The Scientific World Journal

Volume 2014, Article ID 275374, 10 pages

http://dx.doi.org/10.1155/2014/275374

## Parameter Interval Estimation of System Reliability for Repairable Multistate Series-Parallel System with Fuzzy Data

Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi, Bangkok 10140, Thailand

Received 12 February 2014; Revised 10 April 2014; Accepted 24 April 2014; Published 22 May 2014

Academic Editor: Nirupam Chakraborti

Copyright © 2014 Wimonmas Bamrungsetthapong and Adisak Pongpullponsak. 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 purpose of this paper is to create an interval estimation of the fuzzy system reliability for the repairable multistate series–parallel system (RMSS). Two-sided fuzzy confidence interval for the fuzzy system reliability is constructed. The performance of fuzzy confidence interval is considered based on the coverage probability and the expected length. In order to obtain the fuzzy system reliability, the fuzzy sets theory is applied to the system reliability problem when dealing with uncertainties in the RMSS. The fuzzy number with a triangular membership function is used for constructing the fuzzy failure rate and the fuzzy repair rate in the fuzzy reliability for the RMSS. The result shows that the good interval estimator for the fuzzy confidence interval is the obtained coverage probabilities the expected confidence coefficient with the narrowest expected length. The model presented herein is an effective estimation method when the sample size is . In addition, the optimal * α*-cut for the narrowest lower expected length and the narrowest upper expected length are considered.

#### 1. Introduction

Most researches on reliability theory involve traditional binary reliability models where each component in a system basically consists of two functional states, perfect functionality and complete failure. However, in the system reliability of multistate components, the entire system performance will be considered from different performance levels and several failure modes. The evolution of such a system is represented by a continuous-time discrete state stochastic process. The multistate system is widely used in various industrial areas such as power generation systems, computer systems, and transportation systems (Lisnianki and Levitin [1]). Compared with a binary system model, a multistate system (MSS) model provides a more flexible tool for representing engineering systems in real life as first introduced in [2–4]. Recent research has focused on reliability evaluation and optimization of MSS [5–7]. In conventional multistate theory, it is assumed that the exact probability of each component state is given. However, with the progress of modern industrial technologies, the product development cycles have become shorter, while the lifetime of products has become longer [8]. In many highly reliable applications, there may be only a few available observations of the system’s failures. Therefore, it may be difficult to obtain sufficient data to estimate the precise values of the probabilities and performance levels of these systems. Moreover, the inaccuracy of system models, caused by human errors, is difficult to quantify using conventional reliability theory alone [9]. In light of these significant challenges, new techniques are needed to solve these fundamental problems related to reliability.

More recently, fuzzy reliability theory has been developed on the basis of fuzzy theory (Zadeh [10]). Presently, the applications of fuzzy idea in reliability theory that deal with the problem of lacking of inaccuracy or fluctuation data can be seen in many areas. In reliability analysis, many theories and methods have been constructed to facilitate the multistate system reliability assessment such as the universal generating function (Levitin [11]) and the multistate weight system (Li and Zuo [12]). Ding and Lisnianski [13] proposed the fuzzy universal generating function method to derive the fuzzy probability distribution and fuzzy system availability of the overall system when the component’s performance rate and state probabilities take fuzzy values. Liu et al. [14] investigated the dynamic fuzzy system state probabilities, fuzzy availability, and fuzzy performance rewards of a multistate system under a continuous-time Markov model. Liu and Huang [15] proposed the fuzzy multistate system that has extended the multistate system model to the cases that the transition and performance rates of multistate elements are uncertain.

This research considers the problem of interval estimation of fuzzy system reliability when the parameter of interest is fuzzy and the data are observations from fuzzy random variables. A random sample of fuzzy data set is generated in constructing a fuzzy confidence interval. In some studies on fuzzy confidence interval, Corral and Gil [16] considered the problem of constructing confidence interval using fuzzy data without considering any fuzzy random variables. Geyer and Meeden [17] introduced a fuzzy confidence interval that is the optimality of UMP and UMPU test. Viertl [18] investigated statistical inference about an unknown parameter based on fuzzy observations and developed testing hypothesis for crisp parameter based on fuzzy data. Wu [19] proposed an approach based on fuzzy random variables for constructing a fuzzy confidence interval for an unknown fuzzy parameter. Škrjanc [20] introduced a method to define a fuzzy confidence interval that combines a fuzzy identification methodology with some idea from applied statistics in finding the confidence interval defined by the lower and upper fuzzy bounds. Chachi and Taheri [21] proposed a method to construct the one-sided and two-sided fuzzy confidence intervals for an unknown fuzzy parameter based on normal fuzzy random variable. Škrjanc [22] presented a new method of confidence interval identification for Takagi-Sugeno fuzzy models in the case of the data with regionally changeable variance. The method combines a fuzzy identification methodology with some ideas from applied statistics. Some studies showed that much research proposed an approach based on fuzzy data for constructing a fuzzy confidence interval, but without considering the performance analysis of interval estimator. Then the performance of fuzzy confidence interval will be assessed based on the coverage probability and the expected length in this research.

