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Bayesian system reliability assessment under fuzzy environments
Reliability Engineering and System Safety 83 (2004) 277–286 www.elsevier.com/locate/ress
Bayesian system reliability assessment under fuzzy environments Hsien-Chung Wu Department of Information Management, National Chi Nan University, Puli, Nantou 545, Taiwan, ROC Received 13 February 2003; accepted 30 September 2003
Abstract The Bayesian system reliability assessment under fuzzy environments is proposed in this paper. In order to apply the Bayesian approach, the fuzzy parameters are assumed as fuzzy random variables with fuzzy prior distributions. The (conventional) Bayes estimation method will be used to create the fuzzy Bayes point estimator of system reliability by invoking the well-known theorem called ‘Resolution Identity’ in fuzzy sets theory. On the other hand, we also provide the computational procedures to evaluate the membership degree of any given Bayes point estimate of system reliability. In order to achieve this purpose, we transform the original problem into a nonlinear programming problem. This nonlinear programming problem is then divided into four subproblems for the purpose of simplifying computation. Finally, the subproblems can be solved by using any commercial optimizers, e.g. GAMS or LINGO. q 2003 Elsevier Ltd. All rights reserved. Keywords: Bayes point estimators; Confidence degree; Fuzzy real numbers; Nonlinear programming; System reliability
1. Introduction In the real world, the data sometimes cannot be recorded or collected precisely due to human errors, machine errors, or some other unexpected situations. For instance, the water level of a river and the temperature in a room cannot be measured in an exact way because of the fluctuation. The more appropriate way to describe the water level is to say that the water level is around 30 m. The phrase ‘around f Therefore, 30 m’ should be regarded as a fuzzy number 30 30: the fuzzy sets theory naturally provides an appropriate tool in modeling the imprecise data. Let us consider an experiment by tossing a fair coin. It is well known that, from the law of large numbers, the probability for observing ‘head’ of the coin should be 1/2. Now suppose that we toss this fair coin for 1000 times. Then it is rarely possible to observe ‘head’ of the coin for 500 times. The possible outcome should be around 500 times. In this case, the more reasonable way is to regard the probability for observing ‘head’ of the coin as ‘around 1/2’. The phrase ‘around 1/2’ can be modeled as a fuzzy number f by using the fuzzy sets theory. 1=2 Now let us consider another example, considering to ask a group of economists for predicting the rate of economical E-mail address: [email protected] (H.-C. Wu). 0951-8320/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2003.09.021
growth in the next year. Their statements may be like ‘approximately 5%’, ‘should be 4 –6%’ or ‘may be below 7%’, etc. All of those statements are described linguistically, and therefore, should be characterized as fuzzy sets. Suppose that we introduce a selection procedure which will determine the probability that each economist will be selected. Then for each possible selection, their statements are realized as fuzzy sets. Therefore, the association of fuzzy sets and probabilities forms the notion of fuzzy random variable. We know that the observations of (traditional) random variables are real numbers. However, the observations of fuzzy random variables are fuzzy real numbers. The concept of fuzzy random variable was introduced by Kwakernaak [4] and Puri and Ralescu [7]. The occurrence of fuzzy random variable makes the combination of randomness and fuzziness more persuasive, since the probability theory can be used to model uncertainty and the fuzzy sets theory can be used to model imprecision. Sometimes the items may not be failed completely during a test, or the number of survivors cannot be recorded exactly due to the human errors or machine errors on a test of n items. In this case, we may just say that there are around s survivors during the test of n items. Therefore, the component reliability will be around s=n: The phrase ‘around s=n’ can be described by using fuzzy sets theory. That is to say, the component reliabilities will be regarded
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as fuzzy real numbers in this case. The main purpose of this paper is to provide a methodology for discussing the fuzzy Bayesian system reliability from the fuzzy component reliabilities. The reliability analysis is an important research topic in engineering. However, the data sometimes cannot be measured and recorded precisely as described above. Several researchers then pay attention to applying the fuzzy sets theory to reliability analysis. The collection of papers edited by Onisawa and Kacprzyk [6] gave many different approaches to fuzzy reliability. Some other researchers also used the fuzzy probability, linguistic variables and fuzzy logic approach to study the reliability models and risk assessment in Refs. [1,10,11,13]. The applications of fuzzy-Bayesian method in hypotheses testing and structural reliability were proposed by Taheri and Behboodian [9] and Chou and Yuan [2], respectively. Taheri and Behboodian [9] considered the ordinary (crisp) data (the hypotheses were taken in the fuzzy sense). However, the fuzzy data (i.e. fuzzy observations) are taken into account in this paper. Chou and Yuan [2] considered the Bayes’ theorem based on the events that were not defined precisely (i.e. fuzzy events were considered). However, the fuzzyBayesian approach proposed in this paper is completely different from that of Chou and Yuan [2], since the fuzzy events will not be considered here. In this paper, the conventional Bayes point estimator is incorporated into the fuzzy sets theory to deduce the so-called fuzzy Bayes point estimator. In Section 2, we introduce the concept of fuzzy random variable and discuss the fuzzy Bayes point estimator that is constructed from the (conventional) Bayes point estimators by invoking a well-known theorem called ‘Resolution Identity’ in fuzzy sets theory. The fuzzy Bayes point estimator has also been discussed in Ref. [12]. The concept of fuzzy real number proposed in this paper has a little difference from that of Wu [12] (the concept proposed in this paper is more general than that of Wu [12]). However, we shall see that the arguments used in Ref. [12] can also apply to this paper without difficulty. In Section 3, we discuss the fuzzy Bayes point estimators of system reliability for series system, parallel system and k-out-of-n system. In Section 4, the computational procedures and examples are provided in order to clarify the theory discussed in this paper, and to give a possible insight for applying the fuzzy sets theory to Bayesian system reliability assessment.
2. Fuzzy bayes point estimators Let X be a universal set and A be a subset of X: We can define a characteristic function xA : X ! {0; 1} with respect
to A by (
xA ðaÞ ¼
1
if a [ A;
0
if a A:
Zadeh [14] introduced the concept of fuzzy subset A~ of X by extending the characteristic function. A fuzzy subset A~ of X is defined by its membership function jA~ : X ! ½0; 1 which is viewed as an extension of characteristic function. The value jA~ ðaÞ can be interpreted as the membership degree of a point a in the set A: Under some suitable conditions on the membership function, the fuzzy set is then termed as a fuzzy real number (this will be realized in the sequel). Let a be a real number. The fuzzy real number a~ corresponding to a can be interpreted as ‘around a’: The graph of the membership function ja~ ðxÞ is bell-shaped and ja~ ðaÞ ¼ 1: It means that the membership degree ja~ ðxÞ is close to 1 when x is close to a: Let X ¼ R be a real number system and a~ be a fuzzy subset of R: We denote by a~ a ¼ {x : ja~ ðxÞ $ a} the a-level set of a~ for a [ ð0; 1; and a~ 0 is the closure of the set {x : ja~ ðxÞ . 0}: Now a~ is called a normal fuzzy set if there exists an x such that ja~ ðxÞ ¼ 1; and called a convex fuzzy set if ja~ ðlx þ ð1 2 lÞyÞ $ min{ja~ ðxÞ; ja~ ðyÞ} for l [ ½0; 1; i.e. ja~ is a quasi-concave function. Zadeh [14] has shown that a~ is a convex fuzzy subset of R if and only if its a-level set a~ a ¼ {x : ja~ ðxÞ $ a} is a convex set in R for all a [ ½0; 1: Let f be a real-valued function defined on R: Then f is said to be upper semicontinuous, if {x : f ðxÞ $ a} is a closed set in R for each a: Let a~ be a fuzzy subset of R satisfying the following conditions: (i) a~ is a normal and convex fuzzy set; (ii) Its membership function ja~ is upper semicontinuous; (iii) The 0-level set a~ 0 is bounded in R: Then, from the above three conditions, we see that the alevel set a~ a is convex, closed and bounded in R: It shows that a~ a is a closed interval. We then denote it by a~ a ¼ ½~aLa ; a~ U a : Therefore we propose the following definition. Definition 2.1. Let a~ be a fuzzy subset of R: Then a~ is called a fuzzy real number if the following conditions are satisfied: (i) (ii) (iii) (iv) (v)
a~ is a normal and convex fuzzy set; Its membership function ja~ is upper semicontinuous; The 0-level set a~ 0 is bounded in R; The 1-level set a~ 1 is a singleton set, i.e. a~ L1 ¼ a~ U 1; The functions gðaÞ ¼ a~ La and hðaÞ ¼ a~ U are continuous a with respect to a on [0,1].
We denote by FR the set of all fuzzy real numbers. Remark 2.1. Let a~ be a fuzzy subset of R satisfying conditions (i) – (iv) in Definition 2.1. Suppose that its
H.-C. Wu / Reliability Engineering and System Safety 83 (2004) 277–286
membership function ja~ is strictly increasing on the interval ½~aL0 ; a~ L1 and strictly decreasing on the interval ½~aU ~U 1;a 0 : Then a~ is a fuzzy real number, since the fact of strict monotonicity implies that condition (v) is satisfied. This shows that the concept of fuzzy real number proposed in this paper is more general than that of Wu [12]. However, the arguments used in Ref. [12] for deriving the fuzzy Bayes point estimator can also apply to this paper, and finally it leads to the same results for fuzzy Bayes point estimator because only the continuity in Definition 2.1 (v) was taken into account in Ref. [12]. Let X~ : V ! FR be a fuzzy-valued function. Then, according to Puri and Ralescu [7] and Kra¨tschmer [3], X~ is a fuzzy random variable if and only if X~ La and X~ U a are (traditional) random variables for all a [ ½0; 1: Let X be a random variable with pdf f ðx; u1 ; …; un Þ of known functional form but depending on an unknown n-dimensional vector u ¼ ðu1 ; …; un Þ in which ui ; i ¼ 1; …; n; are called parameters. We let Q ¼ ðQ1 ; …; Qn Þ stand for the set of all possible values of u and call it the parameter space. Here we are going to consider the fuzzy parameters. Let Q ¼ ðQ1 ; …; Qn Þ be the parameter space. We write u~ ¼ ðu~1 ; …; u~n Þ; where each fuzzy parameter u~i is a fuzzy subset of Qi for i ¼ 1; …; n; i.e. each u~i is associated with a membership function ju~i : Qi ! ½0; 1 for i ¼ 1; …; n: Here each u~i is assumed to be a fuzzy real number. We also see that ðu~i ÞLa and ðu~i ÞU a are in the parameter space Qi for all a [ ½0; 1: Therefore we can discuss the point estimator of ðu~i ÞLa and ðu~i ÞU a for all a [ ½0; 1 and all i ¼ 1; …; n: It is without loss of generality to assume n ¼ 1: Let u~ be a ~ From the fuzzy parameter of the fuzzy random variable X: proposed concept of fuzzy random variable, we can say that u~La and u~Ua are the parameters of the random variables X~ La and X~ U a ; respectively, for all a [ ½0; 1: The purpose is to estimate u~ under fuzzy environments. In order to apply the Bayesian approach, the fuzzy parameter u~ is assumed as a fuzzy random variable. It says that the random variables u~La and u~U a have distributions with known parameters U ðh~1 ÞLa ; …; ðh~m ÞLa and ðh~1 ÞU a ; …; ðh~m Þa ; respectively, for all a [ ½0; 1; where h~1 ; …; h~m are known fuzzy parameters. Since u~ is a fuzzy real number, we see that u~La and u~U a are continuous with respect to a by definition. Therefore the closed intervals ½u~La ; u~U a ; for a [ ½0; 1; are continuously shrinking with respect to a: Then for any parameter u [ ~L ~U ½u~La ; u~U a ; we have u ¼ ub or u ¼ ub for some b $ a; since L U u~1 ¼ u~1 : Thus for any parameter u [ ½u~La ; u~Ua ; we can find a Bayes point estimator u^ for u: Let ~L ; inf uc ~U }; max{ sup uc ~U }: ~L ; sup uc A ¼ ½min{ inf uc a
a#b#1
b
a#b#1 b
a#b#1
b
a#b#1
b
ð1Þ Then this interval will contain all of the Bayes point estimators for each u [ ½u~La ; u~U a : According to the following
279
Proposition 2.1, we can naturally define the membership function of fuzzy Bayes point estimator.
