Reliability Analysis of Random Fuzzy Unrepairable Systems

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.


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 " 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][16][17][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 fuzzystate 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 -independence associated with any continuous Archimedean -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, parallelseries 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.

Fuzzy Variables and Random Fuzzy Variables
In this section, we first introduce some basic concepts of fuzzy variables based on the credibility measure.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 5 (Liu [45]).Let  be a fuzzy variable and  ∈ (0, 1].Then are called the -pessimistic value and the -optimistic value of , respectively. Definition 6 (B.Liu and Y.-K.Liu [43]).Let  be a fuzzy variable.The expected value [] is defined as provided that at least one of the two integrals is finite.In particular, if  is a positive fuzzy variable, then Proposition 7 (Y.-K.Liu and B. Liu [46]).Let  be a fuzzy variable with finite expected value []; then one has where    and    are the -pessimistic value and the optimistic value of , respectively.Definition 8 (Liu [44]).The fuzzy variables  1 ,  2 , . . .,   are said to be independent if 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.
Example 11.A random fuzzy variable  is said to be exponential if for each , () is an exponentially distributed random variable whose density function is defined as where  is a positive fuzzy variable defined on the space Θ.An exponentially distributed random fuzzy variable is denoted by  ∼ EXP(), and the fuzziness of random fuzzy variable  is said to be characterized by fuzzy variable .It follows from Proposition 10 that Pr{() ≥ } and [()] are fuzzy variables.We can arrive at Pr{() ≥ } = exp(−()) and [()] = 1/().
Then the expected value [] is defined by provided that at least one of the two integrals is finite.In particular, if  is a positive fuzzy variable, then Definition 13 (Y.-K.Liu and B. Liu [47]).Let  be a random fuzzy variable.Then the average chance, denoted by Ch, of random fuzzy event characterized by { ∈ B} is defined as Remark 14.If  degenerates to a random variable, then the average chance degenerates to Pr{ ∈ B}, which is just the probability of random event.If  degenerates to a fuzzy variable, then the average chance degenerates to Cr{ ∈ 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  1 ,  2 ∈ F, and  ∈ R, either or Remark 16 (Ross [48]).For any given  1 ,  2 ∈ F, we have

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  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 Definition 18.Let  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
Remark 20.If   ,  = 1, 2, . . ., , degenerate to random variables, the result in Theorem 19 degenerates to the form which is consistent with the result in stochastic case (see Barlow and Proschan [1]).
Remark 21.If   ,  = 1, 2, . . ., , degenerate to fuzzy variables, the result in Theorem 19 degenerates to the form which is consistent with the result in fuzzy case (see Liu and Zhu [49]).
Proof.By Definition 18 and Proposition 7, we have It follows from ( 23), (29), and (32) that The theorem is proved.
in which () is the reliability of series system in stochastic case.
Remark 24.If   ,  = 1, 2, . . ., , degenerate to fuzzy variables, the result in Theorem 22 degenerates to the form in which () is the reliability of series system in fuzzy case.
By Theorems 19 and 22, we have in which "" is the expected value operator of fuzzy variable.

Reliability Analysis of Random Fuzzy Unrepairable Parallel
Systems.Consider a parallel system composed of  independent components.Let   be the lifetime of component , which is a random fuzzy variable on the credibility space (Θ  , P(Θ  ), Cr  ),  = 1, 2, . . ., .Obviously, the lifetime of the parallel system is  = max{ 1 ,  2 , . . .,   }, which is a random fuzzy variable on the product credibility space (Θ, P(Θ), Cr), where Proof.By Definitions 13 and 17 and Proposition 7, we have Let It is easy to see that that is, On the other hand, by (19), we have By ( 46) and ( 47), we have By ( 39) and ( 48) we have which completes the proof.
Remark 27.If   ,  = 1, 2, . . ., , degenerate to random variables, the result in Theorem 26 degenerates to the form which is consistent with the result in stochastic case (see Barlow and Proschan [1]).
Remark 28.If   ,  = 1, 2, . . ., , degenerate to fuzzy variables, the result in Theorem 26 degenerates to the form which is consistent with the result in fuzzy case (see Liu and Zhu [49]).
Remark 30.If   ,  = 1, 2, . . ., , degenerate to random variables, the result in Theorem 29 degenerates to the form in which () is the reliability of parallel system in stochastic case.
Remark 31.If   ,  = 1, 2, . . ., , degenerate to fuzzy variables, the result in Theorem 29 degenerates to the form in which () is the reliability of parallel system in fuzzy case.

Reliability Analysis of Random Fuzzy Unrepairable
Series-Parallel Systems.Consider a series-parallel system which is a series system of  subsystems; each subsystem is composed of  parallel components.Let   be the lifetime of component  in th subsystem, which is a random fuzzy variable on the credibility space (Θ  , P(Θ  ), Cr  ),  = 1, 2, . . ., ,  = 1, 2, . . ., .We assume the components are mutually independent.It is easy to know that the lifetime of the seriesparallel system is  = min 1≤≤ (max 1≤≤   ), which is a random fuzzy variable on the product credibility space ) . (62) Remark 34.If   ,  = 1, 2, . . ., ,  = 1, 2, . . ., , degenerate to random variables, the result in Theorem 33 degenerates to the form which is consistent with the result in stochastic case (see Barlow and Proschan [1]).
in which () is the reliability of series-parallel system in fuzzy case.

Reliability Analysis of Random Fuzzy Unrepairable
Parallel-Series Systems.Consider a parallel-series system which is a parallel system of  subsystems; each subsystem is composed of  series components.Let   be the lifetime of component  in th subsystem, which is a random fuzzy variable on the credibility space (Θ  , P(Θ  ), Cr  ),  = 1, 2, . . ., ,  = 1, 2, . . ., .We assume the components are mutually independent.It is easy to know that the lifetime of the parallel-series system is  = max 1≤≤ (min 1≤≤   ), which is a random fuzzy variable on the product credibility space (Θ, P(Θ), Cr), where Θ = Θ On the other hand, by (83) and Definition 15, we have Since  , are arbitrary points in   ,  = 1, 2, . . ., , we have It follows from ( 86) and (88) that The theorem is proved.
in which () is the reliability of cold standby system in fuzzy case.

Numerical Examples
In this section, we give the reliability analysis of a lighting lamp system and a hi-fi system with random fuzzy lifetimes.

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, seriesparallel 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.