Selective maintenance optimization for fuzzy multi-state systems

Abstract

This paper addresses a selective maintenance optimization problem for a fuzzy multi-state system composed of fuzzy multi-state elements. Due to insufficient data and unpredictable external working conditions, both the performance capacity and states transition intensities of multi-state elements cannot be known precisely, but are represented by fuzzy numbers. Additionally, both the durations of a break and a succeeding mission are also treated as fuzzy values. To maximize the fuzzy probability of a system successfully completing a succeeding mission, a selective maintenance model is proposed to identify an optimal subset of maintenance activities to be performed on some elements in the system. A solution algorithm containing three rules to eliminate inferior solutions and narrow down elements’ states combinations is proposed to resolve the new selective maintenance model in a computationally efficient manner. An illustrative example of an archibald is presented to demonstrate the effectiveness of the proposed model.

Keywords

Selective maintenance fuzzy multi-state system (FMSS)fuzzy multi-state element (FMSE)fuzzy mission fuzzy Markov model (FMM)fuzzy universal generating function (FUGF)

Notations

MSS

Multi-State System

MSE

Multi-State Element

FMSS

Fuzzy Multi-State System

FMSE

Fuzzy Multi-State Element

FMM

Fuzzy Markov Model

UGF

Universal Generating Function

FUGF

Fuzzy Universal Generating Function

Number of FMSEs in a FMSS

{\tilde{t}}_{b}

Fuzzy duration of a break

{\tilde{t}}_{d}

Fuzzy duration of a suceeding mission

k _i

The best state of element i

{\tilde{g}}_{i}

Set of the possible fuzzy performance capacities of element i

{\tilde{g}}_{(i, j)}

Fuzzy performance capacity of element i in state j

{\tilde{p}}_{i} (\tilde{t})

Set of the fuzzy state probabilities of element i at fuzzy time $\tilde{t}$

{\tilde{p}}_{(i, j)} (\tilde{t})

Fuzzy probability of element i staying in state j at fuzzy time $\tilde{t}$

{\tilde{G}}_{i} (\tilde{t})

Fuzzy performance capacity of element i at fuzzy time $\tilde{t}$

{\tilde{g}}_{s}

Set of the possible fuzzy performance capacities of a system

{\tilde{g}}_{(s, u)}

Fuzzy performance capacity of a system at its state u

{\tilde{p}}_{s} (\tilde{t})

Set of the fuzzy state probabilities of a system at time $\tilde{t}$

{\tilde{p}}_{(s, u)} (\tilde{t})

Fuzzy probability of a system at state u at time $\tilde{t}$

{\tilde{G}}_{s} (\tilde{t})

Fuzzy performance capacity of a system at time $\tilde{t}$

φ (•)

System structure function

(a, b, c)

A triangular fuzzy number

l_{h, j}^{i}

Maintenance activity to restore element i from state h to j (0 ≤ h ≤ j ≤ k_i)

s_{h, j}^{i}

Resources consumption for restoring element i from state h to j (0 ≤ h ≤ j ≤ k_i)

μ_{\tilde{X}} (x)

Membership function of a fuzzy number $\tilde{X}$

{\tilde{X}}_{α}

α-cut level set of a fuzzy number $\tilde{X}$ (0 ≤ α ≤ 1)

{\tilde{λ}}_{(j, h)}^{i}

Fuzzy transition intensity of element i from state j to h

m _i

State of element i after maintenance

{\tilde{Ω}}_{φ} (•)

Fuzzy composition function

\tilde{w}

Fuzzy demand of a suceeding mission

SU_α (L)

Fuzzy probability of a system successfully completing the next mission with the maintenance option L

V _i

Binary variable indicating whether element i is to be maintained

Y _i

State of element i before conducting maintenance

c_{i, l_{Y_{i}, m_{i}}^{i}}

Cost of recovering element i from state Y_i to state m_i

c_{i}^{fix}

Fixed maintenance cost for element i

c_{i, l_{Y_{i}, m_{i}}^{i}}^{var}

Variable maintenance cost of recovering element i from state Y_i to state m_i

c (L)

Total maintenance cost of the maintenance option L

t_{i, l_{Y_{i}, m_{i}}^{i}}

Time of recovering element i from state Y_i to state m_i

t_{i}^{fix}

Fixed maintenance time for element i

t_{i, l_{Y_{i}, m_{i}}^{i}}^{var}

Variable maintenance time of recovering element i from state Y_i to state m_i

t (L)

Total maintenance time of the maintenance option L

1 Introduction

In many practical applications, systems, such as nuclear plants, military equipment, fertilizer plants, aircrafts, are designed to execute a sequence of missions with a break between two adjacent missions [1 –3]. Failures of these systems during missions may cause severe threat to personal security or damage to environment [4, 5]. Therefore, maintenance should be carried out during the breaks between missions to avoid the potential risk of mission failure. However, due to the limited maintenance resources, such as time, budget, spare parts etc., it may not be possible to perform all the desired maintenance actions within a break. In such case, decision-makers have to make a trade-off between the system performance and resources consumption, and identify the optimal subset of maintenance activities to be performed. This sort of maintenance strategy is the so-called selective maintenance [6].

The selective maintenance model was firstly developed by Rice et al. [7] in 1998, and in their work, a series-parallel binary-state system composed of s-independent and exponentially distributed components was investigated. The model can determine a subset of failed components to be replaced before executing the next mission. Motivated by some practical applications, selective maintenance optimization for much more practical cases has been extensively investigated in Ref. [8 –11]. As many systems manifest multi-state nature, the selective maintenance problem for multi-state systems (MSSs) has received an extensive amount of attentions in the past few years. For instance, Chen et al. [12] discussed a selective maintenance problem for multi-state systems and components. They assumed that both systems and components were homogenous with K+1 possible states. A probability distribution matrix was given to evaluate the probabilities of components’ states at the end of the next mission. Liu et al. [13] investigated a selective maintenance problem for a multi-state system composed of binary-state components. The capacity of each individual component in the MSS had a cumulative contribution to the performance capacity of the entire system. The universal generating function (UGF) method was adopted to evaluate the probability of a MSS successfully completing the next mission. Pandey et al. [14] discussed a selective maintenance problem for a multi-state system consisting of multi-state components. The state probability distributions of components at the end of the next mission were assessed by the continuous-time Markov chain. The method proposed by Chen et al. [12] was followed by Dao et al. [15] to investigate the selective maintenance problem in MSSs subject to economic dependence.

Although considerable efforts have been made to study the selective maintenance optimization for MSSs. An implicit hypothesis of these reported works is that the knowledge of systems and mission profiles, such as state performance capacity, state transition intensity, mission durations, etc., is deterministic and precisely known. This assumption may not be always true in engineering practice. Uncertainty is unavoidable and extensively exists in reliability evaluation [16 –20], system operation [21, 22], maintenance optimisation [23, 24], reliability optimisation [25 –27] system diagnosis [28], and state performance assessment [29]. Because (1) it is very difficult to precisely estimate coefficients or parameters due to model simplification or inaccurate and sparse data [30, 31], (2) the performance of systems and elements is vague and also affected by the external uncertain working environment and, (3) the durations of missions and breaks are unpredictable and may be affected by various uncertain factors [32, 33]. To the best of our knowledge, only limited reported works have addressed the selective maintenance problem considering uncertain missions in the context of binary-state systems (see Djelloul et al. [34], Khatab et al. [35] and Cao et al. [36]). In their studies, the durations of missions and/or breaks were assumed to be random variables characterized by probability distributions. Fuzzy set, as an alternative tool to quantify imprecise information, can also be utilized to characterize the uncertainties associated with system deterioration model and the durations of missions and breaks in the case of limited data or vague knowledge. For example, in a battle, the launcher of an archibald system is requested to perform a sequence of missions, and maintenance can only be performed within an uncertain combat mission break. The durations of missions and breaks can only be represented by an ambiguous way, such as “a mission duration is around 10 days”, “a mission break is between 3 and 5 hours, but close to 3.5 hours”. The launcher can work at different states with respect to different shell shooting capacities. Because of the effects of unpredictable external environment and insufficient data, the performance capacity of each state and states transition intensities can only be quantified by an ambiguous way. In this situation, decision-makers have to choose a subset of maintenance actions for some elements, so as to maximize the probability of the system successfully completing the succeeding mission. As fuzziness cannot be straightforwardly tackled by the reported works on the selective maintenance optimization, there is a pressing need to develop a new selective maintenance model to address the aforementioned challenges.

Furthermore, the selective maintenance model is oftentimes a complex, non-linear, discrete problem, and the solution space may be very huge in many practical applications. Some meta-heuristic algorithms, such as the genetic algorithm, the differential evolution algorithm etc., are capable of identifying the optimal global solution in a computationally efficient way. However, the solution efficiency of these algorithms can be further improved from twofold: (1) some inferior solutions (maintenance strategies) in terms of system structure, resources constraints etc. can be eliminated directly before performing an intensive searching. (2) Based on the system structure, some elements’ states combinations can be discarded intuitively when evaluating the probability of the system successfully completing the next mission. Hence, by taking the advantage of these two aspects, the solution space and elements’ states combinations can be firstly reduced before executing a full enumeration method or meta-heuristic algorithms.

Motivated by the real-life applications, a selective maintenance optimization problem for FMSSs confronting fuzzy durations of missions and breaks is investigated in this paper. It can be considered as an important extension of the work in Ref. [14] by taking account of the fuzzy uncertainty in modeling and decision-making. Compared to the exiting works on selective maintenance optimization, the unique features of the present work are threefold: (1) The system and element can operate in partially degraded states between perfectly functioning and completely failed, and the performance capacity of each state and states transition intensities are represented as fuzzy values. (2) The durations of missions and breaks are uncertain, and they are also characterized by fuzzy values instead of crisp values or probability distributions. (3) A solution algorithm containing some rules is proposed to improve the solution efficiency in the fuzzy context.

