Fuzzy Availability Assessment for a Repairable Multistate Series-Parallel System

This paper considers a repairable multistate series-parallel system (RMSSPS) with fuzzy parameters. It is assumed that the system components are independent, and their state transition rates and performance rates are fuzzy values. The fuzzy universal generating function technique is adopted to determine fuzzy state probability and fuzzy performance rate of the system. On the basis of 𝛼 -cut approach and the extension principle, parametric programming technique is employed to obtain the 𝛼 -cuts of some indices for the system. The system fuzzy availability is defined as the ability of the system to satisfy fuzzy consumer demand. A special assessment approach is developed for evaluating the fuzzy steady-state availability of the system with the fuzzy demand. A flow transmission system with three components is presented to demonstrate the validity of the proposed method.


Introduction
The study of repairable systems is an important topic in engineering systems.System availability is very good evaluation for performance of repairable systems and occupies an increasingly important issue in power plants, manufacturing systems, industrial systems, and transportation systems, and so forth.
In the real-world problems, many repairable systems are designed to perform their intended tasks in a given environment.One type of these repairable systems is repairable multistate system (RMSS).The RMSS is able to perform its task with various distinguished levels of efficiency usually referred to as performance rates.Since the number of RMSS states increases very rapidly with the increase in the number of its components, the universal generating function (UGF) technique was introduced and proved to be efficient in evaluating the reliability of the multistate systems [1][2][3][4].
The repairable multistate series-parallel system (RMSSPS) model is frequently used in practice and has been extensively studied for many years.The conventional study for the RMSSPS considers the assumptions of the exact state transition rate and performance rate for each system component.However, in many engineering applications it is very difficult to obtain accurate and sufficient data to estimate the precise values of the state transition rate and performance rate for each system component.For this reason, the concept of fuzzy reliability has been introduced and developed by several authors [5][6][7][8][9].
Fuzzy set theory proposed by Zadeh [10] is a very good approach to deal with fuzzy uncertainty and has gained successful applications in various fields.It provides useful tools to investigate and analyze imprecision phenomena in queuing systems [11], rock engineering classification systems [12], transport systems [13], manufacturing systems [14], supply chain problems [15], and various optimization problems [16][17][18].Wong and Lai [19] provided a survey of applications of the fuzzy set theory technique in production and operations management and pointed out that nearly every application is potentially able to realize some of the benefits of fuzzy set theory.Furthermore, fuzzy set theory also provides useful methodology to analyze the reliability in uncertain systems.It can deal with the problem of lacking of inaccuracy or fluctuation data for system components in reliability analysis of some realistic engineering systems.Thus, it is necessary to introduce fuzzy set theory into the reliability theory to deal with reliability of the system with uncertain parameters.The theory of fuzzy reliability has been developed on the basis of 2 Discrete Dynamics in Nature and Society fuzzy set theory.Many research works on the application of fuzzy set theory to problems in reliability or availability of systems were presented in [20][21][22][23], and a systemic review on fuzzy reliability of systems with binary-state was provided by Cai [24].
Recently, fuzzy reliability research has focused on reliability evaluation of fuzzy multistate systems.Ding et al. [25,26] proposed firstly the concept of fuzzy multistate system and assessed the fuzzy reliability of fuzzy multistate systems with fuzzy demand using fuzzy UGF method.Liu et al. [27,28] investigated the dynamic fuzzy state probabilities, fuzzy performance rates, and fuzzy availability for fuzzy unrepairable multistate elements and fuzzy unrepairable multistate system according to the parametric programming technique and the extension principle.Bamrungsetthapong and Pongpullponsak [29] discussed fuzzy confidence interval for the fuzzy reliability of a RMSSPS with a fuzzy failure rate and a fuzzy repair rate and considered the performance of fuzzy confidence interval based on the coverage probability and the expected length.
Steady-state availability of a repairable system, as a system performance measure, is the probability that the system is performing satisfactorily over a reasonable period of time [30].The RMSS availability is defined as the ability of the system to satisfy consumer demand, which is equal to the sum of the probabilities of occurrence of system states in which the system performance rates satisfy consumer demand.Comparable work on the steady-state availability assessment for the RMSSPS with fuzzy state transition rate and fuzzy performance rate is rarely found in the literature.This motivates us to develop the fuzzy steady-state availability assessment for a RMSSPS with fuzzy parameters.According to [4], in this paper, we assumed that the fuzzy performance rate of each multistate parallel subsystem is the sum of the fuzzy performance rates of its all components, and the fuzzy performance rate of the entire RMSSPS is the minimum of the fuzzy performance rates of all parallel subsystems.The purpose of this study is to utilize the -cut approach, the extension principle, and parametric programming technique to determine the fuzzy state probability and fuzzy performance rate of the RMSSPS and to evaluate the fuzzy steadystate availability of the system with the fuzzy consumer demand.
The rest of the paper is organized as follows.Section 2 introduces the description of the system considered here and analyzes the fuzzy state probability of multistate component.The fuzzy state probabilities and fuzzy performance rates of the repairable parallel subsystem and the RMSSPS are presented in Sections 3 and 4, respectively.The fuzzy availability assessment method for the RMSSPS is given in Section 5.An illustrative example is presented in Section 6.Finally, Section 7 gives conclusions.

