Fuzzy availability assessment for a discrete time repairable multi-state series-parallel system

Abstract

A discrete time repairable multi-state series-parallel system with fuzzy data is considered. The lifetimes and the repair times of the system components are assumed to be geometrically distributed random variables with fuzzy parameters, and the performance levels of the system components are presented as fuzzy values. Fuzzy set theory is employed to deal with the uncertain problem of the system. A discrete time fuzzy Markov model is proposed to assess the fuzzy state probability of binary component. The fuzzy universal generating function technique is adopted to evaluate fuzzy state probability and fuzzy performance level of the entire system. An availability assessment approach is proposed to compute the dynamic fuzzy availability of the system with the fuzzy consumer demand. Based on the extension principle, the α - cuts of some indices for the system are obtained by using parametric programming technique. Finally, a numerical example is presented.

Keywords

Discrete time multi-state system fuzzy availability fuzzy universal generating function parametric programming

1 Introduction

The reliability models for continuous time multi-state systems have been studied extensively in the past, and have been successfully used for solving various reliability problems in many engineering systems. However, in some engineering circumstances, the lifetimes and the repair times of the systems (components) can not be measured with calendar time. In such situations, these lifetimes and repair times can be expressed in terms of the number of working (repairing) periods or the number of working (repairing) cycles. Moreover, reliability data are often grouped or truncated [5]. Therefore, all these lifetimes and repair times are intrinsically discrete random variables.

The discrete time system model widely exists in system engineering domain, e.g. manufacturing systems [13 , 24], queueing systems [38 –40] and control systems [18 –20], etc. Some reliability models for the discrete time systems have been investigated in the previous research. The discrete time reliability modeling for Markov systems can be found in [1 , 35] and the reliability modeling for discrete time semi-Markov systems are investigated in [3 , 9]. Bracquemond and Gaudoin [5] presented a good overview of discrete probability distributions used in reliability for modeling discrete lifetimes of nonrepairable systems. Rocha-Martinez and Shaked [34] investigated a discrete time reliability model for a repairable system with discrete lifetimes and repair times of components. However, most of the reported works mainly focus on the issues of discrete time systems with two possible states (completely working and totally failed). In the real-world, many systems perform their intended tasks in a range of degraded performance levels. This type of system is called multi-state system [30]. The multi-state system has wide spread applications such as computer systems, power generation systems, andtransportation systems.

Multi-state systems are more flexible for reliability modeling in real engineering systems. Since the number of states for multi-state system increases very rapidly with the increase in the number of its components, the universal generating function (UGF) technique was proved to be efficient in evaluating the reliability of the multi-state systems [25 , 36]. It has been extensively studied in the literature for reliability assessment of multi-state system under the assumption that the system state probability does not change throughout system lifetime [15]. As stated in Kolowrocki and Kwiatuszewska-Sarnecka [23], many complex systems are often subjected to aging process which implies that the system state probability may gradually change with time. Therefore, it is of large practical value to investigate the dynamic reliability assessment for discrete time multi-state system with the state probability as a function of time. Chryssaphinou et al. [8] analyzed basic dynamic reliability measures of a multi-state system under discrete time semi-Markovian hypothesis. Eryilmaz [14] studied the mean residual life of a discrete time nonrepairable multi-state system when the states of the system are modeled by a Markov chain.

The conventional reliability study for multi-state system considers the assumptions of the exact state probability and performance level for each system component. However, in many engineering applications, it is very difficult to obtain accurate data to estimate the precise values of these parameters. Therefore, the basic concept and theory of fuzzy reliability based on fuzzy set theory have been introduced and developed by several authors [6 , 37]. Fuzzy set theory proposed by Zadeh [41] provides a useful tool for dealing with the problem of reliability evaluation for many realistic engineering systems with uncertainties. A systemic review on fuzzy reliability of systems with binary-state and uncertainties was provided by Cai [7]. Recently, fuzzy reliability research has focused on reliability evaluation of fuzzy multi-state systems. Ding et al. [11, 12] proposed firstly the concept of fuzzy multi-state system and fuzzy universal generating function (FUGF) method, and discussed the reliability of fuzzy multi-state systems. Liu et al. [30, 31] investigated the dynamic fuzzy state probabilities, fuzzy performance rates and fuzzy availability for fuzzy nonrepairable multi-state elements and fuzzy nonrepairable multi-state systems. Bamrungsetthapong and Pongpullponsak [2] discussed fuzzy confidence interval for the fuzzy reliability of a repairable multi-state system. Li et al. [26] analyzed the interval-valued reliability of nonrepairable multi-state systems. Hu et al. [17] studied the fuzzy steady-state availability of a repairable multi-state series-parallel system with fuzzy parameters according to the parametric programming technique and the extension principle.

However, the dynamic reliability/availability assessment for discrete time multi-state system with fuzzy data has been seldom discussed in the literature. This motivates us to develop the dynamic availability assessment for a discrete time repairable multi-state series-parallel system with fuzzy parameters. In this paper, we assumed that the failure rate, repair rate and performance level for each component are fuzzy values. The contributions of this work are summarized as follows. (1) A discrete time fuzzy Markov model is proposed to analyze the fuzzy state probability of the component. (2) The FUGF technique is employed to evaluate fuzzy state probability and fuzzy performance level of the entire system. (3) A fuzzy availability assessment approach is proposed to compute the dynamic fuzzy availability of the system with the fuzzy consumer demand. (4) Parametric programming technique is applied to obtain the α - cuts and membership functions of some indices for the system.

The remainder of this paper is organized as follows. Section 2 investigates the discrete time Markov model for binary component. The description of the system and the discrete time fuzzy binary Markov model are given in Section 3. The fuzzy state probability, fuzzy performance level and fuzzy availability of the system are also discussed in this section. A numerical example is presented in Section 4. Finally, Section 5 gives conclusions.