The remainder of this paper is organized as follows. Section 2 describes the FMSSs in question and some basic assumptions. Section 3 discusses the proposed selective maintenance model in details. An approach to solve the proposed selective maintenance model is presented in Section 4. In Section 5, an illustrative example together with some discussions is given to illustrate the effectiveness the proposed method. A conclusion and some future works are summarized in Section 6.

2 Fuzzy multi-state systems

Suppose that a FMSS is composed of n s-independent FMSEs. The element i (i = 1, 2, ⋯ , n) has k_i + 1 states, and the corresponding performance capacities are represented by an ordered fuzzy set ${\tilde{g}}_{i} = {{\tilde{g}}_{(i, 0)}, {\tilde{g}}_{(i, 1)}, \dots, {\tilde{g}}_{(i, k_{i})}}$ . 0 corresponds to the completely failed state, whereas k_i is the perfectly functioning state. Other intermediate states, denoted as {1, 2, ⋯ , k_i - 1}, are the partially functioning states between the two extremes, i.e., perfectly functioning and completely failed. As the state transition intensities of FMSEs are fuzzy-valued, the performance capacity ${\tilde{G}}_{i} (t)$ of element i at any time t takes value from ${\tilde{g}}_{i}$ : ${\tilde{G}}_{i} (t) \in {\tilde{g}}_{i}$ . The fuzzy probability of element i staying at each state at time t can be denoted as ${\tilde{p}}_{i} (t) = {{\tilde{p}}_{(i, 0)} (t), {\tilde{p}}_{(i, 1)} (t), \dots, {\tilde{p}}_{(i, k_{i})} (t)}$ . The entire system has k_s + 1 possible performance capacities, which are determined by the performance capacities of all the elements and the system configuration [13]. The system performance capacity at time t is denoted by ${\tilde{G}}_{s} (t) = φ ({\tilde{G}}_{1} (t), {\tilde{G}}_{2} (t), \dots, {\tilde{G}}_{n} (t))$ , where φ (•) is the system structure function and ${\tilde{G}}_{s} (t) \in {\tilde{g}}_{s}$ , ${\tilde{g}}_{s} = {{\tilde{g}}_{(s, 0)}, {\tilde{g}}_{(s, 1)}, \dots, {\tilde{g}}_{(s, k_{s})}}$ . The state distribution of a system is denoted as ${\tilde{p}}_{s} (t) = {{\tilde{p}}_{(s, 0)} (t), {\tilde{p}}_{(s, 1)} (t), \dots, {\tilde{p}}_{(s, k_{s})} (t)}$ .

Given the fuzzy performance capacities and the fuzzy transition intensities of all the FMSEs, the state distribution and the reliability function of a FMSS can be evaluated by the parametric programming and the FUGF. The details of the evaluation process can be found in Ref. [37].

3 The proposed selective maintenance model

3.1 Problem description and maintenance resources consumption

Suppose that a FMSS has just finished the last mission (denoted as the previous mission in Fig. 1) and arrived a depot for maintenance. All the maintenance activities must be finished within a required mission break. However, due to the limited maintenance budget and time, only a subset of elements in the system can be maintained. Both the durations of missions and breaks are fuzzy values, denoted as ${\tilde{t}}_{d}$ and ${\tilde{t}}_{b}$ , respectively, as shown in Fig. 1. Each element has multiple maintenance options (including do nothing, imperfect maintenance and replacement) associated with different performance capacities to be restored to. The decision-makers need to determine a subset of elements to be maintained, as well as the maintenance action to be performed for each element, so as to maximize the fuzzy probability of the system successfully completing the next mission.

Fig.1

An illustration of a system executing the succeeding mission.

After the previous mission, an element can be in any one of the states {0, 1, ⋯ , k_i} due to deterioration. From the perspective of maintenance efficiency, the available maintenance actions to be performed on each component can be categorized as:

Do Nothing. No maintenance action is performed and elements will be left in its present state. For such a case, no maintenance resources (such as time and cost) are consumed.

Imperfect Maintenance. Imperfect maintenance can bring an element from its current state to a better state (not the best state) with a higher performance capacity. In general, maintenance actions with greater maintenance efficiencies will certainly consume more resources.

Replacement. An element after replacement will be in a brand-new condition at the beginning of the next mission.

The state transition and resources consumption associated to each maintenance option are shown in Fig. 2. For element i, all the possible maintenance actions and the corresponding resources consumption can be represented by a set ${(l_{0, 1}^{i}, s_{0, 1}^{i}), \dots, (l_{0, k_{i}}^{i}, s_{0, k_{i}}^{i}), (l_{1, 2}^{i}, s_{1, 2}^{i}), \dots, (l_{k_{i} - 1, k_{i}}^{i}, s_{k_{i} - 1, k_{i}}^{i})}$ , where $l_{h, j}^{i}$ and $s_{h, j}^{i}$ (0 ≤ h ≤ j ≤ k_i) denote the maintenance action recovering element i from state h to state j and the associated maintenance resources consumption, respectively. When h = j, it is “do nothing”, that is, the element is left in its present state. When j = k_i, $l_{h, j}^{i} = l_{h, k_{i}}^{i}$ and it denotes “replacement”.

Fig.2

The maintenance resources consumption and the state transition of element i for each maintenance option.

In addition to the above problem descriptions, all the basic assumptions are summarized as follows:

The elements’ states at the end of the previous mission are observable.

Each element has multiple maintenance options. Maintenance can certainly restore an element to its better state, and the associated resources consumption can be explicitly defined as shown in Fig. 2.

The degradation of elements can be modeled by homogeneous Markov models. The states transition intensities, as well as performance capacities, are represented by fuzzy values.

Maintenance can only be performed within a mission break.

3.2 The state distribution of a MSE based on FMM and FUGF

In this paper, the performance capacities and states transition intensities of all the elements are represented by triangular fuzzy numbers. For simplicity, a triangular fuzzy number $\bar{X}$ can be denoted as (a, b, c), of which the membership function is defined as: $μ_{\tilde{X}} (x) = {\begin{matrix} \frac{x - a}{b - a} & a \leq x \leq b \\ \frac{c - x}{c - b} & b \leq x \leq c \\ 0 & otherwise \end{matrix} .$ (1)

The α-cut level set (0 ≤ α ≤ 1) of the fuzzy number $\tilde{X}$ is defined as ${\tilde{X}}_{α} = {x | μ_{\tilde{X}} (x) \geq α}$ . Hence, the upper and lower bounds of ${\tilde{X}}_{α}$ , as presented in Fig. 3, can be computed by: $[{\tilde{X}}_{α}^{L}, {\tilde{X}}_{α}^{U}] = [\begin{matrix} min (x | μ_{\tilde{X}} (x) \geq α), \\ max (x | μ_{\tilde{X}} (x) \geq α) \end{matrix}] .$ (2)

Fig.3

The α-cut level set of a triangular fuzzy number (a, b, c).

Assume that the state of element i after performing a maintenance option is denoted by m_i (0 ≤ m_i ≤ k_i). The transition intensity between a pair of states during the next mission is denoted by a fuzzy value ${\tilde{λ}}_{(j, h)}^{i}$ (0 ≤ h ≤ j ≤ m_i). The state transitions of element i are presented in Fig. 4. Based on the fuzzy Markov model (FMM) proposed by Liu et al. [37], the state distributions of elements at the end of the next mission are fuzzy values too.

Fig.4

The state transitions of element i in the next mission.

Let the fuzzy transition intensity matrix of element i be denoted as: ${\tilde{λ}}^{i} = [\begin{matrix} {\tilde{λ}}_{(m_{i}, m_{i})}^{i} \dots {\tilde{λ}}_{(m_{i}, 0)}^{i} \\ \dots ⋱ \dots \\ {\tilde{λ}}_{(0, m_{i})}^{i} \dots {\tilde{λ}}_{(0, 0)}^{i} \end{matrix}],$ (3) where ${\tilde{λ}}_{(j, j)}^{i} = - \sum_{\binom{h = 0}{h \neq j}}^{m_{i}} {\tilde{λ}}_{(j, h)}^{i}$ . Based on the assumptions in Section 3.1, one can get ${\tilde{λ}}_{(j, h)}^{i} = 0$ for h > j. If the duration of the next mission ${\tilde{t}}_{d}$ is also a fuzzy number. Then, the Kolmogorov differential equations with fuzzy quantities can be written as: ${\begin{matrix} \frac{d {\tilde{p}}_{(i, m_{i})} ({\tilde{t}}_{d})}{dt} = - {\tilde{p}}_{(i, m_{i})} ({\tilde{t}}_{d}) \sum_{h = 0}^{m_{i} - 1} {\tilde{λ}}_{(m_{i}, h)}^{i} \\ \frac{d {\tilde{p}}_{(i, u_{i})} ({\tilde{t}}_{d})}{dt} = \sum_{j = u_{i} + 1}^{m_{i}} {\tilde{p}}_{(i, j)} ({\tilde{t}}_{d}) {\tilde{λ}}_{(j, u_{i})}^{i} \\ - {\tilde{p}}_{(i, u_{i})} ({\tilde{t}}_{d}) \sum_{h = 0}^{u_{i} - 1} {\tilde{λ}}_{(u_{i}, h)}^{i} \\ when 0 < u_{i} < m_{i} \\ \frac{d {\tilde{p}}_{(i, 0)} ({\tilde{t}}_{d})}{dt} = \sum_{j = 1}^{m_{i}} {\tilde{p}}_{(i, j)} ({\tilde{t}}_{d}) {\tilde{λ}}_{(j, 0)}^{i} \end{matrix},$ (4) where u_i(0 ≤ u_i ≤ m_i) denotes the state of element i at the end of the next mission. The initial conditions are set to ${\tilde{p}}_{(i, m_{i})} (0) = 1$ , and ${\tilde{p}}_{(i, 0)} (0) = \dots = {\tilde{p}}_{(i, u_{i})} (0) = \dots = {\tilde{p}}_{(i, m_{i} - 1)} (0) = 0$ . In order to reduce the complexity, the Laplace transform can be used to transform Equation (4) into a set of linear equations: ${\begin{matrix} s {\tilde{p}}_{(i, m_{i})} (s) - 1 = - {\tilde{p}}_{(i, m_{i})} (s) \sum_{h = 0}^{m_{i} - 1} {\tilde{λ}}_{(m_{i}, h)}^{i} \\ s {\tilde{p}}_{(i, u_{i})} (s) = \sum_{j = u_{i} + 1}^{m_{i}} {\tilde{p}}_{(i, j)} (s) {\tilde{λ}}_{(j, u_{i})}^{i} \\ - {\tilde{p}}_{(i, u_{i})} (s) \sum_{h = 0}^{u_{i} - 1} {\tilde{λ}}_{(u_{i}, h)}^{i} \\ s {\tilde{p}}_{(i, 0)} (s) = \sum_{j = 1}^{m_{i}} {\tilde{p}}_{(i, j)} (s) {\tilde{λ}}_{(j, 0)}^{i} \end{matrix} .$ (5)

