# Reliability Analysis of Random Fuzzy Unrepairable Systems

Authors: Ying Liu; Xiaozhong Li; Jianbin Li
Journal: Discrete Dynamics in Nature and Society
 (2014)

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## Abstract

The lifetimes of components in unrepairable systems are considered as random fuzzy variables
since randomness and fuzziness are often merged with each other. Then we establish the fundamental
mathematical models of random fuzzy unrepairable systems, including series systems, parallel systems,
series-parallel systems, parallel-series systems, and cold standby systems with absolutely reliable conversion
switches. Furthermore, the expressions of reliability and mean time to failure (MTTF) are given for
the above five random fuzzy unrepairable systems, respectively. Finally, numerical examples are given to
show the application in a lighting lamp system and a hi-fi system.


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## Body

## 1. Introduction

The conventional reliability theory has been successfully used for solving various reliability problems, in which the lifetimes of systems are assumed to be random variables and the system behavior can be fully characterized by probability theory. It is well-known that reliability and mean time to failure (MTTF) are good evaluations in unrepairable systems, where the reliability is defined by the probability of the random event “system is functioning at timet” and MTTF is the expected value of random lifetime of the system. The results on classical reliability theory can be referred to studies such as Barlow and Proschan [1], Dhillon and Singh [2], Epstein and Sobel [3], Gnedenko et al. [4], Kaufmann [5], Kapur and Lamberson [6], Natarajan [7], Ross [8], Sharma et al. [9], Lin and Yeh [10], Tian et al. [11], Marquez et al. [12], and Hsu et al. [13].Although the traditional reliability theory has been proved to be effective in many cases, using probability method in engineering problems needs to have three basic premises: firstly, the events should be clearly defined; secondly, there should exist a large number of samples; and thirdly, the samples should have the probability of repetition. If the three premises do not hold, using probability theory to deal with the reliability problems has certain limitations. So fuzzy theory has been introduced to reliability theory by several authors. In 1975, Kaufmann [5] first used fuzzy theory in reliability engineering. Chowdhury and Misra [14] presented a method to find an expression of fuzzy system reliability of a nonseries parallel network taking into consideration the special requirements of fuzzy sets. Cai et al. [15–18] introduced various forms of fuzzy reliability theories, including profust reliability theory, posbist reliability theory, and posfust reliability theory. Their studies can be considered by taking new assumptions, such as the possibility assumption or the fuzzy-state assumption, in place of the probability assumption or the binary-state assumption. Utkin [19, 20] discussed the fuzzy system reliability based on the binary-state assumption and possibility assumption and considered the fuzzy availability and unavailability and the fuzzy operative availability and unavailability. Utkin and Gurov [21] proposed a general approach on the basis of a system of functional equations according to Cai's theory. In Praba et al. [22], a new method for finding fuzzy system reliability using posfust reliability theory was demonstrated, where the system was modelled as a unified fuzzy Markov model. Cooman [23] introduced the notion of possibilistic structure function based on the concept of the classical two-valued structure function and studied the possibilistic uncertainty of the states of a system and its components. Huang [24] developed the fundamental calculation formulas of fuzzy reliability and established the fuzzy reliability models of unrepairable systems. Huang et al. [25] proposed a new method to determine the membership function of the estimates of the parameters and the reliability function of multiparameter lifetime distributions. In Liu et al. [26], reliability and performance assessment for fuzzy multistate elements were considered. Ding and Lisnianski [27] considered a multistate system where performance rates and corresponding state probabilities were presented as fuzzy values. Recently, Jiang and Chen [28] developed a computational model of fuzzy reliability focusing on solving the engineering problems with random general stress and fuzzy general strength. Zhang et al. [29] considered a fuzzy age-dependent replacement policy, in which the lifetimes of components were treated as fuzzy variables. Linda and Manic [30] considered interval type-2 fuzzy voter design for fault tolerant systems.A more general case in practice is that randomness and fuzziness are merged with each other in one unrepairable system. Many researchers have paid attention to these problems. Wang and Watada [31] considered a renewal reward process with fuzzy random interarrival times and rewards under the T-independence associated with any continuous Archimedean t-norm. Based on using fuzzy random variables to characterize the lifetimes, Wang and Watada [32] studied the redundancy allocation problems to a fuzzy random parallel-series system. Adduri and Penmetsa [33] made system reliability analysis for mixed uncertain variables which contained both probability distributions and fuzzy membership functions. Utkin and Coolen [34] gave an overview of a lot of methods and models for reliability problems mixed with randomness and fuzziness. Utkin et al. [35] studied a simple one-unit system description in the probability and possibility contexts. According to the situation of randomness and fuzziness existing in the actual project, Li et al. [36] proposed a reliability-credibility model based on fuzzy theory, possibility theory, and credibility theory. Liu et al. [37] considered the fuzzy random reliability of structures based on fuzzy random variables. Random fuzzy theory proposed by Liu [38] mainly uses the average chance measure to evaluate the random fuzzy events. Although many measures proposed by researchers have been used to deal with the behavior of random fuzzy phenomena, they have no self-duality properties. However, a self-duality measure is absolutely needed in both theory and practice. Until today, few people have used random fuzzy theory as the basic mathematical tool to deal with reliability problems. For example, Zhao et al. [39] used random fuzzy theory into renewal process, which results were very useful in repairable system theory. Zhao and Liu [40] provided three types of system performances, in which the lifetimes of redundant systems were treated as random fuzzy variables. Since the important figures of merit for repairable systems were the limited availability, steady state failure frequency, mean time between failures, and mean time to repair, Liu et al. [41] gave the reliability analysis of a random fuzzy repairable series system with independent components. In most cases, the components in the system were dependent. So Liu et al. [42] considered two dependent components, established a random fuzzy shock model and a random fuzzy fatal shock model, and studied the bivariate random fuzzy exponential distribution.The topic of unrepairable system is an important content in system reliability theory. There are many reasons cannot be repaired, some because of technical reasons, cannot repair; some because of economic reasons, not worth to repair; and some because of making repairable system simplification. In this paper, random fuzzy variables are employed to represent uncertain lifetimes of components in the unrepairable systems. We establish the fundamental mathematical models of random fuzzy unrepairable systems, including series systems, parallel systems, series-parallel systems, parallel-series systems, and cold standby systems with absolutely reliable conversion switches. Furthermore, the expressions of reliability and MTTF are given for the above five systems, respectively. The expressions of reliability and MTTF of the random fuzzy unrepairable systems we arrived at are suitable for stochastic cases and fuzzy cases, which shows that the reliability mathematical models and results in this paper generalize the traditional reliability theory.The rest of this paper is organized as follows. In Section2, we recall some basic concepts on fuzzy variables and random fuzzy variables. In Section 3, we establish the fundamental mathematical models of random fuzzy unrepairable systems and give the expressions of reliability and MTTF for each system. Some examples are also presented to illustrate how to calculate the reliability and MTTF of given unrepairable systems, in which the lifetimes of components follow certain probability distributions with fuzzy parameters. In Section 4, the application in a lighting lamp system and a hi-fi system is presented.


## 2. Fuzzy Variables and Random Fuzzy Variables

In this section, we first introduce some basic concepts of fuzzy variables based on the credibility measure.Definition 1 (B. Liu and Y.-K. Liu [43]).
LetΘ be a nonempty set and let P(Θ) be the power set of Θ. The set function is called a credibility measure if it satisfies the following four axioms.Axiom 1. Consider Cr{Θ}=1.Axiom 2.
Cr is increasing; that is, Cr{A}≤Cr{B} whenever A⊂B.Axiom 3.
Cr is self-dual; that is, Cr{A}+Cr{Ac}=1 for any A∈P(Θ).Axiom 4. Consider Cr{∪iAi}∧0.5=sup⁡iCr{Ai} for any {Ai} with Cr{Ai}≤0.5.Definition 2 (Liu [44]).
A fuzzy variable is defined as a function from the credibility space(Θ,P(Θ),Cr) to the set of real numbers.Definition 3 (Liu [44]).
Letξ be a fuzzy variable defined on the credibility space (Θ,P(Θ),Cr). Then its membership function is derived from the credibility measure by
(1)μ(x)=(2Cr{ξ=x})∧1,x∈R.Definition 4 (Liu [44]).
A fuzzy variableξ is said to be positive if Cr{ξ≤0}=0.Definition 5 (Liu [45]).
Letξ be a fuzzy variable and α∈(0,1]. Then
(2)ξαL=inf⁡{x∣μ(x)≥α},ξαU=sup⁡{x∣μ(x)≥α}
are called the α-pessimistic value and the α-optimistic value of ξ, respectively.Definition 6 (B. Liu and Y.-K. Liu [43]).
Letξ be a fuzzy variable. The expected value E[ξ] is defined as
(3)E[ξ]=∫0+∞‍Cr{ξ≥r}dr-∫-∞0‍Cr{ξ≤r}dr
provided that at least one of the two integrals is finite. In particular, if ξ is a positive fuzzy variable, then E[ξ]=∫0+∞‍Cr{ξ≥r}dr.Proposition 7 (Y.-K. Liu and B. Liu [46]).
Letξ be a fuzzy variable with finite expected value E[ξ]; then one has
(4)E[ξ]=12∫01‍[ξαL+ξαU]dα,
where ξαL and ξαU are the α-pessimistic value and the α-optimistic value of ξ, respectively.Definition 8 (Liu [44]).
The fuzzy variablesξ1,ξ2,…,ξn are said to be independent if
(5)Cr{⋂i=1n‍{ξi∈Bi}}=min⁡1≤i≤nCr{ξi∈Bi}
for any sets B1,B2,…,Bn of R.Proposition 9 (Y.-K. Liu and B. Liu [46] and Zhao et al. [39]).
Letξ and η be two independent fuzzy variables. Then (i)
for anyα∈(0,1],(ξ+η)αL=ξαL+ηαL;(ii)
for anyα∈(0,1],(ξ+η)αU=ξαU+ηαU.
Furthermore, ifξ and η are positive, then (iii)
for anyα∈(0,1],(ξ·η)αL=ξαL·ηαL;(iv)
for anyα∈(0,1],(ξ·η)αU=ξαU·ηαU.The concept of the random fuzzy variable was given by Liu [45]. Let (Ω,A,Pr) be a probability space and F a collection of random variables. A random fuzzy variable is defined as a function from a credibility space (Θ,P(Θ),Cr) to a collection of random variables F.Proposition 10 (Liu [38]).
Letξ be a random fuzzy variable on the credibility space (Θ,P(Θ),Cr). Then, for θ∈Θ, one has (1)
Pr{ξ(θ)∈B}is a fuzzy variable for any Borel setBofR;(2)
E[ξ(θ)]is a fuzzy variable provided thatE[ξ(θ)]is finite for any fixedθ∈Θ.Example 11.
A random fuzzy variableξ is said to be exponential if for each θ, ξ(θ) is an exponentially distributed random variable whose density function is defined as
(6)fξ(θ)(t)={X(θ)exp⁡(-X(θ)t),ift≥0,0,ift<0,
where X is a positive fuzzy variable defined on the space Θ. An exponentially distributed random fuzzy variable is denoted by ξ~EXP(X), and the fuzziness of random fuzzy variable ξ is said to be characterized by fuzzy variable X. It follows from Proposition 10 that Pr{ξ(θ)≥t} and E[ξ(θ)] are fuzzy variables. We can arrive at Pr{ξ(θ)≥t}=exp⁡(-X(θ)t) and E[ξ(θ)]=1/X(θ).Definition 12 (Y.-K. Liu and B. Liu [46]).
Letξ be a random fuzzy variable defined on the credibility space (Θ,P(Θ),Cr). Then the expected value E[ξ] is defined by
(7)E[ξ]=∫0+∞‍Cr{θ∈Θ∣E[ξ(θ)]≥r}dr-∫-∞0‍Cr{θ∈Θ∣E[ξ(θ)]≤r}dr
provided that at least one of the two integrals is finite. In particular, if ξ is a positive fuzzy variable, then E[ξ]=∫0+∞‍Cr{θ∈Θ∣E[ξ(θ)]≥r}dr.Definition 13 (Y.-K. Liu and B. Liu [47]).
Letξ be a random fuzzy variable. Then the average chance, denoted by Ch, of random fuzzy event characterized by {ξ∈B} is defined as
(8)Ch{ξ∈B}=∫01‍Cr{θ∈Θ∣Pr{ξ(θ)∈B}≥p}dp.Remark 14.
Ifξ degenerates to a random variable, then the average chance degenerates to Pr{ξ∈B}, which is just the probability of random event. If ξ degenerates to a fuzzy variable, then the average chance degenerates to Cr{ξ∈B}, which is just the credibility of fuzzy event.Finally, we refer to a definition on the stochastic ordering which is usually employed in the comparison of the lifetimes of systems.Definition 15 (Ross [48]).
A collection of random variablesF is said to be a totally ordered set with stochastic ordering if and only if, for any given ζ1,ζ2∈F, and r∈R, either
(9)Pr{ζ1≤r}≤Pr{ζ2≤r}(denotedbyζ2≤dζ1)
or
(10)Pr{ζ1≤r}≥Pr{ζ2≤r}(denotedbyζ1≤dζ2).Remark 16 (Ross [48]).
For any givenζ1,ζ2∈F, we have
(11)E[ζ1]≤E[ζ2]⟺ζ1≤dζ2.


