 ..... Xn=0
}
i
(since mutually independent)
mI
F U Z Z Y RELIABILITY ANALYSIS
149
Thus n
We regard t: as a fuzzy-probabilistic number by truncation. 4.1.
THE PIVOTAL
DECOMPOSITION
In this subsection, we just consider the Bernoulli random variables, thus ~ is finite. The fuzzy conditional probability can be used in our discussions for the pivotal decomposition by Corollary 3.1. Let (~/i; 0) denote the vector ~ with its ith component replaced by ~. So (li;P):(/~I ..... j0 i 1,],F),+I ..... fin ) and (6~;~)=(/3~ .... ,]~i_l,0, a/)i+ I ..... fin). We also use the following notations, (li,x)
-- (X 1 .....
Xi
i,l,xi~ 1..... xn),
(O~;x) --- ( x , . . . . . x~ t,O,x~+l . . . . . x n ) , ('i;X)
~-(X 1 .....
Xi
l,',Xi,
i .....
Xn).
PROPOSITION 4.1 ([1])(Pivotal decomposition of the structure function).
The following identity holds for any structure function 4) of order n,
~(x) =x~(l~;x) + (I -x~) ~(0i;x) Vx. THEOREM 4.1 (Sub-pivotal decomposition). Let ~-~(~) be the fuzzy reliability of the system and all components be mutually independent. Then,
(i)
~= ~(~) ---c,[ ~, ®P{~(1,; x) = llx~ = q]
(ii)
-~c.[,~ ®,~{~(.,;x)= 1ix,= 1}]. [0 e~,)®,~{~(0i; x)= .l~ = 0}]
150
H.-C. WU
Proof. We have ~(fi) = 6 { ~ ( X ) = 1}
--(~cl= [`6{Xi -~ 1} ®`6{(h(X) = 1]Xi = 1}] ¢ [`6{X i =0} ®`6{6(X) = liXi =0}] (by Propositions 3.4 and 2.7 for commutativity) = [pi ®`6{4)(li;X)= IlX,= 1}]
_~c,[~, ®`6(~(1,;x)= 1)1 * [0 o~,)®`6{~(0,;x)= 1)] (by Propositions 3.3, 2.13 (ii), and 2.14)
This completes the proof.
[]
COROLLARY 4.1 (Pivotal decomposition). If `6{Xi = 1} and `6(X i = 0} are crisp numbers then we have the pivotal decomposition,
r~r(P) ~ [/~i ®F(Xi;P)] ~ [(1 ~/~i)~r([~i;P)] • Proof. By Corollary 3.2 and Proposition 3.3.
[]
LEMMA 4.1. Let ~= ?(~) be the fuzzy reliability of the system. Then we haue ?(~) =`6{X~ = 1, (#(1/;X) = 1 = &(O~;X)} ¢`6{X i = 1, d)(1,; X) = 1, 4)(0i; X) = O}
• `6{x~ =
0, ~ , ( l ~ ; x )
¢`6{x~= 0, 6 ( l i ;
= 1 = 4,(0~; x ) }
x ) = 0, 6 ( % ; x )
= 1}.
151
F U Z Z Y RELIABILITY ANALYSIS Proof. We have
~(~) =/5{ ~b(X) = 1} = / ~ ( X / ~ ( I i ;X) + ( 1 - X i ) 1~)(0i ;X) = 1}
(by Proposition 4.1)
=15{Xiqb(1i; X) = 1, (1 -X,)~b(O,; X) = O}
~/~{X~b(1,;X) = 0 , ( 1 - X , ) ~b(O,;X) = 1} =/5{)(/= 1,&(li;X ) = 1 = &(Oi;X)}
• P{x
=
+P{x
= O, Ob(1,;X) = 1 = ~b(Oi;X)} =
= 1,
=o}
O, ¢b(1,;X) = O, ¢b(Oi; X ) = 1}.
This completes the proof.
