Reliability assessment of systems subject to interval-valued probabilistic common cause failure by evidential networks

Abstract

Reliability assessment of complex engineered systems is challenging as epistemic uncertainty and common cause failure (CCF) are inevitable. The probabilistic common cause failure (PCCF), which characterizes the simultaneous failures of multiple components with distinguished chances, is a generalized model of traditional CCF model. To accurately assess system reliability, it is of great significance to take both the effects of PCCF and the epistemic uncertainty of components’ state probabilities into account. In this paper, an evidential network model is proposed to assess system reliability with interval-valued PCCFs and epistemic uncertainty associated with components’ state probabilities. The procedures of computing the mass distribution of a component suffering from multiple PCCFs are detailed. The inference algorithm in the evidential network is, then, used to calculate the mass distribution of the entire system. The Birnbaum importance measure is also defined to identify the weak components under PCCFs and epistemic uncertainty. A safety instrumented system is exemplified to demonstrate the effectiveness of the proposed evidential network model in terms of coping with PCCFs and epistemic uncertainty. The importance results show that both the epistemic uncertainty associated with components’ state probabilities and PCCFs have impact on components’ importance.

Keywords

Evidence theory evidential networks interval-valued probabilistic common cause failure epistemic uncertainty

1 Introduction

Reliability modeling and assessment of complex engineering systems have become a challenging task due to many factors such as complex dependency among components in a system, impact from external environments, and epistemic uncertainty associated with parameters of components’ failure/degradation models. Various methods and tools have been investigated to analyze the reliability of complex engineering systems, including the fault tree analysis [1], the Markov chains [2], the stochastic petri nets [3], and the Bayesian networks (BN) [4]. Among them, the BN models have been widely implemented in reliability and safety analysis from a frequentist or subjectivist perspective. The BN was first developed by Boudali and Dugan [5] who demonstrated how fault trees can be implemented by using BNs. As compared to the traditional fault tree analysis, the BN models possess some attractive features in representing the functioning and malfunctioning dependency between the system and its components [6]: (i) Logic gates, such as the “AND” gate and “OR” gate, can only represent a deterministic dependency in-between nodes, whereas the conditional probability tables (CPTs) in BN could represent any arbitrary probabilistic relationship; (ii) The forward and backward inferences in BN enable dynamically updating the marginal probability distributions of all nodes, whereas the fault tree can only assess the failure of top event in a bottom-up fashion; (iii) Fault tree represents its events at the same level with the independency assumption, whereas BN releases such assumption. Dependency, such as the failure dependency among components, cascading failure can also be modeled by BN. Due to its powerful capability of probabilistic reasoning, a suite of investigations have been made to implement the BN in reliability assessment of a diversity of engineering systems [4 , 7].

Nevertheless, in BN models, the probability of a node being in a specific proposition can only be represented by a crisp value. The epistemic uncertainty, which arises from lack of knowledge and data and/or vague judgments from experts, is one of the important challenges in system reliability assessment [8, 10]. Some non-probabilistic methods can be used to represent epistemic uncertainty, such as the evidence theory [9], the fuzzy theory [10, 11], the interval theory [12]. The evidence theory has received considerable attentions in the field of uncertainty reasoning [9], decision making analysis [13], information fusion [14] and so forth. Some attempts have been carried out to incorporate the evidence theory into BN models. Simon et al. [15] first analyzed the reliability of complex engineering systems with epistemic uncertainty associated with components’ state probabilities by using of BN and evidence theory. Zhao et al. [16] investigated the BN model combined the evidential inference algorithm to assess the reliability of power distribution system. Simon and Webber [17] extended the CPT under the evidence theory, namely the conditional belief mass table (CBMT), when the mass functions were conditioned on evidential variables. The evidential network (EN) model was developed by them to infer system reliability bounds. From the fault tree perspective, Yang et al. [18] developed an EN model to manipulate the epistemic uncertainty associated with failure probabilities of basic events. The logic gates, such as “AND” gate, “OR” gate, and “NOR” gate, with imprecise knowledge were converted to the EN model. The CBMTs introduced by Simon and Webber [17] were used to quantify the dependency between the outputs and inputs of the logic gates under epistemic uncertainty. As the EN model is capable of reasoning epistemic uncertainty for complex systems as reported in [17, 18], it is used in the present work to quantify the epistemic uncertainty associated with the parameters of components’ reliability models.

It is worth noting that the foregoing studies on system reliability assessment by EN models were based on a strong premise that nodes in the same level are statistically independent. In general, due to physical interaction and external environments/shocks, components or subsystems in a system may fail simultaneously. Such phenomena is termed as common cause failure (CCF) in the reliability community [19]. As EN model inherits the capability of modeling dependency in the BN model [17], some attempts have been made to incorporate CCF into the EN models. Mi et al. [20] investigated an EN model embedded with the CCF to calculate system reliability. As compared to the CBMT defined in [17], the conditional belief/ plausibility tables were developed to calculate the lower and upper bounds of system reliability. More recently, Mi et al. [21] extended their previous work into multi-state systems, some new nodes were added into the EN model to represent the CCFs. Zuo et al. [23] utilized the CBMT defined by Simon and Webber [17] to infer the mass distributions of child nodes and developed another type of evidential network model embedded with the CCF to assess system reliability. However, it bears noting that the above CCF models all assumed that the occurrence of common cause events will definitely cause the failure of all the affected components. Such an assumption may not be necessarily true in practical cases. Put another way, there are many scenarios in which a common cause event causes the failure of affected components with distinct chances, with the guaranteed failure being just a special case. Such phenomena of probabilistic dependency among components are often called the probabilistic common cause failure (PCCF) [22]. It worth mentioning that reliability assessment of systems under both epistemic uncertainty and PCCFs has not been reported to date. In this paper, the probability that a common cause event may result in the failure of a specific component also contains epistemic uncertainty and is represented as an interval value. The mass distribution of a component suffering from multiple interval-valued PCCFs is, then, calculated by the EN model. The CBMTs defined in Ref. [17] is utilized to facilitate the mass inference in EN model. Therefore, the system reliability can be computed by applying the belief and plausibility functions into the leaf node of the EN model. Moreover, to quantitatively measure the contribution of a component’s failure to a system failure, the Birnbaum importance measure is extended under epistemic uncertainty and PCCFs. A safety instrumented system is exemplified to validate the proposed EN model and the extended Birnbaum importance measure.