Based on Equation (5), one can get ${\tilde{p}}_{(i, u_{i})} (s)$ . Then, the state distribution of element i at the end of the next mission with a fuzzy duration ${\tilde{t}}_{d}$ can be evaluated by the inverse Laplace transform as follows: ${\tilde{p}}_{(i, u_{i})} ({\tilde{t}}_{d}) = L^{- 1} ({\tilde{p}}_{(i, u_{i})} (s)),$ (6) where L^-1 (•) is the inverse Laplace operator. By introducing the FUGF [30], the probability distribution of the performance capacity of element i at the end of the next mission with a fuzzy duration of ${\tilde{t}}_{d}$ can be written as: ${\tilde{U}}_{i} (z, {\tilde{t}}_{d}) = \sum_{u_{i} = 0}^{m_{i}} {\tilde{p}}_{(i, u_{i})} ({\tilde{t}}_{d}) \cdot z^{{\tilde{g}}_{(i, u_{i})}} .$ (7)

As ${\tilde{p}}_{(i, u_{i})} ({\tilde{t}}_{d})$ is a function of a set of fuzzy values, i.e., ${\tilde{t}}_{d}$ and ${\tilde{λ}}^{i}$ . Its corresponding α-cut level set is written as:

$\begin{matrix} {\tilde{p}}_{(i, u_{i}) α} ({\tilde{t}}_{d}) \\ = [{\tilde{p}}_{(i, u_{i}) α}^{L} ({\tilde{t}}_{d}), {\tilde{p}}_{(i, u_{i}) α}^{U} ({\tilde{t}}_{d})] \\ = [min {\tilde{p}}_{(i, u_{i})} ({\tilde{t}}_{d α}, {\tilde{λ}}_{α}^{i}), max {\tilde{p}}_{(i, u_{i})} ({\tilde{t}}_{d α}, {\tilde{λ}}_{α}^{i})], \end{matrix}$ (8)

where ${\tilde{t}}_{d α}$ is the α-cut level set of ${\tilde{t}}_{d}$ , whereas ${\tilde{λ}}_{α}^{i}$ indicates the α-cut level set of all the ${\tilde{λ}}^{i}$ of component i and it can be solved by Equation (2). Based on Equation (2) and the parametric programming proposed by Liu et al. in Ref. [37], ${\tilde{p}}_{(i, u_{i}) α} ({\tilde{t}}_{d})$ , representing the state distribution of element i at the end of the next mission, can be readily evaluated.

3.3 The state distribution of a MSS based on FUGF

After executing the next mission, the entire system will be in one of k_s + 1 states ranging from the best state k_s to the worst state 0. Based on the possible combinations of components’ states after ${\tilde{t}}_{d}$ , the system performance capacity and the associated probability at any state u (u ∈ {0, 1, …, k_s}) can be evaluated by the FUGFs of components. For a system with arbitrary system structure φ (·), the FUGF of the system is written as:

$\begin{array}{l} {\tilde{U}}_{s} (z, {\tilde{t}}_{d}) \\ = {\tilde{Ω}}_{ϕ} (\sum_{u_{1} = 0}^{m_{1}} {\tilde{p}}_{(1, u_{1})} ({\tilde{t}}_{d}) \cdot z^{{\tilde{g}}_{(1, u_{_{1}})}}, \dots, \sum_{u_{n} = 0}^{m_{n}} {\tilde{p}}_{(n, u_{n})} ({\tilde{t}}_{d}) \cdot z^{{\tilde{g}}_{(n, u_{n})}}) \\ = \sum_{u_{1} = 0}^{m_{1}} \sum_{u_{2} = 0}^{m_{2}} \dots \sum_{u_{n} = 0}^{m_{n}} (\prod_{i = 1}^{n} {\tilde{p}}_{(i, u_{i})} ({\tilde{t}}_{d}) \cdot z^{ϕ ({\tilde{g}}_{(1, u_{_{1}})}, \dots, {\tilde{g}}_{(n, u_{n})})}) \\ = \sum_{u = 0}^{k_{s}} {\tilde{p}}_{(s, u)} ({\tilde{t}}_{d}) \cdot z^{{\tilde{g}}_{(s, u)}} \end{array}$ (9)

where ${\tilde{Ω}}_{φ} (\cdot)$ denotes the fuzzy composition function. ${\tilde{g}}_{(s, u)}$ and ${\tilde{p}}_{(s, u)} ({\tilde{t}}_{d})$ denote the system performance capacity and the associated probability of the system staying at state u at the end of the next mission, respectively. It can be seen from Equation (9) that the system state distribution is completely determined by the system structure φ (·) and the state distributions of all the elements. Likewise as Equation (8), the α-cut level set of ${\tilde{g}}_{(s, u)}$ and ${\tilde{p}}_{(s, u)} ({\tilde{t}}_{d})$ can be evaluated by the parametric programming proposed by Liu et al. [37], and they are written as: $\begin{matrix} {\tilde{g}}_{(s, u) α} \\ = [{\tilde{g}}_{(s, u) α}^{L}, {\tilde{g}}_{(s, u) α}^{U}] \\ = [\begin{matrix} min φ ({\tilde{g}}_{(1, u_{1}) α}, \dots, {\tilde{g}}_{(n, u_{n}) α}), \\ max φ ({\tilde{g}}_{(1, u_{1}) α}, \dots, {\tilde{g}}_{(n, u_{n}) α}) \end{matrix}], \end{matrix}$ (10) $\begin{matrix} {\tilde{p}}_{(s, u) α} ({\tilde{t}}_{d}) \\ = [{\tilde{p}}_{(s, u) α}^{L} ({\tilde{t}}_{d}), {\tilde{p}}_{(s, u) α}^{U} ({\tilde{t}}_{d})] \\ = [\begin{matrix} min (\sum_{\begin{matrix} {\tilde{g}}_{(s, u) α} = \\ φ ({\tilde{g}}_{(i, u_{i}) α}) \end{matrix}} \prod_{i = 1}^{n} {\tilde{p}}_{(i, u_{i})} ({\tilde{t}}_{d α}, {\tilde{λ}}_{α}^{i})), \\ max (\sum_{\begin{matrix} {\tilde{g}}_{(s, u) α} = \\ φ ({\tilde{g}}_{(i, u_{i}) α}) \end{matrix}} \prod_{i = 1}^{n} {\tilde{p}}_{(i, u_{i})} ({\tilde{t}}_{d α}, {\tilde{λ}}_{α}^{i})) \end{matrix}] . \end{matrix}$ (11) where ${\tilde{g}}_{(i, u_{i}) α}$ and ${\tilde{p}}_{(i, u_{i})} ({\tilde{t}}_{d α}, {\tilde{λ}}_{α}^{i})$ denote the α-cut level set of the performance capacity and the state distribution of element i, respectively.

3.4 The fuzzy probability of a system successfully completing the next mission

Suppose that the state of element i after the previous mission is Y_i (0 ≤ Y_i ≤ k_i). The objective of the selective maintenance problem is to determine a subset of maintenance actions $L = (l_{Y_{1}, m_{1}}^{1}, l_{Y_{2}, m_{2}}^{2}, \dots, l_{Y_{n}, m_{n}}^{n})$ to be performed within a break, so as to maximize the probability of a system successfully completing the next mission, that is, the performance capacity of a maintained FMSS is not less than the required mission demand level at the end of the next mission.

Assume that the total required maintenance time for a maintenance option L is denoted as t (L). The probability that the maintenance option L can be completed within a break with a fuzzy duration is defined as $Pr (t (L) \leq {\tilde{t}}_{b})$ . Obviously, t (L) is a crisp value and ${\tilde{t}}_{b}$ is a fuzzy value parameterized by a triangle fuzzy number, denoted as (a, b, c). It can be seen from Fig. 5 that when t (L₁) < a, all the maintenance activities can be completed within the fuzzy mission break, and thus $Pr (t (L) \leq {\tilde{t}}_{b}) = 1$ . On the other hand, if t (L₃) ≥ c, all the selected maintenance actions cannot be finished and $Pr (t (L) \leq {\tilde{t}}_{b}) = 0$ . However, if a < t (L₂) < c, maintenance actions may be or not be completed. The relative cardinality method proposed by Ding et al. [30], can be used to evaluate the value of $Pr (t (L) \leq {\tilde{t}}_{b})$ . Let ${\tilde{r}}_{t_{b}} = {\tilde{t}}_{b} - t (L)$ , and the α-cut set of $Pr (t (L) \leq {\tilde{t}}_{b})$ is, therefore, equal to the α-cut level relative cardinality of the fuzzy set ${\tilde{r}}_{t_{b}}$ . It is given by: $Pr (t (L) \leq {\tilde{t}}_{b})_{α} = | r_{t_{b}} |_{α}^{rel},$ (12) where the α-cut level relative cardinality of the fuzzy set ${\tilde{r}}_{t_{b}}$ (see Liu et al. [37]) is defined as: $| r_{t_{b}} |_{α}^{rel} = \frac{| {tr}_{t_{b}} |_{α}}{| r_{t_{b}} |_{α}},$ (13) where $t {\tilde{r}}_{t_{b}} = {r_{t_{b}}; μ_{{\tilde{r}}_{t_{b}}} (r_{t_{b}}) | r_{t_{b}} \geq 0}$ . |tr_{t
_b}|_α and |r_{t
_b}|_α are the α-cut level cardinality of $t {\tilde{r}}_{t_{b}}$ and ${\tilde{r}}_{t_{b}}$ , respectively. They are defined as: $| {tr}_{t_{b}} |_{α} = \int_{\begin{matrix} r_{t_{b}} \in R, r_{t_{b}} > 0 \\ μ_{{\tilde{r}}_{t_{b}}} (r_{t_{b}}) \geq α \end{matrix}} μ_{{\tilde{r}}_{t_{b}}} (r_{t_{b}}) {dr}_{t_{b}} (0 \leq α \leq 1) .$ (14) $| r_{t_{b}} |_{α} = \int_{\begin{matrix} r_{t_{b}} \in R \\ μ_{{\tilde{r}}_{t_{b}}} (r_{t_{b}}) \geq α \end{matrix}} μ_{{\tilde{r}}_{t_{b}}} (r_{t_{b}}) {dr}_{t_{b}} (0 \leq α \leq 1) .$ (15)