2 Discrete time Markov model for binary component

Consider a binary repairable component defined on the states {0, 1}, where 0 represents working state and 1 represents failure state. It is assumed that the lifetime and the repair time of the component have geometric distributions with mean values 1/p and 1/q, respectively. Let T_c and T_r denote respectively the lifetime and the repair time of the component, we have $P {T_{c} = k} = p {(1 - p)}^{k - 1}, k = 1, 2, \dots$ and $P {T_{r} = k} = q {(1 - q)}^{k - 1}, k = 1, 2, \dots$

Assume that X_k denotes the state of the component at the end of the kth time period (such as hour, day, month, etc.). Then {X_k, k = 1, 2, … } forms a binary Markov chain with the following one-step transition probability matrix $P = [\begin{matrix} p_{00} & p_{01} \\ p_{10} & p_{11} \end{matrix}] = [\begin{matrix} 1 - p & p \\ q & 1 - q \end{matrix}] .$

Let P_i (k) denote the probability that the component is in state i (i = 0, 1) at time k, we have

$\begin{matrix} [P_{0} (k), P_{1} (k)] & = & [P_{0} (k - 1), P_{1} (k - 1)] \\ \times [\begin{matrix} 1 - p & p \\ q & 1 - q \end{matrix}] \end{matrix}$ (1) with initial conditions P₀ (0) =1 and P₁ (0) =0. Using recursive method, we can obtain

$\begin{matrix} [P_{0} (k), P_{1} (k)] & = & [P_{0} (0), P_{1} (0)] \\ \times {[\begin{matrix} 1 - p & p \\ q & 1 - q \end{matrix}]}^{k}, \end{matrix}$ (2) where $\begin{matrix} {[\begin{matrix} 1 - p & p \\ q & 1 - q \end{matrix}]}^{k} \\ = [\begin{matrix} \frac{q + p (1 - p - q)^{k}}{p + q} & \frac{p - p (1 - p - q)^{k}}{p + q} \\ \frac{q - q (1 - p - q)^{k}}{p + q} & \frac{p + q (1 - p - q)^{k}}{p + q} \end{matrix}] . \end{matrix}$

Therefore, the dynamic state probabilities of the component can be obtained as $P_{0} (k) = \frac{q + p (1 - p - q)^{k}}{p + q}$ (3) and $P_{1} (k) = \frac{p - p (1 - p - q)^{k}}{p + q} .$ (4)

3 Discrete time fuzzy repairable multi-state series-parallel system

3.1 Description for the system

Discrete time fuzzy repairable multi-state series-parallel system is composed of n subsystems connected in series, and each subsystem i (i = 1, 2, …, n) has m_i different parallel components with fuzzy failure rates, fuzzy repair rates and fuzzy performance levels. The structure of the system is shown in Fig. 1. In the subsystem i, each component j has two different states: working state (corresponding to the fuzzy performance level ${\tilde{ω}}_{ij})$ and failure state (corresponding to the performance level 0). For the component j in the subsystem i, it is assumed that the lifetime and the repair time have geometric distributions with fuzzy rates, and the failure rate and the repair rate are determined by fuzzy values ${\tilde{p}}_{ij}$ and ${\tilde{q}}_{ij}$ , respectively.

3.2 Discrete time fuzzy binary Markov model

Based on the description of the discrete time fuzzy multi-state series-parallel system, the state probability of the component j in the subsystem i at time k must also be a fuzzy value denoted as ${\tilde{P}}_{ij, l} (k)$ , l = 0 or 1 (working state or failure state). In order to obtain the fuzzy dynamic probability ${\tilde{P}}_{ij, l} (k)$ , the discrete time fuzzy Markov model for binary component is introduced. The fuzzy one-step transition probability matrix for the component j in the subsystem i is given as $\tilde{P} = [\begin{matrix} 1 - {\tilde{p}}_{ij} & {\tilde{p}}_{ij} \\ {\tilde{q}}_{ij} & 1 - {\tilde{q}}_{ij} \end{matrix}] .$

The fuzzy dynamic probability ${\tilde{P}}_{ij, l} (k)$ can be computed from the following recursive equation

$\begin{matrix} [{\tilde{P}}_{ij, 0} (k), {\tilde{P}}_{ij, 1} (k)] & = & [{\tilde{P}}_{ij, 0} (k - 1), {\tilde{P}}_{ij, 1} (k - 1)] \\ \times [\begin{matrix} 1 - {\tilde{p}}_{ij} & {\tilde{p}}_{ij} \\ {\tilde{q}}_{ij} & 1 - {\tilde{q}}_{ij} \end{matrix}], \end{matrix}$ (5) where the initial conditions are ${\tilde{P}}_{ij},_{0} (0) = 1$ and ${\tilde{P}}_{ij, 1} (0) = 0$ . Then the fuzzy dynamic probabilities of the component j in the subsystem i at time k are ${\tilde{P}}_{ij, 0} (k) = \frac{{\tilde{q}}_{ij} + {\tilde{p}}_{ij} (1 - {\tilde{p}}_{ij} - {\tilde{q}}_{ij})^{k}}{{\tilde{p}}_{ij} + {\tilde{q}}_{ij}}$ (6) and ${\tilde{P}}_{ij, 1} (k) = \frac{{\tilde{p}}_{ij} - {\tilde{p}}_{ij} (1 - {\tilde{p}}_{ij} - {\tilde{q}}_{ij})^{k}}{{\tilde{p}}_{ij} + {\tilde{q}}_{ij}} .$ (7)

Let $η_{{\tilde{p}}_{ij}} (p_{ij})$ and $η_{{\tilde{q}}_{ij}} (q_{ij})$ denote the membership functions of ${\tilde{p}}_{ij}$ and ${\tilde{q}}_{ij}$ , respectively. The α - cuts of ${\tilde{p}}_{ij}$ and ${\tilde{q}}_{ij}$ as crisp intervals can be determined as follows $\begin{matrix} {({\tilde{p}}_{ij})}_{α} & = & [min_{p_{ij} \in P_{ij}} {p_{ij} | η_{{\tilde{p}}_{ij}} (p_{ij}) \geq α}, \\ max_{p_{ij} \in P_{ij}} {p_{ij} | η_{{\tilde{p}}_{ij}} (p_{ij}) \geq α}] \\ = & [{(p_{ij})}_{α}^{L}, {(p_{ij})}_{α}^{U}] \end{matrix}$ and $\begin{matrix} {({\tilde{q}}_{ij})}_{α} & = & [min_{q_{ij} \in Q_{ij}} {q_{ij} | η_{{\tilde{q}}_{ij}} (q_{ij}) \geq α}, \\ max_{q_{ij} \in Q_{ij}} {q_{ij} | η_{{\tilde{q}}_{ij}} (q_{ij}) \geq α}] \\ = & [{(q_{ij})}_{α}^{L}, {(q_{ij})}_{α}^{U}], \end{matrix}$ where P_ij and Q_ij are the crisp universal sets of the failure rate and the repair rate for the component j in the subsystem i, 0 ≤ α ≤ 1.