## 3. Random Fuzzy Unrepairable Systems

In this section, we first define the reliability and MTTF of random fuzzy unrepairable systems. Then the reliability and MTTF of random fuzzy series systems, parallel systems, series-parallel systems, parallel-series systems, and cold standby systems are discussed, respectively.Definition 17.
LetX be the random fuzzy lifetime of an unrepairable system, which is defined on the credibility space (Θ,P(Θ),Cr); then the reliability of the unrepairable system is defined by
(12)R(t)=Ch{X≥t}.Definition 18.
LetX be the random fuzzy lifetime of an unrepairable system, which is defined on the credibility space (Θ,P(Θ),Cr); then MTTF of the unrepairable system is defined by
(13)MTTF=∫0+∞‍Cr{θ∈Θ∣E[X(θ)]≥r}dr.


### 3.1. Reliability Analysis of Random Fuzzy Unrepairable Series Systems

Consider a series system composed ofn independent components. Let Xi be the lifetime of component i, which is a random fuzzy variable on the credibility space (Θi,P(Θi),Cri), i=1,2,…,n. Obviously, the lifetime of the series system is X=min⁡{X1,X2,…,Xn}, which is a random fuzzy variable on the product credibility space (Θ,P(Θ),Cr), where Θ=Θ1×Θ2×⋯×Θn and Cr=Cr1∧Cr2∧⋯∧Crn.Theorem 19.
LetXi, i=1,2,…,n be random fuzzy variables. Assume that the α-pessimistic values and the α-optimistic values of E[Xi(θi)], i=1,2,…,n, are continuous at the point α, α∈(0,1]. The reliability of random fuzzy series system is
(14)R(t)=E[∏i=1n‍Pr{ω∈Ω∣Xi(θi)(ω)≥t}].Proof.
By Definitions13 and 17 and Proposition 7, we have
(15)R(t)=Ch{X≥t}=∫01Cr{θ∈Θ∣Pr⁡{X(θ)≥t}≥p}dp‍=12∫01‍(PrαL{ω∈Ω∣X(θ)(ω)≥t}+PrαU{ω∈Ω∣X(θ)(ω)≥t})dα.
LetAi={θi∈Θi∣μ{θi}≥α}, i=1,2,…,n. Since the α-pessimistic values and the α-optimistic values of fuzzy variables E[Xi(θi)], θi∈Θi, i=1,2,…,n, are continuous at the point α, α∈(0,1], then there exist maximum and minimum values; that is, at least there exist points θi′,θi′′∈Ai such that
(16)E[Xi(θi′)]=E[Xi(θi)]αL,E[Xi(θi′′)]=E[Xi(θi)]αU.
For anyθi,α∈Ai, we have
(17)E[Xi(θi′)]≤E[Xi(θi,α)]≤E[Xi(θi′′)],i=1,2,…,n.
Hence, by Remark16, we have
(18)Xi(θi′)≤dXi(θi,α)≤dXi(θi′′),θi,α∈Ai,i=1,2,…,n.
It follows from Definition15 that
(19)Pr{ω∈Ω∣Xi(θi′)(ω)≥t}≤Pr{ω∈Ω∣Xi(θi,α)(ω)≥t}≤Pr{ω∈Ω∣Xi(θi′′)(ω)≥t},i=1,2,…,n.
It is easy to see that(20)∏i=1n‍Pr{ω∈Ω∣Xi(θi′)(ω)≥t}≤∏i=1n‍Pr{ω∈Ω∣Xi(θi,α)(ω)≥t}≤∏i=1n‍Pr{ω∈Ω∣Xi(θi′′)(ω)≥t};
that is,
(21)Pr{ω∈Ω∣min⁡{X1(θ1′)(ω),…,Xn(θn′)(ω)}≥t}≤Pr{ω∈Ω∣min⁡{X1(θ1,α)(ω),…,Xn(θn,α)(ω)}≥t}≤Pr{ω∈Ω∣min⁡{X1(θ1′′)(ω),…,Xn(θn′′)(ω)}≥t}.
Sinceθi,α are arbitrary points in Ai, i=1,2,…,n, by (21), we have
(22)PrαL{ω∈Ω∣X(θ)(ω)≥t}=Pr{ω∈Ω∣min⁡{X1(θ1′)(ω),…,Xn(θn′)(ω)}≥t}=∏i=1n‍Pr{ω∈Ω∣Xi(θi′)(ω)≥t},PrαU{ω∈Ω∣X(θ)(ω)≥t}=Pr{ω∈Ω∣min⁡{X1(θ1′′)(ω),…,Xn(θn′′)(ω)}≥t}=∏i=1n‍Pr{ω∈Ω∣Xi(θi′′)(ω)≥t}.
On the other hand, by (19), we have
(23)PrαL{ω∈Ω∣Xi(θi)(ω)≥t}=Pr{ω∈Ω∣Xi(θi′)(ω)≥t},PrαU{ω∈Ω∣Xi(θi)(ω)≥t}=Pr{ω∈Ω∣Xi(θi′′)(ω)≥t},i=1,2,…,n.
By (22) and (23), we have
(24)PrαL{ω∈Ω∣X(θ)(ω)≥t}=∏i=1n‍PrαL{ω∈Ω∣Xi(θi)(ω)≥t},PrαU{ω∈Ω∣X(θ)(ω)≥t}=∏i=1n‍PrαU{ω∈Ω∣Xi(θi)(ω)≥t}.
By (15) and (24), we have
(25)R(t)=12∫01(∏i=1n‍PrαL{ω∈Ω∣Xi(θi)(ω)≥t}‍+∏i=1n‍PrαU{ω∈Ω∣Xi(θi)(ω)≥t})dα=12∫01‍([∏i=1n‍Pr{ω∈Ω∣Xi(θi)(ω)≥t}]αL+[∏i=1n‍Pr{ω∈Ω∣Xi(θi)(ω)≥t}]αU)dα=E[∏i=1n‍Pr{ω∈Ω∣Xi(θi)(ω)≥t}],
which completes the proof.Remark 20.
IfXi, i=1,2,…,n, degenerate to random variables, the result in Theorem 19 degenerates to the form
(26)R(t)=Pr{ω∈Ω∣min⁡{X1(ω1),X2(ω2),…,Xn(ωn)}≥t}=∏i=1n‍Pr{Xi≥t},
which is consistent with the result in stochastic case (see Barlow and Proschan [1]).Remark 21.
IfXi, i=1,2,…,n, degenerate to fuzzy variables, the result in Theorem 19 degenerates to the form
(27)R(t)=Cr{θ∈Θ∣min⁡{X1(θ1),X2(θ2),…,Xn(θn)}≥t}=min⁡1≤i≤nCr{Xi>t},
which is consistent with the result in fuzzy case (see Liu and Zhu [49]).Theorem 22.
LetXi, i=1,2,…,n, be random fuzzy variables. Assume that the α-pessimistic values and the α-optimistic values of E[Xi(θi)], i=1,2,…,n, are continuous at the point α, α∈(0,1]. The MTTF of the series system is
(28)MTTF=∫0+∞‍R(t)dt.Proof.
By Definition18 and Proposition 7, we have
(29)MTTF=∫0+∞Cr{θ∈Θ∣E[X(θ)]≥r}dr‍=12∫01‍(E[X(θ)]αL+E[X(θ)]αU)dα.
By (21), we know that
(30)min⁡{X1(θ1′)(ω),…,Xn(θn′)(ω)}≤dmin⁡{X1(θ1,α)(ω),…,Xn(θn,α)(ω)}≤dmin⁡{X1(θ1′′)(ω),…,Xn(θn′′)(ω)}.
It follows from Remark16 that
(31)E[min⁡{X1(θ1′)(ω),…,Xn(θn′)(ω)}]≤E[min⁡{X1(θ1,α)(ω),…,Xn(θn,α)(ω)}]≤E[min⁡{X1(θ1′′)(ω),…,Xn(θn′′)(ω)}].
Sinceθi,α are arbitrary points in Ai, i=1,2,…,n, we have
(32)E[X(θ)]αL=E[min⁡{X1(θ1′)(ω),…,Xn(θn′)(ω)}]=∫0+∞‍∏i=1n‍Pr{ω∈Ω∣Xi(θi′)(ω)>t}dt,E[X(θ)]αU=E[min⁡{X1(θ1′′)(ω),…,Xn(θn′′)(ω)}]=∫0+∞‍∏i=1n‍Pr{ω∈Ω∣Xi(θi′′)(ω)>t}dt.
It follows from (23), (29), and (32) that
(33)MTTF=12∫01‍{∫0+∞‍∏i=1n‍Pr{ω∈Ω∣Xi(θi′)(ω)>t}dt+∫0+∞‍∏i=1n‍Pr{ω∈Ω∣Xi(θi′′)(ω)>t}dt}dα=12∫01‍∫0+∞‍{∏i=1n‍PrαL{ω∈Ω∣Xi(θi)(ω)≥t}+∏i=1n‍PrαU{ω∈Ω∣Xi(θi)(ω)≥t}}dtdα=∫0+∞‍12∫01‍{∏i=1n‍PrαL{ω∈Ω∣Xi(θi)(ω)≥t}+∏i=1n‍PrαU{ω∈Ω∣Xi(θi)(ω)≥t}}dαdt=∫0+∞‍R(t)dt.
The theorem is proved.Remark 23.
IfXi, i=1,2,…,n, degenerate to random variables, the result in Theorem 22 degenerates to the form
(34)MTTF=∫0+∞‍R(t)dt,
in which R(t) is the reliability of series system in stochastic case.Remark 24.
IfXi, i=1,2,…,n, degenerate to fuzzy variables, the result in Theorem 22 degenerates to the form
(35)MTTF=∫0+∞‍R(t)dt,
in which R(t) is the reliability of series system in fuzzy case.Example 25.
If the random fuzzy variableXi~EXP(λi), where λi is a fuzzy variable on (Θi,P(Θi),Cri), i=1,2,…,n, then we can arrive at
(36)PrαL{ω∈Ω∣X(θ)(ω)≥t}=∏i=1n‍e-λi,αUt=e-∑i=1nλi,αUt,PrαU{ω∈Ω∣X(θ)(ω)≥t}=∏i=1n‍e-λi,αLt=e-∑i=1nλi,αLt.
By Theorems19 and 22, we have
(37)R(t)=e-∑i=1nEλit,MTTF=1∑i=1nEλi,
in which “E” is the expected value operator of fuzzy variable.