The remainder of this paper is rolled out as follows. In Section 2, the fundamentals of the evidence theory are briefly reviewed. The procedures of computing the mass distribution of a component suffering from interval-valued PCCFs are provided in Section 3. The Birnbaum importance measure is extended under the interval-valued PCCFs and epistemic uncertainty in Section 4. The proposed method is implemented to a safety instrumented system in Section 5. Some conclusions are drawn in Section 6.

2 Fundamentals of the evidence theory

The evidence theory, also known as Dempster-Shafer evidence theory, was initialized by Dempster [24] and developed by Shafer [25] to characterize the epistemic uncertainty. It has widespread applications in the fields of information fusion [14], fault diagnosis [13] and reliability analysis [9 , 27]. The fundamentals of the evidence theory will be briefly reviewed here.

2.1 Definition of the frame of discernment (FoD)

In the framework of evidence theory, the frame of discernment (FoD) Ω _X is defined as an exhaustive set which contains all the propositions of a random variable X, and it can be expressed as: $Ω_{X} = {x_{1}, x_{2}, x_{3}, . . ., x_{n}}$ (1) where x _i (i = 1, 2, 3, . . . , n) denotes the elementary proposition of X; n is the number of elementary propositions. The FoD is similar to the sample space in probability theory. However, due to lack of knowledge, some quantities have to be assigned to the union sets, and all the possible union sets form a power set:

$\begin{matrix} 2^{Ω_{X}} & = & {\emptyset, {x_{1}}, {x_{2}}, . . ., {x_{n}}, {x_{1}, x_{2}}, \\ {x_{1}, x_{3}}, . . ., Ω_{X}} \end{matrix}$ (2) where ∅ represents the empty set, and 2ⁿ distinct propositions are contained in the power set.

2.2 Construction of the basic belief assignment (BBA)

In the evidence theory, an independent probability-like measurement can be assigned to the power set 2^ΩX. This probability-like description is called the basic belief assignment (BBA) and can be represented by a mass function m : 2^ΩX → [0, 1], mapping from 2^ΩX to the unit interval [0,1]. Similar to the probability measure, the mass function should satisfy three following axioms: $m (A) \geq 0 for A \in 2^{Ω X}; m (\emptyset) = 0; \sum_{A \in 2^{Ω X}} m (A) = 1 .$

The mass function m (A) represents the quantity that the truth lies in the proposition A. If m (A) ≠0 satisfies, A is called a focal set. Note that if all the focal sets are singletons, the mass function, then, degenerates to the Bayesian mass.

2.3 Calculation of the belief and plausibility functions

Unlike to the probability theory which assigns a crisp value to a proposition B, the evidence theory employs an interval measure [Bel(B), Pl(B)] to quantitatively characterize the belief degree of proposition B. The lower bound, called the belief function, is defined as the total amount of the masses that support proposition B: $Bel (A) = \sum_{B \subseteq A} m (B)$ (3)

The upper bound, namely the plausibility function, is defined as the sum of all the masses of propositions that agree with proposition B, either partially or totally: $Pl (A) = \sum_{B \cap A \neq \emptyset} m (B)$ (4)

By introducing $\bar{B}$ , the connection between the belief function Bel (B) and the plausibility function Pl (B) can be formulated as: $Bel (B) = 1 - Pl (\bar{B})$ (5) and $Pl (B) = 1 - B e l (\bar{B})$ (6)

2.4 Relation between the evidence theory and the interval theory

The epistemic uncertainty associated with the probability of proposition B can be represented by an interval of [Pr_min {B} , Pr_max {B}], where Pr_min {B} and Pr_max {B} denote the minimum and maximum probabilities of proposition B, respectively. In the evidence theory, the epistemic uncertainty associated with the probability of proposition B can be quantified by the belief and plausibility functions, and one has Bel (B) ≤ Pr {B} ≤ Pl (B). Hence, the relation between the evidence theory and the interval theory can be readily formulated as: $[Bel (B), Pl (B)] = [\Pr_{min} {B}, \Pr_{max} {B}]$ (7)

3 Evidential network models with interval-valued probabilistic common cause failure

In this section, due to the lack of knowledge, the probabilistic common cause failure (PCCF) is defined as an interval value, the evidential network with interval-valued PCCF is then constructed to assess system reliability.

3.1 Interval-valued probabilistic common cause failure

In the conventional reliability assessment of systems with CCF, it is assumed that the occurrence of a common cause event will result in the guaranteed failure of all the affected components. However, there are many real-world cases in which a common cause event causes the failure of affected components with different chances. The probabilistic dependency among components caused by a common cause event is termed as the probabilistic common cause failure (PCCF) which are to be studied in this paper. However, due to the lack of knowledge and insufficient information, the failure probability of components caused by the common cause events inevitably contains epistemic uncertainty, and it is represented as an interval value.

