Reliability Analysis of Random Fuzzy Unrepairable Systems

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.

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 time 𝑡” 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 Section 2, 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]). Let Θ 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{𝐴^𝑐}=1 for any 𝐴∈𝒫(Θ).

Axiom 4. Consider Cr{∪_𝑖𝐴_𝑖}∧0.5=sup_𝑖Cr{𝐴_𝑖} for any {𝐴_𝑖} with Cr{𝐴_𝑖}≤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]). Let 𝜉 be a fuzzy variable defined on the credibility space (Θ,𝒫(Θ),Cr). Then its membership function is derived from the credibility measure by 𝜇(𝑥)=(2Cr{𝜉=𝑥})∧1,𝑥∈ℜ. (1)

Definition 4 (Liu [44]). A fuzzy variable 𝜉 is said to be positive if Cr{𝜉≤0}=0.

Definition 5 (Liu [45]). Let 𝜉 be a fuzzy variable and 𝛼∈(0,1]. Then 𝜉^𝐿_𝛼=inf𝑥∣𝜇(𝑥)≥𝛼,𝜉^𝑈_𝛼=sup𝑥∣𝜇(𝑥)≥𝛼 (2) are called the 𝛼-pessimistic value and the 𝛼-optimistic value of 𝜉, respectively.

Definition 6 (B. Liu and Y.-K. Liu [43]). Let 𝜉 be a fuzzy variable. The expected value 𝐸[𝜉] is defined as 𝐸𝜉=₀^+∞Cr{𝜉≥𝑟}d𝑟−⁰_−∞Cr{𝜉≤𝑟}d𝑟 (3) 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]). Let 𝜉 be a fuzzy variable with finite expected value 𝐸[𝜉]; then one has 𝐸𝜉=12¹₀𝜉^𝐿_𝛼+𝜉^𝑈_𝛼𝑑𝛼, (4) where 𝜉^𝐿_𝛼 and 𝜉^𝑈_𝛼 are the 𝛼-pessimistic value and the 𝛼-optimistic value of 𝜉, respectively.

Definition 8 (Liu [44]). The fuzzy variables 𝜉₁,𝜉₂,…,𝜉_𝑛 are said to be independent if Cr^𝑛_𝑖=1𝜉_𝑖∈ℬ_𝑖=min_{1≤𝑖≤𝑛}Cr𝜉_𝑖∈ℬ_𝑖 (5) for any sets ℬ₁,ℬ₂,…,ℬ_𝑛 of ℜ.

Proposition 9 (Y.-K. Liu and B. Liu [46] and Zhao et al. [39]). Let 𝜉 and 𝜂 be two independent fuzzy variables. Then (i) for any 𝛼 ∈ (0,1], (𝜉+𝜂)^𝐿_𝛼=𝜉^𝐿_𝛼+𝜂^𝐿_𝛼; (ii) for any 𝛼 ∈ (0,1], (𝜉+𝜂)^𝑈_𝛼=𝜉^𝑈_𝛼+𝜂^𝑈_𝛼.
Furthermore, if 𝜉 and 𝜂 are positive, then (iii) for any 𝛼 ∈(0,1], (𝜉⋅𝜂)^𝐿_𝛼=𝜉^𝐿_𝛼⋅𝜂^𝐿_𝛼;(iv) for any 𝛼 ∈ (0,1], (𝜉⋅𝜂)^𝑈_𝛼=𝜉^𝑈_𝛼⋅𝜂^𝑈_𝛼.

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]). Let 𝜉 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 set ℬ of ℜ;(2)𝐸[𝜉(𝜃)] is a fuzzy variable provided that 𝐸[𝜉(𝜃)] is finite for any fixed 𝜃∈Θ.

Example 11. A random fuzzy variable 𝜉 is said to be exponential if for each 𝜃, 𝜉(𝜃) is an exponentially distributed random variable whose density function is defined as 𝑓_𝜉(𝜃)(𝑡)=𝑋(𝜃)exp(−𝑋(𝜃)𝑡),if𝑡≥0,0,if𝑡<0, (6) 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]). Let 𝜉 be a random fuzzy variable defined on the credibility space (Θ,𝒫(Θ),Cr). Then the expected value 𝐸[𝜉] is defined by 𝐸𝜉=₀^+∞𝜉(𝜃−Cr𝜃∈Θ∣𝐸)≥𝑟d𝑟_0−∞Cr𝜃∈Θ∣𝐸𝜉(𝜃)≤𝑟d𝑟 (7) 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]). Let 𝜉 be a random fuzzy variable. Then the average chance, denoted by Ch, of random fuzzy event characterized by {𝜉∈ℬ} is defined as Ch{𝜉∈ℬ}=¹₀Cr𝜃∈Θ∣Pr{𝜉(𝜃)∈ℬ}≥𝑝d𝑝. (8)

Remark 14. If 𝜉 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 variables ℱ 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 Pr𝜁₁≤𝑟≤Pr𝜁₂≤𝑟denotedby𝜁₂≤_𝑑𝜁₁ (9) or Pr𝜁₁≤𝑟≥Pr𝜁₂≤𝑟denotedby𝜁₁≤_𝑑𝜁₂. (10)

Remark 16 (Ross [48]). For any given 𝜁₁,𝜁₂∈ℱ, we have 𝐸𝜁₁≤𝐸𝜁₂⟺𝜁₁≤_𝑑𝜁₂. (11)

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. Let 𝑋 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 𝑅(𝑡)=Ch{𝑋≥𝑡}. (12)

Definition 18. Let 𝑋 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 MTTF=₀^+∞Cr{𝜃∈Θ∣𝐸[𝑋(𝜃)]≥𝑟}d𝑟. (13)

3.1. Reliability Analysis of Random Fuzzy Unrepairable Series Systems

Consider a series system composed of 𝑛 independent components. Let 𝑋_𝑖 be the lifetime of component 𝑖, which is a random fuzzy variable on the credibility space (Θ_𝑖,𝒫(Θ_𝑖),Cr_𝑖), 𝑖=1,2,…,𝑛. Obviously, the lifetime of the series system is 𝑋=min{𝑋₁,𝑋₂,…,𝑋_𝑛}, which is a random fuzzy variable on the product credibility space (Θ,𝒫(Θ),Cr), where Θ=Θ₁×Θ₂×⋯×Θ_𝑛 and Cr=Cr₁∧Cr₂∧⋯∧Cr_𝑛.