In this research, the fuzzy system reliability of RMSS is constructed, where the fuzzy failure rate and the fuzzy repair rate for each component are the triangular fuzzy number. An approach to construct interval estimation of the fuzzy system reliability of RMSS which subsequently will be used in estimation of the fuzzy confidence interval of fuzzy system reliability is developed. Finally, the analytic expression to find the coverage probability and the expected length that it is used to interpret the efficiency of fuzzy confidence interval are presented.

#### 2. Materials and Methods

##### 2.1. Markov Model for Multistate Element

A Markov model has been used for evaluating the expected number of failures at an arbitrary time interval in many practical cases and can be described as a Poisson process (Lisnianki and Levitin [1]).

*Definition 1 (see Ibe [23]). *A stochastic process is a continuous-time Markov chain if, for all , and nonnegative integers , , ,
This means that in a continuous-time Markov chain the conditional probability of the future state at time given the present state at and all part state depends only on the present state and is independent of the past. If in addition is dependent on , then the process is said to be time homogeneous or have the time homogeneity* property*. Time homogeneous Markov chains have stationary transition probabilities. Let be the probability that a Markov chain in state will be in state after an additional time . Thus, the is the transition probability function that satisfies the condition . Also, .

##### 2.2. Kolmogorov Differential Equations

Using a set of differential equations to find (Rausand and Hoyland [24]), it is start by considering the Chapman-Kolmogorov equations The interval of is split into two parts. First, consider a transition from state to state in the small interval and then a transition from state to state in the rest of the interval. It is seen that where and the following notation for the time derivative is introduced. The differential equation (3) is known as the Kolmogorov backward equations. They are called backward equations because we start with a transition back by the start of the interval. The Kolmogorov backward equations may also be written in matrix format as Likewise, split the time interval into two parts. Consider a transition from to in the interval and then a transition from to in the small interval . It is seen that where, as before, . The differential equation (5) is known as the Kolmogorov forward equations. The interchange of the limit and the sum above does not hold in all cases but is always valid when the state space is finite.

Consider the following: For the Markov processes, the backward and the forward equations have the same unique solution, where for all in .

##### 2.3. Repairable Multistate Elements

In this section, the repairable multistate element assumes that minor failures can happen and minor repairs can be done (Xie et al. [25]). A minor failure is a failure that causes the element transition from state to state denoted by . On the other hand, a minor repair is a repair that causes the element transition from state to state denoted by . It is actually a birth and death process as presented in Figure 1.

The Chapman-Kolmogorov equations for the general case are as follows:

Assume that the initial state is in the state with the best performance. Therefore, by solving (7) of differential equations under the initial condition , . The unreliability function for the multistate element will be a sum of the probabilities of the unacceptable states . Therefore reliability function is given by

##### 2.4. Basic Concept of Fuzzy Set Theory

The basic concepts which are used for analysing the fuzzy system reliability are discussed in this section. In classical set theory, an element in a universe may or may not be a membership of some crisp set . This binary membership can be represented by the following indicator function:

Zadeh’s [10] extended the notation of binary membership to accommodate various degrees of membership on the real continuous interval and defined the fuzzy set by the membership function , given that is the degree of membership of element in fuzzy set . Consider a closed interval of real numbers . The following are formulas for four basic arithmetic operations on closed intervals of real numbers (Ross [26]): The triangular membership function of fuzzy set is given by Let be a universal set of real numbers and a fuzzy subset of . is referred to as triangular fuzzy number (TFN) as shown in Figure 2, denoted by . Let be the -cut level, so we have

##### 2.5. Repairable Fuzzy Multistate Elements

Based on the conventional multistate elements, the state space diagram of a repairable multistate system takes the form presented in Figure 1 where state is the best state and state 1 is the worst state. The minor failures between states and are determined by fuzzy value . And the minor repairs between states and are determined by fuzzy value . With the fuzzy transition intensities, the state probability of elements at time must also be a fuzzy value , and then the Chapman-Kolmogorov equation with fuzzy transition intensities can be written as where the initial conditions are and . The unreliability function for the multistate element will be a sum of the probabilities of the unacceptable states . Laplace transform is adopted to transform (13) into linear equation as follows: where . Given that is a function of , , and , then the inverse Laplace transform is executed to get the in time domain: where is an inverse Laplace operator and is a function in terms of fuzzy variables and at any time . The fuzzy state probabilities can be obtained in the form of at the -cut level: where and .