In addition to the above definitions of PCCF, some general assumptions are itemized as follows:

(1) A system is subject to different common cause events, i.e., CC ₁,CC ₂,...,CC _M and each common cause event may result in different components’ failure. The probability of the occurrence of a CCF CC _i (i = 1, 2, . . . , M) can be represented as Pr {CC _i}. Several parametric methods can be used to model, such as the α -factor model, β -factor model, and the Multiple Greek letters (MGL) model. In this study, the β -factor model [29] is implemented to quantitatively model the probability of the occurrence of CCFs, and β _i represents the probability of the occurrence of the i th CCF, i.e., Pr {CC _i} = β _i.

(2) Once a common cause event has occurred, it may result in the failure of some components in a probabilistic manner. The affected components by the i th CCF can be represented as a probabilistic common cause group PCCG_i. Specifically, PCCG_i,l represents component C _l is probabilistically affected by CC _i. This allows one component belonging to more than one PCCG.

(3) The probability of component C _l affected by the i th CCF is represented as an interval value [Pr _min {PCCG_i,l} , Pr _max {PCCG_i,l}]. Hence, the mass that CC _i (i = 1, 2, . . . , M) causes the failure of component C _l (C _l ∈ PCCG_i) is: $m ({PCCG}_{i, l}) = \Pr_{\min} {{PCCG}_{i, l}} \times β_{i}$ (8) whereas the mass that CC _i (i = 1, 2, . . . , M) cannot cause the failure of component C _l is: $m ({\bar{PCCG}}_{i, l}) = 1 - \Pr_{\max} {{PCCG}_{i, l}} \times β_{i}$ (9)

A mass has to assigned to the FoD of PCCG_i,l to represent the quantity of ignorance that has caused, or not, the failure of component C _l:

$\begin{matrix} m (Ω_{{PCCG}_{i, l}}) & = & (\Pr_{\max} {{PCCG}_{i, l}} \\ - \Pr_{\min} {{PCCG}_{i, l}}) \times β_{i} \end{matrix}$ (10)

It is obvious that $m ({PCCG}_{i, l}) + m ({\bar{PCCG}}_{i, l}) + m (Ω_{{PCCG}_{i, l}}) = 1$ .

3.2 Evidential network model with interval-valued probabilistic common cause failures

The evidential network (EN), which was introduced by Simon [17], is a graphic representation of the mass relationships. Similar to the Bayesian networks, a general EN consists of N nodes, {X ₁, X ₂, . . . , X _n}, and Z directed arcs between two nodes. Fig.1 illustrates an evidential network with 5 nodes and 4 directed arcs. Each nodes X _i (i = 1, 2, 3, 4, 5) represents a random variable which manifests all the propositions of X _i and all its subsets. If a directed arc from X _j to X _i exists, X _i called a “parent” of X _i, denoted as Pa (X _i). A child node may have more than one parent node meanwhile a parent node may contain more than one child node. An arc characterizes the mass dependency of a node on its parent nodes. The conditional belief mass distribution, denoted as m (X _i|Pa (X _i)), is used to quantitatively characterize the mass dependency between node X _i and its parent nodesPa (X _i). All the conditional belief mass distributions of a child node with respect to its parent nodes consist of a conditional belief mass table (CBMT). The mass distributions of all root nodes (nodes without parent nodes) are given in advance, and the evidential inference algorithm [28] is utilized to infer all child nodes’ mass distributions based on CBMTs:

$\begin{matrix} m (X_{i}) = \sum_{Pa (X_{i})} m (X_{i} | Pa (X_{i})) \prod_{Pa (X_{i})} m (Pa (X_{i})) \end{matrix}$ (11)

Fig.1

An illustration of evidential network.

3.3 Modeling interval-valued probabilistic common cause group with “OR” gate

Without loss of generality, for component C _l subject to multiple probabilistic common cause events, PCCG₁, PCCG₂,..., the contributions to the failure of component C _l can be divided into i + 1 parts, i.e., the independent part and the dependent part j (1 ≤ j ≤ i) caused by PCCG_j. The failure of component C _l can be constructed as an “OR” gate, as shown in Fig. 2, and the corresponding evidential network of the “OR” gate can be constructed as shown in Fig. 3.

Fig.2

The fault tree of component C _l with multiple PCCGs.

The steps of computing the mass distribution of nodeC _l are as follows:

Step 1: Constructing the event space of the common cause events in which all combinations of the occurrence and non-occurrence of every elementary common cause events are involved. Given i independent common cause events which may cause the failure of component C _l, then, an event space with 2ⁱ disjoint events is constructed: $\begin{matrix} {CCG}_{1} = \bar{{CC}_{1}} \cap \bar{{CC}_{2}} \cap . . . \cap \bar{{CC}_{i}} \\ {CCG}_{2} = {CC}_{1} \cap \bar{{CC}_{2}} \cap . . . \cap \bar{{CC}_{i}} \\ {CCG}_{3} = \bar{{CC}_{1}} \cap {CC}_{2} \cap . . . \cap \bar{{CC}_{i}} \\ . . . \\ {CCG}_{2^{i}} = {CC}_{1} \cap {CC}_{2} \cap . . . \cap {CC}_{i} \end{matrix}$

Let Pr {CCG _k} (1 ≤ k ≤ 2ⁱ) denotes the probability of the occurrence of CCG _k and $\sum_{k = 1}^{2^{i}} Pr {{CCG}_{k}} = 1 .$

Step 2: In PCCF, the occurrence of CCG _k (1 ≤ k ≤ 2ⁱ) does not guarantee the failure of component C _l. Φ _{CC
_j} denotes the occurrence of CC _j whereas $Φ_{\bar{{CC}_{j}}}$ denotes CC _j doesn’t occur in CCG _k. Then, the mass that component C _l is still functioning corresponds to the mass that component C _l is reliable meanwhile all the occurred common cause events cannot cause the failure of component C _l:

$\begin{matrix} m_{PCCG} (R_{l}) = \underset{Part 1}{\underset{︸}{m (R_{l, ind})}} \times \sum_{{CCG}_{k}} (\underset{Part 2}{\underset{︸}{\prod_{Φ_{\bar{{CC}_{k}}}} (1 - β_{k})}} \\ \times \underset{Part 3}{\underset{︸}{\prod_{Φ_{{CC}_{j}}} (1 - {Pr}_{max} {{PCCG}_{j, l}} \times β_{j})}}) \end{matrix}$ (12)

Fig.3

The corresponding evidential network of component C _l.

where R _l and R _l,ind denote the functioning state with and without PCCFs of component C _l, respectively. “Part 1” in Eq.(12) is the independent mass that component C _l is functioning, whereas “Part 2” is the probability that some common cause events in the k th common cause group (1 ≤ k ≤ 2ⁱ) do not occur. “Part 3” is the mass that the occurred common cause events cannot result in the failure of component C _l. The mass that component C _l is in failure state can be separated into three items. “Item 1” is the mass that component C _l fails independently: $m_{PCCG, Item 1} (F_{l}) = m (F_{l, ind})$ (13) where F _l and F _l,ind denote the failure state with and without PCCFs of component C _l, respectively. “Item 2” corresponds to the mass that component C _l is functioning but at least one of the occurred common cause events result in the failure of component C _l:

$\begin{array}{l} m_{P C C G, I t e m 2} (F_{l}) = m (R_{l, i n d}) \times \sum_{C C G_{k}}^{} (\prod_{Φ_{\bar{C C_{k}}}}^{} (1 - β_{k}) \\ \times \underset{Part 1}{\underset{︸}{\cup_{Φ_{C C_{j}}}^{} (\Pr_{\min} {{PCCG}_{j, l}} \times β_{j})}}) \end{array}$ (14)

“Item 3” is the mass that the state of component C _l is unknown but at least one of the occurred common cause events results in the failure of component C _l:

$\begin{array}{l} m_{P C C G, I t e m 3} (F_{l}) = m ([R_{l, i n d}, F_{l, i n d}]) \times \sum_{C C G_{k}}^{} (\prod_{Φ_{\bar{C C_{k}}}}^{} (1 - β_{k}) \\ \times \underset{Part 1}{\underset{︸}{\cup_{Φ_{C C_{j}}}^{} (\Pr_{\min} {{PCCG}_{j, l}} \times β_{j})}}) \end{array}$ (15)

where [R _l,ind, F _l,ind] denotes the unknown state without PCCFs. “Part 1” in Eqs.(14) and (15) is the mass that at least one of the occurred common cause events result in the failure of component C _l. Note that “Part 1” in Eqs.(14) and (15) can be computed by the Inclusive-Exclusive principle shown as follows: $\begin{matrix} ⋃_{Φ_{{CC}_{j}}} ({Pr}_{min} {{PCCG}_{j, l}} \times β_{j}) = \\ \sum_{Φ_{{CC}_{j}}} ({Pr}_{min} {{PCCG}_{j, l}} \times β_{j}) \\ - \sum_{Φ_{{CC}_{j}}, Φ_{{CC}_{k}}} ({Pr}_{min} {{PCCG}_{j, l}} \times β_{j}) \\ \times ({Pr}_{min} {{PCCG}_{k, l}} \times β_{k}) \\ . . . \\ + (- 1)^{m_{k}} \prod_{Φ_{{CC}_{j}}} ({Pr}_{min} {{PCCG}_{j, l}} \times β_{j}) \end{matrix}$ (16) where m _k denotes the number of occurred common cause events in CCG _k (1 ≤ k ≤ 2ⁱ).

Therefore, the total mass that component C _l fails can be computed by: $\begin{matrix} m_{PCCG} (F_{l}) = m_{PCCG, Item 1} (F_{l}) \\ + m_{PCCG, Item 2} (F_{l}) \\ + m_{PCCG, Item 3} (F_{l}) \end{matrix}$ (17)

The mass that component C _l is in the unknown state (functioning or failed) can also be divided into three items. “Item 1” is the mass that state of component C _l is unknown meanwhile the PCCGs do not affect component C _l:

$\begin{matrix} \begin{matrix} m_{PCCG, Item 1} ([R_{l}, F_{l}]) = m ([R_{l, ind}, F_{l, ind}]) \\ \times \sum_{{CCG}_{k}} (\prod_{Φ_{\bar{{CC}_{k}}}} (1 - β_{k}) \\ \times \prod_{Φ_{{CC}_{j}}} (1 - {Pr}_{max} {{PCCG}_{j, l}} \times β_{j})) \end{matrix} \end{matrix}$ (18)

where [R _l, F _l] denotes the unknown state with PCCFs. “Item 2” is the mass that the state of component C _l is unknown meanwhile one does not know whether the PCCGs affect component C _l or not. For a specific CCG _k (1 ≤ k ≤ 2ⁱ), the mass that one does not know whether the PCCGs affect component C _l or not can be computed by one minus the mass that the PCCGs affect component C _l and the mass the PCCGs do not affect component C _l, and it results in:

$\begin{matrix} \begin{matrix} m (Ω_{{CCG}_{k, l}}) = \prod_{Φ_{\bar{{CC}_{k}}}} (1 - β_{k}) \\ \times (1 - \prod_{Φ_{{CC}_{j}}} (1 - {Pr}_{max} {{PCCG}_{j, l}} \times β_{j}) \\ - ⋃_{Φ_{{CC}_{j}}} ({Pr}_{min} {{PCCG}_{j, l}} \times β_{j})) \end{matrix} \end{matrix}$ (19) Therefore, “Item 2” can be calculated by:

$\begin{matrix} m_{PCCG, Item 2} ([R_{l}, F_{l}]) \\ = m ([R_{l, ind}, F_{l, ind}]) \times \sum_{{CCG}_{k}} m (Ω_{{CCG}_{k, l}}) \end{matrix}$ (20)

“Item 3” is the mass that component C _l is functioning but one does not know whether the PCCGs affect component C _l or not, and it corresponds to:

$\begin{matrix} m_{PCCG, Item 3} ([R_{l}, F_{l}]) \\ = m (R_{l, ind}) \times \sum_{{CCG}_{k}} m (Ω_{{CCG}_{k, l}}) \end{matrix}$ (21)

Therefore, the total mass that the state of component C _l is unknown can be computed by: $\begin{matrix} m_{PCCG} ([R_{l}, F_{l}]) = m_{PCCG, Item 1} ([R_{l}, F_{l}]) \\ + m_{PCCG, Item 2} ([R_{l}, F_{l}]) \\ + m_{PCCG, Item 3} ([R_{l}, F_{l}]) \end{matrix}$ (22)

After the mass distributions of all root nodes were calculated by the above Steps 1 and 2, the evidential inference algorithm, i.e., Eq.(11) can be used to infer the mass distributions of all child nodes. Therefore, the mass distribution of the leaf node, i.e., the system node, can be obtained. Finally, by applying the belief and plausibility functions to the system node, the lower and upper bounds of system reliability can be assessed.

4 Component importance analysis

The importance measures focus on investigating how a specific component reliability affects the reliability of a system. The importance ranking results are significant in identifying the weak components so as to provide useful guideline for maintenance and reliability improvement. A suite of importance measures has been proposed from various perspectives on system reliability/failure, such as the Birnbaum importance measure, the Fussell-Vesely importance measure (FV), the risk reduction worth (RRW), and the risk achievement worth (RAW). These importance measures are, however, developed under the conventional probabilistic framework, and they are incapable of tackling the epistemic uncertainty and PCCFs which are simultaneously studied in this work.

In this work, the well-known probabilistic Birnbaum importance measure is extended to the case where both the epistemic uncertainty and interval-valued PCCFs exist. The Birnbaum importance measure quantifies the most critical component to the system reliability, and it is defined as: $I_{l}^{B} (t) = Pr {R_{S} (t) | R_{l}} - Pr {R_{S} (t) | F_{l}}$ (23) where R _S (t) is the reliability of the entire system at time instant t. Analytically, the Birnbaum importance measure of component C _l when both epistemic uncertainty and PCCFs appear can be defined using the evidence theory by:

$\begin{matrix} \begin{matrix} {\tilde{I}}_{l}^{B} (t) = 1 pt 1 pt 1 pt [{Bel}_{PCCF} (R_{S} (t) | R_{l}), {Pl}_{PCCF} (R_{S} (t) | R_{l})] \\ - [{Bel}_{PCCF} (R_{S} (t) | F_{l}), {Pl}_{PCCF} (R_{S} (t) | F_{l})] \end{matrix} \end{matrix}$ (24) where ${Bel}_{PCCF} (R_{S} (t) | R_{l})$ andPl _PCCF (R _S (t) |R _l) represent respectively the belief and plausibility measures of the system being functioning when it is known that component C _l is working, whereas Bel _PCCF (R _S (t) |F _l) and Pl _PCCF (R _S (t) |F _l) denote respectively the belief and plausibility measures that the system is functioning when component C _l is in the failure state.

$[{Bel}_{PCCF} (R_{S} (t) | R_{l}), {Pl}_{PCCF} (R_{S} (t) | R_{l})]$ can be calculated by putting m (R _l = 1) of node C _l together with the mass distributions of the remaining components into the proposed EN model. Similarly, the interval [Bel _PCCF (R _S (t) |F _l) , Pl _PCCF (R _S (t) |F _l)] can be computed by putting m (F _l = 1) of node C _l together with the mass distributions of the remaining components into the proposed EN model.

To avoid the interval explosion problem of computing Eq.(24), the affined arithmetic (AA) introduced in [30] is implemented to calculate the extended Birnbaum importance measure, and it results in: ${\tilde{I}}_{l}^{B} (t) = I_{l, 0}^{B} (t) + I_{l, 1}^{B} (t) ɛ_{1} + I_{l, 2}^{B} (t) ɛ_{2}$ (25) where $\begin{matrix} I_{l, 0}^{B} (t) = \frac{{Bel}_{PCCF} (R_{S} (t) | R_{l}) + {Pl}_{PCCF} (R_{S} (t) | R_{l})}{2} \\ - \frac{{Bel}_{PCCF} (R_{S} (t) | F_{l}) + {Pl}_{PCCF} (R_{S} (t) | F_{l})}{2} \end{matrix}$ (26)

$\begin{matrix} I_{l, 1}^{B} (t) = \frac{{Pl}_{PCCF} (R_{S} (t) | R_{l}) - {Bel}_{PCCF} (R_{S} (t) | R_{l})}{2} \end{matrix}$ (27)

$\begin{matrix} I_{l, 2}^{B} (t) = \frac{{Pl}_{PCCF} (R_{S} (t) | F_{l}) - {Bel}_{PCCF} (R_{S} (t) | F_{l})}{2} \end{matrix}$ (28) and ɛ ₁, ɛ ₂ are two independent random variables range in [-1, 1].