Fig.5

The break with a fuzzy duration vs. the total maintenance time of L.

Fig.6

The fuzzy performance capacities and the required mission demand level.

Considering a FMSS with k_s (L) +1 possible states during the next mission after performing a maintenance option L, based on the required demand level of the next mission, each state can be categorized into one of the three situations, namely acceptable state, failure state, and the states in between [30]. As shown in Fig. 6, if ${\tilde{g}}_{(s, u)} < \tilde{w}$ ( ${\tilde{g}}_{(s, u_{1})}$ in Fig. 6), the system performance capacity is lower than the mission demand and the state u is a failed state. If ${\tilde{g}}_{(s, u)} > \tilde{w}$ ( ${\tilde{g}}_{(s, u_{3})}$ in Fig. 6), the system performance capacity is greater than the demand level of a mission and the state u is an acceptable state. If the membership functions of ${\tilde{g}}_{(s, u)}$ and $\tilde{w}$ overlap each other ( ${\tilde{g}}_{(s, u_{2})}$ in Fig. 6), the state u may be somewhere between the acceptable state and the failed state. Hence, based on the relative cardinality, the probability that the maintained FMSS can meet the mission demand level at the α-cut level cardinality is equal to [30, 37]:

$\begin{matrix} {\tilde{R}}_{α} ({\tilde{t}}_{d}, \tilde{w} | L) \\ = Pr ({\tilde{G}}_{s} ({\tilde{t}}_{d}) \geq \tilde{w})_{α} \\ = \sum_{u = 0}^{k_{s} (L)} {\tilde{p}}_{(s, u) α} ({\tilde{t}}_{d}) \cdot \frac{| {sr}_{g} |_{α}}{| r_{g} |_{α}} (0 \leq α \leq 1), \end{matrix}$ (16) where ${\tilde{r}}_{g} = {\tilde{g}}_{(s, u)} - \tilde{w}$ and $s {\tilde{r}}_{g} = {r_{g}; μ_{{\tilde{r}}_{g}} (r_{g}) | r_{g} \geq 0}$ . |sr_g|_α and |r_g|_α are the α-cut level cardinality of $s {\tilde{r}}_{g}$ and ${\tilde{r}}_{g}$ , respectively, and they can be computed by Equations (14 and 15), respectively.

Hence, the fuzzy probability of a FMSS successfully completing the succeeding mission after performing a set of maintenance actions is equal to:

$\begin{matrix} {SU}_{α} (L) = {\tilde{R}}_{α} ({\tilde{t}}_{d}, \tilde{w} | L) \cdot Pr (t (L) \leq {\tilde{t}}_{b})_{α} \\ = (\sum_{u = 0}^{k_{s} (L)} {\tilde{p}}_{(s, u) α} ({\tilde{t}}_{d}) \cdot \frac{| {sr}_{g} |_{α}}{| r_{g} |_{α}}) \cdot \frac{| {tr}_{t_{b}} |_{α}}{| r_{t_{b}} |_{α}} (0 \leq α \leq 1) . \end{matrix}$ (17)

The interval of SU_α (L) can be, therefore, computed by:

$\begin{matrix} {SU}_{α} (L) = [{SU}_{α}^{L} (L), {SU}_{α}^{U} (L)] \\ = [\begin{matrix} min ((\sum_{u = 0}^{k_{s} (L)} {\tilde{p}}_{(s, u) α} ({\tilde{t}}_{d}) \cdot \frac{| {sr}_{g} |_{α}}{| r_{g} |_{α}}) \cdot \frac{| {tr}_{t_{b}} |_{α}}{| r_{t_{b}} |_{α}}), \\ max ((\sum_{u = 0}^{k_{s} (L)} {\tilde{p}}_{(s, u) α} ({\tilde{t}}_{d}) \cdot \frac{| {sr}_{g} |_{α}}{| r_{g} |_{α}}) \cdot \frac{| {tr}_{t_{b}} |_{α}}{| r_{t_{b}} |_{α}}) \end{matrix}], \end{matrix}$ (18) where 0 ≤ α ≤ 1.

3.5 Selective maintenance model

Let V_i be a binary variable indicating whether element i is selected to be maintained. It can be written as: $V_{i} = {\begin{matrix} 0 if Y_{i} = m_{i} \\ 1 if Y_{i} < m_{i} \end{matrix} .$ (19)

The maintenance resources considered in this paper include both maintenance cost and time. As defined in Ref. [14], the cost and time incurred by each maintenance action are composed of two parts: the fixed cost/time and the variable cost/time, and they are written as: $c_{i, l_{Y_{i}, m_{i}}^{i}} = c_{i}^{fix} + c_{i, l_{Y_{i}, m_{i}}^{i}}^{var},$ (20) $t_{i, l_{Y_{i}, m_{i}}^{i}} = t_{i}^{fix} + t_{i, l_{Y_{i}, m_{i}}^{i}}^{var},$ (21) where the fixed term, which is a constant for each element and represents resources consumption for a setup, doesn’t vary with maintenance options. The variable term reflects the improvement actions of recovering the element state. Furthermore, when m_i = k_i, $c_{i, l_{Y_{i}, m_{i}}^{i}}^{var} = c_{i}^{R}$ and $t_{i, l_{Y_{i}, m_{i}}^{i}}^{var} = t_{i}^{R}$ denote the variable cost and time corresponding to replacement. For a specific maintenance option L, the total maintenance time and cost of a system composed of n elements are: $c (L) = \sum_{i = 1}^{n} c_{i, l_{Y_{i}, m_{i}}^{i}} V_{i} = \sum_{i = 1}^{n} (c_{i}^{fix} + c_{i, l_{Y_{i}, m_{i}}^{i}}^{var}) V_{i},$ (22) and $t (L) = \sum_{i = 1}^{n} t_{i, l_{Y_{i}, m_{i}}^{i}} V_{i} = \sum_{i = 1}^{n} (t_{i}^{fix} + t_{i, l_{Y_{i}, m_{i}}^{i}}^{var}) V_{i} .$ (23)

Then, the selective maintenance model can be formulated as: $Objective : max {SU}_{α} (L)$ (24) $Subject to : c (L) = \sum_{i = 1}^{n} (c_{i}^{fix} + c_{i, l_{Y_{i}, m_{i}}^{i}}^{var}) V_{i} \leq c_{0}$ (25) $0 \leq Y_{i} \leq m_{i} \leq k_{i}$ (26) $1 \leq i \leq n$ (27) $0 \leq α \leq 1$ (28) $V_{i} is a binary variable$ (29) where the constraint (25) is the limited available budget (c₀). The constraint (26) is the relationship of the element state before and after maintenance. $L = (l_{Y_{1}, m_{1}}^{1}, l_{Y_{2}, m_{2}}^{2}, \dots, l_{Y_{n}, m_{n}}^{n})$ is a set of decision variables to be optimized and it corresponds to the maintenance option to be performed during a break for each element.

4 The solution algorithm

For the specific selective maintenance optimization problem, some inferior solutions can be eliminated by a set of rules. The flowchart of the proposed optimization algorithm is presented in Fig. 7. The solution procedure is:

Step 1. Initialization. Input the parameters associated with system and component states at the end of the previous mission, the performance capacities and the states transition intensities of elements, the available maintenance options, and the durations of the break and the next mission.

Fig.7

The flowchart of the solution algorithm.

Step 2. Enumerate a possible maintenance option L based on the current state and the available maintenance options of each element.

Step 3. Check the feasibility of the option L by Rule 1. If the option L conforms to Rule 1, go to Step 4, otherwise, go to Step 2. The Rule 1 is given from the perspective of the limited resources, that is, the resources consumption of feasible maintenance options must be below the constraint.

Step 4. Check the feasibility of the option L by Rule 2. If the option L meets Rule 2, then, go to Step 5, otherwise, go to Step 2. The Rule 2 is given from the perspective of the system structure, that is, the critical elements must be maintained to improve the system state. The critical elements can be interpreted as the important elements have significant contributions to the state improvement of the system, e.g., the failed elements in a series configuration.

Step 5. Reduce the system state space at the end of the next mission for the option L by Rule 3. Rule 3 is to reduce the computational burden in evaluating the fuzzy probability of a system successfully completing the next mission under a certain maintenance option. Based on the following two criteria, some states of the system at the end of the next mission can be discarded in computation: Criterion (1): when one element in a series configuration fails, the other elements will not work in the next mission, and Criterion (2): the system state of which the performance capacity cannot meet the required mission demand level should be eliminated.