### 3.2. Reliability Analysis of Random Fuzzy Unrepairable Parallel Systems

Consider a parallel system composed ofn independent components. Let Xi be the lifetime of component i, which is a random fuzzy variable on the credibility space (Θi,P(Θi),Cri), i=1,2,…,n. Obviously, the lifetime of the parallel system is X=max⁡{X1,X2,…,Xn}, which is a random fuzzy variable on the product credibility space (Θ,P(Θ),Cr), where Θ=Θ1×Θ2×⋯×Θn and Cr=Cr1∧Cr2∧⋯∧Crn.Theorem 26.
LetXi, i=1,2,…,n, be random fuzzy variables. Assume that the α-pessimistic values and the α-optimistic values of E[Xi(θi)], i=1,2,…,n, are continuous at the point α, α∈(0,1]. The reliability of the random fuzzy parallel system is
(38)R(t)=1-E{∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi)(ω)≥t}]}.Proof.
By Definitions13 and 17 and Proposition 7, we have
(39)R(t)=Ch{X≥t}=∫01Cr{θ∈Θ∣Pr⁡{X(θ)≥t}≥p}dp‍=12∫01‍(PrαL{ω∈Ω∣X(θ)(ω)≥t}+PrαU{ω∈Ω∣X(θ)(ω)≥t})dα.
LetAi={θi∈Θi∣μ{θi}≥α}, i=1,2,…,n. Since the α-pessimistic values and the α-optimistic values of fuzzy variables E[Xi(θi)], θi∈Θi, i=1,2,…,n, are continuous at the point α, α∈(0,1], then there exist maximum and minimum values; that is, at least there exist points θi′,θi′′∈Ai such that
(40)E[Xi(θi′)]=E[Xi(θi)]αL,E[Xi(θi′′)]=E[Xi(θi)]αU.
For anyθi,α∈Ai, we have
(41)E[Xi(θi′)]≤E[Xi(θi,α)]≤E[Xi(θi′′)],i=1,2,…,n.
Hence, by Remark16 we have
(42)Xi(θi′)≤dXi(θi,α)≤dXi(θi′′),θi,α∈Ai,i=1,2,…,n.
It follows from Definition15 that
(43)Pr{ω∈Ω∣Xi(θi′)(ω)≥t}≤Pr{ω∈Ω∣Xi(θi,α)(ω)≥t}≤Pr{ω∈Ω∣Xi(θi′′)(ω)≥t},i=1,2,…,n.
It is easy to see that(44)1-∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi′)(ω)>t}]≤1-∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi,α)(ω)>t}]≤1-∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi′′)(ω)>t}];that is,
(45)Pr{ω∈Ω∣max⁡{X1(θ1′)(ω),…,Xn(θn′)(ω)}>t}≤Pr{ω∈Ω∣max⁡{X1(θ1,α)(ω),…,Xn(θn,α)(ω)}>t}≤Pr{ω∈Ω∣max⁡{X1(θ1′′)(ω),…,Xn(θn′′)(ω)}>t}.
Sinceθi,α are arbitrary points in Ai, i=1,2,…,n, by (45), we have
(46)PrαL{ω∈Ω∣X(θ)(ω)≥t}=Pr{ω∈Ω∣max⁡{X1(θ1′)(ω),…,Xn(θn′)(ω)}>t}=1-∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi′)(ω)>t}],PrαU{ω∈Ω∣X(θ)(ω)≥t}=Pr{ω∈Ω∣max⁡{X1(θ1′′)(ω),…,Xn(θn′′)(ω)}>t}=1-∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi′′)(ω)>t}].
On the other hand, by (19), we have
(47)PrαL{ω∈Ω∣Xi(θi)(ω)≥t}=Pr{ω∈Ω∣Xi(θi′)(ω)>t},PrαU{ω∈Ω∣Xi(θi)(ω)≥t}=Pr{ω∈Ω∣Xi(θi′′)(ω)>t},i=1,2,…,n.
By (46) and (47), we have
(48)PrαL{ω∈Ω∣X(θ)(ω)≥t}=1-∏i=1n‍[1-PrαL{ω∈Ω∣Xi(θi)(ω)≥t}],PrαU{ω∈Ω∣X(θ)(ω)≥t}=1-∏i=1n‍[1-PrαU{ω∈Ω∣Xi(θi)(ω)≥t}].
By (39) and (48) we have
(49)R(t)=12∫01‍(1-∏i=1n‍[1-PrαL{ω∈Ω∣Xi(θi)(ω)≥t}]+1-∏i=1n‍[1-PrαU{ω∈Ω∣Xi(θi)(ω)≥t}])dα=1-12×∫01‍({∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi)(ω)≥t}]}αU+{∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi)(ω)≥t}]}αL)dα=1-E{∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi)(ω)≥t}]},
which completes the proof.Remark 27.
IfXi, i=1,2,…,n, degenerate to random variables, the result in Theorem 26 degenerates to the form
(50)R(t)=Pr{ω∈Ω∣max⁡{X1(ω1),X2(ω2),…,Xn(ωn)}≥t}=1-∏i=1n‍[1-Pr{Xi≥t}],
which is consistent with the result in stochastic case (see Barlow and Proschan [1]).Remark 28.
IfXi, i=1,2,…,n, degenerate to fuzzy variables, the result in Theorem 26 degenerates to the form
(51)R(t)=Cr{θ∈Θ∣max⁡{X1(θ1),X2(θ2),…,Xn(θn)}≥t}=max⁡1≤i≤nCr{Xi>t},
which is consistent with the result in fuzzy case (see Liu and Zhu [49]).Theorem 29.
LetXi, i=1,2,…,n, be random fuzzy variables. Assume that the α-pessimistic values and the α-optimistic values of E[Xi(θi)], i=1,2,…,n, are continuous at the point α, α∈(0,1]. The MTTF of the parallel system is
(52)MTTF=∫0+∞‍R(t)dt.Proof.
By Definition18 and Proposition 7, we can see
(53)MTTF=∫0+∞Cr{θ∈Θ∣E[X(θ)]≥r}dr‍=12∫01‍(E[X(θ)]αL+E[X(θ)]αU)dα.
By (45), we know that
(54)max⁡{X1(θ1′)(ω),…,Xn(θn′)(ω)}≤dmax⁡{X1(θ1,α)(ω),…,Xn(θn,α)(ω)}≤dmax⁡{X1(θ1′′)(ω),…,Xn(θn′′)(ω)}.
It follows from Remark16 that
(55)E[max⁡{X1(θ1′)(ω),…,Xn(θn′)(ω)}]≤E[max⁡{X1(θ1,α)(ω),…,Xn(θn,α)(ω)}]≤E[max⁡{X1(θ1′′)(ω),…,Xn(θn′′)(ω)}].
Sinceθi,α are arbitrary points in Ai, i=1,2,…,n, we have
(56)E[X(θ)]αL=E[max⁡{X1(θ1′)(ω),…,Xn(θn′)(ω)}]=∫0+∞‍{1-∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi′)(ω)>t}]}dt,E[X(θ)]αU=E[max⁡{X1(θ1′′)(ω),…,Xn(θn′′)(ω)}]=∫0+∞‍{1-∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi′′)(ω)>t}]}dt.
It follows from (47), (53), and (56) that
(57)MTTF=12∫01‍{∏i=1n‍∫0+∞‍{1-∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi′)(ω)>t}]}dt+∫0+∞‍{1-∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi′′)×(ω)>tXi(θi′′)}]∏i=1n‍}dt}dα=12∫01‍{∫0+∞‍{1-∏i=1n‍[1-PrαL{ω∈Ω∣Xi(θi)(ω)≥t}]}dt+∫0+∞‍{1-∏i=1n‍[1-PrαU{ω∈Ω∣Xi(θi)×(ω)≥tXi(θi)}]∏i=1n‍}dt}dα=∫0+∞‍12∫01‍{1-∏i=1n‍[1-PrαL{ω∈Ω∣Xi(θi)(ω)≥t}]+1-∏i=1n‍[1-PrαU{ω∈Ω∣Xi(θi)×Xi(θi)(ω)≥t}]∏i=1n‍}dαdt=∫0+∞‍R(t)dt.
The theorem is proved.Remark 30.
IfXi, i=1,2,…,n, degenerate to random variables, the result in Theorem 29 degenerates to the form
(58)MTTF=∫0+∞‍R(t)dt,
in which R(t) is the reliability of parallel system in stochastic case.Remark 31.
IfXi, i=1,2,…,n, degenerate to fuzzy variables, the result in Theorem 29 degenerates to the form
(59)MTTF=∫0+∞‍R(t)dt, in which R(t) is the reliability of parallel system in fuzzy case.Example 32.
If the random fuzzy variableXi(λi)~EXP(λi), where λi is a fuzzy variable on (Θi,P(Θi),Cri), i=1,2,…,n, then we can arrive at
(60)PrαL{ω∈Ω∣X(θ)(ω)≥t}=1-∏i=1n‍[1-e-λi,αUt],PrαU{ω∈Ω∣X(θ)(ω)≥t}=1-∏i=1n‍[1-e-λi,αLt].
By Theorems26 and 29, we have
(61)R(t)=1-∏i=1n‍(1-e-Eλit),MTTF=∫0+∞‍{1-∏i=1n‍(1-e-Eλit)}dt,
in which “E” is the expected value operator of fuzzy variable.