[]
DEFINITION 4.3. The ith component is irrelevant to the structure d~(x) if ~b(x) is constant in x i, that is, ~b(li;x) = ~b(0i;x) for all (.i;x). Otherwise the ith component is relevant to the structure. THEOREM 4.2 (Pivotal decomposition). If the ith component is irrelevant to the structure ~b(x) then we have the pivotal decomposition,
As a matter of fact, we have t:(0i; ~ ) = ~(1i; ~)col Y(O). If fi i & a crisp number then Y(0i; ~) = Y(li; ~) = Y(~). Proof. We have
P(O) =/5{)(/= 1, ~b(li;X) = 1 = ~b(Oi; X)} • lS{Xi = O, ~b(Oi; X) = 1 = 6 ( l i ; X ) }
(by Lemma 4.1 and the irrelevance)
152
H.-C. WU = [15{ X~= 1} ®/5{&(l~;X) = 1}] • [/5[ Xi=O } ®/5{~b(Oi;X ) = 1}] (by the independence of components)
[/~/®r('ii~P)] ~ [(] ~Pi)®r(6i~P)]
(since 7 ( 0 g ; 0 ) = r ( l i ; 0 ) f o r the irrelevance in t h e / t h component and/~i ® (1 O/5i) is nonnegative, then by Proposition 2.11 (ii)) ~-c, ~ ® ?(13g;~) =Y(()i;~)
(by Propositions 2.12 and 2.13 (ii))
(since 1 is a crisp number with value 1).
The crisp case follows from Proposition 2.12.
[]
REMARK 4.2. In the conventional system, if the ith component is irrelevant to the structure &(x) then r=r(p)=r(Oi;p)=r(li;p), i.e., the system reliability is equal to the reliability whenever the ith component is failed or functioning. (It seems that we ignore the ith component because of the irrelevance.) (i) If/5 i is a crisp number then we have 7(13/;~)=~(1i;~)=7(~). However, here 7(~) is still a closed fuzzy number (i.e., the fuzzy reliability) since the others/Sj may not be crisp numbers. (ii) If/5 i is a fuzzy number then the fuzzy reliability, whenever the ith component is failed or functioning, has less fuzziness than that of the fuzzy reliability of the whole system. It is well behaved, since we seem to ignore the fuzziness for the ith component owing to the irrelevance. In practice, we expect that the system reliability r has membership 1 in the fuzzy system reliability. Now if the reliability r has membership 1 in the fuzzy system reliability &((3i;X), then the reliability r also has membership 1 in the original fuzzy system reliability ~(X) since (r(0i;p))l c-(r(P))l (the l-level set). This explains the equality r=r(p)=r(Oi;p)=r(li;p) for the conventional system.
F U Z Z Y RELIABILITY ANALYSIS
153
4.2. IMPROVING THE FUZZY SYSTEM RELIABILITY DEFINITION 4.4. A system with structure function 6(x) is called increasing if it satisfies the following conditions: (i) 6 ( 0 . . . . . 0 ) = 0 ,
6(1 ..... 1)=1
(ii) If x < y then 6(x) ~<6(Y). This definition says that the state of the increasing system cannot get worse if the state of any of its components was improved. The system is up if all its components are up, and down if all its components are down. THEOREM 4.3. Let ~ = ~(~) be the fuzzy reliability of an increasing system 6(X) which consists of independent components. Then we have: (i) if~ic_cl ~ then r(D)--d r(P'), where ~ =(/51 ..... /5, ,,/si,Pi+, ..... /5.) and ~' = (/51 ..... fii-1, fi;, Pi + p "" , /5,). (ii) if" p," ' ~~p~ - and (/5,)u- (/5,)~ = (Pi), -, u --(Pi), -, L for all a (if the membership function of f~ is the translation of the membership function of /si then/5i and/5~ satisfy those conditions) then r(P') ~ r(P). (iii) if /5~ ~ /5i and ~ is a crisp number then 7(~') ~ ~(~)). (iv) if~51 ~ Pi and /51,/si are two crisp numbers then 7(~') ~ r(O).