RMSSPS with Fuzzy Parameters
The RMSSPS we considered here is composed of  subsystems connected in series, and each subsystem  consists of   components in parallel,  = 1, 2, . . ., .The structure of a RMSSPS is shown in Figure 1.
Each component  of subsystem  has   + 1 different states corresponding to the fuzzy performance rates which can be represented by ordered fuzzy values g = ( g,0 , g,1 , . . ., g,  ), and g, ,  = 0, 1, . . .,   , is arbitrary fuzzy number.It is assumed that transitions for each component can only occur between adjacent states, and the state transition diagram is shown in Figure 2. The transition rate from state  to state  − 1 is presented as the fuzzy value λ (,  − 1), and the transition rate from state  − 1 to state  is presented as the fuzzy value μ ( − 1, ),  = 1, 2, . . .,   , and λ (,  − 1) and μ ( − 1, ) are arbitrary fuzzy numbers.
With the fuzzy transition rate, the state probability of the component  of subsystem  for the steady state is also a fuzzy value denoted as p,  ,   = 0, 1, . . .,   ; let p = ( p,0 , p,1 , . . ., p,  ).
The fuzzy transition rate matrix for the component  of subsystem  is given as ) . ( Based on the state transition diagram (Figure 2) and the fuzzy transition rate matrix, the balance equations for steady state of the component  of subsystem  are given by μ (0, 1) p,0 = λ (1, 0) p,1 , According to the state probability distribution of the conventional multistate component, we have where Let {  (,  − 1) |  λ (  (,  − 1)) where Λ  (,  − 1) and   ( − 1, ) are the crisp universal sets of the transition rate from state  to state  − 1 and the transition rate from state  − 1 to state  for the component  of subsystem .
The -cut of p can be determined as The -cut of the fuzzy performance rate g can be determined as where Υ = { ,  ∈  ,  ;  = 1, 2, . . .,   }.The lower bound ()   and the upper bound (  )   of the -cut of g can be obtained using the following parametric programming:

Fuzzy Steady-State Availability Assessment for the RMSSPS
For the RMSSPS, the system availability is defined as the ability of the system to satisfy consumer demand.Here, the consumer demand is presented as fuzzy value ω, and the steady-state availability of the system is the fuzzy probability that the fuzzy performance rate of the system satisfies the fuzzy demand ω.So, we have where ( g ≥ ω) denote the possibility degree of g ≥ ω.
For crisp performance rate   and consumer demand , (  ≥ ) = 1 if   ≥ , and (  ≥ ) = 0 if   < .In the fuzzy system model, the relationship between g and ω is illustrated in Figure 3 under triangular fuzzy number.It can be seen from Figure 3 that ( g ≥ ω1 ) = 0, ( g ≥ ω2 ) = 1, and 0 < (g  ≥ ω3(4) ) < 1.For 0 < (g  ≥ ω3( 4 Based on the analysis above, we can replace g ≥ ω with r ≥ 0; that is ( g ≥ ω) = (r  ≥ 0).According to [27,28], the possibility degree of r ≥ 0 is defined as However, the definition of the possibility degree of r ≥ 0 in ( 21) is given by the membership function of r .When the performance rates of the components and consumer demand are triangular fuzzy numbers, the membership function of r can be easily obtained.If system has a lot of components, and the performance rates of the components and consumer demand are arbitrary fuzzy representation, it is difficult to obtain the explicit expression of the membership function of r , and some approximation techniques may be developed to calculate the membership function.Here, we tackle the problem by using the -cut approach.Based on the -cuts of g and ω, the -cut of r can be easily determined.Let [(  )   , (  )   ] denote the -cut of r ; we define the possibility degree of r ≥ 0 as Hence, the fuzzy steady-state availability of the system for a given fuzzy demand ω can be rewritten as We denote the -cut of Ã( ω) by ( Ã( ω))  = [(())   , (())    ]; the lower and upper bounds of ( Ã( ω))  can be determined using the following parametric programming: In order to obtain rational outcome of the system steadystate availability, the equality constraint ∑  =0   = 1 is added into the parametric programming.

Illustrative Example
Consider a flow transmission multistate series-parallel system with three repairable components.In this example, the system consists of  = 2 parallel subsystems connected in series, and subsystems 1 and 2 consist of  1 = 2 and  2 = 1 components, respectively.The parameters for the components are represented as triangular fuzzy values.The fuzzy transition rates and fuzzy performance rates of each component are presented in Tables 1 and 2, respectively.Suppose that the fuzzy demand ω is a triangular fuzzy number (1.9, 2.1, 2.3).

Figure 2 :
Figure 2: The state-transition diagram for the component  of subsystem .

Figure 3 :
Figure 3: Fuzzy performance rate and fuzzy demand.

Table 1 :
The fuzzy transition rates for the components of each subsystem.

Table 3 :
The -cuts of the fuzzy state probabilities at 11  values.