### 3.3. Reliability Analysis of Random Fuzzy Unrepairable Series-Parallel Systems

Consider a series-parallel system which is a series system ofm subsystems; each subsystem is composed of n parallel components. Let Xij be the lifetime of component j in ith subsystem, which is a random fuzzy variable on the credibility space (Θij,P(Θij),Crij), i=1,2,…,m, j=1,2,…,n. We assume the components are mutually independent. It is easy to know that the lifetime of the series-parallel system is X=min⁡1≤i≤m(max⁡1≤j≤nXij), which is a random fuzzy variable on the product credibility space (Θ,P(Θ),Cr), where Θ=Θ11×Θ12×⋯×Θmn and Cr=Cr11∧Cr12∧⋯∧Crmn.Theorem 33.
LetXij, i=1,2,…,m, j=1,2,…,n, be random fuzzy variables. Assume that the α-pessimistic values and the α-optimistic values of E[Xij(θij)], i=1,2,…,m, j=1,2,…,n, are continuous at the point α, α∈(0,1]. The reliability of the series-parallel system is
(62)R(t)=E(∏i=1m‍{1-∏j=1n‍[1-Pr{ω∈Ω∣Xij(θij)(ω)≥t}]}).Remark 34.
IfXij, i=1,2,…,m, j=1,2,…,n, degenerate to random variables, the result in Theorem 33 degenerates to the form
(63)R(t)=∏i=1m‍{1-∏j=1n‍[1-Pr{Xij≥t}]},
which is consistent with the result in stochastic case (see Barlow and Proschan [1]).Remark 35.
IfXij, i=1,2,…,m, j=1,2,…,n, degenerate to fuzzy variables, the result in Theorem 33 degenerates to the form
(64)R(t)=Cr{min⁡1≤i≤m(max⁡1≤j≤nXij)≥t},
which is consistent with the result in fuzzy case (see Liu and Zhu [49]).Theorem 36.
LetXij, i=1,2,…,m, j=1,2,…,n, be random fuzzy variables. Assume that the α-pessimistic values and the α-optimistic values of E[Xij(θij)], i=1,2,…,m, j=1,2,…,n, are continuous at the point α, α∈(0,1]. The MTTF of the random fuzzy series-parallel system is
(65)MTTF=∫0+∞‍R(t)dt.Remark 37.
IfXij, i=1,2,…,m, j=1,2,…,n, degenerate to random variables, the result in Theorem 36 degenerates to the form
(66)MTTF=∫0+∞‍R(t)dt,
in which R(t) is the reliability of series-parallel system in stochastic case.Remark 38.
IfXij, i=1,2,…,m, j=1,2,…,n, degenerate to fuzzy variables, the result in Theorem 36 degenerates to the form
(67)MTTF=∫0+∞‍R(t)dt,
in which R(t) is the reliability of series-parallel system in fuzzy case.


### 3.4. Reliability Analysis of Random Fuzzy Unrepairable Parallel-Series Systems

Consider a parallel-series system which is a parallel system ofm subsystems; each subsystem is composed of n series components. Let Xij be the lifetime of component j in ith subsystem, which is a random fuzzy variable on the credibility space (Θij,P(Θij),Crij), i=1,2,…,m, j=1,2,…,n. We assume the components are mutually independent. It is easy to know that the lifetime of the parallel-series system is X=max⁡1≤i≤m(min⁡1≤j≤nXij), which is a random fuzzy variable on the product credibility space (Θ,P(Θ),Cr), where Θ=Θ11×Θ12×⋯×Θmn and Cr=Cr11∧Cr12∧⋯∧Crmn.Theorem 39.
LetXij, i=1,2,…,m, j=1,2,…,n, be random fuzzy variables. Assume that the α-pessimistic values and the α-optimistic values of E[Xij(θij)], i=1,2,…,m, j=1,2,…,n, are continuous at the point α, α∈(0,1]. The reliability of the parallel-series system is
(68)R(t)=1-E{∏i=1m‍[1-∏j=1n‍Pr{ω∈Ω∣Xij(θij)(ω)≥t}]}.Remark 40.
IfXij, i=1,2,…,m, j=1,2,…,n, degenerate to random variables, the result in Theorem 39 degenerates to the form
(69)R(t)=1-∏i=1m‍[1-∏j=1n‍Pr{Xij≥t}],
which is consistent with the result in stochastic case (see Barlow and Proschan [1]).Remark 41.
IfXij, i=1,2,…,m, j=1,2,…,n, degenerate to fuzzy variables, the result in Theorem 39 degenerates to the form
(70)R(t)=Cr{max⁡1≤i≤m(min⁡1≤j≤nXij)≥t},
which is consistent with the result in fuzzy case (see Liu and Zhu [49]).Theorem 42.
LetXij, i=1,2,…,m, j=1,2,…,n, be random fuzzy variables. Assume that the α-pessimistic values and the α-optimistic values of E[Xij(θij)], i=1,2,…,m, j=1,2,…,n, are continuous at the point α, α∈(0,1]. The MTTF of the random fuzzy parallel-series system is
(71)MTTF=∫0+∞‍R(t)dt.Remark 43.
IfXij, i=1,2,…,m, j=1,2,…,n, degenerate to random variables, the result in Theorem 42 degenerates to the form
(72)MTTF=∫0+∞‍R(t)dt,
in which R(t) is the reliability of parallel-series system in stochastic case.Remark 44.
IfXij, i=1,2,…,m, j=1,2,…,n, degenerate to fuzzy variables, the result in Theorem 42 degenerates to the form
(73)MTTF=∫0+∞‍R(t)dt,
in which R(t) is the reliability of parallel-series system in fuzzy case.