Theorem 19. Let 𝑋_𝑖, 𝑖=1,2,…,𝑛 be random fuzzy variables. Assume that the 𝛼-pessimistic values and the 𝛼-optimistic values of 𝐸[𝑋_𝑖(𝜃_𝑖)], 𝑖=1,2,…,𝑛, are continuous at the point 𝛼, 𝛼∈(0,1]. The reliability of random fuzzy series system is 𝑅(𝑡)=𝐸^𝑛_𝑖=1𝑃𝑟𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)≥𝑡. (14)

Proof. By Definitions 13 and 17 and Proposition 7, we have 𝑅(𝑡){}==Ch𝑋≥𝑡₁₀{𝑋(𝜃)}=1Cr𝜃∈Θ∣Pr≥𝑡≥𝑝d𝑝2¹₀Pr^𝐿_𝛼{𝜔∈Ω∣𝑋(𝜃)(𝜔)≥𝑡}+Pr^𝑈_𝛼{𝜔∈Ω∣𝑋(𝜃)(𝜔)≥𝑡}d𝛼. (15)
Let 𝐴_𝑖={𝜃_𝑖∈Θ_𝑖∣𝜇{𝜃_𝑖}≥𝛼}, 𝑖=1,2,…,𝑛. Since the 𝛼-pessimistic values and the 𝛼-optimistic values of fuzzy variables 𝐸[𝑋_𝑖(𝜃_𝑖)], 𝜃_𝑖∈Θ_𝑖, 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 𝜃^′_𝑖,𝜃_𝑖^′′∈𝐴_𝑖 such that 𝐸𝑋_𝑖𝜃^′_𝑖=𝐸𝑋_𝑖(𝜃_𝑖)^𝐿_𝛼,𝐸𝑋_𝑖𝜃_𝑖^′′=𝐸𝑋_𝑖(𝜃_𝑖)^𝑈_𝛼. (16)
For any 𝜃_𝑖,𝛼∈𝐴_𝑖, we have 𝐸𝑋_𝑖𝜃^′_𝑖≤𝐸𝑋_𝑖𝜃_𝑖,𝛼≤𝐸𝑋_𝑖𝜃_𝑖^′′,𝑖=1,2,…,𝑛. (17)
Hence, by Remark 16, we have 𝑋_𝑖𝜃^′_𝑖≤_𝑑𝑋_𝑖𝜃_𝑖,𝛼≤_𝑑𝑋_𝑖𝜃_𝑖^′′,𝜃_𝑖,𝛼∈𝐴_𝑖,𝑖=1,2,…,𝑛. (18)
It follows from Definition 15 that Pr𝜔∈Ω∣𝑋_𝑖𝜃^′_𝑖(𝜔)≥𝑡≤Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖,𝛼(𝜔)≥𝑡≤Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖^′′(𝜔)≥𝑡,𝑖=1,2,…,𝑛. (19)
It is easy to see that ^𝑛_𝑖=1Pr𝜔∈Ω∣𝑋_𝑖𝜃^′_𝑖(𝜔)≤≥𝑡_𝑛𝑖=1Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖,𝛼(𝜔)≤≥𝑡_𝑛𝑖=1Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖^′′(𝜔)≥𝑡; (20) that is, Pr𝜔∈Ω∣min𝑋₁𝜃^′₁(𝜔),…,𝑋_𝑛𝜃^′_𝑛(𝜔𝑋)≥𝑡≤Pr𝜔∈Ω∣min₁𝜃_1,𝛼(𝜔),…,𝑋_𝑛𝜃_𝑛,𝛼(𝜔𝑋)≥𝑡≤Pr𝜔∈Ω∣min₁𝜃₁^′′(𝜔),…,𝑋_𝑛𝜃_𝑛^′′(𝜔)≥𝑡. (21)
Since 𝜃_𝑖,𝛼 are arbitrary points in 𝐴_𝑖, 𝑖=1,2,…,𝑛, by (21), we have Pr^𝐿_𝛼{(𝜃𝜔)}𝑋𝜔∈Ω∣𝑋)(≥𝑡=Pr𝜔∈Ω∣min₁𝜃^′₁(𝜔),…,𝑋_𝑛𝜃^′_𝑛(𝜔=)≥𝑡_𝑛𝑖=1Pr𝜔∈Ω∣𝑋_𝑖𝜃^′_𝑖(𝜔)≥𝑡,Pr^𝑈_𝛼{(𝜃𝜔)}𝑋𝜔∈Ω∣𝑋)(≥𝑡=Pr𝜔∈Ω∣min₁𝜃₁^′′(𝜔),…,𝑋_𝑛𝜃_𝑛^′′(𝜔=)≥𝑡_𝑛𝑖=1Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖^′′(𝜔)≥𝑡. (22)
On the other hand, by (19), we have Pr^𝐿_𝛼𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)≥𝑡=Pr𝜔∈Ω∣𝑋_𝑖𝜃^′_𝑖(𝜔)≥𝑡,Pr^𝑈_𝛼𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)≥𝑡=Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖^′′(𝜔)≥𝑡,𝑖=1,2,…,𝑛. (23)
By (22) and (23), we have Pr^𝐿_𝛼{(𝜃𝜔)}=𝜔∈Ω∣𝑋)(≥𝑡_𝑛𝑖=1Pr^𝐿_𝛼𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)≥𝑡,Pr^𝑈_𝛼{(𝜃𝜔)}=𝜔∈Ω∣𝑋)(≥𝑡_𝑛𝑖=1Pr^𝑈_𝛼𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)≥𝑡. (24)
By (15) and (24), we have 𝑅(𝑡)=12¹₀^𝑛_𝑖=1Pr^𝐿_𝛼𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)+≥𝑡_𝑛𝑖=1Pr^𝑈_𝛼𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)=1≥𝑡d𝛼2¹₀^𝑛_𝑖=1Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)≥𝑡^𝐿_𝛼+^𝑛_𝑖=1Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)≥𝑡^𝑈_𝛼d𝛼=𝐸^𝑛_𝑖=1Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)≥𝑡, (25) which completes the proof.

Remark 20. If 𝑋_𝑖, 𝑖=1,2,…,𝑛, degenerate to random variables, the result in Theorem 19 degenerates to the form 𝑅(𝑡)=Pr𝜔∈Ω∣min𝑋₁𝜔₁,𝑋₂𝜔₂,…,𝑋_𝑛𝜔_𝑛=≥𝑡_𝑛𝑖=1Pr𝑋_𝑖≥𝑡, (26) which is consistent with the result in stochastic case (see Barlow and Proschan [1]).

Remark 21. If 𝑋_𝑖, 𝑖=1,2,…,𝑛, degenerate to fuzzy variables, the result in Theorem 19 degenerates to the form 𝑅(𝑡)=Cr𝜃∈Θ∣min𝑋₁𝜃₁,𝑋₂𝜃₂,…,𝑋_𝑛𝜃_𝑛≥𝑡=min_{1≤𝑖≤𝑛}Cr𝑋_𝑖>𝑡, (27) which is consistent with the result in fuzzy case (see Liu and Zhu [49]).