Step 6. Evaluate the fuzzy probability of a system successfully completing the next mission based on Equation (17).

Step 7. The searching iteration ends if the termination criterion reaches; otherwise, go to Step (2).

We take the launcher of an archibald as an example to elaborate the proposed solution algorithm. The launcher is a shell filling and shooting system, and it consists of two subsystems connected in parallel as shown in Fig. 8. Both subsystem 1 and subsystem 2 are composed of two elements connected in series. Element 1 and element 2 are two s-independent and identical elements, namely shell filling. Element 3 and element 4 are two identical shooting parts. The shell filling elements are capable of loading ammunitions automatically to the shooting elements, and then, the shell is launched by the shooting elements.

Fig.8

The structure of the studied launcher system.

To facilitate computation, the maintenance option $L = (l_{Y_{1}, m_{1}}^{1}, l_{Y_{2}, m_{2}}^{2}, \dots, l_{Y_{n}, m_{n}}^{n})$ hereinafter is replaced by the corresponding elements’ states after performing the maintenance option, i.e. L = (m₁, m₂, ⋯ , m_n). For the specific launcher system, the proposed solution algorithm can be performed as follows:

Step 1. Table 1 shows the possible states and the corresponding fuzzy performance capacities of elements. Each element has three possible performance capacities, namely perfectly functioning (state 2), partially failed (state 1), and completely failed (state 0). The performance capacities and the states transition intensities are represented by fuzzy numbers. The maintenance budget is c₀ = 30 units, and the required demand level of the next mission is $\tilde{w} = (75, 80, 85)$ units/hour. The maintenance cost and the time associated with each maintenance action are shown in Table 2.

Table 1

The element states and the fuzzy performance capacities (units/hour)

Elements (i)	${\tilde{g}}_{(i, 0)}$	${\tilde{g}}_{(i, 1)}$	${\tilde{g}}_{(i, 2)}$
1	0	(60,75,85)	(100,115,120)
2	0	(55,70,85)	(105,115,125)
3	0	(70,80,95)	(110,120,125)
4	0	(70,90,100)	(110,120,130)

Table 2

The maintenance cost (units) and the time (hours) associated with each maintenance action

Elements (i)	Y _i	Resources consumption $s_{h, j}^{i}$
		$c_{i}^{fix}$	$c_{i}^{R}$	$c_{i, l_{0, 1}^{i}}^{var}$	$t_{i}^{fix}$	$t_{i}^{R}$	$t_{i, l_{0, 1}^{i}}^{var}$
1	0	1	10	5	0.2	2	1
2	1	1.2	12	—	0.2	2	—
3	1	1.2	14	—	0.3	1.5	—
4	0	1.5	15	10	0.3	1.5	1

Step 2. For element 1, Y₁ = 0 and there are three available maintenance actions to choose, including do nothing, imperfect maintenance, and replacement. Similarly, the numbers of the available maintenance actions for elements 2, 3, and 4 are 2, 2, and 3, respectively. Hence, by considering all the possible maintenance actions of each element, one can get 3 × 2 ×2 × 3 =36 possible combinations of the maintenance options for the whole system. As the solution space is not very huge, the enumerating method can be used to enumerate all the possible maintenance options.

Step 3. Based on Rule 1, the maintenance options exceeding the maintenance budget should be discarded first. When enumerating the possible maintenance options, one can find out some maintenance options satisfying c (L) > c₀, and these options are labeled by $L_{c} = (m_{1}^{c}, m_{2}^{c}, \dots, m_{n}^{c})$ . As the maintenance cost increases strictly with the improvement of element state, the maintenance options hold $m_{1} > m_{1}^{c}$ , $m_{2} > m_{2}^{c}$ , ... ... , and $m_{n} > m_{n}^{c}$ simultaneously can be eliminated. For example, Table 3 lists some possible maintenance options. It can be seen that the cost of the options (0,1,2,2) and (0,2,2,1) are 31.7 and 39.9 units, respectively, exceeding the maintenance budget, i.e., 30 units. Hence, (0,1,2,2) and (0,2,2,1) can be viewed as critical options. Based on Rule 1, the maintenance options highlighted by the underlined texts in Table 3 can be discarded. After Step (3), the number of the remaining possible maintenance options reduces to 19.

Table 3

Eliminating the maintenance options based on Rule 1

Maintenance options				c (L)	Maintenance options				c (L)
m ₁	m ₂	m ₃	m ₄		m ₁	m ₂	m ₃	m ₄
0	1	1	0	0	1	1	1	1	17.5
0	1	1	1	11.5	1	1	1	2	22.5
0	1	1	2	16.5	1	1	2	0	21.2
0	1	2	0	15.2	1	1	2	1	32.7
0	1	2	1	26.7	1	1	2	2	37.7
0	1	2	2	31.7	1	2	1	0	19.2
0	2	1	0	13.2	1	2	1	1	30.7
0	2	1	1	24.7	1	2	1	2	35.7
0	2	1	2	29.7	1	2	2	0	35.4
0	2	2	0	28.4	1	2	2	1	45.9
0	2	2	1	3 9.9	1	2	2	2	50.9
0	2	2	2	44.4	... ... ... ...
1	1	1	0	6

Step 4. Based on Rule 2, the maintenance options corresponding to do nothing on the critical elements should be eliminated. For example, the critical elements of the system in Fig. 8 are elements 1 and 4, because at least one of these two elements must be repaired to recover the system to a functioning state. Hence, the options in which elements 1 and 4 are not to be repaired are eliminated first. While, for subsystem 1, the critical element is 1. It doesn’t make sense to do preventive maintenance on element 3 when element 1 is not selected to be repaired. For example, compared to option (0,1,1,1), the option (0,1,2,1) will not bring any benefit to the system reliability improvement. Hence, the maintenance options have the patterns of (0,-,-,0), (0,-,2,-) and (-,2,-,0) should also be discarded. Based on Rule 2, twelve possible maintenance options are left, and they are tabulated in Table 4.

Table 4

The reduced solution space based on Rule 2

Maintenance options				Performance capacities of subsystem 1	Performance capacities of subsystem 2	System performance capacities	Cost c (L)	Time t (L)
m ₁	m ₂	m ₃	m ₄
0	1	1	1	(0,0,0)	(55,70,85)	(55,70,85)	11.5	1.3
0	1	1	2	(0,0,0)	(55,70,85)	(55,70,85)	16.5	1.8
0	2	1	1	(0,0,0)	(70,90,100)	(70,90,100)	24.7	3.5
0	2	1	2	(0,0,0)	(105,115,125)	(105,115,125)	29.7	4
1	1	1	0	(60,75,85)	(0,0,0)	(60,75,85)	6	1.2
1	1	1	1	(60,75,85)	(55,70,85)	(115,145,170)	17.5	2.5
1	1	1	2	(60,75,85)	(55,70,85)	(115,145,170)	22.5	3
1	1	2	0	(60,75,85)	(0,0,0)	(60,75,85)	21.2	3
2	1	1	0	(70,80,95)	(0,0,0)	(70,80,95)	11	2.2
2	1	1	1	(70,80,95)	(55,70,85)	(125,150,180)	22.5	3.5
2	1	1	2	(70,80,95)	(55,70,85)	(125,150,180)	27.5	4
2	1	2	0	(100,115,120)	(0,0,0)	(100,115,120)	26.2	4

Step 5. Assume that the maintenance option (2,1,1,2) is selected, there are 3 × 2 ×2 × 3 =36 possible states combinations of elements after executing the next mission as shown in Table 5. Based on the Criterion (1) of Rule 3, the elements’ states combinations with the patterns of (0,-,0,-) and (-,0,-,0) are eliminated. Based on the Criterion (2) of Rule 3, the states combinations of elements satisfying ${\tilde{r}}_{g} = {\tilde{g}}_{(s, u)} - \tilde{w} < 0$ are discarded as well. Based on these two criterions, the possible states combinations of elements to be eliminated in the computation of the fuzzy probability of a system successfully completing the next mission are highlighted by underlined texts in Table 5. Hence, the fuzzy probability of a system successfully completing the next mission under maintenance option (2,1,1,2) is the sum of the fuzzy probability of the remaining sixteen combinations of element states (the plain texts in Table 5). Based on Rule 3, the numbers of the elements’ states combinations to be evaluated for each maintenance option are presented in Table 6.

Table 5

The reduced solution space based on Rule 2

Maintenance option				Possible states combinations				Maintenance option				Possible states combinations
m ₁	m ₂	m ₃	m ₄					m ₁	m ₂	m ₃	m ₄
				2	1	1	2					1	0	1	2
				2	1	1	1					1	0	1	1
				2	1	1	0					1	0	1	0
				2	1	0	2					1	0	0	2
				2	1	0	1					1	0	0	1
				2	1	0	0					1	0	0	0
				2	0	1	2					0	1	1	2
				2	0	1	1					0	1	1	1
2	1	1	2	2	0	1	0	2	1	1	2	0	1	1	0
				2	0	0	2					0	1	0	2
				2	0	0	1					0	1	0	1
				2	0	0	0					0	1	0	0
				1	1	1	2					0	0	1	2
				1	1	1	1					0	0	1	1
				1	1	1	0					0	0	1	0
				1	1	0	2					0	0	0	2
				1	1	0	1					0	0	0	1
				1	1	0	0					0	0	0	0

Table 6

The number of the elements’ states combinations to be evaluated under each maintenance option

Maintenance options				No. of elements’ states combinations
0	1	1	1	1
0	1	1	2	2
0	2	1	1	2
0	2	1	2	4
1	1	1	0	1
1	1	1	1	5
1	1	1	2	9
1	1	2	0	2
2	1	1	0	2
2	1	1	1	9
2	1	1	2	16
2	1	2	0	4

Step 6. Evaluating the fuzzy probability of a system successfully completing the next mission under the maintenance options listed in Table 4.

Step 7. Choose the optimal maintenance option with the maximum fuzzy probability of a system successfully completing the next mission.