### 3.5. Reliability Analysis of Random fuzzy Unrepairable Cold Standby Systems

Consider a cold standby system composed ofn independent components, in which only one component is in operation. If the operating component fails, it will be replaced by another component. Suppose that the failed components are unrepairable and the components in standby do not deteriorate. The failure state of the system occurs only when there is no operative component left. We also assume that the switching mechanism is absolutely reliable. Let Xi be the lifetime of component i, which is a random fuzzy variable on the credibility space (Θi,P(Θi),Cri), i=1,2,…,n. The lifetime of the cold standby system can be expressed by the sum of the reliability of n components; that is, X=X1+X2+⋯+Xn, which is a random fuzzy variable on the product credibility space (Θ,P(Θ),Cr), where Θ=Θ1×Θ2×⋯×Θn and Cr=Cr1∧Cr2∧⋯∧Crn.Theorem 45.
LetXi, i=1,2,…,n, be random fuzzy variables. Assume that the α-pessimistic values and the α-optimistic values of E[Xi(θi)], i=1,2,…,n, are continuous at the point α, α∈(0,1]. The reliability of random fuzzy cold standby system is
(74)R(t)=E[Pr{ω∈Ω∣X1(θ1)(ω)+⋯+Xn(θn)(ω)≥t}].Proof.
By Definitions13 and 17 and Proposition 7, we have
(75)R(t)=Ch{X≥t}=∫01Cr{θ∈Θ∣Pr⁡{X(θ)≥t}≥p}dp‍=12∫01(PrαL{ω∈Ω∣X(θ)(ω)≥t}‍+PrαU{ω∈Ω∣X(θ)(ω)≥t})dα=12∫01‍(PrαL{ω∈Ω∣X1(θ1)(ω)+⋯+Xn(θn)(ω)≥t}+PrαU{ω∈Ω∣X1(θ1)(ω)PrαL+⋯+Xn(θn)(ω)≥t})dα=E[Pr{ω∈Ω∣X1(θ1)(ω)+⋯+Xn(θn)(ω)≥t}],
which completes the proof.Remark 46.
IfXi, i=1,2,…,n, degenerate to random variables, the result in Theorem 45 degenerates to the form
(76)R(t)=Pr{X1+X2+⋯+Xn≥t},
which is consistent with the result in stochastic case (see Barlow and Proschan [1]).Remark 47.
IfXi, i=1,2,…,n, degenerate to fuzzy variables, the result in Theorem 45 degenerates to the form
(77)R(t)=Cr{X1+X2+⋯+Xn≥t},
which is consistent with the result in fuzzy case (see Liu and Zhu [49]).Theorem 48.
LetXi, i=1,2,…,n, be random fuzzy variables. Assume that the α-pessimistic values and the α-optimistic values of E[Xi(θi)], i=1,2,…,n, are continuous at the point α, α∈(0,1]. The MTTF of the cold standby system is
(78)MTTF=∫0+∞‍R(t)dt.Proof.
By Definition18 and Proposition 7, we can see
(79)MTTF=∫0+∞Cr{θ∈Θ∣E[X(θ)]≥r}dr‍=12∫01‍(E[X(θ)]αL+E[X(θ)]αU)dα.
LetAi={θi∈Θi∣μ{θi}≥α}, i=1,2,…,n. Since the α-pessimistic values and the α-optimistic values of fuzzy variables E[Xi(θi)], θi∈Θi, i=1,2,…,n, are continuous at the point α, α∈(0,1], then there exist maximum and minimum values; that is, at least there exist points θi′,θi′′∈Ai such that
(80)E[Xi(θi′)]=E[Xi(θi)]αL,E[Xi(θi′′)]=E[Xi(θi)]αU.
For anyθi,α∈Ai, i=1,2,…,n, we have
(81)E[Xi(θi′)]≤E[Xi(θi,α)]≤E[Xi(θi′′)].
Hence, by Remark16 we have
(82)Xi(θi′)≤dXi(θi,α)≤dXi(θi′′),θi,α∈Ai,i=1,2,…,n.
Then(83)X1(θ1′)+⋯+Xn(θn′)≤dX1(θ1,α)+⋯+Xn(θn,α)≤dX1(θ1′′)+⋯+Xn(θn′′).
It follows from Remark 16 that
(84)E[X1(θ1′)+⋯+Xn(θn′)]≤E[X1(θ1,α)+⋯+Xn(θn,α)]≤E[X1(θ1′′)+⋯+Xn(θn′′)].
Sinceθi,α are arbitrary points in Ai, i=1,2,…,n, we have
(85)E[X(θ)]αL=E[X1(θ1′)+⋯+Xn(θn′)]=∫0+∞‍Pr{ω∈Ω∣X1(θ1′)(ω)+⋯+Xn(θn′)(ω)≥t}dt,E[X(θ)]αU=E[X1(θ1′′)+⋯+Xn(θn′′)]=∫0+∞‍Pr{ω∈Ω∣X1(θ1′′)(ω)+⋯+Xn(θn′′)(ω)≥t}dt.
It follows from (79) and (85) that
(86)MTTF=12∫01‍{∫0+∞‍Pr{ω∈Ω∣X1(θ1′)(ω)+⋯+Xn(θn′)(ω)≥t}dt+∫0+∞‍Pr{ω∈Ω∣X1(θ1′′)(ω)+⋯+Xn(θn′′)X1(θ1′′)×(ω)≥t}dt∫0+∞‍}dα=12∫01‍∫0+∞‍{Pr{ω∈Ω∣X1(θ1′)(ω)+⋯+Xn(θn′)(ω)≥t}+Pr{ω∈Ω∣X1(θ1′′)(ω)+⋯+Xn(θn′′)(ω)≥t}}dtdα.
On the other hand, by (83) and Definition 15, we have
(87)Pr{ω∈Ω∣X1(θ1′)+⋯+Xn(θn′)≥t}≤Pr{ω∈Ω∣X1(θ1,α)+⋯+Xn(θn,α)≥t}≤Pr{ω∈Ω∣X1(θ1′′)+⋯+Xn(θn′′)≥t}.
Sinceθi,α are arbitrary points in Ai, i=1,2,…,n, we have
(88)PrαL{ω∈Ω∣X1(θ1)(ω)+⋯+Xn(θn)(ω)≥t}=Pr{ω∈Ω∣X1(θ1′)+⋯+Xn(θn′)≥t},PrαU{ω∈Ω∣X1(θ1)(ω)+⋯+Xn(θn)(ω)≥t}=Pr{ω∈Ω∣X1(θ1′′)+⋯+Xn(θn′′)≥t}.
It follows from (86) and (88) that
(89)MTTF=12∫01‍∫0+∞‍{PrαL{ω∈Ω∣X1(θ1)(ω)+⋯+Xn(θn)(ω)≥t}+PrαU{ω∈Ω∣X1(θ1)(ω)+⋯+Xn(θn)(ω)≥t}PrαL}dtdα=∫0+∞‍12∫01‍{PrαL{ω∈Ω∣X1(θ1)(ω)+⋯+Xn(θn)(ω)≥t}+PrαU{ω∈Ω∣X1(θ1)(ω)+⋯+Xn(θn)(ω)≥t}PrαL}dαdt=∫0+∞‍R(t)dt.
The theorem is proved.Remark 49.
IfXi, i=1,2,…,n, degenerate to random variables, the result in Theorem 48 degenerates to the form
(90)MTTF=∫0+∞‍R(t)dt,
in which R(t) is the reliability of cold standby system in stochastic case.Remark 50.
IfXi, i=1,2,…,n, degenerate to fuzzy variables, the result in Theorem 48 degenerates to the form
(91)MTTF=∫0+∞‍R(t)dt,
in which R(t) is the reliability of cold standby system in fuzzy case.


## 3.1. Reliability Analysis of Random Fuzzy Unrepairable Series Systems

Consider a series system composed ofn independent components. Let Xi be the lifetime of component i, which is a random fuzzy variable on the credibility space (Θi,P(Θi),Cri), i=1,2,…,n. Obviously, the lifetime of the series system is X=min⁡{X1,X2,…,Xn}, which is a random fuzzy variable on the product credibility space (Θ,P(Θ),Cr), where Θ=Θ1×Θ2×⋯×Θn and Cr=Cr1∧Cr2∧⋯∧Crn.Theorem 19.
LetXi, i=1,2,…,n be random fuzzy variables. Assume that the α-pessimistic values and the α-optimistic values of E[Xi(θi)], i=1,2,…,n, are continuous at the point α, α∈(0,1]. The reliability of random fuzzy series system is
(14)R(t)=E[∏i=1n‍Pr{ω∈Ω∣Xi(θi)(ω)≥t}].Proof.
By Definitions13 and 17 and Proposition 7, we have
(15)R(t)=Ch{X≥t}=∫01Cr{θ∈Θ∣Pr⁡{X(θ)≥t}≥p}dp‍=12∫01‍(PrαL{ω∈Ω∣X(θ)(ω)≥t}+PrαU{ω∈Ω∣X(θ)(ω)≥t})dα.
LetAi={θi∈Θi∣μ{θi}≥α}, i=1,2,…,n. Since the α-pessimistic values and the α-optimistic values of fuzzy variables E[Xi(θi)], θi∈Θi, i=1,2,…,n, are continuous at the point α, α∈(0,1], then there exist maximum and minimum values; that is, at least there exist points θi′,θi′′∈Ai such that
(16)E[Xi(θi′)]=E[Xi(θi)]αL,E[Xi(θi′′)]=E[Xi(θi)]αU.
For anyθi,α∈Ai, we have
(17)E[Xi(θi′)]≤E[Xi(θi,α)]≤E[Xi(θi′′)],i=1,2,…,n.
Hence, by Remark16, we have
(18)Xi(θi′)≤dXi(θi,α)≤dXi(θi′′),θi,α∈Ai,i=1,2,…,n.
It follows from Definition15 that
(19)Pr{ω∈Ω∣Xi(θi′)(ω)≥t}≤Pr{ω∈Ω∣Xi(θi,α)(ω)≥t}≤Pr{ω∈Ω∣Xi(θi′′)(ω)≥t},i=1,2,…,n.
It is easy to see that(20)∏i=1n‍Pr{ω∈Ω∣Xi(θi′)(ω)≥t}≤∏i=1n‍Pr{ω∈Ω∣Xi(θi,α)(ω)≥t}≤∏i=1n‍Pr{ω∈Ω∣Xi(θi′′)(ω)≥t};
that is,
(21)Pr{ω∈Ω∣min⁡{X1(θ1′)(ω),…,Xn(θn′)(ω)}≥t}≤Pr{ω∈Ω∣min⁡{X1(θ1,α)(ω),…,Xn(θn,α)(ω)}≥t}≤Pr{ω∈Ω∣min⁡{X1(θ1′′)(ω),…,Xn(θn′′)(ω)}≥t}.
Sinceθi,α are arbitrary points in Ai, i=1,2,…,n, by (21), we have
(22)PrαL{ω∈Ω∣X(θ)(ω)≥t}=Pr{ω∈Ω∣min⁡{X1(θ1′)(ω),…,Xn(θn′)(ω)}≥t}=∏i=1n‍Pr{ω∈Ω∣Xi(θi′)(ω)≥t},PrαU{ω∈Ω∣X(θ)(ω)≥t}=Pr{ω∈Ω∣min⁡{X1(θ1′′)(ω),…,Xn(θn′′)(ω)}≥t}=∏i=1n‍Pr{ω∈Ω∣Xi(θi′′)(ω)≥t}.
On the other hand, by (19), we have
(23)PrαL{ω∈Ω∣Xi(θi)(ω)≥t}=Pr{ω∈Ω∣Xi(θi′)(ω)≥t},PrαU{ω∈Ω∣Xi(θi)(ω)≥t}=Pr{ω∈Ω∣Xi(θi′′)(ω)≥t},i=1,2,…,n.
By (22) and (23), we have
(24)PrαL{ω∈Ω∣X(θ)(ω)≥t}=∏i=1n‍PrαL{ω∈Ω∣Xi(θi)(ω)≥t},PrαU{ω∈Ω∣X(θ)(ω)≥t}=∏i=1n‍PrαU{ω∈Ω∣Xi(θi)(ω)≥t}.
By (15) and (24), we have
(25)R(t)=12∫01(∏i=1n‍PrαL{ω∈Ω∣Xi(θi)(ω)≥t}‍+∏i=1n‍PrαU{ω∈Ω∣Xi(θi)(ω)≥t})dα=12∫01‍([∏i=1n‍Pr{ω∈Ω∣Xi(θi)(ω)≥t}]αL+[∏i=1n‍Pr{ω∈Ω∣Xi(θi)(ω)≥t}]αU)dα=E[∏i=1n‍Pr{ω∈Ω∣Xi(θi)(ω)≥t}],
which completes the proof.Remark 20.
IfXi, i=1,2,…,n, degenerate to random variables, the result in Theorem 19 degenerates to the form
(26)R(t)=Pr{ω∈Ω∣min⁡{X1(ω1),X2(ω2),…,Xn(ωn)}≥t}=∏i=1n‍Pr{Xi≥t},
which is consistent with the result in stochastic case (see Barlow and Proschan [1]).Remark 21.
IfXi, i=1,2,…,n, degenerate to fuzzy variables, the result in Theorem 19 degenerates to the form
(27)R(t)=Cr{θ∈Θ∣min⁡{X1(θ1),X2(θ2),…,Xn(θn)}≥t}=min⁡1≤i≤nCr{Xi>t},
which is consistent with the result in fuzzy case (see Liu and Zhu [49]).Theorem 22.
LetXi, i=1,2,…,n, be random fuzzy variables. Assume that the α-pessimistic values and the α-optimistic values of E[Xi(θi)], i=1,2,…,n, are continuous at the point α, α∈(0,1]. The MTTF of the series system is
(28)MTTF=∫0+∞‍R(t)dt.Proof.
By Definition18 and Proposition 7, we have
(29)MTTF=∫0+∞Cr{θ∈Θ∣E[X(θ)]≥r}dr‍=12∫01‍(E[X(θ)]αL+E[X(θ)]αU)dα.
By (21), we know that
(30)min⁡{X1(θ1′)(ω),…,Xn(θn′)(ω)}≤dmin⁡{X1(θ1,α)(ω),…,Xn(θn,α)(ω)}≤dmin⁡{X1(θ1′′)(ω),…,Xn(θn′′)(ω)}.
It follows from Remark16 that
(31)E[min⁡{X1(θ1′)(ω),…,Xn(θn′)(ω)}]≤E[min⁡{X1(θ1,α)(ω),…,Xn(θn,α)(ω)}]≤E[min⁡{X1(θ1′′)(ω),…,Xn(θn′′)(ω)}].
Sinceθi,α are arbitrary points in Ai, i=1,2,…,n, we have
(32)E[X(θ)]αL=E[min⁡{X1(θ1′)(ω),…,Xn(θn′)(ω)}]=∫0+∞‍∏i=1n‍Pr{ω∈Ω∣Xi(θi′)(ω)>t}dt,E[X(θ)]αU=E[min⁡{X1(θ1′′)(ω),…,Xn(θn′′)(ω)}]=∫0+∞‍∏i=1n‍Pr{ω∈Ω∣Xi(θi′′)(ω)>t}dt.
It follows from (23), (29), and (32) that
(33)MTTF=12∫01‍{∫0+∞‍∏i=1n‍Pr{ω∈Ω∣Xi(θi′)(ω)>t}dt+∫0+∞‍∏i=1n‍Pr{ω∈Ω∣Xi(θi′′)(ω)>t}dt}dα=12∫01‍∫0+∞‍{∏i=1n‍PrαL{ω∈Ω∣Xi(θi)(ω)≥t}+∏i=1n‍PrαU{ω∈Ω∣Xi(θi)(ω)≥t}}dtdα=∫0+∞‍12∫01‍{∏i=1n‍PrαL{ω∈Ω∣Xi(θi)(ω)≥t}+∏i=1n‍PrαU{ω∈Ω∣Xi(θi)(ω)≥t}}dαdt=∫0+∞‍R(t)dt.
The theorem is proved.Remark 23.
IfXi, i=1,2,…,n, degenerate to random variables, the result in Theorem 22 degenerates to the form
(34)MTTF=∫0+∞‍R(t)dt,
in which R(t) is the reliability of series system in stochastic case.Remark 24.
IfXi, i=1,2,…,n, degenerate to fuzzy variables, the result in Theorem 22 degenerates to the form
(35)MTTF=∫0+∞‍R(t)dt,
in which R(t) is the reliability of series system in fuzzy case.Example 25.
If the random fuzzy variableXi~EXP(λi), where λi is a fuzzy variable on (Θi,P(Θi),Cri), i=1,2,…,n, then we can arrive at
(36)PrαL{ω∈Ω∣X(θ)(ω)≥t}=∏i=1n‍e-λi,αUt=e-∑i=1nλi,αUt,PrαU{ω∈Ω∣X(θ)(ω)≥t}=∏i=1n‍e-λi,αLt=e-∑i=1nλi,αLt.
By Theorems19 and 22, we have
(37)R(t)=e-∑i=1nEλit,MTTF=1∑i=1nEλi,
in which “E” is the expected value operator of fuzzy variable.