According to extension principle [41, 42], the α - cuts of ${\tilde{P}}_{ij, 0} (k)$ and ${\tilde{P}}_{ij, 1} (k)$ can be determined as $\begin{matrix} {({\tilde{P}}_{ij, 0} (k))}_{α} \\ = [min_{Ψ_{ij}} \frac{q_{ij} + p_{ij} (1 - p_{ij} - q_{ij})^{k}}{p_{ij} + q_{ij}} | \begin{matrix} η_{{\tilde{p}}_{ij}} (p_{ij}) \geq α \\ η_{{\tilde{q}}_{ij}} (q_{ij}) \geq α \end{matrix}, \\ max_{Ψ_{ij}} \frac{q_{ij} + p_{ij} (1 - p_{ij} - q_{ij})^{k}}{p_{ij} + q_{ij}} | \begin{matrix} η_{{\tilde{p}}_{ij}} (p_{ij}) \geq α \\ η_{{\tilde{q}}_{ij}} (q_{ij}) \geq α \end{matrix}] \\ = [{(P_{ij, 0} (k))}_{α}^{L}, {(P_{ij, 0} (k))}_{α}^{U}] \end{matrix}$

and $\begin{matrix} {({\tilde{P}}_{ij, 1} (k))}_{α} \\ = [min_{Ψ_{ij}} \frac{p_{ij} - p_{ij} (1 - p_{ij} - q_{ij})^{k}}{p_{ij} + q_{ij}} | \begin{matrix} η_{{\tilde{p}}_{ij}} (p_{ij}) \geq α \\ η_{{\tilde{q}}_{ij}} (q_{ij}) \geq α \end{matrix}, \\ max_{Ψ_{ij}} \frac{p_{ij} - p_{ij} (1 - p_{ij} - q_{ij})^{k}}{p_{ij} + q_{ij}} | \begin{matrix} η_{{\tilde{p}}_{ij}} (p_{ij}) \geq α \\ η_{{\tilde{q}}_{ij}} (q_{ij}) \geq α \end{matrix}] \\ = [{(P_{ij, 1} (k))}_{α}^{L}, {(P_{ij, 1} (k))}_{α}^{U}], \end{matrix}$ where Ψ_ij ={ p_ij ∈ P_ij, q_ij ∈ Q_ij }, 0 ≤ α ≤ 1. The lower and upper bounds of the α - cuts of ${\tilde{P}}_{ij, 0} (k)$ and ${\tilde{P}}_{ij, 1} (k)$ can be obtained by using parametric programming technique [21 , 31] as follows $\begin{matrix} {(P_{ij, 0} (k))}_{α}^{L} & = & min \frac{q_{ij} + p_{ij} (1 - p_{ij} - q_{ij})^{k}}{p_{ij} + q_{ij}} \\ s . t . {(p_{ij})}_{α}^{L} \leq p_{ij} \leq {(p_{ij})}_{α}^{U} \end{matrix}$ (8) ${(q_{ij})}_{α}^{L} \leq q_{ij} \leq {(q_{ij})}_{α}^{U},$ $\begin{matrix} {(P_{ij, 0} (k))}_{α}^{U} & = & max \frac{q_{ij} + p_{ij} (1 - p_{ij} - q_{ij})^{k}}{p_{ij} + q_{ij}} \\ s . t . {(p_{ij})}_{α}^{L} \leq p_{ij} \leq {(p_{ij})}_{α}^{U} \end{matrix}$ (9) ${(q_{ij})}_{α}^{L} \leq q_{ij} \leq {(q_{ij})}_{α}^{U},$ $\begin{matrix} {(P_{ij, 1} (k))}_{α}^{L} & = & min \frac{p_{ij} - p_{ij} (1 - p_{ij} - q_{ij})^{k}}{p_{ij} + q_{ij}} \\ s . t . {(p_{ij})}_{α}^{L} \leq p_{ij} \leq {(p_{ij})}_{α}^{U} \end{matrix}$ (10) ${(q_{ij})}_{α}^{L} \leq q_{ij} \leq {(q_{ij})}_{α}^{U},$ and $\begin{matrix} {(P_{ij, 1} (k))}_{α}^{U} & = & max \frac{p_{ij} - p_{ij} (1 - p_{ij} - q_{ij})^{k}}{p_{ij} + q_{ij}} \\ s . t . {(p_{ij})}_{α}^{L} \leq p_{ij} \leq {(p_{ij})}_{α}^{U} \end{matrix}$ (11) ${(q_{ij})}_{α}^{L} \leq q_{ij} \leq {(q_{ij})}_{α}^{U} .$

3.3 Fuzzy state probability and fuzzy performance level for the system

The fuzzy state probability and fuzzy performance level for the discrete time fuzzy multi-state series-parallel system is evaluated through the FUGF method. The concept of the FUGF is introduced in Ding and Lisnianski [11]. The FUGF of a discrete random variable Z is defined as $\begin{matrix} \tilde{u} (z) & = & {\tilde{p}}_{1} \cdot z^{{\tilde{g}}_{1}} + {\tilde{p}}_{2} \cdot z^{{\tilde{g}}_{2}} + \dots + {\tilde{p}}_{M} \cdot z^{{\tilde{g}}_{M}} \\ = & \sum_{j = 1}^{M} {\tilde{p}}_{j} \cdot z^{{\tilde{g}}_{j}}, \end{matrix}$ where the variable Z has M possible values and ${\tilde{p}}_{j}$ is the fuzzy probability that Z is equal to fuzzy value ${\tilde{g}}_{j}$ . The FUGF of the fuzzy multi-state system with n different components is written as $\begin{array}{l} \tilde{U} (z) = Θ_{ϕ} (\sum_{j_{1} = 1}^{M_{1}} {\tilde{p}}_{1 j_{1}} . z^{{\tilde{g}}_{1}, j_{1}}, \dots, \sum_{j_{n} = 1}^{M_{n}} {\tilde{p}}_{n j_{n}} . z^{{\tilde{g}}_{n}, j_{n}}) \\ = \sum_{j_{1} = 1}^{M_{1}} \sum_{j_{2} = 1}^{M_{2}} \dots \sum_{j_{n} = 1}^{M_{n}} (\prod_{l = 1}^{n} \tilde{p} l, j_{l} \cdot z^{ϕ ({\tilde{g}}_{1}, j_{1}, \dots, {\tilde{g}}_{n}, j_{n})}) \\ = \sum_{j_{1} = 1}^{M_{1}} \sum_{j_{2} = 1}^{M_{2}} \dots \sum_{j_{n} = 1}^{M_{n}} ({\tilde{p}}_{i} \cdot z^{{\tilde{g}}_{i}}), \end{array}$ where ${\tilde{g}}_{i}$ is the fuzzy performance level of the system in its state i, ${\tilde{p}}_{i}$ is the fuzzy state probability associated with the state i. The structure function φ (·) represents the fuzzy performance level of the entire system consisting of different components in terms of individual fuzzy performance levels of components, and ${\tilde{g}}_{i} = φ ({\tilde{g}}_{1, j_{1}}, \dots, {\tilde{g}}_{n, j_{n}})$ , ${\tilde{p}}_{i} = \prod_{l = 1}^{n} {\tilde{p}}_{l, j_{l}}$ .

