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

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 n ≥ 100. In addition, the optimal α-cut for the narrowest lower expected length and the narrowest upper expected length are considered.


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][3][4]. Recent research has focused on reliability evaluation and optimization of MSS [5][6][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 2 The Scientific World Journal 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.

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]).

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 (0, + Δ ) is split into two parts. First, consider a transition from state to state in the small interval (0 , Δ ) and then a transition from state to state in the rest of the interval. It is seen thaṫ where = − and the following notation for the time derivativė( ) = ( / ) ( ) 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 asṖ Likewise, split the time interval (0, + Δ ) into two parts. Consider a transition from to in the interval (0, ) and then a transition from to in the small interval ( , + Δ ). It is seen thaṫ 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 P( ), where ∑ =0 ( ) = 1 for all in .

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 − 1 denoted by , −1 . On the other hand, a minor repair is a repair that causes the element transition from state − 1 to state denoted by −1, . 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 (0) = 1, −1 (0) = −2 (0) = ⋅ ⋅ ⋅ = 1 (0) = 0. The unreliability function for the multistate element will be a sum of the probabilities of the unacceptable states 1, . . . , . Therefore reliability function is given by

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: The triangular membership function of fuzzy set̃is given bỹ= Let R be a universal set of real numbers and̃a fuzzy subset of R.̃is referred to as triangular fuzzy number (TFN) as shown in Figure 2, denoted bỹ= ( , , ). Let̃be the -cut level, so we havẽ The Scientific World Journal in Figure 1 where state is the best state and state 1 is the worst state. The minor failures between states and − 1 are determined by fuzzy valuẽ, −1 . And the minor repairs between states − 1 and are determined by fuzzy valuẽ− 1, . With the fuzzy transition intensities, the state probability of elements at time must also be a fuzzy valuẽ( ), and then the Chapman-Kolmogorov equation with fuzzy transition intensities can be written as where the initial conditions arẽ(0) The unreliability function for the multistate element will be a sum of the probabilities of the unacceptable states 1, . . . , (∑ =1 ( )). Laplace transform is adopted to transform (13) into linear equation as follows: wherẽ( ) −̃(0) = [̃( )/ ]. Given that̃( ) is a function of̃,̃, and , then the inverse Laplace transform is executed to get thẽ( ) in time domain: wherẽ( ) = miñ * ( , ; , ( , ) ≥ ) and̃( ) = max̃ * ( , ; , ( , ) ≥ ). Then the fuzzy reliability of repairable multistate element is given by The fuzzy state probabilities can be obtained in the form that

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. Let̃be a fuzzy reliability function and̃a fuzzy unreliability function of subsystem , which is connected in series, and component , which is connected in parallel ( = 1, 2, . . . , and = 1, 2, . . . , ), respectively. Let a failure rate and a repair rate be represented bỹand̃at time , respectively. Then the fuzzy system reliability of repairable multistate series-parallel system (RMSS) at time is given bỹ

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 bỹ= [(̃) , (̃) ]. A method to calculate the fuzzy coverage probability and expected length for̃is demonstrated as follows.
Step 3. Compute fuzzy confidence intervals of̃= where is the lower fuzzy confidence interval and is the upper fuzzy confidence interval at repeated time , = 1, 2, . . . , .
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 = 10, 000. The -test statistic hypothesis testing is used in confirming the level of the coverage probability as follows: 0 : the coverage probability is not less than the expected confidence coefficient ( ≥ 0 ), 1 : the coverage probability is lower than the expected confidence coefficient ( when the significance level is = 0.95 and the test statistic Since the criterion in the test is < − , the hypothesis 0 cannot be rejected if where is the coverage probability,̂is the coverage probability estimated from this study, 0 is the expected confidence coefficient, and is the repeated time.

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.
wherẽ( ) is fuzzy unreliability function of repairable multistate system at subsystem connected in series and component connected in parallel, respectively. Suppose that 2,1 and̃2 ,1 are triangular membership function in each ∈ From the example system in Figure 6, RMSS with fuzzy failure rate (̃2 ,1 ) and a fuzzy repair rate (̃2 ,1 ) are considered. The fuzzy system reliability in each ∈ [0, 1] 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 bỹ= [(̃) , (̃) ]. The confidence coefficient with = 0.95 is defined and fuzzy random variables from normal distribution are generated by using MATLAB [28]. Then the fuzzy coverage probability and the expected length for̃are 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 = 10, 000 and the criterion used in comparing the coverage probability at significance level 95% will bê Considering a fuzzy confidence interval at -cut which gives the coverage probability higher than 0.9464 at significant level 95% 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 − cut 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.

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 ∈ [0, 1] 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 95% 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 = 10, 000.
In Figure 7(a), it is seen that at the sample size = 30, 50 ( < 100) there are some -cut that give lower coverage probability covering in the expected confidence coefficient 8 The Scientific World Journal     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 -cut = 0.9 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 -cut = 0.7 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 -cut 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 coefficient̂≥ 0.9464 with the narrowest expected length.

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 100(1 − )% 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 ≥ 100. This results in lower coverage probability and upper coverage probability which covered in the expected confidence coefficient at ∈ [0, 1] and the narrowest lower expected length when -cut = 0.9 and the narrowest upper expected length when -cut = 0.7. Accordingly, we conclude that the model presented herein is an effective estimation method at -cut = 0.9 and -cut = 0.7. 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.