Proposition 2.1. (Zadeh [15]) (Resolution Identity) Let a~ be a fuzzy subset of R with membership function ja~ : Then
ja~ ðxÞ ¼ sup a·1a~ a ðxÞ; a[½0;1
where 1A is a characteristic function of set A; i.e. 1A ðxÞ ¼ 1 if x [ A and 1A ðxÞ ¼ 0 if x A (note that the a-level set a~ a of a~ is a usual set). The fuzzy Bayes point estimator of u~ is then denoted by ^u~; and the membership function of u^~ is defined by
ju^~ðrÞ ¼ sup a·1Aa ðrÞ;
ð2Þ
0#a#1
via the form of Resolution Identity in Proposition 2.1. As we have known that the Bayes point estimator is a random variable, Wu [12, Theorem 3.1] has also shown that the fuzzy Bayes point estimator is a fuzzy random variable.
3. Fuzzy bayesian system reliability assessment The Bayesian approach to system reliability is the assignment of the prior distribution. From the Bayesian point of view, the goal is to determine the posterior distribution of the system reliability. The prior distribution is assigned to each component of a system. Therefore the posterior distributions of each component reliability can be found by Bayes theorem. In this case, we can derive the posterior distribution of system reliability from the posterior distributions of component reliabilities. However, the key methodology in such a derivation is that system reliability can be expressed as a product of independent random variables each of which corresponds to either component reliability (for series system) or component unreliability (for parallel system). Therefore we need to invoke the method of Mellin integral transform. We describe it as follows. Let X be a non-negative random variable with pdf f ðxÞ: The Mellin transform of f with respect to the complex parameter u is defined by Mðf ; uÞ ¼
ð1
xu21 f ðxÞdx:
0
It is also convenient to regard the Mellin transform as the moments of X; i.e. Mðf ; uÞ ¼ E½X u21 :
ð3Þ
Let X1 ; X2 ; …; Xk be independent non-negative random variables with pdf’s f1 ; f2 ; Q …; fk ; respectively. Let gk ðyÞ be the pdf of the product Y ¼ ki¼1 Xi : Springer and Thompson
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[8] proved that Mðgk ; uÞ ¼
k Y
Mðfi ; uÞ:
ð4Þ
i¼1
With the help of Eq. (4), we can obtain the posterior distribution of system reliability from the posterior distributions of component reliabilities. Then the Bayes point estimator of system reliability is the mean of the posterior distribution under the squared error loss function, and it can be obtained from Eq. (3). 3.1. Series system Suppose that we have n items. Each item i can be represented as a Bernoulli random variable Yi with reliability (survival probability) r: Then the number of P survivors X ¼ ni¼1 Yi is recorded. The probability distribution of the number of survivors is the Binomial distribution given by f ðxlrÞ ¼ Pr{x survivors will occur in n trialslr} ¼
x ¼ 0; 1; …; n and 0 , r , 1: For a series system consisting of k independent components, the system reliabilityQr is the product of component reliabilities ri ; i.e. r ¼ ki¼1 ri : Now suppose that ni items are tested and that si survivors are observed for the ith component, i ¼ 1; …; k: Then the sampling distribution is a Binomial distribution with parameter ri as described above. In order to obtain the Bayes point estimator of the system reliability r; we regard the component reliability ri as a random variable Ri ; for i ¼ 1; …; k; with pdf pi ðri Þ; where pi ðri Þ is termed as a prior distribution of component reliability Ri : Then the posterior distribution of Ri is then given by ð5Þ
for 0 , ri , 1; where si is the observed number of survivors in ni trials. The most widely used prior distribution for Ri is the Beta distribution Bðsi0 ; ni0 Þ with pdf given by pi ðri Þ ¼
Gðni0 Þ r si0 21 ð1 2 ri Þni0 2si0 21 ; Gðsi0 ÞGðni0 2 si0 Þ i
pi ðri lsi Þ ¼
Gðni þ ni0 Þ r si þsi0 21 Gðsi þ si0 ÞGðni þ ni0 2 si 2 si0 Þ i ð1 2 ri Þni þni0 2si 2si0 21 ;
ð7Þ
which is Bðsi þ si0 ; ni þ ni0 Þ distribution. The Mellin transform of the posterior pdf of Ri is found to be M{pi ðri lsi Þ; u} Gðni þ ni0 Þ Gðsi þ si0 þ u 2 1Þ ¼ Gðsi þ si0 Þ Gðni þ ni0 þ u 2 1Þ ðni þ ni0 2 1Þ! ¼ ðsi þ si0 2 1Þ! 1 ; ðu þ si þ si0 2 1Þðu þ si þ si0 Þ· · ·ðu þ ni þ ni0 2 2Þ where ReðuÞ . 2ðsi þ si0 2 1Þ (also see Q Ref. [5]). Since the system reliability is R ¼ ki¼1 Ri ; from Eq. (4), the Mellin transform of the posterior pdf pðrls; nÞ of the system reliability R is given by
n! r x ð1 2 rÞn2x ðn 2 xÞ!x!
r si ð1 2 ri Þn2si pðri Þ pi ðri lsi Þ ¼ Ð1 i s ; n2si i pðyÞdy 0 y ð1 2 yÞ
can be taken from the past tests, e.g. the average of survivors from the previous tests. Thus it is not necessarily to assume si0 as an integer when we consider the prior Beta distribution Bðsi0 ; ni0 Þ: From Eqs. (5) and (6), the posterior distribution of Ri is given by
ð6Þ
where 0 # ri # 1 and u1 ; u2 . 0; since the conjugate prior distribution of the Binomial distribution is Beta distribution. Since the expectation of Beta distribution Bðsi0 ; ni0 Þ is si0 =ni0 ; the numbers ni0 and si0 can be interpreted as ‘pseudo’ number of items that are placed on test and ‘pseudo’ number of survivors, respectively. The ‘pseudo’ number of survivors
M{pðrlsÞ; u} k Y ðni þ ni0 2 1Þ! ¼ ðsi þ si0 2 1Þ! i¼1
1 ; ðu þ si þ si0 2 1Þðu þ si þ si0 Þ· · ·ðu þ ni þ ni0 2 2Þ ð8Þ
where ReðuÞ . 2ðsi þ si0 2 1Þ for all i ¼ 1; …; k: The Bayes point estimator of system reliability R is the mean of the posterior distribution in Eq. (8) under squared error loss function, and it is given by k Y si þ si0 E½Rls ¼ M{pðrlsÞ; u ¼ 2} ¼ ; ð9Þ ni þ ni0 i¼1 from Eq. (3). Suppose that the component reliabilities cannot be sured as exact real numbers as described above. In order to apply the Bayesian approach, the fuzzy component reliabilities are considered as fuzzy random variables R~ i with the assumptions that ni0 is a known integer representing the ‘pseudo’ number of items for the ith component and s~i0 is a known fuzzy real number representing the ‘pseudo’ number of survivors (since the number of survivors cannot be recorded precisely as described above, s~i0 is assumed as a fuzzy real number). Then, from Eq. (9), the Bayes point estimates of r~La and r~U a are ! ! k k L U Y Y ð~ s Þ þ s ð~ s Þ þ s i0 a i i0 a i b b and r~U ; ð10Þ r~La ¼ a ¼ ni þ ni0 ni þ ni0 i¼1 i¼1
H.-C. Wu / Reliability Engineering and System Safety 83 (2004) 277–286
281
respectively, for all a [ ½0; 1: According to the discussions in Section 2, let A ¼ ½min{ min r~bL ; min r~bU }; max{ max r~bL ; max r~bU }:
respectively, for all a [ ½0; 1: The membership function of the fuzzy Bayes point estimate of system reliability r is defined by the same way as discussed above.
ð11Þ Then this interval will contain all of the Bayes point estimates for each r [ ½~rLa ; r~U a : It is obvious that c c c c c c U L U L L U r~a # r~a ; r~a # r~b and r~a $ r~b for a , b; i.e. rc ~aL # rc ~1L # bU U r~c 1 # r~a : Thus Aa can be rewritten as
3.3. k-out-of-m System
a
a#b#1 b a#b#1 b
Aa ¼ ½rc ~aL ; rc ~aU :
a#b#1 b a#b#1 b
ð12Þ
Now the membership function of the fuzzy Bayes point estimate of r~; denoted as r^~; is defined by
jr^~ ðrÞ ¼ sup a·1Aa ðrÞ; 0#a#1
via the form of Resolution Identity in Proposition 2.1. The numerical example will be given in the sequel.