Proof. By Lemma 4.1, we have
P(~) = [/~{X~ = 1} ~ f ( 6 ( 1 , ; X ) = 1 = 6(Oi;X)} ]
[f(x, = 1) ® f(6(l,;X)
= 1,6(o,;x)
=o)]
(~ [ P ( X i=O} ~/t~{6(Oi;X ) : ] = 6(1/;X)}]
(~/~(~)
(the fourth term has empty event since 6 is increasing function)
= [~, ®P(6(1,;x)
= 1= 6(o,;x)}]
[/5/®,5(6(1i;X) = 1,6(O,;X) =0}]
[(i
f(6(o,; x)=1= 6(1,;x))]
154
H.-C. W U
Then, -
~
L
~
L
( r ( p ) ) . = ( Pi)~(/5{(5(1,;X) = 1 = &(0,; X)})~
+(/5,) ~(f{4,(L; x ) .
= 1,4,(o,; x)=o})~
+ ( 1 - ( / 5 , ) i)(/5{(;b (1,;X) = 1 = &(0,;X)})~ L
= [ 1 - ((/5i) ~'-(/5,) .)] (P{~b(1,;X) = 1 = &(0,;X)})~ ~
L
= 1,
x)
and .
~
U
( r ( p ) ) . = [1 + ( ( / s i ) ~ - ( / s i ) L ) ] ( P { d p ( l i ; X )
= 1 = qS(O,;X)})~
U " 1 , '~X ) = l,qb(0,'X)=0}) -t-(/s,),~(P{qb(
U.
(i) We have (f(A))~ ~ >10 and (P(A)) U >10 for any event A. thus (Y(~))~ >t (r(p)),~ and (?(~))v ~<(?(~,))v (since (/5,)~ >/( p,),~ "' L and (/5)~-..<(/5;)~). ~
r
L
(ii), (iii), and (iv) are obvious.
[]
REMARK 4.3. (i) From Theorem 4.3 (i), in the ith component, if we replace the fuzzy probability/51 by another fuzzy probability/5i which has less fuzziness than that of/51 then the fuzzy reliability of the new system has less fuzziness than that of the original system. This seems reasonable from intuition. (ii) From Theorem 4.3 (ii), (iii), and (iv), we can improve the fuzzy system reliability by improving the fuzzy probability in the ith component.
4.3. COMPUTATIONAL METHODS AND EXAMPLE Suppose that each of/5 i is a nonnegative closed fuzzy number. Then by Proposition 2.6, the fuzzy reliability frO) is also a nonnegative closed fuzzy number. The a-level set of F(O) is A s = {r: /~(r) >/a} = [(f(~))~,(f(O))~]. By Proposition 2.2, the membership function of ~(~) is ~(r)=
sup a l A o ( r ) = s u p { a : O < a . . < l , ( 7 ( [ O ) ~ < r . . < ( ? ( ~ ) ) ~ } .
F U Z Z Y R E L I A B I L I T Y ANALYSIS
155
ThUS, p.(r) = M a x
a
subject to a ~< 1
(~(~))~r u
a>~0. METHOD 1. The membership of the reliability r is equal to zero for r ~ [(~(~))0L, (~(~))0v ]. That is, the preceding nonlinear program is infeasible for r~[(~(~))L,(~(~))0U], since (~(~))L is increasing with respect a and (~(~))~ is decreasing with respect 4. As a matter of fact, if we regard the fuzzy reliability ~(~) as a fuzzy-probabilistic number then we are just interested in r E [(~(~))0L,min{1,(fi(~))0u}]. For the further analysis, we can discard one of the constraints in the following ways. (i) If (7(~)) L -%
Thus we solve the relaxed nonlinear program, /x(r) = Max
o(
subject to a~
(F(~))~L<-~r a>~0. (iii) If r/> (f(~))~ then the constraint (7(~)) L -%
(~(~))~L -~ .< (~O))~ .< (~(0))~' ~ r V s Thus we solve the relaxed nonlinear program ~(r) =Max
,~
subject to a ~< 1
U
(~(0))~>r a>~0. We can use the commercial optimizer GAMS (General Algebraic Modeling System) to solve the above nonlinear programs.