## 3.2. Reliability Analysis of Random Fuzzy Unrepairable Parallel Systems

Consider a parallel system composed ofn independent components. Let Xi be the lifetime of component i, which is a random fuzzy variable on the credibility space (Θi,P(Θi),Cri), i=1,2,…,n. Obviously, the lifetime of the parallel system is X=max⁡{X1,X2,…,Xn}, which is a random fuzzy variable on the product credibility space (Θ,P(Θ),Cr), where Θ=Θ1×Θ2×⋯×Θn and Cr=Cr1∧Cr2∧⋯∧Crn.Theorem 26.
LetXi, i=1,2,…,n, be random fuzzy variables. Assume that the α-pessimistic values and the α-optimistic values of E[Xi(θi)], i=1,2,…,n, are continuous at the point α, α∈(0,1]. The reliability of the random fuzzy parallel system is
(38)R(t)=1-E{∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi)(ω)≥t}]}.Proof.
By Definitions13 and 17 and Proposition 7, we have
(39)R(t)=Ch{X≥t}=∫01Cr{θ∈Θ∣Pr⁡{X(θ)≥t}≥p}dp‍=12∫01‍(PrαL{ω∈Ω∣X(θ)(ω)≥t}+PrαU{ω∈Ω∣X(θ)(ω)≥t})dα.
LetAi={θi∈Θi∣μ{θi}≥α}, i=1,2,…,n. Since the α-pessimistic values and the α-optimistic values of fuzzy variables E[Xi(θi)], θi∈Θi, i=1,2,…,n, are continuous at the point α, α∈(0,1], then there exist maximum and minimum values; that is, at least there exist points θi′,θi′′∈Ai such that
(40)E[Xi(θi′)]=E[Xi(θi)]αL,E[Xi(θi′′)]=E[Xi(θi)]αU.
For anyθi,α∈Ai, we have
(41)E[Xi(θi′)]≤E[Xi(θi,α)]≤E[Xi(θi′′)],i=1,2,…,n.
Hence, by Remark16 we have
(42)Xi(θi′)≤dXi(θi,α)≤dXi(θi′′),θi,α∈Ai,i=1,2,…,n.
It follows from Definition15 that
(43)Pr{ω∈Ω∣Xi(θi′)(ω)≥t}≤Pr{ω∈Ω∣Xi(θi,α)(ω)≥t}≤Pr{ω∈Ω∣Xi(θi′′)(ω)≥t},i=1,2,…,n.
It is easy to see that(44)1-∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi′)(ω)>t}]≤1-∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi,α)(ω)>t}]≤1-∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi′′)(ω)>t}];that is,
(45)Pr{ω∈Ω∣max⁡{X1(θ1′)(ω),…,Xn(θn′)(ω)}>t}≤Pr{ω∈Ω∣max⁡{X1(θ1,α)(ω),…,Xn(θn,α)(ω)}>t}≤Pr{ω∈Ω∣max⁡{X1(θ1′′)(ω),…,Xn(θn′′)(ω)}>t}.
Sinceθi,α are arbitrary points in Ai, i=1,2,…,n, by (45), we have
(46)PrαL{ω∈Ω∣X(θ)(ω)≥t}=Pr{ω∈Ω∣max⁡{X1(θ1′)(ω),…,Xn(θn′)(ω)}>t}=1-∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi′)(ω)>t}],PrαU{ω∈Ω∣X(θ)(ω)≥t}=Pr{ω∈Ω∣max⁡{X1(θ1′′)(ω),…,Xn(θn′′)(ω)}>t}=1-∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi′′)(ω)>t}].
On the other hand, by (19), we have
(47)PrαL{ω∈Ω∣Xi(θi)(ω)≥t}=Pr{ω∈Ω∣Xi(θi′)(ω)>t},PrαU{ω∈Ω∣Xi(θi)(ω)≥t}=Pr{ω∈Ω∣Xi(θi′′)(ω)>t},i=1,2,…,n.
By (46) and (47), we have
(48)PrαL{ω∈Ω∣X(θ)(ω)≥t}=1-∏i=1n‍[1-PrαL{ω∈Ω∣Xi(θi)(ω)≥t}],PrαU{ω∈Ω∣X(θ)(ω)≥t}=1-∏i=1n‍[1-PrαU{ω∈Ω∣Xi(θi)(ω)≥t}].
By (39) and (48) we have
(49)R(t)=12∫01‍(1-∏i=1n‍[1-PrαL{ω∈Ω∣Xi(θi)(ω)≥t}]+1-∏i=1n‍[1-PrαU{ω∈Ω∣Xi(θi)(ω)≥t}])dα=1-12×∫01‍({∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi)(ω)≥t}]}αU+{∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi)(ω)≥t}]}αL)dα=1-E{∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi)(ω)≥t}]},
which completes the proof.Remark 27.
IfXi, i=1,2,…,n, degenerate to random variables, the result in Theorem 26 degenerates to the form
(50)R(t)=Pr{ω∈Ω∣max⁡{X1(ω1),X2(ω2),…,Xn(ωn)}≥t}=1-∏i=1n‍[1-Pr{Xi≥t}],
which is consistent with the result in stochastic case (see Barlow and Proschan [1]).Remark 28.
IfXi, i=1,2,…,n, degenerate to fuzzy variables, the result in Theorem 26 degenerates to the form
(51)R(t)=Cr{θ∈Θ∣max⁡{X1(θ1),X2(θ2),…,Xn(θn)}≥t}=max⁡1≤i≤nCr{Xi>t},
which is consistent with the result in fuzzy case (see Liu and Zhu [49]).Theorem 29.
LetXi, i=1,2,…,n, be random fuzzy variables. Assume that the α-pessimistic values and the α-optimistic values of E[Xi(θi)], i=1,2,…,n, are continuous at the point α, α∈(0,1]. The MTTF of the parallel system is
(52)MTTF=∫0+∞‍R(t)dt.Proof.
By Definition18 and Proposition 7, we can see
(53)MTTF=∫0+∞Cr{θ∈Θ∣E[X(θ)]≥r}dr‍=12∫01‍(E[X(θ)]αL+E[X(θ)]αU)dα.
By (45), we know that
(54)max⁡{X1(θ1′)(ω),…,Xn(θn′)(ω)}≤dmax⁡{X1(θ1,α)(ω),…,Xn(θn,α)(ω)}≤dmax⁡{X1(θ1′′)(ω),…,Xn(θn′′)(ω)}.
It follows from Remark16 that
(55)E[max⁡{X1(θ1′)(ω),…,Xn(θn′)(ω)}]≤E[max⁡{X1(θ1,α)(ω),…,Xn(θn,α)(ω)}]≤E[max⁡{X1(θ1′′)(ω),…,Xn(θn′′)(ω)}].
Sinceθi,α are arbitrary points in Ai, i=1,2,…,n, we have
(56)E[X(θ)]αL=E[max⁡{X1(θ1′)(ω),…,Xn(θn′)(ω)}]=∫0+∞‍{1-∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi′)(ω)>t}]}dt,E[X(θ)]αU=E[max⁡{X1(θ1′′)(ω),…,Xn(θn′′)(ω)}]=∫0+∞‍{1-∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi′′)(ω)>t}]}dt.
It follows from (47), (53), and (56) that
(57)MTTF=12∫01‍{∏i=1n‍∫0+∞‍{1-∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi′)(ω)>t}]}dt+∫0+∞‍{1-∏i=1n‍[1-Pr{ω∈Ω∣Xi(θi′′)×(ω)>tXi(θi′′)}]∏i=1n‍}dt}dα=12∫01‍{∫0+∞‍{1-∏i=1n‍[1-PrαL{ω∈Ω∣Xi(θi)(ω)≥t}]}dt+∫0+∞‍{1-∏i=1n‍[1-PrαU{ω∈Ω∣Xi(θi)×(ω)≥tXi(θi)}]∏i=1n‍}dt}dα=∫0+∞‍12∫01‍{1-∏i=1n‍[1-PrαL{ω∈Ω∣Xi(θi)(ω)≥t}]+1-∏i=1n‍[1-PrαU{ω∈Ω∣Xi(θi)×Xi(θi)(ω)≥t}]∏i=1n‍}dαdt=∫0+∞‍R(t)dt.
The theorem is proved.Remark 30.
IfXi, i=1,2,…,n, degenerate to random variables, the result in Theorem 29 degenerates to the form
(58)MTTF=∫0+∞‍R(t)dt,
in which R(t) is the reliability of parallel system in stochastic case.Remark 31.
IfXi, i=1,2,…,n, degenerate to fuzzy variables, the result in Theorem 29 degenerates to the form
(59)MTTF=∫0+∞‍R(t)dt, in which R(t) is the reliability of parallel system in fuzzy case.Example 32.
If the random fuzzy variableXi(λi)~EXP(λi), where λi is a fuzzy variable on (Θi,P(Θi),Cri), i=1,2,…,n, then we can arrive at
(60)PrαL{ω∈Ω∣X(θ)(ω)≥t}=1-∏i=1n‍[1-e-λi,αUt],PrαU{ω∈Ω∣X(θ)(ω)≥t}=1-∏i=1n‍[1-e-λi,αLt].
By Theorems26 and 29, we have
(61)R(t)=1-∏i=1n‍(1-e-Eλit),MTTF=∫0+∞‍{1-∏i=1n‍(1-e-Eλit)}dt,
in which “E” is the expected value operator of fuzzy variable.