It can be observed from the above illustrations that, compared to the method without executing the proposed rules, e.g., a full enumeration method, the solution space of the proposed solution algorithm can be significantly reduced by 66.7%, i.e., (36 - 12)/36. It should be noted that, in practical applications, system may be composed of many elements, and each element may have many maintenance options. It is very time-consuming to enumerate all the possible solutions as the aforementioned illustrative example. Some computational intelligent algorithms, such as the genetic algorithm, the differential evolution algorithm etc., can be, therefore, introduced to resolve the optimization problem in a computationally effective manner. The proposed three rules can still be embedded in these advanced optimization algorithms at the stage of generating new solutions to eliminate the inferior solutions as many as possible.

5 An illustrative example and discussions

5.1 An illustrative example

To demonstrate the effectiveness of the proposed selective maintenance model, we recall the example presented earlier in Section 4. The fuzzy states transition intensities of elements are given in Table 7. Assume that the durations of the break and the next mission are ${\tilde{t}}_{b} = (3, 3.5, 5)$ hours and ${\tilde{t}}_{d} = (0.5, 0.6, 0.65)$ months, respectively. We take the maintenance option (2,1,1,2) as an example to illustrate the process of evaluating the fuzzy probability of the system successfully completing the next mission with a fuzzy duration.

Table 7
The number of the elements’ states combinations to be evaluated under each maintenance option

Elements (i) ${\tilde{λ}}_{(1, 0)}^{i}$ ${\tilde{λ}}_{(2, 1)}^{i}$ ${\tilde{λ}}_{(2, 0)}^{i}$

1 (0.45,0.5,0.55) (0.2,0.3,0.4) (0.1,0.3,0.35)

2 (0.35,0.4,0.5) (0.2,0.25,0.4) (0.1,0.2,0.3)

3 (0.45,0.5,0.6) (0.1,0.2,0.3) (0.25,0.35,0.45)

4 (0.3,0.4,0.5) (0.1,0.2,0.25) (0.15,0.25,0.35)

Elements (i)	${\tilde{λ}}_{(1, 0)}^{i}$	${\tilde{λ}}_{(2, 1)}^{i}$	${\tilde{λ}}_{(2, 0)}^{i}$
1	(0.45,0.5,0.55)	(0.2,0.3,0.4)	(0.1,0.3,0.35)
2	(0.35,0.4,0.5)	(0.2,0.25,0.4)	(0.1,0.2,0.3)
3	(0.45,0.5,0.6)	(0.1,0.2,0.3)	(0.25,0.35,0.45)
4	(0.3,0.4,0.5)	(0.1,0.2,0.25)	(0.15,0.25,0.35)

Firstly, based on Equation (7), one can get the FUGF of each element. For element 1, the FUGF is given as: $\begin{matrix} {\tilde{U}}_{1} (z, {\tilde{t}}_{d}) & = & {\tilde{p}}_{(1, 0)} ({\tilde{t}}_{d}) \cdot z^{{\tilde{g}}_{(1, 0)}} + {\tilde{p}}_{(1, 1)} ({\tilde{t}}_{d}) \cdot z^{{\tilde{g}}_{(1, 1)}} \\ + {\tilde{p}}_{(1, 2)} ({\tilde{t}}_{d}) \cdot z^{{\tilde{g}}_{(1, 2)}} \end{matrix}$

Based on the FMM, the state distribution of element 1 at the end of the next mission is computed by Equation (8), and it equals: $\begin{matrix} {\tilde{p}}_{(1, 2)} ({\tilde{t}}_{d}) & = & e^{- ({\tilde{λ}}_{(2, 1)}^{1} + {\tilde{λ}}_{(2, 0)}^{1}) {\tilde{t}}_{d}}, \\ {\tilde{p}}_{(1, 1)} ({\tilde{t}}_{d}) & = & \frac{{\tilde{λ}}_{(2, 1)}^{1} (e^{- {\tilde{λ}}_{(1, 0)}^{1} {\tilde{t}}_{d}} - e^{- ({\tilde{λ}}_{(2, 1)}^{1} + {\tilde{λ}}_{(2, 0)}^{1}) {\tilde{t}}_{d}})}{{\tilde{λ}}_{(2, 1)}^{1} + {\tilde{λ}}_{(2, 0)}^{1} - {\tilde{λ}}_{(1, 0)}^{1}}, \\ {\tilde{p}}_{(1, 0)} ({\tilde{t}}_{d}) & = & 1 - {\tilde{p}}_{(1, 2)} ({\tilde{t}}_{d}) - {\tilde{p}}_{(1, 1)} ({\tilde{t}}_{d}) . \end{matrix}$

For element 2, the FUGF is written as: ${\tilde{U}}_{2} (z, {\tilde{t}}_{d}) = {\tilde{p}}_{(2, 0)} ({\tilde{t}}_{d}) \cdot z^{{\tilde{g}}_{(2, 0)}} + {\tilde{p}}_{(2, 1)} ({\tilde{t}}_{d}) \cdot z^{{\tilde{g}}_{(2, 1)}},$ and the state distribution of element 2 at the end of the next mission is: $\begin{matrix} {\tilde{p}}_{(2, 1)} ({\tilde{t}}_{d}) & = & e^{- {\tilde{λ}}_{(1, 0)}^{2} {\tilde{t}}_{d}}, \\ {\tilde{p}}_{(2, 0)} ({\tilde{t}}_{d}) & = & 1 - {\tilde{p}}_{(2, 1)} ({\tilde{t}}_{d}) . \end{matrix}$

For element 3, the FUGF is: ${\tilde{U}}_{3} (z, {\tilde{t}}_{d}) = {\tilde{p}}_{(3, 0)} ({\tilde{t}}_{d}) \cdot z^{{\tilde{g}}_{(3, 0)}} + {\tilde{p}}_{(3, 1)} ({\tilde{t}}_{d}) \cdot z^{{\tilde{g}}_{(3, 1)}}$ where, ${\tilde{p}}_{(3, 1)} ({\tilde{t}}_{d}) = e^{- {\tilde{λ}}_{(1, 0)}^{3} {\tilde{t}}_{d}}$ and ${\tilde{p}}_{(3, 0)} ({\tilde{t}}_{d}) = 1 - {\tilde{p}}_{(3, 1)} ({\tilde{t}}_{d})$ .

Likewise, for element 4, the FUGF is: $\begin{matrix} {\tilde{U}}_{4} (z, {\tilde{t}}_{d}) & = & {\tilde{p}}_{(4, 0)} ({\tilde{t}}_{d}) \cdot z^{{\tilde{g}}_{(4, 0)}} + {\tilde{p}}_{(4, 1)} ({\tilde{t}}_{d}) \cdot z^{{\tilde{g}}_{(4, 1)}} \\ + {\tilde{p}}_{(4, 2)} ({\tilde{t}}_{d}) \cdot z^{{\tilde{g}}_{(4, 2)}} \end{matrix}$

The state distribution of element 4 is given by: $\begin{matrix} {\tilde{p}}_{(4, 2)} ({\tilde{t}}_{d}) & = & e^{- ({\tilde{λ}}_{(2, 1)}^{4} + {\tilde{λ}}_{(2, 0)}^{4}) {\tilde{t}}_{d}} \\ {\tilde{p}}_{(4, 1)} ({\tilde{t}}_{d}) & = & \frac{{\tilde{λ}}_{(2, 1)}^{4} (e^{- {\tilde{λ}}_{(1, 0)}^{4} {\tilde{t}}_{d}} - e^{- ({\tilde{λ}}_{(2, 1)}^{4} + {\tilde{λ}}_{(2, 0)}^{4}) {\tilde{t}}_{d}})}{{\tilde{λ}}_{(2, 1)}^{4} + {\tilde{λ}}_{(2, 0)}^{4} - {\tilde{λ}}_{(1, 0)}^{4}} \\ {\tilde{p}}_{(4, 0)} ({\tilde{t}}_{d}) & = & 1 - {\tilde{p}}_{(4, 2)} ({\tilde{t}}_{d}) - {\tilde{p}}_{(4, 1)} ({\tilde{t}}_{d}) \end{matrix}$

Based on Equation (9), the FUGF of the FMSS can be recursively derived by the UGF operators as follows:

${\tilde{U}}_{s} (z, {\tilde{t}}_{d}) = \underset{parallel}{{\tilde{Ω}}_{φ}} (\begin{matrix} \underset{series}{{\tilde{Ω}}_{φ}} ({\tilde{U}}_{1} (z, {\tilde{t}}_{d}), {\tilde{U}}_{3} (z, {\tilde{t}}_{d})), \\ \underset{series}{{\tilde{Ω}}_{φ}} ({\tilde{U}}_{2} (z, {\tilde{t}}_{d}), {\tilde{U}}_{4} (z, {\tilde{t}}_{d})) \end{matrix})$

By applying Rule 3, all the possible combinations of elements’ states with the system performance capacity being greater than the required mission demand level at the end of the next mission are given by:

$\begin{matrix} {\tilde{U}}_{s} (\tilde{w}, z, {\tilde{t}}_{d}) \\ = {\tilde{p}}_{(1, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(2, 0)} ({\tilde{t}}_{d}) {\tilde{p}}_{(3, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(4, 1)} ({\tilde{t}}_{d}) • z^{(60, 75, 85)} \\ + {\tilde{p}}_{(1, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(2, 0)} ({\tilde{t}}_{d}) {\tilde{p}}_{(3, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(4, 2)} ({\tilde{t}}_{d}) • z^{(60, 75, 85)} \\ + {\tilde{p}}_{(1, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(2, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(3, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(4, 0)} ({\tilde{t}}_{d}) • z^{(60, 75, 85)} \\ + {\tilde{p}}_{(1, 0)} ({\tilde{t}}_{d}) {\tilde{p}}_{(2, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(3, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(4, 1)} ({\tilde{t}}_{d}) • z^{(55, 70, 85)} \\ + {\tilde{p}}_{(1, 0)} ({\tilde{t}}_{d}) {\tilde{p}}_{(2, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(3, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(4, 2)} ({\tilde{t}}_{d}) • z^{(55, 70, 85)} \\ + {\tilde{p}}_{(1, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(2, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(3, 0)} ({\tilde{t}}_{d}) {\tilde{p}}_{(4, 1)} ({\tilde{t}}_{d}) • z^{(55, 70, 85)} \\ + {\tilde{p}}_{(1, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(2, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(3, 0)} ({\tilde{t}}_{d}) {\tilde{p}}_{(4, 2)} ({\tilde{t}}_{d}) • z^{(55, 70, 85)} \\ + {\tilde{p}}_{(1, 2)} ({\tilde{t}}_{d}) {\tilde{p}}_{(2, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(3, 0)} ({\tilde{t}}_{d}) {\tilde{p}}_{(4, 1)} ({\tilde{t}}_{d}) • z^{(55, 70, 85)} \\ + {\tilde{p}}_{(1, 2)} ({\tilde{t}}_{d}) {\tilde{p}}_{(2, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(3, 0)} ({\tilde{t}}_{d}) {\tilde{p}}_{(4, 2)} ({\tilde{t}}_{d}) • z^{(55, 70, 85)} \\ + {\tilde{p}}_{(1, 2)} ({\tilde{t}}_{d}) {\tilde{p}}_{(2, 0)} ({\tilde{t}}_{d}) {\tilde{p}}_{(3, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(4, 1)} ({\tilde{t}}_{d}) • z^{(70, 80, 95)} \\ + {\tilde{p}}_{(1, 2)} ({\tilde{t}}_{d}) {\tilde{p}}_{(2, 0)} ({\tilde{t}}_{d}) {\tilde{p}}_{(3, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(4, 2)} ({\tilde{t}}_{d}) • z^{(70, 80, 95)} \\ + {\tilde{p}}_{(1, 2)} ({\tilde{t}}_{d}) {\tilde{p}}_{(2, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(3, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(4, 0)} ({\tilde{t}}_{d}) • z^{(70, 80, 95)} \\ + {\tilde{p}}_{(1, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(2, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(3, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(4, 1)} ({\tilde{t}}_{d}) • z^{(115, 145, 170)} \\ + {\tilde{p}}_{(1, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(2, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(3, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(4, 2)} ({\tilde{t}}_{d}) • z^{(115, 145, 170)} \\ + {\tilde{p}}_{(1, 2)} ({\tilde{t}}_{d}) {\tilde{p}}_{(2, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(3, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(4, 1)} ({\tilde{t}}_{d}) • z^{(125, 150, 180)} \\ + {\tilde{p}}_{(1, 2)} ({\tilde{t}}_{d}) {\tilde{p}}_{(2, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(3, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(4, 2)} ({\tilde{t}}_{d}) • z^{(125, 150, 180)} \end{matrix}$

Table 8

The fuzzy probability, cost, and time of all the feasible maintenance options

Index	Options L	Cost c (L)	Time t (L)	Probability
				α = 0	α = 1
1	(0,1,1,1)	11.5	1.3	[0.0653,0.0903]	0
2	(0,1,1,2)	16.5	1.8	[0.0714,0.0972]	0
3	(0,2,1,1)	24.7	3.5	[0.2784,0.4375]	0.6005
4	(0,2,1,2)	29.7	4	[0.1676,0.2662]	0
5	(1,1,1,0)	6	1.2	[0.0902,0.1215]	0
6	(1,1,1,1)	17.5	2.5	[0.3114,0.5207]	0.3396
7	(1,1,1,2)	22.5	3	[0.3346,0.5526]	0.3697
8	(1,1,2,0)	21.2	3	[0.0986,0.1339]	0
9	(2,1,1,0)	11	2.2	[0.2596,0.4053]	0.5169
10	(2,1,1,1)	22.5	3.5	[0.2854,0.4894]	0.5329
11	(2,1,1,2)	27.5	4	[0.1428,0.2313]	0
12	(2,1,2,0)	26.2	4	[0.1485,0.2522]	0

Thus, the FMSS has five elements’ states combinations to be evaluated. Assume that the cut level α equals to 0. For state 1: ${\tilde{g}}_{(s, 1)} = (60, 75, 85)$ , the fuzzy state probability at the end of the next mission is: $\begin{matrix} {\tilde{p}}_{(s, 1)} ({\tilde{t}}_{d}) \\ = {\tilde{p}}_{(1, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(2, 0)} ({\tilde{t}}_{d}) {\tilde{p}}_{(3, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(4, 1)} ({\tilde{t}}_{d}) \\ + {\tilde{p}}_{(1, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(2, 0)} ({\tilde{t}}_{d}) {\tilde{p}}_{(3, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(4, 2)} ({\tilde{t}}_{d}) \\ + {\tilde{p}}_{(1, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(2, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(3, 1)} ({\tilde{t}}_{d}) {\tilde{p}}_{(4, 0)} ({\tilde{t}}_{d}) \\ = (\begin{matrix} \frac{{\tilde{λ}}_{(2, 1)}^{1} (e^{- {\tilde{λ}}_{(1, 0)}^{1} {\tilde{t}}_{d}} - e^{- ({\tilde{λ}}_{(2, 1)}^{1} + {\tilde{λ}}_{(2, 0)}^{1}) {\tilde{t}}_{d}})}{{\tilde{λ}}_{(2, 1)}^{1} + {\tilde{λ}}_{(2, 0)}^{1} - {\tilde{λ}}_{(1, 0)}^{1}} (1 - e^{- {\tilde{λ}}_{(1, 0)}^{2} {\tilde{t}}_{d}}) \\ e^{- {\tilde{λ}}_{(1, 0)}^{3} {\tilde{t}}_{d}} \frac{{\tilde{λ}}_{(2, 1)}^{4} (e^{- {\tilde{λ}}_{(1, 0)}^{4} {\tilde{t}}_{d}} - e^{- ({\tilde{λ}}_{(2, 1)}^{4} + {\tilde{λ}}_{(2, 0)}^{4}) {\tilde{t}}_{d}})}{{\tilde{λ}}_{(2, 1)}^{4} + {\tilde{λ}}_{(2, 0)}^{4} - {\tilde{λ}}_{(1, 0)}^{4}} \end{matrix}) \\ + (\begin{matrix} \frac{{\tilde{λ}}_{(2, 1)}^{1} (e^{- {\tilde{λ}}_{(1, 0)}^{1} {\tilde{t}}_{d}} - e^{- ({\tilde{λ}}_{(2, 1)}^{1} + {\tilde{λ}}_{(2, 0)}^{1}) {\tilde{t}}_{d}})}{{\tilde{λ}}_{(2, 1)}^{1} + {\tilde{λ}}_{(2, 0)}^{1} - {\tilde{λ}}_{(1, 0)}^{1}} (1 - e^{- {\tilde{λ}}_{(1, 0)}^{2} {\tilde{t}}_{d}}) \\ e^{- {\tilde{λ}}_{(1, 0)}^{3} {\tilde{t}}_{d}} e^{- ({\tilde{λ}}_{(2, 1)}^{4} + {\tilde{λ}}_{(2, 0)}^{4}) {\tilde{t}}_{d}} \end{matrix}) \\ + (\begin{matrix} \frac{{\tilde{λ}}_{(2, 1)}^{1} (e^{- {\tilde{λ}}_{(1, 0)}^{1} {\tilde{t}}_{d}} - e^{- ({\tilde{λ}}_{(2, 1)}^{1} + {\tilde{λ}}_{(2, 0)}^{1}) {\tilde{t}}_{d}})}{{\tilde{λ}}_{(2, 1)}^{1} + {\tilde{λ}}_{(2, 0)}^{1} - {\tilde{λ}}_{(1, 0)}^{1}} e^{- {\tilde{λ}}_{(1, 0)}^{2} {\tilde{t}}_{d}} e^{- {\tilde{λ}}_{(1, 0)}^{3} {\tilde{t}}_{d}} \\ (1 - e^{- ({\tilde{λ}}_{(2, 1)}^{4} + {\tilde{λ}}_{(2, 0)}^{4}) {\tilde{t}}_{d}} - \frac{{\tilde{λ}}_{(2, 1)}^{4} (e^{- {\tilde{λ}}_{(1, 0)}^{4} {\tilde{t}}_{d}} - e^{- ({\tilde{λ}}_{(2, 1)}^{4} + {\tilde{λ}}_{(2, 0)}^{4}) {\tilde{t}}_{d}})}{{\tilde{λ}}_{(2, 1)}^{4} + {\tilde{λ}}_{(2, 0)}^{4} - {\tilde{λ}}_{(1, 0)}^{4}}) \end{matrix}) \\ \approx (\begin{matrix} {\tilde{λ}}_{(2, 1)}^{1} {\tilde{t}}_{d} (1 - e^{- {\tilde{λ}}_{(1, 0)}^{2} {\tilde{t}}_{d}}) e^{- {\tilde{λ}}_{(1, 0)}^{3} {\tilde{t}}_{d}} {\tilde{λ}}_{(2, 1)}^{4} {\tilde{t}}_{d} + \\ {\tilde{λ}}_{(2, 1)}^{1} {\tilde{t}}_{d} (1 - e^{- {\tilde{λ}}_{(1, 0)}^{2} {\tilde{t}}_{d}}) e^{- {\tilde{λ}}_{(1, 0)}^{3} {\tilde{t}}_{d}} e^{- ({\tilde{λ}}_{(2, 1)}^{4} + {\tilde{λ}}_{(2, 0)}^{4}) {\tilde{t}}_{d}} + \\ {\tilde{λ}}_{(2, 1)}^{1} {\tilde{t}}_{d} e^{- {\tilde{λ}}_{(1, 0)}^{2} {\tilde{t}}_{d}} e^{- {\tilde{λ}}_{(1, 0)}^{3} {\tilde{t}}_{d}} (1 - e^{- ({\tilde{λ}}_{(2, 1)}^{4} + {\tilde{λ}}_{(2, 0)}^{4}) {\tilde{t}}_{d}} - {\tilde{λ}}_{(2, 1)}^{4} {\tilde{t}}_{d}) \end{matrix}) \end{matrix}$