156
H.-C. W U
METHOD 2. Since (~(~))~ is increasing with respect to a and (~(~))~ is decreasing with respect to a, we have: (this follows from Method 1) (i) if (7(~))1L 4 r 4 ( 7 ( ~ ) ) ~ then ~z(r)= 1. (ii) if r 4 (~(~))~ then, p~(r) = max{ a : 0 4 a 4 1, a is the root of ( r_( p_ ) ) L~ - r
=
0},
(iii) if r > (F(~))~ then,
U /x(r) = m a x { a : 0 4 a 4 1 , a is the root of ( ~ ( ~ ) ) ~ - r = 0 } . We can use the commercial software MATLAB to find the roots of the preceding equations. EXAMPLE. We consider the following system. All components are mutually independent.
O
?@
®
The structure function is ~(x) =Xm(1- (1 - x 2 ) ( ! - x 3)). We have ~(~) =P{ ,~(X) = 1} =/5{X1(1 - (1 - X 2 ) ( 1 - X 3 ) ) = 1}
=/~{X 1 = 1 , ( 1 - X 2 ) ( 1 = ~ { x , = 1, (1 - x ~ )
- X 3 ) =0} = o, (! - x ~ )
= o}
• /~{X~ = 1,(1 - ) ( 2 ) = O , ( 1 - X 3 ) = 1}
• ,~{x~= 1,(1 -x~) = 1,(1 -x3) =o} =/~{X 1 = 1 , X 2 = i , X 3 = 1} */5{X, = 1 , X 2 = i,)(.3=0} O ]5{Xl = l , X 2 = 0 , X 3 = 1} = [p1~/~2~/~3] ~ [/01~t02~( ~- ~i03)] ~ [i O. ~('1 0/~2)~/~3].
FUZZY
RELIABILITY
ANALYSIS
157
Let /5i have m e m b e r s h i p function (with spread ~ri),
/zq(r) =
\-~/]
+1,
ifxi-tri<~r<~xi+(ri, otherwise.
Let ¢ri=0.2, V i = 1,2,3 and x I = x 2 = x 3 =0.8. T h e n each of/5~ is a closed fuzzy-probabilistic n u m b e r and, (/6i)
={r:/.t~,(r)>~c~}
={r: xi-~ri~/1-cr <~r<~xi+cri l~L--d-~} = [0.8 - 0.2~/1 - or, 0.8 + 0.2 l~-Z~-~ ]. Thus by Proposition 2.8, we have ( i ( ~ ) ) ~ = (/6~')3 + 2((/5~ ) 2 ( 1 - / 5 ~ )) = (0.8 - 0.2 l¢~L~-a ) s + 2(0.8 - 0.2¢1 - c~ )2(0.2 - 0 . 2 f f - c~ ) = 1.008 - 0 . 7 9 2 ¢ ] - - ~ - 0.24~ + 0.024c~v/1 - ce, and ( r ( p ) ) ~ = (p~')3 + 2((/5ff)2(1 - / 5 ~ ) )
=
(0.8
+
0.2 1¢T7~-~ ) 3 + 2(0.8 + 0.2 lOlLS-~ )2(0.2 + 0.2¢1 - ~-)
= 1.008 + 0.792~/1 - ce - 0.24c~ - 0.024c~gl - cr. H e r e , we consider the fuzzy probability, thus we can say that the functioning probability of each c o m p o n e n t is approximately equal to 0.8. F o r the conventional system with Pl = P z = P 3 =0.8, the reliability is r = r ( p ) = (0.8) 3 + 2(0.8)2(0.2) = 0.768. Thus in fuzzy system reliability, we expect that 0.768 will have m e m b e r s h i p 1. As a m a t t e r of fact, we have /z(0.768)= sup0 ,<, ,< 1 a 1A(0.768) = 1, since A 1 = [(?(0))~, (?(0))~ ] = [0.768, 0.768]. Now
158
H.-C. W U TABLE 1 Reliability r
Membership
0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.768 0.80 0.85 0.90 0.95 1.00
0.1729 0.3787 0.5399 0.6667 0.7661 0.8432 0.9016 0.9444 0.9739 0.9917 1.0000 0.9983 0.9893 0.9733 0.9510 0.9231
(r(P))0 = [0.216,1.8], if we regard the fuzzy reliability f(O) as a fuzzy-probabilistic n u m b e r then we are just interested in the reliability r ~ [0.216,1]. We use Method 2 to find the membership. Then we have Table 1. 5.