For a k-out-of-m system consisting of m independent and identical components, the system reliability is given by ! m X m j r¼ r ð1 2 rÞm2j ; j j¼k where r is the reliability of the (common) component. In order to apply Bayesian approach, the (common) component reliability r is assumed as a random variable R with prior Beta distribution Bðs0 ; n0 Þ: Then, from Eq. (7), the posterior distribution of R is a Bðs þ s0 ; n þ n0 Þ distribution with pdf given by pR ðrls; s0 Þ ¼
3.2. Parallel systems For a parallel system consisting of k independent Q components, the system reliability is r ¼ 1 2 ki¼1 ð1 2 ri Þ; where ri is the component reliability of the ith component. Equivalently, the system unreliability q ¼ 1 2 r is the product of component unreliabilities qi ¼ Q 1 2 ri ; i.e. q ¼ ki¼1 qi : In this case, we can use the Mellin transform technique as described in Eq. (4) to obtain the Bayes point estimate of system unreliability. Assuming the prior distribution of component unreliabilities for the ith component is Beta distribution Bðmi0 ; ni0 Þ; where mi0 may be regarded as the (pseudo) number of failures in a test of (pseudo) ni0 items. Therefore ni0 ¼ si0 þ mi0 : According to Eqs. (8) and (9), the Bayes point estimate of the system unreliability is given by k Y mi þ mi0 E½Qlm ¼ : ni þ ni0 i¼1 Thus the Bayes point estimate of system reliability r under the squared error loss function is given by k Y mi þ mi0 E½Rls ¼ 1 2 E½Qlm ¼ 1 2 ni þ ni0 i¼1 k Y ni 2 si þ ni0 2 si0 ¼12 ni þ ni0 i¼1 k Y s þ si0 ¼12 12 i : ni þ ni0 i¼1 Under the fuzzy assumptions as described above, the Bayes point estimates of r~La and r~U a are ! k Y ð~s ÞL þ si and r~bLa ¼ 1 2 1 2 i0 a ni þ ni0 i¼1 ! k U Y ð~ s Þ þ s i0 a i b ; ð13Þ 12 r~U a ¼12 ni þ ni0 i¼1
Gðn þ n0 Þ rsþs0 21 Gðs þ s0 ÞGðn þ n0 2 s 2 s0 Þ ð1 2 rÞnþn0 2s2s0 21 :
The Bayes point estimate of system reliability r is the mean E½Rls; s0 under the squared error loss function, and is given by E½Rls;s0 0 1 m m ð1 X ¼ @ A rj ð1 2 rÞm2j ·pR ðrls;s0 Þdr 0 j j¼k Gðn þ n0 Þ ¼ Gðs þ s0 ÞGðn þ n0 2 s 2 s0 Þ 2 0 1 3 m m Gðj þ s þ s ÞGðm 2 j þ n þ n 2 s 2 s Þ X 0 0 0 5: 4 @ A Gðm þ n þ n0 Þ j j¼k Under the fuzzy assumptions as described above, the Bayes point estimates of r~La and r~U a are given by r~bU a¼
Gðnþn0 Þ L Gðsþð~s0 Þa ÞGðnþn0 2s2ð~s0 ÞLa Þ
2 0 1 3 m m Gðjþsþð~s ÞL ÞGðm2jþnþn 2s2ð~s ÞL Þ X 0 a 0 0 a 5 4 @ A ; Gðmþnþn0 Þ j¼k j ð14Þ and
Gðnþn0 Þ Gðsþð~s0 ÞUa ÞGðnþn0 2s2ð~s0 ÞU0 Þ 2 0 1 3 m m Gðjþsþð~s ÞU ÞGðm2jþnþn 2s2ð~s ÞU Þ X 0 a 0 0 a 5 4 @ A ; Gðmþnþn0 Þ j j¼k
r~bU a¼
ð15Þ respectively, for all a [ ½0; 1: The membership function of the fuzzy Bayes point estimate of system reliability r is defined by the same way as discussed above.
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4. Computational procedures and examples In this section, we shall provide some computational techniques to evaluate the membership degree of the fuzzy Bayes point estimate as presented in Eq. (2). From Eq. (1), we adopt the following notations Aa ¼ ½gðaÞ;hðaÞ ¼ ½min{g1 ðaÞ;g2 ðaÞ};max{h1 ðaÞ; h2 ðaÞ}; ð16Þ where ~U ; h1 ðaÞ ¼ sup uc ~Lb ; g2 ðaÞ ¼ inf uc ~Lb ; g1 ðaÞ ¼ inf uc b a#b#1
a#b#1
a#b#1
~U : h2 ðaÞ ¼ sup uc b a#b#1
the original problem with optimal solution apo (in fact, apo ¼ Zop ). Then Wu [12] has shown that Z p ¼ Zop : In other words, in order to obtain the objective value Zop of the original nonlinear programming problem, it will be enough to just solve the four subproblems I– IV. It is easy to see that g1 and g2 are increasing, and h1 and h2 are decreasing. In order to solve the above four subproblems I – IV, we further consider the following general type of nonlinear programming problem. Let gðaÞ be increasing, hðaÞ be decreasing and gðaÞ # hðaÞ for all a [ ½0; 1: We plan to solve the following problem when r is given and fixed
jðrÞ ¼ max
From Eq. (2), the membership function of the fuzzy Bayes point estimate u^~ is given by ju^~ðrÞ ¼ sup a·1Aa ðrÞ ¼ sup{a : gðaÞ # r # hðaÞ;0 # a # 1}:
a
subject to
hðaÞ $ r
0#a#1
0 # a # 1:
Therefore, we need to solve the following type of nonlinear programming problem max
a
subject to
min{g1 ðaÞ; g2 ðaÞ} # r max{h1 ðaÞ; h2 ðaÞ} $ r 0 # a # 1:
Now we consider the following four subproblems
a
I : max subject to
g1 ðaÞ # r h1 ðaÞ $ r 0 # a # 1;
a
II : max subject to
g 1 ð aÞ # r h2 ð aÞ $ r 0 # a # 1;
III :
max
a
subject to
g 2 ð aÞ # r h 1 ð aÞ $ r 0 # a # 1;
a
IV : max subject to
g 2 ð aÞ $ r h2 ð aÞ $ r 0 # a # 1:
ZIp ;
ZIIp ;
gðaÞ $ r
p ZIII
p Let and ZIV be the objective values of subproblems I – IV, respectively. Let Zp ¼ p p p p p max{ZI ; ZII ; ZIII ; ZIV } and Zo be the objective value of
We see that the subproblems I, II and IV satisfy the above assumptions (in subproblem III, g2 # h1 is not always true). Therefore they can be solved by using the Supplemental Procedure which is proposed in Ref. [12]. Before providing the numerical examples, we introduce a special kind of fuzzy real number. We say that a~ is a triangular fuzzy real number if its membership function is given by (the graph of its membership function will look like a triangle) 8 ðr 2 a1 Þ=ða2 2 a1 Þ if a1 # r # a2 > > < ja~ ðrÞ ¼ ða3 2 rÞ=ða3 2 a2 Þ if a2 , r # a3 > > : 0 otherwise: We denote by a~ ¼ ða1 ; a2 ; a3 Þ: Then a~ is a fuzzy real number and its a-level set is given by a~ a ¼ ½ða2 2 a1 Þa þ a1 ; ða2 2 a3 Þa þ a3 for all a [ ½0; 1:
Example 4.1. A system consists of three independent components in series. Suppose that we collect following data on a test
Component 1 Component 2 Component 3
Successes
Failures
Items tested
8 7 3
2 2 1
10 9 4
. From the past tests and experiences, the percentage of survivors in a system is ‘around’ 80% for component 1, ‘around’ 80% for component 2 and ‘around’ 75% for ~ s~ 20 ¼ component 3. Therefore we can assume that s~ 10 ¼ 8; ~ n10 ¼ 10; n20 ¼ 10 and n30 ¼ 8; where 8~ ¼ ~ s~30 ¼ 6; 8; ð6; 8; 10Þ and 6~ ¼ ð4; 6; 8Þ are triangular fuzzy real numbers. Then the a-level sets of 8~ and 6~ are 8~ a ¼ ½6 þ 2a; 10 2 2a
H.-C. Wu / Reliability Engineering and System Safety 83 (2004) 277–286
and 6~ a ¼ ½4 þ 2a; 8 2 2a; respectively. The Bayes point estimates of r~La and r~U a ; from Eq. (10), are given by 6 þ 2a þ 8 6 þ 2a þ 7 4 þ 2a þ 3 b L r~a ¼ 10 þ 10 10 þ 9 8þ4 ¼ and r~bU a ¼ ¼
ð14 þ 2aÞð13 þ 2aÞð7 þ 2aÞ ; 4560
10 2 2a þ 8 10 þ 10
10 2 2a þ 7 10 þ 9
8 2 2a þ 3 8þ4
283
r~ by evaluating the above formulas. On the other hand, we also have the 0.95-level closed interval A0:95 ¼ ½0:46239; 0:48515: In this case, we may say that the system reliability lies in the interval A0:95 ¼ ½0:46239; 0:48515 with confidence degree 0.95. In general, we can say that the system reliability r lies in the interval Aa with confidence degree a:
ð18 2 2aÞð17 2 2aÞð11 2 2aÞ ; 4560
respectively, for all a [ ½0; 1: Comparing Eqs. (11), (12) and (16), we see that gðaÞ ¼ g1 ðaÞ ¼ r~bLa and hðaÞ ¼ h2 ðaÞ ¼ r~bU a : Therefore, we just need to solve subproblem II, i.e. Zop ¼ ZIIp : Now we are going to use the Supplemental Procedure to solve subproblem II. We see that r~bL1 ¼ 2160=4560 ¼ r~bU 1 : Therefore jr^~ ð2160=4560Þ ¼ 1: Suppose that we reconsider the fuzzy real numbers 8~ and 6~ as the real numbers 8 and 6, respectively. Then the Bayes point estimate of system reliability r will be 2160/4560. This shows our anticipation that the Bayes point estimate will have membership degree 1. Since A0 ¼ ½1274=4560; 3366=4560; we are just interested in considering the system reliability r [ A0 : By applying the Supplemental Procedure, we have, for r [ A0 ; (i) if r , 2160=4560 then we solve the problem
Example 4.2. A system consists of three independent components in parallel. Suppose that we use the same data as in Example 4.1. Then, from Eq. (13), the Bayes point estimates of r~La and r~U a are 6 þ 2a þ 8 6 þ 2a þ 7 r~bLa ¼ 1 2 1 2 12 10 þ 10 10 þ 9 4 þ 2a þ 3 12 ; 8þ4 and 10 2 2a þ 8 10 2 2a þ 7 b U r~a ¼ 1 2 1 2 12 10 þ 10 10 þ 9 8 2 2a þ 3 12 : 8þ4 The numerical results can also be obtained using the same techniques as in Example 4.1.