5 The case study of a safety instrumented system

A safety instrumented system (SIS), as shown in Fig. 4, is exemplified to examine the effectiveness of the proposed evidential network for complex engineering systems. The SIS is used to evaluate the safety target level for the vessels. In this study, we aim at evaluating the reliability of the SIS when both the epistemic uncertainty associated with parameters of components’ reliability models and the PCCFs exist. The fault tree of the SIS can be found in Ref. [31]. The evidential network can be constructed from the fault tree of the SIS by transforming all the basic events into roots nodes and all the intermediate events into child nodes. The DAG of the evidential network for the SIS is shown in Fig. 5. In Fig. 5, nodes X1X14 are the root nodes, and each logic gate in the fault tree is converted into a child node, i.e., the nodes E1, E3 E14, and D1 D10. The mass distribution of the SIS can be obtained by calculating the masses of node D10. Therefore, the reliability bounds of the SIS can be computed by applying the belief and plausibility functions into the mass distribution of node D10.

Fig.4

The configuration of the safety instrumented system [31].

Fig.5

The evidential network of the SIS without PCCFs [31].

It should be noted that even though the reliability assessment of this SIS under fuzzy set theory has been studied in [31], the common cause failure, which related to the simultaneous failure of components due to the physical interactions and some external shocks, inevitably exist in the SIS. Because the SIS consists of many identical components which are more likely to suffer from the common cause failure (CCF) [21]. Moreover, the occurrence of these CCFs does not necessarily result in the guaranteed failure of components/subsystems in the SIS, but may cause the affected components to fail with interval-valued probability (bounded in [0,1]). In this example, the following interval-valued PCCFs are considered:

(1) The pressure transmitters (PTs) X1, X2 and the flow transmitters (FTs) X4, X5, and X6, can fail simultaneously due to external shocks, i.e., PCCG₁ = {X1, X2, X4, X5, X6} . Based on the expert experience, the occurrence probability of this CCF is β ₁ = 0.01, and this CCF results in the failure of the PTs, i.e., X1, X2, with an interval-valued probability [0.6, 0.7], as well as the failure of the FTs, i.e., X4, X5, and X6, with an interval-valued probability [0.7, 0.8].

(2) The temperature switches (TSs) X7, X8, and the level switches (LVs) X13, X14, will fail at the same time due to external shocks, i.e.,PCCG₂ = {X7, X8, X13, X14} . Based on the expert experience, the occurrence of the CCF β ₂ = 0.01 . The occurrence of the CCF results in the failure of both the TSs and LVs with an interval-valued probability [0.7,0.8].

To compute the mass distribution of the SIS, the mass distributions of all the components should be identified first. In this example, as the PTs and FTs are mechanical components, these components are assumed to comply with the Weibull distribution, whereas the remaining components are assumed to obey the exponential distribution as they are electronic components. The model parameter settings for all the components are listed in Table 1 based on the historical data [32].

Table 1

The failure parameters of the components in the oil system (Time unit: hours)

SIS Components	Parameters	Lower bound	Upper bound
X1, X2: Pressure Transmitter (PT)	Scale parameter	2.5 × 10⁵	2.6 × 10⁵
	Shape parameter	1.2	1.4
X3: Logic Solver (LS)	Failure rate	7 × 10^-5	8 × 10^-5
X4, X5, X6: Flow Transmitter (FT)	Scale parameter	4.5 × 10⁵	4.7 × 10⁵
	Shape parameter	1.2	1.4
X7, X8: Temperature Switch (TS)	Failure rate	3 × 10^-5	5 × 10^-5
X9, X11: Solenoid (SOL)	Failure rate	3 × 10^-6	4 × 10^-6
X10, X12: Block Valve (BV)	Failure rate	3 × 10^-6	4 × 10^-6
X13, X14: Level Switch (LV)	Failure rate	3 × 10^-5	5 × 10^-5

For the components complying with the exponential distribution, the mass distribution can be calculated by:

$\begin{matrix} {\begin{matrix} m (R_{l} (t)) = exp (- λ_{l, max} \cdot t) \\ m (F_{l} (t)) = 1 - exp (- λ_{l, min} \cdot t) \\ m ([R_{l} (t), F_{l} (t)]) = exp (- λ_{l, min} \cdot t) - exp (- λ_{l, max} \cdot t) \end{matrix} \end{matrix}$ (29)

where $λ_{l, max}$ and $λ_{l, min}$ represent, respectively, the upper and lower bounds of the failure rate of component C _l. For the components complying with the Weibull distribution, the upper and lower reliability bounds of the components $[R_{l, min} (t), R_{l, max} (t)]$ can be calculated by the following constrained optimization formulization: $\begin{matrix} max (\min) R_{l} (t) = exp ((- t / θ_{l})^{β_{l}}) \\ s . t . {\begin{matrix} θ_{l, min} \leq θ_{l} \leq θ_{l, max} \\ β_{l, min} \leq β_{l} \leq β_{l, max} \end{matrix} \end{matrix}$ (30)

where $θ_{l, max}$ , $θ_{l, min}$ , $β_{l, min}$ , and $β_{l, max}$ represent, respectively, the upper and lower bounds of the scale parameter and shape parameter of component C _l. Hence, the mass distribution of mechanical component C _l can be obtained by: ${\begin{matrix} m (R_{l} (t)) = R_{l, min} (t) \\ m (F_{l} (t)) = 1 - R_{l, max} (t) \\ m ([R_{l} (t), F_{l} (t)]) = R_{l, max} (t) - R_{l, min} (t) \end{matrix}$ (31)

Fig.6

The evidential network of the SIS with PCCFs.

Firstly, without considering the PCCFs, the reliability of the SIS can be assessed by the EN model shown in Fig. 7. The interval-valued reliability results indicate that the epistemic uncertainties associated with the model parameters of components are propagated to the entire system through the EN model.