Theorem 22. Let 𝑋_𝑖, 𝑖=1,2,…,𝑛, be random fuzzy variables. Assume that the 𝛼-pessimistic values and the 𝛼-optimistic values of 𝐸[𝑋_𝑖(𝜃_𝑖)], 𝑖=1,2,…,𝑛, are continuous at the point 𝛼, 𝛼∈(0,1]. The 𝑀𝑇𝑇𝐹 of the series system is 𝑀𝑇𝑇𝐹=₀^+∞𝑅(𝑡)𝑑𝑡. (28)

Proof. By Definition 18 and Proposition 7, we have MTTF=₀^+∞{[𝑋(𝜃}=1Cr𝜃∈Θ∣𝐸)]≥𝑟d𝑟2¹₀𝐸[𝑋(𝜃)]^𝐿_𝛼+𝐸[𝑋(𝜃)]^𝑈_𝛼d𝛼. (29)
By (21), we know that min𝑋₁𝜃^′₁(𝜔),…,𝑋_𝑛𝜃^′_𝑛(𝜔≤_𝑑)min𝑋₁𝜃_1,𝛼(𝜔),…,𝑋_𝑛𝜃_𝑛,𝛼(𝜔≤_𝑑)min𝑋₁𝜃₁^′′(𝜔),…,𝑋_𝑛𝜃_𝑛^′′(𝜔). (30)
It follows from Remark 16 that 𝐸min𝑋₁𝜃^′₁(𝜔),…,𝑋_𝑛𝜃^′_𝑛(𝜔𝑋)≤𝐸min₁𝜃_1,𝛼(𝜔),…,𝑋_𝑛𝜃_𝑛,𝛼(𝜔𝑋)≤𝐸min₁𝜃₁^′′(𝜔),…,𝑋_𝑛𝜃_𝑛^′′(𝜔). (31)
Since 𝜃_𝑖,𝛼 are arbitrary points in 𝐴_𝑖, 𝑖=1,2,…,𝑛, we have 𝐸[𝑋(𝜃)]^𝐿_𝛼=𝐸min𝑋₁𝜃^′₁(𝜔),…,𝑋_𝑛𝜃^′_𝑛(𝜔=_{0𝑛+∞𝑖=1})Pr𝜔∈Ω∣𝑋_𝑖𝜃^′_𝑖(𝜔)𝐸[𝑋(𝜃>𝑡d𝑡,)]^𝑈_𝛼=𝐸min𝑋₁𝜃₁^′′(𝜔),…,𝑋_𝑛𝜃_𝑛^′′(𝜔=_{0𝑛+∞𝑖=1})Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖^′′(𝜔)>𝑡d𝑡. (32)
It follows from (23), (29), and (32) that =1MTTF2¹₀₀^𝑛+∞_𝑖=1Pr𝜔∈Ω∣𝑋_𝑖𝜃^′_𝑖(𝜔)+>𝑡d𝑡_{0𝑛+∞𝑖=1}Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖^′′(𝜔)=1>𝑡d𝑡d𝛼2¹₀₀^+∞𝑛_𝑖=1Pr^𝐿_𝛼𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)+≥𝑡_𝑛𝑖=1Pr^𝑈_𝛼𝜔∈Ω∣𝑋_𝑖𝜃_𝑖𝜔)=(≥𝑡d𝑡d𝛼_0+∞12¹₀^𝑛_𝑖=1Pr^𝐿_𝛼𝜔∈Ω∣𝑋_𝑖𝜃_𝑖𝜔)+(≥𝑡_𝑛𝑖=1Pr^𝑈_𝛼𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)≥𝑡d𝛼d𝑡=₀^+∞𝑅(𝑡)d𝑡. (33)
The theorem is proved.

Remark 23. If 𝑋_𝑖, 𝑖=1,2,…,𝑛, degenerate to random variables, the result in Theorem 22 degenerates to the form MTTF=₀^+∞𝑅(𝑡)d𝑡, (34) in which 𝑅(𝑡) is the reliability of series system in stochastic case.

Remark 24. If 𝑋_𝑖, 𝑖=1,2,…,𝑛, degenerate to fuzzy variables, the result in Theorem 22 degenerates to the form MTTF=₀^+∞𝑅(𝑡)d𝑡, (35) in which 𝑅(𝑡) is the reliability of series system in fuzzy case.

Example 25. If the random fuzzy variable 𝑋_𝑖∼ℰ𝒳𝒫(𝜆_𝑖), where 𝜆_𝑖 is a fuzzy variable on (Θ_𝑖,𝒫(Θ_𝑖),Cr_𝑖), 𝑖=1,2,…,𝑛, then we can arrive at Pr^𝐿_𝛼{𝜔∈Ω∣𝑋(𝜃)(𝜔)≥𝑡}=^𝑛_𝑖=1𝑒^{−𝜆𝑈𝑖,𝛼𝑡}=𝑒^{−∑𝑛𝑖=1𝜆𝑈𝑖,𝛼𝑡},Pr^𝑈_𝛼{𝜔∈Ω∣𝑋(𝜃)(𝜔)≥𝑡}=^𝑛_𝑖=1𝑒^{−𝜆𝐿𝑖,𝛼𝑡}=𝑒^{−∑𝑛𝑖=1𝜆𝐿𝑖,𝛼𝑡}. (36)
By Theorems 19 and 22, we have 𝑅(𝑡)=𝑒^{−∑𝑛𝑖=1𝐸𝜆𝑖𝑡},1MTTF=∑^𝑛_𝑖=1𝐸𝜆_𝑖, (37) 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 of 𝑛 independent components. Let 𝑋_𝑖 be the lifetime of component 𝑖, which is a random fuzzy variable on the credibility space (Θ_𝑖,𝒫(Θ_𝑖),Cr_𝑖), 𝑖=1,2,…,𝑛. Obviously, the lifetime of the parallel system is 𝑋=max{𝑋₁,𝑋₂,…,𝑋_𝑛}, which is a random fuzzy variable on the product credibility space (Θ,𝒫(Θ),Cr), where Θ=Θ₁×Θ₂×⋯×Θ_𝑛 and Cr=Cr₁∧Cr₂∧⋯∧Cr_𝑛.

Theorem 26. Let 𝑋_𝑖, 𝑖=1,2,…,𝑛, be random fuzzy variables. Assume that the 𝛼-pessimistic values and the 𝛼-optimistic values of 𝐸[𝑋_𝑖(𝜃_𝑖)], 𝑖=1,2,…,𝑛, are continuous at the point 𝛼, 𝛼∈(0,1]. The reliability of the random fuzzy parallel system is 𝑅(𝑡)=1−𝐸^𝑛_𝑖=11−𝑃𝑟𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)≥𝑡. (38)