ð14 þ 2aÞð13 þ 2aÞð7 þ 2aÞ #r ; jr^~ ðrÞ ¼ max a [ ½0; 1 : gðaÞ ¼ g1 ðaÞ ¼ r~bLa ¼ 4560 using any commercial optimizers, e.g. GAMS or LINGO. Equivalently, we can also use the Newton method to solve the following problem
jr^~ ðrÞ ¼ {a [ ½0; 1 : a is the root of equation ð14 þ 2aÞ ð13 þ 2aÞð7 þ 2aÞ ¼ 4560r}; (ii) if r . 2160=4560 then we solve the problem
Example 4.3. We consider a 2-out-of-3 system consisting of three independent and identical components. Suppose that there are eight successors (survivors) when 10 items are placed on a test, i.e. n ¼ 10 and s ¼ 8: From the past tests and experiences, the percentage of survivors is ‘around’ ~ 10Þ is 80%. Therefore the fuzzy prior Beta distribution Bð8; used. From Eqs. (14) and (15), the Bayes point estimates of r~La and r~U a are given by
ð18 2 2aÞð17 2 2aÞð11 2 2aÞ $r ; jr^~ ðrÞ ¼ max a [ ½0; 1 : hðaÞ ¼ h1 ðaÞ ¼ r~bLa ¼ 4560 using any commercial optimizers, e.g. GAMS or LINGO. Equivalently, we can also use the Newton method to solve the following problem
jr^~ ðrÞ ¼ {a [ ½0; 1 : a is the root of equation ð18 2 2aÞ ð17 2 2aÞð11 2 2aÞ ¼ 4560r}: Therefore we now can obtain the membership degree for any given Bayes point estimate r of fuzzy system reliability
r~bLa ¼
Gð10þ10Þ Gð8þ6þ2aÞGð10þ10282622aÞ 2 0 1 3 3 3 Gðjþ8þ6þ2aÞGð32jþ10þ10282622aÞ X 5; 4 @ A Gð3þ10þ10Þ j¼2 j
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and 2 3 ! 3 X 3 Gðj þ 8 þ 10 2 2aÞGð3 2 j þ 10 þ 10 2 8 2 10 þ 2aÞ G ð10 þ 10Þ 4 5: r~bU a ¼ Gð8 þ 10 2 2aÞGð10 þ 10 2 8 2 10 þ 2aÞ j¼2 j Gð3 þ 10 þ 10Þ
The numerical results can also be obtained using the same techniques as in Example 4.1. Example 4.4. We consider the following system. All components are mutually independent.
Under the fuzzy assumptions, the Bayes point estimates of r~La and r~U a are r~bLa ¼ MaL {p1 ðr1 ls1 Þ; u ¼ 2}·MaL {pRp ðrp ln2 2 s2 ; n3 2 s3 Þ; u ¼ 2};
ð20Þ
and U U r~bU a ¼ Ma {p1 ðr1 ls1 Þ; u ¼ 2}·Ma {pRp ðrp ln2 2 s2 ; n3 2 s3 Þ;
u ¼ 2}; The posterior distribution of component reliabilities of R1 ; R2 ; R3 are given by Eq. (7). Let Qi ¼ 1 2 Ri be the component unreliabilities of the ith component for i ¼ 2; 3: Assuming the prior distribution of component unreliabilities for the ith component is Beta distribution Bðmi0 ; ni0 Þ; for i ¼ 2; 3; where mi0 may be regarded as the (pseudo) number of failures in a test of (pseudo) ni0 items. Therefore ni0 ¼ si0 þ mi0 : Then, from Eq. (7), the posterior pdf of Qi is given by p i ðqi lmi Þ ¼
Gðni þ ni0 Þ Gðmi þ mi0 ÞGðni þ ni0 2 mi 2 mi0 Þ i þmi0 21 qm ð1 2 qi Þni þni0 2mi 2mi0 21 ; i
ð17Þ
for i ¼ 2; 3: Let Rp be the reliability of the parallel configuration which consists of components 2 and 3. Then Rp ¼ 1 2 Q2 Q3 : We see that the posterior pdf of Rp is pRp ðrp lm2 ; m3 Þ ¼
ð1 12rp
1 2 rp 1 ·p 3 ð 2 ðylm2 Þdy: jm3 ·p y y
It is easy to see that the system reliability is R ¼ R1 Rp with posterior pdf pðrls1 ; s2 ; s3 Þ (note that ni ¼ si þ mi ). Therefore the Mellin transform of pðrls1 ; s2 ; s3 Þ is the product of the Mellin transform of R1 and Rp ; i.e. M{pðrls1 ; s2 ; s3 Þ; u} ¼ M{p1 ðr1 ls1 Þ; u}·M{pRp ðrp ln2 2 s2 ; n3 2 s3 Þ; u}:
ð21Þ
respectively, for all a [ ½0; 1; where MaL ð·Þ and MaU ð·Þ are obtained from Eq. (19) by replacing the variable s10 as ð~s10 ÞLa and ð~s10 ÞU ~ i0 ÞLa and a ; respectively, and the variable mi0 as ðm U ðm ~ i0 Þa ; respectively, for i ¼ 2; 3: Suppose that we collect following data on a test
Component 1 Component 2 Component 3
Successes ðsi Þ
Failures ðmi Þ
Items tested ðni Þ
8 7 3
2 2 1
10 9 4
From the past tests and experiences, the percentage of survivors for component 1 is ‘around’ 80%. The percentage of failures is ‘around’ 20% for component 2 and ‘around’ 25% for component 3. Therefore the fuzzy prior ~ 10Þ is assigned to the component Beta distribution Bð8; ~ 10Þ reliability R1 ; and the fuzzy prior Beta distribution Bð2; ~ and Bð2; 8Þ are assigned to the component unreliabilities Q2 and Q3 ; respectively, where 8~ ¼ ð6; 8; 10Þ and 2~ ¼ ð1; 2; 3Þ are triangular fuzzy real numbers. Then the a-level sets of 8~ and 2~ . are 8~ a ¼ ½6 þ 2a; 10 2 2a and 2~ a ¼ ½1 þ a; 3 2 a; respectively. First of all, we evaluate r~bLa : From Eq. (20), we have MaL {p1 ðr1 ls1 Þ; u ¼ 2} ¼
ð18Þ
Now, from Eq. (17), we also have
The Bayes point estimator of system reliability R is the mean of the posterior distribution in Eq. (18) under squared error loss function, and it is given by
ðp 2 ÞLa ðq2 lm2 Þ ¼
E½Rls1 ; s2 ; s3 ¼ M{pðrls1 ; s2 ; s3 Þ; u ¼ 2} ¼ M{p1 ðr1 ls1 Þ;
and
u ¼ 2}·M{pRp ðrp ln2 2 s2 ; n3 2 s3 Þ; u ¼ 2}: ð19Þ
6 þ 2a þ 8 : 10 þ 10
ðp 3 ÞLa ðq3 lm3 Þ ¼
Gð19Þ q2þa ·ð1 2 q2 Þ152a ; Gð3 þ aÞGð16 2 aÞ 2
Gð12Þ q1þa ·ð1 2 q3 Þ92a : Gð2 þ aÞGð10 2 aÞ 3
H.-C. Wu / Reliability Engineering and System Safety 83 (2004) 277–286
Then
the system reliability assessment is a kind of extension of the conventional Bayesian system reliability assessment. The advantages of proposed methods in this paper are described as follows
ðpRp ÞLa ðrp lm2 ; m3 Þ ¼ K·ð1 2 rp Þ1þa ·
ð1 12rp
ð1 2 yÞ152a ·ya29 ·ðy 2 1 þ rp Þ92a dy; (i)
where
Gð19ÞGð12Þ : Gð3 þ aÞGð16 2 aÞGð2 þ aÞGð10 2 aÞ Now we obtain K¼
MaL {pRp ðrp ln2 2 s2 ; n3 2 s3 Þ; u ¼ 2} " ð1 ð1 ¼ K· ð1 2 rp Þ1þa · ð1 2 yÞ152a ·ya29 0
285
i
12rp
·ðy 2 1 þ rp Þ92a dy drp : According to Eqs. (11), (20), (21) and using the numerical integration (e.g. using the commercial software MATLAB), we can obtain the closed interval Aa : For example, for a ¼ 0:95; we have A0:95 ¼ ½0:6589; 0:7135: It says that the system reliability r lies in the interval Aa with confidence degree a:
5. Conclusions The fuzzy sets theory has successfully applied to the Bayesian system reliability assessment in this paper. Owing to the imprecise occurrence of observations (i.e. fuzzy data) in the real world as presented in this paper, we invoke the concept of fuzzy random variable to conquer this difficulty. On the other hand, in order to interpret the meaning of the membership degree of any given Bayes point estimate (i.e. the fuzzy Bayes point estimate), a nonlinear programming problem has to be solved. Furthermore, this nonlinear programming problem is fortunately divided into four solvable subproblems. Therefore the commercial optimizers are useful for obtaining the results. The four examples presented in this paper make the incorporation of fuzzy sets theory in the Bayesian system reliability assessment to be a more persuasive evidence for considering the fuzzy-Bayesian approach in system reliability assessment. Recall that the real number a can be regarded as a fuzzy real number a~ ; since the characteristic function xa of the real number can be regarded as a membership function ( 1 if x ¼ a; xa ðxÞ ¼ ja~ ðxÞ ¼ 0 if x – a: Therefore those examples show that the membership degree of Bayes point estimate of system reliability will be 1 if the fuzzy data were degenerated as the real data. We may also conclude that the fuzzy-Bayesian approach in
When the imprecise data (fuzzy data) occur in the system, we can handle this kind of situation by considering the fuzzy-Bayesian approach. However, the conventional Bayesian approach may have difficulty for handling this kind of imprecise data. (ii) Even though the data occurring in the system are taken as real numbers, the proposed method in this paper can still handle it as we just described above (the fuzzyBayesian approach in the system reliability assessment is a kind of extension of the conventional Bayesian system reliability assessment). Therefore, the fuzzyBayesian approach proposed in this paper is no loss of generality. (iii) Those examples (e.g. Example 4.1) show that if the value a is fixed then we can say that the system reliability lies in the interval Aa with confidence degree a: Therefore, we may use the similar techniques proposed in this paper (by fixing the value a) to develop the so-called ‘Interval Bayesian Reliability Assessment’ without any difficulty. In this paper, we propose a possible attempt on how to apply the fuzzy sets theory to Bayesian system reliability assessment. Therefore, it is naturally realized that the fuzzy sets theory can also impose upon some known techniques of reliability analysis in the future research.
References [1] Cho HN, Choi HH, Kim YB. A risk assessment methodology for incorporating uncertainties using fuzzy concepts. Reliab Engng Syst Saf 2002;78:173–83. [2] Chou KC, Yuan J. Fuzzy-Bayesian approach to reliability of existing structures. J Struct Engng 1993;119:3276–90. [3] Kra¨tschmer V. A unified approach to fuzzy random variables. Fuzzy Sets Syst 2001;123:1 –9. [4] Kwakernaak H. Fuzzy random variables I: definitions and theorems. Inf Sci 1978;15:1–29. [5] Martz HF, Waller RA. Bayesian reliability analysis. New York: Wiley; 1982. [6] Onisawa T, Kacprzyk J, editors. Reliability and safety analyses under fuzziness. Heidelberg: Physica-Verlag; 1995. [7] Puri ML, Ralescu DA. Fuzzy random variables. J Math Anal Appl 1986;114:409 –22. [8] Springer MD, Thompson WE. The distribution of products of independent random variables. SIAM J Appl Math 1966;14: 511–26. [9] Taheri SM, Behboodian J. A Bayesian approach to fuzzy hypotheses testing. Fuzzy Sets Syst 2001;123:39–48. [10] Wang KS, Po J, Hsu FS, Liu CS. Analysis of equivalent dynamic reliability with repairs under partial information. Reliab Engng Syst Saf 2002;76:29–42.