By taking account of the interval-valued PCCFs in the SIS, another two nodes PCCF₁ and PCCF₂ are additionally added to the evidential network as shown in Fig. 6, these two nodes manifest the influence of the occurrence of PCCFs on the components, and its mass distribution can be calculated by Eqs.(8) (10). In this example, by setting:

m (PCCG_1,X1) = m (PCCG_1,X2) =0.6β ₁ = 0.006; m (PCCG_1,X4) = m (PCCG_1,X5) = m (PCCG_1,X6) = 0.7β ₁ = 0.007; $m ({\bar{PCCG}}_{1, X 1}) = m ({\bar{PCCG}}_{1, X 2}) = 1 - 0.7 β_{1} = 0.993$ ; $m ({\bar{PCCG}}_{1, X 4}) = m ({\bar{PCCG}}_{1, X 5}) = m ({\bar{PCCG}}_{1, X 6}) = 1 - 0.8 β_{1} = 0.992$ ; m (Ω _{PCCG_1,X1}) = m (Ω _{PCCG_1,X2}) =0.7β ₁ - 0.6β ₁ = 0.001; m (Ω _{PCCG_1,X4}) = m (Ω _{PCCG_1,X5}) = m (Ω _{PCCG_1,X6}) =0.8β ₁ - 0.7β ₁ = 0.001; for node PCCF₁, and setting:m (PCCG_2,X7) = m (PCCG_2,X8) = 0.7β ₂ = 0.007; m (PCCG_2,X13) = m (PCCG_2,X14) = 0.7β ₂ = 0.007; $m ({\bar{PCCG}}_{2, X 7}) = m ({\bar{PCCG}}_{2, X 8}) = 1 - 0.8 β_{2} = 0.992$ ; $m ({\bar{PCCG}}_{1, X 13}) = m ({\bar{PCCG}}_{2, X 14}) = 1 - 0.8 β_{2} = 0.992$ ; m (Ω _{PCCG_2,X7}) = m (Ω _{PCCG_2,X8}) =0.8β ₂ - 0.7β ₂ = 0.001; m (Ω _{PCCG_2,X13}) = m (Ω _{PCCG_2,X14}) =0.8β ₂ - 0.7β ₂ = 0.001;

for node PCCF₂. The mass distributions of all the components suffering from interval-valued PCCFs can be computed by Eqs. (12)–(22). The reliability bounds of the SIS with PCCFs can be, then, calculated by the proposed evidential network model, and the result is shown in Fig. 7. As observed in Fig. 7, the interval-valued PCCFs between components lead to a decrement of the reliability bounds as compared with the results without taking account of interval-valued PCCFs. Hence, the interval-valued PCCFs should be properly addressed when conducting the reliability assessment of such complex engineering systems.

Fig.7

The reliability bounds of the SIS with/without PCCFs.

Furthermore, the importance of all the components in SIS can be calculated by Eq. (24). The results are depicted in Fig. 8. As shown in Fig. 8, the Birnbaum importance of each component extends to an interval value. These interval values provide an answer to the influence of epistemic uncertainty associated with model parameters on the importance measures.

Fig.8

The Birnbaum importance of all the components in SIS.

Moreover, the importance of the logic solver starts from a high value at time t=0 and end up with 0 as time goes to infinite. Because the failure of the logic solver at the beginning will directly incur the failure of the entire system. In the same manner, only the logic solver recovering from a worse state to a better state can’t make system work again if the system is very aged. Hence, the importance of the logic solver starts from a high value at the beginning and approaches to zero when time goes to infinite. However, as the failure of other components cannot lead to the failure of the SIS at the beginning, while the recovering of these components cannot make the system work again at a very aged time, hence, the importance of these components begins with zero and ends up with zero. It bears noting that the most critical component to system reliability/failure over time can be identified, i.e., the logic solver. This ranking result is also consistent with the result in Ref. [31] under the fuzzy theory. It indicates that a small variation of the reliability of the logic solver causes a relatively greater change to the reliability of the SIS. This result drives our attentions to the reliability improvement of the logic solver.

6 Conclusions

The reliability assessment of complex engineering systems is critical as reliability-related decisions can be effectively made. The epistemic uncertainty associated with model parameters and the PCCFs are two important issues in system reliability assessment. In this paper, by taking both the epistemic uncertainty associated with model parameters and the interval-valued PCCFs into account, the reliability assessment based on the EN model was conducted. The epistemic uncertainties associated with the parameters of components’ failure models were quantified by interval values. The mass distribution of each component was then computed under the evidence theory. Then, the evidential inference algorithm was implemented to calculate the mass distribution of the system node. Finally, two functions, namely the belief and plausibility functions, were utilized to compute the lower and upper bounds of system reliability. Furthermore, the Birnbaum importance measure was defined under epistemic uncertainty and PCCFs. A safety instrumented system was exemplified to demonstrate the effectiveness of our proposed evidential network model. The results showed that the epistemic uncertainty of model parameters and the interval-valued PCCFs have impact on the system reliability analysis. Moreover, the most critical components of the SIS were identified by our proposed importance measure. The importance results can further provide guideline to the reliability-related decisions, such as reliability improvement and maintenance planning.

Footnotes

Acknowledgements

The authors greatly acknowledge grant support from the Science Challenge Project under contract number TZ2018007, the National Natural Science Foundation of China under contract numbers 71771039 and 61877009, and the Sichuan Provincial Science and Technology Plan Project under contract number 2018GZ0396.

References

Mhalla

, Dutilleul

S.C.

, Craye

, et al., Estimation of failure probability of milk manufacturing unit by fuzzy fault tree analysis, Journal of Intelligent and Fuzzy Systems 26(2) (2014), 741–750.

Beck

J.L.

, Au

S.K.

, Bayesian updating of structural models and reliability using Markov chain Monte Carlo simulation, Journal of Engineering Mechanics 128(4) (2002), 380–391.

Z.W.

