Discrete Dynamics in Nature and Society

Volume 2014 (2014), Article ID 625985, 15 pages

http://dx.doi.org/10.1155/2014/625985

## Reliability Analysis of Random Fuzzy Unrepairable Systems

^{1}Department of Computer Sciences, Tianjin University of Science and Technology, Tianjin 300222, China^{2}School of Management, Huazhong University of Science and Technology, Wuhan 430074, China

Received 15 February 2014; Revised 4 April 2014; Accepted 4 April 2014; Published 2 June 2014

Academic Editor: Beatrice Paternoster

Copyright © 2014 Ying Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 ” 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 -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, 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 be a nonempty set and let be the power set of . The set function is called a credibility measure if it satisfies the following four axioms.

*Axiom 1.* Consider .

*Axiom 2. * is increasing; that is, whenever .

*Axiom 3. * is self-dual; that is, for any .

*Axiom 4.* Consider for any with .

*Definition 2 (Liu [44]). *A fuzzy variable is defined as a function from the credibility space to the set of real numbers.

*Definition 3 (Liu [44]). *Let be a fuzzy variable defined on the credibility space . Then its membership function is derived from the credibility measure by

*Definition 4 (Liu [44]). *A fuzzy variable is said to be positive if .

*Definition 5 (Liu [45]). *Let be a fuzzy variable and . 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 are said to be independent if
for any sets of .

Proposition 9 (Y.-K. Liu and B. Liu [46] and Zhao et al. [39]). *Let and be two independent fuzzy variables. Then **(i)** for any * *, **; **(ii)** for any * *, **.**Furthermore, if and are positive, then **(iii)** for any * *, **;**(iv)** for any * *, **.*

The concept of the random fuzzy variable was given by Liu [45]. Let be a probability space and a collection of random variables. A random fuzzy variable is defined as a function from a credibility space to a collection of random variables .

Proposition 10 (Liu [38]). *Let be a random fuzzy variable on the credibility space . Then, for , one has *(1)* is a fuzzy variable for any Borel set ** of **;*(2)* is a fuzzy variable provided that ** is finite for any fixed **. *

*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 . An exponentially distributed random fuzzy variable is denoted by , and the fuzziness of random fuzzy variable is said to be characterized by fuzzy variable . It follows from Proposition 10 that and are fuzzy variables. We can arrive at and .

*Definition 12 (Y.-K. Liu and B. Liu [46]). *Let be a random fuzzy variable defined on the credibility space . 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 , of random fuzzy event characterized by is defined as

*Remark 14. *If degenerates to a random variable, then the average chance degenerates to , which is just the probability of random event. If degenerates to a fuzzy variable, then the average chance degenerates to , 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 is said to be a totally ordered set with stochastic ordering if and only if, for any given , and , either
or

*Remark 16 (Ross [48]). *For any given , we have

#### 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 be the random fuzzy lifetime of an unrepairable system, which is defined on the credibility space ; 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 ; then MTTF of the unrepairable system is defined by

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

Consider a series system composed of independent components. Let be the lifetime of component , which is a random fuzzy variable on the credibility space , . Obviously, the lifetime of the series system is , which is a random fuzzy variable on the product credibility space , where and .

Theorem 19. *Let , be random fuzzy variables. Assume that the -pessimistic values and the -optimistic values of , , are continuous at the point , . The reliability of random fuzzy series system is
*

*Proof. *By Definitions 13 and 17 and Proposition 7, we have

Let , . Since the -pessimistic values and the -optimistic values of fuzzy variables , , , are continuous at the point , , then there exist maximum and minimum values; that is, at least there exist points such that

For any , we have

Hence, by Remark 16, we have

It follows from Definition 15 that

It is easy to see that
that is,

Since are arbitrary points in , , by (21), we have

On the other hand, by (19), we have

By (22) and (23), we have

By (15) and (24), we have
which completes the proof.

*Remark 20. *If , , 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 , , 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]).

Theorem 22. *Let , , be random fuzzy variables. Assume that the -pessimistic values and the -optimistic values of , , are continuous at the point , . The of the series system is
*

*Proof. *By Definition 18 and Proposition 7, we have

By (21), we know that

It follows from Remark 16 that

Since are arbitrary points in , , we have

It follows from (23), (29), and (32) that

The theorem is proved.

*Remark 23. *If , , degenerate to random variables, the result in Theorem 22 degenerates to the form
in which is the reliability of series system in stochastic case.

*Remark 24. *If , , degenerate to fuzzy variables, the result in Theorem 22 degenerates to the form
in which is the reliability of series system in fuzzy case.

*Example 25. *If the random fuzzy variable , where is a fuzzy variable on , , then we can arrive at

By Theorems 19 and 22, we have
in which “” is the expected value operator of fuzzy variable.

##### 3.2. 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 , . Obviously, the lifetime of the parallel system is , which is a random fuzzy variable on the product credibility space , where and .