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[11] Wu HC. Fuzzy reliability analysis based on closed fuzzy numbers. Inf Sci 1997;103:135 –59. [12] Wu HC. The fuzzy estimators of fuzzy parameters based on fuzzy random variables. Eur J Oper Res 2003;146:101–14. [13] Xu K, Tang LC, Xie M, Ho SL, Zhu ML. Fuzzy assessment of FMEA for engine systems. Reliab Engng Syst Saf 2002;75:17–29. [14] Zadeh LA. Fuzzy sets. Inf Control 1965;8:338–53.
[15] Zadeh LA. The concept of linguistic variable and its application to approximate reasoning I. Inf Sci 1975;8:199–249. Zadeh LA. The concept of linguistic variable and its application to approximate reasoning II. Inf Sci 1975;8:301–57. Zadeh LA. The concept of linguistic variable and its application to approximate reasoning III. Inf Sci 1975;9:43–80.
Bayesian system reliability assessment under fuzzy environments Hsien-Chung Wu Department of Information Management, National Chi Nan University, Puli, Nantou 545, Taiwan, ROC Received 13 February 2003; accepted 30 September 2003
Abstract The Bayesian system reliability assessment under fuzzy environments is proposed in this paper. In order to apply the Bayesian approach, the fuzzy parameters are assumed as fuzzy random variables with fuzzy prior distributions. The (conventional) Bayes estimation method will be used to create the fuzzy Bayes point estimator of system reliability by invoking the well-known theorem called ‘Resolution Identity’ in fuzzy sets theory. On the other hand, we also provide the computational procedures to evaluate the membership degree of any given Bayes point estimate of system reliability. In order to achieve this purpose, we transform the original problem into a nonlinear programming problem. This nonlinear programming problem is then divided into four subproblems for the purpose of simplifying computation. Finally, the subproblems can be solved by using any commercial optimizers, e.g. GAMS or LINGO. q 2003 Elsevier Ltd. All rights reserved. Keywords: Bayes point estimators; Confidence degree; Fuzzy real numbers; Nonlinear programming; System reliability
1. Introduction In the real world, the data sometimes cannot be recorded or collected precisely due to human errors, machine errors, or some other unexpected situations. For instance, the water level of a river and the temperature in a room cannot be measured in an exact way because of the fluctuation. The more appropriate way to describe the water level is to say that the water level is around 30 m. The phrase ‘around f Therefore, 30 m’ should be regarded as a fuzzy number 30 30: the fuzzy sets theory naturally provides an appropriate tool in modeling the imprecise data. Let us consider an experiment by tossing a fair coin. It is well known that, from the law of large numbers, the probability for observing ‘head’ of the coin should be 1/2. Now suppose that we toss this fair coin for 1000 times. Then it is rarely possible to observe ‘head’ of the coin for 500 times. The possible outcome should be around 500 times. In this case, the more reasonable way is to regard the probability for observing ‘head’ of the coin as ‘around 1/2’. The phrase ‘around 1/2’ can be modeled as a fuzzy number f by using the fuzzy sets theory. 1=2 Now let us consider another example, considering to ask a group of economists for predicting the rate of economical E-mail address: [email protected] (H.-C. Wu). 0951-8320/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2003.09.021
growth in the next year. Their statements may be like ‘approximately 5%’, ‘should be 4 –6%’ or ‘may be below 7%’, etc. All of those statements are described linguistically, and therefore, should be characterized as fuzzy sets. Suppose that we introduce a selection procedure which will determine the probability that each economist will be selected. Then for each possible selection, their statements are realized as fuzzy sets. Therefore, the association of fuzzy sets and probabilities forms the notion of fuzzy random variable. We know that the observations of (traditional) random variables are real numbers. However, the observations of fuzzy random variables are fuzzy real numbers. The concept of fuzzy random variable was introduced by Kwakernaak [4] and Puri and Ralescu [7]. The occurrence of fuzzy random variable makes the combination of randomness and fuzziness more persuasive, since the probability theory can be used to model uncertainty and the fuzzy sets theory can be used to model imprecision. Sometimes the items may not be failed completely during a test, or the number of survivors cannot be recorded exactly due to the human errors or machine errors on a test of n items. In this case, we may just say that there are around s survivors during the test of n items. Therefore, the component reliability will be around s=n: The phrase ‘around s=n’ can be described by using fuzzy sets theory. That is to say, the component reliabilities will be regarded
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as fuzzy real numbers in this case. The main purpose of this paper is to provide a methodology for discussing the fuzzy Bayesian system reliability from the fuzzy component reliabilities. The reliability analysis is an important research topic in engineering. However, the data sometimes cannot be measured and recorded precisely as described above. Several researchers then pay attention to applying the fuzzy sets theory to reliability analysis. The collection of papers edited by Onisawa and Kacprzyk [6] gave many different approaches to fuzzy reliability. Some other researchers also used the fuzzy probability, linguistic variables and fuzzy logic approach to study the reliability models and risk assessment in Refs. [1,10,11,13]. The applications of fuzzy-Bayesian method in hypotheses testing and structural reliability were proposed by Taheri and Behboodian [9] and Chou and Yuan [2], respectively. Taheri and Behboodian [9] considered the ordinary (crisp) data (the hypotheses were taken in the fuzzy sense). However, the fuzzy data (i.e. fuzzy observations) are taken into account in this paper. Chou and Yuan [2] considered the Bayes’ theorem based on the events that were not defined precisely (i.e. fuzzy events were considered). However, the fuzzyBayesian approach proposed in this paper is completely different from that of Chou and Yuan [2], since the fuzzy events will not be considered here. In this paper, the conventional Bayes point estimator is incorporated into the fuzzy sets theory to deduce the so-called fuzzy Bayes point estimator. In Section 2, we introduce the concept of fuzzy random variable and discuss the fuzzy Bayes point estimator that is constructed from the (conventional) Bayes point estimators by invoking a well-known theorem called ‘Resolution Identity’ in fuzzy sets theory. The fuzzy Bayes point estimator has also been discussed in Ref. [12]. The concept of fuzzy real number proposed in this paper has a little difference from that of Wu [12] (the concept proposed in this paper is more general than that of Wu [12]). However, we shall see that the arguments used in Ref. [12] can also apply to this paper without difficulty. In Section 3, we discuss the fuzzy Bayes point estimators of system reliability for series system, parallel system and k-out-of-n system. In Section 4, the computational procedures and examples are provided in order to clarify the theory discussed in this paper, and to give a possible insight for applying the fuzzy sets theory to Bayesian system reliability assessment.
2. Fuzzy bayes point estimators Let X be a universal set and A be a subset of X: We can define a characteristic function xA : X ! {0; 1} with respect
to A by (
xA ðaÞ ¼
1
if a [ A;
0
if a A:
Zadeh [14] introduced the concept of fuzzy subset A~ of X by extending the characteristic function. A fuzzy subset A~ of X is defined by its membership function jA~ : X ! ½0; 1 which is viewed as an extension of characteristic function. The value jA~ ðaÞ can be interpreted as the membership degree of a point a in the set A: Under some suitable conditions on the membership function, the fuzzy set is then termed as a fuzzy real number (this will be realized in the sequel). Let a be a real number. The fuzzy real number a~ corresponding to a can be interpreted as ‘around a’: The graph of the membership function ja~ ðxÞ is bell-shaped and ja~ ðaÞ ¼ 1: It means that the membership degree ja~ ðxÞ is close to 1 when x is close to a: Let X ¼ R be a real number system and a~ be a fuzzy subset of R: We denote by a~ a ¼ {x : ja~ ðxÞ $ a} the a-level set of a~ for a [ ð0; 1; and a~ 0 is the closure of the set {x : ja~ ðxÞ . 0}: Now a~ is called a normal fuzzy set if there exists an x such that ja~ ðxÞ ¼ 1; and called a convex fuzzy set if ja~ ðlx þ ð1 2 lÞyÞ $ min{ja~ ðxÞ; ja~ ðyÞ} for l [ ½0; 1; i.e. ja~ is a quasi-concave function. Zadeh [14] has shown that a~ is a convex fuzzy subset of R if and only if its a-level set a~ a ¼ {x : ja~ ðxÞ $ a} is a convex set in R for all a [ ½0; 1: Let f be a real-valued function defined on R: Then f is said to be upper semicontinuous, if {x : f ðxÞ $ a} is a closed set in R for each a: Let a~ be a fuzzy subset of R satisfying the following conditions: (i) a~ is a normal and convex fuzzy set; (ii) Its membership function ja~ is upper semicontinuous; (iii) The 0-level set a~ 0 is bounded in R: Then, from the above three conditions, we see that the alevel set a~ a is convex, closed and bounded in R: It shows that a~ a is a closed interval. We then denote it by a~ a ¼ ½~aLa ; a~ U a : Therefore we propose the following definition. Definition 2.1. Let a~ be a fuzzy subset of R: Then a~ is called a fuzzy real number if the following conditions are satisfied: (i) (ii) (iii) (iv) (v)
a~ is a normal and convex fuzzy set; Its membership function ja~ is upper semicontinuous; The 0-level set a~ 0 is bounded in R; The 1-level set a~ 1 is a singleton set, i.e. a~ L1 ¼ a~ U 1; The functions gðaÞ ¼ a~ La and hðaÞ ¼ a~ U are continuous a with respect to a on [0,1].