, Zhou

M.C.

, Elementary siphons of Petri nets and their application to deadlock prevention in flexible manufacturing systems, IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans 34(1) (2004), 38–51.

Jiang

, Liu

, Parameter inference for non-repairable multi-state system reliability models by multi-level observation sequences, Reliability Engineering and System Safety 166 (2017), 3–15.

Boudali

, Dugan

J.B.

, A discrete-time Bayesian network reliability modeling and analysis framework, Reliability Engineering and System Safety 87 (2005), 337–349.

M.Y.

, Liu

, Li

, et al., Bayesian modeling of multi-state hierarchical systems with multi-level information aggregation, Reliability Engineering and System Safety 124 (2014), 158–164.

Z.P.

, Mo

Y.C.

, Liu

, et al., Reliability assessment of multi-state phased-mission systems by fusing observation data from multiple phases of operation, Mechanical Systems and Signal Processing 118 (2019), 603–622.

Abbasi

, Ghahraman

, Tadayon

F.M.

, Consideration effect of uncertainty in the reliability indices of power systems using a scenario-based approach, Journal of Intelligent and Fuzzy Systems 28(1) (2015), 291–299.

Yang

J.P.

, Huang

H.Z.

, Sun

, et al., Reliability analysis of aircraft servo-actuation systems using evidential networks, International Journal of Turbo and Jet-Engines 29(2) (2012), 59–68.

10.

Mula

, Poler

, Garcia-Sabater

J.P.

, Material requirement planning with fuzzy constraints and fuzzy coefficients, Fuzzy Sets and Systems 158(7) (2007), 783–793.

11.

Muhammad

, Reinhard

, Empirical reliability functions based on fuzzy life time data, Journal of Intelligent and Fuzzy Systems 28(2) (2015), 707–711.

12.

Sankararaman

, Mahadevan

, Likelihood-based representation of epistemic uncertainty due to sparse point data and/or interval data, Reliability Engineering and System Safety 96(7) (2011), 814–824.

13.

Smets

, Decision making in the TBM: the necessity of the pignistic transformation, International Journal of Approximate Reasoning 38(2) (2005), 133–147.

14.

Basir

, Yuan

, Engine fault diagnosis based on multi-sensor information fusion using Dempster-Shafer evidence theory, Information Fusion 8(4) (2007), 379–386.

15.

Simon

, Weber

, Evsukoff

, Bayesian networks inference algorithm to implement Dempster Shafer theory in reliability analysis, Reliability Engineering and System Safety 93(7) (2008), 950–963.

16.

Zhao

, Wang

, Cheng

, Power distribution system reliability evaluation by DS evidence inference and Bayesian network method, In Proceedings of IEEE 11th International Conference on Probabilistic Methods Applied to Power Systems (2010), 654–658.

17.

Simon

, Weber

, Evidential networks for reliability analysis and performance evaluation of systems with imprecise knowledge, IEEE Transactions on Reliability 58(1) (2009), 69–87.

18.

Yang

J.P.

, Huang

H.Z.

, Liu

, et al., Evidential networks for fault tree analysis with imprecise knowledge, International Journal of Turbo and Jet-Engines 29(2) (2012), 111–122.

19.

Fleming

K.N.

, Reliability Model for Common Mode Failures in Redundant Safety Systems, Modeling, and Simulation, General Atomic Report, GA-13284 (1974).

20.

J.H.

, LI

Y. F.

, Huang

H. Z.

, et al., Reliability analysis of multi-state systems with common cause failure based on Bayesian networks, In Proceedings of Quality, Reliability, Risk, Maintenance, and Safety Engineering, International Conference on IEEE, (2012), 1117–1121.

21.

, Li

Y.F.

, Peng

, et al., Reliability analysis of complex multi-state system with common cause failure based on evidential networks, Reliability Engineering and System Safety 174 (2018), 71–81.

22.

Wang

, Xing

, Levitin

, Probabilistic common cause failures in phased-mission systems, Reliability Engineering and System Safety 144 (2015), 53–60.

23.

Zuo

, Xiahou

, Liu

, Evidential network-based failure analysis for systems suffering common cause failure and model parameter uncertainty, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science (2018), 10.1177-0954406218781407.

24.

Dempster

A.P.

, Upper and lower probabilities induced by a multi-valued mapping, Annals Mathematical Statistics 38 (1967), 325–339.

25.

Shafer

, A Mathematical Theory of Evidence, Princeton: Princeton University Press (1976).

26.

Dempster

A.P.

, Kong

, Uncertain evidence and artificial analysis, Journal of Statistical Planning Inference 20 (1988), 355–368.

27.

Martinez

F.A.

, Sallak

, Schön

, An efficient method for reliability analysis of systems under epistemic uncertainty using belief function theory, IEEE Transactions on Reliability 64(3) (2015), 893–909.

28.

Xiahou

, Liu

, Jiang

, Extended composite importance measures of multi-state systems with epistemic uncertainty of state assignment, Mechanical Systems and Signal Processing 109 (2018), 305–329.

29.

Börcsök

, Estimation and evaluation of common cause failures, In Proceeding of Second International Conference on Systems (2007).

30.

Sallak

, Schon

, Aguirre

, Extended component importance measures considering aleatory and epistemic uncertainties, IEEE Transactions on Reliability 62(1) (2013), 49–65.

31.

Sallak

, Simon

, Aubry

J.F.

, A fuzzy probabilistic approach for determining safety integrity level, IEEE Transactions on Fuzzy Systems 16(1) (2008), 239–248.

32.

Liu

, Lin

, Li

Y.F.

, Huang

H.Z.

, Bayesian reliability and performance assessment for multi-state systems, IEEE Transactions on Reliability 64(1) (2015), 394–409.