Proof. By Definitions 13 and 17 and Proposition 7, we have 𝑅(𝑡){}==Ch𝑋≥𝑡₁₀{𝑋(𝜃)}=1Cr𝜃∈Θ∣Pr≥𝑡≥𝑝d𝑝2¹₀Pr^𝐿_𝛼{𝜔∈Ω∣𝑋(𝜃)(𝜔)≥𝑡}+Pr^𝑈_𝛼{𝜔∈Ω∣𝑋(𝜃)(𝜔)≥𝑡}d𝛼. (39)
Let 𝐴_𝑖={𝜃_𝑖∈Θ_𝑖∣𝜇{𝜃_𝑖}≥𝛼}, 𝑖=1,2,…,𝑛. Since the 𝛼-pessimistic values and the 𝛼-optimistic values of fuzzy variables 𝐸[𝑋_𝑖(𝜃_𝑖)], 𝜃_𝑖∈Θ_𝑖, 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 𝜃^′_𝑖,𝜃_𝑖^′′∈𝐴_𝑖 such that 𝐸𝑋_𝑖𝜃^′_𝑖=𝐸𝑋_𝑖(𝜃_𝑖)^𝐿_𝛼,𝐸𝑋_𝑖𝜃_𝑖^′′=𝐸𝑋_𝑖(𝜃_𝑖)^𝑈_𝛼. (40)
For any 𝜃_𝑖,𝛼∈𝐴_𝑖, we have 𝐸𝑋_𝑖𝜃^′_𝑖≤𝐸𝑋_𝑖𝜃_𝑖,𝛼≤𝐸𝑋_𝑖𝜃_𝑖^′′,𝑖=1,2,…,𝑛. (41)
Hence, by Remark 16 we have 𝑋_𝑖𝜃^′_𝑖≤_𝑑𝑋_𝑖𝜃_𝑖,𝛼≤_𝑑𝑋_𝑖𝜃_𝑖^′′,𝜃_𝑖,𝛼∈𝐴_𝑖,𝑖=1,2,…,𝑛. (42)
It follows from Definition 15 that Pr𝜔∈Ω∣𝑋_𝑖𝜃^′_𝑖(𝜔)≥𝑡≤Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖,𝛼(𝜔)≥𝑡≤Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖^′′(𝜔)≥𝑡,𝑖=1,2,…,𝑛. (43)
It is easy to see that 1−^𝑛_𝑖=11−Pr𝜔∈Ω∣𝑋_𝑖𝜃^′_𝑖(𝜔)>𝑡≤1−^𝑛_𝑖=11−Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖,𝛼(𝜔)>𝑡≤1−^𝑛_𝑖=11−Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖^′′(𝜔)>𝑡; (44)that is, Pr𝜔∈Ω∣max𝑋₁𝜃^′₁(𝜔),…,𝑋_𝑛𝜃^′_𝑛(𝜔𝑋)>𝑡≤Pr𝜔∈Ω∣max₁𝜃_1,𝛼(𝜔),…,𝑋_𝑛𝜃_𝑛,𝛼(𝜔𝑋)>𝑡≤Pr𝜔∈Ω∣max₁𝜃₁^′′(𝜔),…,𝑋_𝑛𝜃_𝑛^′′(𝜔)>𝑡. (45)
Since 𝜃_𝑖,𝛼 are arbitrary points in 𝐴_𝑖, 𝑖=1,2,…,𝑛, by (45), we have Pr^𝐿_𝛼{(𝜃𝜔)}𝑋𝜔∈Ω∣𝑋)(≥𝑡=Pr𝜔∈Ω∣max₁𝜃^′₁(𝜔),…,𝑋_𝑛𝜃^′_𝑛(𝜔)>𝑡=1−^𝑛_𝑖=11−Pr𝜔∈Ω∣𝑋_𝑖𝜃^′_𝑖(𝜔)>𝑡,Pr^𝑈_𝛼{(𝜃𝜔)}𝑋𝜔∈Ω∣𝑋)(≥𝑡=Pr𝜔∈Ω∣max₁𝜃₁^′′(𝜔),…,𝑋_𝑛𝜃_𝑛^′′(𝜔)>𝑡=1−^𝑛_𝑖=11−Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖^′′(𝜔)>𝑡. (46)
On the other hand, by (19), we have Pr^𝐿_𝛼𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)≥𝑡=Pr𝜔∈Ω∣𝑋_𝑖𝜃^′_𝑖(𝜔)>𝑡,Pr^𝑈_𝛼𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)≥𝑡=Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖^′′(𝜔)>𝑡,𝑖=1,2,…,𝑛. (47)
By (46) and (47), we have Pr^𝐿_𝛼{𝜔∈Ω∣𝑋(𝜃)(𝜔)≥𝑡}=1−^𝑛_𝑖=11−Pr^𝐿_𝛼𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)≥𝑡,Pr^𝑈_𝛼{𝜔∈Ω∣𝑋(𝜃)(𝜔)≥𝑡}=1−^𝑛_𝑖=11−Pr^𝑈_𝛼𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)≥𝑡. (48)
By (39) and (48) we have 𝑅(𝑡)=12¹₀1−^𝑛_𝑖=11−Pr^𝐿_𝛼𝜔∈Ω∣𝑋_𝑖𝜃_𝑖𝜔)−(≥𝑡+1_𝑛𝑖=11−Pr^𝑈_𝛼𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)1≥𝑡d𝛼=1−2×¹₀^𝑛_𝑖=11−Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)≥𝑡^𝑈_𝛼+^𝑛_𝑖=11−Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)≥𝑡^𝐿_𝛼d𝛼=1−𝐸^𝑛_𝑖=11−Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)≥𝑡, (49)
which completes the proof.

Remark 27. If 𝑋_𝑖, 𝑖=1,2,…,𝑛, degenerate to random variables, the result in Theorem 26 degenerates to the form 𝑅(𝑡)=Pr𝜔∈Ω∣max𝑋₁𝜔₁,𝑋₂𝜔₂,…,𝑋_𝑛𝜔_𝑛≥𝑡=1−^𝑛_𝑖=11−Pr𝑋_𝑖≥𝑡, (50) which is consistent with the result in stochastic case (see Barlow and Proschan [1]).

Remark 28. If 𝑋_𝑖, 𝑖=1,2,…,𝑛, degenerate to fuzzy variables, the result in Theorem 26 degenerates to the form 𝑅(𝑡)=Cr𝜃∈Θ∣max𝑋₁𝜃₁,𝑋₂𝜃₂,…,𝑋_𝑛𝜃_𝑛≥𝑡=max_{1≤𝑖≤𝑛}Cr𝑋_𝑖>𝑡, (51) which is consistent with the result in fuzzy case (see Liu and Zhu [49]).