## 3.3. Reliability Analysis of Random Fuzzy Unrepairable Series-Parallel Systems

Consider a series-parallel system which is a series system ofm subsystems; each subsystem is composed of n parallel components. Let Xij be the lifetime of component j in ith subsystem, which is a random fuzzy variable on the credibility space (Θij,P(Θij),Crij), i=1,2,…,m, j=1,2,…,n. We assume the components are mutually independent. It is easy to know that the lifetime of the series-parallel system is X=min⁡1≤i≤m(max⁡1≤j≤nXij), which is a random fuzzy variable on the product credibility space (Θ,P(Θ),Cr), where Θ=Θ11×Θ12×⋯×Θmn and Cr=Cr11∧Cr12∧⋯∧Crmn.Theorem 33.
LetXij, i=1,2,…,m, j=1,2,…,n, be random fuzzy variables. Assume that the α-pessimistic values and the α-optimistic values of E[Xij(θij)], i=1,2,…,m, j=1,2,…,n, are continuous at the point α, α∈(0,1]. The reliability of the series-parallel system is
(62)R(t)=E(∏i=1m‍{1-∏j=1n‍[1-Pr{ω∈Ω∣Xij(θij)(ω)≥t}]}).Remark 34.
IfXij, i=1,2,…,m, j=1,2,…,n, degenerate to random variables, the result in Theorem 33 degenerates to the form
(63)R(t)=∏i=1m‍{1-∏j=1n‍[1-Pr{Xij≥t}]},
which is consistent with the result in stochastic case (see Barlow and Proschan [1]).Remark 35.
IfXij, i=1,2,…,m, j=1,2,…,n, degenerate to fuzzy variables, the result in Theorem 33 degenerates to the form
(64)R(t)=Cr{min⁡1≤i≤m(max⁡1≤j≤nXij)≥t},
which is consistent with the result in fuzzy case (see Liu and Zhu [49]).Theorem 36.
LetXij, i=1,2,…,m, j=1,2,…,n, be random fuzzy variables. Assume that the α-pessimistic values and the α-optimistic values of E[Xij(θij)], i=1,2,…,m, j=1,2,…,n, are continuous at the point α, α∈(0,1]. The MTTF of the random fuzzy series-parallel system is
(65)MTTF=∫0+∞‍R(t)dt.Remark 37.
IfXij, i=1,2,…,m, j=1,2,…,n, degenerate to random variables, the result in Theorem 36 degenerates to the form
(66)MTTF=∫0+∞‍R(t)dt,
in which R(t) is the reliability of series-parallel system in stochastic case.Remark 38.
IfXij, i=1,2,…,m, j=1,2,…,n, degenerate to fuzzy variables, the result in Theorem 36 degenerates to the form
(67)MTTF=∫0+∞‍R(t)dt,
in which R(t) is the reliability of series-parallel system in fuzzy case.


## 3.4. Reliability Analysis of Random Fuzzy Unrepairable Parallel-Series Systems

Consider a parallel-series system which is a parallel system ofm subsystems; each subsystem is composed of n series components. Let Xij be the lifetime of component j in ith subsystem, which is a random fuzzy variable on the credibility space (Θij,P(Θij),Crij), i=1,2,…,m, j=1,2,…,n. We assume the components are mutually independent. It is easy to know that the lifetime of the parallel-series system is X=max⁡1≤i≤m(min⁡1≤j≤nXij), which is a random fuzzy variable on the product credibility space (Θ,P(Θ),Cr), where Θ=Θ11×Θ12×⋯×Θmn and Cr=Cr11∧Cr12∧⋯∧Crmn.Theorem 39.
LetXij, i=1,2,…,m, j=1,2,…,n, be random fuzzy variables. Assume that the α-pessimistic values and the α-optimistic values of E[Xij(θij)], i=1,2,…,m, j=1,2,…,n, are continuous at the point α, α∈(0,1]. The reliability of the parallel-series system is
(68)R(t)=1-E{∏i=1m‍[1-∏j=1n‍Pr{ω∈Ω∣Xij(θij)(ω)≥t}]}.Remark 40.
IfXij, i=1,2,…,m, j=1,2,…,n, degenerate to random variables, the result in Theorem 39 degenerates to the form
(69)R(t)=1-∏i=1m‍[1-∏j=1n‍Pr{Xij≥t}],
which is consistent with the result in stochastic case (see Barlow and Proschan [1]).Remark 41.
IfXij, i=1,2,…,m, j=1,2,…,n, degenerate to fuzzy variables, the result in Theorem 39 degenerates to the form
(70)R(t)=Cr{max⁡1≤i≤m(min⁡1≤j≤nXij)≥t},
which is consistent with the result in fuzzy case (see Liu and Zhu [49]).Theorem 42.
LetXij, i=1,2,…,m, j=1,2,…,n, be random fuzzy variables. Assume that the α-pessimistic values and the α-optimistic values of E[Xij(θij)], i=1,2,…,m, j=1,2,…,n, are continuous at the point α, α∈(0,1]. The MTTF of the random fuzzy parallel-series system is
(71)MTTF=∫0+∞‍R(t)dt.Remark 43.
IfXij, i=1,2,…,m, j=1,2,…,n, degenerate to random variables, the result in Theorem 42 degenerates to the form
(72)MTTF=∫0+∞‍R(t)dt,
in which R(t) is the reliability of parallel-series system in stochastic case.Remark 44.
IfXij, i=1,2,…,m, j=1,2,…,n, degenerate to fuzzy variables, the result in Theorem 42 degenerates to the form
(73)MTTF=∫0+∞‍R(t)dt,
in which R(t) is the reliability of parallel-series system in fuzzy case.