Based on Equations (6) and (7), the FUGF of the component j in the subsystem i can be written as

$\begin{matrix} {\tilde{u}}_{ij} (z, k) & = & {\tilde{P}}_{ij, 1} (k) \cdot z^{0} + {\tilde{P}}_{ij, 0} (k) \cdot z^{{\tilde{ω}}_{ij}} \\ = & \frac{{\tilde{p}}_{ij} - {\tilde{p}}_{ij} (1 - {\tilde{p}}_{ij} - {\tilde{q}}_{ij})^{k}}{{\tilde{p}}_{ij} + {\tilde{q}}_{ij}} \cdot z^{0} \\ + \frac{{\tilde{q}}_{ij} + {\tilde{p}}_{ij} (1 - {\tilde{p}}_{ij} - {\tilde{q}}_{ij})^{k}}{{\tilde{p}}_{ij} + {\tilde{q}}_{ij}} \cdot z^{{\tilde{ω}}_{ij}} . \end{matrix}$ (12)

Applying the fuzzy composition operator ${\tilde{Θ}}_{φ_{P}}$ [11] and the FUGFs of the components in the subsystem i, the FUGF of the subsystem i can be determined as

$\begin{matrix} {\tilde{U}}_{i} (z, k) & = & {\tilde{Θ}}_{φ_{P}} ({\tilde{u}}_{i 1} (z, k), \dots, {\tilde{u}}_{{im}_{i}} (z, k)) \\ = & \prod_{j = 1}^{m_{i}} (\frac{{\tilde{p}}_{ij} - {\tilde{p}}_{ij} (1 - {\tilde{p}}_{ij} - {\tilde{q}}_{ij})^{k}}{{\tilde{p}}_{ij} + {\tilde{q}}_{ij}} \cdot z^{0} \\ + \frac{{\tilde{q}}_{ij} + {\tilde{p}}_{ij} (1 - {\tilde{p}}_{ij} - {\tilde{q}}_{ij})^{k}}{{\tilde{p}}_{ij} + {\tilde{q}}_{ij}} \cdot z^{{\tilde{ω}}_{ij}}) . \end{matrix}$ (13)

The FUGF ${\tilde{U}}_{i} (z, k)$ of the parallel subsystem i can be represented as follows ${\tilde{U}}_{i} (z, k) = \sum_{r_{i} = 1}^{M_{i}} {\tilde{P}}_{r_{i}} (k) \cdot z^{{\tilde{g}}_{r_{i}}},$ (14) where ${\tilde{P}}_{r_{i}} (k)$ is the fuzzy state probability associated with the state r_i (r_i = 1, 2, …, M_i) of the subsystem i at time k, it is function of ${\tilde{p}}_{ij}$ , ${\tilde{q}}_{ij}$ and k, that is ${\tilde{P}}_{r_{i}} (k) = {\tilde{P}}_{r_{i}} ({\tilde{p}}_{ij}, {\tilde{q}}_{ij}, k)$ , j = 1, 2, …, m_i. ${\tilde{g}}_{r_{i}}$ is the fuzzy performance level of the subsystem i in state r_i, ${\tilde{g}}_{r_{i}} = \sum_{j = 1}^{m_{i}} {\tilde{ω}}_{ij, l} (l = 0, 1)$ , ${\tilde{ω}}_{ij, l}$ is 0 when l = 0 and ${\tilde{ω}}_{ij, l}$ is ${\tilde{ω}}_{ij}$ when l = 1.

Applying the fuzzy composition operator ${\tilde{Θ}}_{φ_{S}}$ [11] and the FUGFs of the subsystems, the FUGF of the entire system can be determined as

$\begin{matrix} \tilde{U} (z, k) & = & {\tilde{Θ}}_{φ_{S}} (\tilde{U_{1}} (z, k), \dots, \tilde{U_{n}} (z, k)) \\ = & \sum_{r_{1} = 1}^{M_{1}} \sum_{r_{2} = 1}^{M_{2}} \dots \sum_{r_{n} = 1}^{M_{n}} (\prod_{i = 1}^{n} {\tilde{P}}_{r_{i}} (k) \\ \times z^{min {{\tilde{g}}_{r_{1}}, {\tilde{g}}_{r_{2}}, \dots, {\tilde{g}}_{r_{n}}}}) . \end{matrix}$ (15)

The FUGF $\tilde{U} (z, k)$ of the system can be represented as the following as $\tilde{U} (z, k) = \sum_{r = 1}^{M} {\tilde{P}}_{r} (k) \cdot z^{{\tilde{g}}_{r}},$ (16) where ${\tilde{P}}_{r} (k)$ is the fuzzy state probability associated with the state r (r = 1, 2, …, M) of the system at time k, and ${\tilde{P}}_{r} (k) = \prod_{i = 1}^{n} {\tilde{P}}_{r_{i}} ({\tilde{p}}_{ij}, {\tilde{q}}_{ij}, k)$ , j = 1, 2, …, m_i. ${\tilde{g}}_{r}$ is the fuzzy performance level of the system in state r, and ${\tilde{g}}_{r} = min {\sum_{j = 1}^{m_{1}} {\tilde{ω}}_{1 j, l}, \dots, \sum_{j = 1}^{m_{n}} {\tilde{ω}}_{nj, l}}$ .

Let $η_{{\tilde{ω}}_{ij, l}} (ω_{ij, l})$ denote the membership function of the fuzzy performance level ${\tilde{ω}}_{ij, l} ({\tilde{ω}}_{ij, l} = 0 or {\tilde{ω}}_{ij})$ for the component j in the subsystem i, the α - cut of ${\tilde{ω}}_{ij, l}$ as crisp interval can be determined as follows $\begin{matrix} {({\tilde{ω}}_{ij, l})}_{α} & = & [min_{ω_{ij, l} \in W_{ij}} {ω_{ij, l} | η_{{\tilde{ω}}_{ij, l}} (ω_{ij, l}) \geq α}, \\ max_{ω_{ij, l} \in W_{ij}} {ω_{ij, l} | η_{{\tilde{ω}}_{ij, l}} (ω_{ij, l}) \geq α}] \\ = & [{(ω_{ij, l})}_{α}^{L}, {(ω_{ij, l})}_{α}^{U}], \end{matrix}$ where W_ij is the crisp set of the performance level of the component j in the subsystem i, 0 ≤ α ≤ 1.