CONCLUSION
The attempt for considering the fuzzy-valued probability measure is proposed in this paper. We take advantage of using the closed fuzzy numbers to develop our techniques by just considering the lower and upper bounds of the closed intervals. In this case, the fuzzy system reliability is reduced to the conventional system reliability, which also makes the theory discussed so far m o r e tractable mathematically. Most of the structural properties of the coherent systems can be similarly extended under the consideration of fuzzy-valued probability measures. The author would like to thank the anonymous referees for their ualuable comments and suggestions.
REFERENCES 1. R. E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing: Probability Models, Holt, Rinehart and Winston, New York, 1975. 2. K. Y. Cai, C. Y. Wen, and M. L. Zhang, Fuzzy variables as a basis for a theory of fuzzy reliability in the possibility context, Fuzzy Sets and Systems 42:145-172 (1991).
FUZZY
RELIABILITY
ANALYSIS
159
3. K. Y. Cai, C. Y. Wen, and M. L. Zhang, Posbist reliability behavior of typical systems with two types of failures, Fuzzy Sets and Systems 43:17-32 (1991). 4. K. Y. Cai, C. Y. Wen, and M. L. Zhang, Fuzzy states as a basis for a theory of fuzzy reliability, Microelectron. Reliab. 33:2253-2263 (1993). 5. S.-M. Chen, Fuzzy system reliability analysis using fuzzy number arithmetic operations, Fuzzy Sets and Systems 64:31-38 (1994). 6. C.-H. Cheng and D.-L. Mon, Fuzzy system reliability analysis by the interval of confidence, Fuzzy Sets and Systems 56:29-35 (1993). 7. D.-L Mon and C.-H. Cheng, Fuzzy system reliability analysis for components with different membership functions, Fuzzy Sets and Systems 64:145-157 (1994). 8. R. E. Moore, lnten~al Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1966. 9. M. Mizumoto and K. Tanaka, The four operations of arithmetic on fuzzy numbers, Syst. Comput. Contr. 7:703-710 (1976). 10. T. Onisawa and J. Kacprzyk (Eds.), Reliability and Safety Analyses under Fuzziness, Physica-Verlag, Heidelberg, 1995. 11. M. L. Puri and D. A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114:409-422 (1986). 12. W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1986. 13. M. Sakawa and H. Yano, Feasibility and Pareto optimality for multiobjective linear and linear fractional programming problems with fuzzy parameters, in: J. L. Verdegacy and M. Delgado (Eds.), The Interface between Artificial Intelligence and Operations Research in Fuzzy Enl,ironment, Verlag TUV Rheinland, 1990, pp. 213-232. 14. L. A. Zadeh, Fuzzy sets, Inf. and Control 8:338-353 (1965). 15. L. A. Zadeh, The concept of linguistic variable and its application to approximate reasoning I, II and III, Inform. Sci. 8:199-249 (1975); 8:301-57 (1975); 9:43-80 (1975).
Receit,ed 19 Nol~ember 1996; ret~ised 22 January 1997