## 3.5. Reliability Analysis of Random fuzzy Unrepairable Cold Standby Systems

Consider a cold standby system composed ofn independent components, in which only one component is in operation. If the operating component fails, it will be replaced by another component. Suppose that the failed components are unrepairable and the components in standby do not deteriorate. The failure state of the system occurs only when there is no operative component left. We also assume that the switching mechanism is absolutely reliable. Let Xi be the lifetime of component i, which is a random fuzzy variable on the credibility space (Θi,P(Θi),Cri), i=1,2,…,n. The lifetime of the cold standby system can be expressed by the sum of the reliability of n components; that is, X=X1+X2+⋯+Xn, which is a random fuzzy variable on the product credibility space (Θ,P(Θ),Cr), where Θ=Θ1×Θ2×⋯×Θn and Cr=Cr1∧Cr2∧⋯∧Crn.Theorem 45.
LetXi, i=1,2,…,n, be random fuzzy variables. Assume that the α-pessimistic values and the α-optimistic values of E[Xi(θi)], i=1,2,…,n, are continuous at the point α, α∈(0,1]. The reliability of random fuzzy cold standby system is
(74)R(t)=E[Pr{ω∈Ω∣X1(θ1)(ω)+⋯+Xn(θn)(ω)≥t}].Proof.
By Definitions13 and 17 and Proposition 7, we have
(75)R(t)=Ch{X≥t}=∫01Cr{θ∈Θ∣Pr⁡{X(θ)≥t}≥p}dp‍=12∫01(PrαL{ω∈Ω∣X(θ)(ω)≥t}‍+PrαU{ω∈Ω∣X(θ)(ω)≥t})dα=12∫01‍(PrαL{ω∈Ω∣X1(θ1)(ω)+⋯+Xn(θn)(ω)≥t}+PrαU{ω∈Ω∣X1(θ1)(ω)PrαL+⋯+Xn(θn)(ω)≥t})dα=E[Pr{ω∈Ω∣X1(θ1)(ω)+⋯+Xn(θn)(ω)≥t}],
which completes the proof.Remark 46.
IfXi, i=1,2,…,n, degenerate to random variables, the result in Theorem 45 degenerates to the form
(76)R(t)=Pr{X1+X2+⋯+Xn≥t},
which is consistent with the result in stochastic case (see Barlow and Proschan [1]).Remark 47.
IfXi, i=1,2,…,n, degenerate to fuzzy variables, the result in Theorem 45 degenerates to the form
(77)R(t)=Cr{X1+X2+⋯+Xn≥t},
which is consistent with the result in fuzzy case (see Liu and Zhu [49]).Theorem 48.
LetXi, i=1,2,…,n, be random fuzzy variables. Assume that the α-pessimistic values and the α-optimistic values of E[Xi(θi)], i=1,2,…,n, are continuous at the point α, α∈(0,1]. The MTTF of the cold standby system is
(78)MTTF=∫0+∞‍R(t)dt.Proof.
By Definition18 and Proposition 7, we can see
(79)MTTF=∫0+∞Cr{θ∈Θ∣E[X(θ)]≥r}dr‍=12∫01‍(E[X(θ)]αL+E[X(θ)]αU)dα.
LetAi={θi∈Θi∣μ{θi}≥α}, i=1,2,…,n. Since the α-pessimistic values and the α-optimistic values of fuzzy variables E[Xi(θi)], θi∈Θi, i=1,2,…,n, are continuous at the point α, α∈(0,1], then there exist maximum and minimum values; that is, at least there exist points θi′,θi′′∈Ai such that
(80)E[Xi(θi′)]=E[Xi(θi)]αL,E[Xi(θi′′)]=E[Xi(θi)]αU.
For anyθi,α∈Ai, i=1,2,…,n, we have
(81)E[Xi(θi′)]≤E[Xi(θi,α)]≤E[Xi(θi′′)].
Hence, by Remark16 we have
(82)Xi(θi′)≤dXi(θi,α)≤dXi(θi′′),θi,α∈Ai,i=1,2,…,n.
Then(83)X1(θ1′)+⋯+Xn(θn′)≤dX1(θ1,α)+⋯+Xn(θn,α)≤dX1(θ1′′)+⋯+Xn(θn′′).
It follows from Remark 16 that
(84)E[X1(θ1′)+⋯+Xn(θn′)]≤E[X1(θ1,α)+⋯+Xn(θn,α)]≤E[X1(θ1′′)+⋯+Xn(θn′′)].
Sinceθi,α are arbitrary points in Ai, i=1,2,…,n, we have
(85)E[X(θ)]αL=E[X1(θ1′)+⋯+Xn(θn′)]=∫0+∞‍Pr{ω∈Ω∣X1(θ1′)(ω)+⋯+Xn(θn′)(ω)≥t}dt,E[X(θ)]αU=E[X1(θ1′′)+⋯+Xn(θn′′)]=∫0+∞‍Pr{ω∈Ω∣X1(θ1′′)(ω)+⋯+Xn(θn′′)(ω)≥t}dt.
It follows from (79) and (85) that
(86)MTTF=12∫01‍{∫0+∞‍Pr{ω∈Ω∣X1(θ1′)(ω)+⋯+Xn(θn′)(ω)≥t}dt+∫0+∞‍Pr{ω∈Ω∣X1(θ1′′)(ω)+⋯+Xn(θn′′)X1(θ1′′)×(ω)≥t}dt∫0+∞‍}dα=12∫01‍∫0+∞‍{Pr{ω∈Ω∣X1(θ1′)(ω)+⋯+Xn(θn′)(ω)≥t}+Pr{ω∈Ω∣X1(θ1′′)(ω)+⋯+Xn(θn′′)(ω)≥t}}dtdα.
On the other hand, by (83) and Definition 15, we have
(87)Pr{ω∈Ω∣X1(θ1′)+⋯+Xn(θn′)≥t}≤Pr{ω∈Ω∣X1(θ1,α)+⋯+Xn(θn,α)≥t}≤Pr{ω∈Ω∣X1(θ1′′)+⋯+Xn(θn′′)≥t}.
Sinceθi,α are arbitrary points in Ai, i=1,2,…,n, we have
(88)PrαL{ω∈Ω∣X1(θ1)(ω)+⋯+Xn(θn)(ω)≥t}=Pr{ω∈Ω∣X1(θ1′)+⋯+Xn(θn′)≥t},PrαU{ω∈Ω∣X1(θ1)(ω)+⋯+Xn(θn)(ω)≥t}=Pr{ω∈Ω∣X1(θ1′′)+⋯+Xn(θn′′)≥t}.
It follows from (86) and (88) that
(89)MTTF=12∫01‍∫0+∞‍{PrαL{ω∈Ω∣X1(θ1)(ω)+⋯+Xn(θn)(ω)≥t}+PrαU{ω∈Ω∣X1(θ1)(ω)+⋯+Xn(θn)(ω)≥t}PrαL}dtdα=∫0+∞‍12∫01‍{PrαL{ω∈Ω∣X1(θ1)(ω)+⋯+Xn(θn)(ω)≥t}+PrαU{ω∈Ω∣X1(θ1)(ω)+⋯+Xn(θn)(ω)≥t}PrαL}dαdt=∫0+∞‍R(t)dt.
The theorem is proved.Remark 49.
IfXi, i=1,2,…,n, degenerate to random variables, the result in Theorem 48 degenerates to the form
(90)MTTF=∫0+∞‍R(t)dt,
in which R(t) is the reliability of cold standby system in stochastic case.Remark 50.
IfXi, i=1,2,…,n, degenerate to fuzzy variables, the result in Theorem 48 degenerates to the form
(91)MTTF=∫0+∞‍R(t)dt,
in which R(t) is the reliability of cold standby system in fuzzy case.


## 4. Numerical Examples

In this section, we give the reliability analysis of a lighting lamp system and a hi-fi system with random fuzzy lifetimes.Example 1.
Consider a lighting lamp system composed by lampL1, lamp L2, and lamp L3; see Figure 1. The lifetimes of lamp L1, lamp L2, and lamp L3 are denoted by X1, X2, and X3, respectively. The lifetime of the lighting lamp system is denoted by X. We also assume X1, X2, and X3 to be random fuzzy variables on (Θ,P(Θ),Cr) and Xi(λi)~EXP(λi), i=1,2,3, where λ1=(1,2,3), λ2=(0,1,2), and λ3=(0,1,2). Then we can arrive at
(92)λ1,αL=1+α,λ1,αU=3-α,λ2,αL=α,λ2,αU=2-α,λ3,αL=α,λ3,αU=2-α.Figure 1
A lighting lamp system.Then we have(93)PrαL{X≥t}=PrαL{X1≥t}×{1-(1-PrαL{X2≥t})(1-PrαL{X3≥t})}=e-(3-α)t{1-(1-e-(2-α)t)(1-e-(2-α)t)}=2e-(5-2α)t-e-(7-3α)t,PrαU{X≥t}=PrαU{X1≥t}×{1-(1-PrαU{X2≥t})(1-PrαU{X3≥t})}=e-(1+α)t{1-(1-e-αt)(1-e-αt)}=2e-(1+2α)t-e-(1+3α)t.So the reliability of the lighting lamp system is(94)R(t)=12∫01‍[2e-(5-2α)t-e-(7-3α)t+2e-(1+2α)t-e-(1+3α)t]dα=16t(2e-t-3e-5t+e-7t).The MTTF of the lighting lamp system is(95)MTTF=∫0+∞‍16t(2e-t-3e-5t+e-7t)dt≈0.4804.Example 2.
A more elaborate example is a stereo hi-fi system with the following components:(1) FM tuner; (2) record changer; (3) amplifier; (4) speaker A; and (5) speaker B. We consider the system functioning if we can obtain music (monaural or stereo) through FM or records. The system structure is illustrated in Figure 2. Let Xi be the random fuzzy lifetime of component i, i=1,2,…,5. The lifetime of the hi-fi system is denoted by X. We assume Xi(λi)~EXP(λi), i=1,2,…,5, where λ1=(1,2,3), λ2=(1.5,2.5,3.5), λ3=(0,1,2), λ4=(1,2,3), and λ5=(0.5,1.5,2.5).Figure 2
A hi-fi system.We first compute(96)λ1,αL=1+α,λ1,αU=3-α,λ2,αL=1.5+α,λ2,αU=3.5-α,λ3,αL=α,λ3,αU=2-α,λ4,αL=1+α,λ4,αU=3-α,λ5,αL=0.5+α,λ5,αU=2.5-α.Then we can arrive at(97)PrαL{X≥t}={1-(1-PrαL{X1≥t})(1-PrαL{X2≥t})}×PrαL{X3≥t}·{1-(1-PrαL{X4≥t})(1-PrαL{X5≥t})}={1-[1-e-(3-α)t][1-e-(3.5-α)t]}e-(2-α)t×{1-[1-e-(3-α)t][1-e-(2.5-α)t]}=e-(2-α)t[e-(3.5-α)t+e-(3-α)t-e-(6.5-2α)t]×[e-(3-α)t+e-(2.5-α)t-e-(5.5-2α)t],PrαU{X≥t}={1-(1-PrαU{X1≥t})(1-PrαU{X2≥t})}×PrαU{X3≥t}·{1-(1-PrαU{X4≥t})(1-PrαU{X5≥t})}={1-[1-e-(1+α)t][1-e-(1.5+α)t]}e-αt×{1-[1-e-(1+α)t][1-e-(0.5+α)t]}=e-αt[e-(1+α)t+e-(1.5+α)t-e-(2.5+2α)t]×[e-(1+α)t+e-(0.5+α)t-e-(1.5+2α)t].The reliability of the hi-fi system is(98)R(t)=12∫01‍{e-(2-α)t[e-(3.5-α)t+e-(3-α)t-e-(6.5-2α)t]×[e-(3-α)t+e-(2.5-α)t-e-(5.5-2α)t]+e-αt[e-(1+α)t+e-(1.5+α)t-e-(2.5+2α)t]×[e-(1+α)t+e-(0.5+α)t-e-(1.5+2α)t]}dα=16t(2e-2t+e-1.5t+e-2.5t-e-8.5t-2e-8t-e-7.5t)+110t(e-4t-e-14t)+18t(2e-10t-2e-6t+e-10.5t+e-11.5t-e-2.5t+2e-7t-2e-3t-e-3.5t).The MTTF of the hi-fi system is(99)MTTF=∫0+∞‍[16t(2e-2t+e-1.5t+e-2.5t-e-8.5t-2e-8t-e-7.5t)+110t(e-4t-e-14t)+18t(2e-10t-2e-6t+e-10.5t+e-11.5t-e-2.5t+2e-7t-2e-3t-e-3.5t)16t]dt≈0.3921.


## 5. Conclusion

In this paper, the random fuzzy theory provides a mathematical foundation for the reliability theory, which makes it possible to solve more complex unrepairable systems with fuzziness and randomness. Based on that, we establish five basic mathematical models of random fuzzy unrepairable systems, including series systems, parallel systems, series-parallel systems, parallel-series systems, and cold standby systems with absolutely reliable conversion switches. Furthermore, the expressions of reliability and MTTF are given for the above five random fuzzy unrepairable systems, respectively. When the random fuzzy lifetimes degenerate to random lifetimes or fuzzy lifetimes, the results we arrived at are also suitable. In future research, continuous attention might be paid to random fuzzy systems, and we should give reliability analysis or discuss the maintenance policy of repairable systems.



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