We denote by FR the set of all fuzzy real numbers. Remark 2.1. Let a~ be a fuzzy subset of R satisfying conditions (i) – (iv) in Definition 2.1. Suppose that its
H.-C. Wu / Reliability Engineering and System Safety 83 (2004) 277–286
membership function ja~ is strictly increasing on the interval ½~aL0 ; a~ L1 and strictly decreasing on the interval ½~aU ~U 1;a 0 : Then a~ is a fuzzy real number, since the fact of strict monotonicity implies that condition (v) is satisfied. This shows that the concept of fuzzy real number proposed in this paper is more general than that of Wu [12]. However, the arguments used in Ref. [12] for deriving the fuzzy Bayes point estimator can also apply to this paper, and finally it leads to the same results for fuzzy Bayes point estimator because only the continuity in Definition 2.1 (v) was taken into account in Ref. [12]. Let X~ : V ! FR be a fuzzy-valued function. Then, according to Puri and Ralescu [7] and Kra¨tschmer [3], X~ is a fuzzy random variable if and only if X~ La and X~ U a are (traditional) random variables for all a [ ½0; 1: Let X be a random variable with pdf f ðx; u1 ; …; un Þ of known functional form but depending on an unknown n-dimensional vector u ¼ ðu1 ; …; un Þ in which ui ; i ¼ 1; …; n; are called parameters. We let Q ¼ ðQ1 ; …; Qn Þ stand for the set of all possible values of u and call it the parameter space. Here we are going to consider the fuzzy parameters. Let Q ¼ ðQ1 ; …; Qn Þ be the parameter space. We write u~ ¼ ðu~1 ; …; u~n Þ; where each fuzzy parameter u~i is a fuzzy subset of Qi for i ¼ 1; …; n; i.e. each u~i is associated with a membership function ju~i : Qi ! ½0; 1 for i ¼ 1; …; n: Here each u~i is assumed to be a fuzzy real number. We also see that ðu~i ÞLa and ðu~i ÞU a are in the parameter space Qi for all a [ ½0; 1: Therefore we can discuss the point estimator of ðu~i ÞLa and ðu~i ÞU a for all a [ ½0; 1 and all i ¼ 1; …; n: It is without loss of generality to assume n ¼ 1: Let u~ be a ~ From the fuzzy parameter of the fuzzy random variable X: proposed concept of fuzzy random variable, we can say that u~La and u~Ua are the parameters of the random variables X~ La and X~ U a ; respectively, for all a [ ½0; 1: The purpose is to estimate u~ under fuzzy environments. In order to apply the Bayesian approach, the fuzzy parameter u~ is assumed as a fuzzy random variable. It says that the random variables u~La and u~U a have distributions with known parameters U ðh~1 ÞLa ; …; ðh~m ÞLa and ðh~1 ÞU a ; …; ðh~m Þa ; respectively, for all a [ ½0; 1; where h~1 ; …; h~m are known fuzzy parameters. Since u~ is a fuzzy real number, we see that u~La and u~U a are continuous with respect to a by definition. Therefore the closed intervals ½u~La ; u~U a ; for a [ ½0; 1; are continuously shrinking with respect to a: Then for any parameter u [ ~L ~U ½u~La ; u~U a ; we have u ¼ ub or u ¼ ub for some b $ a; since L U u~1 ¼ u~1 : Thus for any parameter u [ ½u~La ; u~Ua ; we can find a Bayes point estimator u^ for u: Let ~L ; inf uc ~U }; max{ sup uc ~U }: ~L ; sup uc A ¼ ½min{ inf uc a
a#b#1
b
a#b#1 b
a#b#1
b
a#b#1
b
ð1Þ Then this interval will contain all of the Bayes point estimators for each u [ ½u~La ; u~U a : According to the following
279
Proposition 2.1, we can naturally define the membership function of fuzzy Bayes point estimator.
Proposition 2.1. (Zadeh [15]) (Resolution Identity) Let a~ be a fuzzy subset of R with membership function ja~ : Then
ja~ ðxÞ ¼ sup a·1a~ a ðxÞ; a[½0;1
where 1A is a characteristic function of set A; i.e. 1A ðxÞ ¼ 1 if x [ A and 1A ðxÞ ¼ 0 if x A (note that the a-level set a~ a of a~ is a usual set). The fuzzy Bayes point estimator of u~ is then denoted by ^u~; and the membership function of u^~ is defined by
ju^~ðrÞ ¼ sup a·1Aa ðrÞ;
ð2Þ
0#a#1
via the form of Resolution Identity in Proposition 2.1. As we have known that the Bayes point estimator is a random variable, Wu [12, Theorem 3.1] has also shown that the fuzzy Bayes point estimator is a fuzzy random variable.
3. Fuzzy bayesian system reliability assessment The Bayesian approach to system reliability is the assignment of the prior distribution. From the Bayesian point of view, the goal is to determine the posterior distribution of the system reliability. The prior distribution is assigned to each component of a system. Therefore the posterior distributions of each component reliability can be found by Bayes theorem. In this case, we can derive the posterior distribution of system reliability from the posterior distributions of component reliabilities. However, the key methodology in such a derivation is that system reliability can be expressed as a product of independent random variables each of which corresponds to either component reliability (for series system) or component unreliability (for parallel system). Therefore we need to invoke the method of Mellin integral transform. We describe it as follows. Let X be a non-negative random variable with pdf f ðxÞ: The Mellin transform of f with respect to the complex parameter u is defined by Mðf ; uÞ ¼
ð1
xu21 f ðxÞdx:
0
It is also convenient to regard the Mellin transform as the moments of X; i.e. Mðf ; uÞ ¼ E½X u21 :
ð3Þ
Let X1 ; X2 ; …; Xk be independent non-negative random variables with pdf’s f1 ; f2 ; Q …; fk ; respectively. Let gk ðyÞ be the pdf of the product Y ¼ ki¼1 Xi : Springer and Thompson
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[8] proved that Mðgk ; uÞ ¼
k Y
Mðfi ; uÞ:
ð4Þ
i¼1
With the help of Eq. (4), we can obtain the posterior distribution of system reliability from the posterior distributions of component reliabilities. Then the Bayes point estimator of system reliability is the mean of the posterior distribution under the squared error loss function, and it can be obtained from Eq. (3). 3.1. Series system Suppose that we have n items. Each item i can be represented as a Bernoulli random variable Yi with reliability (survival probability) r: Then the number of P survivors X ¼ ni¼1 Yi is recorded. The probability distribution of the number of survivors is the Binomial distribution given by f ðxlrÞ ¼ Pr{x survivors will occur in n trialslr} ¼
x ¼ 0; 1; …; n and 0 , r , 1: For a series system consisting of k independent components, the system reliabilityQr is the product of component reliabilities ri ; i.e. r ¼ ki¼1 ri : Now suppose that ni items are tested and that si survivors are observed for the ith component, i ¼ 1; …; k: Then the sampling distribution is a Binomial distribution with parameter ri as described above. In order to obtain the Bayes point estimator of the system reliability r; we regard the component reliability ri as a random variable Ri ; for i ¼ 1; …; k; with pdf pi ðri Þ; where pi ðri Þ is termed as a prior distribution of component reliability Ri : Then the posterior distribution of Ri is then given by ð5Þ
for 0 , ri , 1; where si is the observed number of survivors in ni trials. The most widely used prior distribution for Ri is the Beta distribution Bðsi0 ; ni0 Þ with pdf given by pi ðri Þ ¼
Gðni0 Þ r si0 21 ð1 2 ri Þni0 2si0 21 ; Gðsi0 ÞGðni0 2 si0 Þ i
pi ðri lsi Þ ¼
Gðni þ ni0 Þ r si þsi0 21 Gðsi þ si0 ÞGðni þ ni0 2 si 2 si0 Þ i ð1 2 ri Þni þni0 2si 2si0 21 ;
ð7Þ
which is Bðsi þ si0 ; ni þ ni0 Þ distribution. The Mellin transform of the posterior pdf of Ri is found to be M{pi ðri lsi Þ; u} Gðni þ ni0 Þ Gðsi þ si0 þ u 2 1Þ ¼ Gðsi þ si0 Þ Gðni þ ni0 þ u 2 1Þ ðni þ ni0 2 1Þ! ¼ ðsi þ si0 2 1Þ! 1 ; ðu þ si þ si0 2 1Þðu þ si þ si0 Þ· · ·ðu þ ni þ ni0 2 2Þ where ReðuÞ . 2ðsi þ si0 2 1Þ (also see Q Ref. [5]). Since the system reliability is R ¼ ki¼1 Ri ; from Eq. (4), the Mellin transform of the posterior pdf pðrls; nÞ of the system reliability R is given by
n! r x ð1 2 rÞn2x ðn 2 xÞ!x!
r si ð1 2 ri Þn2si pðri Þ pi ðri lsi Þ ¼ Ð1 i s ; n2si i pðyÞdy 0 y ð1 2 yÞ
can be taken from the past tests, e.g. the average of survivors from the previous tests. Thus it is not necessarily to assume si0 as an integer when we consider the prior Beta distribution Bðsi0 ; ni0 Þ: From Eqs. (5) and (6), the posterior distribution of Ri is given by
ð6Þ
where 0 # ri # 1 and u1 ; u2 . 0; since the conjugate prior distribution of the Binomial distribution is Beta distribution. Since the expectation of Beta distribution Bðsi0 ; ni0 Þ is si0 =ni0 ; the numbers ni0 and si0 can be interpreted as ‘pseudo’ number of items that are placed on test and ‘pseudo’ number of survivors, respectively. The ‘pseudo’ number of survivors
M{pðrlsÞ; u} k Y ðni þ ni0 2 1Þ! ¼ ðsi þ si0 2 1Þ! i¼1
1 ; ðu þ si þ si0 2 1Þðu þ si þ si0 Þ· · ·ðu þ ni þ ni0 2 2Þ ð8Þ
where ReðuÞ . 2ðsi þ si0 2 1Þ for all i ¼ 1; …; k: The Bayes point estimator of system reliability R is the mean of the posterior distribution in Eq. (8) under squared error loss function, and it is given by k Y si þ si0 E½Rls ¼ M{pðrlsÞ; u ¼ 2} ¼ ; ð9Þ ni þ ni0 i¼1 from Eq. (3). Suppose that the component reliabilities cannot be sured as exact real numbers as described above. In order to apply the Bayesian approach, the fuzzy component reliabilities are considered as fuzzy random variables R~ i with the assumptions that ni0 is a known integer representing the ‘pseudo’ number of items for the ith component and s~i0 is a known fuzzy real number representing the ‘pseudo’ number of survivors (since the number of survivors cannot be recorded precisely as described above, s~i0 is assumed as a fuzzy real number). Then, from Eq. (9), the Bayes point estimates of r~La and r~U a are ! ! k k L U Y Y ð~ s Þ þ s ð~ s Þ þ s i0 a i i0 a i b b and r~U ; ð10Þ r~La ¼ a ¼ ni þ ni0 ni þ ni0 i¼1 i¼1
H.-C. Wu / Reliability Engineering and System Safety 83 (2004) 277–286
281
respectively, for all a [ ½0; 1: According to the discussions in Section 2, let A ¼ ½min{ min r~bL ; min r~bU }; max{ max r~bL ; max r~bU }:
respectively, for all a [ ½0; 1: The membership function of the fuzzy Bayes point estimate of system reliability r is defined by the same way as discussed above.
ð11Þ Then this interval will contain all of the Bayes point estimates for each r [ ½~rLa ; r~U a : It is obvious that c c c c c c U L U L L U r~a # r~a ; r~a # r~b and r~a $ r~b for a , b; i.e. rc ~aL # rc ~1L # bU U r~c 1 # r~a : Thus Aa can be rewritten as
3.3. k-out-of-m System
a
a#b#1 b a#b#1 b
Aa ¼ ½rc ~aL ; rc ~aU :
a#b#1 b a#b#1 b
ð12Þ
Now the membership function of the fuzzy Bayes point estimate of r~; denoted as r^~; is defined by
jr^~ ðrÞ ¼ sup a·1Aa ðrÞ; 0#a#1
via the form of Resolution Identity in Proposition 2.1. The numerical example will be given in the sequel.