For the maintenance option L = (2, 1, 1, 2), one has ${\tilde{r}}_{t_{b}} = {\tilde{t}}_{b} - t (L) = (- 1, - 0.5, 1)$ , and the α-cut level set of ${\tilde{r}}_{t_{b}}$ is [- 1 +0.5α, 1 - 1.5α]. Based on Equation (13), the probability that the maintenance option can be completed within a break with a fuzzy duration is equal to: $\frac{| {tr}_{t_{b}} |_{α}}{| r_{t_{b}} |_{α}} = {\begin{matrix} 0 & if α \geq \frac{2}{3} \\ \frac{{(1 - 1.5 α)}^{2}}{3 {(1 - α)}^{2}} & if α < \frac{2}{3} \end{matrix} .$

For state 1, ${\tilde{r}}_{1} = {\tilde{g}}_{(s, 1)} - \tilde{w} = (- 25, - 5, 10)$ , and the α-cut level set of ${\tilde{r}}_{1}$ is [- 25 + 20α, 10 - 15α]. The probability of state 1 being able to meet the required mission demand level is: $\frac{| {sr}_{g} |_{α}}{| r_{g} |_{α}} = {\begin{matrix} 0 & if α \geq \frac{2}{3} \\ \frac{{(2 - 3 α)}^{2}}{21 {(1 - α)}^{2}} & if α < \frac{2}{3} \end{matrix} .$

In the same fashion, one can get the probability of other states being able to meet the required mission demand level. Then, based on Equation (18) and the parametric programming, the fuzzy probability of the system successfully completing the next mission under maintenance option L = (2, 1, 1, 2) is: $\begin{matrix} {SU}_{α} (L) & = & \sum_{u = 1}^{5} {\tilde{p}}_{(s, u) α} (t_{d}) • \frac{| {sr}_{g} |_{α}}{| r_{g} |_{α}} • \frac{| {tr}_{t_{b}} |_{α}}{| r_{t_{b}} |_{α}} \\ = & [0.1428, 0.2313] \end{matrix}$ where α = 0. When α = 1, the fuzzy probability of the system successfully completing the next mission degenerates to a crisp value of 0.

The fuzzy probability and the associated membership functions of all the twelve maintenance options narrowed down by Rules 1 and 2 in Section 4 are shown in Table 8 and Fig. 9, respectively. It can be observed from Fig. 9 that the membership functions of the fuzzy probabilities of the system successfully completing the next mission in present work are no longer triangle fuzzy numbers. When α is greater than 0.6, the probabilities of the maintained system successfully completing the next mission under maintenance options 1,2,4,5,8,11 and 12 are zero. Because the fuzzy uncertainty decreases with the increase of α. For a special case, that is α = 1, all the fuzzy numbers reduce to crisp values. In such case, the total maintenance time of options 4, 11 and 12 is 4 hours which is less than the mission break 3.5 hours, and the corresponding maintenance activities cannot be completed in the limited mission break. Similarly, for options 1, 2, 5 and 8, the system performance capacity cannot satisfy the next mission demand when α is very large.

Fig.9

The membership functions of the fuzzy probability of a system successfully completing the next mission under the twelve feasible maintenance options.

Furthermore, it can be observed from Fig. 9 and Table 8 that the membership functions for options 3, 9 and 10 are not continuous, and there is an abrupt change (jump) at α = 1. This is because, in some cases, the α-cut level relative cardinality of the difference of two fuzzy numbers ${\tilde{X}}_{1}$ and ${\tilde{X}}_{2}$ will have a jump if ${\tilde{X}}_{1} = {\tilde{X}}_{2}$ for α = 1. In present work, there are two main reasons about the abrupt change: (1) The probability of the performance capacity satisfying the next mission demand at some degraded system states contributes to the abrupt change. For instance, the performance capacity of the degraded states (2,0,1,1) and (2,1,1,0) under option 10 is (70,80,95). When 0 ≤ α < 1, the probabilities of system performance satisfying the next mission demand is less than one (0.5714), while for α = 1, the probability is equal to one which generate an abrupt increase in the total probability. (2) The probability of the maintenance option completed in the break can also generate an abrupt change. For instance, the total maintenance time for options 3 and 10 is 3.5 hours, and $Pr (t (L) \leq {\tilde{t}}_{b})_{α} = 0.75$ for 0 ≤ α < 1, and $Pr (t (L) \leq {\tilde{t}}_{b})_{α} = 1$ for α = 1.

According to Table 8, the best maintenance strategy is maintenance option (0,2,1,1) when α = 1, that is, doing nothing for elements 1 and 3, replacing element 2, and performing imperfect maintenance on element 4. The associated maintenance cost, and time are 24.7 units and 3.5 hours, respectively. The probability of the system successfully completing the next mission reduced to a crisp value, i.e. 0.6005. When 0 ≤ α < 1, it is not an easy work to identify the optimal maintenance option directly. However, it can be observed from Fig. 9 that the optimal maintenance option for 0 ≤ α < 1 is among the option 3, option 6, option 7, and option 10. For a specific α, the calculating centroid point method proposed by Cheng [38] can be adopted to rank these fuzzy values, and thereby, to identify the optimal maintenance option. Here, we take α = 0 as an example to explain this method. When α = 0, the centroid point of the fuzzy probability of option 3 is $({\bar{x}}_{3}, {\bar{y}}_{3}) = (0.4077, 0.5194)$ , and the distance index between the centroid point $({\bar{x}}_{3}, {\bar{y}}_{3})$ and original point (0,0) is $R_{3} = \sqrt{({\bar{x}}_{3})^{2} + ({\bar{y}}_{3})^{2}} = 0.6603$ . Similarly, the centroid points of option 6, option 7, and option 10 are $({\bar{x}}_{6}, {\bar{y}}_{6}) = (0.3774, 0.5050)$ , $({\bar{x}}_{7}, {\bar{y}}_{7}) = (0 . 4056, 0 . 5064)$ , and $({\bar{x}}_{10}, {\bar{y}}_{10}) = (0.3709, 0.5162)$ , respectively. The corresponding distance indexes are R₆ = 0.6305, R₇ = 0.6488, and R₁₀ = 0.6356. Obviously, one has R₃ > R₇ > R₁₀ > R₆, and the optimal maintenance option for α = 0 is (0,2,1,1), which is the same as that of α = 1.

5.2 Sensitivity analysis

Intuitively, the maintenance budget as an important maintenance resource can significantly affect the selection of maintenance options, as well as the fuzzy probability of a system successfully completing the next mission. In this section, the sensitivity analysis is conducted to examine how the optimal results may change with respect to the setting of the maintenance budget.

The maintenance budget c₀ is changed from 0 to 30 units as depicted in Fig. 10. It can be observed from Fig. 10 that when the maintenance budget is 6 units, the optimal maintenance option includes performing imperfect maintenance action on element 1 and doing nothing on elements 2, 3 and 4 for α = 0, and the corresponding fuzzy probability of the system successfully completing the next mission is [0.0902,0.1215]. When α = 1, there is no feasible maintenance option, and the probability of the system successfully completing the next mission is 0. With the increase of the maintenance budget, the optimal maintenance option changes and a greater fuzzy probability of a system successfully completing the next mission can be achieved. However, it is not always the case that the fuzzy probability of a system successfully completing the next mission is keeping increasing with the augment of the devoted maintenance budget. It can be seen that when the maintenance budget increases to 24.7 units, the optimal maintenance option and the fuzzy probability of a system successfully completing the next mission do not change with the increase of the maintenance budget. Because the duration of the break becomes a critical bottleneck to the improvement of the fuzzy probability of a system successfully completing the next mission.

Fig.10

The fuzzy probability of a system successfully completing the next mission with respect to the different settings of the maintenance budget.

Additionally, it can be observed from Fig. 10 that, for a specific maintenance budget, the optimal maintenance option may vary with α. For instance, when the maintenance cost is 17.5 units, the optimal maintenance option for α = 0 is (1,1,1,1). While, for α = 1, the optimal maintenance option is (2,1,1,0). It can be concluded from the observations that: (1) the result of selective maintenance optimization for fuzzy multi-state systems relies on the α-cut level; (2) The optimal maintenance option in the crisp case, i.e., α = 1, may not be the same as the case with fuzzy uncertainty. Hence, compared to the conventional selective maintenance models (without considering fuzzy property), the proposed selective maintenance model can provide a more effective and reasonable outcome by taking account of the potential fuzzy uncertainty, and thereby offer a more reliable decision.

6 Conclusion

In this paper, a new selective maintenance model for a FMSS executing the next mission with the fuzzy mission and break durations is proposed. The FUGF and FMM are adopted to evaluate the states distributions of elements and systems at the end of the next mission. A solution algorithm containing three rules to reduce the solution space is proposed to solve the selective maintenance model in a computationally efficient manner. To verify the effectiveness of the proposed model, the selective maintenance optimization problem for a launcher system in an archibald is discussed.

This work can be further extended from several angles. The degradation process of each element is assumed to be a homogeneous Markov process in this work. The case of non-Markov degradation process is worth investigating. Another possible extension lies in considering the maintenance personnel as one of limited maintenance resources. In many practical applications, a system is composed of many elements, and many maintenance tasks are required to be performed within a limited break. While, the number of maintenance personnel is not always large enough to perform all the tasks simultaneously. The method of optimally allocating the maintenance tasks to the limited maintenance personnel is meaningful and to be addressed.

Footnotes

Acknowledgments

The authors greatly acknowledge grant support from the National Natural Science Foundation of China under contract numbers 71371042 and 71401173.

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