According to extension principle, the α - cut of the fuzzy performance level ${\tilde{g}}_{r}$ of the system can be obtained as follows $\begin{matrix} {({\tilde{g}}_{r})}_{α} \\ = [min_{W} min {\sum_{j = 1}^{m_{1}} ω_{1 j, l}, \dots, \sum_{j = 1}^{m_{n}} ω_{nj, l}} | \begin{matrix} η_{{\tilde{ω}}_{ij, l}} (ω_{ij, l}) \geq α \end{matrix} \\ max_{W} min {\sum_{j = 1}^{m_{1}} ω_{1 j, l}, \dots, \sum_{j = 1}^{m_{n}} ω_{nj, l}} | \begin{matrix} η_{{\tilde{ω}}_{ij, l}} (ω_{ij, l}) \geq α \end{matrix}] \\ = [{(g_{r})}_{α}^{L}, {(g_{r})}_{α}^{U}], \end{matrix}$

where W = w_ij,l ∈ W_ij,l, i = 1, 2, …, n, j = 1, 2, … , m_i, 0 ≤ α ≤ 1.

By using the parametric programming technique, we have $\begin{matrix} {(g_{r})}_{α}^{L} & = & min {min {\sum_{j = 1}^{m_{1}} ω_{1 j, l}, \dots, \sum_{j = 1}^{m_{n}} ω_{nj, l}}} \\ s . t . {(ω_{ij, l})}_{α}^{L} \leq ω_{ij, l} \leq {(ω_{ij, l})}_{α}^{U} \end{matrix}$ (17) $i = 1, 2, \dots, n, j = 1, 2, \dots, m_{i}$ and $\begin{matrix} {(g_{r})}_{α}^{U} & = & max {min {\sum_{j = 1}^{m_{1}} ω_{1 j, l}, \dots, \sum_{j = 1}^{m_{n}} ω_{nj, l}}} \\ s . t . {(ω_{ij, l})}_{α}^{L} \leq ω_{ij, l} \leq {(ω_{ij, l})}_{α}^{U} \end{matrix}$ (18) $i = 1, 2, \dots, n, j = 1, 2, \dots, m_{i} .$

The α - cut of ${\tilde{P}}_{r} (k) = \prod_{i = 1}^{n} {\tilde{P}}_{r_{i}} ({\tilde{p}}_{ij}, {\tilde{q}}_{ij}, k)$ can be obtained as $\begin{matrix} {({\tilde{P}}_{r} (k))}_{α} & = & [min_{Ψ} \prod_{i = 1}^{n} P_{r_{i}} (p_{ij}, q_{ij}, k) | \begin{matrix} η_{{\tilde{p}}_{ij}} (p_{ij}) \geq α \\ η_{{\tilde{q}}_{ij}} (q_{ij}) \geq α \end{matrix}, \\ max_{Ψ} \prod_{i = 1}^{n} P_{r_{i}} (p_{ij}, q_{ij}, k) | \begin{matrix} η_{{\tilde{p}}_{ij}} (p_{ij}) \geq α \\ η_{{\tilde{q}}_{ij}} (q_{ij}) \geq α \end{matrix}] \\ = & [{(P_{r} (k))}_{α}^{L}, {(P_{r} (k))}_{α}^{U}], \end{matrix}$ where Ψ = {p_ij ∈ P_ij, q_ij ∈ Q_ij, i = 1, 2, …, n, j = 1, 2, …, m_i}, 0 ≤ α ≤ 1.

By using the parametric programming technique, we have

$\begin{matrix} {(P_{r} (k))}_{α}^{L} & = & min \prod_{i = 1}^{n} P_{r_{i}} (p_{ij}, q_{ij}, k) \\ s . t . {(p_{ij})}_{α}^{L} \leq p_{ij} \leq {(p_{ij})}_{α}^{U} \end{matrix}$ (19) $\begin{matrix} {(q_{ij})}_{α}^{L} \leq q_{ij} \leq {(q_{ij})}_{α}^{U} \\ i = 1, 2, \dots, n, j = 1, 2, \dots, m_{i} \end{matrix}$ and

$\begin{matrix} {(P_{r} (k))}_{α}^{U} & = & max \prod_{i = 1}^{n} P_{r_{i}} (p_{ij}, q_{ij}, k) \\ s . t . {(p_{ij})}_{α}^{L} \leq p_{ij} \leq {(p_{ij})}_{α}^{U} \end{matrix}$ (20) $\begin{matrix} {(q_{ij})}_{α}^{L} \leq q_{ij} \leq {(q_{ij})}_{α}^{U} \\ i = 1, 2, \dots, n, j = 1, 2, \dots, m_{i} . \end{matrix}$

3.4 Fuzzy availability assessment for the system

The fuzzy availability of the system, which is denoted by $\tilde{A} (\tilde{ω}, k)$ , is defined as the fuzzy probability that the fuzzy performance level of the system satisfies the fuzzy consumer demand $\tilde{ω}$ at time k. According to the FUGF of the system, we have $\tilde{A} (\tilde{ω}, k) = \sum_{r = 1}^{M} {\tilde{p}}_{r} (k) \cdot p ({\tilde{g}}_{r} \geq \tilde{ω}),$ (21) where $p ({\tilde{g}}_{r} \geq \tilde{ω})$ denote the possibility of ${\tilde{g}}_{r} \geq \tilde{ω}$ .

Let $η_{\tilde{ω}} (ω)$ denote the membership function of the fuzzy demand $\tilde{ω}$ , the α - cut of $\tilde{ω}$ can be obtained as $\begin{matrix} {(\tilde{ω})}_{α} & = & [min_{ω \in ϒ} {ω | η_{\tilde{ω}} (ω) \geq α}, max_{ω \in ϒ} {ω | η_{\tilde{ω}} (ω) \geq α}] \\ = & [{(ω)}_{α}^{L}, {(ω)}_{α}^{U}], \end{matrix}$ where ϒ is the crisp universal set of the consumer demand, 0 ≤ α ≤ 1.