For a k-out-of-m system consisting of m independent and identical components, the system reliability is given by ! m X m j r¼ r ð1 2 rÞm2j ; j j¼k where r is the reliability of the (common) component. In order to apply Bayesian approach, the (common) component reliability r is assumed as a random variable R with prior Beta distribution Bðs0 ; n0 Þ: Then, from Eq. (7), the posterior distribution of R is a Bðs þ s0 ; n þ n0 Þ distribution with pdf given by pR ðrls; s0 Þ ¼
3.2. Parallel systems For a parallel system consisting of k independent Q components, the system reliability is r ¼ 1 2 ki¼1 ð1 2 ri Þ; where ri is the component reliability of the ith component. Equivalently, the system unreliability q ¼ 1 2 r is the product of component unreliabilities qi ¼ Q 1 2 ri ; i.e. q ¼ ki¼1 qi : In this case, we can use the Mellin transform technique as described in Eq. (4) to obtain the Bayes point estimate of system unreliability. Assuming the prior distribution of component unreliabilities for the ith component is Beta distribution Bðmi0 ; ni0 Þ; where mi0 may be regarded as the (pseudo) number of failures in a test of (pseudo) ni0 items. Therefore ni0 ¼ si0 þ mi0 : According to Eqs. (8) and (9), the Bayes point estimate of the system unreliability is given by k Y mi þ mi0 E½Qlm ¼ : ni þ ni0 i¼1 Thus the Bayes point estimate of system reliability r under the squared error loss function is given by k Y mi þ mi0 E½Rls ¼ 1 2 E½Qlm ¼ 1 2 ni þ ni0 i¼1 k Y ni 2 si þ ni0 2 si0 ¼12 ni þ ni0 i¼1 k Y s þ si0 ¼12 12 i : ni þ ni0 i¼1 Under the fuzzy assumptions as described above, the Bayes point estimates of r~La and r~U a are ! k Y ð~s ÞL þ si and r~bLa ¼ 1 2 1 2 i0 a ni þ ni0 i¼1 ! k U Y ð~ s Þ þ s i0 a i b ; ð13Þ 12 r~U a ¼12 ni þ ni0 i¼1
Gðn þ n0 Þ rsþs0 21 Gðs þ s0 ÞGðn þ n0 2 s 2 s0 Þ ð1 2 rÞnþn0 2s2s0 21 :
The Bayes point estimate of system reliability r is the mean E½Rls; s0 under the squared error loss function, and is given by E½Rls;s0 0 1 m m ð1 X ¼ @ A rj ð1 2 rÞm2j ·pR ðrls;s0 Þdr 0 j j¼k Gðn þ n0 Þ ¼ Gðs þ s0 ÞGðn þ n0 2 s 2 s0 Þ 2 0 1 3 m m Gðj þ s þ s ÞGðm 2 j þ n þ n 2 s 2 s Þ X 0 0 0 5: 4 @ A Gðm þ n þ n0 Þ j j¼k Under the fuzzy assumptions as described above, the Bayes point estimates of r~La and r~U a are given by r~bU a¼
Gðnþn0 Þ L Gðsþð~s0 Þa ÞGðnþn0 2s2ð~s0 ÞLa Þ
2 0 1 3 m m Gðjþsþð~s ÞL ÞGðm2jþnþn 2s2ð~s ÞL Þ X 0 a 0 0 a 5 4 @ A ; Gðmþnþn0 Þ j¼k j ð14Þ and
Gðnþn0 Þ Gðsþð~s0 ÞUa ÞGðnþn0 2s2ð~s0 ÞU0 Þ 2 0 1 3 m m Gðjþsþð~s ÞU ÞGðm2jþnþn 2s2ð~s ÞU Þ X 0 a 0 0 a 5 4 @ A ; Gðmþnþn0 Þ j j¼k
r~bU a¼
ð15Þ respectively, for all a [ ½0; 1: The membership function of the fuzzy Bayes point estimate of system reliability r is defined by the same way as discussed above.
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4. Computational procedures and examples In this section, we shall provide some computational techniques to evaluate the membership degree of the fuzzy Bayes point estimate as presented in Eq. (2). From Eq. (1), we adopt the following notations Aa ¼ ½gðaÞ;hðaÞ ¼ ½min{g1 ðaÞ;g2 ðaÞ};max{h1 ðaÞ; h2 ðaÞ}; ð16Þ where ~U ; h1 ðaÞ ¼ sup uc ~Lb ; g2 ðaÞ ¼ inf uc ~Lb ; g1 ðaÞ ¼ inf uc b a#b#1
a#b#1
a#b#1
~U : h2 ðaÞ ¼ sup uc b a#b#1
the original problem with optimal solution apo (in fact, apo ¼ Zop ). Then Wu [12] has shown that Z p ¼ Zop : In other words, in order to obtain the objective value Zop of the original nonlinear programming problem, it will be enough to just solve the four subproblems I– IV. It is easy to see that g1 and g2 are increasing, and h1 and h2 are decreasing. In order to solve the above four subproblems I – IV, we further consider the following general type of nonlinear programming problem. Let gðaÞ be increasing, hðaÞ be decreasing and gðaÞ # hðaÞ for all a [ ½0; 1: We plan to solve the following problem when r is given and fixed
jðrÞ ¼ max
From Eq. (2), the membership function of the fuzzy Bayes point estimate u^~ is given by ju^~ðrÞ ¼ sup a·1Aa ðrÞ ¼ sup{a : gðaÞ # r # hðaÞ;0 # a # 1}:
a
subject to
hðaÞ $ r
0#a#1
0 # a # 1:
Therefore, we need to solve the following type of nonlinear programming problem max
a
subject to
min{g1 ðaÞ; g2 ðaÞ} # r max{h1 ðaÞ; h2 ðaÞ} $ r 0 # a # 1:
Now we consider the following four subproblems
a
I : max subject to
g1 ðaÞ # r h1 ðaÞ $ r 0 # a # 1;
a
II : max subject to
g 1 ð aÞ # r h2 ð aÞ $ r 0 # a # 1;
III :
max
a
subject to
g 2 ð aÞ # r h 1 ð aÞ $ r 0 # a # 1;
a
IV : max subject to
g 2 ð aÞ $ r h2 ð aÞ $ r 0 # a # 1:
ZIp ;
ZIIp ;
gðaÞ $ r
p ZIII
p Let and ZIV be the objective values of subproblems I – IV, respectively. Let Zp ¼ p p p p p max{ZI ; ZII ; ZIII ; ZIV } and Zo be the objective value of
We see that the subproblems I, II and IV satisfy the above assumptions (in subproblem III, g2 # h1 is not always true). Therefore they can be solved by using the Supplemental Procedure which is proposed in Ref. [12]. Before providing the numerical examples, we introduce a special kind of fuzzy real number. We say that a~ is a triangular fuzzy real number if its membership function is given by (the graph of its membership function will look like a triangle) 8 ðr 2 a1 Þ=ða2 2 a1 Þ if a1 # r # a2 > > < ja~ ðrÞ ¼ ða3 2 rÞ=ða3 2 a2 Þ if a2 , r # a3 > > : 0 otherwise: We denote by a~ ¼ ða1 ; a2 ; a3 Þ: Then a~ is a fuzzy real number and its a-level set is given by a~ a ¼ ½ða2 2 a1 Þa þ a1 ; ða2 2 a3 Þa þ a3 for all a [ ½0; 1:
Example 4.1. A system consists of three independent components in series. Suppose that we collect following data on a test
Component 1 Component 2 Component 3
Successes
Failures
Items tested
8 7 3
2 2 1
10 9 4
. From the past tests and experiences, the percentage of survivors in a system is ‘around’ 80% for component 1, ‘around’ 80% for component 2 and ‘around’ 75% for ~ s~ 20 ¼ component 3. Therefore we can assume that s~ 10 ¼ 8; ~ n10 ¼ 10; n20 ¼ 10 and n30 ¼ 8; where 8~ ¼ ~ s~30 ¼ 6; 8; ð6; 8; 10Þ and 6~ ¼ ð4; 6; 8Þ are triangular fuzzy real numbers. Then the a-level sets of 8~ and 6~ are 8~ a ¼ ½6 þ 2a; 10 2 2a
H.-C. Wu / Reliability Engineering and System Safety 83 (2004) 277–286
and 6~ a ¼ ½4 þ 2a; 8 2 2a; respectively. The Bayes point estimates of r~La and r~U a ; from Eq. (10), are given by 6 þ 2a þ 8 6 þ 2a þ 7 4 þ 2a þ 3 b L r~a ¼ 10 þ 10 10 þ 9 8þ4 ¼ and r~bU a ¼ ¼
ð14 þ 2aÞð13 þ 2aÞð7 þ 2aÞ ; 4560
10 2 2a þ 8 10 þ 10
10 2 2a þ 7 10 þ 9
8 2 2a þ 3 8þ4
283
r~ by evaluating the above formulas. On the other hand, we also have the 0.95-level closed interval A0:95 ¼ ½0:46239; 0:48515: In this case, we may say that the system reliability lies in the interval A0:95 ¼ ½0:46239; 0:48515 with confidence degree 0.95. In general, we can say that the system reliability r lies in the interval Aa with confidence degree a:
ð18 2 2aÞð17 2 2aÞð11 2 2aÞ ; 4560
respectively, for all a [ ½0; 1: Comparing Eqs. (11), (12) and (16), we see that gðaÞ ¼ g1 ðaÞ ¼ r~bLa and hðaÞ ¼ h2 ðaÞ ¼ r~bU a : Therefore, we just need to solve subproblem II, i.e. Zop ¼ ZIIp : Now we are going to use the Supplemental Procedure to solve subproblem II. We see that r~bL1 ¼ 2160=4560 ¼ r~bU 1 : Therefore jr^~ ð2160=4560Þ ¼ 1: Suppose that we reconsider the fuzzy real numbers 8~ and 6~ as the real numbers 8 and 6, respectively. Then the Bayes point estimate of system reliability r will be 2160/4560. This shows our anticipation that the Bayes point estimate will have membership degree 1. Since A0 ¼ ½1274=4560; 3366=4560; we are just interested in considering the system reliability r [ A0 : By applying the Supplemental Procedure, we have, for r [ A0 ; (i) if r , 2160=4560 then we solve the problem
Example 4.2. A system consists of three independent components in parallel. Suppose that we use the same data as in Example 4.1. Then, from Eq. (13), the Bayes point estimates of r~La and r~U a are 6 þ 2a þ 8 6 þ 2a þ 7 r~bLa ¼ 1 2 1 2 12 10 þ 10 10 þ 9 4 þ 2a þ 3 12 ; 8þ4 and 10 2 2a þ 8 10 2 2a þ 7 b U r~a ¼ 1 2 1 2 12 10 þ 10 10 þ 9 8 2 2a þ 3 12 : 8þ4 The numerical results can also be obtained using the same techniques as in Example 4.1.