Theorem 29. Let 𝑋_𝑖, 𝑖=1,2,…,𝑛, be random fuzzy variables. Assume that the 𝛼-pessimistic values and the 𝛼-optimistic values of 𝐸[𝑋_𝑖(𝜃_𝑖)], 𝑖=1,2,…,𝑛, are continuous at the point 𝛼, 𝛼∈(0,1]. The 𝑀𝑇𝑇𝐹 of the parallel system is 𝑀𝑇𝑇𝐹=₀^+∞𝑅(𝑡)𝑑𝑡. (52)

Proof. By Definition 18 and Proposition 7, we can see MTTF=₀^+∞{[𝑋(𝜃}=1Cr𝜃∈Θ∣𝐸)]≥𝑟d𝑟2¹₀𝐸[𝑋(𝜃)]^𝐿_𝛼+𝐸[𝑋(𝜃)]^𝑈_𝛼d𝛼. (53)
By (45), we know that max𝑋₁𝜃^′₁(𝜔),…,𝑋_𝑛𝜃^′_𝑛(𝜔≤_𝑑)max𝑋₁𝜃_1,𝛼(𝜔),…,𝑋_𝑛𝜃_𝑛,𝛼(𝜔≤_𝑑)max𝑋₁𝜃₁^′′(𝜔),…,𝑋_𝑛𝜃_𝑛^′′(𝜔). (54)
It follows from Remark 16 that 𝐸max𝑋₁𝜃^′₁(𝜔),…,𝑋_𝑛𝜃^′_𝑛(𝜔𝑋)≤𝐸max₁𝜃_1,𝛼(𝜔),…,𝑋_𝑛𝜃_𝑛,𝛼(𝜔𝑋)≤𝐸max₁𝜃₁^′′(𝜔),…,𝑋_𝑛𝜃_𝑛^′′(𝜔). (55)
Since 𝜃_𝑖,𝛼 are arbitrary points in 𝐴_𝑖, 𝑖=1,2,…,𝑛, we have 𝐸[𝑋(𝜃)]^𝐿_𝛼=𝐸max𝑋₁𝜃^′₁(𝜔),…,𝑋_𝑛𝜃^′_𝑛(𝜔=_0+∞)1−^𝑛_𝑖=11−Pr𝜔∈Ω∣𝑋_𝑖𝜃^′_𝑖(𝜔)𝐸[𝑋(𝜃>𝑡d𝑡,)]^𝑈_𝛼=𝐸max𝑋₁𝜃₁^′′(𝜔),…,𝑋_𝑛𝜃_𝑛^′′(𝜔=_0+∞)1−^𝑛_𝑖=11−Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖^′′(𝜔)>𝑡d𝑡. (56)
It follows from (47), (53), and (56) that =1MTTF2¹₀₀^+∞1−^𝑛_𝑖=11−Pr𝜔∈Ω∣𝑋_𝑖𝜃^′_𝑖(𝜔)+>𝑡d𝑡_0+∞1−^𝑛_𝑖=11−Pr𝜔∈Ω∣𝑋_𝑖𝜃_𝑖^′′×(𝜔)=1>𝑡d𝑡d𝛼2¹₀₀^+∞1−^𝑛_𝑖=11−Pr^𝐿_𝛼𝜔∈Ω∣𝑋_𝑖𝜃_𝑖𝜔)+(≥𝑡d𝑡_0+∞1−^𝑛_𝑖=11−Pr^𝑈_𝛼𝜔∈Ω∣𝑋_𝑖𝜃_𝑖×(𝜔)=≥𝑡d𝑡d𝛼_0+∞12¹₀1−^𝑛_𝑖=11−Pr^𝐿_𝛼𝜔∈Ω∣𝑋_𝑖𝜃_𝑖(𝜔)≥𝑡+1−^𝑛_𝑖=11−Pr^𝑈_𝛼𝜔∈Ω∣𝑋_𝑖𝜃_𝑖×(𝜔)=≥𝑡}]d𝛼d𝑡_0+∞𝑅(𝑡)d𝑡. (57)
The theorem is proved.

Remark 30. If 𝑋_𝑖, 𝑖=1,2,…,𝑛, degenerate to random variables, the result in Theorem 29 degenerates to the form MTTF=₀^+∞𝑅(𝑡)d𝑡, (58) in which 𝑅(𝑡) is the reliability of parallel system in stochastic case.

Remark 31. If 𝑋_𝑖, 𝑖=1,2,…,𝑛, degenerate to fuzzy variables, the result in Theorem 29 degenerates to the form MTTF=₀^+∞𝑅(𝑡)d𝑡, (59) in which 𝑅(𝑡) is the reliability of parallel system in fuzzy case.

Example 32. If the random fuzzy variable 𝑋_𝑖(𝜆_𝑖)∼ℰ𝒳𝒫(𝜆_𝑖), where 𝜆_𝑖 is a fuzzy variable on (Θ_𝑖,𝒫(Θ_𝑖),Cr_𝑖), 𝑖=1,2,…,𝑛, then we can arrive at Pr^𝐿_𝛼{𝜔∈Ω∣𝑋(𝜃)(𝜔)≥𝑡}=1−^𝑛_𝑖=11−𝑒^{−𝜆𝑈𝑖,𝛼𝑡},Pr^𝑈_𝛼{𝜔∈Ω∣𝑋(𝜃)(𝜔)≥𝑡}=1−^𝑛_𝑖=11−𝑒^{−𝜆𝐿𝑖,𝛼𝑡}. (60)
By Theorems 26 and 29, we have 𝑅(𝑡)=1−^𝑛_𝑖=11−𝑒^{−𝐸𝜆𝑖𝑡},MTTF=₀^+∞1−^𝑛_𝑖=11−𝑒^{−𝐸𝜆𝑖𝑡}d𝑡, (61) 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 of 𝑚 subsystems; each subsystem is composed of 𝑛 parallel components. Let 𝑋_𝑖𝑗 be the lifetime of component 𝑗 in 𝑖th subsystem, which is a random fuzzy variable on the credibility space (Θ_𝑖𝑗,𝒫(Θ_𝑖𝑗),Cr_𝑖𝑗), 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≤𝑛}𝑋_𝑖𝑗), which is a random fuzzy variable on the product credibility space (Θ,𝒫(Θ),Cr), where Θ=Θ₁₁×Θ₁₂×⋯×Θ_𝑚𝑛 and Cr=Cr₁₁∧Cr₁₂∧⋯∧Cr_𝑚𝑛.

Theorem 33. Let 𝑋_𝑖𝑗, 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[𝑋_𝑖𝑗(𝜃_𝑖𝑗)], 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 𝑅(𝑡)=𝐸^𝑚_𝑖=11−^𝑛_𝑗=11−𝑃𝑟𝜔∈Ω∣𝑋_𝑖𝑗𝜃_𝑖𝑗(𝜔)≥𝑡. (62)

Remark 34. If 𝑋_𝑖𝑗, 𝑖=1,2,…,𝑚, 𝑗=1,2,…,𝑛, degenerate to random variables, the result in Theorem 33 degenerates to the form 𝑅(𝑡)=^𝑚_𝑖=11−^𝑛_𝑗=11−Pr𝑋_𝑖𝑗≥𝑡, (63) which is consistent with the result in stochastic case (see Barlow and Proschan [1]).

Remark 35. If 𝑋_𝑖𝑗, 𝑖=1,2,…,𝑚, 𝑗=1,2,…,𝑛, degenerate to fuzzy variables, the result in Theorem 33 degenerates to the form 𝑅(𝑡)=Crmin_{1≤𝑖≤𝑚}max_{1≤𝑗≤𝑛}𝑋_𝑖𝑗≥𝑡, (64) which is consistent with the result in fuzzy case (see Liu and Zhu [49]).