Theorem 26. *Let , , be random fuzzy variables. Assume that the -pessimistic values and the -optimistic values of , , are continuous at the point , . The reliability of the random fuzzy parallel system is
*

*Proof. *By Definitions 13 and 17 and Proposition 7, we have

Let , . Since the -pessimistic values and the -optimistic values of fuzzy variables , , , are continuous at the point , , then there exist maximum and minimum values; that is, at least there exist points such that

For any , we have

Hence, by Remark 16 we have

It follows from Definition 15 that

It is easy to see that
that is,

Since are arbitrary points in , , by (45), we have

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 , , 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 , , 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]).

Theorem 29. *Let , , be random fuzzy variables. Assume that the -pessimistic values and the -optimistic values of , , are continuous at the point , . The of the parallel system is
*

*Proof. *By Definition 18 and Proposition 7, we can see

By (45), we know that

It follows from Remark 16 that

Since are arbitrary points in , , we have

It follows from (47), (53), and (56) that

The theorem is proved.

*Remark 30. *If , , 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 , , degenerate to fuzzy variables, the result in Theorem 29 degenerates to the form
in which is the reliability of parallel system in fuzzy case.

*Example 32. *If the random fuzzy variable , where is a fuzzy variable on , , then we can arrive at

By Theorems 26 and 29, we have
in which “” 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 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 , , . We assume the components are mutually independent. It is easy to know that the lifetime of the series-parallel system is , which is a random fuzzy variable on the product credibility space , where and .

Theorem 33. *Let , , , be random fuzzy variables. Assume that the -pessimistic values and the -optimistic values of , , , are continuous at the point , . The reliability of the series-parallel system is
*

*Remark 34. *If , , , 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]).

*Remark 35. *If , , , degenerate to fuzzy variables, the result in Theorem 33 degenerates to the form
which is consistent with the result in fuzzy case (see Liu and Zhu [49]).

Theorem 36. *Let , , , be random fuzzy variables. Assume that the -pessimistic values and the -optimistic values of , , , are continuous at the point , . The of the random fuzzy series-parallel system is
*

*Remark 37. *If , , , degenerate to random variables, the result in Theorem 36 degenerates to the form
in which is the reliability of series-parallel system in stochastic case.

*Remark 38. *If , , , degenerate to fuzzy variables, the result in Theorem 36 degenerates to the form
in which 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 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 , , . We assume the components are mutually independent. It is easy to know that the lifetime of the parallel-series system is , which is a random fuzzy variable on the product credibility space , where and .

Theorem 39. *Let , , , be random fuzzy variables. Assume that the -pessimistic values and the -optimistic values of , , , are continuous at the point , . The reliability of the parallel-series system is
*

*Remark 40. *If , , , degenerate to random variables, the result in Theorem 39 degenerates to the form
which is consistent with the result in stochastic case (see Barlow and Proschan [1]).

*Remark 41. *If , , , degenerate to fuzzy variables, the result in Theorem 39 degenerates to the form
which is consistent with the result in fuzzy case (see Liu and Zhu [49]).

Theorem 42. *Let , , , be random fuzzy variables. Assume that the -pessimistic values and the -optimistic values of , , , are continuous at the point , . The of the random fuzzy parallel-series system is
*

*Remark 43. *If , , , degenerate to random variables, the result in Theorem 42 degenerates to the form
in which is the reliability of parallel-series system in stochastic case.

*Remark 44. *If , , , degenerate to fuzzy variables, the result in Theorem 42 degenerates to the form
in which 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 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 be the lifetime of component , which is a random fuzzy variable on the credibility space , . The lifetime of the cold standby system can be expressed by the sum of the reliability of components; that is, , which is a random fuzzy variable on the product credibility space , where and .

Theorem 45. *Let , , be random fuzzy variables. Assume that the -pessimistic values and the -optimistic values of , , are continuous at the point , . The reliability of random fuzzy cold standby system is
*

*Proof. *By Definitions 13 and 17 and Proposition 7, we have

which completes the proof.

*Remark 46. *If , , degenerate to random variables, the result in Theorem 45 degenerates to the form
which is consistent with the result in stochastic case (see Barlow and Proschan [1]).

*Remark 47. *If , , degenerate to fuzzy variables, the result in Theorem 45 degenerates to the form
which is consistent with the result in fuzzy case (see Liu and Zhu [49]).

Theorem 48. *Let , , be random fuzzy variables. Assume that the -pessimistic values and the -optimistic values of , , are continuous at the point , . The of the cold standby system is
*

*Proof. *By Definition 18 and Proposition 7, we can see

Let , . Since the -pessimistic values and the -optimistic values of fuzzy variables , , , are continuous at the point , , then there exist maximum and minimum values; that is, at least there exist points such that

For any , , we have

Hence, by Remark 16 we have

Then
It follows from Remark 16 that

Since are arbitrary points in , , we have

It follows from (79) and (85) that

On the other hand, by (83) and Definition 15, we have

Since are arbitrary points in , , we have

It follows from (86) and (88) that