Then the fuzzy reliability of repairable multistate element is given by The fuzzy state probabilities can be obtained in the form that at the -cut level where . Let be fuzzy unreliability functions of repairable multistate element in each element .

##### 2.6. Multistate Element under Series-Parallel System

In this section, analysis of the fuzzy system reliability using the repairable multistate element with series-parallel system will be demonstrated. Figure 3 represents a system containing subsystem connection in series where each subsystem consists of components in parallel. Let be a fuzzy reliability function and a fuzzy unreliability function of subsystem , which is connected in series, and component , which is connected in parallel ( and ), respectively. Let a failure rate and a repair rate be represented by and at time , respectively.

Then the fuzzy system reliability of repairable multistate series-parallel system (RMSS) at time is given by

##### 2.7. Fuzzy Confidence Interval Probability for Fuzzy Parameter

The interval estimation of the fuzzy system reliability of RMSS is constructed by extending the concepts of two-sided confidence interval of the real parameters to the case where both parameter and random variables are fuzzy as shown in Figure 4.

According to Wu [19], are independent and identically distributed random variables. Let and be two statistics such that , where . If the random interval satisfies , then is a confidence interval for with confidence coefficient , where and each is an observed value of when . It can be applied to fuzzy confidence interval. Let be independent and identically distributed fuzzy random variables with fuzzy parameter . Let be the observed value of for , where each is a fuzzy number for . Therefore, and are the observed values of and for . Then and are identically distributed fuzzy random variables. Suppose that the distribution of is unknown for . Then the approximate fuzzy confidence interval can be constructed using the central limit theorem when the sample size is sufficiently large (Ross [27]). Let be independent and identically distributed fuzzy random variables. There exists a unique fuzzy number where for all ; then is called the expectation of . Suppose that , , and , respectively. Therefore and have finite expectations and with unknown variances for and .

Let and let an .

It is shown that and as . The central limit theorem gives that as well as has approximately distribution. Therefore the approximate fuzzy confidence interval for and is given by respectively.

We assign as lower fuzzy confidence interval and as upper fuzzy confidence interval.

##### 2.8. Coverage Probability and Expected Length of Fuzzy Confidence Interval

For an interval estimator of a true parameter , the coverage probability of is the probability that the random interval covers . It is denoted by . In this research, the performance of fuzzy confidence interval is assessed based on the coverage probability and the expected length. The analytic expressions for the coverage probability of fuzzy confidence interval of true parameter are derived.

Let and be the coverage probability of fuzzy confidence interval and , respectively. Then the lower coverage probability of fuzzy confidence interval of is given by where where , , and is the standard normal distribution.

Likewise, the upper coverage probability of fuzzy confidence interval of is given by where where , , and is the standard normal distribution.

##### 2.9. General Procedure for Investigating the Coverage Probabilities and Expected Length for Fuzzy Confidence Interval of Fuzzy System Reliability

In this section, the coverage probability and expected length for fuzzy system reliability of RMSS is investigated. Suppose that fuzzy parameter (population mean) is denoted by . A method to calculate the fuzzy coverage probability and expected length for is demonstrated as follows.

*Step 1. *Generate fuzzy random variables from normal distribution with and . The sample sizes of this example are and for .

*Step 2. *Compute fuzzy sample mean and fuzzy sample variance for .

*Step 3. *Compute fuzzy confidence intervals of for , the lower fuzzy confidence interval , and the upper fuzzy confidence interval .

*Step 4. *Compute the fuzzy coverage probability and the expected length for ,
where is the lower fuzzy confidence interval and is the upper fuzzy confidence interval at repeated time , .

*Step 5. *Repeat Steps 2–4 for a given condition.

After receiving the coverage probabilities and the expected length of the fuzzy confidence interval, the next step is to consider the sample size that gives the coverage probability higher than the expected confidence coefficient, where the number of repeatition is . The -test statistic hypothesis testing is used in confirming the level of the coverage probability as follows::the coverage probability is not less than the expected confidence coefficient ,:the coverage probability is lower than the expected confidence coefficient ,

when the significance level is and the test statistic Since the criterion in the test is , the hypothesis cannot be rejected if where is the coverage probability, is the coverage probability estimated from this study, is the expected confidence coefficient, and is the repeated time.

#### 3. Numerical Example and Results

In this section, a RMSS which consists of 3 subsystems in series and 2 components in each subsystem in parallel is considered. In each element, it has repairable multistate with a fuzzy failure rate and a fuzzy repair rate as shown in Figure 5. Since a failure rate and a repair rate cannot be recorded precisely due to human errors, machine errors, or some unexpected situations, triangular fuzzy numbers are used to describe the fuzzy failure rate and fuzzy repair rate. The parameters of these functions are shown in Table 1.