To obtain the fuzzy availability of the system, the possibility of ${\tilde{g}}_{r} \geq \tilde{ω}$ must be defined. Here, we tackle the problem by using the α - cut approach and method of interval number ranking based on possibility [10]. Based on the $α - cuts [{(g_{r})}_{α}^{L}, {(g_{r})}_{α}^{U}]$ and $[{(ω)}_{α}^{L}, {(ω)}_{α}^{U}]$ , the possibility of ${\tilde{g}}_{r} \geq \tilde{ω}$ is defined as

$\begin{matrix} p ({\tilde{g}}_{r} \geq \tilde{ω})_{α} \\ = \frac{max {0, L_{g_{r}} + L_{ω} - max {0, {(ω)}_{α}^{U} - {(g_{r})}_{α}^{L}}}}{L_{g_{r}} + L_{ω}}, \end{matrix}$ (22) where $L_{g_{r}} = {(g_{r})}_{α}^{U} - {(g_{r})}_{α}^{L}$ , $L_{ω} = {(ω)}_{α}^{U} - {(ω)}_{α}^{L}$ , 0 ≤ α ≤ 1.

We denote the α - cut of fuzzy availability of the entire system for a given fuzzy demand $\tilde{ω}$ at time k by ${(\tilde{A} (\tilde{ω}, k))}_{α} = [{(A (ω, k))}_{α}^{L}, {(A (ω, k))}_{α}^{U}]$ . By using the parametric programming technique, ${(A (ω, k))}_{α}^{L}$ and ${(A (ω, k))}_{α}^{U}$ can be determined as $\begin{matrix} {(A (ω, k))}_{α}^{L} & = & min \sum_{r = 1}^{M} P_{r} (k) \cdot p ({\tilde{g}}_{r} \geq \tilde{ω})_{α} \\ s . t . {(P_{r} (k))}_{α}^{L} \leq P_{r} (k) \leq {(P_{r} (k))}_{α}^{U} \end{matrix}$ (23) $\begin{matrix} r = 1, 2, \dots, M \\ \sum_{r = 1}^{M} P_{r} (k) = 1 \end{matrix}$ and $\begin{matrix} {(A (ω, k))}_{α}^{U} & = & max \sum_{r = 1}^{M} P_{r} (k) \cdot p ({\tilde{g}}_{r} \geq \tilde{ω})_{α} \\ s . t . {(P_{r} (k))}_{α}^{L} \leq P_{r} (k) \leq {(P_{r} (k))}_{α}^{U} \end{matrix}$ (24) $\begin{matrix} r = 1, 2, \dots, M \\ \sum_{r = 1}^{M} P_{r} (k) = 1 . \end{matrix}$

4 Numerical example

Consider a repairable flow transmission multi-state series-parallel system with discrete lifetime and repair time. In this example, the system consists of two subsystems in series and two different components in each subsystem in parallel. The fuzzy failure rate, fuzzy repair rate and fuzzy performance level for each component are treated as trapezoidal fuzzy number, and tabulated in Table 1. Suppose that the fuzzy demand $\tilde{ω}$ is a trapezoidal fuzzy number (3,4,5,6).

According to the method of determining α - cut of the trapezoidal fuzzy number, the α - cuts of the fuzzy failure rate and fuzzy repair rate for each componentare $\begin{matrix} [{(p_{11})}_{α}^{L}, {(p_{11})}_{α}^{U}] & = & [0.02 + 0.01 α, 0.05 - 0.01 α], \\ [{(q_{11})}_{α}^{L}, {(q_{11})}_{α}^{U}] & = & [0.2 + 0.1 α, 0.5 - 0.1 α], \\ [{(p_{12})}_{α}^{L}, {(p_{12})}_{α}^{U}] & = & [0.03 + 0.01 α, 0.06 - 0.01 α], \\ [{(q_{12})}_{α}^{L}, {(q_{12})}_{α}^{U}] & = & [0.3 + 0.1 α, 0.6 - 0.1 α], \\ [{(p_{21})}_{α}^{L}, {(p_{21})}_{α}^{U}] & = & [0.04 + 0.01 α, 0.07 - 0.01 α], \\ [{(q_{21})}_{α}^{L}, {(q_{21})}_{α}^{U}] & = & [0.4 + 0.1 α, 0.7 - 0.1 α], \\ [{(p_{22})}_{α}^{L}, {(p_{22})}_{α}^{U}] & = & [0.05 + 0.01 α, 0.08 - 0.01 α], \\ [{(q_{22})}_{α}^{L}, {(q_{22})}_{α}^{U}] & = & [0.5 + 0.1 α, 0.8 - 0.1 α] . \end{matrix}$

The fuzzy dynamic probabilities ${\tilde{P}}_{ij, 0} (k)$ and ${\tilde{P}}_{ij, 1} (k)$ for the components of the subsystems 1 and 2 at time k can be determined according to Equations (6) and (7), i = 1, 2, j = 1, 2. The FUGFs of the components in each subsystem can be described as follows $\begin{matrix} {\tilde{u}}_{11} (z, k) & = & {\tilde{P}}_{11, 1} (k) \cdot z^{0} + {\tilde{P}}_{11, 0} (k) \cdot z^{(1, 2, 3, 4)}, \\ {\tilde{u}}_{12} (z, k) & = & {\tilde{P}}_{12, 1} (k) \cdot z^{0} + {\tilde{P}}_{12, 0} (k) \cdot z^{(5, 6, 7, 8)}, \\ {\tilde{u}}_{21} (z, k) & = & {\tilde{P}}_{21, 1} (k) \cdot z^{0} + {\tilde{P}}_{21, 0} (k) \cdot z^{(10, 11, 12, 13)}, \\ {\tilde{u}}_{22} (z, k) & = & {\tilde{P}}_{22, 1} (k) \cdot z^{0} + {\tilde{P}}_{22, 0} (k) \cdot z^{(9, 10, 11, 12)} . \end{matrix}$

Based on the individual FUGFs of the components in each subsystem, we can get the fuzzy UGFs of the subsystems 1 and 2 by using operator ${\tilde{Θ}}_{φ_{P}}$ as follows $\begin{matrix} {\tilde{U}}_{1} (z, k) & = & {\tilde{P}}_{11, 1} (k) {\tilde{P}}_{12, 1} (k) \cdot z^{0} \\ + {\tilde{P}}_{11, 0} (k) {\tilde{P}}_{12, 1} (k) \cdot z^{(1, 2, 3, 4)} \\ + {\tilde{P}}_{11, 1} (k) {\tilde{P}}_{12, 0} (k) \cdot z^{(5, 6, 7, 8)} \\ + {\tilde{P}}_{11, 0} (k) {\tilde{P}}_{12, 0} (k) \cdot z^{(6, 8, 10, 12)} \end{matrix}$ and $\begin{matrix} {\tilde{U}}_{2} (z, k) & = & {\tilde{P}}_{21, 1} (k) {\tilde{P}}_{22, 1} (k) \cdot z^{0} \\ + {\tilde{P}}_{21, 1} (k) {\tilde{P}}_{22, 0} (k) \cdot z^{(9, 10, 11, 12)} \\ + {\tilde{P}}_{21, 0} (k) {\tilde{P}}_{22, 1} (k) \cdot z^{(10, 11, 12, 13)} \\ + {\tilde{P}}_{21, 0} (k) {\tilde{P}}_{22, 0} (k) \cdot z^{(19, 21, 23, 25)} . \end{matrix}$