ð14 þ 2aÞð13 þ 2aÞð7 þ 2aÞ #r ; jr^~ ðrÞ ¼ max a [ ½0; 1 : gðaÞ ¼ g1 ðaÞ ¼ r~bLa ¼ 4560 using any commercial optimizers, e.g. GAMS or LINGO. Equivalently, we can also use the Newton method to solve the following problem
jr^~ ðrÞ ¼ {a [ ½0; 1 : a is the root of equation ð14 þ 2aÞ ð13 þ 2aÞð7 þ 2aÞ ¼ 4560r}; (ii) if r . 2160=4560 then we solve the problem
Example 4.3. We consider a 2-out-of-3 system consisting of three independent and identical components. Suppose that there are eight successors (survivors) when 10 items are placed on a test, i.e. n ¼ 10 and s ¼ 8: From the past tests and experiences, the percentage of survivors is ‘around’ ~ 10Þ is 80%. Therefore the fuzzy prior Beta distribution Bð8; used. From Eqs. (14) and (15), the Bayes point estimates of r~La and r~U a are given by
ð18 2 2aÞð17 2 2aÞð11 2 2aÞ $r ; jr^~ ðrÞ ¼ max a [ ½0; 1 : hðaÞ ¼ h1 ðaÞ ¼ r~bLa ¼ 4560 using any commercial optimizers, e.g. GAMS or LINGO. Equivalently, we can also use the Newton method to solve the following problem
jr^~ ðrÞ ¼ {a [ ½0; 1 : a is the root of equation ð18 2 2aÞ ð17 2 2aÞð11 2 2aÞ ¼ 4560r}: Therefore we now can obtain the membership degree for any given Bayes point estimate r of fuzzy system reliability
r~bLa ¼
Gð10þ10Þ Gð8þ6þ2aÞGð10þ10282622aÞ 2 0 1 3 3 3 Gðjþ8þ6þ2aÞGð32jþ10þ10282622aÞ X 5; 4 @ A Gð3þ10þ10Þ j¼2 j
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and 2 3 ! 3 X 3 Gðj þ 8 þ 10 2 2aÞGð3 2 j þ 10 þ 10 2 8 2 10 þ 2aÞ G ð10 þ 10Þ 4 5: r~bU a ¼ Gð8 þ 10 2 2aÞGð10 þ 10 2 8 2 10 þ 2aÞ j¼2 j Gð3 þ 10 þ 10Þ
The numerical results can also be obtained using the same techniques as in Example 4.1. Example 4.4. We consider the following system. All components are mutually independent.
Under the fuzzy assumptions, the Bayes point estimates of r~La and r~U a are r~bLa ¼ MaL {p1 ðr1 ls1 Þ; u ¼ 2}·MaL {pRp ðrp ln2 2 s2 ; n3 2 s3 Þ; u ¼ 2};
ð20Þ
and U U r~bU a ¼ Ma {p1 ðr1 ls1 Þ; u ¼ 2}·Ma {pRp ðrp ln2 2 s2 ; n3 2 s3 Þ;
u ¼ 2}; The posterior distribution of component reliabilities of R1 ; R2 ; R3 are given by Eq. (7). Let Qi ¼ 1 2 Ri be the component unreliabilities of the ith component for i ¼ 2; 3: Assuming the prior distribution of component unreliabilities for the ith component is Beta distribution Bðmi0 ; ni0 Þ; for i ¼ 2; 3; where mi0 may be regarded as the (pseudo) number of failures in a test of (pseudo) ni0 items. Therefore ni0 ¼ si0 þ mi0 : Then, from Eq. (7), the posterior pdf of Qi is given by p i ðqi lmi Þ ¼
Gðni þ ni0 Þ Gðmi þ mi0 ÞGðni þ ni0 2 mi 2 mi0 Þ i þmi0 21 qm ð1 2 qi Þni þni0 2mi 2mi0 21 ; i
ð17Þ
for i ¼ 2; 3: Let Rp be the reliability of the parallel configuration which consists of components 2 and 3. Then Rp ¼ 1 2 Q2 Q3 : We see that the posterior pdf of Rp is pRp ðrp lm2 ; m3 Þ ¼
ð1 12rp
1 2 rp 1 ·p 3 ð 2 ðylm2 Þdy: jm3 ·p y y
It is easy to see that the system reliability is R ¼ R1 Rp with posterior pdf pðrls1 ; s2 ; s3 Þ (note that ni ¼ si þ mi ). Therefore the Mellin transform of pðrls1 ; s2 ; s3 Þ is the product of the Mellin transform of R1 and Rp ; i.e. M{pðrls1 ; s2 ; s3 Þ; u} ¼ M{p1 ðr1 ls1 Þ; u}·M{pRp ðrp ln2 2 s2 ; n3 2 s3 Þ; u}:
ð21Þ
respectively, for all a [ ½0; 1; where MaL ð·Þ and MaU ð·Þ are obtained from Eq. (19) by replacing the variable s10 as ð~s10 ÞLa and ð~s10 ÞU ~ i0 ÞLa and a ; respectively, and the variable mi0 as ðm U ðm ~ i0 Þa ; respectively, for i ¼ 2; 3: Suppose that we collect following data on a test
Component 1 Component 2 Component 3
Successes ðsi Þ
Failures ðmi Þ
Items tested ðni Þ
8 7 3
2 2 1
10 9 4
From the past tests and experiences, the percentage of survivors for component 1 is ‘around’ 80%. The percentage of failures is ‘around’ 20% for component 2 and ‘around’ 25% for component 3. Therefore the fuzzy prior ~ 10Þ is assigned to the component Beta distribution Bð8; ~ 10Þ reliability R1 ; and the fuzzy prior Beta distribution Bð2; ~ and Bð2; 8Þ are assigned to the component unreliabilities Q2 and Q3 ; respectively, where 8~ ¼ ð6; 8; 10Þ and 2~ ¼ ð1; 2; 3Þ are triangular fuzzy real numbers. Then the a-level sets of 8~ and 2~ . are 8~ a ¼ ½6 þ 2a; 10 2 2a and 2~ a ¼ ½1 þ a; 3 2 a; respectively. First of all, we evaluate r~bLa : From Eq. (20), we have MaL {p1 ðr1 ls1 Þ; u ¼ 2} ¼
ð18Þ
Now, from Eq. (17), we also have
The Bayes point estimator of system reliability R is the mean of the posterior distribution in Eq. (18) under squared error loss function, and it is given by
ðp 2 ÞLa ðq2 lm2 Þ ¼
E½Rls1 ; s2 ; s3 ¼ M{pðrls1 ; s2 ; s3 Þ; u ¼ 2} ¼ M{p1 ðr1 ls1 Þ;
and
u ¼ 2}·M{pRp ðrp ln2 2 s2 ; n3 2 s3 Þ; u ¼ 2}: ð19Þ
6 þ 2a þ 8 : 10 þ 10
ðp 3 ÞLa ðq3 lm3 Þ ¼
Gð19Þ q2þa ·ð1 2 q2 Þ152a ; Gð3 þ aÞGð16 2 aÞ 2
Gð12Þ q1þa ·ð1 2 q3 Þ92a : Gð2 þ aÞGð10 2 aÞ 3
H.-C. Wu / Reliability Engineering and System Safety 83 (2004) 277–286
Then
the system reliability assessment is a kind of extension of the conventional Bayesian system reliability assessment. The advantages of proposed methods in this paper are described as follows
ðpRp ÞLa ðrp lm2 ; m3 Þ ¼ K·ð1 2 rp Þ1þa ·
ð1 12rp
ð1 2 yÞ152a ·ya29 ·ðy 2 1 þ rp Þ92a dy; (i)
where
Gð19ÞGð12Þ : Gð3 þ aÞGð16 2 aÞGð2 þ aÞGð10 2 aÞ Now we obtain K¼
MaL {pRp ðrp ln2 2 s2 ; n3 2 s3 Þ; u ¼ 2} " ð1 ð1 ¼ K· ð1 2 rp Þ1þa · ð1 2 yÞ152a ·ya29 0
285
i
12rp
·ðy 2 1 þ rp Þ92a dy drp : According to Eqs. (11), (20), (21) and using the numerical integration (e.g. using the commercial software MATLAB), we can obtain the closed interval Aa : For example, for a ¼ 0:95; we have A0:95 ¼ ½0:6589; 0:7135: It says that the system reliability r lies in the interval Aa with confidence degree a:
5. Conclusions The fuzzy sets theory has successfully applied to the Bayesian system reliability assessment in this paper. Owing to the imprecise occurrence of observations (i.e. fuzzy data) in the real world as presented in this paper, we invoke the concept of fuzzy random variable to conquer this difficulty. On the other hand, in order to interpret the meaning of the membership degree of any given Bayes point estimate (i.e. the fuzzy Bayes point estimate), a nonlinear programming problem has to be solved. Furthermore, this nonlinear programming problem is fortunately divided into four solvable subproblems. Therefore the commercial optimizers are useful for obtaining the results. The four examples presented in this paper make the incorporation of fuzzy sets theory in the Bayesian system reliability assessment to be a more persuasive evidence for considering the fuzzy-Bayesian approach in system reliability assessment. Recall that the real number a can be regarded as a fuzzy real number a~ ; since the characteristic function xa of the real number can be regarded as a membership function ( 1 if x ¼ a; xa ðxÞ ¼ ja~ ðxÞ ¼ 0 if x – a: Therefore those examples show that the membership degree of Bayes point estimate of system reliability will be 1 if the fuzzy data were degenerated as the real data. We may also conclude that the fuzzy-Bayesian approach in
When the imprecise data (fuzzy data) occur in the system, we can handle this kind of situation by considering the fuzzy-Bayesian approach. However, the conventional Bayesian approach may have difficulty for handling this kind of imprecise data. (ii) Even though the data occurring in the system are taken as real numbers, the proposed method in this paper can still handle it as we just described above (the fuzzyBayesian approach in the system reliability assessment is a kind of extension of the conventional Bayesian system reliability assessment). Therefore, the fuzzyBayesian approach proposed in this paper is no loss of generality. (iii) Those examples (e.g. Example 4.1) show that if the value a is fixed then we can say that the system reliability lies in the interval Aa with confidence degree a: Therefore, we may use the similar techniques proposed in this paper (by fixing the value a) to develop the so-called ‘Interval Bayesian Reliability Assessment’ without any difficulty. In this paper, we propose a possible attempt on how to apply the fuzzy sets theory to Bayesian system reliability assessment. Therefore, it is naturally realized that the fuzzy sets theory can also impose upon some known techniques of reliability analysis in the future research.
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