Theorem 36. Let 𝑋_𝑖𝑗, 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[𝑋_𝑖𝑗(𝜃_𝑖𝑗)], 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 𝑀𝑇𝑇𝐹=₀^+∞𝑅(𝑡)𝑑𝑡. (65)

Remark 37. If 𝑋_𝑖𝑗, 𝑖=1,2,…,𝑚, 𝑗=1,2,…,𝑛, degenerate to random variables, the result in Theorem 36 degenerates to the form MTTF=₀^+∞𝑅(𝑡)d𝑡, (66) in which 𝑅(𝑡) is the reliability of series-parallel system in stochastic case.

Remark 38. If 𝑋_𝑖𝑗, 𝑖=1,2,…,𝑚, 𝑗=1,2,…,𝑛, degenerate to fuzzy variables, the result in Theorem 36 degenerates to the form MTTF=₀^+∞𝑅(𝑡)d𝑡, (67) 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 of 𝑚 subsystems; each subsystem is composed of 𝑛 series components. Let 𝑋_𝑖𝑗 be the lifetime of component 𝑗 in 𝑖th subsystem, which is a random fuzzy variable on the credibility space (Θ_𝑖𝑗,𝒫(Θ_𝑖𝑗),Cr_𝑖𝑗), 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≤𝑛}𝑋_𝑖𝑗), which is a random fuzzy variable on the product credibility space (Θ,𝒫(Θ),Cr), where Θ=Θ₁₁×Θ₁₂×⋯×Θ_𝑚𝑛 and Cr=Cr₁₁∧Cr₁₂∧⋯∧Cr_𝑚𝑛.

Theorem 39. Let 𝑋_𝑖𝑗, 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[𝑋_𝑖𝑗(𝜃_𝑖𝑗)], 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 𝑅(𝑡)=1−𝐸^𝑚_𝑖=11−^𝑛_𝑗=1𝑃𝑟𝜔∈Ω∣𝑋_𝑖𝑗𝜃_𝑖𝑗(𝜔)≥𝑡. (68)

Remark 40. If 𝑋_𝑖𝑗, 𝑖=1,2,…,𝑚, 𝑗=1,2,…,𝑛, degenerate to random variables, the result in Theorem 39 degenerates to the form 𝑅(𝑡)=1−^𝑚_𝑖=11−^𝑛_𝑗=1Pr𝑋_𝑖𝑗≥𝑡, (69) which is consistent with the result in stochastic case (see Barlow and Proschan [1]).

Remark 41. If 𝑋_𝑖𝑗, 𝑖=1,2,…,𝑚, 𝑗=1,2,…,𝑛, degenerate to fuzzy variables, the result in Theorem 39 degenerates to the form 𝑅(𝑡)=Crmax_{1≤𝑖≤𝑚}min_{1≤𝑗≤𝑛}𝑋_𝑖𝑗≥𝑡, (70) which is consistent with the result in fuzzy case (see Liu and Zhu [49]).

Theorem 42. Let 𝑋_𝑖𝑗, 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[𝑋_𝑖𝑗(𝜃_𝑖𝑗)], 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 𝑀𝑇𝑇𝐹=₀^+∞𝑅(𝑡)𝑑𝑡. (71)

Remark 43. If 𝑋_𝑖𝑗, 𝑖=1,2,…,𝑚, 𝑗=1,2,…,𝑛, degenerate to random variables, the result in Theorem 42 degenerates to the form MTTF=₀^+∞𝑅(𝑡)d𝑡, (72) in which 𝑅(𝑡) is the reliability of parallel-series system in stochastic case.

Remark 44. If 𝑋_𝑖𝑗, 𝑖=1,2,…,𝑚, 𝑗=1,2,…,𝑛, degenerate to fuzzy variables, the result in Theorem 42 degenerates to the form MTTF=₀^+∞𝑅(𝑡)d𝑡, (73) 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 of 𝑛 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 𝑋_𝑖 be the lifetime of component 𝑖, which is a random fuzzy variable on the credibility space (Θ_𝑖,𝒫(Θ_𝑖),Cr_𝑖), 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+⋯+𝑋_𝑛, which is a random fuzzy variable on the product credibility space (Θ,𝒫(Θ),Cr), where Θ=Θ₁×Θ₂×⋯×Θ_𝑛 and Cr=Cr₁∧Cr₂∧⋯∧Cr_𝑛.

Theorem 45. Let 𝑋_𝑖, 𝑖=1,2,…,𝑛, be random fuzzy variables. Assume that the 𝛼-pessimistic values and the 𝛼-optimistic values of 𝐸[𝑋_𝑖(𝜃_𝑖)], 𝑖=1,2,…,𝑛, are continuous at the point 𝛼, 𝛼∈(0,1]. The reliability of random fuzzy cold standby system is 𝑅(𝑡)=𝐸𝑃𝑟𝜔∈Ω∣𝑋₁𝜃₁(𝜔)+⋯+𝑋_𝑛𝜃_𝑛(𝜔)≥𝑡. (74)

Proof. By Definitions 13 and 17 and Proposition 7, we have 𝑅(𝑡){}==Ch𝑋≥𝑡₁₀{𝑋(𝜃)}=1Cr𝜃∈Θ∣Pr≥𝑡≥𝑝d𝑝2¹₀Pr^𝐿_𝛼{𝜔∈Ω∣𝑋(𝜃)(𝜔)≥𝑡}+Pr^𝑈_𝛼{(𝜃𝜔)}=1𝜔∈Ω∣𝑋)(≥𝑡d𝛼2¹₀Pr^𝐿_𝛼𝜔∈Ω∣𝑋₁𝜃₁(𝜔)+⋯+𝑋_𝑛𝜃_𝑛(𝜔)≥𝑡+Pr^𝑈_𝛼𝜔∈Ω∣𝑋₁𝜃₁(𝜔)+⋯+𝑋_𝑛𝜃_𝑛(𝜔)≥𝑡d𝛼=𝐸Pr𝜔∈Ω∣𝑋₁𝜃₁(𝜔)+⋯+𝑋_𝑛𝜃_𝑛(𝜔)≥𝑡, (75)
which completes the proof.

Remark 46. If 𝑋_𝑖, 𝑖=1,2,…,𝑛, degenerate to random variables, the result in Theorem 45 degenerates to the form 𝑅(𝑡)=Pr𝑋₁+𝑋₂+⋯+𝑋_𝑛≥𝑡, (76) which is consistent with the result in stochastic case (see Barlow and Proschan [1]).

Remark 47. If 𝑋_𝑖, 𝑖=1,2,…,𝑛, degenerate to fuzzy variables, the result in Theorem 45 degenerates to the form 𝑅(𝑡)=Cr𝑋₁+𝑋₂+⋯+𝑋_𝑛≥𝑡, (77) which is consistent with the result in fuzzy case (see Liu and Zhu [49]).