Based on ${\tilde{U}}_{1} (z, k)$ and ${\tilde{U}}_{2} (z, k)$ , the FUGF of the entire system can be obtained by using the operator ${\tilde{Θ}}_{φ_{S}}$ as follows $\begin{matrix} \tilde{U} (z, k) & = & {\tilde{P}}_{11, 1} (k) {\tilde{P}}_{12, 1} (k) ({\tilde{P}}_{21, 1} (k) {\tilde{P}}_{22, 1} (k) \\ + {\tilde{P}}_{21, 0} (k) {\tilde{P}}_{22, 1} (k) + {\tilde{P}}_{21, 1} (k) {\tilde{P}}_{22, 0} (k) \\ + {\tilde{P}}_{21, 0} (k) {\tilde{P}}_{22, 0} (k)) \cdot z^{0} \\ + {\tilde{P}}_{21, 1} (k) {\tilde{P}}_{22, 1} (k) ({\tilde{P}}_{11, 0} (k) {\tilde{P}}_{12, 1} (k) \\ + {\tilde{P}}_{11, 1} (k) {\tilde{P}}_{12, 0} (k) + {\tilde{P}}_{11, 0} (k) {\tilde{P}}_{12, 0} (k)) \cdot z^{0} \\ + {\tilde{P}}_{11, 0} (k) {\tilde{P}}_{12, 1} (k) ({\tilde{P}}_{21, 0} (k) {\tilde{P}}_{22, 1} (k) \\ + {\tilde{P}}_{21, 1} (k) {\tilde{P}}_{22, 0} (k) + {\tilde{P}}_{21, 0} (k) {\tilde{P}}_{22, 0} (k)) \\ \times z^{(1, 2, 3, 4)} + {\tilde{P}}_{11, 1} (k) {\tilde{P}}_{12, 0} (k) \\ \times ({\tilde{P}}_{21, 0} (k) {\tilde{P}}_{22, 1} (k) + {\tilde{P}}_{21, 1} (k) {\tilde{P}}_{22, 0} (k) \\ + {\tilde{P}}_{21, 0} (k) {\tilde{P}}_{22, 0} (k)) \cdot z^{(5, 6, 7, 8)} \\ + {\tilde{P}}_{11, 0} (k) {\tilde{P}}_{12, 0} (k) ({\tilde{P}}_{21, 0} (k) {\tilde{P}}_{22, 1} (k) \\ + {\tilde{P}}_{21, 1} (k) {\tilde{P}}_{22, 0} (k) + {\tilde{P}}_{21, 0} (k) {\tilde{P}}_{22, 0} (k)) \\ \times z^{(6, 8, 10, 12)} . \end{matrix}$

The FUGF of the system $\tilde{U} (z, k)$ can be written as $\tilde{U} (z, k) = \sum_{r = 1}^{4} {\tilde{P}}_{r} (k) \cdot z^{{\tilde{g}}_{r}} .$ That is, the fuzzy performance levels of the system are $\begin{matrix} {\tilde{g}}_{1} & = & 0, {\tilde{g}}_{2} = (1, 2, 3, 4), \\ {\tilde{g}}_{3} & = & (5, 6, 7, 8), {\tilde{g}}_{4} = (6, 8, 10, 12) . \end{matrix}$ The fuzzy state probabilities corresponding to the different fuzzy performance levels are $\begin{matrix} {\tilde{P}}_{1} (k) & = & {\tilde{P}}_{11, 1} (k) {\tilde{P}}_{12, 1} (k) ({\tilde{P}}_{21, 1} (k) {\tilde{P}}_{22, 1} (k) \\ + {\tilde{P}}_{21, 0} (k) {\tilde{P}}_{22, 1} (k) + {\tilde{P}}_{21, 1} (k) {\tilde{P}}_{22, 0} (k) \\ + {\tilde{P}}_{21, 0} (k) {\tilde{P}}_{22, 0} (k)) \\ + {\tilde{P}}_{21, 1} (k) {\tilde{P}}_{22, 1} (k) ({\tilde{P}}_{11, 0} (k) {\tilde{P}}_{12, 1} (k) \\ + {\tilde{P}}_{11, 1} (k) {\tilde{P}}_{12, 0} (k) + {\tilde{P}}_{11, 0} (k) {\tilde{P}}_{12, 0} (k)), \end{matrix}$ $\begin{matrix} {\tilde{P}}_{2} (k) & = & {\tilde{P}}_{11, 0} (k) {\tilde{P}}_{12, 1} (k) ({\tilde{P}}_{21, 0} (k) {\tilde{P}}_{22, 1} (k) \\ + {\tilde{P}}_{21, 1} (k) {\tilde{P}}_{22, 0} (k) + {\tilde{P}}_{21, 0} (k) {\tilde{P}}_{22, 0} (k)), \end{matrix}$ $\begin{matrix} {\tilde{P}}_{3} (k) & = & {\tilde{P}}_{11, 1} (k) {\tilde{P}}_{12, 0} (k) ({\tilde{P}}_{21, 0} (k) {\tilde{P}}_{22, 1} (k) \\ + {\tilde{P}}_{21, 1} (k) {\tilde{P}}_{22, 0} (k) + {\tilde{P}}_{21, 0} (k) {\tilde{P}}_{22, 0} (k)), \end{matrix}$ $\begin{matrix} {\tilde{P}}_{4} (k) & = & {\tilde{P}}_{11, 0} (k) {\tilde{P}}_{12, 0} (k) ({\tilde{P}}_{21, 0} (k) {\tilde{P}}_{22, 1} (k) \\ + {\tilde{P}}_{21, 1} (k) {\tilde{P}}_{22, 0} (k) + {\tilde{P}}_{21, 0} (k) {\tilde{P}}_{22, 0} (k)) . \end{matrix}$