For Markov model, the Chapman-Kolmogorov equations of each element can be written as where is probabilities of good function and is probabilities of fail function with fuzzy failure rate and fuzzy repair rate for each element in multistate model. Using the inverse Laplace transform, the fuzzy state probabilities are obtained as functions of time in the form of From Figure 5, fuzzy reliability functions and fuzzy unreliability functions of repairable for each element are given by where is fuzzy unreliability function of repairable multistate system at subsystem connected in series and component connected in parallel, respectively. Suppose that and are triangular membership function in each , as shown in Table 1. It can be written in the form of as follows: From the example system in Figure 6, RMSS with fuzzy failure rate and a fuzzy repair rate are considered. The fuzzy system reliability in each is given by Substituting fuzzy reliability function in each element into (32), then the fuzzy system reliability of RMSS is shown in Table 2.

Suppose that fuzzy system reliability for RMSS in Table 2 is the fuzzy parameter (population mean) denoted by . The confidence coefficient with is defined and fuzzy random variables from normal distribution are generated by using MATLAB [28]. Then the fuzzy coverage probability and the expected length for are shown in Figure 6.

After receiving the coverage probabilities and the expected length of fuzzy confidence interval, the next step is to compare the values between fuzzy coverage probabilities and the expected confidence coefficient by the test of hypothesis. In this example, let and the criterion used in comparing the coverage probability at significance level will be

Considering a fuzzy confidence interval at which gives the coverage probability higher than at significant level will be the coverage probability that is covered in the expected confidence coefficient. Then only the coverage probability that is covered in the expected confidence coefficient will be used in the most appropriate expected length estimation. In addition, the fuzzy confidence interval of the which gives the narrowest expected length is considered as the most appropriate expected length. Both lower and upper fuzzy coverage probabilities and expected lengths results are shown in Tables 3 and 4, respectively.

#### 4. Numerical Results

From the numerical example, the fuzzy reliability for the RMSS is calculated. Estimation of fuzzy confidence interval for fuzzy system reliability model in each revealed that the fuzzy confidence interval can be divided into 2 parts which are lower fuzzy confidence interval and upper fuzzy confidence interval as shown in Figure 6.

From Figure 6, the lower bound and upper bound of the fuzzy parameter are estimated at a significance level. Next, the performance of the estimated parameter of the fuzzy confidence interval of fuzzy system reliability model is considered. The coverage probability and expected length is used in calculation with the above method with repeated time at .

In Figure 7(a), it is seen that at the sample size there are some that give lower coverage probability covering in the expected confidence coefficient when . However, if considering, at sample size , the lower coverage probability of every is covering in the expected confidence coefficient, likewise, Figure 7(b) shows that at the sample size there are only few numbers that contain upper coverage probability in the expected confidence coefficient at , but, at the sample size , the upper coverage probability of every is in the expected confidence coefficient.

Then only the coverage probability that is covered in the expected confidence coefficient will be used in estimation of the most appropriate expected length. The result shows that the fuzzy confidence interval at gives the narrowest lower expected length. It is considered as the most appropriate lower expected length. It is also seen that the fuzzy confidence interval at gives the narrowest upper expected length. It is considered as the most appropriate upper expected length as shown in Tables 3 and 4.

It seems that at the larger sample size more numbers are obtained for the estimated parameter of the fuzzy confidence interval of fuzzy system reliability model. The good interval estimator for the fuzzy confidence interval is the obtained coverage probabilities that can cover the expected confidence coefficient with the narrowest expected length.

#### 5. Conclusions and Discussions

This paper presents an innovative modeling approach when dealing with uncertainties in the RMSS. The Markov process for the RMSS with a fuzzy failure rate and a fuzzy repair rate is considered and the fuzzy system reliability is constructed. Interval estimation of the fuzzy system reliability model is constructed by extending the concepts of two-sided confidence interval of the true parameters to the case where both parameter and random variables are fuzzy based on the central limit theorem. Recently, much research proposed an approach based on fuzzy data for constructing a fuzzy confidence interval, but without considering the performance analysis of interval estimator. In this research, the performance of fuzzy confidence interval is assessed based on the coverage probability and the expected length.

From the study, it is seen that estimation of the fuzzy confidence interval of fuzzy system reliability for RMSS will be effective when the sample size is . This results in lower coverage probability and upper coverage probability which covered in the expected confidence coefficient at and the narrowest lower expected length when and the narrowest upper expected length when . Accordingly, we conclude that the model presented herein is an effective estimation method at and . This study is suitable for the system reliability of multistate system where the accurate data are fuzzy values. In further work, fuzzy confidence intervals of system reliability for more complex systems are created. In particular, performance of interval estimator is also being considered based on the coverage probability and the expected length.

#### Conflict of Interests

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

#### Acknowledgments

This work is supported by the Science Achievement Scholarship of Thailand (SAST) and the Faculty of Science, KMUTT.

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