Theorem 48. Let 𝑋_𝑖, 𝑖=1,2,…,𝑛, be random fuzzy variables. Assume that the 𝛼-pessimistic values and the 𝛼-optimistic values of 𝐸[𝑋_𝑖(𝜃_𝑖)], 𝑖=1,2,…,𝑛, are continuous at the point 𝛼, 𝛼∈(0,1]. The 𝑀𝑇𝑇𝐹 of the cold standby system is 𝑀𝑇𝑇𝐹=₀^+∞𝑅(𝑡)𝑑𝑡. (78)

Proof. By Definition 18 and Proposition 7, we can see MTTF=₀^+∞{[𝑋(𝜃}=1Cr𝜃∈Θ∣𝐸)]≥𝑟d𝑟2¹₀𝐸[𝑋(𝜃)]^𝐿_𝛼+𝐸[𝑋(𝜃)]^𝑈_𝛼d𝛼. (79)
Let 𝐴_𝑖={𝜃_𝑖∈Θ_𝑖∣𝜇{𝜃_𝑖}≥𝛼}, 𝑖=1,2,…,𝑛. Since the 𝛼-pessimistic values and the 𝛼-optimistic values of fuzzy variables 𝐸[𝑋_𝑖(𝜃_𝑖)], 𝜃_𝑖∈Θ_𝑖, 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 𝜃^′_𝑖,𝜃_𝑖^′′∈𝐴_𝑖 such that 𝐸𝑋_𝑖𝜃^′_𝑖=𝐸𝑋_𝑖(𝜃_𝑖)^𝐿_𝛼,𝐸𝑋_𝑖𝜃_𝑖^′′=𝐸𝑋_𝑖(𝜃_𝑖)^𝑈_𝛼. (80)
For any 𝜃_𝑖,𝛼∈𝐴_𝑖, 𝑖=1,2,…,𝑛, we have 𝐸𝑋_𝑖𝜃^′_𝑖≤𝐸𝑋_𝑖𝜃_𝑖,𝛼≤𝐸𝑋_𝑖𝜃_𝑖^′′. (81)
Hence, by Remark 16 we have 𝑋_𝑖𝜃^′_𝑖≤_𝑑𝑋_𝑖𝜃_𝑖,𝛼≤_𝑑𝑋_𝑖𝜃_𝑖^′′,𝜃_𝑖,𝛼∈𝐴_𝑖,𝑖=1,2,…,𝑛. (82)
Then 𝑋₁𝜃^′₁+⋯+𝑋_𝑛𝜃^′_𝑛≤_𝑑𝑋₁𝜃_1,𝛼+⋯+𝑋_𝑛𝜃_𝑛,𝛼≤_𝑑𝑋₁𝜃₁^′′+⋯+𝑋_𝑛𝜃_𝑛^′′. (83) It follows from Remark 16 that 𝐸𝑋₁𝜃^′₁+⋯+𝑋_𝑛𝜃^′_𝑛≤𝐸𝑋₁𝜃_1,𝛼+⋯+𝑋_𝑛𝜃_𝑛,𝛼≤𝐸𝑋₁𝜃₁^′′+⋯+𝑋_𝑛𝜃_𝑛^′′. (84)
Since 𝜃_𝑖,𝛼 are arbitrary points in 𝐴_𝑖, 𝑖=1,2,…,𝑛, we have 𝐸[𝑋(𝜃)]^𝐿_𝛼=𝐸𝑋₁𝜃^′₁+⋯+𝑋_𝑛𝜃^′_𝑛=₀^+∞Pr𝜔∈Ω∣𝑋₁𝜃^′₁(𝜔)+⋯+𝑋_𝑛𝜃^′_𝑛(𝜔)𝐸[𝑋(𝜃≥𝑡d𝑡,)]^𝑈_𝛼=𝐸𝑋₁𝜃₁^′′+⋯+𝑋_𝑛𝜃_𝑛^′′=₀^+∞Pr𝜔∈Ω∣𝑋₁𝜃₁^′′(𝜔)+⋯+𝑋_𝑛𝜃_𝑛^′′(𝜔)≥𝑡d𝑡. (85)
It follows from (79) and (85) that =1MTTF2¹₀₀^+∞Pr𝜔∈Ω∣𝑋₁𝜃^′₁(𝜔)+⋯+𝑋_𝑛𝜃^′_𝑛𝜔)+(≥𝑡d𝑡_0+∞Pr𝜔∈Ω∣𝑋₁𝜃₁^′′(𝜔)+⋯+𝑋_𝑛𝜃_𝑛^′′×(𝜔)=1≥𝑡d𝑡d𝛼2¹₀₀^+∞Pr𝜔∈Ω∣𝑋₁𝜃^′₁(𝜔)+⋯+𝑋_𝑛𝜃^′_𝑛(𝜔)≥𝑡+Pr𝜔∈Ω∣𝑋₁𝜃₁^′′(𝜔)+⋯+𝑋_𝑛𝜃_𝑛^′′(𝜔)≥𝑡d𝑡d𝛼. (86)
On the other hand, by (83) and Definition 15, we have Pr𝜔∈Ω∣𝑋₁𝜃^′₁+⋯+𝑋_𝑛𝜃^′_𝑛≥𝑡≤Pr𝜔∈Ω∣𝑋₁𝜃_1,𝛼+⋯+𝑋_𝑛𝜃_𝑛,𝛼≥𝑡≤Pr𝜔∈Ω∣𝑋₁𝜃₁^′′+⋯+𝑋_𝑛𝜃_𝑛^′′≥𝑡. (87)
Since 𝜃_𝑖,𝛼 are arbitrary points in 𝐴_𝑖, 𝑖=1,2,…,𝑛, we have Pr^𝐿_𝛼𝜔∈Ω∣𝑋₁𝜃₁(𝜔)+⋯+𝑋_𝑛𝜃_𝑛(𝜔)≥𝑡=Pr𝜔∈Ω∣𝑋₁𝜃^′₁+⋯+𝑋_𝑛𝜃^′_𝑛≥𝑡,Pr^𝑈_𝛼𝜔∈Ω∣𝑋₁𝜃₁(𝜔)+⋯+𝑋_𝑛𝜃_𝑛(𝜔)≥𝑡=Pr𝜔∈Ω∣𝑋₁𝜃₁^′′+⋯+𝑋_𝑛𝜃_𝑛^′′≥𝑡. (88)
It follows from (86) and (88) that =1MTTF2¹₀₀^+∞Pr^𝐿_𝛼𝜔∈Ω∣𝑋₁𝜃₁(𝜔)+⋯+𝑋_𝑛𝜃_𝑛(𝜔)≥𝑡+Pr^𝑈_𝛼𝜔∈Ω∣𝑋₁𝜃₁(𝜔)+⋯+𝑋_𝑛𝜃_𝑛(𝜔)≥𝑡d𝑡d𝛼=₀^+∞12¹₀Pr^𝐿_𝛼𝜔∈Ω∣𝑋₁𝜃₁(𝜔)+⋯+𝑋_𝑛𝜃_𝑛(𝜔)≥𝑡+Pr^𝑈_𝛼𝜔∈Ω∣𝑋₁𝜃₁(𝜔)+⋯+𝑋_𝑛𝜃_𝑛(𝜔)=≥𝑡d𝛼d𝑡_0+∞𝑅(𝑡)d𝑡. (89)
The theorem is proved.