The α - cut (0 ≤ α ≤ 1) of ${\tilde{P}}_{r} (k)$ (r = 1, 2, 3, 4) can be obtained by Equations (19) and (20). For example, we denote α - cut of ${\tilde{P}}_{4} (2)$ by $[{(P_{4} (2))}_{α}^{L}, {(P_{4} (2))}_{α}^{U}]$ , the lower bound ${(P_{4} (2))}_{α}^{L}$ and the upper bound ${(P_{4} (2))}_{α}^{U}$ of α - cut for ${\tilde{P}}_{4} (2)$ can be solved using the following parametric programming $\begin{matrix} {(P_{4} (2))}_{α}^{L} & = & min {P_{11, 0} (2) P_{12, 0} (2) (P_{21, 0} (2) P_{22, 1} (2) \\ + P_{21, 1} (2) P_{22, 0} (2) + P_{21, 0} (2) P_{22, 0} (2))} \\ s . t . 0.02 + 0.01 α \leq p_{11} \leq 0.05 - 0.01 α \\ 0.2 + 0.1 α \leq q_{11} \leq 0.5 - 0.1 α \\ 0.03 + 0.01 α \leq p_{12} \leq 0.06 - 0.01 α \\ 0.3 + 0.1 α \leq q_{12} \leq 0.6 - 0.1 α \\ 0.04 + 0.01 α \leq p_{21} \leq 0.07 - 0.01 α \\ 0.4 + 0.1 α \leq q_{21} \leq 0.7 - 0.1 α \\ 0.05 + 0.01 α \leq p_{22} \leq 0.08 - 0.01 α \\ 0.5 + 0.1 α \leq q_{22} \leq 0.8 - 0.1 α \end{matrix}$ and $\begin{matrix} {(P_{4} (2))}_{α}^{U} & = & max {P_{11, 0} (2) P_{12, 0} (2) (P_{21, 0} (2) P_{22, 1} (2) \\ + P_{21, 1} (2) P_{22, 0} (2) + P_{21, 0} (2) P_{22, 0} (2))} \\ s . t . 0.02 + 0.01 α \leq p_{11} \leq 0.05 - 0.01 α \\ 0.2 + 0.1 α \leq q_{11} \leq 0.5 - 0.1 α \\ 0.03 + 0.01 α \leq p_{12} \leq 0.06 - 0.01 α \\ 0.3 + 0.1 α \leq q_{12} \leq 0.6 - 0.1 α \\ 0.04 + 0.01 α \leq p_{21} \leq 0.07 - 0.01 α \\ 0.4 + 0.1 α \leq q_{21} \leq 0.7 - 0.1 α \\ 0.05 + 0.01 α \leq p_{22} \leq 0.08 - 0.01 α \\ 0.5 + 0.1 α \leq q_{22} \leq 0.8 - 0.1 α . \end{matrix}$

With the help of Matlab program system, the α - cuts of ${\tilde{P}}_{r} (k)$ (r = 1, 2, 3, 4) can be computed at some distinct α values. The rough shapes of the membership functions for ${\tilde{P}}_{r} (k)$ can be determined according to different α values. The α - cuts of the fuzzy state probabilities ${\tilde{P}}_{r} (k)$ (r = 1, 2, 3, 4) at 11 distinct α values: 0.0, 0.1, …, 1.0 when k = 2 are presented in Table 2. From Table 2, we can see that: (1) it is impossible for the values of ${\tilde{P}}_{4} (2)$ to fall below 0.8463 or exceed 0.9119 though the ${\tilde{P}}_{4} (2)$ is fuzzy, (2) the α - cut of ${\tilde{P}}_{4} (2)$ at α = 1 contains the values from 0.8636 to 0.8854, which are the most possible values for ${\tilde{P}}_{4} (2)$ . Similarly, one can interpret the rest of the data for ${\tilde{P}}_{r} (2)$ (r = 1, 2, 3, 4) shown in Table 2. The membership functions of fuzzy state probabilities ${\tilde{P}}_{r} (k)$ (r = 1, 2, 3, 4) at time k = 2 are presented in Figures 2–5.

The α - cuts of the fuzzy availability $\tilde{A} (\tilde{ω}, k)$ at different α values can be computed by Equations (23) and (24). Table 3 presents the α - cuts of the fuzzy availability at 11 distinct α values: 0.0, 0.1, …, 1.0 when k = 2, 4, 8, 12. From Table 3, we can see that the ranges of the availability at time k = 2 for α = 0 and α = 1.0 are approximately [0.8824, 0.9794] and [0.9103, 0.9429], respectively. This indicates that the availability cannot exceed 0.9794 or fall below 0.8824 and the most possible values of the availability are 0.9103 to 0.9429. Figures 6–9 illustrate the membership functions of the fuzzy availability when k = 2, 4, 8 and 12, respectively. Figures 10–12 show the system fuzzy availability decreases as time k increases, and the decrease is rapid initially and tends to vanish as k becomes large. Moreover, from the results presented in Table 3 and Figures 10–12, we can see that the interval length of α - cuts for fuzzy availability will decrease with the increase of k. This indicates that the interval length of α - cut for fuzzy availability at the steady state is very small.

5 Conclusions

In this paper, we investigate the fuzzy state probability and fuzzy availability for a discrete time repairable multi-state series-parallel system with uncertainties. The lifetimes and the repair times of the system components have geometric distributions with uncertain or imprecise parameters, where the uncertain or imprecise parameters for each component are presented as fuzzy numbers. A discrete time binary fuzzy Markov model is proposed to analyze the fuzzy state probability for each component. The fuzzy set theory and the FUGF technique are employed to evaluate fuzzy state probability and fuzzy performance level of the system. A fuzzy availability assessment approach is proposed to compute the fuzzy availability of the system with the fuzzy consumer demand. By using parametric programming technique, we obtain the α - cuts and the membership functions of the fuzzy state probability and the fuzzy availability for the system.

In the discrete time reliability model, discrete lifetime or repair time distributions usually contain not only geometric distribution, but also discrete Weibull distribution, geometric-Weibull distribution and other discrete time distributions. In future research, we will concern the development of dynamic fuzzy reliability/availability assessment for the multi-state system with discrete Weibull distribution and geometric-Weibull distribution. We will also investigate fuzzy reliability/availability optimization design for the discrete time multi-state system.

Footnotes

Acknowledgments

This paper is partially supported by the Natural Science Foundation of Hebei Province (No. A2014203096), the National Natural Science Foundation of China (No. 11201408), the Science Research Project of Yanshan University (No. 13LGA017). The authors are grateful to the editor and the reviewers for their insightful comments and suggestions.

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