Remark 49. If 𝑋_𝑖, 𝑖=1,2,…,𝑛, degenerate to random variables, the result in Theorem 48 degenerates to the form MTTF=₀^+∞𝑅(𝑡)d𝑡, (90) in which 𝑅(𝑡) is the reliability of cold standby system in stochastic case.

Remark 50. If 𝑋_𝑖, 𝑖=1,2,…,𝑛, degenerate to fuzzy variables, the result in Theorem 48 degenerates to the form MTTF=₀^+∞𝑅(𝑡)d𝑡, (91) 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 lamp 𝐿₁, 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 𝑋_𝑖(𝜆_𝑖)∼ℰ𝒳𝒫(𝜆_𝑖), 𝑖=1,2,3, where 𝜆₁=(1,2,3), 𝜆₂=(0,1,2), and 𝜆₃=(0,1,2). Then we can arrive at 𝜆^𝐿_1,𝛼=1+𝛼,𝜆^𝑈_1,𝛼𝜆=3−𝛼,_𝐿2,𝛼=𝛼,𝜆^𝑈_2,𝛼𝜆=2−𝛼,_𝐿3,𝛼=𝛼,𝜆^𝑈_3,𝛼=2−𝛼. (92)

Figure 1 

A lighting lamp system.

Then we have Pr^𝐿_𝛼{𝑋≥𝑡}=Pr^𝐿_𝛼𝑋₁×≥𝑡1−1−Pr^𝐿_𝛼𝑋₂≥𝑡1−Pr^𝐿_𝛼𝑋₃≥𝑡=𝑒^{−(3−𝛼)𝑡}1−1−𝑒^{−(2−𝛼)𝑡}1−𝑒^{−(2−𝛼)𝑡}=2𝑒^{−(5−2𝛼)𝑡}−𝑒^{−(7−3𝛼)𝑡},Pr^𝑈_𝛼{𝑋≥𝑡}=Pr^𝑈_𝛼𝑋₁×≥𝑡1−1−Pr^𝑈_𝛼𝑋₂≥𝑡1−Pr^𝑈_𝛼𝑋₃≥𝑡=𝑒^{−(1+𝛼)𝑡}1−1−𝑒^−𝛼𝑡1−𝑒^−𝛼𝑡=2𝑒^{−(1+2𝛼)𝑡}−𝑒^{−(1+3𝛼)𝑡}. (93)

So the reliability of the lighting lamp system is 𝑅(𝑡)=12¹₀2𝑒^{−(5−2𝛼)𝑡}−𝑒^{−(7−3𝛼)𝑡}+2𝑒^{−(1+2𝛼)𝑡}−𝑒^{−(1+3𝛼)𝑡}=1d𝛼6𝑡2𝑒^−𝑡−3𝑒^−5𝑡+𝑒^−7𝑡. (94)

The MTTF of the lighting lamp system is MTTF=₀^+∞16𝑡2𝑒^−𝑡−3𝑒^−5𝑡+𝑒^−7𝑡d𝑡≈0.4804. (95)

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 𝑋_𝑖 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 𝑋_𝑖(𝜆_𝑖)∼ℰ𝒳𝒫(𝜆_𝑖), 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 𝜆^𝐿_1,𝛼=1+𝛼,𝜆^𝑈_1,𝛼𝜆=3−𝛼,_𝐿2,𝛼=1.5+𝛼,𝜆^𝑈_2,𝛼𝜆=3.5−𝛼,_𝐿3,𝛼=𝛼,𝜆^𝑈_3,𝛼𝜆=2−𝛼,_𝐿4,𝛼=1+𝛼,𝜆^𝑈_4,𝛼𝜆=3−𝛼,_𝐿5,𝛼=0.5+𝛼,𝜆^𝑈_5,𝛼=2.5−𝛼. (96)

Then we can arrive at Pr^𝐿_𝛼{}=𝑋≥𝑡1−1−Pr^𝐿_𝛼𝑋₁≥𝑡1−Pr^𝐿_𝛼𝑋₂≥𝑡×Pr^𝐿_𝛼𝑋₃⋅≥𝑡1−1−Pr^𝐿_𝛼𝑋₄≥𝑡1−Pr^𝐿_𝛼𝑋₅=≥𝑡1−1−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},Pr^𝑈_𝛼{}=𝑋≥𝑡1−1−Pr^𝑈_𝛼𝑋₁≥𝑡1−Pr^𝑈_𝛼𝑋₂≥𝑡×Pr^𝑈_𝛼𝑋₃⋅≥𝑡1−1−Pr^𝑈_𝛼𝑋₄≥𝑡1−Pr^𝑈_𝛼𝑋₅=≥𝑡1−1−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^−𝛼𝑡𝑒^{−(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}. (97)

The reliability of the hi-fi system is 𝑅(𝑡)=12¹₀𝑒^{−(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^−𝛼𝑡𝑒^{−(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}=1d𝛼6𝑡2𝑒^−2𝑡+𝑒^−1.5𝑡+𝑒^−2.5𝑡−𝑒^−8.5𝑡−2𝑒^−8𝑡−𝑒^−7.5𝑡+1𝑒10𝑡^−4𝑡−𝑒^−14𝑡+18𝑡2𝑒^−10𝑡−2𝑒^−6𝑡+𝑒^−10.5𝑡+𝑒^−11.5𝑡−𝑒^−2.5𝑡+2𝑒^−7𝑡−2𝑒^−3𝑡−𝑒^−3.5𝑡. (98)

The MTTF of the hi-fi system is =MTTF_0+∞16𝑡2𝑒^−2𝑡+𝑒^−1.5𝑡+𝑒^−2.5𝑡−𝑒^−8.5𝑡−2𝑒^−8𝑡−𝑒^−7.5𝑡+1𝑒10𝑡^−4𝑡−𝑒^−14𝑡+18𝑡2𝑒^−10𝑡−2𝑒^−6𝑡+𝑒^−10.5𝑡+𝑒^−11.5𝑡−𝑒^−2.5𝑡+2𝑒^−7𝑡−2𝑒^−3𝑡−𝑒^−3.5𝑡d𝑡≈0.3921. (99)

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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the National Natural Science Foundation of China, Grant nos. 11301382, 61070021, and 71171088. The research is also supported by NCET-12-0081 and the Fundamental Research Funds for the Central Universities, HUST: CXY12M013.